The squeeze a petrol engine is not allowed

The temperature an engine really takes its heat at found the efficiency of an ideal Otto cycle — the idealised cycle of a petrol engine — to depend only on how much the engine compresses its charge. Compress the gas into a smaller volume before the fuel burns and the heat goes in at a higher mean temperature; the efficiency is $1 - r^{1-\gamma}$, where $r$ is the compression ratio and $\gamma$ the ratio of the gas’s heat capacities. The formula has no maximum. Every increase in $r$ is an increase in efficiency.

Petrol engines nevertheless run at compression ratios between about nine and thirteen, and for most of the twentieth century around eight. Diesel engines run at fifteen to twenty-two. The reason is not in the cycle at all. It is in the chemistry of what is being compressed, and it is a race between two clocks.

Efficiency that keeps rising #

The Otto cycle compresses a charge adiabatically, adds heat at constant volume — the fast burn of the fuel near the top of the stroke — expands it adiabatically, and rejects heat at constant volume as the exhaust is replaced by fresh charge. Its efficiency depends on the ratio of volumes alone, because both adiabats change temperature by the same factor, $r^{\gamma-1}$. The ceiling on every engine set the bound no cycle can pass, Carnot’s, between the hottest and coldest temperatures available; the Otto cycle falls short of it because its heat goes in over a range of temperatures rather than at the top, and compressing harder raises that whole range. Real engines fall short again for the reasons the engine that has to finish gave, since every exchange of heat in finite time needs a temperature difference to drive it, and an engine that runs at three thousand revolutions a minute has milliseconds for each.

The real charge, a mixture of air and fuel vapour, has a $\gamma$ below air’s 1.4, because the fuel’s larger molecules have more ways of holding energy — the counting the ball that weighs how a gas stores heat did with a stopwatch — and its hot combustion products lower it further. The fuel–air curve sits below the air curve, but it rises just as steadily. Nothing in the curve asks for a limit. Real engines lose to friction, heat leakage through the cylinder walls and the incomplete burning of the charge, and those losses grow somewhat at high compression; but not nearly enough to explain why petrol engines stop where they do.

A charge that is already fuel #

A petrol engine mixes its fuel with the air before compressing it. The spark then lights the mixture at one point, and a flame spreads across the chamber in about two milliseconds at three thousand revolutions a minute. The gas the flame has burned expands as it heats, and because the chamber is closed, that expansion squeezes the gas the flame has not yet reached — the end gas — further, adiabatically, raising its pressure and temperature well beyond what the piston’s compression gave it.

The end gas is fuel and air, hot and dense, waiting for the flame. If it ignites by itself before the flame arrives, it burns all at once rather than progressively: a sudden pressure rise in the corner of the chamber, sending a pressure wave ringing round the cylinder at several kilohertz. That is knock, the metallic rattle of an engine labouring uphill on the wrong fuel. Mild knock wastes energy and heats the walls; heavy knock erodes pistons and breaks rings. Every petrol engine is designed so that it does not happen.

The rattle has a pitch, and the pitch identifies it. When the end gas ignites at once in one corner of the chamber, the sudden pressure rise launches a pressure wave that rings across the cylinder in its natural acoustic modes. The lowest is a sloshing mode across the bore, whose frequency is set by the speed of sound in the hot gas and the bore’s diameter through the first zero of a Bessel function’s slope, 1.84 — the same kind of number that the drum that has no harmonics found fixing a circular membrane’s overtones. For a bore of 86 millimetres and gas at two thousand kelvin, where sound travels at about nine hundred metres a second, the mode rings at about six kilohertz. Modern engines carry a knock sensor, an accelerometer bolted to the block and listening in that band, and retard the spark the moment it hears it. The pressure wave itself is not quite a detonation — the self-igniting end gas burns faster than a flame but usually slower than a true detonation wave — but at its worst it steepens into something close to one, the process the front that steepens until it cannot followed in sound, and it scours the boundary layer of cooler gas off the piston crown, which is how heavy knock melts pistons.

