Principles of Heat Transfer in Porous Media (Mechanical Engineering Series)

Convective heat tranfer is the result of fluid flowing between objects of different temperatures. Thus it may be the objective of a process (as in refrigeration) or it may be an incidental aspect of other processes. This monograph reviews in a concise and unified manner recent contributions to the principles of convective heat transfer for single- and multi-phase systems: It summarizes the role of the fundamental mechanism, discusses the governing differential equations, describes approximation schemes and phenomenological models, and examines their solutions and applications. After a review of the basic physics and thermodynamics, the book divides the subject into three parts. Part 1 deals with single-medium transfer, specifically with intraphase transfers in single-phase flows and with intramedium transfers in two-phase flows. Part 2 deals with fluid-solid transfer processes, both in cases where the interface is small and in cases where it is large, as well as liquid-liquid transfer processes. Part 3 considers three media, addressing both liquid-solid-solid and gas-liquid-solid systems.

1 Introduction.- 1.1 Historical Background.- 1.2 Length, Time, and Temperature Scales.- 1.3 Scope.- 1.4 References.- I Single-Phase Flow.- 2 Fluid Mechanics.- 2.1 Stokes Flow and Darcy Equation.- 2.2 Porosity.- 2.3 Pore Structure.- 2.4 Permeability.- 2.4.1 Capillary Models.- 2.4.2 Hydraulic Radius Model.- 2.4.3 Drag Models for Periodic Structures.- 2.5 High Reynolds Number Flows.- 2.5.1 Macroscopic Models.- 2.5.2 Microscopic Fluid Dynamics.- 2.5.3 Turbulence.- 2.6 Brinkman Superposition of Bulk and Boundary Effects.- 2.7 Local Volume-Averaging Method.- 2.7.1 Local Volume Averages.- 2.7.2 Theorems.- 2.7.3 Momentum Equation.- 2.8 Homogenization Method.- 2.8.1 Continuity Equation.- 2.8.2 Momentum Equation.- 2.9 Semiheuristic Momentum Equations.- 2.10 Significance of Macroscopic Forces.- 2.10.1 Macroscopic Hydrodynamic Boundary Layer.- 2.10.2 Macroscopic Entrance Length.- 2.11 Porous Plain Media Interfacial Boundary Conditions.- 2.11.1 Slip Boundary Condition.- 2.11.2 On Beavers-Joseph Slip Coefficient.- 2.11.3 Taylor-Richardson Results for Slip Coefficient.- 2.11.4 Slip Coefficient for a Two-Dimensional Structure.- 2.11.5 No-Slip Models Using Effective Viscosity.- 2.11.6 Variable Effective Viscosity for a Two-Dimensional Structure.- 2.11.7 Variable Permeability for a Two-Dimensional Structure.- 2.12 Variation of Porosity near Bounding Impermeable Surfaces.- 2.12.1 Dependence of Average Porosity on Linear Dimensions of System.- 2.12.2 Local Porosity Variation.- 2.12.3 Velocity Nonuniformities Due to Porosity Variation.- 2.12.4 Velocity Nonuniformity for a Two-Dimensional Structure.- 2.13 Analogy with Magneto-Hydrodynamics.- 2.14 References.- 3 Conduction Heat Transfer.- 3.1 Local Thermal Equilibrium.- 3.2 Local Volume Averaging for Periodic Structures.- 3.2.1 Local Volume Averaging.- 3.2.2 Determination of bf and bs.- 3.2.3 Numerical Values for bf and bs.- 3.3 Particle Concentrations from Dilute to Point Contact.