The air that breaks a siphon nothing else can

Two results about siphons now stand side by side. The siphon that does not break finds that a frictionless siphon’s crown pressure is fixed by the crown’s height above its outlet, so that a siphon which starts cannot break as it drains. The margin friction lends a siphon finds that friction only ever raises that pressure, and that a draining tank hands the raise back until, at the very end, the crown is at the frictionless constant again.

Neither describes how siphons in practice stop. A siphon left running overnight over the rim of a tank, well inside its height limit, is found in the morning to have stopped with water still in the tank, a bubble of air sitting in the crown and the column parted beneath it. The heights did not do that, and friction did not do it. The air did — air that was in the water all along, and that the crown’s low pressure made the water unable to keep.

What a litre of water carries to the crown #

Water that has stood open to the air holds the air’s gases in solution in proportion to their partial pressures, which is Henry’s law. At twenty degrees a litre holds 9.3 milligrams of oxygen and about fifteen of nitrogen, and the amounts are set by the pressure the water last came to equilibrium at — here, one atmosphere, less the part of it that is water vapour.

At a siphon’s crown the water is at a lower pressure, and a lower pressure can hold less gas. The water arrives carrying the amount it held at one atmosphere, so it is supersaturated at the crown: over-full, by the ratio of the partial pressure it was saturated at to the partial pressure a gas phase at the crown would have.

$$S = \frac{p_{\text{atm}} - p_v}{p_c - p_v}.$$

The excess — what the water holds beyond what the crown can hold — is the gas it could release, and released at the crown it expands to the crown’s pressure. At the reference siphon’s starting crown pressure of 51.5 kilopascals the supersaturation is 2.02, and each litre could release 20.6 millilitres of free gas. At the frictionless floor of 23.0 kilopascals the supersaturation is 4.79, and the figure is 76.7 millilitres a litre — nearly an eighth of the water’s own volume, at the crown’s own pressure.

Cold water is worse. Gases are more soluble cold, so water saturated at five degrees carries more of them to the crown and could release 87.4 millilitres a litre there; at thirty degrees the figure is 74.9. The difference is small against the effect of the crown’s pressure, which is the variable that matters.

An upper bound, and why it is one #

Those volumes are what would come out if the water stayed at the crown long enough to reach equilibrium. It does not, and the gap between the bound and what happens is a question the arithmetic cannot close.

Gas leaves a supersaturated liquid through a surface. A bubble that already exists grows by diffusion across its own surface at a rate set by its size and the supersaturation; a liquid with no bubble in it has to nucleate one, and the tension a clean liquid can bear shows how much a liquid with nothing to nucleate on resists doing that. Tap water is not that liquid. It carries microscopic gas nuclei in crevices on particles and on the hose wall, and a supersaturation of two to five is ample to grow them.

So some fraction of the excess comes out in the seconds the water spends near the crown, and that fraction is a mass-transfer rate that depends on the nuclei, the wall, the water’s history and the flow’s turbulence. Nothing here computes it. It is carried as a stated parameter, and the rest of the essay is about why the conclusion hardly depends on what value it has.

The clock friction set #

The two effects now meet. Friction lends a siphon’s crown a margin at the start of a drain and the draining tank takes it back, so the crown pressure falls through the drain, and the gas each litre can release rises with it — from 20.6 millilitres at the start to 76.4 at the end, most of the rise coming late, when the margin is being returned fastest in proportion.

The siphon’s crown is therefore at its most gas-laden exactly when it is at its lowest pressure, and both happen at the end of the drain. That coincidence is not an accident of this installation. Both follow from the same term, $\rho g(1 - \varphi)\Delta$, going to zero with the drop — one because the pressure falls with it, and the other because a lower pressure holds less gas.

The volume of gas released per minute does not grow the same way, because the flow is falling far faster than the gas per litre is rising. Early in the drain the hose passes nearly thirteen litres a minute carrying twenty millilitres each; late in it, under a litre a minute carrying seventy-five. The total gas coming out at the crown per minute is largest at the start. What changes at the end is not how much gas there is but what happens to it.

Where along the hose the water is over-full #

The supersaturation is a ratio of pressures and it follows the pressure along the hose. Near the inlet, below the tank’s surface, the water is above atmospheric pressure and is under-full: it could take gas in rather than give it out. On the rising leg the pressure falls and the water passes through saturation. At the crown it is most over-full. On the falling leg the pressure rises again and the over-fullness declines to exactly one at the outlet, where the jet leaves at atmospheric pressure.

The place in the hose where gas is likeliest to come out is also the highest place in the hose, which is the place a bubble released anywhere nearby rises to. A siphon’s crown is, in the plainest geometric sense, a gas trap placed at its point of greatest supersaturation. The only thing preventing it from filling is the flow.

