The thermometer that reads low when wet

The puddle that could dry in a second took the upper limit on evaporation, set by how many molecules can leave a surface, and it showed that this limit is enormously greater than any real rate. What sets the real rate, in almost all ordinary cases, is the air’s ability to carry vapour off, and also the surface’s ability to get the heat. It closed one of its paragraphs by mentioning that an evaporating drop cools towards the wet-bulb temperature. The thermometer in question has been used for two centuries, and is older than the theory behind it. It tells something very specific about a mixture of air and water, and with a change of temperature of about 12 °C at everyday conditions, this is a very large effect that anyone can check with a wet sock.

Anyone who has stepped out of a pool on a dry day knows the sensation. The water on the skin is colder than the air around it, and it stays cold. It should not be possible, at first sight. A body exposed to warm air should warm up. Instead, the wet surface settles at a steady temperature below the air’s, and it holds that for as long as water is left. No fuel is being burned. The cooling power comes from the air itself, or more precisely from the water’s preference for being a vapour in air that is short of it, a preference with a known name.

What a wet surface wants #

The salt that dissolves in the air’s own water and related essays developed the idea of chemical potential: water will move from where its chemical potential is high to where it is low. Water vapour in air has a chemical potential set by its partial pressure $e$, and liquid water at temperature $T$ has one that corresponds to the saturation vapour pressure $e_s(T)$. Unless the air is saturated, $e_s > e$ at the same temperature and so water evaporates.

Evaporation takes latent heat, and that heat has to come from somewhere. The obvious source is the wet surface itself, which then cools. A colder surface loses less to evaporation, because the saturation pressure falls exponentially with temperature, by the curve in a boiling point is a pressure. A colder surface also gains heat from the warmer air, at a rate proportional to the temperature difference. So two flows meet:

$$h\,(T_{\rm air} - T_w) = \frac{h_m L}{R_v T}\big(e_s(T_w) - e\big).$$

The left side is sensible heat arriving by conduction through a thin air layer. The right side is the latent heat leaving with the vapour through the same layer.

The stability of the crossing is worth seeing. If the surface is warmer than 24 °C, it is losing more to evaporation than it is taking from the air, so it cools. If it is colder, the air supplies more than it can use, so it warms. So any surface placed in this air will head for that temperature and stay there, and that is the wet-bulb temperature. Its value depends on air temperature and vapour pressure, and on the ratio of the two transfer coefficients. It is a property of the air, and it has been the working measure of humidity for meteorologists for most of the time since John Leslie, in about 1799, and Ernst August, who gave the instrument its name, in the 1820s.

A chart of the air #

The result is usually written as

$$e_s(T_w) - e = \gamma\,(T - T_w),$$

where $\gamma$ is the psychrometric constant, about 66 Pa/K at atmospheric pressure. It contains the specific heat of air, the latent heat and the ratio of molecular weights, and has pressure in it because vapour’s concentration depends on pressure. Solving it gives the chart.

Twenty-two degrees of depression in a desert is what makes an unpowered cooler a practical thing. A porous clay pot of water, kept in the shade, cools its contents to near the wet-bulb temperature in this way, and they were used throughout history in dry climates. Modern evaporative or “swamp” coolers do the same to a stream of air. Ancient builders in Iran used wind towers over shaded water to bring air down near the wet-bulb temperature. Mohammed Bah Abba won a Rolex award in 2000 for a clay pot within a pot, with damp sand between, used to keep vegetables in northern Nigeria. All of this depends on low humidity. On a day of 100 per cent humidity, the chart gives no difference at all. A wet towel is then no cooler than a dry one.

The chart also shows how much information a single reading gives. Together with an ordinary thermometer, a wet-bulb reading fixes the humidity and it can be solved for directly, with nothing more than a thermometer and some cloth. Before electronic sensors, that was how humidity was measured nearly everywhere, using a sling psychrometer, a pair of thermometers swung round on a handle. That handle is the key to the next point.

Why the wind does nothing #

The obvious way to make a wet thing colder is to blow on it. Blowing on wet skin makes it feel colder, and it might be expected to take the wet bulb further down. It does not. Skin feels colder under the wind because it gets closer to the wet-bulb temperature, more quickly.

Both flows pass through the same thin boundary layer of air on the surface. Wind thins that layer. In both $h$ and $h_m$, its effect is the same, because the diffusivities of heat and of water vapour in air are nearly equal. Their ratio is the Lewis number, about 0.85 for air and water vapour, close to unity. Since it is a ratio of those two transfer coefficients that enters the balance, the wind’s effect cancels. That is an unusual thing. Much of engineering heat transfer involves terrible empirical functions of wind speed and geometry, and here all of them divide out. It is also a lucky property of water in air. For other liquids in air, or water in another gas, the Lewis number is different and the wet-bulb temperature would depend on the conditions.

