Effect of a pressure surge on a float gauge

Water hammer (or, more generally, fluid hammer) is a pressure surge or wave caused when a fluid (usually a liquid but sometimes also a gas) in motion is forced to stop or change direction suddenly (momentum change). A water hammer commonly occurs when a valve closes suddenly at an end of a pipeline system, and a pressure wave propagates in the pipe. It is also called hydraulic shock.

This pressure wave can cause major problems, from noise and vibration to pipe collapse. It is possible to reduce the effects of the water hammer pulses with accumulators, expansion tanks, surge tanks, and other features.

Rough calculations can be made either using the Zhukovsky (romanised as Joukovsky or Joukowsky) equation, or more accurate ones using the method of characteristics.

## Water hammer: Cause and effect

When a pipe is suddenly closed at the outlet (downstream), the mass of water before the closure is still moving, thereby building up high pressure and a resulting shock wave. In domestic plumbing this is experienced as a loud banging resembling a hammering noise. Water hammer can cause pipelines to break if the pressure is high enough. Air traps or stand pipes (open at the top) are sometimes added as dampers to water systems to absorb the potentially damaging forces caused by the moving water.

In hydroelectric generating stations, the water travelling along the tunnel or pipeline may be prevented from entering a turbine by closing a valve. For example, if there is 14 km of tunnel of 7.7 m diameter full of water travelling at 3.75 m/s, that represents approximately 8000 megajoules of kinetic energy that must be arrested. This arresting is frequently achieved by a surge shaft open at the top, into which the water flows. As the water rises up the shaft its kinetic energy is converted into potential energy, which causes the water in the tunnel to decelerate. At some HEP stations, what looks like a water tower is actually one of these devices, known in these cases as a surge drum.

In the home a water hammer may occur when a dishwasher, washing machine or toilet shuts off water flow. The result may be heard as a loud bang, repetitive banging (as the shock wave travels back and forth in the plumbing system), or as some shuddering.

On the other hand, when an upstream valve in a pipe closes, water downstream of the valve attempts to continue flowing creating a vacuum that may cause the pipe to collapse or implode. This problem can be particularly acute if the pipe is on a downhill slope. To prevent this, air and vacuum relief valves or air vents are installed just downstream of the valve to allow air to enter the line to prevent this vacuum from occurring.

Other causes of water hammer are pump failure and check valve slam (due to sudden deceleration, a check valve may slam shut rapidly, depending on the dynamic characteristic of the check valve and the mass of the water between a check valve and tank).

Expansion joints on a steam line that have been destroyed by steam hammer

Steam distribution systems may also be vulnerable to a situation similar to water hammer, known as steam hammer. In a steam system, a water hammer most often occurs when some of the steam condenses into water in a horizontal section of the piping. Steam picks up the water forming a "slug" and hurls this at high velocity into a pipe fitting, creating a loud hammering noise and greatly stressing the pipe. This condition is usually caused by a poor condensate drainage strategy.

Where air filled traps are used, these eventually become depleted of their trapped air over a long period of time through absorption into the water. This can be cured by shutting off the supply, opening taps at the highest and lowest locations to drain the system (thereby restoring air to the traps), and then closing the taps and re-opening the supply.

## Water hammer: Water hammer from a jet of water

If a stream of high pressure water impinges on a surface, water hammer can quickly erode and destroy it. In the 2009 Sayano–Shushenskaya hydroelectric power station accident, the lid to a 640 MW turbine was ejected upwards, hitting the ceiling above. During the accident, the rotor was seen flying through the air, still spinning, about 3 meters above the floor. Unrestrained, 256 cubic metres (67,600 US gal) per second of water began to spray all over the generator hall. The geyser caused the structural failure of steel ceiling joists, precipitating a roof collapse around the failed turbine.

## Water hammer: Water hammer during an explosion

When an explosion happens in an enclosed space, water hammer can cause the walls of the container to deform. However, it can also impart momentum to the enclosure if it is free to move. An underwater explosion in the SL-1 nuclear reactor vessel caused the water to accelerate upwards through 2.5 feet (0.76 m) of air before it struck the vessel head at 160 feet per second (49 m/s) with a pressure of 10,000 pounds per square inch (69,000 kPa). This pressure wave caused the 26,000 pounds (12,000 kg) steel vessel to jump 9 feet 1 inch (2.77 m) into the air before it dropped into its prior location.

