Otto Cycle.

The spark-ignition cycle — heat added at constant volume, and efficiency that depends on the compression ratio alone.

Mode

Cycle

Otto Diesel Dual Brayton Rankine Carnot Stirling Ericsson Atkinson

The machine — The spark-ignition cycle — heat added at constant volume, and efficiency that depends on the compression ratio alone.

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 -303.1 kJ/kg

Thermal efficiency

58.48%

Carnot limit, same span

83.33%

Net work

452.1 kJ/kg

Heat added

773.1 kJ/kg

Heat rejected

321 kJ/kg

Peak temperature

1,800 K

Lowest temperature

300 K

Mean effective pressure

590.7 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	2,167	0.0957	722	0
3	Peak — spark has fired	5,400	0.0957	1,800	0.655
4	Fully expanded	249.1	0.861	747	0.655

Adiabatic compression · Constant-volume heat addition · Adiabatic expansion · Constant-volume heat rejection

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

A naturally aspirated car engine on ordinary pump fuel. Compression is limited by knock, not by anything the cycle cares about.

Compression ratio r 9

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 Otto cycle works

The Otto 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
Adiabatic expansion
Constant-volume heat rejection

You will find it in petrol engines. At the standard set-up above — compression ratio 9, peak temperature 1,800 K, intake temperature 300 K — it reaches a thermal efficiency of 58.48%, against a Carnot limit of 83.33% for the same temperature span. The net work is 452.1 kJ/kg, from 773.1 kJ/kg of heat in and 321 kJ/kg rejected.

Its efficiency has a closed form: η = 1 − 1/r^(γ−1). 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.

Where the Otto cycle came from

Nikolaus Otto built the first commercially successful four-stroke engine in 1876, and the air-standard cycle that models it carries his name. The four strokes a car engine actually performs — intake, compression, power, exhaust — are not the four processes of the Otto cycle; the cycle is an idealisation of what happens to the gas trapped in the cylinder while the valves are shut, with the gas exchange abstracted away entirely.

That abstraction is why the model has only one meaningful input. A real engine has valve timing, fuel chemistry, turbulence, heat loss to the coolant and a throttle. The air-standard Otto cycle has a compression ratio.

Compression, and what stops you having more of it

Efficiency here depends on the compression ratio alone: η = 1 − 1/r^(γ−1). Nothing about how much fuel you burn appears in it. Set the peak temperature slider anywhere between 700 K and 2600 K and the efficiency card will not move by a hundredth of a point — only the net work will.

So why does no petrol engine run a compression ratio of 20? Because compressing the fuel–air mixture heats it, and past a point it ignites on its own, ahead of the spark, in a violent pressure spike that destroys pistons. That is knock, and it is the real limit — not the thermodynamics. Everything petrol engineers do about compression is a fight with chemistry, not with this cycle.

Why a real engine returns half of this

A modern petrol engine converts roughly a quarter to a third of its fuel energy into work at the crankshaft. The cycle above says 58% at a compression ratio of 9. The gap is not one thing.

Constant specific heats are the largest single error: real combustion gases at 2000 K absorb far more energy per degree than cold air does, so the model over-predicts the temperature rise and with it the work. Then there is heat lost through the cylinder walls during the burn, combustion that takes real time rather than happening instantly at top dead centre, friction, pumping work to breathe through a throttle, and the fact that the working fluid is exhaust rather than air.

Otto cycle — common questions

Why does Otto efficiency not depend on how hot the engine gets?

Because raising the peak temperature increases the heat added and the work produced in the same proportion, and efficiency is their ratio. Try it above: drag peak temperature from 1400 K to 2400 K and watch net work climb while thermal efficiency sits exactly still. Burning hotter buys power, not efficiency.

What compression ratio does a petrol engine actually use?

Naturally aspirated engines on pump fuel typically sit somewhere around 9 to 11, and direct-injection or high-octane engines go higher because both resist knock. The ceiling is set by pre-ignition, not by the cycle — thermodynamically, more is always better.

Is the Otto cycle the same as a four-stroke engine?

Not quite. A four-stroke engine performs intake, compression, power and exhaust. The Otto cycle models only what happens to the trapped gas — two adiabatic strokes and two constant-volume heat exchanges — and treats the breathing as if it were free. That simplification is why the model gives 58% where a real engine gives about 30%.

Why is the heat added at constant volume?

Because a spark ignites the whole charge almost at once, and the flame crosses the cylinder far faster than the piston can move. To a good approximation the volume has not changed by the time the burn is over — which is exactly what makes it the most efficient moment to add heat.

Thermodynamic cycles reference library