Membrane Gas Separation

9.3.3
Time Lag Method and D1 - D8 System 
In the experiments we were able to measure the fl ux and then, after integration with 
respect to time, we obtain a downstream permeation curve (shown in Figure 9.7 ) where 
the typical plot of Q
a
( l , t ) versus time is presented.
The most important parameter is the average diffusion coeffi cient  D
¯  , calculated from 
the stationary permeation data. To calculate this coeffi cient, we have used experimental 
data (the thickness of membrane l , a stationary fl ux J
S
and  
Δ
 c
0
, obtained from an intercept 
of the asymptote to the stationary permeation curve with the Q
a
( l , t ) axis) (Figure 9.7 ).
D
J l
c
= ⋅
s
in cm s
Δ
0
2
(9.18)

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168
Membrane Gas Separation
We can get some insight into the nature of the transport process by comparing  D
¯  with 
the value of diffusion coeffi cient D
L
, calculated by the time lag method, obtained as an 
intercept of the asymptote to the stationary permeation curve with the time axis (Figure 
9.7 ):
D
l
L l
L
a
in cm s
=
( )
2
2
6
(9.19)
If  D
¯  = D
L
, the diffusion is supposed to be ‘ ideal Fickian ’ , unless there are some hidden 
processes, like a drift, which are not included in this simple protocol [31] . In fact, in case 
of the ‘ magnetic membranes ’ the presence of a drift is almost certain. 
Having the drift and time lag values, the average diffusion coeffi cient can be calculated 
from a proper theory from which the following formula for time lag can be derived [37] .
ˆ
L l
D
e
Dwl
w l
w l
a
w
D
l
( )
=
−
−
+
⎛
⎝⎜
⎞
⎠⎟
−
2
2 2
3
2 2
2
2
(9.20)
To understand properly air enrichment by the ‘ magnetic membranes ’ we had to estimate 
the values of  
Δ
 c
0
, i.e. an equilibrium concentration of the air components (no sorption 
experiments have been carried out). Since  
Δ
 c
0
is not measured (in this chapter), we have 
to calculate it. The most realistic value of  
Δ
 c
0
can be estimated by comparing its values 
obtained by the different methods. 
The fi rst method is based on the determination of the permeability coeffi cient:
t
Q
a
(l,t)
point for checking
oxygen content
early time zone
late time zone
stationary state
L
a
(l)
-lc
o
/6
6 L
a
(l)
Figure 9.7 Downstream absorption permeation curve – a schematic view. Reprinted with 
permission from Journal of Membrane Science, On the air enrichment by polymer 
magnetic membranes by A. Rybak, Z. J. Grzywna and W. Kaszuwara, 336, 1 – 2, 79 – 85, 
Copyright (2009) Elsevier Ltd

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Air Enrichment by Polymeric Magnetic Membranes
169
-12
-10
-8
-6
-4
-2
0
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
ln[t
1/2.
J(l,t)]
ln[2c
0
(D
4
/
Π)
1/
2
]
1/t
Figure 9.8 An example of early - time permeation curve according to Equation (9.22). 
 Reprinted with permission from Journal of Membrane Science, On the air enrichment by 
polymer magnetic membranes by A. Rybak, Z. J. Grzywna and W. Kaszuwara, 336, 1 – 2, 
79 – 85, Copyright (2009) Elsevier Ltd
P
D S
D
c
p
= ⋅ = ⋅ Δ
Δ
0
(9.21)
We have to remember that the solubility coeffi cient for a pure EC is known. We do not 
know, however, its dependence on the parameters of the experiments with a magnetic 
fi eld! 
The subsequent three methods are based on analysis of Q
a
( l , t ) (see Figure 9.7 ). If we 
are to deal with an ‘ ideal Fickian ’ description of a permeation process, the value
−
lc
0
6
can be pointed out as an intercept of the asymptote to the stationary permeation curve 
with Q
a
( l , t ) axis. 
The next two ways use a description of transient, early - and late - time zones. 
For an early - time we can use a formula [45] 
ln
,
ln
t
J
l t
c
D
l
D t
a
1
2
0
4
1
2
2
3
2
4
⋅
( )
⎡
⎣⎢
⎤
⎦⎥
=
⎛
⎝
⎞
⎠
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
−
π
(9.22)
which can be presented as a
ln
,
t
J l t
a
1 2
⋅
( )
[
]
versus
1
t
graph (Figure 9.8 ). D
3
could be 
obtained from the slope of the asymptote to the early - time permeation curve, and D
4
, along 
with c
0
, can be obtained from the intercept of the asymptote with
ln
,
t
J l t
a
1 2
⋅
( )
[
]
axis.
While, for the late - time permeation, we get [45] 
ln
,
,
ln
Q l t
Q l t
lc
D
l
t
a
s
a
( )
−
( )
[
]
=
−
2
0
2
5
2
2
π
π
(9.23)

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170
Membrane Gas Separation
and consequently
ln
,
,
Q l t
Q l t
a
s
a
( )
−
( )
[
]
versus t graph (Figure 9.9 ). D
5
could be obtained 
from the slope of the asymptote to the late - time permeation curve, and c
0
from the inter-
cept of the asymptote with
ln
,
,
Q l t
Q l t
a
s
a
( )
−
( )
[
]
axis.
We have collected the data for the oxygen and nitrogen, both for individual gases 
and a mixture of the components of air, and analysed the system using formulas 
(18 – 23).