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WOLFRAM|DEMONSTRATIONS PROJECT

Dynamics of Two Coupled Continuously Stirred Tank Reactors

​
time t
17.
Damköhler number ratio Δ
3.
coupling coefficient μ
10
time plots
This Demonstration shows the dynamic behavior of two coupled continuously stirred tank reactors (CSTRs) in which an autocatalytic reaction with product inhibition takes place. The system is described by six ordinary differential equations, three for each reactor [1].
d
u
1
dt
=
u
0
-
u
1
+
Da
1
(
u
1
-
α
1
u
1
w
1
)+μ(
u
2
-
u
1
)
,
d
v
1
dt
=
v
0
-
v
1
+
Da
1
(
u
1
-
α
2
v
1
)+μ(
v
2
-
v
1
)
,
d
w
1
dt
=
w
0
-
w
1
+
Da
1
(
α
2
v
1
-
α
3
w
1
)+μ(
w
2
-
w
1
)
,
d
u
2
dt
=
u
0
-
u
2
+Δ
Da
1
(
u
2
-
α
1
u
2
w
2
)+μ(
u
1
-
u
2
)
,
d
v
2
dt
=
v
0
-
v
2
+Δ
Da
1
(
u
2
-
α
2
v
2
)+μ(
v
1
-
v
2
)
,
d
w
2
dt
=
w
0
-
w
2
+Δ
Da
1
(
α
2
v
2
-
α
3
w
2
)+μ(
w
1
-
w
2
)
.
Here
u
1
,
v
1
,
w
1
and
u
2
,
v
2
,
w
2
are dimensionless concentrations in reactors 1 and 2,
t
is the dimensionless time, and the other quantities are dimensionless parameters. The parameter values are fixed at
(
α
1
,
α
2
,
α
3
,
u
0
,
v
0
,
w
0
)=(0.2,0.2,0.2,1.0,0.0,0.0)
and
Da
1
=8.0
.
The parameters of interest are
μ
, the coupling coefficient, which may be viewed as the ratio of cross-flow rate to feed rate to reactor 1, and
Δ=
Da
2
Da
1
, the ratio of Damköhler numbers of the reactors. When
Δ=1
, each reactor has a stable periodic solution with a period 2.0917; increasing the value of
Δ
decreases the frequency. Increasing the coupling coefficient
μ
shows interesting states of oscillation that lead to steady states for intermediate values of the coefficient.
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