Problem $2(4$ points$)$: A thin rod of radius $r$ length $3L$ and thermal conductivity $k$ is divided into three regions, each of equal length. Regions $1$ and $2$ are surrounded by a perfectly insulating wall, while Region $3$ protrudes into ambient air at temperature $T_{},$ with convection coefficient $h.$ The tip of the protruding rod is insulated. Uniform heat generation $q_g^{}$ occurs only in region $1.$

a$.$ Choose suitable coordinate system, and for each region, write down the temperature profile in terms of constants. For each region, specify why the temperature profile has the form you have chosen. What would you choose as the direction of heat transfer for each region?

b$.$ Using the temperature profiles from part $($a$),$ qualitatively sketch the temperature profile along the entire length of the rod. Also, specify the nature of the slopes at the start and at the end of the profile, and describe why it is so$.$ Label all regions where temperature changes occur and label the overall nature of the profile at each of regions $1,2$ and $3.$ Finally, do not forget to label the axes.

c$.$ Provide a qualitative sketch of the heat flux profile across all the three regions.

d$.$ Derive an expression for the temperature $T_o$ in terms of the variables of the problem, i$.$e$.,$ temperature $T_o$ should be obtained as a function of $r,h,L,k,q_g^{},T_{}$