Volume-averaging approach

The volume averaging approach discussed in the section on diffusive transport can also be extended to account for electrophoresis [215] and hydrodynamic flow [215,436]. Locke [215] considered the application of volume averaging to the determination of the effective... 

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. 

It has become customary to rewrite the interfacial momentum transfer term M.J in terms of the interfacially averaged pressure pk)Ai and shear stresses Ck)Ai of phase k to separate the mean field effects from local effects [233, 194, 112, 54, 56, 153, 154, 193]. In the volume averaging approach the interfacial area averaged pressure is defined by ... 

To relate the classical mixture theory to the more familiar volume averaging method we may assume that the mixture CV, which is larger than a phase element but smaller than the characteristic domain dimension, coincides with the averaging volume used in the volume averaging approach. 

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. 

Detailed derivations of the porous electrode model and the volume averaging approach for battery modeling can be found in Refs [14,42, 52], respectively. 

In order to overcome these problems, one approach, originally developed by Whitaker [420], Slattery [359], and Anderson and Jackson [17], involves the method of volume averaging. Using volume averaging theory, Whitaker and coworkers [193,264,268,337,436] found the effective diffusion tensor for a two-phase system to be given by... 

Penke, B Kinsey, S Gibbs, SJ Moerland, TS Locke, BR, Proton Diffusion and T1 Relaxation in Polyacrylamide Gels A Unified Approach Using Volume Averaging, Journal of Magnetic Resonance 132, 240, 1998. 

When a sphere of radius a is compared to plane sheets with half-thickness a, the volume-averaged apparent age for the case of sphere is 1/5 of that for the case of plane sheet, and the apparent age at the center for the case of sphere is 1/3 of that for the case of plane sheet. Figure 5-16 shows how the volume-averaged apparent age of the mineral grains approaches the steady-state value for the case of (a) Co = 0 and (b) Co=pa /D. 

Today there are between 45 and 50 plastic materials. Physical volume, averaging a 13% per year growth rate for the past ten years, reached an estimated 16 billion pounds in 1968, higher than that of any metal except iron and steel, and approaching the total for non-ferrous metals. The number of formulas, grades, and types of these materials is greatly expanded by the use of plasticizers, fillers, and polymerization alternatives. All of these formulations are presumably different from one another and offer the user a broad material selection to fit his property and cost requirements. 

Increasing the concentration of particles (roughly for a volume fraction approaching 10%), the stress concentration effects of neighboring particles can overlap (Fig. 13.2). Therefore, a large volume fraction of the matrix supports an average load higher than the applied load and can yield. This stress concentration effect increases when the volume fraction of dispersed particles increases or the interparticle distance decreases. 

Parker, J. C., and van Genuchten, M. Th. (1984). Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport, Water Resources Research, 20(7), 866-872. 

As mentioned, the pseudocontinuum approach can be used for the multiphase flow analysis only when the control volume contains enough particles that the volume average is of statistical significance, and at the same time, the control volume is small enough to ensure the validity of the continuum assumption. Thus, in order to have a statistically meaningful phase average, there exists a minimum averaging volume. 

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