The room that is cold at twenty-one degrees

Anyone who has lived in an old house with solid walls knows the experience. In winter the thermostat is set to twenty-one degrees, the thermometer confirms it, and the room is cold. People sit in jumpers and complain of a chill that the instruments cannot find. Turn the heating up to twenty-four and the room is finally comfortable, at a heating bill that is not. Move to a modern, well-insulated flat, set the same thermostat to twenty-one, and the room is warm. The air is the same temperature in both. The difference is in the walls.

A human body is not an air thermometer. It exchanges heat with its surroundings by several routes at once, and the largest of them, for a person sitting still indoors, is radiation: every surface of the body radiates to the walls, ceiling, floor and windows, and they radiate back. A body therefore responds to the temperature of the surfaces it can see at least as much as to the air it is in, and a room with cold walls is cold whatever its air says.

Two-thirds by radiation #

The two routes can be compared directly. Convection to still air carries heat at a rate of about 3 watts per square metre for each kelvin of difference between a seated person’s clothing and the air. Radiation carries heat at a rate set by the Stefan–Boltzmann law. For two surfaces whose temperatures are close, it is proportional to their difference, with a coefficient $4\varepsilon\sigma T^3$, about 5.6 watts per square metre per kelvin at room temperature for clothing with an emissivity of 0.95.

So about two-thirds of a still, seated person’s dry heat loss goes by radiation. This is a surprise to most people, who think of radiation as something that happens near fires and in sunshine. But every object at room temperature radiates strongly in the far infrared, about 420 watts per square metre at 21 °C — the thermal spectrum whose shape the curve that would not come down traced to Planck — and receives almost as much back from its surroundings. The net exchange is the small difference between two large flows, and it is sensitive to the temperature of whatever the body sees. The cold reservoir overhead followed the same exchange outdoors, where a clear night sky radiates back far less than a body sends to it, and frost forms on a car roof in air above freezing.

What the body responds to is the combination. The temperature that would give the same total heat loss in a room where air and surfaces were equal is called the operative temperature, and with these coefficients it is

$$t_o \approx 0.65\,\bar t_r + 0.35\,t_a,$$

where $\bar t_r$ is the mean radiant temperature, the average temperature of the surroundings weighted by how much of the body’s view each part fills. Comfort standards are written in terms of operative temperature, not air temperature, for this reason. The weights shift with air movement — a fan or a draught raises the convective share — but in a still room, radiation dominates.

Cold walls #

The inside surface of an outside wall is colder than the room, by an amount set by how fast heat leaks through the wall. Heat flows through the wall at a rate $U\,\Delta T$ per square metre, where $U$ is the wall’s thermal transmittance and $\Delta T$ the difference between inside and outside, and the same heat has to cross the thin layer of still air on the inside face, with a resistance of about 0.13 square-metre kelvins per watt. The inside surface therefore sits below the room by $U\,\Delta T \times 0.13$.

An old solid brick wall has a U-value of about 2 watts per square metre per kelvin. On a day of −5 °C its inside face sits at about 14 °C, seven degrees below the air. In a corner room with two such walls filling much of the view, the mean radiant temperature is several degrees below the air and the operative temperature is about 18.4 °C — a room that a thermometer calls 21 feels like one at 18. A modern wall insulated to a U-value of 0.2 has its inside face within half a degree of the air, and the room feels like its thermometer.

This is the effect that people in old houses describe as a chill that the heating cannot reach, and it is real. It is also why the effect of insulation on comfort is often larger than its effect on the heat bill suggests: insulation not only reduces the heat leaving, it warms the inside surfaces, so the same air temperature feels warmer, and the thermostat can be turned down. Occupants of newly insulated houses commonly report exactly that.

A window is a cold patch in the view #

Windows are the coldest surfaces in most rooms, but they are usually small, and what matters is how much of the body’s view they fill.

The inside face of single glazing on a cold day is just above freezing — cold enough to form condensation and, on a hard night, frost on the inside. Yet a person sitting a metre away, with the window filling a good part of one side of their view, loses only about a degree of operative temperature to it, because the window fills well under a tenth of the full sphere of directions around them. The effect is felt mostly close to the glass, and mostly on the side of the body facing it, as a cold cheek and a cold shoulder.

