A cylindrical graphite electrode of diameter D passes through a furnace wall into a water cooler. The length of the electrode between the outside of the furnace wall and its entry into the cooling jacket is L. If the heat lost to the surrounding atmosphere per m2 of the electrode is constant (Q); and the temperature of the electrode just outside the furnace is TH; estimate the amount of heat that must be removed by the water cooler if the temperature of the electrode at the entrance to the cooler is to be Tc. The thermal conductivity of the graphite is kT=k0T where k0 and are constants. The temperature of the electrode may be assumed uniform at any cross-section. a) Show that the temperature distribution in the electrode is given by the following differential equation: (k0T)dx2d2T(dxdT)2D4Q=0 b) Write the appropriate boundary conditions. c) Solve the differential equation given in part (a) using an appropriate analytical technique. (Hint: Let u=dT/dx. At one point in your solution you may need the variable transformation z=(k0T)2.)