- is the heat moved from the cold reservoir (to the hot reservoir).
According to the first law of thermodynamics, in a reversible system we can show that and , where is the heat given off by the hot heat reservoir and is the heat taken in 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 temperatures of the hot and cold heat reservoirs respectively.
Hence, at maximum theoretical efficiency,
It can also be shown that . Note that these equations must use the absolute temperature, such as the Kelvin 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.
A geothermal heat pump operating at 3.5 provides 3.5 units of heat for each unit of energy consumed (e.g. 1 kW consumed would provide 3.5 kW of output heat). The output heat comes from both the heat source and 1 kW of input energy, so the heat-source is cooled by 2.5 kW, not 3.5 kW.
A heat pump of 3.5, such as in the example above, could be less expensive to use than even the most efficient gas furnace.
A heat pump cooler operating at 2.0 removes 2 units of heat for each unit of energy consumed (e.g. such an air conditioner consuming 1 kW would remove heat from a building's air at a rate of 2 kW).
The COP of heat pumps compares favorably with high-efficiency gas-burning furnaces (90-99% efficient), and electric heating (100%), but the full costs of the energy consumed must be considered, and energy from gas is typically much less expensive than that from electricity.