Coefficient of performance

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The coefficient of performance, or COP (sometimes CP), of a heat pump is the ratio of the output heat to the supplied work or

COP = \frac{|Q|}{W}
where Q is the useful heat supplied by the condenser and W is the work consumed by the compressor. (Note: COP has no units, therefore in this equation, heat and work must be expressed in the same units.)

According to the first law of thermodynamics, Qhot = Qcold + W and W = Qhot − Qcold, where Qhot is the heat taken in by the cold heat reservoir and Qcold is the heat given off by the hot heat reservoir.
Therefore, by substituting for W,
COP_{heating}=\frac{Q_{hot}}{Q_{hot}-Q_{cold}}
It can be shown that \frac{Q_{hot}}{T_{hot}}=\frac{Q_{cold}}{T_{cold}} and Q_{cold}=\frac{Q_{hot}T_{cold}}{T_{hot}}, where Thot and Tcold are the temperatures of the hot and cold heat reservoirs respectively.

Hence,
COP_{heating}=\frac{T_{hot}}{T_{hot}-T_{cold}}
Similarly,
COP_{cooling}=\frac{Q_{cold}}{Q_{hot}-Q_{cold}} =\frac{T_{cold}}{T_{hot}-T_{cold}}


It can also be shown that COPcooling = COPheating − 1

A geothermal heat pump heating a building and operating at COPheating 3.5 provides 3.5 units of heat for every unit of energy it consumes (e.g. such a heat pump consuming 1 kW would provide 3.5 kW to heat a building). The "provided" heat comes from both the heat source and the input energy, so this does not mean that the heat source is cooled at a rate of 3.5 kW. The COP of heat pumps seems to compare very 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. Even so, a heat pump of COPheating 3.5, such as in the example above, could be less expensive to use than even the most efficient gas furnace.

A heat pump (air conditioner) cooling a building and operating at COPcooling 2.0 removes 2 units of heat for every unit of energy it consumes (e.g. such an air conditioner consuming 1 kW would remove heat from a building's air at a rate of 2 kW, partly through cooling and partly through dehumidification).

COPheating applies to heat pumps and COPcooling applies to air conditioners or refrigerators. For heat engines, see Efficiency.


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