# Cascade of Two Continuous Stirred-Tank Reactors with Recycle

Cascade of Two Continuous Stirred-Tank Reactors with Recycle

Consider a cascade of two continuous stirred-tank reactors with recycle undergoing exothermic first-order chemical reactions. The steady state is described by four nonlinear equations for the reaction conversions , , dimensionless temperature , , and parameter (Damköhler number). The dimensionless equations that describe the steady state are given by [2] (see also [1])

X

1

X

2

T

1

T

2

α

f(X,X,T,T)=αe(1-X)-X=0

1

1

2

1

2

T

1

1+0.001T

1

1

1

f(X,X,T,T)=22αe(1-X)-3T=0

2

1

2

1

2

T

1

1+0.001T

1

1

1

f(X,X,T,T)=X+αe(1-X)-X=0

3

1

2

1

2

1

T

2

1+0.001T

2

2

2

f(X,X,T,T)=22αe(1-X)+T-3T=0

4

1

2

1

2

T

2

1+0.001T

2

2

1

2

The derivation of these equations (which represent the species and energy balances for each tank) is given in [2]. A steady solution for is , , , and (see snapshot 4). The computational method used to generate is based on arc length continuation.

α=0.01

X(0.01)=0.0107005

1

X(0.01)=0.0215779

2

T(0.01)=0.0784704

1

T(0.01)=0.105924

2

T(α)

2

The Demonstration shows the dependence of on the parameter . The solution path has six turning points (indicated in green in snapshot 3) and for there are seven steady state solutions! (These are indicated in blue in snapshot 1.) Again, not all these steady state solutions are stable to small perturbations. The stability of the steady states is determined by the sign of the real part of the eigenvalues of the Jacobian matrix. The results are in quantitative agreement with the calculations in the references. Though not shown, one can readily plot the dependence of the remaining variables , , and on .

T

2

α

α≈0.039

X

1

X

2

T

1

α