Exercise $5.-$ In a heater tank like the one in the figure, temperature $T(t)$ at the

output may change due to variations in the input stream temperature $T_i(t)$ or in

the injected heat $Q(t).$ The input and output flow rates are assumed to be equal

and constant $(F_i(t)=F(t)-=q).$ The energy balance gives the equation:

$V{C}_v\frac{dT(t)}{dt}=q{C}_p(T_i(t)-T(t))+Q(t)$

$($a$)$ Calculate the transfer function between the increase in input

temperature and the increase in output temperature.

$($b$)$ Calculate the transfer function between the increase in injected heat and the increase in output temperature

$($c$)$ Which one of the two variables, $T_i(t)$ or $Q(t),$ would you choose as the manipulated variable to control

the output temperature in the tank, and which one would act as a disturbance?