Wet-bulb temperature and the limits of human survival

2023 February 26

In 2010, the paper “An adaptability limit to climate change due to heat stress”1source by Sherwood and Huber captured media attention, introducing the concept of wet-bulb temperature (WBT) to the public. The paper presented a very simple argument:

What are those basic physiological reasons? People generate about 100 W of heat while inactive, which must be shed to the environment to maintain homeostasis. Skin temperature must be greater than the air’s WBT for heat to leave the body, but skin temperature is tightly regulated to 35 C, and at higher skin temperatures core temperatures must rise above 37 C for heat to leave the core. While brief excursions above 35 C are survivable by storing excess heat in the body, this is only sustainable on a time scale of hours.2Notably this excess heat must be shed after a heat stress event, which requires significantly cooler environmental temperatures; but WBTs are fairly stable on a time scale of days, so a peak of 35 C often means remaining near that peak for an extended period of time. The most lethal heat waves tend to be those when night-time temperatures are especially elevated, as the body has no relief to get rid of heat accumulated in the day.

This is certainly a naive limit on extremes of human survivability; Sherwood and Huber are climate scientists, so their main interest was in the second bullet point of simulating future Earth climate. They chose 35 C as a threshold which is necessarily lethal knowing that in practice one might expect significant mortality that precludes human habitation at significantly lower WBTs.3For context, WBT very rarely exceeds 30 C, and the majority of humans live in areas that reach a maximum WBT above 26 C annually.

While the paper is quite short and simple, inevitably its central purpose was largely missed and what permeated the public conscious is the idea that people can survive WBTs up to around 35 C – not the converse, that 35 C is unsurvivable, nor the implications this has on human habitation in extreme global warming scenarios. Thus we get this 2022 press release:

It has been widely believed that a 35°C wet-bulb temperature (equal to 95°F at 100% humidity or 115°F at 50% humidity) was the maximum a human could endure before they could no longer adequately regulate their body temperature […] But in their new study, the researchers found that the actual maximum wet-bulb temperature is lower — about 31°C wet-bulb or 87°F at 100% humidity — even for young, healthy subjects.

The 2022 paper in the Journal of Applied Physiology claims to the be first to experimentally test the WBT-limits of human endurance. Healthy young subjects swallowed a remote thermometer to measure the limit of heat stress before their internal gut temperature began to rise.

Despite the oppositional tone of the authors, this study is the natural complement to the 2010 study: physiologists probe the limits of human endurance, and climate scientists determine the implications under various climate scenarios.4To my knowledge no one has yet used the 2022 physiology paper to update the survivability of various scenarios. Also, the limits of human habitability is surely lower than human survivability, so ideally some sociologists would bridge that gap.

We are left with one question, the topic of this article: what is wet-bulb temperature? It turns out there are two different, unrelated definitions of WBT, which by pure coincidence happen to give close to the same numerical values.

Understanding humidity

Air contains a mixture of various gases, one of which is water vapor. Most components of the atmosphere are well-mixed and stable: e.g., air is consistently around 21% oxygen. However water vapor condenses to liquid water in conditions close to atmospheric conditions; as a result, there is a rapid exchange between liquid and gaseous water, and the quantity of water vapor in the air is highly variable depending on local conditions.

We know, of course, that water boils/condenses at 100 C: so if a local parcel of air is, say, 25 C, why doesn’t all the water vapor in it condense to liquid water? The answer is that water prefers5i.e., maximizes entropy to be in a mixture of liquid and gaseous6and solid states, with the amount in each state depending on the temperature.

