Page 115 - gas transport in porous media

108
With this result we can construct a boundary value problem for ˜c Aγ in which the
governing differential equation is given by Whitaker

E=N−1
∂˜c Aγ γ
=∇ · D AE ∇˜c Eγ (6.199)
∂t
E=1

accumulation
diffusion
⎡ ⎤
E=N−1 γ
D AE 1
⎢ ⎥
−∇ · ⎣ V n γκ ˜c Eγ dA⎦
ε γ
E=1
A γκ

non-local diffusion
while the interfacial boundary condition takes the form
E=N−1
γ
BC.1 − n γκ · D AE ∇˜c Eγ (6.200)
E=1

diffusive flux
E=N−1

γ
γ
= n γκ · D AE ∇ c Eγ , at the γ –κ interface
E=1

diffusive source
In addition to the interfacial boundary condition, we require a boundary condition at
the entrances and exits of the macroscopic system illustrated in Figure 6.3 and we
express this boundary condition as
BC.2 ˜ c Aγ = F (r, t), at A γ e (6.201)
To complete our statement of the boundary value problem, we propose an initial
condition of the form

IC. ˜ c Aγ = F (r), at t = 0 (6.202)
In general, both the boundary condition at the entrances and exits of the macroscopic
system and the initial condition are unknown in terms of the spatial deviation con-
centration, ˜c Aγ . However, neither of these is important when the separation of length
scales indicated by Eq. (6.117) is valid. Under these circumstances, the boundary
condition imposed at A γ e influences the ˜c Aγ field only over a negligibly small
region, and the initial condition given by Eq. (6.202) can be discarded because the
closure problem is quasi-steady. Under these circumstances, the closure problem can
be solved in some representative, local region (Quintard and Whitaker, 1994a–e).
In the governing differential equation for ˜c Aγ , we have identified the accumulation
term, the diffusion term, and the so-called non-local diffusion term. In the boundary