Edited by Kenneth J. Korane [email protected]
Water hammer (or steam hammer) is a violent flow transient in piping named for the loud banging it generates. It can affect almost any fluid system that experiences rapidly changing flows, including power-plant piping, watersupply systems, pumped storage facilities, oil pipelines, and hydraulic and general fluid-handling lines.
Water hammer is not just a nuisance. It can rupture or collapse pipes, uproot anchors, and cause other calamities associated with excessive pipe movement. Proper design and operation prevent such destruction.
To design piping systems that stand up to the forces that water hammer generates, engineers first need to recognize pressure-wave propagation in pipes. This includes both the size and swiftness of the pressure surge, and how pressure waves affect pipe.
Suddenly closing a gate or valve builds up pressure by Δp that propagates upstream at the speed of sound. The pressure wave is reflected at the reservoir or junction and travels back to the gate, changing pressure in the pipe by –Δp. The wave reflects off the closed gate, turns into a negative pressure wave and travels toward the reservoir for a second round trip. The pressure wave decays in two to three cycles.
Engineers can determine the magnitude of the pressure surge by considering a layer of fluid adjacent to the gate (as shown in the Control volume illustration). As the gate closes to block flow, the resulting boundary forces on the control volume accelerate the fluid mass inside.
(po + Δp)A + ρ A(Vo + ΔV)2 – poA – ρ AVo2 = ρA(a – Vo)Δt(–ΔV/Δt) Δp = –ρ (a – Vo)ΔV – ρ (Vo + ΔV)2 + ρVo2Δp = – ρaΔV(1 + Vo /a) ≈ –ρ aΔV.
The acoustic velocity of water in Schedule 40 to 60 steel pipes is about 4,000 fps. For steam or gases, calculate acoustic velocity using:
a = (144gckpv)0.5.
The specific heat ratio, k, is 1.25 to 1.3 for steam and 1.4 for air and most gases.
For quick full-gate closings, ΔV = –Vo, the pressure surge and corresponding force on the gate are, respectively:
A safety factor, Sf =1.1, is generally appropriate for surge-pressure calculations. Pressure surges measured in 24-in. main steam pipes of power plants during turbine trip tests are less than 5% above the analytical prediction using these equations. This is a reasonable validation of this calculation method which neglects friction, compressibility, and related factors.
Pressure design for water-hammer load must consider both rupture and buckling failures. Overpressure may rupture the pipe from hoop-tension failure. Hoop-tension stress in a pipe wall from internal pressure is:
Shp = pr/h = pdi /2h, p = p0 + Δpmax
Longitudinal stress in pipe is half of the hoop stress. But it must be combined with tensile and bending stresses from all concurrent loads and could dictate the design.
If pipe pressure (po – Δpmax) becomes negative, the pipe may collapse or buckle from external pressure. The critical net external pressure for buckling a cylindrical shell is:
Low-pressure, large-diameter pipes and bellows are most vulnerable to buckling. The external pressure from underground seepage on buried pipes or tunnel linings can be high enough to buckle pipe during water-hammer events, when drained, or even during construction. Stiffener rings can be used instead of thicker shells. The critical pressure per unit axial length for buckling a ring is:
qcr = 3EI/r3.
A good practice is to weld the stiffener to the outside of the shell wall with full-penetration continuous welds so that the combined moment of inertia of the cross section exceeds the sum of the two individual moments of inertia.
The speed of pressure surge at the gate depends on valve characteristics in terms of mass flow rate versus time. A typical nonlinear valve-characteristic curve may be approximated by a straight line. The linear effective closing time is about one-half to one-third of the nominal stroke time. For linear-valve characteristics, surge pressure ramps up to Δpmax in time, tg, and then remains constant. This is a ramp function, Δp(t), defined as:
Δp(t) = Δpmax(t/tg), 0 ≤ t ≤ tg; Δp(t) = Δpmax, tg ≤ t.
