Preface ..................................................... XIII
List of Contributors .......................................... XV
1 Interfacial Heat Transport in Highly Permeable Media:
A Finite Volume Approach .................................... 1
Marcelo J.S. de Lemos and Marcelo B. Saito
1.1 Introduction ........................................... 1
1.2 Governing Equations .................................... 3
1.2.1 Microscopic Transport Equations ................. 3
1.2.2 Decomposition of Flow Variables in Space and
Time ............................................ 4
1.2.3 Macroscopic Flow and Energy Equations ........... 5
1.2.4 Macroscopic Two-Energy Equation Modeling ........ 8
1.2.5 Interfacial Heat Transfer Coefficient .......... 10
1.3 Numerical Determination of hi ......................... 12
1.3.1 Physical Model ................................. 12
1.3.2 Periodic Flow .................................. 14
1.3.3 Film Coefficient hi ............................ 15
1.4 Results and Discussion ................................ 16
1.4.1 Array of Square Rods ........................... 16
1.4.2 Array of Elliptic Rods ......................... 16
1.4.3 Correlations for Laminar and Turbulent Flows ... 20
1.5 Conclusions ........................................... 27
References ............................................ 27
2 Effective Thermal Properties of Hollow-Sphere-Structures:
A Finite Element Approach .................................. 31
Andreas Öchsner and Thomas Fiedler
2.1 Introduction .......................................... 31
2.1.1 Finite Element Method and Heat Transfer
Problems ....................................... 31
2.1.2 Hollow-Sphere Structures in the Context of
Cellular Metals ................................ 33
2.2 Finite Element Method ................................. 37
2.2.1 Basics of Heat Transfer ........................ 37
2.2.2 Weighted Residual Method ....................... 38
2.2.3 Discretization and Principal Finite Element
Equation ....................................... 39
2.2.4 Four-Node Planar Bilinear Quadrilateral
(Quad4) ........................................ 42
2.2.4.1 General Rectangular Quad4 Element ..... 48
2.2.4.2 Postprocessing ........................ 51
2.2.5 Nonlinearities ................................. 53
2.3 Modelling of Hollow-Sphere-Structures ................. 56
2.3.1 Geometry, Mesh and Boundary Conditions ......... 56
2.3.2 Material Properties ............................ 58
2.4 Determination of the Effective Thermal
Conductivities ........................................ 59
2.4.1 Influence of the Morphology and Joining
Technique ...................................... 60
2.4.2 Influence of the Topology ...................... 62
2.4.3 Temperature-Dependent Material Properties ...... 65
2.4.3.1 Low Temperature Gradient .............. 65
2.4.3.2 High Temperature Gradient ............. 66
2.4.4 Application Example: Sandwich Structure ........ 67
2.5 Conclusions ........................................... 68
References ............................................ 69
3 Thermal Properties of Composite Materials and Porous
Media: Lattice-Based Monte Carlo Approaches ................ 73
Irina V. Belova and Graeme E. Murch
3.1 Introduction .......................................... 73
3.2 Monte Carlo Methods of Calculation of the Effective
Thermal Conductivity .................................. 73
3.2.1 The Einstein Equation .......................... 74
3.2.2 Fick's First Law (Fourier Equation) ............ 80
3.3 Monte Carlo Calculations of the Effective Thermal
Conductivity .......................................... 81
3.3.1 Effective Diffusion in Two-Component
Composites / Porous Media ...................... 81
3.3.2 Effective Diffusion in Three-Component
Composites ..................................... 90
3.4 Determination of Temperature Profiles ................. 91
References ............................................ 94
4 Fluid Dynamics in Porous Media: A Boundary Element
Approach ................................................... 97
Leopold Škerget, Renata Jecl, and Janja Kramer
4.1 Introduction .......................................... 97
4.1.1 Transport Phenomena in Porous Media ............ 97
4.1.2 Boundary Element Method for Fluid Dynamics
in Porous Media ................................ 98
4.2 Governing Equations ................................... 99
4.3 Boundary Element Method .............................. 101
