The building that breathes through its doors

The pressure that only knows depth found the first rule of fluids at rest: pressure grows with depth as the weight of the fluid above, $\rho g$ per metre, and depends on nothing else — not on the shape of the container, not on how much fluid there is beside the point. Air obeys the rule as water does. Its density is about a thousandth of water’s, so its pressure falls about twelve pascals for every metre of height, a tenth of a per cent of the atmosphere for every eight metres climbed.

The balloon that floats at a density used the rule to float a balloon: a body of light gas surrounded by heavier air feels the difference of two columns’ weights as buoyancy. A heated building is the same arrangement with the balloon’s skin replaced by walls with cracks in them, and the consequences are part of every winter in every cold city.

Two columns that cannot agree #

A building at 21 °C on a day at −10 °C holds air that is twelve per cent less dense than the air outside, at the same pressure, because a gas’s density is inversely proportional to its absolute temperature. The pressure in each column falls with height at a rate set by its own density, so the pressure inside the building falls more slowly than the pressure outside. If the two were equal at the ground, the inside would be higher everywhere above it; if they were equal at the roof, the inside would be lower everywhere below.

They cannot be arranged to agree everywhere, and a building with openings at different heights settles at the only compromise: equal pressures at one height, the neutral plane, and differences growing linearly above and below it,

$$\Delta p(z) = (\rho_{\text{out}} - \rho_{\text{in}})\,g\,(z - z_n).$$

The first figure draws the two pressure profiles for a hundred-metre building with its openings evenly spread. The two lines are nearly parallel, each falling by about twelve hundred pascals over the building’s height, and the difference between them is small by comparison — sixty-nine pascals at the roof and at the ground — but sixty-nine pascals is not small for a door. Because pressure pushes equally in every direction, as the push that has no direction established, those pascals press on a door, a window or a wall whatever way it faces. It is the pressure of a column of water seven millimetres deep, or of a strong wind, about forty kilometres an hour, blowing straight at a wall.

The airflow follows the pressure. Below the neutral plane the outside is at the higher pressure and cold air comes in through every gap in the lower floors: under doors, round windows, through the gaps where pipes pass through walls. It rises through the building as it warms, through stairwells, lift shafts and service ducts, and leaves through every gap in the upper floors, where the inside is at the higher pressure. The whole building is a chimney, drawing a steady current of air through itself with its own warmth as the engine.

The doors in a cold snap #

The second figure varies the weather. The pressure difference at each floor is proportional to the difference in density between the two columns, which for air at nearly the same pressure is proportional to the difference between the reciprocals of their absolute temperatures. On a −20 °C day the ground floor of the hundred-metre building is nearly a hundred pascals below the outside. A pair of glass lobby doors two square metres in area is pushed shut with a force of nearly two hundred newtons — the weight of twenty kilograms — and anyone trying to open them pushes against it, and when they do open, the air pours in at several metres a second.

This is the reason tall buildings in cold climates have revolving doors and double sets of doors with lobbies between them: a revolving door is never open straight through, so the building never has a continuous opening at the bottom. It is also why the upper floors of an office block in winter can be stuffy and over-warm while the lobby is cold and draughty, and why smoke from a fire on a low floor rises through the building’s vertical shafts and spreads into the upper floors, the problem that drives the design of fire stairs, which are kept at a slightly higher pressure than the floors they serve so that smoke cannot enter them.

The last line of the figure is the summer case. A building cooled to 21 °C on a 30 °C day holds air denser than the outside, and its column sinks: the pressures reverse, air leaks out of the lower floors and in at the upper ones, and the reverse stack carries outside air and its humidity into the top of the building. The effect is weaker, because the summer temperature difference is smaller, but it is the same hydrostatics running backwards.

