Page 110 - gas transport in porous media

Chapter 6: Conservation Equations
The second of these allows us to discard the last term in Eq. (6.165) so that the local
volume averaged mass transport equation for linear adsorption takes the form 103
γ
a v K eq ∂ c Aγ γ
ε γ 1 + +∇ · v γ c Aγ (6.168)
ε γ ∂t
⎡ ⎛ ⎞⎤

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

diffusion
The closure problem associated with this transport process has been explored in detail
elsewhere (Whitaker, 1997) where it is shown that the spatial deviation concentration
can be expressed as
γ
∂ c Aγ
γ
˜ c Aγ = b γ ·∇ c Aγ + s γ K eq (6.169)
∂t
Here the vector b γ and the scalar s γ are the closure variables or the mapping variables.
γ
The latter description is often used because b γ maps ∇ c Aγ onto ˜c Aγ , and s γ maps
γ
K eq ∂ c Aγ ∂t onto ˜c Aγ . Use of Eq. (6.169) in Eq. (6.168) leads to
γ
a v K eq ∂ c Aγ γ
ε γ 1 + +∇ · v γ c Aγ (6.170)
ε γ ∂t
γ
K eq ∂ c Aγ
∗ γ
+ ε γ d γ ·∇ =∇ · ε γ D ·∇ c Aγ
γ
∂t
∗
in which d γ is a dimensionless vector and D is the total dispersion tensor. These
γ
quantities are defined by
1 1
ε γ d γ = s γ ˜v γ dV − n γκ s γ D Am dA (6.171)
V V
V γ A γκ
⎛ ⎞

1 1
∗ ⎜ ⎟
D = D Am ⎝I + n γκ b γ dA⎠ − ˜ v γ b γ dV (6.172)
γ
V γ V γ
A γκ V γ
and can be calculated by means of the closure problems for b γ and s γ . If the flow
can be approximated as incompressible, the continuity equation is given by
∇· v γ = 0 (6.173)
and the volume averaged form is
γ
∇· ε γ v γ = 0 (6.174)