The fraction of the view a surface fills is its view factor, the share of the radiation leaving a body that would arrive at that surface. For a small body near a large wall, the view factor of the wall approaches a half; for a person in the middle of a room, each wall fills a modest fraction and the floor and ceiling the rest. The view factor is what makes radiation a matter of geometry rather than of distance alone: a cold window two metres away and twice as wide and twice as tall affects a body as much as one a metre away, because both fill the same solid angle.

Why emissivity is the lever #

The radiative coefficient has an emissivity in it, and that is the one number a designer can change without changing a temperature. Most building surfaces — plaster, paint, brick, wood, glass, skin and cloth — are nearly black in the far infrared, with emissivities above 0.9 whatever their visible colour, because the wavelengths at which room-temperature bodies radiate, around ten micrometres, are absorbed strongly by almost every non-metal. The glow that says nothing about the surface followed that distinction between how a surface looks and how it radiates. A white wall and a black one exchange heat with a person almost identically.

Metals are the exception, and they are what low-emissivity glazing exploits. A coating of silver or tin oxide a few tens of nanometres thick, transparent to visible light, reflects most of the far infrared, and on the inner pane of a double-glazed unit it cuts the radiation crossing the gap between the panes by a factor of five or more. The thin foils that stop radiant heat set out how a few reflective layers can block radiant exchange almost entirely, and the low-emissivity window is that idea made invisible. Its effect on comfort is the one drawn here: the inner pane stays warmer, closer to the room, so the window takes less from the mean radiant temperature.

The same lever explains the emergency blanket. A sheet of aluminised plastic wrapped round a cold person adds almost no insulation by conduction, being a fraction of a millimetre thick. What it removes is the body’s radiative exchange with cold surroundings, two-thirds of its dry loss in still air, by replacing a nearly black surface with a nearly perfect reflector. The third route out of a body, evaporation of sweat and moisture from the lungs, is not touched by any of this: it is set by the humidity of the air, as the thermometer that reads low when wet found for any wet surface, and it becomes the dominant route only when the room is too warm for the dry routes to carry the body’s heat.

Paying for cold walls with warm air #

The usual response to a cold room is to turn up the thermostat, which warms the air. Because air carries only about a third of the exchange, this is an expensive way to buy comfort.

The arithmetic follows from the weights. To raise the operative temperature by one degree with the surroundings fixed, the air has to rise by $1/0.35$, nearly three degrees. A metre from a single-glazed window, where the window costs a degree, the air has to be at 24 °C to feel like 21. In a corner room with solid walls, where the cost is nearly three degrees, the air would have to be at about 28 °C — which is roughly what people in such rooms end up living with, in the warmest part of the room near the radiator, with the far corners cold.

Warm air also does not stay where it is put. It rises to the ceiling, where it is least useful, and it leaks out through every gap in the building, carrying its heat with it. The heat lost by ventilation is proportional to the difference between inside air and outside air, so heating the air three degrees more than necessary raises that loss accordingly. A room kept comfortable by hot air around cold walls pays twice: once for the extra air temperature, and again for the extra heat that leaks out with the air.

Warming the surfaces instead #

The same arithmetic run in reverse is the case for radiant heating.

If the surroundings are warmer than the air, the air can be cooler. A floor kept a few degrees above the room, or panels in the ceiling, raise the mean radiant temperature, and a person feels comfortable in air several degrees cooler than a convector-heated room would need. The line in the picture is steep for the same reason the window was costly: every degree of radiant temperature is worth nearly two degrees of air. Underfloor heating, radiant ceiling panels and the old kachelofen, the tiled stove of central Europe whose great warm mass radiates into the room for hours, all work this way. So does a sunlit room in winter, where the sunlit floor and furniture are warmer than the air and the room feels warm with the air cool.

