Multiplicity and Open-Loop Instability

The idea that a unit operation could have two or more steady states for the same values of the input variables is not only confusing in practice but somewhat hard to understand conceptually. We will try to explain the situation, first in words and then graphically. The verbal explanation of multiplicity centers around two of the necessary conditions nonlinearity and process feedback. 

The need for nonlinearity is easy to see. A linear equation has no more than a single solution. A quadratic equation may have two solutions, etc. The describing equations for a reactor must therefore be nonlinear to show output multiplicity. 

We next turn to process feedback. W e mentioned earlier that a plug-flow reactor can be viewed as a string of small batch reactors. We also pointed out that the result of each batch is uniquely determined by the fresh feeds since the solution to the batch equations is a forward integration in time. A plug-flow7 reactor cannot by itself show output multiplicity or open-loop instability. This picture changes when we 

The graphic illustration of output multiplicity focuses on the steady-state solutions of Eq. (4.17). This equation can be viewed as a tradeoff between a nonlinear heat generation term, Q iT), and a linear heat removal expression  

In Fig. 4.6 we have plotted a typical heat generation expression (curve a) along with the heat removal line, b. In this case the two curves intersect at three locations corresponding to three different reactor conditions that are possible for the same operating parameters and feed conditions. The low-temperature steady state is uneconomical since the feeds are virtually unconverted. The highest-temperature steady state has nearly complete conversion but may be too hot. Under those conditions side reactions may set in or the reactor pressure becomes too high. The middle steady state strikes a good compromise and is where 

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We noted earlier in this chapter that many reactions in the chemical industries are exothermic and require heat removal. A simple way of meeting this objective is to design an adiabatic reactor. The reaction heat is then automatically exported with the hot exit stream. No control system is required, making this a preferred way of designing the process. However, adiabatic operation may not always be feasible. In plug-flow systems the exit temperature may be too hot due to a minimum inlet temperature and the adiabatic temperature rise. Systems with baekmixing suffer from other problems in that they face the awkward possibilities of multiplicity and open-loop instability. The net result is that we need external cooling on many industrial reactors. This also carries with it a control system to ensure that the correct amount of heat is removed at all times. 

Inverse response creates control difficulties. Assume, for example, that we wish to control the exit temperature of an adiabatic plug-flow reactor by manipulating the inlet temperature as shown in Fig. 4.13. From a steady-state viewpoint this is a perfectly reasonable thing to consider, since there are no issues of output multiplicity or open-loop instability, assuming the fluid is in perfect plug flow and there is no... 

Chemical reactors are inherently nonlinear in character. This is primarily due to the exponential relationship between reaction rate and temperature but can also stem from nonlinear rate expressions such as Eqs. (4.10) and (4.11). One implication of this nonlinearity for control is the change in process gain with operating conditions. A control loop tuned for one set of conditions can easily go unstable at another operating point. Related to this phenomenon is the possibility of open-loop instability and multiple steady states that can exist when there is material and/or thermal recycle in the reactor. It is essential for the control engineer to understand the implications of nonlinearities and what can be done about them from a control standpoint as well as from a process design standpoint. 

In Chap. 4 we mentioned that the simplest reactor type from a control viewpoint is the adiabatic plug-flow reactor. It does not suffer from output multiplicity, open-loop instability, or hot-spot sensitivity. Furthermore, it is dominated by the inlet temperature that is easy to control for an isolated unit. The only major issue with this reactor type is the risk of achieving high exit temperatures due to a large adiabatic temperature rise. As we recall from Chap. 4, the adiabatic temperature rise is proportional to the inlet concentration of the reactants and inversely proportional to the heat capacity of the feed stream. WTe can therefore limit the temperature rise by diluting the reactants with a heat carrier. 

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