2.3  Fluxes That Comprise Total Diffusion Flux

David B. McWhorter

2.3 Fluxes That Comprise Total Diffusion Flux

Graham’s (1833) experiments clearly demonstrate that the components of a binary gas at uniform pressure in a porous medium diffuse at different rates in general. Consequently, the sum of the total diffusive fluxes of the individual components, $\inline N_{A}^{D}+N_{B}^{D}$ , is not zero. Rather, this sum contributes to the motion of the fluid as a whole—a feature of diffusion in porous solids that is not observable in systems free of solid obstructions. We refer to this net diffusive flux as the non-equimolar flux (Cunningham and Williams, 1980) and write Equation 5.

$\displaystyle N^{D}=N_{A}^{D}+N_{B}^{D}$

(5)

Similar to the viscous flux, the non-equimolar flux imparts motion to the individual species in the mixture by advection (i.e., x_i N^D, i = A, B), but is distinguished from advection via a viscous flux by the fact it arises solely as a result of diffusion. The sum of advection by the non-equimolar flux and by the viscous flux is the total advection by the phase motion.

The increment of motion for component i that is in addition to advection via the phase motion is the equimolar diffusion flux J_i defined by Equation 6.

$\displaystyle J_{i}=N_{i}-x_{i}N$ , $\displaystyle i=A,B$

(6)

Because equimolar diffusion makes no net contribution to motion of the phase, we have Equation 7.

$\displaystyle J_{A}+J_{B}=0$

(7)

Equation 7 expresses a condition that holds under all circumstances treated in this book.

It is common to rearrange Equation 6 so that the mole flux is expressed as the sum of the equimolar and advection fluxes as in Equation 8a for constituent A. We may then express the flux of component A by any one of Equations 8a through 8d. The subscripts can be interchanged to obtain the equivalent expressions for component B.

$\displaystyle N_{A}=J_{A}+x_{A}N$

(8a)

$\displaystyle N_{A}=J_{A}+x_{A}\left ( N_{A}+N_{B} \right )$

(8b)

$\displaystyle N_{A}=J_{A}+x_{A}\left ( N^{D}+N^{v} \right )$

(8c)

$\displaystyle N_{A}=J_{A}+x_{A}\left ( N_{A}^{D}+N_{B}^{D}+N^{v} \right )$

(8d)

The reader is encouraged to become thoroughly familiar with these definitional equations and the various forms they may take. For example, if there is no viscous flux then $N^{v}=0$ , $N_{A}=N_{A}^{D}$ and Equation 8d becomes Equation 9.

$\displaystyle N_{A}^{D}=J_{A}+x_{A}\left ( N_{A}^{D}+N_{B}^{D} \right )$

(9)

Rearranging and solving for J_A gives Equation 10.

$\displaystyle J_{A}=N_{A}^{D}x_{B}-x_{A}N_{B}^{D}$

(10)

This is in the form of a Stefan-Maxwell equation for a binary gas that we soon will have occasion to use in our calculations.

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