Water Hammer Calculator — Joukowsky Surge Pressure and Wave Speed — Sample Report

1 · Design inputs

A water line — NPS 12 Schedule 40 carbon steel, 12.0 in bore on a 0.375 in wall — running 2,000 ft from a supply vessel to a valve. Water at 62.4 lb/ft³ with a bulk modulus of 300,000 psi, steel at 29.5×10⁶ psi, thin-wall anchored restraint (c₁ = 1.0). The valve closes on a flow of 5 ft/s.

2 · Method

Three results, computed in sequence. Pressure wave speed first, with the elastic pipe-wall correction:

a = √( K·144·g / ρ ) ⁄ √( 1 + c₁·(K/E)·(D/t) )

The numerator is the rigid-pipe acoustic speed in the fluid; the denominator is the compliance correction. Note what drives it: the ratio K/E (how soft the pipe material is relative to the fluid) times D/t (how thin-walled the pipe is). A thin-walled plastic line is soft on both counts and its wave speed collapses; a heavy-wall steel line is close to rigid.

Joukowsky surge — the instantaneous pressure rise for a velocity change ΔV:

ΔP = ρ · a · ΔV / (144 · g)

Critical closure period — the wave round-trip time:

t_c = 2 · L / a

Units are US customary throughout: K and E in psi, ρ in lb/ft³, ΔV in ft/s, L in ft, g = 32.174 ft/s². The 144 converts ft² to in² so ΔP lands in psi. The restraint coefficient c₁ depends on how the pipe is anchored — whether it is free to move axially, anchored at one end, or fully restrained against longitudinal movement — and is your input; approximately 1.0 is typical for a thin-walled anchored line.

Entering L = 0 suppresses the critical-time output and returns wave speed and surge only, which is the right mode when you want the surge magnitude for a valve or pump-trip screen and the line length is not yet fixed. Every run carries a standing warning that ΔP is the instantaneous worst case: it is the rise on top of the operating pressure, and taking credit for a slower closure requires a transient analysis, not a rule of thumb.

3 · Calculation

1. Rigid-pipe acoustic speed: a_rigid = √(K·144·g/ρ) = √(300,000·144·32.174/62.4) = √22,274,308 = 4,719.57 ft/s.
2. Elastic correction term: c₁·(K/E)·(D/t) = 1.0·(300,000/29,500,000)·(12/0.375) = 0.0101695·32 = 0.325424.
3. Correction divisor: √(1 + 0.325424) = √1.325424 = 1.151271 — the pipe wall costs about 13% of the wave speed.
4. Wave speed: a = 4,719.57/1.151271 = 4,099.44 ft/s.
5. Joukowsky surge: ΔP = ρ·a·ΔV/(144·g) = 62.4·4,099.44·5/(144·32.174) = 1,279,025/4,633.06 = 276.065 psi.
6. Critical closure period: t_c = 2L/a = 2·2,000/4,099.44 = 4,000/4,099.44 = 0.9757 s.

4 · Results COMPUTED

5 · Verification statement

The result values above are locked as fixture water-hammer.json, case “water in 12 in steel line, dV=5 ft/s, L=2000 ft”, tolerance 0.05, in the calc-core continuous verification gate (see https://pipingtoolset.com/trust/validation.html). Allowable-stress and factor data are user-supplied inputs; citations: N. E. Joukowsky, 'Über den hydraulischen Stoss in Wasserleitungsröhren' (1898) — the momentum-based derivation of ΔP = ρ·a·ΔV. · Standard elastic pipe-wall wave-speed correction, a = a_rigid / √(1 + c₁·(K/E)·(D/t)), with c₁ set by the pipe restraint condition — user-supplied. · ASME B31.4, Pipeline Transportation Systems for Liquids and Slurries — surge pressure provisions; codes commonly permit a bounded overpressure above MAOP for surge, evaluated against the code of record. · Constant: g = 32.174 ft/s²; the factor 144 converts lb/ft² to psi. Physical and unit constants, not code values.