Membrane Gas Separation

16.2
Theory 
16.2.1
Gaussian Distribution of Sweep 
The simulations assuming a Gaussian distribution of sweep fl ow are based on mass 
balances for each fi bre in the module. Key assumptions in the theoretical analysis are as 
follows. 
1 The retentate fl ows in the lumen and the permeate in the shell.
2 Gas mixtures behave ideally.
3 The shell pressure is constant.
4 The lumen pressure drop is described by the Hagen - Poiseuille equation for compress-
ible fl ows.
5 Mass transfer by axial diffusion is small relative to axial convection.
6 The process is isothermal and steady - state.
7 The properties of a given fi bre do not vary along its length. However, properties may 
vary from fi bre to fi bre.
8 Lumen - side mass transfer coeffi cients are calculated using the Leveque correlation and 
shell - side mass transfer coeffi cients are calculated using the Donohue correlation.
The Gaussian sweep distribution is given by Equation (16.1) :
g
φ
σ π
φ φ
σ
( )
=
−
−
(
)
⎛
⎝⎜
⎞
⎠⎟
1
2
2
2
2
exp
(16.1)
where  
φ
  is the sweep fl ow rate around a fi bre,
φ
the mean sweep fl ow rate per fi bre, and 
 
σ
  the standard deviation. The fraction of fi bres for which the sweep fl ow rate falls in the 
interval [  
φ
  ,  
φ
  + d 
φ
  ] is equal to g (  
φ
  ) d 
φ
  . For the bundle of fi bres, the average fl ow rate per 
fi bre is given by Equation (16.2) :
f
f
g
d
=
( ) ( )
∫
φ
φ φ
φ
φ
min
max
(16.2)
where f may be the retentate or permeate fl ow. Gauss - Hermite quadrature [28] is used to 
evaluate Equation (16.2) and determine the overall performance of the fi bre bundle. The 
number of quadrature points is determined by increasing the number until the results 
change by less than the desired solution accuracy. 
The lumen and shell mass balances for individual fi bres in the bundle are solved by 
dividing each fi bre into N Weller - Steiner Case I stages along its length, as illustrated in 
Figure 16.1 .
For stage j and component i , the mass balance equations and permeation relationships 
are given by Equations (16.3) and (16.4) , respectively:
R
x
P y
R x
P y
j
i j
j
i j
j
i j
j
i j
−
−
+
+
+
=
+
1
1
1
1
,
,
,
,
(16.3)

---

The Effect of Sweep Uniformity on Gas Dehydration Module Performance
337
P y
P y
A J
A k p x
p y
j
i j
j
i j
j
i j
j i
j
i j
i j
,
,
,
,
,
,
−
=
=
−
(
)
+
+
1
1
h
l
(16.4)
where R is retentate molar fl ow rate, P permeate molar fl ow rate, x retentate mole fraction, 
y permeate mole fraction, A membrane area, p pressure, J permeation fl ux, and k the 
overall mass transfer coeffi cient. The subscripts h and l denote the high pressure retentate 
and low pressure permeate, respectively. The number of stages N is varied until the results 
obtained by increasing N change by less than 1%. 
Summing the mass balances for each of the n components, Equation (16.5) , gives the 
following expressions for the change in retentate and permeate fl ow rate.
P
P
A k p x
p y
j
j
j i
j
i j
i j
i
n
−
=
−
(
)
+
=
∑
1
1
h
l
,
,
,
(16.5)
R
R
A k p x
p y
j
j
j i
j
i j
i j
i
n
−
=
−
=
−
(
)
∑
1
1
h
l
,
,
,
(16.6)
Pressure drops in the lumen are calculated using the Hagen - Poiseuille equation [29] modi-
fi ed by substituting the product of molar fl ow rate and molar density for the volumetric 
fl ow rate and using the ideal gas law to calculate molar density:
dp
dz
R T
r N
R
h j
j
,
2
4
16
= −
η
π
g
i
f
(16.7)
where z is distance along the module,  
η
  the gas viscosity, R
g
the ideal gas constant, T
temperature, r
i
the fi bre inner radius, and N
f
the number of fi bres. The integral of this 
equation for stage j is given by:
p
p
R T
r N
R
R
L
h j
h j
jm
j
,
,
=
−
+
⎛
⎝⎜
⎞
⎠⎟
−
1
2
4
1
16
2
η
π
g
i
f
Δ
(16.8)
where  
Δ
 L is the length of the stage (i.e. (active fi bre length)/ N
f
). Equation (16.8) is used 
to evaluate the pressure in each stage given the pressure of the feed gas to stage 1. 
Equations
(16.3) 
– 
(16.7) 
are solved with an iterative solution algorithm. Using the 
crossfl ow solution as an initial guess, improved estimates for x 
i,j
  and y 
i,j
  are obtained from 
Equations
(16.9) 
and
(16.10) 
, respectively, which follow from Equations
(16.3) 
and 
(16.4) :
x
R
x
A k p y
R
A k p
i j
j
i j
j i
l
i j
j
j i
h j
,
,
,
,
=
+
+
−
−
1
1
(16.9)
N
j