Principles of Heat Transfer in Porous Media

Examination of transport, reaction, and phase change in natural and engineered porous media relies on the knowledge we have gained in studying these phenomena in otherwise plain media. The presence of a permeable solid (which we assume to be rigid and stationary) influences these phenomena significantly. Due to practical limitations, as a general approach we choose to describe these phenomena at a small length scale which is yet, larger than a fraction of the linear dimension of the pore or the linear dimension of the solid particle (for a particle-based porous medium). This requires the use of the local volume averaging theories. Also, depending on the validity, local mechanical, thermal, and chemical equilibrium or nonequilibrium, may be imposed between the fluid (liquid and/or gas) and/or solid phases.

In this chapter we examine the isothermal, single-phase flow through porous media by starting from the Darcy law. We then examine the permeability tensor and its relation with the matrix structure. The deviation from the Darcy law observed at high velocities and the pore-level fluid dynamics are then examined. Attempts at arriving at the Darcy law (the macroscopic momentum equation) from the point description of the flow field (Navier-Stokes equation) by the local volume-averaging and homogenization techniques are reviewed. Then, a semiempirical momentum equation, which includes the bulk and boundary viscous effects, the flow development in porous media, and the high-velocity effects, is given. The significance of these terms is assessed by the order-of-magnitude analyses and some estimations. When the porous media arc bounded by the fluid occupying them, the hydrodynamic boundary conditions on these interfaces must be specified. The available slip velocity model and the Brinkman no-slip (uniform and variable effective viscosity) model for this interface are examined. The dependence of the slip coefficient (and the interfacial effective viscosity) on the bulk and surface structures of the matrix are studied. The velocity nonuniformities observed near the bounding impermeable surfaces and the various theoretical treatments of them are investigated. The chapter ends with an examination of the analogy between porous media- and magneto- hydrodynamics.

Heat conduction through fully saturated matrices (i.e., a single-phase fluid occupying the pores), as with heat conduction through any heterogeneous media, depends on the structure of the matrix and the thermal conductivity of each phase. When the pore or particle size is smaller than the bulk mean-free path of heat carriers, then pore scattering of the heat carriers must be addressed (Chung and Kaviany, 1999)

As we consider simultaneous fluid flow and heat transfer in porous media, the role of the macroscopic (Darcean) and microscopic (pore-level) velocity fields on the temperature field needs to be examined. Experiments have shown that the mere inclusion of u _D • ∇ 〈T〉 in the energy equation does not accurately account for all the hydrodynamic effects. The pore-level hydrodynamics also influence the temperature field. Inclusion of the effect of the pore-level velocity nonuniformity on the temperature distribution (called the dispersion effect and generally included in the diffusion transport) is the main concern in this chapter.

In this chapter, heat transfer by radiation in porous media is examined. The medium may be treated either as a single continuum or as a collection of particles (i.e., scatterers). In the particle-based analysis, the interaction of radiation with a collection of elements of the solid matrix (e.g., particles in a packed bed) is considered. On the other hand, the continuum treatment attempts to obtain the effective radiative properties of the medium by using the element-based interaction along with a local volume-averaging procedure. This volume averaging is greatly simplified if it is assumed that the interaction of a particle with radiation is not affected by the presence of neighboring particles [i.e., the scattering (or absorption) is independent]. In case the assumption of independent scattering fails, the volume averaging must include dependent effects.

Although throughout the text we consider both liquids and gases, in this chapter we consider mass transfer in gases only. We examine the observed deviations from the viscous flow behavior at low pressures (or for very small pore sizes) and consider chemical reactions. The subject of gas diffusion in porous media has been extensively treated by Jackson (1977), Cunningham and Williams (1980), and Wakao and Kaguei (1982). Jackson gives special attention to the multicomponent gas mixtures (more than two components), even though for more than three components and at low gas pressures where the molecular slip occurs, the treatment becomes very difficult because of the interdependence of the individual mass fluxes.

In this chapter we examine the single-phase flow through solid matrices where the assumption of the local thermal equilibrium between the phases is not valid. When there is a significant heat generation occurring in any one of the phases (solid or fluid), i.e., when the primary heat transfer is by heat generation in a phase and the heat transfer through surfaces bounding the porous medium is less significant, then the local (finite and small) volumes of the solid and fluid phases will be far from the local thermal equilibrium. Also, when the temperature at the bounding surface changes significantly with respect to time, then in the presence of an interstitial flow and when solid and fluid phases have significantly different heat capacities and thermal conductivities, the local rate of change of temperature for the two phases will not be equal.

