Molar-average velocity

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 ... 

J mass-transfer flux relative to molar average velocity, mol/(m -s) ... 

It should be emphasized that the flux vectors for which expressions have been given in Eqs. (28) through (36) are all defined here as fluxes with respect to the mass average velocity. Not all authors use this convention, and considerable confusion has resulted in the definition of the energy flux and the mass flux. Mass fluxes with respect to molar average velocity, stationary coordinates, and the velocity of one component (such as the solvent, for example) are all to be found in the literature on diffusional processes. Research workers in the field of diffusion should be meticulous in specifying the frame of reference for fluxes used in writing up their research work. In the next section this important matter is considered in detail for two-component systems. 

There are also many ways of expressing the velocity of a chemical species present in a flow system. We do not concern ourselves here with the instantaneous velocity of the individual molecules of a species, but rather with the average macroscopic velocities with which the species travel. These may be measured from a stationary coordinate system, but for flow systems the velocities of individual species are frequently measured from a coordinate frame moving with (a) the mass-average velocity of the stream, (b) the molar average velocity of the stream, or (c) the velocity of one particular component. The mass- and the molar-average velocity are defined in Table II and the notation for the various velocities of an individual species is given. 

Sherwood and Pigford (S9) have discussed the problem of the absorption of a solute A by a solvent S upon solution, A may be converted into B according to the reaction A = B (k/ and krf being the forward and reverse reaction-rate constants, and K = k//k/). The concentration of A is maintained at cAo at the surface of the liquid S, and it is assumed that S is semiinfinite in extent. It is further assumed that B is nonvolatile that is, it cannot escape from solvent S. Equation (51) is then used to explain the diffusion of A and B, with DAg and DBs taken as concentration independent, and the term containing the molar average velocity w is neglected. Hence the mathematical statement of the problem is (for very dilute solutions of A and B)... 

Next the dependence of W on xKo has to be determined. Since w depends on time but not on position, it can best be evaluated by getting an expression for wo (the molar-average velocity at z = 0), which is... 

Intermolecular potential energy of interaction A Mass flux of A with respect to molar-average velocity (see Table III) I a Molar flux of A with respect to molar-average velocity (see Table III) (38)... 

The molar flux of species k relative to the molar average velocity is denoted J , calculated as... 

See also Problem 12.12.) The mixture-averaged diffusion coefficient of Eq. 12.176 was derived for use in calculating the molar diffusion velocity J with respect to the molar average velocity V, as in Eq. 12.172. 

Fick s law of molecular diffusion states that, for a binary mixture of components A and B, the molar flux of component A by ordinary molecular diffusion relative to the molar average velocity of the mixture in the positive z direction, is proportional to the concentration gradient dcA/dz, which is negative in the direction of ordinary molecular diffusion ... 

Here the term (v, - v) is called the diffusion velocity. The mass flow jj, is defined analogously. As the two chemical species interdiffiise, there is a shifting of the center of mass in the v-dircc lion if the molecular weights of components i and j differ. The flows j, and j, are measured with respect to the motion of the center of mass, and j, + jj, = 0. Molar average velocity is... 

Here w, is the mass fraction of component i. We can express the molar diffusion flow J(A, based on the molar average velocity vM... 

Equations 3.13 are known as the Stefan-Maxwell equations and are valid when the total pressure and temperature gradients as well as external forces can be neglected. They have the physical meaning that the rate of momentum transfer between two species is proportional to their concentrations and to the difference in their velocities. The molar average velocities of the species v, and v are defined in a such way that the molar fluxes of the various species are... 

This process is called equimolar counterdiffusinn for obvious reasons. The net molar flow rate of the mixiure for such a process, and thus the molar-average velocity, is zero since... 

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

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