A sphere $30$ mm in diameter is initially at $800$ K$.$ It is then quenched in a large bath having a constant temperature of $320$ K with a convection heat transfer coefficient of $75$ W$/$m$\text{'}.$K$.$

The thermo$-$physical properties of the sphere material are: k $=1\mathrm{.}7$ W$/$m$.$K$.$ p $=400$ kg$/$m$\text{'}.$

cp $=1600$ J$/$kg$-$K$.$

$($a$)$ Calculate the time required for the surface of the sphere to reach $415$ K$.$

$($b$)$ Determine the heat flux $($W$/$m$\text{'})$ at the outer surface of the sphere at the time determined in part $($a$).$

$($c$)$ Determine the energy that has been lost by the sphere during the process of cooling to the surface temperature of $415$ K$.$