A degassed siphon is as tall as its largest nucleus allows

The siphon that does not need the air took apart the textbook explanation of a siphon — that the atmosphere pushes the water up the rising leg — and found that the column is held together by the water’s own cohesion, with the atmosphere’s part being to keep the crown’s pressure above the point where that cohesion is tested. The air that breaks a siphon nothing else can found what ends an ordinary siphon long before the cohesion is tested: the dissolved air that comes out of solution at the low pressure of the crown, gathering there until it parts the column. It ended on the siphon that has no air to give up — water degassed so thoroughly that its crown can sit below the vapour pressure and below zero — and asked how far such a siphon’s crown can be taken before something breaks it.

This essay answers that. The answer is not the water’s strength, which where a liquid does pull found to be enormous — thousands of atmospheres in principle, hundreds in the cleanest measurements. It is the largest speck of gas the water still carries, and the calculation says how large a speck each crown height allows.

A running siphon’s crown #

The siphon is a hose of 10 mm bore rising from an upper reservoir over a crown a height $H$ above the reservoir’s surface, and falling to an outlet in a lower reservoir a depth $D$ below it. The flow speed follows from the energy equation: the height $D$ between the two surfaces is spent on the entrance loss, the friction along the whole hose and the kinetic energy thrown away at the outlet. The friction factor comes from Churchill’s correlation, which is continuous through the transition from laminar to turbulent flow — a friction factor that jumps at a Reynolds number of 2300 leaves the energy equation with no solution near the jump, as the first version of this calculation found. With the flow speed known, the crown’s absolute pressure is the atmosphere’s, less the height of water above the reservoir, less the dynamic pressure and the friction spent on the way up:

$$p_c = p_0 - \rho g H - \tfrac12\rho V^2\left(1 + K_\text{in} + f\frac{L_1}{d}\right).$$

For an ordinary siphon the crown pressure is positive and the formula is unremarkable. The question here is what happens when the formula says it is negative.

Water holds a tension until a nucleus lets go #

The first figure follows the crown’s pressure as the crown is raised. It falls by one metre of water for every metre of height — about 9.8 kilopascals — passes the vapour pressure near ten metres, where an ordinary siphon’s water would boil at the crown, and passes zero soon after. Below zero it is a tension, the water being pulled on by the columns on either side, and nothing in the flow’s arithmetic objects.

What objects is a nucleus. Degassed water is not free of gas: it still carries microscopic pockets held in crevices of the hose wall and on the surfaces of motes too small to settle, each a bubble whose surface tension keeps it from growing. A nucleus of radius $R_0$, in equilibrium at the reservoir’s pressure, survives any pressure down to a critical value — Blake’s threshold — below which the balance between its gas, its vapour and its surface tension has no equilibrium left, and it grows without limit. The levels in the figure are Blake’s thresholds for largest nuclei of 5, 2, 1 and 0.3 microns: −3, −16, −41 and −167 kilopascals. A siphon runs until its crown’s pressure line reaches the level of the largest nucleus in its water, and there the column breaks.

The threshold was checked independently of the formula that gives it: a direct search for the minimum of the nucleus’s equilibrium curve lands on Blake’s value to two parts in $10^{10}$.

The largest nucleus decides the height #

The second figure is the answer. For each size of largest nucleus it finds the crown height at which the crown’s pressure reaches that nucleus’s threshold, for three outlet depths. Nuclei of ten microns and more make almost no difference: their threshold is within a fraction of an atmosphere of the vapour pressure, and the siphon stops at the ordinary limit of about ten metres. Smaller nuclei hold much more. With an outlet 0.2 m down, a slow siphon whose largest nucleus is a micron runs over a crown of 14.4 m; two microns, 11.9 m; three-tenths of a micron, 27.3 m. The curve steepens because a nucleus’s breaking tension grows as the inverse of its radius once its surface tension dominates its gas: halve the largest nucleus and the extra height roughly doubles.

The water’s own strength never enters. Even the most pessimistic measurements of pure water’s tensile strength, tens of megapascals, correspond to crowns kilometres tall. Every limit in the figure is set by dirt, and that is the practical content of the result: the height a siphon of degassed water can reach is a measure of how clean its water and its hose are, and nothing about siphons.

