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Water Hammer Calculator

Surge pressure from a sudden velocity change in a pipeline. Computes the pressure wave speed for an elastic pipe wall, the Joukowsky surge head and pressure, the critical closure time 2L/a, and a slow-closure estimate when the valve closes more gradually.

in
in
psi
psi
slug/ft³
— (1.0 = expansion joints)
ft/s
ft
s
—ft/s
—s
—
—ft
—psi
—ft
—psi

Defaults: 12-inch steel main, ¼-inch wall, water, 5 ft/s stopped, 2,000 ft to the reservoir, valve closed in 0.5 s. Surge adds to the steady operating pressure; check the total against the pipe's pressure rating.

Wave speed in an elastic, liquid-filled pipe:
$$ a = \sqrt{\dfrac{K/\rho}{1 + c_1\,\dfrac{K\,D}{E\,e}}} $$
Joukowsky (rapid closure, tc ≤ 2L/a) and slow closure (tc > 2L/a):
$$ \Delta H = \frac{a\,\Delta V}{g} \qquad \Delta P = \rho\,a\,\Delta V \qquad \Delta H_{slow} \approx \frac{2\,L\,\Delta V}{g\,t_c} $$
a pressure wave speed · K fluid bulk modulus · ρ fluid density · D inside diameter · e wall thickness · E pipe Young's modulus · c₁ pipe restraint factor · ΔV velocity change · g gravitational acceleration · L pipe length to the reflecting boundary · tc effective valve closure time.

Why closure time matters more than anything else

Once a valve closes faster than the critical time 2L/a, the surge is set entirely by the wave speed and the velocity change — making the valve close even faster changes nothing, because the full Joukowsky pressure has already formed before any reflected wave can return. Close it more slowly than 2L/a and the low-pressure reflection from the upstream reservoir arrives in time to cancel part of the rise. That is why long pipelines use slow-closing actuators: on 2,000 ft of steel pipe the critical time is about one second, and closing over ten seconds cuts the surge by roughly a factor of ten.

The slow-closure estimate uses the Michaud approximation, which assumes the flow decelerates linearly. Real valves do not — most of the flow reduction in a gate or butterfly valve happens in the last 10–20% of travel — so the effective closure time is shorter than the actuator time. Treat the slow-closure value as an order-of-magnitude check and use a transient analysis for final design.

Typical pressure wave speeds for water-filled pipe

Wave speed a for water (K = 316,000 psi, ρ = 1.94 slug/ft³), c₁ = 1
PipeE (psi)D/ea (ft/s)ΔP per 1 ft/s (psi)
Rigid pipe (theoretical limit)∞—4,84365.2
Ductile iron, 12 in × 0.34 in wall24,000,00035.34,00253.9
Steel, 12 in × 0.25 in wall30,000,000483,94753.2
PVC DR18, 12 in × 0.67 in wall400,00017.91,24416.8

Computed from the wave-speed equation above. K and ρ are for fresh water near 60–70°F; bulk modulus rises slightly with temperature to about 50°C, and entrained air reduces wave speed sharply — a fraction of a percent of free air by volume can cut a in half.

Pipe restraint factor c₁

Restraint factor for thin-walled pipe (ν = Poisson's ratio of the pipe material)
Pipe support conditionc₁
Expansion joints throughout (axial movement free)1.0
Anchored against axial movement throughout1 − ν²
Anchored at the upstream end only1 − ν/2

For steel (ν ≈ 0.30) c₁ ranges only from 0.85 to 1.0, changing wave speed by a few percent. The wall stiffness ratio K·D/(E·e) matters far more.

Worked examples

Example 1 — Rapid valve closure on a steel main

Given: 12-in steel pipe, 0.25-in wall, E = 30×10⁶ psi, water (K = 316,000 psi, ρ = 1.94 slug/ft³), c₁ = 1. Flow of 5 ft/s stopped by a valve 2,000 ft from the reservoir, closing in 0.5 s.
K·D/(E·e) = (316,000 × 12)/(30×10⁶ × 0.25) = 0.5056
K/ρ = 316,000 × 144 / 1.94 = 2.346×10⁷ ft²/s²
a = √(2.346×10⁷ / 1.5056) = 3,947 ft/s
2L/a = 2 × 2,000 / 3,947 = 1.01 s → 0.5 s closure is rapid
ΔH = a·ΔV/g = 3,947 × 5 / 32.174 = 613 ft
ΔP = ρ·a·ΔV = 1.94 × 3,947 × 5 / 144 = 266 psi above operating pressure

Example 2 — Same valve, 10-second closure

Given: As Example 1, but the actuator closes over tc = 10 s.
10 s > 2L/a = 1.01 s → slow closure
ΔH ≈ 2·L·ΔV/(g·tc) = 2 × 2,000 × 5 / (32.174 × 10) = 62.2 ft
ΔP ≈ 62.2 × 62.4 / 144 = 26.9 psi — about one-tenth of the rapid-closure surge

Example 3 — PVC instead of steel

Given: 12-in PVC DR18 (0.67-in wall), E = 400,000 psi, otherwise as Example 1.
K·D/(E·e) = (316,000 × 12)/(400,000 × 0.67) = 14.15
a = √(2.346×10⁷ / 15.15) = 1,244 ft/s; 2L/a = 3.21 s
ΔH = 193 ft · ΔP = 83.8 psi — under one-third of the steel surge

References: Wylie, E.B., Streeter, V.L. (1993). Fluid Transients in Systems. Prentice Hall. Joukowsky, N. (1898). Über den hydraulischen Stoss in Wasserleitungsröhren. AWWA Manual M11, Steel Pipe — A Guide for Design and Installation. AWWA Manual M23, PVC Pipe — Design and Installation.

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