Consider the candle shown in the figure below. After the candle is lit the wax next to the flame soon reaches its melting temperature $($Tm$),$ and the candle begins to melt. Assume that after certain time the candle starts to glow steadily. Let the length of the candle be L and the radius be R$.$ The temperature at the base of the candle is T$($inf$),$ and the temperature of the surrounding air is also T$($inf$).$ The candle losses some heat to the surrounding by convection along the length of the candle. The average convective heat transfer coefficient is h$.$ For simplicity, assume that the melting occurs slowly enough that pseudo steady state approximation can be used.

$($a$)$ Find the temperature profile T$($z$)$ in the candle, assuming that the temperature is approximately independent of radial position.

$($b$)$ Calculate the amount of heat $($Q$)$ enters the candle at z $=0.$

$($c$)$ Assuming that the intial length of the candle is L$0,$ and the heat loss by radiation and convection at z $=0$ is negligible, what is the time required for the candle to melt completely?

HINT: Assume that the candle as our system, write an unsteady state energy balance. An unsteady$-$state balance on the candle gives: Accumulation $=$ rate in$.$ The rate heat into the candle is derived in part $($b$).$ The latend heat of melting of wax is lambda$($sl$)$