Heat-Pump COP + Reversible Limit

Get heating COP from compressor work plus the Carnot ceiling for the same temperature lift.

InputCOP_h = Q̇_H / Ẇ , COP_Carnot = T_H / (T_H − T_C) , η₂ = COP_h / COP_Carnot (T in K)

Heat delivered Q̇_H (kW)or enter BTU/h — converts to kWCompressor work Ẇ (kW)or enter hp — converts to kWCold reservoir temp (°C)or enter °F — converts to °CHot reservoir temp (°C)or enter °F — converts to °C

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The engineering

The coefficient of performance is heat moved per unit of work put in — for a heat pump in heating mode that's the warm-side delivery Q̇_H divided by compressor work Ẇ. Because the pump moves ambient heat rather than making it, COP comfortably exceeds 1: a value of 4 means 4 kW of heat for 1 kW of electricity. Refrigeration mode uses Q̇_C instead, giving a number one lower.

The Carnot column is the thermodynamic ceiling for the temperature lift you asked for — no real machine beats it, and the smaller the lift (T_H − T_C), the higher that ceiling climbs. That's why heat pumps fall off hard on frigid days: as the outdoor coil temperature drops the lift grows and both the ideal and the real COP sag. A good air-source unit runs a second-law efficiency of roughly 40–55% of Carnot; if your η₂ comes out above 100% you've mixed up your temperatures or your power measurement.

Where this math comes from

Sadi Carnot laid the groundwork in 1824 with Réflexions sur la puissance motrice du feu, proving that the maximum work between two reservoirs depends only on their temperatures. William Thomson (Lord Kelvin) turned the idea around in 1852, pointing out that a heat engine run backward would pump heat 'up' a temperature gradient — the theoretical heat pump — and even proposed warming buildings with one.

The practical machine came later: Peter von Rittinger built an early heat-pump evaporator for Austrian salt works in 1855, and Robert Webber cobbled together a ground-source home system in the 1940s. Today ASHRAE and AHRI codify how COP (and its cousins HSPF and SEER) are measured on the test stand, so a rated number means the same thing across manufacturers.

1824Sadi CarnotEstablishes the reversible-cycle temperature limit on heat-to-work conversion.
1852William Thomson (Lord Kelvin)Describes running a heat engine in reverse to heat buildings — the heat pump concept.
1855Peter von RittingerBuilds an early industrial heat-pump evaporator for salt drying.
1948Robert C. WebberConstructs one of the first ground-source residential heat pumps.
2020ASHRAE / AHRIStandardize COP, HSPF and SEER rating methods for equipment comparison.

See the full timeline of the math behind every calculator →

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