Coefficient of performance

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, $Q_{hot}=Q_{cold}+W$ and $W=Q_{hot}-Q_{cold}$ , where $Q_{hot}$ is the heat given off by the warm heat reservoir and $Q_{cold}$ is the heat taken in by the cold 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 $T_{hot}$ and $T_{cold}$ 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 $COP_{cooling}=COP_{heating}-1$

Example

A geothermal heat pump operating at COP 3.5 is able to move 3.5 (11,946 BTUh) kwh 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.)

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

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Example

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