Stability of Operation and Transient Behavior

Polymerization in Perfectly Mixed Flow Reactors Stability of Operation and Transient Behavior... 

Quality windows are also delineated by set values of the damping coefficients and attenuation factors that are computed in stability and frequency analysis. These too can be traced out efficiently in parameter space by augmented continuation schemes. The same is true of the turning points and bifurcation points in parameter space, points of marginal stability. These are the guides to situations in which there is more than one stable operating state. When such situations may arise, it becomes desirable to solve repeatedly the full equation system of flow for transient behavior in order to know how different start-up procedures and upsets select among the multiple stable states. 

Stability analysis could prove to be useful for the identification of stable and unstable steady-state solutions. Obviously, the system will gravitate toward a stable steady-state operating point if there is a choice between stable and unstable steady states. If both steady-state solutions are stable, the actual path followed by the double-pipe reactor depends on the transient response prior to the achievement of steady state. Hill (1977, p. 509) and Churchill (1979a, p. 479 1979b, p. 915 1984 1985) describe multiple steady-state behavior in nonisothermal plug-flow tubular reactors. Hence, the classic phenomenon of multiple stationary (steady) states in perfect backmix CSTRs should be extended to differential reactors (i.e., PFRs). 

The transient behavior of the reactor for finite perturbations was studied by computer simulation of Eqs (1.1)-(1.4). The curves 1 and 2 in the Fig. 6 demonstrate the transition from the lower stable state to stabilized intermediate one after beginning of the proportional control operation at C = 0. The curve 1 corresponds to the case of ideal control (f =0). The curve 2 corresponds to nonideal control with parameters d and Cj corresponding to the stability domain in Fig, 5 One can see that the curve 1 is monotonous and the curve 2 is not monotonous. In the case of nonideal control the damping oscillations of concentration and temperature near the intermediate steady values take place with the amplitude being dependent on the time lag value. The curve 3 corresponds to the case when the point (, d) belongs to the... 

By computer similation the transient behavior on nonlinear system is studied. It is shown that for zero time lag the stabilized intermediate steady state is achieved without oscillations. Under imperfect control operation damping oscillations of concentration profile and temperature near intermediate state can exist the amplitude being dependent on the time-lag value. For certain values of flow-control parameters sustained oscillations of concentration profile and temperature in the reactor are possible. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

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