Coefficient of Performance Calculator

This coefficient of performance calculator determines the coefficient of performance of reversible and irreversible refrigerators and heat pumps.

Coefficient of performance, cooling7.6986a refrigerator moves this much heat per unit of work — it exceeds 1, which is why "efficiency" is the wrong word for it

Carnot efficiency11.496%1 − 277.15 ÷ 313.15 — the absolute ceiling for ANY heat engine between these two temperatures

Hot reservoir313.15 K40 °C

Cold reservoir277.15 K4 °C

Temperature ratio0.885039the fraction of heat that must be rejected

Coefficient of performance, heating8.6986a heat pump delivers this much heat per unit of work, and it is always exactly the cooling COP plus one

Check: COP_heat − COP_cool1exactly 1, always — the heat pump also delivers the work itself as heat

Heat input needed at Carnot efficiency434.9306 kW

Heat rejected to the cold reservoir384.9306 kW88.5% of the input — this is not waste that better engineering can eliminate, it is thermodynamically required

Second-law efficiency278.36%the actual efficiency as a fraction of the Carnot limit — the honest measure of how good a real engine is

That is impossibleabove the Carnot limitno engine can exceed 11.5% between these temperatures — check the reservoir temperatures

Real heat input at that efficiency156.25 kW

Work to move a 3.5 kW cooling load454.6275 Wat the Carnot COP — a real machine needs two to three times this

If the hot side were 50 K hotter23.682%12.186 points better

If the cold side were 50 K colder27.463%15.967 points better — lowering the cold side is usually the stronger lever, and usually the harder one, since it is set by the environment

The formula

η_Carnot = 1 − T_c/T_h; COP_cooling = T_c/(T_h−T_c)

A ceiling nothing can beat

Carnot efficiency depends only on the two absolute temperatures, not on the working fluid, the mechanism or the engineering. No heat engine operating between a given pair of reservoirs can beat it — this is the second law of thermodynamics stated as a number, and the rejected heat is not waste that better design could recover.

Because the formula uses absolute temperatures, real engines are limited far more than people expect. A steam plant at 250 °C rejecting to 25 °C has a Carnot ceiling around 43%, and achieves perhaps 35% in practice. The way to improve it is a bigger temperature ratio, which usually means a hotter source, since the cold side is set by the environment.

Coefficient of performance is the same physics run backwards, and it exceeds 1 — a heat pump can deliver several units of heat per unit of electricity, because it moves heat rather than generating it. That is why "efficiency" is the wrong word and a separate term exists. The heating COP always exceeds the cooling COP by exactly 1, because the work put in also ends up as heat in the warm space.

Second-law efficiency — actual divided by Carnot — is the fair way to judge a machine. A 35% engine sounds poor until you note it is achieving 81% of the maximum physics permits.