THC Science

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 taken in by the cold heat reservoir and $Q_{cold}$ is the heat given off by the hot 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 units of heat for every unit it consumes (e.g. such a heat pump consuming 1 kW would provide 3.5 kW to heat a building). This seems to compare very favorably with high-efficiency gas-burning furnaces (90-99% efficient), and electric heating (100%), but the full costs of the energy consumed must be considered, and gas is typically much less expensive.

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

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