The engine that expands further than it compresses

The ceiling on every engine set Carnot’s limit, the most that any engine running between two temperatures can turn into work, and the temperature an engine really takes its heat at found real cycles falling short of it because they take in heat over a range of temperatures. The squeeze a petrol engine is not allowed followed the petrol engine’s particular constraint. Its ideal efficiency, the Otto formula $1 - r^{1-\gamma}$, rises with the compression ratio $r$, but the fuel–air mixture, squeezed and heated, ignites spontaneously ahead of the flame if $r$ is pushed much above ten. The explosive pressure wave of that knock wrecks engines, and so a petrol engine sits near ten while a diesel, which compresses air alone, reaches eighteen or twenty.

That essay closed by asking how close to a diesel a petrol engine can get. Half the answer comes from noticing that the Otto formula contains one ratio where the engine has two.

Two ratios that happen to be equal #

An engine compresses its charge on one stroke and expands the burned gas on another. In an ordinary piston engine the two strokes use the same cylinder with the same piston travel, so the volume at the start of compression equals the volume at the end of expansion, and the compression ratio equals the expansion ratio. The Otto formula then has one ratio in it, and it is easy to read it as the compression ratio, since that is the ratio the designer chooses. But the formula’s derivation uses the ratio twice: the compression stroke raises the charge’s temperature by $r^{\gamma-1}$ before the heat is added, and the expansion stroke lowers it by the same factor afterwards. Work is the difference between the heat added and the heat thrown away, and the heat thrown away depends on how cool the gas is when the exhaust valve opens — on how far it has expanded.

The first figure separates the two ratios. Both engines compress by ten and add the same heat at the top of the stroke. The Otto engine then expands by ten and opens its exhaust valve while the gas is still at nearly four times atmospheric pressure. That gas rushes out with a bang that a silencer has to tame, and the push it could have given the piston is wasted. The over-expanded engine keeps expanding until the gas is back at the pressure it came in at, which in this ideal cycle takes a volume nearly twenty-seven times the combustion chamber. The extra work is the area between the expansion curve and the intake pressure beyond the Otto engine’s stroke, and it raises the ideal efficiency from 55 to 64 per cent, with exactly the same compression and exactly the same risk of knock.

James Atkinson patented engines built on this idea in the 1880s, with linkages that gave the piston a longer expansion stroke than compression stroke. They were mechanically complicated, made less power than simpler engines of the same size, and were forgotten. The idea returned a century later in a form that needs no linkage, and the second and third figures show why it was worth reviving and why it had been forgotten.

Most of the gain is in the expansion #

The second figure plots the efficiency against the expansion ratio for three compression ratios, beside the Otto line on which the two ratios are equal. The over-expanded lines climb steeply from where they leave the Otto line and run almost alongside it. At an expansion ratio of thirteen, an engine that compresses by only ten reaches 59.0 per cent, against 59.3 for an Otto engine that compresses and expands by thirteen. The three-tenths of a percentage point is all the higher compression buys, and the higher compression is the part that knocks.

This is the decisive observation, and it reframes the petrol engine’s whole limitation. The thermodynamics does not care how hot the charge got during compression; it cares how much the burned gas is allowed to cool by doing work before it leaves. Raising the compression ratio raises the expansion ratio as a side effect, and it is the side effect that pays. The knock limit, which is about the temperature and pressure of the unburned mixture at the end of compression, restricts the compression ratio only. The expansion ratio is restricted by nothing in the combustion and can be chosen separately — if the engine can be built to choose it.

There is a limit, and it is complete expansion: the gas cannot usefully expand below the pressure of the air outside, because below that the atmosphere would be pushing on the piston harder than the gas is. The atmosphere is the engine’s cold reservoir in pressure as well as in temperature, and like the cold reservoir of the cold reservoir overhead, it sets the floor the working fluid can usefully reach. The dots at the ends of the lines mark it. For a compression ratio of ten in this ideal cycle it lies at an expansion ratio near twenty-seven, an engine whose expansion stroke is nearly three times its compression stroke.

Paid for in power #

The cost is in the third figure. An engine’s power for its size is measured by its mean effective pressure, the work done per cycle divided by the volume the piston sweeps: the steady pressure that, pushing on the piston for one stroke, would do the same work. The over-expanded cycle does more work with each charge, but it needs a bigger cylinder for the expansion, and the work per unit of swept volume falls. Expanding to thirteen instead of ten costs a fifth of the engine’s power for its size; expanding completely costs three-fifths. An engine that is more efficient per litre of fuel is less powerful per litre of cylinder.

