Carnot Cycle.

The ceiling. Two isothermals and two adiabatics, and no engine working between the same two temperatures can ever beat it.

Mode

Cycle

Otto Diesel Dual Brayton Rankine Carnot Stirling Ericsson Atkinson

The machine — The ceiling. Two isothermals and two adiabatics, and no engine working between the same two temperatures can ever beat it.

Diagram

Press Run to turn the machine over. Space runs and pauses, R resets, and the arrow keys step through the cycle one process at a time.

isothermal 1→2 · Isothermal expansion, absorbing heat q 0 w 252.2 kJ/kg

Thermal efficiency

62.5%

Carnot limit, same span

62.5%

Net work

157.7 kJ/kg

Heat added

252.2 kJ/kg

Heat rejected

94.6 kJ/kg

Peak temperature

800 K

Lowest temperature

300 K

Mean effective pressure

2 kPa

Every setting is inside its sensible range for this cycle. Push a slider to an extreme and anything worth knowing about will appear here.

State points — the cycle, one corner at a time

State	What just happened	P kPa	v m³/kg	T K	s kJ/kg·K
1	Hot, fully compressed	100	2.296	800	0.985
2	Isothermal expansion complete	33.3	6.888	800	1.301
3	Adiabatic expansion complete	1.1	79.9862	300	1.301
4	Isothermal compression complete	3.2	26.6621	300	0.985

Isothermal expansion · Adiabatic expansion · Isothermal compression · Adiabatic compression

Starting points Pick one, then move any slider from there.

Steam at 800 K rejecting to a cooling tower. The ceiling is 62.5%; real stations reach about 40, and the whole gap is irreversibility.

Hot reservoir T_H 800 K

Cold reservoir T_C 300 K

Isothermal volume ratio r_iso 3

Explore

What a cycle can do, and what nothing can do.

Energy in equals energy out

Over one complete cycle the working fluid comes back to exactly the state it started in, so it has stored nothing. Everything that went in as heat has left again — some as work, the rest as waste heat. That is why the three energy cards on the panel are not three independent readings but two readings and their difference.

Subtract heat rejected from heat added. You get net work, exactly, on every cycle and at every setting.

The ceiling nothing gets past

Heat will not flow from cold to hot on its own, and the price of that is a hard limit on any engine: one minus the cold temperature over the hot one, both absolute. It does not care what the machine is made of, what fuel it burns, or how well it is built.

The Carnot card and the efficiency card read the same number here, because this cycle IS the limit. On every other cycle the first sits below the second.

Why the T–s diagram is worth reading

Entropy is the bookkeeping that makes the second law quantitative. Its usefulness on screen is direct: on a temperature–entropy diagram, the area under a leg is the heat crossing the boundary on it, and the area enclosed by the loop is the net heat — which the first law says must equal the net work.

Switch the diagram to T–s. The enclosed area is the same number as the area inside the P–v loop, in different units of picture.

Where the missing efficiency goes

Every cycle here is drawn reversible — no friction, no turbulence, no temperature difference across a heat exchanger. Real machines lose more on top. The gap between the efficiency card and the Carnot card is not that loss: it is the cost of the cycle’s SHAPE, before anything real has gone wrong at all.

Drag turbine efficiency from 0.88 up to 1. That single slider is the largest single gap between this model and a working power station.

Compression ratio decides an Otto engine

Squeeze harder before you burn, and you expand further afterwards against a lower back pressure. For the air-standard Otto cycle that is the whole story — the efficiency expression contains the compression ratio and the ratio of specific heats, and nothing else at all.

Drag the compression ratio from 6.5 to 14 and watch the efficiency climb from 52.7% to 65.2% — with the peak temperature untouched.

Burning hotter does not make it more efficient

This is the result nobody believes. A hotter peak means more work out of every cycle, and the net-work card does climb — but heat added climbs by exactly the same proportion, so the ratio between them does not move. Peak temperature buys power, not efficiency.

Drag peak temperature from 1400 K to 2400 K. Net work rises a long way. Efficiency does not move a hundredth of a point.

When the heat goes in matters

Add heat at constant volume and it all arrives while the gas is at its most compressed, where it does the most good. Add it at constant pressure and the piston is already moving away, so the later heat arrives at a lower temperature and earns less. That is why Diesel sits below Otto at the same compression ratio — and why real diesels win anyway, by running roughly double the ratio.

Drag the cut-off ratio down towards 1.1 and watch the efficiency rise. Shorter injection means the heat arrives earlier, and earlier heat is worth more.

Expanding further than you compressed

An ordinary engine throws away gas that is still above atmospheric pressure — energy that walks out of the exhaust. Let it expand all the way down and you collect that instead. It costs power density, because the same compression now serves a bigger cylinder, which is exactly the trade a hybrid can afford to make.

These are the Otto default’s exact numbers. Switch between the two cycles: the only difference left is how far the gas is allowed to expand, and it is worth 5.7 points.

The area inside the loop is the work

Pressure times a change in volume is work, so the area enclosed by the P–v loop is the net work per kilogram of working fluid. A tall thin loop and a short fat one can enclose the same area and produce the same work from very different machines.

The shaded region on the P–v diagram is the net-work card, drawn. Drop the compression ratio and watch the loop and the number shrink together.

