Membrane Gas Separation

5.2
Background 
A quantitative measure of gas transport is the fl ux (or permeation rate), which is defi ned 
as the number of molecules that pass through a unit area per unit time [25] . It is believed 
that this molecular fl ux J follows Fick ’ s fi rst law. The fl ux is proportional to the concen-
tration gradient through the membrane, i.e. there is a movement of gas from regions of 
high concentration to regions of low concentration, which may be expressed in the form
J
D
dc
dx
= −
(5.1)
where D is the diffusivity (or diffusion coeffi cient), c ( x ) is the concentration and x is the 
position across the membrane. By assuming a linear concentration gradient across the 
membrane, the fl ux can be approximated as
J
D
C
C
L
=
−
2
1
(5.2)
where C
1
= c (0) and C
2
= c ( L ) are the downstream and upstream concentrations (corre-
sponding to the pressures p
1
and p
2
via sorption isotherm c ( p )), respectively, and L is the 
membrane thickness, as labelled in Figure 5.1 .
The membrane performance of various materials is commonly compared using the 
thickness - independent material property, namely the permeability P , which is related to 
the fl ux J in the following way
P
J L
p
p
C
C
p
p
D
=
−
=
−
−
⎛
⎝⎜
⎞
⎠⎟
2
1
2
1
2
1
(5.3)
In the case where the upstream pressure is much greater than the downstream pressure 
( p
2
>> p
1
and C
2
>> C
1
) the permeability can be simplifi ed to give