That is why Atkinson’s engines lost to Otto’s. In the nineteenth century, and for most of the twentieth, the question an engine had to answer was how much power it could make for its weight and cost, and over-expansion answered it badly. The revival came when the question changed to how little fuel an engine could burn in ordinary driving, and when an electric motor could fill in the power the engine lacked.

The revival also found a way to over-expand without a linkage. An engine with an ordinary crank and an ordinary cylinder can be given an expansion ratio of thirteen, a geometric compression ratio of thirteen too, and then made to compress by less by holding its intake valve open for part of the compression stroke, so that some of the charge is pushed back out into the intake before the valve closes and compression begins in earnest. The effective compression ratio is set by when the valve closes and can be kept below the knock limit, while the expansion ratio is the full geometric ratio. The hybrid cars of the late 1990s, beginning with the Toyota Prius, used exactly this, and engines run on this late-closing cycle — called Atkinson or Miller cycles after their two inventors — now reach brake efficiencies of forty per cent and more, which was diesel territory a generation ago.

What the exhaust no longer carries #

The fourth figure follows the gas to the exhaust valve. The more it expands, the cooler and the lower in pressure it arrives: at complete expansion it is at the pressure of the intake air and a few hundred degrees cooler than an Otto engine’s exhaust. The difference is the extra work, taken out of the exhaust’s energy. The work left in two buckets of water found that any temperature difference holds work that a suitable engine could extract, and an Otto engine’s exhaust holds two kinds: the pressure it still has when the valve opens, which a longer expansion recovers directly, and the heat it carries at atmospheric pressure, which only a second engine could use. Over-expansion takes the first. Thermoelectric generators clamped to an exhaust pipe have been tried on the second, turning a little of its heat into electricity directly; an engine with one number in it found why they recover only a few per cent of it, their efficiency capped by a figure of merit no material has yet pushed far. Turbochargers, which put a turbine in the exhaust to drive a compressor in the intake, take part of the first by a different route, and combined-cycle power stations take the second with a steam turbine heated by a gas turbine’s exhaust.

A cooler exhaust has costs of its own. A catalytic converter must reach a few hundred degrees before it works, and engines that exhaust cooler gas take longer to warm it, which matters most in the first minutes of a journey, when most of a car’s emissions are made. Engine designers trade efficiency against emissions here as elsewhere.

Five engines and the remaining gap #

The last figure sets the cycles side by side, all computed with the same gas properties and the same heat added. The over-expanded petrol engine matches an Otto engine it could never be built as, and the fully expanded cycle passes it. The ideal Diesel cycle, compressing air eighteen times and burning its fuel at nearly constant pressure as it is injected, comes out no better than the over-expanded petrol cycle on these assumptions.

Real diesels are still more efficient than real petrol engines, and the reasons are instructive because none is the compression ratio alone. A diesel runs unthrottled: it controls its power by how much fuel it injects, not by restricting the air, so it does not waste work pumping air past a partly closed throttle, which costs a petrol engine a large fraction of its efficiency at light load. A throttle is an irreversible expansion through a valve, the process the gas that cools by being let go followed for a gas forced through a plug, and the engine must then do work to draw its charge past it. The late-closing intake valve of an over-expanded engine helps here too, since the engine can control its charge by when the valve closes rather than by throttling. A diesel runs lean, with far more air than its fuel needs, and lean gas, mostly air, has a higher ratio of specific heats than a rich mixture, which raises every efficiency in the figure. And a diesel’s high compression is not wasted: its expansion ratio is high too. The over-expanded petrol engine closes the gap from the expansion side; the remaining gap is in throttling and in the mixture. Petrol engines that run lean, inject their fuel directly or ignite by compression, as the squeeze a petrol engine is not allowed noted, are closing it from those sides.

Boost puts the power back #

The late-closing valve leaves an engine with a smaller charge than its cylinder could hold, and the third figure priced that loss in mean effective pressure. There is a way to have the expansion without paying so much of it, which Ralph Miller patented in the 1950s: fill the cylinder with air at more than atmospheric pressure, from a supercharger or a turbocharger, so that the shortened compression stroke starts with a denser charge. The cylinder then holds as much air as an ordinary engine would, and the mean effective pressure comes back.

The reason this does not simply restore the knock is that the compression has been split into two parts with cooling between them. The compressor squeezes the intake air first and heats it, an intercooler takes that heat out at nearly constant pressure, and only then does the piston finish the compression. Knock depends on the temperature and pressure of the unburned mixture at the end of compression, and for a given final pressure a charge compressed in two cooled stages arrives cooler than one compressed in a single stroke from the atmosphere. That is the principle of every multistage compressor, and in an engine it means that the effective compression ratio inside the cylinder can stay below the knock limit while the pressure at the start of combustion is as high as a high-ratio engine’s.

