water-hammer

name	water-hammer
description	Water hammer (hydraulic transients) — Joukowsky equation, wave speed, pressure surge, valve closure time, pipeline protection (surge tank, PRV, air vessel), Allievi equations.
metadata	{"priority":7,"promptSignals":{"phrases":["water hammer","hydraulic transient","surge","Joukowsky","valve closure","pressure wave"],"minScore":3}}

Water Hammer — Complete Skill

Wave Speed

Pressure Wave Celerity

a = 1 / √(ρ(1/(K_w) + D/(e × E_pipe)))

K_w = bulk modulus of water = 2.1 GPa D = pipe internal diameter [m] e = pipe wall thickness [m] E_pipe = pipe material Young's modulus [GPa] ρ = liquid density [kg/m³]

For rigid pipe: a = √(K_w/ρ) ≈ 1430 m/s (water, rigid) Steel pipe (D/e = 50): a ≈ 1300 m/s Plastic pipe (HDPE): a ≈ 300-500 m/s (lower E reduces wave speed) Approximate: a = 1320 / √(1 + (K_w D)/(e E_pipe)) [m/s, units consistent]

Joukowsky Equation (Instantaneous Valve Closure)

Pressure rise: ΔP = ρ a ΔV

ΔV = change in flow velocity [m/s] = V_0 (complete closure) ΔP = pressure increase [Pa]

Head rise: Δh = a × ΔV / g [meters]

Example: V = 2 m/s, a = 1300 m/s → Δh = 1300×2/9.81 = 265 m additional head

Valve Closure Time Criterion

Critical Closure Time

t_c = 2L / a [s, L = pipe length from reservoir to valve]

Rapid (sudden) closure: T_c < t_c → full Joukowsky pressure rise Slow closure: T_c > t_c → reduced pressure rise (linear approx for linear closure): Δh_slow ≈ (2L/g) × (ΔV/T_c) [Allievi simplified]

Design rule: T_c > 5 × t_c for effective pressure reduction

Method of Characteristics (MOC)

Full solution via finite difference: Positive characteristic: H_P + (a/gA) Q_P = H_A + (a/gA) Q_A - (f Δt/(2gDA²)) Q_A |Q_A| Negative characteristic: similar equation propagating other direction

Boundary conditions: reservoir (H = const), valve (Q = f(H, opening)), pump (pump curve) Time step: Δt = Δx/a (Courant condition)

Pipeline Protection Devices

Surge Tank (Air Vessel)

Open surge tank: reflects wave, limits pressure, allows inflow Closed air vessel: compressed air cushion absorbs wave energy Volume: V_vessel = V_air ≥ (ΔQ_pipe × L)/(2 × a/g × (P_max - P_min))

Location: as close to pump/valve as possible Surge tank head oscillation: T_os = 2π√(L_pipe A_pipe/(g A_tank))

Pressure Relief Valve (PRV)

Opens at P_set to relieve excess pressure Response time must be ≤ t_c for effectiveness

Slow-Closing Check Valve

Spring-assisted closure: increases closure time → reduces water hammer

Negative Pressure / Column Separation

Minimum pressure: H_min = H_0 - a V_0/g (at instant of valve closure, at valve) If H_min < vapor pressure (≈ 0.24 bar abs at 20°C): column separation → collapse → secondary surge (can exceed initial)

Prevention: maintain minimum pressure > vapor pressure + 5 m head margin

Output

Provide: wave speed a [m/s], Joukowsky pressure rise ΔH [m] and ΔP [bar], critical closure time t_c [s], recommended closure time T_c [s], maximum pressure at valve [m head], minimum pressure check vs. vapor pressure, protection device recommendation.