This sounds more complicated than it is because it is easy to get distracted by the non-water components of air, which are in fact largely irrelevant:

The vapour saturation curve is not affected by the presence or absence of other gases including dry air. […] The phrase “the air is saturated with water vapour” is often heard. This phrase can be confusing because saturation has nothing to do with the air. A saturated water vapour condition exists when the volume contains the maximum possible number of water vapour molecules. The fact that the dry air and the water vapour occupy the same volume has no bearing on the fact that the water vapour is in a saturated state. Such usage is widespread in phrases such as “the air is saturated” or “the dew-point temperature of the air.” Source (page 31)

At any particular temperature, water desires to have a certain pressure, called the saturation pressure7This is the same information as the boiling point, looked at in reverse: the saturation pressure for a certain temperature is the pressure at which that would be the boiling point.. This can be calculated with the Clausius-Clapeyron relation, which for water gives approximately:

e_s(T) = (610.78 \text{ Pa}) \exp \left( \frac {17.27 T}{T + 237.3} \right)

where T is in Celsius.

Any excess water beyond that needed to reach saturation pressure remains as liquid water. If the partial pressure of water vapor exceeds the saturation pressure, then the excess condenses8assuming nucleation sites are available into liquid water: this is the process that forms clouds9yes, clouds are made of liquid. When water vapor condenses out of air not near any large solid surface, it condenses everywhere onto tiny particles about 0.1 micrometers in size, thus forming countless numbers of microscopic drops which can remain suspended in the air indefinitely. Eventually these droplets collide and merge until they are large enough to fall due to gravity..

Conversely, if the pressure is below the saturation pressure, then liquid water will evaporate. The non-water components of the atmosphere are all well below their saturation pressures10Actually room temperature is above the critical point for many atmospheric gases, e.g. -147 C for nitrogen and -119 C for oxygen. If you tried to raise the pressure of oxygen or nitrogen until they condensed into liquid, they would actually smoothly transition into supercritical fluids which have properties of both gas and liquid, without ever passing through a state transition at some saturation pressure.: carbon dioxide, for example, has a saturation pressure around 74 atm at room temperature – if you were to somehow get liquid carbon dioxide to 1 atm of pressure it would just violently boil into gas. Consider also the example of placing liquid water in a large vacuum: water will rapidly evaporate (boil) to form gas, which will immediately disperse, and so evaporation will continue. Since evaporation consumes heat, the liquid will cool, until some of it freezes into solid ice11about 7/8 of it will freeze if it started near 0 C, the amount varies significantly with temperature.

So far we have spoken of the saturation pressure of water vapor with respect to liquid water, but one can similarly calculate a saturation pressure with respect to ice; and so when ice is in a vacuum, it too will sublimate into gas, though far slower than liquid water would boil. As it sublimates the remaining ice will get colder and colder, and sublimate slower and slower.

The difference between the two saturation pressures at temperatures below 0 C drives the Wegener-Bergeron-Findeisen process:

Here, the saturation pressure for liquid water is greater than for solid water, so the deposition of gaseous water onto ice crystals brings it below the saturation point for liquid water, causing water droplets to evaporate into gas. Thus water goes from liquid to gas to solid until all liquid is gone.

Okay, enough of a digression – we will ignore ice from now on. The ratio of the water vapor pressure to the saturation pressure is called the relative humidity (RH). At RH = 100%, condensation and evaporation are balanced. RH > 100% is very rare in the atmosphere because ample nucleation sites means excess water vapor can condense immediately. However RH < 100% can be maintained if liquid water is absent; thus we recover the obvious fact that humidity tends to be highest just above the ocean surface.

Before we move on, let us consider what makes boiling different from ordinary evaporation. This is the part where the non-water constituents of air are relevant. The total pressure of a parcel of air is tightly controlled by physical processes: any horizontal gradient in air pressure causes strong winds to neutralize it12Vertical gradients exist in balance with gravity, called hydrostatic equilibrium. However deviations from such equilibrium would equally create strong restorative winds.. This contrasts with partial pressures; if two air parcels have different partial pressures of their components but same total pressure, then equilibrium is restored by much slower diffusive forces.

Above 100 C, the saturation pressure of water exceeds 1 atmosphere: thus evaporation continues until the local partial pressure of water is greater than the ambient total air pressure. Necessarily this creates rapid winds expelling the excess water vapor to the ambient air, making it impossible for partial pressure of water vapor to ever reach the saturation pressure.