The longitudinal load on a pipe section is generated by a wave front hitting the two ends with a time lag. Consider a straight pipe section between elbows B1 and B2 shown in the accompanying graphic. The wave front reaches B1 at time t1 and arrives at B2 at t2. Forces on B1 and B2 are time-shifted ramp functions:
ΔF(t) = Δp(t)A; ΔF1(t) = -ΔF(t–t1), ΔF2(t) = –ΔF(t–t2).
The net unbalanced force on the pipe section is a trapezoid pulse:
ΔFs(t) = ΔF1(t) + ΔF2(t) = –ΔF(t–t1) + ΔF(t–t2).
For long pipe sections, defined as Ls ≥ atg , ΔFsmax = ΔFmax. For short pipe sections, where Ls < atg , ΔFsmax = ΔFmax(t2–t1)/tg. Thus, short sections are subjected to smaller forces because the wave front hits B2 before ΔF1(t) reaches the peak. For Ls < Lcr = atg, the load on the pipe section is smaller.
This indicates that unbalanced forces, ΔFsmax, on pipe sections can be reduced by slowing down the valve for Ls < Lcr . Most pipe sections in utilities and other plants are shorter than these critical lengths.
Dynamic-load factor (DLF) is a multiplier to estimate maximum dynamic load from static load considering the input interactions with the single degreeof- freedom dynamic system. For example, in loading an object on a scale, if the scale swings to 15 lb before settling down to 10 lb, the static load on the scale is 10 lb while the maximum dynamic load on the scale is 15 lb. So DLF is 1.5.
DLF depends on the input pulse shape, how rapidly the load is applied, and system stiffness. DLF for a particular input pulse is customarily presented as a function of ˆτ/T, where ˆ τ = the time interval that defines the input pulse and T = the system’s natural period of vibration. Water-hammer load on a pipe section is a trapezoidal or triangular pulse when Ls = atg. The maximum response to a trapezoidal pulse may occur before or after the input decays. In the former, DLF of the ramp function is used; for the latter, the residual DLF applies. Those DLFs are plotted with τ = tg and τ/T = fntg in the chart, DLF for water- hammer loads. The design DLF, an envelope of all the above, is an educated approximation on the part of the design engineer.
Water hammer induces longitudinal loads on pipe sections, resulting in excess pipe movements that can overstress the pipe or break small branches. It may uproot anchors, permanently deform supports, or overload connections to adjacent equipment. Pipes may even ram the surrounding structures or equipment. Therefore, piping must be restrained to handle waterhammer loads at or near the sources and limit movement.
Waterhammer restraints include axial and offset restraints, and equipment connections. In selecting the restraint type, engineers must consider other loads such as thermal movement of the pipe. Typical restraints include:
- Rigid rod, clamp, and bracket devices are common for coldpipe supports.
- Snubbers lock up and function like a rigid strut when subjected to a fastacting axial force, and otherwise allow slow thermal movements with little resistance.
- Ubolt clamps may slip or bend under outofplane load.
- Expansion joints may slip if either end is not rigidly held. (The expansion joint acts like a hydraulic jack.)
The separation force, pA, on an expansion joint may be huge even without water hammer, though water hammer may expose otherwise hidden support weaknesses. Neglecting these forces often leads to support failure around an expansion joint. Small branch pipes must be flexible enough to tolerate movements, or they may be overstressed and even break. Long and large pipes such as penstocks are typically anchored at the bends and have an expansion joint in every section between anchors.
Quick gate opening directly ramps up pressure to accelerate the fluid mass, and the wave front rushes ahead at the speed of sound. A fast actuator-operated valve that taps water from a high-pressure source could cause water hammer in a branch line. Loose parts in a valve assembly may rattle to initiate water hammer under certain conditions.
Slowing the valve’s closing rate reduces water-hammer loads on piping. The wicket gates of hydraulic turbines and main steam-inlet valves of steam turbines in power plants must close quickly to prevent turbine generators from excess overspeeding when the unit at a high output is cut off from the power grid. Here, the quick-closing gate causes water/steam hammer and the piping system must be ready to bear the loads.
