Water Hammer and Pipe Expansion: Method

Wave Speed, Joukowsky, Characteristics, Expansion Bends

Scope

The tool has two independent analyses chosen by the first dropdown. Water hammer treats a single pipe of constant diameter between a reservoir of constant level and a valve that discharges to the atmosphere: the speed of the pressure wave, the Joukowsky rise of a rapid closure, the critical time 2L/a, the classical estimates for a slow closure, Allievi's exact frictionless chain, a method-of-characteristics solution with quasi-steady friction for a linear or an entered closure law, a pump-trip check by rigid column, surge-tank and air-chamber sizing by the rigid-column closed forms, and a plain comparison of the surge pressure with the pipe rating and hoop allowable. Thermal expansion treats a straight run between anchors: its free growth, the force and stress when it is fully restrained, the leg lengths of an L-bend, a Z-bend and a U-loop by the guided-cantilever rule, the ASME B31.3 flexibility screen, expansion-joint travel and support spacing. Branched or looped networks, pump characteristics with inertia, cavity (column-separation) modelling, unsteady friction, and a formal piping flexibility analysis are outside the scope; the Pipe Flow tool covers the steady losses and pump matching.

Conventions and units

Lengths, heads, diameters, velocities, pressures, moduli, forces and areas go through the site's unit dictionary (m, mm, m/s, kPa, GPa, kN in SI; ft, in, ft/s, psi, Msi, kip in USCS, with the dropdown alternatives). Times (closure, delay, periods) and the free-air fraction are plain numbers. The water temperature is a plain number read as °C in SI mode and °F in USCS mode, not converted by the unit switch; the thermal sub-tool's temperature rise and expansion coefficient are dictionary types and do convert (300 °F = 166.7 °C, 6.5 µin/in/°F = 11.7 µm/m/°C). Heads are piezometric heads above the valve, which is taken at the pipe's level; gauge pressures are ρ g H. Everything is computed in SI and displayed in the selected system.

Wave speed

A pressure wave in a liquid-filled elastic pipe travels at a = 1/√[ρ (1/K + ψ/E)]: the compressibility of the liquid (bulk modulus K) and the hoop stretch of the wall (modulus E) both store the water the wave front packs in. For a thin wall ψ = c₁ D/e with the Wylie–Streeter restraint factor c₁ = 1 − ν² for a pipe anchored throughout against axial movement (buried mains, welded lines), 1 − ν/2 for a pipe anchored at its upstream end only and free to stretch, and 1 for a line with expansion joints throughout. For a thick wall the tool uses ψ = 2(1 + ν) + c₁ D²/(e (D + e)), which for c₁ = 1 is exactly twice Lamé's hoop strain at the bore, 2[(R₂² + R₁²)/(R₂² − R₁²) + ν], and tends to D/e as the wall thins. The Bureau of Reclamation's penstock monograph (EM 3, Figure 13) prints the same relation as a = 4,660/√(1 + K D/(E e)) ft/s with K = 294,000 psi and E = 29.4 Msi, i.e. a = 4,660/√(1 + D/(100 e)) for steel: 3,295 ft/s at D/e = 100. King and Wisler (1922) give the identical form with the Joukowsky head written as 145 v/√(1 + 0.01 D/b) ft for steel pipe. For water the tool takes ρ, the speed of sound w and the viscosity from the IAPWS-IF97 formulation at the line pressure and temperature, with K = ρ w² (2.20 GPa = 319 ksi at 20 °C; the 294,000 psi of the older references is a rounded value for cold water); for another liquid enter them.

Free air lowers the wave speed drastically because the gas is a thousand times more compressible than the water: 1/K_eff = (1 − α)/K + α/p_abs with the air isothermal, and ρ_eff = (1 − α) ρ + α ρ_air. A tenth of a percent of air at 2 bar absolute cuts the speed of the default steel main from about 1,300 to about 420 m/s. Dissolved air does not count; only bubbles do, and they come out of solution at low pressure, which is one reason surge pressures in practice are often below the de-aerated prediction.

