Nonisothermal CSTR

In the reactors studied so far, we have shown the effects of variable holdups, variable densities, and higher-order kinetics on the total and component continuity equations. Energy equations were not needed because we assumed isothermal operations. Let us now consider a system in which temperature can change with time. An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in Fig. 3.3. 

The reaction is nth-order in reactant A and has a heat of reaction X (Btu/lb mol of A reacted). Negligible heat losses and constant densities are assumed. 

To remove the heat of reaction, a cooling jacket surrounds the reactor. Cooling water is added to the jacket at a volumetric flow rate Fj and with an 

PERFECTLY MIXED QOOUNG JACKET. We assume that the temperature everywhere in the jacket is 7. The heat transfer between the process at tem-l5efature T and the cooling water at temperature 7 is described by an oyetall heat transfer eoeffieient. 

U = overall heat transfer eoeffieient An = heat transfer area 

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Phenomena of multiple steady states and instabilities occur particularly with nonisothermal CSTRs. Some isothermal processes with hyperbohc rate equations and processes with porous catalysts also can have such behavior. 

The dynamic behavior of nonisothermal CSTRs is extremely complex and has received considerable academic study. Systems exist that have only a metastable state and no stable steady states. Included in this class are some chemical oscillators that operate in a reproducible limit cycle about their metastable... 

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. 

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

Sometimes useful information and insight can be obtained about the dynamics of a system from just the steadystate equations of the system. Van Heerden Ind. Eng. Chem. Vol. 45, 1953, p. 1242) proposed the application of the following steadystate analysis to a continuous perfectly mixed chemical reactor. Consider a nonisothermal CSTR described by the two nonlinear ODEs... 

Example 9Ji. The nonisothermal CSTR modeled in Sec. 3.6 can be linearized (sec Prob. 6.12) to give two linear ODEs in teinis of perturbation variables. 

Show that the linearized nonisothermal CSTR of Example 9.6 can be stable only if... 

The actual stability requirement for the nonisothermal CSTR system is a little more complex than Eq. (11.43) because the concentration C does change. 

These are two coupled algebraic equations, which must be solved simultaneously to determine the solutions Cj(x) and T(t). For multiple reactions the + 1 equations are easily written down, as are the differential equations for the transient situation. However, for these situations the solutions are considerably more difficult to find We will in fact consider theaolutions of the transient CSTR equations in Chapter 6 to describe phase-plane trajectories and the stability of solutions in the nonisothermal CSTR. 

We stiU must use a PFTR in many chemical processes, and we must then determine how to program the cooling or heating to attain a temperature profile in the reactor close to that desired. The subject of this chapter is the proper temperature management to attain desired operation of a PFTR. In the next chapter the nonisothermal CSTR will be considered specifically. 

X(T) still has the same qualitative shape, and if the reaction is reversible and exothermic, X(T) decreases at sufficiently high temperature as the equihhrium X decreases. We can generalize therefore to say that multiple steady states may exist in ary exothermic reaction in a nonisothermal CSTR. 

Stability would occur if there are initial conditions in the reactor Cas T (subscript S for steady state) from which the system will evolve into each of these steady states. Also, if we start the reactor at exactly these steacfy states, the system will remain at that state for long times if they are stable. We will look at the first case later in connection with transients in the nonisothermal CSTR. Here we examine stability by asking if a steady state, once attained, will persist. 

We also studied the forced nonisothermal CSTR with the A —> B reaction [Uppal et al., 1974] ... 

Section 6.3 treats distributed nonreacting systems and specifically packed bed absorption, while Section 6.4 studies a battery of nonisothermal CSTRs and its dynamic behavior. 

Plugging this expression for y into (3.8) gives us the following equivalent nonadiabatic nonisothermal CSTR equation, which depends solely on xa. ... 

To gain further and broader insights into the bifurcation behavior of nonadiabatic, nonisothermal CSTR systems, we again use the level-set method for nonalgebraic surfaces such as z = /(K,., y). This particular surface is defined via equation (3.14) as follows for a given constant value of yc with the bifurcation parameter Kc ... 

Equation (3.17) allows a different interpretation of the underlying system s bifurcation behavior by taking Kc and yc as fixed and letting a vary, for example. We now study the bifurcation behavior of nonadiabatic and nonisothermal CSTR systems via their level-zero curves for the associated transcendental surface z = g(a,y). The surface is defined as before, except that here we treat Kc and yc as constants and vary a and y in the 3D surface equation... 

Nonadiabatic, nonisothermal CSTR bifurcation curve for a Figure 3.18... 

Thus far we have explored the bifurcation behavior of equation (3.14) with respect to Kc via equation (3.17) in Figures 3.14 through 3.16, and with respect to a via (3.19) in Figures 3.17 and 3.18. Since different Kc and a values can lead to bifurcation behavior for the same nonadiabatic, nonisothermal CSTR system, it is of interest and advantageous to be able to plot the joint bifurcation region for the parameters Kc and a as well. 

Nonadiabatic, nonisothermal CSTR joint bifurcation region for Kc and a... 

Nonadiabatic, nonisothermal CSTR system with disjoint multiplicity regions for Kc... 

Our final MATLAB m file in this section rounds out our efforts just as Figure 3.10 did for the adiabatic CSTR problem. It uses the plotting routine for Figure 3.18 in conjunction with a MATLAB interpolator to mark and evaluate the (multiple) steady state(s) graphically for nonadiabatic, nonisothermal CSTR problems. 

Similar behavior to that of the nonisothermal CSTR system will be observed in an isothermal bioreactor with nonmonotonic enzyme reaction, called a continuous stirred tank enzyme reactor (Enzyme CSTR). Figure 3.27 gives a diagram. 

For nonmonotonic kinetics the stability details of the steady states A, B, and C are indicated by arrows in Figure 3.29. The stability behavior of the different steady states is explained below using chemico-physical reasoning. This applies to the earlier-mentioned adiabatic and nonadiabatic nonisothermal CSTRs as well. 

Note that for a nonisothermal CSTR we will always obtain unique solutions (no bifurcation) for endothermic reactions, defined by [3 < 0, for both the adiabatic and the nonadiabatic cases. 

An industrial system consists of a nonisothermal CSTR (with a cooling jacket) and a tubular adiabatic reactor in series. The reaction is a first-order irreversible reaction ... 

The last section of this chapter describes a rather complex multi-stage process involving several nonisothermal CSTRs in series and partially solves it numerically, showing the wide range of dynamic possibilities associated with this unit, including... 

Fig. 3. The three steady states of the nonisothermal CSTR in Example 2.

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