Carnot Efficiency Calculator
Enter the hot-reservoir and cold-reservoir temperatures to find the theoretical maximum efficiency any heat engine can achieve between those two temperatures. Switch between Kelvin, Celsius, and Fahrenheit. Reverse-solve to find the temperature needed for a target efficiency, compare actual vs Carnot efficiency, or calculate the coefficient of performance for a heat pump or refrigerator.
Formula
Worked example
A steam turbine operates with a boiler at 500 C (773.15 K) and a condenser at 25 C (298.15 K). Carnot efficiency = (773.15 - 298.15) / 773.15 = 475 / 773.15 = 61.44%. The Carnot COP for heat pump mode is 1 / 0.6144 = 1.63, and for refrigerator mode is 298.15 / 475 = 0.628.
What is Carnot efficiency and why does it matter?
Carnot efficiency is the theoretical maximum thermal efficiency that any heat engine operating between two fixed-temperature reservoirs can achieve. It was derived in 1824 by French engineer Sadi Carnot and later formalised through the Second Law of Thermodynamics. The formula is simple: eta = (Th - Tc) / Th, where Th is the temperature of the hot source and Tc is the temperature of the cold sink, both measured in absolute (Kelvin) units. The result tells engineers the ceiling that no engine, however cleverly designed, can exceed when converting heat into work between those two temperatures. Real engines always fall short because of friction, heat leaks, finite temperature differences in heat exchangers, and other irreversibilities. Comparing a real engine's efficiency against the Carnot limit shows how much room there is for improvement and where energy is being wasted.
The four phases of the Carnot cycle
The ideal Carnot cycle consists of four reversible processes. In the first phase, isothermal expansion, the working fluid absorbs heat from the hot reservoir at constant temperature Th, expanding and doing work on the surroundings. In the second phase, adiabatic expansion, the fluid expands further with no heat exchange, cooling down to the cold-reservoir temperature Tc. In the third phase, isothermal compression, the surroundings compress the fluid at constant temperature Tc, rejecting heat to the cold reservoir - this step requires less work than was produced in phase one because the temperature is lower. In the fourth phase, adiabatic compression, the fluid is compressed back to its starting temperature Th with no heat exchange, completing the cycle. Because every step is reversible, the cycle wastes nothing to irreversibilities, making it the most efficient possible process between those two temperatures. The impracticality comes from the fact that truly isothermal processes require infinitely slow heat transfer, so a real Carnot engine would have zero power output.
Carnot efficiency and the coefficient of performance
The same temperature ratio that limits heat-engine efficiency also governs the best achievable performance of heat pumps and refrigerators. Running the Carnot cycle in reverse turns it into a heat pump or refrigerator. For a heat pump (which heats a space), the Carnot COP for heating is Th / (Th - Tc). For a refrigerator (which cools a space), the Carnot COP for cooling is Tc / (Th - Tc). Both figures represent the maximum heat or cold that can be moved per unit of work input. A typical air-source heat pump has a Carnot COP for heating of around 8-12 when operating at mild outdoor temperatures, but real units achieve only 3-5 because of compressor inefficiency, fan power, and heat-exchanger losses. Closing the gap between the Carnot COP and the real COP is the central engineering challenge for refrigeration and heat-pump designers.
How to raise Carnot efficiency in practice
Because eta = 1 - Tc/Th, there are exactly two levers: raise Th or lower Tc. Raising the hot-source temperature is usually more effective, which is why modern gas turbines operate at combustion temperatures above 1400 C and why supercritical and ultra-supercritical steam plants push boiler temperatures as high as metallurgy permits. Lowering the cold-sink temperature helps too: once-through cooling with cold river or sea water gives a lower Tc than air cooling, and combined-cycle plants capture exhaust heat in a steam cycle rather than dumping it directly to the environment. Combined-cycle plants therefore get two bites at the same fuel, and their 55-60% actual efficiency approaches the Carnot limit more closely than any other large-scale power technology. The key insight is that every degree of temperature difference at the hot end is worth more in efficiency gain than the same degree at the cold end, because adding to Th also increases the denominator of the formula by the same amount.
