Convection in Porous Media (eBook)

Following five years of service in the Royal New Zealand Navy, Donald Nield has held an appointment at the University of Auckland since 1962, the first 24 years in the Department of Mathematics and the remainder in the Department of Engineering Science, where he is currently an Honorary Academic. He holds the degrees of BD (Otago), MSc (NZ), MA (Cambridge) and PhD (Auckland). He currently serves as an Associate Editor of the journal Transport in Porous Media.Adrian Bejan earned all his degrees at M.I.T.: B.S. (1971, Honors Course), M.S. (1972, Honors Course) and Ph.D. (1975). His work is in engineering science, applied physics, and the Constructal Law of physics, which governs organization and evolution in nature. He is the author of 30 books and over 600 peer-refereed journal articles. In 2001 he was ranked among the 100 most-cited authors in all Engineering worldwide. He is a member of the Academy of Europe, and an honorary member of ASME. He was awarded 18 honorary doctorates from universities in 11 countries.

Preface to the Fifth Edition 6
Preface to the Fourth Edition 8
Preface to the Third Edition 9
Preface to the Second Edition 10
Preface to the First Edition 12
Contents 14
Nomenclature 23
Chapter 1: Mechanics of Fluid Flow Through a Porous Medium 26
1.1 Introduction 26
1.2 Porosity 29
1.3 Seepage Velocity and the Equation of Continuity 30
1.4 Momentum Equation: Darcy´s Law 30
1.4.1 Darcy´s Law: Permeability 30
1.4.2 Deterministic Models Leading to Darcy´s Law 32
1.4.3 Statistical Models Leading to Darcy´s Law 32
1.5 Extensions of Darcy´s Law 33
1.5.1 Acceleration and Other Inertial Effects 33
1.5.2 Quadratic Drag: Forchheimer´s Equation 35
1.5.3 Brinkman´s Equations 40
1.5.4 Non-Newtonian Fluid 44
1.6 Hydrodynamic Boundary Conditions 45
1.7 Effects of Porosity Variation 52
1.8 Turbulence in Porous Media 54
1.9 Fractured Media, Deformable Media, and Complex Porous Structures 57
1.10 Bidisperse Porous Media 58
Chapter 2: Heat Transfer Through a Porous Medium 61
2.1 Energy Equation: Simple Case 61
2.2 Energy Equation: Extensions to More Complex Situations 62
2.2.1 Overall Thermal Conductivity of a Porous Medium 62
2.2.2 Effects of Pressure Changes and Viscous Dissipation 65
2.2.3 Absence of Local Thermal Equilibrium 66
2.2.4 Thermal Dispersion 70
2.2.5 Cellular Porous Media 73
2.2.6 Heat Wave Theory 73
2.3 Oberbeck-Boussinesq Approximation 74
2.4 Thermal Boundary Conditions 74
2.5 Hele-Shaw Analogy 75
2.6 Bioheat Transfer 77
2.7 Other Approaches, Numerical Methods 78
Chapter 3: Mass Transfer in a Porous Medium: Multicomponent and Multiphase Flows 80
3.1 Multicomponent Flow: Basic Concepts 80
3.2 Mass Conservation in a Mixture 82
3.3 Combined Heat and Mass Transfer 84
3.4 Effects of a Chemical Reaction 86
3.5 Multiphase Flow 87
3.5.1 Conservation of Mass 89
3.5.2 Conservation of Momentum 90
3.5.3 Conservation of Energy 92
3.5.4 Summary: Relative Permeabilities 94
3.6 Unsaturated Porous Media 97
3.7 Electrodiffusion Through Porous Media 98
3.8 Nanofluids 100
3.8.1 Property Variations 101
3.8.2 Processes Associated with the Smallness of Nanoparticles 102
3.8.2.1 The Buongiorno Model 102
