The balloon a cold morning lifts

The balloon that floats at a density followed balloons filled with helium, sealed or open at the bottom, as they climbed until something stopped them: bursting, running out of lift as the gas expanded, or floating at a fixed density. The oldest kind of balloon works differently from all of those. The Montgolfier brothers’ balloon that carried two men over Paris on 21 November 1783 held no special gas at all — only the surrounding air, heated by a fire of straw and wool, in an envelope open at the bottom. Hydrogen balloons followed ten days later, and lifted far more for their size. The hot-air balloon was nearly abandoned for a century and a half, and revived in the 1950s and 1960s when propane burners and nylon made it cheap and controllable, which is why the sky over a balloon festival is full of them now.

Its physics is simple and unforgiving. The weight of the water that is not there found Archimedes’ principle: a body immersed in a fluid is pushed up by the weight of the fluid it displaces. A balloon displaces air, and is pushed up by the air’s weight; it falls by the weight of whatever is inside it plus its own. The only question is how much lighter the inside can be made than the outside, and for hot air the answer is: not much.

A third of a kilogram per cubic metre #

The envelope is open at the throat, so the air inside is at the same pressure as the air just outside. At equal pressure, the gas law that falls out of counting says that a gas’s density is inversely proportional to its absolute temperature. Air at 100 °C, 373 kelvin, is lighter than air at 15 °C, 288 kelvin, by a factor of 288 over 373 — it has 77 per cent of the outside air’s density. Every cubic metre of envelope therefore lifts the remaining 23 per cent of a cubic metre of outside air, about 0.28 kilograms.

The first figure draws that for three mornings. The lift grows with the inside temperature but slowly, and it is capped: the nylon or polyester fabric of a modern envelope loses strength above about 120 °C, and an envelope regularly flown hotter wears out in a few dozen flights. At that limit the best a cubic metre can lift is under half a kilogram on a frosty morning and a quarter of a kilogram on a hot afternoon. Helium, by comparison, lifts more than a kilogram per cubic metre at any sensible temperature, because its molecules weigh a seventh of air’s. That ratio, three to five times, is the reason a hot-air balloon carrying four people needs an envelope of about 2,800 cubic metres — the size of a house — while a helium balloon carrying the same load could be a quarter of the size.

Why the cold matters as much as the heat #

The second figure turns the lift into a payload. The outside temperature enters twice: cold air is denser, so each cubic metre displaced weighs more, and cold air makes the ratio of absolute temperatures more favourable, so each cubic metre of hot air is lighter by a larger fraction of it. The two effects add, and the payload falls steeply as the day warms — by more than four hundred kilograms between a frosty dawn and a hot afternoon, with the same envelope at the same temperature. On a hot day in summer a balloon that carried four people at dawn can lift two at noon.

Balloonists fly in the first hours after sunrise and the last before sunset mainly because the air is calm then. Once the sun heats the ground, convection sets in and the air fills with thermals, rising columns of warm air that can carry a balloon up and drop it in ways a pilot cannot control, with gusty winds near the ground. But the dawn flight also buys lift, and so buys a cooler envelope for the same load, which lengthens its life and saves fuel.

The ceiling #

As a balloon climbs, both terms in its lift change. Why the air thins with height found that air pressure and density fall roughly exponentially, by about a tenth for every kilometre near the ground, so the weight of the air displaced by each cubic metre falls. But the air also gets colder — six and a half degrees per kilometre in the standard atmosphere — and that improves the ratio of temperatures for the same inside temperature. The third figure computes the net effect, and the density wins: lift per cubic metre falls with height, at very nearly the same tenth per kilometre as the density.

A balloon of given weight per cubic metre of envelope therefore has a ceiling, where the lift its hottest permitted air can give equals what it needs. For a four-person balloon carrying its people, fuel and basket, the figure puts it at about six kilometres with the envelope at its limit — higher than balloonists normally go, because above four kilometres the pilot and passengers need oxygen and the burners begin to falter in the thin air. The ceiling is not a limit set by the burners: with the envelope at its maximum temperature, more heat would only damage the fabric, and the balloon cannot go higher however much propane it carries.

Above the tropopause the air stops cooling with height, so the improving temperature ratio no longer offsets any of the falling density, and lift per cubic metre falls faster. Climbing into the stratosphere on hot air requires a different scale of balloon altogether.