The same principle runs the other way in summer. Chilled ceilings, cooled by water a few degrees below the room, absorb radiation from occupants and make a warm room comfortable with little air movement. And in hot climates, the traditional thick masonry walls that stay cool through the day keep the mean radiant temperature low even when the air is warm, which is part of why such buildings feel cool when the air in them is not. The summer that reaches the cellar in December followed the slowness with which heat diffuses through masonry; a thick wall’s inside face lags the outside by many hours and damps its swings, which is the property that makes it a stable radiant surface.

A body is a thermometer of the walls #

The deeper point is that the quantity a body measures is not a property of the air. The metal that feels colder than the wood made the same point about touch: the hand does not measure the temperature of what it touches, it measures the rate at which heat leaves the skin, and metal and wood at the same temperature draw heat at different rates. A body in a room does not measure the air temperature either. It measures its own rate of heat loss, by every route, and adjusts its comfort accordingly. Two rooms with the same air and different walls are, to a body, two different temperatures.

That is why physiologists and building engineers developed their own thermometers. The globe thermometer, invented in the 1930s, is a thermometer inside a hollow black sphere about fifteen centimetres across. It exchanges heat with its surroundings by radiation and convection in roughly the proportions a person does, and its reading approximates the operative temperature directly. In a room with cold walls a globe thermometer reads several degrees below an ordinary one beside it. It is a thermometer built to feel what a body feels, which is the opposite of the usual aim of thermometry, to read the air and ignore the walls.

The insulation that makes a wire lose more heat found a related surprise in the other direction: adding a thin layer of insulation to a small wire can increase its heat loss, because it enlarges the surface that convects and radiates. In both cases the outcome is decided by the balance between conduction through solids and transfer at a surface, and in both the intuition that more of something means more of its effect fails at the surface where the routes meet.

What the picture cannot show #

The model is a seated, still, clothed person in still air, with the heat-transfer coefficients of the comfort literature. A person moving about, or in a draught, has a larger convective coefficient, and the weight on the air rises towards a half; the figures would shift towards the air thermometer. The coefficients themselves vary between people, with posture and with clothing, and the 65 per cent radiative share is a representative value, not a constant.

The window calculation treats the person as a small sphere at the height of a seated chest on the window’s centre line, and every surface but the glass as being at the air’s temperature. Real rooms are more complicated: the wall around the window is also cold, the floor near an outside wall is colder than the middle of the room, and radiators under windows warm the surfaces near them. And the most noticeable effect of a cold window is usually not radiation at all, but the downdraught of air cooled against the glass, which falls and spreads across the floor as a cold current around the ankles. That draught, not the radiant exchange, is what comfort standards most often find objectionable near windows, and it is not in these figures.

The wall temperatures use steady-state heat flow. Massive walls are rarely in steady state; they warm and cool slowly over days, and a solid wall’s inside face may be warmer or colder than the steady value depending on the recent weather and on how the room has been heated. A house heated intermittently, only in the evenings, never lets its walls warm through, and feels colder than its thermostat for that reason too.

Still open: how comfort depends on what a body cannot see #

Thermal comfort standards rest on models fitted to experiments in which people in climate chambers reported how warm they felt, mostly in uniform conditions and over hours. Real rooms are not uniform: a warm foot and a cold neck, a sunlit side and a shaded one, a warm floor and a cold window all combine into a single feeling of comfort in ways the uniform models do not capture. How the body integrates such patchy surroundings — whether a warm patch can compensate for a cold one, which parts of the body matter most, and how the integration changes over time — is an active research area, driven by the move to radiant heating and cooling, personal heaters and fans at desks, and buildings designed to keep people comfortable with much less energy. The physics of the exchange is settled; how a person turns it into a sensation is not.

A still, seated person loses about two-thirds of their dry heat by radiation to the surfaces around them, so what they feel is the operative temperature, roughly 0.65 of the mean radiant temperature plus 0.35 of the air’s: a corner room with uninsulated brick walls at 14 °C feels 18.4 °C when its air is 21, and making up each lost degree with warm air costs nearly three degrees of air, while warming the surfaces lets the air be cooler. The habit worth carrying is to ask which temperature a body actually exchanges heat with, because a thermometer in the air answers a question the body did not ask.