HEAT CONDUCTION

HEAT CONDUCTION

Third Edition

DAVID W. HAHNM. NECATI OZISIK

JOHN WILEY & SONS, INC.

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Library of Congress Cataloging-in-Publication Data:

Hahn, David W., 1964-Heat conduction.—3rd ed. / David W. Hahn.

p. cm.Rev. ed. of: Heat conduction / M. Necati Ozisik. 2nd ed. c1993.Includes bibliographical references and index.

ISBN 978-0-470-90293-6 (hardback : acid-free paper); ISBN 978-1-118-32197-3 (ebk); ISBN 978-1-118-32198-0 (ebk); ISBN 978-1-118-33011-1(ebk); ISBN 978-1-118-33285-6 (ebk); ISBN 978-1-118-33450-8 (ebk);ISBN 978-1-118-41128-5

1. Heat–Conduction. I. Title.QC321.O34 2012621.402′2–dc23

2011052322

Printed in the United States of America10 9 8 7 6 5 4 3 2 1

To Allison-DWH

To Gul-MNO

CONTENTS

Preface xiii

Preface to Second Edition xvii

1 Heat Conduction Fundamentals 1

1-1 The Heat Flux, 21-2 Thermal Conductivity, 41-3 Differential Equation of Heat Conduction, 61-4 Fourier’s Law and the Heat Equation in Cylindrical and Spherical

Coordinate Systems, 141-5 General Boundary Conditions and Initial Condition for the Heat

Equation, 161-6 Nondimensional Analysis of the Heat Conduction Equation, 251-7 Heat Conduction Equation for Anisotropic Medium, 271-8 Lumped and Partially Lumped Formulation, 29References, 36Problems, 37

2 Orthogonal Functions, Boundary Value Problems, and theFourier Series 40

2-1 Orthogonal Functions, 402-2 Boundary Value Problems, 412-3 The Fourier Series, 602-4 Computation of Eigenvalues, 632-5 Fourier Integrals, 67

vii

viii CONTENTS

References, 73Problems, 73

3 Separation of Variables in the Rectangular Coordinate System 75

3-1 Basic Concepts in the Separation of Variables Method, 753-2 Generalization to Multidimensional Problems, 853-3 Solution of Multidimensional Homogenous Problems, 863-4 Multidimensional Nonhomogeneous Problems: Method of

Superposition, 983-5 Product Solution, 1123-6 Capstone Problem, 116References, 123Problems, 124

4 Separation of Variables in the Cylindrical Coordinate System 128

4-1 Separation of Heat Conduction Equation in the CylindricalCoordinate System, 128

4-2 Solution of Steady-State Problems, 1314-3 Solution of Transient Problems, 1514-4 Capstone Problem, 167References, 179Problems, 179

5 Separation of Variables in the Spherical Coordinate System 183

5-1 Separation of Heat Conduction Equation in the SphericalCoordinate System, 183

5-2 Solution of Steady-State Problems, 1885-3 Solution of Transient Problems, 1945-4 Capstone Problem, 221References, 233Problems, 233Notes, 235

6 Solution of the Heat Equation for Semi-Infiniteand Infinite Domains 236

6-1 One-Dimensional Homogeneous Problems in a Semi-InfiniteMedium for the Cartesian Coordinate System, 236

6-2 Multidimensional Homogeneous Problems in a Semi-InfiniteMedium for the Cartesian Coordinate System, 247

6-3 One-Dimensional Homogeneous Problems in An Infinite Mediumfor the Cartesian Coordinate System, 255

6-4 One-Dimensional homogeneous Problems in a Semi-InfiniteMedium for the Cylindrical Coordinate System, 260

CONTENTS ix

6-5 Two-Dimensional Homogeneous Problems in a Semi-InfiniteMedium for the Cylindrical Coordinate System, 265

6-6 One-Dimensional Homogeneous Problems in a Semi-InfiniteMedium for the Spherical Coordinate System, 268

References, 271Problems, 271

7 Use of Duhamel’s Theorem 273

7-1 Development of Duhamel’s Theorem for ContinuousTime-Dependent Boundary Conditions, 273

7-2 Treatment of Discontinuities, 2767-3 General Statement of Duhamel’s Theorem, 2787-4 Applications of Duhamel’s Theorem, 2817-5 Applications of Duhamel’s Theorem for Internal Energy

Generation, 294References, 296Problems, 297

8 Use of Green’s Function for Solution of HeatConduction Problems 300

8-1 Green’s Function Approach for Solving Nonhomogeneous TransientHeat Conduction, 300

8-2 Determination of Green’s Functions, 3068-3 Representation of Point, Line, and Surface Heat Sources with Delta

Functions, 3128-4 Applications of Green’s Function in the Rectangular Coordinate

System, 3178-5 Applications of Green’s Function in the Cylindrical Coordinate

System, 3298-6 Applications of Green’s Function in the Spherical Coordinate

System, 3358-7 Products of Green’s Functions, 344References, 349Problems, 349

9 Use of the Laplace Transform 355

9-1 Definition of Laplace Transformation, 3569-2 Properties of Laplace Transform, 3579-3 Inversion of Laplace Transform Using the Inversion Tables, 3659-4 Application of the Laplace Transform in the Solution of

Time-Dependent Heat Conduction Problems, 3729-5 Approximations for Small Times, 382References, 390Problems, 390

x CONTENTS

10 One-Dimensional Composite Medium 393

10-1 Mathematical Formulation of One-Dimensional Transient HeatConduction in a Composite Medium, 393

10-2 Transformation of Nonhomogeneous Boundary Conditions intoHomogeneous Ones, 395

