Solving Direct And Inverse Heat Conduction Problems

by Taler, Jan; Duda, Piotr

• ISBN13: 9783540334705
• ISBN10: 354033470X
• Format: Hardcover
• Copyright: 2006-11-10
• Publisher: Springer Verlag

Summary

The book presents a solution for direct and inverse heat conduction problems. In the first part, the authors discuss the theoretical basis for the heat transfer process. In the second part, they present selected theoretical and numerical problems in the form of exercises with their subsequent solutions. Such layout of the book will allow the reader to become more familiar with step-by-step calculation methods and with the practical application of the equations to the solution of design and utilization problems of thermal machinery. It will also help to master complex mathematics behind the heat transfer theory. The book covers one-, two- and three dimensional problems which are solved by using 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 method. Unlike other books on the subject, the superposition method is thoroughly presented. Particular attention is paid to the solution of inverse heat conduction problems. The authors took special care that the solved inverse problems can be implemented in indirect measurements of boundary heat flux and heat transfer coefficient.

Author Biography

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.

Table of Contents

Part I Heat Conduction Fundamentals	1	(50)
1 Fourier Law	3	(4)
Literature	6	(1)
2 Mass and Energy Balance Equations	7	(22)
2.1 Mass Balance Equation for a Solid that Moves at an Assigned Velocity	7	(2)
2.2 Inner Energy Balance Equation	9	(7)
2.2.1 Energy Balance Equations in Three Basic Coordinate Systems	12	(4)
2.3 Hyperbolic Heat Conduction Equation	16	(1)
2.4 Initial and Boundary Conditions	17	(9)
2.4.1 First Kind Boundary Conditions (Dirichlet Conditions)	18	(1)
2.4.2 Second Kind Boundary Conditions (von Neumann Conditions)	18	(1)
2.4.3 Third Kind Boundary Conditions	19	(2)
2.4.4 Fourth Kind Boundary Conditions	21	(1)
2.4.5 Non-Linear Boundary Conditions	22	(2)
2.4.6 Boundary Conditions on the Phase Boundaries	24	(2)
Literature	26	(3)
3 The Reduction of Transient Heat Conduction Equations and Boundary Conditions	29	(12)
3.1 Linearization of a Heat Conduction Equation	29	(2)
3.2 Spatial Averaging of Temperature	31	(8)
3.2.1 A Body Model with a Lumped Thermal Capacity	31	(2)
3.2.2 Heat Conduction Equation for a Simple Fin with 'inform 'thickness	33	(2)
3.2.3 Heat Conduction Equation for a Round Fin with Uniform Thickness	35	(2)
3.2.4 Heat Conduction Equation for a Circular Rod or a Pipe that Moves at Constant Velocity	37	(2)
Literature	39	(2)
4 Substituting Heat Conduction Equation by Two-Equations System	41	(6)
4.1 Steady-State Heat Conduction in a Circular Fin with Variable Thermal Conductivity and Transfer Coefficient	41	(2)
4.2 One-Dimensional Inverse Transient Heat Conduction Problem	43	(3)
Literature	46	(1)
5 Variable Change	47	(4)
Literature	50	(1)
Part II Exercises. Solving Heat Conduction Problems	51	(780)
6 Heat Transfer Fundamentals	53	(88)
Exercise 6.1 Fourier Law in a Cylindrical Coordinate System	53	(2)
Exercise 6.2 The Equivalent Heat Transfer Coefficient Accounting for Heat Exchange by Convection and Radiation	55	(2)
Exercise 6.3 Heat Transfer Through a Flat Single-Layered and Double-Layered Wall	57	(3)
Exercise 6.4 Overall Heat Transfer Coefficient and Heat Loss Through a Pipeline Wall	60	(2)
Exercise 6.5 Critical Thickness of an Insulation on an Outer Surface of a Pipe	62	(3)
Exercise 6.6 Radiant Tube Temperature	65	(3)
Exercise 6.7 Quasi-Steady-State of Temperature Distribution and Stresses in a Pipeline Wall	68	(2)
Exercise 6.8 Temperature Distribution in a Flat Wall with Constant and Temperature Dependent Thermal Conductivity	70	(4)
