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

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

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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].

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