The one place the atmosphere pushes

The rung below this one refutes the standard account of a siphon carefully, and then lists what it did not compute. One entry in that list is worth taking seriously:

No starting problem. Everything above concerns a siphon already running. Getting one started — filling the tube, or drawing the liquid over — is a separate question, and it is the one place where the atmosphere genuinely does push something.

That sentence is doing more work than it looks. It is not a caveat; it is the location of the entire truth in an account the essay has just spent three thousand words dismantling.

Two devices, sharing a tube #

The thing to notice is that a siphon before it starts and a siphon while it runs are not the same problem, and almost none of the vocabulary carries over.

Running. Liquid is moving, the flow rate is $\sqrt{2g\,\Delta z}$, the outlet is open, both ends are at atmospheric pressure, and the pressure everywhere in the tube is set by the height and the velocity head. That is the rung below.

Priming. Nothing is moving, the outlet is closed, and the pressure inside the tube is whatever a pump or a mouth or a squeezed bulb has made it. There is no flow rate, no Torricelli, no velocity head and no drop to speak of — so none of Bernoulli’s four forms is the one being used, because nothing is on a streamline. It is a hydrostatics problem, and hydrostatics is where the atmosphere is a mechanism.

The lift is one line of hydrostatics:

$$h = \frac{p_{\text{atm}} - p_{\text{inside}}}{\rho g}.$$

At thirty kilopascals inside, the liquid rises 7.29 metres. The atmosphere is doing that, in the straightforward sense that removing it would leave the liquid where it was, and there is no other candidate. It is worth being careful with the word even here: a fluid cannot pull, so what happens is that the atmosphere presses the free surface down and the liquid has nowhere to go but up the tube.

The ceiling, and why a better pump cannot get past it #

The lift expression has a floor under $p_{\text{inside}}$, and the floor is not zero.

Below the liquid’s own vapour pressure it boils. Pull harder than that and what fills the top of the tube is not a better vacuum but water vapour at 2.34 kilopascals, at which point the pressure inside has stopped falling and the column has stopped rising. So the greatest lift available to any pump is

$$h_{\max} = \frac{p_{\text{atm}} - p_v}{\rho g} = 10.11\ \text{metres},$$

which is the barometric height — the same number the running siphon’s ceiling is built on, arrived at from an entirely different direction.

That coincidence is not one. Both are the same statement about how much pressure difference a column of water can be asked to support — and it is the same floor a body moving through water runs into when its own suction peak reaches the vapour pressure, and it shows up once as a limit on how high a running siphon’s crown may be and once as a limit on how high a stopped one can be sucked. The number is a property of the liquid and the atmosphere, and it does not know which process is asking.

So a tall siphon is filled, not sucked #

The practical consequence is a rule of thumb every plumber knows and few state as a limit.

A siphon whose crown is under about ten metres above the source can be started by suction, if a good enough vacuum can be pulled. One above it cannot be started that way at all, and the alternative is to fill the tube — submerge it, or close both ends and fill it from a tap, then place the ends and open them. Filling puts liquid over the crown without ever asking the atmosphere to lift it there.

And filling is subject to no limit at all, which is worth being clear about because it looks like it should be. Once the tube is full and both ends are submerged or closed, opening the outlet starts the flow, and the running siphon’s own arithmetic takes over: the crown pressure is what the heights make it, and the column holds as long as it stays above the vapour pressure. A twelve-metre crown cannot be sucked over and it also cannot be run, but for two different reasons that happen to share a number.

There is a real device that exploits the difference. A siphon spillway on a reservoir has a hooded passage whose crest is above the water level, and it is primed not by suction but by the flow itself: water rising over the crest entrains air out of the passage until the passage runs full, at which point the siphon action starts and the discharge jumps. It is a self-priming arrangement whose priming mechanism is entrainment rather than pressure — the same entrainment a jet pump uses as its whole mechanism, and its failure mode — losing the seal and reverting to a weir — is exactly a loss of prime.

The bubble that is the real enemy #

There is a failure mode common to every siphon and it is a priming problem wearing running clothes.

