Heat Transfer GREGORY NELLIS University of Wisconsin–Madison
SANFORD KLEIN University of Wisconsin–Madison
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Heat Transfer GREGORY NELLIS University of Wisconsin–Madison

SANFORD KLEIN University of Wisconsin–Madison

cambridge university press ˜ Paulo, Delhi Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521881074 c Gregory Nellis and Sanford Klein 2009

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2009 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Nellis, Gregory. Heat transfer / Gregory Nellis, Sanford Klein p. cm. Includes bibliographical references and index. ISBN 978-0-521-88107-4 (hardback) 1. Heat – Transmission. I. Klein, Sanford A., 1950– II. Title. TJ260.N45 2008 621.402 2 – dc22 2008021961 ISBN 978-0-521-88107-4 hardback Additional resources for this publication at www.cambridge.org/nellisandklein Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work are correct at the time of ﬁrst printing, but Cambridge University Press does not guarantee the accuracy of such information thereafter.

This book is dedicated to Stephen H. Nellis . . . thanks Dad.

CONTENTS

Preface Acknowledgments Study guide Nomenclature 1 1.1

1.2

page xix xxi xxiii xxvii

ONE-DIMENSIONAL, STEADY-STATE CONDUCTION r 1 Conduction Heat Transfer 1.1.1 Introduction 1.1.2 Thermal Conductivity Thermal Conductivity of a Gas∗ (E1) Steady-State 1-D Conduction without Generation 1.2.1 Introduction 1.2.2 The Plane Wall 1.2.3 The Resistance Concept 1.2.4 Resistance to Radial Conduction through a Cylinder 1.2.5 Resistance to Radial Conduction through a Sphere 1.2.6 Other Resistance Formulae Convection Resistance Contact Resistance Radiation Resistance EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.3

Steady-State 1-D Conduction with Generation 1.3.1 Introduction 1.3.2 Uniform Thermal Energy Generation in a Plane Wall 1.3.3 Uniform Thermal Energy Generation in Radial Geometries EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3.4

Spatially Non-Uniform Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.4

Numerical Solutions to Steady-State 1-D Conduction Problems (EES) 1.4.1 Introduction 1.4.2 Numerical Solutions in EES 1.4.3 Temperature-Dependent Thermal Conductivity 1.4.4 Alternative Rate Models

1.5

Numerical Solutions to Steady-State 1-D Conduction Problems using MATLAB 1.5.1 Introduction 1.5.2 Numerical Solutions in Matrix Format 1.5.3 Implementing a Numerical Solution in MATLAB

EXAMPLE 1.4-1: FUEL ELEMENT

∗

1 1 1 5 5 5 5 9 10 11 13 14 14 16 17 24 24 24 29 31 37 38 44 44 45 55 60 62 68 68 69 71

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vii

viii

Contents

1.5.4 1.5.5 1.5.6

Functions Sparse Matrices Temperature-Dependent Properties

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.6

Analytical Solutions for Constant Cross-Section Extended Surfaces 1.6.1 Introduction 1.6.2 The Extended Surface Approximation 1.6.3 Analytical Solution 1.6.4 Fin Behavior 1.6.5 Fin Efﬁciency and Resistance EXAMPLE 1.6-1: SOLDERING TUBES

1.6.6

Finned Surfaces

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

1.7

1.6.7 Fin Optimization∗ (E2) Analytical Solutions for Advanced Constant Cross-Section Extended Surfaces 1.7.1 Introduction 1.7.2 Additional Thermal Loads EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7.3

Moving Extended Surfaces

EXAMPLE 1.7-2: DRAWING A WIRE

1.8

Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 1.8.1 Introduction 1.8.2 Series Solutions 1.8.3 Bessel Functions 1.8.4 Rules for Using Bessel Functions EXAMPLE 1.8-1: PIPE IN A ROOF EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

1.9

Numerical Solution to Extended Surface Problems 1.9.1 Introduction EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

Problems References 2 2.1

TWO-DIMENSIONAL, STEADY-STATE CONDUCTION r 202 Shape Factors EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

2.2

∗

77 80 82 84 92 92 92 95 103 105 110 113 117 122 122 122 122 127 133 136 139 139 139 142 150 155 161 164 164 165 171 185 201

Separation of Variables Solutions 2.2.1 Introduction 2.2.2 Separation of Variables Requirements for using Separation of Variables Separate the Variables Solve the Eigenproblem Solve the Non-homogeneous Problem for each Eigenvalue Obtain Solution for each Eigenvalue Create the Series Solution and Enforce the Remaining Boundary Conditions Summary of Steps

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

202 205 207 207 208 209 211 212 213 214 215 222

Contents

2.2.3

ix

Simple Boundary Condition Transformations

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

2.3 2.4

2.5

2.6

2.7

2.8

Advanced Separation of Variables Solutions∗ (E3) Superposition 2.4.1 Introduction 2.4.2 Superposition for 2-D Problems Numerical Solutions to Steady-State 2-D Problems with EES 2.5.1 Introduction 2.5.2 Numerical Solutions with EES Numerical Solutions to Steady-State 2-D Problems with MATLAB 2.6.1 Introduction 2.6.2 Numerical Solutions with MATLAB 2.6.3 Numerical Solution by Gauss-Seidel Iteration∗ (E4) Finite Element Solutions 2.7.1 Introduction to FEHT∗ (E5) 2.7.2 The Galerkin Weighted Residual Method∗ (E6) Resistance Approximations for Conduction Problems 2.8.1 Introduction EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

2.8.2 2.8.3

Isothermal and Adiabatic Resistance Limits Average Area and Average Length Resistance Limits

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

2.9

Conduction through Composite Materials 2.9.1 Effective Thermal Conductivity EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

Problems References 3 3.1

TRANSIENT CONDUCTION r 302 Analytical Solutions to 0-D Transient Problems 3.1.1 Introduction 3.1.2 The Lumped Capacitance Assumption 3.1.3 The Lumped Capacitance Problem 3.1.4 The Lumped Capacitance Time Constant EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.2

Numerical Solutions to 0-D Transient Problems 3.2.1 Introduction 3.2.2 Numerical Integration Techniques Euler’s Method Heun’s Method Runge-Kutta Fourth Order Method Fully Implicit Method Crank-Nicolson Method Adaptive Step-Size and EES’ Integral Command MATLAB’s Ordinary Differential Equation Solvers EXAMPLE 3.2-1(A): OVEN BRAZING (EES) EXAMPLE 3.2-1(B): OVEN BRAZING (MATLAB)

∗

224 225 236 242 242 242 245 250 250 251 260 260 260 268 269 269 269 269 269 270 272 275 276 278 278 282 290 301

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302 302 302 303 304 307 310 317 317 317 318 322 326 328 330 332 335 339 344

x

Contents

3.3

Semi-Inﬁnite 1-D Transient Problems 3.3.1 Introduction 3.3.2 The Diffusive Time Constant EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

3.3.3 3.3.4

The Self-Similar Solution Solutions to other Semi-Inﬁnite Problems

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.4

The Laplace Transform 3.4.1 Introduction 3.4.2 The Laplace Transformation Laplace Transformations with Tables Laplace Transformations with Maple 3.4.3 The Inverse Laplace Transform Inverse Laplace Transform with Tables and the Method of Partial Fractions Inverse Laplace Transformation with Maple 3.4.4 Properties of the Laplace Transformation 3.4.5 Solution to Lumped Capacitance Problems 3.4.6 Solution to Semi-Inﬁnite Body Problems

348 348 348 351 354 361 363 369 369 370 371 371 372 373 376 378 380 386

3.4.7 Numerical Inverse Laplace Transform* (E29) EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

3.5

Separation of Variables for Transient Problems 3.5.1 Introduction 3.5.2 Separation of Variables Solutions for Common Shapes The Plane Wall The Cylinder The Sphere EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

3.5.3

Separation of Variables Solutions in Cartesian Coordinates Requirements for using Separation of Variables Separate the Variables Solve the Eigenproblem Solve the Non-homogeneous Problem for each Eigenvalue Obtain a Solution for each Eigenvalue Create the Series Solution and Enforce the Initial Condition Limits of the Separation of Variables Solution

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.6 3.7 3.8

∗

3.5.4 Separation of Variables Solutions in Cylindrical Coordinates∗ (E7) 3.5.5 Non-homogeneous Boundary Conditions∗ (E8) Duhamel’s Theorem∗ (E9) Complex Combination∗ (E10) Numerical Solutions to 1-D Transient Problems 3.8.1 Introduction 3.8.2 Transient Conduction in a Plane Wall Euler’s Method Fully Implicit Method Heun’s Method Runge-Kutta 4th Order Method Crank-Nicolson Method EES’ Integral Command MATLAB’s Ordinary Differential Equation Solvers

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

391 395 395 396 396 401 403 405 408 409 410 411 413 414 414 417 420 427 428 428 428 428 428 429 432 438 442 445 449 452 453

Contents EXAMPLE 3.8-1: TRANSIENT RESPONSE OF A BENT-BEAM ACTUATOR

3.9

4 4.1

4.2

4.3

3.8.3 Temperature-Dependent Properties Reduction of Multi-Dimensional Transient Problems∗ (E11) Problems References

∗

457 463 468 469 482

EXTERNAL FORCED CONVECTION r 483 Introduction to Laminar Boundary Layers 4.1.1 Introduction 4.1.2 The Laminar Boundary Layer A Conceptual Model of the Laminar Boundary Layer A Conceptual Model of the Friction Coefﬁcient and Heat Transfer Coefﬁcient The Reynolds Analogy 4.1.3 Local and Integrated Quantities The Boundary Layer Equations 4.2.1 Introduction 4.2.2 The Governing Equations for Viscous Fluid Flow The Continuity Equation The Momentum Conservation Equations The Thermal Energy Conservation Equation 4.2.3 The Boundary Layer Simpliﬁcations The Continuity Equation The x-Momentum Equation The y-Momentum Equation The Thermal Energy Equation Dimensional Analysis in Convection 4.3.1 Introduction 4.3.2 The Dimensionless Boundary Layer Equations The Dimensionless Continuity Equation The Dimensionless Momentum Equation in the Boundary Layer The Dimensionless Thermal Energy Equation in the Boundary Layer 4.3.3 Correlating the Solutions of the Dimensionless Equations The Friction and Drag Coefﬁcients The Nusselt Number EXAMPLE 4.3-1: SUB-SCALE TESTING OF A CUBE-SHAPED MODULE

4.4

xi

4.3.4 The Reynolds Analogy (revisited) Self-Similar Solution for Laminar Flow over a Flat Plate 4.4.1 Introduction 4.4.2 The Blasius Solution The Problem Statement The Similarity Variables The Problem Transformation Numerical Solution 4.4.3 The Temperature Solution The Problem Statement The Similarity Variables The Problem Transformation Numerical Solution 4.4.4 The Falkner-Skan Transformation∗ (E12)

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

483 483 484 485 488 492 494 495 495 495 495 496 498 500 500 501 502 503 506 506 508 508 509 509 511 511 513 515 520 521 521 522 522 522 526 530 535 535 536 536 538 542

xii

4.5

4.6

4.7

4.8

Contents

Turbulent Boundary Layer Concepts 4.5.1 Introduction 4.5.2 A Conceptual Model of the Turbulent Boundary Layer The Reynolds Averaged Equations 4.6.1 Introduction 4.6.2 The Averaging Process The Reynolds Averaged Continuity Equation The Reynolds Averaged Momentum Equation The Reynolds Averaged Thermal Energy Equation The Laws of the Wall 4.7.1 Introduction 4.7.2 Inner Variables 4.7.3 Eddy Diffusivity of Momentum 4.7.4 The Mixing Length Model 4.7.5 The Universal Velocity Proﬁle 4.7.6 Eddy Diffusivity of Momentum Models 4.7.7 Wake Region 4.7.8 Eddy Diffusivity of Heat Transfer 4.7.9 The Thermal Law of the Wall Integral Solutions 4.8.1 Introduction 4.8.2 The Integral Form of the Momentum Equation Derivation of the Integral Form of the Momentum Equation Application of the Integral Form of the Momentum Equation EXAMPLE 4.8-1: PLATE WITH TRANSPIRATION

4.8.3

4.9

The Integral Form of the Energy Equation Derivation of the Integral Form of the Energy Equation Application of the Integral Form of the Energy Equation 4.8.4 Integral Solutions for Turbulent Flows External Flow Correlations 4.9.1 Introduction 4.9.2 Flow over a Flat Plate Friction Coefﬁcient Nusselt Number EXAMPLE 4.9-1: PARTIALLY SUBMERGED PLATE

4.9.3

Unheated Starting Length Constant Heat Flux Flow over a Rough Plate Flow across a Cylinder Drag Coefﬁcient Nusselt Number

EXAMPLE 4.9-2: HOT WIRE ANEMOMETER

4.9.4

Flow across a Bank of Cylinders Non-Circular Extrusions Flow past a Sphere

EXAMPLE 4.9-3: BULLET TEMPERATURE

Problems References

∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

542 542 543 548 548 549 550 551 554 556 556 557 560 561 562 565 566 567 568 571 571 571 571 575 580 584 584 587 591 593 593 593 593 598 603 606 606 607 609 611 613 615 617 617 618 620 624 633

Contents

5 5.1

5.2

xiii

INTERNAL FORCED CONVECTION r 635 Internal Flow Concepts 5.1.1 Introduction 5.1.2 Momentum Considerations The Mean Velocity The Laminar Hydrodynamic Entry Length Turbulent Internal Flow The Turbulent Hydrodynamic Entry Length The Friction Factor 5.1.3 Thermal Considerations The Mean Temperature The Heat Transfer Coefﬁcient and Nusselt Number The Laminar Thermal Entry Length Turbulent Internal Flow Internal Flow Correlations 5.2.1 Introduction 5.2.2 Flow Classiﬁcation 5.2.3 The Friction Factor Laminar Flow Turbulent Flow EES’ Internal Flow Convection Library EXAMPLE 5.2-1: FILLING A WATERING TANK

5.2.4

The Nusselt Number Laminar Flow Turbulent Flow

EXAMPLE 5.2-2: DESIGN OF AN AIR HEATER

5.3

The Energy Balance 5.3.1 Introduction 5.3.2 The Energy Balance 5.3.3 Prescribed Heat Flux Constant Heat Flux 5.3.4 Prescribed Wall Temperature Constant Wall Temperature 5.3.5 Prescribed External Temperature

5.4

Analytical Solutions for Internal Flows 5.4.1 Introduction 5.4.2 The Momentum Equation Fully Developed Flow between Parallel Plates The Reynolds Equation∗ (E13) Fully Developed Flow in a Circular Tube∗ (E14) 5.4.3 The Thermal Energy Equation Fully Developed Flow through a Round Tube with a Constant Heat Flux Fully Developed Flow through Parallel Plates with a Constant Heat Flux Numerical Solutions to Internal Flow Problems 5.5.1 Introduction 5.5.2 Hydrodynamically Fully Developed Laminar Flow EES’ Integral Command

EXAMPLE 5.3-1: ENERGY RECOVERY WITH AN ANNULAR JACKET

5.5

∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

635 635 635 637 638 638 640 641 644 644 645 646 648 649 649 650 650 651 654 656 657 661 662 667 668 671 671 671 673 674 674 674 675 677 686 686 686 687 689 689 689 691 695 697 697 698 702

xiv

Contents

The Euler Technique The Crank-Nicolson Technique MATLAB’s Ordinary Differential Equation Solvers 5.5.3 Hydrodynamically Fully Developed Turbulent Flow Problems References 6 6.1

6.2

NATURAL CONVECTION r 735 Natural Convection Concepts 6.1.1 Introduction 6.1.2 Dimensionless Parameters for Natural Convection Identiﬁcation from Physical Reasoning Identiﬁcation from the Governing Equations Natural Convection Correlations 6.2.1 Introduction 6.2.2 Plate Heated or Cooled Vertical Plate Horizontal Heated Upward Facing or Cooled Downward Facing Plate Horizontal Heated Downward Facing or Cooled Upward Facing Plate Plate at an Arbitrary Tilt Angle EXAMPLE 6.2-1: AIRCRAFT FUEL ULLAGE HEATER

6.2.3

Sphere

EXAMPLE 6.2-2: FRUIT IN A WAREHOUSE

6.2.4

6.2.5

Cylinder Horizontal Cylinder Vertical Cylinder Open Cavity Vertical Parallel Plates

EXAMPLE 6.2-3: HEAT SINK DESIGN

6.2.6 6.2.7

Enclosures Combined Free and Forced Convection

EXAMPLE 6.2-4: SOLAR FLUX METER

6.3 6.4

7

Self-Similar Solution∗ (E15) Integral Solution∗ (E16) Problems References

735 735 735 736 739 741 741 741 742 744 745 747 748 752 753 757 757 758 760 761 763 766 768 769 772 772 773 777

BOILING AND CONDENSATION r 778

7.1 7.2

Introduction Pool Boiling 7.2.1 Introduction 7.2.2 The Boiling Curve 7.2.3 Pool Boiling Correlations

7.3

Flow Boiling 7.3.1 Introduction 7.3.2 Flow Boiling Correlations

EXAMPLE 7.2-1: COOLING AN ELECTRONICS MODULE USING NUCLEATE BOILING

EXAMPLE 7.3-1: CARBON DIOXIDE EVAPORATING IN A TUBE ∗

704 706 710 712 723 734

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

778 779 779 780 784 786 790 790 791 794

Contents

7.4

xv

Film Condensation 7.4.1 Introduction 7.4.2 Solution for Inertia-Free Film Condensation on a Vertical Wall 7.4.3 Correlations for Film Condensation Vertical Wall EXAMPLE 7.4-1: WATER DISTILLATION DEVICE

7.5

8 8.1

Horizontal, Downward Facing Plate Horizontal, Upward Facing Plate Single Horizontal Cylinder Bank of Horizontal Cylinders Single Horizontal Finned Tube Flow Condensation 7.5.1 Introduction 7.5.2 Flow Condensation Correlations Problems References HEAT EXCHANGERS r 823 Introduction to Heat Exchangers 8.1.1 Introduction 8.1.2 Applications of Heat Exchangers 8.1.3 Heat Exchanger Classiﬁcations and Flow Paths 8.1.4 Overall Energy Balances 8.1.5 Heat Exchanger Conductance Fouling Resistance EXAMPLE 8.1-1: CONDUCTANCE OF A CROSS-FLOW HEAT EXCHANGER

8.1.6

Compact Heat Exchanger Correlations

EXAMPLE 8.1-2: CONDUCTANCE OF A CROSS-FLOW HEAT EXCHANGER (REVISITED)

8.2

The Log-Mean Temperature Difference Method 8.2.1 Introduction 8.2.2 LMTD Method for Counter-Flow and Parallel-Flow Heat Exchangers 8.2.3 LMTD Method for Shell-and-Tube and Cross-Flow Heat Exchangers

8.3

The Effectiveness-NTU Method 8.3.1 Introduction 8.3.2 The Maximum Heat Transfer Rate 8.3.3 Heat Exchanger Effectiveness

EXAMPLE 8.2-1: PERFORMANCE OF A CROSS-FLOW HEAT EXCHANGER

EXAMPLE 8.3-1: PERFORMANCE OF A CROSS-FLOW HEAT EXCHANGER (REVISITED)

8.3.4

8.4

8.5 ∗

798 798 799 805 805 807 810 811 811 811 811 812 812 813 815 821

Further Discussion of Heat Exchanger Effectiveness Behavior as CR Approaches Zero Behavior as NTU Approaches Zero Behavior as NTU Becomes Inﬁnite Heat Exchanger Design Pinch Point Analysis 8.4.1 Introduction 8.4.2 Pinch Point Analysis for a Single Heat Exchanger 8.4.3 Pinch Point Analysis for a Heat Exchanger Network Heat Exchangers with Phase Change

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

823 823 823 824 828 831 831 832 838 841 841 841 842 847 848 851 851 852 853 858 861 862 863 864 865 867 867 867 872 876

xvi

Contents

8.5.1 Introduction 8.5.2 Sub-Heat Exchanger Model for Phase-Change 8.6 Numerical Model of Parallel- and Counter-Flow Heat Exchangers 8.6.1 Introduction 8.6.2 Numerical Integration of Governing Equations Parallel-Flow Conﬁguration Counter-Flow Conﬁguration∗ (E17) 8.6.3 Discretization into Sub-Heat Exchangers Parallel-Flow Conﬁguration Counter-Flow Conﬁguration∗ (E18) 8.6.4 Solution with Axial Conduction∗ (E19) 8.7 Axial Conduction in Heat Exchangers 8.7.1 Introduction 8.7.2 Approximate Models for Axial Conduction Approximate Model at Low λ Approximate Model at High λ Temperature Jump Model 8.8 Perforated Plate Heat Exchangers 8.8.1 Introduction 8.8.2 Modeling Perforated Plate Heat Exchangers 8.9 Numerical Modeling of Cross-Flow Heat Exchangers 8.9.1 Introduction 8.9.2 Finite Difference Solution Both Fluids Unmixed with Uniform Properties Both Fluids Unmixed with Temperature-Dependent Properties One Fluid Mixed, One Fluid Unmixed∗ (E20) Both Fluids Mixed∗ (E21) 8.10 Regenerators 8.10.1 Introduction 8.10.2 Governing Equations 8.10.3 Balanced, Symmetric Flow with No Entrained Fluid Heat Capacity Utilization and Number of Transfer Units Regenerator Effectiveness 8.10.4 Correlations for Regenerator Matrices Packed Bed of Spheres Screens Triangular Passages EXAMPLE 8.10-1: AN ENERGY RECOVERY WHEEL

8.10.5 Numerical Model of a Regenerator with No Entrained Heat Capacity∗ (E22) Problems References 9

MASS TRANSFER∗ (E23) r 974 Problems

10

974

RADIATION r 979

10.1 Introduction to Radiation 10.1.1 Radiation ∗

876 876 888 888 888 889 896 897 897 902 902 903 903 905 907 907 909 911 911 913 919 919 920 920 927 936 936 937 937 939 942 942 944 948 950 951 952 953 962 962 973

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979 979

Contents

xvii

10.1.2 The Electromagnetic Spectrum 10.2 Emission of Radiation by a Blackbody 10.2.1 Introduction 10.2.2 Blackbody Emission Planck’s Law Blackbody Emission in Speciﬁed Wavelength Bands EXAMPLE 10.2-1: UV RADIATION FROM THE SUN

10.3 Radiation Exchange between Black Surfaces 10.3.1 Introduction 10.3.2 View Factors The Enclosure Rule Reciprocity Other View Factor Relationships The Crossed and Uncrossed String Method EXAMPLE 10.3-1: CROSSED AND UNCROSSED STRING METHOD

View Factor Library EXAMPLE 10.3-2: THE VIEW FACTOR LIBRARY

10.3.3 Blackbody Radiation Calculations The Space Resistance EXAMPLE 10.3-3: APPROXIMATE TEMPERATURE OF THE EARTH

N-Surface Solutions EXAMPLE 10.3-4: HEAT TRANSFER IN A RECTANGULAR ENCLOSURE EXAMPLE 10.3-5: DIFFERENTIAL VIEW FACTORS: RADIATION EXCHANGE BETWEEN PARALLEL PLATES

10.4 Radiation Characteristics of Real Surfaces 10.4.1 Introduction 10.4.2 Emission of Real Materials Intensity Spectral, Directional Emissivity Hemispherical Emissivity Total Hemispherical Emissivity The Diffuse Surface Approximation The Diffuse Gray Surface Approximation The Semi-Gray Surface 10.4.3 Reﬂectivity, Absorptivity, and Transmittivity Diffuse and Specular Surfaces Hemispherical Reﬂectivity, Absorptivity, and Transmittivity Kirchoff’s Law Total Hemispherical Values The Diffuse Surface Approximation The Diffuse Gray Surface Approximation The Semi-Gray Surface EXAMPLE 10.4-1: ABSORPTIVITY AND EMISSIVITY OF A SOLAR SELECTIVE SURFACE

10.5 Diffuse Gray Surface Radiation Exchange 10.5.1 Introduction 10.5.2 Radiosity 10.5.3 Gray Surface Radiation Calculations EXAMPLE 10.5-1: RADIATION SHIELD EXAMPLE 10.5-2: EFFECT OF OVEN SURFACE PROPERTIES ∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

980 981 981 982 982 985 987 989 989 989 990 991 992 992 993 996 998 1001 1001 1002 1006 1007 1009 1012 1012 1012 1012 1014 1014 1015 1016 1016 1016 1018 1019 1020 1020 1022 1023 1023 1023 1024 1027 1027 1028 1029 1032 1037

xviii

Contents

10.5.4 The Fˆ Parameter EXAMPLE 10.5-3: RADIATION HEAT TRANSFER BETWEEN PARALLEL PLATES

10.5.5 Radiation Exchange for Semi-Gray Surfaces EXAMPLE 10.5-4: RADIATION EXCHANGE IN A DUCT WITH SEMI-GRAY SURFACES

10.6 Radiation with other Heat Transfer Mechanisms 10.6.1 Introduction 10.6.2 When Is Radiation Important? 10.6.3 Multi-Mode Problems 10.7 The Monte Carlo Method 10.7.1 Introduction 10.7.2 Determination of View Factors with the Monte Carlo Method Select a Location on Surface 1 Select the Direction of the Ray Determine whether the Ray from Surface 1 Strikes Surface 2 10.7.3 Radiation Heat Transfer Determined by the Monte Carlo Method Problems References

1043 1046 1050 1051 1055 1055 1055 1057 1058 1058 1058 1060 1060 1061 1068 1077 1088

Appendices

1089 1089 1089 1089 1090 1090

Index

1091

A.1: Introduction to EES∗ (E24) A.2: Introduction to Maple∗ (E25) A.3: Introduction to MATLAB∗ (E26) A.4: Introduction to FEHT∗ (E27) A.5: Introduction to Economics ∗ (E28)

∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

PREFACE

The single objective of this book is to provide engineers with the capability, tools, and conﬁdence to solve real-world heat transfer problems. This objective has resulted in a textbook that differs from existing heat transfer textbooks in several ways. First, this textbook includes many topics that are typically not covered in undergraduate heat transfer textbooks. Examples are the detailed presentations of mathematical solution methods such as Bessel functions, Laplace transforms, separation of variables, Duhamel’s theorem, and Monte Carlo methods as well as high order explicit and implicit numerical integration algorithms. These analytical and numerical solution methods are applied to advanced topics that are ordinarily not considered in a heat transfer textbook. Judged by its content, this textbook should be considered as a graduate text. There is sufﬁcient material for two-semester courses in heat transfer. However, the presentation does not presume previous knowledge or expertise. This book can be (and has been) successfully used in a single-semester undergraduate heat transfer course by appropriately selecting from the available topics. Our recommendations on what topics can be included in a ﬁrst heat transfer course are provided in the suggested syllabus. The reason that this book can be used for a ﬁrst course (despite its expanded content) and the reason it is also an effective graduate-level textbook is that all concepts and methods are presented in detail, starting at the beginning. The derivation of important results is presented completely, without skipping steps, in order to improve readability, reduce student frustration, and improve retention. You will not ﬁnd many places in this textbook where it states that “it can be shown that . . . ” The use of examples, solved and explained in detail, is ubiquitous in this textbook. The examples are not trivial, “textbook” exercises, but rather complex and timely real-world problems that are of interest by themselves. As with the presentation, the solutions to these examples are complete and do not skip steps. Another signiﬁcant difference between this textbook and most existing heat transfer textbooks is its integration of modern computational tools. The engineering student and practicing engineer of today is expected to be proﬁcient with engineering computer tools. Engineering education must evolve accordingly. Most real engineering problems cannot be solved using a sequential set of calculations that can be accomplished with a pencil or hand calculator. Engineers must have the ability to quickly solve problems using the powerful computational tools that are available and essential for design, parametric study, and optimization of real-world systems. This book integrates the computational software packages Maple, MATLAB, FEHT, and Engineering Equation Solver (EES) directly with the heat transfer material. The speciﬁc commands and output associated with these software packages are presented as the theory is developed so that the integration is seamless rather than separated. The computational software tools used in this book share some important characteristics. They are used in industry and have existed for more than a decade; therefore, while this software will certainly continue to evolve, it is not likely to disappear. Educational versions of these software packages are available, and therefore the use of these xix

xx

Preface

tools should not represent an economic hardship to any academic institution or student. Useful versions of EES and FEHT are provided on the website that accompanies this textbook (www.cambridge.org/nellisandklein). With the help provided in the book, these tools are easy to learn and use. Students can become proﬁcient with all of them in a reasonable amount of time. Learning the computer tools will not detract signiﬁcantly from material coverage. To facilitate this learning process, tutorials for each of the software packages are provided on the companion website. The book itself is structured so that more advanced features of the software are introduced progressively, allowing students to become increasingly proﬁcient using these tools as they progress through the text. Most (if not all) of the tables and charts that have traditionally been required to solve heat transfer problems (for example, to determine properties, view factors, shape factors, convection relations, etc.) have been made available as functions and procedures in the EES software so that they can be easily accessed and used to solve problems. Indeed, the library of heat transfer functions that has been developed and integrated with EES as part of the preparation of this textbook enables a profound shift in the focus of the educational process. It is trivial to obtain, for example, a shape factor, a view factor, or a convection heat transfer coefﬁcient using the heat transfer library. Therefore, it is possible to assign problems involving design and optimization studies that would be computationally impossible without the computer tools. Integrating the study of heat transfer with computer tools does not diminish the depth of understanding of the underlying physics. Conversely, our experience indicates that the innate understanding of the subject matter is enhanced by appropriate use of these tools for several reasons. First, the software allows the student to tackle practical and relevant problems as opposed to the comparatively simple problems that must otherwise be assigned. Real-world engineering problems are more satisfying to the student. Therefore, the marriage of computer tools with theory motivates students to understand the governing physics as well as learn how to apply the computer tools. The use of these tools allows for coverage of more advanced material and more interesting and relevant problems. When a solution is obtained, students can carry out a more extensive investigation of its behavior and therefore obtain a more intuitive and complete understanding of the subject of heat transfer. This book is unusual in its linking of classical theory and modern computing tools. It ﬁlls an obvious void that we have encountered in teaching both undergraduate and graduate heat transfer courses. The text was developed over many years from our experiences teaching Introduction to Heat Transfer (an undergraduate course) and Heat Transfer (a ﬁrst-year graduate course) at the University of Wisconsin. It is our hope that this text will not only be useful during the heat transfer course, but also provide a lifelong resource for practicing engineers. G. F. Nellis S. A. Klein May, 2008

Acknowledgments

The development of this book has taken several years and a substantial effort. This has only been possible due to the collegial and supportive atmosphere that makes the Mechanical Engineering Department at the University of Wisconsin such a unique and impressive place. In particular, we would like to acknowledge Tim Shedd, Bill Beckman, Doug Reindl, John Pfotenhauer, Roxann Engelstad, and Glen Myers for their encouragement throughout the process. Several years of undergraduate and graduate students have used our initial drafts of this manuscript. They have had to endure carrying two heavy volumes of poorly bound paper with no index and many typographical errors. Their feedback has been invaluable to the development of this book. We have had the extreme good fortune to have had dedicated and insightful teachers. These include Glen Myers, John Mitchell, Bill Beckman, Joseph Smith Jr., John Brisson, Borivoje Mikic, and John Lienhard V. These individuals, among others, have provided us with an indication of the importance of teaching and provided an inspiration to us for writing this book. Preparing this book has necessarily reduced the “quality time” available to spend with our families. We are most grateful to them for this indulgence. In particular, we wish to thank Jill, Jacob, and Spencer and Sharon Nellis and Jan Klein. We could have not completed this book without their continuous support. Finally, we are indebted to Cambridge University Press and in particular Peter Gordon for giving us this opportunity and for helping us with the endless details needed to bring our original idea to this ﬁnal state.

xxi

STUDY GUIDE

This book has been developed for use in either a graduate or undergraduate level course in heat transfer. A sample program of study is laid out below for a one-semester graduate course (consisting of 45 class sessions). Graduate heat transfer class Day 1 2 3 4 5 6 7 8 9 10 11 12

Sections in Book 1.1 1.2 2.8 1.3 1.4, 1.5 1.6 1.7 1.8 2.2 2.2 2.4 3.1

13

3.2

14 15 16 17 18 19 20 21 22 23

3.3 3.3 3.4 3.4 3.5 3.8 4.1 4.2, 4.3 4.4 4.5, 4.6

24 25 26 27 28 29 30 31

4.7 4.8 4.8, 4.9 5.1, 5.2 5.3 5.4 5.5 6.1, 6.2

Topic Conduction heat transfer 1-D steady conduction and resistance concepts Resistance approximations 1-D steady conduction with generation Numerical solutions with EES and MATLAB Fin solution, ﬁn efﬁciency, and ﬁnned surfaces Other constant cross-section extended surface problems Bessel function solutions 2-D conduction, separation of variables 2-D conduction, separation of variables Superposition Transient, lumped capacitance problems – analytical solutions Transient, lumped capacitance problems – numerical solutions Semi-inﬁnite bodies, diffusive time constant Semi-inﬁnite bodies, self-similar solution Laplace transform solutions to lumped capacitance problems Laplace transform solutions to 1-D transient problems Separation of variables for 1-D transient problems Numerical solutions to 1-D transient problems Laminar boundary layer concepts The boundary layer equations & dimensionless parameters Blasius solution for ﬂow over a ﬂat plate Turbulent boundary layer concepts, Reynolds averaged equations Mixing length models and the laws of the wall Integral solutions Integral solutions, external ﬂow correlations Internal ﬂow concepts and correlations The energy balance Analytical solutions to internal ﬂow problems Numerical solutions to internal ﬂow problems Natural convection concepts and correlations xxiii

xxiv

32 33 34 35 36 37 38 39 40 41 42 43 44 45

Study Guide

8.1 8.2, 8.3 8.5 8.7 8.8, 8.10 10.1, 10.2 10.3 10.3 10.4 10.5 10.5 10.5 10.7 10.7

Introduction to heat exchangers The LMTD and ε-NTU forms of the solutions Heat exchangers with phase change Axial conduction in heat exchangers Perforated plate heat exchangers and regenerators Introduction to radiation, Blackbody emissive power View factors and the space resistance Blackbody radiation exchange Real surfaces, Kirchoff’s law Gray surface radiation exchange Gray surface radiation exchange Semi-gray surface radiation exchange Introduction to Monte Carlo techniques Introduction to Monte Carlo techniques

A sample program of study is laid out below for a one-semester undergraduate course (consisting of 45 class sessions). Undergraduate heat transfer class Day 1 2 3

Sections in Book A.1 1.2.2-1.2.3 1.2.4-1.2.6

4 5 6 7 8 9 10 11 12 13 14 15

1.3.1-1.3.3 1.4 1.6.1-1.6.3 1.6.4-1.6.6 1.9.1 2.1 2.8.1-2.8.2 2.9 2.5 3.1 3.2.1, 3.2.2

16 17

3.3.1-3.3.2 3.3.2, 3.3.4

18 19 20 21 22 23 24 25

3.5.1-3.5.2 3.8.1-3.8.2 4.1 4.2, 4.3 4.5 4.9.1-4.9.2 4.9.3-4.9.4 5.1

Topic Review of thermodynamics, Using EES 1-D steady conduction, resistance concepts and circuits 1-D steady conduction in radial systems, other thermal resistance More thermal resistance problems 1-D steady conduction with generation Numerical solutions with EES The extended surface approximation and the ﬁn solution Fin behavior, ﬁn efﬁciency, and ﬁnned surfaces Numerical solutions to extended surface problems 2-D steady-state conduction, shape factors Resistance approximations Conduction through composite materials Numerical solution to 2-D steady-state problems with EES Lumped capacitance assumption, the lumped time constant Numerical solution to lumped problems (Euler’s, Heun’s, Crank-Nicolson) Semi-inﬁnite body, the diffusive time constant Approximate models of diffusion, other semi-inﬁnite solutions Solutions to 1-D transient conduction in a bounded geometry Numerical solution to 1-D transient conduction using EES Introduction to laminar boundary layer concepts Dimensionless numbers Introduction to turbulent boundary layer concepts Correlations for external ﬂow over a plate Correlations for external ﬂow over spheres and cylinders Internal ﬂow concepts

Study Guide

26 27 28 29 30 31

5.2 5.3

32 33

7.1, 7.2 7.3, 7.4.3, 7.5

34

8.1

35 36 37 38

8.2 8.3.1-8.3.3 8.3.4 8.10.1, 8.10.3-4

39 40 41 42 43 44 45

10.1, 10.2 10.3.1-10.3.2 10.3.3 10.4 10.5.1-10.5.3

6.1 6.2.1-6.2.3 6.2.4-6.2.7

10.6

xxv

Internal ﬂow correlations Energy balance for internal ﬂows Internal ﬂow problems Introduction to natural convection Natural convection correlations Natural convection correlations and combined forced/free convection Pool boiling Correlations for ﬂow boiling, ﬂow condensation, and ﬁlm condensation Introduction to heat exchangers, compact heat exchanger correlations The LMTD Method The ε-NTU Method Limiting behaviors of the ε-NTU Method Regenerators, solution for balanced & symmetric regenerator, packings Introduction to radiation, blackbody emission View factors Blackbody radiation exchange Real surfaces and Kirchoff’s law Gray surface radiation exchange Gray surface radiation exchange Radiation with other heat transfer mechanisms

NOMENCLATURE

ith coefﬁcient of a series solution cross-sectional area (m2 ) minimum ﬂow area (m2 ) projected area (m2 ) surface area (m2 ) surface area of a ﬁn (m2 ) prime (total) surface area of a ﬁnned surface (m2 ) aspect ratio of a rectangular duct area ratio of ﬁn tip to ﬁn surface area attenuation (-) parameter in the blowing factor (-) blowing factor (-) Biot number (-) boiling number (-) Brinkman number speciﬁc heat capacity (J/kg-K) concentration (-) speed of light (m/s) speciﬁc heat capacity of an air-water mixture on a unit mass of air basis ca (J/kga -K) speciﬁc heat capacity of an air-water mixture along the saturation line on a ca,sat unit mass of air basis (J/kga -K) effective speciﬁc heat capacity of a composite (J/kg-K) ceff ratio of the energy carried by a micro-scale energy carrier to its cms temperature (J/K) speciﬁc heat capacity at constant volume (J/kg-K) cv C total heat capacity (J/K) ˙ C capacitance rate of a ﬂow (W/K) C1 , C2 , . . . undetermined constants dimensionless coefﬁcient for critical heat ﬂux correlation (-) Ccrit drag coefﬁcient (-) CD friction coefﬁcient (-) Cf Cf average friction coefﬁcient (-) coefﬁcient for laminar plate natural convection correlation (-) Clam dimensionless coefﬁcient for nucleate boiling correlation (-) Cnb capacity ratio (-) CR coefﬁcient for turbulent, horizontal upward plate natural conv. Cturb,U correlation (-) coefﬁcient for turbulent, vertical plate natural convection correlation (-) Cturb,V ai Ac Amin Ap As As,ﬁn Atot AR ARtip Att B BF Bi Bo Br c

xxvii

xxviii

Co CTE D Dh dx dy e err ˙ E E Eb Eλ Eb,λ Ec f

f fl F F 0−λ1 Fi,j Fˆ i,j fd Fo Fr Frmod g g˙ g˙ g˙ eff g˙ v G Gλ Ga Gr Gz h h h˜ hD hD hl

Nomenclature

convection number (-) coefﬁcient of thermal expansion (1/K) diameter (m) diffusion coefﬁcient (m2 /s) hydraulic diameter (m) differential in the x-direction (m) differential in the y-direction (m) size of surface roughness (m) convergence or numerical error rate of thermal energy carried by a mass ﬂow (W) total emissive power (W/m2 ) total blackbody emissive power (W/m2 ) spectral emissive power (W/m2 -μm) blackbody spectral emissive power (W/m2 -μm) Eckert number (-) frequency (Hz) dimensionless stream function, for Blasius solution (-) friction factor (-) average friction factor (-) friction factor for liquid-only ﬂow in ﬂow boiling (-) force (N) correction-factor for log-mean temperature difference (-) external fractional function (-) view factor from surface i to surface j (-) the “F-hat” parameter characterizing radiation from surface i to surface j (-) fractional duty for a pinch-point analysis (-) Fourier number (-) Froude number (-) modiﬁed Froude number (-) acceleration of gravity (m/s2 ) rate of thermal energy generation (W) rate of thermal energy generation per unit volume (W/m3 ) effective rate of generation per unit volume of a composite (W/m3 ) rate of thermal energy generation per unit volume due to viscous dissipation (W/m3 ) mass ﬂux or mass velocity (kg/m2 -s) total irradiation (W/m2 ) spectral irradiation (W/m2 -μm) Galileo number (-) Grashof number (-) Graetz number (-) local heat transfer coefﬁcient (W/m2 -K) average heat transfer coefﬁcient (W/m2 -K) dimensionless heat transfer coefﬁcient for ﬂow boiling correlation (-) mass transfer coefﬁcient (m/s) average mass transfer coefﬁcient (m/s) superﬁcial heat transfer coefﬁcient for the liquid phase (W/m2 -K)

Nomenclature

hrad i

ia I Ie Ii j J jH k kB kc ke keff Kn l1 l1,2 L L+ L∗ Lchar Lchar,vs Lcond Lﬂow Lml Lms Le M m m ˙ m ˙ mms mf MW n nms n˙ N Ns Nu

xxix

the equivalent heat transfer coefﬁcient associated with radiation (W/m2 -K) index of node (-) index of eigenvalue (-) index of term in a series solution (-) speciﬁc enthalpy (J/kg-K) √ square root of negative one, −1 speciﬁc enthalpy of an air-water mixture on a per unit mass of air basis (J/kga ) current (ampere) intensity of emitted radiation (W/m2 -μm-steradian) intensity of incident radiation (W/m2 -μm-steradian) index of node (-) index of eigenvalue (-) radiosity (W/m2 ) Colburn jH factor (-) thermal conductivity (W/m-K) Bolzmann’s constant (J/K) contraction loss coefﬁcient (-) expansion loss coefﬁcient (-) effective thermal conductivity of a composite (W/m-K) Knudsen number (-) Lennard-Jones 12-6 potential characteristic length for species 1 (m) characteristic length of a mixture of species 1 and species 2 (m) length (m) dimensionless length for a hydrodynamically developing internal ﬂow (-) dimensionless length for a thermally developing internal ﬂow (-) characteristic length of the problem (m) the characteristic size of the viscous sublayer (m) length for conduction (m) length in the ﬂow direction (m) mixing length (m) distance between interactions of micro-scale energy or momentum carriers (m) Lewis number (-) number of nodes (-) mass (kg) ﬁn parameter (1/m) mass ﬂow rate (kg/s) mass ﬂow rate per unit area (kg/m2 -s) mass of microscale momentum carrier (kg/carrier) mass fraction (-) molar mass (kg/kgmol) number density (#/m3 ) number density of the micro-scale energy carriers (#/m3 ) molar transfer rate per unit area (kgmol/m2 -s) number of nodes (-) number of moles (kgmol) number of species in a mixture (-) Nusselt number (-)

xxx

Nu NTU p P p∞ p˜ Pe per Pr Prturb q˙ q˙ i to j q˙ max q˙ q˙ s q˙ s,crit Q ˜ Q r r˜ R

RA Rac Rad Rbl Rc Rconv Rcyl Re Rf Rﬁn Ri,j Riso RL Rpw Rrad Rs,i Rsemi-∞ Rsph Rtot Runiv Rc Rf Ra

Nomenclature

average Nusselt number (-) number of transfer units (-) pressure (Pa) pitch (m) LMTD effectiveness (-) probability distribution (-) free-stream pressure (Pa) dimensionless pressure (-) Peclet number (-) perimeter (m) Prandtl number (-) turbulent Prandtl number (-) rate of heat transfer (W) rate of radiation heat transfer from surface i to surface j (W) maximum possible rate of heat transfer, for an effectiveness solution (W) heat ﬂux, rate of heat transfer per unit area (W/m2 ) surface heat ﬂux (W/m2 ) critical heat ﬂux for boiling (W/m2 ) total energy transfer by heat (J) dimensionless total energy transfer by heat (-) radial coordinate (m) radius (m) dimensionless radial coordinate (-) thermal resistance (K/W) ideal gas constant (J/kg-K) LMTD capacitance ratio (-) thermal resistance approximation based on average area limit (K/W) thermal resistance to axial conduction in a heat exchanger (K/W) thermal resistance approximation based on adiabatic limit (K/W) thermal resistance of the boundary layer (K/W) thermal resistance due to solid-to-solid contact (K/W) thermal resistance to convection from a surface (K/W) thermal resistance to radial conduction through a cylindrical shell (K/W) electrical resistance (ohm) thermal resistance due to fouling (K/W) thermal resistance of a ﬁn (K/W) the radiation space resistance between surfaces i and j (1/m2 ) thermal resistance approximation based on isothermal limit (K/W) thermal resistance approximation based on average length limit (K/W) thermal resistance to radial conduction through a plane wall (K/W) thermal resistance to radiation (K/W) the radiation surface resistance for surface i (1/m2 ) thermal resistance approximation for a semi-inﬁnite body (K/W) thermal resistance to radial conduction through a spherical shell (K/W) thermal resistance of a ﬁnned surface (K/W) universal gas constant (8314 J/kgmol-K) area-speciﬁc contact resistance (K-m2 /W) area-speciﬁc fouling resistance (K-m2 /W) Rayleigh number (-)

Nomenclature

Re Recrit RH RR s S Sc Sh Sh St t tsim th tol T Tb Tﬁlm Tm Ts Tsat T∞ T∗ T T TR Tt TX TY th U u uchar uf um u∞ u∗ u+ u˜ u u UA v vδ vms

xxxi

Reynolds number (-) critical Reynold number for transition to turbulence (-) relative humidity (-) radius ratio of an annular duct (-) Laplace transformation variable (1/s) generic coordinate (m) shape factor (m) channel spacing (m) Schmidt number (-) Sherwood number (-) average Sherwood number (-) Stanton number (-) time (s) simulated time (s) thickness (m) convergence tolerance temperature (K) base temperature of ﬁn (K) ﬁlm temperature (K) mean or bulk temperature (K) surface temperature (K) saturation temperature (K) free-stream or ﬂuid temperature (K) eddy temperature ﬂuctuation (K) ﬂuctuating component of temperature (K) average temperature (K) temperature solution that is a function of r, for separation of variables temperature solution that is a function of t, for separation of variables temperature solution that is a function of x, for separation of variables temperature solution that is a function of y, for separation of variables thickness (m) internal energy (J) utilization (-) speciﬁc internal energy (J/kg) velocity in the x-direction (m/s) characteristic velocity (m/s) frontal or upstream velocity (m/s) mean or bulk velocity (m/s) free-stream velocity (m/s) eddy velocity (m/s) inner velocity (-) dimensionless x-velocity (-) ﬂuctuating component of x-velocity (m/s) average x-velocity (m/s) conductance (W/K) velocity in the y- or r-directions (m/s) y-velocity at the outer edge of the boundary layer, approximate scale of y-velocity in a boundary layer (m/s) mean velocity of micro-scale energy or momentum carriers (m/s)

xxxii

v˜ v v V V˙ vf w w ˙ W x x˜ X xfd,h xfd,t Xtt y y+ y˜ Y z

Nomenclature

dimensionless y-velocity (-) ﬂuctuating component of y-velocity (m/s) average y-velocity (m/s) volume (m3 ) voltage (V) volume ﬂow rate (m3 /s) void fraction (-) velocity in the z-direction (m/s) rate of work transfer (W) width (m) total amount of work transferred (J) x-coordinate (m) quality (-) dimensionless x-coordinate (-) particular solution that is only a function of x hydrodynamic entry length (m) thermal entry length (m) Lockhart Martinelli parameter (-) y-coordinate (m) mole fraction (-) inner position (-) dimensionless y-coordinate (-) particular solution that is only a function of y z-coordinate (m)

Greek Symbols α

αeff αλ αλ,θ,φ β δ δd δm δvs δt ifus ivap p r

thermal diffusivity (m2 /s) absorption coefﬁcient (1/m) absorptivity or absorptance (-), total hemispherical absorptivity (-) surface area per unit volume (1/m) effective thermal diffusivity of a composite (m2 /s) hemispherical absorptivity (-) spectral directional absorptivity (-) volumetric thermal expansion coefﬁcient (1/K) ﬁlm thickness for condensation (m) boundary layer thickness (m) mass transfer diffusion penetration depth (m) concentration boundary layer thickness (m) momentum diffusion penetration depth (m) momentum boundary layer thickness (m) viscous sublayer thickness (m) energy diffusion penetration depth (m) thermal boundary layer thickness (m) latent heat of fusion (J/kg) latent heat of vaporization (J/kg) pressure drop (N/m2 ) distance in r-direction between adjacent nodes (m)

Nomenclature

T Te Tlm t tcrit x y ε εﬁn εH ελ ελ,θ,φ εM ε1 ε1,2 φ η ηﬁn ηo κ λ λi μ v θ θ˜ θ+ θR θt θX θXt θY θYt θZt

xxxiii

temperature difference (K) excess temperature (K) log-mean temperature difference (K) time step (s) time period (s) critical time step (s) distance in x-direction between adjacent nodes (m) distance in y-direction between adjacent nodes (m) heat exchanger effectiveness (-) emissivity or emittance (-), total hemispherical emissivity (-) ﬁn effectiveness (-) eddy diffusivity for heat transfer (m2 /s) hemispherical emissivity (-) spectral, directional emissivity (-) eddy diffusivity of momentum (m2 /s) Lennard-Jones 12-6 potential characteristic energy for species 1 (J) characteristic energy parameter for a mixture of species 1 and species 2 (J) porosity (-) phase angle (rad) spherical coordinate (rad) similarity parameter (-) efﬁciency (-) ﬁn efﬁciency (-) overall efﬁciency of a ﬁnned surface (-) ´ an ´ constant von Karm dimensionless axial conduction parameter (-) wavelength of radiation (μm) ith eigenvalue of a solution (1/m) viscosity (N-s/m2 ) frequency of radiation (1/s) temperature difference (K) angle (rad) spherical coordinate (rad) dimensionless temperature difference (-) inner temperature difference (-) temperature difference solution that is only a function of r, for separation of variables temperature difference solution that is only a function of t, for separation of variables temperature difference solution that is only a function of x, for separation of variables temperature difference solution that is only a function of x and t, for reduction of multi-dimensional transient problems temperature difference solution that is only a function of y, for separation of variables temperature difference solution that is only a function of y and t, for reduction of multi-dimensional transient problems temperature difference solution that is only a function of z and t, for reduction of multi-dimensional transient problems

xxxiv

ρ ρe ρeff ρλ ρλ,θ,ϕ σ

τ τdiff τlumped τλ τλ,θ,ϕ τs υ ω D ζ ζi

Nomenclature

density (kg/m3 ) reﬂectivity or reﬂectance (-), total hemispherical reﬂectivity (-) electrical resistivity (ohm-m) effective density of a composite (kg/m3 ) hemispherical reﬂectivity (-) spectral, directional reﬂectivity (-) surface tension (N/m), molecular radius (m) ratio of free-ﬂow to frontal area (-) Stefan-Boltzmann constant (5.67 × 10-8 W/m2 -K4 ) time constant (s) shear stress (Pa) transmittivity or transmittance (-), total hemispherical transmittivity (-) diffusive time constant (s) lumped capacitance time constant (s) hemispherical transmittivity (-) spectral, directional transmittivity (-) shear stress at surface (N/m2 ) kinematic viscosity (m2 /s) angular velocity (rad/s) humidity ratio (kgv /kga ) solid angle (steradian) dimensionless collision integral for diffusion (-) stream function (m2 /s) tilt angle (rad) curvature parameter for vertical cylinder, natural convection correlation (-) the ith dimensionless eigenvalue (-)

Superscripts o

at inﬁnite dilution

Subscripts a abs ac an app b bl bottom c C cc char cf

air absorbed axial conduction (in heat exchangers) analytical apparent approximate blackbody boundary layer bottom condensate ﬁlm corrected cold cold-side of a heat exchanger complex conjugate, for complex combination problems characteristic counter-ﬂow heat exchanger

Nomenclature

cond conv crit CTHB dc df diff eff emit evap ext f fc fd,h fd,t ﬁn h H

hs HTCB i

in ini int

j l lam LHS lumped m max min mod ms n nac nb nc no-ﬁn out p

conduction, conductive convection, convective critical cold-to-hot blow process dry coil downward facing diffusive transfer effective emitted evaporative external ﬂuid forced convection hydrodynamically fully developed thermally fully developed ﬁn, ﬁnned homogeneous solution hot hot-side of a heat exchanger constant heat ﬂux boundary condition on a hemisphere hot-to-cold blow process node i surface i species i inner inlet initial internal interface integration period node j surface j liquid laminar left-hand side lumped-capacitance mean or bulk melting maximum or maximum possible minimum or minimum possible modiﬁed micro-scale carrier normal without axial conduction (in heat exchangers) nucleate boiling natural convection without a ﬁn outer outlet particular (or non-homogeneous) solution

xxxv

xxxvi

pf pp r rad ref RHS s sat sat,l sat,v sc semi-∞ sh sph sur sus T top tot turb uf unﬁn v

w wb wc x x− x+ y ∞ 90◦

Nomenclature

parallel-ﬂow heat exchanger pinch-point regenerator matrix at position r radiation, radiative reference right-hand side at the surface saturated saturated section of a heat exchanger saturated liquid saturated vapor sub-cooled section of a heat exchanger semi-inﬁnite super-heated section of a heat exchanger sphere surroundings sustained solution constant temperature boundary condition at temperature T top total turbulent upward-facing not ﬁnned vapor vertical viscous dissipation water wet-bulb wet coil at position x in the x-direction in the negative x-direction in the positive x-direction at position y in the y-direction free-stream, ﬂuid solution that is 90◦ out of phase, for complex combination problems

Other notes A A A A A Aˆ

A

arbitrary variable ﬂuctuating component of variable A value of variable A on a unit length basis value of variable A on a unit area basis value of variable A on a unit volume basis dimensionless form of variable A a guess value or approximate value for variable A Laplace transform of the function A

Nomenclature

A A A dA δA A O(A)

average of variable A denotes that variable A is a vector denotes that variable A is a matrix differential change in the variable A uncertainty in the variable A ﬁnite change in the variable A order of magnitude of the variable A

xxxvii

1

One-Dimensional, Steady-State Conduction

1.1 Conduction Heat Transfer 1.1.1 Introduction Thermodynamics deﬁnes heat as a transfer of energy across the boundary of a system as a result of a temperature difference. According to this deﬁnition, heat by itself is an energy transfer process and it is therefore redundant to use the expression ‘heat transfer’. Heat has no option but to transfer and the expression ‘heat transfer’ reinforces the incorrect concept that heat is a property of a system that can be ‘transferred’ to another system. This concept was originally proposed in the 1800’s as the caloric theory (Keenan, 1958); heat was believed to be an invisible substance (having mass) that transferred from one system to another as a result of a temperature difference. Although the caloric theory has been disproved, it is still common to refer to ‘heat transfer’. Heat is the transfer of energy due to a temperature gradient. This transfer process can occur by two very different mechanisms, referred to as conduction and radiation. Conduction heat transfer occurs due to the interactions of molecular (or smaller) scale energy carriers within a material. Radiation heat transfer is energy transferred as electromagnetic waves. In a ﬂowing ﬂuid, conduction heat transfer occurs in the presence of energy transfer due to bulk motion (which is not a heat transfer) and this leads to a substantially more complex situation that is referred to as convection.

1.1.2 Thermal Conductivity Conduction heat transfer occurs due to the interactions of micro-scale energy carriers within a material; the type of energy carriers depends upon the structure of the material. For example, in a gas or a liquid, the energy carriers are individual molecules whereas the energy carriers in a solid may be electrons or phonons (i.e., vibrations in the structure of the solid). The transfer of energy by conduction is fundamentally related to the interactions of these energy carriers; more energetic (i.e., higher temperature) energy carriers transfer energy to less energetic (i.e., lower temperature) ones, resulting in a net ﬂow of energy from hot to cold (i.e., heat transfer). Regardless of the type of energy carriers involved, conduction heat transfer can be characterized by Fourier’s law, provided that the length and time scales of the problem are large relative to the distance and time between energy carrier interactions. Fourier’s law relates the heat ﬂux in any direction to the temperature gradient in that direction. For example: q˙ = −k

∂T ∂x

(1-1)

1

2

One-Dimensional, Steady-State Conduction

Aluminum

Water

(a)

(b)

Figure 1-1: Conductivity functions in EES for (a) compressible substances and (b) incompressible substances.

where q˙ is the heat ﬂux in the x-direction and k is the thermal conductivity of the material. Fourier’s law actually provides the deﬁnition of thermal conductivity: k=

−q˙ ∂T ∂x

(1-2)

Thermal conductivity is a material property that varies widely depending on the type of material and its state. Thermal conductivity has been extensively measured and values have been tabulated in various references (e.g., NIST (2005)). The thermal conductivity of many substances is available within the Engineering Equation Solver (EES) program. It is suggested that the reader go through the tutorial that is provided in Appendix A.1 in order to become familiar with EES. Appendix A.1 can be found on the web site associated with this book (www.cambridge.org/nellisandklein). To access the thermal conductivity functions in EES, select Function Info from the Options menu and select the Fluid Properties button; this action displays the properties that are available for compressible ﬂuids. Navigate to Conductivity in the left hand window and select the ﬂuid of interest in the right hand window (e.g., Water), as shown in Figure 1-1(a). Select Paste in order to place the call to the Conductivity function into the Equations Window. Select the Solid/liquid properties button in order to access the properties for incompressible ﬂuids and solids, as shown in Figure 1-1(b). The EES code below speciﬁes the unit system to be used (SI) and then computes the conductivity of water, air, and aluminum (kw , ka , and kal ) at T = 20◦ C and p = 1.0 atm, $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in T=converttemp(C,K,20 [C]) P=1.0 [atm]∗ convert(atm,Pa) k w=Conductivity(Water,T=T,P=P) k a=Conductivity(Air,T=T) k al=k (‘Aluminum’, T)

“temperature” “pressure” “conductivity of water at T and P” “conductivity of air at T and P” “conductivity of aluminum at T”

which leads to kw = 0.59 W/m-K, ka = 0.025 W/m-K, and kal = 236 W/m-K. The conductivity of aluminum (an electrically conductive metal) is approximately 10,000x that

1.1 Conduction Heat Transfer

3 Temperature q⋅ ′′x+

q⋅ ′′x−

Lms

Figure 1-2: Energy ﬂow through a plane at position x.

x+Lms

x-Lms x

Position

of air (a dilute gas, at these conditions), with water (a liquid) falling somewhere between these values. It is possible to understand the thermal conductivity of various materials based on the underlying characteristics of their energy carriers, the microscopic physical entities that are responsible for conduction. For example, the kinetic theory of gases may be used to provide an estimate of the thermal conductivity of a gas and the thermal conductivity of electrically conductive metals can be understood based on a careful study of electron behavior. Consider conduction through a material in which a temperature gradient has been established in the x-direction, as shown in Figure 1-2. We can evaluate (approximately) the net rate of energy transferred through a plane that is located at position x. The ﬂux of energy carriers passing through the plane from left-to-right (i.e., in the positive x-direction) is proportional to the number density of the energy carriers (nms ) and their mean velocity (vms ). The energy carriers that are moving in the positive x-direction experienced their last interaction at approximately x–Lms (on average), where Lms is the distance between energy carrier interactions. (Actually, the last interaction would not occur exactly at this position since the energy carriers are moving relative to each other and also in the y- and z-directions.) The energy associated with these left-to-right moving carriers is proportional to the temperature at position x-Lms (T x−Lms ). The energy per unit area passing through the plane from left-to-right (q˙ x+ ) is given approximately by: q˙ x+ ≈ nms vms cms T x−Lms #carriers area-time

(1-3)

energy carrier

where cms is the ratio of the energy of the carrier to its temperature. Similarly, the energy per unit area carried through the plane in the negative x-direction by the energy carriers that are moving from right-to-left (q˙ x− ) is given approximately by: q˙ x− ≈ nms vms cms T x+Lms

(1-4)

The net conduction heat ﬂux passing through the plane (q˙ ) is the difference between q˙ x+ and q˙ x− , q˙ ≈ nms vms cms (T x−Lms − T x+Lms )

(1-5)

which can be rearranged to yield: q˙ ≈ −nms vms cms Lms

(T x+Lms − T x−Lms ) Lms

(1-6)

Recall from calculus that the deﬁnition of the temperature gradient is: T x+dx − T x−dx ∂T = lim dx→0 ∂x 2 dx

(1-7)

4

One-Dimensional, Steady-State Conduction

Thermal conductivity (W/m-K)

400 pure aluminum

100

304 stainless steel

bronze 10 glass (pyrex) 1

liquid water at 100 bar 0.1

nitrogen gas at 100 bar

nitrogen gas at 1 bar 0.01 275 300 325 350 375 400 425 450 475 500 525 550 Temperature (K)

Figure 1-3: Thermal conductivity of various materials as a function of temperature.

In the limit that the length between energy carrier interactions (Lms ) is much less than the length scale that characterizes the problem (Lchar ): T x+dx − T x−dx ∂T (T x+Lms − T x−Lms ) ≈ lim = dx→0 2 Lms 2 dx ∂x

(1-8)

Equation (1-8) can be substituted into Eq. (1-6) to yield: ∂T q˙ ≈ −2 nms vms cms Lms ∂x

(1-9)

∝k

The ratio of the length between energy carrier interactions to the length scale that characterizes the problem is referred to as the Knudsen number. The Knudsen number (Kn) should be calculated in order to ensure that continuum concepts (like Fourier’s law) are applicable: Kn =

Lms Lchar

(1-10)

If the Knudsen number is not small then continuum theory breaks down. This limit may be reached in micro- and nano-scale systems where Lchar becomes small as well as in problems involving rareﬁed gas where Lms becomes large. Specialized theory for heat transfer is required in these limits and the interested reader is referred to books such as Tien et al. (1998), Chen (2005), and Cercignani (2000). Comparing Eq. (1-9) with Fourier’s law, Eq. (1-1), shows that the thermal conductivity is proportional to the product of the number of energy carriers per unit volume, their average velocity, the mean distance between their interactions, and the ratio of the amount of energy carried by each energy carrier to its temperature: k ∝ nms vms cms Lms

(1-11)

The scaling relation expressed by Eq. (1-11) is informative. Figure 1-3 illustrates the thermal conductivity of several common materials as a function of temperature. Notice that metals have the largest thermal conductivity, followed by other solids and liquids, while gases have the lowest conductivity. Gases are diffuse and thus the number density of the energy carriers (gas molecules) is substantially less than for other

1.2 Steady-State 1-D Conduction without Generation

5

forms of matter. Pure metals have the highest thermal conductivity because energy is carried primarily by electrons which are numerous and fast moving. The thermal conductivity and electrical resistivity of pure metals are related (by the Weidemann-Franz law) because both electricity and thermal energy are transported by the same mechanism, electron ﬂow. Alloys have lower thermal conductivity because the electron motion is substantially impeded by the impurities within the structure of the material; this effect is analogous to reducing the parameter Lms in Eq. (1-11). In non-metals, the energy is carried by phonons (or lattice vibrations), while in liquids the energy is carried by molecules. Thermal Conductivity of a Gas This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein) and discusses the application of Eq. (1-11) to the particular case of an ideal gas where the energy carriers are gas molecules.

1.2 Steady-State 1-D Conduction without Generation 1.2.1 Introduction Chapters 1 through 3 examine conduction problems using a variety of conceptual, analytical, and numerical techniques. We will begin with simple problems and move eventually to complex problems, starting with truly one-dimensional (1-D), steady-state problems and working ﬁnally to two-dimensional and transient problems. Throughout this book, problems will be solved both analytically and numerically. The development of an analytical or a numerical solution is accomplished using essentially the same steps regardless of the complexity of the problem; therefore, each class of problem will be solved in a uniform and rigorous fashion. The use of computer software tools facilitates the development of both analytical and numerical solutions; therefore, these tools are introduced and used side-by-side with the theory.

1.2.2 The Plane Wall In general, the temperature in a material will be a function of position (x, y, and z, in Cartesian coordinates) and time (t). The deﬁnition of steady-state is that the temperature is unchanging with time. There are certain idealized problems in which the temperature varies in only one direction (e.g., the x-direction). These are one-dimensional (1-D), steady-state problems. The classic example is a plane wall (i.e., a wall with a constant cross-sectional area, Ac , in the x-direction) that is insulated around its edges. In order for the temperature distribution to be 1-D, each face of the wall must be subjected to a uniform boundary condition. For example, Figure 1-4 illustrates a plane wall in which the left face (x = 0) is maintained at TH while the right face (x = L) is held at TC . L

Figure 1-4: A plane wall with ﬁxed temperature boundary conditions.

TH

q⋅ x

q⋅ x+dx TC

x dx

6

One-Dimensional, Steady-State Conduction

The ﬁrst step toward developing an analytical solution for this, or any, problem involves the deﬁnition of a differential control volume. The control volume must encompass material at a uniform temperature; therefore, in this case it must be differentially small in the x-direction (i.e., it has width dx, see Figure 1-4) but can extend across the entire cross-sectional area of the wall as there are no temperature gradients in the y- or z-directions. Next, the energy transfers across the control surfaces must be deﬁned as well as any thermal energy generation or storage terms. For the steady-state, 1-D case considered here, there are only two energy transfers, corresponding to the rate of conduction heat transfer into the left side (i.e., at position x, q˙ x) and out of the right side (i.e., at position x + dx, q˙ x+dx) of the control volume. A steady-state energy balance for the differential control volume is therefore: q˙ x = q˙ x+dx

(1-19)

A Taylor series expansion of the term at x + dx leads to: q˙ x+dx = q˙ x +

dq˙ d2 q˙ dx2 d3 q˙ dx3 dx + 2 + 3 + ··· dx dx 2! dx 3!

(1-20)

The analytical solution proceeds by taking the limit as dx goes to zero so that the higher order terms in Eq. (1-20) can be neglected: dq˙ dx (1-21) q˙ x+dx = q˙ x + dx Substituting Eq. (1-21) into Eq. (1-19) leads to: dq˙ dx (1-22) q˙ x = q˙ x + dx or dq˙ =0 (1-23) dx Equation (1-23) is typical of the initial result that is obtained by considering a differential energy balance: a differential equation that is expressed in terms of energy rather than temperature. This form of the differential equation should be checked against your intuition. Equation (1-23) indicates that the rate of conduction heat transfer is not a function of x. For the problem in Figure 1-4, there are no sources or sinks of energy and no energy storage within the wall; therefore, there is no reason for the rate of heat transfer to vary with position. The ﬁnal step in the derivation of the governing equation is to substitute appropriate rate equations that relate energy transfer rates to temperatures. The result of this substitution will be a differential equation expressed in terms of temperature. The rate equation for conduction is Fourier’s law: ∂T (1-24) ∂x For our problem, the temperature is only a function of position, x, and therefore the partial differential in Eq. (1-24) can be replaced with an ordinary differential: q˙ = −k Ac

q˙ = −k Ac

dT dx

Substituting Eq. (1-25) into Eq. (1-23) leads to: dT d −k Ac =0 dx dx

(1-25)

(1-26)

1.2 Steady-State 1-D Conduction without Generation

7

If the thermal conductivity is constant then Eq. (1-26) may be simpliﬁed to: d2 T =0 dx2

(1-27)

The derivation of Eq. (1-27) is trivial and yet the steps are common to the derivation of the governing equation for more complex problems. These steps include: (1) the definition of an appropriate control volume, (2) the development of an energy balance, (3) the expansion of terms, and (4) the substitution of rate equations. In order to completely specify a problem, it is necessary to provide boundary conditions. Boundary conditions are information about the solution at the extents of the computational domain (i.e., the limits of the range of position and/or time over which your solution is valid). A second order differential equation requires two boundary conditions. For the problem shown in Figure 1-4, the boundary conditions are: T x=0 = T H

(1-28)

T x=L = T C

(1-29)

Equations (1-27) through (1-29) represent a well-posed mathematical problem: a second order differential equation with boundary conditions. Equation (1-27) is very simple and can be solved by separation and direct integration: dT =0 (1-30) d dx Equation (1-30) is integrated according to: dT d = 0 dx

(1-31)

Because Eq. (1-31) is an indeﬁnite integral (i.e., there are no limits on the integrals), an undetermined constant (C1 ) results from the integration: dT = C1 dx Equation (1-32) is separated and integrated again: dT = C1 dx

(1-32)

(1-33)

to yield T = C1 x + C2

(1-34)

Equation (1-34) shows that the temperature distribution must be linear; any linear function (i.e., any values of the constants C1 and C2 ) will satisfy the differential equation, Eq. (1-27). The constants of integration are obtained by forcing Eq. (1-34) to also satisfy the two boundary conditions, Eqs. (1-28) and (1-29): T H = C1 0 + C2

(1-35)

T C = C1 L + C2

(1-36)

Equations (1-35) and (1-36) are solved for C1 and C2 and substituted into Eq. (1-34) to provide the solution: T =

(T C − T H ) x + TH L

(1-37)

8

One-Dimensional, Steady-State Conduction

The heat transfer at any location within the wall is obtained by substituting the temperature distribution, Eq. (1-37), into Fourier’s law, Eq. (1-25): q˙ = −k Ac

dT k Ac = (T H − T C) dx L

(1-38)

Equation (1-38) shows that the heat transfer does not change with the position within the wall; this behavior is consistent with Eq. (1-23). The development of analytical solutions is facilitated using a symbolic software package such as Maple. It is suggested that the reader stop and go through the tutorial provided in Appendix A.2 which can be found on the web site associated with the book (www.cambridge.org/nellisandklein) in order to become familiar with Maple. Note that the Maple Command Applet that is discussed in Appendix A.2 is available on the internet and can be used even if you do not have access to the Maple software. The mathematical solution to the 1-D, steady-state conduction problem associated with a plane wall is easy enough that there is no reason to use Maple. However, it is worthwhile to use the problem in order to illustrate some of the basic steps associated with using Maple in anticipation of more difﬁcult problems. Start a new problem in Maple (select New from the File menu). Enter the governing differential equation, Eq. (1-27), and assign it to the function ODE; note that the second derivative of T with respect to x is obtained by applying the diff command twice. > restart; > ODE:=diff(diff(T(x),x),x)=0; ODE :=

d2 T (x) = 0 dx2

The solution to the ordinary differential equation is obtained using the dsolve command and assigned to the function Ts. > Ts:=dsolve(ODE); Ts := T(x) = C1x + C2

The solution identiﬁed by Maple is consistent with Eq. (1-34), except that Maple uses the variables C1 and C2 rather than C1 and C2 to represent the constants of integration. The two boundary conditions, Eqs. (1-35) and (1-36), are obtained symbolically using the eval command to evaluate the solution at a particular position and assigned to the functions BC1 and BC2. > BC1:=eval(Ts,x=0)=T_H; BC1 := (T(0) = C2) = T H > BC2:=eval(Ts,x=L)=T_C; BC2 := (T (L) = C1L + C2) = T C

The result of the eval command is almost, but not quite what is needed to solve for the constants. The expressions include the extraneous statements T(0) and T(L); use the rhs function in order to return just the expression on the right hand side.

1.2 Steady-State 1-D Conduction without Generation

9

> BC1:=rhs(eval(Ts,x=0))=T_H; BC1 := C2 = T H > BC2:=rhs(eval(Ts,x=L))=T_C; BC2 := C1L + C2 = T C

The constants are explicitly determined using the solve command. Note that the solve command requires two arguments; the ﬁrst is the equation or, in this case, set of equations to be solved (the boundary conditions, BC1 and BC2) and the second is the variable or set of variables to solve for (the constants C1 and C2). > constants:=solve({BC1,BC2},{_C1,_C2}); constants := { C2 = T H, C1 = −

T H−T C } L

The constants are substituted into the general solution using the subs command. The subs command requires two arguments; the ﬁrst is the set of deﬁnitions to be substituted and the second is the set of equations to substitute them into. > Ts:=subs(constants,Ts); Ts := T(x) = −

(T H − T C)x +T H L

This result is the same as Eq. (1-37).

1.2.3 The Resistance Concept Equation (1-38) is the solution for the rate of heat transfer through a plane wall. The equation suggests that, under some limiting conditions, conduction of heat through a solid can be thought of as a ﬂow that is driven by a temperature difference and resisted by a thermal resistance, in the same way that electrical current is driven by a voltage difference and resisted by an electrical resistance. Inspection of Eq. (1-38) suggests that the thermal resistance to conduction through a plane wall (R pw ) is given by: R pw =

L k Ac

(1-39)

allowing Eq. (1-38) to be rewritten: q˙ =

(T H − T C) R pw

(1-40)

The concept of a thermal resistance is broadly useful and we will often return to this idea of a thermal resistance in order to help develop a conceptual understanding of various heat transfer processes. The usefulness of Eqs. (1-39) and (1-40) go beyond the simple situation illustrated in Figure 1-4. It is possible to approximately understand conduction heat transfer in most any situation provided that you can identify the distance that heat must be conducted and the cross-sectional area through which the conduction occurs.

10

One-Dimensional, Steady-State Conduction dr q⋅ r TH

rin

TC q⋅ r+dr

L

Figure 1-5: A cylinder with ﬁxed temperature boundary conditions.

r rout

Resistance equations provide a method for succinctly summarizing a particular solution and we will derive resistance solutions for a variety of physical situations. By cataloging these resistance equations, it is possible to quickly use the solution in the context of a particular problem without having to go through all of the steps that were required in the original derivation. For example, if we are confronted with a problem involving steady-state heat transfer through a plane wall then it is not necessary to rederive Eqs. (1-19) through (1-38); instead, Eqs. (1-39) and (1-40) conveniently represent all of this underlying math.

1.2.4 Resistance to Radial Conduction through a Cylinder Figure 1-5 illustrates steady-state, radial conduction through an inﬁnitely long cylinder (or one with insulated ends) without thermal energy generation. The analytical solution to this problem is derived using the steps described in Section 1.2.2. The differential energy balance (see Figure 1-5) leads to: q˙ r = q˙ r+dr

(1-41)

The r + dr term in Eq. (1-41) is expanded and Fourier’s law is substituted in order to reach: dT d −k Ac =0 (1-42) dr dr The difference between the plane wall geometry considered in Section 1.2.2 and the cylindrical geometry considered here is that the cross-sectional area for heat transfer, Ac in Eq. (1-42), is not constant but rather varies with radius: ⎡ ⎤ d ⎣ dT ⎦ −k 2 =0 (1-43) πr L dr dr Ac

where L is the length of the cylinder. Assuming that the thermal conductivity is constant, Eq. (1-43) is simpliﬁed to: d dT r =0 (1-44) dr dr and integrated twice according to the following steps: dT d r = 0 dr r

dT = C1 dr

(1-45) (1-46)

1.2 Steady-State 1-D Conduction without Generation

11

C1 dr r

(1-47)

T = C1 ln (r) + C2

(1-48)

dT =

where C1 and C2 are constants of integration, evaluated by applying the boundary conditions: T H = C1 ln (rin ) + C2

(1-49)

T C = C1 ln (rout ) + C2

(1-50)

After some algebra, the temperature distribution in the cylinder is obtained: r out ln T = T C + (T H − T C) r rout ln rin

(1-51)

The rate of heat transfer is given by: q˙ = −k 2 π r L

dT 2πLk (T H − T C) = rout dr ln r in

(1-52)

1/Rcyl

Therefore, the thermal resistance to radial conduction through a cylinder (Rcyl ) is: rout ln rin Rcyl = (1-53) 2πLk It is worth noting that the thermal resistance to radial conduction through a cylinder must be computed using the ratio of the outer to the inner radii in the numerator, regardless of the direction of the heat transfer. TC dr TH

r q⋅ r+dr

q⋅r

Figure 1-6: A sphere with ﬁxed temperature boundary conditions.

rin rout

1.2.5 Resistance to Radial Conduction through a Sphere Figure 1-6 illustrates steady-state, radial conduction through a sphere without thermal energy generation. The differential energy balance (see Figure 1-6) leads to: q˙ r = q˙ r+dr

(1-54)

12

One-Dimensional, Steady-State Conduction

which is expanded and used with Fourier’s law to reach: dT d −k Ac =0 dr dr

(1-55)

The cross-sectional area for heat transfer is the surface area of a sphere: ⎡ ⎤ dT d ⎣ ⎦ =0 −k 4 π r2 dr dr

(1-56)

Ac

Assuming that k is constant allows Eq. (1-56) to be simpliﬁed: d 2 dT r =0 dr dr

(1-57)

Equation (1-57) is entered in Maple: > restart; > ODE:=diff(rˆ2*diff(T(r),r),r)=0;

ODE := 2r

2 d d T(r) + r2 T(r) =0 dr dr2

and solved: > Ts:=dsolve(ODE); Ts := T(r) = C1 +

C2 r

The boundary conditions are: T r=rin = T H

(1-58)

T r=rout = T C

(1-59)

These equations are entered in Maple: > BC1:=rhs(eval(Ts,r=r_in))=T_H; BC1 := C1 +

C2 =T H r in

> BC2:=rhs(eval(Ts,r=r_out))=T_C; BC2 := C1 +

C2 =T C r out

1.2 Steady-State 1-D Conduction without Generation

13

The constants are obtained by solving this system of two equations and two unknowns: > constants:=solve({BC1,BC2},{_C1,_C2}); T Hr in − T Cr out r in r out(−T C + T H) constants := { C1 = , C2 = − } −r out + r in −r out + r in

and substituted into the general solution: > Ts:=subs(constants,Ts); Ts := T(r) =

r in r out(−T C + T H) T Hr in − T Cr out − −r out + r in (−r out + r in)r

The heat transfer at any radial location is given by Fourier’s law: q˙ = −k 4 π r2

> q_dot:=-k*4*pi*rˆ2*diff(Ts,r); q dot := −4 k π r2

dT dr

(1-60)

d 4 k π r in r out(−T C + T H) T(r) = − dr −r out + r in

and used to compute the thermal resistance for steady-state, radial conduction through a sphere: TH − TC q˙

Rsph =

(1-61)

> R_sph:=(T_H-T_C)/rhs(q_dot); R sph := −

which can be simpliﬁed to:

−r out + r in 4 k π r in r out

Rsph =

1 1 − rin rout 4πk

(1-62)

1.2.6 Other Resistance Formulae Many heat transfer processes may be cast in the form of a resistance formula, allowing problems involving various types of heat transfer to be represented conveniently using thermal resistance networks. Resistance networks can be solved using techniques borrowed from electrical engineering. Also, it is often possible to obtain a physical feel for the problem by inspection of the thermal resistance network. For example, small resistances in series with large ones will tend to be unimportant. Large resistances in parallel

14

One-Dimensional, Steady-State Conduction

with small ones can also be neglected. This type of understanding is important and can be obtained quickly using thermal resistance networks. Convection Resistance Convection is discussed in Chapters 4 through 7 and refers to heat transfer between a surface and a moving ﬂuid. The rate equation that characterizes the rate of convection heat transfer (q˙ conv ) is Newton’s law of cooling: q˙ conv = h As (T s − T ∞ )

(1-63)

1/Rconv

where h is the average heat transfer coefﬁcient, As is the surface area at temperature Ts that is exposed to ﬂuid at temperature T∞ . Note that the heat transfer coefﬁcient is not a material property like thermal conductivity, but rather a complex function of the geometry, ﬂuid properties, and ﬂow conditions. By inspection of Eq. (1-63), the thermal resistance associated with convection (Rconv ) is: R conv =

1 h As

(1-64)

Contact Resistance Contact resistance refers to the complex phenomenon that occurs when two solid surfaces are brought together. Regardless of how well prepared the surfaces are, they are not ﬂat at the micro-scale and therefore energy carriers in either solid cannot pass through the interface unimpeded. The energy carriers in two dissimilar materials may not be the same in any case. The result is a temperature change at the interface that, at the macro-scale, appears to occur over an inﬁnitesimally small spatial extent and grows in proportion to the rate of heat transfer across the interface. In reality, the temperature does not drop discontinuously but rather over some micro-scale distance that depends on the details of the interface. This phenomenon is usually modeled by characterizing the interface as having an area-speciﬁc contact resistance (Rc , often provided in K-m2 /W). The resistance to heat transfer across an interface (Rc ) is: Rc =

Rc As

(1-65)

where As is the contact area (the projected area of the surfaces, ignoring their microstructure). Contact resistance is not a material property but rather a complex function of the micro-structure, the properties of the two materials involved, the contact pressure, the interstitial material, etc. The area-speciﬁc contact resistance for interface conditions that are commonly encountered has been measured and tabulated in various references, for example Schneider (1985). Some representative values are listed in Table 1-1. The area-speciﬁc contact resistance tends to be reduced with increasing clamping pressure and smaller surface roughness. One method for reducing contact resistance is to insert a soft metal (e.g., indium) or grease into the interface in order to improve the heat transfer across the interstitial gap. The values listed in Table 1-1 can be used to determine whether contact resistance is likely to play an important role in a speciﬁc application. However, if contact resistance is important, then more precise data for the interface of interest should be obtained or measurements should be carried out.

Table 1-1: Area-speciﬁc contact resistance for some interfaces, from Schneider (1985) and Fried (1969). Materials

Clamping pressure

Surface roughness

Interstitial material

Temperature

Area-speciﬁc contact resistance

copper-to-copper copper-to-copper aluminum-to-aluminum aluminum-to-aluminum stainless-to-stainless stainless-to-stainless stainless-to-stainless stainless-to-stainless stainless-to-aluminum aluminum-to-aluminum aluminum-to-aluminum aluminum-to-aluminum aluminum-to-aluminum aluminum-to-aluminum

100 kPa 1000 kPa 100 kPa 100 kPa 100 kPa 1000 kPa 100 kPa 1000 kPa 100 kPa 1000 kPa 100 kPa 100 kPa 100 kPa 100 kPa

0.2 μm 0.2 μm 0.3 μm 1.5 μm 1.3 μm 1.3 μm 0.3 μm 0.3 μm 1.2 μm 0.3 μm 10 μm 10 μm 10 μm 10 μm

vacuum vacuum vacuum vacuum vacuum vacuum vacuum vacuum air air air helium hydrogen silicone oil

46◦ C 46◦ C 46◦ C 46◦ C 30◦ C 30◦ C 30◦ C 30◦ C 93◦ C 93◦ C 20◦ C 20◦ C 20◦ C 20◦ C

1.5 × 10−4 K-m2 /W 1.3 × 10−4 K-m2 /W 2.5 × 10−3 K-m2 /W 3.3 × 10−3 K-m2 /W 4.5 × 10−3 K-m2 /W 2.4 × 10−3 K-m2 /W 2.9 × 10−3 K-m2 /W 7.7 × 10−4 K-m2 /W 3.3 × 10−4 K-m2 /W 6.7 × 10−5 K-m2 /W 2.8 × 10−4 K-m2 /W 1.1 × 10−4 K-m2 /W 0.72 × 10−4 K-m2 /W 0.53 × 10−4 K-m2 /W

15

16

One-Dimensional, Steady-State Conduction

Radiation Resistance Radiation heat transfer occurs between surfaces due to the emission and absorption of electromagnetic waves, as described in Chapter 10. Radiation heat transfer is complex when many surfaces at different temperatures are involved; however, in the limit that a single surface at temperature T s interacts with surroundings at temperature T sur then the radiation heat transfer from the surface can be calculated according to: 4 (1-66) q˙ rad = As σ ε T s4 − T sur where As is the area of the surface, σ is the Stefan-Boltzmann constant (5.67 × 10−8 W/m2 -K4 ), and ε is the emissivity of the surface. Emissivity is a parameter that ranges between near 0 (for highly reﬂective surfaces) to near 1 (for highly absorptive surfaces). Note that both Ts and T sur must be expressed as absolute temperature (i.e., in units K rather than ◦ C) in Eq. (1-66). Equation (1-66) does not seem to resemble a resistance equation because the heat transfer is not driven by a difference in temperatures but rather by a difference in temperatures to the fourth power. However, Eq. (1-66) may be expanded to yield: 2 (1-67) q˙ rad = As σ ε T s2 + T sur (T s + T sur ) (T s − T sur ) hrad

1/Rrad

Comparing Eq. (1-67) for radiation to Eq. (1-63) for convection shows that a ‘radiation heat transfer coefﬁcient’, hrad , can be deﬁned as: 2 (1-68) hrad = σ ε T s2 + T sur (T s + T sur ) The radiation heat transfer coefﬁcient is a useful quantity for many problems because it allows convection and radiation to be compared directly, as discussed in Section 10.6.2. Equation (1-67) suggests that an appropriate thermal resistance for radiation heat transfer (Rrad ) is: Rrad =

As σ ε

(T s2

1 2 ) (T + T ) + T sur s sur

(1-69)

Because the absolute surface and surrounding temperatures are both typically large and not too different from each other, Eq. (1-69) can often be approximated by: Rrad ≈

1 As σ ε 4 T

3

(1-70)

where T is the average of the surface and surrounding temperatures: T =

T s + T sur 2

(1-71)

With this approximation, the radiation heat transfer coefﬁcient is: hrad ≈ σ ε 4 T

3

(1-72)

The resistance associated with radiation and the radiation heat transfer coefﬁcient are both clearly temperature-dependent. However, the conductivity, contact resistance, and average heat transfer coefﬁcient that are required to compute other types of resistances are also temperature-dependent and therefore the resistance concept can only be approximate in any case. A summary of the thermal resistance associated with several common situations is presented in Table 1-2.

1.2 Steady-State 1-D Conduction without Generation

17

Table 1-2: A summary of common resistance formulae. Situation Plane wall

Resistance formula R pw =

Rcyl Sphere (radial heat transfer) Convection

Contact between surfaces Radiation (exact)

Radiation (approximate)

L = wall thickness (|| to heat ﬂow) k = conductivity Ac = cross-sectional area (⊥ to heat ﬂow)

L k Ac

rout rin = 2πLk

Cylinder (radial heat transfer)

Rsph =

ln

L = cylinder length k = conductivity rin and rout = inner and outer radii

1 1 1 − 4 π k rin rout

k = conductivity rin and rout = inner and outer radii

Rconv = Rc =

h = average heat transfer coefﬁcient As = surface area exposed to convection

1 ¯h As

Rc = area speciﬁc contact resistance As = surface area in contact

Rc As

Rrad =

Rrad ≈

Nomenclature

1 2 ) (T + T As σ ε (T s2 + T sur s sur )

1 As σ ε 4 T

3

As = radiating surface area σ = Stefan-Boltzmann constant ε = emissivity T s = absolute surface temperature T sur = absolute surroundings temperature As = radiating surface area σ = Stefan-Boltzmann constant ε = emissivity T = average absolute temperature

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR Figure 1 illustrates a spherical dewar containing saturated liquid oxygen that is kept at pressure pLO x = 25 psia; the saturation temperature of oxygen at this pressure is TLO x = 95.6 K. The dewar consists of an inner and outer metal liner separated by polystyrene foam insulation. The inner metal liner has inner radius r mli,in = 10.0 cm and thickness thm = 2.5 mm. The outer metal liner also has thickness thm = 2.5 mm. The conductivity of both metal liners is k m = 15 W/m-K. The heat transfer coefﬁcient between the oxygen within the dewar and the inner surface of the dewar is hin = 150 W/m2 -K. The outer surface of the dewar is surrounded by air at T∞ = 20◦ C and radiates to surroundings that are also at T∞ = 20◦ C. The emissivity of the outer surface of the dewar is ε = 0.7. The heat transfer coefﬁcient between the outer surface of the dewar and the surrounding air is hout = 6 W/m2 -K. The area-speciﬁc

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

Note that useful reference information, such as Table 1-2, is included in the Heat Transfer Reference Section of EES in order to facilitate solving heat transfer problems without requiring that you locate a written reference book. To access this section, select the Reference Material from the Heat Transfer menu. This will open an online document that contains material from this book. Notice that the Heat Transfer menu also includes all of the examples that are associated with the book.

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

18

One-Dimensional, Steady-State Conduction

contact resistance that characterizes the interfaces between the insulation and the adjacent metal liners is Rc = 3.0 × 10−3 K-m2 /W. The thickness of the insulation between the two metal liners is thins = 1.0 cm. You are trying to evaluate the impact of using polystyrene foam insulation in place of the more expensive insulation that is currently used. Flynn (2005) suggests that the conductivity of polystyrene foam at cryogenic temperatures is approximately k ins = 330 μW/cm-K. a) Draw a network that represents this situation using 1-D resistances. saturated liquid oxygen TLOx = 95.6 K, hin = 150 W/m2 -K

Rc′′ = 3x10

rmli, in = 10.0 cm T∞ = 20°C 2 hout = 6 W/m -K

−3

2

K-m /W

km = 15 W/m-K kins = 330 μW/cm-K

k m = 15 W/m-K

ε = 0.7

insulation outer metal liner thm = 2.5 mm

inner metal liner insulation

thins = 1.0 cm thm = 2.5 mm

outer metal liner

Figure 1: Spherical dewar containing saturated liquid oxygen.

The resistance network is illustrated in Figure 2.

K K R cond, mlo = 0.0010 R cond, mli = 0.0013 W W K R cond, ins = 2.09 TLOx = 95.6 K W

Rconv, in = 0.053

K W R c, ins, mli = 0.023

Rc, ins, mlo = 0.053

K W

Rrad = 1.91

K W

T∞ = 293.2 K

Ts, out

K W

The resistances include: Rconv, in = convection from inside surface Rcond, mli = conduction through inner metal liner Rc, ins, mli = contact between inner metal liner & insulation Rcond, ins = conduction through insulation Rc, ins, mlo = contact between outer metal liner & insulation Rcond, mlo = conduction through outer metal liner Rrad = radiation Rconv, out = convection from outer surface Figure 2: Resistance network representing the dewar.

R conv, out = 1.00

K W

19

The resistance network interacts with the surrounding air and surroundings (at T∞ ) and the saturated liquid oxygen (at TLO x ). b) Estimate the rate of heat transfer to the liquid oxygen. The solution will be carried out using EES. It is assumed that you have already been exposed to the EES software by completing the self-guided tutorial contained in Appendix A.1. The ﬁrst step in preparing a successful solution to any problem with EES is to enter the inputs to the problem and set their units. Experience has shown that it is generally best to work exclusively in SI units (m, J, K, kg, Pa, etc.) because this unit system is entirely self-consistent. If the problem statement includes parameters in other units, they can be converted to SI units within the “Inputs” section of the code. The upper section of your EES code should look something like: “EXAMPLE 1.2-1: Liquid Oxygen Dewar” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” “pressure of liquid oxygen” p LOx=25 [psia]∗ convert(psia,Pa) T LOx=95.6 [K] “temperature of liquid oxygen” h bar in =150 [W/mˆ2-K] “heat transfer coefﬁcient between the liquid oxygen and the inner wall” r mli in=10 [cm]∗ convert(cm,m) “inner radius of the inner metal liner” th m=2.5 [mm]∗ convert(mm,m) “thickness of inner metal liner” th ins cm=1.0 [cm] “thickness of insulation, in cm” th ins=th ins cm∗ convert(cm,m) “thickness of insulation” e=0.7 [-] “emissivity of outside surface” T inﬁnity=converttemp(C,K,20 [C]) “temperature of surroundings and surrounding air” R c=3.0e-3 [K-mˆ2/W] “area-speciﬁc contact resistance” k ins=330 [microW/cm-K]∗ convert(microW/cm-K,W/m-K) “mean conductivity of insulation” k m=15 [W/m-K] “conductivity of metal” h bar out=6 [W/mˆ2-K] “heat transfer coefﬁcient between outer wall and surrounding air”

The resistance to convection between the inner surface of the dewar and the oxygen is: Rconv,in =

R conv in=1/(4∗ pi∗ r mli inˆ2∗ h bar in)

1 2 hin 4 π r mli,in

“convection resistance to liquid oxygen”

The inner radius of the insulation is: rins,in = r mli,in + thm The resistance to conduction through the inner metal liner is: 1 1 1 − Rcond ,mli = 4 π k m r mli,in rins,in

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.2 Steady-State 1-D Conduction without Generation

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

20

One-Dimensional, Steady-State Conduction

and the contact resistance resistance between the inner metal liner and the insulation is: Rc Rc,ins,mli = 2 4 π rins,in

r ins in=r mli in+th m “inner radius of insulation” R cond mli=(1/r mli in-1/r ins in)/(4∗ pi∗ k m) “conduction resistance of inner metal liner” R c ins mli=R c/(4∗ pi∗ r ins inˆ2) “contact resistance between inner metal liner and insulation”

The outer radius of the insulation is: rins,out = rins,in + thins The resistance to conduction through the insulation is: 1 1 1 Rcond ,ins = − 4 π k ins rins,in rins,out and the contact resistance resistance between the insulation and the outer metal liner is: Rc Rc,ins,mlo = 2 4 π rins,out

r ins out=r ins in+th ins “outer radius of insulation” R cond ins=(1/r ins in-1/r ins out)/(4∗ pi∗ k ins) “conduction resistance of insulation” R c ins mlo=R c/(4∗ pi∗ r ins outˆ2) “contact resistance between insulation and outer metal liner”

The outer radius of the outer metal liner is: r mlo,out = rins,out + thm The resistance to conduction through the outer metal liner is: 1 1 1 Rcond ,mlo = − 4 π k m rins,out r mlo,out and the convection resistance between the outer surface of the dewar and the air is: 1 Rconv,out = 2 hout 4 π r mli,out

r mlo out=r ins out+th m R cond mlo=(1/r ins out-1/r mlo out)/(4∗ pi∗ k m) R conv out=1/ (4∗ pi∗ r mlo outˆ2∗ h bar out)

“outer radius of outer metal liner” “conduction resistance of outer metal liner” “convection resistance to surrounding air”

The surface temperature on the outside of the dewar (Ts,out in Figure 2) cannot be known until the problem is solved and yet it must be used to calculate the resistance to radiation, Rrad . One of the nice things about using the EES software to solve this problem is that the software can deal with this type of nonlinearity and provide the solution to the implicit equations. It is this capability that simultaneously makes

21

EES so powerful and yet sometimes, ironically, difﬁcult to use. EES should be able to solve equations regardless of the order in which they are entered. However, you should enter equations in a sequence that allows you to solve them as you enter them; this is exactly what you would be forced to do if you were to solve the problem using a typical programming language (e.g., MATLAB, FORTRAN, etc.). This technique of entering your equations in a systematic order provides you with the opportunity to debug each subset of equations as you move along, rather than waiting until all of the equations have been entered before you try to solve them. Another beneﬁt of approaching a problem in this sequential manner is that you can consistently update the guess values associated with the variables in your problem. EES solves your equations using a nonlinear relaxation technique and therefore the closer the guess values of the variables are to “reasonable” values, the more likely it is that EES will ﬁnd the correct solution. To proceed with the solution to this problem using EES, it is helpful to initially assume a reasonable surface temperature (e.g., a reasonable guess might be the average of the surrounding and the liquid oxygen temperatures) so that it is possible to calculate a value of the radiation resistance: Rrad =

1 2 2 2 (T 4 π r mli,out σ ε Ts,out + T∞ s,out + T∞ )

and continue with the solution. T s out = (T LOx+T inﬁnity)/2 “guess for the surface temperature (removed to complete problem)” R rad=1/(4∗ pi∗ r mlo outˆ2∗ sigma#∗ e∗ (T s outˆ2+T inﬁnityˆ2)∗ (T s out+T inﬁnity)) “radiation resistance”

Solve the equations that have been entered (select Calculate from the Solve menu) and check that your answers make sense. Verify that the variables and equations have a consistent set of units by setting the units for each of the variables. The best way to do this is to go to the Variable Information window (select Variable Info from the Options menu) and enter the units for each variable in the Units column. Once this is done, check the units for your problem (select Check Units from the Calculate menu) in order to make sure that all of the units are consistent with the equations. Alternatively, units can be set by right-clicking on the variables in the Solution Window. The total resistance separating the liquid oxygen from the surroundings is: Rt ot al = Rconv,in + Rcond ,mli + Rc,ins,mli +Rcond ,ins + Rc,ins,mlo + Rcond ,mlo +

1 Rconv,out

+

1 Rrad

−1

and the heat transfer rate from the surroundings to the liquid oxygen is: q˙ =

(T∞ − TLO x ) Rt ot al

R total=R conv in+R cond mli+R c ins mli+R cond ins+R c ins mlo& +R cond mlo+(1/R conv out+1/R rad)ˆ(-1) q dot=(T inﬁnity-T LOx)/R total

“total resistance” “heat ﬂow”

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.2 Steady-State 1-D Conduction without Generation

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

22

One-Dimensional, Steady-State Conduction

At this point, we can use the heat transfer rate to recalculate the surface temperature (as opposed to assuming it). Ts,out = TLO x + q˙ (Rconv,in + Rcond ,mli + Rc,ins,mli + Rcond ,ins + Rc,ins,mlo + Rcond ,mlo ) It is necessary to comment out or delete the equation that provided the assumed surface temperature and instead calculate the surface temperature correctly. This step creates an implicit set of nonlinear equations. Before you ask EES to solve the set of equations, it is a good idea to update the guess values for each variable (select Update Guesses from the Calculate menu). {T s out=(T LOx+T inﬁnity)/2} “guess for the surface temperature” T s out=T LOx+q dot∗ (R conv in+ R cond mli+R c ins mli+R cond ins+R c ins mlo+R cond mlo) “surface temperature”

The rate of heat transfer to the liquid oxygen is q˙ = 69.4 W. Resistance networks provide substantial insight into the problem. Figure 2 shows the magnitude of each of the resistances in the network. The resistances associated with conduction through the insulation, radiation from the surface of the dewar, and convection from the surface of the dewar are of the same order of magnitude and large relative to the others in the circuit. Conduction through the insulation is much more important than conduction through the metal liners, convection to the liquid oxygen or the contact resistance; these resistances can probably be neglected in a rough analysis and certainly very little effort should be expended to better understand these aspects of the problem. Both radiation and convection from the outer surface are important, as they are of similar magnitude. The convection resistance is smaller and therefore more heat will be transferred by convection from the surface than is radiated from the surface. If the radiation resistance had been much larger than the convection resistance (as is often the case in forced convection problems where the convection heat transfer coefﬁcient is much larger) then radiation could be neglected. The smallest resistance in a parallel network will dominate because most of the energy will tend to ﬂow through that resistance. It is almost always a good idea to estimate the size of the resistances in a heat transfer problem prior to solving the problem. Often it is possible to simplify the analysis considerably, and the size of the resistances can certainly be used to guide your efforts. For this problem, a detailed analysis of conduction through the metal liner or a lengthy search for the most accurate value of the thermal conductivity of the metal would be a misguided use of time whereas a more accurate measurement of the conductivity of the insulation might be important. c) Plot the rate of heat transfer to the liquid oxygen as a function of the insulation thickness. In order to generate the requested plot, it is necessary to parametrically vary the insulation thickness. The speciﬁed value of the insulation thickness is commented out {th ins cm=1.0 [cm]}

“thickness of insulation, in cm”

23

and a parametric table is generated (select New Parametric Table from the Tables menu) that includes the variables th ins cm and q dot (Figure 3).

Figure 3: New Parametric Table Window.

Right-click on the th_ins_cm column and select Alter Values; vary the thickness from 0 cm to 10 cm and solve the table (select Solve Table from the Calculate menu). Prepare a plot of the results (select New Plot Window from the Plots menu and then select X-Y Plot) by selecting the variable th_ins_cm for the X-Axis and q_dot for the Y-Axis (Figure 4).

Figure 4: New Plot Setup Window.

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.2 Steady-State 1-D Conduction without Generation

One-Dimensional, Steady-State Conduction

Figure 5 illustrates the rate of heat transfer as a function of the insulation thickness. 100 90 80 Heat transfer rate (W)

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

24

70 60 50 40 30 20 10 0 0

1

2

3 4 5 6 7 Insulation thickness (cm)

8

9

10

Figure 5: Heat transfer rate as a function of insulation thickness.

1.3 Steady-State 1-D Conduction with Generation 1.3.1 Introduction The generation of thermal energy within a conductive medium may occur through ohmic dissipation, chemical or nuclear reactions, or absorption of radiation. According to the ﬁrst law of thermodynamics, energy cannot be generated (excluding nuclear reactions); however, it can be converted from other forms (e.g., electrical energy) to thermal energy. The energy balance that we use to solve these problems is then strictly a thermal energy conservation equation. The addition of thermal energy generation to the 1-D steadystate solutions considered in Section 1.2 is straightforward and the steps required to obtain an analytical solution are essentially the same.

1.3.2 Uniform Thermal Energy Generation in a Plane Wall Consider a plane wall with temperatures ﬁxed at either edge that experiences a volumetric generation of thermal energy, as shown in Figure 1-7.

L

TH

q⋅ x

x

q⋅ x+dx g⋅ dx

Figure 1-7: Plane wall with thermal energy generation and ﬁxed temperature boundary conditions.

TC

1.3 Steady-State 1-D Conduction with Generation

25

The problem is assumed to be 1-D in the x-direction and therefore an appropriate differential control volume has width dx (see Figure 1-7). Notice the additional energy term in the control volume that is related to the generation of thermal energy. A steadystate energy balance includes conduction into the left-side of the control volume (q˙ x, at position x), generation within the control volume (g), ˙ and conduction out of the rightside of the control voume (q˙ x+dx, at position x + dx): q˙ x + g˙ = q˙ x+dx

(1-73)

or, after expanding the right side: q˙ x + g˙ = q˙ x +

dq˙ dx dx

(1-74)

The rate of thermal energy generation within the control volume can be expressed as the product of the volume enclosed by the control volume and the rate of thermal energy generation per unit volume, g˙ (which may itself be a function of position or temperature): g˙ = g˙ Ac dx

(1-75)

where Ac is the cross-sectional area of the wall. The conduction term is expressed using Fourier’s law: q˙ = −k Ac

dT dx

Substituting Eqs. (1-76) and (1-75) into Eq. (1-74) results in dT d −k Ac dx g˙ Ac dx = dx dx which can be simpliﬁed (assuming that conductivity is constant): d dT g˙ =− dx dx k Equation (1-78) is separated and integrated: dT g˙ d = − dx dx k

(1-76)

(1-77)

(1-78)

(1-79)

If the volumetric rate of thermal energy generation is spatially uniform, then the integration leads to: g˙ dT = − x + C1 dx k where C1 is a constant of integration. Equation (1-80) is integrated again: g˙ dT = − x + C1 dx k

(1-80)

(1-81)

which leads to: T =−

g˙ 2 x + C1 x + C2 2k

(1-82)

26

One-Dimensional, Steady-State Conduction

where C2 is another constant of integration. Note that the same solution can be obtained using Maple. The governing differential equation, Eq. (1-78), is entered in Maple: > restart; > GDE:=diff(diff(T(x),x),x)=-gv/k; GDE :=

d2 gv T(x) = − dx2 k

and solved: > Ts:=dsolve(GDE); Ts := T(x) = −

gvx2 + C1x + C2 2k

Equation (1-82) satisﬁes the governing differential equation, Eq. (1-78), throughout the computational domain (i.e., from x = 0 to x = L) for arbitrary values of C1 and C2 . It is easy to use Maple to check that this is true: > rhs(diff(diff(Ts,x),x))+gv/k; 0

Note that it is often a good idea to use Maple to quickly doublecheck that an analytical solution does in fact satisfy the original governing differential equation. All that remains is to force the general solution, Eq. (1-82), to satisfy the boundary conditions by adjusting the constants C1 and C2 . The ﬁxed temperature boundary conditions shown in Figure 1-7 correspond to: T x=0 = T H

(1-83)

T x=L = T C

(1-84)

Substituting Eq. (1-82) into Eqs. (1-83) and (1-84) leads to: C2 = T H −

g˙ 2 L + C1 L + T H = T C 2k

(1-85) (1-86)

Solving Eq. (1-86) for C1 leads to: C1 =

g˙ (T H − T C) L− 2k L

(1-87)

1.3 Steady-State 1-D Conduction with Generation

27

Substituting Eqs. (1-85) and (1-87) into Eq. (1-82) leads to: T =

g˙ L2 2k

x x 2 (T H − T C) − x + TH − L L L

(1-88)

Again, Maple can be used to achieve the same result. The boundary condition equations are deﬁned in Maple:

> BC1:=rhs(eval(Ts,x=0))=T_H; BC1 := C2 = T H > BC2:=rhs(eval(Ts,x=L))=T_C; BC2 := −

gvL2 + C1L + C2 = T C 2k

and solved for the two constants:

> constants:=solve({BC1,BC2},{_C1,_C2}); constants := { C2 = T H, C1 =

gvL2 − 2T Hk + 2T Ck } 2 Lk

The constants are substituted into the general equation:

> Ts:=subs(constants,Ts); Ts := T(x) = −

(gvL2 − 2T Hk + 2T Ck)x gvx2 + +T H 2k 2 Lk

It is good practice to examine any solution and verify that it makes sense. By inspection, it is clear that Eq. (1-88) limits to T H at x = 0 and T C at x = L; therefore, the boundary conditions were implemented correctly. Also, in the absence of any generation Eq. (1-88) limits to Eq. (1-37), the solution that was derived in Section 1.2.2 for steady-state conduction through a plane wall without generation. Figure 1-8 illustrates the temperature distribution for TH = 80◦ C and TC = 20◦ C with L = 1.0 cm and k = 1 W/m-K for various values of g˙ . The temperature proﬁle becomes more parabolic as the rate of thermal energy generation increases because energy must be transferred toward the edges of the wall at an increasing rate. The temperature gradient is proportional to the local rate of conduction heat transfer; as g˙ increases, the heat transfer rate at x = L increases while the heat transfer rate entering at x = 0 decreases and eventually becomes negative (i.e., heat actually begins to leave from both edges of the wall). This effect is evident in Figure 1-8 by observing that the temperature gradient at x = 0 changes from a negative to a positive value as g˙ is increased.

28

One-Dimensional, Steady-State Conduction 180

g⋅ ′′′= 1× 107 W/m3

160

Temperature (°C)

140

g⋅ ′′′ = 5 × 106 W/m3

120 100

g⋅ ′′′ = 2 × 106 W/m3

80 60 40 20 0

g⋅ ′′′ = 1× 106 W/m3 g⋅ ′′′ = 5 × 105 W/m3 0.1

0.2

0.3

g⋅ ′′′ = 0 W/m3

0.4 0.5 0.6 0.7 Axial location (cm)

0.8

0.9

1

Figure 1-8: Temperature distribution within a plane wall with thermal energy generation (k = 1 W/m-K, T H = 80◦ C, T C = 20◦ C, L = 1.0 cm).

The rate of heat transfer by conduction in the wall is obtained by applying Fourier’s law to the solution for the temperature distribution: q˙ = −k Ac

dT dx

Substituting Eq. (1-88) into Eq. (1-89) leads to: x 1 k Ac − + q˙ = Ac g˙ L (T H − T C) L 2 L

(1-89)

(1-90)

The heat transfer rate is not constant with position. Therefore, the plane wall with generation cannot be represented as a thermal resistance in the manner discussed in Section 1.2. However, it is always a good idea to carry out a number of sanity checks on your solution and the resistance concepts discussed in Section 1.2 provide an excellent mechanism for doing this. Equation (1-88) and Figure 1-8 both show that there is a maximum temperature elevation (relative to the zero generation case) that occurs at the center of the wall. Substituting x = L/2 into the 1st term in Eq. (1-88) shows that the magnitude of the maximum temperature elevation is: T max =

g˙ L2 8k

(1-91)

Maple can provide this result as well. The zero-generation solution is obtained by substituting g˙ = 0 into the original solution: > Tsng:=subs(gv=0,Ts); Tsng := T(x) =

(−2T Hk + 2T Ck)x +T H 2 Lk

1.3 Steady-State 1-D Conduction with Generation

29

Then the temperature elevation is the difference between the original solution and the zero-generation solution: > DeltaT:=rhs(Ts-Tsng); DeltaT := −

gvx2 (gvL2 − 2T Hk + 2T Ck)x (−2T Hk + 2T Ck)x + − 2k 2 Lk 2 Lk

and the maximum value of the temperature elevation is evaluated at x = L/2: > DeltaTmax=eval(DeltaT,x=L/2); DeltaTmax = −

gvL2 gvL2 − 2T Hk + 2T Ck −2T Hk + 2T Ck + − 8k 4k 4k

This result can be simpliﬁed using the simplify command: > simplify(%); DeltaTmax =

gvL2 8k

Note that the % character refers to the result of the previous command (i.e., the output of the last calculation). The temperature elevation occurs because the energy that is generated within the wall must be conducted to one of the external surfaces. Therefore, the temperature elevation must be consistent, in terms of its order of magnitude if not its exact value, with the temperature rise that is associated with the rate of thermal energy generation passing through an appropriately deﬁned resistance. This is an approximate analysis and is only meant to illustrate the process of providing a “back of the envelope” estimate or a sanity check on a solution. The total energy that is generated in one-half of the wall material must pass to the adjacent edge. A very crude estimate of the temperature elevation is: L g˙ L2 L g˙ Ac = (1-92) T max ∼ 2 2kA 4k c rate of thermal energy generated in half the wall

thermal resistance of half the wall

which, in this case, is within a factor of two of the exact analytical solution (the “back of the envelope” calculation is too large because the energy generated near the surfaces does not need to pass through half of the wall material). Again, the intent of this analysis was not to obtain exact agreement, but rather to provide a quick check that the solution makes sense.

1.3.3 Uniform Thermal Energy Generation in Radial Geometries The area for conduction through the plane wall discussed in Section 1.3.2 is constant in the coordinate direction (x). If the conduction area is a function of position, then it cannot be canceled from each side of Eq. (1-77) and therefore the differential equation becomes more complicated. Figure 1-9 illustrates the differential control volumes that should be deﬁned in order to analyze radial heat transfer in (a) a cylinder and (b) a sphere with thermal energy generation.

30

One-Dimensional, Steady-State Conduction

TC dr dr q⋅ r TH

g⋅

TH

TC L ⋅q r+dr

r q⋅ r

g⋅

q⋅ r+ dr

rin rin

rout

r rout

(a)

(b)

Figure 1-9: Differential control volume for (a) a cylinder and (b) a sphere with volumetric thermal energy generation.

The differential control volume suggested by either Figure 1-9(a) or (b) is: q˙ r + g˙ = q˙ r+dr

(1-93)

which is expanded and simpliﬁed: dq˙ dr dr The rate equations for q˙ and g˙ in a cylindrical geometry, Figure 1-9(a), are: g˙ =

(1-94)

dT dr

(1-95)

g˙ = 2 π r L dr g˙

(1-96)

q˙ = −k 2 π r L

where L is the length of the cylinder and g˙ is the rate of thermal energy generation per unit volume. Substituting Eqs. (1-95) and (1-96) into Eq. (1-94) leads to: d dT r g˙ = − kr (1-97) dr dr which is integrated twice (assuming that k and g˙ are constant) to achieve: g˙ r2 (1-98) + C1 ln (r) + C2 4k where C1 and C2 are constants of integration that depend on the boundary conditions. The rate equations for q˙ and g˙ in a spherical geometry, Figure 1-9(b), are: T =−

dT dr

(1-99)

g˙ = 4 π r2 dr g˙

(1-100)

q˙ = −k 4 π r2

Substituting Eqs. (1-99) and (1-100) into Eq. (1-94) leads to: dT d g˙ r2 = −k r2 dr dr

(1-101)

which is integrated twice to achieve: T =−

g˙ 2 C1 r + + C2 6k r

(1-102)

1.3 Steady-State 1-D Conduction with Generation

31

Table 1-3: Summary of formulae for 1-D uniform thermal energy generation cases. Plane wall

Cylinder

Sphere

Governing differential equation

g˙ d2 T = − dx2 k

d dT g˙ r = −k r dr dr

Temperature gradient

g˙ dT = − x + C1 dx k

dT g˙ r C1 =− + dr 2k r

General solution

T =−

g˙ 2 x + C1 x + C2 2k

d 2 dT g˙ r = −k r dr dr

T =−

2

dT g˙ C1 =− r− 2 dr 3k r

g˙ r2 + C1 ln (r) + C2 4k

T =−

g˙ 2 C1 r + + C2 6k r

EXAMPLE 1.3-1: MAGNETIC ABLATION Thermal ablation is a technique for treating cancerous tissue by heating it to a lethal temperature. A number of thermal ablation techniques have been suggested in order to apply heat locally to the cancerous tissue and therefore spare surrounding healthy tissue. One interesting technique utilizes ferromagnetic thermoseeds, as discussed by Tompkins (1992). Small metallic spheres (thermoseeds) are embedded at precise locations within the cancer tumor and then the region is exposed to an oscillating magnetic ﬁeld. The magnetic ﬁeld does not generate thermal energy in the tissue. However, the spheres experience a volumetric generation of thermal energy that causes their temperature to increase and results in the conduction of heat to the surrounding tissue. Precise placement of the thermoseed can be used to control the application of thermal energy. The concept is shown in Figure 1. ferromagnetic thermoseed kts = 10 W/m-K g⋅ ts = 1.0 W Figure 1: A thermoseed used for ablation of a tumor.

q⋅ r =r + ts

rts = 1.0 mm q⋅ r =r − ts

tissue kt = 0.5 W/m-K body temperature Tr→∞ = Tb = 37°C

It is necessary to determine the temperature ﬁeld associated with a single thermoseed placed in an inﬁnite medium of tissue. The thermoseed has radius rts = 1.0 mm and conductivity kts = 10 W/m-K. A total of g˙ ts = 1.0 W of generation is uniformly distributed throughout the sphere. The temperature far from the thermoseed is the body temperature, Tb = 37◦ C. The tissue has thermal conductivity kt = 0.5 W/m-K and is assumed to be in perfect thermal contact with the thermoseed. The effects of metabolic heat generation (i.e., volumetric generation in the tissue) and blood perfusion (i.e., the heat removed by blood ﬂow in the tissue) are not considered in this problem.

EXAMPLE 1.3-1: MAGNETIC ABLATION

The governing differential equation and general solutions for these 1-D geometries with a uniform rate of thermal energy generation are summarized in Table 1-3.

EXAMPLE 1.3-1: MAGNETIC ABLATION

32

One-Dimensional, Steady-State Conduction

a) Prepare a plot showing the temperature in the thermoseed and in the tissue (i.e., the temperature from r = 0 to r rts ). This problem is 1-D because the temperature varies only in the radial direction. There are no circumferential non-uniformities that would lead to temperature gradients in any dimension except r. However, the problem includes two, separate computational domains that share a common boundary (i.e., the thermoseed and the tissue). Therefore, there will be two different governing equations that must be solved and additional boundary conditions that must be considered. It is always good to start your problem with an input section in which all of the given information is entered and, if necessary, converted to SI units.

“EXAMPLE 1.3-1: Magnetic Ablation” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” r ts=1 [mm]∗ convert(mm,m) k ts=10 [W/m-K] g dot ts=1.0 [W] T b=converttemp(C,K,37 [C]) k t=0.5 [W/m-K]

“radius of the thermoseed” “thermal conductivity of thermoseed” “total generation of thermal energy in thermoseed” “body temperature” “tissue thermal conductivity”

Notice a few things about the EES code. First, comments are provided to deﬁne the nomenclature and make the code understandable; this type of annotation is important for clarity and organization. Also, units are not ignored but rather explicitly speciﬁed and dealt with as the problem is set up, rather than as an afterthought at the end. The unit system that EES will use can be speciﬁed in the Properties Dialog (select Preferences from the Options menu) from the Unit System tab or by using the $UnitSystem directive, as shown in the EES code above. The units of numerical constants can be set directly in square brackets following the value. For example, the statement r ts=1 [mm]∗ convert(mm,m)

“radius of the thermoseed”

tells EES that the constant 1 has units of mm and these should be converted to units of m. Therefore the variable r_ts will have units of m. The units of r_ts are not automatically set, as the equation involving r_ts is not a simple assignment. If you check units at this point (select Check Units from the Calculate menu) then EES will indicate that there is a unit conversion error. This unit error occurs because the variable r_ts has not been assigned any units but the equations are consistent with r_ts having units of m. It is possible to have EES set units automatically with an option in the Options tab in the Preferences Dialog. However, this is not recommended because the engineer doing the problem should know and set the units for each variable. The Formatted Equations window (select Formatted Equations from the Windows menu) shows the equations and their units more clearly (Figure 2).

33

EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-1: Magnetic Ablation Inputs

0.001 .

m

rts =

1

[mm] .

kts =

10

[W/m-K]

g⋅ ts =

1 [W]

Tb =

ConvertTemp (C , K , 37

kt

=

0.5

radius of the thermoseed

mm

thermal conductivity of thermoseed

total generation of thermal energy in thermoseed

[W/m-K]

[C])

body temperature

tissue thermal conductivity

Figure 2: Formatted Equations window.

It is good practice to set the units of all variables. One method of accomplishing this is to right-click on a variable in the Solutions window, which brings up the Format Selected Variables dialog. The units can be typed directly into the Units input box. Note that right-clicking in the Units input box and selecting the Unit List menu item provides a partial list of the SI units that EES recognizes. All of the units that are recognized by EES can be examined by selecting Unit Conversion Info from the Options menu. Once the units of r_ts are set to m, then a unit check (select Check Units from the Calculate menu) should reveal no errors. It is necessary to work with a different governing equation in each of the two computational domains. An appropriate differential control volume for the spherical geometry with uniform thermal energy generation was presented in Section 1.3.3 and leads to the general solution listed in Table 1-3. The general solution that is valid within the thermoseed (i.e., from 0 < r < r ts ) is: Tts = −

g˙ ts C1 r2 + + C2 6 k ts r

(1)

is the volumetric where C1 and C2 are undetermined constants of integration and g˙ ts rate of generation of thermal energy in the thermoseed, which is the ratio of the total rate of thermal energy generation to the volume of the thermoseed: = g˙ ts

g dot ts=3∗ g dot ts/(4∗ pi∗ r tsˆ3)

3 g˙ ts 4 π r ts3

“volumetric rate of generation in the thermoseed”

The general solution that is valid within the tissue (i.e., for r > rts ) is: Tt =

C3 + C4 r

(2)

EXAMPLE 1.3-1: MAGNETIC ABLATION

34

One-Dimensional, Steady-State Conduction

because the volumetric rate of thermal energy generation in the tissue is zero; note that C3 and C4 are undetermined constants of integration that are different from C1 and C2 in Eq. (1). The next step is to deﬁne the boundary conditions. There are four undetermined constants and therefore there must be four boundary conditions. At the center of the thermoseed (r = 0) the temperature must remain ﬁnite. Substituting r = 0 into Eq. (1) leads to: Tts,r =0 = −

02 C1 g˙ + + C2 6 k ts ts k ts 0

which indicates that C1 must be zero: C1 = 0

(3)

Alternatively, specifying that the temperature gradient at the center of the sphere is equal to zero leads to the same conclusion, C1 = 0. As the radius approaches inﬁnity, the tissue temperature must approach the body temperature. Substituting r → ∞ into Eq. (2) leads to: Tb = −

C3 + C4 ∞

which indicates that: C 4 = Tb

(4)

The remaining boundary conditions are deﬁned at the interface between the thermoseed and the tissue. It is assumed that the sphere and tissue are in perfect thermal contact (i.e., there is no contact resistance) so that the temperature must be continuous at the interface: Tts,r =rts = Tt,r =rts or, substituting r = rts into Eqs. (1) and (2): −

r ts2 C 1 C3 g˙ + + C2 = + C4 6 k ts ts r ts r ts

(5)

An energy balance on the interface (see Figure 1) requires that the heat transfer rate at the outer edge of the thermoseed (q˙r =rts− in Figure 1) must equal the heat transfer rate at the inner edge of the tissue (q˙r =rts+ in Figure 1). q˙r =rts− = q˙r =rts+

(6)

According to Fourier’s law, Eq. (6) can be written as: dTts dTt 2 = −4 π r k −4 π r ts2 k ts t ts dr r =rts dr r =rts or k ts

dTts dTt = k t dr r =rts dr r =rts

Substituting Eqs. (1) and (2) into Eq. (7) leads to: r ts C 1 C3 g˙ ts − 2 = −k t − 2 −k ts − 3k ts r ts r ts

(7)

(8)

35

Entering Eqs. (3), (4), (5), and (8) into EES will lead to the solution for the four constants of integration without the algebra and the associated opportunities for error. “Determine constants of integration” C 1=0 “temperature at center must be ﬁnite” C 4=T b “temperature far from the thermoseed” -r tsˆ2∗ g dot ts/(6∗ k ts)+C 1/r ts+C 2=C 3/r ts+C 4 “continuity of temperature at the interface” -k ts∗ (-r ts∗ g dot ts/(3∗ k ts)-C 1/r tsˆ2)=-k t∗ (-C 3/r tsˆ2) “equal heat ﬂux at the interface”

Finally, we can generate a plot using the solution. Displaying the radius in millimeters in the plot will make a much more reasonable scale than in meters and the temperature should be displayed in ◦ C. New variables, r_mm, T_ts_C, and T_t_C, are deﬁned for this purpose. “Prepare a plot” r=r mm∗ convert(mm,m) T ts=-rˆ2∗ g dot ts/(6∗ k ts)+C 1/r+C 2 T ts C=converttemp(K,C,T ts) T t=C 3/r+C 4 T t C=converttemp(K,C,T t)

“radius” “thermoseed temperature” “in C” “tissue temperature” “in C”

The units of the variables C_1, C_2, etc. should be set in the Variable Information window and the set of equations subsequently checked for unit consistency. The relationship between temperature and radial position will be determined using two parametric tables. The ﬁrst table will include variables r_mm and T_ts_C and the second will include the variables r_mm and T_t_C. In the ﬁrst table, r_mm is varied from 0 to 1.0 mm (i.e., within the thermoseed) and in the second it is varied from 1.0 mm to 10.0 mm (i.e., within the tissue). A plot in which the information contained in the two tables is overlaid leads to the temperature distribution shown in Figure 3. 225

Tt

Temperature (°C)

200 175 150 125

Tts

100 75 50 0

1

2

3

4 5 6 7 Radial position (mm)

8

9

Figure 3: Temperature distribution through thermoseed and tissue.

10

EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-1: MAGNETIC ABLATION

36

One-Dimensional, Steady-State Conduction

Figure 3 agrees with physical intuition. The temperature decays toward the body temperature with increasing distance from the thermoseed. The rate of energy being transferred through the tissue is constant, but the area for conduction is growing as r 2 and therefore the gradient in the temperature is dropping. The − ) and conduction heat transfer rates at the outer edge of the sphere (i.e., r = r sp + at the inner edge of the tissue (i.e., r = r sp ) are identical; the discontinuity in slope is related to the fact that the thermoseed is more conductive than the tissue. b) Determine the maximum temperature in the tissue and the extent of the lesion as a function of the rate of thermal energy generation in the thermoseed. The extent of the lesion (rlesion ) is deﬁned as the radial location where the tissue temperature reaches the lethal temperature for tissue, approximately Tlethal = 50◦ C according to Izzo (2003). The maximum tissue temperature (Tt,max ) is the temperature at the interface between the thermoseed and the tissue and is obtained by substituting r = r ts into Eq. (2): Tt,max =

T t max=C 3/r ts+C 4 T t max C=converttemp(K,C,T t max)

C3 + C4 r ts

“maximum tissue temperature” “in C”

The extent of the lesion can be obtained by determining the radial location where Tt = Tlethal : Tlethal =

T lethal=converttemp(C,K, 50 [C]) T lethal=C 3/r lesion+C 4 r lesion mm=r lesion∗ convert(m,mm)

C3 + C4 rlesion

“lethal temperature for cell death” “determine the extent of the lesion” “in mm”

In order to investigate the maximum tissue temperature and the extent of the lesion as a function of the thermal energy generation rate, it is necessary to prepare a parametric table that includes the variables T_t_max_C, r_lesion, and g_dot_sp and then vary the value of g_dot_sp within the table. Figure 4 illustrates the maximum temperature and the lesion extent as a function of the rate of thermal energy generation in the thermoseed. Note that Tt,max and rlesion have very different magnitudes and therefore it is necessary to plot the variable r_lesion on a secondary y-axis.

40

350

35

Tt ,max

300 250

rlesion

30 25

200

20

150

15

100

10

50

5

0 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Rate of thermal energy generation in thermoseed (W)

Extent of lesion (mm)

Maximum tissue temperature (°C)

400

37

0 2

Figure 4: Maximum tissue temperature and the extent of the lesion as a function of the rate of thermal energy generation in the thermoseed.

1.3.4 Spatially Non-Uniform Generation The ﬁrst few steps in solving conduction heat transfer problems include setting up an energy balance on a differential control volume and substituting in the appropriate rate equations. This process results in one or more differential equations that must be solved in order to determine the temperature distribution and heat transfer rates. The governing equations resulting from 1-D conduction with constant properties and constant internal generation are provided in Table 1-3 for the Cartesian, cylindrical, and spherical geometries. These equations are relatively simple and we have demonstrated how to solve them analytically by hand and, in some cases, using Maple. The complexity of the governing equation can increase signiﬁcantly if the thermal conductivity or internal generation depends on position or temperature. In these cases, an analytical solution to the governing equation may not be possible and the numerical solution techniques presented in Sections 1.4 and 1.5 must be used. In many of these cases, however, an analytical solution is possible but may require more mathematical expertise than you have or more effort than you’d like to expend. In these cases, the combined use of a symbolic software tool to identify the solution and an equation solver to manipulate the solution provides a powerful combination of tools. Analytical solutions are concise and elegant as well as being accurate and therefore preferable in many ways to numerical solutions. It is often best to have both an analytical and a numerical solution to a problem; their agreement constitutes the best possible double-check of a solution. EXAMPLE 1.3-2 demonstrates the combined use of the symbolic solver Maple (to derive the symbolic solution and boundary condition equations) and the equation solver EES (to carry out the algebra required to obtain the constants of integration and implement the solution).

EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

38

One-Dimensional, Steady-State Conduction

EXAMPLE 1.3-2: ABSORPTION IN A LENS A lens is used to focus the illumination energy (i.e., radiation) that is required to develop the resist in a lithographic manufacturing process, as shown in Figure 1. The lens can be modeled as a plane wall with thickness L = 1.0 cm and thermal conductivity k = 1.5 W/m-K. The lens is not perfectly transparent but rather absorbs some of the illumination energy that is passed through it. The absorption coefﬁcient of the lens is α = 0.1 mm−1 . The ﬂux of radiant energy that is incident = 0.1 W/cm2 .The top and bottom surfaces of the at the lens surface (x = 0) is q˙ rad lens are exposed to air at T∞ = 20◦ C and the average heat transfer coefﬁcient on these surfaces is h = 20 W/m2 -K. ′′ = 0.1 W/cm incident radiant energy, q⋅ rad

x dx

k = 1.5 W/m-K α = 0.1 mm-1

2

T∞ = 20°C 2 h = 20 W/m -K q⋅ conv, x =0 q⋅cond, x =0 q⋅ x L = 1.0 cm g⋅ ⋅q q⋅ cond, x =L x+dx q⋅ conv, x =L

T∞ = 20°C 2 h = 20 W/m -K

transmitted radiant energy Figure 1: Lens absorbing radiant energy.

The volumetric rate at which absorbed radiation is converted to thermal energy in the lens (g˙ ) is proportional to the local intensity of the radiant energy ﬂux, which is reduced in the x-direction by absorption. The result is an exponentially distributed volumetric generation that can be expressed as: α exp (−α x) g˙ = q˙ rad

a) Determine and plot the temperature distribution within the lens. The inputs are entered into EES: “EXAMPLE 1.3-2: Absorption in a lens” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in k=1.5 [W/m-K] “conductivity” L=1 [cm]∗ convert(cm,m) “lens thickness” alpha=0.1 [1/mm]∗ convert(1/mm,1/m) “absorption coefﬁcient” qf dot rad=0.1 [W/cmˆ2]∗ convert(W/cmˆ2,W/mˆ2) “incident energy ﬂux” h bar=20 [W/mˆ2-K] “average heat transfer coefﬁcient” T inﬁnity=converttemp(C,K,20 [C]) “ambient air temperature” A c=1 [mˆ2] “carry out the problem on a per unit area basis”

(1)

39

An energy balance on an appropriate, differential control volume (see Figure 1) provides: q˙ x + g˙ = q˙ x+d x which is expanded and simpliﬁed: g˙ =

d q˙ dx dx

Substituting the rate equations for q˙ and g˙ leads to: dT d −k Ac g˙ Ac = dx dx

(2)

where Ac is the cross-sectional area of the lens. Substituting Eq. (1) into Eq. (2) and simplifying leads to the governing differential equation for this problem. q˙ α dT d = − rad exp (−α x) (3) dx dx k The governing differential equation is entered in Maple. > restart; > GDE:=diff(diff(T(x),x),x)=-qf_dot_rad*alpha*exp(-alpha*x)/k;

G D E :=

d2 qf d ot r ad α e(−α x) T(x) = − 2 dx k

The general solution to the equation is obtained using the dsolve command: > Ts:=dsolve(GDE); T s := T(x) = −

qf d ot r ad e(−α x) + C1 x + C2 αk

We can check that this solution is correct by integrating Eq. (3) by hand: q˙ rad α dT d =− exp (−α x) d x dx k which leads to: q˙ dT = rad exp (−α x) + C 1 dx k

(4)

Equation (4) is integrated again: q˙ rad dT = exp (−α x) + C 1 d x k which leads to: T =−

q˙ rad exp (−α x) + C 1 x + C 2 kα

(5)

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

40

One-Dimensional, Steady-State Conduction

Note that Eq. (5) is consistent with the result from Maple. The constants of integration, C1 and C2 , are obtained by enforcing the boundary conditions. The boundary conditions for this problem are derived from “interface” energy balances at the two edges of the computational domain (x = 0 and x = L, as shown in Figure 1). It would be correct to include the radiant energy ﬂux in the interface balances. However, the interface thickness is zero and therefore no radiant energy is absorbed at the interface. The amount of radiant energy entering and leaving the interface is the same and these terms would immediately cancel. q˙ conv,x=0 = q˙ cond ,x=0 q˙ cond ,x=L = q˙ conv,x=L Substituting the rate equations for convection and conduction into the interface energy balances leads to the boundary conditions: dT h Ac (T∞ − Tx=0 ) = −k Ac (6) d x x=0 dT −k Ac = h Ac (Tx=L − T∞ ) d x x=L

(7)

Note that it is important to consider the direction of the energy transfers during the substitution of the rate equations. For example, q˙ conv,x=0 is deﬁned in Figure 1 as being into the top surface of the lens and therefore it is driven by (T∞ − Tx =0 ) while q˙ conv,x=L is deﬁned as being out of the bottom surface of the lens and therefore it is driven by (Tx =L − T∞ ). The general solution for the temperature distribution, Eq. (5), must be substituted into the boundary conditions, Eqs. (6) and (7), and solved algebraically to determine the constants C1 and C2 . Maple and EES can be used together in order to solve the differential equation, derive the symbolic expressions for the boundary conditions, carry out the required algebra to obtain C1 and C2 , and manipulate the solution. The temperature gradient is obtained symbolically from Maple using the diff command. > dTdx:=diff(Ts,x);

dT d x :=

d qf d ot r ad e(−α x) T(x) = − + C1 dx k

The ﬁrst boundary condition, Eq. (6), requires both the temperature and the temperature gradient evaluated at x = 0. The eval function in Maple is used to symbolically determine these quantities (T0 and dTdx0): > T0:=eval(Ts,x=0); T0 := T(0) = −

qf d ot r ad + C2 αk

41

Use the rhs function to redeﬁne T0 to be just the expression on the right-hand side of the above result.

> T0:=rhs(T0); T 0 := −

qf d ot r ad + C2 αk

The rhs function can also be applied directly to the eval function. The statement below determines the symbolic expression for the temperature gradient evaluated at x = 0.

> dTdx0:=rhs(eval(dTdx,x=0)); dT d x0 := −

qf d ot r ad + C1 k

These expressions for T0 and dTdx0 are copied and pasted into EES in order to specify the ﬁrst boundary condition. The equation format used by Maple is similar to that used in EES and therefore only minor modiﬁcations are required. Select the desired symbolic expressions in Maple (note that the selected text will appear to be highlighted), use the Copy command from the Edit menu to place the selection on the Clipboard. When pasted into EES, the equations will appear as:

“boundary condition at x=0” T0 := -1/alpha∗ qf dot rad/k+ C2 dTdx0 := qf dot rad/k+ C1

“temperature at x=0, copied from Maple” “temperature gradient at x=0, copied from Maple”

All that is necessary to use this equation in EES is to change the := to = and change the constants _C1 and _C2 to C_1 and C_2, respectively. For lengthier expressions, the search and replace feature in EES (select Replace from the Search manu) facilitates this process. After modiﬁcation, the expressions should be:

T0= -1/alpha∗ qf dot rad/k+C 2 dTdx0= qf dot rad/k+C 1

“temperature at x=0” “temperature gradient at x=0”

The boundary condition at x = 0, Eq. (6), is speciﬁed in EES: h bar∗ A c∗ (T inﬁnity-T0)=-k∗ A c∗ dTdx0

“boundary condition at x=0”

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

42

One-Dimensional, Steady-State Conduction

The same process is used for the boundary condition at x = L, Eq. (7). The temperature and temperature gradient at x = L are determined using Maple: > TL:=rhs(eval(Ts,x=L)); T L := −

qf d ot r ad e(−α αk

L)

+ C1 L + C2

> dTdxL:=rhs(eval(dTdx,x=L)); dT d x L := −

qf d ot r ad e(−α k

L)

+ C1

These expressions are copied into EES: “boundary condition at x=L” TL := -1/alpha∗ qf dot rad∗ exp(-alpha∗ L)/k+ C1∗ L+ C2 “temperature at x=L, copied from Maple” dTdxL := qf dot rad∗ exp(-alpha∗ L)/k+ C1 “temperature gradient at x=L, copied from Maple”

and modiﬁed for consistency with EES: TL= -1/alpha∗ qf dot rad∗ exp(-alpha∗ L)/k+C 1∗ L+C 2 “temperature at x=L, copied from Maple” dTdxL= qf dot rad∗ exp(-alpha∗ L)/k+C 1 “temperature gradient at x=L, copied from Maple”

The boundary condition at x = L, Eq. (7), is speciﬁed in EES: -k∗ A c∗ dTdxL=h bar∗ A c∗ (TL-T inﬁnity)

“boundary condition at x=L”

Solving the EES program will provide numerical values for both of the constants. The units should be set for each of the variables, including the constants, in order to ensure that the expressions are dimensionally consistent. The general solution is copied from Maple to EES: T(x) = -1/alpha∗ qf dot rad∗ exp(-alpha∗ x)/k+ C1∗ x+ C2

“general solution, copied from Maple”

and modiﬁed for consistency with EES: x mm=0 [mm] x=x mm∗ convert(mm,m) T=-1/alpha∗ qf dot rad∗ exp(-alpha∗ x)/k+C 1∗ x+C 2 T C=converttemp(K,C,T)

“x-position, in mm” “x position” “general solution for temperature” “in C”

The temperature as a function of position is shown in Figure 2. Notice the asymmetry that is produced by the non-uniform volumetric generation (i.e.,

43

more thermal energy is generated towards the top of the lens than the bottom and so the temperature is higher near the top surface of the lens). 36.4 36.3

Temperature (°C)

36.2 36.1 36 35.9 35.8 35.7 35.6 0

1

2

3

4 5 6 Position (mm)

7

8

9

10

Figure 2: Temperature distribution in the lens.

b) Determine the location of the maximum temperature (xmax ) and the value of the maximum temperature (Tmax ) in the lens. The location of the maximum temperature in the lens can be determined by setting the temperature gradient, Eq. (4), to zero. The value of the maximum temperature (Tmax ) is obtained by substituting the resulting value of xmax into the equation for temperature, Eq. (5). The expression for the temperature gradient is copied from Maple, pasted into EES and modiﬁed for consistency. qf dot rad∗ exp(-alpha∗ x max)/k+C 1=0 x max mm=x max∗ convert(m,mm)

“temperature gradient is zero at position x max” “in mm”

The temperature at xmax is determined using the general solution. T max=-1/alpha∗ qf dot rad∗ exp(-alpha∗ x max)/k+C 1∗ x max+C 2 “maximum temperature in the lens” T max C=converttemp(K,C,T max) “in C”

It is not likely that solving the code above will immediately result in the correct value of either xmax or Tmax . These are non-linear equations and EES must iterate to ﬁnd a solution. The success of this process depends in large part on the initial guesses and bounds used for the unknown variables. In most cases, any reasonable values of the guess and bounds will work. For this problem, set the lower and upper bounds on xmax to 0 (the top of the lens) and 0.01 m (1.0 cm, the bottom of the lens), respectively, and the guess for xmax to 0.005 m (the middle of the lens) using the

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

44

One-Dimensional, Steady-State Conduction

Variable Information dialog (Figure 3). Upon solving, EES will correctly predict that xmax = 0.38 cm and Tmax = 36.35◦ C.

Figure 3: Variable Information window showing the limits and guess value for the variable x max.

It is also possible to determine the maximum temperature within the lens using EES’ built-in optimization routines. There are several sophisticated single- and multi-variable optimization algorithms included with EES that can be accessed with the Min/Max command in the Calculate menu.

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES) 1.4.1 Introduction The analytical solutions examined in the previous sections are convenient since they provide accurate results for arbitrary inputs with minimal computational effort. However, many problems of practical interest are too complicated to allow an analytical solution. In such cases, numerical solutions are required. Analytical solutions remain useful as a way to test the validity of numerical solutions under limiting conditions. Numerical solutions are generally more computationally complex and are not unconditionally accurate. Numerical solutions are only approximations to a real solution, albeit approximations that are extremely accurate when done correctly. It is relatively straightforward to solve even complicated problems using numerical techniques. The steps required to set up a numerical solution using the ﬁnite difference approach remain the same even as the problems become more complex. The result of a numerical model is not a functional relationship between temperature and position but rather a prediction of the temperatures at many discrete positions. The ﬁrst step is to deﬁne small control volumes that are distributed through the computational domain and to specify the locations where the numerical model will compute the temperatures (i.e., the nodes). The control volumes used in the numerical model are small but ﬁnite, as opposed to the inﬁnitesimally small (differential) control volume that is deﬁned in order to derive an analytical solution. It is necessary to perform an energy balance on each

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

45

control volume. This requirement may seem daunting given that many control volumes will be required to provide an accurate solution. However, computers are very good at repetitious calculations. If your numerical code is designed in a systematic manner, then these operations can be done quickly for 1000’s of control volumes. Once the energy balance equations for each control volume have been set up, it is necessary to include rate equations that approximate each term in the energy balance based upon the nodal temperatures or other input parameters. The result of this step will be a set of algebraic equations (one for each control volume) in an equal number of unknown temperatures (one for each node). This set of equations can be solved in order to provide the numerical prediction of the temperature at each node. It is tempting to declare victory after successfully solving the ﬁnite difference equations and obtaining a set of results that looks reasonable. In reality, your work is only half done and there are several important steps remaining. First, it is necessary to verify that you have chosen a sufﬁciently large number of control volumes (i.e., an adequately reﬁned mesh) so that your numerical solution has converged to a solution that no longer depends on the number of nodes. This veriﬁcation can be accomplished by examining some aspect of your solution (e.g., a temperature or an energy transfer rate that is particularly important) as the number of control volumes increases. You should observe that your solution stops changing (to within engineering relevance) as the number of control volumes (and therefore the computational effort) increases. Some engineering judgment is required for this step. You have to decide what aspect of the solution is most important and how accurately it must be predicted in order to determine the level of grid reﬁnement that is required. Next, you should make sure that the solution makes sense. There are a number of ‘sanity checks’ that can be applied to verify that the numerical model is behaving according to physical expectations. For example, if you change the input parameters, does the solution respond as you would expect? Finally, it is important that you verify the numerical solution against an analytical solution in some appropriate limit. This step may be the most difﬁcult one, but it provides the strongest possible veriﬁcation. If the numerical model is to be used to make decisions that are important (e.g., to your company’s bottom line or to your career) then you should strive to ﬁnd a limit where an analytical solution can be derived and show that your numerical model matches the analytical solution to within numerical error. The numerical solutions considered in this section are for steady-state, 1-D problems. More complex problems (e.g., multi-dimensional and transient) are discussed in subsequent sections and chapters. This section also focuses on solving these problems using EES whereas subsequent sections discuss how these solutions may be implemented using a more formal programming language, speciﬁcally MATLAB.

1.4.2 Numerical Solutions in EES The development of a numerical model is best discussed in the context of a problem. Figure 1-10 illustrates an aluminum oxide cylinder that is exposed to ﬂuid at its internal and external surfaces. The temperature of the ﬂuid exposed to the internal surface is T ∞,in = 20◦ C and the average heat transfer coefﬁcient on the internal surface is hin = 100 W/m2 -K. The temperature of the ﬂuid exposed to the outer surface is T ∞,out = 100◦ C and the average heat transfer coefﬁcient on the outer surface is hout = 200 W/m2 -K. The thermal conductivity of the aluminum oxide is k = 9.0 W/m-K in the temperature range of interest. The rate of thermal energy generation per unit

46

One-Dimensional, Steady-State Conduction k = 9 W/m-K 2 g⋅ ′′′= a + b r + c r rin = 10 cm rout = 20 cm

Figure 1-10: Cylinder with volumetric generation.

T∞, in = 20°C T∞, out = 100°C 2 2 hin = 100 W/m -K hout = 200 W/m -K

volume within the cylinder varies with radius according to: g˙ = a + b r + c r2

(1-103)

where a = 1 × 104 W/m3 , b = 2 × 105 W/m4 , and c = 5 × 107 W/m5 . The inner and outer radii of the cylinder are rin = 10 cm and rout = 20 cm, respectively. The inputs are entered in EES: “Section 1.4.2” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” r in=10 [cm]∗ convert(cm,m) r out=20 [cm]∗ convert(cm,m) L=1 [m] k=9.0 [W/m-K] T inﬁnity in=converttemp(C,K,20 [C]) h bar in= 100 [W/mˆ2-K] T inﬁnity out=converttemp(C,K,100 [C]) h bar out=200 [W/mˆ2-K]

“inner radius of cylinder” “outer radius of cylinder” “unit length of cylinder” “thermal conductivity” “temperature of ﬂuid inside cylinder” “average heat transfer coefﬁcient at inner surface” “temperature of ﬂuid outside cylinder” “average heat transfer coefﬁcient at outer surface”

It is convenient to deﬁne a function that provides the volumetric rate of thermal energy generation speciﬁed by Eq. (1-103). An EES function is a self-contained code segment that is provided with one or more input parameters and returns a single value based on these parameters. Functions in EES must be placed at the top of the Equations Window, before the main body of equations. The EES code required to specify a function for the rate of thermal energy generation per unit volume is shown below. function gen(r) “This function deﬁnes the volumetric heat generation in the cylinder Inputs: r, radius (m) Output: volumetric heat generation at r (W/mˆ3)” a=1e4 [W/mˆ3] “coefﬁcients for generation function” b=2e5 [W/mˆ4] c=5e7 [W/mˆ5] gen=a+b∗ r+c∗ rˆ2 “generation is a quadratic” end

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

47

Figure 1-11: Variable Information Window showing the variables for the function GEN.

The function begins with a header that deﬁnes the name of the function (gen) and the input arguments (r) and is terminated by the statement end. None of the variables in the main EES program are accessible within the function other than those that are explicitly passed to the function as an input. Unlike equations in the main program, the equations within a function are executed in the order that they are entered and all variables on the right hand side of each expression must be deﬁned (i.e., the equations are assignments rather than relationships). The statements within the function are used to deﬁne the value of the function (gen, in this case). Units for the variables in the function should be set using the Variable Information dialog in the same way that units are set for variables in the main program. The most direct way to enter the units for the function is to select Variable Info from the Options menu. Then select Function GEN from the pull down menu at the top of the Variable Information dialog and enter the units for the variables (Figure 1-11). The ﬁnite difference technique divides the continuous medium into a large (but not inﬁnite) number of small control volumes that are treated using simple approximations. The computational domain in this problem lies between the edges of the cylinder (rin < r < rout ). The ﬁrst step in the solution process is to locate nodes (i.e., the positions where the temperature will be predicted) throughout the computational domain. The easiest way to distribute the nodes is uniformly, as shown in Figure 1-12 (only nodes 1, 2, i − 1, i, i + 1, N − 1, and N are shown). The extreme nodes (i.e., nodes 1 and N) are placed on the surfaces of the cylinder. In some problems, it may not be computationally efﬁcient to distribute the nodes uniformly. For example, if there are large temperature gradients at some location then it may be necessary to concentrate nodes in that region. Placing closelyspaced nodes throughout the entire computational domain may be prohibitive from a

qLHS

ri T1

Figure 1-12: Nodes and control volumes for the numerical model.

T2 g

qconv, in

Ti-1

Ti Ti+1

qLHS g qRHS

rin rout

g qconv, out

qRHS

TN-1

TN

L

48

One-Dimensional, Steady-State Conduction

computational viewpoint. For the uniform distribution shown in Figure 1-12, the radial location of each node (ri ) is: ri = rin +

(i − 1) (rout − rin ) (N − 1)

for i = 1..N

(1-104)

where N is the number of nodes. The radial distance between adjacent nodes ( r) is: r =

(rout − rin ) (N − 1)

(1-105)

It is necessary to specify the number of nodes used in the numerical solution. We will start with a small number of nodes, N = 6, and increase the number of nodes when the solution is complete. N=6 [-] DELTAr=(r out-r in)/(N-1)

“number of nodes” “distance between adjacent nodes (m)”

The location of each node will be placed in an array, i.e., a variable that contains more than one element (rather than a scalar, as we’ve used previously). EES recognizes a variable name to be an element of an array if it ends with square brackets surrounding an array index, e.g., r[4]. Array variables are just like any other variable in EES and they can be assigned to values, e.g., r[4]=0.16. Therefore, one way of setting up the position array would be to individually assign each value; r[1]=0.1, r[2]=0.12, r[3]=0.14, etc. This process is tedious, particularly if there are a large number of nodes. It is more convenient to use the duplicate command which literally duplicates the equations in its domain, allowing for varying array index values. The duplicate command must be followed by an integer index, in this case i, that passes through a range of values, in this case 1 to 6. The EES code shown below will copy (duplicate) the statement(s) that are located between duplicate and end N times; each time, the value of i is incremented by 1. It is exactly like writing: r[1]=r in+(r out-r in)∗ 0/(N-1) r[2]=r in+(r out-r in)∗ 1/(N-1) r[3]=r in+(r out-r in)∗ 2/(N-1) etc.

“Set up nodes” duplicate i=1,N r[i]=r in+(r out-r in)∗ (i-1)/(N-1) end

“this loop assigns the radial location to each node”

Be careful not to put statements that you do not want to be duplicated between the duplicate and end statements. For example, if you accidentally placed the statement N=6 inside the duplicate loop it would be like writing N=6 six times, which corresponds to six equations in a single unknown and is not solvable. Units should be assigned to arrays in the same manner as for other variables. The units of all variables in array r can be set within the Variable Information dialog. Note that each element of the array r appears in the dialog and that the units of each element can be set one by one. In most arrays, each element will have the same units and therefore it is not convenient to set the units an element at a time. Instead, deselect the Show array variables box in the upper left corner of the window, as shown in

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

49

De-select the Show array variables button in order to collapse the arrays

Figure 1-13: Collapsing an array in the Variable Information dialog.

Figure 1-13. The array is collapsed onto a single entry r[] and you can assign units to all of the elements in the array r[]. A control volume is deﬁned around each node. You have some freedom relative to the deﬁnition of a control volume, but the best control volume for this problem is deﬁned by bisecting the distance between the nodes, as shown in Figure 1-12. The second step in the numerical solution is to write an energy balance for the control volume associated with every node. The internal nodes (i.e., nodes 2 through N − 1) must be considered separately from the nodes at the edge of the computational domain (i.e., nodes 1 and N). The control volume for an arbitrary, internal node (node i) is shown in Figure 1-12. There are three energy terms associated with each control volume: conduction heat transfer passing through the surface on the left-hand side (q˙ LHS ), conduction heat transfer passing through the surface on the right-hand side (q˙ RHS ), and generation of thermal energy within the control volume (g). ˙ A steady-state energy balance for the internal control volume is: q˙ LHS + q˙ RHS + g˙ = 0

(1-106)

Note that Eq. (1-106) is rigorously correct since no approximations have been used in its development. In the next step, however, each of the terms in the energy balance are modeled using a rate equation that is only approximately valid; it is this step that makes the numerical solution only an approximation of the actual solution. Conduction through the left-hand surface is driven by the temperature difference between nodes i − 1 and i through the material that lies between these nodes. If there are a large number of nodes, then r is small and the effect of curvature within any control volume is small. In this case, q˙ LHS can be modeled using the resistance of a plane wall, given in Table 1-2: kL2π r (1-107) q˙ LHS = ri − (T i−1 − T i ) r 2 where L is the length of the cylinder and k is the thermal conductivity of the material, which is assumed to be constant here. It is relatively easy to consider a material with temperature- or spatially dependent thermal conductivity, as discussed in Section 1.4.3. Note that it does not matter which direction the heat ﬂow arrow associated with q˙ LHS is drawn in Figure 1-12; that is, the heat transfer could have been deﬁned either as an input or an output to the control volume. However, once the direction is deﬁned, it is

50

One-Dimensional, Steady-State Conduction

absolutely necessary that the energy balance on the control volume and the model of the heat transfer rate be consistent with this selection. For the energy balance shown in Figure 1-12, the heat transfer rate was written on the inﬂow side of the energy balance in Eq. (1-106) and the heat transfer was written as being driven by (T i−1 − T i ) in Eq. (1-107). The rate of conduction into the right-hand surface can be approximated in the same manner: kL2π r (1-108) ri + q˙ RHS = (T i+1 − T i ) r 2 The rate of generation of thermal energy is the product of the volume of the control volume and the rate of thermal energy generation per unit volume, which is approximately: g˙ = g˙ r=ri 2 π ri L r

(1-109)

The generation term is spatially dependent in this problem, but it is approximated by assuming that the volumetric generation evaluated at the position of the node (ri ) can be applied throughout the entire control volume. This approximation improves as r is reduced. Equations (1-106) through (1-109) can be conveniently written for each internal node using a duplicate loop: “Internal control volume energy balances” duplicate i=2,(N-1) q dot LHS[i]=2∗ pi∗ L∗ k∗ (r[i]-DELTAr/2)∗ (T[i-1]-T[i])/DELTAr q dot RHS[i]=2∗ pi∗ L∗ k∗ (r[i]+DELTAr/2)∗ (T[i+1]-T[i])/DELTAr g dot[i]=gen(r[i])∗ 2∗ pi∗ r[i]∗ L∗ DELTAr q dot LHS[i]+g dot[i]+q dot RHS[i]=0 end

“conduction in from inner radius” “conduction in from outer radius” “generation” “energy balance”

Attempting to solve the EES program at this point will result in a message indicating that there are two more variables than equations and so the problem is under-speciﬁed. We have not yet written energy balance equations for the two nodes that lie on the boundaries. Figure 1-12 illustrates the control volume associated with the node that is placed on the inner surface of the cylinder (i.e., node 1). The energy balance for the control volume ˙ and a associated with node 1 includes a conduction term (q˙ RHS ), a generation term (g), convection term (q˙ conv,in ). For steady-state conditions and energy terms having the signs indicated in Figure 1-12: q˙ conv,in + q˙ RHS + g˙ = 0 The conduction term model is the same as it was for internal nodes: kL2π r r1 + q˙ RHS = (T 2 − T 1 ) r 2

(1-110)

(1-111)

Even though the control volume for node 1 is half as wide as the others, the distance between nodes 1 and 2 is still r and therefore the resistance to conduction between nodes 1 and 2 does not change. The generation term is slightly different because the control volume is half as large as the internal control volumes. r (1-112) g˙ = g˙ r=r1 2 π r1 L 2

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

51

Convection from the ﬂuid is given by: q˙ conv,in = hin 2 π r1 L (T ∞,in − T 1 )

(1-113)

Equations (1-110) through (1-113) are programmed in EES: “Energy balance for node on internal surface” q dot RHS[1]=2∗ pi∗ L∗ (r[1]+DELTAr/2)∗ k∗ (T[2]-T[1])/DELTAr q dot conv in=2∗ pi∗ L∗ r in∗ h bar in∗ (T inﬁnity in-T[1]) g dot[1]=gen(r[1])∗ 2∗ pi∗ r[1]∗ L∗ DELTAr/2 q dot RHS[1]+q dot conv in+g dot[1]=0

“conduction in from outer radius” “convection from internal ﬂuid” “generation” “energy balance for node 1”

A similar procedure applied to the control volume associated with node N (see Figure 1-12) leads to: q˙ conv,out + q˙ LHS + g˙ = 0

(1-114)

q˙ conv,out = hout 2 π rN L (T ∞,out − T N )

(1-115)

where

q˙ LHS

kL2π = r

r rN − 2

g˙ = g˙ r=rN 2 π rN L

(T N−1 − T N )

(1-116)

r 2

(1-117)

“Energy balance for node on external surface” q dot LHS[N]=2∗ pi∗ L∗ (r[N]-DELTAr/2)∗ k∗ (T[N-1]-T[N])/DELTAr “conduction in from from inner radius” q dot conv out=2∗ pi∗ L∗ r out∗ h bar out∗ (T inﬁnity out-T[N]) “convection from external ﬂuid” g dot[N]=gen(r[N])∗ 2∗ pi∗ r[N]∗ L∗ DELTAr/2 “generation” q dot LHS[N]+q dot conv out+g dot[N]=0 “energy balance for node N”

There are now an equal number of equations as unknowns and therefore solving the problem will provide the temperature at each node. The calculated temperatures are converted to ◦ C: duplicate i=1,N T C[i]=converttemp(K,C,T[i]) end

“temperature in C”

Computers are very good at solving large systems of equations, particularly with linear equations such as these. There are a number of programs other than EES that can be used to solve these equations (e.g., MATLAB, FORTRAN, and C++). With most of these software packages, you must take the system of equations, carefully put them into a matrix format, and then decompose the matrix in order to obtain the solution. This additional step is not necessary with EES, which saves considerable effort for the user (although EES must do this step internally). In Section 1.5, we will look at how these equations have to be set up in a formal programming environment, speciﬁcally MATLAB, in order to obtain a solution.

52

One-Dimensional, Steady-State Conduction 675 N = 20 650

N=6

Temperature (°C)

625 600 575 550 525 500 475 450 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-14: Predicted temperature distribution for N = 6 and N = 20.

It is good practice to assign the units for all variables including the arrays before attempting to solve the equations. The unit consistency of each equation is checked when the equations are solved. The solution is provided in the Solution Window for the scalar quantities, and the Arrays Window for the array of predicted temperatures. A plot showing the predicted temperature as a function of radius is shown in Figure 1-14 for N = 6 and N = 20. Note that EES calculates the individual energy transfer rates for each of the control volumes. To see these energy transfer rates, select Arrays from the Windows menu in order to view the Arrays table (Figure 1-15). It is useful to examine these energy transfer rates and make sure that they agree with your intuition. For example, energy should not be created or destroyed at the interfaces between the control volumes; that is, q˙ LHS for node i should be equal and opposite q˙ RHS for node i − 1. It is easy to inadvertently use incorrect rate equations for the conduction terms where this is not true, and you can quickly identify this type of problem using the Arrays window. Figure 1-15 shows that the rate of energy transfer by conduction in the positive radial direction (i.e., q˙ LHS ) becomes more positive with increasing radius, as it should due to the volumetric generation. Before the numerical solution can be used with conﬁdence, it is necessary to verify its accuracy. The ﬁrst step in this process is to ensure that the mesh is adequately reﬁned. Figure 1-14 shows that the solution becomes smoother and represents the actual

[m]

[K]

[C]

[W]

[W]

[W]

0.1

366.4

93.27

62.83

0.12

369.2

96.03

150.8

-857.9

707.1

0.14

371.1

97.95

175.9

-707.1

531.2

0.16

372.4

99.21

201.1

-531.2

330.1

0.18

373

99.89

226.2

-330.1

103.9

0.2

373.2

100.1

125.7

-103.9

Figure 1-15: Arrays table.

857.9

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

53

660

Maximum temperature (°C)

650 640 630 620 610 600 590 580 570 1

10

100 Number of nodes

1000

Figure 1-16: Predicted maximum temperature as a function of the number of nodes.

temperature distribution better as the number of nodes is increased. The general approach to choosing a mesh is to pick an important characteristic of the solution and examine how this characteristic changes as the number of nodes in the computational domain is increased. In this case, an appropriate characteristic is the maximum predicted temperature within the cylinder. The following EES code extracts this value from the solution using the max function, which returns the maximum of the arguments provided to it. T max C=max(T C[1..N])

“max temperature in the cylinder, in C”

The maximum temperature as a function of the number of nodes is shown in Figure 1-16 (Figure 1-16 was created by making a parametric table that includes the variables N and T_max). Notice that the solution has converged after approximately 20 nodes and further reﬁnement is not likely to be necessary. The next step is to check that the solution agrees with physical intuition. For example, if the heat transfer coefﬁcient on the internal surface is reduced, then the temperatures within the cylinder should increase. Figure 1-17 illustrates the temperature as a function of radius for various values of hin (with N = 20) and shows that reducing the heat transfer coefﬁcient does tend to increase the temperature in the cylinder. There are many additional ‘sanity checks’ that could be tested. For example, decrease the thermal conductivity or increase the generation rate and make sure that the temperatures in the computational domain increase as they should. Finally, it is important that the numerical solution be veriﬁed against an analytical solution in an appropriate limit. In this case, it is not easy (although it is possible) to develop an analytical solution to the problem with the spatially varying volumetric generation. However, it is relatively straightforward to develop an analytical solution for the problem in the limit of a constant volumetric generation rate. It is also very easy to adapt the numerical model to this limiting case so that the analytical and numerical solutions can be compared. The variables b and c in the generation function, Eq. (1-103),

54

One-Dimensional, Steady-State Conduction 900

Temperature (°C)

800 700

hin = 20 W/m2-K

hin = 50 W/m2-K hin = 100 W/m2-K

600 500

hin = 200 W/m2-K

400 300 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-17: Temperature as a function of radius for various values of the heat transfer coefﬁcient.

are temporarily set to 0 in order to implement the numerical solution using a spatially uniform volumetric generation: function gen(r) “This function deﬁnes the volumetric heat generation in the cylinder Inputs: r: radius “Output: volumetric heat generation at r (W/mˆ3)” a=1e4 [W/mˆ3] “coefﬁcients for generation function” b=0{2e5} [W/mˆ4] c=0{5e7} [W/mˆ5] gen=a+b*r+c*rˆ2 “generation is a quadratic” end

The analytical solution is solved using the technique presented in Section 1.3. Table 1-3 provides the temperature distribution and temperature gradient in a cylinder with constant generation, to within the constants of integration: T =−

g˙ r2 + C1 ln (r) + C2 4k

(1-118)

dT g˙ r C1 =− + (1-119) dr 2k r The constants of integration may be determined using the boundary conditions at the inner and outer surfaces, which are obtained from energy balances at these interfaces. At the inner surface, convection and conduction must balance: dT hin (T ∞,in − T r=rin ) = −k (1-120) dr r=rin Substituting Eqs. (1-118) and (1-119) into Eq. (1-120) leads to: 2 g˙ rin C1 g˙ rin hin T ∞,in − − + C1 ln (rin ) + C2 + = −k − 4k 2k rin

(1-121)

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

At the outer surface, the interface energy balance leads to: dT = hout (T r=rout − T ∞,out ) −k dr r=rout Substituting Eqs. (1-118) and (1-119) into Eq. (1-122) leads to: 2 g˙ rout C1 g˙ rout −k − + + C1 ln (rout ) + C2 − T ∞,out = hout − 2k rout 4k

55

(1-122)

(1-123)

The constants of integration are determined by using EES to solve Eqs. (1-121) and (1-123): “Analytical Solution” g dot c=gen(r in) “constant volumetric generation rate for veriﬁcation” h bar in∗ (T inﬁnity in-(-g dot c∗ r inˆ2/(4∗ k)+C 1∗ ln(r in)C 2))=-k∗ (-g dot c∗ r in/(2∗ k)+C 1/r in) “boundary condition at inner surface” -k∗ (-g dot c∗ r out/(2∗ k)+C 1/r out)=h bar out∗ (-g dot c∗ r outˆ2/(4∗ k)+& C 1∗ ln(r out)+C 2-T inﬁnity out) “boundary condition at outer surface”

The ampersand character appearing in the equation above is a line break character that is only needed for formatting (the equation that is terminated with the ampersand continues on the following line). The analytical solution is evaluated at the locations of the nodes in the numerical solution. The absolute error between the analytical and numerical solution is calculated at each position and the maximum error over the computational domain is computed using the max command. duplicate i=1,N “analytical temperature at node i” T an[i]=-g dot c∗ r[i]ˆ2/(4∗ k)+C 1∗ ln(r[i])+C 2 T an C[i]=converttemp(K,C,T an[i]) “in C” err[i]=abs(T an[i]-T[i]) “absolute value of the discrepancy between numerical and analytical temperature” end err max=max(err[1..N]) “maximum error”

Figure 1-18 illustrates the temperature distribution predicted by the analytical and numerical models in the limit where the volumetric generation is constant (i.e., the coefﬁcients b and c in the volumetric generation function are set equal to zero). The agreement is nearly exact for 20 nodes, conﬁrming that the numerical model is valid. Figure 1-19 illustrates the maximum value of the error between the analytical and numerical models as a function of the number of nodes in the solution. Note that the constants b and c in the generation function were set to zero in order to provide a uniform generation. Figure 1-19 provides a more precise method of selecting the number of nodes. A required level of accuracy (i.e., agreement with the analytical model) can be used to specify a required number of nodes. For example, if the problem requires temperature estimates that are accurate to within 0.01 K, then you should use at least 7 nodes. However, predictions accurate to within 0.1 mK will require more than 60 nodes.

1.4.3 Temperature-Dependent Thermal Conductivity The thermal conductivity of most materials is a function of temperature, although it is often approximated to be constant. This approximation is appropriate for situations in

56

One-Dimensional, Steady-State Conduction

101 numerical model (with b = c = 0) analytical model

100

Temperature (°C)

99 98 97 96 95 94 93 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-18: Temperature as a function of position predicted by the analytical model should be

indicated by a line and the numerical model by the dots. which the temperature range is small provided that the thermal conductivity is evaluated at an average temperature. A constant value of thermal conductivity is usually assumed when solving a problem analytically; otherwise the differential equation is intractable. A major advantage of a numerical solution is that the temperature dependence of physical properties can be considered with little additional effort. The consideration of temperature-dependent thermal conductivity in a numerical model is demonstrated in the context of the problem that was considered previously in Section 1.4.2. The thermal conductivity of the aluminum oxide cylinder was assumed to be 9 W/m-K, independent of temperature. However, the temperature of the aluminum

Maximum error (K)

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

1

10

100 Number of nodes

1000

Figure 1-19: Maximum discrepancy between the analytical and numerical solutions (in the limit that b = c = 0) as a function of N.

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

57

40

Conductivity (W/m-K)

35 30 25 20 15 10 5 0 0

100

200

300 400 500 Temperature (°C)

600

700

800

Figure 1-20: Thermal conductivity of polycrystalline aluminum oxide as a function of temperature.

oxide varies by 200◦ C within the cylinder (see Figure 1-14) and the thermal conductivity of aluminum oxide varies substantially over this range of temperatures, as shown in Figure 1-20. It is possible to alter the EES code that was developed in Section 1.4.2 so that the temperature-dependent thermal conductivity of polycrystalline aluminum oxide is considered. A function k is written in EES that returns the conductivity. The function must be placed at the top of the Equations window, either above or below the previously deﬁned function gen, and has a single input (the temperature). EES’ internal function for the thermal conductivity of solids is used in function k to evaluate the thermal conductivity of the polycrystalline aluminum oxide. function k(T) “This function provides the thermal conductivity of the aluminum oxide Inputs: T: temperature (K) Output: thermal conductivity (W/m-K)” k=k (‘Al oxide-polycryst’,T) end

The temperature-dependent conductivity can cause a problem. It is tempting to evaluate the conduction terms for each control volume using the thermal conductivity evaluated at the temperature of the node. However, doing so will result in an error in the energy balance. Each conduction term, for example q˙ LHS , represents an energy exchange between node i and its adjacent node to the left, node i − 1. The value of q˙ LHS evaluated at node i must be equal and opposite to q˙ RHS evaluated at node i − 1; if the thermal conductivity is evaluated using the nodal temperature, then this will not be true and energy will be artiﬁcially generated or destroyed at the boundaries between the nodal control volumes. To avoid this problem, the thermal conductivity should be evaluated at the average temperature of the two nodes that are involved in the conduction heat transfer

58

One-Dimensional, Steady-State Conduction

(i.e., the temperature at the boundary). The energy balance on the internal nodes (see Figure 1-12) remains: q˙ LHS + q˙ RHS + g˙ = 0

(1-124)

However, the conduction terms must be calculated according to: q˙ LHS =

kT =(T i +T i−1 )/2 L 2 π r

q˙ RHS =

kT =(T i +T i+1 )/2 L 2 π r

r (T i−1 − T i ) 2

(1-125)

r ri + (T i+1 − T i ) 2

(1-126)

ri −

where kT =(T i +T i−1 )/2 is the thermal conductivity evaluated at the average of Ti and T i−1 and kT =(T i +T i+1 )/2 is the thermal conductivity evaluated at the average of Ti and T i+1 . A similar process is used for nodes 1 and N. The modiﬁed energy balances in EES are shown below, with the modiﬁed code indicated in bold: “Internal control volume energy balances” duplicate i=2,(N-1) k LHS[i]=k((T[i-1]+T[i])/2) “thermal conductivity at LHS boundary” k RHS[i]=k((T[i+1]+T[i])/2) “thermal conductivity at RHS boundary” q dot LHS[i]=2∗ pi∗ L∗ k LHS[i]∗ (r[i]-DELTAr/2)∗ (T[i-1]-T[i])/DELTAr “conduction in from inner radius” q dot RHS[i]=2∗ pi∗ L∗ k RHS[i]∗ (r[i]+DELTAr/2)∗ (T[i+1]-T[i])/DELTAr “conduction in from outer radius” g dot[i]=gen(r[i])∗ 2∗ pi∗ r[i]∗ L∗ DELTAr “generation” q dot LHS[i]+g dot[i]+q dot RHS[i]=0 “energy balance” end “Energy balance for node on internal surface” “thermal conductivity at RHS boundary” k RHS[1]=k((T[2]+T[1])/2) q dot RHS[1]=2∗ pi∗ L∗ (r[1]+DELTAr/2)∗ k RHS[1]∗ (T[2]-T[1])/DELTAr “conduction in from outer radius” q dot conv in=2∗ pi∗ L∗ r in∗ h bar in∗ (T inﬁnity in-T[1]) “convection from internal ﬂuid” g dot[1]=gen(r[1])∗ 2∗ pi∗ r[1]∗ L∗ DELTAr/2 “generation” q dot RHS[1]+q dot conv in+g dot[1]=0 “energy balance” “Energy balance for node on external surface” “thermal conductivity at LHS boundary” k LHS[N]=k((T[N-1]+T[N])/2) q dot LHS[N]=2∗ pi∗ L∗ (r[N]-DELTAr/2)∗ k LHS[N]∗ (T[N-1]-T[N])/DELTAr “conduction in from from inner radius” q dot conv out=2∗ pi∗ L∗ r out∗ h bar out∗ (T inﬁnity out-T[N]) g dot[N]=gen(r[N])∗ 2∗ pi∗ r[N]∗ L∗ DELTAr/2 “generation in node N” q dot LHS[N]+q dot conv out+g dot[N]=0 “energy balance for node N”

Select Solve from the Calculate menu and you are likely to ﬁnd that the problem either fails to converge or converges to some ridiculously high temperatures. This is not surprising, since the use of a temperature-dependent conductivity has transformed the algebraic equations from a set of equations that are linear in the unknown temperatures to a set of equations that are nonlinear. Therefore, EES must start from some guess value for the unknown temperatures and attempt to iterate until a solution is obtained.

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

59

Figure 1-21: Setting the guess values for the array T[ ] in the Variable Information window.

The success of this process is highly dependent on the guess values that are used. It may be possible to simply set better guess values for the unknown temperatures. Select Variable Info from the Options menu and deselect the Show array variables button. Set the guess value for the array T[ ] to something more reasonable than its default value of 1 K (e.g., 700 K), as shown in Figure 1-21. This strategy of manually setting reasonable guess values will not always work. A more reliable strategy uses the solution with constant conductivity, from Section 1.4.2, in order to provide guess values for the non-linear problem associated with temperaturedependent conductivity. This is an easy process; modify the conductivity function (k) as shown below, so that it returns a constant value rather than the temperature-dependent value:

function k(T) “This function provides the thermal conductivity of the aluminum oxide Inputs: T: temperature (K) Output: thermal conductivity (W/m-K)” {k=k (‘Al oxide-polycryst’,T)} k=9 [W/m-K] end

Solve the problem and EES will converge to a solution. To use this constant conductiviy solution as the guess value for the non-linear problem, select Update Guesses from the Calculate menu and then return the conductivity function to its original form. Solve the problem and your EES code will converge to the actual solution. Examine the Arrays Table (Figure 1-22) and notice that energy is conserved at each boundary (i.e., q˙ RHS for node i is equal and opposite to q˙ LHS for node i + 1 for every node); this simple check shows that your rate equations have been set up appropriately. Figure 1-23 illustrates the temperature of the aluminum oxide as a function of radius for the case in which the thermal conductivity is evaluated as a function of temperature. The temperature distribution calculated in Section 1.4.2 using a constant thermal conductivity of 9 W/m-K is also shown. This example illustrates that the temperature dependence of thermal conductivity may be an important factor in some problems.

60

One-Dimensional, Steady-State Conduction energy is conserved at each boundary

[W]

[W/m-k]

[W/m-k]

[W]

10.12

3330

[W]

[m]

[K]

[C]

27426

0.1

782.7

509.5

11370

10.12

9.353

-27426

16056

0.12

861.1

587.9

17910

9.353

9.113

-16056

-1853

0.14

903.1

629.9

26580

9.113

9.498

1853

-28434

0.16

898.8

625.6

37684

9.498

10.65

28434

-66118

0.18

842.7

569.6

25761

10.65

0.2

738.7

465.6

66118

Figure 1-22: Arrays Window.

675 650

k = 9 W/m-K

Temperature (°C)

625 600 575

k (T )

550 525 500 475 450 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-23: Temperature as a function of radius for the case where conductivity is a function of temperature, k(T), and conductivity is constant, k = 9.0 W/m-K.

1.4.4 Alternative Rate Models The rate equations used to model the conduction between adjacent nodes in Section 1.4.2 were based on separating these nodes with a thin cylindrical shell of material that is modeled as a plane wall. An even better approximation for these conduction terms uses the resistance to conduction through a cylinder. According to the resistance formula listed in Table 1-2, the rate equations become: q˙ LHS =

q˙ RHS =

k L 2 π (T i−1 − T i ) ri ln ri−1 k L 2 π (T i+1 − T i ) ri+1 ln ri

(1-127)

(1-128)

Also, the volume of the control volume may be computed more exactly in order to provide a better estimate of the thermal energy generation term: r 2 r 2 ri + − ri − g˙ = g˙ r=ri π L (1-129) 2 2

Maximum error (K)

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES) 10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

61

original rate model, Section 1.4.2

advanced rate model, Section 1.4.4

1

10

100 Number of nodes

1,000

Figure 1-24: Maximum error between the analytical and numerical models (original and advanced) as a function of the number of nodes.

The portion of the EES code from Section 1.4.2 that is modiﬁed to use these more advanced rate models (with modiﬁcations indicated in bold), is shown below. “Internal control volume energy balances” duplicate i=2,(N-1) q dot LHS[i]=2∗ pi∗ L∗ k∗ (T[i-1]-T[i])/ln(r[i]/r[i-1]) q dot RHS[i]=2∗ pi∗ L∗ k∗ (T[i+1]-T[i])/ln(r[i+1]/r[i]) g dot[i]=gen(r[i])∗ pi∗ ((r[i]+DELTAr/2)ˆ2-(r[i]-DELTAr/2)ˆ2)∗ L q dot LHS[i]+g dot[i]+q dot RHS[i]=0 end

“conduction in from inner radius” “conduction in from outer radius” “generation” “energy balance”

“Energy balance for node on internal surface” q dot RHS[1]=2∗ pi∗ L∗ k∗ (T[2]-T[1])/ln(r[2]/r[1]) q dot conv in=2∗ pi∗ L∗ r in∗ h bar in∗ (T inﬁnity in-T[1]) g dot[1]=gen(r[1])∗ pi∗ ((r[1]+DELTAr/2)ˆ2-r[1]ˆ2)∗ L q dot RHS[1]+q dot conv in+g dot[1]=0

“conduction in from outer radius” “convection from internal ﬂuid” “generation” “energy balance”

“Energy balance for node on external surface” q dot LHS[N]=2∗ pi∗ L∗ k∗ (T[N-1]-T[N])/ln(r[N]/r[N-1]) q dot conv out=2∗ pi∗ L∗ r out∗ h bar out∗ (T inﬁnity out-T[N]) g dot[N]=gen(r[N])∗ pi∗ (r[N]ˆ2-(r[N]-DELTAr/2)ˆ2)∗ L q dot LHS[N]+q dot conv out+g dot[N]=0

“conduction in from from inner radius” “convection from external ﬂuid” “generation in node N” “energy balance for node N”

The more advanced numerical solution will approach the actual solution somewhat more quickly (i.e., with fewer nodes) than the original numerical solution. Figure 1-24 illustrates the difference between the advanced numerical solution (in the limit of a constant generation rate, b = c = 0) and the analytical solution as well as the error associated with the original solution derived in Section 1.4.2. Notice that the use of the advanced rate models provides a substantial improvement in accuracy for any given number of nodes, N. The use of advanced rate models for a spherical problem is illustrated in EXAMPLE 1.4-1.

EXAMPLE 1.4-1: FUEL ELEMENT

62

One-Dimensional, Steady-State Conduction

EXAMPLE 1.4-1: FUEL ELEMENT A nuclear fuel element consists of a sphere of ﬁssionable material (fuel) with radius r fuel = 5.0 cm and conductivity k fuel = 1.0 W/m-K that is surrounded by a shell of metal cladding with outer radius r clad = 7.0 cm and conductivity k clad = 300 W/m-K. The outer surface of the cladding is exposed to helium gas that is being heated by the reactor. The average convection coefﬁcient between the gas and the cladding surface is h = 100 W/m2 -K and the temperature of the gas is T∞ = 500◦ C. Inside the fuel element, ﬁssion fragments are produced that have high velocities. The products collide with the atoms of the material and provide the thermal energy for the reactor. This process can be modeled as a non-uniform volumetric thermal energy generation (g˙ ) that can be approximated by: r g˙ = g˙ 0 exp −b (1) r fuel where g˙ 0 = 5× 105 W/m3 is the volumetric rate of heat generation at the center of the sphere and b = 1.0 is a dimensionless constant that characterizes how quickly the generation rate decays in the radial direction. a) Develop a numerical model for the spherical fuel element using EES. A function is deﬁned that returns the volumetric generation given the radial position and the radius of the fuel element. function gen(r, r fuel) “This function deﬁnes the volumetric heat generation in the fuel element Inputs: r: radius (m) r fuel: radius of fuel sphere (m) Output: volumetric heat generation at r (W/mˆ3)” g dot 0=5e5 [W/mˆ3] “volumetric generation rate at the center” b=1.0 [-] “constant that describes rate of decay” gen=g dot 0∗ exp(-b∗ r/r fuel) “volumetric rate of generation” end

The next section of the EES code provides the problem inputs. “EXAMPLE 1.4-1: Fuel Element” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” r fuel=5.0 [cm]∗ convert(cm,m) r clad=7.0 [cm]∗ convert(cm,m) k fuel=1.0 [W/m-K] k clad=300 [W/m-K] h bar=100 [W/mˆ2-K] T inﬁnity=converttemp(C,K,500 [C])

“fuel radius” “cladding radius” “fuel conductivity” “cladding conductivity” “average convection coefﬁcient” “helium temperature”

The numerical solution proceeds by distributing nodes throughout the computational domain that stretches from r = 0 to r = r fuel . There is no reason to include the metal cladding in the numerical model. The cladding increases the thermal

63

resistance that is already present due to convection with the gas; however, this effect can be included using a conduction thermal resistance. The positions of a uniformly distributed set of nodes are obtained from: ri =

(i − 1) r fuel (N − 1)

for i = 1..N

and the distance between adjacent nodes is: r =

“Setup nodes” N=50 [-] duplicate i=1,N r[i]=r fuel∗ (i-1)/(N-1) end DELTAr=r fuel/(N-1)

r fuel (N − 1)

“number of nodes” “radial position of each node” “distance between adjacent nodes”

A control volume for an arbitrary internal node is shown in Figure 1. rfuel

cladding rclad fuel element ri +

Figure 1: Control volume for an internal node.

q⋅ LHS

g⋅

Δr 2 Δ r ri − 2 q⋅ RHS

Ti-1 Ti T i+1

The energy balances for the internal control volumes are: q˙ LH S + q˙ RH S + g˙ = 0 The control volumes are spherical shells. Therefore, it is appropriate to use a conduction model that is consistent with conduction through a spherical shell (see Table 1-2). Note that this problem could also be solved by approximating the spherical shells as plane walls with different surface areas, as was done in Section 1.4.2. However, building the proper geometry into the control volume energy balances will allow the problem to be solved to a speciﬁed accuracy with fewer nodes. q˙ LH S =

(Ti−1 − Ti ) 1 1 1 − 4 π k fuel ri−1 ri

and q˙ RH S =

(Ti+1 − Ti ) 1 1 1 − 4 π k fuel ri ri+1

The temperature differences used to evaluate q˙ LH S and q˙ RH S are consistent with the direction of the conduction heat transfer terms shown in Figure 1 whereas the

EXAMPLE 1.4-1: FUEL ELEMENT

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

EXAMPLE 1.4-1: FUEL ELEMENT

64

One-Dimensional, Steady-State Conduction

resistance values in the denominators are written in the form of 1/rin − 1/r out (e.g., 1/ri −1 − 1/ri and 1/ri − 1/ri +1 ) so that the resistances are positive. The generation in each control volume is given by: 4 r 3 r 3 g˙ri g˙ i = π − ri − ri + 3 2 2 Combining these equations allows the control volume energy balances for the internal nodes to be written as: 4 π k fuel (Ti−1 − Ti ) 4 π k fuel (Ti+1 − Ti ) 4 r 3 r 3 g˙ri = 0 + + π − ri − ri + 1 1 1 1 3 2 2 − − ri−1 ri ri ri+1 for i = 2.. (N − 1)

(2)

“Internal control volume energy balance” duplicate i=2,(N-1) 4∗ pi∗ k fuel∗ (T[i-1]-T[i])/(1/r[i-1]-1/r[i])+4∗ pi∗ k fuel∗ (T[i+1]-T[i])/(1/r[i]-1/r[i+1])+& 4∗ pi∗ ((r[i]+DELTAr/2)ˆ3-(r[i]-DELTAr/2)ˆ3)∗ gen(r[i],r fuel)/3=0 end

The energy balance for the node that is placed at the outer edge of the fuel (i.e., node N ) is: 4 π k fuel (TN −1 − TN ) 4 r 3 (T∞ − TN ) 3 g˙rN = 0 + + π rN − rN − 1 1 Rcond ,clad + Rconv 3 2 − r N −1 r N combined thermal generation in outer shell resistance of cladding conduction between the outermost and adjoining nodes

and convection

(3)

where Rclad is the resistance to conduction through the cladding: 1 1 1 − Rcond ,clad = 4 π k clad r fuel r clad and Rconv is the resistance to convection from the surface of the cladding to the gas: Rconv =

1 2 4 π r clad h

R cond clad=(1/r fuel-1/r clad)/(4∗ pi∗ k clad) “conduction resistance of cladding” R conv=1/(4∗ pi∗ r cladˆ2∗ h bar) “convection resistance from surface of cladding” 4*pi*k fuel*(T[N-1]-T[N])/(1/r[N-1]-1/r[N])+(T inﬁnity-T[N])/(R cond clad+R conv)+& 4∗ pi∗ (r[N]ˆ3-(r[N]-DELTAr/2)ˆ3)∗ gen(r[N],r fuel)/3=0 “node N energy balance”

If the same conduction model is used, then the energy balance for the node placed at the center of the fuel (i.e., node 1) is: 4 π k fuel (T2 − T1 ) 4 r 3 3 − r 1 g˙r1 = 0 (4) r1 + + π 1 1 3 2 − r1 r2

65

4∗ pi∗ k fuel∗ (T[2]-T[1])/(1/r[1]-1/r[2])+4∗ pi∗ ((r[1]+DELTAr/2)ˆ3-r[1]ˆ3)∗ gen(r[1],r fuel)/3=0 “node 1 energy balance”

Executing the EES code will lead to a division by zero error message. The radial location of node 1, r1 , is equal to 0 and therefore the 1/r1 term in the denominator of Eq. (4) is inﬁnite. (The actual resistance associated with conducting energy to a point is inﬁnite.) A similar error will be encountered when computing q˙ LH S for node 2 in Eq. (2). This problem can be dealt with by calculating the conduction between nodes 1 and 2 using a plane wall approximation. The energy balance for node 1 becomes: 4 π k fuel

r 2 (T2 − T1 ) 4 r 3 3 − r 1 g˙r1 = 0 + π r1 + r1 + 2 r 3 2

{4∗ pi∗ k fuel∗ (T[2]-T[1])/(1/r[1]-1/r[2])+4∗ pi∗ ((r[1]+DELTAr/2)ˆ3-r[1]-ˆ3)∗ gen(r[1],r fuel)/3=0} 4∗ pi∗ k fuel∗ (r[1]+DELTAr/2)ˆ2∗ (T[2]-T[1])/DELTAr+4∗ pi∗ ((r[1]+DELTAr/2)ˆ3-r[1]ˆ3)∗ gen(r[1],r fuel)/3=0 “node 1 energy balance”

The energy balance for node 2 has to be rewritten in the same way: r 2 (T1 − T2 ) 4 π k fuel (T3 − T2 ) + 4 π k fuel r 1 + 1 1 2 r − r2 r3 3 3 4 r r g˙r2 = 0 + π − r2 − r2 + 3 2 2

“Internal control volume energy balance” {duplicate i=2,(N-1) 4∗ pi∗ k fuel∗ (T[i-1]-T[i])/(1/r[i-1]-1/r[i])+4∗ pi∗ k fuel∗ (T[i+1]-T[i])/(1/r[i]-1/r[i+1])+& 4∗ pi∗ ((r[i]+DELTAr/2)ˆ3-(r[i]-DELTAr/2)ˆ3)∗ gen(r[i],r fuel)/3=0 end} duplicate i=3,(N-1) 4∗ pi∗ k fuel∗ (T[i-1]-T[i])/(1/r[i-1]-1/r[i])+4∗ pi∗ k fuel∗ (T[i+1]-T[i])/(1/r[i]-1/r[i+1])+& 4∗ pi∗ ((r[i]+DELTAr/2)ˆ3-(r[i]-DELTAr/2)ˆ3)∗ gen(r[i],r fuel)/3=0 end 4∗ pi∗ k fuel∗ (r[1]+DELTAr/2)ˆ2∗ (T[1]-T[2])/DELTAr+4∗ pi∗ k fuel∗ (T[3]-T[2])/(1/r[2]-1/r[3])+& 4∗ pi∗ ((r[2]+DELTAr/2)ˆ3-(r[2]-DELTAr/2)ˆ3)∗ gen(r[2],r fuel)/3=0 “node 2 energy balance”

With these changes, the program can be solved. The solution is converted to ◦ C:

duplicate i=1,N T C[i]=converttemp(K,C,T[i]) end

“temperature in C”

EXAMPLE 1.4-1: FUEL ELEMENT

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

One-Dimensional, Steady-State Conduction

Figure 2 illustrates the temperature in the fuel element as a function of radius. 660

Temperature (°C)

640 620 600 580 560 540 520 0

0.01

0.02 0.03 Radius (m)

0.04

0.05

Figure 2: Temperature distribution within fuel.

The maximum temperature in the fuel element is obtained with the max function. T max C=max(T C[1..N])

“maximum temperature of the fuel, in C”

Figure 3 shows the maximum temperature in the fuel as a function of the number of nodes and indicates that the solution converges if at least 100 nodes are used. 668 666 Maximum temperature (°C)

EXAMPLE 1.4-1: FUEL ELEMENT

66

664 662 660 658 656 654 652 650 2

10

100 Number of nodes

1000

Figure 3: Maximum temperature within fuel as a function of the number of nodes.

Several sanity checks can be carried out in order to verify that the solution is physically correct. Figure 4 shows the maximum temperature in the fuel as a function of the fuel conductivity for various values of the volumetric generation at the center of

67

the fuel (g˙ 0 ). The maximum temperature increases as either the fuel conductivity decreases or the volumetric rate of generation increases. 800

g⋅ o′′′= 10 × 106 W/m3

Maximum temperature (°C)

750 700 650 600

g⋅ o′′′= 5 × 106 W/m3

550 500 450 0.5

6 3 g⋅ ′′′ o = 2 × 10 W/m

1

1.5

g⋅ o′′′= 1 × 106 W/m 3

2 2.5 3 3.5 Fuel conductivity (W/m-K)

4

4.5

5

Figure 4: Maximum temperature in the fuel as a function of kfuel for various values of g˙ 0.

Finally, we can compare the numerical model with the analytical solution for the limiting case where b = 0 in Eq. (1) (i.e., the fuel experiences a uniform rate of volumetric generation). The general solution for the temperature distribution and temperature gradient within a sphere exposed to a uniform generation rate is given in Table 1-3: T =−

g˙ 2 C 1 r + + C2 6 k fuel r

(5)

and dT C1 g˙ r− 2 =− dr 3 k fuel r

(6)

where C1 and C2 are constants of integration. The temperature at the center of the sphere must be bounded and therefore C1 must be equal to 0 by inspection of Eq. (5); alternatively, the temperature gradient at the center must be zero, which would also require that C1 = 0 according to Eq. (6). The second boundary condition is related to an energy balance at the interface between the cladding and the fuel: Tr =rfuel − T∞ dT 2 = (7) −k fuel 4 π r fuel dr r =rfuel Rcond ,clad + Rconv Combining Equations (5) through (7) leads to: g˙ 2 r + C 2 − T∞ − 6 k fuel fuel g˙ 2 −k fuel 4 π r fuel r fuel = − 3 k fuel Rcond ,clad + Rconv

EXAMPLE 1.4-1: FUEL ELEMENT

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

One-Dimensional, Steady-State Conduction

which can be solved for C2 . “Analytical solution” g dot=gen(0 [m],r fuel) “rate of volumetric generation to use in analytical solution” -k fuel∗ 4∗ pi∗ r fuelˆ2∗ (-g dot∗ r fuel/(3∗ k fuel))= & (-g dot∗ r fuelˆ2/(6∗ k fuel)+C 2-T inﬁnity)/(R cond clad+R conv) “boundary condition at r=r fuel”

The analytical solution is obtained at the same radial locations as the numerical solution. duplicate i=1,N T an[i]=-g dot∗ r[i]ˆ2/(6∗ k fuel)+C 2 T an C[i]=converttemp(K,C,T an[i]) end

“analytical solution” “in C”

Figure 5 shows the analytical and numerical solutions in the limit that b = 0 for 50 nodes; the agreement is nearly exact, indicating that the numerical solution is adequate. 800

predicted by numerical model, with b = 0 predicted by analytical model

750 Temperature (°C)

EXAMPLE 1.4-1: FUEL ELEMENT

68

700 650 600 550 500 0

0.01

0.02 0.03 Radius (m)

0.04

0.05

Figure 5: Temperature as a function of radius predicted by the analytical model should be

indicated by a line and the numerical model by the dots.

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems using MATLAB 1.5.1 Introduction Numerical models of 1-D steady-state conduction problems are introduced and implemented using EES in Section 1.4. EES internally provides all of the numerical manipulations that are needed to solve the system of algebraic equations that constitutes a numerical model. This capability reduces the complexity of the problem. However, there are disadvantages to using EES. For example, EES will generally require signiﬁcantly

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

69 k = 9 W/m-K 5 3 g⋅ ′′′= 1x10 W/m rin = 10 cm rout = 20 cm

Figure 1-25: Cylinder with volumetric generation. T∞, in = 20°C 2 hin = 100 W/m -K

T∞, out = 100°C 2 hout = 200 W/m -K

more time to solve the equations than is required by a compiled computer language. There is an upper limit to the number of variables that EES can handle, which places an upper bound on the number of nodes that can be used in the model. The structure of EES requires that every variable be retained in the ﬁnal solution. Therefore, you cannot deﬁne and then erase intermediate variables in the course of obtaining a solution and, as a result, numerical solutions in EES often require a lot of memory. Finally, some models require logic statements (e.g., if-then-else statements) that are difﬁcult to include in EES. For these reasons, it is useful to learn how to implement numerical models in a formal programming language, e.g., FORTRAN, C++, or MATLAB. The steps required to solve the algebraic equations associated with a numerical model are demonstrated in this section using the MATLAB software. It is suggested that the reader stop and go through the tutorial provided in Appendix A.3 in order to become familiar with MATLAB. Appendix A.3 can be found on the web site associated with this book (www.cambridge.org/nellisandklein).

1.5.2 Numerical Solutions in Matrix Format The cylinder problem that is considered in Section 1.4 in order to illustrate numerical methods using EES is shown again in Figure 1-25. An aluminum oxide cylinder is exposed to ﬂuid on its internal and external surfaces. The temperature of the ﬂuid that is exposed to the internal surface is T ∞,in = 20◦ C and the average heat transfer coefﬁcient on this surface is hin = 100 W/m2 -K. The temperature of the ﬂuid exposed to the external surface is T ∞,out = 100◦ C and the average heat transfer coefﬁcient on this surface is hout = 200 W/m2 -K. The thermal conductivity is assumed to be constant and equal to k = 9.0 W/m-K. We will begin by solving the problem for the case where the rate of volumetric generation of thermal energy within the cylinder is uniform and equal to g˙ = 1 × 105 W/m3 . The inner and outer radii of the cylinder are rin = 10 cm and rout = 20 cm, respectively. The development of the system of equations proceeds as discussed in Section 1.4.2. A uniform distribution of nodes is used and therefore the radial location of each node (ri ) is: ri = rin +

(i − 1) (rout − rin ) (N − 1)

i = 1..N

(1-130)

where N is the number of nodes used for the simulation. The radial distance between adjacent nodes ( r) is: r =

(rout − rin ) (N − 1)

(1-131)

70

One-Dimensional, Steady-State Conduction

The energy balances for the internal nodes are: q˙ LHS + q˙ RHS + g˙ = 0 where

kL2π

ri −

q˙ LHS = kL2π q˙ RHS =

ri +

(1-132)

r (T i−1 − T i ) 2 r r (T i+1 − T i ) 2 r

g˙ = g˙ 2 π ri L r

(1-133)

(1-134)

(1-135)

Equations (1-132) through (1-135) are combined: r r k L 2 π ri + k L 2 π ri − 2 2 (T i−1 − T i ) + (T i+1 − T i ) + g˙ 2 π ri L r = 0 r r for i = 2.. (N − 1) (1-136) The energy balance for the control volume associated with node 1 is: r k L 2 π r1 + r 2 hin 2 π r1 L (T ∞,in − T 1 ) + =0 (T 2 − T 1 ) + g˙ 2 π r1 L r 2 (1-137) The energy balance for the control volume associated with node N is: r k L 2 π rN − r 2 hout 2 π rN L (T ∞,out − T N ) + =0 (T N−1 − T N ) + g˙ 2 π rN L r 2 (1-138) Equations (1-136) through (1-138) represent N linear algebraic equations in an equal number of unknown temperatures. In order to solve these equations using a formal programming language, it is necessary to represent this set of equations as a matrix equation. Recall from linear algebra that a linear system of equations, such as: 2x1 + 3x2 + 1x3 = 1 1x1 + 5x2 + 1x3 = 2 7x1 + 1x2 + 2x3 = 5

(1-139)

can be written as a matrix equation: ⎡ ⎤⎡ ⎤ ⎡ ⎤ 2 3 1 1 x1 ⎣1 5 1 ⎦ ⎣x2 ⎦ = ⎣2 ⎦ 7 1 2 x3 5 A

X

(1-140)

b

or AX = b

(1-141)

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

where A is a matrix and X ⎡ 2 3 A = ⎣1 5 7 1

and b are vectors: ⎡ ⎤ ⎤ 1 x1 1 ⎦ , X = ⎣x2 ⎦ , x3 2

and

⎡ ⎤ 1 b = ⎣2 ⎦ 5

71

(1-142)

Most programming languages, including MATLAB, have built-in or library routines for decomposing the system of equations and solving for the vector of unknowns, X. This is a mature area of research and advanced methods exist for quickly solving matrix equations, particularly when most of the entries in A are 0 (i.e., A is a sparse matrix). MATLAB is speciﬁcally designed to handle large matrix equations. In this section, we will use MATLAB to solve heat transfer problems. This requires an understanding of how to place large systems of equations, corresponding to the energy balances, into a matrix format. Each row of the A matrix and b vector correspond to an equation whereas each column of the A matrix is the coefﬁcient that multiplies the corresponding unknown (typically a nodal temperature) in that equation. To set up a system of equations in matrix format, it is necessary to carefully deﬁne how the rows and energy balances are related and how the columns and unknown temperatures are related. The ﬁrst step is to deﬁne the vector of unknowns, the vector X in Eq. (1-140). It does not really matter what order the unknowns are placed in X, but the implementation of the solution is much easier if a logical order is used. In this problem, the unknowns are the nodal temperatures. Therefore, the most logical technique for ordering the unknown temperatures in the vector X is: ⎡ ⎤ X1 = T1 ⎢ X2 = T2 ⎥ ⎥ (1-143) X=⎢ ⎣ ... ⎦ XN = TN Equation (1-143) shows that the unknown temperature at node i (i.e., Ti ) corresponds to element i of vector X (i.e., Xi ). The next step is to deﬁne how the rows in the matrix A and the vector b correspond to the N control volume energy balances that must be solved. Again, it does not matter what order the equations are placed into the A matrix, but the solution is easiest if a logical order is used: ⎡ ⎤ row 1 = control volume 1 equation ⎢ row 2 = control volume 2 equation ⎥ ⎥ A=⎢ (1-144) ⎣ ⎦ ··· row N = control volume N equation Equation (1-144) shows that the equation for control volume i is placed into row i of matrix A.

1.5.3 Implementing a Numerical Solution in MATLAB We can return to the numerical problem that was discussed in Section 1.5.2. Open a new M-ﬁle (select New M-File from the File menu) which will bring up the M-ﬁle editor. Save the script as cylinder (select Save As from the File menu) in a directory that is in your search path (you can specify the directories in your search path by typing pathtool in the Command window). Enter the inputs to the problem at the top of the script and save it. Note that the % symbol indicates that anything that follows on that line will be a comment.

72

One-Dimensional, Steady-State Conduction

MATLAB will not assign units to any of the variables; they are all dimensionless as far as the software is concerned. This limitation puts the burden squarely on the user to clearly understand the units of each variable and ensure that they are consistent. The use of a semicolon after each assignment statement prevents the variables from being echoed in the working environment. The clear command at the top of the script clears all variables from the workspace. clear;

%clear all variables from the workspace

% Inputs r in=0.1; r out=0.2; g dot tp=1e5; L=1; k=9; T infinity in=20+273.2; h bar in=100; T infinity out=100+273.2; h bar out=200;

%inner radius of cylinder (m) %outer radius of cylinder (m) %constant volumetric generation (W/mˆ3) %unit length of cylinder (m) %thermal conductivity of cylinder material (W/m-K) %average temperature of fluid inside cylinder (K) %heat transfer coefficient inside cylinder (W/mˆ2-K) %temperature of fluid outside cylinder (K) %heat transfer coefficient at outer surface (W/mˆ2-K)

In order to run your script from the MATLAB working environment, type cylinder at the command prompt: >> cylinder

Nothing appears to have happened. However, all of the variables that are deﬁned in the script are now available in the work space. For example, if the name of any variable is entered at the command prompt then its value is displayed. >> h bar out h bar out = 200

For a complete list of variables in the workspace, use the command who. >> who Your variables are: L T infinity out h bar in k r out T infinity in g dot tp h bar out r in

The number and location of the nodes for the solution must be speciﬁed. A vector of radial locations (r) is setup using a for loop. Each of the statements between the for and end statements is executed each time through the loop. Enter the following lines into the cylinder M-ﬁle: % Setup nodes N=10; Dr=(r out-r in)/(N-1); for i=1:N r(i)=r in+(i-1)∗ (r out-r in)/(N-1); end

%number of nodes (-) %distance between nodes (m) %radial position of each node (m)

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

73

The energy balances must be setup in an appropriately sized matrix, the variable A, and vector, the variable b. Recall that our matrix A needs to have as many rows as there are equations (the N control volume energy balances) and as many columns as there are unknowns (the N unknown temperatures) and b is a vector with as many elements as there are equations (N). We’ll start with A and b composed entirely of zeros and subsequently add non-zero elements according to the equations. The A matrix ends up being composed almost entirely of zeros and thus it is referred to as a sparse matrix. Initially we will not take advantage of this sparse characteristic of A. However, in Section 1.5.5, it will be shown that the solution can be accelerated considerably by using specialized matrix solution techniques that are designed for sparse matrices. MATLAB makes it extremely easy to use these sparse matrix solution techniques. The zeros function used below returns a matrix ﬁlled with zeros with its size determined by the input arguments; the variable A will be an N × N matrix ﬁlled with zeros and the variable b will be an N × 1 vector ﬁlled with zeros where the variable N is the number of nodes. %Setup A and b A=zeros(N,N); b=zeros(N,1);

The most difﬁcult step in the process is to ﬁll in the non-zero elements of A and b so that the solution of the system of equations can be obtained through a matrix decomposition process. According to Eq. (1-144), the 1st row in A must correspond to the energy balance for control volume 1, which is given by Eq. (1-137), repeated below: kL2π r r hin 2 π r1 L (T ∞,in − T 1 ) + r1 + =0 (T 2 − T 1 ) + g˙ 2 π r1 L r 2 2 (1-137) It is necessary to algebraically manipulate Eq. (1-137) so that the coefﬁcients that multiply each of the unknowns in this equation (i.e., T1 and T2 ) and the constant term in the equation (i.e., terms that are known) can be identiﬁed. r r kL2π kL2π r1 + − hin 2 π r1 L +T 2 r1 + T1 − r 2 r 2 A1,1

r − hin 2 π r1 L T ∞,in = −g˙ 2 π r1 L 2

A1,2

(1-145)

b1

Equation (1-145) corresponds to the 1st row of A and b. The coefﬁcient in the ﬁrst equation that multiplies the ﬁrst unknown in X, T1 according to Eq. (1-143), must be A1,1 : kL2π r (1-146) r1 + − hin 2 π r1 L A1,1 = − r 2 The coefﬁcient in the ﬁrst equation that multiplies the second unknown in X, T2 according to Eq. (1-143), must be A1,2 : kL2π r r1 + (1-147) A1,2 = r 2

74

One-Dimensional, Steady-State Conduction

Finally, the constant terms associated with the ﬁrst equation must be b1 : r − hin 2 π r1 L T ∞,in 2 These assignments are accomplished in MATLAB: b1 = −g˙ 2 π r1 L

(1-148)

%Energy balance for control volume 1 A(1,1)=-k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr-h bar in∗ 2∗ pi∗ r(1)∗ L; A(1,2)=k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr; b(1)=-h bar in∗ 2∗ pi∗ r(1)∗ L∗ T infinity in-g dot tp∗ 2∗ pi∗ r(1)∗ L∗ Dr/2;

According to Eq. (1-144), rows 2 through N − 1 of matrix A correspond to the energy balances for the corresponding internal control volumes; these equations are given by Eq. (1-136), which is repeated below: kL2π r r kL2π ri − ri + (T i−1 − T i ) + (T i+1 − T i ) + g˙ 2 π ri L r = 0 r 2 r 2 (1-136) for i = 2.. (N − 1) Again, Eq. (1-136) must be rearranged to identify coefﬁcients and constants. r r r kL2π kL2π kL2π +T i−1 ri − − ri + ri − Ti − r 2 r 2 r 2 + T i+1

kL2π r

Ai,i

Ai,i−1

r = −g˙ 2 π ri L r ri + 2 bi

for i = 2.. (N − 1)

(1-149)

Ai,i+1

All of the coefﬁcients for control volume i must go into row i of A, the column depends on which unknown they multiply. Therefore: kL2π r r kL2π ri − − ri + for i = 2 . . . (N − 1) (1-150) Ai,i = − r 2 r 2 Ai,i−1

kL2π = r

Ai,i+1

kL2π = r

r ri − 2 r ri + 2

for i = 2 . . . (N − 1)

(1-151)

for i = 2 . . . (N − 1)

(1-152)

The constant for control volume i must go into row i of b. bi = −g˙ 2 π ri L r

for i = 2 . . . (N − 1)

(1-153)

These equations are programmed most conveniently in MATLAB using a for loop: %Energy balances for internal control volumes for i=2:(N-1) A(i,i)=-k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr-k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; A(i,i-1)=k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr; A(i,i+1)=k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; b(i)=-g dot tp∗ 2∗ pi∗ r(i)∗ L∗ Dr; end

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

75

Finally, the last row of A (i.e., row N) corresponds to the energy balance for the last control volume (node N), which is given by Eq. (1-138), repeated below:

r k L 2 π rN − 2 hout 2 π rN L (T ∞,out − T N ) + r

(T N−1 − T N ) + g˙ 2 π rN L

r =0 2 (1-138)

Equation (1-138) is rearranged: kL2π r r kL2π T N −hout 2 π rN L − + T N−1 rN − rN − r 2 r 2 AN,N

AN,N−1

r = −g˙ 2 π rN L 2

(1-154)

bN

The coefﬁcients in the last row of A and b are: AN,N = −

kL2π r

AN,N−1

rN −

kL2π = r

r 2

− hout 2 π rN L

r rN − 2

(1-155)

(1-156)

and bN = −hout 2 π rN L T ∞,out − g˙ 2 π rN

r L 2

(1-157)

%Energy balance for control volume N A(N,N)=-k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr-h bar out∗ 2∗ pi∗ r(N)∗ L; A(N,N-1)=k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr; b(N)=-h bar out∗ 2∗ pi∗ r(N)∗ L∗ T infinity out-g dot tp∗ 2∗ pi∗ r(N)∗ Dr∗ L/2;

At this point, the matrix A and vector b are completely set up and can be used to determine the unknown temperatures. The solution to the matrix equation, Eq. (1-141), is: X = A−1 b

(1-158)

where A−1 is the inverse of matrix A. The solution is obtained using the backslash operator in MATLAB (note that this is much more efﬁcient than explicitly solving for the inverse of A using the inv command): %Solve for unknowns X=A\b;

%solve for unknown vector, X

76

One-Dimensional, Steady-State Conduction 130

Temperature (°C)

125 120 115 110 105 100 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-26: Temperature as a function of radius predicted by the numerical model.

The vector of unknowns, X, is identical to the temperatures for this problem. T=X; T C=T-273.2;

%assign temperatures from X %in C

The script cylinder can be executed from the working environment by typing cylinder at the command prompt. After execution, the variables that were deﬁned in the M-ﬁle and the solution will reside in the workspace; for example, you can view the solution vector (T C) by typing T C and you can plot temperature as a function of radius using the plot command. >> cylinder >> T C T C= 100.3213 109.0658 115.6711 120.3899 123.4143 124.8938 124.9468 123.6689 121.1383 117.4197 >> plot(r,T C)

Figure 1-26 illustrates the temperature predicted by the MATLAB numerical model as a function of radius. The solution is exactly equal to the solution that would be obtained

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

77

using the EES code in Section 1.4.2 if the same rate of volumetric generation of thermal energy is used.

1.5.4 Functions The problem in Section 1.5.3 was solved using a script, which is a set of MATLAB instructions that can be stored and edited, but otherwise operates on the main workspace just as if you typed the instructions in one at a time. It is often more convenient to use a function. A function will solve the problem in a separate workspace that communicates with the main workspace (or any workspace from which the function is called) only through input and output variables. There are a few advantages to using a function rather than a script. The function may embody a sequence of operations that must be repeated several times within a larger program. For example, suppose that there are multiple sections of a cylinder, but each section has different material properties or boundary conditions. It would be possible to cut and paste the script that was written in Section 1.5.3 over and over again in order to solve this problem. A more attractive option (and the one least likely to result in an error) is to turn the script cylinder.m into a function that can be called whenever you need to consider conduction through a cylinder with generation. The function can be debugged and tested until you are sure that it works and then applied with conﬁdence at any later time. The use of functions provides modularity and elegance to a program and facilitates parametric studies and optimizations. Any computer code that is even moderately complicated should be broken down into smaller, well-deﬁned sub-programs (functions) that can be written and tested separately before they are integrated through well-deﬁned input/output protocols. MATLAB (or EES) programs are no different. It is convenient to develop your code as a script, but you will likely need to turn your script into a function at some point. Let’s turn the script cylinder.m into a function. Save the ﬁle cylinder as cylinderf. The ﬁrst line of the function must declare that it is a function and deﬁne the input/output protocol. For the cylinder problem, the inputs might include the number of nodes (the variable N), the cylinder radii (the variables r in and r out), and the material conductivity (the variable k). Any other parameter that you are interested in varying could also be provided as an input. The logical outputs include the vector of radial positions that deﬁne the nodes (the vector r) and the predicted temperature at these positions (the vector T C). function[r,T C]=cylinderf(N,r in,r out,k)

The keyword function declares the M-ﬁle to be a function and the variables in square brackets are outputs; these variables should be assigned in the body of the function and are passed to the calling workspace. The function name follows the equal sign and the variables in parentheses are the inputs. Note that nothing you do within the function can affect any variable that is external to the function other than those that are explicitly deﬁned as output variables. You should also comment out the clear command that was used to develop the script. The clear statement is not needed since the function operates with its own variable space. The function is terminated with an end statement. The cylinderf function is shown below; the modiﬁcations to the original script cylinder are indicated in bold.

78

One-Dimensional, Steady-State Conduction

function[r,T C]=cylinderf(N,r in,r out,k) %clear;

%clear all variables from the workspace

% Inputs %r in=0.1; %r out=0.2; g dot tp=1e5; L=1; %k=9; T infinity in=20+273.2; h bar in=100; T infinity out=100+273.2; h bar out=200;

%inner radius of cylinder (m) %outer radius of cylinder (m) %constant volumetric generation (W/mˆ3) %unit length of cylinder (m) %thermal conductivity of cylinder material (W/m-K) %average temperature of fluid inside cylinder (K) %heat transfer coefficient inside cylinder (W/mˆ2-K) %temperature of fluid outside cylinder (K) %average heat transfer coefficient at outer surface (W/mˆ2-K)

% Setup nodes %N=10; %number of nodes (-) Dr=(r out-r in)/(N-1); %distance between nodes (m) for i=1:N r(i)=r in+(i-1)∗ (r out-r in)/(N-1); %radial position of each node (m) end %Setup A and b A=zeros(N,N); b=zeros(N,1); %Energy balance for control volume 1 A(1,1)=-k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr-h bar in∗ 2∗ pi∗ r(1)∗ L; A(1,2)=k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr; b(1)=-h bar in∗ 2∗ pi∗ r(1)∗ L∗ T infinity in-g dot tp∗ 2∗ pi∗ r(1)∗ L∗ Dr/2; %Energy balances for internal control volumes for i=2:(N-1) A(i,i)=-k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr-k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; A(i,i-1)=k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr; A(i,i+1)=k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; b(i)=-g dot tp∗ 2∗ pi∗ r(i)∗ L∗ Dr; end %Energy balance for control volume N A(N,N)=-k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr-h bar out∗ 2∗ pi∗ r(N)∗ L; A(N,N-1)=k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr; b(N)=-h bar out∗ 2∗ pi∗ r(N)∗ L∗ T infinity out-g dot tp∗ 2∗ pi∗ r(N)∗ Dr∗ L/2; % Solve for unknowns X=A\b; T=X; T C=T-273.2; end

%solve for unknown vector, X %assign temperatures from X %in C

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

79

130

Maximum temperature (°C)

128 126 124 122 120 118 116 114 0

10

20 30 40 Thermal conductivity (W/m-K)

50

60

Figure 1-27: Maximum temperature as a function of thermal conductivity.

The following code, typed in the main workspace, will call the function cylinderf for a speciﬁc set of values of the input parameters: N = 10, rin = 0.1 m, rout = 0.2 m, and k = 9 W/m-K. >> N=10; >> r in=0.1; >> r out=0.2; >> k=9; >> [r,T C]=cylinderf(N,r in,r out,k);

The vectors r and T C are the same as those determined in Section 1.5.3. It is easy to carry out a parametric study using functions. For example, create a new script called varyk that calls the function cylinderf for a range of conductivity and keeps track of the maximum temperature in the wall that is predicted for each value of conductivity. N=50; %number of nodes r in=0.1; %inner radius (m) r out=0.2; %outer radius (m) Nk=10; %number of values of k to investigate for i=1:Nk k(i,1)=2+i∗ 50/Nk; %a vector consisting of the conductivities to be considered (W/m-K) [r,T C]=cylinderf(N,r in,r out,k(i,1)); %call the cylinderf function T max C(i,1)=max(T C); %determine the maximum temperature end

Call the script varyk from the main workspace and plot the maximum temperature as a function of conductivity (Figure 1-27). >> varyk >> plot(k,T max C)

80

One-Dimensional, Steady-State Conduction

It is possible to call a function from within a function. The volumetric generation in the cylinder was speciﬁed in Section 1.4.2 as a function of radius according to: g˙ = a + b r + c r2

(1-159)

where a = 1 × 104 W/m3 , b = 2 × 105 W/m4 , and c = 5 × 107 W/m5 . Generate a subfunction (a function that is only visible to other functions in the same M-ﬁle) that has one input (r, the radial position) and one output (g, the volumetric rate of thermal energy generation). The function below, generation, placed at the bottom of the cylinderf ﬁle will be callable from within the function cylinderf. function[g]=generation(r) %the generation function returns the volumetric %generation (W/mˆ3) as a function of radius (m) %constants for generation function a=1e4; %W/mˆ3 b=2e5; %W/mˆ4 c=5e7; %W/mˆ5 g=a+b∗ r+c∗ rˆ2; %volumetric generation end

Replacing the constant generation within the cylinderf code with calls to the function generation will implement the solution for non-uniform generation; the altered portion of the code is shown in bold. %Energy balance for control volume 1 A(1,1)=-k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr-h bar in∗ 2∗ pi∗ r(1)∗ L; A(1,2)=k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr; b(1)=-h bar in∗ 2∗ pi∗ r(1)∗ L∗ T infinity in-generation(r(1))∗ 2∗ pi∗ r(1)∗ L∗ Dr/2; %Energy balances for internal control volumes for i=2:(N-1) A(i,i)=-k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr-k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; A(i,i-1)=k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr; A(i,i+1)=k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; b(i)=-generation(r(i))∗ 2∗ pi∗ r(i)∗ L∗ Dr; end %Energy balance for control volume N A(N,N)=-k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr-h bar out∗ 2∗ pi∗ r(N)∗ L; A(N,N-1)=k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr; b(N)=-h bar out∗ 2∗ pi∗ r(N)∗ L∗ T infinity out-generation(r(N))∗ 2∗ pi∗ r(N)∗ Dr∗ L/2;

Figure 1-28 illustrates the temperature as a function of radius predicted by the model, modiﬁed to account for the non-uniform volumetric generation. Figure 1-28 is identical to Figure 1-14, the solution obtained using EES in Section 1.4.2.

1.5.5 Sparse Matrices The problem considered in Section 1.5.3 can be solved more efﬁciently (from a computational time standpoint) using sparse matrix solution techniques. Sparse matrices are matrices with mostly zero elements. The matrix A that was setup in Section 1.5.3 has

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

81

660 640

Temperature (°C)

620 600 580 560 540 520 500 480 460 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-28: Temperature as a function of radius with non-uniform volumetric generation.

many zero elements and the fraction of the entries in the matrix that are zero increases with increasing N. Matrix A is tridiagonal, which means that it only has non-zero values on the diagonal and on the super- and sub-diagonal positions. This type of banded matrix will occur frequently in numerical solutions of conduction heat transfer problems because the nonzero elements are related to the coefﬁcients in the energy equations and therefore represent thermal interactions between different nodes. Typically, only a few nodes can directly interact and therefore there will be only a few non-zero coefﬁcients in any row. The script below (varyN) keeps track of the time required to run the cylinderf function as the number of nodes in the solution increases. The MATLAB function toc returns the elapsed time relative to the time when the function tic was executed. These functions provide a convenient way to keep track of how much time various parts of a MATLAB program are consuming. A more complete delineation of the execution time within a function can be obtained using the profile function in MATLAB. clear; r in=0.1; r out=0.2; k=9; for i=1:9 N(i,1)=2ˆ(i+1); tic; [r,T C]=cylinderf(N(i,1),r in,r out,k); time(i,1)=toc; end

%inner radius (m) %outer radius (m) %thermal conductivity (W/m-K) %number of nodes (-) %start time %call cylinder function %end timer and record time

The elapsed time as a function of the number of nodes is shown in Figure 1-29. The computational time grows approximately with the number of nodes to the second power. There is an upper limit to the number of nodes that can be considered that depends on the amount of memory installed in your personal computer. It is likely that you cannot set N to be greater than a few thousand nodes. The problem can be solved more efﬁciently if sparse matrices are used. Rather than initializing A as a full matrix of zeros, it can be initialized as a sparse matrix using the

Time required for execution (s)

82

One-Dimensional, Steady-State Conduction 10

1

10

0

10

-1

10

-2

10

-3

10

-4

100

without sparse matrices

with sparse matrices

101

102 103 Number of nodes

104

105

Figure 1-29: Time required to run the cylinderf.m function as a function of the number of nodes with and without sparse matrices.

spalloc (sparse matrix allocation) command. The spalloc command requires three arguments, the ﬁrst two are the dimensions of the matrix and the last is the number of nonzero entries. Equations (1-145), (1-149), and (1-154) show that each equation will include at most three unknowns and therefore each row of A will have at most three non-zero entries. Therefore, the matrix A can have no more than 3 N non-zero entries. %A=zeros(N,N); A=spalloc(N,N,3∗ N);

When the variable A is deﬁned by spalloc, only the non-zero entries of the matrix are tracked. MATLAB operates on sparse matrices just as it does on full matrices. The remainder of the function cylinderf does not need to be modiﬁed, however the function is now much more efﬁcient for large numbers of nodes. If the script varyN is run again, you will ﬁnd that you can use much larger values of N before running out of memory and also that the code executes much faster at large values of N. The execution time as a function of the number of nodes for the cylinderf function using sparse matrices is also shown in Figure 1-29. Notice that the sparse matrix code is actually somewhat less efﬁcient for small values of N due to the overhead required to set up the sparse matrices; however, for large values of N, the code is much more efﬁcient. The computation time grows approximately with N to the ﬁrst power when sparse matrices are used. The use of sparse matrices may not be particularly important for the steady-state, 1-D problem investigated in this section. However, the 2-D and transient problems investigated in subsequent chapters require many more nodes in order to obtain accurate solutions and therefore the use of sparse matrices becomes important for these problems.

1.5.6 Temperature-Dependent Properties The cylinder problem considered in Section 1.5.3 is an example of a linear problem. The set of equations in a linear problem can be represented in the form A X = b, where A is a matrix and b is a vector. Linear problems can be solved in MATLAB without iteration. The inclusion of temperature-dependent conductivity (or generation, or

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

83

any aspect of the problem) causes the problem to become non-linear. The introduction of temperature-dependent properties did not cause any apparent problem for the EES model discussed in Section 1.4.3, although it became important to identify a good set of guess values. EES automatically detected the non-linearity and iterated as necessary to solve the non-linear system of equations. However, non-linearity complicates the solution using MATLAB because the equations can no longer be put directly into matrix format. The coefﬁcients multiplying the unknown temperatures themselves depend on the unknown temperatures. It is necessary to use some type of a relaxation process in order to use MATLAB to solve the problem. There are a few options for solving this kind of nonlinear problem; in this section, a technique that is sometimes referred to as successive substitution is discussed. The successive substitution process begins by assuming a temperature distribution throughout the computational domain (i.e., assume a value of temperature for each node, Tˆ i for i = 1..N). The assumed values of temperature are used to compute the coefﬁcients that are required to set up the matrix equation (e.g., the temperature-dependent conductivity). The matrix equation is subsequently solved, as discussed in Section 1.5.3, which results in a prediction for the temperature distribution throughout the computational domain (i.e., a predicted value of the temperature for each node, Ti for i = 1..N). The assumed and predicted temperatures at each node are compared and used to compute an error; for example, the sum of the square of the difference between the value of Tˆ i and Ti at every node. If the error is greater than some threshold value, then the process is repeated, this time using the solution Ti as the assumed temperature distribution Tˆ i in order to calculate the coefﬁcients of the matrix equation. The implementation of the successive substitution process carries out the solution that was developed in Sections 1.5.2 through 1.5.4 within a while loop that terminates when the error becomes sufﬁciently small. This process is illustrated schematically in Figure 1-30 and demonstrated in EXAMPLE 1.5-1. assume a temperature distribution Tˆi for i = 1.. N setup A and b using Tˆi to solve for any temperature-dependent coefficients

Tˆi = Ti for i = 1.. N

solve the matrix equation A X = b in order to predict Ti for i = 1.. N compute the convergence error, err 2 1 N ˆ e.g., err = Ti − Ti ∑ N i =1

(

no

)

err < convergence tolerance?

yes done Figure 1-30: Successive substitution technique for solving problems with temperature-dependent properties.

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

84

One-Dimensional, Steady-State Conduction

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM The kinetic energy associated with the atmospheric entry of a space vehicle results in extremely large heat ﬂuxes, large enough to completely vaporize the vehicle if it were not adequately protected. The outer structure of the vehicle is called its aeroshell and the outer layer of the material on the aeroshell is called the Thermal Protection System (or TPS). The heat ﬂux experienced by the aeroshell can reach 100 W/cm2 , albeit for only a short period of time. Consider a TPS consisting of a non-metallic ablative layer with thickness, thab = 5 cm that is bonded to a layer of steel with thickness, ths = 1 cm, as shown in Figure 1. The outer edge of the ablative heat shield (x = 0) reaches the material’s melting temperature (Tm = 755 K) under the inﬂuence of the heat ﬂux. The melting limits the temperature that is reached at the outer surface of the shield and protects the internal air until the shield is consumed. In this problem, we will assume that the shield is consumed very slowly so that a quasi-steady temperature distribution is set up in the ablative shield. The latent heat of fusion of the ablative shield is ifus,ab = 200 kJ/kg and its density is ρab = 1200 kg/m3 . 2

T∞ = 320 K, h = 10 W/m -K ths = 1 cm

x

steel, ks = 20 W/m-K

thab = 5 cm

ablative shield Tm = 755 K Δifus, ab = 200 kJ/kg

q⋅ ′′ = 100 W/cm

2

ρab = 1200 kg/m3 Figure 1: A Thermal Protection System.

The thermal conductivity of the ablative shield is highly temperature dependent; thermal conductivity values at several temperatures are provided in Table 1. Table 1: Thermal conductivity of ablative shield material in the solid phase Temperature

Thermal conductivity

300 K 350 K 400 K 450 K 500 K 550 K 600 K 650 K 700 K 755 K

0.10 W/m-K 0.15 W/m-K 0.19 W/m-K 0.21 W/m-K 0.22 W/m-K 0.24 W/m-K 0.28 W/m-K 0.33 W/m-K 0.38 W/m-K 0.45 W/m-K

85

The thermal conductivity of the steel may be assumed to be constant at k s = 20 W/m-K. The internal surface of the steel is exposed to air at T∞ = 320 K with average heat transfer coefﬁcient, h = 10 W/m2 -K. Assume that the TPS reaches a quasi-steady-state under the inﬂuence of a heat ﬂux q˙ = 100 W/cm2 and that the surface temperature of the ablation shield reaches its melting point. a) Develop a numerical model using MATLAB that can determine the heat ﬂux that is transferred to the air and the rate that the ablative shield is being consumed. The solution is developed as a MATLAB function called Ablative shield; the input to the function is the number of nodes to use in the solution while the outputs include the position of the nodes and the predicted temperature at each node as well as the two quantities speciﬁcally requested, the heat ﬂux incident on the air and the rate of shield ablation. Select New and M-File from the File menu and save the M-ﬁle as Ablative shield (the .m extension is added automatically). The ﬁrst line of the function establishes the input/output protocol: function[x,T,q flux in Wcm2,dthabdt cms]=Ablative shield(N) %EXAMPLE 1.5-1: Thermal Protection System for Atmospheric Entry % % Inputs: % N: number of nodes in solution (-) % % Outputs: % x: position of nodes (m) % T: temperatures at nodes (K) % q flux in Wcm2: heat flux to air (W/cmˆ2) % dthabdt cms: rate of shield consumption (cm/s)

The next section of the code establishes the remaining input parameters (i.e., those not provided as arguments to the function); note that each input is converted immediately to SI units. th ab=0.05; th s=0.01; k s=20; q flux=100∗ 100ˆ2; T m=755; DELTAi fus ab=200e3; h bar=10; T infinity=320; rho ab=1200; A c=1;

%ablation shield thickness (m) %steel thickness (m) %steel conductivity (W/m-K) %heat flux (W/mˆ2) %melting temperature (K) %latent heat of fusion (J/kg) %heat transfer coefficient (W/mˆ2-K) %internal air temperature (K) %density (kg/mˆ3) %per unit area of wall (mˆ2)

In order to solve this problem, it is necessary to create a function that returns the conductivity of the ablative shield material. The easiest way to do this is to enter the data from Table 1 into a sub-function and interpolate between the data points.

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

86

One-Dimensional, Steady-State Conduction

function[k]=k ab(T) %data Td=[300,350,400,450,500,550,600,650,700,755]; kd=[0.1,0.15,0.19,0.21,0.22,0.24,0.28,0.33,0.38,0.45]; k=interp1(Td,kd,T,’spline’); %interpolate data to obtain conductivity end

The interp1 function in MATLAB is used for the interpolation. The interp1 function requires three arguments; the ﬁrst two are the vectors of the independent and dependent data, respectively, and the third is the value of the independent variable at which you want to ﬁnd the dependent variable. An optional fourth argument speciﬁes the type of interpolation to use. To obtain more detailed help for this (or any) MATLAB function, use the help command: >> help interp1 INTERP1 1-D interpolation (table lookup) YI=INTERP1(X,Y,XI) interpolates to find YI, the values of the underlying function Y at the points in the array XI. X must be a vector of length N. If Y is a vector, then . . .

The numerical model of the TPS will consider the ablative material; the steel will be considered as part of the thermal resistance between the inner surface of the shield and the air and therefore will affect the boundary condition at x = thab . There is no reason to treat the steel with the numerical model since the steel has, by assumption, constant properties and is at steady state. Therefore, the analytical solution derived in Section 1.2.3 for the resistance of a plane wall holds exactly. The nodes are distributed uniformly from x = 0 to x = thab , where x = 0 corresponds to the outer surface of the shield, as shown in Figure 2. xi = (i − 1)

thab (N − 1)

for i = 1 . . . N

The distance between adjacent nodes (x) is: x =

thab (N − 1)

The nodes are setup in the MATLAB code according to: %setup nodes DELTAx=th ab/(N-1); for i=1:N x(i,1)=th ab∗ (i-1)/(N-1); end

%distance between nodes %position of each node

87

An internal control volume, shown in Figure 2, experiences only conduction; therefore, a steady-state energy balance on the control volume is: q˙ t op + q˙ bot t om = 0

(1) q⋅ top

TN

q⋅ bottom q⋅

top

Figure 2: Distribution of nodes and control volumes.