The turbocharger’s turbine is driven by the exhaust, so part of the expansion that a short-stroke Otto engine throws away is done in the turbine instead of in the cylinder — the same work collected in a different machine. The combination of a long expansion in the cylinder, a split and cooled compression, and a turbine on the exhaust is how most efficient modern petrol engines are built, and how large gas engines for power generation reach efficiencies close to those of diesels. Each piece is a way of giving back what one cylinder serving for two strokes took away.

Steam engines and turbines learned it first #

The principle that the work is in the expansion was understood long before Atkinson, in steam. James Watt’s separate condenser of 1769 let the steam in a cylinder expand against a near vacuum instead of against the atmosphere, and the compound and triple-expansion engines of the nineteenth century passed the steam through two or three cylinders of increasing size in succession, each expanding it further, so that a ship’s engine extracted work from steam all the way down from boiler pressure to the condenser’s vacuum. The triple-expansion engine made long-distance steamships economical because it stopped throwing away steam that could still push.

A turbine does the same thing without trying. In a gas turbine the hot gas flows continuously through stage after stage of blades and expands until it reaches the pressure at the exhaust, whatever that takes; there is no cylinder whose size fixes the expansion, and the Brayton cycle that describes it is a fully expanded cycle by construction. The piston engine is the odd one out among heat engines in having its expansion tied to its compression, and it is tied only because one cylinder and one crank serve for both strokes. Atkinson’s linkages and the later trick of the late-closing valve are ways of cutting that tie in an engine that keeps the cylinder.

How ideal the ideal cycle is #

All the figures use the air-standard model: the working gas is treated as a single ideal gas with a fixed ratio of specific heats, heat is added instantaneously at the top of the stroke as though the fuel burned in no time, and the gas is exhausted and replaced with fresh charge at no cost. A real engine burns its fuel over tens of degrees of crank rotation, loses heat to the cylinder walls throughout, does work pumping gas in and out, loses some to friction, and its working gas changes composition and heat capacities as it burns. Real efficiencies are therefore lower than the figures by large factors — the 55 per cent of an ideal Otto cycle at a ratio of ten becomes thirty-something in a car, a gap that the engine that has to finish traced partly to the cost of doing anything in a finite time — and the heat release used here, a 2,000-degree rise at constant volume, is an assumption chosen to be representative, not a property of any fuel.

The comparisons between cycles are more trustworthy than the absolute numbers, because most of the losses affect every cycle alike. The ordering of the bars and the shape of the curves survive the simplifications; the percentages do not.

What a pressure–volume loop does not show #

The pressure–volume diagram is a map of the work and nothing else. It does not show time, so it says nothing about how fast the engine runs or how quickly the fuel burns, and it does not show the flow of heat through the walls, which in a small engine at low load is the largest loss of all. The late-closing intake valve that real over-expanded engines use appears in the figures only as an effective compression ratio; the backflow of charge into the intake and the pumping work it costs are left out. And the figure cannot show knock, the reason for everything: the limit on compression is a matter of chemistry, of how fast the end gas reacts at the temperature compression gives it, which the earlier essay computed and this one takes as given.

Still open: whether a petrol engine can reach a diesel’s efficiency #

Production petrol engines with over-expanded cycles, high compression, direct injection and lean or dilute combustion now reach brake efficiencies near forty-one per cent, and laboratory engines a few points more; the best diesels for cars are a little above forty-five. Whether the remaining gap can be closed in an engine that meets emission limits — lean petrol combustion produces nitrogen oxides that the usual catalytic converter cannot remove, and compression ignition of petrol is hard to control across an engine’s range of speeds and loads — is the subject of active development. The thermodynamics drawn here says the expansion side of the gap is nearly closed. The rest is chemistry and control.

The habit worth carrying away is to ask which of a formula’s parameters is doing the work. An engine’s efficiency is set by how far its burned gas expands, not by how hard its charge was squeezed: compressing by 10 and expanding by 13 gives 59.0 per cent in the ideal cycle, against 59.3 for a knocking Otto engine compressing by 13, and expanding all the way to the intake pressure gives 63.6 — paid for by a mean effective pressure that falls from 11.0 to 4.4 bar. When two quantities are equal because one mechanism happens to set both, a formula written in one of them may be about the other.