Dynamics of a Continuous Stirred-Tank Reactor with Consecutive Exothermic and Endothermic Reactions

maximum time t

7

heat transfer coefficient β

8.995

time plots

This Demonstration illustrates the dynamics of two irreversible consecutive reactions,

XY

, the first exothermic, the second endothermic, in a continuous stirred-tank reactor. The dimensionless equations for this system [1] are

dX

dt

=1-X-DaX

T/(1+ϵT)

e

,

dY

dt

=-Y+DaX

T/(1+ϵT)

e

-DaSY

κT/(1+ϵT)

e

,

dT

dt

=-T-β(T-

T

c

)+DaβX

T/(1+ϵT)

e

-DaBαSY

κT/(1+ϵT)

e

,

where

X

,

Y

, are the concentrations and

T

is the temperature,

Da

is the Damköhler number,

ϵ

is the activation energy,

S

is the ratio of the two rate constants,

κ

is the ratio of activation energies,

β

is the heat transfer coefficient,

T

c

is the coolant temperature,

B

is the adiabatic temperature rise, and

α

is the ratio of enthalpies of reaction. The equations are solved with

(Da,ϵ,S,κ,

T

c

,B,α)=(0.26,0,0.5,1,0,67.77,0.426)

and initial conditions

(X(0),Y(0),T(0))=(0.0213,0.0375,4.629)

. As the heat transfer coefficient

β

is increased, the trajectories change from damped oscillations leading to a steady state to periodic, then to chaotic oscillations; further increases in β lead to damped oscillations and finally to an asymptotic approach to equilibrium.