Solving Direct and Inverse Heat Conduction Problems (eBook)

Professor Jan Taler is a director of the Department of Process and Power Engineering at the Faculty of Mechanical Engineering, Krakow University of Technology. He has lectures on heat transfer processes and thermal power plants at the Faculty of Mechanical Engineering and the Faculty of Computer and Electrical Engineering. His research interests mainly lie in heat transfer, inverse heat conduction problems and monitoring of thermal stresses, which arise during the operations of energy installations and machinery. The results of his research on heat transfer, thermal stresses, optimum heating and cooling of solids and measuring of heat flux were published in well-known international journals, such as Transactions of the ASME, International Journal of Heat and Mass Transfer, Heat and Mass Transfer, Forschung im Ingenieurwessen, Brennstoff-Warme-Kraft, VGB Kraftwerkstechnik and VGB Power Tech. Professor Taler was a research fellow of DAAD in Germany and of Alexander von Humboldt Foundation at the University of Stuttgart. He is also a member of the Committee of Combustion and Thermodynamics at the Polish Academy of Sciences. He is also the author of over 200 publications and 5 monographies, including three in German language. He has received Siemens Award for his achievements in scientific research, the Award of the Minister of Education and the Award of the Rector of the Krakow University of Technology. Dr. Piotr Duda is an associate professor at the Department of Process and Power Engineering of the Faculty of Mechanical Engineering, Krakow University of Technology. Between 1997-1998, he was a research fellow at the Swiss Federal Institute of Technology in Lausanne (EPFL). Between 2002-2003, he was a research fellow of the Alexander von Humboldt Foundation at the University of Stuttgart, Germany. He has published over 50 articles on heat transfer problems, thermal stresses and numerical methods both at home and abroad.

Preface 5
Table of contents 
7 
Nomenclature 23
PART I Heat Conduction Fundamentals 
27 
1 Fourier Law 28
Literature 31
2 Mass and Energy Balance Equations 32
2.1 Mass Balance Equation for a Solid that Moves at an Assigned Velocity 
32 
2.2 Inner Energy Balance Equation 34
2.2.1 Energy Balance Equations in Three Basic Coordinate Systems 
37 
2.3 Hyperbolic Heat Conduction Equation 41
2.4 Initial and Boundary Conditions 42
2.4.1 First Kind Boundary Conditions (Dirichlet Conditions) 
43 
2.4.2 Second Kind Boundary Conditions von Neumann Conditions) 
43 
2.4.3 Third Kind Boundary Conditions 44
2.4.4 Fourth Kind Boundary Conditions 46
2.4.5 Non-Linear Boundary Conditions 47
2.4.6 Boundary Conditions on the Phase Boundaries 49
Literature 51
3 The Reduction of Transient Heat Conduction Equations and Boundary Conditions 
53 
3.1 Linearization of a Heat Conduction Equation 53
3.2 Spatial Averaging of Temperature 55
3.2.1 A Body Model with a Lumped Thermal Capacity 55
3.2.2 Heat Conduction Equation for a Simple Fin with Uniform Thickness 
57 
3.2.3 Heat Conduction Equation for a Circular Fin with Uniform Thickness 
59 
3.2.4 Heat Conduction Equation for a Circular Rod or a Pipe that Moves at Constant Velocity 61
Literature 63
4 Substituting Heat Conduction Equation by Two-Equations System 
64 
4.1 Steady-State Heat Conduction in a Circular Fin with Variable Thermal Conductivity and Transfer Coefficient 
64 
4.2 One-Dimensional Inverse Transient Heat Conduction Problem 
66 
Literature 69
5 Variable Change 70
Literature 73
Part II Exercises. Solving Heat Conduction Problems 
74 
6 Heat Transfer Fundamentals 75
Exercise 6.1 Fourier Law in a Cylindrical Coordinate System 
75 
Solution 76
Exercise 6.2 The Equivalent Heat Transfer Coefficient Accounting for Heat Exchange by Convection and Radiation 
77 
Solution 78
Exercise 6.3 Heat Transfer Through a Flat Single-Layeredand Double-Layered Wall 79
Solution 79
Exercise 6.4 Overall Heat Transfer Coefficient and Heat Loss Through a Pipeline Wall 
82 
Solution 83
Exercise 6.5 Critical Thickness of an Insulation on an Outer Surface of a Pipe 
84 
Solution 85
Exercise 6.6 Radiant TubeTemperature 87
Solution 87
Exercise 6.7 Quasi-Steady-State of Temperature Distribution and Stresses in a Pipeline Wall 
90 
Solution 91
Exercise 6.8 Temperature Distribution in a Flat Wall with Constant and Temperature Dependent Thermal Conductivity 
92 
Solution 93
Exercise 6.9 Determining Heat Flux on the Basis of Measured Temperature at Two Points Using a Flat and Cylindrical Sensor 
96 
Solution 97
