Membrane Gas Separation

206
Membrane Gas Separation
ln
ln
f
f
V
V
0
1
1
1
⎛
⎝⎜
⎞
⎠⎟ =
+ −
(
)
−
+
+
−
−
(
)
(
)
−
(
)
−
A
A
A
B
A
B
AB
B
A
A
B
A
φ
φ
φ
χ φ
χ
φ
φ
φ
χχ φ
φ
φ
B
B
A
B
A
B
1
−
−
(
)
V
V
(11.14)
rearranging assuming  
φ
 
A
 
φ
 
B
→
0:
ln
ln
f
f
V
V
0
1
1
1 2
⎛
⎝⎜
⎞
⎠⎟ =
−
−
+ +
(
)
⎡
⎣⎢
⎤
⎦⎥
+ +
−
+
(
)
A
A
A
AB
B
A
B
B
A
A
A
φ
χ
χ
χ
φ
χ
φ
χ
(11.15)
Substitution of Equations analogous to Equations (11.4) and (11.5) yields:
ln
ln
S
V
V
C V
V
C V
V
V
V
AB
G
A
A
A
A
G
A
B
B
G
A
AB
B
A
B
=
− −
+
+
(
)
+
−
+ +
(
)
⎛
⎝⎜
1
1 2
1
χ
χ
χ
χ
χ
⎞⎞
⎠⎟ −
ln f
A0
(11.16)
where
S
AB
is the solubility of A in the polymer with gas B present. This can be 
simplifi ed:
ln
ln
ln
S
K
C V
V
V
V
C
K
AB
A
B B
G
A
AB
B
A
B
A
A
A
AB
=
( )
+
−
+ +
(
)
⎡
⎣⎢
⎤
⎦⎥
+
=
( )
+
χ
χ
χ
σ
σ
1
C
C
C
B
A
A
+
σ
(11.17)
where
σ
χ
χ
χ
AB
A
AB
B
A
B
B
G
=
−
+ +
(
)
⎛
⎝⎜
⎞
⎠⎟
1
V
V
V
V
(11.18)
Therefore, the solubility of gas A in polymer P with gas B present can be expressed in 
terms of the solubility of gas A in the binary system (Equation 11.8 ):
ln
ln
S
S
C
AB
A
AB
B
(
)
=
( )
+
σ
(11.19)
If gas B concentration is low, then Henry ’ s law allows Equation (11.19) to be approxi-
mated by:
ln
ln
S
S
K p
AB
A
AB
B
B
(
)
=
( )
+
σ
(11.20)
Where p
B
is the pressure of gas B, and K
B
the Henry ’ s law constant. Hence, gas B may 
enhance or decrease the solubility of gas A within the rubbery polymeric membrane, 
depending on the relative strengths of the Flory – Huggins parameters for the gases between 
themselves and for the polymers. 
For a quaternary system, the presence of gases B and C may infl uence the sorption of 
gas A into the polymer, P. The Flory – Huggins equation for such a system is [17] :

---

The Effects of Minor Components on the Gas Separation Performance
207
ln
ln
f
f
V
V
V
V
V
V
0
2
1
⎛
⎝⎜
⎞
⎠⎟ =
+ −
(
)
−
−
−
+
+
A
A
A
B
A
B
C
A
C
P
A
P
AB
B
AC
C
φ
φ
φ
φ
φ
χ φ
χ φ
2
2
2
+
+
+
−
(
)
+
+
−
⎛
⎝⎜
⎞
⎠⎟ +
χ φ
φ φ χ
χ
χ
φ φ χ
χ
χ
φ φ χ
A
P
B C
AB
AC
CB
B
P
AB
A
B
A
B
C
P
AC
V
V
++
−
⎛
⎝⎜
⎞
⎠⎟
χ
χ
A
C
A
C
V
V
(11.21)
Where  
χ

ij
  is the binary interaction parameter between gases i and j . For simplicity this 
model assumes no higher interaction parameters between the gases and polymer, e.g. 
 
