The air a cavity releases is capped by the cavity

Air at a valve softens the hammer only in quantity put a pocket of free gas at the closed valve of a water main and found that its size decides everything. A millionth of the pipe’s volume trims the tallest, briefest pulses. A hundred-thousandth takes a third off them. A ten-thousandth can make the pulse taller than the vapour cavity’s, because it delays the collapse onto a different part of the returning wave. Only past half a thousandth of the pipe does the largest head fall below the Joukowsky rise, and that much gas is a deliberate air vessel, not an accident.

That essay fixed the gas and left open where it comes from. Water drawn from an open reservoir carries air dissolved in it, saturated at atmospheric pressure. When the pressure behind a shut valve falls far enough, the air starts to come out — at a rate set by how much surface it has to leave through — and when the pressure rises again it goes back into solution, more slowly. The question was how much gas a single cavity phase releases, whether that reaches the amounts that matter, and whether the gas left after the first collapse lowers the next pulse or, by the timing effect, raises it.

A pocket that feeds itself #

The line is the one both earlier essays used: 600 metres of 100-millimetre steel pipe holding 4.7 cubic metres of water, a wave speed of 1,200 metres a second, half a metre a second shut off instantly at the valve, no friction. A Joukowsky rise is 61.2 metres and a round trip of the wave one second. The steady head is 25 metres above the atmosphere, a margin of 0.574 rises down to the vapour pressure.

At the valve sits a pocket of gas, obeying Boyle’s law as before and entering the line’s delay equation in the same way, but now its gas content can change. Henry’s law sets how much air the water holds: its dissolved content is proportional to the air’s partial pressure, so water saturated at atmospheric pressure and then exposed to a lower partial pressure p has an excess it can give up, in proportion to the drop. Measured as the gas’s pressure times its volume, a volume of water V gives up $\sigma V(p_s - p)$, where $p_s$ is the saturation pressure and σ, about 0.019 for air in water at room temperature, is how much air water holds. Nothing comes out while the head at the valve stays above atmospheric pressure — that is, until it has fallen 25 metres — and after that the gas approaches its equilibrium at a rate set by a release time τ. When the pressure rises again the gas goes back, ten times more slowly by default, since dissolving needs the gas to diffuse away from the bubble rather than into it. A crown dissolves its nuclei or breaks on them followed that diffusion bubble by bubble; here it is lumped into one rate.

A pressure low enough to cavitate the water is not needed. The dissolved air starts to leave long before the water itself tears at the vapour pressure, and far before it could sustain the tension that a liquid does pull in a clean vessel; what the air needs is only to be below the pressure it was saturated at, and for long enough. As with a cavitation threshold that is also a duration, the question is how far and for how long together, which is why a rate appears.

Two numbers are unknown, and the essay treats them as the variables. One is τ, which for air coming out of still water is seconds to tens of seconds and for turbulent water with many nuclei much less. The other is φ, how much of the pipe’s water exchanges with the pocket at the valve, since air coming out a hundred metres up the pipe forms bubbles there and not in the pocket. The pocket starts from a nucleus of a hundred-millionth of the pipe, which on its own is so small that the cavity behaves as vapour.

Three histories #

The first figure is the head at the valve for three cases. With no release the history is the column-separation essay’s: the rise, the drop to vapour pressure, the cavity held there for two round trips, and a first pulse of 1.30 rises when it collapses.

With all the water giving up its air over ten round trips, the pocket gathers gas all through the cavity phase and still holds most of it at the collapse. The gas cushions the collapse as the fixed pocket of the same size did, and the first pulse falls to 1.16.

With the air coming out in three-hundredths of a round trip, the history looks nothing like a cushioned one. As soon as the head falls below atmospheric pressure the air comes out to hold it there. The cavity is no longer at the vapour pressure but at the saturation pressure, 0.17 rises above it; it lasts a round trip longer, and the pulse that follows is 1.46 rises — taller than with no gas at all. Nothing here is a numerical accident. The same pulse comes out exactly from the column-separation theory, and the section on the margin below says why.

The ceiling on the gas #

The second figure follows the gas itself. The fast release rises to about 1.4 ten-thousandths of the pipe’s volume and falls back when the cavity closes; the slow release reaches 1.2 ten-thousandths and keeps most of it. The dashed line explains why neither goes higher: it is the cavity’s own volume, filled with air at the saturation pressure.