Typical Carnot efficiencies for real heat-engine systems
| Engine type | Th (approx.) | Tc (approx.) | Carnot limit | Typical actual |
|---|---|---|---|---|
| Coal power plant (steam) | 600 C | 30 C | 67% | 33-40% |
| Combined-cycle gas turbine | 1400 C | 30 C | 82% | 55-60% |
| Nuclear power plant | 300 C | 30 C | 48% | 30-35% |
| Automobile petrol engine | 1000 C | 30 C | 76% | 20-35% |
| Geothermal power plant | 150 C | 40 C | 26% | 10-20% |
| Solar thermal plant | 400 C | 25 C | 56% | 20-35% |
| Household refrigerator | 30 C | -18 C | COP ~5.4 | COP ~1.2-1.5 |
| Industrial heat pump | 60 C | -10 C | COP ~4.8 | COP ~2.5-3.5 |
Indicative operating temperatures and Carnot limits. Actual efficiencies are always lower due to friction, heat leaks, and irreversibilities.
Frequently asked questions
Why must temperatures be in Kelvin for the Carnot formula?
The Carnot formula uses the ratio Tc/Th, which only has physical meaning on an absolute temperature scale where zero corresponds to the complete absence of thermal motion. Celsius and Fahrenheit have arbitrary zero points, so the same temperature difference can give wildly different efficiency values depending on what you call zero. Converting to Kelvin (add 273.15 to Celsius, or use (F - 32) x 5/9 + 273.15) is mandatory. This calculator does that conversion automatically whichever unit you choose.
Can any real engine reach the Carnot efficiency?
No. The Carnot cycle requires every process to be fully reversible, which in practice means infinitely slow heat exchange (so there are no temperature gradients) and frictionless pistons. A real engine operating that slowly would produce essentially zero power. Real engines trade some efficiency for finite power output, which is why even the best combined-cycle gas turbines, at around 60%, fall 20 or more percentage points below their Carnot ceiling.
What happens to Carnot efficiency when Tc approaches absolute zero?
As Tc approaches 0 K, eta approaches 1 (100%). However, the Third Law of Thermodynamics states that absolute zero can never actually be reached, so 100% Carnot efficiency is thermodynamically impossible. Practically, cooling a cold reservoir to very low temperatures requires enormous energy expenditure, so the gain in efficiency is usually outweighed by the cost of refrigeration.
How is Carnot efficiency different from thermal efficiency?
Carnot efficiency is a theoretical maximum - the best any heat engine could ever do between two given temperatures. Thermal efficiency is the ratio of actual useful work output to heat input for a real engine. The ratio of thermal efficiency to Carnot efficiency (the performance ratio in the compare mode) tells you how close the real engine comes to the ideal. A performance ratio of 0.7 means the engine recovers 70% of what a perfect Carnot engine would.
What is the Carnot efficiency for a typical car engine?
A petrol engine with a combustion temperature of about 2000 K (around 1727 C) rejecting heat at 350 K (about 77 C) has a Carnot limit of (2000 - 350) / 2000 = 82.5%. Actual petrol engines achieve only 20-35% due to incomplete combustion, friction, pumping losses, and heat transfer through cylinder walls. Diesel engines run hotter and reach 40-45%, still well below their Carnot ceiling.
How do I find the cold-reservoir temperature needed for a target efficiency?
Rearrange the Carnot formula: Tc = Th x (1 - eta). If you want 50% efficiency from a 600 C (873.15 K) boiler, you need Tc = 873.15 x (1 - 0.50) = 436.6 K, which is about 163.4 C. Use the "Find cold-reservoir temperature" mode in this calculator to solve this automatically for any combination of known hot temperature and target efficiency.
What is the performance ratio shown in the compare mode?
The performance ratio is actual efficiency divided by Carnot efficiency. A value of 1.0 would mean a perfect Carnot engine - impossible in practice. Typical large power plants have performance ratios of 0.5 to 0.75, meaning they convert 50-75% of what the Carnot limit says is theoretically available. The power-loss figure shows how many kilowatts are wasted compared to the theoretical ideal at the given heat input rate.