Page 19 - gas transport in porous media

12
Webb
Second Law
Fick’s second law of diffusion for clear fluids is concerned with the temporal evolution
of the concentration, or (BSL, pg. 558)
∂c A
2
= D AB ∇ c A
∂t
which is only strictly applicable if the molar-average velocity is zero, or for equimolar
counterdiffusion. This equation is similar to the heat conduction equation, so many
solutions exist such as in Carslaw and Jaeger (1959).
The rest of the present chapter is concerned with Fick’s first law, not Fick’s second
law. For an excellent discussion of Fick’s second law, see Fen and Abriola (2004).
Abriola et al. (1992) and Sleep (1998) also evaluate Fick’s second law.
2.2.1.2 Stefan-Maxwell Equations
Fick’s first law of diffusion presented above is applicable to binary gases. This restric-
tion is due to the fact that the gradients of the two gases are directly related to each
other, so only a single gradient needs to be specified. For multicomponent gases,
multiple gradients need to be determined. For an ideal mixture, the component mass
flux equations can be manipulated resulting in (BSL, pg. 569)
n
1
∇x i = x i N j − x j N i
cD ij
j=1
whichareknownastheStefan-Maxwellequationsapplicabletostationarycoordinates
in a clear fluid. For a two-component system, the Stefan-Maxwell equations reduce
to Fick’s first law. For application to a porous medium, the diffusion coefficients need
to be modified as discussed above.

2.2.2 Free-Molecule Diffusion
Asdiscussedearlier, whenthegasmolecularmeanfreepathbecomesofthesameorder
as the tube dimensions, free-molecule, or Knudsen, diffusion becomes important. Due
to the influence of walls, Knudsen diffusion and configurational diffusion implicitly
include the effect of the porous medium. Unlike ordinary (continuum) diffusion,
there are no approaches for the free-molecule diffusion regime that use clear fluid
approaches modified to include porous media effects.
The molecular flux of gas i due to Knudsen diffusion is given by the general
diffusion equation (Mason and Malinauskas, 1983, pg. 16)
J iK =−D iK ∇n i

where n i is the molecular density and D iK is the Knudsen diffusion coefficient. The
Knudsendiffusivityofgasi foracapillaryofagivenradiuscanbeestimatedasfollows
(Cunningham and Williams, 1980, eqns. 2.17 and 2.65) assuming a coefficient of