Thermal quadrupoles: solving the heat equation through integral transforms
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Chichester [u.a.]
Wiley
2000
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 370 S. graph. Darst. |
| ISBN: | 0471983209 |
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| 245 | 1 | 0 | |a Thermal quadrupoles |b solving the heat equation through integral transforms |c Denis Maillet ... |
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|---|---|
| adam_text | Titel: Thermal quadrupoles
Autor: Maillet, Denis
Jahr: 2000
Contents
Preface...................................................... xi
1 Interest in the Quadrupole Approach....................... l
1.1 The Quadrupole Method: Use, Origin and Presentation in this Book........ 1
1.2 Definition of the Quadrupole Method ............................. 3
1.3 How Can it be Written?....................................... 4
1.3.1 Case a: Steady-State Transfer.............................. 4
1.3.2 Case b: A Lumped Body in Transient Conditions................ 5
1.3.3 Case c: The General Case ................................ 7
1.3.4 Case d: A One-layer Slab in Transient Transfer - the Analytical
Quadrupole Matrix ..................................... 10
1.4 How Does it Work in Concrete Example? .......................... 12
1.4.1 Heat Pulse on an Insulated Slab............................ 12
1.4.2 Analytical Asymptotic Solutions............................ 14
1.4.3 Other Types of Transient Excitations......................... 16
1.4.4 Quadrupole Toolbox for Multimaterials....................... 18
1.5 Interest of Analytical Methods .................................. 25
1.5.1 The Heat Pulse Experiment with Losses ...................... 25
1.5.2 Non-Destructive Thermal Testing of Delamination in
Stratified Media ....................................... 26
Appendices................................................ 28
Bibliography............................................... 31
2 Linear Conduction and Simple Geometries ................. 37
2.1 Introduction ............................................... 37
2.2 Fundamental Concepts........................................ 37
2.2.1 Heat Flux............................................ 37
2.2.2 Heat Flux Density ...................................... 38
2.2.3 Internal Source ........................................ 39
2.2.4 Fourier s Law ......................................... 40
2.2.5 Homogeneous Medium .................................. 43
2.3 The Heat Equation and Associated Conditions in Rectangular
Coordinates ............................................... 43
2.3.1 The Heat Equation ..................................... 43
2.3.2 Boundaries and Boundary Conditions ........................ 44
2.3.3 The Case of Several Media and Interface Conditions ............. 51
2.3.4 Initial Condition....................................... 55
2.4 Heat Equation and Associated Conditions in Cylindrical and Spherical
Coordinates ............................................... 55
2.4.1 Cylindrical Coordinates.................................. 55
CONTENTS
2.4.2 Spherical Coordinates.................................. 58
2.5 Classification of the Different Problem Types for Quadrupole Processing .... 61
References................................................ 63
One-Dimensional Quadrupoles ........ ................... 65
3.1 Introduction .............................................. 65
3.2 One-Dimensional Steady-State Conduction: A Recall................. 65
3.2.1 Notion of Thermal Resistance ............................ 66
3.2.2 Conduction with an Internal Source........... ............ 69
3.3 Strictly One-Dimensional Transient Transfer ....................... 73
3.3.1 The Notion of a Thermal Quadrupole: the Passive Wall ......... 73
3.3.2 Quadrupole Associated with a Solid-Solid Contact ............. 84
3.3.3 Quadrupole Associated with a Fourier Boundary Condition........ 84
3.3.4 Localized Heat Sources................................. 85
3.3.5 Internal Sources and Initial Temperature Imbalance ............. 87
3.3.6 Application to Multilayered Materials....................... 89
3.3.7 Other Geometries ..................................... 92
3.4 Two- or Three-Dimensional Transfer that can be Modelled by a
Global One-Dimensional Transfer or a Transfer without a
Space Dimension .......................................... 94
3.4.1 The Lumped Body Approach............................. 95
3.4.2 The Fin Approximation................................. 96
3.4.3 Use of Average Temperatures............................. 100
3.5 Extensions of the Method in One-Directional Transfer................. 101
3.5.1 Quadrupole Associated with a Fluid Element................. 102
3.5.2 Passive Quadrupoles Associated with Media with Sources......... 103
3.6 Problems with Solutions...................................... 105
Appendices............................................... 145
Bibliography.............................................. 148
Multidimensional Transfers ............................... 149
