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

xiv PREFACE

heat equation. We now present both differential and integral derivations, whichtogether present additional insight into the governing heat equation, notably withregard to conservation of energy. Extension of the heat equation to other coor-dinate systems, partial lumping of the heat equation, and the detailed treatmentof relevant boundary conditions complete the chapter.

Chapter 2 is completely new in the third edition, presenting the characteristicvalue problem (the Sturm–Liouville problem) and the concept of orthogonalfunctions in considerable detail. We then develop the trigonometric orthogonalfunctions, Bessel functions, and the Legendre polynomials, followed by a rigorouspresentation of the Fourier series, and finally the Fourier integrals. This materialwas dispersed in Chapters 2–4 in the second edition. The current unified treatmentis envisioned to provide a more comprehensive foundation for the followingchapters, while avoiding discontinuity through dispersion of the material overmany chapters.

Chapters 3, 4, and 5 present the separation of variables method for Cartesian,cylindrical, and spherical coordinate systems, respectively. The organization dif-fers from the second edition in that we first emphasize the steady-state solutionsand then the transient solutions, with the concept of superposition presented inChapter 3. In particular, the superposition schemes are presented using a moresystematic approach and added illustrative figures. The many tables of charac-teristic value problems and resulting eigenfunctions and eigenvalues are retainedin this edition in Chapter 2, although the approach from the second editionof developing more generic solutions in conjunction with these tables was de-emphasized. Finally, each of these chapters now ends with a capstone exampleproblem, attempting to illustrate the numerical implementation of representativeanalytic solutions, with goals of discussing the numerical convergence of real-ization of our analytic solutions, as well as emphasizing the concepts of energyconservation. A significant change in the third edition was the removal of allsemi-infinite and infinite domain material from these three chapters and consoli-dating this material into Chapter 6, a new chapter dedicated to the semi-infiniteand infinite domain problems in the context of the Fourier integral. Pedagogi-cally, it was felt that this topic was better suited to a dedicated chapter, ratherthan treated along with separation of variables. Overall, the revised treatment inChapters 2–6 will hopefully assist students in learning the material, while alsogreatly improving the utility of the third edition as a reference book. Additionalhomework problems have been added throughout.

Chapters 7, 8, and 9 focus on the treatment of nonhomogeneities in heat con-duction problems, notably time-dependent nonhomogeneities, using Duhamel’stheorem, the Green’s function approach, and the Laplace transform method,respectively. A new derivation of Duhamel’s theorem is presented along witha new presentation of Duhamel’s various solutions that is intended to clarifythe use and limitations of this method. Also included in Chapter 7 is a newtable giving closed-form solutions for various surface temperature functions. TheGreen’s function chapter contains new sample problems with a greater varietyof boundary condition types. The Laplace transform chapter contains additional

PREFACE xv

problems focusing on the general solution method (i.e., not limited to the small-time approximations), and the Laplace transform tables are greatly expanded tofacilitate such solutions. Together, Chapters 1–9 are considered the backboneof a graduate course on conduction heat transfer, with the remainder of the textproviding additional topics to pursue depending on the scope of the class and theinterests of the instructor and students.

Chapters 10, 11, and 12 cover the one-dimensional composite medium, themoving heat source problem, and phase-change heat transfer, respectively, muchalong the lines of the original treatment by Professor Ozisik, although effortswere made to homogenize the style with the overall revision. Chapter 13 is onapproximate analytic methods and takes a departure from the exact analytic treat-ment of conduction presented in the first dozen chapters. The emphasis on theintegral method and method of residuals has applications to broader techniques(e.g., finite-element methods), although efforts are made in the current text toemphasize conservation of energy and formulation in the context of these meth-ods. Chapter 14 presents the integral transform technique as a means for solutionof the heat equation under a variety of conditions, setting the foundation foruse of the integral transform technique in Chapter 15, which focuses on heatconduction in anisotropic solids.

Chapter 16 is a totally new contribution to the text and presents an introduc-tion to microscale heat conduction. There were several motivating factors forthe inclusion of Chapter 16. First, engineering and science fields are increas-ingly concerned with the micro- and nanoscales, including with regard to energytransfer. Second, given that the solution of the heat equation in conjunction withFourier’s law remains at the core of this book, it is useful to provide a succincttreatment of the limitations of these equations.