- 3.4 Areal Contact Between Particles Caused by Compressive Force.- 3.4.1 Effect of Rarefaction.- 3.4.2 Dependence of Gas Conductivity on Knudsen Number.- 3.5 Statistical Analyses.- 3.5.1 A Variational Formulation.- 3.5.2 A Thermodynamic Analogy.- 3.6 Summary of Correlations.- 3.7 Adjacent to Bounding Surfaces.- 3.7.1 Temperature Slip for a Two-Dimensional Structure.- 3.7.2 Variable Effective Conductivity for a Two-Dimensional Structure.- 3.8 On Generalization.- 3.9 References.- 4 Convection Heat Transfer.- 4.1 Dispersion in a Tube—Hydrodynamic Dispersion.- 4.1.1 No Molecular Diffusion.- 4.1.2 Molecular Diffusion Included.- 4.1.3 Asymptotic Behavior for Large Elapsed Times.- 4.1.4 Turbulent Flow.- 4.2 Dispersion in Porous Media.- 4.3 Local Volume Average for Periodic Structures.- 4.3.1 Local Volume Averaging for ks = 0.- 4.3.2 Reduction to Taylor-Aris Dispersion.- 4.3.3 Evaluation of u’ and b.- 4.3.4 Results for ks = 0 and In-Line Arrangement.- 4.3.5 Results for ks ? 0 and General Arrangements.- 4.4 Three-Dimensional Periodic Structures.- 4.4.1 Unit-Cell Averaging.- 4.4.2 Evaluation of u’, b, and D.- 4.4.3 Comparison with Experimental Results.- 4.4.4 Effect of Darcean Velocity Direction.- 4.5 Dispersion in Disordered Structures—Simplified Hydrodynamics.- 4.5.1 Scheidegger Dynamic and Geometric Models.- 4.5.2 De Josselin De Jong Purely Geometric Model.- 4.5.3 Saffman Inclusion of Molecular Diffusion.- 4.5.4 Horn Method of Moments.- 4.6 Dispersion in Disordered Structures—Particle Hydrodynamics.- 4.6.1 Local Volume Averaging.- 4.6.2 Low Peclet Numbers.- 4.6.3 High Peclet Numbers.- 4.6.4 Contribution of Solid Holdup (Mass Transfer).- 4.6.5 Contribution Due to Thermal Boundary Layer in Fluid.- 4.6.6 Combined Effect of All Contributions.- 4.7 Properties of Dispersion Tensor.- 4.8 Experimental Determination of D.- 4.8.1 Experimental Methods.- 4.8.2 Entrance Effect.- 4.8.3 Effect of Particle Size Distribution.- 4.8.4 Some Experimental Results and Correlations.- 4.9 Dispersion in Oscillating Flow.- 4.9.1 Formulation and Solution.- 4.9.2 Longitudinal Dispersion Coefficient.- 4.10 Dispersion Adjacent to Bounding Surfaces.- 4.10.1 Temperature-Slip Model.- 4.10.2 No-Slip Treatments.- 4.10.3 Models Based on Mixing-Length Theory.- 4.10.4 A Model Using Particle-Based Hydrodynamics.- 4.10.5 Results of a Two-Dimensional Simulation.- 4.11 References.- 5 Radiation Heat Transfer.- 5.1 Continuum Treatment.- 5.2 Radiation Properties of a Single Particle.- 5.2.1 Wavelength Dependence of Optical Properties.- 5.2.2 Solution to Maxwell Equations.- 5.2.3 Scattering Efficiency and Cross Section.- 5.2.4 Mie Scattering.- 5.2.5 Rayleigh Scattering.- 5.2.6 Geometric- or Ray-Optics Scattering.- 5.2.7 Comparison of Predictions.- 5.3 Radiative Properties: Dependent and Independent.- 5.4 Volume Averaging for Independent Scattering.- 5.5 Experimental Determination of Radiative Properties.- 5.5.1 Measurements.- 5.5.2 Models Used to Interpret Experimental Results.- 5.6 Boundary Conditions.- 5.6.1 Transparent Boundaries.- 5.6.2 Opaque Diffuse Emitting/Reflecting Boundaries.