Thermal Efficiency Calculator — actual output against the Carnot limit

The thermal efficiency calculator above does two things that belong together and are usually shown apart. It computes the actual efficiency of a heat engine as work output divided by heat input, and it computes the Carnot limit that the reservoir temperatures impose on any engine operating between them. The ratio of the two — how much of the available maximum the machine is capturing — is the number that tells you whether an engine is poorly built or simply working between temperatures that do not allow much better.

Arb Digital builds free calculators that put the benchmark next to the measurement. The boundary against our Carnot efficiency calculator is straightforward: that page gives the temperature-limited ideal on its own, while this one takes a real work and heat measurement and reports what fraction of that ideal it achieves. If you only have temperatures, use that page. If you have measured output, use this one.

What This Thermal Efficiency Calculator Does

It divides the useful work or power an engine produces by the heat or fuel energy it consumes, and reports the result as a percentage. Because efficiency is a ratio, the units cancel — kilojoules per cycle, kilowatts, megawatts or Btu per hour all work, provided both fields use the same one. The tool also reports the waste heat, which is simply whatever the engine did not convert.

Alongside that it computes the Carnot efficiency from the hot and cold reservoir temperatures you enter, converting from Celsius or Fahrenheit to kelvin first. Dividing actual by Carnot gives the second-law or exergetic efficiency, which is the fairest way to compare engines that operate between different temperatures. A gas turbine at 40 per cent and a low-temperature waste heat engine at 12 per cent may be equally good machines once the limit is accounted for.

Finally it reports heat rate, the power industry's inverted form of the same quantity: the energy input required per kilowatt-hour of output. It is a more intuitive figure for plant operators because it maps directly onto fuel cost, and it is what a plant's performance guarantee is usually written against.

How to Use It

1. Enter the work output and the heat input in the same unit. Per cycle, per second or per hour — it does not matter which, only that both use it.
2. Enter both reservoir temperatures. The hot one is the source, the cold one the sink, and the sink is usually ambient air or cooling water rather than anything designed.
3. Set the temperature unit before reading the Carnot figure. The tool converts to kelvin internally, which is exactly what makes the Carnot result meaningful.
4. Check the fraction of the limit reached. This is the diagnostic number. A low actual efficiency with a high fraction means the temperatures are the constraint, not the machine.
5. Read the bar breakdown. It splits the heat input into useful work and rejected heat, which makes the scale of the waste stream immediately visible.

The Formula: How Thermal Efficiency Is Calculated

Thermal efficiency is e = W ÷ Q_h, the work out over the heat in. OpenStax University Physics Volume 2, section 4.2 on heat engines, defines it as what we get out divided by what we put in during each cycle, and gives the equivalent form e = 1 − Q_c/Q_h, where Q_c is the heat rejected to the cold reservoir.

The Carnot limit for reservoirs at absolute temperatures T_h and T_c is e_Carnot = 1 − T_c/T_h. The HyperPhysics page on heat engines describes this ceiling as the thermal bottleneck, the constraint the second law places on any engine whatever its design. Second-law efficiency is then e ÷ e_Carnot.

Work the default values. An engine producing 100 kW from a 250 kW heat input has an efficiency of 100 ÷ 250 = 0.40, or 40 per cent, and rejects 150 kW as waste heat. With reservoirs at 800 K and 300 K the Carnot limit is 1 − 300/800 = 0.625, or 62.5 per cent. The engine is therefore capturing 40 ÷ 62.5 = 64 per cent of what the temperatures permit. Heat rate is 3,600 ÷ 0.40 = 9,000 kJ per kWh.

Why the Carnot Limit Is Not a Design Target

The Carnot cycle achieves its limit only by running reversibly, which means infinitely slowly. Every real process that happens at a finite rate generates entropy — friction, turbulence, heat crossing a finite temperature difference, unrestrained expansion — and each of those costs work. An engine that actually reached the Carnot efficiency would produce power at a rate of exactly zero, which makes it a bound rather than a goal.

