Membrane Gas Separation

The Effects of Minor Components on the Gas Separation Performance
205
ln
ln
ln
S
V
V
V
V
C
f
A
G
A
A
A
G
A
A
A0
( )
=
− −
+
+
(
)
−
( )
1
1 2
χ
χ
(11.7)
The non - concentration dependent terms can be grouped together to yield:
ln
ln
S
K
C
A
A
A
A
( )
=
( )
+
σ
(11.8)
where
ln
ln
ln
K
V
V
f
A
A
A
G
A0
( )
= − +
(
)
−
−
( )
1
χ
(11.9)
is the Henry ’ s law constant of gas A in polymer P and
σ
χ
A
A
A
G
= +
(
)
1 2
V
V
(11.10)
Hence, the solubility of gas A within a rubbery polymeric membrane can be determined 
from the Flory – Huggins parameter. 
Suwandi and Stern [16] point out that in many sorption studies the polymer swells as 
sorption occurs, but the concentration of A is calculated as the volume of gas dissolving 
in 1 cm 
3
of unswollen polymer. That is, the weight change recorded by the sorption ana-
lyzer is converted to a concentration by division by the density of the unswollen system. 
In this case:
′ =
−
(
)
=
′
(
)
′
(
)
+
C
V
C V V
C V V
V
or
G
A
A
A
A
A
G
A
G
φ
φ
φ
1
1
(11.11)
Mathematical manipulation of equations yields expressions identical to Equations (11.8)
and (11.9) , but importantly in this case:
′ =
+
(
)
σ
χ
A
A
A
G
2 1
V
V
(11.12)
For a tertiary system, the presence of gas B may infl uence the sorption of gas A in the 
polymer. Hence, the Flory – Huggins equation of gas A for a tertiary system is [14,17] :
ln
ln
f
f
V
V
V
V
0
1
⎛
⎝⎜
⎞
⎠⎟ =
+ −
(
)
−
−
+
+
(
)
+
(
)
A
A
A
B A
B
P
A
P
AB B
A
P
B
P
φ
φ
φ
φ
χ φ
χ φ φ
φ −− χ φ φ
B B P
A
B
V
V
(11.13)
Here  
χ
 
AB
characterizes the mixing between gas A and B, while the other parameters 
subscripts denote if it is for gas A, B or the polymer P. Again, the Flory – Huggins equa-
tion for a tertiary system can be simplifi ed by the same assumptions for the binary system. 
That is, V
A
and V
B
<< V
P
, due to difference in gas volume compared to the polymer, and 
at low pressures the volume fraction of gas A and B is small, and
φ
A
2
0
→
and
φ
B
2
0
→
. 
Further,  
φ
 
P
= 1
−
 
φ
 
A
−
 
φ
 
B
. Hence:

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