Coefficient of performance
From Free net encyclopedia
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> </br> where Q is the useful heat supplied by the condenser 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.</br> Therefore, by substituting for W,</br> <math> COP_{heating}=\frac{Q_{hot}}{Q_{hot}-Q_{cold}}</math></br> 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, </br> <math> COP_{heating}=\frac{T_{hot}}{T_{hot}-T_{cold}} </math> </br> Similarly, </br> <math> COP_{cooling}=\frac{Q_{cold}}{Q_{hot}-Q_{cold}} =\frac{T_{cold}}{T_{hot}-T_{cold}}</math></br>
It can also be shown that <math> COP_{cooling}=COP_{heating}-1 </math></br>
<math>COP_{heating}</math> applies to heat pumps and <math>COP_{cooling}</math> applies to air conditioners or refrigerators. For heat engines, see Efficiency.Template:Physics-stub