Q$1.(20$ marks$)$

A completely enclosed stirred$-$tank heating process is used to heat an incoming stream whose flow rate varies. The heating rate from this coil and the volume are both constant.

The volume of the tank $(V)$ as well as the heating rate $(Q)$ both are constant.

a$.$ Develop a mathematical model $($differential and algebraic equations$)$ that describes the exit temperature if heat losses to the ambient surroundings occur, the surrounding temperature being $T_a.$ Assume that the overall heat transfer coefficient used for accounting for this heat loss is $U$ and the area of heat transfer to the surroundings is $A.$

b$.$ Also discuss qualitatively, what to expect to happen to $T$ if $($i$)T_i$ increases keeping $w$ constant and the heat added is greater than the heat losses; and $($ii$)w$ increases keeping $T_i$ constant with heat added being greater than heat lost to surrounding. Justify by referring to your model.