The above material considerably lengthened the manuscript with regard to thecorresponding treatment in the second edition, and therefore, some material wasomitted from the revised edition. It is here that I greatly missed the opportunityto converse with Professor Ozisik, but ultimately, such decisions were minealone. The lengthy treatment of finite-difference methods and the chapter oninverse heat conduction from the second edition were omitted from the thirdedition. With regard to the former, the logic was that the strength of this text isthe analytic treatment of heat conduction, noting that many texts on numericalmethods are available, including treatment of the heat equation. The latter topichas been the subject of entire monographs, and, therefore, the brief treatment ofinverse conduction was essentially replaced with the introduction to microscaleheat transfer.

I would like to express my sincere gratitude to the many heat transfer studentsthat have provided me with their insight into a student’s view of heat conductionand shared their thoughts, both good and bad, on the second edition. You remainmy primary motivation for taking on this project. A few of my former studentshave reviewed chapters of the third edition, and I would like to express my thanksto Fotouh Al-Ragom, Roni Plachta, Lawrence Stratton and Nadim Zgheib for theirfeedback. I would also like to thank my graduate student Philip Jackson for his

xvi PREFACE

invaluable help with the numerical implementation of the Legendre polynomialsfor the capstone problem in Chapter 5. I would like to acknowledge several ofmy colleagues at the University of Florida, including Andreas Haselbacher foruseful discussions and feedback regarding Chapter 1, Renwei Mei for his reviewsof several chapters and his very generous contribution of Table 1 in Chapter 7,Greg Sawyer for his suggestion of the capstone problem in Chapter 5, as wellas for many motivating discussions, and Simon Phillpot for his very insightfulreview of Chapter 16. I would like to thank my editors at John Wiley & Sons,Daniel Magers and Bob Argentieri, for their support and enthusiasm from thevery beginning, as well as for their generous patience as this project nearedcompletion.

On a more personal note, I want to thank my wonderful children, Katherine,William, and Mary-Margaret, for their support throughout this process. Many ofthe hours that I dedicated to this manuscript came at their expense, and I willnever forget their patience with me to the very end. I also thank Mary-Margaretfor her admirable proof-reading skills when needed. The three of you never letme down. Finally, I thank the person that gave me the utmost support throughoutthis project, my wife and dearest partner in life, Allison. This effort would neverhave been completed without your unwavering support, and it is to you that Idedicate this book.

David W. HahnGainesville, Florida

PREFACE TO SECOND EDITION

In preparing the second edition of this book, the changes have been motivatedby the desire to make this edition a more application-oriented book than the firstone in order to better address the needs of the readers seeking solutions to heatconduction problems without going through the details of various mathematicalproofs. Therefore, emphasis is placed on the understanding and use of variousmathematical techniques needed to develop exact, approximate, and numericalsolutions for a broad class of heat conduction problems. Every effort has beenmade to present the material in a clear, systematic, and readily understandablefashion. The book is intended as a graduate-level textbook for use in engineeringschools and a reference book for practicing engineers, scientists and researchers.To achieve such objectives, lengthy mathematical proofs and developments havebeen omitted, instead examples are used to illustrate the applications of varioussolution methodologies.

During the twelve years since the publication of the first edition of this book,changes have occurred in the relative importance of some of the applicationareas and the solution methodologies of heat conduction problems. For example,in recent years, the area of inverse heat conduction problems (IHCP) associatedwith the estimation of unknown thermophysical properties of solids, surface heattransfer rates, or energy sources within the medium has gained significant impor-tance in many engineering applications. To answer the needs in such emergingapplication areas, two new chapters are added, one on the theory and applicationof IHCP and the other on the formulation and solution of moving heat sourceproblems. In addition, the use of enthalpy method in the solution of phase-changeproblems has been expanded by broadening its scope of applications. Also, thechapters on the use of Duhamel’s method, Green’s function, and finite-difference

xvii

xviii PREFACE TO SECOND EDITION

methods have been revised in order to make them application-oriented. Green’sfunction formalism provides an efficient, straightforward approach for developingexact analytic solutions to a broad class of heat conduction problems in the rect-angular, cylindrical, and spherical coordinate systems, provided that appropriateGreen’s functions are available. Green’s functions needed for use in such formalsolutions are constructed by utilizing the tabulated eigenfunctions, eigenvaluesand the normalization integrals presented in the tables in Chapters 2 and 3.

