# Continuously Stirred Tank Reactor Using Arc-Length Parameter

Continuously Stirred Tank Reactor Using Arc-Length Parameter

Consider a first-order, exothermic, irreversible reaction carried out in a well-mixed, continuously stirred tank reactor (CSTR). Assume that the fresh feed of pure is mixed with a recycle stream of unreacted ; then the material and energy balances are given by:

A→B

A

A

V=λF+F(1-λ)-F-Vexp

c

A

t'

c

A,f

c

A

c

A

k

0

-E

RT

c

A

Vρ=ρF(λ+(1-λ)T-T)+V(-ΔH)exp-hA(T-)

C

p

T

t'

C

p

T

f

k

0

-E

RT

c

A

T

c

where and are the reactor temperature and the exit concentration of species , and are the density and heat capacity, and are the activation energy and the heat of reaction, and are the reaction rate constant and the heat transfer coefficient, is the recycle flow rate, and are the feed concentration and temperature, is the temperature of the cooling medium, and finally is the reactor's volume and is the time.

T

c

A

A

ρ

C

p

E

ΔH

k

0

h

(1-λ)F

c

A,f

T

f

T

c

V

t'

Let us define the following nondimensional variables: (dimensionless time) with , =- (dimensionless concentration), = (dimensionless temperature), =V (Damköhler number), (dimensionless activation energy), (dimensionless heat transfer coefficient), (dimensionless adiabatic temperature rise), and finally = (dimensionless cooling medium temperature).

t=

t'

τ

τ=

V

Fλ

x

1

c

A,f

c

A

c

A,f

x

2

T-

T

f

T

f

D

a

k

0

-γ

e

Fλ

γ=

E

R

T

f

β=

hAτ

Vρ

C

p

B=

(-ΔH)

c

a,f

ρ

C

p

T

f

E

R

T

f

x

2c

T

c-

T

f

T

f

E

R

T

f

The governing equations, that is, equations (1) and (2) become:

x

1

t

x

1

D

a

x

1

x

2

1+

x

2

γ

x

2

t

x

2

D

a

x

1

x

2

1+

x

2

γ

x

2

x

2c

This Demonstration plots the locus of reactor steady states (concentration and temperature) versus the Damköhler number when there is multiplicity for user set values of parameters , , and =7. The unstable steady states are denoted by the dashed region of the locus. The blue dots denote the values of at the turning points, a red dot denotes the value of at a Hopf bifurcation, and the green dots denote the values where two real eigenvalues change into a complex conjugate pair.

B

β

x

2c

D

a

D

a

D

a

Time series plots for normalized concentration (in blue) and normalized temperature (in magenta) for a given set of initial conditions ( and ) and parameters are given. One can investigate the basin of attraction for the steady states. The approach to the steady state may be monotonic or oscillatory depending on the associated eigenvalues for that steady state. Initial conditions near the Hopf point are attracted to a stable periodic solution.

x

10

x

20