3.8.2.2 Conservation Equations for a Nanofluid 102
3.8.2.3 Conservation Equations for a Porous Medium Saturated by a Nanofluid 105
Chapter 4: Forced Convection 108
4.1 Plane Wall with Prescribed Temperature 108
4.2 Plane Wall with Constant Heat Flux 112
4.3 Sphere and Cylinder: Boundary Layers 113
4.4 Point Source and Line Source: Thermal Wakes 116
4.5 Confined Flow 118
4.6 Transient Effects 120
4.6.1 Scale Analysis 121
4.6.2 Wall with Constant Temperature 122
4.6.3 Wall with Constant Heat Flux 125
4.6.4 Other Configurations 126
4.7 Effects of Inertia and Thermal Dispersion: External Flow 127
4.8 Effects of Boundary Friction and Porosity Variation: Exterior Flow 129
4.9 Effects of Boundary Friction, Inertia, Porosity Variation, Thermal Dispersion, and Axial Conduction: Confined Flow 133
4.10 Local Thermal Nonequilibrium 142
4.11 Partly Porous Configurations 144
4.12 Transversely Heterogeneous Channels and Pipes 147
4.13 Thermal Development 149
4.14 Surfaces Covered with Porous Layers 150
4.15 Designed Porous Media 154
4.16 Other Configurations or Effects 157
4.16.1 Effect of Temperature-Dependent Viscosity 157
4.16.2 Oscillatory Flows, Counterflows 158
4.16.3 Non-Newtonian Fluids 159
4.16.4 Bidisperse Porous Media 160
4.16.5 Other Flows, Other Effects 161
4.17 Heatlines for Visualizing Convection 163
4.18 Constructal Tree Networks: Flow Access in Volume-to-Point Flow 166
4.18.1 The Fundamental Volume-to-Point Flow Problem 167
4.18.2 The Elemental Volume 168
4.18.3 The First Construct 171
4.18.4 Higher-Order Constructs 172
4.18.5 The Constructal Law of Design and Evolution in Nature 174
4.19 Constructal Multiscale Flow Structures: Vascular Design 177
4.20 Optimal Spacings for Plates Separated by Porous Structures 181
Chapter 5: External Natural Convection 184
5.1 Vertical Plate 184
5.1.1 Power Law Wall Temperature: Similarity Solution 186
5.1.2 Vertical Plate with Lateral Mass Flux 188
5.1.3 Transient Case: Integral Method 189
5.1.4 Effects of Ambient Thermal Stratification 191
5.1.5 Conjugate Boundary Layers 193
5.1.6 Higher-Order Boundary Layer Theory 196
5.1.7 Effects of Boundary Friction, Inertia, and Thermal Dispersion 198
5.1.7.1 Boundary Friction Effects 198
5.1.7.2 Inertial Effects 199
5.1.7.3 Thermal Dispersion Effects 203
5.1.8 Experimental Investigations 205
5.1.9 Further Extensions of the Theory 207
5.1.9.1 Particular Analytical Solutions 207
5.1.9.2 Non-Newtonian Fluids 207
Bingham and other yield-stress fluids 208
5.1.9.3 Local Thermal Nonequilibrium 209
5.1.9.4 Volumetric Heating due to Viscous Dissipation, Radiation, or Otherwise 209
5.1.9.5 Anisotropy and Heterogeneity 210
5.1.9.6 Wavy Surface 211
5.1.9.7 Time-Dependent Gravity or Time-Dependent Heating, Unsteady Flow 211
5.1.9.8 Other Thermal Boundary Conditions 212
5.1.9.9 Moving Plate 212
5.1.9.10 Magnetic Field 212
5.1.9.11 Radiation, Chemical Reaction, Internal Heating 213
5.1.9.12 Other Aspects 214
5.2 Horizontal Plate 216
5.3 Inclined Plate, Wedge 221
5.4 Vortex Instability 223
5.5 Horizontal Cylinder 225
5.5.1 Flow at High Rayleigh Number 225
5.5.2 Flow at Low and Intermediate Rayleigh Number 228
5.6 Sphere or Spherical Annulus 230
5.6.1 Flow at High Rayleigh Number 230
