A specialty chemical is produced in a batch reactor. The tune required to successfully complete one batch of product depends on the amount charged to (and produced from) the reactor. Using reactor data, a correlation is 2.0P0.4, where P is the amount of product in pounds per batch and t is given in hours. A certain amount of non-production time is associated with each batch for charging, discharging and minor maintenance, namely, 14 h/batch. The operating cost for the batch system is $50/h. Other costs including storage depend on the size of each batch and have been estimated to he C1 = $800 P0.7 ($/yr). The required annual production is 300,000 lb/yr, and the process can be operated 320 days/yr (24 h/day). Total raw material cost at this production Level is $400,000/yr.

(a) Formulate an objective function using P as the only variable. (Show algebraic substitution.)

(b) Are there any constraints on P? (Give relations if applicable.)

(c) Solve for the optimum value of P analytically. Check that it is a minimum. Also check applicable constraints.