Page 22 - gas transport in porous media

Chapter 2: Gas Transport Mechanisms
1989a, eqns. 41 and 48). The two expressions are
n
D
x i N − x j N D D 15
j i N i ∇P i P∇x i x i ∇P
− = = +
D ∗ D iK RT RT RT
ij
j=1
j =i
n
T
x i N − x j N T T
j i N i P∇x i k P x i ∇P
− = + 1 +
D ∗ D iK RT D ik µ RT
ij
j=1
j =i
where the second equation simply includes the advective flux on both sides of the
equation. The first term on the LHS considers molecule-molecule interactions and is
immediately recognized as being based on the Stefan–Maxwell equations discussed
earlier. The second term on the LHS considers molecule-particle (Knudsen diffusion)
interactions, while the RHS is the driving force for diffusion and advection, which
includes concentration and pressure gradients.
There are many forms of the DGM. One particularly useful form is for the total mass
flux of component 1 in an isothermal binary system, or (Thorstenson and Pollock,
1989a, eqn. F4)
F 1 = m 1 N T
1

∗
D 1K D (P/RT)∇x 1 + D 1K (D ∗ + D 2K )x 1 (∇P/RT)
12 12
=−m 1 ∗
(D + x 1 D 2K + x 2 D 1K )
12
k 0 P ∇P
− x 1 m 1
µ RT
The flux of component 1 has diffusive (first term) and advective (second term) com-
ponents. The diffusive flux consists of ordinary diffusion (mole fraction gradient) and
Knudsen diffusion (pressure gradient) components.
Note that in the special case of isobaric conditions (∇P = 0), the advective and
Knudsen diffusion fluxes are zero. However, this does not mean that the Knudsen
diffusion coefficients are not important. The ordinary diffusion flux is dependent on
both diffusion (Knudsen and ordinary) coefficients. The Knudsen diffusion coeffi-
cients characterize the impact of the porous media (gas-wall interactions) on ordinary
diffusion. This behavior is absent in the clear fluid formulations, such as Fick’s law,
that are modified for porous media applications.

2.4 COMPARISON TO FUNDAMENTAL RELATIONSHIPS AND
EXPERIMENTAL DATA

In the 1800s, Thomas Graham discovered two important relationships for gas diffu-
sion in a porous media that relate the diffusive fluxes of a binary mixture in a porous
medium (Mason and Malinauskas, 1983, pg. 3). Graham’s law of effusion applies to