Stability unstable CSTR

It is straightforward to show that the desired steady-state (i.e., the origin) is unstable. A robust tracking control law can be constructed to stabilize the CSTR under forced oscillatory operation. That is, we can derive a controller to track an oscillatory temperature profile (say, yr t) = a- - sin(47rt)), which can be generated by the exosystem (3) where... 

The rabbit and l5mx problem does have stable steady states. A stable steady state is insensitive to small perturbations in the system parameters. Specifically, small changes in the initial conditions, inlet concentrations, flow rates, and rate constants lead to small changes in the observed response. It is usually possible to stabilize a reactor by using a control system. Controlhng the input rate of lynx can stabilize the rabbit population. Section 14.1.2 considers the more realistic control problem of stabilizing a nonisothermal CSTR at an unstable steady state. 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

However, more precisely, the stability at the different steady sates can be determined by calculating the eigenvalues of the matrix of the linearized model of the CSTR. If there is an eigenvalue with positive real part, the steady state is unstable, and all eigenvalues with negative real part indicates a stable steady state. Thus, by simulation it can be verified that the steady states Pi and P3 are stable and P2 is unstable. This means that it is impossible to reach the point P2 when the coolant flow rate is constrained. 

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

If it is unstable, develop and use a MATLAB program for a nonadiabatic CSTR and find the cooling jacket parameters Kc and yc that will stabilize the unstable steady state. 

The first reactor in the 3-CSTR process has a conversion rate of 72.8%, and the reactant concentration in this first reactor is 2.18 kmol/m3. The reactor volume is low (14.3 m3), and the jacket heat transfer area is only 24.5 m2. The resulting jacket temperature (300 K) is almost down to the inlet cooling water temperature of 294 K. Linear analysis gives a Nyquist plot that never drops into the third quadrant, so the critical (—1,0) point cannot be encircled in a counterclockwise direction. This is required for closedloop stability because the openloop system is unstable and has a positive pole. Thus a proportional controller cannot stabilize this first reactor. 

Linearized or asymptotic stability analysis examines the stability of a steady state to small perturbations from that state. For example, when heat generation is greater than heat removal (as at points A— and B+ in Fig. 19-4), the temperature will rise until the next stable steady-state temperature is reached (for A— it is A, for B+ it is C). In contrast, when heat generation is less than heat removal (as at points A+ and B— in Fig. 19-4), the temperature will fall to the next-lower stable steady-state temperature (for A+ and B— it is A). A similar analysis can be done around steady-state C, and the result indicates that A and C are stable steady states since small perturbations from the vicinity of these return the system to the corresponding stable points. Point B is an unstable steady state, since a small perturbation moves the system away to either A or C, depending on the direction of the perturbation. Similarly, at conditions where a unique steady state exists, this steady state is always stable for the adiabatic CSTR. Hence, for the adiabatic CSTR considered in Fig. 19-4, the slope condition dQH/dT > dQG/dT is a necessary and sufficient condition for asymptotic stability of a steady state. In general (e.g., for an externally cooled CSTR), however, the slope condition is a necessary but not a sufficient condition for stability i.e., violation of this condition leads to asymptotic instability, but its satisfaction does not ensure asymptotic stability. For example, in select reactor systems even... 

Reactors do not always run at steady state. In fact, many pharmaceuticals are made in a batch mode. Such problems are easily solved using the same techniques presented above because the plug flow reactor equations are identical to the batch reactor equations. Even CSTRs can be run in a transient mode, and it may be necessary to model a time-dependent CSTR to study the stability of steady solutions. When there is more than one solution, one or more of them will be unstable. Thus, this section considers a time-dependent CSTR as described by Eq. (8.51) ... 

