Convection in Porous Media

By a porous medium, we mean a material consisting of a solid matrix with an interconnected void. We suppose that the solid matrix is either rigid (the usual situation) or it undergoes small deformation. The interconnectedness of the void (the pores) allows the flow of one or more fluids through the material. In the simplest situation (“single-phase flow”), the void is saturated by a single fluid. In “two-phase flow,” a liquid and a gas share the void space.

In this chapter we focus on the equation that expresses the first law of thermodynamics in a porous medium. We start with a simple situation in which the medium is isotropic and where radiative effects, viscous dissipation, and the work done by pressure changes are negligible. Very shortly we shall assume that there is local thermal equilibrium so that T _s = T _f = T, where T _s and T _f are the temperatures of the solid and fluid phases, respectively. Here we also assume that heat conduction in the solid and fluid phases takes place in parallel so that there is no net heat transfer from one phase to the other. More complex situations will be considered in Sect. 6.5. The fundamentals of heat transfer in porous media are also presented in Bejan et al. (2004) and Bejan (2004a).

The term “mass transfer” is used here in a specialized sense, namely the transport of a substance that is involved as a component (constituent, species) in a fluid mixture. An example is the transport of salt in saline water. As we shall see below, convective mass transfer is analogous to convective heat transfer.

The fundamental question in heat transfer engineering is to determine the relationship between the heat transfer rate and the driving temperature difference. In nature, many saturated porous media interact thermally with one another and with solid surfaces that confine them or are embedded in them. In this chapter we analyze the basic heat transfer question by looking only at forced-convection situations, in which the fluid flow is caused (forced) by an external agent unrelated to the heating effect. First we discuss the results that have been developed based on the Darcy flow model and later we address the more recent work on the non-Darcy effects. We end this chapter with a review of current engineering applications of the method of forced convection through porous media. Some fundamental aspects of the subject have been discussed by Lage and Narasimhan (2000), and the topic has been reviewed by Lauriat and Ghafir (2000).

Numerical calculation from the full differential equations for convection in an unbounded region is expensive, and hence approximate solutions are important. For small values of the Rayleigh number Ra, perturbation methods are appropriate. At large values of Ra thermal boundary layers are formed, and boundary-layer theory is the obvious method of investigation. This approach forms the subject of much of this chapter. We follow to a large extent the discussion by Cheng (1985a), supplemented by recent surveys by Pop and Ingham (2000, 2001) and Pop (2004).

We start with the simplest case, that of zero flow through the fluid-saturated porous medium. For an equilibrium state, the momentum equation is satisfied if

Enclosures heated from the side are most representative of porous systems that function while oriented vertically, as in the insulations for buildings, industrial cold-storage installations, and cryogenics. As in the earlier chapters, we begin with the most fundamental aspects of the convection heat transfer process when the flow is steady and in the Darcy regime. Later, we examine the special features of flows that deviate from the Darcy regime, flows that are time dependent, and flows that are confined in geometries more complicated than the two-dimensional rectangular space shown in Fig. 7.1.

We already have discussed one form of mixed convection in a horizontal layer, namely, the onset of convection with throughflow when the heating is from below (see Sect 6.10). In this chapter, we discuss some more general aspects of mixed convection. Since we have dealt with natural convection and forced convection in some detail, our treatment of mixed convection in a porous medium [first treated by Wooding (1960)] can be brief. It is guided by the surveys by Lai et al. (1991a) and Lai (2000). We endorse the statement by Lai (2000) that despite the increased volume of research in this field, experimental results are still very few. In particular experimental data on thermal dispersion are very scarce, and this is hindering the study of the functional relationship between effective thermal conductivity and thermal dispersion.

In this chapter, we turn our attention to processes of combined (simultaneous) heat and mass transfer that are driven by buoyancy. The density gradients that provide the driving buoyancy force are induced by the combined effects of temperature and species concentration nonuniformities present in the fluid-saturated medium. The present chapter is guided by the review of Trevisan and Bejan (1990), which began by showing that the conservation statements for mass, momentum, energy, and chemical species are the equations that have been presented here in Chaps. 1, 2, and 3. In particular, the material in Sect. 3.3 is relevant. The new feature is that beginning with Eq. (3.26), the buoyancy effect in the momentum equation is represented by two terms, one due to temperature gradients and the other to concentration gradients. Useful review articles on double-diffusive convection include those by Mojtabi and Charrier-Mojtabi (2000, 2005), Mamou (2002b), and Diersch and Kolditz (2002).

In the examples of forced and natural convection discussed until now, the fluid that flowed through the pores did not experience a change of phase, no matter how intense the heating or cooling effect. In this chapter, we turn our attention to situations in which a change of phase occurs, for example, melting or evaporation upon heating and solidification or condensation upon cooling. These convection problems constitute a relatively new and active area in the field of convection in porous media.

Most of the studies of convection in porous media published before 1970 were motivated by geophysical applications, and many published since have geophysical ramifications; see, for example, the reviews by Cheng (1978, 1985b). On the other hand, geothermal reservoir modeling involves several features that are outside the scope of this book. Relevant reviews include those by Donaldson (1982), Grant (1983), O’Sullivan (1985a), Bodvarsson et al. (1986), Bjornsson and Stefansson (1987), McKibbin (1998, 2005), and O’Sullivan et al. (2000, 2001). An important book dealing with geological fluid dynamics is that by Phillips (2009). In this book, the emphasis is on flow patterns and specifically geological processes, involving dissolution, chemical reaction, and deposition. Some examples are discussed below in Sect. 11.12.