5.6.2 Flow at Low Rayleigh Number 232
5.6.3 Flow at Intermediate Rayleigh Number 233
5.7 Vertical Cylinder 234
5.8 Cone or Wedge 236
5.9 General Two-Dimensional or Axisymmetric Surface 240
5.10 Horizontal Line Heat Source 242
5.10.1 Flow at High Rayleigh Number 242
5.10.1.1 Darcy Model 242
5.10.1.2 Forchheimer Model 244
5.10.2 Flow at Low Rayleigh Number 247
5.11 Point Heat Source 249
5.11.1 Flow at High Rayleigh Number 249
5.11.2 Flow at Low Rayleigh Number 252
5.11.3 Flow at Intermediate Rayleigh Number 256
5.12 Other Configurations 258
5.12.1 Fins Projecting from a Heated Base 258
5.12.2 Flows in Regions Bounded by Two Planes 259
5.12.3 Other Situations 259
5.13 Surfaces Covered with Hair 260
Chapter 6: Internal Natural Convection: Heating from Below 263
6.1 Horton-Rogers-Lapwood Problem 263
6.2 Linear Stability Analysis 265
6.3 Weak Nonlinear Theory: Energy and Heat Transfer Results 269
6.4 Weak Nonlinear Theory: Further Results 273
6.5 Effects of Solid-Fluid Heat Transfer: Local Thermal Nonequilibrium 280
6.6 Non-Darcy, Dispersion, and Viscous Dissipation Effects 282
6.7 Non-Boussinesq Effects 286
6.8 Finite-Amplitude Convection: Numerical Computation and Higher-Order Transitions 288
6.9 Experimental Observations 291
6.9.1 Observations of Flow Patterns and Heat Transfer 291
6.9.2 Correlations of the Heat Transfer Data 294
6.9.3 Further Experimental Observations 299
6.10 Effect of Net Mass Flow 301
6.10.1 Horizontal Throughflow 301
6.10.2 Vertical Throughflow 302
6.11 Effect of Nonlinear Basic Temperature Profiles 305
6.11.1 General Theory 305
6.11.2 Internal Heating 306
6.11.3 Time-Dependent Heating 310
6.11.4 Penetrative Convection, Icy Water, Quadratic Density Model, Resonance 317
6.11.5 Imperfect Heat Transfer 318
6.12 Effects of Anisotropy 318
6.13 Effects of Heterogeneity 322
6.13.1 General Considerations 322
6.13.2 Layered Porous Media 323
6.13.3 Analogy Between Layering and Anisotropy 326
6.13.4 Heterogeneity in the Horizontal Direction 326
6.13.5 Heterogeneity in Both Horizontal and Vertical Directions 331
6.13.6 Strong Heterogeneity 331
6.14 Effects of Nonuniform Heating 331
6.15 Rectangular Box or Channel 334
6.15.1 Linear Stability Analysis, Bifurcation Theory, and Numerical Studies 334
6.15.2 Thin Box or Slot 339
6.15.3 Additional Effects 340
6.16 Cylinder or Annulus 343
6.16.1 Vertical Cylinder or Annulus 343
6.16.2 Horizontal Cylinder or Annulus or Spherical Annulus 345
6.17 Internal Heating in Other Geometries 346
6.18 Localized Heating 348
6.19 Superposed Fluid and Porous Layers, Partly Porous Configurations 353
6.19.1 Onset of Convection 353
6.19.1.1 Formulation 353
6.19.1.2 Results 358
6.19.2 Flow Patterns and Heat Transfer 361
6.19.3 Other Configurations and Effects 362
6.20 Layer Saturated with Water Near 4C 363
6.21 Effects of a Magnetic Field or Electric Field, Ferromagnetic Fluid 364
6.21.1 MHD Effects 364
6.21.2 Ferrofluid 366
6.21.3 Electroconvection 366
6.22 Effects of Rotation 367
6.22.1 Coriolis and Centrifugal Effects 367
6.22.2 Rotating Non-Newtonian Fluids 370
6.23 Non-Newtonian and Other Types of Fluids 370
6.23.1 Power-Law Fluids 370
6.23.2 Micropolar Fluids 370
6.23.3 Viscoelastic Fluids 371
6.23.4 Couple-Stress Fluids 371
