The dynamic response of a stirred-tank bioreactor can be represented by the transfer function C′ (s) C′ F(s) = 4 2s + 1 where C′ is the exit substrate concentration, mol/L, and C′ F is the feed substrate concentration, mol/L. (a) Derive an expression for c′ (t) if cF(t) is a rectangular pulse (Fig. 5.2) with the following characteristics: cF(t) = ⎧ ⎪ ⎨ ⎪ ⎩ 2 t < 0 4 0 ≤ t < 2 2 2 ≤ t < [infinity] (b) What is the maximum value of c′ (t)? When does it occur? What is the final value of c′ (t)? (c) If the initial value is c(0) = 1, how long does it take for c(t) to return to a value of 1.05 after it has reached its maximum value? 5.5 A th