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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