Biomedical Engineering Reference
In-Depth Information
Since the volume is constant (constant volume batch reactor), we further obtain
d
C
A
d
n
A
kC
O
RA
O
RB
A
C
B
¼
(E4-6.4)
t
Based on stoichiometry, we have
n
A
n
A;0
n
A
¼
n
B
n
B;0
n
B
(E4-6.5)
For constant volume reactor, Eqn
(E4-6.5)
is reduced to
C
B
¼ C
B0
þ
n
B
n
A
ðC
A
C
A0
Þ
(E4-6.6)
Therefore, C
B
and C
A
are related. As the rate constant
k
is only a function of temperature
(Arrhenius relationship), separation of variables for Eqn
(E4.3-4)
leads to
d
C
A
C
O
RA
d
E
RT
¼ n
A
k
d
t ¼ n
A
k
0
exp
t
(E4-6.7)
A
C
O
RB
B
For a desired reaction end condition or conversion, the reaction time can be obtained by
integrating Eqn
(E4-6.7)
.
d
Z
C
A
Z
t
d
C
A
C
O
RA
E
RT
¼
n
A
k
0
exp
t
(E4-6.8)
A
C
O
RB
B
C
A0
0
where C
B
is related to C
A
via Eqn
(E4-6.6)
. Equation
(E4.3-8)
is applicable to reaction occur-
ring isothermally at a constant temperature,
T
R
as well:
Z
C
A
d
Z
t
R
d
C
A
C
O
RA
A
E
RT
R
E
RT
R
¼
n
A
k
0
exp
t ¼ n
A
k
0
exp
t
R
(E4-6.9)
C
O
RB
B
C
A0
0
where
t
R
is the time required to reach the reaction end condition (concentration C
A
from C
A0
)
isothermally at a reaction temperature of
T
R
. Substituting Eqn
(E4-6.9)
into Eqn
(E4-6.8)
,we
obtain
d
Z
C
A
Z
t
E
RT
R
d
C
A
C
O
RA
E
RT
n
A
k
0
exp
t
R
¼
¼
n
A
k
0
exp
t
(E4-6.10)
A
C
O
RB
B
C
A0
0
which can be reduced to
E=R
d
Z
t
T
R
E=R
t
R
¼
exp
t
(E4-6.11)
T
0
Thus, we have obtained a timescale that would give rise to the same conversion for
different temperatures and/or different temperature progressions. Integration for
t
R
can be
achieved numerically (Liu, S. 2010, J. Biotech. Adv., 28: 563
e
582.)
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