Membrane Gas Separation

5.7.2
Resistance in Series Transport Model 
Resistance model for transport in composite hollow fi bre membranes based on polysul-
fone with siloxane coating has been described in a classical work by Henis and Tripody 
[51] . The resistance in series model assumes that the gas molecules encounter constric-
tions at certain positions throughout the pore which control the rate of diffusion [20,27,33] . 
For this scenario the total permeability is inversely related to the total resistance, thus
R
l
P
x l
P
x
l
P
tot
tot
K
K
K
A
=
=
+ −
(
)
τ
τ
1
(5.19)
where x
K
is the fraction of the total pore length l with pore size d
p2
in which Knudsen 
diffusion dominates while (1 – x
K
) is the fraction of the total pore length l with pore size 
d
p1
in which activated diffusion dominates and  
τ
  is the pore tortuosity (see Figure 5.5 b). 
The total permeability simplifi es to give
P
P P
x P
x
P
tot
A
K
K
A
K
K
=
+ −
(
)
(
)
τ
1
(5.20)
where P
A
is the activation diffusion permeability and P
K
is the Knudsen diffusion perme-
ability as defi ned earlier.
5.8
Pore Size, Shape and Composition 
To understand gas transport phenomenon it is critical to consider the interactions between 
the gas molecules and the pore wall. The van der Waals interactions between particles 
are explained well by the Lennard - Jones function containing two parameters, the kinetic 
diameter  
σ
  (the distance where the potential energy between the particles is zero) and the 
well depth
ε
(the deepest potential minimum between the particles). These parameters 
were used, e.g. by Freeman [52] , to establish a theoretical basis for the relationship 
between selectivity and permeability for a range of polymers and gases (so - called fi rst 
upper bound empirically determined by Robeson in 1991 [53] ; see also more recent paper 
by Robeson [54] ). The rate of diffusion of a gas is dependent on its kinetic diameter while 
its solubility mainly depends on the condensability of the gas and consequently on the 
well depth for gas - gas interactions. 
Gas – pore wall interactions have been considered to identify different pore size regimes 
for gas adsorption by Everett and Powl [31] and later modifi ed to determine gas separa-
tion scenarios by de Lange et al. [30] . Figure 5.6 shows the potential energy within slit -
shaped pores. A deep single minimum occurs within small pores and the shallower double 
minimum occurs in larger pores, as calculated by Everett and Powl [31] . This potential 
energy can be thought of as the adsorption energy which is enhanced at an optimal pore 
size, indicated by the peaks in Figure 5.7 for cylindrical and slit - shaped pores. Everett 
and Powl [31] used these calculations in order to further understand adsorption of the 
noble gases within microporous carbons. One of the key points outlined in Everett and 
Powl ’ s work is that the separation outcomes may be predicted by comparing the potential 
energy curves of the particular gases.

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