Singularity theory

The CSTR is, in many ways, the easier to set up and operate, and to analyse theoretically. Figure 6.1 shows a typical CSTR, appropriate for solution-phase reactions. In the next three chapters we will look at the wide range of behaviour which chemical systems can show when operated in this type of reactor. In this chapter we concentrate on stationary-state aspects of isothermal autocatalytic reactions similar to those introduced in chapter 2. In chapter 7, we turn to non-isothermal systems similar to the model of chapter 4. There we also draw on a mathematical technique known as singularity theory to explain the many similarities (and some differences) between chemical autocatalysis and thermal feedback. Non-stationary aspects such as oscillations appear in chapter 8. 

REACTION IN A NON-ISOTHERMAL CSTR STATIONARY STATES AND SINGULARITY THEORY... 

Chapter 6 considered isothermal autocatalysis in an open system here we study a classic case of thermal feedback. A rich variety of stationary-state patterns (bifurcation diagrams) are generated and considered here alongside those of the previous isothermal example. Flow diagrams are again illuminating and singularity theory provides a systematic approach. After study a reader should be able to ... 

Singularity theory approach to stationary-state loci... 

Singularity theory for cubic autocatalysis with uncatalysed reaction... 

Golubitsky, M. and Schaeffer, D. (1979). A theory for imperfect bifurcation via singularity theory. Comm Pure Appl. Math., 32, 21-98. 

The equivalence between a vanishing eigenvalue and a turning point in the stationary-state locus can be made firmer using the ideas of the singularity theory introduced in the previous chapter. The stationary-state condition, in general terms, for eqn (8.14) is... 

For larger attractive interactions, however, the stationary-state locus becomes S-shaped with a hysteresis loop (Fig. 12.2(b)). The critical value of a at which the hysteresis loop first appears can be determined using the singularity theory equations of 7.3.1. The appropriate form of the stationary-state condition F is... 

The condition for the appearance of a hysteresis loop is again most easily obtained using the singularity theory recipe, from which we find the relationship... 

Singularity theory can be used again to find the conditions at which a hysteresis loop first appears in the 0p — (p/K) locus. Setting F = F0r = Fg g = 0, some algebra yields... 

Using the singularity theory approach, we find that the stationary-state locus will display a hysteresis loop provided that... 

The literature on this model reaction is already vast and a complete bibliography would be of great use to the mathematical modeler. Of particular interest are A. d Anna, P. G. Lignola, and S. K. Scott. The application of singularity theory to isothermal autocatalytic open systems The elementary scheme A + mB = (m + 1) B. Proc. Roy. Soc. Lond. A 403, 341-363 (1986) and S. R. Kay, S. K. Scott, and P. G. Lignola. The application of singularity theory to isothermal autocatalytic open systems The influence of uncatalyzed reactions. Proc. Roy. Soc. Lond A 409, 433-448 (1987). 

Abstract—The question of the multiplicity of the steady states of a chemical reactor was one of the concerns in the pioneering work ofBilous and Amundson. Their diagrams showed quite clearly the geometry of the situation, and this kind of analysis sufficed for many years. It remained for Balakotaiah and Luss, using the methods of singularity theory, to give a comprehensive treatment of the question. After a brief survey, we take up the case of consecutive first-order reactions and show that up to seven steady states are possible. 

In this section we will develop the equations needed to apply singularity theory to the problem of two sequential reactions in a CSTR. The well-known equations for cA, cB and T, the concentrations of A, B and the temperature in a reactor of volume V through which a homogeneous stream of rate q flows, are ... 

We now begin the singularity theory analysis of eq. (18). This equation may be regarded for our purposes as the function... 

This is the equation to which we will apply singularity theory (or at least its mechanics), though when certain delicate mathematical questions encountered in the next section arise we will be forced to return to eq. (18). 

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