Page 249 - gas transport in porous media

Šolcová and Schneider
246
Dusty Gas Model (DGM) visualizes the porous medium as a collection of giant
spherical molecules (dust particles) kept in space by external force. The movement of
gas molecules in the spaces between dust particles is described by the kinetic theory of
2
gases. Formally, two of MTPM transport parameters, r and r , can be used also in
DGM. The third DGM transport parameter, B 0 , characterizes the viscous (Poiseuille)
2
gas flow in pores and can be, formally, replaced by r ψ/8.
14.1 CONSTITUTIVE EQUATIONS
The net molar flux density of component i in a n-component gas mixture per unit total
cross-section of the porous solid, N i , due to the combined influence of composition
gradients and total pressure gradient in a porous solids is given as the sum of the
p
permeation molar flux density of component i, N , and the diffusion molar flux
i
d
density of this component, N , (Mason et al., 1967):
i
p d
N i = N + N i = 1, ... , n (14.1)
i i
For the net mixture molar flux density, N, it follows
p
N = N + N d i = 1, ... , n (14.2)
n
d d 6 d p
where N is the mixture molar diffusion flux density (N = N ) and N is the
i
i=1
n
p 6 p
mixture molar permeation flux density (N = N ).
i
i=1
In MTPM and DGM the steady-state isothermal diffusion transport in multicom-
ponent gas mixtures, for cylindrical pores with diameter r and the transition region
(i.e., when the pore diameter, 2 r , is comparable with the mean free-path length of
gas molecules, λ;2 r ≈ λ), is described by the modified Maxwell-Stefan equation
(Rothfeld, 1963):
d
n
N d y j N − y i N j d
i
i + = (df ) i i = 1, ... , n (14.3)
D k D m
i j = 1 ij
j = i
m
(df ) is the driving force, y i the mole fraction of component i, D is the effective
ij
diffusion coefficient of the pair i−j in the bulk diffusion region:
m m
D = ψD ij (14.4)
ij
m k
with D the binary bulk diffusion coefficient and D , is the effective Knudsen
ij i
diffusion coefficient of component i:
k
D = ψ r K i (14.5)
i