Efficiency Calculator

The efficiency calculator finds the ratio of energy output to energy input.

Second-law efficiency74.4%the actual efficiency as a fraction of the Carnot limit — the honest measure of how good a real engine is

Carnot efficiency43.009%1 − 298.15 ÷ 523.15 — the absolute ceiling for ANY heat engine between these two temperatures

Hot reservoir523.15 K250 °C

Cold reservoir298.15 K25 °C

Temperature ratio0.569913the fraction of heat that must be rejected

Coefficient of performance, cooling1.3251a refrigerator moves this much heat per unit of work — it exceeds 1, which is why "efficiency" is the wrong word for it

Coefficient of performance, heating2.3251a 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 efficiency116.2556 kW

Heat rejected to the cold reservoir66.2556 kW56.99% of the input — this is not waste that better engineering can eliminate, it is thermodynamically required

Real heat input at that efficiency156.25 kW

Work to move a 3.5 kW cooling load2.6413 kWat the Carnot COP — a real machine needs two to three times this

If the hot side were 50 K hotter47.98%4.972 points better

If the cold side were 50 K colder52.566%9.557 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.