Stability analysis, CSTR

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... 

They treated the system much like a CSTR, with the balance for the gas-phase concentration substituted by the coverage equation for the catalyst. Ray and Hastings then applied the analytical treatment that they had developed for the CSTR in this same publication. Stability analysis revealed that the critical Lewis numbers for oscillations were in a range that did not allow for oscillations on normal nonporous catalytic surfaces. However, as Jensen and Ray 243) showed, a certain model for catalytic surfaces, the fuzzy wire model, with the assumption of a very rough surface with protrusions is able to produce Lewis numbers in the proper range for the occurrence of oscillations. This model, however, included both mass and heat balances as well as coverage equations, thus combining the two classes of reactor-reaction models discussed above. 

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. 

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). 

Multiple Steady States and Local Stability in CSTR.—In the two decades since the seminal work of van Heerden and Amimdson, there has been vast output of papers conoemed with the dynamic behaviour of stirred-tank reactors. Bilous and Amundson put the van He den analysis of local stability of the equilibrium state on a rigorous basis by use of linear stability theory. Their method is similar to the phase-plane treatments of thermokinetic ignitions and oscillations discussed here in Sections 4 and 3 (and preceded them dironologically). The mass and energy balance for the CSTR having a single reactant as feedstock may be expressed as ... 

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. 

We conclude with an interesting question on the operation of our CSTR. Recall that we wish to react as much CO as possible. When we start our reactor, [CO]om is equal to [CO]in. The reaction starts and [CO] decreases, but only to the third operating point. Whenever [CO] decreases below the third steady-state point, our stability analysis predicts that [CO] will return to the third steady-state point. How does one reach the desirable operating point at the lower [CO] ... 

This analysis is limited, since it is based on a steady-state criterion. The linearisation approach, outlined above, also fails in that its analysis is restricted to variations, which are very close to the steady state. While this provides excellent information on the dynamic stability, it cannot predict the actual trajectory of the reaction, once this departs from the near steady state. A full dynamic analysis is, therefore, best considered in terms of the full dynamic model equations and this is easily effected, using digital simulation. The above case of the single CSTR, with a single exothermic reaction, is covered by the simulation examples, THERMPLOT and THERM. Other simulation examples, covering aspects of stirred-tank reactor stability are COOL, OSCIL, REFRIG and STABIL. 

Prom the results presented in this chapter, it has been shown that the first step in the control problem of a CSTR should be the use of an appropriate mathematical model of the reactor. The analysis of the stability condition at the steady states is a previous consideration to obtain a linearised model for control purposes. The analysis of a CSTR linear model is carried out trough a scaling up reactor s volume in order to investigate the difference between the reactor and jacket equilibrium temperatures as the volume of the reactor changes from small to high value. 

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. 

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. 

Parametric Sensitivity and Dynamics The global stability and sensitivity to abrupt changes in parameters cannot be determined from an asymptotic analysis. For instance, for the simple CSTR, a key question is whether the temperature can run away from a lower stable... 

In reactor startup it is often very important /tow temperature and concentrations approach their steady-state values. For example, a significant overshoot in temperature may cause a reactant or product to degrade, or the overshoot may be imacceptable for safe operation. If either case were to occur, we would say that the system exceeded its practical stability limit. Although we can solve the imsteady temperature-time and concentration-time equations numerically to see if such a limit is exceeded, it is often more insightful to study the approach to steady state by using the temperature-concentration phase plane. To illustrate these concepts we shall confine our analysis to a liquid-phase reaction carried out in 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. 

The left side of Eq. (9.43) represents the heat generation due to reaction. The right side represents heat removal due to sensible heat and the heat transfer to the jacket. Thus, our simple linear analysis tells us that the heat removal capacity must be greater than the heat generation if the system is to be stable. The actual stability requirement for the nonisothermal CSTR system is a little more complex than Eq. (9.43) because the concentration Ca does change. 

The problem of existence and stability of limit cycles has been attadred in many ways. Luus and Lapidus have used an averaging technique, Douglas et employ a perturbation analysis, and Gilles and Hyon and Aris use Fourier transform methods to examine the occurrence of limit cycles, all with moderate successes. More recently Uppal et a/. have discussed the dynamic behaviour of the CSTR in terms of the bifurcation of the steady state, devdoping critmia for the existence and stability of oscillatory states as a function of system parameters, and... 

Tubular Reactok. —Perhaps the most significant distinction between the CSTR and tubular reactors is one of dimensionality. The CSTR is represented by two (or more) ordinary differential equations corresponding to the energy and mass balances in the reactor. The methods of non-linear analysis have been successfully applied in determining the multiplicity and stability of the reactor s operating states. 

IjekI, M., O.A. Asbjomsen, and H.J. Astrom, Reaction Invariants dflieir Importance in the Analysis of Eigenvectors, Stability and Con-lolliMity of CSTRs, Chem. Eng. Sci., 30,1917 (1974). 

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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