Thermodynamic Cycles Simulator.

Nine cycles, three machines, one stage. Set the compression ratio and watch the loop change shape — the area inside it is the work you get out.

Mode

Cycle

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 to read a thermodynamic cycle

Pick a cycle. Otto is the petrol engine, Diesel the compression-ignition one, Brayton the gas turbine and Rankine the steam power station; Carnot, Stirling and Ericsson are the ideal limits, and Dual and Atkinson are what real engines actually approach.
Set the compression or pressure ratio. Watch the loop on the P–v diagram change shape — the area it encloses is the net work per kilogram, so a fatter loop is a more powerful engine.
Compare the thermal efficiency with the Carnot limit beside it. No cycle can beat that number, and how far short it falls is the honest measure of the design.
Switch to the T–s diagram. There the area inside the loop is the net heat instead, and the shape tells you at what temperature the heat went in — which is what really sets the efficiency.
Read the state-point table underneath. Every corner of the loop is a real pressure, volume, temperature and entropy, and the arithmetic joining them is the whole subject.

What a cycle is, and why it has to close

An engine is not a thing that burns fuel. It is a thing that takes a working fluid — air, steam, helium — around a loop of processes and returns it to exactly the state it started in, over and over. That closing is not a detail; it is the entire constraint. Because the fluid comes back to where it began, its internal energy is unchanged over a full loop, and the first law collapses to something remarkably simple: the net work out equals the net heat in minus the net heat rejected.

That is why the loop on a P–v diagram matters so much. The work done in any process is the area under its path, so going around a closed loop leaves you with the area enclosed by it. A fat loop is a powerful engine and a thin one is a feeble engine, and you can see which you have at a glance without computing anything. Run the compression-ratio slider above and watch the loop stretch.

The T–s diagram does the same job for heat. Area under a path there is heat transferred, so the enclosed area is the net heat — which, by the first law, is the same number as the net work. Two completely different pictures of the same cycle, and they have to agree. That they do is one of the more satisfying facts in the subject, and switching between the two views above is the fastest way to feel it.

The nine cycles, and what each one is for

They are not nine variations on a theme. Each isolates a different idea, and the differences are almost entirely about when the heat goes in.

Cycle	Where it lives	η at our defaults	Carnot limit
Otto cycle	Petrol engines	58.5%	83.3%
Diesel cycle	Lorry, ship and locomotive engines	63.2%	84.3%
Dual cycle	Modern high-speed diesel engines	65.0%	86.3%
Brayton cycle	Jet engines and power-station gas turbines	50.8%	80.0%
Rankine cycle	Coal, gas, nuclear and solar-thermal power stations	40.2%	63.5%
Carnot cycle	Nowhere — it is a limit, not a machine	62.5%	62.5%
Stirling cycle	Submarine power, cryocoolers, solar dishes	66.7%	66.7%
Ericsson cycle	Intercooled, reheated and regenerated gas turbines approximate it	66.7%	66.7%
Atkinson cycle	Hybrid petrol engines	65.4%	83.3%

Those efficiency figures are computed by the same solver that drives the machine above, at each cycle's own standard set-up — so they move if the model moves, and they cannot quietly disagree with what the instrument shows.

Why the Carnot limit is not a suggestion

Beside every efficiency on this page sits a second number: the Carnot limit for that cycle's own temperature span. It is the most important number here, and it is worth being precise about what it claims.

Carnot's 1824 result says that no heat engine operating between a hot reservoir at T_H and a cold one at T_C can exceed 1 − T_C/T_H, with both temperatures absolute. Not "no engine we have built yet" — no engine, ever, of any design, using any working fluid, made of anything. It falls out of the second law, and an engine that beat it could be run backwards to move heat from cold to hot for free.

Two consequences catch people out. First, the limit depends only on the two temperatures, so the way to a better engine is almost always to run it hotter, not to be cleverer about the middle. Second, real efficiency is bounded well below Carnot anyway, because a real cycle does not add all its heat at T_H — the Otto cycle adds heat over a rising temperature, and the average temperature of heat addition is what actually counts. Watch the gap between the two figures above as you move the sliders: it never closes, and understanding why is most of a thermodynamics course.

