Membrane Gas Separation

9.2.1
One - component Permeation Process – Fick ’ s and Smoluchowski Equation 
The second of Fick ’ s law (Equation 9.2 ) can be used as a sort of the reference point for 
describing a permeation process of one component gas (N 
2
, O 
2
) through a dense polymeric 
membrane with no external fi eld. Functional dependence of the diffusion coeffi cient on 
coordinate, time or concentration can result from different types of circumstances, i.e. 
when the membrane is heterogeneous [28] , is subjected to some relaxation phenomena 
[29,30] , or membrane transport is accompanied by some other processes, like adsorption, 
etc. [9,12] .
∂
∂
c x t
t
div D
grad c
,
( )
=
⋅
( )
(
)
(9.2)
where D ( · ) takes the D ( c ), D ( x ) or D ( t ) form. To provide a coherent frame for this sort 
of approach we repeat several formulae for Fick ’ s permeation. 
The simplest case of the diffusion equation with constant diffusion coeffi cient D with 
initial and boundary conditions for permeation reads
∂
∂
∂
∂
c
t
D
c
x
x
l
t
R
c x
c
t
c
c l t
=
∈
( )
∈
(
)
=
( )
=
( )
=
⎧
⎨
⎪
⎪
⎩
⎪
⎪
+
2
2
0
0
0
0
0
0
,
,
,
,
,
,
(9.3)
C (0,t) = C
0
FEED
PERMEATE
l
C (l,t) = 0
Figure 9.1 Schematic membrane with boundary conditions and direction of fl ow

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162
Membrane Gas Separation
1
C(x,t)
0,8
t
1
t
2
t
3...
t
n
0,6
0,4
0,2
0
–0,2
0,2
0,4
0,6
0,8
1
x
1,2
0
Figure 9.2 Solution of Equation (9.3) , i.e. concentration profi les, as a function of x
and t . Reprinted with permission from Journal of Membrane Science, Studies on the air 
membrane separation in the presence of a magnetic fi eld by Anna Strzelewicz and 
Zbigniew J. Grzywna, 294, 1 – 2, 60 – 67 Copyright (2007) Elsevier Ltd 
Solution of Equation (9.3) in the form of a Fourier series discussed by Crank [31] has a 
form
c x t
c
x
l
c
n
n x
l
n
l
Dt
n
,
sin
exp
( )
=
−
⎛
⎝
⎞
⎠ −
−
⎛
⎝⎜
⎞
⎠⎟
=
∞
∑
0
0
2
2
2
1
1
2
1
π
π
π
(9.4)
The diffusive fl ux J ( x , t ) can be obtained from Equation (9.4) using the defi nition of fl ux:
J x t
D
c
x
,
( )
= − ∂
∂
(9.5)
to get a form
J x t
Dc
l
Dc
l
n x
l
n
l
Dt
n
,
cos
exp
( )
=
+
−
⎛
⎝⎜
⎞
⎠⎟
=
∞
∑
0
0
2
2
2
1
2
π
π
(9.6)
In case of functional dependence of diffusion coeffi cients, the system (Equation 9.3 ) can 
be solved numerically if an analytical solution is not known. To start with, the diffusion 
equation is rewritten as a difference quotient [32] :
c
c
t
D
c
c
c
x
i j
i j
i
j
i j
i
j
,
,
,
,
,
+
+
−
−
=
−
+
( )
1
1
1
2
2
Δ
Δ
(9.7)
Figure 9.2 shows the solution of Equation (9.3) , i.e. a concentration profi le as a function 
of x for some chosen values of t .
When other processes accompany diffusion (for example reaction) or when an external 
fi eld is present, Equation (9.3) must be modifi ed by adding the appropriate terms. If a 
potential fi eld acts on a system (in our case magnetic), we add the ‘ drift term ’ to Equation 
(9.3) , i.e.
w
c
x
∂
∂
, to get fi nally the Smoluchowski equation [33] :

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Air Enrichment by Polymeric Magnetic Membranes
163
∂
∂
∂
∂
∂
∂
c
t
D
c
x
w
c
x
=
−
2
2
(9.8)
where D is the constant diffusion coeffi cient, and w is the constant drift coeffi cient. 
One of the ways to get an analytical solution of the Smoluchowski equation is to use 
some transformations, which reduce the Smoluchowski equation into Fick ’ s equation 
[34,35] 
. The interesting fact is that for suffi ciently large
‘ 
w 
’ 
, the solution of the 
Smoluchowski equation behaves like a ‘ travelling wave ’ , i.e. it starts to behave like a 
solution of the following equation:
∂
∂
∂
∂
c
t
w
c
x
= −
(9.9)
and represents a unidirectional wave motion with velocity w . Figure 9.3 shows the con-
centration profi les for steady state of Smoluchowski equation for different values of the 
drift coeffi cient.
For the steady state, Equation (9.8) takes the form
D
d c
dx
w
dc
dx
2
2
0
s
s
−
=
(9.10)
which gives the formula for concentration profi le c
s
( x )
c x
c
e
e
e
w
D
x
wl
D
wl
D
s
( )
=
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
0
1
(9.11)
shown in Figure 9.3 . 
1
C
S
(x)
0.8
0.6
0.4
0.2
1
x
0.8
0.6
0.4
w=10
w=5
0.2
w=0.1
w=1
Figure 9.3 Concentration profi les for steady state of Smoluchowski equation for different 
values of the drift coeffi cient: w = 0.1, w = 1, w = 5 and w = 10

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164
Membrane Gas Separation
We might also consider the problem of diffusion with a chemical reaction. In a suffi -
ciently strong magnetic fi eld, molecular clusters can be formed. For the case of air, the 
clusters N 
2
- O 
2
- O 
2
are preferable [36] . This can be regarded as a problem of diffusion with 
chemical reaction, which has described [31,37] 
∂
∂
∂
∂
∂
∂
c
t
x
D
c
x
wc
k k x f c
=
⋅
( )
−
⎡
⎣⎢
⎤
⎦⎥
−
( ) ( )
0
(9.12)
where k
0
is the rate constant, k ( x ) is the distribution function, and f ( c ) is the reaction 
kinetic term, ‘ w ’ , means the drift coeffi cient, as before. The term ‘ chemical reaction ’ has 
a rather formal meaning, and can represent also adsorption or ‘ trapping ’ processes.