Gas comes out of solution at the crown, because the crown is where the pressure is lowest. Over hours a bubble accumulates there, and as it grows it reduces the effective cross-section, and eventually it breaks the column. The siphon stops with liquid still on both sides and it will not restart, because restarting means re-priming and the crown may be higher than the atmosphere can lift.

Which is the interesting connection to the rung below. That essay’s whole argument is that the atmosphere holds the column together rather than driving it; here is the failure of exactly that job, happening slowly, in a device that was running perfectly. The gas that comes out of solution is the same gas whose absence lets degassed water be siphoned over fifteen metres, and it is what decides a cavitation threshold everywhere else in this collection, and what makes a pump’s suction margin a design constraint — so the property that makes tap water easy to prime is the property that makes it hard to keep primed.

Long siphons in service have a vent and a small vacuum pump at the crown for this reason, and running one is a maintenance task rather than a mechanism.

Why the mouth works and the pump often does not #

There is a domestic observation that sharpens the arithmetic, because it looks like it contradicts it.

A person can start a garden siphon by mouth, over a hump of half a metre, with no equipment. A hand vacuum pump on a laboratory bench often struggles to start a siphon over two metres. The mouth appears to beat the pump, which is absurd.

It does not, and the reason is that the two are being asked different questions. The mouth is not evacuating the tube at all — it is drawing liquid through it, using the tube already full of liquid behind the lips, so the process is a flow rather than a hydrostatic lift and the pressure differences involved are tiny. A human can generate perhaps five to eight kilopascals of suction, which is seventy or eighty centimetres of lift, and that is quite enough for a hump of half a metre.

The pump is being asked to hold a static column at height with a proper vacuum behind it, which needs a much larger pressure difference and a tube that does not leak. Leakage is usually the real answer: a tube that admits a trickle of air can never reach a low pressure, and a prime that fails at two metres usually fails at the joints rather than at the physics.

So the arithmetic here is a ceiling and not a prediction, which is the honest position. It says what cannot be done at any quality of apparatus. What is achieved with real apparatus is nearly always less and is limited by things this model has none of.

The reverse operation, and the same number again #

Every argument above runs the other way, and the mirror case is worth stating because it is the one most people have actually performed.

Stopping a siphon by lifting the outlet is the reverse of priming. Raise the outlet above the source and the drop reverses; the flow reverses with it and then stops when the levels equalise. Nothing about the barometric height enters, because the column never has to be supported anywhere higher than it already is.

Stopping it by lifting the inlet out of the liquid is different and is where the number comes back. Air enters, the column parts at the inlet end, and the liquid on the crown’s far side runs out while the liquid on the near side falls back — and how much falls back is decided by whether the crown is above the barometric height of the remaining column. That is why a siphon that has swallowed air sometimes drains its short leg completely and sometimes leaves it standing.

The general rule is the one this ladder keeps arriving at. Whenever the question is how high can a column of this liquid stand, the answer is the barometric height and it does not care what caused the column. Whenever the question is how fast does it flow, the answer is the drop between the ends and it does not care about the height at all. Almost every confusion about siphons is one of those questions being answered with the other’s number.

What the running siphon’s argument was actually refusing #

It is worth returning to the rung below with the priming calculation in hand, because the two together say something neither says alone.

That essay’s refutation is that the crown height is not in the flow rate. What it establishes is that the atmosphere is not a driver of a steady flow — and the reason, stated in its own terms, is that a support force does no net work on a steady flow: the liquid rises through the same height it later falls through, and the pressure field that held it up returns everything it took.

Priming is not a steady flow. The liquid goes up and does not come down; the process has a net displacement in the direction the atmosphere is pushing; and work is genuinely done. So the objection that dismisses the atmospheric account of the running device does not apply to the starting one at all, and it fails to apply for a reason that is stated in the very sentence that makes it.

That is a satisfying place for a refutation to end up. The popular account is not simply wrong — it is a correct description of one phase of the device’s operation, generalised to a phase where its own justification has evaporated. The same shape appears in every entry in this field: something real, given a job it does not do, in a situation adjacent to one where it does.

