The throttle plate is a wet bulb for petrol

Carburettor ice is made by the fuel, not the venturi takes the usual account apart. The venturi’s expansion cools the moving air by five kelvin at a hundred metres a second, but the metal the ice grows on sits at the air’s recovery temperature and barely feels it; the cold is the petrol’s, twenty-eight kelvin if all of it evaporates. That essay then puts the fuel’s latent heat into the whole of the air, adds back the heat of the water that condenses out of humid air, and finds that the mixture is below freezing, with water to freeze, only on days up to about fifteen to twenty degrees.

The icing charts pilots use go to thirty degrees, and the flight-training handbook goes further: carburettor ice is possible, it says, at temperatures as high as 38 °C and humidity as low as 50 per cent. The earlier essay names the gap and the likely reason for it — the fuel does not evaporate evenly into the air, it evaporates from wet metal — and leaves the calculation open. This essay does it.

A wet bulb for petrol #

A thermometer with a wet wick round its bulb, swung in air, reads lower than a dry one. Water evaporating from the wick takes its latent heat from the bulb; the bulb cools until the air flowing past it brings heat in as fast as the evaporation takes it away. The temperature at which that happens, the wet-bulb temperature, is a property of the air — how warm it is, and how dry — and of the liquid: how readily it evaporates, and how much heat each kilogram takes.

A throttle plate in a carburettor is a wet bulb with petrol on the wick. The fuel sprayed from the nozzle wets it, the mixture flows across it at tens of metres a second, and the plate settles where the heat the stream brings in balances the heat the evaporating fuel takes out. Everything that decides a wet bulb decides the plate, with two differences that turn out to matter a great deal. Petrol is far more volatile than water, so it evaporates much faster at the same temperature and drives the plate much colder. And petrol is not one liquid, so how volatile the film is depends on how long it has been evaporating.

The balance is written with film theory. Heat reaches the plate at a coefficient $h$, and fuel vapour leaves it at a mass-transfer coefficient that the Chilton–Colburn analogy ties to $h$ — the same boundary layer carries both, and they differ by the ratio of the thermal diffusivity to the vapour’s, the Lewis number, to the two-thirds power. That ratio is one for water in air to within fifteen per cent and between two and three for petrol vapours, which are heavy molecules and diffuse slowly; the heat the eddies do not carry is the same comparison between two diffusivities in a single layer. The fuel is volatile enough that its outward flow thickens the layer and cuts the heat reaching the plate, and the balance allows for that too:

$$\dot m_f = g_f \ln(1 + B_f), \qquad B_f = \frac{Y_{fs} - Y_{f\infty}}{1 - Y_{fs}}, \qquad \dot m_f\,h_{fg} = q + \dot m_w L_w .$$

The surface’s fuel-vapour fraction $Y_{fs}$ is set by the film’s vapour pressure at the plate’s temperature, and $\dot m_w L_w$ is the heat given up by any water vapour the cold plate pulls out of the stream as frost.

The balance checked on water #

Before the plate, the balance was given water and humid air, where the answer is in every psychrometric table. At 20 °C and 50 per cent it returns a wet bulb of 13.51 °C against the tables’ 13.70; at 30 °C and 70 per cent, 25.36 against 25.60; at 10 °C and 30 per cent, 3.21 against 3.26. The largest error is 0.24 kelvin, and it comes from taking the Lewis number for water vapour as a constant. Against petrol’s drops of ten and twenty kelvin, that is the size of the uncertainty the method brings.

Petrol was represented by five liquids spanning its distillation range, by mass: five per cent butane, fifteen each of pentane and hexane, thirty-five of iso-octane and thirty of toluene, with the film’s vapour pressure given by Raoult’s law. The vapour-pressure fits reproduce each liquid’s boiling point to 0.03 K, and the blend’s Reid vapour pressure — its vapour pressure at 37.8 °C, the number a refinery certifies petrol by — comes out at 58.9 kilopascals, inside the range of a summer petrol.

How cold the plate gets #

The first figure is the plate’s temperature across a range of days at 80 per cent humidity. The stream over it is the mixture after half the fuel has evaporated into the air — the earlier essay’s middling case — at the plate’s recovery temperature. A plate kept wet with fresh petrol sits 8.6 kelvin below that stream on a −10 °C day, 13.6 at 0 °C, 20.9 at 20 °C and 23.1 at 30 °C. It is below freezing on every day up to about 34 °C.

