Page 163 - gas transport in porous media

156
∂P
(8.27)
=− 1 Sahimi et al.
∂ζ λ(C)
where ζ = (x − υt)/L is a moving coordinate, λ denotes a normalized mobility, erfc
denotes the complementary error function, and all the lengths are scaled with D L0 /υ,

where D L0 is the base-state dispersion coefficient. Writing C = C + C , υ =¯υ + υ ,

and, P = P+P , and using normal modes for concentration and flow rate, respectively

(C , υ ) = ( , ) exp(ωt + iαy) (8.28)
x
the following equations are obtained
2
−(εα + ω) = C ζ − L c ( C ζ )ζ (8.29)
ζζ

−1 2 2
λ(λ ζ ) ζ − α =−α R (8.30)
where the concentration dependence of the mobility is taken to be,
λ(C) = exp(RC) (8.31)
with R = lnM. The subscripts denote derivatives with respect to the variables. Two
important quantites,
D m + α T υ
ε = (8.32)
D m + α L υ

α L υ
L c = (8.33)
D m + α L υ
appear in Eq. (8.29). One, ε, is a measure of flow-induced anisotropic dispersion and
is characteristic of porous media, while the other, L c , is a measure of the contribution
of longitudinal dispersion to total dispersion.
Based on our discussion in the previous sections, one expects that the onset of
instability and related features should be dictated by the sharpest mobility contrast,
namely, those associated with a step concentration profile, which also allow for an
analytical solution given by (Yortsos and Zeybek, 1988),

R
αR 1 + L c γ 0 tanh = 2γ 0 (α + γ 0 ) (8.34)
2
where
9
2
γ 0 = εα + ω> 0 (8.35)
In general, the solution of (8.29) leads to parabolic-like profiles, examples of which
are shown in Figure 8.4. In the limit L c = 0 and for an unfavorable mobility ratio