EngineeringFormulaThermodynamics
Carnot Efficiency Calculator
Calculate the maximum theoretical heat-engine efficiency, or solve for hot or cold reservoir temperature using η = 1 − Tc/Th.
Carnot efficiency is the maximum theoretical efficiency any heat engine can achieve when operating between two thermal reservoirs. It depends only on the absolute hot and cold reservoir temperatures, not on the engine's working fluid or hardware details. This calculator lets you find the Carnot efficiency directly, or back-solve the hot or cold reservoir temperature when the other quantities are known.
Th · Absolute temperature of the high-temperature reservoir supplying heat to the engine.
Tc · Absolute temperature of the low-temperature reservoir receiving rejected heat.
Carnot efficiency uses absolute temperature. Celsius and Fahrenheit inputs are fine here because the calculator converts them to kelvin internally before applying the formula.
Solution
Enter the required values to calculate carnot efficiency.
η = 1 − Tc / Th
Formula Sheet
- ηCarnot Efficiency
- ThHot Reservoir Temperature
- TcCold Reservoir Temperature
Variables & Units
| Symbol | Variable | Description | Common Units |
|---|---|---|---|
| η | Carnot Efficiency | Maximum theoretical heat-engine efficiency between the two reservoir temperatures. | 1, % |
| Th | Hot Reservoir Temperature | Absolute temperature of the high-temperature reservoir supplying heat to the engine. | K, °C, °F |
| Tc | Cold Reservoir Temperature | Absolute temperature of the low-temperature reservoir receiving rejected heat. | K, °C, °F |
How to Use This Calculator
- 01Choose which variable to solve for first. The calculator then shows only the two inputs needed for that form of η = 1 − Tc/Th.
- 02Reservoir temperatures must be absolute in the formula, but you can enter K, °C, or °F. The calculator converts Celsius and Fahrenheit to kelvin internally before solving.
- 03If you enter efficiency, use the unit selector to choose either a decimal ratio or percent. For example, 0.4 and 40% represent the same Carnot efficiency.
- 04Use this page only for the ideal Carnot limit of a heat engine. Actual engines and full plant thermal-efficiency problems are different and always perform worse than the Carnot limit.
- 05After calculating the Carnot limit, compare any real engine efficiency against it as a ceiling. If a real claimed efficiency is above the Carnot value for the same reservoir temperatures, one of the inputs or assumptions is wrong.
How the Formula Works
For a reversible heat engine operating between a hot reservoir at Th and a cold reservoir at Tc, Carnot efficiency is η = 1 − Tc/Th. Because the formula uses a temperature ratio, the temperatures must be on an absolute scale such as kelvin or rankine.
The result is a theoretical upper bound, not an actual machine rating. Raising the hot-reservoir temperature or lowering the cold-reservoir temperature increases the maximum possible efficiency, but 100% efficiency would require the cold reservoir to be at absolute zero, which is not physically achievable.
Worked Example 01
Carnot efficiency from hot and cold reservoir temperatures
Known
- Hot Reservoir Temperature (Th): 300 °C
- Cold Reservoir Temperature (Tc): 27 °C
Formula
η = 1 − Tc / Th
Substitution
η = 1 − 300.15 / 573.15
Result
η ≈ 0.476 (47.6%)
A reversible engine working between 300 °C and 27 °C has a maximum theoretical efficiency of about 47.6%.
Worked Example 02
Hot reservoir temperature from target Carnot efficiency
Known
- Carnot Efficiency (η): 60%
- Cold Reservoir Temperature (Tc): 300 K
Formula
Th = Tc / (1 − η)
Substitution
Th = 300 / (1 − 0.6)
Result
Th = 750 K
To reach a 60% Carnot limit with a 300 K cold reservoir, the hot reservoir must be 750 K.
Worked Example 03
Cold reservoir temperature from hot reservoir temperature and efficiency
Known
- Hot Reservoir Temperature (Th): 500 K
- Carnot Efficiency (η): 40%
Formula
Tc = Th (1 − η)
Substitution
Tc = 500 × (1 − 0.4)
Result
Tc = 300 K
If the hot reservoir is 500 K and the Carnot limit is 40%, the corresponding cold reservoir temperature is 300 K.
Applications
- 01Estimating the theoretical upper limit for a heat engine operating between two temperature levels
- 02Benchmarking an actual engine or power cycle against the ideal Carnot limit
- 03Back-solving the reservoir temperature needed to support a target theoretical efficiency in conceptual studies
Assumptions
- 01The cycle is an ideal reversible Carnot heat engine operating between two uniform reservoir temperatures.
- 02The result is a maximum theoretical efficiency limit, not an actual engine efficiency.
- 03Only the reservoir temperatures matter in this idealized model; irreversibilities and component losses are neglected.
Where This Model Stops
- 01Does not calculate actual thermal efficiency from fuel input, shaft work, cycle details, or component losses.
- 02Does not apply directly to refrigerators or heat pumps; those use Carnot coefficient of performance relations instead.
- 03The formula requires absolute temperatures, so very small temperature gaps can produce unrealistic-looking high limits that real equipment cannot reach.
- 04Does not include finite heat-transfer rates, pressure drops, combustion limits, turbine/compressor efficiency, condenser performance, or working-fluid property limits.
- 05Back-solving reservoir temperature is conceptual only. A feasible machine also needs material, safety, pressure, and heat-exchanger checks.
References
- [1]3.3 The Carnot Cycle
MIT Unified Engineering Notes
States the Carnot cycle efficiency in temperature-ratio form and explains why the result is the ideal limit for reversible heat engines.
- [2]4.5 The Carnot Cycle
OpenStax University Physics Volume 2
Provides worked examples using η = 1 − Tc/Th and emphasizes the use of absolute temperature.
- [3]15.4 Carnot's Perfect Heat Engine
OpenStax College Physics
Explains the Carnot efficiency limit and shows why 100 percent efficiency would require a cold reservoir at absolute zero.
Frequently Asked Questions
Why do I need absolute temperature for Carnot efficiency?
Because the formula uses a temperature ratio. Celsius and Fahrenheit have arbitrary zero points, so they must be converted to an absolute scale such as kelvin before taking the ratio Tc/Th.
Is Carnot efficiency the same as actual thermal efficiency?
No. Carnot efficiency is the ideal upper bound for a reversible heat engine between two reservoir temperatures. Real engines are always less efficient because of irreversibilities, pressure drops, friction, finite-rate heat transfer, and other losses.
Can Carnot efficiency ever reach 100%?
Not for a physically realizable engine. The formula would require the cold reservoir to be at absolute zero, which cannot be achieved in practice.
Why does entering Celsius directly by hand give the wrong answer?
Because 300 °C and 27 °C are not a valid ratio scale. Convert them to kelvin first: 573.15 K and 300.15 K. This calculator performs that conversion internally when you choose °C or °F units.
How is this different from a heat energy or heat conduction calculator?
Carnot efficiency is about the maximum fraction of heat that could become work in an ideal engine. Heat Energy calculates Q = mcΔT, and Heat Conduction calculates heat flow through a material; neither gives an engine efficiency limit.