Stability of a CSTR

THERM and THERM PLOT - Thermal Stability of a CSTR System... 

FIGURE 14.4 Stabilization of a nonisothermal CSTR near a metastable steady state. 

The stability of a first-order exothermic reaction A—>B, in a single CSTR with jacket cooling has been studied by Seborg (1971), and the usefulness of simulations for this type of investigation has been emphasised by Luus (1972). The influence of sinusoidal, feed-temperature variations is corrected by simple... 

It is well known, and it has been seen in the previous example, that to scale-up the size of a CSTR affects the reactor stability, because the ratio of heat transfer area to reactor volume decreases as far as the size of the reactor is increased (they are proportional to the square and power of three, respectively). 

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. 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

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 local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

Independently of enzyme stability an enzyme reaction at constant temperature and pH should be run with the smallest possible contribution by inhibition. For this reason, a CSTR is most suitable in the case of substrate inhibition because the substrate concentration is evened out across the reactor volume and thus minimized. In the case of product inhibition, however, a batch reactor or a PFR is preferred, as the reactor volume required for complete or nearly complete conversion is much smaller than in the case of a 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... 

Startup of a CSTR (Figure S-1) and the approach to the steady state (CD-ROM). By mapping out regions of the concentration-temperature phase plane, one can view the approach to steady state and learn if the practical stability limit is exceeded. 

Nonlinear equations may admit no real solntions or mnltiple real solutions. For example, the quadratic equation can have no real solutions or two real solutions. Thus, it is important to know whether a given equation governing the behavior of an engineering system can admit more than one solution, since it is related to the issue of operation and performance of the system. In this subsection, criteria for the existence of multiple steady-state solutions to the governing equations of a CSTR and tubular reactors and, subsequently, the stability of these multiple steady states are presented. 

Startup of a CSTR In reactor startup it is often very important /ttm lenriperature 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 unacceptable for safe operation. If cither case were to occur, we would say that the system exceeded its practical stability limit. Although we can. solve the unsteady 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. 

Figure 1.9 Dynamic response of a CSTR (a) and (b) indicate the instability of the middle steady state, while (c) and (b) demonstrate the stability of the other two.

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. 

D. S. Sabo and J. S. Dranoff [AIChE J., 16, 211 (1970)] considered the question of the stability of a single CSTR in which the following consecutive first-order reactions occur 1 h... 

Professor Viejo Dinosaurio wishes to utilize this reaction as the basis for a laboratory demonstration that would illustrate both autothermal operation of a CSTR and reactor stability concepts. He plans to use a well-insulated reactor with an effective liquid volume of 1.0 L. The feed is to consist of a mixture of the hydroperoxide (HP), acetone (A), and acid catalyst at 25°C. Initial concentrations of hydroperoxide and acetone are 0.5 and 1.0 mol/L, respectively. The following property values are available the standard heat of reaction at 25°C (estimated) = -60.5 kcal/g-mol and the heat capacity per liter of fluid (estimated) = 406 cal/(L-°C). This value may be taken as independent of the fraction conversion. 

The continuous loop reactor is likely to be the only tubular reactor used on commercial production of emulsion polymers [71], although its use is limited to production of vinyl acetate homopolymers and copolymers (with ethylene and VeovalO) [77-79]. A continuous-loop reactor consists of a tubular loop that connects the inlet and the outlet of a recycle pump. These reactors combine the heat transfer characteristics of a tubular reactor with the RTD of a CSTR. The main drawback of this reactor is that the requirements for the mechanical stability of the latex are stringent because the recycling pump may induce shear coagulation. 

Recycling cells from a CSTR effluent provides a means to continually inoculate the vessel and to add stability to the reactor, minimizing the effect of process perturbation. The productivity of a CSTR may be increased remarkably by recycling cells when the cells to be recycled are first concentrated by the factor P = XJX in a sedimentation unit. With a recycling stream and with r = FJF, the conservation of mass equations are... 

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... 

We wish to determine the local stability of the CSTR system of section 4.3. To do this, we first linearize the nonlinear problem about the steady-state condition. You might want to use the symbolic toolbox of MATLAB to help in this linearization. This converts the problem to a linear set of differential equations... 

Figure 4.10.28 Static stability of a cooled CSTR forthe example of styrene polymerization.

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