Question$-3$

A high$-$temperature, gas$-$cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element $(k$~~$57\frac{W}{m}K)$ is encased in graphite $(k$~~$3\frac{W}{m}K)$ and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium temperature is $T_{}=600K$ and the convection coefficient at the outer surface of the graphite is $h=2000\frac{W}{m^2}K.$

a$)$ If thermal energy is uniformly generated in the fuel element at a rate $q={10}^8\frac{W}{m^3},$ what are the temperatures $T1$ and $T2$ at the inner and outer surfaces, respectively, of the fuel element?

b$)$ Compute and plot the temperature distribution in the composite wall for selected values of $q.$ What is the maximum allowable value of $q?$