$13\mathrm{.}6$ A four$-$cylinder, four$-$stroke diesel engine has a bore of $212mm$ and a stroke of $292mm.$

At full load at $720re\frac{v}{m}in$ the bmep is $\frac{5\mathrm{.}93}{b}ar$ and the specific fuel consumption is

$0\mathrm{.}226k\frac{g}{k}Wh.$ The air$-$fuel ratio as determined by exhaust gas analysis is $\frac{25}{1}.$ Calculate

the brake thermal efficiency and the volumetric efficiency of the engine. Atmospheric

conditions are $1\mathrm{.}01$ bar and $15C,$ and $Q_{\text{n}\text{e}\text{t}\text{,},},$ for the fuel may be taken as $44200k\frac{J}{k}g.$

$(36\%$;$76\mathrm{.}1\%)$

$13\mathrm{.}7$ The engine of Problem $13\mathrm{.}6$ is to be used as a dual$-$fuel engine. It is to burn methane

$($calorific value $33480k\frac{J}{m^3}$ at $1\mathrm{.}013$ bar and $15C),$ and has a pilot injection of of

$10\%$ of the input when running as a diesel engine. The air$-$fuel ratio for the oil is $\frac{25}{1}$

as before, and for the methane $\frac{8\mathrm{.}5}{1}.$ If the volumetric efficiency and the power output

remain the same, what is the brake thermal efficiency of the engine when running on

the dual fuel?