H EAT CONDUCTION

M. Necati k>Zl§Ik Department of Mechanical and Aerospace Engineering North Carolina State University, Raleigh

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY AND SONS, New York· Chichester · Brisbane ·Toronto

CoNTENTs

Chapter 1

HEAT-CONDUCTION FUNDAMENTALS

1-1 The Heat Flux, 1 1-2 The Differential Equation of Heat Conduction, 4 1-3 Heat-Conduction Equation in Different

Orthogonal Coordinate Systems, 7 1-4 Boundary Conditions, 12 1-5 Dimensionless Heat-Conduction Parameters, 15 1-6 Homogeneous and Nonhomogeneous Problems, 17 1-7 Methods of Solution of Heat-Conduction Problems, 18

References, 21 Problems, 22

Chapter 2

THE SEPARATION OF VARIABLES IN THE RECTANGULAR COORDINATE SYSTEM

2-1 Method of Separation ofVariables, 25 2-2 Separation of The Heat-Conduction Equation

in the Rectangular Coordinate System, 30 2-3 One-Dimensional Homogeneous Problems

in a Finite Medium, 32 2-4 One-Dimensional Homogeneous Problems

in a Semi-Infinite Medium, 39 2-5 One-Dimensional Homogeneous Problems

in an Infinite Medium, 43 2-6 Multidimensional Homogeneous Problems, 46 2-7 Product Solution, 54

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2-8 Multidimensional Steady-State Problems with No Heat Generation, 57

2-9 Multidimensional Steady-State Problems with Heat Generation, 66

2-10 Splitting Up of N onhomogeneous Problems into Simpler Problems, 69

2-l l Useful Transformations, 74 References, 76 Problems, 77 Notes, 79

Chapter 3

THE SEPARATION OF VARIABLES IN TIIE CYLINDRICAL COORDINATE SYSTEM

3-1 Separation of Heat-Conduction Equaticm in the Cylindrical Coordinate System, 83

3-2 Representation of an Arbitrary Function in Terms of Bessel Functions, 88

3-3 Homogeneous Problems in (r, t) Variables, 100 3-4 Homogeneous Problems in (r, z, t) Variables, 110 3-5 Homogeneous Problems in (r. </J, t) Variables, 114 3-6 Homogeneous Problems in (r, <fJ ,z, t) Variables, 123 3-7 Product Solution, 127 3-8 Multidimensional Steady-State Problems with

No Heat Generation, 129 3-9 Multidimensional Steady-State Problems with

Heat Generation, 133 3-10 Splitting Up ofNonhomogeneous Problems into

Simpler Problems, 136 References, 138 Problems, 139 Notes, 141

Chapter 4

THE SEPARATION OF VARIABLES IN TIIE SPHERICAL COORDINATE SYSTEM

4-1 Separation of The Heat-Conduction Equation in the Spherical Coordinate System, 144

CONTENTS

83

144

CONTENTS

4-2 Legendre Functions and Legendre's Associated Functions, 148

4-3 Representation of an Arbitrary Function in Terms of Legendre Functions, 154

4-4 Homogeneous Problems in (r, t) Variables, 162 4-5 Homogeneous Problems in (r, µ, t) Variables, 168 4-6 Homogeneous Problems in (r, µ, </>, t) Variables, 175 4-7 M ultidimensional Steady-State Problems, 182 4-8 Splitting Up of Nonhomogeneous Problems into

Simpler Problems, 185 References, 187 Problems, 187 Notes, 189

Chapter 5

TUE USE OF DUHAMEL'S THEOREM

5-1 The Statement of Duhamel's Theorem, 194 5-2 A Proof of Duhamel 's Theorem, 197 5-3 Applications of Duhamel's Theorem, 199

References, 206 Problems, 206 Notes, 208

Chapter 6

TUE USE OF GREEN'S FUNCTION

6-1 Green's Function in the Solution ofNonhomogeneous, , Time-Dependent Heat-Conduction Problems, 209

6-2 Determination of Green's Function, 216 6-3 Application of Green's Function in the Rectangular

Coordinate System, 219 6-4 Applications of Green's Function in the Cylindrical

Coordinate System, 226 6-5 Applications of Green's Function in the Spherical

Coordinate System, 232 6-6 Product of Green's Functions, 239

References, 240 Problems, 240 Notes, 245

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Chapter 7

CONTENTS

THE USE OF LAPLACE TRANSFORM

7-1 Definition of Laplace Transformation, 246 7-2 Properties of Laplace Transform, 248 7-3 The Inversion of Laplace Transform Using the

Inversion Tables, 258 7-4 The Inversion of Laplace Transform by the

Contour Integration Technique, 263 7-5 Application ofLaplace Transform in the Solution of

Time-Dependent Heat-Conduction Problems, 273 7-6 Approximations for Small and Large Times, 283

