Membrane Gas Separation

5.9.2
Determining Diffusion Regime from Experimental Flux 
A widely used experimental approach that involves the determination of the mode of 
transport is performed by fi tting permeation data to the Arrhenius equation for the fl ux 
Table 5.3  Minimum pore size for barrier - free transport ( d
min
 ) and minimum pore size for 
Knudsen diffusion ( d
K
 ) at room temperature (298 K). All pore sizes are given as the PALS 
pore size d 
 * 

(A) Cylindrical pore
Gas
Carbon
Silica
d
min
( Å )
d
K
( Å )
d
min
( Å )
d
K
( Å )
He
2.30
5.69
2.53
6.37
H 
2
2.57
8.92
2.79
9.79
CO 
2
2.95
12.30
3.17
13.35
O 
2
3.10
11.68
3.31
12.68
N 
2
3.27
11.49
3.48
12.46
CH 
4
3.49
13.74
3.69
14.84
CO
3.32
12.14
3.52
13.16
Ar
3.18
11.68
3.39
12.67
n - C 
5
H 
12
5.27
23.51
5.45
25.04
C 
2
H 
6
4.02
16.69
4.21
17.93
SF 
6
4.66
19.46
4.85
20.81
(B) Slit - shaped pore
Gas
Carbon
Silica
d
min
( Å )
d
K
( Å )
d
min
( Å )
d
K
( Å )
He
1.86
2.19
2.06
4.35
H 
2
2.10
6.52
2.31
7.22
CO 
2
2.46
9.27
2.65
10.12
O 
2
2.59
8.76
2.79
9.58
N 
2
2.75
8.60
2.94
9.39
CH 
4
2.95
10.43
3.13
11.32
CO
2.79
9.14
2.98
10.00
Ar
2.66
8.76
2.86
9.56
n - C 
5
H 
12
4.59
18.33
4.75
19.57
C 
2
H 
6
3.44
12.82
3.62
13.82
SF 
6
4.03
15.06
4.20
16.15

---

102
Membrane Gas Separation
or permeance J = J
0
exp( –
Δ
E / RT ). As the fl ux J and permeability coeffi cient P are related 
as J = P
Δ
p / L , where  
Δ
 p is the pressure gradient and L is the membrane thickness, the 
above permeability expressions in the previous section provide the decision criteria. If 
Δ
E is positive then the dominant transport mechanism is size - sieving activated diffusion 
with
Δ
E representing the energy barrier | W| for d 
 * 
 
<
d
min
. If
Δ
E is negative then the 
dominant transport mechanism is surface diffusion with
Δ
E representing the weighted 
contribution of adsorption energy (or potential energy difference) for the energy barrier 
and surface concentration ( a – 1) W . If
Δ
E is zero, i.e. no change in permeability with 
varying temperature, then either the size 
- 
sieving energy barrier and the enthalpy of 
adsorption are equal to zero ( W = 0) or the surface diffusion energy barrier and the heat 
of adsorption are both equal ( a = 1). If the Arrhenius equation does not fi t the data then 
the mode of transport could be Knudsen fl ow, or a combination of all mechanisms since 
the dominant mode of transport can change as the temperature is varied. The above criteria 
will be a useful tool in understanding the mode of transport for each gas in a range of 
porous membrane materials.