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

<math>COP = \frac{|Q|}{W}</math> 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, <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 taken in 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>

It can also be shown that <math> COP_{cooling}=COP_{heating}-1 </math>

[edit] Example

A geothermal heat pump operating at COP 3.5 is able to move 3.5 kwh (11,946 BTUh) of heat for every 1 kwh it consumes. This can also be viewed as an efficiency of 350%, which compares very favorably to high efficiency (condensing) gas burning furnaces (90-99% efficient), and electric heating (100%). (The COP of an air source heat pump may be 2.0 (200% efficiency) at low outdoor air temperatures before its backup electric resistance heating coils are turned on.) {{{author}}}, {{{title}}}, [[{{{publisher}}}]], [[{{{date}}}]].

<math>COP_{heating}</math> applies to heat pumps and <math>COP_{cooling}</math> applies to air conditioners or refrigerators. For heat engines, see Efficiency.