Membrane Gas Separation

9.2.2
Diffusion Equation for Two - component Gas Mixture (Without and With 
a Potential Field) 
A diffusion process, in the simplest case described by the Fick ’ s law, for the multicom-
ponent diffusion is described by its generalization to an n - component system [38,39] . The 
fl ux has then a form:
− =
∇
=
∑
j
D
c
i
ij
j
j
n
1
(9.13)
In this chapter we consider the separation of a binary mixture by a dense membrane. The 
simplest case i.e. when D 
ij
  is constant, is represented by a system (14):
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
c
t
D
c
x
D
c
x
c
t
D
c
x
D
c
x
1
11
2
1
2
12
2
2
2
2
21
2
1
2
22
2
2
2
=
+
=
+
⎧
⎨
⎪⎪
⎩⎩
⎪
⎪
∈
( )
∈
+
x
0,
, t
R
l
(9.14)
with initial and boundary conditions:
c x
c x
c
t
c
c
t
c
c l t
c l t
1
2
1
01
2
02
1
2
0
0
0
0
0
0
0
0
,
,
,
,
,
,
(
)
=
(
)
=
( )
=
( )
=
( )
=
( )
=
(9.15)
In general, the coeffi cients are not symmetric, i.e. D 
ij
 
≠ 
D 
ji
  . The diagonal terms are called 
the ‘ main term ’ coeffi cients, while off diagonal terms, i.e. D 
ij, i
 
≠ 
 
j
  , are called the ‘ cross - term ’
coeffi cients [38,39] . In fact, they represent the degree of coupling between the diffusing 
gas components. The set (Equation 9.14 ) can be solved numerically. The fi nite - difference 
method gives the following schemes:
c
c
t
D
c
c
c
x
D
c
i j
i j
i
j
i j
i
j
1
1
1
11
1
1
1
1
1
2
12
2
2
, ,
, ,
,
,
, ,
,
,
,
+
+
−
−
=
−
+
( )
+
Δ
Δ
ii
j
i j
i
j
i j
i j
i
j
c
c
x
c
c
t
D
c
+
−
+
+
−
+
( )
−
=
−
1
2
2
1
2
2
1
2
21
1
1
2
,
, ,
,
,
, ,
, ,
,
,
Δ
Δ
2
2
2
1
1
1
2
22
2
1
2
2
1
2
c
c
x
D
c
c
c
x
i j
i
j
i
j
i j
i
j
, ,
,
,
,
,
, ,
,
,
+
( )
+
−
+
( )
⎧
⎨
−
+
−
Δ
Δ
⎪⎪⎪
⎩
⎪
⎪
(9.16)

---

Air Enrichment by Polymeric Magnetic Membranes
165
1
"
a
"
"
b
"
C
1
(x,t)
0,8
0,6
0,4
0,2
0
1
C
2
(x,t)
0,8
0,6
0,4
0,2
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
0,8
1
x
x
0
t
1
t
2...
t
n
t
1
t
2...
t
n
Figure 9.4 Concentration profi les – solutions of Equation (9.14) for constant coeffi cients 
 D
11
= 1, D
12
= 0.13, D
21
= 0.14, D
22
= 1.1, for different times: (a) concentration profi les 
for component 1 of the mixture; (b) concentration profi les for component 2 of the 
mixture. 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
Figure
9.4 
shows the concentration profi les of
c
1
( x , t 
), and
c
2
( x , t 
), i.e. solution of 
Equation (9.14) with conditions shown in Equation (9.15) .
In the case of the presence of an additional force which acts on one component of a 
permeating gas mixture, the system shown in Equation (9.14) must be modifi ed. We add 
the drift term, i.e.
w
c
x
∂
∂
, to the fi rst equation of Equation (9.14) , and now it reads:
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
∂
c
t
D
c
x
D
c
x
w
c
x
c
t
D
c
x
D
c
1
11
2
1
2
12
2
2
2
1
2
21
2
1
2
22
2
2
=
+
−
=
+
∂∂
x
l
2
⎧
⎨
⎪⎪
⎩
⎪
⎪
∈
( )
∈
+
x
0,
, t
R
(9.17)
along with the initial and boundary conditions shown in Equation (9.15) . 
The resulting concentration profi les for large values of the drift term are presented in 
Figure 9.5 .
Since our system (Equation 9.17 ) is weakly coupled, we can collect rough, but still 
essential information, about separation by analysis of just a single equation addressed to 
the particular air components, i.e. Fick ’ s equation to the nitrogen, and the Smoluchowski 
equation to the oxygen, respectively.
9.3
Experimental 
9.3.1
Membrane Preparation 
Flat ethylcellulose membranes of thickness 28 – 80
μ
m and EC magnetic membranes of 
thickness 90 – 202
μ
m (depending on magnetic powder granulation and on the amount of 
added powder) were used. Both types of fi lms were made by casting. The solution in 

---

166
Membrane Gas Separation
40:60 ethanol/toluene with EC concentration of 3% was poured into a Petri dish, and 
evaporated at room temperature for 24 h. Magnetic membranes were made by pouring the 
3% ethylcellulose solution with a dispersed magnetic neodymium powder (of appropriate 
granulation 20 – 50
μ
m) into a Petri dish, and then evaporated in external fi eld of a coil 
(stable magnetic fi eld with range of induction 0 – 40 mT) for 24 h. Magnetic membranes 
with 1.0 – 10.5% of neodymium powder content in polymer solution and 19.1 – 73.0% in 
dry membrane, were obtained [25,26] . 
The membranes were removed from the Petri dish (with some distilled water), and 
dried at 40 ° C. For membranes with dispersed magnetic powder, the permeation measure-
ments were carried out before (just powder) and after magnetization (magnetic mem-
branes). Magnetic induction of this fi eld was approximately of 2 T. Teslameter FH 54 was 
used for measurement of magnetic induction. The membranes were stored in an desiccator 
under vacuum conditions. We have also tested polyphenyleneoxide (PPO) membranes 
[40 – 44] , courtesy of B. Kruczek (Faculty of Engineering, University of Ottawa, Canada).