Carnot Efficiency Calculator — Heat Engine
Solve the Carnot efficiency relation η = 1 − Tc / Th for any of its three quantities — thermal efficiency, hot reservoir temperature, or cold reservoir temperature — with step-by-step math, a live T–s diagram, a Carnot cycle P–V plot, and companion thermodynamics metrics including the coefficient of performance.
Efficiency is bounded by the temperature ratio
No heat engine operating between reservoirs at absolute temperatures Th and Tc can exceed the Carnot efficiency η = 1 − Tc/Th. This is a fundamental consequence of the second law of thermodynamics, not an engineering limitation. Even a frictionless, perfectly insulated, infinitely slow engine cannot beat this bound. Real engines approach it only asymptotically, and modern power plants typically reach 60–85% of their Carnot limit — impressive in practice, but the remaining gap is the price of doing anything on a finite timescale.
Raising Th improves efficiency most
Because the cold reservoir is usually the environment and its temperature is fixed by nature, engineers almost always improve efficiency by raising Th. The improvement is significant but requires better materials: a gas turbine blade that can survive at 1500 K instead of 1200 K gains roughly 15 percentage points of Carnot efficiency, but requires single-crystal nickel superalloys with ceramic thermal barrier coatings and internal cooling channels. The 60-year history of gas turbine efficiency — from 20% to nearly 45% in simple cycle — is largely a history of high-temperature materials.
Lowering Tc helps but is expensive
The same formula shows that lowering Tc also raises efficiency — but the effect is smaller per kelvin and much harder to achieve. Lowering Tc by 30 K (from 300 K to 270 K) improves Carnot efficiency from 75% to 77.5% at Th = 1200 K. But reaching 270 K requires refrigeration, which consumes work, and the work spent may exceed the efficiency gain. This is why power plants almost never refrigerate their cold reservoir — they use whatever cooling the environment provides, whether river water, cooling towers, or (in some cases) dry air.
Carnot works in reverse as a heat pump
The same cycle run backwards becomes a heat pump or refrigerator. The coefficient of performance for cooling is COPc = Tc/(Th − Tc), and for heating is COPh = Th/(Th − Tc). Both can be much greater than 1 — a heat pump moving heat from 0°C outside to 20°C inside has a theoretical COPh of 293/20 = 14.65, meaning it delivers 14.65 units of heat per unit of work consumed. This is why heat pumps are so attractive for building heating: their effective efficiency, while not comparable to a heat engine, is far better than any direct electric heating.
How to Use This Calculator
The Carnot efficiency relation links three quantities: η, Th, and Tc. Pick which one you want the tool to solve, then fill in the other two.
1. Choose what to solve for
Pick η, Th, or Tc from the toggle. The calculator rearranges η = 1 − Tc/Th for whichever variable you choose, and flags the singular cases (η = 100% or η = 0%) that correspond to infinite or zero temperature ratios.
2. Use realistic temperature values
Presets cover the practical range: a coal plant at 800 K / 300 K, a gas turbine at 1200 K / 300 K, a combined cycle at 1500 K / 300 K, a nuclear plant at 600 K / 300 K, geothermal at 450 K / 320 K, and a solar thermal field at 700 K / 300 K.
3. Always use absolute temperatures
The formula requires Kelvin, not Celsius. To convert from °C, add 273.15. Using Celsius by mistake gives dramatically wrong results — a 27°C hot reservoir and a 27°C cold reservoir give η = 1 − 27/27 = 0, which is clearly wrong; converting to Kelvin gives 300 K and 300 K, still zero (correctly identifying that no work is available).
4. Enter Qin for the energy picture
Entering the heat input per cycle unlocks the maximum work output Wmax = η · Qin and the heat that must be rejected Qout = Qin − Wmax. This is the same energy balance you would use for any engine, but with the Carnot efficiency plugged in.
5. Enter actual efficiency for the reality check
If you know the real efficiency of a specific engine, entering it computes the second-law efficiency: ηII = ηactual/ηCarnot. This tells you how close the engine is to the theoretical ideal. A value of 0.60 means the engine achieves 60% of Carnot — a good figure for a real machine. A value above 1.0 indicates an error, since no real engine can exceed Carnot.
