An exothermic reaction, A ? 213, takes place adiabatically in a stirred-tank reactor. This liquid reaction occurs at constant volume in a 1000-gal reactor. The reaction can be considered to he first order and irreversible with the rate constant given by

k = 2.4 x 10¹⁵e^-20.000/T?(min^?1)

where T is in ^oR.

(a) Using the information below, derive a transfer function relating the exit temperature T to the inlet concentration c_At. State all assumptions that you make,

(b) How sensitive is the transfer function gain K to the operating conditions? Find an expression for the gain in terms q, T, and c_Ai evaluate the sensitivities. (That is, ?k/?q, etc.)

Available Information

(i) Nominal steady-state conditions are:

T = 150 ^oF, c_Ai = 0.8 mol/ft³

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

(ii) Physical property data (or the mixture at the nominal steady state:

-AHR = 500 kJ/mol C= 0.8- Btu P= 52 lb/t, Ib