Coefficient of performance
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The coefficient of performance of a heat pump is the ratio of the output heat to the supplied work or
<math>COP = \frac{|Q|}{W}<math> where Q is the useful heat supplied by the condensator and W is the work consumed by the compressor.
According to the first law of thermodynamics, <math>Q_{hot}=Q_{cold}+W <math> and <math>W=Q_{hot}-Q_{cold}<math>, where <math>Q_{hot}<math> is the heat given off by the warm heat reservoir and <math>Q_{cold}<math> is the heat received by the cold heat reservoir. Therefore, by substituting for W, <math> COP_{heating}=\frac{Q_{hot}}{Q_{hot}-Q_{cold}}<math> It can be shown that <math> \frac{Q_{hot}}{T_{hot}}=\frac{Q_{cold}}{T_{cold}}<math> and <math>Q_{cold}=\frac{Q_{hot}T_{cold}}{T_{hot}}<math>, where <math>T_{hot} <math> and <math>T_{cold}<math> are the temperatures of the hot and cold heat reservoirs respectively.
Hence, <math> COP_{heating}=\frac{T_{hot}}{T_{hot}-T_{cold}} <math> Similarly, <math> COP_{cooling}=\frac{Q_{cold}}{Q_{hot}-Q_{cold}} =\frac{T_{cold}}{T_{hot}-T_{cold}}<math>
<math>COP_{heating}<math> applies to heat pumps and <math>COP_{cooling}<math> applies to air conditioners or refrigerators. For heat engines, see Efficiency.