Diesel Cycle.

Compression ignition — no spark, and heat added at constant pressure while the fuel is still being injected.

Mode

Cycle

Otto Diesel Dual Brayton Rankine Carnot Stirling Ericsson Atkinson

The machine — Compression ignition — no spark, and heat added at constant pressure while the fuel is still being injected.

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 -468.7 kJ/kg

Thermal efficiency

63.16%

Carnot limit, same span

84.27%

Net work

604.8 kJ/kg

Heat added

957.6 kJ/kg

Heat rejected

352.8 kJ/kg

Peak temperature

1,907 K

Lowest temperature

300 K

Mean effective pressure

743.8 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 — injection begins	5,720	0.0478	953	0
3	Cut-off — injection ends	5,720	0.0957	1,907	0.696
4	Fully expanded	263.9	0.861	792	0.696

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

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

A small high-speed diesel at a moderate load. Note the compression ratio is roughly double a petrol engine’s — there is no spark, so the air itself has to reach ignition temperature.

Compression ratio r 18

Cut-off ratio r_c 2

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 Diesel cycle works

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

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

You will find it in lorry, ship and locomotive engines. At the standard set-up above — compression ratio 18, cut-off ratio 2, intake temperature 300 K — it reaches a thermal efficiency of 63.16%, against a Carnot limit of 84.27% for the same temperature span. The net work is 604.8 kJ/kg, from 957.6 kJ/kg of heat in and 352.8 kJ/kg rejected.

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

Rudolf Diesel’s idea

Rudolf Diesel patented his engine in the 1890s explicitly as an attempt to build something closer to the Carnot ideal than the spark-ignition engines of his day. He did not achieve that — nothing does — but he achieved something better than what existed, and by a route nobody else had taken: compress the air so hard that it becomes hot enough to ignite fuel on contact, and there is no need for a spark at all.

That removes the knock limit at a stroke. Nothing is in the cylinder to pre-ignite during compression except air, so the compression ratio can go as high as the structure will bear.

Why it looks worse on paper and wins in practice

At the same compression ratio, the Diesel cycle is less efficient than the Otto cycle — always, at every setting. Fuel is injected progressively while the piston is already moving away, so heat arrives at constant pressure rather than constant volume, and the later it arrives the lower the temperature it arrives at and the less of it can become work. The cut-off ratio measures how far that injection continues, and the efficiency falls as it lengthens.

Real diesels beat real petrol engines anyway, by roughly doubling the compression ratio. They win the comparison the cycle loses. Set the ratio here to 18 and then look at what an Otto engine on pump fuel manages at 9 — that is the whole argument, and it is why lorries, ships and locomotives are diesel.

The part-load result that surprises people

Notice which way this cycle moves with load. Less fuel means a shorter injection, a smaller cut-off ratio, and higher efficiency. A diesel is at its most efficient when it is working lightly.

A petrol engine does the opposite: throttling it down forces the pistons to pull against a partial vacuum, and that pumping work is pure loss. A diesel has no throttle — it always breathes a full cylinder of air and simply injects less fuel into it. That single architectural difference is why diesels dominate anything that spends its life at part load.

Diesel cycle — common questions

Why does the Diesel cycle have no peak-temperature slider here?

Because fixing the compression ratio and the cut-off ratio fixes the peak temperature — it is an output, not an input. Offering a slider for it would be offering a control that does nothing. At r = 18 and a cut-off of 2 the peak lands near 1900 K, and you can read it off the panel above.

Why is Diesel less efficient than Otto at the same compression ratio?

Because heat added at constant pressure arrives while the piston is already descending, so it enters at a lower temperature than heat added at constant volume would. Lower-temperature heat is worth less work. The gap widens as the cut-off ratio grows — drag the slider above and watch.

Do diesel engines really not have spark plugs?

Correct — the air is compressed until it is hot enough to ignite fuel on contact, typically well above 800 K. Many diesels do carry glow plugs, but those are heaters used only to warm the combustion chamber for starting from cold, not igniters.

Why do diesels use such high compression ratios?

Two reasons that reinforce each other. Nothing combustible is present during compression, so there is no knock limit to respect. And the air must reach ignition temperature by compression alone, which requires a high ratio to begin with. Ratios in the high teens to low twenties are ordinary.

Thermodynamic cycles reference library