Analytical Heat Diffusion Theory – Luikov

In this post, we will see the book Analytical Heat Diffusion Theory by A. V. Luikov.

About the book

This work is a revised edition of an earlier book by Academician Luikov which was widely used throughout the Soviet Union and the surrounding socialist countries. The presentation is unique in that it not only treats heat conduction problems by the classical methods such as separation of variables, but, in addition, it emphasizes the advantages of the transform method, particularly in obtaining short time solutions of many transient problems. In such cases, the long time solution may be obtained from the classical approach, and by interpolation, a very good estimate is obtained for intermediate times. The text is also noteworthy in that it covers a wide variety of geometrical shapes and treats boundary conditions of constant surface temperature, and constant surface heat flux, as well as the technically important case of a convective boundary condition.

The level of the book is advanced undergraduate or graduate. In addition to its value as a textbook, the availability of many technically important results in the form of tables and curves should make the book a valuable asset to the practicing engineers.


The book was translated from Russian (translator name is not mentioned) and was edited by James Hartnett. The book was published in 1968.

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Editor’s Preface v

Introduction xiii

Chapter 1. Physical Fundamentals of Meat Transfer 1

1.1 Temperature Field 1
1.2 The Fundamental Fourier Heat Conduction Law 3
1.3 Heat Distribution in the High Rate Processes 9
1.4 Heat Distribution Equation in Liquid and Gas Mixtures 12
1.5 Differential Heat Conduction Equation 15
1.6 Hyperbolic Heat Conduction Equation 20
1.7. A System of Differential Heat and Mass Transfer Equations 22
1.8 End Conditions 24
1.9 Methods for Calculating the Heat Flow 31

Chapter 2. Theory of Generalized Variables 35

Introduction 35
2.1 Dimensionless Quantities 36
2.2 Operational Calculus and Similarity Theory 44

Chapter 3. Basic Methods for Solution of Boundary Value Problems 48

3.1 Analysis of a Differential Equation for Heat Conduction 48
3.2 Solution of the Equation by Classical Methods 50
3.3 Integral Transform Methods 57
3.4 Methods of Numerical Solution of Heat Conduction Problems 67

Chapter 4 Nonstationary Temperature Field without Heat Sources: Boundary Condition of the First Kind 81

41 Infinite Body 82
42 Semi-Infinite Body 85
43 Infinite Plate 97
44 Sphere (Symmetrical Problem) 119
45 Infinite Cylinder 131
46 Infinite Hollow Cylinder 148
4.7 Parallelepiped 160
4.8 Finite Cylinder 164
4.9 Heating Problems 166

Chapter 5. Boundary Condition of the Second Kind 167

5.1 Semi-infinite Body 168
5.2 Infinite Plate 172
5.3 Sphere (Symmetrical Problem) 182
5.4 Infinite Cylinder 190
5.5 Hollow Infinite Cylinder 197

Chapter 6. Boundary Condition of the Third Kind 201

6.1 Semi-Infinite Body 203
6.2 Semi-Infinite Rod without Thermal Insulation of Its Surface 208
6.3 Infinite Plate 214
6.4 Finite Rod without Thermal Insulation of Its Lateral Surface 240
6.5 Sphere (Symmetrical Problem) 247
6.6 Infinite Cylinder 265
6.7 Infinite Hollow Cylinder 281
6.8 Finite Cylinder 283
6.9 Finite Plate 286
6.10 Analysis of the Generalized Solution 288
6.11 Estimation of Approximation 295

Chapter 7. Temperature Fields without Heat Sources with Variable Temperature of the Surrounding Medium 300

7.1 Infinite Plate. Ambient Temperature as a Linear Function of Time 300
7.2. Sphere. Ament Temperature 2s a Linear Function of Time 306
7.3. Infinite Cylinder. Ambient Temperature as a Linear Function of Time 310
7.4 Infinite Plate, Sphere, and Cylinder, Ambient Temperature as an Ex
potential Function of Time 314
7.5  Heating of Moot Nodes (afinite Plate, Sphere, and Infinite Cylinder) 317
7.6 Thermal Wases, lutinite Plate, Semi-infinite Body, Sphere, and Infinite Cylinder, Ambient Temperature as a Simple Harmonic Function of Time 325
7.7 Semi-infinite Body, Ambient Temperature as a Function of Time 342
7.8 Generalized Solution, Dubamel’s Theorem 344
7.9 Hollow Cylinder 348
7.10 Parallelepiped, Ambient Temperature as a Linear Function of Time 350

