# Dynamics of a Coupled Reactor-Separator System with Time Delay

Dynamics of a Coupled Reactor-Separator System with Time Delay

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 takes place in the reactor and there is a time delay 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

A→B

τ

dz

dt

x

af

z(t-τ)-

y

e

x

e

y

e

x

e

x

af

D

a

T

e

dT

dt

1

γ

D

a

T

e

z(-τ≤t≤0)=

z

0

T(-τ≤t≤0)=

T

0

In these equations, represents the mole fraction of species in the reactor, is the reactor dimensionless temperature, and is the dimensionless time. , , and 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.

z

A

T

t

D

a

β

γ

The mole fractions of species in the reactor fresh feed, the distillate stream, and the recycle stream are , , . The equations are solved with and . 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.

A

x

af

y

e

x

e

(β,γ)=(4.2,14)

(,,,,)=(0.45,0,0.9,0.8,0.2)

z

0

T

0

x

af

x

e

y

e