WOLFRAM|DEMONSTRATIONS PROJECT

Dynamics of a Coupled Reactor-Separator System with Time Delay

​
time t
50.
delay τ
0
Damköhler number
D
a
0.4
This Demonstration analyzes the effect of time delay on the behavior of a coupled non-isothermal continuous-flow stirred tank reactor (CSTR) with a separator.
The effluent of the reactor is fed to an isothermal separator and the liquid stream of the separator is recycled to the reactor. A first-order exothermic irreversible reaction
A→B
takes place in the reactor and there is a time delay
τ
in the transport from the reactor to the separator. The dimensionless delay-differential equations that describe the system (equations 8 and 9 in [1]) are
dz
dt
=(
x
af
-z)+
z(t-τ)-
y
e
x
e
-
y
e
(
x
e
-
x
af
)-
D
a
z
T
e
,​
dT
dt
=-T(
1
-β)+
​
γ
D
a
z
T
e
,​
z(-τ≤t≤0)=
z
0
,​
T(-τ≤t≤0)=
T
0
.
In these equations,
z
represents the mole fraction of species
A
in the reactor,
T
is the reactor dimensionless temperature, and
t
is the dimensionless time.
D
a
,
β
, and
γ
are the Damköhler number, the dimensionless heat transfer coefficient, and the dimensionless adiabatic temperature rise; these dimensionless numbers are defined in terms of system variables in the reference.
The mole fractions of species
A
in the reactor fresh feed, the distillate stream, and the recycle stream are
x
af
,
y
e
,
x
e
. The equations are solved with
(β,γ)=(4.2,14)
and
(
z
0
,
T
0
,
x
af
,
x
e
,
y
e
)=(0.45,0,0.9,0.8,0.2)
. In the absence of delay, the coupled system exhibits damped oscillations leading to a steady state for low and high values of the Damköhler number and oscillations without a steady state for intermediate values of the Damköhler number. Delay induces new regions of dynamic instability: increasing the delay beyond a lower threshold value can either destabilize the system or lead to isolated states of stability.