## Water hammer: Mitigating measures

Water hammer has caused accidents and fatalities, but usually damage is limited to breakage of pipes or appendages. An engineer should always assess the risk of a pipeline burst. Pipelines transporting hazardous liquids or gases warrant special care in design, construction, and operation. Hydroelectric power plants especially must be carefully designed and maintained because the water hammer can cause water pipes to fail catastrophically.

The following characteristics may reduce or eliminate water hammer:

• Reduce the pressure of the water supply to the building by fitting a regulator.
• Lower fluid velocities. To keep water hammer low, pipe-sizing charts for some applications recommend flow velocity at or below 1.5 m/s (4.9 ft/s)
• Fit slowly closing valves. Toilet fill valves are available in a quiet fill type that closes quietly.
• High pipeline pressure rating (expensive).
• Good pipeline control (start-up and shut-down procedures).
• Water towers (used in many drinking water systems) help maintain steady flow rates and trap large pressure fluctuations.
• Air vessels work in much the same way as water towers, but are pressurized. They typically have an air cushion above the fluid level in the vessel, which may be regulated or separated by a bladder. Sizes of air vessels may be up to hundreds of cubic meters on large pipelines. They come in many shapes, sizes and configurations. Such vessels often are called accumulators or expansion tanks.
• A hydropneumatic device similar in principle to a shock absorber called a 'Water Hammer Arrestor' can be installed between the water pipe and the machine, to absorb the shock and stop the banging.
• Air valves often remediate low pressures at high points in the pipeline. Though effective, sometimes large numbers of air valves need be installed. These valves also allow air into the system, which is often unwanted.
• Shorter branch pipe lengths.
• Shorter lengths of straight pipe, i.e. add elbows, expansion loops. Water hammer is related to the speed of sound in the fluid, and elbows reduce the influences of pressure waves.
• Arranging the larger piping in loops that supply shorter smaller run-out pipe branches. With looped piping, lower velocity flows from both sides of a loop can serve a branch.
• Flywheel on pump.
• Pumping station bypass.
Typical pressure wave caused by closing a valve in a pipeline

## Water hammer: The magnitude of the pulse

One of the first to successfully investigate the water hammer problem was the Italian engineer Lorenzo Allievi.

Water hammer can be analyzed by two different approaches-rigid column theory, which ignores compressibility of the fluid and elasticity of the walls of the pipe, or by a full analysis that includes elasticity. When the time it takes a valve to close is long compared to the propagation time for a pressure wave to travel the length of the pipe, then rigid column theory is appropriate; otherwise considering elasticity may be necessary. Below are two approximations for the peak pressure, one that considers elasticity, but assumes the valve closes instantaneously, and a second that neglects elasticity but includes a finite time for the valve to close.

### Water hammer: Instant valve closure; compressible fluid

The pressure profile of the water hammer pulse can be calculated from the Joukowsky equation

So for a valve closing instantaneously, the maximum magnitude of the water hammer pulse is:

where ΔP is the magnitude of the pressure wave (Pa), ρ is the density of the fluid (kg m), a0 is the speed of sound in the fluid (ms), and Δv is the change in the fluid's velocity (ms). The pulse comes about due to Newton's laws of motion and the continuity equation applied to the deceleration of a fluid element.

#### Water hammer: Equation for wave speed

As the speed of sound in a fluid is ${\displaystyle a={\sqrt {\frac {B}{\rho }}}}$, the peak pressure depends on the fluid compressibility if the valve is closed abruptly.

where

• a = wave speed
• B = equivalent bulk modulus of elasticity of the system fluid-pipe
• ρ = density of the fluid
• K = bulk modulus of elasticity of the fluid
• E = elastic modulus of the pipe
• D = internal pipe diameter
• t = pipe wall thickness
• c = dimensionless parameter due to system pipe-constraint condition on wave speed

### Water hammer: Slow valve closure; incompressible fluid

When the valve is closed slowly compared to the transit time for a pressure wave to travel the length of the pipe, the elasticity can be neglected, and the phenomenon can be described in terms of inertance or rigid column theory:

Assuming constant deceleration of the water column (dv/dt = v/t), gives:

where:

• F = force, N
• m = mass of the fluid column, kg
• a = acceleration, m/s
• P = pressure, Pa
• A = pipe cross section, m
• ρ = fluid density, kg/m
• L = pipe length, m
• v = flow velocity, m/s
• t = valve closure time, s

The above formula becomes, for water and with imperial unit: P = 0.0135 V L/t. For practical application, a safety factor of about 5 is recommended:

where P1 is the inlet pressure in psi, V is the flow velocity in ft/sec, t is the valve closing time in seconds and L is the upstream pipe length in feet.

