The complete mixed tank system is given in the figure below. The mass flow rate of the inlet stream is w $($kg$/$min$)$ and the temperature is Ti $(\backslash$deg C$).$ given

It is desired to control the tank outlet temperature To $(\backslash$deg C$)$ for the system. Since a homogeneous mixture is provided in the tank, the outlet temperature is the same.

is equal to the temperature in the tank at the same time.

a$)$ Derive an equation containing the expressions Ti$\text{'}($s$)$ and Q$($s$)$ for the tank outlet temperature To$\text{'}($s$).$

b$)$ Simplify the equation you derived in part $($a$)$ for the regulator problem.

c$)$ Simplify the equation you derived in $($a$)$ for the servo problem.

d$)$ Draw the block diagram of the system.

e$)$ Simplify the block diagram you drew in $($d$)$ for the regulator problem.

f$)$ Simplify the block diagram you drew in $($d$)$ for the servo problem.