Dual Cycle.

The limited-pressure cycle — part of the burn at constant volume, the rest at constant pressure. The closest simple model to what a real engine does.

Mode

Cycle

Otto Diesel Dual Brayton Rankine Carnot Stirling Ericsson Atkinson

The machine — The limited-pressure cycle — part of the burn at constant volume, the rest at constant pressure. The closest simple model to what a real engine does.

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 -437.3 kJ/kg

Thermal efficiency

65.04%

Carnot limit, same span

86.26%

Net work

730 kJ/kg

Heat added

1,122 kJ/kg

Heat rejected

392.3 kJ/kg

Peak temperature

2,183 K

Lowest temperature

300 K

Mean effective pressure

904.4 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	4,850	0.0538	909	0
3	Constant-volume burn complete	7,760	0.0538	1,455	0.337
4	Constant-pressure burn complete	7,760	0.0807	2,183	0.745
5	Fully expanded	282.3	0.861	847	0.745

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

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

Part of the charge burns almost instantly at constant volume, the rest as injection continues. This is the closest any simple cycle gets to a measured indicator diagram.

Compression ratio r 16

Pressure-rise ratio rₚ 1.6

Cut-off ratio r_c 1.5

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 Dual cycle works

The Dual cycle takes its working fluid through 5 processes and returns it to the state it began in. In order, those are:

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

You will find it in modern high-speed diesel engines. At the standard set-up above — compression ratio 16, pressure-rise ratio 1.6, cut-off ratio 1.5, intake temperature 300 K — it reaches a thermal efficiency of 65.04%, against a Carnot limit of 86.26% for the same temperature span. The net work is 730 kJ/kg, from 1,122 kJ/kg of heat in and 392.3 kJ/kg rejected.

Its efficiency has a closed form: η = 1 − (1/r^(γ−1)) · (rₚr_c^γ − 1)/((rₚ−1) + γrₚ(r_c−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.

The cycle that admits engines are not tidy

The Dual cycle — also called the limited-pressure, Seiliger or Sabathé cycle — is not named for an inventor because it does not describe an invention. It describes a measurement. When engineers took indicator diagrams from real compression-ignition engines, the burn matched neither the Otto model nor the Diesel model: some of the charge went up almost instantly at constant volume, and the rest burned progressively at roughly constant pressure as injection continued.

So the Dual cycle simply has both, in sequence: a constant-volume rise as the premixed portion goes up, then a constant-pressure stretch as injection continues into the descending piston. It is the closest any simple air-standard model gets to a measured pressure trace from a working engine, and the extra state point it needs is the price of that fidelity.

One model, both parents

Two of its three sliders are the interesting ones. Set the pressure-rise ratio to 1 and nothing burns at constant volume — the cycle becomes the Diesel cycle exactly, to machine precision, and the test suite behind this page asserts that. Push the cut-off ratio down towards 1 and almost nothing burns at constant pressure, and it approaches the Otto cycle just as closely.

That degeneracy is the point. Otto and Diesel are not two different theories of combustion; they are the two ends of one, and every real engine lives somewhere between them.

Why the pressure rise is limited at all

The name "limited-pressure cycle" is the honest one. Left alone, injecting all the fuel at constant volume would give the best efficiency the model can offer — that is simply the Otto limit. What stops it is that peak pressure has to be carried by the connecting rod, the bearings and the block.

So the constant-volume portion is deliberately curtailed, and the rest of the fuel is fed in while the piston descends and the pressure holds roughly steady. The cycle is shaped by a structural constraint rather than a thermodynamic one, which is unusual and worth noticing.

You can watch that trade directly. Raise the pressure-rise ratio on the panel above and the efficiency climbs — and so does the peak pressure at state 3, which is the number a designer is actually rationing. Every point of efficiency won this way is paid for in bearing loads and metal.

Dual cycle — common questions

What is the Dual cycle used to model?

Modern high-speed compression-ignition engines, where part of the injected fuel burns almost instantly on ignition and the rest burns progressively. It is also called the limited-pressure, Seiliger or Sabathé cycle.

How is it different from the Diesel cycle?

The Diesel cycle adds all of its heat at constant pressure. The Dual cycle splits the addition: some at constant volume, the rest at constant pressure. Set the pressure-rise ratio slider above to 1 and the two become the same cycle exactly.

Why does it have five state points instead of four?

Because the heat addition is split into two processes, so it needs a state point between them. That is the only structural difference — compression and expansion are still adiabatic, and rejection is still at constant volume.

Which is more realistic, Dual or Diesel?

Dual, for any engine running at speed. Injection and combustion take real time, and at a few thousand revolutions per minute the piston has moved appreciably before the burn finishes — so neither pure constant-volume nor pure constant-pressure describes it, and the split does.

Thermodynamic cycles reference library