## Water hammer: Expression for the excess pressure due to water hammer

When a valve with a volumetric flow rate Q is closed, an excess pressure ΔP is created upstream of the valve, whose value is given by the Joukowsky equation:

In this expression:

• overpressurization ΔP is expressed in Pa;
• Q is the volumetric flow in m/s;
• Z is the hydraulic impedance, expressed in kg/m/s.

The hydraulic impedance Z of the pipeline determines the magnitude of the water hammer pulse. It is itself defined by:

with:

• ρ the density of the liquid, expressed in kg/m;
• A cross sectional area of the pipe, m;
• B equivalent modulus of compressibility of the liquid in the pipe, expressed in Pa.

The latter follows from a series of hydraulic concepts:

• compressibility of the liquid, defined by its adiabatic compressibility modulus Bl, resulting from the equation of state of the liquid generally available from thermodynamic tables;
• the elasticity of the walls of the pipe, which defines an equivalent bulk modulus of compressibility for the solid Bs. In the case of a pipe of circular cross section whose thickness t is small compared to the diameter D, the equivalent modulus of compressibility is given by the following formula: ${\displaystyle B={\frac {t}{D}}E}$; in which E is the Young's modulus (in Pa) of the material of the pipe;
• possibly compressibility Bg of gas dissolved in the liquid, defined by: ${\displaystyle B_{g}={\frac {\gamma }{\alpha }}P}$
• γ being the specific heat ratio of the gas
• α the rate of ventilation (the volume fraction of undissolved gas)
• and P the pressure (in Pa).

Thus, the equivalent elasticity is the sum of the original elasticities:

As a result, we see that we can reduce the water hammer by:

• increasing the pipe diameter at constant flow, which reduces the flow velocity and hence the deceleration of the liquid column;
• employing the solid material as tight as possible with respect to the internal fluid bulk (solid Young modulus low with respect to fluid bulk modulus);
• introducing a device that increases the flexibility of the entire hydraulic system, such as a hydraulic accumulator;
• where possible, increasing the percentage of undissolved gases in the liquid.

## Water hammer: Dynamic equations

The water hammer effect can be simulated by solving the following partial differential equations.

where V is the fluid velocity inside pipe, ${\displaystyle \rho }$ is the fluid density and B is the equivalent bulk modulus, f is the friction factor.

## Water hammer: Column separation

Column separation is a phenomenon that can occur during a water-hammer event. If the pressure in a pipeline drops below the vapor pressure of the liquid, cavitation will occur (some of the liquid vaporizes, forming a bubble in the pipeline, keeping the pressure close to the vapor pressure). This is most likely to occur at specific locations such as closed ends, high points or knees (changes in pipe slope). When subcooled liquid flows into the space previously occupied by vapor the area of contact between the vapor and the liquid increases. This causes the vapor to condense into the liquid reducing the pressure in the vapor space. The liquid on either side of the vapor space is then accelerated into this space by the pressure difference. The collision of the two columns of liquid (or of one liquid column if at a closed end) causes a large and nearly instantaneous rise in pressure. This pressure rise can damage hydraulic machinery, individual pipes and supporting structures. Many repetitions of cavity formation and collapse may occur in a single water-hammer event.

## Water hammer: Simulation software

Most water hammer software packages use the method of characteristics to solve the differential equations involved. This method works well if the wave speed does not vary in time due to either air or gas entrainment in a pipeline. The Wave Method (WM) is also used in various software packages. WM lets operators analyze large networks efficiently. Many commercial and non commercial packages are available.

Software packages vary in complexity, dependent on the processes modeled. The more sophisticated packages may have any of the following features:

## Water hammer: Applications

• The water hammer principle can be used to create a simple water pump called a hydraulic ram.
• Leaks can sometimes be detected using water hammer.
• Enclosed air pockets can be detected in pipelines.

## Water hammer: History

Marcus Vitruvius Pollio described in the 1st century B.C. the effect of water hammer in lead pipes and stone tubes of the Roman public water supply. Water hammer was exploited before there was even a word for it: In 1772, Englishman John Whitehurst built a hydraulic ram for a home in Cheshire, England. In 1796, French inventor Joseph Michel Montgolfier (1740–1810) built a hydraulic ram for his paper mill in Voiron. In French and Italian, the terms for "water hammer" come from the hydraulic ram: coup de bélier (French) and colpo d'ariete (Italian) both mean "blow of the ram". As the 19th century witnessed the installation of municipal water supplies, water hammer became a concern to civil engineers. Water hammer also interested physiologists who were studying the circulatory system.