4.3.1 Velocity-Vorticity Formulation ................ 102
4.3.2 Boundary Domain Integral Equations ............ 102
4.3.3 Discretized Boundary Domain Integral
Equations ..................................... 205
4.3.4 Solution Procedure ............................ 106
4.4 Numerical Examples ................................... 107
4.4.1 Double-Diffusive Natural Convection in
Vertical Cavity ............................... 107
4.4.2 Double-Diffusive Natural Convection in
a Horizontal Porous Layer ..................... 113
4.5 Conclusion ........................................... 117
References ........................................... 117
5 Analytical Methods for Heat Conduction in Composites and
Porous Media .............................................. 121
Vladimir V. Mityushev, Ekaterina Pesetskaya, and Sergei
V. Rogosin
5.1 Introduction ......................................... 121
5.2 Mathematical Models for Heat Conduction .............. 122
5.2.1 General ....................................... 122
5.2.2 Boundary Value Problems ....................... 127
5.2.3 Conjugation Problem ........................... 128
5.2.4 Complex Potentials ............................ 129
5.2.5 Periodic Problems ............................. 132
5.3 Effective Conductivity Tensor ........................ 134
5.4 Review of Known Formulas ............................. 137
5.4.1 Laminates ..................................... 137
5.4.2 Clausius-Mossotti Approximation (CMA) ......... 137
5.4.3 Effective Medium Theory (EMT) ................. 141
5.4.4 Duality Theory for 2D Media ................... 144
5.5 Network Approximations ............................... 146
5.6 Doubly Periodic Problems ............................. 149
5.6.1 Introduction to Elliptic Function Theory ...... 149
5.6.2 Method of Functional Equations ................ 154
5.7 Representative Cell .................................. 156
5.8 Nonlinear Heat Conduction ............................ 159
References ................................................ 160
6 Modeling of Composite Heat Transfer in Open-Cellular
Porous Materials at High Temperatures ..................... 165
Kouichi Kamiuto
6.1 Introduction ......................................... 165
6.2 Governing Equations .................................. 166
6.3 Transport Properties and Heat Transfer Correlation ... 168
6.3.1 Effective Thermal Conductivities .............. 168
6.3.2 Thermal Dispersion Conductivities ............. 171
6.3.3 Radiative Properties .......................... 173
6.3.4 Fluid Mechanical Properties ................... 174
6.3.5 Volumetric Heat Transfer Coefficient .......... 178
6.4 Radiative Transfer ................................... 179
6.5 Combined Conductive and Radiative Heat Transfer ...... 183
6.6 Combined Forced-Convective and Radiative Heat
Transfer ............................................. 186
6.6.1 Analysis of Gas Enthalpy-Radiation
Conversion System ............................. 187
6.6.2 Analysis of Transpiration Cooling System in
a Radiative Environment ....................... 189
6.7 Conclusions and Recommendations ...................... 194
References ........................................... 197
7 Thermal Conduction Through Porous Systems ................. 199
Ramvir Singh
7.1 Introduction ......................................... 199
7.2 Theoretical Models ................................... 201
7.2.1 Models for Thermal Conductivity ............... 201
7.2.2 Discussion .................................... 219
7.3 Experimental Techniques .............................. 221
7.3.1 Thermal Conductivity Probe .................... 221
7.3.1.1 Theory ............................... 223
7.3.2 Differential Temperature Sensor Technique ..... 224
7.3.2.1 Mathematical Analysis ................ 225
7.3.3 Probe-Controlled Transient Technique .......... 227
7.3.3.1 Mathematical Analysis ................ 227
7.3.4 Plane Heat Source ............................. 230
7.3.4.1 Theory ............................... 230
7.3.5 Transient Plane Source (TPS) .................. 234
7.3.5.1 Theory ............................... 234
7.3.6 Discussion .................................... 236
References ........................................... 237
8 Thermal Property of Lotus-Type Porous Copper and
Application to Heat Sinks ................................. 239
Tetsuro Ogushi, Hiroshi Chiba, Masakazu Tane, and Hideo
Nakajima
8.1 Introduction ......................................... 239
