In a ship with a main engine of $8000$ kW power and $30\%$ thermal efficiency, $60\%$ of the total waste heat is discharged through the chimney. It is desired to obtain hot water at $80$ C from the exhaust gas. Distilled water is taken into the boiler at a temperature of $30$ C$.$ The main engine exhaust gas outlet temperature is $600$ C and the chimney atmosphere outlet temperature is $220$ C$.$ The boiler working order is considered as $2$ passes and $2$ stages of plate separation. When the body diameter is $150$ cm$,$ the inner pipe diameter is $10$ mm$,$ the inner pipe wall thickness is $2$ mm and the body length is $3$ meters,

a$)$ Find the total heat transfer amount.

Find the total heat transfer coefficient.

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

NOTE:

When calculating the convection coefficient, ignore the area blocked by the internal pipes for the area on the body side. Cp$($water$)=4\mathrm{.}18$ kJ$/$kg$.$K$,$ Cp$($air$)=1\mathrm{.}006$ kJ$/$kg$.$ k$($internal pipes$)=385$ W$/$mK$,$ k$($water$)=0\mathrm{.}65$ W$/$mK$,$ k$($air$)=0\mathrm{.}026$ W$/$mK$,$ Pr$($water$)=6\mathrm{.}7,$ Pr$($air$)=0\mathrm{.}7,\backslash$mu $($water$)=8\mathrm{.}9*10^-4,\backslash$mu $($air$)=1\mathrm{.}85*10^-5$ Pa$.$s$,\backslash$rho $($water$)=997$ kg$/$m$^3,\backslash$rho $($air$)=1\mathrm{.}25$ kg$/$m$^3$