Optimal designs for tubular reactors

We shall consider only the most elementary formulations of optimal design and avoid the heavy mathematics that is often associated with the more sophisticated problems. For a single reaction, the basic problem is to get as large a conversion in as short a reactor as possible. For simultaneous reactions, the problem is to get as good a yield of the main product as possible. 

Consider first an isothermal tubular reactor with constant density of reactants for which the equation is 

If the integral in Eq. (9.6.2) can be worked out explicitly, then it may be possible to obtain an equation for the optimal temperature. For example, consider the first order reversible reaction A B, with rate law r = ka — taking place in an isothermal reactor whose feed is pure A. If the required fractional conversion is Y, then the feed concentration can be written the current concentrations are a = Gq — and 6 = f, and 

Now in looking for the minimum of the right-hand side it is convenient to remember that k is proportional to a power of k, for 

Let us rearrange this equation so that all the terms involving X arc on one side and note that 

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