In this module we have only been able to cover only a small part
of the interesting behavior that can occur in CSTRís, due
to nonlinear dynamics. In this section we provide a concise overview
of other interesting topics in CSTR analysis. First (section 8.1)
we will draw a connection between our physical-based approach
(heat removal and generation curves) for analyzing multiplicity
behavior, and the more mathematical bifurcation theory. Then (section
8.2) we will discuss the effect of cooling jacket flowrate on
the behavior of a CSTR. Finally (section 8.3) we discuss limit
cycle behavior and Hopf bifurcations.

**8.1 Connecting Bifucation Theory with Physical Reasoning
**

In chapter 15 we presented mathematical conditions for bifurcation
behavior, while we have focused on physical arguments in this
module. The goal of this section is to draw a connection between
the mathematical conditions and the physical conditions for multiple
steady-state behavior.

First, recall the mathematical conditions for a bifurcation. Let
x represent the state variable, m a
bifurcation parameter and g(x,m) a
single nonlinear algebraic equation which must be satisfied at
steady-state. If the following bifurcation conditions are satisfied

and

then there exists k steady-state solutions in the vicinity of
the bifurcation point.

In our case, we were able to write a single nonlinear algebraic
equation that must be satisfied for a steady-state solution

For simplicity, we write this equation as

g(Ts,m)= Qrem(Ts,m)-Qgen(Ts,m)
= 0

where Ts is the state variable (steady-state reactor temperature)
and m is used to represent the vector
of physical parameters that can possibly be varied.

The steady-state solution is obtained by solving

g(Ts,m)=0

or

Qrem(Ts,m) = Qgen(Ts,m)

which is simply the intersection of the heat removal and heat
generation curves. The first derivative condition

is simply

These conditions are satisfied by curves B and D in Figure 12
below. The slope conditions are satisfied at a high temperature
for curve B and a low temperature for curve D, and we directly
see how a ìlimit pointî on the input-output curve
(Figure 12b) corresponds to the heat generation and removal curves
(Figure 12a). We also note that the following condition also holds

since, although the second derivative of the heat removal line
is always zero

the heat generation curve is not at an inflection point

and we see that the conditions for saddle-node behavior are satisfied.

Now we illustrate the conditions for a hysteresis bifurcation,
which are

All of these conditions are satisfied in Figure 13b below. We show Figures 13a and 13c to illustrate the progression to satisfying the bifurcation conditions.

The progression shown in Figure 13 involved a decrease in the heat transfer capability from a to c. This is also shown in the cusp diagram of Figure 14, where there is no hysteresis at high UA values, the onset of hysteresis (bifurcation point) at an intermediate value of UA, and complete hysteresis behavior at low values of UA.