Porous transport coefficient

Many investigators have studied diffusion in systems composed of a stationary porous solid phase and a continuous fluid phase in which the solute diffuses. The effective transport coefficients in porous media have often been estimated using the following expression ... 

In order to be useful in practice, the effective transport coefficients have to be determined for a porous medium of given morphology. For this purpose, a broad class of methods is available (for an overview, see [191]). A very straightforward approach is to assume a periodic structure of the porous medium and to compute numerically the flow, concentration or temperature field in a unit cell [117]. Two very general and powerful methods are the effective-medium approximation (EMA) and the position-space renormalization group method. 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

As a second method to determine effective transport coefficients in porous media, the position-space renormalization group method will be briefly discussed. 

Permeability (k) is the transport coefficient for the flow of fluids through a porous medium and has the units of length squared. NMR measures the porosity and the... 

One must understand the physical mechanisms by which mass transfer takes place in catalyst pores to comprehend the development of mathematical models that can be used in engineering design calculations to estimate what fraction of the catalyst surface is effective in promoting reaction. There are several factors that complicate efforts to analyze mass transfer within such systems. They include the facts that (1) the pore geometry is extremely complex, and not subject to realistic modeling in terms of a small number of parameters, and that (2) different molecular phenomena are responsible for the mass transfer. Consequently, it is often useful to characterize the mass transfer process in terms of an effective diffusivity, i.e., a transport coefficient that pertains to a porous material in which the calculations are based on total area (void plus solid) normal to the direction of transport. For example, in a spherical catalyst pellet, the appropriate area to use in characterizing diffusion in the radial direction is 47ir2. 

In an aquifer, the total Fickian transport coefficient of a chemical is the sum of the dispersion coefficient and the effective molecular diffusion coefficient. For use in the groundwater regime, the molecular diffusion coefficient of a chemical in free water must be corrected to account for tortuosity and porosity. Commonly, the free-water molecular diffusion coefficient is divided by an estimate of tortuosity (sometimes taken as the square root of two) and multiplied by porosity to estimate an effective molecular diffusion coefficient in groundwater. Millington (1959) and Millington and Quirk (1961) provide a review of several approaches to the estimation of effective molecular diffusion coefficients in porous media. Note that mixing by molecular diffusion of chemicals dissolved in pore waters always occurs, even if mechanical dispersion becomes zero as a consequence of no seepage velocity. 

A nonconventional view of membrane microstructure, which neither conforms with the solution nor with the porous rock picture, was recently suggested in Ref. 84. Classical MDs simulations on microstructure and molecular mobility in swollen Nation membranes revealed a picture of a rather dynamic structure of water clusters with temporary formation and break-up of water bridges between them. The frequency of intercluster bridge formation was found to be consistent with the experimental transport coefficients through the membrane. 

Brenner (1980) has explored the subject of solute dispersion in spatially periodic porous media in considerable detail. Brenner s analysis makes use of the method of moments developed by Aris (1956) and later extended by Horn (1971). Carbonell and Whitaker (1983) and Koch et al. (1989) have addressed the same problem using the method of volume averaging, whereby mesoscopic transport coefficients are derived by averaging the basic conservation equations over a single unit cell. Numerical simulations of solute dispersion, based on lattice scale calculations of the Navier-Stokes velocity fields in spatially periodic structures, have also been performed (Eidsath et al., 1983 Edwards et al., 1991 Salles et al., 1993). These simulations are discussed in detail in the Emerging Areas section. 

Reyes, S. and K. Jensen, Estimation of effective transport coefficient in porous solids based on percolation concepts. Chemical Engineering Science, 1985, 40, 1723-1734. 

In addition to the kinetic parameters such as rate constant and activation energy that we have become accustomed to dealing with, the analysis of this section has introduced some very important newcomers. Two of these, the effective transport properties within the porous matrix of the catalyst, and k ff, differ in substance from the transport coefficients in homogeneous phases with which we are familiar, and warrant some special discussion. 

However, note that D (in Eq. 1) is an effective transport coefficient and not a physical property. The r value cannot usually be identified precisely and is therefore approximated as the average size of pores in porous materials. 

Jonsson, G. and Benavente, J. 1992. Determination of some transport coefficients for the skin and porous layer of a composite membrane. J. Memb. ScL 69 29-42. 

The usual orders of magnitude for effective transport coefficients of gases, liquids, and porous solid particles are [13] ... 

In spite of all the difficulties caused by the two-phase effects, channel flows in fuel cells are understood better than the flows in porous layers. Channel flows are subject to fluid dynamics equations with known transport coefficients. Modern commercially available CFD packages... 

D macroscopic transport equations are then solved in this domain with the transport coefficients, which take into account the nature of every computational cell (direct numerical simulation (DNS) model). Physically, this approach enables to calculate transport parameters of a porous media, for example, porosity or the Bruggemann exponent (Mukherjee and Wang, 2006 Wang et al., 2006a). 

Then in Section 10.3, we address the problem of computing mass transport coefficients in porous materials called zeolites. Zeolites are materials with a wide range of applications, such as petrochemical separation, water purification, and catalysis. Understanding and predictably computing mass transport coefficients for a variety of molecules in these molecular sieves instruct optimal use of the appropriate zeolite for an application. We describe the methodology used to compute... 

Pharoah, J. G., Karan, K., and Sun, W. 2006. On effective transport coefficients in PEM fuel cell electrodes Anisotropy of the porous transport layers. Journal of Power Sources 161 214-224. 

A common difficulty in all the aforementioned cases is the irregular geometry of the porous medium. In addition, a precise analysis will have to consider that the diffusion in the pores is modified by surface diffusion and surface reactions. A completely phenomenological approach based on irreversible thermodynamics would give theoretically consistent transport expressions but would be too complicated for experimental determination of the transport coefficients and for the solution of the conservation equations. 

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