The height a siphon cannot pass

A tube filled with water, one end in a full bucket and the other in an empty one below it, moves the water from the first to the second and keeps doing it until the first is empty. It works over a hill in the tube, and it does not work over a hill much more than ten metres high. Both facts have been known for two thousand years, and the explanation usually given for the second is offered as the explanation for the first, which cannot be right.

The ceiling is real and the arithmetic behind it is straightforward. What is not straightforward is the inference usually drawn from it, which is that the atmosphere is what makes the siphon go.

The pressure profile, which knows only height #

The pressure in a still fluid depends only on depth, and inside a siphon the fluid is not still — but it is moving slowly, and the pressure at any point is very nearly the hydrostatic value.

Follow the tube. At the upper surface the pressure is atmospheric — the same hydrostatic statement that decides how high water will climb a narrow tube, with the sign the other way up. Climbing to the crown, it falls by $\rho g h$. Descending the far side to the outlet, it rises again by $\rho g$ times the drop. That is the whole profile, and the only quantity in it is height above the upper reservoir’s surface.

The liquid flows because the two ends are at different heights, and the drive is the difference between the two column weights. Nothing in that statement mentions air. What the air does is set the baseline of the pressure profile, and therefore how far the profile may drop before it hits something.

It is worth being explicit about what drives the flow, because the pressure profile does not by itself say. Take the two surfaces — the one in the upper reservoir and the one at the outlet — and note that both are at atmospheric pressure, since both are open to the air. Between them the liquid forms a continuous column, and the two arms of that column have different weights: the long arm hanging below the crown outweighs the short one. The unbalanced weight is what accelerates the liquid, and it is proportional to the height difference between the two surfaces, not to the height of the crown. That is the drive, and it is why a siphon with its outlet level with its source does nothing at all however deep the reservoir or however high the crown.

The role the crown plays is different in kind. It is where the column is most nearly pulled apart, and the whole question of whether the siphon can run at all is the question of whether the column survives there.

What it hits #

A liquid held below its own vapour pressure boils. That is what a boiling point is — a pressure, not a temperature — and the crown of a siphon is the lowest-pressure point in the system.

At the crown of a tall siphon the pressure is a few kilopascals, which is where the vapour-pressure curve becomes the binding constraint. Water boils at 20 °C below about 2.3 kPa — so a siphon does not fail because the column breaks under tension but because the liquid at the top starts to boil, and the height at which that happens depends on the temperature rather than on anything about the tube.

Setting the crown pressure equal to the vapour pressure gives the ceiling:

$$h_{\max} = \frac{p_{\text{ambient}} - p_{\text{vapour}}}{\rho g},$$

which for water at 20 °C is 10.11 m and for mercury is 0.76 m — the height of a barometer, which is the same calculation with the flow left out.

Two features of that expression are worth pointing at. The vapour pressure appears with a minus sign, so a more volatile liquid siphons less far — and it appears as a pressure, so what matters is how it compares with the ambient one rather than how volatile the liquid is in absolute terms. And the density appears in the denominator, so a dense liquid siphons less far even though it is no more prone to boiling. Ethanol beats water on both counts at once — a fifth less dense but three times more volatile — and the two nearly cancel, which is a good example of a comparison that cannot be made by naming one property.

The experiment that separates the two claims #

If the atmosphere were what drives the siphon, a siphon in a vacuum would not run. If the atmosphere merely sets a ceiling, a siphon in a vacuum should run perfectly well as long as the liquid can be held together.

It runs. The experiment needs water that has been degassed — boiled under reduced pressure to drive out dissolved air — in a tube that has been cleaned so that no crevice on its wall can hold a bubble, and under those conditions a siphon works in a vacuum chamber, and it works over crowns well above the ten-metre figure. The column is then held together not by atmospheric pressure but by the liquid’s own cohesion, and the pressure at the crown is negative: the water is in tension.

Whether a liquid under tension survives is decided by a curvature. A bubble smaller than a critical radius collapses under its own surface tension and one larger than it grows without limit — so a degassed liquid in a clean tube can be pulled far below zero pressure and an ordinary one cannot, because ordinary water is full of nucleation sites. That is the experiment which separates the two explanations: the barometric limit is about pressure and the tensile limit is about nucleation.

The strength of that pull is remarkable. Water in inclusions in quartz has been held at tensions of tens of megapascals — hundreds of atmospheres — and the theoretical limit from the intermolecular forces is higher again. Against those numbers the single atmosphere the air provides is small, and the ten-metre ceiling is a statement about nucleation rather than about the water.

In a fine enough tube the meniscus can hold a column far taller than the atmosphere could push, and capillary rise going as the inverse radius is what makes trees possible. That is worth setting against the siphon: both are columns of water held up, one by the atmosphere and one by surface tension, and only the second has no ten-metre limit.