4.1 Introduction and Principle..................................... 149
4.1.1 Introduction ......................................... 149
4.1.2 Principle............................................ 151
4.2 Choice of the Integral Transformations........................... 152
4.2.1 General Method ...................................... 152
4.2.2 Example in Two-Dimensional Steady-State Transfer............. 153
4.2.3 Some Usual Integral Transforms........................... 156
4.3 Transfer in a Homogeneous Wall Without an Internal Source and
Without Initial Temperature Imbalance - the Case of Homogeneous
Boundary Conditions........................................ 159
4.3.1 A Simple Case in Two Dimensions......................... 159
4.3.2 The General Case..................................... 161
4.3.3 An Example in the Cylindrical Coordinate System.............. 164
4.3.4 Heat Transfer in a Conveyor Belt.......................... 166
4.3.5 Radial and Azimuthal Heat Transfer in a Cylinder .............. 169
4.3.6 Convective Boundary Conditions and Surface Heat Source ........ 171
CONTENTS vii
4.4 Study of Some Non-Homogeneous Problems ....................... 172
4.4.1 A Two-Dimensional Steady-State Example.................... 173
4.4.2 Non-zero Initial Temperature Terms and Internal
Power Generation..................................... 174
4.4.3 Quadrupole Formulation of a Source Term ................... 176
4.5 Two- or Three-Dimensional Transfers in Multilayered Materials.......... 178
4.5.1 A Stack of Several Materials in Perfect Contact................ 178
4.5.2 Multilayer with Contact Resistance and Internal Sources.......... 180
4.5.3 Multilayer with Contact Resistances, Internal Sources and
Surface Heat Generation ................................ 180
4.5.4 Example: Two-layer Material with Interface Resistance Heated by a
Non-uniform Heat Pulse ................................ 181
References ............................................... 182
5 Time-Dependent Periodic Regimes......................... 185
5.1 Introduction .............................................. 185
5.2 Semi-Infinite Medium with Sine Temperature Variation at the Front
Surface - Resolution from the Initial State with Laplace Transforms....... 185
5.3 Semi-Infinite Medium with Sine Temperature Variation at the Front
Surface - Resolution at the Steady State with a Complex Amplitude....... 187
5.4 Generalization to the Fourier Transform and Implementation of the
Quadrupole Method......................................... 190
5.4.1 Different Forms of a Steady Periodic Temperature .............. 191
5.4.2 A Semi-Infinite or Finite Slab in a Steady Periodic Regime........ 192
5.5 Local Surface Excitation of a Rotating Hollow Cylinder................ 197
5.6 Problem with Solution....................................... 203
Problem 5.1: Multilayered Slab in a Periodic Regime and
Equivalent Homogeneous Medium............................... 203
Appendix 5.1 Code for the Rotating Cylinder Problem in MATLAB*
Language................................................ 208
References................................................... 208
6 Advanced Quadrupoles.................................... 211
6.1 The Notion of the Constriction of Flux Lines....................... 211
6.1.1 The Steady State...................................... 211
6.1.2 The Transient Regime.................................. 220
6.1.3 Applications......................................... 233
6.2 Space-Varying Interface Resistance and Modal Approach .............. 241
6.2.1 Non-Uniform Interface Resistance Under Steady-State
Conditions.......................................... 242
6.2.2 Non-Uniform Interface Resistance under Transient Conditions...... 249
6.3 Intermittent Contact and the Modal Approach ...................... 252
6.4 Numerical Quadrupoles ...................................... 257
6.5 Problem with Solution....................................... 262
Problem 6.1: Steady-state Cooling of a Power Electronics Board ......... 262
Appendices............................................... 265
Bibliography.............................................. 273
vili CONTENTS
7 Mass Transfer in a Porous Medium........................ 275
7.1 Introduction .............................................. 275
7.2 Diffusion and Dispersion in a Porous Medium ...................... 275
7.2.1 Dispersion in a Saturated Porous Medium.................... 275
7.2.2 Solution in Terms of Quadrupoles.......................... 277
7.2.3 Representation in Terms of Impedances...................... 278
7.2.4. The Case of a Mass Source in an Infinite Medium.............. 280
7.2.5 The Case of a Finite Domain............................. 282
7.3 Examples................................................ 283
Appendices............................................... 287
Bibliography.............................................. 293
8 The Quadrupole Approach Applied to Heat Transfer in
Semi-Transparent Materials............................... 295
8.1 Conductive and Radiative Transfer .............................. 295
