Problem

A system with cooling jacket is shown in the figure. The heat is transferred from the wall of the tank to the fluid in the jacket by convection. The amount of heat transferred from the tank wall to the jacket is formulated as $Q=2Fc(T-T_{c_0}).$ Tank is well mixed. It is desired to keep the output temperature at $30C$ by manipulating the flow rate of the cooling water.

The volume of fluid in the tank is $V=10m^3$; the fluid density is $=800k\frac{g}{m^3}$; heat capacity of the fluid is $Cp=1\mathrm{.}4$kca$\frac{l}{k}g.C.$ The flow rate of the fluid fed to the system is $F=0\mathrm{.}5\frac{m^3}{h}$ and the temperature of feed stream is $T_0=80C.$ The fed temperature of cooling water is $T_{co}=20C.$

a$)$ Derive the transfer function gives the change in fluid temperature $($T$).$ Define the process and disturbance transfer function.

b$)$ If the flow rate of the cooling water $($Fc$)$ is decreased $500\frac{m^3}{h},$ find the output temperature $(T)$ after $2,5,10$ and $20h.$