Chapter 8. Temperature field with Continuous Heat Sources 351

8.1 Semi-infinite Body 351
8.2 Infinte Plate 356
8.3 Sphere (Symmetrical Problem) 365
8.4 Infinite Cylinder 371

Chapter 9. Temperature Field with Pulse-Type Heat Sources 377

Introduction 377
9.1 Semi-infinite Body 381
9.2 Infinite Plate 384
9.3 Sphere (Symmetrical Problem) 388
9.4 Infinite Cylinder 391
9.8 Regular Thermal Regime 394

Chapter 10. Boundary Conditions of the Fourth Kind 399

10.1 System of Two Bodies (Two Semi-Infinite Rods) 401
10.2 System of Two Bodies (Finite and Semi-infinite Rods) 406
10.3 System of Two Bodies (Two Infinite Plates) 411
10.4 System af Two Spherical Bodies {Sphere inside Sphere) 417
10.5. System of Two Cylindrical Bodies 420
10.6 Infinite Plate 422
10.7 Sphere (Symmetrical Problem) 428
10.8 Infinite Cylinder 431
10.9 Heat Transfer between a Body and a Liquid Flow 434
10.10 Symmetrical System of Bodies Consisting of Three Infinite Plates 440

Chapter 11. Temperature Field of Body with Changing State of Aggregation 443

11.1 Freezing of Wet Ground 443
11.2 Approximate Solutions of Problems af Solidification of a Semi-Infinite Body, an Infinite Plate, a Sphere, and an Infinite Cylinder 451
11.3 Metal Solidification with the Heat Conduction Coefficient and Heat Capacity as Functions of Temperature 456

Chapter 12. Two-Dimensional Temperature Field: Particular Problems 460

12.1 Semi-Infinite Plate 460
12.2 Two-Dimensional Plate 463
12.3 Semi-Infinite Cylinder 465
12.4 Heat Transfer in Cylindrical Regions 467

Chapter 13 Heat Conduction with Variable Transfer Coefficients 478

13.1 Semi-lnfinite Body, Heat Conductivity, and Heat Capacity as Power
Functions of Coordinates 479
13.2 Finite Plate Thermal Conductivity as an Exponential Function of the Coordinate 479
13.3 Nonstationary Temperature Fields in Nonlinear Temperature Processes 486
13.4 Boundary-Value Problems for the Heat Conduction Equation with the Coefficients Dependent upon the Coordinate 506

Chapter 14. Fundamentals of the Integral Transforms 520

14.1 Definitions 523
14.2 Laplace Transformation Properties 526
14.3 Method of Solution for Simplest Differential Equations 532
14.4 Other Properties of the Laplace Transformation 535
14.5 Solution of the Linear Differential Equation with Constant Coefficients by Operational Methods 543
14.6 Expansion Theorems 544
14.7 Solution of Some Differential Equations with Variable Coefficients 552
14.8 Integral Transformations and Operational Methods 555
14.9 Inversion of the Transform 560
14.40 Integral Fourier and Hankel Transforms 568
14.15 Finite Integral Fourier and Hankel Transforms 575
14.12 Kernels of Finite Integral Transforms 583

Chapter 15. Elements of the Theory of Analytic Functions and Its
Applications 589

15.1 Analytic Functions 589
15.2 Contour Integration of Complex Variable Functions 591
15.3 Representation of Analytic Functions by Series 596
15.4 Classification of Analytic Functions by Their Singularities. The Concept of Analytical Continuation 602
15.5 Residue Theory and Its Application to Calculating Integrals and Summing Up Series 607
15.6 Some Analytical Properties of Laplace Transforms and Asymptotic Estimates 624

Appendix 1. Some Reference Formulas 649
Appendix 2. The Uniqueness Theorem 656
Appendix 3. Differential Heat Conduction Equation in Various Coordinate Systems 658
Appendix 4. Main Rules and Theorems of the Laplace Transformation 660
Appendix 5. Transforms of Some Functions 662
Appendix 6. Values of Functions i^{n} erfc x 669


Autor Index 679

Subject Index 682

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