Analysis of Multiplicity by Singularity Theory

Techniques based on the implicit function theorem have been used to predict the existence of multiple solutions in a CSTR (Chang and Calo, 1979). An extension of catastrophe theory known as singularity theory has also been effectively used to determine the conditions for the existence of multiple solutions in a CSTR and a tubular reactor (Balakotaiah and Luss, 1981, 1982 Witmer et al., 1986). In this subsection, the technique of singularity to find the maximum number of solutions of a single mathematical equation and its application to analysis of the multiplicity of a CSTR are presented (Luss, 1986 Balakotaiah at al., 1985). The details of singularity theory can be found in Golubitsky and Schaeffer (1985). 

For the first-order reaction, the steady-state equations for mass and energy balance in a CSTR can be combined into a single equation represented as 

The application of catastrophe theory (Hofp, 1960) predicts that in the neighborhood of a singular point of codimension k, the qualitative features represented by Equation (2.284) are similar to that of the polynomial. 

In the neighborhood of such a singular point. Equation (2.284) admits k+ solutions if 

the singular points of Equation (2.284) characterized by Equation (2.285) allow one to analyze the multiplicity of Equation (2.284) from the knowledge of the polynomial Equation 

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