On a ship with a main engine of $8000kW$ power and $30\%$ thermal efficiency, $60\%$ of the total waste heat is emitted from the exhaust. Hot water is desired to be obtained from the exhaust gas at a temperature of $80C.$ Distilled water is taken into the boiler at a temperature of $30C.$ The main engine exhaust gas exit temperature is $600C,$ and the atmospheric exit temperature from the chimney is $220C.$ The boiler operates with a $2-$pass$-2-$stage plate separation design. Considering a body diameter of $150cm,$ an inner tube diameter of $10mm,$ an inner tube wall thickness of $2mm,$ and a body length of $3$ meters:

a$)$ Determine the total heat transfer amount.

b$)$ Determine the overall heat transfer coefficient.

c$)$ Find the number of pipes to be used. $(2$ passes $=1$ pipe$).$

NOTE:

When calculating the convective heat transfer coefficient, neglect the area blocked by the inner tubes.

Cp water$=4\mathrm{.}18$ kJ $\frac{?}{kg.K},Cp$ air $=1\mathrm{.}006k\frac{J}{k}g.K,$ tailpipe $=385\frac{W}{m}K,k$ water $=0\mathrm{.}65\frac{W}{m}K,k$ air$=0\mathrm{.}026$ W$/$mK$,Pr$ water$=6\mathrm{.}7,Pr$ hava$=0\mathrm{.}7,$ water$=8\mathrm{.}9{10}^{{?}^{{?}^{?}}}-4,$ air $=1\mathrm{.}85{10}^{{?}^{{?}^{?}}}-5$ Pa$.$s$,$ water $=997k\frac{g}{m^{{?}^{{?}^{?}}}}3,$ air $=1\mathrm{.}25k\frac{g}{m^{{?}^{{?}^{?}}}}3.$