Biomedical Engineering Reference
In-Depth Information
Substituting the rate expression into the mole balance equation, we obtain
d
n
A
d
¼ r
A
V ¼kC
2
A
V ¼ kC
A
n
A
(E4-2.5)
t
Substituting Eqn
(E4-2.2)
into Eqn
(E4-2.4)
, we obtain
n
2
A
2n
A0
n
A
d
n
A
d
t
¼kC
A0
(E4-2.6)
Separation of variables,
2n
A0
n
A
n
2
A
d
n
A
¼kC
A0
d
t
(E4-2.7)
Integration between
t
¼
0 and
t
¼
t
, we obtain
2n
A0
n
A
þ
2n
A0
n
A0
ln
n
A
n
A0
¼kC
A0
t
(E4-2.8)
which is reduced to
kC
A0
t ¼ 2
n
A0
n
A
2 ln
n
A0
(E4-2.9)
n
A
Substituting Eqn
(E4-2.3)
into Eqn
(E4-2.9)
, we obtain
2
2 V=V
0
2 þ ln
2
V
V
0
t ¼
(E4-2.10)
kC
A0
Thus, the time required for 50% expansion of the reactor volume is given by
2
2 V=V
0
2 þ ln
2
V
V
0
2
2 V=V
0
2 þ ln
2
V
V
0
t ¼
¼
k
P
RT
kC
A0
2
2 1:5
2 þ lnð2 1:5Þ
¼
s ¼ 405:3 s
1 10
5
8:314 373
0:1 10
3
Therefore
;
it takes 405
:
3 s for the reactor volume to expand 50%
:
Example 4-3. Concentration profile for a series reaction. For the following elementary reac-
tion system
k
1
k
2
A !
B
!
C
Find the change of concentrations of A, B, and C as a function of time starting with pure A
initially at C
A
¼ C
A
0
in a constant volume isothermal batch reactor.
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