Although it was prefigured in work by Thomas Young, the theory of water hammer is generally considered to have begun in 1883 with the work of German physiologist Johannes von Kries (1853–1928), who was investigating the pulse in blood vessels. However, his findings went unnoticed by civil engineers. Kries's findings were subsequently derived independently in 1898 by the Russian fluid dynamicist Nikolay Yegorovich Zhukovsky (1847–1921), in 1898 by the American civil engineer Joseph Palmer Frizell (1832–1910), and in 1902 by the Italian engineer Lorenzo Allievi (1856–1941).

• Blood hammer
• Cavitation
• Fluid dynamics
• Hydraulic ram – makes constructive use of the water hammer effect
• Hydraulophone – musical instruments employing water and other fluids
• Impact force
• Watson's water hammer pulse

## Water hammer: References

1. Kay, Melvyn (2008). Practical Hydraulics (2nd ed.). Taylor & Francis. ISBN 0-415-35115-4.
2. Shu, Jian-Jun (2003). "Modelling vaporous cavitation on fluid transients". International Journal of Pressure Vessels and Piping. 80 (3): 187–195. doi:10.1016/S0308-0161(03)00025-5.
3. http://communities.bentley.com/products/hydraulics___hydrology/f/5925/p/60896/147250.aspx#147250
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6. Flight Propulsion Laboratory Department, General Electric Company, Idaho Falls, Idaho (November 21, 1962), Additional Analysis of the SL-1 Excursion: Final Report of Progress July through October 1962 (PDF), U.S. Atomic Energy Commission, Division of Technical Information, IDO-19313 ; also TM-62-11-707
7. Bruce, S.; Larock, E.; Jeppson, R. W.; Watters, G. Z. (2000), Hydraulics of Pipeline Systems, CRC Press, ISBN 0-8493-1806-8
8. Thorley, A. R. D. (2004), Fluid Transients in Pipelines (2nd ed.), Professional Engineering Publishing, ISBN 0-79180210-8
9. Streeter, V. L.; Wylie, E. B. (1998), Fluid Mechanics (International 9th Revised ed.), McGraw-Hill Higher Education
10. "Water Hammer & Pulsation"
11. Faisandier, J., Hydraulic and Pneumatic Mechanisms, 8th edition, Dunod, Paris, 1999 (Buy book ISBN 2100499483)
12. Bergeron, L., 1950. Du Coup de Bélier en Hydraulique - Au Coup de Foudre en Electricité. (Waterhammer in hydraulics and wave surges in electricity.) Paris: Dunod (in French). (English translation by ASME Committee, New York: John Wiley & Sons, 1961.)
13. Vitruvius Pollio with Morris Hicky Morgan, trans. The Ten Books on Architecture (Cambridge, Massachusetts: Harvard University Press, 1914) ; Book 8, Chapter 6, sections 5-8 , pp. 245-246. Vitruvius states that when a water pipe crosses a wide valley, it must sometimes be constructed as an inverted siphon. He states that cavities ("venters") must be constructed periodically along the pipe "and in the venter, water cushions must be constructed to relieve the pressure of the air." "But if there is no such venter made in the valleys, nor any substructure built on a level, but merely an elbow, the water will break out, and burst the joints of the pipes." Swiss engineer Martin Schwarz - Martin Schwarz, "Neue Forschungsergebnisse zu Vitruvs colliviaria" [New research results on Vitruvius' colliviaria], pp. 353-357, in: Christoph Ohlig, ed., Cura Aquarum in Jordanien (Siegburg, Germany: Deutschen Wasserhistorischen Gesellschaft, 2008) - argues that Vitruvius' phrase vis spiritus referred not to air pressure, but to pressure transients (water hammer) in the water pipes. He found stone plugs (colliviaria) in Roman water pipes, which could be expelled by water hammer, allowing water in the pipe to flood the air chamber above the pipe, instead of rupturing the pipe.
14. Ismaier, Andreas (2011), Untersuchung der fluiddynamischen Wechselwirkung zwischen Druckstößen und Anlagenkomponenten in Kreiselpumpensystemen [Investigation of the fluid dynamic interaction between pressure surges and system components in centrifugal pumping systems], Schriftenreihe des Lehrstuhls für Prozessmaschinen und Anlagentechnik, Universität Erlangen; Nürnberg Lehrstuhl für Prozessmaschinen und Anlagentechnik (in German), 11, Shaker, ISBN 978-3-8322-9779-4
