6. Phase-plane Analysis

In section 4 we provided the results of a few dynamic simulations, noting that different initial conditions caused the system to converge to different steady-state operating points. In this section we construct a phase-plane plot by performing simulations for a large number of initial conditions.

The phase-plane plot shown in Figure 6 was generated using cstr_run.m and cstr.m from the appendix. Three steady-state values are clearly shown; 2 are stable (the high and low temperature steady-states, shown as ëoí), while one is unstable (the intermediate temperature steady-state, shown as ë+í). Notice that initial conditions of low concentration (0.5 kgmol/m3) and relatively low-to-intermediate temperatures (300 to 365 K) all converge to the low temperature steady-state. When the initial temperature is increased above 365 K, convergence to the high temperature steady-state is achieved.

Now, consider initial conditions with a high concentration (9.5 kgmol/m3) and low temperature (300 to 325 K); these converge to the low temperature steady-state. Once the initial temperature is increased to above 325 K, convergence to the high temperature steady-state is achieved. Also notice that, once the initial temperature is increased to around 340 K, a very high overshoot to above 425 K occurs, before the system settles down to the high temperature steady-state. Although not shown on this phase-plane plot, higher initial temperatures can have overshoot to over 500 K before settling to the high temperature steady-state. This could cause potential safety problems if, for example, secondary decomposition reactions occur at high temperatures. The phase plane analysis then, is able to ìpoint-outî problem initial conditions.

Also notice that no initial conditions have converged to the intermediate temperature steady-state, since it is unstable. The reader should perform an eigenvalue/eigenvector analysis for the A matrix at each steady-state (low, intermediate and high temperature) (see exercise 3). You will find that the low, intermediate and high temperature steady-states have stable node, saddle point (unstable) and stable focus behavior (see chapter 13), respectively.

Figure 5. Phase-plane plot for case 2. Stable points (conditions 1 and 3) are marked with ëoí. The unstable point (condition 2) is marked with ë+í.

It should be noted that feedback control can be used to operate at the unstable intermediate temperature steady-state. The feedback controller would measure the reactor temperature and manipulate the cooling jacket temperature (or flowrate) to maintain the intermediate temperature steady-state. Also, a feedback controller could be used to make certain that the large overshoot to high temperatures does not occur from certain initial conditions.