A cylindrical rod of length 2L, diameter D, and thermal conductivity k is inserted into a wall of insulation. One half of the rod is exposed to air with a temperature t_infinity and a convective heat transfer coefficient of h at the surface of the rod. The embedded portion of the rod experiences a uniform heat generation of: (q-dot_v''')(in W/m^3). The exposed portion of the rod does not experience heat generation.

For 1 and 2, the expressions need to be in terms of the known variables, T_b and T_0 are unknown, everything else labeled in the figure is known.

1. Derive an expression for the steady-state temperature, T_b, at the base of the rod (x=0). The exposed portion of the rod should be modeled as an infinitely long fin.

2. Derive an expression for the steady-state temperature, T_0, at the end of the embedded portion of the rod (x=-L). You can treat T_b as a known variable since you derived it above. (Hint: begin with the heat diffusion equation)

3. Find the temperature at the end of the exposed portion of the rod (x=L). Use: L=50mm, D=5mm, k=25 W/mK, q-dot_v''' = 1*10^6 W/m^3, T_infinity=20 degrees Celsius, h=100 W/m^2-K

4. Is the infinitely long fin approximation valid? Why or why not?