Exercise 6.10 Determining Heat FluxBy Means of Gardon Sensor with a Temperature Dependent Thermal Conductivity 
99 
Solution 100
Exercise 6.11 One-Dimensional Steady-State Plate Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources 
102 
Solution 102
Exercise 6.12 One-Dimensional Steady-State Pipe Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources 
104 
Solution 105
Exercise 6.13 Inverse Steady-State Heat Conduction Problem in a Pipe 
107 
Solution 107
Exercise 6.14 General Equation of Heat Conduction in Fins 
109 
Solution 109
Exercise 6.15 Temperature Distribution and Efficiency of a Straight Fin with Constant Thickness 
111 
Solution 111
Exercise 6.16 Temperature Measurement Error Caused by Thermal Conduction Through Steel Casing that Contains a Thermoelement as a Measuring Device 
114 
Solution 115
Exercise 6.17 Temperature Distribution and Efficiency of a Circular Fin of Constant Thickness 
117 
Solution 117
Exercise 6.18 Approximated Calculation of a Circular Fin Efficiency 
120 
Solution 120
Exercise 6.19 Calculating Efficiency of Square and Hexagonal Fins 
121 
Solution 122
Exercise 6.20 Calculating Efficiency of Hexagonal Fins by Means of an Equivalent Circular Fin Method and Sector Method 
124 
Solution 125
Exercise 6.21 Calculating Rectangular Fin Efficiency 130
Solution 130
Exercise 6.22 HeatTransfer Coefficient in Exchangers with Extended Surfaces 
131 
Solution 131
Exercise 6.23 Calculating Overall HeatTransfer Coefficient in a Fin Plate Exchanger 
136 
Solution 136
Exercise 6.24 Overall HeatTransfer Coefficient for a Longitudinally Finned Pipewith a Scale Layer on an Inner Surface 
137 
Solution 139
Exercise 6.25 Overall Heat Transfer Coefficient for a Longitudinally Finned Pipe 
141 
Solution 141
Exercise 6.26 Determining One-Dimensional Temperature Distribution in a Flat Wall by Means of Finite Volume Method 
144 
Solution 144
Exercise 6.27 Determining One-Dimensional Temperature Distribution in a Cylindrical Wall By Means of Finite Volume Method 
149 
Solution 149
Exercise 6.28 Inverse Steady-State Heat Conduction Problem for a Pipe Solved by Space-Marching Method 
153 
Solution 153
Exercise 6.29 Temperature Distribution and Efficiency of a Circular Fin with Temperature-Dependent Thermal Conductivity 
156 
Solution 156
Literature 160
7 Two-Dimensional Steady-State Heat Conduction. Analytical Solutions 
162 
Exercise 7.1 Temperature Distribution in an Infinitely Long Fin with Constant Thickness 
162 
Solution 163
Exercise 7.2 Temperature Distribution in a Straight Fin with Constant Thickness and Insulated Tip 
166 
Solution 166
Exercise 7.3 Calculating Temperature Distribution and Heat Flux in a Straight Fin with Constant Thickness and Insulated Tip 
169 
Solution 170
Exercise 7.4 Temperature Distribution in a Radiant Tube of a Boiler 
177 
Solution 178
Literature 181
8 Analytical Approximation Methods. Integral Heat Balance Method 
182 
Exercise 8.1 Temperature Distribution within a Rectangular Cross-Section of a Bar 
182 
Solution 182
Exercise 8.2 Temperature Distribution in an Infinitely Long Fin of Constant Thickness 
184 
Solution 184
Exercise 8.3 Determining Temperature Distribution in a Boiler's Water-Wall Tube by Means of Functional Correction Method 
186 
Solution 186
Literature 190
9 Two-Dimensional Steady-State Heat Conduction. Graphical Method 
191 
Exercise 9.1 Temperature Gradient and Surface-Transmitted Heat Flow 
191 
Solution 191
Exercise 9.2 Orthogonality of Constant Temperature Line and Constant Heat Flux 
193 
Solution 193
Exercise 9.3 Determining Heat Flow between Isothermal Surfaces 
196 
Solution 196
Exercise 9.4 Determining Heat Loss Through a Chimney Wall Combustion Channel (Chimney) with Square Cross-Section
199 
Solution 199
Exercise 9.5 Determining Heat Loss Through Chimney Wall with a Circular Cross-Section 
201 
Solution 201
Literature 202
10 Two-Dimensional Steady-State Problems. The Shape Coefficient 
203 
Exercise 10.1 Buried Pipe-to-Ground Surface Heat Flow 203
Solution 203
Exercise 10.2 Floor Heating 205
Solution 205
Exercise 10.3 Temperature of a Radioactive Waste Container Buried Underground 
206 
Solution 206
Literature 207
11 Solving Steady-State Heat Conduction Problems by Means of Numerical Methods 
208 
Exercise 11.1 Description of the Control Volume Method 208
Solution 209
a) Heat balance equation- Cartesian coordinates 210
b) Heat balance equation-cylindrical coordinates 211
Exercise 11.2 Determining Temperature Distribution in a Square Cross-Section of a Long Rod by Means of the Finite Volume Method 
213 
Solution 213