Exercise 6.9 Determining Heat Flux on the Basis of Measured Temperature at Two Points Using a Flat and Cylindrical Sensor	74	(3)
Exercise 6.10 Determining Heat Flux By Means of Gardon Sensor with a Temperature Dependent Thermal Conductivity	77	(3)
Exercise 6.11 One-Dimensional Steady-State Plate Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources	80	(2)
Exercise 6.12 One-Dimensional Steady-State Pipe Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources	82	(3)
Exercise 6.13 Inverse Steady-State Heat Conduction Problem in a Pipe	85	(2)
Exercise 6.14 General Equation of Heat Conduction in Fins	87	(2)
Exercise 6.15 Temperature Distribution and Efficiency of a Straight Fin with Constant Thickness	89	(3)
Exercise 6.16 Temperature Measurement Error Caused by Thermal Conduction Through Steel Casing that Contains a Thermoelement as a Measuring Device	92	(3)
Exercise 6.17 Temperature Distribution and Efficiency of a Circular Fin of Constant Thickness	95	(3)
Exercise 6.18 Approximated Calculation of a Circular Fin Efficiency	98	(1)
Exercise 6.19 Calculating Efficiency of Square and Hexagonal Fins	99	(3)
Exercise 6.20 Calculating Efficiency of Hexagonal Fins by Means of an Equivalent Circular Fin Method and Sector Method	102	(6)
Exercise 6.21 Calculating Rectangular Fin Efficiency	108	(1)
Exercise 6.22 Heat Transfer Coefficient in Exchangers with Extended Surfaces	109	(5)
Exercise 6.23 Calculating Overall Heat Transfer Coefficient in a Fin Plate Exchanger	114	(1)
Exercise 6.24 Overall Heat Transfer Coefficient for a Longitudinally Finned Pipe with a Scale Layer on an Inner Surface	115	(4)
Exercise 6.25 Overall Heat Transfer Coefficient for a Longitudinally Finned Pipe	119	(3)
Exercise 6.26 Determining One-Dimensional Temperature Distribution in a Flat Wall by Means of Finite Volume Method	122	(5)
Exercise 6.27 Determining One-Dimensional Temperature Distribution in a Cylindrical Wall by Means of Finite Volume Method	127	(4)
Exercise 6.28 Inverse Steady-State Heat Conduction Problem for a Pipe Solved by Space-Marching Method	131	(3)
Exercise 6.29 Temperature Distribution and Efficiency of a Circular Fin with Temperature-Dependent Thermal Conductivity	134	(4)
Literature	138	(3)
7 Two-Dimensional Steady-State Heat Conduction. Analytical Solutions	141	(20)
Exercise 7.1 Temperature Distribution in an Infinitely Long Fin with Constant Thickness	141	(4)
Exercise 7.2 Temperature Distribution in a Straight Fin with Constant Thickness and Insulated Tip	145	(3)
Exercise 7.3 Calculating Temperature Distribution and Heat Flux in a Straight Fin with Constant Thickness and Insulated Tip	148	(8)
Exercise 7.4 Temperature Distribution in a Radiant Tube of a Boiler	156	(4)
Literature	160	(1)
8 Analytical Approximation Methods. Integral Heat Balance Method	161	(10)
Exercise 8.1 Temperature Distribution within a Rectangular Cross-Section of a Bar	161	(2)
Exercise 8.2 Temperature Distribution in an Infinitely Long Fin of Constant Thickness	163	(2)
Exercise 8.3 Determining Temperature Distribution in a Boiler's Water-Wall Tube by Means of Functional Correction Method	165	(4)
Literature	169	(2)
9 Two-Dimensional Steady-State Heat Conduction. Graphical Method	171	(12)
Exercise 9.1 Temperature Gradient and Surface-Transmitted Heat Flow	171	(2)
Exercise 9.2 Orthogonality of Constant Temperature Line and Constant Heat Flux	173	(3)
Exercise 9.3 Determining Heat Flow between Isothermal Surfaces	176	(3)
Exercise 9.4 Determining Heat Loss Through a Chimney Wall; Combustion Channel (Chimney) with Square Cross-Section	179	(2)
Exercise 9.5 Determining Heat Loss Through Chimney Wall with a Circular Cross-Section	181	(1)
Literature	182	(1)
10 Two-Dimensional Steady-State Problems. The Shape Coefficient	183	(6)
Exercise 10.1 Buried Pipe-to-Ground Surface Heat Flow	183	(2)
Exercise 10.2 Floor Heating	185	(1)
Exercise 10.3 Temperature of a Radioactive Waste Container Buried Underground	186	(1)
