Taler / Duda Solving Direct and Inverse Heat Conduction Problems

1. Auflage 2010
ISBN: 978-3-540-33471-2
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 890 Seiten

This book presents a solution for direct and inverse heat conduction problems, discussing the theoretical basis for the heat transfer process and presenting selected theoretical and numerical problems in the form of exercises with solutions. The book covers one-, two- and three dimensional problems which are solved by using exact and approximate analytical methods and numerical methods. An accompanying CD-Rom includes computational solutions of the examples and extensive FORTRAN code.

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.

Autoren/Hrsg.

Taler, Jan

Duda, Piotr

Inhaltsverzeichnis

1;Preface;5
2;Table of contents
;7
3;Nomenclature;23
4;PART I Heat Conduction Fundamentals
;27
4.1;1 Fourier Law;28
4.1.1;Literature;31
4.2;2 Mass and Energy Balance Equations;32
4.2.1;2.1 Mass Balance Equation for a Solid that Moves at an Assigned Velocity
;32
4.2.2;2.2 Inner Energy Balance Equation;34
4.2.2.1;2.2.1 Energy Balance Equations in Three Basic Coordinate Systems
;37
4.2.3;2.3 Hyperbolic Heat Conduction Equation;41
4.2.4;2.4 Initial and Boundary Conditions;42
4.2.4.1;2.4.1 First Kind Boundary Conditions (Dirichlet Conditions)
;43
4.2.4.2;2.4.2 Second Kind Boundary Conditions von Neumann Conditions)
;43
4.2.4.3;2.4.3 Third Kind Boundary Conditions;44
4.2.4.4;2.4.4 Fourth Kind Boundary Conditions;46
4.2.4.5;2.4.5 Non-Linear Boundary Conditions;47
4.2.4.6;2.4.6 Boundary Conditions on the Phase Boundaries;49
4.2.5;Literature;51
4.3;3 The Reduction of Transient Heat Conduction Equations and Boundary Conditions
;53
4.3.1;3.1 Linearization of a Heat Conduction Equation;53
4.3.2;3.2 Spatial Averaging of Temperature;55
4.3.2.1;3.2.1 A Body Model with a Lumped Thermal Capacity;55
4.3.2.2;3.2.2 Heat Conduction Equation for a Simple Fin with Uniform Thickness
;57
4.3.2.3;3.2.3 Heat Conduction Equation for a Circular Fin with Uniform Thickness
;59
4.3.2.4;3.2.4 Heat Conduction Equation for a Circular Rod or a Pipe that Moves at Constant Velocity ;61
4.3.3;Literature;63
4.4;4 Substituting Heat Conduction Equation by Two-Equations System
;64
4.4.1;4.1 Steady-State Heat Conduction in a Circular Fin with Variable Thermal Conductivity and Transfer Coefficient
;64
4.4.2;4.2 One-Dimensional Inverse Transient Heat Conduction Problem
;66
4.4.3;Literature;69
4.5;5 Variable Change;70
4.5.1;Literature;73
5;Part II Exercises. Solving Heat Conduction Problems
;74
5.1;6 Heat Transfer Fundamentals;75
5.1.1;Exercise 6.1 Fourier Law in a Cylindrical Coordinate System
;75
5.1.1.1;Solution;76
5.1.2;Exercise 6.2 The Equivalent Heat Transfer Coefficient Accounting for Heat Exchange by Convection and Radiation
;77
5.1.2.1;Solution;78
5.1.3;Exercise 6.3 Heat Transfer Through a Flat Single-Layeredand Double-Layered Wall;79
5.1.3.1;Solution;79
5.1.4;Exercise 6.4 Overall Heat Transfer Coefficient and Heat Loss Through a Pipeline Wall
;82
5.1.4.1;Solution;83
5.1.5;Exercise 6.5 Critical Thickness of an Insulation on an Outer Surface of a Pipe
;84
5.1.5.1;Solution;85
5.1.6;Exercise 6.6 Radiant TubeTemperature;87
5.1.6.1;Solution;87
5.1.7;Exercise 6.7 Quasi-Steady-State of Temperature Distribution and Stresses in a Pipeline Wall
;90
5.1.7.1;Solution;91
5.1.8;Exercise 6.8 Temperature Distribution in a Flat Wall with Constant and Temperature Dependent Thermal Conductivity
;92
5.1.8.1;Solution;93
5.1.9;Exercise 6.9 Determining Heat Flux on the Basis of Measured Temperature at Two Points Using a Flat and Cylindrical Sensor
;96
5.1.9.1;Solution;97
5.1.10;Exercise 6.10 Determining Heat FluxBy Means of Gardon Sensor with a Temperature Dependent Thermal Conductivity
;99
5.1.10.1;Solution;100
5.1.11;Exercise 6.11 One-Dimensional Steady-State Plate Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources
;102
5.1.11.1;Solution;102
5.1.12;Exercise 6.12 One-Dimensional Steady-State Pipe Temperature Distribution Produced by Uniformly Distributed Volumetric Heat Sources