Chapter 1 reviews the pertinent background material related to the heat conduc-tion equation, boundary conditions, and important system parameters. Chapters 2,3, and 4 are devoted to the solution of time-dependent homogeneous heat conduc-tion problems in the rectangular, cylindrical, and spherical coordinates, respec-tively, by the application of the classical method of separation of variables andorthogonal expansion technique. The resulting eigenfunctions, eigenconditions,and the normalization integrals are systematically tabulated for various combina-tions of the boundary conditions in Tables 2-2, 2-3, 3-1, 3-2, and 3-3. The resultsfrom such tables are used to construct the Green functions needed in solutionsutilizing Green’s function formalism.

Chapters 5 and 6 are devoted to the use of Duhamel’s method and Green’sfunction, respectively. Chapter 7 presents the use of Laplace transform techniquein the solution of one-dimensional transient heat conduction problems.

Chapter 8 is devoted to the solution of one-dimensional, time-dependent heatconduction problems in parallel layers of slabs and concentric cylinders andspheres. A generalized orthogonal expansion technique is used to solve thehomogeneous problems, and Green’s function approach is used to generalizethe analysis to the solution of problems involving energy generation.

Chapter 9 presents approximate analytical methods of solving heat conductionproblems by the integral and Galerkin methods. The accuracy of approximateresults are illustrated by comparing with the exact solutions. Chapter 10 isdevoted to the formulation and the solution of moving heat source problems,while Chapter 11 is concerned with the exact, approximate, and numerical meth-ods of solution of phase-change problems.

Chapter 12 presents the use of finite difference methods for solving the steady-state and time-dependent heat conduction problems. Chapter 13 introduces theuse of integral transform technique in the solution of general time-dependentheat conduction equations. The application of this technique for the solutionof heat conduction problems in rectangular, cylindrical, and spherical coordi-nates requires no additional background, since all basic relationships needed forconstructing the integral transform pairs have already been developed and sys-tematically tabulated in Chapters 2 to 4. Chapter 14 presents the formulation andmethods of solution of inverse heat conduction problems and some backgroundinformation on statistical material needed in the inverse analysis. Finally, Chapter15 presents the analysis of heat conduction in anisotropic solids. A host of use-ful information, such as the roots of transcendental equations, some propertiesof Bessel functions, and the numerical values of Bessel functions and Legendrepolynomials are included in Appendixes IV and V for ready reference.

PREFACE TO SECOND EDITION xix

I would like to express my thanks to Professors J. P. Bardon and Y. Jarnyof University of Nantes, France, J. V. Beck of Michigan State University, andWoo Seung Kim of Hanyang University, Korea, for valuable discussions andsuggestions in the preparation of the second edition.

M. Necati OzisikRaleigh, North CarolinaDecember 1992

HEAT CONDUCTION

1HEAT CONDUCTIONFUNDAMENTALS

No subject has more extensive relations with the progress of industry andthe natural sciences; for the action of heat is always present, it penetrates allbodies and spaces, it influences the processes of the arts, and occurs in allthe phenomena of the universe.

—Joseph Fourier, Theorie Analytique de la Chaleur, 1822 [1]

All matter when considered at the macroscopic level has a definite and preciseenergy. Such a state of energy may be quantified in terms of a thermodynamicenergy function, which partitions energy at the atomic level among, for example,electronic, vibrational, and rotational states. Under local equilibrium, the energyfunction may be characterized by a measurable scalar quantity called tempera-ture. The energy exchanged by the constituent particles (e.g., atoms, molecules,or free electrons) from a region with a greater local temperature (i.e., greaterthermodynamic energy function) to a region with a lower local temperature iscalled heat . The transfer of heat is classically considered to take place by conduc-tion, convection, and radiation, and although it cannot be measured directly, theconcept has physical meaning because of the direct relationship to temperature.Conduction is a specific mode of heat transfer in which this energy exchangetakes place in solids or quiescent fluids (i.e., no convective motion resulting fromthe macroscopic displacement of the medium) from the region of high temper-ature to the region of low temperature due to the presence of a temperaturegradient within the system. Once the temperature distribution T (r, t) is knownwithin the medium as a function of space (defined by the position vector r) andtime (defined by scalar t), the flow of heat is then prescribed from the gov-erning heat transfer laws. The study of heat conduction provides an enriching