8.2 Effective Thermal Conductivity of Lotus-Type Porous
Copper ............................................... 241
8.2.1 Measurement ................................... 241
8.2.1.1 Definition of Effective Thermal
Conductivity ......................... 241
8.2.1.2 Experimental Method .................. 242
8.2.1.3 Specimen Preparation ................. 243
8.2.2 Thermal Conductivity Parallel to Pores ........ 244
8.2.3 Thermal Conductivity Perpendicular to Pores ... 245
8.2.4 Effect of Pore Shape on Thermal
Conductivity .................................. 248
8.2.5 Effect of Pore Orientation on Thermal
Conductivity .................................. 251
8.2.5.1 Introduction ......................... 251
8.2.5.2 EMF Theory ........................... 251
8.2.5.3 Application of Extended EMF Theory
to Lotus Metals ...................... 252
8.3 Application of Lotus-Type Porous Copper to Heat
Sinks ................................................ 255
8.3.1 Analysis of Fin Efficiency .................... 255
8.3.1.1 Straight Fin Model ................... 255
8.3.1.2 Numerical Analysis ................... 256
8.3.2 Experiments of Heat Transfer
Characteristics ............................... 258
8.3.2.1 Experimental Method .................. 258
8.3.2.2 Investigated Heat Sinks .............. 259
8.3.3 Predictions of Heat Transfer
Characteristics ............................... 260
8.3.3.1 Conventional Groove Fins and
Microchannels ........................ 260
8.3.3.2 Lotus-Type Porous Copper Fins ........ 260
8.3.4 Comparison of Experiments with Predictions .... 261
8.4 Conclusions .......................................... 264
References ........................................... 265
9 Thermal Characterization of Open-Celled Metal Foams
by Direct Simulation ...................................... 267
Shankar Krishnan, Suresh V. Garimella, and Jayathi
Y. Murthy
9.1 Introduction ......................................... 267
9.2 Foam Geometry ........................................ 269
9.3 Mathematical Modeling ................................ 271
9.3.1 Effective Thermal Conductivity ................ 271
9.3.2 Computation of Flow and Heat Transfer
Through Foam .................................. 272
9.3.2.1 Flow and Temperature Periodicity ..... 272
9.3.2.2 Governing Equations .................. 273
9.3.2.3 Computational Details ................ 274
9.4 Results and Discussion ............................... 274
9.4.1 Direct Simulations of Foams: BCC Model ........ 275
9.4.1.1 Effective Thermal Conductivity ....... 276
9.4.1.2 Pressure Drop and Heat Transfer
Coefficient .......................... 278
9.4.2 Direct Simulations of Foams: Effect of Unit
Cell Structure ................................ 283
9.4.2.1 Effective Thermal Conductivity ....... 284
9.4.2.2 Pressure Drop and Nusselt Number ..... 285
9.5 Conclusion ........................................... 286
References ........................................... 288
10 Heat Transfer in Open-Cell Metal Foams Subjected
to Oscillating Flow ....................................... 291
Kai Choong Leong and Liwen Jin
10.1 Introduction ......................................... 291
10.1.1 Fluid Flow and Heat Transfer in Open-Cell
Foams ......................................... 292
10.1.2 Oscillating Flow Through Porous Media ......... 295
10.2 Fluid Behavior of Oscillatory Flow in Open-Cell
Metal Foams .......................................... 296
10.2.1 Critical Properties of Open-Cell Foams ........ 297
10.2.2 Analysis of Similarity Parameters ............. 299
10.2.3 Oscillatory Flow Through a Channel Filled
with Open-Cell Foams .......................... 302
10.2.3.1 Effects of Kinetic Reynolds Number
and Dimensionless Flow Amplitude ..... 303
10.2.3.2 Friction Factor in Metal Foam ....... 306
10.3 Heat Transfer Characteristics of Oscillatory Flow
in Open-Cell Foams ................................... 309
10.3.1 Theoretical Analysis of Forced Convection in
Oscillating Flow .............................. 309
10.3.2 Oscillatory Heat Transfer in Open-Cell Metal
Foams ......................................... 313
10.3.3 Effects of Oscillation Frequency and Flow
Amplitude ..................................... 315
10.3.4 Heat Transfer Rate in Metal Foams ............. 318
10.4 Thermal Management Using Highly Conductive Metal
Foams ................................................ 323
10.4.1 Steady and Oscillating Flows in Open-Cell