Exercise 11.3A Two-Dimensional Inverse Steady-State Heat Conduction Problem 
218 
Solution 218
Exercise 11.4 Gauss-Seidel Method and Over-Relaxation Method 
223 
Solution 223
Exercise 11.5 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Uniform Thickness by Means of the Finite Volume Method 
227 
Solution 227
Exercise 11.6 Determining Two-Dimensional Temperature Distribution in a Square Cross-Section of a Chimney 
234 
Solution 235
Exercise 11.7 Pseudo-Transient Determination of Steady StateTemperatureDistribution in a Square Cross-Section of a Chimney Heat Transfer by Convection and Radiation on an Outer Surface of a Chimney
240 
Solution 240
Exercise 11.8 Finite Element Method 249
Historical Development of FEM 249
Solution 250
Exercise 11.9 Linear Functions That Interpolate Temperature Distribution (Shape Functions) Inside Triangular and Rectangular Elements 
253 
Solution 253
Exercise 11.10 Description of FEM Based on Galerkin Method 
257 
Solution 257
Exercise 11.11 Determining Conductivity Matrix for a Rectangular and Triangular Element 
264 
Solution 264
a) Conductivity matrix [Kec] for a finite rectangular element 
264 
b) Conductivity matrix [Kec] for a finite triangular element 
266 
Exercise 11.12 Determining Matrix [Kae] in Terms of Convective Boundary Conditions for a Rectangular and Triangular Element 
268 
Solution 268
a) Rectangular finite element 268
b) Triangular finite element 270
Exercise 11.13 Determining Vector {fqe} with Respect to Volumetric and Point Heat Sources in a Rectangular and Triangular Element 
272 
Solution 272
a) Rectangular element 272
b) Triangular element 273
Exercise 11.14 Determining Vectors {fqe} and {fae} withRespect to Boundary Conditions of 2nd and 3rd Kind on the Boundary of a Rectangular or Triangular Element 
275 
Solution 275
a) Finite rectangular element 275
b) Finite triangular element 277
Exercise 11.15 Methods for Building Global Equation System in FEM 
278 
Solution 279
Exercise 11.16 Determining Temperature Distribution in a Square Cross-Section of an Infinitely Long Rod by Means of FEM, in which the Global Equation System is Constructed using Method I (from Ex. 11.15) 
283 
Solution 284
Exercise 11.17 Determining Temperature Distributionin an In finitely Long Rod with Square Cross-Sectionby Means of FEM, in which the Global Equation System is Constructed using Method II (from Ex. 11.15) 
290 
Solution 290
Exercise 11.18 Determining Temperature Distribution by Means of FEM in an Infinitely Long Rod with Square Cross-Section, in which Volumetric Heat Sources Operate 
294 
Solution 294
Exercise 11.19 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Constant Thickness by Means of FEM 
304 
Solution 304
Exercise 11.20 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Straight Fin with Constant Thickness (ANSYS Program) 
316 
Solution 316
Exercise 11.21 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Hexagonal Fin with Constant Thickness (ANSYS Program) 
319 
Solution 319
Exercise 11.22 Determining Axisymmetrical Temperature Distribution in a Cylindrical and Conical Pin by Means of FEM (ANSYS Program) 
322 
Solution 323
Literature 326
12 Finite Element Balance Method and Boundary Element Method 
327 
Exercise 12.1 Finite Element Balance Method 327
Solution 327
Exercise 12.2 Boundary Element Method 332
Solution 332
Exercise 12.3 Determining Temperature Distribution in Square Region by Means of FEM Balance Method 
341 
Solution 341
Exercise 12.4 Determining Temperature Distribution in a Square Region using Boundary Element Method 
345 
Solution 346
Literature 349
13 Transient Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings 
350 
Exercise 13.1 Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings 
350 
Solution 350
Exercise 13.2 Heat Exchange between a Body with Lumped Thermal Capacity and Surroundings with Time-Dependent Temperature 
353 
Solution 353
Exercise 13.3 Determining Temperature Distribution of a Body with Lumped Thermal Capacity, when the Temperature of a Medium Changes Periodically 
356 
Solution 356
Exercise 13.4 Inverse Problem: Determining Temperature of a Medium on the Basis of Temporal Thermometer Indicated Temperature History 
357 
Solution 357
Exercise 13.5 Calculating Dynamic Temperature Measurement Error by Means of a Thermocouple 
359 
Solution 360
Exercise 13.6 Determining the Time It Takes to Cool Body Down to a Given Temperature 
361 
Solution 361
Exercise 13.7 Temperature Measurement Error of a Medium whoseTemperature Changes at Constant Rate 
362 
Solution 362