1

2 HEAT CONDUCTION FUNDAMENTALS

combination of fundamental science and mathematics. As the prominent ther-modynamicist H. Callen wrote: “The history of the concept of heat as a formof energy transfer is unsurpassed as a case study in the tortuous developmentof scientific theory, as an illustration of the almost insuperable inertia presentedby accepted physical doctrine, and as a superb tale of human ingenuity appliedto a subtle and abstract problem” [2]. The science of heat conduction is princi-pally concerned with the determination of the temperature distribution and flowof energy within solids. In this chapter, we present the basic laws relating theheat flux to the temperature gradient in the medium, the governing differentialequation of heat conduction, the boundary conditions appropriate for the analysisof heat conduction problems, the rules of coordinate transformation needed forworking in different orthogonal coordinate systems, and a general discussion ofthe various solution methods applicable to the heat conduction equation.

1-1 THE HEAT FLUX

Laws of nature provide accepted descriptions of natural phenomena based onobserved behavior. Such laws are generally formulated based on a large bodyof empirical evidence accepted within the scientific community, although theyusually can be neither proven nor disproven. To quote Joseph Fourier from theopening sentence of his Analytical Theory of Heat : “Primary causes are unknownto us; but are subject to simple and constant laws, which may be discovered byobservation” [1]. These laws are considered general laws , as their applicationis independent of the medium. Well-known examples include Newton’s laws ofmotion and the laws of thermodynamics. Problems that can be solved using onlygeneral laws of nature are referred to as deterministic and include, for example,simple projectile motion.

Other problems may require supplemental relationships in addition to thegeneral laws. Such problems may be referred to as nondeterministic, and theirsolution requires laws that apply to the specific medium in question. These addi-tional laws are referred to as particular laws or constitutive relations . Well-knownexamples include the ideal gas law, the relationship between shear stress and thevelocity gradient for a Newtonian fluid, and the relationship between stress andstrain for a linear-elastic material (Hooke’s law).

The particular law that governs the relationship between the flow of heat andthe temperature field is named after Joseph Fourier. For a homogeneous, isotropicsolid (i.e., material in which thermal conductivity is independent of direction),Fourier’s law may be given in the form

q ′′(r, t) = −k∇T (r, t) W/m2 (1-1)

where the temperature gradient ∇T (r, t) is a vector normal to the isothermalsurface, the heat flux vector q ′′(r, t) represents the heat flow per unit time, per unitarea of the isothermal surface in the direction of decreasing temperature gradient,

THE HEAT FLUX 3

and k is the thermal conductivity of the material. The thermal conductivity isa positive, scalar quantity for a homogeneous, isotropic material. The minussign is introduced in equation (1-1) to make the heat flow a positive quantityin the positive coordinate direction (i.e., opposite of the temperature gradient),as described below. This text will consider the heat flux in the SI units W/m2

and the temperature gradient in K/m (equivalent to the unit oC/m), giving thethermal conductivity the units of W/(m · K). In the Cartesian coordinate system(i.e., rectangular system), equation (1-1) is written as

q ′′(x, y, z, t) = − ik∂T

∂x− j k

∂T

∂y− kk

∂T

∂z(1-2)

where i,j , and k are the unit direction vectors along the x , y , and z directions,respectively. One may consider the three components of the heat flux vector inthe x , y , and z directions, respectively, as given by

q ′′x = −k

∂T

∂xq ′′

y = −k∂T

∂yand q ′′

z = −k∂T

∂z(1-3a,b,c)

Clearly, the flow of heat for a given temperature gradient is directly propor-tional to the thermal conductivity of the material. Equation (1-3a) is generallyused for one-dimensional (1-D) heat transfer in a rectangular coordinate system.Figure 1-1 illustrates the sign convention of Fourier’s law for the 1-D Carte-sian coordinate system. Both plots depict the heat flux (W/m2) through the planeat x = x0 based on the local temperature gradient. In Figure 1-1(a), the gradi-ent dT/dx is negative with regard to the Cartesian coordinate system; hence theresulting flux is mathematically positive, and by convention is in the positive xdirection , as shown in the figure. In contrast, Figure 1-1(b) depicts a positive gra-dient dT/dx . This yields a mathematically negative heat flux, which by convention

″ ″

(a) (b)

Figure 1-1 Fourier’s law illustrated for a (a) positive heat flux and (b) a negative heatflux.