What wind does change is how fast it gets there, and how fast the water runs out. That matters for how quickly the reading settles, and for how quickly a damp cloth dries. The rate of evaporation at the fixed wet-bulb temperature rises as wind speed does, while the temperature stays. It also explains why a mountain hut’s water bag hung in a breeze cools more quickly than one out of the wind, and why the final temperature is the same.

Radiation is the exception, and an interesting one. The surroundings radiate heat into the bulb in proportion to a temperature difference, at a rate that wind cannot scale, about 6 W/m²/K. In still air, that is a noticeable part of the heat budget, and the bulb reads high. In a fan, radiation’s fraction is small. That is why the proper instrument is whirled round or aspirated by a fan at a few metres per second: not to cool it further but to swamp radiation, so that the reading depends on the air alone.

The line a sweating body cannot cross #

Human skin is a wet bulb, at least potentially. A sweating person loses heat by exactly this process. The core has to be near 37 °C, and heat flows outward through the skin at a temperature near 35 °C. If the wet-bulb temperature of the air exceeds about 35 °C, no amount of sweating can carry away any metabolic heat. The skin’s wet surface settles at a temperature above the body’s own. Heat would have to flow in the wrong direction. And a person sitting in the shade, fanned and with unlimited water, in these conditions, would keep warming up and would ultimately die. Steven Sherwood and Matthew Huber put this forward in 2010 as an absolute limit to habitability.

The physics here is a matter of a boundary, and it does not depend on any details of physiology. It would apply to anyone with a body temperature of 37 °C, whatever they do. But that makes it an upper bound, and people actually fall well short of it. Daniel Vecellio and colleagues at Penn State measured in 2022 at what conditions young, healthy, lightly active people could no longer keep a stable core temperature. In humid environments it was about 31 °C wet-bulb. In hot, dry conditions, it was lower still in terms of wet bulb, because very high air temperature heats the body directly at a rate that sweating cannot follow. So there is no one wet-bulb limit for real bodies, and a simple threshold of 35 °C would be seriously optimistic for most people, especially the old and unwell, who sweat less. Heat waves have killed tens of thousands at wet-bulb temperatures below 30 °C.

Weather records rarely go near 35 °C. Colin Raymond and colleagues in 2020 reported a few brief instances in the Persian Gulf and in the Indus valley at the level of hours, a finding that is partly a question of how they are measured, as station data are often averaged. Wet-bulb temperatures in the tropics seem to be limited by atmospheric convection, which transports the excess away once the surface becomes too warm and moist. A warming climate raises that limit. Current projections suggest that the areas that reach 35 °C for hours every year might expand considerably through this century. Unlike the direct effect on air temperature, the wet-bulb trend is not something that can be solved by a shaded patio. The air conditioner is the remaining solution, and the heat it dumps outside helps very little.

Salt changes what the surface wants #

The balance also has something to say about a solution. A surface of salt water has a lower vapour pressure than pure water at the same temperature, by the factor of water activity $a$. So in the balance $e_s$ is replaced by $a\, e_s$.

The chemical potential of the water is what has moved. It is the same drop that causes a freezing point depression and an osmotic pressure, and the degree a spoonful of solute buys worked through it. A saturated salt solution in air at 75 per cent humidity is in balance with it. There is then no evaporation and no cooling at any temperature, and that is what gives a laboratory a means of fixing the humidity of a sealed box, with a dish of saturated salt. In air at higher humidity the solution would actually draw water out of the air, warming as it goes. That is deliquescence, and it was the subject of the salt that dissolves in the air’s own water.

So sweat has a very slightly higher wet-bulb reading than pure water, since it is a dilute salt solution. Sea spray too has a slightly higher reading than a fresh drop, a small effect, and it matters in the way that sea-salt aerosols are studied. For a drying salt solution, as water leaves, it concentrates, and its reading rises towards that of a saturated solution. That matters in a solar pond, in processing salts and in concentration-driven desalination.