Literature	187	(2)
11 Solving Steady-State Heat Conduction Problems by Means of Numerical Methods	189	(120)
Exercise 11.1 Description of the Control Volume Method	189	(5)
Exercise 11.2 Determining Temperature Distribution in a Square Cross-Section of a Long Rod by Means of the Finite Volume Method	194	(5)
Exercise 11.3 A Two-Dimensional Inverse Steady-State Heat Conduction Problem	199	(5)
Exercise 11.4 Gauss-Seidel Method and Over-Relaxation Method	204	(4)
Exercise 11.5 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Uniform Thickness by Means of the Finite Volume Method	208	(7)
Exercise 11.6 Determining Two-Dimensional Temperature Distribution in a Square Cross-Section of a Chimney	215	(6)
Exercise 11.7 Pseudo-Transient Determination of Steady-State Temperature Distribution in a Square Cross-Section of a Chimney; Heat Transfer by Convection and Radiation on an Outer Surface of a Chimney	221	(9)
Exercise 11.8 Finite Element Method	230	(4)
Exercise 11.9 Linear Functions That Interpolate Temperature Distribution (Shape Functions) Inside Triangular and Rectangular Elements	234	(4)
Exercise 11.10 Description of FEM Based on Galerkin Method	238	(7)
Exercise 11.11 Determining Conductivity Matrix for a Rectangular and Triangular Element	245	(4)
Exercise 11.12 Determining Matrix [K&aLPHA;E in Terms of Convective Boundary Conditions for a Rectangular and Triangular Element	249	(4)
Exercise 11.13 Determining Vector {fQe} with Respect to Volumetric and Point Heat Sources in a Rectangular and Triangular Element	253	(3)
Exercise 11.14 Determining Vectors {fqe} and {fαe} with Respect to Boundary Conditions of 2nd and 3rd Kind on the Boundary of a Rectangular or Triangular Element	256	(3)
Exercise 11.15 Methods for Building Global Equation System in FEM	259	(5)
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)	264	(7)
Exercise 11.17 Determining Temperature Distribution in an Infinitely Long Rod with Square Cross-Section by Means of FEM, in which the Global Equation System is Constructed using Method II (from Ex. 11.15)	271	(4)
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	275	(10)
Exercise 11.19 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Constant Thickness by Means of FEM	285	(12)
Exercise 11.20 Determining Two-Dimensional Temperature Distribution by Means of I-EM in a Straight Fin with Constant Thickness (ANSYS Program)	297	(3)
Exercise 11.21 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Hexagonal Fin with Constant Thickness (ANSYS Program)	300	(3)
Exercise 11.22 Determining Axisymmetrical Temperature Distribution in a Cylindrical and Conical Pin by Means of FEM (ANSYS program)	303	(4)
Literature	307	(2)
12 Finite Element Balance Method and Boundary Element Method	309	(24)
Exercise 12.1 Finite Element Balance Method	309	(5)
Exercise 12.2 Boundary Element Method.	314	(9)
Exercise 12.3 Determining Temperature Distribution in Square Region by Means of FEM Balance Method	323	(4)
Exercise 12.4 Determining Temperature Distribution in a Square Region Using Boundary Element Method	327	(4)
Literature	331	(2)
13 Transient Heat Exchange Between a Body with Lumped Thermal Capacity and Its Surroundings	333	(20)
Exercise 13.1 Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings	333	(3)
Exercise 13.2 Heat Exchange between a Body with Lumped Thermal Capacity and Surroundings with Time-Dependent Temperature	336	(3)
Exercise 13.3 Determining Temperature Distribution of a Body with Lumped Thermal Capacity, when the Temperature of a Medium Changes Periodically	339	(1)
Exercise 13.4 Inverse Problem: Determining Temperature of a Medium on the Basis of Temporal Thermometer-Indicated Temperature History	340	(2)
Exercise 13.5 Calculating Dynamic Temperature Measurement Error by Means of a Thermocouple	342	(2)
Exercise 13.6 Determining the Time It Takes to Cool Body Down to a Given Temperature	344	(1)