The distinction between what a liquid can sustain and what a real column does sustain is the same distinction that separates the strength of a perfect crystal from the strength of a piece of metal. In both cases the ideal number is enormous, the observed number is far smaller, and the gap is entirely accounted for by defects that let failure begin somewhere rather than everywhere at once. Removing the defects moves the observed number toward the ideal one, which is what degassing does for water and what a whisker of pure silicon does for a crystal.

A tall tree is therefore not a puzzle, and it is often presented as one. Water in the xylem is under tension of about a megapascal at the top of a hundred-metre tree, which is well within what water can sustain when it is held in vessels a few tens of micrometres across with no nucleation sites — and the tension is generated by evaporation from the leaves pulling on menisci in the cell walls. The mechanism is capillarity and cohesion, not pumping, and the ten-metre figure never applies because nothing in the tree is relying on atmospheric pressure.

The dip that has no ceiling #

The ceiling belongs entirely to the crown, and the cleanest way to see that is a device that has no crown at all.

Turn the siphon upside down: instead of a tube rising over a hill between two reservoirs, run one that dips into a valley. The liquid enters at one end, descends, crosses the bottom, and climbs to the other end, which is a little lower than the first. It flows for the same reason — the two surfaces are at different heights — and the pressure profile is the same hydrostatic one with the sign of the excursion reversed: every point in the tube is above atmospheric pressure rather than below it, by $\rho g$ times its depth below the upper surface.

Which means there is no ceiling. There is nowhere the pressure can approach the vapour pressure, nothing can boil, and the depth of the dip is limited only by the strength of the pipe. That is an inverted siphon, and the Romans built them: where an aqueduct met a valley too deep to bridge, the channel was taken into a closed pipe, down one side, across the floor, and up the other, under pressures of many atmospheres in lead or stone.

The contrast makes the argument in this essay unmistakable. Two devices, driven by the same thing — the weight difference between two arms of a continuous column — differing only in whether the excursion is up or down. One is limited to ten metres and the other is not limited at all. If the atmosphere were the agent, both would be limited; if the crown’s pressure is the constraint, only the one with a crown is, and only the one with a crown is.

It also explains why the practical engineering answer to a hill is usually to pump rather than to siphon, and to a valley is usually to run a pressure pipe. The first has a hard ten-metre bound; the second has a bound set by materials, which can be pushed.

What the disagreement was actually about #

It is worth recording that this was a live argument recently rather than an ancient settled one. In 2010 a dictionary’s definition of “siphon” — which attributed the effect to atmospheric pressure — was pointed out as wrong in a letter to a newspaper, and the correction ran through the physics teaching literature for several years afterwards. Both sides of that argument were partly right, which is why it lasted.

The atmospheric account is right that a siphon at ordinary pressures, with ordinary water, will not run over a crown above about ten metres, and right that the number is set by the atmosphere. It is wrong that the atmosphere is pushing the liquid up the short arm in any sense that makes the atmosphere necessary — the vacuum experiment settles that. The cohesion account is right that the column is held together by the liquid’s own tensile strength and that this is what carries it over the crown, and it is wrong if it suggests that the ten-metre figure is unimportant, since for untreated water in an ordinary tube it is exactly where the thing fails.

What the argument really needed was the distinction between a mechanism and a bound, and that distinction is not visible in any single experiment done at atmospheric pressure — which is why it took a vacuum chamber to settle it.

How the tension was measured #

The claim that water sustains tens of megapascals of tension is a large one and it has been measured several ways, each ingenious for the same reason: there is no way to attach a pull to a liquid.

The neatest is a spinning tube. Take a tube bent into a Z, fill it with degassed water, seal it, and spin it about its centre. The liquid in the two arms is flung outward, and since it cannot leave, the water at the centre is stretched — the tension there is set by the rotation rate and the arms’ length, and it can be raised smoothly until the column breaks. The moment it breaks is unmistakable, because the water snaps apart with an audible click and a visible cavity.

Done carefully in 1950, that experiment held water at about 28 megapascals of tension before it broke — some 280 atmospheres, and roughly thirty times what the whole atmosphere could ever have supplied. It has a strange temperature dependence: the tension the water withstands rises as it is cooled from room temperature, peaks around 10 °C, and falls again below that, which is one of the several anomalies traceable to water’s hydrogen-bonded structure.

The other route measures tension that nature has already applied. Water trapped in a cavity inside a quartz crystal, cooled after the crystal formed, contracts less than the quartz around it and ends up stretched. Heating such an inclusion until the vapour bubble in it redissolves, then cooling it and watching the bubble reappear, gives a reading of the tension it was under — and inclusions have been found at over a hundred megapascals.

Both experiments make the same point in different ways. The number that limits a real column is set by whatever nucleation site is available, and in a system with none — a spun tube of degassed water, a sealed cavity in a crystal — the number is a property of the liquid and is enormous. Ten metres is a statement about ordinary plumbing.

Torricelli’s column, which is the same figure #

The barometer is the siphon’s arithmetic with the flow removed, and it is worth putting the two side by side, because the barometer is where the argument this essay is about was first had.