Exercise 13.8 Temperature Measurement Error of a Medium whose Temperature Changes Periodically 
363 
Solution 363
Exercise 13.9 Inverse Problem: Calculating Temperature of a Medium whose Temperature Changes Periodically, on the Basis of Temporal Temperature History Indicated by a Thermometer 
364 
Solution 364
Exercise 13.10 Measuring Heat Flux 366
Solution 367
Literature 368
14 Transient Heat Conduction in Half-Space 369
Exercise 14.1 Laplace Transform 369
Solution 369
Exercise 14.2 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Surface Temperature 
371 
Solution 372
Exercise 14.3 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increasein Heat Flux 
374 
Solution 374
Exercise 14.4 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Temperature of a Medium 
376 
Solution 376
Exercise 14.5 Formula Derivation for Temperature Distribution in a Half-Space when Surface Temperature isTime-Dependent 380
Solution 380
Exercise 14.6 Formula Derivation for a Quasi-Steady State Temperature Field in a Half-Space when Surface Temperature Changes Periodically 
382 
Solution 382
Exercise 14.7 Formula Derivation for Temperature of Two Contacting Semi-Infinite Bodies 
390 
Solution 390
Exercise 14.8 Depth of Heat Penetration 391
Solution 392
Exercise 14.9 Calculating Plate Surface Temperature Under the Assumption that the Plate is a Semi-Infinite Body 
393 
Solution 393
Exercise 14.10 Calculating Ground Temperature at a Specific Depth 
394 
Solution 394
Exercise 14.11 Calculating the Depth of Heat Penetration in the Wall of a Combustion Engine 
395 
Solution 395
Exercise 14.12 Calculating auasi-Steady-State Ground Temperature at a Specific Depth when Surface Temperature Changes Periodically 
396 
Solution 396
Exercise 14.13 Calculating Surface Temperature at the Contact Point of Two Objects 
398 
Solution 398
Literature 399
15 Transient Heat Conduction in Simple-Shape Elements 
400 
Exercise 15.1 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 3rd Kind 
400 
Solution 400
Exercise 15.2 A Program for Calculating Temperature Distribution and Its Change Rate in a Plate with Boundary Conditions of 3rd Kind 
409 
Solution 410
Exercise 15.3 Calculating Plate Surface Temperature and Average Temperature Across the Plate Thickness by Means of the Provided Graphs 
413 
Solution 413
Exercise 15.4 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind 
417 
Solution 417
Exercise 15.5 A Program for Calculating Temperature Distribution and Its Change Rate in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind 
427 
Solution 428
Exercise 15.6 Calculating Temperature in an Infinitely Long Cylinder using the Annexed Diagrams 
431 
Solution 431
Exercise 15.7 Formula Derivation for a Temperature Distribution in a Sphere with Boundary Conditionsof 3rd Kind 
435 
Solution 435
Exercise 15.8 A Program for Calculating Temperature Distribution and Its Change Rate in a Sphere with Boundary Conditions of 3rd Kind 
443 
Solution 443
Exercise 15.9 Calculating Temperature of a Sphere using the Diagrams Provided 
447 
Solution 447
Exercise 15.10 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind 
451 
Solution 451
Exercise 15.11 A Program and Calculation Results for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind 
456 
Solution 456
Exercise 15.12 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind 
459 
Solution 459
Exercise 15.13 Program and Calculation Results for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind 
463 
Solution 463
Exercise 15.14 Formula Derivation for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind 
467 
Solution 467
Exercise 15.15 Program and Calculation Results for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind 
471 
Solution 471
Exercise 15.16 Heating Rate Calculations for a Thick-Walled Plate 
475 
Solution 475
Exercise 15.17 Calculating the Heating Rate of a Steel Shaft 
476 
Solution 477
Exercise 15.18 Determining Transients of Thermal Stresses in a Cylinder and a Sphere 
478 
Solution 478
Exercise 15.19 Calculating Temperature and Temperature Change Rate in a Sphere 
479 
Solution 479
Exercise 15.20 Calculating Sensor Thickness for Heat Flux Measuring 
480 
Solution 480
Literature 482
16 Superposition Method in One-Dimensional Transient Heat Conduction Problems 
483 
Exercise 16.1 Derivation of Duhamel Integral 483
Solution 483
Exercise 16.2 Derivation of an Analytical Formula for a Half-Space Surface Temperature when Medium's Temperature Undergoes a Linear Change in the Function of Time 
486 
Solution 487
