Biomedical Engineering Reference
In-Depth Information
and the concentration of A in the feed into PFR is given by
QC
A
0
þ
RQC
Ae
ð1 þ RÞQ
C
A
0
þ
RC
Ae
1 þ R
C
A
i
¼
¼
(E5-6.4)
Mole balance of A in a differential volume within the PFR (
V
,
V
þ
d
V
) leads to
d
½ð1 þ
R
Þ
QC
A
þr
A
d
V ¼ 0
(E5-6.5)
Since
Q
is constant, substituting
Eqn (E5-6.2)
into
Eqn (E5-6.5)
and rearranging, we obtain
1
C
A
þ
ð
1
þ
R
Þ
Q
d
C
A
C
A
0
k
d
V ¼
ð1 þ RÞQ
d
C
A
1
C
A
0
C
A
C
A
ðC
A
0
C
A
Þ
¼
(E5-6.6)
Integrating
Eqn (E5-6.6)
around the PFR,
1
C
A
þ
ð
1
þ
R
Þ
Q
d
C
A
C
A
0
Z
V
Z
C
Ae
1
C
A
0
C
A
k
d
V ¼
(E5-6.7)
0
C
Ai
we obtain
kV ¼
ð
1
þ
R
Þ
Q
C
A
0
ln
C
Ai
ð
C
A
0
C
Ae
Þ
C
Ae
ðC
A
0
C
A
i
Þ
(E5-6.8)
Substituting
Eqn (E5-6.4)
into
Eqn (E5-6.8)
, we obtain
kV ¼
ð
1
þ
R
Þ
Q
C
A
0
ln
C
A
0
þ
RC
Ae
RC
Ae
(E5-6.9)
Equation
(E5-6.9)
can be rearranged to give
C
A
0
R exp
kVC
A
0
ð1 þ RÞQ
C
Ae
¼
(E5-6.10)
R
or
C
Ae
¼ R exp
kVC
A
0
C
A
0
R
(E5-6.11)
ð1 þ RÞQ
The optimum recycle ratio is one that minimizes
C
Ae
or maximizes
C
A0
/
C
Ae
. Substituting in
the numbers in
Eqn (E5-6.11)
, we have
C
Ae
¼ R exp
0
:
02
200
5
R ¼ R exp
h
10ð1 þ RÞ
1
i
C
A
0
R
(E5-6.12)
ð1 þ RÞ120=60
Maximizing the right-hand side, we obtain
R
¼
0.126994 and
C
Ae
¼
0.005517 mol/L. The
optimum conversion is thus
QC
A
0
QC
Ae
QC
A
0
C
A
0
C
Ae
C
A
0
5
0
:
005517
5
f
Ae
¼
¼
¼
¼ 0:9989
The conversion is at 99.89%.
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