Bigger balloons fly higher #

The way past the ceiling is size. An envelope’s lift grows with its volume, the cube of its size, while its own weight grows only with its surface area, the square; the payload is fixed. So a larger envelope carries the same load with a smaller weight per cubic metre and climbs higher. The fourth figure shows the ceiling rising steadily with volume, by several kilometres for each tenfold increase.

The altitude record for a hot-air balloon is 21,027 metres, set by Vijaypat Singhania over India in 2005 with an envelope of about 45,000 cubic metres — sixteen times the volume of a sport balloon — and a pressurised gondola. The figure’s estimate for an envelope of that size with a heavy gondola falls a little short of the record, which is fair warning about the round numbers in the model: the real record envelope was made of lighter fabric than a sport balloon’s, and its pilot dropped every kilogram he could on the way up. At those heights the air has a sixteenth of its sea-level density, and keeping a propane flame alight in it was one of the expedition’s main problems.

The lag a pilot steers by #

A hot-air balloon has no controls but the burner and a vent at the top, and both act slowly. The last figure integrates what happens after a single short burn. Three megawatts for five seconds — the output of a typical double burner — warms the nearly three tonnes of air in the envelope by under six degrees. Nothing visible happens at first: the balloon, with its passengers, basket and envelope, and with the air it must push out of the way as it moves, has a large inertia, and the extra lift is a few per cent of its weight. It begins to rise perceptibly seven seconds after the burn starts, and then keeps rising for minutes, because the heat it was given leaks out through the fabric slowly, with a time constant of several minutes, and until it has leaked away the balloon is lighter than its surroundings.

The balloon is heading for the height at which its warmer air would just float it — several hundred metres up — and it would reach that height and then sink again as it cooled, if the pilot did nothing. So a pilot flies ahead of the balloon, judging each burn by where the balloon will be in half a minute, and stops a climb by letting the air cool long before the intended height is reached. It is the same discipline as steering a ship or flying an airship, and it has the same cause: a big mass, a small push and a long delay between them. The building that breathes through its doors met the same air-temperature buoyancy at a smaller scale, as a pressure difference pushing air through a tall building; the balloon is the same arithmetic with the warm air free to move.

The fuel a hovering balloon burns #

Holding a balloon level costs a steady supply of heat, because the envelope is always losing it, and the arithmetic of the loss explains why the burners are so powerful and the fuel bill so large. The envelope’s fabric is a few tenths of a millimetre thick and is no insulator; heat leaves through its whole surface by convection to the outside air and by radiation, at a rate roughly proportional to the temperature difference across it. For a 2,800 cubic metre envelope, nearly a thousand square metres of fabric, held seventy degrees above the outside air, the loss comes to about half a megawatt. Propane releases about 46 megajoules per kilogram, so hovering burns on the order of forty kilograms of fuel an hour, and a typical flight of an hour or so uses most of the three or four cylinders a basket carries.

That loss is set by the area, which grows more slowly than the volume, so a larger balloon is more economical per kilogram lifted as well as able to fly higher. And it explains why pilots burn in bursts at full power rather than continuously at low power. The burner’s job is to replace the loss on average while keeping the air inside within a few degrees of the temperature that holds the balloon level, and a big burner fired briefly lets the pilot choose the moment. Insulating the envelope would cut the fuel bill, but every kilogram of insulation must itself be lifted, and as the insulation that makes a wire lose more heat found for a thin wire, adding a layer to a surface does not always reduce what it loses. Envelopes have stayed as thin and light as their strength allows.

Hot air and helium in one envelope #

The two kinds of balloon have opposite strengths, and the balloons that first flew round the world used both. A helium balloon lifts three to five times more per cubic metre but can climb only by dropping ballast and descend only by venting gas, and it is at the mercy of the sun: by day its gas warms and expands, by night it cools and contracts, and every cycle costs gas or ballast. A hot-air balloon can go up and down as often as its fuel allows, but carries little for its size and burns a great deal. In 1785 Jean-François Pilâtre de Rozier, who had made the first hot-air flight two years before, combined them, with a hydrogen cell above a hot-air cone; the balloon caught fire and he was killed. The design, made safe with helium and named after him, was revived in the 1970s, and it is what Bertrand Piccard and Brian Jones flew round the world non-stop in 1999, and Steve Fossett alone in 2002.