Joukowsky rise and the critical time

Joukowsky (1898; translated by Simin, 1904) showed by his Moscow experiments that a sudden stop of a column moving at V₀ raises the head at the valve by a V₀/g (the pressure by ρ a V₀), independent of the pipe length. The wave runs to the reservoir and back in 2L/a, the critical time; a closure completed in less than that time produces the full rise whatever the pipe, because no relief returns before the flow is stopped. The wave then reflects as a rarefaction and the head swings below the steady value by the same amount, with a period of 4L/a, decaying through friction. If the downswing takes the head below the vapor-pressure head, the water column separates, a vapor cavity opens and its collapse can produce a second, sometimes larger, blow; the tool flags this and stops short of modelling the cavity.

Slow closure: the classical estimates

For a closure slower than 2L/a the relief waves return during the closure and the rise is smaller. The older estimates treat the water as a rigid column: a uniform deceleration from V₀ to zero in time T needs a head L V₀/(g T) at the valve; Joukowsky's 1898 construction with a constant rate of rise, reproduced by King and Wisler as their eq. 49 and usually called Michaud's formula, gives twice that, 2 L V₀/(g T); Allievi's quasi-steady form for a linear gate law, with N = L V₀/(g H₀ T), gives Δh/H₀ = (N/2)(N + √(N² + 4)), which is what the exact rigid column with the gate law V = V₀ τ √(H/H₀) reaches at the end of the closure; the tool integrates that rigid column exactly and prints both. King and Wisler already remarked in 1922 that the slow-closure formulas "give results twice as great as others" because of their different assumptions; the characteristics solution below replaces them all, and they are printed only when the closure is slower than 2L/a.

Allievi's parameters ρ_A = a V₀/(2 g H₀) (the pipeline constant, Quick's K) and θ = a T/(2L) (the closure time in intervals of 2L/a) govern the frictionless elastic solution. His interlocking equations, ζ_i−1² + ζ_i² − 2 = 2 ρ_A (η_i−1 ζ_i−1 − η_i ζ_i) with ζ = √(H/H₀) at the valve and η the gate opening at the end of each interval, give the head at the valve at every multiple of 2L/a exactly; the Bureau of Reclamation's Figure 13 is Quick's chart of this solution. The tool tabulates the chain for the linear closure; its characteristics solution reproduces the chain to round-off when friction is removed, which is the check that the two agree.

Method of characteristics

The continuity and momentum equations of the transient become ordinary differential equations along the characteristic lines dx/dt = ±a. The pipe is divided into N reaches of Δx = L/N and marched with Δt = Δx/a, so every characteristic lands exactly on a grid node (Courant number 1, no interpolation). At an interior node the C⁺ line from the upstream neighbour gives H_P = C_P − B Q_P and the C⁻ line from the downstream neighbour H_P = C_M + B Q_P, with B = a/(g A), C_P = H_i−1 + B Q_i−1 − R Q_i−1|Q_i−1|, C_M = H_i+1 − B Q_i+1 + R Q_i+1|Q_i+1| and R = f Δx/(2 g D A²), the Darcy friction taken quasi-steady at the friction factor of the initial flow (Colebrook, from the Pipe Flow library). The reservoir fixes H = H₀; the valve is an orifice whose effective opening τ(t) falls from 1 to 0, Q = Q₀ τ √(H/H_v0), solved with the C⁺ equation as Q_P = −B C_v + √[(B C_v)² + 2 C_v C_P], C_v = (Q₀ τ)²/(2 H_v0) (Wylie and Streeter). The results are the head at the valve and at mid-length against time, the envelope of the highest and lowest head at every node, and the closure law used. The default of 20 reaches is enough for a single pipe; more reaches only sharpen the corners. Twenty periods of 2L/a on 80 reaches is a run of a few hundred thousand steps and takes a second or two; beyond 400,000 steps the grid is coarsened.

The closure law matters more than anything else. A real gate or butterfly valve loses most of its flow in the last 10–20 % of its travel, so a linear stem motion is far from a linear τ; paste the valve's τ(t) as pairs of t/T and τ (τ = C_d A relative to fully open, from the manufacturer's C_v curve) to see the difference. Column separation is flagged when the minimum head falls below the vapor-pressure head; the solution after that instant is not physical.