The hydrodynamics of the two-phase (liquid-gas) flow in porous media is addressed in this chapter and very briefly thefluid-solid two-phase flow is mentioned at the end of the chapter. Before introducing the volume-averaged momentum and continuity equations for each phase, we review the elements of the hydrodynamics of three-phase systems(solid-liquid- gas). These elements are the interfacial tensions, the static contact angle, the moving contact angle, and the van der Waals interfacial-layer forces. We examine the interfacial tension between a liquid and another fluid. For the case of astatic equilibrium at this interface, we discuss the effect of thecurvature for the simple problem of ring formation between spheres (and cylinders). For dynamic aspects, we examine the combined effect of capillarity and buoyancy by discussing the rise of a bubble in a capillary tube. Then we consider more realistic conditions and examine the effects of various factors on the phase distributions and the existing results for the phase distributions in flow through packed beds. The moving contact line and the effects of solid surface tension and thesurface roughness and heterogeneities will then be discussed. For the perfectly wetting liquids at equilibrium, a thin extension of the liquid is present on the surface. We will examine the thickness and stability of this thin layer. After the phase-volume averaging of the momentum equation, we discuss the various coefficients that appear in the two momentum equations (one for the wetting phase and one for the nonwetting phase).

In this chapter we examine the thermodynamics of the liquid-gas systems in porous media. The treatment centers around the examination of two effects, namely, the effect of the liquid-gas interfacial tension and curvature (when the radius of curvature becomes very small), and the effect of the solid-fluid interfacial forces which results in a significant surface adsorption (when specific interfacial area becomes very large for small pores). First the classical treatments, which are centered around the effect of the meniscuscurvature on the thermodynamic state, are discussed. Both single- and multicomponent systems are considered. Then we examine the thermodynamics of thin liquid-film extensions of perfectly wetting liquids and the role of the van der Waals forces on the equilibrium state. Next, the capillary condensation (adsorption) and evaporation (desorption) in small pores is discussed. The hysteresis in the adsorption isotherm, as well as the other features of adsorption and desorption are discussed. The curvature arguments made for the existence of the hysteresis are discussed before the introduction of the more modern descriptions based on the molecular interaction theories. Then two of these modern theories and their predictions of the phase change and the stability of the thin liquid films in small pores are discussed. Some thermodynamic aspects of solid-liquid phase change are discussed in Section 12.6.

As we discussed in Chapter 8, our knowledge of the pore-level fluid mechanics in two-phase flow through porous media is rather incomplete. In this chapter, we discuss thermal dispersion, i.e., convective heat transfer at the pore level, using the available knowledge about the subject. This knowledge is even more inconclusive. We begin with the local volume averaging of the energy equation, and then we arrive at the effective thermal conductivity tensor and the thermal dispersion tensor for the three-phase system (liquid-gas-solid). The same closure conditions used in the single- phase flow treatments are used. Then we examine the various features of these tensors such as their anisotropy, and we discuss some of the available models and empirical relations for the various elements of these tensors. We conclude by noting that near the bounding surfaces, the phase distribution nonuniformities lead to substantial variations in the magnitude of the components of the effective thermal conductivity and the dispersion tensors.

When a gaseous plain medium bounds a partially saturated porous medium, then interfacial heat and mass transfer can occur across this interface. This interfacial convective heat and mass transfer requires temperature and concentration gradients and fluid motions. Figure 11.1 depicts this interface A_pa between a porous and a plain (ambient) medium. A discussion of this interfacial transport is given by van Brakel (1982). This interface is not planar and the distribution of the phases on this interface is very complex and depends on the surface saturation (defined for a thin interfacial layer), the solid topology, the wettability, and the history. There is, in general, some motion within the interfacial liquid phase. In addition to this and the motion in the gaseous plain medium, the liquid and gaseous phases in the bulk of the porous medium experience motion. The porous medium gaseous phase moves due to the concentration and the total gas- phase pressure gradients. The liquid phase, in addition to the saturation gradient, can undergo motion due to the gradient of the surface-tension, buoyancy, wind-shear, solid-fluid surface forces (as discussed in Chapter 9), or combinations of these forces. In this chapter, we review the elements and existing treatments of the interfacial transport across partially saturated surfaces. Since the problem involves the simultaneous heat and mass transfer, a complete treatment is very involved. However, as with other phenomena we examine some simple systems in detail and evaluate the experimental results for the more complex systems.

In this chapter we examine evaporation and condensation in porous media in detail and briefly review melting and solidification in porous media. The heat supply or removal causing these to occur is generally through the bounding surfaces and these surfaces can be impermeable or permeable. We begin by considering condensation and evaporation adjacent to vertical impermeable surfaces. These are the counterparts of the film condensation and evaporation in plain media. The presence of the solid matrix results in the occurrence of a two-phase flow region governed by gravity and capillarity. The study of this two-phase flow and its effect on the condensation or evaporation rate (i.e., the heat transfer rate) has begun recently. Evaporation from horizontal impermeable surfaces is considered next. Because the evaporation is mostly from thin-liquid films forming on the solid matrix (in the evaporation zone), the evaporation does not require a significant superheat. The onset of dryout, i.e., the failure of the gravity and capillarity to keep the surface wet, occurs at a critical heat flux but only small superheat is required. We examine the predictions of the critical heat flux and the treatment of the vapor-film and the two-phase regions. We also examine the case of thin porous-layer coating of horizontal surfaces and review the limited data on the porous-layer thickness dependence of the heat flow rate versus the superheat curve.