A Rozière balloon’s helium provides most of the lift; its small hot-air cone, burned gently at night, keeps the helium from cooling and contracting and so saves the ballast a pure gas balloon would drop. The gas cell has its own slow losses as well, since helium atoms are small and fast enough to diffuse through any fabric, and the gas that leaves is not the gas inside found that a gas escaping through small openings is enriched in its lightest, fastest molecules — which, for a helium cell, means it leaks the lifting gas preferentially. Long flights are a budget of heat, gas and ballast kept balanced for days.

A buoyant body in a layered sky #

A balloon at its equilibrium height is held there by a restoring force: displaced upward, it finds thinner air and less lift and sinks back; displaced downward, more lift. The layer a parcel cannot leave found the same restoring force acting on a parcel of air displaced in a stratified atmosphere, with an oscillation at the buoyancy frequency, a period of several minutes. A balloon whose air temperature was held fixed would bob about its level with a period of the same order, set by how fast its lift changes with height compared with its total mass, including the air inside it and the air it drags along.

The comparison also says when a balloon is not like a parcel at all. A parcel of the atmosphere exchanges heat with its surroundings slowly and is carried by the wind; so is a balloon, which is why balloons cannot be steered sideways except by finding winds blowing in different directions at different heights — the pilot’s only real means of navigation. But a parcel that rises cools as it expands, while a balloon’s air is held at whatever temperature the burner gives it. That is why the balloon is a controllable vehicle and the parcel is weather.

Where the model is rough #

The figures treat the envelope as a sphere of fixed volume at a uniform temperature, with the inside at exactly the outside pressure. A real envelope is a teardrop, its air is hottest at the crown and coolest near the mouth, and the burner’s flame heats a layer near the top more than the whole volume; pilots read a temperature from a sensor at the crown, which is why the limit is quoted as a crown temperature. The heat loss through the fabric is taken as a single coefficient, and in reality depends on wind, sunshine and the fabric’s coating. The standard atmosphere is an average, and on any given day the temperature profile, inversions and all, changes both the lift and the ceiling. None of these change the shape of the arguments, which rest only on the gas law and Archimedes, but all of them move the numbers by a few per cent.

The balloon also does not float at a perfectly stable height. The balloon that floats at a density found that a sealed balloon finds a stable level and bobs about it; a hot-air balloon at fixed temperature does too, since its lift falls with height, but its temperature is not fixed, and the slow loss of heat makes its free flight a slow descent that the pilot counters with burns. Equilibrium is maintained, not found.

The flame, the fuel and the fabric #

The figures do not show the flame, the fuel or the fabric, and they treat the burner as a simple source of heat. A propane burner’s efficiency in putting heat into the envelope is well below one, since part of the flame’s heat escapes through the mouth and part warms fabric that then radiates it away, and the figures quietly fold all of that into the heat-loss coefficient. Nor do they show the solar balloon, a dark envelope heated by sunlight rather than flame, which works on exactly the same arithmetic with the sun as the burner and is a favourite of children’s science fairs and, in larger forms, of some high-altitude research flights.

Still open: how much the sun and the fabric add #

The largest uncertainties in predicting a hot-air balloon’s behaviour are in its heat budget: how much heat the fabric radiates to a clear sky, how much sunlight it absorbs, how the inside air circulates and stratifies, and how all of these change as the envelope rises into thinner, colder, sunnier air. Long-duration flights, including the hybrid balloons that combine a helium cell with a hot-air cone and were used for the first round-the-world balloon flights in 1999 and 2002, depend on getting that budget right over days, through repeated cycles of daytime heating and night-time cooling. The physics is all classical; the difficulty is in the many small heat flows, and models of them are still compared with flight data rather than trusted on their own.

The habit worth carrying away is to ask what a ratio of temperatures can buy. An open balloon’s air is lighter than the air outside only by the ratio of absolute temperatures, so at the fabric’s limit of 120 °C a cubic metre lifts under half a kilogram on a frosty morning and a quarter on a hot afternoon — a 2,800 m³ balloon’s load falling from 980 kg at 0 °C to 569 kg at 35 °C — and as the air thins by a tenth per kilometre, the same balloon runs out of lift near six kilometres, while its air, minutes slow to warm and cool, makes it answer every burn half a minute late.