Compression ratio, and the one result nobody believes

Set the Otto cycle running above and change the peak temperature. The net work moves. The efficiency does not move at all.

That is not a bug in this simulator. For an ideal Otto cycle the efficiency is 1 − 1/r^(γ−1), and there is no temperature in that expression anywhere. Adding more fuel makes the engine more powerful — a bigger loop, more work per cycle — without making it any more efficient, because the extra heat you put in comes back out of the exhaust in the same proportion. Efficiency belongs to the geometry of the engine, not to how hard you drive it.

Which leads to the obvious question: why not just raise the compression ratio forever? Two reasons, and both are real. In a petrol engine the mixture ignites on its own once it gets hot enough, and that pre-ignition — knock — destroys engines; it is why road petrol engines sit around 9 to 12 and why higher-octane fuel allows more. In a diesel there is no mixture to pre-ignite, since only air is compressed and fuel arrives at the end, which is exactly why diesels run at 16 to 22 and why they are more efficient in practice despite the Diesel cycle being less efficient than Otto at the same compression ratio. Compare them above at equal r and you will see the crossover.

Reading the two diagrams

The diagram pane switches between P–v and T–s, and the same cycle changes 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 reversible heat is the integral of T ds.

Those two areas are the same number, and that is the first law applied to a closed loop: the fluid ends where it started, its internal energy has not changed, so everything that entered as heat left as work. Two pictures, one quantity.

Every leg on both diagrams is sampled along its true path rather than drawn corner to corner. An adiabatic expansion follows Pv^γ = constant and is visibly bowed; the gap between that curve and a straight chord between the same two points is where the work actually is. The constant-volume leg on a T–s plot climbs faster than the constant-pressure one because c_v < c_p — and that gap, drawn, is the entire difference between the Otto and Diesel cycles.

Thermodynamic cycles — common questions

What is a thermodynamic cycle?

A sequence of processes that takes a working fluid through changes of pressure, volume and temperature and returns it to exactly the state it started in. Because it closes, its internal energy is unchanged over the loop, so every joule of heat that went in has left again — as work, or as waste heat. That is why net work equals heat in minus heat out on the panel above, exactly, for every cycle here.

Which cycle is the most efficient?

Carnot, and it is not a machine — it is the limit. Of the buildable ones, the Stirling and Ericsson cycles reach that limit with a perfect regenerator, which no real regenerator is. Comparing cycles by efficiency alone is misleading anyway: they are quoted at different temperature spans, and a cycle's efficiency only means something against its own Carnot ceiling.

Why is efficiency compared to a Carnot limit rather than to 100%?

Because 100% is not the relevant bar. No engine working between a hot and a cold reservoir can exceed 1 − T_C/T_H, whatever it is made of. A power station at 40% against a ceiling of 62% is doing well; a waste-heat engine at 12% against a ceiling of 20% is doing well too. The ratio between the two cards on the panel is the honest measure.

What does the area inside the loop mean?

On the P–v diagram it is the net work per kilogram of working fluid, because work is the integral of P dv. On the T–s diagram it is the net heat, because reversible heat is the integral of T ds. They are the same number — two pictures of one quantity. Switch the diagram pane above and compare.

Are these ideal cycles or real engines?

Ideal, and deliberately so. These are air-standard cycles: no friction, no heat loss through the walls, constant specific heats, and air as the working fluid throughout. A real petrol engine returns roughly half what the Otto cycle predicts. The model's job is to show which variables matter and in which direction, not to predict a dynamometer reading.

Why do the diagrams here show curves rather than straight lines?

Because the processes are curves. An adiabatic expansion follows Pv^γ = constant, which is visibly bowed, and the gap between that curve and a straight chord between the same two corners is where the work actually is. Every leg on both diagrams is sampled along its true path rather than drawn corner to corner.