Page 107 - gas transport in porous media

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Eq. (6.154) in the form (Whitaker, 1999)
γ Whitaker
∂ c Aγ γ
ε γ +∇ · v γ c Aγ (6.155)
∂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 + a v R As γκ
∂t

dispersive adsorption heterogeneous
transport reaction
The first two terms on the left hand side of this result require only the knowledge
of the superficial mass average velocity, and when Eqs. (6.138) and (6.139) are
applicable this represents a straightforward matter. The last four terms all depend,
directly or indirectly, on the spatial deviation concentration, ˜c Aγ , and are thus sensitive
to the approximations made in the closure problem. In the absence of a closure
problem, these four terms become sensitive to the intuition associated with a particular
model.

6.4.1 Passive Transport

When there is negligible adsorption and/or heterogeneous reaction, the spatial
deviation concentration takes the form (Whitaker, 1999)

γ
˜ c Aγ = b γ ·∇ c Aγ (6.156)
and this leads to a transport equation of the form
γ
∂ c Aγ γ
γ
ε γ +∇ · v γ c Aγ =∇ · ε γ D eff + ε γ D γ ·∇ c Aγ (6.157)
∂t
Here D eff is the effective diffusivity tensor defined by
⎛ ⎞
1
⎜ ⎟
D eff = D Am ⎝I + n γκ b γ dA⎠ (6.158)
V γ
A γκ
For purely diffusive transport in porous media that are isotropic at the Darcy scale,
good agreement between theory and experiment can be obtained using very simple
geometrical models (Quintard, 1993). For porous media that are anisotropic at the
Darcy scale, simple models are not satisfactory and more work is required in order