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.

Note: scan quality is not good.

Credits to original uploader.

You can get the book here.

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# Contents

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

REFERENCES 672

Autor Index 679

Subject Index 682