The ceiling follows from Henry’s law in one line. Air comes out only while its partial pressure in the pocket, the gas content over the volume, is below $p_s$. Once the pocket holds as much air as would fill it at $p_s$, the water around it is in equilibrium and nothing more comes out. So the free gas can never exceed $p_s$ times the cavity’s volume, whatever σ is, whatever φ is and however short τ is. The fast release rides along that ceiling for the whole cavity phase; the slow one lags below it.

This turns a question about chemistry into one about the hammer. The cavity’s size is set by the water’s momentum and the margin, the things twice the margin, on top of the hammer computed; on this main it grows to about half a round trip’s delivery, four or five ten-thousandths of the pipe’s volume. Measured at the steady head’s pressure, 3.5 times the saturation pressure, the air that can fill it is a fraction $p_s/\varepsilon$ of that — a seven-thousandth of the pipe, 0.67 litres.

However much water, the cavity caps it #

The third figure sweeps the exchanging water across six decades. When little water takes part, the gas is simply what that water holds: $\sigma\phi p_s/\varepsilon$, the faint dashed line, reached if the release is fast enough. When more water takes part, every curve bends over onto the same ceiling, 1.42 ten-thousandths of the pipe, and a hundred times the pipe’s own water would give no more than the pipe’s own water does.

The previous essay’s thresholds are drawn across the figure. A hundred-thousandth trims the tall pulses, and released air easily gets there once a hundredth of the pipe’s water takes part. A ten-thousandth re-times the collapse, and released air can just about reach that. Half a thousandth removes the hammer, and released air cannot reach it on this main at all. Dissolved air can never do what an air vessel does, because the space it would need is space the cavity does not open. A line protected by air needs the air put there on purpose, which is why air vessels and air-admission valves exist. An air vessel is the pocket made large deliberately, the gas-filled cousin of the open surge tank that turns a hammer into a swing, and the same air chamber smooths the delivery of the hydraulic ram; what all three share is that their gas, or their free surface, is there before the transient begins, and is not asked to arrive during it.

Slow air cushions, fast air arms #

The fourth figure asks the question that matters for a real line, where τ is uncertain by a factor of a hundred: what does the release time do to the first pulse? At long release times nothing comes out in time and the pulse is the vapour cavity’s, 1.30 rises. At release times of a few to tens of round trips — seconds on this main, the range measured for air coming out of still water — the gas released during the cavity phase cushions the collapse, and the pulse falls, to 1.06 for a tenth of the pipe’s water at τ = 10. At release times shorter than a round trip the pulse rises instead, towards the air cavity’s 1.45; and for the smallest exchanging volume it overshoots to 1.69 at τ = 0.01, because a small amount of gas released fast re-times the collapse in the way the fixed ten-thousandth pocket did.

The same two regimes appear in a single bubble. The bubble that hammers collapses with nothing inside to resist, and a trace of gas inside it stops the collapse short and makes it rebound. Gas that diffuses into a bubble while it grows and has not diffused out by the time it collapses is the cushion; gas that has gone back into the liquid is not there to help. The pipe’s cavity is the bubble drawn out along the line, and the release and return times are the bubble’s diffusion, lumped.

So released air does two opposite things, and which one it does depends on a rate. Released slowly, it acts as the previous essay’s pocket, whose effect grows with its size. Released fast, it stops acting as a pocket and starts acting as a change in the vapour pressure — and what that does is written down exactly by the theory of column separation.

The margin, measured to the atmosphere #

When the air comes out as fast as the pressure falls and goes back as fast as it rises, the pocket is always in equilibrium. Its pressure can then never fall below $p_s$ while there is a cavity, because any lower pressure would bring more air out at once, and above $p_s$ it holds no gas at all. That is a vapour cavity with a vapour pressure of $p_s$. Its first pulse is the column-separation essay’s exact result with one change: the margin is measured from the steady head down to the saturation pressure instead of to the vapour pressure. For this main that margin is 0.41 rises instead of 0.57, and the pulse 2ε′⌈1/ε′⌉ − 1 is 1.451 rises instead of 1.296. The computed pocket with fast release in unlimited water reproduces 1.451 to two parts in a million, at this margin and at a second one.

The fifth figure draws both sawtooths against the margin. The vapour sawtooth is the column-separation essay’s: its teeth rise linearly and drop wherever 1/ε passes a whole number. The air sawtooth is the same shape moved right by $p_s$. On this main the move carries the line from low on one tooth to high on the next, which is why the fast release made the pulse taller; at other margins it moves it from a tall tooth to a short one, and fast release lowers the pulse. Released air does not cushion the collapse. It relocates the line on the sawtooth, by an amount that depends only on the pressure the water was saturated at.