Exercise 16.3 Derivation of an Approximate Formula for a Half-Space Surface Temperature with an Arbitrary Change in Medium's Temperature in the Function of Time 
490 
Solution 491
Exercise 16.4 Definition of an Approximate Formulafor a Half-Space Surface Temperature when Temperature of a Medium Undergoes a Linear Change in the Function of Time 
493 
Solution 493
Exercise 16.5 Application of the Superposition Method when Initial Body Temperature is Non-Uniform 
495 
Solution 495
Exercise 16.6 Description of the Superposition Method Applied to Heat Transfer Problems with Time-Dependent Boundary Conditions 
498 
Solution 498
Example 1 499
Example 2 500
Example 3 501
Example 4 501
Exercise 16.7 Formula Derivation for a Half-Space Surface Temperature with a Change in Surface Heat Flux in the Form of a Triangular Pulse 
502 
Solution 503
Exercise 16.8 Formula Derivation for a Half-Space Surface Temperature with a Mixed Step-Variable Boundary Condition in Time 
505 
Solution 505
Exercise 16.9 Formula Derivation for a Plate Surface Temperature with a Surface Heat Flux Change in the Form of a Triangular Pulse and the Calculationof This Temperature 
509 
Solution 510
Exercise 16.10 Formula Derivation for a Plate Surface Temperature with a Surface Heat Flux Change in the Form of a Rectangular Pulse Temperature Calculation
514 
Solution 514
Exercise 16.11 A Program and Calculation Results for a Half-Space Surface Temperature with a Change in Surface Heat Flux in the Form of a Triangular Pulse 
517 
Solution 517
Exercise 16.12 Calculation of a Half-Space Temperature with a Mixed Step-Variable Boundary Condition in Time 
520 
Solution 521
Exercise 16.13 Calculating Plate Temperature by Means of the Superposition Method with Diagrams Provided 
521 
Solution 522
Exercise 16.14 Calculating the Temperature of a Paper in an Electrostatic Photocopier 
523 
Solution 525
Literature 527
17 Transient Heat Conduction in a Semi-Infinite Body. The Inverse Problem 
528 
Exercise 17.1 Measuring Heat Transfer Coefficient. The Transient Method 
528 
Solution 528
Exercise 17.2 Deriving a Formula for Heat Fluxon the Basis of Measured Half-Space Surface Temperature Transient Interpolated by a Piecewise Linear Function 
531 
Solution 532
Exercise 17.3 Deriving Heat Flux Formula on the Basis of a Measured and Polynomial-Approximated Half-Space Surface Temperature Transient 
534 
Solution 534
Exercise 17.4 Formula Derivation for a Heat Flux Periodically Changing in Time on the Basis of a Measured Temperature Transient at a Point Located under the Semi-Space Surface 
536 
Solution 536
Exercise 17.5 Deriving a Heat Flux Formula on the Basis of Measured Half-Space Surface Temperature Transient, Approximated by a Linear and Square Function 
540 
Solution 540
Exercise 17.6 Determining Heat Transfer Coefficient on the Plexiglass Plate Surface using the Transient Method 
541 
Solution 542
Exercise 17.7 Determining Heat Fluxon the Basis of a Measured Time Transient of the Half-Space Temperature, Approximated by a Piecewise Linear Function 
545 
Solution 545
Exercise 17.8 Determining Heat Flux on the Basis of Measured Time Transient of a Polynomial-Approximated Half-Space Temperature 
548 
Solution 549
Literature 552
18 Inverse Transient Heat Conduction Problems 
553 
Exercise 18.1 Derivation of Formulas for Temperature Distribution and Heat Flux in a Simple-Shape Bodies on the Basis of a Measured Temperature Transient in a Single Point 
553 
Solution 554
Exercise 18.2 Formula Derivation for a Temperature of a Medium when Linear Time Change in Plate Surface Temperature is Assigned 
557 
Solution 557
Exercise 18.3 Determining Temperature Transient of a Medium for Which Plate Temperature at a Point with a Given Coordinate Changes According to the Prescribed Function 
559 
Solution 559
Exercise 18.4 Formula Derivation for a Temperature of a Medium, which is Warming an Infinite Plate Plate Temperature at a Point with a Given Coordinate Changes at Constant Rate
561 
Solution 561
Exercise 18.5 Determining Temperature and Heat Flux on the Plate Front Face on the Basis of a measured Temperature Transient on an Insulated BackSurface Heat Flowon the Plate Surface is in the Form of a Triangular Pulse
567 
Solution 567
Exercise 18.6 Determining Temperature and Heat Flux on the Surface of a Plate Front Faceon the Basis of a Measured Temperature Transient on an Insulated Back Surface Heat Flowon the Plate Surface is in the Form of a Rectangular Pulse
574 
Solution 575
Exercise 18.7 Determining Time-Temperature Transient of a Medium, for which the Plate Temperature at a Point with a Given Coordinate Changes in a Linear Way 
