1 Introduction

Thermodynamics imposes fundamental limits on the efficiency of a refrigerator, air conditioner, or heat pump.

1.1 Air Conditioner or Refrigerator

To understand where this comes from, start with the left panel in figure ‍1, which shows the entropy balance for a refrigerator or air conditioner. The device takes in a certain amount of entropy dS from the cold side, i.e. the space you are trying to cool. It then pushes that entropy out the hot side. The second law of thermodynamics says entropy can never be destroyed. You can always create entropy, but that would be unnecessarily wasteful, so under ideal conditions we won’t do that. Therefore, ideally, entropy out is the same as entropy in.

Figure ‍1: Air Conditioner or Refrigerator

Now we consider the right panel, which shows the energy balance. To push the entropy out the hot side requires an energy T_H dS. We get most of that energy for free from the cold side, namely T_C dS. That leaves us with a shortfall, namely dE_X = T_H dS − T_C dS that must be supplied from an external source.

The Carnot efficiency is the bang for the buck, ı.e. the amount of energy removed from the space we care about, divided by the amount of energy we have to pay for.

1.2 Heat Pump

The same logic applies to a heat pump, used to provide warmth in winter. All we have to do is flip the diagram upside down.

Figure ‍2: Heat Pump

Once again, the Carnot efficiency is the bang for the buck, ı.e. the amount of energy provided to the space we care about, divided by the amount of energy we have to pay for. The formula is not quite the same, because the space we care about this time is the hot side. So we wind up with T_H in the numerator:

Note that when the temperature difference is small, the efficiency is tremendously large. That is to say, the heat pump is vastly cheaper to operate compared to a plain old electric heater.