$2$ Two engine options are to be offered in a new automobile model. Engine A is naturally aspirated with a compression ratio of $10\mathrm{.}5$:$1$ and cylinder inlet conditions of $60C$ and $96$ kPa $.$ Engine B is supercharged with aftercooling and has cylinder inlet conditions of $80C$ and $130$ kPa $.$ To avoid knock problems, it is desirable to have the air$-$fuel temperature at the start of combustion in engine B be the same as in engine A$.$

Calculate:

$($a$)$ Temperature at start of combustion in engine A$,$ using air$-$standard Otto cycle analysis. $C$

$($b$)$ Compression ratio of engine B that would give the same temperature at the start of combustion.

$($c$)$ Temperature reduction in the aftercooler of engine B if the compressor has an isentropic efficiency of $82\%$ and inlet conditions are the same as in engine $A.[C]$