10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 6: Concentration that Optimizes a Desired Rate
This lecture covers: Selectivity vs. conversion and combining reactors with
separations.
Optimal
A
- What is the target? goal? Æ Often it is to maximize profit or
production
recycle A (U, P)
Separator
P, A, U
product
Purer P
(A, U)
byproducts
Figure 1. Schematic of a reacting system with a recycle stream.
const. (vol.)
Fp = rp ⋅V
Simple Target: maximize
rp (concentrations, T)
Constraint: T ≤ Tmax
High temperature Æ maximum rate constant
Simplest Case:
[ A] ⇒ [ A] feed
rp = k [ A]
A→ P
τ →0
residence
time in reaction
Fp = [ P ]V0 = [ P ]
V
τ
(1− e − Da )
= Vk [ A] feed
Da
Constraint: [ P ] ≥ [ P ]min
(purity constraint )
Æ X ≥ X min conversion
{
}
Fp = k (Tmax ) [ A]0 − [ P ]min V = k (Tmax ) [ A] feed V (1− X )
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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A + B → 2C
rc = 2k [ A][ B ]
2nd order
A, B
[ A] = [ A] feed (1 − X A )
[ B ] = [ B ] feed (1 − X B )
.1
.1
rc = 2k [ A] feed [ B ] feed (1 − X A )(1 − X B )
[ A][ B ][C ]
.01
Figure 2. Schematic
of a CSTR.
A, B
B
[ B ] [ A]
A, B, C
Separator
Figure 3. A reacting system with
recycle stream.
(A)C
A→ B
B→C
A →U
product may
also react
undesirable
FB = f ([ A] , [ B ] , [C ] , [U ] ,τ , T )
6 variables
Æ fsolve (Matlab)
SS.
0 = Fin − Fout + rAV
A
A
0 = Fin − Fout + rBV
B
B
0 = ...
0 = ...
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture #6
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Simplest Case:
k1 (T )
A→B
k2 (T )
B→C
k3 (T )
FA
A→U
in
A
=
[ ]
V (k1 + k3 + 1 )
τ
[U ] = k3 [ A]τ
[C ] = k2τ [ B ]
k [ A]
[ B] = 1 1
k2 +
τ
FB (τ , T ) =
FAo k1τ
N
(k1τ + k3τ + 1)(1+ k2τ )
N
N N
optimize
Da1
2 variables
Æ contour plot
Da2
Da3
Figure 4. Sample contour plot for the 2variable optimization.
matlabÆfmincon
(allows for constraints)
∂FB
∂FB
= 0,
=0
∂T
∂τ Don’t use these – you may not find an actual optima.
Yield ≡
FP
FA0
Yield A→ P
≡ X A S A→ P
Selectivity =
S A→ P
FP
( FA0 − FA )
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture #6
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(especially important when A is expensive)
Y
A
→
P
A → P →U
P →U
τ
Figure 5. Yield versus residence time. Intermediate P rises in
concentration and then falls off as it is converted to U.
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture #6
Page 4 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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