Where the neutral plane sits #

The neutral plane’s height is not fixed by the temperatures. It is fixed by the requirement that as much air leaves as enters, and the third figure shows how the openings decide it. The flow through an opening grows as the square root of the pressure difference across it, the result Torricelli found for water leaving a tank, which the vessel whose level falls evenly used for a water clock. Air entering through the lower openings must leave through the upper ones, and if the lower openings are larger, a smaller pressure difference suffices to drive the same flow through them; the neutral plane moves down, closer to the large openings, and the upper openings, now further above it, face a larger difference. Balancing the two flows puts the plane at a fraction $1/(1 + r^2)$ of the height, where $r$ is the ratio of the lower openings’ area to the upper ones’.

A building whose ground floor stands open — a loading bay, a busy entrance — has its neutral plane pulled down towards the ground, which relieves the doors and puts the whole pressure difference across the upper floors, where windows rattle and offices are draughty. A building with vents in its roof has its plane pushed up and its ground-floor doors pressed shut harder. The designers of tall buildings work with this deliberately: tight construction at the base, pressurised stairwells and lift lobbies, and in very tall buildings sealed floors every few dozen storeys that break the building into several shorter chimneys, each with its own small neutral plane and its own small pressure difference.

The chimney itself #

A chimney is the stack effect concentrated. Its column of hot flue gas is much lighter than the outside air, and the difference in weight between the two columns, over the chimney’s height, is the draught that pulls air into the fire at the bottom and pushes smoke out at the top. The fourth figure computes it. Ten metres of flue gas at 200 °C against outside air at 0 °C gives fifty-four pascals, which drives the gas up at about seven metres a second. The draught is proportional to the chimney’s height, which is why furnaces, smelters and power stations built before fans were cheap had tall chimneys: a hundred-metre stack gives ten times the draught of a ten-metre one.

The temperature of the gas helps less than one might expect. The draught depends on one minus the ratio of the outside air’s absolute temperature to the gas’s, and that approaches one as the gas gets hotter: raising the gas temperature from 200 to 400 °C raises the draught by only about two-fifths. A chimney’s height is worth more than its heat. Mine ventilation in the nineteenth century used exactly this, with furnaces at the foot of an upcast shaft heating the air in it and drawing fresh air down another shaft and through the workings, a scheme that depended on the shaft’s depth and became dangerous when the air in the mine carried methane.

The same weight difference drives a singing pipe. The tube that sings when heated low down needed a steady upward flow of air through a vertical tube with a hot gauze inside it, and that flow was the stack effect of the warm air in the tube; with the tube horizontal, there is no flow and no song.

How much a house breathes on a still day #

A house is a short chimney, and on a still winter day the stack effect is most of its ventilation. The last figure takes a two-storey house with openings totalling five hundred square centimetres near the ground and as much near the roof, six metres apart — a gap under a door and some cracks round windows, roughly — and computes how much air it exchanges with no wind. On a −20 °C day the house changes most of its air every hour; on a mild day, a third of it; on a day as warm outside as in, none. The flow grows only as the square root of the temperature difference, because the pressure difference is proportional to the temperature difference and the flow through an opening to the square root of the pressure.

Two consequences follow, and both are familiar to anyone who has lived in an old house. The house is draughtiest in the coldest weather, when the draught is least welcome, and it loses heat by ventilation in proportion to the flow times the temperature difference, so the loss grows as the temperature difference to the power three halves. And a house built tight to stop draughts needs another source of fresh air, because on a still day its stack effect was the only thing exchanging its air. Passive ventilation designs use the effect on purpose, with high vents and solar chimneys warmed by the sun to strengthen it, and work best in exactly the conditions — cold, still days — when opening windows is least attractive.

Hydrostatics, not heat rising #

The usual phrase, warm air rises, describes the result and hides the cause. Warm air does not rise by itself; it is pushed up by the colder, denser air around it, exactly as a cork is pushed up through water. The weight of the water that is not there found buoyancy as the missing weight of displaced fluid, and the stack effect is buoyancy with the floating body replaced by a column of warm air confined by walls. The walls matter. In the open, warm air rises as a plume and mixes with its surroundings within a few metres. Confined in a building or a chimney, it cannot mix, and the full weight difference of the whole column acts across the openings at its top and bottom. A hundred-metre building makes the difference large because it is tall, not because it is hot: the same 30-degree difference across a single-storey shed gives a few pascals.

