Background


The problem of chemical reactor stability has received considerable attention in the chemical engineering literature since the early 1950s. The first work discussing the dynamic behavior of continuous stirred tank reactors appeared in 1953 by Van Heerden [1]. The analysis of local stability of the reactor steady states was not undertaken on a rigorous basis until Biolous and Amundson [1] . Aris and Amundson [2] were the first to show that in addition to multiple steady states, undamped oscillations in the form of limit cycles can occur for some combination of system parameters. They realized that these limit cycles bifurcated at the threshold value of some system parameter when the stable steady state became unstable.
The continuous stirred tank reactor is the simplest flow reactor, in which the reactor contents are kept well mixed. Therefore, the concentration of species A and the temperature do not vary with position in the reactor, and the concentration and temperature of the effluent stream is the same as in the reactor. A study of the dynamical behavior of a CSTR will show that this system can exhibit rather complex and interesting behavior.

The dynamic model of a first order reaction A -> B, in a CSTR consists of differential material and energy balance. For simplicity, the balances have been written in a dimensionless and autonomous form. The material balance in terms of the dimensionless concentration of A, u is

      (1)


and the transient energy balance in terms of a dimensionless temperature, u is

      (2)

b and k are kinetic parameters, while a is a dimensionless parameter reflecting reactor size and residence time of reaction mixture. The value of a is the parameter that will determine the dynamic behavior of CSTR because the values of the kinetic parameters b and k are set by the reaction, A -> B.
The heat of reaction intensity is represented by the dimensionless variable, b, which is

      (3)

where DH is the heat of reaction, Cf is the concentration of A in the feed, Cp is the heat capacity of the reaction mixture, r is the density of reaction mixture and Tf is the temperature of the feed. The parameter b is positive if the reaction is exothermic and negative is the reaction is endothermic. The value of b will be restricted to values between 2 and 20. Therefore, this report will only analyze the model when the reaction is exothermic. The parameter of k has been fixed at a value of 4.
It will be shown in the results and analysis section, that there exists three steady states, two of which are stable and correspond to high and low conversions. The middle conversion corresponds to a steady state that is always unstable. In the analysis section, we will also show a scenario when system parameters allow limit cycles to occur.





References
1. Uppal, A., Ray, W.H., Poore, A.B. Chemical Engineering Science, 29, 967 (1974).
2. Aris, R., Amundson, N.R. AIChE Journal, 1, 513 (1955).



[ Index ] [ Introduction ] [ Analysis ] [ Conclusions ]
Date: October 11, 1998
Time: 7:05 PM