Page 93 - gas transport in porous media

Whitaker
86
Dilute Solution Convective-Diffusion Equation
6.1.8
If species A is dilute, that is, x A 1, one can make use of Eqs. (6.19) and (6.87) to
obtain a classic form of the convective-diffusion equation. One begins with the use
of Eq. (6.37) in order to express Eq. (6.87) in the form
B=N
x A x B (u B − u A )
0 =−∇x A + , A = 1, 2, ... ., N − 1 (6.88)
D AB
B = 1
B = A
which can be rearranged to obtain
⎧ ⎫
⎪ ⎪
⎪
B=N ⎪ B=N ⎪
⎪
x A c B u B 1 ⎨ x B ⎬
0 =−∇x A + − c A u A (6.89)
c D AB c ⎪ ⎪
B = 1 ⎪ B = 1 D AB ⎪
⎪
⎪
⎩ ⎭
B = A B = A
The mass diffusion velocities are constrained by
B=N

0 = ω B u B (6.90)
B=1
and from this we can conclude that diffusion velocities tend to be of the same order of
magnitude. This suggests that Eq. (6.89) can be simplified by imposing the restriction
⎧ ⎫
⎪ ⎪
⎪ ⎪
B=N ⎪ B=N ⎪
⎬
x A c B u B 1 ⎨ x B
c A u A (6.91)
c D AB c ⎪ ⎪
B = 1 ⎪ D AB ⎪
⎪ B = 1
⎪
⎩ ⎭
B = A B = A
whenever the mole fraction of species A is small compared to one, that is,
x A 1 (6.92)
On the basis of Eq. (6.91) we simplify Eq. (6.89) to the form
⎧ ⎫
⎪ ⎪
⎪ B=N ⎪
⎪
⎪
1 ⎨ x B ⎬
0 =−∇x A − c A u A (6.93)
c ⎪ ⎪
⎪ D AB ⎪
⎪ B = 1 ⎪
⎩ ⎭
B = A
and we define the mixture diffusivity for species A according to
B=N
1 x B
= (6.94)
D Am D AB
B = 1
B = A