;104
5.1.12.1;Solution;105
5.1.13;Exercise 6.13 Inverse Steady-State Heat Conduction Problem in a Pipe
;107
5.1.13.1;Solution;107
5.1.14;Exercise 6.14 General Equation of Heat Conduction in Fins
;109
5.1.14.1;Solution;109
5.1.15;Exercise 6.15 Temperature Distribution and Efficiency of a Straight Fin with Constant Thickness
;111
5.1.15.1;Solution;111
5.1.16;Exercise 6.16 Temperature Measurement Error Caused by Thermal Conduction Through Steel Casing that Contains a Thermoelement as a Measuring Device
;114
5.1.16.1;Solution;115
5.1.17;Exercise 6.17 Temperature Distribution and Efficiency of a Circular Fin of Constant Thickness
;117
5.1.17.1;Solution;117
5.1.18;Exercise 6.18 Approximated Calculation of a Circular Fin Efficiency
;120
5.1.18.1;Solution;120
5.1.19;Exercise 6.19 Calculating Efficiency of Square and Hexagonal Fins
;121
5.1.19.1;Solution;122
5.1.20;Exercise 6.20 Calculating Efficiency of Hexagonal Fins by Means of an Equivalent Circular Fin Method and Sector Method
;124
5.1.20.1;Solution;125
5.1.21;Exercise 6.21 Calculating Rectangular Fin Efficiency;130
5.1.21.1;Solution;130
5.1.22;Exercise 6.22 HeatTransfer Coefficient in Exchangers with Extended Surfaces
;131
5.1.22.1;Solution;131
5.1.23;Exercise 6.23 Calculating Overall HeatTransfer Coefficient in a Fin Plate Exchanger
;136
5.1.23.1;Solution;136
5.1.24;Exercise 6.24 Overall HeatTransfer Coefficient for a Longitudinally Finned Pipewith a Scale Layer on an Inner Surface
;137
5.1.24.1;Solution;139
5.1.25;Exercise 6.25 Overall Heat Transfer Coefficient for a Longitudinally Finned Pipe
;141
5.1.25.1;Solution;141
5.1.26;Exercise 6.26 Determining One-Dimensional Temperature Distribution in a Flat Wall by Means of Finite Volume Method
;144
5.1.26.1;Solution;144
5.1.27;Exercise 6.27 Determining One-Dimensional Temperature Distribution in a Cylindrical Wall By Means of Finite Volume Method
;149
5.1.27.1;Solution;149
5.1.28;Exercise 6.28 Inverse Steady-State Heat Conduction Problem for a Pipe Solved by Space-Marching Method
;153
5.1.28.1;Solution;153
5.1.29;Exercise 6.29 Temperature Distribution and Efficiency of a Circular Fin with Temperature-Dependent Thermal Conductivity
;156
5.1.29.1;Solution;156
5.1.30;Literature;160
5.2;7 Two-Dimensional Steady-State Heat Conduction. Analytical Solutions
;162
5.2.1;Exercise 7.1 Temperature Distribution in an Infinitely Long Fin with Constant Thickness
;162
5.2.1.1;Solution;163
5.2.2;Exercise 7.2 Temperature Distribution in a Straight Fin with Constant Thickness and Insulated Tip
;166
5.2.2.1;Solution;166
5.2.3;Exercise 7.3 Calculating Temperature Distribution and Heat Flux in a Straight Fin with Constant Thickness and Insulated Tip
;169
5.2.3.1;Solution;170
5.2.4;Exercise 7.4 Temperature Distribution in a Radiant Tube of a Boiler
;177
5.2.4.1;Solution;178
5.2.5;Literature;181
5.3;8 Analytical Approximation Methods. Integral Heat Balance Method
;182
5.3.1;Exercise 8.1 Temperature Distribution within a Rectangular Cross-Section of a Bar
;182
5.3.1.1;Solution;182
5.3.2;Exercise 8.2 Temperature Distribution in an Infinitely Long Fin of Constant Thickness
;184
5.3.2.1;Solution;184
5.3.3;Exercise 8.3 Determining Temperature Distribution in a Boiler's Water-Wall Tube by Means of Functional Correction Method
;186
5.3.3.1;Solution;186
5.3.4;Literature;190
5.4;9 Two-Dimensional Steady-State Heat Conduction. Graphical Method
;191
5.4.1;Exercise 9.1 Temperature Gradient and Surface-Transmitted Heat Flow
;191
5.4.1.1;Solution;191
5.4.2;Exercise 9.2 Orthogonality of Constant Temperature Line and Constant Heat Flux
;193
5.4.2.1;Solution;193
5.4.3;Exercise 9.3 Determining Heat Flow between Isothermal Surfaces
;196
5.4.3.1;Solution;196
5.4.4;Exercise 9.4 Determining Heat Loss Through a Chimney Wall; Combustion Channel (Chimney) with Square Cross-Section
;199
5.4.4.1;Solution;199
5.4.5;Exercise 9.5 Determining Heat Loss Through Chimney Wall with a Circular Cross-Section
;201
5.4.5.1;Solution;201
5.4.6;Literature;202