The drop grows with the day’s temperature because a volatile film’s vapour pressure grows exponentially with its own temperature, while the heat the stream can supply grows only in proportion to the difference. On a warm day the film would evaporate enormously at the stream’s temperature, and the plate has to fall a long way before evaporation has slowed to what the stream can pay for.

The second curve is the same plate wet with weathered petrol, the same fuel after its butane, pentane and hexane have gone. It is only one to six kelvin below the stream. Heavy fuel is not much more volatile than water at these temperatures, and a plate wet with it is barely a wet bulb at all.

Where the plate’s heat comes from #

The second figure opens up the balance on one day, 20 °C at 80 per cent. A plate wet with fresh fuel is evaporating nine kilowatts’ worth of petrol from every square metre. The stream supplies four of them by conduction across a twenty-kelvin difference. The other five come from frost: the plate is at −10.5 °C, the stream’s water vapour is far above saturation over ice at that temperature, and every gram that deposits gives up 2.8 kilojoules of latent heat to the plate it lands on.

That is the part the well-mixed account has no room for. Humid air warms the mixture as its water condenses — the earlier essay’s finding that the worst day is not a cold one — and it warms the plate as its water freezes, by more than the stream does. Frost growth heats the surface it grows on, which makes it self-limiting: a plate cannot fall far below the frost point while the air is carrying much water, because the frost it collects warms it back. The plate on the weathered side of the figure is evaporating a third as much and collects a third as much frost, for the same reason in reverse.

Volatility is the whole difference #

The third figure puts the two petrols beside the pure liquids that make them up. Toluene, with a vapour pressure of three kilopascals at 20 °C, holds a plate 4.1 kelvin below the stream; iso-octane, at five, 5.7 kelvin; hexane, at sixteen, 14; pentane, at fifty-six, 30. Fresh petrol, with its butane and pentane, sits between hexane and pentane at 21 kelvin, and the weathered petrol sits with its heavy components at 4.8.

A carburettor has both. The fuel arrives fresh from the float bowl on every cycle of the nozzle, but a wetted surface loses its light ends within a fraction of a second, and a film that is not being renewed is a heavy film. Where the spray keeps the plate flooded — its upstream face, the edge nearest the nozzle, low throttle when the plate sits across the spray — the surface is close to fresh. Where it is wetted only occasionally, it is weathered. The two curves in the first figure are the two limits of a real plate, and the ice grows first where the fuel is freshest.

The icing boundary moves by fifteen degrees #

The fourth figure is the question the earlier essay left: the warmest day on which ice can grow, against the humidity. Ice needs the surface below freezing and the stream supersaturated with respect to ice at the surface’s temperature. For the well-mixed mixture with all the fuel evaporated, the earlier essay’s model, that is 20.4 °C at 50 per cent humidity, 15.6 at 80 and 13.7 at 100. For a plate wet with weathered fuel it is lower still, 18.1, 14.2 and 12.3, because only half the fuel is in the stream and the plate’s own evaporation is weak.

For a plate kept wet with fresh petrol it is 40.4 °C at 50 per cent humidity, 33.7 at 80 and 30.4 at 100. At full humidity that is the upper edge of the icing charts. At half humidity it is two degrees above the handbook’s 38 °C at 50 per cent, which is as close as a calculation with a representative petrol and a representative plate can claim to come to a warning distilled from incident reports.

The boundary falls as the humidity rises, which looks backwards and is not. Humid air warms both the stream and the plate through the water it gives up, so the plate is warmer on a humid day than on a dry one at the same temperature, and the warmest icing day is lower. What the humidity buys is the rate — the amount of water available to freeze — and that is the next figure.

How fast it builds #

The fifth figure is the frost’s growth rate, as solid ice. At 0 °C and 80 per cent it is 0.02 millimetres a minute; at 10 °C, 0.05; at 20 °C, 0.12; at 25 °C, a sixth of a millimetre. The rate rises through the warm days because warm humid air carries far more water, and the fresh-wet plate stays cold enough to take it out. It stops only when the plate itself reaches freezing, which on a saturated day is just above 30 °C.

A throttle plate’s clearance at idle is a fraction of a millimetre. At a sixth of a millimetre a minute it closes in a few minutes, and the engine note falls before the pilot has thought of ice — the experience the charts were drawn from. Frost is less dense than solid ice, so the build-up in thickness is faster than the figure’s still.

