Porous media cell model

Aluminum foam can be used as a porous medium in the model of a heat sink with inner heat generation (Hetsroni et al. 2006a). Open-cell metal foam has a good effective thermal conductivity and a high specific solid-fluid interfacial surface area. 

Figure 2.68 Grid model of a porous medium (left) and renormalization group transformation replacing a cluster of grid cells by a unit cell of larger scale (right).

A novel flow cell has been developed to observe on a microscopic level the steady state, cocurrent flow of two pre-equilibrated phases in a porous medium. It consists of a rectangular capillary tube packed with a bilayer of monodisperse glass beads 109 microns in diameter. The pore sizes in the model are of the order of magnitude of those in petroleum reservoirs. An enhanced videomicroscopy and digital imaging system is used to record and analyze the flow data. 

The mechanisms of steady state, cocurrent, two-phase flow through a model porous medium have been established for the complete range of capillary numbers of interest in petroleum recovery. A fundamental understanding of the mobile ganglia behavior observed requires a knowledge of how phases break up during flow through porous media. Several mechanisms have been reported in the literature and two have been observed in this flow cell. 

Fig. 23, Model structure of porous reaction medium (a) Overall view, (b) schematic diagram of cell model structure.

In order to include the coupling between the rugged laminar flow in a porous medium and the molecular diffusion, Horvath and Lin [50] used a model in which each particle is supposed to be surrounded by a stagnant film of thickness 5. Axial dispersion occurs only in the fluid outside this stagnant film, whose thickness decreases with increasing velocity. In order to obtain an expression for S, they used the Pfeffer and Happel "free-surface" cell model [52] for the mass transfer in a bed of spherical particles. According to the Pfeffer equation, at high values of the reduced velocity the Sherwood number, and therefore the film mass transfer coefficient, is proportional to... 

Through the UBE models the basic deposition mechanisms of fines can be probed. However these models center their analysis at one or a series of unit cells. Furthermore, the predictions focus on the deposition and entrainment of particles. Often fines are released from substrate of the porous medium, and it becomes necessary to understand the releasing mechanism to avoid potential plugging problems. 

In Happel s model the porous medium is taken to be a random assemblage that is assumed to consist of a number of cells, each of which contains a particle surrounded by a fluid envelope. The fluid envelope is assumed to contain the same volumetric proportion of fluid to solid as exists in the entire assemblage. This determines the envelope radius of each cell. For illustration we... 

According to the model of porous medium with resistance, the pressure drop per unit length of the medium is equal to the force F divided by the cell volume... 

Assume that the porous medium is composed of a series of parallel tubes (bundle of tubes model) each of diameter 2a (Fig. 2c). The unit cell is now formed by the region 0 < r < a, 0 < r < L, where L is an arbitrary length. The external field E and the pressure gradient VP are applied parallel to the tube axis. The capillary axis is z, and the distance to this axis is r. 

Tindy and Raynal (1966) measured the bubblepoint pressure of two reservoir crude oils in both an open space (PVT cell) and a porous medium with grain sizes in the range of 160 to 200 microns. The bubble-point pressures of those two crude oils were higher in the porous medium than in a PVT cell by 7 and 4 kg/cm, respectively. Specifically, the bubblepoint pressure of one of the two crude oils measured at 80 C in a PVT cell was 121 kg/cm and the bubblepoint pressure at the same temperature in a porous medium of 160 to 20 microns was 128 kg/cm. On the other hand, when these authors used a mixture of methane and n-heptane, they observed no differences in the saturation pressure. Sigmund et aL (1973) have also investigated the effect of the porous medium on phase behavior of model fluids. Their measurements on dewpoint and bubblepoint pressures showed no effect of the porous medium. The fluid systems used by these authors were Cj/nC. and Ci/nCs. The smallest bead size used was 30 to 40 U.S. mesh. In Example 2.3 presented at the end of this chapter, the effect of interface curvature on dewpoint pressure and equilibrium phase composition will be examined. 

Rigid random arrays have generally been simulated by cell models that have not been limited to dilute suspensions. An early example of a cell model is that of Brinkman (1947), who eonsidered flow past a single sphere in a porous medium of permeability k. The flow is deseribed by an equation that collapses to Darcy s (1856) law (in its post-Darcy form, which includes viscosity) for low values of and to the creeping flow version of the Navier Stokes equation for high values of K. His solution is... 

The thermal conductivity of a porous medium depends on the material composition and sttucture. There has been a number of works on modeling the thermal conductivity of porous GDLs (Sadeghi et al., 2008, 2011). Cell and stack modeling usually employ experimental data on Xp, which can be found in Khandelwal and Mench (2006). 

The fluid velocity distribution given by Eqs. (93)-(96) are only valid for an isolated particle. However, there are a number of practically important situations, like the deep-bed filtration process, when the flow past an assembly of spheres (forming a porous mediiun) takes place. In this case, the flow field around a single sphere is influenced by the presence of other spheres. Various models that describe the flow field in the packed bed consisting of spheres are available. The sphere in cell models [81-83] assume that each sphere in the packed bed is surrounded by the spherical cavity filled with fluid. The size of the cavity is determined by the overall average porosity of the medium. The general solution of the Navier-Stokes equation for the stream function inside the cavity may be written as [7]... 

In the development of an expression for Et, a more detailed model of the porous medium is needed. Of the three types of models of a granular filter as a porous medium, the capillaric model is not preferred. For the sake of simplicity, we will consider one of the other two, namely the spherical collector model. In this model, the filter grain is assumed to be a sphere. There are a number of alternative approaches based on a spherical collector. We will illustrate the approach by Happel (1958). In Happel s model, the granular porous medium is assumed to consist of a large collection of identical cells, where each cell consists of a spherical particle of radius ((dgr)/2) (i.e. half of the average grain diameter) surrounded by a liquid envelope of radius b, such that the void volume of this cell is identical to the void volume of the porous medium ... 

Two different types of dynamic test have been devised to exploit this possibility. The first and more easily interpretable, used by Gibilaro et al [62] and by Dogu and Smith [63], employs a cell geometrically similar to the Wicke-Kallenbach apparatus, with a flow of carrier gas past each face of the porous septum. A sharp pulse of tracer is injected into the carrier stream on one side, and the response of the gas stream composition on the other side is then monitored as a function of time. Interpretation is based on the first two moments of the measured response curve, and Gibilaro et al refer explicitly to a model of the medium with a blmodal pore... 

Going from planar to porous electrode introduces another length scale, the electrode thickness. In the case of a PEM fuel cell catalyst layer, the thickness lies in the range of IcL — 5-10 pm. The objective of porous electrode theory is to describe distributions of electrostatic potentials, concentrations of reactant and product species, and rates of electrochemical reactions at this scale. An accurate description of a potential distribution that accounts explicitly for the potential drop at the metal/electrolyte interface would require spatial resolution in the order of 1 A. This resolution is hardly feasible (and in most cases not necessary) in electrode modeling because of the huge disparity of length scales. The simplified description of a porous electrode as an effective medium with two continuous potential distributions for the metal and electrolyte phases appears to be a consistent and practicable option for modeling these stmctures. 

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