Zone refining of metals is a materials processing technology based on the fact that when a molten metal crystallizes on cooling, impurities are automatically expelled since they do not form part of the pure crystal. Germanium, silicon, and gallium are examples of metals refined in this manner. As shown in the sketch below, impure germanium (Ge), for example, is made into a rod. A heater of circular geometry heats the rod radially, thereby producing a region in which the temperature is uniformly elevated to the melting point in a plane perpendicular to the axis of the rod. The heater can be moved over the length of the rod so that the molten zone propagates down the rod. In a typical embodiment of this process, a piece of ultra-pure Ge (seed) is attached to the rod on the left hand side. The heater is placed over it. The Ge seed and the impure Ge rod melt. The heater is moved to the right slowly. The molten seed and the Ge from the impure rod re- crystallize as the temperature drops as the heater moves to the right away from the left-hand end of the rod. The ultra-pure Ge crystal grows in size as the molten zone progresses down the rod. The ultra-pure Ge moves to the left and the impurities move in the opposite direction. The net result of the process is the concentration of the impurities in the right-hand end of the rod with the major portion of the rod to the left pure Ge. For Ge: k = 59 W/m-K, = 5323.4 kg/m3, c = 322.22 J/kg-K, Tmelting = 1231 K Now consider a Ge rod of length L = 1 m and diameter D = 12 cm. For the case in which the molten zone is located midway between the two ends of the rod, estimate the power necessary to heat the rod if the heat transfer coefficient from the surface of the rod is h = 5 W/m2K. The molten zone has a thickness of 1 cm measured along the axis of the rod, and the temperature of the air is Tair = 300 K