Classification of steady-state and dynamic behavior of distributed reactor models

Abstract

Singularity theory is combined with the Liapunov-Schmidt reduction and the shooting method to present a systematic procedure that may be used to classify the steady-state and dynamic behavior of distributed reactor models. The method is applicable to problems of diffusion-reaction, convection-reaction and diffusion-convection-reaction, dependent on one spatial coordinate and involves the numerical computation of various singularities (hysteresis, isola, double zero and double or degenerate Hopf loci). The usefulness of the present approach is demonstrated by applying it to four classical examples from chemical reactor theory. The first example deals with the problem of diffusion and reaction in a porous catalyst particle (slab geometry) with no external heat and mass transfer resistances (Dirichlet model). Here, the different loci divide the parameter space into six regions corresponding to six different bifurcation diagrams. It is observed that in the physically feasible parameter region, the catalyst particle model is more likely to exhibit oscillatory behavior when the reacting fluid is a liquid. The second example is the convection-reaction model of a cooled tubular reactor with recycle. It is shown that for the non-adiabatic case, the hysteresis, isola, double-zero and double Hopf loci divide the parameter space into thirty three different regions, each corresponding to a different bifurcation diagram. The cooled recycle reactor can exhibit both low and high temperature oscillations for realistic parameter values. The third example considers a diffusion-convection-reaction model of a tubular reactor with axial dispersion of heat and mass. For the adiabatic case, the parameter space of the model is divided, as in the catalyst particle model, into six regions corresponding to different bifurcation diagrams. The steady-state model of cooled tubular reactor is shown to be described by a single intrinsic state variable only in the limiting case of negligible axial dispersion of mass. This limiting model is examined and it is shown that oscillatory solutions are possible on the ignited high conversion branch for realistic values of the parameters. Finally, two autothermal reactor models with internal and external heat exchange are considered. Once again the parameter space is divided into six regions and for realistic parameter values no oscillatory behavior is exhibited.