Carnot Efficiency Calculator
Calculate the maximum theoretical efficiency of a heat engine operating between two temperature reservoirs using the Carnot formula η = 1 - Tc/Th. Also shows COP for heat pumps and refrigerators. See also our Heat Transfer Calculator and Ideal Gas Law Calculator.
How to Calculate Carnot Efficiency
The Carnot efficiency represents the absolute maximum efficiency that any heat engine can achieve when operating between two temperature reservoirs. It was derived by Sadi Carnot in 1824 and is a cornerstone of thermodynamics. No real engine can exceed this efficiency — it represents the theoretical limit imposed by the second law of thermodynamics. Real engines typically achieve 30-60% of their Carnot efficiency due to friction, irreversibilities, and practical limitations.
The formula is remarkably simple: η = 1 - Tc/Th, where Tc is the absolute temperature of the cold reservoir and Th is the absolute temperature of the hot reservoir (both must be in Kelvin). The efficiency depends only on the temperature ratio — not on the working fluid, engine design, or any other factor. To increase Carnot efficiency, you must either raise Th or lower Tc. This is why power plants use superheated steam (high Th) and why efficiency improves in cold weather (lower Tc).
The Carnot efficiency can never reach 100% because that would require either Tc = 0 K (absolute zero, unattainable) or Th = infinity. Even the sun's surface temperature (5778 K) with a cold reservoir at room temperature (300 K) gives only 94.8% Carnot efficiency. Real power plants achieve 33-45% actual efficiency, which is 50-65% of their Carnot limit.
Carnot Efficiency Formula
Carnot Efficiency (Heat Engine):
η = 1 - Tc/Th = (Th - Tc)/Th
Work Output:
W = η × Qh = Qh - Qc
Heat Rejected:
Qc = Qh × (Tc/Th) = Qh × (1 - η)
COP — Heat Pump (Carnot):
COP_HP = Th/(Th - Tc) = 1/η
COP — Refrigerator (Carnot):
COP_ref = Tc/(Th - Tc)
Entropy Relationship:
Qh/Th = Qc/Tc (reversible cycle)
Example Calculation
A power plant operates with steam at 500 K and rejects heat to a river at 300 K:
Given: Th = 500 K, Tc = 300 K
η = 1 - Tc/Th = 1 - 300/500 = 1 - 0.6 = 0.4 = 40%
For 1000 MW thermal input:
Maximum work output: W = 0.4 × 1000 = 400 MW
Heat rejected to river: Qc = 1000 - 400 = 600 MW
Carnot COP as heat pump: 500/(500-300) = 2.5
Carnot COP as refrigerator: 300/(500-300) = 1.5
Real plant efficiency might be 60% of Carnot:
Actual η ≈ 0.6 × 40% = 24%
Carnot Efficiency Reference Table
| Heat Source | Th (K) | Tc (K) | Carnot η |
|---|---|---|---|
| Nuclear power plant | 600 | 300 | 50.0% |
| Coal power plant | 810 | 300 | 63.0% |
| Gas turbine | 1500 | 300 | 80.0% |
| Car engine | 2500 | 300 | 88.0% |
| Steam engine | 500 | 373 | 25.4% |
| Ocean thermal (OTEC) | 298 | 278 | 6.7% |
| Geothermal | 450 | 300 | 33.3% |
| Solar thermal | 700 | 300 | 57.1% |
| Diesel engine | 2000 | 300 | 85.0% |
| Stirling engine | 900 | 300 | 66.7% |
Frequently Asked Questions
What is Carnot efficiency?
Carnot efficiency is the maximum possible efficiency of any heat engine operating between two temperature reservoirs. It is given by η = 1 - Tc/Th, where temperatures must be in Kelvin. No real engine can exceed this limit — it is a fundamental consequence of the second law of thermodynamics. The Carnot cycle (two isothermal + two adiabatic processes) is the only cycle that achieves this maximum efficiency, but it requires infinitely slow (reversible) processes, making it impractical.
Why can't any engine be 100% efficient?
100% efficiency would require Tc = 0 K (absolute zero) or Th = infinity, both physically impossible. The second law of thermodynamics states that heat cannot be completely converted to work in a cyclic process — some heat must always be rejected to a cold reservoir. This is not an engineering limitation but a fundamental law of nature. Even a perfect, frictionless engine must reject heat proportional to Tc/Th. Entropy must increase (or stay constant) in any real process.
What is COP and how does it relate to Carnot efficiency?
COP (Coefficient of Performance) measures the effectiveness of heat pumps and refrigerators. For a heat pump, COP = Qh/W = Th/(Th-Tc) = 1/η_Carnot. For a refrigerator, COP = Qc/W = Tc/(Th-Tc). COP can be greater than 1 (unlike efficiency) because you are not converting heat to work — you are moving heat using work. A heat pump with COP = 3 delivers 3 kW of heating for every 1 kW of electricity consumed. Real heat pumps achieve 50-60% of Carnot COP.
Why must temperatures be in Kelvin?
The Carnot formula requires absolute temperature (Kelvin) because it is derived from the ratio of heat flows, which are proportional to absolute temperature. Using Celsius would give incorrect results because 0°C is not zero energy — it is 273.15 K. For example, Th=100°C, Tc=50°C in Celsius would incorrectly give η=50%, but the correct answer using Kelvin (373K, 323K) is η=13.4%. The Kelvin scale starts at absolute zero where molecular motion ceases.
How efficient are real power plants?
Real power plants achieve 30-60% of their Carnot efficiency due to irreversibilities. Typical actual efficiencies: coal plants 33-40%, nuclear plants 33-37%, combined-cycle gas turbines 55-62%, diesel generators 35-45%. The gap between Carnot and actual efficiency comes from friction, heat losses, finite-rate heat transfer, pressure drops, and incomplete combustion. Combined-cycle plants are most efficient because they use exhaust heat from a gas turbine to power a steam turbine, effectively using two temperature stages.
How can I increase engine efficiency?
To increase Carnot efficiency, either raise Th or lower Tc. In practice: (1) Use higher combustion temperatures (limited by material strength); (2) Use superheated/supercritical steam; (3) Reject heat to colder sinks (cold water, cold air); (4) Use combined cycles (gas + steam turbine); (5) Reduce irreversibilities (better insulation, less friction, slower processes). Modern research focuses on high-temperature materials (ceramics, superalloys) that allow higher Th, and on waste heat recovery to capture rejected heat.
Significance of the Carnot Cycle
The Carnot cycle is more than a theoretical curiosity — it establishes the fundamental limits of energy conversion and defines the thermodynamic temperature scale. It proves that all reversible engines operating between the same temperatures have the same efficiency, regardless of working fluid or design. It also shows that perpetual motion machines of the second kind (converting heat entirely to work) are impossible. Every real heat engine, from car engines to power plants, is benchmarked against its Carnot limit.