Singularity theory for non-isothermal CSTR

We now turn to the non-isothermal reaction system in a non-adiabatic CSTR, as studied in 7.2.4—6. We begin with the simplified model with exponential approximation to the Arrhenius law, and to systems for which the inflow and ambient temperatures are the same (y = 0 and gc = 0), This system has two unfolding parameters gad and rN. The stationary-state equation and its various derivatives are 

Although these seem a formidable array of equations, they yield remarkably to elimination and simplification when various combinations are required to be zero simultaneously. 

For the hysteresis limit we require, as usual, that F = Fx = Fxx = 0. Two equalities give x and ires. The third then leads to eqn (7.34) relating gad to rN. The isola condition F = Fx = Ft = 0 is best handled parametrically, as x cannot be eliminated so readily (it is given by the solution of a cubic equation in terms of gad). The parametric forms have been given as eqns (7.35) and (7.36). 

The highest-order singularity in this system cannot satisfy the winged cusp condition F = Fx = Fxx = Fz = Fxz = 0. For one thing the system of equations only has four quantities x, ires, gad, and tn, whereas the winged cusp 

With the full Arrhenius rate law, an extra unfolding parameter y is introduced. Even then, however, the appropriate stationary-state condition and its derivatives for the winged cusp cannot be satisfied simultaneously (at least not for positive values of the various parameters). Thus we do not expect to find all seven patterns. 

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