Water Hammer: Critical Closure Time and the Joukowsky Rise
If a valve closes faster than 2L/c, you get the full acoustic surge. Celerity, t_c, slow-closure estimate, surge tanks, and column separation.
Key takeaways
- c is typically 1000–1400 m/s in steel water pipes and much lower in plastic.
- t_c = 2L/c is the round-trip; faster effective closure is “rapid.”
- Full Joukowsky: ΔP = ρ c ΔV. Slow linear stroke: scale by t_c/t.
- Downsurge below vapor pressure needs a full transient model, not a spreadsheet.
Steady-state head loss is metres. A slammed valve can add tens or hundreds of metres in milliseconds. The screening tools are wave speed, the round-trip time 2L/c, and the Joukowsky formula.
Wave speed
| System | Typical c (m/s) |
|---|---|
| Water, thick steel | 1200–1400 |
| Water, thin steel / DI | 1000–1200 |
| Water, PVC / PE | 300–500 |
| Unconfined water (pure acoustic) | ~1480 |
Rapid vs slow
Example: L = 800 m, c = 1100 m/s, V = 1.8 m/s, water. t_c = 1.45 s. Joukowsky ΔP = 998 × 1100 × 1.8 ≈ 2.0 MPa (about 200 m of head). If the effective closure is 1 s, you take the full wave. If it is 6 s, a linear estimate is ΔP_J × 1.45/6 ≈ 0.48 MPa — still a serious surge, but not the full slam.
Open solver: Critical closure / JoukowskyOpen solver: Water hammer analysisWhat actually reduces ΔV
- Slower valves, or two-stage closure that crawls through the last 20% of travel.
- A surge tank or air vessel that stores the rejected mass as a free-surface rise instead of an acoustic spike.
- Lower normal velocity. ΔP scales with ΔV; fat pipes at 1 m/s are kinder than skinny pipes at 3 m/s.
Frequently asked questions
Usually yes. Most of the ΔV happens near seat. A 30 s stroke can still be “rapid” if the last effective 2 s are shorter than t_c. Use the effective closure time, not the actuator nameplate.