CSTR optimal operation

The following example illustrates a simple case of optimal operation of a multistage CSTR to minimize the total volume. We continue to assume a constant-density system with isothermal operation. 

Figure 1-3 Example of unrestricted optimization in a train of two CSTRs that operate at the same temperature. This graph illustrates the effect of residence time for each reactor (i.e., ri = T2) on the yield of intermediate product D in the exit stream of the second reactor.

The optimal operating point must lie on the intersection of the AR boundary and the plane described by c, = 0.3 moI/L. Note that the concentration at point B (the critical CSTR point) is approximately [0.223,0.08686,0.316] moI/L. Although there are a number of points that intersect this plane with the AR, we wish to determine the point that maximizes the concentration of component B. This occurs on the AR boundary at point 1. Note that this point corresponds the point of maximum Cg when Cg = 0.3 mol/L, and it is different from point B in Figure 7.5(b). Point I is located below point B (for point I contains less Cg than point B), and therefore it is obtained by structure 2 (given by Figure 7.6) and not by structure 1. A CSTR operated at the equilibrium point at point D from the feed, followed by a critical DSR to point H, terminated by a PFR to point 1 is therefore the required reactor structure. The path ADHl given in Figure 7.7(a) must therefore be followed in order to achieve the desired concentration. 

Usually, the design goal of such a chemical reaction is to maximize the yield, i.e. to get the most possible product B for the amount of reactant A fed into the CSTR. In the case of only one manipulated variable q, the operating point of the input variable can be chosen such that this yield is maximized, denoted as the optimal operating point (OF). 

Since a CSTR operates at or close to uniform conditions of temperature and composition, its kinetic and product parameters can usually be predicted more accurately and controlled with greater ease. The CSTR can often be operated at a selected conversion level to optimize space-time yield, or where a particular product parameter is especially favored. 

The results of Example 5.2 apply to a reactor with a fixed reaction time, i or thatch- Equation (5.5) shows that the optimal temperature in a CSTR decreases as the mean residence time increases. This is also true for a PFR or a batch reactor. There is no interior optimum with respect to reaction time for a single, reversible reaction. When Ef < Ef, the best yield is obtained in a large reactor operating at low temperature. Obviously, the kinetic model ceases to apply when the reactants freeze. More realistically, capital and operating costs impose constraints on the design. 

Example 6.7 Determine optimal reactor volumes and operating temperatures for the three ideal reactors a single CSTR, an isothermal PER, and an adiabatic PER. 

For exothermic, reversible reactions, the existence of a locus of maximum rates, as shown in Section 5.3.4, and illustrated in Figures 5.2(a) and 18.3, introduces the opportunity to optimize (minimize) the reactor volume or mean residence time for a specified throughput and fractional conversion of reactant. This is done by choice of an appropriate T (for a CSTR) or T profile (for a PFR) so that the rate is a maximum at each point. The mode of operation (e.g., adiabatic operation for a PFR) may not allow a faithful interpretation of this requirement. For illustration, we consider the optimization of both a CSTR and a PFR for the model reaction... 

For the optimal strategy of maintaining operational stability, Lee et al. have calculated the optimal profile of addition of fresh, non-deactivated enzyme into a CSTR under different deactivation kinetics. If a CSTR is charged initially with an amount of enzyme of initial activity N0, at time t under deactivation, the amount remaining is given by Eq. (5.83), where k(t) denotes an arbitrary deactivation function (J. Y. Lee, 1990). 

The primary reason for choosing a particular reactor type is the influence of mixing on the reaction rates. Since the rates affect conversion, yield, and selectivity we can select a reactor that optimizes the steady-state economics of the process. For example, the plug-flow reactor has a smaller volume than the CSTR for the same production rate under isothermal conditions and kinetics dominated by the reactant concentrations. The opposite may be true for adiabatic operation or autocata-lytic reactions. For those situations, the CSTR would have the smaller volume since it could operate at the exit conditions of a plug-flow reactor and thus achieve a higher overall rate of reaction. 

Other recent work in the field of optimization of catalytic reactors experiencing catalyst decay includes the work of Romero e/ n/. (1981 a) who carried out an analysis of the temperature-time sequence for deactivating isothermal catalyst bed. Sandana (1982) investigated the optimum temperature policy for a deactivating catalytic packed bed reactor which is operated isothermally. Promanik and Kunzru (1984) obtained the optimal policy for a consecutive reaction in a CSTR with concentration dependent catalyst deactivation. Ferraris ei al. (1984) suggested an approximate method to obtain the optimal control policy for tubular catalytic reactors with catalyst decay. 

For the great majority of reaction schemes, piston flow is optimal. Thus the reactor designer normally wants to build a tubular reactor and to operate it at high Reynolds numbers so that piston flow is closely approximated. This may not be possible. There are many situations where a tubular reactor is infeasible and where CSTRs are used instead. Typical examples are reactions involving suspended solids and autorefrigerated reactors where the reaction mass is held at its boiling point. 

B) Having solved the batch case, now set up the same kinetics for the case of a CSTR operated at steady state. Use NSolve to find the optimal flow rate and holding time for the production of D (and E) assuming that the inlet concentrations of A and B are each 1 and that the volume is 100. 

Helpful hints. Use the conjugate gradient method of optimization with 2 degrees of freedom. In other words, you should develop a set of n equations in terms of n + 2 variables that describe the steady-state operation of three independent chemical reactions in a train of two chemical reactors. Maximization algorithms implicitly use two additional equations to determine optimum performance of the CSTR train ... 

TABLE 1-2 Restricted Residence-Time Optimization for Two CSTRs in Series Operating at the Same Temperature ... 

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