5.5;10 Two-Dimensional Steady-State Problems. The Shape Coefficient
;203
5.5.1;Exercise 10.1 Buried Pipe-to-Ground Surface Heat Flow;203
5.5.1.1;Solution;203
5.5.2;Exercise 10.2 Floor Heating;205
5.5.2.1;Solution;205
5.5.3;Exercise 10.3 Temperature of a Radioactive Waste Container Buried Underground
;206
5.5.3.1;Solution;206
5.5.4;Literature;207
5.6;11 Solving Steady-State Heat Conduction Problems by Means of Numerical Methods
;208
5.6.1;Exercise 11.1 Description of the Control Volume Method;208
5.6.1.1;Solution;209
5.6.1.1.1;a) Heat balance equation- Cartesian coordinates;210
5.6.1.1.2;b) Heat balance equation-cylindrical coordinates;211
5.6.2;Exercise 11.2 Determining Temperature Distribution in a Square Cross-Section of a Long Rod by Means of the Finite Volume Method
;213
5.6.2.1;Solution;213
5.6.3;Exercise 11.3A Two-Dimensional Inverse Steady-State Heat Conduction Problem
;218
5.6.3.1;Solution;218
5.6.4;Exercise 11.4 Gauss-Seidel Method and Over-Relaxation Method
;223
5.6.4.1;Solution;223
5.6.5;Exercise 11.5 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Uniform Thickness by Means of the Finite Volume Method
;227
5.6.5.1;Solution;227
5.6.6;Exercise 11.6 Determining Two-Dimensional Temperature Distribution in a Square Cross-Section of a Chimney
;234
5.6.6.1;Solution;235
5.6.7;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
5.6.7.1;Solution;240
5.6.8;Exercise 11.8 Finite Element Method;249
5.6.8.1;Historical Development of FEM;249
5.6.8.2;Solution;250
5.6.9;Exercise 11.9 Linear Functions That Interpolate Temperature Distribution (Shape Functions) Inside Triangular and Rectangular Elements
;253
5.6.9.1;Solution;253
5.6.10;Exercise 11.10 Description of FEM Based on Galerkin Method
;257
5.6.10.1;Solution;257
5.6.11;Exercise 11.11 Determining Conductivity Matrix for a Rectangular and Triangular Element
;264
5.6.11.1;Solution;264
5.6.11.1.1;a) Conductivity matrix [Kec] for a finite rectangular element
;264
5.6.11.1.2;b) Conductivity matrix [Kec] for a finite triangular element
;266
5.6.12;Exercise 11.12 Determining Matrix [Kae] in Terms of Convective Boundary Conditions for a Rectangular and Triangular Element
;268
5.6.12.1;Solution;268
5.6.12.1.1;a) Rectangular finite element;268
5.6.12.1.2;b) Triangular finite element;270
5.6.13;Exercise 11.13 Determining Vector {fqe} with Respect to Volumetric and Point Heat Sources in a Rectangular and Triangular Element
;272
5.6.13.1;Solution;272
5.6.13.1.1;a) Rectangular element;272
5.6.13.1.2;b) Triangular element;273
5.6.14;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
5.6.14.1;Solution;275
5.6.14.1.1;a) Finite rectangular element;275
5.6.14.1.2;b) Finite triangular element;277
5.6.15;Exercise 11.15 Methods for Building Global Equation System in FEM
;278
5.6.15.1;Solution;279
5.6.16;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
5.6.16.1;Solution;284
5.6.17;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
5.6.17.1;Solution;290
5.6.18;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
5.6.18.1;Solution;294
5.6.19;Exercise 11.19 Determining Two-Dimensional Temperature Distribution in a Straight Fin with Constant Thickness by Means of FEM
;304
5.6.19.1;Solution;304
5.6.20;Exercise 11.20 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Straight Fin with Constant Thickness (ANSYS Program)
;316
5.6.20.1;Solution;316
5.6.21;Exercise 11.21 Determining Two-Dimensional Temperature Distribution by Means of FEM in a Hexagonal Fin with Constant Thickness (ANSYS Program)
;319
5.6.21.1;Solution;319
5.6.22;Exercise 11.22 Determining Axisymmetrical Temperature Distribution in a Cylindrical and Conical Pin by Means of FEM (ANSYS Program)
;322
5.6.22.1;Solution;323
5.6.23;Literature;326
5.7;12 Finite Element Balance Method and Boundary Element Method
;327
5.7.1;Exercise 12.1 Finite Element Balance Method;327
5.7.1.1;Solution;327
5.7.2;Exercise 12.2 Boundary Element Method;332
5.7.2.1;Solution;332
5.7.3;Exercise 12.3 Determining Temperature Distribution in Square Region by Means of FEM Balance Method