The steady states which are unstable using the static analysis discussed above are always unstable. However, steady states that are stable from a static point of view may prove to be unstable when the full dynamic analysis is performed. That is to say simply that branch 2 in Figure 4.8 is always unstable, while branches 1,3 in Figure 4.8 and branch 4 in Figure 4.8 can be stable or unstable depending upon the dynamic stability analysis of the system. As mentioned earlier, the analysis for the CSTR presented here is mathematically equivalent to that of a catalyst pellet using lumped parameter models or a distributed parameter model made discrete by a technique such as the orthogonal collocation technique. However, in the latter case, the system dimensionality will increase considerably, with n dimensions for each state variable, where n is the number of internal collocation points. 

For the CSTR system discussed in Example 1.2 (Figure 1.7), the control objective (qualitatively defined) is to ensure the stability of the middle, unstable steady state. But such a qualitative description of the control objectives is not useful for the design of a control system and must be quantified. A quantitative translation of the qualitative control objective requires that the temperature (an output variable) not deviate more than 5% from its nominal value at the unstable steady state. 

In Section 1.2 we introduced a simpleminded notion of stability. A system was considered unstable if, after it had been disturbed by an input change, its output took off and did not return to the initial state of rest. Figure 1.6 shows typical outputs for unstable processes. Example 1.2 also described the unstable operation of a CSTR. 

Before performing a controllability analysis, ensure the stability of the plant. The first step is to close all inventory control loops, by means of level and pressure controllers. Then, check the stability, by dynamic simulation. If the plant is unstable, it will drift away from the nominal operating point. Eventually, the dynamic simulator will report variables exceeding bounds, or will fail due to numerical errors. Try to Identify the reasons and add stabilizing control loops. Often a simple explanation can be found in uncontrolled inventories. In other situations the origin is subtler. Some units are inherently unstable, as with CSTR s or the heat-integrated reactors. The special case when the instability has a plantwide origin will be discussed in Chapter 13. 

Figure 6.1 illustrates the fact that for various ranges of kinetic and reactor parameters it is possible for the mass and energy conservation relations for a CSTR to be in stable balance at more than one condition. This may imply that there are other balance conditions that are unstable the point needs to be examined. Which of the stable balances is attained in actual operation may be dependent on the details of startup procedure, for example, which are not subject to the control of the designer. Thus, it is important to investigate reactor stability using unsteady-state rather than steady-state models. 

On an operating diagram oiy — T, equation (6-89) is given by a straight line of slope 7, as seen in Figure 6.13. We observe from this figure that it is again possible to have three steady states, at b, bj, and 3, and the relative stability of these is as found for the three states in a nonunique CSTR thus, bj is unstable to small perturbations. 

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). 

Global StaMlity in the CSTR.— The failure of linear stability analysis to cover the macroscopic behaviour of the CSTR is well illustrated by the oscillatory states computed by Aris and Amundson for such a reactor operating with feedback control. Local stability analysis indicates an unstable equilibrium state but in the large this is surrounded by a stable limit cycle and the resultant behaviour is one of temperatures and concentrations oscillating about an unstable state, rather than approaching a stable one. 

The existence of oscillatory behaviour for the recycled CSTR was demonstrated by Aris and Amundson." Undamped oscillations in the fonn of limit cycles occur under particular parameter values for the system and Aris and Amundson recogniied that these limit cycles should originate at the critical value when the stable steady state becomes an unstable one. Such an instance is called a bifurcation point. They developed a criterion for the direction of bifurcation and gave plausible arguments for the stability of the limit cycle. 

It is seen that (Figure 3.35) the slope of the Qc versus T plot (dQc/dTj) is greater than the slope of the Qp. versus T line (dQj /dTf) at unstable state and is less than dQ /dT at stable states. Thus, the condition for stability of a non-isothermal CSTR is... 

A more rigorous analysis would show that the first inequality is a sufficient criterion for instability. If this inequality is satisfied, the operating point will be intrinsically unstable. The rigorous analysis also shows that the second criterion is a sufficient criterion for stability, provided that the CSTR is adiabatic. If the reactor is not adiabatic, the second condition is necessary, but not sufficient. 

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