This is where the misconception and the physics meet. The venturi story put the cold in the air and so made cold days the danger. The mixture story put it in the fuel and moved the danger to mild humid days. The wet-plate story puts it on the metal the ice grows on, and finds the danger extending to days that are warm by any pilot’s reckoning, and the ice growing fastest on the warmest of them.

Why the stream’s temperature is the wrong one #

The mistake the well-mixed account makes is the same one the thermometer that heats itself is about: a surface in a moving gas is not at the gas’s temperature. There the difference is kinetic — a probe recovers most of the stream’s kinetic energy as heat and reads above the static temperature. Here it is latent: a wet surface pays for its own evaporation and reads below. In both cases the number that decides what happens at the surface is the surface’s own equilibrium, and in both cases it can be computed from the boundary layer alone. The venturi that turned out not to be half an explanation made the same point first about a streamline: the stream is not where the answer is.

The venturi itself has a smaller part in this than its name suggests, and two essays about its pressure say why: the venturi that stops listening downstream chokes, and a choked throat buys time, not silence follows what a choked throat can and cannot shield. A carburettor never gets near either. At a hundred metres a second its throat runs at a Mach number of 0.3, and its pressure drop matters to the ice only as the suction that draws fuel out of the nozzle. That is the venturi’s real role in icing, and it is not a small one: it meters the fuel, and the fuel does the cooling, on the metal the fuel lands on.

Evaporative cooling is not special to carburettors. A cavity cools the water it came from by the same exchange, latent heat taken from the liquid that supplies the vapour, and the gradient the heat never hears is the analogy between momentum and heat on which the whole coefficient $g_f$ rests.

What was checked #

The sixth figure is the ledger. The balance given water reproduces the psychrometric wet bulb to 0.24 K at three conditions. The Antoine fits reproduce four boiling points to 0.03 K. The petrol stand-in’s Reid vapour pressure is 58.9 kPa. And on the reference day the fresh-wet plate is at −10.5 °C against a stream at 10.4, the weathered one above freezing.

What the picture cannot show #

The plate conducts. A throttle plate is a disc of brass or aluminium on a shaft through a body bolted to a hot engine, and the heat it receives by conduction is not in the balance. The model’s plate is an insulated one, which makes it the coldest a plate can be; a plate well sunk in a warm body would sit higher, and so would its icing limit.

One film, one composition. A real plate carries a film that is fresh at one edge and weathered at the other, with a composition that changes continuously as it evaporates. The two limits bracket it.

Frost as solid ice, and only from vapour. The rate is given as solid ice; frost is lighter and grows thicker for the same mass. And the fog of droplets that forms when the stream itself falls below its dew point can strike the plate too, adding to the rate.

A flat plate. The coefficient is a laminar flat plate’s, three centimetres long at a hundred metres a second. The edge of a real plate, where the gap is, has a thinner boundary layer and a higher coefficient: both the heat in and the evaporation out are larger, and the plate’s temperature, which depends on their ratio, changes little.

The convention the numbers depend on #

Temperatures are in °C; the day’s humidity is relative, measured at ambient pressure, and the carburettor is at 85 kPa. The stream is the mixture at an air–fuel ratio of 12.5 with half the fuel evaporated, at the plate’s recovery temperature for a throat speed of a hundred metres a second and a recovery factor of 0.89. Fresh petrol is the five-component blend above; weathered petrol is its iso-octane and toluene in their original proportion.

Who found it, and when #

The wet-bulb balance is the psychrometer’s, worked out by Carrier in 1911 for water, and the logarithmic mass-transfer driving force Spalding’s of the 1950s, from fuel-droplet combustion. The analogy between heat and mass transfer is Chilton and Colburn’s of 1934. Carburettor icing charts come from engine test-cell and flight trials, and the training handbook’s warning of icing up to 38 °C at 50 per cent humidity from the incident record rather than from a calculation.

Still open: how much of the plate is fresh #

The answer turns on a fraction the model takes at its two limits: how much of the plate’s surface the spray keeps wet with fresh fuel, and how fast a film weathers once the spray has moved on. The next calculation follows a film laid down by one pulse of spray as it evaporates on the plate — its composition distilling from fresh to weathered, its surface temperature rising as its light ends go — and asks how long it stays below freezing, so that the fraction of a nozzle cycle a plate spends cold can be set against the fraction it spends wet, and the charts’ two regimes, cruise power and idle, read as two different spray patterns on the same metal.