A completely enclosed stirred$-$tank heating process is used to heat an incoming stream whose flow rate varies as shown in the figure below. The heating rate from this coil and the volume are both constant. We have the following assumptions$/$parameters:

The density $$ and heat capacity $C_p$ are constants

The overall heat transfer coefficient $U$ is constant

$w$ is the mass flow rate and $A_s$ is the surface area for heat losses to ambient

$T_i>T_a($inlet temperature is higher than ambient temperature$)$

$($a$)$ Develop a mathematical model $($differential and algebraic equations$)$ that describes the exit temperature if heat losses to the ambient occur and if the ambient temperature $(T_a)$ and the incoming stream's temperature $(T_i)$ both can vary.

$($b$)$ Discuss qualitatively what you expect to happen as $T_i$ and $w$ increase $($or decrease$).$ Justify based on your model.

$($c$)$ If the variations in the ambient temperature $(T_a)$ and the incoming stream's temperature $(T_i)$ are negligible $($i$.$e$.,T_a$ and $T_i$ can be considered as constants$),$ linearize the model obtained in $($a$)$ using $Q$ as the input variable $($i$.$e$.,y=T,u=Q).$

$($d$)$ If only the variation in the incoming stream's temperature $(T_i)$ is negligible, linearize the model obtained in $($a$)$ using both $T_a$ and $Q$ as the input variables $($i$.$e$.,y=T,u=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{c}Q\\ T_a\end{array}\right]).$