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 tank $T,$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 $G_s(s)=\frac{\frac{16}{100}\frac{mA}{}C}{(s+1)}$ and $G_v(s)=\frac{\frac{-500}{16}\frac{\text{k}\text{c}\text{a}\frac{l}{d}k}{mA}}{(s+1)},$ 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.