Nonisothermal reactors CSTRs

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

The question of choosing a PFTR or a CSTR will occur throughout this book. From the preceding arguments it is clear that the PFTR usuaUy requires a smaller reactor volume for a given conversion, but even here the CSTR may be preferred because it may have lower material cost (pipe is more expensive than a pot). We will later see other situations where a CSTR is clearly preferred, for example, in some situations to maximize reaction selectivity, in most nonisothermal reactors, and in polymerization processes where plugging a tube with overpolymerized solid polymer could be disastrous. 

Thus we see that for nonisothermal reactors this 1/r versus Cao Ca curve is not always an increasing function of conversion as it was for isothermal reactors even with positive-order kinetics. Since the 1/r curve can have a rninimum for the nonisothermal reactor, we confirm the possibility that the CSTR requires a smaller volume than the PFTR for positive-order kinetics. This is hue even before the multiple steady-state possibilities are accounted for, which we will discuss in the next chapter. This is evident from our 1 /r plot for the PFTR and CSTR and will occur whenever r has a sufficiently large maximum that the area under the rectangle is less than the area under the curve of 1/r versus Cao Ca-... 

In the previous chapter we showed how nonisothermal reactors can exhibit much more complex behavior than isothermal reactors. This occurs basically because k(T) is strongly temperature dependent Only a single steacfy state is possible in the PFTR, but the CSTR, although (or because) it is described by algebraic equations, can exhibit even more interesting (and potentially even more dangerous) behavior. 

We have already dealt with nonisothermal reactors and their effects in Chapters 5 and 6. The major and potentially most dangerous application of these effects is in combustion reactors. Consider our standard CSTR with parameters as indicated in Figure 10-10. 

One of the simplest practical examples is the homogeneous nonisothermal and adiabatic continuous stirred tank reactor (CSTR), whose steady state is described by nonlinear transcendental equations and whose unsteady state is described by nonlinear ordinary differential equations. 

Example 2—Unstable CSTR with bounded output. Consider the reaction R P occurring in a nonisothermal jacket-cooled continuous stirred tank reactor (CSTR) with three steady states. A, B, C, corresponding to the intersection points of the two lines shown in Fig. 3 (Stephanopoulos,... 

Up to now we have focused on the steady-state operation of nonisothermal reactors. In this section the unsteady-state energy balance wtU be developed and then applied to CSTRs, plug-flow reactors, and well-mixed batch and semibateh reactors. 

Steady-State Nonisothermal Reactor Design Chap. 8 Table 8-1. CSTR Algorithm... 

The influence of activity changes on the dynamic behavior of nonisothermal pseudohomogeneoiis CSTR and axial dispersion tubular reactor (ADTR) with first order catalytic reaction and reversible deactivation due to adsorption and desorption of a poison or inert compound is considered. The mathematical models of these systems are described by systems of differential equations with a small time parameter. Thereforej the singular perturbation methods is used to study several features of their behavior. Its limitations are discussed and other, more general methods are developed. 

Example 14. /. Multiple steady states and hysteresis in a nonisothermal continuous stirred-tank reactor (CSTR) [1,2]. In a CSTR, the curve for the temperature dependence of heat loss to the cooling coil is linear (loss proportional to temperature difference) while that for heat generation by the reaction is S-shaped (Arrhenius ex-... 

The difficulty in this is the awkward form of equation (6-140) with respect to 7). One likes to compute in sequence through the series of cells, but here we face the implicit form of Ti as a function of r, i. This is the same basic difficulty that limits the utility of the CSTR sequence as an analytical model for nonisothermal reactors. For the case here though, where we employ a relatively large value of the index n in approximation of a plug-flow reactor, and where the solution will be via numerical methods anyway, we will strong-arm the problem with the approximation T,- r,- ] in the exponentials, so that... 

Now, we will apply these simple principles to batch reactors, CSTRs, and tubular reactors (distributed systems), starting with isothermal systems followed by nonisothermal systems. 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

The design equations for a chemical reactor contain several parameters that are functions of temperature. Equation (7.17) applies to a nonisothermal batch reactor and is exemplary of the physical property variations that can be important even for ideal reactors. Note that the word ideal has three uses in this chapter. In connection with reactors, ideal refers to the quality of mixing in the vessel. Ideal batch reactors and CSTRs have perfect internal mixing. Ideal PFRs are perfectly mixed in the radial direction and have no mixing in the axial direction. These ideal reactors may be nonisothermal and may have physical properties that vary with temperature, pressure, and composition. 

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. 

In order to reduce the disparities in volume or space time requirements between an individual CSTR and a plug flow reactor, batteries or cascades of stirred tank reactors ard employed. These reactor networks consist of a number of stirred tank reactors confiected in series with the effluent from one reactor serving as the input to the next. Although the concentration is uniform within any one reactor, there is a progressive decrease in reactant concentration as ohe moves from the initial tank to the final tank in the cascade. In effect one has stepwise variations in composition as he moves from onfe CSTR to another. Figure 8.9 illustrates the stepwise variations typical of reactor cascades for different numbers of CSTR s in series. In the general nonisothermal case one will also en-... 

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... 

J. Hamer, T. Akramov, and W. Ray. The dynamic behavior of continuous polymerization reactors II, nonisothermal solution homopolymerization and copolymerization in a CSTR. Chem. Eng. Sci., 36 1897-1914, 1981. 

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. 

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. 

In this and the previous chapters we considered the effects of nonisothermal operation on reactor behavior. The effects of nonisothermal operation can be dramatic, especially for exothermic reactions, often leading to reactor volumes many times smaller than if isothermal and often leading to the possibility of multiple steady states. Further, in nonisothermal operation, the CSTR can require a smaller volume for a given conversion than a PFTR. In this section we summarize some of these characteristics and modes of operation. For endothermic reactions, nonisothermal operation cools the reactor, and this reduces the rate, so that these reactors are inherently stable. The modes of operation can be classified as follows ... 

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