Two clocks #

Fuel and air at high temperature do not ignite instantly. A chain of chemical reactions has to build up a population of reactive fragments first, and the time it takes — the ignition delay — falls steeply with temperature, as the rate of any thermally activated process does, through the Arrhenius factor that the exponential that decides everything followed, and less steeply with pressure. The delay of petrol-like fuels is described for engineering purposes by a correlation fitted by Douaud and Eyzat in 1978,

$$\tau = 17.68\left(\frac{\mathrm{ON}}{100}\right)^{3.402} p^{-1.7}\,e^{3800/T}\ \text{milliseconds},$$

with $p$ in atmospheres, $T$ in kelvin and ON the fuel’s octane number.

At the end of compression alone, the charge is already above the autoignition temperature that a laboratory measures for petrol, but its delay is several milliseconds to hundreds, depending on the ratio, and the flame comfortably wins. It is the extra squeeze from the burning gas that changes the race. At the peak of that squeeze the end gas’s delay has fallen to around a millisecond, comparable to the time the flame takes to cross the chamber. The outcome is decided by which clock runs out first, and both clocks run faster as the compression ratio rises: the end gas gets hotter, and its delay shortens far faster than the flame speeds up.

Counting down the delay #

Since the end gas’s temperature and pressure change throughout the burn, its delay is not a single number. The standard treatment, due to Livengood and Wu in 1955, adds up the fraction of the delay used in each instant, $\int dt/\tau(p, T)$: when the sum reaches one, the end gas ignites. Knock happens if it does so while a significant part of the charge is still unburned.

The integral climbs slowly at first and steeply at the end, because the end gas is hottest and most compressed when the flame is nearly through. At a compression ratio of nine on 95-octane fuel the flame finishes first; at thirteen the end gas ignites with more than a fifth of the charge unburned. The knock limit lies between, near eleven and a half for the engine modelled. It is not a sharp line in a real engine — it moves with speed, load, intake temperature, the shape of the chamber and the moment of the spark — but it is always there, and engine designers set the compression ratio just below it for the worst conditions the engine will meet.

The structure is the one the heap that sets itself alight followed in a pile of coal or hay: a reacting material whose rate of heating rises exponentially with temperature, which ignites if it is held hot long enough and does not if it is cooled first. In the heap the cooling was conduction to the surroundings; in the engine it is the flame arriving and consuming the gas before its chemistry has run its course.

A race the chamber can rig #

Because knock is a race, anything that speeds the flame or slows the end gas’s chemistry moves the limit. Turbulence speeds the flame, and engine designers make chambers whose shape forces the charge to swirl and tumble as it is compressed, so that the flame front is wrinkled and crosses the chamber faster; the compact, central-spark chambers of modern engines owe much of their higher compression to that. A spark placed centrally shortens the distance the flame must travel. Cool walls near the far side of the chamber cool the end gas and lengthen its delay. A leaner or recirculated-exhaust charge burns cooler. Each of these buys a fraction of a unit of compression, and together they took petrol engines from about eight in the 1970s to eleven or more today on the same fuels.

The gain is worth chasing. Near a ratio of ten, each additional unit raises the ideal efficiency by about one and a half percentage points — roughly three per cent less fuel for the same work — which over the life of a car is a large quantity of petrol. Harry Ricardo, who in the 1920s first studied knock systematically in a test engine whose compression ratio could be changed while it ran, put the problem in exactly these terms: a fuel was to be judged by its highest useful compression ratio, the ratio at which it began to knock, and that single number told an engine designer what the fuel was worth.

What an octane number buys #

The octane number is a measure of the ignition delay. It is defined by comparison with mixtures of two reference fuels, iso-octane (rated 100), which resists autoignition, and normal heptane (rated 0), which does not, run in a standard test engine; a fuel of octane 95 knocks under the same test conditions as a mixture of 95 per cent iso-octane and 5 per cent heptane. In the correlation it enters as a factor in the delay.

The whole value of a higher octane number is permission to compress further. A fuel of higher octane does not contain more energy — high-octane fuels often contain slightly less per litre — and an engine designed for lower-octane fuel gains nothing from a better one, because it is not compressing enough to knock in the first place. An engine designed for high octane, run on low, knocks, and modern engines with knock sensors respond by retarding the spark, which lowers the end gas’s peak pressure and costs efficiency. For most of the twentieth century the octane of petrol was raised with tetraethyl lead, which interferes with the chain reactions that lead to autoignition; its removal for health reasons from the 1970s onward was offset by refining fuels with more resistant molecules, and by engines whose chambers burn faster and cooler.