577 
Solution 578
Exercise 18.8 Determining Time-Temperature Transient of a Medium, for which the Plate Temperature at a Point with a Given Coordinate Changes According to the Square Function Assigned 
581 
Solution 581
Literature 583
19 Multidimensional Problems. The Superposition Method 
585 
Exercise 19.1 The Application of the Superposition Method to Multidimensional Problems 
585 
Solution 585
Boundary Conditions of 1st and 3rd Kind 
585 
Boundary Conditions of 2nd Kind 
586 
Exercise 19.2 Formula Derivation for Temperature Distribution in a Rectangular Region with a Boundary Condition of 3rd Kind 
589 
Solution 589
Exercise 19.3 Formula Derivation for Temperature Distribution in a Rectangular Region with Boundary Conditions of 2nd Kind 
592 
Solution 592
Exercise 19.4 Calculating Temperature in a Steel Cylinder of a Finite Height 
594 
Solution 594
Exercise 19.5 Calculating Steel Block Temperature 596
Solution 596
20 Approximate Analytical Methods for Solving Transient Heat Conduction Problems 
599 
Exercise 20.1 Description of an Integral Heat Balance Method by Means of a One-Dimensional Transient Heat Conduction Example 
599 
Solution 599
Exercise 20.2 Determining Transient Temperature Distribution in a Flat Wall with Assigned Conditions of 1st, 2nd and 3rd Kind 
602 
Solution 602
Example 1 606
Example 2 607
Example 3 610
Example 4 611
Exercise 20.3 Determining Thermal Stresses in a Flat Wall 612
Solution 612
Literature 614
21 Finite Difference Method 617
Exercise 21.1 Methods of Heat Flux Approximation on the Plate Surface 
618 
Solution 618
Exercise 21.2 Explicit Finite Difference Methodwith Boundary Conditions of 1st, 2nd and 3rd Kind 622
Solution 623
Exercise 21.3 Solving Two-Dimensional Problems by Means of the Explicit Difference Method 
628 
Solution 629
Exercise 21.4 Solving Two-Dimensional Problems by Means of the Implicit Difference Method 
634 
Solution 634
Exercise 21.5 Algorithm and a Program for Solving a Tridiagonal Equation System by Thomas Method 
638 
Solution 638
Exercise 21.6 Stability Analysis of the Explicit Finite Difference Method by Means of the von Neumann Method 
642 
Solution 642
Exercise 21.7 Calculating One-Dimensional Transient Temperature Field by Means of the Explicit Method and a Computational Program 
646 
Solution 647
Exercise 21.8 Calculating One-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program 
651 
Solution 651
Exercise 21.9 Calculating Two-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program Algebraic Equation System is Solved by Gaussian Elimination Method
656 
Solution 656
Exercise 21.10 Calculating Two-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program Algebraic Equation System Solved by Over-Relaxation Method
664 
Solution 664
Literature 668
22 Solving Transient Heat Conduction Problems by Means of Finite Element Method (FEM) 
670 
Exercise 22.1 Description of FEM Based on GalerkinMethod Used for Solving Two-Dimensional Transient Heat Conduction Problems 
670 
Solution 670
Exercise 22.2 Concentrated (Lumped) Thermal Finite Element Capacity in FEM 
673 
Solution 673
Exercise 22.3 Methods for Integrating Ordinary Differential Equations with Respect to Time Used in FEM 
679 
Solution 679
Exercise 22.4 Comparison of FEM Based on Galerkin Method and Heat Balance Method with Finite Volume Method 
682 
Solution 682
Exercise 22.5 Natural Coordinate System for One-Dimensional, Two-Dimensional Triangular and Two-Dimensional Rectangular Elements 
685 
Solution 685
a. One-dimensional elements 685
b. Two-dimensional tetragonal elements 686
c. Two-dimensional triangular elements 687
Exercise 22.6 Coordinate System Transformations and Integral Calculations by Means of the Gauss-Legendre Quadratures 
689 
Solution 689
a. One-dimensional elements 693
b. Tetragonal elements 695
c. Triangular elements 696
Exercise 22.7 Calculating Temperature in a Complex ShapeFin by Means of the ANSVS Program 
698 
Solution 700
Literature 701
23 Numerical-Analytical Methods 703
Explicit Method 704
Implicit Method 704
Crank-Nicolson Method 704
Exercise 23.1 Integration of the Ordinary Differential Equation System by Means of the Runge-Kutta Method 
705 
Solution 706
Exercise 23.2 Numerical-Analytical Method for Integrating a Linear Ordinary Differential Equation System 
708 
Solution 708
a. Approximating u(t) with a step function 
709 
b. Approximating u(t) with a piecewise linear function 
711 
Exercise 23.3 Determining Steel Plate Temperature by Means of the Method of Lines, while the Plate is Cooled by Air and Boiling Water 
713 
