Page 108 - gas transport in porous media

Chapter 6: Conservation Equations
to develop a predictive theory of diffusion (Ochoa-Tapia et al., 1994). In Eq. (6.157)
we have used D γ to represent the hydrodynamic dispersion tensor defined by 101

1
D γ =− ˜ v γ b γ dV (6.159)
V γ
V γ
In general, these two tensor coefficients are combined so that Eq. (6.157) takes
the form
γ
∂ c Aγ γ
∗ γ
ε γ +∇ · v γ c Aγ =∇ · ε γ D ·∇ c Aγ (6.160)
γ
∂t
∗
in which D is referred to as the total dispersion tensor. Simple geometric models fail
γ
to provide good agreement with experimental results for both lateral and longitudinal
dispersion (Eidsath et al., 1983); however, complex, two-dimensional unit cells for
a spatially periodic model of a porous medium can be used to provide attractive
agreement (Whitaker, 1999, Sec. 3.4). At this point in time, it would appear that
three-dimensional unit cells containing an appropriate degree of randomness will be
required to accurately predict the dispersion tensor for porous media that are isotropic
at the Darcy scale.

6.4.2 Active Transport: Adsorption
When adsorption occurs in the absence of heterogeneous reaction, Eq. (6.155)
reduces to
γ
∂ c Aγ γ
ε γ +∇ · v γ c Aγ (6.161)
∂t
⎡ ⎛ ⎞⎤

1
γ
⎢ ⎜ ⎟⎥
=∇ · ⎣ε γ D Am ⎝∇ c Aγ + n γκ ˜c Aγ ,dA⎠⎦
V γ
A γκ

diffusion
∂ c As γκ
+∇ · ˜v γ ˜c Aγ − a v
∂t

dispersive
adsorption
transport
Formanydynamic processes, the condition of local adsorption equilibrium(Whitaker,
1999, Problem 1-3) can be used as a reasonable approximation (Wood et al., 2004).
For a linear adsorption isotherm, this allows us to use
c As = K eq c Aγ (6.162)