Fill a metre-long tube with mercury, invert it in a dish, and the mercury falls until it stands 760 millimetres above the dish’s surface, with a space above it. That height is exactly the ceiling formula of this essay evaluated for mercury: atmospheric pressure over $\rho g$, with mercury’s tiny vapour pressure making no difference.

Torricelli’s account in 1643 — that the column is held up by the weight of the air, and that the space above it is a vacuum — was not accepted. The rival account had the space filled with some rarefied substance and the column held by nature’s abhorrence of a void, which explains the same observation.

The experiment that separated them is Pascal’s: carry a barometer up a mountain and see whether the column falls. If the air’s weight holds it, less air above means a shorter column; if a void is being abhorred, altitude should make no difference. Done on the Puy-de-Dôme in 1648, the column fell by about three inches over a thousand metres of climb.

That is exactly the manoeuvre this essay’s vacuum siphon performs, three hundred and fifty years earlier and on the other side of the question. In both cases two accounts fit the observations at one pressure, and in both cases the way to separate them is to change the pressure and see which prediction moves.

Where the model stops #

The flow is not computed. Everything here is the pressure profile of a nearly static column. How fast a siphon actually runs depends on the tube’s bore, its length and its roughness, and computing that is a question about flow, which belongs to the collection that owns it. The one thing the hydrostatic argument does say about the rate is that it is set by the height difference between the two surfaces and not by the crown height, which is why a siphon over a high hill runs at the same speed as one over a low hill with the same drop.

A hydraulic press is the contrast. There a pressure is transmitted through a fluid to multiply a force, and the fluid is everywhere in compression — so nothing about the ten-metre limit applies. The siphon is the case where the fluid is asked to go up before it comes down, and what limits it is that the atmosphere can only push.

The liquid is assumed continuous. Once a bubble does nucleate at the crown the analysis fails completely — the column separates, the two arms drain independently, and the siphon stops. That failure is abrupt rather than gradual, which is why the ten-metre figure behaves like a hard limit even though the physics behind it is a metastability.

The ambient pressure is assumed the same at both ends. For a siphon a few metres tall that is true to within a fraction of a per cent, since the air’s own pressure falls by about 12 pascals per metre against the water’s 9800. For a siphon over a hill tens of metres high — which needs a liquid that can take tension, or a shorter crown than the water case allows — the difference in atmospheric pressure between the two reservoirs enters, and it works against the siphon when the outlet is lower, because the air there pushes back harder.

Dissolved gas is ignored. Ordinary tap water carries dissolved air, and at low pressure that air comes out of solution long before the water boils — so a real siphon with untreated water fails somewhat below the vapour-pressure ceiling, and the failure is bubbles of air rather than of steam.

The whole of the atmosphere weighs the same as ten metres of water, which is what makes the two numbers in this essay coincide. That is not a coincidence to be explained but a definition to be noticed: the barometric height is the atmospheric pressure expressed in metres of the working fluid, so the same siphon runs to 13.6 times shorter a height in mercury and would run further in something lighter.

One more limit belongs here because it is where the whole hydrostatic treatment stops being about a siphon at all. Everything above assumes a tube of a bore large enough for surface tension to be irrelevant — millimetres or more. Below a fraction of a millimetre the meniscus at each end exerts its own pressure jump, the capillary rise in the tube becomes comparable with the heights being discussed, and the device stops being a siphon and becomes a wick. That crossover is at the capillary length, 2.7 mm for water, and it is the same length that separates a drop from a puddle.

What the pictures cannot show #

The siphon figures draw a pressure against a coordinate along the tube, and the path from surface to crown to outlet has been drawn as though height varies linearly with distance along it. It does not, in any real siphon, and it does not need to: the pressure at every point is $\rho g$ times the height whatever the shape of the tube, so the curve’s horizontal axis is a bookkeeping device and the vertical axis is the physics.

Nothing here shows the liquid moving, and the whole subject is a device for moving liquid. The pressures drawn are the pressures of a column that is running slowly enough for the dynamic contribution to be negligible, which is the ordinary case, and a fast siphon has a lower pressure at its crown than the figure shows — so the ceiling in practice is a little lower than the static calculation gives.

Where the ladder goes next #

The hydrostatics ladder began with pressure depending only on depth, continued with what that lets a press do, and with the shape a spin gives a free surface. This rung takes the pressure below zero. The rungs above it are the metastability of a stretched liquid treated properly, with the nucleation rate as a function of tension; the cavitation that follows, which is destructive enough to erode a bronze propeller; and negative pressures in confined geometries, where a liquid in a pore of a few nanometres behaves differently again.

The habit worth carrying away is about explanations that fit too well. A quantity that sets the limit of a mechanism is not thereby the mechanism, and the two are constantly confused because the same number appears in both stories. The way to tell them apart is to remove the candidate and see whether the thing still works — and a siphon in a vacuum still works.