The coldest thing a power station can reach #

The largest users of this number are not meteorologists. A steam power station has to dump about half or more of its heat somewhere, and its efficiency depends on how cold the place is, by the rule that the ceiling on every engine gave. A river is one option. Where there is no river, the great concrete towers do the job, and a cooling tower is a wet bulb the size of a cathedral. Hot water runs in a thin film over packing, while air is drawn upward past it by the buoyancy of the warmed plume, and the water evaporates. It can cool to within a few degrees of the wet-bulb temperature of the incoming air, and never below. That difference, the approach, is what a tower’s designer quotes. So the cold reservoir of a power station in Arizona is not the air at 45 °C. It is the wet bulb at about 25 °C, and that 20 degrees is worth several per cent of thermodynamic efficiency, paid for with water. Each megawatt-hour takes about two tonnes of evaporated water. This is the same work that the cold reservoir overhead tried to get out of the sky, by another route.

The atmosphere works the same arithmetic. An unsaturated parcel of air lifted upward cools until it is saturated, at what is called its condensation level. If rain falling through a dry layer evaporates, it cools the air towards that layer’s wet-bulb temperature, the temperature to which evaporation alone can take it. Air cooled in that way is denser, so it falls as a downdraught and arrives on the ground as a cold gust front, the sudden blast before a storm. The temperature a person feels in that gust is close to the wet-bulb temperature of the mid-level air. For a downdraught of the kind that the plume that stops below the surface would treat as a negatively buoyant plume, the wet-bulb temperature also determines whether falling snow will melt on the way to the ground: snow can reach the surface at air temperatures of a few degrees above freezing if the air is dry enough, because it is the wet-bulb reading of the air which sets the temperature of the wet snowflake. Forecasters use this to tell where the snow line will fall. In the same sense, snow-making machines can operate at air temperatures above zero if the air is dry, which any ski resort can explain.

A similar balance sets the temperature of a cloud droplet growing or shrinking. The ice that grows by stealing from the droplets relies on the vapour pressure difference between water and ice, in the same way as here. A drop in a cloud that has just begun to evaporate settles at its own wet-bulb temperature in seconds. It is the same reason that the cloud that cools more slowly than the air has its slope: latent heat has been brought into a balance. So a thermometer wrapped in a wet sock happens to be measuring what clouds do and what power stations pay for.

Where the balance is simple and where it is not #

The formula treats the surface as being at one temperature, with air far from the surface at uniform conditions. Those are reasonable for a thermometer and much less so for a large wet object or a person, where temperature varies across the surface, where clothing changes the transfer and where sweat does not all evaporate. The psychrometric constant was taken as fixed; it changes modestly with temperature and pressure. In very dry and very hot air, the proper equations include a correction for the vapour’s own contribution to heat and for the bulk motion of the air outward with the vapour. All these are small at ordinary conditions.

The figure of the body’s limit is a line in the air’s properties. It says nothing about what a person is doing, what they are wearing, what they have to drink or how long they are exposed. A body’s own mass delays its heating for hours. A short excursion above a line would not be fatal, and what does most of the killing in actual heat waves is cardiac strain well below any of these lines. So a wet-bulb number is a physical limit and a useful index. It is not a forecast of mortality.

Radiation in the wind figure has been treated as a simple linear term with a fixed coefficient and surroundings at the air’s temperature. In direct sunlight, where the radiative gain is larger, or indoors beside a hot wall, the reading would change in a different way, and for this reason a related index, the wet-bulb globe temperature, combines a shaded wet bulb with a black globe’s reading and is widely used for athletic and military guidance.

Still open: how close the tropics can get #

The upper limit on tropical wet-bulb temperatures seems to be tied to the moist convective stability of the atmosphere: air rising from a hot, humid surface has to be warmer than the free troposphere above it. As the troposphere warms, the limit moves up, in step. Whether wet-bulb temperatures in the deep tropics will follow that upper limit closely enough to exceed 35 °C at a large scale, within the next few decades, or whether the land surface and aerosols will buy some delay, is the subject of a major research effort. The issue involves measurement as well as modelling, as many of the world’s most affected regions have few proper stations, and small errors in a humidity reading change the wet bulb by a degree or more. A simple instrument that has been in use since the 1800s, and whose physics needs only a pair of transfer coefficients, is now one of the numbers on which assessments of habitable land rest.

A wet surface settles where heat arriving from the air balances latent heat leaving with the vapour: 24.0 °C in air at 35 °C and 40 per cent, 23.4 °C at 45 °C and 15 per cent, independent of wind because heat and vapour diffuse nearly alike, and limited by physics at about 35 °C for any body that must lose heat by sweating, with real people failing well below that at about 31 °C. A wet sock measures the chemical potential of water in the air. The habit worth carrying is to ask what cancels when two rates change together, because a quantity that does not depend on the obvious control is usually a property of something more fundamental.