A siphon of degassed water running over a crown near fifteen metres — the height reported for such experiments — implies, by this figure, water whose largest nucleus is about a micron. That is a demanding standard of cleanliness but not an exotic one: filtered, boiled and handled without contact with air, water reaches it.

A faster siphon cannot stand as tall #

The third figure adds what the flow does. A deeper outlet makes the siphon run faster, and faster water arrives at the crown with more of its pressure spent — on the dynamic pressure of its speed and on the friction of the hose it has climbed. With the outlet 5 cm down the water crawls at five to seven centimetres a second and the crown feels almost the static pressure; with it ten metres down the water runs at a metre and a half a second, and every limit falls together, by three to three and a half metres. The friction is the larger share for a long thin hose; the margin friction lends a siphon found the same trade from the other side, as a reserve an ordinary siphon has against its crown’s pressure.

So the tallest siphons are slow ones. A 10 mm hose running at five centimetres a second carries about four millilitres a second — a litre in four minutes — and the tallest siphon a given water allows is correspondingly a trickle. Every litre per minute more that is asked of it is paid for in centimetres of crown, and the trade is steepest for the cleanest water, whose limit is furthest above the ordinary ten metres and whose extra height is all tension the flow’s speed eats into. A demonstration of a tall crown is best run with the outlet barely below the upper reservoir, where the flow is a trickle and the crown’s pressure is set almost entirely by its height.

A siphon breaks the water gently #

Blake’s threshold is a statement about equilibrium, and a threshold that is also a duration found that a cavitation threshold quoted as one number is really the long-exposure end of a curve: a tension applied for a short pulse must be much larger to break the water, because a nucleus needs time to grow. The question for a siphon is which end of that curve it lives on.

A nucleus rising towards the crown feels its pressure fall steadily as it climbs, and crosses the region below its threshold in about a second at the flow speeds here. The fourth figure integrates the Rayleigh–Plesset equation for a one-micron nucleus whose surrounding pressure falls over a stated time and then stays down. When the fall takes hundreds of the nucleus’s growth times — a growth time is a fifth of a microsecond — the nucleus follows its equilibrium all the way and breaks at Blake’s static tension, to two parts in ten thousand. Only when the fall is nearly sudden does anything change, and then in the other direction: the nucleus is flung past its new equilibrium by its own inertia and breaks at 97.6 per cent of the static tension.

A siphon is at the gentle end by a factor of millions. Its crown’s limit is the static threshold, which is why the calculation can use it; and the water hammer of a valve slammed on a tall siphon, which drops the pressure suddenly, would break the column at a crown a few centimetres lower than the steady limit.

A siphon is a nucleus meter #

The first figure can be read backwards, and read that way it is an instrument. Raise a siphon of a given water slowly until it breaks, and the crown height at which it breaks gives the radius of the largest nucleus the water carried: fourteen metres, a micron; twelve, two microns; ten and a half, five. It is the same inference a breaking strength that is the size of a flaw made about water’s measured tensile strengths, where every measurement was really a measurement of the largest flaw in the sample; a siphon makes it at the gentle end, with tensions of an atmosphere rather than hundreds, and for the largest nuclei rather than the smallest.

Water tunnels need exactly this number. Whether a propeller or a hydrofoil cavitates in a test depends on the nuclei in the tunnel’s water as much as on the flow, and a body tears the water at a cavitation number that shifts with them; tunnels are fitted with cavitation susceptibility meters, small venturis whose throat pressure is lowered until the water breaks, which are siphons’ crowns in miniature. The arithmetic that turns a breaking pressure into a nucleus size is the Blake threshold used here.

Trees are siphons that exclude everything #

The tallest trees raise water more than a hundred metres, and they do it with a siphon’s physics rather than a pump’s: evaporation from the leaves pulls on continuous columns of sap in the xylem, and the sap near the top of a tall tree is under a tension of one to two megapascals — ten to twenty atmospheres below zero. By the threshold used here, a column under two megapascals of tension survives only if its largest nucleus is smaller than about 30 nanometres, thirty times smaller than the siphon of the fourth section needed. A nucleus of a tenth of a micron would break it at 0.6 megapascals.