On the other hand, many valves close much faster than necessary simply because they’re driven by fast actuators. If the hydraulic or pneumatic actuator speed is not adjustable, inserting an improvised flow restrictor (tube connector with a small flow area) on the actuator control fluid line slows the actuator without impairing its force capability.
How slow is a slow valve? Pick the pressure-wave round trip time, tr = 2L/a, as a benchmark. By slowing the valve to tg> tr, the –Δ‰p wave returns to the gate before full gate closure to cancel further pressure surges. This proportionally reduces the crest of the pressure surge.
By setting an acceptable pressure-surge magnitude, one can calculate a necessary tg. Actual valve-closing time required, (2 to 3)tg, depends on the valve characteristics.
Entrapped air may also cause annoying water hammer. Air gets trapped by filling an empty pipe with liquid too quickly while impeding orderly evacuation of the air. An unusually low static head at the bottom of a vertical pipe may indicate an entrapped air column. When fluid flows, air is pushed up and down in elevation, and pressure at particular locations fluctuates and disturbs flow and jolts the pipes. The compressed, entrapped air expands as it approaches the exit to atmosphere and thrusts the water slug in front at high speed. The air and water with a huge density disparity alternately exit the pipe and the reactive momentum ρAV2 changes abruptly to jolt the pipe and shake the entire line.
Because filling operations are infrequent and can be planned, it is more cost effective to give sufficient time to fill or discharge the line slowly to alleviate the impact of air entrapment, rather than fortify the supports. Long and large pipes like penstock use a bypass valve to slowly fill the line and evacuate air through an air-release valve at the highest elevation. The air-release valve stays open when dry and shuts while flooded. Shutoff force come from the buoyancy of a bulky hollow disk.
An engineer’s priority for handling water hammer is to remove the sources or minimize the effects. Slowing valves and avoiding air entrapment can eliminate many headaches. But when quick valve action is mandatory and water hammer is inevitable, piping systems must be designed to accommodate the loads.
Calculating water-hammer loads
Here are typical calculations to determine water-hammer loads on the pipe section.
For 24-in. Sch. 60 pipe, A = 382.35 in.2, di = 22.064 in., and h = 0.968 in.
Saturated steam is at p = 900 psia (po = 885.3 psig) and speciŠfic volume v = 0.50091‹ft3/lb.
SpeciŠfic heat ratio k = 1.265 (Ref. Flow of Fluids Through Valves, Fittings and Pipes, Crane Technical Paper 410).
Acoustic velocity in steam a = (144gckpv)0.5 = 1,626 fps.
Safety factor Sf = 1.1.
Pressure surge ‰Δpmax = Sf × aW/(Agc) = 1.1 [1,626(2.8 × 106/3,600)]/(382.35 × 32.2) = 1.1 × 102.73 = 113 psi.
Maximum internal pressure pmax = po + Δ‰pmax = 998.3 psig.
Minimum internal pressure pmin = po – ‰Δpmax =772 psig. Buckling is not an issue.
Hoop stress in pipe St = (pmaxdi )/(2h) = 11.38 ksi.
Maximum force from the pressure surge on the gate ‰ΔFmax = ‰ΔpmaxA = 43.2 kip.
Critical pipe length Lcr = atg = 1,626 × 0.050 = 81.3 ft.
Dynamic load factor (DLF) = 1.3 from the chart for žτ/T = fntg ≥‚ 20 × 0.05 = 1.
Longitudinal loads on the pipe sections are calculated as follows:
Ls > 81.3 ft, R = ‰ΔFmax × DLF = 43.2 × 1.3 = 56.2 kips.
Ls < 81.3 ft, Rs1 = (56.2 /81.3) = 0.691 kip/ft (load per foot of pipe length). For example, for Ls = 50 ft, R = Rs1Ls = 0.691 × 50 = 34.6 kip.
Pressure wave traveling frequency f = a/(4L) = 1 Hz.
Given that the minimum tensile strength of typical power-piping ASTM A106 steels is at least 48 ksi (yield stress about 30 ksi), it appears this pipe would handle water-hammer loads.