Water Resources HubWater topic 11 of 18 — free theory
Water Hammer & Surge
Slam a valve shut and the moving water has nowhere to go — its momentum converts into a pressure wave that races up and down the pipe. Here is how fast that wave travels, how big the pressure spike gets, and how designers keep it from bursting the pipe.
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How fast the pressure wave travels
When the flow velocity changes suddenly, the disturbance propagates as a pressure wave with celerity a. The wave speed depends on how compressible the water is and how much the pipe wall stretches:
Rigid pipe (reference value): a0 = √(K/ρ)
| K | bulk modulus of the liquid (water ≈ 2.2 GPa) |
| ρ | liquid density (water ≈ 1000 kg/m³) |
| a0 ≈ 1480 m/s for water | the absolute ceiling — a real pipe always gives a slower wave |
Elastic pipe: a = a0√[1 + (K/E)(D/e)]
| E | elastic modulus of the pipe wall (steel ≈ 200 GPa, ductile iron ≈ 170 GPa, PVC ≈ 2.8 GPa, HDPE ≈ 0.8 GPa) |
| D, e | pipe internal diameter and wall thickness (same units!) |
| (K/E)(D/e) | dimensionless — check that D/e is dimensionless before plugging in |
A flexible plastic pipe stretches far more than steel, so its wave crawls: a few hundred m/s is typical for PVC/HDPE, versus ~1200–1400 m/s for steel. Stiffer pipe = faster wave = bigger surge for the same velocity change.
The Joukowsky surge — how big the spike is
If the valve closes instantly (or faster than the wave can make a round trip), the full pressure rise is given by the Joukowsky equation. It is just momentum written as a pressure:
Δp = ρ · a · ΔV ΔH = a · ΔVg
| ΔV | magnitude of the velocity change (Vinitial − Vfinal), not the velocity itself |
| Δp, ΔH | surge pressure rise (Pa) and equivalent head rise (m) — Δp = ρgΔH |
Critical closure time: tc = 2La
| L | pipe length from the valve to the nearest free surface (reservoir, tank) |
| Rapid closure: t < 2L/a | full Joukowsky surge applies |
| Gradual closure: t > 2L/a | the reflected wave relieves the valve before it finishes closing — surge is reduced |
Gradual-closure (Michaud) estimate, for t > 2L/a: ΔH ≈ 2LΔV/(g t). It assumes the velocity falls off roughly linearly with time — a handy approximation, not an exact solution.
Keeping the surge under control
Real systems never rely on hoping the operator closes the valve gently. The standard toolkit:
Surge tank · air chamber · slow-closing valve · pressure-relief valve · pump flywheel · combination air valve
| Surge tank | an open standpipe near the valve/pump giving a free water surface — the transient reflects off it instead of building up |
| Air chamber | a closed vessel with a trapped compressed-air cushion that absorbs the surge; needs an air compressor to maintain the cushion |
| Slow-closing valve / two-stage closure | stretches the closure past 2L/a so the surge stays in the gradual regime |
| Pressure-relief valve | opens at a set pressure and vents flow — limits the up-surge only |
| Pump flywheel / air-vacuum valves | flywheel keeps the pump spinning through a power trip; air valves admit air to prevent column separation (cavitation collapse) on the down-surge |
Upsurge bursts pipes; downsurge can pull a vacuum, collapse thin-walled pipe, and — when the separated water columns slam back together — cause an even bigger secondary upsurge. Protection must handle both directions.
PE depth: transient analysis and column separation
PE problems treat the transient as a wave-timing problem, not just a formula. The key picture: the surge wave travels to the reservoir, reflects as a negative (relief) wave, and returns after 2L/a. Anything the valve does after that return is partially cancelled.
Wave round trip: 2L/a Slow pipe (plastic): long tc — even multi-second closures can be “rapid”
Column separation is the PE favourite: when the downsurge drops the hydraulic grade line below the pipe profile at a high point, the water column parts, vapour cavities form, and their collapse drives a secondary upsurge that can exceed the original Joukowsky value. That is why air-vacuum valves sit at high points — admitting a little air is far cheaper than repairing a burst main.
PE trap: surge pressure adds to the steady-state operating pressure. A pipe rated for the static head alone can still fail when Joukowsky is stacked on top. Always check total pressure = operating + surge.
Worked example Surge from an instant valve closure
Given:
- Steel water main: L = 500 m, D = 300 mm, e = 8 mm, E = 200 GPa.
- Water: K = 2.2 GPa, ρ = 1000 kg/m³.
- Flow velocity 1.5 m/s; the valve is slammed shut (t ≈ 0).
Solution:
- Wave celerity: a0 = √(2.2×109/1000) = 1483 m/s. (K/E)(D/e) = (2.2/200)(300/8) = 0.4125, so a = 1483/√1.4125 = 1248 m/s.
- Critical time 2L/a = 1000/1248 = 0.80 s. An instant closure is far shorter, so full Joukowsky applies.
- ΔH = aΔV/g = 1248 × 1.5/9.81 = 191 m of head.
- Δp = ρaΔV = 1000 × 1248 × 1.5 = 1.87×106 Pa = 1.87 MPa.
Answer: The instant closure adds ≈ 191 m of head — about 1.87 MPa — on top of the steady operating pressure.