Metal Foams ................................... 323
10.4.1.1 Thermal Performance of Open-Cell
Metal Foams .......................... 323
10.4.1.2 Comparison of Steady and
Oscillating Flows .................... 326
10.4.2 Pumping Power of Oscillatory Cooling System ... 331
10.5 Conclusions .......................................... 333
References ........................................... 337
11 Radiative and Conductive Thermal Properties of Foams ...... 343
Dominique Baillis and Rémi Coquard
11.1 Introduction ......................................... 343
11.2 Description of Cellular Foam Structure ............... 344
11.2.1 Open-Cell Foams ............................... 344
11.2.2 Closed-Cell Foams ............................. 344
11.3 Modeling of Foam Structure ........................... 346
11.3.1 Cell Modeling ................................. 346
11.3.2 Particle Modeling ............................. 347
11.4 Determination of Foam Conductive Properties .......... 347
11.4.1 Analytical / Semi-analytical Models ........... 348
11.4.1.1 Polymer Foams ........................ 348
11.4.1.2 Ceramic, Metallic and Carbon Foams ... 350
11.4.2 Numerical Models .............................. 352
11.4.2.1 Polymer Foams ............................... 352
11.4.2.2 Ceramic, Metallic and Carbon Foams .......... 353
11.5 Determination of Cellular Foam Radiative
Properties ........................................... 355
11.5.1 Theoretical Prediction of Radiative
Properties of Particulate Media ............... 356
11.5.1.1 Single-Particle Properties ........... 356
11.5.1.2 Dispersion Properties ................ 357
11.5.2 Parameter Identification Method ............... 357
11.5.3 Application to Open-Cell and Closed-Cell
Foams ......................................... 359
11.5.3.1 Open-Cell Carbon Foam ................ 359
11.5.3.2 Metallic Foam ........................ 361
11.5.3.3 Closed-Cell Foam: Case of Low-
Density EPS Foams .................... 362
11.5.3.4 Closed-Cell Foam: Case of XPS and
PUR Foams ............................ 367
11.6 Combined Conductive and Radiative Heat Transfer in
Foam ................................................. 369
11.6.1 Heat Transfer Equations for Cellular Foam
Insulation .................................... 369
11.6.2 Resolution of the Heat Transfer Equations ..... 370
11.6.2.1 Resolution of the Radiative
Transfer Equation / Rosseland
Approximation ........................ 370
11.6.2.2 Resolution of the Radiative
Transfer Equation / Discrete
Ordinates Method ..................... 371
11.6.2.3 Resolution of the Energy Equation .... 372
11.6.3 Equivalent Thermal Conductivity Results ....... 372
11.6.3.1 Closed-Cell EPS Foams ................ 372
11.6.3.2 Closed-Cell XPS and PUR Foams ........ 375
11.6.3.3 Metallic Open-Cell Foams ............. 376
11.6.3.4 Open-Cell Carbon Foams ............... 380
11.7 Conclusions .......................................... 381
References ........................................... 382
12 On the Application of Optimization Techniques to Heat
Transfer in Cellular Materials ............................ 385
Pablo A. Muñoz-Rojas, Emilio C. Nelli Silva, Eduardo
L. Cardoso, and Miguel Vaz Junior
12.1 Introduction ......................................... 385
12.2 Optimization Approaches .............................. 386
12.2.1 Evolutionary Algorithms (EAs) ................. 387
12.2.1.1 Basic Concepts in Evolutionary
Algorithms .......................... 387
12.2.2 Mathematical Programming using Gradient-
Based Procedures .............................. 389
12.3 Periodic Composite Materials ......................... 389
12.3.1 Homogenization of Heat Properties in
Periodic Composite Materials .................. 390
12.3.2 Functionally Graded Materials ................. 394
12.3.3 Numerical Implementation of Homogenization .... 395
12.3.4 Material Design: Shape and Topology
Optimization of a Unit Cell ................... 397
12.3.4.1 Shape Optimization ................... 398
12.3.4.2 Topology Optimization ................ 401
12.4 General Applications Review .......................... 403
12.5 Results Obtained with the FGM Approach in this
Work ................................................. 410
12.6 Conclusions .......................................... 413
References ................................................ 414
Index ........................................................ 419
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