In an iron foundry cupola, $14$ metric tons of hematite is smelted in one hour, using coke $(90$ percent carbon$).$ The exit gas contains by volume CO $13\%,$ CO$213\%,$ N$274\%,$ and leave the cupola at $500$oC$.$ Assumptions: pig iron contains $95\%$ Fe and $5\%$ C$.$

Calculate:

$($a$)$The proportion of the calorific power of the coke lost.

$($b$)$ By the sensible heat of the hot gases escaping.

$($c$)$ by the imperfect $($incomplete$)$ combustion of the coke