Page 111 - gas transport in porous media

104
For the special case of a homogeneous porous medium, Eq. (6.170) can be
expressed as Whitaker
γ
a v K eq ∂ c Aγ γ γ
ε γ 1 + + ε γ v γ ·∇ c Aγ (6.175)
ε γ ∂t
γ
K eq ∂ c Aγ γ
∗
+ ε γ d γ ·∇ = ε γ D : ∇∇ c Aγ
γ
∂t
provided one is willing to ignore the variations in the total dispersion tensor that
result from variations in the velocity field. For a homogeneous porous medium and a
constant velocity, Eq. (6.175) is exact. If the adsorption isotherm is nonlinear, K eq is
γ
a function of c Aγ ; however, for linear adsorption K eq is a constant and Eq. (6.175)
simplifies to
γ
a v K eq ∂ c Aγ γ γ
ε γ 1 + + ε γ v γ ·∇ c Aγ (6.176)
ε γ ∂t
γ
∂ c Aγ γ
∗
+ ε γ d γ K eq ·∇ = ε γ D : ∇∇ c Aγ
γ
∂t
For chromatographic processes, this result can be simplified following the original
analysis of Golay (1958). For pulsed systems, we identify the pulse velocity by u p
and express the time derivative of the concentration as
γ γ
∂ c Aγ d c Aγ γ
= − u p ·∇ c Aγ (6.177)
∂t dt
u p
Here the subscript u p is used to indicate the time derivative as determined by an
observer moving at the velocity u p . When the following restriction is valid
γ γ
∂ c Aγ d c Aγ
(6.178)
∂t dt
u p
the mixed derivative term in Eq. (6.176) can be expressed as
γ
∂ c Aγ γ
ε γ d γ K eq ·∇ =−ε γ K eq d γ u p : ∇∇ c Aγ (6.179)
∂t
Use of this result in Eq. (6.175) leads to a chromatographic equation in the form
γ γ
∂ c Aγ v γ γ
·∇ c Aγ (6.180)
+
∂t −1
1 + ε γ a v K eq

∗
D + K eq d γ u p γ
γ
= : ∇∇ c Aγ
−1
1 + ε γ a v K eq