Local volume averaging

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 

A number of different volume averages can be defined. First we will consider a spatial average, which is the average value of a quantity within V, 

This average describes the combined property of both phases. A second type of average can be defined which is specific to a given phase. This is called the phase average, 

We have chosen to call the two phases resin and fiber. Each phase will be denoted by subscript r and respectively. A similar phase function (i.e., Yf) can be defined for the fiber phase. It should be noted that if the fiber phase is stationary Y is not a function of time. 

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The porosity and permeability depend solely on the structure of the medium. In principle, if the entire geometry of the solid surface were known, these properties could be calculated by local volume-averaging [1]. As mentioned previously, this is not the case, except possibly for certain idealized situations. Arguably, there should... 

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. 

As we do for all mass transfer problems, we must satisfy the differential equation of continuity for each species as well as the differential momentum balance. Since we are dealing with a porous medium having a complex and normally unknown geometry, we choose to work in terms of the local volume averaged forms of these relations. Reviews of local volume averaging are available elsewhere (23-25). 

We will limit our attention here to the creeping flow of a Newtonian fluid in a rigid, isotropic, permeable medium having a uniform porosity f, and we will assume that the total mass density p of the fluid is nearly a constant within the averaging region. Under these conditions, the local volume average of the ecjuation of continuity may be written as (23)... 

Let us begin with the first of these cases, binary diffusion of species A and B with a constant diffusion coefficient. If there are no chemical reactions and neglecting any adsorption on the pore walls, the local volume-averaged ecjuation of continuity for a particular species A may be written as [the more general case is treated by Slattery (23)]... 

With the exceptions of Schowalter (38) and of Hickernell and Yortsos (22), all previous linear stability analyses (34-37, 39-42) have used the local volume-averaged ecjuation of continuity for an incompressible fluid, although they assumed that density was a function of concentration and therefore position and time. This is... 

As discussed in Section 1, the absolute permeability is a constitutive macroscopic property which arises in the local volume-averaged momentum balance, which is Darcy s law (Eq. (2)). It is interesting that, while the porosity can be defined independently of the equation of continuity, the permeability is... 

The calculated values required to evaluate the performance index are determined from the solution of differential equations describing flow. Pressure and velocity values are obtained by solving the locally volume-averaged equation of continuity and differential momentum balance (Eqs (1) and (2)). For steady-state conditions, and incompressible flow, the equation of continuity becomes... 

Area averaging can be considered to be a limiting case of local volume averaging [43, 47, 189]. Thus the phrase limiting form refers to the modified forms of the averaging theorems which are applicable to the governing 3D equations to derive a set of equations valid for ID problems. 

Gray WG, Lee PCY (1977) On the Theorems for Local Volume Averaging of Multiphase Systems. Int J Multiphase Flow 3 333-340... 

Gray WG (1983) Local Volume Averaging of Multiphase Systems Using A Non-Constant Averaging Volume. Int J Multiphase Flow 9(6) 755-761... 

Plumb, O.A., and S. Whitaker. 1988. Dispersion in heterogeneous porous media I. Local volume averaging and large-scale averaging. Water Resour. Res. 24 913 926. 

For the analysis of the macroscopic heat flow through heterogeneous media, the local volume-averaged (or effective) properties such as the effective thermal conductivity k) = k, are used. These local effective properties such as the heat capacity (pcp), thermal conductivity (k), and radiation absorption and scattering coefficients (a ) and (a,) need to be arrived at from the application of the first principles to the volume over which these local properties are averaged, that is, the representative elementary volume. 

Local Volume Averaging. The principle of volume averaging and the requirement of existence of the local thermal equilibrium between the fluid and solid phases was discussed in the section entitled Conduction Heat Transfer. In addition to the diffusion time and length scale requirements for the existence of the local thermal equilibrium, the residence timescales (the time it takes for a fluid particle to cover the length scales d, i, and L) must be included in the length and timescale requirements. 

Local Volume Averaging. The local volume-averaging treatment leading to the coupling between the energy equation for each phase is formulated by Carbonell and Whitaker [81] and is given in Zanotti and Carbonell [82], Levee and Carbonell [83], and Quintard et al. [84]. Their development for the transient heat transfer with a steady flow is reviewed here. Some of the features of their treatment are discussed first. 

In order to arrive at a local volume-averaged momentum equation for each phase, the effect of the preceding parameters on the microscopic hydrodynamics must be examined. This is done to an extent through the particular forces that appear in the momentum equations. 

In order to study two-phase flow and heat transfer for this phase change problem, we assume that the local volume-averaged conservation equations (including the assumption of local thermal equilibrium) are applicable. For this problem, we note the following. 

For plain media, the film evaporation adjacent to a heated vertical surface is similar to the film condensation. In porous media, we also expect some similarity between these two processes. For the reasons given in the last section, we do not discuss the cases where 8gld — 1, where 8g is the vapor-film region thickness. When 8gld 1 and because the liquid flows (due to capillarity) toward the surface located at y = 8g, we also expect a large two-phase region, that is, ld 1. Then a local volume-averaged treatment can be applied. 

PLUMB, O.A. 8c WHITAKER, S. 1990. Diffusion, adsorption and dispersion in porous media Small-scale averaging and local volume averaging. In Dynamics of Fluids in Hierarchical Porous Media (ed. J.H. Cushman), pp. 97-149. San Diego Academic Press. 

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