Au exothermic reaction, A → 2B, takes place adiabatically in a stirred-tank system. This liquid phase reaction occurs at constant volume in a 100-gal reactor. The reaction can be considered to be first order and irreversible with the rate constant given by k = 2.4 X 1015e–20,000/T (min–1) where T is in °R. Using the information below, derive a transfer function relating the exit temperature T to the inlet concentration cAi. State any assumptions that you make. Simplify the transfer function by making a first-order approximation and show that the approximation is valid by comparing the step responses of both the original and the approximate models.

Available Information

(i) Nominal steady-state conditions are:

T = 150 °F, cA1 = 0.8 lb mole/ft

q 20 gal/min = flow rate in and out of the reactor

(ii) Physical property data for the mixture at the nominal steady state: Cp = 0.8 Btu/lb °F,

p = 52 lb/ft3, —∆HR = 500 kJ/lb mole