6.23.5 Other Fluids 372
6.24 Effects of Vertical Vibration and Variable Gravity 372
6.25 Bioconvection 375
6.26 Constructal Theory of Bénard Convection 376
6.26.1 The Many Counterflows Regime 377
6.26.2 The Few Plumes Regime 379
6.26.3 The Intersection of Asymptotes 382
6.27 Bidisperse Porous Media, Cellular Porous Media 383
Chapter 7: Internal Natural Convection: Heating from the Side 384
7.1 Darcy Flow Between Isothermal Side Walls 384
7.1.1 Heat Transfer Regimes 384
7.1.2 Boundary Layer Regime 389
7.1.3 Shallow Layer 394
7.1.4 Stability of Flow 398
7.1.5 Conjugate Convection 400
7.1.6 Non-Newtonian Fluid 401
7.1.7 Other Situations 402
7.2 Side Walls with Uniform Flux and Other Thermal Conditions 404
7.3 Other Configurations and Effects of Property Variation 406
7.3.1 Internal Partitions 406
7.3.2 Effects of Heterogeneity and Anisotropy 407
7.3.3 Cylindrical or Annular Enclosure 412
7.3.3.1 Horizontal Cylinder 412
7.3.3.2 Vertical Cylinder 412
7.3.3.3 Horizontal Annulus 413
7.3.3.4 Vertical Annulus 416
7.3.3.5 Other Annuli 417
7.3.4 Spherical Enclosure 417
7.3.5 Porous Medium Saturated with Water Near 4C 418
7.3.6 Triangular Enclosure 420
7.3.7 Other Enclosures 421
7.3.7.1 Loops 424
7.3.8 Internal Heating 424
7.3.9 Bidisperse Porous Media, Other Situations 425
7.4 Penetrative Convection 425
7.4.1 Lateral Penetration 426
7.4.2 Vertical Penetration 428
7.4.3 Other Penetrative Flows 430
7.5 Transient Effects 431
7.6 Departure from Darcy Flow 435
7.6.1 Inertial Effects 435
7.6.2 Boundary Friction, Variable Porosity, Local Thermal Nonequilibrium, Viscous Dissipation, and Thermal Dispersion Effects 438
7.7 Fluid and Porous Regions 439
7.8 Sloping Porous Layer or Enclosures 444
7.9 Inclined Temperature Gradient 451
7.10 Periodic Heating 453
7.11 Sources in Confined or Partly Confined Regions 455
7.12 Effects of Rotation 457
Chapter 8: Mixed Convection 459
8.1 External Flow 459
8.1.1 Inclined or Vertical Plane Wall 459
8.1.1.1 Stagnation Point Flow 465
8.1.1.2 Magnetic Field 465
8.1.1.3 Transient Convection 466
8.1.1.4 Inclined Plate 466
8.1.1.5 Non-Newtonian Fluids 467
8.1.2 Horizontal Wall 467
8.1.3 Cylinder or Sphere 468
8.1.4 Other Geometries 472
8.1.5 Unified Theory 474
8.1.6 Other Aspects of External Flow 478
8.2 Internal Flow: Horizontal Channel 478
8.2.1 Horizontal Layer: Uniform Heating 478
8.2.2 Horizontal Layer: Localized Heating 480
8.2.3 Horizontal Cylinder or Annulus 482
8.2.4 Horizontal Layer: Lateral Heating 482
8.3 Internal Flow: Vertical Channel 483
8.3.1 Vertical Layer: Uniform Heating 483
8.3.2 Vertical Layer: Localized Heating 485
8.3.3 Vertical Cylinder or Annulus: Uniform Heating 485
8.3.4 Vertical Annulus: Localized Heating 487
8.4 Other Geometries and Other Effects 488
8.4.1 Partly Porous Configurations 488
8.4.2 Jet Impingement 489
8.4.3 Other Aspects 489
Chapter 9: Double-Diffusive Convection 492
9.1 Vertical Heat and Mass Transfer 492
9.1.1 Horton-Rogers-Lapwood Problem 492
9.1.2 Nonlinear Initial Profiles 497
9.1.3 Finite-Amplitude Effects 497
9.1.4 Soret and Dufour Cross-Diffusion Effects 501
9.1.5 Flow at High Rayleigh Number 504
9.1.6 Other Effects 506
9.1.6.1 Dispersion 506
9.1.6.2 Anisotropy and Heterogeneity 507
