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
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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
1
25
ix
X
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
xi
194
209
xH
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
246
294
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
xiii
397
439
471
xiv CONTENTS
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
'
522
594
XV CONTENTS
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
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