Coefficient of performance

The coefficient of performance or COP (sometimes CP or CoP) of a heat pump, refrigerator or air conditioning system is a ratio of useful heating or cooling provided to work required. Higher COPs equate to lower operating costs. The COP usually exceeds 1, especially in heat pumps, because, instead of just converting work to heat (which, if 100% efficient, would be a COP_hp of 1), it pumps additional heat from a heat source to where the heat is required. For complete systems, COP calculations should include energy consumption of all power consuming auxiliaries. COP is highly dependent on operating conditions, especially absolute temperature and relative temperature between sink and system, and is often graphed or averaged against expected conditions.

The equation is:

where

The COP for heating and cooling are thus different, because the heat reservoir of interest is different. When one is interested in how well a machine cools, the COP is the ratio of the heat removed from the cold reservoir to input work. However, for heating, the COP is the ratio of the heat removed from the cold reservoir plus the input work to the input work:

where

Both means the same for understanding.

According to the first law of thermodynamics, in a reversible system we can show that ${\displaystyle Q_{hot}=Q_{cold}+W}$ and ${\displaystyle W=Q_{hot}-Q_{cold}}$, where ${\displaystyle Q_{hot}}$ is the heat transferred to the hot reservoir and ${\displaystyle Q_{cold}}$ is the heat collected from the cold reservoir.
Therefore, by substituting for W,

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