TN-1 Ti+1

Ti ⋅q bottom Ti-1 x T1

The conductivity used to approximate the conduction heat transfer rates in Eq. (1) must be evaluated at the temperature of the boundaries, i.e., the average of the temperatures of the nodes involved in the conduction process, as discussed previously in Section 1.4.3. With this understanding, these rate equations become: q˙ t op = k ab,T =(Ti+1 +Ti )/2

Ac (Ti+1 − Ti ) x

q˙ bot t om = k ab,T =(Ti−1 +Ti )/2

Ac (Ti−1 − Ti ) x

(2) (3)

where Ac is the cross-sectional area. Substituting Eqs. (2) and (3) into Eq. (1) leads to: k ab,T =(Ti+1 +Ti )/2

Ac Ac (Ti+1 − Ti ) + k ab,T =(Ti−1 +Ti )/2 (Ti−1 − Ti ) = 0 for i = 2.. (N − 1) x x (4)

The node on the outer surface (i.e., node 1) has a speciﬁed temperature, the melting temperature of the ablative material: T1 = Tm

(5)

The energy balance for the node on the inner surface (i.e., node N) is also shown in Figure 2: q˙ t op + q˙ bot t om = 0

(6)

where q˙ bot t om = k ab,T =(TN −1 +TN )/2 q˙ t op =

Ac (TN −1 − TN ) x

(T∞ − TN ) Rcond,s + Rconv

(7)

(8)

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

88

One-Dimensional, Steady-State Conduction

where Rcond,s and Rconv are the thermal resistances associated with conduction through the steel and convection from the internal surface of the steel to the air.

R cond s=th s/(A c∗ k s); R conv=1/(A c∗ h bar);

Rcond,s =

ths k s Ac

Rconv =

1 h Ac

%conduction resistance of steel (K/W) %convection resistance (K/W)

Substituting Eqs. (7) and (8) into Eq. (6) leads to: Ac (T∞ − TN ) + k ab,T =(TN −1 +TN )/2 (TN −1 − TN ) = 0 Rcond,s + Rconv x

(9)

Note that Eqs. (4), (5), and (9) are a complete set of equations in the unknown temperatures Ti for i = 1..N; however, these equations cannot be written as a linear combination of the unknown temperatures because the conductivity of the ablative shield depends on temperature. In order to apply successive substitution, the conductivity will be evaluated using guess values for these temperatures (Tˆ ). A linear variation in temperature from Tm to T∞ is used as the guess values to start the process: (i − 1) Tˆi = Tm + (T∞ − Tm) (N − 1)

for i = 1..N

%initial guess for temperature distribution for i=1:N Tg(i,1)=T m+(T infinity-T m)∗ (i-1)/(N-1); %linear from melting to air (K) end

The matrix A and vector b are initialized according to: %setup matrices A=spalloc(N,N,3∗ N); b=zeros(N,1);

Equation (5) is rewritten to make it clear what the coefﬁcient and the constants are: T1 [1] = Tm A1,1

%node 1 A(1,1)=1; b(1,1)=T m;

b1

89

Equation (4) is rewritten, using the guess temperatures to compute the conductivity of the ablative shield and also to clearly identify the coefﬁcients and constants: Ac Ac Ac +Ti+1 k ab,T =(Tˆi+1 +Tˆi )/2 Ti −k ab,T =(Tˆi+1 +Tˆi )/2 − k ab,T =(Tˆi−1 +Tˆi )/2 x x x Ai,i

Ai,i+1

Ac + Ti−1 k ab,T =(Tˆi−1 +Tˆi )/2 = 0 for i = 2.. (N − 1) x Ai,i−1

%internal nodes for i=2:(N-1) A(i,i)=-k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx-k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; A(i,i+1)=k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx; A(i,i-1)=k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; end

Equation (9) is rewritten: Ac 1 TN − − k ab,T =(TˆN −1 +TˆN )/2 Rcond,s + Rconv x

AN ,N

+ TN −1 k ab,T =(TˆN −1 +TˆN )/2 AN ,N −1

Ac x

T∞ =− R + Rconv cond,s bN

%node N A(N,N)=-k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx-1/(R cond s+R conv); A(N,N-1)=k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx; b(N,1)=-T infinity/(R cond s+R conv);

The matrix equation is solved: X=A\b; T=X;

%solve matrix equation

The solution is not complete, it is necessary to iterate until the solution (T ) matches the assumed temperature (Tˆ ). This is accomplished by placing the commands that setup and solve the matrix equation within a while loop that terminates when the rms error (err) is below some tolerance (tol). The rms error is computed according to: N 1 (Ti − Tˆi )2 err = N i=1

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

90

One-Dimensional, Steady-State Conduction

err=sqrt(sum((T-Tg).ˆ2)/N)

%compute rms error

Note that the sum command computes the sum of all of the elements in the vector provided to it and the use of .ˆ2 indicates that each element in the vector should be squared (as opposed to ˆ2 which would multiply the vector by itself ). Also notice that the error computation is not terminated with a semicolon so that the value of the rms error will be reported after each iteration. To start the iteration process, the value of the error is set to a large number (larger than tol) in order to ensure that the while loop executes at least once. After the solution has been obtained, the rms error is computed and the vector Tg is reset to the vector T. The result is shown below, with the new lines highlighted in bold. err=999; tol=0.01; while (err>tol)

%initial value of error (K), must be larger than tol %tolerance for convergence (K)

%node 1 A(1,1)=1; b(1,1)=T m; %internal nodes for i=2:(N-1) A(i,i)=-k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx-k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; A(i,i+1)=k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx; A(i,i-1)=k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; end %node N A(N,N)=-k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx-1/(R cond s+R conv); A(N,N-1)=k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx; b(N,1)=-T infinity/(R cond s+R conv); X=A\b; T=X;

%solve matrix equation

err=sqrt(sum((T-Tg).ˆ2)/N) Tg=T; end

%compute rms error %reset guess values used to setup A and b

The heat ﬂux to the air (q˙ in ) is computed. q˙ in =

(TN − T∞ ) Ac (Rs + Rconv )

The rate at which the ablative shield is consumed can be determined using an energy balance at the outer surface; the heat ﬂux related to re-entry either consumes the shield or is transferred to the air. Note that this is actually a simpliﬁcation of the problem; this is a moving boundary problem and this solution is valid only in the limit that the energy carried by the motion of interface is small relative to the energy removed by its vaporization. q˙ = ρab ifus,ab

dthab + q˙ in dt

91

or dthab q˙ − q˙ in = dt ρab ifus,ab

These calculations are provided by adding the following lines to the Ablative shield function: q flux in=(T(N)-T infinity)/(R cond s+R conv)/A c; q flux in Wcm2=q flux in/100ˆ2; dthabdt=(q flux-q flux in)/(DELTAi fus ab∗ rho ab); dthabdt cms=dthabdt∗ 100; end

%heat flux to air (W/mˆ2) %heat flux to air (W/cmˆ2) %rate of shield consumption (m/s) %rate of shield consumption (cm/s)

Calling the function Ablative_shield from the workspace leads to: >> [x,T,q flux in Wcm2,dthabdt cms]=Ablative shield(100); err = 105.7742 err = 9.0059 err = 0.8979 err = 0.1798 err = 0.0238 err = 0.0035

800

Temperature (K)

750 700 650 600 550 500 450 0

0.01

0.02 0.03 Position (m)

0.04

0.05

Figure 3: Temperature as a function of position within the ablative shield.

Figure 3 shows the temperature as a function of position within the shield. The temperature gradient agrees with intuition; the temperature gradient is smaller where the conductivity is largest (i.e., at higher temperatures), which is consistent

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

One-Dimensional, Steady-State Conduction

with a constant heat ﬂow. It is important to verify that the number of nodes used in the solution is adequate. Figure 4 shows the heat ﬂux to the air as a function of the number of nodes in the solution and indicates that at least 20 nodes are required. The heat ﬂux to the air at the inner surface of the TPS is 0.166 W/cm2 , nearly three orders of magnitude less than the heat ﬂux at the outer surface. The TPS is being consumed at a rate of 0.416 cm/s suggesting that the atmospheric entry process cannot last more than ten seconds without consuming the entire shield. The thermal analysis of this problem does not consider the loss of ablative material with time and it is therefore a very simpliﬁed model of the TPS. 0.166 0.1655 Heat flux to air (W/cm2)

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

92

0.165 0.1645 0.164 0.1635 0.163 0.1625 0.162 1

10

100 Number of nodes

1000

Figure 4: Heat ﬂux to the air as a function of the number of nodes.

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces 1.6.1 Introduction The situations that were examined in Sections 1.2 through 1.5 were truly onedimensional; that is, the geometry and boundary conditions dictated that the temperature could only vary in one direction. In this section, problems that are only approximately 1-D, referred to as extended surfaces, are considered. Extended surfaces are thin pieces of conductive material that can be approximated as being isothermal in two dimensions with temperature variations in only one direction. Extended surfaces are particularly relevant to a large number of thermal engineering applications because the ﬁns that are used to enhance heat transfer in heat exchangers can often be treated as extended surfaces.

1.6.2 The Extended Surface Approximation An extended surface is not truly 1-D; however, it is often approximated as being such in order to simplify the analysis. Figure 1-31 shows a simple extended surface, sometimes called a ﬁn. The ﬁn length (in the x-direction) is L and its thickness (in the y-direction) is th. The width of the ﬁn in the z-direction, W, is assumed to be much larger than its thickness in the y-direction, th. The conductivity of the ﬁn material is k. The base of

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

93

W

Figure 1-31: A constant cross-sectional ﬁn.

y

T∞, h

x

Tb

L th

the ﬁn (at x = 0) is maintained at temperature Tb and the ﬁn is surrounded by ﬂuid at temperature T∞ with heat transfer coefﬁcient h. Energy is conducted axially from the base of the ﬁn. As energy moves along the ﬁn in the x-direction, it is also conducted laterally to the ﬁn surface where it is ﬁnally transferred by convection to the surrounding ﬂuid. Temperature gradients always accompany the transfer of energy by conduction through a material; thus, the temperature must vary in both the x- and y-directions and the temperature distribution in the extended surface must be 2-D. However, there are many situations where the temperature gradient in the y-direction is small and therefore can be neglected in the solution without signiﬁcant loss of accuracy. Figure 1-32 illustrates the temperature as a function of lateral position (y) at various axial locations (x) for an arbitrary set of conditions. (The 2-D solution for this ﬁn is derived in EXAMPLE 2.2-1.) Notice that the temperature decreases in both the x- and y-directions as conduction occurs in both of these directions. At every value of axial location, x, there is a temperature drop through the material in the y-direction due to conduction. (For x/L = 0.25, this temperature drop is labeled T cond,y in Figure 1-32.) There is another temperature drop from the surface of the material to the surrounding ﬂuid due to convection (labeled T conv in Figure 1-32 for x/L = 0.25). The extended surface approximation refers to the assumption that the temperature in the material is a function only of x and not of y; this approximation turns a 2-D problem into a 1-D problem, which is easier to solve. The extended surface approximation is valid when the temperature drop due to conduction in the y-direction is much less than the temperature drop due to convection (i.e, T cond,y restart; > ODE:=diff(diff(T(x),x),x)-per∗ h_bar∗ T(x)/(k∗ A_c)=-per∗ h_bar∗ T_infinity/(k∗ A_c); ODE :=

d2 per h bar T inf inity per h bar T(x) =− T(x) − dx2 kA c kA c

and obtain the solution using the dsolve command:

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

99

> Ts:=dsolve(ODE); √

Ts := T(x) = e

√ per h bar x √ √ k A c

C2 + e

√ √ − per h bar x √ √ k A c

C1 + T inf inity

The solution identiﬁed by Maple is identical to Eq. (1-192). The constants C1 and C2 are obtained by enforcing the boundary conditions. One boundary condition is clear; the base temperature is speciﬁed and therefore: T x=0 = T b

(1-194)

Substituting Eq. (1-192) into Eq. (1-194) leads to: C1 + C2 + T ∞ = T b

(1-195)

The boundary condition at the tip of the ﬁn is less clear and there are several possibilities. The most common model assumes that the tip is adiabatic. In this case, an interface balance at the tip (see Figure 1-35) leads to: q˙ x=L = 0 Substituting Fourier’s law into Eq. (1-196) leads to: dT =0 dx x=L

(1-196)

(1-197)

Substituting Eq. (1-192) into Eq. (1-197) leads to: C1 m exp (m L) − C2 m exp (−m L) = 0

(1-198)

Note that Eqs. (1-195) and (1-198) are together sufﬁcient to determine C1 and C2 . If the solution is implemented in EES then no further algebra is required. However, it is worthwhile to obtain the explicit form of the solution for this common problem. Equation (1-195) is multiplied by m exp(m L) and rearranged: C1 m exp (m L) + C2 m exp (m L) = (T b − T ∞ ) m exp (m L)

(1-199)

Equation (1-198) is added to Eq. (1-199): C1 m exp (m L) − C2 m exp (−m L) = 0 + [C1 m exp(m L) + C2 m exp(m L) = (T b − T ∞ ) m exp(m L)] −C2 m exp(−m L) − C2 m exp(m L) = −(T b − T ∞ ) m exp(m L)

(1-200)

Equation (1-200) can be solved for C2 : C2 =

(T b − T ∞ ) exp (m L) exp (−m L) + exp (m L)

(1-201)

A similar sequence of operations leads to: C1 =

(T b − T ∞ ) exp (−m L) exp (−m L) + exp (m L)

(1-202)

100

One-Dimensional, Steady-State Conduction

These constants can also be obtained from Maple; the governing differential equation, Eq. (1-173) is entered and solved, this time in terms of m: > restart; > ODE:=diff(diff(T(x),x),x)-mˆ2∗ T(x)=-mˆ2∗ T_infinity; ODE :=

d2 T(x) − m2 T(x) = −m2 T inf inity dx2

> Ts:=dsolve(ODE); Ts := T(x) = e(−mx) C2 + e(mx) C1 + T inf inity

The boundary conditions, Eqs. (1-194) and (1-197), are deﬁned: > BC1:=rhs(eval(Ts,x=0))=T_b; BC1 := C2 + C1 + T inf inity = T b > BC2:=rhs(eval(diff(Ts,x),x=L))=0; BC2 := −m e(−m L) C2 + m e(m L) C1 = 0

and solved symbolically: > constants:=solve({BC1,BC2},{_C1,_C2});

constants :=

C2 = −

e(m L) (T inf inity − T b) e(−m L) (T inf inity − T b) , C1 = − (−m L) (m L) e +e e(−m L) + e(m L)

!

The constants identiﬁed by Maple are identical to Eqs. (1-201) and (1-202). Substituting the constants of integration into the general solution, Eq. (1-192), leads to: (T b − T ∞ ) exp (m L) (T b − T ∞ ) exp (−m L) exp (m x) + exp (−m x) + T ∞ exp (−m L) + exp (m L) exp (−m L) + exp (m L) (1-203) or, using Maple: T =

> Ts:=subs(constants,Ts); Ts := T(x) = e(mx) e(−m L) (T inf inity − T b) e(−mx) e(m L) (T inf inity − T b) − + T inf inity − (−m L) (m L) e +e e(−m L) + e(m L)

Equation (1-203) can be simpliﬁed to: T = (T b − T ∞ )

[exp (−m (L − x)) + exp (m (L − x))] + T∞ [exp (−m L) + exp (m L)]

(1-204)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

101

Equation (1-204) can be stated more concisely using hyperbolic functions as opposed to exponentials; hyperbolic functions are functions that have been deﬁned in terms of exponentials. The combinations: 1 [exp (A) + exp (−A)] 2

(1-205)

1 [exp (A) − exp (−A)] 2

(1-206)

occur so frequently in math and science that they are given special names, the hyperbolic cosine (cosh, pronounced “kosh”) and the hyperbolic sine (sinh, pronounced “cinch”), respectively. cosh (A) =

1 [exp (A) + exp (−A)] 2

(1-207)

sinh (A) =

1 [exp (A) − exp (−A)] 2

(1-208)

These hyperbolic functions behave in much the same way that the cosine and sine functions do. For example: cosh2 (A) − sinh2 (A) 1 1 [exp (A) + exp (−A)]2 − [exp (A) − exp (−A)]2 4 4 1 2 = [exp (A) + 2 exp (A) exp (−A) + exp2 (−A)] 4 1 − [exp2 (A) − 2 exp (A) exp (−A) + exp2 (−A)] 4 = exp (A) exp (−A) = 1 =

(1-209)

or cosh2 (A) − sinh2 (A) = 1

(1-210)

which is analogous to the trigonometric identity: cos2 (A) + sin2 (A) = 1

(1-211)

Furthermore, the derivative of cosh is sinh and vice versa, which is analogous to derivatives of sine and cosine (albeit, without the sign change): " # d d 1 [sinh (A)] = [exp (A) − exp (−A)] dx dx 2 (1-212) dA dA 1 = cosh (A) = [exp (A) + exp (−A)] 2 dx dx or d dA [sinh (A)] = cosh (A) dx dx

(1-213)

A similar set of operations leads to: d dA [cosh (A)] = sinh (A) dx dx

(1-214)

102

One-Dimensional, Steady-State Conduction

Equation (1-204) is rearranged so that it can be expressed in terms of hyperbolic cosines: T = (T b − T ∞ )

2 [exp (−m (L − x)) + exp (m (L − x))] +T ∞ 2 [exp L) + exp (m L)] (−m cosh(m(L−x))

1/ cosh(m L)

(1-215) T = (T b − T ∞ )

cosh (m (L − x)) + T∞ cosh (m L)

(1-216)

Equation (1-216) is much more concise but functionally identical to Eq. (1-204). Note that Maple can convert from exponential to hyperbolic form as well, using the convert command with the ‘trigh’ identiﬁer: > T_s:=convert(Ts,‘trigh’); T s := T(x) = (−T inf inity + T b) cosh (m x) + T inf inity +

sinh (m x) sinh (m L) (T inf inity − T b) cosh (m L)

The rate of heat transfer to the base of the ﬁn (q˙ ﬁn ) is obtained from Fourier’s law evaluated at x = 0: dT (1-217) q˙ ﬁn = −k Ac dx x=0 Substituting Eq. (1-216) into Eq. (1-217) leads to: cosh (m (L − x)) d + T∞ q˙ ﬁn = −k Ac (T b − T ∞ ) dx cosh (m L) x=0 =−

k Ac (T b − T ∞ ) d [cosh (m (L − x))]x=0 cosh (m L) dx

(1-218)

Recalling that the derivative of cosh is sinh, according to Eq. (1-214): q˙ ﬁn =

k Ac (T b − T ∞ ) m [sinh (m (L − x))]x=0 cosh (m L)

(1-219)

or q˙ ﬁn = (T b − T ∞ ) k Ac m

sinh (m L) cosh (m L)

The same result may be obtained from Maple: > q_dot_fin:=-k∗ A_c∗ eval(diff(T_s,x),x=0); q dot f in := −k A c

d k A c m sinh(m L)(T inf inity − T b) T(x) x=0 = − dx cosh(m L)

(1-220)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces 1

mL = 0.1

0.9 Dimensionless temperature

103

mL = 0.5

0.8 0.7

mL = 1

0.6 0.5 0.4

mL = 2

0.3 mL = 5 mL = 10 0.1 mL = 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L 0.2

0.8

0.9

1

Figure 1-36: Dimensionless ﬁn temperature as a function of dimensionless position for various values of the parameter m L.

The ratio of sinh to cosh is the hyperbolic tangent ( just as the ratio of sine to cosine is tangent); therefore, Eq. (1-220) may be written as: $ (1-221) q˙ ﬁn = (T b − T ∞ ) h per k Ac tanh (m L) The temperature distribution and heat transfer rate provided by Eqs. (1-216) and (1-221) represent the most important aspects of the solution. The solutions for ﬁns with other types of boundary conditions at the tip are summarized in Table 1-4.

1.6.4 Fin Behavior The temperature distribution within a ﬁn with an adiabatic tip, Eq. (1-216), can be expressed as a ratio of the temperature elevation with respect to the ﬂuid temperature to the base-to-ﬂuid temperature difference: x cosh m L 1 − T − T∞ L = (1-222) Tb − T∞ cosh (m L) The dimensionless temperature as a function of dimensionless position (x/L) is shown in Figure 1-36 for various values of m L. Regardless of the value of m L, the solutions satisfy the boundary conditions; the curves intersect at (T − T∞ )/(Tb − T∞ ) = 1.0 at x/L = 0 and the slope of each curve is zero at x/L = 1.0. However, the shape of the curves changes with m L. Smaller values of m L result in a smaller temperature drop due to conduction along the ﬁn (and therefore more due to convection from the ﬁn surface) whereas large values of m L have a corresponding large temperature drop due to conduction and little for convection. The functionality of an extended surface is governed by two processes; conduction along the ﬁn (in the x-direction) and convection from its surface. (Conduction in the y-direction was neglected in the derivation of the solution.) The parameter m L represents the balance of these two effects. The resistance to conduction along the ﬁn (Rcond,x )

104

One-Dimensional, Steady-State Conduction

Table 1-4: Solutions for constant cross-section extended surfaces with different end conditions. Tip condition

Solution cosh (m (L − x)) T − T∞ = Tb − T∞ cosh (m L) $ q˙ ﬁn = (T b − T ∞ ) h per k Ac tanh (m L)

h, T∞

Adiabatic tip Tb

x

ηﬁn = tanh (m L) / (m L) h, T∞

Convection from tip Tb

x

h, T∞

Speciﬁed tip temperature

TL Tb

x

h, T∞

Inﬁnitely long

to ∞ Tb

x

where: T b = base temperature T ∞ = ﬂuid temperature per = perimeter L = length T = temperature per h L = ﬁn constant mL = k Ac

h sinh (m (L − x)) cosh (m (L − x)) + T − T∞ m k = Tb − T∞ h sinh (m L) cosh (m L) + mk h $ cosh (m L) sinh (m L) + m k q˙ ﬁn = (T b − T ∞ ) h per k Ac h sinh (m L) cosh (m L) + mk [tanh (m L) + m L ARtip ] ηﬁn = m L [1 + m L ARtip tanh (m L)] (1 + ARtip ) TL − T∞ sinh (m x) + sinh (m (L − x)) T − T∞ Tb − T∞ = Tb − T∞ sinh (m L) TL − T∞ cosh (m L) − $ Tb − T∞ q˙ ﬁn = (T b − T ∞ ) h per k Ac sinh (m L) T − T∞ = exp (−m x) Tb − T∞ $ q˙ ﬁn = (T b − T ∞ ) h per k Ac h = heat transfer coefﬁcient Ac = cross-sectional area k = thermal conductivity q˙ ﬁn = ﬁn heat transfer rate x = position (relative to base of ﬁn) ARtip =

Ac = tip area ratio per L

is given by: Rcond,x =

L k Ac

(1-223)

and the resistance to convection from the surface (Rconv ) is: Rconv =

1 h per L

(1-224)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

105

Note that the resistance in Eq. (1-223) is related to conduction in the x-direction and should not be confused with Rcond,y in Eq. (1-160), which was used to deﬁne the Biot number. Clearly the behavior of the ﬁn cannot be exactly represented using these thermal resistances; the analysis in Section 1.6.3 was complex and showed that the conduction along the ﬁn is gradually reduced by convection. Nevertheless, the relative value of these resistances provides substantial insight into the qualitative characteristics of the ﬁn: per h 2 Rcond,x = L Rconv k Ac The resistance ratio in Eq. (1-225) is related to the parameter m L: ⎡ ⎤2 per h per h 2 Rcond,x 2 L⎦ = L = (m L) = ⎣ k Ac k Ac Rconv

(1-225)

(1-226)

In the light of Eq. (1-226), Figure 1-36 begins to make sense. A small value of m L represents a ﬁn with a small resistance to conduction in the x-direction relative to the resistance to convection. The temperature drop due to the conduction heat transfer along the ﬁn must therefore be small. At the other extreme, a large value of m L indicates that the resistance to conduction in the x-direction is much larger than the resistance to convection and therefore most of the temperature drop is related to the conduction heat transfer. Before starting an analysis of an extended surface, it is helpful to calculate the two dimensionless parameters discussed thus far. The value of the Biot number will indicate whether it is possible to treat the situation as a 1-D problem and the value of m L will determine whether it is even worth the time. If m L is either very small or very large, then the behavior can be understood with no analysis: the ﬁn temperature will be very close to the base temperature or the ﬂuid temperature, respectively.

1.6.5 Fin Efﬁciency and Resistance The ﬁn efﬁciency is deﬁned as the ratio of the heat transfer to the ﬁn (q˙ ﬁn ) to the heat transfer to an ideal ﬁn. An ideal ﬁn is made of an inﬁnitely conductive material and therefore this limit corresponds to a ﬁn that is everywhere at a temperature of Tb . Note that an ideal ﬁn with inﬁnite conductivity corresponds to the limit of m L = 0 in Figure 1-36. ηﬁn =

heat transfer to ﬁn heat transfer to ﬁn as k → ∞

(1-227)

or ηﬁn =

q˙ ﬁn h As,ﬁn (T b − T ∞ )

(1-228)

where the denominator of Eq. (1-228) is the product of the average heat transfer coefﬁcient, the surface area of the ﬁn that is exposed to ﬂuid, and the base-to-ﬂuid temperature difference. The ﬁn efﬁciency represents the degree to which the temperature drop along the ﬁn due to conduction has reduced the average temperature difference driving convection from the ﬁn surface. The ﬁn efﬁciency depends on the boundary condition at the tip and the geometry of the ﬁn. For a constant cross-sectional area ﬁn with an adiabatic tip, Eq. (1-221) can be

106

One-Dimensional, Steady-State Conduction 1 0.9 0.8

Fin efficiency

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2 2.5 3 Fin constant, mL

3.5

4

4.5

5

Figure 1-37: Fin efﬁciency for a constant cross-section, adiabatic tipped ﬁn as a function of the parameter m L.

substituted into Eq. (1-228): ηﬁn =

(T b − T ∞ )

$ h per k Ac tanh (m L)

h per L (T b − T ∞ )

(1-229)

or tanh (m L) ηﬁn = hper L kAc

(1-230)

which can be simpliﬁed by substituting in the deﬁnition of m: ηﬁn =

tanh (m L) mL

(1-231)

Figure 1-37 illustrates the ﬁn efﬁciency for a ﬁn having a constant cross-sectional area and an adiabatic tip as a function of m L. Notice that the ﬁn efﬁciency drops as m L increases. This is consistent with the discussion in Section 1.6.4; a large value of m L corresponds to a large temperature drop due to conduction along the ﬁn, as seen in Figure 1-36. The ﬁn efﬁciency is the most useful format for presenting the results of a ﬁn solution because it allows the calculation of a ﬁn resistance (Rﬁn ). The ﬁn resistance is the thermal resistance that opposes heat transfer from the base of the ﬁn to the surrounding ﬂuid. Equation (1-228) can be rearranged: q˙ ﬁn = ηﬁn h As,ﬁn (T b − T ∞ )

(1-232)

1/Rﬁn

where As,ﬁn is the surface area of the ﬁn exposed to the ﬂuid. Note that without the ﬁn efﬁciency, Eq. (1-232) is equivalent to Newton’s law of cooling and therefore the thermal resistance is deﬁned in basically the same way: Rﬁn =

1 ηﬁn h As,ﬁn

(1-233)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

107

The ﬁn efﬁciency is always less than one and therefore the ﬁn resistance will be larger than the corresponding convection resistance; this increase in resistance is related to the conduction resistance within the ﬁn. The concept of a ﬁn resistance is convenient since it allows the effect of ﬁns to be incorporated into a more complex problem (for example, one in which ﬁns are attached to other structures) as additional resistances in a network. For most extended surfaces, the surface area that is available for convection at the tip is insigniﬁcant relative to the total area for convection and therefore the adiabatic tip solution for ﬁn efﬁciency is sufﬁcient. However, the solution for the heat transfer from a ﬁn that experiences convection from its tip (see Table 1-4) can also be used to provide an expression for ﬁn efﬁciency: $ q˙ ﬁn = (T b − T ∞ )

h cosh (m L) mk h per k Ac h cosh (m L) + sinh (m L) mk sinh (m L) +

(1-234)

So the ﬁn efﬁciency is: $

ηﬁn

h cosh (m L) m k = = h (per L + Ac ) h (per L + Ac ) h cosh (m L) + sinh (m L) mk h per k Ac

q˙ ﬁn

sinh (m L) +

which can be simpliﬁed somewhat to: & % tanh (m L) + m L ARtip % & ηﬁn = m L 1 + m L ARtip tanh (m L) 1 + ARtip

(1-235)

(1-236)

where ARtip is the ratio of the area for convection from the tip to the surface area along the length of the ﬁn: ARtip =

Ac per L

(1-237)

1 0.9 0.8 Fin efficiency

0.7

AR tip = 0.0 AR tip = 0.1 AR tip = 0.2

0.6 0.5 0.4 0.3 0.2

AR tip = 0.3 AR tip = 0.4 AR tip = 0.5

0.1 0 0

0.5

1

1.5

2 2.5 3 3.5 Fin constant, m L

4

4.5

5

Figure 1-38: Fin efﬁciency for a constant cross-section ﬁn with convection from the tip as a function of the parameter m L and various values of the tip area ratio.

108

One-Dimensional, Steady-State Conduction

Figure 1-38 illustrates the ﬁn efﬁciency associated with a ﬁn with a convective tip as a function of the ﬁn parameter (m L) for various values of the tip area ratio (ARtip ). Note that the ﬁn efﬁciency is reduced as the tip area is larger. This counterintuitive result is related to the fact that the tip area is included in the surface area that is available for convection from an ideal ﬁn in the deﬁnition of the ﬁn efﬁciency. The heat transfer rate from the ﬁn will increase as the tip area is increased; however, the rate that heat could be transferred from an ideal (i.e., isothermal) ﬁn would increase by a larger amount. It is possible to approximately correct for convection from the tip and use the simpler adiabatic tip ﬁn efﬁciency equation by modifying the length of the ﬁn slightly (e.g., adding the half-thickness of a ﬁn with a rectangular cross-section). In most cases the correction associated with the tip convection is so small that it is not worth considering. In any case, neglecting convection from the tip is slightly conservative and other uncertainties in the problem (e.g., the value of the heat transfer coefﬁcient) are likely to be more important. The ﬁn efﬁciency solutions for many common ﬁn geometries have been determined. For ﬁns without a constant cross-section, the solution requires the use of more advanced techniques, such as Bessel functions, which are covered in Section 1.8. Several common ﬁn solutions are listed in Table 1-5. A more comprehensive set of ﬁn efﬁciency solutions has been programmed in EES. To access these solutions, select Function Info from the Options menu and then select the radio button in the lower right side of the top box and scroll to the Fin Efﬁciency category (Figure 1-39). It is possible to scroll through the various functions that are available or see more detailed information about any of these functions by pressing the Info button. Note that the ﬁn efﬁciency can be accessed either in dimensional form (in which case the geometric parameters, conductivity, and heat transfer coefﬁcient must be supplied) or nondimensional form (in which case the nondimensional parameters, such as m L, must be supplied).

Figure 1-39: Fin efﬁciency function information in EES.

Table 1-5: Solutions for extended surfaces. Shape

Solution ηﬁn =

As,ﬁn = 2 W L 2h L mL = k th

Straight rectangular W

th

tanh (m L) mL

L

BesselI (1, 2 m L) m L BesselI (0, 2 m L) 2 th As,ﬁn = 2 W L2 + 2 2h L mL = k th ηﬁn =

Straight triangular

th

W L

2

ηﬁn = $

Straight parabolic

th

4 (m L)2 + 1 + 1 L2 th ln + C1 As,ﬁn = W C1 L + th L 2 2h th L, C1 = 1 + mL = k th L

W L

ηﬁn = D

Spine rectangular

tanh (m L) mL

As,ﬁn = π D L 4h L kD

mL =

L

2BesselI (2, 2 m L) m L BesselI (1, 2 m L) 2 πD D As,ﬁn = L2 + 2 2 4h L mL = kD ηﬁn =

D

Spine triangular L

2 2 As,ﬁn = 2 π rout − rin

Rectangular annular

rout

th

mrout =

rin

mrin =

2h rout k th 2h rin k th

2 mrin [BesselK (1,mrin ) BesselI (1,mrout ) − BesselI (1,mrin ) BesselK (1,mrout )] ( (mrout )2 −(mrin )2 [BesselI (0,mrin ) BesselK (1,mrout ) + BesselK (0,mrin ) BesselI (1,mrout )]

ηﬁn = ' where

h = heat transfer coefﬁcient

k = thermal conductivity

EXAMPLE 1.6-1: SOLDERING TUBES

110

One-Dimensional, Steady-State Conduction

EXAMPLE 1.6-1: SOLDERING TUBES Two large pipes must be soldered together using a propane torch, as shown in Figure 1. Din = 4.0 inch L = 2.5 ft

th = 0.375 inch

Tm = 230°C

heat from torch, q⋅

k = 150 W/m-K

T∞ = 20°C, h = 20 W/m2 -K Figure 1: Two bare pipes being soldered together.

Each of the two pipes is L = 2.5 ft long with inner diameter Din = 4.0 inches and a thickness th = 0.375 inch. The pipe material has conductivity k = 150 W/m-K. The surrounding air is at T∞ = 20◦ C and the heat transfer coefﬁcient between the external surface of the pipe and the air is h = 20 W/m2 -K. Assume that convection from the internal surface of the pipe can be neglected. a) The temperature of the interface between the two pipes must be elevated to ˙ that Tm = 230◦ C in order to melt the solder; estimate the heat transfer rate, q, must be applied by the propane torch in order to accomplish this process. This problem is solved using EES. The initial section of the code provides the stated inputs (converted to SI units). “EXAMPLE 1.6-1: Soldering Tubes” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” D in=4.0 [inch]∗ convert(inch,m) th=0.375 [inch]∗ convert(inch,m) k=150 [W/m-K] h bar=20 [W/mˆ2-K] T infinity=converttemp(C,K,20 [C]) L=2.5 [ft]∗ convert(ft,m) T m=converttemp(C,K,230 [C])

“Inner diameter” “Pipe thickness” “Conductivity” “Heat transfer coefficient” “Air temperature” “Pipe length” “Melt temperature”

The two pipes can be treated as constant cross-sectional area ﬁns; the solutions obtained in Section 1.6 are valid provided that the Biot number characterizing the temperature gradient within the pipe in the radial direction is sufﬁciently small: Bi =

h th k

Bi=h bar∗ th/k

111

“Biot number”

The Biot number is 0.0013, which is much less than unity. The cross-sectional area for conduction (Ac ) is: & π % 2 (Din + 2 th)2 − Din Ac = 4 and the perimeter exposed to air (per) is: per = π(Din + 2 th) Notice that the internal surface of the pipe (which is assumed to be adiabatic) is not included in the perimeter. The ratio of the area of the exposed ends of the pipe to the external surface area (ARtip ) is calculated according to: ARtip =

A c=pi∗ ((D in+2∗ th)ˆ2-D inˆ2)/4 per=pi∗ (D in+2∗ th) AR tip=A c/(per∗ L)

Ac per L

“area for conduction” “perimeter” “area ratio”

The tip area ratio is ARtip = 0.012 and therefore, according to Figure 1-38, the adiabatic tip ﬁn solution can be used with no loss of accuracy. The ﬁn constant (m L) is: per h L mL = k Ac and the ﬁn efﬁciency (ηﬁn ) is: ηﬁn =

tanh (m L) mL

Therefore, the resistance of each ﬁn (Rﬁn ) is: Rﬁn =

mL=sqrt(h bar∗ per/(k∗ A c))∗ L eta fin=tanh(mL)/mL R fin=1/(eta fin∗ h bar∗ per∗ L)

1 ηﬁn h per L

“fin parameter” “fin efficiency” “fin resistance”

The problem can be represented by the resistance network shown in Figure 2; the two pipes correspond to the two resistors connecting the interface to the air and the

EXAMPLE 1.6-1: SOLDERING TUBES

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

EXAMPLE 1.6-1: SOLDERING TUBES

112

One-Dimensional, Steady-State Conduction

heat input from the propane torch enters at the interface. In order for the solder to melt, the interface temperature must reach Tm . q⋅ T∞

T∞ R fin

Tm

Figure 2: Resistance network associated with soldering two bare pipes.

R fin

The heat transfer required from the torch is therefore: q˙ = 2

(Tm − T∞ ) Rﬁn

q dot=2∗ (T m-T infinity)/R fin

“required torch heat transfer rate”

The factor 2 appears because there are two pipes, each of which acts as a ﬁn. The EES solution indicates that the propane torch must provide at least 812 W to accomplish this job. b) Unfortunately, the propane torch cannot provide 812 W and it is not possible to melt the solder. Therefore, you decide to place insulating sleeves over the pipes adjacent to the soldering torch, as shown in Figure 3. If the insulation is perfect (i.e., convection is eliminated from the section of the pipe covered by the insulating sleeves), then how long must the sleeves be (Lins ) in order to reduce the heat required to q˙ = 500 W? insulating sleeves Lins

Figure 3: Pipes with insulating sleeves placed over them to reduce the heat transfer required.

Tm = 230° C

heat from torch, q⋅

The pipes with insulating sleeves can be represented by a resistance network similar to the one shown in Figure 2, but with additional resistances inserted between the interface and the base of the ﬁns. These additional resistances correspond to the insulated sections of pipes, as shown in Figure 4. q⋅ T∞ R fin

R cond, ins Tm

T∞ R cond, ins

R fin

Figure 4: Resistance network with additional resistors associated with the insulated sections of the pipe.

113

The resistance of the insulated sections of the pipe is: Rcond,ins =

R cond ins=L ins/(k∗ A c)

L ins k Ac

“resistance of insulated portion of pipe”

The length of the un-insulated section of pipe is reduced and therefore the ﬁn efﬁciency and ﬁn resistance must be recalculated. The ﬁn efﬁciency (ηﬁn ) becomes: ηﬁn =

tanh [m (L − L ins )] m (L − L ins )

and the resistance of each ﬁn (Rﬁn ) is: Rﬁn =

1 ηﬁn h per (L − L ins )

mL=sqrt(h bar∗ per/(k∗ A c))∗ (L-L ins) eta fin=tanh(mL)/mL R fin=1/(eta fin∗ h bar∗ per∗ (L-L ins))

“fin parameter” “fin efficiency” “fin resistance”

Using the resistance network shown in Figure 4, the required heat transfer rate can be expressed as: q˙ = 2

(Tm − T∞ ) Rcond,ins + Rﬁn

EES can solve for the length of insulation that is required by setting the heat transfer rate to the available heat transfer rate, q dot=2∗ (T m-T infinity)/(R fin+R cond ins) q dot=500 [W] L ins ft=L ins∗ convert(m,ft)

“required torch heat transfer” “available torch heat transfer” “length of insulation in ft”

The solution indicates that the length of the insulating sleeves must be at least 0.16 m (0.52 ft) in order to reduce the required heat transfer rate to the point where 500 W will sufﬁce.

1.6.6 Finned Surfaces Fins are often placed on surfaces in order to improve their heat transfer capability. Examples of ﬁnned surfaces can be found within nearly every appliance in your house, from the evaporator and condenser on your refrigerator and air conditioner to the processor in your personal computer. Fins are essential to the design of economical but high-performance thermal devices. Figure 1-40 illustrates a single ﬁn installed on a surface; the temperature of the surface is the base temperature of the ﬁn, Tb . The ﬁn and surface are surrounded by ﬂuid at T∞ with heat transfer coefﬁcient h. The ﬁn has perimeter per, length L, conductivity k, and cross-sectional area Ac .

EXAMPLE 1.6-1: SOLDERING TUBES

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

114

T∞ , h

One-Dimensional, Steady-State Conduction single fin: cross-sectional area, Ac perimeter, per conductivity, k Figure 1-40: Single ﬁn placed on a surface.

surface at Tb

It is of interest to determine the heat transfer rate from an area of the surface that is equal to the base area of the ﬁn, both with and without the ﬁn installed. If there were no ﬁn, then the heat transfer rate from area Ac is: q˙ no−ﬁn = h Ac (T b − T ∞ )

(1-238)

while the heat transfer rate from the ﬁn, assuming an adiabatic tip and the same heat transfer coefﬁcient, is given by Eq. (1-221): ⎛ ⎞ $ h per ⎠ (1-239) q˙ ﬁn = (T b − T ∞ ) h per k Ac tanh ⎝ L k Ac The ﬁn effectiveness (εﬁn ) is deﬁned as the ratio of the heat transfer rate from the ﬁn (q˙ ﬁn ) to the heat transfer rate that would have occurred from the surface area occupied by the ﬁn without the ﬁn attached (q˙ no−ﬁn ): ⎛ ⎞ $ h per L⎠ (T b − T ∞ ) h per k Ac tanh ⎝ k Ac q˙ ﬁn εﬁn = = (1-240) q˙ no−f in h Ac (T b − T ∞ ) which can be simpliﬁed to: εﬁn =

⎛ ⎞ k per h A 1 c ⎠ tanh ⎝ h Ac k per ARtip

(1-241)

where ARtip is the ratio of the area of the tip to the exposed surface area of the ﬁn. The ﬁn effectiveness predicted by Eq. (1-241) is illustrated in Figure 1-41 as a function of the dimensionless group (k per) /(h Ac ) for various values of the area ratio ARtip . The effectiveness of the ﬁn provides a measure of the improvement in thermal performance that is achieved by placing ﬁns onto the surface. Equation (1-241) and Figure 1-41 are useful in that they clarify the characteristics of an application that would beneﬁt substantially from the use of ﬁns. If the heat transfer coefﬁcient is low, then the group (k per) /(h Ac ) is large and there is a substantial beneﬁt associated with the addition of ﬁns. This explains why many devices that transfer thermal energy to air or other low conductivity gases with correspondingly low heat transfer coefﬁcients are ﬁnned. For example, the air-side of a domestic refrigerator condenser will certainly be ﬁnned while the refrigerant side is typically not ﬁnned, since the heat transfer coefﬁcient associated with the refrigerant condensation process is very high (as discussed in Chapter 7). Fins are typically thin structures (with a large perimeter to cross-sectional area ratio, per/Ac ) made of high conductivity material; these features enhance the ﬁn effectiveness by increasing the parameter (k per)/(h Ac ).

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

115

Fin effectiveness

10 AR tip AR tip AR tip AR tip AR tip AR tip AR tip

5

→ 0

= = = = = =

0.1 0.15 0.2 0.25 0.3 0.4

2

1 1

10 (k per)/(h Ac)

100

Figure 1-41: Fin effectiveness as a function of (k per)/(h Ac ) for various values of ARtip .

The concept of a ﬁn resistance makes it possible to approximately consider the thermal performance of an array of ﬁns that are placed on a surface. For example, Figure 1-42 illustrates an array of square ﬁns placed on a base with area As,b at temperature T b. The surface is partially covered with ﬁns; in Figure 1-42, the number of ﬁns is Nﬁn = 16. Each ﬁn has a cross-sectional area at its base of Ac,b and surface area As,ﬁn . The surfaces are exposed to a surrounding ﬂuid with temperature T ∞ and average heat transfer coefﬁcient h. The heat transferred from the base can either pass through one of the Nﬁn ﬁns (each with resistance, Rﬁn ) or from the un-ﬁnned surface area on the base (with resistance, Run-ﬁnned ). The resistance of a single ﬁn is given by Eq. (1-233): Rﬁn =

1

(1-242)

ηﬁn h As,ﬁn

where ηﬁn is the ﬁn efﬁciency, computed using the formulae or function speciﬁc to the geometry of the ﬁn. The resistance of the un-ﬁnned surface of the base is: Run−ﬁnned =

T∞ , h

h As,b − Nﬁn Ac,b

Nfin fins, each with base cross-sectional area, Ac,b , and surface area, As,fin

base surface area, As,b Figure 1-42: An array of ﬁns on a base.

1

(1-243)

116

One-Dimensional, Steady-State Conduction

Run−finned =

Tb

1 h (As,b − N fin Ac,b )

R fin =

1 ηfin h As, fin

R fin =

1 ηfin h As, fin

Run−finned =

Tb

T∞

T∞

R fins = R fin =

1 h (As,b − N fin Ac,b )

1 N fin η fin h As, fin

1 ηfin h As, fin

Figure 1-43: Resistance network associated with a ﬁnned surface.

where the area in the denominator of Eq. (1-243) is the area of the exposed portion of the base, i.e., the area not covered by ﬁns. These heat transfer paths are in parallel and therefore the thermal resistance network that represents the situation is shown in Figure 1-43. The total resistance of the ﬁnned surface is therefore: Nﬁn −1 1 + (1-244) Rtot = Run−ﬁnned Rﬁn or, substituting Eqs. (1-242) and (1-243) into Eq. (1-244): Rtot = [h (As,b − Nﬁn Ac,b) + Nﬁn ηﬁn h As,ﬁn ]−1

(1-245)

The total rate of heat transfer is: q˙ tot =

(T b − T ∞ ) = [h (As,b − Nﬁn Ac,b) + Nﬁn ηﬁn h As,ﬁn ](T b − T ∞ ) Rtot

(1-246)

The overall surface efﬁciency (ηo ) is deﬁned as the ratio of the total heat transfer rate from the surface to the heat transfer rate that would result if the entire surface (the exposed base and the ﬁns) were at the base temperature; as with the ﬁn efﬁciency, this limit corresponds to using a material with an inﬁnite conductivity. ηo =

q˙ tot h [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ] (T b − T ∞ ) prime surface area, Atot

(1-247)

The area in the denominator of Eq. (1-247) is often referred to as the prime surface area (Atot ): Atot = As,b − Nﬁn Ac,b + Nﬁn As,ﬁn

(1-248)

Substituting Eq. (1-246) into Eq. (1-247) leads to: ηo =

[(As,b − Nﬁn Ac,b) + Nﬁn ηﬁn As,ﬁn ] [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ]

(1-249)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

117

Equation (1-249) can be rearranged: ηo =

[(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn + Nﬁn ηﬁn As,ﬁn − Nﬁn As,ﬁn ] [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ]

(1-250)

or ηo = 1 −

Nﬁn As,ﬁn (1 − ηﬁn ) [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ]

(1-251)

which can be expressed in terms of the prime surface area: ηo = 1 −

Nﬁn As,ﬁn (1 − ηﬁn ) Atot

(1-252)

Rearranging Eq. (1-247), the total resistance to heat transfer from a ﬁnned surface can be expressed in terms of the overall surface efﬁciency and the prime surface area: 1 ηo h Atot

(1-253)

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK Heat rejection from a thermoelectric cooling device is accomplished using a 10 × 10 array of Dﬁn = 1.5 mm diameter pin ﬁns that are L ﬁn = 15 mm long. The ﬁns are attached to a square base plate that is W b = 3 cm on a side and thb = 2 mm thick, as shown in Figure 1. The conductivity of the ﬁn material is k ﬁn = 70 W/m-K and the thermal conductivity of the base material is k b = 25 W/m-K. There is a contact resistance of Rc = 1 × 10−4 m2 -K/W at the interface between the base of the ﬁns and the base plate. The hot end of the thermoelectric cooler is at Thot = 30◦ C and the surrounding air temperature is T∞ = 20◦ C. The average heat transfer coefﬁcient between the air and the surface of the heat sink is h = 50 W/m2 -K. 2 T∞ = 20°C, h = 50 W/m -K

10x10 array of fins kfin = 70 W/m-K

Dfin =1.5 mm Lfin =15 mm

thb =2.0 mm kb =25 W/m-K Wb =3.0 cm Thot = 30°C Rc′′ = 1x10

-4

m2 -K W

Figure 1: Heat sink mounted on a thermoelectric cooler.

a) What is the total thermal resistance between the hot end of the thermoelectric cooler and the air? What is the rate of heat rejection that can be accomplished under these conditions?

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

Rtot =

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

118

One-Dimensional, Steady-State Conduction

The ﬁrst section of the EES code provides the inputs for the problem. “EXAMPLE 1.6-2: Thermoelectric Heat Sink” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” T infinity=converttemp(C,K,20 [C]) T hot=converttemp(C,K,30 [C]) D fin=1.5[mm]∗ convert(mm,m) L fin=15[mm]∗ convert(mm,m) N fin=100 W b=3 [cm] ∗ convert(cm,m) th b=2[mm]∗ convert(mm,m) k fin=70 [W/m-K] k b=25 [W/m-K] h bar=50 [W/mˆ2-K] R c=1e-4 [mˆ2-K/W]

“Air temperature” “Hot end of thermoelectric cooler” “Fin diameter” “Fin length” “Number of fins” “Width of base (square)” “Thickness of base” “Conductivity of fin” “Conductivity of base” “Heat transfer coefficient” “Contact resistance”

The constant cross-sectional area ﬁns can be treated using the solutions presented in Section 1.6. The perimeter (per), cross-sectional area (Ac ), and surface area for convection (As,ﬁn, assuming adiabatic ends) associated with each ﬁn are calculated according to: per = π Dﬁn Ac =

π 2 D 4 ﬁn

As,ﬁn = π L Dﬁn

per=pi∗ D fin A c=pi∗ D finˆ2/4 A s fin=pi∗ L fin∗ D fin

“Perimeter of fin” “Cross-sectional area for conduction” “Surface area of fin for convection”

The ﬁn constant and ﬁn efﬁciency for an adiabatic tip, constant cross-sectional area ﬁn are computed according to: per h m= k ﬁn Ac ηﬁn =

tanh(m L ﬁn ) m L ﬁn

The resistance of any type of ﬁn (Rﬁn ) can be obtained from its efﬁciency: Rﬁn =

1 ηﬁn h As,ﬁn

mL=sqrt(h bar∗ per/(k fin∗ A c))∗ L fin eta fin=tanh(mL)/mL R fin=1/(h bar∗ A s fin∗ eta fin)

119

“Fin parameter” “Fin efficiency” “Fin resistance”

The resistance network that represents the entire heat sink (Figure 2) extends from the hot end of the cooler to the air and includes conduction through the base (Rcond,b ) followed by two paths in parallel, corresponding to the heat that is transferred by convection from the unﬁnned upper surface of the base (Run−ﬁnned ) and the heat that is transferred through the contact resistance at the base of the ﬁns (Rc ) and then through the resistance associated with the ﬁn itself (Rﬁn ). Note that Rc and Rﬁn are in parallel Nﬁn times and therefore the value these resistances in the circuit is reduced by 1/Nﬁn . R fin

Rc N fin

N fin T∞

Figure 2: Resistance network representing the heat sink.

Thot

R cond, b

0.088 K/W

0.57 K/W

3.23 K/W

R un−finned T∞ 27.7 K/W

The resistance to conduction through the base is: thb Rcond,b = k b W b2 The contact resistance associated with each ﬁn-to-base interface is: R Rc = c Ac The resistance of the unﬁnned upper surface of the base is: 1 Run−ﬁnned = 2 h W b − N ﬁn Ac These resistances are calculated in EES: R b=th b/(k b∗ W bˆ2) R unfinned=1/((W bˆ2-N fin∗ A c)∗ h bar) R c=R c/A c

“Resistance due to conduction through the base” “Resistance of unfinned base” “Fin-to-base contact resistance”

The total resistance (Rtot ) and heat transfer (q˙ tot ) from the heat sink are obtained according to: ⎞−1 ⎛ ⎟ ⎜ 1 1 ⎟ ⎜ + Rtot = Rb + ⎜ ⎟ Rﬁn ⎝ Rc Run−ﬁnned ⎠ + N ﬁn N ﬁn q˙ tot =

(Thot − T∞ ) Rtot

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

120

One-Dimensional, Steady-State Conduction

and calculated in EES.

R tot=R b+(1/(R c/N fin+R fin/N fin)+1/R unfinned)ˆ(-1) q dot tot=(T hot-T infinity)/R tot

“Total resistance” “Total rate of heat transfer”

The total resistance is 3.42 K/W and the rate of heat transfer is 2.92 W. The numerical values of each resistance are included in Figure 2 in order to understand the mechanisms that are governing the behavior of the heat sink. Notice that the resistance of the base is not very important, as it is a small resistor in series with larger ones. The resistance of the unﬁnned portion of the base is also not critical, since it is a large resistor in parallel with smaller ones. On the other hand, both the contact resistance and the ﬁn resistance are important as these two resistors dominate the problem and are of the same order of magnitude. The ﬁn resistance is the most critical parameter in the problem and any attempt to improve performance should focus on this element of the heat sink. b) Through material selection and manipulation of the air ﬂow across the heat sink, it is possible to affect design changes to kﬁn and h. Generate a contour plot that illustrates contours of constant heat rejection in the parameter space of kﬁn (ranging from 5 W/m-K to 150 W/m-K) and h (ranging from 10 W/m2 -K to 200 W/m2 -K). One of the nice things about solving problems using a computer program as opposed to pencil and paper is that parametric studies and optimization are relatively straightforward. In order to prepare a contour plot with EES, it is necessary to setup a parametric table in which both of the parameters of interest vary over a speciﬁed range. Open a new parametric table and include the two independent variables (the variables k fin and h bar) as well as the dependent variable of interest (the variable q dot tot). In order to run the simulation for 20 values of kﬁn and 20 values of h, 20 × 20 = 400 runs must be included in the table. (Add runs using the Insert/Delete Runs option from the Tables menu.) It is necessary to set the values of k fin and h bar in the table. It is possible to vary k fin from 5 to 150 W/m-K, 20 times by using the “Repeat pattern every” option in the Alter Values dialog that appears when you right-click on the k fin column, as shown in Figure 3.

Figure 3: Vary kﬁn from 5 to 150 W/m-K 20 times.

121

In order to completely cover the parameter space, it is necessary to evaluate the solution over a range of h bar at each unique value of k fin; this can be accomplished using the “Apply pattern every” option in the Alter Values dialog for the h bar column of the table, see Figure 4.

Figure 4: Vary h from 10 to 200 W/m-K with 20 runs for each of 20 values.

2

Average heat transfer coefficient (W/m-K)

When the speciﬁed values of the variables k fin and h bar are commented out in the Equations window, it is possible to run the parametric table using the Solve Table command in the Calculate menu (F3); 400 values of q˙ are determined, one for each combination of kﬁn and h set in the parametric table. To generate a contour plot, select X-Y-Z plot from the New Plot Window option in the Plots menu. Select k fin as the variable on the x-axis, h bar as the y-axis variable and q dot tot as the contour variable. The appearance of the resulting contour plot can be adjusted by altering the resolution, smoothing, color options, and the type of function used for interpolation. A contour plot generated using isometric lines is shown in Figure 5.

200 q tot = 7 W

180 160 140

q tot = 6 W

120 q tot = 5 W

100 80

q tot = 4 W

60

q tot = 3 W

40 nominal design

20 0 0

25

q tot = 2 W q tot = 1 W

50 75 100 Fin conductivity (W/m-K)

125

150

Figure 5: Contours of constant heat transfer rate in the parameter space of ﬁn material conductivity and heat transfer coefﬁcient.

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

EXAMPLE 1.6-2

122

One-Dimensional, Steady-State Conduction

The nominal design point shown in Figure 1 is also indicated in Figure 5. Contour plots are useful in that they can clarify the impact of design changes. For example, Figure 5 shows that it would be more beneﬁcial to explore methods to increase the heat transfer coefﬁcient than the ﬁn conductivity at the nominal design conditions (i.e., moving from the nominal design point towards higher heat transfer will result in much larger performance gains than moving toward higher ﬁn conductivity).

1.6.7 Fin Optimization This extended section of the book, which can be found on the website (www. cambridge.org/nellisandklein), presents an optimization of a constant cross-sectional area ﬁn in order to maximize the rate of heat transfer per unit volume of ﬁn material. The process illustrates the use of EES’ single-variable optimization capability and shows that a well-optimized ﬁn is characterized by m L that is approximately equal to 1.4. Fins with m L much less than 1.4 are shorter than optimal and therefore have very small temperature gradients due to conduction; additional length will provide a substantial beneﬁt and therefore the available volume of ﬁn material should be stretched, providing additional length at the expense of cross-sectional area. Fins with m L much greater than 1.4 are longer than optimal and therefore have large temperature gradients due to conduction; additional length will not provide much beneﬁt as the tip temperature is approaching the ambient temperature. Therefore, the available volume should be compressed, reducing the length but providing more cross-sectional area for conduction.

1.7 Analytical Solutions for Advanced Constant Cross-Section Extended Surfaces 1.7.1 Introduction The constant cross-section ﬁns that were investigated in Section 1.6 are certainly the most common type of extended surface used in practice. However, other extended surface problems (with alternative boundary conditions, more complex thermal loadings, multiple computational domains, etc.) are also encountered. Extended surfaces represent 2-D heat transfer situations that can be approximated as being 1-D and these problems can be solved analytically using the techniques that were introduced in Section 1.6.

1.7.2 Additional Thermal Loads An extended surface can be subjected to additional thermal loads such as thermal energy generation (due to ohmic heating, for example) or an external heat ﬂux. These additional effects show up in the governing differential equation but do not affect the character of the solution. Figure 1-46 illustrates an extended surface with cross-sectional area Ac and perimeter per that has a uniform volumetric generation (g˙ ) and is exposed to a uniform heat ﬂux (q˙ ext , for example from solar radiation). The extended surface is surrounded by ﬂuid at T ∞ with average heat transfer coefﬁcient h. A differential control volume is used to derive the governing differential equation (see Figure 1-46) and provides the energy balance: q˙ x + g˙ + q˙ ext = q˙ conv + q˙ x+dx

(1-267)

1.7 Analytical Solutions for Advanced Constant Cross-Section

123 ′′ q⋅ ext

T∞ , h x

dx

g⋅ ′′′

Figure 1-46: Extended surface with additional thermal loads related to generation and an external heat ﬂux.

q⋅ conv q⋅ x

per g⋅

q⋅ext

q⋅ x + dx Ac

The ﬁnal term can be expanded: g˙ + q˙ ext = q˙ conv +

dq˙ dx dx

Substituting the appropriate rate equation for each term results in: d dT g˙ Ac dx + q˙ ext per dx = h per dx (T − T ∞ ) + −k Ac dx dx dx

(1-268)

(1-269)

where k is the conductivity of the material and T is the temperature at any axial position. Note that temperature is assumed to be only a function of x, which is consistent with the extended surface approximation. This assumption should be veriﬁed using an appropriately deﬁned Biot number, as discussed in Section 1.6.2. After some simpliﬁcation, the governing differential equation for the extended surface becomes: h per h per g˙ q˙ ext per d2 T − T = − T − − ∞ dx2 k Ac k Ac k k Ac

(1-270)

Equation (1-270) is a nonhomogeneous, linear, second order ODE. The solution is assumed to be the sum of a homogeneous and particular solution: T = Th + T p

(1-271)

Equation (1-271) is substituted into Eq. (1-270): d2 T p h per h per h per g˙ q˙ per d2 T h − Th + − Tp = − T∞ − − ext 2 2 dx kA dx k Ac k Ac k k Ac c =0 for homogeneous differential equation

(1-272)

whatever is left over must be the particular differential equation

The homogeneous differential equation is: h per d2 T h − Th = 0 dx2 k Ac

(1-273)

T h = C1 exp (m x) + C2 exp (−m x)

(1-274)

which is solved by:

where

m =

per h k Ac

(1-275)

124

One-Dimensional, Steady-State Conduction

The particular differential equation is: d2 T p h per h per g˙ q˙ per − Tp = − T∞ − − ext 2 dx k Ac k Ac k k Ac

(1-276)

Notice that the right side of Eq. (1-276) is a constant; therefore, the particular solution is a constant: T p = C3

(1-277)

Substituting Eq. (1-277) into Eq. (1-276) leads to: −

h per h per g˙ q˙ per C3 = − T∞ − − ext k Ac k Ac k k Ac

(1-278)

Solving for C3 : C3 = T ∞ +

g˙ Ac h per

+

q˙ ext

(1-279)

h

Substituting Eq. (1-279) into Eq. (1-277) leads to: T p = T∞ +

g˙ Ac h per

+

q˙ ext

(1-280)

h

Substituting Eqs. (1-280) and (1-274) into Eq. (1-271) leads to: T = C1 exp (m x) + C2 exp (−m x) + T ∞ +

g˙ Ac h per

+

q˙ ext h

(1-281)

The boundary conditions at either edge of the extended surface should be used to evaluate C1 and C2 for a speciﬁc situation. It is possible to use Maple to solve this problem (and therefore avoid the mathematical steps discussed above). Enter the governing differential equation: > restart; > ODE:=diff(diff(T(x),x),x)-h_bar∗ per∗ T(x)/(k∗ A_c)=-h_bar∗ per∗ T_infinity/ (k∗ A_c)-gv/k-qf_ext∗ per/(k∗ A_c); ODE :=

d2 h bar per T inf inity gv qf extper h bar per T(x) =− − − T(x) − dx2 kA c kA c k kA c

and solve it: > Ts:=dsolve(ODE); Ts := T(x) = √ e

√ h bar per x √ √ x A c

C2 + e

−

√ √ h bar per x √ √ x A c

C1 +

(h bar T inf inity + qf ext) per + gv A c h bar per

The solution identiﬁed by Maple is functionally identical to Eq. (1-281).

1.7 Analytical Solutions for Advanced Constant Cross-Section

125

For situations where the volumetric generation or external heat ﬂux is not spatially uniform, it will not be as easy to identify the particular solution. For example, suppose that the volumetric generation varies sinusoidally from x = 0 to x = L, where L is the length of the extended surface. x (1-282) g˙ = g˙ max sin π L where g˙ max is the volumetric generation at the center of the extended surface. The resulting governing differential equation is: x q˙ per h per h per g˙ d2 T max − T = − T − (1-283) sin π − ext ∞ dx2 k Ac k Ac k L k Ac The solution is assumed to be the sum of a homogeneous and particular solution. Substituting Eq. (1-271) into Eq. (1-283) leads to: x q˙ per d2 T p h per h per h per g˙ d2 T h max − T + − T = − T − sin π − ext h p ∞ dx2 k Ac dx2 k Ac k Ac k L k Ac =0 for homogeneous differential equation

the particular differential equation

(1-284) The homogeneous differential equation has not changed and therefore the homogeneous solution remains Eq. (1-274). However, the particular differential equation has become more complex: x q˙ per d2 T p h per h per g˙ max − T = − T − (1-285) sin π − ext p ∞ dx2 k Ac k Ac k L k Ac Identifying the particular solution can take some skill. Equation (1-285) involves both a constant and a sinusoidal term on the right hand side and the governing differential equation involves both the solution and its derivatives. Therefore, it seems likely that a particular solution that includes sines, cosines, and constants as well as their derivatives (cosines, sines, and 0) might work. One method of obtaining the particular solution is to assume such a solution with appropriate, undetermined constants (C3 , C4 , and C5 ): x x (1-286) + C4 cos π + C5 T p = C3 sin π L L and substitute it into the particular differential equation. The ﬁrst and second derivatives of the particular solution, Eq. (1-286), are: x C π x dT p C3 π 4 = cos π − sin π dx L L L L x C π2 x d2 T p C3 π2 4 = − sin π cos π − dx2 L2 L L2 L

(1-287) (1-288)

Substituting Eqs. (1-288) and (1-286) into Eq. (1-285) leads to: x C π2 x h per ' x x ( C3 π2 4 cos π sin π cos π − − + C + C C − 2 sin π 3 4 5 L L L2 L k Ac L L h per g˙ x q˙ per =− T ∞ − max sin π (1-289) − ext k Ac k L k Ac In order for the particular solution to work, the sine, cosine and constant terms in Eq. (1-289) must add up correctly. By considering the coefﬁcients of the sine terms, it is

126

One-Dimensional, Steady-State Conduction

possible to obtain the equation: − which can be solved for C3 :

C3 π2 h per g˙ − C3 = − max 2 L k Ac k g˙ max k C3 = h per π2 + L2 k Ac

(1-290)

(1-291)

The sum of the coefﬁcients of the cosine terms provides an additional equation: −

C4 π2 h per − C4 = 0 2 L k Ac

(1-292)

which indicates that C4 = 0

(1-293)

Finally, the sum of the coefﬁcients for the constant terms leads to: −

h per h per q˙ per C5 = − T ∞ − ext k Ac k Ac k Ac

(1-294)

so C5 = T ∞ +

q˙ ext

h Substituting Eqs. (1-291), (1-293), and (1-295) into Eq. (1-286) leads to: g˙ max x q˙ k sin π + T ∞ + ext Tp = L h h per π2 + 2 L k Ac

(1-295)

(1-296)

The solution to the differential equation is the sum of the homogeneous and the particular solutions. g˙ max x q˙ k sin π + T ∞ + ext T = C1 exp (m x) + C2 exp (−m x) + L h h per π2 + 2 L k Ac (1-297) where the boundary conditions determine the values of C1 and C2 . It is somewhat easier to obtain the solution using Maple. Enter the governing differential equation, Eq. (1-283): > restart; > ODE:=diff(diff(T(x),x),x)-h_bar∗ per∗ T(x)/(k∗ A_c)=-h_bar∗ per∗ T_infinity/ (k∗ A_c)-gv_max∗ sin(pi∗ x/L)/k-qf_ext∗ per/(k∗ A_c);

d2 h bar per T(x) T(x) − dx2 kA c πx max sin gv h bar per T inf inity L − qf extper − =− kA c k kA c

ODE : =

1.7 Analytical Solutions for Advanced Constant Cross-Section

127

and then solve the differential equation:

> Ts:=dsolve(ODE); √

Ts := T(x) = e

+

√ h bar per x √ √ k A c

√

C2 + e

gv max A c L2 h bar sin

√ h bar per x √ √ k A c

C1

πx

+(h bar T inf inity+qf ext) (h bar per L2 +π2 k A c) L h bar (h bar per L2 +π2 k A c)

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR One design of a micro-scale, lithographically fabricated (i.e., MEMS) device that can produce in-plane motion is called a bent-beam actuator (Que (2000)). A V-shaped structure (the bent-beam in Figure 1) is suspended between two anchors. The anchors are thermally staked to the underlying substrate and therefore keep the ends of the bent-beam at room temperature (Ta = 20◦ C). An elevated voltage is applied to one pillar and the other is grounded. The voltage difference causes current (I) to ﬂow through the bent-beam structure. The temperature of the bent-beam rises as a result of ohmic heating and the thermally induced expansion causes the apex of the bent-beam to move outwards. The result is a voltage-controlled actuator capable of producing in-plane motion.

substrate tip motion current flow through bent-beam

anchor post, kept at Ta Figure 1: Bent-beam actuator.

The anchors of the bent-beam actuator are placed La = 1 mm apart and the beam structure has a cross-section of W = 10 μm by th = 5 μm. The slope of the beams (with respect to a line connecting the two pillars) is θ = 0.5 rad, as shown in Figure 2. The bent-beam material has conductivity k = 80 W/m-K, electrical resistivity ρe = 1 × 10−5 ohm-m and coefﬁcient of thermal expansion CTE = 3.5 × 10−6 K−1 . You may neglect radiation from the beam and assume all of the heat that is generated is convected to the surrounding air at temperature T∞ = 20◦ C with average heat transfer coefﬁcient h = 100 W/m2 -K or transferred conductively to the pillars (which remain at Ta = 20◦ C). The actuator is activated with I = 10 mA of current.

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

The solution identiﬁed by Maple is functionally equivalent to Eq. (1-297). This result from Maple can be copied and pasted directly into EES for evaluation and manipulation.

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

128

One-Dimensional, Steady-State Conduction 2 T∞ = 20°C, h = 100 W/m -K

k = 80 W/m-K ρe = 1x10-5 ohm-m CTE = 3.5x10-6 K-1

L s Ta = 20°C

0.5 rad La=1.0 mm

th = 5 μm Ta = 20°C

W = 10μm (into page) Figure 2: Dimensions and conditions associated with bent-beam actuator.

a) Is it appropriate to treat the bent-beam as an extended surface? The input parameters for the problem are entered into EES: “EXAMPLE 1.7-1: Bent-beam Actuator” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” L a=1 [mm]∗ convert(mm,m) w=10 [micron]∗ convert(micron,m) th=5 [micron]∗ convert(micron,m) I=0.010 [Amp] theta=0.5 [rad] T a=converttemp(C,K,20 [C]) T infinity=converttemp(C,K,20 [C]) h bar=100 [W/mˆ2-K] k=80 [W/m-K] rho e=1e-5 [ohm-m] CTE=3.5e-6 [1/K]

“distance between anchors” “width of beam” “thickness of beam” “current” “slope of beam” “temperature of pillars” “temperature of air” “heat transfer coefficient” “conductivity” “electrical resistivity” “coefficient of thermal expansion”

The extended surface approximation requires that the 3-D temperature distribution within the bent-beam be approximated as 1-D; that is, temperature gradients within the beam that are perpendicular to the surface will be ignored so that the temperature may be approximated as a function only of s, the coordinate that follows the beam (see Figure 2). The resistance that must be neglected in order to use the extended surface approximation is conduction in the lateral direction (Rcond,lat ). The extended surface approximation is justiﬁed provided that the lateral conduction resistance is small relative to the resistance that is being considered, convection from the outer surface (Rconv ). The Biot number is therefore: Bi =

Rcond,lat Rconv

The heat transfer will take the shortest path to the surface and therefore it is appropriate to use the smallest lateral dimension (th/2) to compute the lateral conduction resistance. hW L th h th = Bi = 2kW L 1 2k

129

where L is the length of the beam from pillar to apex (see Figure 2). Bi=th∗ h bar/(2∗ k)

“Biot number”

The Biot number is small (3 × 10−6 ) and therefore the extended surface approximation is justiﬁed. b) Develop an analytical solution that can predict the temperature of one leg of the bent-beam as a function of position along the beam, s. The general solution for an extended surface with a constant cross-sectional area and spatially uniform generation is derived in Section 1.7.2: T = C 1 exp (m s) + C 2 exp (−m s) + T∞ +

g˙ Ac h per

(1)

For the bent-beam actuator, the perimeter (per), cross-sectional area (Ac ), and ﬁn parameter (m) are per = 2 (W + th) Ac = W th h per m= k Ac

per=2∗ (W+th) A c=W∗ th m=sqrt(h bar∗ per/(k∗ A c))

“perimeter” “area” “fin parameter”

The volumetric generation, g˙ , is related to ohmic heating. The electrical resistance of the bent-beam structure (Re ) is: Re =

ρe 2 L Ac

where L=

La 2 cos (θ)

The volumetric rate of electrical dissipation is the ratio of ohmic dissipation to the volume of the structure: g˙ =

L=L a/(2∗ cos(theta)) R e=rho e∗ L∗ 2/A c g dot=Iˆ2∗ R e/(2∗ L∗ A c)

I 2 Re 2 L Ac

“length of half-beam” “resistance of beam structure” “volumetric generation”

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7 Analytical Solutions for Advanced Constant Cross-Section

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

130

One-Dimensional, Steady-State Conduction

The constants C1 and C2 in Eq. (1) are determined using the boundary conditions. The temperature of the beam where it meets the pillar is speciﬁed: Ts=0 = Ta

(2)

Substituting Eq. (1) into Eq. (2) leads to: C 1 + C 2 + T∞ +

g˙ Ac h per

= Ta

(3)

A half-symmetry model of the bent-beam actuator considers only one leg. Because both legs of the bent-beam see identical conditions, there is nothing to drive heat from one leg to the other and therefore there will be no conduction through the end of the leg (at s = L): q˙ s=L = −k

dT =0 d s s=L

or dT =0 d s s=L

(4)

Substituting Eq. (1) into Eq. (4) leads to: C 1 m exp (m L) − C 2 m exp (−m L) = 0

(5)

Equations (3) and (5) can be entered in EES and used to determine C1 and C2 . T infinity+C 1+C 2+g dot∗ A c/(h bar∗ per)=T a C 1∗ m∗ exp(m∗ L)-C 2∗ m∗ exp(-m∗ L)=0

“from boundary condition at s=0” “from boundary condition at s=L”

A variable s_bar is deﬁned as s/L so that s_bar = 0 corresponds to the pillar and s_bar = 1 to the apex. The variable s_bar is deﬁned for convenience, so that it is easy to generate a parametric table in which s is varied from 0 to L even if parameters such as θ and La change. s bar=s/L

“non-dimensional position”

The temperature is evaluated using Eq. (1). T=T infinity+C 1∗ exp(m∗ s)+C 2∗ exp(-m∗ s)+g dot∗ A c/(h bar∗ per) T C=converttemp(K,C,T)

“temperature” “in C”

131

A parametric table is generated that includes the variables s bar and T C. The temperature distribution through one leg of the beam is shown in Figure 3. 800 700

Temperature (°C)

600 500 400 300 200 100 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, s/L

0.8

0.9

1

Figure 3: Temperature as a function of dimensionless position along one leg of beam.

c) The thermally induced elongation of a differential segment of the beam (of length ds) is given by: dL = CTE ( T − Ta) d s Estimate the displacement of the apex of the beam. Plot the displacement as a function of voltage.

The total elongation of the beam (L) is obtained by integrating the differential elongation along the beam: L CTE (T − Ta ) d s

L =

(6)

0

Substituting the solution for the temperature distribution, Eq. (1) into Eq. (6) leads to:

L L =

C 1 exp (m s) + C 2 exp (−m s) + T∞ +

CTE 0

g˙ Ac h per

− Ta

ds

Evaluating the integral: L = CTE

T∞ − Ta +

g˙ Ac h per

L

C2 C1 exp (m s) − exp (−m s) s+ m m

0

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7 Analytical Solutions for Advanced Constant Cross-Section

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

132

One-Dimensional, Steady-State Conduction

Substituting the integration limits: L = CT E

T∞ − Ta +

g˙ Ac

h per

C2 C1 [exp (m L) − 1] − [exp (−m L) − 1] L+ m m

!

DELTAL=CTE∗ ((T infinity-T a+g dot∗ A c/(h bar∗ per))∗ L+C 1∗ (exp(m∗ L)-1)/m-C 2∗ (exp(-m∗ L)-1)/m) “displacement of beam”

Assuming that the joint associated with the apex does not provide a torque on either leg of the beam, the displacement of the apex can be estimated using trigonometry (Figure 4). L+ΔL

Δy heated beam unheated beam

L La 2

Figure 4: Trigonometry associated with apex motion.

y

The original position of the apex (y) is given by: y =

L2

−

La 2

2

therefore, the motion of the apex (y ) is: y =

(L + L) − 2

La 2

2 −

DELTAy=sqrt((L+DELTAL)ˆ2-(L a/2)ˆ2)-sqrt(Lˆ2-(L a/2)ˆ2) DELTAy micron=DELTAy∗ convert(m,micron)

L2

−

La 2

2

“displacement of apex” “in μm”

The voltage across the beam (V) is: V = I Re

V=I∗ R e

“voltage”

Figure 5 illustrates the actuator displacement as a function of voltage. This plot was generated using a parametric table including the variables DELTAy micron and V; the variable I was commented out in order to make the table.

133

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7 Analytical Solutions for Advanced Constant Cross-Section 10 9 Actuator motion (μm)

8 7 6 5 4 3 2 1 0 0

0.5

1

1.5

2 2.5 3 Voltage (V)

3.5

4

4.5

5

Figure 5: Actuator displacement as a function of the applied voltage.

1.7.3 Moving Extended Surfaces An interesting class of problems arises in situations where an extended surface is moving with respect to the frame of reference of the problem. Problems of this type occur in rotating systems, (such as in drum and disk brakes), extrusions, and in manufacturing systems. The energy carried by the moving material represents an additional energy transfer into and out of the differential control volume and provides an additional term in the governing differential equation. Figure 1-47 illustrates an extended surface (i.e., a material for which temperature is only a function of x) that is moving with velocity u through ﬂuid with temperature T ∞ and h. An energy balance on the differential control volume (Figure 1-47) includes con˙ at either edge, as well as conduction and energy transport due to material motion (E) vection to the surrounding ﬂuid. q˙ x + E˙ x = q˙ conv + q˙ x+dx + E˙ x+dx

(1-298)

or 0 = q˙ conv +

dq˙ dE˙ dx + dx dx dx

(1-299)

material is moving with velocity u dx

Figure 1-47: An extended surface moving with velocity u.

x

⋅ Ex q⋅ x

⋅ E x+dx

q⋅ conv

q⋅ x+dx

T∞ , h

134

One-Dimensional, Steady-State Conduction

The conduction and convection terms are represented by the familiar rate equations: q˙ cond = −k Ac

dT dx

q˙ conv = per dx h (T − T ∞ )

(1-300) (1-301)

where k is the conductivity of the material and per and Ac are the perimeter and crosssectional area of the extended surface, respectively. The rate of energy transfer due to ˙ is the product of the enthalpy of the material (i) and its the motion of the material, E, mass ﬂow rate. The mass ﬂow rate is the product of the velocity, density (ρ), and crosssectional area. E˙ = u Ac ρ i Substituting Eqs. (1-300) through (1-302) into Eq. (1-299) leads to: d dT d 0 = per dx h (T − T ∞ ) + −k Ac dx + [u Ac ρ i] dx dx dx dx

(1-302)

(1-303)

Assuming constant properties: d2 T di + u Ac ρ dx2 dx

(1-304)

d2 T di dT + u Ac ρ 2 dx dT dx

(1-305)

0 = per h (T − T ∞ ) − k Ac The enthalpy gradient is expanded: 0 = per h (T − T ∞ ) − k Ac

Assuming that the material is incompressible, the derivative of enthalpy with respect to temperature is the speciﬁc heat capacity (c): 0 = per h (T − T ∞ ) − k Ac

d2 T dT + u Ac ρ c 2 dx dx

(1-306)

Equation (1-306) is rearranged in order to obtain the governing differential equation: u ρ c dT per h per h d2 T − T =− T∞ − 2 dx k dx k Ac k Ac

(1-307)

The solution is again divided into a homogeneous and particular solution: T = Th + T p

(1-308)

Substituting Eq. (1-308) into Eq. (1-307) leads to: d2 T p u ρ c dT p d2 T h u ρ c dT h per h per h per h − T + − Tp = − T∞ − − h dx2 k dx k Ac dx2 k dx k Ac k Ac

= 0 for homogeneous differential equation

whatever is left is the particular differential equation

(1-309)

1.7 Analytical Solutions for Advanced Constant Cross-Section

135

The particular differential equation: d2 T p u ρ c dT p per h per h − Tp = − T∞ − 2 dx k dx k Ac k Ac

(1-310)

is solved by a constant: T p = T∞

(1-311)

The homogeneous differential equation is: d2 T h u ρ c dT h per h − Th = 0 − dx2 k dx k Ac The ﬁn parameter (m) is deﬁned as in Section 1.6: h per m= k Ac

(1-312)

(1-313)

The group of properties, k/ρ c, appearing in Eq. (1-312) is encountered often in heat transfer and is deﬁned as the thermal diffusivity (α): α=

k ρc

(1-314)

With these deﬁnitions, Eq. (1-312) can be written as: u dT h d2 T h − − m2 T h = 0 dx2 α dx

(1-315)

Equation (1-315) is solved by an exponential: T h = C exp (λ x)

(1-316)

where C and λ are both arbitrary constants. Equation (1-316) is substituted into Eq. (1-315): C λ2 exp (λ x) −

u C λ exp (λ x) − m2 C exp (λ x) = 0 α

(1-317)

u λ − m2 = 0 α

(1-318)

which can be simpliﬁed: λ2 −

Equation (1-318) is quadratic and therefore has two solutions (λ1 and λ2 ): / 1 u 2 u + m2 + λ1 = 2α 4 α u λ2 = − 2α

/

1 u 2 + m2 4 α

(1-319)

(1-320)

Because the governing equation is linear, the sum of the two solutions (Th,1 and Th,2 ): T h,1 = C1 exp (λ1 x)

(1-321)

T h,2 = C2 exp (λ2 x)

(1-322)

136

One-Dimensional, Steady-State Conduction

is also a solution and therefore the general solution to the homogeneous governing differential equation is: T h = C1 exp (λ1 x) + C2 exp (λ2 x)

(1-323)

The solution to the differential equation is the sum of the homogeneous and particular solutions: T = C1 exp (λ1 x) + C2 exp (λ2 x) + T ∞

(1-324)

where the constants C1 and C2 are determined according to the boundary conditions. The solution may also be obtained using Maple by entering and solving the governing differential equation: > restart; > ODE:=diff(diff(T(x),x),x)-u∗ diff(T(x),x)/alpha-mˆ2∗ T(x)=-mˆ2∗ T_infinity; ODE :=

d u T(x) d dx T(x) − − m2 T(x) = −m2 T inf inity dx2 α 2

> T_s:=dsolve(ODE);

T s := T(x) = e

(u +

√

u2 + 4m2 α2 ) x 2α

C2 + e

(u −

√

u2 + 4m2 α2 ) x 2α

C1 + T inf inity

EXAMPLE 1.7-2: DRAWING A WIRE

which is equivalent to Eq. (1-324). EXAMPLE 1.7-2: DRAWING A WIRE Figure 1 illustrates a wire drawn from a die. The wire diameter is D = 0.5 mm. The temperature of the material at the exit of the die is Tdr aw = 600◦ C and it has a draw velocity of u = 10 mm/s. The properties of the wire are ρ = 2700 kg/m3 , k = 230 W/m-K, and c = 1000 J/kg-K. The wire is surrounded by air at T∞ = 20◦ C with an average heat transfer coefﬁcient of h = 25 W/m2 -K. The wire travels for L = 25 cm before entering a pool of water that is kept at Tw = 20◦ C; you may assume that the water-to-wire heat transfer coefﬁcient is very high so that the wire equilibrates essentially instantaneously with the water as it enters the pool. h = 25 W/m2-K T∞ = 20°C

Tdraw = 600°C u = 10 mm/s D = 0.5 mm ρ = 2700 kg/m3 L = 25 cm k = 230 W/m-K c = 1000 J/kg-K Tw = 20°C

Figure 1: Wire drawn from a die.

a) Develop an analytical model that can predict the temperature distribution in the wire.

1.7 Analytical Solutions for Advanced Constant Cross-Section

137

EXAMPLE 1.7-2: DRAWING A WIRE

The input parameters are entered in EES: “EXAMPLE 1.7-2: Drawing a Wire” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” D=0.5 [mm]∗ convert(mm,m) u=10 [mm/s]∗ convert(mm/s,m/s) c=1000 [J/kg-K] k=230 [W/m-K] rho=2700 [kg/mˆ3] h bar=25 [W/mˆ2-K] T infinity=converttemp(C,K,20 [C]) T draw=converttemp(C,K,600 [C]) T w=converttemp(C,K,20 [C]) L=25 [cm]∗ convert(cm,m)

“diameter” “draw velocity” “specific heat capacity” “conductivity” “density” “heat transfer coefficient” “air temperature” “draw temperature” “water temperature” “length of wire”

The governing differential equation for a moving extended surface was derived in Section 1.7.3: d 2T u dT − − m2 T = −m2 T∞ d x2 α dx where α is the thermal diffusivity: α= m is the ﬁn constant:

k ρc

m=

h per k Ac

and per and Ac are the perimeter and cross-sectional area, respectively, of the moving surface: per = π D Ac = π

A c=pi∗ Dˆ2/4 per=pi∗ D alpha=k/(rho∗ c) m=sqrt(h bar∗ per/(k∗ A c))

D2 4

“cross-sectional area” “perimeter” “thermal diffusivity” “fin parameter”

The general solution derived in Section 1.7.3 is: T = C 1 exp (λ1 x) + C 2 exp (λ2 x) + T∞

(1)

EXAMPLE 1.7-2: DRAWING A WIRE

138

One-Dimensional, Steady-State Conduction

where C1 and C2 are undetermined constants and: / 1 u 2 u λ1 = + m2 + 2α 4 α u λ2 = − 2α

/

1 u 2 + m2 4 α

lambda 1=u/(2∗ alpha)+sqrt((u/alpha)ˆ2/4+mˆ2) lambda 2=u/(2∗ alpha)-sqrt((u/alpha)ˆ2/4+mˆ2)

“solution parameter 1” “solution parameter 2”

The constants are evaluated using the boundary conditions. The temperatures at x = 0 and x = L are speciﬁed: Tx=L = Tw

(2)

Tx=0 = Tdr aw

(3)

Substituting Eq. (1) into Eqs. (2) and (3) leads to two algebraic equations for C1 and C2 : C 1 exp (λ1 L) + C 2 exp (λ2 L) + T∞ = Tw C 1 + C 2 + T∞ = Tdr aw which are entered in EES: C 1∗ exp(lambda 1∗ L)+C 2∗ exp(lambda 2∗ L)+T infinity=T w C 1+C 2+T infinity=T draw

“boundary condition at x=L” “boundary condition at x=0”

The solution is evaluated in EES and converted to Celsius. x=x bar∗ L T=C 1∗ exp(lambda 1∗ x)+C 2∗ exp(lambda 2∗ x)+T infinity T C=converttemp(K,C,T)

“position” “solution” “in C”

A parametric table in EES can be used to provide the temperature as a function of position. It is convenient to deﬁne the variable x bar, the axial position normalized by the length of the wire. Including x_bar in the table and varying it from 0 to 1 is equivalent to varying the position from 0 to L. One advantage of using the variable x_bar is that as the length of the wire is changed, it is not necessary to adjust the parametric table, only to re-run it. Figure 2 illustrates the temperature as a function of dimensionless position for various values of the length.

139

EXAMPLE 1.7-2: DRAWING A WIRE

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 700

Temperature (°C)

600 500 L = 5 cm

400

L = 15 cm

300

L =25 cm L =35 cm L =45 cm

200 100 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless position, x/L Figure 2: Temperature as a function of dimensionless position for various values of length.

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 1.8.1 Introduction Sections 1.6 and 1.7 showed how the differential equation describing constant crosssection ﬁns and other extended surfaces is derived. Analytical solutions for these differential equations take the form of an exponential function. In this section, extended surface problems are considered for which the cross-sectional area for conduction and the wetted perimeter for convection are not constant. The resulting differential equation is solved by Bessel functions.

1.8.2 Series Solutions It is worthwhile asking what the “exponential function” really is; we take it for granted in terms of its properties (i.e., how it can be integrated and differentiated). With some experience, it is possible to see that it solves a certain type of differential equation. In fact, that is its purpose: the exponential is really a polynomial series that has been deﬁned so that it solves a commonly encountered differential equation. There are other types of differential equations that appear in engineering problems; series solutions to these differential equations have been deﬁned and given formal names like “Bessel function” and “Kelvin function”. The homogeneous differential equation that results from the analysis of a constant cross-sectional area ﬁn is derived in Section 1.6.3 d2 T h − m2 T h = 0 dx2

(1-325)

Provided that the solution to Eq. (1-325) is continuous, it can be represented by a series of the form: T h = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · =

∞ i=0

ai xi

(1-326)

140

One-Dimensional, Steady-State Conduction

By substituting Eq. (1-326) into Eq. (1-325), it is possible to identify the characteristics of the series that solves this class of differential equation. The second derivative of the solution is required: ∞

dT h ai i xi−1 = a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + · · · = dx

(1-327)

i=1

d2 T h = 2 (1) a2 + 3 (2) a3 x + 4 (3) a4 x2 + 5 (4) a5 x3 + 6 (5) a6 x4 + · · · dx2 ∞ = ai i (i − 1) xi−2 (1-328) i=2

Substituting Eqs. (1-328) and (1-326) into Eq. (1-325) leads to: 2 (1) a2 + 3 (2) a3 x + 4 (3) a4 x2 + 5 (4) a5 x3 + 6 (5) a6 x4 + · · · −m2 [a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · ·] = 0

(1-329)

or ∞

ai i (i − 1) xi−2 − m2

i=2

∞

ai xi = 0

(1-330)

i=0

Since x is an independent variable that can assume any value, Eqs. (1-329) or (1-330) can only be generally satisﬁed if the coefﬁcients that multiply each term of the series (i.e., each power of x) each sum to zero. Examining Eq. (1-329), this requirement leads to: 2 (1) a2 − m2 a0 = 0 3 (2) a3 − m2 a1 = 0 4 (3) a4 − m2 a2 = 0 5 (4) a5 − m2 a3 = 0

(1-331)

6 (5) a6 − m2 a4 = 0 ··· The even coefﬁcients are therefore related according to: a2 =

m2 a0 2 (1)

a4 =

m2 a2 m4 a0 = 4 (3) 4 (3) (2) (1)

a6 =

m2 a4 m6 a0 = 6 (5) 6 (5) (4) (3) (2) (1)

(1-332)

··· or, more generally a2 i =

m2 i a0 (2i)!

where i = 0 · · · ∞

(1-333)

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

141

The odd coefﬁcients are also related: a3 =

m2 a1 3 (2)

a5 =

m2 a3 m4 a1 = 5 (4) 5 (4) (3) (2)

a7 =

m2 a5 m6 a1 = 7 (6) 7 (6) (5) (4) (3) (2)

(1-334)

··· or, more generally a2i+1 =

m2i a1 where i = 0 · · · ∞ (2i + 1)!

(1-335)

Therefore, we have determined two functions that both solve Eq. (1-325), related to the even and odd terms of the series; let’s call them F even and F odd : F even = a0

∞ (m x)2 i i=0

F odd = a1

∞ i=0

(1-336)

(2i)! m2i x2i+1 (2i + 1)!

(1-337)

Feven and Fodd are two solutions to the governing equation regardless of the particular values of the constants a0 and a1 in the same way that the functions C1 exp(m x) and C2 exp(-m x) (or, equivalently, C1 sinh(m x) and C2 cosh(m x)) were identiﬁed in Section 1.6 as solutions to Eq. (1-325) regardless of the values of C1 and C2 . The constants are determined in order to make the solution match the boundary conditions. In fact, if Eqs. (1-336) and (1-337) are rearranged slightly we see that they are identical to the series expansion of the functions sinh(m x) and cosh(m x): C1 cosh (m x) = C1

∞ (m x)2 i i=0

C2 sinh (m x) = C2

(2i)!

∞ (m x)2i+1 i=0

(2i + 1)!

=

a0 F even C1

(1-338)

=

a1 F odd m C2

(1-339)

The functions cosh and sinh are useful because they solve a particular differential equation, Eq. (1-325), which appears in many engineering problems; they are in fact nothing more than shorthand for the series given by Eqs. (1-336) and (1-337). To see this clearly, compute each of the terms in Eqs. (1-336) and (1-337) using EES: mx=1 Nterm=10 duplicate i=0,Nterm F even[i]=(mx)ˆ(2∗ i)/Factorial(2∗ i) F odd[i]=(mx)ˆ(2∗ i+1)/Factorial(2∗ i+1) end

“argument of function” “number of terms in series” “term in F even” “term in F odd”

142

One-Dimensional, Steady-State Conduction 4

3.5

Feven series Fodd series

Function value

3 2.5 2 hyperbolic cosine 1.5 1 hyperbolic sine

0.5 0 0

0.2

0.4

0.6

0.8 1 1.2 1.4 Argument of function

1.6

1.8

2

Figure 1-48: Comparison of the functions Feven and Fodd to the functions sinh and cosh.

and sum these terms using the sum command: F even=sum(F even[i],i=0,Nterm) F odd=sum(F odd[i],i=0,Nterm)

“sum of all terms in F even” “sum of all terms in F odd”

The results can be compared to the functions sinh and cosh: sinh=sinh(mx) cosh=cosh(mx)

“sinh function” “cosh function”

A parametric table is created that includes the variables mx, F_even, F_odd, sinh and cosh; the variable mx is varied from 0 to 2.0 and the results are shown in Figure 1-48. This exercise is meant to show that “solving” the homogeneous ordinary differential equation, Eq. (1-325), was really a matter of recognizing that it is solved by the series solutions that we call sinh and cosh (or equivalently exponentials with positive and negative arguments). Maple is very good at recognizing the solutions to differential equations. In this section, extended surfaces that do not have a constant cross-sectional area are considered. The governing differential equations that apply to these problems are more complex than Eq. (1-325). However, these differential equations are also solved by correctly deﬁned series that are given different names: Bessel functions.

1.8.3 Bessel Functions Extended surfaces with non-uniform cross-section may arise in many engineering applications. For example, tapered ﬁns are of interest since they may provide heat transfer rates comparable to constant cross-section ﬁns but require less material. Figure 1-49 illustrates a wedge ﬁn, an extended surface that has a thickness that varies linearly from its value at the base (th) to zero at the tip. The ﬁn is surrounded by ﬂuid at T ∞ with average heat transfer coefﬁcient h. The ﬁn material has conductivity k and the base of the ﬁn is kept at Tb .

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

143

T∞ , h

Figure 1-49: Wedge ﬁn.

th

k

W x

Tb

L

The width of the ﬁn is W and its length is L. We will assume that W th so that convection from the edges of the ﬁn may be ignored. Also, we will assume that the criteria for the extended approximation is satisﬁed (in this case, the Biot number h th/ (2 k) 1) so that the temperature can be assumed to be spatially uniform at any axial location x. It is convenient to deﬁne the origin of the axial coordinate at the tip of the ﬁn (see Figure 1-49) so that the cross-sectional area for conduction (Ac ) can be expressed as: Ac = th W

x L

(1-340)

The differential control volume used to derive the governing differential equation is shown in Figure 1-50 and suggests the energy balance: q˙ x = q˙ x+dx + q˙ conv

(1-341)

or, after expanding the x + dx term and simplifying: 0=

dq˙ dx + q˙ conv dx

(1-342)

The convection heat transfer rate is, approximately q˙ conv = 2 W dx h (T − T ∞ )

(1-343)

Note that Eq. (1-343) is only valid if th/L 1 so that the surface area within the control volume that is exposed to ﬂuid is approximately 2 W dx. The conduction heat transfer rate is: q˙ = −k Ac

dT x dT = −k th W dx L dx

(1-344)

Substituting Eqs. (1-343) and (1-344) into Eq. (1-342) leads to: d x dT 0= −k th W dx + 2 W dx h (T − T ∞ ) dx L dx

(1-345)

dx

Figure 1-50: Differential control volume.

q⋅ x

q⋅ x+dx

x q⋅ conv

144

One-Dimensional, Steady-State Conduction

which can be simpliﬁed to: d dT 2hL 2hL x − T =− T∞ dx dx k th k th

(1-346)

The solution is divided into a homogeneous and particular component: T = Th + T p

(1-347)

Substituting Eq. (1-347) into Eq. (1-346) leads to: dT p d d 2hL dT h 2hL 2hL x − Th + x − Tp = − T∞ dx dx k th dx dx k th k th = 0 for homogeneous differential equation

(1-348)

whatever is left is the particular differential equation

The solution to the particular differential equation is: T p = T∞

(1-349)

The homogeneous differential equation is: d dT h x − β Th = 0 dx dx

(1-350)

where the parameter β is deﬁned for convenience to be: β=

2hL k th

(1-351)

Note that the homogeneous differential equation, Eq. (1-350), is fundamentally different from the homogeneous differential equation that was obtained for a constant crosssection ﬁn, Eq. (1-179). Equation (1-350) is not solved by exponentials; to make this clear, assume an exponential solution: T h = C exp (m x)

(1-352)

and substitute it into the homogeneous differential equation: d [x C m exp (m x)] − β C exp (m x) = 0 dx

(1-353)

or, using the chain rule: C m exp (m x) + x C m2 exp (m x) − β C exp (m x) = 0

(1-354)

which can be simpliﬁed to: m + x m2 − β = 0

(1-355)

Unfortunately, there is no value of m that will satisfy Eq. (1-355) for all values of x. There must be some other function that solves Eq. (1-350). A series solution is again assumed: T h = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · =

∞

ai xi

(1-356)

i=0

The series is substituted into the governing differential equation in order to identify the characteristics of the series that solves this new class of differential equation. Expanding

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

145

Eq. (1-350) using the chain rule shows that both the ﬁrst and second derivatives are required: x

dT h d2 T h + − β Th = 0 dx2 dx

(1-357)

These derivatives were derived in Section 1.8.2: ∞

dT h ai i xi−1 = a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + · · · = dx

(1-358)

i=1

d2 T h = 2 (1) a2 + 3 (2) a3 x + 4 (3) a4 x2 + 5 (4) a5 x3 + 6 (5) a6 x4 + · · · dx2 ∞ = ai i (i − 1) xi−2 (1-359) i=2

Substituting Eqs. (1-356), (1-358), and (1-359) into Eq. (1-357) leads to: d2 T h → 2 (1) a2 x + 3 (2) a3 x2 + 4 (3) a4 x3 + 5 (4) a5 x4 + 6 (5) a6 x5 + · · · dx2 dT h + → a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + · · · dx −β T h → −β a0 − β a1 x − β a2 x2 − β a3 x3 − β a4 x4 + · · · =0 x

(1-360)

or, collecting like terms: [a1 − β a0 ] + [2 (1) a2 + 2 a2 − β a1 ] x + [3 (2) a3 + 3 a3 − β a2 ] x2 + [4 (3) a4 + 4 a4 − β a3 ] x3 + [5 (4) a5 + 5 a5 − β a4 ] x4 · · · = 0

(1-361)

For the series to solve the differential equation, each of the coefﬁcients must be zero; again, considering the coefﬁcients one at a time leads to a recursive formula that deﬁnes the series. a1 = β a0 (2 (1) + 2) a2 = β a1 22

(3 (2) + 3) a3 = β a2 32

(4 (3) + 4) a4 = β a3 42

(5 (4) + 5) a5 = β a4 52

···

(1-362)

146

One-Dimensional, Steady-State Conduction

or a1 = β a0 β β2 a2 = 2 a1 = 2 a0 2 2 β β3 a3 = 2 a2 = a0 3 [3 (2)]2 β β4 a4 = 2 a3 = a0 4 [4 (3) (2)]2 a5 =

(1-363)

β β5 a = a0 4 52 [5 (4) (3) (2)]2

··· More generally, the coefﬁcients are deﬁned by the equation: ai =

βi (i!)2

a0 where i = 1 · · · ∞

(1-364)

Equation (1-364) deﬁnes a function (let’s call it F) that provides a general solution to the homogeneous differential equation, Eq. (1-350). F = a0

∞ (β x) i i=0

(i!)2

(1-365)

The function F is actually a combination of Bessel functions; this is simply the name given to the series solutions of a particular class of differential equations ( just as hyperbolic sine and hyperbolic cosine are names given to series solutions of a different class of differential equations). The Bessel functions behave according to a set of formalized rules (just as hyperbolic sines and cosines do) that must be carefully obeyed when using them to solve problems. Bessel functions and the rules for manipulating them are completely recognized by Maple. Therefore, the combination of Maple and EES together allow you to avoid much of tedium associated with recognizing the correct Bessel function and then manipulating it to satisfy the boundary conditions of the problem. For the wedge ﬁn problem considered here, it is possible to enter the governing differential equation, Eq. (1-346), in Maple: > restart; > ODE:=diff(x∗ diff(T(x),x),x)-beta∗ T(x)=-beta∗ T_inﬁnity; ODE :=

2 d d T(x) + x T(x) − β T(x) = −β T inf inity dx dx2

and solve it: > Ts:=dsolve(ODE); 0 0 √ √ Ts := T(x) = BesselJ(0, 2, −β x) C2 + BesselY (0, 2 −β x) C1 + T inf inity

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

147

Maple has recognized that the solution to the governing differential equation includes two Bessel functions: the functions BesselJ and BesselY corresponding to Bessel functions of the ﬁrst and second kind, respectively. The ﬁrst argument indicates the order of the Bessel function and the second indicates the argument. The parameter β was deﬁned in Eq. (1-351) and involves the product of only positive quantities. Therefore, β must be positive and the arguments of both of the Bessel functions are complex (i.e., they involve the square root of a negative number); Bessel functions evaluated with a complex argument result in what are called modiﬁed Bessel functions (BesselI and BesselK are the modiﬁed Bessel functions of the ﬁrst and second kind, respectively). Maple will identify this fact for you, provided that you specify that the variable beta must be positive (using the assume command) and solve the equation again: > assume(beta>0); > Ts:=dsolve(ODE); 0 0 √ √ Ts := T(x) = BesselI(0, 2, β ∼ x) C2 + BesselK (0, 2 β ∼ x) C1 + T inf inity

The trailing tilde (∼) notation is used in Maple to indicate that the variable is associated with an assumption regarding its value; this convention can be changed in the preferences dialog in Maple. The solution is expressed in terms of modiﬁed Bessel functions with real arguments rather than Bessel functions with complex arguments. It is worth noting that if the argument of the function cosh is complex, then the result is cosine; thus the cosine function can be thought of as the modiﬁed hyperbolic cosine. The same behavior occurs for the sine and hyperbolic sine functions. All of the Bessel functions are built into EES and therefore the solution from Maple can be copied and pasted into EES for evaluation and manipulation. Maple has identiﬁed the general solution to the wedge ﬁn problem: 0 0 (1-366) T = C2 BesselI(0, 2 β x) + C1 BesselK(0, 2 β x) + T ∞ All that remains is to determine the constants C1 and C2 so that Eq. (1-366) also satisﬁes the boundary conditions. It is tempting to assume that one boundary condition must force the rate of heat transfer at the tip to be zero; however, the fact that the cross-sectional area at the tip is zero guarantees this fact, provided that the temperature gradient and therefore the temperature at the tip (i.e., at x = 0) is ﬁnite. T x=0 < ∞

(1-367)

Substituting Eq. (1-366) into Eq. (1-367) leads to: C2 BesselI (0, 0) + C1 BesselK (0, 0) + T ∞ < ∞

(1-368)

Figure 1-51 illustrates the behavior of the zeroth order modiﬁed Bessel function of the ﬁrst (BesselI) and second (BesselK) kind. Notice that the zeroth order modiﬁed Bessel function of the second kind is unbounded at zero and therefore the solution cannot include BesselK; the constant C1 must be zero. The remaining boundary condition corresponds to the speciﬁed base temperature of the ﬁn: T x=L = T b Substituting Eq. (1-366) into Eq. (1-369) leads to: 0 C2 BesselI(0, 2 β L) + T ∞ = T b

(1-369)

(1-370)

148

One-Dimensional, Steady-State Conduction 5

Modified Bessel function

4.5 4 3.5 3 st

1 kind

2.5 2 1.5 1 2

0.5 0 0

0.5

nd

kind

1 1.5 2 Argument of the function

2.5

3

Figure 1-51: Modiﬁed zeroth order Bessel functions of the ﬁrst and second kind.

Solving for the constant C2 leads to: C2 =

(T b − T ∞ ) √ BesselI 0, 2 β L

(1-371)

Substituting Eq. (1-371) into Eq. (1-366) (with C1 = 0) leads to the solution for the temperature distribution for a wedge ﬁn: √ BesselI(0, 2 β x) (1-372) + T∞ T = (T b − T ∞ ) √ BesselI(0, 2 β L) which can be expressed as:

/ x BesselI 0, 2 β L (T − T ∞ ) L = √ (T b − T ∞ ) BesselI(0, 2 β L)

(1-373)

Figure 1-52 illustrates the dimensionless temperature, (T − T ∞ )/(T b − T ∞ ), as a function of the dimensionless position, x/L, for various values of β L. Note that according to Eq. (1-351), the dimensionless parameter β L is: 2 h L2 (1-374) k th and resembles the ﬁn parameter, m L, for a constant cross-sectional area ﬁn. The parameter β L plays a similar role in the solution. The resistance to conduction in the x-direction is approximately: βL =

2L k W th and the resistance to convection is approximately: Rcond,x ≈

Rconv ≈

The ratio of Rcond,x

1

h2LW to Rconv is related to β L: h L2 2L h2LW Rcond,x ≈ =4 = 2βL Rconv k W th 1 k th

(1-375)

(1-376)

(1-377)

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

149

1

Dimensionless temperature

0.9 0.8

β L = 0.1 βL = 0.2 βL = 0.5

0.7 0.6

βL = 1

0.5

βL = 2

0.4 0.3

βL = 5

0.2

β L = 10

0.1 0 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position

0.8

0.9

1

Figure 1-52: Dimensionless temperature distribution as a function of dimensionless position for various values of β L.

As β L is reduced, the resistance to conduction in the x-direction becomes small relative to the resistance to convection and so the ﬁn becomes nearly isothermal at the base temperature. If β L is large, then the convection resistance is large relative to the conduction resistance and so the ﬁn temperature approaches the ﬂuid temperature. The rate of heat transfer to the ﬁn is given by: dT (1-378) q˙ ﬁn = k W th dx x=L Substituting Eq. (1-372) into Eq. (1-378) leads to: √ BesselI(0, 2 β x) d + T∞ q˙ ﬁn = k W th (T b − T ∞ ) √ dx BesselI(0, 2 β L) x=L

(1-379)

or q˙ ﬁn =

0 k W th (T b − T ∞ ) d √ [BesselI(0, 2 β x)]x=L BesselI 0, 2 β L dx

(1-380)

Rules for manipulating Bessel functions are presented in Section 1.8.4; however, Maple can be used to work with Bessel functions. The derivative required by Eq. (1-380) is computed easily using Maple: > restart; > diff(BesselI(0,2∗ sqrt(beta∗ x)),x); √ BesselI(1, 2 β x) β √ βx

so that Eq. (1-380) can be written as: q˙ ﬁn

/ √ BesselI(1, 2 β L) β = k W th (T b − T ∞ ) √ BesselI(0, 2 β L) L

(1-381)

150

One-Dimensional, Steady-State Conduction 1 0.9 0.8

Fin efficiency

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4 5 6 7 Fin parameter, βL

8

9

10

Figure 1-53: Fin efﬁciency of a wedge ﬁn as a function of β L.

The efﬁciency of the wedge ﬁn is deﬁned as discussed in Section 1.6.5: ηﬁn =

q˙ ﬁn h 2 W L (T b − T ∞ )

(1-382)

Substituting Eq. (1-381) into Eq. (1-382) leads to: ηﬁn

/ √ k th BesselI(1, 2 β L) β = √ h 2 L BesselI(0, 2 β L) L

or ηﬁn =

√ BesselI(1, 2 β L) √ √ BesselI(0, 2 β L) β L

(1-383)

(1-384)

Figure 1-53 illustrates the ﬁn efﬁciency of a wedge ﬁn as a function of the parameter β L. Notice that the ﬁn efﬁciency approaches unity when β L approaches zero because the ﬁn is nearly isothermal and the efﬁciency decreases as β L increases.

1.8.4 Rules for using Bessel Functions Bessel functions are well-deﬁned functions with speciﬁc rules for integration and differentiation. These rules are summarized in this section; however, the use of Maple will greatly reduce the need to know these rules. The differential equation: d p dθ (1-385) x ± c2 xs θ = 0 dx dx or, equivalently xp

d2 θ dθ + pxp−1 ± c2 xs θ = 0 dx2 dx

(1-386)

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

151

Governing differential equation: d ⎛ p dθ ⎞ 2 s ⎜x ⎟±c x θ =0 dx ⎝ dx ⎠ s− p+2≠0 Calculate solution parameters: 2 n (1 − p ) = n= a= 2 (s − p + 2 ) a (s − p + 2 )

(1 − p )

s− p+2=0

( p − 1)

2

2

− 4c > 0

( p − 1)

2

Calculate solution parameters:

Last term in dif. eq. is negative: d ⎛ p dθ ⎞ 2 s ⎜x ⎟−c x θ =0 dx ⎝ dx ⎠

r1 =

(1 − p ) + ( p − 1)

−c

r2 =

(1 − p ) − ( p − 1)

−c

Calculate solution parameters: (1 − p ) d= 2 2 p − 1) ( 2 e= c − 4

2

2

4

2

2

n

( ) BesselK (n, c a x )

θ = C1 x a BesselI n, c a x n

+C2 x

a

2

− 4c < 0

1 a

1 a

2

4

2

θ = C1 x r1 + C2 x r2

θ = C1 x d cos ⎣⎡e ln (x )⎦⎤

+C2 x sin ⎡⎣e ln (x )⎤⎦ d

( p − 1)

2

Last term in dif. eq. is positive: d ⎛ p dθ ⎞ 2 s ⎜x ⎟+c x θ = 0 dx ⎝ dx ⎠ n

+C2 x

n

a

Calculate solution parameter: (1 − p ) d= 2

( ) BesselY (n, c a x )

θ = C1 x a BesselJ n, c a x

2

− 4c = 0

1 a

θ = C1 x d + C2 x d ln (x )

1 a

Figure 1-54: Flowchart illustrating the steps involved with identifying the correct solution to Bessel’s equation.

where θ is a function of x and p, c, and s are constants is a form of Bessel’s equation that has been solved using power series. The rules for identifying the appropriate solution given the form of the equation are laid out in ﬂowchart form in Figure 1-54. Following the path outlined in Figure 1-54, the ﬁrst step is to evaluate the quantity s − p + 2; if s − p + 2 is not equal to zero, then the intermediate solution parameters n and a should be calculated. n=

1− p s − p+ 2

(1-387)

a=

2 s − p+ 2

(1-388)

1− p n = a 2

(1-389)

The solution depends on the sign of the last term in Eq. (1-385); if the sign of the last term is negative, then the solution is expressed as: n 1 n 1 θ = C1 x /a BesselI n, c a x /a + C2 x /a BesselK n, c a x /a (1-390)

152

One-Dimensional, Steady-State Conduction

where C1 and C2 are the undetermined constants that depend on the boundary conditions. The functions BesselI and BesselK are modiﬁed Bessel functions of the ﬁrst and second kind, respectively. The ﬁrst parameter in the function is the order of the modiﬁed Bessel function and the second parameter is the argument of the function. The EES code below provides the zeroth order modiﬁed Bessel function of the second kind evaluated at 2.5 (0.06235). y=BesselK(0,2.5)

The order of the Bessel function can either be integer (e.g., 0, 1, 2, . . . ) or fractional (e.g. 0.5). If the sign of the last term in Eq. (1-385) is positive, then the solution is: n 1 n 1 (1-391) θ = C1 x /a BesselJ n, c a x /a + C2 x /a BesselY n, c a x /a where the functions BesselJ and BesselY are Bessel functions of the ﬁrst and second kind, respectively. If s − p + 2 is equal to zero, then the solution depends on the sign of the parameter (p − 1)2 − 4c2 . If (p − 1)2 − 4c2 is positive, then the solution is: θ = C1 xr1 + C2 xr2 where

(1-392)

(1 − p) + r1 = 2

and

(p − 1)2 − c2 4

(1-393)

(p − 1)2 − c2 4

(1-394)

(1 − p) − r2 = 2

If (p − 1)2 − 4c2 is zero, then the solution is: θ = C1 xd + C2 xd ln (x)

(1-395)

where d=

(1 − p) 2

(1-396)

Finally, if (p − 1)2 − 4c2 is negative, then the solution is: θ = C1 xd cos (e ln (x)) + C2 xd sin (e ln (x)) where

(1-397)

e=

c2 −

(p − 1)2 4

(1-398)

The zeroth and ﬁrst order modiﬁed Bessel functions of the ﬁrst and second kind are shown in Figure 1-55. Notice that the modiﬁed Bessel functions of the second kind are unbounded at zero while the modiﬁed Bessel functions of the ﬁrst kind are unbounded as the argument tends towards inﬁnity; this characteristic can be helpful to determine the undetermined constants.

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

153

2.5 nd

Modified Bessel function

th

2 kind, 0 order BesselK(0, x)

2.25 2

nd

st

2 kind, 1 order BesselK(1, x)

1.75 1.5

st

th

1 kind, 0 order BesselI(0, x)

1.25 1

st

st

1 kind, 1 order BesselI(1, x)

0.75 0.5 0.25 0 0

0.5 1 1.5 2 2.5 Argument of modified Bessel function

3

Figure 1-55: Modiﬁed Bessel functions of the ﬁrst and second kinds and the zeroth and ﬁrst orders.

The zeroth and ﬁrst order Bessel functions of the ﬁrst and second kind are shown in Figure 1-56. Notice that the Bessel functions of the second kind, like the modiﬁed Bessel functions of the second kind, are unbounded at zero. The rules for differentiating zeroth order Bessel and zeroth order modiﬁed Bessel functions are:

1

st

du d [BesselI(0, u)] = BesselI(1, u) dx dx

(1-399)

d du [BesselK(0, u)] = −BesselK(1, u) dx dx

(1-400)

th

1 kind, 0 order BesselJ(0, x) 0.75

st

st

1 kind, 1 order BesselJ(1, x)

Bessel function

0.5 0.25 0 -0.25 -0.5

2

-0.75 -1 0

2 1

nd

nd

2

th

kind, 0 order BesselY(0, x) st

kind, 1 order BesselY(1, x) 3 4 5 6 7 Argument of Bessel function

8

9

10

Figure 1-56: Bessel functions of the ﬁrst and second kinds and the zeroth and ﬁrst orders.

154

One-Dimensional, Steady-State Conduction

d du [BesselJ(0, u)] = −BesselJ(1, u) dx dx

(1-401)

d du [BesselY(0, u)] = − BesselY(1, u) dx dx

(1-402)

For arbitrary order Bessel and modiﬁed Bessel functions with positive integer order n, the rules for differentiation are: d n BesselI(n, m x) = m BesselI(n − 1, m x) − BesselI(n, m x) dx x

(1-403)

d n BesselK(n, m x) = −m BesselK(n − 1, m x) − BesselK(n, m x) dx x

(1-404)

d n BesselJ(n, m x) = m BesselJ(n − 1, m x) − BesselJ(n, m x) dx x

(1-405)

d n BesselY(n, m x) = m BesselY(n − 1, m x) − BesselY(n, m x) dx x

(1-406)

Finally, the following differentials are also sometimes useful: d n [x BesselI(n, m x)] = m xn BesselI(n − 1, m x) dx d n [x BesselK(n, m x)] = −m xn BesselK(n − 1, m x) dx d n [x BesselJ(n, m x)] = m xn BesselJ(n − 1, m x) dx

(1-407)

(1-408)

(1-409)

d n [x BesselY(n, m x)] = −m xn BesselY(n − 1, m x) dx

(1-410)

d −n [x BesselI(n, m x)] = m x−n BesselI(n + 1, m x) dx

(1-411)

d −n [x BesselK(n, m x)] = −m x−n BesselK(n + 1, m x) dx

(1-412)

d −n [x BesselJ(n, m x)] = −m x−n BesselJ(n + 1, m x) dx

(1-413)

d −n [x BesselY(n, m x)] = −m x−n BesselY(n + 1, m x) dx

(1-414)

155

EXAMPLE 1.8-1: PIPE IN A ROOF A pipe with outer radius rp = 5.0 cm emerges from a metal roof carrying hot gas at Thot = 90◦ C. The pipe is welded to the roof, as shown in Figure 1. Assume that the temperature at the interface between the pipe and the roof is equal to the gas temperature, Thot . The inside of the roof is well-insulated, but the outside of the roof is exposed to ambient air at T∞ = 20◦ C. The average heat transfer coefﬁcient between the outside of the roof and the ambient air is h = 50 W/m2 -K. The outside of the roof is also exposed to a uniform heat ﬂux due to the incident solar radiation, q˙ s = 800 W/m2 . The spatial extent of the roof is large with respect to the outer radius of the pipe. The metal roof has thickness th = 2.0 cm and thermal conductivity k = 50 W/m-K. 2 q⋅ s′′ = 800 W/m T∞ = 20°C 2 h = 50 W/m -K Thot = 90°C th = 2 cm k = 50 W/m-K

Figure 1: Pipe passing through a roof exposed to solar radiation.

rp = 5 cm adiabatic

a) Can the roof be modeled using an extended surface approximation? The input parameters are entered in EES: “EXAMPLE 1.8-1: Pipe in a Roof” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Input Parameters” r p=5.0 [cm]∗ convert(cm,m) T hot=converttemp(C,K,90[C]) T inﬁnity=converttemp(C,K,20[C]) h bar=50 [W/mˆ2-K] qf s=800 [W/mˆ2] th=2.0 [cm]∗ convert(cm,m) k=50 [W/m-K]

“Pipe radius” “Hot gas temperature” “Air temperature” “Heat transfer coefﬁcient” “Solar ﬂux” “Roof thickness” “Roof conductivity”

The extended surface approximation ignores any temperature gradients across the thickness of the roof. This is equivalent to ignoring the resistance to conduction across the thickness of the roof while considering the resistance associated with convection from the top surface of roof. The ratio of these resistances is calculated using an appropriately deﬁned Biot number: Bi =

th h k

EXAMPLE 1.8-1: PIPE IN A ROOF

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

EXAMPLE 1.8-1: PIPE IN A ROOF

156

One-Dimensional, Steady-State Conduction

which is calculated in EES: Bi=h bar∗ th/k

“Biot number to check extended surface approximation”

The Biot number is 0.02, which is sufﬁciently less than 1 to justify the extended surface approximation. b) Develop an analytical model for the roof that can be used to predict the temperature distribution in the roof and also determine the rate of heat loss from the pipe by conduction to the roof. Because the roof is large relative to the spatial extent of our problem, the edge of the roof will have no effect on the temperature distribution in the metal around the pipe and the temperature distribution will be axisymmetric; the problem can be solved in radial coordinates. An energy balance on a differential segment of the roof is shown in Figure 2. q⋅ s q⋅ conv q⋅ r r

q⋅ r + dr

Figure 2: Differential energy balance.

dr

The energy balance includes conduction, convection and solar irradiation: q˙r + q˙ s = q˙r +dr + q˙ conv or q˙ s =

d q˙r dr + q˙ conv dr

Substituting the rate equations: q˙r = −k 2 π r th

dT dr

q˙ s = q˙ s 2 π r dr q˙ conv = 2 π r dr h (T − T∞ ) into the energy balance leads to: d dT −k 2 π r th dr + 2 π r dr h (T − T∞ ) q˙ s 2 π r dr = dr dr Simplifying leads to: h h d q˙ dT r − rT =− r T∞ − s r dr dr k th k th k th

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

157

EXAMPLE 1.8-1: PIPE IN A ROOF

The solution is split into its homogeneous and particular components: T = Th + Tp which leads to: h dTh d r − r Th dr dr k th

dTp h h d q˙ + r − r Tp = − r T∞ − s r dr dr k th k th k th

= 0 for homogeneous differential equation

whatever is left is the particular differential equation

The solution to the particular differential equation: dTp h h q˙ d r − r Tp = − r T∞ − s r dr dr k th k th k th is a constant: Tp = T∞ +

q˙ s h

The homogeneous differential equation is: d dTh r − m2 r Th = 0 dr dr where

(1)

m=

h k th

Equation (1) is a form of Bessel’s equation: d dθ xp ± c2 x s θ = 0 dx dx

(2)

where p = 1, c = m, and s = 1. Referring to the ﬂow chart presented in Figure 1-54, the value of s − p + 2 is equal to 2 and therefore the solution parameters n and a must be computed: n=

1−1 =0 1−1+2

a=

2 =1 1−1+2

The last term in Eq. (1) is negative and therefore the solution to Eq. (2), as indicated by Figure 1-54, is given by: n 1 n 1 θ = C 1 x /a BesselI n, c a x /a + C 2 x /a BesselK n, c a x /a where x = r and c = m for this problem. The homogeneous solution is: Th = C 1 BesselI (0, mr ) + C 2 BesselK (0, mr ) The temperature distribution is the sum of the homogeneous and particular solutions: q˙ (3) T = C 1 BesselI (0, mr ) + C 2 BesselK (0, mr ) + T∞ + s h

EXAMPLE 1.8-1: PIPE IN A ROOF

158

One-Dimensional, Steady-State Conduction

Maple can be used to obtain the same result: > restart; > ODE:=diff(r∗ diff(T(r),r),r)-mˆ2∗ r∗ T(r)=-mˆ2∗ r∗ T_inﬁnity-qf_s∗ r/(k∗ th); O D E :=

d T(r ) dr

+r

d2 T(r ) dr 2

− m2 r T(r ) = −m2 r T inﬁnity −

qf sr k th

> Ts:=dsolve(ODE); Ts := T(r ) = BesselI (0, mr ) C 2 + BesselK (0, mr ) C 1 +

m2 T inﬁnity k th + qf s m2 k th

Note that the constants C1 and C2 are interchanged in the Maple solution but it is otherwise the same as Eq. (3). The boundary conditions must be used to obtain C1 and C2 . As r approaches ∞, the effect of the pipe disappears. In this limit, the heat gain from the sun exactly balances convection, therefore: q˙ s = h (Tr →∞ − T∞ )

(4)

Substituting Eq. (3) into Eq. (4) leads to: q˙ q˙ s = h C 1 BesselI (0, ∞) + C 2 BesselK (0, ∞) + T∞ + s − T∞ h or C 1 BesselI (0, ∞) + C 2 BesselK (0, ∞) = 0 Figure 1-55 shows that the zeroth order modiﬁed Bessel function of the ﬁrst kind (i.e., BesselI(0,x)) limits to ∞ as x approaches ∞ while the zeroth order modiﬁed Bessel function of the second kind (i.e., BesselK(0,x)) approaches 0 as x approaches ∞. This information can also be obtained using Maple and the limit command: > limit(BesselI(0,x),x=inﬁnity); ∞ > limit(BesselK(0,x),x=inﬁnity); 0

Therefore, C1 must be zero while C2 can be any ﬁnite value. T = C 2 BesselK (0, mr ) + T∞ +

q˙ s h

The temperature where the roof meets the pipe is speciﬁed: Tr =r p = Thot or q˙ C 2 BesselK 0, mr p + T∞ + s = Thot h

(3)

159

The solution is programmed in EES: m=sqrt(h bar/(k∗ th)) C 2∗ BesselK(0,m∗ r p)=T hot-T inﬁnity-qf s/h bar T=C 2∗ BesselK(0,m∗ r)+T inﬁnity+qf s/h bar T C=converttemp(K,C,T)

“ﬁn parameter” “boundary condition” “solution” “in C”

The temperature in the roof as a function of position is shown in Figure 3 for h = 50 W/m2 -K (as speciﬁed in the problem statement) and also for h = 5 W/m2 -K. 180 2

h = 5 W/m -K 160

Temperature (°C)

140 120 100 80 60

2

h = 50 W/m -K 40 20 0.05

0.1

0.15

0.2

0.25 0.3 0.35 Radius (m)

0.4

0.45

0.5

Figure 3: Temperature as a function of radius for h = 50 W/m2 -K and h = 5 W/m2 -K with q˙ s = 800 W/m2 .

The heat transfer between the pipe and the roof (q˙ p ) is evaluated using Fourier’s law at r = r p : dT q˙ p = −k th 2 π r p (4) dr r =r p Substituting Eq. (3) into Eq. (4) leads to: q˙ p = −k th 2 π r p C 2

d [BesselK (0, mr )]r =r p dr

which can be evaluated using the differentiation rule provided by Eq. (1-400): q˙ p = k th 2 π r p C 2 m BesselK(1, mr p ) or using Maple: > q_dot_p:=-k∗ th∗ 2∗ pi∗ r_p∗ C_2∗ eval(diff(BesselK(0,m∗ r),r),r=r_p); q d ot p := 2 k th π r p C 2 BesselK (1, mr p) m

EXAMPLE 1.8-1: PIPE IN A ROOF

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

One-Dimensional, Steady-State Conduction

The solution is programmed in EES: q dot p=k∗ th∗ 2∗ pi∗ r p∗ C 2∗ m∗ BesselK(1,m∗ r p)

“heat transfer into pipe”

Figure 4 illustrates the rate of heat transfer into the pipe as a function of the heat transfer coefﬁcient and for various values of the solar ﬂux. 400 300 Heat transfer from pipe (W)

EXAMPLE 1.8-1: PIPE IN A ROOF

160

q⋅ s′′= 0 W/m2 q⋅ s′′= 200 W/m2 q⋅s′′= 500 W/m2 ⋅q′′= 800 W/m2 s

200 100 0 -100 -200 -300 -400 0

10

20 30 40 50 60 70 80 2 Heat transfer coefficient (W/m-K)

90

100

Figure 4: Heat transfer from pipe to roof as a function of the heat transfer coefﬁcient for various values of the solar ﬂux.

It is always important to understand your solution after it has been obtained. Notice in Figure 4 that the rate of heat transfer to the roof tends to increase with increasing heat transfer coefﬁcient. This makes sense, as the temperature gradient at the interface between the roof and the pipe will increase as the heat transfer coefﬁcient increases. However, when there is a non-zero solar ﬂux, the heat transfer rate will change direction (i.e., become negative) at low values of the heat transfer coefﬁcient indicating that the heat ﬂow is into the pipe under these conditions. This effect occurs when the solar ﬂux elevates the temperature of the roof to the point that it is above the hot gas temperature. Figure 4 shows that we can expect this behavior for h = 5 W/m2 -K and q˙ s = 800 W/m2 and Figure 3 illustrates the temperature distribution under these conditions.

161

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION EXAMPLE 1.3-1 examined an ablative technique for locally heating cancerous tissue using small, conducting spheres (thermoseeds) that are embedded at precise locations and exposed to a magnetic ﬁeld. Each thermoseed experiences a volumetric generation of thermal energy that causes its temperature and the temperature of the adjacent tissue to rise. In EXAMPLE 1.3-1, blood perfusion in the tissue was neglected; blood perfusion refers to the volumetric removal of energy in the tissue by the blood ﬂowing in the microvascular structure. The blood perfusion may be modeled as a volumetric heat sink that is proportional to the difference between the local temperature and the normal body temperature (Tb = 37◦ C); the constant of proportionality, β, is nominally 20,000 W/m3 -K. The thermoseed has a radius r t s = 1.0 mm and it experiences a total rate of thermal energy generation of g˙ t s = 1.0 W. The temperature far from the thermoseed is the body temperature, Tb . The tissue has thermal conductivity k t = 0.5 W/m-K. a) Determine the steady-state temperature distribution in the tissue associated with a single sphere placed in an inﬁnite medium of tissue considering blood perfusion. The input parameters are entered in EES: “EXAMPLE 1.8-2: Magnetic Ablation with Blood Perfusion” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” r ts=1.0 [mm]∗ convert(mm,m) T b=converttemp(C,K,37 [C]) g dot ts=1.0 [W] beta=20000 [W/mˆ3-K] k t=0.5 [W/m-K]

“radius of thermoseed” “blood and body temperature” “generation in the thermoseed” “blood perfusion constant” “tissue conductivity”

Figure 1 illustrates a differential control volume in the tissue that balances conduction with blood perfusion. The energy balance on the control volume is: q˙r = q˙r +dr + g˙ where q˙ is conduction and g˙ is the rate of energy removed by blood perfusion.

rts Figure 1: Differential control volume in the tissue.

g⋅ ts

⋅ ⋅ q⋅ r g qr+dr

dr

The conduction through the tissue is given by: q˙r = −k t 4 π r 2

dT dr

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

162

One-Dimensional, Steady-State Conduction

and the rate of energy removal by blood perfusion is: g˙ = 4 π r 2 dr β (T − Tb ) Combining these equations leads to: d 2 dT 0= −k t 4 π r dr + 4 π r 2 dr β (T − Tb ) dr dr which can be simpliﬁed:

dT β β 2 d r T = − r 2 Tb r2 − dr dr kt kt

The solution is divided into its homogeneous and particular components: T = Th + Tp so that:

d dTh β 2 r Th r2 − dr dr kt

+

= 0 for homogeneous differential equation

dTp d β β 2 r Tp = − r 2 Tb r2 − dr dr kt kt whatever is left is the particular differential equation

The particular solution is: Tp = Tb The homogeneous differential equation is: dTh d r2 − m2 r 2 Th = 0 dr dr where

m=

(1)

β kt

Equation (1) is a form of Bessel’s equation: dθ d xp ± c2 x s θ = 0 dx dx where p = 2, c = m, and s = 2. Referring to the ﬂow chart presented in Figure 1-54, the value of s − p + 2 is equal to 2 and therefore the solution parameters n and a must be computed: n=

1 1−2 =− 2−2+2 2

a=

2 =1 2−2+2

The last term in Eq. (1) is negative and therefore the solution is given by: n 1 n 1 θ = C 1 x /a BesselI n, c a x /a + C 2 x /a BesselK n, c a x /a or

1 1 1 1 Th = C 1 r − /2 BesselI − , mr + C 2 r − /2 BesselK − , mr 2 2

163

The solution is the sum of the homogeneous and particular solutions: 1 1 1 1 T = C 1 r − /2 BesselI − , mr + C 2 r − /2 BesselK − , mr + Tb 2 2

(2)

The constants are obtained by applying the boundary conditions. As r approaches ∞, the temperature must approach the body temperature: Tr →∞ = Tb

(3)

Substituting Eq. (2) into Eq. (3) leads to: 1 1 BesselI − , ∞ BesselK − , ∞ 2 2 C1 + C2 =0 √ √ ∞ ∞

(4)

At ﬁrst glance it is unclear how Eq. (4) helps to establish the constants; however, Maple can be used to show that C1 must be zero because the ﬁrst term limits to ∞ while the second term limits to 0: > limit(BesselI(-1/2,r)/sqrt(r),r=inﬁnity); ∞ > limit(BesselK(-1/2,r)/sqrt(r),r=inﬁnity); 0

The second boundary condition is obtained from an interface energy balance at r = rts ; the rate of conduction heat transfer into the tissue must equal the rate of generation within the thermoseed: −4 π r t2s k t

dT = g˙ t s dr r =rt s

(5)

Substituting Eq. (2) with C1 = 0 into Eq. (5) leads to: −4 π

r t2s

d 1 −1/2 kt C2 BesselK − , mr = g˙ t s r dr 2 r =r t s

Using Eq. (1-408) leads to: −1/2

−C 2 r t s

3 g˙ t s m BesselK − , mr t s = − 2 4 π r t2s k t

The constant C2 is evaluated in EES:

“Determine constant” m=sqrt(beta/k t) -C 2∗ m∗ BesselK(-3/2,m∗ r ts)/sqrt(r ts)=-g dot ts/(4∗ pi∗ r tsˆ2∗ k t)

“solution parameter” “determine constant”

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

One-Dimensional, Steady-State Conduction

The solution is programmed in EES and converted to Celsius: “Solution” T=C 2∗ BesselK(-0.5,m∗ r)/sqrt(r)+T b T C=converttemp(K,C,T) r mm=r∗ convert(m,mm)

“temperature” “in C” “radius in mm”

Figure 2 illustrates the temperature in the tissue as a function of radial position for various values of blood perfusion. Note that the temperature distribution as β → 0 (i.e., in the absence of blood perfusion) agrees exactly with the solution for the tissue temperature obtained in EXAMPLE 1.3-1 (which is overlaid onto Figure 2) although the mathematical form of the solution looks very different. Figure 2 shows that the effect of blood perfusion is to reduce the extent of the elevated temperature region and therefore diminish the amount of tissue killed by the thermoseed. 200 EXAMPLE 1.3-1 result

180 160 Temperature (°C)

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

164

3

β → 0 W/m -K β = 5,000 W/m 3-K β = 20,000 W/m 3-K 3 β = 50,000 W/m -K

140 120 100 80 60 40 20 1

1.5

2

2.5

3 3.5 4 Radius (mm)

4.5

5

5.5

6

Figure 2: Temperature in the tissue as a function of radius for various values of blood perfusion; also shown is the result from EXAMPLE 1.3-1 which was derived for the same problem in the absence of blood perfusion (β = 0).

1.9 Numerical Solution to Extended Surface Problems 1.9.1 Introduction Sections 1.6 through 1.8 present analytical solutions to extended surface problems. Only simple problems with constant properties can be considered analytically. There will be situations where these simpliﬁcations are not justiﬁed and it will be necessary to use a numerical model. Numerical modeling of extended surface problems is a straightforward extension of the numerical modeling techniques that are described in Sections 1.4 and 1.5. If the extended surface approximation discussed in Section 1.6.2 is valid, then it is possible to obtain a numerical solution by dividing the computational domain into many small (but ﬁnite) one-dimensional control volumes. Energy balances are written for each control volume; the energy balances can include convective and/or radiative terms in addition to the conductive and generation terms that are considered in Sections 1.4 and

1.9 Numerical Solution to Extended Surface Problems

165

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING A resistance temperature detector (RTD) utilizes a material that has an electrical resistivity that is a strong function of temperature. The temperature of the RTD is inferred by measuring its electrical resistance. Figure 1 shows an RTD that is mounted at the end of a metal rod and inserted into a pipe in order to measure the temperature of a ﬂowing liquid. The RTD is monitored by passing a known current through it and measuring the voltage drop across it. This process results in a constant amount of ohmic heating that will cause the RTD temperature to rise relative to the temperature of the surrounding liquid; this effect is referred to as a self-heating measurement error. Also, conduction from the wall of the pipe to the temperature sensor through the metal rod can result in a temperature difference between the RTD and the liquid; this effect is referred to as a mounting measurement error. Tw = 20°C

L = 5.0 cm

pipe

D = 0.5 mm x

Figure 1: Temperature sensor mounted in a ﬂowing liquid.

T∞ = 5.0°C

k = 10 W/m-K q⋅ sh = 2.5 mW RTD

The thermal energy generation associated with ohmic heating is q˙ sh = 2.5 mW. All of this ohmic heating is assumed to be transferred from the RTD into the end of the rod at x = L. The rod has a thermal conductivity k = 10 W/m-K, diameter D = 0.5 mm, and length L = 5.0 cm. The end of the rod that is connected to the pipe wall (at x = 0) is maintained at a temperature of Tw = 20◦ C. The liquid is at a uniform temperature, T∞ = 5◦ C. However, the local heat transfer coefﬁcient between the liquid and the rod (h) varies with x due to the variation of the liquid velocity in the pipe. This problem resembles external ﬂow over a cylinder, which will be discussed in Chapter 4; however, you may assume that the heat transfer coefﬁcient between the rod surface and the ﬂuid varies according to: h = 2000

W m2.8 K

x 0.8

(1)

where h is the heat transfer coefﬁcient in W/m2 -K and x is position along the rod in m.

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.5. Each term in the energy balance is represented by a rate equation that reﬂects the governing heat transfer mechanism; the result is a system of algebraic equations that can be solved using EES or MATLAB. The solution should be checked for convergence, checked against your physical intuition, and compared with an analytical solution in the limit where one is valid.

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

166

One-Dimensional, Steady-State Conduction

a) Can the rod be treated as an extended surface? The input parameters are entered in EES; note that the heat transfer coefﬁcient is computed using a function deﬁned at the top of the EES code. “EXAMPLE 1.9-1: Temperature Sensor Error due to Mounting and Self Heating” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Function for heat transfer coefﬁcient” function h(x) h=2000 [W/mˆ2.8-K]∗ xˆ0.8 end “Inputs” q dot sh=2.5 [milliW]∗ convert(milliW,W) k=10 [W/m-K] D=0.5 [mm]∗ convert(mm,m) L=5.0 [cm]∗ convert(cm,m) T w=converttemp(C,K,20 [C]) T inﬁnity=converttemp(C,K,5 [C])

“self-heating power” “conductivity of mounting rod” “diameter of mounting rod” “length of mounting rod” “temperature of wall” “temperature of liquid”

The appropriate Biot number for this case is: Bi =

hD 2k

The Biot number will be largest (and therefore the extended surface approximation least valid) when the heat transfer coefﬁcient is largest. According to Eq. (1), the highest heat transfer coefﬁcient occurs at the tip of the rod; therefore, the Biot number is calculated according to: Bi=h(L)∗ D/(2∗ k)

“Biot number”

The Biot number calculated by EES is 0.0046, which is much less than 1.0 and therefore the extended surface approximation is justiﬁed. b) Develop a numerical model of the rod that will predict the temperature distribution in the rod and therefore the error in the temperature measurement; this error is the difference between the temperature at the tip of the rod (i..e, the temperature of the RTD) and the liquid. The development of the numerical model follows the same steps that are discussed in Section 1.4. Nodes (i.e., locations where the temperature will be determined) are positioned uniformly along the length of the rod, as shown in Figure 2. The location of each node (xi ) is: xi =

(i − 1) L (N − 1)

i = 1..N

167

where N is the number of nodes used for the simulation. The distance between adjacent nodes (x) is: x =

L (N − 1)

This distribution is entered in EES:

N=100 duplicate i=1,N x[i]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1)

“number of nodes” “position of each node” “distance between adjacent nodes”

A control volume is deﬁned around each node; the control surface bisects the distance between the nodes, as shown in Figure 2. T1 ⋅

Figure 2: Control volume for an internal node.

q⋅ top

Ti-1

qconv q⋅ bottom

Ti

q⋅ top q⋅ conv ⋅ qsh

TN-1

Ti+1 TN

The control volume for internal node i shown in Figure 2 is subject to conduction heat transfer at each edge (q˙ t op and q˙ bottom ) and convection (q˙ conv ). The energy balance is: q˙ t op + q˙ bottom + q˙ conv = 0 The conduction terms are approximated according to:

q˙ t op =

k π D2 (Ti−1 − Ti ) 4 x

q˙ bottom =

k π D2 (Ti+1 − Ti ) 4 x

The convection term is modeled using the convection coefﬁcient evaluated at the position of the node: q˙ conv = hxi π D x (T∞ − Ti )

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

168

One-Dimensional, Steady-State Conduction

Combining these equations leads to: k π D2 k π D2 (Ti−1 −Ti ) + (Ti+1 −Ti ) + hxi π Dx(T∞ − Ti ) = 0 for i = 2.. (N −1) 4 x 4 x (2) “internal control volume energy balances” duplicate i=2,(N-1) k∗ pi∗ Dˆ2∗ (T[i-1]-T[i])/(4∗ DELTAx)+k∗ pi∗ Dˆ2∗ (T[i+1]-T[i])/(4∗ DELTAx)+ & pi∗ D∗ DELTAx∗ h(x[i])∗ (T inﬁnity-T[i])=0 end

The nodes at the edges of the domain must be treated separately. At the pipe wall, the temperature is speciﬁed: T1 = Tw

T[1]=T w

(3)

“boundary condition at wall”

The ohmic dissipation, q˙ sh is assumed to enter the half-node at the tip (i.e., node N) and therefore is included in the energy balance for this node (see Figure 2): hx π D x k π D2 (TN −1 − TN ) + N (T∞ − TN ) + q˙ sh = 0 4 x 2

(4)

Note the factor of 2 in the denominator of the convection term that arises because the half-node has half the surface area of the internal nodes. k∗ pi∗ Dˆ2∗ (T[N-1]-T[N])/(4∗ DELTAx)+pi∗ D∗ DELTAx∗ h(x[N])∗ (T inﬁnity-T[N])/2+q dot sh=0 “boundary condition at tip”

Equations (2) through (4) are a system of N equations in an equal number of unknown temperatures that are entered in EES. The solution is converted to Celsius:

duplicate i=1,N T C[i]=converttemp(K,C,T[i]) end

“solution in Celsius”

Figure 3 illustrates the temperature distribution in the rod for N = 100 nodes. The temperature elevation of the tip relative to the ﬂuid is about 3.4 K and represents the measurement error. For the conditions in the problem statement, it is clear that the measurement error is primarily due to self-heating because the effect of the wall (the temperature elevation at the base) has died off after about 2.0 cm.

169

20 18

Temperature (°C)

16 14 12 10 8 measurement error 6 4 0

0.01

0.02

0.03

0.04

0.05

Axial position (m)

Figure 3: Temperature distribution in the mounting rod.

As with any numerical solution, it is important to verify that a sufﬁcient number of nodes have been used so that the numerical solution has converged. The key result of the solution is the tip-to-ﬂuid temperature difference, which is the measurement error for the sensor (δT ): δT = TN − T∞ deltaT=T[N]-T inﬁnity

“measurement error”

Figure 4 illustrates the tip-to-ﬂuid temperature difference as a function of the number of nodes and shows that the solution has converged for N greater than 100 nodes.

Temperature measurement error (K)

3.5 3 2.5 2 1.5 1 0.5 1

10

100

1000

Number of nodes (-)

Figure 4: Tip-to-ﬂuid temperature difference as a function of the number of nodes.

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction

The analytical solution for this problem in the limit of a constant heat transfer coefﬁcient and an adiabatic tip was derived in Section 1.6.3 and is included in Table 1-4: cosh (m (L − x)) T − T∞ = Tw − T∞ cosh (m L) where 4h m= kD The analytical solution is programmed in EES: “Analytical solution for veriﬁcation in the limit q dot sh=0 and h=constant” m=sqrt(4∗ h(L)/(k∗ D)) “ﬁn parameter” duplicate i=1,N T an[i]=T inﬁnity+(T w-T inﬁnity)∗ cosh(m∗ (L-x[i]))/cosh(m∗ L) “analytical solution” T an C[i]=converttemp(K,C,T an[i]) “in C” end

The numerical solution is obtained in this limit by setting the variable q_dot_sh equal to zero and modifying the function h so that it returns 100 W/m2 -K regardless of position. “Function for heat transfer coefﬁcient” function h(x) {h=2000 [W/mˆ2.8-K]∗ xˆ0.8} h=100 [W/mˆ2-K] end “Inputs” q dot sh=0 [W] {2.5 [milliW]∗ convert(milliW,W)}

“self-heating power”

The temperature distribution predicted by the numerical model is compared with the analytical solution in Figure 5. 22.5 Analytical solution Numerical model

20 17.5

Temperature (°C)

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

170

15 12.5 10 7.5 5 2.5 0 0

0.01

0.02

0.03

0.04

0.05

Axial position (m) Figure 5: Veriﬁcation of the numerical model against the analytical solution in the limit that the heat transfer coefﬁcient is constant at h = 100 W/m2 -K and there is no self-heating, q˙ sh = 0 W.

171

c) Investigate the effect of thermal conductivity on the temperature measurement error. Identify the optimal thermal conductivity and explain why an optimal thermal conductivity exists. Figure 6 illustrates the temperature measurement error as a function of the thermal conductivity of the rod material; note that the function h has been set back to its original form and the variable q_dot_sh restored to 2.5 mW. Figure 6 shows that the optimal thermal conductivity, corresponding to the minimum measurement error, is around 100 W/m-K. Below the optimal value, the self-heating error dominates as the local temperature rise at the tip of the rod is large. Above the optimal value, the conduction from the wall dominates.

Temperature measurement error (K)

10 9 8 7 6 5 self heating dominates

4

conduction from the wall dominates

3 2 1 0 0

25

50

75

100

125

150

175

200

225

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.9 Numerical Solution to Extended Surface Problems

250

Rod thermal conductivity (W/m-K)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS It is often necessary to supply a cryogenic experiment or apparatus with electrical current. Some examples include current for superconducting electronics and magnets, to energize a resistance-based temperature sensor, and to energize a heater used for temperature control. In any of these cases, careful design of the wires that are used to supply and return the current to the facility is important. The heat transfer to the cryogenic device from these wires should be minimized as this energy must be removed either by a refrigeration system (i.e., a cryocooler) or by consumption of a relatively expensive cryogen (e.g., by the boil off of liquid helium or liquid nitrogen). There is an optimal wire diameter for any given application that minimizes this parasitic heat transfer to the device. Figure 1 illustrates two current leads, each carrying I = 100 ampere (one supply and the other return). These current leads extend from the room temperature wall of the vacuum vessel, where the wire material is at TH = 20◦ C, to the experiment, where the wire material is at TC = 50 K. The length of both current leads is L = 1.0 m and their diameter, D, should be optimized. The vacuum in the vessel prevents any convection heat transfer from the surface of the wires. However, the surface of the

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

Figure 6: Temperature measurement error as a function of rod thermal conductivity.

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

172

One-Dimensional, Steady-State Conduction

wires radiate to their surroundings, which may be assumed to be at TH = 20◦ C. The external surface of the wires has emissivity, ε = 0.5. TH = 20°C

2 current leads, each carrying I =100 ampere with emissivity ε = 0.5

L = 1.0 m

TH = 20°C TC = 50 K

D

Figure 1: Cryogenic current leads.

The leads are made of oxygen free, high-conductivity copper; the thermal conductivity and resistivity of copper can vary substantially at cryogenic temperatures depending on the purity and history of the material (e.g., whether it has been annealed or not). The purity of the metal is often expressed as the Residual Resistivity Ratio (RRR), which is deﬁned as the ratio of the metal’s electrical resistivity at 273 K to that at 4.2 K. Oxygen free, high conductivity copper (OFHC) has an RRR of approximately 200. The thermal conductivity and electrical resistivity of RRR 200 copper as a function of temperature is provided in Table 1 (Iwasa (1994)). Table 1: Thermal conductivity and electrical resistivity of OFHC copper. Temperature

Thermal conductivity

Electrical resistivity

500 K 400 K 300 K 250 K 200 K 150 K 125 K 100 K 90 K 80 K 70 K 60 K 55 K 50 K

4.31 W/cm-K 4.15 W/cm-K 3.99 W/cm-K 4.04 W/cm-K 4.11 W/cm-K 4.24 W/cm-K 4.34 W/cm-K 4.71 W/cm-K 4.98 W/cm-K 5.43 W/cm-K 6.25 W/cm-K 7.83 W/cm-K 9.11 W/cm-K 11.0 W/cm-K

3.19 μohm-cm 2.49 μohm-cm 1.73 μohm-cm 1.39 μohm-cm 1.06 μohm-cm 0.72 μohm-cm 0.54 μohm-cm 0.36 μohm-cm 0.29 μohm-cm 0.22 μohm-cm 0.15 μohm-cm 0.098 μohm-cm 0.076 μohm-cm 0.057 μohm-cm

a) Develop a numerical model in MATLAB that can predict the rate of heat transfer to the cryogenic experiment from the pair of current leads. The input conditions are entered in a MATLAB function EXAMPLE1p9 2.m; the two arguments to the function are diameter (the variable D) and number of nodes (the variable N) as we know that these parameters will be varied during the veriﬁcation

173

and optimization process. Any of the other parameters can be added in order to facilitate additional parametric studies or optimization.

function [ ]=EXAMPLE1p9 2(D,N) I=100; T H=20+273.2; T C=50; L=1; eps=0.5; sigma=5.67e-8;

%current (amp) %hot temperature (K) %cold temperature (K) %length of lead (m) %emissivity of lead surface (-) %Stefan-Boltzmann constant (W/mˆ2-Kˆ4)

Notice that we have not, to this point, speciﬁed what parameters are returned when the function executes (i.e., there are no variables listed between the square brackets in the function header). Functions are deﬁned (at the bottom of the M-ﬁle) that return the conductivity and electrical resistivity of the OFHC copper; the interp1 function is used to carry out interpolation on the data provided in Table 1 using a cubic spline technique.

%——–Property functions————function[k]=k cu(T) %returns the thermal conductivity (W/m-K) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) kd=[4.31,4.15,3.99,4.04,4.11,4.24,4.34,4.71,4.98,5.43,6.25,7.83,9.11,11.0]∗ 100; %conductivity data (W/m-K) k=interp1(Td,kd,T); end function[rho e]=rho e cu(T) %returns the electrical resistivity (ohm-m) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) rho ed=[3.19,2.49,1.73,1.39,1.06,0.72,0.54,0.36,0.29,0.22,0.15,0.098,0.076,0.057]/(1e6∗ 100); %electrical resistivity data (ohm-m) rho e=interp1(Td,rho ed,T); end

The ﬁrst step is to position the nodes throughout the computational domain. For this problem, the nodes will be distributed uniformly, as shown in Figure 2: xi =

(i − 1) L for i = 1..N (N − 1)

The distance between adjacent nodes (x) is: x =

L (N − 1)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

174

One-Dimensional, Steady-State Conduction

The MATLAB code that accomplishes these assignments is: %Position nodes for i=1:N x(i,1)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

%position of each node (m) %distance between adjacent nodes (m)

A control volume for an internal node is shown in Figure 2; the control volume experiences conduction heat transfer from the adjacent nodes above and below (q˙ t op and q˙ bottom , respectively), as well as radiation (q˙r ad ) and generation due to the ˙ An energy balance for the control ohmic dissipation associated with the current (g). volume is: q˙ t op + q˙ bottom + g˙ = q˙r ad

(1)

T1

q⋅ top q⋅ rad q⋅

bottom

Ti-1 T g⋅ i Ti+1

Figure 2: Control volume for an internal node and associated energy terms.

TN

The conductivity used to approximate the conduction heat transfer rates must be evaluated at the temperature of the boundaries in order to avoid energy balance violations, as discussed in Section 1.4.3. With this understanding, these rate equations become: q˙ t op = kT =(Ti +Ti−1 )/2

π D2 (Ti−1 − Ti ) 4 x

q˙ bottom = kT =(Ti +Ti+1 )/2

π D2 (Ti+1 − Ti ) 4 x

(2)

(3)

The rate of thermal energy generation is calculated using the resistivity evaluated at the temperature of each node: g˙ = ρe,T =Ti

4 x 2 I π D2

The rate of radiation heat transfer is approximately given by: q˙r ad = ε σ π D x Ti 4 − TH4

(4)

(5)

175

where ε is the emissivity of the surface of the leads and σ is the Stefan-Boltzmann constant. Substituting Eqs. (2) through (5) into Eq. (1) leads to: π D2 π D2 4 x 2 I (Ti−1 − Ti ) + kT =(Ti +Ti+1 )/2 (Ti+1 − Ti ) + ρe,T =Ti 4 x 4 x π D2 = ε σ π D x Ti 4 − TH4 for i = 2.. (N − 1) (6)

kT = (Ti +Ti−1 )/2

The remaining equations specify the boundary temperatures: T1 = TH

(7)

TN = TC

(8)

Equations (6) through (8) are a set of N equations in the N unknown temperatures; however, the temperature dependence of the material properties (k and ρe ) as well as the non-linear rate equation associated with radiation heat transfer cause the system of equations to be non-linear. Therefore, a relaxation technique will be employed in order to obtain the solution; the relaxation process is discussed in Section 1.5.6. The assumed solution (Tˆ ) will be successively substituted with the predicted solution. This process will continue until the assumed and predicted solutions agree to within an acceptable tolerance. The solution proceeds by assuming a temperature distribution that can be used to evaluate the coefﬁcients in the linearized equations. A linear temperature distribution provides a reasonable start for the iteration: (i − 1) Tˆi = TH + (TH − TC ) for i = 1..N (N − 1)

%Start relaxation with a linear temperature distribution for i=1:N Tg(i,1)=T H-(T H-T C)∗ (i-1)/(N-1); end

In order to solve this problem using MATLAB, we will need a set of linear equations; linear equations cannot contain products of the unknown temperatures with other unknown temperatures or functions of the unknown temperatures. Therefore, the temperature-dependent material properties must be evaluated at the assumed temperatures (Tˆ ): q˙ t op = kT =(Tˆi +Tˆi−1 )/2

π D2 (Ti−1 − Ti ) 4 x

q˙ bottom = kT =(Tˆi +Tˆi+1 )/2 g˙ = ρe,T =Tˆi

π D2 (Ti+1 − Ti ) 4 x

4 x 2 I π D2

(9)

(10) (11)

The fourth power temperature terms cause the radiation equation, Eq. (5), to be nonlinear. Therefore, it is necessary to linearize the radiation equation so that it can be placed in matrix format, as was done for the material properties. The radiation

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

176

One-Dimensional, Steady-State Conduction

terms can be linearized most conveniently using the same factorization that was previously used to deﬁne a radiation resistance in Section 1.2.6: (12) q˙r ad = σ ε π D x Tˆi 2 + TH2 (Tˆi + TH )(Ti − TH ) Substituting the linearized rate equations, Eqs. (9) through (12), into the energy balance for an internal node, Eq. (1), leads to: π D2 π D2 4 x 2 I (Ti−1 − Ti ) + kT =(Tˆi +Tˆi+1 )/2 (Ti+1 − Ti ) + ρe,T =Tˆi 4 x 4 x π D2 = σ ε π D x (Tˆi 2 + TH2 )(Tˆi + TH )(Ti − TH ) for i = 2.. (N − 1) (13)

kT =(Tˆi +Tˆi−1 )/2

Equations (7), (8), and (13) must be placed in matrix format: AX = b where X is a vector of unknown temperatures, A is a matrix containing the coefﬁcients of each equation, and b is a vector containing the constant terms for each equation. The matrix A is declared as sparse in MATLAB; note that Eq. (13) indicates that there are at most three nonzero entries in each row of A: %Setup A and b A=spalloc(N,N,3∗ N); b=zeros(N,1);

Equation (7) can be placed in row 1 of the matrix equation: T1 [1] = TH A1,1

(14)

b1

and Eq. (8) can be placed in row N of the matrix equation: TN [1] = TC AN ,N

(15)

bN

Equation (13) must be rearranged to make it clear which row and column each coefﬁcient should be entered into the matrix: 2 π D2 π D2 2 ˆ ˆ − kT =(Tˆi +Tˆi+1 )/2 − σ ε π D x Ti + TH Ti + TH Ti −kT =(Tˆi +Tˆi−1 )/2 4 x 4 x + Ti−1 kT =(Tˆi +Tˆi−1 )/2

2

πD 4 x

Ai,i

+Ti+1 kT =(Tˆi +Tˆi+1 )/2

Ai,i−1

π D2 4 x

(16)

Ai,i+1

4 x 2 = −σ ε π D x Tˆi 2 + TH2 Tˆi + TH TH − ρe,T =Tˆi I π D2

for i = 2.. (N − 1)

bi

The numerical solution is placed within a while loop that checks for convergence of the relaxation scheme. The variable err is used to terminate the while loop and represents the average, absolute error between the assumed and predicted temperature distribution. (There are other criteria that could be used, but this is sufﬁcient for most problems.) The while loop is terminated when the variable err decreases to less than the input parameter tol, which represents the convergence

177

tolerance for the relaxation process. Initially, the value of err is set to a value greater than tol to ensure that the while loop executes at least one time. err=999; tol=0.1; while(err>tol)

%error that terminates the while loop (K) %criteria for terminating the while loop (K)

end end

Within the while loop, the matrix is ﬁlled in using the coefﬁcients suggested by Eq. (14), %specify the hot end temperature A(1,1)=1; b(1,1)=T h;

Eq. (15), %specify the cold end temperature A(N,N)=1; b(N,1)=T C;

and Eq. (16). %internal nodes for i=2:(N-1) A(i,i)=-k cu((Tg(i+1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx)- . . . k cu((Tg(i-1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx)-. . . sigma*eps*pi*D*DELTAx*(Tg(i,1)ˆ2+T Hˆ2)*(Tg(i,1)+T H); A(i,i-1)=k cu((Tg(i-1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx); A(i,i+1)=k cu((Tg(i+1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx); b(i,1)=-rho e cu(Tg(i,1))*4*DELTAx*Iˆ2/(pi*Dˆ2)- . . . sigma*eps*pi*D*DELTAx*(Tg(i,1)ˆ2+T Hˆ2)*(Tg(i,1)+T H)*T H; end

Note that the three periods in the above code is a line break; it indicates that the code is continued on the subsequent line. The matrix equation is solved and the error between the assumed and predicted temperature is computed. err =

N 1 |Ti − Tˆi | N i=1

The ﬁnal step in the while loop is to update the assumed temperature distribution with the predicted temperature distribution. T=full(A/b); err=sum(abs(T-Tg))/N Tg=T;

%compute the error %update the guess temperature array

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction

Note that the full command in the above code converts the sparse matrix, T, that results from the operation on the sparse matrix A into a full matrix. The header of the function is modiﬁed to specify the output arguments, x and T: function[x,T]=EXAMPLE1p9 2(D,N)

Because the statement that calculates the variable err is not terminated with a semicolon, the result of the calculation will be echoed in the workspace allowing you to keep track of the progress. If you call this program with a diameter of 5.0 mm you should see: >> [x,T]=EXAMPLE1p9 2(0.005,100); err = 41.2789 err = 16.1165 err = 6.2792 err = 2.4864 err = 1.0023 err = 0.4110 err = 0.1685 err = 0.0693

The relaxation process had to iterate several times in order to converge due to the nonlinearity of the problem. Figure 3 illustrates the temperature distribution in the current lead for several different values of the diameter. 500 450 D=4.0 mm

400 Temperature (K)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

178

350 D=4.5 mm 300

D=5.0 mm

250

D=6.0 mm

200

D=8.0 mm D=10.0 mm

150 100 50 0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Axial position (m)

0.8

0.9

1

Figure 3: Temperature distribution in the current lead for various values of the diameter.

179

Notice that the smaller diameter leads result in large amounts of ohmic dissipation and therefore the wire tends to become hotter and the temperature gradient at the cold end increases. For a given temperature gradient, larger diameter leads will result in a higher rate of heat transfer to the cold end due to the larger area for conduction. There is a balance between these effects that results in an optimal diameter. The heat transferred to the cold end (q˙ c ) is calculated using an energy balance on node N: q˙ c = kT =(TN +TN −1 )/2

π D2 2 x 2 x 4 I −σ επ D TH − TN4 (TN −1 − TN ) + ρe,T =TN 2 4 x πD 2

or, in MATLAB: q dot c=k cu((T(N)+T(N-1))/2)∗ pi∗ Dˆ2∗ (T(N-1)-T(N))/(4∗ DELTAx)+... rho e cu(T(N))∗ 2∗ DELTAx∗ Iˆ2/(pi∗ Dˆ2)-sigma∗ eps∗ pi∗ D∗ DELTAx∗ (T Hˆ4-T(N)ˆ4)/2; %heat transfer to cold end

The function header is modiﬁed so that q˙ c is also returned:

function[q dot c,x,T]=EXAMPLE1p9 2(D,N)

It is necessary to verify that the solution has a sufﬁcient number of nodes and, if possible, verify the result against an analytical solution. The critical parameter for the solution is the rate of heat transfer to the experiment per current lead; therefore Figure 4 illustrates q˙ c as a function of the number of nodes in the solution, N. The information shown in Figure 4 was generated quickly using the script varyN (below), which calls the function EXAMPLE1p9_2 multiple times with varying values of N:

%Script varyN.m clear all; D=0.005; N=[2,5,10,20,50,100,200,500,1000,2000]’; for i=1:10 i [q dot c(i,1),x,T]=EXAMPLE1p9 2(D,N(i)); end

The clear all statement at the beginning of the script clears all variables from memory. Using the clear all statement is often a good idea as it prevents previous elements (e.g., from previous runs) of the variables q_dot_c or N from being retained. If you had previously run the script varyN with more than 10 runs, then N and q_dot_c would exist in memory with more than 10 elements. Running the script varyN as shown above would overwrite the ﬁrst 10 elements of these variables, but leave all subsequent elements which could lead to confusion.

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction 6

5.5 Heat transfer to cold end (W)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

180

5 4.5 4 3.5 3 2.5 2 1.5 10 0

10 1

10 2 Number of nodes

10 3

Figure 4: Heat transferred to the cold end per lead as a function of the number of nodes for a 5.0 mm lead.

Figure 4 suggests that at least 100 nodes should be used for sufﬁcient accuracy. In the absence of any radiation heat transfer (ε = 0) and with constant resistivity and conductivity, it is possible to compare the numerical solution to the analytical solution for the temperature in a generating wall with ﬁxed end conditions. This result was derived in Section 1.3.2 and is repeated below: g˙ L 2 x x 2 (TH − TC ) − x + TH − T = 2k L L L where the volumetric generation is given by: g˙ =

16 I 2 ρe π 2 D4

%constant property analytical solution g dot vol=16∗ Iˆ2∗ rho e cu(T H)/(piˆ2∗ Dˆ4); for i=1:N T an(i,1)=g dot vol∗ Lˆ2∗ ((x(i)/L)-(x(i)/L)ˆ2)/(2∗ k cu(T H))- . . . (T H-T C)∗ x(i)/L+T H; end

The property functions in MATLAB are modiﬁed to return, temporarily, constant values of k = 200 W/m-K and ρe = 1 × 10−8 ohm-m. %—-Property functions————function[k]=k cu(T) %returns the thermal conductivity (W/m-K) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) kd=[4.31,4.15,3.99,4.04,4.11,4.24,4.34,4.71,4.98,5.43,6.25,7.83,9.11,11.0]*100; %conductivity data (W/m-K) %k=interp1(Td,kd,T); k=200; end

181

function[rho e]=rho e cu(T) %returns the electrical resistivity (ohm-m) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) rho ed=[3.19,2.49,1.73,1.39,1.06,0.72,0.54,0.36,0.29,0.22,0.15,0.098,0.076,0.057]/(1e6*100); %electrical resistivity data (ohm-m) %rho e=interp1(Td,rho ed,T); rho e=1e-8; end

The emissivity is set to 0 and the MATLAB code is run for a 5.0 mm diameter lead. The temperature distribution predicted by the MATLAB code is compared with the analytical solution in Figure 5. Note that with these modiﬁcations (i.e., constant k and ρe and ε = 0), the problem becomes linear and therefore a single iteration is required in order to reduce the relaxation error to 0. 400 350

Temperature (K)

300 250 200 150 100 50 0

numerical solution analytical solution 0.1

0.2

0.3

0.4 0.5 0.6 0.7 Axial position (m)

0.8

0.9

1

Figure 5: Comparison of the analytical and numerical solutions in the limit that k = 200 W/m-K (constant), ρe = 1e-8 ohm-m (constant) and ε = 0 for a 5.0 mm diameter wire.

Finally, it is possible to parametrically vary the wire diameter, D, in order to minimize the heat ﬂow to the cold end of the current lead. The code is returned to its original, non-linear form. A script (varyd) is used to call the function multiple times with various diameters in order to carry out a parametric study of this parameter:

%Script varyd.m clear all; N=100; D=linspace(0.0038,0.01,100)’; %generate 100 values of D between 0.0038 and 0.01 [m] for i=1:100 [q dot c(i,1),x,T]=EXAMPLE1p9 2(D(i),N); end

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction

Figure 6 illustrates the heat leak to the cold end as a function of wire diameter (note that the functions for k and ρe were reset and the value of ε was reset to 0.50) and shows that there is a clear optimal diameter around 5.1 mm for this application. Smaller values of D lead to excessive self-heating whereas larger values provide a large path for conduction heat transfer. 13 12 Heat leak per lead (W)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

182

11 10 9 8

optimal lead diameter

7 6 5 0.003

0.004

0.005

0.006 0.007 Diameter (m)

0.008

0.009

0.01

Figure 6: Heat leak to the cold end of each current lead as a function of diameter.

MATLAB has powerful, built-in optimization algorithms that allow you to automate the process of determining the optimal diameter. The MATLAB function fminbnd is the simplest available and carries out a bounded, 1-D minimization. The function fminbnd is called according to: x opt=fminbnd(function,x1,x2)

where function is the name of a function that requires a single argument and provides a single output (that should be minimized) and x1 and x2 are the lower and upper bounds of the argument to use for the minimization. First, it is necessary to modify the function EXAMPLE1p9_2 so that it takes a single argument (D) and returns a single output (q_dot_c):

function[q dot c]=EXAMPLE1p9 2(D) N=100;

%number of nodes (-)

Then the fminbnd function can be called directly from the workspace:

>> D opt=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01);

in order to identify the optimal diameter.

183

>> D opt D opt = 0.0051

Note that the calculation of the variable err in the function EXAMPLE1p9_2 is terminated with a semicolon so that the error is not echoed to the workspace during each iteration. The function fminbnd will return the optimized value of the heat leak as well by adding an additional output argument to the fminbnd call: >>[D opt,q dot c min]=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01);

which indicates that the optimal value of the heat leak is 5.57 W. >> q dot c min q dot c min = 5.6815

It is possible to control the details of the optimization using an optional fourth input argument to the function fminbnd that sets the optimization parameters; the easiest way to set this last argument is using the optimset command. If you enter >> help optimset

into the workspace then a complete list of the parameters that can be controlled is returned. It is possible, for example, to display the progress of the optimization using:

[D opt,q dot c min]=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01,optimset(‘Display’,‘iter )) Func-count 1 2 3 4 5 6 7 8

x 0.0062918 0.0077082 0.00541641 0.0043798 0.00523819 0.00515231 0.00511897 0.00508564

f(x) 6.35363 8.12533 5.73753 6.14038 5.68985 5.6824 5.68148 5.68212

Procedure initial golden golden parabolic parabolic parabolic parabolic parabolic

Optimization terminated: the current x satisﬁes the termination criteria using OPTIONS.TolX of 1.000000e-004 D opt = 0.0051 q dot c min = 5.6815

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

184

One-Dimensional, Steady-State Conduction

The ﬁrst argument to optimset speciﬁes the parameter to be controlled (‘Display’, which controls the level of display) and the second indicates its new value (‘iter’, which indicates that the progress should be displayed after each iteration). It would be inconvenient to use the fminbnd function to carry out a parametric variation of how the optimal value of the variables D and q_dot_c are affected by current or some other parameter. In its current format, it is not possible to pass the value of the current (I) to the fminbnd function and therefore the study would have to be carried out manually by running the fminbnd function and then changing the value of the variable I within the function EXAMPLE1p9_2. This process would become tedious and can be avoided by parameterizing the function. For example, suppose you want to determine how the optimal value of diameter changes with current. First, include current as an additional argument to the function: function[q dot c]=EXAMPLE1p9_2(D,I) N=100; % I=100;

%number of nodes (-) %current (amp)

If you try to repeat the optimization using the previous protocol you will receive an error: >> [D opt,q dot c min]=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01) ??? Input argument “I” is undeﬁned. Error in ==> EXAMPLE9 2 at 42 b(i,1)=-rho e cu(Tg(i,1))∗ 4∗ DELTAx∗ Iˆ2/(pi∗ Dˆ2)-. . . Error in ==> fminbnd at 182 x=xf; fx=funfcn(x,varargin{:});

However, you can parameterize the function using a one-argument anonymous function that captures the value of I (set in the workspace) and calls EXAMPLE1p9_2 with two arguments: >> I=100; >> [D opt,q dot c min]=fminbnd(@(D) EXAMPLE1p9 2(D,I),0.004,0.01) D opt = 0.0051 q dot c min = 5.6815

Now it is possible to generate a script, varyI, that evaluates the optimized diameter and heat ﬂow as a function of current. %Script varyI.m clear all; I=linspace(1,100,10)’; for i=1:10 [d opt(i,1),q dot min(i,1)]=fminbnd(@(d) EXAMPLE1p9 2(d,I(i,1)),0.0025,0.01) end

185

Figure 7 illustrates the optimal diameter and the associated heat leak as a function of current. Note that the optimal diameter and minimized heat leak are both approximately linear functions of the current. 6

0.0055

5.5

0.005

5

0.0045

4.5 minimum heat leak

0.004

4

0.0035 0.003 50

3.5

optimal diameter

55

60

65

70 75 80 85 Current (ampere)

90

95

Minimum heat leak per lead (W)

Optimal current lead diameter (m)

0.006

3 100

Figure 7: Optimal diameter and minimized heat leak to cold end of each current lead as a function of current.

Chapter 1: One-Dimensional, Steady-State Conduction The website associated with this book (www.cambridge.org/nellisandklein) provides many more problems than are included here. Conduction Heat Transfer 1–1 Section 1.1.2 provides an approximation for the thermal conductivity of a monatomic gas at ideal gas conditions. Test the validity of this approximation by comparing the conductivity estimated using Eq. (1-18) to the value of thermal conductivity for a monotonic ideal gas (e.g., low pressure argon) provided by the internal function in EES. Note that the molecular radius, σ, is provided in EES by the Lennard-Jones potential using the function sigma_LJ. a.) What are the value and units of the proportionality constant required to make Eq. (1-18) an equality? b.) Plot the value of the proportionality constant for 300 K argon at pressures between 0.01 and 100 MPa on a semi-log plot with pressure on the log scale. At what pressure does the approximation given in Eq. (1-18) begin to fail? Steady-State 1-D Conduction without Generation 1–2 Figure P1-2 illustrates a plane wall made of a thin (thw = 0.001 m) and conductive (k = 100 W/m-K) material that separates two ﬂuids. Fluid A is at TA = 100◦ C and the heat transfer coefﬁcient between the ﬂuid and the wall is hA= 10 W/m2 -K while ﬂuid B is at T B = 0◦ C with hB = 100 W/m2 -K.

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

Chapter 1: One-Dimensional, Steady-State Conduction

186

One-Dimensional, Steady-State Conduction thw = 0.001 m TA = 100° C

TB = 0°C

hA = 10 W/m2 -K

hB = 100 W/m2 -K k = 100 W/m-K

Figure P1-2: Plane wall separating two ﬂuids.

a.) Draw a resistance network that represents this situation and calculate the value of each resistor (assuming a unit area for the wall, A = 1 m2 ). b.) If you wanted to predict the heat transfer rate from ﬂuid A to ﬂuid B very accurately then which parameters (e.g., thw , k, etc.) would you try to understand/measure very carefully and which parameters are not very important? Justify your answer. 1–3 You have a problem with your house. Every spring at some point the snow immediately adjacent to your roof melts and runs along the roof line until it reaches the gutter. The water in the gutter is exposed to air at temperature less than 0◦ C and therefore freezes, blocking the gutter and causing water to run into your attic. The situation is shown in Figure P1-3. snow melts at this surface Tout , hout = 15 W/m2 -K Ls = 2.5 inch

snow, ks = 0.08 W/m-K insulation, kins = 0.05 W/m-K Tin = 22°C, hin = 10 W/m2 -K Lins = 3 inch plywood, L p = 0.5 inch, k p = 0.2 W/m-K

Figure P1-3: Roof of your house.

The air in the attic is at T in = 22◦ C and the heat transfer coefﬁcient between the inside air and the inner surface of the roof is hin = 10 W/m2 -K. The roof is composed of a Lins = 3.0 inch thick piece of insulation with conductivity kins = 0.05 W/m-K that is sandwiched between two L p = 0.5 inch thick pieces of plywood with conductivity k p = 0.2 W/m-K. There is an Ls = 2.5 inch thick layer of snow on the roof with conductivity ks = 0.08 W/m-K. The heat transfer coefﬁcient between the outside air at temperature Tout and the surface of the snow is hout = 15 W/m2 -K. Neglect radiation and contact resistances for part (a) of this problem. a.) What is the range of outdoor air temperatures where you should be concerned that your gutters will become blocked by ice? b.) Would your answer change much if you considered radiation from the outside surface of the snow to surroundings at Tout ? Assume that the emissivity of snow is εs = 0.82. 1–4 Figure P1-4(a) illustrates a composite wall. The wall is composed of two materials (A with kA = 1 W/m-K and B with kB = 5 W/m-K), each has thickness L =

Chapter 1: One-Dimensional, Steady-State Conduction

187

1.0 cm. The surface of the wall at x = 0 is perfectly insulated. A very thin heater is placed between the insulation and material A; the heating element provides q˙ = 5000 W/m2 of heat. The surface of the wall at x = 2L is exposed to ﬂuid at T f ,in = 300 K with heat transfer coefﬁcient hin = 100 W/m2 -K. q⋅ ′′ = 5000 W/m 2

material A kA = 1 W/m-K

insulated

L = 1 cm

Tf, in = 300 K hin = 100 W/m2 -K

x

L = 1 cm

material B kB = 5 W/m-K

Figure P1-4 (a): Composite wall with a heater.

You may neglect radiation and contact resistance for parts (a) through (c) of this problem. a.) Draw a resistance network to represent this problem; clearly indicate what each resistance represents and calculate the value of each resistance. b.) Use your resistance network from (a) to determine the temperature of the heating element. c.) Sketch the temperature distribution through the wall. Make sure that the sketch is consistent with your solution from (b). Figure P1-4(b) illustrates the same composite wall shown in Figure P1-4(a), but there is an additional layer added to the wall, material C with kC = 2.0 W/m-K and L = 1.0 cm. material C kC = 2 W/m-K

q⋅ ′′ = 5000 W/m 2

material A kA = 1 W/m-K L = 1 cm

insulated x

L = 1 cm

Tf, in = 300 K hin = 100 W/m2 -K material B L = 1 cm kB = 5 W/m-K

Figure P1-4 (b): Composite wall with material C.

Neglect radiation and contact resistance for parts (d) through (f) of this problem. d.) Draw a resistance network to represent the problem shown in Figure P1-4(b); clearly indicate what each resistance represents and calculate the value of each resistance. e.) Use your resistance network from (d) to determine the temperature of the heating element. f.) Sketch the temperature distribution through the wall. Make sure that the sketch is consistent with your solution from (e).

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One-Dimensional, Steady-State Conduction

Figure P1-4(c) illustrates the same composite wall shown in Figure P1-4(b), but there is a contact resistance between materials A and B, Rc = 0.01 K-m2 /W, and the surface of the wall at x = −L is exposed to ﬂuid at T f ,out = 400 K with a heat transfer coefﬁcient hout = 10 W/m2 -K. material C kC = 2 W/m-K Tf, out = 400 K hout = 10 W/m2 -K

q⋅ ′′ = 5000 W/m2

material A kA = 1 W/m-K L = 1 cm

x

Tf, in = 300 K hin = 100 W/m2 -K material B kB = 5 W/m-K

L = 1 cm L = 1 cm

R′′c = 0.01 K-m2 /W

Figure P1-4 (c): Composite wall with convection at the outer surface and contact resistance.

Neglect radiation for parts (g) through (i) of this problem. g.) Draw a resistance network to represent the problem shown in Figure P1-4(c); clearly indicate what each resistance represents and calculate the value of each resistance. h.) Use your resistance network from (g) to determine the temperature of the heating element. i.) Sketch the temperature distribution through the wall. 1–5 You have decided to install a strip heater under the linoleum in your bathroom in order to keep your feet warm on cold winter mornings. Figure P1-5 illustrates a cross-section of the bathroom ﬂoor. The bathroom is located on the ﬁrst story of your house and is W = 2.5 m wide × L = 2.5 m long. The linoleum thickness is thL = 5.0 mm and has conductivity kL = 0.05 W/m-K. The strip heater under the linoleum is negligibly thin. Beneath the heater is a piece of plywood with thickness thP = 5 mm and conductivity kP = 0.4 W/m-K. The plywood is supported by ths = 6.0 cm thick studs that are Ws = 4.0 cm wide with thermal conductivity ks = 0.4 W/m-K. The center-to-center distance between studs is ps = 25.0 cm. Between each stud are pockets of air that can be considered to be stagnant with conductivity ka = 0.025 W/m-K. A sheet of drywall is nailed to the bottom of the studs. The thickness of the drywall is thd = 9.0 mm and the conductivity of drywall is kd = 0.1 W/m-K. The air above in the bathroom is at T air,1 = 15◦ C while the air in the basement is at T air,2 = 5◦ C. The heat transfer coefﬁcient on both sides of the ﬂoor is h = 15 W/m2 -K. You may neglect radiation and contact resistance for this problem. a.) Draw a thermal resistance network that can be used to represent this situation. Be sure to label the temperatures of the air above and below the ﬂoor (Tair,1 and Tair,2 ), the temperature at the surface of the linoleum (TL ), the temperature of the strip heater (Th ), and the heat input to the strip heater (q˙ h ) on your diagram. b.) Compute the value of each of the resistances from part (a). c.) How much heat must be added by the heater to raise the temperature of the ﬂoor to a comfortable 20◦ C? d.) What physical quantities are most important to your analysis? What physical quantities are unimportant to your analysis?

Chapter 1: One-Dimensional, Steady-State Conduction Tair,1 = 15°C, h = 15 W/m2 -K

linoleum, kL = 0.05 W/m-K plywood, kp = 0.4 W/m-K

strip heater thp = 5 mm

189

thL = 5 mm

ps = 25 cm

thd = 9 mm

ths = 6 cm Ws = 4 cm studs, ks = 0.4 W/m-K drywall, k d = 0.1 W/m-K air, ka = 0.025 W/m-K

Tair, 2 = 5°C, h = 15 W/m2 -K

Figure P1-5: Bathroom ﬂoor with heater.

e.) Discuss at least one technique that could be used to substantially reduce the amount of heater power required while still maintaining the ﬂoor at 20◦ C. Note that you have no control over T air,1 or h. 1–6 You are a fan of ice ﬁshing but don’t enjoy the process of augering out your ﬁshing hole in the ice. Therefore, you want to build a device, the super ice-auger, that melts a hole in the ice. The device is shown in Figure P1-6.

h = 50 W/m 2 -K T∞ = 5 ° C

ε = 0.9 insulation, kins = 2.2 W/m-K

heater, activated with V = 12 V and I = 150 A plate, kp = 10 W/m-K D = 10 inch

thp = 0.75 inch

thins = 0.5 inch thice = 5 inch ρice = 920 kg/m3 Δifus = 333.6 kJ/kg

Figure P1-6: The super ice-auger.

A heater is attached to the back of a D = 10 inch plate and electrically activated by your truck battery, which is capable of providing V = 12 V and I = 150 A. The plate is thp = 0.75 inch thick and has conductivity k p = 10 W/m-K. The back of the heater is insulated; the thickness of the insulation is thins = 0.5 inch and the insulation has conductivity kins = 2.2 W/m-K. The surface of the insulation experiences convection with surrounding air at T ∞ = 5◦ C and radiation with surroundings also at T ∞ = 5◦ C. The emissivity of the surface of the insulation is ε = 0.9 and the heat transfer coefﬁcient between the surface and the air is h = 50 W/m2 -K. The super ice-auger is placed on the ice and activated, causing a heat transfer to the plate-ice interface that melts the ice. Assume that the water under the ice is at T ice = 0◦ C so that no heat is conducted away from the plate-ice interface; all of the energy transferred to the plate-ice interface goes into melting the ice. The thickness of the ice is

190

One-Dimensional, Steady-State Conduction

thice = 5 inch and the ice has density ρice = 920 kg/m3 . The latent heat of fusion for the ice is if us = 333.6 kJ/kg. a.) Determine the heat transfer rate to the plate-ice interface. b.) How long will it take to melt a hole in the ice? c.) What is the efﬁciency of the melting process? d.) If your battery is rated at 100 amp-hr at 12 V then what fraction of the battery’s charge is depleted by running the super ice-auger. Steady-State 1-D Conduction with Generation 1–7 One of the engineers that you supervise has been asked to simulate the heat transfer problem shown in Figure P1-7(a). This is a 1-D, plane wall problem (i.e., the temperature varies only in the x-direction and the area for conduction is constant with x). Material A (from 0 < x < L) has conductivity kA and experiences a uniform rate of volumetric thermal energy generation, g˙ . The left side of material A (at x = 0) is completely insulated. Material B (from L < x < 2L) has lower conductivity, kB < kA. The right side of material B (at x = 2L) experiences convection with ﬂuid at room temperature (20◦ C). Based on the facts above, critically examine the solution that has been provided to you by the engineer and is shown in Figure P1-7(b). There should be a few characteristics of the solution that do not agree with your knowledge of heat transfer; list as many of these characteristics as you can identify and provide a clear reason why you think the engineer’s solution must be wrong. 250

L

L

material A

material B kB < kA g⋅ ′′′= 0

kA = g⋅′′′ g⋅′′′ A

B

x

(a)

Temperature (°C)

200

h , Tf = 20°C

150 100 50 0 -50 Material A -100

0

Material B L Position (m)

2L

(b)

Figure P1-7 (a): Heat transfer problem and (b) “solution” provided by the engineer.

1–8 Freshly cut hay is not really dead; chemical reactions continue in the plant cells and therefore a small amount of heat is released within the hay bale. This is an example of the conversion of chemical to thermal energy. The amount of thermal energy generation within a hay bale depends on the moisture content of the hay when it is baled. Baled hay can become a ﬁre hazard if the rate of volumetric generation is sufﬁciently high and the hay bale sufﬁciently large so that the interior temperature of the bale reaches 170◦ F, the temperature at which self-ignition can occur. Here, we will model a round hay bale that is wrapped in plastic to protect it from the rain. You may assume that the bale is at steady state and is sufﬁciently long that it can be treated as a one-dimensional, radial conduction problem. The radius of the hay bale is Rbale = 5 ft and the bale is wrapped in plastic that is t p = 0.045 inch thick with conductivity k p = 0.15 W/m-K. The bale is surrounded by air at T ∞ = 20◦ C with h = 10 W/m2 -K. You may neglect radiation. The conductivity of the hay is k = 0.04 W/m-K.

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191

a.) If the volumetric rate of thermal energy generation is constant and equal to g˙ = 2 W/m3 then determine the maximum temperature in the hay bale. b.) Prepare a plot showing the maximum temperature in the hay bale as a function of the hay bale radius. How large can the hay bale be before there is a problem with self-ignition? Prepare a model that can consider temperature-dependent volumetric generation. Increasing temperature tends to increase the rate of chemical reaction and therefore increases the rate of generation of thermal energy according to: g˙ = a + b T where a = −1 W/m3 and b = 0.01 W/m3 -K and T is in K. c.) Enter the governing equation into Maple and obtain the general solution (i.e., a solution that includes two constants). d.) Use the boundary conditions to obtain values for the two constants in your general solution. (hint: one of the two constants must be zero in order to keep the temperature at the center of the hay bale ﬁnite). You should obtain a symbolic expression for the boundary condition in Maple that can be evaluated in EES. e.) Overlay on your plot from part (b) a plot of the maximum temperature in the hay bale as a function of bale radius when the volumetric generation is a function of temperature. 1–9 Figure P1-9 illustrates a simple mass ﬂow meter for use in an industrial reﬁnery. T∞ = 20°C hout = 20 W/m2 -K rout = 1 inch rin = 0.75 inch

insulation kins = 1.5 W/m-K test section g⋅ ′′′= 1x107 W/m3 k = 10 W/m-K m⋅ = 0.75kg/s Tf = 18°C

L = 3 inch

thins = 0.25 inch

Figure P1-9: A simple mass ﬂow meter.

A ﬂow of liquid passes through a test section consisting of an L = 3 inch section of pipe with inner and outer radii, rin = 0.75 inch and rout = 1.0 inch, respectively. The test section is uniformly heated by electrical dissipation at a rate g˙ = 1×107 W/m3 and has conductivity k = 10 W/m-K. The pipe is surrounded with insulation that is thins = 0.25 inch thick and has conductivity kins = 1.5 W/m-K. The external surface of the insulation experiences convection with air at T ∞ = 20◦ C. The heat transfer coefﬁcient on the external surface is hout = 20 W/m2 -K. A thermocouple is embedded at the center of the pipe wall. By measuring the temperature of the thermocouple, it is possible to infer the mass ﬂow rate of ﬂuid because the heat transfer coefﬁcient on the inner surface of the pipe (hin ) is strongly related to mass ﬂow rate (m). ˙ Testing has shown that the heat transfer coefﬁcient and mass ﬂow rate are related according to: hin = C

m ˙ 1 [kg/s]

0.8

where C = 2500 W/m2 -K. Under nominal conditions, the mass ﬂow rate through the meter is m ˙ = 0.75 kg/s and the ﬂuid temperature is T f = 18◦ C. Assume that the

192

One-Dimensional, Steady-State Conduction

ends of the test section are insulated so that the problem is 1-D. Neglect radiation and assume that the problem is steady state. a.) Develop an analytical model in EES that can predict the temperature distribution in the test section. Plot the temperature as a function of radial position for the nominal conditions. b.) Using your model, develop a calibration curve for the meter; that is, prepare a plot of the mass ﬂow rate as a function of the measured temperature at the mid-point of the pipe. The range of the instrument is 0.2 kg/s to 2.0 kg/s. The meter must be robust to changes in the ﬂuid temperature. That is, the calibration curve developed in (b) must continue to be valid even as the ﬂuid temperature changes by as much as 10◦ C. c.) Overlay on your plot from (b) the mass ﬂow rate as a function of the measured temperature for T f = 8◦ C and T f = 28◦ C. Is your meter robust to changes in Tf ? In order to improve the meters ability to operate over a range of ﬂuid temperature, a temperature sensor is installed in the ﬂuid in order to measure Tf during operation. d.) Using your model, develop a calibration curve for the meter in terms of the mass ﬂow rate as a function of T , the difference between the measured temperatures at the mid-point of the pipe wall and the ﬂuid. e.) Overlay on your plot from (d) the mass ﬂow rate as a function of the difference between the measured temperatures at the mid-point of the pipe wall and the ﬂuid if the ﬂuid temperature is T f = 8◦ C and T f = 28◦ C. Is the meter robust to changes in Tf ? f.) If you can measure the temperature difference to within δ T = 1 K then what is the uncertainty in the mass ﬂow rate measurement? (Use your plot from part (d) to answer this question.) g.) Set the temperature difference to the value you calculated at the nominal conditions and allow EES to calculate the associated mass ﬂow rate. Now, select Uncertainty Propagation from the Calculate menu and specify that the mass ﬂow rate as the calculated variable while the temperature difference is the measured variable. Set the uncertainty in the temperature difference to 1 K and verify that EES obtains an answer that is approximately consistent with part (f). h.) The nice thing about using EES to determine the uncertainty is that it becomes easy to assess the impact of multiple sources of uncertainty. In addition to the uncertainty δ T , the constant C has relative uncertainty of δC = 5% and the conductivity of the material is only known to within δk = 3%. Use EES’ builtin uncertainty propagation to assess the resulting uncertainty in the mass ﬂow rate measurement. Which source of uncertainty is the most important? i.) The meter must be used in areas where the ambient temperature and heat transfer coefﬁcient may vary substantially. Prepare a plot showing the mass ﬂow rate predicted by your model for T = 50 K as a function of T ∞ for various values of hout . If the operating range of your meter must include −5◦ C < T ∞ < 35◦ C then use your plot to determine the range of hout that can be tolerated without substantial loss of accuracy. Numerical Solutions to Steady-State 1-D Conduction Problems using EES 1–10 Reconsider the mass ﬂow meter that was investigated in Problem 1-9. The conductivity of the material that is used to make the test section is not actually constant,

Chapter 1: One-Dimensional, Steady-State Conduction

193

as was assumed in Problem 1-9, but rather depends on temperature according to:

k = 10

W W + 0.035 (T − 300 [K]) m-K m-K2

a.) Develop a numerical model of the mass ﬂow meter using EES. Plot the temperature as a function of radial position for the conditions shown in Figure P1-9 with the temperature-dependent conductivity. b.) Verify that your numerical solution limits to the analytical solution from Problem 1-9 in the limit that the conductivity is constant. c.) What effect does the temperature dependent conductivity have on the calibration curve that you generated in part (d) of Problem 1-9. Numerical Solutions to Steady-State 1-D Conduction Problems using MATLAB 1–11 Reconsider Problem 1-8, but obtain a solution numerically using MATLAB. The description of the hay bale is provided in Problem 1-8. Prepare a model that can consider the effect of temperature on the volumetric generation. Increasing temperature tends to increase the rate of reaction and therefore increase the rate of generation of thermal energy; the volumetric rate of generation can be approximated by: g˙ = a + b T where a = −1 W/m3 and b = 0.01 W/m3 -K and T is in K. a.) Prepare a numerical model of the hay bale. Plot the temperature as a function of position within the hay bale. b.) Show that your model has numerically converged; that is, show some aspect of your solution as a function of the number of nodes and discuss an appropriate number of nodes to use. c.) Verify your numerical model by comparing your answer to an analytical solution in some, appropriate limit. The result of this step should be a plot that shows the temperature as a function of radius predicted by both your numerical solution and the analytical solution and demonstrates that they agree. 1–12 Reconsider the mass ﬂow meter that was investigated in Problem 1-9. Assume that the conductivity of the material that is used to make the test section is not actually constant, as was assumed in Problem 1-9, but rather depends on temperature according to:

k = 10

W W + 0.035 (T − 300 [K]) m-K m-K2

a.) Develop a numerical model of the mass ﬂow meter using MATLAB. Plot the temperature as a function of radial position for the conditions shown in Figure P1-9 with the temperature-dependent conductivity. b.) Verify that your numerical solution limits to the analytical solution from Problem 1-9 in the limit that the conductivity is constant.

194

One-Dimensional, Steady-State Conduction

Analytical Solutions for Constant Cross-Section Extended Surfaces 1–13 A resistance temperature detector (RTD) utilizes a material that has a resistivity that is a strong function of temperature. The temperature of the RTD is inferred by measuring its electrical resistance. Figure P1-13 shows an RTD that is mounted at the end of a metal rod and inserted into a pipe in order to measure the temperature of a ﬂowing liquid. The RTD is monitored by passing a known current through it and measuring the voltage across it. This process results in a constant amount of ohmic heating that may tend to cause the RTD temperature to rise relative to the temperature of the surrounding liquid; this effect is referred to as a selfheating error. Also, conduction from the wall of the pipe to the temperature sensor through the metal rod can result in a temperature difference between the RTD and the liquid; this effect is referred to as a mounting error.

Tw = 20°C

L = 5.0 cm

pipe

D = 0.5 mm h = 150 W/m2 -K

x

T∞ = 5.0°C

k = 10 W/m-K q⋅sh = 2.5 mW RTD

Figure P1-13: Temperature sensor mounted in a ﬂowing liquid.

The thermal energy generation associated with ohmic heating is q˙ sh = 2.5 mW. All of this ohmic heating is assumed to be transferred from the RTD into the end of the rod at x = L. The rod has a thermal conductivity k = 10 W/m-K, diameter D = 0.5 mm, and length L = 5.0 cm. The end of the rod that is connected to the pipe wall (at x = 0) is maintained at a temperature of T w = 20◦ C. The liquid is at a uniform temperature, T ∞ = 50◦ C and the heat transfer coefﬁcient between the liquid and the rod is h = 150 W/m2 -K. a.) Is it appropriate to treat the rod as an extended surface (i.e., can we assume that the temperature in the rod is a function only of x)? Justify your answer. b.) Develop an analytical model of the rod that will predict the temperature distribution in the rod and therefore the error in the temperature measurement; this error is the difference between the temperature at the tip of the rod and the liquid. c.) Prepare a plot of the temperature as a function of position and compute the temperature error. d.) Investigate the effect of thermal conductivity on the temperature measurement error. Identify the optimal thermal conductivity and explain why an optimal thermal conductivity exists. 1–14 Your company has developed a micro-end milling process that allows you to easily fabricate an array of very small ﬁns in order to make heat sinks for various types of electrical equipment. The end milling process removes material in order to generate the array of ﬁns. Your initial design is the array of pin ﬁns shown in Figure P1-14. You have been asked to optimize the design of the ﬁn array for a particular application where the base temperature is T base = 120◦ C and the air temperature is T air = 20◦ C. The heat sink is square; the size of the heat sink is W = 10 cm.

Chapter 1: One-Dimensional, Steady-State Conduction

195

The conductivity of the material is k = 70 W/m-K. The distance between the edges of two adjacent ﬁns is a, the diameter of a ﬁn is D, and the length of each ﬁn is L.

array of fins k = 70 W/m-K

Tair = 20°C, h D a L

W = 10 cm

Tbase = 120°C

Figure P1-14: Pin ﬁn array.

Air is forced to ﬂow through the heat sink by a fan. The heat transfer coefﬁcient between the air and the surface of the ﬁns as well as the unﬁnned region of the base, h, has been measured for the particular fan that you plan to use and can be calculated according to: h = 40

W m2 K

a 0.005 [m]

0.4

D 0.01 [m]

−0.3

Mass is not a concern for this heat sink; you are only interested in maximizing the heat transfer rate from the heat sink to the air given the operating temperatures. Therefore, you will want to make the ﬁns as long as possible. However, in order to use the micro-end milling process you cannot allow the ﬁns to be longer than 10x the distance between two adjacent ﬁns. That is, the length of the ﬁns may be computed according to: L = 10 a. You must choose the optimal values of a and D for this application. a.) Prepare a model using EES that can predict the heat transfer coefﬁcient for a given value of a and D. Use this model to predict the heat transfer rate from the heat sink for a = 0.5 cm and D = 0.75 cm. b.) Prepare a plot that shows the heat transfer rate from the heat sink as a function of the distance between adjacent ﬁns, a, for a ﬁxed value of D = 0.75 cm. Be sure that the ﬁn length is calculated using L = 10 a. Your plot should exhibit a maximum value, indicating that there is an optimal value of a. c.) Prepare a plot that shows the heat transfer rate from the heat sink as a function of the diameter of the ﬁns, D, for a ﬁxed value of a = 0.5 cm. Be sure that the ﬁn length is calculated using L = 10 a. Your plot should exhibit a maximum value, indicating that there is an optimal value of D. d.) Determine the optimal values of a and D using EES’ built-in optimization capability. Analytical Solutions for Advanced Constant Cross-Section Extended Surfaces 1–15 Figure P1-15 illustrates a material processing system.

196

One-Dimensional, Steady-State Conduction oven wall temperature varies with x gap filled with gas th = 0.6 mm kg = 0.03 W/m-K

u = 0.75 m/s Tin = 300 K

D = 5 cm x extruded material k = 40 W/m-K α = 0.001 m2 /s Figure P1-15: Material processing system.

Material is extruded and enters the oven at T in = 300 K with velocity u = 0.75 m/s. The material has diameter D = 5 cm. The conductivity of the material is k = 40 W/m-K and the thermal diffusivity is α = 0.001 m2 /s. In order to precisely control the temperature of the material, the oven wall is placed very close to the outer diameter of the extruded material and the oven wall temperature distribution is carefully controlled. The gap between the oven wall and the material is th = 0.6 mm and the oven-to-material gap is ﬁlled with gas that has conductivity kg = 0.03 W/m-K. Radiation can be neglected in favor of convection through the gas from the oven wall to the material. For this situation, the heat ﬂux experienced by the material surface can be approximately modeled according to: kg q˙ conv ≈ (T w − T ) th where T w and T are the oven wall and material temperatures at that position, respectively. The oven wall temperature varies with position x according to: x T w = T f − (T f − T w,0 ) exp − Lc where T w,0 is the temperature of the wall at the inlet (at x = 0), Tf = 1000 K is the temperature of the wall far from the inlet, and Lc is a characteristic length that dictates how quickly the oven wall temperature approaches Tf . Initially, assume that T w,0 = 500 K, T f = 1000 K, and Lc = 1 m. Assume that the oven can be approximated as being inﬁnitely long. a.) Is an extended surface model appropriate for this problem? b.) Assume that your answer to (a) was yes. Develop an analytical solution that can be used to predict the temperature of the material as a function of x. c.) Plot the temperature of the material and the temperature of the wall as a function of position for 0 < x < 20 m. Plot the temperature gradient experienced by the material as a function of position for 0 < x < 20 m. The parameter Lc can be controlled in order to control the maximum temperature gradient and therefore the thermal stress experienced by the material as it moves through the oven. d.) Prepare a plot showing the maximum temperature gradient as a function of Lc . Overlay on your plot the distance required to heat the material to T p = 800 K (L p). If the maximum temperature gradient that is allowed is 60 K/m, then what is the appropriate value of Lc and the corresponding value of Lp ?

Chapter 1: One-Dimensional, Steady-State Conduction

197

1–16 The receiver tube of a concentrating solar collector is shown in Figure P1-16. q⋅ ′′s

Ta = 25°C ha = 25 W/m2 -K r = 5 cm th = 2.5 mm k = 10 W/m-K

φ

Tw = 80°C hw = 100 W/m2 -K

Figure P1-16: A solar collector.

The receiver tube is exposed to solar radiation that has been reﬂected from a concentrating mirror. The heat ﬂux received by the tube is related to the position of the sun and the geometry and efﬁciency of the concentrating mirrors. For this problem, you may assume that all of the radiation heat ﬂux is absorbed by the collector and neglect the radiation emitted by the collector to its surroundings. The ﬂux received at the collector surface (q˙ s ) is not circumferentially uniform but rather varies with angular position; the ﬂux is uniform along the top of the collector, π < φ < 2π rad, and varies sinusoidally along the bottom, 0 < φ < π rad, with a peak at φ = π/2 rad. q˙ t + q˙ p − q˙ t sin (φ) for 0 < φ < π q˙ s (φ) = for π < φ < 2 π q˙ t where q˙ t = 1000 W/m2 is the uniform heat ﬂux along the top of the collector tube and q˙ p = 5000 W/m2 is the peak heat ﬂux along the bottom. The receiver tube has an inner radius of r = 5.0 cm and thickness of th = 2.5 mm (because th/r 1 it is possible to ignore the small difference in convection area on the inner and outer surfaces of the tube). The thermal conductivity of the tube material is k = 10 W/m-K. The solar collector is used to heat water, which is at T w = 80◦ C at the axial position of interest. The average heat transfer coefﬁcient between the water and the internal surface of the collector is hw = 100 W/m2 -K. The external surface of the collector is exposed to air at T a = 25◦ C. The average heat transfer coefﬁcient between the air and the external surface of the collector is ha = 25 W/m2 -K. a.) Can the collector be treated as an extended surface for this problem (i.e., can the temperature gradients in the radial direction in the collector material be neglected)? b.) Develop an analytical model that will allow the temperature distribution in the collector wall to be determined as a function of circumferential position. Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 1–17 Figure P1-17 illustrates a disk brake for a rotating machine. The temperature distribution within the brake can be assumed to be a function of radius only. The brake is divided into two regions. In the outer region, from R p = 3.0 cm to Rd = 4.0 cm, the stationary brake pads create frictional heating and the disk is not exposed to convection. The clamping pressure applied to the pads is P = 1.0 MPa and the coefﬁcient of friction between the pad and the disk is μ = 0.15. You may

198

One-Dimensional, Steady-State Conduction

assume that the pads are not conductive and therefore all of the frictional heating is conducted into the disk. The disk rotates at N = 3600 rev/min and is b = 5.0 mm thick. The conductivity of the disk is k = 75 W/m-K and you may assume that the outer rim of the disk is adiabatic. coefficient of friction, μ = 0.15

stationary brake pads

clamping pressure P = 1 MPa Ta = 30°C, h

b = 5 mm

Rp = 3 cm

Rd = 4 cm

center line k = 75 W/m-K disk, rotates at N = 3600 rev/min Figure P1-17: Disk brake.

The inner region of the disk, from 0 to Rp , is exposed to air at T a = 30◦ C. The heat transfer coefﬁcient between the air and disk surface depends on the angular velocity of the disk, ω, according to: h = 20

1.25 W ω W + 1500 m2 -K m2 -K 100 [rad/s]

a.) Develop an analytical model of the temperature distribution in the disk brake; prepare a plot of the temperature as a function of radius for r = 0 to r = Rd . b.) If the disk material can withstand a maximum safe operating temperature of 750◦ C then what is the maximum allowable clamping pressure that can be applied? Plot the temperature distribution in the disk at this clamping pressure. What is the braking torque that results? c.) Assume that you can control the clamping pressure so that as the machine slows down the maximum temperature is always kept at the maximum allowable temperature, 750◦ C. Plot the torque as a function of rotational speed for 100 rev/min to 3600 rev/min. 1–18 Figure P1-18 illustrates a ﬁn that is to be used in the evaporator of a space conditioning system for a space-craft. The ﬁn is a plate with a triangular shape. The thickness of the plate is th = 1 mm and the width of the ﬁn at the base is Wb = 1 cm. The length of the ﬁn is L = 2 cm. The ﬁn material has conductivity k = 50 W/m-K. The average heat transfer coefﬁcient between the ﬁn surface and the air in the space-craft is h = 120 W/m2 -K. The air is at T ∞ = 20◦ C and the base of the ﬁn is at T b = 10◦ C. Assume that the temperature distribution in the ﬁn is 1-D in x. Neglect convection from the edges of the ﬁn. a.) Obtain an analytical solution for the temperature distribution in the ﬁn. Plot the temperature as a function of position. b.) Calculate the rate of heat transfer to the ﬁn. c.) Determine the ﬁn efﬁciency.

Chapter 1: One-Dimensional, Steady-State Conduction h = 120 W/m2 -K T∞ = 20°C

199

th = 1 mm

x

L = 2 cm

k = 50 W/m-K ρ= 3000 kg/m3

ρb = 8000 kg/m3

Tb = 10°C thg = 2 mm

thb = 2 mm

Wb = 1 cm Figure P1-18: Fin on an evaporator.

The ﬁn has density ρ = 3000 kg/m3 and is installed on a base material with thickness thb = 2 mm and density ρb = 8000 kg/m3 . The half-width of the gap between adjacent ﬁns is thg = 2 mm. Therefore, the volume of the base material associated with each ﬁn is thbWb(th + 2thg ). d.) Determine the ratio of the absolute value of the rate of heat transfer to the ﬁn to the total mass of material (ﬁn and base material associated with the ﬁn). e.) Prepare a contour plot that shows the ratio of the heat transfer to the ﬁn to the total mass of material as a function of the length of the ﬁn (L) and the ﬁn thickness (th). f.) What is the optimal value of L and th that maximizes the absolute value of the ﬁn heat transfer rate to the mass of material? Numerical Solution of Extended Surface Problems 1–19 A ﬁber optic bundle (FOB) is shown in Figure P1-19 and used to transmit the light for a building application. h = 5 W/m2 -K T∞ = 20°C

rout = 2 cm

q⋅ ′′ = 1x105 W/m 2

x

fiber optic bundle

Figure P1-19: Fiber optic bundle used to transmit light.

The ﬁber optic bundle is composed of several, small-diameter ﬁbers that are each coated with a thin layer of polymer cladding and packed in approximately a hexagonal close-packed array. The porosity of the FOB is the ratio of the open area of the FOB face to its total area. The porosity of the FOB face is an important characteristic because any radiation that does not fall directly upon the ﬁbers will not be transmitted and instead contributes to a thermal load on the FOB. The ﬁbers are designed so that any radiation that strikes the face of a ﬁber is “trapped” by total internal reﬂection. However, radiation that strikes the interstitial areas between the ﬁbers will instead be absorbed in the cladding very close to the FOB face. The volumetric generation of thermal energy associated with this radiation can be represented by: φ q˙ x exp − g˙ = Lch Lch

200

One-Dimensional, Steady-State Conduction

where q˙ = 1 × 105 W/m2 is the energy ﬂux incident on the face, φ = 0.05 is the porosity of the FOB, x is the distance from the face, and Lch = 0.025 m is the characteristic length for absorption of the energy. The outer radius of the FOB is rout = 2 cm. The face of the FOB as well as its outer surface are exposed to air at T ∞ = 20◦ C with heat transfer coefﬁcient h = 5 W/m2 -K. The FOB is a composite structure and therefore conduction through the FOB is a complicated problem involving conduction through several different media. Section 2.9 discusses methods for computing the effective thermal conductivity for a composite. The effective thermal conductivity of the FOB in the radial direction is keff ,r = 2.7 W/m-K. In order to control the temperature of the FOB near the face, where the volumetric generation of thermal energy is largest, it has been suggested that high conductivity ﬁller material be inserted in the interstitial regions between the ﬁbers. The result of the ﬁller material is that the effective conductivity of the FOB in the axial direction varies with position according to: x keff ,x = keff ,x,∞ + keff ,x exp − Lk where keff ,x,∞ = 2.0 W/m-K is the effective conductivity of the FOB in the xdirection without ﬁller material, keff ,x = 28 W/m-K is the augmentation of the conductivity near the face, and Lk = 0.05 m is the characteristic length over which the effect of the ﬁller material decays. The length of the FOB is effectively inﬁnite. Assume that the volumetric generation is unaffected by the ﬁller material. a.) Is it appropriate to use a 1-D model of the FOB? b.) Assume that your answer to (a) was yes. Develop a numerical model of the FOB. c.) Overlay on a single plot the temperature distribution within the FOB for the case where the ﬁller material is present ( keff ,x = 28 W/m-K) and the case where no ﬁller material is present ( keff ,x = 0). 1–20 An expensive power electronics module normally receives only a moderate current. However, under certain conditions it might experience currents in excess of 100 amps. The module cannot survive such a high current and therefore, you have been asked to design a fuse that will protect the module by limiting the current that it can experience, as shown in Figure P1-20. L = 2.5 cm ε = 0.9 Tend = 20°C

Tend = 20°C D = 0.9 mm T∞ = 20°C h = 5 W/m2 -K

k = 150 W/m-K ρr = 1x10 -7 ohm-m I = 100 amp

Figure P1-20: A fuse that protects a power electronics module from high current.

The space available for the fuse allows a wire that is L = 2.5 cm long to be placed between the module and the surrounding structure. The surface of the fuse wire is exposed to air at T ∞ = 20◦ C. The heat transfer coefﬁcient between the surface of the fuse and the air is h = 5.0 W/m2 -K. The fuse surface has an emissivity of ε = 0.90. The fuse is made of an aluminum alloy with conductivity

One-Dimensional, Steady-State Conduction

201

k = 150 W/m-K. The electrical resistivity of the aluminum alloy is ρe = 1 × 10−7 ohm-m and the alloy melts at approximately 500◦ C. Assume that the properties of the alloy do not depend on temperature. The ends of the fuse (i.e., at x = 0 and x = L) are maintained at T end = 20◦ C by contact with the surrounding structure and the module. The current passing through the fuse, I, results in a uniform volumetric generation within the fuse material. If the fuse operates properly, then it will melt (i.e., at some location within the fuse, the temperature will exceed 500◦ C) when the current reaches 100 amp. Your job will be to select the fuse diameter; to get your model started, you may assume a diameter of D = 0.9 mm. Assume that the volumetric rate of thermal energy generation due to ohmic dissipation is uniform throughout the fuse volume. a.) Prepare a numerical model of the fuse that can predict the steady-state temperature distribution within the fuse material. Plot the temperature as a function of position within the wire when the current is 100 amp and the diameter is 0.9 mm. b.) Verify that your model has numerically converged by plotting the maximum temperature in the wire as a function of the number of nodes in your model. c.) Prepare a plot of the maximum temperature in the wire as a function of the diameter of the wire for I = 100 amp. Use your plot to select an appropriate fuse diameter.

REFERENCES

Cercignani, C., Rareﬁed Gas Dynamics: From Basic Concepts to Actual Calculations, Cambridge University Press, Cambridge, U.K., (2000). Chen, G., Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons, Oxford University Press, Oxford, U.K., (2005). Flynn, T. M., Cryogenic Engineering, 2nd Edition, Revised and Expanded, Marcel Dekker, New York, (2005). Fried, E., Thermal Conduction Contribution to Heat Transfer at Contacts, in Thermal Conductivity, Volume 2, R. P. Tye, ed., Academic Press, London, (1969). Iwasa, Y., Case Studies in Superconducting Magnets: Design and Operational Issues, Plenum Press, New York, (1994). Izzo, F., “Other Thermal Ablation Techniques: Microwave and Interstitial Laser Ablation of Liver Tumors,” Annals of Surgical Oncology, Vol. 10, pp. 491–497, (2003). Keenan, J. H., “Adventures in Science,” Mechanical Engineering, May, p. 79, (1958). NIST Standard Reference Database Number 69, http://webbook.nist.gov/chemistry, June 2005 Release, (2005). Que, L., Micromachined Sensors and Actuators Based on Bent-Beam Suspensions, Ph.D. Thesis, Electrical and Computer Engineering Dept., University of Wisconsin-Madison, (2000). Schneider, P. J., Conduction, in Handbook of Heat Transfer, 2nd Edition, W.M. Rohsenow et al., eds., McGraw-Hill, New York, (1985). Tien, C.-L., A. Majumdar, and F. M. Gerner, eds., Microscale Energy Transport, Taylor & Francis, Washington (1998). Tompkins, D. T., A Finite Element Heat Transfer Model of Ferromagnetic Thermoseeds and a Physiologically-Based Objective Function for Pretreatment Planning of Ferromagnetic Hypothermia, Ph.D. Thesis, Mechanical Engineering, University of Wisconsin at Madison, (1992). Walton, A. J., Three Phases of Matter, Oxford University Press, Oxford, U.K., (1989).

2

Two-Dimensional, Steady-State Conduction

Chapter 1 discussed the analytical and numerical solution of 1-D, steady-state problems. These are problems where the temperature within the material is independent of time and varies in only one spatial dimension (e.g., x). Examples of such problems are the plane wall studied in Section 1.2, which is truly a 1-D problem, and the constant cross section ﬁn studied in Section 1.6, which is approximately 1-D. The governing differential equation for these problems is an ordinary differential equation and the mathematics required to solve the problem are straightforward. In this chapter, more complex, 2-D steady-state conduction problems are considered where the temperature varies in multiple spatial dimensions (e.g., x and y). These can be problems where the temperature actually varies in only two coordinates or approximately varies in only two coordinates (e.g., the temperature gradient in the third direction is negligible, as justiﬁed by an appropriate Biot number). The governing differential equation is a partial differential equation and therefore the mathematics required to analytically solve these problems are more advanced and the bookkeeping required to solve these problems numerically is more cumbersome. However, many of the concepts that were covered in the context of 1-D problems continue to apply.

2.1 Shape Factors There are many 2-D and 3-D conduction problems involving heat transfer between two well-deﬁned surfaces (surface 1 and surface 2) that commonly appear in heat transfer applications and have previously been solved analytically and/or numerically. The solution to these problems is conveniently expressed in the form of a shape factor, S, which is deﬁned as: S=

1 kR

(2-1)

where k is the conductivity of the material separating the surfaces and R is the thermal resistance between surfaces 1 and 2. Solving Eq. (2-1) for the thermal resistance leads to: R=

1 kS

(2-2)

Recall that the resistance of a plane wall, derived in Section 1.2, is given by: R pw =

L k Ac

(2-3)

where L is the length of conduction path and Ac is the area for conduction. Comparing Eqs. (2-2) and (2-3) suggests that: S≈ 202

Ac L

(2-4)

2.1 Shape Factors

203 T2 q⋅ W

Figure 2-1: Sphere buried in a semi-inﬁnite medium.

T1

D

The shape factor has units of length and represents the ratio of the effective area for conduction to the effective length for conduction. Any shape factor solution should be checked against your intuition using Eq. (2-4). Given a problem, it should be possible to approximately identify the area and length that characterize the conduction process; the ratio of these quantities should have the same order of magnitude as the shape factor solution. One example of a shape factor solution is for a sphere buried in a semi-inﬁnite medium (i.e., a medium that extends forever in one direction but is bounded in the other) as shown in Figure 2-1. In this case, the surface of the sphere is surface 1 (assumed to be isothermal, at T1 ) while the surface of the medium is surface 2 (assumed to be isothermal, at T2 ). The shape factor solution for a completely buried sphere in a semi-inﬁnite medium is: S=

2πD D 1− 4W

(2-5)

where D is the diameter of the sphere and W is the distance between the center of the sphere and the surface. The thermal resistance characterizing conduction between the surface of the sphere and the surface of the medium is: R=

1 = kS

D 4W 2πDk

1−

(2-6)

The rate of conductive heat transfer between the sphere and the surface (q) ˙ is: q˙ =

T1 − T2 2 π D k (T 1 − T 2 ) = D R 1− 4W

(2-7)

There are numerous formulae for shape factors that have been tabulated in various references; for example, Rohsenow et al. (1998). Table 2-1 summarizes a few shape factor solutions. A library of shape factors, including those shown in Table 2-1 as well as others, has been integrated with EES. To access the shape factor library, select Function Information from the Options menu and then scroll through the list to Conduction Shape Factors, as shown in Figure 2-2. The shape factor functions that are available can be selected by moving the scroll bar below the picture.

Table 2-1: Shape factors. Buried beam (L a,b)

Buried sphere

Circular extrusion with off-center hole (L Dout )

T2

T2

L T1 W

W

W

T2

L

Din Dout

a T1

D

Buried cylinder (L D)

Parallel cylinders (L D1 , D2 ) T1

−1

cosh

W −0 59 W −0.078 S = 2.756 L ln 1 + a b

2πD D 1− 4W

S=

S=

T1

b

T2

T2 W

D1

W

S= cosh−1

2πL 4 W 2 − D21 − D22 2 D1 D2

W

T1

L

Square extrusion with a centered circular hole (L W)

L

D

S=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

L D

2πL −1

T1

W

if W ≤ 3 D/2

(2 W/D) 2πL if W > 3 D/2 ln (4 W/D) cosh

Disk on surface of semi-inﬁnite body

T2

W −0 59 W −0.078 S = 2.756 L ln 1 + a b Square extrusion (L a) T1 T2

L

T1

L T2 W

T1

2πL D2out + D2in − 4 W 2 2 Dout Din

Cylinder half-way between parallel plates (W D & L W)

T2

D2

T2

D

a

b b

D

S = 2D

a

W

2πL S= ln (1.08 W/D)

Figure 2-2: Accessing the shape factor library from EES.

S=

⎧ ⎪ ⎪ ⎨

2πL 0.785 ln (a/b)

⎪ ⎪ ⎩

2πL if a/b > 0.25 [0.93 ln (a/b) − 0.0502]

if a/b ≤ 0.25

205

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT This example revisits the magnetic ablation concept that was previously described in EXAMPLES 1.3-1 and 1.8-2. You want to measure the power generated by the thermoseed that is used for the ablation process. The radius of the thermoseed is rts = 1.0 mm. The sphere is placed W = 5 cm below the surface of a solution of agar, as shown in Figure 1. Agar is a material with well-known thermal properties that resembles gelatin and is sometimes used as a surrogate for tissue in biological experiments. The agar is allowed to solidify around the sphere and the container of agar is large enough to be considered semi-inﬁnite. The surface of the agar is exposed to an ice-water bath in order to keep it at a constant temperature, Tice = 0◦ C. The sphere is heated using an oscillating magnetic ﬁeld and its surface temperature is measured using a thermocouple. The conductivity of agar is k = 0.35 W/m-K. Tice = 0°C agar, k = 0.35 W/m-K W = 5.0 cm Figure 1: Test setup to measure the power generated by the thermoseed.

Ts = 95°C

rts = 1.0 mm

a) If the measured surface temperature of the sphere is Ts = 95◦ C, how much energy is generated in the thermoseed? The inputs are entered in EES: “EXAMPLE 2.1-1: Magnetic Ablative Power Measurement” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in

“Inputs” r ts=1.0 [mm]∗ convert(mm,m) W=5 [cm]∗ convert(cm,m) k=0.35 [W/m-K] T ice=converttemp(C,K,0 [C]) T s=converttemp(C,K,95 [C])

“thermoseed radius” “depth of sphere” “conductivity of agar” “ice bath temperature” “surface temperature”

The shape factor associated with the buried sphere (S) is accessed from the EES library of shape factors: S=SF 1(2∗ r ts,W)

“shape factor for buried sphere”

The thermal resistance between the surface of the sphere and the semi-inﬁnite body (R) is: R=

1 kS

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

2.1 Shape Factors

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

206

Two-Dimensional, Steady-State Conduction

˙ is computed using the thermal resistance and the known The heat transfer rate (q) temperatures: q˙ =

R=1/(k∗ S) q dot=(T s-T ice)/R

Ts − Tice R

“thermal resistance” “heat transfer rate”

which leads to a generation rate of 0.422 W. b) Estimate the uncertainty in your measurement of the power. Assume that your temperature measurements are accurate to δT = 1.0◦ C, the conductivity of agar is known to within 10% (i.e., δk = 0.035 W/m-K), the depth measurement has an uncertainty of δW = 2.0 mm, and the sphere radius is known to within δrts = 0.1 mm. It is possible to separately estimate the uncertainty introduced by each of the parameters listed above. For example, to evaluate the effect of the uncertainty in the conductivity, simply increase the conductivity by δk: deltak=0.035 [W/m-K] k=0.35 [W/m-K]+deltak

“uncertainty in conductivity” “conductivity of agar”

and re-run the model. The result is a change in the generation rate from 0.422 W to 0.464 W which translates into an uncertainty in the power of 0.042 W or 10%. This process can be repeated for each of the independent variables in order to identify the uncertainty that is introduced into the dependent variable calculation. These contributions should be combined using the root-sum-square technique in order to obtain an overall uncertainty. This process can be carried out automatically in EES; select Uncertainty Propagation from the Calculate Menu to access the dialog shown in Figure 2.

Figure 2: Propagation of uncertainty dialog.

The possible dependent (calculated) and independent (measured) variables are listed; highlight the independent variables that have some uncertainty (all of them for this problem). The calculated variable that you want to examine is q_dot. Select

207

the Set uncertainties button to reach the window shown in Figure 3. Set each of the uncertainties either in absolute or relative terms; note that each of these quantities can also be set using variables that are deﬁned in the main equation window.

Figure 3: Uncertainty of measured variables dialog.

Select OK twice in order to initiate the calculations; the results appear the Uncertainty Results window (Figure 4) which shows the total uncertainty in the variable q_dot (0.06 W or 14%) as well as a delineation of the source of the uncertainty.

[W] [W/m-K] [m] [K] [K] [m] Figure 4: Uncertainty of Results window.

Notice that the dominant sources of uncertainty for this problem are the conductivity of agar and the radius of the sphere (each contributing approximately 50% of the total). The temperature measurements are adequate and the depth does not matter at all since the shape factor becomes nearly independent of the depth provided W is much larger than the diameter (see Eq. (2-5)).

2.2 Separation of Variables Solutions 2.2.1 Introduction Two-dimensional steady-state conduction problems are governed by partial rather than ordinary differential equations; the analytical solution to partial differential equations

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

2.2 Separation of Variables Solutions

208

Two-Dimensional, Steady-State Conduction q⋅ y+ dy

q⋅ x dy q⋅ y Tx =0 = 0

q⋅ x + dx

Figure 2-3: Plate.

y x

Ty=0 = Tb (x )

Ty →∞ = 0

dx

th

W Tx =W = 0

is somewhat more involved. Separation of variables is a common technique that is used to solve the partial differential equations that arise in many areas of science and engineering. A complete understanding of separation of variables requires a substantial mathematical background; in this section, the technique is introduced and used to solve several problems. In the subsequent section, more difﬁcult problems are solved using separation of variables. It is not possible to cover separation of variables thoroughly in this book and the interested reader is referred to the textbook by G. E. Myers (1998).

2.2.2 Separation of Variables The method of separation of variables is most conveniently discussed in the context of a speciﬁc problem. In this section, the ﬂat plate shown in Figure 2-3 is considered. The top and bottom surfaces of the plate are insulated and therefore there is no temperature variation in the z-direction; the problem is truly two-dimensional, as the temperature depends on x and y but not z. If the top and bottom surfaces were not insulated (e.g., they experienced convection to a surrounding ﬂuid) but the plate was sufﬁciently thin and conductive, then it still might be possible to ignore temperature gradients in the zdirection and treat the problem as being two-dimensional. This assumption is equivalent to the extended surface assumption that was discussed in Section 1.6.2 and should be justiﬁed using an appropriately deﬁned Biot number. The plate in Figure 2-3 has conductivity k, thickness th, width (in the x-direction) W, and extends to inﬁnity in the y-direction. The governing differential equation for the problem is derived in a manner that is analogous to the 1-D problems that have been previously considered. A differential control volume is deﬁned (see Figure 2-3) and used to develop a steady-state energy balance. Note that the control volume must be differential in both the x- and y-directions because there are temperature gradients in both of these directions. q˙ x + q˙ y = q˙ x+dx + q˙ y+dy

(2-8)

The x + dx and y + dy terms are expanded as usual: q˙ x + q˙ y = q˙ x +

∂ q˙ y ∂ q˙ x dx + q˙ y + dy ∂x ∂y

(2-9)

Equation (2-9) can be simpliﬁed to: ∂ q˙ y ∂ q˙ x dx + dy = 0 ∂x ∂y

(2-10)

2.2 Separation of Variables Solutions

209

Fourier’s law is used to determine the conduction heat transfer rates in the x- and y-directions: q˙ x = −k th dy

∂T ∂x

(2-11)

q˙ y = −k th dx

∂T ∂y

(2-12)

Equations (2-11) and (2-12) are substituted into Eq. (2-10): ∂T ∂ ∂T ∂ −k th dy dx + −k th dx dy = 0 ∂x ∂x ∂y ∂y

(2-13)

If the thermal conductivity and plate thickness are both constant, then Eq. (2-13) can be simpliﬁed to: ∂2T ∂2T + 2 =0 2 ∂x ∂y

(2-14)

which is the governing partial differential equation for this problem. Equation (2-14) is called Laplace’s equation. Equation (2-14) is second order in both the x- and y-directions and therefore two boundary conditions are required in each of these directions. The left and right edges of the plate have a temperature of zero: T x=0 = 0

(2-15)

T x=W = 0

(2-16)

The zero temperature boundaries are necessary here to ensure that the boundary conditions are homogeneous, as explained below. However, these boundary conditions are likely not of general interest. Techniques that allow the solution of problems with more realistic boundary conditions are presented in subsequent sections. The edge of the plate at y = 0 has a speciﬁed temperature that is an arbitrary function of position x: T y=0 = T b(x)

(2-17)

The plate is inﬁnitely long in the y-direction and the temperature approaches 0 as y becomes inﬁnite: T y→∞ = 0

(2-18)

Requirements for using Separation of Variables The method of separation of variables will not work for every problem; there are some fairly restrictive conditions that limit where it can be applied. First, the governing equation must be linear; that is, the equation cannot contain any products of the dependent variable or its derivative. Equation (2-14) is certainly linear. An example of a non-linear equation might be: T

∂T ∂ 2 T ∂2T + =0 ∂x2 ∂x ∂y2

(2-19)

The governing equation must also be homogeneous, which is a more restrictive condition. If T is a solution to a homogeneous equation then C T is also a solution, where C

210

Two-Dimensional, Steady-State Conduction

is an arbitrary constant. Equation (2-14) is homogeneous; to prove this, simply check if C T can be substituted into the equation and still satisfy the equality: ∂ 2 (C T ) ∂ 2 (C T ) + =0 ∂x2 ∂y2 or

(2-20)

C

∂2T ∂2T + =0 ∂x2 ∂y2 =0 according to Eq. (2-14)

(2-21)

A non-homogeneous equation would result, for example, if the plate were exposed to a volumetric generation of thermal energy (g˙ ); the governing differential equation for this situation would be: ∂2T g˙ ∂2T + 2 + =0 2 ∂x ∂y k Substituting C T into Eq. (2-22) leads to: 2 ∂2T ∂ T + 2 C ∂x2 ∂y

+

g˙ =0 k

(2-22)

(2-23)

= − g˙k according to Eq. (2-22)

Substituting Eq. (2-22) into Eq. (2-23) leads to: g˙ g˙ C − + =0 k k

(2-24)

Equation (2-24) shows that C T is a solution to Eq. (2-22) only if g˙ = 0. We will show how some non-homogeneous problems can be solved using separation of variables in Section 2.3. In order to apply the separation of variables method, the boundary conditions must also be linear with respect to the dependent variable. Linear has the same deﬁnition for the boundary conditions that it does for the differential equation; i.e., the boundary condition cannot involve products of the dependent variable or its derivatives. The boundary conditions for the plate in Figure 2-3 are given by Eqs. (2-15) through (2-18) and are all linear. A non-linear boundary condition would result from, for example, radiation. If the right edge of the plate were radiating to surroundings at T = 0 then Eq. (2-15) should be replaced with: ∂T 4 = σ ε T x=W (2-25) −k ∂x x=W which is non-linear. Finally, both of the boundary conditions in one direction must be homogeneous (i.e., either both boundary conditions in the x-direction or both boundary conditions in the y-direction). Again, the meaning of homogeneity for a boundary condition is analogous to its meaning for the differential equation. If a boundary condition is homogeneous then any multiple of a solution also satisﬁes the boundary condition. Examination shows that all of the boundary conditions except for Eq. (2-17) are homogeneous. Therefore, both boundary conditions in the x-direction, Eqs. (2-15) and (2-16), are homogeneous. For this problem, x is therefore the homogeneous direction; this will be important to keep in mind as we solve the problem. A ﬁnal criterion for the use of separation of variables is that the computational domain must be simple; that is, it must have boundaries that lie along constant values of

2.2 Separation of Variables Solutions

211

the coordinate axes. For a Cartesian coordinate system, we are restricted to rectangular problems. The problem shown in Figure 2-3 meets all of the criteria and should therefore be solvable using separation of variables. Separate the Variables The name of the technique, separation of variables, is related to the next step in the solution; it is assumed that the solution T, which is a function of both x and y, can be expressed as the product of two functions, TX which is only a function of x and TY which is only a function of y: T (x, y) = TX (x) TY (y)

(2-26)

Substituting Eq. (2-26) into Eq. (2-14) leads to: ∂2 ∂2 [TX TY ] + 2 [TX TY ] = 0 2 ∂x ∂y

(2-27)

or TY

d2 TY d2 TX + TX =0 2 dx dy2

(2-28)

Dividing through by the product TY TX leads to: d2 TY d2 TX 2 dx2 + dy =0 TX TY function of x

(2-29)

function of y

The ﬁrst term in Eq. (2-29) is a function only of x while the second is a function only of y. Therefore, Eq. (2-29) can only be satisﬁed if both terms are equal and opposite and constant. To see this clearly, imagine moving along a line of constant y (i.e., across the plate in Figure 2-3 in the x-direction from one side to the other). If the ﬁrst term were not constant then, by deﬁnition, its value would change as x changes; however, the second term is not a function of x and therefore it cannot change in response. Clearly then the sum of the two terms could not continue to be zero in this situation. Equation (2-29) can be expressed as two statements: d2 TX dx2 = ± λ2 TX

(2-30)

d2 TY dy2 (2-31) = ∓ λ2 TY where λ2 is a constant that must be positive. Notice that there is a choice that must be made at this point. The TX group can either be set equal to a positive constant (λ2 ) or a negative constant (−λ2 ). Depending on this choice, the TY group must be set equal to a negative constant (−λ2 ) or a positive constant (λ2 ), in order to satisfy Eq. (2-29). The choice at this point seems arbitrary but in fact it is important. Recall that one condition for using separation of variables is that one of the coordinate directions must have homogeneous boundary conditions; this was referred to as the homogeneous direction. In this problem, the x-direction is the homogeneous direction because the boundary conditions at x = 0, Eq. (2-15), and at x = W, Eq. (2-16), are both homogeneous. It is necessary to choose the negative constant for the group associated

212

Two-Dimensional, Steady-State Conduction

with the homogeneous direction (i.e., for Eq. (2-30) in this problem). With this choice, rearranging Eqs. (2-30) and (2-31) leads to the two ordinary differential equations: d2 TX + λ2 TX = 0 dx2

(2-32)

d2 TY − λ2 TY = 0 dy2

(2-33)

We have effectively converted our partial differential equation, Eq. (2-14), into two ordinary differential equations, Eqs. (2-32) and (2-33). The solutions to Eqs. (2-32) and (2-33) can be identiﬁed using Maple: > restart; > ODEX:=diff(diff(TX(x),x),x)+lambdaˆ2∗ TX(x)=0; 2 d ODEX = TX(x) + λ2 TX(x) = 0 dx2 > Xs:=dsolve(ODEX); Xs = TX(x) = C1 sin(λx) + C2 cos(λx) > ODEY:=diff(diff(TY(y),y),y)-lambdaˆ2∗ TY(y)=0; 2 d ODEY := TY (Y ) − λ2 TY (Y ) = 0 dy2 > Ys:=dsolve(ODEY); Ys := TY (y) = C1e(−λY ) + C2e(λY )

So the solution for TX (i.e., the solution in the homogeneous direction) is: TX = C1 sin (λ x) + C2 cos (λ x)

(2-34)

where C1 and C2 are undetermined constants. The solution for TY (i.e., in the nonhomogeneous direction) is: TY = C3 exp (−λ y) + C4 exp (λ y)

(2-35)

where C3 and C4 are undetermined constants. Note that Eq. (2-35) could equivalently be expressed in terms of hyperbolic sines and cosines: TY = C3 sinh (λ y) + C4 cosh (λ y)

(2-36)

where C3 and C4 are undetermined constants (different from those in Eq. (2-35)). The choice of the negative value of the constant for the homogeneous direction (i.e., for Eq. (2-30)) has led directly to sine/cosine solutions in the homogeneous direction; this result is necessary in order to use separation of variables. Solve the Eigenproblem It is necessary to address the solution in the homogeneous direction (in this problem, TX) before moving on to the non-homogeneous direction. This portion of the problem is often called the eigenproblem and the solutions are referred to as eigenfunctions. The boundary conditions for TX can be obtained by revisiting the original boundary conditions for the problem in the x-direction using the assumed, separated form of the solution. Equation (2-15) becomes: TX x=0 TY = 0

(2-37)

2.2 Separation of Variables Solutions

213

which can only be true at an arbitrary location y if: TX x=0 = 0

(2-38)

The remaining boundary condition in the x-direction is given by Eq. (2-16) and leads to: TX x=W = 0

(2-39)

Substituting the solution to the ordinary differential equation in the homogeneous direction, Eq. (2-34), into Eq. (2-38) leads to: C1 sin (λ 0) + C2 cos (λ 0) = 0 0

(2-40)

1

or C2 = 0

(2-41)

TX = C1 sin (λ x)

(2-42)

So that:

Substituting Eq. (2-42) into Eq. (2-39) leads to: C1 sin (λ W) = 0

(2-43)

Equation (2-43) could be satisﬁed if C1 is 0, but that would lead to TX = 0 (and therefore T = 0) everywhere, which is not a useful solution. However, Eq. (2-43) is also satisﬁed whenever the sine function becomes zero; this occurs whenever the argument of sine is an integer multiple of π: λi W = i π

where i = 0, 1, 2, . . . ∞

(2-44)

Equation (2-43) satisﬁes the eigenproblem (i.e., the ordinary differential equation in the x-direction, Eq. (2-32), and both boundary conditions in the x-direction, Eqs. (2-38) and (2-39)) for each value of λi identiﬁed by Eq. (2-44). TX i = C1,i sin (λi x)

where λi =

iπ W

i = 1, 2, . . . ∞

(2-45)

Note that the i = 0 case is not included in Eq. (2-45) because the sine of 0 is zero; therefore this solution does not provide any useful information. The functions TXi given by Eq. (2-45) are referred to as the eigenfunctions that solve the linear, homogeneous problem for TX and the values λi are the eigenvalues associated with each eigenfunction. The function sin(i πx/W) is referred to as the ith eigenfunction and λi = i π/W is the ith eigenvalue. Solve the Non-homogeneous Problem for each Eigenvalue With the eigenproblem solved, it is necessary to return to the non-homogeneous portion of the problem, TY . Each of the eigenvalues identiﬁed by Eq. (2-44) is associated with an ordinary differential equation in the y-direction according to Eq. (2-33): d2 TYi − λ2i TYi = 0 dy2

(2-46)

These ordinary differential equations have either an exponential or hyperbolic solution according to Eqs. (2-35) or (2-36). The choice of one form over the other is arbitrary and either will lead to the same solution. Because one of the boundary conditions is at y → ∞, the exponentials will provide a more concise solution for this problem.

