8. Further Complexities

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.

a. Heat Removed and Generatedb. Input/Ouput

Figure 12. Illustration of Saddle-node (limit point) Bifurcation.

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.

a.

b.

c.

Figure 13. Illustration of a Hysteresis Bifurcation. Bifurcation conditions are satisfied in b.

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.

Figure 14. Cusp Diagram.