The bird that drinks on the wet-bulb gap

A drinking bird is a sealed glass vessel shaped like a bird: a bulb at the bottom for a body, a long glass neck running up to a second bulb for a head, the head covered in felt and given a beak. It pivots on a crossbar at the middle of its neck. Inside is a volatile liquid, traditionally dichloromethane, and its vapour, with no air. Dip the beak in water once and set it upright, and it begins to rock, slowly leaning forward until it tips, drinks, rights itself, and starts again. It will do this for as long as there is water in the glass, every ten or twenty seconds, day and night.

It looks like perpetual motion, and in the 1940s and 1950s when it was sold as a novelty, it was often described as one. It is nothing of the kind. It is a heat engine, with a hot reservoir and a cold one and a working fluid that carries heat from one to the other, and like every heat engine it is bounded by the efficiency that the ceiling on every engine derived for any machine running between two temperatures. What makes it charming is that its cold reservoir is something it carries on its own head.

The reservoir on its head #

The felt on the head is wet. Water evaporates from it, and evaporation takes heat: about 2.4 megajoules for every kilogram of water. The head therefore cools, and it cools until the heat flowing into it from the warmer room exactly balances the heat carried off by evaporation. The temperature at which that balance is struck is the wet-bulb temperature, which the thermometer that reads low when wet traced to the competition between heat arriving and vapour leaving, and showed to depend on the air’s humidity and not on how hard it blows.

So the bird has two reservoirs: the room, at 22 °C, warming its body; and its head, held at perhaps 15.5 °C on an ordinary day. The difference, the wet-bulb depression, is what drives everything. Carnot’s limit for an engine between those temperatures is

$$\eta_C = 1 - \frac{T_w}{T_a} = \frac{T_a - T_w}{T_a},$$

about 2.2 per cent at 50 per cent humidity. It rises to over 3 per cent in dry air and falls to nothing as the air approaches saturation. A drinking bird in a desert is a better engine than one in a greenhouse, for the same reason that the wet-bulb thermometer reads lower there.

How a degree becomes a lift #

The working fluid turns the temperature difference into motion. Dichloromethane boils at 40 °C at atmospheric pressure, so at room temperature its vapour pressure is high, about 47 kilopascals, and steep: it changes by about two kilopascals for every kelvin. Inside the bird, the vapour in the warm body is at the body’s temperature and pressure, the vapour in the cool head at the head’s. The difference in pressure pushes liquid from the body up the neck towards the head.

The column rises until its weight balances the pressure difference. A kilopascal holds up about 7.7 cm of liquid of density 1.33 g/cm³, so two kilopascals per kelvin is about 15 cm per kelvin. The neck is roughly 10 cm long, and two-thirds of a kelvin is enough to fill it. As the liquid rises, the bird’s centre of mass moves upward and forward, past the pivot, and the bird tips. In the tipped position the bottom of the neck comes clear of the liquid in the body, a bubble of warm vapour slips up the tube, the pressures equalise, the liquid drains back, the centre of mass returns, and the bird swings upright. Its beak, meanwhile, has been dipped in the water, rewetting the felt.

That is a complete cycle. Heat flows from the room into the body, raising the vapour pressure there; heat flows from the head into the evaporating water; the difference does work lifting liquid; and the tipping resets the state. The bird is a closed-cycle vapour engine with a cycle time of tens of seconds, and its design is forgiving because two-thirds of a kelvin is so little. Almost any room gives it several kelvins to spare.

Why the bird is filled with something volatile #

The choice of working fluid is not arbitrary, and the reason is the slope of a vapour-pressure curve. A boiling point is a pressure traced that slope to the Clausius–Clapeyron relation,

$$\frac{dp}{dT} = \frac{L\,p}{R\,T^2},$$

with $L$ the molar latent heat. The slope is proportional to the vapour pressure itself. For dichloromethane, with a latent heat of about 28 kilojoules per mole and a vapour pressure of 47 kilopascals, it comes to just under two kilopascals per kelvin, matching the curve the figure uses. For water at room temperature, whose vapour pressure is only about 2.6 kilopascals, the slope is about 0.16 kilopascals per kelvin, more than ten times smaller, and a water column of the same height would need a temperature difference of six or seven kelvins to lift it. A bird filled with water would work only in dry rooms, and stall in an ordinary one on a damp day.