The same arithmetic governs engines that force more air in rather than squeezing it harder. A turbocharger or supercharger raises the pressure of the charge before the piston compresses it, which raises its temperature and pressure at every point of the cycle, and the end gas reaches its short delay sooner. Turbocharged petrol engines therefore run at lower compression ratios than unboosted ones, trading compression for boost, and the boost they can use is limited by knock in exactly the same way. The availability of 100-octane aviation fuel to British fighters in 1940, which allowed their supercharged engines to run at substantially higher boost without knocking, is often credited with a meaningful part of their performance — an octane rating turned directly into horsepower.

The engine that compresses only air #

Rudolf Diesel’s engine escapes the race by not running it. It compresses air alone, to ratios of fifteen to twenty-two, raising it to eight or nine hundred kelvin; the fuel is sprayed in only near the top of the stroke, and the ignition delay that is the petrol engine’s enemy becomes the diesel engine’s means of ignition. A short delay is now wanted, so that the fuel lights promptly once it is injected, and diesel fuels are rated by a cetane number that measures how readily they ignite — the opposite property to octane.

The ideal Diesel cycle adds its heat at constant pressure while the piston moves, rather than at constant volume at the top, so at the same compression ratio it is less efficient than the Otto cycle: some heat goes in at a lower temperature. But it is not limited by knock, so it runs at much higher compression, and there it beats a petrol engine at its knock limit. That, more than anything about the fuel’s energy content, is why diesel engines are more efficient. The diesel pays in other coin. Compressing to eighteen means peak pressures two or three times a petrol engine’s, and a heavier block and crankshaft to hold them; and because its fuel burns as it is sprayed, in a spray that is rich at its core and lean at its edges, it makes soot where the mixture is rich and nitrogen oxides where it is hot, problems a premixed petrol charge largely avoids. The diesel’s efficiency is won by refusing to premix, and its emissions are the price of the same refusal. Modern petrol engines borrow from the diesel’s trick: direct injection, which sprays the fuel into the cylinder late in the compression stroke, cools the charge as the fuel evaporates and shortens the time the end gas is exposed, allowing compression ratios of twelve or thirteen on ordinary fuel.

What the model leaves out #

The figures use the simplest model that shows the race: a charge compressed adiabatically from 320 kelvin with a single $\gamma$, a burn that raises the pressure to three and a half times the compression pressure along a standard burn curve over forty degrees of crank rotation at three thousand revolutions a minute, a single delay correlation, and the criterion that knock occurs if the end gas ignites with more than a tenth of the charge unburned. Real engines have heat losses to the walls, which cool the end gas and help; pistons that move during the burn, which changes the pressure history; and chambers whose shape and turbulence decide how fast the flame crosses them, which is why the knock limits of real engines on the same fuel differ by two or three units of compression ratio. Modern fuels also show negative-temperature-coefficient chemistry, in which the delay lengthens over a range of rising temperature, which a single Arrhenius correlation cannot capture. The figures show the mechanism and its scale, not the limit of any particular engine. Their domain is a homogeneous charge, compression ratios of four to twenty-four, and octane numbers of eighty to a hundred and ten.

Still open: how close to a diesel a petrol engine can get #

The most efficient engines now being developed try to combine the two cycles’ advantages: to burn a premixed, lean charge throughout the cylinder at once, by compression ignition rather than by a spreading flame, with no knock because there is no end gas — the whole charge ignites together. Homogeneous-charge compression ignition can reach diesel-like efficiencies with petrol-like cleanliness, but controlling exactly when the whole charge ignites, when the timing is set by chemistry rather than by a spark or an injector, has proved very hard across the range of speeds and loads a road engine meets. How to hold such an engine on the edge of ignition, cycle after cycle, is an open engineering problem, and it is the same race the ordinary petrol engine loses, run deliberately and with no flame to win it.

The thermodynamics and the chemistry fit in one sentence. An ideal Otto cycle’s efficiency rises without limit as 1 − r^(1−γ), but the unburned charge ahead of the flame is squeezed until its ignition delay — about a millisecond at a ratio of 10 — is no longer than the flame’s crossing time, and once it ignites first, knock caps the ratio near 11.4 on 95-octane fuel; octane buys about a tenth of a unit of compression per point, and a diesel, compressing only air, runs at 18 instead. A petrol engine is limited not by Carnot but by a race against its own fuel’s chemistry.