Preface xiii
Preface to Second Edition xvii
1 Heat Conduction Fundamentals 1
1-1 The Heat Flux 2
1-2 Thermal Conductivity 4
1-3 Differential Equation of Heat Conduction 6
1-4 Fourier's Law and the Heat Equation in Cylindrical and Spherical Coordinate Systems 14
1-5 General Boundary Conditions and Initial Condition for the Heat Equation 16
1-6 Nondimensional Analysis of the Heat Conduction Equation 25
1-7 Heat Conduction Equation for Anisotropic Medium 27
1-8 Lumped and Partially Lumped Formulation 29
References 36
Problems 37
2 Orthogonal Functions, Boundary Value Problems, and the Fourier Series 40
2-1 Orthogonal Functions 40
2-2 Boundary Value Problems 41
2-3 The Fourier Series 60
2-4 Computation of Eigenvalues 63
2-5 Fourier Integrals 67
References 73
Problems 73
3 Separation of Variables in the Rectangular Coordinate System 75
3-1 Basic Concepts in the Separation of Variables Method 75
3-2 Generalization to Multidimensional Problems 85
3-3 Solution of Multidimensional Homogenous Problems 86
3-4 Multidimensional Nonhomogeneous Problems: Method of Superposition 98
3-5 Product Solution 112
3-6 Capstone Problem 116
References 123
Problems 124
4 Separation of Variables in the Cylindrical Coordinate System 128
4-1 Separation of Heat Conduction Equation in the Cylindrical Coordinate System 128
4-2 Solution of Steady-State Problems 131
4-3 Solution of Transient Problems 151
4-4 Capstone Problem 167
References 179
Problems 179
5 Separation of Variables in the Spherical Coordinate System 183
5-1 Separation of Heat Conduction Equation in the Spherical Coordinate System 183
5-2 Solution of Steady-State Problems 188
5-3 Solution of Transient Problems 194
5-4 Capstone Problem 221
References 233
Problems 233
Notes 235
6 Solution of the Heat Equation for Semi-Infinite and Infinite Domains 236
6-1 One-Dimensional Homogeneous Problems in a Semi-Infinite Medium for the Cartesian Coordinate System 236
6-2 Multidimensional Homogeneous Problems in a Semi-Infinite Medium for the Cartesian Coordinate System 247
6-3 One-Dimensional Homogeneous Problems in An Infinite Medium for the Cartesian Coordinate System 255
6-4 One-Dimensional homogeneous Problems in a Semi-Infinite Medium for the Cylindrical Coordinate System 260
6-5 Two-Dimensional Homogeneous Problems in a Semi-Infinite Medium for the Cylindrical Coordinate System 265
6-6 One-Dimensional Homogeneous Problems in a Semi-Infinite Medium for the Spherical Coordinate System 268
References 271
Problems 271
7 Use of Duhamel's Theorem 273
7-1 Development of Duhamel's Theorem for Continuous Time-Dependent Boundary Conditions 273
7-2 Treatment of Discontinuities 276
7-3 General Statement of Duhamel's Theorem 278
7-4 Applications of Duhamel's Theorem 281
7-5 Applications of Duhamel's Theorem for Internal Energy Generation 294
References 296
Problems 297
8 Use of Green's Function for Solution of Heat Conduction Problems 300
8-1 Green's Function Approach for Solving Nonhomogeneous Transient Heat Conduction 300
8-2 Determination of Green's Functions 306
8-3 Representation of Point, Line, and Surface Heat Sources with Delta Functions 312
8-4 Applications of Green's Function in the Rectangular Coordinate System 317
8-5 Applications of Green's Function in the Cylindrical Coordinate System 329
8-6 Applications of Green's Function in the Spherical Coordinate System 335
8-7 Products of Green's Functions 344
References 349
Problems 349
9 Use of the Laplace Transform 355
9-1 Definition of Laplace Transformation 356
9-2 Properties of Laplace Transform 357
9-3 Inversion of Laplace Transform Using the Inversion Tables 365
9-4 Application of the Laplace Transform in the Solution of Time-Dependent Heat Conduction Problems 372
9-5 Approximations for Small Times 382
References 390
Problems 390
