Atkinson Cycle.

An Otto engine that keeps expanding until the pressure is back to atmospheric — more efficient, less powerful, and why hybrids use it.

Mode

Cycle

Otto Diesel Dual Brayton Rankine Carnot Stirling Ericsson Atkinson

The machine — An Otto engine that keeps expanding until the pressure is back to atmospheric — more efficient, less powerful, and why hybrids use 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.

isentropic 1→2 · Adiabatic compression q 0 w -325.4 kJ/kg

Thermal efficiency

65.38%

Carnot limit, same span

83.33%

Net work

490.9 kJ/kg

Heat added

750.8 kJ/kg

Heat rejected

259.9 kJ/kg

Peak temperature

1,800 K

Lowest temperature

300 K

Mean effective pressure

323.5 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	Start of compression	100	0.861	300	0
2	Compressed	2,512	0.0861	754	0
3	Peak — spark has fired	6,000	0.0861	1,800	0.625
4	Expanded back to intake pressure	100	1.6037	559	0.625

Adiabatic compression · Constant-volume heat addition · Full adiabatic expansion · Constant-pressure heat rejection

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

Late inlet-valve closing gives a short compression and a full-length expansion. The motor covers the power the engine gives up.

Compression ratio r 10

Peak temperature T₃ 1,800 K

Intake temperature T₁ 300 K

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 Atkinson cycle works

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

Adiabatic compression
Constant-volume heat addition
Full adiabatic expansion
Constant-pressure heat rejection

You will find it in hybrid petrol engines. At the standard set-up above — compression ratio 10, peak temperature 1,800 K, intake temperature 300 K — it reaches a thermal efficiency of 65.38%, against a Carnot limit of 83.33% for the same temperature span. The net work is 490.9 kJ/kg, from 750.8 kJ/kg of heat in and 259.9 kJ/kg rejected.

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 linkage that outlived its engine

James Atkinson patented his engine in 1882, at a time when Otto held patents that made building conventional four-strokes awkward. His answer was a multi-link mechanism that gave the piston four different stroke lengths in one revolution of the crank — in particular, an expansion stroke longer than the compression stroke.

The linkage was complicated and it did not last. The idea did. What everybody calls an Atkinson-cycle engine today has an ordinary crank and achieves the same effect with valve timing instead.

Collecting what the exhaust normally throws away

An ordinary Otto engine opens its exhaust valve while the gas inside is still well above atmospheric pressure. That pressure represents work you could have had and did not take — it leaves as noise and heat. Let the gas expand all the way down instead and you collect it.

Set the compression ratio here to 9 and compare with the Otto cycle at the same 9: the over-expansion is worth about 5.7 percentage points. The cost is power density. The extra expansion means a larger cylinder for the same amount of trapped charge, so the engine is bigger and heavier for the power it makes.

Why hybrids can afford it and other cars cannot

A modern implementation holds the inlet valve open into the compression stroke, pushing some of the charge back out. The gas that remains is compressed over a shorter effective stroke and then expanded over the full one. Same crank, same block, different cam.

That costs power, and in a conventional car power is what you are buying. In a hybrid it is not: the electric motor supplies acceleration, so the engine is free to be optimised for efficiency at the steady cruise where it actually runs. That is the whole reason this nineteenth-century idea came back.

The model on this page treats the compression ratio as the effective one — what the trapped charge is actually squeezed by — and derives the expansion from it, which is why there is no separate expansion slider. A real engine sets that split with cam timing, and can vary it as conditions change, which is a freedom the ideal cycle does not have.

Atkinson cycle — common questions

What makes the Atkinson cycle more efficient than Otto?

The expansion is longer than the compression, so the gas is taken down closer to atmospheric pressure before the exhaust opens. An Otto engine dumps gas that still has usable pressure in it. At a compression ratio of 9 the difference is worth about 5.7 percentage points — check it above by switching between the two cycles at the same settings.

Why do hybrids use it?

Because it trades power for efficiency, and a hybrid has an electric motor to cover the power. The engine can then be tuned for the steady, moderate load where it spends most of its life. In a conventional car the same trade would just make the car feel slow.

Is the Miller cycle the same thing?

Closely related. Both shorten the effective compression stroke relative to expansion by altering valve timing. The usual distinction is that a Miller-cycle engine adds forced induction to recover the lost charge density, while an Atkinson-cycle engine accepts the lower output.

Why does the advantage shrink at high compression ratios?

Because a high-compression Otto engine is already expanding a long way, so there is less pressure left in the cylinder for over-expansion to collect. The gap is about 5.7 points at r = 9 and around 4.1 at r = 13 — compression and expansion are two routes to the same place.

Thermodynamic cycles reference library