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Process Dynamics 2
ü
Copyright Brian G. Higgins (2004)
Overview
In these notes we show how to derive the governing equations for a perfectly mixed flow through reactor, also
known as a CSTR. A schematic of the reactor (taken from Ref 1) is shown below
Figure 1
The reactor displayed in Figure 1 has a single entrance stream at 1 and an exit stream at 2. The overall mass
balance for this vessel is
(1)
„
ÅÅÅÅÅÅÅ
„
t
‡
V
H
t
L
r „
V
=
‡
A
1
r
v
ÿ
n
„
A

‡
A
2
r
v
ÿ
n
„
A
Here
v
is the velocity of the fluid,
r
is the density of the fluid, and
n
is the outward directed unit normal at the
entrance and exits of the control volume
V
H
t
L
. The crosssectional areas at positions 1 and 2 are denoted by
A
1
and
A
2
. Note that
r
v
ÿ
n
„
A is the mass flux passing through the differential area
„
A. Thus we can write (1) as
(2)
„
M
ÅÅÅÅÅÅÅ
„
t
=
F
°
1

F
°
2
where M is the mass of fluid in the tank at time t and
F
°
i
is the mass flow rate across exit/entrance i. In liquid
systems it is reasonably (at low to moderate pressures) to assume that the density
r
is constant so that
(3)
M
= r
V ,
F
°
i
= r
Q
°
i
where now
Q
°
i
is the volumetric flow rate across exit/entrance i. Thus (2) becomes
(4)
„
V
ÅÅÅÅÅÅÅ
„
t
=
Q
°
1

Q
°
2
Now consider a species A in the inlet stream with concentration C
H
kg
ê
m
3
L
that undergoes a irreversible reaction
with a rate constant k:
(5)
A
ö
k
B,
r
A
= 
kC
A
, r
B
=
kC
A
A species balance over the reactor gives
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„
ÅÅÅÅÅÅÅ
„
t
‡
V
H
t
L
C
i
„
V
=
‡
A
1
C
i
v
i
ÿ
n
„
A

‡
A
2
C
i
v
i
ÿ
n
„
A
+
‡
V
r
i
„
V, i
=
A, B
Here
v
i
is the
i
th
species velocity which at entrances and exits can be approximated by the mass average velocity
v
(since diffusion effects are negligible at entrances and exits). Thus the two species balances become
(7)
„
C
A
ÅÅÅÅÅÅÅÅÅ
„
t
=
Q
°
ÅÅÅÅ
V
C
Af

Q
°
ÅÅÅÅ
V
C
A
+
r
A
,
r
A
= 
k C
A
„
C
B
ÅÅÅÅÅÅÅÅÅ
„
t
=
Q
°
ÅÅÅÅ
V
C
Bf

Q
°
ÅÅÅÅ
V
C
B
+
r
B
r
B
=
kC
a
We have assumed that the exit concentration of the species are the same as the species concentrations in the
reactor and the volume V of the reactor remains constant. The latter assumption implies that the flow rates
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