Velocity volume average

The first equahty (on the left-hand side) corresponds to the molar flux with respect to the volume average velocity while the equahty in the center represents the molar flux with respect to the molar average velocity and the one on the right is the mass flux with respect to the mass average velocity These must be used with consistent flux expressions for fixed coordinates and for Nc components, such as ... 

The choice of vx is a matter of convenience for the system of interest. Table 1 summarizes the various definitions of vx and corresponding, /Y, commonly in use [3], The various diffusion coefficients listed in Table 1 are interconvertible, and formulas have been derived. For polymer-solvent systems, the volume average velocity, vv, is generally used, resulting in the simplest form of Jx,i- Assuming that this vv = 0, implying that the volume of the system does not change, the equation of continuity reduces to the common form of Fick s second law. In one dimension, this is... 

The first term on the right-hand side of Eqs. (145) and (146) is a pressure term shared by both phases. The purpose of this term (when ps and pg are constant) is to ensure that the volume-average velocity, defined by... 

The first term on the right-hand side is the diffusive flux relative to the volume average velocity. The second term represents a contribution due to bulk flow. It should be emphasized here that the separation of the total flux into two contributions is always possible regardless of the actual transport mechanism through the membrane. In other words, Eq. (7) is purely phenomenological and does not require any specific transport model. 

The losses associated with the flow over the particles of the porous material are proportional to the volume-averaged velocity. 

For a bounded suspension subject to an external electric field E,, the volume-average velocity and electric field obey the following ... 

V porous medium volume average velocity of liquid. Figure 5... 

These are the most commonly encountered sets of fluxes other sets could be defined. We could, for example, define a mass diffusion flux relative to the molar average velocity or a molar diffusion flux relative to the mass average velocity. Still other choices of reference velocity are sometimes used for example, the volume average velocity... 

To relate the molar diffusion flux relative to the volume average velocity to the molar diffusion flux relative to the molar average reference velocity we use the transformation... 

If we choose to represent the diffusion fluxes with respect to the volume average velocity... 

Molar flux with respect to volume average velocity. 

To relate [D] to the volume average velocity reference frame we use another... 

Starting from the definitions of the molar diffusion fluxes in the molar average velocity reference frame and in volume average velocity reference frame (Table 1.3), verify Eqs. 1.2.20 and 1.2.22. These equations may be written in n — 1 dimensional matrix form as... 

Matrix of Fick diffusion coefficients in volume average velocity reference frame [mVs]... 

Figure 2-1. We consider a surface 5 drawn in a fluid that is modeled as a billiard-ball gas. Initially, when viewed at a macroscopic level, there is a discontinuity across the surface. The fluid above is white and the fluid below is black. The macroscopic (volume average) velocity is parallel to S so that u n — 0. Thus there is no transfer of black fluid to the white zone, or vice versa, because of the macroscopic motion u. At the molecular (billiard-ball) level, however, all of the molecules undergo a random motion (it is the average of this motion that we denote as u). This random motion produces no net transport of billiard balls across S when viewed at the macroscopic scale because u n = 0. However, it does produce a net flux of color. On average there is a net flux of black balls across S into the white region and vice versa. In a macroscopic theory designed to describe the transport of white and black fluid, this net flux would appear as a surface contribution and will be described in the theory as a diffusive flux. The presence of this flux would gradually smear the initial step change in color until eventually the average color on both sides of. S would be the same mixture of white and black.

Strictly speaking, for a multicomponenet fluid, this should be the mass average velocity v as defined in Note 2. However, in this section we restrict our discussion to a single-component fluid for which the volume average velocity u and the mass average velocity v are equal. 

We should note that this description of the system as being two fluids separated by a flat interface already has inherent in it the spatial averaging to a scale of resolution that is much larger than the individual pore level of description. The volume-averaged velocity in each fluid is determined by Darcy s law. As noted earlier, we assume that the fluids (and the interface between them) move with a uniform velocity V in the positive z direction. It is therefore convenient to consider the problem with respect to a moving reference frame that is fixed at the unperturbed fluid interface, i.e., we introduce z, which is related to the original laboratory frame of reference as... 

Difliisivities are defined, not with respect to a stationary plane, but relative to a plane moving at the volume-average velocity Mq. By definition there is no net volumetric flow across this reference plane, although in some cases there is a net molar flow or a net mass flow. The molar flux of component A through this reference plane is a diffusion flux designated and is equal to the flux of A for a stationary plane [Eq. (21.2)] minus the flux due to the total flow at velocity Mq and concentration... 

Liquid-Film Region. The single-phase flow and heat transfer in this region can be described by the continuity equation, the momentum equation in Eqs. 9.76 and 9.77, and the energy equation (Eq. 9.80). We deal only with the volume-averaged velocities, such as (ue) = U(, therefore, we drop the averaging symbol from the superficial (or Darcean) velocities. For the two-dimensional steady-state boundary-layer flow and heat transfer, we have (the coordinates are those shown in Fig. 9.18) the following ... 

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