- 5.6.3 Opaque Diffusely Emitting Specularly Reflecting Boundaries.- 5.6.4 Semitransparent Nonemitting Specularly Reflecting Boundaries.- 5.7 Solution Methods for Equation of Radiative Transfer.- 5.7.1 Two-Flux Approximations, Quasi-Isotropic Scattering.- 5.7.2 Diffusion (Differential) Approximation.- 5.7.3 Spherical Harmonics-Moment (P-N) Approximation.- 5.7.4 Discretc-Ordinates (S-N) Approximation.- 5.7.5 Finite-Volume Method.- 5.8 Scaling (Similarity) in Radiative Heat Transfer.- 5.8.1 Similarity Between Phase Functions.- 5.8.2 Similarity Between Anisotropic and Isotropic Scattering.- 5.9 Noncontinuum Treatment: Monte Carlo Simulation.- 5.9.1 Opaque Particles.- 5.9.2 Semitransparent Particles.- 5.9.3 Emitting Particles.- 5.10 Geometric, Layered Model.- 5.11 Radiant Conductivity Model.- 5.11.1 Calculation of F.- 5.11.2 Effect of Solid Conductivity.- 5.12 Modeling Dependent Scattering.- 5.12.1 Modeling Dependent Scattering for Large Particles.- 5.13 Summary.- 5.14 References.- 6 Mass Transfer in Gases.- 6.1 Knudsen Flows.- 6.2 Fick Diffusion.- 6.3 Knudsen Diffusion.- 6.4 Crossed Diffusion.- 6.5 Prediction of Transport Coefficients from Kinetic Theory.- 6.5.1 Fick Diffusivity in Plain Media.- 6.5.2 Knudsen Diffusivity for Tube Flows.- 6.5.3 Slip Self-Diffusivity for Tube Flows.- 6.5.4 Adsorption and Surface Flux.- 6.6 Dusty-Gas Model for Transition Flows.- 6.7 Local Volume-Averaged Mass Conservation Equation.- 6.8 Chemical Reactions.- 6.9 Evaluation of Total Effective Mass Diffusivity Tensor.- 6.9.1 Effective Mass Diffusivity.- 6.9.2 Mass Dispersion Tensor.- 6.10 Evaluation of Local Volume-Averaged Source Terms.- 6.10.1 Homogeneous Reaction.- 6.10.2 Heterogeneous Reaction.- 6.11 Local Chemical Noncquilibrium.- 6.12 Modifications to Energy Equation.- 6.13 References.- 7 Thermal Nonequilibrium Between Fluid and Solid Phases.- 7.1 Local Phase Volume Averaging for Steady Flows.- 7.1.1 Allowing for Difference in Average Local Temperatures.- 7.1.2 Evaluation of [b] and [?].- 7.1.3 Energy Equation for Each Phase.- 7.1.4 Example: Axial Travel of Thermal Pulses.- 7.2 Interfacial Convective Heat Transfer Coefficient hsf.- 7.2.1 Models Based on hsf.- 7.2.2 Experimental Determination of hsf.- 7.3 Distributed Treatment of Oscillating Flow.- 7.4 Chemical Reaction.- 7.4.1 Two-Dimensional Direct Simulation.- 7.4.2 Volume-Averaged Models.- 7.4.3 Interfacial Nusselt Number.- 7.4.4 Comparison of Results of Various Treatments.- 7.5 References.- II Two-Phase Flow.- 8 Fluid Mechanics.- 8.1 Elements of Pore-Level Flow Structure.- 8.1.1 Surface Tension.- 8.1.2 Continuous Phase Distribution.- 8.1.3 Discontinuous Phase Distributions.- 8.1.4 Contact Line.- 8.1.5 Thin Extension of Meniscus.- 8.2 Local Volume Averaging.- 8.2.1 Effect of Surface Tension Gradient.- 8.3 A Semiheuristic Momentum Equation.- 8.3.1 Inertial Regime.- 8.3.2 Liquid-Gas Interfacial Drag.- 8.3.3 Coefficients in Momentum Equations.- 8.4 Capillary Pressure.- 8.4.1 Hysteresis.- 8.4.2 Models.- 8.5 Relative Permeability.- 8.5.1 Constraint on Applicability.- 8.5.2 Influencing Factors.- 8.5.3 Models.