A bubble in a falling leg #

Gas released at the crown enters the falling leg as bubbles, and a bubble in a vertical tube of water rises. Whether it goes up or down is a comparison of two speeds: the water’s, carrying it down, and its own, carrying it up relative to the water.

A small bubble rises at a speed set by its own size and the water’s viscosity. A bubble large enough to fill the bore — the kind that collects at a crown and then slips back down into the falling leg — does something remarkable. It rises at a speed that does not depend on its length at all, only on the tube:

$$U = 0.351\,\sqrt{gD}.$$

That coefficient is Dumitrescu’s, from 1943, and it comes from potential flow — from an ideal-flow solution for the round nose of the bubble, matched to a thin film of liquid falling down the wall past it. The theory that predicts no drag on a body at all predicts the rise of a long bubble to within a few per cent of measurement, because a bubble’s nose is a surface the liquid slips over freely, and that is the one kind of surface ideal flow describes well.

In a 12.5 millimetre hose the drift speed is 0.123 metres a second. At the start of the drain the flow is 1.75 metres a second, fourteen times faster, and any bubble released at the crown is carried down the falling leg and out of the outlet with the water. The gas the crown releases is being cleared as fast as it appears.

As the drain proceeds the flow slows, and after 30.9 hours, with 4.7 centimetres of level left in the tank and the hose passing 0.9 litres a minute, the flow falls below the drift speed. From then on a bubble released at the crown and slipping into the falling leg rises back up against the flow and returns to the crown. Gas that was being swept away now accumulates, and a pocket forms at the top of the hose.

Narrow hoses that cannot trap a bubble #

The drift speed depends on the bore, and below a certain bore it does something the formula does not show.

In a narrow enough tube the bubble’s own surface tension matters. The meniscus stretched across its nose resists being deformed, and when the Eötvös number — the weight of liquid across the bore against the surface tension holding its surface, $\rho g D^2/\sigma$ — falls below about 3.37, the bubble cannot rise at all. For water that is a bore of 5.0 millimetres. The same competition between weight and tension sets how large a drop can hang before it flattens, and here it sets whether a bubble can climb a tube.

The consequence is that a siphon in a hose narrower than five millimetres cannot gather gas at its crown by this route: a bubble in it moves with the water as a plug, at any flow speed whatever, and is carried out. A siphon in a wider hose gathers gas as soon as its flow is slower than its bore’s drift speed, and a wider hose’s drift speed is larger, so it starts gathering sooner in the drain for the same flow. The aquarium siphon in narrow tubing and the tank siphon in garden hose are, on this account, different devices.

Just above the threshold the formula is optimistic — surface tension slows a bubble well before it stops one — and the model carries Dumitrescu’s number down to 5.0 millimetres and then cuts it to zero. The cut is a threshold taken from measurement and drawn as one.

How long a pocket takes to part the column #

Once the sweep stops, the pocket at the crown grows by the gas each passing litre releases, and it breaks the siphon when it spans the bore over enough of the crown to part the column. Taking fifteen centimetres of crown as that length, the pocket is 18.4 millilitres.

If half a per cent of the equilibrium excess comes out, the pocket fills in 74 minutes. If two per cent, 13. If ten per cent, 2. The drain had 162 minutes still to run when the sweep stopped, so in every case the siphon breaks with water still in the tank and with the source still above the outlet.

This is the essay’s result, and it is robust in an unusual way. A factor of twenty in the one number the model cannot compute — how much of the dissolved gas actually comes out — moves the break by a little over an hour, inside a drain of a day and a half. When the siphon breaks is decided by the speed comparison, which is computed; only how soon after is decided by the chemistry, which is not. A result that survives twenty-fold ignorance of its most uncertain input is a result about the mechanism rather than about the number.

Water that warms on its way to the crown #

Every figure so far saturates the water and delivers it to the crown at one temperature. A tank is rarely that obliging. Water drawn from a cold main, or left standing through a cool night, comes to equilibrium with the air at the temperature it had then — and warms through the next day, in the tank, in the hose and in the sun.

Warming does to dissolved gas what the crown’s low pressure does. Gases are less soluble warm, so water saturated cold and then warmed carries more gas than its new temperature can hold, at atmospheric pressure, before it has climbed an inch. Saturated at five degrees, a litre holds 12.4 milligrams of oxygen; at twenty it can hold 9.3. Water saturated at five that has warmed to twenty is already a third over-full of oxygen at the surface of the tank.

The crown then multiplies it. The over-fullness the crown imposes is a ratio of pressures and the one warming imposes is a ratio of solubilities, and they act on the same dissolved gas, so they compound: water arriving at a 23-kilopascal crown after warming fifteen degrees on the way is over-full by the product of the two. That is why a siphon, like a water main, collects air on a warm afternoon that it did not collect on the cold morning it was set running — a daily cycle that belongs to Henry’s constants rather than to the hydraulics.

