A reactor process system is shown below. The reaction taking place is a

simple irreversible, first$-$order reaction of $AB$ that is exothermic in nature. The heat

generated by the reaction is removed using a jacket with cooling water running through

it$."*$ote: variables associated with the jacket are denoted by the subscript $"J".$

Units are:

$T[=]C$

$q[=]\frac{m^3}{m}in$

$C_1[=]k\frac{g}{m^3}$

$V[=]m^3$

$Q[=]k\frac{J}{m}in$

Heat removed through the jacket

is represented by:

$Q(t)=UA(T(t)-T_J(t))$

Variables dependent on time are

indicated with $(t).$ All others are

constant.

a$)$ Using the variables given, derive the dynamic balance for the system. Be sure to state any

assumptions that you make. Hint: you should end up with four equations.

b$)$ Using the dynamic energy balance on the tank contents, find each of the transfer functions that

are associated with the temperature leaving the reactor, $T(t).$ Note: I only want you to find the

transfer functions for this equation.