214

Two-Dimensional, Steady-State Conduction

However, in most other cases, the sinh and cosh solution will be easier to work with. The solution for TYi is: TYi = C3,i exp (−λi y) + C4,i exp (λi y)

(2-47)

Obtain Solution for each Eigenvalue According to Eq. (2-26), the solution for temperature associated with the ith eigenvalue is: T i = TX i TYi = C1,i sin (λi x) [C3,i exp (−λi y) + C4,i exp (λi y)]

(2-48)

The products of the undetermined constants C1,i C3,i and C1,i C4,i are also undetermined constants and therefore Eq. (2-48) can be written as: T i = sin (λi x) [C3,i exp (−λi y) + C4,i exp (λi y)]

(2-49)

Equation (2-49) will, for any value of i, satisfy the governing differential equation, Eq. (2-14), and satisfy both of the boundary conditions in the homogeneous direction, Eqs. (2-15) and Eq. (2-16). This is a typical outcome of solving the eigenproblem: a set of solutions that each satisfy the governing partial differential equation and all of the boundary conditions in the homogeneous direction. It is worth checking that your solution has these properties using Maple. First, it is necessary to let Maple know that i is an integer using the assume command. > restart; > assume(i,integer);

Next, deﬁne λi according to Eq. (2-44): > lambda:=i∗ Pi/W; λ :=

i∼π W

Note the use of Pi rather than pi in the Maple code; Pi indicates that π should be evaluated symbolically whereas pi is the numerical value of π. Create a function T in the independent variables x and y according to Eq. (2-49): > T:=(x,y)->sin(lambda∗ x)∗ (C3∗ exp(-lambda∗ y)+C4∗ exp(lambda∗ y)); T := (x, y) → sin(λx)(C3e(−λy) + C4e(λy) )

You can verify that the two homogeneous direction boundary conditions, Eqs. (2-15) and (2-16), are satisﬁed: > T(0,y);

0

> T(W,y);

0

and also that the partial differential equation, Eq. (2-14), is satisﬁed;

2.2 Separation of Variables Solutions

215

> diff(diff(T(x,y),x),x)+diff(diff(T(x,y),y),y); i∼πy i∼πy i ∼ πx i ∼2 π2 C3e(− W ) + C4e( W ) sin W − W2 i∼πy i∼πy C3 i ∼2 π2 e(− W ) i ∼ πx C4 i ∼2 π2 e( W ) + sin + W W2 W2 > simplify(%); 0

Create the Series Solution and Enforce the Remaining Boundary Conditions Because the partial differential equation is linear, the sum of the solutions for each eigenvalue, Ti given by Eq. (2-49), is itself a solution: T =

∞

Ti =

i=1

∞

sin (λi x) [C3,i exp (−λi y) + C4,i exp (λi y)]

(2-50)

i=1

The ﬁnal step of the solution selects the constants so that the boundary conditions in the non-homogeneous direction are satisﬁed. Equation (2-18) provides the boundary condition as y approaches inﬁnity; substituting Eq. (2-50) into Eq. (2-18) leads to: T y→∞ =

∞ i=1

sin (λi x) C3,i exp (−∞) + C4,i exp (∞) = 0

(2-51)

∞

0

or ∞

sin (λi x) C4,i ∞ = 0

(2-52)

i=1

Equation (2-52) can be solved by inspection; the equality can only be satisﬁed if C4,i = 0 for all i. T =

∞

C3,i sin (λi x) exp (−λi y)

(2-53)

i=1

Because only C3,i remains in our solution it is no longer necessary to designate it as the third undetermined constant: T =

∞

Ci sin (λi x) exp (−λi y)

(2-54)

i=1

Equation (2-17) provides the boundary condition at y = 0; substituting Eq. (2-54) into Eq. (2-17) leads to: T y=0 =

∞ i=1

Ci sin (λi x) exp (−λi 0) = T b (x) 1

(2-55)

216

Two-Dimensional, Steady-State Conduction

or ∞

Ci sin (λi x) = T b (x)

(2-56)

i=1

Equation (2-56) deﬁnes the constants in the solution; they are the Fourier coefﬁcients of the non-homogeneous boundary condition. At ﬁrst glance, it may seem like we have not really come very far. The solution to the problem is certainly provided by Eqs. (2-54) and (2-56), however an inﬁnite number of unknown constants, Ci , are needed to evaluate this solution and it is not clear how Eq. (2-56) can be manipulated in order to evaluate these constants. Fortunately the eigenfunctions have the property of orthogonality, which makes it relatively easy to determine the constants Ci . The meaning of orthogonality becomes evident when Eq. (2-56) is multiplied by a single eigenfunction, say the jth one, and then integrated in the homogeneous direction from one boundary to the other (i.e., from x = 0 to x = W): ∞

W Ci

i=1

W jπ iπ jπ sin x sin x dx = T b (x) sin x dx W W W

0

(2-57)

0

The property of orthogonality guarantees that the only term in the summation on the left side of Eq. (2-57) that will not integrate to zero is the one where i = j. We can verify this result by consulting a table of integrals. The integral of the product of two sine functions is: W 0

⎤W ⎡ π (j − i) x π (i + j) x sin sin ⎥ iπ W ⎢ jπ W W ⎢ ⎥ − sin x sin x dx = ⎦ W W 2π ⎣ (j − i) (i + j)

(2-58)

0

This result can be obtained using Maple: > assume(j,integer); > assume(i,integer); > int(sin(i∗ Pi∗ x/W)∗ sin(j∗ Pi∗ x/W),x);

1 2

π(−i + j)x W π(−i + j)

W sin

−

1 2

π(i + j)x W π(i + j)

W sin

Applying the limits of integration to Eq. (2-58) leads to:

W sin 0

iπ jπ W sin (π (j − i)) sin (π (i + j)) − x sin x dx = W W 2π (j − i) (i + j)

(2-59)

The term involving sin (π (i + j)) must always be zero for any positive integer values of i and j because the sine of any integer multiple of π is zero. The ﬁrst term on the right side of Eq. (2-59) will also be zero provided j = i. However, if j = i then both the numerator

2.2 Separation of Variables Solutions

217

and denominator of this term are zero and the value of the integral is not obvious. In the limit that j = i, the value of the integral in Eq. (2-59) is:

W 2

sin 0

W iπ sin (π (j − i)) x dx = lim W 2 π (j−i)→0 (j − i)

(2-60)

π

The limit of the bracketed term in Eq. (2-60) can be evaluated using Maple: > restart; > limit(sin(Pi∗ x)/x,x=0); π

which leads to:

W sin2

W iπ x dx = W 2

(2-61)

0

This behavior lies at the heart of orthogonality: the integral of the product of two different eigenfunctions between the homogeneous boundary conditions will always be zero while the integral of any eigenfunction multiplied by itself between the same boundary conditions will not be zero. It is possible to prove that this behavior is generally true for any solution in the homogeneous direction (see Myers (1987)). The orthogonality of the eigenfunctions simpliﬁes the problem considerably because it allows the summation in Eq. (2-57) to be replaced by a single integration. (All of the terms on the left hand side where j = i must integrate to zero and disappear.) W Ci 0

W iπ iπ iπ sin x sin x dx = T b (x) sin x dx for i = 1..∞ W W W

(2-62)

0

The only term in the sum that has been retained is the integration of the eigenfunction sin(iπ x/W) multiplied by itself; notice that Eq. (2-62) provides a single equation for each of the constants Ci and therefore completes the solution. A more physical feel for the orthogonality of the eigenfunctions can be obtained by examining various eigenfunctions and their products. For example, Figure 2-4 shows the ﬁrst eigenfunction, sin (π x/W), and the second eigenfunction, sin (2 π x/W). Also shown in Figure 2-4 is the product of these two eigenfunctions; notice that the integral of the product must be zero as the areas above and below the axis are equal. Figure 2-5 illustrates the behavior of the second and fourth eigenfunctions and their product; while their oscillations are more complex, it is clear that the product of these eigenfunctions must also integrate to zero. Finally, Figure 2-6 shows the behavior of the third eigenfunction and its value squared; the square of any real valued function is never negative and therefore it cannot integrate to zero. The eigenfunctions that are appropriate for other problems with different boundary conditions will not be sin(i π x/W). However, they will be orthogonal functions. As a result, the sum that deﬁnes the constants in the series can always be reduced to one equation that allows each constant to be evaluated; the equivalent of Eq. (2-57) can always be reduced to the equivalent of Eq. (2-62).

218

Two-Dimensional, Steady-State Conduction

Eigenfunctions and their product

1 0.8

st

1 eigenfunction, sin(π x)

0.6 0.4 0.2 0 product of 1 st & 2 nd eigenfunctions

-0.2 -0.4 -0.6

nd

2 eigenfunction, sin(2π x)

-0.8 -1 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/W

0.8

0.9

1

Figure 2-4: Behavior of the ﬁrst and second eigenfunctions and their product.

Eigenfunctions and their product

1.5 1

nd

th

2 eigenfunction, sin(2 π x)

4 eigenfunction, sin(4π x)

0.5 0 -0.5 -1 -1.5 0

nd

product of 2 th & 4 eigenfunctions 0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/W

0.8

0.9

1

Figure 2-5: Behavior of the second and fourth eigenfunctions and their product.

Equation (2-62) provides an integral equation that can be used to evaluate each of the coefﬁcients:

W T b (x) sin Ci =

iπ x dx W

0

W sin2

iπ x dx W

i = 1..∞

(2-63)

0

Still, it is necessary to determine both of the integrals in Eq. (2-63) in order to evaluate each coefﬁcient. In some cases, the integrals can be evaluated by inspection or by the use of mathematical tables; usually these integrals can be evaluated easily with the aid of Maple. The integral in the denominator of Eq. (2-63) was previously evaluated in

Eigenfunctions and their product

2.2 Separation of Variables Solutions 1.2 1 0.8 0.6 0.4 0.2 0 rd -0.2 3 eigenfunction squared -0.4 -0.6 -0.8 -1 -1.2 0 0.1 0.2 0.3

219

rd

3 eigenfunction, sin(3 π x) 0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless position, x/W Figure 2-6: Behavior of the third eigenfunction and its square.

Eq. (2-61). The integral could also be evaluated with the aid of trigonometric identities and integral tables:

W 2

sin 0

W W W W 2iπ x 2iπ iπ 1 = x dx = − cos x dx = − sin x W 2 W 2 2iπ W 2 0 0

(2-64) Maple makes this process much easier: > int((sin(i∗ Pi∗ x/W))ˆ2,x=0..W); W 2

The ith coefﬁcient is therefore: 2 Ci = W

W T b (x) sin

iπ x dx W

(2-65)

0

The remaining integral in Eq. (2-65) depends on the functional form of the boundary condition. The simplest possibility is a constant temperature, Tb , which leads to: 2 Tb Ci = W

W 0

W iπ 2 Tb 2 Tb iπ sin = x dx = − cos x [1 − cos (i π)] W iπ W iπ 0

or, using Maple: > 2∗ Tb∗ int(sin(i∗ pi∗ x/W),x=0..W)/W; −

2Tb(−1 + cos(i π)) iπ

(2-66)

220

Two-Dimensional, Steady-State Conduction

Substituting Eq. (2-66) into Eq. (2-54) leads to: T =

∞ 2 Tb i=1

iπ

[1 − cos (i π)] sin

iπ iπ x exp − y W W

(2-67)

It is usually more convenient to let Maple carry out the symbolic math and then evaluate the solution using EES. The input parameters are entered in EES: $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” W=1.0 [m] k=10 [W/m-K] T b=1 [K]

“width of plate” “conductivity of plate” “base temperature”

The temperature is evaluated at an arbitrary location: “position to evaluate temperature” x=0.1 [m] y=0.25 [m]

“x-position” “y-position”

Each of the ﬁrst N terms of the series solution in Eq. (2-67) are evaluated using a duplicate loop. The coefﬁcient for each term is evaluated using the formula obtained in Maple by copying and pasting it into EES. N=10 [-] duplicate i=1,N C[i]=-2∗ T b∗ (-1+cos(i∗ pi))/(i∗ pi) T[i]=C[i]∗ sin(i∗ pi∗ x/W)∗ exp(-i∗ pi∗ y/W) end

“number of terms in the solution to evaluate” “constant for i’th term in the series” “i’th term in the series”

The terms in the series are summed using the sum function in EES: T=sum(T[1..N])

“sum of N terms in the series”

A parametric table is created in order to examine the temperature as a function of position along the bottom of the plate, y = 0. The ability of the solution to match the imposed boundary condition depends on the number of terms that are used. Figure 2-7 illustrates the solution with 5, 10, and 100 terms. A contour plot of the temperature distribution is generated by creating a parametric table containing the variables x, y, and T that includes 400 runs. Click on the arrow in the x-column header in order to bring up the dialog that allows values to be automatically entered into the table. Click the check box for the option to repeat the pattern of running x from 0 to 1 every 20 rows, as shown in Figure 2-8(a). Repeat the process for the y column, but in this case apply the pattern of running y from 0 to 1 every 20 rows, as shown in Figure 2-8(b). When the table is solved, the solution is obtained over a 20 × 20 grid ranging from 0 to 1 in both x and y. Select X-Y-Z Plot from the New Plot Window selection under the Plots menu and select Isometric Lines to generate the contour plot shown in Figure 2-9.

2.2 Separation of Variables Solutions

221

1.4 N = 100 terms

N = 5 terms

Temperature (K)

1.2 1 0.8

N = 10 terms

0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Position, x (m)

0.7

0.8

0.9

1

Figure 2-7: Temperature as a function of x at y = 0 for different values of N.

(a)

(b)

Figure 2-8: Automatically entering repeating values for (a) x and (b) y.

More complicated boundary conditions can be considered at y = 0. For example, the temperature may vary linearly from 0 to 1 K according to: T b (x) = T b

x W

(2-68)

The coefﬁcients in the general solution, Eq. (2-54) are obtained by substituting Eq. (2-68) into Eq. (2-65): 2 Tb Ci = W2

W

iπ x sin x dx W

0

which can be evaluated using Maple: > restart; > assume(i,integer); > C[i]:=2∗ T_b∗ int(x∗ sin(i∗ pi∗ x/W),x=0..W)/Wˆ2; Ci∼ := −

2 T b(−sin(i ∼ π) + cos(i ∼ π)i ∼ π) i ∼2 π2

(2-69)

222

Two-Dimensional, Steady-State Conduction 1 0.9 0.8

Position, y (m)

0.7 0.6 0.5 0.4

0.1 0.4

0.5

0.2

0.6

0.7

0.1 0 0

0.2

0.3

0.3

0.9 0.1

0.2

0.3

0.8

0.4 0.5 0.6 Position, x (m)

0.7

0.8

0.9

1

Figure 2-9: Contour plot of temperature distribution for constant temperature boundary.

The Maple result is copied and pasted into EES to achieve: N=100 [-] “number of terms in the solution to evaluate” duplicate i=1,N { C[i]=-2∗ T b∗ (-1+cos(i∗ pi))/(i∗ pi) “constant for i’th term in the series, constant temp. boundary”} C[i]=-2∗ T b∗ (-sin(i∗ pi)+cos(i∗ pi)∗ i∗ pi)/iˆ2/piˆ2 “constant for i’th term in the series, linear variation in boundary temp.” “i’th term in the series” T[i]=C[i]∗ sin(i∗ pi∗ x/W)∗ exp(-i∗ pi∗ y/W) end T=sum(T[1..N]) “sum of N terms in the series”

It is obviously not possible to include an inﬁnite number of terms in the solution and therefore a natural question is: how many terms are sufﬁcient? The magnitude of the neglected terms can be assessed by considering the magnitude of the last term that was included. For example, Figure 2-10 shows the Arrays Table that results when the solution is evaluated at x = 0.1 m and y = 0.25 m with N = 11 terms. The size of the terms in the solution drop dramatically as the index of the term increases. The accuracy of the solution computed using 11 terms is within 3.2 × 10-6 K of the actual solution and therefore it is clear that only a few terms are required at this position. However, the number of terms that are required depends on the position within the computational domain. More terms are typically required to resolve the solution near the boundary and, in particular, near boundaries where non-physical conditions are being enforced. For example, in this problem we are requiring that an edge at T = 0 (i.e., the left edge) intersect with an edge at T = 1 (i.e., the bottom edge) which results in an inﬁnite temperature gradient at x = y = 0 that cannot physically exist. Summary of Steps The steps required to solve a problem using separation of variables are summarized below: 1. Verify that the problem satisﬁes all of the conditions that are required for separation of variables. The partial differential equation must be linear and homogeneous,

2.2 Separation of Variables Solutions

Figure 2-10: Arrays Table containing the solution terms for x = 0.1 m and y = 0.25 m and N = 11 terms.

223

0.6366

0.0897

-0.3183

-0.03889

0.2122

0.01627

-0.1592

-0.006541

0.1273

0.002509

-0.1061

-0.0009065

0.09095

0.0003014

-0.07958

-0.00008735

0.07074

0.00001861

-0.06366

5.110E-18

0.05787 -0.000003165

2.

3.

4. 5.

6. 7.

all boundary conditions must be linear, and both boundary conditions in one direction (the homogeneous direction) must be homogeneous. If the problem does not meet these requirements then it may be possible to apply a simple transformation to the boundary conditions (as discussed in Section 2.2.3), use superposition (as discussed in Section 2.4), or carefully divide the problem into its homogeneous and non-homogeneous parts (as discussed in Section 2.3.2). Separate the variables; that is, express the solution (T) as the product of a function of x (TX) and a function of y (TY ). Use this approach to split the partial differential equation into two ordinary differential equations; the ordinary differential equation in the homogeneous direction should be selected so that it is solved by a function involving sines and cosines. Solve the ordinary differential equation in the homogeneous direction (the eigenproblem) and apply the boundary conditions in this direction in order to obtain the eigenfunctions and eigenvalues. Solve the ordinary differential equation in the non-homogeneous direction for each eigenvalue. Determine a solution for temperature associated with each eigenvalue, Ti , using the results from steps 3 and 4. This solution should satisfy the partial differential equation and both of the homogeneous direction boundary conditions. It is helpful to use Maple to check this solution. Express your general solution as a series composed of the solutions for each eigenvalue that resulted from step 5. Enforce the boundary conditions in the non-homogeneous direction in order to determine the constants in the series. Note that this step will require that the property of the orthogonality of the eigenfunctions be utilized at one or both of the two non-homogeneous direction boundary conditions. The property of orthogonality is utilized by multiplying the series solution by an arbitrary eigenfunction and integrating between the two homogeneous boundaries. This mathematical operation will reduce the series to a single equation involving the constants for only one of the terms in the series. The integration required to carry out this step can often be facilitated using Maple and the resulting equation can often be solved using EES.

224

Two-Dimensional, Steady-State Conduction ∂T ∂y

h (T∞ − Tx=0) = −k

∂T ∂x

=0 y =H

H ∂ 2T ∂ 2T + =0 ∂x 2 ∂y 2 W

x =0

y

∂T ∂x

x =W

∂θ ∂y

y =W

=0

x Ty =0 = Tb

(a) ∂θ ∂y h (T∞ − Tb − θ x=0 ) = −k

∂θ ∂x

=0 y =H

H ∂2θ ∂2θ + =0 ∂x 2 ∂y 2

x=0

=0

W

y x θ y=0 = 0

(b) Figure 2-11: Rectangular plate problem stated (a) in terms of temperature, T, and (b) in terms of temperature difference, θ.

2.2.3 Simple Boundary Condition Transformations The separation of variables technique discussed in Section 2.2.2 can be applied to a linear and homogeneous problem that has linear boundary conditions. In addition, both of the boundary conditions in one direction must be homogeneous; that is, any solution that satisﬁes the boundary condition must still satisfy the boundary condition if it is multiplied by a constant. Three types of linear boundary conditions are often encountered in heat transfer problems: (1) speciﬁed temperature, (2) speciﬁed heat ﬂux, and (3) convection to a ﬂuid of speciﬁed temperature. Direct application of any of these conditions generally results in a non-homogeneous boundary condition. In general, it is possible to deal with non-homogeneous boundary conditions through superposition, as discussed in Section 2.4, or by breaking a solution into its particular and homogeneous components, as discussed in Section 2.3.2. However, it is often possible to apply a relatively simple transformation in order to reduce the number of non-homogeneous boundary conditions by at least one, thereby (possibly) avoiding the need to use these advanced techniques. Consider the rectangular plate shown in Figure 2-11(a). The governing differential equation (assuming that there is no convection from the top and bottom surfaces or thermal energy generation within the plate material) is: ∂2T ∂2T + 2 =0 2 ∂x ∂y

(2-70)

2.2 Separation of Variables Solutions

225

The differential equation is linear and homogeneous. The plate has a speciﬁed temperature-type boundary condition at the bottom edge: T y=0 = T b

(2-71)

Equation (2-71) is not homogeneous unless Tb = 0. The plate has a convection-type boundary condition applied to the left edge: ∂T h (T ∞ − T x=0 ) = −k (2-72) ∂x x=0

Equation (2-72) is not homogeneous unless T∞ = 0. The plate has speciﬁed heat ﬂuxtype boundary conditions applied to the remaining two edges; these boundaries are adiabatic and therefore the speciﬁed heat ﬂux is equal to 0: ∂T =0 (2-73) ∂x x=W ∂T =0 (2-74) ∂y y=H Because the speciﬁed heat ﬂux is zero (i.e., the boundaries are adiabatic), Eqs. (2-73) and (2-74) are homogeneous. The problem posed by Figure 2-11(a) cannot be directly solved using separation of variables as neither direction is characterized by two non-homogeneous boundary conditions. However, it is possible to reduce the number of non-homogeneous boundary conditions by one. The problem is transformed and solved for the temperature difference relative to a boundary temperature; that is, the problem is solved in terms of either θ = T – T∞ or θ = T – Tb rather than T. The governing differential equation that results is unaffected by this modiﬁcation (the derivatives of θ are the same as those of T) and it is easy to re-state the remaining boundary conditions in terms of θ rather than T. For example, transforming the problem shown in Figure 2-11(a) to solve for: θ = T − Tb

(2-75)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN In Section 1.6 the constant cross-section, straight ﬁn shown in Figure 1 was analyzed under the assumption that it could be treated as an extended surface (i.e., temperature gradients in the y direction are neglected). In this example, the 2-D temperature distribution within the ﬁn will be determined using separation of variables. W

Figure 1: Straight, constant cross-sectional area ﬁn. y Tb

T∞, h

x L th

Assume that the tip of the ﬁn is insulated and that the width (W) is much larger than the thickness (th) so that convection from the edges can be neglected. The length of

E 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

results in the problem shown in Figure 2-11(b). The problem posed in terms of θ can be solved directly by separation of variables because both boundary conditions in the y-direction are homogeneous.

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

226

Two-Dimensional, Steady-State Conduction

the ﬁn is L. The ﬁn base temperature is Tb and the ﬁn experiences convection with ﬂuid at T∞ with average heat transfer coefﬁcient, h. a) Develop an analytical solution for the temperature distribution in the ﬁn using separation of variables. The upper and lower halves of the ﬁn are symmetric; that is, there is no difference between the upper and lower portions of the ﬁn and therefore no heat transfer across the mid-plane of the ﬁn. The mid-plane of the ﬁn (i.e., the surface at y = 0) can therefore be treated as if it were adiabatic. The computational domain including the boundary conditions is shown in Figure 2(a). −k

∂T ∂y

y=th/2

Tx=0 = Tb

= h (Ty=th/2 − T∞ ) th/2

2

2

∂T ∂T + =0 ∂x 2 ∂y 2

∂T ∂x

x =L

=0

L

y x

∂T ∂y

=0 y =0

(a) −k

∂θ ∂y

y=th/2

= h (θy=th/2 − θ ∞ )

θx=0 = 0

th/2

∂2θ ∂2θ + =0 ∂x 2 ∂y 2

∂θ ∂x

=0 x=L

L

y x

∂θ ∂y

=0 y=0

(b) Figure 2: Problem statement posed in terms of (a) temperature, T, and (b) temperature difference, θ.

The governing equation within the ﬁn can be derived using the process described in Section 2.2.2: ∂ 2T ∂ 2T + =0 2 ∂x ∂y2 Figure 2(a) indicates that the problem stated in terms of T has two non-homogeneous boundary conditions (the base and the top surface). However, the boundary condition at the base can be made homogeneous by deﬁning: θ = T − Tb

227

so that the governing equation becomes: ∂ 2θ ∂ 2θ + =0 2 ∂x ∂y2

(1)

The boundary conditions for the transformed problem, illustrated in Figure 2(b), are: θx=0 = 0

(2)

∂θ =0 ∂ x x=L

(3)

∂θ =0 ∂ y y =0

(4)

∂θ = h (θ y =t h/2 − θ∞ ) −k ∂ y y =t h/2

(5)

where θ∞ = T∞ − Tb The problem stated in terms of θ satisﬁes all of the requirements discussed in Section 2.2.2 with x being the homogeneous direction. Therefore, the separation of variables solution proceeds using the steps laid out in Section 2.2.2. The solution for the temperature difference (θ ) is expressed as the product of a function only of x (θ X) and a function only of y (θ Y): θ (x, y ) = θX (x) θ Y (y )

(6)

Substitution of Eq. (6) into Eq. (1) leads to two ordinary differential equations, as shown in Section 2.2.2: d 2θ X ± λ2 θ X = 0 d x2 d 2θ Y ∓ λ2 θ Y = 0 dy2 It is necessary to determine the sign of the constant λ2 in the ordinary differential equations. Recall that it is necessary to have the sine/cosine eigenfunctions in the homogeneous direction. Therefore, it is necessary to select the positive sign for the ordinary differential equation for θ X and the negative sign for the ordinary differential equation for θ Y: d 2θ X + λ2 θ X = 0 d x2

(7)

d 2θ Y − λ2 θ Y = 0 dy2

(8)

The next step is to solve the eigenproblem (i.e., the problem for θ X); the solution to the ordinary differential equation for θ X, Eq. (7), is: θ X = C 1 sin (λ x) + C 2 cos (λ x)

(9)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

228

Two-Dimensional, Steady-State Conduction

The boundary conditions for θ X are obtained by substituting Eq. (9) into Eqs. (2) and (3): θ X x=0 = 0

(10)

d θ X =0 d x x=L

(11)

Substituting Eq. (9) into Eq. (10) leads to: θ X x=0 = C 1 sin (λ 0) + C 2 cos (λ 0) = 0 0

1

which can only be true if C2 = 0. Substituting Eq. (9), with C2 = 0, into Eq. (11) leads to: d θX = C 1 λ cos (λ L) = 0 dx x=L

which can only be true if the argument of the cosine function is π /2, 3π /2, 5π /2, etc. Therefore, the argument of the cosine function must be: λi L =

(1 + 2 i) π 2

where i = 0, 1, 2, . . .

The eigenfunctions of the problem are: θ X i = C 1,i sin (λi x)

where i = 0, 1, 2, . . .

(12)

and the eigenvalues of the problem are: λi =

(1 + 2 i) π 2L

(13)

The next step is to solve the problem in the non-homogeneous direction. The ordinary differential equation in the y-direction that is associated with each eigenvalue is: d 2θ Y i − λi2 θ Y i = 0 dy2 which is solved by either θ Y i = C 3,i exp (λi y ) + C 4,i exp (−λi y ) or θ Y i = C 3,i cosh (λi y ) + C 4,i sinh (λi y )

(14)

The choice of either exponentials or sinh and cosh is arbitrary in that both will lead to the correct solution. However, the proper choice often makes the solution process easier. The plate in Figure 2-3 extended to inﬁnity where the temperature became zero. As a result, the constant multiplying the positive exponential was forced to be zero, which made the problem easier to solve. Looking ahead for this ﬁn problem, we see that the gradient of temperature at y = 0 must be 0. This boundary condition would not eliminate either of the exponential terms. On the other hand, the boundary condition will force the constant C4,i in Eq. (14) to be zero and therefore the sinh term will be eliminated. Clearly then, Eq. (14) is the better choice; a little insight early in the problem can make the solution process easier.

229

The next step is to determine the temperature difference solution associated with each eigenvalue: θi = θX i θ Y i = sin (λi x) [C 3,i cosh (λi y ) + C 4,i sinh (λi y )] where the constant C1,i was absorbed into the constants C3,i and C4,i . This solution should satisfy both of the homogeneous boundary conditions as well as the partial differential equation for all values of i; it is worthwhile using Maple to verify that this is true. Specify that i is an integer and enter the deﬁnition of the eigenvalues: > restart; > assume(i,integer); > lambda:=(1+2∗ i)∗ Pi/(2∗ L); λ :=

(1 + 2i ∼)π 2L

Enter the solution for each eigenvalue: > T:=(x,y)->sin(lambda∗ x)∗ (C3∗ cosh(lambda∗ y)+C4∗ sinh(lambda∗ y)); T := (x, y ) → sin(λx)(C 3 cosh(λy ) + C 4 sinh(λy ))

Verify that the solution satisﬁes the two boundary conditions in the x-direction, Eqs. (2) and (3): > T(0,y); 0 > eval(diff(T(x,y),x),x=L); 0

and the partial differential equation, Eq. (1): > diff(diff(T(x,y),x),x)+diff(diff(T(x,y),y),y); (1 + 2i ∼)π y (1 + 2i ∼)π x (1 + 2i ∼)π y sin (1 + 2i ∼)2 π 2 C 3 cosh + C 4 sinh 1 2L 2L 2L − 4 L2 ⎛ (1 + 2i ∼)π y C 3 cosh (1 + 2i ∼)2 π 2 1 (1 + 2i ∼)π x ⎜ 2L ⎜ + sin ⎝4 2L L2

+

1 4

C 3 sinh

(1 + 2i ∼)π y 2L L2

(1 + 2i ∼)2 π 2

⎞ ⎟ ⎟ ⎠

> simplify(%); 0

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

230

Two-Dimensional, Steady-State Conduction

The sum of the solutions for each eigenvalue becomes the general solution to the problem: θ=

∞

θi = θ X i θ Y i =

i=0

∞

sin (λi x) [C 3,i cosh (λi y ) + C 4,i sinh (λi y )]

(15)

i=0

The boundary conditions in the non-homogeneous directions are enforced. Substituting Eq. (15) into Eq. (4) leads to: ⎡ ⎤ ∞ ∂θ = sin(λi x) ⎣C 3,i λi sinh(λi 0) + C 4,i λi cosh(λi 0)⎦ = 0 ∂ y y =0 i=0 0

1

The cosh(0) = 1 and the sinh(0) = 0 (much like the cos(0) = 1 and the sin(0) = 0) and therefore this boundary condition can be written as: ∞

sin (λi x) C 4,i λi = 0

i=0

which can only be true if C4,i = 0, therefore: ∞

θ=

C i sin (λi x) cosh (λi y )

(16)

i=0

where the subscript 3 has been removed from C3,i as it is the only remaining undetermined constant. Equation (16) is substituted into the boundary condition at y = th/2, Eq. (5):

∞

th C i sin (λi x) λi sinh λi −k 2 i=0

th C i sin (λi x) cosh λi =h − θ∞ 2 i=0 ∞

which can be rearranged: ∞

C i sin (λi x)

i=0

th th sinh λi + cosh λi = θ∞ 2 2 h

k λi

(17)

The eigenfunctions must be orthogonal between x = 0 and x = L (it is not necessary to prove this for each problem) and therefore Eq. (17) can be converted into an algebraic equation for each individual constant. Equation (17) is multiplied by one eigenfunction, sin(λj x), and integrated from x = 0 to x = L: ∞ i=0

Ci

L L th th sin (λi x) sin λ j x d x = θ∞ sin λ j x d x + cosh λi sinh λi 2 2 h

k λi

0

0

231

Orthogonality guarantees that the integral on the left side of this equation will be zero for every term in the summation except the one where i = j; therefore, the series equation can be rewritten as: Ci

L L th th 2 sinh λi sin (λi x) d x = θ∞ sin (λi x) d x + cosh λi 2 2 h

k λi

0

0

The coefﬁcients are evaluated according to: L θ∞ Ci =

sin (λi x) d x 0

k λi h

sinh λi

th th + cosh λi 2 2

(18)

L sin2 (λi x) d x 0

The integrals in Eq. (18) can be evaluated either using math tables or, more easily, using Maple:

> restart; > assume(i,integer); > lambda:=(1+2∗ i)∗ Pi/(2∗ L); λ := > int(sin(lambda∗ x),x=0..L);

> int(sin(lambda∗ x)∗ sin(lambda∗ x),x=0..L);

(1 + 2i ∼)π 2L

2L (1 + 2i ∼)π L 2

The constants can therefore be written as:

Ci = L λi

2 θ∞ th th sinh λi + cosh λi 2 2 h

k λi

(19)

Equations (16) and (19) together provide the analytical solution for the temperature distribution within the ﬁn. b) Use the analytical solution to predict and plot the temperature distribution in a ﬁn that is L = 5.0 cm long, th = 4 cm thick, with conductivity k = 0.5 W/m-K, and h = 100 W/m2 -K. The base temperature is Tb = 200◦ C and the ﬂuid temperature is T∞ = 20◦ C.

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

232

Two-Dimensional, Steady-State Conduction

The inputs are entered in EES: “EXAMPLE 2.2-1: 2-D Fin” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” th cm=4 [cm] th=th cm∗ convert(cm,m) L=5 [cm]∗ convert(cm,m) k=0.5 [W/m-K] h bar=100 [W/mˆ2-K] T b=converttemp(C,K,200[C]) T inﬁnity=converttemp(C,K,20[C])

“thickness of ﬁn in cm” “thickness of ﬁn” “length of ﬁn” “thermal conductivity” “heat transfer coefﬁcient” “base temperature” “ﬂuid temperature”

Dimensionless coordinates within the ﬁn are deﬁned in order to facilitate plotting the temperature distribution: y bar=0.5 x bar=0.5 y=y bar∗ th x=x bar∗ L

“dimensionless y-position” “dimensionless x-position” “y-position” “x-position”

The solution is implemented using a duplicate loop that calculates the ﬁrst N terms of the series. The number of terms that is required for accuracy should be checked by exploring the sensitivity of the calculation to the number of terms in the same way that a numerical model should be checked for grid convergence. N=100 “number of terms in series” duplicate i=0,N “eigenvalues” lambda[i]=(1+2∗ i)∗ pi/(2∗ L) C[i]=2∗ (T inﬁnity-T b)/(L∗ lambda[i]∗ (k∗ lambda[i]∗ sinh(lambda[i]∗ th/2)/h bar+cosh(lambda[i]∗ th/2))) “constants” “term in summation” theta[i]=C[i]∗ sin(lambda[i]∗ x)∗ cosh(lambda[i]∗ y) end theta=sum(theta[0..N]) “temperature difference” “temperature” T=theta+T b “in C” T C=converttemp(K,C,T)

Figure 3 shows the temperature distribution as a function of x/L for various values of y/th. Notice that for these conditions, an extended surface (i.e., 1-D) model of the ﬁn would not be justiﬁed because there is a substantial difference between the temperature at the center of the ﬁn (y/th = 0) and the edge (y/th = 0.5). This is evident from the Biot number: Bi =

h th 2k

233

Bi=h bar∗ th/(2k)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

“Biot number”

which leads to Bi = 4.0. 200 180

Temperature (°C)

160 140 y/th = 0 y/th = 0.1 y/th = 0.2 y/th = 0.3 y/th = 0.4 y/th = 0.5

120 100 80 60 40 20 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L

0.8

0.9

1

Figure 3: Temperature as a function of x/L for various values of y/th.

c) Use the analytical solution to predict the ﬁn efﬁciency of the 2-D ﬁn. The rate of conductive heat transfer into the base of the ﬁn is: t h/2

q˙ ﬁn = −2 k W 0

∂θ dy ∂ x x=0

Substituting Eqs. (16) and (19) into Eq. (20) leads to:

q˙ ﬁn = −4 θ∞

th/2 cosh (λi y ) d y

∞ kW L i=0 k λi

0

th th sinh λi + cosh λi 2 2 h

The integral can be accomplished using Maple: > restart; > int(cosh(lambda∗ y),y=0..th/2);

sinh λ

λth 2

(20)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

234

Two-Dimensional, Steady-State Conduction

so that the rate of conductive heat transfer to the ﬁn is: th sinh λ ∞ i kW 2 q˙ ﬁn = −4 θ∞ th th k λi L i=0 sinh λi λi + cosh λi 2 2 h

(21)

The ﬁn efﬁciency, discussed in Section 1.6.5, is deﬁned as the ratio of the rate of heat transfer to the maximum possible rate of heat transfer rate that is obtained with an inﬁnitely conductive ﬁn: ηﬁn =

q˙ ﬁn 2 hW L (Tb − T∞ )

(22)

Substituting Eq. (21) into Eq. (22) leads to the ﬁn efﬁciency predicted by the 2-D analytical solution:

ηﬁn,2D

th sinh λ ∞ i 2k 2 = 2 th th k λ i h L i=0 λ sinh λ + cosh λ i i i 2 2 h

which is evaluated in EES according to:

duplicate i=0,N eta ﬁn[i]=(2∗ k/(h bar∗ lambda[i]∗ Lˆ2))∗ sinh(lambda[i]∗ th/2)/(k∗ lambda[i]∗ sinh(lambda[i]∗ th/2)/h bar+& cosh(lambda[i]∗ th/2)) end eta ﬁn=sum(eta ﬁn[0..N])

d) Plot the ﬁn efﬁciency predicted by the 2-D analytical solution as a function of the ﬁn thickness and overlay on the plot the ﬁn efﬁciency predicted using the extended surface approximation, developed in Section 1.6. Figure 4 illustrates the ﬁn efﬁciency predicted by the 2-D model as a function of ﬁn thickness. Overlaid on Figure 4 is the solution from Section 1.6 that is listed in Table 1-4 for a ﬁn with an adiabatic tip: ηﬁn,1D =

tanh (m L) mL

where mL =

h2 L k th

As the ﬁn becomes thicker, the impact of the temperature gradients in the y-direction, neglected in the 1-D solution, become larger and therefore the 1-D

235

and 2-D solutions diverge, with the 1-D solution always over-predicting the performance. 0.35 0.3 1-D solution

Fin efficiency

0.25 0.2

2-D solution

0.15 0.1 0.05 0 0

1

2

3

4 5 6 7 Fin thickness (cm)

8

9

10

Figure 4: Fin efﬁciency as a function of the ﬁn thickness predicted by the 2-D solution and the 1-D solution.

Ratio of 2-D to 1-D fin efficiency solutions

The ratio ηﬁn,2D /ηﬁn,1D is shown in Figure 5 as a function of the Biot number; recall that the Biot number was used to justify the extended surface approximation in Section 1.6. Note that 1-D model is quite accurate (better than 2%) provided the Biot number is less than 0.1 and, surprisingly, remains reasonably accurate (10%) even up to a Biot number of 1.0. 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.05

0.1

1 Biot number

10

Figure 5: Ratio of the ﬁn efﬁciency predicted by the 2-D solution to the ﬁn efﬁciency predicted by the 1-D solution as a function of the Biot number.

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

236

Two-Dimensional, Steady-State Conduction

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE Figure 1 illustrates the situation where energy is transferred by conduction through a structure that suddenly changes cross-sectional area. The conduction resistance associated with this structure can be computed using separation of variables. This problem also illustrates an issue that is often confusing for separation of variables problems; speciﬁcally, the zeroth term in a cosine series must often be treated separately from the rest of the series. The proper methodology for dealing with this situation is demonstrated in this example. 2 q⋅ ′′ = 10, 000 W/m

c = 1.5 cm b = 5.0 cm k = 50 W/m-K

a = 10 cm

Figure 1: A constriction in a conduction path.

y x Tb = 0

The width of the larger cross-sectional area is b = 5.0 cm and its length is a = 10 cm. The heat ﬂux, q˙ = 10,000 W/m2 , is applied to the upper surface over a smaller width, c = 1.5 cm. The conductivity of the material is k = 50 W/m-K. The bottom surface of the object is maintained at some reference temperature, taken to be 0. a) Develop a solution for the temperature distribution in the material. The partial differential equation for the problem is: ∂ 2T ∂ 2T + =0 ∂x2 ∂y2 The boundary conditions in the x-direction are: ∂T =0 ∂ x x=0 ∂T =0 ∂ x x=b

(1)

(2)

(3)

and the boundary conditions in the y-direction are: Ty =0 = 0 " ∂T q˙ k = 0 ∂ y y =a

(4) x limit(q_dot_ﬂux∗ c∗ sinh(i∗ Pi∗ y/b)/(i∗ Pi∗ k),i=0); y q d ot f lux c bk

Substituting this result into Eq. (18) leads to: ∞

T =

q˙ c y C i cos (λi x) sinh (λi y ) + bk i=1

(21)

Substituting Eq. (21) into Eq. (5) leads to: " ∞ q˙ c q˙ x < c C i λi cos (λi x) cosh (λi a) = +k 0 x ≥c b i=1

(22)

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

2.2 Separation of Variables Solutions

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

240

Two-Dimensional, Steady-State Conduction

Next, we will deal with the non-zero terms in the series. Both sides of Eq. (22) are multiplied by cos(λj x) and integrated from x = 0 to x = b. q˙ c b

b cos(λ j x)d x + k

∞

0

i=1

c

b

=

q˙ cos(λ j x)d x +

b C i λi cosh(λi a)

cos(λi x) cos(λ j x)d x 0

0 cos(λ j x)d x c

0

The zeroth order term integrates to zero for any j > 0 and the only term of the summation that does not integrate to zero is the one where i = j: b k C i λi cosh (λi a)

cos (λi x) d x = q˙ 2

0

c cos (λi x) d x 0

The integrals in Eq. (23) are computed using Maple: > int((cos(lambda∗ x))ˆ2,x=0..b); b 2

> int(cos(lambda∗ x),x=0..c);

b sin

i ∼ πc b i∼π

in order to obtain an equation for the undetermined coefﬁcients: k C i λi cosh (λi a)

sin (λi c) b = q˙ 2 λi

The solution is programmed in EES:

“EXAMPLE 2.2-2: Constriction Resistance” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” q dot ﬂux=10000 [W/mˆ2] k=50 [W/m-K] c=1.5 [cm]∗ convert(cm,m) a=10 [cm]∗ convert(cm,m) b=5.0 [cm]∗ convert(cm,m)

“Heat ﬂux” “Conductivity” “width of applied ﬂux” “length of object” “width of object”

(23)

241

x bar=0.75 y bar=1 x=x bar∗ b y=y bar∗ a

“dimensionless x-position” “dimensionless y-position” “x-position” “y-position”

N=400 “number of terms” duplicate i=1,N “evaluate coefﬁcients for N terms” lambda[i]=i∗ pi/b k∗ C[i]∗ lambda[i]∗ cosh(lambda[i]∗ a)∗ b/2=q dot ﬂux∗ sin(lambda[i]∗ c)/lambda[i] T[i]=C[i]∗ cos(lambda[i]∗ x)∗ sinh(lambda[i]∗ y) end T=q dot ﬂux∗ c∗ y/(k∗ b)+sum(T[1..N])

A parametric table is generated and used to generate the contour plot of temperature shown in Figure 2. 1

7K

0.9

6K 5K

0.8 Position, y (m)

0.7

4K

0.6 3K

0.5 0.4

2K

0.3 0.2

1K

0.1 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Position, x (m)

0.7

0.8

0.9

1

Figure 2: Contour plot of temperature distribution in constriction.

It is worth comparing the answer with physical intuition. The temperature elevation at the constriction relative to the base is approximately 8 K according to Figure 2; does this value make sense? In Section 2.8, methods for estimating the conduction resistance of 2-D geometries using 1-D models are discussed. However, it is clear that the resistance of the constriction cannot be greater than the resistance to conduction through the material if the heat ﬂux is applied uniformly at the top surface: a Rnc = bLk where L is the length of the material. The temperature elevation at the constriction in this limit is: a q˙ c Tnc = Rnc q˙ c L = bk DeltaT nc=a∗ q dot ﬂux∗ c/(b∗ k)

“temperature rise without constriction”

This leads to Tnc = 6.0 K, which has the same magnitude as the observed temperature rise but is smaller, as expected.

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

2.2 Separation of Variables Solutions

242

Two-Dimensional, Steady-State Conduction

2.3 Advanced Separation of Variables Solutions This extended section of the book can be found on the website www.cambridge.org/ nellisandklein. Section 2.2 provides an introduction to the technique of separation of variables and discusses its application in the context of a few examples. The separation of variables method, as it is presented in Section 2.2, is rather limited as it does not allow, for example, non-homogeneous terms that might arise from effects such as volumetric generation or for problems that are in cylindrical coordinates. One technique for solving non-homogeneous partial differential equations is discussed in Section 2.3.2. The extension of separation of variables to cylindrical coordinates is presented in Section 2.3.3 and demonstrated in EXAMPLE 2.3-1. L TLHS

g⋅ ′′′

TRHS

Figure 2-15: Plane wall with uniform volumetric generation and speciﬁed edge temperatures.

x

2.4 Superposition 2.4.1 Introduction Many conduction heat transfer problems are governed by linear differential equations; in some cases, many different functions will all satisfy the differential equation. The sum of all of the functions that separately satisfy a linear differential equation will itself be a solution. This property is used in separation of variables when the solutions associated with each of the eigenfunctions are added together in order to obtain a series solution to the problem. Superposition uses this property to determine the solution to a complex problem by breaking it into several, simpler problems that are solved individually and then added together. Care must be taken to ensure that the boundary conditions for the individual problems properly add to satisfy the desired boundary condition. A series of 1-D steady-state problems appears in Chapter 1; although superposition was not used to solve these problems, it would have been possible to apply this methodology. For example, consider a plane wall with thickness (L) and conductivity (k) experiencing a constant volumetric rate of thermal energy generation (g˙ ). The edges of the wall have speciﬁed temperatures, TLHS and TRHS , as shown in Figure 2-15. The governing differential equation for this problem is: g˙ d2 T = − dx2 k

(2-126)

T x=0 = T LHS

(2-127)

T x=L = T RHS

(2-128)

and the boundary conditions are:

Notice that the governing differential equation and both boundary conditions are linear but they are not homogeneous. It is possible to solve the problem posed by Eqs. (2-126) through (2-128) without resorting to superposition; indeed there is no advantage to using

2.4 Superposition

243

2 g⋅ ′′′ d T =− dx 2 k

Tx=0 = TLHS

Tx=L = TRHS

x complete problem for T

TA, x=L = 0

2

d TA =0 dx 2

TA, x=0 = TLHS

TB, x=0 = 0

x

TB, x=L = TRHS

2

d TB =0 dx2

x

sub-problem B for TB

sub-problem A for TA

T

T

T

2

TC, x=0 = 0

x

TC, x=L = 0

d TC g⋅ ′′′ =− dx2 k

sub-problem C for TC

TLHS TA

TC

TRHS TB

0

0

L

x

0

L

0

T

0

x

0

L

x

T=TA + TB + TC

TLHS

T

TA

TC

TRHS TB 0

L

0

x

Figure 2-16: Principle of superposition applied to the plane wall of Figure 2-15.

superposition for such a simple problem. The solution was found in Section 1.3 to be: g˙ L2 x x 2 (T LHS − T RHS ) (2-129) − x + T RHS − T = 2k L L L Nevertheless, the problem shown in Figure 2-15 provides a useful introduction to superposition. The problem can be broken into three, simpler sub-problems each of which retains only one of the non-homogeneities that are inherent in the total problem. The complete problem is solved by T and is broken into sub-problems A, B, and C which are solved by the functions TA , TB and TC , respectively, as shown in Figure 2-16. Sub-problem A retains the non-homogeneous boundary condition at x = 0, but uses the homogeneous version of the differential equation (i.e., the generation term is dropped) and the boundary condition at x = L. d2 T A =0 dx2

(2-130)

T A,x=0 = T LHS

(2-131)

T A,x=L = 0

(2-132)

The solution to sub-problem A, TA , is linear from TLHS to 0 as shown in Figure 2-16. x T A = T LHS 1 − (2-133) L

244

Two-Dimensional, Steady-State Conduction

Sub-problem B retains the non-homogeneous boundary condition at x = L, but uses the homogeneous version of the differential equation and the boundary condition at x = 0: d2 T B =0 dx2

(2-134)

T B,x=0 = 0

(2-135)

T B,x=L = T RHS

(2-136)

The solution, TB , is linear from 0 to TRHS . x (2-137) L Finally, sub-problem C retains the non-homogeneous differential equation but uses the homogeneous versions of both boundary conditions: T B = T RHS

g˙ d2 T C = − dx2 k

(2-138)

T C,x=0 = 0

(2-139)

T C,x=L = 0

(2-140)

The solution, TC , is a quadratic with a maximum at the center of the wall: g˙ L2 x x 2 TC = − 2k L L

(2-141)

The solution to the complete problem is the sum of the solutions to the three subproblems: T = TA + TB + TC or T = T LHS

x g˙ L2 x + T RHS + 1− L L 2 k

TA

TB

(2-142)

x x 2 − L L

(2-143)

TC

which is identical to Eq. (2-129), the solution obtained in Section 1.3. It is easy to see from Figure 2-16 that this process of superposition must work; the differential equations for the sub-problems, Eqs. (2-130), (2-134), and (2-138) can be added together to recover Eq. (2-126): g˙ g˙ g˙ d2 (T A + T B + T C) d2 T d2 T A d2 T B d2 T C + + = − = − = − → → dx2 dx2 dx2 k dx2 k dx2 k (2-144) and the boundary conditions for the sub-problems can be added together to recover Eqs. (2-127) and (2-128): T A,x=0 + T B,x=0 + T C,x=0 = T LHS → (T A + T B + T C)x=0 = T LHS → T x=0 = T LHS (2-145) T A,x=L + T B,x=L + T C,x=L = T RHS → (T A + T B + T C)x=L = T RHS → T x=L = T RHS (2-146)

2.4 Superposition

245 Ts = 100°C Tb = 20°C

Figure 2-17: Rectangular plate used to illustrate superposition for 2-D problems.

H=1m

Ts = 100°C

W=1m y x Tb = 20°C

2.4.2 Superposition for 2-D Problems Superposition becomes much more useful for 2-D problems because the separation of variables technique is restricted to problems that have homogeneous boundary conditions in one direction. Most real problems will not satisfy this condition and therefore it is absolutely necessary to use superposition to solve these problems. The solution can be developed by superimposing several solutions, each constructed so that they are tractable using separation of variables. The process is only slightly more complex than the 1-D problem that is discussed in the previous section. For example, consider the plate with height H = 1 m and width W = 1 m, shown in Figure 2-17. The top and right sides are kept at Ts = 100◦ C while the bottom and left sides are kept at Tb = 20◦ C. The temperature distribution is a function only of x and y. The complete problem is shown in Figure 2-18(a); notice that all four boundary conditions are non-homogeneous and even transforming the problem by subtracting Tb or Ts will not result in two homogeneous boundary conditions in either the x- or y-direction. Therefore, the problem cannot be solved directly using separation of variables. Figure 2-18(b) illustrates the problem transformed by deﬁning the temperature difference relative to Tb : θ = T − Tb

(2-147)

It is necessary to break the problem for θ into two sub-problems: θ = θA + θB

(2-148)

Each sub-problem is characterized by a homogeneous direction, as shown in Figure 2-18(c). Note that for each sub-problem, the homogeneous boundary conditions that are selected are analogous to the original, non-homogeneous boundary conditions. In this problem, the speciﬁed temperature boundaries are replaced with a temperature of zero. A speciﬁed heat ﬂux boundary should be replaced with an adiabatic boundary and a boundary with convection to ﬂuid at T∞ should be replaced by convection to ﬂuid at zero temperature. Each of the two sub-problems are solved using separation of variables, as discussed in Section 2.2. The governing partial differential equation for sub-problem A: ∂ 2 θA ∂ 2 θA + =0 ∂x2 ∂y2

(2-149)

is separated into two ordinary differential equations; note that the x-direction is homogeneous for sub-problem A and therefore the ordinary differential equation

246

Two-Dimensional, Steady-State Conduction T y =H = Ts Tx=0 = Tb

Tx=W = Ts ∂2T + ∂2T = 0 ∂x 2 ∂y 2 Ty =0 = Tb problem for T (a) θy =H = Ts − Tb θx=W = Ts − Tb

θx=0 = 0 2

2

∂θ +∂θ = 0 ∂x 2 ∂y 2 θy=0 = 0 problem for θ

(b) θB, y=H = 0

θA, y =H = Ts − Tb θA, x =0 = 0

θB, x=0 ∂2θA ∂x 2

+

∂2θA ∂y 2

∂2θB

=0

∂x

θA, x =W = 0

2

+

∂2θB ∂y 2

=0 θB, x =W = Ts − Tb

θB, y=0 = 0 sub-problem for θB

θA, y=0 = 0 sub-problem for θA

(c) Figure 2-18: Mathematical description of (a) the problem for temperature T, (b) the problem for temperature difference θ, and (c) the two sub-problems θA and θB that can be solved using separation of variables.

for θX A is selected so that it is solved by sines and cosines. d2 θX A + λ2A θX A = 0 dx2

(2-150)

d2 θYA − λ2A θYA = 0 dx2 The eigenproblem is solved ﬁrst; the solution to Eq. (2-150) is: θX A = CA,1 sin (λA x) + CA,2 cos (λA x)

(2-151)

(2-152)

The homogeneous boundary condition at x = 0 leads to: θX A,x=0 = CA,1 sin (λA 0) +CA,2 cos (λA 0) = 0 0

1

(2-153)

2.4 Superposition

247

which can only be true if CA,2 = 0: θX A = CA,1 sin (λA x)

(2-154)

The homogeneous boundary condition at x = W leads to: θX A,x=W = CA,1 sin (λA W) = 0

(2-155)

which leads to the eigenvalues: iπ for i = 1, 2..∞ (2-156) W The solution to the ordinary differential equation in the non-homogeneous direction, Eq. (2-151), is: λA,i =

θYA,i = CA,3,i sinh (λA,i y) + CA,4,i cosh (λA,i y)

(2-157)

The solution for each eigenvalue is: θA,i = θX A,i θYA,i = sin (λA,i x) [CA,3,i sinh (λA,i y) + CA,4,i cosh (λA,i y)]

(2-158)

The general solution is the sum of the solutions for each eigenvalue: θA =

∞

θA,i =

i=1

∞

sin (λA,i x) [CA,3,i sinh (λA,i y) + CA,4,i cosh (λA,i y)]

(2-159)

i=1

The general solution is required to satisfy the boundary condition at y = 0: ⎤ ⎡ ∞ sin (λA,i x) ⎣CA,3,i sinh (λA,i 0) + CA,4,i cosh(λA,i 0)⎦ = 0 θA,y=0 = i=1

=0

(2-160)

=1

which can only be true if CA,4,i = 0: θA =

∞

CA,i sin (λA,i x) sinh (λA,i y)

(2-161)

i=1

The solution is required to satisfy the boundary condition at y = H: θA,y=H =

∞

CA,i sin (λA,i x) sinh (λA,i H) = T s − T b

(2-162)

i=1

The orthogonality property of the eigenfunctions is used: W

W sin (λA,i x) dx = (T s − T b) 2

CA,i sinh (λA,i H) 0

0

The integrals in Eq. (2-163) are evaluated in Maple: > restart; > assume(i,integer); > lambda:=i∗ Pi/W; λ := > int((sin(lambda∗ x))ˆ2,x=0..W);

sin (λA,i x) dx

i∼π W

W 2

(2-163)

248

Two-Dimensional, Steady-State Conduction

> int(sin(lambda∗ x),x=0..W); −

W(−1 + (−1)i∼ ) i∼π

which leads to:

CA,i sinh (λA,i H)

W = − (T s − T b) 2

( ' W −1 + (−1)i iπ

(2-164)

or: CA,i = − (T s − T b)

2 [−1 + (−1)i ] i π sinh (λA,i H)

(2-165)

When solving a problem using superposition, it is useful to separately implement and examine the solution to each of the sub-problems and verify that they separately satisfy the boundary conditions and satisfy our physical intuition. The inputs are entered in EES: $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” H=1 [m] W=1 [m] T s=converttemp(C,K,100 [C]) T b=converttemp(C,K,20 [C])

“height of plate” “width of plate” “temperature of right and top of plate” “temperature of left and bottom of plate”

The position to evaluate the temperature is speciﬁed in terms of dimensionless variables:

x=x bar∗ W y=y bar∗ H

“x-position” “y-position”

The solution to sub-problem A is implemented in EES: N=100 [-] duplicate i=1,N lambda A[i]=i∗ pi/W C A[i]=2∗ (T s-T b)∗ (-(-1+(-1)ˆi)/i/Pi)/sinh(lambda A[i]∗ H) theta A[i]=C A[i]∗ sin(lambda A[i]∗ x)∗ sinh(lambda A[i]∗ y) end theta A=sum(theta A[1..N])

The solution to sub-problem A is shown in Figure 2-19(a).

“number of terms” “eigenvalue” “evaluate constants”

“sub-problem A”

Next Page 2.4 Superposition

249

y x

(a)

y x

(b)

y x

(c) Figure 2-19: Solution for (a) sub-problem A (θA ), (b) sub-problem B (θB ), (c) and temperature difference (θ).

250

Two-Dimensional, Steady-State Conduction

A similar process leads to the solution for sub-problem B: λB,i =

iπ H

for i = 1, 2..∞ 2 [−1 + (−1)i ] i π sinh (λB,i W)

(2-167)

CB,i sin (λB,i y) sinh (λB,i x)

(2-168)

CB,i = − (T s − T b)

θB =

∞

(2-166)

i=1

which is also implemented in EES: duplicate i=1,N lambda B[i]=i∗ pi/H C B[i]=2∗ (T s-T b)∗ (-(-1+(-1)ˆi)/i/Pi)/sinh(lambda B[i]∗ W) theta B[i]=C B[i]∗ sin(lambda B[i]∗ y)∗ sinh(lambda B[i]∗ x) end theta B=sum(theta B[1..N])

“eigenvalue” “evaluate constants”

“sub-problem B”

The solution to sub-problem B is shown in Figure 2-19(b). The temperature difference solution is obtained by superposition, Eq. (2-148): theta=theta A+theta B

“temperature difference, from superposition”

and shown in Figure 2-19(c). The temperature solution (in ◦ C) is obtained with the following code: T=theta+T b T C=converttemp(K,C,T)

“temperature” “in C”

The process of superposition was illustrated in this section for a simple problem. However, it is possible to use superposition to break a relatively complicated problem with multiple, non-homogeneous and complex boundary conditions into a series of problems that are each tractable, allowing the problem to be solved and veriﬁed one sub-problem at a time.

2.5 Numerical Solutions to Steady-State 2-D Problems with EES 2.5.1 Introduction Sections 2.1 through 2.4 present analytical techniques that are useful for solving 2-D conduction problems. These solution techniques have some fairly severe limitations. For example, the shape factors discussed in Section 2.1 can be used only in those situations where the problem can be represented using one of the limited set of shape factor solutions that are available. The separation of variables techniques coupled with

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

251

superposition presented in Sections 2.2 through 2.4 can be applied to a more general set of problems; however, they are still limited to linear problems (e.g., radiation and temperature dependent properties cannot be explicitly considered) with simple boundaries. To consider a problem of any real complexity would require the superposition of many solutions and that would be somewhat time consuming. Also, the solution is speciﬁc to the problem; if any aspect of the problem changes then the solution must be re-derived. These analytical solutions are most useful for verifying numerical solutions or, in some cases, creating multi-scale models. This section begins the discussion of numerical solutions to 2-D problems. There are two techniques that are used to solve 2-D problems: ﬁnite difference solutions and ﬁnite element solutions. Finite difference solutions are discussed in this section as well as in Section 2.6. The application of the ﬁnite difference approach to 2-D problems is a natural extension of the 1-D ﬁnite difference solutions that were studied in Chapter 1. Finite difference solutions are intuitive and powerful, but difﬁcult to apply to complex geometries. Finite element solutions are dramatically different from ﬁnite difference solutions and can be applied more easily to complex geometries. A complete description of the ﬁnite element technique is beyond the scope of this book; however, ﬁnite element solutions to heat transfer problems are extremely powerful and many commercial packages are available for this purpose. Section 2.7 provides a discussion of the ﬁnite element technique followed by an introduction to the ﬁnite element package FEHT. An academic version of FEHT can be downloaded from the website www.cambridge.org/ nellisandklein. Both ﬁnite difference and ﬁnite element techniques break a large computational domain into many smaller ones that are referred to as control volumes for the ﬁnite difference technique and elements for the ﬁnite element technique. The control volumes or elements are modeled approximately in order to generate a system of equations that can be efﬁciently solved using a computer. The approximate, numerical solution will approach the actual solution as the number of control volumes or elements is increased. It is important to remember that it is not sufﬁcient to obtain a solution. Regardless of what technique you are using (including the use of a pre-packaged piece of software, such as FEHT), you must still: 1. verify that your solution has an adequately large number of control volumes or elements, 2. verify that your solution makes physical sense and obeys your intuition, and 3. verify your solution against an analytical solution in an appropriate limit. These steps are widely accepted as being “best practice” when working with numerical solutions of any type.

2.5.2 Numerical Solutions with EES Finite difference solutions to 1-D steady-state problems are presented in Sections 1.4 and 1.5. The steps required to set up a numerical solution to a 2-D problem are essentially the same; however, the bookkeeping process (i.e., the process of entering the algebraic equations into the computer) may be somewhat more cumbersome. The ﬁrst step is to deﬁne small control volumes that are distributed through the computational domain and to precisely deﬁne the locations at which the numerical model will compute the temperatures (i.e., the locations of the nodes). The control volumes are

252

Two-Dimensional, Steady-State Conduction W

Figure 2-20: Straight, constant cross-sectional area ﬁn.

y Tb

T∞ , h

x L th

small but ﬁnite; for the 1-D problems that were investigated in Chapter 1, the control volumes were small in a single dimension whereas they must be small in two dimensions for a 2-D problem. It is necessary to perform an energy balance on each differential control volume and provide rate equations that approximate each term in the energy balance based upon the nodal temperatures or other input parameters. The result of this step will be a set of equations (one for each control volume) in an equal number of unknown temperatures (one for each node). This set of equations can be solved in order to provide the numerical prediction of the temperature at each node. In this section, EES is used to solve the system of equations. In the next section, MATLAB is used to solve these types of problems. In Section 1.6, the constant cross-section, straight ﬁn shown in Figure 2-20 is analyzed under the assumption that it could be treated as an extended surface (i.e., temperature gradients in the y direction could be neglected). The ﬁn is reconsidered using separation of variables in EXAMPLE 2.2-1 without making the extended surface approximation. In this section the problem is revisited again, this time using a ﬁnite difference technique to obtain a solution for the temperature distribution. The tip of the ﬁn is insulated and the width (W) is much larger than its thickness (th) so that convection from the edges of the ﬁn can be neglected; therefore, the problem is 2-D. The length of the ﬁn is L = 5.0 cm and its thickness is th = 4.0 cm. The ﬁn base temperature is Tb = 200◦ C and it transfers heat to the surrounding ﬂuid at T∞ = 20◦ C with average heat transfer coefﬁcient, h = 100 W/m2 -K. The conductivity of the ﬁn material is k = 0.5 W/m-K. The inputs are entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” L=5.0 [cm]∗ convert(cm,m) th=4.0 [cm]∗ convert(cm,m) k=0.5 [W/m-K] h bar=100 [W/mˆ2-K] T b=converttemp(C,K,200 [C]) T inﬁnity=converttemp(C,K,20 [C])

“length of ﬁn” “width of ﬁn” “thermal conductivity” “heat transfer coefﬁcient” “base temperature” “ﬂuid temperature”

The computational domain associated with a half-symmetry model of the ﬁn is shown in Figure 2-21. The ﬁrst step in obtaining a numerical solution is to position the nodes throughout the computational domain. A regularly spaced grid of nodes is uniformly distributed, with the ﬁrst and last nodes in each dimension placed on the boundaries of the domain

2.5 Numerical Solutions to Steady-State 2-D Problems with EES T∞ , h T1, N T2, N

253

TM-1, N TM, N

Ti-1, N Ti, N Ti+1, N

T1, N-1

control volume for T control volume for top Ti, N-1 corner node M, N M, N-1 boundary node i, N Ti, j+1

Tb

Ti-1, j T1, 3 T1, 2 T2, 2 y

Ti, j

th/2

Ti+1, j

control volume for T i, j-1 internal node i, j

TM, 2

T1, 1 T2, 1 T3, 1

TM-1, 1 TM, 1

x L Figure 2-21: The computational domain associated with the constant cross-sectional area ﬁn and the regularly spaced grid used to obtain a numerical solution.

as shown in Figure 2-21. The x- and y-positions of any node (i, j) are given by: xi =

(i − 1) L (M − 1)

(2-169)

yj =

(j − 1) th 2 (N − 1)

(2-170)

where M and N are the number of nodes used in the x- and y-directions, respectively. The x- and y-distance between adjacent nodes ( x and y, respectively) are: x = y =

L (M − 1)

(2-171)

th 2 (N − 1)

(2-172)

This information is entered in EES: “Setup grid” M=40 [-] N=21 [-] duplicate i=1,M x[i]=(i-1)∗ L/(M-1) x bar[i]=x[i]/L end DELTAx=L/(M-1) duplicate j=1,N y[j]=(j-1)∗ th/(2∗ (N-1)) y bar[j]=y[j]/th end DELTAy=th/(2∗ (N-1))

“number of x-nodes” “number of y-nodes” “x-position of each node” “dimensionless x-position of each node” “x-distance between adjacent nodes” “y-position of each node” “dimensionless y-position of each node” “y-distance between adjacent nodes”

254

Two-Dimensional, Steady-State Conduction

Ti, j+1 q⋅top Ti-1, j

q⋅ LHS Δy

Ti, j q⋅ RHS Ti+1, j Δx

Figure 2-22: Energy balance for an internal node.

q⋅bottom

Ti, j-1

The next step in the solution is to write an energy balance for each node. Figure 2-22 illustrates a control volume and the associated energy transfers for an internal node (see Figure 2-21); each energy balance includes conduction from each side (q˙ RHS and q˙ LHS ), the top (q˙ top ), and the bottom (q˙ bottom ). Note that the direction associated with these energy transfers is arbitrary (i.e., they could have been taken as positive if energy leaves the control volume), but it is important to write the energy balance and rate equations in a manner that is consistent with the directions chosen in Figure 2-22. The energy balance suggested by Figure 2-22 is: q˙ RHS + q˙ LHS + q˙ top + q˙ bottom = 0

(2-173)

The next step is to approximate each of the terms in the energy balance. The material separating the nodes is assumed to behave as a plane wall thermal resistance. Therefore, y W (where W is the width of the ﬁn into the page) is the area for conduction between nodes (i, j) and (i + 1, j) and x is the distance over which the conduction heat transfer occurs. k y W (T i+1,j − T i,j ) x

q˙ RHS =

(2-174)

Note that the temperature difference in Eq. (2-174) is consistent with the direction of the arrow in Figure 2-22. The other conductive heat transfers are approximated using a similar model: q˙ LHS =

k y W (T i−1,j − T i,j ) x

(2-175)

q˙ top =

k x W (T i,j+1 − T i,j ) y

(2-176)

q˙ bottom =

k x W (T i,j−1 − T i,j ) y

(2-177)

Substituting the rate equations, Eqs. (2-174) through (2-177), into the energy balance, Eq. (2-173) written for all of the internal nodes in Figure 2-21, leads to: k y W k x W k y W (T i+1,j − T i,j ) + (T i−1,j − T i,j ) + (T i,j+1 − T i,j ) x x y +

k x W (T i,j−1 − T i,j ) = 0 y

for i = 2 . . . (M − 1)

and

j = 2 . . . (N − 1)

(2-178)

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

255 T∞ , h

Ti-1, N Figure 2-23: Energy balance for a node on the top boundary.

q⋅ LHS

Δx q⋅ conv

Ti, N

q⋅ bottom

Ti+1, N q⋅ RHS Δy

Ti, N-1

which can be simpliﬁed to: y x x y (T i+1,j − T i,j ) + (T i−1,j − T i,j ) + (T i,j+1 − T i,j ) + (T i,j−1 − T i,j ) = 0 x x y y for i = 2 . . . (M − 1) and j = 2 . . . (N − 1) (2-179) These equations are entered in EES using nested duplicate loops: “Internal node energy balances” duplicate i=2,(M-1) duplicate j=2,(N-1) DELTAy∗ (T[i+1,j]-T[i,j])/DELTAx+DELTAy∗ (T[i-1,j]-T[i,j])/DELTAx& +DELTAx∗ (T[i,j+1]-T[i,j])/DELTAy+DELTAx∗ (T[i,j-1]-T[i,j])/DELTAy=0 end end

Note that each time the outer duplicate statement iterates once (i.e., i is increased by 1), the inner duplicate statement iterates (N − 2) times (i.e., j runs from 2 to N − 1). Therefore, all of the internal nodes are considered with these two nested duplicate loops. Also note that the unknowns are placed in an array rather than a vector. The entries in the array T are accessed using two indices that are contained in square brackets. Boundary nodes must be treated separately from internal nodes, just as they are in the 1-D problems that are considered in Section 1.4; however, 2-D problems have many more boundary nodes than 1-D problems. The left boundary (x = 0) is easy because the temperature is speciﬁed: T 1,j = T b

for j = 1 . . . N

(2-180)

where Tb is the base temperature. These equations are entered in EES: “left boundary” duplicate j=1,N T[1,j]=T_b end

The remaining boundary nodes do not have speciﬁed temperatures and therefore must be treated using energy balances. Figure 2-23 illustrates an energy balance associated with a node that is located on the top boundary (at y = th/2, see Figure 2-21). The

256

Two-Dimensional, Steady-State Conduction

energy balance suggested by Figure 2-23 is: q˙ RHS + q˙ LHS + q˙ bottom + q˙ conv = 0

(2-181)

The conduction terms in the x-direction must be approximated slightly differently than for the internal nodes: q˙ RHS =

k y W (T i+1,N − T i,N ) 2 x

(2-182)

q˙ LHS =

k y W (T i−1,N − T i,N ) 2 x

(2-183)

The factor of 2 in the denominator of Eqs. (2-182) and (2-183) appears because there is half the area available for conduction through the sides of the control volumes located on the top boundary. The conduction term in the y-direction is approximated as before: q˙ bottom =

k x W (T i,N−1 − T i,N ) y

(2-184)

The convection term is: q˙ conv = h x W (T ∞ − T i,N )

(2-185)

Substituting Eqs. (2-182) through (2-185) into Eq. (2-181) for all of the nodes on the upper boundary leads to: k y W k x W k y W (T i+1,N − T i,N ) + (T i−1,N − T i,N ) + (T i,N−1 − T i,N ) 2 x 2 x y + W x h (T ∞ − T i,N ) = 0

(2-186)

for i = 2 . . . (M − 1)

which can be simpliﬁed to: y x y (T i+1,N − T i,N ) + (T i−1,N − T i,N ) + (T i,N−1 − T i,N ) 2 x 2 x y +

x h (T ∞ − T i,N ) = 0 k

(2-187)

for i = 2 . . . (M − 1)

These equations are entered in EES using a single duplicate statement: “top boundary” duplicate i=2,(m-1) DELTAy∗ (T[i+1,n]-T[i,n])/(2∗ DELTAx)+DELTAy∗ (T[i-1,n]-T[i,n])/(2∗ DELTAx)+& DELTAx∗ (T[i,n-1]-T[i,n])/DELTAy+DELTAx∗ h_bar∗ (T_inﬁnity-T[i,n])/k=0 end

Notice that the control volume at the top left corner, node (1, N), has already been speciﬁed by the equations for the left boundary, Eq. (2-180). It is important not to write an additional equation related to this node, or the problem will be over-speciﬁed. Therefore, the equations for the top boundary should only be written for i = 2 . . . (M − 1).

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

257 T∞ , h Δx q⋅ conv

TM-1, N q⋅ LHS q⋅

Figure 2-24: Energy balance for a node on the top right corner.

TM, N Δy

bottom

TM, N-1

A similar procedure for the nodes on the lower boundary leads to: y x y (T i+1,1 − T i,1 ) + (T i−1,1 − T i,1 ) + (T i,2 − T i,1 ) = 0 2 x 2 x y

for i = 2 . . . (M − 1) (2-188)

Notice that there is no convection term in Eq. (2-188) because the lower boundary is adiabatic. These equations are entered into EES: “bottom boundary” duplicate i=2,(M-1) DELTAy∗ (T[i+1,1]-T[i,1])/(2∗ DELTAx)+DELTAy∗ (T[i-1,1]-T[i,1])/(2∗ DELTAx)& +DELTAx∗ (T[i,2]-T[i,1])/DELTAy=0 end

Energy balances for the nodes on the right-hand boundary (x = L) lead to: x y x (T M,j+1 − T M,j ) + (T M,j−1 − T M,j ) + (T M−1,j − T M,j ) = 0 2 y 2 y x (2-189) for j = 2 . . . (N − 1) “right boundary” duplicate j=2,(n-1) DELTAx∗ (T[M,j+1]-T[M,j])/(2∗ DELTAy)+DELTAx∗ (T[M,j-1]-T[M,j])/(2∗ DELTAy)& +DELTAy∗ (T[M-1,j]-T[M,j])/DELTAx=0 end

The two corners (right upper and right lower) have to be considered separately. A control volume and energy balance for node (M, N), which is at the right upper corner (see Figure 2-21), is shown in Figure 2-24. The energy balance suggested by Figure 2-24 is: k y W x W k x W (T M,N−1 − T M,N ) + (T M−1,N − T M,N ) + h (T ∞ − T M,N ) = 0 2 y 2 x 2 (2-190) which can be simpliﬁed to: y h x x (T M,N−1 − T M,N ) + (T M−1,N − T M,N ) + (T ∞ − T M,N ) = 0 2 y 2 x 2k

(2-191)

258

Two-Dimensional, Steady-State Conduction

and entered into EES: “upper right corner” DELTAx∗ (T[M,N-1]-T[M,N])/(2∗ DELTAy)+DELTAy∗ (T[M-1,N]-T[M,N])/(2∗ DELTAx)+& h_bar∗ DELTAx∗ (T_inﬁnity-T[M,N])/(2∗ k)=0

The energy balance for the right lower boundary, node (M, 1), leads to: y x (T M,2 − T M,1 ) + (T M−1,1 − T M,1 ) = 0 2 y 2 x

(2-192)

“lower right corner” DELTAx∗ (T[M,2]-T[M,1])/(2∗ DELTAy)+DELTAy∗ (T[M-1,1]-T[M,1])/(2∗ DELTAx)=0

We have derived a total of M × N equations in the M × N unknown temperatures; these equations completely specify the problem and they have now all been entered in EES. Therefore, a solution can be obtained by solving the EES code. The solution is contained in the Arrays window; each column of the table corresponds to the temperatures associated with one value of i and all of the values of j (i.e., the temperatures in a column are at a constant value of y and varying values of x). The temperature solution is converted from K to ◦ C with the following equations. duplicate i=1,M duplicate j=1,N T_C[i,j]=converttemp(K,C,T[i,j]) end end

The solution is obtained for N = 21 and M = 40; the columns Ti,1 (corresponding to y/th = 0), Ti,5 (corresponding to y/th = 0.10), Ti,9 (corresponding to y/th = 0.2), etc. to Ti,21 (corresponding to y/th = 0.50) are plotted in Figure 2-25 as a function of the dimensionless x-position. The solution corresponds to our physical intuition as it exhibits temperature gradients in the x- and y-directions that correspond to conduction in these directions. The results from the analytical solution derived in EXAMPLE 2.2-1 are overlaid onto the plot and show nearly exact agreement with the numerical solution. The ﬁn efﬁciency will be used to verify that the grid is adequately reﬁned. The ﬁn efﬁciency is the ratio of the actual to the maximum possible heat transfer rates. The actual heat transfer rate per unit width (q˙ ﬁn ) is computed by evaluating the conductive heat transfer rate into the left hand side of each of the nodes that are located on the left boundary (i.e., all of the nodes where i = 1). ⎡ q˙ ﬁn = 2 ⎣k

y (T 1,1 − T 2,1 ) + 2 x

m−1 j=2

⎤ k

y y (T 1,j − T 2,j ) + k (T 1,N − T 2,N )⎦ x 2 x (2-193)

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

259

200 numerical nu ri l solution o i analytical solution ana

180 y/th 0 0.1 0.2 0.3 0.4 0.5

Temperature (°C)

160 140 120 100 80 60 40 20 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L

0.8

0.9

1

Figure 2-25: Temperature predicted by numerical model and analytical model (from EXAMPLE 2.2-1) as a function of x/L for various values of y/th.

while the maximum heat transfer rate per unit width (q˙ max) is associated with an isothermal ﬁn: q˙ max = 2 L h (T b − T ∞ )

(2-194)

Note that the corner nodes must be considered outside of the duplicate loop as they have 1/ the cross-sectional area available for conduction. Also, the factor of 2 in Eq. (2-193) 2 appears because the numerical model only considers one-half of the ﬁn. “calculate ﬁn efﬁciency” q_dot_ﬁn[1]=(T[1,1]-T[2,1])∗ k∗ DELTAy/(2∗ DELTAx) duplicate j=2,(N-1) q_dot_ﬁn[j]=(T[1,j]-T[2,j])∗ k∗ DELTAy/DELTAx end q_dot_ﬁn[N]=(T[1,N]-T[2,N])∗ k∗ DELTAy/(2∗ DELTAx) q_dot_ﬁn=2∗ sum(q_dot_ﬁn[1..N]) q_dot_max=2∗ L∗ h_bar∗ (T_b-T_inﬁnity) eta_ﬁn=q_dot_ﬁn/q_dot_max

Figure 2-26 illustrates the ﬁn efﬁciency as a function of the number of nodes in the xdirection (M) for various values of the number of nodes in the y direction (N). The solution is more sensitive to M than it is to N, but appears to converge (for the conditions considered here) when M is greater than 80 and N is greater than 10. (The solution for N = 10 is not shown in Figure 2-26, because it is nearly identical to the solution for N = 20.) EES can solve up to 6,000 simultaneous equations, so a reasonably large problem can be considered using EES. However EES is not really the best tool for dealing with very large sets of equations. In the next section, we will look at how MATLAB can be used to solve this type of 2-D problem.

260

Two-Dimensional, Steady-State Conduction 0.2

Fin efficiency

0.15

N=2 N=5 N=20

0.1

0.05

0 1

10 Number of nodes in x-direction, M

100

200

Figure 2-26: Fin efﬁciency as a function of M for various values of N predicted by the numerical model.

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB 2.6.1 Introduction Section 2.5 describes how 2-D, steady-state problems can be solved using a ﬁnite difference solution implemented in EES. This process is intuitive and easy because EES will automatically solve a set of implicit equations. However, EES is not well-suited for problems that involve very large numbers of equations. The ﬁnite difference method results in a system of algebraic equations that can be solved in a number of ways using different computer tools. Large problems will normally be implemented in a formal programming language such as C++, FORTRAN, or MATLAB. This section describes the methodology associated with solving the system of equations using MATLAB; however, the process is similar in any programming language.

2.6.2 Numerical Solutions with MATLAB The system of equations that results from applying the ﬁnite difference technique to a steady-state problem can be solved by placing these equations into a matrix format: AX = b

(2-195)

where the vector X contains the unknown temperatures. Each row of the A matrix and b vector corresponds to an equation (for one of the control volumes in the computational domain) whereas each column of the A matrix holds the coefﬁcients that multiply the corresponding unknown (the nodal temperature) in that equation. To place a system of equations in matrix format, it is necessary to carefully deﬁne how the rows and energy balances are related and how the columns and unknown temperatures are related. This process is easy for the 1-D steady-state problems considered in Section 1.5, but it becomes somewhat more difﬁcult for 2-D problems. The basic steps associated with carrying out a 2-D ﬁnite difference solution using MATLAB remain the same as those discussed in Section 1.5 for a 1-D problem. The ﬁrst step is to deﬁne the structure of the vector of unknowns, the vector X in Eq. (2-195). It doesn’t really matter what order the unknowns are placed in X, but the implementation

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

261

of the solution is easier if a logical order is used. For a 2-D problem with M nodes in one dimension and N in the other, a logical technique for ordering the unknown temperatures in the vector X is: ⎤ ⎡ X 1 = T 1,1 ⎥ ⎢ ⎢ X 2 = T 2,1 ⎥ ⎥ ⎢ ⎢ X 3 = T 3,1 ⎥ ⎥ ⎢ ⎥ ⎢ ... ⎥ ⎢ ⎥ ⎢ X = ⎢ X M = T M,1 ⎥ (2-196) ⎥ ⎢ ⎢ X M+1 = T 1,2 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ X M+2 = T 2,2 ⎥ ⎥ ⎢ ... ⎦ ⎣ X M N = T M,N Equation (2-196) indicates that temperature T i,j corresponds to element X M(j−1)+i of the vector X; this mapping is important to keep in mind as you work towards implementing a solution in MATLAB. The next step is to deﬁne how the control volume equations will be placed into each row of the matrix A. For a 2-D problem with M nodes in one dimension and N in the other, a logical technique is: ⎡ ⎤ row 1 = control volume equation for node (1, 1) ⎢ row 2 = control volume equation for node (2, 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ row 3 = control volume equation for node (3, 1) ⎥ ⎢ ⎥ ⎢ ⎥ ... ⎢ ⎥ row M = control volume equation for node (M, 1) ⎥ A=⎢ (2-197) ⎢ ⎥ ⎢ row M + 1 = control volume equation for node (1, 2) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ row M + 2 = control volume equation for node (2, 2) ⎥ ⎢ ⎥ ⎣ ⎦ ... row M N = control volume equation for node (M, N) Equation (2-197) indicates that the equation for the control volume around node (i, j) is placed into row M(j − 1) + i of matrix A. 2

T∞ = 20°C, h = 20 W/m -K ′′ = 5000 W/m 2 q⋅ inc mask holder, Tmh = 20°C pattern density, pd

th = 0.75 μm thin membrane, k = 150 W/m-K

W = 1.0 mm W = 1.0 mm y x

Figure 2-27: EPL Mask.

The process of implementing a numerical solution in MATLAB is illustrated in the context of the problem shown in Figure 2-27. An electron projection lithography (EPL)

262

Two-Dimensional, Steady-State Conduction

mask may be used to generate the next generation of computer chips. The EPL mask is used to reﬂect an electron beam onto a resist-covered wafer. The EPL mask consists of a thin membrane that extends between the much larger struts of a mask holder. The membrane absorbs the electron beam in some regions and reﬂects it in others so that the resist on the wafer is developed (i.e., exposed to energy) only in certain locations. The developing process changes the chemical structure of the resist so that it is selectively etched away during subsequent processes; thus, the pattern on the mask is transferred to the wafer. The portion of the membrane that absorbs the incident electron beam is heated. A precise calculation of the temperature rise in the mask is critical as any heating will result in thermally induced distortion that causes imaging errors in the printed features. Very small temperature rises can result in errors that are large relative to the printed features, as these features are themselves on the order of 100 nm or less. A membrane that is th = 0.75 μm thick and W = 1.0 mm on a side contains the pattern to be written. The membrane is supported by the mask holder; this is a substantially thicker piece of material that can be assumed to be at a constant temperature, T mh = 20◦ C. The thermal conductivity of the membrane is k = 150 W/m-K. The pattern density, pd (i.e., the fraction of the area of the mask that absorbs the incident radiation) can vary spatially across the EPL mask and therefore the thermal load applied to the surface of the mask area is non-uniform in x and y. The pattern density in this case is given by: pd = 0.1 + 0.5

xy W2

(2-198)

The thermal load on the mask per unit area at any location is equal to the product of the incident energy ﬂux, q˙ inc = 5000 W/m2 , and the pattern density pd at that location. The mask is exposed to ambient air on both sides. The air temperature is T ∞ = 20◦ C and the average heat transfer coefﬁcient is h = 20 W/m2 -K. The input parameters are entered in the MATLAB function EPL_Mask. The input arguments are M and N, the number of nodes in the x- and y-coordinates, while the output arguments are not speciﬁed yet. function[ ]=EPL_Mask(M,N) %[ ]=EPL_Mask % % This function determines the temperature distribution in an EPL_Mask % % Inputs: % M - number of nodes in the x-direction (-) % N - number of nodes in the y-direction (-) %INPUTS W=0.001; th=0.75e-6; q dot ﬂux=5000; k=150; T mh=20+273.2; T inﬁnity=20+273.2; h bar=20;

% width of membrane (m) % thickness of membrane (m) % incident energy (W/mˆ2) % conductivity (W/m-K) % mask holder temperature (K) % ambient air temperature (K) % heat transfer coefﬁcient (W/mˆ2-K)

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

263

T1, N

TM, N T q⋅conv i, j+1⋅ qtop q⋅LHS Ti, j Ti-1, j Ti+1, j g⋅ abs q⋅ RHS q⋅bottom

Figure 2-28: Numerical grid and control volume for an internal node (i, j).

Ti, j-1 T1, 2

T2, 2

T1, 1

T2, 1

TM, 1

A sub-function pd_f is created (at the bottom of the function EPL-Mask) in order to provide the pattern density: function[pd]=pd_f(x,y,W) % [pd]=pd_f(x,y,W) % % This sub-function returns the pattern density of the EPL mask % % Inputs: % x - x-position (m) % y - y-position (m) % W - dimension of mask (m) % Output: % pd - pattern density (-) pd=0.1+0.5∗ x∗ y/Wˆ2; end

The problem is two-dimensional; the membrane is sufﬁciently thin that temperature gradients in the z-direction can be neglected. This assumption can be veriﬁed by calculating an appropriate Biot number: Bi =

h th = 1 × 10−7 k

(2-199)

Therefore, a 2-D numerical model will be generated using the grid shown in Figure 2-28. The x- and y-coordinates of each node are provided by: xi =

(i − 1) W (M − 1)

for i = 1..M

(2-200)

yi =

(j − 1) W (N − 1)

for j = 1..N

(2-201)

264

Two-Dimensional, Steady-State Conduction

The distance between adjacent nodes is: x =

W (M − 1)

(2-202)

y =

W (N − 1)

(2-203)

The grid is set up in the MATLAB function: %Setup grid for i=1:M x(i,1)=(i-1)∗ W/(M-1); end DELTAx=W/(M-1); for j=1:N y(j,1)=(j-1)∗ W/(N-1); end DELTAy=W/(N-1);

The problem will be solved by placing the system of equations that result from considering each control volume into matrix format. A control volume for an internal node is shown in Figure 2-28. The energy balance for this control volume includes conduction from the left and right sides (q˙ LHS and q˙ RHS ) and the top and bottom (q˙ top and q˙ bottom ) as well as generation of thermal energy due to the absorbed illumination (g˙ abs ) and heat loss due to convection (q˙ conv ). The energy balance suggested by Figure 2-28 is: q˙ RHS + q˙ LHS + q˙ top + q˙ bottom + g˙ abs = q˙ conv

(2-204)

The conduction terms are approximated using the technique discussed in Section 2.5: q˙ RHS =

k y th (T i+1,j − T i,j ) x

(2-205)

q˙ LHS =

k y th (T i−1,j − T i,j ) x

(2-206)

q˙ bottom = q˙ top =

k x th (T i,j−1 − T i,j ) y

(2-207)

k x th (T i,j+1 − T i,j ) y

(2-208)

The absorbed energy is: g˙ abs = q˙ inc pd x y

(2-209)

The rate of convection heat transfer is: q˙ conv = 2 h x y (T i,j − T ∞ )

(2-210)

Substituting Eqs. (2-205) to (2-210) into Eq. (2-204) for all internal nodes leads to: k y th k x th k x th k y th (T i−1,j −T i,j ) + (T i+1,j −T i,j ) + (T i,j−1 −T i,j ) + (T i,j+1 −T i,j ) x x y y + q˙ inc pd x y = 2 h x y (T i,j −T ∞ )

for i = 2..(M − 1)

and

j = 2..(N − 1) (2-211)

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

265

Equation (2-211) is rearranged to identify the coefﬁcients that multiply each unknown temperature: k x th k y th k y th k y th T i,j −2 + T i+1,j −2 − 2 h x y + T i−1,j x y x x

+ T i,j−1

AM(j−1)+i,M(j−1)+i

k x th y

+ T i,j+1

AM(j−1)+i,M(j−1−1)+i

AM(j−1)+i,M(j−1)+i−1

AM(j−1)+i,M(j−1)+i+1

k x th = −2 h x y T ∞ − q˙ inc pd x y y bM(j−1)+i

(2-212)

AM(j−1)+i,M(j+1−1)+i

for i = 2..(M − 1)

and

j = 2..(N − 1)

The control volume equations must be placed into the matrix equation: AX = b

(2-213)

where the equation for the control volume around node (i, j) is placed into row M(j − 1) + i of A and T i,j corresponds to element X M(j−1)+i in the vector X, as required by Eqs. (2-197) and (2-196), respectively. Each coefﬁcient in Eq. (2-212) (i.e., each term multiplying an unknown temperature on the left side of the equation) must be placed in the row of A corresponding to the control volume being examined and the column of A corresponding to the unknown in X. The matrix assignments consistent with Eq. (2-212) are: AM(j−1)+i,M(j−1)+i = −2

k y th k x th −2 − 2 h x y x y

for i = 2..(M − 1)

and

(2-214)

j = 2..(N − 1)

AM(j−1)+i,M(j−1)+i−1 =

k y th x

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-215)

AM(j−1)+i,M(j−1)+i+1 =

k y th x

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-216)

AM(j−1)+i,M(j−1−1)+i =

k x th y

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-217)

AM(j−1)+i,M(j+1−1)+i =

k x th y

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-218)

bM(j−1)+i = −2 h x y T ∞ − q˙ inc pd x y for i = 2..(M − 1)

and

j = 2..(N − 1) (2-219)

A sparse matrix is allocated in MATLAB for A and the equations derived above are implemented using nested for loops. The spalloc command requires the number of rows and columns and the maximum number of non-zero elements in the matrix. Note that there are at most ﬁve non-zero entries in each row of A, corresponding to Eqs. (2-214) through (2-218); thus the last argument in the spalloc command is 5 M N.

266

Two-Dimensional, Steady-State Conduction

A=spalloc(M∗ N,M∗ N,5∗ M∗ N); %allocate a sparse matrix for A %energy balances for internal nodes for i=2:(M-1) for j=2:(N-1) A(M∗ (j-1)+i,M∗ (j-1)+i)=-2∗ k∗ DELTAy∗ th/DELTAx-2∗ k∗ DELTAx∗ th/DELTAy-... 2∗ h_bar∗ DELTAx∗ DELTAy; A(M∗ (j-1)+i,M∗ (j-1)+i-1)=k∗ DELTAy∗ th/DELTAx; A(M∗ (j-1)+i,M∗ (j-1)+i+1)=k∗ DELTAy∗ th/DELTAx; A(M∗ (j-1)+i,M∗ (j-1-1)+i)=k∗ DELTAx∗ th/DELTAy; A(M∗ (j-1)+i,M∗ (j+1-1)+i)=k∗ DELTAx∗ th/DELTAy; b(M∗ (j-1)+i,1)=-2∗ h_bar∗ DELTAx∗ DELTAy∗ T_inﬁnity-... q_dot_ﬂux∗ pd_f(x(i,1),y(j,1),W)∗ DELTAx∗ DELTAy; end end

The boundary nodes have speciﬁed temperature: T 1,j

= T mh

[1]

= T mh

[1] AM(j−1)+M,M(j−1)+M

T i,1

[1] AM(1−1)+i,M(1−1)+i

T i,N

[1] AM(N−1)+i,M(N−1)+i

(2-220)

bM(j−1)+1

AM(j−1)+1,M(j−1)+1

T M,j

for j = 1..N

for j = 1..N

(2-221)

bM(j−1)+M

= T mh

for i = 2..(M − 1)

(2-222)

bM(1−1)+i

= T mh

for i = 2..(M − 1)

(2-223)

bM(N−1)+i

Note that Eqs. (2-220) through (2-223) are written so that the corner nodes (e.g. node (1, 1)) are not speciﬁed twice. The matrix assignments suggested by Eqs. (2-220) through (2-223) are: AM(j−1)+1,M(j−1)+1 = 1 bM(j−1)+1 = T mh

for j = 1..N

AM(j−1)+M,M(j−1)+M = 1 bM(j−1)+M = T mh

for j = 1..N

for j = 1..N

for j = 1..N

AM(1−1)+i,M(1−1)+i = 1 for i = 2..(M − 1) bM(1−1)+i = T mh

for i = 2..(M − 1)

AM(N−1)+i,M(N−1)+i = 1 bM(N−1)+i = T mh

for i = 2..(M − 1)

for i = 2..(M − 1)

(2-224) (2-225) (2-226) (2-227) (2-228) (2-229) (2-230) (2-231)

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

267

These assignments are implemented in MATLAB: %speciﬁed temperatures around all edges for j=1:N A(M∗ (j-1)+1,M∗ (j-1)+1)=1; b(M∗ (j-1)+1,1)=T_mh; A(M∗ (j-1)+M,M∗ (j-1)+M)=1; b(M∗ (j-1)+M,1)=T_mh; end for i=2:(M-1) A(M∗ (1-1)+i,M∗ (1-1)+i)=1; b(M∗ (1-1)+i,1)=T_mh; A(M∗ (N-1)+i,M∗ (N-1)+i)=1; b(M∗ (N-1)+i,1)=T_mh; end

The vector X is obtained using MATLAB’s backslash command and the temperature of each node in degrees Celsius is placed in the matrix T_C. X=A\b; for i=1:M for j=1:N T_C(i,j)=X(M∗ (j-1)+i)-273.2; end end end

The function header is modiﬁed so that running the MATLAB function provides the temperature prediction (the matrix T_C) as well as the vectors x and y that contain the x- and y-positions of each node in the matrix. function[x,y,T_C]=EPL_Mask(M,N) % [x,y,T_C]=EPL_Mask % % This function determines the temperature distribution in an EPL_Mask % % Inputs: % M - number of nodes in the x-direction (-) % N - number of nodes in the y-direction (-) % Outputs: % x - Mx1 vector of x-positions of each node (m) % y - Nx1 vector of y-positions of each node (m) % T_C - MxN matrix of temperature at each node (C)

The solution should be examined for grid convergence. Figure 2-29 illustrates the maximum temperature in the EPL mask as a function of M and N (the two parameters are set equal for this analysis). The analysis is carried out using the script varyM, below, which deﬁnes a vector Mv that contains a range of values of the number of nodes, M, and runs

268

Two-Dimensional, Steady-State Conduction 20.76

Maximum temperature (°C)

20.75 20.74 20.73 20.72 20.71 20.7 20.69 20.68 4

10

100

300

Number of nodes Figure 2-29: Maximum predicted temperature as a function of the number of nodes (M and N).

the function EPL_Mask for each value. The command max(max(T_C)) computes the maximum value of each column and then the maximum value of the resulting vector in order to obtain the maximum nodal temperature in the mask. clear all; Mv=[5;10;20;30;50;70;100;200]; % values of M to use for i=1:8 [x,y,T_C]=EPL_Mask(Mv(i),Mv(i)); % obtain temperature distribution T_Cmaxv(i,1)=max(max(T_C)) % obtain maximum temperature end

There is a variety of 3-D plotting functions in MATLAB; these can be investigated by typing help graph3d at the command window. For example, >> mesh(x,y,T_C’); >> colorbar;

produces a mesh plot indicating the temperature, as shown in Figure 2-30. Note that the matrix T C has to be transposed (by adding the ’ character after the variable name) in order to match the dimensions of the x and y vectors.

2.6.3 Numerical Solution by Gauss-Seidel Iteration This extended section of the book can be found on the website www.cambridge.org/ nellisandklein. A ﬁnite difference solution results in a system of algebraic equations that must be solved simultaneously. In Section 2.6.2, we looked at placing these equations into a matrix equation that was solved by a single matrix inversion (or the equivalent mathematical manipulation). An alternative technique, Gauss-Seidel iteration, can also be used to approximately solve the system of equations using an iterative technique that requires much less memory than the direct matrix solution method. In some cases

2.8 Resistance Approximations for Conduction Problems

269

Temperature (deg. C)

21 20.8 20.6 20.4 20.2 20 1 0.8 x 10

-3

0.6 0.4 0.2 Position, x (m)

0

0

0.4

0.2

0.6

1

0.8

-3

x 10

Position, y (m)

Figure 2-30: Mesh plot of temperature distribution.

it can require less computational effort as well. The Gauss-Seidel iteration process is illustrated using the EPL mask problem that was discussed in Section 2.6.2.

2.7 Finite Element Solutions This extended section of the book can be found on the website www.cambridge.org/ nellisandklein. Sections 2.5 and 2.6 present the ﬁnite difference method for solving 2D steady-state conduction problems. In Section 2.7.1, the FEHT (Finite Element Heat Transfer) program is discussed and used to solve EXAMPLE 2.7-1. FEHT implements the ﬁnite element technique to solve 2-D steady-state conduction problems. A version of FEHT that is limited to 1000 nodes can be downloaded from www.cambridge.org/ nellisandklein. In order to become familiar with FEHT it is suggested that the reader go through the tutorial provided in Appendix A.4 which can be found on the web site associated with the book (www.cambridge.org/nellisandklein). In Section 2.7.2, the theory behind ﬁnite element techniques is presented.

2.8 Resistance Approximations for Conduction Problems 2.8.1 Introduction The resistance to conduction through a plane wall (R pw ) was derived in Section 1.2 and is given by: R pw =

L k Ac

(2-232)

The concept of a thermal resistance is a broadly useful and practical idea that goes beyond the simple situation for which Eq. (2-232) was derived. It is possible to understand conduction heat transfer in most situations if you can identify the appropriate distance that heat must be conducted (L) and the area through which that conduction occurs (Ac ). EXAMPLE 2.8-1 illustrates this type of “back-of-the-envelope” calculation.

Two-Dimensional, Steady-State Conduction

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET You may be faced with trying to understand the heat transfer through a complex, 2-D or 3-D geometry, such as the bracket illustrated in Figure 1. The bracket is made of steel having a thermal conductivity k = 14 W/m-K. One surface of the bracket is held at TH = 200◦ C and the other is at TC = 20◦ C. It is beyond the scope of any technique discussed in this book to analytically determine the heat ﬂow through this geometry and therefore it will be necessary to use a ﬁnite element software package for this purpose. However, it is possible to use the resistance concept represented by Eq. (2-232) in order to bound and estimate the heat ﬂow through the bracket. If you determine that the heat ﬂow through the bracket cannot possibly be important to the larger application (whatever that is) then the time and money required to generate the ﬁnite element model can be saved. If a ﬁnite element model is generated, then the simple thermal resistance estimate can provide a sanity check on the results.

1 cm

surface held at 200°C

1 cm

surface held at 20°C

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

270

Figure 1: A bracket with a complex, 2-D geometry made of steel with k = 14 W/m-K and thickness 1 cm (into the page).

a) Estimate the rate of heat transfer through the bracket using a resistance approximation. The length that heat must be conducted in order to go from the surface at TH to the surface at TC is approximately L = 14 cm and the area for conduction is approximately Ac = 1 cm2 . Clearly these are not exact values because the problem is two-dimensional; some energy must ﬂow a longer distance to reach the more proximal regions of the bracket and there are several portions of the bracket where the area is larger than 1 cm2 . However, it is possible to estimate the resistance of the bracket with these approximations: 5 100 cm L 14 cm m K K 5 = = 100 Rbr acket ≈ 5 2 kA 14 W 1 cm m W

271

which provides an estimate of the heat ﬂow:

q˙ ≈

(TH − TC ) (200◦ C − 20◦ C) W = 100 K = 1.8 W Rbr acket

It may be that 1.8 W is a trivial rate of energy loss from whatever is being supported by the bracket and therefore the bracket does not require a more detailed analysis. However, if a more exact answer is required then a ﬁnite element solution is necessary. b) Use FEHT to determine the rate of heat transfer through the bracket. The geometry from Figure 1 can be entered in FEHT and solved, as discussed in Section 2.7.1 and Appendix A.4. Set a scale where 1 cm on the screen corresponds to 0.01 m and use the Outline selection from the Draw menu to approximately trace out the bracket. Then, right-click on each of the corner nodes and enter the exact position in the Node Information Dialog. The boundary conditions should be set as well as the material properties. Create a crude mesh and reﬁne it. The problem is solved and the solution is shown in Figure 2.

Figure 2: Solution.

The total heat ﬂux at either the 200◦ C or the 20◦ C boundaries can be obtained by selecting Heat Flows from the View menu and then selecting all of the nodal boundaries along these boundaries. (Left-click and drag a selection rectangle.) At the TH boundary, the total heat ﬂow is reported as 249.8 W/m (Figure 3) or, for a 1 cm thick bracket, 2.5 W.

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

2.8 Resistance Approximations for Conduction Problems

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

272

Two-Dimensional, Steady-State Conduction

Figure 3: Heat ﬂow along the top boundary.

The same calculation along the TC boundary leads to 2.5 W. Therefore, the total heat ﬂow is within 40% of the 1.8 W value predicted by the simple resistance approximation. The sanity check is valuable; if the ﬁnite element model had predicted 10’s or 100’s of W then it would be almost certain that there is an error in the solution (perhaps a unit conversion or a material property entered incorrectly). Furthermore, the 1-D solution was an underestimate of the heat transfer because it did not account for the regions of the bracket that have larger cross-section. In the next sections, several methods are presented that can be used to bound the thermal resistance of a multi-dimensional object using 1-D resistances that are calculated with speciﬁc assumptions.

2.8.2 Isothermal and Adiabatic Resistance Limits Figure 2-31(a) illustrates a composite structure made from four materials (A through D). The composite structure experiences convection on its left and right sides to ﬂuid temperatures T C = 0◦ C and T H = 100◦ C, respectively, with average heat transfer coefﬁcient h = 500 W/m2 -K. The other surfaces are insulated. The problem represented by Figure 2-31(a) is 2-D; to see this clearly, imagine the situation where material B has very low conductivity, kB = 1.0 W/m-K, and material C has very high conductivity, kC = 100.0 W/m-K. Material A and D both have intermediate conductivity, kA = kD = 10 W/m-K. In this limit, thermal energy will transfer primarily through material C with very little passing through material B. The problem was solved in FEHT and the resulting temperature distribution is shown in Figure 2-31(b); the heat transfer rate is 281.9 W (assuming unit width into the page). The composite structure cannot be represented exactly with a 1-D resistance network due to the temperature gradients in the y-direction. In order to estimate the behavior of the system using 1-D resistance concepts, it is necessary to either allow unrestricted heat ﬂow in the y-direction (referred to as the isothermal limit) or completely eliminate heat ﬂow in the y-direction (referred to as the adiabatic limit). The temperature distribution is substantially simpliﬁed in these two limiting cases, as shown in Figure 2-32(a) and Figure 2-32(b), respectively. Note that Figure 2-32(a) and (b) can be obtained using FEHT; the material properties provides an Anisotropic ky/kx Type. For Figure 2-32(a), the ky/kx value is set to a large value (10,000) which effectively eliminates any resistance to heat ﬂow in the y-direction. In Figure 2-32(b), the ky/kx value is set to a small value (0.0001) which essentially prevents any heat ﬂow in the y-direction. The 1-D resistance network that corresponds to the isothermal limit is shown in Figure 2-33. The isothermal limit implies that there are no temperature gradients in the y-direction and therefore the temperature at any axial location may be represented by a single node. Note that in Figure 2-33, Ac is the area of the composite (0.02 m2 assuming

2.8 Resistance Approximations for Conduction Problems

k A = 10

W m-K

LD= 1 cm 1 cm

TC = 0°C W h = 500 2 m -K

2 cm

273

y x

kB = 1

W m-K

W kC = 100 m-K 1 cm

TH = 100°C W h = 500 2 m -K k D = 10

LA= 1 cm

W m-K

LB = LC = 2 cm

(a)

(b) Figure 2-31: (a) A composite structure consisting of four materials with convection from each edge and (b) the temperature distribution (◦ C) that will occur if kC kB .

a unit width into the page) and LA, LB, etc. are the thicknesses of the materials in the x-direction. The total resistance associated with the resistance network shown in Figure 2-33 is 0.32 K/W and therefore the rate of heat transfer through the composite structure predicted in the isothermal limit is 313 W. The isothermal limit corresponds to a lower bound on the thermal resistance, since the resistance to heat ﬂow in the y-direction is neglected. The adiabatic limit assumes that there is no heat transfer in the y-direction. Thermal energy can only pass through the composite axially and therefore the resistance network corresponding to the adiabatic limit consists of parallel paths for heat ﬂow by convection and through materials B and C (these parallel paths are labeled 1 and 2), as shown in Figure 2-34. The total thermal resistance associated with the resistance network shown in Figure 2-34 is 0.50 K/W and the rate of heat transfer through the composite structure predicted in the adiabatic limit is 200 W. This adiabatic limit corresponds to an upper bound on the resistance since the thermal energy is prohibited from spreading in the y-direction. The true solution lies somewhere between the isothermal (313 W) and adiabatic (200 W) limits; the FEHT model in Figure 2-31 predicted 282 W. Thus, the limits

274

Two-Dimensional, Steady-State Conduction

Figure 2-32: The temperature distribution in (a) the isothermal limit (kA,y , kB,y , kC,y , and kD,y → ∞) and (b) the adiabatic limit (kA,y , kB,y , kC,y , and kD,y → 0).

(a)

(b) 2.0

TC = 0°C

0.1

K W

Rconv =

0.05

K W

K W 2 LB k B Ac

R cond, B =

L 1 Rcond, A = A h Ac k A Ac

0.02

K W

Rcond, C =

0.05

K W

Rcond, D =

0.1

K W T = 100°C H

LD 1 R conv = h Ac k D Ac

2 LC kC Ac

Figure 2-33: Resistance network representing the isothermal limit of the behavior of the composite structure.

0.2

TC = 0°C

K W

R conv,1 = 0.2

K W

2.0

K W

0.1

K W

0.2

K W

2 2 LA 2 LB 2 LD 2 Rcond, A,1 = Rcond, B = R cond, D,1 = Rconv,1 = A k D Ac h Ac k A Ac kB c h Ac TH = 100°C

K W

R conv, 2 =

0.1

0.1

K W

0.02

K W

0.1

K W

0.2

K W

2 2 LA 2 LD 2 2 LC Rcond, D, 2 = R conv, 2 = R cond, A, 2 = R cond, C = k A Ac k D Ac h Ac h Ac kC Ac

Figure 2-34: Resistance network representing the adiabatic limit of the behavior of the composite structure.

2.8 Resistance Approximations for Conduction Problems

275 adiabatic surfaces

area Ac, 2 at TC

Figure 2-35: An example of a geometry with constant length and varying area. L area Ac,1 at TH L

are useful for determining the validity of a 2-D solution as well as bounding the problem without requiring a 2-D solution.

2.8.3 Average Area and Average Length Resistance Limits There are conduction problems in which the length for conduction is known but the area of the conduction path varies along this length; a simple example is shown in Figure 2-35. The adiabatic approximation discussed in Section 2.8.2 suggests that the resistance of this shape is: Rad =

2L k Ac,1

(2-233)

where k is the conductivity of the material. The isothermal approximation for the resistance yields: Riso =

L L + k Ac,1 k Ac,2

(2-234)

An alternative technique for estimating the resistance in this situation is to use the average area for conduction: RA =

4L k (Ac,1 + Ac,2 )

(2-235)

The resistance based on the average area underestimates the actual resistance to a greater extent than even the isothermal approximation. To see that this is so, imagine the case where Ac,1 approaches zero; clearly the actual resistance will become inﬁnite and both the adiabatic and isothermal approximations provided by Eqs. (2-233) and (2-234), respectively, predict this. However, the average area estimate remains ﬁnite and therefore substantially under-predicts the resistance. The alternative situation may occur, where the area for conduction is essentially constant but the length varies (perhaps randomly, as in a contact resistance problem). In this case, it is natural to use an average length to compute the resistance as shown in Figure 2-36. The average length estimate of the resistance (RL) is: RL =

L k Ac

(2-236)

where L is the average length for conduction. The average length model overestimates the resistance to a greater extent than the adiabatic approximation. Consider the case where the length anywhere within the shape shown in Figure 2-36 approaches zero, which would cause the actual resistance to become zero. The adiabatic approximation will faithfully predict a zero resistance while the average length estimate will remain

276

Two-Dimensional, Steady-State Conduction

L

TH

TC

Figure 2-36: An example of a geometry with constant area and varying length.

Ac

ﬁnite. In terms of accuracy, the various 1-D estimates that have been discussed can be arranged in the following order: RA ≤ Riso ≤ R ≤ Rad ≤ RL

(2-237)

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

where R is the actual thermal resistance. EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL Figure 1 illustrates a square channel. The inner and outer surfaces are held at different temperatures, T1 = 250◦ C and T2 = 50◦ C, respectively. T2 = 50°C T1 = 250°C

L=1m b = 5 cm

a = 10 cm b = 5 cm

Figure 1: Square channel with heat transfer from inner to outer surface.

a = 10 cm

The outer dimension of the square channel is a = 10 cm and the inner dimension is b = 5.0 cm. The thermal conductivity of the material is k = 100 W/m-K and the length of the channel is L = 1 m. a) Using an appropriate shape factor, determine the actual rate of heat transfer through the square channel. The inputs are entered in EES: “EXAMPLE 2.8-2: Resistance of a Square Channel” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” a=10[cm]∗ convert(cm,m) b=5[cm]∗ convert(cm,m) L=1[m] k=100[W/m-K] T 1=converttemp(C,K,250 [C]) T 2=converttemp(C,K,50 [C])

“outer dimension of square channel” “inner dimension” “length” “material conductivity” “inner wall temperature” “outer wall temperature”

277

This problem is a 2-D conduction problem; however, the solution for this 2-D problem is correlated in the form of a shape factor, S. Shape factors are discussed in Section 2.1. The shape factor (S) is deﬁned according to: q˙ cond = S k (T1 − T2 ) where q˙ cond is the rate of conductive heat transfer between the two surfaces at T1 and T2 . The particular shape factor for a square channel can be accessed from EES’ built-in shape factor library. Select Function Info from the Options menu and select Shape Factors from the lower-right pull down menu. Use the scroll-bar to select the shape factor function for a square channel. The function SF_7 is pasted into the Equation Window using the Paste button and used to calculate the actual heat transfer rate: “Actual heat transfer rate” SF=SF 7(b,a,L) q dot=SF∗ k∗ (T 1-T 2) q dot kW=q dot∗ convert(W,kW)

“shape factor for square channel” “heat transfer” “heat transfer in kW”

The actual heat transfer rate predicted using a shape factor solution is 211.4 kW. b) Provide a lower bound on the heat transfer through the square channel using an appropriate 1-D model. According to Eq. (2-237), the adiabatic or average length models can be used to provide an upper bound on the resistance of the square channel. The adiabatic model does not allow the heat to spread as it moves across the channel, as shown in Figure 2(a).

(a − b )

(a − b )

2 b

2

(a + b ) 2

(a)

(b)

Figure 2: 1-D models based on the (a) adiabatic limit which allows no heat spreading and (b) the average area limit which uses the average area along the heat transfer path.

The resistance in the adiabatic limit is equal to the resistance of a plane wall with an area equal to the internal surface area of the channel and length equal to the channel thickness: a−b Rad = 8k L b The rate of heat transfer predicted by the adiabatic limit is: q˙ ad =

(T1 − T2 ) Rad

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

2.8 Resistance Approximations for Conduction Problems

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

278

Two-Dimensional, Steady-State Conduction

“Adiabatic limit” R ad=(a-b)/(8∗ k∗ L∗ b) q dot ad=(T 1-T 2)/R ad q dot ad kW=q dot ad∗ convert(W,kW)

“thermal resistance in the adiabatic limit” “heat transfer in the adiabatic limit”

The adiabatic model predicts 160.0 kW and therefore leads to a 32% underestimate of the actual heat transfer rate (211.4 kW from part (a)). c) Provide an upper bound on the heat transfer rate using an appropriate 1-D model. Equation (2-237) indicates that either the isothermal or average area approach can be used to establish a lower bound on the thermal resistance and therefore an upper bound on the heat transfer. The average area along the heat transfer path is shown in Figure 2(b). The resistance calculated according to the average area model is: RA =

(a − b) 4 a+b Lk

The heat transfer in this limit is: q˙ A =

(T1 − T2 ) RA

“Average area limit” R A bar=(a-b)/(4∗ (a+b)∗ k∗ L) “thermal resistance in the average area limit” “heat transfer in the average area limit” q dot A bar=(T 1-T 2)/R A bar q dot A bar kW=q dot A bar∗ convert(W,kW)

The average area approximation predicts a heat transfer rate of 240 kW and is therefore a 12% overestimate of the actual heat transfer rate.

2.9 Conduction through Composite Materials 2.9.1 Effective Thermal Conductivity Composite structures are made by joining different materials to create a structure with beneﬁcial properties. Composites are often encountered in engineering applications; for example, motor laminations and windings, screens, and woven fabric composites. The length scale associated with the underlying structure of the composite material is often much smaller than the length scale associated with the overall problem of interest and therefore the details of the local energy ﬂow through the materials that make up the structure are not important. In this case, the composite material can be modeled as a single, equivalent material with an effective conductivity that reﬂects the more complex behavior of the underlying structure. If the composite structure is not isotropic (i.e., it is anisotropic) then the effective conductivity may also be anisotropic; that is, the effective conductivity may depend on direction, reﬂecting some underlying characteristic of the composite structure that allows heat to ﬂow more easily in certain directions. The effective thermal conductivity of the composite structure must be determined by considering the details associated with heat transfer through the structure. The process of determining the effective conductivity involves (theoretically) imposing a temperature gradient in one direction and evaluating the resulting heat transfer rate. The

2.9 Conduction through Composite Materials epoxy kep = 2 W/m-K thep = 0.2 mm

Ttop = 40°C

279

iron 3 g⋅ ′′′= 100,000 W/m thlam = 0.5 mm klam = 10 W/m-K

H = 6 cm W q⋅ ′′ = 5000 2 m

W q⋅ ′′= 5000 2 m y x

L = 4 cm

Figure 2-37: Motor pole.

effective conductivity in that direction is the conductivity of a homogeneous material that would yield the same heat transfer rate. It is often possible to determine the effective conductivity by inspection; however, for complex structures it will be necessary to generate a detailed, numerical model of a unit cell of the structure using a ﬁnite difference or ﬁnite element technique. The combination of a local model of the very small scale features of the underlying structure (in order to determine the effective conductivity) and a larger scale model of the global problem is sometimes referred to as multi-scale modeling. Other effective characteristics of the composite may also be important. For example, an effective rate of volumetric generation or, for transient problems, an effective speciﬁc heat capacity and density. An effective property is the property that a homogenous material must have if it is to behave in the same way as the composite. The process of estimating and using an effective conductivity to model a composite structure is illustrated in the context of the motor pole shown in Figure 2-37. The pole is composed of laminations of iron that are separated by an epoxy coating. Each iron lamination is thlam = 0.5 mm thick and has conductivity klam = 10 W/m-K while the epoxy coating is approximately thep = 0.2 mm thick and has conductivity kep = 2.0 W/m-K. The motor pole is adiabatic on its bottom surface and experiences a heat ﬂux of q˙ = 5000 W/m2 from the windings on the sides. The top surface is maintained at a temperature of T top = 40◦ C. The pole is L = 4.0 cm long and H = 6.0 cm high. The temperature distribution in the pole is 2-D in the x- and y-directions. The iron laminations are generating thermal energy due to eddy current heating at a volumetric rate of g˙ = 100,000 W/m3 . There is no thermal energy generation in the epoxy. The inputs required to determine the effective conductivity are entered into EES:

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” k lam=10 [W/m-K] k ep=2 [W/m-K] th lam=0.5 [mm]∗ convert(mm,m) th ep=0.2 [mm]∗ convert(mm,m) g dot=100000 [W/mˆ3]

“lamination conductivity” “epoxy conductivity” “lamination thickness” “epoxy thickness” “rate of volumetric generation in the laminations”

280

Two-Dimensional, Steady-State Conduction

Heat transfer in the axial (x) direction occurs through the laminations and epoxy in parallel. The methodology for calculating the effective conductivity in the axial direction consists of imposing a temperature difference ( T) in the x-direction (i.e., across the width of the pole) and calculating the heat transfer rate through the two parallel paths. The effective conductivity in the x-direction (keff,x ) is the conductivity of a homogeneous material that would provide the same heat transfer rate. The rate of heat transfer through the iron laminations is: q˙ lam =

klam thlam H W T (thlam + thep) L

(2-238)

where W is the width of the pole (into the page). The heat transfer rate through the epoxy is: q˙ ep =

kep thep H W T (thlam + thep) L

(2-239)

The total heat transfer rate in the x-direction through the equivalent material (q˙ eff ,x ) must therefore be: keff ,x H W kep thep H W klam thlam H W T + T = T q˙ eff ,x = q˙ lam + q˙ ep = (thlam + thep) L (thlam + thep) L L (2-240) or, solving for keff ,x : keff ,x =

kep thep klam thlam + (thlam + thep) (thlam + thep)

(2-241)

The effective conductivity in the x-direction is the thickness (or area) weighted average of the conductivity of the two parallel paths. The effective conductivity calculated using Eq. (2-241) depends only on the details of the microstructure (e.g., the conductivity of the laminations and their thickness) and not the macroscopic details of the problem (e.g., the size of the motor pole and the boundary conditions on the problem). k_eff_x=k_lam∗ th_lam/(th_lam+th_ep)+k_ep∗ th_ep/(th_lam+th_ep) “eff. conductivity in the x-direction”

which leads to keff ,x = 7.71 W/m-K. Note that the effective conductivity must lie between the conductivity of the epoxy (2 W/m-K) and the laminations (10 W/m-K). If the thickness of the lamination becomes large relative to the thickness of the epoxy then the effective conductivity will approach the lamination conductivity. If the conductivity of one material (for example, the lamination) is substantially greater than the other (for example, the epoxy) then the effective conductivity in the x-direction will approach the product of the conductivity of the more conductive material and the fraction of the thickness that is occupied by that material. This behavior is typical of any parallel resistance network because the rate of heat transfer is more sensitive to the smaller resistance. Heat transferred in the y-direction must pass through the laminations and epoxy in series. A temperature difference is applied across the pole in the y-direction. The heat transfer rate is calculated and used to establish the effective conductivity in the y-direction: q˙ eff ,y =

keff ,y W L T T = H thep H thlam H + (thlam + thep) klam W L (thlam + thep) kep W L

(2-242)

2.9 Conduction through Composite Materials

281

or, solving for keff ,y : keff ,y =

1 thep thlam + (thlam + thep) klam (thlam + thep) kep

(2-243)

k_eff_y=1/(th_lam/((th_lam+th_ep)∗ k_lam)+th_ep/((th_lam+th_ep)∗ k_ep)) “eff. conductivity in the y-direction”

which leads to keff ,y = 4.67 W/m-K. The effective conductivity is again bounded by kep and klam . The effective conductivity in the y-direction is dominated by the thicker and less conductive of the two materials. This behavior is typical of a series resistance network, where the larger resistance is the most important. In addition to the effective conductivity, it is necessary to determine an effective rate of volumetric generation (g˙ eff ) that characterizes the motor pole. This is the volumetric generation rate for an equivalent piece of homogeneous material that produces the same total rate of generation. The total rate of energy generation in the motor pole is: g˙ = g˙

H thlam W L = g˙ eff H W L (thlam + thep)

(2-244)

and therefore: ˙ g˙ eff = g

g _dot_eff=g _dot∗ th_lam/(th_lam+th_ep)

thlam (thlam + thep)

(2-245)

“effective rate of volumetric generation”

3 which leads to g˙ eff = 71,400 W/m . The effective properties of the motor pole, keff ,x , keff ,y , and g˙ eff , can be used to generate a model of the motor pole that does not explicitly consider the microscale features of the composite structure but does capture the overall geometry and boundary conditions of the problem shown in Figure 2-37. The geometry is entered in FEHT as discussed in Section 2.7.1 and Appendix A.4. A grid is used where 1 cm of the screen corresponds to 1 cm and the corner nodes are approximately positioned using the Outline command in the Draw menu. The corner nodes are then precisely positioned by right-clicking on each in turn. The material properties are set by clicking on the outline and selecting Material Properties from the Specify menu. Create a new material (select “not speciﬁed” from the list of materials) and rename it lamination. Click on the box next to Type until the choice is Anisotropic; set the x-conductivity to 7.71 W/m-K and the ratio ky/kx to 0.605 (the ratio of keff ,x to keff ,y ). Set the effective rate of volumetric generation by clicking on the outline and selecting Generation from the Specify menu. Set the boundary conditions according to Figure 2-37 and draw a crude grid (two triangles formed by a single element line across the pole will do). Reﬁne the grid multiple times and solve. The temperature distribution in the pole is shown in Figure 2-38.

282

Two-Dimensional, Steady-State Conduction

Figure 2-38: Temperature distribution in the motor pole (◦ C).

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

Note that the 2-D temperature distribution is adequately captured using the ﬁnite element model with equivalent properties; however, the actual temperature distribution would include “ripples” corresponding to the effects of the individual laminations. Unless the characteristics of these very small-scale effects are important, it is convenient to consider the effect of the micro-structure on the larger-scale problem using the effective conductivity concept. EXAMPLE 2.9-1: FIBER OPTIC BUNDLE Lighting represents one of the largest uses of electrical energy in residential and commercial buildings; lighting loads are highest during on-peak hours when electrical energy is most costly. Also, the thermal energy deposited into conditioned space by electrical lighting adds to the air conditioning load on the building which, in turn, adds to the electrical energy required to run the air conditioning system.

Figure 1: Hybrid lighting system (Cheadle, 2006).

A novel lighting system consists of a sunlight collector and a light distribution system, as shown in Figure 1. The sunlight collector tracks the sun and collects and concentrates solar radiation. The light distribution system receives the concentrated solar radiation and distributes it into a building where it is ﬁnally dispensed

283

in ﬁxtures that are referred to as luminaires. Sunlight contains both visible and invisible energy; only the visible portion of the sunlight is useful for lighting and therefore the collector gathers the visible portion of the incident solar radiation while eliminating the invisible ultraviolet and infrared portions of the spectrum. (We will learn more about these characteristics of radiation in Chapter 10.) The ﬁber optic bundle used to transmit the visible light is composed of many, small diameter optical ﬁbers that are packed in approximately a hexagonal close-packed array. A conductive ﬁller material is wrapped around each ﬁber and the entire structure is simultaneously heated and compressed so that the ﬁbers becomes hexagonal shaped with a thin layer of conductive ﬁller separating each hexagon (Figure 2). The dimension of each face of the hexagon is d = 1.0 mm and the thickness of the ﬁller that separates the hexagons is a = 50 μm thick. The ﬁber conductivity is k f b = 1.5 W/m-K while the ﬁller conductivity is k fl = 50.0 W/m-K.

d = 1 mm optical fiber, kfb = 1.5 W/m-K unit cell used for FEHT model (Figure 3)

a = 50 μm

r

filler material, kfl = 50 W/m-K

x

fiber optic bundle

Figure 2: Array of optical ﬁbers packed together and compressed in order to form a ﬁber optic bundle that has hexagonal units.

a) Determine the effective radial and axial conductivity associated with the bundle. The inputs are entered in EES:

“EXAMPLE 2.9-1: Fiber Optic Bundle” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Composite Structure Inputs” d=1 [mm]∗ convert(mm,m) a=0.05 [mm]∗ convert(mm,m) k ﬂ=50 [W/m-K] k fb=1.5 [W/m-K]

“face dimension of hexagon” “ﬁller thickness” “conductivity of ﬁller” “conductivity of ﬁber”

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

284

Two-Dimensional, Steady-State Conduction

The heat transfer in the axial direction can travel through two parallel paths, the ﬁller and the ﬁber. The area of a single ﬁber (a hexagon with each side having length d) is: π ' π ( 1 + cos Af b = 2 d 2 sin 3 3 A fb=2∗ dˆ2∗ sin(pi/3)∗ (1+cos(pi/3))

“area of a ﬁber”

The heat transfer through the ﬁber in the axial direction for a given temperature difference (T ) is: T k f b Af b L where L is the length of the ﬁber. The area of the ﬁller material associated with a single ﬁber (which has thickness a/2 due to sharing of the ﬁller with the neighboring ﬁbers) is: q˙ f b =

Afl = 3 d a A ﬂ=3∗ d∗ a

“area of ﬁller associated with a ﬁber”

The heat transfer through a unit length of the ﬁller in the axial direction for the same temperature difference is: T k fl Afl L The equivalent homogenous material will have conductivity k eff ,x and area Afl + Af b ; therefore, the heat transfer through the equivalent material (q˙ eff ,x ) is: q˙ fl =

T k eff ,x (Afl + Af b ) L The effective conductivity is deﬁned so that the heat transfer through the equivalent material is equal to the sum of the heat transfer through the ﬁber and the ﬁller: q˙ eff ,x =

T k eff ,x (Afl + Af b ) T k f b Af b T k fl Afl = + L L L q˙ eff ,x

q˙ fb

q˙ fl

or, solving for k eff ,x : k eff ,x =

k f b Af b + k fl Afl Afl + Af b

The effective conductivity in the axial direction is the area-weighted conductivity of the two parallel paths: k eff x=(k fb∗ A fb+k ﬂ∗ A ﬂ)/(A ﬂ+A fb)

“effective conductivity in the axial direction”

which leads to k eff ,x = 4.1 W/m-K. The radial conductivity cannot be evaluated using a simple parallel or series resistance circuit because the heat ﬂow across the bundle is complex and 2-D.

285

Therefore, a 2-D ﬁnite element model of a unit cell of the structure (shown in Figure 2) must be generated. Figure 3 illustrates the details of a unit cell and includes the coordinates of the points (in mm) that deﬁne the geometry. (0.75,1.782) (0,1.782)

(0.808,1.782) (1.558,1.782)

Region 3

(1.058, 1.324) (1.0, 1.324) (1.058, 1.299)

(1.558, 1.349) (1.558, 1.299)

Region 2 (0, 0.483) (0, 0.433)

Region 1 (0, 0)

(0.5,0.483) (0.558,0.458) (0.5,0.433)

Region 5 Region 4

(1.558, 0) (0.75, 0)

(0.808, 0)

Figure 3: A unit cell of the ﬁber optic bundle structure; the points that deﬁne the structure are shown (dimensions are in mm).

The ﬁnite element model is generated using FEHT as discussed in Section 2.7.1 and Appendix A.4. A grid is speciﬁed where 1 cm of screen dimension corresponds to 0.2 mm. The ﬁve regions in Figure 3 are generated using ﬁve outlines; the points are initially placed approximately and then precisely positioned by double-clicking on each one in turn. The conductivity for regions 1 through 4 are set to 1.5 W/m-K (consistent with the optical ﬁber) and the conductivity for region 5 is set to 50.0 W/m-K (consistent with the ﬁller material). The boundary conditions along the upper and lower edges are set as adiabatic. In order to set a temperature difference across the unit cell (from left to right) it would seem logical to set the temperature at the left hand side to 1.0◦ C and the right side to 0.0◦ C. However, due to the manner in which ﬁnite element techniques determine the heat ﬂux at a surface, a more accurate answer is obtained if a convective boundary condition is set with a very high heat transfer coefﬁcient; for example, 1 × 105 W/m2 -K. A relatively crude mesh is generated and then reﬁned, particularly in the ﬁller material where most of the heat ﬂow is expected. The result is shown in Figure 4(a). The ﬁnite element model is solved and the temperature contours are shown in Figure 4(b). The heat ﬂow through the unit cell can be determined by selecting Heat Flows from the View menu and then selecting all of the boundaries on either the left or right side. The selection process can be accomplished by using the mouse to ‘drag’ a selection rectangle around the lines on the boundary. The total heat ﬂow is q˙ eff ,r /L = 3.070 W/m; this result can be used to compute the effective conductivity of the composite. The effective conductivity across the bundle (k eff ,r ) is deﬁned as the conductivity of a homogeneous material that would provide the same heat transfer per length into the page as the composite structure simulated by the ﬁnite element model. q˙ eff ,r H T = k eff ,r L W

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

Two-Dimensional, Steady-State Conduction

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

286

(a)

(b)

Figure 4: Finite element model showing (a) the reﬁned grid and (b) the temperature distribution.

where H is the height of the unit cell (1.782 mm, from Figure 3), W is the width of the unit cell (1.558 mm, from Figure 3), and T = 1.0 K (the imposed temperature difference). “k eff r calculation using inputs from FEHT” H=1.782 [mm]∗ convert(mm,m) “height of unit cell” W=1.558 [mm]∗ convert(mm,m) “width of unit cell” q dot eff r\L=3.070 [W/m] “rate of heat transfer per unit length predicted by FE model” DT=1 [K] “temperature difference imposed on FE model” q dot eff r\L=k eff r∗ H∗ DT/W “effective conductivity in the radial direction”

which leads to k eff ,r = 2.7 W/m-K. The effective conductivity in the x-direction is 4.1 W/m-K while the effective conductivity in the radial direction is 2.7 W/m-K. These values make sense. Both lie between the conductivity of the ﬁber and the ﬁller and both are closer to the conductivity of the ﬁber because it occupies most of the space. Further, the conductivity in the x-direction is larger because the path through the high conductivity ﬁller is more direct in this direction. The advantage of the hexagonal pattern in Figure 2 is that is has the lowest possible porosity (φ). The porosity is deﬁned as the fraction of the area of the face of the bundle that is occupied by the ﬁller. The optical ﬁbers are designed so that any radiation that strikes the face of a ﬁber is “trapped” by total internal reﬂection and transmitted without substantial loss. However, the radiation that strikes the opaque ﬁller in the interstitial areas between the ﬁbers will be absorbed and result in a thermal load on the bundle that manifests itself as a heat ﬂux on the surface. This heat ﬂux can lead to elevated temperatures and thermal failure. Assume that the outer edge of the ﬁber optic bundle is exposed to air at T∞ = ◦ 20 C with a heat transfer coefﬁcient hout = 5.0 W/m2 -K. The front face of the bundle (at x = 0) is exposed to air at T∞ = 20◦ C with a heat transfer coefﬁcient hf =

287

10.0 W/m2 -K. The radius of the bundle is r out = 2.0 cm and its length is L = 2.5 m. The end of the bundle at x = L is maintained at a temperature of TL = 20◦ C. = 1 × 105 W/m2 . A The radiant heat ﬂux incident on the face of the bundle is q˙inc schematic of the problem is shown in Figure 5. 2

T∞ = 20°C, hout = 5 W/m -K q⋅

keff, x and keff, r

T∞ = 20°C 2 h f = 10 W/m -K

conv

r = 2.0 cm

TL = 20°C

r x

q⋅ x

W ′′ = 1x10 2 q⋅inc m on the filler material 5

q⋅x+dx

dx L = 2.5 m

Figure 5: Schematic of the ﬁber optic bundle problem.

b) Is it appropriate to treat the bundle as an extended surface? Justify your answer. The additional inputs for the problem are entered in EES: “Problem Inputs” r out=2 [cm]∗ convert(cm,m) L=2.5 [m] h bar out=5 [W/mˆ2-K] T inﬁnity=converttemp(C,K,20[C]) h bar f=10 [W/mˆ2-K] T L=converttemp(C,K,20[C]) q dot ﬂux inc=100000 [W/mˆ2]

“outer radius” “bundle length” “heat transfer coefﬁcient on outer surface” “air temperature” “heat transfer coefﬁcient on the face” “temperature at x=L” “incident radiant heat ﬂux on the face”

The extended surface approximation neglects temperature gradients in the radial direction within the bundle. The conduction resistance in the radial direction (Rcond,r ) must be neglected in order to treat the bundle as an extended surface. This assumption is justiﬁed provided that Rcond,r is small relative to the resistance that is being considered, convection from the outer surface of the bundle (Rconv ). The appropriate Biot number is therefore: Bi =

Rcond ,r Rconv

It is not possible to precisely compute a resistance that characterizes conduction in the radial direction within the bundle; conduction from the center of the bundle is characterized by an inﬁnite resistance. Instead, an approximate conduction length (r out /2) and area (πr out L) are used: r out hout 2 π r out L hout r out Bi = = 2 k eff ,r π r out L 1 k eff ,r Rcond ,r

Bi=h bar out∗ r out/k eff r

Rconv

“Biot number”

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

288

Two-Dimensional, Steady-State Conduction

The Biot number is 0.037 which is much less than 1 and therefore the extended surface approximation is justiﬁed. c) Develop an analytical model for the temperature distribution in the bundle. The differential control volume used to derive the governing equation is shown in Figure 5 and leads to the energy balance: q˙ x = q˙ x+d x + q˙ conv which can be expanded and simpliﬁed: 0=

d q˙ x d x + q˙ conv dx

(1)

The rate equations are: 2 q˙ x = −k eff ,x π r out

dT dx

q˙ conv = 2 π r out hout d x (T − T∞ )

(2) (3)

Substituting Eqs. (1) and (2) into Eq. (3) leads to: d 2 dT 0= −k eff ,x π r out d x + 2 π r out hout d x (T − T∞ ) dx dx or d 2T − m2 T = −m2 T∞ d x2

(4)

where m2 =

2 hout k eff ,x r out

The governing differential equation, Eq. (4), is satisﬁed by exponential functions. The differential equation can also be entered in Maple and solved. > restart; > ODE:=diff(diff(T(x),x),x)-mˆ2∗ T(x)=-mˆ2∗ T_inﬁnity; 2 d T(x) − m2T(x) = −m2T inﬁnity O D E := d x2 > Ts:=dsolve(ODE); Ts := T(x) = e(−mx) C 2 + e(mx) C 1 + T inﬁnity

The solution is copied and pasted into EES. “Solution” m=sqrt(2∗ h bar out/(k eff x∗ r out)) T=exp(-m∗ x)∗ C 2+exp(m∗ x)∗ C 1+T inﬁnity

“solution parameter” “solution from Maple”

The ﬁrst boundary condition is the speciﬁed temperature at x = L: Tx=L = TL

289

A symbolic expression for this boundary condition is obtained in Maple: > rhs(eval(Ts,x=L))=T_L; e(−mL) C 2 + e(mL) C 1 + T inﬁnity = T L

and pasted into EES: exp(-m∗ L)∗ C 2+exp(m∗ L)∗ C 1+T inﬁnity=T L

“boundary condition at x=L”

The second boundary condition is obtained from an interface energy balance at x = 0. Recall that only the ﬂux incident on the ﬁller material results in a heat load. Therefore, the absorbed ﬂux is the product of incident heat ﬂux and the porosity (φ), which is the ratio of the area of the ﬁller to the total area: φ=

phi=A ﬂ/(A ﬂ+A fb)

Afl Afl + Af b

“porosity”

The absorbed ﬂux must either be transferred by conduction to the bundle or convection to the air: dT + hf (Tx=0 − Tair ) q˙inc φ = −k eff ,x d x x=0 A symbolic expression for this boundary condition is obtained in Maple: > q_dot_ﬂux_inc∗ phi=-k_eff_x∗ rhs(eval(diff(Ts,x),x=0))+h_bar_f∗ (rhs(eval(Ts,x=0))-T_inﬁnity);

q d ot f lux inc φ = k eff x(−m C 2 + m C 1) + h bar f ( C 2 + C 1)

and pasted into EES: q_dot_ﬂux_inc∗ phi = -k_eff_x∗ (-m∗ C_2+m∗ C_1)+h_bar_f∗ (C_2+C_1) “boundary condition at x=0”

The solution is converted to Celsius: T C=converttemp(K,C,T)

“temperature in C”

Figure 6 illustrates the temperature distribution near the face of the ﬁber optic bundle.

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

Two-Dimensional, Steady-State Conduction

120

Temperature (°C)

100 80 60

40 20 0

0.1

0.2

0.3

0.4 0.5 0.6 Position (m)

0.7

0.8

0.9

1

Figure 6: Temperature distribution in ﬁber optic bundle.

The model can be used to assess alternative methods of thermal management. For example, the heat transfer coefﬁcient at the face might be increased by adding a fan or the conductivity of the ﬁller material might be increased through material selection. Figure 7 illustrates the maximum temperature (the temperature at the face) as a function of hf for various values of k f . 130 120 Temperature at the face (°C)

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

290

110 100 90

k f [W/m-K]

80

50 100

70

150

60 50 0

10 20 30 40 50 60 70 80 90 Heat transfer coefficient on the face (W/m2-K)

100

Figure 7: Temperature at the face as a function of the face heat transfer coefﬁcient for various values of the ﬁller material conductivity.

Chapter 2: Two-Dimensional, Steady-State Conduction The website associated with this book (www.cambridge.org/nellisandklein) provides many more problems than are included here.

Chapter 2: Two-Dimensional, Steady-State Conduction

291

Shape Factors 2–1 Figure P2-1 illustrates two tubes that are buried in the ground behind your house and transfer water to and from a wood burner. The left tube carries hot water from the burner back to your house at T w,h = 135◦ F while the right tube carries cold water from your house to the burner at T w,c = 70◦ F. Both tubes have outer diameter Do = 0.75 inch and thickness th = 0.065 inch. The conductivity of the tubing material is kt = 0.22 W/m-K. The heat transfer coefﬁcient between the water and the tube internal surface (in both tubes) is hw = 250 W/m2 -K. The center to center distance between the tubes is w = 1.25 inch and the length of the tubes is L = 20 ft (into the page). The tubes are buried in soil that has conductivity ks = 0.30 W/m-K. ks = 0.30 W/m-K th = 0.065 inch

kt = 0.22 W/m-K

Tw, c = 70°F hw = 250 W/m2 -K

Tw, h = 135°F h w = 250 W/m2 -K w = 1.25 inch Do = 0.75 inch Figure P2-1: Tubes buried in soil.

a.) Estimate the heat transfer from the hot water to the cold water due to the proximity of the tubes to one another. b.) To do part (a) you should have needed to determine a shape factor; calculate an approximate value of the shape factor and compare it to the accepted value. c.) Plot the rate of heat transfer from the hot water to the cold water as a function of the center to center distance between the tubes. 2–2 A solar electric generation system (SEGS) employs molten salt as both the energy transport and storage ﬂuid. The molten salt is heated to 500◦ C and stored in a buried hemispherical tank. The top (ﬂat) surface of the tank is at ground level. The diameter of the tank before insulation is applied is 14 m. The outside surfaces of the tank are insulated with 0.30 m thick ﬁberglass having a thermal conductivity of 0.035 W/m-K. Sand having a thermal conductivity of 0.27 W/m-K surrounds the tank, except on its top surface. Estimate the rate of heat loss from this storage unit to the 25◦ C surroundings. Separation of Variables Solutions 2–3 You are the engineer responsible for a simple device that is used to measure the heat transfer coefﬁcient as a function of position within a tank of liquid (Figure P2-3). The heat transfer coefﬁcient can be correlated against vapor quality, ﬂuid composition, and other useful quantities. The measurement device is composed of many thin plates of low conductivity material that are interspersed with large, copper interconnects. Heater bars run along both edges of the thin plates. The heater bars are insulated and can only transfer energy to the plate; the heater bars are conductive and can therefore be assumed to come to a uniform temperature as a current is applied. This uniform temperature is assumed to be applied to the top and bottom

292

Two-Dimensional, Steady-State Conduction

edges of the plates. The copper interconnects are thermally well-connected to the ﬂuid; therefore, the temperature of the left and right edges of each plate are equal to the ﬂuid temperature. This is convenient because it isolates the effect of adjacent plates from one another, allowing each plate to measure the local heat transfer coefﬁcient. Both surfaces of the plate are exposed to the ﬂuid temperature via a heat transfer coefﬁcient. It is possible to infer the heat transfer coefﬁcient by measuring heat transfer required to elevate the heater bar temperature to a speciﬁed temperature above the ﬂuid temperature. top and bottom surfaces exposed to fluid T∞ = 20°C, h = 50 W/m2 -K copper interconnect, T∞ = 20°C

a = 20 mm b = 15 mm

plate: k = 20 W/m-K th = 0.5 mm

heater bar, Th = 40°C

Figure P2-3: Device to measure local heat transfer coefﬁcient.

The nominal design of an individual heater plate utilizes metal with k = 20 W/m-K, th = 0.5 mm, a = 20 mm, and b = 15 mm (note that a and b are deﬁned as the half-width and half-height of the heater plate, respectively, and th is the thickness as shown in Figure P2-3). The heater bar temperature is maintained at T h = 40◦ C and the ﬂuid temperature is T ∞ = 20◦ C. The nominal value of the heat transfer coefﬁcient is h = 50 W/m2 -K. a.) Develop an analytical model that can predict the temperature distribution in the plate under these nominal conditions. b.) The measured quantity is the rate of heat transfer to the plate from the heater (q˙ h ) and therefore the relationship between q˙ h and h (the quantity that is inferred from the heater power) determines how useful the instrument is. Determine the heater power and plot the heat transfer coefﬁcient as a function of heater power. c.) If the uncertainty in the measurement of the heater power is δq˙ h = 0.01 W, estimate the uncertainty in the measured heat transfer coefﬁcient (δh). 2–4 A laminated composite structure is shown in Figure P2-4. H = 3 cm

q⋅ ″ = 10000 W/m2

Tset = 20°C Tset = 20°C

W = 6 cm k x = 50 W/m-K k y = 4 W/m-K

Figure P2-4: Composite structure exposed to a heat ﬂux.

Chapter 2: Two-Dimensional, Steady-State Conduction

293

The structure is anisotropic. The effective conductivity of the composite in the xdirection is kx = 50 W/m-K and in the y-direction it is ky = 4 W/m-K. The top of the structure is exposed to a heat ﬂux of q˙ = 10,000 W/m2 . The other edges are maintained at T set = 20◦ C. The height of the structure is H = 3 cm and the halfwidth is W = 6 cm. a.) Develop a separation of variables solution for the 2-D steady-state temperature distribution in the composite. b.) Prepare a contour plot of the temperature distribution. Advanced Separation of Variables Solutions 2–5 Figure P2-5 illustrates a pipe that connects two tanks of liquid oxygen on a spacecraft. The pipe is subjected to a heat ﬂux, q˙ = 8,000 W/m2 , which can be assumed to be uniformly applied to the outer surface of the pipe and is entirely absorbed. Neglect radiation from the surface of the pipe to space. The inner radius of the pipe is rin = 6.0 cm, the outer radius of the pipe is rout = 10.0 cm, and the half-length of the pipe is L = 10.0 cm. The ends of the pipe are attached to the liquid oxygen tanks and therefore are at a uniform temperature of T LOx = 125 K. The pipe is made of a material with a conductivity of k = 10 W/m-K. The pipe is empty and therefore the internal surface can be assumed to be adiabatic. a.) Develop an analytical model that can predict the temperature distribution within the pipe. Prepare a contour plot of the temperature distribution within the pipe. rout = 10 cm rin = 6 cm

k = 10 W/m-K

L = 10 cm TLOx = 125 K

q⋅ ″s = 8,000 W/m2

Figure P2-5: Cryogen transfer pipe connecting two liquid oxygen tanks.

2–6 Figure P2-6 illustrates a cylinder that is exposed to a concentrated heat ﬂux at one end. extends to infinity

k = 168 W/m-K rout = 200 μm x

Ts = 20° C rexp = 21 μm

q⋅ ″ = 1500 W/cm2

adiabatic

Figure P2-6: Cylinder exposed to a concentrated heat ﬂux at one end.

294

Two-Dimensional, Steady-State Conduction

The cylinder extends inﬁnitely in the x-direction. The surface at x = 0 experiences a uniform heat ﬂux of q˙ = 1500 W/cm2 for r < rexp = 21 μm and is adiabatic for rexp < r < rout where rout = 200 μm is the outer radius of the cylinder. The outer surface of the cylinder is maintained at a uniform temperature of T s = 20◦ C. The conductivity of the cylinder material is k = 168 W/m-K. a.) Develop a separation of variables solution for the temperature distribution within the cylinder. Plot the temperature as a function of radius for various values of x. b.) Determine the average temperature of the cylinder at the surface exposed to the heat ﬂux. c.) Deﬁne a dimensionless thermal resistance between the surface exposed to the heat ﬂux and Ts . Plot the dimensionless thermal resistance as a function of rout /rin . d.) Show that your plot from (c) does not change if the problem parameters (e.g., Ts , k, etc.) are changed. Superposition 2–7 The plate shown in Figure P2-7 is exposed to a uniform heat ﬂux q˙ = 1 × 105 W/m2 along its top surface and is adiabatic at its bottom surface. The left side of the plate is kept at TL = 300 K and the right side is at TR = 500 K. The height and width of the plate are H = 1 cm and W = 5 cm, respectively. The conductivity of the plate is k = 10 W/m-K. 5 q⋅ ′′ = 1x10 W/m2

W = 5 cm

TL = 300 K

H = 1 cm

y

TR = 500 K

x k = 10 W/m-K Figure P2-7: Plate.

a.) Derive an analytical solution for the temperature distribution in the plate. b.) Implement your solution in EES and prepare a contour plot of the temperature. Numerical Solutions to Steady-State 2-D Problems using EES 2–8 Figure P2-8 illustrates an electrical heating element that is afﬁxed to the wall of a chemical reactor. The element is rectangular in cross-section and very long (into the page). The temperature distribution within the element is therefore twodimensional, T(x, y). The width of the element is a = 5.0 cm and the height is b = 10.0 cm. The three edges of the element that are exposed to the chemical (at x = 0, y = 0, and x = a) are maintained at a temperature Tc = 200◦ C while the upper edge (at y = b) is afﬁxed to the well-insulated wall of the reactor and can therefore be considered adiabatic. The element experiences a uniform volumetric rate of thermal energy generation, g˙ = 1 × 106 W/m3 . The conductivity of the material is k = 0.8 W/m-K.

Chapter 2: Two-Dimensional, Steady-State Conduction

295

reactor wall k = 0.8 W/m-K g⋅ ′′′= 1x106 W/m3

Tc = 200°C a = 5 cm y

Tc = 200°C

x

b = 10 cm

Tc = 200°C

Figure P2-8: Electrical heating element.

a.) Develop a 2-D numerical model of the element using EES. b.) Plot the temperature as a function of x at various values of y. What is the maximum temperature within the element and where is it located? c.) Prepare a sanily check to show that your solution behaves according to your physical intuition. That is, change some aspect of your program and show that the results behave as you would expect (clearly describe the change that you made and show the result). Finite-Difference Solutions to Steady-State 2-D Problems using MATLAB 2–9 Figure P2-9 illustrates a cut-away view of two plates that are being welded together. Both edges of the plate are clamped and held at temperature Ts = 25◦ C. The top of the plate is exposed to a heat ﬂux that varies with position x, measured from joint, according to: q˙ m (x) = q˙ j exp (−x/Lj ) where q˙ j = 1 × 106 W/m2 is the maximum heat ﬂux (at the joint, x = 0) and Lj = 2.0 cm is a measure of the extent of both edges are clamped and held at fixed temperature

heat flux joint

impingement cooling with liquid jets ⋅ q′′ m

k = 38 W/m-K

W = 8.5 cm b = 3.5 cm

y

Ts = 25°C

x Tf = −35°C h = 5000 W/m2 -K Figure P2-9: Welding process and half-symmetry model of the welding process.

296

Two-Dimensional, Steady-State Conduction

the heat ﬂux. The back side of the plates are exposed to liquid cooling by a jet of ﬂuid at T f = −35◦ C with h = 5000 W/m2 -K. A half-symmetry model of the problem is shown in Figure P2-9. The thickness of the plate is b = 3.5 cm and the width of a single plate is W = 8.5 cm. You may assume that the welding process is steady-state and 2-D. You may neglect convection from the top of the plate. The conductivity of the plate material is k = 38 W/m-K. a.) Develop a separation of variables solution to the problem. Implement the solution in EES and prepare a plot of the temperature as a function of x at y = 0, 1.0, 2.0, 3.0, and 3.5 cm. b.) Prepare a contour plot of the temperature distribution. c.) Develop a numerical model of the problem. Implement the solution in MATLAB and prepare a contour or surface plot of the temperature in the plate. d.) Plot the temperature as a function of x at y = 0, b/2, and b and overlay on this plot the separation of variables solution obtained in part (a) evaluated at the same locations.

Finite Element Solutions to Steady-State 2-D Problems using FEHT 2–10 Figure P2-10(a) illustrates a double paned window. The window consists of two panes of glass each of which is tg = 0.95 cm thick and W = 4 ft wide by H = 5 ft high. The glass panes are separated by an air gap of g = 1.9 cm. You may assume that the air is stagnant with ka = 0.025 W/m-K. The glass has conductivity kg = 1.4 W/m-K. The heat transfer coefﬁcient between the inner surface of the inner pane and the indoor air is hin = 10 W/m2 -K and the heat transfer coefﬁcient between the outer surface of the outer pane and the outdoor air is hout = 25 W/m2 -K. You keep your house heated to Tin = 70◦ F. width of window, W = 4 ft Tin = 70°F hin = 10 W/m2 -K H = 5 ft

tg = 0.95cm tg = 0.95 cm g = 1.9 cm Tout = 23°F hout = 25 W/m2 -K ka = 0.025 W/m-K kg = 1.4 W/m-K

casing shown in P2-10(b) Figure P2-10(a): Double paned window.

The average heating season lasts about time = 130 days and the average outdoor temperature during this time is T out = 23◦ F. You heat with natural gas and pay, on average, ec = 1.415 $/therm (a therm is an energy unit = 1.055 × 108 J).

Chapter 2: Two-Dimensional, Steady-State Conduction

297

a.) Calculate the average rate of heat transfer through the double paned window during the heating season using a 1-D resistance model. b.) How much does the energy lost through the window cost during a single heating season? There is a metal casing that holds the panes of glass and connects them to the surrounding wall, as shown in Figure P2-10(b). Because the metal casing has high conductivity, it seems likely that you could lose a substantial amount of heat by conduction through the casing (potentially negating the advantage of using a double paned window). The geometry of the casing is shown in Figure P2-10(b); note that the casing is symmetric about the center of the window. All surfaces of the casing that are adjacent to glass, wood, or the air between the glass panes can be assumed to be adiabatic. The other surfaces are exposed to either the indoor or outdoor air. glass panes 1.9 cm Tin = 70°F hin = 10 W/m2 -K

0.95 cm air

Tout = 23°F hout = 25 W/m2 -K

2 cm 4 cm

metal casing k m = 25 W/m-K

0.5 cm

3 cm 0.4 cm

wood

Figure P2-10(b): Metal casing.

c.) Prepare a 2-D thermal analysis of the casing using FEHT. Turn in a print out of your geometry as well as a contour plot of the temperature distribution. What is the rate of energy lost via conduction through the casing per unit length (W/m)? d.) Show that your numerical model has converged by recording the rate of heat transfer per length for several values of the number of nodes. e.) How much does the casing add to the cost of heating your house? 2–11 A radiator panel extends from a spacecraft; both surfaces of the radiator are exposed to space (for the purposes of this problem it is acceptable to assume that space is at 0 K); the emissivity of the surface is ε = 1.0. The plate is made of aluminum (k = 200 W/m-K and ρ = 2700 kg/m3 ) and has a ﬂuid line attached to it, as shown in Figure 2-11(a). The half-width of the plate is a = 0.5 m wide and the height of the plate is b = 0.75 m. The thickness of the plate is th = 1.0 cm. The ﬂuid line carries coolant at Tc = 320 K. Assume that the ﬂuid temperature is constant, although the ﬂuid temperature will actually decrease as it transfers heat to the radiator. The combination of convection and conduction through the panel-to-ﬂuid line mounting leads to an effective heat transfer coefﬁcient of h = 1,000 W/m2 -K over the 3.0 cm strip occupied by the ﬂuid line.

298

Two-Dimensional, Steady-State Conduction k = 200 W/m-K ρ = 2700 kg/m 3 ε = 1.0

space at 0 K

a = 0.5 m

3 cm th = 1 cm

b = 0.75 m

fluid at Tc = 320 K

half-symmetry model of panel, Figure P2-11(b) Figure 2-11(a): Radiator panel.

The radiator panel is symmetric about its half-width and the critical dimensions that are required to develop a half-symmetry model of the radiator are shown in Figure 2-11(b). There are three regions associated with the problem that must be deﬁned separately so that the surface conditions can be set differently. Regions 1 and 3 are exposed to space on both sides while Region 2 is exposed to the coolant ﬂuid one side and space on the other; for the purposes of this problem, the effect of radiation to space on the back side of Region 2 is neglected.

Region 1 (both sides exposed to space) Region 2 (exposed to fluid - neglect radiation to space) Region 3 (both sides exposed to space) (0.50,0.75) (0.50,0.55) (0.50,0.52)

y x (0.50,0)

(0,0) (0.22,0)

(0.25,0)

line of symmetry

Figure 2-11(b): Half-symmetry model (coordinates are in m).

a.) Prepare a FEHT model that can predict the temperature distribution over the radiator panel. b.) Export the solution to EES and calculate the total heat transferred from the radiator and the radiator efﬁciency (deﬁned as the ratio of the radiator heat transfer to the heat transfer from the radiator if it were isothermal and at the coolant temperature). c.) Explore the effect of thickness on the radiator efﬁciency and mass.

Chapter 2: Two-Dimensional, Steady-State Conduction

299

Resistance Approximations for Conduction Problems 2–12 There are several cryogenic systems that require a “thermal switch,” a device that can be used to control the thermal resistance between two objects. One class of thermal switch is activated mechanically and an attractive method of providing mechanical actuation at cryogenic temperatures is with a piezoelectric stack. Unfortunately, the displacement provided by a piezoelectric stack is very small, typically on the order of 10 microns. A company has proposed an innovative design for a thermal switch, shown in Figure P2-12(a). Two blocks are composed of th = 10 μm laminations that are alternately copper (kCu = 400 W/m-K) and plastic (k p = 0.5 W/m-K). The thickness of each block is L = 2.0 cm in the direction of the heat ﬂow. One edge of each block is carefully polished and these edges are pressed together; the contact resistance associated with this joint is Rc = 5 × 10−4 K-m2/ W. th = 10 μm plastic laminations k p = 0.5 W/m-K L = 2 cm L = 2 cm −4 m2 -K/W R′′= c 5x10

direction of actuation

TC

TH

Figure 2-12(b) “off ” position

“on” position th = 10 μm copper laminations kCu = 400 W/m-K

Figure P2-12(a): Thermal switch in the “on” and “off” positions.

Figure P2-12(a) shows the orientation of the two blocks when the switch is in the “on” position; notice that the copper laminations are aligned with one another in this conﬁguration in order to provide a continuous path for heat through high conductivity copper (with the exception of the contact resistance at the interface). The vertical location of the right-hand block is shifted by 10 μm to turn the switch “off”. In the “off” position, the copper laminations are aligned with the plastic laminations; therefore, the heat transfer is inhibited by low conductivity plastic. Figure P2-12(b) illustrates a closer view of half (in the vertical direction) of two adjacent laminations in the “on” and “off” conﬁgurations. Note that the repeating nature of the geometry means that it is sufﬁcient to analyze a single lamination set and assume that the upper and lower boundaries are adiabatic. L

th/2

k p kCu

L

R′′c “on” position

th/2 TC

TH “off ” position

Figure P2-12(b): A single set consisting of half of two adjacent laminations in the “on” and “off” positions.

300

Two-Dimensional, Steady-State Conduction

The key parameter that characterizes the performance of a thermal switch is the resistance ratio (RR) which is deﬁned as the ratio of the resistance of the switch in the “off” position to its resistance in the “on” position. The company claims that they can achieve a resistance ratio of more than 100 for this switch. a.) Estimate upper and lower bounds for the resistance ratio for the proposed thermal switch using 1-D conduction network approximations. Be sure to draw and clearly label the resistance networks that are used to provide the estimates. Use your results to assess the company’s claim of a resistance ratio of 100. b.) Provide one or more suggestions for design changes that would improve the performance of the switch (i.e., increase the resistance ratio). Justify your suggestions. c.) Sketch the temperature distribution through the two parallel paths associated with the adiabatic limit of the switch’s operation in the “off” position. Do not worry about the quantitative details of the sketch, just make sure that the qualitative features are correct. d.) Sketch the temperature distribution through the two parallel paths associated with the adiabatic limit in the “on” position. Again, do not worry about the quantitative details of your sketch, just make sure that the qualitative features are correct. 2–13 Figure P2-13 illustrates a thermal bus bar that has width W = 2 cm (into the page). H1 = 5 cm

L2 = 7 cm h = 10 W/m2 -K T∞ = 20°C

TH = 80°C

L1 = 3 cm

H2 = 1 cm

k = 1 W/m-K

Figure P2-13: Thermal bus bar.

The bus bar is made of a material with conductivity k = 1 W/m-K. The middle section is L2 = 7 cm long with thickness H2 = 1 cm. The two ends are each L1 = 3 cm long with thickness H1 = 3 cm. One end of the bar is held at T H = 80◦ C and the other is exposed to air at T ∞ = 20◦ C with h = 10 W/m2 -K. a.) Use FEHT to predict the rate of heat transfer through the bus bar. b.) Obtain upper and lower bounds for the rate of heat transfer through the bus bar using appropriately deﬁned resistance approximations. Conduction through Composite Materials 2–14 A laminated stator is shown in Figure P2-14. The stator is composed of laminations with conductivity klam = 10 W/m-K that are coated with a very thin layer of epoxy with conductivity kepoxy = 2.0 W/m-K in order to prevent eddy current losses. The laminations are thlam = 0.5 mm thick and the epoxy coating is 0.1 mm thick (the total amount of epoxy separating each lamination is thepoxy = 0.2 mm). The inner radius of the laminations is rin = 8.0 mm and the outer radius of the laminations is ro,lam = 20 mm. The laminations are surrounded by a cylinder of plastic with conductivity kp = 1.5 W/m-K that has an outer radius of ro,p = 25 mm. The motor casing surrounds the plastic. The motor casing has an outer radius of ro,c = 35 mm and is composed of aluminum with conductivity kc = 200 W/m-K.

Conduction through Composite Materials

301

laminations, thlam = 0.5 mm, klam = 10 W/m-K epoxy coating, th epoxy = 0.2 mm, k epoxy = 2.0 W/m-K k p = 1.5 W/m-K kc = 200 W/m-K q⋅ ′′ = 5x10 4 W/m2

T∞ = 20°C h = 40 W/m2 -K -4 R′′c = 1x10 K-m2 /W

rin = 8 mm ro,lam = 20 mm ro,p = 25 mm ro,c = 35 mm Figure P2-14: Laminated stator.

The heat ﬂux due to the windage loss associated with the drag on the shaft is q˙ = 5 × 104 W/m2 and is imposed on the internal surface of the laminations. The outer surface of the motor is exposed to air at T ∞ = 20◦ C with a heat transfer coefﬁcient h = 40 W/m2 -K. There is a contact resistance Rc = 1 × 10-4 K-m2 /W between the outer surface of the laminations and the inner surface of the plastic and the outer surface of the plastic and the inner surface of the motor housing. a.) Determine an upper and lower bound for the temperature at the inner surface of the laminations (Tin ). b.) You need to reduce the internal surface temperature of the laminations and there are a few design options available, including: (1) increase the lamination thickness (up to 0.7 mm), (2) reduce the epoxy thickness (down to 0.05 mm), (3) increase the epoxy conductivity (up to 2.5 W/m-K), or (4) increase the heat transfer coefﬁcient (up to 100 W/m-K). Which of these options do you suggest and why?

REFERENCES

Cheadle, M., A Predictive Thermal Model of Heat Transfer in a Fiber Optic Bundle for a Hybrid Solar Lighting System, M.S. Thesis, University of Wisconsin, Dept. of Mechanical Engineering, (2006). Moaveni, S., Finite Element Analysis, Theory and Application with ANSYS, 2nd Edition, Pearson Education, Inc., Upper Saddle River, (2003). Myers, G. E., Analytical Methods in Conduction Heat Transfer, 2nd Edition, AMCHT Publications, Madison, WI, (1998). Rohsenow, W. M., J. P. Hartnett, and Y. I. Cho, eds., Handbook of Heat Transfer, 3rd Edition, McGraw-Hill, New York, (1998).

3

Transient Conduction

3.1 Analytical Solutions to 0-D Transient Problems 3.1.1 Introduction Chapters 1 and 2 discuss steady-state problems, i.e., problems in which temperature depends on position (e.g, x and y) but does not change with time (t). This chapter discusses transient conduction problems, where temperature depends on time.

3.1.2 The Lumped Capacitance Assumption The simplest situation is zero-dimensional (0-D); that is, the temperature does not vary with position but only with time. This approximation is often referred to as the lumped capacitance assumption and it is appropriate for an object that is thin and conductive so that it can be assumed to be at a uniform temperature at any time. The lumped capacitance approximation is similar to the extended surface approximation that was discussed in Section 1.6. The resistance to conduction within the object is neglected as being small relative to the resistance to heat transfer from the surface of the object. Therefore, the lumped capacitance approximation is justiﬁed in the same way as the extended surface approximation, by deﬁning an appropriate Biot number. For the case where only convection occurs from the surface of the object, the Biot number is: Bi =

Lcond h Rcond,int = Rconv k

(3-1)

where Lcond is the conduction length within the object, h is the average heat transfer coefﬁcient, and k is the conductivity of the material. Note that Eq. (3-1) is the simplest possible Biot number; it is often the case that heat transfer from the surface of the object will be resisted by mechanisms other than or in addition to convection (e.g., radiation or conduction through a thin insulating layer). These additional resistances should be included in the denominator of an appropriately deﬁned Biot number. The conduction length that characterizes an irregular shaped object can be ambiguous. Thermal energy will conduct out of the object along the easiest (i.e., shortest) path. For a thin plate, Lcond should be the half-width of the plate. For other shapes, Lcond should be selected so that it characterizes the minimum conduction length; the ratio of the volume of the object (V) to its surface area (As ) is often used for this purpose: Lcond =

V As

(3-2)

If the Biot number is much less than unity, then the lumped capacitance assumption is justiﬁed. A common criteria is that the Bi must be less than 0.1. However, this is certainly a case where engineering judgment is required based on the level of accuracy that is required for the model. 302

3.1 Analytical Solutions to 0-D Transient Problems

303 q⋅ conv

Figure 3-1: An object exposed to a time varying ﬂuid temperature.

dU dt

3.1.3 The Lumped Capacitance Problem The lumped capacitance approximation reduces the spatial dimensionality of the problem to zero. Therefore, a control volume can be placed around the entire object (as all of the material is assumed to be at the same temperature at a given time). An energy balance will include all of the relevant energy transfers (e.g., convection and radiation) as well as the rate of energy storage. The result will be a ﬁrst order differential equation that governs the temperature of the object. A single boundary condition, typically the initial temperature of the object, is required to obtain a solution using either analytical (as discussed in this section) or numerical (as discussed in Section 3.2) techniques. Figure 3-1 illustrates an object with a small Biot number that can be considered to be lumped. The object is initially at a uniform temperature Tini and is exposed to a time varying ﬂuid temperature, T∞ , through a heat transfer coefﬁcient, h. The control volume in Figure 3-1 suggests the energy balance: 0 = q˙ conv +

dU dt

(3-3)

where q˙ conv is the rate of convective heat transfer from the surface and U is the total energy stored in the object. The convective heat transfer is: q˙ conv = h As (T − T ∞ )

(3-4)

where As is the surface area of the object exposed to the ﬂuid. The rate of change of the total energy of the object can be expressed in terms of the speciﬁc energy of the material (u): du dU =M dt dt

(3-5)

where M is the mass of the object. For solids or incompressible ﬂuids, Eq. (3-5) can be written as: du dT dT dU =M = Mc dt dT dt dt

(3-6)

c

where c is the speciﬁc heat capacity of the material. Substituting Eqs. (3-6) and (3-4) into Eq. (3-3) leads to: 0 = h As (T − T ∞ ) + M c

dT dt

(3-7)

304

Transient Conduction

which is the ﬁrst order differential equation that governs the problem. Equation (3-7) can be rearranged: h As h As dT T = + T∞ dt M c Mc

(3-8)

1/τlumped

Note that the group of variables that multiply the temperatures on the left and right sides of Eq. (3-8) must have units of inverse time; this group is used to deﬁne a lumped time constant, τlumped , that governs the problem: τlumped =

Mc h As

(3-9)

Substituting Eq. (3-9) into Eq. (3-8) leads to: dT T∞ T = + dt τlumped τlumped

(3-10)

Equation (3-10) can be solved either analytically, using the same techniques discussed in Chapter 1 for 1-D steady-state conduction, or numerically (see Section 3.2). If additional heat transfer mechanisms or thermal loads are included in the problem (e.g., radiation or volumetric generation of thermal energy) then the governing differential equation will be different than Eq. (3-10). However, the steps associated with the derivation of the differential equation will remain the same and it will be possible to identify an appropriate lumped capacitance time constant.

3.1.4 The Lumped Capacitance Time Constant The concept of a lumped capacitance time concept is useful even in the absence of an analytical solution to the problem. The lumped capacitance time constant is the product of the thermal resistance to heat transfer from the surface of the object (R) and the thermal capacitance of the object (C). For the object in Figure 3-1, the only resistance to heat transfer from the surface of the object is due to convection: R=

1 h As

(3-11)

The thermal capacitance of the object is the product of its mass and speciﬁc heat capacity. The thermal capacitance provides a measure of how much energy is required to change the temperature of the object: C = Mc

(3-12)

The lumped time constant identiﬁed in Eq. (3-9) is the product of R, Eq. (3-11), and C, Eq. (3-12): RC =

1

M c = τlumped h A s C

(3-13)

R

The lumped capacitance time constant for other situations can be calculated by modifying the resistance and capacitance terms in Eq. (3-13) appropriately. For example, if the object experiences both convection and radiation from its surface, then an

3.1 Analytical Solutions to 0-D Transient Problems ambient temperature

object temperature

Temperature

Temperature

ambient temperature

τ

305

object temperature

Time Time (b)

(a) ambient temperature

object temperature

Temperature

Temperature

ambient temperature

τ

object temperature

Time

Time (d)

(c)

Figure 3-2: Approximate temperature response for an object subjected to (a) a step change in its ambient temperature, (b) and (c) an oscillatory ambient temperature where (b) the frequency of the oscillation is much less than the inverse of the time constant and (c) the frequency of the oscillation is much greater than the inverse of the time constant, and (d) a ramped ambient temperature.

appropriate thermal resistance is the parallel combination of a convective and radiative resistance: −1 3 R = h As + σ εs 4 T As

(3-14)

The lumped time constant is analogous to the electrical time constant associated with an R-C circuit and many of the concepts that may be familiar from electrical circuits can also be applied to transient heat transfer problems. A quick estimate of the lumped time constant for a problem can provide substantial insight into the behavior of the system. The lumped time constant is, approximately,

306

Transient Conduction

Table 3-1: Summary of lumped capacitance solutions to some typical variations in the ambient temperature. Situation

Solution

Nomenclature

Step change in ambient temperature

T = T ∞ + (T ini − T ∞ ) exp −

Oscillatory ambient temperature (after initial transient has decayed)

T =T +'

Ramped ambient temperature

t τlumped

T ini = initial temperature of environment and object T ∞ = temperature of environment after step τlumped = lumped time constant

T ( sin (ω t − φ) 1 + (ω τlumped )2 where φ = tan−1 (ω τlumped )

T = T ini + β t + β τlumped exp −

t τlumped

T = average ambient temperature T = amplitude of temperature oscillation ω = angular frequency of temperature oscillation (rad/s) τlumped = lumped time constant

−1

T ini = initial temperature of environment and object β = rate of ambient temperature change τlumped = lumped time constant

the amount of time that it will take the object to respond to any change in its thermal environment. For example, if the object is subjected to a step change in the ambient temperature, T∞ in Eq. (3-10), then its temperature will be within 5% of the new temperature after 3 time constants, as shown in Figure 3-2(a). If the object is subjected to an oscillatory ambient temperature (e.g., within an engine cylinder or some other cyclic device) then the temperature of the object will follow the ambient temperature nearly exactly if the period of oscillation is much greater than the time constant (i.e., if the frequency is much less than the inverse of the time constant). In the opposite extreme, the object’s temperature will be essentially constant if the period of oscillation is much less than the the time constant (i.e., if the frequency is much greater than the inverse of the time constant). These extremes in behavior are shown in Figure 3-2(b) and (c). If the object is subjected to a ramped (i.e., linearly increasing) ambient temperature, then its temperature will tend to increase linearly as well, but its response will be delayed by approximately one time constant, as shown in Figure 3-2(d). Some of the situations illustrated in Figure 3-2 will be investigated more completely in the following examples. The analytical solutions to these problems are summarized in Table 3-1. However, it is clear that simply knowing the time constant and its physical signiﬁcance is sufﬁcient in many cases.

307

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT Plastic parts are formed in an injection mold and dropped (ﬂat) onto a conveyor belt (Figure 1). The parts are disk-shaped with thickness th = 2.0 mm and diameter D = 10.0 cm. The plastic has thermal conductivity k = 0.35 W/m-K, density ρ = 1100 kg/m3 , and speciﬁc heat capacity c = 1900 J/kg-K. The side of the part that faces the conveyor belt is adiabatic. The top surface of the part is exposed to air at T∞ = 20◦ C with a heat transfer coefﬁcient h = 15 W/m2 -K. The temperature of the part immediately after it is formed is Tini = 180◦ C. The part must be cooled to Tmax = 80◦ C before it can be stacked and packaged. The packaging system is positioned L = 15 ft away from the molding machine. k = 0.35 W/m-K ρ = 1100 kg/m3 c = 1900 J/kg-K Figure 1: Plastic piece.

T∞ = 20°C 2 h = 15 W/m -K th = 2 mm

D =10 cm

adiabatic initial temperature Tini = 180°C

a) Is a lumped capacitance model of the part justiﬁed for this situation? The inputs are entered in EES: “EXAMPLE 3.1-1: Design of a Conveyor Belt” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” th=2.0 [mm]∗ convert(mm,m) k=0.35 [W/m-K] D=10 [cm]∗ convert(cm,m) rho=1100 [kg/mˆ3] c=1900 [J/kg-K] h bar=15 [W/mˆ2-K] T ini=converttemp(C,K,180 [C]) T inﬁnity=converttemp(C,K,20 [C]) T max=converttemp(C,K,80 [C]) L=15 [ft]∗ convert(ft,m)

“thickness” “conductivity” “diameter” “density” “speciﬁc heat capacity” “heat transfer coefﬁcient” “initial temperature of part” “ambient temperature” “maximum handling temperature” “conveyor length”

A lumped capacitance model of the part can be justiﬁed by examining the Biot number, the ratio of the resistance to internal conduction to the resistance to heat transfer from the surface of the object. In this problem, the resistance to heat transfer from the surface is due only to convection and therefore Eq. (3-1) is valid, where the conduction length is intuitively the thickness of the object (it would be the halfwidth if the conveyor side of the part were not adiabatic). Note that the characteristic

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT

308

Transient Conduction

length deﬁned by Eq. (3-2) as the ratio of the volume (V) to the exposed area for heat transfer (As ) is equal to the thickness of the part: V =

π D 2 th 4

As = L cond =

V=pi∗ Dˆ2∗ th/4 As=pi∗ Dˆ2/4 L cond=V/As Bi=h bar∗ L cond/k

π D2 4

V π D 2 th 4 = = th As 4 π D2

“Volume” “Surface area exposed to cooling” “Conduction length” “Biot number based on conduction length”

The Biot number predicted by EES is 0.09, which is sufﬁciently small to use the lumped capacitance model unless very high accuracy is required. b) What is the maximum acceptable conveyor velocity so that the parts arrive at the packaging station below Tmax ? The governing differential equation is obtained by considering a control volume that encloses the entire plastic part; the energy balance is: 0 = q˙ conv +

dU dt

The rate of convection heat transfer is: q˙ conv = h As (T − T∞ ) and the rate of energy storage is: dU dT =ρV c dt dt Combining these equations leads to: 0 = h As (T − T∞ ) + ρ V c

dT dt

which can be rearranged: dT (T − T∞ ) =− dt τlumped

(1)

where τlumped is the time constant for this problem: τlumped =

tau lumped=V∗ rho∗ c/(h bar∗ As)

ρV c h As

“time constant”

The time constant for the part is 279 s. We should keep in mind that it will take on the order of 5 minutes to cool the plastic piece substantially and use this insight to check the more precise analytical solution that is obtained.

309

Equation (1) is a ﬁrst order differential equation with the boundary condition: Tt =0 = Tini

(2)

The differential equation is separable; that is, all of the terms involving the dependent variable, T, can be placed on one side while the terms involving the independent variable, t, can be placed on the other: dt dT =− τlumped (T − T∞ ) The separated equation can be directly integrated: T Tini

dT =− (T − T∞ )

t

dt

(3)

τlumped

0

This integration is most easily accomplished by deﬁning the temperature difference (θ ): θ = T − T∞

(4)

d θ = dT

(5)

so that:

Substituting Eqs. (4) and (5) into Eq. (3) leads to: T −T∞

Tini −T∞

dθ =− θ

t 0

dt τlumped

Carrying out the integration leads to: T − T∞ t ln =− Tini − T∞ τlumped Solving for T leads to:

T = T∞ + (Tini − T∞ ) exp −

t

τlumped

(6)

which is equivalent to the entry in Table 3-1 for a step change in ambient temperature. The time required to cool the part from Tini to Tmax can be computed using Eq. (6): T max=T inﬁnity+(T ini-T inﬁnity)∗ exp(-t cool/tau lumped)

“time required to cool part”

and is found to be tcool = 273 s; note that this value is in good agreement with the previously calculated time constant. The linear velocity of the conveyor that is required so that it takes at least 273 s for the part to travel the 15 ft between the molding machine and the packaging station is: uc =

u c=L/t cool u c fpm=u c∗ convert(m/s,ft/min)

L tcool “conveyor velocity” “conveyor velocity in ft/min”

The maximum allowable velocity uc is found to be 0.0167 m/s (3.29 ft/min).

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

310

Transient Conduction

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT A temperature sensor is installed in a chemical reactor that operates in a cyclic fashion. The temperature of the ﬂuid in the reactor varies in an approximately sinusoidal manner with a mean temperature T ∞ = 320◦ C, an amplitude T∞ = 50◦ C, and a frequency f = 0.5 Hz. The sensor can be modeled as a sphere with diameter D = 1.0 mm. The sensor is made of a material with conductivity ks = 50 W/m-K, speciﬁc heat capacity cs = 150 J/kg-K, and density ρs = 16000 kg/m3 . In order to provide corrosion resistance, the sensor has been coated with a thin layer of plastic; the coating is thc = 100 μm thick with conductivity kc = 0.2 W/m-K and has negligible heat capacity relative to the sensor itself. The heat transfer coefﬁcient between the surface of the coating and the ﬂuid is h = 500 W/m2 -K. The sensor is initially at Tini = 260◦ C. a) Is a lumped capacitance model of the temperature sensor appropriate? The inputs are entered in EES: “EXAMPLE 3.1-2: Sensor in an Oscillating Temperature Environment” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” T inﬁnity bar=converttemp(C,K,320[C]) T ini=converttemp(C,K,260[C]) DT inﬁnity=50[K] f=0.5 [Hz] D=1.0 [mm]∗ convert(mm,m) k s=50 [W/m-K] c s=150 [J/kg-K] rho s=16000 [kg/mˆ3] th c=100 [micron]∗ convert(micron,m) k c=0.2 [W/m-K] h bar=500 [W/mˆ2-K]

“average temperature of reactor” “initial temperature of sensor” “amplitude of reactor temperature change” “frequency of reactor temperature change” “diameter of sensor” “conductivity of sensor material” “speciﬁc heat capacity of sensor material” “density of sensor material” “thickness of coating” “conductivity of coating” “heat transfer coefﬁcient”

The Biot number is the ratio of the internal conduction resistance to the resistance to heat transfer from the surface of the object. In this problem, the resistance to heat transfer from the surface is the series combination of convection (Rconv ): Rconv =

h4 π

1 D + t hc 2

2

and the conduction resistance of the coating (Rcond,c , from Table 1-2): 2 2 − D D + 2 t hc Rcond ,c = 4 π kc

R conv=1/(h bar∗ 4∗ pi∗ (D/2+th c)ˆ2) R cond c=(1/(D/2)-1/(D/2+th c))/(4∗ pi∗ k c)

311

“convective resistance” “conduction resistance of coating”

The resistance to internal conduction (Rcond,int ) is approximated according to: Rcond ,int =

L cond k s As

where As is the surface area of the sensor: As = 4 π

D 2

2

and Lcond is the conduction length, approximated according to Eq. (3-2): L cond =

V As

where: 4π V = 3

V=4∗ pi∗ (D/2)ˆ3/3 A s=4∗ pi∗ (D/2)ˆ2 L cond=V/A s R cond int=L cond/(k s∗ A s)

D 2

3

“volume of sensor” “surface area of sensor” “approximate conduction length” “internal conduction resistance”

The Biot number that characterizes this problem is therefore: Bi =

Bi=R cond int/(R conv+R cond c)

Rcond Rc + Rconv

“Biot number”

which leads to Bi = 0.0018; this is sufﬁciently less than 1 to justify a lumped capacitance model. b) What is the time constant associated with the sensor? Do you expect there to be a substantial temperature measurement error related to the dynamic response of the sensor? The time constant (τlumped ) is the product of the resistance to heat transfer from the surface of the sensor (which is related to conduction through the coating and convection) and the thermal mass of the sensor (C): τlumped = (Rcond ,c + Rconv )C

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

312

Transient Conduction

where C = V ρs cs C=V∗ rho s∗ c s tau=(R conv+R cond c)∗ C

“capacitance of the sensor” “time constant of the sensor”

The time constant is 0.72 s and the time per cycle (the inverse of the frequency) is 2 s. These quantities are on the same order and therefore it is not likely that the temperature sensor will be able to faithfully follow the reactor temperature. c) Develop an analytical model of the temperature response of the sensor. The temperature sensor is exposed to a sinusoidally varying temperature: T∞ = T ∞ + T∞ sin 2 π f t

(1)

The governing differential equation for the sensor balances heat transfer to ambient against energy storage: 0=

dT [T − T∞ ] +C Rcond ,c + Rconv dt

(2)

Substituting Eq. (1) into Eq. (2) leads to: 0=

[T − T ∞ − T∞ sin(2 π f t )] dT + τlumped dt

which is rearranged: T∞ T∞ sin(2 π f t ) T dT = + + dt τlumped τlumped τlumped

(3)

Equation (3) is a non-homogeneous ordinary differential equation. The solution is assumed to consist of a homogeneous and particular solution: T = Th + Tp

(4)

Substituting Eq. (4) into Eq. (3) leads to: dTp Tp dTh T∞ T∞ sin(2 π f t ) Th + = + + + dt τlumped dt τlumped τlumped τlumped =0 for homogeneous differential equation

particular differential equation

The homogeneous differential equation is: dTh Th =0 + dt τlumped The solution to the homogeneous differential equation can be obtained by separating variables and integrating: dt dTh =− Th τlumped Carrying out the indeﬁnite integral leads to: ln (Th) = −

t τlumped

+ C1

(5)

313

where C1 is a constant of integration. Equation (5) can be rearranged: t t = C 1∗ exp − Th = exp (C 1 ) exp − τlumped τlumped C 1∗

where C 1∗ is an also undetermined constant that will subsequently be referred to as C1 : t Th = C 1 exp − (6) τlumped Notice that the homogeneous solution provided by Eq. (6) dies off after about three time constants. The particular solution (Tp ) is obtained by identifying any function that satisﬁes the particular differential equation: Tp dTp T∞ T∞ sin(2 π f t ) = + + dt τlumped τlumped τlumped

(7)

By inspection, the sum of a constant and a sine and cosine with the same frequency can be made to solve Eq. (7): Tp = C 2 sin(2 π f t ) + C 3 cos(2 π f t ) + C 4

(8)

Substituting Eq. (8) into Eq. (7) leads to: C2 sin(2 π f t ) τlumped

C 2 2 π f cos(2 π f t ) − C 3 2 π f sin(2 π f t ) + +

C3 τlumped

cos(2 π f t ) +

C4 τlumped

=

T∞ τlumped

+

T∞ sin(2 π f t ) τlumped

(9)

Equation (9) can only be true if the constant, sine and cosine terms each separately add to zero: C4 τlumped

=

C2 2 π f + −C 3 2 π f +

T∞ τlumped C3

τlumped

=0

C2 T∞ = τlumped τlumped

Solving for C2 , C3 , and C4 leads to: C2 =

T∞ 1 + (2 π f τlumped )2

C3 = −

2 π f τ T∞ 1 + (2 π f τlumped )2 C4 = T ∞

so that the particular solution is: Tp = T ∞ +

T∞ [sin(2 π f t ) − (2 π f τlumped ) cos(2 π f t )] 1 + (2 π f τlumped )2

(10)

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

314

Transient Conduction

The solution is the sum of the particular and homogeneous solutions, Eqs. (6) and (10): t T = C 1 exp − τlumped (11) T∞ +T∞ + [sin(2 π f t ) − (2 π f τlumped ) cos(2 π f t )] 2 1 + (2 π f τlumped ) Note that the same conclusion can be reached using two lines of Maple code; the governing differential equation, Eq. (3), is entered and solved: > restart; > ODE:=diff(T(t),t)+T(t)/tau=T_inﬁnity_bar/tau+DT_inﬁnity*sin(2*pi*f*t)/tau; d T (t ) T infinity bar DT infinity sin(2πf t ) O D E := T (t ) + = + dt τ τ τ > Ts:=dsolve(ODE); t T s := T (t ) = e(− τ ) C 1 + (T infinity bar + 4 T infinity barf 2 π 2 τ 2

−2DT infinity cos(2 πf t )f π τ + DT infinity sin(2 π f t ))/(1 + 4f 2 π 2 τ 2 )

The solution identiﬁed by Maple is the equivalent to Eq. (11); the solution is copied into EES, with minor editing: “Solution” Temp = exp(-1/tau∗ t)∗ C1-(-T_inﬁnity_bar-4∗ T_inﬁnity_bar∗ piˆ2∗ fˆ2∗ tauˆ2+& 2∗ DT_inﬁnity∗ cos(2∗ pi∗ f∗ t)∗ pi∗ f∗ tau& -DT_inﬁnity*sin(2∗ pi∗ f∗ t))/(1+4∗ piˆ2∗ fˆ2∗ tauˆ2)

The constant C1 must be selected so that the boundary condition is satisﬁed: Tt =0 = Tini

(12)

Substituting Eq. (11) into Eq. (12) leads to: Tt =0 = C 1 + T ∞ −

T∞ 2 π f τlumped = Tini 1 + (2 π f τlumped )2

which leads to: C1 =

T∞ 2 π f τlumped + Tini − T ∞ 1 + (2 π f τlumped )2

The symbolic expression for the boundary condition can also be found using Maple: > rhs(eval(Ts,t=0))=T_ini; T infinity bar + 4T infinity barf 2 π 2 τ 2 − 2 DT infinityf π τ C1 + = T ini 1 + 4f 2 π 2 τ 2

315

which is copied and pasted into EES: C_1+(T_inﬁnity_bar+4∗ T_inﬁnity_bar∗ fˆ2∗ piˆ2∗ tauˆ2-2∗ DT_inﬁnity∗ f∗ pi∗ tau)/(1+4∗ fˆ2∗ piˆ2∗ tauˆ2) = T_ini “initial condition”

The sensor temperature and ﬂuid temperature are converted to Celsius. T inﬁnity=T inﬁnity bar+DT inﬁnity∗ sin(2∗ pi∗ f∗ t)

“ambient temperature”

T C=converttemp(K,C,Temp)

“sensor temperature in C”

T inﬁnity C=converttemp(K,C,T inﬁnity)

“ambient temperature in C”

The temperatures are computed in a parametric table in which time is set so that it ranges from 0 to 10 s. The ﬂuid and sensor temperature variation are shown as a function of time in Figure 1. 380

fluid temperature

Temperature (°C)

360 340 320 300 280 260 0

sensor temperature 1

2

3

4

5 6 Time (s)

7

8

9

10

Figure 1: Temperature sensor and ﬂuid temperature as a function of time.

Note that after approximately 3 seconds (i.e., a few time constants) the homogeneous solution has decayed to zero and the temperature response of the sensor is given entirely by the particular solution, Eq. (10): Tp = T ∞ +

T∞ [sin(2 π f t ) − (2 π f τlumped ) cos(2 π f t )] 1 + (2 π f τlumped )2

(10)

Equation (10) can be rewritten in terms of an attenuation of the amplitude of the oscillation (Att) and a phase lag relative to the ﬂuid temperature variation (φ) Tp = T ∞ + Att T∞ sin(2 π f t − φ)

(13)

Equation (13) is rewritten using the trigonometric identity: Tp = T ∞ + Att T∞ [sin(2 π f t ) cos(φ) − cos(2 π f t ) sin(φ)]

(14)

Comparing Eq. (10) with Eq. (14) leads to: Att cos (φ) =

1 1 + (2 π f τ )2

(15)

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.1 Analytical Solutions to 0-D Transient Problems

Transient Conduction

and Att sin (φ) =

2 π f τlumped 1 + (2 π f τlumped )2

(16)

Dividing Eq. (16) by Eq. (15) leads to: tan (φ) = 2 π f τlumped so the phase (the lag) between the sensor and ﬂuid temperature is: φ = tan−1 (2 π f τlumped ) and the attenuation is: 1 Att = 0 1 + (2 π f τlumped )2 Figure 2 shows the attenuation and phase angle as a function of the product of the frequency and the time constant (f τlumped ). 1.6 Attenuation (-) and phase (rad)

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

316

1.4 1.2

φ

1 0.8 0.6 Att 0.4 0.2 0 0.001

0.01

0.1

1

10

100

Frequency time constant product, f τ Figure 2: Attenuation and phase angle as a function of the product of the frequency and the time constant.

Notice that if either the frequency or the time constant is small, then the attenuation goes to unity and the phase goes to zero. In this limit, the temperature sensor will faithfully follow the ﬂuid temperature with little error related to the transient response characteristics of the sensor; this is the situation that was illustrated earlier in Figure 3-2(b). In the other limit, if either the frequency or time constant of the sensor are very large then the attenuation will approach zero and the phase will approach π/2 rad (90 deg.). This situation corresponds to Figure 3-2(c) where the sensor cannot respond to the temperature oscillations. The dynamic characteristics of a temperature sensor are important in many applications and should be carefully considered when selecting an instrument for a transient temperature measurement.

3.2 Numerical Solutions to 0-D Transient Problems

317

3.2 Numerical Solutions to 0-D Transient Problems 3.2.1 Introduction Section 3.1 discussed the lumped capacitance model, which neglects any spatial temperature gradients within an object and therefore approximates the temperature in a transient problem as being only a function only of time. The analytical solution to such 0-D (or lumped capacitance) problems was examined in Section 3.1. In this section, lumped capacitance problems will be solved numerically. The numerical solution to any transient problem begins with the derivation of the governing differential equation, which allows the calculation of the temperature rate of change as a function of the current temperature and time. The solution to the problem is therefore a matter of integrating the governing differential equation forward in time. There are a number of techniques available for numerical integration, each with its own characteristic level of accuracy, stability, and complexity. These numerical integration techniques are discussed in this section and revisited in Section 3.8 in order to solve 1-D transient problems and again in Chapter 5 in order to solve convection problems. The governing differential equation that provides the rate of change of a variable (or several variables) given its own value (or values) is sometimes referred to as the state equation for the dynamic system. State equations characterize the transient behavior of problems in the areas of controls, dynamics, kinematics, ﬂuids, electrical circuits, etc. Therefore, the numerical solution techniques provided in this section are generally relevant to a wide range of engineering problems. The same caveats that were discussed previously in Sections 1.4 and 2.5 with regard to the numerical solution of steady-state conduction problems also apply to numerical solutions of transient conduction problems. Any numerical solution should be evaluated for numerical convergence (i.e., to ensure that you are using a sufﬁcient number of time steps), examined against your physical intuition, and compared to an analytical solution in some limit.

3.2.2 Numerical Integration Techniques In this section, the simplest lumped capacitance problem is solved numerically in order to illustrate the various options for numerical integration. An object initially in equilibrium with its environment at Tini is subjected to a step change in the ambient temperature from T ini to T ∞ . The governing differential equation provides the temperature rate of change given the current value of the temperature; the governing differential equation is derived from an energy balance on the object (see EXAMPLE 3.1-1): dT (T − T ∞ ) =− dt τlumped

(3-15)

where τlumped is the lumped time constant for the object. The analytical solution to the problem is derived in EXAMPLE 3.1-1, providing a basis for evaluating the accuracy of the numerical solutions that are derived in this section: t (3-16) T an = T ∞ + (T ini − T ∞ ) exp − τlumped The solution can also be obtained numerically using one of several available numerical integration techniques. Each numerical technique requires that the total simulation time

318

Transient Conduction

(tsim ) be broken into small time steps. The simplest option is to use equal-sized steps, each with duration t: tsim (3-17) t = (M − 1) where M is the number of times at which the temperature will be evaluated. The temperature at each time step (Tj ) is computed by the numerical model, where j indicates the time step (T1 is the initial temperature of the object and TM is the temperature at the end of a simulation). The time corresponding to each time step is therefore: tj =

(j − 1) tsim (M − 1)

for j = 1..M

(3-18)

Euler’s Method The temperature at the end of each time step is computed based on the temperature at the beginning of the time step and the governing differential equation. The simplest (and generally the worst) technique for numerical integration is Euler’s method. Euler’s method approximates the rate of temperature change within the time step as being constant and equal to its value at the beginning of the time step. Therefore, for any time step j: dT t (3-19) T j+1 = T j + dt T =T j ,t=tj Because the temperature at the end of the time step (Tj+1 ) can be calculated explicitly using information that is available at the beginning of the time step (Tj ), Euler’s method is referred to as an explicit numerical technique. The temperature at the end of the ﬁrst time step (T2 ) is given by: dT T2 = T1 + t (3-20) dt T =T 1 ,t=0 Substituting the state equation for our problem, Eq. (3-15), into Eq. (3-20) leads to: T2 = T1 −

(T 1 − T ∞ ) t τlumped

(3-21)

Equation (3-21) is written for every time step, resulting in a set of explicit equations that can be solved sequentially for the temperature at each time. As an example, consider the case where τlumped = 100 s, T ini = 300 K, and T ∞ = 400 K. These inputs are entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

The time step duration and array of times for which the solution will be computed is speciﬁed using Eqs. (3-17) and (3-18):

3.2 Numerical Solutions to 0-D Transient Problems

t sim=500 [s] M=501 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

319

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

For the values of tsim and M entered above, t = 1 s and the temperature predicted by the numerical model at the end of the ﬁrst timestep, T2 , computed using Eq. (3-21) is 301 K, as obtained from: “Euler’s Method” T[1]=T ini T[2]=T[1]-(T[1]-T inﬁnity)∗ DELTAt/tau

“estimate of temperature at the 1st timestep”

The actual temperature at t = 1 s is obtained using the analytical solution to the problem, Eq. (3-16): T an[2]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[2]/tau) “analytical solution for 1st time step”

and found to be T an,t=1s = 300.995 K. The numerical technique is not exact; the error between the numerical and analytical solutions at the end of the ﬁrst time step is err = 0.005 K. If the time step duration is increased by a factor of 10, to t = 10 s (by reducing M to 51), then the numerical solution at the end of the ﬁrst time step becomes T 2 = 310 K and the analytical solution is T an,t=10s = 309.52◦ C. Therefore, the error between the numerical and analytical solutions increases by approximately a factor of 100, to err = 0.48 K. This behavior is a characteristic of the Euler technique; the local error is proportional to the square of the size of the time step. err ∝ t2

(3-22)

This characteristic can be inferred by examination of Eq. (3-20), which is essentially the ﬁrst two terms of a Taylor series expansion of the temperature about time t = 0: t2 t3 dT d2 T d3 T T2 = T1 + t + (3-23) + + ··· dt T =T 1 ,t=0 dt2 T =T 1 ,t=0 2! dt3 T =T 1 ,t=0 3! Euler’s approximation

err ≈neglected terms

Examination of Eq. (3-23) shows that the error, corresponding to the neglected terms in the Taylor’s series, is proportional to t2 (neglecting the smaller, higher order terms). Therefore, Euler’s technique is referred to as a ﬁrst order method because it retains the ﬁrst term in the Taylor’s series approximation. The error associated with a ﬁrst order technique is proportional to the second power of the time step duration. Most numerical techniques that are commonly used have higher order and therefore achieve higher accuracy. The other drawback of Euler’s technique (and any explicit numerical technique) is that it may become unstable if the duration of the time step becomes too large. For our example, if the time step duration is increased beyond the time constant of the object, τlumped = 100 s, then the problem becomes unstable.

320

Transient Conduction 600 550

Temperature (K)

500 450 400 350 300 250 200 150 100 0

Δt = 250 s Δt = 200 s Δt = 143 s Δt = 50 s analytical solution th Δt = 5 s (every 4 point shown) 100 200 300 400 500 600 700 800 900 1000 Time (s)

Figure 3-3: Numerical solution obtained using Euler’s method with various values of t. Also shown is the analytical solution to the same problem.

The process of moving forward through all of the time steps may be automated using a duplicate loop: “Euler’s Method” T[1]=T ini “initial temperature” duplicate j=1,(M-1) “Euler solution” T[j+1]=T[j]-(T[j]-T inﬁnity)∗ DELTAt/tau T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “error between numerical and analytical solution” err[j+1]=abs(T an[j+1]-T[j+1]) end “maximum error” err max=max(err[2..M])

Note that the analytical solution and the absolute value of the error between the numerical and analytical solutions are also obtained at each time step. The maximum error is computed at the conclusion of the duplicate loop. Figure 3-3 illustrates the numerical solution for various values of the time step duration. The analytical solution provided by Eq. (3-16) is also shown in Figure 3-3. Notice that if the simulation is carried out with a time step duration that is less than t = τlumped = 100 s, then the solution is stable and follows the analytical solution more precisely as the time step is reduced. If the numerical solution is carried out with a time step duration that is between t = τlumped = 100 s and t = 2τlumped = 200 s then the prediction oscillates about the actual temperature but remains bounded. For time step durations greater than t = 2τlumped = 200 s, the solution oscillates in an unbounded manner. Figure 3-3 shows that any solution in which t > τlumped is unstable. This threshold time step duration that governs the stability of the solution is called the critical time step, tcrit , and its value can be determined from the details of the problem. Substituting the

Maximum numerical error (K)

3.2 Numerical Solutions to 0-D Transient Problems 10

2

10

1

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

Euler Heun RK4 Fully implicit Crank-Nicholson

-8

10 2 x 10-1

321

100

101 Duration of time step (s)

102

Figure 3-4: Maximum numerical error as a function of t for various numerical integration techniques.

governing equation, Eq. (3-15) into the deﬁnition of Euler’s method, Eq. (3-19), leads to: T j+1 = T j −

(T j − T ∞ ) t τlumped

(3-24)

Rearranging Eq. (3-24) leads to: T j+1 = T j 1 −

t τlumped

+ T∞

t τlumped

(3-25)

The solution becomes unstable when the coefﬁcient multiplying the temperature at the beginning of the time step (Tj ) becomes negative; therefore: tcrit = τlumped

(3-26)

Not surprisingly, the numerical simulation of objects that have small time constants will require the use of small time steps. The accuracy of the solution (i.e., the degree to which the numerical solution agrees with the analytical solution) improves as the duration of the time step is reduced. Figure 3-4 illustrates the global error, deﬁned as the maximum error between the numerical and analytical solution, as a function of the duration of the time step. The global error is related to the local error associated with any one time step. Note that while the local error is proportional to t2 , the global error is proportional to t. Also shown in Figure 3-4 is the global accuracy of several alternative numerical integration techniques that are discussed in subsequent sections. Any of the numerical integration techniques discussed in this section can be implemented in most software. As the problems become more complex (e.g., 1-D transient problems), the number of equations and the amount of data that must be stored increases and therefore it will be appropriate to utilize a formal programming language (e.g., MATLAB) for the solution.

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400

Temperature (K)

380 360 340 320 300 0

100

200 300 Time (s)

400

500

Figure 3-5: Temperature as a function of time predicted by MATLAB using Euler’s method.

The numerical solution using Euler’s method is implemented in MATLAB using a script with the inputs provided as the ﬁrst lines: clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400;

% time constant (s) % initial temperature (K) % ambient temperature (K)

The array of times at which the solution will be determined is speciﬁed: t sim=500; M=101; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% simulation time (s) % number of time steps (-) % time step duration (s) % time associated with each step

Because Euler’s method is explicit, the solution is implemented in MATLAB using exactly the same technique that is used in EES; this is possible because each of the predictions for Tj+1 require only knowledge of Tj , as shown by Eq. (3-25). T(1)=T ini; for j=1:(M-1) T(j+1)=T(j)-(T(j)-T inﬁnity)∗ DELTAt/tau; end

% initial temperature % Euler method

Running the script will place the vectors T and time in the command space. The numerical solution obtained by MATLAB is shown in Figure 3-5. Heun’s Method Euler’s method is the simplest example of a numerical integration technique; it is a ﬁrst order explicit technique. In this section, Heun’s method is presented. Heun’s method is

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323

a second order explicit technique (but with the same stability characteristics as Euler’s method). Heun’s method is an example of a predictor-corrector technique. In order to simulate any time step j, Heun’s method begins with an Euler step to obtain an initial prediction for the temperature at the conclusion of the time step (Tˆ j+1 ). This ﬁrst step in the solution is referred to as the predictor step and the details are essentially identical to Euler’s method: dT t (3-27) Tˆ j+1 = T j + dt T =T j ,t=tj However, Heun’s method uses the results of the predictor step to carry out a corrector step. The temperature predicted at the end of the time step (Tˆ j+1 ) is used to predict the | ). The corrector temperature rate of change at the end of the time step ( dT dt T =Tˆ j+1 ,t=t j+1 step predicts the temperature at the end of the time step (Tj+1 ) based on the average of the time rates of change at the beginning and end of the time step. dT dT t + (3-28) T j+1 = T j + dt T =T j ,t=tj dt T =Tˆ ,t=tj+1 2 j+1

Heun’s method is illustrated in the context of the same simple problem that was used to demonstrate Euler’s technique. Initially, the technique will be implemented using EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

t sim=500 [s] M=51 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

The process of moving through the ﬁrst time step begins with the predictor step: dT t (3-29) Tˆ 2 = T 1 + dt T =T 1 ,t=t1 or, substituting Eq. (3-15) into Eq. (3-29): (T 1 − T ∞ ) t Tˆ 2 = T 1 − τlumped T[1]=T ini T hat[2]=T[1]-(T[1]-T inﬁnity)∗ DELTAt/tau

(3-30)

“initial temperature” “predictor step”

Note that for the same conditions considered previously (i.e., τlumped = 100 s, T ini = 300 K, and T ∞ = 400 K), the predictor step using t = 10 s leads to Tˆ 2 = 310 K, which is about 0.5 K in error relative to the actual temperature calculated using Eq. (3-16), T an,t=10s = 309.52 K.

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Transient Conduction

The corrector step follows: T2 = T1 +

dT dT t + dt T =T 1 ,t=t1 dt T =Tˆ 2 ,t=t2 2

(3-31)

or, substituting Eq. (3-15) into Eq. (3-31): (T 1 − T ∞ ) (Tˆ 2 − T ∞ ) t T2 = T1 − + τlumped τlumped 2 T[2]=T[1]-((T[1]-T inﬁnity)/tau+(T hat[2]-T inﬁnity)/tau)∗ DELTAt/2

(3-32)

“corrector step”

The corrector step predicts T 2 = 9.50◦ C, which is only 0.02◦ C in error with respect to T an,t=10s = 9.52◦ C. This result illustrates the power of the predictor/corrector process; a single-step correction (which approximately doubles the computational time) can lead to more than an order of magnitude increase in accuracy. The two-step predictor/corrector process is a second order method; the local error is proportional to the duration of the time step to the third power. The EES code below implements a numerical solution to the problem using a duplicate loop and computes the numerical error. (Note that the EES code corresponding to the Euler solution must be commented out or deleted.) “Heun’s Method” T[1]=T ini duplicate j=1,(M-1) “predictor step” T hat[j+1]=T[j]-(T[j]-T inﬁnity)∗ DELTAt/tau T[j+1]=T[j]-((T[j]-T inﬁnity)/tau+(T hat[j+1]-T inﬁnity)/tau)∗ DELTAt/2 “corrector step” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M]) “maximum numerical error”

Figure 3-6 illustrates the numerical solution obtained using Heun’s method with time step durations of t = 50 s, t = 83.3 s, t = 167.7 s and t = 250 s. Also shown in Figure 3-6 are the analytical solution and the solution using Euler’s method with t = 50 s. Notice that for the same time step duration ( t = 50 s), the numerical result obtained with Heun’s method is much closer to the analytical solution than the numerical result obtained with Euler’s method. The maximum error between the numerical and analytical solutions is shown in Figure 3-4 for Heun’s technique and the improvement relative to Euler’s technique is obvious. Also note in Figure 3-6 that Heun’s method is unstable for time step durations that are greater than τlumped = 100 s and temperature actually decreases for t > 2τlumped = 200 s. The instability is not oscillatory; however, the numerical result begins to diverge from the analytical results and provides a non-physical solution. Both Euler’s and Heun’s methods are explicit techniques and both will therefore become unstable when the time step duration exceeds the critical time step. Heun’s method is explicit and therefore its implementation in MATLAB is straightforward. The script begins by specifying the inputs and setting up the time steps and initial conditions:

3.2 Numerical Solutions to 0-D Transient Problems

325

400

Temperature (K)

350 Heun's method, Δt = 50 s Heun's method, Δt = 83.3 s Heun's method, Δt = 167.7 s Heun's method, Δt = 250 s Euler's method, Δt = 50 s analytical solution

300

250

200 0

50

100 150 200 250 300 350 400 450 500 Time (s)

Figure 3-6: Numerical solution obtained using Heun’s method for various values of t. Also shown is the analytical solution and the solution using Euler’s method with t = 50 s.

clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400; t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% time constant (s) % initial temperature (K) % ambient temperature (K) % simulation time (s) % number of time steps (-) % time step duration (s)

The integration follows the form provided by Eqs. (3-27) and (3-28). One advantage of using MATLAB over EES is that MATLAB utilizes assignment statements. Therefore, it is not necessary to store the intermediate variable Tˆ j+1 (i.e., the result of the predictor step) for each time step in an array. Instead, the value of the variable Tˆ j+1 (the variable T_hat) is over-written during each iteration of the for loop; this saves memory and time. EES uses equality statements rather than assignment statements and so it will not allow the value of a variable to be overwritten in the main body of the program. (It is possible to overwrite values in an EES function or procedure.)

T(1)=T ini; % initial temperature for j=1:(M-1) % predictor step T hat=T(j)-(T(j)-T inﬁnity)∗ DELTAt/tau; T(j+1)=T(j)-((T(j)-T inﬁnity)/tau+(T hat-T inﬁnity)/tau)∗ DELTAt/2; % corrector step end

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Transient Conduction

Runge-Kutta Fourth Order Method Heun’s method is a two-step predictor/corrector technique that improves the order of accuracy by one (i.e., Heun’s method is second order whereas Euler’s method is ﬁrst order). It is possible to carry out additional predictor/corrector steps and further improve the accuracy of the numerical solution. One of the most popular, higher order techniques is the fourth order Runge-Kutta method which involves four predictor/corrector steps. The Runge-Kutta fourth order method (referred to subsequently as the RK4 method) estimates the time rate of change of the state variable four times; recall that Euler’s method did this only once (at the beginning of time step) and Heun’s method did this twice (at the beginning and end of the time step). The RK4 method begins by estimating the time rate of change at the beginning of the time step (referred to, for convenience, as aa): dT (3-33) aa = dt T =T j ,t=tj The ﬁrst estimate, aa, is used to predict the temperature half-way through the time step (Tˆ j+ 1 ): 2

t Tˆ j+ 1 = T j + aa 2 2

(3-34)

which is used to obtain the second estimate of the time rate of change, bb, at the midpoint of the time step: dT (3-35) bb = dt T =Tˆ ,t=tj + t j+ 1 2

2

The second estimate is used to re-compute the temperature half-way through the time step (Tˆˆ j+ 1 ): 2

t Tˆˆ j+ 1 = T j + bb 2 2

(3-36)

which is used to obtain a third estimate of the time rate of change, cc, also at the midpoint of the time step: dT (3-37) cc = dt T =Tˆ ,t=tj + t j+ 1 2

2

The third estimate is used to predict the temperature at the end of the time step (Tˆ j+1 ): Tˆ j+1 = T j + cc t

(3-38)

which is used to obtain the fourth and ﬁnal estimate of the time rate of change, dd, at the end of the time step: dT (3-39) dd = dt T =Tˆ ,t=tj+1 j+1

The integration is ﬁnally carried out using the weighted average of these four separate estimates of the time rate of change: T j+1 = T j + (aa + 2 bb + 2 cc + dd)

t 6

(3-40)

3.2 Numerical Solutions to 0-D Transient Problems

327

The RK4 technique is implemented in EES in order to solve the problem discussed previously: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K] t sim=500 [s] M=4 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“time constant” “initial temperature” “ambient temperature” “simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

The code below implements the RK4 technique: “Runge-Kutta 4th Order Method” T[1]=T_ini duplicate j=1,(M-1) “1st estimate of time-rate of change” aa[j]=-(T[j]-T inﬁnity)/tau T hat1[j]=T[j]+aa[j]∗ DELTAt/2 “1st predictor step” bb[j]=-(T hat1[j]-T inﬁnity)/tau “2nd estimate of time-rate of change” “2nd predictor step” T hat2[j]=T[j]+bb[j]∗ DELTAt/2 cc[j]=-(T hat2[j]-T inﬁnity)/tau “3rd estimate of time-rate of change” T hat3[j]=T[j]+cc[j]∗ DELTAt “3rd predictor step” “4th estimate of time-rate of change” dd[j]=-(T hat3[j]-T inﬁnity)/tau T[j+1]=T[j]+(aa[j]+2∗ bb[j]+2∗ cc[j]+dd[j])∗ DELTAt/6 “ﬁnal integration uses all 4 estimates” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M]) “maximum numerical error”

Figure 3-4 illustrates the maximum numerical error for the RK4 solution as a function of the time step duration and clearly indicates the improvement that can be obtained by using a higher order method. If a certain level of accuracy is required, then much larger time steps (and therefore many fewer computations) can be used if a higher order technique is used. For example, if an accuracy of 0.1 K is required then t = 0.4 s must be used with Euler’s method whereas t = 70 s can be used with the RK4 method. This difference represents more than two orders of magnitude of reduction in the number of time steps required while the number of computations per time step has only increased by a factor of four. The slope of the curves shown in Figure 3-4 is related to the order of the technique with respect to the global error as opposed to the local error. The maximum error associated with Euler’s method increases by an order of magnitude as the time step duration increases by an order of magnitude; therefore, while the local error associated with each

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Transient Conduction

time step has order two, the global error has order one. The maximum error associated with Heun’s method changes by two orders of magnitude for every order of magnitude change in t and therefore Heun’s method is order two with respect to the global error. The RK4 method is nominally order four with respect to global error. The MATLAB code to implement the RK4 method follows naturally from the EES code because the method is explicit. Note that the intermediate variables (e.g., aa, bb, etc.) are not stored and therefore the MATLAB program is less memory intensive.

clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400;

% time constant (s) % initial temperature (K) % ambient temperature (K)

t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% simulation time (s) % number of time steps (-) % time step duration (s)

T(1)=T ini; for j=1:(M-1) aa=-(T(j)-T inﬁnity)/tau; T hat=T(j)+aa∗ DELTAt/2; bb=-(T hat-T inﬁnity)/tau; T hat=T(j)+bb∗ DELTAt/2; cc=-(T hat-T inﬁnity)/tau; T hat=T(j)+cc∗ DELTAt; dd=-(T hat-T inﬁnity)/tau; T(j+1)=T(j)+(aa+2∗ bb+2∗ cc+dd)∗ DELTAt/6;

% initial temperature % 1st estimate of time-rate of change % 1st predictor step % 2nd estimate of time-rate of change % 2nd predictor step % 3rd estimate of time-rate of change % 3rd predictor step % 4th estimate of time-rate of change % ﬁnal integration uses all 4 estimates

end

Fully Implicit Method The methods discussed thus far are explicit; they all therefore share the characteristic of becoming unstable when the time step exceeds a critical value. An implicit technique avoids this problem. The fully implicit method is similar to Euler’s method in that the time rate of change is assumed to be constant throughout the time step. However, the time rate of change is computed at the end of the time step rather than the beginning. Therefore, for any time step j: T j+1

dT = Tj + t dt T =T j+1 ,t=tj+1

(3-41)

The time rate of change at the end of the time step depends on the temperature at the end of the time step (Tj+1 ). Therefore, Tj+1 cannot be calculated explicitly using information at the beginning of the time step (Tj ) and instead an implicit equation is obtained for Tj+1 . For the example problem that has been considered in previous sections, the

3.2 Numerical Solutions to 0-D Transient Problems

329

implicit equation is obtained by substituting Eq. (3-15) into Eq. (3-41): T j+1 = T j −

(T j+1 − T ∞ ) t τlumped

(3-42)

Notice that Tj+1 appears on both sides of Eq. (3-42). Because EES solves implicit equations, it is not necessary to rearrange Eq. (3-42). The implicit solution is obtained using the following EES code: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

t sim=500 [s] M=4 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

“Fully Implicit Method” T[1]=T ini duplicate j=1,(M-1) T[j+1]=T[j]-(T[j+1]-T inﬁnity)∗ DELTAt/tau

“initial temperature”

“implicit equation for temperature” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M])

Figure 3-7 illustrates the fully implicit solution for various values of the time step; also shown is the analytical solution, Eq. (3-16). The accuracy of the fully implicit solution is reduced as the duration of the time step increases. Figure 3-4 illustrates the maximum error associated with the fully implicit technique as a function of the time step duration; notice that the fully implicit technique is no more accurate than Euler’s technique. However, Figure 3-7 shows that the fully implicit solution does not become unstable even when the duration of the time step is greater than the critical time step. The implementation of the fully implicit method cannot be accomplished in the same way in MATLAB that it is in EES because Eq. (3-42) is not explicit for Tj+1 ; while EES can solve this implicit equation, MATLAB cannot. It is necessary to solve Eq. (3-42) for Tj+1 in order to carry out the integration step in MATLAB:

T j+1

t T j + T∞ τ = t 1+ τ

(3-43)

330

Transient Conduction 400

Temperature (K)

380

360

340 analytical solution Δt = 1 s (every 15 th point shown) Δt = 50 s Δt = 150 s Δt = 300 s

320

300 0

100

200

300 Time (s)

400

500

600

Figure 3-7: Fully implicit solution for various values of the time step duration as well as the analytical solution.

clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400;

% time constant (s) % initial temperature (K) % ambient temperature (K)

t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% simulation time (s) % number of time steps (-) % time step duration (s)

T(1)=T ini; for j=1:(M-1) T(j+1)=(T(j)+T inﬁnity∗ DELTAt/tau)/(1+DELTAt/tau); end

% initial temperature % implicit step

Crank-Nicolson Method The Crank-Nicolson method combines Euler’s method with the fully implicit method. The time rate of change for the time step is estimated based on the average of its values at the beginning and end of the time step. Therefore, for any time step j: T j+1 = T j +

dT dT t + dt T =T j ,t=tj dt T =T j+1 ,t=tj+1 2

(3-44)

Notice that the Crank-Nicolson is an implicit method because the solution for T j+1 involves a time rate of change that must be evaluated based on T j+1 . Therefore, the technique will have the stability characteristics of the fully implicit method. The solution

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331

also involves two estimates for the time rate of change and is therefore second order and more accurate than the fully implicit technique. Substituting Eq. (3-15) into Eq. (3-44) leads to:

T j+1

(T j − T ∞ ) (T j+1 − T ∞ ) t = Tj − + τlumped τlumped 2

(3-45)

The EES code below implements the Crank-Nicolson solution for the example problem:

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

t sim=600 [s] M=601 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

“Crank Nicholson Method” T[1]=T ini duplicate j=1,(M-1) T[j+1]=T[j]-((T[j]-T inﬁnity)/tau+(T[j+1]-T inﬁnity)/tau)∗ DELTAt/2 “C-N equation for temperature” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M])

The maximum numerical error associated with the Crank-Nicolson technique is shown in Figure 3-4. Notice that the fully implicit method has accuracy that is nominally equivalent to Euler’s method while the Crank-Nicolson method has accuracy slightly better than Heun’s method. The Crank-Nicolson method is a popular choice because it combines high accuracy with stability. To implement the Crank-Nicolson technique using MATLAB, it is necessary to solve the implicit Eq. (3-45) for T j+1 :

T j+1

t t + T∞ Tj 1 − 2τ τ = t 1+ 2τ

(3-46)

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Transient Conduction

so the MATLAB script becomes: clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400; t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% time constant (s) % initial temperature (K) % ambient temperature (K) % simulation time (s) % number of time steps (-) % time step duration (s)

T(1)=T ini; % initial temperature for j=1:(M-1) T(j+1)=(T(j)∗ (1-DELTAt/(2∗ tau))+T inﬁnity∗ DELTAt/tau)/(1+DELTAt/(2∗ tau));

% C-N step

end

Adaptive Step-Size and EES’ Integral Command The implementation of the techniques that have been discussed to this point is accomplished using a ﬁxed duration time step for the entire simulation. This implementation is often not efﬁcient because there are regions of time during the simulation where the solution is not changing substantially and therefore large time steps could be taken with little loss of accuracy. Adaptive step-size solutions adjust the size of the time step used based on the local characteristics of the state equation. Typically, the absolute value of the local time rate of change or the second derivative of the time rate of change is used to set a step-size that guarantees a certain level of accuracy. For the current example, shown in Figure 3-6, smaller time steps would be used near t = 0 s because the rate of temperature change is largest at this time. Later in the simulation, for t > 300 s, the temperature is not changing substantially and therefore large time steps could be used to obtain a solution more rapidly. The implementation of adaptive step-size solutions is beyond the scope of this book. However, a third order integration routine that optionally uses an adaptive step-size is provided with EES and can be accessed using the Integral command. EES’ Integral command requires four arguments and allows an optional ﬁfth argument: F = Integral(Integrand,VarName,LowerLimit,UpperLimit,StepSize) where Integrand is the EES variable or expression that must be integrated, VarName is the integration variable, LowerLimit and UpperLimit deﬁne the limits of integration, and StepSize provides the duration of the time step. When using the Integral technique (or indeed any numerical integration technique) it is useful to ﬁrst verify that, given temperature and time, the EES code is capable of computing the time rate of change of the temperature. Therefore, the ﬁrst step is to implement Eq. (3-15) in EES for an arbitrary (but reasonable) value of T and t:

3.2 Numerical Solutions to 0-D Transient Problems

333

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K] t sim=600 [s]

“time constant” “initial temperature” “ambient temperature” “simulation time”

“EES’ Integral Method” T=350 [K] time = 0 [s] dTdt=-(T-T inﬁnity)/tau

“Temperature, arbitrary - used to test dTdt” “Time, arbitrary - used to test dTdt” “Time rate of change”

which leads to dTdt = 0.5 K/s. The next step is to comment out the arbitrary values of temperature and time that are used to test the computation of the state equation and instead let EES’ Integral function control these variables for the numerical integration. The temperature of the object is given by: tsim T = T ini +

dT dt dt

(3-47)

0

Therefore, the solution to our example problem is obtained by calling the Integral function; Integrand is replaced with the variable dTdt, VarName with time, LowerLimit with 0, UpperLimit with the variable t_sim, and StepSize with DELTAt, the speciﬁed duration of the time step: “EES’ Integral Method” {T=350 [K] “Temperature, arbitrary - used to test dTdt” time=0 [s] “Time, arbitrary - used to test dTdt”} dTdt=-(T-T inﬁnity)/tau “Time rate of change” DELTAt=1 [s] “Duration of the time step” T=T ini+INTEGRAL(dTdt,time,0,t sim,DELTAt) “Call EES’ Integral function”

In order to accomplish the numerical integration, EES will adjust the value of the variable time from 0 to t_sim in increments of DELTAt. At each value of time, EES will iteratively solve all of the equations in the Equations window that depend on time. For the example above, this process will result in the variable T being evaluated at each value time. When the solution converges, the value of the variable time is incremented and the process is repeated until time is equal to t_sim. The results shown in the Solution window will provide the temperature at the end of the process (i.e., the result of the integration, which is the temperature at time = t_sim). Often it is most interesting to know the temperature variation with time during the process. This information can be provided by including the $IntegralTable directive in the ﬁle. The format of the $IntegralTable directive is: $IntegralTable VarName: Step, x,y,z

334

Transient Conduction

[K]

[s] 0

300

1

301

2

302

3

303

4

303.9

Figure 3-8: Integral Table generated by EES.

where VarName is the integration variable; the ﬁrst column in the Integral Table will hold values of this variable. The colon followed by the parameter Step (which can either be a number or a variable name) and list of variables (x, y, z) are optional. If these values are provided, then the value of Step will be used as the output step size and the integration variables will be reported in the Integral Table at the speciﬁed output step size. The output step size may be a variable name, rather than a number, provided that the variable has been set to a constant value preceding the $IntegralTable directive. The step size that is used to report integration results is totally independent of the duration of the time step that is used in the numerical integration. If the numerical integration step size and output step size are not the same, then linear interpolation is used to determine the integrated quantities at the speciﬁed output steps. If an output step size is not speciﬁed, then EES will output all speciﬁed variables at every time step. The variables x, y, z . . . must correspond to variables in the EES program. Algebraic equations involving variables are not accepted. A separate column will be created in the Integral Table for each speciﬁed variable. The variables must be separated by a space or list delimiter (comma or semicolon). Solving the EES code will result in the generation of an Integral Table that is ﬁlled with intermediate values resulting from the numerical integration. The values in the Integral Table can be plotted, printed, saved, and copied in exactly the same manner as for other tables. The Integral Table is saved when the EES ﬁle is saved and the table is restored when the EES ﬁle is loaded. If an Integral Table exists when calculations are initiated, it will be deleted if a new Integral Table is created. Use the EES code below to generate an Integral Table containing the results of the numerical simulation and the analytical solution.

$IntegralTable time, T

After running the code with a time step duration of 1 s, the Integral Table shown in Figure 3-8 will be generated. The results of the integration can be plotted by selecting the Integral Table as the source of the data to plot. The StepSize input to the Integral command is optional. If the parameter StepSize is not included or if it is set to a value of 0, then EES will use an adaptive step-size algorithm that maintains accuracy while maximizing computational speed. The parameters used to control the adaptive step-size algorithm can be accessed and adjusted by selecting Preferences from the Options menu and selecting the Integration Tab. A complete description of these parameters can be obtained from the EES Help menu. To run the program using an adaptive step-size, remove the ﬁfth argument from the Integral command (or set its value to 0):

3.2 Numerical Solutions to 0-D Transient Problems

335

400

Temperature (K)

380

360

340

320

300 0

100

200

300 Time (s)

400

500

600

Figure 3-9: Numerical solution predicted using EES’ internal Integration function with an adaptive time step.