;341
5.7.3.1;Solution;341
5.7.4;Exercise 12.4 Determining Temperature Distribution in a Square Region using Boundary Element Method
;345
5.7.4.1;Solution;346
5.7.5;Literature;349
5.8;13 Transient Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings
;350
5.8.1;Exercise 13.1 Heat Exchange between a Body with Lumped Thermal Capacity and Its Surroundings
;350
5.8.1.1;Solution;350
5.8.2;Exercise 13.2 Heat Exchange between a Body with Lumped Thermal Capacity and Surroundings with Time-Dependent Temperature
;353
5.8.2.1;Solution;353
5.8.3;Exercise 13.3 Determining Temperature Distribution of a Body with Lumped Thermal Capacity, when the Temperature of a Medium Changes Periodically
;356
5.8.3.1;Solution;356
5.8.4;Exercise 13.4 Inverse Problem: Determining Temperature of a Medium on the Basis of Temporal Thermometer Indicated Temperature History
;357
5.8.4.1;Solution;357
5.8.5;Exercise 13.5 Calculating Dynamic Temperature Measurement Error by Means of a Thermocouple
;359
5.8.5.1;Solution;360
5.8.6;Exercise 13.6 Determining the Time It Takes to Cool Body Down to a Given Temperature
;361
5.8.6.1;Solution;361
5.8.7;Exercise 13.7 Temperature Measurement Error of a Medium whoseTemperature Changes at Constant Rate
;362
5.8.7.1;Solution;362
5.8.8;Exercise 13.8 Temperature Measurement Error of a Medium whose Temperature Changes Periodically
;363
5.8.8.1;Solution;363
5.8.9;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
5.8.9.1;Solution;364
5.8.10;Exercise 13.10 Measuring Heat Flux;366
5.8.10.1;Solution;367
5.8.11;Literature;368
5.9;14 Transient Heat Conduction in Half-Space;369
5.9.1;Exercise 14.1 Laplace Transform;369
5.9.1.1;Solution;369
5.9.2;Exercise 14.2 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Surface Temperature
;371
5.9.2.1;Solution;372
5.9.3;Exercise 14.3 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increasein Heat Flux
;374
5.9.3.1;Solution;374
5.9.4;Exercise 14.4 Formula Derivation for Temperature Distribution in a Half-Space with a Step Increase in Temperature of a Medium
;376
5.9.4.1;Solution;376
5.9.5;Exercise 14.5 Formula Derivation for Temperature Distribution in a Half-Space when Surface Temperature isTime-Dependent;380
5.9.5.1;Solution;380
5.9.6;Exercise 14.6 Formula Derivation for a Quasi-Steady State Temperature Field in a Half-Space when Surface Temperature Changes Periodically
;382
5.9.6.1;Solution;382
5.9.7;Exercise 14.7 Formula Derivation for Temperature of Two Contacting Semi-Infinite Bodies
;390
5.9.7.1;Solution;390
5.9.8;Exercise 14.8 Depth of Heat Penetration;391
5.9.8.1;Solution;392
5.9.9;Exercise 14.9 Calculating Plate Surface Temperature Under the Assumption that the Plate is a Semi-Infinite Body
;393
5.9.9.1;Solution;393
5.9.10;Exercise 14.10 Calculating Ground Temperature at a Specific Depth
;394
5.9.10.1;Solution;394
5.9.11;Exercise 14.11 Calculating the Depth of Heat Penetration in the Wall of a Combustion Engine
;395
5.9.11.1;Solution;395
5.9.12;Exercise 14.12 Calculating auasi-Steady-State Ground Temperature at a Specific Depth when Surface Temperature Changes Periodically
;396
5.9.12.1;Solution;396
5.9.13;Exercise 14.13 Calculating Surface Temperature at the Contact Point of Two Objects
;398
5.9.13.1;Solution;398
5.9.14;Literature;399
5.10;15 Transient Heat Conduction in Simple-Shape Elements
;400
5.10.1;Exercise 15.1 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 3rd Kind
;400
5.10.1.1;Solution;400
5.10.2;Exercise 15.2 A Program for Calculating Temperature Distribution and Its Change Rate in a Plate with Boundary Conditions of 3rd Kind
;409
5.10.2.1;Solution;410
5.10.3;Exercise 15.3 Calculating Plate Surface Temperature and Average Temperature Across the Plate Thickness by Means of the Provided Graphs
;413
5.10.3.1;Solution;413
5.10.4;Exercise 15.4 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 3rd Kind
;417
5.10.4.1;Solution;417
5.10.5;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
5.10.5.1;Solution;428