Moreover, this process takes place even in the interior of any body of liquid water. For a shallow body of water, the pressure inside the liquid is equal to the external air pressure. If some of this liquid water were to convert to gas, it would indirectly push on and displace some of the external air. When the saturation pressure is greater than the external air pressure, this is a net movement from high to low pressure and thus favored.13An alternative perspective is that inside of a volume of liquid water, the surrounding water “shields” it from “knowing” the composition of any external air. Effectively it is like the interior of the water had been exposed to 1 atm of pressure of pure water vapor. Thus evaporation only takes place in the interior when the saturation vapor pressure is greater than 1 atm, i.e., when temperature is above 100 C. Therefore evaporation takes place not only on the surface of any liquids but in their whole interiors. This is why boiling is a much faster process than evaporation at temperatures below the boiling point.

If elements of this discussion seem familiar, it is not the first time it has appeared on my site – but hopefully much clearer this time!

Wet-bulb temperature

The human body is able to maintain a temperature cooler than its environment through sweating. Evaporation of water on the surface of the skin consumes heat, cooling the body; however this is only possible if RH is less than 100%. If RH is greater than 100%, in fact the reverse happens, and condensation appears, warming the surface.14This is how dew or condensation on cold drinks occurs; warm, humid air cools down when in contact with a cold surface, lowering the saturation pressure, and so raising RH above 100%. Dew is more likely after cloudless nights because it requires a large disparity between ground and air temperatures, which is enabled by infrared radiation from the ground into space.

WBT is an effective temperature that takes into account evaporative cooling: it is the minimum temperature one can reach in an environment assuming one gets the maximum possible benefit from evaporative cooling – sometimes described as the temperature of a thermometer with a soaking wet bulb exposed to gale winds.15Of course, sweating is not perfect, so using WBT slightly overestimates human tolerance to dry extreme heat. That is, WBT is the temperature air would have if it had 100% RH.

The heat index and humidex are in principle very similar, in that they give equivalent temperatures based on normalizing the amount of humidity in the air. The heat index gives the equivalent temperature if water vapor pressure were 1.6 kPa, or 0.0158 atm; this is a fairly typical atmospheric value for water pressure, which makes the heat index useful for relating to typical conditions that people have experience with. In contrast 100% RH is quite far from people’s typical experience, so WBT is not so easily relatable.

This leaves us only the question of what an “equivalent” of “effective” temperature is when normalizing for water vapor pressure. We will start with what is sometimes called the thermodynamic WBT:

Consider a parcel of air with a (dry-bulb) temperature of T and WBT of T_w, which we wish to calculate. If a wet surface with temperature T_w is brought into contact with the air, there is no exchange of heat into or out of the surface by definition. However to reach equilibrium the air needs to become fully saturated; since no heat enters or leaves the surface, the heat of evaporation is supplied by the air parcel.

Let us write q_0 and q_1 for the initial and final specific humidities16mass of water content divided by total mass of the air parcel, and L for the latent heat of vaporization of water17measured at T_w, as it varies slightly with temperature. Then the heat absorbed by the water is (q_1 - q_0) L. Similarly, the heat expelled by the air mass is (T - T_w) c_p where c_p is the specific heat capacity of the air parcel; we use the heat capacity at constant pressure as the process is isobaric (the initial and final pressures are the same, since in both cases it equals the pressure of the surrounding air masses). Therefore

 (q_1 - q_0) L = (T - T_w) c_p.

Since the air parcel ends up saturated,

q_1 = \frac {e_s(T_w)}{p} \cdot \frac{\rho_{\text{water}}}{\rho_{\text{air}}}

where e_s(T_w) is the saturation pressure of the parcel at the end, p is the initial (and final) air pressure, and the factor of \rho / \rho is to convert from volumetric ratio to mass ratio (it is the ratio of their molecular weights). The equation can then be solved (numerically) for T_w in terms of p, q_0, T, and physical constants L, c_p, \rho_{\text{water}}, and \rho_{\text{air}}.

