Complete Mixing Model for Gas Separation by Membranes ​​Authors: Housam Binous and Ahmed Bellagi

reference : H . Binous, Gas Permeation Computations with Mathematica®, Chemical Engineering Education, 40 (2), pp . 140 - 144, 2006.
We compute the permeate and residue compositions obtained using a membrane separation module . Membrane area is also computed . Feed is air and is considered as a binary mixture . Complete mixing model is used . This problem was proposed by C . J . Geankoplis in Transport Processes and Unit Operations, 3 rd edition, Prentice Hall, 1993 (examples 13.4 - 2 and 13.6 - 1 in pages 767 and 775) . We use a different approach to solve this air separation problem than the one used by Professor Geankoplis . Indeed, we solve a quadratic equation using Solve in order to determine the permeate composition.
Reference:C. J. Geankoplis, Transport Processes and Unit Operations, 3rd edition, Prentice Hall, 1993.
The separation factor for this binary mixture (components A and B) that we want to separate by membrane permeation is
α
=
P'
A
P'
B
=
10.
In[]:=
α=10;
The pressure of the feed side is
p
h
=190cmHg
while
p
l
=
19 cmHg is the pressure on the permeate side. The pressure ratio is R=
p
l
p
h
=0.1.
In[]:=
p
l
=19;
In[]:=
p
h
=190;
In[]:=
R=0.1;
Stage cut is equal to θ=0.2.
In[]:=
θ=0.2;
t= 2.54
-3
10
cm is membrane thickness.
q
O
=(1-θ)
q
f
is reject out flow rate.
q
f
=1
6
10
3
cm
(STP)/s
is feed flow rate. Feed mole fraction is xin=0.209.
P'
A
=500
-10
10
3
cm
(STP)cm(s
2
cm
cmHg)
is the permeability of component A. The mole fraction y in the permeate side is given by:
In[]:=
t=2.54 10^-3;
In[]:=
q
f
=10^6;
In[]:=
P'
A
=50010^-10;
In[]:=
xin=0.209;
In[]:=
sol2=Solve[(α-1) (θ+R (1-θ)) y^2-(α θ+(α-1) (xin+R(1-θ))+(1-θ)) y+α xin0,y]
Out[]=
{{y0.506809},{y1.63644}}
In[]:=
yout=0.506809;
A material balance gives the reject out mole fraction
In[]:=
Off[General::spell1]
In[]:=
xout=(xin-θ yout)/(1-θ)
Out[]=
0.134548
Membrane area is given by the following relation:
In[]:=
xout=0.1346;yout=0.5067;(*ItookGeankoplisapproximatevaluesinsteadofthosecomputedabove*)
In[]:=
A
m
=
θ
q
f
yout
P'
A
t
(
p
h
xout-
p
l
yout)
(*expressedincm2*)
Out[]=
3.2283×
8
10