Transient pressure analysis: the Joukowsky surge equation, pressure-wave speed and the critical time 2L/a, surge mitigation (slow valves, VFD ramps, surge vessels, air valves) and a worked valve-slam example.
When flow in a pipeline is stopped or started quickly — a valve slamming, a pump tripping — the moving column of water cannot stop instantly. Its momentum converts to a pressure wave that races up and down the line: water hammer. The transient pressure can be many times the steady operating pressure, bursting pipes, wrecking joints and collapsing them under the following vacuum. Every pumped main and fast-acting valve needs a surge check.
For an instantaneous flow change, the peak pressure rise is the Joukowsky equation:
ΔP = pressure rise (Pa), ΔH = head rise (m)a = pressure-wave speed in the pipe (m/s)Δv = change in velocity (m/s), ρ = densityThe wave speed depends on fluid compressibility and pipe elasticity, and the change is “sudden” only if it happens faster than the pipe’s characteristic time:
K = bulk modulus of the fluid, E = pipe Young’s modulusD, e = pipe diameter and wall thickness, L = pipe lengthtc = 2L/a gives a milder “slow” surge| Measure | What it does | Best for |
|---|---|---|
| Slow-closing / actuated valves | Extend closure beyond 2L/a so the surge is “slow” | Most pumped mains |
| Soft starters / VFD ramps | Ramp pump speed so Δv is gradual | Pump start/stop transients |
| Surge / air vessels | Cushion the column with a gas pocket | Long high-energy mains |
| Air-release / vacuum valves | Admit air to prevent column separation & collapse | Profiles with high points |
| Flywheels on pumps | Add inertia so a trip decelerates slowly | Pump-trip protection |
Steel main, L = 800 m, wave speed a ≈ 1200 m/s, flow velocity v = 1.8 m/s suddenly stopped (Δv = 1.8 m/s).
Confirm wall thickness for the rated/transient pressure and support spacing for the thrust.
Reynolds & Bauhm sizes pipework, pumps and valves and closes the head budget as part of a gated, fully documented bespoke design — every figure traceable from flow to fabrication.
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