12.5.3 Ignition-Extinction Curve
The points of intersection of R(T) and G(T) give us the temperature at which the reactor can operate at steady state. Suppose that we begin to feed our reactor at some relatively low temperature, T01. If we construct our G(T) and R(T) curves, illustrated by curves y and a, respectively, in Figure 12-11, we see that there will be only one point of intersection, point 1. From this point of intersection, one can find the steady-state temperature in the reactor, Ts1, by following a vertical line down to the T-axis and reading off the temperature, Ts1, as shown in Figure 12-11.
Figure 12-11. Finding multiple steady states with T0 varied.
If one were now to increase the entering temperature to T02, the G(T) curve, y, would remain unchanged, but the R(T) curve would move to the right, as shown by line b in Figure 12-11, and will now intersect the G(T) at point 2 and be tangent at point 3. Consequently, we see from Figure 12-11 that there are two steady-state temperatures, Ts2 and Ts3, that can be realized in the CSTR for an entering temperature T02. If the entering temperature is increased to T03, the R(T) curve, line c (Figure 12-12), intersects the G(T) curve three times and there are three steady-state temperatures, Ts4, Ts5, and Ts6. As we continue to increase T0, we finally reach line e, in which there are only two steady-state temperatures, Ts10 and Ts11. By further increasing we reach line f, corresponding to T06, in which we have only one reactor temperature that will satisfy both the mole and energy balances, Ts12. For the six entering temperatures, we can form Table 12-4, relating the entering temperature to the possible reactor operating temperatures.
Figure 12-12. Finding multiple steady states with T0 varied.
Table 12-4. Multiple Steady-State Temperatures
By plotting Ts as a function of T0, we obtain the well-known ignition-extinction curve shown in Figure 12-13. From this figure we see that as the entering temperature is increased, the steady-state temperature increases along the bottom line until T05 is reached. Any fraction of a degree increase in temperature beyond T05 and the steady-state reactor temperature will jump up to Ts11, as shown in Figure 12-13. The temperature at which this jump occurs is called the ignition temperature. That is, we must exceed a certain feed temperature, T05, to operate at the upper steady state where the temperature and conversion are higher.
Figure 12-13. Temperature ignition-extinction curve.
If a reactor were operating at Ts12 and we began to cool the entering temperature down from T06, the steady-state reactor temperature Ts3 would eventually be reached, corresponding to an entering temperature T02. Any slight decrease below T02 would drop the steady-state reactor temperature to the lower steady state value Ts2. Consequently, T02 is called the extinction temperature.
The middle points 5 and 8 in Figures 12-12 and 12-13 represent unstable steady-state temperatures. Consider the heat removal line d in Figure 12-12, along with the heat-generated curve, which is replotted in Figure 12-14. If we were operating at the middle steady state temperature Ts8, for example, and a pulse increase in reactor temperature occurred, we would find ourselves at the temperature shown by vertical line 
between points 8 and 9. We see that along this vertical line the heat-generated curve, y ≡ G(T), is greater than the heat-removed line d ≡ R(T) i.e., (G > R). Consequently, the temperature in the reactor would continue to increase until point 9 is reached at the upper steady state. On the other hand, if we had a pulse decrease in temperature from point 8, we would find ourselves on a vertical line 
between points 7 and 8. Here we see that the heat-removed curve d is greater than the heat-generated curve y, (R > G), so the temperature will continue to decrease until the lower steady state is reached. That is, a small change in temperature either above or below the middle steady-state temperature, Ts8, will cause the reactor temperature to move away from this middle steady state. Steady states that behave in this manner are said to be unstable.
Figure 12-14. Stability of multiple steady state temperatures.
In contrast to these unstable operating points, there are stable operating points. Consider what would happen if a reactor operating at Ts9 were subjected to a pulse increase in reactor temperature indicated by line 
in Figure 12-14. We see that the heat-removed line d is greater than the heat-generated curve y, (R > G), so that the reactor temperature will decrease and return to Ts9. On the other hand, if there is a sudden drop in temperature below Ts9, as indicated by line , we see the heat-generated curve y is greater than the heat-removed line d, (G > R), and the reactor temperature will increase and return to the upper steady state at Ts9. Consequently, Ts9 is a stable steady state.
Next let’s look at what happens when the lower steady-state temperature at Ts7 is subjected to pulse increase to the temperature shown as line in Figure 12-14. Here we again see that the heat removed, R, is greater than the heat generated, G, so that the reactor temperature will drop and return to Ts7. If there is a sudden decrease in temperature below Ts7 to the temperature indicated by line 
, we see that the heat generated is greater than the heat removed, (G > R), and that the reactor temperature will increase until it returns to Ts7. Consequently, Ts7 is a stable steady state. A similar analysis could be carried out for temperature Ts1, Ts2, Ts4, Ts6, Ts11, and Ts12, and one would find that reactor temperatures would always return to locally stable steady-state values, when subjected to both positive and negative fluctuations.
While these points are locally stable, they are not necessarily globally stable. That is, a perturbation in temperature or concentration, while small, may be sufficient to cause the reactor to fall from the upper steady state (corresponding to high conversion and temperature, such as point 9 in Figure 12-14) to the lower steady state (corresponding to low temperature and conversion, point 7).
12.6 Nonisothermal Multiple Chemical Reactions
Most reacting systems involve more than one reaction and do not operate isothermally. This section is one of the most important sections of the book. It ties together all the previous chapters to analyze multiple reactions that do not take place isothermally.