5.10.6;Exercise 15.6 Calculating Temperature in an Infinitely Long Cylinder using the Annexed Diagrams
;431
5.10.6.1;Solution;431
5.10.7;Exercise 15.7 Formula Derivation for a Temperature Distribution in a Sphere with Boundary Conditionsof 3rd Kind
;435
5.10.7.1;Solution;435
5.10.8;Exercise 15.8 A Program for Calculating Temperature Distribution and Its Change Rate in a Sphere with Boundary Conditions of 3rd Kind
;443
5.10.8.1;Solution;443
5.10.9;Exercise 15.9 Calculating Temperature of a Sphere using the Diagrams Provided
;447
5.10.9.1;Solution;447
5.10.10;Exercise 15.10 Formula Derivation for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind
;451
5.10.10.1;Solution;451
5.10.11;Exercise 15.11 A Program and Calculation Results for Temperature Distribution in a Plate with Boundary Conditions of 2nd Kind
;456
5.10.11.1;Solution;456
5.10.12;Exercise 15.12 Formula Derivation for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind
;459
5.10.12.1;Solution;459
5.10.13;Exercise 15.13 Program and Calculation Results for Temperature Distribution in an Infinitely Long Cylinder with Boundary Conditions of 2nd Kind
;463
5.10.13.1;Solution;463
5.10.14;Exercise 15.14 Formula Derivation for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind
;467
5.10.14.1;Solution;467
5.10.15;Exercise 15.15 Program and Calculation Results for Temperature Distribution in a Sphere with Boundary Conditions of 2nd Kind
;471
5.10.15.1;Solution;471
5.10.16;Exercise 15.16 Heating Rate Calculations for a Thick-Walled Plate
;475
5.10.16.1;Solution;475
5.10.17;Exercise 15.17 Calculating the Heating Rate of a Steel Shaft
;476
5.10.17.1;Solution;477
5.10.18;Exercise 15.18 Determining Transients of Thermal Stresses in a Cylinder and a Sphere
;478
5.10.18.1;Solution;478
5.10.19;Exercise 15.19 Calculating Temperature and Temperature Change Rate in a Sphere
;479
5.10.19.1;Solution;479
5.10.20;Exercise 15.20 Calculating Sensor Thickness for Heat Flux Measuring
;480
5.10.20.1;Solution;480
5.10.21;Literature;482
5.11;16 Superposition Method in One-Dimensional Transient Heat Conduction Problems
;483
5.11.1;Exercise 16.1 Derivation of Duhamel Integral;483
5.11.1.1;Solution;483
5.11.2;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
5.11.2.1;Solution;487
5.11.3;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
5.11.3.1;Solution;491
5.11.4;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
5.11.4.1;Solution;493
5.11.5;Exercise 16.5 Application of the Superposition Method when Initial Body Temperature is Non-Uniform
;495
5.11.5.1;Solution;495
5.11.6;Exercise 16.6 Description of the Superposition Method Applied to Heat Transfer Problems with Time-Dependent Boundary Conditions
;498
5.11.6.1;Solution;498
5.11.6.1.1;Example 1;499
5.11.6.1.2;Example 2;500
5.11.6.1.3;Example 3;501
5.11.6.1.4;Example 4;501
5.11.7;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
5.11.7.1;Solution;503
5.11.8;Exercise 16.8 Formula Derivation for a Half-Space Surface Temperature with a Mixed Step-Variable Boundary Condition in Time
;505
5.11.8.1;Solution;505
5.11.9;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
5.11.9.1;Solution;510
5.11.10;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
5.11.10.1;Solution;514
5.11.11;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
5.11.11.1;Solution;517
5.11.12;Exercise 16.12 Calculation of a Half-Space Temperature with a Mixed Step-Variable Boundary Condition in Time
;520
5.11.12.1;Solution;521
5.11.13;Exercise 16.13 Calculating Plate Temperature by Means of the Superposition Method with Diagrams Provided
;521
5.11.13.1;Solution;522
5.11.14;Exercise 16.14 Calculating the Temperature of a Paper in an Electrostatic Photocopier
;523
5.11.14.1;Solution;525
5.11.15;Literature;527
5.12;17 Transient Heat Conduction in a Semi-Infinite Body. The Inverse Problem
;528
5.12.1;Exercise 17.1 Measuring Heat Transfer Coefficient. The Transient Method
;528
5.12.1.1;Solution;528
5.12.2;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
5.12.2.1;Solution;532