6. Switch to Advanced for depth
Advanced mode adds a chart of η versus Th for three cold reservoir temperatures, a chart of η versus Tc for three hot reservoir temperatures, a sensitivity panel showing the ±1% effect of each input, and companion metrics including the Carnot COP for cooling and heating, the entropy change per cycle, and the fraction of Qin rejected as Qout.
What This Tool Offers
Built around the classic Carnot efficiency relation η = 1 − Tc/Th, the fundamental thermodynamic bound on any heat engine's performance.
Three-way solving
Solve for Carnot efficiency, hot reservoir temperature, or cold reservoir temperature. The calculator rearranges the relation for whichever variable you choose.
Dual cycle diagrams
A T–s diagram shows the Carnot rectangle with hot and cold isotherms and isentropic legs; a P–V diagram shows the same cycle with its work area.
Efficiency verdict
Each result is classified from very low through low, moderate, high, very high, to theoretical — matching the range from ocean thermal (5%) to advanced power plants (60%+) and approaching the ideal.
Second-law efficiency
Enter the actual engine efficiency to see how close it comes to the Carnot bound. The ratio ηactual/ηCarnot is the standard measure of how well an engine performs relative to its thermodynamic ideal.
Energy balance metrics
With Qin entered, the tool computes maximum work output Wmax, rejected heat Qout, and the entropy change per cycle — the complete energy accounting for a Carnot cycle.
Reference tables
Typical efficiency by engine type, a Carnot efficiency lookup grid across four reservoir combinations, a unit converter, and a full variable reference with Kelvin/Celsius conversion notes.
Reference Tables & Conversions
A temperature and efficiency converter, a Carnot efficiency lookup grid, and typical parameters by engine type.
Temperature & efficiency converter
Carnot Efficiency Lookup
Computed for the hot and cold reservoir combinations shown. Values in percent.
| Th / Tc | Tc = 250 K | Tc = 300 K | Tc = 350 K | Tc = 400 K | Tc = 500 K |
|---|
Typical Carnot and Actual Efficiencies by Engine Type
Representative values. Actual efficiencies vary with load, ambient temperature, and maintenance.
| Engine type | Hot T (K) | Cold T (K) | ηCarnot | ηactual | ηII = act/Carnot |
|---|---|---|---|---|---|
| Ocean thermal (OTEC) | 300 | 280 | 6.7% | 2–3% | ~0.4 |
| Geothermal (binary cycle) | 450 | 310 | 31.1% | 10–12% | ~0.35 |
| Nuclear steam cycle | 600 | 300 | 50.0% | 33–37% | ~0.70 |
| Coal-fired steam plant | 800 | 300 | 62.5% | 38–42% | ~0.65 |
| Supercritical coal | 870 | 300 | 65.5% | 42–45% | ~0.67 |
| Gas turbine (simple cycle) | 1200 | 300 | 75.0% | 35–42% | ~0.50 |
| Combined cycle (gas + steam) | 1500 | 300 | 80.0% | 58–62% | ~0.75 |
| Diesel engine (vehicle) | 2200 | 350 | 84.1% | 38–42% | ~0.48 |
| Spark-ignition gasoline | 2500 | 350 | 86.0% | 25–30% | ~0.33 |
| Stirling engine (lab) | 1200 | 300 | 75.0% | 35–40% | ~0.50 |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| η | Carnot efficiency | 0.05 – 0.90 | — (or %) |
| Th | Hot reservoir temperature | 300 – 5500 | K |
| Tc | Cold reservoir temperature | 4 – 400 | K |
| Tc/Th | Temperature ratio | 0 – 1 | — |
| Qin | Heat input per cycle | 0 – 10⁶ | kJ |
| Wmax | Maximum work output = η·Qin | — | kJ |
| Qout | Rejected heat = Qin − W | — | kJ |
| ηII | Second-law efficiency = ηact/ηCarnot | 0.2 – 0.85 | — |
| COP | Coefficient of performance (reversed cycle) | 1 – 20 | — |
What Does η = 1 − Tc / Th Measure?