χ
 
ABC
= 0. Again, Equation (11.21) can be simplifi ed by the same assumptions used in the 
binary and tertiary systems. The solution, assuming gases B and C obey Henry ’ s law, is:
ln
ln
S
S
K p
K p
ABC
A
AB
B
B
AC
C
C
(
)
=
( )
+
+
σ
σ
(11.22)
where
σ
χ
χ
χ
AC
A
AC
C
A
C
C
G
=
−
+ +
(
)
⎛
⎝⎜
⎞
⎠⎟
1
V
V
V
V
(11.23)
where
K
C
is the Henry 
’ 
s law constant of gas C. Hence, in a quaternary system the 
presence of gas B and C may enhance or decrease the solubility of gas A in the membrane, 
which can be determined from the various Flory – Huggins parameters. 
Impurities can also affect the diffusivity of a gas through polymer swelling or dilation. 
These effects can be modelled using the well - known expression originally proposed by 
Fujita [18] 
D
A
B v
=
−
(
)
exp
f
(11.24)
Where v
f
is the fractional free volume in the polymer. The effect of plasticization by 
penetrants A and B on this fractional free volume is given by [19] :
v
v
f
A
A
B B
f
=
+
+
0
γ φ
γ φ
(11.25)
Where
v
f
0
is the fractional free volume in the polymer at the same temperature and pres-
sure in the absence of plasticization and  
γ
 
A
and  
γ
 
B
are positive constants characteristic of 
the system.
11.2.2
Glassy Membranes 
Glassy membranes operate below the glass transition temperature, and therefore polymer 
chain rearrangement is on an extraordinary long timescale, meaning the membrane never 
reaches thermodynamic equilibrium. The polymer chains are packed imperfectly, leading 
to excess free volume in the form of microscopic voids in the polymeric matrix. Within 
these voids Langmuir adsorption of gases occurs that increases the solubility. Examples 
of glassy membranes are polysulfone and Matrimid 
© 
polyimide. 
The considerable free volume within glassy polymeric membranes, due to the pre-
sence of microvoids, generally means this class of membranes are diffusivity selectivity 

---

208
Membrane Gas Separation
controlled [12] . That is, the membrane is selective towards the smaller gas molecules. 
Because of this, glassy polymeric membranes have been suggested for post - combustion 
capture as well as natural gas sweetening. 
Gas concentration within glassy membranes consists of gas within the polymeric matrix 
as well as adsorbed in the microvoids. Therefore, the total concentration of absorbed gas 
within a glassy membrane (C) can be described by [15] :
C
C
C
=
+
D
H
(11.26)
where C
H
is the standard Langmuir adsorption relationship
C
C bf
bf
H
H
=
′
+
(
)
1
(11.27)
C
′
H
is the maximum adsorption capacity while b is the ratio of rate coeffi cients of adsorp-
tion and desorption, or the Langmuir affi nity constant, defi ned as:
b
C
C
C
f
=
′ −
(
)
H
H
H
(11.28)
Hence, the dual - mode sorption for glassy membranes is written as:
C
K f
bf
bf
=
+
′
+
(
)
D
H
C
1
(11.29)
Petropoulos [20] and Paul and Koros [21] independently developed models where the 
diffusion of the gas species adsorbed in the Langmuir region is partially, or even totally, 
immobilized. In this case, a parameter F
A
is introduced, defi ned either as the ratio of dif-
fusion coeffi cients in the Langmuir ( D
H
) and Henry ’ s Law region ( D
D
) or as the fraction 
of the Langmuir species that are fully mobile. In this case, the concentration of mobile 
species is given by:
C
K f
F C b
K
bf
m
D
A
H
D
=
+
′
+
(
)
⎡
⎣⎢
⎤
⎦⎥
1
1
(11.30)
To determine the permeability across the membrane from this expression, it is necessary 
to account for the fugacity gradient by integration between the upstream and downstream 
values, f
0
and f
1
[20] :
P
f
f
P f
f
f
f
=
−
( )
∫
1
1
1
0
d
0
(11.31)
The result is the mean or integral permeability:
P
K D
C F D
f
f
bf
bf
=
+ ′
−
(
)
+
+
⎛
⎝⎜
⎞
⎠⎟
D
D
H
A
D
0
1
0
1
1
1
ln
(11.32)

