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, , 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.
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(5) |
Similar to the viscous flux, the non-equimolar flux imparts motion to the individual species in the mixture by advection (i.e., xi ND, 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 Ji defined by Equation 6.
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(6) |
Because equimolar diffusion makes no net contribution to motion of the phase, we have Equation 7.
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(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.
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(8a) |
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(8b) |
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(8c) |
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(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 ,
and Equation 8d becomes Equation 9.
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(9) |
Rearranging and solving for JA gives Equation 10.
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(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.