Pump trip

When the power fails at a pump feeding a rising main, the pump head collapses and the column decelerates under the static lift. Its first effect at the pump discharge is a downsurge bounded by Joukowsky's a V₀/g; if the absolute head left there, H_s + h_f + p_atm/(ρ g) − a V₀/g, is below the vapor-pressure head, the column separates at the pump. The rigid column then stops in t_s = L V₀ atan(√(h_f/H_s)) / (g √(H_s h_f)), the exact integral of dV/dt = −(g/L)(H_s + h_f (V/V₀)²), and reverses. A check valve that closes at the instant of reversal destroys no velocity; one that closes t_d later slams against a reverse velocity of about g H_s t_d/L and produces an upsurge a V_r/g, which is why slow-closing or spring-assisted check valves are specified on long mains. The inertia of the pump and motor, which the Bureau of Reclamation's pump monograph (EM 40) and Parmakian's charts account for, makes the real run-down slower and the downsurge smaller than this bound.

Surge tank and air chamber

Both devices turn the elastic shock into a slow mass oscillation. For a simple surge tank of area A_t at the valve end after a sudden stoppage, the frictionless rigid column gives an upsurge Z = V₀ √(L A/(g A_t)) and a period 2π √(L A_t/(g A)); Thoma's condition for a stable oscillation under governor control needs an area of at least L A V₀²/(2 g h_f (H₀ − h_f)), in practice multiplied by 1.5 to 2. For an air chamber the kinetic energy of the column, ½ ρ L A V₀², is absorbed by the air compressing from the line pressure p₀ (absolute) to the allowed maximum along p Vⁿ = const; the work balance ∫(p − p₀) dV = KE gives the initial air volume V_a = KE / (p₀ F) with F = (r^((n−1)/n) − 1)/(n − 1) − (1 − r^(−1/n)), r = p_max/p₀, and F = ln r − 1 + 1/r for isothermal air. On a pump trip the chamber expands, feeding the line while the column slows, and the balance is V_a = KE / (p₀ F_e) with F_e = (s − 1) − (1 − s^(1−n))/(n − 1), s = (p₀/p_min)^(1/n). Friction is neglected, which overstates the upsurge and the volume; n = 1.2 is the usual design exponent between isothermal and adiabatic. The chamber must hold the larger of the two initial volumes plus the water that enters or leaves.

Pipe rating and hoop stress

The maximum transient gauge pressure at the valve, ρ g H_max, is compared with the entered pressure class and the hoop stress p D/(2e) with the entered allowable, as plain ratios. Standards differ on surge: AWWA C150 (ductile iron) allows a surge of 100 psi above the working pressure, AWWA C900 / C905 (PVC) set a surge allowance by pressure class, ASME B31.4 allows a 10 % overpressure for transients and B31.3 a 33 % one for events of limited duration, and the Bureau of Reclamation's penstock criteria (EM 3, Table 1) allow stresses approaching yield for emergency surges with a factor of safety of 2 on the ultimate, 4 to 5 for normal operation. The entered rating should be the limit the governing standard sets for the surge condition.

Thermal expansion

A run of length L heated by ΔT grows ΔL = α L ΔT. The coefficient of carbon steel rises with temperature, from about 6.3 µin/in/°F over the first 100 °F to 6.9 over 500 °F; MIL-HDBK-1003/8A Table 12 tabulates the expansion of steel and copper pipe directly, in inches per 100 ft against the temperature change, and the tool prints that value beside the α L ΔT result for those two materials (the table's 660 °F steel cell is a misprint, 5.583 for 5.683, corrected). The material dropdown fills α and E with typical values that stay editable.

If the run is held between two rigid anchors with no bend, the growth is suppressed and the pipe carries the stress σ = E α ΔT, independent of its length (390 MPa for steel heated 167 °C, above the yield of most pipe steels), and the anchors the force F = E A α ΔT with A the wall area. Real anchors yield a little, which is why a straight restrained steam line is never built without an expansion bend, loop or joint.

