1. (50 points) A pebble bed reactor is a new type of nuclear reactor that was developed in the early 2000 s. The fuel for these reactors are "pebbles" of nuclear material. Specifically, they are nuclear material made into a sphere. As with all nuclear material, there is a constant nuclear reaction within the material, which is exothermic. The nuclear fuel is kept in the reactor and is cooled by a constant flow of inert gas held at a temperature of T. The outer radius of the fuel sphere is R. Find the specific temperature profile for the fuel sphere and the heat flux at the interface for the system at steady state