Q$5.$ A four$-$stroke $3$ litre V$6$ spark ignition engine has a maximum power output of $100$ kW at $5500$ rpm $,$ and a maximum torque of $236$ Nm at $3000$ rpm $.$ The minimum bsfc is $0\mathrm{.}090k\frac{g}{M}J$ at $3000$ rpm $,$ and the air flow rate is $0\mathrm{.}068\frac{m^3}{s}.$ The compression ratio is $8\mathrm{.}9$:$1$ and the mechanical efficiency is $90$ per cent. The engine was tested under ambient conditions of $20C$ and $1$ bar ; take the calorific value of the fuel to be $44M\frac{J}{k}g.$

$($a$)$ Calculate the power output at $3000$ rpm and the torque output at $5500$ rpm $.$

$($b$)$ Calculate for both speeds the bmep and the imep.

$($c$)$ How does the arbitrary overall efficiency at $3000$ rpm compare with the corresponding air standard Otto cycle efficiency?

$($d$)$ What is the volumetric efficiency and air$/$fuel ratio at $3000$ rpm $?$