The Carnot efficiency calculator computes the maximum theoretical efficiency of a heat engine operating between a hot reservoir (T_hot) and a cold reservoir (T_cold). Named after Sadi Carnot, this upper bound — η = 1 − T_cold/T_hot — is set by the second law of thermodynamics. The tool also calculates the Coefficient of Performance (COP) for heat pumps and refrigerators.
Temperature Inputs
Enter temperatures and calculate
Coefficient of Performance (COP)
Real Engine Efficiency Comparison
| Engine Type | Typical Efficiency | vs. Carnot Limit |
|---|---|---|
| Carnot (ideal) | — | 100% of limit |
| Combined cycle gas | 55–60% | — |
| Fuel cell | 40–60% | — |
| Diesel engine | 35–45% | — |
| Gas turbine | 35–40% | — |
| Nuclear power plant | ~33% | — |
| Gasoline engine | 25–30% | — |
How to Use the Carnot Efficiency Calculator
The Carnot efficiency formula η = 1 − T_cold/T_hot gives the absolute maximum efficiency any heat engine can achieve. It was derived by Sadi Carnot in 1824 before thermodynamics was formally developed. The result is remarkably general: no engine, however clever, can exceed this limit when operating between two given temperature reservoirs.
Power Plant Example
A coal power plant uses steam at 600°C (873 K) and exhausts to a 30°C (303 K) environment. Carnot limit: η = 1 − 303/873 = 1 − 0.347 = 65.3%. Real plants only achieve 33-40% due to friction, heat loss, and non-ideal steam behavior. Combined cycle gas plants reach 55-60% by using waste heat from a gas turbine to power a steam turbine.
Heat Pump COP
A heat pump moves heat from cold outside air (say −5°C = 268 K) to a warm house (20°C = 293 K). Carnot COP_heat = 293/(293−268) = 293/25 = 11.7. Real heat pumps achieve COP 3-5, meaning they deliver 3-5 kWh of heat for every 1 kWh of electricity consumed. This is why heat pumps are more efficient than direct electric heating.
Temperature Entry
You can enter temperatures in Kelvin, Celsius, or Fahrenheit — the calculator converts automatically. The Carnot formula requires temperatures in Kelvin (absolute scale). Remember: T_hot must be greater than T_cold, and both must be positive Kelvin values.
FAQ
What is Carnot efficiency?
Carnot efficiency is the maximum theoretical efficiency any heat engine can achieve operating between two temperature reservoirs. η = 1 − T_cold/T_hot, where temperatures must be in Kelvin. No real engine can exceed this limit — it represents an upper bound set by the second law of thermodynamics.
Why must temperatures be in Kelvin?
The Carnot formula η = 1 − Tc/Th requires absolute temperatures. Using Celsius or Fahrenheit gives incorrect results because these are relative scales (0°C is not 'no temperature'). This calculator automatically converts your input to Kelvin before computing. 0°C = 273.15 K, 100°C = 373.15 K.
What is the coefficient of performance (COP)?
COP measures how efficiently a heat pump or refrigerator moves heat. For a heat pump: COP_heat = T_hot/(T_hot − T_cold) — values greater than 1 are normal (you get more heat energy out than electricity in). For a refrigerator: COP_cool = T_cold/(T_hot − T_cold). A home heat pump might have COP_heat = 3-4.
What is the efficiency of a typical power plant?
A coal power plant running between ~600°C (873 K) and ~30°C (303 K) has Carnot limit of 1 − 303/873 ≈ 65.3%. Real plants achieve only 33-40% due to irreversibilities like friction, heat loss, and non-ideal working fluid behavior. Combined cycle gas plants reach 55-60% efficiency.
Is this tool free?
Yes, completely free with no signup required. All calculations run locally in your browser.
What happens if T_cold equals T_hot?
If both reservoirs are the same temperature, η = 1 − 1 = 0. No work can be extracted from two reservoirs at the same temperature — this is a statement of the second law. You need a temperature difference to run a heat engine, just as you need a pressure difference for water to flow.