It is also why the ordinary advice to use water that has stood for a while is sound and slightly misapplied. Standing lets the larger bubbles rise out, and it lets the water reach the room’s temperature so that it will not be over-filled by warming later. It does not remove the dissolved air the crown will release, and nothing short of boiling or a vacuum does.

What a pocket does before it parts the column #

A pocket at the crown does not break the siphon the moment it forms. It grows through a stage that is visible in a clear hose and worth describing, because what happens in it is not what the phrase “air lock” suggests.

A small pocket sits at the top of the bore and the water runs beneath it through a passage narrowed by the pocket’s depth. That is a constriction, and a constriction in a hose is a loss: the flow slows a little, which — by the argument about friction’s margin — raises the crown pressure a little and makes the water slightly less over-full. The pocket’s first effect is mildly self-limiting.

The limit does not hold, because the flow it slows is already below the drift speed. A slower flow carries bubbles away even less well, and the pocket keeps growing. When its depth reaches the bore, the water in the falling leg is no longer joined to the water in the rising leg, and each drains back towards its own end: the falling leg empties through the outlet and the rising leg falls back to the level of the tank. What is left is a hose holding two separate columns with a gas space between them, and nothing about it will restart without being primed again — which is the one process the atmosphere is genuinely the mechanism of.

What the gas budget leaves out #

The released fraction is prescribed. How much gas comes out of a supersaturated liquid in a few seconds is a mass-transfer problem with nucleation in it, and it is not solved here. The conclusion above was built to survive that, and it does; the minutes do not.

Bubbles in the falling leg lighten it. Before the sweep stops, the released gas travels down the falling leg as bubbles, and a column of bubbly water is lighter than one of water. A lighter falling leg has less weight to drive the siphon, so the flow is a little slower than computed — which brings the sweep’s end a little earlier. Air lift, the pump that raises water by injecting air into a rising pipe, is the same effect working the other way.

The crown is treated as a vertical tube. A real crown is a bend or a horizontal run, and whether a pocket in a horizontal or gently sloping pipe is swept away depends on a critical velocity for which there are correlations rather than theory. The vertical drift speed is the cleanest available criterion and is used as one.

The released gas is taken to have air’s composition. Oxygen is roughly twice as soluble as nitrogen and comes out of solution enriched, so the pocket is richer in oxygen than air. That changes the gas and not its volume, to the accuracy anything here is claimed.

And gas can go back in. Bubbles carried down the falling leg reach higher pressures and begin to redissolve, and water in the rising leg below the tank’s surface is under-full and would take gas in. Neither is followed.

The two claims the figures rest on are computed a second way. Water saturated at the crown’s own pressure has exactly nothing to release and a supersaturation of exactly one; and the constants give 9.28 milligrams of oxygen a litre at twenty degrees, which is within the range the tables quote and is the check on the constants rather than on the arithmetic. The drift speed above the threshold is Dumitrescu’s number to machine precision, and zero below it.

Air valves, and an old habit #

Henry’s law is from 1803. Dumitrescu’s long-bubble result is from 1943, and Davies and Taylor’s, which reached nearly the same coefficient from a different argument, from 1950. None of it was developed for siphons.

What does exist for siphons and pipelines is a habit, and the habit is the result of this essay stated as practice. A long water main that rises over a hill has an air valve at the top — a float valve that lets gas out of the pipe and keeps water in — because engineers have known for as long as there have been pressure mains that air gathers at high points, reduces the bore, and in the worst case stops the flow. Siphon spillways on dams are designed with their air entrainment in mind for the same reason. A siphon draining a tank has neither, and it fails the way an unvented main would, at its high point, when its flow is slowest.

The habit came first and the arithmetic explains it: the high point is where the water is most over-full, where a released bubble rises to, and where a slow enough flow cannot stop it staying.

Still open: whether a crown can be kept from gathering #

Every siphon so far has been made of ordinary water, and the gas has been its undoing. Water that has been degassed carries nothing to release, and a crown full of it could sit at pressures the heights alone would forbid — below the vapour pressure, and, with nothing in it to boil on, below zero. Siphons made that way have been run over crowns taller than the barometric height, and the atmosphere was never what held them up. How far a degassed siphon’s crown can be taken before its own nuclei break it is the question this essay’s gas and the breaking strength’s flaws share.

Beside it is the other place in pumps and propellers where a falling pressure meets dissolved gas at speed rather than at a trickle: the suction side of a propeller, where the water spends milliseconds rather than seconds at its lowest pressure and where the distinction between gas coming out and the liquid itself tearing is the whole of the difference between a nuisance and a bubble that hammers.