Editing Coefficient of performance (section)

==Theoretical performance limits==
According to the [[first law of thermodynamics]], after a full cycle of the process <math>Q_{\rm H}+Q_{\rm C}+W = \Delta_{\rm cycle}U = 0 </math> and thus <math>W=-\ Q_{\rm H}-Q_{\rm C}</math>.<br>
Since <math>  |Q_{\rm H}| = -Q_{\rm H} \ </math>, we obtain
:<math> {\rm COP}_{\rm heating}=\frac{Q_{\rm H}}{Q_{\rm H}-Q_{\rm C}}</math>

For a heat pump operating at maximum theoretical efficiency (i.e. [[Carnot efficiency]]), it can be shown<ref name="FermiBook">{{cite book |last=Fermi |first=E. |title=Thermodynamics |page=48 |quote= eq.(64) |publisher=Dover Publications (still in print) |year=1956}}.</ref><ref name="PlanckBook"/> that 
:<math> \frac{Q_{\rm H}}{T_{\rm H}}+ \frac{Q_{\rm C}}{T_{\rm C}}=0</math>   and thus   <math>Q_{\rm C}=-\frac{Q_{\rm H}T_{\rm C}}{T_{\rm H}}</math>
where <math>T_{\rm H} </math> and <math>T_{\rm C}</math> are the [[thermodynamic temperature]]s of the hot and cold heat reservoirs, respectively.

At maximum theoretical efficiency, therefore
:<math> {\rm COP}_{\rm heating}=\frac{T_{\rm H}}{T_{\rm H}-T_{\rm C}} </math>
which is equal to the reciprocal of the [[thermal efficiency]] of an ideal [[heat engine]], because a heat pump is a heat engine operating in reverse.<ref>Borgnakke, C., & Sonntag, R. (2013). The Second Law of Thermodynamics. In Fundamentals of Thermodynamics (8th ed., pp. 244-245). Wiley.</ref>

Similarly, the COP of a refrigerator or air conditioner operating at maximum theoretical efficiency,
:<math> {\rm COP}_{\rm cooling}=\frac{Q_{\rm C}}{\ Q_{\rm H}-Q_{\rm C}} =\frac{T_{\rm C}}{T_{\rm H}-T_{\rm C}}</math>

<math>{\rm COP}_{\rm heating}</math> applies to heat pumps and <math>{\rm COP}_{\rm cooling}</math> applies to air conditioners and refrigerators. 
Measured values for actual systems will always be significantly less than these theoretical maxima.

In Europe, the standard test conditions for ground source heat pump units use 308&nbsp;K (35&nbsp;°C; 95&nbsp;°F) for <math>{T_{\rm H}}</math> and 273&nbsp;K (0&nbsp;°C; 32&nbsp;°F) for <math>{T_{\rm C}}</math>. According to the above formula, the maximum theoretical COPs would be  <br>
:<math> {\rm COP}_{\rm heating}=\frac{308}{308-273} = 8.8</math><br>
:<math> {\rm COP}_{\rm cooling}=\frac{273}{308-273} = 7.8</math>

Test results of the best systems are around 4.5. When measuring installed units over a whole season and accounting for the energy needed to pump water through the piping systems, seasonal COP's for heating are around 3.5 or less. This indicates room for further improvement.

The EU standard test conditions for an air source heat pump is at [[dry-bulb temperature]] of 20&nbsp;°C (68&nbsp;°F) for <math>{T_{\rm H}}</math> and 7&nbsp;°C (44.6&nbsp;°F) for <math>{T_{\rm C}}</math>.<ref>According to European Union COMMISSION DELEGATED REGULATION (EU) No 626/2011 ANNEX VII Table 2</ref> Given sub-zero European winter temperatures, real world heating performance is significantly poorer than such standard COP figures imply.