The Carnot efficiency is the maximum possible thermal efficiency of any heat engine operating between a hot reservoir at temperature Th and a cold reservoir at temperature Tc, with both temperatures measured on the absolute (Kelvin) scale. It is the theoretical ceiling on how much work you can extract from a given amount of heat, and it depends only on the two temperatures — not on the working fluid, the mechanical design, the fuel, or the speed of operation. This remarkable independence is what makes Carnot's result so powerful.
Sadi Carnot derived this result in 1824, before the first law of thermodynamics was even formulated, in a small book called Réflexions sur la puissance motrice du feu. He imagined an ideal cycle — now called the Carnot cycle — consisting of two reversible isothermal processes and two reversible adiabatic (isentropic) processes. The cycle is fully reversible and, by a simple argument, any engine more efficient than it would violate the second law by allowing heat to flow spontaneously from cold to hot. The Carnot efficiency is therefore not just a practical target — it is a fundamental thermodynamic bound.
Carnot Efficiency Formula
The defining relationship is:
where Th and Tc are the absolute temperatures of the hot and cold reservoirs in kelvin. Rearranged to solve for the other two quantities:
Tc = Th · (1 − η)
Companion relations that follow from the same cycle:
Maximum work: Wmax = η · Qin
Rejected heat: Qout = (1 − η) · Qin
Carnot COP (refrigerator): COPc = Tc / (Th − Tc)
Carnot COP (heat pump): COPh = Th / (Th − Tc)
Second-law efficiency: ηII = ηactual / ηCarnot
The Celsius-to-Kelvin conversion is essential: T[K] = T[°C] + 273.15. Using Celsius in the formula is the single most common mistake and it produces nonsense — for example, an engine operating between 25°C and 0°C would give η = 1 − 0/25 = 1 (100%) if you accidentally used Celsius, whereas the correct calculation with 298 K and 273 K gives η = 1 − 273/298 = 8.4%.
Carnot Efficiency Variables
| Symbol | Meaning | Notes |
|---|---|---|
| η | Carnot efficiency | Maximum theoretical efficiency; dimensionless, often shown as % |
| Th | Hot reservoir temperature | Absolute temperature of the heat source; in kelvin |
| Tc | Cold reservoir temperature | Absolute temperature of the heat sink; in kelvin |
| Qin | Heat input | Energy supplied to the engine per cycle |
| Wmax | Maximum work output | The work the ideal cycle would produce |
| Qout | Rejected heat | The unavoidable heat that must be dumped to the cold reservoir |
| ηII | Second-law efficiency | Ratio of actual to Carnot efficiency; the "effectiveness" of the engine |
How Each Variable Affects the Efficiency
The formula shows that efficiency increases as the temperature ratio Tc/Th decreases. Because the cold reservoir is almost always at or near ambient temperature — which is fixed by climate and cannot easily be lowered — the practical lever is to raise Th. The sensitivity of η to Th is proportional to Tc/Th², so improvements are largest when Th is small: going from 400 K to 500 K at Tc = 300 K raises efficiency from 25% to 40%, a 15-point gain; going from 1200 K to 1300 K raises it from 75% to 76.9%, only a 1.9-point gain. This diminishing return is why the pursuit of ever-higher temperatures is so difficult in practice — the big wins are already captured.
The dependence on Tc is weaker and in the opposite direction. Lower Tc always helps, but each kelvin of cooling delivers less improvement than the previous one, and reaching sub-ambient temperatures requires refrigeration work that eats into the net efficiency. This is why cooling towers are designed to bring the cooling water just close to ambient — not far below it — and why plants in hot climates have slightly lower efficiency than identical plants in cold climates.
In real engines, additional inefficiencies reduce the practical efficiency far below the Carnot bound. Friction, heat leaks, finite-rate heat transfer, combustion irreversibility, and turbulence all consume work that the ideal cycle would convert into output. The ratio of actual to Carnot efficiency — the second-law efficiency — is typically 0.5–0.75 for well-designed engines, and it is the more meaningful measure of engineering quality. A small improvement in the second-law efficiency (say from 0.65 to 0.70) is usually easier to achieve than an equivalent improvement in Carnot efficiency, because it does not require raising temperatures or changing materials.
