Water is boiling in a $12-$cm$-$deep pan with an outer diameter of $25$ cm that is placed on top of a stove. The ambient air and the surrounding surfaces are at a temperature of $25$oC$,$ and the emissivity of the outer surface of the pan is $0\mathrm{.}80.$ Assuming the entire pan to be at an average temperature of $98$oC$,$ determine the rate of heat loss from the cylindrical side surface of the pan to the surroundings by $($a$)$ natural convection and $($b$)$ radiation. $($c$)$ If water is boiling at a rate of $1\mathrm{.}5$ kg$/$h at $100$oC$,$ determine the ratio of the heat lost from the side surfaces of the pan to that by the evaporation of water. The enthalpy of vaporization of water at $100$oC is $2257$ kJ$/$kg