Joukowsky equation — the ceiling of the surge
A sudden velocity change ΔV converts kinetic energy into elastic energy of the fluid and pipe wall; the resulting pressure jump travels along the pipe at wave speed a. Anchor it: ρ = 1000 kg/m³, a = 1200 m/s, ΔV = 1 m/s → Δp = 1.2 MPa, ΔH = 1200/9.807 ≈ 122 m. Water columns of that size appear from thin air when a fast valve or a slammed check valve stops the flow — which is why closure time, not just velocity, is the design variable.
Wave speed — where the pipe softens the wave
In an infinitely rigid conduit the liquid alone carries the wave at the acoustic speed a₀ = √(K/ρ) (water: K = 2.19 GPa = 318,000 psi, ρ ≈ 998 kg/m³ → ≈ 1483 m/s as tabulated). Real thin-wall pipes expand circumferentially under the pressure wave; the term K·D/(E·e) divides the acoustic speed by the square root of the total stiffness ratio, so a soft or large-diameter thin-wall pipe slows the wave down — steel at D/e ≈ 50 lands near 1200 m/s, PVC can sit at 300–400 m/s. Slower waves are not safer: the 2L/a window widens with them.
Closure class — the 2L/a rule
te is the round-trip time of the pressure wave from valve to the first reflecting boundary. The reflected wave carries the opposite sign; it can only cancel the ongoing rise if it returns before the valve finishes closing:
$$t_c \le t_e:\ \Delta H_{max}=\frac{a\,\Delta V}{g}\ \text{(rapid — full Joukowsky)}\qquad t_c > t_e:\ \Delta H_{max}\approx\frac{2L\,\Delta V}{g\,t_c}\ \text{(gradual, first-order)}$$The gradual-case estimate is the classic first-order correction: the reflected negative wave eats into the rise while the valve is still moving. When tc ≫ te the system stops being elastic at all — the inertia-dominated rigid-column regime takes over and the event becomes slow surging rather than water hammer.
| Closure time tc | Class | Max head rise | Regime |
|---|---|---|---|
| = 0 | Instantaneous | a·ΔV/g | Water hammer |
| ≤ 2L/a | Rapid | a·ΔV/g | Water hammer |
| > 2L/a | Gradual | < a·ΔV/g | Water hammer (attenuated) |
| ≫ 2L/a | Slow | ≪ a·ΔV/g | Surging (rigid column) |
Column separation — the line you must not cross
Wherever the transient head drops to vapor pressure, the water column tears open and a vapor cavity forms; when the reflected wave returns, the cavity collapses and the two columns slam together at well above the original Joukowsky magnitude — burst-pipe territory. Screening rule: keep the computed minimum head above −10.3 m (−34 ft) gauge; anything lower means vapor pressure has been reached and separation must be assumed. Also keep every point above the network minimum of 20 psig (1.38 bar) where the standard applies.
Surge-protection checklist
| Device | What it does | Watch out for |
|---|---|---|
| Slow-closing check valve (damped) | Stops backflow before it accelerates | Sluggish discs in short systems slam; near-pump air chambers need a dashpot on the disc |
| Two-stage pump-discharge valve stroke | Fast first stroke to low flow, slow second to seat | Optimal single-pump strokes are not automatically optimal for multi-pump power failure |
| Surge relief valve | Dumps the rising wave at a set pressure | Relieved flow needs a safe destination |
| Surge anticipation valve | Opens on pump power loss, ahead of the returning wave | Never enable anticipation where negative pressures already exist — it deepens the downsurge |
| Simple / one-way surge tank | Feeds water into the pipe at the high point during downsurge | One-way tanks need a responsive check; sizing is site-specific |
| Air chamber | Cushions both directions; keeps the whole line positive | Air volume must stay near 50 % of the vessel — starved chambers have burst lines |
| Vacuum breaking / air-valve | Admits air to break the column instead of vapor | Re-admission slam; field case: ~60 % peak reduction on a 60-in line |
| Flywheel (increased WR²) | Slows pump rundown after power failure, softening the downsurge | Motor cost and bearing load grow; motors already carry 75–90 % of the train WR² |