5.12.3;Exercise 17.3 Deriving Heat Flux Formula on the Basis of a Measured and Polynomial-Approximated Half-Space Surface Temperature Transient
;534
5.12.3.1;Solution;534
5.12.4;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
5.12.4.1;Solution;536
5.12.5;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
5.12.5.1;Solution;540
5.12.6;Exercise 17.6 Determining Heat Transfer Coefficient on the Plexiglass Plate Surface using the Transient Method
;541
5.12.6.1;Solution;542
5.12.7;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
5.12.7.1;Solution;545
5.12.8;Exercise 17.8 Determining Heat Flux on the Basis of Measured Time Transient of a Polynomial-Approximated Half-Space Temperature
;548
5.12.8.1;Solution;549
5.12.9;Literature;552
5.13;18 Inverse Transient Heat Conduction Problems
;553
5.13.1;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
5.13.1.1;Solution;554
5.13.2;Exercise 18.2 Formula Derivation for a Temperature of a Medium when Linear Time Change in Plate Surface Temperature is Assigned
;557
5.13.2.1;Solution;557
5.13.3;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
5.13.3.1;Solution;559
5.13.4;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
5.13.4.1;Solution;561
5.13.5;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
5.13.5.1;Solution;567
5.13.6;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
5.13.6.1;Solution;575
5.13.7;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
5.13.7.1;Solution;578
5.13.8;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
5.13.8.1;Solution;581
5.13.9;Literature;583
5.14;19 Multidimensional Problems. The Superposition Method
;585
5.14.1;Exercise 19.1 The Application of the Superposition Method to Multidimensional Problems
;585
5.14.1.1;Solution;585
5.14.1.1.1;Boundary Conditions of 1st and 3rd Kind
;585
5.14.1.1.2;Boundary Conditions of 2nd Kind
;586
5.14.2;Exercise 19.2 Formula Derivation for Temperature Distribution in a Rectangular Region with a Boundary Condition of 3rd Kind
;589
5.14.2.1;Solution;589
5.14.3;Exercise 19.3 Formula Derivation for Temperature Distribution in a Rectangular Region with Boundary Conditions of 2nd Kind
;592
5.14.3.1;Solution;592
5.14.4;Exercise 19.4 Calculating Temperature in a Steel Cylinder of a Finite Height
;594
5.14.4.1;Solution;594
5.14.5;Exercise 19.5 Calculating Steel Block Temperature;596
5.14.5.1;Solution;596
5.15;20 Approximate Analytical Methods for Solving Transient Heat Conduction Problems
;599
5.15.1;Exercise 20.1 Description of an Integral Heat Balance Method by Means of a One-Dimensional Transient Heat Conduction Example
;599
5.15.1.1;Solution;599
5.15.2;Exercise 20.2 Determining Transient Temperature Distribution in a Flat Wall with Assigned Conditions of 1st, 2nd and 3rd Kind
;602
5.15.2.1;Solution;602
5.15.2.1.1;Example 1;606
5.15.2.1.2;Example 2;607
5.15.2.1.3;Example 3;610
5.15.2.1.4;Example 4;611
5.15.3;Exercise 20.3 Determining Thermal Stresses in a Flat Wall;612
5.15.3.1;Solution;612
5.15.4;Literature;614
5.16;21 Finite Difference Method;617
5.16.1;Exercise 21.1 Methods of Heat Flux Approximation on the Plate Surface
;618
5.16.1.1;Solution;618
5.16.2;Exercise 21.2 Explicit Finite Difference Methodwith Boundary Conditions of 1st, 2nd and 3rd Kind;622
5.16.2.1;Solution;623
5.16.3;Exercise 21.3 Solving Two-Dimensional Problems by Means of the Explicit Difference Method
;628
5.16.3.1;Solution;629
5.16.4;Exercise 21.4 Solving Two-Dimensional Problems by Means of the Implicit Difference Method
;634
5.16.4.1;Solution;634
5.16.5;Exercise 21.5 Algorithm and a Program for Solving a Tridiagonal Equation System by Thomas Method
;638
5.16.5.1;Solution;638
5.16.6;Exercise 21.6 Stability Analysis of the Explicit Finite Difference Method by Means of the von Neumann Method
;642
5.16.6.1;Solution;642
5.16.7;Exercise 21.7 Calculating One-Dimensional Transient Temperature Field by Means of the Explicit Method and a Computational Program
;646
5.16.7.1;Solution;647
5.16.8;Exercise 21.8 Calculating One-Dimensional Transient Temperature Field by Means of the Implicit Method and a Computational Program
