Page 94 - gas transport in porous media

Chapter 6: Conservation Equations
87
so that the mixed-mode diffusive flux can be expressed as
c A u A =− c D Am ∇x A (6.95)
We refer to c A u A as a mixed-mode diffusive flux because it is constructed in terms
of a molar concentration, c A , and a mass diffusion velocity, u A . Given the dilute
solution constraint for the diffusing species indicated by Eq. (6.92), one can think of
cases in which the temperature and pressure gradients are small enough so that the
following restriction is satisfied.

x A ∇c c ∇x A (6.96)
Under these conditions the mixed-mode diffusive flux takes the form

c A u A =−D Am ∇c A (6.97)
and use of this result with Eqs. (6.19) and (6.37) yields

∂c A
+∇ · (c A v) =∇ · D Am ∇c A + R A (6.98)
∂t
This result is ubiquitous in the chemical engineering literature; however, the
limitations imposed by Eqs. (6.86), (6.92) and (6.96) are not always made clear.

6.1.9 Non-Dilute Solutions
When the simplification indicated by Eq. (6.91) is not valid, we must work directly
with Eq. (6.89) which can be arranged in the form
⎧ ⎫
⎪ ⎪
B=N ⎪ B=N ⎪
⎪
⎪
J B x B
⎨ ⎬
0 =−c∇x A + x A − J A , (6.99)
D AB ⎪ ⎪
B = 1 ⎪ B = 1 D AB ⎪
⎪
⎪
⎩ ⎭
B = A B = A
A = 1, 2, ... , N − 1
where the mixed-mode diffusive flux is given by
J B = c B u B (6.100)
In terms of this diffusive flux, Eq. (6.19) takes the form
∂c A
+∇ · (c A v) =−∇ · J A + R A , A = 1, 2, ... , N (6.101)
∂t