And it explains the account’s durability better than calling it a mistake does. Anybody who has started a siphon has felt the atmosphere do the work, because on that occasion it did.

A number worth having beside the ten metres #

The barometric height is quoted for water and the quantity it is really about is the liquid, so the range across liquids is worth a paragraph. It is wider than the water figure suggests and it changes what is possible.

$h_{\max} = (p_{\text{atm}} - p_v)/\rho g$ has the density on the bottom, so a dense liquid is lifted less far. Mercury is the extreme: thirteen and a half times denser than water and with a vapour pressure of essentially nothing, it stands at 0.76 metres — which is not a coincidence but the whole of why a barometer is that tall. A mercury siphon can have a crown three-quarters of a metre above its source and no more.

Petrol goes the other way and is worse than water rather than better, despite being lighter: its density is about 0.73 of water’s, which helps, but its vapour pressure at summer temperatures is fifty or sixty kilopascals, which hurts far more. The available margin is $p_{\text{atm}} - p_v$ rather than $p_{\text{atm}}$, so a hot fuel siphon has perhaps five or six metres rather than ten, and it is the vapour pressure rather than the density that took them.

And liquid at its boiling point has none at all. A liquid whose vapour pressure equals the ambient cannot be lifted a millimetre, in a tube or a pump, at any vacuum — which is the same statement as a centrifugal pump’s net positive suction head being zero, and it is why a boiling liquid is fed to a pump by gravity rather than drawn.

So the ten metres is water’s number and not the physics’, and reading it as a universal constant is a smaller version of the error this whole field is about.

What the picture cannot show #

No flow anywhere. Every figure is hydrostatic, which is the correct model for the priming problem and is exactly why the running siphon’s expressions do not appear. A reader looking for a flow rate here is looking in the wrong essay.

The prime is treated as instantaneous. How long a pump takes to evacuate a tube depends on its displacement and on the tube’s volume, and none of that is computed. What is computed is whether the final pressure is low enough.

No surface tension. In a narrow tube capillary rise adds a few millimetres to the lift, and in a very narrow one it dominates — which is a different subject entirely and a different regime.

And the vapour pressure is at one temperature. Water at twenty degrees has a vapour pressure of 2.34 kilopascals and at sixty degrees it is twenty, so the barometric height falls to about eight metres for warm water and to nothing at all at the boiling point. A siphon of hot liquid is a much harder device than a siphon of cold, and the temperature enters only through this one number.

The assertion behind the figures is the one that could reject: a perfect vacuum must give exactly the barometric height and not a millimetre more, an eleven-metre crown must be reported as unprimeable, and a tube pressurised above the atmosphere must be refused rather than returning a negative lift. A model that quietly reported a lift of minus two metres would be a model with no physics in it.

Who found it, and when #

The lift limit is Torricelli’s and Pascal’s, from the 1640s, and it arrived as a genuine surprise: suction pumps in Florentine mines would not raise water more than about ten metres and nobody knew why. Galileo had considered the possibility that the water column broke under its own weight; Torricelli’s answer — that the atmosphere has a weight, and the column stands as high as that weight supports — is one of the founding measurements of the subject.

The historical irony is exact. The atmospheric explanation of the siphon, which the rung below spends an essay refuting, is a straightforward generalisation of the atmospheric explanation of the pump, which is correct. Torricelli and his successors were right about the pump, right about the barometer, and then applied the same picture to a device where it does not hold — and the mistake has outlived them by four centuries because its one testable consequence, the ten-metre ceiling, is right for a reason that has nothing to do with driving anything.

Where the ladder goes next #

This rung has taken a siphon from stopped to running. The rung above takes it from running to stopped, and the arithmetic there produces a result that is the opposite of what the rung below’s coupling suggests.

A siphon draining a real reservoir has a falling source level, which shrinks the drop and grows the rise at the same time. The first slows the flow and the second brings the crown pressure down — so a reader would expect a draining siphon to break as it empties, and the previous rung’s finding that a metre of extra drop costs a metre of hump makes that expectation look well founded.

It is exactly wrong, and the reason is one line of cancellation. The essay above computes it.