;651
5.16.8.1;Solution;651
5.16.9;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
5.16.9.1;Solution;656
5.16.10;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
5.16.10.1;Solution;664
5.16.11;Literature;668
5.17;22 Solving Transient Heat Conduction Problems by Means of Finite Element Method (FEM)
;670
5.17.1;Exercise 22.1 Description of FEM Based on GalerkinMethod Used for Solving Two-Dimensional Transient Heat Conduction Problems
;670
5.17.1.1;Solution;670
5.17.2;Exercise 22.2 Concentrated (Lumped) Thermal Finite Element Capacity in FEM
;673
5.17.2.1;Solution;673
5.17.3;Exercise 22.3 Methods for Integrating Ordinary Differential Equations with Respect to Time Used in FEM
;679
5.17.3.1;Solution;679
5.17.4;Exercise 22.4 Comparison of FEM Based on Galerkin Method and Heat Balance Method with Finite Volume Method
;682
5.17.4.1;Solution;682
5.17.5;Exercise 22.5 Natural Coordinate System for One-Dimensional, Two-Dimensional Triangular and Two-Dimensional Rectangular Elements
;685
5.17.5.1;Solution;685
5.17.5.1.1;a. One-dimensional elements;685
5.17.5.1.2;b. Two-dimensional tetragonal elements;686
5.17.5.1.3;c. Two-dimensional triangular elements;687
5.17.6;Exercise 22.6 Coordinate System Transformations and Integral Calculations by Means of the Gauss-Legendre Quadratures
;689
5.17.6.1;Solution;689
5.17.6.1.1;a. One-dimensional elements;693
5.17.6.1.2;b. Tetragonal elements;695
5.17.6.1.3;c. Triangular elements;696
5.17.7;Exercise 22.7 Calculating Temperature in a Complex ShapeFin by Means of the ANSVS Program
;698
5.17.7.1;Solution;700
5.17.8;Literature;701
5.18;23 Numerical-Analytical Methods;703
5.18.1;Explicit Method;704
5.18.2;Implicit Method;704
5.18.3;Crank-Nicolson Method;704
5.18.4;Exercise 23.1 Integration of the Ordinary Differential Equation System by Means of the Runge-Kutta Method
;705
5.18.4.1;Solution;706
5.18.5;Exercise 23.2 Numerical-Analytical Method for Integrating a Linear Ordinary Differential Equation System
;708
5.18.5.1;Solution;708
5.18.5.1.1;a. Approximating u(t) with a step function
;709
5.18.5.1.2;b. Approximating u(t) with a piecewise linear function
;711
5.18.6;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
5.18.6.1;Solution;713
5.18.7;Exercise 23.4 Using the Exact Analytical Method and the Method of Lines to Determine Temperature of a Cylindrical Chamber
;719
5.18.7.1;Solution;719
5.18.8;Exercise 23.5 Determining Thermal Stresses in a Cylindrical Chamber using the Exact Analytical Method and the Method of Lines
;724
5.18.8.1;Solution;725
5.18.9;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
5.18.9.1;Solution;730
5.18.10;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
5.18.10.1;Solution;735
5.18.11;Literature;739
5.19;24 Solving Inverse Heat Conduction Problems by Means of Numerical Methods
;742
5.19.1;Exercise 24.1 Numerical-Analytical Method for Solving Inverse Problems
;742
5.19.1.1;Solution;743
5.19.1.1.1;a. Division of an inverse region into two control volumes (Fig. 24.2a)
;745
5.19.1.1.2;b. Division of an inverse region into three control volumes (Fig. 24.2b)
;746
5.19.1.1.3;c. Division of an inverse region into four control volumes (Fig. 24.2c)
;747
5.19.2;Exercise 24.2 Step-Marching Method in Time Used for Solving Non-Linear Transient Inverse Heat Conduction Problems
;748
5.19.2.1;Solution;749
5.19.3;Exercise 24.3 Weber Method Step-Marching Methods in Space
;755
5.19.3.1;Solution;755
5.19.4;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
5.19.4.1;Solution;761
5.19.5;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
5.19.5.1;Solution;768
5.19.6;Literature;772
5.20;25 Heat Sources;773
5.20.1;Exercise 25.1 Determining Formula for Transient Temperature Distribution Around an Instantaneous (Impulse) Point Heat Source Active in an Infinite Space
;775
5.20.1.1;Solution;775
5.20.2;Exercise 25.2 Determining Formula for Transient Temperature Distribution in an Infinite Body Produced by an Impulse Surface Heat Source
;778
5.20.2.1;Solution;778