8.1.1 The Radiative Transfer Equation (RTE)...................... 295
8.1.2 The Grey Medium .................................... 298
8.1.3 Expression for a Plane-Parallel Medium ..................... 299
8.1.4 The Radiative Flux Vector............................... 300
8.1.5 The Radiative Boundary Conditions ........................ 301
8.1.6 Category of Solutions of the RTE.......................... 303
8.1.7 Transient Energy Transfer by Combined Conduction and Radiation .. . 306
8.1.8 The Introduction of Dimensionless Quantities ................. 308
8.2 The Semi-Transparent Quadrupoles.............................. 311
8.2.1 Basis of Derivation.................................... 311
8.2.2 The Absorbing-Emitting Quadrupole (Model 2) ................ 313
8.2.3 The Participating Medium Quadrupole (Model 3)............... 317
8.2.4 The Purely Scattering Quadrupole (Model 1).................. 318
8.2.5 Schematic Representation of the Various Quadrupoles............ 319
8.3 Examples................................................ 320
8.3.1 Example 1: The Flash Method............................ 321
8.3.2 Example 2.......................................... 323
Appendices............................................... 325
Bibliography.............................................. 330
9 Inverse Laplace Transform................................ 333
9.1 Introduction .............................................. 333
9.2 Brief Description of the Laplace Transform ........................ 333
9.2.1 Definition........................................... 333
9.2.2 Existence Conditions and Inversion Equations ................. 334
9.2.3 Usual Properties...................................... 335
9.2.4 Usual Laplace Transforms and Explicit Inversion............... 337
9.3 Inverse Transform.......................................... 340
9.3.1 Numerical Evaluation of the Mellin-Fourier Integral
with Fourier Transforms ................................ 340
9.3.2 The Gaver-Stehfest Method.............................. 342
CONTENTS ix
9.4 Some Examples and a Comparison of the Inversion Methods............ 345
9.5 Guidelines for the Laplace Inversion............................. 351
9.5.1 First Step: Decomposition of F(p) into Simpler Functions........ 352
9.5.2 Second Step: Application of the Fourier Transform
or the Stehfest Numerical Algorithm........................ 352
9.5.3 Third Step: Verification of the Inversion Results................ 353
Appendices............................................... 353
Bibliography.............................................. 355
Appendix A The Sturm-Liouville Problem....................... 357
Appendix B Bessel Functions.................................. 361
Index....................................................... 366
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| spelling | Thermal quadrupoles solving the heat equation through integral transforms Denis Maillet ... Chichester [u.a.] Wiley 2000 XIII, 370 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Transformations intégrales Équation de la chaleur - Solutions numériques Heat equation Numerical solutions Integral transforms Wärmeleitungsgleichung (DE-588)4188859-5 gnd rswk-swf Integraltransformation (DE-588)4027235-7 gnd rswk-swf Wärmeleitungsgleichung (DE-588)4188859-5 s Integraltransformation (DE-588)4027235-7 s DE-604 Maillet, Denis Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008494826&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Thermal quadrupoles solving the heat equation through integral transforms Transformations intégrales Équation de la chaleur - Solutions numériques Heat equation Numerical solutions Integral transforms Wärmeleitungsgleichung (DE-588)4188859-5 gnd Integraltransformation (DE-588)4027235-7 gnd |
| subject_GND | (DE-588)4188859-5 (DE-588)4027235-7 |
| title | Thermal quadrupoles solving the heat equation through integral transforms |
| title_auth | Thermal quadrupoles solving the heat equation through integral transforms |
| title_exact_search | Thermal quadrupoles solving the heat equation through integral transforms |
| title_full | Thermal quadrupoles solving the heat equation through integral transforms Denis Maillet ... |
| title_fullStr | Thermal quadrupoles solving the heat equation through integral transforms Denis Maillet ... |
| title_full_unstemmed | Thermal quadrupoles solving the heat equation through integral transforms Denis Maillet ... |
| title_short | Thermal quadrupoles |
| title_sort | thermal quadrupoles solving the heat equation through integral transforms |
| title_sub | solving the heat equation through integral transforms |
| topic | Transformations intégrales Équation de la chaleur - Solutions numériques Heat equation Numerical solutions Integral transforms Wärmeleitungsgleichung (DE-588)4188859-5 gnd Integraltransformation (DE-588)4027235-7 gnd |
| topic_facet | Transformations intégrales Équation de la chaleur - Solutions numériques Heat equation Numerical solutions Integral transforms Wärmeleitungsgleichung Integraltransformation |
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