Porous volume-averaged transport

The continuum form of the bubble population balance, applicable to flow of foams in porous media, can be obtained by volume averaging. Bubble generation, coalescence, mobilization, trapping, condensation, and evaporation are accounted for in the volume averaged transport equations of the flowing and stationary foam texture. 

The dynamics of a heterogeneously catalyzed gas-phase reaction occurring in a nanoporous medium in combination with heat and mass transfer was simulated using a finite-volume approach. In contrast to other studies of similar type, heat and mass transfer in the nanoporous medium was explicitly accounted for by solving volume-averaged transport equations in the porous medium. Such an approach made it possible to compare the transport resistances in the gas phase and in the porous medium and to study the tradeoff between maximization of... 

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. 

It is extremely difficult to model macroscopic transport of mass, energy, and momentum in porous media commonly encountered in various fields of science and engineering based on microscopic transport models that account for variation of velocity and temperature as well as other quantities of interest past individual solid particles. The basic idea of porous media theory, therefore, is to volume average the quantities of interest and develop field equations based on these average quantities. 

The first step in applying volume averaging is to consider a representative volume for every point A in the porous medium. This volume must be large enough to contain sufficient amount of each phase such that continuum theory for transport of mass, energy, and... 

In this review, a set of balance equations for transport of heat, mass, and momentum in stationary and moving porous media has been derived based on a local volume averaging approach. The advantage of this method is that it allows precise definition of average temperature, velocity, and pressure. Moreover, equations are derived rigorously from first principles. 

Volume averaging is a technique in which the fundamental equations are spatially averaged over a representative elementary volume (REV) of porous media.f ° This approach has provided insight into the relationship between fundamental physics and larger-scale behavior but is rarely used for studying transport in specific media in a deterministic sense. 

Whitaker, S. Volume averaging of the transport equations. In Fluid Transport in Porous Media] du Plessis, P., Ed. Computational Mechanics Publications Southampton, U.K., 1997. 

Quintard M, Whitaker S (1993) Transport in Ordered and Disordered Porous Media Volume-Averaged Equations, Closure Problems, and Comparisons with Experiments. Chem Eng Sci 48(14) 2537-2564... 

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. 

Here we have emphasized the intrinsic nature of our area-averaged transport equation, and this is especially clear with respect to the last term which represents the rate of reaction per unit volume of the fluid phase. In the study of diffusion and reaction in real porous media (Whitaker, 1986a, 1987), it is traditional to work with the rate of reaction per unit volume of the porous medium. Since the ratio of the fluid volume to the volume of the porous medium is the porosity, i.e. 

Quintard M. and Whitaker S. 1993a. Transport in ordered and disordered porous media Volume averaged equations, closure problems, and comparison with experiment, Chem. Eng. Sci., 48, 2537-2564. 

The fourth type of electrode models [39, 46] is based on the volume-averaging approach for porous media [51-55]. As indicated in Figure 28.4, it is similar to the agglomerate model, but no macropores are considered. Hence the main direction of mass transport in these micropores is along the depth of the electrode. 

Whitaker, S. 1998. The Method of Volume Averaging. Theory and Applications of Transport in Porous Media. Dordrecht, the Netherlands Kluwer Academic Publ. 

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