$5-21.$ A plane wall of thickness $L$ is insulated at the surface $x=0$ and maintained at a constant temperature $T_L$ at the surface $x=L.$ Energy is generated in the wall at a constant rate of $g\frac{W}{m^3}.$

$($a$)$ Develop an expression for the steady$-$state temperature distribution in the slab.

$($b$)$ Determine the maximum amount of energy that can be generated in the wall if the temperature of the insulated surface should not exceed $200C$ when plate thickness $L=15cm,$ plate thermal conductivity $k=30\frac{W}{m*C},$ and the temperature of the surface at $x=L$ is $T_L=50C.$

Answer: $6{10}^4\frac{W}{m^3}$