Membrane Gas Separation

…
…
P
1
, y
1,i
Feed
P
2
, y
2,i
R
1
, x
1,i
1
…
…
P
j
, y
j,i
R
j-1
, x
j-1,i
P
j+1
, y
j+1,i
R
j
, x
j,i
P
N
, y
N,i
R
N-1
, x
N-1,i
Sweep
R
N
, x
N,i
Figure 16.1 Countercurrent module divided into N stages

---

338
Membrane Gas Separation
y
P y
A k p x
P
A k p
i j
j
i j
j i
h j
i j
j
j i
l
,
,
,
,
=
+
+
+
+
1
1
(16.10)
Improved estimates for P 
j
  , R 
j
  , and p 
h,j
  are obtained using Equations (16.5) , (16.6) , and 
(16.8) , respectively. The iterative procedure is stopped when the variables change by less 
than 1% from one iteration to the next. This direct substitution iterative algorithm proved 
to be robust and converged quickly for all cases considered. The converged solution for 
each fi bre is used to evaluate overall module performance by using Equation (16.2) to 
determine average fl ow rates. 
The overall mass transfer coeffi cient for species i is calculated by summing the lumen -
side, membrane, and shell - side mass transfer resistances as given by Equation (16.11) :
1
1
1
k
r
k r
Q
k
i
l i i
i
s i
=
+
+
o
,
,
(16.11)
where r
o
is the fi bre outer radius and Q the membrane permeance based on the fi bre outer 
radius. As defi ned in Equation (16.11) , the overall mass transfer coeffi cient is based on 
the fi bre outer radius. 
The shell and lumen side boundary resistances are calculated using the Donohue and 
Leveque equations, Equations (16.12) and (16.13) respectively, as suggested by Wang 
[4,5] .
Sh
k L
D
Re Sc
r
L
i
=
=
⎛
⎝
⎞
⎠
⎡
⎣⎢
⎤
⎦⎥
l
1 62
2
0 33
.
.
(16.12)
Sh
k L
D
D Re Sc
=
=
s
h
0 0732
0 6
0 6
0 33
.
.
.
.
(16.13)
where k
l
is the lumen - side mass transfer coeffi cient, k
s
the shell - side mass transfer coef-
fi cient, L the length of the fi bres, D the diffusion coeffi cient, Re the Reynolds number, 
Sc the Schmidt number, and D
h
the hydraulic diameter given by:
D
r
N
r
r
N r
i
i
h
m
f
m
f
=
(
)
−
( )
+
2
2
2
2
2
2
(16.14)
where r
m
is the module radius. Note the Reynolds number is calculated based on the fi bre 
inner radius for Equation (16.12) and the fi bre outer radius for Equation (16.13) . 
Results obtained for a Gaussian sweep distribution are compared to results obtained 
for a Gaussian fi bre inner radius distribution. Previous work [30,31] has shown that vari-
ation in fi bre inner radius has the largest impact on module performance when the variabil-
ity in fi bre properties is included in module performance simulations.