- 8.6 Microscopic Inertial Coefficient.- 8.7 Liquid-Gas Interfacial Drag.- 8.8 Immiscible Displacement.- 8.8.1 Interfacial Instabilities.- 8.8.2 Buckley-Leverett Front.- 8.8.3 Stability of Buckley-Leverett Front.- 8.9 Fluid-Solid Two-Phase Flow.- 8.10 References.- 9 Thermodynamics.- 9.1 Thermodynamics of Single-Component Capillary Systems.- 9.1.1 Work of Surface Formation.- 9.1.2 First and Second Laws of Thermodynamics.- 9.1.3 Thickness of Interfacial Layer.- 9.2 Effect of Curvature in Single-Component Systems.- 9.2.1 Vapor Pressure Reduction.- 9.2.2 Reduction of Chemical Potential.- 9.2.3 Increase in Heat of Evaporation.- 9.2.4 Liquid Superheat.- 9.2.5 Change in Freezing Temperature.- 9.2.6 Change in Triple-Point Temperature.- 9.3 Multicomponcnt Systems.- 9.3.1 Surface Tension of Solution.- 9.3.2 Vapor Pressure Reduction.- 9.4 Interfacial Thermodynamics of Meniscus Extension.- 9.5 Capillary Condensation.- 9.5.1 Adsorption by Solid Surface.- 9.5.2 Condensation in a Mesoporous Solid.- 9.6 Prediction of Fluid Behavior in Small Pores.- 9.6.1 Phase Transition in Small Pores: Hysteresis.- 9.6.2 Stability of Liquid Film in Small Pores: Hysteresis.- 9.7 References.- 10 Conduction and Convection.- 10.1 Local Volume Averaging of Energy Equation.- 10.1.1 Averaging.- 10.1.2 Effective Thermal Conductivity and Dispersion Tensors.- 10.2 Effective Thermal Conductivity.- 10.2.1 Anisotropy.- 10.2.2 Correlations.- 10.3 Thermal Dispersion.- 10.3.1 Anisotropy.- 10.3.2 Models.- 10.3.3 Correlations for Lateral Dispersion Coefficient.- 10.3.4 Dispersion near Bounding Surfaces.- 10.4 References.- 11 Transport Through Bounding Surfaces.- 11.1 Evaporation from Heated Liquid Film.- 11.1.1 Simple Model for Transition Region.- 11.1.2 Inclusion of Capillary Meniscus.- 11.2 Mass Diffusion Adjacent to a Partially Saturated Surface.- 11.2.1 Large Knudsen Number Model.- 11.2.2 Small Knudsen Number Model.- 11.3 Convection from Heterogeneous Planar Surfaces.- 11.3.1 Mass Transfer from a Single Strip.- 11.3.2 Simultaneous Heat and Mass Transfer from Multiple Surface Sources.- 11.4 Convection from Heterogeneous Two-Dimensional Surfaces.- 11.4.1 A Simple Surface Model.- 11.4.2 Experimental Observation on Simultaneous Heat and Mass Transfer.- 11.5 Simultaneous Heat and Mass Transfer from Packed Beds.- 11.6 References.- 12 Phase Change.- 12.1 Condensation at Vertical Impermeable Bounding Surfaces.- 12.1.1 Thick Liquid-Film Region (?? / d ” 1).- 12.1.2 Thin Liquid-Film Region (?? / d ” 1).- 12.2 Evaporation at Vertical Impermeable Bounding Surfaces.- 12.3 Evaporation at Horizontal Impermeable Bounding Surfaces.- 12.3.1 Effect of Bond Number.- 12.3.2 A One-Dimensional Analysis for Bo ” 1.- 12.4 Evaporation at Thin Porous-Layer Coated Surfaces.- 12.5 Moving Evaporation or Condensation Front.- 12.5.1 Temperatures Equal to or Larger than Saturation Temperature.- 12.5.2 Temperatures Below Saturation Temperature.- 12.5.3 Condensation Front Moving into Dry Porous Media.- 12.6 Melting and Solidification.- 12.6.1 Single-Component Systems.- 12.6.2 Multicomponent Systems.- 12.7 References.- Nomenclature.- Citation Index.