The compressor eats first

In a gas turbine the compressor is driven by the turbine, so a large slice of the output never leaves the machine. The back work ratio is that slice. It is why gas turbines were useless until compressors got good — and why a steam plant, which pumps an almost incompressible liquid instead, gets away with a back work ratio near zero.

Compare this with the Rankine cycle’s back work ratio. A pump moving water costs almost nothing; a compressor moving air costs a third of the machine.

Mean effective pressure, and why it exists

Efficiency says nothing about size. Mean effective pressure is the net work divided by the swept volume — the steady pressure that would do the same job over one stroke. It is how you compare two engines of different capacity, and it is the number that decides how big the engine has to be.

Raise the peak temperature: efficiency sits still while mean effective pressure climbs. More power from the same engine, at the same efficiency.

Why a steam turbine has a wetness limit

Expand steam far enough and it starts condensing inside the turbine. Water droplets at those speeds erode the last-stage blades, so plants hold the exhaust above roughly 88% dry. Superheat is the usual answer — and the reason real stations reheat halfway down instead of simply superheating harder.

Read the warning under the panel, then drag turbine inlet temperature up as far as it goes. It helps, and it does not get you to 88% — which is the honest reason reheat exists.

How the Carnot cycle works

The Carnot cycle takes its working fluid through 4 processes and returns it to the state it began in. In order, those are:

Isothermal expansion
Adiabatic expansion
Isothermal compression
Adiabatic compression

You will find it in nowhere — it is a limit, not a machine. At the standard set-up above — hot reservoir 800 K, cold reservoir 300 K, isothermal volume ratio 3 — it reaches a thermal efficiency of 62.50%, against a Carnot limit of 62.50% for the same temperature span. The net work is 157.7 kJ/kg, from 252.2 kJ/kg of heat in and 94.6 kJ/kg rejected.

Its efficiency has a closed form: η = 1 − T_C/T_H. The simulator above does not use it — it solves the state points and divides net work by heat in — but the two agree to machine precision, and the test suite behind this page checks that on every build.

Reading the two diagrams

Switch the diagram pane between P–v and T–s and watch the same cycle change costume. On the pressure–volume plot the area inside the loop is the net work per kilogram of working fluid, because work is the integral of P dv. On the temperature–entropy plot the area inside the loop is the net heat, because heat in a reversible process is the integral of T ds.

Those two areas are the same number. That is the first law applied to a closed loop: the fluid ends where it started, so its internal energy has not changed, so everything that went in as heat came out as work. Two different pictures, one arithmetic.

The T–s view is the more revealing of the two once you are comfortable with it, because it shows the temperature at which heat crossed the boundary — and that, not the amount, is what decides how much of it can become work.

A book about steam engines that founded a science

Sadi Carnot published Réflexions sur la puissance motrice du feu in 1824, trying to answer a practical question: is there a limit to how much work you can get from a given quantity of heat, and does it depend on what the engine is made of? He was working before energy conservation was understood and before the word thermodynamics existed.

His answer was that there is a limit, that it depends only on the two temperatures between which the engine works, and that no engine of any construction can beat it. He was right, and the reasoning survives intact today even though the caloric theory he used to reach it does not.

The one cycle nobody is trying to build

Every other cycle on this page describes a machine somebody makes. This one describes a bound. Its two isothermal processes require heat to cross a boundary at zero temperature difference, which means at zero rate — a genuinely reversible engine would take infinitely long to produce any work at all.

That is why the isothermal volume ratio slider is worth playing with. Move it and the work per cycle changes while the efficiency card does not move at all. Carnot efficiency depends on the two temperatures and on nothing else — not the working fluid, not the size, not the shape of the loop.

How to use a limit you cannot reach

The Carnot figure is not a target. It is a measuring stick, and its practical use is diagnostic: it tells you whether an engine is disappointing because it is badly built or because it is working across too narrow a temperature span.

A power station at 40% against a Carnot ceiling near 62% is doing genuinely well. A waste-heat engine at 12% against a ceiling of 20% is also doing well, and no amount of engineering will make it good — the answer there is to find hotter heat, not a better machine. Set the hot reservoir to 450 K above and see how little is available even in principle.

Carnot cycle — common questions

Why can no engine beat the Carnot efficiency?

Because if one could, you could run it forward and a Carnot engine backward between the same two reservoirs and end up moving heat from cold to hot with no net work input — which the second law forbids. The argument needs no detail about either machine, which is exactly what makes the limit universal.

Do I have to use Kelvin?

Yes, always. The expression is η = 1 − T_C/T_H and it is a ratio of absolute temperatures. Using Celsius produces nonsense — a cold reservoir at 0 °C would appear to give 100% efficiency. This page shows temperatures in kelvin for SI and degrees Rankine for imperial, because both are absolute.

Why is the Carnot cycle impossible to build?

Its isothermal legs need heat to flow across zero temperature difference, and heat driven by zero difference flows at zero rate. Any real machine needs a finite temperature gap to move heat in finite time, and that gap is itself an irreversibility. The limit is approached, never met.

Which real cycles get closest to it?

The Stirling and Ericsson cycles, both on this page. With a perfect regenerator each reaches the Carnot efficiency exactly — you can verify it above by setting regenerator effectiveness to 1 and comparing the two cards. Real regenerators are not perfect, and the gap opens quickly as effectiveness falls.

Thermodynamic cycles reference library