A more realistic benchmark for a power-producing engine is the endoreversible limit, sometimes called the Curzon-Ahlborn efficiency, which is one minus the square root of the temperature ratio rather than one minus the ratio. For the default 800 K and 300 K reservoirs, that gives about 39 per cent against Carnot's 62.5 per cent — remarkably close to the 40 per cent the example engine actually achieves. Real plants cluster far nearer this figure than the Carnot one, which is a good sanity check on any efficiency claim.

The practical implication is that the fraction-of-limit number in the grid should not be read as a grade out of a hundred. Something in the 50 to 70 per cent range is typical of good machinery. A figure above about 80 per cent should prompt you to check the measurements, and a figure above 100 per cent means the inputs are inconsistent, which the tool flags explicitly.

Where the Heat Actually Goes

The waste heat figure is not a loss in the sense of an inefficiency to be engineered away — it is a thermodynamic requirement. An engine must reject heat to a cold reservoir to complete its cycle, and the second law sets the minimum amount. In the default case, 150 of every 250 units of heat leave the engine at low temperature because there is no legal way for them to leave as work instead.

That is why combined heat and power changes the arithmetic so dramatically. A plant with 40 per cent electrical efficiency that also captures 45 per cent of the input as useful process or space heat has a total fuel utilisation near 85 per cent, while its thermal efficiency as a heat engine is still exactly 40 per cent. The two figures answer different questions and neither is wrong. Combined cycle plants exploit the same idea differently, using the gas turbine's exhaust as the hot reservoir for a steam cycle, which is how modern plants pass 60 per cent electrical efficiency.

Raising the hot reservoir temperature is the other main lever, and it is limited by materials rather than by thermodynamics. Turbine blade alloys and their cooling schemes set the ceiling on combustion temperature, which is why turbine inlet temperature is the metric that gas turbine development is measured by. Cooling the sink helps too but far less, because ambient conditions leave little room to move.

Reading Published Efficiency Figures Carefully

Two engines quoted at the same efficiency may not be comparable. The first question is which heating value the fuel energy was measured against: higher heating value counts the latent heat released when water vapour in the exhaust condenses, lower heating value does not. The same plant looks several percentage points more efficient on the lower basis, and both conventions are in common use in different countries.

The second is where the boundary was drawn. Gross output at the generator terminals is higher than net output after the plant's own pumps, fans and controls have taken their share, and the gap can be several per cent. The third is whether the figure is a peak or an average — an engine's best point is not where it spends most of its life, and part-load efficiency is usually much worse.

None of this changes the arithmetic on this page. It changes what you should feed into it, which is a more common source of wrong answers than the calculation. Use the energy converter to convert an energy total into the units this page wants, the electrical power calculator for the output side of a generator, and the horsepower calculator if the output is quoted as shaft power.

Common Mistakes to Avoid

• Using Celsius in the Carnot expression — the ratio of temperatures is only meaningful on an absolute scale, and 300 °C over 800 °C is a completely different number from 573 K over 1,073 K.
• Mixing units between work and heat — kilowatts against kilojoules gives a ratio that means nothing. Both fields must be on the same basis.
• Comparing efficiencies across different heating values — the same plant reads several points higher on a lower-heating-value basis than on a higher one.
• Treating the Carnot limit as a design target — reaching it requires a reversible cycle running infinitely slowly, and therefore producing no power at all.
• Confusing thermal efficiency with fuel utilisation — a combined heat and power plant can use 85 per cent of its fuel usefully while its efficiency as a heat engine is still 40 per cent.

Related Free Tools From Arb Digital

For the temperature-limited ideal on its own, use the Carnot efficiency calculator. Convert energy totals with the energy converter, handle generator output with the electrical power calculator or shaft output with the horsepower calculator, and work out the heat needed to raise a working fluid's temperature with the specific heat calculator or the sensible heat calculator. For the waste heat leaving through a surface, the heat transfer calculator covers conduction. Everything else is in the free online tools hub.

Frequently Asked Questions

This tool is provided for educational and estimating use. It applies idealised first and second law relationships and takes no account of part-load behaviour, heating-value conventions, parasitic loads or measurement boundaries, so treat its output as a physics result rather than a plant performance figure.