Molar average bulk velocity

Reference Frames and Fluxes, in speaking of a diffusional flux, it is necessary to specify a reference frame from which the diffusion process is to be observed. Generally, a reference velocity such as the mass, molar, or volume average bulk velocity (v,v, or v, respectively) in the system is selected, and movement of the component of interest relative to this reference velocity is defined to be true diffusion. For practical reasons, a fixed reference frame is generally also considered to relate the mathematical treatment of this molecular scale transport process to an actual physical system with well-defined dimensions. In this case, the total flux relative to the fixed reference frame is partitioned into two parts bulk flow and true molecular diffusion. This partitioning is necessary, because even in the absence of an externally imposed bulk flow, interdiflfusion of molecules with respect to each other produces an effective bulk flow relative to fixed coordinates if the molecules have different masses (1,2). 

The definitions of the mass, molar, and volume average bulk velocities are given in Table 1 along with selected mass and molar flux expressions related to each of the specified reference velocities for a binary system of components A and B. The mutual diffusion coefficient Dab is the diffusivity of component A and B in the mixture as defined below. The mutual diffusion coefficients appearing in Table 1 are identical in all of the expressions and Dab = DBA-... 

When the total pressures at the two ends of the pore are different, there will be bulk flow. If conditions for Knudsen diffusion exist, then tbe molar average bulk flow velocity (v ) may be estimated from... 

The bulk flow term Ba can also be expressed in terms of the concentration of A and the molar average velocity V... 

In Section 6.2B the flux was defined as the molar flux of A in kg moM/s-m relative to the molar average velocity of the whole or bulk stream. Also, was defined as the molar flux of A relative to stationary coordinates. Fluxes and velocities can also be defined in other ways. Table 7.5-1 lists the different types of fluxes and velocities often used in binary systems. 

Velocity (m/s) or Molar internal energy, (kj/kmol) Averaged axial bulk velocity, m/s (ft/s)... 

Calculate now the molar flux of the ith species (in a binary system) in the positive z-direction across a plane fixed with respect to stationary axes (Bird et al., 1960, 2002). Assume temporarily that there can be bulk flow the molar average and mass average velocities in the z-direction are and v, respectively. If the averaged (over one mole, say) z-directional velocity of molecules of the ith species is Vj2, then the flux Niz is defined by (see Figure 3.1.3B)... 

This implies that the total mass flux of the ith species relative to a fixed coordinate system consists of two fluxes one is due to the molecular diffusion (diffusion velocity v,- — v) and the other is due to bulk movement (bulk velocity v). Similar arguments hold for the total molar and volume fluxes. This partitioning is necessary, because interdiffusion of unequal size molecules of two components causes bulk flow, even in the absence of an external bulk flow. Table 4.2 summarizes the definitions of the average velocities and their relations. The following example (basic features of which are drawn... 

While the flux relative to the average molar velocity J always means transfer down a concentration gradient, the flux N need not. For example, consider the dissolution of a hydrated salt ciystal such as NajCOj IOH2O into pure water at 20 C. The solution in contact with the crystal surface contains NajCOj and HjO at a concentration corresponding to the solubility of Na2C03 in HjO, or 0.0353 mole fraction NajCOj and 0.9647 mole fraction HjO. For the Na2C03, transfer is from the crystal surface at a concentration 0.0353 outward to 0 mole fraction Na2C03 in the bulk liquid. But the water of crystallization which dissolves must transfer outward in the ratio 10 mol HjO to 1 mol NajCOj from a concentration... 

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