Expansion bends by the guided cantilever

A leg perpendicular to the run absorbs its growth by bending. With both of its ends held against rotation (the run on one side, the anchor or next run on the other) and one end displaced by Δ, the leg is a guided cantilever: its end moment is 6 E I Δ/L², its end force 12 E I Δ/L³ and the outer-fibre stress 3 E D_o Δ/L², so the leg that keeps the stress within the allowable displacement-stress range S_A is L = √(3 E D_o Δ/S_A). The tool checks this against a beam stiffness-matrix solution. A Z-bend's offset leg takes the growth of both runs it joins; a U-loop's two legs share the run's growth, Δ/2 each, and the loop width is taken as half the leg height by the usual proportion. The rule neglects the flexibility of the elbows and of the run, so its legs are on the long side; the M. W. Kellogg Design of Piping Systems and the ASHRAE pipe-design chapter present it as the quick method, and the charts in the older Navy and Army utility manuals are the same formula drawn. S_A is the ASME B31.1 / B31.3 allowable displacement-stress range f (1.25 S_c + 0.25 S_h), about 20 ksi (140 MPa) for carbon steel at moderate temperature, entered.

ASME B31.3 (319.4.1) lets a two-anchor system of uniform size with no intermediate restraints skip a formal flexibility analysis when D y/(L − U)² ≤ 30 S_A/E_a, with D and the resultant movement y in inches and the developed length L and anchor distance U in feet; with every length in metres the limit is 0.2083 S_A/E_a. The tool evaluates it for the sized L-bend. It is a sufficiency screen, not a stress: for a short leg on a long run, L − U is nearly the leg's length squared over twice the run and the screen is far stricter than the guided-cantilever stress, so a bend that fails it is not thereby too short, but the code then requires a formal analysis. The screen is also known to be unconservative for some unequal-leg configurations, which is why the code restricts its use.

Expansion joints and supports

An expansion joint in place of a bend must travel the full ΔL plus the installation tolerance and must be guided and anchored to take the pressure thrust; MIL-HDBK-1003/8A allows bellows joints only where space prevents a bend, with travel under 4 in, and slip joints under 8 in. The support spacing takes the weight per unit length of the pipe, its water-filled contents and the insulation entered on a simple span: w L²/8 ≤ S_b Z for the bending allowable and 5 w L⁴/(384 E I) ≤ δ for the sag limit, the shorter governing. A continuous run would allow w L²/10 and sag less, so the result is conservative; the 2,300 psi bending allowable and 0.1 in sag of common practice (MSS SP-58, B31.1) are the entered defaults.

Reading the default cases

The surge default is a 1 km 12-in Schedule 40 steel main at 2 m/s under 50 m of head, closed in 1 s. Its wave speed is 1,305 m/s and its critical time 1.53 s, so the 1 s closure is rapid: the head at the valve rises by the full Joukowsky value of 266 m to 312 m (3.06 MPa, 1.9 times a PN 16 rating) and then swings below the vapor pressure, column separation. Lengthening the closure to twenty times 2L/a brings the rise down to the rigid-column estimates of a few metres. The thermal default is a 100 ft 6-in Schedule 40 steel steam line heated 300 °F: it grows 2.34 in by α L ΔT (2.38 in by the handbook table), would load rigid anchors with 316 kips at 56.6 ksi, needs an L-bend leg of 21.5 ft (a U-loop 15.2 ft high), and can span 17.1 ft between supports with the sag governing.

Limits

Single pipe of uniform properties between a constant-level reservoir and a valve to atmosphere; quasi-steady friction at the initial friction factor (unsteady friction damps real transients faster); no cavity model after column separation; the pump trip by rigid column without the pump's inertia or characteristic; surge tank and air chamber without friction, inlet throttling or the chamber's own connection losses; the thin-wall hoop stress; the guided-cantilever bends without elbow flexibility; the B31.3 screen within its own restrictions; support spacing on a simple span with no concentrated loads. Water properties from IF97 up to 350 °C.