Exercise 13.7 Temperature Measurement Error of a Medium whose Temperature Changes at Constant Rate	345	(1)
Exercise 13.8 Temperature Measurement Error of a Medium whose Temperature Changes Periodically	346	(1)
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	347	(2)
Exercise 13.10 Measuring Heat Flux	349	(2)
Literature	351	(2)
14 Transient Heat Conduction in Half-Space	353	(32)
Exercise 14.1 Laplace Transform	353	(2)
Exercise 14.2 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Surface Temperature	355	(3)
Exercise 14.3 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Heat Flux	358	(2)
Exercise 14.4 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Temperature of a Medium	360	(4)
Exercise 14.5 Formula Derivation for Temperature Distribution in a Half-Space when Surface Temperature is Time-Dependent	364	(2)
Exercise 14.6 Formula Derivation for a Quasi-Steady State Temperature Field in a Half-Space when Surface Temperature Changes Periodically	366	(8)
Exercise 14.7 Formula Derivation for Temperature of Two Contacting Semi-Infinite Bodies	374	(1)
Exercise 14.8 Depth of Heat Penetration	375	(2)
Exercise 14.9 Calculating Plate Surface Temperature under the Assumption that the Plate is a Semi-Infinite Body	377	(1)
Exercise 14.10 Calculating Ground Temperature at a Specific Depth	378	(1)
Exercise 14.11 Calculating the Depth of Heat Penetration in the Wall of a Combustion Engine	379	(1)
Exercise 14.12 Calculating Quasi-Steady-State Ground Temperature at a Specific Depth when Surface Temperature Changes Periodically	380	(2)
Exercise 14.13 Calculating Surface Temperature at the Contact Point of Two Objects	382	(1)
Literature	383	(2)
15 Transient Heat Conduction in Simple-Shape Elements	385	(84)
Exercise 15.1 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 3rd Kind	385	(9)
Exercise 15.2 A Program for Calculating Temperature Distribution and Its Change Rate in a Plate with Boundary Conditions of 3rd Kind	394	(4)
Exercise 15.3 Calculating Plate Surface Temperature and Average Temperature Across the Plate Thickness by Means of the Provided Graphs	398	(4)
Exercise 15.4 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind	402	(10)
Exercise 15.5 A Program for Calculating Temperature Distribution and Its Change Rate in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind	412	(4)
Exercise 15.6 Calculating Temperature in an Infinitely Long Cylinder using the Annexed Diagrams	416	(4)
Exercise 15.7 Formula Derivation for a Temperature Distribution in a Sphere with Boundary Conditions of 3rd Kind	420	(8)
Exercise 15.8 A Program for Calculating Temperature Distribution and Its Change Rate in a Sphere with Boundary Conditions of 3rd Kind	428	(4)
Exercise 15.9 Calculating Temperature of a Sphere using the Diagrams Provided	432	(4)
Exercise 15.10 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind	436	(5)
Exercise 15.11 A Program and Calculation Results for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind	441	(3)
Exercise 15.12 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind	444	(4)
Exercise 15.13 Program and Calculation Results for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind	448	(4)
Exercise 15.14 Formula Derivation for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind	452	(4)
Exercise 15.15 Program and Calculation Results for Temperature Distribution in a Sphere with Boundary Conditions of 2nd kind	456	(4)
Exercise 15.16 Heating Rate Calculations for a Thick-Walled Plate	460	(1)
Exercise 15.17 Calculating the Heating Rate of a Steel Shaft	461	(2)
Exercise 15.18 Determining Transients of Thermal Stresses in a Cylinder and a Sphere	463	(1)
Exercise 15.19 Calculating Temperature and Temperature Change Rate in a Sphere	464	(1)
Exercise 15.20 Calculating Sensor Thickness for Heat Flux Measuring	465	(2)