That is the design problem a tree’s plumbing solves. The conduits are sealed from each other and from the air by pit membranes whose pores are tens of nanometres across, which let sap through and stop any bubble larger than the pore from being pulled in — and the tension at which air is pulled through the largest pore, which is the same surface-tension arithmetic, is what sets how dry a tree can get before its columns break. A siphon of degassed water over fifteen metres is a demonstration of what a redwood does every day with water it has filtered through a membrane.

What “about ten metres” was #

The ordinary limit has a precise form in this calculation, and it is worth stating because it is where the textbook’s number comes from. With nuclei large enough that their threshold is the vapour pressure, and a flow slow enough that nothing is spent on speed, the tallest crown is the atmosphere’s pressure less the vapour pressure, divided by the water’s weight per unit volume: 10.11 metres at 20 °C, which the calculation reproduces to five parts in ten thousand. It is not a limit on how high a siphon can lift water; it is the height at which ordinary water, full of nuclei, boils at the crown. The one place the atmosphere pushes is the reservoir’s surface, and ten metres is how far that push can hold a column whose water cannot be pulled.

What was checked #

The fifth figure is the ledger. Blake’s breaking pressure matches a direct search of the equilibrium curve to two parts in $10^{10}$ for three nuclei. The tallest crown satisfies both the energy equation for the flow and the breaking condition at the crown to machine precision in four cases. With large nuclei and a slow flow it reproduces the vapour-pressure limit, 10.11 m, to five parts in ten thousand. And a one-micron nucleus whose pressure falls over a ten-thousandth of a second breaks at Blake’s tension to two parts in ten thousand, while a nucleus crossing a real siphon’s crown takes 0.7 s to do it.

What the calculation leaves out #

Where the nuclei are. A nucleus in the bulk and a nucleus in a crevice of the hose wall do not have the same threshold; a crevice can hold gas stably at a threshold that depends on its angle and its size. The calculation treats every nucleus as a free spherical bubble, which is the least stable kind.

Gas that is not quite gone. Degassing is never complete. The small amount of gas left in solution diffuses into any nucleus held under tension, and a nucleus that grows by diffusion over the minutes a siphon runs lowers its own threshold. The air that breaks a siphon is the same mechanism at full strength.

Temperature. The vapour pressure and the surface tension both depend on it; warm water loses a metre of every limit.

The hose itself. A crown below zero absolute pressure has the whole atmosphere, and more, pressing it inwards. A garden hose collapses long before its water breaks, which is why demonstrations of tall degassed siphons use rigid tubing; the calculation assumes a bore that does not change.

The wall’s chemistry. A patch of wall that water does not wet holds gas in its crevices more stubbornly than a wetted one, and behaves as a larger nucleus than its size suggests. Clean glass is the best case; plastic tubing, whose surface is partly hydrophobic, is worse, and the largest nucleus may be the wall’s rather than the water’s.

The rest of the hose. The crown is the lowest pressure but not the only low one; a hose that rises and falls more than once has several crowns, and the tallest decides.

The convention the numbers depend on #

Crown heights are above the upper reservoir’s surface; outlet depths below it, to the lower reservoir’s surface. Pressures are absolute, the atmosphere at 101.3 kPa, water at 20 °C with a vapour pressure of 2.34 kPa and a surface tension of 0.0725 N/m. A nucleus’s radius is its equilibrium radius at the reservoir’s pressure, its gas isothermal. Growth time is Blake’s critical radius over the square root of the tension over the density.

Who found it, and when #

Blake gave the threshold for a gas nucleus in 1949. The tensile strength of water has been measured since Berthelot’s sealed tubes of 1850; Briggs’s spinning tubes of 1950 and the inclusion measurements of the 1990s gave values from tens to over a hundred megapascals, far beyond any siphon. A siphon of degassed water over a crown taller than the barometric height has been demonstrated more than once; Boatwright, Hughes and Barry reported one near fifteen metres in 2015, run in the open air with water degassed in advance.

Still open: the nucleus that grows while it waits #

The calculation takes the largest nucleus as given and fixed. A real one is not: held under tension at the crown, it gathers whatever gas remains dissolved in the water around it, and its threshold falls as it grows. The next calculation lets a nucleus sitting at the crown grow by diffusion from water holding a stated residual gas content, and asks how long a siphon running at a given height has before its largest nucleus has grown to the size that breaks it — whether a tall degassed siphon has a height limit at all, or only a lifetime.