Diffusional Fluxes in Multicomponent Mixtures

all tenns in the mass balance are expressed as volume integrals. When these three terms are moved to the left-hand side of the mass balance, given by (9-3), the following result is obtained  

No assumptions have been invoked to obtain this result. As illustrated below, the mass flux term with respect to a stationary reference frame, V p, v, contains contributions from bulk fluid flow (i.e., convection) and molecular mass transfer via diffusion. In fact, whenever the divergence of a flux appears in a microscopic balance expression, its origin was a dot product of that flux with the outward-directed unit normal vector on the surface of the control volume, accounting for input and output due to flux across the surface that bounds V(t). The divergence of a flux actually represents a surface-related phenomenon that has been transformed into a volume integral via Gauss s law. 

which suggests that the sum of all diffusional mass fluxes in the mixture with respect to the mass-averaged frame is zero. For binary mixtures, this statement is known as equimolar counterdijfusion. 

With the foregoing definitions of diffusional fluxes in a multicomponent mixture, it is possible to manipulate the mass transfer eqnation. 

The net input due to mass flux acting across the surface of the control volume is split into contribntions from convection and diffusion via (9-8)  

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