Literature	467	(2)
16 Superposition Method in One-Dimensional Transient Heat Conduction Problems	469	(46)
Exercise 16.1 Derivation of Duhamel Integral	469	(3)
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	472	(4)
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	476	(3)
Exercise 16.4 Derivation of an Approximate Formula for a Half-Space Surface Temperature when Temperature of a Medium Undergoes a Linear Change in the Function of Time	479	(2)
Exercise 16.5 Application of the Superposition Method when Initial Body Temperature is Non-Uniform	481	(3)
Exercise 16.6 Description of the Superposition Method Applied to Heat Transfer Problems with Time-Dependent Boundary Conditions	484	(4)
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	488	(3)
Exercise 16.8 Formula Derivation for a Half-Space Surface Temperature with a Mixed Step-Variable Boundary Condition in Time	491	(4)
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 Calculation of this Temperature	495	(5)
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	500	(3)
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	503	(3)
Exercise 16.12 Calculation of a Half-Space Temperature with a Mixed Step-Variable Boundary Condition in Time	506	(1)
Exercise 16.13 Calculating Plate Temperature by Means of the Superposition Method with Diagrams Provided	507	(2)
Exercise 16.14 Calculating the Temperature of a Paper in an Electrostatic Photocopier	509	(4)
Literature	513	(2)
17 Transient Heat Conduction in a Semi-Infinite body. The Inverse Problem	515	(26)
Exercise 17.1 Measuring Heat Transfer Coefficient. The Transient Method	515	(3)
Exercise 17.2 Deriving a Formula for Heat Flux on the Basis of Measured Half-Space Surface Temperature Transient Interpolated by a Piecewise Linear Function	518	(3)
Exercise 17.3 Deriving Heat Flux Formula on the Basis of a Measured and Polynomial-Approximated Half-Space Surface Temperature Transient	521	(2)
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	523	(4)
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	527	(1)
Exercise 17.6 Determining Heat Transfer Coefficient on the Plexiglass Plate Surface using the Transient Method	528	(4)
Graphical Method	529	(1)
Numerical Method	529	(3)
Exercise 17.7 Determining Heat Flux on the Basis of a Measured Time Transient of the Half-Space Temperature, Approximated by a Piecewise Linear Function	532	(3)
Exercise 17.8 Determining Heat Flux on the Basis of Measured Time Transient of a Polynomial-Approximated Half-Space Temperature	535	(4)
Literature	539	(2)
18 Inverse Transient Heat Conduction Problems	541	(32)
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	541	(4)
Plate	543	(1)
Cylinder	543	(1)
Sphere	544	(1)
Exercise 18.2 Formula Derivation for a Temperature of a Medium when Linear Time Change in Plate Surface Temperature is Assigned	545	(2)
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	547	(2)
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	549	(6)
Exercise 18.5 Determining Temperature and Heat Flux on the Plate Front Face on the Basis of a Measured Temperature Transient on an Insulated Back Surface; Heat Flow on the Plate Surface is in the Form of a Triangular Pulse	555	(7)
Exercise 18.6 Determining Temperature and Heat Flux on the Surface of a Plate Front Face on the Basis of a Measured Temperature Transient on an Insulated Back Surface; Heat Flow on the Plate Surface is in the Form of a Rectangular Pulse	562	(3)
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	565	(4)
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	569	(2)
Literature	571	(2)
19 Multidimensional Problems. The Superposition Method	573	(14)
Exercise 19.1 The Application of the Superposition Method to Multidimensional Problems	573	(4)
Exercise 19.2 Formula Derivation for Temperature Distribution in a Rectangular Region with a Boundary Condition of 3rd Kind	577	(3)
