Primary aluminum production is the making of metallic aluminum from bauxite ore. The method used in all primary aluminum production is the Hall-Héroult process, in which an electric current is run through a high-temperature molten-salt bath, and bauxite is electrochemically converted to aluminum at the cathode. This process requires large amounts of electrical energy, as well as process heat to maintain the bath at a sufficiently high temperature (over 900°C). A key problem in this process is that the cathodic current efficiency in industrial-scale processes is only between 87% and 96%. (Perfect efficiency, 100%, would be achieved if every mole of electrons produced one-third mole of metallic aluminum. Note that the "valence" of Al is +3.). A bright young electrochemical engineer has realized that much of this inefficiency is due to inadequate mass transfer at the cathode (where molten aluminum is produced), and that if the boundary layer thickness can be reduced, mass transfer will be enhanced and the cathodic current efficiency can be increased.

The process is very complex, and involves use of a sacrificial cathode (made of coke, which reacts with oxygen to make CO2), gravitational settling of the molten aluminum formed at the cathode, and other complications. In addition, heat is added through one surface of the cell (SA), and lost from other surfaces (SB). In what follows, you should make the following simplifying assumptions. First, you should assume that the process operates at steady state.

Second, you should assume that the rate of heat loss from SB (in terms of energy per unit time) depends only on the liquid temperature and the temperature of the surroundings, which are assumed to be the same with and without the improved mixing. Third, you should neglect the power required to improve the mixing. (The electrochemical engineer knows a good bit of fluid mechanics, and has a clever way to do that, which requires very little power.) Fourth, you should assume that the electrical current input is the same with and without the improved mixing. Fifth, you should treat this as a thermodynamically closed system. Finally, you should neglect the change in chemical enthalpy associated with the process.

When the proposal to increase cathodic current efficiency is evaluated by a senior process engineer, he is concerned that higher cathodic current efficiency will lead to less Joulean dissipation of current into heat, and a requirement for additional process heat. He therefore poses the question "For each kilogram of metallic aluminum produced, how much more process heat will we need to add, in order to maintain the temperature of the salt bath?"

a) If the rate of heat loss from SB to the surroundings is 800 kW, and the DC electrical input is 130 A at 240 V, how much heat must be added through SA? (Make the assumptions stated above.)

b) Making the assumptions stated above, explain why the answer to part a) does or does not depend on the current efficiency.

c) Explain why one or more of the assumptions above causes he answer to part b) to be incorrect in the "real world."