9.1.6.3 Brinkman Model 508
9.1.6.4 Additional Effects 509
Multicomponent Convection 509
Magnetic Field 509
Rotation 510
Non-Newtonian Fluid 510
Local Thermal Nonequilibrium 511
Throughflow 511
Thermal Modulation 512
Vibration 512
Groundwater Studies 512
Chemical Reaction 513
Internal Heating 514
Composite Domains 514
Other Studies 514
9.2 Horizontal Heat and Mass Transfer 515
9.2.1 Boundary Layer Flow and External Natural Convection 515
9.2.1.1 Magnetic Field 519
9.2.1.2 Non-Newtonian Fluid 521
9.2.1.3 Cross-Diffusion 522
9.2.1.4 Moving Surface, Stretching Sheet 522
9.2.1.5 Horizontal or Inclined Wall 522
9.2.1.6 Wavy Surface 523
9.2.1.7 Cone or Wedge or Cylinder or Sphere 523
9.2.1.8 Other Situations 524
9.2.2 Enclosed Porous Medium: Channel or Box 524
9.2.3 Transient Effects 532
9.2.4 Stability of Flow 534
9.3 Concentrated Heat and Mass Sources 535
9.3.1 Point Source 535
9.3.2 Horizontal Line Source 538
9.4 Other Configurations and Effects 538
9.5 Inclined and Crossed Gradients 541
9.6 Mixed Double-Diffusive Convection 542
9.6.1 Mixed External Convection 542
9.6.1.1 Vertical Plate 542
9.6.1.2 Other Surfaces 543
9.6.2 Mixed Internal Convection 544
9.7 Nanofluids 545
9.7.1 Forced Convection 545
9.7.2 Internal Natural Convection 547
9.7.2.1 Horizontal Layer 547
9.7.2.2 Rectangular Box 549
9.7.2.3 Vertical or Inclined Channel, Vertical Pipe 549
9.7.2.4 Other Cavities 550
9.7.3 External Natural Convection 551
9.7.3.1 Vertical Plate 551
9.7.3.2 Horizontal or Inclined Plate or Wedge 553
9.7.3.3 Curved Surface 554
9.7.4 Mixed Convection 554
9.7.4.1 Vertical Plate 554
9.7.4.2 Horizontal or Inclined Plate 555
9.7.4.3 Curved Surfaces 555
9.7.4.4 Vertical, Horizontal, or Inclined Channel 556
9.7.4.5 Other Cavities 556
Chapter 10: Convection with Change of Phase 557
10.1 Melting 557
10.1.1 Enclosure Heated from the Side 557
10.1.2 Scale Analysis 563
10.1.3 Effect of Liquid Superheating 566
10.1.4 Horizontal Liquid Layer 574
10.1.5 Vertical Melting Front in an Infinite Porous Medium 576
10.1.6 A More General Model 577
10.1.7 Further Studies 580
10.2 Freezing and Solidification 583
10.2.1 Cooling from the Side 583
10.2.1.1 Steady State 583
10.2.1.2 Other Studies 586
10.2.2 Cooling from Above 587
10.2.3 Solidification of Binary Alloys 588
10.3 Boiling and Evaporation 595
10.3.1 Boiling and Evaporation Produced by Heating from Below 595
10.3.2 Film Boiling and Evaporation 601
10.4 Condensation 606
10.5 Spaces Filled with Fluid and Fibers Coated with a Phase-Change Material 609
Chapter 11: Geophysical Aspects 612
11.1 Snow 612
11.2 Patterned Ground 614
11.3 Thawing Subsea Permafrost 616
11.4 Magma Production and Magma Chambers 618
11.5 Diagenetic Processes 619
11.6 Oceanic Crust 621
11.6.1 Heat Flux Distribution 621
11.6.2 Topographical Forcing 622
11.7 Geothermal Reservoirs: Injection and Withdrawal 624
11.8 Other Aspects of Single-Phase Flow 625
11.9 Two-Phase Flow 629
11.9.1 Vapor-Liquid Counterflow 629
11.9.2 Heat Pipes 634
11.9.3 Other Aspects 636
11.10 Cracks in Shrinking Solids 636
11.11 Carbon Dioxide Sequestration 639
11.12 Reaction Scenarios 642
11.12.1 Reaction Fronts 642
11.12.2 Gradient Reactions 644
11.12.3 Mixing Zones 645
References 646
Index 1000