Solution 713
Exercise 23.4 Using the Exact Analytical Method and the Method of Lines to Determine Temperature of a Cylindrical Chamber 
719 
Solution 719
Exercise 23.5 Determining Thermal Stresses in a Cylindrical Chamber using the Exact Analytical Method and the Method of Lines 
724 
Solution 725
Exercise 23.6 Determining Temperature Distribution in a Cylindrical Chamber with Constant and Temperature Dependent Thermo-Physical Properties by Means of the Method of Lines 
728 
Solution 730
Exercise 23.7 Determining Transient Temperature Distribution in an Infinitely Long Rod with a Rectangular Cross-Section by Means of the Method of Lines 
734 
Solution 735
Literature 739
24 Solving Inverse Heat Conduction Problems by Means of Numerical Methods 
742 
Exercise 24.1 Numerical-Analytical Method for Solving Inverse Problems 
742 
Solution 743
a. Division of an inverse region into two control volumes (Fig. 24.2a) 
745 
b. Division of an inverse region into three control volumes (Fig. 24.2b) 
746 
c. Division of an inverse region into four control volumes (Fig. 24.2c) 
747 
Exercise 24.2 Step-Marching Method in Time Used for Solving Non-Linear Transient Inverse Heat Conduction Problems 
748 
Solution 749
Exercise 24.3 Weber Method Step-Marching Methods in Space 
755 
Solution 755
Exercise 24.4 Determining Temperature and Heat Flux Distribution in a Plate on the Basis of a Measured Temperature on a Thermally Insulated Back Plate Surface Heat Flux is in the Shapeof a Rectangular Pulse
760 
Solution 761
Exercise 24.5 Determining Temperature and Heat Flux Distribution in a Plate on the Basis of a Temperature Measurement on an Insulated Back Plate Surface Heat Flux is in the Shape of a Triangular Pulse
768 
Solution 768
Literature 772
25 Heat Sources 773
Exercise 25.1 Determining Formula for Transient Temperature Distribution Around an Instantaneous (Impulse) Point Heat Source Active in an Infinite Space 
775 
Solution 775
Exercise 25.2 Determining Formula for Transient Temperature Distribution in an Infinite Body Produced by an Impulse Surface Heat Source 
778 
Solution 778
Exercise 25.3 Determining Formula for Transient Temperature Distribution Around Instantaneous Linear Impulse Heat Source Active in an Infinite Space 
780 
Solution 780
Exercise 25.4 Determining Formula for Transient Temperature Distribution Around a Point Heat Source, which Lies in an Infinite Space and is Continuously Active 
782 
Solution 782
Exercise 25.5 Determining Formula for a Transient Temperature Distribution Triggered by a Surface Heat Source Continuously Active in an Infinite Space 
785 
Solution 785
Exercise 25.6 Determining Formula for a Transient Temperature Distribution Around a Continuously Active Linear Heat Source with Assigned Power q1 Per Unit of Length 
787 
Solution 787
Exercise 25.7 Determining Formula for Quasi-Steady StateTemperature Distribution Caused by a Point Heat Source with a Power Q0 that Moves at Constant Velocity v in Infinite Space or on the Half Space Surface 
789 
Solution 789
Exercise 25.8 Determining Formula for Transient Temperature Distribution Produced by a Point Heat Source with Power Qo that Moves at Constant Velocity v in Infinite Spaceor on the Half Space Surface 
793 
Solution 793
Exercise 25.9 Calculating Temperature Distribution along a Straight Line Traversed by a Laser Beam 
797 
Solution 797
Exercise 25.10 Quasi-Steady State Temperature Distribution in a Plate During the Welding Process A Comparison between the Analytical Solution and FEM
800 
Solution 800
Literature 804
26 Melting and Solidification (Freezing) 806
Exercise 26.1 Determination of a Formula which Describes the Solidification (Freezing) and Melting of a Semi-Infinite Body (the Stefan Problem) 
810 
Solution 810
Exercise 26.2 Derivation of a Formula that Describes the Solidification (Freezing) of a Semi-Infinite Body Under the Assumption that the Temperature of a Liquid is Non-Uniform 
815 
Solution 815
Exercise 26.3 Derivation of a Formula that Describe Quasi-Steady-State Solidification (Freezing) of a Flat Liquid Layer 
818 
Solution 819
Exercise 26.4 Derivation of Formulas that Describe Solidification (Freezing) of Simple-Shape Bodies: Plate, Cylinder and Sphere 
823 
Solution 823
a. Plate 825
b. Cylinder 825
Exercise 26.5 Ablation of a Semi-Infinite Body 827
Solution 828
Exercise 26.6 Solidification of a Falling Droplet of Lead 830
Solution 830
Exercise 26.7 Calculating the Thickness of an Ice Layer After the Assigned Time 
832 
Solution 832
Exercise 26.8 Calculating Accumulated Energy in a Melted Wax 
833 