Exercise 19.3 Formula Derivation for Temperature Distribution in a Rectangular Region with Boundary Conditions of 2nd Kind	580	(2)
Exercise 19.4 Calculating Temperature in a Steel Cylinder of a Finite Height	582	(2)
Exercise 19.5 Calculating Steel Block Temperature	584	(3)
20 Approximate Analytical Methods for Solving Transient Heat Conduction Problems	587	(18)
Exercise 20.1 Description of an Integral Heat Balance Method by Means of a One-Dimensional Transient Heat Conduction Example	587	(3)
Exercise 20.2 Determining Transient Temperature Distribution in a Flat Wall with Assigned Conditions of 1st, 2nd and 3rd Kind	590	(10)
Exercise 20.3 Determining Thermal Stresses in a Flat Wall	600	(1)
Literature	600	(5)
21 Finite Difference Method	605	(54)
Exercise 21.1 Methods of Heat Flux Approximation on the Plate surface	606	(4)
Exercise 21.2 Explicit Finite Difference Method with Boundary Conditions of 1st, 2nd and 3rd Kind	610	(6)
Exercise 21.3 Solving Two-Dimensional Problems by Means of the Explicit Difference Method	616	(6)
Exercise 21.4 Solving Two-Dimensional Problems by Means of the Implicit Difference Method	622	(4)
Exercise 21.5 Algorithm and a Program for Solving a Tridiagonal Equation System by Thomas Method	626	(4)
Exercise 21.6 Stability Analysis of the Explicit Finite Difference Method by Means of the von Neumann Method	630	(4)
Exercise 21.7 Calculating One-Dimensional Transient Temperature Field by Means of the Explicit Method and a Computational Program	634	(5)
Exercise 21.8 Calculating One-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program	639	(5)
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	644	(8)
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	652	(4)
Literature	656	(3)
22 Solving Transient Heat Conduction Problems by Means of Finite Element Method (FEM)	659	(34)
Exercise 22.1 Description of FEM Based on Galerkin Method Used for Solving Two-Dimensional Transient Heat Conduction Problems	659	(3)
Exercise 22.2 Concentrating (Lumped) Thermal Finite Element Capacity in FEM	662	(6)
Exercise 22.3 Methods for Integrating Ordinary Differential Equations with Respect to Time Used in FEM	668	(3)
Exercise 22.4 Comparison of FEM Based on Galerkin Method and Heat Balance Method with Finite Volume Method	671	(3)
Exercise 22.5 Natural Coordinate System for One-Dimensional, Two-Dimensional Triangular and Two-Dimensional Rectangular Elements	674	(4)
Exercise 22.6 Coordinate System Transformations and Integral Calculations by Means of the Gauss-Legendre Quadratures	678	(9)
Exercise 22.7 Calculating Temperature in a Complex-Shape Fin by Means of the ANSYS Program	687	(3)
Literature	690	(3)
23 Numerical-Analytical Methods	693	(40)
Explicit Method	694	(1)
Implicit Method	694	(1)
Crank-Nicolson Method	694	(1)
Exercise 23.1 Integration of the Ordinary Differential Equation System by Means of the Runge-Kutta Method	695	(3)
Exercise 23.2 Numerical-Analytical Method for Integrating a Linear Ordinary Differential Equation System	698	(5)
Exercise 23.3 Determining Steel Plate Temperature by Means of the Method of Lines, while the Plate is Cooled by air and Boiling Water	703	(6)
Exercise 23.4 Using the Exact Analytical Method and the Method of lines to Determine Temperature of a Cylindrical Chamber	709	(5)
Exercise 23.5 Determining Thermal Stresses in a Cylindrical Chamber using the Exact Analytical Method and the Method of Lines	714	(4)
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	718	(6)
Exercise 23.7 Determining Transient Temperature Distribution in an Infinitely Long Rod with a Rectangular Cross-Section by Means of the Method of Lines	724	(5)
Literature	729	(4)
24 Solving Inverse Heat Conduction Problems by Means of Numerical Methods	733	(32)
Exercise 24.1 Numerical-Analytical Method for Solving Inverse Problems	733	(6)
