Exercise $2.-$ Consider the stirred$-$tank heating system shown in the following figure. The liquid inlet stream

consists of a single component with a mass flow rate $w_i(t)$ and input temperature $T_i(t).$ The tank contents are

agitated and heated using an electrical heater that provides a heating rate $Q(t).$

Assuming $(1)$ perfect mixing, so the exit temperature $T(t)$ is also the temperature of the tank content; $(2)$ the

inlet and outlet flow rates are equal $(w(t)=w_i(t))$; $(3)$ the density $$ and heat capacity $C$ of the liquid are

constant, thus their temperature dependence is neglected and $(4)$ heat losses are negligible; then the dynamic

model of the stirred tank heating system is the following:

$VC\frac{dT}{dt}=wC(T_i-T)+Q$

$($a$)$ Determine the steady$-$state equilibrium relationships among the different variables.

$($b$)$ Calculate the corresponding linearized equation around these steady$-$state values.