Thus if a wet surface with temperature T_s > T_w is brought into contact with the air parcel, we can imagine the air cooling to T_w as it saturates, and thus heat can transfer from the surface into the air (via the liquid interface). Conversely if T_s < T_w we cannot transfer heat out of the surface. Thus T_w really is the coldest temperature we can achieve through ideal evaporation alone.

Now let us consider the more standard, and less elegant, definition for psychrometric WBT. A typical wet-bulb thermometer consists of a thermometer whose bulb is covered by a fabric wick dipped in a reservoir of water, and has air blown over it. The temperature the thermometer reads at equilibrium is then T_w.18Note that some length of the fabric wick should be exposed to moving air before reaching the thermometer so that equilibrium can be reached; without this readings will be slightly high, assuming the reservoir is at the dry-bulb temperature. Heat flows into the thermometer through conduction across the wet surface at a rate proportional to the difference T - T_w in temperatures, and (latent) heat is brought out through evaporation which occurs at this same surface at a rate proportional to the difference q_1 - q_0 in specific humidities.19This can be argued by supposing there is a static mass of air just touching the wet thermometer, and considering the boundary between this static air mass and the flowing air mass. If heat and humidity are exchanged through turbulent mixing then in both cases the rate of exchange will be proportional to the differences as in Fourier’s Law. It is the fact that heat and humidity are transferred across the same interface as each other that allows us to ignore the shape and size of the thermometer and wick. Then at equilibrium we have

 (q_1 - q_0) L k_y = (T - T_w) h_c

where k_y is the mass transfer coefficient (kg / m^2 / s) and h_c is the heat conduction coefficient (W / m^2 / K). Both sides have units of W / m^2, a rate of heat flow per area of surface of contact. As before, this equation can be solved for T_w. Note that real-world hygrometers (at least until the 1980s) usually work in reverse: they measure T and T_w and then infer the initial humidity q_0 of the air.

The two definitions for T_w are in agreement when h_c = c_p k_y.

Psychrometric ratio

We define the psychrometric ratio (or psychrometric coefficient)

\beta = \frac {h_c}{c_p k_y}

which is equal to 1 exactly when the two definitions given above are in agreement. Fortunately, for water evaporating into air (as opposed to other liquid-gas combinations), \beta happens to be close to 1, and we can use the two definitions interchangeably. The deviation from 1 measures how good (or bad) of an approximation this is.

So, what is the value of \beta? If someone knows, they aren’t telling. Most resources I could find online simply say “about 1”, and leave it at that. These were often engineering handbooks for which the approximation was serviceable for practical use; they would also use terrible units like “btu per foot lb second” (when they used units at all) and have the bad habit of leaving numerical constants in their equations.

Albright’s Chemical Engineering Handbook (page 1670) says that \beta is between 0.96 and 1.005 before helpfully pointing out that “thus it is nearly equal to the value of the humid heat c_s” (i.e., heat capacity c_p, which has different units from \beta). I found a 1959 paper that gives (equation 10):

 \beta = 0.91 \sqrt{Le}

for Schmidt numbers from 0.26 to 2.65, where Le = Sc / Pr is the Lewis number, Sc is the Schmidt number, and Pr is the Prandtl number. In principle one can thus just look up the values for these quantities and substitute, but in my searching these unitless numbers seemed equally if not more obscure than the psychrometric ratio.

The 1988 Phd thesis by Andre Dreyer says that Lewis (for whom the Lewis number is named) tried to prove in 1922 that the psychrometric ratio20which Dreyer calls the “Lewis factor”, before continuing that many authors erroneously refer to it as the “Lewis number” is 1, but I could not find the original article21Lewis, W.K., The Evaporation of a Liquid into a Gas, Transactions of ASME, Vol. 44, pp. 325-340, May 1922; Lewis corrected it in 1933 to show that the psychrometric ratio happened to be approximately equal to 1 for water-air but again I cannot find the article22Lewis, W.K., The Evaporation of a Liquid into a Gas - A Correction, Mechanical Engineering, Vol. 55, pp. 567-573, 1933..