References, 290 Problems, 290 Notes, 292

Chapter 8

ONE-DIMENSIONAL COMPOSITE MEDIUM

8-1 Solution of the Homogeneous Problem by the Generalized Orthogonal Expansion Technique, 295

8-2 Determination of Eigenfunctions and Eigenvalues, 300 8-3 Transformation of Nonhomogeneous Outer Boundary

Conditions into Homogeneous Ones, 311 8-4 The Use of Green's Functions in the Solution of

Nonhomogeneous Problems, 317 8-5 The Use of Laplace Transformation, 323

Ref erences, 328 Problems, 329 Notes, 331

Chapter 9

APPROXIMATE ANALYTICAL METHODS

9-1 The Integral Method-Basic Concepts, 335 9-2 The Integral Method-Various Applications, 341 9-3 The Variational Principles, 358 9-4 The Ritz Method, 367 9-5 The Galerkin Method, 372

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335

CONTENTS

9-6 Partial Integration, 380 9-7 Time-Dependent Problems, 386

References, 391 Problems, 393 Notes, 395

Chapter 10

PHASE-CHANGE PROBLEMS

10-1 Boundary Conditions at the Moving Interface, 399 10-2 Exact Solution of Phase-Change Problems, 406 10-3 Integral Method of Solution of Phase-Change Problems, 416 10-4 Moving Heat Source Method for the Solution of

Phase-Change Problems, 423 10-5 Phase Change over a Temperature Range, 430

References, 432 Problems, 434 Notes, 435

Chapter 11

NONLINEAR PROBLEMS

11-1 Transformation of a Dependent Variable-The Kirchhoff Transformation, 440

11-2 Linearization of a One-Dimensional Nonlinear Heat-Conduction Problem, 443

11-3 Transformation of an Independent Variable-The Boltzmann transformation, 448

11-4 Similarity Transformation via One-Parameter Group Theory, 452

11-5 Transformation into Integral Equation, 460 ' References, 464

Problems, 466 Notes, 468

Chapter 12

NUMERICAL METHODS OF SOLUTION

12-1 Finite Diff erence Approximation of Derivatives Through Taylor's Series, 471

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12-2 Finite-Difference Representation of Steady-State Heat-Conduction Problems, 477

12-3 Methods ofSolving Simultaneous Linear Algebraic Equations, 484

12-4 Errors Involved in Numerical Solutions, 486 12-5 Finite Difference Representation ofTime-Dependent

Heat-Conduction Equation, 487 12-6 Applications of Finite-Difference Methods to

Time-Dependent Heat Conduction Problems, 496 12-7 Finite Diff erence in Cylindrical and Spherical

Coordinate Systems, 503 12-8 Variable Thermal Properties, 511 12-9 Curved Boundaries, 513

References, 516 Problems, 518

Cbapter 13

INTEGRAL-TRANSFORM TECHNIQUE

13-1 The Use oflntegral Transform in the Solution of Heat-Conduction Problems in Finite Regions, 523

13-2 Alternative Form of General Solution for Finite Regions, 532 13-3 Applications in the Rectangular Coordinate System, 536 13-4 Applications in the Cylindrical Coordinate System, 551 13-5 Applications in the Spherical Coordinate System, 568 13-6 Applications in the Solution of Steady-State Problems, 579

References, 582 Problems, 583 Notes, 587

Cbapter 14

INTEGRAL-TRANSFORM TECHNIQUE FOR COMPOSITE MEDIUM

14-1 The Use of Integral Transform in the Solution of Heat-Conduction Problems in Finite Composite Regions, 594

14-2 One-Dimensional Case, 601 References, 607 Problems, 608 Notes, 608

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Chapter 15

HEAT CONDUCTION IN ANISOTROPIC MEDIUM

15-1 Heat Flux for Anisotropie Solids, 612 15-2 Heat-Conduetion Equation for Anisotropie Solids, 614 15-3 Boundary Conditions, 615 15-4 Thermal-Resistivity Coefficients, 617 15-5 Transformation of Axes and Conduetivity Coeffieients, 618 15-6 Geometrieal Interpretation of Conduetivity Coefficients, 620 15-7 The Symmetry of Crystals, 625 15-8 One-Dimensional Steady-State Heat Conduetion in

Anisotropie Solids, 626 15-9 One-Dimensional Time-Dependent Heat Conduetion in

Anisotropie Solids, 629 15-10 Heat Conduetion in an Orthotropie Medium, 631 15-11 Multidimensional Heat Conduetion in an

Anisotropie Medium, 638 References, 646 Problems, 647 Notes, 649

Appendices 651

Appendix I Appendix II Appendix III Appendix IV

Index

Roots of Transeendental Equations, 653 Error Funetions, 656 Bessel Funetions, 659 Numerieal Values of Legendre Polynomials of the First Kind, 674

611

679