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.
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.