This thesis continues with listing multiple empirical relations for the psychrometric ratio:

    \beta &= Le^{2/3} && \qquad \text{Chilton and Colburn 1934} \\
    \beta &= Le^{0.56} && \qquad \text{Bedingfield and Drew 1950} \\
    \beta &= Le^{0.5} && \qquad \text{Mizush...

among other, more complicated relations. The Chilton-Colburn exponent was not truly the best fit for their data but rather chosen for ease of computation on slide rules; the equation apparently is referred to as the Chilton-Colburn “analogy”, which does not imbue me with confidence.

Principles and Modern Applications of Mass Transfer Operations (page 486) does better, saying

 \beta = Le^{0.567}

and citing a 1961 paper by Parker and Treybal (which again I could not find23Parker, R.O., and Treybal, R.E., The Heat, Mass Transfer Characteristics of Evaporative Coolers, Chemical Engineering Progress Symposium Series, Vol. 57, pp. 138-149, 1961.) as giving

 \frac {h_c}{k_y} = 0.950 \text{ kJ} / \text{kg} \cdot \text{K}

which with the heat capacity c_p = 1.005 \text{ kJ} / \text{kg} \cdot \text{K} at 300 K finally yields

 \beta = 0.94527.

One cannot help but notice that this actually somewhat far from 1, and outside of the range that Albright’s had quoted.

The truth, of course, is that the value of \beta depends in some way on atmospheric conditions. I did at long last find in Perry’s Chemical Engineering Handbook (9th edition, page 1133) a worked example of numerically computing the two values for WBT at a specific temperature, pressure, and humidity, from which one can infer an intended way to compute the psychrometric ratio. They write

For relatively dry air the Schmidt number \mu / \rho D_\nu is 0.60, and from Eq. (12-4) h_c / k' = 0.294(0.60)^{0.56} = 0.221.

The left side, what we were calling h_c / k_y, has units of heat capacity while the right side is missing any units. None-the-less they compute a heat capacity of 0.24624presumably the missing units are horsepower per fortnight hogshead or something which corresponds to

 \beta = 0.221 / 0.246 = 0.8984.

We seem to be drifting further afield, and what happened to the dependence on environmental conditions? Inferring from their calculations and from the language used in other sources25Almost every exposition of wet-bulb temperatures I saw followed much the same structure and language, leading me to suspect that they are not independent but rather all stem from a single source that predates the 1950s; however the calculations are simple enough that some similarity would be inevitable., the dependence comes through heat capacity varying with water content – air with more water content will have a higher heat capacity per mass of dry air. Previously I had used c_p as the specific heat capacity per mass of moist air, which seems more appropriate, and varies half as much with humidity as the former. One also wonders whether the Schmidt number would likewise vary with humidity, and – most irksome – what is the point in a computation of WBT that is only good at low humidity?! The whole idea behind WBT is it adjusts for variations in humidity by converting to an equivalent temperature that doesn’t depend on humidity. How does it make logical sense to have it still depend on humidity after adjusting for humidity?

Heat capacity, at least, is well-understood: these mysterious Schmidt and Prandtl numbers appear to exist only to befuddle, and little should be staked on having accurate values for them. An ideal gas should have a Schmidt number of 1, but supposedly the Schmidt and Prandtl numbers of air are typically around 0.7. I have also seen 0.66 suggested for the Schmidt number of water in air at 300 K. Using 0.7 instead of 0.6 above improves the value of \beta only modestly.

It is an unsatisfying end to this mess, and many hours of hunting through old papers, but I think the truth is that if you want to know the psychrometric ratio, either say “about 1” like everyone else, or you have to go out and measure it yourself.

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