15. Whitehurst, John (1775), "Account of a machine for raising water, executed at Oulton, in Cheshire, in 1772", Philosophical Transactions of the Royal Society of London, 65: 277–279, doi:10.1098/rstl.1775.0026 See also plate preceding page 277.
16. Montgolfier, J. M. de (1803), "Note sur le bélier hydraulique, et sur la manière d'en calculer les effets" [Note on the hydraulic ram, and on the method of calculating its effects] (PDF), Journal des Mines (in French), 13 (73): 42–51
17. Tijsseling, A. S.; Anderson, A. (2008), "Thomas Young's research on fluid transients: 200 years on" (PDF), Proceedings of the 10th International Conference on Pressure Surges, Edinburgh, UK: 21–33 see page 22.
18. Ménabréa, L. F. (1858), "Note sur les effects de choc de l'eau dans les conduites" [Note on the effects of water shocks in pipes], Comptes rendus (in French), 47: 221–224
19. Michaud, J. (1878), "Coups de bélier dans les conduites. Étude des moyens employés pour en atténeur les effects" [Water hammer in pipes. Study of means used to mitigate its effects], Bulletin de la Société Vaudoise des Ingénieurs et des Architects (in French), 4 (3,4): 56–64, 65–77
20. Tijsseling, A. S.; Anderson, A. (2008). Hunt, S., ed. "Thomas Young's research on fluid transients: 200 years on". Proc. of the 10th Int. Conf. on Pressure Surges. Edinburgh, United Kingdom: BHR Group: 21–33. ISBN 978-1-85598-095-2.
21. Young, Thomas (1808). "Hydraulic investigations, subservient to an intended Croonian lecture on the motion of the blood". Philosophical Transactions of the Royal Society of London. 98: 164–186.
22. von Kries, J. (1883), "Ueber die Beziehungen zwischen Druck und Geschwindigkeit, welche bei der Wellenbewegung in elastischen Schläuchen bestehen" [On the relationships between pressure and velocity, which exist in connection with wave motion in elastic tubing], Festschrift der 56. Versammlung Deutscher Naturforscher und Ärzte (Festschrift of the 56th Convention of German Scientists and Physicians) (in German), Tübingen, Germany: Akademische Verlagsbuchhandlung: 67–88
23. von Kries, J. (1892), Studien zur Pulslehre [Studies in Pulse Science] (in German), Tübingen, Germany: Akademische Verlagsbuchhandlung
24. Tijsseling, Arris S.; Anderson, Alexander (2004), "A precursor in waterhammer analysis – rediscovering Johannes von Kries" (PDF), Proceedings of the 9th International Conference on Pressure Surges, Chester, UK: 739–751
25. Tijsseling, Arris S.; Anderson, Alexander (2007), "Johannes von Kries and the history of water hammer", Journal of Hydraulic Engineering, 133 (1): 1–8
26. Joukowsky, N. (1898), "Über den hydraulischen Stoss in Wasserleitungsröhren" [On the hydraulic hammer in water supply pipes], Mémoires de l'Académie Impériale des Sciences de St.-Pétersbourg (1900), series 8 (in German), 9 (5): 1–71
27. Tijsseling, Arris S.; Anderson, Alexander (2006), The Joukowsky equation for fluids and solids (PDF)
28. Frizell, J. P. (1898), "Pressures resulting from changes of velocity of water in pipes", Transactions of the American Society of Civil Engineers, 39: 1–18
29. Hale, R. A. (September 1911), "Obituary: Joseph Palmer Frizell, M. Am. Soc. C. E.", Transactions of the American Society of Civil Engineers, 73: 501–503
30. Allievi, L. (1902), "Teoria generale del moto perturbato dell'acqua nei tubi in pressione (colpo d'ariete)" [General theory of the perturbed motion of water in pipes under pressure (water hammer)], Annali della Società degli Ingegneri ed Architetti Italiani (Annals of the Society of Italian Engineers and Architects) (in Italian), 17 (5): 285–325
• What is Water Hammer and Why is it Important That You Prevent it?
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