“EES’ Integral Method” dTdt=-(T-T inﬁnity)/tau T=T ini+INTEGRAL(dTdt,time,0,t sim)

“Time rate of change” “Call EES’ Integral function”

$IntegralTable time, T

The results are shown in Figure 3-9. Note the concentration of time steps near t = 0 s. Also note that the Adaptive Step-Size parameters had to be adjusted in order for the algorithm to operate for such a simple problem; speciﬁcally, the minimum number of steps had to be reduced to 10 and the criteria for increasing the step-size had to be increased to 0.01. MATLAB’s Ordinary Differential Equation Solvers MATLAB has a suite of solvers for initial value problems that implement advanced numerical integration algorithms (e.g., the functions ode45, ode23, etc.). These ordinary differential equation solvers all have a similar protocol; for example: [ time ,

T

vector temperature of time at the times

] = ode45(

‘dTdt’ function that returns the derivative of temperature

,

tspan time span to be integrated

,

T0

)

initial temperature

The ode solvers require three inputs, the name of the function that returns the derivative of temperature with respect to time given the current value of temperature and time (i.e., the function that implements the state equation for the system), the simulation time, and the initial temperature. The ode solver returns two vectors containing the solution times and the temperatures at the solution times. The ode solvers require that you have created a function that implements the state equation for the system. MATLAB assumes that this function will accept two inputs, corresponding to the current values of time and temperature, and return a single output, the rate at which temperature is changing with

336

Transient Conduction

400

Temperature (K)

380 360 340 320 300 0

100

200 300 Time (s)

400

500

Figure 3-10: Predicted temperature as a function of time using MATLAB’s ode45 solver.

time. The function dTdt_function, shown below, implements the state equation for the example considered previously: function[dTdt]=dTdt function(time,T) % this function computes the rate of change of temperature % Inputs % T - temperature (K) % time - time (s) % % Outputs % dTdt - time rate of change (K/s) tau=100; % time constant (s) T inﬁnity=500; % ambient temperature (K) dTdt=-(T-T inﬁnity)/tau; end

The script that solves the problem calls the integration routine ode45 and speciﬁes that the state equations are determined in the function dTdt_function, the simulation time is from 0 to t_sim and the initial condition is T_ini. clear all; %INPUTS T ini=300; t sim=500; [time,T]=ode45(‘dTdt function’,t sim,T ini);

% initial temperature (K) % simulation time (s) % use ode45 to solve problem

The vectors time and T are the times and temperatures used to carry out the simulation. Figure 3-10 shows the temperature predicted using the ode45 solver as a function of time. Figure 3-10 shows that the MATLAB solver is using an adaptive step-size algorithm because the duration of the time step varies throughout the computational domain. If the variable t_sim is speciﬁed as a vector of times, then MATLAB will return the solution at these speciﬁc times rather than the times that are actually used for the integration

3.2 Numerical Solutions to 0-D Transient Problems

337

(although the integration process itself does not change). For example, in order to obtain a solution every 10 s the script should be changed as shown: t span=linspace(0,500,51); [time,T]=ode45(‘dTdt function’,t span,T ini);

% specify times to return the solution at

It is not convenient that the state equation function, dTdt_function, only allows two inputs. It is difﬁcult to run parametric studies or optimization if the values of tau and T_inﬁnity must be changed manually within the function. This requirement can be avoided by modifying the function dTdt_function so that it accepts all of the inputs of interest: function[dTdt]=dTdt_function(time, T, tau, T_inﬁnity) % this function computes the rate of change of temperature % Inputs % T - temperature (K) % time - time (s) % tau - time constant (s) % T_inﬁnity - ambient temperature (K) % % Outputs % dTdt - time rate of change (K/s) dTdt=-(T-T_inﬁnity)/tau; end

The call to the function dTdt_function with the two inputs required by the function ode45 (time and T) is “mapped” to the full function call. This mapping has the form: @(time, T) dTdt function(time, T, tau, T amb) 2 inputs that the full function call, including the 2 required inputs are required by ode solver

The revised script becomes: clear all; %INPUTS T ini=300; t sim=500; tau=100; T inﬁnity=400;

% initial temperature (K) % simulation time (s) % time constant (s) % ambient temperature (K)

[time,T]=ode45(@(time,T) dTdt function(time,T, tau, T inﬁnity),[0,t sim],T ini);

A fourth argument (which is optional) can be added to the function ode45: [time,T ]=ode45(‘dTdt’, tspan, T0, OPTIONS)

where OPTIONS is a vector that controls the integration properties. In the absence of a fourth argument, MATLAB will use the default settings for the integration properties. The most convenient way to adjust the integration settings is with the odeset function,

338

Transient Conduction

400

Temperature (K)

380 360 340 320 300 0

100

200 300 Time (s)

400

500

Figure 3-11: Predicted temperature as a function of time using MATLAB’s ode45 solver with increased accuracy.

which has the format: OPTIONS = odeset(‘property1’, value1, ‘property2’, value2, . . .) where property1 refers to one of the integration properties and value1 refers to its speciﬁed value, property2 refers to a different integration property and value2 refers to its speciﬁed value, etc. The properties that are not explicitly set in the odeset command remain at their default values. Type help odeset at the command line prompt in order to see a list and description of the integration properties with their default values. The property RelTol refers to the relative error tolerance and has a default value of 0.001 (0.1% accuracy). The relative error tolerance of the integration process can be reduced to 1×10−6 by changing the script according to: OPTIONS=odeset(‘RelTol’,1e-6); % change the relative error tolerance used in the integration [time,T]=ode45(@(time,T) dTdt function(time,T, tau, T inﬁnity),[0,t sim],T ini,OPTIONS);

The results of the integration with the improved accuracy are shown in Figure 3-11. Notice that substantially more integration steps were required (compared with Figure 3-10) in order to obtain the improved accuracy. The integration function ode45 was used in the discussion in this section. However, MATLAB provides other differential equation solvers that are specialized based on the stiffness and other aspects of the problem. To examine these solvers, type help funfun at the command line. The differential equation solvers have similar calling protocol, so it is easy to switch between them. For example, to use the ode15s rather than the ode45 function it is only necessary to change the name of the function call: OPTIONS=odeset(‘RelTol’,1e-6); [time,T]=ode15s(@(time,T) dTdt_function(time,T, tau, T_inﬁnity),[0,t_sim],T_ini,OPTIONS);

339

EXAMPLE 3.2-1(a): OVEN BRAZING (EES) A brazing operation is carried out in an evacuated oven. The metal pieces to be brazed have a complex geometry; they are made of bronze (with k = 50 W/m-K, c = 500 J/kg-K, and ρ = 8700 kg/m3 ) and have total volume V = 10 cm3 and total surface area As = 35 cm2 . The pieces are heated by radiation heat transfer from the walls of the oven. A detailed presentation of radiation heat transfer is presented in Chapter 10. For this problem, assume that the emissivity of the surface of the piece is ε = 0.8 and that the wall of the oven is black. In this limit, the rate of radiation heat transfer from the wall to the piece (q˙r ad ) may be written as: q˙r ad = As ε σ Tw4 − T 4 where Tw is the temperature of the wall, T is the temperature of the surface of the piece, and σ is the Stefan-Boltzmann constant. The temperature of the oven wall is increased at a constant rate β = 1 K/s from its initial temperature Tini = 20◦ C to its ﬁnal temperature Tf = 470◦ C which is held for thold = 1000 s before the temperature of the wall is reduced at the same constant rate back to its initial temperature. The pieces and the oven are initially in thermal equilibrium at Tt =0 = Tini . a) Is the lumped capacitance assumption appropriate for this problem? The inputs are entered in EES. “EXAMPLE 3.2-1(a): Oven Brazing (EES)” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” V=10 [cmˆ3]∗ convert(cmˆ3,mˆ3) A s=35 [cmˆ2]∗ convert(cmˆ2,mˆ2) e=0.8 [-] T ini=converttemp(C,K,20 [C]) T f=converttemp(C,K,470 [C]) t hold=1000 [s] beta=1 [K/s] c=500 [J/kg-K] k=50 [W/m-K] rho=8700 [kg/mˆ3]

“volume” “surface area” “emissivity of surface” “initial temperature” “ﬁnal oven temperature” “oven hold time” “oven ramp rate” “speciﬁc heat capacity” “conductivity” “density”

The lumped capacitance assumption ignores the internal resistance to conduction as being small relative to the resistance to heat transfer from the surface of the object. The radiation heat transfer coefﬁcient, deﬁned in Section 1.2.6, is: hrad = σ ε Ts2 + Tw2 (Ts + Tw ) where Ts is the surface temperature of the object. The Biot number deﬁned based on this radiation heat transfer coefﬁcient is: Bi =

hrad L cond k

where L cond =

V As

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

340

Transient Conduction

The Biot number is largest (and therefore the lumped capacitance model is least justiﬁed) when the value of hrad is largest, which occurs if both Ts and Tw achieve their maximum possible value (Tf ). h rad max=sigma# ∗ e∗ (T fˆ2+T fˆ2)∗ (T f+T f) L cond=V/A s Bi=h rad max∗ L cond/k

“maximum radiation coefﬁcient” “characteristic length for conduction” “maximum Biot number”

The Biot number is calculated to be 0.004 at this upper limit, which indicates that the lumped capacitance assumption is valid. b) Calculate a lumped capacitance time constant that characterizes the brazing process. It is useful to calculate a time constant even when the problem is solved numerically. The value of the time constant provides guidance relative to the time step that will be required and it also allows a sanity check on your results. The time constant, discussed in Section 3.1.4, is the product of the thermal capacitance of the object and the net resistance from the surface. Using the concept of the radiation heat transfer coefﬁcient allows the time constant to be written as: τlumped =

V ρc hrad As

The minimum value of the time-constant (again, corresponding to the object and the oven wall being at their maximum temperature) is computed in EES: tau=rho∗ c∗ V/(h rad max∗ A s)

“time constant”

and found to be 167 s. c) Develop a numerical solution based on Heun’s method that predicts the temperature of the object for 3000 s after the oven is activated. A function T_w is deﬁned which returns the wall temperature as a function of time; the function is placed at the top of the EES ﬁle.

“Oven temperature function” function T w(time,T ini,T f,beta,t hold) “INPUTS: time - time relative to initiation of process (s) T ini - initial temperature (K) T f - ﬁnal temperature (K)

341

beta - ramp rate (K/s) t hold - hold time (s) OUTPUTS T w - wall temperature (K)” T w=T ini+beta∗ time t ramp=(T f-T ini)/beta if (time>t ramp) then T w=T f endif if (time>(t ramp+t hold)) then T w=T f-beta∗ (time-t ramp-t hold) endif if (time>(2∗ t ramp+t hold)) then T w=T ini endif end

“temperature of wall during ramp up period” “duration of ramp period” “temperature of wall during hold period”

“temperature during ramp down period”

“temperature after ramp down period”

Note the use of the if-then-endif statements in the function to activate different equations for the wall temperature based on the time of the simulation. The function T_w be called from the Equation Window: T_w=T_w(time,T_i,T_f,beta,tau_hold)

It is wise to check that the function is working correctly by setting up a parametric table that includes time and the wall temperature; the result is shown in Figure 1. 800

Wall temperature (K)

700 600 500 400 300 200 0

500

1000

1500 Time (s)

2000

Figure 1: Oven wall temperature as a function of time.

2500

3000

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

342

Transient Conduction

The simulation time and number of time steps are deﬁned and used to compute the time and wall temperature at each time step: t sim=3000 [s] M=101 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) T w[j]=T w(time[j],T ini,T f,beta,t hold) end

“simulation time” “number of time-steps” “time-step duration” “time” “wall temperature”

The governing differential equation is obtained from an energy balance on the piece: q˙ rad =

dU dt

or dT As ε σ Tw4 − T 4 = V ρ c dt

(1)

Rearranging Eq. (1) leads to the state equation that provides the time rate of change of the temperature: As ε σ Tw4 − T 4 dT = (2) dt V ρc The temperature at the beginning of the ﬁrst time step is the initial condition: T1 = Tini

T[1]=T ini

“initial temperature”

Heun’s method consists of an initial, predictor step: dT t Tˆ j+1 = T j + d t T =T j ,t =t j

(3)

Substituting the state equation, Eq. (2), into Eq. (3) leads to: 4 4 A ε σ T − T s w,t =t j dT j = d t T =T j ,t =t j V ρc The corrector step is:

T j+1 = T j +

dT dT + d t T =T j ,t =t j d t T =Tˆ

where dT d t T =Tˆ

j+1

,t =t j+1

=

j+1

,t =t j+1

4 ˆ4 As ε σ Tw,t =t j+1 − T j+1 V ρc

t 2

343

Heun’s method is implemented in EES: T[1]=T ini “initial temperature” duplicate j=1,(M-1) dTdt[j]=e∗ A s∗ sigma # ∗ (T w[j]ˆ4-T[j]ˆ4)/(rho∗ V∗ c) “Temperature rate of change at the beginning of the time-step” “Predictor step” T hat[j+1]=T[j]+dTdt[j]∗ DELTAt dTdt hat[j+1]=e∗ A s∗ sigma #∗ (T w[j+1]ˆ4-T hat[j+1]ˆ4)/(rho∗ V∗ c) “Temperature rate of change at the end of the time-step” “Corrector step” T[j+1]=T[j]+(dTdt[j]+dTdt hat[j+1])∗ DELTAt/2 end

The solution for 100 time steps is shown in Figure 2. 800 Heun's method EES Integral function th (every 10 point)

Temperature (K)

700 600 500 400 300 200 0

Oven wall temperature 500

1000

1500 Time (s)

2000

2500

3000

Figure 2: Oven wall and piece temperature as a function of time, predicted by Heun’s method with 100 time steps and using EES’ built-in Integral function.

Note that the piece temperature lags the oven wall temperature by 100’s of seconds, which is consistent with the time constant calculated in (b). d) Develop a numerical solution using EES’ Integral function that predicts the temperature of the object for 3000 s after the oven is activated. The adaptive time step algorithm is used within EES’ Integral function and an integral table is created that holds the results of the integration: “EES’ Integral function” dTdt=e∗ A s∗ sigma #∗ (T w(time,T ini,T f,beta,t hold)ˆ4-Tˆ4)/(rho∗ V∗ c) T=T ini+INTEGRAL(dTdt,time,0,t sim) “Call EES’ Integral function” $IntegralTable time, T

The results are included in Figure 2 and agree with Heun’s method; note that only 1 out of every 10 points are included in the EES solution in order to show the results of the adaptive step-size.

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

344

Transient Conduction

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB) Repeat EXAMPLE 3.2-1(a) using MATLAB rather than EES to do the calculations. a) Develop a numerical solution that is based on Heun’s method and implemented in MATLAB that predicts the temperature of the object for 3000 s after the oven is activated. The solution will be obtained in a function called Oven_brazing that can be called from the command space. The function will accept the number of time steps (M) and the total simulation time (tsim ) and return three arrays that are the values of time, the wall temperature, and the surface temperature at each time step. function [time, T_w, T]=Oven_Brazing(M, t_sim) % EXAMPLE 3.2-1(b) Oven Brazing % INPUTS % M - number of time steps (-) % t_sim - duration of simulation (s) % OUTPUTS % time - array with the values of time for each time step (s) % T_w - array containing values of the wall temperature at each time (K) % T - array containing values of the surface temperature at each time (K)

Next, the known information from the problem statement is entered into the function. %Known information V=10/100000; A s=35/1000; e=0.8; T ini=293.15; T f=743.15; t hold=1000; beta=1; c=500; k=50; rho=8700; sigma=5.67e-8;

% volume (mˆ3) % surface area (mˆ2) % emissivity of surface (-) % initial temperature (K) % ﬁnal oven temperature (K) % oven hold temperature (s) % oven ramp rate (K/s) % speciﬁc heat capacity (J/kg-K) % conductivity (W/m-K) % density (kg/mˆ3) % Stefan-Boltzmann constant (W/mˆ2-Kˆ4)

A function is needed to return the wall temperature as a function of time. The following code implements this function. Place this function at the bottom of the ﬁle, after an end statement that terminates the function Oven_Brazing. function[T_w]=T_wf(time, T_ini, T_f, beta, t_hold) % Oven temperature function % Inputs % time - current time value (s) % T_ini - initial value of the wall temperature (K)

345

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

3.2 Numerical Solutions to 0-D Transient Problems

% T_f - ﬁnal value of the wall temperature (K) % beta - rate of increase in temperature of the wall (K/s) % t_hold - time period in which the wall temperature is held constant (s) % Output % T_w - wall temperature at specﬁed time (K) T_w=T_ini+beta∗ time; t_ramp=(T_f-T_ini)/beta; if (time>t_ramp) T_w=T_f; end if (time>(t_ramp+t_hold)) T_w=T_f-beta∗ (time-t_ramp-t_hold); end if (time>(2∗ t_ramp+t_hold)) T_w=T_ini; end end

Note the use of the if-end clauses to activate different equations for the wall temperature based on the time of the simulation. The temperature of the wall at any time can be evaluated by a call to the function T_w. The following lines ﬁll the time and T_w vectors with the values of time and the wall temperature at each time step. DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); T w(j)=T wf(time(j),T ini,T f,beta,t hold);

% time step duration % value of time for each step % value of the wall temperature at each step

end

The last task is to enter the equations that implement Heun’s method for solving the differential equation. The governing equation is obtained from an energy balance on the piece: dT q˙r ad = As ε σ Tw4 − T 4 = V ρ c dt or

As ε σ Tw4 − T 4 dT = dt V ρc

Heun’s method consists of an initial, predictor step: dT Tˆ j+1 = T j + d t T =T j ,t =t j where:

4 4 A ε σ T − T s w,t =t j j dT = d t T =T j ,t =t j V ρc

(1)

Transient Conduction

The corrector step is:

T j+1 = T j +

dT dT + d t T =T j ,t =t j d t T =Tˆ

where, from Eq. (1), dT d t T =T ∗

j+1

,t =t j+1

=

j+1

,t =t j+1

t 2

4 ˆ4 As ε σ Tw,t − T =t j+1 j+1 V ρc

The following equations implement Heun’s method in MATLAB: T(1)=T ini; for j=1:(M-1) dTdt=e∗ A s∗ sigma∗ (T w(j)ˆ4-T(j)ˆ4)/(rho∗ V∗ c); %Temp deriv. at the start of the time step %Predictor step T hat=T(j)+dTdt∗ DELTAt; dTdt hat=e∗ A s∗ sigma∗ (T w(j+1)ˆ4-T hatˆ4)/(rho∗ V∗ c); %deriv. at end of time step %Corrector step” T(j+1)=T(j)+(dTdt+dTdt hat)∗ DELTAt/2; end

The Oven_Brazing function is terminated with an end statement and saved. end

The function can now be called from the command window >> [time, T_w, T]=Oven_Brazing(101, 3000);

The temperature of the wall and the piece is shown in Figure 1. 800 oven wall temperature Heun's method ode45 solver

700

Temperature (K)

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

346

600 500 400 300 200 0

500

1000

1500

2000

2500

3000

Time (s) Figure 1: Temperature of the wall and work piece, predicted using Heun’s method and the ode45 solver, as a function of time.

347

b) Develop a numerical solution using MATLAB’s ode45 function that predicts the temperature of the object for 3000 s after the oven is activated. The ode solvers are designed to call a function the returns the derivative of dependent variable with respect to the independent variable. In our case, the dependent variable is the temperature of the work piece and time is the independent variable. A function must be provided that returns the time derivative of the temperature at a speciﬁed time and temperature. The MATLAB function dTdt_f accepts all of the input parameters that are needed to determine the derivative; according to Eq. (1), these include As , ε, σ, V , ρ, and c as well as parameters needed determine the wall temperature at any time. Place the function dTdt_f below the function T_wf. Note that the function dTdt_f calls the function T_wf in order to determine the wall temperature at a speciﬁed time. function[dTdt]=dTdt_f(time,T,e,A_s,rho,V,c,T_f,T_ini,beta,t_hold) % dTdt is called by the ode45 solver to evaluate the derivative dTdt % INPUTS % time - time relative to start of process (s) % T - temperature of piece (K) % e - emissivity of piece (-) % A_s - surface area of piece (mˆ2) % rho - density (kg/mˆ3) % V - volume of piece (mˆ3) % c - speciﬁc heat capacity of piece (J/kg-K) % T_f - ﬁnal oven temperature (K) % T_ini - initial oven temperature (K) % beta - oven ramp rate (K/s) % t_hold - oven hold time (s) % OUTPUTS % dTdt - rate of change of temperature of the piece (K/s)

sigma=5.67e-8; T w=T wf(time, T ini, T f, beta, t hold); dTdt=e∗ A s∗ sigma∗ (T wˆ4-Tˆ4)/(rho∗ V∗ c); end

% Stefan-Boltzmann constant (W/mˆ2-Kˆ4) % wall temperature % energy balance

Now, all that is necessary is to comment out the code in the function Oven brazing from part (a) and instead call the MATLAB ode45 solver function to determine the temperatures as a function of time. % T(1)=T_ini; % for j=1:(M-1) % dTdt=e∗ A_s∗ sigma∗ (T_w(j)ˆ4-T(j)ˆ4)/(rho∗ V∗ c); % T_hat=T(j)+dTdt∗ DELTAt; % dTdt_hat=e∗ A_s∗ sigma∗ (T_w(j+1)ˆ4-T_hatˆ4)/(rho∗ V∗ c); % T(j+1)=T(j)+(dTdt+dTdt_hat)∗ DELTAt/2;

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(b)

348

Transient Conduction

% end %Solution determined by ode solver [time, T]=ode45(@(time,T) dTdt_f(time,T,e,A_s,rho,V,c,T_f,T_ini,beta,t_hold),time,T_ini);

The time span for the integration is provided by supplying the time array to the ode45 function. As a result, MATLAB will evaluate the temperatures at the same times as were used with Heun’s method. A plot of the ode solver results (identiﬁed with circles) is superimposed onto the results obtained using Heun’s method in Figure 1.

3.3 Semi-Inﬁnite 1-D Transient Problems 3.3.1 Introduction Sections 3.1 and 3.2 present analytical and numerical solutions to transient problems in which the spatial temperature gradients within the solid object can be neglected. Therefore, the problem is zero-dimensional; the transient solution is a function only of time. This section begins the discussion of transient problems where internal temperature gradients related to conduction are non-negligible (i.e., the Biot number is not much less than unity).

3.3.2 The Diffusive Time Constant The simplest case for which the temperature gradients are one-dimensional (i.e., temperature varies in only one spatial dimension) occurs when the object itself is semi-inﬁnite. A semi-inﬁnite body is shown in Figure 3-12; semi-inﬁnite means that the material is bounded on one edge (at x = 0) but extends to inﬁnity in the other. No object is truly semi-inﬁnite although many approach this limit; the earth, for example, is semi-inﬁnite relative to most surface phenomena. Furthermore, we will see that every object is essentially semi-inﬁnite with respect to surface processes that occur over a sufﬁciently “small” time scale. Figure 3-12 shows a semi-inﬁnite body that is initially at a uniform temperature, Tini , when at time t = 0 the temperature of the surface of the body (at x = 0) is raised to Ts . The temperature as a function of position is shown in Figure 3-12 for various times. The transient response can be characterized as a “thermal wave” that penetrates into the solid from the surface. The temperature of the solid is, at ﬁrst, affected by the semi-infinite body with initial temperature Tini Ts x Ts

Tini

T increasing time t1 t2 t3 δt,t1

δt,t 2

δt,t 3

x

Figure 3-12: Semi-inﬁnite body subjected to a sudden change in the surface temperature.

3.3 Semi-Inﬁnite 1-D Transient Problems

349

T Ts

Figure 3-13: Control volume used to develop a simple model of the semi-inﬁnite body.

control volume at t=t1 control volume at t=t3 dU dt

q⋅ cond t1 Tini

δt, t1

t2

t3 δt, t3

x

surface change only at positions that are very near the surface. As time increases, the thermal wave penetrates deeper into the solid; the depth of the penetration (δt ) grows and therefore the amount of material affected by the surface change increases. The analytical and numerical methods discussed in this section as well as in Sections 3.4 and 3.8 can be used to provide the solution to the problem posed in Figure 3-12. However, before the problem is solved exactly, it is worthwhile to pause and understand the behavior of the thermal wave that characterizes this and all transient conduction problems. There are two phenomena occurring in Figure 3-12: (1) thermal energy is conducted from the surface into the body, and (2) the energy is stored by the temperature rise associated with the material that lies within the ever-growing thermal wave. By developing simple models for these two processes, it is possible to understand, to a ﬁrst approximation, how the thermal wave behaves. A control volume is drawn from the surface to the outer edge of the thermal wave, as shown in Figure 3-13; the outer edge of the thermal wave is deﬁned as the position within the material where the conduction heat transfer is small. The control volume must grow with time as the extent of the thermal wave increases. An energy balance on the control volume shown in Figure 3-13 includes conduction into the surface (q˙ cond ) and energy storage: q˙ cond =

dU dt

(3-48)

The thermal resistance to conduction into the thermally affected region through the material that lies within the thermal wave (Rw ) is approximated as a plane wall with the thickness of the thermal wave. Rw ≈

δt k Ac

(3-49)

where k is the conductivity of the material and Ac is the cross-sectional area of the material. This approximation is, of course, not exact as the temperature distribution within the thermal wave is not linear (see Figure 3-13). However, this approach is approximately correct; certainly it is more difﬁcult to conduct heat into the solid as the thermal wave grows and Eq. (3-49) reﬂects this fact. The rate of conduction heat transfer is approximately: q˙ cond =

T s − T ini k Ac (T s − T ini ) ≈ Rw δt

(3-50)

350

Transient Conduction

The thermal energy stored in the material (U) relative to its initial state is the product of the average temperature elevation of the material within the thermal wave ( T ): T ≈

(T s − T ini ) 2

(3-51)

and the heat capacity of the material within the control volume: C ≈ ρ c δt Ac

(3-52)

where ρ and c are the density and speciﬁc heat capacity of the material, respectively. Note that Eqs. (3-51) and (3-52) are also only approximate. U≈

(T s − T ini ) ρ c δt Ac 2

(3-53)

C

T

Substituting Eqs. (3-50) and (3-53) into Eq. (3-48) leads to: k Ac (T s − T ini ) d (T s − T ini ) ≈ ρ c δt Ac δt dt 2

(3-54)

Only the penetration depth (δt ) varies with time in Eq. (3-54) and therefore Eq. (3-54) can be rearranged to provide an ordinary differential equation for δt : k ρ c dδt ≈ δt 2 dt

(3-55)

dδt 2k ≈ δt ρc dt

(3-56)

Equation (3-55) is rearranged:

Equation (3-56) can be simpliﬁed somewhat using the deﬁnition of the thermal diffusivity (α): α=

k ρc

(3-57)

Substituting Eq. (3-57) into Eq. (3-56) leads to: 2 α ≈ δt

dδt dt

(3-58)

Equation (3-58) is separated and integrated: t

δt 2 α dt ≈

0

δt dδt

(3-59)

0

in order to obtain an approximate expression for the penetration of the thermal wave as a function of time: 2αt ≈

δ2t 2

(3-60)

3.3 Semi-Inﬁnite 1-D Transient Problems

351

or: √ δt ≈ 2 α t

(3-61)

√ Equation (3-61) indicates that the thermal wave will grow in proportion to α t. This is a very important and practical (but not an exact) result that governs transient conduction problems. The constant in Eq. (3-61) may change depending on the precise nature of the problem and the deﬁnition of the thermal penetration depth. However, Eq. (3-61) will be approximately correct for a large variety of transient conduction problems. For example, if you heat one side of a plate of stainless steel (α = 1.5 × 10−5 m2 /s) that is L = 1.0 cm thick, then the temperature at the opposite side of the plate will begin to change in about τdiff = 1.7 s; this result follows directly from Eq. (3-61), rearranged to solve for time: τdiff ≈

L2 4α

(3-62)

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL A metal wall (Figure 1) is used to separate two tanks of liquid at different temperatures, Thot = 500 K and Tcold = 400 K. The thickness of the wall is th = 0.8 cm and its area is Ac = 1.0 m2 . The properties of the wall material are ρ = 8000 kg/m3 , c = 400 J/kg-K, and k = 20 W/m-K. The average heat transfer coefﬁcient between the wall and the liquid in either tank is hliq = 5000 W/m2 -K. th = 0.8 cm liquid at Tcold = 400 K hliq = 5000 W/m2 -K

liquid at Thot = 500 K hliq = 5000 W/m2 -K x k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K

Figure 1: Tank wall exposed to ﬂuid.

a) Initially, the wall is at steady-state. That is, the wall has been exposed to the ﬂuid in the tanks for a long time and therefore the temperature within the wall is not changing in time. What is the rate of heat transfer through the wall? What are the temperatures of the two surfaces of the wall (i.e., what is Tx=0 and Tx=th )?

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

The time required for the thermal wave to pass across the extent of a body is referred to as the diffusive time constant (τdiff ) and it is a broadly useful concept in the same way the lumped capacitance time constant (introduced in Section 3.1) is useful. The diffusive time constant characterizes, approximately, how long it takes for an object to equilibrate internally by conduction. The lumped capacitance time constant characterizes, approximately, how long it takes for an object to equilibrate externally with its environment. The ﬁrst step in understanding any transient heat transfer problem involves the calculation of these two time constants.

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

352

Transient Conduction

The known information is entered in EES: “EXAMPLE 3.3-1: Transient Response of a Tank Wall” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” k=20 [W/m-K] c=400 [J/kg-K] rho=8000 [kg/mˆ3] T cold=400 [K] T hot=500 [K] h bar liq=5000 [W/mˆ2-K] th=0.8 [cm]∗ convert(cm,m) A c=1 [mˆ2]

“thermal conductivity” “speciﬁc heat capacity” “density” “cold ﬂuid temperature” “hot ﬂuid temperature” “liquid-to-wall heat transfer coefﬁcient” “wall thickness” “wall area”

There are three thermal resistances governing this problem, convection from the surface on either side (Rconv,liq ) and conduction through the wall (Rcond ): Rconv,liq =

Rcond =

“Steady-state solution, part (a)” R conv liq=1/(h bar liq∗ A c) R cond=th/(k∗ A c)

1 hliq Ac th k Ac

“convection resistance with liquid” “conduction resistance”

The rate of heat transfer is: q˙ =

Thot − Tcold 2 Rconv,liq + Rcond

and the temperatures at x = 0 and x = th are: Tx=0 = Tcold + q˙ Rconv,liq Tx=t h = Thot − q˙ Rconv,liq q dot=(T hot-T cold)/(2∗ R conv liq+R cond) T 0=T cold+q dot∗ R conv liq T L=T hot-q dot∗ R conv liq

“heat transfer” “temperature of cold side of wall” “temperature of hot side of wall”

The rate of heat transfer through the wall is q˙ = 125 kW and the edges of the wall are at Tx=0 = 425 K and Tx=t h = 475 K. At time, t = 0, both tanks are drained and then both sides of the wall are exposed to gas at Tgas = 300 K (Figure 2). The average heat transfer coefﬁcient between the walls and the gas is hgas = 100 W/m2 -K. Assume that the process

353

of draining the tanks and ﬁlling them with gas occurs instantaneously so that the wall has the linear temperature distribution from part (a) at time t = 0. th = 0.8 cm gas at Tgas = 300 K hgas = 100 W/m2 -K

gas at Tgas = 300 K hgas = 100 W/m 2 -K

x

k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K Figure 2: Tank wall exposed to gas.

b) On the axes in Figure 3, sketch the temperature distribution in the wall (i.e., the temperature as a function of position) at t = 0 s (i.e., immediately after the process starts) and also at t = 0.5 s, 5 s, 50 s, 500 s, and 5000 s. Clearly label these different sketches. Be sure that you have the qualitative features of the temperature distribution drawn correctly. There are two processes that occur after the tank is drained. The tank material is not in equilibrium with itself due to its internal temperature gradient. Therefore, there is an internal equilibration process that will cause the wall material to come to a uniform temperature. Also, there is an external equilibration process as the wall transfers heat with its environment. The internal equilibration process is governed by a diffusive time constant, τdiff , discussed in Section 3.3.2 and provided, approximately, by Eq. (3-62): τdiff =

th2 4α

where α is the thermal diffusivity of the wall material: α=

“Internal equilibration process” alpha=k/(rho∗ c) tau diff=thˆ2/(4∗ alpha)

k ρc

“thermal diffusivity” “diffusive time constant”

The diffusive time constant is about 2.6 seconds; this is, approximately, how long it will take for a thermal wave to pass from one side of the wall to the other and therefore this is the amount of time that is required for the wall to internally equilibrate. If the edges of the wall were adiabatic, then the wall will be at a nearly uniform temperature after a few seconds. The external equilibration process is governed by a lumped time constant, discussed previously in Section 3.1 and deﬁned for this problem as: τlumped = Rconv,gas C

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

3.3 Semi-Inﬁnite 1-D Transient Problems

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

354

Transient Conduction

where Rconv,gas is the thermal resistance to convection between the wall and the gas: Rconv,gas =

1 2 hgas Ac

and C is the thermal capacitance of the wall: C = Ac th ρ c “External equilibration process” h bar gas=100 [W/mˆ2-K] R conv gas=1/(2∗ h bar gas∗ A c) C total=A c∗ th∗ rho∗ c tau lumped=R conv gas∗ C total

“gas-to-wall heat transfer coefﬁcient” “convection resistance with gas” “capacity of wall” “lumped time constant”

The lumped time constant is about 130 s. Because the diffusive time constant is two orders of magnitude smaller than the lumped time constant, the wall will initially internally equilibrate rapidly and subsequently externally equilibrate more slowly. The initial internal equilibration process will be completed after about 5 to 10 s. Subsequently, the wall will equilibrate externally with the surrounding gas; this external equilibration process will be completed after 200 to 400 s. The temperature distributions sketched in Figure 3 are consistent with these time constants; they do not represent an exact solution, but they are consistent with the physical intuition that was gained through knowledge of the two time constants. Temperature (K) 500 t=0s t = 0.5s t = 5s t = 50 s

475 450 425 400

t = 500s t = 5000s

300 0

0.8

Position (cm)

Figure 3: Temperature distributions in the wall as it equilibrates.

A more exact solution to this problem can be obtained using the techniques discussed in subsequent sections (see EXAMPLE 3.5-2). However, calculation of the diffusive and lumped time constants provide important physical intuition about the problem.

3.3.3 The Self-Similar Solution The governing differential equation for the semi-inﬁnite solid is derived using a control volume that is differential in x, as shown in Figure 3-14.

3.3 Semi-Inﬁnite 1-D Transient Problems

355

semi-infinite body with initial temperature Tini Ts x

q⋅ x

∂U ∂t

q⋅ x+dx

dx Figure 3-14: Differential control volume within semi-inﬁnite body.

The energy balance suggested by Figure 3-14 is: q˙ x = q˙ x+dx +

∂U ∂t

(3-63)

Expanding the x + dx term in Eq. (3-63) leads to: q˙ x = q˙ x +

∂ q˙ x ∂U dx + ∂x ∂t

(3-64)

The conduction term is evaluated using Fourier’s law: q˙ x = −k Ac

∂T ∂x

(3-65)

where Ac is the area of the wall. The internal energy contained in the differential control volume is: U = ρ c Ac dx T

(3-66)

where ρ and c are the density and speciﬁc heat capacity of the wall material. Assuming that the speciﬁc heat capacity is constant, the time rate of change of the internal energy is: ∂T ∂U = ρ c Ac dx ∂t ∂t Substituting Eqs. (3-65) and (3-67) into Eq. (3-64) leads to: ∂T ∂T ∂ −k Ac dx + ρ c Ac dx 0= ∂x ∂x ∂t

(3-67)

(3-68)

which, for a constant thermal conductivity, can be simpliﬁed to: α

∂2T ∂T = ∂x2 ∂t

(3-69)

where α is the thermal diffusivity. For the situation shown in Figure 3-14 in which the initial temperature of the solid is uniform (Tini ) and the surface temperature is suddenly elevated (to Ts ), the boundary conditions are: T x=0,t = T s

(3-70)

T x,t=0 = T ini

(3-71)

T x→∞,t = T ini

(3-72)

The solution to the partial differential equation, Eq. (3-69), subject to the boundary conditions, Eqs. (3-70) through (3-72), is not obvious. Fortunately, the partial differential

356

Ts

Transient Conduction

T T

Ts increasing time

t1, t2, and t3 Tini

t1 δt, t1

t2 δt, t2

t3 δt, t3

Tini η=

x

(b)

(a)

x x = δt 2 α t

Figure 3-15: Temperature distribution in the semi-inﬁnite body as (a) a function of position for various times, and (b) a function of position normalized by the thermal penetration depth for various times.

equation can be reduced to an ordinary differential by using the concept of the thermal penetration depth introduced in Section 3.3.2. Figure 3-12 shows the temperature distribution at different times; these distributions are repeated in Figure 3-15(a). Figure 3-15(b) shows that when the temperature distribution is plotted against x/δt where √ δt is 2 α t, according to Eq. (3-61), the temperature proﬁles for each time collapse onto a single curve. The temperature in Figure 3-15(a) is a function of two independent variables x and t. In this problem, the two independent variables can be combined in order to express the temperature as a function of a single independent variable, deﬁned as η: η=

x √ 2 αt

(3-73)

This process is often called combination of variables and the resulting solution is referred to as a self-similar solution. The ﬁrst step is to transform the governing partial differential equation in x and t to an ordinary differential equation in η. The similarity parameter η, given by Eq. (3-73), is substituted into Eq. (3-69). The temperature is expressed functionally as: T (η (x, t))

(3-74)

Therefore, according to the chain rule, the partial derivative of temperature with respect to x is: dT ∂η ∂T = ∂x dη ∂x

(3-75)

The partial derivative of η with respect to x is obtained by inspection of Eq. (3-73) or using Maple: > eta:=x/(2∗ sqrt(alpha∗ t)); x η := √ 2 αt > diff(eta,x); 1 √ 2 αt

3.3 Semi-Inﬁnite 1-D Transient Problems

357

So that: 1 ∂η = √ ∂x 2 αt

(3-76)

∂T dT 1 = √ ∂x dη 2 α t

(3-77)

and therefore Eq. (3-75) becomes:

The same process is used to calculate the second derivative of T with respect to x: d ∂2T = 2 ∂x dη

∂T ∂x

∂η ∂x

(3-78)

Substituting Eq. (3-77) into Eq. (3-78) leads to: ∂2T d = ∂x2 dη

dT 1 √ dη 2 α t

∂η ∂x

(3-79)

Substituting Eq. (3-76) into Eq. (3-79) leads to: ∂2T d2 T 1 = 2 ∂x dη2 4 α t

(3-80)

The partial derivative of the temperature, Eq. (3-74), with respect to time is also obtained using the chain rule: dT ∂η ∂T = ∂t dη ∂t

(3-81)

where the partial derivative of η with respect to t is evaluated using Maple: > diff(eta,t); 1 xα 4 (α t)(3/2)

So that: x ∂η =− √ ∂t 4t αt

(3-82)

∂T x dT =− √ ∂t 4 t α t dη

(3-83)

and therefore Eq. (3-81) becomes:

Substituting Eqs. (3-80) and (3-83) into Eq. (3-69) leads to: α

d2 T 1 x dT =− √ 2 dη 4 α t 4 t α t dη

(3-84)

358

Transient Conduction

which can be rearranged: x dT d2 T = −2 √ dη2 2 α t dη

(3-85)

d2 T dT = −2 η dη2 dη

(3-86)

η

or

which completes the transformation of the partial differential equation, Eq. (3-69), into an ordinary differential equation, Eq. (3-86). The boundary conditions must also be transformed. Equations (3-70) through (3-72) are transformed from expressions in x and t to expressions in η: T x=0,t = T s ⇒ T η=0 = T s

(3-87)

T x,t=0 = T ini ⇒ T η→∞ = T ini

(3-88)

T x→∞,t = T ini ⇒ T η→∞ = T ini

(3-89)

Notice that Eqs. (3-88) and (3-89) are identical and so the partial differential equation has been transformed into a second order ordinary differential equation with two boundary conditions. It is not always possible to accomplish this transformation; if either x or t had been retained in the partial differential equation or any of the boundary conditions, then a self-similar solution would not be possible. In this case, an alternative analytical technique such as the Laplace transform (Section 3.4) or a numerical solution technique (Section 3.8) is required. Equation (3-86) can be separated and solved. The variable w is deﬁned as: w=

dT dη

(3-90)

and substituted into Eq. (3-86): dw = −2 η w dη Equation (3-91) can be rearranged and integrated: dw = −2 η dη w

(3-91)

(3-92)

which leads to: ln (w) = −η2 + C1

(3-93)

where C1 is a constant of integration that is necessary because Eq. (3-92) is an indeﬁnite integral. Solving Eq. (3-93) for w leads to: w = exp(−η2 + C1 )

(3-94)

Substituting Eq. (3-90) into Eq. (3-94) leads to: dT = exp(−η2 + C1 ) dη

(3-95)

3.3 Semi-Inﬁnite 1-D Transient Problems

359

Gaussian error function and complementary error function

1 0.9 0.8

erf (η )

0.7 0.6 0.5 0.4 0.3

erfc (η )

0.2 0.1 0 0

0.5

1

1.5

2

2.5

Figure 3-16: Gaussian error function and the complementary Gaussian error function.

or dT = C2 exp(−η2 ) dη

(3-96)

where C2 is another undetermined constant (equal to the exponential of C1 ). Equation (3-96) can be integrated:

dT =

C2 exp(−η2 ) dη

(3-97)

exp(−η2 ) dη + C3

(3-98)

or η T = C2 0

where C3 is an undetermined constant. The integral in Eq. (3-98) cannot be evaluated analytically and yet it shows up often in engineering problems. The integral is deﬁned in terms of the Gaussian error function (typically called the erf function, which is pronounced so that it rhymes with smurf). The Gaussian error function is deﬁned as: 2 erf(η) = √ π

η exp(−η2 ) dη

(3-99)

0

and illustrated in Figure 3-16. The complementary error function (erfc) is deﬁned as: erfc (η) = 1 − erf (η)

(3-100)

and is also illustrated in Figure 3-16. Equation (3-98) can be written in terms of the erf function: √ π (3-101) erf (η) + C3 T = C2 2

360

Transient Conduction

The solution to the ordinary differential equation, Eq. (3-86), can also be obtained using Maple: > GDE:=diff(diff(T(eta),eta),eta)=-2∗ eta∗ diff(T(eta),eta); GDE :=

d2 d T (η) T (η) = −2 η 2 dη dη

> Ts:=dsolve(GDE); Ts := T (η) = C1 + erf (η) C2

The two constants in Eq. (3-101) are obtained using the boundary conditions. Substituting Eq. (3-101) into (3-87) leads to: √ π (3-102) erf (0) +C3 = T s T η=0 = C2 2 0

Figure 3-16 shows that erf(0) = 0 so Eq. (3-102) becomes: C3 = T s Substituting Eq. (3-103) into Eq. (3-101) leads to: √ π erf (η) + T s T = C2 2 Substituting Eq. (3-104) into Eq. (3-88) leads to: √ π erf (∞) +T s = T ini T η→∞ = C2 2

(3-103)

(3-104)

(3-105)

=1

Figure 3-16 shows that erf(∞) = 1 so Eq. (3-105) becomes: 2 C2 = √ (T ini − T s ) π The temperature distribution within the semi-inﬁnite body is therefore: x T = T s + (T ini − T s ) erf √ 2 αt

(3-106)

(3-107)

The heat transfer into the surface of the body, q˙ x=0 , is evaluated using Fourier’s law: ∂T q˙ x=0 = −k Ac (3-108) ∂x x=0 which can be expressed in terms of η by substituting Eq. (3-77) into Eq. (3-108): q˙ x=0 = −k Ac

dT 1 √ dη 2 α t

(3-109)

Substituting Eq. (3-107) into Eq. (3-109) leads to: q˙ x=0

d 1 = −k Ac √ [T s + (T ini − T s ) erf (η)] dη η=0 2 α t

(3-110)

3.3 Semi-Inﬁnite 1-D Transient Problems

or: q˙ x=0

361

d k Ac = − √ (T ini − T s ) [erf (η)] dη 2 αt η=0

(3-111)

Substituting the deﬁnition of the erf function, Eq. (3-99), into Eq. (3-111) leads to: ⎤ ⎡ η d ⎣ 2 k Ac exp(−η2 ) dη⎦ (3-112) q˙ x=0 = − √ (T ini − T s ) √ dη π 2 αt 0

η=0

The derivative of an integral is the integrand, and therefore Eq. (3-112) becomes: 2 k Ac q˙ x=0 = − √ (T ini − T s ) √ [exp(−η2 )]η=0 π 2 αt

(3-113)

k Ac q˙ x=0 = √ (T s − T ini ) παt

(3-114)

or

1/Rsemi−∞

This result is extremely useful and intuitive; the material within the thermal wave in Figure 3-15 acts like a thermal resistance to heat transfer from the surface (Rsemi−∞ ): q˙ x=0 =

(T s − T ini ) Rsemi−∞

(3-115)

where Rsemi−∞ increases with time as the thermal wave grows (and therefore the distance over which the conduction occurs increases) according to: Rsemi−∞ =

√ παt k Ac

(3-116)

Equation (3-116) is similar to the thermal √ resistance to conduction through a plane wall with thickness that grows according to π α t. This concept is useful for understanding the physics associated with transient conduction problems. Of course, as soon as the thermal wave reaches a boundary, the problem is no longer semi-inﬁnite and therefore the concepts of the semi-inﬁnite resistance and thermal penetration wave are no longer valid.

3.3.4 Solution to other Semi-Inﬁnite Problems Analytical solutions to a semi-inﬁnite body exposed to other surface boundary conditions have been developed and are summarized in Table 3-2. These analytical solutions can often be applied to various processes that occur over very short time-scales where the thermal penetration wave (δt ) is small relative to the spatial extent of the object. Functions that return the temperature at a speciﬁed position and time for the semiinﬁnite body problems in Table 3-2 are available in EES. Select Function Info from the Options menu and select Transient Conduction from the pull-down menu at the lower right corner of the upper box, toggle to the Semi-Inﬁnite Body library and scroll across to ﬁnd the function of interest.

362

Transient Conduction

Table 3-2: Solutions to semi-inﬁnite body problems. Boundary Condition

Solution T − T ini x = 1 − erf √ T s − T ini 2 αt k (T s − T ini ) q ′′x =0 = √ παt

Ts T increasing time

Tini

x step change in surface temp.: Tx=0 = Ts

T

q⋅′′s

T − T ini

q˙ = s k

/

x2 4αt x exp − − x erfc √ π 4αt 4αt

increasing time Tini

x surface heat flux: q⋅s′′ = − k ∂T ∂x x=0

T∞ increasing time

h

x

convection to fluid: h (T∞ −Tx = 0 ) = − k ∂T ∂x x =0 T surface energy per unit area released at t = 0 wall adiabatic for t>0

E ′′

T − T ini x = erfc √ T ∞ − T ini 2 αt 2 hx h αt h√ x erfc √ + + − exp αt k k2 k 2 αt

T − T ini

x2 E exp − = √ 4αt ρc παt

increasing time Tini

x

surface energy pulse: lim q⋅′′s Δt = E ′′ t, Δt→0

T Tx =0 = Tini + ΔT sin (ωt)

ΔT

/ / ω ω sin ω t − x T − T ini = T exp −x 2α 2α

Tini

x periodic surface temperature xB

T Tini, B solid A

increasing time

Tint increasing time

solid B

Tini, A

0 T ini,A − T int kA ρA cA = 0 T int − T ini,B kB ρB cB T B − T ini,B T A − T ini,A xA xB , = 1 − erf = 1 − erf √ √ T int − T ini,A T int − T ini,B 2 αt 2 αt

xA

contact between two semi-infinite solids

Tini = initial temperature x = position from surface α = thermal diffusivity

t = time relative to surface disturbance ρ = density

k = conductivity c = speciﬁc heat capacity

363

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE A laminated structure is fabricated by diffusion bonding alternating layers of high conductivity silicon (k s = 150 W/m-K, ρs = 2300 kg/m3 , cs = 700 J/kg-K) and low conductivity pyrex (k p = 1.4 W/m-K, ρ p = 2200 kg/m3 , c p = 800 J/kg-K). The thickness of each layer is ths = thp = 0.5 mm, as shown in Figure 1. silicon layers ks = 150 W/m-K

pyrex layers kp = 1.4 W/m-K

ρs = 2300 kg/m3 cs = 700 J/kg-K

ρp = 2200 kg/m3 cp = 800 J/kg-K

thp = 0.5 mm

ths = 0.5 mm x-direction Figure 1: A composite structure formed from silicon and glass.

a) Determine an effective conductivity that can be used to characterize the composite structure with respect to heat transfer in the x-direction (see Figure 1). The known information is entered in EES: “EXAMPLE 3.3-2: Quenching a Composite Structure” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 0.8 3.5 in “Inputs” k s=150 [W/m-K] rho s=2300 [kg/mˆ3] c s=700 [J/kg-K] k p=1.4 [W/m-K] rho p=2200 [kg/mˆ3] c p=800 [J/kg-K] th s=0.5 [mm]∗ convert(mm,m) th p=0.5 [mm]∗ convert(mm,m)

“conductivity of silicon” “density of silicon” “speciﬁc heat capacity of silicon” “conductivity of pyrex” “density of pyrex” “speciﬁc heat capacity of pyrex” “thickness of silicon lamination” “thickness of pyrex lamination”

The method discussed in Section 2.9 is used to determine the effective thermal conductivity of the composite structure. The heat transfer through a thickness (L, in the x-direction) with cross-sectional area (Ac ) of the composite structure when it is subjected to a given temperature difference (T ) is calculated as the series combination of the thermal resistance of the silicon and pyrex laminations according to: q˙ =

T L th p L ths + ths + th p k s Ac ths + th p k p Ac

(1)

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.3 Semi-Inﬁnite 1-D Transient Problems

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

364

Transient Conduction

The effective conductivity in the x-direction (keff ) is equal to the conductivity of a homogeneous material that would result in the same heat transfer rate: k eff Ac T L Setting Eq. (1) equal to Eq. (2) leads to: q˙ =

(2)

k eff Ac T T = L th p L ths L + ths + th p k s Ac ths + th p k p Ac Solving for keff leads to:

k eff =

th p ths + ths + th p k s ths + th p k p

−1

“Part a: effective conductivity for heat transfer across laminations” k eff=1/(th s/((th s+th p)∗ k s)+th p/((th s+th p)∗ k p)) “effective conductivity”

The effective conductivity is k eff = 2.8 W/m-K, which is between the conductivity of pyrex and silicon. Because the conductivity of silicon is high, the silicon laminations contribute essentially no thermal resistance; therefore, because the laminations are of equal size, the effective conductivity is twice that of pyrex. b) Determine an effective heat capacity (ceff ) and density (ρ eff ) that can be used characterize the composite structure. The process of determining an effective density and speciﬁc heat capacity is conceptually similar to the calculation of an effective thermal conductivity. The effective property is chosen so that the composite, when modeled as a homogeneous material using the effective property, behaves as the actual composite does. The effective density is deﬁned so that material has the same mass as the composite. The mass of the composite (with thickness L and cross-sectional area Ac ) is: th p ths L Ac ρs + L Ac ρ p th p + ths th p + ths

M =

(3)

The mass of the homogeneous material with an effective density (ρeff ) is: M = L Ac ρeff

(4)

Setting Eq. (3) equal to Eq. (4) and solving for ρeff leads to: th p ths ρs + ρp th p + t hs th p + ths

ρeff =

“Part b: effective density and heat capacity” rho eff=th s∗ rho s/(th p+th s)+th p∗ rho p/(th p+th s)

“effective density”

The effective speciﬁc heat capacity is deﬁned so that the homogeneous material model has the same total heat capacity as the composite. The total heat capacity of the composite (again with thickness L and area Ac ) is: th p ths L Ac ρs cs + L Ac ρ p c p th p + ths th p + ths

C=

(5)

365

The total heat capacity of the homogeneous material with effective density (ρeff ) and effective heat capacity (ceff ) is: C = L Ac ρeff ceff

(6)

Setting Eq. (5) equal to Eq. (6) and solving for ceff leads to: th p ρ p ths ρs cs + cp th p + ths ρeff th p + ths ρeff

ceff =

c_eff=th_s∗ rho_s∗ c_s/((th_p+th_s)∗ rho_eff)_th_p∗ rho_p∗ c_p/((th_p+th_s)∗ rho_eff) “effective speciﬁc heat capacity”

The effective density and speciﬁc heat capacity of the composite structure are ρeff = 2250 kg/m3 and ceff = 749 J/kg-K. The diffusion bonding of the composite structure occurs at high temperature, Tbond = 750◦ C. When the bonding process is complete, the manufacturing process is terminated by quenching the composite structure from both sides with water at Tw = 20◦ C, as shown in Figure 2. Tw = 20°C, h w → ∞ L = 10 cm

Figure 2: Quenching a composite structure with water.

Tw = 20°C, hw → ∞ composite structure (Figure 1) initial temperature, Tbond = 750°C

The heat transfer coefﬁcient between the water and the surface of the structure is very high because the water is vaporizing. Therefore the quenching process corresponds approximately to applying a step change in the surface temperature of the composite. c) The laminated structure is L = 10 cm thick, for approximately how long will it be appropriate to model the composite as a semi-inﬁnite body? The thermal wave moves from the top and bottom surfaces of the structure approximately according to: $ (7) δt = 2 αeff t where the effective thermal diffusivity of the composite structure is: αeff =

“Part c” alpha eff=k eff/(rho eff∗ c eff)

k eff ρeff ceff

“effective diffusivity”

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.3 Semi-Inﬁnite 1-D Transient Problems

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

366

Transient Conduction

When the thermal wave reaches the half-thickness of the structure then it will no longer behave as a semi-inﬁnite body but rather as a bounded, 1-D transient problem. Substituting δt = L/2 into Eq. (7) leads to: $ L = 2 αeff tsemi−∞ 2 where tsemi−∞ is the time at which the semi-inﬁnite model is no longer appropriate. L=10[cm]∗ convert(cm,m) 2∗ sqrt(alpha eff∗ t semi inﬁnite)=L/2

“width of structure” “time that semi-inﬁnite solution is valid”

The semi-inﬁnite body solution is valid for approximately 380 s. It has been observed that the laminations that are within xfail = 1.0 cm of the two surfaces tend to de-bond during the quenching process. You suspect that the large spatial temperature gradients near the surface during the quench are causing thermally induced stresses that are responsible for this failure. In the presence of large temperature gradients, two laminations that are adjacent to one another will experience very different thermally induced expansions that result in a shear stress. d) Prepare a plot of the temperature gradient as a function of time (up to the time at which the semi-inﬁnite solution is no longer valid, from part (c)) at x = xfail as well as other positions that are greater than and less than xfail . Determine the critical spatial temperature gradient that causes failure (i.e., the maximum spatial temperature gradient experienced at x = xfail ). Table 3-2 or Eq. (3-107) provides the temperature within the semi-inﬁnite body: x 0 (8) T = Tw + (Tbond − Tw ) erf 2 αeff t The spatial temperature gradient can be obtained by differentiating Eq. (8) using Maple: > restart; > T(x,time):=T_w+(T_bond-T_w)∗ erf(x/(2∗ sqrt(alpha_eff∗ time))); x 0 T (x, time) := T w + T bond − T w erf 2 alpha eff time > dTdx:=diff(T(x,time),x);

⎛ ⎝−

dT d x :=

⎞

x2 ⎠ 4alpha eff time

T bond − T w e √ 0 π alpha eff time

so the temperature gradient is: ∂T x2 (Tbond − Tw ) = 0 exp − ∂x 4 αeff t π αeff t

(9)

367

The result from Maple is copied and pasted into EES and then modiﬁed slightly for compatibility:

“Part d - analytical differentiation” T bond=converttemp(C,K,750 [C]) “bond temperature” T w=converttemp(C,K,20 [C]) “water temperature” “observed position of failure” x fail=1 [cm]∗ convert(cm,m) x=x fail “vary x from 0.5 to 4 ∗ x fail” dTdx=(T bond-T w)/Piˆ(1/2)∗ exp(-1/4∗ xˆ2/alpha eff/time)/(alpha eff∗ time)ˆ(1/2)

A parametric table is used to generate Figure 3. The independent variable time is varied from 1 × 10−6 s to 380 s. A value of 1 × 10−6 s rather than 0 is used to avoid division by zero in Eq. (9). The value of x is changed in the Equation window in order to produce the different curves that are shown in Figure 3.

Spatial temperature gradient (K/m)

4 x10 4

3 x10 4 x=0.5 cm 2 x10 4

10

x = x fail = 1 cm x = 2 cm x = 3 cm x = 4 cm

4

0 x10 0 0

100

200 300 Time (s)

400

500

Figure 3: Spatial temperature gradient as a function of time for various locations in the composite structure.

Examination of Figure 3 suggests that the critical spatial temperature gradient for failure is about 3.5 × 104 K/m. Locations exposed to a spatial gradient greater than this value are likely to fail. A more exact solution can be obtained by determining the time at which the spatial gradient is maximized when x = xf ail . This result can be obtained by selecting Min/Max from the Calculate menu and maximizing dTdx by varying the independent variable time. Provide limits on time from slightly greater than 0 (to prevent a division by 0 error) to 380 s. The result will be dTdx = 3.53 × 104 K/m which occurs at 30.4 s. In order to reduce the temperature gradients within the laminations, you suggest that the current water quenching process be replaced by a gas cooling process in which the surfaces of the composite are exposed to a gas at Tgas = 20◦ C with a lower heat transfer coefﬁcient, hgas = 400 W/m2 -K.

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.3 Semi-Inﬁnite 1-D Transient Problems

Transient Conduction

e) Determine whether there will be any de-lamination using this process and, if so, what the thickness of the damaged region (xfail ) will be. Rather than use the analytical solution for the temperature distribution within a semi-inﬁnite body that is subjected to surface convection, from Table 3-2, the gradient will be evaluated numerically using the built-in function SemiInf3 in EES. The numerical derivative is obtained by adding and subtracting a very small value, x, to/from the nominal value of x according to: Tx+x − Tx−x ∂T ≈ ∂x 2 x The value of x must be small in order for this approach to work, but not so small that numerical precision is exceeded. EES provides about 20 digits of numerical precision, so accurate determination of numerical derivatives is usually not a problem. The EES code to compute the temperature gradient for a given value of position and time is: “Part e - gas quenching process” “bond temperature” T bond=converttemp(C,K,750 [C]) T gas=converttemp(C,K,20 [C]) “temperature of gas” h bar gas=400 [W/mˆ2-K] “heat transfer coefﬁcient” DELTAx=1e-5 [m] “differential change used to evaluate numerical derivative” “position” x=2.0 [cm]∗ convert(cm,m) T xplusdx=SemiInf3(T bond,T gas,h bar gas,k eff,alpha eff,x+DELTAx,time) T xminusdx=SemiInf3(T bond,T gas,h bar gas,k eff,alpha eff,x-DELTAx,time) dTdx=(T xplusdx-T xminusdx)/(2∗ DELTAx) “numerical temperature gradient”

In order for this code to run, it is necessary to comment out the EES code from (d). Figure 4 illustrates the spatial temperature gradient as a function of time for various values of position. 5 x10 4 Spatial temperature gradient (K/m)

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

368

4 x10 4

x=0.25 cm

3 x10 4

x=0.5 cm x=0.75 cm x=1.0 cm

2 x10 4

x=1.5 cm 10

x=2.0 cm

4

0 x10 0 0

25

50

75 Time (s)

100

125

150

Figure 4: Spatial temperature gradient as a function of time for various values of position using the gas quenching process.

369

Maximum spatial temperature gradient (K/m)

Figure 4 shows that there is a maximum temperature gradient experienced at each value of x that occurs as the thermal wave passes by that position. The maximum temperature gradient experienced as a function of position can be obtained by using the Min/Max Table option from the Calculate menu. Comment out the speciﬁed value of position and generate a parametric table that includes the variable x as well as the dependent variable to be maximized (the variable dTdx) and the independent variable to vary (the variable time). Vary x from 0.1 mm to 1.45 cm in the parametric table. Select Min/Max Table from the Calculate menu. Maximize the value of the temperature gradient by varying the independent variable time. The maximum temperature gradient as a function of position is shown in Figure 5. 60,000 50,000 40,000 critical temperature gradient 30,000

2

h gas [W/m -K] 20,000

400

10,000

200 100

0 0

0.002 0.004 0.006 0.008 0.01 Axial position (m)

0.012 0.014

Figure 5: Maximum temperature gradient as a function of position for various values of the heat transfer coefﬁcient.

Figure 5 suggests that for hgas = 400 W/m2 -K, the extent of the damaged region will be reduced to only 0.42 cm. Figure 5 also shows that if hgas is reduced to 100 W/m2 -K then the spatial temperature gradient will not exceed 25,000 K/m anywhere in the composite. Because 25,000 K/m is less than the critical temperature gradient identiﬁed in part (d) (35,300 K/m), it is likely that no damage would occur for hgas = 100 W/m2 -K.

3.4 The Laplace Transform 3.4.1 Introduction Sections 3.1 and 3.3 present techniques for obtaining analytical solutions to 0-D (lumped) and 1-D transient conduction problems. This section presents an alternative method for obtaining an analytical solution to these problems: the Laplace transform. The Laplace transform is a mathematical technique that is used to solve differential equations that arise in many engineering disciplines. The mathematical speciﬁcation of a transient conduction problem will include a differential equation and boundary conditions that involve time. In the case of the lumped capacitance problems discussed in Section 3.1, an ordinary differential equation in time was obtained. The semi-inﬁnite problems in Section 3.3 result in a partial differential equation in time and position. The Laplace transform maps a problem involving time,

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.4 The Laplace Transform

370

Transient Conduction

t, onto a problem involving a different variable, typically called s. The attractive feature of the Laplace transform is that the mapping process removes time derivatives from the problem. Therefore, if an ordinary differential equation in t is Laplace-transformed, it becomes an algebraic equation in s. A partial differential equation in t and x becomes an ordinary differential equation in x that is algebraic in s. The complexity of the differential equation is reduced by one level and the problem is correspondingly easier to solve in the s domain than in the t domain. The Laplace transform solution proceeds by obtaining the differential equation and boundary conditions as usual and transforming them into the Laplace (s) domain. The solution is obtained in the s domain and then transformed back to the time domain. The process of transforming between the s and t domains is facilitated by extensive tables that exist as well as symbolic software packages such as Maple. This section is by no means a comprehensive review of Laplace transform theory and application; the interested reader is referred to more complete coverage which can be found in Arpaci (1966) and Myers (1998). However, the introduction provided here is sufﬁcient to allow the solution of many interesting heat transfer problems and provide some insight into the technique. The Laplace transform is particularly useful for short time-scale and semi-infinite body problems.

3.4.2 The Laplace Transformation

The Laplace transform of a function of time, T(t), is indicated by T (s) and obtained according to: ∞

T (s) = T (t) =

exp (−s t) T (t) dt 0

(3-117)

the Laplace transform operation

where the notation T (t) denotes the Laplace transform operation. As an example, consider the Laplace transform of a constant, C: T (t) = C

(3-118)

Substituting Eq. (3-118) into Eq. (3-117) leads to: ∞

T (s) = C =

exp (−s t) C dt

(3-119)

0

Carrying out the integral:

T (s) = −

C [exp (−s t)]∞ 0 s

(3-120)

and evaluating the limits:

T (s) =

C s

(3-121)

Thus the Laplace transform of C is C/s. There are several techniques that can be used to transform a function from the time to the s domain. The integration expressed by Eq. (3-117) can be carried out explicitly. More commonly, one of the extensive tables of Laplace transforms that have been published (e.g., Abramowitz and Stegun, (1964)) is used to obtain the transform. Recently, symbolic software packages such as Maple have become available that are capable of identifying most Laplace transforms automatically.

3.4 The Laplace Transform

371

Table 3-3: Some common Laplace transforms. Function in t

Function in s

Function in t C erfc √ 2 t √ C 2 t C2 − C erfc √ exp − √ 4t π 2 t C C2 t+ erfc √ 2 2 t √ C2 C t − √ exp − 4t π

C

C s

Ct

C s2

exp (C t)

1 s−C

sin (C t)

C s2 + C2

C21 C1 exp − − C2 t √ 4t 2 π t3

cos (C t)

s s2 + C2

exp (−C1 t) − exp (−C2 t) C2 − C1

sinh (C t)

C s2 − C2

t exp (−C1 t)

s s2 − C2

√ C C2 exp − exp −C s √ 3 4 t 2 πt √ exp −C s C2 1 √ exp − √ 4t s πt

√ exp −C s s2

exp(−C1

0 s + C2 )

1 (s + C1 ) (s + C2 ) 1

⎤ (C3 − C2 ) exp (−C1 t) ⎥ ⎢ ⎣ + (C1 − C3 ) exp (−C2 t) ⎦ + (C2 − C1 ) exp (−C3 t) (C1 − C2 ) (C2 − C3 ) (C3 − C1 ) ⎡

cosh (C t)

Function in s √ exp −C s s √ exp −C s √ s s

(s + C1 )2

1 (s + C1 ) (s + C2 ) (s + C3 )

exp (−C2 t) − exp (−C1 t) [1 − (C2 − C1 ) t]

1

(C2 − C1 )2

(s + C1 )2 (s + C2 )

t2 exp (−C t) 2

Function in t √ C2 1 C − a exp(a C) exp(a2 t) erfc a t + √ √ exp − 4t πt 2 t √ C C erfc √ − exp (a C) exp(a2 t) erfc a t + √ 2 t 2 t √ C exp(a C) exp(a2 t) erfc a t + √ 2 t

1 (s + C)3 Function in s √ exp(−C s) √ a+ s √ a exp(−C s) √ s(a + s) √ exp(−C s) √ √ s(a + s)

Laplace Transformations with Tables Tables that provide Laplace transforms can be found in many mathematical references, a few common Laplace transforms are summarized in Table 3-3. Laplace Transformations with Maple The Laplace and inverse Laplace transforms can be obtained using Maple. To access the integral transform library in Maple, it is necessary to activate the inttrans package; this is accomplished using the with command: > restart: > with(inttrans):

372

Transient Conduction

The Laplace transform of an arbitrary function can be obtained using the laplace command. The laplace command requires three arguments; the ﬁrst is the expression to be transformed, the second is the variable to transform from (typically t), and the third is the variable to transform to (typically s). To obtain the Laplace transform of a constant, as we did in Eqs. (3-118) through (3-121) > laplace(C,t,s); C s

More complex functions can also be transformed: > laplace(sin(C∗ t),t,s); C s2 + C2

> laplace(sin(C∗ t)∗ exp(-t/tau),t,s);

C 2

1 s+ τ

+ C2

which indicates that: sin (C t) = and

C s2 + C 2

6 7 t sin (C t) exp − = τ

C 1 2 + C2 s+ τ

(3-122)

(3-123)

3.4.3 The Inverse Laplace Transform Once the solution to a problem has been obtained in terms of s, it is necessary to obtain the inverse Laplace transform of the solution in order to express the result in terms of t. There are several techniques for obtaining the inverse Laplace transform. The most general technique is to mathematically invert the Laplace transform, Eq. (3-117), but this operation requires integration in the complex plane. The computational effort required to obtain the inverse transform in this way defeats the purpose of using Laplace transforms to simplify the solution of heat transfer problems. However, many inverse transforms have been determined and tabulated. If the solution in the s domain appears in a table, such as Table 3-3, then it is a simple matter to obtain the inverse transform from the table (e.g., the inverse transform of C/s is C). More often, the particular transform that is needed will not be found in exactly the right form in a table and it will be necessary to break the solution into simpler pieces using the method of partial fractions. The typical form of the solution in the s domain is a complicated fraction in s; for example:

T (s) =

s2 − 3 s + 4 (s + 1) (s − 1) (s + 2)

(3-124)

3.4 The Laplace Transform

373

The inverse Laplace transform for Eq. (3-124) cannot be found in Table 3-3; however, the transform for simpler fractions are included in the table. The method of partial fractions is therefore required to effectively use tables of Laplace transforms. Many common inverse Laplace transforms can be obtained automatically using Maple. Several additional Laplace transforms that are not normally available with Maple have been added to a ﬁle that can be downloaded from the text website (www.cambridge.org/nellisandklein), as discussed in Section 3.4.6. Inverse Laplace Transform with Tables and the Method of Partial Fractions Equation (3-124) can be reduced using the method of partial fractions, as discussed in Myers (1998). The method of partial fractions requires that the order of the numerator is less than that of the denominator. Distinct Factors. For cases like Eq. (3-124) where there are distinct factors in the denominator, the fraction can be expressed as the sum of individual, lower order fractions each of which has one of the distinct factors in the denominator. For example, Eq. (3-124) can be written as: C1 C2 C3 s2 − 3 s + 4 = + + (s + 1) (s − 1) (s + 2) (s + 1) (s − 1) (s + 2)

(3-125)

where C1 , C2 , and C3 are unknown constants. Both sides of Eq. (3-125) are multiplied by the denominator (s + 1)(s − 1)(s + 2): s2 − 3 s + 4 = C1 (s − 1) (s + 2) + C2 (s + 1) (s + 2) + C3 (s + 1) (s − 1)

(3-126)

Eq. (3-126) must be valid for any value of s since s is the independent variable. If s = −1 is substituted into Eq. (3-126) then the terms involving C2 and C3 become zero (as they both involve s + 1) and an equation is obtained that involves only C1 : (−1)2 − 3 (−1) + 4 = C1 ((−1) − 1) ((−1) + 2)

(3-127)

Equation (3-127) can easily be solved for C1 : 1 + 3 + 4 = C1 (−2) (1)

(3-128)

or C1 = −4. A similar process can be used to obtain C2 (i.e., eliminate C1 and C3 by substituting s = 1 into Eq. (3-126) and solve for C2 ) and C3 . The result is C2 = 1/3 (or 0.333) and C3 = 14/3 (or 4.667). Therefore, Eq. (3-124) can be written as:

T (s) =

0.333 4.667 −4 + + (s + 1) (s − 1) (s + 2)

(3-129)

Expressed in this form, the inverse transform of Eq. (3-129) can be obtained by inspection from Table 3-3: T (t) = −4 exp (−t) + 0.333 exp (t) + 4.667 exp (−2 t)

(3-130)

An alternative and more general method for obtaining C1 , C2 , and C3 is to carry out the multiplications in Eq. (3-126): s2 − 3 s + 4 = C1 (s2 + s − 2) + C2 (s2 + 3 s + 2) + C3 (s2 − 1)

(3-131)

374

Transient Conduction

and then require that the coefﬁcients multiplying like powers of s on either side of Eq. (3-131) must be equal in order to obtain three equations (one for each power of s) in three unknowns (C1 , C2 , and C3 ): 1 = C1 + C2 + C3

(3-132)

−3 = C1 + 3 C2

(3-133)

4 = −2 C1 + 2 C2 − C3

(3-134)

The solution of these simultaneous equations in EES: 1=C_1+C_2+C_3 -3=C_1+3∗ C_2 4=-2∗ C_1+2∗ C_2-C_3

yields C1 = −4, C2 = 0.333, and C3 = 4.667. Note that Maple can be used to quickly convert an expression to its partial fraction form using the convert command with the parfrac identiﬁer. To convert Eq. (3-124) into its partial fraction form, enter the expression and then convert it using the convert command; note that the ﬁrst argument is the expression to be converted, the second identiﬁes the type of conversion, and the third identiﬁes the name of the independent variable in the expression. > restart; > TS:=(sˆ2-3∗ s+4)/((s+1)∗ (s-1)∗ (s+2)); #expression to be converted s2 − 3s + 4 (s + 1) (s − 1) (s + 2) > TSpf:=convert(TS,parfrac,s); #expression converted 4 14 1 − + TSpf := 3 (s − 1) s+1 3 (s + 2) TS :=

Notice that the coefﬁcients identiﬁed by Maple (−4, 1/3, and 14/3) agree with the coefﬁcients identiﬁed manually. Repeated Factors. In the instance that the terms in the denominator of the expression to be converted are repeated, it is necessary to include each power of the term. For example, if the expression in the s domain is:

T (s) =

s2 − 3 s + 4 (s + 1)2 (s − 1)

(3-135)

then the partial fraction form must include fractions with both (s + 1) and (s + 1)2 : s2 − 3 s + 4 2

(s + 1) (s − 1)

=

C2 C1 C3 + + 2 (s + 1) (s + 1) (s − 1)

(3-136)

Otherwise, the solution proceeds as before. Both sides of Eq. (3-136) are multiplied by (s + 1)2 (s − 1): s2 − 3 s + 4 = C1 (s + 1) (s − 1) + C2 (s − 1) + C3 (s + 1)2

(3-137)

3.4 The Laplace Transform

375

or s2 − 3 s + 4 = C1 (s2 − 1) + C2 (s − 1) + C3 (s2 + 2 s + 1)

(3-138)

1 = C1 + C3

(3-139)

−3 = C2 + 2 C3

(3-140)

4 = −C1 − C2 + C3

(3-141)

which leads to:

The solution to these three equations: 1=C_1+C_3 -3=C_2+2∗ C_3 4=C_1-C_2+C_3

leads to C1 = 0.5, C2 = −4, and C3 = 0.5. Therefore:

T (s) =

−4 0.5 0.5 + + 2 (s + 1) (s + 1) (s − 1)

(3-142)

Maple can provide the same result using the convert command: > restart; > TS:=(sˆ2-3∗ s+4)/((s+1)ˆ2∗ (s-1)); #expression to be converted TS :=

s2 − 3 s + 4

(s + 1)2 (s − 1) > TSpf:=convert(TS,parfrac,s); #expression converted 1 1 4 TSpf := + − 2 (s − 1) 2 (s + 1) (s + 1)2

The inverse transform of Eq. (3-142) can be obtained using Table 3-3:

T (t) = 0.5 exp (t) + 0.5 exp (−t) − 4 t exp (−t)

(3-143)

Polynomial Factors. In the instance that the terms in the denominator of the expression include a polynomial factor, it is necessary to include a polynomial numerator with the order of the polynomial reduced by 1. For example, if the expression in the s domain is:

T (s) =

s2 − 3 s + 4 + 2) (s − 1)

(s2

(3-144)

then the partial fraction form must include fractions: C1 s + C2 C3 s2 − 3 s + 4 = + 2 + 2) (s − 1) (s + 2) (s − 1)

(s2

(3-145)

376

Transient Conduction

Otherwise, the solution proceeds as before. Both sides of Eq. (3-145) are multiplied by (s2 + 2) (s − 1): s2 − 3 s + 4 = (C1 s + C2 ) (s − 1) + C3 (s2 + 2)

(3-146)

s2 − 3 s + 4 = C1 s2 − C1 s + C2 s − C2 + C3 s2 + 2 C3

(3-147)

1 = C1 + C3

(3-148)

−3 = −C1 + C2

(3-149)

4 = −C2 + 2 C3

(3-150)

or

which leads to:

The solution to these three equations: 1=C_1+C_3 -3= -C_1+C_2 4= -C_2+2∗ C_3

leads to C1 = 0.333, C2 = −2.667, and C3 = 0.667. Therefore: 0.333 s − 2.667 0.667 + 2 (s + 2) (s − 1)

(3-151)

0.333 s 2.667 0.667 − + (s2 + 2) (s2 + 2) (s − 1)

(3-152)

T (s) = or

T (s) =

Maple can provide the same result: > restart; > TS:=(sˆ2-3∗ s+4)/((sˆ2+2)∗ (s-1)); #expression to be converted s2 − 3s + 4 (s2 + 2) (s − 1) > TSpf:=convert(TS,parfrac,s); #expression converted 2 s−8 + TSpf := 3 (s2 + 2) 3 (s − 1) TS :=

The inverse transform of Eq. (3-152) can be obtained using Table 3-3: √ √ √ sin( 2 t) + 0.667 exp (t) T (t) = 0.333 cos( 2 t) − 2.667 2

(3-153)

Inverse Laplace Transformation with Maple Maple can also be used to transform from a function in the s domain to a function in the time domain using the invlaplace command. The invlaplace command has the same basic calling protocol as the laplace command. The ﬁrst argument is the expression to be inverse transformed, the second argument is the variable to transform from (typically s),

3.4 The Laplace Transform

377

and the third argument is the variable to transform to (typically t). To obtain the inverse Laplace transform of C/s: > restart: > with(inttrans): > invlaplace(C/s,s,t); C

The inverse Laplace transforms that are obtained in Section 3.4.3 could also be obtained using the invlaplace command in Maple. For example, the inverse Laplace transform of Eq. (3-124) is: > restart: > with(inttrans): > TS:=(sˆ2-3∗ s+4)/((s+1)∗ (s-1)∗ (s+2)); TS :=

s2 − 3s + 4 (s + 1) (s − 1) (s + 2)

> Tt:=invlaplace(TS,s,t); Tt :=

13 14 (−2t) 11 cosh (t) + sinh (t) + e 3 3 3

The solution identiﬁed by Maple looks different from Eq. (3-130), the solution identiﬁed using partial fractions and Table 3-3. However, if the hyperbolic cosine and sine terms are written in terms of exponentials then the result is the same. (This can be accomplished using the convert function with the exp identiﬁer.) > Tt:=convert(Tt,exp); Tt :=

4 14 (−2t) 1 t e − t + e 3 e 3

The inverse Laplace transforms of Eqs. (3-135) and (3-144) can also be identiﬁed using Maple: > restart: > with(inttrans): > TS:=(sˆ2-3∗ s+4)/((s+1)ˆ2∗ (s-1)); TS :=

s2 − 3 s + 4 (s + 1)2 (s − 1)

> Tt:=invlaplace(TS,s,t); Tt := 4 t sinh (t) − cosh (t) (−1 + 4 t) > Tt:=simplify(convert(Tt,exp)); Tt := 4 t e(−t) +

1 t 1 (−t) e + e 2 2

378

Transient Conduction

> TS:=(sˆ2-3∗ s+4)/((sˆ2+2)∗ (s-1)); TS :=

s2 − 3 s + 4 (s2 + 2) (s − 1)

> Tt:=invlaplace(TS,s,t); Tt :=

√ √ 4√ 2 1 cos( 2 t) − 2 sin( 2 t) + et 3 3 3

The library of inverse Laplace transforms that is available in Maple is limited and several transforms that are useful for solving heat transfer problems are not available. Additional inverse Laplace transforms are available from the website associated with this book (www.cambridge.org/nellisandklein). The ﬁle that includes these inverse Laplace transforms is titled Inverse Laplace Transforms. The top of this Maple ﬁle includes a section of code that adds a series of entries to the existing invlaplace table in Maple; these entries symbolically deﬁne some additional transforms that are commonly encountered in heat transfer problems and therefore augments Maple’s functionality.

3.4.4 Properties of the Laplace Transformation The Laplace transform is linear. Therefore, the transform of the sum of two functions is the sum of their individual transforms:

T 1 (t) + T 2 (t) = T 1 (s) + T 2 (s)

(3-154)

and the transform of the product of a constant and a function is the product of the constant and the transform:

C T (t) = C T (s)

(3-155)

The transform of a time derivative of a function T(t) is the product of the transform of the function and s less the initial condition in the time domain: 7 6 dT (t) (3-156) = s T (s) − T t=0 dt This property of the Laplace transform is its primary feature and the reason it is useful for solving differential equations. Equation (3-156) can be proven. Apply the Laplace transform to the time derivative of T: 7 ∞ 6 dT dT (t) = exp(−s t) dt (3-157) dt dt 0

Equation (3-157) can be simpliﬁed through integration by parts. We will encounter integration by parts again elsewhere in this book and therefore it is worth spending some time understanding the process. Integration by parts is based on the chain rule for differentiation; the differential of the product of two functions (u and v) is: d (u v) = u dv + v du

(3-158)

which can be integrated: (u v)2

v2

d (u v) = (u v)1

u2 u dv +

v1

v du u1

(3-159)

3.4 The Laplace Transform

379

The left side of Eq. (3-159) is an exact differential and therefore: v2 (u v)2 − (u v)1 =

u2 u dv +

v1

v du

(3-160)

v du

(3-161)

u1

or, rearranging: v2

u2 u dv = (u v)2 − (u v)1 −

v1

u1

Successful use of integration by parts requires that the functions u and v are identiﬁed and substituted into Eq. (3-161) in order to simplify the expression of interest, in this case Eq. (3-157): 6

7 ∞ dT dT = exp(−s t) dt 0

(3-162)

dv

u

By inspection of Eqs. (3-161) and (3-162), we will deﬁne: u = exp(−s t)

(3-163)

dv = dT

(3-164)

du = −s exp(−s t)dt

(3-165)

v=T

(3-166)

therefore

Substituting Eqs. (3-163) through (3-166) into Eq. (3-161) leads to: 6

t=∞ 7 dT T (−s exp(−s t)dt) = [exp(−s t) T ]t=∞ − [exp(−s t) T ]t=0 − dt (u v)2

(u v)1

t=0

v

(3-167)

du

or 6

7 ∞ dT = −T t=0 + s T exp(−s t)dt dt 0

(3-168)

T (s)

The ﬁnal term in Eq. (3-168) is the product of s and the Laplace transform of T: 7 6 dT (3-169) = s T (s) − T t=0 dt

380

Transient Conduction

Table 3-4: Useful properties of the Laplace transforms. 8

9 T 1 (t) + T 2 (t) = T 1 (s) + T 2 (s) 8

9 C T (t) = C T (s) 7 6 dT (t) = s T (s) − T t=0 dt 7 6 ∂T (x, t) = s T (x, s) − T x,t=0 ∂t 6

7 ∂ n T (x, s) ∂ n T (x, t) = ∂xn ∂xn

The transform of a derivative is not, itself, a derivative. This is a very useful property, as it turns a differential equation in t into an algebraic equation in s. It is possible to transform a partial derivative of temperature with respect to time (for example, when temperature depends on both position and time, T (x, t)): 6

7 ∞ ∂T ∂T (x, t) = exp (−s t) dt ∂t ∂t

(3-170)

0

Carrying out the same process of integration by parts leads to a similar conclusion: 7 6 ∂T (x, t) (3-171) = s T (x, s) − T x,t=0 ∂t The Laplace transform of a partial derivative of temperature with respect to position is the partial derivative of the transformed function with respect to position: 6

7 ∂ T (x, s) ∂T (x, t) = ∂x ∂x

(3-172)

This is true for all higher derivatives as well: 6

7 ∂ n T (x, t) ∂ n T (x, s) = ∂xn ∂xn

(3-173)

These properties of the Laplace transform are summarized in Table 3-4.

3.4.5 Solution to Lumped Capacitance Problems The Laplace transform can be used to obtain analytical solutions to the 0-D transient problems that are considered in Section 3.1. The process will be illustrated using the problem discussed in EXAMPLE 3.1-2. A temperature sensor is exposed to an oscillating temperature environment (T∞ ): T ∞ = T ∞ + T ∞ sin (2 π f t)

(3-174)

where T ∞ = 320◦ C is the average temperature of the ﬂuid and T ∞ = 50K and f = 0.5 Hz are the amplitude and frequency of the temperature oscillation. The governing

3.4 The Laplace Transform

381

differential equation for the problem, determined in EXAMPLE 3.1-2, is: dT T∞ T ∞ sin (2 π f t) T = + + dt τlumped τlumped τlumped

(3-175)

where τlumped = 0.72 s is the lumped capacitance time constant of the sensor. The initial condition is: T t=0 = T ini

(3-176)

where Tini = 260◦ C is the initial temperature of the sensor. The known information is entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” T inﬁnity bar=converttemp(C,K,320 [C]) T ini=converttemp(C,K,260 [C]) DELTAT inﬁnity=50 [K] f=0.5 [Hz] tau=0.72 [s]

“average environmental temperature” “initial temperature” “amplitude of oscillation” “frequency of oscillation” “time constant of temperature sensor”

In order to solve this problem using the Laplace transform approach it is necessary to transform the governing differential equation from the t domain to the s domain. The ﬁrst term in the governing equation, Eq. (3-175), is transformed using Eq. (3-169): 6

7 dT (t) = s T (s) − T t=0 dt

(3-177)

Substituting Eq. (3-176) into Eq. (3-177) leads to: 6

7 dT = s T (s) − T ini dt

(3-178)

The second term in Eq. (3-175) becomes: 6

7 T (s) T (t) = τlumped τlumped

(3-179)

which follows from the fact that the Laplace transform is linear, see Eq. (3-155). The ﬁnal two terms in Eq. (3-175) are obtained from Table 3-3: :

T∞ τlumped

; =

T∞ s τlumped

(3-180)

and 6

7 T ∞ 2πf T ∞ sin (2 π f t) = τlumped τlumped s2 + (2 π f )2

(3-181)

382

Transient Conduction

Substituting Eqs. (3-178) through (3-181) into Eq. (3-175) leads to the transformed governing equation:

T (s) T∞ T ∞ = + s T (s) − T ini + τlumped s τlumped τlumped

2πf

s2 + (2 π f )2

(3-182)

Notice that the differential equation in t, Eq. (3-175), has been transformed to an algebraic equation in s. The same result can be obtained using Maple. The governing differential equation is entered: > restart:with(inttrans): > ODEt:=diff(T(time),time)+T(time)/tau=T_inﬁnity_bar/tau+DELTAT_inﬁnity∗ sin (2∗ pi∗ f∗ time)/tau; d T (time) dtime T (time) T infinity bar DELTAT infinity sin (2 π f time) + = + τ τ τ

ODEt : =

and the laplace command is used to obtain the Laplace transform of the entire differential equation: > AEs:=laplace(ODEt,time,s); laplace(T (time), time, s) τ T infinity bar 2 DELTAT infinity π f = + τs τ(s2 + 4π2 f 2 )

AEs : = s laplace(T(time), time, s) − T (0) +

where laplace(T(time),time,s) indicates the Laplace transform of the function T. The transformed equation can be made more concise by using the subs command to substitute a single variable, T(s), for the laplace() result. Recall that the ﬁrst argument of the subs command is the substitution you want to make while the second argument is the expression that you want to make the substitution into: > AEs:=subs(laplace(T(time),time,s)=T(s),AEs); T (s) T infinity bar 2 DELTAT infinity π f AEs := s T (s) − T (0) + = + τ τs τ (s2 + 4 π2 f 2 )

The solution includes the initial condition, T(0), which can be eliminated using the subs command again: > AEs:=subs(T(0)=T_ini,AEs); AEs := s T (s) − T ini +

T (s) T infinity bar 2 DELTAT inf inity π f = + τ τs τ (s2 + 4 π2 f 2 )

3.4 The Laplace Transform

383

This result is identical to the result that was obtained manually, Eq. (3-182); the algebraic

equation in the s domain, Eq. (3-182), can be solved for T (s): T ini 1 s+ τlumped

T (s) =

+

T∞ τlumped s s +

(3-183) ⎡ 1

+

2 π f T ∞ ⎢ ⎢ τlumped ⎣

τlumped

⎤ 1 2 2 (s + (2 π f ) ) s +

1

⎥ ⎥ ⎦

τlumped

The inverse Laplace transform of Eq. (3-183) does not appear in Table 3-3 and therefore it is necessary to use the method of partial fractions to reduce Eq. (3-183) to simpler fractions that do appear in Table 3-3. The second term in Eq. (3-183) can be written as: T∞ τlumped s s + Multiplying through by s (s +

=

1

C2

C1 + s

s+

τlumped

1

(3-184)

τlumped

1 ) leads to: τlumped T∞

τlumped

= C1 s +

C1 τlumped

+ C2 s

(3-185)

which leads to: 0 = C1 + C2

(3-186)

T∞ C1 = τlumped τlumped

(3-187)

C1 = T ∞

(3-188)

and

Solving Eq. (3-187) leads to:

Substituting Eq. (3-188) into Eq. (3-186) leads to: C2 = −T ∞

(3-189)

“coefﬁcients from partial fraction expansion” C_1=T_inﬁnity_bar C_2=-T_inﬁnity_bar

The third term in Eq. (3-183) has a second order polynomial term (s2 ) in the denominator and therefore it can be expressed as: ⎡ ⎤ 2 π f T ∞ ⎢ ⎢ τlumped ⎣

1 (s2 + (2 π f )2 ) s +

1 τlumped

⎥ ⎥ ⎦=

C3 s + C4 + + (2 π f )2 )

(s2

C5 s+

1

τlumped (3-190)

384

Transient Conduction

Multiplying Eq. (3-190) through by (s2 + (2 π f )2 )(s + 1τ ) leads to: 2 π f T ∞ 1 = (C3 s + C4 ) s + + C5 (s2 + (2 π f )2 ) τlumped τlumped

(3-191)

or C3 s C4 2 π f T ∞ = C3 s2 + + C4 s + + C5 s2 + C5 (2 π f )2 τlumped τlumped τlumped

(3-192)

which leads to: 0 = C3 + C5 0=

(3-193)

C3 + C4 τlumped

(3-194)

2 π f T ∞ C4 = + C5 (2 π f )2 τlumped τlumped

(3-195)

Equations (3-193) through (3-195) are 3 equations in the 3 unknowns C1 , C2 , and C3 : 0=C_3+C_5 0=C_3/tau+C_4 2∗ pi∗ f∗ DELTAT_inﬁnity/tau=C_4/tau+C_5∗ (2∗ pi∗ f)ˆ2

With the coefﬁcients for the partial fractions now identiﬁed, Eq. (3-183) can be expressed as:

T (s) =

+

T ini C2 C1 + + 1 1 s s+ s+ τlumped τlumped (s2

C4 C3 s + + + (2 π f )2 ) (s2 + (2 π f )2 )

C5 (s +

(3-196)

1 ) τlumped

The inverse Laplace transform of each of the terms in Eq. (3-196) is obtained using Table 3-3. t t T (t) = T ini exp − + C1 + C2 exp − + C3 cos (2 π f t) τlumped τlumped t C4 sin (2 π f t) + C5 exp − (3-197) + 2πf τlumped The solution is programmed in EES: T=T ini∗ exp(-time/tau)+C 1+C 2∗ exp(-time/tau)+C 3∗ cos(2∗ pi∗ f∗ t+ & “sensor temp., obtained manually” C 4∗ sin(2∗ pi∗ f∗ time)/(2∗ pi∗ f)+C 5∗ exp(-time/tau) T C=converttemp(K,C,T) “in C” T inﬁnity=T inﬁnity bar+DELTAT inﬁnity∗ sin(2∗ pi∗ f∗ time) “ﬂuid temperature” “in C” T inﬁnity C=converttemp(K,C,T inﬁnity)

3.4 The Laplace Transform 380

385

fluid temperature

Temperature (°C)

360 340 320 300 280 260 0

sensor temperature (from Example 3.1-2 and Laplace transform) 2

1

3

4 Time (s)

5

6

7

8

Figure 3-17: Fluid and sensor temperature as a function of time.

The solution obtained using the Laplace transform is identical to the solution obtained in EXAMPLE 3.1-2, as shown in Figure 3-17. Maple could also be used to obtain the solution. The solve command can be used to carry out the algebra required to solve for T(s): > Ts:=solve(AEs,T(s)); Ts := (T ini τ s3 + 4 T ini τ s π2 f 2 + T inf inity bars2 + 4 T inf inity barπ2 f 2 + 2 DELTAT inf inity π f s )/(s(s3 τ + 4 s τ π2 f 2 + s2 + 4 π2 f 2 ))

And the invlaplace command can be used to carry out the inverse Laplace transform: > Tt:=invlaplace(Ts,s,time); ime Tt := e(− τ ) T ini + DELTAT inf inity < $ $ $ ime − sinh(2 −π2 f 2 time) −π2 f 2 + 2 τπ2 f 2 e(− τ ) − cosh(2 −π2 f 2 time) ime (πf (1 + 4 π2 f 2 τ2 )) + T inf inity bar 1 − e(− τ )

Notice that the inverse Laplace transform identiﬁed by Maple includes the hyperbolic sine and cosine (rather than the sine and cosine); √ however, the argument of the sinh and cosh functions are complex (i.e., they involve −1) and therefore these functions become sin and cos. It is possible to specify that the frequency, f, is positive using the assume command: > assume(f,positive);

386

Transient Conduction

With this stipulation, Maple will correctly identify the solution in terms of sine and cosine: > Tt:=invlaplace(Ts,s,time); ime Tt : = e(− τ ) T ini ime sin(2f ∼ π time) + 2τπf ∼ − cos(2f π˜ time) + e(− τ ) DELTAT inf inity + 1 + 4 π 2 f ∼2 τ 2 (− ime +T inf inity bar 1 − e τ )

The solution can be copied and pasted into EES: T_maple=exp(-time/tau)∗ T_ini+(sin(2∗ f∗ pi∗ time)+2∗ tau∗ pi∗ f∗ (-cos(2∗ f∗ pi∗ time)+& exp(-time/tau)))∗ DELTAT_inﬁnity/(1+4∗ piˆ2∗ fˆ2∗ tauˆ2)+T_inﬁnity_bar∗ (1-exp(-time/tau)) “solution from Maple” T_maple_C=converttemp(K,C,T_maple) “in C”

where it provides an identical solution to the one obtained manually (see Figure 3-17). There is not usually a clear advantage associated with using the Laplace transform to solve the ordinary differential equations in time that result from lumped capacitance problems in heat transfer over the analytical techniques discussed in Section 3.1. However, the Laplace transform technique provides another useful tool and possibly a method for double-checking an important solution. The Laplace transform is very useful for certain types of 1-D transient problems, discussed in Section 3.4.6.

3.4.6 Solution to Semi-Inﬁnite Body Problems The Laplace transform can be used to obtain solutions to partial differential equations as well as to ordinary differential equations. The solution steps are basically the same; however, the Laplace transform directly incorporates the initial condition (see Eq. (3-169)) and therefore the initial condition does not have to be transformed. The Laplace transform of a partial differential equation in x and t will result in an ordinary differential equation in x in the s domain. Therefore, it is necessary to transform the boundary conditions involving x into the s domain so that the ordinary differential equation can be solved. The solution must be converted to a function of x and t using the inverse Laplace transform. The process will be illustrated using the problem that was discussed in Section 3.3.3 in which a semi-inﬁnite body that is initially at a uniform temperature Tini is exposed to a step change in its surface temperature, from Tini to Ts . The governing partial differential equation for this situation is: ∂T ∂2T = ∂x2 ∂t where α is the thermal diffusivity. The boundary conditions are provided by: α

(3-198)

T x=0 = T s

(3-199)

T t=0 = T ini

(3-200)

T x→∞ = T ini

(3-201)

3.4 The Laplace Transform

387

The governing differential equation is transformed from the x, t domain to the x, s. The ﬁrst term in Eq. (3-198) is transformed using Eq. (3-173): 7 6 2 ∂ 2 T (x, s) ∂ T (x, t) (3-202) =α α ∂x2 ∂x2 and the second term is transformed using Eq. (3-171): 7 6 ∂T (x, t) = s T (x, s) − T ini ∂t

(3-203)

so that the transformed differential equation is:

∂ 2 T (x, s) α = s T (x, s) − T ini (3-204) 2 ∂x Maple can identify the same transformed governing differential equation using basically the same steps discussed Section 3.4.5:

> restart:with(inttrans): > PDE:=alpha∗ diff(diff(T(x,time),x),x)=diff(T(x,time),time); 2 ∂ ∂ PDE := α T (x, time) T (x, time) = ∂x2 ∂time > ODE:=laplace(PDE,time,s); 2 ∂ ODE := α laplace(T (x,time), time, s) = s laplace(T (x, time), time, s) − T (x, 0) 2 ∂x > ODE:=subs(T(x,0)=T_ini,ODE); 2 ∂ = s laplace(T (x, time), time, s) − T ini ODE := α ∂x 2 laplace(T (x, time), time, s) > ODE:=subs(laplace(T(x,time),time,s)=Ts(x),ODE); 2 d ODE := α Ts(x) = s Ts(x) − T ini 2 dx

Equation (3-204) does not involve any derivative with respect to s and therefore it is an ordinary differential equation in x; the partial differential in Eq. (3-204) can be changed to an ordinary differential:

d2 T (s, x) = s T (s, x) − T ini α 2 dx

(3-205)

which can be rearranged:

s T ini d2 T − T = − (3-206) 2 dx α α Equation (3-206) is a second order, non-homogeneous equation and therefore requires two boundary conditions; these are obtained from Eqs. (3-199) and (3-201), which must also be transformed to the s domain. Ts T x=0 = (3-207) s

T x→∞ =

T ini s

(3-208)

388

Transient Conduction

The second order differential equation is split into homogeneous and particular components:

T = Th + T p

(3-209)

Equation (3-209) substituted into Eq. (3-206):

d2 T p d2 T h s s T ini − T + − Tp= − h 2 2 dx α dx α α =0 for homogeneous differential equation

(3-210)

particular differential equation

The homogeneous differential equation is:

s d2 T h − Th = 0 2 dx α

(3-211)

which has the general solution: / T h = C1 exp

/ s s x + C2 exp − x α α

(3-212)

where C1 and C2 are undetermined constants. The solution to the particular differential equation

d2 T p s T ini − Tp= − 2 dx α α

(3-213)

is, by inspection:

Tp=

T ini s

(3-214)

Substituting Eqs. (3-212) and (3-214) into Eq. (3-209) leads to: / / s s T ini T = C1 exp x + C2 exp − x + α α s

(3-215)

Maple can be used to obtain the same solution: > dsolve(ODE);

√

Ts(x) = e

sx √ α

− C2 + e

−

√ sx √ α

C1 +

T ini s

The constants C1 and C2 are obtained from the boundary conditions. The boundary condition at x→ ∞, Eq. (3-208), leads to: / / s s T ini T ini T x→∞ = C1 exp ∞ + C2 exp − ∞ + = (3-216) α α s s or

/ C1 exp

s ∞ =0 α

(3-217)

3.4 The Laplace Transform

389

which can only be true if C1 = 0:

/ s T ini T = C2 exp − x + α s

(3-218)

The boundary condition at x = 0, Eq. (3-207), leads to: / s T ini Ts T x=0 = C2 exp − 0 + = α s s

(3-219)

or C2 =

(T s − T ini ) s

(3-220)

Substituting Eq. (3-220) into Eq. (3-218) leads to the solution to the problem in the x, s domain: / s T ini (T s − T ini ) T (x, s) = exp − x + (3-221) s α s The solution in the x, t domain can be obtained using the inverse Laplace transforms contained in Table 3-3: x erfc − T (3-222) + T ini T (x, t) = (T s √ ini ) 2 αt Recall that the complementary error function (erfc) is deﬁned as: erfc (x) = 1 − erf (x)

(3-223)

so Eq. (3-222) can be rewritten as:

x T (x, t) = (T s − T ini ) 1 − erf √ + T ini 2 αt

or

T (x, t) = T s + (T ini − T s ) erf

x √ 2 αt

(3-224)

(3-225)

which is identical to the similarity solution obtained in Section 3.3.3. Unfortunately, Maple is not able to automatically identify the inverse Laplace transform of Eq. (3-221): > restart:with(inttrans): > Ts:=(T_s-T_ini)∗ exp(-sqrt(s)∗ x/sqrt(alpha))/s+T_ini/s;

−

(T s − T ini) e Ts := s > invlaplace(Ts,s,time); (T s − T ini) invlaplace

⎛

−

⎝e

√ sx α

√ sx √ α

s

+

T ini s ⎞

, s, time ⎠ + T ini

The indicator invlaplace() is a placeholder that shows that the inverse Laplace transform was not found in Maple’s library. It is possible to add entries to Maple’s library, as discussed by Aziz (2006), in order to allow Maple to solve this and other semiinﬁnite body heat transfer problems. Several of the most common inverse transforms

390

Transient Conduction

that are encountered for heat transfer problems have been added to the ﬁle Inverse Laplace Transforms, which can be downloaded from the website associated with the text (www.cambridge.org/nellisandklein). Rename the ﬁle after you download it and place the Maple code associated with this problem at the bottom of the ﬁle. Run the entire ﬁle to obtain: > #Inverse Laplace transforms > restart:with(inttrans): > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/s,erfc(p/(2∗ sqrt(t))),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/sˆ(3/2),2∗ sqrt(t)∗ exp(-pˆ2/(4∗ t))/sqrt (pi)-p∗ erfc(p/(2∗ sqrt(t))),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic),p∗ exp(-pˆ2/(4∗ t))/(2∗ sqrt(pi∗ tˆ3)),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/sqrt(s),exp(-pˆ2/(4∗ t))/sqrt(pi∗ tˆ3),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/sˆ2,(t+pˆ2/2)∗ erfc(p/(2∗ sqrt(t)))-p∗ sqrt(t)∗ exp(-pˆ2/(4∗ t))/sqrt(pi),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ C1::algebraic)/(C2::algebraic+(C3::algebraic)∗ sqrt(s)),(exp(-C1ˆ2/(4∗ t))/sqrt(Pi∗ t)-(C2/C3)∗ exp((C2/C3)∗ C1)∗ exp((C2/C3)ˆ2∗ t) ∗ erfc((C2/C3)∗ sqrt(t)+C1/(2∗ sqrt(t))))/C3,s,t); > addtable(invlaplace,exp(-sqrt(s)∗ C1::algebraic)/(s∗ (C2::algebraic+(C3::algebraic) ∗ sqrt(s))),erfc(1/2∗ C1/tˆ(1/2))-exp(C2/C3∗ C1)∗ exp(C2ˆ2/C3ˆ2∗ t)∗ erfc(C2/C3∗ tˆ(1/2) +1/2∗ C1/tˆ(1/2)),s,t;) > addtable(invlaplace,exp(-sqrt(s)∗ C1::algebraic)/(sqrt(s)∗ (C2::algebraic+ (C3::algebraic)∗ sqrt(s))),(exp((C2/C3)∗ C1)∗ exp((C2/C3)ˆ2∗ t)∗ erfc((C2/C3)∗ sqrt(t) +C1/(2∗ sqrt(t))))/C3,s,t); > #place your code below this line > Ts:=(T_s-T_ini)∗ exp(-sqrt(s)∗ x/sqrt(alpha))/s+T_ini/s; √ − √sαx

(T s − T ini) e Ts := s > invlaplace(Ts,s,time);

T s + erf

√

x

2 α time

+

T ini s

(−T s + T ini)

391

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR When a superconducting conductor “quenches,” it goes from having no resistance (i.e., a superconducting state) to being resistive (i.e., a normal state). When the conductor is carrying current, the quenching process is accompanied by a step change in the rate of volumetric generation of thermal energy within the material (from nearly zero to a relatively high value, depending on the amount of current that is being carried). The result can be disastrous. This problem examines the temperature distribution in the conductor during the initial stages of a quenching process, shown schematically in Figure 1. initial temperature, Tini = 4.2 K Ts = 4.2 K q⋅ x

Figure 1: Quenching of a conductor.

∂U ∂t g⋅

q⋅ x+dx

x k = 500 W/m-K -4 2 α = 6.25x10 m /s ⋅g ′′′ = 1x106 W/m3

dx

The conductor is initially at a uniform temperature of Tini = 4.2 K when the quench process occurs, resulting in a uniform rate of volumetric generation, g˙ = 1 × 106 W/m3 . The conductor has conductivity k = 500 W/m-K and thermal diffusivity α = 0.000625 m2 /s. The surface of the conductor is maintained at Ts = 4.2 K by boiling liquid helium. a) Develop an analytical model of the quench process that is valid for short times, while the conductor behaves as a semi-inﬁnite body. The governing differential equation for the semi-inﬁnite solid is derived by focusing on a differential control volume (see Figure 1). The energy balance suggested by Figure 1 is: ∂U ∂t expanding the x + d x term and simplifying leads to: g˙ + q˙ x = q˙ x+d x +

∂ q˙ x ∂U dx + ∂x ∂t The conduction term is evaluated using Fourier’s law: g˙ =

(1)

∂T (2) ∂x where Ac is the cross-sectional area of the conductor perpendicular to the xdirection. The time rate of change of the internal energy of the material in the control volume is: ∂T ∂U = ρ c Ac d x (3) ∂t ∂t The rate of generation of thermal energy in the control volume is: q˙ x = −k Ac

g˙ = Ac d x g˙

(4)

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

3.4 The Laplace Transform

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

392

Transient Conduction

Substituting Eqs. (2) through (4) into Eq. (1) leads to: ∂T ∂ ∂T Ac d x g˙ = −k Ac d x + ρ c Ac d x ∂x ∂x ∂t or ∂ 2T 1 ∂T g˙ − = − ∂x2 α ∂t k The boundary conditions for the problem include the initial condition: Tt =0 = Tini

(5)

(6)

the surface temperature is speciﬁed: Tx=0 = Ts

(7)

and the temperature gradient must approach zero as you move away from the surface: dT =0 (8) dx x→∞ To solve this problem using the Laplace transform it is necessary to transform the governing differential equation, Eq. (5): 1 g˙ d 2T − s T − Tx,t =0 = − 2 dx α ks Substituting Eq. (6) into Eq. (9) and rearranging:

(9)

s Tini g˙ d 2T − T = − − (10) d x2 α α ks The spatial boundary conditions, Eqs. (7) and (8), are transformed to the s domain in order to provide the boundary conditions for the ordinary differential equation in the s domain, Eq. (10):

T x=0 = d T d x

Ts s

(11)

=0

(12)

x→∞

The solution to the ordinary differential equation, Eq. (10), is: / / s s g˙ α Tini T = C 1 exp + x + C 2 exp − x + α α k s2 s The boundary condition at x → ∞ leads to: / / / / d T s s s s = C1 exp ∞ − C2 exp − ∞ =0 d x α α α α x→∞

which can only be true if C1 = 0:

/ s g˙ α Tini T = C 2 exp − + x + α k s2 s

393

The boundary condition at x = 0 leads to: / s g˙ α Tini Ts T x=0 = C 2 exp − + 0 + = α k s2 s s which leads to: C2 =

(Ts − Tini ) g˙ α − s k s2

The solution in the s domain is: / s g˙ α Tini (Ts − Tini ) g˙ α T = + − x + exp − s k s2 α k s2 s which can be rearranged: / / s s g˙ α g˙ α Tini (Ts − Tini ) T = exp − + exp − x − x + 2 s α ks α k s2 s The inverse Laplace transform can be obtained using Table 3-3: x T = (Ts − Tini ) erfc √ 2 αt / 2 g˙ α t x x x2 − −x t+ erfc exp − − t + Tini √ k 2α απ 4αt 2 αt

(13)

(14)

The known information is entered in EES: “EXAMPLE 3.4-1: Quenching of a Superconductor” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” T ini=4.2 [K] T s=4.2 [K] g dot=1e6 [W/mˆ3] k=500 [W/m-K] alpha=0.000625 [mˆ2/s]

“initial temperature of superconductor” “surface temperature” “volumetric rate of generation” “conductivity” “thermal diffusivity”

and the solution is programmed: “Solution” time=0.1 [s] “time” x mm=10 [mm] “position in mm” “position” x=x mm∗ convert(mm,m) T=(T s-T ini)∗ erfc(x/(2∗ sqrt(alpha∗ time)))+(g dot∗ alpha/k)∗ (time-(time+xˆ2/(2∗ alpha))∗ & erfc(x(2∗ sqrt(alpha∗ time)))+x∗ sqrt(time/(pi∗ alpha))∗ exp(-xˆ2/(4∗ alpha∗ time)))+T ini

Figure 2 illustrates the temperature as a function of position for various times relative to the start of the quench process.

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

3.4 The Laplace Transform

Transient Conduction 5.2 t = 0.75 s 5

Temperature (K)

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

394

t = 0.5 s

4.8 4.6

t = 0.25 s 4.4 t = 0.1 s t=0s

4.2 0

10

20

30

40 50 60 Position (mm)

70

80

90

100

Figure 2: Temperature as a function of position at various times relative to the start of the quench process.

Notice that the portion of the conductor removed from the edge increases linearly in temperature. (This behavior corresponds to the term in Eq. (14) that is linear with time.) However, the effect of the cooled edge propagates into the conductor at a rate that increases with time. The solution can be checked against our physical intuition by verifying that the speed of the thermal wave agrees, approximately, with the diffusive time constant that was discussed in Section 3.3.2. At time = 0.5 s the thermal wave should have moved approximately: √ (15) δt = 2 α t EES’ calculator window can be used to carry out supplementary calculations such as this. EES’ calculator window is accessed by selecting Calculator from the Windows menu (Figure 3).

0.000625 0.035355

Figure 3: Calculator window.

To enter a command in the Calculator window, enter ? followed by the command and terminate the command with the enter key. The variables from the last time that the EES code was run are accessible from the Calculator; for example, typing ?alpha will provide the value of the thermal diffusivity of the superconductor. To determine the size of the thermal wave at 0.5 s using Eq. (15), type ?2∗ sqrt(alpha∗ 0.5), as shown in Figure 3. Notice that the answer, 0.035 m or 35 mm, is consistent with the thermal wave thickness at 0.5 s in Figure 2.

3.5 Separation of Variables for Transient Problems

395

The analytical solution of partial differential equations with the Laplace transform is convenient in some situations. For example, it is not possible to solve semi-inﬁnite problems or (with some exceptions) problems with non-homogeneous boundary conditions in space using the method of separation of variables discussed in Section 3.5. Therefore, the Laplace transform solution or a self-similar solution (Section 3.3.3) may be the best alternative. However, the process of obtaining the inverse Laplace transform can be difﬁcult and it is often easier to develop numerical solutions to these problems, as discussed in Section 3.8.

3.5 Separation of Variables for Transient Problems 3.5.1 Introduction Section 3.3 discussed the behavior of a thermal wave within an un-bounded (i.e., semiinﬁnite) solid and introduced the concept of a diffusion time constant. The self-similar solution to a semi-inﬁnite solid was presented in Section 3.3.3 and the solution to this type of problem using the Laplace transform approach was discussed in Section 3.4.6. This section examines the analytical solution to 1-D transient problems that are bounded using the method of separation of variables. Figure 3-18 illustrates, qualitatively, the temperature distribution at various times that will be present in a plane wall that is initially at a uniform temperature, Tini , when the temperature of one surface (at x = L) is suddenly increased to Ts while the other surface (at x = 0) is adiabatic. adiabatic surface

surface exposed to Ts increasing time

Temperature

Ts

T ini Position Figure 3-18: Temperature as a function of position at various times for a plane wall initially at Tini that is subjected to a sudden change in the surface temperature to Ts .

The behavior illustrated in Figure 3-18 is initially consistent with the behavior of a semi-inﬁnite body, discussed in Section 3.3. A thermal wave emanates from the surface at x = L and penetrates into the solid; the depth of the thermal wave (δt ) is approximately given by: √ δt = 2 α t

(3-226)

396

Transient Conduction

where α is the thermal diffusivity and t is time. When the thermal wave reaches the adiabatic edge at x = 0, it becomes bounded and cannot grow further. The character of the problem changes at this time. The temperature distributions will no longer collapse when plotted as a function of x/δt , as they did in Figure 3-15, and therefore a self-similar solution to the bounded problem is not possible. The method of separation of variables can be used to analytically solve the problem shown in Figure 3-18. The solutions associated with some common shapes (a plane wall, cylinder, and sphere initially at a uniform temperature and exposed to a convective boundary condition) are presented in Section 3.5.2 without derivation. Sections 3.5.3 and 3.5.4 provide an introduction to the application of the method of separation of variables for 1-D transient problems in Cartesian and cylindrical coordinates, respectively. A more thorough discussion can be found in Myers (1998). The concepts and steps used to obtain separation of variables solutions to 1-D transient problems are quite similar to those discussed for steady-state, 2-D problems in Section 2.2 and 2.3. The partial differential equation in position x (or r for cylindrical or spherical problems) and time t is transformed (by separation of variables) into a second order ordinary differential equation in position and a ﬁrst order ordinary differential equation in time. The sub-problem involving x must be the eigenproblem and result in eigenfunctions. Therefore, both spatial boundary conditions must be homogeneous in order to apply separation of variables to a 1-D transient problem. Recall that a homogeneous boundary condition is one where any solution that satisﬁes the boundary condition must still satisfy the boundary condition if it is multiplied by a constant. Three types of homogeneous linear boundary conditions are encountered in heat transfer problems: (1) a speciﬁed temperature of zero, (2) an adiabatic boundary, and (3) convection to a ﬂuid with a temperature of zero. Section 2.2.3 shows how it is possible to transform some non-homogeneous boundary conditions into homogeneous boundary conditions by subtracting the speciﬁed temperature or ﬂuid temperature. Section 2.4 discusses the superposition of different solutions in order to accommodate multiple nonhomogeneous boundary conditions. These techniques can also be used for 1-D transient problems.

3.5.2 Separation of Variables Solutions for Common Shapes The separation of variables solutions for the plane wall, cylinder, and sphere are available in most textbooks as approximate formulae and in graphical format. The solutions are also available within EES both in dimensional and non-dimensional forms using the Transient Conduction library. This section provides, without derivation, the solutions to the basic problems associated with a plane wall, cylinder, and sphere initially at a uniform temperature when, at time t = 0, the surface is exposed via convection to a step change in the surrounding ﬂuid temperature. These problems are summarized in Figure 3-19. The solutions to this set of problems are obtained using the separation of variables techniques discussed in Sections 3.5.3 and 3.5.4 and are widely applicable to 1-D transient problems. The Plane Wall Governing Equation and Boundary Conditions. Section 3.5.3 provides the solution to a plane wall subjected to a step change in the ﬂuid temperature at a convective boundary. The partial differential equation associated with this problem is: 1∂T ∂2T = 2 ∂x α ∂t

(3-227)

3.5 Separation of Variables for Transient Problems

rout

L

T∞ , h

T∞ , h

397

rout

T∞, h

x

(a)

(b)

(c)

Figure 3-19: 1-D transient problems associated with (a) a plane wall, (b) a cylinder, and (c) a sphere, initially at a uniform temperature (T ini ) subjected to a step change in the convective boundary condition (h, T ∞ ).

The boundary conditions for the problem are: ∂T =0 ∂x x=0

(3-228)

∂T = h [T x=L − T ∞ ] −k ∂x x=L

(3-229)

T t=0 = T ini

(3-230)

Exact Solution. The solution derived in Section 3.5.3 for the plane wall problem shown in Figure 3-19(a) can be rearranged: θ˜ (x, ˜ Fo) =

∞

% & Ci cos(ζi x) ˜ exp −ζ2i Fo

(3-231)

i=1

where x, ˜ Fo, and θ˜ are the dimensionless position, Fourier number, and dimensionless temperature difference, deﬁned as: x x˜ = (3-232) L Fo = θ˜ =

tα L2

T − T∞ T ini − T ∞

(3-233) (3-234)

The dimensionless eigenvalues, ζi in Eq. (3-231), correspond to the product of the dimensional eigenvalues λi and the wall thickness (L) and are therefore provided by the multiple roots of the eigencondition: tan(ζi ) =

Bi ζi

(3-235)

where Bi is the Biot number: Bi =

hL k

(3-236)

398

Transient Conduction

The constants in Eq. (3-231) are: Ci =

2 sin(ζi ) ζi + cos(ζi ) sin(ζi )

(3-237)

The exact solution (the ﬁrst 20 terms of Eq. (3-231)) is programmed in EES in its dimen˜ as a function of the dimensionless sionless format (i.e., the dimensionless temperature, θ, independent variables Bi, Fo, and x) ˜ as the function planewall_T_ND. The solution can be accessed by selecting Function Info from the Options menu and selecting Transient Conduction from the pull-down menu. The different transient conduction functions that are available can be seen using the scroll-bar. The exact solutions for the cylinder and sphere are also programmed in EES as the functions cylinder_T_ND and sphere_T_ND. Dimensional versions of each function are also available; these functions (planewall_T, cylinder_T, and sphere_T) return the temperature at a given position and time as a function of the dimensional independent variables ( i.e., L, α, k, h, T i , T ∞ ). It is often important to calculate the total amount of energy transferred to the wall. An energy balance on the wall indicates that the total energy transfer (Q) is the difference between the energy stored in the wall (U) and the energy stored in the wall at its initial condition (U t=0 ): Q = U − U t=0

(3-238)

˜ by normalizing it against the maxiThe total energy transfer is made dimensionless (Q) mum amount of energy that could be transferred to the wall (Qmax ): Q˜ =

Q Qmax

(3-239)

The maximum energy transfer would occur if the process continued to t → ∞ and therefore the wall material equilibrates completely with the ﬂuid temperature at T∞ . Qmax = ρ c L Ac (T ∞ − T ini )

(3-240)

The dimensionless energy transfer is related to the dimensionless, volume average tem˜ according to: perature difference in the wall (θ) Q˜ = 1 − θ˜

(3-241)

where 1 θ˜ =

θ˜ dx˜

(3-242)

0

Substituting the exact solution, Eq. (3-231), into the Eqs. (3-241) and (3-242) leads to: Q˜ = 1 −

∞ i=1

Ci

sin(ζi ) exp(−ζ2i Fo) ζi

(3-243)

The solution for the dimensionless total energy transfer for the plane wall, cylinder, and sphere have been programmed in EES as the functions planewall_Q_ND, cylinder_ Q_ND, and sphere_Q_ND. The exact solutions for the dimensionless temperature and heat transfer are often presented graphically in a form that was initially published by Heisler (1947) and is now referred to as the Heisler charts; these charts were convenient when access to computer solutions was not readily available. Heisler charts typically include the dimensionless center temperature difference (θ˜x˜ =0 ) as a function of the Fourier number for various values of the inverse of the Biot number, Figure 3-20(a). The temperature information

Dimensionless center temperature

1 100

Bi −1

50 20

0.1

10 5 0.2 0.1 0.01

0.01

0.5

1

2

lines of constant Bi −1

0.001 0.1

1

1

10 Fourier number (a)

100

1000

10

100

x/L= 0.2

0.9

0.4

0.8 Ratio of θ/θx=0

0.7 0.6

0.6 0.5 0.4

0.8

0.3 0.2

x/L = 1.0 (surface)

0.9

0.1 0 0.01

0.1

1 Inverse Biot number

(b)

Dimensionless energy transfer

1

0.8

0.6

Bi=0.001 0.002 0.005 0.01 0.02 0.05 0.1

0.2 0.5 1 2 5 10

0.4

20 50

0.2

0 10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

2

Bi Fo (c) Figure 3-20: Heisler charts for a plane wall including (a) θ˜x=0 as a function of Fo for various values ˜ ˜ θ˜x˜ as a function of Bi−1 , and (c) Q˜ as a function of Bi2 Fo for various values of Bi. of Bi−1 , (b) θ/

400

Transient Conduction

presented in this ﬁgure corresponds to the position of the adiabatic boundary (x = 0). This is referred to as the center temperature because the center-line would be adiabatic if convection occured on both sides of the wall. The temperature at other locations can be obtained using Figure 3-20(b), which shows the ratio of the dimensionless temper˜ θ˜x˜ =0 ) as a funcature difference to the dimensionless center temperature difference (θ/ tion of the inverse of the Biot number for various dimensionless positions; note that the Fourier number does not effect this ratio. Figure 3-20(c) illustrates the dimensionless total energy transfer, Q˜ as a function of Bi2 Fo for various values of the Biot number. The Heisler charts shown in Figure 3-20 were generated using the solution programmed in EES. Figure 3-20(a) was generated by accessing the planewall_ND function using the following EES code (varying Fo for different values of invBi): invBi=10 theta hat 0=planewall T ND(0,Fo,1/invBi)

“Inverse Biot number” “Dimensionless center temperature”

Figure 3-20(b) was generated by accessing planewall_ND twice using the following EES code (varying x_hat for different values of invBi): x hat=0.2 “Dimensionless position” Fo=1 “Fourier No. (doesn’t effect ratio)” theta hat 0=planewall T ND(0,Fo,1/invBi) “Dimensionless center temperature” thetatotheta hat 0=planewall T ND(x hat,Fo,1/invBi)/theta hat 0 “Dimensionless center temperature”

Figure 3-20(c) was generated by accessing the planewall_Q_ND function (varying Bi2Fo for different values of Bi): Bi=50 Fo=Bi2Fo/Biˆ2 Q hat=planewall Q ND(Fo, Bi)

“Biot number” “Fourier number (doesn’t change ratio)” “Dimensionless energy transfer”

Approximate Solution for Large Fourier Number. When Fo 0.2, the series solution given by Eq. (3-231) can be adequately approximated using only the ﬁrst term. Only the ﬁrst eigenvalue (i.e., ζi , the ﬁrst root of Eq. (3-235)) is required to implement this approximate solution and therefore many textbooks will tabulate the ﬁrst eigenvalue as a function of the Biot number. The value of the ﬁrst eigenvalue as a function Biot number is shown in Figure 3-21 for the plane wall, as well as the cylinder and sphere solutions, discussed subsequently. Using ζ1 , the approximate solutions for the dimensionless temperature difference and dimensionless energy transfer become: θ˜ (x, ˜ Fo) ≈

% & 2 sin(ζ1 ) ˜ exp −ζ21 Fo cos(ζ1 x) ζ1 + cos(ζ1 ) sin(ζ1 )

⎡ ⎤ sin (ζ 1 ) 2sin (ζ 1 ) Q ≈ 1 − ⎢ exp ( −ζ 12 Fo ) ⎥ ζ cos ζ sin ζ ζ + ( ) ( ) 1 1 ⎦ 1 ⎣ 1

(3-244)

(3-245)

First dimensionless eigenvalue, ζ 1

3.5 Separation of Variables for Transient Problems

401

sphere

3

2.5

cylinder

2 plane wall

1.5 1 0.5 0 0.001

0.01

0.1

1 10 Biot number

100

1000

10000

Figure 3-21: First eigenvalue for use in the approximate solution to the problems illustrated in Figure 3-19.

The Cylinder Governing Equation and Boundary Conditions. The partial differential equation associated with the cylinder is derived by carrying out an energy balance on a control volume that is a differentially small cylindrical shell with thickness dr. ∂U ∂t

(3-246)

∂U ∂ q˙ r dr + ∂r ∂t

(3-247)

q˙ r = q˙ r+dr + or 0= The rate of conductive heat transfer is:

q˙ r = −k 2 π r L

∂T ∂r

(3-248)

where L is the length of the cylinder. The rate of energy storage is: ∂U ∂T = 2 π r L dr ρ c ∂t ∂t Substituting Eqs. (3-248) and (3-249) into Eq. (3-247) leads to: ∂ ∂T ∂T 0= −k 2 π r L dr + 2 π r L dr ρ c ∂r ∂r ∂t

(3-249)

(3-250)

or with k constant, α ∂ ∂T ∂T r = r ∂r ∂r ∂t

(3-251)

T t=0 = T ini

(3-252)

The initial condition is:

402

Transient Conduction

and the spatial boundary conditions are:

∂T =0 ∂r r=0

(3-253)

∂T = h (T r=rout − T ∞ ) −k ∂r r=rout

(3-254)

Exact Solution. The exact solution for the cylinder problem shown in Figure 3-19(b) can be derived using the methods discussed in Section 3.5.4: θ˜ (˜r, Fo) =

∞

% & Ci BesselJ (0, ζi r˜ ) exp −ζ2i Fo

(3-255)

i=1

where r˜ , Fo, and θ˜ are the dimensionless position, Fourier number, and dimensionless temperature difference, deﬁned as: r (3-256) r˜ = rout Fo =

θ˜ =

tα 2 rout

T − T∞ T ini − T ∞

(3-257)

(3-258)

The dimensionless eigenvalues, ζi in Eq. (3-255) are the roots of the eigencondition: ζi BesselJ (1, ζi ) − Bi BesselJ (0, ζi ) = 0

(3-259)

where Bi is the Biot number, deﬁned as: Bi =

h rout k

(3-260)

The constants in Eq. (3-255) are given by: Ci =

2 BesselJ (1, ζi ) ζi [BesselJ (0, ζi ) + BesselJ2 (1, ζi )] 2

(3-261)

The dimensionless energy transfer is related to the dimensionless, volume average tem˜ according to: perature in the cylinder (θ) Q˜ = 1 − θ˜

(3-262)

where 1 θ˜ = 2

θ˜ r˜ dr˜

(3-263)

0

Substituting the exact solution, Eq. (3-255), into Eqs. (3-262) and (3-263) leads to: Q˜ = 1 −

∞ i=1

Ci

2 BesselJ(1, ζi ) exp −ζ2i Fo ζi

(3-264)

3.5 Separation of Variables for Transient Problems

403

Approximate Solution for Large Fourier Number. The series solution for the cylinder, like the plane wall, can be adequately approximated using only the ﬁrst term when Fo > 0.2. The value of the ﬁrst eigenvalue as a function Biot number is shown in Figure 3-21. Using ζ1 , the approximate solutions for the dimensionless temperature difference and energy transfer become: θ˜ (r, ˜ Fo) =

Q˜ = 1 −

˜ 2 BesselJ(1, ζ1 ) BesselJ(0, ζ1 r) ζ1 [BesselJ2 (0, ζ1 ) + BesselJ2 (1, ζ1 )] 4 BesselJ2 (1, ζ1 )

ζ21 [BesselJ2 (0, ζ1 )

+ BesselJ (1, ζ1 )] 2

exp[−ζ21 Fo]

exp −ζ2i Fo

(3-265)

(3-266)

The Sphere Governing Equation and Boundary Conditions. The partial differential equation associated with the sphere is derived by carrying out an energy balance on a control volume that is a differentially small spherical shell with thickness r. ∂U ∂t

(3-267)

∂U ∂ q˙ r dr + ∂r ∂t

(3-268)

∂T ∂r

(3-269)

q˙ r = q˙ r+dr + or 0= The rate of conductive heat transfer is: q˙ r = −k 4 π r2 The rate of energy storage is:

∂T ∂U = 4 π r2 dr ρ c ∂t ∂t Substituting Eqs. (3-269) and (3-270) into Eq. (3-268) leads to: ∂T ∂ 2 ∂T −k 4 π r dr + 4 π r2 dr ρ c 0= ∂r ∂r ∂t

(3-270)

(3-271)

or α ∂ 2 ∂T ∂T r = r2 ∂r ∂r ∂t

(3-272)

T t=0 = T ini

(3-273)

The initial condition is:

and the spatial boundary conditions are:

∂T =0 ∂r r=0

(3-274)

∂T = h (T r=rout − T ∞ ) −k ∂r r=rout

(3-275)

404

Transient Conduction

Exact Solution. The exact solution for the spherical problem shown in Figure 3-19(c) is: ∞ & % sin(ζi r) ˜ Ci (3-276) exp −ζ2i Fo θ˜ (r, ˜ Fo) = ζi r˜ i=1

where r, ˜ Fo, and θ˜ are the dimensionless position, Fourier number, and dimensionless temperature difference, deﬁned as: r (3-277) r˜ = rout Fo = θ˜ =

tα 2 rout

T − T∞ T ini − T ∞

(3-278)

(3-279)

The dimensionless eigenvalues, ζi in Eq. (3-276) are the roots of the eigencondition: ζi cos (ζi ) + (Bi − 1) sin (ζi ) = 0

(3-280)

where Bi is the Biot number, deﬁned as: h rout k

(3-281)

2 [sin (ζi ) − ζi cos (ζi )] ζi − sin (ζi ) cos (ζi )

(3-282)

Bi = and the constants in Eq. (3-276) are: Ci =

The dimensionless energy transfer is related to the dimensionless, volume average tem˜ according to: perature in the sphere (θ) Q˜ = 1 − θ˜

(3-283)

where 1 θ˜ = 3

θ˜ r˜2 dr˜

(3-284)

0

Substituting the exact solution, Eq. (3-276) into Eqs. (3-283) and (3-284) leads to: Q˜ = 1 −

∞ i=1

Ci

3 [sin (ζi ) − ζi cos (ζi )] exp −ζ2i Fo 3 ζi

(3-285)

Approximate Solution for Large Fourier Number. The series solution for the sphere can be adequately approximated using only the ﬁrst term when Fo is > 0.2. The value of the ﬁrst eigenvalue as a function Biot number is shown in Figure 3-21. Using ζ1 , the approximate solutions for the dimensionless temperature distribution and energy transfer become: & % ˜ 2 [sin (ζ1 ) − ζ1 cos(ζ1 )] sin(ζ1 r) (3-286) exp −ζ21 Fo θ˜ (r, ˜ Fo) = ζ1 r˜ [ζ1 − sin(ζ1 ) cos(ζ1 )] 6 [sin(ζ1 ) − ζ1 cos(ζ1 )]2 exp −ζ21 Fo Q˜ = 1 − 3 ζ1 [ζ1 − sin(ζ1 ) cos(ζ1 )]

(3-287)

405

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN As part of a manufacturing process, long cylindrical pieces of material with radius r out = 5.0 cm are placed into a radiant oven. The initial temperature of the material is Tini = 20◦ C. The walls of the oven are maintained at a temperature of Twall = 750◦ C and the oven is evacuated so that the outer edge of the cylinder is exposed only to radiation heat transfer. The emissivity of the surface of the cylinder is ε = 0.95. The material is considered to be completely processed when the temperature everywhere is at least Tp = 250◦ C. The properties of the material are k = 1.4 W/m-K, ρ = 2500 kg/m3 , c = 700 J/kg-K. a) Is a lumped capacitance model appropriate for this problem? The known inputs are entered in EES: “EXAMPLE 3.5-1: Material in a Radiant Oven” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” r out=5.0 [cm]∗ convert(cm,m) T ini=converttemp(C,K,20[C]) T wall=converttemp(C,K,750[C]) e=0.95 T p=converttemp(C,K,250[C]) k=1.4 [W/m-K] rho=2500 [kg/mˆ3] c=700 [J/kg-K] alpha=k/(rho∗ c)

“radius” “initial temperature” “wall temperature” “emissivity” “processing temperature” “conductivity” “density” “speciﬁc heat capacity” “thermal diffusivity”

The effective heat transfer coefﬁcient associated with radiation (hr ad , discussed in Section 1.2.6) is: 2 hr ad = σ ε Twall + Ts2 (Twall + Ts ) where Ts is the surface temperature, which is not known but will certainly not be less then Tini and will likely not be much higher than Tp . An average of these values is used to evaluate the radiation heat transfer coefﬁcient. T s=(T ini+T p)/2 “average temperature used for surface temp.” h bar rad=sigma# ∗ e∗ (T wallˆ2+T sˆ2)∗ (T wall+T s) “effective heat transfer coefﬁcient due to radiation”

Because Twall is so much higher than Ts , the value of hr ad is not affected signiﬁcantly by the choice of Ts . Using hr ad , it is possible to calculate the Biot number: Bi =

hr ad r out k

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

3.5 Separation of Variables for Transient Problems

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

406

Transient Conduction

Bi=h bar rad∗ r out/k

“Biot number”

which provides a Biot number of 3.3. Therefore, a lumped capacitance model is not valid. b) How long will the processing require? Use an effective, radiation heat transfer coefﬁcient for this calculation. The dimensionless temperature difference at which the processing is complete can be calculated using Eq. (3-258): θ˜p =

theta hat p=(T p-T wall)/(T ini-T wall)

Tp − Twall Tini − Twall

“dimensionless temperature for processing”

The function cylinder_ND implements the exact solution for a cylinder, Eq. (3-255), and provides θ˜ given r˜, Fo, Bi. In this case, we know θ˜ = θ˜p at r˜ = 0 (the center of the cylinder, which will be the lowest temperature part of the material), and the value of Bi is also known. We would like to solve for the corresponding value of Fo (and therefore time). EES will solve this implicit equation in order to determine Fo: theta hat p=cylinder T ND(0, Fo, Bi) “implements the dimensionless solution to a cylinder”

Initially you are likely to obtain an error in EES related to the value of Fo being negative. This error condition can be easily overcome by setting appropriate limits (e.g., 0.001 to 1000) on the value of Fo in the Variable Information window. The Fourier number is used to calculate the processing time (tprocess ), according to Eq. (3-257): t pr ocess = Fo

t process=Fo∗ r outˆ2/alpha

2 r out α

“processing time”

which leads to a processing time of 678 s. c) What is the minimum amount of energy per unit length that is required to process the material? That is, how much energy would be required to bring the material to a uniform temperature of Tp ? An energy balance on the material shows that the minimum amount of energy required to bring the material to a uniform temperature (Qmin ) is: 2 L ρ c Tp − Tini Qmin = π r out

L=1 [m] Q min=pi∗ r outˆ2∗ L∗ rho∗ c∗ (T p-T ini)

407

“per unit length of material” “minimum amount of energy transfer required”

which leads to Qmin = 3.16 MJ. d) How much energy is actually required per unit length of material? ˜ is obtained using the cylinThe dimensionless energy transfer to the cylinder (Q) der_Q_ND function in EES: Q hat=cylinder Q ND(Fo, Bi)

“dimensionless heat transfer”

The dimensionless energy transfer is deﬁned according to Eq. (3-239) Q˜ =

Q Qmax

where Qmax is the energy transfer that occurs if the process is continued until the cylinder reaches equilibrium with the wall: 2 Qmax = ρ c L π r out (Twall − Tini )

Q max=pi∗ r outˆ2∗ L∗ rho∗ c∗ (T wall-T ini) “maximum amount of energy transfer that could occur” “actual energy transfer” Q=Q max∗ Q hat

which leads to Q = 5.62 MJ. e) The efﬁciency of the process (η) is deﬁned as the ratio of the minimum possible energy transfer required to process the material (from part (b)) to the actual energy transfer (from part (c)). Plot the efﬁciency of the process as a function of the radius of the material for various values of Twall . The efﬁciency is deﬁned as: η=

eta=Q min/Q

Qmin Q

“Process efﬁciency”

The efﬁciency is computed over a range of radii, rout , using a parametric table at several values of Twall and the results are shown in Figure 1.

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

3.5 Separation of Variables for Transient Problems

Transient Conduction

1 0.9 0.8 Efficiency

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

408

T wall

0.7

600°C

0.6 750°C 0.5 900°C 0.4 0.3 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Outer radius (m) Figure 1: Efﬁciency of the process as a function of the radius of the material for various values of the oven wall temperature.

Notice that the efﬁciency improves with reduced radius because the temperature gradients within the material are reduced and so the amount of energy wasted in ‘overheating’ the material toward the outer radius of the cylinder is reduced. Also, reducing the oven temperature tends to improve the efﬁciency for the same reason, but at the expense of increased processing time. The most efﬁcient process is associated with processing a very small amount of material (with small radius) very slowly (at low oven temperature); this is a typical result: very efﬁcient processes are not usually very practical.

3.5.3 Separation of Variables Solutions in Cartesian Coordinates The solution to 1-D transient problems using separation of variables is illustrated in the context of the problem shown in Figure 3-22. A plane wall is initially at a uniform temperature, T ini = 100 K, when the surface is exposed to a step change in the surrounding ﬂuid temperature to T ∞ = 200 K. The average heat transfer coefﬁcient between the surface and the ﬂuid is h = 200 W/m2 -K. The wall has thermal diffusivity α = 5 × 10−6 m2 /s

initial temperature, Tini = 100 K L = 5 cm

T∞ = 200 K 2 h = 200 W/m -K x k = 10 W/m-K α = 5x10-6 m2/s Figure 3-22: Plane wall initially at a uniform temperature that is exposed to convection at the surface.

3.5 Separation of Variables for Transient Problems

409

and thermal conductivity k = 10 W/m-K. The thickness of the wall is L = 5 cm. The known information is entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” k=10 [W/m-K] alpha=5e-6 [mˆ2/s] T ini=100 [K] T inﬁnity=200 [K] h bar=200 [W/mˆ2-K] L=5.0 [cm]∗ convert(cm,m)

“conductivity” “thermal diffusivity” “initial temperature” “ﬂuid temperature” “heat transfer coefﬁcient” “thickness”

The governing partial differential equation for this situation is identical to the one derived for a semi-inﬁnite body in Section 3.3.3: ∂2T ∂T −α 2 =0 ∂t ∂x

(3-288)

The partial differential equation is ﬁrst order in time and therefore it requires one boundary condition with respect to time; this is the initial condition: T t=0 = T ini

(3-289)

Equation (3-288) is second order in space and therefore two boundary conditions are required with respect to x. At the adiabatic wall, the temperature gradient must be zero: ∂T =0 (3-290) ∂x x=0 An interface energy balance at the surface (x = L) balances conduction with convection: ∂T = h [T x=L − T ∞ ] −k (3-291) ∂x x=L The steps required to solve 1-D transient problems using separation of variables follow naturally from those presented in Section 2.2.2 for 2-D steady-state problems. Requirements for using Separation of Variables In order to apply separation of variables, it is necessary that the partial differential equation and all of the boundary condition be linear; these criteria are satisﬁed for the problem shown in Figure 3-22 by inspection of Eqs. (3-288) through (3-291). It is also necessary that the partial differential equation and both boundary conditions in space be homogeneous (there is only one direction associated with a 1-D transient problem and therefore it must be homogeneous). The partial differential equation, Eq. (3-288) is homogeneous and the boundary condition associated with the adiabatic wall, Eq. (3-290), is also homogeneous. However, the convective boundary condition at x = L, Eq. (3-291), is not homogeneous. Fortunately, it is possible to transform this problem, as discussed in Section 2.2.3. The temperature difference relative to the ﬂuid temperature is deﬁned: θ = T − T∞

(3-292)

410

Transient Conduction

The transformed partial differential equation becomes: α

∂θ ∂2θ = 2 ∂x ∂t

(3-293)

and the boundary conditions become: θt=0 = T ini − T ∞

(3-294)

∂θ =0 ∂x x=0

(3-295)

∂θ = h θx=L −k ∂x x=L

(3-296)

Notice that both spatial boundary conditions for the transformed problem, Eqs. (3-295) and (3-296), are homogeneous and therefore it will be possible to obtain a set of orthogonal eigenfunctions in x. The initial condition, Eq. (3-294), is not homogeneous but it doesn’t have to be. (Recall that the boundary conditions in one direction of the 2-D conduction problems, considered in Sections 2.2 and 2.3, did not have to be homogeneous.) As an aside, if the dimension of the problem were extended to inﬁnity (L → ∞) then the resulting spatial boundary condition, Eq. (3-296), would not be homogeneous and therefore the semiinﬁnite problem cannot be solved using separation of variables. One of the reasons that the Laplace transform is a valuable tool for heat transfer problems is that it can be used to analytically solve semi-inﬁnite problems. Separate the Variables The temperature difference, θ, is a function of axial position and time. The separation of variables approach assumes that the solution can be expressed as the product of a function only of time, θt(t), and a function only of position, θX(x): θ (x, t) = θX (x) θt (t)

(3-297)

Substituting Eq. (3-297) into Eq. (3-293) leads to: α θt

dθt d2 θX = θX dx2 dt

(3-298)

Equation (3-298) is divided through by α θXθt in order to obtain: d2 θX dθt dx2 = dt θX α θt

(3-299)

Both sides of Eq. (3-299) must be equal to a constant in order for the solution to be valid at an arbitrary time and position. We know from our experience with 2-D separation of variables that Eq. (3-299) will lead to two ordinary differential equations (one in x and the other in t). Furthermore, the ODE in x must lead to a set of eigenfunctions. This foresight motivates the choice of a negative constant −λ2 : dθt d2 θX 2 dx = dt = −λ2 θX α θt

(3-300)

3.5 Separation of Variables for Transient Problems

411

The ODE in x that is obtained from Eq. (3-300) is: d2 θX + λ2 θX = 0 dx2

(3-301)

which is solved by sines and cosines. The ODE in time suggested by Eq. (3-300) is: dθt + λ2 α θt = 0 dt

(3-302)

Solve the Eigenproblem It is always necessary to address the solution to the homogeneous sub-problem (in this problem, θX) before moving on to the non-homogeneous sub-problem (for θt). The homogeneous sub-problem is called the eigenproblem. The general solution to Eq. (3-301) is: θX = C1 sin(λ x) + C2 cos(λ x)

(3-303)

where C1 and C2 are unknown constants. Substituting Eq. (3-297) and Eq. (3-303) into the spatial boundary condition at x = 0, Eq. (3-295), leads to: ⎡ ⎤ dθX ∂θ = θt = θt ⎣C1 λ cos(λ 0) −C2 λ sin(λ 0)⎦ = 0 ∂x x=0 dx x=0 =1

(3-304)

=0

or θt C1 λ = 0

(3-305)

which can only be true (for a non-trivial solution) if C1 = 0: θX = C2 cos(λ x)

(3-306)

Substituting Eq. (3-297) into the spatial boundary condition at x = L, Eq. (3-296), leads to: dθX = h θt θX x=L (3-307) −k θt dx x=L or dθX = h θX x=L −k dx x=L

(3-308)

Substituting Eq. (3-306) into Eq. (3-308) leads to: k C2 λ sin(λ L) = h C2 cos(λ L)

(3-309)

Equation (3-309) provides the eigencondition for the problem, which deﬁnes multiple eigenvalues: sin(λ L) h = cos(λ L) kλ

(3-310)

412

Transient Conduction

or, multiplying and dividing the right side of Eq. (3-310) by L: hL sin(λ L) = cos(λ L) kλL

(3-311)

The dimensionless group h L/k is the Biot number (Bi) that was encountered in Section 1.6 and elsewhere. In this context, the Biot number represents the ratio of the resistance to conduction heat transfer within the wall to convective heat transfer from the surface. Writing Eq. (3-311) in terms of the Biot number leads to: tan(λ L) =

Bi λL

(3-312)

Equation (3-312) is analogous to the eigenconditions for the problems considered in Section 2.2. There are an inﬁnite number of values of λ that will satisfy Eq. (3-312). However, Eq. (3-312) provides an implicit rather than an explicit equation for these eigenvalues. This situation was encountered previously in EXAMPLE 2.3-1; the eigencondition involved the zeroes of the Bessel function. In EXAMPLE 2.3-1, the process of calculating the eigenvalues was automated by using EES to determine the roots of the eigencondition within speciﬁed ranges. We can use the same procedure for any problem with an implicit eigencondition, such as is given by Eq. (3-312). Figure 3-23 illustrates the left and right sides of Eq. (3-312) as a function of λL for the case where Bi = 1.0. tan( λL) Bi/( λL )

Tan (λL) and Bi/(λL)

1.25 1 0.75 0.5

λ1 L 0.25

λ2L

0 0

π/2

π

3π/2

λ3L 2π

λ4L 5π/2

λL

3π

7π/2

λ5L 4π

9π/2

Figure 3-23: The left and right sides of the eigencondition equation for Bi = 1.0 ; the intersections correspond to eigenvalues for the problem, λi L.

Figure 3-23 shows that each successive value of λi L can be found in a well-deﬁned interval; λ1 L lies between 0 and π/2, λ2 L lies between π and 3π/2, etc.; this will be true regardless of the value of the Biot number. The number of terms to use in the solution is speciﬁed and arrays of appropriate guess values and upper and lower bounds for each eigenvalue are generated.

3.5 Separation of Variables for Transient Problems

413

Nterm=10 [-] “number of terms to use in the solution” “Setup guess values and lower and upper bounds for eigenvalues” duplicate i=1,Nterm lowerlimit[i]=(i-1)∗ pi upperlimit[i]=lowerlimit[i]+pi/2 guess[i]=lowerlimit[i]+pi/4 end

The eigencondition is programmed using a duplicate loop: Bi=h bar∗ L/k “Identify eigenvalues” duplicate i=1,Nterm tan(lambdaL[i])=Bi/lambdaL[i] lambda[i]=lambdaL[i]/L end

“Biot number”

“eigencondition” “eigenvalue”

The solution obtained at this point will provide the same value for all of the eigenvalues. (Select Arrays from the Windows menu and you will see that each value of the array lambdaL[i] is the same, probably equal to whichever root of Eq. (3-312) lies closest to the default guess value of 1.0.) The interval for each eigenvalue can be controlled by selecting Variable Info from the Options menu. Deselect the Show array variables check box at the upper left so that the arrays are collapsed to a single entry and use the guess[ ], upperlimit[ ], and lowerlimit[ ] arrays to control the process of identifying the eigenvalues in the array lambdaL[ ]. The solution will now identify the ﬁrst 10 eigenvalues of the problem (more can be obtained by changing the value of Nterm). The number of terms required depends on the time and position where you need the solution; this is discussed at the end of this section. At this point, each of the eigenfunctions of the problem have been obtained. The ith eigenfunction is: θX i = C2,i cos(λi x)

(3-313)

where λi is the ith eigenvalue, identiﬁed by the eigencondition: tan(λi L) =

Bi λi L

(3-314)

Solve the Non-homogeneous Problem for each Eigenvalue The solution to the non-homogeneous ordinary differential equation corresponding to the ith eigenvalue, Eq. (3-302): dθti + λ2i α θti = 0 dt

(3-315)

θti = C3,i exp(−λ2i α t)

(3-316)

is

where C3,i is an undetermined constant.

414

Transient Conduction

Obtain a Solution for each Eigenvalue According to Eq. (3-297), the solution associated with the ith eigenvalue is: θi = θX i θti = Ci cos(λi x) exp −λ2i α t

(3-317)

where the constants C2,i and C3,i have been combined to form a single undetermined constant Ci . Equation (3-317) will, for any value of i, satisfy the governing differential equation, Eq. (3-293), throughout the domain and satisfy all of the boundary conditions in the x-direction, Eqs. (3-295) and (3-296). It is worth checking that the solution has these properties using Maple before proceeding. Enter the solution as a function of x and t: > restart; > theta:=(x,t)->C∗ cos(lambda∗ x)∗ exp(-lambdaˆ2∗ alpha∗ t); θ := (x, t) → C cos(λx)e(−x αt) 2

Verify that it satisﬁes Eq. (3-295): > eval(diff(theta(x,t),x),x=0); 0

and Eq. (3-296): > -k∗ eval(diff(theta(x,t),x),x=L)-h_bar∗ theta(L,t); 2

k C sin(λ L) λ e(−λ

αt)

2

− h bar C cos(λL) e(−λ

αt)

> simplify(%); 2

C e(−λ

αt)

(k sin(λ L) λ − h bar cos(λL))

Note that the eigencondition cannot be enforced in Maple using the assume command as it was in the problems in Section 2.2. However, it is clear that the eigencondition, Eq. (3-310), requires that the term within the parentheses must be zero and therefore the second spatial boundary condition will be satisﬁed for any eigenvalue. Finally, verify that the partial differential equation, Eq. (3-293), is satisﬁed: > alpha∗ diff(diff(theta(x,t),x),x)-diff(theta(x,t),t); 0

Create the Series Solution and Enforce the Initial Condition Because the partial differential equation is linear, the sum of the solution θi for each eigenvalue, Eq. (3-317), is itself a solution: θ=

∞ i=1

θi =

∞ i=1

Ci cos(λi x) exp −λ2i α t

(3-318)

3.5 Separation of Variables for Transient Problems

415

The ﬁnal step of the problem selects the constants so that the series solution satisﬁes the initial condition, Eq. (3-294): θt=0 =

∞

Ci cos(λi x) = T ini − T ∞

(3-319)

i=1

We found in Section 2.2 that the eigenfunctions are orthogonal to one another. The property of orthogonality ensures that when any two, different eigenfunctions are multiplied together and integrated from one homogeneous boundary to the other, the result will necessarily be zero. Each side of Eq. (3-319) is multiplied by cos(λ j x) and integrated from x = 0 to x = L: ∞

L cos(λi x) cos(λ j x)dx =

Ci

i=1

L

0

(T ini − T ∞ ) cos(λ j x)dx

(3-320)

0

The property of orthogonality ensures that the only term on the left side of Eq. (3-320) that is not zero is the one for which j = i: L

L cos2 (λi x)dx = (T ini − T ∞ )

Ci 0

cos(λi x) dx 0

Integral 1

(3-321)

Integral 2

The integrals in Eq. (3-321) can be evaluated conveniently using either integral tables or Maple: > Integral1:=int((cos(lambda∗ x))ˆ2,x=0..L); Integral1 := > Integral2:=int(cos(lambda∗ x),x=0..L);

1 cos(λL) sin(λL) + λL 2 λ

Integral 2 :=

sin(λL) λ

Substituting these results into Eq. (3-321) leads to: Ci

(T ini − T ∞ ) sin(λi L) [cos(λi L) sin(λi L) + λi L] = 2 λi λi

(3-322)

or Ci =

2 (T ini − T ∞ ) sin(λi L) [cos(λi L) sin(λi L) + λi L]

(3-323)

The result is used to evaluate each constant in EES: “Evaluate constants” duplicate i=1,Nterm C[i]=2∗ (T_ini-T_inﬁnity)∗ sin(lambda[i]∗ L)/(cos(lambda[i]∗ L)∗ sin(lambda[i]∗ L)+lambda[i]∗ L) end

416

Transient Conduction

The solution at a speciﬁc time and position is evaluated using Eq. (3-318): x=0.01 [m] time=1000 [s] duplicate i=1,Nterm theta[i]=C[i]∗ cos(lambda[i]∗ x)∗ exp(-lambda[i]ˆ2∗ alpha∗ time) end T=T_inﬁnity+sum(theta[1..Nterm])

The solution to this bounded, 1-D transient problem is often expressed in terms of a dimensionless position (˜x), deﬁned as: x L

x˜ =

(3-324)

A dimensionless time can also be deﬁned. There is no characteristic time that appears in the problem statement. However, the diffusive time constant, discussed in Section 3.3.1, provides a convenient characteristic time for the problem that is related to the time required for a thermal wave to penetrate from the surface of the wall to the adiabatic boundary. The diffusive time constant is: τdiff =

L2 4α

(3-325)

and so an appropriate dimensionless time would be: t˜ =

t τdiff

=

4tα L2

(3-326)

The dimensionless time is typically referred to as the Fourier number (Fo) and, in most textbooks, the factor of 4 is removed from the deﬁnition: Fo =

tα L2

(3-327)

The dimensionless position and Fourier number can be used to more conveniently specify the position and time in the EES code: x hat=0.5 [-] x=x hat∗ L Fo=0.2 [-] time=Fo∗ Lˆ2/alpha duplicate i=1,Nterm theta[i]=C[i]∗ cos(lambda[i]∗ x)∗ exp(-lambda[i]ˆ2∗ alpha∗ time) end T=T inﬁnity+sum(theta[1..Nterm])

“dimensionless position” “position” “Fourier number” “time”

Figure 3-24 illustrates the temperature as a function of dimensionless position for various values of the Fourier number. The transient response of a plane wall (as well as the response of a cylinder and sphere) that is initially at a uniform temperature and is subjected to a convective boundary condition can be accessed from the EES library of heat transfer functions. Select Function Info from the Options menu and select Transient Conduction from the

3.5 Separation of Variables for Transient Problems

417

210 Fo=5.0

Temperature (K)

190 170 150

Fo =2.0

Fo=1.0 Fo=0.1

130

Fo=0.05

Fo=0.5 110 90 0

Fo=0.02

Fo=0.2 Fo=0.005 0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L

0.8

0.9

1

Figure 3-24: Temperature as a function of dimensionless position for various values of the Fourier number (dimensionless time).

pulldown menu in order to access these solutions. Note that the solutions programmed in EES are the ﬁrst 20 terms of the inﬁnite series solution (i.e., the ﬁrst 20 terms of Eq. (3-318) for a plane wall). Access and use of these library functions is discussed in more detail in Section 3.5.2. Limit Behaviors of the Separation of Variables Solution It is worthwhile spending some time understanding the behavior of the separation of variables solution in order to reinforce some of the concepts that were introduced in earlier sections. Figure 3-24 shows that for Fo less than about 0.20, the wall behaves as a semi-inﬁnite body. For Fo < 0.2 (approximately), the semi-inﬁnite body solution listed in Table 3-2 or accessed from the EES function SemiInf3 will provide accurate results. Figure 3-25 illustrates the temperature at x˜ = 0.25 as a function of Fo using the separation of variables solution developed in this section (100 terms are used to evaluate the series, so it is close to being exact) and obtained from the SemiInf3 function: T_semiinf=SemiInf3(T_ini,T_inﬁnity,h_bar,k,alpha,L-x,time) “temperature evaluated using the semi-inﬁnite body assumption”

Note that the semi-inﬁnite body solution is expressed in terms of the distance from the surface that is exposed to the ﬂuid whereas the separation of variables solution is expressed in terms of the distance from the adiabatic surface; therefore, the coordinate transformation (L-x) is required in the call to the SemiInf3 function. Figure 3-25 shows that the 100 term separation of variables solution agrees extremely well with the semi-inﬁnite body solution until Fo reaches about 0.20, at which point the thermal wave encounters the adiabatic wall and the problem becomes bounded. The solution to the problem, Eq. (3-318), expressed in terms of x˜ and Fo, is: θ (x, ˜ Fo) =

∞ i=1

Ci cos(λi L x) ˜ exp[− (λi L)2 Fo]

(3-328)

418

Transient Conduction

150 100 term separation of variables solution

Temperature (K)

140 130 120 110 semi-infinite body solution 100 90 single term separation of variables solution 80 0

0.1

0.2

0.3

0.4 0.5 0.6 Fourier number

0.7

0.8

0.9

1

Figure 3-25: Temperature as a function of Fo at x˜ = 0.25 evaluated using the separation of variables solution with 100 terms, with 1 term, and using the semi-inﬁnite body function.

where absolute value is ≤1

Ci =

2(T ini − T ∞ ) sin(λi L) cos(λi L) sin(λi L) + λi L

absolute value is ≤1 regardless of i

(3-329)

grows with i

It is interesting to look at how quickly the inﬁnite series converges to a solution (i.e., how many terms are actually required?). The value of the constants will decrease in magnitude as they increase in index. Examination of Eq. (3-329) shows that the denominator increases as i increases. Therefore, regardless of x, ˜ Fo, and Bi, the terms in Eq. (3-328) corresponding to larger values of i will have a decreasing value. This can also be seen by examining the array theta[i] in the EES solution. Further, examination of Eq. (3-328) shows that the value of each term in the series decays in time as exp(−(λi L)2 Fo). Because the value of λi L increases with i, this decay will occur much more rapidly for the terms with larger values of i. As a result, as the Fourier number increases, fewer and fewer terms are required to achieve an accurate solution and eventually a single term will sufﬁce. Many textbooks tabulate the ﬁrst constant and ﬁrst eigenvalue in Eq. (3-328) as a function of Bi and advocate using the “single-term approximation” for large values of Fourier number. ˜ exp[− (λ1 L)2 Fo] if Fo > 0.2 θ(x, ˜ Fo) ≈ C1 cos (λ1 L x)

(3-330)

The single term solution is also shown in Figure 3-25 and is clearly quite accurate for Fo > 0.2. Finally, it is worth revisiting the concept of the Biot number. If the Biot number is very small, then we would expect that temperature gradients within the wall will be negligible and the lumped capacitance model, discussed in Section 3.1, will be accurate. The solution for a lumped capacitance subjected to a step-change in the ﬂuid temperature is provided in Table 3-1: t (3-331) T = T ∞ + (T ini − T ∞ ) exp − τlumped

3.5 Separation of Variables for Transient Problems

419

where the lumped capacitance time constant (τlumped ) is: τlumped = Rconv C = Substituting Eq. (3-332) into Eq. (3-331) leads to:

Lρc

(3-332)

h

ht T = T ∞ + (T ini − T ∞ ) exp − Lρc

(3-333)

Multiplying and dividing the argument of the exponential by k L allows it to be rewritten as: ⎞ ⎛ ⎜ hL tk ⎟ ⎟ T = T ∞ + (T ini − T ∞ ) exp ⎜ ⎝− k L2 ρ c ⎠ = T ∞ + (T ini − T ∞ ) exp (−Bi Fo) Bi

Fo

(3-334) The lumped capacitance solution as a function of the product of the Biot number and the Fourier number, Eq. (3-334), is implemented in EES: T_lumped=T_inﬁnity+(T_ini-T_inﬁnity)∗ exp(-Bi∗ Fo) “temperature evaluated using the lumped capacitance assumption”

Figure 3-26 illustrates the temperature predicted by the lumped capacitance model, Eq. (3-334), as a function of the product Fo Bi. The temperature at the center of the wall (ˆx = 0) predicted by the separation of variables solution, Eq. (3-328), with 100 terms is also shown in Figure 3-26 for various values of the Biot number. 210

Temperature (K)

190 170

100 term separation of variables solution lumped capacitance solution Bi=0.1 Bi=0.2 Bi=0.5

Bi=1 Bi=2

150 Bi=5 130 Bi=10 110 90 0

Bi=20 0.5 1 1.5 2 2.5 3 3.5 Product of the Biot number and the Fourier number

4

Figure 3-26: Temperature at the center of the wall as a function of the product Fo Bi predicted by the lumped capacitance solution and by the separation of variables solution with 100 terms at x = 0.

Notice that the separation of variables solution approaches the lumped capacitance solution for Bi < 0.2. The separation of variables solution is an exact but complex method for analyzing a transient problem. However, Figure 3-25 and Figure 3-26 show that the solution limits to more simple expressions under certain conditions.

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

420

Transient Conduction

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED) The separation of variables technique provides a tool that can be used to provide a more quantitative solution to the problem posed in EXAMPLE 3.3-1, which is re-stated here. Figure 1(a) shows a metal wall that separates two tanks of liquid at different temperatures, Thot = 500 K and Tcold = 400 K; initially the wall is at steady state and therefore it has the linear temperature distribution shown in Figure 1(b). The thickness of the wall is th = 0.8 cm and its cross-sectional area is Ac = 1.0 m2 . The properties of the wall material are ρ = 8000 kg/m3 , c = 400 J/kg-K, and k = 20 W/m-K. The heat transfer coefﬁcient between the wall and the liquid in either tank is hliq = 5000 W/m2 -K. At time, t = 0, both tanks are drained and then exposed to gas at Tgas = 300 K, as shown in Figure 1(c). The heat transfer coefﬁcient between the walls and the gas is hgas = 100 W/m2 -K. Assume that the process of draining the tanks and ﬁlling them with air occurs instantaneously so that the wall has the linear temperature distribution shown in Figure 1(b) at time t = 0. th = 0.8 cm liquid at Tcold = 400 K hliq = 5000 W/m2 -K

liquid at Thot = 500 K 2 hliq = 5000 W/m -K x k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K (a)

Temperature (K) 500 475 450 425 400 0.8 Position (cm)

0

(b) th = 0.8 cm gas at Tgas = 300 K 2 h gas = 100 W/m -K

x

gas at Tgas = 300 K hgas = 100 W/m2 -K k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K (c)

Figure 1: (a) Tank wall exposed to liquid at two temperatures with (b) a linear steady state initial temperature distribution, when (c) at time t = 0 both walls are exposed to low temperature gas.

421

a) In EXAMPLE 3.3-1, we calculated a diffusive and lumped time constant for this problem (τdiff = 2.6 s and τlumped = 130 s) and used these values to sketch the expected temperature distribution at various times. Use the method of separation of variables to obtain a more precise solution. Note that this problem can not be solved using the planewall T function in EES because of the non-uniform initial temperature distribution. The separation of variables method must be applied. The known information is entered in EES: “EXAMPLE 3.5-2: Transient Response of a Tank Wall (Revisited)” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” k=20 [W/m-K] c=400 [J/kg-K] rho=8000 [kg/mˆ3] T cold=400 [K] T hot=500 [K] h bar liq=5000 [W/mˆ2-K] th=0.8[cm]∗ convert(cm,m) A c=1 [mˆ2] h bar gas=100 [W/mˆ2-K] T gas=300 [K]

“thermal conductivity” “speciﬁc heat capacity” “density” “cold ﬂuid temperature” “hot ﬂuid temperature” “liquid-to-wall heat transfer coefﬁcient” “wall thickness” “wall area” “gas-to-wall heat transfer coefﬁcient” “gas temperature”

The steady state temperature distribution provides the initial condition for the transient problem: Tt =0 = Tx=0,t =0 + (Tx=t h,t =0 − Tx=0,t =0 )

x th

(1)

where Tx=0,t =0 and Tx=t h,t =0 are the steady-state temperatures at either edge of the wall (Figure 1(b)): Tx=0,t =0 = Tcold +

(Thot − Tcold ) Rconv,liq 2 Rconv,liq + Rcond

and Tx=t h,t =0 = Tcold +

(Thot − Tcold ) (Rconv,liq + Rcond ) 2 Rconv,liq + Rcond

where Rconv,liq and Rcond are the convective and conductive resistances: Rconv,liq =

Rcond =

1 hliq Ac th k Ac

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

422

Transient Conduction

“Initial condition” “convection resistance with liquid” R conv liq=1/(h bar liq∗ A c) R cond=th/(k∗ A c) “conduction resistance” T x0 t0=T cold+(T hot-T cold)∗ R conv liq/(2∗ R conv liq+R cond) “initial temperature of cold side of wall” T xth t0=T cold+(T hot-T cold)∗ (R conv liq+R cond)/(2∗ R conv liq+R cond) “initial temperature of hot side of wall”

The governing differential equation is derived in Section 3.5.3: ∂T ∂ 2T =0 −α ∂t ∂x2 The spatial boundary conditions are provided by interface balances at either edge of the wall: ∂T hgas (Tgas − Tx=0 ) = −k ∂x x=0

∂T −k = hgas (Tx=t h − Tgas ) ∂ x x=t h The solution follows the steps that were outlined in Section 3.5.3. The problem as stated cannot be solved using separation of variables because neither of the two spatial boundary conditions are homogeneous. However, the problem can be transformed by deﬁning the temperature difference relative to the gas temperature: θ = T − Tgas The transformed partial differential equation becomes: ∂ 2θ ∂θ = 2 ∂x ∂t and the initial and boundary conditions become: α

θt =0 = (Tx=0,t =0 − Tgas ) + (Tx=t h,t =0 − Tx=0,t =0 )

hgas θx=0

∂θ =k ∂ x x=0

∂θ −k = hgas θx=t h ∂ x x=t h

(2)

x th

(3)

(4)

(5)

Both spatial boundary conditions, Eqs. (4) and (5), are homogeneous and therefore separation of variables can be used on the transformed problem. The solution is assumed to be the product of θX , a function of x, and θt , a function of t. The two ordinary differential equations that result are: d 2θ X + λ2 θ X = 0 d x2

(6)

dθt + λ2 α θ t = 0 dt

(7)

and

423

The eigenproblem (the problem in x) will be solved ﬁrst. The general solution to Eq. (6) is: θ X = C 1 sin (λ x) + C 2 cos (λ x)

(8)

Substituting Eq. (8) into the boundary condition at x = 0, Eq. (4), leads to: hgas [C 1 sin (λ 0) + C 2 cos (λ 0)] = k [C 1 λ cos (λ 0) − C 2 λ sin (λ 0)] or C 2 hgas = k C 1 λ Therefore, Eq. (8) can be written as: θ X = C1

sin (λ x) +

kλ

cos (λ x)

hgas

(9)

Substituting Eq. (9) into the boundary condition at x = th, Eq. (5), leads to the eigencondition for the problem: −k

λ cos (λ th) −

k λ2 hgas

sin (λ th) = hgas sin (λ th) +

kλ hgas

cos (λ th)

(10)

The eigenvalues are identiﬁed automatically using EES, as discussed in Section 3.5.3. The left side of Eq. (10) is moved to the right side in order to identify a residual, Res, that must be zero at each eigenvalue: Res = sin (λ th) +

k (λ th) hgas th

cos (λ th) +

k (λ th) hgas th

cos (λ th) −

k (λ th) hgas th

sin (λ th) (11)

Equation (11) can be simpliﬁed by deﬁning the Biot number in the usual way: Bi =

hgas th k

which leads to: Res = sin (λ th) +

(λ th) (λ th) (λ th) cos (λ th) + cos (λ th) − sin (λ th) Bi Bi Bi

Equation (12) is programmed in EES: “Eigenvalues” Bi=h bar gas∗ th/k “Biot number” Res=sin(lambdath)+lambdath∗ cos(lambdath)/Bi+lambdath∗ (cos(lambdath)& -lambdath∗ sin(lambdath)/Bi)/Bi “Residual”

(12)

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

Transient Conduction

and used to generate Figure 2, which shows the residual as a function of λ th. 10,000 8,000 6,000 4,000 Residual

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

424

2,000 0 -2,000 -4,000 -6,000 -8,000

-10,000

0

π/2

π

3 /2

π2 π

5π/2 λ th

π3

7π /2 π4

9 /2

Figure 2: Residual as a function of λ th. Notice that the residual crosses zero between every interval of π.

Figure 2 shows that the roots of Eq. (12) lie in each interval of π . Therefore, upper and lower bounds and appropriate guess values can be identiﬁed for each eigenvalue and assigned to arrays that are subsequently used to constrain each value of λ th (the entries in the array lambdath[ ]) in the Variable Information window: “Eigenvalues” Bi=h bar gas∗ th/k “Biot number” {Res=sin(lambdath)+lambdath∗ cos(lambdath)/Bi+lambdath∗ (cos(lambdath)& -lambdath∗ sin(lambdath)/Bi)/Bi “Residual”} Nterm=10 duplicate i=1,Nterm lowerlimit[i]=(i-1)∗ pi “lower limit” upperlimit[i]=i∗ pi “upper limit” guess[i]=(i-0.5)∗ pi “guess value” sin(lambdath[i])+lambdath[i]∗ cos(lambdath[i])/Bi+lambdath[i]∗ (cos(lambdath[i])& “eigencondition” -lambdath[i]∗ sin(lambdath[i])/Bi)/Bi=0 end

The EES code will identify each of the eigenvalues. The solution for each eigenvalue is therefore: k λi cos (λi x) θX i = C 1,i sin (λi x) + (13) hgas The solution to the ordinary differential equation in time, Eq. (7), for each eigenvalue is: θ ti = C 3,i exp −λi2 α t

425

and therefore the solution for each eigenvalue is: k λi θi = θ xi θ ti = C i sin (λi x) + cos (λi x) exp −λi2 α t hgas The sum of the solutions for each eigenvalue is the series solution to the problem: ∞ ∞ k λi θi = C i sin (λi x) + cos (λi x) exp −λi2 α t (14) θ= hgas i=1 i=1 ˜ and The series solution can be expressed in terms of dimensionless position (x) Fourier number (Fo ): θ=

& % λi th ˜ + ˜ exp − (λi th)2 F o C i sin (λi th x) cos (λi th x) Bi i=1

∞

(15)

where x˜ and Fo are deﬁned as: x˜ = Fo =

x th αt th2

The constants must be selected so that the initial condition, Eq. (3), is satisﬁed: θt =0

∞

λi th ˜ = Ci cos (λi th x) Bi i=1 = Tx=0,t =0 − Tgas + Tx=t h − Tss,x=0 xˆ ˆ + sin (λi th x)

(16)

The eigenfunctions must be orthogonal. Therefore, multiplying both sides of Eq. (16) by the j th eigenfunction and integrating from x = 0 to x = th (or x˜ = 0 to x˜ = 1) leads to: ∞

1

i=1 0

Ci

˜ + sin (λi th x)

λ j th ˜ ˜ cos λ j th x d x˜ sin λ j th x + Bi

λi th ˜ cos (λi th x) Bi

% & λ j th Tx=0,t =0 − Tgas + Tx=t h,t =0 − Tx=0,t =0 x˜ sin λ j th x˜ + cos λ j th x˜ d x˜ Bi

1

= 0

The only term on the left side of the equation that does not integrate to zero is i = j and therefore: 1

2

λi th ˜ cos (λi th x) Bi (Integral1)i

˜ + sin (λi th x)

Ci 0

1

=

d x˜ (17)

% & λi th ˜ + ˜ d x˜ cos (λi th x) Tx=0,t =0 − Tgas + Tx=t h,t =0 − Tx=0,t =0 x˜ sin (λi th x) Bi 0 (Integral 2)i

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

426

Transient Conduction

The integrals in Eq. (17) seem imposing, but they can be accomplished relatively easily using Maple and then copied into EES for evaluation. Equation (17) is written in terms of the two integrals: C i Integral 1 i = Integral 2 i where

Integral 1 i =

1

2

˜ + sin (λi th x) 0

λi th ˜ cos (λi th x) Bi

d x˜

and Integral 2 i =

1

%

(Tx=0,t =0 − Tgas ) + (Tx=t h,t =0 − Tx=0,t =0 )x˜

&

0

λi th ˜ + ˜ d x˜ × sin(λi th x) cos(λi th x) Bi These integrals are entered in Maple: > restart; > Integral_1[i]:=int((sin(lambdath[i]∗ x_hat)+lambdath[i]∗ cos(lambdath[i]∗ x_hat)/Bi)ˆ2,x_hat=0..1); Integral 1i := 12 (2 Bi lambdathi − Bi2 cos(lambdathi ) sin(lambdathi ) + Bi 2 (lambdathi ) − 2 Bi lambdathi cos(lambdathi )2 + lambdathi2 cos(lambdathi ) sin(lambdathi ) + lambdathi3 )/(Bi 2 lambdathi ) > Integral_2[i]:=int((T_x0_t0-T_gas+(T_xth_t0 T_x0_t0)∗ x_hat)∗ (sin(lambdath[i]∗ x_hat)+lambdath[i]∗ cos(lambdath[i]∗ x_hat)/Bi),x_hat=0..1); Integral 2i := −(−T x0 t 0 Bi lambdathi + T gasBi lambdathi + T xt h t 0 lambdathi − T x0 t 0 lambdathi − T gas Bi cos(lambdathi ) lambdathi + T gas lambdathi2 sin(lambdathi ) − T xt h t 0 Bi sin(lambdathi ) + T xt h t 0 Bi lambdathi cos(lambdathi ) − T xt h t 0 lambdathi cos(lambdathi ) − T xt h t 0 lambdathi2 sin(lambdathi ) + T x0 t 0 Bi sin(lambdathi ) + T x0 t 0 lambdathi cos(lambdathi )) / (Bi lambdathi2 )

and copied into EES to evaluate each of the constants. The only modiﬁcation required to the Maple results is to change the := to = “Evaluate Constants” duplicate i=1,Nterm Integral_1[i] = 1/2∗ (2∗ Bi∗ lambdath[i]-Biˆ2∗ cos(lambdath[i])∗ sin(lambdath[i])& +Biˆ2∗ lambdath[i]-2∗ Bi∗ lambdath[i]∗ cos(lambdath[i])ˆ2+lambdath[i]ˆ2∗ & cos(lambdath[i])∗ sin(lambdath[i])+lambdath[i]ˆ3)/Biˆ2/lambdath[i] Integral_2[i] = -(-T_x0_t0∗ Bi∗ lambdath[i]+T_gas∗ Bi∗ lambdath[i]+T_xth_t0∗ lambdath[i]& -T_x0_t0∗ lambdath[i]-T_gas∗ Bi∗ cos(lambdath[i])∗ lambdath[i]+T_gas∗ lambdath[i]ˆ2& ∗ sin(lambdath[i])-T_xth_t0∗ Bi∗ sin(lambdath[i])+T_xth_t0∗ Bi∗ lambdath[i]∗ cos(lambdath[i])& -T_xth_t0∗ lambdath[i]∗ cos(lambdath[i])-T_xth_t0∗ lambdath[i]ˆ2∗ sin(lambdath[i])& +T_x0_t0∗ Bi∗ sin(lambdath[i])+T_x0_t0∗ lambdath[i]∗ cos(lambdath[i]))/Bi/lambdath[i]ˆ2 C[i]∗ Integral_1[i]=Integral_2[i] end

427

The solution can be obtained at arbitrary values of position and time: “Solution” alpha=k/(rho∗ c) “thermal diffusivity” x hat=0 “dimensionless position” x hat=x/th Fo=alpha∗ time/thˆ2 “Fourier number” time=5 [s] duplicate i=1,Nterm theta[i]=C[i]∗ (sin(lambdath[i]∗ x hat)+lambdath[i]∗ cos(lambdath[i]∗ x hat)/Bi)∗ exp(-lambdath[i]ˆ2∗ Fo) end T=T gas+sum(theta[1..Nterm])

Figure 3 shows the temperature as a function of position in the wall for the same times (t = 0 s, 0.5 s, 5 s, 50 s, 500 s, and 5000 s) that were sketched in EXAMPLE 3.3-1. 475 t =0.5 s t=5 s

450

Temperature (K)

425

t =0 s t=50 s

400 375 350 325

t =500 s

300 t =5000 s 275 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 Dimensionless axial position, x/L

0.9

1

Figure 3: Temperature distribution within the wall at various times as it equilibrates, predicted by the separation of variables model. The times indicated are the same as those considered in EXAMPLE 3.3 -1.

Figure 3 shows the same trends that were discussed in EXAMPLE 3.3-1; there is an internal, conductive equilibration process that occurs very quickly (with a time scale that is similar to τdiff = 2.6 s). The Fourier number associated with this internal equilibration process will be on the order of 1.0, because the Fourier number is deﬁned as the ratio of time to the time required for a thermal wave to move across the wall. There is subsequently an external, convective equilibration process that occurs much more slowly (with a time scale that is similar to τlumped = 130 s). The Fourier-Biot number product associated with this external equilibration process will be on the order of 1.0 because the Fourier-Biot number product characterizes the ratio of time to the lumped time constant.

3.5.4 Separation of Variables Solutions in Cylindrical Coordinates This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). The separation of variables solution for transient problems can be

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

428

Transient Conduction

applied in cylindrical coordinates. The solution for a cylinder exposed to a step-change in ambient conditions that is presented in Section 3.5.2 was derived using separation of variables in cylindrical coordinates. In this section, the techniques required to solve this problem are presented.

3.5.5 Non-homogeneous Boundary Conditions This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). The method of separation of variables can only be applied to 1-D transient problems where both spatial boundary conditions are homogeneous. In Sections 3.5.3 and 3.5.4, a single, obvious transformation is sufﬁcient to make both spatial boundary conditions homogeneous. In many problems this will not be the case and more advanced techniques will be required. Section 2.3.2 discusses methods for breaking 2D steady problems with non-homogeneous terms into sub-problems that can be solved either by separation of variables or by the solution of an ordinary differential equation. In Section 2.4, superposition for 2-D steady-state problems is discussed. These techniques for solving problems with non-homogeneous boundary conditions using separation of variables remain valid for 1-D transient problems and are presented in Section 3.5.5.

3.6 Duhamel’s Theorem This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). The separation of variables technique discussed in Section 3.5 is not capable of solving problems with time-dependent spatial boundary conditions (e.g., an ambient temperature or heat ﬂux that varies with time). However, problems with time dependent spatial boundary conditions are common. Duhamel’s theorem provides one method of extending an analytical solution that is derived (for example, using separation of variables) assuming a time-invariant boundary condition in order to consider the temperature response to an arbitrary time variation of that boundary condition.

3.7 Complex Combination This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). Complex combination is a useful technique for solving problems that have periodic (i.e., oscillating) boundary conditions or forcing functions. This type of problem was encountered in EXAMPLE 3.1-2 where a temperature sensor (treated as a lumped capacitance) was exposed to an oscillating ﬂuid temperature. In EXAMPLE 3.1-2, the problem is solved analytically and the temperature response of the sensor is found to be the sum of a homogeneous solution that decays to zero (as time became sufﬁciently greater than the time constant of the sensor), and a particular solution that is the sustained response. Complex combination is a convenient method for obtaining only this sustained solution, which is often the only portion of the solution that is of interest. Complex combination can be used for transient problems that are 0-D (i.e., lumped), 1-D, and even 2-D or 3-D.

3.8 Numerical Solutions to 1-D Transient Problems 3.8.1 Introduction Sections 1.4 and 1.5 discuss numerical solutions to steady-state 1-D problems and Section 3.2 discusses the numerical solution to 0-D (lumped) transient problems. In this section, these concepts are extended in order to numerically solve 1-D transient problems.

3.8 Numerical Solutions to 1-D Transient Problems

429

The underlying methodology is the same; a set of equations is obtained from energy balances on small (but not differentially small) control volumes that are distributed throughout the computational domain. These equations will contain energy storage terms in addition to energy transfer terms (due to conduction, convection, and/or radiation) because of the transient nature of the problem. The energy storage terms involve the time rate of temperature change and therefore must be numerically integrated forward in time using one of the techniques that was discussed previously in Section 3.2 (e.g., the Crank-Nicolson technique). Most software applications, including EES and MATLAB, provide advanced numerical integration capabilities that can be applied to this problem.

3.8.2 Transient Conduction in a Plane Wall The process of obtaining a numerical solution to a 1-D, transient conduction problem will be illustrated in the context of a plane wall subjected to a convective boundary condition on one surface, as shown in Figure 3-41. initial temperature, Tini = 20°C L = 5 cm

Figure 3-41: A plane wall exposed to a convective boundary condition at time t = 0.

T∞ = 200°C 2 h = 500 W/m -K x k = 5 W/m-K ρ = 2000 kg/m3 c = 200 J/kg-K

The plane wall has thickness L = 5.0 cm and properties k = 5.0 W/m-K, ρ = 2000 kg/m3 , and c = 200 J/kg-K. The wall is initially at T ini = 20◦ C when at time t = 0, the surface (at x = L) is exposed to ﬂuid at T ∞ = 200◦ C with average heat transfer coefﬁcient h = 500 W/m2 -K. The wall at x = 0 is adiabatic. The known information is entered into EES. $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “inputs" L=5 [cm]∗ convert(cm,m) k=5.0 [W/m-K] rho=2000 [kg/mˆ3] c=200 [J/kg-K] T ini=converttemp(C,K,20 [C]) T inﬁnity=converttemp(C,K,200 [C]) h bar=500 [W/mˆ2-K]

“wall thickness” “conductivity” “density” “speciﬁc heat capacity” “initial temperature” “ﬂuid temperature” “heat transfer coefﬁcient”

The plane wall problem in Figure 3-41 corresponds to the problem that is solved using separation of variables in Section 3.5.3. This problem was selected so that the solution that is obtained numerically can be directly compared to the analytical solution accessed by the planewall_T function in EES that is described in Section 3.5.2. However, the major advantage of using a numerical technique is the capability to easily solve complex problems that may not have an analytical solution.

430

Transient Conduction Δx 2

T1 T2

Ti-1

Ti

q⋅RHS q⋅LHS dU dt x

Δx

Δx 2

Ti+1

TN-1

q⋅RHS

TN

q⋅conv

q⋅LHS

dU dt

Figure 3-42: Nodes and control volumes distributed uniformly throughout computational domain.

dU dt

The ﬁrst step in developing the numerical solution is to partition the continuous medium into a large number of small volumes that are analyzed using energy balances in order to develop the set of equations that must be solved. The nodes (i.e., the positions at which the temperature will be obtained) are distributed uniformly through the wall, as shown in Figure 3-42. For the uniform distribution of nodes that is shown in Figure 3-42, the location of each node (xi ) is: xi =

(i − 1) L for i = 1..N (N − 1)

(3-495)

where N is the number of nodes used for the simulation. The distance between adjacent nodes ( x) is: x =

L (N − 1)

(3-496)

This node spacing is speciﬁed in EES: “Setup grid” N=6 [-] duplicate i=1,N x[i]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1)

“number of nodes” “position of each node” “distance between adjacent nodes”

A control volume is deﬁned around each node. The control surface bisects the distance between the nodes, as shown in Figure 3-42. An energy balance must be written for the control volume associated with every node. The control volume for an arbitrary, internal node experiences conduction heat transfer with the adjacent nodes as well as energy storage (as shown in Figure 3-42): q˙ LHS + q˙ RHS =

dU dt

(3-497)

Each term in Eq. (3-497) must be approximated. The conduction terms from the adjacent nodes are modeled according to: q˙ LHS =

k Ac (T i−1 − T i ) x

(3-498)

q˙ RHS =

k Ac (T i+1 − T i ) x

(3-499)

3.8 Numerical Solutions to 1-D Transient Problems

431

where Ac is the cross-sectional area of the wall. The rate of energy storage is the product of the time rate of change of the nodal temperature and the thermal mass of the control volume: dT i dU = Ac x ρ c dt dt

(3-500)

Substituting Eqs. (3-498) through (3-500) into Eq. (3-497) leads to: Ac x ρ c

dT i k Ac (T i−1 − T i ) k Ac (T i+1 − T i ) = + dt x x

for i = 2... (N − 1)

(3-501)

Solving for the time rate of the temperature change: k dT i = (T i−1 + T i+1 − 2 T i ) dt x2 ρ c

for i = 2... (N − 1)

(3-502)

The control volumes at the boundaries must be treated separately because they have a smaller volume and experience different energy transfers. An energy balance on the control volume at the adiabatic wall (node 1 in Figure 3-42) leads to: q˙ RHS =

dU dt

(3-503)

or Ac x ρ c dT 1 k Ac (T 2 − T 1 ) = 2 dt x

(3-504)

The factor of two on the left side of Eq. (3-504) results because the control volume around node 1 has half the width and thus half the heat capacity of the other nodes. Solving for the time rate of temperature change for node 1: 2k dT 1 = (T 2 − T 1 ) dt ρ c x2

(3-505)

An energy balance on the control volume for the node located at the outer surface (node N in Figure 3-42) leads to: dU = q˙ LHS + q˙ conv dt

(3-506)

or Ac x ρ c dT N k Ac (T N−1 − T N ) = + h Ac (T ∞ − T N ) 2 dt x

(3-507)

Solving for the time rate of temperature change for node N: dT N 2h 2k = (T N−1 − T N ) + (T ∞ − T N ) 2 dt ρ c x x ρ c

(3-508)

Equations (3-502), (3-505), and (3-508) provide the time rate of change for the temperature of every node, given the temperatures of the nodes. This result is similar to the situation encountered in Section 3.2 when developing numerical solutions for lumped capacitance problems. In that case, the energy balance for the single control volume (around the object being studied) provided an equation for the time rate of change of the temperature in terms of the temperature. Here, the energy balance written for each of the lumped capacitances (i.e., each of the control volumes) has provided a set of

432

Transient Conduction

equations for the time rates of change of each of the nodal temperatures. In order to solve the problem, it is necessary to integrate these time derivatives. All of the numerical integration techniques that were discussed in Section 3.2 to solve lumped capacitance problems can be applied here to solve 1-D transient problems. The temperature of each node is a function both of position (x) and time (t). The index that speciﬁes the node’s position is i where i = 1 corresponds to the adiabatic wall and i = N corresponds to the surface of the wall (see Figure 3-42). A second index, j, is added to each nodal temperature in order to indicate the time (Ti,j ), where j = 1 corresponds to the beginning of the simulation and j = M corresponds to the end of the simulation. The total simulation time, tsim , is divided into M time steps. Most of the techniques discussed here will divide the simulation time into time steps of equal duration, t, although other distributions may be used. t =

tsim (M − 1)

(3-509)

The time associated with any time step is: t j = (j − 1) t

for j = 1...M

(3-510)

An array (time []) that provides the time for each step is deﬁned: “Setup time steps” M=21 [-] t sim=40 [s] DELTAtime=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ DELTAtime end

“number of time steps” “simulation time” “time step duration”

The initial conditions for this problem speciﬁes that all of the temperatures at t = 0 are equal to T ini . T i,1 = T ini

duplicate i=1,N T[i,1]=T ini end

for i = 1...N

(3-511)

“initial condition”

Note that the variable T is a two-dimensional array (i.e., a matrix) and calculated results will be displayed in the Arrays Window. The use of a 2-D array simpliﬁes the presentation of the methodology, but it requires unnecessary variable storage space in EES. MATLAB is a more appropriate tool for these types of simulations and the implementation of the numerical simulation in MATLAB will be discussed subsequently. Euler’s Method The temperature of all of the nodes at the end of each time step (i.e., T i,j+1 for all i = 1 to N) must be computed given the temperatures at the beginning of the time step (i.e., T i,j for all i=1 to N) and the algebraic equations derived from the energy balances (i.e., Eqs. (3-502), (3-505), and (3-508)). The simplest technique for numerical integration is

3.8 Numerical Solutions to 1-D Transient Problems

433

Euler’s Method, which approximates the time rate of temperature change within each time step as being constant and equal to its value at the beginning of the time step. Therefore, for any node i during time step j: dT t for i = 1...N (3-512) T i,j+1 = T i,j + dt T =T i,j ,t=tj Note that it is often useful to develop a numerical simulation of a transient process by initially taking only a single step and then, once that works, automating the process of simulating all of the time steps. For example, the temperatures of all N nodes at the end of the ﬁrst time step (i.e., j = 2) are determined from: dT t for i = 1...N (3-513) T i,2 = T i,1 + dt T =T i,1 ,t=t1 Substituting Eqs. (3-502), (3-505), and (3-508), into Eq. (3-513) leads to: T 1,2 = T 1,1 +

T i,2 = T i,1 +

2k (T 2,1 − T 1,1 ) t ρ c x2

k (T i−1,1 + T i+1,1 − 2 T i,1 ) t x2 ρ c

(3-514)

for i = 2... (N − 1)

(3-515)

T N,2

2h 2k = T N,1 + (T N−1,1 − T N,1 ) + (T f − T N,1 ) t ρ c x2 x ρ c

(3-516)

Equations (3-514) through (3-516) are programmed in EES: “Take a single Euler step” T[1,2]=T[1,1]+2∗ k∗ (T[2,1]-T[1,1])∗ DELTAtime/(rho∗ c∗ DELTAxˆ2) duplicate i=2,(N-1) T[i,2]=T[i,1]+k∗ (T[i-1,1]+T[i+1,1]-2∗ T[i,1])∗ DELTAtime/(rho∗ c∗ DELTAxˆ2)

“node 1”

“internal nodes” end T[N,2]=T[N,1]+(2∗ k∗ (T[N-1,1]-T[N,1])/(rho∗ c∗ DELTAxˆ2)+& 2∗ h bar∗ (T inﬁnity-T[N,1])/(rho∗ c∗ DELTAx))∗ DELTAtime

“node N”

Solving the program should provide a solution for the temperature at each node at the end of the ﬁrst time step. The solution can be examined by selecting Arrays from the Windows menu. The equations used to move through any arbitrary time step j follow logically from Eqs. (3-514) through (3-516): T 1,j+1 = T 1,j +

T i,j+1 = T i,j +

2k (T 2,j − T 1,j ) t ρ c x2

k (T i−1,j + T i+1,j − 2 T i,j ) t x2 ρ c

for i = 2... (N − 1)

(3-517)

(3-518)

T N,j+1

2h 2k = T N,j + (T N−1,j − T N,j ) + (T ∞ − T N,j ) t ρ c x2 x ρ c

(3-519)

434

Transient Conduction 450

Temperature (K)

425

numerical solution analytical solution ana

400 40 s

375

30 s 350

20 s

325

10 s

300

4s

275 0

0s 0.01

0.02

0.03

0.04

0.05

Position (m) Figure 3-43: Temperature as a function of position predicted by Euler’s method at various times.

Equations (3-517) through (3-519) are automated in order to simulate all of the time steps by placing the EES code within a second duplicate loop that steps from j = 1 to j = (M − 1). This is accomplished by nesting the entire EES code associated with the ﬁrst time step within an outer duplicate loop; wherever the second index was 2 (i.e., the end of time step 1) we replace it with j + 1 and wherever the second index was 1 (i.e., the beginning of time step 1) we replace it with j. The revised code is shown below; the additional or changed code is shown in bold:

“Move through all of the time steps” duplicate j=1,(M-1) T[1,j+1]=T[1,j]+2∗ k∗ (T[2,j]-T[1,j])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2) duplicate i=2,(N-1) T[i,j+1]=T[i,j]+k∗ (T[i-1,j]+T[i+1,j]-2∗ T[i,j])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2)

“node 1”

“internal nodes” end T[N,j+1]=T[N,j]+(2∗ k∗ (T[N-1,j]-T[N,j])/(rho∗ c∗ DELTAxˆ2)+& 2∗ h bar∗ (T inﬁnity-T[N,j])/(rho∗ c∗ DELTAx))∗ DELTAtime end

“node N”

Figure 3-43 illustrates the temperature as a function of position at t = 0 s, t = 4 s, t = 10 s, t = 20 s, t = 30 s, and t = 40 s. Figure 3-43 was generated by selecting New Plot Window from the Plots menu and selecting the array x[ ] for the X-Axis and the arrays T[i,1], T[i,3], T[i,6], etc. for the Y-Axis (note that time[1] = 0, time[3] = 4 s, time[6] = 10 s, etc. You will be prompted that you are missing data in various rows because the time[ ] array is longer than the x[ ] or any of the T[i,j] arrays. Select Yes in response to this message in order to continue plotting the points. The numerical solution can be compared to the analytical solution derived in Section 3.5.3 and accessed using the planewall_T function.

3.8 Numerical Solutions to 1-D Transient Problems

435

“Analytical solution” alpha=k/(rho∗ c) duplicate j=1,M duplicate i=1,N T_an[i,j]=planewall_T(x[i], time[j], T_ini, T_inﬁnity, alpha, k, h_bar, L) end end

The numerical and analytical solutions agree, as shown in Figure 3-43. If the duration of the time step is increased from 2.0 s to 4.0 s (by reducing M from 21 to 11) then the solution becomes unstable (try it and see; the solution will oscillate between large positive and negative temperatures). The existence of a stability limit is one of the key disadvantages associated with the Euler technique, as we saw in Section 3.2 for lumped capacitance problems. The maximum time step that can be used before the solution becomes unstable (i.e., the critical time step, tcrit ) can be determined by examining the algebraic equations that are used to step through time. Rearranging Eq. (3-517), which governs the behavior of the node at the adiabatic edge, leads to: 2k t 2k t T 2,j (3-520) + T 1,j+1 = T 1,j 1 − ρ c x2 ρ c x2 The solution will become unstable when the coefﬁcient multiplying T1,j becomes negative. Therefore, Eq. (3-520) shows that the solution will tend to become unstable as t becomes larger or x becomes smaller. The critical time step associated with node 1 is: tcrit,1 =

ρ c x2 2k

(3-521)

Applying the same process to Eq. (3-518), which governs the behavior of the internal nodes, leads to: k t 2 k t (3-522) + T i,j+1 = T i,j 1 − (T i−1,j + T i+1,j ) for i = 2... (N − 1) x2 ρ c x2 ρ c According to Eq. (3-522), the critical time step for the internal nodes is the same as for node 1: tcrit,i =

ρ c x2 2k

for i = 2... (N − 1)

(3-523)

Equation (3-519), which governs the behavior of the node placed on the surface of the wall, is rearranged: 2 h t 2 h t 2 k t 2 k t T N,j+1 = T N,j 1 − − T N−1,j + (3-524) + Tf ρ c x2 x ρ c ρ c x2 x ρ c Equation (3-524) leads to a different critical time step for node N: tcrit,N =

x ρ c k 2 +h x

(3-525)

436

Transient Conduction

Comparing Eq. (3-525) with Eq. (3-523) indicates that the critical time step for node N will always be somewhat less than the critical time step for the other nodes and therefore Eq. (3-525) will govern the stability of the problem. The critical time step is calculated according to: “Critical time step for explicit techniques” Deltatime crit N=Deltax∗ rho∗ c/(2∗ (k/Deltax+h bar))

“critical time step for node N”

For the problem considered here, the critical time step tcrit,N = 2.0 s. Note that even a stable solution may not be sufﬁciently accurate. We followed a similar process in Section 3.2 in order to identify the critical time step associated with the explicit integration of a lumped capacitance problem and found that the critical time step was related to the time constant of the object. Equations (3-521), (3-523), and (3-525) essentially restate this conclusion, but for an individual node rather than a lumped object. Recall that the time constant of an object is equal to the product of the heat capacity of the object and its thermal resistance to the environment. The heat capacity of an internal node (Ci ) is: Ci = x Ac ρ c

(3-526)

and the resistance (Ri ) between the nodal temperature of an internal node and its environment (i.e., the adjacent nodes) is: k Ac k Ac −1 Ri = (3-527) + x x or Ri =

x 2 k Ac

(3-528)

The lumped time constant of an internal node (τlumped,i ) is therefore: τlumped,i = Ri Ci =

x ρ c x2 x Ac ρ c = 2 k Ac 2k

(3-529)

and therefore the time constant of an internal node is exactly equal to its critical time step, Eq. (3-523). The time constant for the node located on the surface of the wall (node N) is smaller and therefore its critical time constant will be smaller. The heat capacity of node N (CN ) is: CN =

x Ac ρ c 2

(3-530)

and RN , the thermal resistance between node N and its environment (i.e., node N − 1 and the ﬂuid), is: −1 k Ac RN = (3-531) + h Ac x The time constant for node N is: τlumped,N = RN CN =

x Ac ρ c ρ c x = k Ac k 2 2 + h Ac +h x x

which is identical to the critical time step for node N, Eq. (3-525).

(3-532)

3.8 Numerical Solutions to 1-D Transient Problems

437

In Section 1.4 we found that the accuracy of a numerical solution is related to the size of the control volumes. Smaller values of x will provide more accurate solutions. However, smaller values of x will simultaneously reduce the thermal mass of each node as well as the resistance between the node and its adjacent nodes. Thus the time constant and therefore the critical time step duration for any node scales with x2 (see Eq. (3-529)). As a result, reducing x will greatly increase the number of time steps required (by an explicit numerical technique) in order to maintain stability. It is often the case that the number of time steps required by stability is much larger than would be required for sufﬁcient accuracy and therefore the use of explicit techniques can be computationally intensive. Implementing Euler’s Method within MATLAB is a straightforward extension of the EES code. The simulation in MATLAB is setup in a script. The inputs are provided as the ﬁrst lines: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T inﬁnity=473.2; h bar=500;

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % speciﬁc heat capacity (J/kg-K) % initial temperature (K) % ﬂuid temperature (K) % heat transfer coefﬁcient (W/mˆ2-K)

The axial location of each node and the time steps are speciﬁed: %Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1); %Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% number of time steps (-) % simulation time (s) % time step duration (s)

The initial conditions for each node are inserted into the ﬁrst column of the matrix T: % Initial condition for i=1:N T(i,1)=T_ini; end

438

Transient Conduction

The remaining columns of T are ﬁlled in by stepping through time using the Euler approach: % Step through time for j=1:(M-1) T(1,j+1)=T(1,j)+2∗ k∗ (T(2,j)-T(1,j))∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) T(i,j+1)=T(i,j)+k∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); end T(N,j+1)=T(N,j)+(2∗ k∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+ . . . 2∗ h_bar∗ (T_inﬁnity-T(N,j))/(rho∗ c∗ DELTAx))∗ DELTAtime; end

One advantage of using MATLAB is that it is easy to manipulate the solution. For example, it is possible to plot the temperature as a function of time for various positions by entering plot(time,T) in the command window (Figure 3-44). 440 x (cm) 0

420

Temperature (K)

400 380 1

360 340

2

320

3

300 280

4 0

5

10

15

20 Time (s)

25

30

5 35

40

Figure 3-44: Temperature as a function of time for various positions.

Fully Implicit Method Euler’s method is an explicit technique. Therefore, it has the characteristic of becoming unstable when the time step exceeds a critical value. An implicit technique avoids this problem. The fully implicit method is similar to Euler’s method in that the time rate of change is assumed to be constant throughout the time step. The difference is that the time rate of change is computed at the end of the time step rather than the beginning. Therefore, for any node i and time step j: dT t for i = 1 . . . N (3-533) T i,j+1 = T i,j + dt T =T i,j+1 ,t=tj+1 The temperature rate of change at the end of the time step depends on the temperatures at the end of the time step (T i,j+1 ). The temperatures T i,j+1 cannot be calculated explicitly using information at the beginning of the time step (Ti,j ) and instead Eq. (3-533) provides an implicit set of equations for T i,j+1 where i = 1 to i = N. For the example

3.8 Numerical Solutions to 1-D Transient Problems

439

problem shown in Figure 3-41, the implicit equations are obtained by substituting Eqs. (3-502), (3-505), and (3-508) into Eq.(3-533):

T 1,j+1 = T 1,j +

T i,j+1 = T i,j + T N,j+1 = T N,j +

2k (T 2,j+1 − T 1,j+1 ) t ρ c x2

k (T i−1,j+1 + T i+1,j+1 − 2 T i,j+1 ) t x2 ρ c

(3-534)

for i = 2... (N − 1) (3-535)

2h 2k (T N−1,j+1 − T N,j+1 ) + (T f − T N,j+1 ) t ρ c x2 x ρ c

(3-536)

Because EES solves implicit equations, it is not necessary to rearrange Eqs. (3-534) through (3-536) in order to solve them. The implicit solution is obtained using the following EES code. (Note that the modiﬁcations to the Euler method code are indicated in bold.)

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” L=5 [cm]∗ convert(cm,m) k=5.0 [W/m-K] rho=2000 [kg/mˆ3] c=200 [J/kg-K] T ini=converttemp(C,K,20 [C]) T inﬁnity=converttemp(C,K,200 [C]) h bar=500 [W/mˆ2-K] “Setup grid” N=5 [-] duplicate i=1,N x[i]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1) “Setup time steps” M=21 [-] t sim=40 [s] DELTAtime=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ DELTAtime end duplicate i=1,N T[i,1]=T ini end

“wall thickness” “conductivity” “density” “speciﬁc heat capacity” “initial temperature” “ﬂuid temperature” “heat transfer coefﬁcient”

“number of nodes” “position of each node” “distance between adjacent nodes”

“number of time steps” “simulation time” “time step duration”

“initial condition”

440

Transient Conduction

“Move through all of the time steps” duplicate j=1,(M-1) T[1,j+1]=T[1,j]+2∗ k∗ (T[2,j+1]-T[1,j+1])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2) “node 1” duplicate i=2,(N-1) T[i,j+1]=T[i,j]+k∗ (T[i-1,j+1]+T[i+1,j+1]-2∗ T[i,j+1])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2) “internal nodes” end T[N,j+1]=T[N,j]+(2∗ k∗ (T[N-1,j+1]-T[N,j+1])/(rho∗ c∗ DELTAxˆ2)+& 2∗ h bar∗ (T inﬁnity-T[N,j+1])/(rho∗ c∗ DELTAx))∗ DELTAtime “node N” end

An attractive feature of the implicit technique is that it is possible to vary M and N independently in order to achieve sufﬁcient accuracy without being constrained by stability considerations. The implementation of the implicit technique in MATLAB is not a straightforward extension of the EES code because MATLAB cannot directly solve a set of implicit equations. Instead, Eqs. (3-534) through (3-536) must be placed in matrix format before they can be solved. This operation is similar to the solution of 1-D and 2-D steady-state problems using MATLAB, discussed in Sections 1.5, 1.9, and 2.6. The inputs are entered in MATLAB and the grid and time steps are setup: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T inﬁnity=473.2; h bar=500; % Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1); % Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end % Initial condition for i=1:N T(i,1)=T ini; end

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % speciﬁc heat capacity (J/kg-K) % initial temperature (K) % ﬂuid temperature (K) % heat transfer coefﬁcient (W/mˆ2-K)

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% number of time steps (-) % simulation time (s) % time step duration (s)

3.8 Numerical Solutions to 1-D Transient Problems

441

For any time step j, Eqs. (3-534) through (3-536) represent N linear equations for the N unknown temperatures T i,j+1 for i = 1 to i = N. In order to solve these implicit equations, they must be placed in matrix format: AX = b

(3-537)

It is important to clearly specify the order that the equations are placed into the matrix A and the order that the unknown temperatures are placed into the vector X. The most logical method to setup the vector X is: ⎤ ⎡ X 1 = T 1,j+1 ⎥ ⎢X = T 2,j+1 ⎥ ⎢ 2 X=⎢ (3-538) ⎥ ⎦ ⎣··· X N = T N,j+1 so that T i,j+1 corresponds to element i of X. The most logical method for placing the equations into A is: ⎡ ⎤ row 1 = control volume 1 equation ⎢ row 2 = control volume 2 equation ⎥ ⎥ A=⎢ (3-539) ⎣··· ⎦ row N = control volume N equation so that the equation derived based on the control volume for node i corresponds to row i of A. Equations (3-534) through (3-536) are rearranged so that the coefﬁcients multiplying the unknowns and the constants for the linear equations are clear: 2k t 2k t + T 2,j+1 − = T 1,j (3-540) T 1,j+1 1 + ρ c x2 ρ c x2 b1 A1,1

A1,2

2 k t k t k t T i,j+1 1+ 2 + T i−1,j+1 − 2 + T i+1,j+1 − 2 = T i,j x ρ c x ρ c x ρ c bi Ai,i

Ai,i−1

for i = 2 . . . (N − 1)

Ai,i+1

(3-541)

2 h t 2 h t 2 k t 2 k t T N,j+1 1 + + + T N−1,j+1 − = T N,j + T∞ ρ c x2 x ρ c ρ c x2 x ρ c

AN,N

AN,N−1

(3-542)

bN

The matrix A and vector b are initialized: A=spalloc(N,N,3∗ N); b=zeros(N,1);

% initialize A % initialize b

Note that the variable A is deﬁned as being a sparse matrix with at most three nonzero entries per row. The maximum number of nonzero elements can be determined based on examination of Eqs. (3-540) through (3-542). The values of the elements in the matrix A are all initialized to zero. The matrix A does not change depending on the particular time step being considered (at least for this problem with constant properties). Therefore, the matrix A can be constructed just one time and used without modiﬁcation for each time step, which saves computational effort.

442

Transient Conduction

% Setup A matrix A(1,1)=1+2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(1,2)=-2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) A(i,i)=1+2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(i,i-1)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(i,i+1)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); end A(N,N)=1+2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ DELTAtime/(DELTAx∗ rho∗ c); A(N,N-1)=-2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2);

On the other hand, the vector b does change depending on the time step because it includes the current value of the temperatures. Therefore, the vector b will need to be reconstructed before the simulation of each time step. for j=1:(M-1) % Setup b matrix for i=1:(N-1) b(i)=T(i,j); end b(N)=T(N,j)+2∗ h_bar∗ DELTAtime∗ T_inﬁnity/(DELTAx∗ rho∗ c); % Simulate time step T(:,j+1)=A\b; end

Note that T(:,j+1) is the MATLAB syntax for specifying the entire column j+1 in the matrix T. Running the script at this point will provide the matrix T, which contains the temperatures for each node (corresponding to each row of T) and each time step (corresponding to each column of T). This information can be plotted against either time or position by either typing: plot(time,T);

or plot(x,T);

in the command window, respectively. Heun’s Method Heun’s method was discussed previously in Section 3.2.2 as a simple example of a class of numerical integration methods referred to as predictor-corrector techniques. Predictorcorrector techniques take an initial predictor step (based on Euler’s technique) followed by one or more corrector steps, in which the knowledge obtained from the predictor step is used to improve the integration process. Because the predictor-corrector techniques rely on an explicit predictor step, they suffer from the same limitations related to stability that were discussed in the context of Euler’s method. The implementation of Heun’s method is illustrated using MATLAB. The problem is set up as before:

3.8 Numerical Solutions to 1-D Transient Problems

443

clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T infinity=473.2; h bar=500;

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % specific heat capacity (J/kg-K) % initial temperature (K) % fluid temperature (K) % heat transfer coefficient (W/mˆ2-K)

% Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of time steps (-) % simulation time (s) % time step duration (s)

% Initial condition for i=1:N T(i,1)=T ini; end

In order to simulate any time step j for any node i, Heun’s method begins with an Euler step to obtain an initial prediction for the temperature at the conclusion of the time step (Tˆ i,j+1 ). This ﬁrst step is referred to as the predictor step and the details are identical to Euler’s method: dT ˆ t for i = 1 . . . N (3-543) T i,j+1 = T i,j + dt T =T i,j ,t=tj The equations used to predict the time rate of change are speciﬁc to the problem being studied even though the methodology is broadly applicable. The results of the predictor step are used to carry out a subsequent corrector step. The predicted values of the temperatures at the end of the time step are used to predict the temperature rate of | ). The corrector step predicts the change at the end of the time step ( dT dt T =Tˆ i,j+1 ,t=t j + t | and temperature at the end of the time step (Ti,j+1 ) based on the average of dT dt T =T i,j ,t=t j dT | according to: dt T =Tˆ i,j+1 ,t=t j + t T i,j+1 = T i,j +

dT dT t + dt T =T i,j ,t=tj dt T =Tˆ i,j+1 ,t=tj + t 2

for i = 1 . . . N

(3-544)

444

Transient Conduction

Heun’s method is illustrated in the context of the problem that is illustrated in Figure 3-41. Equations (3-502), (3-505), and (3-508) are used to compute the time rate of change for each node at the beginning of a time step: 2k dT (3-545) = (T 2,j − T 1,j ) dt T =T 1,j ,t=tj ρ c x2 dT k = (T i−1,j + T i+1,j − 2 T i,j ) dt T =T i,j ,t=tj x2 ρ c

for i = 2 . . . (N − 1)

dT 2h 2k = (T ∞ − T N,j ) (T N−1,j − T N,j ) + 2 dt T =T N,j ,t=tj ρ c x x ρ c

(3-546)

(3-547)

The time rate of change for the temperature of each node at the beginning of the time step is stored in a temporary vector, dTdt0, which is overwritten during each time step: % Step through time for j=1:(M-1) % compute time rates of change at the beginning of the time step dTdt0(1)=2∗ k∗ (T(2,j)-T(1,j))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) dTdt0(i)=k∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))/(rho∗ c∗ DELTAxˆ2); end dTdt0(N)=2∗ k∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-T(N,j))/(rho∗ c∗ DELTAx);

A ﬁrst estimate of the temperatures at the end of the time step is obtained using Eq. (3-543); these values are also stored in a temporary vector, Ts, which is also not saved beyond the time step being simulated: % compute first estimate of temperatures at the end of the time step for i=1:N Ts(i)=T(i,j)+dTdt0(i)∗ DELTAtime; end

The results of the predictor step are used to carry out a corrector step. The time rates of change for each temperature at the end of the time step are computed using Eqs. (3-502), (3-505), and (3-508) with the time derivative evaluated using the predicted temperatures. 2k ˆ dT T 2,j+1 − Tˆ 1,j+1 (3-548) = dt T =Tˆ 1,j+1 ,t=tj + t ρ c x2 k ˆ dT = T i−1,j+1 + Tˆ i+1,j+1 − 2 Tˆ i,j+1 2 dt T =Tˆ i,j+1 ,t=tj + t x ρ c

for i = 2 . . . (N − 1) (3-549)

dT 2h 2k ˆ T N−1,j+1 − Tˆ N,j+1 + = (T ∞ − Tˆ N,j+1 ) 2 dt T =Tˆ N,j+1 ,t=tj + t ρ c x x ρ c

(3-550)

3.8 Numerical Solutions to 1-D Transient Problems

445

% compute time rates of change at the end of the time step dTdts(1)=2∗ k∗ (Ts(2)-Ts(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) dTdts(i)=k∗ (Ts(i-1)+Ts(i+1)-2∗ Ts(i))/(rho∗ c∗ DELTAxˆ2); end dTdts(N)=2∗ k∗ (Ts(N-1)-Ts(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Ts(N))/(rho∗ c∗ DELTAx);

The corrector step is based on the average of the two estimates of the time rate of change: dT dT t + for i = 1 . . . N (3-551) T i,j+1 = T i,j + dt T =T i,j ,t=tj dt T =Tˆ i,j+1 ,t=tj + t 2 % corrector step for i=1:N T(i,j+1)=T(i,j)+(dTdt0(i)+dTdts(i))∗ DELTAtime/2; end end

The numerical error associated with Heun’s method is substantially less than Euler’s method or the fully implicit method because Heun’s method is a second-order technique, as discussed in Section 3.2. Also note that if M is reduced from 21 to 11 (i.e., the time step is increased from 2.0 s to 4.0 s) then Heun’s method becomes unstable, just as Euler’s method did. Runge-Kutta 4th Order Method Heun’s method is a two-step predictor/corrector technique, which improves the accuracy of the solution but not the stability characteristics. The fourth order Runge-Kutta method involves four predictor/corrector steps and therefore improves the accuracy to an even larger extent. Because the Runge-Kutta technique is explicit, the stability characteristics of the solution are not affected. The Runge-Kutta fourth order method (or RK4 method) was previously discussed in Section 3.2.2. The RK4 technique estimates the time rate of change of the temperature of each node four times in order to simulate a single time step. (Contrast this method with Euler’s method where the time rate of change was computed only once, at the beginning of time step.) The implementation of the RK4 method is illustrated using MATLAB. The problem is setup as before: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T infinity=473.2; h bar=500;

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % specific heat capacity (J/kg-K) % initial temperature (K) % fluid temperature (K) % heat transfer coefficient (W/mˆ2-K)

446

Transient Conduction

% Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of time steps (-) % simulation time (s) % time step duration (s)

% Initial condition for i=1:N T(i,1)=T ini; end

The RK4 method begins by estimating the time rate of change of the temperature of each node at the beginning of the time step (referred to as aai ): dT for i = 1 . . . N (3-552) aai = dt T =T i,j ,t=tj or, for this problem: aa1 = aai =

2k (T 2,j − T 1,j ) ρ c x2

k (T i−1,j + T i+1,j − 2 T i,j ) x2 ρ c

aaN =

for i = 2 . . . (N − 1)

2h 2k (T ∞ − T N,j ) (T N−1,j − T N,j ) + 2 ρ c x x ρ c

(3-553)

(3-554)

(3-555)

% Step through time for j=1:(M-1) % compute the 1st estimate of the time rate of change aa(1)=2∗ k∗ (T(2,j)-T(1,j))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) aa(i)=k∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))/(rho∗ c∗ DELTAxˆ2); end aa(N)=2∗ k∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-T(N,j))/(rho∗ c∗ DELTAx);

The ﬁrst estimate of the time rate of change is used to predict the temperature of each node half-way through the time step (Tˆ i,j+ 1 ) in the ﬁrst predictor step: 2

t Tˆ i,j+ 1 = T i,j + aai 2 2

(3-556)

3.8 Numerical Solutions to 1-D Transient Problems

447

% 1st predictor step for i=1:N Ts(i)=T(i,j)+aa(i)∗ DELTAtime/2; end

The estimated temperatures are used to obtain the second estimate of the time rate of change (bbi ), this time at the midpoint of the time step: dT for i = 1 . . . N (3-557) bbi = dt T =Tˆ 1 ,t=tj + t i,j+

2

2

or, for this problem: bb1 =

bbi =

2k ˆ ˆ 1 − T 1 T 2,j+ 2 1,j+ 2 ρ c x2

k ˆ ˆ i+1,j+ 1 − 2 Tˆ i,j+ 1 1 + T T i−1,j+ 2 2 2 x2 ρ c

bbN =

(3-558)

for i = 2 . . . (N − 1)

2k ˆ 2h ˆ ˆ 1 − T 1 1 − T T T + ∞ N−1,j+ 2 N,j+ 2 N,j+ 2 ρ c x2 x ρ c

(3-559)

(3-560)

% compute the 2nd estimate of the time rate of change bb(1)=2∗ k∗ (Ts(2)-Ts(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) bb(i)=k∗ (Ts(i-1)+Ts(i+1)-2∗ Ts(i))/(rho∗ c∗ DELTAxˆ2); end bb(N)=2∗ k∗ (Ts(N-1)-Ts(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Ts(N))/(rho∗ c∗ DELTAx);

The second estimate of the time rate of change is used to obtain a new prediction of the temperature of each node, again half-way through the time step (Tˆˆ i,j+ 1 ): 2

t Tˆˆ i,j+ 1 = T i,j + bbi 2 2

for i = 1 . . . N

(3-561)

% 2nd predictor step for i=1:N Tss(i)=T(i,j)+bb(i)∗ DELTAtime/2; end

The third estimate of the time rate of change of each node (cci ) is also obtained at the mid-point of the time step: dT for i = 1 . . . N (3-562) cci = dt T =Tˆˆ 1 ,t=tj + t i,j+

2

2

or, for this problem: cc1 =

2k ˆˆ T 2,j+ 1 − Tˆˆ 1,j+ 1 2 2 2 ρ c x

(3-563)

448

Transient Conduction

cci =

k ˆˆ ˆˆ ˆˆ 1 + T 1 − 2T 1 T i−1,j+ 2 i+1,j+ 2 i,j+ 2 x2 ρ c

ccN =

for i = 2 . . . (N − 1)

2 k ˆˆ 2h ˆˆ ˆˆ 1 − T 1 1 − T T T + ∞ N−1,j+ 2 N,j+ 2 N,j+ 2 ρ c x2 x ρ c

(3-564)

(3-565)

% compute the 3rd estimate of the time rate of change cc(1)=2∗ k∗ (Tss(2)-Tss(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) cc(i)=k∗ (Tss(i-1)+Tss(i+1)-2∗ Tss(i))/(rho∗ c∗ DELTAxˆ2); end cc(N)=2∗ k∗ (Tss(N-1)-Tss(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Tss(N))/(rho∗ c∗ DELTAx);

The third estimate of the time rate of change is used to predict the temperature at the end of the time step (Tˆ i,j+1 ): Tˆ i,j+1 = T i,j + cci t

for i = 1 . . . N

(3-566)

% 3rd predictor step for i=1:N Tsss(i)=T(i,j)+cc(i)∗ DELTAtime; end

Finally, the fourth estimate of the time rate of change (ddi ) is obtained at the end of the time step: dT for i = 1 . . . N (3-567) ddi = dt Tˆ i,j+1 ,t=tj + t or, for this problem: dd1 = ddi =

2k ˆ T 2,j+1 − Tˆ 1,j+1 2 ρ c x

k ˆ T i−1,j+1 + Tˆ i+1,j+1 − 2 Tˆ i,j+1 2 x ρ c

ddN =

for i = 2 . . . (N − 1)

2k ˆ 2h T N−1,j+1 − Tˆ N,j+1 + T ∞ − Tˆ N,j+1 2 ρ c x x ρ c

(3-568) (3-569)

(3-570)

% compute the 4th estimate of the time rate of change dd(1)=2∗ k∗ (Tsss(2)-Tsss(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) dd(i)=k∗ (Tsss(i-1)+Tsss(i+1)-2∗ Tsss(i))/(rho∗ c∗ DELTAxˆ2); end dd(N)=2∗ k∗ (Tsss(N-1)-Tsss(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Tsss(N))/(rho∗ c∗ DELTAx);

3.8 Numerical Solutions to 1-D Transient Problems

449

The integration through the time step is ﬁnally carried out using the weighted average of these four separate estimates of the time rate of change for each node: T i,j+1 = T i,j + (aai + 2 bbi + 2 cci + ddi)

t 6

(3-571)

% corrector step for i=1:N T(i,j+1)=T(i,j)+(aa(i)+2∗ bb(i)+2∗ cc(i)+dd(i))∗ DELTAtime/6; end end

Crank-Nicolson Method The Crank-Nicolson method combines Euler’s method with the fully implicit method. The time rate of change for each time step is estimated based on the average of its values at the beginning and the end of the time step. T i,j+1 = T i,j +

dT dT t + dt T =T i,j ,t=tj dt T =T i,j+1 ,t=tj+1 2

for i = 1 . . . N

(3-572)

Notice that Eq. (3-572) is not a predictor-corrector method, like Heun’s method, because the temperature at the end of the time step is not predicted with an explicit predictor step (i.e., there is no Tˆ i,j+1 in Eq. (3-572)). Rather, the unknown solution for the temperatures at the end of the time step (T i,j+1 ) is substituted directly into Eq. (3-572), resulting in a set of implicit equations for T i,j+1 where i = 1 to i = N. Therefore, the Crank-Nicolson technique has stability characteristics that are similar to the fully implicit method. The solution involves two estimates for the time rate of change of each node and therefore has higher order accuracy than either Euler’s method or the fully implicit technique. The Crank-Nicolson method is illustrated using MATLAB. The problem is setup as before: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T infinity=473.2; h bar=500; % Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % specific heat capacity (J/kg-K) % initial temperature (K) % fluid temperature (K) % heat transfer coefficient (W/mˆ2-K)

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

450

Transient Conduction

% Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of time steps (-) % simulation time (s) % time step duration (s)

% Initial conditions for i=1:N T(i,1)=T ini; end

Substituting Eqs. (3-502), (3-505), and (3-508) into Eq. (3-572) leads to:

T 1,j+1

2k 2k t = T 1,j + (T 2,j − T 1,j ) + (T 2,j+1 − T 1,j+1 ) 2 2 ρ c x ρ c x 2

T i,j+1 = T i,j +

(3-573)

k k t + + T − 2 T + T − 2 T ) ) (T (T i−1,j i+1,j i,j i−1,j+1 i+1,j+1 i,j+1 x2 ρ c x2 ρ c 2 for i = 2 . . . (N − 1) (3-574)

⎡

T N,j+1

⎤ 2h 2k + + − T − T (T ) ) (T N−1,j N,j ∞ N,j ⎢ ρ c x2 ⎥ t x ρ c ⎥ = T N,j + ⎢ ⎣ 2k ⎦ 2 2h + − T − T (T ) ) (T N−1,j+1 N,j+1 ∞ N,j+1 ρ c x2 x ρ c

(3-575)

Equations (3-573) through (3-575) are a set of N equations in the unknown temperatures T i,j+1 for i = 1 to i = N. These equations must be placed into matrix format in order to be solved in MATLAB. This is similar to the fully implicit method and the same process is used. Equations (3-573) through (3-575) are rearranged so that the coefﬁcients multiplying the unknowns and the constants for the linear equations are clear: k t k t k t + T = T 1,j + (3-576) − T 1,j+1 1 + (T 2,j − T 1,j ) 2,j+1 ρ c x2 ρ c x2 ρ c x2 A1,1

b1

A1,2

k t k t k t T i,j+1 1 + + T + T − − i−1,j+1 i+1,j+1 x2 ρ c 2 x2 ρ c 2 x2 ρ c Ai,i

Ai,i−1

k t = T i,j + (T i−1,j + T i+1,j − 2 T i,j ) 2 x2 ρ c bi

Ai,i+1

(3-577) for i = 2 . . . (N − 1)

3.8 Numerical Solutions to 1-D Transient Problems

451

h t k t k t T N,j+1 1 + + +T N−1,j+1 − ρ c x2 x ρ c ρ c x2

AN,N−1

AN,N

(3-578) h t h t k t T∞ = T N,j + (T N−1,j − T N,j ) + (T ∞ − T N,j ) + ρ c x2 x ρ c x ρ c bN

The variables A and b are initialized: A=spalloc(N,N,3∗ N); b=zeros(N,1);

% initialize A % initialize b

The matrix A does not depend on the time step and can therefore be constructed just once: % Setup A matrix A(1,1)=1+k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(1,2)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) A(i,i)=1+k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(i,i-1)=-k∗ DELTAtime/(2∗ rho∗ c∗ DELTAxˆ2); A(i,i+1)=-k∗ DELTAtime/(2∗ rho∗ c∗ DELTAxˆ2); end A(N,N)=1+k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2)+h_bar∗ DELTAtime/(DELTAx∗ rho∗ c); A(N,N-1)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2);

while the vector b must be reconstructed during each time step: for j=1:(M-1) % Setup b matrix b(1)=T(1,j)+k∗ DELTAtime∗ (T(2,j)-T(1,j))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) b(i)=T(i,j)+k∗ DELTAtime∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))/(2∗ rho∗ c∗ DELTAxˆ2); end b(N)=T(N,j)+k∗ DELTAtime∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+ . . . h_bar∗ DELTAtime∗ (T_infinity-T(N,j))/(DELTAx∗ rho∗ c)+ . . . h_bar∗ DELTAtime∗ T_infinity/(DELTAx∗ rho∗ c); % Simulate time step T(:,j+1)=A\b; end

The numerical error associated with the Crank-Nicolson technique is substantially less than either Euler’s method or the fully implicit method and it retains the stability characteristics of the fully implicit technique. The Crank-Nicolson technique is likely to be the most attractive option for many problems due to its superior accuracy combined with its stability.

452

Transient Conduction

EES’ Integral Command The EES Integral function was introduced in Section 3.2.2 and used to solve lumped capacitance problems. EES is used in this section to illustrate Euler’s method and the fully implicit method. However, these are not optimal methods for using EES to solve 1-D transient problems. Instead, the Integral command provides a more powerful and computationally efﬁcient method to solve this class of problem. The Integral command implements a third order accurate integration scheme that uses an (optional) adaptive step-size in order to minimize computation time while maintaining accuracy. The problem is set up in EES by entering the inputs: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” L=5 [cm]∗ convert(cm,m) k=5.0 [W/m-K] rho=2000 [kg/mˆ3] c=200 [J/kg-K] T ini=converttemp(C,K,20 [C]) T infinity=converttemp(C,K,200 [C]) h bar=500 [W/mˆ2-K] t sim=40 [s]

“wall thickness” “conductivity” “density” “specific heat capacity” “initial temperature” “fluid temperature” “heat transfer coefficient” “simulation time”

and the spatial grid is setup: “Setup grid” N=6 [-] duplicate i=1,N x[ ]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1)

“number of nodes” “position of each node” “distance between adjacent nodes”

The EES Integral command requires four arguments (with an optional ﬁfth argument to specify the integration step size): F=INTEGRAL(Integrand,VarName,LowerLimit,UpperLimit, Stepsize)

where Integrand is the EES variable or expression that is be integrated (which, in this case, is each of the time derivatives of the nodal temperatures), VarName is the integration variable (time), LowerLimit and UpperLimit deﬁne the limits of integration (0 and t_sim), and Stepsize is the optional size of the time step. The solution to our example problem is obtained by replacing Integrand with the variable that speciﬁes the time rate of change of the temperature for each of the nodes; i.e., Eqs. (3-502), (3-505), and (3-508). To solve this problem using the Integral command, it is ﬁrst necessary to set up the equations that calculate the time rates of temperature change within a one-dimensional array. Note that two-dimensional arrays are not needed when the problem is solved in this manner, since EES will automatically keep track of the integrated values as they change with time. The use of the Integral

3.8 Numerical Solutions to 1-D Transient Problems

453

command requires many fewer variables and less computation time than the numerical schemes presented in earlier sections require when they are implemented in EES. “time rate of change” dTdt[1]=2∗ k∗ (T[2]-T[1])/(rho∗ c∗ Deltaxˆ2) “node 1” duplicate i=2,(N-1) “internal nodes” dTdt[i]=k∗ (T[i-1]+T[i+1]-2∗ T[i])/(rho∗ c∗ Deltaxˆ2) end dTdt[N]=2∗ k∗ (T[N-1]-T[N])/(rho∗ c∗ DELTAxˆ2)+2∗ h bar∗ (T infinity-T[N])/(rho∗ c∗ DELTAx) “node N”

The following code integrates the time rates of change for each node forward through time: “integrate using the INTEGRAL command” duplicate i=1,N T[i]=T_ini+INTEGRAL(dTdt[i],time,0,t_sim) end

Note that after the calculations are completed, the array T[i] will provide the temperature of each node at the end of the time that is simulated, i.e., at time t = tsim . The time variation in the temperature at each node can be recorded by including the $IntegralTable directive in the ﬁle. The format of the $IntegralTable directive is: $IntegralTable VarName:Interval, x,y,z

where VarName is the integration variable (time), Interval is the output step size in the integration variable at which results will be placed in the integral table (which is completely i

SANFORD KLEIN University of Wisconsin–Madison

cambridge university press ˜ Paulo, Delhi Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521881074 c Gregory Nellis and Sanford Klein 2009

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2009 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Nellis, Gregory. Heat transfer / Gregory Nellis, Sanford Klein p. cm. Includes bibliographical references and index. ISBN 978-0-521-88107-4 (hardback) 1. Heat – Transmission. I. Klein, Sanford A., 1950– II. Title. TJ260.N45 2008 621.402 2 – dc22 2008021961 ISBN 978-0-521-88107-4 hardback Additional resources for this publication at www.cambridge.org/nellisandklein Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work are correct at the time of ﬁrst printing, but Cambridge University Press does not guarantee the accuracy of such information thereafter.

This book is dedicated to Stephen H. Nellis . . . thanks Dad.

