Page 109 - gas transport in porous media

102
in Eq. (6.161) to obtain
γ Whitaker
∂ c Aγ γ
ε γ +∇ · v γ c Aγ (6.163)
∂t
⎡ ⎛ ⎞⎤
1
γ
⎢ ⎜ ⎟⎥
=∇ · ⎣ε γ D Am ⎝∇ c Aγ + n γκ ˜c Aγ dA⎠⎦
V γ
A γκ

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

dispersive
adsorption
transport
Use of the first of Eqs. (6.131) along with the approximation (Whitaker, 1999, Sec.
1.3.3) given by
+ γ , γ
c Aγ = c Aγ (6.164)
γκ
allows us to express Eq. (6.163) as

γ
a v K eq ∂ c Aγ γ
ε γ 1 + +∇ · v γ c Aγ (6.165)
ε γ ∂t
⎡ ⎛ ⎞⎤

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

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

dispersive
transport
When the length scale constraints indicated by Eq. (6.117) are valid one can show
that the spatial deviation concentration is constrained by

γ
˜ c Aγ c Aγ (6.166)
This, in turn, leads to the inequalities given by

γ
γ ∂ ˜c Aγ γκ ∂ c Aγ
˜c Aγ γκ ≪ c Aγ , ≪ (6.167)
∂t ∂t