Derivation
According to the first law of thermodynamics, in a reversible system we can show that and, where is the heat taken in by the hot heat reservoir and is the heat given off by the cold heat reservoir.
Therefore, by substituting for W,
For a heat pump operating at maximum theoretical efficiency (i.e. Carnot efficiency), it can be shown that and, where and are the absolute temperatures of the hot and cold heat reservoirs respectively.
At maximum theoretical efficiency,
Which is equal to the inverse of the ideal Carnot cycle efficiency because a heat pump is a heat engine operating in reverse. Similarly,
It can also be shown that . Note that these equations must use the absolute temperature (the Kelvin or Rankine scale.)
applies to heat pumps and applies to air conditioners or refrigerators. For heat engines, see Efficiency. Values for actual systems will always be less than these theoretical maximums. In Europe, ground source heat pump units are standard tested at is 35 °C (95 °F) and is 0 °C (32 °F). According to the above formula, the maximum achievable COP would be 8.8. Test results of the best systems are around 4.5. When measuring installed units over a whole season and one also counts the energy needed to pump water through the piping systems, then seasonal COP's are around 3.5 or less. This indicates room for improvement.
Read more about this topic: Coefficient Of Performance