Membrane Gas Separation

5.9.1
Enhanced Separation by Tailoring Pore Size 
The three most common diffusion mechanisms known as activated diffusion, surface 
diffusion and Knudsen diffusion, usually dominate in small pores ( d 
 * 
 
<
3 Å ), medium 
pores (3 Å
<
d 
 * 
 
<
10 Å ) and large pores (10 Å
<
d 
 * 
 
<
500 Å ) for light gases, respectively. 
Separation by differences in diffusivity and/or differences in solubility can be enhanced 
Table 5.2  Lennard - Jones constants, molecular masses and average velocities at room 
temperature used throughout this chapter [57 – 59]
Gas / Pore atoms
 
σ
  ( Å )
ε
/ k
B
(K)
m (g/mol)
 v
¯
  (m/s)
from UFF [59]
C
3.43
53
12.01
–
H
2.57
22
1.01
–
O
3.12
30
16.00
–
N
3.26
35
14.01
–
Si
3.83
202
28.09
–
from Breck [57]
He
2.60
10
4.00
1277
H 
2
2.89
60
2.02
1800
CO 
2
3.30
195
44.01
385
O 
2
3.46
107
32.00
452
N 
2
3.64
71
28.01
483
CH 
4
3.87
149
16.04
638
from Poling [58]
CO
3.69
92
25.01
476
Ar
3.54
93
39.95
399
n - C 
5
H 
12
5.78
341
72.15
297
C 
2
H 
6
4.44
216
30.07
455
SF 
6
5.13
222
146.06
209

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100
Membrane Gas Separation
by tailoring the pore size such that the differences are maximized. The greatest separations 
are usually achieved when the competing gases are in different modes of transport. Thus 
it is important to know the critical pore sizes that distinguish the different diffusion 
mechanisms for each gas. 
The critical pore sizes are summarized in Table 5.3 for the light gases He, H 
2
, O 
2
, N 
2
CO 
2
, and CH 
4
, entering carbon and silica pores of cylindrical and slit shape. Additionally, 
Table 5.3 includes the results for carbon monoxide (a key component of synthesis gas), 
argon (an inert gas frequently used in industrial processes), ethane and n - pentane (hydro-
carbons present in fossil fuels), and sulfur hexafl uoride (the most potent greenhouse gas 
according to the Intergovernmental Panel on Climate Change [60] ; in permeation studies 
SF 
6
is often considered as a penetrant with an unusually large size). The results can be 
used as a guide for pore size design of a membrane according to the desired gas separa-
tion application. For example, if the application was natural gas purifi cation (separation 
of CO 
2
from CH 
4
) then the pore size range that allows CO 
2
through while rejecting CH 
4
can be found from Table 5.3 (carbon tube: 2.95 – 3.49 Å ; silica tube: 3.17 – 3.69 Å ; carbon 
slit: 2.46 – 2.95 Å ; silica slit: 2.65 – 3.13 Å ). Further, by using the transport equations (out-
lined earlier) it is possible to determine the pore size necessary to achieve a desired 
permeability and selectivity at the specifi ed operating temperature, demonstrated later in 
this chapter.
The fi rst observation to be made from the results in Table 5.3 is that the minimum pore 
sizes for barrier - free transport d
min
of each gas are in the same order as the kinetic diameter 
with slightly different values because the model takes into account the interaction with 
Pore size(Å)
W (eV)
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
d
min
d
K
Activated 
diffusion 
Surface 
diffusion
Knudsen 
diffusion 
Figure 5.10 Potential energy difference ( W ) for a single oxygen molecule at the entrance 
of a carbon cylindrical pore of diameter d . The pore regions where the diffusion 
mechanisms (activated diffusion, surface, and Knudsen fl ow) dominate are separated by 
the critical pore sizes d
min
(where W = 0) and d
K
(where W = 0.04 eV), indicated by 
dashed lines

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Modelling Gas Separation in Porous Membranes 
101
the pore wall and not kinetic size only. This means that the model is a more accurate 
method for predicting whether a gas molecule will experience an energy barrier or not, 
consequently predicting the mode of transport. Another important observation is that the 
model predicts that Knudsen diffusion occurs in different pore size regions for each gas. 
For example, within a 12 Å sized pore, the model predicts that helium and hydrogen will 
be in Knudsen fl ow while all the other light gases will not. Excellent agreement between 
experimental separation results and the model predictions is found [23] .