10 One-Dimensional Composite Medium 393
10-1 Mathematical Formulation of One-Dimensional Transient Heat Conduction in a Composite Medium 393
10-2 Transformation of Nonhomogeneous Boundary Conditions into Homogeneous Ones 395
10-3 Orthogonal Expansion Technique for Solving M-Layer Homogeneous Problems 401
10-4 Determination of Eigenfunctions and Eigenvalues 407
10-5 Applications of Orthogonal Expansion Technique 410
10-6 Green's Function Approach for Solving Nonhomogeneous Problems 418
10-7 Use of Laplace Transform for Solving Semi-Infinite and Infinite Medium Problems 424
References 429
Problems 430
11 Moving Heat Source Problems 433
11-1 Mathematical Modeling of Moving Heat Source Problems 434
11-2 One-Dimensional Quasi-Stationary Plane Heat Source Problem 439
11-3 Two-Dimensional Quasi-Stationary Line Heat Source Problem 443
11-4 Two-Dimensional Quasi-Stationary Ring Heat Source Problem 445
References 449
Problems 450
12 Phase-Change Problems 452
12-1 Mathematical Formulation of Phase-Change Problems 454
12-2 Exact Solution of Phase-Change Problems 461
12-3 Integral Method of Solution of Phase-Change Problems 474
12-4 Variable Time Step Method for Solving Phase-Change Problems: A Numerical Solution 478
12-5 Enthalpy Method for Solution of Phase-Change Problems: A Numerical Solution 484
References 490
Problems 493
Note 495
13 Approximate Analytic Methods 496
13-1 Integral Method: Basic Concepts 496
13-2 Integral Method: Application to Linear Transient Heat Conduction in a Semi-Infinite Medium 498
13-3 Integral Method: Application to Nonlinear Transient Heat Conduction 508
13-4 Integral Method: Application to a Finite Region 512
13-5 Approximate Analytic Methods of Residuals 516
13-6 The Galerkin Method 521
13-7 Partial Integration 533
13-8 Application to Transient Problems 538
References 542
Problems 544
14 Integral Transform Technique 547
14-1 Use of Integral Transform in the Solution of Heat Conduction Problems 548
14-2 Applications in the Rectangular Coordinate System 556
14-3 Applications in the Cylindrical Coordinate System 572
14-4 Applications in the Spherical Coordinate System 589
14-5 Applications in the Solution of Steady-state problems 599
References 602
Problems 603
Notes 607
15 Heat Conduction in Anisotropic Solids 614
15-1 Heat Flux for Anisotropic Solids 615
15-2 Heat Conduction Equation for Anisotropic Solids 617
15-3 Boundary Conditions 618
15-4 Thermal Resistivity Coefficients 620
15-5 Determination of Principal Conductivities and Principal Axes 621
15-6 Conductivity Matrix for Crystal Systems 623
15-7 Transformation of Heat Conduction Equation for Orthotropic Medium 624
15-8 Some Special Cases 625
15-9 Heat Conduction in an Orthotropic Medium 628
15-10 Multidimensional Heat Conduction in an Anisotropic Medium 637
References 645
Problems 647
Notes 649
16 Introduction to Microscale Heat Conduction 651
16-1 Microstructure and Relevant Length Scales 652
16-2 Physics of Energy Carriers 656
16-3 Energy Storage and Transport 661
16-4 Limitations of Fourier's Law and the First Regime of Microscale Heat Transfer 667
16-5 Solutions and Approximations for the First Regime of Microscale Heat Transfer 672
16-6 Second and Third Regimes of Microscale Heat Transfer 676
16-7 Summary Remarks 676
References 676
Appendixes 679
Appendix I Physical Properties 681
Table I-1 Physical Properties of Metals 681
Table I-2 Physical Properties of Nonmetals 683
Table 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 696
Table IV-2 First 10 Roots of Jn(z) = 0, n = 0,1,2,3,4,5 704
Table IV-3 First Six Roots of ßJ1(ß) ¿ cJ0(ß) = 0 705
Table IV-4 First Five Roots of J0(ß)Y0(cß) ¿ Y0(ß)J0(cß) = 0 706
Appendix V Numerical Values of Legendre Polynomials of the First Kind 707
Appendix VI Properties of Delta Functions 710
Index 713