10-3 Orthogonal Expansion Technique for Solving M-LayerHomogeneous Problems, 401

10-4 Determination of Eigenfunctions and Eigenvalues, 40710-5 Applications of Orthogonal Expansion Technique, 41010-6 Green’s Function Approach for Solving Nonhomogeneous

Problems, 41810-7 Use of Laplace Transform for Solving Semi-Infinite and Infinite

Medium Problems, 424References, 429Problems, 430

11 Moving Heat Source Problems 433

11-1 Mathematical Modeling of Moving Heat Source Problems, 43411-2 One-Dimensional Quasi-Stationary Plane Heat Source Problem, 43911-3 Two-Dimensional Quasi-Stationary Line Heat Source Problem, 44311-4 Two-Dimensional Quasi-Stationary Ring Heat Source Problem, 445References, 449Problems, 450

12 Phase-Change Problems 452

12-1 Mathematical Formulation of Phase-Change Problems, 45412-2 Exact Solution of Phase-Change Problems, 46112-3 Integral Method of Solution of Phase-Change Problems, 47412-4 Variable Time Step Method for Solving Phase-Change Problems:

A Numerical Solution, 47812-5 Enthalpy Method for Solution of Phase-Change Problems:

A Numerical Solution, 484References, 490Problems, 493Note, 495

13 Approximate Analytic Methods 496

13-1 Integral Method: Basic Concepts, 49613-2 Integral Method: Application to Linear Transient Heat Conduction

in a Semi-Infinite Medium, 49813-3 Integral Method: Application to Nonlinear Transient Heat

Conduction, 508

CONTENTS xi

13-4 Integral Method: Application to a Finite Region, 51213-5 Approximate Analytic Methods of Residuals, 51613-6 The Galerkin Method, 52113-7 Partial Integration, 53313-8 Application to Transient Problems, 538References, 542Problems, 544

14 Integral Transform Technique 547

14-1 Use of Integral Transform in the Solution of Heat ConductionProblems, 548

14-2 Applications in the Rectangular Coordinate System, 55614-3 Applications in the Cylindrical Coordinate System, 57214-4 Applications in the Spherical Coordinate System, 58914-5 Applications in the Solution of Steady-state problems, 599References, 602Problems, 603Notes, 607

15 Heat Conduction in Anisotropic Solids 614

15-1 Heat Flux for Anisotropic Solids, 61515-2 Heat Conduction Equation for Anisotropic Solids, 61715-3 Boundary Conditions, 61815-4 Thermal Resistivity Coefficients, 62015-5 Determination of Principal Conductivities and Principal Axes, 62115-6 Conductivity Matrix for Crystal Systems, 62315-7 Transformation of Heat Conduction Equation for Orthotropic

Medium, 62415-8 Some Special Cases, 62515-9 Heat Conduction in an Orthotropic Medium, 62815-10 Multidimensional Heat Conduction in an Anisotropic Medium, 637References, 645Problems, 647Notes, 649

16 Introduction to Microscale Heat Conduction 651

16-1 Microstructure and Relevant Length Scales, 65216-2 Physics of Energy Carriers, 65616-3 Energy Storage and Transport, 66116-4 Limitations of Fourier’s Law and the First Regime of Microscale

Heat Transfer, 66716-5 Solutions and Approximations for the First Regime of Microscale

Heat Transfer, 672

xii CONTENTS

16-6 Second and Third Regimes of Microscale Heat Transfer, 67616-7 Summary Remarks, 676References, 676

APPENDIXES 679

Appendix I Physical Properties 681

Table I-1 Physical Properties of Metals, 681Table I-2 Physical Properties of Nonmetals, 683Table I-3 Physical Properties of Insulating Materials, 684

Appendix II Roots of Transcendental Equations 685

Appendix III Error Functions 688

Appendix IV Bessel Functions 691

Table IV-1 Numerical Values of Bessel Functions, 696Table IV-2 First 10 Roots of Jn(z) = 0, n = 0, 1, 2, 3, 4, 5, 704Table IV-3 First Six Roots of βJ1(β) − cJ0(β) = 0, 705Table IV-4 First Five Roots of J0(β)Y0(cβ) − Y0(β)J0(cβ) = 0, 706

Appendix V Numerical Values of Legendre Polynomials of theFirst Kind 707

Appendix VI Properties of Delta Functions 710

Index 713

PREFACE

The decision to take on the third edition of Professor Ozisik’s book was notone that I considered lightly. Having taught from the second edition for morethan a dozen years and to nearly 500 graduate students, I was intimately familiarwith the text. For the last few years I had considered approaching ProfessorOzisik with the idea for a co-authored third edition. However, with his passingin October 2008 at the age of 85, before any contact between us, I was facedwith the decision of moving forward with a revision on my own. Given my deepfamiliarity and appreciation for Professor Ozisik’s book, it was ultimately aneasy decision to proceed with the third edition.

My guiding philosophy to the third edition was twofold: first, to preserve thespirit of the second edition as the standard contemporary analytic treatment ofconduction heat transfer, and, second, to write a truly major revision with goalsto improve and advance the presentation of the covered material. The feedbackfrom literally hundreds of my students over the years combined with my ownpedagogical ideas served to shape my overall approach to the revision. At the endof this effort, I sincerely feel that the result is a genuine collaboration betweenme and Professor Ozisik, equally combining our approaches to conduction heattransfer. As noted in the second edition, this book is meant to serve as a graduate-level textbook on conduction, as well as a comprehensive reference for practicingengineers and scientists. The third edition remains true to these goals.

I will now attempt to summarize the changes and additions to the third edition,providing some commentary and motivating thoughts. Chapter 1 is very muchin the spirit of a collaborative effort, combining the framework from the secondedition with significant revision. Chapter 1 presents the concepts of conductionstarting with the work of Fourier and providing a detailed derivation of the

xiii