Solution 834
Exercise 26.9 Calculating Fish Freezing Time 835
Solution 835
Literature 836
Appendix A Basic Mathematical Functions 837
A.1. Gauss Error Function 837
A.2. Hyperbolic Functions 839
A.3. Bessel Functions 840
Literature 841
Appendix B Thermo-Physical Properties of Solids 842
B.1. Tables of Thermo-Physical Properties of Solids 842
B.2. Diagrams 861
B.3. Approximated Dependencies for Calculating Thermo PhysicalProperties of a Steel [8] 
863 
Densityp at temperature 20°C 863
Specific heat capacity c in a temperature function 864
Longitudinal elasticity module (Young's modulus) E in function of temperature 
865 
Average temperature expansion coefficient ß within temperature interval from 20°C to a given temperature T expressed in [°C] 
865 
Poisson ratio v in function of temperature 865
Literature 866
Appendix C Fin Efficiency Diagrams (for Chap. 6, part II) 
867 
Literature 869
Appendix D Shape Coefficients for Isothermal Surfaces with Different Geometry (for Chap. 10, Part II) 
870 
Appendix E Subprogram for Solving Linear Algebraic Equations System using Gauss Elimination Method (for Chap. 6, Part II) 
882 
Subprogram for solving linearalgebraic equations system using Gauss method 
882 
Appendix F Subprogram for Solving a Linear Algebraic Equations System by Means of Over Relaxation Method 
884 
Subprogram SOR section appendix f subprogram,for solving a linear algebraic equations system by means of over-relaxation method 
884 
Appendix G Subprogram for Solving an Ordinary Differential Equations System of 1st Order using Runge-Kutta Method of 4th Order (for Chap. 11, Part II) 
885 
Subprogram for solving an ordinary differential equations system of 1st order using Runge-Kutta method of 4th order 
885 
Appendix H Determining inverse Laplace Transform (for Chap. 15, part II) 
886 
Literature 890

"Preface (p. v-vi)

This book is devoted to the concept of simple and inverse heat conduction problems. The process of solving direct problems is based on the temperature determination when initial and boundary conditions are known, while the solving of inverse problems is based on the search for boundary conditions when temperature properties are known, provided that temperature is the function of time, at the selected inner points of a body. In the first part of the book (Chaps. 1-5), we have discussed theoretical basis for thermal conduction in solids, motionless liquids and liquids that move in time.

In the second part of the book, (Chapters 6-26), we have discussed at great length different engineering problems, which we have presented together with the proposed solutions in the form of theoretical and mathematical examples. It was our intention to acquaint the reader in a step-by-step fashion with all the mathematical derivations and solutions to some of the more significant transient and steady-state heat conduction problems with respect to both, the movable and immovable heat sources and the phenomena of melting and freezing. Lots of attention was paid to non-linear problems.

The methods for solving heat conduction problems, i.e. the exact and approximate analytical methods and numerical methods, such as the finite difference method, the finite volume method, the finite element method and the boundary element method are discussed in great detail. Aside from algorithms, applicable computational programs, written in a FORTRAN language, were given. The accuracy of the results obtained by means of various numerical methods was evaluated by way of comparison with accurate analytical solutions.

The presented solutions not only allow to illustrate mathematical methods used in thermal conduction but also show the methods one can use to solve concrete practical problems, for example during the designing and life-time calculations of industrial machinery, combustion engines and in refrigerating and air conditioning engineering. Many examples refer to the topic of heating and thermo-renovation of apartment buildings.

The methods for solving problems involved with welding and laser technology are also discussed in great detail. This book is addressed to undergraduate and PhD students of mechanical, power, process and environmental engineering. Due to the complexity of the heat conduction problems elaborated in this book, this edition can also serve as a reference book that can be used by nuclear, industrial and civil engineers. Jan Taler is the author of the theoretical part of this book, mathematical exercises (excluding 12.1 & 12.3), and C, D & H attachments (found at the back of this book)."