Carnot Efficiency Calculation Examples
η = 1 − 300 / 1200 = 1 − 0.25 = 0.75 = 75%
Th = 300 / (1 − 0.75) = 300 / 0.25 = 1200 K
Tc = 1200 · (1 − 0.75) = 1200 · 0.25 = 300 K
η = 1 − 300 / 800 = 0.625 = 62.5%
If the actual efficiency is 40%, second-law efficiency = 40/62.5 = 0.64
η = 1 − 280 / 300 = 0.0667 = 6.67%
This shows why ocean thermal energy conversion is difficult — the tiny temperature difference makes the theoretical efficiency very low.
η = 0.75
Wmax = 0.75 × 1000 = 750 kJ
Qout = 1000 − 750 = 250 kJ
COPh = Th / (Th − Tc) = 300 / 20 = 15.0
Every unit of work delivers 15 units of heat — a strong argument for heat pump heating.
Frequently Asked Questions
Because the second law of thermodynamics forbids it. A heat engine works by allowing heat to flow from hot to cold through a working fluid, and the flow itself is what produces work. To reach 100% efficiency you would have to convert all of the heat into work and reject none — but then there would be no cold reservoir to absorb heat, no temperature difference to drive the flow, and the cycle could not close. Equivalently, reaching 100% would require the cold reservoir to be at absolute zero (Tc = 0), which is unattainable; or the hot reservoir to be at infinite temperature, which is also unattainable. The formula captures both: η = 1 − Tc/Th approaches 1 as Tc→0 or as Th→∞, but neither limit is physical.
No — that is one of the remarkable implications of Carnot's theorem. The maximum efficiency between two given reservoirs is independent of the working fluid, whether it is steam, air, helium, ammonia, or anything else. What matters is only the two temperatures. This is because the argument that establishes the Carnot limit is a thermodynamic one: if any engine could beat the Carnot efficiency, you could use it to move heat from cold to hot, violating the second law. The result holds for any reversible cycle between the same two reservoirs, regardless of what substance performs the cycle. In practice the working fluid does matter for the real efficiency and power density of a given engine — some fluids are better suited to some temperature ranges — but not for the theoretical bound itself.
Second-law efficiency (also called effectiveness or exergy efficiency) is the ratio of the actual efficiency of an engine to its Carnot efficiency: ηII = ηactual/ηCarnot. It measures how close the engine comes to the theoretical ideal for its operating temperatures. First-law efficiency tells you the fraction of fuel energy converted to useful work; second-law efficiency tells you how much of the theoretical maximum work you are capturing. A gas turbine with 40% first-law efficiency and 75% Carnot efficiency has ηII = 0.53, meaning it captures about half of the available thermodynamic potential. Modern combined-cycle plants reach ηII ≈ 0.75, which is why they are considered close to practical limits. There is essentially no room for improvement without changing the operating temperatures themselves.
Real engines cannot reach the Carnot efficiency, but the Carnot efficiency still applies to them as an upper bound. No real engine has ever exceeded it, and no engine ever will — it is a theorem of thermodynamics, not a design target. Real engines fall short because they operate on finite timescales (finite-rate heat transfer), have friction between moving parts, leak heat to the surroundings, and undergo irreversible processes such as combustion, mixing, and throttling. Each of these consumes work that the ideal cycle would convert to output. A well-designed engine typically reaches 50–75% of its Carnot limit; the remaining gap is the irreducible cost of doing useful work in the real world.
The same four-process cycle run in reverse: instead of taking heat from a hot source and dumping part of it to a cold sink, the reversed cycle takes heat from the cold source, uses work to pump it "uphill" to the hot sink, and delivers more heat there than it consumed in work. This is the principle behind every refrigerator and air conditioner, and it is the same cycle that heat pumps use for building heating. The coefficient of performance for cooling is COPc = Tc/(Th − Tc), and for heating is COPh = Th/(Th − Tc). Both can be much greater than 1 — a 15-kW heat pump delivering 45 kW of heat has COPh = 3.0, which is far better than any direct electric heater. The Carnot COP is the theoretical maximum for these devices, and real heat pumps reach 40–60% of it.
