numvana

Water Hammer Surge Pressure Calculator (Joukowsky Equation)

Estimate the pressure surge from suddenly closing a valve in a pipe, using the Joukowsky equation for fast closures and Michaud's formula for slower ones.

Surge pressure
2,400 kPa
Surge pressure (psi)
348.091 psi
Critical (reflection) time
0.8333 s
Closure regime
Rapid (Joukowsky)

How it works

Closing a valve suddenly stops the moving fluid's momentum, sending a pressure shock wave back up the pipe at the fluid/pipe's characteristic wave speed (celerity) c — the "water hammer" effect. When the valve closes faster than the pipe's critical time Tc = 2L/c (the round trip for the wave to travel the pipe length L and reflect back), the full surge applies: the Joukowsky equation, ΔP = ρ × c × Δv, where ρ is fluid density and Δv is the change in flow velocity.

If the actual closure time exceeds Tc, part of the reflected wave arrives back at the valve before it finishes closing, which relieves some of the pressure buildup. The standard approximation for that slower case is Michaud's formula, ΔP = 2 × ρ × L × Δv ÷ T, where T is the closure time. The two formulas agree exactly at T = Tc, so the result is continuous across the boundary rather than jumping between regimes.

Fluid density is left in the universally-used kg/m³ regardless of the unit toggle, the same convention this site's other fluid-mechanics calculators (Venturi Flow Rate, Darcy-Weisbach) use; wave speed, velocity change, and pipe length are converted to canonical meters/meters-per-second internally before either formula runs. Closure time is entered in seconds either way, since time isn't a metric/imperial distinction.

FAQ

Where does the wave speed (celerity) value come from?

It depends on the fluid's bulk modulus, the pipe material's elasticity, and the pipe's diameter-to-wall-thickness ratio — rigid pipes (like steel) carry the pressure wave faster than flexible ones (like PVC or rubber hose). For water in a steel pipe it's commonly in the 1000-1400 m/s range, which is why that's this calculator's default; this calculator doesn't look that value up for you, since it depends on which pipe you actually have.

Why does closing the valve more slowly reduce the surge?

Once the closure takes longer than the pipe's round-trip reflection time (Tc = 2L/c), the pressure wave that reflected off the far end has time to arrive back at the valve and partially cancel the buildup before the valve fully shuts — which is exactly what Michaud's formula captures. This is also the practical reason slow-closing valves (and surge tanks, in real systems) are used to protect pipelines from hammer damage.

Related calculators