Optimal Temperature Policies

The case of reversible exothermic reactions is more complicated, because even though the rate may increase with temperature, as the equilibrium conversion is reached, higher temperatures have an adverse effect of decreased equilibrium conversion. Thus, there is an optimum intermediate temperature where reasonably rapid rates are obtained together with a sufficiently large equilibrium conversion. The precise value of the optimal temperature can be found with use of Eq. 8.1-4 

This can always—in principle, and usually in practice—be integrated for a constant value of temperature, and then the best temperature found for a given conversion, It can be shown that this is exactly equivalent to the problem of choosing the optimal temperature for the maximum conversion for a given reaction time. Or. 

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Eqs. (1) and (7) will be used to derive optimal temperature policies. 

In this type of problem, the objective is to compute the optimal temperature policy maximizing the amount of a desired product concentration for a given fixed batch time subject to bounds on the reactor temperature. The problem can be written mathematically as... 

The Disjoint Character of the Optimal Temperature Policy with a Single Reaction... 

In this chapter we pass from the discrete to the continuous deterministic process, and the difference equations derived for the stages now become differential equations. It is scarcely surprising to find many features of the discrete process are retained by the continuous process in particular we know from the work of Denbigh (1944), and will prove here afresh, that the optimal temperature policy is disjoint in the case of a single reaction. However, this simplicity is lost when more than one reaction is taking place and we shall do well to examine the simple consecutive system A B C with some care, as it opens up the principal features of the general case. 

OPTIMAL TEMPERATURE POLICY WITH A SINGLE REACTION... 

It is instructive to approach the same problem from another angle and ask for the optimal temperature policy that will achieve the... 

In this simple problem not all these equations are needed. Dividing the first by the second gives dcf/dd = —R cf) which is simply the defining Eq. (7.1.1) with the optimal policy T = Tn in use. This is important for it shows that the characteristics of the partial differential equation (3) are actual trajectories or reaction paths. The principle of optimality thus shows that the optimal temperature policy for any reaction is just the temperature specified in the integration of (6) along the characteristic. Separating variables we have... 

The optimal temperature profiles obtained pose the practical problem of how to implement this optimal temperature policy. The optimal temperature profiles obtained suggest that the reactor should be operated with quite a high feed temperature followed by a very efficient cooling to decrease the temperature sharply at the first 15% of the reactor depth. The implementation of this optimal policy is technically difficult and expensive. A number of suboptimal policies were suggested and discussed (Elnashaie et ai, 1987b) to overcome the technical difficulties associated with the implementation of the absolute optimal policy. One of the main technical problems associated with optimal temperature is the very high temperature at the reactor inlet part which the catalyst may not be able to withstand. 

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. 

An equation relating temperature T and conversion Xa is required to design the non-isothermal reactors. This relationship between temperature T and conversion is obtained by setting up a heat balance equation around the reactor (Section 3.1.5.3). In certain cases, reactor temperature T is deliberately varied with conversion by regulating the heat supply to the reactor or heat removal from the reactor. One such case is the non-isothermal reactor in which a reversible exothermic reaction is carried out. In the case of a reversible exothermic reaction, there is an optimum temperature T for every value of conversion x at which the rate is maximum. A specified conversion Xaj will be achieved in a CSTR or a PFR with the smallest volume or in a batch reactor in the shortest reaction time if the temperature in the reaction vessel is maintained at the optimum level. This optimal temperature policy in which temperature is varied as a function of conversion x,i is known as the optimal progression of temperature presented in the following section. 

Is carried out in an ideal PFR in which optimal temperature policy is maintained with the maximum value of the reactor temperature restricted to 900 K. Calculate the space time required to achieve 70% conversion of A. The concentration of A in the feed solution is 0.5 kmol/m. ... 

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