What is the efficency of a heat pump driving a phase change? If a heat pump is positioned between two reservoirs of 100C liquid water and drives heat from one reservoir to the other in order to boil the "hot" side what is the coefficient of performance of the pump? The Carnot equation simplifies to 1/0 when the temperatures are equal but that doesn't make physical sense here.
Edit:
Phrasing the problem differently; how much energy does a heat pump require to boil 1kg of 100C water if it's pumping heat from a 100C temperature reservoir?
 A: This isn't an answer in the typical sense, but some engineering concepts to consider.  For your heat pump to work, it must contain a working fluid.  That working fluid will exist as a liquid and will boil as it absorbs heat from the "cold" reservoir.  When the working fluid gets to the "hot" reservoir, it must condense to release its heat.  Condensation requires a temperature difference, and that temperature difference depends on the heat transfer rate, the area of the heat exchanger that exists on the "hot" end of the heat pump, and a few other factors that have to be considered in heat exchanger design, as pointed out by @Bob D (e.g., overall heat transfer coefficient, log mean temperature difference, etc.).  A typical temperature difference in practice is 10 deg C, so you would have a heat pump that condenses its working fluid at 110 deg C, meaning that the "hot" side temperature would be the condensing temperature of the working fluid.  This situation is definitely not typical for heat pumps, but considering the condensing temperature of the working fluid as your "hot" side temperature would avoid the division by zero that you noted in your question.
