Membrane Gas Separation

C
2
C
1
Upstream 
Downstream 
Membrane 
L
p
2
p
1
Figure 5.1 Gas separation membrane with a constant concentration gradient across 
membrane thickness L

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Modelling Gas Separation in Porous Membranes 
87
P
C
p
D
=
2
2
(5.4)
By introducing a solubility coeffi cient S , that is, the ratio of concentration over pressure 
C
2
/ p
2
, when sorption isotherm can be represented by the Henry ’ s law, the permeability 
coeffi cient may be expressed simply as
P
S D
=
(5.5)
This form is useful as it facilitates the understanding of this physical property by repre-
senting it in terms of two components: 
• solubility, S , which is an equilibrium component describing the concentration of gas 
molecules within the membrane, that is the driving force, and
• diffusivity, D , which is a dynamic component describing the mobility of the gas mol-
ecules within the membrane.
The separation of a mixture of molecules A and B is characterized by the selectivity 
or ideal separation factor  
α
 
A/B
= P (A)/ P (B), i.e. the ratio of permeability of the molecule 
A over the permeability of the molecule B. According to Equation (5.5) , it is possible to 
make separations by diffusivity selectivity D (A)/ D (B) or solubility selectivity S (A)/ S (B) 
[25,26] . This formalism is known in membrane science as the solution - diffusion mecha-
nism. Since the limiting stage of the mass transfer is overcoming of the diffusion energy 
barrier, this mechanism implies the activated diffusion. Because of this, the temperature 
dependences of the diffusion coeffi cients and permeability coeffi cients are described by 
the Arrhenius equations. 
Gas molecules that encounter geometric constrictions experience an energy barrier such 
that suffi cient kinetic energy of the diffusing molecule, or the groups that form this barrier, 
in the membrane is required in order to overcome the barrier and make a successful dif-
fusive jump. The common form of the Arrhenius dependence for the diffusion coeffi cient 
can be expressed as
D
D
E RT
A
A
a
=
−
(
)
* exp
Δ
(5.6)
For the solubility coeffi cient the van ’ t Hoff equation holds
S
S
H
RT
A
A
a
=
−
(
)
* exp
Δ
(5.7)
where
Δ
H
a
<
0 is the enthalpy of sorption. 
Bearing in mind Equation (5.5) 
P
P
E
RT
A
A
P
=
−
(
)
* exp
Δ
(5.8)
where
Δ
E
p
=
Δ
E
a
+
Δ
H
a
Gases are known to diffuse within non 
- 
porous or porous membranes according 
to various transport mechanisms [20,25,32,34] . Table 5.1 illustrates the mechanism of 
transport depending on the size of pores. For very narrow pores, size sieving mechanism 

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88
Membrane Gas Separation
is realized that can be considered as a case of activated diffusion. This mechanism of 
diffusion is most common in the case of extensively studied non - porous polymeric mem-
branes. For wider pores, the surface diffusion (also an activated diffusion process) and 
the Knudsen diffusion are observed.