So the right working fluid is one that is nearly boiling at room temperature, with a high vapour pressure and therefore a steep curve. Dichloromethane, boiling at 40 °C, is close to ideal: volatile enough to give a steep slope, not so volatile that its pressure would burst thin glass on a hot day. Ether, used in some early birds, boils at 35 °C and works similarly. The bird’s sensitivity to a fraction of a kelvin is borrowed entirely from the liquid’s closeness to its boiling point.

The same physics runs the heat pipe, which the pipe that carries heat as vapour followed: a sealed tube with a working fluid near its boiling point, in which the smallest temperature difference between ends drives vapour from the warm end to the cool one and carries heat far better than solid copper. A drinking bird is a heat pipe that has been made to rock. The vapour carries heat from body to head, the liquid’s weight is lifted against gravity by the difference in vapour pressure, and the bird’s geometry turns that lift into a tip that resets the cycle. Take away the pivot and fix the bird upright, and it becomes a small, slow heat pipe, carrying heat from the room into its wet head with the column standing steadily in the neck.

There is a pleasing symmetry in the two ends. The head is cooled by water evaporating at the wet-bulb temperature, and the bird’s own fluid condenses inside the head at nearly the same temperature. The body is warmed by the room, and the fluid inside it evaporates. The bird thus contains a vapour engine running between the same two temperatures as the evaporation outside it, the one inside the glass driven by the one outside.

Warm and dry, cold and damp #

The Carnot limit depends on both temperature and humidity, and the dependence is strong.

Warm air holds more water vapour, and its capacity rises steeply with temperature, so a given relative humidity leaves more room for evaporation in warm air than in cold. The wet-bulb depression at 32 °C and low humidity exceeds fifteen kelvins, and the Carnot efficiency exceeds 5 per cent. At 12 °C the same relative humidity gives a smaller depression. The efficiency is small everywhere, as it must be for an engine working across a few kelvins — but it is not zero, and the source of heat is free and inexhaustible as long as there is water and unsaturated air.

This is the same source that the cold reservoir overhead found in the clear night sky, which offers a radiative cold reservoir tens of kelvins below the air. The atmosphere is full of such small temperature differences, between air and water, between ground and sky, between sun and shade, and each could in principle drive an engine. The drinking bird is the only commercial product built on the wet-bulb one, and it is a toy.

A ceiling of milliwatts #

How much power can the bird deliver? The ceiling is the heat flowing into the head times the Carnot efficiency. The heat flow is set by how well the air delivers heat to the head, a heat-transfer coefficient times the area times the temperature difference. Both factors are proportional to the depression, so the ceiling goes as its square.

A few milliwatts is the most the bird could ever deliver, and the real bird delivers far less, most of its heat flow passing through the glass and the vapour without doing anything useful, and most of the work it does being spent in the friction of its pivot and the sloshing of its liquid. Measurements of instrumented birds have put their mechanical output at tens of microwatts, around one per cent of the Carnot ceiling. But the square law is the important lesson for any engine with a small temperature difference: halving the gap quarters the power. That is why such engines struggle commercially, from ocean thermal energy conversion, which runs on the twenty kelvins between warm surface water and cold deep water, to thermoelectric generators on a warm pipe. An engine with one number in it found the efficiency at maximum power, $1 - \sqrt{T_c/T_h}$, which for a small gap is about half of Carnot’s; the power at that efficiency still goes as the gap squared.

The square law also says that a fan helps the bird a great deal. Air movement raises the heat-transfer coefficient without changing the wet-bulb temperature, which, as the wet-bulb thermometer showed, is independent of the wind. So a fan doubles or triples the heat flow at the same efficiency, and the bird drinks faster. That observation is a clean experimental separation of the two factors: blowing on the bird changes its speed but not the gap it runs on.

The humidity that stops it #

The bird stops when the wet-bulb depression falls below the two-thirds of a kelvin needed to fill its neck.

In a room at 22 °C the threshold is about 95 per cent humidity. Anywhere drier, it drinks. That is why the bird seems tireless: almost every indoor climate is far from saturation. It also gives the bird a second career as an instrument. Its drinking rate rises with the wet-bulb depression, so a bird timed in a room is a crude hygrometer, and it stops abruptly as the air approaches saturation. Placed under a bell jar with its glass of water, it slows and stops as its own evaporation fills the jar with vapour — the clearest demonstration there is that it was never running on nothing.