The computed points with all the pipe’s water releasing its air fast sit on the air sawtooth, within a few hundredths of a rise, everywhere except at the smallest margins, where the cavity is large and the pipe’s air runs short. Past a margin of one the line never reaches the vapour pressure and the vapour sawtooth is flat at the Joukowsky rise. The air sawtooth is not: between ε = 1 and $1 + p_s$ the head falls below atmospheric pressure, the air comes out, and the computed pulses reach 2.7 to 2.9 rises. Those pulses are brief — 6 to 34 milliseconds wide at half their height — and they are close cousins of the spike that the fixed pocket sent back on a line that never cavitated. A lossless line keeps every such spike sharp, and a real one rounds it; but the order of events is real. A line designed with margin to spare against vapour pressure is not designed with margin against its own air.

What the gas does after the first collapse #

The last question was about the pulses after the first. The sixth figure varies how much more slowly the air goes back into the water than it came out. When it goes back as fast, the pocket empties at every collapse, each cavity starts from nothing, and each pulse is an air-cavity pulse: 1.46, then 1.09, then 1.42 rises. When it goes back a hundred times more slowly, the gas from the first cavity is still in the pocket when the second begins, and the pulses fall to 1.15, 1.02 and 0.97. The asymmetry between coming out and going back in — which is real, and large, because a bubble of air in water takes far longer to dissolve than to form — is what turns released air into a cushion for the pulses that follow.

The answer to the still-open question is therefore both. The gas left after the first collapse makes the later pulses lower, provided it stays; if the first collapse has put it back into solution, the next cavity arms itself afresh.

What was checked #

The pocket with release is the fixed pocket’s delay equation with one more unknown, and its checks follow from that. With no release it reproduces the fixed pocket step for step, to rounding. With fast release in unlimited water it reproduces the exact column-separation pulse at the margin measured to the saturation pressure, to a few parts in a million at two margins — the check that the whole mechanism of the margin section is in the equations and not in an interpretation. In the first cavity the gas stays below the release curve at zero pressure, as it must since any pressure in the pocket lowers the equilibrium, and within 2.2 per cent of it for a slow release. Halving the step moves the first pulse and its gas by a part in ten thousand.

One trap is worth recording. The release is a relaxation towards an equilibrium that itself depends on the gas, through the pocket’s pressure, and with a large exchanging volume the effective rate is far faster than 1/τ. The first version froze the equilibrium over each step and rang, and its pulses were wrong by a rise; the step now solves the relaxation exactly for the linear law, at the pocket’s volume.

What the model leaves out #

Friction. The line is lossless, as in both essays before this one, so every spike stays sharp and every tooth of every sawtooth stands at its full height. A real line rounds the brief pulses and damps the later ones within a few round trips.

Air released along the pipe. The low-pressure wave covers the whole pipe while the cavity lives, and the air that comes out far from the valve forms bubbles where it is, lowering the wave speed there and spreading every front. That distributed gas is what the discrete gas cavity model is for, and a single pocket cannot represent it; it is the reason φ, the water that feeds the valve’s pocket, is a variable here rather than a known.

One release time. Air comes out onto nuclei and bubble surfaces, and the rate depends on how many there are, on the turbulence, and on how supersaturated the water is, so a single τ is a summary. The essay’s range of τ spans the measured range, and the ceiling does not depend on it.

Saturation at atmospheric pressure. Water drawn from a pressurised source may be saturated at a higher pressure, which raises $p_s$, moves the air sawtooth further, and raises the ceiling in proportion.

Who worked it out #

Henry’s law is from 1803. Gaseous cavitation in pipelines — air coming out of solution in a transient, as distinct from vaporous cavitation — was studied experimentally and in models in the 1970s, among others by Kranenburg in 1974 and by Wiggert and Sundquist in 1979. The broad finding, that air leaves still water slowly compared with a transient and goes back more slowly still, is what the release and return times here stand for. Wylie’s discrete gas cavity model of 1984 is how most transient codes include gas today, and Bergant, Simpson and Tijsseling’s review of 2006 surveys the measurements.

Still open: the pump that trips #

Every history here, like the two before it, begins with a valve shut on a flowing line. Twice the margin named the case most lines are actually destroyed by, which begins with a down-surge instead: a pump that loses power. There the first wave is the low one, the cavity forms at the pump end or at a high point before any valve closes, and the non-return valve that shuts on the returning column is the reflecting end the collapse lands against. The next calculation follows that order of events, with the air-saturated water of this one, and asks whether the air sawtooth still predicts the first pulse, and whether the high point — where the pressure is lowest and the air comes out first — is where a line’s own air protects it or where it arms it.