4 HEAT CONDUCTION FUNDAMENTALS

is in the negative x direction, as indicated in the figure. As defined, Fourier’s lawis directly tied to the coordinate system, with positive heat flux always flowing inthe positive coordinate direction. While determining the actual direction of heatflow is often trivial for 1-D problems, multidimensional problems, and notablytransient problems, can present considerable difficulty in determining the direc-tion of the local heat flux terms. Adherence to the sign convention of Fourier’slaw will avoid any such difficulties of flux determination, which is useful in thecontext of overall energy conservation for a given heat transfer problem.

In addition to the heat flux, which is the flow of heat per unit area normal tothe direction of flow (e.g., a plane perpendicular to the page in Fig. 1-1), one maydefine the total heat flow, often called the heat rate, in the unit of watts (W). Theheat rate is calculated by multiplying the heat flux by the total cross-sectionalarea through which the heat flows for a 1-D problem or by integrating over thearea of flow for a multidimensional problem. The heat rate in the x directionfor one-, two-, and three-dimensional (1-D, 2-D, and 3-D) Cartesian problems isgiven by

qx = −kAx

dT

dxW (1-4)

qx = −kH∫ L

y=0

∂T (x, y)

∂xdy W (1-5)

qx = −k

∫ L

y=0

∫ H

z=0

∂T (x, y, z)

∂xdz dy W (1-6)

where Ax is the total cross-sectional area for the 1-D problem in equation (1-4).The total cross-sectional area for the 2-D problem in equation (1-5) is defined bythe surface from y = 0 to L in the second spatial dimension and by the length Hin the z direction, for which there is no temperature dependence [i.e., T �= f (z)].The total cross-sectional area for the 3-D problem in equation (1-6) is definedby the surface from y = 0 to L and z = 0 to H in the second and third spatialdimensions, noting that T = f (x, y, z).

1-2 THERMAL CONDUCTIVITY

Given the direct dependency of the heat flux on the thermal conductivity viaFourier’s law, the thermal conductivity is an important parameter in the analysisof heat conduction. There is a wide range in the thermal conductivities of variousengineering materials. Generally, the highest values are observed for pure metalsand the lowest value by gases and vapors, with the amorphous insulating materialsand inorganic liquids having thermal conductivities that lie in between. There areimportant exceptions. For example, natural type IIa diamond (nitrogen free) hasthe highest thermal conductivity of any bulk material (∼2300 W/m · K at ambient

THERMAL CONDUCTIVITY 5

temperature), due to the ability of the well-ordered crystal lattice to transmitthermal energy via vibrational quanta called phonons . In Chapter 16, we willexplore in depth the physics of energy carriers to gain further insight into thisimportant material property.

To give some idea of the order of magnitude of thermal conductivity forvarious materials, Figure 1-2 illustrates the typical range for various materialclasses. Thermal conductivity also varies with temperature and may change withorientation for nonisotropic materials. For most pure metals the thermal conduc-tivity decreases with increasing temperature, whereas for gases it increases withincreasing temperature. For most insulating materials it increases with increasingtemperature. Figure 1-3 provides the effect of temperature on the thermal conduc-tivity for a range of materials. At very low temperatures, thermal conductivityincreases rapidly and then exhibits a sharp decrease as temperatures approachabsolute zero, as shown in Figure 1-4, due to the dominance of energy carrierscattering from defects at extreme low temperatures. A comprehensive compila-tion of thermal conductivities of materials may be found in references 3–6. Wepresent in Appendix I the thermal conductivity of typical engineering materialstogether with the specific heat cp, density ρ, and the thermal diffusivity α. Theselatter properties are discussed in more detail in the following section.

Figure 1-2 Typical range of thermal conductivity of various material classes.

6 HEAT CONDUCTION FUNDAMENTALS

Figure 1-3 Effect of temperature on thermal conductivity.

1-3 DIFFERENTIAL EQUATION OF HEAT CONDUCTION

We now derive the differential equation of heat conduction, often called the heatequation , for a stationary, homogeneous, isotropic solid with heat generationwithin the body. Internal heat generation may be due to nuclear or chemicalreactivity, electrical current (i.e., Joule heating), absorption of laser light, orother sources that may in general be a function of time and/or position. The heatequation may be derived using either a differential control volume approach oran integral approach. The former is perhaps more intuitive and will be presentedfirst, while the latter approach is more general and readily extends the derivationto moving solids.