$(40P)$ In a beer factory, the waste water is being heated by a heat exchanger. The temperature

of the heating water is $45C$ and its flow rate is $25\frac{m^3}{h}.$ The inlet and outlet temperatures

of waste water recorded as $10$ and $30C,$ respectively and its flow rate is $30\frac{m^3}{h}.$

a$)$ Calculate $K$ and $$ values for this heating system.

b$)$ If the temperature of heating water is increased to $55C$ at $t=0,$ what will be the

response equation of the output variable, $y(t)=?$

c$)$ What will be outlet temperature of waste water at $5.$ hour?

Assumptions:

Cp$($water$)=$ Cp $($waste water$)=4\mathrm{.}18$ kJ$/$kg$.$K$.$ Density of water $=$ density of waste water. Basis: $1$ hour