The figure shows a system with a cooling jacket. Heat is transferred from the wall of the tank to the fluid in the jacket by convection. The heat transfer from the fluid in the tank to the fluid in the jacket is formulated as $=2*F_c*(T-$Tco$).$ The tank is mixed very well. The output temperature of the $tankT,$is desired to be kept at $30C$ by adjusting the flow rate of the cooling water with a $P$ controller. The volume of fluid in the inner tank is $V=10m^3$; the density of the fluid is $=800k\frac{g}{m^3}$; the heat capacity of the fluid is $Cp=1\mathrm{.}4$kca$\frac{l}{k}gC.$ The feed flow rate of the fluid fed to the system is $F=0\mathrm{.}5\frac{m^3}{m}in$ and the feed temperature is $T_0=80C.$ The feed temperature of the cooling fluid $T{c}_0=20C.$

The transfer functions of thermocouple and pneumatic valve used in the proposed feedback control loop are given as $G_s(s)=\frac{\frac{16}{100}\frac{mA}{}C}{(0\mathrm{.}5s+1)}$ and $G_y(s)=\frac{\frac{-75000\text{k}\text{c}\text{a}\text{l}}{16(mA)}}{(2s+1)},$ respectively.

a$)$ Derive the transfer function of the system $($without control loop$).$

b$)$ Calculate the $P$ controller parameters in order to hold the output temperature of fluid at $30C.$

c$)$ On the tank controlled with $P,$ if the temperature of $T_0$ decreases to $50C,$ find the last temperature of the fluid and flow rate of the cooling fluid.