CONTENTS

Preface Acknowledgments Study guide Nomenclature 1 1.1

1.2

page xix xxi xxiii xxvii

ONE-DIMENSIONAL, STEADY-STATE CONDUCTION r 1 Conduction Heat Transfer 1.1.1 Introduction 1.1.2 Thermal Conductivity Thermal Conductivity of a Gas∗ (E1) Steady-State 1-D Conduction without Generation 1.2.1 Introduction 1.2.2 The Plane Wall 1.2.3 The Resistance Concept 1.2.4 Resistance to Radial Conduction through a Cylinder 1.2.5 Resistance to Radial Conduction through a Sphere 1.2.6 Other Resistance Formulae Convection Resistance Contact Resistance Radiation Resistance EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.3

Steady-State 1-D Conduction with Generation 1.3.1 Introduction 1.3.2 Uniform Thermal Energy Generation in a Plane Wall 1.3.3 Uniform Thermal Energy Generation in Radial Geometries EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3.4

Spatially Non-Uniform Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.4

Numerical Solutions to Steady-State 1-D Conduction Problems (EES) 1.4.1 Introduction 1.4.2 Numerical Solutions in EES 1.4.3 Temperature-Dependent Thermal Conductivity 1.4.4 Alternative Rate Models

1.5

Numerical Solutions to Steady-State 1-D Conduction Problems using MATLAB 1.5.1 Introduction 1.5.2 Numerical Solutions in Matrix Format 1.5.3 Implementing a Numerical Solution in MATLAB

EXAMPLE 1.4-1: FUEL ELEMENT

∗

1 1 1 5 5 5 5 9 10 11 13 14 14 16 17 24 24 24 29 31 37 38 44 44 45 55 60 62 68 68 69 71

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

vii

viii

Contents

1.5.4 1.5.5 1.5.6

Functions Sparse Matrices Temperature-Dependent Properties

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.6

Analytical Solutions for Constant Cross-Section Extended Surfaces 1.6.1 Introduction 1.6.2 The Extended Surface Approximation 1.6.3 Analytical Solution 1.6.4 Fin Behavior 1.6.5 Fin Efﬁciency and Resistance EXAMPLE 1.6-1: SOLDERING TUBES

1.6.6

Finned Surfaces

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

1.7

1.6.7 Fin Optimization∗ (E2) Analytical Solutions for Advanced Constant Cross-Section Extended Surfaces 1.7.1 Introduction 1.7.2 Additional Thermal Loads EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7.3

Moving Extended Surfaces

EXAMPLE 1.7-2: DRAWING A WIRE

1.8

Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 1.8.1 Introduction 1.8.2 Series Solutions 1.8.3 Bessel Functions 1.8.4 Rules for Using Bessel Functions EXAMPLE 1.8-1: PIPE IN A ROOF EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

1.9

Numerical Solution to Extended Surface Problems 1.9.1 Introduction EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

Problems References 2 2.1

TWO-DIMENSIONAL, STEADY-STATE CONDUCTION r 202 Shape Factors EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

2.2

∗

77 80 82 84 92 92 92 95 103 105 110 113 117 122 122 122 122 127 133 136 139 139 139 142 150 155 161 164 164 165 171 185 201

Separation of Variables Solutions 2.2.1 Introduction 2.2.2 Separation of Variables Requirements for using Separation of Variables Separate the Variables Solve the Eigenproblem Solve the Non-homogeneous Problem for each Eigenvalue Obtain Solution for each Eigenvalue Create the Series Solution and Enforce the Remaining Boundary Conditions Summary of Steps

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

202 205 207 207 208 209 211 212 213 214 215 222

Contents

2.2.3

ix

Simple Boundary Condition Transformations

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

2.3 2.4

2.5

2.6

2.7

2.8

Advanced Separation of Variables Solutions∗ (E3) Superposition 2.4.1 Introduction 2.4.2 Superposition for 2-D Problems Numerical Solutions to Steady-State 2-D Problems with EES 2.5.1 Introduction 2.5.2 Numerical Solutions with EES Numerical Solutions to Steady-State 2-D Problems with MATLAB 2.6.1 Introduction 2.6.2 Numerical Solutions with MATLAB 2.6.3 Numerical Solution by Gauss-Seidel Iteration∗ (E4) Finite Element Solutions 2.7.1 Introduction to FEHT∗ (E5) 2.7.2 The Galerkin Weighted Residual Method∗ (E6) Resistance Approximations for Conduction Problems 2.8.1 Introduction EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

2.8.2 2.8.3

Isothermal and Adiabatic Resistance Limits Average Area and Average Length Resistance Limits

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

2.9

Conduction through Composite Materials 2.9.1 Effective Thermal Conductivity EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

Problems References 3 3.1

TRANSIENT CONDUCTION r 302 Analytical Solutions to 0-D Transient Problems 3.1.1 Introduction 3.1.2 The Lumped Capacitance Assumption 3.1.3 The Lumped Capacitance Problem 3.1.4 The Lumped Capacitance Time Constant EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.2

Numerical Solutions to 0-D Transient Problems 3.2.1 Introduction 3.2.2 Numerical Integration Techniques Euler’s Method Heun’s Method Runge-Kutta Fourth Order Method Fully Implicit Method Crank-Nicolson Method Adaptive Step-Size and EES’ Integral Command MATLAB’s Ordinary Differential Equation Solvers EXAMPLE 3.2-1(A): OVEN BRAZING (EES) EXAMPLE 3.2-1(B): OVEN BRAZING (MATLAB)

∗

224 225 236 242 242 242 245 250 250 251 260 260 260 268 269 269 269 269 269 270 272 275 276 278 278 282 290 301

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

302 302 302 303 304 307 310 317 317 317 318 322 326 328 330 332 335 339 344

x

Contents

3.3

Semi-Inﬁnite 1-D Transient Problems 3.3.1 Introduction 3.3.2 The Diffusive Time Constant EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

3.3.3 3.3.4

The Self-Similar Solution Solutions to other Semi-Inﬁnite Problems

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.4

The Laplace Transform 3.4.1 Introduction 3.4.2 The Laplace Transformation Laplace Transformations with Tables Laplace Transformations with Maple 3.4.3 The Inverse Laplace Transform Inverse Laplace Transform with Tables and the Method of Partial Fractions Inverse Laplace Transformation with Maple 3.4.4 Properties of the Laplace Transformation 3.4.5 Solution to Lumped Capacitance Problems 3.4.6 Solution to Semi-Inﬁnite Body Problems

348 348 348 351 354 361 363 369 369 370 371 371 372 373 376 378 380 386

3.4.7 Numerical Inverse Laplace Transform* (E29) EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

3.5

Separation of Variables for Transient Problems 3.5.1 Introduction 3.5.2 Separation of Variables Solutions for Common Shapes The Plane Wall The Cylinder The Sphere EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

3.5.3

Separation of Variables Solutions in Cartesian Coordinates Requirements for using Separation of Variables Separate the Variables Solve the Eigenproblem Solve the Non-homogeneous Problem for each Eigenvalue Obtain a Solution for each Eigenvalue Create the Series Solution and Enforce the Initial Condition Limits of the Separation of Variables Solution

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.6 3.7 3.8

∗

3.5.4 Separation of Variables Solutions in Cylindrical Coordinates∗ (E7) 3.5.5 Non-homogeneous Boundary Conditions∗ (E8) Duhamel’s Theorem∗ (E9) Complex Combination∗ (E10) Numerical Solutions to 1-D Transient Problems 3.8.1 Introduction 3.8.2 Transient Conduction in a Plane Wall Euler’s Method Fully Implicit Method Heun’s Method Runge-Kutta 4th Order Method Crank-Nicolson Method EES’ Integral Command MATLAB’s Ordinary Differential Equation Solvers

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

391 395 395 396 396 401 403 405 408 409 410 411 413 414 414 417 420 427 428 428 428 428 428 429 432 438 442 445 449 452 453

Contents EXAMPLE 3.8-1: TRANSIENT RESPONSE OF A BENT-BEAM ACTUATOR

3.9

4 4.1

4.2

4.3

3.8.3 Temperature-Dependent Properties Reduction of Multi-Dimensional Transient Problems∗ (E11) Problems References

∗

457 463 468 469 482

EXTERNAL FORCED CONVECTION r 483 Introduction to Laminar Boundary Layers 4.1.1 Introduction 4.1.2 The Laminar Boundary Layer A Conceptual Model of the Laminar Boundary Layer A Conceptual Model of the Friction Coefﬁcient and Heat Transfer Coefﬁcient The Reynolds Analogy 4.1.3 Local and Integrated Quantities The Boundary Layer Equations 4.2.1 Introduction 4.2.2 The Governing Equations for Viscous Fluid Flow The Continuity Equation The Momentum Conservation Equations The Thermal Energy Conservation Equation 4.2.3 The Boundary Layer Simpliﬁcations The Continuity Equation The x-Momentum Equation The y-Momentum Equation The Thermal Energy Equation Dimensional Analysis in Convection 4.3.1 Introduction 4.3.2 The Dimensionless Boundary Layer Equations The Dimensionless Continuity Equation The Dimensionless Momentum Equation in the Boundary Layer The Dimensionless Thermal Energy Equation in the Boundary Layer 4.3.3 Correlating the Solutions of the Dimensionless Equations The Friction and Drag Coefﬁcients The Nusselt Number EXAMPLE 4.3-1: SUB-SCALE TESTING OF A CUBE-SHAPED MODULE

4.4

xi

4.3.4 The Reynolds Analogy (revisited) Self-Similar Solution for Laminar Flow over a Flat Plate 4.4.1 Introduction 4.4.2 The Blasius Solution The Problem Statement The Similarity Variables The Problem Transformation Numerical Solution 4.4.3 The Temperature Solution The Problem Statement The Similarity Variables The Problem Transformation Numerical Solution 4.4.4 The Falkner-Skan Transformation∗ (E12)

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

483 483 484 485 488 492 494 495 495 495 495 496 498 500 500 501 502 503 506 506 508 508 509 509 511 511 513 515 520 521 521 522 522 522 526 530 535 535 536 536 538 542

xii

4.5

4.6

4.7

4.8

Contents

Turbulent Boundary Layer Concepts 4.5.1 Introduction 4.5.2 A Conceptual Model of the Turbulent Boundary Layer The Reynolds Averaged Equations 4.6.1 Introduction 4.6.2 The Averaging Process The Reynolds Averaged Continuity Equation The Reynolds Averaged Momentum Equation The Reynolds Averaged Thermal Energy Equation The Laws of the Wall 4.7.1 Introduction 4.7.2 Inner Variables 4.7.3 Eddy Diffusivity of Momentum 4.7.4 The Mixing Length Model 4.7.5 The Universal Velocity Proﬁle 4.7.6 Eddy Diffusivity of Momentum Models 4.7.7 Wake Region 4.7.8 Eddy Diffusivity of Heat Transfer 4.7.9 The Thermal Law of the Wall Integral Solutions 4.8.1 Introduction 4.8.2 The Integral Form of the Momentum Equation Derivation of the Integral Form of the Momentum Equation Application of the Integral Form of the Momentum Equation EXAMPLE 4.8-1: PLATE WITH TRANSPIRATION

4.8.3

4.9

The Integral Form of the Energy Equation Derivation of the Integral Form of the Energy Equation Application of the Integral Form of the Energy Equation 4.8.4 Integral Solutions for Turbulent Flows External Flow Correlations 4.9.1 Introduction 4.9.2 Flow over a Flat Plate Friction Coefﬁcient Nusselt Number EXAMPLE 4.9-1: PARTIALLY SUBMERGED PLATE

4.9.3

Unheated Starting Length Constant Heat Flux Flow over a Rough Plate Flow across a Cylinder Drag Coefﬁcient Nusselt Number

EXAMPLE 4.9-2: HOT WIRE ANEMOMETER

4.9.4

Flow across a Bank of Cylinders Non-Circular Extrusions Flow past a Sphere

EXAMPLE 4.9-3: BULLET TEMPERATURE

Problems References

∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

542 542 543 548 548 549 550 551 554 556 556 557 560 561 562 565 566 567 568 571 571 571 571 575 580 584 584 587 591 593 593 593 593 598 603 606 606 607 609 611 613 615 617 617 618 620 624 633

Contents

5 5.1

5.2

xiii

INTERNAL FORCED CONVECTION r 635 Internal Flow Concepts 5.1.1 Introduction 5.1.2 Momentum Considerations The Mean Velocity The Laminar Hydrodynamic Entry Length Turbulent Internal Flow The Turbulent Hydrodynamic Entry Length The Friction Factor 5.1.3 Thermal Considerations The Mean Temperature The Heat Transfer Coefﬁcient and Nusselt Number The Laminar Thermal Entry Length Turbulent Internal Flow Internal Flow Correlations 5.2.1 Introduction 5.2.2 Flow Classiﬁcation 5.2.3 The Friction Factor Laminar Flow Turbulent Flow EES’ Internal Flow Convection Library EXAMPLE 5.2-1: FILLING A WATERING TANK

5.2.4

The Nusselt Number Laminar Flow Turbulent Flow

EXAMPLE 5.2-2: DESIGN OF AN AIR HEATER

5.3

The Energy Balance 5.3.1 Introduction 5.3.2 The Energy Balance 5.3.3 Prescribed Heat Flux Constant Heat Flux 5.3.4 Prescribed Wall Temperature Constant Wall Temperature 5.3.5 Prescribed External Temperature

5.4

Analytical Solutions for Internal Flows 5.4.1 Introduction 5.4.2 The Momentum Equation Fully Developed Flow between Parallel Plates The Reynolds Equation∗ (E13) Fully Developed Flow in a Circular Tube∗ (E14) 5.4.3 The Thermal Energy Equation Fully Developed Flow through a Round Tube with a Constant Heat Flux Fully Developed Flow through Parallel Plates with a Constant Heat Flux Numerical Solutions to Internal Flow Problems 5.5.1 Introduction 5.5.2 Hydrodynamically Fully Developed Laminar Flow EES’ Integral Command

EXAMPLE 5.3-1: ENERGY RECOVERY WITH AN ANNULAR JACKET

5.5

∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

635 635 635 637 638 638 640 641 644 644 645 646 648 649 649 650 650 651 654 656 657 661 662 667 668 671 671 671 673 674 674 674 675 677 686 686 686 687 689 689 689 691 695 697 697 698 702

xiv

Contents

The Euler Technique The Crank-Nicolson Technique MATLAB’s Ordinary Differential Equation Solvers 5.5.3 Hydrodynamically Fully Developed Turbulent Flow Problems References 6 6.1

6.2

NATURAL CONVECTION r 735 Natural Convection Concepts 6.1.1 Introduction 6.1.2 Dimensionless Parameters for Natural Convection Identiﬁcation from Physical Reasoning Identiﬁcation from the Governing Equations Natural Convection Correlations 6.2.1 Introduction 6.2.2 Plate Heated or Cooled Vertical Plate Horizontal Heated Upward Facing or Cooled Downward Facing Plate Horizontal Heated Downward Facing or Cooled Upward Facing Plate Plate at an Arbitrary Tilt Angle EXAMPLE 6.2-1: AIRCRAFT FUEL ULLAGE HEATER

6.2.3

Sphere

EXAMPLE 6.2-2: FRUIT IN A WAREHOUSE

6.2.4

6.2.5

Cylinder Horizontal Cylinder Vertical Cylinder Open Cavity Vertical Parallel Plates

EXAMPLE 6.2-3: HEAT SINK DESIGN

6.2.6 6.2.7

Enclosures Combined Free and Forced Convection

EXAMPLE 6.2-4: SOLAR FLUX METER

6.3 6.4

7

Self-Similar Solution∗ (E15) Integral Solution∗ (E16) Problems References

735 735 735 736 739 741 741 741 742 744 745 747 748 752 753 757 757 758 760 761 763 766 768 769 772 772 773 777

BOILING AND CONDENSATION r 778

7.1 7.2

Introduction Pool Boiling 7.2.1 Introduction 7.2.2 The Boiling Curve 7.2.3 Pool Boiling Correlations

7.3

Flow Boiling 7.3.1 Introduction 7.3.2 Flow Boiling Correlations

EXAMPLE 7.2-1: COOLING AN ELECTRONICS MODULE USING NUCLEATE BOILING

EXAMPLE 7.3-1: CARBON DIOXIDE EVAPORATING IN A TUBE ∗

704 706 710 712 723 734

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

778 779 779 780 784 786 790 790 791 794

Contents

7.4

xv

Film Condensation 7.4.1 Introduction 7.4.2 Solution for Inertia-Free Film Condensation on a Vertical Wall 7.4.3 Correlations for Film Condensation Vertical Wall EXAMPLE 7.4-1: WATER DISTILLATION DEVICE

7.5

8 8.1

Horizontal, Downward Facing Plate Horizontal, Upward Facing Plate Single Horizontal Cylinder Bank of Horizontal Cylinders Single Horizontal Finned Tube Flow Condensation 7.5.1 Introduction 7.5.2 Flow Condensation Correlations Problems References HEAT EXCHANGERS r 823 Introduction to Heat Exchangers 8.1.1 Introduction 8.1.2 Applications of Heat Exchangers 8.1.3 Heat Exchanger Classiﬁcations and Flow Paths 8.1.4 Overall Energy Balances 8.1.5 Heat Exchanger Conductance Fouling Resistance EXAMPLE 8.1-1: CONDUCTANCE OF A CROSS-FLOW HEAT EXCHANGER

8.1.6

Compact Heat Exchanger Correlations

EXAMPLE 8.1-2: CONDUCTANCE OF A CROSS-FLOW HEAT EXCHANGER (REVISITED)

8.2

The Log-Mean Temperature Difference Method 8.2.1 Introduction 8.2.2 LMTD Method for Counter-Flow and Parallel-Flow Heat Exchangers 8.2.3 LMTD Method for Shell-and-Tube and Cross-Flow Heat Exchangers

8.3

The Effectiveness-NTU Method 8.3.1 Introduction 8.3.2 The Maximum Heat Transfer Rate 8.3.3 Heat Exchanger Effectiveness

EXAMPLE 8.2-1: PERFORMANCE OF A CROSS-FLOW HEAT EXCHANGER

EXAMPLE 8.3-1: PERFORMANCE OF A CROSS-FLOW HEAT EXCHANGER (REVISITED)

8.3.4

8.4

8.5 ∗

798 798 799 805 805 807 810 811 811 811 811 812 812 813 815 821

Further Discussion of Heat Exchanger Effectiveness Behavior as CR Approaches Zero Behavior as NTU Approaches Zero Behavior as NTU Becomes Inﬁnite Heat Exchanger Design Pinch Point Analysis 8.4.1 Introduction 8.4.2 Pinch Point Analysis for a Single Heat Exchanger 8.4.3 Pinch Point Analysis for a Heat Exchanger Network Heat Exchangers with Phase Change

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

823 823 823 824 828 831 831 832 838 841 841 841 842 847 848 851 851 852 853 858 861 862 863 864 865 867 867 867 872 876

xvi

Contents

8.5.1 Introduction 8.5.2 Sub-Heat Exchanger Model for Phase-Change 8.6 Numerical Model of Parallel- and Counter-Flow Heat Exchangers 8.6.1 Introduction 8.6.2 Numerical Integration of Governing Equations Parallel-Flow Conﬁguration Counter-Flow Conﬁguration∗ (E17) 8.6.3 Discretization into Sub-Heat Exchangers Parallel-Flow Conﬁguration Counter-Flow Conﬁguration∗ (E18) 8.6.4 Solution with Axial Conduction∗ (E19) 8.7 Axial Conduction in Heat Exchangers 8.7.1 Introduction 8.7.2 Approximate Models for Axial Conduction Approximate Model at Low λ Approximate Model at High λ Temperature Jump Model 8.8 Perforated Plate Heat Exchangers 8.8.1 Introduction 8.8.2 Modeling Perforated Plate Heat Exchangers 8.9 Numerical Modeling of Cross-Flow Heat Exchangers 8.9.1 Introduction 8.9.2 Finite Difference Solution Both Fluids Unmixed with Uniform Properties Both Fluids Unmixed with Temperature-Dependent Properties One Fluid Mixed, One Fluid Unmixed∗ (E20) Both Fluids Mixed∗ (E21) 8.10 Regenerators 8.10.1 Introduction 8.10.2 Governing Equations 8.10.3 Balanced, Symmetric Flow with No Entrained Fluid Heat Capacity Utilization and Number of Transfer Units Regenerator Effectiveness 8.10.4 Correlations for Regenerator Matrices Packed Bed of Spheres Screens Triangular Passages EXAMPLE 8.10-1: AN ENERGY RECOVERY WHEEL

8.10.5 Numerical Model of a Regenerator with No Entrained Heat Capacity∗ (E22) Problems References 9

MASS TRANSFER∗ (E23) r 974 Problems

10

974

RADIATION r 979

10.1 Introduction to Radiation 10.1.1 Radiation ∗

876 876 888 888 888 889 896 897 897 902 902 903 903 905 907 907 909 911 911 913 919 919 920 920 927 936 936 937 937 939 942 942 944 948 950 951 952 953 962 962 973

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

979 979

Contents

xvii

10.1.2 The Electromagnetic Spectrum 10.2 Emission of Radiation by a Blackbody 10.2.1 Introduction 10.2.2 Blackbody Emission Planck’s Law Blackbody Emission in Speciﬁed Wavelength Bands EXAMPLE 10.2-1: UV RADIATION FROM THE SUN

10.3 Radiation Exchange between Black Surfaces 10.3.1 Introduction 10.3.2 View Factors The Enclosure Rule Reciprocity Other View Factor Relationships The Crossed and Uncrossed String Method EXAMPLE 10.3-1: CROSSED AND UNCROSSED STRING METHOD

View Factor Library EXAMPLE 10.3-2: THE VIEW FACTOR LIBRARY

10.3.3 Blackbody Radiation Calculations The Space Resistance EXAMPLE 10.3-3: APPROXIMATE TEMPERATURE OF THE EARTH

N-Surface Solutions EXAMPLE 10.3-4: HEAT TRANSFER IN A RECTANGULAR ENCLOSURE EXAMPLE 10.3-5: DIFFERENTIAL VIEW FACTORS: RADIATION EXCHANGE BETWEEN PARALLEL PLATES

10.4 Radiation Characteristics of Real Surfaces 10.4.1 Introduction 10.4.2 Emission of Real Materials Intensity Spectral, Directional Emissivity Hemispherical Emissivity Total Hemispherical Emissivity The Diffuse Surface Approximation The Diffuse Gray Surface Approximation The Semi-Gray Surface 10.4.3 Reﬂectivity, Absorptivity, and Transmittivity Diffuse and Specular Surfaces Hemispherical Reﬂectivity, Absorptivity, and Transmittivity Kirchoff’s Law Total Hemispherical Values The Diffuse Surface Approximation The Diffuse Gray Surface Approximation The Semi-Gray Surface EXAMPLE 10.4-1: ABSORPTIVITY AND EMISSIVITY OF A SOLAR SELECTIVE SURFACE

10.5 Diffuse Gray Surface Radiation Exchange 10.5.1 Introduction 10.5.2 Radiosity 10.5.3 Gray Surface Radiation Calculations EXAMPLE 10.5-1: RADIATION SHIELD EXAMPLE 10.5-2: EFFECT OF OVEN SURFACE PROPERTIES ∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

980 981 981 982 982 985 987 989 989 989 990 991 992 992 993 996 998 1001 1001 1002 1006 1007 1009 1012 1012 1012 1012 1014 1014 1015 1016 1016 1016 1018 1019 1020 1020 1022 1023 1023 1023 1024 1027 1027 1028 1029 1032 1037

xviii

Contents

10.5.4 The Fˆ Parameter EXAMPLE 10.5-3: RADIATION HEAT TRANSFER BETWEEN PARALLEL PLATES

10.5.5 Radiation Exchange for Semi-Gray Surfaces EXAMPLE 10.5-4: RADIATION EXCHANGE IN A DUCT WITH SEMI-GRAY SURFACES

10.6 Radiation with other Heat Transfer Mechanisms 10.6.1 Introduction 10.6.2 When Is Radiation Important? 10.6.3 Multi-Mode Problems 10.7 The Monte Carlo Method 10.7.1 Introduction 10.7.2 Determination of View Factors with the Monte Carlo Method Select a Location on Surface 1 Select the Direction of the Ray Determine whether the Ray from Surface 1 Strikes Surface 2 10.7.3 Radiation Heat Transfer Determined by the Monte Carlo Method Problems References

1043 1046 1050 1051 1055 1055 1055 1057 1058 1058 1058 1060 1060 1061 1068 1077 1088

Appendices

1089 1089 1089 1089 1090 1090

Index

1091

A.1: Introduction to EES∗ (E24) A.2: Introduction to Maple∗ (E25) A.3: Introduction to MATLAB∗ (E26) A.4: Introduction to FEHT∗ (E27) A.5: Introduction to Economics ∗ (E28)

∗

Section can be found on the website that accompanies this book (www.cambridge.org/nellisandklein)

PREFACE

The single objective of this book is to provide engineers with the capability, tools, and conﬁdence to solve real-world heat transfer problems. This objective has resulted in a textbook that differs from existing heat transfer textbooks in several ways. First, this textbook includes many topics that are typically not covered in undergraduate heat transfer textbooks. Examples are the detailed presentations of mathematical solution methods such as Bessel functions, Laplace transforms, separation of variables, Duhamel’s theorem, and Monte Carlo methods as well as high order explicit and implicit numerical integration algorithms. These analytical and numerical solution methods are applied to advanced topics that are ordinarily not considered in a heat transfer textbook. Judged by its content, this textbook should be considered as a graduate text. There is sufﬁcient material for two-semester courses in heat transfer. However, the presentation does not presume previous knowledge or expertise. This book can be (and has been) successfully used in a single-semester undergraduate heat transfer course by appropriately selecting from the available topics. Our recommendations on what topics can be included in a ﬁrst heat transfer course are provided in the suggested syllabus. The reason that this book can be used for a ﬁrst course (despite its expanded content) and the reason it is also an effective graduate-level textbook is that all concepts and methods are presented in detail, starting at the beginning. The derivation of important results is presented completely, without skipping steps, in order to improve readability, reduce student frustration, and improve retention. You will not ﬁnd many places in this textbook where it states that “it can be shown that . . . ” The use of examples, solved and explained in detail, is ubiquitous in this textbook. The examples are not trivial, “textbook” exercises, but rather complex and timely real-world problems that are of interest by themselves. As with the presentation, the solutions to these examples are complete and do not skip steps. Another signiﬁcant difference between this textbook and most existing heat transfer textbooks is its integration of modern computational tools. The engineering student and practicing engineer of today is expected to be proﬁcient with engineering computer tools. Engineering education must evolve accordingly. Most real engineering problems cannot be solved using a sequential set of calculations that can be accomplished with a pencil or hand calculator. Engineers must have the ability to quickly solve problems using the powerful computational tools that are available and essential for design, parametric study, and optimization of real-world systems. This book integrates the computational software packages Maple, MATLAB, FEHT, and Engineering Equation Solver (EES) directly with the heat transfer material. The speciﬁc commands and output associated with these software packages are presented as the theory is developed so that the integration is seamless rather than separated. The computational software tools used in this book share some important characteristics. They are used in industry and have existed for more than a decade; therefore, while this software will certainly continue to evolve, it is not likely to disappear. Educational versions of these software packages are available, and therefore the use of these xix

xx

Preface

tools should not represent an economic hardship to any academic institution or student. Useful versions of EES and FEHT are provided on the website that accompanies this textbook (www.cambridge.org/nellisandklein). With the help provided in the book, these tools are easy to learn and use. Students can become proﬁcient with all of them in a reasonable amount of time. Learning the computer tools will not detract signiﬁcantly from material coverage. To facilitate this learning process, tutorials for each of the software packages are provided on the companion website. The book itself is structured so that more advanced features of the software are introduced progressively, allowing students to become increasingly proﬁcient using these tools as they progress through the text. Most (if not all) of the tables and charts that have traditionally been required to solve heat transfer problems (for example, to determine properties, view factors, shape factors, convection relations, etc.) have been made available as functions and procedures in the EES software so that they can be easily accessed and used to solve problems. Indeed, the library of heat transfer functions that has been developed and integrated with EES as part of the preparation of this textbook enables a profound shift in the focus of the educational process. It is trivial to obtain, for example, a shape factor, a view factor, or a convection heat transfer coefﬁcient using the heat transfer library. Therefore, it is possible to assign problems involving design and optimization studies that would be computationally impossible without the computer tools. Integrating the study of heat transfer with computer tools does not diminish the depth of understanding of the underlying physics. Conversely, our experience indicates that the innate understanding of the subject matter is enhanced by appropriate use of these tools for several reasons. First, the software allows the student to tackle practical and relevant problems as opposed to the comparatively simple problems that must otherwise be assigned. Real-world engineering problems are more satisfying to the student. Therefore, the marriage of computer tools with theory motivates students to understand the governing physics as well as learn how to apply the computer tools. The use of these tools allows for coverage of more advanced material and more interesting and relevant problems. When a solution is obtained, students can carry out a more extensive investigation of its behavior and therefore obtain a more intuitive and complete understanding of the subject of heat transfer. This book is unusual in its linking of classical theory and modern computing tools. It ﬁlls an obvious void that we have encountered in teaching both undergraduate and graduate heat transfer courses. The text was developed over many years from our experiences teaching Introduction to Heat Transfer (an undergraduate course) and Heat Transfer (a ﬁrst-year graduate course) at the University of Wisconsin. It is our hope that this text will not only be useful during the heat transfer course, but also provide a lifelong resource for practicing engineers. G. F. Nellis S. A. Klein May, 2008

Acknowledgments

The development of this book has taken several years and a substantial effort. This has only been possible due to the collegial and supportive atmosphere that makes the Mechanical Engineering Department at the University of Wisconsin such a unique and impressive place. In particular, we would like to acknowledge Tim Shedd, Bill Beckman, Doug Reindl, John Pfotenhauer, Roxann Engelstad, and Glen Myers for their encouragement throughout the process. Several years of undergraduate and graduate students have used our initial drafts of this manuscript. They have had to endure carrying two heavy volumes of poorly bound paper with no index and many typographical errors. Their feedback has been invaluable to the development of this book. We have had the extreme good fortune to have had dedicated and insightful teachers. These include Glen Myers, John Mitchell, Bill Beckman, Joseph Smith Jr., John Brisson, Borivoje Mikic, and John Lienhard V. These individuals, among others, have provided us with an indication of the importance of teaching and provided an inspiration to us for writing this book. Preparing this book has necessarily reduced the “quality time” available to spend with our families. We are most grateful to them for this indulgence. In particular, we wish to thank Jill, Jacob, and Spencer and Sharon Nellis and Jan Klein. We could have not completed this book without their continuous support. Finally, we are indebted to Cambridge University Press and in particular Peter Gordon for giving us this opportunity and for helping us with the endless details needed to bring our original idea to this ﬁnal state.

xxi

STUDY GUIDE

This book has been developed for use in either a graduate or undergraduate level course in heat transfer. A sample program of study is laid out below for a one-semester graduate course (consisting of 45 class sessions). Graduate heat transfer class Day 1 2 3 4 5 6 7 8 9 10 11 12

Sections in Book 1.1 1.2 2.8 1.3 1.4, 1.5 1.6 1.7 1.8 2.2 2.2 2.4 3.1

13

3.2

14 15 16 17 18 19 20 21 22 23

3.3 3.3 3.4 3.4 3.5 3.8 4.1 4.2, 4.3 4.4 4.5, 4.6

24 25 26 27 28 29 30 31

4.7 4.8 4.8, 4.9 5.1, 5.2 5.3 5.4 5.5 6.1, 6.2

Topic Conduction heat transfer 1-D steady conduction and resistance concepts Resistance approximations 1-D steady conduction with generation Numerical solutions with EES and MATLAB Fin solution, ﬁn efﬁciency, and ﬁnned surfaces Other constant cross-section extended surface problems Bessel function solutions 2-D conduction, separation of variables 2-D conduction, separation of variables Superposition Transient, lumped capacitance problems – analytical solutions Transient, lumped capacitance problems – numerical solutions Semi-inﬁnite bodies, diffusive time constant Semi-inﬁnite bodies, self-similar solution Laplace transform solutions to lumped capacitance problems Laplace transform solutions to 1-D transient problems Separation of variables for 1-D transient problems Numerical solutions to 1-D transient problems Laminar boundary layer concepts The boundary layer equations & dimensionless parameters Blasius solution for ﬂow over a ﬂat plate Turbulent boundary layer concepts, Reynolds averaged equations Mixing length models and the laws of the wall Integral solutions Integral solutions, external ﬂow correlations Internal ﬂow concepts and correlations The energy balance Analytical solutions to internal ﬂow problems Numerical solutions to internal ﬂow problems Natural convection concepts and correlations xxiii

xxiv

32 33 34 35 36 37 38 39 40 41 42 43 44 45

Study Guide

8.1 8.2, 8.3 8.5 8.7 8.8, 8.10 10.1, 10.2 10.3 10.3 10.4 10.5 10.5 10.5 10.7 10.7

Introduction to heat exchangers The LMTD and ε-NTU forms of the solutions Heat exchangers with phase change Axial conduction in heat exchangers Perforated plate heat exchangers and regenerators Introduction to radiation, Blackbody emissive power View factors and the space resistance Blackbody radiation exchange Real surfaces, Kirchoff’s law Gray surface radiation exchange Gray surface radiation exchange Semi-gray surface radiation exchange Introduction to Monte Carlo techniques Introduction to Monte Carlo techniques

A sample program of study is laid out below for a one-semester undergraduate course (consisting of 45 class sessions). Undergraduate heat transfer class Day 1 2 3

Sections in Book A.1 1.2.2-1.2.3 1.2.4-1.2.6

4 5 6 7 8 9 10 11 12 13 14 15

1.3.1-1.3.3 1.4 1.6.1-1.6.3 1.6.4-1.6.6 1.9.1 2.1 2.8.1-2.8.2 2.9 2.5 3.1 3.2.1, 3.2.2

16 17

3.3.1-3.3.2 3.3.2, 3.3.4

18 19 20 21 22 23 24 25

3.5.1-3.5.2 3.8.1-3.8.2 4.1 4.2, 4.3 4.5 4.9.1-4.9.2 4.9.3-4.9.4 5.1

Topic Review of thermodynamics, Using EES 1-D steady conduction, resistance concepts and circuits 1-D steady conduction in radial systems, other thermal resistance More thermal resistance problems 1-D steady conduction with generation Numerical solutions with EES The extended surface approximation and the ﬁn solution Fin behavior, ﬁn efﬁciency, and ﬁnned surfaces Numerical solutions to extended surface problems 2-D steady-state conduction, shape factors Resistance approximations Conduction through composite materials Numerical solution to 2-D steady-state problems with EES Lumped capacitance assumption, the lumped time constant Numerical solution to lumped problems (Euler’s, Heun’s, Crank-Nicolson) Semi-inﬁnite body, the diffusive time constant Approximate models of diffusion, other semi-inﬁnite solutions Solutions to 1-D transient conduction in a bounded geometry Numerical solution to 1-D transient conduction using EES Introduction to laminar boundary layer concepts Dimensionless numbers Introduction to turbulent boundary layer concepts Correlations for external ﬂow over a plate Correlations for external ﬂow over spheres and cylinders Internal ﬂow concepts

Study Guide

26 27 28 29 30 31

5.2 5.3

32 33

7.1, 7.2 7.3, 7.4.3, 7.5

34

8.1

35 36 37 38

8.2 8.3.1-8.3.3 8.3.4 8.10.1, 8.10.3-4

39 40 41 42 43 44 45

10.1, 10.2 10.3.1-10.3.2 10.3.3 10.4 10.5.1-10.5.3

6.1 6.2.1-6.2.3 6.2.4-6.2.7

10.6

xxv

Internal ﬂow correlations Energy balance for internal ﬂows Internal ﬂow problems Introduction to natural convection Natural convection correlations Natural convection correlations and combined forced/free convection Pool boiling Correlations for ﬂow boiling, ﬂow condensation, and ﬁlm condensation Introduction to heat exchangers, compact heat exchanger correlations The LMTD Method The ε-NTU Method Limiting behaviors of the ε-NTU Method Regenerators, solution for balanced & symmetric regenerator, packings Introduction to radiation, blackbody emission View factors Blackbody radiation exchange Real surfaces and Kirchoff’s law Gray surface radiation exchange Gray surface radiation exchange Radiation with other heat transfer mechanisms

NOMENCLATURE

ith coefﬁcient of a series solution cross-sectional area (m2 ) minimum ﬂow area (m2 ) projected area (m2 ) surface area (m2 ) surface area of a ﬁn (m2 ) prime (total) surface area of a ﬁnned surface (m2 ) aspect ratio of a rectangular duct area ratio of ﬁn tip to ﬁn surface area attenuation (-) parameter in the blowing factor (-) blowing factor (-) Biot number (-) boiling number (-) Brinkman number speciﬁc heat capacity (J/kg-K) concentration (-) speed of light (m/s) speciﬁc heat capacity of an air-water mixture on a unit mass of air basis ca (J/kga -K) speciﬁc heat capacity of an air-water mixture along the saturation line on a ca,sat unit mass of air basis (J/kga -K) effective speciﬁc heat capacity of a composite (J/kg-K) ceff ratio of the energy carried by a micro-scale energy carrier to its cms temperature (J/K) speciﬁc heat capacity at constant volume (J/kg-K) cv C total heat capacity (J/K) ˙ C capacitance rate of a ﬂow (W/K) C1 , C2 , . . . undetermined constants dimensionless coefﬁcient for critical heat ﬂux correlation (-) Ccrit drag coefﬁcient (-) CD friction coefﬁcient (-) Cf Cf average friction coefﬁcient (-) coefﬁcient for laminar plate natural convection correlation (-) Clam dimensionless coefﬁcient for nucleate boiling correlation (-) Cnb capacity ratio (-) CR coefﬁcient for turbulent, horizontal upward plate natural conv. Cturb,U correlation (-) coefﬁcient for turbulent, vertical plate natural convection correlation (-) Cturb,V ai Ac Amin Ap As As,ﬁn Atot AR ARtip Att B BF Bi Bo Br c

xxvii

xxviii

Co CTE D Dh dx dy e err ˙ E E Eb Eλ Eb,λ Ec f

f fl F F 0−λ1 Fi,j Fˆ i,j fd Fo Fr Frmod g g˙ g˙ g˙ eff g˙ v G Gλ Ga Gr Gz h h h˜ hD hD hl

Nomenclature

convection number (-) coefﬁcient of thermal expansion (1/K) diameter (m) diffusion coefﬁcient (m2 /s) hydraulic diameter (m) differential in the x-direction (m) differential in the y-direction (m) size of surface roughness (m) convergence or numerical error rate of thermal energy carried by a mass ﬂow (W) total emissive power (W/m2 ) total blackbody emissive power (W/m2 ) spectral emissive power (W/m2 -μm) blackbody spectral emissive power (W/m2 -μm) Eckert number (-) frequency (Hz) dimensionless stream function, for Blasius solution (-) friction factor (-) average friction factor (-) friction factor for liquid-only ﬂow in ﬂow boiling (-) force (N) correction-factor for log-mean temperature difference (-) external fractional function (-) view factor from surface i to surface j (-) the “F-hat” parameter characterizing radiation from surface i to surface j (-) fractional duty for a pinch-point analysis (-) Fourier number (-) Froude number (-) modiﬁed Froude number (-) acceleration of gravity (m/s2 ) rate of thermal energy generation (W) rate of thermal energy generation per unit volume (W/m3 ) effective rate of generation per unit volume of a composite (W/m3 ) rate of thermal energy generation per unit volume due to viscous dissipation (W/m3 ) mass ﬂux or mass velocity (kg/m2 -s) total irradiation (W/m2 ) spectral irradiation (W/m2 -μm) Galileo number (-) Grashof number (-) Graetz number (-) local heat transfer coefﬁcient (W/m2 -K) average heat transfer coefﬁcient (W/m2 -K) dimensionless heat transfer coefﬁcient for ﬂow boiling correlation (-) mass transfer coefﬁcient (m/s) average mass transfer coefﬁcient (m/s) superﬁcial heat transfer coefﬁcient for the liquid phase (W/m2 -K)

Nomenclature

hrad i

ia I Ie Ii j J jH k kB kc ke keff Kn l1 l1,2 L L+ L∗ Lchar Lchar,vs Lcond Lﬂow Lml Lms Le M m m ˙ m ˙ mms mf MW n nms n˙ N Ns Nu

xxix

the equivalent heat transfer coefﬁcient associated with radiation (W/m2 -K) index of node (-) index of eigenvalue (-) index of term in a series solution (-) speciﬁc enthalpy (J/kg-K) √ square root of negative one, −1 speciﬁc enthalpy of an air-water mixture on a per unit mass of air basis (J/kga ) current (ampere) intensity of emitted radiation (W/m2 -μm-steradian) intensity of incident radiation (W/m2 -μm-steradian) index of node (-) index of eigenvalue (-) radiosity (W/m2 ) Colburn jH factor (-) thermal conductivity (W/m-K) Bolzmann’s constant (J/K) contraction loss coefﬁcient (-) expansion loss coefﬁcient (-) effective thermal conductivity of a composite (W/m-K) Knudsen number (-) Lennard-Jones 12-6 potential characteristic length for species 1 (m) characteristic length of a mixture of species 1 and species 2 (m) length (m) dimensionless length for a hydrodynamically developing internal ﬂow (-) dimensionless length for a thermally developing internal ﬂow (-) characteristic length of the problem (m) the characteristic size of the viscous sublayer (m) length for conduction (m) length in the ﬂow direction (m) mixing length (m) distance between interactions of micro-scale energy or momentum carriers (m) Lewis number (-) number of nodes (-) mass (kg) ﬁn parameter (1/m) mass ﬂow rate (kg/s) mass ﬂow rate per unit area (kg/m2 -s) mass of microscale momentum carrier (kg/carrier) mass fraction (-) molar mass (kg/kgmol) number density (#/m3 ) number density of the micro-scale energy carriers (#/m3 ) molar transfer rate per unit area (kgmol/m2 -s) number of nodes (-) number of moles (kgmol) number of species in a mixture (-) Nusselt number (-)

xxx

Nu NTU p P p∞ p˜ Pe per Pr Prturb q˙ q˙ i to j q˙ max q˙ q˙ s q˙ s,crit Q ˜ Q r r˜ R

RA Rac Rad Rbl Rc Rconv Rcyl Re Rf Rﬁn Ri,j Riso RL Rpw Rrad Rs,i Rsemi-∞ Rsph Rtot Runiv Rc Rf Ra

Nomenclature

average Nusselt number (-) number of transfer units (-) pressure (Pa) pitch (m) LMTD effectiveness (-) probability distribution (-) free-stream pressure (Pa) dimensionless pressure (-) Peclet number (-) perimeter (m) Prandtl number (-) turbulent Prandtl number (-) rate of heat transfer (W) rate of radiation heat transfer from surface i to surface j (W) maximum possible rate of heat transfer, for an effectiveness solution (W) heat ﬂux, rate of heat transfer per unit area (W/m2 ) surface heat ﬂux (W/m2 ) critical heat ﬂux for boiling (W/m2 ) total energy transfer by heat (J) dimensionless total energy transfer by heat (-) radial coordinate (m) radius (m) dimensionless radial coordinate (-) thermal resistance (K/W) ideal gas constant (J/kg-K) LMTD capacitance ratio (-) thermal resistance approximation based on average area limit (K/W) thermal resistance to axial conduction in a heat exchanger (K/W) thermal resistance approximation based on adiabatic limit (K/W) thermal resistance of the boundary layer (K/W) thermal resistance due to solid-to-solid contact (K/W) thermal resistance to convection from a surface (K/W) thermal resistance to radial conduction through a cylindrical shell (K/W) electrical resistance (ohm) thermal resistance due to fouling (K/W) thermal resistance of a ﬁn (K/W) the radiation space resistance between surfaces i and j (1/m2 ) thermal resistance approximation based on isothermal limit (K/W) thermal resistance approximation based on average length limit (K/W) thermal resistance to radial conduction through a plane wall (K/W) thermal resistance to radiation (K/W) the radiation surface resistance for surface i (1/m2 ) thermal resistance approximation for a semi-inﬁnite body (K/W) thermal resistance to radial conduction through a spherical shell (K/W) thermal resistance of a ﬁnned surface (K/W) universal gas constant (8314 J/kgmol-K) area-speciﬁc contact resistance (K-m2 /W) area-speciﬁc fouling resistance (K-m2 /W) Rayleigh number (-)

Nomenclature

Re Recrit RH RR s S Sc Sh Sh St t tsim th tol T Tb Tﬁlm Tm Ts Tsat T∞ T∗ T T TR Tt TX TY th U u uchar uf um u∞ u∗ u+ u˜ u u UA v vδ vms

xxxi

Reynolds number (-) critical Reynold number for transition to turbulence (-) relative humidity (-) radius ratio of an annular duct (-) Laplace transformation variable (1/s) generic coordinate (m) shape factor (m) channel spacing (m) Schmidt number (-) Sherwood number (-) average Sherwood number (-) Stanton number (-) time (s) simulated time (s) thickness (m) convergence tolerance temperature (K) base temperature of ﬁn (K) ﬁlm temperature (K) mean or bulk temperature (K) surface temperature (K) saturation temperature (K) free-stream or ﬂuid temperature (K) eddy temperature ﬂuctuation (K) ﬂuctuating component of temperature (K) average temperature (K) temperature solution that is a function of r, for separation of variables temperature solution that is a function of t, for separation of variables temperature solution that is a function of x, for separation of variables temperature solution that is a function of y, for separation of variables thickness (m) internal energy (J) utilization (-) speciﬁc internal energy (J/kg) velocity in the x-direction (m/s) characteristic velocity (m/s) frontal or upstream velocity (m/s) mean or bulk velocity (m/s) free-stream velocity (m/s) eddy velocity (m/s) inner velocity (-) dimensionless x-velocity (-) ﬂuctuating component of x-velocity (m/s) average x-velocity (m/s) conductance (W/K) velocity in the y- or r-directions (m/s) y-velocity at the outer edge of the boundary layer, approximate scale of y-velocity in a boundary layer (m/s) mean velocity of micro-scale energy or momentum carriers (m/s)

xxxii

v˜ v v V V˙ vf w w ˙ W x x˜ X xfd,h xfd,t Xtt y y+ y˜ Y z

Nomenclature

dimensionless y-velocity (-) ﬂuctuating component of y-velocity (m/s) average y-velocity (m/s) volume (m3 ) voltage (V) volume ﬂow rate (m3 /s) void fraction (-) velocity in the z-direction (m/s) rate of work transfer (W) width (m) total amount of work transferred (J) x-coordinate (m) quality (-) dimensionless x-coordinate (-) particular solution that is only a function of x hydrodynamic entry length (m) thermal entry length (m) Lockhart Martinelli parameter (-) y-coordinate (m) mole fraction (-) inner position (-) dimensionless y-coordinate (-) particular solution that is only a function of y z-coordinate (m)

Greek Symbols α

αeff αλ αλ,θ,φ β δ δd δm δvs δt ifus ivap p r

thermal diffusivity (m2 /s) absorption coefﬁcient (1/m) absorptivity or absorptance (-), total hemispherical absorptivity (-) surface area per unit volume (1/m) effective thermal diffusivity of a composite (m2 /s) hemispherical absorptivity (-) spectral directional absorptivity (-) volumetric thermal expansion coefﬁcient (1/K) ﬁlm thickness for condensation (m) boundary layer thickness (m) mass transfer diffusion penetration depth (m) concentration boundary layer thickness (m) momentum diffusion penetration depth (m) momentum boundary layer thickness (m) viscous sublayer thickness (m) energy diffusion penetration depth (m) thermal boundary layer thickness (m) latent heat of fusion (J/kg) latent heat of vaporization (J/kg) pressure drop (N/m2 ) distance in r-direction between adjacent nodes (m)

Nomenclature

T Te Tlm t tcrit x y ε εﬁn εH ελ ελ,θ,φ εM ε1 ε1,2 φ η ηﬁn ηo κ λ λi μ v θ θ˜ θ+ θR θt θX θXt θY θYt θZt

xxxiii

temperature difference (K) excess temperature (K) log-mean temperature difference (K) time step (s) time period (s) critical time step (s) distance in x-direction between adjacent nodes (m) distance in y-direction between adjacent nodes (m) heat exchanger effectiveness (-) emissivity or emittance (-), total hemispherical emissivity (-) ﬁn effectiveness (-) eddy diffusivity for heat transfer (m2 /s) hemispherical emissivity (-) spectral, directional emissivity (-) eddy diffusivity of momentum (m2 /s) Lennard-Jones 12-6 potential characteristic energy for species 1 (J) characteristic energy parameter for a mixture of species 1 and species 2 (J) porosity (-) phase angle (rad) spherical coordinate (rad) similarity parameter (-) efﬁciency (-) ﬁn efﬁciency (-) overall efﬁciency of a ﬁnned surface (-) ´ an ´ constant von Karm dimensionless axial conduction parameter (-) wavelength of radiation (μm) ith eigenvalue of a solution (1/m) viscosity (N-s/m2 ) frequency of radiation (1/s) temperature difference (K) angle (rad) spherical coordinate (rad) dimensionless temperature difference (-) inner temperature difference (-) temperature difference solution that is only a function of r, for separation of variables temperature difference solution that is only a function of t, for separation of variables temperature difference solution that is only a function of x, for separation of variables temperature difference solution that is only a function of x and t, for reduction of multi-dimensional transient problems temperature difference solution that is only a function of y, for separation of variables temperature difference solution that is only a function of y and t, for reduction of multi-dimensional transient problems temperature difference solution that is only a function of z and t, for reduction of multi-dimensional transient problems

xxxiv

ρ ρe ρeff ρλ ρλ,θ,ϕ σ

τ τdiff τlumped τλ τλ,θ,ϕ τs υ ω D ζ ζi

Nomenclature

density (kg/m3 ) reﬂectivity or reﬂectance (-), total hemispherical reﬂectivity (-) electrical resistivity (ohm-m) effective density of a composite (kg/m3 ) hemispherical reﬂectivity (-) spectral, directional reﬂectivity (-) surface tension (N/m), molecular radius (m) ratio of free-ﬂow to frontal area (-) Stefan-Boltzmann constant (5.67 × 10-8 W/m2 -K4 ) time constant (s) shear stress (Pa) transmittivity or transmittance (-), total hemispherical transmittivity (-) diffusive time constant (s) lumped capacitance time constant (s) hemispherical transmittivity (-) spectral, directional transmittivity (-) shear stress at surface (N/m2 ) kinematic viscosity (m2 /s) angular velocity (rad/s) humidity ratio (kgv /kga ) solid angle (steradian) dimensionless collision integral for diffusion (-) stream function (m2 /s) tilt angle (rad) curvature parameter for vertical cylinder, natural convection correlation (-) the ith dimensionless eigenvalue (-)

Superscripts o

at inﬁnite dilution

Subscripts a abs ac an app b bl bottom c C cc char cf

air absorbed axial conduction (in heat exchangers) analytical apparent approximate blackbody boundary layer bottom condensate ﬁlm corrected cold cold-side of a heat exchanger complex conjugate, for complex combination problems characteristic counter-ﬂow heat exchanger

Nomenclature

cond conv crit CTHB dc df diff eff emit evap ext f fc fd,h fd,t ﬁn h H

hs HTCB i

in ini int

j l lam LHS lumped m max min mod ms n nac nb nc no-ﬁn out p

conduction, conductive convection, convective critical cold-to-hot blow process dry coil downward facing diffusive transfer effective emitted evaporative external ﬂuid forced convection hydrodynamically fully developed thermally fully developed ﬁn, ﬁnned homogeneous solution hot hot-side of a heat exchanger constant heat ﬂux boundary condition on a hemisphere hot-to-cold blow process node i surface i species i inner inlet initial internal interface integration period node j surface j liquid laminar left-hand side lumped-capacitance mean or bulk melting maximum or maximum possible minimum or minimum possible modiﬁed micro-scale carrier normal without axial conduction (in heat exchangers) nucleate boiling natural convection without a ﬁn outer outlet particular (or non-homogeneous) solution

xxxv

xxxvi

pf pp r rad ref RHS s sat sat,l sat,v sc semi-∞ sh sph sur sus T top tot turb uf unﬁn v

w wb wc x x− x+ y ∞ 90◦

Nomenclature

parallel-ﬂow heat exchanger pinch-point regenerator matrix at position r radiation, radiative reference right-hand side at the surface saturated saturated section of a heat exchanger saturated liquid saturated vapor sub-cooled section of a heat exchanger semi-inﬁnite super-heated section of a heat exchanger sphere surroundings sustained solution constant temperature boundary condition at temperature T top total turbulent upward-facing not ﬁnned vapor vertical viscous dissipation water wet-bulb wet coil at position x in the x-direction in the negative x-direction in the positive x-direction at position y in the y-direction free-stream, ﬂuid solution that is 90◦ out of phase, for complex combination problems

Other notes A A A A A Aˆ

A

arbitrary variable ﬂuctuating component of variable A value of variable A on a unit length basis value of variable A on a unit area basis value of variable A on a unit volume basis dimensionless form of variable A a guess value or approximate value for variable A Laplace transform of the function A

Nomenclature

A A A dA δA A O(A)

average of variable A denotes that variable A is a vector denotes that variable A is a matrix differential change in the variable A uncertainty in the variable A ﬁnite change in the variable A order of magnitude of the variable A

xxxvii

1

One-Dimensional, Steady-State Conduction

1.1 Conduction Heat Transfer 1.1.1 Introduction Thermodynamics deﬁnes heat as a transfer of energy across the boundary of a system as a result of a temperature difference. According to this deﬁnition, heat by itself is an energy transfer process and it is therefore redundant to use the expression ‘heat transfer’. Heat has no option but to transfer and the expression ‘heat transfer’ reinforces the incorrect concept that heat is a property of a system that can be ‘transferred’ to another system. This concept was originally proposed in the 1800’s as the caloric theory (Keenan, 1958); heat was believed to be an invisible substance (having mass) that transferred from one system to another as a result of a temperature difference. Although the caloric theory has been disproved, it is still common to refer to ‘heat transfer’. Heat is the transfer of energy due to a temperature gradient. This transfer process can occur by two very different mechanisms, referred to as conduction and radiation. Conduction heat transfer occurs due to the interactions of molecular (or smaller) scale energy carriers within a material. Radiation heat transfer is energy transferred as electromagnetic waves. In a ﬂowing ﬂuid, conduction heat transfer occurs in the presence of energy transfer due to bulk motion (which is not a heat transfer) and this leads to a substantially more complex situation that is referred to as convection.

1.1.2 Thermal Conductivity Conduction heat transfer occurs due to the interactions of micro-scale energy carriers within a material; the type of energy carriers depends upon the structure of the material. For example, in a gas or a liquid, the energy carriers are individual molecules whereas the energy carriers in a solid may be electrons or phonons (i.e., vibrations in the structure of the solid). The transfer of energy by conduction is fundamentally related to the interactions of these energy carriers; more energetic (i.e., higher temperature) energy carriers transfer energy to less energetic (i.e., lower temperature) ones, resulting in a net ﬂow of energy from hot to cold (i.e., heat transfer). Regardless of the type of energy carriers involved, conduction heat transfer can be characterized by Fourier’s law, provided that the length and time scales of the problem are large relative to the distance and time between energy carrier interactions. Fourier’s law relates the heat ﬂux in any direction to the temperature gradient in that direction. For example: q˙ = −k

∂T ∂x

(1-1)

1

2

One-Dimensional, Steady-State Conduction

Aluminum

Water

(a)

(b)

Figure 1-1: Conductivity functions in EES for (a) compressible substances and (b) incompressible substances.

where q˙ is the heat ﬂux in the x-direction and k is the thermal conductivity of the material. Fourier’s law actually provides the deﬁnition of thermal conductivity: k=

−q˙ ∂T ∂x

(1-2)

Thermal conductivity is a material property that varies widely depending on the type of material and its state. Thermal conductivity has been extensively measured and values have been tabulated in various references (e.g., NIST (2005)). The thermal conductivity of many substances is available within the Engineering Equation Solver (EES) program. It is suggested that the reader go through the tutorial that is provided in Appendix A.1 in order to become familiar with EES. Appendix A.1 can be found on the web site associated with this book (www.cambridge.org/nellisandklein). To access the thermal conductivity functions in EES, select Function Info from the Options menu and select the Fluid Properties button; this action displays the properties that are available for compressible ﬂuids. Navigate to Conductivity in the left hand window and select the ﬂuid of interest in the right hand window (e.g., Water), as shown in Figure 1-1(a). Select Paste in order to place the call to the Conductivity function into the Equations Window. Select the Solid/liquid properties button in order to access the properties for incompressible ﬂuids and solids, as shown in Figure 1-1(b). The EES code below speciﬁes the unit system to be used (SI) and then computes the conductivity of water, air, and aluminum (kw , ka , and kal ) at T = 20◦ C and p = 1.0 atm, $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in T=converttemp(C,K,20 [C]) P=1.0 [atm]∗ convert(atm,Pa) k w=Conductivity(Water,T=T,P=P) k a=Conductivity(Air,T=T) k al=k (‘Aluminum’, T)

“temperature” “pressure” “conductivity of water at T and P” “conductivity of air at T and P” “conductivity of aluminum at T”

which leads to kw = 0.59 W/m-K, ka = 0.025 W/m-K, and kal = 236 W/m-K. The conductivity of aluminum (an electrically conductive metal) is approximately 10,000x that

1.1 Conduction Heat Transfer

3 Temperature q⋅ ′′x+

q⋅ ′′x−

Lms

Figure 1-2: Energy ﬂow through a plane at position x.

x+Lms

x-Lms x

Position

of air (a dilute gas, at these conditions), with water (a liquid) falling somewhere between these values. It is possible to understand the thermal conductivity of various materials based on the underlying characteristics of their energy carriers, the microscopic physical entities that are responsible for conduction. For example, the kinetic theory of gases may be used to provide an estimate of the thermal conductivity of a gas and the thermal conductivity of electrically conductive metals can be understood based on a careful study of electron behavior. Consider conduction through a material in which a temperature gradient has been established in the x-direction, as shown in Figure 1-2. We can evaluate (approximately) the net rate of energy transferred through a plane that is located at position x. The ﬂux of energy carriers passing through the plane from left-to-right (i.e., in the positive x-direction) is proportional to the number density of the energy carriers (nms ) and their mean velocity (vms ). The energy carriers that are moving in the positive x-direction experienced their last interaction at approximately x–Lms (on average), where Lms is the distance between energy carrier interactions. (Actually, the last interaction would not occur exactly at this position since the energy carriers are moving relative to each other and also in the y- and z-directions.) The energy associated with these left-to-right moving carriers is proportional to the temperature at position x-Lms (T x−Lms ). The energy per unit area passing through the plane from left-to-right (q˙ x+ ) is given approximately by: q˙ x+ ≈ nms vms cms T x−Lms #carriers area-time

(1-3)

energy carrier

where cms is the ratio of the energy of the carrier to its temperature. Similarly, the energy per unit area carried through the plane in the negative x-direction by the energy carriers that are moving from right-to-left (q˙ x− ) is given approximately by: q˙ x− ≈ nms vms cms T x+Lms

(1-4)

The net conduction heat ﬂux passing through the plane (q˙ ) is the difference between q˙ x+ and q˙ x− , q˙ ≈ nms vms cms (T x−Lms − T x+Lms )

(1-5)

which can be rearranged to yield: q˙ ≈ −nms vms cms Lms

(T x+Lms − T x−Lms ) Lms

(1-6)

Recall from calculus that the deﬁnition of the temperature gradient is: T x+dx − T x−dx ∂T = lim dx→0 ∂x 2 dx

(1-7)

4

One-Dimensional, Steady-State Conduction

Thermal conductivity (W/m-K)

400 pure aluminum

100

304 stainless steel

bronze 10 glass (pyrex) 1

liquid water at 100 bar 0.1

nitrogen gas at 100 bar

nitrogen gas at 1 bar 0.01 275 300 325 350 375 400 425 450 475 500 525 550 Temperature (K)

Figure 1-3: Thermal conductivity of various materials as a function of temperature.

In the limit that the length between energy carrier interactions (Lms ) is much less than the length scale that characterizes the problem (Lchar ): T x+dx − T x−dx ∂T (T x+Lms − T x−Lms ) ≈ lim = dx→0 2 Lms 2 dx ∂x

(1-8)

Equation (1-8) can be substituted into Eq. (1-6) to yield: ∂T q˙ ≈ −2 nms vms cms Lms ∂x

(1-9)

∝k

The ratio of the length between energy carrier interactions to the length scale that characterizes the problem is referred to as the Knudsen number. The Knudsen number (Kn) should be calculated in order to ensure that continuum concepts (like Fourier’s law) are applicable: Kn =

Lms Lchar

(1-10)

If the Knudsen number is not small then continuum theory breaks down. This limit may be reached in micro- and nano-scale systems where Lchar becomes small as well as in problems involving rareﬁed gas where Lms becomes large. Specialized theory for heat transfer is required in these limits and the interested reader is referred to books such as Tien et al. (1998), Chen (2005), and Cercignani (2000). Comparing Eq. (1-9) with Fourier’s law, Eq. (1-1), shows that the thermal conductivity is proportional to the product of the number of energy carriers per unit volume, their average velocity, the mean distance between their interactions, and the ratio of the amount of energy carried by each energy carrier to its temperature: k ∝ nms vms cms Lms

(1-11)

The scaling relation expressed by Eq. (1-11) is informative. Figure 1-3 illustrates the thermal conductivity of several common materials as a function of temperature. Notice that metals have the largest thermal conductivity, followed by other solids and liquids, while gases have the lowest conductivity. Gases are diffuse and thus the number density of the energy carriers (gas molecules) is substantially less than for other

1.2 Steady-State 1-D Conduction without Generation

5

forms of matter. Pure metals have the highest thermal conductivity because energy is carried primarily by electrons which are numerous and fast moving. The thermal conductivity and electrical resistivity of pure metals are related (by the Weidemann-Franz law) because both electricity and thermal energy are transported by the same mechanism, electron ﬂow. Alloys have lower thermal conductivity because the electron motion is substantially impeded by the impurities within the structure of the material; this effect is analogous to reducing the parameter Lms in Eq. (1-11). In non-metals, the energy is carried by phonons (or lattice vibrations), while in liquids the energy is carried by molecules. Thermal Conductivity of a Gas This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein) and discusses the application of Eq. (1-11) to the particular case of an ideal gas where the energy carriers are gas molecules.

1.2 Steady-State 1-D Conduction without Generation 1.2.1 Introduction Chapters 1 through 3 examine conduction problems using a variety of conceptual, analytical, and numerical techniques. We will begin with simple problems and move eventually to complex problems, starting with truly one-dimensional (1-D), steady-state problems and working ﬁnally to two-dimensional and transient problems. Throughout this book, problems will be solved both analytically and numerically. The development of an analytical or a numerical solution is accomplished using essentially the same steps regardless of the complexity of the problem; therefore, each class of problem will be solved in a uniform and rigorous fashion. The use of computer software tools facilitates the development of both analytical and numerical solutions; therefore, these tools are introduced and used side-by-side with the theory.

1.2.2 The Plane Wall In general, the temperature in a material will be a function of position (x, y, and z, in Cartesian coordinates) and time (t). The deﬁnition of steady-state is that the temperature is unchanging with time. There are certain idealized problems in which the temperature varies in only one direction (e.g., the x-direction). These are one-dimensional (1-D), steady-state problems. The classic example is a plane wall (i.e., a wall with a constant cross-sectional area, Ac , in the x-direction) that is insulated around its edges. In order for the temperature distribution to be 1-D, each face of the wall must be subjected to a uniform boundary condition. For example, Figure 1-4 illustrates a plane wall in which the left face (x = 0) is maintained at TH while the right face (x = L) is held at TC . L

Figure 1-4: A plane wall with ﬁxed temperature boundary conditions.

TH

q⋅ x

q⋅ x+dx TC

x dx

6

One-Dimensional, Steady-State Conduction

The ﬁrst step toward developing an analytical solution for this, or any, problem involves the deﬁnition of a differential control volume. The control volume must encompass material at a uniform temperature; therefore, in this case it must be differentially small in the x-direction (i.e., it has width dx, see Figure 1-4) but can extend across the entire cross-sectional area of the wall as there are no temperature gradients in the y- or z-directions. Next, the energy transfers across the control surfaces must be deﬁned as well as any thermal energy generation or storage terms. For the steady-state, 1-D case considered here, there are only two energy transfers, corresponding to the rate of conduction heat transfer into the left side (i.e., at position x, q˙ x) and out of the right side (i.e., at position x + dx, q˙ x+dx) of the control volume. A steady-state energy balance for the differential control volume is therefore: q˙ x = q˙ x+dx

(1-19)

A Taylor series expansion of the term at x + dx leads to: q˙ x+dx = q˙ x +

dq˙ d2 q˙ dx2 d3 q˙ dx3 dx + 2 + 3 + ··· dx dx 2! dx 3!

(1-20)

The analytical solution proceeds by taking the limit as dx goes to zero so that the higher order terms in Eq. (1-20) can be neglected: dq˙ dx (1-21) q˙ x+dx = q˙ x + dx Substituting Eq. (1-21) into Eq. (1-19) leads to: dq˙ dx (1-22) q˙ x = q˙ x + dx or dq˙ =0 (1-23) dx Equation (1-23) is typical of the initial result that is obtained by considering a differential energy balance: a differential equation that is expressed in terms of energy rather than temperature. This form of the differential equation should be checked against your intuition. Equation (1-23) indicates that the rate of conduction heat transfer is not a function of x. For the problem in Figure 1-4, there are no sources or sinks of energy and no energy storage within the wall; therefore, there is no reason for the rate of heat transfer to vary with position. The ﬁnal step in the derivation of the governing equation is to substitute appropriate rate equations that relate energy transfer rates to temperatures. The result of this substitution will be a differential equation expressed in terms of temperature. The rate equation for conduction is Fourier’s law: ∂T (1-24) ∂x For our problem, the temperature is only a function of position, x, and therefore the partial differential in Eq. (1-24) can be replaced with an ordinary differential: q˙ = −k Ac

q˙ = −k Ac

dT dx

Substituting Eq. (1-25) into Eq. (1-23) leads to: dT d −k Ac =0 dx dx

(1-25)

(1-26)

1.2 Steady-State 1-D Conduction without Generation

7

If the thermal conductivity is constant then Eq. (1-26) may be simpliﬁed to: d2 T =0 dx2

(1-27)

The derivation of Eq. (1-27) is trivial and yet the steps are common to the derivation of the governing equation for more complex problems. These steps include: (1) the definition of an appropriate control volume, (2) the development of an energy balance, (3) the expansion of terms, and (4) the substitution of rate equations. In order to completely specify a problem, it is necessary to provide boundary conditions. Boundary conditions are information about the solution at the extents of the computational domain (i.e., the limits of the range of position and/or time over which your solution is valid). A second order differential equation requires two boundary conditions. For the problem shown in Figure 1-4, the boundary conditions are: T x=0 = T H

(1-28)

T x=L = T C

(1-29)

Equations (1-27) through (1-29) represent a well-posed mathematical problem: a second order differential equation with boundary conditions. Equation (1-27) is very simple and can be solved by separation and direct integration: dT =0 (1-30) d dx Equation (1-30) is integrated according to: dT d = 0 dx

(1-31)

Because Eq. (1-31) is an indeﬁnite integral (i.e., there are no limits on the integrals), an undetermined constant (C1 ) results from the integration: dT = C1 dx Equation (1-32) is separated and integrated again: dT = C1 dx

(1-32)

(1-33)

to yield T = C1 x + C2

(1-34)

Equation (1-34) shows that the temperature distribution must be linear; any linear function (i.e., any values of the constants C1 and C2 ) will satisfy the differential equation, Eq. (1-27). The constants of integration are obtained by forcing Eq. (1-34) to also satisfy the two boundary conditions, Eqs. (1-28) and (1-29): T H = C1 0 + C2

(1-35)

T C = C1 L + C2

(1-36)

Equations (1-35) and (1-36) are solved for C1 and C2 and substituted into Eq. (1-34) to provide the solution: T =

(T C − T H ) x + TH L

(1-37)

8

One-Dimensional, Steady-State Conduction

The heat transfer at any location within the wall is obtained by substituting the temperature distribution, Eq. (1-37), into Fourier’s law, Eq. (1-25): q˙ = −k Ac

dT k Ac = (T H − T C) dx L

(1-38)

Equation (1-38) shows that the heat transfer does not change with the position within the wall; this behavior is consistent with Eq. (1-23). The development of analytical solutions is facilitated using a symbolic software package such as Maple. It is suggested that the reader stop and go through the tutorial provided in Appendix A.2 which can be found on the web site associated with the book (www.cambridge.org/nellisandklein) in order to become familiar with Maple. Note that the Maple Command Applet that is discussed in Appendix A.2 is available on the internet and can be used even if you do not have access to the Maple software. The mathematical solution to the 1-D, steady-state conduction problem associated with a plane wall is easy enough that there is no reason to use Maple. However, it is worthwhile to use the problem in order to illustrate some of the basic steps associated with using Maple in anticipation of more difﬁcult problems. Start a new problem in Maple (select New from the File menu). Enter the governing differential equation, Eq. (1-27), and assign it to the function ODE; note that the second derivative of T with respect to x is obtained by applying the diff command twice. > restart; > ODE:=diff(diff(T(x),x),x)=0; ODE :=

d2 T (x) = 0 dx2

The solution to the ordinary differential equation is obtained using the dsolve command and assigned to the function Ts. > Ts:=dsolve(ODE); Ts := T(x) = C1x + C2

The solution identiﬁed by Maple is consistent with Eq. (1-34), except that Maple uses the variables C1 and C2 rather than C1 and C2 to represent the constants of integration. The two boundary conditions, Eqs. (1-35) and (1-36), are obtained symbolically using the eval command to evaluate the solution at a particular position and assigned to the functions BC1 and BC2. > BC1:=eval(Ts,x=0)=T_H; BC1 := (T(0) = C2) = T H > BC2:=eval(Ts,x=L)=T_C; BC2 := (T (L) = C1L + C2) = T C

The result of the eval command is almost, but not quite what is needed to solve for the constants. The expressions include the extraneous statements T(0) and T(L); use the rhs function in order to return just the expression on the right hand side.

1.2 Steady-State 1-D Conduction without Generation

9

> BC1:=rhs(eval(Ts,x=0))=T_H; BC1 := C2 = T H > BC2:=rhs(eval(Ts,x=L))=T_C; BC2 := C1L + C2 = T C

The constants are explicitly determined using the solve command. Note that the solve command requires two arguments; the ﬁrst is the equation or, in this case, set of equations to be solved (the boundary conditions, BC1 and BC2) and the second is the variable or set of variables to solve for (the constants C1 and C2). > constants:=solve({BC1,BC2},{_C1,_C2}); constants := { C2 = T H, C1 = −

T H−T C } L

The constants are substituted into the general solution using the subs command. The subs command requires two arguments; the ﬁrst is the set of deﬁnitions to be substituted and the second is the set of equations to substitute them into. > Ts:=subs(constants,Ts); Ts := T(x) = −

(T H − T C)x +T H L

This result is the same as Eq. (1-37).

1.2.3 The Resistance Concept Equation (1-38) is the solution for the rate of heat transfer through a plane wall. The equation suggests that, under some limiting conditions, conduction of heat through a solid can be thought of as a ﬂow that is driven by a temperature difference and resisted by a thermal resistance, in the same way that electrical current is driven by a voltage difference and resisted by an electrical resistance. Inspection of Eq. (1-38) suggests that the thermal resistance to conduction through a plane wall (R pw ) is given by: R pw =

L k Ac

(1-39)

allowing Eq. (1-38) to be rewritten: q˙ =

(T H − T C) R pw

(1-40)

The concept of a thermal resistance is broadly useful and we will often return to this idea of a thermal resistance in order to help develop a conceptual understanding of various heat transfer processes. The usefulness of Eqs. (1-39) and (1-40) go beyond the simple situation illustrated in Figure 1-4. It is possible to approximately understand conduction heat transfer in most any situation provided that you can identify the distance that heat must be conducted and the cross-sectional area through which the conduction occurs.

10

One-Dimensional, Steady-State Conduction dr q⋅ r TH

rin

TC q⋅ r+dr

L

Figure 1-5: A cylinder with ﬁxed temperature boundary conditions.

r rout

Resistance equations provide a method for succinctly summarizing a particular solution and we will derive resistance solutions for a variety of physical situations. By cataloging these resistance equations, it is possible to quickly use the solution in the context of a particular problem without having to go through all of the steps that were required in the original derivation. For example, if we are confronted with a problem involving steady-state heat transfer through a plane wall then it is not necessary to rederive Eqs. (1-19) through (1-38); instead, Eqs. (1-39) and (1-40) conveniently represent all of this underlying math.

1.2.4 Resistance to Radial Conduction through a Cylinder Figure 1-5 illustrates steady-state, radial conduction through an inﬁnitely long cylinder (or one with insulated ends) without thermal energy generation. The analytical solution to this problem is derived using the steps described in Section 1.2.2. The differential energy balance (see Figure 1-5) leads to: q˙ r = q˙ r+dr

(1-41)

The r + dr term in Eq. (1-41) is expanded and Fourier’s law is substituted in order to reach: dT d −k Ac =0 (1-42) dr dr The difference between the plane wall geometry considered in Section 1.2.2 and the cylindrical geometry considered here is that the cross-sectional area for heat transfer, Ac in Eq. (1-42), is not constant but rather varies with radius: ⎡ ⎤ d ⎣ dT ⎦ −k 2 =0 (1-43) πr L dr dr Ac

where L is the length of the cylinder. Assuming that the thermal conductivity is constant, Eq. (1-43) is simpliﬁed to: d dT r =0 (1-44) dr dr and integrated twice according to the following steps: dT d r = 0 dr r

dT = C1 dr

(1-45) (1-46)

1.2 Steady-State 1-D Conduction without Generation

11

C1 dr r

(1-47)

T = C1 ln (r) + C2

(1-48)

dT =

where C1 and C2 are constants of integration, evaluated by applying the boundary conditions: T H = C1 ln (rin ) + C2

(1-49)

T C = C1 ln (rout ) + C2

(1-50)

After some algebra, the temperature distribution in the cylinder is obtained: r out ln T = T C + (T H − T C) r rout ln rin

(1-51)

The rate of heat transfer is given by: q˙ = −k 2 π r L

dT 2πLk (T H − T C) = rout dr ln r in

(1-52)

1/Rcyl

Therefore, the thermal resistance to radial conduction through a cylinder (Rcyl ) is: rout ln rin Rcyl = (1-53) 2πLk It is worth noting that the thermal resistance to radial conduction through a cylinder must be computed using the ratio of the outer to the inner radii in the numerator, regardless of the direction of the heat transfer. TC dr TH

r q⋅ r+dr

q⋅r

Figure 1-6: A sphere with ﬁxed temperature boundary conditions.

rin rout

1.2.5 Resistance to Radial Conduction through a Sphere Figure 1-6 illustrates steady-state, radial conduction through a sphere without thermal energy generation. The differential energy balance (see Figure 1-6) leads to: q˙ r = q˙ r+dr

(1-54)

12

One-Dimensional, Steady-State Conduction

which is expanded and used with Fourier’s law to reach: dT d −k Ac =0 dr dr

(1-55)

The cross-sectional area for heat transfer is the surface area of a sphere: ⎡ ⎤ dT d ⎣ ⎦ =0 −k 4 π r2 dr dr

(1-56)

Ac

Assuming that k is constant allows Eq. (1-56) to be simpliﬁed: d 2 dT r =0 dr dr

(1-57)

Equation (1-57) is entered in Maple: > restart; > ODE:=diff(rˆ2*diff(T(r),r),r)=0;

ODE := 2r

2 d d T(r) + r2 T(r) =0 dr dr2

and solved: > Ts:=dsolve(ODE); Ts := T(r) = C1 +

C2 r

The boundary conditions are: T r=rin = T H

(1-58)

T r=rout = T C

(1-59)

These equations are entered in Maple: > BC1:=rhs(eval(Ts,r=r_in))=T_H; BC1 := C1 +

C2 =T H r in

> BC2:=rhs(eval(Ts,r=r_out))=T_C; BC2 := C1 +

C2 =T C r out

1.2 Steady-State 1-D Conduction without Generation

13

The constants are obtained by solving this system of two equations and two unknowns: > constants:=solve({BC1,BC2},{_C1,_C2}); T Hr in − T Cr out r in r out(−T C + T H) constants := { C1 = , C2 = − } −r out + r in −r out + r in

and substituted into the general solution: > Ts:=subs(constants,Ts); Ts := T(r) =

r in r out(−T C + T H) T Hr in − T Cr out − −r out + r in (−r out + r in)r

The heat transfer at any radial location is given by Fourier’s law: q˙ = −k 4 π r2

> q_dot:=-k*4*pi*rˆ2*diff(Ts,r); q dot := −4 k π r2

dT dr

(1-60)

d 4 k π r in r out(−T C + T H) T(r) = − dr −r out + r in

and used to compute the thermal resistance for steady-state, radial conduction through a sphere: TH − TC q˙

Rsph =

(1-61)

> R_sph:=(T_H-T_C)/rhs(q_dot); R sph := −

which can be simpliﬁed to:

−r out + r in 4 k π r in r out

Rsph =

1 1 − rin rout 4πk

(1-62)

1.2.6 Other Resistance Formulae Many heat transfer processes may be cast in the form of a resistance formula, allowing problems involving various types of heat transfer to be represented conveniently using thermal resistance networks. Resistance networks can be solved using techniques borrowed from electrical engineering. Also, it is often possible to obtain a physical feel for the problem by inspection of the thermal resistance network. For example, small resistances in series with large ones will tend to be unimportant. Large resistances in parallel

14

One-Dimensional, Steady-State Conduction

with small ones can also be neglected. This type of understanding is important and can be obtained quickly using thermal resistance networks. Convection Resistance Convection is discussed in Chapters 4 through 7 and refers to heat transfer between a surface and a moving ﬂuid. The rate equation that characterizes the rate of convection heat transfer (q˙ conv ) is Newton’s law of cooling: q˙ conv = h As (T s − T ∞ )

(1-63)

1/Rconv

where h is the average heat transfer coefﬁcient, As is the surface area at temperature Ts that is exposed to ﬂuid at temperature T∞ . Note that the heat transfer coefﬁcient is not a material property like thermal conductivity, but rather a complex function of the geometry, ﬂuid properties, and ﬂow conditions. By inspection of Eq. (1-63), the thermal resistance associated with convection (Rconv ) is: R conv =

1 h As

(1-64)

Contact Resistance Contact resistance refers to the complex phenomenon that occurs when two solid surfaces are brought together. Regardless of how well prepared the surfaces are, they are not ﬂat at the micro-scale and therefore energy carriers in either solid cannot pass through the interface unimpeded. The energy carriers in two dissimilar materials may not be the same in any case. The result is a temperature change at the interface that, at the macro-scale, appears to occur over an inﬁnitesimally small spatial extent and grows in proportion to the rate of heat transfer across the interface. In reality, the temperature does not drop discontinuously but rather over some micro-scale distance that depends on the details of the interface. This phenomenon is usually modeled by characterizing the interface as having an area-speciﬁc contact resistance (Rc , often provided in K-m2 /W). The resistance to heat transfer across an interface (Rc ) is: Rc =

Rc As

(1-65)

where As is the contact area (the projected area of the surfaces, ignoring their microstructure). Contact resistance is not a material property but rather a complex function of the micro-structure, the properties of the two materials involved, the contact pressure, the interstitial material, etc. The area-speciﬁc contact resistance for interface conditions that are commonly encountered has been measured and tabulated in various references, for example Schneider (1985). Some representative values are listed in Table 1-1. The area-speciﬁc contact resistance tends to be reduced with increasing clamping pressure and smaller surface roughness. One method for reducing contact resistance is to insert a soft metal (e.g., indium) or grease into the interface in order to improve the heat transfer across the interstitial gap. The values listed in Table 1-1 can be used to determine whether contact resistance is likely to play an important role in a speciﬁc application. However, if contact resistance is important, then more precise data for the interface of interest should be obtained or measurements should be carried out.

Table 1-1: Area-speciﬁc contact resistance for some interfaces, from Schneider (1985) and Fried (1969). Materials

Clamping pressure

Surface roughness

Interstitial material

Temperature

Area-speciﬁc contact resistance

copper-to-copper copper-to-copper aluminum-to-aluminum aluminum-to-aluminum stainless-to-stainless stainless-to-stainless stainless-to-stainless stainless-to-stainless stainless-to-aluminum aluminum-to-aluminum aluminum-to-aluminum aluminum-to-aluminum aluminum-to-aluminum aluminum-to-aluminum

100 kPa 1000 kPa 100 kPa 100 kPa 100 kPa 1000 kPa 100 kPa 1000 kPa 100 kPa 1000 kPa 100 kPa 100 kPa 100 kPa 100 kPa

0.2 μm 0.2 μm 0.3 μm 1.5 μm 1.3 μm 1.3 μm 0.3 μm 0.3 μm 1.2 μm 0.3 μm 10 μm 10 μm 10 μm 10 μm

vacuum vacuum vacuum vacuum vacuum vacuum vacuum vacuum air air air helium hydrogen silicone oil

46◦ C 46◦ C 46◦ C 46◦ C 30◦ C 30◦ C 30◦ C 30◦ C 93◦ C 93◦ C 20◦ C 20◦ C 20◦ C 20◦ C

1.5 × 10−4 K-m2 /W 1.3 × 10−4 K-m2 /W 2.5 × 10−3 K-m2 /W 3.3 × 10−3 K-m2 /W 4.5 × 10−3 K-m2 /W 2.4 × 10−3 K-m2 /W 2.9 × 10−3 K-m2 /W 7.7 × 10−4 K-m2 /W 3.3 × 10−4 K-m2 /W 6.7 × 10−5 K-m2 /W 2.8 × 10−4 K-m2 /W 1.1 × 10−4 K-m2 /W 0.72 × 10−4 K-m2 /W 0.53 × 10−4 K-m2 /W

15

16

One-Dimensional, Steady-State Conduction

Radiation Resistance Radiation heat transfer occurs between surfaces due to the emission and absorption of electromagnetic waves, as described in Chapter 10. Radiation heat transfer is complex when many surfaces at different temperatures are involved; however, in the limit that a single surface at temperature T s interacts with surroundings at temperature T sur then the radiation heat transfer from the surface can be calculated according to: 4 (1-66) q˙ rad = As σ ε T s4 − T sur where As is the area of the surface, σ is the Stefan-Boltzmann constant (5.67 × 10−8 W/m2 -K4 ), and ε is the emissivity of the surface. Emissivity is a parameter that ranges between near 0 (for highly reﬂective surfaces) to near 1 (for highly absorptive surfaces). Note that both Ts and T sur must be expressed as absolute temperature (i.e., in units K rather than ◦ C) in Eq. (1-66). Equation (1-66) does not seem to resemble a resistance equation because the heat transfer is not driven by a difference in temperatures but rather by a difference in temperatures to the fourth power. However, Eq. (1-66) may be expanded to yield: 2 (1-67) q˙ rad = As σ ε T s2 + T sur (T s + T sur ) (T s − T sur ) hrad

1/Rrad

Comparing Eq. (1-67) for radiation to Eq. (1-63) for convection shows that a ‘radiation heat transfer coefﬁcient’, hrad , can be deﬁned as: 2 (1-68) hrad = σ ε T s2 + T sur (T s + T sur ) The radiation heat transfer coefﬁcient is a useful quantity for many problems because it allows convection and radiation to be compared directly, as discussed in Section 10.6.2. Equation (1-67) suggests that an appropriate thermal resistance for radiation heat transfer (Rrad ) is: Rrad =

As σ ε

(T s2

1 2 ) (T + T ) + T sur s sur

(1-69)

Because the absolute surface and surrounding temperatures are both typically large and not too different from each other, Eq. (1-69) can often be approximated by: Rrad ≈

1 As σ ε 4 T

3

(1-70)

where T is the average of the surface and surrounding temperatures: T =

T s + T sur 2

(1-71)

With this approximation, the radiation heat transfer coefﬁcient is: hrad ≈ σ ε 4 T

3

(1-72)

The resistance associated with radiation and the radiation heat transfer coefﬁcient are both clearly temperature-dependent. However, the conductivity, contact resistance, and average heat transfer coefﬁcient that are required to compute other types of resistances are also temperature-dependent and therefore the resistance concept can only be approximate in any case. A summary of the thermal resistance associated with several common situations is presented in Table 1-2.

1.2 Steady-State 1-D Conduction without Generation

17

Table 1-2: A summary of common resistance formulae. Situation Plane wall

Resistance formula R pw =

Rcyl Sphere (radial heat transfer) Convection

Contact between surfaces Radiation (exact)

Radiation (approximate)

L = wall thickness (|| to heat ﬂow) k = conductivity Ac = cross-sectional area (⊥ to heat ﬂow)

L k Ac

rout rin = 2πLk

Cylinder (radial heat transfer)

Rsph =

ln

L = cylinder length k = conductivity rin and rout = inner and outer radii

1 1 1 − 4 π k rin rout

k = conductivity rin and rout = inner and outer radii

Rconv = Rc =

h = average heat transfer coefﬁcient As = surface area exposed to convection

1 ¯h As

Rc = area speciﬁc contact resistance As = surface area in contact

Rc As

Rrad =

Rrad ≈

Nomenclature

1 2 ) (T + T As σ ε (T s2 + T sur s sur )

1 As σ ε 4 T

3

As = radiating surface area σ = Stefan-Boltzmann constant ε = emissivity T s = absolute surface temperature T sur = absolute surroundings temperature As = radiating surface area σ = Stefan-Boltzmann constant ε = emissivity T = average absolute temperature

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR Figure 1 illustrates a spherical dewar containing saturated liquid oxygen that is kept at pressure pLO x = 25 psia; the saturation temperature of oxygen at this pressure is TLO x = 95.6 K. The dewar consists of an inner and outer metal liner separated by polystyrene foam insulation. The inner metal liner has inner radius r mli,in = 10.0 cm and thickness thm = 2.5 mm. The outer metal liner also has thickness thm = 2.5 mm. The conductivity of both metal liners is k m = 15 W/m-K. The heat transfer coefﬁcient between the oxygen within the dewar and the inner surface of the dewar is hin = 150 W/m2 -K. The outer surface of the dewar is surrounded by air at T∞ = 20◦ C and radiates to surroundings that are also at T∞ = 20◦ C. The emissivity of the outer surface of the dewar is ε = 0.7. The heat transfer coefﬁcient between the outer surface of the dewar and the surrounding air is hout = 6 W/m2 -K. The area-speciﬁc

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

Note that useful reference information, such as Table 1-2, is included in the Heat Transfer Reference Section of EES in order to facilitate solving heat transfer problems without requiring that you locate a written reference book. To access this section, select the Reference Material from the Heat Transfer menu. This will open an online document that contains material from this book. Notice that the Heat Transfer menu also includes all of the examples that are associated with the book.

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

18

One-Dimensional, Steady-State Conduction

contact resistance that characterizes the interfaces between the insulation and the adjacent metal liners is Rc = 3.0 × 10−3 K-m2 /W. The thickness of the insulation between the two metal liners is thins = 1.0 cm. You are trying to evaluate the impact of using polystyrene foam insulation in place of the more expensive insulation that is currently used. Flynn (2005) suggests that the conductivity of polystyrene foam at cryogenic temperatures is approximately k ins = 330 μW/cm-K. a) Draw a network that represents this situation using 1-D resistances. saturated liquid oxygen TLOx = 95.6 K, hin = 150 W/m2 -K

Rc′′ = 3x10

rmli, in = 10.0 cm T∞ = 20°C 2 hout = 6 W/m -K

−3

2

K-m /W

km = 15 W/m-K kins = 330 μW/cm-K

k m = 15 W/m-K

ε = 0.7

insulation outer metal liner thm = 2.5 mm

inner metal liner insulation

thins = 1.0 cm thm = 2.5 mm

outer metal liner

Figure 1: Spherical dewar containing saturated liquid oxygen.

The resistance network is illustrated in Figure 2.

K K R cond, mlo = 0.0010 R cond, mli = 0.0013 W W K R cond, ins = 2.09 TLOx = 95.6 K W

Rconv, in = 0.053

K W R c, ins, mli = 0.023

Rc, ins, mlo = 0.053

K W

Rrad = 1.91

K W

T∞ = 293.2 K

Ts, out

K W

The resistances include: Rconv, in = convection from inside surface Rcond, mli = conduction through inner metal liner Rc, ins, mli = contact between inner metal liner & insulation Rcond, ins = conduction through insulation Rc, ins, mlo = contact between outer metal liner & insulation Rcond, mlo = conduction through outer metal liner Rrad = radiation Rconv, out = convection from outer surface Figure 2: Resistance network representing the dewar.

R conv, out = 1.00

K W

19

The resistance network interacts with the surrounding air and surroundings (at T∞ ) and the saturated liquid oxygen (at TLO x ). b) Estimate the rate of heat transfer to the liquid oxygen. The solution will be carried out using EES. It is assumed that you have already been exposed to the EES software by completing the self-guided tutorial contained in Appendix A.1. The ﬁrst step in preparing a successful solution to any problem with EES is to enter the inputs to the problem and set their units. Experience has shown that it is generally best to work exclusively in SI units (m, J, K, kg, Pa, etc.) because this unit system is entirely self-consistent. If the problem statement includes parameters in other units, they can be converted to SI units within the “Inputs” section of the code. The upper section of your EES code should look something like: “EXAMPLE 1.2-1: Liquid Oxygen Dewar” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” “pressure of liquid oxygen” p LOx=25 [psia]∗ convert(psia,Pa) T LOx=95.6 [K] “temperature of liquid oxygen” h bar in =150 [W/mˆ2-K] “heat transfer coefﬁcient between the liquid oxygen and the inner wall” r mli in=10 [cm]∗ convert(cm,m) “inner radius of the inner metal liner” th m=2.5 [mm]∗ convert(mm,m) “thickness of inner metal liner” th ins cm=1.0 [cm] “thickness of insulation, in cm” th ins=th ins cm∗ convert(cm,m) “thickness of insulation” e=0.7 [-] “emissivity of outside surface” T inﬁnity=converttemp(C,K,20 [C]) “temperature of surroundings and surrounding air” R c=3.0e-3 [K-mˆ2/W] “area-speciﬁc contact resistance” k ins=330 [microW/cm-K]∗ convert(microW/cm-K,W/m-K) “mean conductivity of insulation” k m=15 [W/m-K] “conductivity of metal” h bar out=6 [W/mˆ2-K] “heat transfer coefﬁcient between outer wall and surrounding air”

The resistance to convection between the inner surface of the dewar and the oxygen is: Rconv,in =

R conv in=1/(4∗ pi∗ r mli inˆ2∗ h bar in)

1 2 hin 4 π r mli,in

“convection resistance to liquid oxygen”

The inner radius of the insulation is: rins,in = r mli,in + thm The resistance to conduction through the inner metal liner is: 1 1 1 − Rcond ,mli = 4 π k m r mli,in rins,in

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.2 Steady-State 1-D Conduction without Generation

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

20

One-Dimensional, Steady-State Conduction

and the contact resistance resistance between the inner metal liner and the insulation is: Rc Rc,ins,mli = 2 4 π rins,in

r ins in=r mli in+th m “inner radius of insulation” R cond mli=(1/r mli in-1/r ins in)/(4∗ pi∗ k m) “conduction resistance of inner metal liner” R c ins mli=R c/(4∗ pi∗ r ins inˆ2) “contact resistance between inner metal liner and insulation”

The outer radius of the insulation is: rins,out = rins,in + thins The resistance to conduction through the insulation is: 1 1 1 Rcond ,ins = − 4 π k ins rins,in rins,out and the contact resistance resistance between the insulation and the outer metal liner is: Rc Rc,ins,mlo = 2 4 π rins,out

r ins out=r ins in+th ins “outer radius of insulation” R cond ins=(1/r ins in-1/r ins out)/(4∗ pi∗ k ins) “conduction resistance of insulation” R c ins mlo=R c/(4∗ pi∗ r ins outˆ2) “contact resistance between insulation and outer metal liner”

The outer radius of the outer metal liner is: r mlo,out = rins,out + thm The resistance to conduction through the outer metal liner is: 1 1 1 Rcond ,mlo = − 4 π k m rins,out r mlo,out and the convection resistance between the outer surface of the dewar and the air is: 1 Rconv,out = 2 hout 4 π r mli,out

r mlo out=r ins out+th m R cond mlo=(1/r ins out-1/r mlo out)/(4∗ pi∗ k m) R conv out=1/ (4∗ pi∗ r mlo outˆ2∗ h bar out)

“outer radius of outer metal liner” “conduction resistance of outer metal liner” “convection resistance to surrounding air”

The surface temperature on the outside of the dewar (Ts,out in Figure 2) cannot be known until the problem is solved and yet it must be used to calculate the resistance to radiation, Rrad . One of the nice things about using the EES software to solve this problem is that the software can deal with this type of nonlinearity and provide the solution to the implicit equations. It is this capability that simultaneously makes

21

EES so powerful and yet sometimes, ironically, difﬁcult to use. EES should be able to solve equations regardless of the order in which they are entered. However, you should enter equations in a sequence that allows you to solve them as you enter them; this is exactly what you would be forced to do if you were to solve the problem using a typical programming language (e.g., MATLAB, FORTRAN, etc.). This technique of entering your equations in a systematic order provides you with the opportunity to debug each subset of equations as you move along, rather than waiting until all of the equations have been entered before you try to solve them. Another beneﬁt of approaching a problem in this sequential manner is that you can consistently update the guess values associated with the variables in your problem. EES solves your equations using a nonlinear relaxation technique and therefore the closer the guess values of the variables are to “reasonable” values, the more likely it is that EES will ﬁnd the correct solution. To proceed with the solution to this problem using EES, it is helpful to initially assume a reasonable surface temperature (e.g., a reasonable guess might be the average of the surrounding and the liquid oxygen temperatures) so that it is possible to calculate a value of the radiation resistance: Rrad =

1 2 2 2 (T 4 π r mli,out σ ε Ts,out + T∞ s,out + T∞ )

and continue with the solution. T s out = (T LOx+T inﬁnity)/2 “guess for the surface temperature (removed to complete problem)” R rad=1/(4∗ pi∗ r mlo outˆ2∗ sigma#∗ e∗ (T s outˆ2+T inﬁnityˆ2)∗ (T s out+T inﬁnity)) “radiation resistance”

Solve the equations that have been entered (select Calculate from the Solve menu) and check that your answers make sense. Verify that the variables and equations have a consistent set of units by setting the units for each of the variables. The best way to do this is to go to the Variable Information window (select Variable Info from the Options menu) and enter the units for each variable in the Units column. Once this is done, check the units for your problem (select Check Units from the Calculate menu) in order to make sure that all of the units are consistent with the equations. Alternatively, units can be set by right-clicking on the variables in the Solution Window. The total resistance separating the liquid oxygen from the surroundings is: Rt ot al = Rconv,in + Rcond ,mli + Rc,ins,mli +Rcond ,ins + Rc,ins,mlo + Rcond ,mlo +

1 Rconv,out

+

1 Rrad

−1

and the heat transfer rate from the surroundings to the liquid oxygen is: q˙ =

(T∞ − TLO x ) Rt ot al

R total=R conv in+R cond mli+R c ins mli+R cond ins+R c ins mlo& +R cond mlo+(1/R conv out+1/R rad)ˆ(-1) q dot=(T inﬁnity-T LOx)/R total

“total resistance” “heat ﬂow”

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.2 Steady-State 1-D Conduction without Generation

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

22

One-Dimensional, Steady-State Conduction

At this point, we can use the heat transfer rate to recalculate the surface temperature (as opposed to assuming it). Ts,out = TLO x + q˙ (Rconv,in + Rcond ,mli + Rc,ins,mli + Rcond ,ins + Rc,ins,mlo + Rcond ,mlo ) It is necessary to comment out or delete the equation that provided the assumed surface temperature and instead calculate the surface temperature correctly. This step creates an implicit set of nonlinear equations. Before you ask EES to solve the set of equations, it is a good idea to update the guess values for each variable (select Update Guesses from the Calculate menu). {T s out=(T LOx+T inﬁnity)/2} “guess for the surface temperature” T s out=T LOx+q dot∗ (R conv in+ R cond mli+R c ins mli+R cond ins+R c ins mlo+R cond mlo) “surface temperature”

The rate of heat transfer to the liquid oxygen is q˙ = 69.4 W. Resistance networks provide substantial insight into the problem. Figure 2 shows the magnitude of each of the resistances in the network. The resistances associated with conduction through the insulation, radiation from the surface of the dewar, and convection from the surface of the dewar are of the same order of magnitude and large relative to the others in the circuit. Conduction through the insulation is much more important than conduction through the metal liners, convection to the liquid oxygen or the contact resistance; these resistances can probably be neglected in a rough analysis and certainly very little effort should be expended to better understand these aspects of the problem. Both radiation and convection from the outer surface are important, as they are of similar magnitude. The convection resistance is smaller and therefore more heat will be transferred by convection from the surface than is radiated from the surface. If the radiation resistance had been much larger than the convection resistance (as is often the case in forced convection problems where the convection heat transfer coefﬁcient is much larger) then radiation could be neglected. The smallest resistance in a parallel network will dominate because most of the energy will tend to ﬂow through that resistance. It is almost always a good idea to estimate the size of the resistances in a heat transfer problem prior to solving the problem. Often it is possible to simplify the analysis considerably, and the size of the resistances can certainly be used to guide your efforts. For this problem, a detailed analysis of conduction through the metal liner or a lengthy search for the most accurate value of the thermal conductivity of the metal would be a misguided use of time whereas a more accurate measurement of the conductivity of the insulation might be important. c) Plot the rate of heat transfer to the liquid oxygen as a function of the insulation thickness. In order to generate the requested plot, it is necessary to parametrically vary the insulation thickness. The speciﬁed value of the insulation thickness is commented out {th ins cm=1.0 [cm]}

“thickness of insulation, in cm”

23

and a parametric table is generated (select New Parametric Table from the Tables menu) that includes the variables th ins cm and q dot (Figure 3).

Figure 3: New Parametric Table Window.

Right-click on the th_ins_cm column and select Alter Values; vary the thickness from 0 cm to 10 cm and solve the table (select Solve Table from the Calculate menu). Prepare a plot of the results (select New Plot Window from the Plots menu and then select X-Y Plot) by selecting the variable th_ins_cm for the X-Axis and q_dot for the Y-Axis (Figure 4).

Figure 4: New Plot Setup Window.

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

1.2 Steady-State 1-D Conduction without Generation

One-Dimensional, Steady-State Conduction

Figure 5 illustrates the rate of heat transfer as a function of the insulation thickness. 100 90 80 Heat transfer rate (W)

EXAMPLE 1.2-1: LIQUID OXYGEN DEWAR

24

70 60 50 40 30 20 10 0 0

1

2

3 4 5 6 7 Insulation thickness (cm)

8

9

10

Figure 5: Heat transfer rate as a function of insulation thickness.

1.3 Steady-State 1-D Conduction with Generation 1.3.1 Introduction The generation of thermal energy within a conductive medium may occur through ohmic dissipation, chemical or nuclear reactions, or absorption of radiation. According to the ﬁrst law of thermodynamics, energy cannot be generated (excluding nuclear reactions); however, it can be converted from other forms (e.g., electrical energy) to thermal energy. The energy balance that we use to solve these problems is then strictly a thermal energy conservation equation. The addition of thermal energy generation to the 1-D steadystate solutions considered in Section 1.2 is straightforward and the steps required to obtain an analytical solution are essentially the same.

1.3.2 Uniform Thermal Energy Generation in a Plane Wall Consider a plane wall with temperatures ﬁxed at either edge that experiences a volumetric generation of thermal energy, as shown in Figure 1-7.

L

TH

q⋅ x

x

q⋅ x+dx g⋅ dx

Figure 1-7: Plane wall with thermal energy generation and ﬁxed temperature boundary conditions.

TC

1.3 Steady-State 1-D Conduction with Generation

25

The problem is assumed to be 1-D in the x-direction and therefore an appropriate differential control volume has width dx (see Figure 1-7). Notice the additional energy term in the control volume that is related to the generation of thermal energy. A steadystate energy balance includes conduction into the left-side of the control volume (q˙ x, at position x), generation within the control volume (g), ˙ and conduction out of the rightside of the control voume (q˙ x+dx, at position x + dx): q˙ x + g˙ = q˙ x+dx

(1-73)

or, after expanding the right side: q˙ x + g˙ = q˙ x +

dq˙ dx dx

(1-74)

The rate of thermal energy generation within the control volume can be expressed as the product of the volume enclosed by the control volume and the rate of thermal energy generation per unit volume, g˙ (which may itself be a function of position or temperature): g˙ = g˙ Ac dx

(1-75)

where Ac is the cross-sectional area of the wall. The conduction term is expressed using Fourier’s law: q˙ = −k Ac

dT dx

Substituting Eqs. (1-76) and (1-75) into Eq. (1-74) results in dT d −k Ac dx g˙ Ac dx = dx dx which can be simpliﬁed (assuming that conductivity is constant): d dT g˙ =− dx dx k Equation (1-78) is separated and integrated: dT g˙ d = − dx dx k

(1-76)

(1-77)

(1-78)

(1-79)

If the volumetric rate of thermal energy generation is spatially uniform, then the integration leads to: g˙ dT = − x + C1 dx k where C1 is a constant of integration. Equation (1-80) is integrated again: g˙ dT = − x + C1 dx k

(1-80)

(1-81)

which leads to: T =−

g˙ 2 x + C1 x + C2 2k

(1-82)

26

One-Dimensional, Steady-State Conduction

where C2 is another constant of integration. Note that the same solution can be obtained using Maple. The governing differential equation, Eq. (1-78), is entered in Maple: > restart; > GDE:=diff(diff(T(x),x),x)=-gv/k; GDE :=

d2 gv T(x) = − dx2 k

and solved: > Ts:=dsolve(GDE); Ts := T(x) = −

gvx2 + C1x + C2 2k

Equation (1-82) satisﬁes the governing differential equation, Eq. (1-78), throughout the computational domain (i.e., from x = 0 to x = L) for arbitrary values of C1 and C2 . It is easy to use Maple to check that this is true: > rhs(diff(diff(Ts,x),x))+gv/k; 0

Note that it is often a good idea to use Maple to quickly doublecheck that an analytical solution does in fact satisfy the original governing differential equation. All that remains is to force the general solution, Eq. (1-82), to satisfy the boundary conditions by adjusting the constants C1 and C2 . The ﬁxed temperature boundary conditions shown in Figure 1-7 correspond to: T x=0 = T H

(1-83)

T x=L = T C

(1-84)

Substituting Eq. (1-82) into Eqs. (1-83) and (1-84) leads to: C2 = T H −

g˙ 2 L + C1 L + T H = T C 2k

(1-85) (1-86)

Solving Eq. (1-86) for C1 leads to: C1 =

g˙ (T H − T C) L− 2k L

(1-87)

1.3 Steady-State 1-D Conduction with Generation

27

Substituting Eqs. (1-85) and (1-87) into Eq. (1-82) leads to: T =

g˙ L2 2k

x x 2 (T H − T C) − x + TH − L L L

(1-88)

Again, Maple can be used to achieve the same result. The boundary condition equations are deﬁned in Maple:

> BC1:=rhs(eval(Ts,x=0))=T_H; BC1 := C2 = T H > BC2:=rhs(eval(Ts,x=L))=T_C; BC2 := −

gvL2 + C1L + C2 = T C 2k

and solved for the two constants:

> constants:=solve({BC1,BC2},{_C1,_C2}); constants := { C2 = T H, C1 =

gvL2 − 2T Hk + 2T Ck } 2 Lk

The constants are substituted into the general equation:

> Ts:=subs(constants,Ts); Ts := T(x) = −

(gvL2 − 2T Hk + 2T Ck)x gvx2 + +T H 2k 2 Lk

It is good practice to examine any solution and verify that it makes sense. By inspection, it is clear that Eq. (1-88) limits to T H at x = 0 and T C at x = L; therefore, the boundary conditions were implemented correctly. Also, in the absence of any generation Eq. (1-88) limits to Eq. (1-37), the solution that was derived in Section 1.2.2 for steady-state conduction through a plane wall without generation. Figure 1-8 illustrates the temperature distribution for TH = 80◦ C and TC = 20◦ C with L = 1.0 cm and k = 1 W/m-K for various values of g˙ . The temperature proﬁle becomes more parabolic as the rate of thermal energy generation increases because energy must be transferred toward the edges of the wall at an increasing rate. The temperature gradient is proportional to the local rate of conduction heat transfer; as g˙ increases, the heat transfer rate at x = L increases while the heat transfer rate entering at x = 0 decreases and eventually becomes negative (i.e., heat actually begins to leave from both edges of the wall). This effect is evident in Figure 1-8 by observing that the temperature gradient at x = 0 changes from a negative to a positive value as g˙ is increased.

28

One-Dimensional, Steady-State Conduction 180

g⋅ ′′′= 1× 107 W/m3

160

Temperature (°C)

140

g⋅ ′′′ = 5 × 106 W/m3

120 100

g⋅ ′′′ = 2 × 106 W/m3

80 60 40 20 0

g⋅ ′′′ = 1× 106 W/m3 g⋅ ′′′ = 5 × 105 W/m3 0.1

0.2

0.3

g⋅ ′′′ = 0 W/m3

0.4 0.5 0.6 0.7 Axial location (cm)

0.8

0.9

1

Figure 1-8: Temperature distribution within a plane wall with thermal energy generation (k = 1 W/m-K, T H = 80◦ C, T C = 20◦ C, L = 1.0 cm).

The rate of heat transfer by conduction in the wall is obtained by applying Fourier’s law to the solution for the temperature distribution: q˙ = −k Ac

dT dx

Substituting Eq. (1-88) into Eq. (1-89) leads to: x 1 k Ac − + q˙ = Ac g˙ L (T H − T C) L 2 L

(1-89)

(1-90)

The heat transfer rate is not constant with position. Therefore, the plane wall with generation cannot be represented as a thermal resistance in the manner discussed in Section 1.2. However, it is always a good idea to carry out a number of sanity checks on your solution and the resistance concepts discussed in Section 1.2 provide an excellent mechanism for doing this. Equation (1-88) and Figure 1-8 both show that there is a maximum temperature elevation (relative to the zero generation case) that occurs at the center of the wall. Substituting x = L/2 into the 1st term in Eq. (1-88) shows that the magnitude of the maximum temperature elevation is: T max =

g˙ L2 8k

(1-91)

Maple can provide this result as well. The zero-generation solution is obtained by substituting g˙ = 0 into the original solution: > Tsng:=subs(gv=0,Ts); Tsng := T(x) =

(−2T Hk + 2T Ck)x +T H 2 Lk

1.3 Steady-State 1-D Conduction with Generation

29

Then the temperature elevation is the difference between the original solution and the zero-generation solution: > DeltaT:=rhs(Ts-Tsng); DeltaT := −

gvx2 (gvL2 − 2T Hk + 2T Ck)x (−2T Hk + 2T Ck)x + − 2k 2 Lk 2 Lk

and the maximum value of the temperature elevation is evaluated at x = L/2: > DeltaTmax=eval(DeltaT,x=L/2); DeltaTmax = −

gvL2 gvL2 − 2T Hk + 2T Ck −2T Hk + 2T Ck + − 8k 4k 4k

This result can be simpliﬁed using the simplify command: > simplify(%); DeltaTmax =

gvL2 8k

Note that the % character refers to the result of the previous command (i.e., the output of the last calculation). The temperature elevation occurs because the energy that is generated within the wall must be conducted to one of the external surfaces. Therefore, the temperature elevation must be consistent, in terms of its order of magnitude if not its exact value, with the temperature rise that is associated with the rate of thermal energy generation passing through an appropriately deﬁned resistance. This is an approximate analysis and is only meant to illustrate the process of providing a “back of the envelope” estimate or a sanity check on a solution. The total energy that is generated in one-half of the wall material must pass to the adjacent edge. A very crude estimate of the temperature elevation is: L g˙ L2 L g˙ Ac = (1-92) T max ∼ 2 2kA 4k c rate of thermal energy generated in half the wall

thermal resistance of half the wall

which, in this case, is within a factor of two of the exact analytical solution (the “back of the envelope” calculation is too large because the energy generated near the surfaces does not need to pass through half of the wall material). Again, the intent of this analysis was not to obtain exact agreement, but rather to provide a quick check that the solution makes sense.

1.3.3 Uniform Thermal Energy Generation in Radial Geometries The area for conduction through the plane wall discussed in Section 1.3.2 is constant in the coordinate direction (x). If the conduction area is a function of position, then it cannot be canceled from each side of Eq. (1-77) and therefore the differential equation becomes more complicated. Figure 1-9 illustrates the differential control volumes that should be deﬁned in order to analyze radial heat transfer in (a) a cylinder and (b) a sphere with thermal energy generation.

30

One-Dimensional, Steady-State Conduction

TC dr dr q⋅ r TH

g⋅

TH

TC L ⋅q r+dr

r q⋅ r

g⋅

q⋅ r+ dr

rin rin

rout

r rout

(a)

(b)

Figure 1-9: Differential control volume for (a) a cylinder and (b) a sphere with volumetric thermal energy generation.

The differential control volume suggested by either Figure 1-9(a) or (b) is: q˙ r + g˙ = q˙ r+dr

(1-93)

which is expanded and simpliﬁed: dq˙ dr dr The rate equations for q˙ and g˙ in a cylindrical geometry, Figure 1-9(a), are: g˙ =

(1-94)

dT dr

(1-95)

g˙ = 2 π r L dr g˙

(1-96)

q˙ = −k 2 π r L

where L is the length of the cylinder and g˙ is the rate of thermal energy generation per unit volume. Substituting Eqs. (1-95) and (1-96) into Eq. (1-94) leads to: d dT r g˙ = − kr (1-97) dr dr which is integrated twice (assuming that k and g˙ are constant) to achieve: g˙ r2 (1-98) + C1 ln (r) + C2 4k where C1 and C2 are constants of integration that depend on the boundary conditions. The rate equations for q˙ and g˙ in a spherical geometry, Figure 1-9(b), are: T =−

dT dr

(1-99)

g˙ = 4 π r2 dr g˙

(1-100)

q˙ = −k 4 π r2

Substituting Eqs. (1-99) and (1-100) into Eq. (1-94) leads to: dT d g˙ r2 = −k r2 dr dr

(1-101)

which is integrated twice to achieve: T =−

g˙ 2 C1 r + + C2 6k r

(1-102)

1.3 Steady-State 1-D Conduction with Generation

31

Table 1-3: Summary of formulae for 1-D uniform thermal energy generation cases. Plane wall

Cylinder

Sphere

Governing differential equation

g˙ d2 T = − dx2 k

d dT g˙ r = −k r dr dr

Temperature gradient

g˙ dT = − x + C1 dx k

dT g˙ r C1 =− + dr 2k r

General solution

T =−

g˙ 2 x + C1 x + C2 2k

d 2 dT g˙ r = −k r dr dr

T =−

2

dT g˙ C1 =− r− 2 dr 3k r

g˙ r2 + C1 ln (r) + C2 4k

T =−

g˙ 2 C1 r + + C2 6k r

EXAMPLE 1.3-1: MAGNETIC ABLATION Thermal ablation is a technique for treating cancerous tissue by heating it to a lethal temperature. A number of thermal ablation techniques have been suggested in order to apply heat locally to the cancerous tissue and therefore spare surrounding healthy tissue. One interesting technique utilizes ferromagnetic thermoseeds, as discussed by Tompkins (1992). Small metallic spheres (thermoseeds) are embedded at precise locations within the cancer tumor and then the region is exposed to an oscillating magnetic ﬁeld. The magnetic ﬁeld does not generate thermal energy in the tissue. However, the spheres experience a volumetric generation of thermal energy that causes their temperature to increase and results in the conduction of heat to the surrounding tissue. Precise placement of the thermoseed can be used to control the application of thermal energy. The concept is shown in Figure 1. ferromagnetic thermoseed kts = 10 W/m-K g⋅ ts = 1.0 W Figure 1: A thermoseed used for ablation of a tumor.

q⋅ r =r + ts

rts = 1.0 mm q⋅ r =r − ts

tissue kt = 0.5 W/m-K body temperature Tr→∞ = Tb = 37°C

It is necessary to determine the temperature ﬁeld associated with a single thermoseed placed in an inﬁnite medium of tissue. The thermoseed has radius rts = 1.0 mm and conductivity kts = 10 W/m-K. A total of g˙ ts = 1.0 W of generation is uniformly distributed throughout the sphere. The temperature far from the thermoseed is the body temperature, Tb = 37◦ C. The tissue has thermal conductivity kt = 0.5 W/m-K and is assumed to be in perfect thermal contact with the thermoseed. The effects of metabolic heat generation (i.e., volumetric generation in the tissue) and blood perfusion (i.e., the heat removed by blood ﬂow in the tissue) are not considered in this problem.

EXAMPLE 1.3-1: MAGNETIC ABLATION

The governing differential equation and general solutions for these 1-D geometries with a uniform rate of thermal energy generation are summarized in Table 1-3.

EXAMPLE 1.3-1: MAGNETIC ABLATION

32

One-Dimensional, Steady-State Conduction

a) Prepare a plot showing the temperature in the thermoseed and in the tissue (i.e., the temperature from r = 0 to r rts ). This problem is 1-D because the temperature varies only in the radial direction. There are no circumferential non-uniformities that would lead to temperature gradients in any dimension except r. However, the problem includes two, separate computational domains that share a common boundary (i.e., the thermoseed and the tissue). Therefore, there will be two different governing equations that must be solved and additional boundary conditions that must be considered. It is always good to start your problem with an input section in which all of the given information is entered and, if necessary, converted to SI units.

“EXAMPLE 1.3-1: Magnetic Ablation” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” r ts=1 [mm]∗ convert(mm,m) k ts=10 [W/m-K] g dot ts=1.0 [W] T b=converttemp(C,K,37 [C]) k t=0.5 [W/m-K]

“radius of the thermoseed” “thermal conductivity of thermoseed” “total generation of thermal energy in thermoseed” “body temperature” “tissue thermal conductivity”

Notice a few things about the EES code. First, comments are provided to deﬁne the nomenclature and make the code understandable; this type of annotation is important for clarity and organization. Also, units are not ignored but rather explicitly speciﬁed and dealt with as the problem is set up, rather than as an afterthought at the end. The unit system that EES will use can be speciﬁed in the Properties Dialog (select Preferences from the Options menu) from the Unit System tab or by using the $UnitSystem directive, as shown in the EES code above. The units of numerical constants can be set directly in square brackets following the value. For example, the statement r ts=1 [mm]∗ convert(mm,m)

“radius of the thermoseed”

tells EES that the constant 1 has units of mm and these should be converted to units of m. Therefore the variable r_ts will have units of m. The units of r_ts are not automatically set, as the equation involving r_ts is not a simple assignment. If you check units at this point (select Check Units from the Calculate menu) then EES will indicate that there is a unit conversion error. This unit error occurs because the variable r_ts has not been assigned any units but the equations are consistent with r_ts having units of m. It is possible to have EES set units automatically with an option in the Options tab in the Preferences Dialog. However, this is not recommended because the engineer doing the problem should know and set the units for each variable. The Formatted Equations window (select Formatted Equations from the Windows menu) shows the equations and their units more clearly (Figure 2).

33

EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-1: Magnetic Ablation Inputs

0.001 .

m

rts =

1

[mm] .

kts =

10

[W/m-K]

g⋅ ts =

1 [W]

Tb =

ConvertTemp (C , K , 37

kt

=

0.5

radius of the thermoseed

mm

thermal conductivity of thermoseed

total generation of thermal energy in thermoseed

[W/m-K]

[C])

body temperature

tissue thermal conductivity

Figure 2: Formatted Equations window.

It is good practice to set the units of all variables. One method of accomplishing this is to right-click on a variable in the Solutions window, which brings up the Format Selected Variables dialog. The units can be typed directly into the Units input box. Note that right-clicking in the Units input box and selecting the Unit List menu item provides a partial list of the SI units that EES recognizes. All of the units that are recognized by EES can be examined by selecting Unit Conversion Info from the Options menu. Once the units of r_ts are set to m, then a unit check (select Check Units from the Calculate menu) should reveal no errors. It is necessary to work with a different governing equation in each of the two computational domains. An appropriate differential control volume for the spherical geometry with uniform thermal energy generation was presented in Section 1.3.3 and leads to the general solution listed in Table 1-3. The general solution that is valid within the thermoseed (i.e., from 0 < r < r ts ) is: Tts = −

g˙ ts C1 r2 + + C2 6 k ts r

(1)

is the volumetric where C1 and C2 are undetermined constants of integration and g˙ ts rate of generation of thermal energy in the thermoseed, which is the ratio of the total rate of thermal energy generation to the volume of the thermoseed: = g˙ ts

g dot ts=3∗ g dot ts/(4∗ pi∗ r tsˆ3)

3 g˙ ts 4 π r ts3

“volumetric rate of generation in the thermoseed”

The general solution that is valid within the tissue (i.e., for r > rts ) is: Tt =

C3 + C4 r

(2)

EXAMPLE 1.3-1: MAGNETIC ABLATION

34

One-Dimensional, Steady-State Conduction

because the volumetric rate of thermal energy generation in the tissue is zero; note that C3 and C4 are undetermined constants of integration that are different from C1 and C2 in Eq. (1). The next step is to deﬁne the boundary conditions. There are four undetermined constants and therefore there must be four boundary conditions. At the center of the thermoseed (r = 0) the temperature must remain ﬁnite. Substituting r = 0 into Eq. (1) leads to: Tts,r =0 = −

02 C1 g˙ + + C2 6 k ts ts k ts 0

which indicates that C1 must be zero: C1 = 0

(3)

Alternatively, specifying that the temperature gradient at the center of the sphere is equal to zero leads to the same conclusion, C1 = 0. As the radius approaches inﬁnity, the tissue temperature must approach the body temperature. Substituting r → ∞ into Eq. (2) leads to: Tb = −

C3 + C4 ∞

which indicates that: C 4 = Tb

(4)

The remaining boundary conditions are deﬁned at the interface between the thermoseed and the tissue. It is assumed that the sphere and tissue are in perfect thermal contact (i.e., there is no contact resistance) so that the temperature must be continuous at the interface: Tts,r =rts = Tt,r =rts or, substituting r = rts into Eqs. (1) and (2): −

r ts2 C 1 C3 g˙ + + C2 = + C4 6 k ts ts r ts r ts

(5)

An energy balance on the interface (see Figure 1) requires that the heat transfer rate at the outer edge of the thermoseed (q˙r =rts− in Figure 1) must equal the heat transfer rate at the inner edge of the tissue (q˙r =rts+ in Figure 1). q˙r =rts− = q˙r =rts+

(6)

According to Fourier’s law, Eq. (6) can be written as: dTts dTt 2 = −4 π r k −4 π r ts2 k ts t ts dr r =rts dr r =rts or k ts

dTts dTt = k t dr r =rts dr r =rts

Substituting Eqs. (1) and (2) into Eq. (7) leads to: r ts C 1 C3 g˙ ts − 2 = −k t − 2 −k ts − 3k ts r ts r ts

(7)

(8)

35

Entering Eqs. (3), (4), (5), and (8) into EES will lead to the solution for the four constants of integration without the algebra and the associated opportunities for error. “Determine constants of integration” C 1=0 “temperature at center must be ﬁnite” C 4=T b “temperature far from the thermoseed” -r tsˆ2∗ g dot ts/(6∗ k ts)+C 1/r ts+C 2=C 3/r ts+C 4 “continuity of temperature at the interface” -k ts∗ (-r ts∗ g dot ts/(3∗ k ts)-C 1/r tsˆ2)=-k t∗ (-C 3/r tsˆ2) “equal heat ﬂux at the interface”

Finally, we can generate a plot using the solution. Displaying the radius in millimeters in the plot will make a much more reasonable scale than in meters and the temperature should be displayed in ◦ C. New variables, r_mm, T_ts_C, and T_t_C, are deﬁned for this purpose. “Prepare a plot” r=r mm∗ convert(mm,m) T ts=-rˆ2∗ g dot ts/(6∗ k ts)+C 1/r+C 2 T ts C=converttemp(K,C,T ts) T t=C 3/r+C 4 T t C=converttemp(K,C,T t)

“radius” “thermoseed temperature” “in C” “tissue temperature” “in C”

The units of the variables C_1, C_2, etc. should be set in the Variable Information window and the set of equations subsequently checked for unit consistency. The relationship between temperature and radial position will be determined using two parametric tables. The ﬁrst table will include variables r_mm and T_ts_C and the second will include the variables r_mm and T_t_C. In the ﬁrst table, r_mm is varied from 0 to 1.0 mm (i.e., within the thermoseed) and in the second it is varied from 1.0 mm to 10.0 mm (i.e., within the tissue). A plot in which the information contained in the two tables is overlaid leads to the temperature distribution shown in Figure 3. 225

Tt

Temperature (°C)

200 175 150 125

Tts

100 75 50 0

1

2

3

4 5 6 7 Radial position (mm)

8

9

Figure 3: Temperature distribution through thermoseed and tissue.

10

EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-1: MAGNETIC ABLATION

36

One-Dimensional, Steady-State Conduction

Figure 3 agrees with physical intuition. The temperature decays toward the body temperature with increasing distance from the thermoseed. The rate of energy being transferred through the tissue is constant, but the area for conduction is growing as r 2 and therefore the gradient in the temperature is dropping. The − ) and conduction heat transfer rates at the outer edge of the sphere (i.e., r = r sp + at the inner edge of the tissue (i.e., r = r sp ) are identical; the discontinuity in slope is related to the fact that the thermoseed is more conductive than the tissue. b) Determine the maximum temperature in the tissue and the extent of the lesion as a function of the rate of thermal energy generation in the thermoseed. The extent of the lesion (rlesion ) is deﬁned as the radial location where the tissue temperature reaches the lethal temperature for tissue, approximately Tlethal = 50◦ C according to Izzo (2003). The maximum tissue temperature (Tt,max ) is the temperature at the interface between the thermoseed and the tissue and is obtained by substituting r = r ts into Eq. (2): Tt,max =

T t max=C 3/r ts+C 4 T t max C=converttemp(K,C,T t max)

C3 + C4 r ts

“maximum tissue temperature” “in C”

The extent of the lesion can be obtained by determining the radial location where Tt = Tlethal : Tlethal =

T lethal=converttemp(C,K, 50 [C]) T lethal=C 3/r lesion+C 4 r lesion mm=r lesion∗ convert(m,mm)

C3 + C4 rlesion

“lethal temperature for cell death” “determine the extent of the lesion” “in mm”

In order to investigate the maximum tissue temperature and the extent of the lesion as a function of the thermal energy generation rate, it is necessary to prepare a parametric table that includes the variables T_t_max_C, r_lesion, and g_dot_sp and then vary the value of g_dot_sp within the table. Figure 4 illustrates the maximum temperature and the lesion extent as a function of the rate of thermal energy generation in the thermoseed. Note that Tt,max and rlesion have very different magnitudes and therefore it is necessary to plot the variable r_lesion on a secondary y-axis.

40

350

35

Tt ,max

300 250

rlesion

30 25

200

20

150

15

100

10

50

5

0 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Rate of thermal energy generation in thermoseed (W)

Extent of lesion (mm)

Maximum tissue temperature (°C)

400

37

0 2

Figure 4: Maximum tissue temperature and the extent of the lesion as a function of the rate of thermal energy generation in the thermoseed.

1.3.4 Spatially Non-Uniform Generation The ﬁrst few steps in solving conduction heat transfer problems include setting up an energy balance on a differential control volume and substituting in the appropriate rate equations. This process results in one or more differential equations that must be solved in order to determine the temperature distribution and heat transfer rates. The governing equations resulting from 1-D conduction with constant properties and constant internal generation are provided in Table 1-3 for the Cartesian, cylindrical, and spherical geometries. These equations are relatively simple and we have demonstrated how to solve them analytically by hand and, in some cases, using Maple. The complexity of the governing equation can increase signiﬁcantly if the thermal conductivity or internal generation depends on position or temperature. In these cases, an analytical solution to the governing equation may not be possible and the numerical solution techniques presented in Sections 1.4 and 1.5 must be used. In many of these cases, however, an analytical solution is possible but may require more mathematical expertise than you have or more effort than you’d like to expend. In these cases, the combined use of a symbolic software tool to identify the solution and an equation solver to manipulate the solution provides a powerful combination of tools. Analytical solutions are concise and elegant as well as being accurate and therefore preferable in many ways to numerical solutions. It is often best to have both an analytical and a numerical solution to a problem; their agreement constitutes the best possible double-check of a solution. EXAMPLE 1.3-2 demonstrates the combined use of the symbolic solver Maple (to derive the symbolic solution and boundary condition equations) and the equation solver EES (to carry out the algebra required to obtain the constants of integration and implement the solution).

EXAMPLE 1.3-1: MAGNETIC ABLATION

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

38

One-Dimensional, Steady-State Conduction

EXAMPLE 1.3-2: ABSORPTION IN A LENS A lens is used to focus the illumination energy (i.e., radiation) that is required to develop the resist in a lithographic manufacturing process, as shown in Figure 1. The lens can be modeled as a plane wall with thickness L = 1.0 cm and thermal conductivity k = 1.5 W/m-K. The lens is not perfectly transparent but rather absorbs some of the illumination energy that is passed through it. The absorption coefﬁcient of the lens is α = 0.1 mm−1 . The ﬂux of radiant energy that is incident = 0.1 W/cm2 .The top and bottom surfaces of the at the lens surface (x = 0) is q˙ rad lens are exposed to air at T∞ = 20◦ C and the average heat transfer coefﬁcient on these surfaces is h = 20 W/m2 -K. ′′ = 0.1 W/cm incident radiant energy, q⋅ rad

x dx

k = 1.5 W/m-K α = 0.1 mm-1

2

T∞ = 20°C 2 h = 20 W/m -K q⋅ conv, x =0 q⋅cond, x =0 q⋅ x L = 1.0 cm g⋅ ⋅q q⋅ cond, x =L x+dx q⋅ conv, x =L

T∞ = 20°C 2 h = 20 W/m -K

transmitted radiant energy Figure 1: Lens absorbing radiant energy.

The volumetric rate at which absorbed radiation is converted to thermal energy in the lens (g˙ ) is proportional to the local intensity of the radiant energy ﬂux, which is reduced in the x-direction by absorption. The result is an exponentially distributed volumetric generation that can be expressed as: α exp (−α x) g˙ = q˙ rad

a) Determine and plot the temperature distribution within the lens. The inputs are entered into EES: “EXAMPLE 1.3-2: Absorption in a lens” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in k=1.5 [W/m-K] “conductivity” L=1 [cm]∗ convert(cm,m) “lens thickness” alpha=0.1 [1/mm]∗ convert(1/mm,1/m) “absorption coefﬁcient” qf dot rad=0.1 [W/cmˆ2]∗ convert(W/cmˆ2,W/mˆ2) “incident energy ﬂux” h bar=20 [W/mˆ2-K] “average heat transfer coefﬁcient” T inﬁnity=converttemp(C,K,20 [C]) “ambient air temperature” A c=1 [mˆ2] “carry out the problem on a per unit area basis”

(1)

39

An energy balance on an appropriate, differential control volume (see Figure 1) provides: q˙ x + g˙ = q˙ x+d x which is expanded and simpliﬁed: g˙ =

d q˙ dx dx

Substituting the rate equations for q˙ and g˙ leads to: dT d −k Ac g˙ Ac = dx dx

(2)

where Ac is the cross-sectional area of the lens. Substituting Eq. (1) into Eq. (2) and simplifying leads to the governing differential equation for this problem. q˙ α dT d = − rad exp (−α x) (3) dx dx k The governing differential equation is entered in Maple. > restart; > GDE:=diff(diff(T(x),x),x)=-qf_dot_rad*alpha*exp(-alpha*x)/k;

G D E :=

d2 qf d ot r ad α e(−α x) T(x) = − 2 dx k

The general solution to the equation is obtained using the dsolve command: > Ts:=dsolve(GDE); T s := T(x) = −

qf d ot r ad e(−α x) + C1 x + C2 αk

We can check that this solution is correct by integrating Eq. (3) by hand: q˙ rad α dT d =− exp (−α x) d x dx k which leads to: q˙ dT = rad exp (−α x) + C 1 dx k

(4)

Equation (4) is integrated again: q˙ rad dT = exp (−α x) + C 1 d x k which leads to: T =−

q˙ rad exp (−α x) + C 1 x + C 2 kα

(5)

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

40

One-Dimensional, Steady-State Conduction

Note that Eq. (5) is consistent with the result from Maple. The constants of integration, C1 and C2 , are obtained by enforcing the boundary conditions. The boundary conditions for this problem are derived from “interface” energy balances at the two edges of the computational domain (x = 0 and x = L, as shown in Figure 1). It would be correct to include the radiant energy ﬂux in the interface balances. However, the interface thickness is zero and therefore no radiant energy is absorbed at the interface. The amount of radiant energy entering and leaving the interface is the same and these terms would immediately cancel. q˙ conv,x=0 = q˙ cond ,x=0 q˙ cond ,x=L = q˙ conv,x=L Substituting the rate equations for convection and conduction into the interface energy balances leads to the boundary conditions: dT h Ac (T∞ − Tx=0 ) = −k Ac (6) d x x=0 dT −k Ac = h Ac (Tx=L − T∞ ) d x x=L

(7)

Note that it is important to consider the direction of the energy transfers during the substitution of the rate equations. For example, q˙ conv,x=0 is deﬁned in Figure 1 as being into the top surface of the lens and therefore it is driven by (T∞ − Tx =0 ) while q˙ conv,x=L is deﬁned as being out of the bottom surface of the lens and therefore it is driven by (Tx =L − T∞ ). The general solution for the temperature distribution, Eq. (5), must be substituted into the boundary conditions, Eqs. (6) and (7), and solved algebraically to determine the constants C1 and C2 . Maple and EES can be used together in order to solve the differential equation, derive the symbolic expressions for the boundary conditions, carry out the required algebra to obtain C1 and C2 , and manipulate the solution. The temperature gradient is obtained symbolically from Maple using the diff command. > dTdx:=diff(Ts,x);

dT d x :=

d qf d ot r ad e(−α x) T(x) = − + C1 dx k

The ﬁrst boundary condition, Eq. (6), requires both the temperature and the temperature gradient evaluated at x = 0. The eval function in Maple is used to symbolically determine these quantities (T0 and dTdx0): > T0:=eval(Ts,x=0); T0 := T(0) = −

qf d ot r ad + C2 αk

41

Use the rhs function to redeﬁne T0 to be just the expression on the right-hand side of the above result.

> T0:=rhs(T0); T 0 := −

qf d ot r ad + C2 αk

The rhs function can also be applied directly to the eval function. The statement below determines the symbolic expression for the temperature gradient evaluated at x = 0.

> dTdx0:=rhs(eval(dTdx,x=0)); dT d x0 := −

qf d ot r ad + C1 k

These expressions for T0 and dTdx0 are copied and pasted into EES in order to specify the ﬁrst boundary condition. The equation format used by Maple is similar to that used in EES and therefore only minor modiﬁcations are required. Select the desired symbolic expressions in Maple (note that the selected text will appear to be highlighted), use the Copy command from the Edit menu to place the selection on the Clipboard. When pasted into EES, the equations will appear as:

“boundary condition at x=0” T0 := -1/alpha∗ qf dot rad/k+ C2 dTdx0 := qf dot rad/k+ C1

“temperature at x=0, copied from Maple” “temperature gradient at x=0, copied from Maple”

All that is necessary to use this equation in EES is to change the := to = and change the constants _C1 and _C2 to C_1 and C_2, respectively. For lengthier expressions, the search and replace feature in EES (select Replace from the Search manu) facilitates this process. After modiﬁcation, the expressions should be:

T0= -1/alpha∗ qf dot rad/k+C 2 dTdx0= qf dot rad/k+C 1

“temperature at x=0” “temperature gradient at x=0”

The boundary condition at x = 0, Eq. (6), is speciﬁed in EES: h bar∗ A c∗ (T inﬁnity-T0)=-k∗ A c∗ dTdx0

“boundary condition at x=0”

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

42

One-Dimensional, Steady-State Conduction

The same process is used for the boundary condition at x = L, Eq. (7). The temperature and temperature gradient at x = L are determined using Maple: > TL:=rhs(eval(Ts,x=L)); T L := −

qf d ot r ad e(−α αk

L)

+ C1 L + C2

> dTdxL:=rhs(eval(dTdx,x=L)); dT d x L := −

qf d ot r ad e(−α k

L)

+ C1

These expressions are copied into EES: “boundary condition at x=L” TL := -1/alpha∗ qf dot rad∗ exp(-alpha∗ L)/k+ C1∗ L+ C2 “temperature at x=L, copied from Maple” dTdxL := qf dot rad∗ exp(-alpha∗ L)/k+ C1 “temperature gradient at x=L, copied from Maple”

and modiﬁed for consistency with EES: TL= -1/alpha∗ qf dot rad∗ exp(-alpha∗ L)/k+C 1∗ L+C 2 “temperature at x=L, copied from Maple” dTdxL= qf dot rad∗ exp(-alpha∗ L)/k+C 1 “temperature gradient at x=L, copied from Maple”

The boundary condition at x = L, Eq. (7), is speciﬁed in EES: -k∗ A c∗ dTdxL=h bar∗ A c∗ (TL-T inﬁnity)

“boundary condition at x=L”

Solving the EES program will provide numerical values for both of the constants. The units should be set for each of the variables, including the constants, in order to ensure that the expressions are dimensionally consistent. The general solution is copied from Maple to EES: T(x) = -1/alpha∗ qf dot rad∗ exp(-alpha∗ x)/k+ C1∗ x+ C2

“general solution, copied from Maple”

and modiﬁed for consistency with EES: x mm=0 [mm] x=x mm∗ convert(mm,m) T=-1/alpha∗ qf dot rad∗ exp(-alpha∗ x)/k+C 1∗ x+C 2 T C=converttemp(K,C,T)

“x-position, in mm” “x position” “general solution for temperature” “in C”

The temperature as a function of position is shown in Figure 2. Notice the asymmetry that is produced by the non-uniform volumetric generation (i.e.,

43

more thermal energy is generated towards the top of the lens than the bottom and so the temperature is higher near the top surface of the lens). 36.4 36.3

Temperature (°C)

36.2 36.1 36 35.9 35.8 35.7 35.6 0

1

2

3

4 5 6 Position (mm)

7

8

9

10

Figure 2: Temperature distribution in the lens.

b) Determine the location of the maximum temperature (xmax ) and the value of the maximum temperature (Tmax ) in the lens. The location of the maximum temperature in the lens can be determined by setting the temperature gradient, Eq. (4), to zero. The value of the maximum temperature (Tmax ) is obtained by substituting the resulting value of xmax into the equation for temperature, Eq. (5). The expression for the temperature gradient is copied from Maple, pasted into EES and modiﬁed for consistency. qf dot rad∗ exp(-alpha∗ x max)/k+C 1=0 x max mm=x max∗ convert(m,mm)

“temperature gradient is zero at position x max” “in mm”

The temperature at xmax is determined using the general solution. T max=-1/alpha∗ qf dot rad∗ exp(-alpha∗ x max)/k+C 1∗ x max+C 2 “maximum temperature in the lens” T max C=converttemp(K,C,T max) “in C”

It is not likely that solving the code above will immediately result in the correct value of either xmax or Tmax . These are non-linear equations and EES must iterate to ﬁnd a solution. The success of this process depends in large part on the initial guesses and bounds used for the unknown variables. In most cases, any reasonable values of the guess and bounds will work. For this problem, set the lower and upper bounds on xmax to 0 (the top of the lens) and 0.01 m (1.0 cm, the bottom of the lens), respectively, and the guess for xmax to 0.005 m (the middle of the lens) using the

EXAMPLE 1.3-2: ABSORPTION IN A LENS

1.3 Steady-State 1-D Conduction with Generation

EXAMPLE 1.3-2: ABSORPTION IN A LENS

44

One-Dimensional, Steady-State Conduction

Variable Information dialog (Figure 3). Upon solving, EES will correctly predict that xmax = 0.38 cm and Tmax = 36.35◦ C.

Figure 3: Variable Information window showing the limits and guess value for the variable x max.

It is also possible to determine the maximum temperature within the lens using EES’ built-in optimization routines. There are several sophisticated single- and multi-variable optimization algorithms included with EES that can be accessed with the Min/Max command in the Calculate menu.

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES) 1.4.1 Introduction The analytical solutions examined in the previous sections are convenient since they provide accurate results for arbitrary inputs with minimal computational effort. However, many problems of practical interest are too complicated to allow an analytical solution. In such cases, numerical solutions are required. Analytical solutions remain useful as a way to test the validity of numerical solutions under limiting conditions. Numerical solutions are generally more computationally complex and are not unconditionally accurate. Numerical solutions are only approximations to a real solution, albeit approximations that are extremely accurate when done correctly. It is relatively straightforward to solve even complicated problems using numerical techniques. The steps required to set up a numerical solution using the ﬁnite difference approach remain the same even as the problems become more complex. The result of a numerical model is not a functional relationship between temperature and position but rather a prediction of the temperatures at many discrete positions. The ﬁrst step is to deﬁne small control volumes that are distributed through the computational domain and to specify the locations where the numerical model will compute the temperatures (i.e., the nodes). The control volumes used in the numerical model are small but ﬁnite, as opposed to the inﬁnitesimally small (differential) control volume that is deﬁned in order to derive an analytical solution. It is necessary to perform an energy balance on each

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

45

control volume. This requirement may seem daunting given that many control volumes will be required to provide an accurate solution. However, computers are very good at repetitious calculations. If your numerical code is designed in a systematic manner, then these operations can be done quickly for 1000’s of control volumes. Once the energy balance equations for each control volume have been set up, it is necessary to include rate equations that approximate each term in the energy balance based upon the nodal temperatures or other input parameters. The result of this step will be a set of algebraic equations (one for each control volume) in an equal number of unknown temperatures (one for each node). This set of equations can be solved in order to provide the numerical prediction of the temperature at each node. It is tempting to declare victory after successfully solving the ﬁnite difference equations and obtaining a set of results that looks reasonable. In reality, your work is only half done and there are several important steps remaining. First, it is necessary to verify that you have chosen a sufﬁciently large number of control volumes (i.e., an adequately reﬁned mesh) so that your numerical solution has converged to a solution that no longer depends on the number of nodes. This veriﬁcation can be accomplished by examining some aspect of your solution (e.g., a temperature or an energy transfer rate that is particularly important) as the number of control volumes increases. You should observe that your solution stops changing (to within engineering relevance) as the number of control volumes (and therefore the computational effort) increases. Some engineering judgment is required for this step. You have to decide what aspect of the solution is most important and how accurately it must be predicted in order to determine the level of grid reﬁnement that is required. Next, you should make sure that the solution makes sense. There are a number of ‘sanity checks’ that can be applied to verify that the numerical model is behaving according to physical expectations. For example, if you change the input parameters, does the solution respond as you would expect? Finally, it is important that you verify the numerical solution against an analytical solution in some appropriate limit. This step may be the most difﬁcult one, but it provides the strongest possible veriﬁcation. If the numerical model is to be used to make decisions that are important (e.g., to your company’s bottom line or to your career) then you should strive to ﬁnd a limit where an analytical solution can be derived and show that your numerical model matches the analytical solution to within numerical error. The numerical solutions considered in this section are for steady-state, 1-D problems. More complex problems (e.g., multi-dimensional and transient) are discussed in subsequent sections and chapters. This section also focuses on solving these problems using EES whereas subsequent sections discuss how these solutions may be implemented using a more formal programming language, speciﬁcally MATLAB.

1.4.2 Numerical Solutions in EES The development of a numerical model is best discussed in the context of a problem. Figure 1-10 illustrates an aluminum oxide cylinder that is exposed to ﬂuid at its internal and external surfaces. The temperature of the ﬂuid exposed to the internal surface is T ∞,in = 20◦ C and the average heat transfer coefﬁcient on the internal surface is hin = 100 W/m2 -K. The temperature of the ﬂuid exposed to the outer surface is T ∞,out = 100◦ C and the average heat transfer coefﬁcient on the outer surface is hout = 200 W/m2 -K. The thermal conductivity of the aluminum oxide is k = 9.0 W/m-K in the temperature range of interest. The rate of thermal energy generation per unit

46

One-Dimensional, Steady-State Conduction k = 9 W/m-K 2 g⋅ ′′′= a + b r + c r rin = 10 cm rout = 20 cm

Figure 1-10: Cylinder with volumetric generation.

T∞, in = 20°C T∞, out = 100°C 2 2 hin = 100 W/m -K hout = 200 W/m -K

volume within the cylinder varies with radius according to: g˙ = a + b r + c r2

(1-103)

where a = 1 × 104 W/m3 , b = 2 × 105 W/m4 , and c = 5 × 107 W/m5 . The inner and outer radii of the cylinder are rin = 10 cm and rout = 20 cm, respectively. The inputs are entered in EES: “Section 1.4.2” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” r in=10 [cm]∗ convert(cm,m) r out=20 [cm]∗ convert(cm,m) L=1 [m] k=9.0 [W/m-K] T inﬁnity in=converttemp(C,K,20 [C]) h bar in= 100 [W/mˆ2-K] T inﬁnity out=converttemp(C,K,100 [C]) h bar out=200 [W/mˆ2-K]

“inner radius of cylinder” “outer radius of cylinder” “unit length of cylinder” “thermal conductivity” “temperature of ﬂuid inside cylinder” “average heat transfer coefﬁcient at inner surface” “temperature of ﬂuid outside cylinder” “average heat transfer coefﬁcient at outer surface”

It is convenient to deﬁne a function that provides the volumetric rate of thermal energy generation speciﬁed by Eq. (1-103). An EES function is a self-contained code segment that is provided with one or more input parameters and returns a single value based on these parameters. Functions in EES must be placed at the top of the Equations Window, before the main body of equations. The EES code required to specify a function for the rate of thermal energy generation per unit volume is shown below. function gen(r) “This function deﬁnes the volumetric heat generation in the cylinder Inputs: r, radius (m) Output: volumetric heat generation at r (W/mˆ3)” a=1e4 [W/mˆ3] “coefﬁcients for generation function” b=2e5 [W/mˆ4] c=5e7 [W/mˆ5] gen=a+b∗ r+c∗ rˆ2 “generation is a quadratic” end

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

47

Figure 1-11: Variable Information Window showing the variables for the function GEN.

The function begins with a header that deﬁnes the name of the function (gen) and the input arguments (r) and is terminated by the statement end. None of the variables in the main EES program are accessible within the function other than those that are explicitly passed to the function as an input. Unlike equations in the main program, the equations within a function are executed in the order that they are entered and all variables on the right hand side of each expression must be deﬁned (i.e., the equations are assignments rather than relationships). The statements within the function are used to deﬁne the value of the function (gen, in this case). Units for the variables in the function should be set using the Variable Information dialog in the same way that units are set for variables in the main program. The most direct way to enter the units for the function is to select Variable Info from the Options menu. Then select Function GEN from the pull down menu at the top of the Variable Information dialog and enter the units for the variables (Figure 1-11). The ﬁnite difference technique divides the continuous medium into a large (but not inﬁnite) number of small control volumes that are treated using simple approximations. The computational domain in this problem lies between the edges of the cylinder (rin < r < rout ). The ﬁrst step in the solution process is to locate nodes (i.e., the positions where the temperature will be predicted) throughout the computational domain. The easiest way to distribute the nodes is uniformly, as shown in Figure 1-12 (only nodes 1, 2, i − 1, i, i + 1, N − 1, and N are shown). The extreme nodes (i.e., nodes 1 and N) are placed on the surfaces of the cylinder. In some problems, it may not be computationally efﬁcient to distribute the nodes uniformly. For example, if there are large temperature gradients at some location then it may be necessary to concentrate nodes in that region. Placing closelyspaced nodes throughout the entire computational domain may be prohibitive from a

qLHS

ri T1

Figure 1-12: Nodes and control volumes for the numerical model.

T2 g

qconv, in

Ti-1

Ti Ti+1

qLHS g qRHS

rin rout

g qconv, out

qRHS

TN-1

TN

L

48

One-Dimensional, Steady-State Conduction

computational viewpoint. For the uniform distribution shown in Figure 1-12, the radial location of each node (ri ) is: ri = rin +

(i − 1) (rout − rin ) (N − 1)

for i = 1..N

(1-104)

where N is the number of nodes. The radial distance between adjacent nodes ( r) is: r =

(rout − rin ) (N − 1)

(1-105)

It is necessary to specify the number of nodes used in the numerical solution. We will start with a small number of nodes, N = 6, and increase the number of nodes when the solution is complete. N=6 [-] DELTAr=(r out-r in)/(N-1)

“number of nodes” “distance between adjacent nodes (m)”

The location of each node will be placed in an array, i.e., a variable that contains more than one element (rather than a scalar, as we’ve used previously). EES recognizes a variable name to be an element of an array if it ends with square brackets surrounding an array index, e.g., r[4]. Array variables are just like any other variable in EES and they can be assigned to values, e.g., r[4]=0.16. Therefore, one way of setting up the position array would be to individually assign each value; r[1]=0.1, r[2]=0.12, r[3]=0.14, etc. This process is tedious, particularly if there are a large number of nodes. It is more convenient to use the duplicate command which literally duplicates the equations in its domain, allowing for varying array index values. The duplicate command must be followed by an integer index, in this case i, that passes through a range of values, in this case 1 to 6. The EES code shown below will copy (duplicate) the statement(s) that are located between duplicate and end N times; each time, the value of i is incremented by 1. It is exactly like writing: r[1]=r in+(r out-r in)∗ 0/(N-1) r[2]=r in+(r out-r in)∗ 1/(N-1) r[3]=r in+(r out-r in)∗ 2/(N-1) etc.

“Set up nodes” duplicate i=1,N r[i]=r in+(r out-r in)∗ (i-1)/(N-1) end

“this loop assigns the radial location to each node”

Be careful not to put statements that you do not want to be duplicated between the duplicate and end statements. For example, if you accidentally placed the statement N=6 inside the duplicate loop it would be like writing N=6 six times, which corresponds to six equations in a single unknown and is not solvable. Units should be assigned to arrays in the same manner as for other variables. The units of all variables in array r can be set within the Variable Information dialog. Note that each element of the array r appears in the dialog and that the units of each element can be set one by one. In most arrays, each element will have the same units and therefore it is not convenient to set the units an element at a time. Instead, deselect the Show array variables box in the upper left corner of the window, as shown in

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

49

De-select the Show array variables button in order to collapse the arrays

Figure 1-13: Collapsing an array in the Variable Information dialog.

Figure 1-13. The array is collapsed onto a single entry r[] and you can assign units to all of the elements in the array r[]. A control volume is deﬁned around each node. You have some freedom relative to the deﬁnition of a control volume, but the best control volume for this problem is deﬁned by bisecting the distance between the nodes, as shown in Figure 1-12. The second step in the numerical solution is to write an energy balance for the control volume associated with every node. The internal nodes (i.e., nodes 2 through N − 1) must be considered separately from the nodes at the edge of the computational domain (i.e., nodes 1 and N). The control volume for an arbitrary, internal node (node i) is shown in Figure 1-12. There are three energy terms associated with each control volume: conduction heat transfer passing through the surface on the left-hand side (q˙ LHS ), conduction heat transfer passing through the surface on the right-hand side (q˙ RHS ), and generation of thermal energy within the control volume (g). ˙ A steady-state energy balance for the internal control volume is: q˙ LHS + q˙ RHS + g˙ = 0

(1-106)

Note that Eq. (1-106) is rigorously correct since no approximations have been used in its development. In the next step, however, each of the terms in the energy balance are modeled using a rate equation that is only approximately valid; it is this step that makes the numerical solution only an approximation of the actual solution. Conduction through the left-hand surface is driven by the temperature difference between nodes i − 1 and i through the material that lies between these nodes. If there are a large number of nodes, then r is small and the effect of curvature within any control volume is small. In this case, q˙ LHS can be modeled using the resistance of a plane wall, given in Table 1-2: kL2π r (1-107) q˙ LHS = ri − (T i−1 − T i ) r 2 where L is the length of the cylinder and k is the thermal conductivity of the material, which is assumed to be constant here. It is relatively easy to consider a material with temperature- or spatially dependent thermal conductivity, as discussed in Section 1.4.3. Note that it does not matter which direction the heat ﬂow arrow associated with q˙ LHS is drawn in Figure 1-12; that is, the heat transfer could have been deﬁned either as an input or an output to the control volume. However, once the direction is deﬁned, it is

50

One-Dimensional, Steady-State Conduction

absolutely necessary that the energy balance on the control volume and the model of the heat transfer rate be consistent with this selection. For the energy balance shown in Figure 1-12, the heat transfer rate was written on the inﬂow side of the energy balance in Eq. (1-106) and the heat transfer was written as being driven by (T i−1 − T i ) in Eq. (1-107). The rate of conduction into the right-hand surface can be approximated in the same manner: kL2π r (1-108) ri + q˙ RHS = (T i+1 − T i ) r 2 The rate of generation of thermal energy is the product of the volume of the control volume and the rate of thermal energy generation per unit volume, which is approximately: g˙ = g˙ r=ri 2 π ri L r

(1-109)

The generation term is spatially dependent in this problem, but it is approximated by assuming that the volumetric generation evaluated at the position of the node (ri ) can be applied throughout the entire control volume. This approximation improves as r is reduced. Equations (1-106) through (1-109) can be conveniently written for each internal node using a duplicate loop: “Internal control volume energy balances” duplicate i=2,(N-1) q dot LHS[i]=2∗ pi∗ L∗ k∗ (r[i]-DELTAr/2)∗ (T[i-1]-T[i])/DELTAr q dot RHS[i]=2∗ pi∗ L∗ k∗ (r[i]+DELTAr/2)∗ (T[i+1]-T[i])/DELTAr g dot[i]=gen(r[i])∗ 2∗ pi∗ r[i]∗ L∗ DELTAr q dot LHS[i]+g dot[i]+q dot RHS[i]=0 end

“conduction in from inner radius” “conduction in from outer radius” “generation” “energy balance”

Attempting to solve the EES program at this point will result in a message indicating that there are two more variables than equations and so the problem is under-speciﬁed. We have not yet written energy balance equations for the two nodes that lie on the boundaries. Figure 1-12 illustrates the control volume associated with the node that is placed on the inner surface of the cylinder (i.e., node 1). The energy balance for the control volume ˙ and a associated with node 1 includes a conduction term (q˙ RHS ), a generation term (g), convection term (q˙ conv,in ). For steady-state conditions and energy terms having the signs indicated in Figure 1-12: q˙ conv,in + q˙ RHS + g˙ = 0 The conduction term model is the same as it was for internal nodes: kL2π r r1 + q˙ RHS = (T 2 − T 1 ) r 2

(1-110)

(1-111)

Even though the control volume for node 1 is half as wide as the others, the distance between nodes 1 and 2 is still r and therefore the resistance to conduction between nodes 1 and 2 does not change. The generation term is slightly different because the control volume is half as large as the internal control volumes. r (1-112) g˙ = g˙ r=r1 2 π r1 L 2

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

51

Convection from the ﬂuid is given by: q˙ conv,in = hin 2 π r1 L (T ∞,in − T 1 )

(1-113)

Equations (1-110) through (1-113) are programmed in EES: “Energy balance for node on internal surface” q dot RHS[1]=2∗ pi∗ L∗ (r[1]+DELTAr/2)∗ k∗ (T[2]-T[1])/DELTAr q dot conv in=2∗ pi∗ L∗ r in∗ h bar in∗ (T inﬁnity in-T[1]) g dot[1]=gen(r[1])∗ 2∗ pi∗ r[1]∗ L∗ DELTAr/2 q dot RHS[1]+q dot conv in+g dot[1]=0

“conduction in from outer radius” “convection from internal ﬂuid” “generation” “energy balance for node 1”

A similar procedure applied to the control volume associated with node N (see Figure 1-12) leads to: q˙ conv,out + q˙ LHS + g˙ = 0

(1-114)

q˙ conv,out = hout 2 π rN L (T ∞,out − T N )

(1-115)

where

q˙ LHS

kL2π = r

r rN − 2

g˙ = g˙ r=rN 2 π rN L

(T N−1 − T N )

(1-116)

r 2

(1-117)

“Energy balance for node on external surface” q dot LHS[N]=2∗ pi∗ L∗ (r[N]-DELTAr/2)∗ k∗ (T[N-1]-T[N])/DELTAr “conduction in from from inner radius” q dot conv out=2∗ pi∗ L∗ r out∗ h bar out∗ (T inﬁnity out-T[N]) “convection from external ﬂuid” g dot[N]=gen(r[N])∗ 2∗ pi∗ r[N]∗ L∗ DELTAr/2 “generation” q dot LHS[N]+q dot conv out+g dot[N]=0 “energy balance for node N”

There are now an equal number of equations as unknowns and therefore solving the problem will provide the temperature at each node. The calculated temperatures are converted to ◦ C: duplicate i=1,N T C[i]=converttemp(K,C,T[i]) end

“temperature in C”

Computers are very good at solving large systems of equations, particularly with linear equations such as these. There are a number of programs other than EES that can be used to solve these equations (e.g., MATLAB, FORTRAN, and C++). With most of these software packages, you must take the system of equations, carefully put them into a matrix format, and then decompose the matrix in order to obtain the solution. This additional step is not necessary with EES, which saves considerable effort for the user (although EES must do this step internally). In Section 1.5, we will look at how these equations have to be set up in a formal programming environment, speciﬁcally MATLAB, in order to obtain a solution.

52

One-Dimensional, Steady-State Conduction 675 N = 20 650

N=6

Temperature (°C)

625 600 575 550 525 500 475 450 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-14: Predicted temperature distribution for N = 6 and N = 20.

It is good practice to assign the units for all variables including the arrays before attempting to solve the equations. The unit consistency of each equation is checked when the equations are solved. The solution is provided in the Solution Window for the scalar quantities, and the Arrays Window for the array of predicted temperatures. A plot showing the predicted temperature as a function of radius is shown in Figure 1-14 for N = 6 and N = 20. Note that EES calculates the individual energy transfer rates for each of the control volumes. To see these energy transfer rates, select Arrays from the Windows menu in order to view the Arrays table (Figure 1-15). It is useful to examine these energy transfer rates and make sure that they agree with your intuition. For example, energy should not be created or destroyed at the interfaces between the control volumes; that is, q˙ LHS for node i should be equal and opposite q˙ RHS for node i − 1. It is easy to inadvertently use incorrect rate equations for the conduction terms where this is not true, and you can quickly identify this type of problem using the Arrays window. Figure 1-15 shows that the rate of energy transfer by conduction in the positive radial direction (i.e., q˙ LHS ) becomes more positive with increasing radius, as it should due to the volumetric generation. Before the numerical solution can be used with conﬁdence, it is necessary to verify its accuracy. The ﬁrst step in this process is to ensure that the mesh is adequately reﬁned. Figure 1-14 shows that the solution becomes smoother and represents the actual

[m]

[K]

[C]

[W]

[W]

[W]

0.1

366.4

93.27

62.83

0.12

369.2

96.03

150.8

-857.9

707.1

0.14

371.1

97.95

175.9

-707.1

531.2

0.16

372.4

99.21

201.1

-531.2

330.1

0.18

373

99.89

226.2

-330.1

103.9

0.2

373.2

100.1

125.7

-103.9

Figure 1-15: Arrays table.

857.9

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

53

660

Maximum temperature (°C)

650 640 630 620 610 600 590 580 570 1

10

100 Number of nodes

1000

Figure 1-16: Predicted maximum temperature as a function of the number of nodes.

temperature distribution better as the number of nodes is increased. The general approach to choosing a mesh is to pick an important characteristic of the solution and examine how this characteristic changes as the number of nodes in the computational domain is increased. In this case, an appropriate characteristic is the maximum predicted temperature within the cylinder. The following EES code extracts this value from the solution using the max function, which returns the maximum of the arguments provided to it. T max C=max(T C[1..N])

“max temperature in the cylinder, in C”

The maximum temperature as a function of the number of nodes is shown in Figure 1-16 (Figure 1-16 was created by making a parametric table that includes the variables N and T_max). Notice that the solution has converged after approximately 20 nodes and further reﬁnement is not likely to be necessary. The next step is to check that the solution agrees with physical intuition. For example, if the heat transfer coefﬁcient on the internal surface is reduced, then the temperatures within the cylinder should increase. Figure 1-17 illustrates the temperature as a function of radius for various values of hin (with N = 20) and shows that reducing the heat transfer coefﬁcient does tend to increase the temperature in the cylinder. There are many additional ‘sanity checks’ that could be tested. For example, decrease the thermal conductivity or increase the generation rate and make sure that the temperatures in the computational domain increase as they should. Finally, it is important that the numerical solution be veriﬁed against an analytical solution in an appropriate limit. In this case, it is not easy (although it is possible) to develop an analytical solution to the problem with the spatially varying volumetric generation. However, it is relatively straightforward to develop an analytical solution for the problem in the limit of a constant volumetric generation rate. It is also very easy to adapt the numerical model to this limiting case so that the analytical and numerical solutions can be compared. The variables b and c in the generation function, Eq. (1-103),

54

One-Dimensional, Steady-State Conduction 900

Temperature (°C)

800 700

hin = 20 W/m2-K

hin = 50 W/m2-K hin = 100 W/m2-K

600 500

hin = 200 W/m2-K

400 300 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-17: Temperature as a function of radius for various values of the heat transfer coefﬁcient.

are temporarily set to 0 in order to implement the numerical solution using a spatially uniform volumetric generation: function gen(r) “This function deﬁnes the volumetric heat generation in the cylinder Inputs: r: radius “Output: volumetric heat generation at r (W/mˆ3)” a=1e4 [W/mˆ3] “coefﬁcients for generation function” b=0{2e5} [W/mˆ4] c=0{5e7} [W/mˆ5] gen=a+b*r+c*rˆ2 “generation is a quadratic” end

The analytical solution is solved using the technique presented in Section 1.3. Table 1-3 provides the temperature distribution and temperature gradient in a cylinder with constant generation, to within the constants of integration: T =−

g˙ r2 + C1 ln (r) + C2 4k

(1-118)

dT g˙ r C1 =− + (1-119) dr 2k r The constants of integration may be determined using the boundary conditions at the inner and outer surfaces, which are obtained from energy balances at these interfaces. At the inner surface, convection and conduction must balance: dT hin (T ∞,in − T r=rin ) = −k (1-120) dr r=rin Substituting Eqs. (1-118) and (1-119) into Eq. (1-120) leads to: 2 g˙ rin C1 g˙ rin hin T ∞,in − − + C1 ln (rin ) + C2 + = −k − 4k 2k rin

(1-121)

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

At the outer surface, the interface energy balance leads to: dT = hout (T r=rout − T ∞,out ) −k dr r=rout Substituting Eqs. (1-118) and (1-119) into Eq. (1-122) leads to: 2 g˙ rout C1 g˙ rout −k − + + C1 ln (rout ) + C2 − T ∞,out = hout − 2k rout 4k

55

(1-122)

(1-123)

The constants of integration are determined by using EES to solve Eqs. (1-121) and (1-123): “Analytical Solution” g dot c=gen(r in) “constant volumetric generation rate for veriﬁcation” h bar in∗ (T inﬁnity in-(-g dot c∗ r inˆ2/(4∗ k)+C 1∗ ln(r in)C 2))=-k∗ (-g dot c∗ r in/(2∗ k)+C 1/r in) “boundary condition at inner surface” -k∗ (-g dot c∗ r out/(2∗ k)+C 1/r out)=h bar out∗ (-g dot c∗ r outˆ2/(4∗ k)+& C 1∗ ln(r out)+C 2-T inﬁnity out) “boundary condition at outer surface”

The ampersand character appearing in the equation above is a line break character that is only needed for formatting (the equation that is terminated with the ampersand continues on the following line). The analytical solution is evaluated at the locations of the nodes in the numerical solution. The absolute error between the analytical and numerical solution is calculated at each position and the maximum error over the computational domain is computed using the max command. duplicate i=1,N “analytical temperature at node i” T an[i]=-g dot c∗ r[i]ˆ2/(4∗ k)+C 1∗ ln(r[i])+C 2 T an C[i]=converttemp(K,C,T an[i]) “in C” err[i]=abs(T an[i]-T[i]) “absolute value of the discrepancy between numerical and analytical temperature” end err max=max(err[1..N]) “maximum error”

Figure 1-18 illustrates the temperature distribution predicted by the analytical and numerical models in the limit where the volumetric generation is constant (i.e., the coefﬁcients b and c in the volumetric generation function are set equal to zero). The agreement is nearly exact for 20 nodes, conﬁrming that the numerical model is valid. Figure 1-19 illustrates the maximum value of the error between the analytical and numerical models as a function of the number of nodes in the solution. Note that the constants b and c in the generation function were set to zero in order to provide a uniform generation. Figure 1-19 provides a more precise method of selecting the number of nodes. A required level of accuracy (i.e., agreement with the analytical model) can be used to specify a required number of nodes. For example, if the problem requires temperature estimates that are accurate to within 0.01 K, then you should use at least 7 nodes. However, predictions accurate to within 0.1 mK will require more than 60 nodes.

1.4.3 Temperature-Dependent Thermal Conductivity The thermal conductivity of most materials is a function of temperature, although it is often approximated to be constant. This approximation is appropriate for situations in

56

One-Dimensional, Steady-State Conduction

101 numerical model (with b = c = 0) analytical model

100

Temperature (°C)

99 98 97 96 95 94 93 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-18: Temperature as a function of position predicted by the analytical model should be

indicated by a line and the numerical model by the dots. which the temperature range is small provided that the thermal conductivity is evaluated at an average temperature. A constant value of thermal conductivity is usually assumed when solving a problem analytically; otherwise the differential equation is intractable. A major advantage of a numerical solution is that the temperature dependence of physical properties can be considered with little additional effort. The consideration of temperature-dependent thermal conductivity in a numerical model is demonstrated in the context of the problem that was considered previously in Section 1.4.2. The thermal conductivity of the aluminum oxide cylinder was assumed to be 9 W/m-K, independent of temperature. However, the temperature of the aluminum

Maximum error (K)

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

1

10

100 Number of nodes

1000

Figure 1-19: Maximum discrepancy between the analytical and numerical solutions (in the limit that b = c = 0) as a function of N.

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

57

40

Conductivity (W/m-K)

35 30 25 20 15 10 5 0 0

100

200

300 400 500 Temperature (°C)

600

700

800

Figure 1-20: Thermal conductivity of polycrystalline aluminum oxide as a function of temperature.

oxide varies by 200◦ C within the cylinder (see Figure 1-14) and the thermal conductivity of aluminum oxide varies substantially over this range of temperatures, as shown in Figure 1-20. It is possible to alter the EES code that was developed in Section 1.4.2 so that the temperature-dependent thermal conductivity of polycrystalline aluminum oxide is considered. A function k is written in EES that returns the conductivity. The function must be placed at the top of the Equations window, either above or below the previously deﬁned function gen, and has a single input (the temperature). EES’ internal function for the thermal conductivity of solids is used in function k to evaluate the thermal conductivity of the polycrystalline aluminum oxide. function k(T) “This function provides the thermal conductivity of the aluminum oxide Inputs: T: temperature (K) Output: thermal conductivity (W/m-K)” k=k (‘Al oxide-polycryst’,T) end

The temperature-dependent conductivity can cause a problem. It is tempting to evaluate the conduction terms for each control volume using the thermal conductivity evaluated at the temperature of the node. However, doing so will result in an error in the energy balance. Each conduction term, for example q˙ LHS , represents an energy exchange between node i and its adjacent node to the left, node i − 1. The value of q˙ LHS evaluated at node i must be equal and opposite to q˙ RHS evaluated at node i − 1; if the thermal conductivity is evaluated using the nodal temperature, then this will not be true and energy will be artiﬁcially generated or destroyed at the boundaries between the nodal control volumes. To avoid this problem, the thermal conductivity should be evaluated at the average temperature of the two nodes that are involved in the conduction heat transfer

58

One-Dimensional, Steady-State Conduction

(i.e., the temperature at the boundary). The energy balance on the internal nodes (see Figure 1-12) remains: q˙ LHS + q˙ RHS + g˙ = 0

(1-124)

However, the conduction terms must be calculated according to: q˙ LHS =

kT =(T i +T i−1 )/2 L 2 π r

q˙ RHS =

kT =(T i +T i+1 )/2 L 2 π r

r (T i−1 − T i ) 2

(1-125)

r ri + (T i+1 − T i ) 2

(1-126)

ri −

where kT =(T i +T i−1 )/2 is the thermal conductivity evaluated at the average of Ti and T i−1 and kT =(T i +T i+1 )/2 is the thermal conductivity evaluated at the average of Ti and T i+1 . A similar process is used for nodes 1 and N. The modiﬁed energy balances in EES are shown below, with the modiﬁed code indicated in bold: “Internal control volume energy balances” duplicate i=2,(N-1) k LHS[i]=k((T[i-1]+T[i])/2) “thermal conductivity at LHS boundary” k RHS[i]=k((T[i+1]+T[i])/2) “thermal conductivity at RHS boundary” q dot LHS[i]=2∗ pi∗ L∗ k LHS[i]∗ (r[i]-DELTAr/2)∗ (T[i-1]-T[i])/DELTAr “conduction in from inner radius” q dot RHS[i]=2∗ pi∗ L∗ k RHS[i]∗ (r[i]+DELTAr/2)∗ (T[i+1]-T[i])/DELTAr “conduction in from outer radius” g dot[i]=gen(r[i])∗ 2∗ pi∗ r[i]∗ L∗ DELTAr “generation” q dot LHS[i]+g dot[i]+q dot RHS[i]=0 “energy balance” end “Energy balance for node on internal surface” “thermal conductivity at RHS boundary” k RHS[1]=k((T[2]+T[1])/2) q dot RHS[1]=2∗ pi∗ L∗ (r[1]+DELTAr/2)∗ k RHS[1]∗ (T[2]-T[1])/DELTAr “conduction in from outer radius” q dot conv in=2∗ pi∗ L∗ r in∗ h bar in∗ (T inﬁnity in-T[1]) “convection from internal ﬂuid” g dot[1]=gen(r[1])∗ 2∗ pi∗ r[1]∗ L∗ DELTAr/2 “generation” q dot RHS[1]+q dot conv in+g dot[1]=0 “energy balance” “Energy balance for node on external surface” “thermal conductivity at LHS boundary” k LHS[N]=k((T[N-1]+T[N])/2) q dot LHS[N]=2∗ pi∗ L∗ (r[N]-DELTAr/2)∗ k LHS[N]∗ (T[N-1]-T[N])/DELTAr “conduction in from from inner radius” q dot conv out=2∗ pi∗ L∗ r out∗ h bar out∗ (T inﬁnity out-T[N]) g dot[N]=gen(r[N])∗ 2∗ pi∗ r[N]∗ L∗ DELTAr/2 “generation in node N” q dot LHS[N]+q dot conv out+g dot[N]=0 “energy balance for node N”

Select Solve from the Calculate menu and you are likely to ﬁnd that the problem either fails to converge or converges to some ridiculously high temperatures. This is not surprising, since the use of a temperature-dependent conductivity has transformed the algebraic equations from a set of equations that are linear in the unknown temperatures to a set of equations that are nonlinear. Therefore, EES must start from some guess value for the unknown temperatures and attempt to iterate until a solution is obtained.

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

59

Figure 1-21: Setting the guess values for the array T[ ] in the Variable Information window.

The success of this process is highly dependent on the guess values that are used. It may be possible to simply set better guess values for the unknown temperatures. Select Variable Info from the Options menu and deselect the Show array variables button. Set the guess value for the array T[ ] to something more reasonable than its default value of 1 K (e.g., 700 K), as shown in Figure 1-21. This strategy of manually setting reasonable guess values will not always work. A more reliable strategy uses the solution with constant conductivity, from Section 1.4.2, in order to provide guess values for the non-linear problem associated with temperaturedependent conductivity. This is an easy process; modify the conductivity function (k) as shown below, so that it returns a constant value rather than the temperature-dependent value:

function k(T) “This function provides the thermal conductivity of the aluminum oxide Inputs: T: temperature (K) Output: thermal conductivity (W/m-K)” {k=k (‘Al oxide-polycryst’,T)} k=9 [W/m-K] end

Solve the problem and EES will converge to a solution. To use this constant conductiviy solution as the guess value for the non-linear problem, select Update Guesses from the Calculate menu and then return the conductivity function to its original form. Solve the problem and your EES code will converge to the actual solution. Examine the Arrays Table (Figure 1-22) and notice that energy is conserved at each boundary (i.e., q˙ RHS for node i is equal and opposite to q˙ LHS for node i + 1 for every node); this simple check shows that your rate equations have been set up appropriately. Figure 1-23 illustrates the temperature of the aluminum oxide as a function of radius for the case in which the thermal conductivity is evaluated as a function of temperature. The temperature distribution calculated in Section 1.4.2 using a constant thermal conductivity of 9 W/m-K is also shown. This example illustrates that the temperature dependence of thermal conductivity may be an important factor in some problems.

60

One-Dimensional, Steady-State Conduction energy is conserved at each boundary

[W]

[W/m-k]

[W/m-k]

[W]

10.12

3330

[W]

[m]

[K]

[C]

27426

0.1

782.7

509.5

11370

10.12

9.353

-27426

16056

0.12

861.1

587.9

17910

9.353

9.113

-16056

-1853

0.14

903.1

629.9

26580

9.113

9.498

1853

-28434

0.16

898.8

625.6

37684

9.498

10.65

28434

-66118

0.18

842.7

569.6

25761

10.65

0.2

738.7

465.6

66118

Figure 1-22: Arrays Window.

675 650

k = 9 W/m-K

Temperature (°C)

625 600 575

k (T )

550 525 500 475 450 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-23: Temperature as a function of radius for the case where conductivity is a function of temperature, k(T), and conductivity is constant, k = 9.0 W/m-K.

1.4.4 Alternative Rate Models The rate equations used to model the conduction between adjacent nodes in Section 1.4.2 were based on separating these nodes with a thin cylindrical shell of material that is modeled as a plane wall. An even better approximation for these conduction terms uses the resistance to conduction through a cylinder. According to the resistance formula listed in Table 1-2, the rate equations become: q˙ LHS =

q˙ RHS =

k L 2 π (T i−1 − T i ) ri ln ri−1 k L 2 π (T i+1 − T i ) ri+1 ln ri

(1-127)

(1-128)

Also, the volume of the control volume may be computed more exactly in order to provide a better estimate of the thermal energy generation term: r 2 r 2 ri + − ri − g˙ = g˙ r=ri π L (1-129) 2 2

Maximum error (K)

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES) 10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

61

original rate model, Section 1.4.2

advanced rate model, Section 1.4.4

1

10

100 Number of nodes

1,000

Figure 1-24: Maximum error between the analytical and numerical models (original and advanced) as a function of the number of nodes.

The portion of the EES code from Section 1.4.2 that is modiﬁed to use these more advanced rate models (with modiﬁcations indicated in bold), is shown below. “Internal control volume energy balances” duplicate i=2,(N-1) q dot LHS[i]=2∗ pi∗ L∗ k∗ (T[i-1]-T[i])/ln(r[i]/r[i-1]) q dot RHS[i]=2∗ pi∗ L∗ k∗ (T[i+1]-T[i])/ln(r[i+1]/r[i]) g dot[i]=gen(r[i])∗ pi∗ ((r[i]+DELTAr/2)ˆ2-(r[i]-DELTAr/2)ˆ2)∗ L q dot LHS[i]+g dot[i]+q dot RHS[i]=0 end

“conduction in from inner radius” “conduction in from outer radius” “generation” “energy balance”

“Energy balance for node on internal surface” q dot RHS[1]=2∗ pi∗ L∗ k∗ (T[2]-T[1])/ln(r[2]/r[1]) q dot conv in=2∗ pi∗ L∗ r in∗ h bar in∗ (T inﬁnity in-T[1]) g dot[1]=gen(r[1])∗ pi∗ ((r[1]+DELTAr/2)ˆ2-r[1]ˆ2)∗ L q dot RHS[1]+q dot conv in+g dot[1]=0

“conduction in from outer radius” “convection from internal ﬂuid” “generation” “energy balance”

“Energy balance for node on external surface” q dot LHS[N]=2∗ pi∗ L∗ k∗ (T[N-1]-T[N])/ln(r[N]/r[N-1]) q dot conv out=2∗ pi∗ L∗ r out∗ h bar out∗ (T inﬁnity out-T[N]) g dot[N]=gen(r[N])∗ pi∗ (r[N]ˆ2-(r[N]-DELTAr/2)ˆ2)∗ L q dot LHS[N]+q dot conv out+g dot[N]=0

“conduction in from from inner radius” “convection from external ﬂuid” “generation in node N” “energy balance for node N”

The more advanced numerical solution will approach the actual solution somewhat more quickly (i.e., with fewer nodes) than the original numerical solution. Figure 1-24 illustrates the difference between the advanced numerical solution (in the limit of a constant generation rate, b = c = 0) and the analytical solution as well as the error associated with the original solution derived in Section 1.4.2. Notice that the use of the advanced rate models provides a substantial improvement in accuracy for any given number of nodes, N. The use of advanced rate models for a spherical problem is illustrated in EXAMPLE 1.4-1.

EXAMPLE 1.4-1: FUEL ELEMENT

62

One-Dimensional, Steady-State Conduction

EXAMPLE 1.4-1: FUEL ELEMENT A nuclear fuel element consists of a sphere of ﬁssionable material (fuel) with radius r fuel = 5.0 cm and conductivity k fuel = 1.0 W/m-K that is surrounded by a shell of metal cladding with outer radius r clad = 7.0 cm and conductivity k clad = 300 W/m-K. The outer surface of the cladding is exposed to helium gas that is being heated by the reactor. The average convection coefﬁcient between the gas and the cladding surface is h = 100 W/m2 -K and the temperature of the gas is T∞ = 500◦ C. Inside the fuel element, ﬁssion fragments are produced that have high velocities. The products collide with the atoms of the material and provide the thermal energy for the reactor. This process can be modeled as a non-uniform volumetric thermal energy generation (g˙ ) that can be approximated by: r g˙ = g˙ 0 exp −b (1) r fuel where g˙ 0 = 5× 105 W/m3 is the volumetric rate of heat generation at the center of the sphere and b = 1.0 is a dimensionless constant that characterizes how quickly the generation rate decays in the radial direction. a) Develop a numerical model for the spherical fuel element using EES. A function is deﬁned that returns the volumetric generation given the radial position and the radius of the fuel element. function gen(r, r fuel) “This function deﬁnes the volumetric heat generation in the fuel element Inputs: r: radius (m) r fuel: radius of fuel sphere (m) Output: volumetric heat generation at r (W/mˆ3)” g dot 0=5e5 [W/mˆ3] “volumetric generation rate at the center” b=1.0 [-] “constant that describes rate of decay” gen=g dot 0∗ exp(-b∗ r/r fuel) “volumetric rate of generation” end

The next section of the EES code provides the problem inputs. “EXAMPLE 1.4-1: Fuel Element” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” r fuel=5.0 [cm]∗ convert(cm,m) r clad=7.0 [cm]∗ convert(cm,m) k fuel=1.0 [W/m-K] k clad=300 [W/m-K] h bar=100 [W/mˆ2-K] T inﬁnity=converttemp(C,K,500 [C])

“fuel radius” “cladding radius” “fuel conductivity” “cladding conductivity” “average convection coefﬁcient” “helium temperature”

The numerical solution proceeds by distributing nodes throughout the computational domain that stretches from r = 0 to r = r fuel . There is no reason to include the metal cladding in the numerical model. The cladding increases the thermal

63

resistance that is already present due to convection with the gas; however, this effect can be included using a conduction thermal resistance. The positions of a uniformly distributed set of nodes are obtained from: ri =

(i − 1) r fuel (N − 1)

for i = 1..N

and the distance between adjacent nodes is: r =

“Setup nodes” N=50 [-] duplicate i=1,N r[i]=r fuel∗ (i-1)/(N-1) end DELTAr=r fuel/(N-1)

r fuel (N − 1)

“number of nodes” “radial position of each node” “distance between adjacent nodes”

A control volume for an arbitrary internal node is shown in Figure 1. rfuel

cladding rclad fuel element ri +

Figure 1: Control volume for an internal node.

q⋅ LHS

g⋅

Δr 2 Δ r ri − 2 q⋅ RHS

Ti-1 Ti T i+1

The energy balances for the internal control volumes are: q˙ LH S + q˙ RH S + g˙ = 0 The control volumes are spherical shells. Therefore, it is appropriate to use a conduction model that is consistent with conduction through a spherical shell (see Table 1-2). Note that this problem could also be solved by approximating the spherical shells as plane walls with different surface areas, as was done in Section 1.4.2. However, building the proper geometry into the control volume energy balances will allow the problem to be solved to a speciﬁed accuracy with fewer nodes. q˙ LH S =

(Ti−1 − Ti ) 1 1 1 − 4 π k fuel ri−1 ri

and q˙ RH S =

(Ti+1 − Ti ) 1 1 1 − 4 π k fuel ri ri+1

The temperature differences used to evaluate q˙ LH S and q˙ RH S are consistent with the direction of the conduction heat transfer terms shown in Figure 1 whereas the

EXAMPLE 1.4-1: FUEL ELEMENT

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

EXAMPLE 1.4-1: FUEL ELEMENT

64

One-Dimensional, Steady-State Conduction

resistance values in the denominators are written in the form of 1/rin − 1/r out (e.g., 1/ri −1 − 1/ri and 1/ri − 1/ri +1 ) so that the resistances are positive. The generation in each control volume is given by: 4 r 3 r 3 g˙ri g˙ i = π − ri − ri + 3 2 2 Combining these equations allows the control volume energy balances for the internal nodes to be written as: 4 π k fuel (Ti−1 − Ti ) 4 π k fuel (Ti+1 − Ti ) 4 r 3 r 3 g˙ri = 0 + + π − ri − ri + 1 1 1 1 3 2 2 − − ri−1 ri ri ri+1 for i = 2.. (N − 1)

(2)

“Internal control volume energy balance” duplicate i=2,(N-1) 4∗ pi∗ k fuel∗ (T[i-1]-T[i])/(1/r[i-1]-1/r[i])+4∗ pi∗ k fuel∗ (T[i+1]-T[i])/(1/r[i]-1/r[i+1])+& 4∗ pi∗ ((r[i]+DELTAr/2)ˆ3-(r[i]-DELTAr/2)ˆ3)∗ gen(r[i],r fuel)/3=0 end

The energy balance for the node that is placed at the outer edge of the fuel (i.e., node N ) is: 4 π k fuel (TN −1 − TN ) 4 r 3 (T∞ − TN ) 3 g˙rN = 0 + + π rN − rN − 1 1 Rcond ,clad + Rconv 3 2 − r N −1 r N combined thermal generation in outer shell resistance of cladding conduction between the outermost and adjoining nodes

and convection

(3)

where Rclad is the resistance to conduction through the cladding: 1 1 1 − Rcond ,clad = 4 π k clad r fuel r clad and Rconv is the resistance to convection from the surface of the cladding to the gas: Rconv =

1 2 4 π r clad h

R cond clad=(1/r fuel-1/r clad)/(4∗ pi∗ k clad) “conduction resistance of cladding” R conv=1/(4∗ pi∗ r cladˆ2∗ h bar) “convection resistance from surface of cladding” 4*pi*k fuel*(T[N-1]-T[N])/(1/r[N-1]-1/r[N])+(T inﬁnity-T[N])/(R cond clad+R conv)+& 4∗ pi∗ (r[N]ˆ3-(r[N]-DELTAr/2)ˆ3)∗ gen(r[N],r fuel)/3=0 “node N energy balance”

If the same conduction model is used, then the energy balance for the node placed at the center of the fuel (i.e., node 1) is: 4 π k fuel (T2 − T1 ) 4 r 3 3 − r 1 g˙r1 = 0 (4) r1 + + π 1 1 3 2 − r1 r2

65

4∗ pi∗ k fuel∗ (T[2]-T[1])/(1/r[1]-1/r[2])+4∗ pi∗ ((r[1]+DELTAr/2)ˆ3-r[1]ˆ3)∗ gen(r[1],r fuel)/3=0 “node 1 energy balance”

Executing the EES code will lead to a division by zero error message. The radial location of node 1, r1 , is equal to 0 and therefore the 1/r1 term in the denominator of Eq. (4) is inﬁnite. (The actual resistance associated with conducting energy to a point is inﬁnite.) A similar error will be encountered when computing q˙ LH S for node 2 in Eq. (2). This problem can be dealt with by calculating the conduction between nodes 1 and 2 using a plane wall approximation. The energy balance for node 1 becomes: 4 π k fuel

r 2 (T2 − T1 ) 4 r 3 3 − r 1 g˙r1 = 0 + π r1 + r1 + 2 r 3 2

{4∗ pi∗ k fuel∗ (T[2]-T[1])/(1/r[1]-1/r[2])+4∗ pi∗ ((r[1]+DELTAr/2)ˆ3-r[1]-ˆ3)∗ gen(r[1],r fuel)/3=0} 4∗ pi∗ k fuel∗ (r[1]+DELTAr/2)ˆ2∗ (T[2]-T[1])/DELTAr+4∗ pi∗ ((r[1]+DELTAr/2)ˆ3-r[1]ˆ3)∗ gen(r[1],r fuel)/3=0 “node 1 energy balance”

The energy balance for node 2 has to be rewritten in the same way: r 2 (T1 − T2 ) 4 π k fuel (T3 − T2 ) + 4 π k fuel r 1 + 1 1 2 r − r2 r3 3 3 4 r r g˙r2 = 0 + π − r2 − r2 + 3 2 2

“Internal control volume energy balance” {duplicate i=2,(N-1) 4∗ pi∗ k fuel∗ (T[i-1]-T[i])/(1/r[i-1]-1/r[i])+4∗ pi∗ k fuel∗ (T[i+1]-T[i])/(1/r[i]-1/r[i+1])+& 4∗ pi∗ ((r[i]+DELTAr/2)ˆ3-(r[i]-DELTAr/2)ˆ3)∗ gen(r[i],r fuel)/3=0 end} duplicate i=3,(N-1) 4∗ pi∗ k fuel∗ (T[i-1]-T[i])/(1/r[i-1]-1/r[i])+4∗ pi∗ k fuel∗ (T[i+1]-T[i])/(1/r[i]-1/r[i+1])+& 4∗ pi∗ ((r[i]+DELTAr/2)ˆ3-(r[i]-DELTAr/2)ˆ3)∗ gen(r[i],r fuel)/3=0 end 4∗ pi∗ k fuel∗ (r[1]+DELTAr/2)ˆ2∗ (T[1]-T[2])/DELTAr+4∗ pi∗ k fuel∗ (T[3]-T[2])/(1/r[2]-1/r[3])+& 4∗ pi∗ ((r[2]+DELTAr/2)ˆ3-(r[2]-DELTAr/2)ˆ3)∗ gen(r[2],r fuel)/3=0 “node 2 energy balance”

With these changes, the program can be solved. The solution is converted to ◦ C:

duplicate i=1,N T C[i]=converttemp(K,C,T[i]) end

“temperature in C”

EXAMPLE 1.4-1: FUEL ELEMENT

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

One-Dimensional, Steady-State Conduction

Figure 2 illustrates the temperature in the fuel element as a function of radius. 660

Temperature (°C)

640 620 600 580 560 540 520 0

0.01

0.02 0.03 Radius (m)

0.04

0.05

Figure 2: Temperature distribution within fuel.

The maximum temperature in the fuel element is obtained with the max function. T max C=max(T C[1..N])

“maximum temperature of the fuel, in C”

Figure 3 shows the maximum temperature in the fuel as a function of the number of nodes and indicates that the solution converges if at least 100 nodes are used. 668 666 Maximum temperature (°C)

EXAMPLE 1.4-1: FUEL ELEMENT

66

664 662 660 658 656 654 652 650 2

10

100 Number of nodes

1000

Figure 3: Maximum temperature within fuel as a function of the number of nodes.

Several sanity checks can be carried out in order to verify that the solution is physically correct. Figure 4 shows the maximum temperature in the fuel as a function of the fuel conductivity for various values of the volumetric generation at the center of

67

the fuel (g˙ 0 ). The maximum temperature increases as either the fuel conductivity decreases or the volumetric rate of generation increases. 800

g⋅ o′′′= 10 × 106 W/m3

Maximum temperature (°C)

750 700 650 600

g⋅ o′′′= 5 × 106 W/m3

550 500 450 0.5

6 3 g⋅ ′′′ o = 2 × 10 W/m

1

1.5

g⋅ o′′′= 1 × 106 W/m 3

2 2.5 3 3.5 Fuel conductivity (W/m-K)

4

4.5

5

Figure 4: Maximum temperature in the fuel as a function of kfuel for various values of g˙ 0.

Finally, we can compare the numerical model with the analytical solution for the limiting case where b = 0 in Eq. (1) (i.e., the fuel experiences a uniform rate of volumetric generation). The general solution for the temperature distribution and temperature gradient within a sphere exposed to a uniform generation rate is given in Table 1-3: T =−

g˙ 2 C 1 r + + C2 6 k fuel r

(5)

and dT C1 g˙ r− 2 =− dr 3 k fuel r

(6)

where C1 and C2 are constants of integration. The temperature at the center of the sphere must be bounded and therefore C1 must be equal to 0 by inspection of Eq. (5); alternatively, the temperature gradient at the center must be zero, which would also require that C1 = 0 according to Eq. (6). The second boundary condition is related to an energy balance at the interface between the cladding and the fuel: Tr =rfuel − T∞ dT 2 = (7) −k fuel 4 π r fuel dr r =rfuel Rcond ,clad + Rconv Combining Equations (5) through (7) leads to: g˙ 2 r + C 2 − T∞ − 6 k fuel fuel g˙ 2 −k fuel 4 π r fuel r fuel = − 3 k fuel Rcond ,clad + Rconv

EXAMPLE 1.4-1: FUEL ELEMENT

1.4 Numerical Solutions to Steady-State 1-D Conduction Problems (EES)

One-Dimensional, Steady-State Conduction

which can be solved for C2 . “Analytical solution” g dot=gen(0 [m],r fuel) “rate of volumetric generation to use in analytical solution” -k fuel∗ 4∗ pi∗ r fuelˆ2∗ (-g dot∗ r fuel/(3∗ k fuel))= & (-g dot∗ r fuelˆ2/(6∗ k fuel)+C 2-T inﬁnity)/(R cond clad+R conv) “boundary condition at r=r fuel”

The analytical solution is obtained at the same radial locations as the numerical solution. duplicate i=1,N T an[i]=-g dot∗ r[i]ˆ2/(6∗ k fuel)+C 2 T an C[i]=converttemp(K,C,T an[i]) end

“analytical solution” “in C”

Figure 5 shows the analytical and numerical solutions in the limit that b = 0 for 50 nodes; the agreement is nearly exact, indicating that the numerical solution is adequate. 800

predicted by numerical model, with b = 0 predicted by analytical model

750 Temperature (°C)

EXAMPLE 1.4-1: FUEL ELEMENT

68

700 650 600 550 500 0

0.01

0.02 0.03 Radius (m)

0.04

0.05

Figure 5: Temperature as a function of radius predicted by the analytical model should be

indicated by a line and the numerical model by the dots.

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems using MATLAB 1.5.1 Introduction Numerical models of 1-D steady-state conduction problems are introduced and implemented using EES in Section 1.4. EES internally provides all of the numerical manipulations that are needed to solve the system of algebraic equations that constitutes a numerical model. This capability reduces the complexity of the problem. However, there are disadvantages to using EES. For example, EES will generally require signiﬁcantly

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

69 k = 9 W/m-K 5 3 g⋅ ′′′= 1x10 W/m rin = 10 cm rout = 20 cm

Figure 1-25: Cylinder with volumetric generation. T∞, in = 20°C 2 hin = 100 W/m -K

T∞, out = 100°C 2 hout = 200 W/m -K

more time to solve the equations than is required by a compiled computer language. There is an upper limit to the number of variables that EES can handle, which places an upper bound on the number of nodes that can be used in the model. The structure of EES requires that every variable be retained in the ﬁnal solution. Therefore, you cannot deﬁne and then erase intermediate variables in the course of obtaining a solution and, as a result, numerical solutions in EES often require a lot of memory. Finally, some models require logic statements (e.g., if-then-else statements) that are difﬁcult to include in EES. For these reasons, it is useful to learn how to implement numerical models in a formal programming language, e.g., FORTRAN, C++, or MATLAB. The steps required to solve the algebraic equations associated with a numerical model are demonstrated in this section using the MATLAB software. It is suggested that the reader stop and go through the tutorial provided in Appendix A.3 in order to become familiar with MATLAB. Appendix A.3 can be found on the web site associated with this book (www.cambridge.org/nellisandklein).

1.5.2 Numerical Solutions in Matrix Format The cylinder problem that is considered in Section 1.4 in order to illustrate numerical methods using EES is shown again in Figure 1-25. An aluminum oxide cylinder is exposed to ﬂuid on its internal and external surfaces. The temperature of the ﬂuid that is exposed to the internal surface is T ∞,in = 20◦ C and the average heat transfer coefﬁcient on this surface is hin = 100 W/m2 -K. The temperature of the ﬂuid exposed to the external surface is T ∞,out = 100◦ C and the average heat transfer coefﬁcient on this surface is hout = 200 W/m2 -K. The thermal conductivity is assumed to be constant and equal to k = 9.0 W/m-K. We will begin by solving the problem for the case where the rate of volumetric generation of thermal energy within the cylinder is uniform and equal to g˙ = 1 × 105 W/m3 . The inner and outer radii of the cylinder are rin = 10 cm and rout = 20 cm, respectively. The development of the system of equations proceeds as discussed in Section 1.4.2. A uniform distribution of nodes is used and therefore the radial location of each node (ri ) is: ri = rin +

(i − 1) (rout − rin ) (N − 1)

i = 1..N

(1-130)

where N is the number of nodes used for the simulation. The radial distance between adjacent nodes ( r) is: r =

(rout − rin ) (N − 1)

(1-131)

70

One-Dimensional, Steady-State Conduction

The energy balances for the internal nodes are: q˙ LHS + q˙ RHS + g˙ = 0 where

kL2π

ri −

q˙ LHS = kL2π q˙ RHS =

ri +

(1-132)

r (T i−1 − T i ) 2 r r (T i+1 − T i ) 2 r

g˙ = g˙ 2 π ri L r

(1-133)

(1-134)

(1-135)

Equations (1-132) through (1-135) are combined: r r k L 2 π ri + k L 2 π ri − 2 2 (T i−1 − T i ) + (T i+1 − T i ) + g˙ 2 π ri L r = 0 r r for i = 2.. (N − 1) (1-136) The energy balance for the control volume associated with node 1 is: r k L 2 π r1 + r 2 hin 2 π r1 L (T ∞,in − T 1 ) + =0 (T 2 − T 1 ) + g˙ 2 π r1 L r 2 (1-137) The energy balance for the control volume associated with node N is: r k L 2 π rN − r 2 hout 2 π rN L (T ∞,out − T N ) + =0 (T N−1 − T N ) + g˙ 2 π rN L r 2 (1-138) Equations (1-136) through (1-138) represent N linear algebraic equations in an equal number of unknown temperatures. In order to solve these equations using a formal programming language, it is necessary to represent this set of equations as a matrix equation. Recall from linear algebra that a linear system of equations, such as: 2x1 + 3x2 + 1x3 = 1 1x1 + 5x2 + 1x3 = 2 7x1 + 1x2 + 2x3 = 5

(1-139)

can be written as a matrix equation: ⎡ ⎤⎡ ⎤ ⎡ ⎤ 2 3 1 1 x1 ⎣1 5 1 ⎦ ⎣x2 ⎦ = ⎣2 ⎦ 7 1 2 x3 5 A

X

(1-140)

b

or AX = b

(1-141)

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

where A is a matrix and X ⎡ 2 3 A = ⎣1 5 7 1

and b are vectors: ⎡ ⎤ ⎤ 1 x1 1 ⎦ , X = ⎣x2 ⎦ , x3 2

and

⎡ ⎤ 1 b = ⎣2 ⎦ 5

71

(1-142)

Most programming languages, including MATLAB, have built-in or library routines for decomposing the system of equations and solving for the vector of unknowns, X. This is a mature area of research and advanced methods exist for quickly solving matrix equations, particularly when most of the entries in A are 0 (i.e., A is a sparse matrix). MATLAB is speciﬁcally designed to handle large matrix equations. In this section, we will use MATLAB to solve heat transfer problems. This requires an understanding of how to place large systems of equations, corresponding to the energy balances, into a matrix format. Each row of the A matrix and b vector correspond to an equation whereas each column of the A matrix is the coefﬁcient that multiplies the corresponding unknown (typically a nodal temperature) in that equation. To set up a system of equations in matrix format, it is necessary to carefully deﬁne how the rows and energy balances are related and how the columns and unknown temperatures are related. The ﬁrst step is to deﬁne the vector of unknowns, the vector X in Eq. (1-140). It does not really matter what order the unknowns are placed in X, but the implementation of the solution is much easier if a logical order is used. In this problem, the unknowns are the nodal temperatures. Therefore, the most logical technique for ordering the unknown temperatures in the vector X is: ⎡ ⎤ X1 = T1 ⎢ X2 = T2 ⎥ ⎥ (1-143) X=⎢ ⎣ ... ⎦ XN = TN Equation (1-143) shows that the unknown temperature at node i (i.e., Ti ) corresponds to element i of vector X (i.e., Xi ). The next step is to deﬁne how the rows in the matrix A and the vector b correspond to the N control volume energy balances that must be solved. Again, it does not matter what order the equations are placed into the A matrix, but the solution is easiest if a logical order is used: ⎡ ⎤ row 1 = control volume 1 equation ⎢ row 2 = control volume 2 equation ⎥ ⎥ A=⎢ (1-144) ⎣ ⎦ ··· row N = control volume N equation Equation (1-144) shows that the equation for control volume i is placed into row i of matrix A.

1.5.3 Implementing a Numerical Solution in MATLAB We can return to the numerical problem that was discussed in Section 1.5.2. Open a new M-ﬁle (select New M-File from the File menu) which will bring up the M-ﬁle editor. Save the script as cylinder (select Save As from the File menu) in a directory that is in your search path (you can specify the directories in your search path by typing pathtool in the Command window). Enter the inputs to the problem at the top of the script and save it. Note that the % symbol indicates that anything that follows on that line will be a comment.

72

One-Dimensional, Steady-State Conduction

MATLAB will not assign units to any of the variables; they are all dimensionless as far as the software is concerned. This limitation puts the burden squarely on the user to clearly understand the units of each variable and ensure that they are consistent. The use of a semicolon after each assignment statement prevents the variables from being echoed in the working environment. The clear command at the top of the script clears all variables from the workspace. clear;

%clear all variables from the workspace

% Inputs r in=0.1; r out=0.2; g dot tp=1e5; L=1; k=9; T infinity in=20+273.2; h bar in=100; T infinity out=100+273.2; h bar out=200;

%inner radius of cylinder (m) %outer radius of cylinder (m) %constant volumetric generation (W/mˆ3) %unit length of cylinder (m) %thermal conductivity of cylinder material (W/m-K) %average temperature of fluid inside cylinder (K) %heat transfer coefficient inside cylinder (W/mˆ2-K) %temperature of fluid outside cylinder (K) %heat transfer coefficient at outer surface (W/mˆ2-K)

In order to run your script from the MATLAB working environment, type cylinder at the command prompt: >> cylinder

Nothing appears to have happened. However, all of the variables that are deﬁned in the script are now available in the work space. For example, if the name of any variable is entered at the command prompt then its value is displayed. >> h bar out h bar out = 200

For a complete list of variables in the workspace, use the command who. >> who Your variables are: L T infinity out h bar in k r out T infinity in g dot tp h bar out r in

The number and location of the nodes for the solution must be speciﬁed. A vector of radial locations (r) is setup using a for loop. Each of the statements between the for and end statements is executed each time through the loop. Enter the following lines into the cylinder M-ﬁle: % Setup nodes N=10; Dr=(r out-r in)/(N-1); for i=1:N r(i)=r in+(i-1)∗ (r out-r in)/(N-1); end

%number of nodes (-) %distance between nodes (m) %radial position of each node (m)

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

73

The energy balances must be setup in an appropriately sized matrix, the variable A, and vector, the variable b. Recall that our matrix A needs to have as many rows as there are equations (the N control volume energy balances) and as many columns as there are unknowns (the N unknown temperatures) and b is a vector with as many elements as there are equations (N). We’ll start with A and b composed entirely of zeros and subsequently add non-zero elements according to the equations. The A matrix ends up being composed almost entirely of zeros and thus it is referred to as a sparse matrix. Initially we will not take advantage of this sparse characteristic of A. However, in Section 1.5.5, it will be shown that the solution can be accelerated considerably by using specialized matrix solution techniques that are designed for sparse matrices. MATLAB makes it extremely easy to use these sparse matrix solution techniques. The zeros function used below returns a matrix ﬁlled with zeros with its size determined by the input arguments; the variable A will be an N × N matrix ﬁlled with zeros and the variable b will be an N × 1 vector ﬁlled with zeros where the variable N is the number of nodes. %Setup A and b A=zeros(N,N); b=zeros(N,1);

The most difﬁcult step in the process is to ﬁll in the non-zero elements of A and b so that the solution of the system of equations can be obtained through a matrix decomposition process. According to Eq. (1-144), the 1st row in A must correspond to the energy balance for control volume 1, which is given by Eq. (1-137), repeated below: kL2π r r hin 2 π r1 L (T ∞,in − T 1 ) + r1 + =0 (T 2 − T 1 ) + g˙ 2 π r1 L r 2 2 (1-137) It is necessary to algebraically manipulate Eq. (1-137) so that the coefﬁcients that multiply each of the unknowns in this equation (i.e., T1 and T2 ) and the constant term in the equation (i.e., terms that are known) can be identiﬁed. r r kL2π kL2π r1 + − hin 2 π r1 L +T 2 r1 + T1 − r 2 r 2 A1,1

r − hin 2 π r1 L T ∞,in = −g˙ 2 π r1 L 2

A1,2

(1-145)

b1

Equation (1-145) corresponds to the 1st row of A and b. The coefﬁcient in the ﬁrst equation that multiplies the ﬁrst unknown in X, T1 according to Eq. (1-143), must be A1,1 : kL2π r (1-146) r1 + − hin 2 π r1 L A1,1 = − r 2 The coefﬁcient in the ﬁrst equation that multiplies the second unknown in X, T2 according to Eq. (1-143), must be A1,2 : kL2π r r1 + (1-147) A1,2 = r 2

74

One-Dimensional, Steady-State Conduction

Finally, the constant terms associated with the ﬁrst equation must be b1 : r − hin 2 π r1 L T ∞,in 2 These assignments are accomplished in MATLAB: b1 = −g˙ 2 π r1 L

(1-148)

%Energy balance for control volume 1 A(1,1)=-k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr-h bar in∗ 2∗ pi∗ r(1)∗ L; A(1,2)=k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr; b(1)=-h bar in∗ 2∗ pi∗ r(1)∗ L∗ T infinity in-g dot tp∗ 2∗ pi∗ r(1)∗ L∗ Dr/2;

According to Eq. (1-144), rows 2 through N − 1 of matrix A correspond to the energy balances for the corresponding internal control volumes; these equations are given by Eq. (1-136), which is repeated below: kL2π r r kL2π ri − ri + (T i−1 − T i ) + (T i+1 − T i ) + g˙ 2 π ri L r = 0 r 2 r 2 (1-136) for i = 2.. (N − 1) Again, Eq. (1-136) must be rearranged to identify coefﬁcients and constants. r r r kL2π kL2π kL2π +T i−1 ri − − ri + ri − Ti − r 2 r 2 r 2 + T i+1

kL2π r

Ai,i

Ai,i−1

r = −g˙ 2 π ri L r ri + 2 bi

for i = 2.. (N − 1)

(1-149)

Ai,i+1

All of the coefﬁcients for control volume i must go into row i of A, the column depends on which unknown they multiply. Therefore: kL2π r r kL2π ri − − ri + for i = 2 . . . (N − 1) (1-150) Ai,i = − r 2 r 2 Ai,i−1

kL2π = r

Ai,i+1

kL2π = r

r ri − 2 r ri + 2

for i = 2 . . . (N − 1)

(1-151)

for i = 2 . . . (N − 1)

(1-152)

The constant for control volume i must go into row i of b. bi = −g˙ 2 π ri L r

for i = 2 . . . (N − 1)

(1-153)

These equations are programmed most conveniently in MATLAB using a for loop: %Energy balances for internal control volumes for i=2:(N-1) A(i,i)=-k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr-k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; A(i,i-1)=k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr; A(i,i+1)=k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; b(i)=-g dot tp∗ 2∗ pi∗ r(i)∗ L∗ Dr; end

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

75

Finally, the last row of A (i.e., row N) corresponds to the energy balance for the last control volume (node N), which is given by Eq. (1-138), repeated below:

r k L 2 π rN − 2 hout 2 π rN L (T ∞,out − T N ) + r

(T N−1 − T N ) + g˙ 2 π rN L

r =0 2 (1-138)

Equation (1-138) is rearranged: kL2π r r kL2π T N −hout 2 π rN L − + T N−1 rN − rN − r 2 r 2 AN,N

AN,N−1

r = −g˙ 2 π rN L 2

(1-154)

bN

The coefﬁcients in the last row of A and b are: AN,N = −

kL2π r

AN,N−1

rN −

kL2π = r

r 2

− hout 2 π rN L

r rN − 2

(1-155)

(1-156)

and bN = −hout 2 π rN L T ∞,out − g˙ 2 π rN

r L 2

(1-157)

%Energy balance for control volume N A(N,N)=-k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr-h bar out∗ 2∗ pi∗ r(N)∗ L; A(N,N-1)=k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr; b(N)=-h bar out∗ 2∗ pi∗ r(N)∗ L∗ T infinity out-g dot tp∗ 2∗ pi∗ r(N)∗ Dr∗ L/2;

At this point, the matrix A and vector b are completely set up and can be used to determine the unknown temperatures. The solution to the matrix equation, Eq. (1-141), is: X = A−1 b

(1-158)

where A−1 is the inverse of matrix A. The solution is obtained using the backslash operator in MATLAB (note that this is much more efﬁcient than explicitly solving for the inverse of A using the inv command): %Solve for unknowns X=A\b;

%solve for unknown vector, X

76

One-Dimensional, Steady-State Conduction 130

Temperature (°C)

125 120 115 110 105 100 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-26: Temperature as a function of radius predicted by the numerical model.

The vector of unknowns, X, is identical to the temperatures for this problem. T=X; T C=T-273.2;

%assign temperatures from X %in C

The script cylinder can be executed from the working environment by typing cylinder at the command prompt. After execution, the variables that were deﬁned in the M-ﬁle and the solution will reside in the workspace; for example, you can view the solution vector (T C) by typing T C and you can plot temperature as a function of radius using the plot command. >> cylinder >> T C T C= 100.3213 109.0658 115.6711 120.3899 123.4143 124.8938 124.9468 123.6689 121.1383 117.4197 >> plot(r,T C)

Figure 1-26 illustrates the temperature predicted by the MATLAB numerical model as a function of radius. The solution is exactly equal to the solution that would be obtained

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

77

using the EES code in Section 1.4.2 if the same rate of volumetric generation of thermal energy is used.

1.5.4 Functions The problem in Section 1.5.3 was solved using a script, which is a set of MATLAB instructions that can be stored and edited, but otherwise operates on the main workspace just as if you typed the instructions in one at a time. It is often more convenient to use a function. A function will solve the problem in a separate workspace that communicates with the main workspace (or any workspace from which the function is called) only through input and output variables. There are a few advantages to using a function rather than a script. The function may embody a sequence of operations that must be repeated several times within a larger program. For example, suppose that there are multiple sections of a cylinder, but each section has different material properties or boundary conditions. It would be possible to cut and paste the script that was written in Section 1.5.3 over and over again in order to solve this problem. A more attractive option (and the one least likely to result in an error) is to turn the script cylinder.m into a function that can be called whenever you need to consider conduction through a cylinder with generation. The function can be debugged and tested until you are sure that it works and then applied with conﬁdence at any later time. The use of functions provides modularity and elegance to a program and facilitates parametric studies and optimizations. Any computer code that is even moderately complicated should be broken down into smaller, well-deﬁned sub-programs (functions) that can be written and tested separately before they are integrated through well-deﬁned input/output protocols. MATLAB (or EES) programs are no different. It is convenient to develop your code as a script, but you will likely need to turn your script into a function at some point. Let’s turn the script cylinder.m into a function. Save the ﬁle cylinder as cylinderf. The ﬁrst line of the function must declare that it is a function and deﬁne the input/output protocol. For the cylinder problem, the inputs might include the number of nodes (the variable N), the cylinder radii (the variables r in and r out), and the material conductivity (the variable k). Any other parameter that you are interested in varying could also be provided as an input. The logical outputs include the vector of radial positions that deﬁne the nodes (the vector r) and the predicted temperature at these positions (the vector T C). function[r,T C]=cylinderf(N,r in,r out,k)

The keyword function declares the M-ﬁle to be a function and the variables in square brackets are outputs; these variables should be assigned in the body of the function and are passed to the calling workspace. The function name follows the equal sign and the variables in parentheses are the inputs. Note that nothing you do within the function can affect any variable that is external to the function other than those that are explicitly deﬁned as output variables. You should also comment out the clear command that was used to develop the script. The clear statement is not needed since the function operates with its own variable space. The function is terminated with an end statement. The cylinderf function is shown below; the modiﬁcations to the original script cylinder are indicated in bold.

78

One-Dimensional, Steady-State Conduction

function[r,T C]=cylinderf(N,r in,r out,k) %clear;

%clear all variables from the workspace

% Inputs %r in=0.1; %r out=0.2; g dot tp=1e5; L=1; %k=9; T infinity in=20+273.2; h bar in=100; T infinity out=100+273.2; h bar out=200;

%inner radius of cylinder (m) %outer radius of cylinder (m) %constant volumetric generation (W/mˆ3) %unit length of cylinder (m) %thermal conductivity of cylinder material (W/m-K) %average temperature of fluid inside cylinder (K) %heat transfer coefficient inside cylinder (W/mˆ2-K) %temperature of fluid outside cylinder (K) %average heat transfer coefficient at outer surface (W/mˆ2-K)

% Setup nodes %N=10; %number of nodes (-) Dr=(r out-r in)/(N-1); %distance between nodes (m) for i=1:N r(i)=r in+(i-1)∗ (r out-r in)/(N-1); %radial position of each node (m) end %Setup A and b A=zeros(N,N); b=zeros(N,1); %Energy balance for control volume 1 A(1,1)=-k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr-h bar in∗ 2∗ pi∗ r(1)∗ L; A(1,2)=k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr; b(1)=-h bar in∗ 2∗ pi∗ r(1)∗ L∗ T infinity in-g dot tp∗ 2∗ pi∗ r(1)∗ L∗ Dr/2; %Energy balances for internal control volumes for i=2:(N-1) A(i,i)=-k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr-k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; A(i,i-1)=k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr; A(i,i+1)=k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; b(i)=-g dot tp∗ 2∗ pi∗ r(i)∗ L∗ Dr; end %Energy balance for control volume N A(N,N)=-k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr-h bar out∗ 2∗ pi∗ r(N)∗ L; A(N,N-1)=k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr; b(N)=-h bar out∗ 2∗ pi∗ r(N)∗ L∗ T infinity out-g dot tp∗ 2∗ pi∗ r(N)∗ Dr∗ L/2; % Solve for unknowns X=A\b; T=X; T C=T-273.2; end

%solve for unknown vector, X %assign temperatures from X %in C

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

79

130

Maximum temperature (°C)

128 126 124 122 120 118 116 114 0

10

20 30 40 Thermal conductivity (W/m-K)

50

60

Figure 1-27: Maximum temperature as a function of thermal conductivity.

The following code, typed in the main workspace, will call the function cylinderf for a speciﬁc set of values of the input parameters: N = 10, rin = 0.1 m, rout = 0.2 m, and k = 9 W/m-K. >> N=10; >> r in=0.1; >> r out=0.2; >> k=9; >> [r,T C]=cylinderf(N,r in,r out,k);

The vectors r and T C are the same as those determined in Section 1.5.3. It is easy to carry out a parametric study using functions. For example, create a new script called varyk that calls the function cylinderf for a range of conductivity and keeps track of the maximum temperature in the wall that is predicted for each value of conductivity. N=50; %number of nodes r in=0.1; %inner radius (m) r out=0.2; %outer radius (m) Nk=10; %number of values of k to investigate for i=1:Nk k(i,1)=2+i∗ 50/Nk; %a vector consisting of the conductivities to be considered (W/m-K) [r,T C]=cylinderf(N,r in,r out,k(i,1)); %call the cylinderf function T max C(i,1)=max(T C); %determine the maximum temperature end

Call the script varyk from the main workspace and plot the maximum temperature as a function of conductivity (Figure 1-27). >> varyk >> plot(k,T max C)

80

One-Dimensional, Steady-State Conduction

It is possible to call a function from within a function. The volumetric generation in the cylinder was speciﬁed in Section 1.4.2 as a function of radius according to: g˙ = a + b r + c r2

(1-159)

where a = 1 × 104 W/m3 , b = 2 × 105 W/m4 , and c = 5 × 107 W/m5 . Generate a subfunction (a function that is only visible to other functions in the same M-ﬁle) that has one input (r, the radial position) and one output (g, the volumetric rate of thermal energy generation). The function below, generation, placed at the bottom of the cylinderf ﬁle will be callable from within the function cylinderf. function[g]=generation(r) %the generation function returns the volumetric %generation (W/mˆ3) as a function of radius (m) %constants for generation function a=1e4; %W/mˆ3 b=2e5; %W/mˆ4 c=5e7; %W/mˆ5 g=a+b∗ r+c∗ rˆ2; %volumetric generation end

Replacing the constant generation within the cylinderf code with calls to the function generation will implement the solution for non-uniform generation; the altered portion of the code is shown in bold. %Energy balance for control volume 1 A(1,1)=-k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr-h bar in∗ 2∗ pi∗ r(1)∗ L; A(1,2)=k∗ L∗ 2∗ pi∗ (r(1)+Dr/2)/Dr; b(1)=-h bar in∗ 2∗ pi∗ r(1)∗ L∗ T infinity in-generation(r(1))∗ 2∗ pi∗ r(1)∗ L∗ Dr/2; %Energy balances for internal control volumes for i=2:(N-1) A(i,i)=-k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr-k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; A(i,i-1)=k∗ L∗ 2∗ pi∗ (r(i)-Dr/2)/Dr; A(i,i+1)=k∗ L∗ 2∗ pi∗ (r(i)+Dr/2)/Dr; b(i)=-generation(r(i))∗ 2∗ pi∗ r(i)∗ L∗ Dr; end %Energy balance for control volume N A(N,N)=-k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr-h bar out∗ 2∗ pi∗ r(N)∗ L; A(N,N-1)=k∗ L∗ 2∗ pi∗ (r(N)-Dr/2)/Dr; b(N)=-h bar out∗ 2∗ pi∗ r(N)∗ L∗ T infinity out-generation(r(N))∗ 2∗ pi∗ r(N)∗ Dr∗ L/2;

Figure 1-28 illustrates the temperature as a function of radius predicted by the model, modiﬁed to account for the non-uniform volumetric generation. Figure 1-28 is identical to Figure 1-14, the solution obtained using EES in Section 1.4.2.

1.5.5 Sparse Matrices The problem considered in Section 1.5.3 can be solved more efﬁciently (from a computational time standpoint) using sparse matrix solution techniques. Sparse matrices are matrices with mostly zero elements. The matrix A that was setup in Section 1.5.3 has

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

81

660 640

Temperature (°C)

620 600 580 560 540 520 500 480 460 0.1

0.12

0.14 0.16 Radius (m)

0.18

0.2

Figure 1-28: Temperature as a function of radius with non-uniform volumetric generation.

many zero elements and the fraction of the entries in the matrix that are zero increases with increasing N. Matrix A is tridiagonal, which means that it only has non-zero values on the diagonal and on the super- and sub-diagonal positions. This type of banded matrix will occur frequently in numerical solutions of conduction heat transfer problems because the nonzero elements are related to the coefﬁcients in the energy equations and therefore represent thermal interactions between different nodes. Typically, only a few nodes can directly interact and therefore there will be only a few non-zero coefﬁcients in any row. The script below (varyN) keeps track of the time required to run the cylinderf function as the number of nodes in the solution increases. The MATLAB function toc returns the elapsed time relative to the time when the function tic was executed. These functions provide a convenient way to keep track of how much time various parts of a MATLAB program are consuming. A more complete delineation of the execution time within a function can be obtained using the profile function in MATLAB. clear; r in=0.1; r out=0.2; k=9; for i=1:9 N(i,1)=2ˆ(i+1); tic; [r,T C]=cylinderf(N(i,1),r in,r out,k); time(i,1)=toc; end

%inner radius (m) %outer radius (m) %thermal conductivity (W/m-K) %number of nodes (-) %start time %call cylinder function %end timer and record time

The elapsed time as a function of the number of nodes is shown in Figure 1-29. The computational time grows approximately with the number of nodes to the second power. There is an upper limit to the number of nodes that can be considered that depends on the amount of memory installed in your personal computer. It is likely that you cannot set N to be greater than a few thousand nodes. The problem can be solved more efﬁciently if sparse matrices are used. Rather than initializing A as a full matrix of zeros, it can be initialized as a sparse matrix using the

Time required for execution (s)

82

One-Dimensional, Steady-State Conduction 10

1

10

0

10

-1

10

-2

10

-3

10

-4

100

without sparse matrices

with sparse matrices

101

102 103 Number of nodes

104

105

Figure 1-29: Time required to run the cylinderf.m function as a function of the number of nodes with and without sparse matrices.

spalloc (sparse matrix allocation) command. The spalloc command requires three arguments, the ﬁrst two are the dimensions of the matrix and the last is the number of nonzero entries. Equations (1-145), (1-149), and (1-154) show that each equation will include at most three unknowns and therefore each row of A will have at most three non-zero entries. Therefore, the matrix A can have no more than 3 N non-zero entries. %A=zeros(N,N); A=spalloc(N,N,3∗ N);

When the variable A is deﬁned by spalloc, only the non-zero entries of the matrix are tracked. MATLAB operates on sparse matrices just as it does on full matrices. The remainder of the function cylinderf does not need to be modiﬁed, however the function is now much more efﬁcient for large numbers of nodes. If the script varyN is run again, you will ﬁnd that you can use much larger values of N before running out of memory and also that the code executes much faster at large values of N. The execution time as a function of the number of nodes for the cylinderf function using sparse matrices is also shown in Figure 1-29. Notice that the sparse matrix code is actually somewhat less efﬁcient for small values of N due to the overhead required to set up the sparse matrices; however, for large values of N, the code is much more efﬁcient. The computation time grows approximately with N to the ﬁrst power when sparse matrices are used. The use of sparse matrices may not be particularly important for the steady-state, 1-D problem investigated in this section. However, the 2-D and transient problems investigated in subsequent chapters require many more nodes in order to obtain accurate solutions and therefore the use of sparse matrices becomes important for these problems.

1.5.6 Temperature-Dependent Properties The cylinder problem considered in Section 1.5.3 is an example of a linear problem. The set of equations in a linear problem can be represented in the form A X = b, where A is a matrix and b is a vector. Linear problems can be solved in MATLAB without iteration. The inclusion of temperature-dependent conductivity (or generation, or

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

83

any aspect of the problem) causes the problem to become non-linear. The introduction of temperature-dependent properties did not cause any apparent problem for the EES model discussed in Section 1.4.3, although it became important to identify a good set of guess values. EES automatically detected the non-linearity and iterated as necessary to solve the non-linear system of equations. However, non-linearity complicates the solution using MATLAB because the equations can no longer be put directly into matrix format. The coefﬁcients multiplying the unknown temperatures themselves depend on the unknown temperatures. It is necessary to use some type of a relaxation process in order to use MATLAB to solve the problem. There are a few options for solving this kind of nonlinear problem; in this section, a technique that is sometimes referred to as successive substitution is discussed. The successive substitution process begins by assuming a temperature distribution throughout the computational domain (i.e., assume a value of temperature for each node, Tˆ i for i = 1..N). The assumed values of temperature are used to compute the coefﬁcients that are required to set up the matrix equation (e.g., the temperature-dependent conductivity). The matrix equation is subsequently solved, as discussed in Section 1.5.3, which results in a prediction for the temperature distribution throughout the computational domain (i.e., a predicted value of the temperature for each node, Ti for i = 1..N). The assumed and predicted temperatures at each node are compared and used to compute an error; for example, the sum of the square of the difference between the value of Tˆ i and Ti at every node. If the error is greater than some threshold value, then the process is repeated, this time using the solution Ti as the assumed temperature distribution Tˆ i in order to calculate the coefﬁcients of the matrix equation. The implementation of the successive substitution process carries out the solution that was developed in Sections 1.5.2 through 1.5.4 within a while loop that terminates when the error becomes sufﬁciently small. This process is illustrated schematically in Figure 1-30 and demonstrated in EXAMPLE 1.5-1. assume a temperature distribution Tˆi for i = 1.. N setup A and b using Tˆi to solve for any temperature-dependent coefficients

Tˆi = Ti for i = 1.. N

solve the matrix equation A X = b in order to predict Ti for i = 1.. N compute the convergence error, err 2 1 N ˆ e.g., err = Ti − Ti ∑ N i =1

(

no

)

err < convergence tolerance?

yes done Figure 1-30: Successive substitution technique for solving problems with temperature-dependent properties.

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

84

One-Dimensional, Steady-State Conduction

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM The kinetic energy associated with the atmospheric entry of a space vehicle results in extremely large heat ﬂuxes, large enough to completely vaporize the vehicle if it were not adequately protected. The outer structure of the vehicle is called its aeroshell and the outer layer of the material on the aeroshell is called the Thermal Protection System (or TPS). The heat ﬂux experienced by the aeroshell can reach 100 W/cm2 , albeit for only a short period of time. Consider a TPS consisting of a non-metallic ablative layer with thickness, thab = 5 cm that is bonded to a layer of steel with thickness, ths = 1 cm, as shown in Figure 1. The outer edge of the ablative heat shield (x = 0) reaches the material’s melting temperature (Tm = 755 K) under the inﬂuence of the heat ﬂux. The melting limits the temperature that is reached at the outer surface of the shield and protects the internal air until the shield is consumed. In this problem, we will assume that the shield is consumed very slowly so that a quasi-steady temperature distribution is set up in the ablative shield. The latent heat of fusion of the ablative shield is ifus,ab = 200 kJ/kg and its density is ρab = 1200 kg/m3 . 2

T∞ = 320 K, h = 10 W/m -K ths = 1 cm

x

steel, ks = 20 W/m-K

thab = 5 cm

ablative shield Tm = 755 K Δifus, ab = 200 kJ/kg

q⋅ ′′ = 100 W/cm

2

ρab = 1200 kg/m3 Figure 1: A Thermal Protection System.

The thermal conductivity of the ablative shield is highly temperature dependent; thermal conductivity values at several temperatures are provided in Table 1. Table 1: Thermal conductivity of ablative shield material in the solid phase Temperature

Thermal conductivity

300 K 350 K 400 K 450 K 500 K 550 K 600 K 650 K 700 K 755 K

0.10 W/m-K 0.15 W/m-K 0.19 W/m-K 0.21 W/m-K 0.22 W/m-K 0.24 W/m-K 0.28 W/m-K 0.33 W/m-K 0.38 W/m-K 0.45 W/m-K

85

The thermal conductivity of the steel may be assumed to be constant at k s = 20 W/m-K. The internal surface of the steel is exposed to air at T∞ = 320 K with average heat transfer coefﬁcient, h = 10 W/m2 -K. Assume that the TPS reaches a quasi-steady-state under the inﬂuence of a heat ﬂux q˙ = 100 W/cm2 and that the surface temperature of the ablation shield reaches its melting point. a) Develop a numerical model using MATLAB that can determine the heat ﬂux that is transferred to the air and the rate that the ablative shield is being consumed. The solution is developed as a MATLAB function called Ablative shield; the input to the function is the number of nodes to use in the solution while the outputs include the position of the nodes and the predicted temperature at each node as well as the two quantities speciﬁcally requested, the heat ﬂux incident on the air and the rate of shield ablation. Select New and M-File from the File menu and save the M-ﬁle as Ablative shield (the .m extension is added automatically). The ﬁrst line of the function establishes the input/output protocol: function[x,T,q flux in Wcm2,dthabdt cms]=Ablative shield(N) %EXAMPLE 1.5-1: Thermal Protection System for Atmospheric Entry % % Inputs: % N: number of nodes in solution (-) % % Outputs: % x: position of nodes (m) % T: temperatures at nodes (K) % q flux in Wcm2: heat flux to air (W/cmˆ2) % dthabdt cms: rate of shield consumption (cm/s)

The next section of the code establishes the remaining input parameters (i.e., those not provided as arguments to the function); note that each input is converted immediately to SI units. th ab=0.05; th s=0.01; k s=20; q flux=100∗ 100ˆ2; T m=755; DELTAi fus ab=200e3; h bar=10; T infinity=320; rho ab=1200; A c=1;

%ablation shield thickness (m) %steel thickness (m) %steel conductivity (W/m-K) %heat flux (W/mˆ2) %melting temperature (K) %latent heat of fusion (J/kg) %heat transfer coefficient (W/mˆ2-K) %internal air temperature (K) %density (kg/mˆ3) %per unit area of wall (mˆ2)

In order to solve this problem, it is necessary to create a function that returns the conductivity of the ablative shield material. The easiest way to do this is to enter the data from Table 1 into a sub-function and interpolate between the data points.

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

86

One-Dimensional, Steady-State Conduction

function[k]=k ab(T) %data Td=[300,350,400,450,500,550,600,650,700,755]; kd=[0.1,0.15,0.19,0.21,0.22,0.24,0.28,0.33,0.38,0.45]; k=interp1(Td,kd,T,’spline’); %interpolate data to obtain conductivity end

The interp1 function in MATLAB is used for the interpolation. The interp1 function requires three arguments; the ﬁrst two are the vectors of the independent and dependent data, respectively, and the third is the value of the independent variable at which you want to ﬁnd the dependent variable. An optional fourth argument speciﬁes the type of interpolation to use. To obtain more detailed help for this (or any) MATLAB function, use the help command: >> help interp1 INTERP1 1-D interpolation (table lookup) YI=INTERP1(X,Y,XI) interpolates to find YI, the values of the underlying function Y at the points in the array XI. X must be a vector of length N. If Y is a vector, then . . .

The numerical model of the TPS will consider the ablative material; the steel will be considered as part of the thermal resistance between the inner surface of the shield and the air and therefore will affect the boundary condition at x = thab . There is no reason to treat the steel with the numerical model since the steel has, by assumption, constant properties and is at steady state. Therefore, the analytical solution derived in Section 1.2.3 for the resistance of a plane wall holds exactly. The nodes are distributed uniformly from x = 0 to x = thab , where x = 0 corresponds to the outer surface of the shield, as shown in Figure 2. xi = (i − 1)

thab (N − 1)

for i = 1 . . . N

The distance between adjacent nodes (x) is: x =

thab (N − 1)

The nodes are setup in the MATLAB code according to: %setup nodes DELTAx=th ab/(N-1); for i=1:N x(i,1)=th ab∗ (i-1)/(N-1); end

%distance between nodes %position of each node

87

An internal control volume, shown in Figure 2, experiences only conduction; therefore, a steady-state energy balance on the control volume is: q˙ t op + q˙ bot t om = 0

(1) q⋅ top

TN

q⋅ bottom q⋅

top

Figure 2: Distribution of nodes and control volumes.

TN-1 Ti+1

Ti ⋅q bottom Ti-1 x T1

The conductivity used to approximate the conduction heat transfer rates in Eq. (1) must be evaluated at the temperature of the boundaries, i.e., the average of the temperatures of the nodes involved in the conduction process, as discussed previously in Section 1.4.3. With this understanding, these rate equations become: q˙ t op = k ab,T =(Ti+1 +Ti )/2

Ac (Ti+1 − Ti ) x

q˙ bot t om = k ab,T =(Ti−1 +Ti )/2

Ac (Ti−1 − Ti ) x

(2) (3)

where Ac is the cross-sectional area. Substituting Eqs. (2) and (3) into Eq. (1) leads to: k ab,T =(Ti+1 +Ti )/2

Ac Ac (Ti+1 − Ti ) + k ab,T =(Ti−1 +Ti )/2 (Ti−1 − Ti ) = 0 for i = 2.. (N − 1) x x (4)

The node on the outer surface (i.e., node 1) has a speciﬁed temperature, the melting temperature of the ablative material: T1 = Tm

(5)

The energy balance for the node on the inner surface (i.e., node N) is also shown in Figure 2: q˙ t op + q˙ bot t om = 0

(6)

where q˙ bot t om = k ab,T =(TN −1 +TN )/2 q˙ t op =

Ac (TN −1 − TN ) x

(T∞ − TN ) Rcond,s + Rconv

(7)

(8)

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

88

One-Dimensional, Steady-State Conduction

where Rcond,s and Rconv are the thermal resistances associated with conduction through the steel and convection from the internal surface of the steel to the air.

