Water hammer

Why can closing a valve slowly protect a pipe that closing it quickly would burst?

A pipe carries water at a couple of metres per second under, say, six bar. Close the valve over ten seconds and nothing happens. Close it in a tenth of a second and the pipe bangs, the brackets shift, and a joint may split.

The water was moving at the same speed either way, and the pump held the same pressure either way. So what did the fast closure do that the slow one did not?

Start with what must be true. The water has momentum, and bringing it to rest takes an impulse — a force acting over a time. Halve the time and you double the force. That is the shape of the answer, but the size is the surprise, so let us calculate it.

The trick that makes it calculable is this. When the valve shuts, the water does not stop all at once along the pipe — it cannot, because news of the closure has to travel. A pressure front moves back upstream at the speed of sound in the fluid-and-pipe system, call it c, and only the water it has already passed is at rest.

So consider one instant of that front's travel. In a short time dt it sweeps a length c dt of pipe, bringing that slug from speed v to zero. The slug's mass is rho times A times c dt, so the momentum destroyed is rho A c v dt. The only thing that can have destroyed it is the pressure step across the front acting on the area, delta-p times A, over the same time dt. Set them equal and the areas and times cancel:

delta-p = rho times c times v.

That is the Joukowsky equation, and notice what is not in it. Not the pipe's length. Not the pressure the system was already running at. Not the valve. Only density, wave speed, and the change in velocity.

Now put numbers to it. Water's density is 1000 kilograms per cubic metre and its bulk modulus about 2.2 gigapascals, so the sound speed in unconfined water is the square root of 2.2 billion over 1000: about 1480 metres per second. A steel pipe stretches as pressure rises, dropping the effective wave speed to somewhere around 1000 to 1300. Take 1200. Then stopping 2 metres per second gives 1000 times 1200 times 2, which is 2.4 million pascals — 24 bar, on top of whatever was already there, from a valve that stopped a modest flow.

Compare that with the folk account. "The water is a slug hitting the valve, so it delivers its kinetic energy" — but a moving fluid's dynamic pressure is only half rho v squared, which for 2 metres per second is 2000 pascals. Two thousand against two point four million: out by three orders of magnitude, because bringing a whole column to rest is nothing like a streamline stagnating where the fluid can go elsewhere. The second folk account, "the banging is the pipe hitting its brackets", has the causality backwards — that knock is the wave's unbalanced axial force shaking the pipe, a symptom rather than a source.

So why does slow closing help at all, if the peak does not contain the closing time? Because of the pipe's length after all — not in the equation, but in the clock. The front runs upstream to the reservoir, which cannot support the raised pressure and reflects a relief wave back. That round trip takes 2L over c: for a 300-metre line at 1200 metres per second, half a second. Close in less than that and the valve is fully shut before any relief arrives, so you get the whole Joukowsky rise. Close over five seconds and relief keeps returning throughout, cancelling most of the build-up — roughly in the ratio of the round-trip time to the closure time.

So "fast" and "slow" are not clock words. They are comparisons against 2L over c, a number that belongs to the pipe.

The wave speed is the softest quantity in the chain. The 1480 figure for open water is firm arithmetic from the bulk modulus; the 1200 for a steel line is typical rather than derived, and in plastic pipe it can fall to a few hundred metres per second, which is why polyethylene mains are far more forgiving. Entrained air lowers it further, since the mixture's compressibility is dominated by the gas.

The weakest link, though, is the closure time itself. A valve's flow area does not fall linearly with stem position: for many types most of the flow is still passing when the stem is nearly shut, so the last fraction of travel does nearly all the stopping. The effective closure time can therefore be a small fraction of the actuator's stroke, and a valve nominally closing over five seconds can behave as though it slammed. Slow-closing hardware is not the same thing as a slow closure, and the difference has burst real pipes.

The load-bearing claim is the derivation itself: the surge is set by the change in velocity times the wave speed times the density, and closure counts as slow only relative to the pipe's own round-trip time 2L over c. Surge vessels and air chambers follow directly: they give the column somewhere compliant to decelerate into, extending the stopping time.

• Why column separation and cavity collapse can produce a second surge above the Joukowsky value.
• How surge vessels, air chambers and relief valves each buy stopping time by a different route.

Every term the collection defines is gathered in the glossary.