WOLFRAM|DEMONSTRATIONS PROJECT

Exothermic Continuous Stirred-Tank Reactor with Dynamically Varying Cooling Jacket

​
controller
concentration
temperature
dimensionless time
30
Damköhler number
0.15
dimensionless coolant flow rate
1
dimensionless initial coolant temperature
0
This Demonstration compares the kinetics of an exothermic reaction with a cooling jacket at constant temperature to a system in which the dynamics of the cooling jacket are taken into account.
The standard two-state CSTR (continuous stirred-tank reactor) model describing an exothermic diabatic first-order irreversible reaction
AB
can be represented by a pair of ordinary dimensionless differential equations [1]
d
C
A
dt
=-
D
a
C
A
T/(1+T/γ)
e
+Q
C
A
0
-
C
A

and
dT
dt
=β
D
a
C
A
T/(1+T/γ)
e
-(Q+δ)T+δ
T
c
+Q
C
0
,
where
C
A
and
T
are the concentration of reactant
A
and reactor temperature, respectively,
T
c
is the constant coolant temperature,
D
a
is the Damköhler number,
γ
is the activation energy,
Q
is the reactor flow rate,
δ
is the heat transfer coefficient, and
t
is time.
When the dynamics of the cooling jacket are considered,
T
c
is a variable and an additional energy balance around the cooling jacket yields a third dimensionless differential equation
d
T
c
dt
=
δ
1
q
T
c
0
-
T
c
+δ
δ
2
(T-
T
c
)
,
where
δ
1
and
δ
2
are the reactor-to-cooling-jacket volume ratio and density heat capacity ratio, respectively, and
q
is the cooling jacket flow rate.
Plots of the functions
C
A
and
T
are shown for the standard two-equation system and for the three-equation system that incorporates the dynamics of the cooling jacket for user-selected values of time
t
, Damköhler number
D
a
, coolant flow rate
q
, and initial coolant temperature
T
c
0
. The parameters are taken as
(Q,β,δ,
δ
1
,
δ
2
,γ)=(1,11,1.5,10,0.11,20)
with initial conditions

C
A
0
,
T
0
=(1,0)
. The addition of the third equation shows that much more complex behavior than a simple time lag can occur and this can significantly affect the behavior and stability of a reactor.