No. As long as Tc > 0 and Th > Tc, the ratio Tc/Th is between 0 and 1, so η is between 0 and 1, or 0% and 100%. At Th = Tc, both the efficiency and the available work are zero — you cannot extract work from two reservoirs at the same temperature, even if that temperature is very high. As Tc → 0 the efficiency approaches 100%, but reaching absolute zero is forbidden by the third law of thermodynamics. If you ever calculate an efficiency of 100% or higher, either the temperatures are wrong (perhaps entered in Celsius instead of Kelvin), or Tc is negative (impossible on the Kelvin scale), or Th is less than Tc (which would mean the "hot" reservoir is actually colder than the "cold" one).
If the two reservoirs are at the same temperature, the Carnot efficiency is exactly zero. No work can be extracted from heat when there is no temperature difference to drive it — even if the temperature itself is very high, like 1000 K. This is a direct consequence of the second law: to convert heat to work, you need a temperature gradient. In practice, of course, this is rarely a design constraint since the cold reservoir is usually the environment at a fixed and much lower temperature than any engine's hot source. But the fact that efficiency depends on the ratio rather than the absolute temperature is a key insight — a "hot" source at 320 K with a "cold" sink at 300 K gives only 6.25% efficiency, whereas a "hot" source at 1200 K with the same 300 K sink gives 75%.
The Carnot cycle is reversible, which means the total change in entropy of the universe during a complete cycle is zero. Heat Qh enters the engine from the hot reservoir at temperature Th, so the entropy change of the hot reservoir is −Qh/Th. Heat Qc leaves to the cold reservoir at Tc, so the entropy change of the cold reservoir is +Qc/Tc. For the cycle to be reversible, these must sum to zero: Qh/Th = Qc/Tc. Combining this with the first law Qh = W + Qc gives the Carnot efficiency directly: η = W/Qh = 1 − Tc/Th. The Carnot efficiency is therefore a restatement of the second law in terms of the entropy balance — the most elegant and fundamental form of the result.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| eta | Carnot (maximum) thermal efficiency (dimensionless) | |
| Tc | Absolute temperature of the cold reservoir | K |
| Th | Absolute temperature of the hot reservoir | K |
What it means
The Carnot efficiency is the theoretical maximum efficiency of any heat engine operating between two thermal reservoirs at temperatures T_hot (heat source) and T_cold (heat sink), both measured in kelvin. It is given by η_Carnot = 1 − (T_c / T_h). This result comes from the Carnot cycle, which consists of two isothermal and two adiabatic processes. The Carnot engine is reversible, and all reversible engines operating between the same reservoirs have the same efficiency. No real engine can exceed this efficiency due to the second law of thermodynamics. The Carnot efficiency increases as T_h increases or T_c decreases. In practice, engine efficiencies are lower due to irreversibilities (friction, heat losses, etc.). The concept is used to evaluate the quality of energy conversion and to guide design improvements. It also applies to refrigerators and heat pumps, where the coefficient of performance is bounded by the Carnot COP.
Worked example
Carnot Efficiency – Two Examples
Real‑World| Parameter | Value |
|---|---|
| Th | 600 K |
| Tc | 300 K |
| Parameter | Value |
|---|---|
| Th | 900 K |
| Tc | 300 K |
Common mistakes
- Temperature ratio: Use absolute temperatures (Kelvin) – not °C or °F.
- Hot and cold reservoirs: Th is the high‑temperature reservoir, Tc the low‑temperature one.
- Idealised efficiency: This is the maximum possible efficiency – real engines are always lower.
- Efficiency decimal vs. percentage: The formula gives a fraction; multiply by 100 for %.
- Reversible cycle: Carnot efficiency applies only to reversible (Carnot) cycles.
Applications
Carnot efficiency represents the maximum possible efficiency of a heat engine operating between two thermal reservoirs. Although idealised, it provides a theoretical benchmark against which real engines are compared. The formula is used to assess the potential of thermodynamic cycles and to identify areas for improvement in power generation and refrigeration. In practical engineering, it guides the selection of operating temperatures and working fluids. For example, increasing the hot‑side temperature or decreasing the cold‑side temperature improves efficiency, but material and economic constraints limit the extremes. The Carnot efficiency also educates engineers on the fundamental limits of energy conversion, reinforcing the importance of thermodynamics in sustainable energy systems.
- Performance benchmarking of real heat engines
- Optimisation of power plant operating conditions
- Design of high‑efficiency refrigeration cycles
- Thermodynamic education and concept illustration
- Feasibility studies for energy‑conversion systems
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