R cond s=th s/(A c∗ k s); R conv=1/(A c∗ h bar);

Rcond,s =

ths k s Ac

Rconv =

1 h Ac

%conduction resistance of steel (K/W) %convection resistance (K/W)

Substituting Eqs. (7) and (8) into Eq. (6) leads to: Ac (T∞ − TN ) + k ab,T =(TN −1 +TN )/2 (TN −1 − TN ) = 0 Rcond,s + Rconv x

(9)

Note that Eqs. (4), (5), and (9) are a complete set of equations in the unknown temperatures Ti for i = 1..N; however, these equations cannot be written as a linear combination of the unknown temperatures because the conductivity of the ablative shield depends on temperature. In order to apply successive substitution, the conductivity will be evaluated using guess values for these temperatures (Tˆ ). A linear variation in temperature from Tm to T∞ is used as the guess values to start the process: (i − 1) Tˆi = Tm + (T∞ − Tm) (N − 1)

for i = 1..N

%initial guess for temperature distribution for i=1:N Tg(i,1)=T m+(T infinity-T m)∗ (i-1)/(N-1); %linear from melting to air (K) end

The matrix A and vector b are initialized according to: %setup matrices A=spalloc(N,N,3∗ N); b=zeros(N,1);

Equation (5) is rewritten to make it clear what the coefﬁcient and the constants are: T1 [1] = Tm A1,1

%node 1 A(1,1)=1; b(1,1)=T m;

b1

89

Equation (4) is rewritten, using the guess temperatures to compute the conductivity of the ablative shield and also to clearly identify the coefﬁcients and constants: Ac Ac Ac +Ti+1 k ab,T =(Tˆi+1 +Tˆi )/2 Ti −k ab,T =(Tˆi+1 +Tˆi )/2 − k ab,T =(Tˆi−1 +Tˆi )/2 x x x Ai,i

Ai,i+1

Ac + Ti−1 k ab,T =(Tˆi−1 +Tˆi )/2 = 0 for i = 2.. (N − 1) x Ai,i−1

%internal nodes for i=2:(N-1) A(i,i)=-k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx-k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; A(i,i+1)=k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx; A(i,i-1)=k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; end

Equation (9) is rewritten: Ac 1 TN − − k ab,T =(TˆN −1 +TˆN )/2 Rcond,s + Rconv x

AN ,N

+ TN −1 k ab,T =(TˆN −1 +TˆN )/2 AN ,N −1

Ac x

T∞ =− R + Rconv cond,s bN

%node N A(N,N)=-k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx-1/(R cond s+R conv); A(N,N-1)=k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx; b(N,1)=-T infinity/(R cond s+R conv);

The matrix equation is solved: X=A\b; T=X;

%solve matrix equation

The solution is not complete, it is necessary to iterate until the solution (T ) matches the assumed temperature (Tˆ ). This is accomplished by placing the commands that setup and solve the matrix equation within a while loop that terminates when the rms error (err) is below some tolerance (tol). The rms error is computed according to: N 1 (Ti − Tˆi )2 err = N i=1

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

90

One-Dimensional, Steady-State Conduction

err=sqrt(sum((T-Tg).ˆ2)/N)

%compute rms error

Note that the sum command computes the sum of all of the elements in the vector provided to it and the use of .ˆ2 indicates that each element in the vector should be squared (as opposed to ˆ2 which would multiply the vector by itself ). Also notice that the error computation is not terminated with a semicolon so that the value of the rms error will be reported after each iteration. To start the iteration process, the value of the error is set to a large number (larger than tol) in order to ensure that the while loop executes at least once. After the solution has been obtained, the rms error is computed and the vector Tg is reset to the vector T. The result is shown below, with the new lines highlighted in bold. err=999; tol=0.01; while (err>tol)

%initial value of error (K), must be larger than tol %tolerance for convergence (K)

%node 1 A(1,1)=1; b(1,1)=T m; %internal nodes for i=2:(N-1) A(i,i)=-k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx-k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; A(i,i+1)=k ab((Tg(i)+Tg(i+1))/2)∗ A c/DELTAx; A(i,i-1)=k ab((Tg(i)+Tg(i-1))/2)∗ A c/DELTAx; end %node N A(N,N)=-k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx-1/(R cond s+R conv); A(N,N-1)=k ab((Tg(N)+Tg(N-1))/2)∗ A c/DELTAx; b(N,1)=-T infinity/(R cond s+R conv); X=A\b; T=X;

%solve matrix equation

err=sqrt(sum((T-Tg).ˆ2)/N) Tg=T; end

%compute rms error %reset guess values used to setup A and b

The heat ﬂux to the air (q˙ in ) is computed. q˙ in =

(TN − T∞ ) Ac (Rs + Rconv )

The rate at which the ablative shield is consumed can be determined using an energy balance at the outer surface; the heat ﬂux related to re-entry either consumes the shield or is transferred to the air. Note that this is actually a simpliﬁcation of the problem; this is a moving boundary problem and this solution is valid only in the limit that the energy carried by the motion of interface is small relative to the energy removed by its vaporization. q˙ = ρab ifus,ab

dthab + q˙ in dt

91

or dthab q˙ − q˙ in = dt ρab ifus,ab

These calculations are provided by adding the following lines to the Ablative shield function: q flux in=(T(N)-T infinity)/(R cond s+R conv)/A c; q flux in Wcm2=q flux in/100ˆ2; dthabdt=(q flux-q flux in)/(DELTAi fus ab∗ rho ab); dthabdt cms=dthabdt∗ 100; end

%heat flux to air (W/mˆ2) %heat flux to air (W/cmˆ2) %rate of shield consumption (m/s) %rate of shield consumption (cm/s)

Calling the function Ablative_shield from the workspace leads to: >> [x,T,q flux in Wcm2,dthabdt cms]=Ablative shield(100); err = 105.7742 err = 9.0059 err = 0.8979 err = 0.1798 err = 0.0238 err = 0.0035

800

Temperature (K)

750 700 650 600 550 500 450 0

0.01

0.02 0.03 Position (m)

0.04

0.05

Figure 3: Temperature as a function of position within the ablative shield.

Figure 3 shows the temperature as a function of position within the shield. The temperature gradient agrees with intuition; the temperature gradient is smaller where the conductivity is largest (i.e., at higher temperatures), which is consistent

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

1.5 Numerical Solutions to Steady-State 1-D Conduction Problems

One-Dimensional, Steady-State Conduction

with a constant heat ﬂow. It is important to verify that the number of nodes used in the solution is adequate. Figure 4 shows the heat ﬂux to the air as a function of the number of nodes in the solution and indicates that at least 20 nodes are required. The heat ﬂux to the air at the inner surface of the TPS is 0.166 W/cm2 , nearly three orders of magnitude less than the heat ﬂux at the outer surface. The TPS is being consumed at a rate of 0.416 cm/s suggesting that the atmospheric entry process cannot last more than ten seconds without consuming the entire shield. The thermal analysis of this problem does not consider the loss of ablative material with time and it is therefore a very simpliﬁed model of the TPS. 0.166 0.1655 Heat flux to air (W/cm2)

EXAMPLE 1.5-1: THERMAL PROTECTION SYSTEM

92

0.165 0.1645 0.164 0.1635 0.163 0.1625 0.162 1

10

100 Number of nodes

1000

Figure 4: Heat ﬂux to the air as a function of the number of nodes.

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces 1.6.1 Introduction The situations that were examined in Sections 1.2 through 1.5 were truly onedimensional; that is, the geometry and boundary conditions dictated that the temperature could only vary in one direction. In this section, problems that are only approximately 1-D, referred to as extended surfaces, are considered. Extended surfaces are thin pieces of conductive material that can be approximated as being isothermal in two dimensions with temperature variations in only one direction. Extended surfaces are particularly relevant to a large number of thermal engineering applications because the ﬁns that are used to enhance heat transfer in heat exchangers can often be treated as extended surfaces.

1.6.2 The Extended Surface Approximation An extended surface is not truly 1-D; however, it is often approximated as being such in order to simplify the analysis. Figure 1-31 shows a simple extended surface, sometimes called a ﬁn. The ﬁn length (in the x-direction) is L and its thickness (in the y-direction) is th. The width of the ﬁn in the z-direction, W, is assumed to be much larger than its thickness in the y-direction, th. The conductivity of the ﬁn material is k. The base of

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

93

W

Figure 1-31: A constant cross-sectional ﬁn.

y

T∞, h

x

Tb

L th

the ﬁn (at x = 0) is maintained at temperature Tb and the ﬁn is surrounded by ﬂuid at temperature T∞ with heat transfer coefﬁcient h. Energy is conducted axially from the base of the ﬁn. As energy moves along the ﬁn in the x-direction, it is also conducted laterally to the ﬁn surface where it is ﬁnally transferred by convection to the surrounding ﬂuid. Temperature gradients always accompany the transfer of energy by conduction through a material; thus, the temperature must vary in both the x- and y-directions and the temperature distribution in the extended surface must be 2-D. However, there are many situations where the temperature gradient in the y-direction is small and therefore can be neglected in the solution without signiﬁcant loss of accuracy. Figure 1-32 illustrates the temperature as a function of lateral position (y) at various axial locations (x) for an arbitrary set of conditions. (The 2-D solution for this ﬁn is derived in EXAMPLE 2.2-1.) Notice that the temperature decreases in both the x- and y-directions as conduction occurs in both of these directions. At every value of axial location, x, there is a temperature drop through the material in the y-direction due to conduction. (For x/L = 0.25, this temperature drop is labeled T cond,y in Figure 1-32.) There is another temperature drop from the surface of the material to the surrounding ﬂuid due to convection (labeled T conv in Figure 1-32 for x/L = 0.25). The extended surface approximation refers to the assumption that the temperature in the material is a function only of x and not of y; this approximation turns a 2-D problem into a 1-D problem, which is easier to solve. The extended surface approximation is valid when the temperature drop due to conduction in the y-direction is much less than the temperature drop due to convection (i.e, T cond,y restart; > ODE:=diff(diff(T(x),x),x)-per∗ h_bar∗ T(x)/(k∗ A_c)=-per∗ h_bar∗ T_infinity/(k∗ A_c); ODE :=

d2 per h bar T inf inity per h bar T(x) =− T(x) − dx2 kA c kA c

and obtain the solution using the dsolve command:

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

99

> Ts:=dsolve(ODE); √

Ts := T(x) = e

√ per h bar x √ √ k A c

C2 + e

√ √ − per h bar x √ √ k A c

C1 + T inf inity

The solution identiﬁed by Maple is identical to Eq. (1-192). The constants C1 and C2 are obtained by enforcing the boundary conditions. One boundary condition is clear; the base temperature is speciﬁed and therefore: T x=0 = T b

(1-194)

Substituting Eq. (1-192) into Eq. (1-194) leads to: C1 + C2 + T ∞ = T b

(1-195)

The boundary condition at the tip of the ﬁn is less clear and there are several possibilities. The most common model assumes that the tip is adiabatic. In this case, an interface balance at the tip (see Figure 1-35) leads to: q˙ x=L = 0 Substituting Fourier’s law into Eq. (1-196) leads to: dT =0 dx x=L

(1-196)

(1-197)

Substituting Eq. (1-192) into Eq. (1-197) leads to: C1 m exp (m L) − C2 m exp (−m L) = 0

(1-198)

Note that Eqs. (1-195) and (1-198) are together sufﬁcient to determine C1 and C2 . If the solution is implemented in EES then no further algebra is required. However, it is worthwhile to obtain the explicit form of the solution for this common problem. Equation (1-195) is multiplied by m exp(m L) and rearranged: C1 m exp (m L) + C2 m exp (m L) = (T b − T ∞ ) m exp (m L)

(1-199)

Equation (1-198) is added to Eq. (1-199): C1 m exp (m L) − C2 m exp (−m L) = 0 + [C1 m exp(m L) + C2 m exp(m L) = (T b − T ∞ ) m exp(m L)] −C2 m exp(−m L) − C2 m exp(m L) = −(T b − T ∞ ) m exp(m L)

(1-200)

Equation (1-200) can be solved for C2 : C2 =

(T b − T ∞ ) exp (m L) exp (−m L) + exp (m L)

(1-201)

A similar sequence of operations leads to: C1 =

(T b − T ∞ ) exp (−m L) exp (−m L) + exp (m L)

(1-202)

100

One-Dimensional, Steady-State Conduction

These constants can also be obtained from Maple; the governing differential equation, Eq. (1-173) is entered and solved, this time in terms of m: > restart; > ODE:=diff(diff(T(x),x),x)-mˆ2∗ T(x)=-mˆ2∗ T_infinity; ODE :=

d2 T(x) − m2 T(x) = −m2 T inf inity dx2

> Ts:=dsolve(ODE); Ts := T(x) = e(−mx) C2 + e(mx) C1 + T inf inity

The boundary conditions, Eqs. (1-194) and (1-197), are deﬁned: > BC1:=rhs(eval(Ts,x=0))=T_b; BC1 := C2 + C1 + T inf inity = T b > BC2:=rhs(eval(diff(Ts,x),x=L))=0; BC2 := −m e(−m L) C2 + m e(m L) C1 = 0

and solved symbolically: > constants:=solve({BC1,BC2},{_C1,_C2});

constants :=

C2 = −

e(m L) (T inf inity − T b) e(−m L) (T inf inity − T b) , C1 = − (−m L) (m L) e +e e(−m L) + e(m L)

!

The constants identiﬁed by Maple are identical to Eqs. (1-201) and (1-202). Substituting the constants of integration into the general solution, Eq. (1-192), leads to: (T b − T ∞ ) exp (m L) (T b − T ∞ ) exp (−m L) exp (m x) + exp (−m x) + T ∞ exp (−m L) + exp (m L) exp (−m L) + exp (m L) (1-203) or, using Maple: T =

> Ts:=subs(constants,Ts); Ts := T(x) = e(mx) e(−m L) (T inf inity − T b) e(−mx) e(m L) (T inf inity − T b) − + T inf inity − (−m L) (m L) e +e e(−m L) + e(m L)

Equation (1-203) can be simpliﬁed to: T = (T b − T ∞ )

[exp (−m (L − x)) + exp (m (L − x))] + T∞ [exp (−m L) + exp (m L)]

(1-204)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

101

Equation (1-204) can be stated more concisely using hyperbolic functions as opposed to exponentials; hyperbolic functions are functions that have been deﬁned in terms of exponentials. The combinations: 1 [exp (A) + exp (−A)] 2

(1-205)

1 [exp (A) − exp (−A)] 2

(1-206)

occur so frequently in math and science that they are given special names, the hyperbolic cosine (cosh, pronounced “kosh”) and the hyperbolic sine (sinh, pronounced “cinch”), respectively. cosh (A) =

1 [exp (A) + exp (−A)] 2

(1-207)

sinh (A) =

1 [exp (A) − exp (−A)] 2

(1-208)

These hyperbolic functions behave in much the same way that the cosine and sine functions do. For example: cosh2 (A) − sinh2 (A) 1 1 [exp (A) + exp (−A)]2 − [exp (A) − exp (−A)]2 4 4 1 2 = [exp (A) + 2 exp (A) exp (−A) + exp2 (−A)] 4 1 − [exp2 (A) − 2 exp (A) exp (−A) + exp2 (−A)] 4 = exp (A) exp (−A) = 1 =

(1-209)

or cosh2 (A) − sinh2 (A) = 1

(1-210)

which is analogous to the trigonometric identity: cos2 (A) + sin2 (A) = 1

(1-211)

Furthermore, the derivative of cosh is sinh and vice versa, which is analogous to derivatives of sine and cosine (albeit, without the sign change): " # d d 1 [sinh (A)] = [exp (A) − exp (−A)] dx dx 2 (1-212) dA dA 1 = cosh (A) = [exp (A) + exp (−A)] 2 dx dx or d dA [sinh (A)] = cosh (A) dx dx

(1-213)

A similar set of operations leads to: d dA [cosh (A)] = sinh (A) dx dx

(1-214)

102

One-Dimensional, Steady-State Conduction

Equation (1-204) is rearranged so that it can be expressed in terms of hyperbolic cosines: T = (T b − T ∞ )

2 [exp (−m (L − x)) + exp (m (L − x))] +T ∞ 2 [exp L) + exp (m L)] (−m cosh(m(L−x))

1/ cosh(m L)

(1-215) T = (T b − T ∞ )

cosh (m (L − x)) + T∞ cosh (m L)

(1-216)

Equation (1-216) is much more concise but functionally identical to Eq. (1-204). Note that Maple can convert from exponential to hyperbolic form as well, using the convert command with the ‘trigh’ identiﬁer: > T_s:=convert(Ts,‘trigh’); T s := T(x) = (−T inf inity + T b) cosh (m x) + T inf inity +

sinh (m x) sinh (m L) (T inf inity − T b) cosh (m L)

The rate of heat transfer to the base of the ﬁn (q˙ ﬁn ) is obtained from Fourier’s law evaluated at x = 0: dT (1-217) q˙ ﬁn = −k Ac dx x=0 Substituting Eq. (1-216) into Eq. (1-217) leads to: cosh (m (L − x)) d + T∞ q˙ ﬁn = −k Ac (T b − T ∞ ) dx cosh (m L) x=0 =−

k Ac (T b − T ∞ ) d [cosh (m (L − x))]x=0 cosh (m L) dx

(1-218)

Recalling that the derivative of cosh is sinh, according to Eq. (1-214): q˙ ﬁn =

k Ac (T b − T ∞ ) m [sinh (m (L − x))]x=0 cosh (m L)

(1-219)

or q˙ ﬁn = (T b − T ∞ ) k Ac m

sinh (m L) cosh (m L)

The same result may be obtained from Maple: > q_dot_fin:=-k∗ A_c∗ eval(diff(T_s,x),x=0); q dot f in := −k A c

d k A c m sinh(m L)(T inf inity − T b) T(x) x=0 = − dx cosh(m L)

(1-220)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces 1

mL = 0.1

0.9 Dimensionless temperature

103

mL = 0.5

0.8 0.7

mL = 1

0.6 0.5 0.4

mL = 2

0.3 mL = 5 mL = 10 0.1 mL = 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L 0.2

0.8

0.9

1

Figure 1-36: Dimensionless ﬁn temperature as a function of dimensionless position for various values of the parameter m L.

The ratio of sinh to cosh is the hyperbolic tangent ( just as the ratio of sine to cosine is tangent); therefore, Eq. (1-220) may be written as: $ (1-221) q˙ ﬁn = (T b − T ∞ ) h per k Ac tanh (m L) The temperature distribution and heat transfer rate provided by Eqs. (1-216) and (1-221) represent the most important aspects of the solution. The solutions for ﬁns with other types of boundary conditions at the tip are summarized in Table 1-4.

1.6.4 Fin Behavior The temperature distribution within a ﬁn with an adiabatic tip, Eq. (1-216), can be expressed as a ratio of the temperature elevation with respect to the ﬂuid temperature to the base-to-ﬂuid temperature difference: x cosh m L 1 − T − T∞ L = (1-222) Tb − T∞ cosh (m L) The dimensionless temperature as a function of dimensionless position (x/L) is shown in Figure 1-36 for various values of m L. Regardless of the value of m L, the solutions satisfy the boundary conditions; the curves intersect at (T − T∞ )/(Tb − T∞ ) = 1.0 at x/L = 0 and the slope of each curve is zero at x/L = 1.0. However, the shape of the curves changes with m L. Smaller values of m L result in a smaller temperature drop due to conduction along the ﬁn (and therefore more due to convection from the ﬁn surface) whereas large values of m L have a corresponding large temperature drop due to conduction and little for convection. The functionality of an extended surface is governed by two processes; conduction along the ﬁn (in the x-direction) and convection from its surface. (Conduction in the y-direction was neglected in the derivation of the solution.) The parameter m L represents the balance of these two effects. The resistance to conduction along the ﬁn (Rcond,x )

104

One-Dimensional, Steady-State Conduction

Table 1-4: Solutions for constant cross-section extended surfaces with different end conditions. Tip condition

Solution cosh (m (L − x)) T − T∞ = Tb − T∞ cosh (m L) $ q˙ ﬁn = (T b − T ∞ ) h per k Ac tanh (m L)

h, T∞

Adiabatic tip Tb

x

ηﬁn = tanh (m L) / (m L) h, T∞

Convection from tip Tb

x

h, T∞

Speciﬁed tip temperature

TL Tb

x

h, T∞

Inﬁnitely long

to ∞ Tb

x

where: T b = base temperature T ∞ = ﬂuid temperature per = perimeter L = length T = temperature per h L = ﬁn constant mL = k Ac

h sinh (m (L − x)) cosh (m (L − x)) + T − T∞ m k = Tb − T∞ h sinh (m L) cosh (m L) + mk h $ cosh (m L) sinh (m L) + m k q˙ ﬁn = (T b − T ∞ ) h per k Ac h sinh (m L) cosh (m L) + mk [tanh (m L) + m L ARtip ] ηﬁn = m L [1 + m L ARtip tanh (m L)] (1 + ARtip ) TL − T∞ sinh (m x) + sinh (m (L − x)) T − T∞ Tb − T∞ = Tb − T∞ sinh (m L) TL − T∞ cosh (m L) − $ Tb − T∞ q˙ ﬁn = (T b − T ∞ ) h per k Ac sinh (m L) T − T∞ = exp (−m x) Tb − T∞ $ q˙ ﬁn = (T b − T ∞ ) h per k Ac h = heat transfer coefﬁcient Ac = cross-sectional area k = thermal conductivity q˙ ﬁn = ﬁn heat transfer rate x = position (relative to base of ﬁn) ARtip =

Ac = tip area ratio per L

is given by: Rcond,x =

L k Ac

(1-223)

and the resistance to convection from the surface (Rconv ) is: Rconv =

1 h per L

(1-224)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

105

Note that the resistance in Eq. (1-223) is related to conduction in the x-direction and should not be confused with Rcond,y in Eq. (1-160), which was used to deﬁne the Biot number. Clearly the behavior of the ﬁn cannot be exactly represented using these thermal resistances; the analysis in Section 1.6.3 was complex and showed that the conduction along the ﬁn is gradually reduced by convection. Nevertheless, the relative value of these resistances provides substantial insight into the qualitative characteristics of the ﬁn: per h 2 Rcond,x = L Rconv k Ac The resistance ratio in Eq. (1-225) is related to the parameter m L: ⎡ ⎤2 per h per h 2 Rcond,x 2 L⎦ = L = (m L) = ⎣ k Ac k Ac Rconv

(1-225)

(1-226)

In the light of Eq. (1-226), Figure 1-36 begins to make sense. A small value of m L represents a ﬁn with a small resistance to conduction in the x-direction relative to the resistance to convection. The temperature drop due to the conduction heat transfer along the ﬁn must therefore be small. At the other extreme, a large value of m L indicates that the resistance to conduction in the x-direction is much larger than the resistance to convection and therefore most of the temperature drop is related to the conduction heat transfer. Before starting an analysis of an extended surface, it is helpful to calculate the two dimensionless parameters discussed thus far. The value of the Biot number will indicate whether it is possible to treat the situation as a 1-D problem and the value of m L will determine whether it is even worth the time. If m L is either very small or very large, then the behavior can be understood with no analysis: the ﬁn temperature will be very close to the base temperature or the ﬂuid temperature, respectively.

1.6.5 Fin Efﬁciency and Resistance The ﬁn efﬁciency is deﬁned as the ratio of the heat transfer to the ﬁn (q˙ ﬁn ) to the heat transfer to an ideal ﬁn. An ideal ﬁn is made of an inﬁnitely conductive material and therefore this limit corresponds to a ﬁn that is everywhere at a temperature of Tb . Note that an ideal ﬁn with inﬁnite conductivity corresponds to the limit of m L = 0 in Figure 1-36. ηﬁn =

heat transfer to ﬁn heat transfer to ﬁn as k → ∞

(1-227)

or ηﬁn =

q˙ ﬁn h As,ﬁn (T b − T ∞ )

(1-228)

where the denominator of Eq. (1-228) is the product of the average heat transfer coefﬁcient, the surface area of the ﬁn that is exposed to ﬂuid, and the base-to-ﬂuid temperature difference. The ﬁn efﬁciency represents the degree to which the temperature drop along the ﬁn due to conduction has reduced the average temperature difference driving convection from the ﬁn surface. The ﬁn efﬁciency depends on the boundary condition at the tip and the geometry of the ﬁn. For a constant cross-sectional area ﬁn with an adiabatic tip, Eq. (1-221) can be

106

One-Dimensional, Steady-State Conduction 1 0.9 0.8

Fin efficiency

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2 2.5 3 Fin constant, mL

3.5

4

4.5

5

Figure 1-37: Fin efﬁciency for a constant cross-section, adiabatic tipped ﬁn as a function of the parameter m L.

substituted into Eq. (1-228): ηﬁn =

(T b − T ∞ )

$ h per k Ac tanh (m L)

h per L (T b − T ∞ )

(1-229)

or tanh (m L) ηﬁn = hper L kAc

(1-230)

which can be simpliﬁed by substituting in the deﬁnition of m: ηﬁn =

tanh (m L) mL

(1-231)

Figure 1-37 illustrates the ﬁn efﬁciency for a ﬁn having a constant cross-sectional area and an adiabatic tip as a function of m L. Notice that the ﬁn efﬁciency drops as m L increases. This is consistent with the discussion in Section 1.6.4; a large value of m L corresponds to a large temperature drop due to conduction along the ﬁn, as seen in Figure 1-36. The ﬁn efﬁciency is the most useful format for presenting the results of a ﬁn solution because it allows the calculation of a ﬁn resistance (Rﬁn ). The ﬁn resistance is the thermal resistance that opposes heat transfer from the base of the ﬁn to the surrounding ﬂuid. Equation (1-228) can be rearranged: q˙ ﬁn = ηﬁn h As,ﬁn (T b − T ∞ )

(1-232)

1/Rﬁn

where As,ﬁn is the surface area of the ﬁn exposed to the ﬂuid. Note that without the ﬁn efﬁciency, Eq. (1-232) is equivalent to Newton’s law of cooling and therefore the thermal resistance is deﬁned in basically the same way: Rﬁn =

1 ηﬁn h As,ﬁn

(1-233)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

107

The ﬁn efﬁciency is always less than one and therefore the ﬁn resistance will be larger than the corresponding convection resistance; this increase in resistance is related to the conduction resistance within the ﬁn. The concept of a ﬁn resistance is convenient since it allows the effect of ﬁns to be incorporated into a more complex problem (for example, one in which ﬁns are attached to other structures) as additional resistances in a network. For most extended surfaces, the surface area that is available for convection at the tip is insigniﬁcant relative to the total area for convection and therefore the adiabatic tip solution for ﬁn efﬁciency is sufﬁcient. However, the solution for the heat transfer from a ﬁn that experiences convection from its tip (see Table 1-4) can also be used to provide an expression for ﬁn efﬁciency: $ q˙ ﬁn = (T b − T ∞ )

h cosh (m L) mk h per k Ac h cosh (m L) + sinh (m L) mk sinh (m L) +

(1-234)

So the ﬁn efﬁciency is: $

ηﬁn

h cosh (m L) m k = = h (per L + Ac ) h (per L + Ac ) h cosh (m L) + sinh (m L) mk h per k Ac

q˙ ﬁn

sinh (m L) +

which can be simpliﬁed somewhat to: & % tanh (m L) + m L ARtip % & ηﬁn = m L 1 + m L ARtip tanh (m L) 1 + ARtip

(1-235)

(1-236)

where ARtip is the ratio of the area for convection from the tip to the surface area along the length of the ﬁn: ARtip =

Ac per L

(1-237)

1 0.9 0.8 Fin efficiency

0.7

AR tip = 0.0 AR tip = 0.1 AR tip = 0.2

0.6 0.5 0.4 0.3 0.2

AR tip = 0.3 AR tip = 0.4 AR tip = 0.5

0.1 0 0

0.5

1

1.5

2 2.5 3 3.5 Fin constant, m L

4

4.5

5

Figure 1-38: Fin efﬁciency for a constant cross-section ﬁn with convection from the tip as a function of the parameter m L and various values of the tip area ratio.

108

One-Dimensional, Steady-State Conduction

Figure 1-38 illustrates the ﬁn efﬁciency associated with a ﬁn with a convective tip as a function of the ﬁn parameter (m L) for various values of the tip area ratio (ARtip ). Note that the ﬁn efﬁciency is reduced as the tip area is larger. This counterintuitive result is related to the fact that the tip area is included in the surface area that is available for convection from an ideal ﬁn in the deﬁnition of the ﬁn efﬁciency. The heat transfer rate from the ﬁn will increase as the tip area is increased; however, the rate that heat could be transferred from an ideal (i.e., isothermal) ﬁn would increase by a larger amount. It is possible to approximately correct for convection from the tip and use the simpler adiabatic tip ﬁn efﬁciency equation by modifying the length of the ﬁn slightly (e.g., adding the half-thickness of a ﬁn with a rectangular cross-section). In most cases the correction associated with the tip convection is so small that it is not worth considering. In any case, neglecting convection from the tip is slightly conservative and other uncertainties in the problem (e.g., the value of the heat transfer coefﬁcient) are likely to be more important. The ﬁn efﬁciency solutions for many common ﬁn geometries have been determined. For ﬁns without a constant cross-section, the solution requires the use of more advanced techniques, such as Bessel functions, which are covered in Section 1.8. Several common ﬁn solutions are listed in Table 1-5. A more comprehensive set of ﬁn efﬁciency solutions has been programmed in EES. To access these solutions, select Function Info from the Options menu and then select the radio button in the lower right side of the top box and scroll to the Fin Efﬁciency category (Figure 1-39). It is possible to scroll through the various functions that are available or see more detailed information about any of these functions by pressing the Info button. Note that the ﬁn efﬁciency can be accessed either in dimensional form (in which case the geometric parameters, conductivity, and heat transfer coefﬁcient must be supplied) or nondimensional form (in which case the nondimensional parameters, such as m L, must be supplied).

Figure 1-39: Fin efﬁciency function information in EES.

Table 1-5: Solutions for extended surfaces. Shape

Solution ηﬁn =

As,ﬁn = 2 W L 2h L mL = k th

Straight rectangular W

th

tanh (m L) mL

L

BesselI (1, 2 m L) m L BesselI (0, 2 m L) 2 th As,ﬁn = 2 W L2 + 2 2h L mL = k th ηﬁn =

Straight triangular

th

W L

2

ηﬁn = $

Straight parabolic

th

4 (m L)2 + 1 + 1 L2 th ln + C1 As,ﬁn = W C1 L + th L 2 2h th L, C1 = 1 + mL = k th L

W L

ηﬁn = D

Spine rectangular

tanh (m L) mL

As,ﬁn = π D L 4h L kD

mL =

L

2BesselI (2, 2 m L) m L BesselI (1, 2 m L) 2 πD D As,ﬁn = L2 + 2 2 4h L mL = kD ηﬁn =

D

Spine triangular L

2 2 As,ﬁn = 2 π rout − rin

Rectangular annular

rout

th

mrout =

rin

mrin =

2h rout k th 2h rin k th

2 mrin [BesselK (1,mrin ) BesselI (1,mrout ) − BesselI (1,mrin ) BesselK (1,mrout )] ( (mrout )2 −(mrin )2 [BesselI (0,mrin ) BesselK (1,mrout ) + BesselK (0,mrin ) BesselI (1,mrout )]

ηﬁn = ' where

h = heat transfer coefﬁcient

k = thermal conductivity

EXAMPLE 1.6-1: SOLDERING TUBES

110

One-Dimensional, Steady-State Conduction

EXAMPLE 1.6-1: SOLDERING TUBES Two large pipes must be soldered together using a propane torch, as shown in Figure 1. Din = 4.0 inch L = 2.5 ft

th = 0.375 inch

Tm = 230°C

heat from torch, q⋅

k = 150 W/m-K

T∞ = 20°C, h = 20 W/m2 -K Figure 1: Two bare pipes being soldered together.

Each of the two pipes is L = 2.5 ft long with inner diameter Din = 4.0 inches and a thickness th = 0.375 inch. The pipe material has conductivity k = 150 W/m-K. The surrounding air is at T∞ = 20◦ C and the heat transfer coefﬁcient between the external surface of the pipe and the air is h = 20 W/m2 -K. Assume that convection from the internal surface of the pipe can be neglected. a) The temperature of the interface between the two pipes must be elevated to ˙ that Tm = 230◦ C in order to melt the solder; estimate the heat transfer rate, q, must be applied by the propane torch in order to accomplish this process. This problem is solved using EES. The initial section of the code provides the stated inputs (converted to SI units). “EXAMPLE 1.6-1: Soldering Tubes” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” D in=4.0 [inch]∗ convert(inch,m) th=0.375 [inch]∗ convert(inch,m) k=150 [W/m-K] h bar=20 [W/mˆ2-K] T infinity=converttemp(C,K,20 [C]) L=2.5 [ft]∗ convert(ft,m) T m=converttemp(C,K,230 [C])

“Inner diameter” “Pipe thickness” “Conductivity” “Heat transfer coefficient” “Air temperature” “Pipe length” “Melt temperature”

The two pipes can be treated as constant cross-sectional area ﬁns; the solutions obtained in Section 1.6 are valid provided that the Biot number characterizing the temperature gradient within the pipe in the radial direction is sufﬁciently small: Bi =

h th k

Bi=h bar∗ th/k

111

“Biot number”

The Biot number is 0.0013, which is much less than unity. The cross-sectional area for conduction (Ac ) is: & π % 2 (Din + 2 th)2 − Din Ac = 4 and the perimeter exposed to air (per) is: per = π(Din + 2 th) Notice that the internal surface of the pipe (which is assumed to be adiabatic) is not included in the perimeter. The ratio of the area of the exposed ends of the pipe to the external surface area (ARtip ) is calculated according to: ARtip =

A c=pi∗ ((D in+2∗ th)ˆ2-D inˆ2)/4 per=pi∗ (D in+2∗ th) AR tip=A c/(per∗ L)

Ac per L

“area for conduction” “perimeter” “area ratio”

The tip area ratio is ARtip = 0.012 and therefore, according to Figure 1-38, the adiabatic tip ﬁn solution can be used with no loss of accuracy. The ﬁn constant (m L) is: per h L mL = k Ac and the ﬁn efﬁciency (ηﬁn ) is: ηﬁn =

tanh (m L) mL

Therefore, the resistance of each ﬁn (Rﬁn ) is: Rﬁn =

mL=sqrt(h bar∗ per/(k∗ A c))∗ L eta fin=tanh(mL)/mL R fin=1/(eta fin∗ h bar∗ per∗ L)

1 ηﬁn h per L

“fin parameter” “fin efficiency” “fin resistance”

The problem can be represented by the resistance network shown in Figure 2; the two pipes correspond to the two resistors connecting the interface to the air and the

EXAMPLE 1.6-1: SOLDERING TUBES

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

EXAMPLE 1.6-1: SOLDERING TUBES

112

One-Dimensional, Steady-State Conduction

heat input from the propane torch enters at the interface. In order for the solder to melt, the interface temperature must reach Tm . q⋅ T∞

T∞ R fin

Tm

Figure 2: Resistance network associated with soldering two bare pipes.

R fin

The heat transfer required from the torch is therefore: q˙ = 2

(Tm − T∞ ) Rﬁn

q dot=2∗ (T m-T infinity)/R fin

“required torch heat transfer rate”

The factor 2 appears because there are two pipes, each of which acts as a ﬁn. The EES solution indicates that the propane torch must provide at least 812 W to accomplish this job. b) Unfortunately, the propane torch cannot provide 812 W and it is not possible to melt the solder. Therefore, you decide to place insulating sleeves over the pipes adjacent to the soldering torch, as shown in Figure 3. If the insulation is perfect (i.e., convection is eliminated from the section of the pipe covered by the insulating sleeves), then how long must the sleeves be (Lins ) in order to reduce the heat required to q˙ = 500 W? insulating sleeves Lins

Figure 3: Pipes with insulating sleeves placed over them to reduce the heat transfer required.

Tm = 230° C

heat from torch, q⋅

The pipes with insulating sleeves can be represented by a resistance network similar to the one shown in Figure 2, but with additional resistances inserted between the interface and the base of the ﬁns. These additional resistances correspond to the insulated sections of pipes, as shown in Figure 4. q⋅ T∞ R fin

R cond, ins Tm

T∞ R cond, ins

R fin

Figure 4: Resistance network with additional resistors associated with the insulated sections of the pipe.

113

The resistance of the insulated sections of the pipe is: Rcond,ins =

R cond ins=L ins/(k∗ A c)

L ins k Ac

“resistance of insulated portion of pipe”

The length of the un-insulated section of pipe is reduced and therefore the ﬁn efﬁciency and ﬁn resistance must be recalculated. The ﬁn efﬁciency (ηﬁn ) becomes: ηﬁn =

tanh [m (L − L ins )] m (L − L ins )

and the resistance of each ﬁn (Rﬁn ) is: Rﬁn =

1 ηﬁn h per (L − L ins )

mL=sqrt(h bar∗ per/(k∗ A c))∗ (L-L ins) eta fin=tanh(mL)/mL R fin=1/(eta fin∗ h bar∗ per∗ (L-L ins))

“fin parameter” “fin efficiency” “fin resistance”

Using the resistance network shown in Figure 4, the required heat transfer rate can be expressed as: q˙ = 2

(Tm − T∞ ) Rcond,ins + Rﬁn

EES can solve for the length of insulation that is required by setting the heat transfer rate to the available heat transfer rate, q dot=2∗ (T m-T infinity)/(R fin+R cond ins) q dot=500 [W] L ins ft=L ins∗ convert(m,ft)

“required torch heat transfer” “available torch heat transfer” “length of insulation in ft”

The solution indicates that the length of the insulating sleeves must be at least 0.16 m (0.52 ft) in order to reduce the required heat transfer rate to the point where 500 W will sufﬁce.

1.6.6 Finned Surfaces Fins are often placed on surfaces in order to improve their heat transfer capability. Examples of ﬁnned surfaces can be found within nearly every appliance in your house, from the evaporator and condenser on your refrigerator and air conditioner to the processor in your personal computer. Fins are essential to the design of economical but high-performance thermal devices. Figure 1-40 illustrates a single ﬁn installed on a surface; the temperature of the surface is the base temperature of the ﬁn, Tb . The ﬁn and surface are surrounded by ﬂuid at T∞ with heat transfer coefﬁcient h. The ﬁn has perimeter per, length L, conductivity k, and cross-sectional area Ac .

EXAMPLE 1.6-1: SOLDERING TUBES

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

114

T∞ , h

One-Dimensional, Steady-State Conduction single fin: cross-sectional area, Ac perimeter, per conductivity, k Figure 1-40: Single ﬁn placed on a surface.

surface at Tb

It is of interest to determine the heat transfer rate from an area of the surface that is equal to the base area of the ﬁn, both with and without the ﬁn installed. If there were no ﬁn, then the heat transfer rate from area Ac is: q˙ no−ﬁn = h Ac (T b − T ∞ )

(1-238)

while the heat transfer rate from the ﬁn, assuming an adiabatic tip and the same heat transfer coefﬁcient, is given by Eq. (1-221): ⎛ ⎞ $ h per ⎠ (1-239) q˙ ﬁn = (T b − T ∞ ) h per k Ac tanh ⎝ L k Ac The ﬁn effectiveness (εﬁn ) is deﬁned as the ratio of the heat transfer rate from the ﬁn (q˙ ﬁn ) to the heat transfer rate that would have occurred from the surface area occupied by the ﬁn without the ﬁn attached (q˙ no−ﬁn ): ⎛ ⎞ $ h per L⎠ (T b − T ∞ ) h per k Ac tanh ⎝ k Ac q˙ ﬁn εﬁn = = (1-240) q˙ no−f in h Ac (T b − T ∞ ) which can be simpliﬁed to: εﬁn =

⎛ ⎞ k per h A 1 c ⎠ tanh ⎝ h Ac k per ARtip

(1-241)

where ARtip is the ratio of the area of the tip to the exposed surface area of the ﬁn. The ﬁn effectiveness predicted by Eq. (1-241) is illustrated in Figure 1-41 as a function of the dimensionless group (k per) /(h Ac ) for various values of the area ratio ARtip . The effectiveness of the ﬁn provides a measure of the improvement in thermal performance that is achieved by placing ﬁns onto the surface. Equation (1-241) and Figure 1-41 are useful in that they clarify the characteristics of an application that would beneﬁt substantially from the use of ﬁns. If the heat transfer coefﬁcient is low, then the group (k per) /(h Ac ) is large and there is a substantial beneﬁt associated with the addition of ﬁns. This explains why many devices that transfer thermal energy to air or other low conductivity gases with correspondingly low heat transfer coefﬁcients are ﬁnned. For example, the air-side of a domestic refrigerator condenser will certainly be ﬁnned while the refrigerant side is typically not ﬁnned, since the heat transfer coefﬁcient associated with the refrigerant condensation process is very high (as discussed in Chapter 7). Fins are typically thin structures (with a large perimeter to cross-sectional area ratio, per/Ac ) made of high conductivity material; these features enhance the ﬁn effectiveness by increasing the parameter (k per)/(h Ac ).

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

115

Fin effectiveness

10 AR tip AR tip AR tip AR tip AR tip AR tip AR tip

5

→ 0

= = = = = =

0.1 0.15 0.2 0.25 0.3 0.4

2

1 1

10 (k per)/(h Ac)

100

Figure 1-41: Fin effectiveness as a function of (k per)/(h Ac ) for various values of ARtip .

The concept of a ﬁn resistance makes it possible to approximately consider the thermal performance of an array of ﬁns that are placed on a surface. For example, Figure 1-42 illustrates an array of square ﬁns placed on a base with area As,b at temperature T b. The surface is partially covered with ﬁns; in Figure 1-42, the number of ﬁns is Nﬁn = 16. Each ﬁn has a cross-sectional area at its base of Ac,b and surface area As,ﬁn . The surfaces are exposed to a surrounding ﬂuid with temperature T ∞ and average heat transfer coefﬁcient h. The heat transferred from the base can either pass through one of the Nﬁn ﬁns (each with resistance, Rﬁn ) or from the un-ﬁnned surface area on the base (with resistance, Run-ﬁnned ). The resistance of a single ﬁn is given by Eq. (1-233): Rﬁn =

1

(1-242)

ηﬁn h As,ﬁn

where ηﬁn is the ﬁn efﬁciency, computed using the formulae or function speciﬁc to the geometry of the ﬁn. The resistance of the un-ﬁnned surface of the base is: Run−ﬁnned =

T∞ , h

h As,b − Nﬁn Ac,b

Nfin fins, each with base cross-sectional area, Ac,b , and surface area, As,fin

base surface area, As,b Figure 1-42: An array of ﬁns on a base.

1

(1-243)

116

One-Dimensional, Steady-State Conduction

Run−finned =

Tb

1 h (As,b − N fin Ac,b )

R fin =

1 ηfin h As, fin

R fin =

1 ηfin h As, fin

Run−finned =

Tb

T∞

T∞

R fins = R fin =

1 h (As,b − N fin Ac,b )

1 N fin η fin h As, fin

1 ηfin h As, fin

Figure 1-43: Resistance network associated with a ﬁnned surface.

where the area in the denominator of Eq. (1-243) is the area of the exposed portion of the base, i.e., the area not covered by ﬁns. These heat transfer paths are in parallel and therefore the thermal resistance network that represents the situation is shown in Figure 1-43. The total resistance of the ﬁnned surface is therefore: Nﬁn −1 1 + (1-244) Rtot = Run−ﬁnned Rﬁn or, substituting Eqs. (1-242) and (1-243) into Eq. (1-244): Rtot = [h (As,b − Nﬁn Ac,b) + Nﬁn ηﬁn h As,ﬁn ]−1

(1-245)

The total rate of heat transfer is: q˙ tot =

(T b − T ∞ ) = [h (As,b − Nﬁn Ac,b) + Nﬁn ηﬁn h As,ﬁn ](T b − T ∞ ) Rtot

(1-246)

The overall surface efﬁciency (ηo ) is deﬁned as the ratio of the total heat transfer rate from the surface to the heat transfer rate that would result if the entire surface (the exposed base and the ﬁns) were at the base temperature; as with the ﬁn efﬁciency, this limit corresponds to using a material with an inﬁnite conductivity. ηo =

q˙ tot h [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ] (T b − T ∞ ) prime surface area, Atot

(1-247)

The area in the denominator of Eq. (1-247) is often referred to as the prime surface area (Atot ): Atot = As,b − Nﬁn Ac,b + Nﬁn As,ﬁn

(1-248)

Substituting Eq. (1-246) into Eq. (1-247) leads to: ηo =

[(As,b − Nﬁn Ac,b) + Nﬁn ηﬁn As,ﬁn ] [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ]

(1-249)

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

117

Equation (1-249) can be rearranged: ηo =

[(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn + Nﬁn ηﬁn As,ﬁn − Nﬁn As,ﬁn ] [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ]

(1-250)

or ηo = 1 −

Nﬁn As,ﬁn (1 − ηﬁn ) [(As,b − Nﬁn Ac,b) + Nﬁn As,ﬁn ]

(1-251)

which can be expressed in terms of the prime surface area: ηo = 1 −

Nﬁn As,ﬁn (1 − ηﬁn ) Atot

(1-252)

Rearranging Eq. (1-247), the total resistance to heat transfer from a ﬁnned surface can be expressed in terms of the overall surface efﬁciency and the prime surface area: 1 ηo h Atot

(1-253)

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK Heat rejection from a thermoelectric cooling device is accomplished using a 10 × 10 array of Dﬁn = 1.5 mm diameter pin ﬁns that are L ﬁn = 15 mm long. The ﬁns are attached to a square base plate that is W b = 3 cm on a side and thb = 2 mm thick, as shown in Figure 1. The conductivity of the ﬁn material is k ﬁn = 70 W/m-K and the thermal conductivity of the base material is k b = 25 W/m-K. There is a contact resistance of Rc = 1 × 10−4 m2 -K/W at the interface between the base of the ﬁns and the base plate. The hot end of the thermoelectric cooler is at Thot = 30◦ C and the surrounding air temperature is T∞ = 20◦ C. The average heat transfer coefﬁcient between the air and the surface of the heat sink is h = 50 W/m2 -K. 2 T∞ = 20°C, h = 50 W/m -K

10x10 array of fins kfin = 70 W/m-K

Dfin =1.5 mm Lfin =15 mm

thb =2.0 mm kb =25 W/m-K Wb =3.0 cm Thot = 30°C Rc′′ = 1x10

-4

m2 -K W

Figure 1: Heat sink mounted on a thermoelectric cooler.

a) What is the total thermal resistance between the hot end of the thermoelectric cooler and the air? What is the rate of heat rejection that can be accomplished under these conditions?

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

Rtot =

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

118

One-Dimensional, Steady-State Conduction

The ﬁrst section of the EES code provides the inputs for the problem. “EXAMPLE 1.6-2: Thermoelectric Heat Sink” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” T infinity=converttemp(C,K,20 [C]) T hot=converttemp(C,K,30 [C]) D fin=1.5[mm]∗ convert(mm,m) L fin=15[mm]∗ convert(mm,m) N fin=100 W b=3 [cm] ∗ convert(cm,m) th b=2[mm]∗ convert(mm,m) k fin=70 [W/m-K] k b=25 [W/m-K] h bar=50 [W/mˆ2-K] R c=1e-4 [mˆ2-K/W]

“Air temperature” “Hot end of thermoelectric cooler” “Fin diameter” “Fin length” “Number of fins” “Width of base (square)” “Thickness of base” “Conductivity of fin” “Conductivity of base” “Heat transfer coefficient” “Contact resistance”

The constant cross-sectional area ﬁns can be treated using the solutions presented in Section 1.6. The perimeter (per), cross-sectional area (Ac ), and surface area for convection (As,ﬁn, assuming adiabatic ends) associated with each ﬁn are calculated according to: per = π Dﬁn Ac =

π 2 D 4 ﬁn

As,ﬁn = π L Dﬁn

per=pi∗ D fin A c=pi∗ D finˆ2/4 A s fin=pi∗ L fin∗ D fin

“Perimeter of fin” “Cross-sectional area for conduction” “Surface area of fin for convection”

The ﬁn constant and ﬁn efﬁciency for an adiabatic tip, constant cross-sectional area ﬁn are computed according to: per h m= k ﬁn Ac ηﬁn =

tanh(m L ﬁn ) m L ﬁn

The resistance of any type of ﬁn (Rﬁn ) can be obtained from its efﬁciency: Rﬁn =

1 ηﬁn h As,ﬁn

mL=sqrt(h bar∗ per/(k fin∗ A c))∗ L fin eta fin=tanh(mL)/mL R fin=1/(h bar∗ A s fin∗ eta fin)

119

“Fin parameter” “Fin efficiency” “Fin resistance”

The resistance network that represents the entire heat sink (Figure 2) extends from the hot end of the cooler to the air and includes conduction through the base (Rcond,b ) followed by two paths in parallel, corresponding to the heat that is transferred by convection from the unﬁnned upper surface of the base (Run−ﬁnned ) and the heat that is transferred through the contact resistance at the base of the ﬁns (Rc ) and then through the resistance associated with the ﬁn itself (Rﬁn ). Note that Rc and Rﬁn are in parallel Nﬁn times and therefore the value these resistances in the circuit is reduced by 1/Nﬁn . R fin

Rc N fin

N fin T∞

Figure 2: Resistance network representing the heat sink.

Thot

R cond, b

0.088 K/W

0.57 K/W

3.23 K/W

R un−finned T∞ 27.7 K/W

The resistance to conduction through the base is: thb Rcond,b = k b W b2 The contact resistance associated with each ﬁn-to-base interface is: R Rc = c Ac The resistance of the unﬁnned upper surface of the base is: 1 Run−ﬁnned = 2 h W b − N ﬁn Ac These resistances are calculated in EES: R b=th b/(k b∗ W bˆ2) R unfinned=1/((W bˆ2-N fin∗ A c)∗ h bar) R c=R c/A c

“Resistance due to conduction through the base” “Resistance of unfinned base” “Fin-to-base contact resistance”

The total resistance (Rtot ) and heat transfer (q˙ tot ) from the heat sink are obtained according to: ⎞−1 ⎛ ⎟ ⎜ 1 1 ⎟ ⎜ + Rtot = Rb + ⎜ ⎟ Rﬁn ⎝ Rc Run−ﬁnned ⎠ + N ﬁn N ﬁn q˙ tot =

(Thot − T∞ ) Rtot

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

120

One-Dimensional, Steady-State Conduction

and calculated in EES.

R tot=R b+(1/(R c/N fin+R fin/N fin)+1/R unfinned)ˆ(-1) q dot tot=(T hot-T infinity)/R tot

“Total resistance” “Total rate of heat transfer”

The total resistance is 3.42 K/W and the rate of heat transfer is 2.92 W. The numerical values of each resistance are included in Figure 2 in order to understand the mechanisms that are governing the behavior of the heat sink. Notice that the resistance of the base is not very important, as it is a small resistor in series with larger ones. The resistance of the unﬁnned portion of the base is also not critical, since it is a large resistor in parallel with smaller ones. On the other hand, both the contact resistance and the ﬁn resistance are important as these two resistors dominate the problem and are of the same order of magnitude. The ﬁn resistance is the most critical parameter in the problem and any attempt to improve performance should focus on this element of the heat sink. b) Through material selection and manipulation of the air ﬂow across the heat sink, it is possible to affect design changes to kﬁn and h. Generate a contour plot that illustrates contours of constant heat rejection in the parameter space of kﬁn (ranging from 5 W/m-K to 150 W/m-K) and h (ranging from 10 W/m2 -K to 200 W/m2 -K). One of the nice things about solving problems using a computer program as opposed to pencil and paper is that parametric studies and optimization are relatively straightforward. In order to prepare a contour plot with EES, it is necessary to setup a parametric table in which both of the parameters of interest vary over a speciﬁed range. Open a new parametric table and include the two independent variables (the variables k fin and h bar) as well as the dependent variable of interest (the variable q dot tot). In order to run the simulation for 20 values of kﬁn and 20 values of h, 20 × 20 = 400 runs must be included in the table. (Add runs using the Insert/Delete Runs option from the Tables menu.) It is necessary to set the values of k fin and h bar in the table. It is possible to vary k fin from 5 to 150 W/m-K, 20 times by using the “Repeat pattern every” option in the Alter Values dialog that appears when you right-click on the k fin column, as shown in Figure 3.

Figure 3: Vary kﬁn from 5 to 150 W/m-K 20 times.

121

In order to completely cover the parameter space, it is necessary to evaluate the solution over a range of h bar at each unique value of k fin; this can be accomplished using the “Apply pattern every” option in the Alter Values dialog for the h bar column of the table, see Figure 4.

Figure 4: Vary h from 10 to 200 W/m-K with 20 runs for each of 20 values.

2

Average heat transfer coefficient (W/m-K)

When the speciﬁed values of the variables k fin and h bar are commented out in the Equations window, it is possible to run the parametric table using the Solve Table command in the Calculate menu (F3); 400 values of q˙ are determined, one for each combination of kﬁn and h set in the parametric table. To generate a contour plot, select X-Y-Z plot from the New Plot Window option in the Plots menu. Select k fin as the variable on the x-axis, h bar as the y-axis variable and q dot tot as the contour variable. The appearance of the resulting contour plot can be adjusted by altering the resolution, smoothing, color options, and the type of function used for interpolation. A contour plot generated using isometric lines is shown in Figure 5.

200 q tot = 7 W

180 160 140

q tot = 6 W

120 q tot = 5 W

100 80

q tot = 4 W

60

q tot = 3 W

40 nominal design

20 0 0

25

q tot = 2 W q tot = 1 W

50 75 100 Fin conductivity (W/m-K)

125

150

Figure 5: Contours of constant heat transfer rate in the parameter space of ﬁn material conductivity and heat transfer coefﬁcient.

EXAMPLE 1.6-2: THERMOELECTRIC HEAT SINK

1.6 Analytical Solutions for Constant Cross-Section Extended Surfaces

EXAMPLE 1.6-2

122

One-Dimensional, Steady-State Conduction

The nominal design point shown in Figure 1 is also indicated in Figure 5. Contour plots are useful in that they can clarify the impact of design changes. For example, Figure 5 shows that it would be more beneﬁcial to explore methods to increase the heat transfer coefﬁcient than the ﬁn conductivity at the nominal design conditions (i.e., moving from the nominal design point towards higher heat transfer will result in much larger performance gains than moving toward higher ﬁn conductivity).

1.6.7 Fin Optimization This extended section of the book, which can be found on the website (www. cambridge.org/nellisandklein), presents an optimization of a constant cross-sectional area ﬁn in order to maximize the rate of heat transfer per unit volume of ﬁn material. The process illustrates the use of EES’ single-variable optimization capability and shows that a well-optimized ﬁn is characterized by m L that is approximately equal to 1.4. Fins with m L much less than 1.4 are shorter than optimal and therefore have very small temperature gradients due to conduction; additional length will provide a substantial beneﬁt and therefore the available volume of ﬁn material should be stretched, providing additional length at the expense of cross-sectional area. Fins with m L much greater than 1.4 are longer than optimal and therefore have large temperature gradients due to conduction; additional length will not provide much beneﬁt as the tip temperature is approaching the ambient temperature. Therefore, the available volume should be compressed, reducing the length but providing more cross-sectional area for conduction.

1.7 Analytical Solutions for Advanced Constant Cross-Section Extended Surfaces 1.7.1 Introduction The constant cross-section ﬁns that were investigated in Section 1.6 are certainly the most common type of extended surface used in practice. However, other extended surface problems (with alternative boundary conditions, more complex thermal loadings, multiple computational domains, etc.) are also encountered. Extended surfaces represent 2-D heat transfer situations that can be approximated as being 1-D and these problems can be solved analytically using the techniques that were introduced in Section 1.6.

1.7.2 Additional Thermal Loads An extended surface can be subjected to additional thermal loads such as thermal energy generation (due to ohmic heating, for example) or an external heat ﬂux. These additional effects show up in the governing differential equation but do not affect the character of the solution. Figure 1-46 illustrates an extended surface with cross-sectional area Ac and perimeter per that has a uniform volumetric generation (g˙ ) and is exposed to a uniform heat ﬂux (q˙ ext , for example from solar radiation). The extended surface is surrounded by ﬂuid at T ∞ with average heat transfer coefﬁcient h. A differential control volume is used to derive the governing differential equation (see Figure 1-46) and provides the energy balance: q˙ x + g˙ + q˙ ext = q˙ conv + q˙ x+dx

(1-267)

1.7 Analytical Solutions for Advanced Constant Cross-Section

123 ′′ q⋅ ext

T∞ , h x

dx

g⋅ ′′′

Figure 1-46: Extended surface with additional thermal loads related to generation and an external heat ﬂux.

q⋅ conv q⋅ x

per g⋅

q⋅ext

q⋅ x + dx Ac

The ﬁnal term can be expanded: g˙ + q˙ ext = q˙ conv +

dq˙ dx dx

Substituting the appropriate rate equation for each term results in: d dT g˙ Ac dx + q˙ ext per dx = h per dx (T − T ∞ ) + −k Ac dx dx dx

(1-268)

(1-269)

where k is the conductivity of the material and T is the temperature at any axial position. Note that temperature is assumed to be only a function of x, which is consistent with the extended surface approximation. This assumption should be veriﬁed using an appropriately deﬁned Biot number, as discussed in Section 1.6.2. After some simpliﬁcation, the governing differential equation for the extended surface becomes: h per h per g˙ q˙ ext per d2 T − T = − T − − ∞ dx2 k Ac k Ac k k Ac

(1-270)

Equation (1-270) is a nonhomogeneous, linear, second order ODE. The solution is assumed to be the sum of a homogeneous and particular solution: T = Th + T p

(1-271)

Equation (1-271) is substituted into Eq. (1-270): d2 T p h per h per h per g˙ q˙ per d2 T h − Th + − Tp = − T∞ − − ext 2 2 dx kA dx k Ac k Ac k k Ac c =0 for homogeneous differential equation

(1-272)

whatever is left over must be the particular differential equation

The homogeneous differential equation is: h per d2 T h − Th = 0 dx2 k Ac

(1-273)

T h = C1 exp (m x) + C2 exp (−m x)

(1-274)

which is solved by:

where

m =

per h k Ac

(1-275)

124

One-Dimensional, Steady-State Conduction

The particular differential equation is: d2 T p h per h per g˙ q˙ per − Tp = − T∞ − − ext 2 dx k Ac k Ac k k Ac

(1-276)

Notice that the right side of Eq. (1-276) is a constant; therefore, the particular solution is a constant: T p = C3

(1-277)

Substituting Eq. (1-277) into Eq. (1-276) leads to: −

h per h per g˙ q˙ per C3 = − T∞ − − ext k Ac k Ac k k Ac

(1-278)

Solving for C3 : C3 = T ∞ +

g˙ Ac h per

+

q˙ ext

(1-279)

h

Substituting Eq. (1-279) into Eq. (1-277) leads to: T p = T∞ +

g˙ Ac h per

+

q˙ ext

(1-280)

h

Substituting Eqs. (1-280) and (1-274) into Eq. (1-271) leads to: T = C1 exp (m x) + C2 exp (−m x) + T ∞ +

g˙ Ac h per

+

q˙ ext h

(1-281)

The boundary conditions at either edge of the extended surface should be used to evaluate C1 and C2 for a speciﬁc situation. It is possible to use Maple to solve this problem (and therefore avoid the mathematical steps discussed above). Enter the governing differential equation: > restart; > ODE:=diff(diff(T(x),x),x)-h_bar∗ per∗ T(x)/(k∗ A_c)=-h_bar∗ per∗ T_infinity/ (k∗ A_c)-gv/k-qf_ext∗ per/(k∗ A_c); ODE :=

d2 h bar per T inf inity gv qf extper h bar per T(x) =− − − T(x) − dx2 kA c kA c k kA c

and solve it: > Ts:=dsolve(ODE); Ts := T(x) = √ e

√ h bar per x √ √ x A c

C2 + e

−

√ √ h bar per x √ √ x A c

C1 +

(h bar T inf inity + qf ext) per + gv A c h bar per

The solution identiﬁed by Maple is functionally identical to Eq. (1-281).

1.7 Analytical Solutions for Advanced Constant Cross-Section

125

For situations where the volumetric generation or external heat ﬂux is not spatially uniform, it will not be as easy to identify the particular solution. For example, suppose that the volumetric generation varies sinusoidally from x = 0 to x = L, where L is the length of the extended surface. x (1-282) g˙ = g˙ max sin π L where g˙ max is the volumetric generation at the center of the extended surface. The resulting governing differential equation is: x q˙ per h per h per g˙ d2 T max − T = − T − (1-283) sin π − ext ∞ dx2 k Ac k Ac k L k Ac The solution is assumed to be the sum of a homogeneous and particular solution. Substituting Eq. (1-271) into Eq. (1-283) leads to: x q˙ per d2 T p h per h per h per g˙ d2 T h max − T + − T = − T − sin π − ext h p ∞ dx2 k Ac dx2 k Ac k Ac k L k Ac =0 for homogeneous differential equation

the particular differential equation

(1-284) The homogeneous differential equation has not changed and therefore the homogeneous solution remains Eq. (1-274). However, the particular differential equation has become more complex: x q˙ per d2 T p h per h per g˙ max − T = − T − (1-285) sin π − ext p ∞ dx2 k Ac k Ac k L k Ac Identifying the particular solution can take some skill. Equation (1-285) involves both a constant and a sinusoidal term on the right hand side and the governing differential equation involves both the solution and its derivatives. Therefore, it seems likely that a particular solution that includes sines, cosines, and constants as well as their derivatives (cosines, sines, and 0) might work. One method of obtaining the particular solution is to assume such a solution with appropriate, undetermined constants (C3 , C4 , and C5 ): x x (1-286) + C4 cos π + C5 T p = C3 sin π L L and substitute it into the particular differential equation. The ﬁrst and second derivatives of the particular solution, Eq. (1-286), are: x C π x dT p C3 π 4 = cos π − sin π dx L L L L x C π2 x d2 T p C3 π2 4 = − sin π cos π − dx2 L2 L L2 L

(1-287) (1-288)

Substituting Eqs. (1-288) and (1-286) into Eq. (1-285) leads to: x C π2 x h per ' x x ( C3 π2 4 cos π sin π cos π − − + C + C C − 2 sin π 3 4 5 L L L2 L k Ac L L h per g˙ x q˙ per =− T ∞ − max sin π (1-289) − ext k Ac k L k Ac In order for the particular solution to work, the sine, cosine and constant terms in Eq. (1-289) must add up correctly. By considering the coefﬁcients of the sine terms, it is

126

One-Dimensional, Steady-State Conduction

possible to obtain the equation: − which can be solved for C3 :

C3 π2 h per g˙ − C3 = − max 2 L k Ac k g˙ max k C3 = h per π2 + L2 k Ac

(1-290)

(1-291)

The sum of the coefﬁcients of the cosine terms provides an additional equation: −

C4 π2 h per − C4 = 0 2 L k Ac

(1-292)

which indicates that C4 = 0

(1-293)

Finally, the sum of the coefﬁcients for the constant terms leads to: −

h per h per q˙ per C5 = − T ∞ − ext k Ac k Ac k Ac

(1-294)

so C5 = T ∞ +

q˙ ext

h Substituting Eqs. (1-291), (1-293), and (1-295) into Eq. (1-286) leads to: g˙ max x q˙ k sin π + T ∞ + ext Tp = L h h per π2 + 2 L k Ac

(1-295)

(1-296)

The solution to the differential equation is the sum of the homogeneous and the particular solutions. g˙ max x q˙ k sin π + T ∞ + ext T = C1 exp (m x) + C2 exp (−m x) + L h h per π2 + 2 L k Ac (1-297) where the boundary conditions determine the values of C1 and C2 . It is somewhat easier to obtain the solution using Maple. Enter the governing differential equation, Eq. (1-283): > restart; > ODE:=diff(diff(T(x),x),x)-h_bar∗ per∗ T(x)/(k∗ A_c)=-h_bar∗ per∗ T_infinity/ (k∗ A_c)-gv_max∗ sin(pi∗ x/L)/k-qf_ext∗ per/(k∗ A_c);

d2 h bar per T(x) T(x) − dx2 kA c πx max sin gv h bar per T inf inity L − qf extper − =− kA c k kA c

ODE : =

1.7 Analytical Solutions for Advanced Constant Cross-Section

127

and then solve the differential equation:

> Ts:=dsolve(ODE); √

Ts := T(x) = e

+

√ h bar per x √ √ k A c

√

C2 + e

gv max A c L2 h bar sin

√ h bar per x √ √ k A c

C1

πx

+(h bar T inf inity+qf ext) (h bar per L2 +π2 k A c) L h bar (h bar per L2 +π2 k A c)

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR One design of a micro-scale, lithographically fabricated (i.e., MEMS) device that can produce in-plane motion is called a bent-beam actuator (Que (2000)). A V-shaped structure (the bent-beam in Figure 1) is suspended between two anchors. The anchors are thermally staked to the underlying substrate and therefore keep the ends of the bent-beam at room temperature (Ta = 20◦ C). An elevated voltage is applied to one pillar and the other is grounded. The voltage difference causes current (I) to ﬂow through the bent-beam structure. The temperature of the bent-beam rises as a result of ohmic heating and the thermally induced expansion causes the apex of the bent-beam to move outwards. The result is a voltage-controlled actuator capable of producing in-plane motion.

substrate tip motion current flow through bent-beam

anchor post, kept at Ta Figure 1: Bent-beam actuator.

The anchors of the bent-beam actuator are placed La = 1 mm apart and the beam structure has a cross-section of W = 10 μm by th = 5 μm. The slope of the beams (with respect to a line connecting the two pillars) is θ = 0.5 rad, as shown in Figure 2. The bent-beam material has conductivity k = 80 W/m-K, electrical resistivity ρe = 1 × 10−5 ohm-m and coefﬁcient of thermal expansion CTE = 3.5 × 10−6 K−1 . You may neglect radiation from the beam and assume all of the heat that is generated is convected to the surrounding air at temperature T∞ = 20◦ C with average heat transfer coefﬁcient h = 100 W/m2 -K or transferred conductively to the pillars (which remain at Ta = 20◦ C). The actuator is activated with I = 10 mA of current.

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

The solution identiﬁed by Maple is functionally equivalent to Eq. (1-297). This result from Maple can be copied and pasted directly into EES for evaluation and manipulation.

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

128

One-Dimensional, Steady-State Conduction 2 T∞ = 20°C, h = 100 W/m -K

k = 80 W/m-K ρe = 1x10-5 ohm-m CTE = 3.5x10-6 K-1

L s Ta = 20°C

0.5 rad La=1.0 mm

th = 5 μm Ta = 20°C

W = 10μm (into page) Figure 2: Dimensions and conditions associated with bent-beam actuator.

a) Is it appropriate to treat the bent-beam as an extended surface? The input parameters for the problem are entered into EES: “EXAMPLE 1.7-1: Bent-beam Actuator” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” L a=1 [mm]∗ convert(mm,m) w=10 [micron]∗ convert(micron,m) th=5 [micron]∗ convert(micron,m) I=0.010 [Amp] theta=0.5 [rad] T a=converttemp(C,K,20 [C]) T infinity=converttemp(C,K,20 [C]) h bar=100 [W/mˆ2-K] k=80 [W/m-K] rho e=1e-5 [ohm-m] CTE=3.5e-6 [1/K]

“distance between anchors” “width of beam” “thickness of beam” “current” “slope of beam” “temperature of pillars” “temperature of air” “heat transfer coefficient” “conductivity” “electrical resistivity” “coefficient of thermal expansion”

The extended surface approximation requires that the 3-D temperature distribution within the bent-beam be approximated as 1-D; that is, temperature gradients within the beam that are perpendicular to the surface will be ignored so that the temperature may be approximated as a function only of s, the coordinate that follows the beam (see Figure 2). The resistance that must be neglected in order to use the extended surface approximation is conduction in the lateral direction (Rcond,lat ). The extended surface approximation is justiﬁed provided that the lateral conduction resistance is small relative to the resistance that is being considered, convection from the outer surface (Rconv ). The Biot number is therefore: Bi =

Rcond,lat Rconv

The heat transfer will take the shortest path to the surface and therefore it is appropriate to use the smallest lateral dimension (th/2) to compute the lateral conduction resistance. hW L th h th = Bi = 2kW L 1 2k

129

where L is the length of the beam from pillar to apex (see Figure 2). Bi=th∗ h bar/(2∗ k)

“Biot number”

The Biot number is small (3 × 10−6 ) and therefore the extended surface approximation is justiﬁed. b) Develop an analytical solution that can predict the temperature of one leg of the bent-beam as a function of position along the beam, s. The general solution for an extended surface with a constant cross-sectional area and spatially uniform generation is derived in Section 1.7.2: T = C 1 exp (m s) + C 2 exp (−m s) + T∞ +

g˙ Ac h per

(1)

For the bent-beam actuator, the perimeter (per), cross-sectional area (Ac ), and ﬁn parameter (m) are per = 2 (W + th) Ac = W th h per m= k Ac

per=2∗ (W+th) A c=W∗ th m=sqrt(h bar∗ per/(k∗ A c))

“perimeter” “area” “fin parameter”

The volumetric generation, g˙ , is related to ohmic heating. The electrical resistance of the bent-beam structure (Re ) is: Re =

ρe 2 L Ac

where L=

La 2 cos (θ)

The volumetric rate of electrical dissipation is the ratio of ohmic dissipation to the volume of the structure: g˙ =

L=L a/(2∗ cos(theta)) R e=rho e∗ L∗ 2/A c g dot=Iˆ2∗ R e/(2∗ L∗ A c)

I 2 Re 2 L Ac

“length of half-beam” “resistance of beam structure” “volumetric generation”

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7 Analytical Solutions for Advanced Constant Cross-Section

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

130

One-Dimensional, Steady-State Conduction

The constants C1 and C2 in Eq. (1) are determined using the boundary conditions. The temperature of the beam where it meets the pillar is speciﬁed: Ts=0 = Ta

(2)

Substituting Eq. (1) into Eq. (2) leads to: C 1 + C 2 + T∞ +

g˙ Ac h per

= Ta

(3)

A half-symmetry model of the bent-beam actuator considers only one leg. Because both legs of the bent-beam see identical conditions, there is nothing to drive heat from one leg to the other and therefore there will be no conduction through the end of the leg (at s = L): q˙ s=L = −k

dT =0 d s s=L

or dT =0 d s s=L

(4)

Substituting Eq. (1) into Eq. (4) leads to: C 1 m exp (m L) − C 2 m exp (−m L) = 0

(5)

Equations (3) and (5) can be entered in EES and used to determine C1 and C2 . T infinity+C 1+C 2+g dot∗ A c/(h bar∗ per)=T a C 1∗ m∗ exp(m∗ L)-C 2∗ m∗ exp(-m∗ L)=0

“from boundary condition at s=0” “from boundary condition at s=L”

A variable s_bar is deﬁned as s/L so that s_bar = 0 corresponds to the pillar and s_bar = 1 to the apex. The variable s_bar is deﬁned for convenience, so that it is easy to generate a parametric table in which s is varied from 0 to L even if parameters such as θ and La change. s bar=s/L

“non-dimensional position”

The temperature is evaluated using Eq. (1). T=T infinity+C 1∗ exp(m∗ s)+C 2∗ exp(-m∗ s)+g dot∗ A c/(h bar∗ per) T C=converttemp(K,C,T)

“temperature” “in C”

131

A parametric table is generated that includes the variables s bar and T C. The temperature distribution through one leg of the beam is shown in Figure 3. 800 700

Temperature (°C)

600 500 400 300 200 100 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, s/L

0.8

0.9

1

Figure 3: Temperature as a function of dimensionless position along one leg of beam.

c) The thermally induced elongation of a differential segment of the beam (of length ds) is given by: dL = CTE ( T − Ta) d s Estimate the displacement of the apex of the beam. Plot the displacement as a function of voltage.

The total elongation of the beam (L) is obtained by integrating the differential elongation along the beam: L CTE (T − Ta ) d s

L =

(6)

0

Substituting the solution for the temperature distribution, Eq. (1) into Eq. (6) leads to:

L L =

C 1 exp (m s) + C 2 exp (−m s) + T∞ +

CTE 0

g˙ Ac h per

− Ta

ds

Evaluating the integral: L = CTE

T∞ − Ta +

g˙ Ac h per

L

C2 C1 exp (m s) − exp (−m s) s+ m m

0

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7 Analytical Solutions for Advanced Constant Cross-Section

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

132

One-Dimensional, Steady-State Conduction

Substituting the integration limits: L = CT E

T∞ − Ta +

g˙ Ac

h per

C2 C1 [exp (m L) − 1] − [exp (−m L) − 1] L+ m m

!

DELTAL=CTE∗ ((T infinity-T a+g dot∗ A c/(h bar∗ per))∗ L+C 1∗ (exp(m∗ L)-1)/m-C 2∗ (exp(-m∗ L)-1)/m) “displacement of beam”

Assuming that the joint associated with the apex does not provide a torque on either leg of the beam, the displacement of the apex can be estimated using trigonometry (Figure 4). L+ΔL

Δy heated beam unheated beam

L La 2

Figure 4: Trigonometry associated with apex motion.

y

The original position of the apex (y) is given by: y =

L2

−

La 2

2

therefore, the motion of the apex (y ) is: y =

(L + L) − 2

La 2

2 −

DELTAy=sqrt((L+DELTAL)ˆ2-(L a/2)ˆ2)-sqrt(Lˆ2-(L a/2)ˆ2) DELTAy micron=DELTAy∗ convert(m,micron)

L2

−

La 2

2

“displacement of apex” “in μm”

The voltage across the beam (V) is: V = I Re

V=I∗ R e

“voltage”

Figure 5 illustrates the actuator displacement as a function of voltage. This plot was generated using a parametric table including the variables DELTAy micron and V; the variable I was commented out in order to make the table.

133

EXAMPLE 1.7-1: BENT-BEAM ACTUATOR

1.7 Analytical Solutions for Advanced Constant Cross-Section 10 9 Actuator motion (μm)

8 7 6 5 4 3 2 1 0 0

0.5

1

1.5

2 2.5 3 Voltage (V)

3.5

4

4.5

5

Figure 5: Actuator displacement as a function of the applied voltage.

1.7.3 Moving Extended Surfaces An interesting class of problems arises in situations where an extended surface is moving with respect to the frame of reference of the problem. Problems of this type occur in rotating systems, (such as in drum and disk brakes), extrusions, and in manufacturing systems. The energy carried by the moving material represents an additional energy transfer into and out of the differential control volume and provides an additional term in the governing differential equation. Figure 1-47 illustrates an extended surface (i.e., a material for which temperature is only a function of x) that is moving with velocity u through ﬂuid with temperature T ∞ and h. An energy balance on the differential control volume (Figure 1-47) includes con˙ at either edge, as well as conduction and energy transport due to material motion (E) vection to the surrounding ﬂuid. q˙ x + E˙ x = q˙ conv + q˙ x+dx + E˙ x+dx

(1-298)

or 0 = q˙ conv +

dq˙ dE˙ dx + dx dx dx

(1-299)

material is moving with velocity u dx

Figure 1-47: An extended surface moving with velocity u.

x

⋅ Ex q⋅ x

⋅ E x+dx

q⋅ conv

q⋅ x+dx

T∞ , h

134

One-Dimensional, Steady-State Conduction

The conduction and convection terms are represented by the familiar rate equations: q˙ cond = −k Ac

dT dx

q˙ conv = per dx h (T − T ∞ )

(1-300) (1-301)

where k is the conductivity of the material and per and Ac are the perimeter and crosssectional area of the extended surface, respectively. The rate of energy transfer due to ˙ is the product of the enthalpy of the material (i) and its the motion of the material, E, mass ﬂow rate. The mass ﬂow rate is the product of the velocity, density (ρ), and crosssectional area. E˙ = u Ac ρ i Substituting Eqs. (1-300) through (1-302) into Eq. (1-299) leads to: d dT d 0 = per dx h (T − T ∞ ) + −k Ac dx + [u Ac ρ i] dx dx dx dx

(1-302)

(1-303)

Assuming constant properties: d2 T di + u Ac ρ dx2 dx

(1-304)

d2 T di dT + u Ac ρ 2 dx dT dx

(1-305)

0 = per h (T − T ∞ ) − k Ac The enthalpy gradient is expanded: 0 = per h (T − T ∞ ) − k Ac

Assuming that the material is incompressible, the derivative of enthalpy with respect to temperature is the speciﬁc heat capacity (c): 0 = per h (T − T ∞ ) − k Ac

d2 T dT + u Ac ρ c 2 dx dx

(1-306)

Equation (1-306) is rearranged in order to obtain the governing differential equation: u ρ c dT per h per h d2 T − T =− T∞ − 2 dx k dx k Ac k Ac

(1-307)

The solution is again divided into a homogeneous and particular solution: T = Th + T p

(1-308)

Substituting Eq. (1-308) into Eq. (1-307) leads to: d2 T p u ρ c dT p d2 T h u ρ c dT h per h per h per h − T + − Tp = − T∞ − − h dx2 k dx k Ac dx2 k dx k Ac k Ac

= 0 for homogeneous differential equation

whatever is left is the particular differential equation

(1-309)

1.7 Analytical Solutions for Advanced Constant Cross-Section

135

The particular differential equation: d2 T p u ρ c dT p per h per h − Tp = − T∞ − 2 dx k dx k Ac k Ac

(1-310)

is solved by a constant: T p = T∞

(1-311)

The homogeneous differential equation is: d2 T h u ρ c dT h per h − Th = 0 − dx2 k dx k Ac The ﬁn parameter (m) is deﬁned as in Section 1.6: h per m= k Ac

(1-312)

(1-313)

The group of properties, k/ρ c, appearing in Eq. (1-312) is encountered often in heat transfer and is deﬁned as the thermal diffusivity (α): α=

k ρc

(1-314)

With these deﬁnitions, Eq. (1-312) can be written as: u dT h d2 T h − − m2 T h = 0 dx2 α dx

(1-315)

Equation (1-315) is solved by an exponential: T h = C exp (λ x)

(1-316)

where C and λ are both arbitrary constants. Equation (1-316) is substituted into Eq. (1-315): C λ2 exp (λ x) −

u C λ exp (λ x) − m2 C exp (λ x) = 0 α

(1-317)

u λ − m2 = 0 α

(1-318)

which can be simpliﬁed: λ2 −

Equation (1-318) is quadratic and therefore has two solutions (λ1 and λ2 ): / 1 u 2 u + m2 + λ1 = 2α 4 α u λ2 = − 2α

/

1 u 2 + m2 4 α

(1-319)

(1-320)

Because the governing equation is linear, the sum of the two solutions (Th,1 and Th,2 ): T h,1 = C1 exp (λ1 x)

(1-321)

T h,2 = C2 exp (λ2 x)

(1-322)

136

One-Dimensional, Steady-State Conduction

is also a solution and therefore the general solution to the homogeneous governing differential equation is: T h = C1 exp (λ1 x) + C2 exp (λ2 x)

(1-323)

The solution to the differential equation is the sum of the homogeneous and particular solutions: T = C1 exp (λ1 x) + C2 exp (λ2 x) + T ∞

(1-324)

where the constants C1 and C2 are determined according to the boundary conditions. The solution may also be obtained using Maple by entering and solving the governing differential equation: > restart; > ODE:=diff(diff(T(x),x),x)-u∗ diff(T(x),x)/alpha-mˆ2∗ T(x)=-mˆ2∗ T_infinity; ODE :=

d u T(x) d dx T(x) − − m2 T(x) = −m2 T inf inity dx2 α 2

> T_s:=dsolve(ODE);

T s := T(x) = e

(u +

√

u2 + 4m2 α2 ) x 2α

C2 + e

(u −

√

u2 + 4m2 α2 ) x 2α

C1 + T inf inity

EXAMPLE 1.7-2: DRAWING A WIRE

which is equivalent to Eq. (1-324). EXAMPLE 1.7-2: DRAWING A WIRE Figure 1 illustrates a wire drawn from a die. The wire diameter is D = 0.5 mm. The temperature of the material at the exit of the die is Tdr aw = 600◦ C and it has a draw velocity of u = 10 mm/s. The properties of the wire are ρ = 2700 kg/m3 , k = 230 W/m-K, and c = 1000 J/kg-K. The wire is surrounded by air at T∞ = 20◦ C with an average heat transfer coefﬁcient of h = 25 W/m2 -K. The wire travels for L = 25 cm before entering a pool of water that is kept at Tw = 20◦ C; you may assume that the water-to-wire heat transfer coefﬁcient is very high so that the wire equilibrates essentially instantaneously with the water as it enters the pool. h = 25 W/m2-K T∞ = 20°C

Tdraw = 600°C u = 10 mm/s D = 0.5 mm ρ = 2700 kg/m3 L = 25 cm k = 230 W/m-K c = 1000 J/kg-K Tw = 20°C

Figure 1: Wire drawn from a die.

a) Develop an analytical model that can predict the temperature distribution in the wire.

1.7 Analytical Solutions for Advanced Constant Cross-Section

137

EXAMPLE 1.7-2: DRAWING A WIRE

The input parameters are entered in EES: “EXAMPLE 1.7-2: Drawing a Wire” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” D=0.5 [mm]∗ convert(mm,m) u=10 [mm/s]∗ convert(mm/s,m/s) c=1000 [J/kg-K] k=230 [W/m-K] rho=2700 [kg/mˆ3] h bar=25 [W/mˆ2-K] T infinity=converttemp(C,K,20 [C]) T draw=converttemp(C,K,600 [C]) T w=converttemp(C,K,20 [C]) L=25 [cm]∗ convert(cm,m)

“diameter” “draw velocity” “specific heat capacity” “conductivity” “density” “heat transfer coefficient” “air temperature” “draw temperature” “water temperature” “length of wire”

The governing differential equation for a moving extended surface was derived in Section 1.7.3: d 2T u dT − − m2 T = −m2 T∞ d x2 α dx where α is the thermal diffusivity: α= m is the ﬁn constant:

k ρc

m=

h per k Ac

and per and Ac are the perimeter and cross-sectional area, respectively, of the moving surface: per = π D Ac = π

A c=pi∗ Dˆ2/4 per=pi∗ D alpha=k/(rho∗ c) m=sqrt(h bar∗ per/(k∗ A c))

D2 4

“cross-sectional area” “perimeter” “thermal diffusivity” “fin parameter”

The general solution derived in Section 1.7.3 is: T = C 1 exp (λ1 x) + C 2 exp (λ2 x) + T∞

(1)

EXAMPLE 1.7-2: DRAWING A WIRE

138

One-Dimensional, Steady-State Conduction

where C1 and C2 are undetermined constants and: / 1 u 2 u λ1 = + m2 + 2α 4 α u λ2 = − 2α

/

1 u 2 + m2 4 α

lambda 1=u/(2∗ alpha)+sqrt((u/alpha)ˆ2/4+mˆ2) lambda 2=u/(2∗ alpha)-sqrt((u/alpha)ˆ2/4+mˆ2)

“solution parameter 1” “solution parameter 2”

The constants are evaluated using the boundary conditions. The temperatures at x = 0 and x = L are speciﬁed: Tx=L = Tw

(2)

Tx=0 = Tdr aw

(3)

Substituting Eq. (1) into Eqs. (2) and (3) leads to two algebraic equations for C1 and C2 : C 1 exp (λ1 L) + C 2 exp (λ2 L) + T∞ = Tw C 1 + C 2 + T∞ = Tdr aw which are entered in EES: C 1∗ exp(lambda 1∗ L)+C 2∗ exp(lambda 2∗ L)+T infinity=T w C 1+C 2+T infinity=T draw

“boundary condition at x=L” “boundary condition at x=0”

The solution is evaluated in EES and converted to Celsius. x=x bar∗ L T=C 1∗ exp(lambda 1∗ x)+C 2∗ exp(lambda 2∗ x)+T infinity T C=converttemp(K,C,T)

“position” “solution” “in C”

A parametric table in EES can be used to provide the temperature as a function of position. It is convenient to deﬁne the variable x bar, the axial position normalized by the length of the wire. Including x_bar in the table and varying it from 0 to 1 is equivalent to varying the position from 0 to L. One advantage of using the variable x_bar is that as the length of the wire is changed, it is not necessary to adjust the parametric table, only to re-run it. Figure 2 illustrates the temperature as a function of dimensionless position for various values of the length.

139

EXAMPLE 1.7-2: DRAWING A WIRE

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 700

Temperature (°C)

600 500 L = 5 cm

400

L = 15 cm

300

L =25 cm L =35 cm L =45 cm

200 100 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless position, x/L Figure 2: Temperature as a function of dimensionless position for various values of length.

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 1.8.1 Introduction Sections 1.6 and 1.7 showed how the differential equation describing constant crosssection ﬁns and other extended surfaces is derived. Analytical solutions for these differential equations take the form of an exponential function. In this section, extended surface problems are considered for which the cross-sectional area for conduction and the wetted perimeter for convection are not constant. The resulting differential equation is solved by Bessel functions.

1.8.2 Series Solutions It is worthwhile asking what the “exponential function” really is; we take it for granted in terms of its properties (i.e., how it can be integrated and differentiated). With some experience, it is possible to see that it solves a certain type of differential equation. In fact, that is its purpose: the exponential is really a polynomial series that has been deﬁned so that it solves a commonly encountered differential equation. There are other types of differential equations that appear in engineering problems; series solutions to these differential equations have been deﬁned and given formal names like “Bessel function” and “Kelvin function”. The homogeneous differential equation that results from the analysis of a constant cross-sectional area ﬁn is derived in Section 1.6.3 d2 T h − m2 T h = 0 dx2

(1-325)

Provided that the solution to Eq. (1-325) is continuous, it can be represented by a series of the form: T h = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · =

∞ i=0

ai xi

(1-326)

140

One-Dimensional, Steady-State Conduction

By substituting Eq. (1-326) into Eq. (1-325), it is possible to identify the characteristics of the series that solves this class of differential equation. The second derivative of the solution is required: ∞

dT h ai i xi−1 = a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + · · · = dx

(1-327)

i=1

d2 T h = 2 (1) a2 + 3 (2) a3 x + 4 (3) a4 x2 + 5 (4) a5 x3 + 6 (5) a6 x4 + · · · dx2 ∞ = ai i (i − 1) xi−2 (1-328) i=2

Substituting Eqs. (1-328) and (1-326) into Eq. (1-325) leads to: 2 (1) a2 + 3 (2) a3 x + 4 (3) a4 x2 + 5 (4) a5 x3 + 6 (5) a6 x4 + · · · −m2 [a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · ·] = 0

(1-329)

or ∞

ai i (i − 1) xi−2 − m2

i=2

∞

ai xi = 0

(1-330)

i=0

Since x is an independent variable that can assume any value, Eqs. (1-329) or (1-330) can only be generally satisﬁed if the coefﬁcients that multiply each term of the series (i.e., each power of x) each sum to zero. Examining Eq. (1-329), this requirement leads to: 2 (1) a2 − m2 a0 = 0 3 (2) a3 − m2 a1 = 0 4 (3) a4 − m2 a2 = 0 5 (4) a5 − m2 a3 = 0

(1-331)

6 (5) a6 − m2 a4 = 0 ··· The even coefﬁcients are therefore related according to: a2 =

m2 a0 2 (1)

a4 =

m2 a2 m4 a0 = 4 (3) 4 (3) (2) (1)

a6 =

m2 a4 m6 a0 = 6 (5) 6 (5) (4) (3) (2) (1)

(1-332)

··· or, more generally a2 i =

m2 i a0 (2i)!

where i = 0 · · · ∞

(1-333)

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

141

The odd coefﬁcients are also related: a3 =

m2 a1 3 (2)

a5 =

m2 a3 m4 a1 = 5 (4) 5 (4) (3) (2)

a7 =

m2 a5 m6 a1 = 7 (6) 7 (6) (5) (4) (3) (2)

(1-334)

··· or, more generally a2i+1 =

m2i a1 where i = 0 · · · ∞ (2i + 1)!

(1-335)

Therefore, we have determined two functions that both solve Eq. (1-325), related to the even and odd terms of the series; let’s call them F even and F odd : F even = a0

∞ (m x)2 i i=0

F odd = a1

∞ i=0

(1-336)

(2i)! m2i x2i+1 (2i + 1)!

(1-337)

Feven and Fodd are two solutions to the governing equation regardless of the particular values of the constants a0 and a1 in the same way that the functions C1 exp(m x) and C2 exp(-m x) (or, equivalently, C1 sinh(m x) and C2 cosh(m x)) were identiﬁed in Section 1.6 as solutions to Eq. (1-325) regardless of the values of C1 and C2 . The constants are determined in order to make the solution match the boundary conditions. In fact, if Eqs. (1-336) and (1-337) are rearranged slightly we see that they are identical to the series expansion of the functions sinh(m x) and cosh(m x): C1 cosh (m x) = C1

∞ (m x)2 i i=0

C2 sinh (m x) = C2

(2i)!

∞ (m x)2i+1 i=0

(2i + 1)!

=

a0 F even C1

(1-338)

=

a1 F odd m C2

(1-339)

The functions cosh and sinh are useful because they solve a particular differential equation, Eq. (1-325), which appears in many engineering problems; they are in fact nothing more than shorthand for the series given by Eqs. (1-336) and (1-337). To see this clearly, compute each of the terms in Eqs. (1-336) and (1-337) using EES: mx=1 Nterm=10 duplicate i=0,Nterm F even[i]=(mx)ˆ(2∗ i)/Factorial(2∗ i) F odd[i]=(mx)ˆ(2∗ i+1)/Factorial(2∗ i+1) end

“argument of function” “number of terms in series” “term in F even” “term in F odd”

142

One-Dimensional, Steady-State Conduction 4

3.5

Feven series Fodd series

Function value

3 2.5 2 hyperbolic cosine 1.5 1 hyperbolic sine

0.5 0 0

0.2

0.4

0.6

0.8 1 1.2 1.4 Argument of function

1.6

1.8

2

Figure 1-48: Comparison of the functions Feven and Fodd to the functions sinh and cosh.

and sum these terms using the sum command: F even=sum(F even[i],i=0,Nterm) F odd=sum(F odd[i],i=0,Nterm)

“sum of all terms in F even” “sum of all terms in F odd”

The results can be compared to the functions sinh and cosh: sinh=sinh(mx) cosh=cosh(mx)

“sinh function” “cosh function”

A parametric table is created that includes the variables mx, F_even, F_odd, sinh and cosh; the variable mx is varied from 0 to 2.0 and the results are shown in Figure 1-48. This exercise is meant to show that “solving” the homogeneous ordinary differential equation, Eq. (1-325), was really a matter of recognizing that it is solved by the series solutions that we call sinh and cosh (or equivalently exponentials with positive and negative arguments). Maple is very good at recognizing the solutions to differential equations. In this section, extended surfaces that do not have a constant cross-sectional area are considered. The governing differential equations that apply to these problems are more complex than Eq. (1-325). However, these differential equations are also solved by correctly deﬁned series that are given different names: Bessel functions.

1.8.3 Bessel Functions Extended surfaces with non-uniform cross-section may arise in many engineering applications. For example, tapered ﬁns are of interest since they may provide heat transfer rates comparable to constant cross-section ﬁns but require less material. Figure 1-49 illustrates a wedge ﬁn, an extended surface that has a thickness that varies linearly from its value at the base (th) to zero at the tip. The ﬁn is surrounded by ﬂuid at T ∞ with average heat transfer coefﬁcient h. The ﬁn material has conductivity k and the base of the ﬁn is kept at Tb .

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

143

T∞ , h

Figure 1-49: Wedge ﬁn.

th

k

W x

Tb

L

The width of the ﬁn is W and its length is L. We will assume that W th so that convection from the edges of the ﬁn may be ignored. Also, we will assume that the criteria for the extended approximation is satisﬁed (in this case, the Biot number h th/ (2 k) 1) so that the temperature can be assumed to be spatially uniform at any axial location x. It is convenient to deﬁne the origin of the axial coordinate at the tip of the ﬁn (see Figure 1-49) so that the cross-sectional area for conduction (Ac ) can be expressed as: Ac = th W

x L

(1-340)

The differential control volume used to derive the governing differential equation is shown in Figure 1-50 and suggests the energy balance: q˙ x = q˙ x+dx + q˙ conv

(1-341)

or, after expanding the x + dx term and simplifying: 0=

dq˙ dx + q˙ conv dx

(1-342)

The convection heat transfer rate is, approximately q˙ conv = 2 W dx h (T − T ∞ )

(1-343)

Note that Eq. (1-343) is only valid if th/L 1 so that the surface area within the control volume that is exposed to ﬂuid is approximately 2 W dx. The conduction heat transfer rate is: q˙ = −k Ac

dT x dT = −k th W dx L dx

(1-344)

Substituting Eqs. (1-343) and (1-344) into Eq. (1-342) leads to: d x dT 0= −k th W dx + 2 W dx h (T − T ∞ ) dx L dx

(1-345)

dx

Figure 1-50: Differential control volume.

q⋅ x

q⋅ x+dx

x q⋅ conv

144

One-Dimensional, Steady-State Conduction

which can be simpliﬁed to: d dT 2hL 2hL x − T =− T∞ dx dx k th k th

(1-346)

The solution is divided into a homogeneous and particular component: T = Th + T p

(1-347)

Substituting Eq. (1-347) into Eq. (1-346) leads to: dT p d d 2hL dT h 2hL 2hL x − Th + x − Tp = − T∞ dx dx k th dx dx k th k th = 0 for homogeneous differential equation

(1-348)

whatever is left is the particular differential equation

The solution to the particular differential equation is: T p = T∞

(1-349)

The homogeneous differential equation is: d dT h x − β Th = 0 dx dx

(1-350)

where the parameter β is deﬁned for convenience to be: β=

2hL k th

(1-351)

Note that the homogeneous differential equation, Eq. (1-350), is fundamentally different from the homogeneous differential equation that was obtained for a constant crosssection ﬁn, Eq. (1-179). Equation (1-350) is not solved by exponentials; to make this clear, assume an exponential solution: T h = C exp (m x)

(1-352)

and substitute it into the homogeneous differential equation: d [x C m exp (m x)] − β C exp (m x) = 0 dx

(1-353)

or, using the chain rule: C m exp (m x) + x C m2 exp (m x) − β C exp (m x) = 0

(1-354)

which can be simpliﬁed to: m + x m2 − β = 0

(1-355)

Unfortunately, there is no value of m that will satisfy Eq. (1-355) for all values of x. There must be some other function that solves Eq. (1-350). A series solution is again assumed: T h = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 + · · · =

∞

ai xi

(1-356)

i=0

The series is substituted into the governing differential equation in order to identify the characteristics of the series that solves this new class of differential equation. Expanding

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

145

Eq. (1-350) using the chain rule shows that both the ﬁrst and second derivatives are required: x

dT h d2 T h + − β Th = 0 dx2 dx

(1-357)

These derivatives were derived in Section 1.8.2: ∞

dT h ai i xi−1 = a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + · · · = dx

(1-358)

i=1

d2 T h = 2 (1) a2 + 3 (2) a3 x + 4 (3) a4 x2 + 5 (4) a5 x3 + 6 (5) a6 x4 + · · · dx2 ∞ = ai i (i − 1) xi−2 (1-359) i=2

Substituting Eqs. (1-356), (1-358), and (1-359) into Eq. (1-357) leads to: d2 T h → 2 (1) a2 x + 3 (2) a3 x2 + 4 (3) a4 x3 + 5 (4) a5 x4 + 6 (5) a6 x5 + · · · dx2 dT h + → a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + · · · dx −β T h → −β a0 − β a1 x − β a2 x2 − β a3 x3 − β a4 x4 + · · · =0 x

(1-360)

or, collecting like terms: [a1 − β a0 ] + [2 (1) a2 + 2 a2 − β a1 ] x + [3 (2) a3 + 3 a3 − β a2 ] x2 + [4 (3) a4 + 4 a4 − β a3 ] x3 + [5 (4) a5 + 5 a5 − β a4 ] x4 · · · = 0

(1-361)

For the series to solve the differential equation, each of the coefﬁcients must be zero; again, considering the coefﬁcients one at a time leads to a recursive formula that deﬁnes the series. a1 = β a0 (2 (1) + 2) a2 = β a1 22

(3 (2) + 3) a3 = β a2 32

(4 (3) + 4) a4 = β a3 42

(5 (4) + 5) a5 = β a4 52

···

(1-362)

146

One-Dimensional, Steady-State Conduction

or a1 = β a0 β β2 a2 = 2 a1 = 2 a0 2 2 β β3 a3 = 2 a2 = a0 3 [3 (2)]2 β β4 a4 = 2 a3 = a0 4 [4 (3) (2)]2 a5 =

(1-363)

β β5 a = a0 4 52 [5 (4) (3) (2)]2

··· More generally, the coefﬁcients are deﬁned by the equation: ai =

βi (i!)2

a0 where i = 1 · · · ∞

(1-364)

Equation (1-364) deﬁnes a function (let’s call it F) that provides a general solution to the homogeneous differential equation, Eq. (1-350). F = a0

∞ (β x) i i=0

(i!)2

(1-365)

The function F is actually a combination of Bessel functions; this is simply the name given to the series solutions of a particular class of differential equations ( just as hyperbolic sine and hyperbolic cosine are names given to series solutions of a different class of differential equations). The Bessel functions behave according to a set of formalized rules (just as hyperbolic sines and cosines do) that must be carefully obeyed when using them to solve problems. Bessel functions and the rules for manipulating them are completely recognized by Maple. Therefore, the combination of Maple and EES together allow you to avoid much of tedium associated with recognizing the correct Bessel function and then manipulating it to satisfy the boundary conditions of the problem. For the wedge ﬁn problem considered here, it is possible to enter the governing differential equation, Eq. (1-346), in Maple: > restart; > ODE:=diff(x∗ diff(T(x),x),x)-beta∗ T(x)=-beta∗ T_inﬁnity; ODE :=

2 d d T(x) + x T(x) − β T(x) = −β T inf inity dx dx2

and solve it: > Ts:=dsolve(ODE); 0 0 √ √ Ts := T(x) = BesselJ(0, 2, −β x) C2 + BesselY (0, 2 −β x) C1 + T inf inity

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

147

Maple has recognized that the solution to the governing differential equation includes two Bessel functions: the functions BesselJ and BesselY corresponding to Bessel functions of the ﬁrst and second kind, respectively. The ﬁrst argument indicates the order of the Bessel function and the second indicates the argument. The parameter β was deﬁned in Eq. (1-351) and involves the product of only positive quantities. Therefore, β must be positive and the arguments of both of the Bessel functions are complex (i.e., they involve the square root of a negative number); Bessel functions evaluated with a complex argument result in what are called modiﬁed Bessel functions (BesselI and BesselK are the modiﬁed Bessel functions of the ﬁrst and second kind, respectively). Maple will identify this fact for you, provided that you specify that the variable beta must be positive (using the assume command) and solve the equation again: > assume(beta>0); > Ts:=dsolve(ODE); 0 0 √ √ Ts := T(x) = BesselI(0, 2, β ∼ x) C2 + BesselK (0, 2 β ∼ x) C1 + T inf inity

The trailing tilde (∼) notation is used in Maple to indicate that the variable is associated with an assumption regarding its value; this convention can be changed in the preferences dialog in Maple. The solution is expressed in terms of modiﬁed Bessel functions with real arguments rather than Bessel functions with complex arguments. It is worth noting that if the argument of the function cosh is complex, then the result is cosine; thus the cosine function can be thought of as the modiﬁed hyperbolic cosine. The same behavior occurs for the sine and hyperbolic sine functions. All of the Bessel functions are built into EES and therefore the solution from Maple can be copied and pasted into EES for evaluation and manipulation. Maple has identiﬁed the general solution to the wedge ﬁn problem: 0 0 (1-366) T = C2 BesselI(0, 2 β x) + C1 BesselK(0, 2 β x) + T ∞ All that remains is to determine the constants C1 and C2 so that Eq. (1-366) also satisﬁes the boundary conditions. It is tempting to assume that one boundary condition must force the rate of heat transfer at the tip to be zero; however, the fact that the cross-sectional area at the tip is zero guarantees this fact, provided that the temperature gradient and therefore the temperature at the tip (i.e., at x = 0) is ﬁnite. T x=0 < ∞

(1-367)

Substituting Eq. (1-366) into Eq. (1-367) leads to: C2 BesselI (0, 0) + C1 BesselK (0, 0) + T ∞ < ∞

(1-368)

Figure 1-51 illustrates the behavior of the zeroth order modiﬁed Bessel function of the ﬁrst (BesselI) and second (BesselK) kind. Notice that the zeroth order modiﬁed Bessel function of the second kind is unbounded at zero and therefore the solution cannot include BesselK; the constant C1 must be zero. The remaining boundary condition corresponds to the speciﬁed base temperature of the ﬁn: T x=L = T b Substituting Eq. (1-366) into Eq. (1-369) leads to: 0 C2 BesselI(0, 2 β L) + T ∞ = T b

(1-369)

(1-370)

148

One-Dimensional, Steady-State Conduction 5

Modified Bessel function

4.5 4 3.5 3 st

1 kind

2.5 2 1.5 1 2

0.5 0 0

0.5

nd

kind

1 1.5 2 Argument of the function

2.5

3

Figure 1-51: Modiﬁed zeroth order Bessel functions of the ﬁrst and second kind.

Solving for the constant C2 leads to: C2 =

(T b − T ∞ ) √ BesselI 0, 2 β L

(1-371)

Substituting Eq. (1-371) into Eq. (1-366) (with C1 = 0) leads to the solution for the temperature distribution for a wedge ﬁn: √ BesselI(0, 2 β x) (1-372) + T∞ T = (T b − T ∞ ) √ BesselI(0, 2 β L) which can be expressed as:

/ x BesselI 0, 2 β L (T − T ∞ ) L = √ (T b − T ∞ ) BesselI(0, 2 β L)

(1-373)

Figure 1-52 illustrates the dimensionless temperature, (T − T ∞ )/(T b − T ∞ ), as a function of the dimensionless position, x/L, for various values of β L. Note that according to Eq. (1-351), the dimensionless parameter β L is: 2 h L2 (1-374) k th and resembles the ﬁn parameter, m L, for a constant cross-sectional area ﬁn. The parameter β L plays a similar role in the solution. The resistance to conduction in the x-direction is approximately: βL =

2L k W th and the resistance to convection is approximately: Rcond,x ≈

Rconv ≈

The ratio of Rcond,x

1

h2LW to Rconv is related to β L: h L2 2L h2LW Rcond,x ≈ =4 = 2βL Rconv k W th 1 k th

(1-375)

(1-376)

(1-377)

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

149

1

Dimensionless temperature

0.9 0.8

β L = 0.1 βL = 0.2 βL = 0.5

0.7 0.6

βL = 1

0.5

βL = 2

0.4 0.3

βL = 5

0.2

β L = 10

0.1 0 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position

0.8

0.9

1

Figure 1-52: Dimensionless temperature distribution as a function of dimensionless position for various values of β L.

As β L is reduced, the resistance to conduction in the x-direction becomes small relative to the resistance to convection and so the ﬁn becomes nearly isothermal at the base temperature. If β L is large, then the convection resistance is large relative to the conduction resistance and so the ﬁn temperature approaches the ﬂuid temperature. The rate of heat transfer to the ﬁn is given by: dT (1-378) q˙ ﬁn = k W th dx x=L Substituting Eq. (1-372) into Eq. (1-378) leads to: √ BesselI(0, 2 β x) d + T∞ q˙ ﬁn = k W th (T b − T ∞ ) √ dx BesselI(0, 2 β L) x=L

(1-379)

or q˙ ﬁn =

0 k W th (T b − T ∞ ) d √ [BesselI(0, 2 β x)]x=L BesselI 0, 2 β L dx

(1-380)

Rules for manipulating Bessel functions are presented in Section 1.8.4; however, Maple can be used to work with Bessel functions. The derivative required by Eq. (1-380) is computed easily using Maple: > restart; > diff(BesselI(0,2∗ sqrt(beta∗ x)),x); √ BesselI(1, 2 β x) β √ βx

so that Eq. (1-380) can be written as: q˙ ﬁn

/ √ BesselI(1, 2 β L) β = k W th (T b − T ∞ ) √ BesselI(0, 2 β L) L

(1-381)

150

One-Dimensional, Steady-State Conduction 1 0.9 0.8

Fin efficiency

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4 5 6 7 Fin parameter, βL

8

9

10

Figure 1-53: Fin efﬁciency of a wedge ﬁn as a function of β L.

The efﬁciency of the wedge ﬁn is deﬁned as discussed in Section 1.6.5: ηﬁn =

q˙ ﬁn h 2 W L (T b − T ∞ )

(1-382)

Substituting Eq. (1-381) into Eq. (1-382) leads to: ηﬁn

/ √ k th BesselI(1, 2 β L) β = √ h 2 L BesselI(0, 2 β L) L

or ηﬁn =

√ BesselI(1, 2 β L) √ √ BesselI(0, 2 β L) β L

(1-383)

(1-384)

Figure 1-53 illustrates the ﬁn efﬁciency of a wedge ﬁn as a function of the parameter β L. Notice that the ﬁn efﬁciency approaches unity when β L approaches zero because the ﬁn is nearly isothermal and the efﬁciency decreases as β L increases.

1.8.4 Rules for using Bessel Functions Bessel functions are well-deﬁned functions with speciﬁc rules for integration and differentiation. These rules are summarized in this section; however, the use of Maple will greatly reduce the need to know these rules. The differential equation: d p dθ (1-385) x ± c2 xs θ = 0 dx dx or, equivalently xp

d2 θ dθ + pxp−1 ± c2 xs θ = 0 dx2 dx

(1-386)

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

151

Governing differential equation: d ⎛ p dθ ⎞ 2 s ⎜x ⎟±c x θ =0 dx ⎝ dx ⎠ s− p+2≠0 Calculate solution parameters: 2 n (1 − p ) = n= a= 2 (s − p + 2 ) a (s − p + 2 )

(1 − p )

s− p+2=0

( p − 1)

2

2

− 4c > 0

( p − 1)

2

Calculate solution parameters:

Last term in dif. eq. is negative: d ⎛ p dθ ⎞ 2 s ⎜x ⎟−c x θ =0 dx ⎝ dx ⎠

r1 =

(1 − p ) + ( p − 1)

−c

r2 =

(1 − p ) − ( p − 1)

−c

Calculate solution parameters: (1 − p ) d= 2 2 p − 1) ( 2 e= c − 4

2

2

4

2

2

n

( ) BesselK (n, c a x )

θ = C1 x a BesselI n, c a x n

+C2 x

a

2

− 4c < 0

1 a

1 a

2

4

2

θ = C1 x r1 + C2 x r2

θ = C1 x d cos ⎣⎡e ln (x )⎦⎤

+C2 x sin ⎡⎣e ln (x )⎤⎦ d

( p − 1)

2

Last term in dif. eq. is positive: d ⎛ p dθ ⎞ 2 s ⎜x ⎟+c x θ = 0 dx ⎝ dx ⎠ n

+C2 x

n

a

Calculate solution parameter: (1 − p ) d= 2

( ) BesselY (n, c a x )

θ = C1 x a BesselJ n, c a x

2

− 4c = 0

1 a

θ = C1 x d + C2 x d ln (x )

1 a

Figure 1-54: Flowchart illustrating the steps involved with identifying the correct solution to Bessel’s equation.

where θ is a function of x and p, c, and s are constants is a form of Bessel’s equation that has been solved using power series. The rules for identifying the appropriate solution given the form of the equation are laid out in ﬂowchart form in Figure 1-54. Following the path outlined in Figure 1-54, the ﬁrst step is to evaluate the quantity s − p + 2; if s − p + 2 is not equal to zero, then the intermediate solution parameters n and a should be calculated. n=

1− p s − p+ 2

(1-387)

a=

2 s − p+ 2

(1-388)

1− p n = a 2

(1-389)

The solution depends on the sign of the last term in Eq. (1-385); if the sign of the last term is negative, then the solution is expressed as: n 1 n 1 θ = C1 x /a BesselI n, c a x /a + C2 x /a BesselK n, c a x /a (1-390)

152

One-Dimensional, Steady-State Conduction

where C1 and C2 are the undetermined constants that depend on the boundary conditions. The functions BesselI and BesselK are modiﬁed Bessel functions of the ﬁrst and second kind, respectively. The ﬁrst parameter in the function is the order of the modiﬁed Bessel function and the second parameter is the argument of the function. The EES code below provides the zeroth order modiﬁed Bessel function of the second kind evaluated at 2.5 (0.06235). y=BesselK(0,2.5)

The order of the Bessel function can either be integer (e.g., 0, 1, 2, . . . ) or fractional (e.g. 0.5). If the sign of the last term in Eq. (1-385) is positive, then the solution is: n 1 n 1 (1-391) θ = C1 x /a BesselJ n, c a x /a + C2 x /a BesselY n, c a x /a where the functions BesselJ and BesselY are Bessel functions of the ﬁrst and second kind, respectively. If s − p + 2 is equal to zero, then the solution depends on the sign of the parameter (p − 1)2 − 4c2 . If (p − 1)2 − 4c2 is positive, then the solution is: θ = C1 xr1 + C2 xr2 where

(1-392)

(1 − p) + r1 = 2

and

(p − 1)2 − c2 4

(1-393)

(p − 1)2 − c2 4

(1-394)

(1 − p) − r2 = 2

If (p − 1)2 − 4c2 is zero, then the solution is: θ = C1 xd + C2 xd ln (x)

(1-395)

where d=

(1 − p) 2

(1-396)

Finally, if (p − 1)2 − 4c2 is negative, then the solution is: θ = C1 xd cos (e ln (x)) + C2 xd sin (e ln (x)) where

(1-397)

e=

c2 −

(p − 1)2 4

(1-398)

The zeroth and ﬁrst order modiﬁed Bessel functions of the ﬁrst and second kind are shown in Figure 1-55. Notice that the modiﬁed Bessel functions of the second kind are unbounded at zero while the modiﬁed Bessel functions of the ﬁrst kind are unbounded as the argument tends towards inﬁnity; this characteristic can be helpful to determine the undetermined constants.

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

153

2.5 nd

Modified Bessel function

th

2 kind, 0 order BesselK(0, x)

2.25 2

nd

st

2 kind, 1 order BesselK(1, x)

1.75 1.5

st

th

1 kind, 0 order BesselI(0, x)

1.25 1

st

st

1 kind, 1 order BesselI(1, x)

0.75 0.5 0.25 0 0

0.5 1 1.5 2 2.5 Argument of modified Bessel function

3

Figure 1-55: Modiﬁed Bessel functions of the ﬁrst and second kinds and the zeroth and ﬁrst orders.

The zeroth and ﬁrst order Bessel functions of the ﬁrst and second kind are shown in Figure 1-56. Notice that the Bessel functions of the second kind, like the modiﬁed Bessel functions of the second kind, are unbounded at zero. The rules for differentiating zeroth order Bessel and zeroth order modiﬁed Bessel functions are:

1

st

du d [BesselI(0, u)] = BesselI(1, u) dx dx

(1-399)

d du [BesselK(0, u)] = −BesselK(1, u) dx dx

(1-400)

th

1 kind, 0 order BesselJ(0, x) 0.75

st

st

1 kind, 1 order BesselJ(1, x)

Bessel function

0.5 0.25 0 -0.25 -0.5

2

-0.75 -1 0

2 1

nd

nd

2

th

kind, 0 order BesselY(0, x) st

kind, 1 order BesselY(1, x) 3 4 5 6 7 Argument of Bessel function

8

9

10

Figure 1-56: Bessel functions of the ﬁrst and second kinds and the zeroth and ﬁrst orders.

154

One-Dimensional, Steady-State Conduction

d du [BesselJ(0, u)] = −BesselJ(1, u) dx dx

(1-401)

d du [BesselY(0, u)] = − BesselY(1, u) dx dx

(1-402)

For arbitrary order Bessel and modiﬁed Bessel functions with positive integer order n, the rules for differentiation are: d n BesselI(n, m x) = m BesselI(n − 1, m x) − BesselI(n, m x) dx x

(1-403)

d n BesselK(n, m x) = −m BesselK(n − 1, m x) − BesselK(n, m x) dx x

(1-404)

d n BesselJ(n, m x) = m BesselJ(n − 1, m x) − BesselJ(n, m x) dx x

(1-405)

d n BesselY(n, m x) = m BesselY(n − 1, m x) − BesselY(n, m x) dx x

(1-406)

Finally, the following differentials are also sometimes useful: d n [x BesselI(n, m x)] = m xn BesselI(n − 1, m x) dx d n [x BesselK(n, m x)] = −m xn BesselK(n − 1, m x) dx d n [x BesselJ(n, m x)] = m xn BesselJ(n − 1, m x) dx

(1-407)

(1-408)

(1-409)

d n [x BesselY(n, m x)] = −m xn BesselY(n − 1, m x) dx

(1-410)

d −n [x BesselI(n, m x)] = m x−n BesselI(n + 1, m x) dx

(1-411)

d −n [x BesselK(n, m x)] = −m x−n BesselK(n + 1, m x) dx

(1-412)

d −n [x BesselJ(n, m x)] = −m x−n BesselJ(n + 1, m x) dx

(1-413)

d −n [x BesselY(n, m x)] = −m x−n BesselY(n + 1, m x) dx

(1-414)

155

EXAMPLE 1.8-1: PIPE IN A ROOF A pipe with outer radius rp = 5.0 cm emerges from a metal roof carrying hot gas at Thot = 90◦ C. The pipe is welded to the roof, as shown in Figure 1. Assume that the temperature at the interface between the pipe and the roof is equal to the gas temperature, Thot . The inside of the roof is well-insulated, but the outside of the roof is exposed to ambient air at T∞ = 20◦ C. The average heat transfer coefﬁcient between the outside of the roof and the ambient air is h = 50 W/m2 -K. The outside of the roof is also exposed to a uniform heat ﬂux due to the incident solar radiation, q˙ s = 800 W/m2 . The spatial extent of the roof is large with respect to the outer radius of the pipe. The metal roof has thickness th = 2.0 cm and thermal conductivity k = 50 W/m-K. 2 q⋅ s′′ = 800 W/m T∞ = 20°C 2 h = 50 W/m -K Thot = 90°C th = 2 cm k = 50 W/m-K

Figure 1: Pipe passing through a roof exposed to solar radiation.

rp = 5 cm adiabatic

a) Can the roof be modeled using an extended surface approximation? The input parameters are entered in EES: “EXAMPLE 1.8-1: Pipe in a Roof” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Input Parameters” r p=5.0 [cm]∗ convert(cm,m) T hot=converttemp(C,K,90[C]) T inﬁnity=converttemp(C,K,20[C]) h bar=50 [W/mˆ2-K] qf s=800 [W/mˆ2] th=2.0 [cm]∗ convert(cm,m) k=50 [W/m-K]

“Pipe radius” “Hot gas temperature” “Air temperature” “Heat transfer coefﬁcient” “Solar ﬂux” “Roof thickness” “Roof conductivity”

The extended surface approximation ignores any temperature gradients across the thickness of the roof. This is equivalent to ignoring the resistance to conduction across the thickness of the roof while considering the resistance associated with convection from the top surface of roof. The ratio of these resistances is calculated using an appropriately deﬁned Biot number: Bi =

th h k

EXAMPLE 1.8-1: PIPE IN A ROOF

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

EXAMPLE 1.8-1: PIPE IN A ROOF

156

One-Dimensional, Steady-State Conduction

which is calculated in EES: Bi=h bar∗ th/k

“Biot number to check extended surface approximation”

The Biot number is 0.02, which is sufﬁciently less than 1 to justify the extended surface approximation. b) Develop an analytical model for the roof that can be used to predict the temperature distribution in the roof and also determine the rate of heat loss from the pipe by conduction to the roof. Because the roof is large relative to the spatial extent of our problem, the edge of the roof will have no effect on the temperature distribution in the metal around the pipe and the temperature distribution will be axisymmetric; the problem can be solved in radial coordinates. An energy balance on a differential segment of the roof is shown in Figure 2. q⋅ s q⋅ conv q⋅ r r

q⋅ r + dr

Figure 2: Differential energy balance.

dr

The energy balance includes conduction, convection and solar irradiation: q˙r + q˙ s = q˙r +dr + q˙ conv or q˙ s =

d q˙r dr + q˙ conv dr

Substituting the rate equations: q˙r = −k 2 π r th

dT dr

q˙ s = q˙ s 2 π r dr q˙ conv = 2 π r dr h (T − T∞ ) into the energy balance leads to: d dT −k 2 π r th dr + 2 π r dr h (T − T∞ ) q˙ s 2 π r dr = dr dr Simplifying leads to: h h d q˙ dT r − rT =− r T∞ − s r dr dr k th k th k th

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

157

EXAMPLE 1.8-1: PIPE IN A ROOF

The solution is split into its homogeneous and particular components: T = Th + Tp which leads to: h dTh d r − r Th dr dr k th

dTp h h d q˙ + r − r Tp = − r T∞ − s r dr dr k th k th k th

= 0 for homogeneous differential equation

whatever is left is the particular differential equation

The solution to the particular differential equation: dTp h h q˙ d r − r Tp = − r T∞ − s r dr dr k th k th k th is a constant: Tp = T∞ +

q˙ s h

The homogeneous differential equation is: d dTh r − m2 r Th = 0 dr dr where

(1)

m=

h k th

Equation (1) is a form of Bessel’s equation: d dθ xp ± c2 x s θ = 0 dx dx

(2)

where p = 1, c = m, and s = 1. Referring to the ﬂow chart presented in Figure 1-54, the value of s − p + 2 is equal to 2 and therefore the solution parameters n and a must be computed: n=

1−1 =0 1−1+2

a=

2 =1 1−1+2

The last term in Eq. (1) is negative and therefore the solution to Eq. (2), as indicated by Figure 1-54, is given by: n 1 n 1 θ = C 1 x /a BesselI n, c a x /a + C 2 x /a BesselK n, c a x /a where x = r and c = m for this problem. The homogeneous solution is: Th = C 1 BesselI (0, mr ) + C 2 BesselK (0, mr ) The temperature distribution is the sum of the homogeneous and particular solutions: q˙ (3) T = C 1 BesselI (0, mr ) + C 2 BesselK (0, mr ) + T∞ + s h

EXAMPLE 1.8-1: PIPE IN A ROOF

158

One-Dimensional, Steady-State Conduction

Maple can be used to obtain the same result: > restart; > ODE:=diff(r∗ diff(T(r),r),r)-mˆ2∗ r∗ T(r)=-mˆ2∗ r∗ T_inﬁnity-qf_s∗ r/(k∗ th); O D E :=

d T(r ) dr

+r

d2 T(r ) dr 2

− m2 r T(r ) = −m2 r T inﬁnity −

qf sr k th

> Ts:=dsolve(ODE); Ts := T(r ) = BesselI (0, mr ) C 2 + BesselK (0, mr ) C 1 +

m2 T inﬁnity k th + qf s m2 k th

Note that the constants C1 and C2 are interchanged in the Maple solution but it is otherwise the same as Eq. (3). The boundary conditions must be used to obtain C1 and C2 . As r approaches ∞, the effect of the pipe disappears. In this limit, the heat gain from the sun exactly balances convection, therefore: q˙ s = h (Tr →∞ − T∞ )

(4)

Substituting Eq. (3) into Eq. (4) leads to: q˙ q˙ s = h C 1 BesselI (0, ∞) + C 2 BesselK (0, ∞) + T∞ + s − T∞ h or C 1 BesselI (0, ∞) + C 2 BesselK (0, ∞) = 0 Figure 1-55 shows that the zeroth order modiﬁed Bessel function of the ﬁrst kind (i.e., BesselI(0,x)) limits to ∞ as x approaches ∞ while the zeroth order modiﬁed Bessel function of the second kind (i.e., BesselK(0,x)) approaches 0 as x approaches ∞. This information can also be obtained using Maple and the limit command: > limit(BesselI(0,x),x=inﬁnity); ∞ > limit(BesselK(0,x),x=inﬁnity); 0

Therefore, C1 must be zero while C2 can be any ﬁnite value. T = C 2 BesselK (0, mr ) + T∞ +

q˙ s h

The temperature where the roof meets the pipe is speciﬁed: Tr =r p = Thot or q˙ C 2 BesselK 0, mr p + T∞ + s = Thot h

(3)

159

The solution is programmed in EES: m=sqrt(h bar/(k∗ th)) C 2∗ BesselK(0,m∗ r p)=T hot-T inﬁnity-qf s/h bar T=C 2∗ BesselK(0,m∗ r)+T inﬁnity+qf s/h bar T C=converttemp(K,C,T)

“ﬁn parameter” “boundary condition” “solution” “in C”

The temperature in the roof as a function of position is shown in Figure 3 for h = 50 W/m2 -K (as speciﬁed in the problem statement) and also for h = 5 W/m2 -K. 180 2

h = 5 W/m -K 160

Temperature (°C)

140 120 100 80 60

2

h = 50 W/m -K 40 20 0.05

0.1

0.15

0.2

0.25 0.3 0.35 Radius (m)

0.4

0.45

0.5

Figure 3: Temperature as a function of radius for h = 50 W/m2 -K and h = 5 W/m2 -K with q˙ s = 800 W/m2 .

The heat transfer between the pipe and the roof (q˙ p ) is evaluated using Fourier’s law at r = r p : dT q˙ p = −k th 2 π r p (4) dr r =r p Substituting Eq. (3) into Eq. (4) leads to: q˙ p = −k th 2 π r p C 2

d [BesselK (0, mr )]r =r p dr

which can be evaluated using the differentiation rule provided by Eq. (1-400): q˙ p = k th 2 π r p C 2 m BesselK(1, mr p ) or using Maple: > q_dot_p:=-k∗ th∗ 2∗ pi∗ r_p∗ C_2∗ eval(diff(BesselK(0,m∗ r),r),r=r_p); q d ot p := 2 k th π r p C 2 BesselK (1, mr p) m

EXAMPLE 1.8-1: PIPE IN A ROOF

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

One-Dimensional, Steady-State Conduction

The solution is programmed in EES: q dot p=k∗ th∗ 2∗ pi∗ r p∗ C 2∗ m∗ BesselK(1,m∗ r p)

“heat transfer into pipe”

Figure 4 illustrates the rate of heat transfer into the pipe as a function of the heat transfer coefﬁcient and for various values of the solar ﬂux. 400 300 Heat transfer from pipe (W)

EXAMPLE 1.8-1: PIPE IN A ROOF

160

q⋅ s′′= 0 W/m2 q⋅ s′′= 200 W/m2 q⋅s′′= 500 W/m2 ⋅q′′= 800 W/m2 s

200 100 0 -100 -200 -300 -400 0

10

20 30 40 50 60 70 80 2 Heat transfer coefficient (W/m-K)

90

100

Figure 4: Heat transfer from pipe to roof as a function of the heat transfer coefﬁcient for various values of the solar ﬂux.

It is always important to understand your solution after it has been obtained. Notice in Figure 4 that the rate of heat transfer to the roof tends to increase with increasing heat transfer coefﬁcient. This makes sense, as the temperature gradient at the interface between the roof and the pipe will increase as the heat transfer coefﬁcient increases. However, when there is a non-zero solar ﬂux, the heat transfer rate will change direction (i.e., become negative) at low values of the heat transfer coefﬁcient indicating that the heat ﬂow is into the pipe under these conditions. This effect occurs when the solar ﬂux elevates the temperature of the roof to the point that it is above the hot gas temperature. Figure 4 shows that we can expect this behavior for h = 5 W/m2 -K and q˙ s = 800 W/m2 and Figure 3 illustrates the temperature distribution under these conditions.

161

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION EXAMPLE 1.3-1 examined an ablative technique for locally heating cancerous tissue using small, conducting spheres (thermoseeds) that are embedded at precise locations and exposed to a magnetic ﬁeld. Each thermoseed experiences a volumetric generation of thermal energy that causes its temperature and the temperature of the adjacent tissue to rise. In EXAMPLE 1.3-1, blood perfusion in the tissue was neglected; blood perfusion refers to the volumetric removal of energy in the tissue by the blood ﬂowing in the microvascular structure. The blood perfusion may be modeled as a volumetric heat sink that is proportional to the difference between the local temperature and the normal body temperature (Tb = 37◦ C); the constant of proportionality, β, is nominally 20,000 W/m3 -K. The thermoseed has a radius r t s = 1.0 mm and it experiences a total rate of thermal energy generation of g˙ t s = 1.0 W. The temperature far from the thermoseed is the body temperature, Tb . The tissue has thermal conductivity k t = 0.5 W/m-K. a) Determine the steady-state temperature distribution in the tissue associated with a single sphere placed in an inﬁnite medium of tissue considering blood perfusion. The input parameters are entered in EES: “EXAMPLE 1.8-2: Magnetic Ablation with Blood Perfusion” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” r ts=1.0 [mm]∗ convert(mm,m) T b=converttemp(C,K,37 [C]) g dot ts=1.0 [W] beta=20000 [W/mˆ3-K] k t=0.5 [W/m-K]

“radius of thermoseed” “blood and body temperature” “generation in the thermoseed” “blood perfusion constant” “tissue conductivity”

Figure 1 illustrates a differential control volume in the tissue that balances conduction with blood perfusion. The energy balance on the control volume is: q˙r = q˙r +dr + g˙ where q˙ is conduction and g˙ is the rate of energy removed by blood perfusion.

rts Figure 1: Differential control volume in the tissue.

g⋅ ts

⋅ ⋅ q⋅ r g qr+dr

dr

The conduction through the tissue is given by: q˙r = −k t 4 π r 2

dT dr

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

162

One-Dimensional, Steady-State Conduction

and the rate of energy removal by blood perfusion is: g˙ = 4 π r 2 dr β (T − Tb ) Combining these equations leads to: d 2 dT 0= −k t 4 π r dr + 4 π r 2 dr β (T − Tb ) dr dr which can be simpliﬁed:

dT β β 2 d r T = − r 2 Tb r2 − dr dr kt kt

The solution is divided into its homogeneous and particular components: T = Th + Tp so that:

d dTh β 2 r Th r2 − dr dr kt

+

= 0 for homogeneous differential equation

dTp d β β 2 r Tp = − r 2 Tb r2 − dr dr kt kt whatever is left is the particular differential equation

The particular solution is: Tp = Tb The homogeneous differential equation is: dTh d r2 − m2 r 2 Th = 0 dr dr where

m=

(1)

β kt

Equation (1) is a form of Bessel’s equation: dθ d xp ± c2 x s θ = 0 dx dx where p = 2, c = m, and s = 2. Referring to the ﬂow chart presented in Figure 1-54, the value of s − p + 2 is equal to 2 and therefore the solution parameters n and a must be computed: n=

1 1−2 =− 2−2+2 2

a=

2 =1 2−2+2

The last term in Eq. (1) is negative and therefore the solution is given by: n 1 n 1 θ = C 1 x /a BesselI n, c a x /a + C 2 x /a BesselK n, c a x /a or

1 1 1 1 Th = C 1 r − /2 BesselI − , mr + C 2 r − /2 BesselK − , mr 2 2

163

The solution is the sum of the homogeneous and particular solutions: 1 1 1 1 T = C 1 r − /2 BesselI − , mr + C 2 r − /2 BesselK − , mr + Tb 2 2

(2)

The constants are obtained by applying the boundary conditions. As r approaches ∞, the temperature must approach the body temperature: Tr →∞ = Tb

(3)

Substituting Eq. (2) into Eq. (3) leads to: 1 1 BesselI − , ∞ BesselK − , ∞ 2 2 C1 + C2 =0 √ √ ∞ ∞

(4)

At ﬁrst glance it is unclear how Eq. (4) helps to establish the constants; however, Maple can be used to show that C1 must be zero because the ﬁrst term limits to ∞ while the second term limits to 0: > limit(BesselI(-1/2,r)/sqrt(r),r=inﬁnity); ∞ > limit(BesselK(-1/2,r)/sqrt(r),r=inﬁnity); 0

The second boundary condition is obtained from an interface energy balance at r = rts ; the rate of conduction heat transfer into the tissue must equal the rate of generation within the thermoseed: −4 π r t2s k t

dT = g˙ t s dr r =rt s

(5)

Substituting Eq. (2) with C1 = 0 into Eq. (5) leads to: −4 π

r t2s

d 1 −1/2 kt C2 BesselK − , mr = g˙ t s r dr 2 r =r t s

Using Eq. (1-408) leads to: −1/2

−C 2 r t s

3 g˙ t s m BesselK − , mr t s = − 2 4 π r t2s k t

The constant C2 is evaluated in EES:

“Determine constant” m=sqrt(beta/k t) -C 2∗ m∗ BesselK(-3/2,m∗ r ts)/sqrt(r ts)=-g dot ts/(4∗ pi∗ r tsˆ2∗ k t)

“solution parameter” “determine constant”

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

1.8 Analytical Solutions for Non-Constant Cross-Section Extended Surfaces

One-Dimensional, Steady-State Conduction

The solution is programmed in EES and converted to Celsius: “Solution” T=C 2∗ BesselK(-0.5,m∗ r)/sqrt(r)+T b T C=converttemp(K,C,T) r mm=r∗ convert(m,mm)

“temperature” “in C” “radius in mm”

Figure 2 illustrates the temperature in the tissue as a function of radial position for various values of blood perfusion. Note that the temperature distribution as β → 0 (i.e., in the absence of blood perfusion) agrees exactly with the solution for the tissue temperature obtained in EXAMPLE 1.3-1 (which is overlaid onto Figure 2) although the mathematical form of the solution looks very different. Figure 2 shows that the effect of blood perfusion is to reduce the extent of the elevated temperature region and therefore diminish the amount of tissue killed by the thermoseed. 200 EXAMPLE 1.3-1 result

180 160 Temperature (°C)

EXAMPLE 1.8-2: MAGNETIC ABLATION WITH BLOOD PERFUSION

164

3

β → 0 W/m -K β = 5,000 W/m 3-K β = 20,000 W/m 3-K 3 β = 50,000 W/m -K

140 120 100 80 60 40 20 1

1.5

2

2.5

3 3.5 4 Radius (mm)

4.5

5

5.5

6

Figure 2: Temperature in the tissue as a function of radius for various values of blood perfusion; also shown is the result from EXAMPLE 1.3-1 which was derived for the same problem in the absence of blood perfusion (β = 0).

1.9 Numerical Solution to Extended Surface Problems 1.9.1 Introduction Sections 1.6 through 1.8 present analytical solutions to extended surface problems. Only simple problems with constant properties can be considered analytically. There will be situations where these simpliﬁcations are not justiﬁed and it will be necessary to use a numerical model. Numerical modeling of extended surface problems is a straightforward extension of the numerical modeling techniques that are described in Sections 1.4 and 1.5. If the extended surface approximation discussed in Section 1.6.2 is valid, then it is possible to obtain a numerical solution by dividing the computational domain into many small (but ﬁnite) one-dimensional control volumes. Energy balances are written for each control volume; the energy balances can include convective and/or radiative terms in addition to the conductive and generation terms that are considered in Sections 1.4 and

1.9 Numerical Solution to Extended Surface Problems

165

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING A resistance temperature detector (RTD) utilizes a material that has an electrical resistivity that is a strong function of temperature. The temperature of the RTD is inferred by measuring its electrical resistance. Figure 1 shows an RTD that is mounted at the end of a metal rod and inserted into a pipe in order to measure the temperature of a ﬂowing liquid. The RTD is monitored by passing a known current through it and measuring the voltage drop across it. This process results in a constant amount of ohmic heating that will cause the RTD temperature to rise relative to the temperature of the surrounding liquid; this effect is referred to as a self-heating measurement error. Also, conduction from the wall of the pipe to the temperature sensor through the metal rod can result in a temperature difference between the RTD and the liquid; this effect is referred to as a mounting measurement error. Tw = 20°C

L = 5.0 cm

pipe

D = 0.5 mm x

Figure 1: Temperature sensor mounted in a ﬂowing liquid.

T∞ = 5.0°C

k = 10 W/m-K q⋅ sh = 2.5 mW RTD

The thermal energy generation associated with ohmic heating is q˙ sh = 2.5 mW. All of this ohmic heating is assumed to be transferred from the RTD into the end of the rod at x = L. The rod has a thermal conductivity k = 10 W/m-K, diameter D = 0.5 mm, and length L = 5.0 cm. The end of the rod that is connected to the pipe wall (at x = 0) is maintained at a temperature of Tw = 20◦ C. The liquid is at a uniform temperature, T∞ = 5◦ C. However, the local heat transfer coefﬁcient between the liquid and the rod (h) varies with x due to the variation of the liquid velocity in the pipe. This problem resembles external ﬂow over a cylinder, which will be discussed in Chapter 4; however, you may assume that the heat transfer coefﬁcient between the rod surface and the ﬂuid varies according to: h = 2000

W m2.8 K

x 0.8

(1)

where h is the heat transfer coefﬁcient in W/m2 -K and x is position along the rod in m.

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.5. Each term in the energy balance is represented by a rate equation that reﬂects the governing heat transfer mechanism; the result is a system of algebraic equations that can be solved using EES or MATLAB. The solution should be checked for convergence, checked against your physical intuition, and compared with an analytical solution in the limit where one is valid.

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

166

One-Dimensional, Steady-State Conduction

a) Can the rod be treated as an extended surface? The input parameters are entered in EES; note that the heat transfer coefﬁcient is computed using a function deﬁned at the top of the EES code. “EXAMPLE 1.9-1: Temperature Sensor Error due to Mounting and Self Heating” $UnitSystem SI MASS DEG PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Function for heat transfer coefﬁcient” function h(x) h=2000 [W/mˆ2.8-K]∗ xˆ0.8 end “Inputs” q dot sh=2.5 [milliW]∗ convert(milliW,W) k=10 [W/m-K] D=0.5 [mm]∗ convert(mm,m) L=5.0 [cm]∗ convert(cm,m) T w=converttemp(C,K,20 [C]) T inﬁnity=converttemp(C,K,5 [C])

“self-heating power” “conductivity of mounting rod” “diameter of mounting rod” “length of mounting rod” “temperature of wall” “temperature of liquid”

The appropriate Biot number for this case is: Bi =

hD 2k

The Biot number will be largest (and therefore the extended surface approximation least valid) when the heat transfer coefﬁcient is largest. According to Eq. (1), the highest heat transfer coefﬁcient occurs at the tip of the rod; therefore, the Biot number is calculated according to: Bi=h(L)∗ D/(2∗ k)

“Biot number”

The Biot number calculated by EES is 0.0046, which is much less than 1.0 and therefore the extended surface approximation is justiﬁed. b) Develop a numerical model of the rod that will predict the temperature distribution in the rod and therefore the error in the temperature measurement; this error is the difference between the temperature at the tip of the rod (i..e, the temperature of the RTD) and the liquid. The development of the numerical model follows the same steps that are discussed in Section 1.4. Nodes (i.e., locations where the temperature will be determined) are positioned uniformly along the length of the rod, as shown in Figure 2. The location of each node (xi ) is: xi =

(i − 1) L (N − 1)

i = 1..N

167

where N is the number of nodes used for the simulation. The distance between adjacent nodes (x) is: x =

L (N − 1)

This distribution is entered in EES:

N=100 duplicate i=1,N x[i]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1)

“number of nodes” “position of each node” “distance between adjacent nodes”

A control volume is deﬁned around each node; the control surface bisects the distance between the nodes, as shown in Figure 2. T1 ⋅

Figure 2: Control volume for an internal node.

q⋅ top

Ti-1

qconv q⋅ bottom

Ti

q⋅ top q⋅ conv ⋅ qsh

TN-1

Ti+1 TN

The control volume for internal node i shown in Figure 2 is subject to conduction heat transfer at each edge (q˙ t op and q˙ bottom ) and convection (q˙ conv ). The energy balance is: q˙ t op + q˙ bottom + q˙ conv = 0 The conduction terms are approximated according to:

q˙ t op =

k π D2 (Ti−1 − Ti ) 4 x

q˙ bottom =

k π D2 (Ti+1 − Ti ) 4 x

The convection term is modeled using the convection coefﬁcient evaluated at the position of the node: q˙ conv = hxi π D x (T∞ − Ti )

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

168

One-Dimensional, Steady-State Conduction

Combining these equations leads to: k π D2 k π D2 (Ti−1 −Ti ) + (Ti+1 −Ti ) + hxi π Dx(T∞ − Ti ) = 0 for i = 2.. (N −1) 4 x 4 x (2) “internal control volume energy balances” duplicate i=2,(N-1) k∗ pi∗ Dˆ2∗ (T[i-1]-T[i])/(4∗ DELTAx)+k∗ pi∗ Dˆ2∗ (T[i+1]-T[i])/(4∗ DELTAx)+ & pi∗ D∗ DELTAx∗ h(x[i])∗ (T inﬁnity-T[i])=0 end

The nodes at the edges of the domain must be treated separately. At the pipe wall, the temperature is speciﬁed: T1 = Tw

T[1]=T w

(3)

“boundary condition at wall”

The ohmic dissipation, q˙ sh is assumed to enter the half-node at the tip (i.e., node N) and therefore is included in the energy balance for this node (see Figure 2): hx π D x k π D2 (TN −1 − TN ) + N (T∞ − TN ) + q˙ sh = 0 4 x 2

(4)

Note the factor of 2 in the denominator of the convection term that arises because the half-node has half the surface area of the internal nodes. k∗ pi∗ Dˆ2∗ (T[N-1]-T[N])/(4∗ DELTAx)+pi∗ D∗ DELTAx∗ h(x[N])∗ (T inﬁnity-T[N])/2+q dot sh=0 “boundary condition at tip”

Equations (2) through (4) are a system of N equations in an equal number of unknown temperatures that are entered in EES. The solution is converted to Celsius:

duplicate i=1,N T C[i]=converttemp(K,C,T[i]) end

“solution in Celsius”

Figure 3 illustrates the temperature distribution in the rod for N = 100 nodes. The temperature elevation of the tip relative to the ﬂuid is about 3.4 K and represents the measurement error. For the conditions in the problem statement, it is clear that the measurement error is primarily due to self-heating because the effect of the wall (the temperature elevation at the base) has died off after about 2.0 cm.

169

20 18

Temperature (°C)

16 14 12 10 8 measurement error 6 4 0

0.01

0.02

0.03

0.04

0.05

Axial position (m)

Figure 3: Temperature distribution in the mounting rod.

As with any numerical solution, it is important to verify that a sufﬁcient number of nodes have been used so that the numerical solution has converged. The key result of the solution is the tip-to-ﬂuid temperature difference, which is the measurement error for the sensor (δT ): δT = TN − T∞ deltaT=T[N]-T inﬁnity

“measurement error”

Figure 4 illustrates the tip-to-ﬂuid temperature difference as a function of the number of nodes and shows that the solution has converged for N greater than 100 nodes.

Temperature measurement error (K)

3.5 3 2.5 2 1.5 1 0.5 1

10

100

1000

Number of nodes (-)

Figure 4: Tip-to-ﬂuid temperature difference as a function of the number of nodes.

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction

The analytical solution for this problem in the limit of a constant heat transfer coefﬁcient and an adiabatic tip was derived in Section 1.6.3 and is included in Table 1-4: cosh (m (L − x)) T − T∞ = Tw − T∞ cosh (m L) where 4h m= kD The analytical solution is programmed in EES: “Analytical solution for veriﬁcation in the limit q dot sh=0 and h=constant” m=sqrt(4∗ h(L)/(k∗ D)) “ﬁn parameter” duplicate i=1,N T an[i]=T inﬁnity+(T w-T inﬁnity)∗ cosh(m∗ (L-x[i]))/cosh(m∗ L) “analytical solution” T an C[i]=converttemp(K,C,T an[i]) “in C” end

The numerical solution is obtained in this limit by setting the variable q_dot_sh equal to zero and modifying the function h so that it returns 100 W/m2 -K regardless of position. “Function for heat transfer coefﬁcient” function h(x) {h=2000 [W/mˆ2.8-K]∗ xˆ0.8} h=100 [W/mˆ2-K] end “Inputs” q dot sh=0 [W] {2.5 [milliW]∗ convert(milliW,W)}

“self-heating power”

The temperature distribution predicted by the numerical model is compared with the analytical solution in Figure 5. 22.5 Analytical solution Numerical model

20 17.5

Temperature (°C)

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

170

15 12.5 10 7.5 5 2.5 0 0

0.01

0.02

0.03

0.04

0.05

Axial position (m) Figure 5: Veriﬁcation of the numerical model against the analytical solution in the limit that the heat transfer coefﬁcient is constant at h = 100 W/m2 -K and there is no self-heating, q˙ sh = 0 W.

171

c) Investigate the effect of thermal conductivity on the temperature measurement error. Identify the optimal thermal conductivity and explain why an optimal thermal conductivity exists. Figure 6 illustrates the temperature measurement error as a function of the thermal conductivity of the rod material; note that the function h has been set back to its original form and the variable q_dot_sh restored to 2.5 mW. Figure 6 shows that the optimal thermal conductivity, corresponding to the minimum measurement error, is around 100 W/m-K. Below the optimal value, the self-heating error dominates as the local temperature rise at the tip of the rod is large. Above the optimal value, the conduction from the wall dominates.

Temperature measurement error (K)

10 9 8 7 6 5 self heating dominates

4

conduction from the wall dominates

3 2 1 0 0

25

50

75

100

125

150

175

200

225

EXAMPLE 1.9-1: TEMPERATURE SENSOR ERROR DUE TO MOUNTING & SELF HEATING

1.9 Numerical Solution to Extended Surface Problems

250

Rod thermal conductivity (W/m-K)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS It is often necessary to supply a cryogenic experiment or apparatus with electrical current. Some examples include current for superconducting electronics and magnets, to energize a resistance-based temperature sensor, and to energize a heater used for temperature control. In any of these cases, careful design of the wires that are used to supply and return the current to the facility is important. The heat transfer to the cryogenic device from these wires should be minimized as this energy must be removed either by a refrigeration system (i.e., a cryocooler) or by consumption of a relatively expensive cryogen (e.g., by the boil off of liquid helium or liquid nitrogen). There is an optimal wire diameter for any given application that minimizes this parasitic heat transfer to the device. Figure 1 illustrates two current leads, each carrying I = 100 ampere (one supply and the other return). These current leads extend from the room temperature wall of the vacuum vessel, where the wire material is at TH = 20◦ C, to the experiment, where the wire material is at TC = 50 K. The length of both current leads is L = 1.0 m and their diameter, D, should be optimized. The vacuum in the vessel prevents any convection heat transfer from the surface of the wires. However, the surface of the

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

Figure 6: Temperature measurement error as a function of rod thermal conductivity.

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

172

One-Dimensional, Steady-State Conduction

wires radiate to their surroundings, which may be assumed to be at TH = 20◦ C. The external surface of the wires has emissivity, ε = 0.5. TH = 20°C

2 current leads, each carrying I =100 ampere with emissivity ε = 0.5

L = 1.0 m

TH = 20°C TC = 50 K

D

Figure 1: Cryogenic current leads.

The leads are made of oxygen free, high-conductivity copper; the thermal conductivity and resistivity of copper can vary substantially at cryogenic temperatures depending on the purity and history of the material (e.g., whether it has been annealed or not). The purity of the metal is often expressed as the Residual Resistivity Ratio (RRR), which is deﬁned as the ratio of the metal’s electrical resistivity at 273 K to that at 4.2 K. Oxygen free, high conductivity copper (OFHC) has an RRR of approximately 200. The thermal conductivity and electrical resistivity of RRR 200 copper as a function of temperature is provided in Table 1 (Iwasa (1994)). Table 1: Thermal conductivity and electrical resistivity of OFHC copper. Temperature

Thermal conductivity

Electrical resistivity

500 K 400 K 300 K 250 K 200 K 150 K 125 K 100 K 90 K 80 K 70 K 60 K 55 K 50 K

4.31 W/cm-K 4.15 W/cm-K 3.99 W/cm-K 4.04 W/cm-K 4.11 W/cm-K 4.24 W/cm-K 4.34 W/cm-K 4.71 W/cm-K 4.98 W/cm-K 5.43 W/cm-K 6.25 W/cm-K 7.83 W/cm-K 9.11 W/cm-K 11.0 W/cm-K

3.19 μohm-cm 2.49 μohm-cm 1.73 μohm-cm 1.39 μohm-cm 1.06 μohm-cm 0.72 μohm-cm 0.54 μohm-cm 0.36 μohm-cm 0.29 μohm-cm 0.22 μohm-cm 0.15 μohm-cm 0.098 μohm-cm 0.076 μohm-cm 0.057 μohm-cm

a) Develop a numerical model in MATLAB that can predict the rate of heat transfer to the cryogenic experiment from the pair of current leads. The input conditions are entered in a MATLAB function EXAMPLE1p9 2.m; the two arguments to the function are diameter (the variable D) and number of nodes (the variable N) as we know that these parameters will be varied during the veriﬁcation

173

and optimization process. Any of the other parameters can be added in order to facilitate additional parametric studies or optimization.

function [ ]=EXAMPLE1p9 2(D,N) I=100; T H=20+273.2; T C=50; L=1; eps=0.5; sigma=5.67e-8;

%current (amp) %hot temperature (K) %cold temperature (K) %length of lead (m) %emissivity of lead surface (-) %Stefan-Boltzmann constant (W/mˆ2-Kˆ4)

Notice that we have not, to this point, speciﬁed what parameters are returned when the function executes (i.e., there are no variables listed between the square brackets in the function header). Functions are deﬁned (at the bottom of the M-ﬁle) that return the conductivity and electrical resistivity of the OFHC copper; the interp1 function is used to carry out interpolation on the data provided in Table 1 using a cubic spline technique.

%——–Property functions————function[k]=k cu(T) %returns the thermal conductivity (W/m-K) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) kd=[4.31,4.15,3.99,4.04,4.11,4.24,4.34,4.71,4.98,5.43,6.25,7.83,9.11,11.0]∗ 100; %conductivity data (W/m-K) k=interp1(Td,kd,T); end function[rho e]=rho e cu(T) %returns the electrical resistivity (ohm-m) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) rho ed=[3.19,2.49,1.73,1.39,1.06,0.72,0.54,0.36,0.29,0.22,0.15,0.098,0.076,0.057]/(1e6∗ 100); %electrical resistivity data (ohm-m) rho e=interp1(Td,rho ed,T); end

The ﬁrst step is to position the nodes throughout the computational domain. For this problem, the nodes will be distributed uniformly, as shown in Figure 2: xi =

(i − 1) L for i = 1..N (N − 1)

The distance between adjacent nodes (x) is: x =

L (N − 1)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

174

One-Dimensional, Steady-State Conduction

The MATLAB code that accomplishes these assignments is: %Position nodes for i=1:N x(i,1)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

%position of each node (m) %distance between adjacent nodes (m)

A control volume for an internal node is shown in Figure 2; the control volume experiences conduction heat transfer from the adjacent nodes above and below (q˙ t op and q˙ bottom , respectively), as well as radiation (q˙r ad ) and generation due to the ˙ An energy balance for the control ohmic dissipation associated with the current (g). volume is: q˙ t op + q˙ bottom + g˙ = q˙r ad

(1)

T1

q⋅ top q⋅ rad q⋅

bottom

Ti-1 T g⋅ i Ti+1

Figure 2: Control volume for an internal node and associated energy terms.

TN

The conductivity used to approximate the conduction heat transfer rates must be evaluated at the temperature of the boundaries in order to avoid energy balance violations, as discussed in Section 1.4.3. With this understanding, these rate equations become: q˙ t op = kT =(Ti +Ti−1 )/2

π D2 (Ti−1 − Ti ) 4 x

q˙ bottom = kT =(Ti +Ti+1 )/2

π D2 (Ti+1 − Ti ) 4 x

(2)

(3)

The rate of thermal energy generation is calculated using the resistivity evaluated at the temperature of each node: g˙ = ρe,T =Ti

4 x 2 I π D2

The rate of radiation heat transfer is approximately given by: q˙r ad = ε σ π D x Ti 4 − TH4

(4)

(5)

175

where ε is the emissivity of the surface of the leads and σ is the Stefan-Boltzmann constant. Substituting Eqs. (2) through (5) into Eq. (1) leads to: π D2 π D2 4 x 2 I (Ti−1 − Ti ) + kT =(Ti +Ti+1 )/2 (Ti+1 − Ti ) + ρe,T =Ti 4 x 4 x π D2 = ε σ π D x Ti 4 − TH4 for i = 2.. (N − 1) (6)

kT = (Ti +Ti−1 )/2

The remaining equations specify the boundary temperatures: T1 = TH

(7)

TN = TC

(8)

Equations (6) through (8) are a set of N equations in the N unknown temperatures; however, the temperature dependence of the material properties (k and ρe ) as well as the non-linear rate equation associated with radiation heat transfer cause the system of equations to be non-linear. Therefore, a relaxation technique will be employed in order to obtain the solution; the relaxation process is discussed in Section 1.5.6. The assumed solution (Tˆ ) will be successively substituted with the predicted solution. This process will continue until the assumed and predicted solutions agree to within an acceptable tolerance. The solution proceeds by assuming a temperature distribution that can be used to evaluate the coefﬁcients in the linearized equations. A linear temperature distribution provides a reasonable start for the iteration: (i − 1) Tˆi = TH + (TH − TC ) for i = 1..N (N − 1)

%Start relaxation with a linear temperature distribution for i=1:N Tg(i,1)=T H-(T H-T C)∗ (i-1)/(N-1); end

In order to solve this problem using MATLAB, we will need a set of linear equations; linear equations cannot contain products of the unknown temperatures with other unknown temperatures or functions of the unknown temperatures. Therefore, the temperature-dependent material properties must be evaluated at the assumed temperatures (Tˆ ): q˙ t op = kT =(Tˆi +Tˆi−1 )/2

π D2 (Ti−1 − Ti ) 4 x

q˙ bottom = kT =(Tˆi +Tˆi+1 )/2 g˙ = ρe,T =Tˆi

π D2 (Ti+1 − Ti ) 4 x

4 x 2 I π D2

(9)

(10) (11)

The fourth power temperature terms cause the radiation equation, Eq. (5), to be nonlinear. Therefore, it is necessary to linearize the radiation equation so that it can be placed in matrix format, as was done for the material properties. The radiation

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

176

One-Dimensional, Steady-State Conduction

terms can be linearized most conveniently using the same factorization that was previously used to deﬁne a radiation resistance in Section 1.2.6: (12) q˙r ad = σ ε π D x Tˆi 2 + TH2 (Tˆi + TH )(Ti − TH ) Substituting the linearized rate equations, Eqs. (9) through (12), into the energy balance for an internal node, Eq. (1), leads to: π D2 π D2 4 x 2 I (Ti−1 − Ti ) + kT =(Tˆi +Tˆi+1 )/2 (Ti+1 − Ti ) + ρe,T =Tˆi 4 x 4 x π D2 = σ ε π D x (Tˆi 2 + TH2 )(Tˆi + TH )(Ti − TH ) for i = 2.. (N − 1) (13)

kT =(Tˆi +Tˆi−1 )/2

Equations (7), (8), and (13) must be placed in matrix format: AX = b where X is a vector of unknown temperatures, A is a matrix containing the coefﬁcients of each equation, and b is a vector containing the constant terms for each equation. The matrix A is declared as sparse in MATLAB; note that Eq. (13) indicates that there are at most three nonzero entries in each row of A: %Setup A and b A=spalloc(N,N,3∗ N); b=zeros(N,1);

Equation (7) can be placed in row 1 of the matrix equation: T1 [1] = TH A1,1

(14)

b1

and Eq. (8) can be placed in row N of the matrix equation: TN [1] = TC AN ,N

(15)

bN

Equation (13) must be rearranged to make it clear which row and column each coefﬁcient should be entered into the matrix: 2 π D2 π D2 2 ˆ ˆ − kT =(Tˆi +Tˆi+1 )/2 − σ ε π D x Ti + TH Ti + TH Ti −kT =(Tˆi +Tˆi−1 )/2 4 x 4 x + Ti−1 kT =(Tˆi +Tˆi−1 )/2

2

πD 4 x

Ai,i

+Ti+1 kT =(Tˆi +Tˆi+1 )/2

Ai,i−1

π D2 4 x

(16)

Ai,i+1

4 x 2 = −σ ε π D x Tˆi 2 + TH2 Tˆi + TH TH − ρe,T =Tˆi I π D2

for i = 2.. (N − 1)

bi

The numerical solution is placed within a while loop that checks for convergence of the relaxation scheme. The variable err is used to terminate the while loop and represents the average, absolute error between the assumed and predicted temperature distribution. (There are other criteria that could be used, but this is sufﬁcient for most problems.) The while loop is terminated when the variable err decreases to less than the input parameter tol, which represents the convergence

177

tolerance for the relaxation process. Initially, the value of err is set to a value greater than tol to ensure that the while loop executes at least one time. err=999; tol=0.1; while(err>tol)

%error that terminates the while loop (K) %criteria for terminating the while loop (K)

end end

Within the while loop, the matrix is ﬁlled in using the coefﬁcients suggested by Eq. (14), %specify the hot end temperature A(1,1)=1; b(1,1)=T h;

Eq. (15), %specify the cold end temperature A(N,N)=1; b(N,1)=T C;

and Eq. (16). %internal nodes for i=2:(N-1) A(i,i)=-k cu((Tg(i+1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx)- . . . k cu((Tg(i-1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx)-. . . sigma*eps*pi*D*DELTAx*(Tg(i,1)ˆ2+T Hˆ2)*(Tg(i,1)+T H); A(i,i-1)=k cu((Tg(i-1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx); A(i,i+1)=k cu((Tg(i+1,1)+Tg(i,1))/2)*pi*Dˆ2/(4*DELTAx); b(i,1)=-rho e cu(Tg(i,1))*4*DELTAx*Iˆ2/(pi*Dˆ2)- . . . sigma*eps*pi*D*DELTAx*(Tg(i,1)ˆ2+T Hˆ2)*(Tg(i,1)+T H)*T H; end

Note that the three periods in the above code is a line break; it indicates that the code is continued on the subsequent line. The matrix equation is solved and the error between the assumed and predicted temperature is computed. err =

N 1 |Ti − Tˆi | N i=1

The ﬁnal step in the while loop is to update the assumed temperature distribution with the predicted temperature distribution. T=full(A/b); err=sum(abs(T-Tg))/N Tg=T;

%compute the error %update the guess temperature array

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction

Note that the full command in the above code converts the sparse matrix, T, that results from the operation on the sparse matrix A into a full matrix. The header of the function is modiﬁed to specify the output arguments, x and T: function[x,T]=EXAMPLE1p9 2(D,N)

Because the statement that calculates the variable err is not terminated with a semicolon, the result of the calculation will be echoed in the workspace allowing you to keep track of the progress. If you call this program with a diameter of 5.0 mm you should see: >> [x,T]=EXAMPLE1p9 2(0.005,100); err = 41.2789 err = 16.1165 err = 6.2792 err = 2.4864 err = 1.0023 err = 0.4110 err = 0.1685 err = 0.0693

The relaxation process had to iterate several times in order to converge due to the nonlinearity of the problem. Figure 3 illustrates the temperature distribution in the current lead for several different values of the diameter. 500 450 D=4.0 mm

400 Temperature (K)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

178

350 D=4.5 mm 300

D=5.0 mm

250

D=6.0 mm

200

D=8.0 mm D=10.0 mm

150 100 50 0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Axial position (m)

0.8

0.9

1

Figure 3: Temperature distribution in the current lead for various values of the diameter.

179

Notice that the smaller diameter leads result in large amounts of ohmic dissipation and therefore the wire tends to become hotter and the temperature gradient at the cold end increases. For a given temperature gradient, larger diameter leads will result in a higher rate of heat transfer to the cold end due to the larger area for conduction. There is a balance between these effects that results in an optimal diameter. The heat transferred to the cold end (q˙ c ) is calculated using an energy balance on node N: q˙ c = kT =(TN +TN −1 )/2

π D2 2 x 2 x 4 I −σ επ D TH − TN4 (TN −1 − TN ) + ρe,T =TN 2 4 x πD 2

or, in MATLAB: q dot c=k cu((T(N)+T(N-1))/2)∗ pi∗ Dˆ2∗ (T(N-1)-T(N))/(4∗ DELTAx)+... rho e cu(T(N))∗ 2∗ DELTAx∗ Iˆ2/(pi∗ Dˆ2)-sigma∗ eps∗ pi∗ D∗ DELTAx∗ (T Hˆ4-T(N)ˆ4)/2; %heat transfer to cold end

The function header is modiﬁed so that q˙ c is also returned:

function[q dot c,x,T]=EXAMPLE1p9 2(D,N)

It is necessary to verify that the solution has a sufﬁcient number of nodes and, if possible, verify the result against an analytical solution. The critical parameter for the solution is the rate of heat transfer to the experiment per current lead; therefore Figure 4 illustrates q˙ c as a function of the number of nodes in the solution, N. The information shown in Figure 4 was generated quickly using the script varyN (below), which calls the function EXAMPLE1p9_2 multiple times with varying values of N:

%Script varyN.m clear all; D=0.005; N=[2,5,10,20,50,100,200,500,1000,2000]’; for i=1:10 i [q dot c(i,1),x,T]=EXAMPLE1p9 2(D,N(i)); end

The clear all statement at the beginning of the script clears all variables from memory. Using the clear all statement is often a good idea as it prevents previous elements (e.g., from previous runs) of the variables q_dot_c or N from being retained. If you had previously run the script varyN with more than 10 runs, then N and q_dot_c would exist in memory with more than 10 elements. Running the script varyN as shown above would overwrite the ﬁrst 10 elements of these variables, but leave all subsequent elements which could lead to confusion.

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction 6

5.5 Heat transfer to cold end (W)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

180

5 4.5 4 3.5 3 2.5 2 1.5 10 0

10 1

10 2 Number of nodes

10 3

Figure 4: Heat transferred to the cold end per lead as a function of the number of nodes for a 5.0 mm lead.

Figure 4 suggests that at least 100 nodes should be used for sufﬁcient accuracy. In the absence of any radiation heat transfer (ε = 0) and with constant resistivity and conductivity, it is possible to compare the numerical solution to the analytical solution for the temperature in a generating wall with ﬁxed end conditions. This result was derived in Section 1.3.2 and is repeated below: g˙ L 2 x x 2 (TH − TC ) − x + TH − T = 2k L L L where the volumetric generation is given by: g˙ =

16 I 2 ρe π 2 D4

%constant property analytical solution g dot vol=16∗ Iˆ2∗ rho e cu(T H)/(piˆ2∗ Dˆ4); for i=1:N T an(i,1)=g dot vol∗ Lˆ2∗ ((x(i)/L)-(x(i)/L)ˆ2)/(2∗ k cu(T H))- . . . (T H-T C)∗ x(i)/L+T H; end

The property functions in MATLAB are modiﬁed to return, temporarily, constant values of k = 200 W/m-K and ρe = 1 × 10−8 ohm-m. %—-Property functions————function[k]=k cu(T) %returns the thermal conductivity (W/m-K) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) kd=[4.31,4.15,3.99,4.04,4.11,4.24,4.34,4.71,4.98,5.43,6.25,7.83,9.11,11.0]*100; %conductivity data (W/m-K) %k=interp1(Td,kd,T); k=200; end

181

function[rho e]=rho e cu(T) %returns the electrical resistivity (ohm-m) given temperature (K) Td=[500,400,300,250,200,150,125,100,90,80,70,60,55,50]; %temperature data (K) rho ed=[3.19,2.49,1.73,1.39,1.06,0.72,0.54,0.36,0.29,0.22,0.15,0.098,0.076,0.057]/(1e6*100); %electrical resistivity data (ohm-m) %rho e=interp1(Td,rho ed,T); rho e=1e-8; end

The emissivity is set to 0 and the MATLAB code is run for a 5.0 mm diameter lead. The temperature distribution predicted by the MATLAB code is compared with the analytical solution in Figure 5. Note that with these modiﬁcations (i.e., constant k and ρe and ε = 0), the problem becomes linear and therefore a single iteration is required in order to reduce the relaxation error to 0. 400 350

Temperature (K)

300 250 200 150 100 50 0

numerical solution analytical solution 0.1

0.2

0.3

0.4 0.5 0.6 0.7 Axial position (m)

0.8

0.9

1

Figure 5: Comparison of the analytical and numerical solutions in the limit that k = 200 W/m-K (constant), ρe = 1e-8 ohm-m (constant) and ε = 0 for a 5.0 mm diameter wire.

Finally, it is possible to parametrically vary the wire diameter, D, in order to minimize the heat ﬂow to the cold end of the current lead. The code is returned to its original, non-linear form. A script (varyd) is used to call the function multiple times with various diameters in order to carry out a parametric study of this parameter:

%Script varyd.m clear all; N=100; D=linspace(0.0038,0.01,100)’; %generate 100 values of D between 0.0038 and 0.01 [m] for i=1:100 [q dot c(i,1),x,T]=EXAMPLE1p9 2(D(i),N); end

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

One-Dimensional, Steady-State Conduction

Figure 6 illustrates the heat leak to the cold end as a function of wire diameter (note that the functions for k and ρe were reset and the value of ε was reset to 0.50) and shows that there is a clear optimal diameter around 5.1 mm for this application. Smaller values of D lead to excessive self-heating whereas larger values provide a large path for conduction heat transfer. 13 12 Heat leak per lead (W)

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

182

11 10 9 8

optimal lead diameter

7 6 5 0.003

0.004

0.005

0.006 0.007 Diameter (m)

0.008

0.009

0.01

Figure 6: Heat leak to the cold end of each current lead as a function of diameter.

MATLAB has powerful, built-in optimization algorithms that allow you to automate the process of determining the optimal diameter. The MATLAB function fminbnd is the simplest available and carries out a bounded, 1-D minimization. The function fminbnd is called according to: x opt=fminbnd(function,x1,x2)

where function is the name of a function that requires a single argument and provides a single output (that should be minimized) and x1 and x2 are the lower and upper bounds of the argument to use for the minimization. First, it is necessary to modify the function EXAMPLE1p9_2 so that it takes a single argument (D) and returns a single output (q_dot_c):

function[q dot c]=EXAMPLE1p9 2(D) N=100;

%number of nodes (-)

Then the fminbnd function can be called directly from the workspace:

>> D opt=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01);

in order to identify the optimal diameter.

183

>> D opt D opt = 0.0051

Note that the calculation of the variable err in the function EXAMPLE1p9_2 is terminated with a semicolon so that the error is not echoed to the workspace during each iteration. The function fminbnd will return the optimized value of the heat leak as well by adding an additional output argument to the fminbnd call: >>[D opt,q dot c min]=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01);

which indicates that the optimal value of the heat leak is 5.57 W. >> q dot c min q dot c min = 5.6815

It is possible to control the details of the optimization using an optional fourth input argument to the function fminbnd that sets the optimization parameters; the easiest way to set this last argument is using the optimset command. If you enter >> help optimset

into the workspace then a complete list of the parameters that can be controlled is returned. It is possible, for example, to display the progress of the optimization using:

[D opt,q dot c min]=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01,optimset(‘Display’,‘iter )) Func-count 1 2 3 4 5 6 7 8

x 0.0062918 0.0077082 0.00541641 0.0043798 0.00523819 0.00515231 0.00511897 0.00508564

f(x) 6.35363 8.12533 5.73753 6.14038 5.68985 5.6824 5.68148 5.68212

Procedure initial golden golden parabolic parabolic parabolic parabolic parabolic

Optimization terminated: the current x satisﬁes the termination criteria using OPTIONS.TolX of 1.000000e-004 D opt = 0.0051 q dot c min = 5.6815

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

1.9 Numerical Solution to Extended Surface Problems

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

184

One-Dimensional, Steady-State Conduction

The ﬁrst argument to optimset speciﬁes the parameter to be controlled (‘Display’, which controls the level of display) and the second indicates its new value (‘iter’, which indicates that the progress should be displayed after each iteration). It would be inconvenient to use the fminbnd function to carry out a parametric variation of how the optimal value of the variables D and q_dot_c are affected by current or some other parameter. In its current format, it is not possible to pass the value of the current (I) to the fminbnd function and therefore the study would have to be carried out manually by running the fminbnd function and then changing the value of the variable I within the function EXAMPLE1p9_2. This process would become tedious and can be avoided by parameterizing the function. For example, suppose you want to determine how the optimal value of diameter changes with current. First, include current as an additional argument to the function: function[q dot c]=EXAMPLE1p9_2(D,I) N=100; % I=100;

%number of nodes (-) %current (amp)

If you try to repeat the optimization using the previous protocol you will receive an error: >> [D opt,q dot c min]=fminbnd(‘EXAMPLE1p9 2’,0.004,0.01) ??? Input argument “I” is undeﬁned. Error in ==> EXAMPLE9 2 at 42 b(i,1)=-rho e cu(Tg(i,1))∗ 4∗ DELTAx∗ Iˆ2/(pi∗ Dˆ2)-. . . Error in ==> fminbnd at 182 x=xf; fx=funfcn(x,varargin{:});

However, you can parameterize the function using a one-argument anonymous function that captures the value of I (set in the workspace) and calls EXAMPLE1p9_2 with two arguments: >> I=100; >> [D opt,q dot c min]=fminbnd(@(D) EXAMPLE1p9 2(D,I),0.004,0.01) D opt = 0.0051 q dot c min = 5.6815

Now it is possible to generate a script, varyI, that evaluates the optimized diameter and heat ﬂow as a function of current. %Script varyI.m clear all; I=linspace(1,100,10)’; for i=1:10 [d opt(i,1),q dot min(i,1)]=fminbnd(@(d) EXAMPLE1p9 2(d,I(i,1)),0.0025,0.01) end

185

Figure 7 illustrates the optimal diameter and the associated heat leak as a function of current. Note that the optimal diameter and minimized heat leak are both approximately linear functions of the current. 6

0.0055

5.5

0.005

5

0.0045

4.5 minimum heat leak

0.004

4

0.0035 0.003 50

3.5

optimal diameter

55

60

65

70 75 80 85 Current (ampere)

90

95

Minimum heat leak per lead (W)

Optimal current lead diameter (m)

0.006

3 100

Figure 7: Optimal diameter and minimized heat leak to cold end of each current lead as a function of current.

Chapter 1: One-Dimensional, Steady-State Conduction The website associated with this book (www.cambridge.org/nellisandklein) provides many more problems than are included here. Conduction Heat Transfer 1–1 Section 1.1.2 provides an approximation for the thermal conductivity of a monatomic gas at ideal gas conditions. Test the validity of this approximation by comparing the conductivity estimated using Eq. (1-18) to the value of thermal conductivity for a monotonic ideal gas (e.g., low pressure argon) provided by the internal function in EES. Note that the molecular radius, σ, is provided in EES by the Lennard-Jones potential using the function sigma_LJ. a.) What are the value and units of the proportionality constant required to make Eq. (1-18) an equality? b.) Plot the value of the proportionality constant for 300 K argon at pressures between 0.01 and 100 MPa on a semi-log plot with pressure on the log scale. At what pressure does the approximation given in Eq. (1-18) begin to fail? Steady-State 1-D Conduction without Generation 1–2 Figure P1-2 illustrates a plane wall made of a thin (thw = 0.001 m) and conductive (k = 100 W/m-K) material that separates two ﬂuids. Fluid A is at TA = 100◦ C and the heat transfer coefﬁcient between the ﬂuid and the wall is hA= 10 W/m2 -K while ﬂuid B is at T B = 0◦ C with hB = 100 W/m2 -K.

EXAMPLE 1.9-2: CRYOGENIC CURRENT LEADS

Chapter 1: One-Dimensional, Steady-State Conduction

186

One-Dimensional, Steady-State Conduction thw = 0.001 m TA = 100° C

TB = 0°C

hA = 10 W/m2 -K

hB = 100 W/m2 -K k = 100 W/m-K

Figure P1-2: Plane wall separating two ﬂuids.

a.) Draw a resistance network that represents this situation and calculate the value of each resistor (assuming a unit area for the wall, A = 1 m2 ). b.) If you wanted to predict the heat transfer rate from ﬂuid A to ﬂuid B very accurately then which parameters (e.g., thw , k, etc.) would you try to understand/measure very carefully and which parameters are not very important? Justify your answer. 1–3 You have a problem with your house. Every spring at some point the snow immediately adjacent to your roof melts and runs along the roof line until it reaches the gutter. The water in the gutter is exposed to air at temperature less than 0◦ C and therefore freezes, blocking the gutter and causing water to run into your attic. The situation is shown in Figure P1-3. snow melts at this surface Tout , hout = 15 W/m2 -K Ls = 2.5 inch

snow, ks = 0.08 W/m-K insulation, kins = 0.05 W/m-K Tin = 22°C, hin = 10 W/m2 -K Lins = 3 inch plywood, L p = 0.5 inch, k p = 0.2 W/m-K

Figure P1-3: Roof of your house.

The air in the attic is at T in = 22◦ C and the heat transfer coefﬁcient between the inside air and the inner surface of the roof is hin = 10 W/m2 -K. The roof is composed of a Lins = 3.0 inch thick piece of insulation with conductivity kins = 0.05 W/m-K that is sandwiched between two L p = 0.5 inch thick pieces of plywood with conductivity k p = 0.2 W/m-K. There is an Ls = 2.5 inch thick layer of snow on the roof with conductivity ks = 0.08 W/m-K. The heat transfer coefﬁcient between the outside air at temperature Tout and the surface of the snow is hout = 15 W/m2 -K. Neglect radiation and contact resistances for part (a) of this problem. a.) What is the range of outdoor air temperatures where you should be concerned that your gutters will become blocked by ice? b.) Would your answer change much if you considered radiation from the outside surface of the snow to surroundings at Tout ? Assume that the emissivity of snow is εs = 0.82. 1–4 Figure P1-4(a) illustrates a composite wall. The wall is composed of two materials (A with kA = 1 W/m-K and B with kB = 5 W/m-K), each has thickness L =

Chapter 1: One-Dimensional, Steady-State Conduction

187

1.0 cm. The surface of the wall at x = 0 is perfectly insulated. A very thin heater is placed between the insulation and material A; the heating element provides q˙ = 5000 W/m2 of heat. The surface of the wall at x = 2L is exposed to ﬂuid at T f ,in = 300 K with heat transfer coefﬁcient hin = 100 W/m2 -K. q⋅ ′′ = 5000 W/m 2

material A kA = 1 W/m-K

insulated

L = 1 cm

Tf, in = 300 K hin = 100 W/m2 -K

x

L = 1 cm

material B kB = 5 W/m-K

Figure P1-4 (a): Composite wall with a heater.

You may neglect radiation and contact resistance for parts (a) through (c) of this problem. a.) Draw a resistance network to represent this problem; clearly indicate what each resistance represents and calculate the value of each resistance. b.) Use your resistance network from (a) to determine the temperature of the heating element. c.) Sketch the temperature distribution through the wall. Make sure that the sketch is consistent with your solution from (b). Figure P1-4(b) illustrates the same composite wall shown in Figure P1-4(a), but there is an additional layer added to the wall, material C with kC = 2.0 W/m-K and L = 1.0 cm. material C kC = 2 W/m-K

q⋅ ′′ = 5000 W/m 2

material A kA = 1 W/m-K L = 1 cm

insulated x

L = 1 cm

Tf, in = 300 K hin = 100 W/m2 -K material B L = 1 cm kB = 5 W/m-K

Figure P1-4 (b): Composite wall with material C.

Neglect radiation and contact resistance for parts (d) through (f) of this problem. d.) Draw a resistance network to represent the problem shown in Figure P1-4(b); clearly indicate what each resistance represents and calculate the value of each resistance. e.) Use your resistance network from (d) to determine the temperature of the heating element. f.) Sketch the temperature distribution through the wall. Make sure that the sketch is consistent with your solution from (e).

188

One-Dimensional, Steady-State Conduction

Figure P1-4(c) illustrates the same composite wall shown in Figure P1-4(b), but there is a contact resistance between materials A and B, Rc = 0.01 K-m2 /W, and the surface of the wall at x = −L is exposed to ﬂuid at T f ,out = 400 K with a heat transfer coefﬁcient hout = 10 W/m2 -K. material C kC = 2 W/m-K Tf, out = 400 K hout = 10 W/m2 -K

q⋅ ′′ = 5000 W/m2

material A kA = 1 W/m-K L = 1 cm

x

Tf, in = 300 K hin = 100 W/m2 -K material B kB = 5 W/m-K

L = 1 cm L = 1 cm

R′′c = 0.01 K-m2 /W

Figure P1-4 (c): Composite wall with convection at the outer surface and contact resistance.

Neglect radiation for parts (g) through (i) of this problem. g.) Draw a resistance network to represent the problem shown in Figure P1-4(c); clearly indicate what each resistance represents and calculate the value of each resistance. h.) Use your resistance network from (g) to determine the temperature of the heating element. i.) Sketch the temperature distribution through the wall. 1–5 You have decided to install a strip heater under the linoleum in your bathroom in order to keep your feet warm on cold winter mornings. Figure P1-5 illustrates a cross-section of the bathroom ﬂoor. The bathroom is located on the ﬁrst story of your house and is W = 2.5 m wide × L = 2.5 m long. The linoleum thickness is thL = 5.0 mm and has conductivity kL = 0.05 W/m-K. The strip heater under the linoleum is negligibly thin. Beneath the heater is a piece of plywood with thickness thP = 5 mm and conductivity kP = 0.4 W/m-K. The plywood is supported by ths = 6.0 cm thick studs that are Ws = 4.0 cm wide with thermal conductivity ks = 0.4 W/m-K. The center-to-center distance between studs is ps = 25.0 cm. Between each stud are pockets of air that can be considered to be stagnant with conductivity ka = 0.025 W/m-K. A sheet of drywall is nailed to the bottom of the studs. The thickness of the drywall is thd = 9.0 mm and the conductivity of drywall is kd = 0.1 W/m-K. The air above in the bathroom is at T air,1 = 15◦ C while the air in the basement is at T air,2 = 5◦ C. The heat transfer coefﬁcient on both sides of the ﬂoor is h = 15 W/m2 -K. You may neglect radiation and contact resistance for this problem. a.) Draw a thermal resistance network that can be used to represent this situation. Be sure to label the temperatures of the air above and below the ﬂoor (Tair,1 and Tair,2 ), the temperature at the surface of the linoleum (TL ), the temperature of the strip heater (Th ), and the heat input to the strip heater (q˙ h ) on your diagram. b.) Compute the value of each of the resistances from part (a). c.) How much heat must be added by the heater to raise the temperature of the ﬂoor to a comfortable 20◦ C? d.) What physical quantities are most important to your analysis? What physical quantities are unimportant to your analysis?

Chapter 1: One-Dimensional, Steady-State Conduction Tair,1 = 15°C, h = 15 W/m2 -K

linoleum, kL = 0.05 W/m-K plywood, kp = 0.4 W/m-K

strip heater thp = 5 mm

189

thL = 5 mm

ps = 25 cm

thd = 9 mm

ths = 6 cm Ws = 4 cm studs, ks = 0.4 W/m-K drywall, k d = 0.1 W/m-K air, ka = 0.025 W/m-K

Tair, 2 = 5°C, h = 15 W/m2 -K

Figure P1-5: Bathroom ﬂoor with heater.

e.) Discuss at least one technique that could be used to substantially reduce the amount of heater power required while still maintaining the ﬂoor at 20◦ C. Note that you have no control over T air,1 or h. 1–6 You are a fan of ice ﬁshing but don’t enjoy the process of augering out your ﬁshing hole in the ice. Therefore, you want to build a device, the super ice-auger, that melts a hole in the ice. The device is shown in Figure P1-6.

h = 50 W/m 2 -K T∞ = 5 ° C

ε = 0.9 insulation, kins = 2.2 W/m-K

heater, activated with V = 12 V and I = 150 A plate, kp = 10 W/m-K D = 10 inch

thp = 0.75 inch

thins = 0.5 inch thice = 5 inch ρice = 920 kg/m3 Δifus = 333.6 kJ/kg

Figure P1-6: The super ice-auger.

A heater is attached to the back of a D = 10 inch plate and electrically activated by your truck battery, which is capable of providing V = 12 V and I = 150 A. The plate is thp = 0.75 inch thick and has conductivity k p = 10 W/m-K. The back of the heater is insulated; the thickness of the insulation is thins = 0.5 inch and the insulation has conductivity kins = 2.2 W/m-K. The surface of the insulation experiences convection with surrounding air at T ∞ = 5◦ C and radiation with surroundings also at T ∞ = 5◦ C. The emissivity of the surface of the insulation is ε = 0.9 and the heat transfer coefﬁcient between the surface and the air is h = 50 W/m2 -K. The super ice-auger is placed on the ice and activated, causing a heat transfer to the plate-ice interface that melts the ice. Assume that the water under the ice is at T ice = 0◦ C so that no heat is conducted away from the plate-ice interface; all of the energy transferred to the plate-ice interface goes into melting the ice. The thickness of the ice is

190

One-Dimensional, Steady-State Conduction

thice = 5 inch and the ice has density ρice = 920 kg/m3 . The latent heat of fusion for the ice is if us = 333.6 kJ/kg. a.) Determine the heat transfer rate to the plate-ice interface. b.) How long will it take to melt a hole in the ice? c.) What is the efﬁciency of the melting process? d.) If your battery is rated at 100 amp-hr at 12 V then what fraction of the battery’s charge is depleted by running the super ice-auger. Steady-State 1-D Conduction with Generation 1–7 One of the engineers that you supervise has been asked to simulate the heat transfer problem shown in Figure P1-7(a). This is a 1-D, plane wall problem (i.e., the temperature varies only in the x-direction and the area for conduction is constant with x). Material A (from 0 < x < L) has conductivity kA and experiences a uniform rate of volumetric thermal energy generation, g˙ . The left side of material A (at x = 0) is completely insulated. Material B (from L < x < 2L) has lower conductivity, kB < kA. The right side of material B (at x = 2L) experiences convection with ﬂuid at room temperature (20◦ C). Based on the facts above, critically examine the solution that has been provided to you by the engineer and is shown in Figure P1-7(b). There should be a few characteristics of the solution that do not agree with your knowledge of heat transfer; list as many of these characteristics as you can identify and provide a clear reason why you think the engineer’s solution must be wrong. 250

L

L

material A

material B kB < kA g⋅ ′′′= 0

kA = g⋅′′′ g⋅′′′ A

B

x

(a)

Temperature (°C)

200

h , Tf = 20°C

150 100 50 0 -50 Material A -100

0

Material B L Position (m)

2L

(b)

Figure P1-7 (a): Heat transfer problem and (b) “solution” provided by the engineer.

1–8 Freshly cut hay is not really dead; chemical reactions continue in the plant cells and therefore a small amount of heat is released within the hay bale. This is an example of the conversion of chemical to thermal energy. The amount of thermal energy generation within a hay bale depends on the moisture content of the hay when it is baled. Baled hay can become a ﬁre hazard if the rate of volumetric generation is sufﬁciently high and the hay bale sufﬁciently large so that the interior temperature of the bale reaches 170◦ F, the temperature at which self-ignition can occur. Here, we will model a round hay bale that is wrapped in plastic to protect it from the rain. You may assume that the bale is at steady state and is sufﬁciently long that it can be treated as a one-dimensional, radial conduction problem. The radius of the hay bale is Rbale = 5 ft and the bale is wrapped in plastic that is t p = 0.045 inch thick with conductivity k p = 0.15 W/m-K. The bale is surrounded by air at T ∞ = 20◦ C with h = 10 W/m2 -K. You may neglect radiation. The conductivity of the hay is k = 0.04 W/m-K.

Chapter 1: One-Dimensional, Steady-State Conduction

191

a.) If the volumetric rate of thermal energy generation is constant and equal to g˙ = 2 W/m3 then determine the maximum temperature in the hay bale. b.) Prepare a plot showing the maximum temperature in the hay bale as a function of the hay bale radius. How large can the hay bale be before there is a problem with self-ignition? Prepare a model that can consider temperature-dependent volumetric generation. Increasing temperature tends to increase the rate of chemical reaction and therefore increases the rate of generation of thermal energy according to: g˙ = a + b T where a = −1 W/m3 and b = 0.01 W/m3 -K and T is in K. c.) Enter the governing equation into Maple and obtain the general solution (i.e., a solution that includes two constants). d.) Use the boundary conditions to obtain values for the two constants in your general solution. (hint: one of the two constants must be zero in order to keep the temperature at the center of the hay bale ﬁnite). You should obtain a symbolic expression for the boundary condition in Maple that can be evaluated in EES. e.) Overlay on your plot from part (b) a plot of the maximum temperature in the hay bale as a function of bale radius when the volumetric generation is a function of temperature. 1–9 Figure P1-9 illustrates a simple mass ﬂow meter for use in an industrial reﬁnery. T∞ = 20°C hout = 20 W/m2 -K rout = 1 inch rin = 0.75 inch

insulation kins = 1.5 W/m-K test section g⋅ ′′′= 1x107 W/m3 k = 10 W/m-K m⋅ = 0.75kg/s Tf = 18°C

L = 3 inch

thins = 0.25 inch

Figure P1-9: A simple mass ﬂow meter.

A ﬂow of liquid passes through a test section consisting of an L = 3 inch section of pipe with inner and outer radii, rin = 0.75 inch and rout = 1.0 inch, respectively. The test section is uniformly heated by electrical dissipation at a rate g˙ = 1×107 W/m3 and has conductivity k = 10 W/m-K. The pipe is surrounded with insulation that is thins = 0.25 inch thick and has conductivity kins = 1.5 W/m-K. The external surface of the insulation experiences convection with air at T ∞ = 20◦ C. The heat transfer coefﬁcient on the external surface is hout = 20 W/m2 -K. A thermocouple is embedded at the center of the pipe wall. By measuring the temperature of the thermocouple, it is possible to infer the mass ﬂow rate of ﬂuid because the heat transfer coefﬁcient on the inner surface of the pipe (hin ) is strongly related to mass ﬂow rate (m). ˙ Testing has shown that the heat transfer coefﬁcient and mass ﬂow rate are related according to: hin = C

m ˙ 1 [kg/s]

0.8

where C = 2500 W/m2 -K. Under nominal conditions, the mass ﬂow rate through the meter is m ˙ = 0.75 kg/s and the ﬂuid temperature is T f = 18◦ C. Assume that the

192

One-Dimensional, Steady-State Conduction

ends of the test section are insulated so that the problem is 1-D. Neglect radiation and assume that the problem is steady state. a.) Develop an analytical model in EES that can predict the temperature distribution in the test section. Plot the temperature as a function of radial position for the nominal conditions. b.) Using your model, develop a calibration curve for the meter; that is, prepare a plot of the mass ﬂow rate as a function of the measured temperature at the mid-point of the pipe. The range of the instrument is 0.2 kg/s to 2.0 kg/s. The meter must be robust to changes in the ﬂuid temperature. That is, the calibration curve developed in (b) must continue to be valid even as the ﬂuid temperature changes by as much as 10◦ C. c.) Overlay on your plot from (b) the mass ﬂow rate as a function of the measured temperature for T f = 8◦ C and T f = 28◦ C. Is your meter robust to changes in Tf ? In order to improve the meters ability to operate over a range of ﬂuid temperature, a temperature sensor is installed in the ﬂuid in order to measure Tf during operation. d.) Using your model, develop a calibration curve for the meter in terms of the mass ﬂow rate as a function of T , the difference between the measured temperatures at the mid-point of the pipe wall and the ﬂuid. e.) Overlay on your plot from (d) the mass ﬂow rate as a function of the difference between the measured temperatures at the mid-point of the pipe wall and the ﬂuid if the ﬂuid temperature is T f = 8◦ C and T f = 28◦ C. Is the meter robust to changes in Tf ? f.) If you can measure the temperature difference to within δ T = 1 K then what is the uncertainty in the mass ﬂow rate measurement? (Use your plot from part (d) to answer this question.) g.) Set the temperature difference to the value you calculated at the nominal conditions and allow EES to calculate the associated mass ﬂow rate. Now, select Uncertainty Propagation from the Calculate menu and specify that the mass ﬂow rate as the calculated variable while the temperature difference is the measured variable. Set the uncertainty in the temperature difference to 1 K and verify that EES obtains an answer that is approximately consistent with part (f). h.) The nice thing about using EES to determine the uncertainty is that it becomes easy to assess the impact of multiple sources of uncertainty. In addition to the uncertainty δ T , the constant C has relative uncertainty of δC = 5% and the conductivity of the material is only known to within δk = 3%. Use EES’ builtin uncertainty propagation to assess the resulting uncertainty in the mass ﬂow rate measurement. Which source of uncertainty is the most important? i.) The meter must be used in areas where the ambient temperature and heat transfer coefﬁcient may vary substantially. Prepare a plot showing the mass ﬂow rate predicted by your model for T = 50 K as a function of T ∞ for various values of hout . If the operating range of your meter must include −5◦ C < T ∞ < 35◦ C then use your plot to determine the range of hout that can be tolerated without substantial loss of accuracy. Numerical Solutions to Steady-State 1-D Conduction Problems using EES 1–10 Reconsider the mass ﬂow meter that was investigated in Problem 1-9. The conductivity of the material that is used to make the test section is not actually constant,

Chapter 1: One-Dimensional, Steady-State Conduction

193

as was assumed in Problem 1-9, but rather depends on temperature according to:

k = 10

W W + 0.035 (T − 300 [K]) m-K m-K2

a.) Develop a numerical model of the mass ﬂow meter using EES. Plot the temperature as a function of radial position for the conditions shown in Figure P1-9 with the temperature-dependent conductivity. b.) Verify that your numerical solution limits to the analytical solution from Problem 1-9 in the limit that the conductivity is constant. c.) What effect does the temperature dependent conductivity have on the calibration curve that you generated in part (d) of Problem 1-9. Numerical Solutions to Steady-State 1-D Conduction Problems using MATLAB 1–11 Reconsider Problem 1-8, but obtain a solution numerically using MATLAB. The description of the hay bale is provided in Problem 1-8. Prepare a model that can consider the effect of temperature on the volumetric generation. Increasing temperature tends to increase the rate of reaction and therefore increase the rate of generation of thermal energy; the volumetric rate of generation can be approximated by: g˙ = a + b T where a = −1 W/m3 and b = 0.01 W/m3 -K and T is in K. a.) Prepare a numerical model of the hay bale. Plot the temperature as a function of position within the hay bale. b.) Show that your model has numerically converged; that is, show some aspect of your solution as a function of the number of nodes and discuss an appropriate number of nodes to use. c.) Verify your numerical model by comparing your answer to an analytical solution in some, appropriate limit. The result of this step should be a plot that shows the temperature as a function of radius predicted by both your numerical solution and the analytical solution and demonstrates that they agree. 1–12 Reconsider the mass ﬂow meter that was investigated in Problem 1-9. Assume that the conductivity of the material that is used to make the test section is not actually constant, as was assumed in Problem 1-9, but rather depends on temperature according to:

k = 10

W W + 0.035 (T − 300 [K]) m-K m-K2

a.) Develop a numerical model of the mass ﬂow meter using MATLAB. Plot the temperature as a function of radial position for the conditions shown in Figure P1-9 with the temperature-dependent conductivity. b.) Verify that your numerical solution limits to the analytical solution from Problem 1-9 in the limit that the conductivity is constant.

194

One-Dimensional, Steady-State Conduction

Analytical Solutions for Constant Cross-Section Extended Surfaces 1–13 A resistance temperature detector (RTD) utilizes a material that has a resistivity that is a strong function of temperature. The temperature of the RTD is inferred by measuring its electrical resistance. Figure P1-13 shows an RTD that is mounted at the end of a metal rod and inserted into a pipe in order to measure the temperature of a ﬂowing liquid. The RTD is monitored by passing a known current through it and measuring the voltage across it. This process results in a constant amount of ohmic heating that may tend to cause the RTD temperature to rise relative to the temperature of the surrounding liquid; this effect is referred to as a selfheating error. Also, conduction from the wall of the pipe to the temperature sensor through the metal rod can result in a temperature difference between the RTD and the liquid; this effect is referred to as a mounting error.

Tw = 20°C

L = 5.0 cm

pipe

D = 0.5 mm h = 150 W/m2 -K

x

T∞ = 5.0°C

k = 10 W/m-K q⋅sh = 2.5 mW RTD

Figure P1-13: Temperature sensor mounted in a ﬂowing liquid.

The thermal energy generation associated with ohmic heating is q˙ sh = 2.5 mW. All of this ohmic heating is assumed to be transferred from the RTD into the end of the rod at x = L. The rod has a thermal conductivity k = 10 W/m-K, diameter D = 0.5 mm, and length L = 5.0 cm. The end of the rod that is connected to the pipe wall (at x = 0) is maintained at a temperature of T w = 20◦ C. The liquid is at a uniform temperature, T ∞ = 50◦ C and the heat transfer coefﬁcient between the liquid and the rod is h = 150 W/m2 -K. a.) Is it appropriate to treat the rod as an extended surface (i.e., can we assume that the temperature in the rod is a function only of x)? Justify your answer. b.) Develop an analytical model of the rod that will predict the temperature distribution in the rod and therefore the error in the temperature measurement; this error is the difference between the temperature at the tip of the rod and the liquid. c.) Prepare a plot of the temperature as a function of position and compute the temperature error. d.) Investigate the effect of thermal conductivity on the temperature measurement error. Identify the optimal thermal conductivity and explain why an optimal thermal conductivity exists. 1–14 Your company has developed a micro-end milling process that allows you to easily fabricate an array of very small ﬁns in order to make heat sinks for various types of electrical equipment. The end milling process removes material in order to generate the array of ﬁns. Your initial design is the array of pin ﬁns shown in Figure P1-14. You have been asked to optimize the design of the ﬁn array for a particular application where the base temperature is T base = 120◦ C and the air temperature is T air = 20◦ C. The heat sink is square; the size of the heat sink is W = 10 cm.

Chapter 1: One-Dimensional, Steady-State Conduction

195

The conductivity of the material is k = 70 W/m-K. The distance between the edges of two adjacent ﬁns is a, the diameter of a ﬁn is D, and the length of each ﬁn is L.

array of fins k = 70 W/m-K

Tair = 20°C, h D a L

W = 10 cm

Tbase = 120°C

Figure P1-14: Pin ﬁn array.

Air is forced to ﬂow through the heat sink by a fan. The heat transfer coefﬁcient between the air and the surface of the ﬁns as well as the unﬁnned region of the base, h, has been measured for the particular fan that you plan to use and can be calculated according to: h = 40

W m2 K

a 0.005 [m]

0.4

D 0.01 [m]

−0.3

Mass is not a concern for this heat sink; you are only interested in maximizing the heat transfer rate from the heat sink to the air given the operating temperatures. Therefore, you will want to make the ﬁns as long as possible. However, in order to use the micro-end milling process you cannot allow the ﬁns to be longer than 10x the distance between two adjacent ﬁns. That is, the length of the ﬁns may be computed according to: L = 10 a. You must choose the optimal values of a and D for this application. a.) Prepare a model using EES that can predict the heat transfer coefﬁcient for a given value of a and D. Use this model to predict the heat transfer rate from the heat sink for a = 0.5 cm and D = 0.75 cm. b.) Prepare a plot that shows the heat transfer rate from the heat sink as a function of the distance between adjacent ﬁns, a, for a ﬁxed value of D = 0.75 cm. Be sure that the ﬁn length is calculated using L = 10 a. Your plot should exhibit a maximum value, indicating that there is an optimal value of a. c.) Prepare a plot that shows the heat transfer rate from the heat sink as a function of the diameter of the ﬁns, D, for a ﬁxed value of a = 0.5 cm. Be sure that the ﬁn length is calculated using L = 10 a. Your plot should exhibit a maximum value, indicating that there is an optimal value of D. d.) Determine the optimal values of a and D using EES’ built-in optimization capability. Analytical Solutions for Advanced Constant Cross-Section Extended Surfaces 1–15 Figure P1-15 illustrates a material processing system.

196

One-Dimensional, Steady-State Conduction oven wall temperature varies with x gap filled with gas th = 0.6 mm kg = 0.03 W/m-K

u = 0.75 m/s Tin = 300 K

D = 5 cm x extruded material k = 40 W/m-K α = 0.001 m2 /s Figure P1-15: Material processing system.

Material is extruded and enters the oven at T in = 300 K with velocity u = 0.75 m/s. The material has diameter D = 5 cm. The conductivity of the material is k = 40 W/m-K and the thermal diffusivity is α = 0.001 m2 /s. In order to precisely control the temperature of the material, the oven wall is placed very close to the outer diameter of the extruded material and the oven wall temperature distribution is carefully controlled. The gap between the oven wall and the material is th = 0.6 mm and the oven-to-material gap is ﬁlled with gas that has conductivity kg = 0.03 W/m-K. Radiation can be neglected in favor of convection through the gas from the oven wall to the material. For this situation, the heat ﬂux experienced by the material surface can be approximately modeled according to: kg q˙ conv ≈ (T w − T ) th where T w and T are the oven wall and material temperatures at that position, respectively. The oven wall temperature varies with position x according to: x T w = T f − (T f − T w,0 ) exp − Lc where T w,0 is the temperature of the wall at the inlet (at x = 0), Tf = 1000 K is the temperature of the wall far from the inlet, and Lc is a characteristic length that dictates how quickly the oven wall temperature approaches Tf . Initially, assume that T w,0 = 500 K, T f = 1000 K, and Lc = 1 m. Assume that the oven can be approximated as being inﬁnitely long. a.) Is an extended surface model appropriate for this problem? b.) Assume that your answer to (a) was yes. Develop an analytical solution that can be used to predict the temperature of the material as a function of x. c.) Plot the temperature of the material and the temperature of the wall as a function of position for 0 < x < 20 m. Plot the temperature gradient experienced by the material as a function of position for 0 < x < 20 m. The parameter Lc can be controlled in order to control the maximum temperature gradient and therefore the thermal stress experienced by the material as it moves through the oven. d.) Prepare a plot showing the maximum temperature gradient as a function of Lc . Overlay on your plot the distance required to heat the material to T p = 800 K (L p). If the maximum temperature gradient that is allowed is 60 K/m, then what is the appropriate value of Lc and the corresponding value of Lp ?

Chapter 1: One-Dimensional, Steady-State Conduction

197

1–16 The receiver tube of a concentrating solar collector is shown in Figure P1-16. q⋅ ′′s

Ta = 25°C ha = 25 W/m2 -K r = 5 cm th = 2.5 mm k = 10 W/m-K

φ

Tw = 80°C hw = 100 W/m2 -K

Figure P1-16: A solar collector.

The receiver tube is exposed to solar radiation that has been reﬂected from a concentrating mirror. The heat ﬂux received by the tube is related to the position of the sun and the geometry and efﬁciency of the concentrating mirrors. For this problem, you may assume that all of the radiation heat ﬂux is absorbed by the collector and neglect the radiation emitted by the collector to its surroundings. The ﬂux received at the collector surface (q˙ s ) is not circumferentially uniform but rather varies with angular position; the ﬂux is uniform along the top of the collector, π < φ < 2π rad, and varies sinusoidally along the bottom, 0 < φ < π rad, with a peak at φ = π/2 rad. q˙ t + q˙ p − q˙ t sin (φ) for 0 < φ < π q˙ s (φ) = for π < φ < 2 π q˙ t where q˙ t = 1000 W/m2 is the uniform heat ﬂux along the top of the collector tube and q˙ p = 5000 W/m2 is the peak heat ﬂux along the bottom. The receiver tube has an inner radius of r = 5.0 cm and thickness of th = 2.5 mm (because th/r 1 it is possible to ignore the small difference in convection area on the inner and outer surfaces of the tube). The thermal conductivity of the tube material is k = 10 W/m-K. The solar collector is used to heat water, which is at T w = 80◦ C at the axial position of interest. The average heat transfer coefﬁcient between the water and the internal surface of the collector is hw = 100 W/m2 -K. The external surface of the collector is exposed to air at T a = 25◦ C. The average heat transfer coefﬁcient between the air and the external surface of the collector is ha = 25 W/m2 -K. a.) Can the collector be treated as an extended surface for this problem (i.e., can the temperature gradients in the radial direction in the collector material be neglected)? b.) Develop an analytical model that will allow the temperature distribution in the collector wall to be determined as a function of circumferential position. Analytical Solutions for Non-Constant Cross-Section Extended Surfaces 1–17 Figure P1-17 illustrates a disk brake for a rotating machine. The temperature distribution within the brake can be assumed to be a function of radius only. The brake is divided into two regions. In the outer region, from R p = 3.0 cm to Rd = 4.0 cm, the stationary brake pads create frictional heating and the disk is not exposed to convection. The clamping pressure applied to the pads is P = 1.0 MPa and the coefﬁcient of friction between the pad and the disk is μ = 0.15. You may

198

One-Dimensional, Steady-State Conduction

assume that the pads are not conductive and therefore all of the frictional heating is conducted into the disk. The disk rotates at N = 3600 rev/min and is b = 5.0 mm thick. The conductivity of the disk is k = 75 W/m-K and you may assume that the outer rim of the disk is adiabatic. coefficient of friction, μ = 0.15

stationary brake pads

clamping pressure P = 1 MPa Ta = 30°C, h

b = 5 mm

Rp = 3 cm

Rd = 4 cm

center line k = 75 W/m-K disk, rotates at N = 3600 rev/min Figure P1-17: Disk brake.

The inner region of the disk, from 0 to Rp , is exposed to air at T a = 30◦ C. The heat transfer coefﬁcient between the air and disk surface depends on the angular velocity of the disk, ω, according to: h = 20

1.25 W ω W + 1500 m2 -K m2 -K 100 [rad/s]

a.) Develop an analytical model of the temperature distribution in the disk brake; prepare a plot of the temperature as a function of radius for r = 0 to r = Rd . b.) If the disk material can withstand a maximum safe operating temperature of 750◦ C then what is the maximum allowable clamping pressure that can be applied? Plot the temperature distribution in the disk at this clamping pressure. What is the braking torque that results? c.) Assume that you can control the clamping pressure so that as the machine slows down the maximum temperature is always kept at the maximum allowable temperature, 750◦ C. Plot the torque as a function of rotational speed for 100 rev/min to 3600 rev/min. 1–18 Figure P1-18 illustrates a ﬁn that is to be used in the evaporator of a space conditioning system for a space-craft. The ﬁn is a plate with a triangular shape. The thickness of the plate is th = 1 mm and the width of the ﬁn at the base is Wb = 1 cm. The length of the ﬁn is L = 2 cm. The ﬁn material has conductivity k = 50 W/m-K. The average heat transfer coefﬁcient between the ﬁn surface and the air in the space-craft is h = 120 W/m2 -K. The air is at T ∞ = 20◦ C and the base of the ﬁn is at T b = 10◦ C. Assume that the temperature distribution in the ﬁn is 1-D in x. Neglect convection from the edges of the ﬁn. a.) Obtain an analytical solution for the temperature distribution in the ﬁn. Plot the temperature as a function of position. b.) Calculate the rate of heat transfer to the ﬁn. c.) Determine the ﬁn efﬁciency.

Chapter 1: One-Dimensional, Steady-State Conduction h = 120 W/m2 -K T∞ = 20°C

199

th = 1 mm

x

L = 2 cm

k = 50 W/m-K ρ= 3000 kg/m3

ρb = 8000 kg/m3

Tb = 10°C thg = 2 mm

thb = 2 mm

Wb = 1 cm Figure P1-18: Fin on an evaporator.

The ﬁn has density ρ = 3000 kg/m3 and is installed on a base material with thickness thb = 2 mm and density ρb = 8000 kg/m3 . The half-width of the gap between adjacent ﬁns is thg = 2 mm. Therefore, the volume of the base material associated with each ﬁn is thbWb(th + 2thg ). d.) Determine the ratio of the absolute value of the rate of heat transfer to the ﬁn to the total mass of material (ﬁn and base material associated with the ﬁn). e.) Prepare a contour plot that shows the ratio of the heat transfer to the ﬁn to the total mass of material as a function of the length of the ﬁn (L) and the ﬁn thickness (th). f.) What is the optimal value of L and th that maximizes the absolute value of the ﬁn heat transfer rate to the mass of material? Numerical Solution of Extended Surface Problems 1–19 A ﬁber optic bundle (FOB) is shown in Figure P1-19 and used to transmit the light for a building application. h = 5 W/m2 -K T∞ = 20°C

rout = 2 cm

q⋅ ′′ = 1x105 W/m 2

x

fiber optic bundle

Figure P1-19: Fiber optic bundle used to transmit light.

The ﬁber optic bundle is composed of several, small-diameter ﬁbers that are each coated with a thin layer of polymer cladding and packed in approximately a hexagonal close-packed array. The porosity of the FOB is the ratio of the open area of the FOB face to its total area. The porosity of the FOB face is an important characteristic because any radiation that does not fall directly upon the ﬁbers will not be transmitted and instead contributes to a thermal load on the FOB. The ﬁbers are designed so that any radiation that strikes the face of a ﬁber is “trapped” by total internal reﬂection. However, radiation that strikes the interstitial areas between the ﬁbers will instead be absorbed in the cladding very close to the FOB face. The volumetric generation of thermal energy associated with this radiation can be represented by: φ q˙ x exp − g˙ = Lch Lch

200

One-Dimensional, Steady-State Conduction

where q˙ = 1 × 105 W/m2 is the energy ﬂux incident on the face, φ = 0.05 is the porosity of the FOB, x is the distance from the face, and Lch = 0.025 m is the characteristic length for absorption of the energy. The outer radius of the FOB is rout = 2 cm. The face of the FOB as well as its outer surface are exposed to air at T ∞ = 20◦ C with heat transfer coefﬁcient h = 5 W/m2 -K. The FOB is a composite structure and therefore conduction through the FOB is a complicated problem involving conduction through several different media. Section 2.9 discusses methods for computing the effective thermal conductivity for a composite. The effective thermal conductivity of the FOB in the radial direction is keff ,r = 2.7 W/m-K. In order to control the temperature of the FOB near the face, where the volumetric generation of thermal energy is largest, it has been suggested that high conductivity ﬁller material be inserted in the interstitial regions between the ﬁbers. The result of the ﬁller material is that the effective conductivity of the FOB in the axial direction varies with position according to: x keff ,x = keff ,x,∞ + keff ,x exp − Lk where keff ,x,∞ = 2.0 W/m-K is the effective conductivity of the FOB in the xdirection without ﬁller material, keff ,x = 28 W/m-K is the augmentation of the conductivity near the face, and Lk = 0.05 m is the characteristic length over which the effect of the ﬁller material decays. The length of the FOB is effectively inﬁnite. Assume that the volumetric generation is unaffected by the ﬁller material. a.) Is it appropriate to use a 1-D model of the FOB? b.) Assume that your answer to (a) was yes. Develop a numerical model of the FOB. c.) Overlay on a single plot the temperature distribution within the FOB for the case where the ﬁller material is present ( keff ,x = 28 W/m-K) and the case where no ﬁller material is present ( keff ,x = 0). 1–20 An expensive power electronics module normally receives only a moderate current. However, under certain conditions it might experience currents in excess of 100 amps. The module cannot survive such a high current and therefore, you have been asked to design a fuse that will protect the module by limiting the current that it can experience, as shown in Figure P1-20. L = 2.5 cm ε = 0.9 Tend = 20°C

Tend = 20°C D = 0.9 mm T∞ = 20°C h = 5 W/m2 -K

k = 150 W/m-K ρr = 1x10 -7 ohm-m I = 100 amp

Figure P1-20: A fuse that protects a power electronics module from high current.

The space available for the fuse allows a wire that is L = 2.5 cm long to be placed between the module and the surrounding structure. The surface of the fuse wire is exposed to air at T ∞ = 20◦ C. The heat transfer coefﬁcient between the surface of the fuse and the air is h = 5.0 W/m2 -K. The fuse surface has an emissivity of ε = 0.90. The fuse is made of an aluminum alloy with conductivity

One-Dimensional, Steady-State Conduction

201

k = 150 W/m-K. The electrical resistivity of the aluminum alloy is ρe = 1 × 10−7 ohm-m and the alloy melts at approximately 500◦ C. Assume that the properties of the alloy do not depend on temperature. The ends of the fuse (i.e., at x = 0 and x = L) are maintained at T end = 20◦ C by contact with the surrounding structure and the module. The current passing through the fuse, I, results in a uniform volumetric generation within the fuse material. If the fuse operates properly, then it will melt (i.e., at some location within the fuse, the temperature will exceed 500◦ C) when the current reaches 100 amp. Your job will be to select the fuse diameter; to get your model started, you may assume a diameter of D = 0.9 mm. Assume that the volumetric rate of thermal energy generation due to ohmic dissipation is uniform throughout the fuse volume. a.) Prepare a numerical model of the fuse that can predict the steady-state temperature distribution within the fuse material. Plot the temperature as a function of position within the wire when the current is 100 amp and the diameter is 0.9 mm. b.) Verify that your model has numerically converged by plotting the maximum temperature in the wire as a function of the number of nodes in your model. c.) Prepare a plot of the maximum temperature in the wire as a function of the diameter of the wire for I = 100 amp. Use your plot to select an appropriate fuse diameter.

REFERENCES

Cercignani, C., Rareﬁed Gas Dynamics: From Basic Concepts to Actual Calculations, Cambridge University Press, Cambridge, U.K., (2000). Chen, G., Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons, Oxford University Press, Oxford, U.K., (2005). Flynn, T. M., Cryogenic Engineering, 2nd Edition, Revised and Expanded, Marcel Dekker, New York, (2005). Fried, E., Thermal Conduction Contribution to Heat Transfer at Contacts, in Thermal Conductivity, Volume 2, R. P. Tye, ed., Academic Press, London, (1969). Iwasa, Y., Case Studies in Superconducting Magnets: Design and Operational Issues, Plenum Press, New York, (1994). Izzo, F., “Other Thermal Ablation Techniques: Microwave and Interstitial Laser Ablation of Liver Tumors,” Annals of Surgical Oncology, Vol. 10, pp. 491–497, (2003). Keenan, J. H., “Adventures in Science,” Mechanical Engineering, May, p. 79, (1958). NIST Standard Reference Database Number 69, http://webbook.nist.gov/chemistry, June 2005 Release, (2005). Que, L., Micromachined Sensors and Actuators Based on Bent-Beam Suspensions, Ph.D. Thesis, Electrical and Computer Engineering Dept., University of Wisconsin-Madison, (2000). Schneider, P. J., Conduction, in Handbook of Heat Transfer, 2nd Edition, W.M. Rohsenow et al., eds., McGraw-Hill, New York, (1985). Tien, C.-L., A. Majumdar, and F. M. Gerner, eds., Microscale Energy Transport, Taylor & Francis, Washington (1998). Tompkins, D. T., A Finite Element Heat Transfer Model of Ferromagnetic Thermoseeds and a Physiologically-Based Objective Function for Pretreatment Planning of Ferromagnetic Hypothermia, Ph.D. Thesis, Mechanical Engineering, University of Wisconsin at Madison, (1992). Walton, A. J., Three Phases of Matter, Oxford University Press, Oxford, U.K., (1989).

2

Two-Dimensional, Steady-State Conduction

Chapter 1 discussed the analytical and numerical solution of 1-D, steady-state problems. These are problems where the temperature within the material is independent of time and varies in only one spatial dimension (e.g., x). Examples of such problems are the plane wall studied in Section 1.2, which is truly a 1-D problem, and the constant cross section ﬁn studied in Section 1.6, which is approximately 1-D. The governing differential equation for these problems is an ordinary differential equation and the mathematics required to solve the problem are straightforward. In this chapter, more complex, 2-D steady-state conduction problems are considered where the temperature varies in multiple spatial dimensions (e.g., x and y). These can be problems where the temperature actually varies in only two coordinates or approximately varies in only two coordinates (e.g., the temperature gradient in the third direction is negligible, as justiﬁed by an appropriate Biot number). The governing differential equation is a partial differential equation and therefore the mathematics required to analytically solve these problems are more advanced and the bookkeeping required to solve these problems numerically is more cumbersome. However, many of the concepts that were covered in the context of 1-D problems continue to apply.

2.1 Shape Factors There are many 2-D and 3-D conduction problems involving heat transfer between two well-deﬁned surfaces (surface 1 and surface 2) that commonly appear in heat transfer applications and have previously been solved analytically and/or numerically. The solution to these problems is conveniently expressed in the form of a shape factor, S, which is deﬁned as: S=

1 kR

(2-1)

where k is the conductivity of the material separating the surfaces and R is the thermal resistance between surfaces 1 and 2. Solving Eq. (2-1) for the thermal resistance leads to: R=

1 kS

(2-2)

Recall that the resistance of a plane wall, derived in Section 1.2, is given by: R pw =

L k Ac

(2-3)

where L is the length of conduction path and Ac is the area for conduction. Comparing Eqs. (2-2) and (2-3) suggests that: S≈ 202

Ac L

(2-4)

2.1 Shape Factors

203 T2 q⋅ W

Figure 2-1: Sphere buried in a semi-inﬁnite medium.

T1

D

The shape factor has units of length and represents the ratio of the effective area for conduction to the effective length for conduction. Any shape factor solution should be checked against your intuition using Eq. (2-4). Given a problem, it should be possible to approximately identify the area and length that characterize the conduction process; the ratio of these quantities should have the same order of magnitude as the shape factor solution. One example of a shape factor solution is for a sphere buried in a semi-inﬁnite medium (i.e., a medium that extends forever in one direction but is bounded in the other) as shown in Figure 2-1. In this case, the surface of the sphere is surface 1 (assumed to be isothermal, at T1 ) while the surface of the medium is surface 2 (assumed to be isothermal, at T2 ). The shape factor solution for a completely buried sphere in a semi-inﬁnite medium is: S=

2πD D 1− 4W

(2-5)

where D is the diameter of the sphere and W is the distance between the center of the sphere and the surface. The thermal resistance characterizing conduction between the surface of the sphere and the surface of the medium is: R=

1 = kS

D 4W 2πDk

1−

(2-6)

The rate of conductive heat transfer between the sphere and the surface (q) ˙ is: q˙ =

T1 − T2 2 π D k (T 1 − T 2 ) = D R 1− 4W

(2-7)

There are numerous formulae for shape factors that have been tabulated in various references; for example, Rohsenow et al. (1998). Table 2-1 summarizes a few shape factor solutions. A library of shape factors, including those shown in Table 2-1 as well as others, has been integrated with EES. To access the shape factor library, select Function Information from the Options menu and then scroll through the list to Conduction Shape Factors, as shown in Figure 2-2. The shape factor functions that are available can be selected by moving the scroll bar below the picture.

Table 2-1: Shape factors. Buried beam (L a,b)

Buried sphere

Circular extrusion with off-center hole (L Dout )

T2

T2

L T1 W

W

W

T2

L

Din Dout

a T1

D

Buried cylinder (L D)

Parallel cylinders (L D1 , D2 ) T1

−1

cosh

W −0 59 W −0.078 S = 2.756 L ln 1 + a b

2πD D 1− 4W

S=

S=

T1

b

T2

T2 W

D1

W

S= cosh−1

2πL 4 W 2 − D21 − D22 2 D1 D2

W

T1

L

Square extrusion with a centered circular hole (L W)

L

D

S=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

L D

2πL −1

T1

W

if W ≤ 3 D/2

(2 W/D) 2πL if W > 3 D/2 ln (4 W/D) cosh

Disk on surface of semi-inﬁnite body

T2

W −0 59 W −0.078 S = 2.756 L ln 1 + a b Square extrusion (L a) T1 T2

L

T1

L T2 W

T1

2πL D2out + D2in − 4 W 2 2 Dout Din

Cylinder half-way between parallel plates (W D & L W)

T2

D2

T2

D

a

b b

D

S = 2D

a

W

2πL S= ln (1.08 W/D)

Figure 2-2: Accessing the shape factor library from EES.

S=

⎧ ⎪ ⎪ ⎨

2πL 0.785 ln (a/b)

⎪ ⎪ ⎩

2πL if a/b > 0.25 [0.93 ln (a/b) − 0.0502]

if a/b ≤ 0.25

205

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT This example revisits the magnetic ablation concept that was previously described in EXAMPLES 1.3-1 and 1.8-2. You want to measure the power generated by the thermoseed that is used for the ablation process. The radius of the thermoseed is rts = 1.0 mm. The sphere is placed W = 5 cm below the surface of a solution of agar, as shown in Figure 1. Agar is a material with well-known thermal properties that resembles gelatin and is sometimes used as a surrogate for tissue in biological experiments. The agar is allowed to solidify around the sphere and the container of agar is large enough to be considered semi-inﬁnite. The surface of the agar is exposed to an ice-water bath in order to keep it at a constant temperature, Tice = 0◦ C. The sphere is heated using an oscillating magnetic ﬁeld and its surface temperature is measured using a thermocouple. The conductivity of agar is k = 0.35 W/m-K. Tice = 0°C agar, k = 0.35 W/m-K W = 5.0 cm Figure 1: Test setup to measure the power generated by the thermoseed.

Ts = 95°C

rts = 1.0 mm

a) If the measured surface temperature of the sphere is Ts = 95◦ C, how much energy is generated in the thermoseed? The inputs are entered in EES: “EXAMPLE 2.1-1: Magnetic Ablative Power Measurement” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in

“Inputs” r ts=1.0 [mm]∗ convert(mm,m) W=5 [cm]∗ convert(cm,m) k=0.35 [W/m-K] T ice=converttemp(C,K,0 [C]) T s=converttemp(C,K,95 [C])

“thermoseed radius” “depth of sphere” “conductivity of agar” “ice bath temperature” “surface temperature”

The shape factor associated with the buried sphere (S) is accessed from the EES library of shape factors: S=SF 1(2∗ r ts,W)

“shape factor for buried sphere”

The thermal resistance between the surface of the sphere and the semi-inﬁnite body (R) is: R=

1 kS

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

2.1 Shape Factors

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

206

Two-Dimensional, Steady-State Conduction

˙ is computed using the thermal resistance and the known The heat transfer rate (q) temperatures: q˙ =

R=1/(k∗ S) q dot=(T s-T ice)/R

Ts − Tice R

“thermal resistance” “heat transfer rate”

which leads to a generation rate of 0.422 W. b) Estimate the uncertainty in your measurement of the power. Assume that your temperature measurements are accurate to δT = 1.0◦ C, the conductivity of agar is known to within 10% (i.e., δk = 0.035 W/m-K), the depth measurement has an uncertainty of δW = 2.0 mm, and the sphere radius is known to within δrts = 0.1 mm. It is possible to separately estimate the uncertainty introduced by each of the parameters listed above. For example, to evaluate the effect of the uncertainty in the conductivity, simply increase the conductivity by δk: deltak=0.035 [W/m-K] k=0.35 [W/m-K]+deltak

“uncertainty in conductivity” “conductivity of agar”

and re-run the model. The result is a change in the generation rate from 0.422 W to 0.464 W which translates into an uncertainty in the power of 0.042 W or 10%. This process can be repeated for each of the independent variables in order to identify the uncertainty that is introduced into the dependent variable calculation. These contributions should be combined using the root-sum-square technique in order to obtain an overall uncertainty. This process can be carried out automatically in EES; select Uncertainty Propagation from the Calculate Menu to access the dialog shown in Figure 2.

Figure 2: Propagation of uncertainty dialog.

The possible dependent (calculated) and independent (measured) variables are listed; highlight the independent variables that have some uncertainty (all of them for this problem). The calculated variable that you want to examine is q_dot. Select

207

the Set uncertainties button to reach the window shown in Figure 3. Set each of the uncertainties either in absolute or relative terms; note that each of these quantities can also be set using variables that are deﬁned in the main equation window.

Figure 3: Uncertainty of measured variables dialog.

Select OK twice in order to initiate the calculations; the results appear the Uncertainty Results window (Figure 4) which shows the total uncertainty in the variable q_dot (0.06 W or 14%) as well as a delineation of the source of the uncertainty.

[W] [W/m-K] [m] [K] [K] [m] Figure 4: Uncertainty of Results window.

Notice that the dominant sources of uncertainty for this problem are the conductivity of agar and the radius of the sphere (each contributing approximately 50% of the total). The temperature measurements are adequate and the depth does not matter at all since the shape factor becomes nearly independent of the depth provided W is much larger than the diameter (see Eq. (2-5)).

2.2 Separation of Variables Solutions 2.2.1 Introduction Two-dimensional steady-state conduction problems are governed by partial rather than ordinary differential equations; the analytical solution to partial differential equations

EXAMPLE 2.1-1: MAGNETIC ABLATIVE POWER MEASUREMENT

2.2 Separation of Variables Solutions

208

Two-Dimensional, Steady-State Conduction q⋅ y+ dy

q⋅ x dy q⋅ y Tx =0 = 0

q⋅ x + dx

Figure 2-3: Plate.

y x

Ty=0 = Tb (x )

Ty →∞ = 0

dx

th

W Tx =W = 0

is somewhat more involved. Separation of variables is a common technique that is used to solve the partial differential equations that arise in many areas of science and engineering. A complete understanding of separation of variables requires a substantial mathematical background; in this section, the technique is introduced and used to solve several problems. In the subsequent section, more difﬁcult problems are solved using separation of variables. It is not possible to cover separation of variables thoroughly in this book and the interested reader is referred to the textbook by G. E. Myers (1998).

2.2.2 Separation of Variables The method of separation of variables is most conveniently discussed in the context of a speciﬁc problem. In this section, the ﬂat plate shown in Figure 2-3 is considered. The top and bottom surfaces of the plate are insulated and therefore there is no temperature variation in the z-direction; the problem is truly two-dimensional, as the temperature depends on x and y but not z. If the top and bottom surfaces were not insulated (e.g., they experienced convection to a surrounding ﬂuid) but the plate was sufﬁciently thin and conductive, then it still might be possible to ignore temperature gradients in the zdirection and treat the problem as being two-dimensional. This assumption is equivalent to the extended surface assumption that was discussed in Section 1.6.2 and should be justiﬁed using an appropriately deﬁned Biot number. The plate in Figure 2-3 has conductivity k, thickness th, width (in the x-direction) W, and extends to inﬁnity in the y-direction. The governing differential equation for the problem is derived in a manner that is analogous to the 1-D problems that have been previously considered. A differential control volume is deﬁned (see Figure 2-3) and used to develop a steady-state energy balance. Note that the control volume must be differential in both the x- and y-directions because there are temperature gradients in both of these directions. q˙ x + q˙ y = q˙ x+dx + q˙ y+dy

(2-8)

The x + dx and y + dy terms are expanded as usual: q˙ x + q˙ y = q˙ x +

∂ q˙ y ∂ q˙ x dx + q˙ y + dy ∂x ∂y

(2-9)

Equation (2-9) can be simpliﬁed to: ∂ q˙ y ∂ q˙ x dx + dy = 0 ∂x ∂y

(2-10)

2.2 Separation of Variables Solutions

209

Fourier’s law is used to determine the conduction heat transfer rates in the x- and y-directions: q˙ x = −k th dy

∂T ∂x

(2-11)

q˙ y = −k th dx

∂T ∂y

(2-12)

Equations (2-11) and (2-12) are substituted into Eq. (2-10): ∂T ∂ ∂T ∂ −k th dy dx + −k th dx dy = 0 ∂x ∂x ∂y ∂y

(2-13)

If the thermal conductivity and plate thickness are both constant, then Eq. (2-13) can be simpliﬁed to: ∂2T ∂2T + 2 =0 2 ∂x ∂y

(2-14)

which is the governing partial differential equation for this problem. Equation (2-14) is called Laplace’s equation. Equation (2-14) is second order in both the x- and y-directions and therefore two boundary conditions are required in each of these directions. The left and right edges of the plate have a temperature of zero: T x=0 = 0

(2-15)

T x=W = 0

(2-16)

The zero temperature boundaries are necessary here to ensure that the boundary conditions are homogeneous, as explained below. However, these boundary conditions are likely not of general interest. Techniques that allow the solution of problems with more realistic boundary conditions are presented in subsequent sections. The edge of the plate at y = 0 has a speciﬁed temperature that is an arbitrary function of position x: T y=0 = T b(x)

(2-17)

The plate is inﬁnitely long in the y-direction and the temperature approaches 0 as y becomes inﬁnite: T y→∞ = 0

(2-18)

Requirements for using Separation of Variables The method of separation of variables will not work for every problem; there are some fairly restrictive conditions that limit where it can be applied. First, the governing equation must be linear; that is, the equation cannot contain any products of the dependent variable or its derivative. Equation (2-14) is certainly linear. An example of a non-linear equation might be: T

∂T ∂ 2 T ∂2T + =0 ∂x2 ∂x ∂y2

(2-19)

The governing equation must also be homogeneous, which is a more restrictive condition. If T is a solution to a homogeneous equation then C T is also a solution, where C

210

Two-Dimensional, Steady-State Conduction

is an arbitrary constant. Equation (2-14) is homogeneous; to prove this, simply check if C T can be substituted into the equation and still satisfy the equality: ∂ 2 (C T ) ∂ 2 (C T ) + =0 ∂x2 ∂y2 or

(2-20)

C

∂2T ∂2T + =0 ∂x2 ∂y2 =0 according to Eq. (2-14)

(2-21)

A non-homogeneous equation would result, for example, if the plate were exposed to a volumetric generation of thermal energy (g˙ ); the governing differential equation for this situation would be: ∂2T g˙ ∂2T + 2 + =0 2 ∂x ∂y k Substituting C T into Eq. (2-22) leads to: 2 ∂2T ∂ T + 2 C ∂x2 ∂y

+

g˙ =0 k

(2-22)

(2-23)

= − g˙k according to Eq. (2-22)

Substituting Eq. (2-22) into Eq. (2-23) leads to: g˙ g˙ C − + =0 k k

(2-24)

Equation (2-24) shows that C T is a solution to Eq. (2-22) only if g˙ = 0. We will show how some non-homogeneous problems can be solved using separation of variables in Section 2.3. In order to apply the separation of variables method, the boundary conditions must also be linear with respect to the dependent variable. Linear has the same deﬁnition for the boundary conditions that it does for the differential equation; i.e., the boundary condition cannot involve products of the dependent variable or its derivatives. The boundary conditions for the plate in Figure 2-3 are given by Eqs. (2-15) through (2-18) and are all linear. A non-linear boundary condition would result from, for example, radiation. If the right edge of the plate were radiating to surroundings at T = 0 then Eq. (2-15) should be replaced with: ∂T 4 = σ ε T x=W (2-25) −k ∂x x=W which is non-linear. Finally, both of the boundary conditions in one direction must be homogeneous (i.e., either both boundary conditions in the x-direction or both boundary conditions in the y-direction). Again, the meaning of homogeneity for a boundary condition is analogous to its meaning for the differential equation. If a boundary condition is homogeneous then any multiple of a solution also satisﬁes the boundary condition. Examination shows that all of the boundary conditions except for Eq. (2-17) are homogeneous. Therefore, both boundary conditions in the x-direction, Eqs. (2-15) and (2-16), are homogeneous. For this problem, x is therefore the homogeneous direction; this will be important to keep in mind as we solve the problem. A ﬁnal criterion for the use of separation of variables is that the computational domain must be simple; that is, it must have boundaries that lie along constant values of

2.2 Separation of Variables Solutions

211

the coordinate axes. For a Cartesian coordinate system, we are restricted to rectangular problems. The problem shown in Figure 2-3 meets all of the criteria and should therefore be solvable using separation of variables. Separate the Variables The name of the technique, separation of variables, is related to the next step in the solution; it is assumed that the solution T, which is a function of both x and y, can be expressed as the product of two functions, TX which is only a function of x and TY which is only a function of y: T (x, y) = TX (x) TY (y)

(2-26)

Substituting Eq. (2-26) into Eq. (2-14) leads to: ∂2 ∂2 [TX TY ] + 2 [TX TY ] = 0 2 ∂x ∂y

(2-27)

or TY

d2 TY d2 TX + TX =0 2 dx dy2

(2-28)

Dividing through by the product TY TX leads to: d2 TY d2 TX 2 dx2 + dy =0 TX TY function of x

(2-29)

function of y

The ﬁrst term in Eq. (2-29) is a function only of x while the second is a function only of y. Therefore, Eq. (2-29) can only be satisﬁed if both terms are equal and opposite and constant. To see this clearly, imagine moving along a line of constant y (i.e., across the plate in Figure 2-3 in the x-direction from one side to the other). If the ﬁrst term were not constant then, by deﬁnition, its value would change as x changes; however, the second term is not a function of x and therefore it cannot change in response. Clearly then the sum of the two terms could not continue to be zero in this situation. Equation (2-29) can be expressed as two statements: d2 TX dx2 = ± λ2 TX

(2-30)

d2 TY dy2 (2-31) = ∓ λ2 TY where λ2 is a constant that must be positive. Notice that there is a choice that must be made at this point. The TX group can either be set equal to a positive constant (λ2 ) or a negative constant (−λ2 ). Depending on this choice, the TY group must be set equal to a negative constant (−λ2 ) or a positive constant (λ2 ), in order to satisfy Eq. (2-29). The choice at this point seems arbitrary but in fact it is important. Recall that one condition for using separation of variables is that one of the coordinate directions must have homogeneous boundary conditions; this was referred to as the homogeneous direction. In this problem, the x-direction is the homogeneous direction because the boundary conditions at x = 0, Eq. (2-15), and at x = W, Eq. (2-16), are both homogeneous. It is necessary to choose the negative constant for the group associated

212

Two-Dimensional, Steady-State Conduction

with the homogeneous direction (i.e., for Eq. (2-30) in this problem). With this choice, rearranging Eqs. (2-30) and (2-31) leads to the two ordinary differential equations: d2 TX + λ2 TX = 0 dx2

(2-32)

d2 TY − λ2 TY = 0 dy2

(2-33)

We have effectively converted our partial differential equation, Eq. (2-14), into two ordinary differential equations, Eqs. (2-32) and (2-33). The solutions to Eqs. (2-32) and (2-33) can be identiﬁed using Maple: > restart; > ODEX:=diff(diff(TX(x),x),x)+lambdaˆ2∗ TX(x)=0; 2 d ODEX = TX(x) + λ2 TX(x) = 0 dx2 > Xs:=dsolve(ODEX); Xs = TX(x) = C1 sin(λx) + C2 cos(λx) > ODEY:=diff(diff(TY(y),y),y)-lambdaˆ2∗ TY(y)=0; 2 d ODEY := TY (Y ) − λ2 TY (Y ) = 0 dy2 > Ys:=dsolve(ODEY); Ys := TY (y) = C1e(−λY ) + C2e(λY )

So the solution for TX (i.e., the solution in the homogeneous direction) is: TX = C1 sin (λ x) + C2 cos (λ x)

(2-34)

where C1 and C2 are undetermined constants. The solution for TY (i.e., in the nonhomogeneous direction) is: TY = C3 exp (−λ y) + C4 exp (λ y)

(2-35)

where C3 and C4 are undetermined constants. Note that Eq. (2-35) could equivalently be expressed in terms of hyperbolic sines and cosines: TY = C3 sinh (λ y) + C4 cosh (λ y)

(2-36)

where C3 and C4 are undetermined constants (different from those in Eq. (2-35)). The choice of the negative value of the constant for the homogeneous direction (i.e., for Eq. (2-30)) has led directly to sine/cosine solutions in the homogeneous direction; this result is necessary in order to use separation of variables. Solve the Eigenproblem It is necessary to address the solution in the homogeneous direction (in this problem, TX) before moving on to the non-homogeneous direction. This portion of the problem is often called the eigenproblem and the solutions are referred to as eigenfunctions. The boundary conditions for TX can be obtained by revisiting the original boundary conditions for the problem in the x-direction using the assumed, separated form of the solution. Equation (2-15) becomes: TX x=0 TY = 0

(2-37)

2.2 Separation of Variables Solutions

213

which can only be true at an arbitrary location y if: TX x=0 = 0

(2-38)

The remaining boundary condition in the x-direction is given by Eq. (2-16) and leads to: TX x=W = 0

(2-39)

Substituting the solution to the ordinary differential equation in the homogeneous direction, Eq. (2-34), into Eq. (2-38) leads to: C1 sin (λ 0) + C2 cos (λ 0) = 0 0

(2-40)

1

or C2 = 0

(2-41)

TX = C1 sin (λ x)

(2-42)

So that:

Substituting Eq. (2-42) into Eq. (2-39) leads to: C1 sin (λ W) = 0

(2-43)

Equation (2-43) could be satisﬁed if C1 is 0, but that would lead to TX = 0 (and therefore T = 0) everywhere, which is not a useful solution. However, Eq. (2-43) is also satisﬁed whenever the sine function becomes zero; this occurs whenever the argument of sine is an integer multiple of π: λi W = i π

where i = 0, 1, 2, . . . ∞

(2-44)

Equation (2-43) satisﬁes the eigenproblem (i.e., the ordinary differential equation in the x-direction, Eq. (2-32), and both boundary conditions in the x-direction, Eqs. (2-38) and (2-39)) for each value of λi identiﬁed by Eq. (2-44). TX i = C1,i sin (λi x)

where λi =

iπ W

i = 1, 2, . . . ∞

(2-45)

Note that the i = 0 case is not included in Eq. (2-45) because the sine of 0 is zero; therefore this solution does not provide any useful information. The functions TXi given by Eq. (2-45) are referred to as the eigenfunctions that solve the linear, homogeneous problem for TX and the values λi are the eigenvalues associated with each eigenfunction. The function sin(i πx/W) is referred to as the ith eigenfunction and λi = i π/W is the ith eigenvalue. Solve the Non-homogeneous Problem for each Eigenvalue With the eigenproblem solved, it is necessary to return to the non-homogeneous portion of the problem, TY . Each of the eigenvalues identiﬁed by Eq. (2-44) is associated with an ordinary differential equation in the y-direction according to Eq. (2-33): d2 TYi − λ2i TYi = 0 dy2

(2-46)

These ordinary differential equations have either an exponential or hyperbolic solution according to Eqs. (2-35) or (2-36). The choice of one form over the other is arbitrary and either will lead to the same solution. Because one of the boundary conditions is at y → ∞, the exponentials will provide a more concise solution for this problem.