Not perpetual, and not free #

The second law has been tested against the drinking bird more often than against almost any other toy, because it looks so much like a counterexample. Every test comes out the same way. The bird takes heat from the room, rejects a slightly smaller amount into the evaporating water at a lower temperature, and converts the difference into motion. The water evaporated carries away the entropy. When the water runs out, the head dries, warms to the room’s temperature, and the bird stops with its column at rest in the body. The heat that changes no temperature described latent heat as energy that disappears into a change of phase; in the bird, that disappearing energy is the cold reservoir, and the bird is an engine that runs on the room’s ability to dry things.

The accounting can be made per gram. A gram of water evaporated from the felt removes about 2.45 kilojoules from the head, and at a Carnot efficiency of 2.2 per cent the most work that heat can have paid for is about 54 joules. A glass holding a hundred grams of water therefore stands for at most five kilojoules of work over the days the bird spends drinking it, roughly the energy needed to lift a kilogram five hundred metres, and in practice a hundredth of that. The glass of water is the bird’s fuel tank, and its contents are measured not in joules of chemical energy but in grams of water the room’s air can still absorb. There have been proposals to scale it up, mostly in the mid-twentieth century, for pumping water in arid regions where dry air is abundant. They founder on the square law. A machine running on a few kelvins has to move enormous amounts of heat for a little work, which means enormous areas of wet surface and large flows of air, and the work it delivers is very small for the size of the apparatus. Large evaporative coolers, which use the same wet-bulb gap to cool air rather than to make work, are commercially successful precisely because they do not try to extract work from it.

What the picture cannot show #

The wet-bulb temperatures are computed from the psychrometric equation with the coefficient for a well-ventilated surface. A felt head in still air sits somewhat warmer than that, because radiation from the room adds heat that evaporation must also remove; the real depression may be a kelvin or two smaller in a still room, and all the figures would shift slightly. The heat-transfer coefficients and the head’s area are representative values, and real birds differ in size, felt and fluid. Some modern birds are filled with other liquids, whose vapour pressure curves differ from dichloromethane’s, and their threshold gaps differ correspondingly.

The power figure is a ceiling, not an estimate. The bird is not a reversible engine: heat flows through finite temperature differences in its glass and through its vapour, liquid sloshes irreversibly, and its pivot dissipates energy. The cycle is intermittent rather than steady, with the tipping event lasting a second or so in a cycle of tens of seconds. A full model of the bird is a dynamical problem involving the motion of the liquid, the rocking of the body and the timing of the tip, and it has been studied in detail only recently, with instrumented birds and high-speed cameras.

And the treatment ignores the water supply. The glass of water slowly cools as it supplies evaporation, the felt can dry between drinks if the cycle is slow, and dissolved salts or dust on the felt change how freely it wets. A bird drinking from salty water slows, because salt lowers the water’s vapour pressure and so reduces the wet-bulb depression, the same effect that the salt that dissolves in the air’s own water followed for salts that pull water out of the air.

Still open: how far small-gap engines can be pushed #

The bird is the visible tip of a serious question. Very small temperature differences are everywhere — between day and night, between air and soil, between wet and dry surfaces — and the energy in them is large in total. Engines that can harvest them efficiently at useful power would matter for sensors, remote instruments and arid-region water supply. Approaches being studied include materials that change shape with humidity, which bend and straighten as the air dries and moistens and so extract work from evaporation directly, and microbial spores that swell and shrink with humidity, which have been built into small engines and shown to deliver measurable power from an evaporating water surface. How close such devices can come to the wet-bulb Carnot limit, and whether any can do so at a power worth having, is being worked out.

The drinking bird is a heat engine between the room and its own wet head, which evaporation holds at the wet-bulb temperature, 6.5 K below a 22 °C room at 50 per cent humidity; that gap bounds its efficiency at 2.2 per cent and its power at a few milliwatts, falling as the gap’s square, and since dichloromethane’s vapour lifts 15 cm of liquid per kelvin, two-thirds of a kelvin tips it, so it drinks in any room drier than about 95 per cent. The habit worth carrying is to look for the reservoirs before deciding a machine runs on nothing, because a cold one can be something as ordinary as wet felt.