Membrane Gas Separation

Membrane Gas Separation
9:3
3
2
2
3
planes
cylinders
R
/
r
o
(
d
/
r
o
)
10:4
ε
*=/
ε
1
*
Figure 5.7 Scaled potential energy minima
ε ε
=
* *
/
1
 within cylindrical and slit - shaped pores 
with varying radius R and slit size 2 d , respectively, where
ε
=
*
 is the minimum potential 
within the pore and
ε
1
*
 is the minimum potential with a single fl at surface. Curves that go 
below the horizontal axis are the scaled potentials within the centre of the pore
ε
ε
0
1
( )
/ *
 
where the potential in the centre
ε
 (0) becomes less than the minimum potential with a 
single fl at surface
ε
1
*
, i.e.
ε
ε
0
1
1
( )
<
/ *
, within larger pores. Reprinted with permission from 
Journal of the Chemical Society; Faraday Transactions 1, Adsorption in slit - like and 
cylindrical micropores in the Henry ’ s law region. A model for the microporosity of carbons 
by D. H. Everett and J. C. Powl, 72, 619 – 636, Copyright (1976) Royal Society of Chemistry
–1
9:3 10:4
(a)
(b)
(c)
9:3
z
/
r
0
10:4
9:3 10:4
–1
ε 
= (
z
)/
ε
1
*
–2
+1
–1
–1
–2
+1
–1
–1
–2
+1
Figure 5.6 Potential energy  
ε
=  ( z ) between two parallel planes (10:4) and two 
parallel slabs (9:3) at a distance apart of 2 d for (a) d / r
0
= 1.60, (b) d / r
0
= 1.14 and 
(c) d / r
0
= 1.00, normalized by the energy minimum
ε
1
*
 located at a distance of r
0
from a 
single slab . Reprinted with permission from Journal of the Chemical Society; Faraday 
Transactions 1, Adsorption in slit - like and cylindrical micropores in the Henry ’ s law region. 
A model for the microporosity of carbons, by D. H. Everett and J. C. Powl, 72, 619 – 636, 
Copyright (1976) Royal Society of Chemistry

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Modelling Gas Separation in Porous Membranes 
97
A
2
a
b1
b2
c1
c2
0.9
R/
s
A
=
1.086
1.239
2
3
Z
0
–2
–4
B
є
A
=(z)/є
A1
*
Figure 5.8 Separation regimes determined by the potential energies within pores of 
different sizes. Potential energy
ε
A
z
=
( )
 for molecule A within cylindrical pores with radius R , 
scaled by the potential minimum
ε
A1
*
 for molecule A with a single free surface and the 
Lennard - Jones kinetic diameter parameter  
σ

A
[30] . Reprinted from Journal of Membrane 
Science, 104 , R. S. A. de Lange, K. Keizer and A. J. Burggraaf, Analysis and theory of gas 
transport in microporous sol - gel derived ceramic membranes, 81 – 100, Copyright (1995), 
with permission from Elsevier
de Lange et al. [30] later extended the work of Everett and Powl [31] by relating 
transport mechanisms to potential energy calculations. Figure 5.8 demonstrates the sepa-
ration scenarios within cylindrical shaped pores. Situation ‘ a ’ is where molecule A is 
accepted within the pore while larger molecule B is rejected by the repulsive forces 
experienced. This refers to true molecular sieving or size - sieving. Situations ‘ b1 ’ and ‘ b2 ’
are where both molecules are accepted within the pore but molecule A has a much deeper 
potential than molecule B. Since the pore is cylindrical, molecules may not pass each 
other and therefore the rate of diffusion is governed by the slowest component. Situations 
‘ c1 ’ and ‘ c2 ’ are where molecules may pass each other and the potential energy becomes 
weaker having less infl uence on transport. In Ref. 30 these scenarios are combined with 
an extensive model that incorporates different stages of transport through the membrane 
and existing transport equations.
The extensive model considers the total fl ux as the sum of contributions of the fl ux at 
different stages, indicated schematically in Figure 5.9 and composed of the following. 
1 Adsorption onto surface and fl ux from position  
θ
 
0,surf
to  
θ
 
0
at the pore entrance via 
surface diffusion (f2.J).
2 Adsorption directly at the pore entrance at position  
θ
 
0
(f1.J).
3 Flux directly to pore entrance with no adsorption taking place (F1.J).
4 Entrance of adsorbed molecules at position  
θ
 
0
to position  
θ
 
1
within the pore (F2.J).

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98
Membrane Gas Separation
5 Micropore diffusion through the pores (J).
6 Desorption of the molecules from within the pore to the external surface or directly to 
the gas phase.
7 Desorption from the external surface to the gas phase.