A reactor converts reactant BC to product CB by heating the material in the presence of an additive A (mole fraction = x_A). The additive can be injected into the reactor, while steam can be injected into a heating coil inside the reactor to provide heal. Some conversion can be obtained by heating without addition of A, and vice versa. The product CB can be sold for $50 per lb-mol. For 1 lb-mol of feed, the cost of the additive (in dollars per lb-mol feed) as a function of X_A is given by the formula 2.0 + l0x_A + 20x²_A. The cost of the steam (in dollars per lb-mal feed) as a function of S is 1.0 + 0.003S + 2.0 ? 10^-6S² (S = lb steam/lb-mol feed). The yield equation is y_CB?= 0.1?+ 0.3 x_A?+ 0.0001S ? 0.0001x_AS.

(a) Formulate the profit function (basis of 1.0 lb-mol feed) in terms of x_A and S.

f = income ? costs

(b) Maximize f subject to the constraints

0 ? x_A?? 1 ?S ? 0

Ib-mol product CB Ib-mol feed