Corelation (Lecture - 6) - Notes, Chemical Engineering, Semester Notes - Chemical Engineering

 Page 1


Consider the unconstrained optimization of a CSTR with volume V. 
A ? B

r = k[A]

q [A]
0 
[A] [B] V 
The goal is to maximize F
B
 with respect to changes in the volumetric flow rate, q. 
F
B 
= q[B] 
Steady state material balances on species A and B give: 
0 = F
A0 
- F
A 
- rV = q ([A]
0 
- [A] ) - kV[A] 
0 = F
B0 
- F
B 
+ rV = -q[B] + kV[A] 
Hence, 
[B] = k[A] (V / q ) 
and 
F
B 
= rV = k[A]V ; 
thus production of B is maximized when [A] takes its maximum value, which is [A]
0
. 
Continuing with the material balances, we find: 
[A] = 
[A]
0 
= 
[A]
0

1 + (kV / q ) 1 + k t

When Da = k t << 1, [A] goes to [A]
0
. 
F
B 
= rV = kV[A] = 
kV[A]
o 
= 
kV[A]
o

1 + k t 1 + kV / q

o
lim F
B 
= lim
?
?
? kV[A] 
?
?
? 
= kV[A]
0

q ? 8 q ? 8
?
1 + kV / q 
?

Unfortunately, in the limiting case of infinite flow rate, the concentration of B in the 
output solution is vanishingly small: 
lim[B] = lim (k[A] (V / q ) ) = lim
?
?
? 
k 
[A]
0 
(V / q )
?
?
? 
= 0 . 
q ? 8 q ? 8 q ? 8
? 
1 + (kV / q )
? 
Cite as: William Green, Jr., and K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering,

Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].