Constant conversion policy

Comparability depends on the policy of measuring half-lives the mode of keeping a constant degree of conversion at variable flow rate or enzyme concentration (constant conversion policy) gives better results than the mode of constant flow rate accompanied by varying degree of conversion (constant flow rate policy). 

If pore diffusion is unimportant, i.e., if the effectiveness factor r is equal to 1, then with constant conversion policy both CSTR and PFR yield half-lives identical to Eq. (5.74) with arbitrary kinetics. With constant flow rate policy, the measured half-life is identical to that obtained through Eq. (5.74) only if the enzyme is saturated, i.e., [S] fCM, and the reaction is zeroth order. 

Right Profiles of decrease in F(t)/F(0) for intraparticle diffusion-influenced zero-order reaction with spherical immobilized enzyme particles packed in the reactor operated under a constant conversion policy (x = 0.99). Enzyme activity decays as E(t)/E(0) = exp ( kd t). 

Ogunye and Ray (1971a,b) have formulated the optimal control problem for tubular reactors with catalyst decay via a weak maximum principle for this distributed system. Detailed numerical examples have been calculated for both adiabatic and isothermal reactors. For irreversible reactions, constant conversion policies are found to not always be optimal. A practical technique for on-line optimal control for fixed bed catalytic reactors, has been suggested by Brisk and Barton (1977). Lovland (1977) derived a simple maximum principle for the optimal flow control of plug flow processes. 

The weak maximum principle can certainly be used to maximize the performance index of Eq. 13.33, for instance, for the independent deactivation problem given by Eqs. 13.40 through 13.45 with the addition of a heat balance at the pellet-bulk fluid interface for the pellet surface temperature 7 [( )s]. Nevertheless, the optimal inlet temperature will be a function of time. Thus, the inlet temperature is to be manipulated in a prescribed manner in time without any regard to what happens to the reactor. This open-loop control is rarely practiced in actual applications because of the uncertainties regarding the model, measurements, and disturbances. Rather, closed-loop control using feedback from the process is usually practiced. This fact should not discourage one from using the maximum principle for optimization, for it will at least indicate what the best possible performance is in a relative sense. At the same time it can yield in some simple cases a very powerful optimal policy such as the constant conversion policy, which can be implemented by a feedback control scheme. 

The single cycle, falling conversion TIR experiment seems to be a reliable measure of decay rate. However, for multiple cycles, this system appears to be vulnerable to problems similar to those of the constant conversion TIR policy. Elimination of temperature upsets and regulated temperature increases need to be addressed in the future. 

The optimal temperature policy in a batch reactor, for a first order irreversible reaction was formulated by Szepe and Levenspiel (1968). The optimal situation was found to be either operating at the maximum allowable temperature, or with a rising temperature policy, Chou el al. (1967) have discussed the problem of simple optimal control policies of isothermal tubular reactors with catalyst decay. They found that the optimal policy is to maintain a constant conversion assuming that the decay is dependent on temperature. Ogunye and Ray (1968) found that, for both reversible and irreversible reactions, the simple optimal policies for the maximization of a total yield of a reactor over a period of catalyst decay were not always optimal. The optimal policy can be mixed containing both constrained and unconstrained parts as well as being purely constrained. 

In commercial operation of fixed-bed reactors in which the catalyst is deactivating, it is often necessary to maintain constant conversion by increasing the temperature of the reactor to compensate for catalyst decay. One can think of this as a constant activity policy in which the bed may be considered isothermal at any given time with the overall temperature level increasing with time of operation. 

When unconstrained temperature control is distributed over the entire reactor, the optimal policy is to maintain a constant exit conversion if the activation energy for the deactivation reaction is the same as that of the main reaction if not, it is constant conversion over the entire reactor. It should be pointed out here that the optimal policy based on the pseudo-steady state assumption regarding Eqs. 13.32 and 13.38 is valid (Gruyaert and Crowe 1976) only when the change of temperature is felt at the outlet before another change is made at the inlet. 

It is immediately seen that the condition of separable temperature dependence, which is necessary for the optimal policy of constant conversion, cannot be satisfied. 

It is seen that the optimal policy would be that of maintaining the conversion constant if A xa- is zero or a function solely of x since then the left hand side of Eq. 13.37 should be constant, which implies constant x. It can be shown (Chou et al. 1967) that Eq. 13.37 leads to the optimal policy of maintaining a constant conversion for an irreversible reaction in which only one rate constant and temperature dependence is involved, and that the policy is also applicable to nonisothermal reactors. The limitations under which the policy of a constant conversion is valid are embodied in Eq. 13.37 x should be a function of A and A and not dependent explicitly on t, and A xa should be a function solely of x. The policy gives the maximum of P when an optimal temperature is chosen and hence requires a search for the optimal temperature. 

Even for this simplest case of deactivation, the condition is not met because the rate constant kt, for instance, is not separable as apparent from Eq. 13.42, not to mention the separability of the temperature dependence of (/ )s> and y. This does not mean that realistic problems cannot be solved. They can still be solved by the maximum principle, for instance, although the results are not as elegant as the policy of constant conversion. 

Equilibrium constraints and catalyst deactivation lead naturally to optimization problems. Simple maximization, dynamic programming, maximum principles, and other techniques can be used to solve the optimization problems. Optimization is usually carried out with respect to operating conditions. As pointed out repeatedly in this chapter, it should also be done with respect to reactor size. Perhaps the most powerful optimal policy for a reactor affected by deactivation is that of a constant conversion. However, it is not usually applicable to realistic systems and more work is needed for simple and yet general policies applicable to realistic systems. Means for estimating the extent of deactivation from process measurements and their use for optimization is another area that needs further work since detailed knowledge of deactivation is usually unavailable. An extensive review of reactor... 

Figure 4.46 gives results for a range of feed flowrates when a constant feed flowrate policy is employed. The batch time is fixed in these results at 300 min. High feed flow-rates produce high concentrations of B, high maximum temperatures, and high conversions, but result in low selectivities. 

A practical application of such a parametric study might be the determination of optimal conditions for a particular batch of catalyst. For example the Temkin equation, which is of the form (10), is used with some confidence for the ammonia synthesis reaction (Annable, 1952). However, the values of the constants k, k%, Ei, and E2 may vary with different catalysts and so the constants p, K, and A and the consequent optimal policy. As we shall now show the results of this section could be used to determine the maximum conversion from a fixed length of converter with the current brand of catalyst. 

The overall activity factor can be used in a straightforward manner for a piecewise feedback control. Suppose that the control policy is to maintain a constant outlet conversion. Suppose further that the inlet temperature is manipulated intermittently, as in the usual operation of an adiabatic reactor, so as to maintain the constant outlet conversion. If one lets the subscript c denote the current quantities... 

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