Question $3-(30$ Points$)$: An extended surface $($rod$)$ with square cross section area is protruding from a large wall at temperature $T_b,$ as shown below. The rod has uniform volumetric heat generation $q_g^{}$ throughout experiences a constant surface heat flux $q_s^{\text{'}\text{'}}$ on the top surface and convection on one side surface, while the other side surface is radiating energy to the surrounding with a constant radiating surface heat flux $q_{\text{r}\text{a}\text{d}\text{.}}^{\text{'}\text{'}}.$ Its bottom surface is perfectly insulated. Under steady state conditions, the temperature is only function of location along the rod,$x(T=T(x)),$

a$.(12$ points$)$ Write the differential equation, using a differential control volume $($slicing method$),$ and boundary conditions to solve for temperature profile as a function of $x,$ if we assume that the rod is infinitely long.

b$.(6$ points$)$ Solve the differential equation in $($a$)$ and provide the formulation for the temperature profile, $T(x).$

c$.(4$ points$)$ How long should $L$ be for the infinitely long assumption in $($a$)$ to be valid and why?

d$.(8$ points$)$ What would the rod temperature be if it is much longer than the length you found in part $($c$)?$ HINT: AT THIS POINT, $T=T(x)$~$=$ constant.