Carnot Efficiency & COP Calculator

Understanding the Carnot efficiency and COP calculator — concept and principle

The Carnot efficiency is the highest fraction of heat that any engine can turn into work when it runs between a hot and a cold reservoir. The same two temperatures also set the best possible coefficient of performance (COP) for a refrigerator and a heat pump. Enter the hot and cold temperatures and the calculator returns all three limits. They serve as benchmarks for engines, fridges and heat pumps.

Between 500 °C and 25 °C, at most 61.44% of the heat can become work.

How it is calculated

With both temperatures converted to kelvin (T = °C + 273.15):

Carnot efficiency: η = 1 − T_c / T_h
Maximum refrigerator COP: COP_R = T_c / (T_h − T_c)
Maximum heat pump COP: COP_HP = T_h / (T_h − T_c)

T_h is the hot reservoir, T_c the cold one. Note that COP_HP = COP_R + 1 = 1 / η.

The principle

Sadi Carnot showed in 1824 that an ideal reversible engine, running on a cycle of two isothermal and two adiabatic steps, gives the maximum possible efficiency. In the language of the second law of thermodynamics, a reversible cycle creates no entropy: the entropy taken in with heat Q_h at T_h must equal the entropy rejected with Q_c at T_c, so Q_c / Q_h = T_c / T_h. The work is Q_h − Q_c, which gives η = 1 − T_c/T_h. Any real engine generates extra entropy through friction and heat transfer across temperature differences, and so achieves less. Running the cycle backwards gives the refrigerator and heat pump limits.

Worked example

Inputs: T_h = 500 °C, T_c = 25 °C.
In kelvin: T_h = 773.15 K, T_c = 298.15 K, difference 475 K.
Efficiency: 1 − 298.15 / 773.15 = 1 − 0.38563 = 61.44%.
Refrigerator COP: 298.15 / 475 = 0.6276842105.
Heat pump COP: 773.15 / 475 = 1.627684211.

Each point is the calculator re-run with only “Hot reservoir (°C)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Hot reservoir (°C)	Carnot efficiency
125	25.12%
250	43.01%
375	54%
500	61.44%
625	66.8%
750	70.86%
1,000	76.58%

How to read the result

The efficiency rises when the hot side gets hotter or the cold side colder, which is why power plants push steam and gas temperatures as high as materials allow. For fridges and heat pumps the COP is large when the temperature lift is small: the default case, pumping heat across 475 K, gives very low COPs, while a heat pump lifting heat from 0 °C outdoors to 35 °C indoor water would have a Carnot COP of about 8.8. Real heat pumps reach roughly half of that or less.

Common mistakes

Using °C in the ratio; 25/500 would give a meaningless 95%.
Expecting real machines to reach the Carnot value; it is an upper bound, not a prediction.
Swapping the two inputs; T_h must be the higher temperature.

Keep in mind: Carnot limits assume perfectly reversible processes and reservoirs at constant temperature. They tell you what is impossible, not what a specific machine will do. For real equipment rely on rated or measured performance.

Frequently asked questions

Can an engine ever be 100% efficient?

Only with a cold reservoir at absolute zero, which cannot be reached. The second law forbids converting all heat to work in a cycle.

Why is a heat pump COP above 1 not a violation of energy conservation?

The heat pump moves existing heat from a cold place to a warm one; the electricity only drives the transfer, so delivered heat can exceed electrical input.

Sources: Y. A. Çengel and M. A. Boles, Thermodynamics: An Engineering Approach, McGraw-Hill · S. Carnot, Réflexions sur la puissance motrice du feu, Paris, 1824

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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