Tools › Fluids and Heat Transfer › Pipe Flow › Water Hammer and Pipe Expansion › Example
1 km 12-in steel main closed in 1 s: wave speed, Joukowsky rise and the characteristics solution; a 100 ft steam line heated 300 °F: expansion, anchor force and bends
| Tool | Water Hammer and Pipe Expansion |
| Source | U.S. Bureau of Reclamation (1966). Welded Steel Penstocks. Engineering Monograph No. 3, Denver, CO (eq. 4, Figure 13); King, H. W., and Wisler, C. O. (1922). Hydraulics. Wiley, New York (eqs. 46–49); Naval Facilities Engineering Command (1990). MIL-HDBK-1003/8A (Table 12); Wylie, E. B., Streeter, V. L., and Suo, L. (1993). Fluid Transients in Systems. Prentice Hall (as mathematics). |
| Level | Undergraduate |
Problem
A 1,000 m 12-in Schedule 40 carbon-steel main (ID 303.2 mm, wall 10.3 mm, anchored throughout) carries water at 20 °C at 2 m/s under a static head of 50 m at a valve discharging to the atmosphere; the valve closes linearly in 1 s. Find the pressure-wave speed, the critical time 2L/a, the Joukowsky rise, and the maximum and minimum head at the valve by the method of characteristics with Darcy friction. Separately, a 100 ft run of 6-in Schedule 40 steel steam line is heated 300 °F: find its growth, the force and stress if it were fully restrained, and the guided-cantilever leg of an L-bend for an allowable stress range of 20 ksi.
Inputs to Enter
- Surge: water 20 °C (IF97: ρ 998.4 kg/m³, K = ρ w² 2.199 GPa), 12 in Sch 40 (303.2 / 10.3 mm), L 1,000 m, E 200 GPa, ν 0.30, anchored throughout (c₁ = 1 − ν²), thin wall, roughness 0.045 mm, H₀ 50 m, V₀ 2 m/s, T 1 s linear, 20 reaches, 6 periods, rating 1,600 kPa, hoop allowable 140 MPa
- Thermal: carbon steel, 6 in Sch 40 (168.3 / 7.11 mm), L 100 ft (30.48 m), ΔT 300 °F (166.7 °C), α 6.5 µin/in/°F (11.7 µm/m/°C), E 29 Msi (200 GPa), S_A 20.3 ksi (140 MPa), insulation 150 N/m, bending allowable 15.9 MPa, sag 2.5 mm
Answers
- ψ = c₁ D/e = 0.91 × 29.40 = 26.76; a = 1/√[ρ (1/K + ψ/E)] = 1,304.6 m/s (4,280 ft/s); 2L/a = 1.533 s, so the 1 s closure is rapid
- Steady: Re 6.05 × 10⁵, f 0.01467, h_f 9.87 m, H_v0 40.13 m; Joukowsky a V₀/g = 266.07 m (ρ a V₀ = 2,605 kPa); head with the rise 306.2 m
- Characteristics (20 reaches, Δt 0.0383 s): maximum head at the valve 312.2 m (3,057 kPa, 1.91 × the PN 16 rating; hoop stress 44.9 MPa), minimum −204.6 m, below the vapor-pressure head of −10.1 m: column separation flagged; Allievi's frictionless chain gives 306.2 / −225.9 m
- Thermal: ΔL = α L ΔT = 2.340 in (59.4 mm); MIL-HDBK-1003/8A Table 12 gives 2.376 in/100 ft at 300 °F → 2.376 in; restrained stress E α ΔT = 56.6 ksi (390 MPa), anchor force E A α ΔT = 315.7 kip (1,404 kN) on a wall area of 5.58 in²
- L-bend leg √(3 E D_o ΔL / S_A) = 21.48 ft (6.547 m), end moment 19.5 kN·m; U-loop legs 15.2 ft; support spacing 17.1 ft (5.21 m, sag governs)
Notes
The tool's default inputs are the surge case; choosing the thermal analysis loads the steam line. Both cases are illustrative (stated on the page); every number is checked against closed forms, Allievi's interlocking equations and an independent numpy / scipy implementation by test_water_hammer_expansion.php, and the wave-speed formula against USBR EM 3 Figure 13 (a = 4,660/√(1 + d/(100 b)) ft/s for steel).