5.20.3;Exercise 25.3 Determining Formula for Transient Temperature Distribution Around Instantaneous Linear Impulse Heat Source Active in an Infinite Space
;780
5.20.3.1;Solution;780
5.20.4;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
5.20.4.1;Solution;782
5.20.5;Exercise 25.5 Determining Formula for a Transient Temperature Distribution Triggered by a Surface Heat Source Continuously Active in an Infinite Space
;785
5.20.5.1;Solution;785
5.20.6;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
5.20.6.1;Solution;787
5.20.7;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
5.20.7.1;Solution;789
5.20.8;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
5.20.8.1;Solution;793
5.20.9;Exercise 25.9 Calculating Temperature Distribution along a Straight Line Traversed by a Laser Beam
;797
5.20.9.1;Solution;797
5.20.10;Exercise 25.10 Quasi-Steady State Temperature Distribution in a Plate During the Welding Process; A Comparison between the Analytical Solution and FEM
;800
5.20.10.1;Solution;800
5.20.11;Literature;804
5.21;26 Melting and Solidification (Freezing);806
5.21.1;Exercise 26.1 Determination of a Formula which Describes the Solidification (Freezing) and Melting of a Semi-Infinite Body (the Stefan Problem)
;810
5.21.1.1;Solution;810
5.21.2;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
5.21.2.1;Solution;815
5.21.3;Exercise 26.3 Derivation of a Formula that Describe Quasi-Steady-State Solidification (Freezing) of a Flat Liquid Layer
;818
5.21.3.1;Solution;819
5.21.4;Exercise 26.4 Derivation of Formulas that Describe Solidification (Freezing) of Simple-Shape Bodies: Plate, Cylinder and Sphere
;823
5.21.4.1;Solution;823
5.21.4.1.1;a. Plate;825
5.21.4.1.2;b. Cylinder;825
5.21.5;Exercise 26.5 Ablation of a Semi-Infinite Body;827
5.21.5.1;Solution;828
5.21.6;Exercise 26.6 Solidification of a Falling Droplet of Lead;830
5.21.6.1;Solution;830
5.21.7;Exercise 26.7 Calculating the Thickness of an Ice Layer After the Assigned Time
;832
5.21.7.1;Solution;832
5.21.8;Exercise 26.8 Calculating Accumulated Energy in a Melted Wax
;833
5.21.8.1;Solution;834
5.21.9;Exercise 26.9 Calculating Fish Freezing Time;835
5.21.9.1;Solution;835
5.21.10;Literature;836
6;Appendix A Basic Mathematical Functions;837
6.1;A.1. Gauss Error Function;837
6.2;A.2. Hyperbolic Functions;839
6.3;A.3. Bessel Functions;840
6.4;Literature;841
7;Appendix B Thermo-Physical Properties of Solids;842
7.1;B.1. Tables of Thermo-Physical Properties of Solids;842
7.2;B.2. Diagrams;861
7.3;B.3. Approximated Dependencies for Calculating Thermo PhysicalProperties of a Steel [8]
;863
7.3.1;Densityp at temperature 20°C;863
7.3.2;Specific heat capacity c in a temperature function;864
7.3.3;Longitudinal elasticity module (Young's modulus) E in function of temperature
;865
7.3.4;Average temperature expansion coefficient ß within temperature interval from 20°C to a given temperature T expressed in [°C]
;865
7.3.5;Poisson ratio v in function of temperature;865
7.4;Literature;866
8;Appendix C Fin Efficiency Diagrams (for Chap. 6, part II)
;867
8.1;Literature;869
9;Appendix D Shape Coefficients for Isothermal Surfaces with Different Geometry (for Chap. 10, Part II)
;870
10;Appendix E Subprogram for Solving Linear Algebraic Equations System using Gauss Elimination Method (for Chap. 6, Part II)
;882
10.1;Subprogram for solving linearalgebraic equations system using Gauss method
;882
11;Appendix F Subprogram for Solving a Linear Algebraic Equations System by Means of Over Relaxation Method
;884
11.1;Subprogram SOR section appendix f subprogram,for solving a linear algebraic equations system by means of over-relaxation method
;884
12;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
12.1;Subprogram for solving an ordinary differential equations system of 1st order using Runge-Kutta method of 4th order
;885
13;Appendix H Determining inverse Laplace Transform (for Chap. 15, part II)
;886
13.1;Literature;890

Leseproben

"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)."