$3-(35$ pts $)$ Consider a long $r_1=100mm$ radius full inner cylindrical rod made up of a material of thermal conductivity $k_1=0\mathrm{.}05\frac{W}{m}K$ in which a uniform heat generation of $24000\frac{W}{m^3}$ occurs. This rod is placed in a tube of outer radius $r_2=200mm,$ whose thermal conductivity is $k_2=4\frac{W}{m}K.$ Outer surface is exposed to a liquid of temperature $100C$ and combined heat transfer coefficient of $h=20\frac{W}{m^2}K.$ Schematic illustration is shown in the figure.

a$)$ Find interface temperature $T_1$ and outer surface temperature $T_2.$

b$)$ Center temperature of inner cylinder.