$($Fall $2023$ exam question$)$ A plane layer of coal

$(k_{\text{c}\text{o}\text{a}\text{l}}=0\mathrm{.}26\frac{W}{m}*K)$ of thickness $L=1m$ experiences

uniform volumetric generation at a rate of $q^{}=10\frac{W}{m^3}$

due to slow oxidation of the coal particles. Averaged over

a daily period, the top surface of the layer transfers heat

by convection to ambient air for which $h=8\frac{W}{m^2}*K$ and

$T_{}=30C,$ while receiving solar irradiation in the amount

$G_s=400\frac{W}{m^2}.$ Irradiation from the atmosphere may be

neglected. The solar absorptivity and emissivity of the

surface are each ${}_s==0\mathrm{.}95.$ Assume that the bottom of

the coal layer is insulated and the surface temperature $T_s$ is known.

a$)$ Starting with an energy balance around a suitable control volume, derive the differential equation

describing the temperature distribution in the coal layer and write the boundary conditions.

b$)$ Solve the differential equation and obtain the temperature profile. Sketch the profile.

c$)$ Obtain an expression for the rate of heat transfer by conduction per unit area at $x=L.$

d$)$ Obtain an expression for the surface temperature $T_s$ and evaluate $T_s.$

e$)$ Find the temperature of the insulated bottom for the prescribed conditions.