214

Two-Dimensional, Steady-State Conduction

However, in most other cases, the sinh and cosh solution will be easier to work with. The solution for TYi is: TYi = C3,i exp (−λi y) + C4,i exp (λi y)

(2-47)

Obtain Solution for each Eigenvalue According to Eq. (2-26), the solution for temperature associated with the ith eigenvalue is: T i = TX i TYi = C1,i sin (λi x) [C3,i exp (−λi y) + C4,i exp (λi y)]

(2-48)

The products of the undetermined constants C1,i C3,i and C1,i C4,i are also undetermined constants and therefore Eq. (2-48) can be written as: T i = sin (λi x) [C3,i exp (−λi y) + C4,i exp (λi y)]

(2-49)

Equation (2-49) will, for any value of i, satisfy the governing differential equation, Eq. (2-14), and satisfy both of the boundary conditions in the homogeneous direction, Eqs. (2-15) and Eq. (2-16). This is a typical outcome of solving the eigenproblem: a set of solutions that each satisfy the governing partial differential equation and all of the boundary conditions in the homogeneous direction. It is worth checking that your solution has these properties using Maple. First, it is necessary to let Maple know that i is an integer using the assume command. > restart; > assume(i,integer);

Next, deﬁne λi according to Eq. (2-44): > lambda:=i∗ Pi/W; λ :=

i∼π W

Note the use of Pi rather than pi in the Maple code; Pi indicates that π should be evaluated symbolically whereas pi is the numerical value of π. Create a function T in the independent variables x and y according to Eq. (2-49): > T:=(x,y)->sin(lambda∗ x)∗ (C3∗ exp(-lambda∗ y)+C4∗ exp(lambda∗ y)); T := (x, y) → sin(λx)(C3e(−λy) + C4e(λy) )

You can verify that the two homogeneous direction boundary conditions, Eqs. (2-15) and (2-16), are satisﬁed: > T(0,y);

0

> T(W,y);

0

and also that the partial differential equation, Eq. (2-14), is satisﬁed;

2.2 Separation of Variables Solutions

215

> diff(diff(T(x,y),x),x)+diff(diff(T(x,y),y),y); i∼πy i∼πy i ∼ πx i ∼2 π2 C3e(− W ) + C4e( W ) sin W − W2 i∼πy i∼πy C3 i ∼2 π2 e(− W ) i ∼ πx C4 i ∼2 π2 e( W ) + sin + W W2 W2 > simplify(%); 0

Create the Series Solution and Enforce the Remaining Boundary Conditions Because the partial differential equation is linear, the sum of the solutions for each eigenvalue, Ti given by Eq. (2-49), is itself a solution: T =

∞

Ti =

i=1

∞

sin (λi x) [C3,i exp (−λi y) + C4,i exp (λi y)]

(2-50)

i=1

The ﬁnal step of the solution selects the constants so that the boundary conditions in the non-homogeneous direction are satisﬁed. Equation (2-18) provides the boundary condition as y approaches inﬁnity; substituting Eq. (2-50) into Eq. (2-18) leads to: T y→∞ =

∞ i=1

sin (λi x) C3,i exp (−∞) + C4,i exp (∞) = 0

(2-51)

∞

0

or ∞

sin (λi x) C4,i ∞ = 0

(2-52)

i=1

Equation (2-52) can be solved by inspection; the equality can only be satisﬁed if C4,i = 0 for all i. T =

∞

C3,i sin (λi x) exp (−λi y)

(2-53)

i=1

Because only C3,i remains in our solution it is no longer necessary to designate it as the third undetermined constant: T =

∞

Ci sin (λi x) exp (−λi y)

(2-54)

i=1

Equation (2-17) provides the boundary condition at y = 0; substituting Eq. (2-54) into Eq. (2-17) leads to: T y=0 =

∞ i=1

Ci sin (λi x) exp (−λi 0) = T b (x) 1

(2-55)

216

Two-Dimensional, Steady-State Conduction

or ∞

Ci sin (λi x) = T b (x)

(2-56)

i=1

Equation (2-56) deﬁnes the constants in the solution; they are the Fourier coefﬁcients of the non-homogeneous boundary condition. At ﬁrst glance, it may seem like we have not really come very far. The solution to the problem is certainly provided by Eqs. (2-54) and (2-56), however an inﬁnite number of unknown constants, Ci , are needed to evaluate this solution and it is not clear how Eq. (2-56) can be manipulated in order to evaluate these constants. Fortunately the eigenfunctions have the property of orthogonality, which makes it relatively easy to determine the constants Ci . The meaning of orthogonality becomes evident when Eq. (2-56) is multiplied by a single eigenfunction, say the jth one, and then integrated in the homogeneous direction from one boundary to the other (i.e., from x = 0 to x = W): ∞

W Ci

i=1

W jπ iπ jπ sin x sin x dx = T b (x) sin x dx W W W

0

(2-57)

0

The property of orthogonality guarantees that the only term in the summation on the left side of Eq. (2-57) that will not integrate to zero is the one where i = j. We can verify this result by consulting a table of integrals. The integral of the product of two sine functions is: W 0

⎤W ⎡ π (j − i) x π (i + j) x sin sin ⎥ iπ W ⎢ jπ W W ⎢ ⎥ − sin x sin x dx = ⎦ W W 2π ⎣ (j − i) (i + j)

(2-58)

0

This result can be obtained using Maple: > assume(j,integer); > assume(i,integer); > int(sin(i∗ Pi∗ x/W)∗ sin(j∗ Pi∗ x/W),x);

1 2

π(−i + j)x W π(−i + j)

W sin

−

1 2

π(i + j)x W π(i + j)

W sin

Applying the limits of integration to Eq. (2-58) leads to:

W sin 0

iπ jπ W sin (π (j − i)) sin (π (i + j)) − x sin x dx = W W 2π (j − i) (i + j)

(2-59)

The term involving sin (π (i + j)) must always be zero for any positive integer values of i and j because the sine of any integer multiple of π is zero. The ﬁrst term on the right side of Eq. (2-59) will also be zero provided j = i. However, if j = i then both the numerator

2.2 Separation of Variables Solutions

217

and denominator of this term are zero and the value of the integral is not obvious. In the limit that j = i, the value of the integral in Eq. (2-59) is:

W 2

sin 0

W iπ sin (π (j − i)) x dx = lim W 2 π (j−i)→0 (j − i)

(2-60)

π

The limit of the bracketed term in Eq. (2-60) can be evaluated using Maple: > restart; > limit(sin(Pi∗ x)/x,x=0); π

which leads to:

W sin2

W iπ x dx = W 2

(2-61)

0

This behavior lies at the heart of orthogonality: the integral of the product of two different eigenfunctions between the homogeneous boundary conditions will always be zero while the integral of any eigenfunction multiplied by itself between the same boundary conditions will not be zero. It is possible to prove that this behavior is generally true for any solution in the homogeneous direction (see Myers (1987)). The orthogonality of the eigenfunctions simpliﬁes the problem considerably because it allows the summation in Eq. (2-57) to be replaced by a single integration. (All of the terms on the left hand side where j = i must integrate to zero and disappear.) W Ci 0

W iπ iπ iπ sin x sin x dx = T b (x) sin x dx for i = 1..∞ W W W

(2-62)

0

The only term in the sum that has been retained is the integration of the eigenfunction sin(iπ x/W) multiplied by itself; notice that Eq. (2-62) provides a single equation for each of the constants Ci and therefore completes the solution. A more physical feel for the orthogonality of the eigenfunctions can be obtained by examining various eigenfunctions and their products. For example, Figure 2-4 shows the ﬁrst eigenfunction, sin (π x/W), and the second eigenfunction, sin (2 π x/W). Also shown in Figure 2-4 is the product of these two eigenfunctions; notice that the integral of the product must be zero as the areas above and below the axis are equal. Figure 2-5 illustrates the behavior of the second and fourth eigenfunctions and their product; while their oscillations are more complex, it is clear that the product of these eigenfunctions must also integrate to zero. Finally, Figure 2-6 shows the behavior of the third eigenfunction and its value squared; the square of any real valued function is never negative and therefore it cannot integrate to zero. The eigenfunctions that are appropriate for other problems with different boundary conditions will not be sin(i π x/W). However, they will be orthogonal functions. As a result, the sum that deﬁnes the constants in the series can always be reduced to one equation that allows each constant to be evaluated; the equivalent of Eq. (2-57) can always be reduced to the equivalent of Eq. (2-62).

218

Two-Dimensional, Steady-State Conduction

Eigenfunctions and their product

1 0.8

st

1 eigenfunction, sin(π x)

0.6 0.4 0.2 0 product of 1 st & 2 nd eigenfunctions

-0.2 -0.4 -0.6

nd

2 eigenfunction, sin(2π x)

-0.8 -1 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/W

0.8

0.9

1

Figure 2-4: Behavior of the ﬁrst and second eigenfunctions and their product.

Eigenfunctions and their product

1.5 1

nd

th

2 eigenfunction, sin(2 π x)

4 eigenfunction, sin(4π x)

0.5 0 -0.5 -1 -1.5 0

nd

product of 2 th & 4 eigenfunctions 0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/W

0.8

0.9

1

Figure 2-5: Behavior of the second and fourth eigenfunctions and their product.

Equation (2-62) provides an integral equation that can be used to evaluate each of the coefﬁcients:

W T b (x) sin Ci =

iπ x dx W

0

W sin2

iπ x dx W

i = 1..∞

(2-63)

0

Still, it is necessary to determine both of the integrals in Eq. (2-63) in order to evaluate each coefﬁcient. In some cases, the integrals can be evaluated by inspection or by the use of mathematical tables; usually these integrals can be evaluated easily with the aid of Maple. The integral in the denominator of Eq. (2-63) was previously evaluated in

Eigenfunctions and their product

2.2 Separation of Variables Solutions 1.2 1 0.8 0.6 0.4 0.2 0 rd -0.2 3 eigenfunction squared -0.4 -0.6 -0.8 -1 -1.2 0 0.1 0.2 0.3

219

rd

3 eigenfunction, sin(3 π x) 0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless position, x/W Figure 2-6: Behavior of the third eigenfunction and its square.

Eq. (2-61). The integral could also be evaluated with the aid of trigonometric identities and integral tables:

W 2

sin 0

W W W W 2iπ x 2iπ iπ 1 = x dx = − cos x dx = − sin x W 2 W 2 2iπ W 2 0 0

(2-64) Maple makes this process much easier: > int((sin(i∗ Pi∗ x/W))ˆ2,x=0..W); W 2

The ith coefﬁcient is therefore: 2 Ci = W

W T b (x) sin

iπ x dx W

(2-65)

0

The remaining integral in Eq. (2-65) depends on the functional form of the boundary condition. The simplest possibility is a constant temperature, Tb , which leads to: 2 Tb Ci = W

W 0

W iπ 2 Tb 2 Tb iπ sin = x dx = − cos x [1 − cos (i π)] W iπ W iπ 0

or, using Maple: > 2∗ Tb∗ int(sin(i∗ pi∗ x/W),x=0..W)/W; −

2Tb(−1 + cos(i π)) iπ

(2-66)

220

Two-Dimensional, Steady-State Conduction

Substituting Eq. (2-66) into Eq. (2-54) leads to: T =

∞ 2 Tb i=1

iπ

[1 − cos (i π)] sin

iπ iπ x exp − y W W

(2-67)

It is usually more convenient to let Maple carry out the symbolic math and then evaluate the solution using EES. The input parameters are entered in EES: $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 3.5 in “Inputs” W=1.0 [m] k=10 [W/m-K] T b=1 [K]

“width of plate” “conductivity of plate” “base temperature”

The temperature is evaluated at an arbitrary location: “position to evaluate temperature” x=0.1 [m] y=0.25 [m]

“x-position” “y-position”

Each of the ﬁrst N terms of the series solution in Eq. (2-67) are evaluated using a duplicate loop. The coefﬁcient for each term is evaluated using the formula obtained in Maple by copying and pasting it into EES. N=10 [-] duplicate i=1,N C[i]=-2∗ T b∗ (-1+cos(i∗ pi))/(i∗ pi) T[i]=C[i]∗ sin(i∗ pi∗ x/W)∗ exp(-i∗ pi∗ y/W) end

“number of terms in the solution to evaluate” “constant for i’th term in the series” “i’th term in the series”

The terms in the series are summed using the sum function in EES: T=sum(T[1..N])

“sum of N terms in the series”

A parametric table is created in order to examine the temperature as a function of position along the bottom of the plate, y = 0. The ability of the solution to match the imposed boundary condition depends on the number of terms that are used. Figure 2-7 illustrates the solution with 5, 10, and 100 terms. A contour plot of the temperature distribution is generated by creating a parametric table containing the variables x, y, and T that includes 400 runs. Click on the arrow in the x-column header in order to bring up the dialog that allows values to be automatically entered into the table. Click the check box for the option to repeat the pattern of running x from 0 to 1 every 20 rows, as shown in Figure 2-8(a). Repeat the process for the y column, but in this case apply the pattern of running y from 0 to 1 every 20 rows, as shown in Figure 2-8(b). When the table is solved, the solution is obtained over a 20 × 20 grid ranging from 0 to 1 in both x and y. Select X-Y-Z Plot from the New Plot Window selection under the Plots menu and select Isometric Lines to generate the contour plot shown in Figure 2-9.

2.2 Separation of Variables Solutions

221

1.4 N = 100 terms

N = 5 terms

Temperature (K)

1.2 1 0.8

N = 10 terms

0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Position, x (m)

0.7

0.8

0.9

1

Figure 2-7: Temperature as a function of x at y = 0 for different values of N.

(a)

(b)

Figure 2-8: Automatically entering repeating values for (a) x and (b) y.

More complicated boundary conditions can be considered at y = 0. For example, the temperature may vary linearly from 0 to 1 K according to: T b (x) = T b

x W

(2-68)

The coefﬁcients in the general solution, Eq. (2-54) are obtained by substituting Eq. (2-68) into Eq. (2-65): 2 Tb Ci = W2

W

iπ x sin x dx W

0

which can be evaluated using Maple: > restart; > assume(i,integer); > C[i]:=2∗ T_b∗ int(x∗ sin(i∗ pi∗ x/W),x=0..W)/Wˆ2; Ci∼ := −

2 T b(−sin(i ∼ π) + cos(i ∼ π)i ∼ π) i ∼2 π2

(2-69)

222

Two-Dimensional, Steady-State Conduction 1 0.9 0.8

Position, y (m)

0.7 0.6 0.5 0.4

0.1 0.4

0.5

0.2

0.6

0.7

0.1 0 0

0.2

0.3

0.3

0.9 0.1

0.2

0.3

0.8

0.4 0.5 0.6 Position, x (m)

0.7

0.8

0.9

1

Figure 2-9: Contour plot of temperature distribution for constant temperature boundary.

The Maple result is copied and pasted into EES to achieve: N=100 [-] “number of terms in the solution to evaluate” duplicate i=1,N { C[i]=-2∗ T b∗ (-1+cos(i∗ pi))/(i∗ pi) “constant for i’th term in the series, constant temp. boundary”} C[i]=-2∗ T b∗ (-sin(i∗ pi)+cos(i∗ pi)∗ i∗ pi)/iˆ2/piˆ2 “constant for i’th term in the series, linear variation in boundary temp.” “i’th term in the series” T[i]=C[i]∗ sin(i∗ pi∗ x/W)∗ exp(-i∗ pi∗ y/W) end T=sum(T[1..N]) “sum of N terms in the series”

It is obviously not possible to include an inﬁnite number of terms in the solution and therefore a natural question is: how many terms are sufﬁcient? The magnitude of the neglected terms can be assessed by considering the magnitude of the last term that was included. For example, Figure 2-10 shows the Arrays Table that results when the solution is evaluated at x = 0.1 m and y = 0.25 m with N = 11 terms. The size of the terms in the solution drop dramatically as the index of the term increases. The accuracy of the solution computed using 11 terms is within 3.2 × 10-6 K of the actual solution and therefore it is clear that only a few terms are required at this position. However, the number of terms that are required depends on the position within the computational domain. More terms are typically required to resolve the solution near the boundary and, in particular, near boundaries where non-physical conditions are being enforced. For example, in this problem we are requiring that an edge at T = 0 (i.e., the left edge) intersect with an edge at T = 1 (i.e., the bottom edge) which results in an inﬁnite temperature gradient at x = y = 0 that cannot physically exist. Summary of Steps The steps required to solve a problem using separation of variables are summarized below: 1. Verify that the problem satisﬁes all of the conditions that are required for separation of variables. The partial differential equation must be linear and homogeneous,

2.2 Separation of Variables Solutions

Figure 2-10: Arrays Table containing the solution terms for x = 0.1 m and y = 0.25 m and N = 11 terms.

223

0.6366

0.0897

-0.3183

-0.03889

0.2122

0.01627

-0.1592

-0.006541

0.1273

0.002509

-0.1061

-0.0009065

0.09095

0.0003014

-0.07958

-0.00008735

0.07074

0.00001861

-0.06366

5.110E-18

0.05787 -0.000003165

2.

3.

4. 5.

6. 7.

all boundary conditions must be linear, and both boundary conditions in one direction (the homogeneous direction) must be homogeneous. If the problem does not meet these requirements then it may be possible to apply a simple transformation to the boundary conditions (as discussed in Section 2.2.3), use superposition (as discussed in Section 2.4), or carefully divide the problem into its homogeneous and non-homogeneous parts (as discussed in Section 2.3.2). Separate the variables; that is, express the solution (T) as the product of a function of x (TX) and a function of y (TY ). Use this approach to split the partial differential equation into two ordinary differential equations; the ordinary differential equation in the homogeneous direction should be selected so that it is solved by a function involving sines and cosines. Solve the ordinary differential equation in the homogeneous direction (the eigenproblem) and apply the boundary conditions in this direction in order to obtain the eigenfunctions and eigenvalues. Solve the ordinary differential equation in the non-homogeneous direction for each eigenvalue. Determine a solution for temperature associated with each eigenvalue, Ti , using the results from steps 3 and 4. This solution should satisfy the partial differential equation and both of the homogeneous direction boundary conditions. It is helpful to use Maple to check this solution. Express your general solution as a series composed of the solutions for each eigenvalue that resulted from step 5. Enforce the boundary conditions in the non-homogeneous direction in order to determine the constants in the series. Note that this step will require that the property of the orthogonality of the eigenfunctions be utilized at one or both of the two non-homogeneous direction boundary conditions. The property of orthogonality is utilized by multiplying the series solution by an arbitrary eigenfunction and integrating between the two homogeneous boundaries. This mathematical operation will reduce the series to a single equation involving the constants for only one of the terms in the series. The integration required to carry out this step can often be facilitated using Maple and the resulting equation can often be solved using EES.

224

Two-Dimensional, Steady-State Conduction ∂T ∂y

h (T∞ − Tx=0) = −k

∂T ∂x

=0 y =H

H ∂ 2T ∂ 2T + =0 ∂x 2 ∂y 2 W

x =0

y

∂T ∂x

x =W

∂θ ∂y

y =W

=0

x Ty =0 = Tb

(a) ∂θ ∂y h (T∞ − Tb − θ x=0 ) = −k

∂θ ∂x

=0 y =H

H ∂2θ ∂2θ + =0 ∂x 2 ∂y 2

x=0

=0

W

y x θ y=0 = 0

(b) Figure 2-11: Rectangular plate problem stated (a) in terms of temperature, T, and (b) in terms of temperature difference, θ.

2.2.3 Simple Boundary Condition Transformations The separation of variables technique discussed in Section 2.2.2 can be applied to a linear and homogeneous problem that has linear boundary conditions. In addition, both of the boundary conditions in one direction must be homogeneous; that is, any solution that satisﬁes the boundary condition must still satisfy the boundary condition if it is multiplied by a constant. Three types of linear boundary conditions are often encountered in heat transfer problems: (1) speciﬁed temperature, (2) speciﬁed heat ﬂux, and (3) convection to a ﬂuid of speciﬁed temperature. Direct application of any of these conditions generally results in a non-homogeneous boundary condition. In general, it is possible to deal with non-homogeneous boundary conditions through superposition, as discussed in Section 2.4, or by breaking a solution into its particular and homogeneous components, as discussed in Section 2.3.2. However, it is often possible to apply a relatively simple transformation in order to reduce the number of non-homogeneous boundary conditions by at least one, thereby (possibly) avoiding the need to use these advanced techniques. Consider the rectangular plate shown in Figure 2-11(a). The governing differential equation (assuming that there is no convection from the top and bottom surfaces or thermal energy generation within the plate material) is: ∂2T ∂2T + 2 =0 2 ∂x ∂y

(2-70)

2.2 Separation of Variables Solutions

225

The differential equation is linear and homogeneous. The plate has a speciﬁed temperature-type boundary condition at the bottom edge: T y=0 = T b

(2-71)

Equation (2-71) is not homogeneous unless Tb = 0. The plate has a convection-type boundary condition applied to the left edge: ∂T h (T ∞ − T x=0 ) = −k (2-72) ∂x x=0

Equation (2-72) is not homogeneous unless T∞ = 0. The plate has speciﬁed heat ﬂuxtype boundary conditions applied to the remaining two edges; these boundaries are adiabatic and therefore the speciﬁed heat ﬂux is equal to 0: ∂T =0 (2-73) ∂x x=W ∂T =0 (2-74) ∂y y=H Because the speciﬁed heat ﬂux is zero (i.e., the boundaries are adiabatic), Eqs. (2-73) and (2-74) are homogeneous. The problem posed by Figure 2-11(a) cannot be directly solved using separation of variables as neither direction is characterized by two non-homogeneous boundary conditions. However, it is possible to reduce the number of non-homogeneous boundary conditions by one. The problem is transformed and solved for the temperature difference relative to a boundary temperature; that is, the problem is solved in terms of either θ = T – T∞ or θ = T – Tb rather than T. The governing differential equation that results is unaffected by this modiﬁcation (the derivatives of θ are the same as those of T) and it is easy to re-state the remaining boundary conditions in terms of θ rather than T. For example, transforming the problem shown in Figure 2-11(a) to solve for: θ = T − Tb

(2-75)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN In Section 1.6 the constant cross-section, straight ﬁn shown in Figure 1 was analyzed under the assumption that it could be treated as an extended surface (i.e., temperature gradients in the y direction are neglected). In this example, the 2-D temperature distribution within the ﬁn will be determined using separation of variables. W

Figure 1: Straight, constant cross-sectional area ﬁn. y Tb

T∞, h

x L th

Assume that the tip of the ﬁn is insulated and that the width (W) is much larger than the thickness (th) so that convection from the edges can be neglected. The length of

E 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

results in the problem shown in Figure 2-11(b). The problem posed in terms of θ can be solved directly by separation of variables because both boundary conditions in the y-direction are homogeneous.

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

226

Two-Dimensional, Steady-State Conduction

the ﬁn is L. The ﬁn base temperature is Tb and the ﬁn experiences convection with ﬂuid at T∞ with average heat transfer coefﬁcient, h. a) Develop an analytical solution for the temperature distribution in the ﬁn using separation of variables. The upper and lower halves of the ﬁn are symmetric; that is, there is no difference between the upper and lower portions of the ﬁn and therefore no heat transfer across the mid-plane of the ﬁn. The mid-plane of the ﬁn (i.e., the surface at y = 0) can therefore be treated as if it were adiabatic. The computational domain including the boundary conditions is shown in Figure 2(a). −k

∂T ∂y

y=th/2

Tx=0 = Tb

= h (Ty=th/2 − T∞ ) th/2

2

2

∂T ∂T + =0 ∂x 2 ∂y 2

∂T ∂x

x =L

=0

L

y x

∂T ∂y

=0 y =0

(a) −k

∂θ ∂y

y=th/2

= h (θy=th/2 − θ ∞ )

θx=0 = 0

th/2

∂2θ ∂2θ + =0 ∂x 2 ∂y 2

∂θ ∂x

=0 x=L

L

y x

∂θ ∂y

=0 y=0

(b) Figure 2: Problem statement posed in terms of (a) temperature, T, and (b) temperature difference, θ.

The governing equation within the ﬁn can be derived using the process described in Section 2.2.2: ∂ 2T ∂ 2T + =0 2 ∂x ∂y2 Figure 2(a) indicates that the problem stated in terms of T has two non-homogeneous boundary conditions (the base and the top surface). However, the boundary condition at the base can be made homogeneous by deﬁning: θ = T − Tb

227

so that the governing equation becomes: ∂ 2θ ∂ 2θ + =0 2 ∂x ∂y2

(1)

The boundary conditions for the transformed problem, illustrated in Figure 2(b), are: θx=0 = 0

(2)

∂θ =0 ∂ x x=L

(3)

∂θ =0 ∂ y y =0

(4)

∂θ = h (θ y =t h/2 − θ∞ ) −k ∂ y y =t h/2

(5)

where θ∞ = T∞ − Tb The problem stated in terms of θ satisﬁes all of the requirements discussed in Section 2.2.2 with x being the homogeneous direction. Therefore, the separation of variables solution proceeds using the steps laid out in Section 2.2.2. The solution for the temperature difference (θ ) is expressed as the product of a function only of x (θ X) and a function only of y (θ Y): θ (x, y ) = θX (x) θ Y (y )

(6)

Substitution of Eq. (6) into Eq. (1) leads to two ordinary differential equations, as shown in Section 2.2.2: d 2θ X ± λ2 θ X = 0 d x2 d 2θ Y ∓ λ2 θ Y = 0 dy2 It is necessary to determine the sign of the constant λ2 in the ordinary differential equations. Recall that it is necessary to have the sine/cosine eigenfunctions in the homogeneous direction. Therefore, it is necessary to select the positive sign for the ordinary differential equation for θ X and the negative sign for the ordinary differential equation for θ Y: d 2θ X + λ2 θ X = 0 d x2

(7)

d 2θ Y − λ2 θ Y = 0 dy2

(8)

The next step is to solve the eigenproblem (i.e., the problem for θ X); the solution to the ordinary differential equation for θ X, Eq. (7), is: θ X = C 1 sin (λ x) + C 2 cos (λ x)

(9)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

228

Two-Dimensional, Steady-State Conduction

The boundary conditions for θ X are obtained by substituting Eq. (9) into Eqs. (2) and (3): θ X x=0 = 0

(10)

d θ X =0 d x x=L

(11)

Substituting Eq. (9) into Eq. (10) leads to: θ X x=0 = C 1 sin (λ 0) + C 2 cos (λ 0) = 0 0

1

which can only be true if C2 = 0. Substituting Eq. (9), with C2 = 0, into Eq. (11) leads to: d θX = C 1 λ cos (λ L) = 0 dx x=L

which can only be true if the argument of the cosine function is π /2, 3π /2, 5π /2, etc. Therefore, the argument of the cosine function must be: λi L =

(1 + 2 i) π 2

where i = 0, 1, 2, . . .

The eigenfunctions of the problem are: θ X i = C 1,i sin (λi x)

where i = 0, 1, 2, . . .

(12)

and the eigenvalues of the problem are: λi =

(1 + 2 i) π 2L

(13)

The next step is to solve the problem in the non-homogeneous direction. The ordinary differential equation in the y-direction that is associated with each eigenvalue is: d 2θ Y i − λi2 θ Y i = 0 dy2 which is solved by either θ Y i = C 3,i exp (λi y ) + C 4,i exp (−λi y ) or θ Y i = C 3,i cosh (λi y ) + C 4,i sinh (λi y )

(14)

The choice of either exponentials or sinh and cosh is arbitrary in that both will lead to the correct solution. However, the proper choice often makes the solution process easier. The plate in Figure 2-3 extended to inﬁnity where the temperature became zero. As a result, the constant multiplying the positive exponential was forced to be zero, which made the problem easier to solve. Looking ahead for this ﬁn problem, we see that the gradient of temperature at y = 0 must be 0. This boundary condition would not eliminate either of the exponential terms. On the other hand, the boundary condition will force the constant C4,i in Eq. (14) to be zero and therefore the sinh term will be eliminated. Clearly then, Eq. (14) is the better choice; a little insight early in the problem can make the solution process easier.

229

The next step is to determine the temperature difference solution associated with each eigenvalue: θi = θX i θ Y i = sin (λi x) [C 3,i cosh (λi y ) + C 4,i sinh (λi y )] where the constant C1,i was absorbed into the constants C3,i and C4,i . This solution should satisfy both of the homogeneous boundary conditions as well as the partial differential equation for all values of i; it is worthwhile using Maple to verify that this is true. Specify that i is an integer and enter the deﬁnition of the eigenvalues: > restart; > assume(i,integer); > lambda:=(1+2∗ i)∗ Pi/(2∗ L); λ :=

(1 + 2i ∼)π 2L

Enter the solution for each eigenvalue: > T:=(x,y)->sin(lambda∗ x)∗ (C3∗ cosh(lambda∗ y)+C4∗ sinh(lambda∗ y)); T := (x, y ) → sin(λx)(C 3 cosh(λy ) + C 4 sinh(λy ))

Verify that the solution satisﬁes the two boundary conditions in the x-direction, Eqs. (2) and (3): > T(0,y); 0 > eval(diff(T(x,y),x),x=L); 0

and the partial differential equation, Eq. (1): > diff(diff(T(x,y),x),x)+diff(diff(T(x,y),y),y); (1 + 2i ∼)π y (1 + 2i ∼)π x (1 + 2i ∼)π y sin (1 + 2i ∼)2 π 2 C 3 cosh + C 4 sinh 1 2L 2L 2L − 4 L2 ⎛ (1 + 2i ∼)π y C 3 cosh (1 + 2i ∼)2 π 2 1 (1 + 2i ∼)π x ⎜ 2L ⎜ + sin ⎝4 2L L2

+

1 4

C 3 sinh

(1 + 2i ∼)π y 2L L2

(1 + 2i ∼)2 π 2

⎞ ⎟ ⎟ ⎠

> simplify(%); 0

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

230

Two-Dimensional, Steady-State Conduction

The sum of the solutions for each eigenvalue becomes the general solution to the problem: θ=

∞

θi = θ X i θ Y i =

i=0

∞

sin (λi x) [C 3,i cosh (λi y ) + C 4,i sinh (λi y )]

(15)

i=0

The boundary conditions in the non-homogeneous directions are enforced. Substituting Eq. (15) into Eq. (4) leads to: ⎡ ⎤ ∞ ∂θ = sin(λi x) ⎣C 3,i λi sinh(λi 0) + C 4,i λi cosh(λi 0)⎦ = 0 ∂ y y =0 i=0 0

1

The cosh(0) = 1 and the sinh(0) = 0 (much like the cos(0) = 1 and the sin(0) = 0) and therefore this boundary condition can be written as: ∞

sin (λi x) C 4,i λi = 0

i=0

which can only be true if C4,i = 0, therefore: ∞

θ=

C i sin (λi x) cosh (λi y )

(16)

i=0

where the subscript 3 has been removed from C3,i as it is the only remaining undetermined constant. Equation (16) is substituted into the boundary condition at y = th/2, Eq. (5):

∞

th C i sin (λi x) λi sinh λi −k 2 i=0

th C i sin (λi x) cosh λi =h − θ∞ 2 i=0 ∞

which can be rearranged: ∞

C i sin (λi x)

i=0

th th sinh λi + cosh λi = θ∞ 2 2 h

k λi

(17)

The eigenfunctions must be orthogonal between x = 0 and x = L (it is not necessary to prove this for each problem) and therefore Eq. (17) can be converted into an algebraic equation for each individual constant. Equation (17) is multiplied by one eigenfunction, sin(λj x), and integrated from x = 0 to x = L: ∞ i=0

Ci

L L th th sin (λi x) sin λ j x d x = θ∞ sin λ j x d x + cosh λi sinh λi 2 2 h

k λi

0

0

231

Orthogonality guarantees that the integral on the left side of this equation will be zero for every term in the summation except the one where i = j; therefore, the series equation can be rewritten as: Ci

L L th th 2 sinh λi sin (λi x) d x = θ∞ sin (λi x) d x + cosh λi 2 2 h

k λi

0

0

The coefﬁcients are evaluated according to: L θ∞ Ci =

sin (λi x) d x 0

k λi h

sinh λi

th th + cosh λi 2 2

(18)

L sin2 (λi x) d x 0

The integrals in Eq. (18) can be evaluated either using math tables or, more easily, using Maple:

> restart; > assume(i,integer); > lambda:=(1+2∗ i)∗ Pi/(2∗ L); λ := > int(sin(lambda∗ x),x=0..L);

> int(sin(lambda∗ x)∗ sin(lambda∗ x),x=0..L);

(1 + 2i ∼)π 2L

2L (1 + 2i ∼)π L 2

The constants can therefore be written as:

Ci = L λi

2 θ∞ th th sinh λi + cosh λi 2 2 h

k λi

(19)

Equations (16) and (19) together provide the analytical solution for the temperature distribution within the ﬁn. b) Use the analytical solution to predict and plot the temperature distribution in a ﬁn that is L = 5.0 cm long, th = 4 cm thick, with conductivity k = 0.5 W/m-K, and h = 100 W/m2 -K. The base temperature is Tb = 200◦ C and the ﬂuid temperature is T∞ = 20◦ C.

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

232

Two-Dimensional, Steady-State Conduction

The inputs are entered in EES: “EXAMPLE 2.2-1: 2-D Fin” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” th cm=4 [cm] th=th cm∗ convert(cm,m) L=5 [cm]∗ convert(cm,m) k=0.5 [W/m-K] h bar=100 [W/mˆ2-K] T b=converttemp(C,K,200[C]) T inﬁnity=converttemp(C,K,20[C])

“thickness of ﬁn in cm” “thickness of ﬁn” “length of ﬁn” “thermal conductivity” “heat transfer coefﬁcient” “base temperature” “ﬂuid temperature”

Dimensionless coordinates within the ﬁn are deﬁned in order to facilitate plotting the temperature distribution: y bar=0.5 x bar=0.5 y=y bar∗ th x=x bar∗ L

“dimensionless y-position” “dimensionless x-position” “y-position” “x-position”

The solution is implemented using a duplicate loop that calculates the ﬁrst N terms of the series. The number of terms that is required for accuracy should be checked by exploring the sensitivity of the calculation to the number of terms in the same way that a numerical model should be checked for grid convergence. N=100 “number of terms in series” duplicate i=0,N “eigenvalues” lambda[i]=(1+2∗ i)∗ pi/(2∗ L) C[i]=2∗ (T inﬁnity-T b)/(L∗ lambda[i]∗ (k∗ lambda[i]∗ sinh(lambda[i]∗ th/2)/h bar+cosh(lambda[i]∗ th/2))) “constants” “term in summation” theta[i]=C[i]∗ sin(lambda[i]∗ x)∗ cosh(lambda[i]∗ y) end theta=sum(theta[0..N]) “temperature difference” “temperature” T=theta+T b “in C” T C=converttemp(K,C,T)

Figure 3 shows the temperature distribution as a function of x/L for various values of y/th. Notice that for these conditions, an extended surface (i.e., 1-D) model of the ﬁn would not be justiﬁed because there is a substantial difference between the temperature at the center of the ﬁn (y/th = 0) and the edge (y/th = 0.5). This is evident from the Biot number: Bi =

h th 2k

233

Bi=h bar∗ th/(2k)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

“Biot number”

which leads to Bi = 4.0. 200 180

Temperature (°C)

160 140 y/th = 0 y/th = 0.1 y/th = 0.2 y/th = 0.3 y/th = 0.4 y/th = 0.5

120 100 80 60 40 20 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L

0.8

0.9

1

Figure 3: Temperature as a function of x/L for various values of y/th.

c) Use the analytical solution to predict the ﬁn efﬁciency of the 2-D ﬁn. The rate of conductive heat transfer into the base of the ﬁn is: t h/2

q˙ ﬁn = −2 k W 0

∂θ dy ∂ x x=0

Substituting Eqs. (16) and (19) into Eq. (20) leads to:

q˙ ﬁn = −4 θ∞

th/2 cosh (λi y ) d y

∞ kW L i=0 k λi

0

th th sinh λi + cosh λi 2 2 h

The integral can be accomplished using Maple: > restart; > int(cosh(lambda∗ y),y=0..th/2);

sinh λ

λth 2

(20)

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

234

Two-Dimensional, Steady-State Conduction

so that the rate of conductive heat transfer to the ﬁn is: th sinh λ ∞ i kW 2 q˙ ﬁn = −4 θ∞ th th k λi L i=0 sinh λi λi + cosh λi 2 2 h

(21)

The ﬁn efﬁciency, discussed in Section 1.6.5, is deﬁned as the ratio of the rate of heat transfer to the maximum possible rate of heat transfer rate that is obtained with an inﬁnitely conductive ﬁn: ηﬁn =

q˙ ﬁn 2 hW L (Tb − T∞ )

(22)

Substituting Eq. (21) into Eq. (22) leads to the ﬁn efﬁciency predicted by the 2-D analytical solution:

ηﬁn,2D

th sinh λ ∞ i 2k 2 = 2 th th k λ i h L i=0 λ sinh λ + cosh λ i i i 2 2 h

which is evaluated in EES according to:

duplicate i=0,N eta ﬁn[i]=(2∗ k/(h bar∗ lambda[i]∗ Lˆ2))∗ sinh(lambda[i]∗ th/2)/(k∗ lambda[i]∗ sinh(lambda[i]∗ th/2)/h bar+& cosh(lambda[i]∗ th/2)) end eta ﬁn=sum(eta ﬁn[0..N])

d) Plot the ﬁn efﬁciency predicted by the 2-D analytical solution as a function of the ﬁn thickness and overlay on the plot the ﬁn efﬁciency predicted using the extended surface approximation, developed in Section 1.6. Figure 4 illustrates the ﬁn efﬁciency predicted by the 2-D model as a function of ﬁn thickness. Overlaid on Figure 4 is the solution from Section 1.6 that is listed in Table 1-4 for a ﬁn with an adiabatic tip: ηﬁn,1D =

tanh (m L) mL

where mL =

h2 L k th

As the ﬁn becomes thicker, the impact of the temperature gradients in the y-direction, neglected in the 1-D solution, become larger and therefore the 1-D

235

and 2-D solutions diverge, with the 1-D solution always over-predicting the performance. 0.35 0.3 1-D solution

Fin efficiency

0.25 0.2

2-D solution

0.15 0.1 0.05 0 0

1

2

3

4 5 6 7 Fin thickness (cm)

8

9

10

Figure 4: Fin efﬁciency as a function of the ﬁn thickness predicted by the 2-D solution and the 1-D solution.

Ratio of 2-D to 1-D fin efficiency solutions

The ratio ηﬁn,2D /ηﬁn,1D is shown in Figure 5 as a function of the Biot number; recall that the Biot number was used to justify the extended surface approximation in Section 1.6. Note that 1-D model is quite accurate (better than 2%) provided the Biot number is less than 0.1 and, surprisingly, remains reasonably accurate (10%) even up to a Biot number of 1.0. 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.05

0.1

1 Biot number

10

Figure 5: Ratio of the ﬁn efﬁciency predicted by the 2-D solution to the ﬁn efﬁciency predicted by the 1-D solution as a function of the Biot number.

EXAMPLE 2.2-1: TEMPERATURE DISTRIBUTION IN A 2-D FIN

2.2 Separation of Variables Solutions

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

236

Two-Dimensional, Steady-State Conduction

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE Figure 1 illustrates the situation where energy is transferred by conduction through a structure that suddenly changes cross-sectional area. The conduction resistance associated with this structure can be computed using separation of variables. This problem also illustrates an issue that is often confusing for separation of variables problems; speciﬁcally, the zeroth term in a cosine series must often be treated separately from the rest of the series. The proper methodology for dealing with this situation is demonstrated in this example. 2 q⋅ ′′ = 10, 000 W/m

c = 1.5 cm b = 5.0 cm k = 50 W/m-K

a = 10 cm

Figure 1: A constriction in a conduction path.

y x Tb = 0

The width of the larger cross-sectional area is b = 5.0 cm and its length is a = 10 cm. The heat ﬂux, q˙ = 10,000 W/m2 , is applied to the upper surface over a smaller width, c = 1.5 cm. The conductivity of the material is k = 50 W/m-K. The bottom surface of the object is maintained at some reference temperature, taken to be 0. a) Develop a solution for the temperature distribution in the material. The partial differential equation for the problem is: ∂ 2T ∂ 2T + =0 ∂x2 ∂y2 The boundary conditions in the x-direction are: ∂T =0 ∂ x x=0 ∂T =0 ∂ x x=b

(1)

(2)

(3)

and the boundary conditions in the y-direction are: Ty =0 = 0 " ∂T q˙ k = 0 ∂ y y =a

(4) x limit(q_dot_ﬂux∗ c∗ sinh(i∗ Pi∗ y/b)/(i∗ Pi∗ k),i=0); y q d ot f lux c bk

Substituting this result into Eq. (18) leads to: ∞

T =

q˙ c y C i cos (λi x) sinh (λi y ) + bk i=1

(21)

Substituting Eq. (21) into Eq. (5) leads to: " ∞ q˙ c q˙ x < c C i λi cos (λi x) cosh (λi a) = +k 0 x ≥c b i=1

(22)

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

2.2 Separation of Variables Solutions

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

240

Two-Dimensional, Steady-State Conduction

Next, we will deal with the non-zero terms in the series. Both sides of Eq. (22) are multiplied by cos(λj x) and integrated from x = 0 to x = b. q˙ c b

b cos(λ j x)d x + k

∞

0

i=1

c

b

=

q˙ cos(λ j x)d x +

b C i λi cosh(λi a)

cos(λi x) cos(λ j x)d x 0

0 cos(λ j x)d x c

0

The zeroth order term integrates to zero for any j > 0 and the only term of the summation that does not integrate to zero is the one where i = j: b k C i λi cosh (λi a)

cos (λi x) d x = q˙ 2

0

c cos (λi x) d x 0

The integrals in Eq. (23) are computed using Maple: > int((cos(lambda∗ x))ˆ2,x=0..b); b 2

> int(cos(lambda∗ x),x=0..c);

b sin

i ∼ πc b i∼π

in order to obtain an equation for the undetermined coefﬁcients: k C i λi cosh (λi a)

sin (λi c) b = q˙ 2 λi

The solution is programmed in EES:

“EXAMPLE 2.2-2: Constriction Resistance” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” q dot ﬂux=10000 [W/mˆ2] k=50 [W/m-K] c=1.5 [cm]∗ convert(cm,m) a=10 [cm]∗ convert(cm,m) b=5.0 [cm]∗ convert(cm,m)

“Heat ﬂux” “Conductivity” “width of applied ﬂux” “length of object” “width of object”

(23)

241

x bar=0.75 y bar=1 x=x bar∗ b y=y bar∗ a

“dimensionless x-position” “dimensionless y-position” “x-position” “y-position”

N=400 “number of terms” duplicate i=1,N “evaluate coefﬁcients for N terms” lambda[i]=i∗ pi/b k∗ C[i]∗ lambda[i]∗ cosh(lambda[i]∗ a)∗ b/2=q dot ﬂux∗ sin(lambda[i]∗ c)/lambda[i] T[i]=C[i]∗ cos(lambda[i]∗ x)∗ sinh(lambda[i]∗ y) end T=q dot ﬂux∗ c∗ y/(k∗ b)+sum(T[1..N])

A parametric table is generated and used to generate the contour plot of temperature shown in Figure 2. 1

7K

0.9

6K 5K

0.8 Position, y (m)

0.7

4K

0.6 3K

0.5 0.4

2K

0.3 0.2

1K

0.1 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Position, x (m)

0.7

0.8

0.9

1

Figure 2: Contour plot of temperature distribution in constriction.

It is worth comparing the answer with physical intuition. The temperature elevation at the constriction relative to the base is approximately 8 K according to Figure 2; does this value make sense? In Section 2.8, methods for estimating the conduction resistance of 2-D geometries using 1-D models are discussed. However, it is clear that the resistance of the constriction cannot be greater than the resistance to conduction through the material if the heat ﬂux is applied uniformly at the top surface: a Rnc = bLk where L is the length of the material. The temperature elevation at the constriction in this limit is: a q˙ c Tnc = Rnc q˙ c L = bk DeltaT nc=a∗ q dot ﬂux∗ c/(b∗ k)

“temperature rise without constriction”

This leads to Tnc = 6.0 K, which has the same magnitude as the observed temperature rise but is smaller, as expected.

EXAMPLE 2.2-2: CONSTRICTION RESISTANCE

2.2 Separation of Variables Solutions

242

Two-Dimensional, Steady-State Conduction

2.3 Advanced Separation of Variables Solutions This extended section of the book can be found on the website www.cambridge.org/ nellisandklein. Section 2.2 provides an introduction to the technique of separation of variables and discusses its application in the context of a few examples. The separation of variables method, as it is presented in Section 2.2, is rather limited as it does not allow, for example, non-homogeneous terms that might arise from effects such as volumetric generation or for problems that are in cylindrical coordinates. One technique for solving non-homogeneous partial differential equations is discussed in Section 2.3.2. The extension of separation of variables to cylindrical coordinates is presented in Section 2.3.3 and demonstrated in EXAMPLE 2.3-1. L TLHS

g⋅ ′′′

TRHS

Figure 2-15: Plane wall with uniform volumetric generation and speciﬁed edge temperatures.

x

2.4 Superposition 2.4.1 Introduction Many conduction heat transfer problems are governed by linear differential equations; in some cases, many different functions will all satisfy the differential equation. The sum of all of the functions that separately satisfy a linear differential equation will itself be a solution. This property is used in separation of variables when the solutions associated with each of the eigenfunctions are added together in order to obtain a series solution to the problem. Superposition uses this property to determine the solution to a complex problem by breaking it into several, simpler problems that are solved individually and then added together. Care must be taken to ensure that the boundary conditions for the individual problems properly add to satisfy the desired boundary condition. A series of 1-D steady-state problems appears in Chapter 1; although superposition was not used to solve these problems, it would have been possible to apply this methodology. For example, consider a plane wall with thickness (L) and conductivity (k) experiencing a constant volumetric rate of thermal energy generation (g˙ ). The edges of the wall have speciﬁed temperatures, TLHS and TRHS , as shown in Figure 2-15. The governing differential equation for this problem is: g˙ d2 T = − dx2 k

(2-126)

T x=0 = T LHS

(2-127)

T x=L = T RHS

(2-128)

and the boundary conditions are:

Notice that the governing differential equation and both boundary conditions are linear but they are not homogeneous. It is possible to solve the problem posed by Eqs. (2-126) through (2-128) without resorting to superposition; indeed there is no advantage to using

2.4 Superposition

243

2 g⋅ ′′′ d T =− dx 2 k

Tx=0 = TLHS

Tx=L = TRHS

x complete problem for T

TA, x=L = 0

2

d TA =0 dx 2

TA, x=0 = TLHS

TB, x=0 = 0

x

TB, x=L = TRHS

2

d TB =0 dx2

x

sub-problem B for TB

sub-problem A for TA

T

T

T

2

TC, x=0 = 0

x

TC, x=L = 0

d TC g⋅ ′′′ =− dx2 k

sub-problem C for TC

TLHS TA

TC

TRHS TB

0

0

L

x

0

L

0

T

0

x

0

L

x

T=TA + TB + TC

TLHS

T

TA

TC

TRHS TB 0

L

0

x

Figure 2-16: Principle of superposition applied to the plane wall of Figure 2-15.

superposition for such a simple problem. The solution was found in Section 1.3 to be: g˙ L2 x x 2 (T LHS − T RHS ) (2-129) − x + T RHS − T = 2k L L L Nevertheless, the problem shown in Figure 2-15 provides a useful introduction to superposition. The problem can be broken into three, simpler sub-problems each of which retains only one of the non-homogeneities that are inherent in the total problem. The complete problem is solved by T and is broken into sub-problems A, B, and C which are solved by the functions TA , TB and TC , respectively, as shown in Figure 2-16. Sub-problem A retains the non-homogeneous boundary condition at x = 0, but uses the homogeneous version of the differential equation (i.e., the generation term is dropped) and the boundary condition at x = L. d2 T A =0 dx2

(2-130)

T A,x=0 = T LHS

(2-131)

T A,x=L = 0

(2-132)

The solution to sub-problem A, TA , is linear from TLHS to 0 as shown in Figure 2-16. x T A = T LHS 1 − (2-133) L

244

Two-Dimensional, Steady-State Conduction

Sub-problem B retains the non-homogeneous boundary condition at x = L, but uses the homogeneous version of the differential equation and the boundary condition at x = 0: d2 T B =0 dx2

(2-134)

T B,x=0 = 0

(2-135)

T B,x=L = T RHS

(2-136)

The solution, TB , is linear from 0 to TRHS . x (2-137) L Finally, sub-problem C retains the non-homogeneous differential equation but uses the homogeneous versions of both boundary conditions: T B = T RHS

g˙ d2 T C = − dx2 k

(2-138)

T C,x=0 = 0

(2-139)

T C,x=L = 0

(2-140)

The solution, TC , is a quadratic with a maximum at the center of the wall: g˙ L2 x x 2 TC = − 2k L L

(2-141)

The solution to the complete problem is the sum of the solutions to the three subproblems: T = TA + TB + TC or T = T LHS

x g˙ L2 x + T RHS + 1− L L 2 k

TA

TB

(2-142)

x x 2 − L L

(2-143)

TC

which is identical to Eq. (2-129), the solution obtained in Section 1.3. It is easy to see from Figure 2-16 that this process of superposition must work; the differential equations for the sub-problems, Eqs. (2-130), (2-134), and (2-138) can be added together to recover Eq. (2-126): g˙ g˙ g˙ d2 (T A + T B + T C) d2 T d2 T A d2 T B d2 T C + + = − = − = − → → dx2 dx2 dx2 k dx2 k dx2 k (2-144) and the boundary conditions for the sub-problems can be added together to recover Eqs. (2-127) and (2-128): T A,x=0 + T B,x=0 + T C,x=0 = T LHS → (T A + T B + T C)x=0 = T LHS → T x=0 = T LHS (2-145) T A,x=L + T B,x=L + T C,x=L = T RHS → (T A + T B + T C)x=L = T RHS → T x=L = T RHS (2-146)

2.4 Superposition

245 Ts = 100°C Tb = 20°C

Figure 2-17: Rectangular plate used to illustrate superposition for 2-D problems.

H=1m

Ts = 100°C

W=1m y x Tb = 20°C

2.4.2 Superposition for 2-D Problems Superposition becomes much more useful for 2-D problems because the separation of variables technique is restricted to problems that have homogeneous boundary conditions in one direction. Most real problems will not satisfy this condition and therefore it is absolutely necessary to use superposition to solve these problems. The solution can be developed by superimposing several solutions, each constructed so that they are tractable using separation of variables. The process is only slightly more complex than the 1-D problem that is discussed in the previous section. For example, consider the plate with height H = 1 m and width W = 1 m, shown in Figure 2-17. The top and right sides are kept at Ts = 100◦ C while the bottom and left sides are kept at Tb = 20◦ C. The temperature distribution is a function only of x and y. The complete problem is shown in Figure 2-18(a); notice that all four boundary conditions are non-homogeneous and even transforming the problem by subtracting Tb or Ts will not result in two homogeneous boundary conditions in either the x- or y-direction. Therefore, the problem cannot be solved directly using separation of variables. Figure 2-18(b) illustrates the problem transformed by deﬁning the temperature difference relative to Tb : θ = T − Tb

(2-147)

It is necessary to break the problem for θ into two sub-problems: θ = θA + θB

(2-148)

Each sub-problem is characterized by a homogeneous direction, as shown in Figure 2-18(c). Note that for each sub-problem, the homogeneous boundary conditions that are selected are analogous to the original, non-homogeneous boundary conditions. In this problem, the speciﬁed temperature boundaries are replaced with a temperature of zero. A speciﬁed heat ﬂux boundary should be replaced with an adiabatic boundary and a boundary with convection to ﬂuid at T∞ should be replaced by convection to ﬂuid at zero temperature. Each of the two sub-problems are solved using separation of variables, as discussed in Section 2.2. The governing partial differential equation for sub-problem A: ∂ 2 θA ∂ 2 θA + =0 ∂x2 ∂y2

(2-149)

is separated into two ordinary differential equations; note that the x-direction is homogeneous for sub-problem A and therefore the ordinary differential equation

246

Two-Dimensional, Steady-State Conduction T y =H = Ts Tx=0 = Tb

Tx=W = Ts ∂2T + ∂2T = 0 ∂x 2 ∂y 2 Ty =0 = Tb problem for T (a) θy =H = Ts − Tb θx=W = Ts − Tb

θx=0 = 0 2

2

∂θ +∂θ = 0 ∂x 2 ∂y 2 θy=0 = 0 problem for θ

(b) θB, y=H = 0

θA, y =H = Ts − Tb θA, x =0 = 0

θB, x=0 ∂2θA ∂x 2

+

∂2θA ∂y 2

∂2θB

=0

∂x

θA, x =W = 0

2

+

∂2θB ∂y 2

=0 θB, x =W = Ts − Tb

θB, y=0 = 0 sub-problem for θB

θA, y=0 = 0 sub-problem for θA

(c) Figure 2-18: Mathematical description of (a) the problem for temperature T, (b) the problem for temperature difference θ, and (c) the two sub-problems θA and θB that can be solved using separation of variables.

for θX A is selected so that it is solved by sines and cosines. d2 θX A + λ2A θX A = 0 dx2

(2-150)

d2 θYA − λ2A θYA = 0 dx2 The eigenproblem is solved ﬁrst; the solution to Eq. (2-150) is: θX A = CA,1 sin (λA x) + CA,2 cos (λA x)

(2-151)

(2-152)

The homogeneous boundary condition at x = 0 leads to: θX A,x=0 = CA,1 sin (λA 0) +CA,2 cos (λA 0) = 0 0

1

(2-153)

2.4 Superposition

247

which can only be true if CA,2 = 0: θX A = CA,1 sin (λA x)

(2-154)

The homogeneous boundary condition at x = W leads to: θX A,x=W = CA,1 sin (λA W) = 0

(2-155)

which leads to the eigenvalues: iπ for i = 1, 2..∞ (2-156) W The solution to the ordinary differential equation in the non-homogeneous direction, Eq. (2-151), is: λA,i =

θYA,i = CA,3,i sinh (λA,i y) + CA,4,i cosh (λA,i y)

(2-157)

The solution for each eigenvalue is: θA,i = θX A,i θYA,i = sin (λA,i x) [CA,3,i sinh (λA,i y) + CA,4,i cosh (λA,i y)]

(2-158)

The general solution is the sum of the solutions for each eigenvalue: θA =

∞

θA,i =

i=1

∞

sin (λA,i x) [CA,3,i sinh (λA,i y) + CA,4,i cosh (λA,i y)]

(2-159)

i=1

The general solution is required to satisfy the boundary condition at y = 0: ⎤ ⎡ ∞ sin (λA,i x) ⎣CA,3,i sinh (λA,i 0) + CA,4,i cosh(λA,i 0)⎦ = 0 θA,y=0 = i=1

=0

(2-160)

=1

which can only be true if CA,4,i = 0: θA =

∞

CA,i sin (λA,i x) sinh (λA,i y)

(2-161)

i=1

The solution is required to satisfy the boundary condition at y = H: θA,y=H =

∞

CA,i sin (λA,i x) sinh (λA,i H) = T s − T b

(2-162)

i=1

The orthogonality property of the eigenfunctions is used: W

W sin (λA,i x) dx = (T s − T b) 2

CA,i sinh (λA,i H) 0

0

The integrals in Eq. (2-163) are evaluated in Maple: > restart; > assume(i,integer); > lambda:=i∗ Pi/W; λ := > int((sin(lambda∗ x))ˆ2,x=0..W);

sin (λA,i x) dx

i∼π W

W 2

(2-163)

248

Two-Dimensional, Steady-State Conduction

> int(sin(lambda∗ x),x=0..W); −

W(−1 + (−1)i∼ ) i∼π

which leads to:

CA,i sinh (λA,i H)

W = − (T s − T b) 2

( ' W −1 + (−1)i iπ

(2-164)

or: CA,i = − (T s − T b)

2 [−1 + (−1)i ] i π sinh (λA,i H)

(2-165)

When solving a problem using superposition, it is useful to separately implement and examine the solution to each of the sub-problems and verify that they separately satisfy the boundary conditions and satisfy our physical intuition. The inputs are entered in EES: $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” H=1 [m] W=1 [m] T s=converttemp(C,K,100 [C]) T b=converttemp(C,K,20 [C])

“height of plate” “width of plate” “temperature of right and top of plate” “temperature of left and bottom of plate”

The position to evaluate the temperature is speciﬁed in terms of dimensionless variables:

x=x bar∗ W y=y bar∗ H

“x-position” “y-position”

The solution to sub-problem A is implemented in EES: N=100 [-] duplicate i=1,N lambda A[i]=i∗ pi/W C A[i]=2∗ (T s-T b)∗ (-(-1+(-1)ˆi)/i/Pi)/sinh(lambda A[i]∗ H) theta A[i]=C A[i]∗ sin(lambda A[i]∗ x)∗ sinh(lambda A[i]∗ y) end theta A=sum(theta A[1..N])

The solution to sub-problem A is shown in Figure 2-19(a).

“number of terms” “eigenvalue” “evaluate constants”

“sub-problem A”

Next Page 2.4 Superposition

249

y x

(a)

y x

(b)

y x

(c) Figure 2-19: Solution for (a) sub-problem A (θA ), (b) sub-problem B (θB ), (c) and temperature difference (θ).

250

Two-Dimensional, Steady-State Conduction

A similar process leads to the solution for sub-problem B: λB,i =

iπ H

for i = 1, 2..∞ 2 [−1 + (−1)i ] i π sinh (λB,i W)

(2-167)

CB,i sin (λB,i y) sinh (λB,i x)

(2-168)

CB,i = − (T s − T b)

θB =

∞

(2-166)

i=1

which is also implemented in EES: duplicate i=1,N lambda B[i]=i∗ pi/H C B[i]=2∗ (T s-T b)∗ (-(-1+(-1)ˆi)/i/Pi)/sinh(lambda B[i]∗ W) theta B[i]=C B[i]∗ sin(lambda B[i]∗ y)∗ sinh(lambda B[i]∗ x) end theta B=sum(theta B[1..N])

“eigenvalue” “evaluate constants”

“sub-problem B”

The solution to sub-problem B is shown in Figure 2-19(b). The temperature difference solution is obtained by superposition, Eq. (2-148): theta=theta A+theta B

“temperature difference, from superposition”

and shown in Figure 2-19(c). The temperature solution (in ◦ C) is obtained with the following code: T=theta+T b T C=converttemp(K,C,T)

“temperature” “in C”

The process of superposition was illustrated in this section for a simple problem. However, it is possible to use superposition to break a relatively complicated problem with multiple, non-homogeneous and complex boundary conditions into a series of problems that are each tractable, allowing the problem to be solved and veriﬁed one sub-problem at a time.

2.5 Numerical Solutions to Steady-State 2-D Problems with EES 2.5.1 Introduction Sections 2.1 through 2.4 present analytical techniques that are useful for solving 2-D conduction problems. These solution techniques have some fairly severe limitations. For example, the shape factors discussed in Section 2.1 can be used only in those situations where the problem can be represented using one of the limited set of shape factor solutions that are available. The separation of variables techniques coupled with

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

251

superposition presented in Sections 2.2 through 2.4 can be applied to a more general set of problems; however, they are still limited to linear problems (e.g., radiation and temperature dependent properties cannot be explicitly considered) with simple boundaries. To consider a problem of any real complexity would require the superposition of many solutions and that would be somewhat time consuming. Also, the solution is speciﬁc to the problem; if any aspect of the problem changes then the solution must be re-derived. These analytical solutions are most useful for verifying numerical solutions or, in some cases, creating multi-scale models. This section begins the discussion of numerical solutions to 2-D problems. There are two techniques that are used to solve 2-D problems: ﬁnite difference solutions and ﬁnite element solutions. Finite difference solutions are discussed in this section as well as in Section 2.6. The application of the ﬁnite difference approach to 2-D problems is a natural extension of the 1-D ﬁnite difference solutions that were studied in Chapter 1. Finite difference solutions are intuitive and powerful, but difﬁcult to apply to complex geometries. Finite element solutions are dramatically different from ﬁnite difference solutions and can be applied more easily to complex geometries. A complete description of the ﬁnite element technique is beyond the scope of this book; however, ﬁnite element solutions to heat transfer problems are extremely powerful and many commercial packages are available for this purpose. Section 2.7 provides a discussion of the ﬁnite element technique followed by an introduction to the ﬁnite element package FEHT. An academic version of FEHT can be downloaded from the website www.cambridge.org/ nellisandklein. Both ﬁnite difference and ﬁnite element techniques break a large computational domain into many smaller ones that are referred to as control volumes for the ﬁnite difference technique and elements for the ﬁnite element technique. The control volumes or elements are modeled approximately in order to generate a system of equations that can be efﬁciently solved using a computer. The approximate, numerical solution will approach the actual solution as the number of control volumes or elements is increased. It is important to remember that it is not sufﬁcient to obtain a solution. Regardless of what technique you are using (including the use of a pre-packaged piece of software, such as FEHT), you must still: 1. verify that your solution has an adequately large number of control volumes or elements, 2. verify that your solution makes physical sense and obeys your intuition, and 3. verify your solution against an analytical solution in an appropriate limit. These steps are widely accepted as being “best practice” when working with numerical solutions of any type.

2.5.2 Numerical Solutions with EES Finite difference solutions to 1-D steady-state problems are presented in Sections 1.4 and 1.5. The steps required to set up a numerical solution to a 2-D problem are essentially the same; however, the bookkeeping process (i.e., the process of entering the algebraic equations into the computer) may be somewhat more cumbersome. The ﬁrst step is to deﬁne small control volumes that are distributed through the computational domain and to precisely deﬁne the locations at which the numerical model will compute the temperatures (i.e., the locations of the nodes). The control volumes are

252

Two-Dimensional, Steady-State Conduction W

Figure 2-20: Straight, constant cross-sectional area ﬁn.

y Tb

T∞ , h

x L th

small but ﬁnite; for the 1-D problems that were investigated in Chapter 1, the control volumes were small in a single dimension whereas they must be small in two dimensions for a 2-D problem. It is necessary to perform an energy balance on each differential control volume and provide rate equations that approximate each term in the energy balance based upon the nodal temperatures or other input parameters. The result of this step will be a set of equations (one for each control volume) in an equal number of unknown temperatures (one for each node). This set of equations can be solved in order to provide the numerical prediction of the temperature at each node. In this section, EES is used to solve the system of equations. In the next section, MATLAB is used to solve these types of problems. In Section 1.6, the constant cross-section, straight ﬁn shown in Figure 2-20 is analyzed under the assumption that it could be treated as an extended surface (i.e., temperature gradients in the y direction could be neglected). The ﬁn is reconsidered using separation of variables in EXAMPLE 2.2-1 without making the extended surface approximation. In this section the problem is revisited again, this time using a ﬁnite difference technique to obtain a solution for the temperature distribution. The tip of the ﬁn is insulated and the width (W) is much larger than its thickness (th) so that convection from the edges of the ﬁn can be neglected; therefore, the problem is 2-D. The length of the ﬁn is L = 5.0 cm and its thickness is th = 4.0 cm. The ﬁn base temperature is Tb = 200◦ C and it transfers heat to the surrounding ﬂuid at T∞ = 20◦ C with average heat transfer coefﬁcient, h = 100 W/m2 -K. The conductivity of the ﬁn material is k = 0.5 W/m-K. The inputs are entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” L=5.0 [cm]∗ convert(cm,m) th=4.0 [cm]∗ convert(cm,m) k=0.5 [W/m-K] h bar=100 [W/mˆ2-K] T b=converttemp(C,K,200 [C]) T inﬁnity=converttemp(C,K,20 [C])

“length of ﬁn” “width of ﬁn” “thermal conductivity” “heat transfer coefﬁcient” “base temperature” “ﬂuid temperature”

The computational domain associated with a half-symmetry model of the ﬁn is shown in Figure 2-21. The ﬁrst step in obtaining a numerical solution is to position the nodes throughout the computational domain. A regularly spaced grid of nodes is uniformly distributed, with the ﬁrst and last nodes in each dimension placed on the boundaries of the domain

2.5 Numerical Solutions to Steady-State 2-D Problems with EES T∞ , h T1, N T2, N

253

TM-1, N TM, N

Ti-1, N Ti, N Ti+1, N

T1, N-1

control volume for T control volume for top Ti, N-1 corner node M, N M, N-1 boundary node i, N Ti, j+1

Tb

Ti-1, j T1, 3 T1, 2 T2, 2 y

Ti, j

th/2

Ti+1, j

control volume for T i, j-1 internal node i, j

TM, 2

T1, 1 T2, 1 T3, 1

TM-1, 1 TM, 1

x L Figure 2-21: The computational domain associated with the constant cross-sectional area ﬁn and the regularly spaced grid used to obtain a numerical solution.

as shown in Figure 2-21. The x- and y-positions of any node (i, j) are given by: xi =

(i − 1) L (M − 1)

(2-169)

yj =

(j − 1) th 2 (N − 1)

(2-170)

where M and N are the number of nodes used in the x- and y-directions, respectively. The x- and y-distance between adjacent nodes ( x and y, respectively) are: x = y =

L (M − 1)

(2-171)

th 2 (N − 1)

(2-172)

This information is entered in EES: “Setup grid” M=40 [-] N=21 [-] duplicate i=1,M x[i]=(i-1)∗ L/(M-1) x bar[i]=x[i]/L end DELTAx=L/(M-1) duplicate j=1,N y[j]=(j-1)∗ th/(2∗ (N-1)) y bar[j]=y[j]/th end DELTAy=th/(2∗ (N-1))

“number of x-nodes” “number of y-nodes” “x-position of each node” “dimensionless x-position of each node” “x-distance between adjacent nodes” “y-position of each node” “dimensionless y-position of each node” “y-distance between adjacent nodes”

254

Two-Dimensional, Steady-State Conduction

Ti, j+1 q⋅top Ti-1, j

q⋅ LHS Δy

Ti, j q⋅ RHS Ti+1, j Δx

Figure 2-22: Energy balance for an internal node.

q⋅bottom

Ti, j-1

The next step in the solution is to write an energy balance for each node. Figure 2-22 illustrates a control volume and the associated energy transfers for an internal node (see Figure 2-21); each energy balance includes conduction from each side (q˙ RHS and q˙ LHS ), the top (q˙ top ), and the bottom (q˙ bottom ). Note that the direction associated with these energy transfers is arbitrary (i.e., they could have been taken as positive if energy leaves the control volume), but it is important to write the energy balance and rate equations in a manner that is consistent with the directions chosen in Figure 2-22. The energy balance suggested by Figure 2-22 is: q˙ RHS + q˙ LHS + q˙ top + q˙ bottom = 0

(2-173)

The next step is to approximate each of the terms in the energy balance. The material separating the nodes is assumed to behave as a plane wall thermal resistance. Therefore, y W (where W is the width of the ﬁn into the page) is the area for conduction between nodes (i, j) and (i + 1, j) and x is the distance over which the conduction heat transfer occurs. k y W (T i+1,j − T i,j ) x

q˙ RHS =

(2-174)

Note that the temperature difference in Eq. (2-174) is consistent with the direction of the arrow in Figure 2-22. The other conductive heat transfers are approximated using a similar model: q˙ LHS =

k y W (T i−1,j − T i,j ) x

(2-175)

q˙ top =

k x W (T i,j+1 − T i,j ) y

(2-176)

q˙ bottom =

k x W (T i,j−1 − T i,j ) y

(2-177)

Substituting the rate equations, Eqs. (2-174) through (2-177), into the energy balance, Eq. (2-173) written for all of the internal nodes in Figure 2-21, leads to: k y W k x W k y W (T i+1,j − T i,j ) + (T i−1,j − T i,j ) + (T i,j+1 − T i,j ) x x y +

k x W (T i,j−1 − T i,j ) = 0 y

for i = 2 . . . (M − 1)

and

j = 2 . . . (N − 1)

(2-178)

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

255 T∞ , h

Ti-1, N Figure 2-23: Energy balance for a node on the top boundary.

q⋅ LHS

Δx q⋅ conv

Ti, N

q⋅ bottom

Ti+1, N q⋅ RHS Δy

Ti, N-1

which can be simpliﬁed to: y x x y (T i+1,j − T i,j ) + (T i−1,j − T i,j ) + (T i,j+1 − T i,j ) + (T i,j−1 − T i,j ) = 0 x x y y for i = 2 . . . (M − 1) and j = 2 . . . (N − 1) (2-179) These equations are entered in EES using nested duplicate loops: “Internal node energy balances” duplicate i=2,(M-1) duplicate j=2,(N-1) DELTAy∗ (T[i+1,j]-T[i,j])/DELTAx+DELTAy∗ (T[i-1,j]-T[i,j])/DELTAx& +DELTAx∗ (T[i,j+1]-T[i,j])/DELTAy+DELTAx∗ (T[i,j-1]-T[i,j])/DELTAy=0 end end

Note that each time the outer duplicate statement iterates once (i.e., i is increased by 1), the inner duplicate statement iterates (N − 2) times (i.e., j runs from 2 to N − 1). Therefore, all of the internal nodes are considered with these two nested duplicate loops. Also note that the unknowns are placed in an array rather than a vector. The entries in the array T are accessed using two indices that are contained in square brackets. Boundary nodes must be treated separately from internal nodes, just as they are in the 1-D problems that are considered in Section 1.4; however, 2-D problems have many more boundary nodes than 1-D problems. The left boundary (x = 0) is easy because the temperature is speciﬁed: T 1,j = T b

for j = 1 . . . N

(2-180)

where Tb is the base temperature. These equations are entered in EES: “left boundary” duplicate j=1,N T[1,j]=T_b end

The remaining boundary nodes do not have speciﬁed temperatures and therefore must be treated using energy balances. Figure 2-23 illustrates an energy balance associated with a node that is located on the top boundary (at y = th/2, see Figure 2-21). The

256

Two-Dimensional, Steady-State Conduction

energy balance suggested by Figure 2-23 is: q˙ RHS + q˙ LHS + q˙ bottom + q˙ conv = 0

(2-181)

The conduction terms in the x-direction must be approximated slightly differently than for the internal nodes: q˙ RHS =

k y W (T i+1,N − T i,N ) 2 x

(2-182)

q˙ LHS =

k y W (T i−1,N − T i,N ) 2 x

(2-183)

The factor of 2 in the denominator of Eqs. (2-182) and (2-183) appears because there is half the area available for conduction through the sides of the control volumes located on the top boundary. The conduction term in the y-direction is approximated as before: q˙ bottom =

k x W (T i,N−1 − T i,N ) y

(2-184)

The convection term is: q˙ conv = h x W (T ∞ − T i,N )

(2-185)

Substituting Eqs. (2-182) through (2-185) into Eq. (2-181) for all of the nodes on the upper boundary leads to: k y W k x W k y W (T i+1,N − T i,N ) + (T i−1,N − T i,N ) + (T i,N−1 − T i,N ) 2 x 2 x y + W x h (T ∞ − T i,N ) = 0

(2-186)

for i = 2 . . . (M − 1)

which can be simpliﬁed to: y x y (T i+1,N − T i,N ) + (T i−1,N − T i,N ) + (T i,N−1 − T i,N ) 2 x 2 x y +

x h (T ∞ − T i,N ) = 0 k

(2-187)

for i = 2 . . . (M − 1)

These equations are entered in EES using a single duplicate statement: “top boundary” duplicate i=2,(m-1) DELTAy∗ (T[i+1,n]-T[i,n])/(2∗ DELTAx)+DELTAy∗ (T[i-1,n]-T[i,n])/(2∗ DELTAx)+& DELTAx∗ (T[i,n-1]-T[i,n])/DELTAy+DELTAx∗ h_bar∗ (T_inﬁnity-T[i,n])/k=0 end

Notice that the control volume at the top left corner, node (1, N), has already been speciﬁed by the equations for the left boundary, Eq. (2-180). It is important not to write an additional equation related to this node, or the problem will be over-speciﬁed. Therefore, the equations for the top boundary should only be written for i = 2 . . . (M − 1).

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

257 T∞ , h Δx q⋅ conv

TM-1, N q⋅ LHS q⋅

Figure 2-24: Energy balance for a node on the top right corner.

TM, N Δy

bottom

TM, N-1

A similar procedure for the nodes on the lower boundary leads to: y x y (T i+1,1 − T i,1 ) + (T i−1,1 − T i,1 ) + (T i,2 − T i,1 ) = 0 2 x 2 x y

for i = 2 . . . (M − 1) (2-188)

Notice that there is no convection term in Eq. (2-188) because the lower boundary is adiabatic. These equations are entered into EES: “bottom boundary” duplicate i=2,(M-1) DELTAy∗ (T[i+1,1]-T[i,1])/(2∗ DELTAx)+DELTAy∗ (T[i-1,1]-T[i,1])/(2∗ DELTAx)& +DELTAx∗ (T[i,2]-T[i,1])/DELTAy=0 end

Energy balances for the nodes on the right-hand boundary (x = L) lead to: x y x (T M,j+1 − T M,j ) + (T M,j−1 − T M,j ) + (T M−1,j − T M,j ) = 0 2 y 2 y x (2-189) for j = 2 . . . (N − 1) “right boundary” duplicate j=2,(n-1) DELTAx∗ (T[M,j+1]-T[M,j])/(2∗ DELTAy)+DELTAx∗ (T[M,j-1]-T[M,j])/(2∗ DELTAy)& +DELTAy∗ (T[M-1,j]-T[M,j])/DELTAx=0 end

The two corners (right upper and right lower) have to be considered separately. A control volume and energy balance for node (M, N), which is at the right upper corner (see Figure 2-21), is shown in Figure 2-24. The energy balance suggested by Figure 2-24 is: k y W x W k x W (T M,N−1 − T M,N ) + (T M−1,N − T M,N ) + h (T ∞ − T M,N ) = 0 2 y 2 x 2 (2-190) which can be simpliﬁed to: y h x x (T M,N−1 − T M,N ) + (T M−1,N − T M,N ) + (T ∞ − T M,N ) = 0 2 y 2 x 2k

(2-191)

258

Two-Dimensional, Steady-State Conduction

and entered into EES: “upper right corner” DELTAx∗ (T[M,N-1]-T[M,N])/(2∗ DELTAy)+DELTAy∗ (T[M-1,N]-T[M,N])/(2∗ DELTAx)+& h_bar∗ DELTAx∗ (T_inﬁnity-T[M,N])/(2∗ k)=0

The energy balance for the right lower boundary, node (M, 1), leads to: y x (T M,2 − T M,1 ) + (T M−1,1 − T M,1 ) = 0 2 y 2 x

(2-192)

“lower right corner” DELTAx∗ (T[M,2]-T[M,1])/(2∗ DELTAy)+DELTAy∗ (T[M-1,1]-T[M,1])/(2∗ DELTAx)=0

We have derived a total of M × N equations in the M × N unknown temperatures; these equations completely specify the problem and they have now all been entered in EES. Therefore, a solution can be obtained by solving the EES code. The solution is contained in the Arrays window; each column of the table corresponds to the temperatures associated with one value of i and all of the values of j (i.e., the temperatures in a column are at a constant value of y and varying values of x). The temperature solution is converted from K to ◦ C with the following equations. duplicate i=1,M duplicate j=1,N T_C[i,j]=converttemp(K,C,T[i,j]) end end

The solution is obtained for N = 21 and M = 40; the columns Ti,1 (corresponding to y/th = 0), Ti,5 (corresponding to y/th = 0.10), Ti,9 (corresponding to y/th = 0.2), etc. to Ti,21 (corresponding to y/th = 0.50) are plotted in Figure 2-25 as a function of the dimensionless x-position. The solution corresponds to our physical intuition as it exhibits temperature gradients in the x- and y-directions that correspond to conduction in these directions. The results from the analytical solution derived in EXAMPLE 2.2-1 are overlaid onto the plot and show nearly exact agreement with the numerical solution. The ﬁn efﬁciency will be used to verify that the grid is adequately reﬁned. The ﬁn efﬁciency is the ratio of the actual to the maximum possible heat transfer rates. The actual heat transfer rate per unit width (q˙ ﬁn ) is computed by evaluating the conductive heat transfer rate into the left hand side of each of the nodes that are located on the left boundary (i.e., all of the nodes where i = 1). ⎡ q˙ ﬁn = 2 ⎣k

y (T 1,1 − T 2,1 ) + 2 x

m−1 j=2

⎤ k

y y (T 1,j − T 2,j ) + k (T 1,N − T 2,N )⎦ x 2 x (2-193)

2.5 Numerical Solutions to Steady-State 2-D Problems with EES

259

200 numerical nu ri l solution o i analytical solution ana

180 y/th 0 0.1 0.2 0.3 0.4 0.5

Temperature (°C)

160 140 120 100 80 60 40 20 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L

0.8

0.9

1

Figure 2-25: Temperature predicted by numerical model and analytical model (from EXAMPLE 2.2-1) as a function of x/L for various values of y/th.

while the maximum heat transfer rate per unit width (q˙ max) is associated with an isothermal ﬁn: q˙ max = 2 L h (T b − T ∞ )

(2-194)

Note that the corner nodes must be considered outside of the duplicate loop as they have 1/ the cross-sectional area available for conduction. Also, the factor of 2 in Eq. (2-193) 2 appears because the numerical model only considers one-half of the ﬁn. “calculate ﬁn efﬁciency” q_dot_ﬁn[1]=(T[1,1]-T[2,1])∗ k∗ DELTAy/(2∗ DELTAx) duplicate j=2,(N-1) q_dot_ﬁn[j]=(T[1,j]-T[2,j])∗ k∗ DELTAy/DELTAx end q_dot_ﬁn[N]=(T[1,N]-T[2,N])∗ k∗ DELTAy/(2∗ DELTAx) q_dot_ﬁn=2∗ sum(q_dot_ﬁn[1..N]) q_dot_max=2∗ L∗ h_bar∗ (T_b-T_inﬁnity) eta_ﬁn=q_dot_ﬁn/q_dot_max

Figure 2-26 illustrates the ﬁn efﬁciency as a function of the number of nodes in the xdirection (M) for various values of the number of nodes in the y direction (N). The solution is more sensitive to M than it is to N, but appears to converge (for the conditions considered here) when M is greater than 80 and N is greater than 10. (The solution for N = 10 is not shown in Figure 2-26, because it is nearly identical to the solution for N = 20.) EES can solve up to 6,000 simultaneous equations, so a reasonably large problem can be considered using EES. However EES is not really the best tool for dealing with very large sets of equations. In the next section, we will look at how MATLAB can be used to solve this type of 2-D problem.

260

Two-Dimensional, Steady-State Conduction 0.2

Fin efficiency

0.15

N=2 N=5 N=20

0.1

0.05

0 1

10 Number of nodes in x-direction, M

100

200

Figure 2-26: Fin efﬁciency as a function of M for various values of N predicted by the numerical model.

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB 2.6.1 Introduction Section 2.5 describes how 2-D, steady-state problems can be solved using a ﬁnite difference solution implemented in EES. This process is intuitive and easy because EES will automatically solve a set of implicit equations. However, EES is not well-suited for problems that involve very large numbers of equations. The ﬁnite difference method results in a system of algebraic equations that can be solved in a number of ways using different computer tools. Large problems will normally be implemented in a formal programming language such as C++, FORTRAN, or MATLAB. This section describes the methodology associated with solving the system of equations using MATLAB; however, the process is similar in any programming language.

2.6.2 Numerical Solutions with MATLAB The system of equations that results from applying the ﬁnite difference technique to a steady-state problem can be solved by placing these equations into a matrix format: AX = b

(2-195)

where the vector X contains the unknown temperatures. Each row of the A matrix and b vector corresponds to an equation (for one of the control volumes in the computational domain) whereas each column of the A matrix holds the coefﬁcients that multiply the corresponding unknown (the nodal temperature) in that equation. To place a system of equations in matrix format, it is necessary to carefully deﬁne how the rows and energy balances are related and how the columns and unknown temperatures are related. This process is easy for the 1-D steady-state problems considered in Section 1.5, but it becomes somewhat more difﬁcult for 2-D problems. The basic steps associated with carrying out a 2-D ﬁnite difference solution using MATLAB remain the same as those discussed in Section 1.5 for a 1-D problem. The ﬁrst step is to deﬁne the structure of the vector of unknowns, the vector X in Eq. (2-195). It doesn’t really matter what order the unknowns are placed in X, but the implementation

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

261

of the solution is easier if a logical order is used. For a 2-D problem with M nodes in one dimension and N in the other, a logical technique for ordering the unknown temperatures in the vector X is: ⎤ ⎡ X 1 = T 1,1 ⎥ ⎢ ⎢ X 2 = T 2,1 ⎥ ⎥ ⎢ ⎢ X 3 = T 3,1 ⎥ ⎥ ⎢ ⎥ ⎢ ... ⎥ ⎢ ⎥ ⎢ X = ⎢ X M = T M,1 ⎥ (2-196) ⎥ ⎢ ⎢ X M+1 = T 1,2 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ X M+2 = T 2,2 ⎥ ⎥ ⎢ ... ⎦ ⎣ X M N = T M,N Equation (2-196) indicates that temperature T i,j corresponds to element X M(j−1)+i of the vector X; this mapping is important to keep in mind as you work towards implementing a solution in MATLAB. The next step is to deﬁne how the control volume equations will be placed into each row of the matrix A. For a 2-D problem with M nodes in one dimension and N in the other, a logical technique is: ⎡ ⎤ row 1 = control volume equation for node (1, 1) ⎢ row 2 = control volume equation for node (2, 1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ row 3 = control volume equation for node (3, 1) ⎥ ⎢ ⎥ ⎢ ⎥ ... ⎢ ⎥ row M = control volume equation for node (M, 1) ⎥ A=⎢ (2-197) ⎢ ⎥ ⎢ row M + 1 = control volume equation for node (1, 2) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ row M + 2 = control volume equation for node (2, 2) ⎥ ⎢ ⎥ ⎣ ⎦ ... row M N = control volume equation for node (M, N) Equation (2-197) indicates that the equation for the control volume around node (i, j) is placed into row M(j − 1) + i of matrix A. 2

T∞ = 20°C, h = 20 W/m -K ′′ = 5000 W/m 2 q⋅ inc mask holder, Tmh = 20°C pattern density, pd

th = 0.75 μm thin membrane, k = 150 W/m-K

W = 1.0 mm W = 1.0 mm y x

Figure 2-27: EPL Mask.

The process of implementing a numerical solution in MATLAB is illustrated in the context of the problem shown in Figure 2-27. An electron projection lithography (EPL)

262

Two-Dimensional, Steady-State Conduction

mask may be used to generate the next generation of computer chips. The EPL mask is used to reﬂect an electron beam onto a resist-covered wafer. The EPL mask consists of a thin membrane that extends between the much larger struts of a mask holder. The membrane absorbs the electron beam in some regions and reﬂects it in others so that the resist on the wafer is developed (i.e., exposed to energy) only in certain locations. The developing process changes the chemical structure of the resist so that it is selectively etched away during subsequent processes; thus, the pattern on the mask is transferred to the wafer. The portion of the membrane that absorbs the incident electron beam is heated. A precise calculation of the temperature rise in the mask is critical as any heating will result in thermally induced distortion that causes imaging errors in the printed features. Very small temperature rises can result in errors that are large relative to the printed features, as these features are themselves on the order of 100 nm or less. A membrane that is th = 0.75 μm thick and W = 1.0 mm on a side contains the pattern to be written. The membrane is supported by the mask holder; this is a substantially thicker piece of material that can be assumed to be at a constant temperature, T mh = 20◦ C. The thermal conductivity of the membrane is k = 150 W/m-K. The pattern density, pd (i.e., the fraction of the area of the mask that absorbs the incident radiation) can vary spatially across the EPL mask and therefore the thermal load applied to the surface of the mask area is non-uniform in x and y. The pattern density in this case is given by: pd = 0.1 + 0.5

xy W2

(2-198)

The thermal load on the mask per unit area at any location is equal to the product of the incident energy ﬂux, q˙ inc = 5000 W/m2 , and the pattern density pd at that location. The mask is exposed to ambient air on both sides. The air temperature is T ∞ = 20◦ C and the average heat transfer coefﬁcient is h = 20 W/m2 -K. The input parameters are entered in the MATLAB function EPL_Mask. The input arguments are M and N, the number of nodes in the x- and y-coordinates, while the output arguments are not speciﬁed yet. function[ ]=EPL_Mask(M,N) %[ ]=EPL_Mask % % This function determines the temperature distribution in an EPL_Mask % % Inputs: % M - number of nodes in the x-direction (-) % N - number of nodes in the y-direction (-) %INPUTS W=0.001; th=0.75e-6; q dot ﬂux=5000; k=150; T mh=20+273.2; T inﬁnity=20+273.2; h bar=20;

% width of membrane (m) % thickness of membrane (m) % incident energy (W/mˆ2) % conductivity (W/m-K) % mask holder temperature (K) % ambient air temperature (K) % heat transfer coefﬁcient (W/mˆ2-K)

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

263

T1, N

TM, N T q⋅conv i, j+1⋅ qtop q⋅LHS Ti, j Ti-1, j Ti+1, j g⋅ abs q⋅ RHS q⋅bottom

Figure 2-28: Numerical grid and control volume for an internal node (i, j).

Ti, j-1 T1, 2

T2, 2

T1, 1

T2, 1

TM, 1

A sub-function pd_f is created (at the bottom of the function EPL-Mask) in order to provide the pattern density: function[pd]=pd_f(x,y,W) % [pd]=pd_f(x,y,W) % % This sub-function returns the pattern density of the EPL mask % % Inputs: % x - x-position (m) % y - y-position (m) % W - dimension of mask (m) % Output: % pd - pattern density (-) pd=0.1+0.5∗ x∗ y/Wˆ2; end

The problem is two-dimensional; the membrane is sufﬁciently thin that temperature gradients in the z-direction can be neglected. This assumption can be veriﬁed by calculating an appropriate Biot number: Bi =

h th = 1 × 10−7 k

(2-199)

Therefore, a 2-D numerical model will be generated using the grid shown in Figure 2-28. The x- and y-coordinates of each node are provided by: xi =

(i − 1) W (M − 1)

for i = 1..M

(2-200)

yi =

(j − 1) W (N − 1)

for j = 1..N

(2-201)

264

Two-Dimensional, Steady-State Conduction

The distance between adjacent nodes is: x =

W (M − 1)

(2-202)

y =

W (N − 1)

(2-203)

The grid is set up in the MATLAB function: %Setup grid for i=1:M x(i,1)=(i-1)∗ W/(M-1); end DELTAx=W/(M-1); for j=1:N y(j,1)=(j-1)∗ W/(N-1); end DELTAy=W/(N-1);

The problem will be solved by placing the system of equations that result from considering each control volume into matrix format. A control volume for an internal node is shown in Figure 2-28. The energy balance for this control volume includes conduction from the left and right sides (q˙ LHS and q˙ RHS ) and the top and bottom (q˙ top and q˙ bottom ) as well as generation of thermal energy due to the absorbed illumination (g˙ abs ) and heat loss due to convection (q˙ conv ). The energy balance suggested by Figure 2-28 is: q˙ RHS + q˙ LHS + q˙ top + q˙ bottom + g˙ abs = q˙ conv

(2-204)

The conduction terms are approximated using the technique discussed in Section 2.5: q˙ RHS =

k y th (T i+1,j − T i,j ) x

(2-205)

q˙ LHS =

k y th (T i−1,j − T i,j ) x

(2-206)

q˙ bottom = q˙ top =

k x th (T i,j−1 − T i,j ) y

(2-207)

k x th (T i,j+1 − T i,j ) y

(2-208)

The absorbed energy is: g˙ abs = q˙ inc pd x y

(2-209)

The rate of convection heat transfer is: q˙ conv = 2 h x y (T i,j − T ∞ )

(2-210)

Substituting Eqs. (2-205) to (2-210) into Eq. (2-204) for all internal nodes leads to: k y th k x th k x th k y th (T i−1,j −T i,j ) + (T i+1,j −T i,j ) + (T i,j−1 −T i,j ) + (T i,j+1 −T i,j ) x x y y + q˙ inc pd x y = 2 h x y (T i,j −T ∞ )

for i = 2..(M − 1)

and

j = 2..(N − 1) (2-211)

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

265

Equation (2-211) is rearranged to identify the coefﬁcients that multiply each unknown temperature: k x th k y th k y th k y th T i,j −2 + T i+1,j −2 − 2 h x y + T i−1,j x y x x

+ T i,j−1

AM(j−1)+i,M(j−1)+i

k x th y

+ T i,j+1

AM(j−1)+i,M(j−1−1)+i

AM(j−1)+i,M(j−1)+i−1

AM(j−1)+i,M(j−1)+i+1

k x th = −2 h x y T ∞ − q˙ inc pd x y y bM(j−1)+i

(2-212)

AM(j−1)+i,M(j+1−1)+i

for i = 2..(M − 1)

and

j = 2..(N − 1)

The control volume equations must be placed into the matrix equation: AX = b

(2-213)

where the equation for the control volume around node (i, j) is placed into row M(j − 1) + i of A and T i,j corresponds to element X M(j−1)+i in the vector X, as required by Eqs. (2-197) and (2-196), respectively. Each coefﬁcient in Eq. (2-212) (i.e., each term multiplying an unknown temperature on the left side of the equation) must be placed in the row of A corresponding to the control volume being examined and the column of A corresponding to the unknown in X. The matrix assignments consistent with Eq. (2-212) are: AM(j−1)+i,M(j−1)+i = −2

k y th k x th −2 − 2 h x y x y

for i = 2..(M − 1)

and

(2-214)

j = 2..(N − 1)

AM(j−1)+i,M(j−1)+i−1 =

k y th x

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-215)

AM(j−1)+i,M(j−1)+i+1 =

k y th x

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-216)

AM(j−1)+i,M(j−1−1)+i =

k x th y

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-217)

AM(j−1)+i,M(j+1−1)+i =

k x th y

for i = 2..(M − 1)

and

j = 2..(N − 1)

(2-218)

bM(j−1)+i = −2 h x y T ∞ − q˙ inc pd x y for i = 2..(M − 1)

and

j = 2..(N − 1) (2-219)

A sparse matrix is allocated in MATLAB for A and the equations derived above are implemented using nested for loops. The spalloc command requires the number of rows and columns and the maximum number of non-zero elements in the matrix. Note that there are at most ﬁve non-zero entries in each row of A, corresponding to Eqs. (2-214) through (2-218); thus the last argument in the spalloc command is 5 M N.

266

Two-Dimensional, Steady-State Conduction

A=spalloc(M∗ N,M∗ N,5∗ M∗ N); %allocate a sparse matrix for A %energy balances for internal nodes for i=2:(M-1) for j=2:(N-1) A(M∗ (j-1)+i,M∗ (j-1)+i)=-2∗ k∗ DELTAy∗ th/DELTAx-2∗ k∗ DELTAx∗ th/DELTAy-... 2∗ h_bar∗ DELTAx∗ DELTAy; A(M∗ (j-1)+i,M∗ (j-1)+i-1)=k∗ DELTAy∗ th/DELTAx; A(M∗ (j-1)+i,M∗ (j-1)+i+1)=k∗ DELTAy∗ th/DELTAx; A(M∗ (j-1)+i,M∗ (j-1-1)+i)=k∗ DELTAx∗ th/DELTAy; A(M∗ (j-1)+i,M∗ (j+1-1)+i)=k∗ DELTAx∗ th/DELTAy; b(M∗ (j-1)+i,1)=-2∗ h_bar∗ DELTAx∗ DELTAy∗ T_inﬁnity-... q_dot_ﬂux∗ pd_f(x(i,1),y(j,1),W)∗ DELTAx∗ DELTAy; end end

The boundary nodes have speciﬁed temperature: T 1,j

= T mh

[1]

= T mh

[1] AM(j−1)+M,M(j−1)+M

T i,1

[1] AM(1−1)+i,M(1−1)+i

T i,N

[1] AM(N−1)+i,M(N−1)+i

(2-220)

bM(j−1)+1

AM(j−1)+1,M(j−1)+1

T M,j

for j = 1..N

for j = 1..N

(2-221)

bM(j−1)+M

= T mh

for i = 2..(M − 1)

(2-222)

bM(1−1)+i

= T mh

for i = 2..(M − 1)

(2-223)

bM(N−1)+i

Note that Eqs. (2-220) through (2-223) are written so that the corner nodes (e.g. node (1, 1)) are not speciﬁed twice. The matrix assignments suggested by Eqs. (2-220) through (2-223) are: AM(j−1)+1,M(j−1)+1 = 1 bM(j−1)+1 = T mh

for j = 1..N

AM(j−1)+M,M(j−1)+M = 1 bM(j−1)+M = T mh

for j = 1..N

for j = 1..N

for j = 1..N

AM(1−1)+i,M(1−1)+i = 1 for i = 2..(M − 1) bM(1−1)+i = T mh

for i = 2..(M − 1)

AM(N−1)+i,M(N−1)+i = 1 bM(N−1)+i = T mh

for i = 2..(M − 1)

for i = 2..(M − 1)

(2-224) (2-225) (2-226) (2-227) (2-228) (2-229) (2-230) (2-231)

2.6 Numerical Solutions to Steady-State 2-D Problems with MATLAB

267

These assignments are implemented in MATLAB: %speciﬁed temperatures around all edges for j=1:N A(M∗ (j-1)+1,M∗ (j-1)+1)=1; b(M∗ (j-1)+1,1)=T_mh; A(M∗ (j-1)+M,M∗ (j-1)+M)=1; b(M∗ (j-1)+M,1)=T_mh; end for i=2:(M-1) A(M∗ (1-1)+i,M∗ (1-1)+i)=1; b(M∗ (1-1)+i,1)=T_mh; A(M∗ (N-1)+i,M∗ (N-1)+i)=1; b(M∗ (N-1)+i,1)=T_mh; end

The vector X is obtained using MATLAB’s backslash command and the temperature of each node in degrees Celsius is placed in the matrix T_C. X=A\b; for i=1:M for j=1:N T_C(i,j)=X(M∗ (j-1)+i)-273.2; end end end

The function header is modiﬁed so that running the MATLAB function provides the temperature prediction (the matrix T_C) as well as the vectors x and y that contain the x- and y-positions of each node in the matrix. function[x,y,T_C]=EPL_Mask(M,N) % [x,y,T_C]=EPL_Mask % % This function determines the temperature distribution in an EPL_Mask % % Inputs: % M - number of nodes in the x-direction (-) % N - number of nodes in the y-direction (-) % Outputs: % x - Mx1 vector of x-positions of each node (m) % y - Nx1 vector of y-positions of each node (m) % T_C - MxN matrix of temperature at each node (C)

The solution should be examined for grid convergence. Figure 2-29 illustrates the maximum temperature in the EPL mask as a function of M and N (the two parameters are set equal for this analysis). The analysis is carried out using the script varyM, below, which deﬁnes a vector Mv that contains a range of values of the number of nodes, M, and runs

268

Two-Dimensional, Steady-State Conduction 20.76

Maximum temperature (°C)

20.75 20.74 20.73 20.72 20.71 20.7 20.69 20.68 4

10

100

300

Number of nodes Figure 2-29: Maximum predicted temperature as a function of the number of nodes (M and N).

the function EPL_Mask for each value. The command max(max(T_C)) computes the maximum value of each column and then the maximum value of the resulting vector in order to obtain the maximum nodal temperature in the mask. clear all; Mv=[5;10;20;30;50;70;100;200]; % values of M to use for i=1:8 [x,y,T_C]=EPL_Mask(Mv(i),Mv(i)); % obtain temperature distribution T_Cmaxv(i,1)=max(max(T_C)) % obtain maximum temperature end

There is a variety of 3-D plotting functions in MATLAB; these can be investigated by typing help graph3d at the command window. For example, >> mesh(x,y,T_C’); >> colorbar;

produces a mesh plot indicating the temperature, as shown in Figure 2-30. Note that the matrix T C has to be transposed (by adding the ’ character after the variable name) in order to match the dimensions of the x and y vectors.

2.6.3 Numerical Solution by Gauss-Seidel Iteration This extended section of the book can be found on the website www.cambridge.org/ nellisandklein. A ﬁnite difference solution results in a system of algebraic equations that must be solved simultaneously. In Section 2.6.2, we looked at placing these equations into a matrix equation that was solved by a single matrix inversion (or the equivalent mathematical manipulation). An alternative technique, Gauss-Seidel iteration, can also be used to approximately solve the system of equations using an iterative technique that requires much less memory than the direct matrix solution method. In some cases

2.8 Resistance Approximations for Conduction Problems

269

Temperature (deg. C)

21 20.8 20.6 20.4 20.2 20 1 0.8 x 10

-3

0.6 0.4 0.2 Position, x (m)

0

0

0.4

0.2

0.6

1

0.8

-3

x 10

Position, y (m)

Figure 2-30: Mesh plot of temperature distribution.

it can require less computational effort as well. The Gauss-Seidel iteration process is illustrated using the EPL mask problem that was discussed in Section 2.6.2.

2.7 Finite Element Solutions This extended section of the book can be found on the website www.cambridge.org/ nellisandklein. Sections 2.5 and 2.6 present the ﬁnite difference method for solving 2D steady-state conduction problems. In Section 2.7.1, the FEHT (Finite Element Heat Transfer) program is discussed and used to solve EXAMPLE 2.7-1. FEHT implements the ﬁnite element technique to solve 2-D steady-state conduction problems. A version of FEHT that is limited to 1000 nodes can be downloaded from www.cambridge.org/ nellisandklein. In order to become familiar with FEHT it is suggested that the reader go through the tutorial provided in Appendix A.4 which can be found on the web site associated with the book (www.cambridge.org/nellisandklein). In Section 2.7.2, the theory behind ﬁnite element techniques is presented.

2.8 Resistance Approximations for Conduction Problems 2.8.1 Introduction The resistance to conduction through a plane wall (R pw ) was derived in Section 1.2 and is given by: R pw =

L k Ac

(2-232)

The concept of a thermal resistance is a broadly useful and practical idea that goes beyond the simple situation for which Eq. (2-232) was derived. It is possible to understand conduction heat transfer in most situations if you can identify the appropriate distance that heat must be conducted (L) and the area through which that conduction occurs (Ac ). EXAMPLE 2.8-1 illustrates this type of “back-of-the-envelope” calculation.

Two-Dimensional, Steady-State Conduction

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET You may be faced with trying to understand the heat transfer through a complex, 2-D or 3-D geometry, such as the bracket illustrated in Figure 1. The bracket is made of steel having a thermal conductivity k = 14 W/m-K. One surface of the bracket is held at TH = 200◦ C and the other is at TC = 20◦ C. It is beyond the scope of any technique discussed in this book to analytically determine the heat ﬂow through this geometry and therefore it will be necessary to use a ﬁnite element software package for this purpose. However, it is possible to use the resistance concept represented by Eq. (2-232) in order to bound and estimate the heat ﬂow through the bracket. If you determine that the heat ﬂow through the bracket cannot possibly be important to the larger application (whatever that is) then the time and money required to generate the ﬁnite element model can be saved. If a ﬁnite element model is generated, then the simple thermal resistance estimate can provide a sanity check on the results.

1 cm

surface held at 200°C

1 cm

surface held at 20°C

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

270

Figure 1: A bracket with a complex, 2-D geometry made of steel with k = 14 W/m-K and thickness 1 cm (into the page).

a) Estimate the rate of heat transfer through the bracket using a resistance approximation. The length that heat must be conducted in order to go from the surface at TH to the surface at TC is approximately L = 14 cm and the area for conduction is approximately Ac = 1 cm2 . Clearly these are not exact values because the problem is two-dimensional; some energy must ﬂow a longer distance to reach the more proximal regions of the bracket and there are several portions of the bracket where the area is larger than 1 cm2 . However, it is possible to estimate the resistance of the bracket with these approximations: 5 100 cm L 14 cm m K K 5 = = 100 Rbr acket ≈ 5 2 kA 14 W 1 cm m W

271

which provides an estimate of the heat ﬂow:

q˙ ≈

(TH − TC ) (200◦ C − 20◦ C) W = 100 K = 1.8 W Rbr acket

It may be that 1.8 W is a trivial rate of energy loss from whatever is being supported by the bracket and therefore the bracket does not require a more detailed analysis. However, if a more exact answer is required then a ﬁnite element solution is necessary. b) Use FEHT to determine the rate of heat transfer through the bracket. The geometry from Figure 1 can be entered in FEHT and solved, as discussed in Section 2.7.1 and Appendix A.4. Set a scale where 1 cm on the screen corresponds to 0.01 m and use the Outline selection from the Draw menu to approximately trace out the bracket. Then, right-click on each of the corner nodes and enter the exact position in the Node Information Dialog. The boundary conditions should be set as well as the material properties. Create a crude mesh and reﬁne it. The problem is solved and the solution is shown in Figure 2.

Figure 2: Solution.

The total heat ﬂux at either the 200◦ C or the 20◦ C boundaries can be obtained by selecting Heat Flows from the View menu and then selecting all of the nodal boundaries along these boundaries. (Left-click and drag a selection rectangle.) At the TH boundary, the total heat ﬂow is reported as 249.8 W/m (Figure 3) or, for a 1 cm thick bracket, 2.5 W.

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

2.8 Resistance Approximations for Conduction Problems

EXAMPLE 2.8-1: RESISTANCE OF A BRACKET

272

Two-Dimensional, Steady-State Conduction

Figure 3: Heat ﬂow along the top boundary.

The same calculation along the TC boundary leads to 2.5 W. Therefore, the total heat ﬂow is within 40% of the 1.8 W value predicted by the simple resistance approximation. The sanity check is valuable; if the ﬁnite element model had predicted 10’s or 100’s of W then it would be almost certain that there is an error in the solution (perhaps a unit conversion or a material property entered incorrectly). Furthermore, the 1-D solution was an underestimate of the heat transfer because it did not account for the regions of the bracket that have larger cross-section. In the next sections, several methods are presented that can be used to bound the thermal resistance of a multi-dimensional object using 1-D resistances that are calculated with speciﬁc assumptions.

2.8.2 Isothermal and Adiabatic Resistance Limits Figure 2-31(a) illustrates a composite structure made from four materials (A through D). The composite structure experiences convection on its left and right sides to ﬂuid temperatures T C = 0◦ C and T H = 100◦ C, respectively, with average heat transfer coefﬁcient h = 500 W/m2 -K. The other surfaces are insulated. The problem represented by Figure 2-31(a) is 2-D; to see this clearly, imagine the situation where material B has very low conductivity, kB = 1.0 W/m-K, and material C has very high conductivity, kC = 100.0 W/m-K. Material A and D both have intermediate conductivity, kA = kD = 10 W/m-K. In this limit, thermal energy will transfer primarily through material C with very little passing through material B. The problem was solved in FEHT and the resulting temperature distribution is shown in Figure 2-31(b); the heat transfer rate is 281.9 W (assuming unit width into the page). The composite structure cannot be represented exactly with a 1-D resistance network due to the temperature gradients in the y-direction. In order to estimate the behavior of the system using 1-D resistance concepts, it is necessary to either allow unrestricted heat ﬂow in the y-direction (referred to as the isothermal limit) or completely eliminate heat ﬂow in the y-direction (referred to as the adiabatic limit). The temperature distribution is substantially simpliﬁed in these two limiting cases, as shown in Figure 2-32(a) and Figure 2-32(b), respectively. Note that Figure 2-32(a) and (b) can be obtained using FEHT; the material properties provides an Anisotropic ky/kx Type. For Figure 2-32(a), the ky/kx value is set to a large value (10,000) which effectively eliminates any resistance to heat ﬂow in the y-direction. In Figure 2-32(b), the ky/kx value is set to a small value (0.0001) which essentially prevents any heat ﬂow in the y-direction. The 1-D resistance network that corresponds to the isothermal limit is shown in Figure 2-33. The isothermal limit implies that there are no temperature gradients in the y-direction and therefore the temperature at any axial location may be represented by a single node. Note that in Figure 2-33, Ac is the area of the composite (0.02 m2 assuming

2.8 Resistance Approximations for Conduction Problems

k A = 10

W m-K

LD= 1 cm 1 cm

TC = 0°C W h = 500 2 m -K

2 cm

273

y x

kB = 1

W m-K

W kC = 100 m-K 1 cm

TH = 100°C W h = 500 2 m -K k D = 10

LA= 1 cm

W m-K

LB = LC = 2 cm

(a)

(b) Figure 2-31: (a) A composite structure consisting of four materials with convection from each edge and (b) the temperature distribution (◦ C) that will occur if kC kB .

a unit width into the page) and LA, LB, etc. are the thicknesses of the materials in the x-direction. The total resistance associated with the resistance network shown in Figure 2-33 is 0.32 K/W and therefore the rate of heat transfer through the composite structure predicted in the isothermal limit is 313 W. The isothermal limit corresponds to a lower bound on the thermal resistance, since the resistance to heat ﬂow in the y-direction is neglected. The adiabatic limit assumes that there is no heat transfer in the y-direction. Thermal energy can only pass through the composite axially and therefore the resistance network corresponding to the adiabatic limit consists of parallel paths for heat ﬂow by convection and through materials B and C (these parallel paths are labeled 1 and 2), as shown in Figure 2-34. The total thermal resistance associated with the resistance network shown in Figure 2-34 is 0.50 K/W and the rate of heat transfer through the composite structure predicted in the adiabatic limit is 200 W. This adiabatic limit corresponds to an upper bound on the resistance since the thermal energy is prohibited from spreading in the y-direction. The true solution lies somewhere between the isothermal (313 W) and adiabatic (200 W) limits; the FEHT model in Figure 2-31 predicted 282 W. Thus, the limits

274

Two-Dimensional, Steady-State Conduction

Figure 2-32: The temperature distribution in (a) the isothermal limit (kA,y , kB,y , kC,y , and kD,y → ∞) and (b) the adiabatic limit (kA,y , kB,y , kC,y , and kD,y → 0).

(a)

(b) 2.0

TC = 0°C

0.1

K W

Rconv =

0.05

K W

K W 2 LB k B Ac

R cond, B =

L 1 Rcond, A = A h Ac k A Ac

0.02

K W

Rcond, C =

0.05

K W

Rcond, D =

0.1

K W T = 100°C H

LD 1 R conv = h Ac k D Ac

2 LC kC Ac

Figure 2-33: Resistance network representing the isothermal limit of the behavior of the composite structure.

0.2

TC = 0°C

K W

R conv,1 = 0.2

K W

2.0

K W

0.1

K W

0.2

K W

2 2 LA 2 LB 2 LD 2 Rcond, A,1 = Rcond, B = R cond, D,1 = Rconv,1 = A k D Ac h Ac k A Ac kB c h Ac TH = 100°C

K W

R conv, 2 =

0.1

0.1

K W

0.02

K W

0.1

K W

0.2

K W

2 2 LA 2 LD 2 2 LC Rcond, D, 2 = R conv, 2 = R cond, A, 2 = R cond, C = k A Ac k D Ac h Ac h Ac kC Ac

Figure 2-34: Resistance network representing the adiabatic limit of the behavior of the composite structure.

2.8 Resistance Approximations for Conduction Problems

275 adiabatic surfaces

area Ac, 2 at TC

Figure 2-35: An example of a geometry with constant length and varying area. L area Ac,1 at TH L

are useful for determining the validity of a 2-D solution as well as bounding the problem without requiring a 2-D solution.

2.8.3 Average Area and Average Length Resistance Limits There are conduction problems in which the length for conduction is known but the area of the conduction path varies along this length; a simple example is shown in Figure 2-35. The adiabatic approximation discussed in Section 2.8.2 suggests that the resistance of this shape is: Rad =

2L k Ac,1

(2-233)

where k is the conductivity of the material. The isothermal approximation for the resistance yields: Riso =

L L + k Ac,1 k Ac,2

(2-234)

An alternative technique for estimating the resistance in this situation is to use the average area for conduction: RA =

4L k (Ac,1 + Ac,2 )

(2-235)

The resistance based on the average area underestimates the actual resistance to a greater extent than even the isothermal approximation. To see that this is so, imagine the case where Ac,1 approaches zero; clearly the actual resistance will become inﬁnite and both the adiabatic and isothermal approximations provided by Eqs. (2-233) and (2-234), respectively, predict this. However, the average area estimate remains ﬁnite and therefore substantially under-predicts the resistance. The alternative situation may occur, where the area for conduction is essentially constant but the length varies (perhaps randomly, as in a contact resistance problem). In this case, it is natural to use an average length to compute the resistance as shown in Figure 2-36. The average length estimate of the resistance (RL) is: RL =

L k Ac

(2-236)

where L is the average length for conduction. The average length model overestimates the resistance to a greater extent than the adiabatic approximation. Consider the case where the length anywhere within the shape shown in Figure 2-36 approaches zero, which would cause the actual resistance to become zero. The adiabatic approximation will faithfully predict a zero resistance while the average length estimate will remain

276

Two-Dimensional, Steady-State Conduction

L

TH

TC

Figure 2-36: An example of a geometry with constant area and varying length.

Ac

ﬁnite. In terms of accuracy, the various 1-D estimates that have been discussed can be arranged in the following order: RA ≤ Riso ≤ R ≤ Rad ≤ RL

(2-237)

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

where R is the actual thermal resistance. EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL Figure 1 illustrates a square channel. The inner and outer surfaces are held at different temperatures, T1 = 250◦ C and T2 = 50◦ C, respectively. T2 = 50°C T1 = 250°C

L=1m b = 5 cm

a = 10 cm b = 5 cm

Figure 1: Square channel with heat transfer from inner to outer surface.

a = 10 cm

The outer dimension of the square channel is a = 10 cm and the inner dimension is b = 5.0 cm. The thermal conductivity of the material is k = 100 W/m-K and the length of the channel is L = 1 m. a) Using an appropriate shape factor, determine the actual rate of heat transfer through the square channel. The inputs are entered in EES: “EXAMPLE 2.8-2: Resistance of a Square Channel” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” a=10[cm]∗ convert(cm,m) b=5[cm]∗ convert(cm,m) L=1[m] k=100[W/m-K] T 1=converttemp(C,K,250 [C]) T 2=converttemp(C,K,50 [C])

“outer dimension of square channel” “inner dimension” “length” “material conductivity” “inner wall temperature” “outer wall temperature”

277

This problem is a 2-D conduction problem; however, the solution for this 2-D problem is correlated in the form of a shape factor, S. Shape factors are discussed in Section 2.1. The shape factor (S) is deﬁned according to: q˙ cond = S k (T1 − T2 ) where q˙ cond is the rate of conductive heat transfer between the two surfaces at T1 and T2 . The particular shape factor for a square channel can be accessed from EES’ built-in shape factor library. Select Function Info from the Options menu and select Shape Factors from the lower-right pull down menu. Use the scroll-bar to select the shape factor function for a square channel. The function SF_7 is pasted into the Equation Window using the Paste button and used to calculate the actual heat transfer rate: “Actual heat transfer rate” SF=SF 7(b,a,L) q dot=SF∗ k∗ (T 1-T 2) q dot kW=q dot∗ convert(W,kW)

“shape factor for square channel” “heat transfer” “heat transfer in kW”

The actual heat transfer rate predicted using a shape factor solution is 211.4 kW. b) Provide a lower bound on the heat transfer through the square channel using an appropriate 1-D model. According to Eq. (2-237), the adiabatic or average length models can be used to provide an upper bound on the resistance of the square channel. The adiabatic model does not allow the heat to spread as it moves across the channel, as shown in Figure 2(a).

(a − b )

(a − b )

2 b

2

(a + b ) 2

(a)

(b)

Figure 2: 1-D models based on the (a) adiabatic limit which allows no heat spreading and (b) the average area limit which uses the average area along the heat transfer path.

The resistance in the adiabatic limit is equal to the resistance of a plane wall with an area equal to the internal surface area of the channel and length equal to the channel thickness: a−b Rad = 8k L b The rate of heat transfer predicted by the adiabatic limit is: q˙ ad =

(T1 − T2 ) Rad

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

2.8 Resistance Approximations for Conduction Problems

EXAMPLE 2.8-2: RESISTANCE OF A SQUARE CHANNEL

278

Two-Dimensional, Steady-State Conduction

“Adiabatic limit” R ad=(a-b)/(8∗ k∗ L∗ b) q dot ad=(T 1-T 2)/R ad q dot ad kW=q dot ad∗ convert(W,kW)

“thermal resistance in the adiabatic limit” “heat transfer in the adiabatic limit”

The adiabatic model predicts 160.0 kW and therefore leads to a 32% underestimate of the actual heat transfer rate (211.4 kW from part (a)). c) Provide an upper bound on the heat transfer rate using an appropriate 1-D model. Equation (2-237) indicates that either the isothermal or average area approach can be used to establish a lower bound on the thermal resistance and therefore an upper bound on the heat transfer. The average area along the heat transfer path is shown in Figure 2(b). The resistance calculated according to the average area model is: RA =

(a − b) 4 a+b Lk

The heat transfer in this limit is: q˙ A =

(T1 − T2 ) RA

“Average area limit” R A bar=(a-b)/(4∗ (a+b)∗ k∗ L) “thermal resistance in the average area limit” “heat transfer in the average area limit” q dot A bar=(T 1-T 2)/R A bar q dot A bar kW=q dot A bar∗ convert(W,kW)

The average area approximation predicts a heat transfer rate of 240 kW and is therefore a 12% overestimate of the actual heat transfer rate.

2.9 Conduction through Composite Materials 2.9.1 Effective Thermal Conductivity Composite structures are made by joining different materials to create a structure with beneﬁcial properties. Composites are often encountered in engineering applications; for example, motor laminations and windings, screens, and woven fabric composites. The length scale associated with the underlying structure of the composite material is often much smaller than the length scale associated with the overall problem of interest and therefore the details of the local energy ﬂow through the materials that make up the structure are not important. In this case, the composite material can be modeled as a single, equivalent material with an effective conductivity that reﬂects the more complex behavior of the underlying structure. If the composite structure is not isotropic (i.e., it is anisotropic) then the effective conductivity may also be anisotropic; that is, the effective conductivity may depend on direction, reﬂecting some underlying characteristic of the composite structure that allows heat to ﬂow more easily in certain directions. The effective thermal conductivity of the composite structure must be determined by considering the details associated with heat transfer through the structure. The process of determining the effective conductivity involves (theoretically) imposing a temperature gradient in one direction and evaluating the resulting heat transfer rate. The

2.9 Conduction through Composite Materials epoxy kep = 2 W/m-K thep = 0.2 mm

Ttop = 40°C

279

iron 3 g⋅ ′′′= 100,000 W/m thlam = 0.5 mm klam = 10 W/m-K

H = 6 cm W q⋅ ′′ = 5000 2 m

W q⋅ ′′= 5000 2 m y x

L = 4 cm

Figure 2-37: Motor pole.

effective conductivity in that direction is the conductivity of a homogeneous material that would yield the same heat transfer rate. It is often possible to determine the effective conductivity by inspection; however, for complex structures it will be necessary to generate a detailed, numerical model of a unit cell of the structure using a ﬁnite difference or ﬁnite element technique. The combination of a local model of the very small scale features of the underlying structure (in order to determine the effective conductivity) and a larger scale model of the global problem is sometimes referred to as multi-scale modeling. Other effective characteristics of the composite may also be important. For example, an effective rate of volumetric generation or, for transient problems, an effective speciﬁc heat capacity and density. An effective property is the property that a homogenous material must have if it is to behave in the same way as the composite. The process of estimating and using an effective conductivity to model a composite structure is illustrated in the context of the motor pole shown in Figure 2-37. The pole is composed of laminations of iron that are separated by an epoxy coating. Each iron lamination is thlam = 0.5 mm thick and has conductivity klam = 10 W/m-K while the epoxy coating is approximately thep = 0.2 mm thick and has conductivity kep = 2.0 W/m-K. The motor pole is adiabatic on its bottom surface and experiences a heat ﬂux of q˙ = 5000 W/m2 from the windings on the sides. The top surface is maintained at a temperature of T top = 40◦ C. The pole is L = 4.0 cm long and H = 6.0 cm high. The temperature distribution in the pole is 2-D in the x- and y-directions. The iron laminations are generating thermal energy due to eddy current heating at a volumetric rate of g˙ = 100,000 W/m3 . There is no thermal energy generation in the epoxy. The inputs required to determine the effective conductivity are entered into EES:

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” k lam=10 [W/m-K] k ep=2 [W/m-K] th lam=0.5 [mm]∗ convert(mm,m) th ep=0.2 [mm]∗ convert(mm,m) g dot=100000 [W/mˆ3]

“lamination conductivity” “epoxy conductivity” “lamination thickness” “epoxy thickness” “rate of volumetric generation in the laminations”

280

Two-Dimensional, Steady-State Conduction

Heat transfer in the axial (x) direction occurs through the laminations and epoxy in parallel. The methodology for calculating the effective conductivity in the axial direction consists of imposing a temperature difference ( T) in the x-direction (i.e., across the width of the pole) and calculating the heat transfer rate through the two parallel paths. The effective conductivity in the x-direction (keff,x ) is the conductivity of a homogeneous material that would provide the same heat transfer rate. The rate of heat transfer through the iron laminations is: q˙ lam =

klam thlam H W T (thlam + thep) L

(2-238)

where W is the width of the pole (into the page). The heat transfer rate through the epoxy is: q˙ ep =

kep thep H W T (thlam + thep) L

(2-239)

The total heat transfer rate in the x-direction through the equivalent material (q˙ eff ,x ) must therefore be: keff ,x H W kep thep H W klam thlam H W T + T = T q˙ eff ,x = q˙ lam + q˙ ep = (thlam + thep) L (thlam + thep) L L (2-240) or, solving for keff ,x : keff ,x =

kep thep klam thlam + (thlam + thep) (thlam + thep)

(2-241)

The effective conductivity in the x-direction is the thickness (or area) weighted average of the conductivity of the two parallel paths. The effective conductivity calculated using Eq. (2-241) depends only on the details of the microstructure (e.g., the conductivity of the laminations and their thickness) and not the macroscopic details of the problem (e.g., the size of the motor pole and the boundary conditions on the problem). k_eff_x=k_lam∗ th_lam/(th_lam+th_ep)+k_ep∗ th_ep/(th_lam+th_ep) “eff. conductivity in the x-direction”

which leads to keff ,x = 7.71 W/m-K. Note that the effective conductivity must lie between the conductivity of the epoxy (2 W/m-K) and the laminations (10 W/m-K). If the thickness of the lamination becomes large relative to the thickness of the epoxy then the effective conductivity will approach the lamination conductivity. If the conductivity of one material (for example, the lamination) is substantially greater than the other (for example, the epoxy) then the effective conductivity in the x-direction will approach the product of the conductivity of the more conductive material and the fraction of the thickness that is occupied by that material. This behavior is typical of any parallel resistance network because the rate of heat transfer is more sensitive to the smaller resistance. Heat transferred in the y-direction must pass through the laminations and epoxy in series. A temperature difference is applied across the pole in the y-direction. The heat transfer rate is calculated and used to establish the effective conductivity in the y-direction: q˙ eff ,y =

keff ,y W L T T = H thep H thlam H + (thlam + thep) klam W L (thlam + thep) kep W L

(2-242)

2.9 Conduction through Composite Materials

281

or, solving for keff ,y : keff ,y =

1 thep thlam + (thlam + thep) klam (thlam + thep) kep

(2-243)

k_eff_y=1/(th_lam/((th_lam+th_ep)∗ k_lam)+th_ep/((th_lam+th_ep)∗ k_ep)) “eff. conductivity in the y-direction”

which leads to keff ,y = 4.67 W/m-K. The effective conductivity is again bounded by kep and klam . The effective conductivity in the y-direction is dominated by the thicker and less conductive of the two materials. This behavior is typical of a series resistance network, where the larger resistance is the most important. In addition to the effective conductivity, it is necessary to determine an effective rate of volumetric generation (g˙ eff ) that characterizes the motor pole. This is the volumetric generation rate for an equivalent piece of homogeneous material that produces the same total rate of generation. The total rate of energy generation in the motor pole is: g˙ = g˙

H thlam W L = g˙ eff H W L (thlam + thep)

(2-244)

and therefore: ˙ g˙ eff = g

g _dot_eff=g _dot∗ th_lam/(th_lam+th_ep)

thlam (thlam + thep)

(2-245)

“effective rate of volumetric generation”

3 which leads to g˙ eff = 71,400 W/m . The effective properties of the motor pole, keff ,x , keff ,y , and g˙ eff , can be used to generate a model of the motor pole that does not explicitly consider the microscale features of the composite structure but does capture the overall geometry and boundary conditions of the problem shown in Figure 2-37. The geometry is entered in FEHT as discussed in Section 2.7.1 and Appendix A.4. A grid is used where 1 cm of the screen corresponds to 1 cm and the corner nodes are approximately positioned using the Outline command in the Draw menu. The corner nodes are then precisely positioned by right-clicking on each in turn. The material properties are set by clicking on the outline and selecting Material Properties from the Specify menu. Create a new material (select “not speciﬁed” from the list of materials) and rename it lamination. Click on the box next to Type until the choice is Anisotropic; set the x-conductivity to 7.71 W/m-K and the ratio ky/kx to 0.605 (the ratio of keff ,x to keff ,y ). Set the effective rate of volumetric generation by clicking on the outline and selecting Generation from the Specify menu. Set the boundary conditions according to Figure 2-37 and draw a crude grid (two triangles formed by a single element line across the pole will do). Reﬁne the grid multiple times and solve. The temperature distribution in the pole is shown in Figure 2-38.

282

Two-Dimensional, Steady-State Conduction

Figure 2-38: Temperature distribution in the motor pole (◦ C).

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

Note that the 2-D temperature distribution is adequately captured using the ﬁnite element model with equivalent properties; however, the actual temperature distribution would include “ripples” corresponding to the effects of the individual laminations. Unless the characteristics of these very small-scale effects are important, it is convenient to consider the effect of the micro-structure on the larger-scale problem using the effective conductivity concept. EXAMPLE 2.9-1: FIBER OPTIC BUNDLE Lighting represents one of the largest uses of electrical energy in residential and commercial buildings; lighting loads are highest during on-peak hours when electrical energy is most costly. Also, the thermal energy deposited into conditioned space by electrical lighting adds to the air conditioning load on the building which, in turn, adds to the electrical energy required to run the air conditioning system.

Figure 1: Hybrid lighting system (Cheadle, 2006).

A novel lighting system consists of a sunlight collector and a light distribution system, as shown in Figure 1. The sunlight collector tracks the sun and collects and concentrates solar radiation. The light distribution system receives the concentrated solar radiation and distributes it into a building where it is ﬁnally dispensed

283

in ﬁxtures that are referred to as luminaires. Sunlight contains both visible and invisible energy; only the visible portion of the sunlight is useful for lighting and therefore the collector gathers the visible portion of the incident solar radiation while eliminating the invisible ultraviolet and infrared portions of the spectrum. (We will learn more about these characteristics of radiation in Chapter 10.) The ﬁber optic bundle used to transmit the visible light is composed of many, small diameter optical ﬁbers that are packed in approximately a hexagonal close-packed array. A conductive ﬁller material is wrapped around each ﬁber and the entire structure is simultaneously heated and compressed so that the ﬁbers becomes hexagonal shaped with a thin layer of conductive ﬁller separating each hexagon (Figure 2). The dimension of each face of the hexagon is d = 1.0 mm and the thickness of the ﬁller that separates the hexagons is a = 50 μm thick. The ﬁber conductivity is k f b = 1.5 W/m-K while the ﬁller conductivity is k fl = 50.0 W/m-K.

d = 1 mm optical fiber, kfb = 1.5 W/m-K unit cell used for FEHT model (Figure 3)

a = 50 μm

r

filler material, kfl = 50 W/m-K

x

fiber optic bundle

Figure 2: Array of optical ﬁbers packed together and compressed in order to form a ﬁber optic bundle that has hexagonal units.

a) Determine the effective radial and axial conductivity associated with the bundle. The inputs are entered in EES:

“EXAMPLE 2.9-1: Fiber Optic Bundle” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Composite Structure Inputs” d=1 [mm]∗ convert(mm,m) a=0.05 [mm]∗ convert(mm,m) k ﬂ=50 [W/m-K] k fb=1.5 [W/m-K]

“face dimension of hexagon” “ﬁller thickness” “conductivity of ﬁller” “conductivity of ﬁber”

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

284

Two-Dimensional, Steady-State Conduction

The heat transfer in the axial direction can travel through two parallel paths, the ﬁller and the ﬁber. The area of a single ﬁber (a hexagon with each side having length d) is: π ' π ( 1 + cos Af b = 2 d 2 sin 3 3 A fb=2∗ dˆ2∗ sin(pi/3)∗ (1+cos(pi/3))

“area of a ﬁber”

The heat transfer through the ﬁber in the axial direction for a given temperature difference (T ) is: T k f b Af b L where L is the length of the ﬁber. The area of the ﬁller material associated with a single ﬁber (which has thickness a/2 due to sharing of the ﬁller with the neighboring ﬁbers) is: q˙ f b =

Afl = 3 d a A ﬂ=3∗ d∗ a

“area of ﬁller associated with a ﬁber”

The heat transfer through a unit length of the ﬁller in the axial direction for the same temperature difference is: T k fl Afl L The equivalent homogenous material will have conductivity k eff ,x and area Afl + Af b ; therefore, the heat transfer through the equivalent material (q˙ eff ,x ) is: q˙ fl =

T k eff ,x (Afl + Af b ) L The effective conductivity is deﬁned so that the heat transfer through the equivalent material is equal to the sum of the heat transfer through the ﬁber and the ﬁller: q˙ eff ,x =

T k eff ,x (Afl + Af b ) T k f b Af b T k fl Afl = + L L L q˙ eff ,x

q˙ fb

q˙ fl

or, solving for k eff ,x : k eff ,x =

k f b Af b + k fl Afl Afl + Af b

The effective conductivity in the axial direction is the area-weighted conductivity of the two parallel paths: k eff x=(k fb∗ A fb+k ﬂ∗ A ﬂ)/(A ﬂ+A fb)

“effective conductivity in the axial direction”

which leads to k eff ,x = 4.1 W/m-K. The radial conductivity cannot be evaluated using a simple parallel or series resistance circuit because the heat ﬂow across the bundle is complex and 2-D.

285

Therefore, a 2-D ﬁnite element model of a unit cell of the structure (shown in Figure 2) must be generated. Figure 3 illustrates the details of a unit cell and includes the coordinates of the points (in mm) that deﬁne the geometry. (0.75,1.782) (0,1.782)

(0.808,1.782) (1.558,1.782)

Region 3

(1.058, 1.324) (1.0, 1.324) (1.058, 1.299)

(1.558, 1.349) (1.558, 1.299)

Region 2 (0, 0.483) (0, 0.433)

Region 1 (0, 0)

(0.5,0.483) (0.558,0.458) (0.5,0.433)

Region 5 Region 4

(1.558, 0) (0.75, 0)

(0.808, 0)

Figure 3: A unit cell of the ﬁber optic bundle structure; the points that deﬁne the structure are shown (dimensions are in mm).

The ﬁnite element model is generated using FEHT as discussed in Section 2.7.1 and Appendix A.4. A grid is speciﬁed where 1 cm of screen dimension corresponds to 0.2 mm. The ﬁve regions in Figure 3 are generated using ﬁve outlines; the points are initially placed approximately and then precisely positioned by double-clicking on each one in turn. The conductivity for regions 1 through 4 are set to 1.5 W/m-K (consistent with the optical ﬁber) and the conductivity for region 5 is set to 50.0 W/m-K (consistent with the ﬁller material). The boundary conditions along the upper and lower edges are set as adiabatic. In order to set a temperature difference across the unit cell (from left to right) it would seem logical to set the temperature at the left hand side to 1.0◦ C and the right side to 0.0◦ C. However, due to the manner in which ﬁnite element techniques determine the heat ﬂux at a surface, a more accurate answer is obtained if a convective boundary condition is set with a very high heat transfer coefﬁcient; for example, 1 × 105 W/m2 -K. A relatively crude mesh is generated and then reﬁned, particularly in the ﬁller material where most of the heat ﬂow is expected. The result is shown in Figure 4(a). The ﬁnite element model is solved and the temperature contours are shown in Figure 4(b). The heat ﬂow through the unit cell can be determined by selecting Heat Flows from the View menu and then selecting all of the boundaries on either the left or right side. The selection process can be accomplished by using the mouse to ‘drag’ a selection rectangle around the lines on the boundary. The total heat ﬂow is q˙ eff ,r /L = 3.070 W/m; this result can be used to compute the effective conductivity of the composite. The effective conductivity across the bundle (k eff ,r ) is deﬁned as the conductivity of a homogeneous material that would provide the same heat transfer per length into the page as the composite structure simulated by the ﬁnite element model. q˙ eff ,r H T = k eff ,r L W

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

Two-Dimensional, Steady-State Conduction

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

286

(a)

(b)

Figure 4: Finite element model showing (a) the reﬁned grid and (b) the temperature distribution.

where H is the height of the unit cell (1.782 mm, from Figure 3), W is the width of the unit cell (1.558 mm, from Figure 3), and T = 1.0 K (the imposed temperature difference). “k eff r calculation using inputs from FEHT” H=1.782 [mm]∗ convert(mm,m) “height of unit cell” W=1.558 [mm]∗ convert(mm,m) “width of unit cell” q dot eff r\L=3.070 [W/m] “rate of heat transfer per unit length predicted by FE model” DT=1 [K] “temperature difference imposed on FE model” q dot eff r\L=k eff r∗ H∗ DT/W “effective conductivity in the radial direction”

which leads to k eff ,r = 2.7 W/m-K. The effective conductivity in the x-direction is 4.1 W/m-K while the effective conductivity in the radial direction is 2.7 W/m-K. These values make sense. Both lie between the conductivity of the ﬁber and the ﬁller and both are closer to the conductivity of the ﬁber because it occupies most of the space. Further, the conductivity in the x-direction is larger because the path through the high conductivity ﬁller is more direct in this direction. The advantage of the hexagonal pattern in Figure 2 is that is has the lowest possible porosity (φ). The porosity is deﬁned as the fraction of the area of the face of the bundle that is occupied by the ﬁller. The optical ﬁbers are designed so that any radiation that strikes the face of a ﬁber is “trapped” by total internal reﬂection and transmitted without substantial loss. However, the radiation that strikes the opaque ﬁller in the interstitial areas between the ﬁbers will be absorbed and result in a thermal load on the bundle that manifests itself as a heat ﬂux on the surface. This heat ﬂux can lead to elevated temperatures and thermal failure. Assume that the outer edge of the ﬁber optic bundle is exposed to air at T∞ = ◦ 20 C with a heat transfer coefﬁcient hout = 5.0 W/m2 -K. The front face of the bundle (at x = 0) is exposed to air at T∞ = 20◦ C with a heat transfer coefﬁcient hf =

287

10.0 W/m2 -K. The radius of the bundle is r out = 2.0 cm and its length is L = 2.5 m. The end of the bundle at x = L is maintained at a temperature of TL = 20◦ C. = 1 × 105 W/m2 . A The radiant heat ﬂux incident on the face of the bundle is q˙inc schematic of the problem is shown in Figure 5. 2

T∞ = 20°C, hout = 5 W/m -K q⋅

keff, x and keff, r

T∞ = 20°C 2 h f = 10 W/m -K

conv

r = 2.0 cm

TL = 20°C

r x

q⋅ x

W ′′ = 1x10 2 q⋅inc m on the filler material 5

q⋅x+dx

dx L = 2.5 m

Figure 5: Schematic of the ﬁber optic bundle problem.

b) Is it appropriate to treat the bundle as an extended surface? Justify your answer. The additional inputs for the problem are entered in EES: “Problem Inputs” r out=2 [cm]∗ convert(cm,m) L=2.5 [m] h bar out=5 [W/mˆ2-K] T inﬁnity=converttemp(C,K,20[C]) h bar f=10 [W/mˆ2-K] T L=converttemp(C,K,20[C]) q dot ﬂux inc=100000 [W/mˆ2]

“outer radius” “bundle length” “heat transfer coefﬁcient on outer surface” “air temperature” “heat transfer coefﬁcient on the face” “temperature at x=L” “incident radiant heat ﬂux on the face”

The extended surface approximation neglects temperature gradients in the radial direction within the bundle. The conduction resistance in the radial direction (Rcond,r ) must be neglected in order to treat the bundle as an extended surface. This assumption is justiﬁed provided that Rcond,r is small relative to the resistance that is being considered, convection from the outer surface of the bundle (Rconv ). The appropriate Biot number is therefore: Bi =

Rcond ,r Rconv

It is not possible to precisely compute a resistance that characterizes conduction in the radial direction within the bundle; conduction from the center of the bundle is characterized by an inﬁnite resistance. Instead, an approximate conduction length (r out /2) and area (πr out L) are used: r out hout 2 π r out L hout r out Bi = = 2 k eff ,r π r out L 1 k eff ,r Rcond ,r

Bi=h bar out∗ r out/k eff r

Rconv

“Biot number”

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

288

Two-Dimensional, Steady-State Conduction

The Biot number is 0.037 which is much less than 1 and therefore the extended surface approximation is justiﬁed. c) Develop an analytical model for the temperature distribution in the bundle. The differential control volume used to derive the governing equation is shown in Figure 5 and leads to the energy balance: q˙ x = q˙ x+d x + q˙ conv which can be expanded and simpliﬁed: 0=

d q˙ x d x + q˙ conv dx

(1)

The rate equations are: 2 q˙ x = −k eff ,x π r out

dT dx

q˙ conv = 2 π r out hout d x (T − T∞ )

(2) (3)

Substituting Eqs. (1) and (2) into Eq. (3) leads to: d 2 dT 0= −k eff ,x π r out d x + 2 π r out hout d x (T − T∞ ) dx dx or d 2T − m2 T = −m2 T∞ d x2

(4)

where m2 =

2 hout k eff ,x r out

The governing differential equation, Eq. (4), is satisﬁed by exponential functions. The differential equation can also be entered in Maple and solved. > restart; > ODE:=diff(diff(T(x),x),x)-mˆ2∗ T(x)=-mˆ2∗ T_inﬁnity; 2 d T(x) − m2T(x) = −m2T inﬁnity O D E := d x2 > Ts:=dsolve(ODE); Ts := T(x) = e(−mx) C 2 + e(mx) C 1 + T inﬁnity

The solution is copied and pasted into EES. “Solution” m=sqrt(2∗ h bar out/(k eff x∗ r out)) T=exp(-m∗ x)∗ C 2+exp(m∗ x)∗ C 1+T inﬁnity

“solution parameter” “solution from Maple”

The ﬁrst boundary condition is the speciﬁed temperature at x = L: Tx=L = TL

289

A symbolic expression for this boundary condition is obtained in Maple: > rhs(eval(Ts,x=L))=T_L; e(−mL) C 2 + e(mL) C 1 + T inﬁnity = T L

and pasted into EES: exp(-m∗ L)∗ C 2+exp(m∗ L)∗ C 1+T inﬁnity=T L

“boundary condition at x=L”

The second boundary condition is obtained from an interface energy balance at x = 0. Recall that only the ﬂux incident on the ﬁller material results in a heat load. Therefore, the absorbed ﬂux is the product of incident heat ﬂux and the porosity (φ), which is the ratio of the area of the ﬁller to the total area: φ=

phi=A ﬂ/(A ﬂ+A fb)

Afl Afl + Af b

“porosity”

The absorbed ﬂux must either be transferred by conduction to the bundle or convection to the air: dT + hf (Tx=0 − Tair ) q˙inc φ = −k eff ,x d x x=0 A symbolic expression for this boundary condition is obtained in Maple: > q_dot_ﬂux_inc∗ phi=-k_eff_x∗ rhs(eval(diff(Ts,x),x=0))+h_bar_f∗ (rhs(eval(Ts,x=0))-T_inﬁnity);

q d ot f lux inc φ = k eff x(−m C 2 + m C 1) + h bar f ( C 2 + C 1)

and pasted into EES: q_dot_ﬂux_inc∗ phi = -k_eff_x∗ (-m∗ C_2+m∗ C_1)+h_bar_f∗ (C_2+C_1) “boundary condition at x=0”

The solution is converted to Celsius: T C=converttemp(K,C,T)

“temperature in C”

Figure 6 illustrates the temperature distribution near the face of the ﬁber optic bundle.

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

2.9 Conduction through Composite Materials

Two-Dimensional, Steady-State Conduction

120

Temperature (°C)

100 80 60

40 20 0

0.1

0.2

0.3

0.4 0.5 0.6 Position (m)

0.7

0.8

0.9

1

Figure 6: Temperature distribution in ﬁber optic bundle.

The model can be used to assess alternative methods of thermal management. For example, the heat transfer coefﬁcient at the face might be increased by adding a fan or the conductivity of the ﬁller material might be increased through material selection. Figure 7 illustrates the maximum temperature (the temperature at the face) as a function of hf for various values of k f . 130 120 Temperature at the face (°C)

EXAMPLE 2.9-1: FIBER OPTIC BUNDLE

290

110 100 90

k f [W/m-K]

80

50 100

70

150

60 50 0

10 20 30 40 50 60 70 80 90 Heat transfer coefficient on the face (W/m2-K)

100

Figure 7: Temperature at the face as a function of the face heat transfer coefﬁcient for various values of the ﬁller material conductivity.

Chapter 2: Two-Dimensional, Steady-State Conduction The website associated with this book (www.cambridge.org/nellisandklein) provides many more problems than are included here.

Chapter 2: Two-Dimensional, Steady-State Conduction

291

Shape Factors 2–1 Figure P2-1 illustrates two tubes that are buried in the ground behind your house and transfer water to and from a wood burner. The left tube carries hot water from the burner back to your house at T w,h = 135◦ F while the right tube carries cold water from your house to the burner at T w,c = 70◦ F. Both tubes have outer diameter Do = 0.75 inch and thickness th = 0.065 inch. The conductivity of the tubing material is kt = 0.22 W/m-K. The heat transfer coefﬁcient between the water and the tube internal surface (in both tubes) is hw = 250 W/m2 -K. The center to center distance between the tubes is w = 1.25 inch and the length of the tubes is L = 20 ft (into the page). The tubes are buried in soil that has conductivity ks = 0.30 W/m-K. ks = 0.30 W/m-K th = 0.065 inch

kt = 0.22 W/m-K

Tw, c = 70°F hw = 250 W/m2 -K

Tw, h = 135°F h w = 250 W/m2 -K w = 1.25 inch Do = 0.75 inch Figure P2-1: Tubes buried in soil.

a.) Estimate the heat transfer from the hot water to the cold water due to the proximity of the tubes to one another. b.) To do part (a) you should have needed to determine a shape factor; calculate an approximate value of the shape factor and compare it to the accepted value. c.) Plot the rate of heat transfer from the hot water to the cold water as a function of the center to center distance between the tubes. 2–2 A solar electric generation system (SEGS) employs molten salt as both the energy transport and storage ﬂuid. The molten salt is heated to 500◦ C and stored in a buried hemispherical tank. The top (ﬂat) surface of the tank is at ground level. The diameter of the tank before insulation is applied is 14 m. The outside surfaces of the tank are insulated with 0.30 m thick ﬁberglass having a thermal conductivity of 0.035 W/m-K. Sand having a thermal conductivity of 0.27 W/m-K surrounds the tank, except on its top surface. Estimate the rate of heat loss from this storage unit to the 25◦ C surroundings. Separation of Variables Solutions 2–3 You are the engineer responsible for a simple device that is used to measure the heat transfer coefﬁcient as a function of position within a tank of liquid (Figure P2-3). The heat transfer coefﬁcient can be correlated against vapor quality, ﬂuid composition, and other useful quantities. The measurement device is composed of many thin plates of low conductivity material that are interspersed with large, copper interconnects. Heater bars run along both edges of the thin plates. The heater bars are insulated and can only transfer energy to the plate; the heater bars are conductive and can therefore be assumed to come to a uniform temperature as a current is applied. This uniform temperature is assumed to be applied to the top and bottom

292

Two-Dimensional, Steady-State Conduction

edges of the plates. The copper interconnects are thermally well-connected to the ﬂuid; therefore, the temperature of the left and right edges of each plate are equal to the ﬂuid temperature. This is convenient because it isolates the effect of adjacent plates from one another, allowing each plate to measure the local heat transfer coefﬁcient. Both surfaces of the plate are exposed to the ﬂuid temperature via a heat transfer coefﬁcient. It is possible to infer the heat transfer coefﬁcient by measuring heat transfer required to elevate the heater bar temperature to a speciﬁed temperature above the ﬂuid temperature. top and bottom surfaces exposed to fluid T∞ = 20°C, h = 50 W/m2 -K copper interconnect, T∞ = 20°C

a = 20 mm b = 15 mm

plate: k = 20 W/m-K th = 0.5 mm

heater bar, Th = 40°C

Figure P2-3: Device to measure local heat transfer coefﬁcient.

The nominal design of an individual heater plate utilizes metal with k = 20 W/m-K, th = 0.5 mm, a = 20 mm, and b = 15 mm (note that a and b are deﬁned as the half-width and half-height of the heater plate, respectively, and th is the thickness as shown in Figure P2-3). The heater bar temperature is maintained at T h = 40◦ C and the ﬂuid temperature is T ∞ = 20◦ C. The nominal value of the heat transfer coefﬁcient is h = 50 W/m2 -K. a.) Develop an analytical model that can predict the temperature distribution in the plate under these nominal conditions. b.) The measured quantity is the rate of heat transfer to the plate from the heater (q˙ h ) and therefore the relationship between q˙ h and h (the quantity that is inferred from the heater power) determines how useful the instrument is. Determine the heater power and plot the heat transfer coefﬁcient as a function of heater power. c.) If the uncertainty in the measurement of the heater power is δq˙ h = 0.01 W, estimate the uncertainty in the measured heat transfer coefﬁcient (δh). 2–4 A laminated composite structure is shown in Figure P2-4. H = 3 cm

q⋅ ″ = 10000 W/m2

Tset = 20°C Tset = 20°C

W = 6 cm k x = 50 W/m-K k y = 4 W/m-K

Figure P2-4: Composite structure exposed to a heat ﬂux.

Chapter 2: Two-Dimensional, Steady-State Conduction

293

The structure is anisotropic. The effective conductivity of the composite in the xdirection is kx = 50 W/m-K and in the y-direction it is ky = 4 W/m-K. The top of the structure is exposed to a heat ﬂux of q˙ = 10,000 W/m2 . The other edges are maintained at T set = 20◦ C. The height of the structure is H = 3 cm and the halfwidth is W = 6 cm. a.) Develop a separation of variables solution for the 2-D steady-state temperature distribution in the composite. b.) Prepare a contour plot of the temperature distribution. Advanced Separation of Variables Solutions 2–5 Figure P2-5 illustrates a pipe that connects two tanks of liquid oxygen on a spacecraft. The pipe is subjected to a heat ﬂux, q˙ = 8,000 W/m2 , which can be assumed to be uniformly applied to the outer surface of the pipe and is entirely absorbed. Neglect radiation from the surface of the pipe to space. The inner radius of the pipe is rin = 6.0 cm, the outer radius of the pipe is rout = 10.0 cm, and the half-length of the pipe is L = 10.0 cm. The ends of the pipe are attached to the liquid oxygen tanks and therefore are at a uniform temperature of T LOx = 125 K. The pipe is made of a material with a conductivity of k = 10 W/m-K. The pipe is empty and therefore the internal surface can be assumed to be adiabatic. a.) Develop an analytical model that can predict the temperature distribution within the pipe. Prepare a contour plot of the temperature distribution within the pipe. rout = 10 cm rin = 6 cm

k = 10 W/m-K

L = 10 cm TLOx = 125 K

q⋅ ″s = 8,000 W/m2

Figure P2-5: Cryogen transfer pipe connecting two liquid oxygen tanks.

2–6 Figure P2-6 illustrates a cylinder that is exposed to a concentrated heat ﬂux at one end. extends to infinity

k = 168 W/m-K rout = 200 μm x

Ts = 20° C rexp = 21 μm

q⋅ ″ = 1500 W/cm2

adiabatic

Figure P2-6: Cylinder exposed to a concentrated heat ﬂux at one end.

294

Two-Dimensional, Steady-State Conduction

The cylinder extends inﬁnitely in the x-direction. The surface at x = 0 experiences a uniform heat ﬂux of q˙ = 1500 W/cm2 for r < rexp = 21 μm and is adiabatic for rexp < r < rout where rout = 200 μm is the outer radius of the cylinder. The outer surface of the cylinder is maintained at a uniform temperature of T s = 20◦ C. The conductivity of the cylinder material is k = 168 W/m-K. a.) Develop a separation of variables solution for the temperature distribution within the cylinder. Plot the temperature as a function of radius for various values of x. b.) Determine the average temperature of the cylinder at the surface exposed to the heat ﬂux. c.) Deﬁne a dimensionless thermal resistance between the surface exposed to the heat ﬂux and Ts . Plot the dimensionless thermal resistance as a function of rout /rin . d.) Show that your plot from (c) does not change if the problem parameters (e.g., Ts , k, etc.) are changed. Superposition 2–7 The plate shown in Figure P2-7 is exposed to a uniform heat ﬂux q˙ = 1 × 105 W/m2 along its top surface and is adiabatic at its bottom surface. The left side of the plate is kept at TL = 300 K and the right side is at TR = 500 K. The height and width of the plate are H = 1 cm and W = 5 cm, respectively. The conductivity of the plate is k = 10 W/m-K. 5 q⋅ ′′ = 1x10 W/m2

W = 5 cm

TL = 300 K

H = 1 cm

y

TR = 500 K

x k = 10 W/m-K Figure P2-7: Plate.

a.) Derive an analytical solution for the temperature distribution in the plate. b.) Implement your solution in EES and prepare a contour plot of the temperature. Numerical Solutions to Steady-State 2-D Problems using EES 2–8 Figure P2-8 illustrates an electrical heating element that is afﬁxed to the wall of a chemical reactor. The element is rectangular in cross-section and very long (into the page). The temperature distribution within the element is therefore twodimensional, T(x, y). The width of the element is a = 5.0 cm and the height is b = 10.0 cm. The three edges of the element that are exposed to the chemical (at x = 0, y = 0, and x = a) are maintained at a temperature Tc = 200◦ C while the upper edge (at y = b) is afﬁxed to the well-insulated wall of the reactor and can therefore be considered adiabatic. The element experiences a uniform volumetric rate of thermal energy generation, g˙ = 1 × 106 W/m3 . The conductivity of the material is k = 0.8 W/m-K.

Chapter 2: Two-Dimensional, Steady-State Conduction

295

reactor wall k = 0.8 W/m-K g⋅ ′′′= 1x106 W/m3

Tc = 200°C a = 5 cm y

Tc = 200°C

x

b = 10 cm

Tc = 200°C

Figure P2-8: Electrical heating element.

a.) Develop a 2-D numerical model of the element using EES. b.) Plot the temperature as a function of x at various values of y. What is the maximum temperature within the element and where is it located? c.) Prepare a sanily check to show that your solution behaves according to your physical intuition. That is, change some aspect of your program and show that the results behave as you would expect (clearly describe the change that you made and show the result). Finite-Difference Solutions to Steady-State 2-D Problems using MATLAB 2–9 Figure P2-9 illustrates a cut-away view of two plates that are being welded together. Both edges of the plate are clamped and held at temperature Ts = 25◦ C. The top of the plate is exposed to a heat ﬂux that varies with position x, measured from joint, according to: q˙ m (x) = q˙ j exp (−x/Lj ) where q˙ j = 1 × 106 W/m2 is the maximum heat ﬂux (at the joint, x = 0) and Lj = 2.0 cm is a measure of the extent of both edges are clamped and held at fixed temperature

heat flux joint

impingement cooling with liquid jets ⋅ q′′ m

k = 38 W/m-K

W = 8.5 cm b = 3.5 cm

y

Ts = 25°C

x Tf = −35°C h = 5000 W/m2 -K Figure P2-9: Welding process and half-symmetry model of the welding process.

296

Two-Dimensional, Steady-State Conduction

the heat ﬂux. The back side of the plates are exposed to liquid cooling by a jet of ﬂuid at T f = −35◦ C with h = 5000 W/m2 -K. A half-symmetry model of the problem is shown in Figure P2-9. The thickness of the plate is b = 3.5 cm and the width of a single plate is W = 8.5 cm. You may assume that the welding process is steady-state and 2-D. You may neglect convection from the top of the plate. The conductivity of the plate material is k = 38 W/m-K. a.) Develop a separation of variables solution to the problem. Implement the solution in EES and prepare a plot of the temperature as a function of x at y = 0, 1.0, 2.0, 3.0, and 3.5 cm. b.) Prepare a contour plot of the temperature distribution. c.) Develop a numerical model of the problem. Implement the solution in MATLAB and prepare a contour or surface plot of the temperature in the plate. d.) Plot the temperature as a function of x at y = 0, b/2, and b and overlay on this plot the separation of variables solution obtained in part (a) evaluated at the same locations.

Finite Element Solutions to Steady-State 2-D Problems using FEHT 2–10 Figure P2-10(a) illustrates a double paned window. The window consists of two panes of glass each of which is tg = 0.95 cm thick and W = 4 ft wide by H = 5 ft high. The glass panes are separated by an air gap of g = 1.9 cm. You may assume that the air is stagnant with ka = 0.025 W/m-K. The glass has conductivity kg = 1.4 W/m-K. The heat transfer coefﬁcient between the inner surface of the inner pane and the indoor air is hin = 10 W/m2 -K and the heat transfer coefﬁcient between the outer surface of the outer pane and the outdoor air is hout = 25 W/m2 -K. You keep your house heated to Tin = 70◦ F. width of window, W = 4 ft Tin = 70°F hin = 10 W/m2 -K H = 5 ft

tg = 0.95cm tg = 0.95 cm g = 1.9 cm Tout = 23°F hout = 25 W/m2 -K ka = 0.025 W/m-K kg = 1.4 W/m-K

casing shown in P2-10(b) Figure P2-10(a): Double paned window.

The average heating season lasts about time = 130 days and the average outdoor temperature during this time is T out = 23◦ F. You heat with natural gas and pay, on average, ec = 1.415 $/therm (a therm is an energy unit = 1.055 × 108 J).

Chapter 2: Two-Dimensional, Steady-State Conduction

297

a.) Calculate the average rate of heat transfer through the double paned window during the heating season using a 1-D resistance model. b.) How much does the energy lost through the window cost during a single heating season? There is a metal casing that holds the panes of glass and connects them to the surrounding wall, as shown in Figure P2-10(b). Because the metal casing has high conductivity, it seems likely that you could lose a substantial amount of heat by conduction through the casing (potentially negating the advantage of using a double paned window). The geometry of the casing is shown in Figure P2-10(b); note that the casing is symmetric about the center of the window. All surfaces of the casing that are adjacent to glass, wood, or the air between the glass panes can be assumed to be adiabatic. The other surfaces are exposed to either the indoor or outdoor air. glass panes 1.9 cm Tin = 70°F hin = 10 W/m2 -K

0.95 cm air

Tout = 23°F hout = 25 W/m2 -K

2 cm 4 cm

metal casing k m = 25 W/m-K

0.5 cm

3 cm 0.4 cm

wood

Figure P2-10(b): Metal casing.

c.) Prepare a 2-D thermal analysis of the casing using FEHT. Turn in a print out of your geometry as well as a contour plot of the temperature distribution. What is the rate of energy lost via conduction through the casing per unit length (W/m)? d.) Show that your numerical model has converged by recording the rate of heat transfer per length for several values of the number of nodes. e.) How much does the casing add to the cost of heating your house? 2–11 A radiator panel extends from a spacecraft; both surfaces of the radiator are exposed to space (for the purposes of this problem it is acceptable to assume that space is at 0 K); the emissivity of the surface is ε = 1.0. The plate is made of aluminum (k = 200 W/m-K and ρ = 2700 kg/m3 ) and has a ﬂuid line attached to it, as shown in Figure 2-11(a). The half-width of the plate is a = 0.5 m wide and the height of the plate is b = 0.75 m. The thickness of the plate is th = 1.0 cm. The ﬂuid line carries coolant at Tc = 320 K. Assume that the ﬂuid temperature is constant, although the ﬂuid temperature will actually decrease as it transfers heat to the radiator. The combination of convection and conduction through the panel-to-ﬂuid line mounting leads to an effective heat transfer coefﬁcient of h = 1,000 W/m2 -K over the 3.0 cm strip occupied by the ﬂuid line.

298

Two-Dimensional, Steady-State Conduction k = 200 W/m-K ρ = 2700 kg/m 3 ε = 1.0

space at 0 K

a = 0.5 m

3 cm th = 1 cm

b = 0.75 m

fluid at Tc = 320 K

half-symmetry model of panel, Figure P2-11(b) Figure 2-11(a): Radiator panel.

The radiator panel is symmetric about its half-width and the critical dimensions that are required to develop a half-symmetry model of the radiator are shown in Figure 2-11(b). There are three regions associated with the problem that must be deﬁned separately so that the surface conditions can be set differently. Regions 1 and 3 are exposed to space on both sides while Region 2 is exposed to the coolant ﬂuid one side and space on the other; for the purposes of this problem, the effect of radiation to space on the back side of Region 2 is neglected.

Region 1 (both sides exposed to space) Region 2 (exposed to fluid - neglect radiation to space) Region 3 (both sides exposed to space) (0.50,0.75) (0.50,0.55) (0.50,0.52)

y x (0.50,0)

(0,0) (0.22,0)

(0.25,0)

line of symmetry

Figure 2-11(b): Half-symmetry model (coordinates are in m).

a.) Prepare a FEHT model that can predict the temperature distribution over the radiator panel. b.) Export the solution to EES and calculate the total heat transferred from the radiator and the radiator efﬁciency (deﬁned as the ratio of the radiator heat transfer to the heat transfer from the radiator if it were isothermal and at the coolant temperature). c.) Explore the effect of thickness on the radiator efﬁciency and mass.

Chapter 2: Two-Dimensional, Steady-State Conduction

299

Resistance Approximations for Conduction Problems 2–12 There are several cryogenic systems that require a “thermal switch,” a device that can be used to control the thermal resistance between two objects. One class of thermal switch is activated mechanically and an attractive method of providing mechanical actuation at cryogenic temperatures is with a piezoelectric stack. Unfortunately, the displacement provided by a piezoelectric stack is very small, typically on the order of 10 microns. A company has proposed an innovative design for a thermal switch, shown in Figure P2-12(a). Two blocks are composed of th = 10 μm laminations that are alternately copper (kCu = 400 W/m-K) and plastic (k p = 0.5 W/m-K). The thickness of each block is L = 2.0 cm in the direction of the heat ﬂow. One edge of each block is carefully polished and these edges are pressed together; the contact resistance associated with this joint is Rc = 5 × 10−4 K-m2/ W. th = 10 μm plastic laminations k p = 0.5 W/m-K L = 2 cm L = 2 cm −4 m2 -K/W R′′= c 5x10

direction of actuation

TC

TH

Figure 2-12(b) “off ” position

“on” position th = 10 μm copper laminations kCu = 400 W/m-K

Figure P2-12(a): Thermal switch in the “on” and “off” positions.

Figure P2-12(a) shows the orientation of the two blocks when the switch is in the “on” position; notice that the copper laminations are aligned with one another in this conﬁguration in order to provide a continuous path for heat through high conductivity copper (with the exception of the contact resistance at the interface). The vertical location of the right-hand block is shifted by 10 μm to turn the switch “off”. In the “off” position, the copper laminations are aligned with the plastic laminations; therefore, the heat transfer is inhibited by low conductivity plastic. Figure P2-12(b) illustrates a closer view of half (in the vertical direction) of two adjacent laminations in the “on” and “off” conﬁgurations. Note that the repeating nature of the geometry means that it is sufﬁcient to analyze a single lamination set and assume that the upper and lower boundaries are adiabatic. L

th/2

k p kCu

L

R′′c “on” position

th/2 TC

TH “off ” position

Figure P2-12(b): A single set consisting of half of two adjacent laminations in the “on” and “off” positions.

300

Two-Dimensional, Steady-State Conduction

The key parameter that characterizes the performance of a thermal switch is the resistance ratio (RR) which is deﬁned as the ratio of the resistance of the switch in the “off” position to its resistance in the “on” position. The company claims that they can achieve a resistance ratio of more than 100 for this switch. a.) Estimate upper and lower bounds for the resistance ratio for the proposed thermal switch using 1-D conduction network approximations. Be sure to draw and clearly label the resistance networks that are used to provide the estimates. Use your results to assess the company’s claim of a resistance ratio of 100. b.) Provide one or more suggestions for design changes that would improve the performance of the switch (i.e., increase the resistance ratio). Justify your suggestions. c.) Sketch the temperature distribution through the two parallel paths associated with the adiabatic limit of the switch’s operation in the “off” position. Do not worry about the quantitative details of the sketch, just make sure that the qualitative features are correct. d.) Sketch the temperature distribution through the two parallel paths associated with the adiabatic limit in the “on” position. Again, do not worry about the quantitative details of your sketch, just make sure that the qualitative features are correct. 2–13 Figure P2-13 illustrates a thermal bus bar that has width W = 2 cm (into the page). H1 = 5 cm

L2 = 7 cm h = 10 W/m2 -K T∞ = 20°C

TH = 80°C

L1 = 3 cm

H2 = 1 cm

k = 1 W/m-K

Figure P2-13: Thermal bus bar.

The bus bar is made of a material with conductivity k = 1 W/m-K. The middle section is L2 = 7 cm long with thickness H2 = 1 cm. The two ends are each L1 = 3 cm long with thickness H1 = 3 cm. One end of the bar is held at T H = 80◦ C and the other is exposed to air at T ∞ = 20◦ C with h = 10 W/m2 -K. a.) Use FEHT to predict the rate of heat transfer through the bus bar. b.) Obtain upper and lower bounds for the rate of heat transfer through the bus bar using appropriately deﬁned resistance approximations. Conduction through Composite Materials 2–14 A laminated stator is shown in Figure P2-14. The stator is composed of laminations with conductivity klam = 10 W/m-K that are coated with a very thin layer of epoxy with conductivity kepoxy = 2.0 W/m-K in order to prevent eddy current losses. The laminations are thlam = 0.5 mm thick and the epoxy coating is 0.1 mm thick (the total amount of epoxy separating each lamination is thepoxy = 0.2 mm). The inner radius of the laminations is rin = 8.0 mm and the outer radius of the laminations is ro,lam = 20 mm. The laminations are surrounded by a cylinder of plastic with conductivity kp = 1.5 W/m-K that has an outer radius of ro,p = 25 mm. The motor casing surrounds the plastic. The motor casing has an outer radius of ro,c = 35 mm and is composed of aluminum with conductivity kc = 200 W/m-K.

Conduction through Composite Materials

301

laminations, thlam = 0.5 mm, klam = 10 W/m-K epoxy coating, th epoxy = 0.2 mm, k epoxy = 2.0 W/m-K k p = 1.5 W/m-K kc = 200 W/m-K q⋅ ′′ = 5x10 4 W/m2

T∞ = 20°C h = 40 W/m2 -K -4 R′′c = 1x10 K-m2 /W

rin = 8 mm ro,lam = 20 mm ro,p = 25 mm ro,c = 35 mm Figure P2-14: Laminated stator.

The heat ﬂux due to the windage loss associated with the drag on the shaft is q˙ = 5 × 104 W/m2 and is imposed on the internal surface of the laminations. The outer surface of the motor is exposed to air at T ∞ = 20◦ C with a heat transfer coefﬁcient h = 40 W/m2 -K. There is a contact resistance Rc = 1 × 10-4 K-m2 /W between the outer surface of the laminations and the inner surface of the plastic and the outer surface of the plastic and the inner surface of the motor housing. a.) Determine an upper and lower bound for the temperature at the inner surface of the laminations (Tin ). b.) You need to reduce the internal surface temperature of the laminations and there are a few design options available, including: (1) increase the lamination thickness (up to 0.7 mm), (2) reduce the epoxy thickness (down to 0.05 mm), (3) increase the epoxy conductivity (up to 2.5 W/m-K), or (4) increase the heat transfer coefﬁcient (up to 100 W/m-K). Which of these options do you suggest and why?

REFERENCES

Cheadle, M., A Predictive Thermal Model of Heat Transfer in a Fiber Optic Bundle for a Hybrid Solar Lighting System, M.S. Thesis, University of Wisconsin, Dept. of Mechanical Engineering, (2006). Moaveni, S., Finite Element Analysis, Theory and Application with ANSYS, 2nd Edition, Pearson Education, Inc., Upper Saddle River, (2003). Myers, G. E., Analytical Methods in Conduction Heat Transfer, 2nd Edition, AMCHT Publications, Madison, WI, (1998). Rohsenow, W. M., J. P. Hartnett, and Y. I. Cho, eds., Handbook of Heat Transfer, 3rd Edition, McGraw-Hill, New York, (1998).

3

Transient Conduction

3.1 Analytical Solutions to 0-D Transient Problems 3.1.1 Introduction Chapters 1 and 2 discuss steady-state problems, i.e., problems in which temperature depends on position (e.g, x and y) but does not change with time (t). This chapter discusses transient conduction problems, where temperature depends on time.

3.1.2 The Lumped Capacitance Assumption The simplest situation is zero-dimensional (0-D); that is, the temperature does not vary with position but only with time. This approximation is often referred to as the lumped capacitance assumption and it is appropriate for an object that is thin and conductive so that it can be assumed to be at a uniform temperature at any time. The lumped capacitance approximation is similar to the extended surface approximation that was discussed in Section 1.6. The resistance to conduction within the object is neglected as being small relative to the resistance to heat transfer from the surface of the object. Therefore, the lumped capacitance approximation is justiﬁed in the same way as the extended surface approximation, by deﬁning an appropriate Biot number. For the case where only convection occurs from the surface of the object, the Biot number is: Bi =

Lcond h Rcond,int = Rconv k

(3-1)

where Lcond is the conduction length within the object, h is the average heat transfer coefﬁcient, and k is the conductivity of the material. Note that Eq. (3-1) is the simplest possible Biot number; it is often the case that heat transfer from the surface of the object will be resisted by mechanisms other than or in addition to convection (e.g., radiation or conduction through a thin insulating layer). These additional resistances should be included in the denominator of an appropriately deﬁned Biot number. The conduction length that characterizes an irregular shaped object can be ambiguous. Thermal energy will conduct out of the object along the easiest (i.e., shortest) path. For a thin plate, Lcond should be the half-width of the plate. For other shapes, Lcond should be selected so that it characterizes the minimum conduction length; the ratio of the volume of the object (V) to its surface area (As ) is often used for this purpose: Lcond =

V As

(3-2)

If the Biot number is much less than unity, then the lumped capacitance assumption is justiﬁed. A common criteria is that the Bi must be less than 0.1. However, this is certainly a case where engineering judgment is required based on the level of accuracy that is required for the model. 302

3.1 Analytical Solutions to 0-D Transient Problems

303 q⋅ conv

Figure 3-1: An object exposed to a time varying ﬂuid temperature.

dU dt

3.1.3 The Lumped Capacitance Problem The lumped capacitance approximation reduces the spatial dimensionality of the problem to zero. Therefore, a control volume can be placed around the entire object (as all of the material is assumed to be at the same temperature at a given time). An energy balance will include all of the relevant energy transfers (e.g., convection and radiation) as well as the rate of energy storage. The result will be a ﬁrst order differential equation that governs the temperature of the object. A single boundary condition, typically the initial temperature of the object, is required to obtain a solution using either analytical (as discussed in this section) or numerical (as discussed in Section 3.2) techniques. Figure 3-1 illustrates an object with a small Biot number that can be considered to be lumped. The object is initially at a uniform temperature Tini and is exposed to a time varying ﬂuid temperature, T∞ , through a heat transfer coefﬁcient, h. The control volume in Figure 3-1 suggests the energy balance: 0 = q˙ conv +

dU dt

(3-3)

where q˙ conv is the rate of convective heat transfer from the surface and U is the total energy stored in the object. The convective heat transfer is: q˙ conv = h As (T − T ∞ )

(3-4)

where As is the surface area of the object exposed to the ﬂuid. The rate of change of the total energy of the object can be expressed in terms of the speciﬁc energy of the material (u): du dU =M dt dt

(3-5)

where M is the mass of the object. For solids or incompressible ﬂuids, Eq. (3-5) can be written as: du dT dT dU =M = Mc dt dT dt dt

(3-6)

c

where c is the speciﬁc heat capacity of the material. Substituting Eqs. (3-6) and (3-4) into Eq. (3-3) leads to: 0 = h As (T − T ∞ ) + M c

dT dt

(3-7)

304

Transient Conduction

which is the ﬁrst order differential equation that governs the problem. Equation (3-7) can be rearranged: h As h As dT T = + T∞ dt M c Mc

(3-8)

1/τlumped

Note that the group of variables that multiply the temperatures on the left and right sides of Eq. (3-8) must have units of inverse time; this group is used to deﬁne a lumped time constant, τlumped , that governs the problem: τlumped =

Mc h As

(3-9)

Substituting Eq. (3-9) into Eq. (3-8) leads to: dT T∞ T = + dt τlumped τlumped

(3-10)

Equation (3-10) can be solved either analytically, using the same techniques discussed in Chapter 1 for 1-D steady-state conduction, or numerically (see Section 3.2). If additional heat transfer mechanisms or thermal loads are included in the problem (e.g., radiation or volumetric generation of thermal energy) then the governing differential equation will be different than Eq. (3-10). However, the steps associated with the derivation of the differential equation will remain the same and it will be possible to identify an appropriate lumped capacitance time constant.

3.1.4 The Lumped Capacitance Time Constant The concept of a lumped capacitance time concept is useful even in the absence of an analytical solution to the problem. The lumped capacitance time constant is the product of the thermal resistance to heat transfer from the surface of the object (R) and the thermal capacitance of the object (C). For the object in Figure 3-1, the only resistance to heat transfer from the surface of the object is due to convection: R=

1 h As

(3-11)

The thermal capacitance of the object is the product of its mass and speciﬁc heat capacity. The thermal capacitance provides a measure of how much energy is required to change the temperature of the object: C = Mc

(3-12)

The lumped time constant identiﬁed in Eq. (3-9) is the product of R, Eq. (3-11), and C, Eq. (3-12): RC =

1

M c = τlumped h A s C

(3-13)

R

The lumped capacitance time constant for other situations can be calculated by modifying the resistance and capacitance terms in Eq. (3-13) appropriately. For example, if the object experiences both convection and radiation from its surface, then an

3.1 Analytical Solutions to 0-D Transient Problems ambient temperature

object temperature

Temperature

Temperature

ambient temperature

τ

305

object temperature

Time Time (b)

(a) ambient temperature

object temperature

Temperature

Temperature

ambient temperature

τ

object temperature

Time

Time (d)

(c)

Figure 3-2: Approximate temperature response for an object subjected to (a) a step change in its ambient temperature, (b) and (c) an oscillatory ambient temperature where (b) the frequency of the oscillation is much less than the inverse of the time constant and (c) the frequency of the oscillation is much greater than the inverse of the time constant, and (d) a ramped ambient temperature.

appropriate thermal resistance is the parallel combination of a convective and radiative resistance: −1 3 R = h As + σ εs 4 T As

(3-14)

The lumped time constant is analogous to the electrical time constant associated with an R-C circuit and many of the concepts that may be familiar from electrical circuits can also be applied to transient heat transfer problems. A quick estimate of the lumped time constant for a problem can provide substantial insight into the behavior of the system. The lumped time constant is, approximately,

306

Transient Conduction

Table 3-1: Summary of lumped capacitance solutions to some typical variations in the ambient temperature. Situation

Solution

Nomenclature

Step change in ambient temperature

T = T ∞ + (T ini − T ∞ ) exp −

Oscillatory ambient temperature (after initial transient has decayed)

T =T +'

Ramped ambient temperature

t τlumped

T ini = initial temperature of environment and object T ∞ = temperature of environment after step τlumped = lumped time constant

T ( sin (ω t − φ) 1 + (ω τlumped )2 where φ = tan−1 (ω τlumped )

T = T ini + β t + β τlumped exp −

t τlumped

T = average ambient temperature T = amplitude of temperature oscillation ω = angular frequency of temperature oscillation (rad/s) τlumped = lumped time constant

−1

T ini = initial temperature of environment and object β = rate of ambient temperature change τlumped = lumped time constant

the amount of time that it will take the object to respond to any change in its thermal environment. For example, if the object is subjected to a step change in the ambient temperature, T∞ in Eq. (3-10), then its temperature will be within 5% of the new temperature after 3 time constants, as shown in Figure 3-2(a). If the object is subjected to an oscillatory ambient temperature (e.g., within an engine cylinder or some other cyclic device) then the temperature of the object will follow the ambient temperature nearly exactly if the period of oscillation is much greater than the time constant (i.e., if the frequency is much less than the inverse of the time constant). In the opposite extreme, the object’s temperature will be essentially constant if the period of oscillation is much less than the the time constant (i.e., if the frequency is much greater than the inverse of the time constant). These extremes in behavior are shown in Figure 3-2(b) and (c). If the object is subjected to a ramped (i.e., linearly increasing) ambient temperature, then its temperature will tend to increase linearly as well, but its response will be delayed by approximately one time constant, as shown in Figure 3-2(d). Some of the situations illustrated in Figure 3-2 will be investigated more completely in the following examples. The analytical solutions to these problems are summarized in Table 3-1. However, it is clear that simply knowing the time constant and its physical signiﬁcance is sufﬁcient in many cases.

307

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT Plastic parts are formed in an injection mold and dropped (ﬂat) onto a conveyor belt (Figure 1). The parts are disk-shaped with thickness th = 2.0 mm and diameter D = 10.0 cm. The plastic has thermal conductivity k = 0.35 W/m-K, density ρ = 1100 kg/m3 , and speciﬁc heat capacity c = 1900 J/kg-K. The side of the part that faces the conveyor belt is adiabatic. The top surface of the part is exposed to air at T∞ = 20◦ C with a heat transfer coefﬁcient h = 15 W/m2 -K. The temperature of the part immediately after it is formed is Tini = 180◦ C. The part must be cooled to Tmax = 80◦ C before it can be stacked and packaged. The packaging system is positioned L = 15 ft away from the molding machine. k = 0.35 W/m-K ρ = 1100 kg/m3 c = 1900 J/kg-K Figure 1: Plastic piece.

T∞ = 20°C 2 h = 15 W/m -K th = 2 mm

D =10 cm

adiabatic initial temperature Tini = 180°C

a) Is a lumped capacitance model of the part justiﬁed for this situation? The inputs are entered in EES: “EXAMPLE 3.1-1: Design of a Conveyor Belt” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” th=2.0 [mm]∗ convert(mm,m) k=0.35 [W/m-K] D=10 [cm]∗ convert(cm,m) rho=1100 [kg/mˆ3] c=1900 [J/kg-K] h bar=15 [W/mˆ2-K] T ini=converttemp(C,K,180 [C]) T inﬁnity=converttemp(C,K,20 [C]) T max=converttemp(C,K,80 [C]) L=15 [ft]∗ convert(ft,m)

“thickness” “conductivity” “diameter” “density” “speciﬁc heat capacity” “heat transfer coefﬁcient” “initial temperature of part” “ambient temperature” “maximum handling temperature” “conveyor length”

A lumped capacitance model of the part can be justiﬁed by examining the Biot number, the ratio of the resistance to internal conduction to the resistance to heat transfer from the surface of the object. In this problem, the resistance to heat transfer from the surface is due only to convection and therefore Eq. (3-1) is valid, where the conduction length is intuitively the thickness of the object (it would be the halfwidth if the conveyor side of the part were not adiabatic). Note that the characteristic

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT

308

Transient Conduction

length deﬁned by Eq. (3-2) as the ratio of the volume (V) to the exposed area for heat transfer (As ) is equal to the thickness of the part: V =

π D 2 th 4

As = L cond =

V=pi∗ Dˆ2∗ th/4 As=pi∗ Dˆ2/4 L cond=V/As Bi=h bar∗ L cond/k

π D2 4

V π D 2 th 4 = = th As 4 π D2

“Volume” “Surface area exposed to cooling” “Conduction length” “Biot number based on conduction length”

The Biot number predicted by EES is 0.09, which is sufﬁciently small to use the lumped capacitance model unless very high accuracy is required. b) What is the maximum acceptable conveyor velocity so that the parts arrive at the packaging station below Tmax ? The governing differential equation is obtained by considering a control volume that encloses the entire plastic part; the energy balance is: 0 = q˙ conv +

dU dt

The rate of convection heat transfer is: q˙ conv = h As (T − T∞ ) and the rate of energy storage is: dU dT =ρV c dt dt Combining these equations leads to: 0 = h As (T − T∞ ) + ρ V c

dT dt

which can be rearranged: dT (T − T∞ ) =− dt τlumped

(1)

where τlumped is the time constant for this problem: τlumped =

tau lumped=V∗ rho∗ c/(h bar∗ As)

ρV c h As

“time constant”

The time constant for the part is 279 s. We should keep in mind that it will take on the order of 5 minutes to cool the plastic piece substantially and use this insight to check the more precise analytical solution that is obtained.

309

Equation (1) is a ﬁrst order differential equation with the boundary condition: Tt =0 = Tini

(2)

The differential equation is separable; that is, all of the terms involving the dependent variable, T, can be placed on one side while the terms involving the independent variable, t, can be placed on the other: dt dT =− τlumped (T − T∞ ) The separated equation can be directly integrated: T Tini

dT =− (T − T∞ )

t

dt

(3)

τlumped

0

This integration is most easily accomplished by deﬁning the temperature difference (θ ): θ = T − T∞

(4)

d θ = dT

(5)

so that:

Substituting Eqs. (4) and (5) into Eq. (3) leads to: T −T∞

Tini −T∞

dθ =− θ

t 0

dt τlumped

Carrying out the integration leads to: T − T∞ t ln =− Tini − T∞ τlumped Solving for T leads to:

T = T∞ + (Tini − T∞ ) exp −

t

τlumped

(6)

which is equivalent to the entry in Table 3-1 for a step change in ambient temperature. The time required to cool the part from Tini to Tmax can be computed using Eq. (6): T max=T inﬁnity+(T ini-T inﬁnity)∗ exp(-t cool/tau lumped)

“time required to cool part”

and is found to be tcool = 273 s; note that this value is in good agreement with the previously calculated time constant. The linear velocity of the conveyor that is required so that it takes at least 273 s for the part to travel the 15 ft between the molding machine and the packaging station is: uc =

u c=L/t cool u c fpm=u c∗ convert(m/s,ft/min)

L tcool “conveyor velocity” “conveyor velocity in ft/min”

The maximum allowable velocity uc is found to be 0.0167 m/s (3.29 ft/min).

EXAMPLE 3.1-1: DESIGN OF A CONVEYOR BELT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

310

Transient Conduction

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT A temperature sensor is installed in a chemical reactor that operates in a cyclic fashion. The temperature of the ﬂuid in the reactor varies in an approximately sinusoidal manner with a mean temperature T ∞ = 320◦ C, an amplitude T∞ = 50◦ C, and a frequency f = 0.5 Hz. The sensor can be modeled as a sphere with diameter D = 1.0 mm. The sensor is made of a material with conductivity ks = 50 W/m-K, speciﬁc heat capacity cs = 150 J/kg-K, and density ρs = 16000 kg/m3 . In order to provide corrosion resistance, the sensor has been coated with a thin layer of plastic; the coating is thc = 100 μm thick with conductivity kc = 0.2 W/m-K and has negligible heat capacity relative to the sensor itself. The heat transfer coefﬁcient between the surface of the coating and the ﬂuid is h = 500 W/m2 -K. The sensor is initially at Tini = 260◦ C. a) Is a lumped capacitance model of the temperature sensor appropriate? The inputs are entered in EES: “EXAMPLE 3.1-2: Sensor in an Oscillating Temperature Environment” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” T inﬁnity bar=converttemp(C,K,320[C]) T ini=converttemp(C,K,260[C]) DT inﬁnity=50[K] f=0.5 [Hz] D=1.0 [mm]∗ convert(mm,m) k s=50 [W/m-K] c s=150 [J/kg-K] rho s=16000 [kg/mˆ3] th c=100 [micron]∗ convert(micron,m) k c=0.2 [W/m-K] h bar=500 [W/mˆ2-K]

“average temperature of reactor” “initial temperature of sensor” “amplitude of reactor temperature change” “frequency of reactor temperature change” “diameter of sensor” “conductivity of sensor material” “speciﬁc heat capacity of sensor material” “density of sensor material” “thickness of coating” “conductivity of coating” “heat transfer coefﬁcient”

The Biot number is the ratio of the internal conduction resistance to the resistance to heat transfer from the surface of the object. In this problem, the resistance to heat transfer from the surface is the series combination of convection (Rconv ): Rconv =

h4 π

1 D + t hc 2

2

and the conduction resistance of the coating (Rcond,c , from Table 1-2): 2 2 − D D + 2 t hc Rcond ,c = 4 π kc

R conv=1/(h bar∗ 4∗ pi∗ (D/2+th c)ˆ2) R cond c=(1/(D/2)-1/(D/2+th c))/(4∗ pi∗ k c)

311

“convective resistance” “conduction resistance of coating”

The resistance to internal conduction (Rcond,int ) is approximated according to: Rcond ,int =

L cond k s As

where As is the surface area of the sensor: As = 4 π

D 2

2

and Lcond is the conduction length, approximated according to Eq. (3-2): L cond =

V As

where: 4π V = 3

V=4∗ pi∗ (D/2)ˆ3/3 A s=4∗ pi∗ (D/2)ˆ2 L cond=V/A s R cond int=L cond/(k s∗ A s)

D 2

3

“volume of sensor” “surface area of sensor” “approximate conduction length” “internal conduction resistance”

The Biot number that characterizes this problem is therefore: Bi =

Bi=R cond int/(R conv+R cond c)

Rcond Rc + Rconv

“Biot number”

which leads to Bi = 0.0018; this is sufﬁciently less than 1 to justify a lumped capacitance model. b) What is the time constant associated with the sensor? Do you expect there to be a substantial temperature measurement error related to the dynamic response of the sensor? The time constant (τlumped ) is the product of the resistance to heat transfer from the surface of the sensor (which is related to conduction through the coating and convection) and the thermal mass of the sensor (C): τlumped = (Rcond ,c + Rconv )C

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

312

Transient Conduction

where C = V ρs cs C=V∗ rho s∗ c s tau=(R conv+R cond c)∗ C

“capacitance of the sensor” “time constant of the sensor”

The time constant is 0.72 s and the time per cycle (the inverse of the frequency) is 2 s. These quantities are on the same order and therefore it is not likely that the temperature sensor will be able to faithfully follow the reactor temperature. c) Develop an analytical model of the temperature response of the sensor. The temperature sensor is exposed to a sinusoidally varying temperature: T∞ = T ∞ + T∞ sin 2 π f t

(1)

The governing differential equation for the sensor balances heat transfer to ambient against energy storage: 0=

dT [T − T∞ ] +C Rcond ,c + Rconv dt

(2)

Substituting Eq. (1) into Eq. (2) leads to: 0=

[T − T ∞ − T∞ sin(2 π f t )] dT + τlumped dt

which is rearranged: T∞ T∞ sin(2 π f t ) T dT = + + dt τlumped τlumped τlumped

(3)

Equation (3) is a non-homogeneous ordinary differential equation. The solution is assumed to consist of a homogeneous and particular solution: T = Th + Tp

(4)

Substituting Eq. (4) into Eq. (3) leads to: dTp Tp dTh T∞ T∞ sin(2 π f t ) Th + = + + + dt τlumped dt τlumped τlumped τlumped =0 for homogeneous differential equation

particular differential equation

The homogeneous differential equation is: dTh Th =0 + dt τlumped The solution to the homogeneous differential equation can be obtained by separating variables and integrating: dt dTh =− Th τlumped Carrying out the indeﬁnite integral leads to: ln (Th) = −

t τlumped

+ C1

(5)

313

where C1 is a constant of integration. Equation (5) can be rearranged: t t = C 1∗ exp − Th = exp (C 1 ) exp − τlumped τlumped C 1∗

where C 1∗ is an also undetermined constant that will subsequently be referred to as C1 : t Th = C 1 exp − (6) τlumped Notice that the homogeneous solution provided by Eq. (6) dies off after about three time constants. The particular solution (Tp ) is obtained by identifying any function that satisﬁes the particular differential equation: Tp dTp T∞ T∞ sin(2 π f t ) = + + dt τlumped τlumped τlumped

(7)

By inspection, the sum of a constant and a sine and cosine with the same frequency can be made to solve Eq. (7): Tp = C 2 sin(2 π f t ) + C 3 cos(2 π f t ) + C 4

(8)

Substituting Eq. (8) into Eq. (7) leads to: C2 sin(2 π f t ) τlumped

C 2 2 π f cos(2 π f t ) − C 3 2 π f sin(2 π f t ) + +

C3 τlumped

cos(2 π f t ) +

C4 τlumped

=

T∞ τlumped

+

T∞ sin(2 π f t ) τlumped

(9)

Equation (9) can only be true if the constant, sine and cosine terms each separately add to zero: C4 τlumped

=

C2 2 π f + −C 3 2 π f +

T∞ τlumped C3

τlumped

=0

C2 T∞ = τlumped τlumped

Solving for C2 , C3 , and C4 leads to: C2 =

T∞ 1 + (2 π f τlumped )2

C3 = −

2 π f τ T∞ 1 + (2 π f τlumped )2 C4 = T ∞

so that the particular solution is: Tp = T ∞ +

T∞ [sin(2 π f t ) − (2 π f τlumped ) cos(2 π f t )] 1 + (2 π f τlumped )2

(10)

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.1 Analytical Solutions to 0-D Transient Problems

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

314

Transient Conduction

The solution is the sum of the particular and homogeneous solutions, Eqs. (6) and (10): t T = C 1 exp − τlumped (11) T∞ +T∞ + [sin(2 π f t ) − (2 π f τlumped ) cos(2 π f t )] 2 1 + (2 π f τlumped ) Note that the same conclusion can be reached using two lines of Maple code; the governing differential equation, Eq. (3), is entered and solved: > restart; > ODE:=diff(T(t),t)+T(t)/tau=T_inﬁnity_bar/tau+DT_inﬁnity*sin(2*pi*f*t)/tau; d T (t ) T infinity bar DT infinity sin(2πf t ) O D E := T (t ) + = + dt τ τ τ > Ts:=dsolve(ODE); t T s := T (t ) = e(− τ ) C 1 + (T infinity bar + 4 T infinity barf 2 π 2 τ 2

−2DT infinity cos(2 πf t )f π τ + DT infinity sin(2 π f t ))/(1 + 4f 2 π 2 τ 2 )

The solution identiﬁed by Maple is the equivalent to Eq. (11); the solution is copied into EES, with minor editing: “Solution” Temp = exp(-1/tau∗ t)∗ C1-(-T_inﬁnity_bar-4∗ T_inﬁnity_bar∗ piˆ2∗ fˆ2∗ tauˆ2+& 2∗ DT_inﬁnity∗ cos(2∗ pi∗ f∗ t)∗ pi∗ f∗ tau& -DT_inﬁnity*sin(2∗ pi∗ f∗ t))/(1+4∗ piˆ2∗ fˆ2∗ tauˆ2)

The constant C1 must be selected so that the boundary condition is satisﬁed: Tt =0 = Tini

(12)

Substituting Eq. (11) into Eq. (12) leads to: Tt =0 = C 1 + T ∞ −

T∞ 2 π f τlumped = Tini 1 + (2 π f τlumped )2

which leads to: C1 =

T∞ 2 π f τlumped + Tini − T ∞ 1 + (2 π f τlumped )2

The symbolic expression for the boundary condition can also be found using Maple: > rhs(eval(Ts,t=0))=T_ini; T infinity bar + 4T infinity barf 2 π 2 τ 2 − 2 DT infinityf π τ C1 + = T ini 1 + 4f 2 π 2 τ 2

315

which is copied and pasted into EES: C_1+(T_inﬁnity_bar+4∗ T_inﬁnity_bar∗ fˆ2∗ piˆ2∗ tauˆ2-2∗ DT_inﬁnity∗ f∗ pi∗ tau)/(1+4∗ fˆ2∗ piˆ2∗ tauˆ2) = T_ini “initial condition”

The sensor temperature and ﬂuid temperature are converted to Celsius. T inﬁnity=T inﬁnity bar+DT inﬁnity∗ sin(2∗ pi∗ f∗ t)

“ambient temperature”

T C=converttemp(K,C,Temp)

“sensor temperature in C”

T inﬁnity C=converttemp(K,C,T inﬁnity)

“ambient temperature in C”

The temperatures are computed in a parametric table in which time is set so that it ranges from 0 to 10 s. The ﬂuid and sensor temperature variation are shown as a function of time in Figure 1. 380

fluid temperature

Temperature (°C)

360 340 320 300 280 260 0

sensor temperature 1

2

3

4

5 6 Time (s)

7

8

9

10

Figure 1: Temperature sensor and ﬂuid temperature as a function of time.

Note that after approximately 3 seconds (i.e., a few time constants) the homogeneous solution has decayed to zero and the temperature response of the sensor is given entirely by the particular solution, Eq. (10): Tp = T ∞ +

T∞ [sin(2 π f t ) − (2 π f τlumped ) cos(2 π f t )] 1 + (2 π f τlumped )2

(10)

Equation (10) can be rewritten in terms of an attenuation of the amplitude of the oscillation (Att) and a phase lag relative to the ﬂuid temperature variation (φ) Tp = T ∞ + Att T∞ sin(2 π f t − φ)

(13)

Equation (13) is rewritten using the trigonometric identity: Tp = T ∞ + Att T∞ [sin(2 π f t ) cos(φ) − cos(2 π f t ) sin(φ)]

(14)

Comparing Eq. (10) with Eq. (14) leads to: Att cos (φ) =

1 1 + (2 π f τ )2

(15)

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

3.1 Analytical Solutions to 0-D Transient Problems

Transient Conduction

and Att sin (φ) =

2 π f τlumped 1 + (2 π f τlumped )2

(16)

Dividing Eq. (16) by Eq. (15) leads to: tan (φ) = 2 π f τlumped so the phase (the lag) between the sensor and ﬂuid temperature is: φ = tan−1 (2 π f τlumped ) and the attenuation is: 1 Att = 0 1 + (2 π f τlumped )2 Figure 2 shows the attenuation and phase angle as a function of the product of the frequency and the time constant (f τlumped ). 1.6 Attenuation (-) and phase (rad)

EXAMPLE 3.1-2: SENSOR IN AN OSCILLATING TEMPERATURE ENVIRONMENT

316

1.4 1.2

φ

1 0.8 0.6 Att 0.4 0.2 0 0.001

0.01

0.1

1

10

100

Frequency time constant product, f τ Figure 2: Attenuation and phase angle as a function of the product of the frequency and the time constant.

Notice that if either the frequency or the time constant is small, then the attenuation goes to unity and the phase goes to zero. In this limit, the temperature sensor will faithfully follow the ﬂuid temperature with little error related to the transient response characteristics of the sensor; this is the situation that was illustrated earlier in Figure 3-2(b). In the other limit, if either the frequency or time constant of the sensor are very large then the attenuation will approach zero and the phase will approach π/2 rad (90 deg.). This situation corresponds to Figure 3-2(c) where the sensor cannot respond to the temperature oscillations. The dynamic characteristics of a temperature sensor are important in many applications and should be carefully considered when selecting an instrument for a transient temperature measurement.

3.2 Numerical Solutions to 0-D Transient Problems

317

3.2 Numerical Solutions to 0-D Transient Problems 3.2.1 Introduction Section 3.1 discussed the lumped capacitance model, which neglects any spatial temperature gradients within an object and therefore approximates the temperature in a transient problem as being only a function only of time. The analytical solution to such 0-D (or lumped capacitance) problems was examined in Section 3.1. In this section, lumped capacitance problems will be solved numerically. The numerical solution to any transient problem begins with the derivation of the governing differential equation, which allows the calculation of the temperature rate of change as a function of the current temperature and time. The solution to the problem is therefore a matter of integrating the governing differential equation forward in time. There are a number of techniques available for numerical integration, each with its own characteristic level of accuracy, stability, and complexity. These numerical integration techniques are discussed in this section and revisited in Section 3.8 in order to solve 1-D transient problems and again in Chapter 5 in order to solve convection problems. The governing differential equation that provides the rate of change of a variable (or several variables) given its own value (or values) is sometimes referred to as the state equation for the dynamic system. State equations characterize the transient behavior of problems in the areas of controls, dynamics, kinematics, ﬂuids, electrical circuits, etc. Therefore, the numerical solution techniques provided in this section are generally relevant to a wide range of engineering problems. The same caveats that were discussed previously in Sections 1.4 and 2.5 with regard to the numerical solution of steady-state conduction problems also apply to numerical solutions of transient conduction problems. Any numerical solution should be evaluated for numerical convergence (i.e., to ensure that you are using a sufﬁcient number of time steps), examined against your physical intuition, and compared to an analytical solution in some limit.

3.2.2 Numerical Integration Techniques In this section, the simplest lumped capacitance problem is solved numerically in order to illustrate the various options for numerical integration. An object initially in equilibrium with its environment at Tini is subjected to a step change in the ambient temperature from T ini to T ∞ . The governing differential equation provides the temperature rate of change given the current value of the temperature; the governing differential equation is derived from an energy balance on the object (see EXAMPLE 3.1-1): dT (T − T ∞ ) =− dt τlumped

(3-15)

where τlumped is the lumped time constant for the object. The analytical solution to the problem is derived in EXAMPLE 3.1-1, providing a basis for evaluating the accuracy of the numerical solutions that are derived in this section: t (3-16) T an = T ∞ + (T ini − T ∞ ) exp − τlumped The solution can also be obtained numerically using one of several available numerical integration techniques. Each numerical technique requires that the total simulation time

318

Transient Conduction

(tsim ) be broken into small time steps. The simplest option is to use equal-sized steps, each with duration t: tsim (3-17) t = (M − 1) where M is the number of times at which the temperature will be evaluated. The temperature at each time step (Tj ) is computed by the numerical model, where j indicates the time step (T1 is the initial temperature of the object and TM is the temperature at the end of a simulation). The time corresponding to each time step is therefore: tj =

(j − 1) tsim (M − 1)

for j = 1..M

(3-18)

Euler’s Method The temperature at the end of each time step is computed based on the temperature at the beginning of the time step and the governing differential equation. The simplest (and generally the worst) technique for numerical integration is Euler’s method. Euler’s method approximates the rate of temperature change within the time step as being constant and equal to its value at the beginning of the time step. Therefore, for any time step j: dT t (3-19) T j+1 = T j + dt T =T j ,t=tj Because the temperature at the end of the time step (Tj+1 ) can be calculated explicitly using information that is available at the beginning of the time step (Tj ), Euler’s method is referred to as an explicit numerical technique. The temperature at the end of the ﬁrst time step (T2 ) is given by: dT T2 = T1 + t (3-20) dt T =T 1 ,t=0 Substituting the state equation for our problem, Eq. (3-15), into Eq. (3-20) leads to: T2 = T1 −

(T 1 − T ∞ ) t τlumped

(3-21)

Equation (3-21) is written for every time step, resulting in a set of explicit equations that can be solved sequentially for the temperature at each time. As an example, consider the case where τlumped = 100 s, T ini = 300 K, and T ∞ = 400 K. These inputs are entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

The time step duration and array of times for which the solution will be computed is speciﬁed using Eqs. (3-17) and (3-18):

3.2 Numerical Solutions to 0-D Transient Problems

t sim=500 [s] M=501 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

319

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

For the values of tsim and M entered above, t = 1 s and the temperature predicted by the numerical model at the end of the ﬁrst timestep, T2 , computed using Eq. (3-21) is 301 K, as obtained from: “Euler’s Method” T[1]=T ini T[2]=T[1]-(T[1]-T inﬁnity)∗ DELTAt/tau

“estimate of temperature at the 1st timestep”

The actual temperature at t = 1 s is obtained using the analytical solution to the problem, Eq. (3-16): T an[2]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[2]/tau) “analytical solution for 1st time step”

and found to be T an,t=1s = 300.995 K. The numerical technique is not exact; the error between the numerical and analytical solutions at the end of the ﬁrst time step is err = 0.005 K. If the time step duration is increased by a factor of 10, to t = 10 s (by reducing M to 51), then the numerical solution at the end of the ﬁrst time step becomes T 2 = 310 K and the analytical solution is T an,t=10s = 309.52◦ C. Therefore, the error between the numerical and analytical solutions increases by approximately a factor of 100, to err = 0.48 K. This behavior is a characteristic of the Euler technique; the local error is proportional to the square of the size of the time step. err ∝ t2

(3-22)

This characteristic can be inferred by examination of Eq. (3-20), which is essentially the ﬁrst two terms of a Taylor series expansion of the temperature about time t = 0: t2 t3 dT d2 T d3 T T2 = T1 + t + (3-23) + + ··· dt T =T 1 ,t=0 dt2 T =T 1 ,t=0 2! dt3 T =T 1 ,t=0 3! Euler’s approximation

err ≈neglected terms

Examination of Eq. (3-23) shows that the error, corresponding to the neglected terms in the Taylor’s series, is proportional to t2 (neglecting the smaller, higher order terms). Therefore, Euler’s technique is referred to as a ﬁrst order method because it retains the ﬁrst term in the Taylor’s series approximation. The error associated with a ﬁrst order technique is proportional to the second power of the time step duration. Most numerical techniques that are commonly used have higher order and therefore achieve higher accuracy. The other drawback of Euler’s technique (and any explicit numerical technique) is that it may become unstable if the duration of the time step becomes too large. For our example, if the time step duration is increased beyond the time constant of the object, τlumped = 100 s, then the problem becomes unstable.

320

Transient Conduction 600 550

Temperature (K)

500 450 400 350 300 250 200 150 100 0

Δt = 250 s Δt = 200 s Δt = 143 s Δt = 50 s analytical solution th Δt = 5 s (every 4 point shown) 100 200 300 400 500 600 700 800 900 1000 Time (s)

Figure 3-3: Numerical solution obtained using Euler’s method with various values of t. Also shown is the analytical solution to the same problem.

The process of moving forward through all of the time steps may be automated using a duplicate loop: “Euler’s Method” T[1]=T ini “initial temperature” duplicate j=1,(M-1) “Euler solution” T[j+1]=T[j]-(T[j]-T inﬁnity)∗ DELTAt/tau T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “error between numerical and analytical solution” err[j+1]=abs(T an[j+1]-T[j+1]) end “maximum error” err max=max(err[2..M])

Note that the analytical solution and the absolute value of the error between the numerical and analytical solutions are also obtained at each time step. The maximum error is computed at the conclusion of the duplicate loop. Figure 3-3 illustrates the numerical solution for various values of the time step duration. The analytical solution provided by Eq. (3-16) is also shown in Figure 3-3. Notice that if the simulation is carried out with a time step duration that is less than t = τlumped = 100 s, then the solution is stable and follows the analytical solution more precisely as the time step is reduced. If the numerical solution is carried out with a time step duration that is between t = τlumped = 100 s and t = 2τlumped = 200 s then the prediction oscillates about the actual temperature but remains bounded. For time step durations greater than t = 2τlumped = 200 s, the solution oscillates in an unbounded manner. Figure 3-3 shows that any solution in which t > τlumped is unstable. This threshold time step duration that governs the stability of the solution is called the critical time step, tcrit , and its value can be determined from the details of the problem. Substituting the

Maximum numerical error (K)

3.2 Numerical Solutions to 0-D Transient Problems 10

2

10

1

10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

Euler Heun RK4 Fully implicit Crank-Nicholson

-8

10 2 x 10-1

321

100

101 Duration of time step (s)

102

Figure 3-4: Maximum numerical error as a function of t for various numerical integration techniques.

governing equation, Eq. (3-15) into the deﬁnition of Euler’s method, Eq. (3-19), leads to: T j+1 = T j −

(T j − T ∞ ) t τlumped

(3-24)

Rearranging Eq. (3-24) leads to: T j+1 = T j 1 −

t τlumped

+ T∞

t τlumped

(3-25)

The solution becomes unstable when the coefﬁcient multiplying the temperature at the beginning of the time step (Tj ) becomes negative; therefore: tcrit = τlumped

(3-26)

Not surprisingly, the numerical simulation of objects that have small time constants will require the use of small time steps. The accuracy of the solution (i.e., the degree to which the numerical solution agrees with the analytical solution) improves as the duration of the time step is reduced. Figure 3-4 illustrates the global error, deﬁned as the maximum error between the numerical and analytical solution, as a function of the duration of the time step. The global error is related to the local error associated with any one time step. Note that while the local error is proportional to t2 , the global error is proportional to t. Also shown in Figure 3-4 is the global accuracy of several alternative numerical integration techniques that are discussed in subsequent sections. Any of the numerical integration techniques discussed in this section can be implemented in most software. As the problems become more complex (e.g., 1-D transient problems), the number of equations and the amount of data that must be stored increases and therefore it will be appropriate to utilize a formal programming language (e.g., MATLAB) for the solution.

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400

Temperature (K)

380 360 340 320 300 0

100

200 300 Time (s)

400

500

Figure 3-5: Temperature as a function of time predicted by MATLAB using Euler’s method.

The numerical solution using Euler’s method is implemented in MATLAB using a script with the inputs provided as the ﬁrst lines: clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400;

% time constant (s) % initial temperature (K) % ambient temperature (K)

The array of times at which the solution will be determined is speciﬁed: t sim=500; M=101; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% simulation time (s) % number of time steps (-) % time step duration (s) % time associated with each step

Because Euler’s method is explicit, the solution is implemented in MATLAB using exactly the same technique that is used in EES; this is possible because each of the predictions for Tj+1 require only knowledge of Tj , as shown by Eq. (3-25). T(1)=T ini; for j=1:(M-1) T(j+1)=T(j)-(T(j)-T inﬁnity)∗ DELTAt/tau; end

% initial temperature % Euler method

Running the script will place the vectors T and time in the command space. The numerical solution obtained by MATLAB is shown in Figure 3-5. Heun’s Method Euler’s method is the simplest example of a numerical integration technique; it is a ﬁrst order explicit technique. In this section, Heun’s method is presented. Heun’s method is

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323

a second order explicit technique (but with the same stability characteristics as Euler’s method). Heun’s method is an example of a predictor-corrector technique. In order to simulate any time step j, Heun’s method begins with an Euler step to obtain an initial prediction for the temperature at the conclusion of the time step (Tˆ j+1 ). This ﬁrst step in the solution is referred to as the predictor step and the details are essentially identical to Euler’s method: dT t (3-27) Tˆ j+1 = T j + dt T =T j ,t=tj However, Heun’s method uses the results of the predictor step to carry out a corrector step. The temperature predicted at the end of the time step (Tˆ j+1 ) is used to predict the | ). The corrector temperature rate of change at the end of the time step ( dT dt T =Tˆ j+1 ,t=t j+1 step predicts the temperature at the end of the time step (Tj+1 ) based on the average of the time rates of change at the beginning and end of the time step. dT dT t + (3-28) T j+1 = T j + dt T =T j ,t=tj dt T =Tˆ ,t=tj+1 2 j+1

Heun’s method is illustrated in the context of the same simple problem that was used to demonstrate Euler’s technique. Initially, the technique will be implemented using EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

t sim=500 [s] M=51 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

The process of moving through the ﬁrst time step begins with the predictor step: dT t (3-29) Tˆ 2 = T 1 + dt T =T 1 ,t=t1 or, substituting Eq. (3-15) into Eq. (3-29): (T 1 − T ∞ ) t Tˆ 2 = T 1 − τlumped T[1]=T ini T hat[2]=T[1]-(T[1]-T inﬁnity)∗ DELTAt/tau

(3-30)

“initial temperature” “predictor step”

Note that for the same conditions considered previously (i.e., τlumped = 100 s, T ini = 300 K, and T ∞ = 400 K), the predictor step using t = 10 s leads to Tˆ 2 = 310 K, which is about 0.5 K in error relative to the actual temperature calculated using Eq. (3-16), T an,t=10s = 309.52 K.

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Transient Conduction

The corrector step follows: T2 = T1 +

dT dT t + dt T =T 1 ,t=t1 dt T =Tˆ 2 ,t=t2 2

(3-31)

or, substituting Eq. (3-15) into Eq. (3-31): (T 1 − T ∞ ) (Tˆ 2 − T ∞ ) t T2 = T1 − + τlumped τlumped 2 T[2]=T[1]-((T[1]-T inﬁnity)/tau+(T hat[2]-T inﬁnity)/tau)∗ DELTAt/2

(3-32)

“corrector step”

The corrector step predicts T 2 = 9.50◦ C, which is only 0.02◦ C in error with respect to T an,t=10s = 9.52◦ C. This result illustrates the power of the predictor/corrector process; a single-step correction (which approximately doubles the computational time) can lead to more than an order of magnitude increase in accuracy. The two-step predictor/corrector process is a second order method; the local error is proportional to the duration of the time step to the third power. The EES code below implements a numerical solution to the problem using a duplicate loop and computes the numerical error. (Note that the EES code corresponding to the Euler solution must be commented out or deleted.) “Heun’s Method” T[1]=T ini duplicate j=1,(M-1) “predictor step” T hat[j+1]=T[j]-(T[j]-T inﬁnity)∗ DELTAt/tau T[j+1]=T[j]-((T[j]-T inﬁnity)/tau+(T hat[j+1]-T inﬁnity)/tau)∗ DELTAt/2 “corrector step” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M]) “maximum numerical error”

Figure 3-6 illustrates the numerical solution obtained using Heun’s method with time step durations of t = 50 s, t = 83.3 s, t = 167.7 s and t = 250 s. Also shown in Figure 3-6 are the analytical solution and the solution using Euler’s method with t = 50 s. Notice that for the same time step duration ( t = 50 s), the numerical result obtained with Heun’s method is much closer to the analytical solution than the numerical result obtained with Euler’s method. The maximum error between the numerical and analytical solutions is shown in Figure 3-4 for Heun’s technique and the improvement relative to Euler’s technique is obvious. Also note in Figure 3-6 that Heun’s method is unstable for time step durations that are greater than τlumped = 100 s and temperature actually decreases for t > 2τlumped = 200 s. The instability is not oscillatory; however, the numerical result begins to diverge from the analytical results and provides a non-physical solution. Both Euler’s and Heun’s methods are explicit techniques and both will therefore become unstable when the time step duration exceeds the critical time step. Heun’s method is explicit and therefore its implementation in MATLAB is straightforward. The script begins by specifying the inputs and setting up the time steps and initial conditions:

3.2 Numerical Solutions to 0-D Transient Problems

325

400

Temperature (K)

350 Heun's method, Δt = 50 s Heun's method, Δt = 83.3 s Heun's method, Δt = 167.7 s Heun's method, Δt = 250 s Euler's method, Δt = 50 s analytical solution

300

250

200 0

50

100 150 200 250 300 350 400 450 500 Time (s)

Figure 3-6: Numerical solution obtained using Heun’s method for various values of t. Also shown is the analytical solution and the solution using Euler’s method with t = 50 s.

clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400; t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% time constant (s) % initial temperature (K) % ambient temperature (K) % simulation time (s) % number of time steps (-) % time step duration (s)

The integration follows the form provided by Eqs. (3-27) and (3-28). One advantage of using MATLAB over EES is that MATLAB utilizes assignment statements. Therefore, it is not necessary to store the intermediate variable Tˆ j+1 (i.e., the result of the predictor step) for each time step in an array. Instead, the value of the variable Tˆ j+1 (the variable T_hat) is over-written during each iteration of the for loop; this saves memory and time. EES uses equality statements rather than assignment statements and so it will not allow the value of a variable to be overwritten in the main body of the program. (It is possible to overwrite values in an EES function or procedure.)

T(1)=T ini; % initial temperature for j=1:(M-1) % predictor step T hat=T(j)-(T(j)-T inﬁnity)∗ DELTAt/tau; T(j+1)=T(j)-((T(j)-T inﬁnity)/tau+(T hat-T inﬁnity)/tau)∗ DELTAt/2; % corrector step end

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Transient Conduction

Runge-Kutta Fourth Order Method Heun’s method is a two-step predictor/corrector technique that improves the order of accuracy by one (i.e., Heun’s method is second order whereas Euler’s method is ﬁrst order). It is possible to carry out additional predictor/corrector steps and further improve the accuracy of the numerical solution. One of the most popular, higher order techniques is the fourth order Runge-Kutta method which involves four predictor/corrector steps. The Runge-Kutta fourth order method (referred to subsequently as the RK4 method) estimates the time rate of change of the state variable four times; recall that Euler’s method did this only once (at the beginning of time step) and Heun’s method did this twice (at the beginning and end of the time step). The RK4 method begins by estimating the time rate of change at the beginning of the time step (referred to, for convenience, as aa): dT (3-33) aa = dt T =T j ,t=tj The ﬁrst estimate, aa, is used to predict the temperature half-way through the time step (Tˆ j+ 1 ): 2

t Tˆ j+ 1 = T j + aa 2 2

(3-34)

which is used to obtain the second estimate of the time rate of change, bb, at the midpoint of the time step: dT (3-35) bb = dt T =Tˆ ,t=tj + t j+ 1 2

2

The second estimate is used to re-compute the temperature half-way through the time step (Tˆˆ j+ 1 ): 2

t Tˆˆ j+ 1 = T j + bb 2 2

(3-36)

which is used to obtain a third estimate of the time rate of change, cc, also at the midpoint of the time step: dT (3-37) cc = dt T =Tˆ ,t=tj + t j+ 1 2

2

The third estimate is used to predict the temperature at the end of the time step (Tˆ j+1 ): Tˆ j+1 = T j + cc t

(3-38)

which is used to obtain the fourth and ﬁnal estimate of the time rate of change, dd, at the end of the time step: dT (3-39) dd = dt T =Tˆ ,t=tj+1 j+1

The integration is ﬁnally carried out using the weighted average of these four separate estimates of the time rate of change: T j+1 = T j + (aa + 2 bb + 2 cc + dd)

t 6

(3-40)

3.2 Numerical Solutions to 0-D Transient Problems

327

The RK4 technique is implemented in EES in order to solve the problem discussed previously: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K] t sim=500 [s] M=4 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“time constant” “initial temperature” “ambient temperature” “simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

The code below implements the RK4 technique: “Runge-Kutta 4th Order Method” T[1]=T_ini duplicate j=1,(M-1) “1st estimate of time-rate of change” aa[j]=-(T[j]-T inﬁnity)/tau T hat1[j]=T[j]+aa[j]∗ DELTAt/2 “1st predictor step” bb[j]=-(T hat1[j]-T inﬁnity)/tau “2nd estimate of time-rate of change” “2nd predictor step” T hat2[j]=T[j]+bb[j]∗ DELTAt/2 cc[j]=-(T hat2[j]-T inﬁnity)/tau “3rd estimate of time-rate of change” T hat3[j]=T[j]+cc[j]∗ DELTAt “3rd predictor step” “4th estimate of time-rate of change” dd[j]=-(T hat3[j]-T inﬁnity)/tau T[j+1]=T[j]+(aa[j]+2∗ bb[j]+2∗ cc[j]+dd[j])∗ DELTAt/6 “ﬁnal integration uses all 4 estimates” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M]) “maximum numerical error”

Figure 3-4 illustrates the maximum numerical error for the RK4 solution as a function of the time step duration and clearly indicates the improvement that can be obtained by using a higher order method. If a certain level of accuracy is required, then much larger time steps (and therefore many fewer computations) can be used if a higher order technique is used. For example, if an accuracy of 0.1 K is required then t = 0.4 s must be used with Euler’s method whereas t = 70 s can be used with the RK4 method. This difference represents more than two orders of magnitude of reduction in the number of time steps required while the number of computations per time step has only increased by a factor of four. The slope of the curves shown in Figure 3-4 is related to the order of the technique with respect to the global error as opposed to the local error. The maximum error associated with Euler’s method increases by an order of magnitude as the time step duration increases by an order of magnitude; therefore, while the local error associated with each

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Transient Conduction

time step has order two, the global error has order one. The maximum error associated with Heun’s method changes by two orders of magnitude for every order of magnitude change in t and therefore Heun’s method is order two with respect to the global error. The RK4 method is nominally order four with respect to global error. The MATLAB code to implement the RK4 method follows naturally from the EES code because the method is explicit. Note that the intermediate variables (e.g., aa, bb, etc.) are not stored and therefore the MATLAB program is less memory intensive.

clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400;

% time constant (s) % initial temperature (K) % ambient temperature (K)

t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% simulation time (s) % number of time steps (-) % time step duration (s)

T(1)=T ini; for j=1:(M-1) aa=-(T(j)-T inﬁnity)/tau; T hat=T(j)+aa∗ DELTAt/2; bb=-(T hat-T inﬁnity)/tau; T hat=T(j)+bb∗ DELTAt/2; cc=-(T hat-T inﬁnity)/tau; T hat=T(j)+cc∗ DELTAt; dd=-(T hat-T inﬁnity)/tau; T(j+1)=T(j)+(aa+2∗ bb+2∗ cc+dd)∗ DELTAt/6;

% initial temperature % 1st estimate of time-rate of change % 1st predictor step % 2nd estimate of time-rate of change % 2nd predictor step % 3rd estimate of time-rate of change % 3rd predictor step % 4th estimate of time-rate of change % ﬁnal integration uses all 4 estimates

end

Fully Implicit Method The methods discussed thus far are explicit; they all therefore share the characteristic of becoming unstable when the time step exceeds a critical value. An implicit technique avoids this problem. The fully implicit method is similar to Euler’s method in that the time rate of change is assumed to be constant throughout the time step. However, the time rate of change is computed at the end of the time step rather than the beginning. Therefore, for any time step j: T j+1

dT = Tj + t dt T =T j+1 ,t=tj+1

(3-41)

The time rate of change at the end of the time step depends on the temperature at the end of the time step (Tj+1 ). Therefore, Tj+1 cannot be calculated explicitly using information at the beginning of the time step (Tj ) and instead an implicit equation is obtained for Tj+1 . For the example problem that has been considered in previous sections, the

3.2 Numerical Solutions to 0-D Transient Problems

329

implicit equation is obtained by substituting Eq. (3-15) into Eq. (3-41): T j+1 = T j −

(T j+1 − T ∞ ) t τlumped

(3-42)

Notice that Tj+1 appears on both sides of Eq. (3-42). Because EES solves implicit equations, it is not necessary to rearrange Eq. (3-42). The implicit solution is obtained using the following EES code: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

t sim=500 [s] M=4 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

“Fully Implicit Method” T[1]=T ini duplicate j=1,(M-1) T[j+1]=T[j]-(T[j+1]-T inﬁnity)∗ DELTAt/tau

“initial temperature”

“implicit equation for temperature” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M])

Figure 3-7 illustrates the fully implicit solution for various values of the time step; also shown is the analytical solution, Eq. (3-16). The accuracy of the fully implicit solution is reduced as the duration of the time step increases. Figure 3-4 illustrates the maximum error associated with the fully implicit technique as a function of the time step duration; notice that the fully implicit technique is no more accurate than Euler’s technique. However, Figure 3-7 shows that the fully implicit solution does not become unstable even when the duration of the time step is greater than the critical time step. The implementation of the fully implicit method cannot be accomplished in the same way in MATLAB that it is in EES because Eq. (3-42) is not explicit for Tj+1 ; while EES can solve this implicit equation, MATLAB cannot. It is necessary to solve Eq. (3-42) for Tj+1 in order to carry out the integration step in MATLAB:

T j+1

t T j + T∞ τ = t 1+ τ

(3-43)

330

Transient Conduction 400

Temperature (K)

380

360

340 analytical solution Δt = 1 s (every 15 th point shown) Δt = 50 s Δt = 150 s Δt = 300 s

320

300 0

100

200

300 Time (s)

400

500

600

Figure 3-7: Fully implicit solution for various values of the time step duration as well as the analytical solution.

clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400;

% time constant (s) % initial temperature (K) % ambient temperature (K)

t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% simulation time (s) % number of time steps (-) % time step duration (s)

T(1)=T ini; for j=1:(M-1) T(j+1)=(T(j)+T inﬁnity∗ DELTAt/tau)/(1+DELTAt/tau); end

% initial temperature % implicit step

Crank-Nicolson Method The Crank-Nicolson method combines Euler’s method with the fully implicit method. The time rate of change for the time step is estimated based on the average of its values at the beginning and end of the time step. Therefore, for any time step j: T j+1 = T j +

dT dT t + dt T =T j ,t=tj dt T =T j+1 ,t=tj+1 2

(3-44)

Notice that the Crank-Nicolson is an implicit method because the solution for T j+1 involves a time rate of change that must be evaluated based on T j+1 . Therefore, the technique will have the stability characteristics of the fully implicit method. The solution

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331

also involves two estimates for the time rate of change and is therefore second order and more accurate than the fully implicit technique. Substituting Eq. (3-15) into Eq. (3-44) leads to:

T j+1

(T j − T ∞ ) (T j+1 − T ∞ ) t = Tj − + τlumped τlumped 2

(3-45)

The EES code below implements the Crank-Nicolson solution for the example problem:

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K]

“time constant” “initial temperature” “ambient temperature”

t sim=600 [s] M=601 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) end

“simulation time” “number of time steps” “duration of time step” “time associated with each temperature”

“Crank Nicholson Method” T[1]=T ini duplicate j=1,(M-1) T[j+1]=T[j]-((T[j]-T inﬁnity)/tau+(T[j+1]-T inﬁnity)/tau)∗ DELTAt/2 “C-N equation for temperature” T an[j+1]=T inﬁnity+(T ini-T inﬁnity)∗ exp(-time[j+1]/tau) “analytical solution” “numerical error” err[j+1]=abs(T an[j+1]-T[j+1]) end err max=max(err[2..M])

The maximum numerical error associated with the Crank-Nicolson technique is shown in Figure 3-4. Notice that the fully implicit method has accuracy that is nominally equivalent to Euler’s method while the Crank-Nicolson method has accuracy slightly better than Heun’s method. The Crank-Nicolson method is a popular choice because it combines high accuracy with stability. To implement the Crank-Nicolson technique using MATLAB, it is necessary to solve the implicit Eq. (3-45) for T j+1 :

T j+1

t t + T∞ Tj 1 − 2τ τ = t 1+ 2τ

(3-46)

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Transient Conduction

so the MATLAB script becomes: clear all; %INPUTS tau=100; T ini=300; T inﬁnity=400; t sim=500; M=51; DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); end

% time constant (s) % initial temperature (K) % ambient temperature (K) % simulation time (s) % number of time steps (-) % time step duration (s)

T(1)=T ini; % initial temperature for j=1:(M-1) T(j+1)=(T(j)∗ (1-DELTAt/(2∗ tau))+T inﬁnity∗ DELTAt/tau)/(1+DELTAt/(2∗ tau));

% C-N step

end

Adaptive Step-Size and EES’ Integral Command The implementation of the techniques that have been discussed to this point is accomplished using a ﬁxed duration time step for the entire simulation. This implementation is often not efﬁcient because there are regions of time during the simulation where the solution is not changing substantially and therefore large time steps could be taken with little loss of accuracy. Adaptive step-size solutions adjust the size of the time step used based on the local characteristics of the state equation. Typically, the absolute value of the local time rate of change or the second derivative of the time rate of change is used to set a step-size that guarantees a certain level of accuracy. For the current example, shown in Figure 3-6, smaller time steps would be used near t = 0 s because the rate of temperature change is largest at this time. Later in the simulation, for t > 300 s, the temperature is not changing substantially and therefore large time steps could be used to obtain a solution more rapidly. The implementation of adaptive step-size solutions is beyond the scope of this book. However, a third order integration routine that optionally uses an adaptive step-size is provided with EES and can be accessed using the Integral command. EES’ Integral command requires four arguments and allows an optional ﬁfth argument: F = Integral(Integrand,VarName,LowerLimit,UpperLimit,StepSize) where Integrand is the EES variable or expression that must be integrated, VarName is the integration variable, LowerLimit and UpperLimit deﬁne the limits of integration, and StepSize provides the duration of the time step. When using the Integral technique (or indeed any numerical integration technique) it is useful to ﬁrst verify that, given temperature and time, the EES code is capable of computing the time rate of change of the temperature. Therefore, the ﬁrst step is to implement Eq. (3-15) in EES for an arbitrary (but reasonable) value of T and t:

3.2 Numerical Solutions to 0-D Transient Problems

333

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” tau=100 [s] T ini=300 [K] T inﬁnity=400 [K] t sim=600 [s]

“time constant” “initial temperature” “ambient temperature” “simulation time”

“EES’ Integral Method” T=350 [K] time = 0 [s] dTdt=-(T-T inﬁnity)/tau

“Temperature, arbitrary - used to test dTdt” “Time, arbitrary - used to test dTdt” “Time rate of change”

which leads to dTdt = 0.5 K/s. The next step is to comment out the arbitrary values of temperature and time that are used to test the computation of the state equation and instead let EES’ Integral function control these variables for the numerical integration. The temperature of the object is given by: tsim T = T ini +

dT dt dt

(3-47)

0

Therefore, the solution to our example problem is obtained by calling the Integral function; Integrand is replaced with the variable dTdt, VarName with time, LowerLimit with 0, UpperLimit with the variable t_sim, and StepSize with DELTAt, the speciﬁed duration of the time step: “EES’ Integral Method” {T=350 [K] “Temperature, arbitrary - used to test dTdt” time=0 [s] “Time, arbitrary - used to test dTdt”} dTdt=-(T-T inﬁnity)/tau “Time rate of change” DELTAt=1 [s] “Duration of the time step” T=T ini+INTEGRAL(dTdt,time,0,t sim,DELTAt) “Call EES’ Integral function”

In order to accomplish the numerical integration, EES will adjust the value of the variable time from 0 to t_sim in increments of DELTAt. At each value of time, EES will iteratively solve all of the equations in the Equations window that depend on time. For the example above, this process will result in the variable T being evaluated at each value time. When the solution converges, the value of the variable time is incremented and the process is repeated until time is equal to t_sim. The results shown in the Solution window will provide the temperature at the end of the process (i.e., the result of the integration, which is the temperature at time = t_sim). Often it is most interesting to know the temperature variation with time during the process. This information can be provided by including the $IntegralTable directive in the ﬁle. The format of the $IntegralTable directive is: $IntegralTable VarName: Step, x,y,z

334

Transient Conduction

[K]

[s] 0

300

1

301

2

302

3

303

4

303.9

Figure 3-8: Integral Table generated by EES.

where VarName is the integration variable; the ﬁrst column in the Integral Table will hold values of this variable. The colon followed by the parameter Step (which can either be a number or a variable name) and list of variables (x, y, z) are optional. If these values are provided, then the value of Step will be used as the output step size and the integration variables will be reported in the Integral Table at the speciﬁed output step size. The output step size may be a variable name, rather than a number, provided that the variable has been set to a constant value preceding the $IntegralTable directive. The step size that is used to report integration results is totally independent of the duration of the time step that is used in the numerical integration. If the numerical integration step size and output step size are not the same, then linear interpolation is used to determine the integrated quantities at the speciﬁed output steps. If an output step size is not speciﬁed, then EES will output all speciﬁed variables at every time step. The variables x, y, z . . . must correspond to variables in the EES program. Algebraic equations involving variables are not accepted. A separate column will be created in the Integral Table for each speciﬁed variable. The variables must be separated by a space or list delimiter (comma or semicolon). Solving the EES code will result in the generation of an Integral Table that is ﬁlled with intermediate values resulting from the numerical integration. The values in the Integral Table can be plotted, printed, saved, and copied in exactly the same manner as for other tables. The Integral Table is saved when the EES ﬁle is saved and the table is restored when the EES ﬁle is loaded. If an Integral Table exists when calculations are initiated, it will be deleted if a new Integral Table is created. Use the EES code below to generate an Integral Table containing the results of the numerical simulation and the analytical solution.

$IntegralTable time, T

After running the code with a time step duration of 1 s, the Integral Table shown in Figure 3-8 will be generated. The results of the integration can be plotted by selecting the Integral Table as the source of the data to plot. The StepSize input to the Integral command is optional. If the parameter StepSize is not included or if it is set to a value of 0, then EES will use an adaptive step-size algorithm that maintains accuracy while maximizing computational speed. The parameters used to control the adaptive step-size algorithm can be accessed and adjusted by selecting Preferences from the Options menu and selecting the Integration Tab. A complete description of these parameters can be obtained from the EES Help menu. To run the program using an adaptive step-size, remove the ﬁfth argument from the Integral command (or set its value to 0):

3.2 Numerical Solutions to 0-D Transient Problems

335

400

Temperature (K)

380

360

340

320

300 0

100

200

300 Time (s)

400

500

600

Figure 3-9: Numerical solution predicted using EES’ internal Integration function with an adaptive time step.