When the outside column stops cooperating #

A chimney works only if the outside column is denser than the gas in it all the way up, and the atmosphere does not always oblige. Normally the air cools with height, and a parcel of warm flue gas leaving the top of a stack keeps rising. Under a temperature inversion — a layer where the air grows warmer with height, common on still winter nights over valleys — the gas reaches air as warm as itself within a few tens of metres and stops. The layer a parcel cannot leave found the stability condition for a displaced parcel in a stratified atmosphere, and an inversion is the case in which it holds strongly: the smoke spreads sideways under the warm lid instead of rising, and the city’s chimneys fill the valley with their own exhaust. The London smog of December 1952, which killed thousands of people, was coal smoke trapped under such a lid for five days. Tall chimneys were built afterwards partly to punch their gas through the inversion before it could stop.

There is a quieter contrast with an atmosphere that is not moving at all. The column that is hotter at the bottom found that a column of gas in true thermal equilibrium in a gravitational field is not at one temperature but very slightly warmer at the bottom, by an amount set by relativity and far too small to matter here. The stack effect needs nothing so subtle: it is a column held out of equilibrium by a heater, and the flow it drives is the atmosphere’s attempt to restore the equilibrium the heater keeps breaking.

Where the simple columns are too simple #

The figures treat each column as a single uniform temperature and the openings as concentrated at the top and bottom. A real building has openings spread over its height, temperatures that differ from floor to floor, internal walls and floors that resist the vertical flow, and lifts whose motion pumps air between floors. Its stack effect is a network of flows through many resistances, computed by engineers with programs that solve for the pressure on every floor. Wind adds a pressure of its own on every face of the building, often larger than the stack effect on a windy day, so the measured pressures on any particular day are a sum of the two. And the outside temperature itself varies with height, by about six and a half degrees per kilometre in a typical atmosphere, which hardly matters over a hundred metres but does over a few hundred, in the tallest buildings and in deep mines.

The flows through the openings are treated with a single discharge coefficient, an empirical factor that depends on the shape of each gap and on the flow speed. A crack under a door and an open window have different coefficients, and real leakage is characterised by blowing air through a building with a fan and measuring how the flow depends on the pressure — a blower-door test — rather than by calculating it.

What a pressure profile cannot show #

The figures draw pressures and flows, and a building’s air is neither static nor uniform. Warm air from the heaters rises in plumes and spreads under ceilings, cold air pours in under doors and slides along floors, and the vertical shafts carry the stack flow in turbulent currents that mix as they rise. A pressure profile is the average of all that, and it predicts the forces on doors and the leakage through walls well, while saying little about where in a room the cold draught is felt. Nor do the figures show moisture: the air leaving through the upper walls of a heated building in winter carries the building’s humidity into cold wall cavities, where it condenses, and that, more than heat loss, is what makes the stack effect a concern for the people who design walls.

Still open: how a building should be pressurised #

The stack effect is understood well enough to design by, and the open questions are engineering rather than physics. How to keep the neutral plane where it does least harm in buildings of several hundred metres, whose stack pressures reach hundreds of pascals, without fans that cost as much energy as the leakage they prevent; how airtight a building can be made before its air quality suffers; and how to use the stack effect for ventilation in a building whose occupants open and close windows at will — each is argued case by case, and the best arrangement for the tallest towers is still being worked out as they are built.

The habit worth carrying away is to treat air as a fluid with weight. A building warmer than its surroundings is a column of light air beside a column of heavy air, and their pressures, falling at different rates with height, agree only at one neutral plane: the difference grows as $(\rho_{\text{out}} - \rho_{\text{in}})g$ per metre from it — 69 Pa at the roof and the ground of a 100 m building at 21 °C in −10 °C air — pushing air in below and out above, and pressing a pair of lobby doors shut with nearly two hundred newtons on a cold day. Wherever two fluids of different density stand side by side and are connected at different heights, a flow is driven, and its strength is the height times the difference in their weights.