Unstable operating point

In cases of multiplicity (that is more than one solution to Equation 4.10), of the three intersection points only those at the lowest and highest pellet temperature represent stable conditions. These are therefore called the stable operating points. The intermediate intersection point is referred to as an unstable operating point. 

Figure 4.4 Relative heat production and heat withdrawal rates versus dimensionless pellet temperature 0p A) exothermic reactions, 0p, 0t and An > 0 B) endothermic reactions, 0p, 0t and An < 0 ( ) stable operating point (o) unstable operating point.

Fig. 4. PolynKTizalion me curve showing stable and unstable operating points for a semibalch emulsion potymerizatioQ-...

In contrast to these unstable operating points, consider what would happen to the reactor temperature if a reactor operating at Tji2 were subjected to very small temperature fluctuations. From Figure 8-24 we observe that a pulse increase in reactor temperature would make the heat of removal greater than the heat of generation [the i [J) curve would be above the G(T) curve] and the temperature would drop back down to. If a small pulse decrease in the reactor... 

Figure 5.11 Stable and unstable operating points for an exothermic reaction In a...

This elaboration makes it obvious that the heat removal line represents a very special case. It describes the limiting case of the first occurrence of an unstable operating point. A mathematical description of this limiting case, which contains all relevant operating and reaction parameters represents the safety technical design limit of this model case. The tangential point of heat removal and heat production ftmction may be described as follovre ... 

Fig. 4-16. Cooled CSTR with statically unstable operating points...

Fig. 4-23. Cooled CSTR with statically stable but dynamically unstable operating points Parameters E/R = 10500 K, In Da., = 32.5, ATad = 135 K, St = 1.5...

Determination of stable and unstable operating points can be found by computing the eigenvalues of the system Strogatz (2001). 

Hidalgo PM, BrosUow CB. Non-linear model predictive control of styrene polymerization at unstable operating points. Comp Chem Eng 1990 14 481 94. 

Both Eqn. (12.48) and (12.49) can be shown in the C-T plane, a stable operating point is obtained when the produced heat is equal to the heat that is removed by the cooling coil (points A and B). This is shown in Fig. 12.11. Point C is an unstable operating point, when a disturbance acts upon the system, the reactor will either move to operating point A or B, depending on the direction of the disturbance. 

One possibility of maintaining the reactor temperature is by adjusting the coolant flow through the jacket. In this case the measuring device could be a thermocouple, protected by a thermowell. As ejq)lained in chapter 12, there are three possible operating points of which oidy two are stable. Operating point P2 is an unstable operating point, the reactor can only be kept in this point by a temperature controller. 

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