---

The Effects of Minor Components on the Gas Separation Performance
209
When two gas species are present, competition can restrict both the solubility within 
the polymer matrix and the amount adsorbed in the Langmuir free volume. Competition 
in the former case is best modelled by adjustments to K
D
based on Equations (11.20) or 
(11.22) . To account for changes to the occupancy of the Langmuir sites for a binary 
mixture of gases A and B, the mobile concentration of gas A becomes [22] :
C
K
f
F C
b
K
b f
b f
mA
DA A
A
HA A
DA
A A
B B
=
+
′
+
+
(
)
⎡
⎣⎢
⎤
⎦⎥
1
1
(11.33)
Similarly, the mobile concentration of gas B is:
C
K
f
F C b
K
b f
b f
mB
DB B
B
HB B
DB
A A
B B
=
+
′
+
+
(
)
⎡
⎣⎢
⎤
⎦⎥
1
1
(11.34)
Each parameter has the same defi nition as in the single gas case with the subscript denot-
ing whether it is a property of gas A or B. The solubility of both gases A and B is reduced 
compared to the single gas case (Equation 11.30 ), and is heavily dependent on the rela-
tionship between b , the affi nity constant, and fugacity. 
When three components or more are present, the mobile concentration of gas A 
becomes:
C
K
f
F C
b
K
b f
b f
b f
mA
DA A
A
HA A
DA
A A
B B
C C
=
+
′
+
+
+
+
(
)
⎡
⎣⎢
⎤
⎦⎥
1
1
…
(11.35)
The Langmuir affi nity constant is generally proportional to the critical temperature of the 
gas, and Figure 11.1 demonstrates the relationship between Langmuir affi nity constant 
and critical temperature [23] . For example, water has a very high critical temperature 
compared to N 
2
and CO 
2
. This means water is more condensable within the free volume 
and correspondingly a higher Langmuir affi nity constant is observed. Hence, the presence 
of water even in trace amounts may dominate observed gas permeabilities, because even 
though the partial pressure is low, water will successfully compete for sorption sites in 
the membrane.
It is common for gases at high pressure, especially acidic gases such as CO 
2
and SO 
2
, 
to also plasticize glassy polymeric membranes. This results in an increase in the diffusiv-
ity of gases through the membrane due to disruption to the polymer chain – chain inter-
molecular bonding network, leading to an increase in the fractional of free volume 
between polymer chains. When the penetrant gas causes plasticization of a glassy mem-
brane, the diffusion coeffi cient becomes concentration dependent, and a simple model 
based on Equations (11.24) and (11.25) can be written as [24] :
D C
D
C
( )
=
⋅
⋅
(
)
0
exp
β
(11.36)
where C is the concentration of the plasticizing gas, D
0
is the diffusion coeffi cient in the 
limit C
→
0, and  
β
  is an empirical constant, known as the plasticization potential, that 
depends on the nature of the gas – polymer system and the temperature. For the specifi c 

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210
Membrane Gas Separation
case when the permeate fugacity is zero, the mean or integral permeability for a ternary 
system can then be determined from:
P
D
f
K
f
F C
b
b f
b f
b f
A
A
A A
A
DA A
A
HA A
A A
B B
C C
=
( )
⋅
+
⋅ ′
+
+
+
⎛
⎝⎜
0
1
1
0
0
0
0
0
β
β
exp
⎞⎞
⎠⎟
⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜
⎞
⎠⎟
1
(11.37)
Hence, the effect of competitive sorption on glassy membranes is to reduce the permea-
bility of all gases. This in turns alters the selectivity of the membrane. The relative mag-
nitude of the Langmuir constants for each component dictates whether the selectivity 
decreases or increases.