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Sequential Bifurcations in Continuous Stirred Tank Chemical Reactors Coupled in Series

Gerhard Dangelmayr and Ian Stewart
SIAM Journal on Applied Mathematics
Vol. 45, No. 6 (Dec., 1985), pp. 895-918
Stable URL: http://www.jstor.org/stable/2101511
Page Count: 24
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Sequential Bifurcations in Continuous Stirred Tank Chemical Reactors Coupled in Series
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Abstract

We apply the theory of sequential bifurcation problems (SBP's) to the static bifurcations of two stirred tank chemical reactors coupled in series. Following the usual line of argument, we seek the most degenerate singularities that can occur, by varying auxiliary parameters. The unfolding theory of singularities then provides a complete description of the perturbations of these most-degenerate problems, giving a qualitative picture of the possible local bifurcation diagrams. We find two most-degenerate singularities of codimension three, occurring along two disjoint lines in the parameter space. They are of type (2)23 and (3)23 in our classification. We obtained similar results in a preliminary study by making the Frank-Kameneckij or infinite activation energy approximation. Here we confirm the occurrence of SBP's of types (2)23 and (3)23 in the more difficult case of Arrhenius kinetics; indeed kinetics "close to" Arrhenius kinetics. The unfolding and bifurcation geometry of these singularities is obtained. In particular, we find that bifurcation diagrams for the system of two stirred tanks can, for suitable parameter values in the "physical" range, exhibit the following phenomena: (a) local multiplicity 6 and global multiplicity at least 7, (b) triple limit points, (c) the occurrence of isolas when the bifurcation parameter is the Damköhler number and (d) more complex isolas with "dents" leading to double limit points. We compare our results with the numerical work of Kubicek et al. Items (b), (c), (d) appear to be new, although Kubicek et al. find isolas if the residence time is used as bifurcation parameter. The lower bound of 7 on the multiplicity was observed by Kubicek et al. in numerical plots; it is here verified rigorously.

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