In chemical engineering, plug flow reactors (that is, those in which fluid flows from one end to the other with minimal mixing along the longitudinal axis) are often used to convert reactants into products. It has been determined that the efficiency of the conversion can sometimes be improved by recycling a portion of the product stream so that it returns to the entrance for an additional pass through the reactor (Fig. P8.2). The recycle rate is defined as R = volume of fluid returned to entrance volume leaving the system Suppose that we are processing a chemical A to generate a product B. For the case where A forms B according to an autocatalytic reaction (that is, in which one of the products acts as a catalyst or stimulus for the reaction), it can be shown that an optimal recycle rate must satisfy In- 1 + R(1-X) R(1 - XA) R+1 R[1 + R(1 - XA)] where X++= the fraction of reactant A that is converted to product B. The optimal recycle rate corresponds to the minimum-sized reactor needed to attain the desired level of conversion. Use a numerical method to determine the recycle ratios needed to minimize reactor size for a fractional conversion of XAF = 0.9. FIGURE P8.2 Schematic representation of a plug flow reactor with recycle. Feed Plug flow reactor Recycle Product