“EES’ Integral Method” dTdt=-(T-T inﬁnity)/tau T=T ini+INTEGRAL(dTdt,time,0,t sim)

“Time rate of change” “Call EES’ Integral function”

$IntegralTable time, T

The results are shown in Figure 3-9. Note the concentration of time steps near t = 0 s. Also note that the Adaptive Step-Size parameters had to be adjusted in order for the algorithm to operate for such a simple problem; speciﬁcally, the minimum number of steps had to be reduced to 10 and the criteria for increasing the step-size had to be increased to 0.01. MATLAB’s Ordinary Differential Equation Solvers MATLAB has a suite of solvers for initial value problems that implement advanced numerical integration algorithms (e.g., the functions ode45, ode23, etc.). These ordinary differential equation solvers all have a similar protocol; for example: [ time ,

T

vector temperature of time at the times

] = ode45(

‘dTdt’ function that returns the derivative of temperature

,

tspan time span to be integrated

,

T0

)

initial temperature

The ode solvers require three inputs, the name of the function that returns the derivative of temperature with respect to time given the current value of temperature and time (i.e., the function that implements the state equation for the system), the simulation time, and the initial temperature. The ode solver returns two vectors containing the solution times and the temperatures at the solution times. The ode solvers require that you have created a function that implements the state equation for the system. MATLAB assumes that this function will accept two inputs, corresponding to the current values of time and temperature, and return a single output, the rate at which temperature is changing with

336

Transient Conduction

400

Temperature (K)

380 360 340 320 300 0

100

200 300 Time (s)

400

500

Figure 3-10: Predicted temperature as a function of time using MATLAB’s ode45 solver.

time. The function dTdt_function, shown below, implements the state equation for the example considered previously: function[dTdt]=dTdt function(time,T) % this function computes the rate of change of temperature % Inputs % T - temperature (K) % time - time (s) % % Outputs % dTdt - time rate of change (K/s) tau=100; % time constant (s) T inﬁnity=500; % ambient temperature (K) dTdt=-(T-T inﬁnity)/tau; end

The script that solves the problem calls the integration routine ode45 and speciﬁes that the state equations are determined in the function dTdt_function, the simulation time is from 0 to t_sim and the initial condition is T_ini. clear all; %INPUTS T ini=300; t sim=500; [time,T]=ode45(‘dTdt function’,t sim,T ini);

% initial temperature (K) % simulation time (s) % use ode45 to solve problem

The vectors time and T are the times and temperatures used to carry out the simulation. Figure 3-10 shows the temperature predicted using the ode45 solver as a function of time. Figure 3-10 shows that the MATLAB solver is using an adaptive step-size algorithm because the duration of the time step varies throughout the computational domain. If the variable t_sim is speciﬁed as a vector of times, then MATLAB will return the solution at these speciﬁc times rather than the times that are actually used for the integration

3.2 Numerical Solutions to 0-D Transient Problems

337

(although the integration process itself does not change). For example, in order to obtain a solution every 10 s the script should be changed as shown: t span=linspace(0,500,51); [time,T]=ode45(‘dTdt function’,t span,T ini);

% specify times to return the solution at

It is not convenient that the state equation function, dTdt_function, only allows two inputs. It is difﬁcult to run parametric studies or optimization if the values of tau and T_inﬁnity must be changed manually within the function. This requirement can be avoided by modifying the function dTdt_function so that it accepts all of the inputs of interest: function[dTdt]=dTdt_function(time, T, tau, T_inﬁnity) % this function computes the rate of change of temperature % Inputs % T - temperature (K) % time - time (s) % tau - time constant (s) % T_inﬁnity - ambient temperature (K) % % Outputs % dTdt - time rate of change (K/s) dTdt=-(T-T_inﬁnity)/tau; end

The call to the function dTdt_function with the two inputs required by the function ode45 (time and T) is “mapped” to the full function call. This mapping has the form: @(time, T) dTdt function(time, T, tau, T amb) 2 inputs that the full function call, including the 2 required inputs are required by ode solver

The revised script becomes: clear all; %INPUTS T ini=300; t sim=500; tau=100; T inﬁnity=400;

% initial temperature (K) % simulation time (s) % time constant (s) % ambient temperature (K)

[time,T]=ode45(@(time,T) dTdt function(time,T, tau, T inﬁnity),[0,t sim],T ini);

A fourth argument (which is optional) can be added to the function ode45: [time,T ]=ode45(‘dTdt’, tspan, T0, OPTIONS)

where OPTIONS is a vector that controls the integration properties. In the absence of a fourth argument, MATLAB will use the default settings for the integration properties. The most convenient way to adjust the integration settings is with the odeset function,

338

Transient Conduction

400

Temperature (K)

380 360 340 320 300 0

100

200 300 Time (s)

400

500

Figure 3-11: Predicted temperature as a function of time using MATLAB’s ode45 solver with increased accuracy.

which has the format: OPTIONS = odeset(‘property1’, value1, ‘property2’, value2, . . .) where property1 refers to one of the integration properties and value1 refers to its speciﬁed value, property2 refers to a different integration property and value2 refers to its speciﬁed value, etc. The properties that are not explicitly set in the odeset command remain at their default values. Type help odeset at the command line prompt in order to see a list and description of the integration properties with their default values. The property RelTol refers to the relative error tolerance and has a default value of 0.001 (0.1% accuracy). The relative error tolerance of the integration process can be reduced to 1×10−6 by changing the script according to: OPTIONS=odeset(‘RelTol’,1e-6); % change the relative error tolerance used in the integration [time,T]=ode45(@(time,T) dTdt function(time,T, tau, T inﬁnity),[0,t sim],T ini,OPTIONS);

The results of the integration with the improved accuracy are shown in Figure 3-11. Notice that substantially more integration steps were required (compared with Figure 3-10) in order to obtain the improved accuracy. The integration function ode45 was used in the discussion in this section. However, MATLAB provides other differential equation solvers that are specialized based on the stiffness and other aspects of the problem. To examine these solvers, type help funfun at the command line. The differential equation solvers have similar calling protocol, so it is easy to switch between them. For example, to use the ode15s rather than the ode45 function it is only necessary to change the name of the function call: OPTIONS=odeset(‘RelTol’,1e-6); [time,T]=ode15s(@(time,T) dTdt_function(time,T, tau, T_inﬁnity),[0,t_sim],T_ini,OPTIONS);

339

EXAMPLE 3.2-1(a): OVEN BRAZING (EES) A brazing operation is carried out in an evacuated oven. The metal pieces to be brazed have a complex geometry; they are made of bronze (with k = 50 W/m-K, c = 500 J/kg-K, and ρ = 8700 kg/m3 ) and have total volume V = 10 cm3 and total surface area As = 35 cm2 . The pieces are heated by radiation heat transfer from the walls of the oven. A detailed presentation of radiation heat transfer is presented in Chapter 10. For this problem, assume that the emissivity of the surface of the piece is ε = 0.8 and that the wall of the oven is black. In this limit, the rate of radiation heat transfer from the wall to the piece (q˙r ad ) may be written as: q˙r ad = As ε σ Tw4 − T 4 where Tw is the temperature of the wall, T is the temperature of the surface of the piece, and σ is the Stefan-Boltzmann constant. The temperature of the oven wall is increased at a constant rate β = 1 K/s from its initial temperature Tini = 20◦ C to its ﬁnal temperature Tf = 470◦ C which is held for thold = 1000 s before the temperature of the wall is reduced at the same constant rate back to its initial temperature. The pieces and the oven are initially in thermal equilibrium at Tt =0 = Tini . a) Is the lumped capacitance assumption appropriate for this problem? The inputs are entered in EES. “EXAMPLE 3.2-1(a): Oven Brazing (EES)” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” V=10 [cmˆ3]∗ convert(cmˆ3,mˆ3) A s=35 [cmˆ2]∗ convert(cmˆ2,mˆ2) e=0.8 [-] T ini=converttemp(C,K,20 [C]) T f=converttemp(C,K,470 [C]) t hold=1000 [s] beta=1 [K/s] c=500 [J/kg-K] k=50 [W/m-K] rho=8700 [kg/mˆ3]

“volume” “surface area” “emissivity of surface” “initial temperature” “ﬁnal oven temperature” “oven hold time” “oven ramp rate” “speciﬁc heat capacity” “conductivity” “density”

The lumped capacitance assumption ignores the internal resistance to conduction as being small relative to the resistance to heat transfer from the surface of the object. The radiation heat transfer coefﬁcient, deﬁned in Section 1.2.6, is: hrad = σ ε Ts2 + Tw2 (Ts + Tw ) where Ts is the surface temperature of the object. The Biot number deﬁned based on this radiation heat transfer coefﬁcient is: Bi =

hrad L cond k

where L cond =

V As

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

340

Transient Conduction

The Biot number is largest (and therefore the lumped capacitance model is least justiﬁed) when the value of hrad is largest, which occurs if both Ts and Tw achieve their maximum possible value (Tf ). h rad max=sigma# ∗ e∗ (T fˆ2+T fˆ2)∗ (T f+T f) L cond=V/A s Bi=h rad max∗ L cond/k

“maximum radiation coefﬁcient” “characteristic length for conduction” “maximum Biot number”

The Biot number is calculated to be 0.004 at this upper limit, which indicates that the lumped capacitance assumption is valid. b) Calculate a lumped capacitance time constant that characterizes the brazing process. It is useful to calculate a time constant even when the problem is solved numerically. The value of the time constant provides guidance relative to the time step that will be required and it also allows a sanity check on your results. The time constant, discussed in Section 3.1.4, is the product of the thermal capacitance of the object and the net resistance from the surface. Using the concept of the radiation heat transfer coefﬁcient allows the time constant to be written as: τlumped =

V ρc hrad As

The minimum value of the time-constant (again, corresponding to the object and the oven wall being at their maximum temperature) is computed in EES: tau=rho∗ c∗ V/(h rad max∗ A s)

“time constant”

and found to be 167 s. c) Develop a numerical solution based on Heun’s method that predicts the temperature of the object for 3000 s after the oven is activated. A function T_w is deﬁned which returns the wall temperature as a function of time; the function is placed at the top of the EES ﬁle.

“Oven temperature function” function T w(time,T ini,T f,beta,t hold) “INPUTS: time - time relative to initiation of process (s) T ini - initial temperature (K) T f - ﬁnal temperature (K)

341

beta - ramp rate (K/s) t hold - hold time (s) OUTPUTS T w - wall temperature (K)” T w=T ini+beta∗ time t ramp=(T f-T ini)/beta if (time>t ramp) then T w=T f endif if (time>(t ramp+t hold)) then T w=T f-beta∗ (time-t ramp-t hold) endif if (time>(2∗ t ramp+t hold)) then T w=T ini endif end

“temperature of wall during ramp up period” “duration of ramp period” “temperature of wall during hold period”

“temperature during ramp down period”

“temperature after ramp down period”

Note the use of the if-then-endif statements in the function to activate different equations for the wall temperature based on the time of the simulation. The function T_w be called from the Equation Window: T_w=T_w(time,T_i,T_f,beta,tau_hold)

It is wise to check that the function is working correctly by setting up a parametric table that includes time and the wall temperature; the result is shown in Figure 1. 800

Wall temperature (K)

700 600 500 400 300 200 0

500

1000

1500 Time (s)

2000

Figure 1: Oven wall temperature as a function of time.

2500

3000

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

342

Transient Conduction

The simulation time and number of time steps are deﬁned and used to compute the time and wall temperature at each time step: t sim=3000 [s] M=101 [-] DELTAt=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ t sim/(M-1) T w[j]=T w(time[j],T ini,T f,beta,t hold) end

“simulation time” “number of time-steps” “time-step duration” “time” “wall temperature”

The governing differential equation is obtained from an energy balance on the piece: q˙ rad =

dU dt

or dT As ε σ Tw4 − T 4 = V ρ c dt

(1)

Rearranging Eq. (1) leads to the state equation that provides the time rate of change of the temperature: As ε σ Tw4 − T 4 dT = (2) dt V ρc The temperature at the beginning of the ﬁrst time step is the initial condition: T1 = Tini

T[1]=T ini

“initial temperature”

Heun’s method consists of an initial, predictor step: dT t Tˆ j+1 = T j + d t T =T j ,t =t j

(3)

Substituting the state equation, Eq. (2), into Eq. (3) leads to: 4 4 A ε σ T − T s w,t =t j dT j = d t T =T j ,t =t j V ρc The corrector step is:

T j+1 = T j +

dT dT + d t T =T j ,t =t j d t T =Tˆ

where dT d t T =Tˆ

j+1

,t =t j+1

=

j+1

,t =t j+1

4 ˆ4 As ε σ Tw,t =t j+1 − T j+1 V ρc

t 2

343

Heun’s method is implemented in EES: T[1]=T ini “initial temperature” duplicate j=1,(M-1) dTdt[j]=e∗ A s∗ sigma # ∗ (T w[j]ˆ4-T[j]ˆ4)/(rho∗ V∗ c) “Temperature rate of change at the beginning of the time-step” “Predictor step” T hat[j+1]=T[j]+dTdt[j]∗ DELTAt dTdt hat[j+1]=e∗ A s∗ sigma #∗ (T w[j+1]ˆ4-T hat[j+1]ˆ4)/(rho∗ V∗ c) “Temperature rate of change at the end of the time-step” “Corrector step” T[j+1]=T[j]+(dTdt[j]+dTdt hat[j+1])∗ DELTAt/2 end

The solution for 100 time steps is shown in Figure 2. 800 Heun's method EES Integral function th (every 10 point)

Temperature (K)

700 600 500 400 300 200 0

Oven wall temperature 500

1000

1500 Time (s)

2000

2500

3000

Figure 2: Oven wall and piece temperature as a function of time, predicted by Heun’s method with 100 time steps and using EES’ built-in Integral function.

Note that the piece temperature lags the oven wall temperature by 100’s of seconds, which is consistent with the time constant calculated in (b). d) Develop a numerical solution using EES’ Integral function that predicts the temperature of the object for 3000 s after the oven is activated. The adaptive time step algorithm is used within EES’ Integral function and an integral table is created that holds the results of the integration: “EES’ Integral function” dTdt=e∗ A s∗ sigma #∗ (T w(time,T ini,T f,beta,t hold)ˆ4-Tˆ4)/(rho∗ V∗ c) T=T ini+INTEGRAL(dTdt,time,0,t sim) “Call EES’ Integral function” $IntegralTable time, T

The results are included in Figure 2 and agree with Heun’s method; note that only 1 out of every 10 points are included in the EES solution in order to show the results of the adaptive step-size.

EXAMPLE 3.2-1(a): OVEN BRAZING (EES)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

344

Transient Conduction

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB) Repeat EXAMPLE 3.2-1(a) using MATLAB rather than EES to do the calculations. a) Develop a numerical solution that is based on Heun’s method and implemented in MATLAB that predicts the temperature of the object for 3000 s after the oven is activated. The solution will be obtained in a function called Oven_brazing that can be called from the command space. The function will accept the number of time steps (M) and the total simulation time (tsim ) and return three arrays that are the values of time, the wall temperature, and the surface temperature at each time step. function [time, T_w, T]=Oven_Brazing(M, t_sim) % EXAMPLE 3.2-1(b) Oven Brazing % INPUTS % M - number of time steps (-) % t_sim - duration of simulation (s) % OUTPUTS % time - array with the values of time for each time step (s) % T_w - array containing values of the wall temperature at each time (K) % T - array containing values of the surface temperature at each time (K)

Next, the known information from the problem statement is entered into the function. %Known information V=10/100000; A s=35/1000; e=0.8; T ini=293.15; T f=743.15; t hold=1000; beta=1; c=500; k=50; rho=8700; sigma=5.67e-8;

% volume (mˆ3) % surface area (mˆ2) % emissivity of surface (-) % initial temperature (K) % ﬁnal oven temperature (K) % oven hold temperature (s) % oven ramp rate (K/s) % speciﬁc heat capacity (J/kg-K) % conductivity (W/m-K) % density (kg/mˆ3) % Stefan-Boltzmann constant (W/mˆ2-Kˆ4)

A function is needed to return the wall temperature as a function of time. The following code implements this function. Place this function at the bottom of the ﬁle, after an end statement that terminates the function Oven_Brazing. function[T_w]=T_wf(time, T_ini, T_f, beta, t_hold) % Oven temperature function % Inputs % time - current time value (s) % T_ini - initial value of the wall temperature (K)

345

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

3.2 Numerical Solutions to 0-D Transient Problems

% T_f - ﬁnal value of the wall temperature (K) % beta - rate of increase in temperature of the wall (K/s) % t_hold - time period in which the wall temperature is held constant (s) % Output % T_w - wall temperature at specﬁed time (K) T_w=T_ini+beta∗ time; t_ramp=(T_f-T_ini)/beta; if (time>t_ramp) T_w=T_f; end if (time>(t_ramp+t_hold)) T_w=T_f-beta∗ (time-t_ramp-t_hold); end if (time>(2∗ t_ramp+t_hold)) T_w=T_ini; end end

Note the use of the if-end clauses to activate different equations for the wall temperature based on the time of the simulation. The temperature of the wall at any time can be evaluated by a call to the function T_w. The following lines ﬁll the time and T_w vectors with the values of time and the wall temperature at each time step. DELTAt=t sim/(M-1); for j=1:M time(j)=(j-1)∗ t sim/(M-1); T w(j)=T wf(time(j),T ini,T f,beta,t hold);

% time step duration % value of time for each step % value of the wall temperature at each step

end

The last task is to enter the equations that implement Heun’s method for solving the differential equation. The governing equation is obtained from an energy balance on the piece: dT q˙r ad = As ε σ Tw4 − T 4 = V ρ c dt or

As ε σ Tw4 − T 4 dT = dt V ρc

Heun’s method consists of an initial, predictor step: dT Tˆ j+1 = T j + d t T =T j ,t =t j where:

4 4 A ε σ T − T s w,t =t j j dT = d t T =T j ,t =t j V ρc

(1)

Transient Conduction

The corrector step is:

T j+1 = T j +

dT dT + d t T =T j ,t =t j d t T =Tˆ

where, from Eq. (1), dT d t T =T ∗

j+1

,t =t j+1

=

j+1

,t =t j+1

t 2

4 ˆ4 As ε σ Tw,t − T =t j+1 j+1 V ρc

The following equations implement Heun’s method in MATLAB: T(1)=T ini; for j=1:(M-1) dTdt=e∗ A s∗ sigma∗ (T w(j)ˆ4-T(j)ˆ4)/(rho∗ V∗ c); %Temp deriv. at the start of the time step %Predictor step T hat=T(j)+dTdt∗ DELTAt; dTdt hat=e∗ A s∗ sigma∗ (T w(j+1)ˆ4-T hatˆ4)/(rho∗ V∗ c); %deriv. at end of time step %Corrector step” T(j+1)=T(j)+(dTdt+dTdt hat)∗ DELTAt/2; end

The Oven_Brazing function is terminated with an end statement and saved. end

The function can now be called from the command window >> [time, T_w, T]=Oven_Brazing(101, 3000);

The temperature of the wall and the piece is shown in Figure 1. 800 oven wall temperature Heun's method ode45 solver

700

Temperature (K)

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

346

600 500 400 300 200 0

500

1000

1500

2000

2500

3000

Time (s) Figure 1: Temperature of the wall and work piece, predicted using Heun’s method and the ode45 solver, as a function of time.

347

b) Develop a numerical solution using MATLAB’s ode45 function that predicts the temperature of the object for 3000 s after the oven is activated. The ode solvers are designed to call a function the returns the derivative of dependent variable with respect to the independent variable. In our case, the dependent variable is the temperature of the work piece and time is the independent variable. A function must be provided that returns the time derivative of the temperature at a speciﬁed time and temperature. The MATLAB function dTdt_f accepts all of the input parameters that are needed to determine the derivative; according to Eq. (1), these include As , ε, σ, V , ρ, and c as well as parameters needed determine the wall temperature at any time. Place the function dTdt_f below the function T_wf. Note that the function dTdt_f calls the function T_wf in order to determine the wall temperature at a speciﬁed time. function[dTdt]=dTdt_f(time,T,e,A_s,rho,V,c,T_f,T_ini,beta,t_hold) % dTdt is called by the ode45 solver to evaluate the derivative dTdt % INPUTS % time - time relative to start of process (s) % T - temperature of piece (K) % e - emissivity of piece (-) % A_s - surface area of piece (mˆ2) % rho - density (kg/mˆ3) % V - volume of piece (mˆ3) % c - speciﬁc heat capacity of piece (J/kg-K) % T_f - ﬁnal oven temperature (K) % T_ini - initial oven temperature (K) % beta - oven ramp rate (K/s) % t_hold - oven hold time (s) % OUTPUTS % dTdt - rate of change of temperature of the piece (K/s)

sigma=5.67e-8; T w=T wf(time, T ini, T f, beta, t hold); dTdt=e∗ A s∗ sigma∗ (T wˆ4-Tˆ4)/(rho∗ V∗ c); end

% Stefan-Boltzmann constant (W/mˆ2-Kˆ4) % wall temperature % energy balance

Now, all that is necessary is to comment out the code in the function Oven brazing from part (a) and instead call the MATLAB ode45 solver function to determine the temperatures as a function of time. % T(1)=T_ini; % for j=1:(M-1) % dTdt=e∗ A_s∗ sigma∗ (T_w(j)ˆ4-T(j)ˆ4)/(rho∗ V∗ c); % T_hat=T(j)+dTdt∗ DELTAt; % dTdt_hat=e∗ A_s∗ sigma∗ (T_w(j+1)ˆ4-T_hatˆ4)/(rho∗ V∗ c); % T(j+1)=T(j)+(dTdt+dTdt_hat)∗ DELTAt/2;

EXAMPLE 3.2-1(b): OVEN BRAZING (MATLAB)

3.2 Numerical Solutions to 0-D Transient Problems

EXAMPLE 3.2-1(b)

348

Transient Conduction

% end %Solution determined by ode solver [time, T]=ode45(@(time,T) dTdt_f(time,T,e,A_s,rho,V,c,T_f,T_ini,beta,t_hold),time,T_ini);

The time span for the integration is provided by supplying the time array to the ode45 function. As a result, MATLAB will evaluate the temperatures at the same times as were used with Heun’s method. A plot of the ode solver results (identiﬁed with circles) is superimposed onto the results obtained using Heun’s method in Figure 1.

3.3 Semi-Inﬁnite 1-D Transient Problems 3.3.1 Introduction Sections 3.1 and 3.2 present analytical and numerical solutions to transient problems in which the spatial temperature gradients within the solid object can be neglected. Therefore, the problem is zero-dimensional; the transient solution is a function only of time. This section begins the discussion of transient problems where internal temperature gradients related to conduction are non-negligible (i.e., the Biot number is not much less than unity).

3.3.2 The Diffusive Time Constant The simplest case for which the temperature gradients are one-dimensional (i.e., temperature varies in only one spatial dimension) occurs when the object itself is semi-inﬁnite. A semi-inﬁnite body is shown in Figure 3-12; semi-inﬁnite means that the material is bounded on one edge (at x = 0) but extends to inﬁnity in the other. No object is truly semi-inﬁnite although many approach this limit; the earth, for example, is semi-inﬁnite relative to most surface phenomena. Furthermore, we will see that every object is essentially semi-inﬁnite with respect to surface processes that occur over a sufﬁciently “small” time scale. Figure 3-12 shows a semi-inﬁnite body that is initially at a uniform temperature, Tini , when at time t = 0 the temperature of the surface of the body (at x = 0) is raised to Ts . The temperature as a function of position is shown in Figure 3-12 for various times. The transient response can be characterized as a “thermal wave” that penetrates into the solid from the surface. The temperature of the solid is, at ﬁrst, affected by the semi-infinite body with initial temperature Tini Ts x Ts

Tini

T increasing time t1 t2 t3 δt,t1

δt,t 2

δt,t 3

x

Figure 3-12: Semi-inﬁnite body subjected to a sudden change in the surface temperature.

3.3 Semi-Inﬁnite 1-D Transient Problems

349

T Ts

Figure 3-13: Control volume used to develop a simple model of the semi-inﬁnite body.

control volume at t=t1 control volume at t=t3 dU dt

q⋅ cond t1 Tini

δt, t1

t2

t3 δt, t3

x

surface change only at positions that are very near the surface. As time increases, the thermal wave penetrates deeper into the solid; the depth of the penetration (δt ) grows and therefore the amount of material affected by the surface change increases. The analytical and numerical methods discussed in this section as well as in Sections 3.4 and 3.8 can be used to provide the solution to the problem posed in Figure 3-12. However, before the problem is solved exactly, it is worthwhile to pause and understand the behavior of the thermal wave that characterizes this and all transient conduction problems. There are two phenomena occurring in Figure 3-12: (1) thermal energy is conducted from the surface into the body, and (2) the energy is stored by the temperature rise associated with the material that lies within the ever-growing thermal wave. By developing simple models for these two processes, it is possible to understand, to a ﬁrst approximation, how the thermal wave behaves. A control volume is drawn from the surface to the outer edge of the thermal wave, as shown in Figure 3-13; the outer edge of the thermal wave is deﬁned as the position within the material where the conduction heat transfer is small. The control volume must grow with time as the extent of the thermal wave increases. An energy balance on the control volume shown in Figure 3-13 includes conduction into the surface (q˙ cond ) and energy storage: q˙ cond =

dU dt

(3-48)

The thermal resistance to conduction into the thermally affected region through the material that lies within the thermal wave (Rw ) is approximated as a plane wall with the thickness of the thermal wave. Rw ≈

δt k Ac

(3-49)

where k is the conductivity of the material and Ac is the cross-sectional area of the material. This approximation is, of course, not exact as the temperature distribution within the thermal wave is not linear (see Figure 3-13). However, this approach is approximately correct; certainly it is more difﬁcult to conduct heat into the solid as the thermal wave grows and Eq. (3-49) reﬂects this fact. The rate of conduction heat transfer is approximately: q˙ cond =

T s − T ini k Ac (T s − T ini ) ≈ Rw δt

(3-50)

350

Transient Conduction

The thermal energy stored in the material (U) relative to its initial state is the product of the average temperature elevation of the material within the thermal wave ( T ): T ≈

(T s − T ini ) 2

(3-51)

and the heat capacity of the material within the control volume: C ≈ ρ c δt Ac

(3-52)

where ρ and c are the density and speciﬁc heat capacity of the material, respectively. Note that Eqs. (3-51) and (3-52) are also only approximate. U≈

(T s − T ini ) ρ c δt Ac 2

(3-53)

C

T

Substituting Eqs. (3-50) and (3-53) into Eq. (3-48) leads to: k Ac (T s − T ini ) d (T s − T ini ) ≈ ρ c δt Ac δt dt 2

(3-54)

Only the penetration depth (δt ) varies with time in Eq. (3-54) and therefore Eq. (3-54) can be rearranged to provide an ordinary differential equation for δt : k ρ c dδt ≈ δt 2 dt

(3-55)

dδt 2k ≈ δt ρc dt

(3-56)

Equation (3-55) is rearranged:

Equation (3-56) can be simpliﬁed somewhat using the deﬁnition of the thermal diffusivity (α): α=

k ρc

(3-57)

Substituting Eq. (3-57) into Eq. (3-56) leads to: 2 α ≈ δt

dδt dt

(3-58)

Equation (3-58) is separated and integrated: t

δt 2 α dt ≈

0

δt dδt

(3-59)

0

in order to obtain an approximate expression for the penetration of the thermal wave as a function of time: 2αt ≈

δ2t 2

(3-60)

3.3 Semi-Inﬁnite 1-D Transient Problems

351

or: √ δt ≈ 2 α t

(3-61)

√ Equation (3-61) indicates that the thermal wave will grow in proportion to α t. This is a very important and practical (but not an exact) result that governs transient conduction problems. The constant in Eq. (3-61) may change depending on the precise nature of the problem and the deﬁnition of the thermal penetration depth. However, Eq. (3-61) will be approximately correct for a large variety of transient conduction problems. For example, if you heat one side of a plate of stainless steel (α = 1.5 × 10−5 m2 /s) that is L = 1.0 cm thick, then the temperature at the opposite side of the plate will begin to change in about τdiff = 1.7 s; this result follows directly from Eq. (3-61), rearranged to solve for time: τdiff ≈

L2 4α

(3-62)

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL A metal wall (Figure 1) is used to separate two tanks of liquid at different temperatures, Thot = 500 K and Tcold = 400 K. The thickness of the wall is th = 0.8 cm and its area is Ac = 1.0 m2 . The properties of the wall material are ρ = 8000 kg/m3 , c = 400 J/kg-K, and k = 20 W/m-K. The average heat transfer coefﬁcient between the wall and the liquid in either tank is hliq = 5000 W/m2 -K. th = 0.8 cm liquid at Tcold = 400 K hliq = 5000 W/m2 -K

liquid at Thot = 500 K hliq = 5000 W/m2 -K x k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K

Figure 1: Tank wall exposed to ﬂuid.

a) Initially, the wall is at steady-state. That is, the wall has been exposed to the ﬂuid in the tanks for a long time and therefore the temperature within the wall is not changing in time. What is the rate of heat transfer through the wall? What are the temperatures of the two surfaces of the wall (i.e., what is Tx=0 and Tx=th )?

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

The time required for the thermal wave to pass across the extent of a body is referred to as the diffusive time constant (τdiff ) and it is a broadly useful concept in the same way the lumped capacitance time constant (introduced in Section 3.1) is useful. The diffusive time constant characterizes, approximately, how long it takes for an object to equilibrate internally by conduction. The lumped capacitance time constant characterizes, approximately, how long it takes for an object to equilibrate externally with its environment. The ﬁrst step in understanding any transient heat transfer problem involves the calculation of these two time constants.

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

352

Transient Conduction

The known information is entered in EES: “EXAMPLE 3.3-1: Transient Response of a Tank Wall” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” k=20 [W/m-K] c=400 [J/kg-K] rho=8000 [kg/mˆ3] T cold=400 [K] T hot=500 [K] h bar liq=5000 [W/mˆ2-K] th=0.8 [cm]∗ convert(cm,m) A c=1 [mˆ2]

“thermal conductivity” “speciﬁc heat capacity” “density” “cold ﬂuid temperature” “hot ﬂuid temperature” “liquid-to-wall heat transfer coefﬁcient” “wall thickness” “wall area”

There are three thermal resistances governing this problem, convection from the surface on either side (Rconv,liq ) and conduction through the wall (Rcond ): Rconv,liq =

Rcond =

“Steady-state solution, part (a)” R conv liq=1/(h bar liq∗ A c) R cond=th/(k∗ A c)

1 hliq Ac th k Ac

“convection resistance with liquid” “conduction resistance”

The rate of heat transfer is: q˙ =

Thot − Tcold 2 Rconv,liq + Rcond

and the temperatures at x = 0 and x = th are: Tx=0 = Tcold + q˙ Rconv,liq Tx=t h = Thot − q˙ Rconv,liq q dot=(T hot-T cold)/(2∗ R conv liq+R cond) T 0=T cold+q dot∗ R conv liq T L=T hot-q dot∗ R conv liq

“heat transfer” “temperature of cold side of wall” “temperature of hot side of wall”

The rate of heat transfer through the wall is q˙ = 125 kW and the edges of the wall are at Tx=0 = 425 K and Tx=t h = 475 K. At time, t = 0, both tanks are drained and then both sides of the wall are exposed to gas at Tgas = 300 K (Figure 2). The average heat transfer coefﬁcient between the walls and the gas is hgas = 100 W/m2 -K. Assume that the process

353

of draining the tanks and ﬁlling them with gas occurs instantaneously so that the wall has the linear temperature distribution from part (a) at time t = 0. th = 0.8 cm gas at Tgas = 300 K hgas = 100 W/m2 -K

gas at Tgas = 300 K hgas = 100 W/m 2 -K

x

k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K Figure 2: Tank wall exposed to gas.

b) On the axes in Figure 3, sketch the temperature distribution in the wall (i.e., the temperature as a function of position) at t = 0 s (i.e., immediately after the process starts) and also at t = 0.5 s, 5 s, 50 s, 500 s, and 5000 s. Clearly label these different sketches. Be sure that you have the qualitative features of the temperature distribution drawn correctly. There are two processes that occur after the tank is drained. The tank material is not in equilibrium with itself due to its internal temperature gradient. Therefore, there is an internal equilibration process that will cause the wall material to come to a uniform temperature. Also, there is an external equilibration process as the wall transfers heat with its environment. The internal equilibration process is governed by a diffusive time constant, τdiff , discussed in Section 3.3.2 and provided, approximately, by Eq. (3-62): τdiff =

th2 4α

where α is the thermal diffusivity of the wall material: α=

“Internal equilibration process” alpha=k/(rho∗ c) tau diff=thˆ2/(4∗ alpha)

k ρc

“thermal diffusivity” “diffusive time constant”

The diffusive time constant is about 2.6 seconds; this is, approximately, how long it will take for a thermal wave to pass from one side of the wall to the other and therefore this is the amount of time that is required for the wall to internally equilibrate. If the edges of the wall were adiabatic, then the wall will be at a nearly uniform temperature after a few seconds. The external equilibration process is governed by a lumped time constant, discussed previously in Section 3.1 and deﬁned for this problem as: τlumped = Rconv,gas C

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

3.3 Semi-Inﬁnite 1-D Transient Problems

EXAMPLE 3.3-1: TRANSIENT RESPONSE OF A TANK WALL

354

Transient Conduction

where Rconv,gas is the thermal resistance to convection between the wall and the gas: Rconv,gas =

1 2 hgas Ac

and C is the thermal capacitance of the wall: C = Ac th ρ c “External equilibration process” h bar gas=100 [W/mˆ2-K] R conv gas=1/(2∗ h bar gas∗ A c) C total=A c∗ th∗ rho∗ c tau lumped=R conv gas∗ C total

“gas-to-wall heat transfer coefﬁcient” “convection resistance with gas” “capacity of wall” “lumped time constant”

The lumped time constant is about 130 s. Because the diffusive time constant is two orders of magnitude smaller than the lumped time constant, the wall will initially internally equilibrate rapidly and subsequently externally equilibrate more slowly. The initial internal equilibration process will be completed after about 5 to 10 s. Subsequently, the wall will equilibrate externally with the surrounding gas; this external equilibration process will be completed after 200 to 400 s. The temperature distributions sketched in Figure 3 are consistent with these time constants; they do not represent an exact solution, but they are consistent with the physical intuition that was gained through knowledge of the two time constants. Temperature (K) 500 t=0s t = 0.5s t = 5s t = 50 s

475 450 425 400

t = 500s t = 5000s

300 0

0.8

Position (cm)

Figure 3: Temperature distributions in the wall as it equilibrates.

A more exact solution to this problem can be obtained using the techniques discussed in subsequent sections (see EXAMPLE 3.5-2). However, calculation of the diffusive and lumped time constants provide important physical intuition about the problem.

3.3.3 The Self-Similar Solution The governing differential equation for the semi-inﬁnite solid is derived using a control volume that is differential in x, as shown in Figure 3-14.

3.3 Semi-Inﬁnite 1-D Transient Problems

355

semi-infinite body with initial temperature Tini Ts x

q⋅ x

∂U ∂t

q⋅ x+dx

dx Figure 3-14: Differential control volume within semi-inﬁnite body.

The energy balance suggested by Figure 3-14 is: q˙ x = q˙ x+dx +

∂U ∂t

(3-63)

Expanding the x + dx term in Eq. (3-63) leads to: q˙ x = q˙ x +

∂ q˙ x ∂U dx + ∂x ∂t

(3-64)

The conduction term is evaluated using Fourier’s law: q˙ x = −k Ac

∂T ∂x

(3-65)

where Ac is the area of the wall. The internal energy contained in the differential control volume is: U = ρ c Ac dx T

(3-66)

where ρ and c are the density and speciﬁc heat capacity of the wall material. Assuming that the speciﬁc heat capacity is constant, the time rate of change of the internal energy is: ∂T ∂U = ρ c Ac dx ∂t ∂t Substituting Eqs. (3-65) and (3-67) into Eq. (3-64) leads to: ∂T ∂T ∂ −k Ac dx + ρ c Ac dx 0= ∂x ∂x ∂t

(3-67)

(3-68)

which, for a constant thermal conductivity, can be simpliﬁed to: α

∂2T ∂T = ∂x2 ∂t

(3-69)

where α is the thermal diffusivity. For the situation shown in Figure 3-14 in which the initial temperature of the solid is uniform (Tini ) and the surface temperature is suddenly elevated (to Ts ), the boundary conditions are: T x=0,t = T s

(3-70)

T x,t=0 = T ini

(3-71)

T x→∞,t = T ini

(3-72)

The solution to the partial differential equation, Eq. (3-69), subject to the boundary conditions, Eqs. (3-70) through (3-72), is not obvious. Fortunately, the partial differential

356

Ts

Transient Conduction

T T

Ts increasing time

t1, t2, and t3 Tini

t1 δt, t1

t2 δt, t2

t3 δt, t3

Tini η=

x

(b)

(a)

x x = δt 2 α t

Figure 3-15: Temperature distribution in the semi-inﬁnite body as (a) a function of position for various times, and (b) a function of position normalized by the thermal penetration depth for various times.

equation can be reduced to an ordinary differential by using the concept of the thermal penetration depth introduced in Section 3.3.2. Figure 3-12 shows the temperature distribution at different times; these distributions are repeated in Figure 3-15(a). Figure 3-15(b) shows that when the temperature distribution is plotted against x/δt where √ δt is 2 α t, according to Eq. (3-61), the temperature proﬁles for each time collapse onto a single curve. The temperature in Figure 3-15(a) is a function of two independent variables x and t. In this problem, the two independent variables can be combined in order to express the temperature as a function of a single independent variable, deﬁned as η: η=

x √ 2 αt

(3-73)

This process is often called combination of variables and the resulting solution is referred to as a self-similar solution. The ﬁrst step is to transform the governing partial differential equation in x and t to an ordinary differential equation in η. The similarity parameter η, given by Eq. (3-73), is substituted into Eq. (3-69). The temperature is expressed functionally as: T (η (x, t))

(3-74)

Therefore, according to the chain rule, the partial derivative of temperature with respect to x is: dT ∂η ∂T = ∂x dη ∂x

(3-75)

The partial derivative of η with respect to x is obtained by inspection of Eq. (3-73) or using Maple: > eta:=x/(2∗ sqrt(alpha∗ t)); x η := √ 2 αt > diff(eta,x); 1 √ 2 αt

3.3 Semi-Inﬁnite 1-D Transient Problems

357

So that: 1 ∂η = √ ∂x 2 αt

(3-76)

∂T dT 1 = √ ∂x dη 2 α t

(3-77)

and therefore Eq. (3-75) becomes:

The same process is used to calculate the second derivative of T with respect to x: d ∂2T = 2 ∂x dη

∂T ∂x

∂η ∂x

(3-78)

Substituting Eq. (3-77) into Eq. (3-78) leads to: ∂2T d = ∂x2 dη

dT 1 √ dη 2 α t

∂η ∂x

(3-79)

Substituting Eq. (3-76) into Eq. (3-79) leads to: ∂2T d2 T 1 = 2 ∂x dη2 4 α t

(3-80)

The partial derivative of the temperature, Eq. (3-74), with respect to time is also obtained using the chain rule: dT ∂η ∂T = ∂t dη ∂t

(3-81)

where the partial derivative of η with respect to t is evaluated using Maple: > diff(eta,t); 1 xα 4 (α t)(3/2)

So that: x ∂η =− √ ∂t 4t αt

(3-82)

∂T x dT =− √ ∂t 4 t α t dη

(3-83)

and therefore Eq. (3-81) becomes:

Substituting Eqs. (3-80) and (3-83) into Eq. (3-69) leads to: α

d2 T 1 x dT =− √ 2 dη 4 α t 4 t α t dη

(3-84)

358

Transient Conduction

which can be rearranged: x dT d2 T = −2 √ dη2 2 α t dη

(3-85)

d2 T dT = −2 η dη2 dη

(3-86)

η

or

which completes the transformation of the partial differential equation, Eq. (3-69), into an ordinary differential equation, Eq. (3-86). The boundary conditions must also be transformed. Equations (3-70) through (3-72) are transformed from expressions in x and t to expressions in η: T x=0,t = T s ⇒ T η=0 = T s

(3-87)

T x,t=0 = T ini ⇒ T η→∞ = T ini

(3-88)

T x→∞,t = T ini ⇒ T η→∞ = T ini

(3-89)

Notice that Eqs. (3-88) and (3-89) are identical and so the partial differential equation has been transformed into a second order ordinary differential equation with two boundary conditions. It is not always possible to accomplish this transformation; if either x or t had been retained in the partial differential equation or any of the boundary conditions, then a self-similar solution would not be possible. In this case, an alternative analytical technique such as the Laplace transform (Section 3.4) or a numerical solution technique (Section 3.8) is required. Equation (3-86) can be separated and solved. The variable w is deﬁned as: w=

dT dη

(3-90)

and substituted into Eq. (3-86): dw = −2 η w dη Equation (3-91) can be rearranged and integrated: dw = −2 η dη w

(3-91)

(3-92)

which leads to: ln (w) = −η2 + C1

(3-93)

where C1 is a constant of integration that is necessary because Eq. (3-92) is an indeﬁnite integral. Solving Eq. (3-93) for w leads to: w = exp(−η2 + C1 )

(3-94)

Substituting Eq. (3-90) into Eq. (3-94) leads to: dT = exp(−η2 + C1 ) dη

(3-95)

3.3 Semi-Inﬁnite 1-D Transient Problems

359

Gaussian error function and complementary error function

1 0.9 0.8

erf (η )

0.7 0.6 0.5 0.4 0.3

erfc (η )

0.2 0.1 0 0

0.5

1

1.5

2

2.5

Figure 3-16: Gaussian error function and the complementary Gaussian error function.

or dT = C2 exp(−η2 ) dη

(3-96)

where C2 is another undetermined constant (equal to the exponential of C1 ). Equation (3-96) can be integrated:

dT =

C2 exp(−η2 ) dη

(3-97)

exp(−η2 ) dη + C3

(3-98)

or η T = C2 0

where C3 is an undetermined constant. The integral in Eq. (3-98) cannot be evaluated analytically and yet it shows up often in engineering problems. The integral is deﬁned in terms of the Gaussian error function (typically called the erf function, which is pronounced so that it rhymes with smurf). The Gaussian error function is deﬁned as: 2 erf(η) = √ π

η exp(−η2 ) dη

(3-99)

0

and illustrated in Figure 3-16. The complementary error function (erfc) is deﬁned as: erfc (η) = 1 − erf (η)

(3-100)

and is also illustrated in Figure 3-16. Equation (3-98) can be written in terms of the erf function: √ π (3-101) erf (η) + C3 T = C2 2

360

Transient Conduction

The solution to the ordinary differential equation, Eq. (3-86), can also be obtained using Maple: > GDE:=diff(diff(T(eta),eta),eta)=-2∗ eta∗ diff(T(eta),eta); GDE :=

d2 d T (η) T (η) = −2 η 2 dη dη

> Ts:=dsolve(GDE); Ts := T (η) = C1 + erf (η) C2

The two constants in Eq. (3-101) are obtained using the boundary conditions. Substituting Eq. (3-101) into (3-87) leads to: √ π (3-102) erf (0) +C3 = T s T η=0 = C2 2 0

Figure 3-16 shows that erf(0) = 0 so Eq. (3-102) becomes: C3 = T s Substituting Eq. (3-103) into Eq. (3-101) leads to: √ π erf (η) + T s T = C2 2 Substituting Eq. (3-104) into Eq. (3-88) leads to: √ π erf (∞) +T s = T ini T η→∞ = C2 2

(3-103)

(3-104)

(3-105)

=1

Figure 3-16 shows that erf(∞) = 1 so Eq. (3-105) becomes: 2 C2 = √ (T ini − T s ) π The temperature distribution within the semi-inﬁnite body is therefore: x T = T s + (T ini − T s ) erf √ 2 αt

(3-106)

(3-107)

The heat transfer into the surface of the body, q˙ x=0 , is evaluated using Fourier’s law: ∂T q˙ x=0 = −k Ac (3-108) ∂x x=0 which can be expressed in terms of η by substituting Eq. (3-77) into Eq. (3-108): q˙ x=0 = −k Ac

dT 1 √ dη 2 α t

(3-109)

Substituting Eq. (3-107) into Eq. (3-109) leads to: q˙ x=0

d 1 = −k Ac √ [T s + (T ini − T s ) erf (η)] dη η=0 2 α t

(3-110)

3.3 Semi-Inﬁnite 1-D Transient Problems

or: q˙ x=0

361

d k Ac = − √ (T ini − T s ) [erf (η)] dη 2 αt η=0

(3-111)

Substituting the deﬁnition of the erf function, Eq. (3-99), into Eq. (3-111) leads to: ⎤ ⎡ η d ⎣ 2 k Ac exp(−η2 ) dη⎦ (3-112) q˙ x=0 = − √ (T ini − T s ) √ dη π 2 αt 0

η=0

The derivative of an integral is the integrand, and therefore Eq. (3-112) becomes: 2 k Ac q˙ x=0 = − √ (T ini − T s ) √ [exp(−η2 )]η=0 π 2 αt

(3-113)

k Ac q˙ x=0 = √ (T s − T ini ) παt

(3-114)

or

1/Rsemi−∞

This result is extremely useful and intuitive; the material within the thermal wave in Figure 3-15 acts like a thermal resistance to heat transfer from the surface (Rsemi−∞ ): q˙ x=0 =

(T s − T ini ) Rsemi−∞

(3-115)

where Rsemi−∞ increases with time as the thermal wave grows (and therefore the distance over which the conduction occurs increases) according to: Rsemi−∞ =

√ παt k Ac

(3-116)

Equation (3-116) is similar to the thermal √ resistance to conduction through a plane wall with thickness that grows according to π α t. This concept is useful for understanding the physics associated with transient conduction problems. Of course, as soon as the thermal wave reaches a boundary, the problem is no longer semi-inﬁnite and therefore the concepts of the semi-inﬁnite resistance and thermal penetration wave are no longer valid.

3.3.4 Solution to other Semi-Inﬁnite Problems Analytical solutions to a semi-inﬁnite body exposed to other surface boundary conditions have been developed and are summarized in Table 3-2. These analytical solutions can often be applied to various processes that occur over very short time-scales where the thermal penetration wave (δt ) is small relative to the spatial extent of the object. Functions that return the temperature at a speciﬁed position and time for the semiinﬁnite body problems in Table 3-2 are available in EES. Select Function Info from the Options menu and select Transient Conduction from the pull-down menu at the lower right corner of the upper box, toggle to the Semi-Inﬁnite Body library and scroll across to ﬁnd the function of interest.

362

Transient Conduction

Table 3-2: Solutions to semi-inﬁnite body problems. Boundary Condition

Solution T − T ini x = 1 − erf √ T s − T ini 2 αt k (T s − T ini ) q ′′x =0 = √ παt

Ts T increasing time

Tini

x step change in surface temp.: Tx=0 = Ts

T

q⋅′′s

T − T ini

q˙ = s k

/

x2 4αt x exp − − x erfc √ π 4αt 4αt

increasing time Tini

x surface heat flux: q⋅s′′ = − k ∂T ∂x x=0

T∞ increasing time

h

x

convection to fluid: h (T∞ −Tx = 0 ) = − k ∂T ∂x x =0 T surface energy per unit area released at t = 0 wall adiabatic for t>0

E ′′

T − T ini x = erfc √ T ∞ − T ini 2 αt 2 hx h αt h√ x erfc √ + + − exp αt k k2 k 2 αt

T − T ini

x2 E exp − = √ 4αt ρc παt

increasing time Tini

x

surface energy pulse: lim q⋅′′s Δt = E ′′ t, Δt→0

T Tx =0 = Tini + ΔT sin (ωt)

ΔT

/ / ω ω sin ω t − x T − T ini = T exp −x 2α 2α

Tini

x periodic surface temperature xB

T Tini, B solid A

increasing time

Tint increasing time

solid B

Tini, A

0 T ini,A − T int kA ρA cA = 0 T int − T ini,B kB ρB cB T B − T ini,B T A − T ini,A xA xB , = 1 − erf = 1 − erf √ √ T int − T ini,A T int − T ini,B 2 αt 2 αt

xA

contact between two semi-infinite solids

Tini = initial temperature x = position from surface α = thermal diffusivity

t = time relative to surface disturbance ρ = density

k = conductivity c = speciﬁc heat capacity

363

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE A laminated structure is fabricated by diffusion bonding alternating layers of high conductivity silicon (k s = 150 W/m-K, ρs = 2300 kg/m3 , cs = 700 J/kg-K) and low conductivity pyrex (k p = 1.4 W/m-K, ρ p = 2200 kg/m3 , c p = 800 J/kg-K). The thickness of each layer is ths = thp = 0.5 mm, as shown in Figure 1. silicon layers ks = 150 W/m-K

pyrex layers kp = 1.4 W/m-K

ρs = 2300 kg/m3 cs = 700 J/kg-K

ρp = 2200 kg/m3 cp = 800 J/kg-K

thp = 0.5 mm

ths = 0.5 mm x-direction Figure 1: A composite structure formed from silicon and glass.

a) Determine an effective conductivity that can be used to characterize the composite structure with respect to heat transfer in the x-direction (see Figure 1). The known information is entered in EES: “EXAMPLE 3.3-2: Quenching a Composite Structure” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 0.8 3.5 in “Inputs” k s=150 [W/m-K] rho s=2300 [kg/mˆ3] c s=700 [J/kg-K] k p=1.4 [W/m-K] rho p=2200 [kg/mˆ3] c p=800 [J/kg-K] th s=0.5 [mm]∗ convert(mm,m) th p=0.5 [mm]∗ convert(mm,m)

“conductivity of silicon” “density of silicon” “speciﬁc heat capacity of silicon” “conductivity of pyrex” “density of pyrex” “speciﬁc heat capacity of pyrex” “thickness of silicon lamination” “thickness of pyrex lamination”

The method discussed in Section 2.9 is used to determine the effective thermal conductivity of the composite structure. The heat transfer through a thickness (L, in the x-direction) with cross-sectional area (Ac ) of the composite structure when it is subjected to a given temperature difference (T ) is calculated as the series combination of the thermal resistance of the silicon and pyrex laminations according to: q˙ =

T L th p L ths + ths + th p k s Ac ths + th p k p Ac

(1)

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.3 Semi-Inﬁnite 1-D Transient Problems

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

364

Transient Conduction

The effective conductivity in the x-direction (keff ) is equal to the conductivity of a homogeneous material that would result in the same heat transfer rate: k eff Ac T L Setting Eq. (1) equal to Eq. (2) leads to: q˙ =

(2)

k eff Ac T T = L th p L ths L + ths + th p k s Ac ths + th p k p Ac Solving for keff leads to:

k eff =

th p ths + ths + th p k s ths + th p k p

−1

“Part a: effective conductivity for heat transfer across laminations” k eff=1/(th s/((th s+th p)∗ k s)+th p/((th s+th p)∗ k p)) “effective conductivity”

The effective conductivity is k eff = 2.8 W/m-K, which is between the conductivity of pyrex and silicon. Because the conductivity of silicon is high, the silicon laminations contribute essentially no thermal resistance; therefore, because the laminations are of equal size, the effective conductivity is twice that of pyrex. b) Determine an effective heat capacity (ceff ) and density (ρ eff ) that can be used characterize the composite structure. The process of determining an effective density and speciﬁc heat capacity is conceptually similar to the calculation of an effective thermal conductivity. The effective property is chosen so that the composite, when modeled as a homogeneous material using the effective property, behaves as the actual composite does. The effective density is deﬁned so that material has the same mass as the composite. The mass of the composite (with thickness L and cross-sectional area Ac ) is: th p ths L Ac ρs + L Ac ρ p th p + ths th p + ths

M =

(3)

The mass of the homogeneous material with an effective density (ρeff ) is: M = L Ac ρeff

(4)

Setting Eq. (3) equal to Eq. (4) and solving for ρeff leads to: th p ths ρs + ρp th p + t hs th p + ths

ρeff =

“Part b: effective density and heat capacity” rho eff=th s∗ rho s/(th p+th s)+th p∗ rho p/(th p+th s)

“effective density”

The effective speciﬁc heat capacity is deﬁned so that the homogeneous material model has the same total heat capacity as the composite. The total heat capacity of the composite (again with thickness L and area Ac ) is: th p ths L Ac ρs cs + L Ac ρ p c p th p + ths th p + ths

C=

(5)

365

The total heat capacity of the homogeneous material with effective density (ρeff ) and effective heat capacity (ceff ) is: C = L Ac ρeff ceff

(6)

Setting Eq. (5) equal to Eq. (6) and solving for ceff leads to: th p ρ p ths ρs cs + cp th p + ths ρeff th p + ths ρeff

ceff =

c_eff=th_s∗ rho_s∗ c_s/((th_p+th_s)∗ rho_eff)_th_p∗ rho_p∗ c_p/((th_p+th_s)∗ rho_eff) “effective speciﬁc heat capacity”

The effective density and speciﬁc heat capacity of the composite structure are ρeff = 2250 kg/m3 and ceff = 749 J/kg-K. The diffusion bonding of the composite structure occurs at high temperature, Tbond = 750◦ C. When the bonding process is complete, the manufacturing process is terminated by quenching the composite structure from both sides with water at Tw = 20◦ C, as shown in Figure 2. Tw = 20°C, h w → ∞ L = 10 cm

Figure 2: Quenching a composite structure with water.

Tw = 20°C, hw → ∞ composite structure (Figure 1) initial temperature, Tbond = 750°C

The heat transfer coefﬁcient between the water and the surface of the structure is very high because the water is vaporizing. Therefore the quenching process corresponds approximately to applying a step change in the surface temperature of the composite. c) The laminated structure is L = 10 cm thick, for approximately how long will it be appropriate to model the composite as a semi-inﬁnite body? The thermal wave moves from the top and bottom surfaces of the structure approximately according to: $ (7) δt = 2 αeff t where the effective thermal diffusivity of the composite structure is: αeff =

“Part c” alpha eff=k eff/(rho eff∗ c eff)

k eff ρeff ceff

“effective diffusivity”

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.3 Semi-Inﬁnite 1-D Transient Problems

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

366

Transient Conduction

When the thermal wave reaches the half-thickness of the structure then it will no longer behave as a semi-inﬁnite body but rather as a bounded, 1-D transient problem. Substituting δt = L/2 into Eq. (7) leads to: $ L = 2 αeff tsemi−∞ 2 where tsemi−∞ is the time at which the semi-inﬁnite model is no longer appropriate. L=10[cm]∗ convert(cm,m) 2∗ sqrt(alpha eff∗ t semi inﬁnite)=L/2

“width of structure” “time that semi-inﬁnite solution is valid”

The semi-inﬁnite body solution is valid for approximately 380 s. It has been observed that the laminations that are within xfail = 1.0 cm of the two surfaces tend to de-bond during the quenching process. You suspect that the large spatial temperature gradients near the surface during the quench are causing thermally induced stresses that are responsible for this failure. In the presence of large temperature gradients, two laminations that are adjacent to one another will experience very different thermally induced expansions that result in a shear stress. d) Prepare a plot of the temperature gradient as a function of time (up to the time at which the semi-inﬁnite solution is no longer valid, from part (c)) at x = xfail as well as other positions that are greater than and less than xfail . Determine the critical spatial temperature gradient that causes failure (i.e., the maximum spatial temperature gradient experienced at x = xfail ). Table 3-2 or Eq. (3-107) provides the temperature within the semi-inﬁnite body: x 0 (8) T = Tw + (Tbond − Tw ) erf 2 αeff t The spatial temperature gradient can be obtained by differentiating Eq. (8) using Maple: > restart; > T(x,time):=T_w+(T_bond-T_w)∗ erf(x/(2∗ sqrt(alpha_eff∗ time))); x 0 T (x, time) := T w + T bond − T w erf 2 alpha eff time > dTdx:=diff(T(x,time),x);

⎛ ⎝−

dT d x :=

⎞

x2 ⎠ 4alpha eff time

T bond − T w e √ 0 π alpha eff time

so the temperature gradient is: ∂T x2 (Tbond − Tw ) = 0 exp − ∂x 4 αeff t π αeff t

(9)

367

The result from Maple is copied and pasted into EES and then modiﬁed slightly for compatibility:

“Part d - analytical differentiation” T bond=converttemp(C,K,750 [C]) “bond temperature” T w=converttemp(C,K,20 [C]) “water temperature” “observed position of failure” x fail=1 [cm]∗ convert(cm,m) x=x fail “vary x from 0.5 to 4 ∗ x fail” dTdx=(T bond-T w)/Piˆ(1/2)∗ exp(-1/4∗ xˆ2/alpha eff/time)/(alpha eff∗ time)ˆ(1/2)

A parametric table is used to generate Figure 3. The independent variable time is varied from 1 × 10−6 s to 380 s. A value of 1 × 10−6 s rather than 0 is used to avoid division by zero in Eq. (9). The value of x is changed in the Equation window in order to produce the different curves that are shown in Figure 3.

Spatial temperature gradient (K/m)

4 x10 4

3 x10 4 x=0.5 cm 2 x10 4

10

x = x fail = 1 cm x = 2 cm x = 3 cm x = 4 cm

4

0 x10 0 0

100

200 300 Time (s)

400

500

Figure 3: Spatial temperature gradient as a function of time for various locations in the composite structure.

Examination of Figure 3 suggests that the critical spatial temperature gradient for failure is about 3.5 × 104 K/m. Locations exposed to a spatial gradient greater than this value are likely to fail. A more exact solution can be obtained by determining the time at which the spatial gradient is maximized when x = xf ail . This result can be obtained by selecting Min/Max from the Calculate menu and maximizing dTdx by varying the independent variable time. Provide limits on time from slightly greater than 0 (to prevent a division by 0 error) to 380 s. The result will be dTdx = 3.53 × 104 K/m which occurs at 30.4 s. In order to reduce the temperature gradients within the laminations, you suggest that the current water quenching process be replaced by a gas cooling process in which the surfaces of the composite are exposed to a gas at Tgas = 20◦ C with a lower heat transfer coefﬁcient, hgas = 400 W/m2 -K.

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.3 Semi-Inﬁnite 1-D Transient Problems

Transient Conduction

e) Determine whether there will be any de-lamination using this process and, if so, what the thickness of the damaged region (xfail ) will be. Rather than use the analytical solution for the temperature distribution within a semi-inﬁnite body that is subjected to surface convection, from Table 3-2, the gradient will be evaluated numerically using the built-in function SemiInf3 in EES. The numerical derivative is obtained by adding and subtracting a very small value, x, to/from the nominal value of x according to: Tx+x − Tx−x ∂T ≈ ∂x 2 x The value of x must be small in order for this approach to work, but not so small that numerical precision is exceeded. EES provides about 20 digits of numerical precision, so accurate determination of numerical derivatives is usually not a problem. The EES code to compute the temperature gradient for a given value of position and time is: “Part e - gas quenching process” “bond temperature” T bond=converttemp(C,K,750 [C]) T gas=converttemp(C,K,20 [C]) “temperature of gas” h bar gas=400 [W/mˆ2-K] “heat transfer coefﬁcient” DELTAx=1e-5 [m] “differential change used to evaluate numerical derivative” “position” x=2.0 [cm]∗ convert(cm,m) T xplusdx=SemiInf3(T bond,T gas,h bar gas,k eff,alpha eff,x+DELTAx,time) T xminusdx=SemiInf3(T bond,T gas,h bar gas,k eff,alpha eff,x-DELTAx,time) dTdx=(T xplusdx-T xminusdx)/(2∗ DELTAx) “numerical temperature gradient”

In order for this code to run, it is necessary to comment out the EES code from (d). Figure 4 illustrates the spatial temperature gradient as a function of time for various values of position. 5 x10 4 Spatial temperature gradient (K/m)

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

368

4 x10 4

x=0.25 cm

3 x10 4

x=0.5 cm x=0.75 cm x=1.0 cm

2 x10 4

x=1.5 cm 10

x=2.0 cm

4

0 x10 0 0

25

50

75 Time (s)

100

125

150

Figure 4: Spatial temperature gradient as a function of time for various values of position using the gas quenching process.

369

Maximum spatial temperature gradient (K/m)

Figure 4 shows that there is a maximum temperature gradient experienced at each value of x that occurs as the thermal wave passes by that position. The maximum temperature gradient experienced as a function of position can be obtained by using the Min/Max Table option from the Calculate menu. Comment out the speciﬁed value of position and generate a parametric table that includes the variable x as well as the dependent variable to be maximized (the variable dTdx) and the independent variable to vary (the variable time). Vary x from 0.1 mm to 1.45 cm in the parametric table. Select Min/Max Table from the Calculate menu. Maximize the value of the temperature gradient by varying the independent variable time. The maximum temperature gradient as a function of position is shown in Figure 5. 60,000 50,000 40,000 critical temperature gradient 30,000

2

h gas [W/m -K] 20,000

400

10,000

200 100

0 0

0.002 0.004 0.006 0.008 0.01 Axial position (m)

0.012 0.014

Figure 5: Maximum temperature gradient as a function of position for various values of the heat transfer coefﬁcient.

Figure 5 suggests that for hgas = 400 W/m2 -K, the extent of the damaged region will be reduced to only 0.42 cm. Figure 5 also shows that if hgas is reduced to 100 W/m2 -K then the spatial temperature gradient will not exceed 25,000 K/m anywhere in the composite. Because 25,000 K/m is less than the critical temperature gradient identiﬁed in part (d) (35,300 K/m), it is likely that no damage would occur for hgas = 100 W/m2 -K.

3.4 The Laplace Transform 3.4.1 Introduction Sections 3.1 and 3.3 present techniques for obtaining analytical solutions to 0-D (lumped) and 1-D transient conduction problems. This section presents an alternative method for obtaining an analytical solution to these problems: the Laplace transform. The Laplace transform is a mathematical technique that is used to solve differential equations that arise in many engineering disciplines. The mathematical speciﬁcation of a transient conduction problem will include a differential equation and boundary conditions that involve time. In the case of the lumped capacitance problems discussed in Section 3.1, an ordinary differential equation in time was obtained. The semi-inﬁnite problems in Section 3.3 result in a partial differential equation in time and position. The Laplace transform maps a problem involving time,

EXAMPLE 3.3-2: QUENCHING A COMPOSITE STRUCTURE

3.4 The Laplace Transform

370

Transient Conduction

t, onto a problem involving a different variable, typically called s. The attractive feature of the Laplace transform is that the mapping process removes time derivatives from the problem. Therefore, if an ordinary differential equation in t is Laplace-transformed, it becomes an algebraic equation in s. A partial differential equation in t and x becomes an ordinary differential equation in x that is algebraic in s. The complexity of the differential equation is reduced by one level and the problem is correspondingly easier to solve in the s domain than in the t domain. The Laplace transform solution proceeds by obtaining the differential equation and boundary conditions as usual and transforming them into the Laplace (s) domain. The solution is obtained in the s domain and then transformed back to the time domain. The process of transforming between the s and t domains is facilitated by extensive tables that exist as well as symbolic software packages such as Maple. This section is by no means a comprehensive review of Laplace transform theory and application; the interested reader is referred to more complete coverage which can be found in Arpaci (1966) and Myers (1998). However, the introduction provided here is sufﬁcient to allow the solution of many interesting heat transfer problems and provide some insight into the technique. The Laplace transform is particularly useful for short time-scale and semi-infinite body problems.

3.4.2 The Laplace Transformation

The Laplace transform of a function of time, T(t), is indicated by T (s) and obtained according to: ∞

T (s) = T (t) =

exp (−s t) T (t) dt 0

(3-117)

the Laplace transform operation

where the notation T (t) denotes the Laplace transform operation. As an example, consider the Laplace transform of a constant, C: T (t) = C

(3-118)

Substituting Eq. (3-118) into Eq. (3-117) leads to: ∞

T (s) = C =

exp (−s t) C dt

(3-119)

0

Carrying out the integral:

T (s) = −

C [exp (−s t)]∞ 0 s

(3-120)

and evaluating the limits:

T (s) =

C s

(3-121)

Thus the Laplace transform of C is C/s. There are several techniques that can be used to transform a function from the time to the s domain. The integration expressed by Eq. (3-117) can be carried out explicitly. More commonly, one of the extensive tables of Laplace transforms that have been published (e.g., Abramowitz and Stegun, (1964)) is used to obtain the transform. Recently, symbolic software packages such as Maple have become available that are capable of identifying most Laplace transforms automatically.

3.4 The Laplace Transform

371

Table 3-3: Some common Laplace transforms. Function in t

Function in s

Function in t C erfc √ 2 t √ C 2 t C2 − C erfc √ exp − √ 4t π 2 t C C2 t+ erfc √ 2 2 t √ C2 C t − √ exp − 4t π

C

C s

Ct

C s2

exp (C t)

1 s−C

sin (C t)

C s2 + C2

C21 C1 exp − − C2 t √ 4t 2 π t3

cos (C t)

s s2 + C2

exp (−C1 t) − exp (−C2 t) C2 − C1

sinh (C t)

C s2 − C2

t exp (−C1 t)

s s2 − C2

√ C C2 exp − exp −C s √ 3 4 t 2 πt √ exp −C s C2 1 √ exp − √ 4t s πt

√ exp −C s s2

exp(−C1

0 s + C2 )

1 (s + C1 ) (s + C2 ) 1

⎤ (C3 − C2 ) exp (−C1 t) ⎥ ⎢ ⎣ + (C1 − C3 ) exp (−C2 t) ⎦ + (C2 − C1 ) exp (−C3 t) (C1 − C2 ) (C2 − C3 ) (C3 − C1 ) ⎡

cosh (C t)

Function in s √ exp −C s s √ exp −C s √ s s

(s + C1 )2

1 (s + C1 ) (s + C2 ) (s + C3 )

exp (−C2 t) − exp (−C1 t) [1 − (C2 − C1 ) t]

1

(C2 − C1 )2

(s + C1 )2 (s + C2 )

t2 exp (−C t) 2

Function in t √ C2 1 C − a exp(a C) exp(a2 t) erfc a t + √ √ exp − 4t πt 2 t √ C C erfc √ − exp (a C) exp(a2 t) erfc a t + √ 2 t 2 t √ C exp(a C) exp(a2 t) erfc a t + √ 2 t

1 (s + C)3 Function in s √ exp(−C s) √ a+ s √ a exp(−C s) √ s(a + s) √ exp(−C s) √ √ s(a + s)

Laplace Transformations with Tables Tables that provide Laplace transforms can be found in many mathematical references, a few common Laplace transforms are summarized in Table 3-3. Laplace Transformations with Maple The Laplace and inverse Laplace transforms can be obtained using Maple. To access the integral transform library in Maple, it is necessary to activate the inttrans package; this is accomplished using the with command: > restart: > with(inttrans):

372

Transient Conduction

The Laplace transform of an arbitrary function can be obtained using the laplace command. The laplace command requires three arguments; the ﬁrst is the expression to be transformed, the second is the variable to transform from (typically t), and the third is the variable to transform to (typically s). To obtain the Laplace transform of a constant, as we did in Eqs. (3-118) through (3-121) > laplace(C,t,s); C s

More complex functions can also be transformed: > laplace(sin(C∗ t),t,s); C s2 + C2

> laplace(sin(C∗ t)∗ exp(-t/tau),t,s);

C 2

1 s+ τ

+ C2

which indicates that: sin (C t) = and

C s2 + C 2

6 7 t sin (C t) exp − = τ

C 1 2 + C2 s+ τ

(3-122)

(3-123)

3.4.3 The Inverse Laplace Transform Once the solution to a problem has been obtained in terms of s, it is necessary to obtain the inverse Laplace transform of the solution in order to express the result in terms of t. There are several techniques for obtaining the inverse Laplace transform. The most general technique is to mathematically invert the Laplace transform, Eq. (3-117), but this operation requires integration in the complex plane. The computational effort required to obtain the inverse transform in this way defeats the purpose of using Laplace transforms to simplify the solution of heat transfer problems. However, many inverse transforms have been determined and tabulated. If the solution in the s domain appears in a table, such as Table 3-3, then it is a simple matter to obtain the inverse transform from the table (e.g., the inverse transform of C/s is C). More often, the particular transform that is needed will not be found in exactly the right form in a table and it will be necessary to break the solution into simpler pieces using the method of partial fractions. The typical form of the solution in the s domain is a complicated fraction in s; for example:

T (s) =

s2 − 3 s + 4 (s + 1) (s − 1) (s + 2)

(3-124)

3.4 The Laplace Transform

373

The inverse Laplace transform for Eq. (3-124) cannot be found in Table 3-3; however, the transform for simpler fractions are included in the table. The method of partial fractions is therefore required to effectively use tables of Laplace transforms. Many common inverse Laplace transforms can be obtained automatically using Maple. Several additional Laplace transforms that are not normally available with Maple have been added to a ﬁle that can be downloaded from the text website (www.cambridge.org/nellisandklein), as discussed in Section 3.4.6. Inverse Laplace Transform with Tables and the Method of Partial Fractions Equation (3-124) can be reduced using the method of partial fractions, as discussed in Myers (1998). The method of partial fractions requires that the order of the numerator is less than that of the denominator. Distinct Factors. For cases like Eq. (3-124) where there are distinct factors in the denominator, the fraction can be expressed as the sum of individual, lower order fractions each of which has one of the distinct factors in the denominator. For example, Eq. (3-124) can be written as: C1 C2 C3 s2 − 3 s + 4 = + + (s + 1) (s − 1) (s + 2) (s + 1) (s − 1) (s + 2)

(3-125)

where C1 , C2 , and C3 are unknown constants. Both sides of Eq. (3-125) are multiplied by the denominator (s + 1)(s − 1)(s + 2): s2 − 3 s + 4 = C1 (s − 1) (s + 2) + C2 (s + 1) (s + 2) + C3 (s + 1) (s − 1)

(3-126)

Eq. (3-126) must be valid for any value of s since s is the independent variable. If s = −1 is substituted into Eq. (3-126) then the terms involving C2 and C3 become zero (as they both involve s + 1) and an equation is obtained that involves only C1 : (−1)2 − 3 (−1) + 4 = C1 ((−1) − 1) ((−1) + 2)

(3-127)

Equation (3-127) can easily be solved for C1 : 1 + 3 + 4 = C1 (−2) (1)

(3-128)

or C1 = −4. A similar process can be used to obtain C2 (i.e., eliminate C1 and C3 by substituting s = 1 into Eq. (3-126) and solve for C2 ) and C3 . The result is C2 = 1/3 (or 0.333) and C3 = 14/3 (or 4.667). Therefore, Eq. (3-124) can be written as:

T (s) =

0.333 4.667 −4 + + (s + 1) (s − 1) (s + 2)

(3-129)

Expressed in this form, the inverse transform of Eq. (3-129) can be obtained by inspection from Table 3-3: T (t) = −4 exp (−t) + 0.333 exp (t) + 4.667 exp (−2 t)

(3-130)

An alternative and more general method for obtaining C1 , C2 , and C3 is to carry out the multiplications in Eq. (3-126): s2 − 3 s + 4 = C1 (s2 + s − 2) + C2 (s2 + 3 s + 2) + C3 (s2 − 1)

(3-131)

374

Transient Conduction

and then require that the coefﬁcients multiplying like powers of s on either side of Eq. (3-131) must be equal in order to obtain three equations (one for each power of s) in three unknowns (C1 , C2 , and C3 ): 1 = C1 + C2 + C3

(3-132)

−3 = C1 + 3 C2

(3-133)

4 = −2 C1 + 2 C2 − C3

(3-134)

The solution of these simultaneous equations in EES: 1=C_1+C_2+C_3 -3=C_1+3∗ C_2 4=-2∗ C_1+2∗ C_2-C_3

yields C1 = −4, C2 = 0.333, and C3 = 4.667. Note that Maple can be used to quickly convert an expression to its partial fraction form using the convert command with the parfrac identiﬁer. To convert Eq. (3-124) into its partial fraction form, enter the expression and then convert it using the convert command; note that the ﬁrst argument is the expression to be converted, the second identiﬁes the type of conversion, and the third identiﬁes the name of the independent variable in the expression. > restart; > TS:=(sˆ2-3∗ s+4)/((s+1)∗ (s-1)∗ (s+2)); #expression to be converted s2 − 3s + 4 (s + 1) (s − 1) (s + 2) > TSpf:=convert(TS,parfrac,s); #expression converted 4 14 1 − + TSpf := 3 (s − 1) s+1 3 (s + 2) TS :=

Notice that the coefﬁcients identiﬁed by Maple (−4, 1/3, and 14/3) agree with the coefﬁcients identiﬁed manually. Repeated Factors. In the instance that the terms in the denominator of the expression to be converted are repeated, it is necessary to include each power of the term. For example, if the expression in the s domain is:

T (s) =

s2 − 3 s + 4 (s + 1)2 (s − 1)

(3-135)

then the partial fraction form must include fractions with both (s + 1) and (s + 1)2 : s2 − 3 s + 4 2

(s + 1) (s − 1)

=

C2 C1 C3 + + 2 (s + 1) (s + 1) (s − 1)

(3-136)

Otherwise, the solution proceeds as before. Both sides of Eq. (3-136) are multiplied by (s + 1)2 (s − 1): s2 − 3 s + 4 = C1 (s + 1) (s − 1) + C2 (s − 1) + C3 (s + 1)2

(3-137)

3.4 The Laplace Transform

375

or s2 − 3 s + 4 = C1 (s2 − 1) + C2 (s − 1) + C3 (s2 + 2 s + 1)

(3-138)

1 = C1 + C3

(3-139)

−3 = C2 + 2 C3

(3-140)

4 = −C1 − C2 + C3

(3-141)

which leads to:

The solution to these three equations: 1=C_1+C_3 -3=C_2+2∗ C_3 4=C_1-C_2+C_3

leads to C1 = 0.5, C2 = −4, and C3 = 0.5. Therefore:

T (s) =

−4 0.5 0.5 + + 2 (s + 1) (s + 1) (s − 1)

(3-142)

Maple can provide the same result using the convert command: > restart; > TS:=(sˆ2-3∗ s+4)/((s+1)ˆ2∗ (s-1)); #expression to be converted TS :=

s2 − 3 s + 4

(s + 1)2 (s − 1) > TSpf:=convert(TS,parfrac,s); #expression converted 1 1 4 TSpf := + − 2 (s − 1) 2 (s + 1) (s + 1)2

The inverse transform of Eq. (3-142) can be obtained using Table 3-3:

T (t) = 0.5 exp (t) + 0.5 exp (−t) − 4 t exp (−t)

(3-143)

Polynomial Factors. In the instance that the terms in the denominator of the expression include a polynomial factor, it is necessary to include a polynomial numerator with the order of the polynomial reduced by 1. For example, if the expression in the s domain is:

T (s) =

s2 − 3 s + 4 + 2) (s − 1)

(s2

(3-144)

then the partial fraction form must include fractions: C1 s + C2 C3 s2 − 3 s + 4 = + 2 + 2) (s − 1) (s + 2) (s − 1)

(s2

(3-145)

376

Transient Conduction

Otherwise, the solution proceeds as before. Both sides of Eq. (3-145) are multiplied by (s2 + 2) (s − 1): s2 − 3 s + 4 = (C1 s + C2 ) (s − 1) + C3 (s2 + 2)

(3-146)

s2 − 3 s + 4 = C1 s2 − C1 s + C2 s − C2 + C3 s2 + 2 C3

(3-147)

1 = C1 + C3

(3-148)

−3 = −C1 + C2

(3-149)

4 = −C2 + 2 C3

(3-150)

or

which leads to:

The solution to these three equations: 1=C_1+C_3 -3= -C_1+C_2 4= -C_2+2∗ C_3

leads to C1 = 0.333, C2 = −2.667, and C3 = 0.667. Therefore: 0.333 s − 2.667 0.667 + 2 (s + 2) (s − 1)

(3-151)

0.333 s 2.667 0.667 − + (s2 + 2) (s2 + 2) (s − 1)

(3-152)

T (s) = or

T (s) =

Maple can provide the same result: > restart; > TS:=(sˆ2-3∗ s+4)/((sˆ2+2)∗ (s-1)); #expression to be converted s2 − 3s + 4 (s2 + 2) (s − 1) > TSpf:=convert(TS,parfrac,s); #expression converted 2 s−8 + TSpf := 3 (s2 + 2) 3 (s − 1) TS :=

The inverse transform of Eq. (3-152) can be obtained using Table 3-3: √ √ √ sin( 2 t) + 0.667 exp (t) T (t) = 0.333 cos( 2 t) − 2.667 2

(3-153)

Inverse Laplace Transformation with Maple Maple can also be used to transform from a function in the s domain to a function in the time domain using the invlaplace command. The invlaplace command has the same basic calling protocol as the laplace command. The ﬁrst argument is the expression to be inverse transformed, the second argument is the variable to transform from (typically s),

3.4 The Laplace Transform

377

and the third argument is the variable to transform to (typically t). To obtain the inverse Laplace transform of C/s: > restart: > with(inttrans): > invlaplace(C/s,s,t); C

The inverse Laplace transforms that are obtained in Section 3.4.3 could also be obtained using the invlaplace command in Maple. For example, the inverse Laplace transform of Eq. (3-124) is: > restart: > with(inttrans): > TS:=(sˆ2-3∗ s+4)/((s+1)∗ (s-1)∗ (s+2)); TS :=

s2 − 3s + 4 (s + 1) (s − 1) (s + 2)

> Tt:=invlaplace(TS,s,t); Tt :=

13 14 (−2t) 11 cosh (t) + sinh (t) + e 3 3 3

The solution identiﬁed by Maple looks different from Eq. (3-130), the solution identiﬁed using partial fractions and Table 3-3. However, if the hyperbolic cosine and sine terms are written in terms of exponentials then the result is the same. (This can be accomplished using the convert function with the exp identiﬁer.) > Tt:=convert(Tt,exp); Tt :=

4 14 (−2t) 1 t e − t + e 3 e 3

The inverse Laplace transforms of Eqs. (3-135) and (3-144) can also be identiﬁed using Maple: > restart: > with(inttrans): > TS:=(sˆ2-3∗ s+4)/((s+1)ˆ2∗ (s-1)); TS :=

s2 − 3 s + 4 (s + 1)2 (s − 1)

> Tt:=invlaplace(TS,s,t); Tt := 4 t sinh (t) − cosh (t) (−1 + 4 t) > Tt:=simplify(convert(Tt,exp)); Tt := 4 t e(−t) +

1 t 1 (−t) e + e 2 2

378

Transient Conduction

> TS:=(sˆ2-3∗ s+4)/((sˆ2+2)∗ (s-1)); TS :=

s2 − 3 s + 4 (s2 + 2) (s − 1)

> Tt:=invlaplace(TS,s,t); Tt :=

√ √ 4√ 2 1 cos( 2 t) − 2 sin( 2 t) + et 3 3 3

The library of inverse Laplace transforms that is available in Maple is limited and several transforms that are useful for solving heat transfer problems are not available. Additional inverse Laplace transforms are available from the website associated with this book (www.cambridge.org/nellisandklein). The ﬁle that includes these inverse Laplace transforms is titled Inverse Laplace Transforms. The top of this Maple ﬁle includes a section of code that adds a series of entries to the existing invlaplace table in Maple; these entries symbolically deﬁne some additional transforms that are commonly encountered in heat transfer problems and therefore augments Maple’s functionality.

3.4.4 Properties of the Laplace Transformation The Laplace transform is linear. Therefore, the transform of the sum of two functions is the sum of their individual transforms:

T 1 (t) + T 2 (t) = T 1 (s) + T 2 (s)

(3-154)

and the transform of the product of a constant and a function is the product of the constant and the transform:

C T (t) = C T (s)

(3-155)

The transform of a time derivative of a function T(t) is the product of the transform of the function and s less the initial condition in the time domain: 7 6 dT (t) (3-156) = s T (s) − T t=0 dt This property of the Laplace transform is its primary feature and the reason it is useful for solving differential equations. Equation (3-156) can be proven. Apply the Laplace transform to the time derivative of T: 7 ∞ 6 dT dT (t) = exp(−s t) dt (3-157) dt dt 0

Equation (3-157) can be simpliﬁed through integration by parts. We will encounter integration by parts again elsewhere in this book and therefore it is worth spending some time understanding the process. Integration by parts is based on the chain rule for differentiation; the differential of the product of two functions (u and v) is: d (u v) = u dv + v du

(3-158)

which can be integrated: (u v)2

v2

d (u v) = (u v)1

u2 u dv +

v1

v du u1

(3-159)

3.4 The Laplace Transform

379

The left side of Eq. (3-159) is an exact differential and therefore: v2 (u v)2 − (u v)1 =

u2 u dv +

v1

v du

(3-160)

v du

(3-161)

u1

or, rearranging: v2

u2 u dv = (u v)2 − (u v)1 −

v1

u1

Successful use of integration by parts requires that the functions u and v are identiﬁed and substituted into Eq. (3-161) in order to simplify the expression of interest, in this case Eq. (3-157): 6

7 ∞ dT dT = exp(−s t) dt 0

(3-162)

dv

u

By inspection of Eqs. (3-161) and (3-162), we will deﬁne: u = exp(−s t)

(3-163)

dv = dT

(3-164)

du = −s exp(−s t)dt

(3-165)

v=T

(3-166)

therefore

Substituting Eqs. (3-163) through (3-166) into Eq. (3-161) leads to: 6

t=∞ 7 dT T (−s exp(−s t)dt) = [exp(−s t) T ]t=∞ − [exp(−s t) T ]t=0 − dt (u v)2

(u v)1

t=0

v

(3-167)

du

or 6

7 ∞ dT = −T t=0 + s T exp(−s t)dt dt 0

(3-168)

T (s)

The ﬁnal term in Eq. (3-168) is the product of s and the Laplace transform of T: 7 6 dT (3-169) = s T (s) − T t=0 dt

380

Transient Conduction

Table 3-4: Useful properties of the Laplace transforms. 8

9 T 1 (t) + T 2 (t) = T 1 (s) + T 2 (s) 8

9 C T (t) = C T (s) 7 6 dT (t) = s T (s) − T t=0 dt 7 6 ∂T (x, t) = s T (x, s) − T x,t=0 ∂t 6

7 ∂ n T (x, s) ∂ n T (x, t) = ∂xn ∂xn

The transform of a derivative is not, itself, a derivative. This is a very useful property, as it turns a differential equation in t into an algebraic equation in s. It is possible to transform a partial derivative of temperature with respect to time (for example, when temperature depends on both position and time, T (x, t)): 6

7 ∞ ∂T ∂T (x, t) = exp (−s t) dt ∂t ∂t

(3-170)

0

Carrying out the same process of integration by parts leads to a similar conclusion: 7 6 ∂T (x, t) (3-171) = s T (x, s) − T x,t=0 ∂t The Laplace transform of a partial derivative of temperature with respect to position is the partial derivative of the transformed function with respect to position: 6

7 ∂ T (x, s) ∂T (x, t) = ∂x ∂x

(3-172)

This is true for all higher derivatives as well: 6

7 ∂ n T (x, t) ∂ n T (x, s) = ∂xn ∂xn

(3-173)

These properties of the Laplace transform are summarized in Table 3-4.

3.4.5 Solution to Lumped Capacitance Problems The Laplace transform can be used to obtain analytical solutions to the 0-D transient problems that are considered in Section 3.1. The process will be illustrated using the problem discussed in EXAMPLE 3.1-2. A temperature sensor is exposed to an oscillating temperature environment (T∞ ): T ∞ = T ∞ + T ∞ sin (2 π f t)

(3-174)

where T ∞ = 320◦ C is the average temperature of the ﬂuid and T ∞ = 50K and f = 0.5 Hz are the amplitude and frequency of the temperature oscillation. The governing

3.4 The Laplace Transform

381

differential equation for the problem, determined in EXAMPLE 3.1-2, is: dT T∞ T ∞ sin (2 π f t) T = + + dt τlumped τlumped τlumped

(3-175)

where τlumped = 0.72 s is the lumped capacitance time constant of the sensor. The initial condition is: T t=0 = T ini

(3-176)

where Tini = 260◦ C is the initial temperature of the sensor. The known information is entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” T inﬁnity bar=converttemp(C,K,320 [C]) T ini=converttemp(C,K,260 [C]) DELTAT inﬁnity=50 [K] f=0.5 [Hz] tau=0.72 [s]

“average environmental temperature” “initial temperature” “amplitude of oscillation” “frequency of oscillation” “time constant of temperature sensor”

In order to solve this problem using the Laplace transform approach it is necessary to transform the governing differential equation from the t domain to the s domain. The ﬁrst term in the governing equation, Eq. (3-175), is transformed using Eq. (3-169): 6

7 dT (t) = s T (s) − T t=0 dt

(3-177)

Substituting Eq. (3-176) into Eq. (3-177) leads to: 6

7 dT = s T (s) − T ini dt

(3-178)

The second term in Eq. (3-175) becomes: 6

7 T (s) T (t) = τlumped τlumped

(3-179)

which follows from the fact that the Laplace transform is linear, see Eq. (3-155). The ﬁnal two terms in Eq. (3-175) are obtained from Table 3-3: :

T∞ τlumped

; =

T∞ s τlumped

(3-180)

and 6

7 T ∞ 2πf T ∞ sin (2 π f t) = τlumped τlumped s2 + (2 π f )2

(3-181)

382

Transient Conduction

Substituting Eqs. (3-178) through (3-181) into Eq. (3-175) leads to the transformed governing equation:

T (s) T∞ T ∞ = + s T (s) − T ini + τlumped s τlumped τlumped

2πf

s2 + (2 π f )2

(3-182)

Notice that the differential equation in t, Eq. (3-175), has been transformed to an algebraic equation in s. The same result can be obtained using Maple. The governing differential equation is entered: > restart:with(inttrans): > ODEt:=diff(T(time),time)+T(time)/tau=T_inﬁnity_bar/tau+DELTAT_inﬁnity∗ sin (2∗ pi∗ f∗ time)/tau; d T (time) dtime T (time) T infinity bar DELTAT infinity sin (2 π f time) + = + τ τ τ

ODEt : =

and the laplace command is used to obtain the Laplace transform of the entire differential equation: > AEs:=laplace(ODEt,time,s); laplace(T (time), time, s) τ T infinity bar 2 DELTAT infinity π f = + τs τ(s2 + 4π2 f 2 )

AEs : = s laplace(T(time), time, s) − T (0) +

where laplace(T(time),time,s) indicates the Laplace transform of the function T. The transformed equation can be made more concise by using the subs command to substitute a single variable, T(s), for the laplace() result. Recall that the ﬁrst argument of the subs command is the substitution you want to make while the second argument is the expression that you want to make the substitution into: > AEs:=subs(laplace(T(time),time,s)=T(s),AEs); T (s) T infinity bar 2 DELTAT infinity π f AEs := s T (s) − T (0) + = + τ τs τ (s2 + 4 π2 f 2 )

The solution includes the initial condition, T(0), which can be eliminated using the subs command again: > AEs:=subs(T(0)=T_ini,AEs); AEs := s T (s) − T ini +

T (s) T infinity bar 2 DELTAT inf inity π f = + τ τs τ (s2 + 4 π2 f 2 )

3.4 The Laplace Transform

383

This result is identical to the result that was obtained manually, Eq. (3-182); the algebraic

equation in the s domain, Eq. (3-182), can be solved for T (s): T ini 1 s+ τlumped

T (s) =

+

T∞ τlumped s s +

(3-183) ⎡ 1

+

2 π f T ∞ ⎢ ⎢ τlumped ⎣

τlumped

⎤ 1 2 2 (s + (2 π f ) ) s +

1

⎥ ⎥ ⎦

τlumped

The inverse Laplace transform of Eq. (3-183) does not appear in Table 3-3 and therefore it is necessary to use the method of partial fractions to reduce Eq. (3-183) to simpler fractions that do appear in Table 3-3. The second term in Eq. (3-183) can be written as: T∞ τlumped s s + Multiplying through by s (s +

=

1

C2

C1 + s

s+

τlumped

1

(3-184)

τlumped

1 ) leads to: τlumped T∞

τlumped

= C1 s +

C1 τlumped

+ C2 s

(3-185)

which leads to: 0 = C1 + C2

(3-186)

T∞ C1 = τlumped τlumped

(3-187)

C1 = T ∞

(3-188)

and

Solving Eq. (3-187) leads to:

Substituting Eq. (3-188) into Eq. (3-186) leads to: C2 = −T ∞

(3-189)

“coefﬁcients from partial fraction expansion” C_1=T_inﬁnity_bar C_2=-T_inﬁnity_bar

The third term in Eq. (3-183) has a second order polynomial term (s2 ) in the denominator and therefore it can be expressed as: ⎡ ⎤ 2 π f T ∞ ⎢ ⎢ τlumped ⎣

1 (s2 + (2 π f )2 ) s +

1 τlumped

⎥ ⎥ ⎦=

C3 s + C4 + + (2 π f )2 )

(s2

C5 s+

1

τlumped (3-190)

384

Transient Conduction

Multiplying Eq. (3-190) through by (s2 + (2 π f )2 )(s + 1τ ) leads to: 2 π f T ∞ 1 = (C3 s + C4 ) s + + C5 (s2 + (2 π f )2 ) τlumped τlumped

(3-191)

or C3 s C4 2 π f T ∞ = C3 s2 + + C4 s + + C5 s2 + C5 (2 π f )2 τlumped τlumped τlumped

(3-192)

which leads to: 0 = C3 + C5 0=

(3-193)

C3 + C4 τlumped

(3-194)

2 π f T ∞ C4 = + C5 (2 π f )2 τlumped τlumped

(3-195)

Equations (3-193) through (3-195) are 3 equations in the 3 unknowns C1 , C2 , and C3 : 0=C_3+C_5 0=C_3/tau+C_4 2∗ pi∗ f∗ DELTAT_inﬁnity/tau=C_4/tau+C_5∗ (2∗ pi∗ f)ˆ2

With the coefﬁcients for the partial fractions now identiﬁed, Eq. (3-183) can be expressed as:

T (s) =

+

T ini C2 C1 + + 1 1 s s+ s+ τlumped τlumped (s2

C4 C3 s + + + (2 π f )2 ) (s2 + (2 π f )2 )

C5 (s +

(3-196)

1 ) τlumped

The inverse Laplace transform of each of the terms in Eq. (3-196) is obtained using Table 3-3. t t T (t) = T ini exp − + C1 + C2 exp − + C3 cos (2 π f t) τlumped τlumped t C4 sin (2 π f t) + C5 exp − (3-197) + 2πf τlumped The solution is programmed in EES: T=T ini∗ exp(-time/tau)+C 1+C 2∗ exp(-time/tau)+C 3∗ cos(2∗ pi∗ f∗ t+ & “sensor temp., obtained manually” C 4∗ sin(2∗ pi∗ f∗ time)/(2∗ pi∗ f)+C 5∗ exp(-time/tau) T C=converttemp(K,C,T) “in C” T inﬁnity=T inﬁnity bar+DELTAT inﬁnity∗ sin(2∗ pi∗ f∗ time) “ﬂuid temperature” “in C” T inﬁnity C=converttemp(K,C,T inﬁnity)

3.4 The Laplace Transform 380

385

fluid temperature

Temperature (°C)

360 340 320 300 280 260 0

sensor temperature (from Example 3.1-2 and Laplace transform) 2

1

3

4 Time (s)

5

6

7

8

Figure 3-17: Fluid and sensor temperature as a function of time.

The solution obtained using the Laplace transform is identical to the solution obtained in EXAMPLE 3.1-2, as shown in Figure 3-17. Maple could also be used to obtain the solution. The solve command can be used to carry out the algebra required to solve for T(s): > Ts:=solve(AEs,T(s)); Ts := (T ini τ s3 + 4 T ini τ s π2 f 2 + T inf inity bars2 + 4 T inf inity barπ2 f 2 + 2 DELTAT inf inity π f s )/(s(s3 τ + 4 s τ π2 f 2 + s2 + 4 π2 f 2 ))

And the invlaplace command can be used to carry out the inverse Laplace transform: > Tt:=invlaplace(Ts,s,time); ime Tt := e(− τ ) T ini + DELTAT inf inity < $ $ $ ime − sinh(2 −π2 f 2 time) −π2 f 2 + 2 τπ2 f 2 e(− τ ) − cosh(2 −π2 f 2 time) ime (πf (1 + 4 π2 f 2 τ2 )) + T inf inity bar 1 − e(− τ )

Notice that the inverse Laplace transform identiﬁed by Maple includes the hyperbolic sine and cosine (rather than the sine and cosine); √ however, the argument of the sinh and cosh functions are complex (i.e., they involve −1) and therefore these functions become sin and cos. It is possible to specify that the frequency, f, is positive using the assume command: > assume(f,positive);

386

Transient Conduction

With this stipulation, Maple will correctly identify the solution in terms of sine and cosine: > Tt:=invlaplace(Ts,s,time); ime Tt : = e(− τ ) T ini ime sin(2f ∼ π time) + 2τπf ∼ − cos(2f π˜ time) + e(− τ ) DELTAT inf inity + 1 + 4 π 2 f ∼2 τ 2 (− ime +T inf inity bar 1 − e τ )

The solution can be copied and pasted into EES: T_maple=exp(-time/tau)∗ T_ini+(sin(2∗ f∗ pi∗ time)+2∗ tau∗ pi∗ f∗ (-cos(2∗ f∗ pi∗ time)+& exp(-time/tau)))∗ DELTAT_inﬁnity/(1+4∗ piˆ2∗ fˆ2∗ tauˆ2)+T_inﬁnity_bar∗ (1-exp(-time/tau)) “solution from Maple” T_maple_C=converttemp(K,C,T_maple) “in C”

where it provides an identical solution to the one obtained manually (see Figure 3-17). There is not usually a clear advantage associated with using the Laplace transform to solve the ordinary differential equations in time that result from lumped capacitance problems in heat transfer over the analytical techniques discussed in Section 3.1. However, the Laplace transform technique provides another useful tool and possibly a method for double-checking an important solution. The Laplace transform is very useful for certain types of 1-D transient problems, discussed in Section 3.4.6.

3.4.6 Solution to Semi-Inﬁnite Body Problems The Laplace transform can be used to obtain solutions to partial differential equations as well as to ordinary differential equations. The solution steps are basically the same; however, the Laplace transform directly incorporates the initial condition (see Eq. (3-169)) and therefore the initial condition does not have to be transformed. The Laplace transform of a partial differential equation in x and t will result in an ordinary differential equation in x in the s domain. Therefore, it is necessary to transform the boundary conditions involving x into the s domain so that the ordinary differential equation can be solved. The solution must be converted to a function of x and t using the inverse Laplace transform. The process will be illustrated using the problem that was discussed in Section 3.3.3 in which a semi-inﬁnite body that is initially at a uniform temperature Tini is exposed to a step change in its surface temperature, from Tini to Ts . The governing partial differential equation for this situation is: ∂T ∂2T = ∂x2 ∂t where α is the thermal diffusivity. The boundary conditions are provided by: α

(3-198)

T x=0 = T s

(3-199)

T t=0 = T ini

(3-200)

T x→∞ = T ini

(3-201)

3.4 The Laplace Transform

387

The governing differential equation is transformed from the x, t domain to the x, s. The ﬁrst term in Eq. (3-198) is transformed using Eq. (3-173): 7 6 2 ∂ 2 T (x, s) ∂ T (x, t) (3-202) =α α ∂x2 ∂x2 and the second term is transformed using Eq. (3-171): 7 6 ∂T (x, t) = s T (x, s) − T ini ∂t

(3-203)

so that the transformed differential equation is:

∂ 2 T (x, s) α = s T (x, s) − T ini (3-204) 2 ∂x Maple can identify the same transformed governing differential equation using basically the same steps discussed Section 3.4.5:

> restart:with(inttrans): > PDE:=alpha∗ diff(diff(T(x,time),x),x)=diff(T(x,time),time); 2 ∂ ∂ PDE := α T (x, time) T (x, time) = ∂x2 ∂time > ODE:=laplace(PDE,time,s); 2 ∂ ODE := α laplace(T (x,time), time, s) = s laplace(T (x, time), time, s) − T (x, 0) 2 ∂x > ODE:=subs(T(x,0)=T_ini,ODE); 2 ∂ = s laplace(T (x, time), time, s) − T ini ODE := α ∂x 2 laplace(T (x, time), time, s) > ODE:=subs(laplace(T(x,time),time,s)=Ts(x),ODE); 2 d ODE := α Ts(x) = s Ts(x) − T ini 2 dx

Equation (3-204) does not involve any derivative with respect to s and therefore it is an ordinary differential equation in x; the partial differential in Eq. (3-204) can be changed to an ordinary differential:

d2 T (s, x) = s T (s, x) − T ini α 2 dx

(3-205)

which can be rearranged:

s T ini d2 T − T = − (3-206) 2 dx α α Equation (3-206) is a second order, non-homogeneous equation and therefore requires two boundary conditions; these are obtained from Eqs. (3-199) and (3-201), which must also be transformed to the s domain. Ts T x=0 = (3-207) s

T x→∞ =

T ini s

(3-208)

388

Transient Conduction

The second order differential equation is split into homogeneous and particular components:

T = Th + T p

(3-209)

Equation (3-209) substituted into Eq. (3-206):

d2 T p d2 T h s s T ini − T + − Tp= − h 2 2 dx α dx α α =0 for homogeneous differential equation

(3-210)

particular differential equation

The homogeneous differential equation is:

s d2 T h − Th = 0 2 dx α

(3-211)

which has the general solution: / T h = C1 exp

/ s s x + C2 exp − x α α

(3-212)

where C1 and C2 are undetermined constants. The solution to the particular differential equation

d2 T p s T ini − Tp= − 2 dx α α

(3-213)

is, by inspection:

Tp=

T ini s

(3-214)

Substituting Eqs. (3-212) and (3-214) into Eq. (3-209) leads to: / / s s T ini T = C1 exp x + C2 exp − x + α α s

(3-215)

Maple can be used to obtain the same solution: > dsolve(ODE);

√

Ts(x) = e

sx √ α

− C2 + e

−

√ sx √ α

C1 +

T ini s

The constants C1 and C2 are obtained from the boundary conditions. The boundary condition at x→ ∞, Eq. (3-208), leads to: / / s s T ini T ini T x→∞ = C1 exp ∞ + C2 exp − ∞ + = (3-216) α α s s or

/ C1 exp

s ∞ =0 α

(3-217)

3.4 The Laplace Transform

389

which can only be true if C1 = 0:

/ s T ini T = C2 exp − x + α s

(3-218)

The boundary condition at x = 0, Eq. (3-207), leads to: / s T ini Ts T x=0 = C2 exp − 0 + = α s s

(3-219)

or C2 =

(T s − T ini ) s

(3-220)

Substituting Eq. (3-220) into Eq. (3-218) leads to the solution to the problem in the x, s domain: / s T ini (T s − T ini ) T (x, s) = exp − x + (3-221) s α s The solution in the x, t domain can be obtained using the inverse Laplace transforms contained in Table 3-3: x erfc − T (3-222) + T ini T (x, t) = (T s √ ini ) 2 αt Recall that the complementary error function (erfc) is deﬁned as: erfc (x) = 1 − erf (x)

(3-223)

so Eq. (3-222) can be rewritten as:

x T (x, t) = (T s − T ini ) 1 − erf √ + T ini 2 αt

or

T (x, t) = T s + (T ini − T s ) erf

x √ 2 αt

(3-224)

(3-225)

which is identical to the similarity solution obtained in Section 3.3.3. Unfortunately, Maple is not able to automatically identify the inverse Laplace transform of Eq. (3-221): > restart:with(inttrans): > Ts:=(T_s-T_ini)∗ exp(-sqrt(s)∗ x/sqrt(alpha))/s+T_ini/s;

−

(T s − T ini) e Ts := s > invlaplace(Ts,s,time); (T s − T ini) invlaplace

⎛

−

⎝e

√ sx α

√ sx √ α

s

+

T ini s ⎞

, s, time ⎠ + T ini

The indicator invlaplace() is a placeholder that shows that the inverse Laplace transform was not found in Maple’s library. It is possible to add entries to Maple’s library, as discussed by Aziz (2006), in order to allow Maple to solve this and other semiinﬁnite body heat transfer problems. Several of the most common inverse transforms

390

Transient Conduction

that are encountered for heat transfer problems have been added to the ﬁle Inverse Laplace Transforms, which can be downloaded from the website associated with the text (www.cambridge.org/nellisandklein). Rename the ﬁle after you download it and place the Maple code associated with this problem at the bottom of the ﬁle. Run the entire ﬁle to obtain: > #Inverse Laplace transforms > restart:with(inttrans): > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/s,erfc(p/(2∗ sqrt(t))),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/sˆ(3/2),2∗ sqrt(t)∗ exp(-pˆ2/(4∗ t))/sqrt (pi)-p∗ erfc(p/(2∗ sqrt(t))),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic),p∗ exp(-pˆ2/(4∗ t))/(2∗ sqrt(pi∗ tˆ3)),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/sqrt(s),exp(-pˆ2/(4∗ t))/sqrt(pi∗ tˆ3),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ p::algebraic)/sˆ2,(t+pˆ2/2)∗ erfc(p/(2∗ sqrt(t)))-p∗ sqrt(t)∗ exp(-pˆ2/(4∗ t))/sqrt(pi),s,t); > addtable(invlaplace,exp(-sqrt(s)∗ C1::algebraic)/(C2::algebraic+(C3::algebraic)∗ sqrt(s)),(exp(-C1ˆ2/(4∗ t))/sqrt(Pi∗ t)-(C2/C3)∗ exp((C2/C3)∗ C1)∗ exp((C2/C3)ˆ2∗ t) ∗ erfc((C2/C3)∗ sqrt(t)+C1/(2∗ sqrt(t))))/C3,s,t); > addtable(invlaplace,exp(-sqrt(s)∗ C1::algebraic)/(s∗ (C2::algebraic+(C3::algebraic) ∗ sqrt(s))),erfc(1/2∗ C1/tˆ(1/2))-exp(C2/C3∗ C1)∗ exp(C2ˆ2/C3ˆ2∗ t)∗ erfc(C2/C3∗ tˆ(1/2) +1/2∗ C1/tˆ(1/2)),s,t;) > addtable(invlaplace,exp(-sqrt(s)∗ C1::algebraic)/(sqrt(s)∗ (C2::algebraic+ (C3::algebraic)∗ sqrt(s))),(exp((C2/C3)∗ C1)∗ exp((C2/C3)ˆ2∗ t)∗ erfc((C2/C3)∗ sqrt(t) +C1/(2∗ sqrt(t))))/C3,s,t); > #place your code below this line > Ts:=(T_s-T_ini)∗ exp(-sqrt(s)∗ x/sqrt(alpha))/s+T_ini/s; √ − √sαx

(T s − T ini) e Ts := s > invlaplace(Ts,s,time);

T s + erf

√

x

2 α time

+

T ini s

(−T s + T ini)

391

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR When a superconducting conductor “quenches,” it goes from having no resistance (i.e., a superconducting state) to being resistive (i.e., a normal state). When the conductor is carrying current, the quenching process is accompanied by a step change in the rate of volumetric generation of thermal energy within the material (from nearly zero to a relatively high value, depending on the amount of current that is being carried). The result can be disastrous. This problem examines the temperature distribution in the conductor during the initial stages of a quenching process, shown schematically in Figure 1. initial temperature, Tini = 4.2 K Ts = 4.2 K q⋅ x

Figure 1: Quenching of a conductor.

∂U ∂t g⋅

q⋅ x+dx

x k = 500 W/m-K -4 2 α = 6.25x10 m /s ⋅g ′′′ = 1x106 W/m3

dx

The conductor is initially at a uniform temperature of Tini = 4.2 K when the quench process occurs, resulting in a uniform rate of volumetric generation, g˙ = 1 × 106 W/m3 . The conductor has conductivity k = 500 W/m-K and thermal diffusivity α = 0.000625 m2 /s. The surface of the conductor is maintained at Ts = 4.2 K by boiling liquid helium. a) Develop an analytical model of the quench process that is valid for short times, while the conductor behaves as a semi-inﬁnite body. The governing differential equation for the semi-inﬁnite solid is derived by focusing on a differential control volume (see Figure 1). The energy balance suggested by Figure 1 is: ∂U ∂t expanding the x + d x term and simplifying leads to: g˙ + q˙ x = q˙ x+d x +

∂ q˙ x ∂U dx + ∂x ∂t The conduction term is evaluated using Fourier’s law: g˙ =

(1)

∂T (2) ∂x where Ac is the cross-sectional area of the conductor perpendicular to the xdirection. The time rate of change of the internal energy of the material in the control volume is: ∂T ∂U = ρ c Ac d x (3) ∂t ∂t The rate of generation of thermal energy in the control volume is: q˙ x = −k Ac

g˙ = Ac d x g˙

(4)

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

3.4 The Laplace Transform

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

392

Transient Conduction

Substituting Eqs. (2) through (4) into Eq. (1) leads to: ∂T ∂ ∂T Ac d x g˙ = −k Ac d x + ρ c Ac d x ∂x ∂x ∂t or ∂ 2T 1 ∂T g˙ − = − ∂x2 α ∂t k The boundary conditions for the problem include the initial condition: Tt =0 = Tini

(5)

(6)

the surface temperature is speciﬁed: Tx=0 = Ts

(7)

and the temperature gradient must approach zero as you move away from the surface: dT =0 (8) dx x→∞ To solve this problem using the Laplace transform it is necessary to transform the governing differential equation, Eq. (5): 1 g˙ d 2T − s T − Tx,t =0 = − 2 dx α ks Substituting Eq. (6) into Eq. (9) and rearranging:

(9)

s Tini g˙ d 2T − T = − − (10) d x2 α α ks The spatial boundary conditions, Eqs. (7) and (8), are transformed to the s domain in order to provide the boundary conditions for the ordinary differential equation in the s domain, Eq. (10):

T x=0 = d T d x

Ts s

(11)

=0

(12)

x→∞

The solution to the ordinary differential equation, Eq. (10), is: / / s s g˙ α Tini T = C 1 exp + x + C 2 exp − x + α α k s2 s The boundary condition at x → ∞ leads to: / / / / d T s s s s = C1 exp ∞ − C2 exp − ∞ =0 d x α α α α x→∞

which can only be true if C1 = 0:

/ s g˙ α Tini T = C 2 exp − + x + α k s2 s

393

The boundary condition at x = 0 leads to: / s g˙ α Tini Ts T x=0 = C 2 exp − + 0 + = α k s2 s s which leads to: C2 =

(Ts − Tini ) g˙ α − s k s2

The solution in the s domain is: / s g˙ α Tini (Ts − Tini ) g˙ α T = + − x + exp − s k s2 α k s2 s which can be rearranged: / / s s g˙ α g˙ α Tini (Ts − Tini ) T = exp − + exp − x − x + 2 s α ks α k s2 s The inverse Laplace transform can be obtained using Table 3-3: x T = (Ts − Tini ) erfc √ 2 αt / 2 g˙ α t x x x2 − −x t+ erfc exp − − t + Tini √ k 2α απ 4αt 2 αt

(13)

(14)

The known information is entered in EES: “EXAMPLE 3.4-1: Quenching of a Superconductor” $UnitSystem SI MASS RAD PA C J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” T ini=4.2 [K] T s=4.2 [K] g dot=1e6 [W/mˆ3] k=500 [W/m-K] alpha=0.000625 [mˆ2/s]

“initial temperature of superconductor” “surface temperature” “volumetric rate of generation” “conductivity” “thermal diffusivity”

and the solution is programmed: “Solution” time=0.1 [s] “time” x mm=10 [mm] “position in mm” “position” x=x mm∗ convert(mm,m) T=(T s-T ini)∗ erfc(x/(2∗ sqrt(alpha∗ time)))+(g dot∗ alpha/k)∗ (time-(time+xˆ2/(2∗ alpha))∗ & erfc(x(2∗ sqrt(alpha∗ time)))+x∗ sqrt(time/(pi∗ alpha))∗ exp(-xˆ2/(4∗ alpha∗ time)))+T ini

Figure 2 illustrates the temperature as a function of position for various times relative to the start of the quench process.

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

3.4 The Laplace Transform

Transient Conduction 5.2 t = 0.75 s 5

Temperature (K)

EXAMPLE 3.4-1: QUENCHING OF A SUPERCONDUCTOR

394

t = 0.5 s

4.8 4.6

t = 0.25 s 4.4 t = 0.1 s t=0s

4.2 0

10

20

30

40 50 60 Position (mm)

70

80

90

100

Figure 2: Temperature as a function of position at various times relative to the start of the quench process.

Notice that the portion of the conductor removed from the edge increases linearly in temperature. (This behavior corresponds to the term in Eq. (14) that is linear with time.) However, the effect of the cooled edge propagates into the conductor at a rate that increases with time. The solution can be checked against our physical intuition by verifying that the speed of the thermal wave agrees, approximately, with the diffusive time constant that was discussed in Section 3.3.2. At time = 0.5 s the thermal wave should have moved approximately: √ (15) δt = 2 α t EES’ calculator window can be used to carry out supplementary calculations such as this. EES’ calculator window is accessed by selecting Calculator from the Windows menu (Figure 3).

0.000625 0.035355

Figure 3: Calculator window.

To enter a command in the Calculator window, enter ? followed by the command and terminate the command with the enter key. The variables from the last time that the EES code was run are accessible from the Calculator; for example, typing ?alpha will provide the value of the thermal diffusivity of the superconductor. To determine the size of the thermal wave at 0.5 s using Eq. (15), type ?2∗ sqrt(alpha∗ 0.5), as shown in Figure 3. Notice that the answer, 0.035 m or 35 mm, is consistent with the thermal wave thickness at 0.5 s in Figure 2.

3.5 Separation of Variables for Transient Problems

395

The analytical solution of partial differential equations with the Laplace transform is convenient in some situations. For example, it is not possible to solve semi-inﬁnite problems or (with some exceptions) problems with non-homogeneous boundary conditions in space using the method of separation of variables discussed in Section 3.5. Therefore, the Laplace transform solution or a self-similar solution (Section 3.3.3) may be the best alternative. However, the process of obtaining the inverse Laplace transform can be difﬁcult and it is often easier to develop numerical solutions to these problems, as discussed in Section 3.8.

3.5 Separation of Variables for Transient Problems 3.5.1 Introduction Section 3.3 discussed the behavior of a thermal wave within an un-bounded (i.e., semiinﬁnite) solid and introduced the concept of a diffusion time constant. The self-similar solution to a semi-inﬁnite solid was presented in Section 3.3.3 and the solution to this type of problem using the Laplace transform approach was discussed in Section 3.4.6. This section examines the analytical solution to 1-D transient problems that are bounded using the method of separation of variables. Figure 3-18 illustrates, qualitatively, the temperature distribution at various times that will be present in a plane wall that is initially at a uniform temperature, Tini , when the temperature of one surface (at x = L) is suddenly increased to Ts while the other surface (at x = 0) is adiabatic. adiabatic surface

surface exposed to Ts increasing time

Temperature

Ts

T ini Position Figure 3-18: Temperature as a function of position at various times for a plane wall initially at Tini that is subjected to a sudden change in the surface temperature to Ts .

The behavior illustrated in Figure 3-18 is initially consistent with the behavior of a semi-inﬁnite body, discussed in Section 3.3. A thermal wave emanates from the surface at x = L and penetrates into the solid; the depth of the thermal wave (δt ) is approximately given by: √ δt = 2 α t

(3-226)

396

Transient Conduction

where α is the thermal diffusivity and t is time. When the thermal wave reaches the adiabatic edge at x = 0, it becomes bounded and cannot grow further. The character of the problem changes at this time. The temperature distributions will no longer collapse when plotted as a function of x/δt , as they did in Figure 3-15, and therefore a self-similar solution to the bounded problem is not possible. The method of separation of variables can be used to analytically solve the problem shown in Figure 3-18. The solutions associated with some common shapes (a plane wall, cylinder, and sphere initially at a uniform temperature and exposed to a convective boundary condition) are presented in Section 3.5.2 without derivation. Sections 3.5.3 and 3.5.4 provide an introduction to the application of the method of separation of variables for 1-D transient problems in Cartesian and cylindrical coordinates, respectively. A more thorough discussion can be found in Myers (1998). The concepts and steps used to obtain separation of variables solutions to 1-D transient problems are quite similar to those discussed for steady-state, 2-D problems in Section 2.2 and 2.3. The partial differential equation in position x (or r for cylindrical or spherical problems) and time t is transformed (by separation of variables) into a second order ordinary differential equation in position and a ﬁrst order ordinary differential equation in time. The sub-problem involving x must be the eigenproblem and result in eigenfunctions. Therefore, both spatial boundary conditions must be homogeneous in order to apply separation of variables to a 1-D transient problem. Recall that a homogeneous boundary condition is one where any solution that satisﬁes the boundary condition must still satisfy the boundary condition if it is multiplied by a constant. Three types of homogeneous linear boundary conditions are encountered in heat transfer problems: (1) a speciﬁed temperature of zero, (2) an adiabatic boundary, and (3) convection to a ﬂuid with a temperature of zero. Section 2.2.3 shows how it is possible to transform some non-homogeneous boundary conditions into homogeneous boundary conditions by subtracting the speciﬁed temperature or ﬂuid temperature. Section 2.4 discusses the superposition of different solutions in order to accommodate multiple nonhomogeneous boundary conditions. These techniques can also be used for 1-D transient problems.

3.5.2 Separation of Variables Solutions for Common Shapes The separation of variables solutions for the plane wall, cylinder, and sphere are available in most textbooks as approximate formulae and in graphical format. The solutions are also available within EES both in dimensional and non-dimensional forms using the Transient Conduction library. This section provides, without derivation, the solutions to the basic problems associated with a plane wall, cylinder, and sphere initially at a uniform temperature when, at time t = 0, the surface is exposed via convection to a step change in the surrounding ﬂuid temperature. These problems are summarized in Figure 3-19. The solutions to this set of problems are obtained using the separation of variables techniques discussed in Sections 3.5.3 and 3.5.4 and are widely applicable to 1-D transient problems. The Plane Wall Governing Equation and Boundary Conditions. Section 3.5.3 provides the solution to a plane wall subjected to a step change in the ﬂuid temperature at a convective boundary. The partial differential equation associated with this problem is: 1∂T ∂2T = 2 ∂x α ∂t

(3-227)

3.5 Separation of Variables for Transient Problems

rout

L

T∞ , h

T∞ , h

397

rout

T∞, h

x

(a)

(b)

(c)

Figure 3-19: 1-D transient problems associated with (a) a plane wall, (b) a cylinder, and (c) a sphere, initially at a uniform temperature (T ini ) subjected to a step change in the convective boundary condition (h, T ∞ ).

The boundary conditions for the problem are: ∂T =0 ∂x x=0

(3-228)

∂T = h [T x=L − T ∞ ] −k ∂x x=L

(3-229)

T t=0 = T ini

(3-230)

Exact Solution. The solution derived in Section 3.5.3 for the plane wall problem shown in Figure 3-19(a) can be rearranged: θ˜ (x, ˜ Fo) =

∞

% & Ci cos(ζi x) ˜ exp −ζ2i Fo

(3-231)

i=1

where x, ˜ Fo, and θ˜ are the dimensionless position, Fourier number, and dimensionless temperature difference, deﬁned as: x x˜ = (3-232) L Fo = θ˜ =

tα L2

T − T∞ T ini − T ∞

(3-233) (3-234)

The dimensionless eigenvalues, ζi in Eq. (3-231), correspond to the product of the dimensional eigenvalues λi and the wall thickness (L) and are therefore provided by the multiple roots of the eigencondition: tan(ζi ) =

Bi ζi

(3-235)

where Bi is the Biot number: Bi =

hL k

(3-236)

398

Transient Conduction

The constants in Eq. (3-231) are: Ci =

2 sin(ζi ) ζi + cos(ζi ) sin(ζi )

(3-237)

The exact solution (the ﬁrst 20 terms of Eq. (3-231)) is programmed in EES in its dimen˜ as a function of the dimensionless sionless format (i.e., the dimensionless temperature, θ, independent variables Bi, Fo, and x) ˜ as the function planewall_T_ND. The solution can be accessed by selecting Function Info from the Options menu and selecting Transient Conduction from the pull-down menu. The different transient conduction functions that are available can be seen using the scroll-bar. The exact solutions for the cylinder and sphere are also programmed in EES as the functions cylinder_T_ND and sphere_T_ND. Dimensional versions of each function are also available; these functions (planewall_T, cylinder_T, and sphere_T) return the temperature at a given position and time as a function of the dimensional independent variables ( i.e., L, α, k, h, T i , T ∞ ). It is often important to calculate the total amount of energy transferred to the wall. An energy balance on the wall indicates that the total energy transfer (Q) is the difference between the energy stored in the wall (U) and the energy stored in the wall at its initial condition (U t=0 ): Q = U − U t=0

(3-238)

˜ by normalizing it against the maxiThe total energy transfer is made dimensionless (Q) mum amount of energy that could be transferred to the wall (Qmax ): Q˜ =

Q Qmax

(3-239)

The maximum energy transfer would occur if the process continued to t → ∞ and therefore the wall material equilibrates completely with the ﬂuid temperature at T∞ . Qmax = ρ c L Ac (T ∞ − T ini )

(3-240)

The dimensionless energy transfer is related to the dimensionless, volume average tem˜ according to: perature difference in the wall (θ) Q˜ = 1 − θ˜

(3-241)

where 1 θ˜ =

θ˜ dx˜

(3-242)

0

Substituting the exact solution, Eq. (3-231), into the Eqs. (3-241) and (3-242) leads to: Q˜ = 1 −

∞ i=1

Ci

sin(ζi ) exp(−ζ2i Fo) ζi

(3-243)

The solution for the dimensionless total energy transfer for the plane wall, cylinder, and sphere have been programmed in EES as the functions planewall_Q_ND, cylinder_ Q_ND, and sphere_Q_ND. The exact solutions for the dimensionless temperature and heat transfer are often presented graphically in a form that was initially published by Heisler (1947) and is now referred to as the Heisler charts; these charts were convenient when access to computer solutions was not readily available. Heisler charts typically include the dimensionless center temperature difference (θ˜x˜ =0 ) as a function of the Fourier number for various values of the inverse of the Biot number, Figure 3-20(a). The temperature information

Dimensionless center temperature

1 100

Bi −1

50 20

0.1

10 5 0.2 0.1 0.01

0.01

0.5

1

2

lines of constant Bi −1

0.001 0.1

1

1

10 Fourier number (a)

100

1000

10

100

x/L= 0.2

0.9

0.4

0.8 Ratio of θ/θx=0

0.7 0.6

0.6 0.5 0.4

0.8

0.3 0.2

x/L = 1.0 (surface)

0.9

0.1 0 0.01

0.1

1 Inverse Biot number

(b)

Dimensionless energy transfer

1

0.8

0.6

Bi=0.001 0.002 0.005 0.01 0.02 0.05 0.1

0.2 0.5 1 2 5 10

0.4

20 50

0.2

0 10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

2

Bi Fo (c) Figure 3-20: Heisler charts for a plane wall including (a) θ˜x=0 as a function of Fo for various values ˜ ˜ θ˜x˜ as a function of Bi−1 , and (c) Q˜ as a function of Bi2 Fo for various values of Bi. of Bi−1 , (b) θ/

400

Transient Conduction

presented in this ﬁgure corresponds to the position of the adiabatic boundary (x = 0). This is referred to as the center temperature because the center-line would be adiabatic if convection occured on both sides of the wall. The temperature at other locations can be obtained using Figure 3-20(b), which shows the ratio of the dimensionless temper˜ θ˜x˜ =0 ) as a funcature difference to the dimensionless center temperature difference (θ/ tion of the inverse of the Biot number for various dimensionless positions; note that the Fourier number does not effect this ratio. Figure 3-20(c) illustrates the dimensionless total energy transfer, Q˜ as a function of Bi2 Fo for various values of the Biot number. The Heisler charts shown in Figure 3-20 were generated using the solution programmed in EES. Figure 3-20(a) was generated by accessing the planewall_ND function using the following EES code (varying Fo for different values of invBi): invBi=10 theta hat 0=planewall T ND(0,Fo,1/invBi)

“Inverse Biot number” “Dimensionless center temperature”

Figure 3-20(b) was generated by accessing planewall_ND twice using the following EES code (varying x_hat for different values of invBi): x hat=0.2 “Dimensionless position” Fo=1 “Fourier No. (doesn’t effect ratio)” theta hat 0=planewall T ND(0,Fo,1/invBi) “Dimensionless center temperature” thetatotheta hat 0=planewall T ND(x hat,Fo,1/invBi)/theta hat 0 “Dimensionless center temperature”

Figure 3-20(c) was generated by accessing the planewall_Q_ND function (varying Bi2Fo for different values of Bi): Bi=50 Fo=Bi2Fo/Biˆ2 Q hat=planewall Q ND(Fo, Bi)

“Biot number” “Fourier number (doesn’t change ratio)” “Dimensionless energy transfer”

Approximate Solution for Large Fourier Number. When Fo 0.2, the series solution given by Eq. (3-231) can be adequately approximated using only the ﬁrst term. Only the ﬁrst eigenvalue (i.e., ζi , the ﬁrst root of Eq. (3-235)) is required to implement this approximate solution and therefore many textbooks will tabulate the ﬁrst eigenvalue as a function of the Biot number. The value of the ﬁrst eigenvalue as a function Biot number is shown in Figure 3-21 for the plane wall, as well as the cylinder and sphere solutions, discussed subsequently. Using ζ1 , the approximate solutions for the dimensionless temperature difference and dimensionless energy transfer become: θ˜ (x, ˜ Fo) ≈

% & 2 sin(ζ1 ) ˜ exp −ζ21 Fo cos(ζ1 x) ζ1 + cos(ζ1 ) sin(ζ1 )

⎡ ⎤ sin (ζ 1 ) 2sin (ζ 1 ) Q ≈ 1 − ⎢ exp ( −ζ 12 Fo ) ⎥ ζ cos ζ sin ζ ζ + ( ) ( ) 1 1 ⎦ 1 ⎣ 1

(3-244)

(3-245)

First dimensionless eigenvalue, ζ 1

3.5 Separation of Variables for Transient Problems

401

sphere

3

2.5

cylinder

2 plane wall

1.5 1 0.5 0 0.001

0.01

0.1

1 10 Biot number

100

1000

10000

Figure 3-21: First eigenvalue for use in the approximate solution to the problems illustrated in Figure 3-19.

The Cylinder Governing Equation and Boundary Conditions. The partial differential equation associated with the cylinder is derived by carrying out an energy balance on a control volume that is a differentially small cylindrical shell with thickness dr. ∂U ∂t

(3-246)

∂U ∂ q˙ r dr + ∂r ∂t

(3-247)

q˙ r = q˙ r+dr + or 0= The rate of conductive heat transfer is:

q˙ r = −k 2 π r L

∂T ∂r

(3-248)

where L is the length of the cylinder. The rate of energy storage is: ∂U ∂T = 2 π r L dr ρ c ∂t ∂t Substituting Eqs. (3-248) and (3-249) into Eq. (3-247) leads to: ∂ ∂T ∂T 0= −k 2 π r L dr + 2 π r L dr ρ c ∂r ∂r ∂t

(3-249)

(3-250)

or with k constant, α ∂ ∂T ∂T r = r ∂r ∂r ∂t

(3-251)

T t=0 = T ini

(3-252)

The initial condition is:

402

Transient Conduction

and the spatial boundary conditions are:

∂T =0 ∂r r=0

(3-253)

∂T = h (T r=rout − T ∞ ) −k ∂r r=rout

(3-254)

Exact Solution. The exact solution for the cylinder problem shown in Figure 3-19(b) can be derived using the methods discussed in Section 3.5.4: θ˜ (˜r, Fo) =

∞

% & Ci BesselJ (0, ζi r˜ ) exp −ζ2i Fo

(3-255)

i=1

where r˜ , Fo, and θ˜ are the dimensionless position, Fourier number, and dimensionless temperature difference, deﬁned as: r (3-256) r˜ = rout Fo =

θ˜ =

tα 2 rout

T − T∞ T ini − T ∞

(3-257)

(3-258)

The dimensionless eigenvalues, ζi in Eq. (3-255) are the roots of the eigencondition: ζi BesselJ (1, ζi ) − Bi BesselJ (0, ζi ) = 0

(3-259)

where Bi is the Biot number, deﬁned as: Bi =

h rout k

(3-260)

The constants in Eq. (3-255) are given by: Ci =

2 BesselJ (1, ζi ) ζi [BesselJ (0, ζi ) + BesselJ2 (1, ζi )] 2

(3-261)

The dimensionless energy transfer is related to the dimensionless, volume average tem˜ according to: perature in the cylinder (θ) Q˜ = 1 − θ˜

(3-262)

where 1 θ˜ = 2

θ˜ r˜ dr˜

(3-263)

0

Substituting the exact solution, Eq. (3-255), into Eqs. (3-262) and (3-263) leads to: Q˜ = 1 −

∞ i=1

Ci

2 BesselJ(1, ζi ) exp −ζ2i Fo ζi

(3-264)

3.5 Separation of Variables for Transient Problems

403

Approximate Solution for Large Fourier Number. The series solution for the cylinder, like the plane wall, can be adequately approximated using only the ﬁrst term when Fo > 0.2. The value of the ﬁrst eigenvalue as a function Biot number is shown in Figure 3-21. Using ζ1 , the approximate solutions for the dimensionless temperature difference and energy transfer become: θ˜ (r, ˜ Fo) =

Q˜ = 1 −

˜ 2 BesselJ(1, ζ1 ) BesselJ(0, ζ1 r) ζ1 [BesselJ2 (0, ζ1 ) + BesselJ2 (1, ζ1 )] 4 BesselJ2 (1, ζ1 )

ζ21 [BesselJ2 (0, ζ1 )

+ BesselJ (1, ζ1 )] 2

exp[−ζ21 Fo]

exp −ζ2i Fo

(3-265)

(3-266)

The Sphere Governing Equation and Boundary Conditions. The partial differential equation associated with the sphere is derived by carrying out an energy balance on a control volume that is a differentially small spherical shell with thickness r. ∂U ∂t

(3-267)

∂U ∂ q˙ r dr + ∂r ∂t

(3-268)

∂T ∂r

(3-269)

q˙ r = q˙ r+dr + or 0= The rate of conductive heat transfer is: q˙ r = −k 4 π r2 The rate of energy storage is:

∂T ∂U = 4 π r2 dr ρ c ∂t ∂t Substituting Eqs. (3-269) and (3-270) into Eq. (3-268) leads to: ∂T ∂ 2 ∂T −k 4 π r dr + 4 π r2 dr ρ c 0= ∂r ∂r ∂t

(3-270)

(3-271)

or α ∂ 2 ∂T ∂T r = r2 ∂r ∂r ∂t

(3-272)

T t=0 = T ini

(3-273)

The initial condition is:

and the spatial boundary conditions are:

∂T =0 ∂r r=0

(3-274)

∂T = h (T r=rout − T ∞ ) −k ∂r r=rout

(3-275)

404

Transient Conduction

Exact Solution. The exact solution for the spherical problem shown in Figure 3-19(c) is: ∞ & % sin(ζi r) ˜ Ci (3-276) exp −ζ2i Fo θ˜ (r, ˜ Fo) = ζi r˜ i=1

where r, ˜ Fo, and θ˜ are the dimensionless position, Fourier number, and dimensionless temperature difference, deﬁned as: r (3-277) r˜ = rout Fo = θ˜ =

tα 2 rout

T − T∞ T ini − T ∞

(3-278)

(3-279)

The dimensionless eigenvalues, ζi in Eq. (3-276) are the roots of the eigencondition: ζi cos (ζi ) + (Bi − 1) sin (ζi ) = 0

(3-280)

where Bi is the Biot number, deﬁned as: h rout k

(3-281)

2 [sin (ζi ) − ζi cos (ζi )] ζi − sin (ζi ) cos (ζi )

(3-282)

Bi = and the constants in Eq. (3-276) are: Ci =

The dimensionless energy transfer is related to the dimensionless, volume average tem˜ according to: perature in the sphere (θ) Q˜ = 1 − θ˜

(3-283)

where 1 θ˜ = 3

θ˜ r˜2 dr˜

(3-284)

0

Substituting the exact solution, Eq. (3-276) into Eqs. (3-283) and (3-284) leads to: Q˜ = 1 −

∞ i=1

Ci

3 [sin (ζi ) − ζi cos (ζi )] exp −ζ2i Fo 3 ζi

(3-285)

Approximate Solution for Large Fourier Number. The series solution for the sphere can be adequately approximated using only the ﬁrst term when Fo is > 0.2. The value of the ﬁrst eigenvalue as a function Biot number is shown in Figure 3-21. Using ζ1 , the approximate solutions for the dimensionless temperature distribution and energy transfer become: & % ˜ 2 [sin (ζ1 ) − ζ1 cos(ζ1 )] sin(ζ1 r) (3-286) exp −ζ21 Fo θ˜ (r, ˜ Fo) = ζ1 r˜ [ζ1 − sin(ζ1 ) cos(ζ1 )] 6 [sin(ζ1 ) − ζ1 cos(ζ1 )]2 exp −ζ21 Fo Q˜ = 1 − 3 ζ1 [ζ1 − sin(ζ1 ) cos(ζ1 )]

(3-287)

405

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN As part of a manufacturing process, long cylindrical pieces of material with radius r out = 5.0 cm are placed into a radiant oven. The initial temperature of the material is Tini = 20◦ C. The walls of the oven are maintained at a temperature of Twall = 750◦ C and the oven is evacuated so that the outer edge of the cylinder is exposed only to radiation heat transfer. The emissivity of the surface of the cylinder is ε = 0.95. The material is considered to be completely processed when the temperature everywhere is at least Tp = 250◦ C. The properties of the material are k = 1.4 W/m-K, ρ = 2500 kg/m3 , c = 700 J/kg-K. a) Is a lumped capacitance model appropriate for this problem? The known inputs are entered in EES: “EXAMPLE 3.5-1: Material in a Radiant Oven” $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” r out=5.0 [cm]∗ convert(cm,m) T ini=converttemp(C,K,20[C]) T wall=converttemp(C,K,750[C]) e=0.95 T p=converttemp(C,K,250[C]) k=1.4 [W/m-K] rho=2500 [kg/mˆ3] c=700 [J/kg-K] alpha=k/(rho∗ c)

“radius” “initial temperature” “wall temperature” “emissivity” “processing temperature” “conductivity” “density” “speciﬁc heat capacity” “thermal diffusivity”

The effective heat transfer coefﬁcient associated with radiation (hr ad , discussed in Section 1.2.6) is: 2 hr ad = σ ε Twall + Ts2 (Twall + Ts ) where Ts is the surface temperature, which is not known but will certainly not be less then Tini and will likely not be much higher than Tp . An average of these values is used to evaluate the radiation heat transfer coefﬁcient. T s=(T ini+T p)/2 “average temperature used for surface temp.” h bar rad=sigma# ∗ e∗ (T wallˆ2+T sˆ2)∗ (T wall+T s) “effective heat transfer coefﬁcient due to radiation”

Because Twall is so much higher than Ts , the value of hr ad is not affected signiﬁcantly by the choice of Ts . Using hr ad , it is possible to calculate the Biot number: Bi =

hr ad r out k

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

3.5 Separation of Variables for Transient Problems

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

406

Transient Conduction

Bi=h bar rad∗ r out/k

“Biot number”

which provides a Biot number of 3.3. Therefore, a lumped capacitance model is not valid. b) How long will the processing require? Use an effective, radiation heat transfer coefﬁcient for this calculation. The dimensionless temperature difference at which the processing is complete can be calculated using Eq. (3-258): θ˜p =

theta hat p=(T p-T wall)/(T ini-T wall)

Tp − Twall Tini − Twall

“dimensionless temperature for processing”

The function cylinder_ND implements the exact solution for a cylinder, Eq. (3-255), and provides θ˜ given r˜, Fo, Bi. In this case, we know θ˜ = θ˜p at r˜ = 0 (the center of the cylinder, which will be the lowest temperature part of the material), and the value of Bi is also known. We would like to solve for the corresponding value of Fo (and therefore time). EES will solve this implicit equation in order to determine Fo: theta hat p=cylinder T ND(0, Fo, Bi) “implements the dimensionless solution to a cylinder”

Initially you are likely to obtain an error in EES related to the value of Fo being negative. This error condition can be easily overcome by setting appropriate limits (e.g., 0.001 to 1000) on the value of Fo in the Variable Information window. The Fourier number is used to calculate the processing time (tprocess ), according to Eq. (3-257): t pr ocess = Fo

t process=Fo∗ r outˆ2/alpha

2 r out α

“processing time”

which leads to a processing time of 678 s. c) What is the minimum amount of energy per unit length that is required to process the material? That is, how much energy would be required to bring the material to a uniform temperature of Tp ? An energy balance on the material shows that the minimum amount of energy required to bring the material to a uniform temperature (Qmin ) is: 2 L ρ c Tp − Tini Qmin = π r out

L=1 [m] Q min=pi∗ r outˆ2∗ L∗ rho∗ c∗ (T p-T ini)

407

“per unit length of material” “minimum amount of energy transfer required”

which leads to Qmin = 3.16 MJ. d) How much energy is actually required per unit length of material? ˜ is obtained using the cylinThe dimensionless energy transfer to the cylinder (Q) der_Q_ND function in EES: Q hat=cylinder Q ND(Fo, Bi)

“dimensionless heat transfer”

The dimensionless energy transfer is deﬁned according to Eq. (3-239) Q˜ =

Q Qmax

where Qmax is the energy transfer that occurs if the process is continued until the cylinder reaches equilibrium with the wall: 2 Qmax = ρ c L π r out (Twall − Tini )

Q max=pi∗ r outˆ2∗ L∗ rho∗ c∗ (T wall-T ini) “maximum amount of energy transfer that could occur” “actual energy transfer” Q=Q max∗ Q hat

which leads to Q = 5.62 MJ. e) The efﬁciency of the process (η) is deﬁned as the ratio of the minimum possible energy transfer required to process the material (from part (b)) to the actual energy transfer (from part (c)). Plot the efﬁciency of the process as a function of the radius of the material for various values of Twall . The efﬁciency is deﬁned as: η=

eta=Q min/Q

Qmin Q

“Process efﬁciency”

The efﬁciency is computed over a range of radii, rout , using a parametric table at several values of Twall and the results are shown in Figure 1.

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

3.5 Separation of Variables for Transient Problems

Transient Conduction

1 0.9 0.8 Efficiency

EXAMPLE 3.5-1: MATERIAL PROCESSING IN A RADIANT OVEN

408

T wall

0.7

600°C

0.6 750°C 0.5 900°C 0.4 0.3 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Outer radius (m) Figure 1: Efﬁciency of the process as a function of the radius of the material for various values of the oven wall temperature.

Notice that the efﬁciency improves with reduced radius because the temperature gradients within the material are reduced and so the amount of energy wasted in ‘overheating’ the material toward the outer radius of the cylinder is reduced. Also, reducing the oven temperature tends to improve the efﬁciency for the same reason, but at the expense of increased processing time. The most efﬁcient process is associated with processing a very small amount of material (with small radius) very slowly (at low oven temperature); this is a typical result: very efﬁcient processes are not usually very practical.

3.5.3 Separation of Variables Solutions in Cartesian Coordinates The solution to 1-D transient problems using separation of variables is illustrated in the context of the problem shown in Figure 3-22. A plane wall is initially at a uniform temperature, T ini = 100 K, when the surface is exposed to a step change in the surrounding ﬂuid temperature to T ∞ = 200 K. The average heat transfer coefﬁcient between the surface and the ﬂuid is h = 200 W/m2 -K. The wall has thermal diffusivity α = 5 × 10−6 m2 /s

initial temperature, Tini = 100 K L = 5 cm

T∞ = 200 K 2 h = 200 W/m -K x k = 10 W/m-K α = 5x10-6 m2/s Figure 3-22: Plane wall initially at a uniform temperature that is exposed to convection at the surface.

3.5 Separation of Variables for Transient Problems

409

and thermal conductivity k = 10 W/m-K. The thickness of the wall is L = 5 cm. The known information is entered in EES: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” k=10 [W/m-K] alpha=5e-6 [mˆ2/s] T ini=100 [K] T inﬁnity=200 [K] h bar=200 [W/mˆ2-K] L=5.0 [cm]∗ convert(cm,m)

“conductivity” “thermal diffusivity” “initial temperature” “ﬂuid temperature” “heat transfer coefﬁcient” “thickness”

The governing partial differential equation for this situation is identical to the one derived for a semi-inﬁnite body in Section 3.3.3: ∂2T ∂T −α 2 =0 ∂t ∂x

(3-288)

The partial differential equation is ﬁrst order in time and therefore it requires one boundary condition with respect to time; this is the initial condition: T t=0 = T ini

(3-289)

Equation (3-288) is second order in space and therefore two boundary conditions are required with respect to x. At the adiabatic wall, the temperature gradient must be zero: ∂T =0 (3-290) ∂x x=0 An interface energy balance at the surface (x = L) balances conduction with convection: ∂T = h [T x=L − T ∞ ] −k (3-291) ∂x x=L The steps required to solve 1-D transient problems using separation of variables follow naturally from those presented in Section 2.2.2 for 2-D steady-state problems. Requirements for using Separation of Variables In order to apply separation of variables, it is necessary that the partial differential equation and all of the boundary condition be linear; these criteria are satisﬁed for the problem shown in Figure 3-22 by inspection of Eqs. (3-288) through (3-291). It is also necessary that the partial differential equation and both boundary conditions in space be homogeneous (there is only one direction associated with a 1-D transient problem and therefore it must be homogeneous). The partial differential equation, Eq. (3-288) is homogeneous and the boundary condition associated with the adiabatic wall, Eq. (3-290), is also homogeneous. However, the convective boundary condition at x = L, Eq. (3-291), is not homogeneous. Fortunately, it is possible to transform this problem, as discussed in Section 2.2.3. The temperature difference relative to the ﬂuid temperature is deﬁned: θ = T − T∞

(3-292)

410

Transient Conduction

The transformed partial differential equation becomes: α

∂θ ∂2θ = 2 ∂x ∂t

(3-293)

and the boundary conditions become: θt=0 = T ini − T ∞

(3-294)

∂θ =0 ∂x x=0

(3-295)

∂θ = h θx=L −k ∂x x=L

(3-296)

Notice that both spatial boundary conditions for the transformed problem, Eqs. (3-295) and (3-296), are homogeneous and therefore it will be possible to obtain a set of orthogonal eigenfunctions in x. The initial condition, Eq. (3-294), is not homogeneous but it doesn’t have to be. (Recall that the boundary conditions in one direction of the 2-D conduction problems, considered in Sections 2.2 and 2.3, did not have to be homogeneous.) As an aside, if the dimension of the problem were extended to inﬁnity (L → ∞) then the resulting spatial boundary condition, Eq. (3-296), would not be homogeneous and therefore the semiinﬁnite problem cannot be solved using separation of variables. One of the reasons that the Laplace transform is a valuable tool for heat transfer problems is that it can be used to analytically solve semi-inﬁnite problems. Separate the Variables The temperature difference, θ, is a function of axial position and time. The separation of variables approach assumes that the solution can be expressed as the product of a function only of time, θt(t), and a function only of position, θX(x): θ (x, t) = θX (x) θt (t)

(3-297)

Substituting Eq. (3-297) into Eq. (3-293) leads to: α θt

dθt d2 θX = θX dx2 dt

(3-298)

Equation (3-298) is divided through by α θXθt in order to obtain: d2 θX dθt dx2 = dt θX α θt

(3-299)

Both sides of Eq. (3-299) must be equal to a constant in order for the solution to be valid at an arbitrary time and position. We know from our experience with 2-D separation of variables that Eq. (3-299) will lead to two ordinary differential equations (one in x and the other in t). Furthermore, the ODE in x must lead to a set of eigenfunctions. This foresight motivates the choice of a negative constant −λ2 : dθt d2 θX 2 dx = dt = −λ2 θX α θt

(3-300)

3.5 Separation of Variables for Transient Problems

411

The ODE in x that is obtained from Eq. (3-300) is: d2 θX + λ2 θX = 0 dx2

(3-301)

which is solved by sines and cosines. The ODE in time suggested by Eq. (3-300) is: dθt + λ2 α θt = 0 dt

(3-302)

Solve the Eigenproblem It is always necessary to address the solution to the homogeneous sub-problem (in this problem, θX) before moving on to the non-homogeneous sub-problem (for θt). The homogeneous sub-problem is called the eigenproblem. The general solution to Eq. (3-301) is: θX = C1 sin(λ x) + C2 cos(λ x)

(3-303)

where C1 and C2 are unknown constants. Substituting Eq. (3-297) and Eq. (3-303) into the spatial boundary condition at x = 0, Eq. (3-295), leads to: ⎡ ⎤ dθX ∂θ = θt = θt ⎣C1 λ cos(λ 0) −C2 λ sin(λ 0)⎦ = 0 ∂x x=0 dx x=0 =1

(3-304)

=0

or θt C1 λ = 0

(3-305)

which can only be true (for a non-trivial solution) if C1 = 0: θX = C2 cos(λ x)

(3-306)

Substituting Eq. (3-297) into the spatial boundary condition at x = L, Eq. (3-296), leads to: dθX = h θt θX x=L (3-307) −k θt dx x=L or dθX = h θX x=L −k dx x=L

(3-308)

Substituting Eq. (3-306) into Eq. (3-308) leads to: k C2 λ sin(λ L) = h C2 cos(λ L)

(3-309)

Equation (3-309) provides the eigencondition for the problem, which deﬁnes multiple eigenvalues: sin(λ L) h = cos(λ L) kλ

(3-310)

412

Transient Conduction

or, multiplying and dividing the right side of Eq. (3-310) by L: hL sin(λ L) = cos(λ L) kλL

(3-311)

The dimensionless group h L/k is the Biot number (Bi) that was encountered in Section 1.6 and elsewhere. In this context, the Biot number represents the ratio of the resistance to conduction heat transfer within the wall to convective heat transfer from the surface. Writing Eq. (3-311) in terms of the Biot number leads to: tan(λ L) =

Bi λL

(3-312)

Equation (3-312) is analogous to the eigenconditions for the problems considered in Section 2.2. There are an inﬁnite number of values of λ that will satisfy Eq. (3-312). However, Eq. (3-312) provides an implicit rather than an explicit equation for these eigenvalues. This situation was encountered previously in EXAMPLE 2.3-1; the eigencondition involved the zeroes of the Bessel function. In EXAMPLE 2.3-1, the process of calculating the eigenvalues was automated by using EES to determine the roots of the eigencondition within speciﬁed ranges. We can use the same procedure for any problem with an implicit eigencondition, such as is given by Eq. (3-312). Figure 3-23 illustrates the left and right sides of Eq. (3-312) as a function of λL for the case where Bi = 1.0. tan( λL) Bi/( λL )

Tan (λL) and Bi/(λL)

1.25 1 0.75 0.5

λ1 L 0.25

λ2L

0 0

π/2

π

3π/2

λ3L 2π

λ4L 5π/2

λL

3π

7π/2

λ5L 4π

9π/2

Figure 3-23: The left and right sides of the eigencondition equation for Bi = 1.0 ; the intersections correspond to eigenvalues for the problem, λi L.

Figure 3-23 shows that each successive value of λi L can be found in a well-deﬁned interval; λ1 L lies between 0 and π/2, λ2 L lies between π and 3π/2, etc.; this will be true regardless of the value of the Biot number. The number of terms to use in the solution is speciﬁed and arrays of appropriate guess values and upper and lower bounds for each eigenvalue are generated.

3.5 Separation of Variables for Transient Problems

413

Nterm=10 [-] “number of terms to use in the solution” “Setup guess values and lower and upper bounds for eigenvalues” duplicate i=1,Nterm lowerlimit[i]=(i-1)∗ pi upperlimit[i]=lowerlimit[i]+pi/2 guess[i]=lowerlimit[i]+pi/4 end

The eigencondition is programmed using a duplicate loop: Bi=h bar∗ L/k “Identify eigenvalues” duplicate i=1,Nterm tan(lambdaL[i])=Bi/lambdaL[i] lambda[i]=lambdaL[i]/L end

“Biot number”

“eigencondition” “eigenvalue”

The solution obtained at this point will provide the same value for all of the eigenvalues. (Select Arrays from the Windows menu and you will see that each value of the array lambdaL[i] is the same, probably equal to whichever root of Eq. (3-312) lies closest to the default guess value of 1.0.) The interval for each eigenvalue can be controlled by selecting Variable Info from the Options menu. Deselect the Show array variables check box at the upper left so that the arrays are collapsed to a single entry and use the guess[ ], upperlimit[ ], and lowerlimit[ ] arrays to control the process of identifying the eigenvalues in the array lambdaL[ ]. The solution will now identify the ﬁrst 10 eigenvalues of the problem (more can be obtained by changing the value of Nterm). The number of terms required depends on the time and position where you need the solution; this is discussed at the end of this section. At this point, each of the eigenfunctions of the problem have been obtained. The ith eigenfunction is: θX i = C2,i cos(λi x)

(3-313)

where λi is the ith eigenvalue, identiﬁed by the eigencondition: tan(λi L) =

Bi λi L

(3-314)

Solve the Non-homogeneous Problem for each Eigenvalue The solution to the non-homogeneous ordinary differential equation corresponding to the ith eigenvalue, Eq. (3-302): dθti + λ2i α θti = 0 dt

(3-315)

θti = C3,i exp(−λ2i α t)

(3-316)

is

where C3,i is an undetermined constant.

414

Transient Conduction

Obtain a Solution for each Eigenvalue According to Eq. (3-297), the solution associated with the ith eigenvalue is: θi = θX i θti = Ci cos(λi x) exp −λ2i α t

(3-317)

where the constants C2,i and C3,i have been combined to form a single undetermined constant Ci . Equation (3-317) will, for any value of i, satisfy the governing differential equation, Eq. (3-293), throughout the domain and satisfy all of the boundary conditions in the x-direction, Eqs. (3-295) and (3-296). It is worth checking that the solution has these properties using Maple before proceeding. Enter the solution as a function of x and t: > restart; > theta:=(x,t)->C∗ cos(lambda∗ x)∗ exp(-lambdaˆ2∗ alpha∗ t); θ := (x, t) → C cos(λx)e(−x αt) 2

Verify that it satisﬁes Eq. (3-295): > eval(diff(theta(x,t),x),x=0); 0

and Eq. (3-296): > -k∗ eval(diff(theta(x,t),x),x=L)-h_bar∗ theta(L,t); 2

k C sin(λ L) λ e(−λ

αt)

2

− h bar C cos(λL) e(−λ

αt)

> simplify(%); 2

C e(−λ

αt)

(k sin(λ L) λ − h bar cos(λL))

Note that the eigencondition cannot be enforced in Maple using the assume command as it was in the problems in Section 2.2. However, it is clear that the eigencondition, Eq. (3-310), requires that the term within the parentheses must be zero and therefore the second spatial boundary condition will be satisﬁed for any eigenvalue. Finally, verify that the partial differential equation, Eq. (3-293), is satisﬁed: > alpha∗ diff(diff(theta(x,t),x),x)-diff(theta(x,t),t); 0

Create the Series Solution and Enforce the Initial Condition Because the partial differential equation is linear, the sum of the solution θi for each eigenvalue, Eq. (3-317), is itself a solution: θ=

∞ i=1

θi =

∞ i=1

Ci cos(λi x) exp −λ2i α t

(3-318)

3.5 Separation of Variables for Transient Problems

415

The ﬁnal step of the problem selects the constants so that the series solution satisﬁes the initial condition, Eq. (3-294): θt=0 =

∞

Ci cos(λi x) = T ini − T ∞

(3-319)

i=1

We found in Section 2.2 that the eigenfunctions are orthogonal to one another. The property of orthogonality ensures that when any two, different eigenfunctions are multiplied together and integrated from one homogeneous boundary to the other, the result will necessarily be zero. Each side of Eq. (3-319) is multiplied by cos(λ j x) and integrated from x = 0 to x = L: ∞

L cos(λi x) cos(λ j x)dx =

Ci

i=1

L

0

(T ini − T ∞ ) cos(λ j x)dx

(3-320)

0

The property of orthogonality ensures that the only term on the left side of Eq. (3-320) that is not zero is the one for which j = i: L

L cos2 (λi x)dx = (T ini − T ∞ )

Ci 0

cos(λi x) dx 0

Integral 1

(3-321)

Integral 2

The integrals in Eq. (3-321) can be evaluated conveniently using either integral tables or Maple: > Integral1:=int((cos(lambda∗ x))ˆ2,x=0..L); Integral1 := > Integral2:=int(cos(lambda∗ x),x=0..L);

1 cos(λL) sin(λL) + λL 2 λ

Integral 2 :=

sin(λL) λ

Substituting these results into Eq. (3-321) leads to: Ci

(T ini − T ∞ ) sin(λi L) [cos(λi L) sin(λi L) + λi L] = 2 λi λi

(3-322)

or Ci =

2 (T ini − T ∞ ) sin(λi L) [cos(λi L) sin(λi L) + λi L]

(3-323)

The result is used to evaluate each constant in EES: “Evaluate constants” duplicate i=1,Nterm C[i]=2∗ (T_ini-T_inﬁnity)∗ sin(lambda[i]∗ L)/(cos(lambda[i]∗ L)∗ sin(lambda[i]∗ L)+lambda[i]∗ L) end

416

Transient Conduction

The solution at a speciﬁc time and position is evaluated using Eq. (3-318): x=0.01 [m] time=1000 [s] duplicate i=1,Nterm theta[i]=C[i]∗ cos(lambda[i]∗ x)∗ exp(-lambda[i]ˆ2∗ alpha∗ time) end T=T_inﬁnity+sum(theta[1..Nterm])

The solution to this bounded, 1-D transient problem is often expressed in terms of a dimensionless position (˜x), deﬁned as: x L

x˜ =

(3-324)

A dimensionless time can also be deﬁned. There is no characteristic time that appears in the problem statement. However, the diffusive time constant, discussed in Section 3.3.1, provides a convenient characteristic time for the problem that is related to the time required for a thermal wave to penetrate from the surface of the wall to the adiabatic boundary. The diffusive time constant is: τdiff =

L2 4α

(3-325)

and so an appropriate dimensionless time would be: t˜ =

t τdiff

=

4tα L2

(3-326)

The dimensionless time is typically referred to as the Fourier number (Fo) and, in most textbooks, the factor of 4 is removed from the deﬁnition: Fo =

tα L2

(3-327)

The dimensionless position and Fourier number can be used to more conveniently specify the position and time in the EES code: x hat=0.5 [-] x=x hat∗ L Fo=0.2 [-] time=Fo∗ Lˆ2/alpha duplicate i=1,Nterm theta[i]=C[i]∗ cos(lambda[i]∗ x)∗ exp(-lambda[i]ˆ2∗ alpha∗ time) end T=T inﬁnity+sum(theta[1..Nterm])

“dimensionless position” “position” “Fourier number” “time”

Figure 3-24 illustrates the temperature as a function of dimensionless position for various values of the Fourier number. The transient response of a plane wall (as well as the response of a cylinder and sphere) that is initially at a uniform temperature and is subjected to a convective boundary condition can be accessed from the EES library of heat transfer functions. Select Function Info from the Options menu and select Transient Conduction from the

3.5 Separation of Variables for Transient Problems

417

210 Fo=5.0

Temperature (K)

190 170 150

Fo =2.0

Fo=1.0 Fo=0.1

130

Fo=0.05

Fo=0.5 110 90 0

Fo=0.02

Fo=0.2 Fo=0.005 0.1

0.2

0.3 0.4 0.5 0.6 0.7 Dimensionless position, x/L

0.8

0.9

1

Figure 3-24: Temperature as a function of dimensionless position for various values of the Fourier number (dimensionless time).

pulldown menu in order to access these solutions. Note that the solutions programmed in EES are the ﬁrst 20 terms of the inﬁnite series solution (i.e., the ﬁrst 20 terms of Eq. (3-318) for a plane wall). Access and use of these library functions is discussed in more detail in Section 3.5.2. Limit Behaviors of the Separation of Variables Solution It is worthwhile spending some time understanding the behavior of the separation of variables solution in order to reinforce some of the concepts that were introduced in earlier sections. Figure 3-24 shows that for Fo less than about 0.20, the wall behaves as a semi-inﬁnite body. For Fo < 0.2 (approximately), the semi-inﬁnite body solution listed in Table 3-2 or accessed from the EES function SemiInf3 will provide accurate results. Figure 3-25 illustrates the temperature at x˜ = 0.25 as a function of Fo using the separation of variables solution developed in this section (100 terms are used to evaluate the series, so it is close to being exact) and obtained from the SemiInf3 function: T_semiinf=SemiInf3(T_ini,T_inﬁnity,h_bar,k,alpha,L-x,time) “temperature evaluated using the semi-inﬁnite body assumption”

Note that the semi-inﬁnite body solution is expressed in terms of the distance from the surface that is exposed to the ﬂuid whereas the separation of variables solution is expressed in terms of the distance from the adiabatic surface; therefore, the coordinate transformation (L-x) is required in the call to the SemiInf3 function. Figure 3-25 shows that the 100 term separation of variables solution agrees extremely well with the semi-inﬁnite body solution until Fo reaches about 0.20, at which point the thermal wave encounters the adiabatic wall and the problem becomes bounded. The solution to the problem, Eq. (3-318), expressed in terms of x˜ and Fo, is: θ (x, ˜ Fo) =

∞ i=1

Ci cos(λi L x) ˜ exp[− (λi L)2 Fo]

(3-328)

418

Transient Conduction

150 100 term separation of variables solution

Temperature (K)

140 130 120 110 semi-infinite body solution 100 90 single term separation of variables solution 80 0

0.1

0.2

0.3

0.4 0.5 0.6 Fourier number

0.7

0.8

0.9

1

Figure 3-25: Temperature as a function of Fo at x˜ = 0.25 evaluated using the separation of variables solution with 100 terms, with 1 term, and using the semi-inﬁnite body function.

where absolute value is ≤1

Ci =

2(T ini − T ∞ ) sin(λi L) cos(λi L) sin(λi L) + λi L

absolute value is ≤1 regardless of i

(3-329)

grows with i

It is interesting to look at how quickly the inﬁnite series converges to a solution (i.e., how many terms are actually required?). The value of the constants will decrease in magnitude as they increase in index. Examination of Eq. (3-329) shows that the denominator increases as i increases. Therefore, regardless of x, ˜ Fo, and Bi, the terms in Eq. (3-328) corresponding to larger values of i will have a decreasing value. This can also be seen by examining the array theta[i] in the EES solution. Further, examination of Eq. (3-328) shows that the value of each term in the series decays in time as exp(−(λi L)2 Fo). Because the value of λi L increases with i, this decay will occur much more rapidly for the terms with larger values of i. As a result, as the Fourier number increases, fewer and fewer terms are required to achieve an accurate solution and eventually a single term will sufﬁce. Many textbooks tabulate the ﬁrst constant and ﬁrst eigenvalue in Eq. (3-328) as a function of Bi and advocate using the “single-term approximation” for large values of Fourier number. ˜ exp[− (λ1 L)2 Fo] if Fo > 0.2 θ(x, ˜ Fo) ≈ C1 cos (λ1 L x)

(3-330)

The single term solution is also shown in Figure 3-25 and is clearly quite accurate for Fo > 0.2. Finally, it is worth revisiting the concept of the Biot number. If the Biot number is very small, then we would expect that temperature gradients within the wall will be negligible and the lumped capacitance model, discussed in Section 3.1, will be accurate. The solution for a lumped capacitance subjected to a step-change in the ﬂuid temperature is provided in Table 3-1: t (3-331) T = T ∞ + (T ini − T ∞ ) exp − τlumped

3.5 Separation of Variables for Transient Problems

419

where the lumped capacitance time constant (τlumped ) is: τlumped = Rconv C = Substituting Eq. (3-332) into Eq. (3-331) leads to:

Lρc

(3-332)

h

ht T = T ∞ + (T ini − T ∞ ) exp − Lρc

(3-333)

Multiplying and dividing the argument of the exponential by k L allows it to be rewritten as: ⎞ ⎛ ⎜ hL tk ⎟ ⎟ T = T ∞ + (T ini − T ∞ ) exp ⎜ ⎝− k L2 ρ c ⎠ = T ∞ + (T ini − T ∞ ) exp (−Bi Fo) Bi

Fo

(3-334) The lumped capacitance solution as a function of the product of the Biot number and the Fourier number, Eq. (3-334), is implemented in EES: T_lumped=T_inﬁnity+(T_ini-T_inﬁnity)∗ exp(-Bi∗ Fo) “temperature evaluated using the lumped capacitance assumption”

Figure 3-26 illustrates the temperature predicted by the lumped capacitance model, Eq. (3-334), as a function of the product Fo Bi. The temperature at the center of the wall (ˆx = 0) predicted by the separation of variables solution, Eq. (3-328), with 100 terms is also shown in Figure 3-26 for various values of the Biot number. 210

Temperature (K)

190 170

100 term separation of variables solution lumped capacitance solution Bi=0.1 Bi=0.2 Bi=0.5

Bi=1 Bi=2

150 Bi=5 130 Bi=10 110 90 0

Bi=20 0.5 1 1.5 2 2.5 3 3.5 Product of the Biot number and the Fourier number

4

Figure 3-26: Temperature at the center of the wall as a function of the product Fo Bi predicted by the lumped capacitance solution and by the separation of variables solution with 100 terms at x = 0.

Notice that the separation of variables solution approaches the lumped capacitance solution for Bi < 0.2. The separation of variables solution is an exact but complex method for analyzing a transient problem. However, Figure 3-25 and Figure 3-26 show that the solution limits to more simple expressions under certain conditions.

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

420

Transient Conduction

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED) The separation of variables technique provides a tool that can be used to provide a more quantitative solution to the problem posed in EXAMPLE 3.3-1, which is re-stated here. Figure 1(a) shows a metal wall that separates two tanks of liquid at different temperatures, Thot = 500 K and Tcold = 400 K; initially the wall is at steady state and therefore it has the linear temperature distribution shown in Figure 1(b). The thickness of the wall is th = 0.8 cm and its cross-sectional area is Ac = 1.0 m2 . The properties of the wall material are ρ = 8000 kg/m3 , c = 400 J/kg-K, and k = 20 W/m-K. The heat transfer coefﬁcient between the wall and the liquid in either tank is hliq = 5000 W/m2 -K. At time, t = 0, both tanks are drained and then exposed to gas at Tgas = 300 K, as shown in Figure 1(c). The heat transfer coefﬁcient between the walls and the gas is hgas = 100 W/m2 -K. Assume that the process of draining the tanks and ﬁlling them with air occurs instantaneously so that the wall has the linear temperature distribution shown in Figure 1(b) at time t = 0. th = 0.8 cm liquid at Tcold = 400 K hliq = 5000 W/m2 -K

liquid at Thot = 500 K 2 hliq = 5000 W/m -K x k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K (a)

Temperature (K) 500 475 450 425 400 0.8 Position (cm)

0

(b) th = 0.8 cm gas at Tgas = 300 K 2 h gas = 100 W/m -K

x

gas at Tgas = 300 K hgas = 100 W/m2 -K k = 20 W/m-K ρ = 8000 kg/m3 c = 400 J/kg-K (c)

Figure 1: (a) Tank wall exposed to liquid at two temperatures with (b) a linear steady state initial temperature distribution, when (c) at time t = 0 both walls are exposed to low temperature gas.

421

a) In EXAMPLE 3.3-1, we calculated a diffusive and lumped time constant for this problem (τdiff = 2.6 s and τlumped = 130 s) and used these values to sketch the expected temperature distribution at various times. Use the method of separation of variables to obtain a more precise solution. Note that this problem can not be solved using the planewall T function in EES because of the non-uniform initial temperature distribution. The separation of variables method must be applied. The known information is entered in EES: “EXAMPLE 3.5-2: Transient Response of a Tank Wall (Revisited)” $UnitSystem SI MASS RAD PA K J $Tabstops 0.2 0.4 0.6 0.8 3.5 “Inputs” k=20 [W/m-K] c=400 [J/kg-K] rho=8000 [kg/mˆ3] T cold=400 [K] T hot=500 [K] h bar liq=5000 [W/mˆ2-K] th=0.8[cm]∗ convert(cm,m) A c=1 [mˆ2] h bar gas=100 [W/mˆ2-K] T gas=300 [K]

“thermal conductivity” “speciﬁc heat capacity” “density” “cold ﬂuid temperature” “hot ﬂuid temperature” “liquid-to-wall heat transfer coefﬁcient” “wall thickness” “wall area” “gas-to-wall heat transfer coefﬁcient” “gas temperature”

The steady state temperature distribution provides the initial condition for the transient problem: Tt =0 = Tx=0,t =0 + (Tx=t h,t =0 − Tx=0,t =0 )

x th

(1)

where Tx=0,t =0 and Tx=t h,t =0 are the steady-state temperatures at either edge of the wall (Figure 1(b)): Tx=0,t =0 = Tcold +

(Thot − Tcold ) Rconv,liq 2 Rconv,liq + Rcond

and Tx=t h,t =0 = Tcold +

(Thot − Tcold ) (Rconv,liq + Rcond ) 2 Rconv,liq + Rcond

where Rconv,liq and Rcond are the convective and conductive resistances: Rconv,liq =

Rcond =

1 hliq Ac th k Ac

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

422

Transient Conduction

“Initial condition” “convection resistance with liquid” R conv liq=1/(h bar liq∗ A c) R cond=th/(k∗ A c) “conduction resistance” T x0 t0=T cold+(T hot-T cold)∗ R conv liq/(2∗ R conv liq+R cond) “initial temperature of cold side of wall” T xth t0=T cold+(T hot-T cold)∗ (R conv liq+R cond)/(2∗ R conv liq+R cond) “initial temperature of hot side of wall”

The governing differential equation is derived in Section 3.5.3: ∂T ∂ 2T =0 −α ∂t ∂x2 The spatial boundary conditions are provided by interface balances at either edge of the wall: ∂T hgas (Tgas − Tx=0 ) = −k ∂x x=0

∂T −k = hgas (Tx=t h − Tgas ) ∂ x x=t h The solution follows the steps that were outlined in Section 3.5.3. The problem as stated cannot be solved using separation of variables because neither of the two spatial boundary conditions are homogeneous. However, the problem can be transformed by deﬁning the temperature difference relative to the gas temperature: θ = T − Tgas The transformed partial differential equation becomes: ∂ 2θ ∂θ = 2 ∂x ∂t and the initial and boundary conditions become: α

θt =0 = (Tx=0,t =0 − Tgas ) + (Tx=t h,t =0 − Tx=0,t =0 )

hgas θx=0

∂θ =k ∂ x x=0

∂θ −k = hgas θx=t h ∂ x x=t h

(2)

x th

(3)

(4)

(5)

Both spatial boundary conditions, Eqs. (4) and (5), are homogeneous and therefore separation of variables can be used on the transformed problem. The solution is assumed to be the product of θX , a function of x, and θt , a function of t. The two ordinary differential equations that result are: d 2θ X + λ2 θ X = 0 d x2

(6)

dθt + λ2 α θ t = 0 dt

(7)

and

423

The eigenproblem (the problem in x) will be solved ﬁrst. The general solution to Eq. (6) is: θ X = C 1 sin (λ x) + C 2 cos (λ x)

(8)

Substituting Eq. (8) into the boundary condition at x = 0, Eq. (4), leads to: hgas [C 1 sin (λ 0) + C 2 cos (λ 0)] = k [C 1 λ cos (λ 0) − C 2 λ sin (λ 0)] or C 2 hgas = k C 1 λ Therefore, Eq. (8) can be written as: θ X = C1

sin (λ x) +

kλ

cos (λ x)

hgas

(9)

Substituting Eq. (9) into the boundary condition at x = th, Eq. (5), leads to the eigencondition for the problem: −k

λ cos (λ th) −

k λ2 hgas

sin (λ th) = hgas sin (λ th) +

kλ hgas

cos (λ th)

(10)

The eigenvalues are identiﬁed automatically using EES, as discussed in Section 3.5.3. The left side of Eq. (10) is moved to the right side in order to identify a residual, Res, that must be zero at each eigenvalue: Res = sin (λ th) +

k (λ th) hgas th

cos (λ th) +

k (λ th) hgas th

cos (λ th) −

k (λ th) hgas th

sin (λ th) (11)

Equation (11) can be simpliﬁed by deﬁning the Biot number in the usual way: Bi =

hgas th k

which leads to: Res = sin (λ th) +

(λ th) (λ th) (λ th) cos (λ th) + cos (λ th) − sin (λ th) Bi Bi Bi

Equation (12) is programmed in EES: “Eigenvalues” Bi=h bar gas∗ th/k “Biot number” Res=sin(lambdath)+lambdath∗ cos(lambdath)/Bi+lambdath∗ (cos(lambdath)& -lambdath∗ sin(lambdath)/Bi)/Bi “Residual”

(12)

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

Transient Conduction

and used to generate Figure 2, which shows the residual as a function of λ th. 10,000 8,000 6,000 4,000 Residual

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

424

2,000 0 -2,000 -4,000 -6,000 -8,000

-10,000

0

π/2

π

3 /2

π2 π

5π/2 λ th

π3

7π /2 π4

9 /2

Figure 2: Residual as a function of λ th. Notice that the residual crosses zero between every interval of π.

Figure 2 shows that the roots of Eq. (12) lie in each interval of π . Therefore, upper and lower bounds and appropriate guess values can be identiﬁed for each eigenvalue and assigned to arrays that are subsequently used to constrain each value of λ th (the entries in the array lambdath[ ]) in the Variable Information window: “Eigenvalues” Bi=h bar gas∗ th/k “Biot number” {Res=sin(lambdath)+lambdath∗ cos(lambdath)/Bi+lambdath∗ (cos(lambdath)& -lambdath∗ sin(lambdath)/Bi)/Bi “Residual”} Nterm=10 duplicate i=1,Nterm lowerlimit[i]=(i-1)∗ pi “lower limit” upperlimit[i]=i∗ pi “upper limit” guess[i]=(i-0.5)∗ pi “guess value” sin(lambdath[i])+lambdath[i]∗ cos(lambdath[i])/Bi+lambdath[i]∗ (cos(lambdath[i])& “eigencondition” -lambdath[i]∗ sin(lambdath[i])/Bi)/Bi=0 end

The EES code will identify each of the eigenvalues. The solution for each eigenvalue is therefore: k λi cos (λi x) θX i = C 1,i sin (λi x) + (13) hgas The solution to the ordinary differential equation in time, Eq. (7), for each eigenvalue is: θ ti = C 3,i exp −λi2 α t

425

and therefore the solution for each eigenvalue is: k λi θi = θ xi θ ti = C i sin (λi x) + cos (λi x) exp −λi2 α t hgas The sum of the solutions for each eigenvalue is the series solution to the problem: ∞ ∞ k λi θi = C i sin (λi x) + cos (λi x) exp −λi2 α t (14) θ= hgas i=1 i=1 ˜ and The series solution can be expressed in terms of dimensionless position (x) Fourier number (Fo ): θ=

& % λi th ˜ + ˜ exp − (λi th)2 F o C i sin (λi th x) cos (λi th x) Bi i=1

∞

(15)

where x˜ and Fo are deﬁned as: x˜ = Fo =

x th αt th2

The constants must be selected so that the initial condition, Eq. (3), is satisﬁed: θt =0

∞

λi th ˜ = Ci cos (λi th x) Bi i=1 = Tx=0,t =0 − Tgas + Tx=t h − Tss,x=0 xˆ ˆ + sin (λi th x)

(16)

The eigenfunctions must be orthogonal. Therefore, multiplying both sides of Eq. (16) by the j th eigenfunction and integrating from x = 0 to x = th (or x˜ = 0 to x˜ = 1) leads to: ∞

1

i=1 0

Ci

˜ + sin (λi th x)

λ j th ˜ ˜ cos λ j th x d x˜ sin λ j th x + Bi

λi th ˜ cos (λi th x) Bi

% & λ j th Tx=0,t =0 − Tgas + Tx=t h,t =0 − Tx=0,t =0 x˜ sin λ j th x˜ + cos λ j th x˜ d x˜ Bi

1

= 0

The only term on the left side of the equation that does not integrate to zero is i = j and therefore: 1

2

λi th ˜ cos (λi th x) Bi (Integral1)i

˜ + sin (λi th x)

Ci 0

1

=

d x˜ (17)

% & λi th ˜ + ˜ d x˜ cos (λi th x) Tx=0,t =0 − Tgas + Tx=t h,t =0 − Tx=0,t =0 x˜ sin (λi th x) Bi 0 (Integral 2)i

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

426

Transient Conduction

The integrals in Eq. (17) seem imposing, but they can be accomplished relatively easily using Maple and then copied into EES for evaluation. Equation (17) is written in terms of the two integrals: C i Integral 1 i = Integral 2 i where

Integral 1 i =

1

2

˜ + sin (λi th x) 0

λi th ˜ cos (λi th x) Bi

d x˜

and Integral 2 i =

1

%

(Tx=0,t =0 − Tgas ) + (Tx=t h,t =0 − Tx=0,t =0 )x˜

&

0

λi th ˜ + ˜ d x˜ × sin(λi th x) cos(λi th x) Bi These integrals are entered in Maple: > restart; > Integral_1[i]:=int((sin(lambdath[i]∗ x_hat)+lambdath[i]∗ cos(lambdath[i]∗ x_hat)/Bi)ˆ2,x_hat=0..1); Integral 1i := 12 (2 Bi lambdathi − Bi2 cos(lambdathi ) sin(lambdathi ) + Bi 2 (lambdathi ) − 2 Bi lambdathi cos(lambdathi )2 + lambdathi2 cos(lambdathi ) sin(lambdathi ) + lambdathi3 )/(Bi 2 lambdathi ) > Integral_2[i]:=int((T_x0_t0-T_gas+(T_xth_t0 T_x0_t0)∗ x_hat)∗ (sin(lambdath[i]∗ x_hat)+lambdath[i]∗ cos(lambdath[i]∗ x_hat)/Bi),x_hat=0..1); Integral 2i := −(−T x0 t 0 Bi lambdathi + T gasBi lambdathi + T xt h t 0 lambdathi − T x0 t 0 lambdathi − T gas Bi cos(lambdathi ) lambdathi + T gas lambdathi2 sin(lambdathi ) − T xt h t 0 Bi sin(lambdathi ) + T xt h t 0 Bi lambdathi cos(lambdathi ) − T xt h t 0 lambdathi cos(lambdathi ) − T xt h t 0 lambdathi2 sin(lambdathi ) + T x0 t 0 Bi sin(lambdathi ) + T x0 t 0 lambdathi cos(lambdathi )) / (Bi lambdathi2 )

and copied into EES to evaluate each of the constants. The only modiﬁcation required to the Maple results is to change the := to = “Evaluate Constants” duplicate i=1,Nterm Integral_1[i] = 1/2∗ (2∗ Bi∗ lambdath[i]-Biˆ2∗ cos(lambdath[i])∗ sin(lambdath[i])& +Biˆ2∗ lambdath[i]-2∗ Bi∗ lambdath[i]∗ cos(lambdath[i])ˆ2+lambdath[i]ˆ2∗ & cos(lambdath[i])∗ sin(lambdath[i])+lambdath[i]ˆ3)/Biˆ2/lambdath[i] Integral_2[i] = -(-T_x0_t0∗ Bi∗ lambdath[i]+T_gas∗ Bi∗ lambdath[i]+T_xth_t0∗ lambdath[i]& -T_x0_t0∗ lambdath[i]-T_gas∗ Bi∗ cos(lambdath[i])∗ lambdath[i]+T_gas∗ lambdath[i]ˆ2& ∗ sin(lambdath[i])-T_xth_t0∗ Bi∗ sin(lambdath[i])+T_xth_t0∗ Bi∗ lambdath[i]∗ cos(lambdath[i])& -T_xth_t0∗ lambdath[i]∗ cos(lambdath[i])-T_xth_t0∗ lambdath[i]ˆ2∗ sin(lambdath[i])& +T_x0_t0∗ Bi∗ sin(lambdath[i])+T_x0_t0∗ lambdath[i]∗ cos(lambdath[i]))/Bi/lambdath[i]ˆ2 C[i]∗ Integral_1[i]=Integral_2[i] end

427

The solution can be obtained at arbitrary values of position and time: “Solution” alpha=k/(rho∗ c) “thermal diffusivity” x hat=0 “dimensionless position” x hat=x/th Fo=alpha∗ time/thˆ2 “Fourier number” time=5 [s] duplicate i=1,Nterm theta[i]=C[i]∗ (sin(lambdath[i]∗ x hat)+lambdath[i]∗ cos(lambdath[i]∗ x hat)/Bi)∗ exp(-lambdath[i]ˆ2∗ Fo) end T=T gas+sum(theta[1..Nterm])

Figure 3 shows the temperature as a function of position in the wall for the same times (t = 0 s, 0.5 s, 5 s, 50 s, 500 s, and 5000 s) that were sketched in EXAMPLE 3.3-1. 475 t =0.5 s t=5 s

450

Temperature (K)

425

t =0 s t=50 s

400 375 350 325

t =500 s

300 t =5000 s 275 0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 Dimensionless axial position, x/L

0.9

1

Figure 3: Temperature distribution within the wall at various times as it equilibrates, predicted by the separation of variables model. The times indicated are the same as those considered in EXAMPLE 3.3 -1.

Figure 3 shows the same trends that were discussed in EXAMPLE 3.3-1; there is an internal, conductive equilibration process that occurs very quickly (with a time scale that is similar to τdiff = 2.6 s). The Fourier number associated with this internal equilibration process will be on the order of 1.0, because the Fourier number is deﬁned as the ratio of time to the time required for a thermal wave to move across the wall. There is subsequently an external, convective equilibration process that occurs much more slowly (with a time scale that is similar to τlumped = 130 s). The Fourier-Biot number product associated with this external equilibration process will be on the order of 1.0 because the Fourier-Biot number product characterizes the ratio of time to the lumped time constant.

3.5.4 Separation of Variables Solutions in Cylindrical Coordinates This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). The separation of variables solution for transient problems can be

EXAMPLE 3.5-2: TRANSIENT RESPONSE OF A TANK WALL (REVISITED)

3.5 Separation of Variables for Transient Problems

428

Transient Conduction

applied in cylindrical coordinates. The solution for a cylinder exposed to a step-change in ambient conditions that is presented in Section 3.5.2 was derived using separation of variables in cylindrical coordinates. In this section, the techniques required to solve this problem are presented.

3.5.5 Non-homogeneous Boundary Conditions This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). The method of separation of variables can only be applied to 1-D transient problems where both spatial boundary conditions are homogeneous. In Sections 3.5.3 and 3.5.4, a single, obvious transformation is sufﬁcient to make both spatial boundary conditions homogeneous. In many problems this will not be the case and more advanced techniques will be required. Section 2.3.2 discusses methods for breaking 2D steady problems with non-homogeneous terms into sub-problems that can be solved either by separation of variables or by the solution of an ordinary differential equation. In Section 2.4, superposition for 2-D steady-state problems is discussed. These techniques for solving problems with non-homogeneous boundary conditions using separation of variables remain valid for 1-D transient problems and are presented in Section 3.5.5.

3.6 Duhamel’s Theorem This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). The separation of variables technique discussed in Section 3.5 is not capable of solving problems with time-dependent spatial boundary conditions (e.g., an ambient temperature or heat ﬂux that varies with time). However, problems with time dependent spatial boundary conditions are common. Duhamel’s theorem provides one method of extending an analytical solution that is derived (for example, using separation of variables) assuming a time-invariant boundary condition in order to consider the temperature response to an arbitrary time variation of that boundary condition.

3.7 Complex Combination This extended section of the book can be found on the website (www.cambridge.org/ nellisandklein). Complex combination is a useful technique for solving problems that have periodic (i.e., oscillating) boundary conditions or forcing functions. This type of problem was encountered in EXAMPLE 3.1-2 where a temperature sensor (treated as a lumped capacitance) was exposed to an oscillating ﬂuid temperature. In EXAMPLE 3.1-2, the problem is solved analytically and the temperature response of the sensor is found to be the sum of a homogeneous solution that decays to zero (as time became sufﬁciently greater than the time constant of the sensor), and a particular solution that is the sustained response. Complex combination is a convenient method for obtaining only this sustained solution, which is often the only portion of the solution that is of interest. Complex combination can be used for transient problems that are 0-D (i.e., lumped), 1-D, and even 2-D or 3-D.

3.8 Numerical Solutions to 1-D Transient Problems 3.8.1 Introduction Sections 1.4 and 1.5 discuss numerical solutions to steady-state 1-D problems and Section 3.2 discusses the numerical solution to 0-D (lumped) transient problems. In this section, these concepts are extended in order to numerically solve 1-D transient problems.

3.8 Numerical Solutions to 1-D Transient Problems

429

The underlying methodology is the same; a set of equations is obtained from energy balances on small (but not differentially small) control volumes that are distributed throughout the computational domain. These equations will contain energy storage terms in addition to energy transfer terms (due to conduction, convection, and/or radiation) because of the transient nature of the problem. The energy storage terms involve the time rate of temperature change and therefore must be numerically integrated forward in time using one of the techniques that was discussed previously in Section 3.2 (e.g., the Crank-Nicolson technique). Most software applications, including EES and MATLAB, provide advanced numerical integration capabilities that can be applied to this problem.

3.8.2 Transient Conduction in a Plane Wall The process of obtaining a numerical solution to a 1-D, transient conduction problem will be illustrated in the context of a plane wall subjected to a convective boundary condition on one surface, as shown in Figure 3-41. initial temperature, Tini = 20°C L = 5 cm

Figure 3-41: A plane wall exposed to a convective boundary condition at time t = 0.

T∞ = 200°C 2 h = 500 W/m -K x k = 5 W/m-K ρ = 2000 kg/m3 c = 200 J/kg-K

The plane wall has thickness L = 5.0 cm and properties k = 5.0 W/m-K, ρ = 2000 kg/m3 , and c = 200 J/kg-K. The wall is initially at T ini = 20◦ C when at time t = 0, the surface (at x = L) is exposed to ﬂuid at T ∞ = 200◦ C with average heat transfer coefﬁcient h = 500 W/m2 -K. The wall at x = 0 is adiabatic. The known information is entered into EES. $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “inputs" L=5 [cm]∗ convert(cm,m) k=5.0 [W/m-K] rho=2000 [kg/mˆ3] c=200 [J/kg-K] T ini=converttemp(C,K,20 [C]) T inﬁnity=converttemp(C,K,200 [C]) h bar=500 [W/mˆ2-K]

“wall thickness” “conductivity” “density” “speciﬁc heat capacity” “initial temperature” “ﬂuid temperature” “heat transfer coefﬁcient”

The plane wall problem in Figure 3-41 corresponds to the problem that is solved using separation of variables in Section 3.5.3. This problem was selected so that the solution that is obtained numerically can be directly compared to the analytical solution accessed by the planewall_T function in EES that is described in Section 3.5.2. However, the major advantage of using a numerical technique is the capability to easily solve complex problems that may not have an analytical solution.

430

Transient Conduction Δx 2

T1 T2

Ti-1

Ti

q⋅RHS q⋅LHS dU dt x

Δx

Δx 2

Ti+1

TN-1

q⋅RHS

TN

q⋅conv

q⋅LHS

dU dt

Figure 3-42: Nodes and control volumes distributed uniformly throughout computational domain.

dU dt

The ﬁrst step in developing the numerical solution is to partition the continuous medium into a large number of small volumes that are analyzed using energy balances in order to develop the set of equations that must be solved. The nodes (i.e., the positions at which the temperature will be obtained) are distributed uniformly through the wall, as shown in Figure 3-42. For the uniform distribution of nodes that is shown in Figure 3-42, the location of each node (xi ) is: xi =

(i − 1) L for i = 1..N (N − 1)

(3-495)

where N is the number of nodes used for the simulation. The distance between adjacent nodes ( x) is: x =

L (N − 1)

(3-496)

This node spacing is speciﬁed in EES: “Setup grid” N=6 [-] duplicate i=1,N x[i]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1)

“number of nodes” “position of each node” “distance between adjacent nodes”

A control volume is deﬁned around each node. The control surface bisects the distance between the nodes, as shown in Figure 3-42. An energy balance must be written for the control volume associated with every node. The control volume for an arbitrary, internal node experiences conduction heat transfer with the adjacent nodes as well as energy storage (as shown in Figure 3-42): q˙ LHS + q˙ RHS =

dU dt

(3-497)

Each term in Eq. (3-497) must be approximated. The conduction terms from the adjacent nodes are modeled according to: q˙ LHS =

k Ac (T i−1 − T i ) x

(3-498)

q˙ RHS =

k Ac (T i+1 − T i ) x

(3-499)

3.8 Numerical Solutions to 1-D Transient Problems

431

where Ac is the cross-sectional area of the wall. The rate of energy storage is the product of the time rate of change of the nodal temperature and the thermal mass of the control volume: dT i dU = Ac x ρ c dt dt

(3-500)

Substituting Eqs. (3-498) through (3-500) into Eq. (3-497) leads to: Ac x ρ c

dT i k Ac (T i−1 − T i ) k Ac (T i+1 − T i ) = + dt x x

for i = 2... (N − 1)

(3-501)

Solving for the time rate of the temperature change: k dT i = (T i−1 + T i+1 − 2 T i ) dt x2 ρ c

for i = 2... (N − 1)

(3-502)

The control volumes at the boundaries must be treated separately because they have a smaller volume and experience different energy transfers. An energy balance on the control volume at the adiabatic wall (node 1 in Figure 3-42) leads to: q˙ RHS =

dU dt

(3-503)

or Ac x ρ c dT 1 k Ac (T 2 − T 1 ) = 2 dt x

(3-504)

The factor of two on the left side of Eq. (3-504) results because the control volume around node 1 has half the width and thus half the heat capacity of the other nodes. Solving for the time rate of temperature change for node 1: 2k dT 1 = (T 2 − T 1 ) dt ρ c x2

(3-505)

An energy balance on the control volume for the node located at the outer surface (node N in Figure 3-42) leads to: dU = q˙ LHS + q˙ conv dt

(3-506)

or Ac x ρ c dT N k Ac (T N−1 − T N ) = + h Ac (T ∞ − T N ) 2 dt x

(3-507)

Solving for the time rate of temperature change for node N: dT N 2h 2k = (T N−1 − T N ) + (T ∞ − T N ) 2 dt ρ c x x ρ c

(3-508)

Equations (3-502), (3-505), and (3-508) provide the time rate of change for the temperature of every node, given the temperatures of the nodes. This result is similar to the situation encountered in Section 3.2 when developing numerical solutions for lumped capacitance problems. In that case, the energy balance for the single control volume (around the object being studied) provided an equation for the time rate of change of the temperature in terms of the temperature. Here, the energy balance written for each of the lumped capacitances (i.e., each of the control volumes) has provided a set of

432

Transient Conduction

equations for the time rates of change of each of the nodal temperatures. In order to solve the problem, it is necessary to integrate these time derivatives. All of the numerical integration techniques that were discussed in Section 3.2 to solve lumped capacitance problems can be applied here to solve 1-D transient problems. The temperature of each node is a function both of position (x) and time (t). The index that speciﬁes the node’s position is i where i = 1 corresponds to the adiabatic wall and i = N corresponds to the surface of the wall (see Figure 3-42). A second index, j, is added to each nodal temperature in order to indicate the time (Ti,j ), where j = 1 corresponds to the beginning of the simulation and j = M corresponds to the end of the simulation. The total simulation time, tsim , is divided into M time steps. Most of the techniques discussed here will divide the simulation time into time steps of equal duration, t, although other distributions may be used. t =

tsim (M − 1)

(3-509)

The time associated with any time step is: t j = (j − 1) t

for j = 1...M

(3-510)

An array (time []) that provides the time for each step is deﬁned: “Setup time steps” M=21 [-] t sim=40 [s] DELTAtime=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ DELTAtime end

“number of time steps” “simulation time” “time step duration”

The initial conditions for this problem speciﬁes that all of the temperatures at t = 0 are equal to T ini . T i,1 = T ini

duplicate i=1,N T[i,1]=T ini end

for i = 1...N

(3-511)

“initial condition”

Note that the variable T is a two-dimensional array (i.e., a matrix) and calculated results will be displayed in the Arrays Window. The use of a 2-D array simpliﬁes the presentation of the methodology, but it requires unnecessary variable storage space in EES. MATLAB is a more appropriate tool for these types of simulations and the implementation of the numerical simulation in MATLAB will be discussed subsequently. Euler’s Method The temperature of all of the nodes at the end of each time step (i.e., T i,j+1 for all i = 1 to N) must be computed given the temperatures at the beginning of the time step (i.e., T i,j for all i=1 to N) and the algebraic equations derived from the energy balances (i.e., Eqs. (3-502), (3-505), and (3-508)). The simplest technique for numerical integration is

3.8 Numerical Solutions to 1-D Transient Problems

433

Euler’s Method, which approximates the time rate of temperature change within each time step as being constant and equal to its value at the beginning of the time step. Therefore, for any node i during time step j: dT t for i = 1...N (3-512) T i,j+1 = T i,j + dt T =T i,j ,t=tj Note that it is often useful to develop a numerical simulation of a transient process by initially taking only a single step and then, once that works, automating the process of simulating all of the time steps. For example, the temperatures of all N nodes at the end of the ﬁrst time step (i.e., j = 2) are determined from: dT t for i = 1...N (3-513) T i,2 = T i,1 + dt T =T i,1 ,t=t1 Substituting Eqs. (3-502), (3-505), and (3-508), into Eq. (3-513) leads to: T 1,2 = T 1,1 +

T i,2 = T i,1 +

2k (T 2,1 − T 1,1 ) t ρ c x2

k (T i−1,1 + T i+1,1 − 2 T i,1 ) t x2 ρ c

(3-514)

for i = 2... (N − 1)

(3-515)

T N,2

2h 2k = T N,1 + (T N−1,1 − T N,1 ) + (T f − T N,1 ) t ρ c x2 x ρ c

(3-516)

Equations (3-514) through (3-516) are programmed in EES: “Take a single Euler step” T[1,2]=T[1,1]+2∗ k∗ (T[2,1]-T[1,1])∗ DELTAtime/(rho∗ c∗ DELTAxˆ2) duplicate i=2,(N-1) T[i,2]=T[i,1]+k∗ (T[i-1,1]+T[i+1,1]-2∗ T[i,1])∗ DELTAtime/(rho∗ c∗ DELTAxˆ2)

“node 1”

“internal nodes” end T[N,2]=T[N,1]+(2∗ k∗ (T[N-1,1]-T[N,1])/(rho∗ c∗ DELTAxˆ2)+& 2∗ h bar∗ (T inﬁnity-T[N,1])/(rho∗ c∗ DELTAx))∗ DELTAtime

“node N”

Solving the program should provide a solution for the temperature at each node at the end of the ﬁrst time step. The solution can be examined by selecting Arrays from the Windows menu. The equations used to move through any arbitrary time step j follow logically from Eqs. (3-514) through (3-516): T 1,j+1 = T 1,j +

T i,j+1 = T i,j +

2k (T 2,j − T 1,j ) t ρ c x2

k (T i−1,j + T i+1,j − 2 T i,j ) t x2 ρ c

for i = 2... (N − 1)

(3-517)

(3-518)

T N,j+1

2h 2k = T N,j + (T N−1,j − T N,j ) + (T ∞ − T N,j ) t ρ c x2 x ρ c

(3-519)

434

Transient Conduction 450

Temperature (K)

425

numerical solution analytical solution ana

400 40 s

375

30 s 350

20 s

325

10 s

300

4s

275 0

0s 0.01

0.02

0.03

0.04

0.05

Position (m) Figure 3-43: Temperature as a function of position predicted by Euler’s method at various times.

Equations (3-517) through (3-519) are automated in order to simulate all of the time steps by placing the EES code within a second duplicate loop that steps from j = 1 to j = (M − 1). This is accomplished by nesting the entire EES code associated with the ﬁrst time step within an outer duplicate loop; wherever the second index was 2 (i.e., the end of time step 1) we replace it with j + 1 and wherever the second index was 1 (i.e., the beginning of time step 1) we replace it with j. The revised code is shown below; the additional or changed code is shown in bold:

“Move through all of the time steps” duplicate j=1,(M-1) T[1,j+1]=T[1,j]+2∗ k∗ (T[2,j]-T[1,j])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2) duplicate i=2,(N-1) T[i,j+1]=T[i,j]+k∗ (T[i-1,j]+T[i+1,j]-2∗ T[i,j])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2)

“node 1”

“internal nodes” end T[N,j+1]=T[N,j]+(2∗ k∗ (T[N-1,j]-T[N,j])/(rho∗ c∗ DELTAxˆ2)+& 2∗ h bar∗ (T inﬁnity-T[N,j])/(rho∗ c∗ DELTAx))∗ DELTAtime end

“node N”

Figure 3-43 illustrates the temperature as a function of position at t = 0 s, t = 4 s, t = 10 s, t = 20 s, t = 30 s, and t = 40 s. Figure 3-43 was generated by selecting New Plot Window from the Plots menu and selecting the array x[ ] for the X-Axis and the arrays T[i,1], T[i,3], T[i,6], etc. for the Y-Axis (note that time[1] = 0, time[3] = 4 s, time[6] = 10 s, etc. You will be prompted that you are missing data in various rows because the time[ ] array is longer than the x[ ] or any of the T[i,j] arrays. Select Yes in response to this message in order to continue plotting the points. The numerical solution can be compared to the analytical solution derived in Section 3.5.3 and accessed using the planewall_T function.

3.8 Numerical Solutions to 1-D Transient Problems

435

“Analytical solution” alpha=k/(rho∗ c) duplicate j=1,M duplicate i=1,N T_an[i,j]=planewall_T(x[i], time[j], T_ini, T_inﬁnity, alpha, k, h_bar, L) end end

The numerical and analytical solutions agree, as shown in Figure 3-43. If the duration of the time step is increased from 2.0 s to 4.0 s (by reducing M from 21 to 11) then the solution becomes unstable (try it and see; the solution will oscillate between large positive and negative temperatures). The existence of a stability limit is one of the key disadvantages associated with the Euler technique, as we saw in Section 3.2 for lumped capacitance problems. The maximum time step that can be used before the solution becomes unstable (i.e., the critical time step, tcrit ) can be determined by examining the algebraic equations that are used to step through time. Rearranging Eq. (3-517), which governs the behavior of the node at the adiabatic edge, leads to: 2k t 2k t T 2,j (3-520) + T 1,j+1 = T 1,j 1 − ρ c x2 ρ c x2 The solution will become unstable when the coefﬁcient multiplying T1,j becomes negative. Therefore, Eq. (3-520) shows that the solution will tend to become unstable as t becomes larger or x becomes smaller. The critical time step associated with node 1 is: tcrit,1 =

ρ c x2 2k

(3-521)

Applying the same process to Eq. (3-518), which governs the behavior of the internal nodes, leads to: k t 2 k t (3-522) + T i,j+1 = T i,j 1 − (T i−1,j + T i+1,j ) for i = 2... (N − 1) x2 ρ c x2 ρ c According to Eq. (3-522), the critical time step for the internal nodes is the same as for node 1: tcrit,i =

ρ c x2 2k

for i = 2... (N − 1)

(3-523)

Equation (3-519), which governs the behavior of the node placed on the surface of the wall, is rearranged: 2 h t 2 h t 2 k t 2 k t T N,j+1 = T N,j 1 − − T N−1,j + (3-524) + Tf ρ c x2 x ρ c ρ c x2 x ρ c Equation (3-524) leads to a different critical time step for node N: tcrit,N =

x ρ c k 2 +h x

(3-525)

436

Transient Conduction

Comparing Eq. (3-525) with Eq. (3-523) indicates that the critical time step for node N will always be somewhat less than the critical time step for the other nodes and therefore Eq. (3-525) will govern the stability of the problem. The critical time step is calculated according to: “Critical time step for explicit techniques” Deltatime crit N=Deltax∗ rho∗ c/(2∗ (k/Deltax+h bar))

“critical time step for node N”

For the problem considered here, the critical time step tcrit,N = 2.0 s. Note that even a stable solution may not be sufﬁciently accurate. We followed a similar process in Section 3.2 in order to identify the critical time step associated with the explicit integration of a lumped capacitance problem and found that the critical time step was related to the time constant of the object. Equations (3-521), (3-523), and (3-525) essentially restate this conclusion, but for an individual node rather than a lumped object. Recall that the time constant of an object is equal to the product of the heat capacity of the object and its thermal resistance to the environment. The heat capacity of an internal node (Ci ) is: Ci = x Ac ρ c

(3-526)

and the resistance (Ri ) between the nodal temperature of an internal node and its environment (i.e., the adjacent nodes) is: k Ac k Ac −1 Ri = (3-527) + x x or Ri =

x 2 k Ac

(3-528)

The lumped time constant of an internal node (τlumped,i ) is therefore: τlumped,i = Ri Ci =

x ρ c x2 x Ac ρ c = 2 k Ac 2k

(3-529)

and therefore the time constant of an internal node is exactly equal to its critical time step, Eq. (3-523). The time constant for the node located on the surface of the wall (node N) is smaller and therefore its critical time constant will be smaller. The heat capacity of node N (CN ) is: CN =

x Ac ρ c 2

(3-530)

and RN , the thermal resistance between node N and its environment (i.e., node N − 1 and the ﬂuid), is: −1 k Ac RN = (3-531) + h Ac x The time constant for node N is: τlumped,N = RN CN =

x Ac ρ c ρ c x = k Ac k 2 2 + h Ac +h x x

which is identical to the critical time step for node N, Eq. (3-525).

(3-532)

3.8 Numerical Solutions to 1-D Transient Problems

437

In Section 1.4 we found that the accuracy of a numerical solution is related to the size of the control volumes. Smaller values of x will provide more accurate solutions. However, smaller values of x will simultaneously reduce the thermal mass of each node as well as the resistance between the node and its adjacent nodes. Thus the time constant and therefore the critical time step duration for any node scales with x2 (see Eq. (3-529)). As a result, reducing x will greatly increase the number of time steps required (by an explicit numerical technique) in order to maintain stability. It is often the case that the number of time steps required by stability is much larger than would be required for sufﬁcient accuracy and therefore the use of explicit techniques can be computationally intensive. Implementing Euler’s Method within MATLAB is a straightforward extension of the EES code. The simulation in MATLAB is setup in a script. The inputs are provided as the ﬁrst lines: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T inﬁnity=473.2; h bar=500;

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % speciﬁc heat capacity (J/kg-K) % initial temperature (K) % ﬂuid temperature (K) % heat transfer coefﬁcient (W/mˆ2-K)

The axial location of each node and the time steps are speciﬁed: %Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1); %Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% number of time steps (-) % simulation time (s) % time step duration (s)

The initial conditions for each node are inserted into the ﬁrst column of the matrix T: % Initial condition for i=1:N T(i,1)=T_ini; end

438

Transient Conduction

The remaining columns of T are ﬁlled in by stepping through time using the Euler approach: % Step through time for j=1:(M-1) T(1,j+1)=T(1,j)+2∗ k∗ (T(2,j)-T(1,j))∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) T(i,j+1)=T(i,j)+k∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); end T(N,j+1)=T(N,j)+(2∗ k∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+ . . . 2∗ h_bar∗ (T_inﬁnity-T(N,j))/(rho∗ c∗ DELTAx))∗ DELTAtime; end

One advantage of using MATLAB is that it is easy to manipulate the solution. For example, it is possible to plot the temperature as a function of time for various positions by entering plot(time,T) in the command window (Figure 3-44). 440 x (cm) 0

420

Temperature (K)

400 380 1

360 340

2

320

3

300 280

4 0

5

10

15

20 Time (s)

25

30

5 35

40

Figure 3-44: Temperature as a function of time for various positions.

Fully Implicit Method Euler’s method is an explicit technique. Therefore, it has the characteristic of becoming unstable when the time step exceeds a critical value. An implicit technique avoids this problem. The fully implicit method is similar to Euler’s method in that the time rate of change is assumed to be constant throughout the time step. The difference is that the time rate of change is computed at the end of the time step rather than the beginning. Therefore, for any node i and time step j: dT t for i = 1 . . . N (3-533) T i,j+1 = T i,j + dt T =T i,j+1 ,t=tj+1 The temperature rate of change at the end of the time step depends on the temperatures at the end of the time step (T i,j+1 ). The temperatures T i,j+1 cannot be calculated explicitly using information at the beginning of the time step (Ti,j ) and instead Eq. (3-533) provides an implicit set of equations for T i,j+1 where i = 1 to i = N. For the example

3.8 Numerical Solutions to 1-D Transient Problems

439

problem shown in Figure 3-41, the implicit equations are obtained by substituting Eqs. (3-502), (3-505), and (3-508) into Eq.(3-533):

T 1,j+1 = T 1,j +

T i,j+1 = T i,j + T N,j+1 = T N,j +

2k (T 2,j+1 − T 1,j+1 ) t ρ c x2

k (T i−1,j+1 + T i+1,j+1 − 2 T i,j+1 ) t x2 ρ c

(3-534)

for i = 2... (N − 1) (3-535)

2h 2k (T N−1,j+1 − T N,j+1 ) + (T f − T N,j+1 ) t ρ c x2 x ρ c

(3-536)

Because EES solves implicit equations, it is not necessary to rearrange Eqs. (3-534) through (3-536) in order to solve them. The implicit solution is obtained using the following EES code. (Note that the modiﬁcations to the Euler method code are indicated in bold.)

$UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” L=5 [cm]∗ convert(cm,m) k=5.0 [W/m-K] rho=2000 [kg/mˆ3] c=200 [J/kg-K] T ini=converttemp(C,K,20 [C]) T inﬁnity=converttemp(C,K,200 [C]) h bar=500 [W/mˆ2-K] “Setup grid” N=5 [-] duplicate i=1,N x[i]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1) “Setup time steps” M=21 [-] t sim=40 [s] DELTAtime=t sim/(M-1) duplicate j=1,M time[j]=(j-1)∗ DELTAtime end duplicate i=1,N T[i,1]=T ini end

“wall thickness” “conductivity” “density” “speciﬁc heat capacity” “initial temperature” “ﬂuid temperature” “heat transfer coefﬁcient”

“number of nodes” “position of each node” “distance between adjacent nodes”

“number of time steps” “simulation time” “time step duration”

“initial condition”

440

Transient Conduction

“Move through all of the time steps” duplicate j=1,(M-1) T[1,j+1]=T[1,j]+2∗ k∗ (T[2,j+1]-T[1,j+1])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2) “node 1” duplicate i=2,(N-1) T[i,j+1]=T[i,j]+k∗ (T[i-1,j+1]+T[i+1,j+1]-2∗ T[i,j+1])∗ DELTAtime/(rho∗ c∗ Deltaxˆ2) “internal nodes” end T[N,j+1]=T[N,j]+(2∗ k∗ (T[N-1,j+1]-T[N,j+1])/(rho∗ c∗ DELTAxˆ2)+& 2∗ h bar∗ (T inﬁnity-T[N,j+1])/(rho∗ c∗ DELTAx))∗ DELTAtime “node N” end

An attractive feature of the implicit technique is that it is possible to vary M and N independently in order to achieve sufﬁcient accuracy without being constrained by stability considerations. The implementation of the implicit technique in MATLAB is not a straightforward extension of the EES code because MATLAB cannot directly solve a set of implicit equations. Instead, Eqs. (3-534) through (3-536) must be placed in matrix format before they can be solved. This operation is similar to the solution of 1-D and 2-D steady-state problems using MATLAB, discussed in Sections 1.5, 1.9, and 2.6. The inputs are entered in MATLAB and the grid and time steps are setup: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T inﬁnity=473.2; h bar=500; % Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1); % Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end % Initial condition for i=1:N T(i,1)=T ini; end

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % speciﬁc heat capacity (J/kg-K) % initial temperature (K) % ﬂuid temperature (K) % heat transfer coefﬁcient (W/mˆ2-K)

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% number of time steps (-) % simulation time (s) % time step duration (s)

3.8 Numerical Solutions to 1-D Transient Problems

441

For any time step j, Eqs. (3-534) through (3-536) represent N linear equations for the N unknown temperatures T i,j+1 for i = 1 to i = N. In order to solve these implicit equations, they must be placed in matrix format: AX = b

(3-537)

It is important to clearly specify the order that the equations are placed into the matrix A and the order that the unknown temperatures are placed into the vector X. The most logical method to setup the vector X is: ⎤ ⎡ X 1 = T 1,j+1 ⎥ ⎢X = T 2,j+1 ⎥ ⎢ 2 X=⎢ (3-538) ⎥ ⎦ ⎣··· X N = T N,j+1 so that T i,j+1 corresponds to element i of X. The most logical method for placing the equations into A is: ⎡ ⎤ row 1 = control volume 1 equation ⎢ row 2 = control volume 2 equation ⎥ ⎥ A=⎢ (3-539) ⎣··· ⎦ row N = control volume N equation so that the equation derived based on the control volume for node i corresponds to row i of A. Equations (3-534) through (3-536) are rearranged so that the coefﬁcients multiplying the unknowns and the constants for the linear equations are clear: 2k t 2k t + T 2,j+1 − = T 1,j (3-540) T 1,j+1 1 + ρ c x2 ρ c x2 b1 A1,1

A1,2

2 k t k t k t T i,j+1 1+ 2 + T i−1,j+1 − 2 + T i+1,j+1 − 2 = T i,j x ρ c x ρ c x ρ c bi Ai,i

Ai,i−1

for i = 2 . . . (N − 1)

Ai,i+1

(3-541)

2 h t 2 h t 2 k t 2 k t T N,j+1 1 + + + T N−1,j+1 − = T N,j + T∞ ρ c x2 x ρ c ρ c x2 x ρ c

AN,N

AN,N−1

(3-542)

bN

The matrix A and vector b are initialized: A=spalloc(N,N,3∗ N); b=zeros(N,1);

% initialize A % initialize b

Note that the variable A is deﬁned as being a sparse matrix with at most three nonzero entries per row. The maximum number of nonzero elements can be determined based on examination of Eqs. (3-540) through (3-542). The values of the elements in the matrix A are all initialized to zero. The matrix A does not change depending on the particular time step being considered (at least for this problem with constant properties). Therefore, the matrix A can be constructed just one time and used without modiﬁcation for each time step, which saves computational effort.

442

Transient Conduction

% Setup A matrix A(1,1)=1+2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(1,2)=-2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) A(i,i)=1+2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(i,i-1)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(i,i+1)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); end A(N,N)=1+2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ DELTAtime/(DELTAx∗ rho∗ c); A(N,N-1)=-2∗ k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2);

On the other hand, the vector b does change depending on the time step because it includes the current value of the temperatures. Therefore, the vector b will need to be reconstructed before the simulation of each time step. for j=1:(M-1) % Setup b matrix for i=1:(N-1) b(i)=T(i,j); end b(N)=T(N,j)+2∗ h_bar∗ DELTAtime∗ T_inﬁnity/(DELTAx∗ rho∗ c); % Simulate time step T(:,j+1)=A\b; end

Note that T(:,j+1) is the MATLAB syntax for specifying the entire column j+1 in the matrix T. Running the script at this point will provide the matrix T, which contains the temperatures for each node (corresponding to each row of T) and each time step (corresponding to each column of T). This information can be plotted against either time or position by either typing: plot(time,T);

or plot(x,T);

in the command window, respectively. Heun’s Method Heun’s method was discussed previously in Section 3.2.2 as a simple example of a class of numerical integration methods referred to as predictor-corrector techniques. Predictorcorrector techniques take an initial predictor step (based on Euler’s technique) followed by one or more corrector steps, in which the knowledge obtained from the predictor step is used to improve the integration process. Because the predictor-corrector techniques rely on an explicit predictor step, they suffer from the same limitations related to stability that were discussed in the context of Euler’s method. The implementation of Heun’s method is illustrated using MATLAB. The problem is set up as before:

3.8 Numerical Solutions to 1-D Transient Problems

443

clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T infinity=473.2; h bar=500;

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % specific heat capacity (J/kg-K) % initial temperature (K) % fluid temperature (K) % heat transfer coefficient (W/mˆ2-K)

% Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of time steps (-) % simulation time (s) % time step duration (s)

% Initial condition for i=1:N T(i,1)=T ini; end

In order to simulate any time step j for any node i, Heun’s method begins with an Euler step to obtain an initial prediction for the temperature at the conclusion of the time step (Tˆ i,j+1 ). This ﬁrst step is referred to as the predictor step and the details are identical to Euler’s method: dT ˆ t for i = 1 . . . N (3-543) T i,j+1 = T i,j + dt T =T i,j ,t=tj The equations used to predict the time rate of change are speciﬁc to the problem being studied even though the methodology is broadly applicable. The results of the predictor step are used to carry out a subsequent corrector step. The predicted values of the temperatures at the end of the time step are used to predict the temperature rate of | ). The corrector step predicts the change at the end of the time step ( dT dt T =Tˆ i,j+1 ,t=t j + t | and temperature at the end of the time step (Ti,j+1 ) based on the average of dT dt T =T i,j ,t=t j dT | according to: dt T =Tˆ i,j+1 ,t=t j + t T i,j+1 = T i,j +

dT dT t + dt T =T i,j ,t=tj dt T =Tˆ i,j+1 ,t=tj + t 2

for i = 1 . . . N

(3-544)

444

Transient Conduction

Heun’s method is illustrated in the context of the problem that is illustrated in Figure 3-41. Equations (3-502), (3-505), and (3-508) are used to compute the time rate of change for each node at the beginning of a time step: 2k dT (3-545) = (T 2,j − T 1,j ) dt T =T 1,j ,t=tj ρ c x2 dT k = (T i−1,j + T i+1,j − 2 T i,j ) dt T =T i,j ,t=tj x2 ρ c

for i = 2 . . . (N − 1)

dT 2h 2k = (T ∞ − T N,j ) (T N−1,j − T N,j ) + 2 dt T =T N,j ,t=tj ρ c x x ρ c

(3-546)

(3-547)

The time rate of change for the temperature of each node at the beginning of the time step is stored in a temporary vector, dTdt0, which is overwritten during each time step: % Step through time for j=1:(M-1) % compute time rates of change at the beginning of the time step dTdt0(1)=2∗ k∗ (T(2,j)-T(1,j))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) dTdt0(i)=k∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))/(rho∗ c∗ DELTAxˆ2); end dTdt0(N)=2∗ k∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-T(N,j))/(rho∗ c∗ DELTAx);

A ﬁrst estimate of the temperatures at the end of the time step is obtained using Eq. (3-543); these values are also stored in a temporary vector, Ts, which is also not saved beyond the time step being simulated: % compute first estimate of temperatures at the end of the time step for i=1:N Ts(i)=T(i,j)+dTdt0(i)∗ DELTAtime; end

The results of the predictor step are used to carry out a corrector step. The time rates of change for each temperature at the end of the time step are computed using Eqs. (3-502), (3-505), and (3-508) with the time derivative evaluated using the predicted temperatures. 2k ˆ dT T 2,j+1 − Tˆ 1,j+1 (3-548) = dt T =Tˆ 1,j+1 ,t=tj + t ρ c x2 k ˆ dT = T i−1,j+1 + Tˆ i+1,j+1 − 2 Tˆ i,j+1 2 dt T =Tˆ i,j+1 ,t=tj + t x ρ c

for i = 2 . . . (N − 1) (3-549)

dT 2h 2k ˆ T N−1,j+1 − Tˆ N,j+1 + = (T ∞ − Tˆ N,j+1 ) 2 dt T =Tˆ N,j+1 ,t=tj + t ρ c x x ρ c

(3-550)

3.8 Numerical Solutions to 1-D Transient Problems

445

% compute time rates of change at the end of the time step dTdts(1)=2∗ k∗ (Ts(2)-Ts(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) dTdts(i)=k∗ (Ts(i-1)+Ts(i+1)-2∗ Ts(i))/(rho∗ c∗ DELTAxˆ2); end dTdts(N)=2∗ k∗ (Ts(N-1)-Ts(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Ts(N))/(rho∗ c∗ DELTAx);

The corrector step is based on the average of the two estimates of the time rate of change: dT dT t + for i = 1 . . . N (3-551) T i,j+1 = T i,j + dt T =T i,j ,t=tj dt T =Tˆ i,j+1 ,t=tj + t 2 % corrector step for i=1:N T(i,j+1)=T(i,j)+(dTdt0(i)+dTdts(i))∗ DELTAtime/2; end end

The numerical error associated with Heun’s method is substantially less than Euler’s method or the fully implicit method because Heun’s method is a second-order technique, as discussed in Section 3.2. Also note that if M is reduced from 21 to 11 (i.e., the time step is increased from 2.0 s to 4.0 s) then Heun’s method becomes unstable, just as Euler’s method did. Runge-Kutta 4th Order Method Heun’s method is a two-step predictor/corrector technique, which improves the accuracy of the solution but not the stability characteristics. The fourth order Runge-Kutta method involves four predictor/corrector steps and therefore improves the accuracy to an even larger extent. Because the Runge-Kutta technique is explicit, the stability characteristics of the solution are not affected. The Runge-Kutta fourth order method (or RK4 method) was previously discussed in Section 3.2.2. The RK4 technique estimates the time rate of change of the temperature of each node four times in order to simulate a single time step. (Contrast this method with Euler’s method where the time rate of change was computed only once, at the beginning of time step.) The implementation of the RK4 method is illustrated using MATLAB. The problem is setup as before: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T infinity=473.2; h bar=500;

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % specific heat capacity (J/kg-K) % initial temperature (K) % fluid temperature (K) % heat transfer coefficient (W/mˆ2-K)

446

Transient Conduction

% Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

% Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of time steps (-) % simulation time (s) % time step duration (s)

% Initial condition for i=1:N T(i,1)=T ini; end

The RK4 method begins by estimating the time rate of change of the temperature of each node at the beginning of the time step (referred to as aai ): dT for i = 1 . . . N (3-552) aai = dt T =T i,j ,t=tj or, for this problem: aa1 = aai =

2k (T 2,j − T 1,j ) ρ c x2

k (T i−1,j + T i+1,j − 2 T i,j ) x2 ρ c

aaN =

for i = 2 . . . (N − 1)

2h 2k (T ∞ − T N,j ) (T N−1,j − T N,j ) + 2 ρ c x x ρ c

(3-553)

(3-554)

(3-555)

% Step through time for j=1:(M-1) % compute the 1st estimate of the time rate of change aa(1)=2∗ k∗ (T(2,j)-T(1,j))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) aa(i)=k∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))/(rho∗ c∗ DELTAxˆ2); end aa(N)=2∗ k∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-T(N,j))/(rho∗ c∗ DELTAx);

The ﬁrst estimate of the time rate of change is used to predict the temperature of each node half-way through the time step (Tˆ i,j+ 1 ) in the ﬁrst predictor step: 2

t Tˆ i,j+ 1 = T i,j + aai 2 2

(3-556)

3.8 Numerical Solutions to 1-D Transient Problems

447

% 1st predictor step for i=1:N Ts(i)=T(i,j)+aa(i)∗ DELTAtime/2; end

The estimated temperatures are used to obtain the second estimate of the time rate of change (bbi ), this time at the midpoint of the time step: dT for i = 1 . . . N (3-557) bbi = dt T =Tˆ 1 ,t=tj + t i,j+

2

2

or, for this problem: bb1 =

bbi =

2k ˆ ˆ 1 − T 1 T 2,j+ 2 1,j+ 2 ρ c x2

k ˆ ˆ i+1,j+ 1 − 2 Tˆ i,j+ 1 1 + T T i−1,j+ 2 2 2 x2 ρ c

bbN =

(3-558)

for i = 2 . . . (N − 1)

2k ˆ 2h ˆ ˆ 1 − T 1 1 − T T T + ∞ N−1,j+ 2 N,j+ 2 N,j+ 2 ρ c x2 x ρ c

(3-559)

(3-560)

% compute the 2nd estimate of the time rate of change bb(1)=2∗ k∗ (Ts(2)-Ts(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) bb(i)=k∗ (Ts(i-1)+Ts(i+1)-2∗ Ts(i))/(rho∗ c∗ DELTAxˆ2); end bb(N)=2∗ k∗ (Ts(N-1)-Ts(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Ts(N))/(rho∗ c∗ DELTAx);

The second estimate of the time rate of change is used to obtain a new prediction of the temperature of each node, again half-way through the time step (Tˆˆ i,j+ 1 ): 2

t Tˆˆ i,j+ 1 = T i,j + bbi 2 2

for i = 1 . . . N

(3-561)

% 2nd predictor step for i=1:N Tss(i)=T(i,j)+bb(i)∗ DELTAtime/2; end

The third estimate of the time rate of change of each node (cci ) is also obtained at the mid-point of the time step: dT for i = 1 . . . N (3-562) cci = dt T =Tˆˆ 1 ,t=tj + t i,j+

2

2

or, for this problem: cc1 =

2k ˆˆ T 2,j+ 1 − Tˆˆ 1,j+ 1 2 2 2 ρ c x

(3-563)

448

Transient Conduction

cci =

k ˆˆ ˆˆ ˆˆ 1 + T 1 − 2T 1 T i−1,j+ 2 i+1,j+ 2 i,j+ 2 x2 ρ c

ccN =

for i = 2 . . . (N − 1)

2 k ˆˆ 2h ˆˆ ˆˆ 1 − T 1 1 − T T T + ∞ N−1,j+ 2 N,j+ 2 N,j+ 2 ρ c x2 x ρ c

(3-564)

(3-565)

% compute the 3rd estimate of the time rate of change cc(1)=2∗ k∗ (Tss(2)-Tss(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) cc(i)=k∗ (Tss(i-1)+Tss(i+1)-2∗ Tss(i))/(rho∗ c∗ DELTAxˆ2); end cc(N)=2∗ k∗ (Tss(N-1)-Tss(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Tss(N))/(rho∗ c∗ DELTAx);

The third estimate of the time rate of change is used to predict the temperature at the end of the time step (Tˆ i,j+1 ): Tˆ i,j+1 = T i,j + cci t

for i = 1 . . . N

(3-566)

% 3rd predictor step for i=1:N Tsss(i)=T(i,j)+cc(i)∗ DELTAtime; end

Finally, the fourth estimate of the time rate of change (ddi ) is obtained at the end of the time step: dT for i = 1 . . . N (3-567) ddi = dt Tˆ i,j+1 ,t=tj + t or, for this problem: dd1 = ddi =

2k ˆ T 2,j+1 − Tˆ 1,j+1 2 ρ c x

k ˆ T i−1,j+1 + Tˆ i+1,j+1 − 2 Tˆ i,j+1 2 x ρ c

ddN =

for i = 2 . . . (N − 1)

2k ˆ 2h T N−1,j+1 − Tˆ N,j+1 + T ∞ − Tˆ N,j+1 2 ρ c x x ρ c

(3-568) (3-569)

(3-570)

% compute the 4th estimate of the time rate of change dd(1)=2∗ k∗ (Tsss(2)-Tsss(1))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) dd(i)=k∗ (Tsss(i-1)+Tsss(i+1)-2∗ Tsss(i))/(rho∗ c∗ DELTAxˆ2); end dd(N)=2∗ k∗ (Tsss(N-1)-Tsss(N))/(rho∗ c∗ DELTAxˆ2)+2∗ h_bar∗ (T_infinity-Tsss(N))/(rho∗ c∗ DELTAx);

3.8 Numerical Solutions to 1-D Transient Problems

449

The integration through the time step is ﬁnally carried out using the weighted average of these four separate estimates of the time rate of change for each node: T i,j+1 = T i,j + (aai + 2 bbi + 2 cci + ddi)

t 6

(3-571)

% corrector step for i=1:N T(i,j+1)=T(i,j)+(aa(i)+2∗ bb(i)+2∗ cc(i)+dd(i))∗ DELTAtime/6; end end

Crank-Nicolson Method The Crank-Nicolson method combines Euler’s method with the fully implicit method. The time rate of change for each time step is estimated based on the average of its values at the beginning and the end of the time step. T i,j+1 = T i,j +

dT dT t + dt T =T i,j ,t=tj dt T =T i,j+1 ,t=tj+1 2

for i = 1 . . . N

(3-572)

Notice that Eq. (3-572) is not a predictor-corrector method, like Heun’s method, because the temperature at the end of the time step is not predicted with an explicit predictor step (i.e., there is no Tˆ i,j+1 in Eq. (3-572)). Rather, the unknown solution for the temperatures at the end of the time step (T i,j+1 ) is substituted directly into Eq. (3-572), resulting in a set of implicit equations for T i,j+1 where i = 1 to i = N. Therefore, the Crank-Nicolson technique has stability characteristics that are similar to the fully implicit method. The solution involves two estimates for the time rate of change of each node and therefore has higher order accuracy than either Euler’s method or the fully implicit technique. The Crank-Nicolson method is illustrated using MATLAB. The problem is setup as before: clear all; % Inputs L=0.05; k=5.0; rho=2000; c=200; T ini=293.2; T infinity=473.2; h bar=500; % Setup grid N=11; for i=1:N x(i)=(i-1)∗ L/(N-1); end DELTAx=L/(N-1);

% wall thickness (m) % conductivity (W/m-K) % density (kg/mˆ3) % specific heat capacity (J/kg-K) % initial temperature (K) % fluid temperature (K) % heat transfer coefficient (W/mˆ2-K)

% number of nodes (-) % position of each node (m) % distance between adjacent nodes (m)

450

Transient Conduction

% Setup time steps M=81; t sim=40; DELTAtime=t sim/(M-1); for j=1:M time(j)=(j-1)∗ DELTAtime; end

% number of time steps (-) % simulation time (s) % time step duration (s)

% Initial conditions for i=1:N T(i,1)=T ini; end

Substituting Eqs. (3-502), (3-505), and (3-508) into Eq. (3-572) leads to:

T 1,j+1

2k 2k t = T 1,j + (T 2,j − T 1,j ) + (T 2,j+1 − T 1,j+1 ) 2 2 ρ c x ρ c x 2

T i,j+1 = T i,j +

(3-573)

k k t + + T − 2 T + T − 2 T ) ) (T (T i−1,j i+1,j i,j i−1,j+1 i+1,j+1 i,j+1 x2 ρ c x2 ρ c 2 for i = 2 . . . (N − 1) (3-574)

⎡

T N,j+1

⎤ 2h 2k + + − T − T (T ) ) (T N−1,j N,j ∞ N,j ⎢ ρ c x2 ⎥ t x ρ c ⎥ = T N,j + ⎢ ⎣ 2k ⎦ 2 2h + − T − T (T ) ) (T N−1,j+1 N,j+1 ∞ N,j+1 ρ c x2 x ρ c

(3-575)

Equations (3-573) through (3-575) are a set of N equations in the unknown temperatures T i,j+1 for i = 1 to i = N. These equations must be placed into matrix format in order to be solved in MATLAB. This is similar to the fully implicit method and the same process is used. Equations (3-573) through (3-575) are rearranged so that the coefﬁcients multiplying the unknowns and the constants for the linear equations are clear: k t k t k t + T = T 1,j + (3-576) − T 1,j+1 1 + (T 2,j − T 1,j ) 2,j+1 ρ c x2 ρ c x2 ρ c x2 A1,1

b1

A1,2

k t k t k t T i,j+1 1 + + T + T − − i−1,j+1 i+1,j+1 x2 ρ c 2 x2 ρ c 2 x2 ρ c Ai,i

Ai,i−1

k t = T i,j + (T i−1,j + T i+1,j − 2 T i,j ) 2 x2 ρ c bi

Ai,i+1

(3-577) for i = 2 . . . (N − 1)

3.8 Numerical Solutions to 1-D Transient Problems

451

h t k t k t T N,j+1 1 + + +T N−1,j+1 − ρ c x2 x ρ c ρ c x2

AN,N−1

AN,N

(3-578) h t h t k t T∞ = T N,j + (T N−1,j − T N,j ) + (T ∞ − T N,j ) + ρ c x2 x ρ c x ρ c bN

The variables A and b are initialized: A=spalloc(N,N,3∗ N); b=zeros(N,1);

% initialize A % initialize b

The matrix A does not depend on the time step and can therefore be constructed just once: % Setup A matrix A(1,1)=1+k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(1,2)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) A(i,i)=1+k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2); A(i,i-1)=-k∗ DELTAtime/(2∗ rho∗ c∗ DELTAxˆ2); A(i,i+1)=-k∗ DELTAtime/(2∗ rho∗ c∗ DELTAxˆ2); end A(N,N)=1+k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2)+h_bar∗ DELTAtime/(DELTAx∗ rho∗ c); A(N,N-1)=-k∗ DELTAtime/(rho∗ c∗ DELTAxˆ2);

while the vector b must be reconstructed during each time step: for j=1:(M-1) % Setup b matrix b(1)=T(1,j)+k∗ DELTAtime∗ (T(2,j)-T(1,j))/(rho∗ c∗ DELTAxˆ2); for i=2:(N-1) b(i)=T(i,j)+k∗ DELTAtime∗ (T(i-1,j)+T(i+1,j)-2∗ T(i,j))/(2∗ rho∗ c∗ DELTAxˆ2); end b(N)=T(N,j)+k∗ DELTAtime∗ (T(N-1,j)-T(N,j))/(rho∗ c∗ DELTAxˆ2)+ . . . h_bar∗ DELTAtime∗ (T_infinity-T(N,j))/(DELTAx∗ rho∗ c)+ . . . h_bar∗ DELTAtime∗ T_infinity/(DELTAx∗ rho∗ c); % Simulate time step T(:,j+1)=A\b; end

The numerical error associated with the Crank-Nicolson technique is substantially less than either Euler’s method or the fully implicit method and it retains the stability characteristics of the fully implicit technique. The Crank-Nicolson technique is likely to be the most attractive option for many problems due to its superior accuracy combined with its stability.

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Transient Conduction

EES’ Integral Command The EES Integral function was introduced in Section 3.2.2 and used to solve lumped capacitance problems. EES is used in this section to illustrate Euler’s method and the fully implicit method. However, these are not optimal methods for using EES to solve 1-D transient problems. Instead, the Integral command provides a more powerful and computationally efﬁcient method to solve this class of problem. The Integral command implements a third order accurate integration scheme that uses an (optional) adaptive step-size in order to minimize computation time while maintaining accuracy. The problem is set up in EES by entering the inputs: $UnitSystem SI MASS RAD PA K J $TABSTOPS 0.2 0.4 0.6 0.8 3.5 in “Inputs” L=5 [cm]∗ convert(cm,m) k=5.0 [W/m-K] rho=2000 [kg/mˆ3] c=200 [J/kg-K] T ini=converttemp(C,K,20 [C]) T infinity=converttemp(C,K,200 [C]) h bar=500 [W/mˆ2-K] t sim=40 [s]

“wall thickness” “conductivity” “density” “specific heat capacity” “initial temperature” “fluid temperature” “heat transfer coefficient” “simulation time”

and the spatial grid is setup: “Setup grid” N=6 [-] duplicate i=1,N x[ ]=(i-1)∗ L/(N-1) end DELTAx=L/(N-1)

“number of nodes” “position of each node” “distance between adjacent nodes”

The EES Integral command requires four arguments (with an optional ﬁfth argument to specify the integration step size): F=INTEGRAL(Integrand,VarName,LowerLimit,UpperLimit, Stepsize)

where Integrand is the EES variable or expression that is be integrated (which, in this case, is each of the time derivatives of the nodal temperatures), VarName is the integration variable (time), LowerLimit and UpperLimit deﬁne the limits of integration (0 and t_sim), and Stepsize is the optional size of the time step. The solution to our example problem is obtained by replacing Integrand with the variable that speciﬁes the time rate of change of the temperature for each of the nodes; i.e., Eqs. (3-502), (3-505), and (3-508). To solve this problem using the Integral command, it is ﬁrst necessary to set up the equations that calculate the time rates of temperature change within a one-dimensional array. Note that two-dimensional arrays are not needed when the problem is solved in this manner, since EES will automatically keep track of the integrated values as they change with time. The use of the Integral

3.8 Numerical Solutions to 1-D Transient Problems

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command requires many fewer variables and less computation time than the numerical schemes presented in earlier sections require when they are implemented in EES. “time rate of change” dTdt[1]=2∗ k∗ (T[2]-T[1])/(rho∗ c∗ Deltaxˆ2) “node 1” duplicate i=2,(N-1) “internal nodes” dTdt[i]=k∗ (T[i-1]+T[i+1]-2∗ T[i])/(rho∗ c∗ Deltaxˆ2) end dTdt[N]=2∗ k∗ (T[N-1]-T[N])/(rho∗ c∗ DELTAxˆ2)+2∗ h bar∗ (T infinity-T[N])/(rho∗ c∗ DELTAx) “node N”

The following code integrates the time rates of change for each node forward through time: “integrate using the INTEGRAL command” duplicate i=1,N T[i]=T_ini+INTEGRAL(dTdt[i],time,0,t_sim) end

Note that after the calculations are completed, the array T[i] will provide the temperature of each node at the end of the time that is simulated, i.e., at time t = tsim . The time variation in the temperature at each node can be recorded by including the $IntegralTable directive in the ﬁle. The format of the $IntegralTable directive is: $IntegralTable VarName:Interval, x,y,z

where VarName is the integration variable (time), Interval is the output step size in the integration variable at which results will be placed in the integral table (which is completely i