Exercise 24.2 Step-Marching Method in Time Used for Solving Non-Linear Transient Inverse Heat Conduction Problems	739	(7)
Exercise 24.3 Weber Method Step-Marching Methods in Space	746	(5)
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 Shape of a Rectangular Pulse	751	(8)
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	759	(4)
Literature	763	(2)
25 Heat Sources	765	(34)
Exercise 25.1 Determining Formula for Transient Temperature Distribution Around an Instantaneous (Impulse) Point Heat Source Active in an Infinite Space	767	(3)
Exercise 25.2 Determining Formula for Transient Temperature Distribution in an Infinite Body Produced by an Impulse Surface Heat Source	770	(2)
Exercise 25.3 Determining Formula for Transient Temperature Distribution Around Instantaneous Linear Impulse Heat Source Active in an Infinite Space	772	(2)
Exercise 25.4 Determining Formula for Transient Temperature Distribution Around a Point Heat Source, which lies in an Infinite Space and is Continuously Active	774	(3)
Exercise 25.5 Determining Formula for a Transient Temperature Distribution Triggered by a Surface Heat Source Continuously Active in an Infinite Space	777	(2)
Exercise 25.6 Determining Formula for a Transient Temperature Distribution Around a Continuously Active Linear Heat Source with Assigned Power ·q per Unit of Length	779	(2)
Exercise 25.7 Determining Formula for Quasi-Steady-State Temperature 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	781	(4)
Exercise 25.8 Determining Formula for Transient Temperature Distribution Produced by a Point Heat Source with Power Q0 that Moves At Constant Velocity v in Infinite Space or on the Half Space Surface	785	(4)
Exercise 25.9 Calculating Temperature Distribution along a Straight Line Traversed by a Laser Beam	789	(3)
Exercise 25.10 Quasi-Steady State Temperature Distribution in a Plate During the Welding Process; a Comparison between the Analytical Solution and FEM	792	(4)
Literature	796	(3)
26 Melting and Solidification (Freezing)	799	(32)
Exercise 26.1 Determination of a Formula which Describes the Solidification (Freezing) and Melting of a Semi-Infinite Body (the Stefan Problem)	803	(5)
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	808	(3)
Exercise 26.3 Derivation of a Formula that Describes Quasi-Steady-State Solidification (Freezing) of a Flat Liquid Layer	811	(5)
Exercise 26.4 Derivation of Formulas that Describe Solidification (Freezing) of Simple-Shape Bodies: Plate, Cylinder and Sphere	816	(4)
Exercise 26.5 Ablation of a Semi-Infinite Body	820	(3)
Exercise 26.6 Solidification of a Falling Droplet of Lead	823	(2)
Exercise 26.7 Calculating the Thickness of an Ice Layer After the Assigned Time	825	(1)
Exercise 26.8 Calculating Accumulated Energy in a Melted Wax	826	(2)
Exercise 26.9 Calculating Fish Freezing Time	828	(1)
Literature	829	(2)
Appendix A Basic Mathematical Functions	831	(6)
A.1. Gauss Error Function	831	(2)
A.2. Hyperbolic Functions	833	(1)
A.3. Bessel Functions	834	(1)
Literature	835	(2)
Appendix B Thermo-Physical Properties of Solids	837	(26)
B.1. Tables of Thermo-Physical Properties of Solids	837	(19)
B.2. Diagrams	856	(2)
B.3. Approximated Dependencies for Calculating Thermo-Physical Properties of a Steel [8]	858	(3)
Literature	861	(2)
Appendix C Fin Efficiency Diagrams (for Chap. 6, Part II)	863	(4)
Literature	865	(2)
Appendix D Shape Coefficients for Isothermal Surfaces with Different Geometry (for Chap. 10, part II)	867	(12)
Appendix E Subprogram for Solving Linear Algebraic Equations System using Gauss Elimination Method (for Chap. 6, Part II)	879	(2)
Appendix F Subprogram for Solving a Linear Algebraic Equations System by Means of Over-Relaxation Method	881	(2)
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)	883	(2)
Appendix H Determining Inverse Laplace Transform for Chap. 15, Part II)	885
Literature	889