Cavitation in Control Valves

Technical Information

Part

3C

ontro

lVal

ves

Cavitation in Control Valves

3

Part 1: Fundamentals

Part 2: Self-operated Regulators

Part 3: Control Valves

Part 4: Communication

Part 5: Building Automation

Part 6: Process Automation

Should you have any questions or comments, please contact:

SAMSON AG Phone: +49 69 4009-1467V74 / Training Fax: +49 69 4009-1716Weismüllerstraße 3 E-mail: [email protected] Frankfurt Internet: http://www.samson.de

Technical Information

Cavitation in Control Valves

Symbols and units. . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Incipient cavitation . . . . . . . . . . . . . . . . . . . . . . . . . 10

Bubble implosion . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Cavitation intensity . . . . . . . . . . . . . . . . . . . . . . . . . 23

Cavitation erosion . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Cavitation resistance. . . . . . . . . . . . . . . . . . . . . . . . . 30

Avoiding cavitation . . . . . . . . . . . . . . . . . . . . . . . . . 34

Operation with cavitation . . . . . . . . . . . . . . . . . . . . . . 40

Influence on the hydraulic characteristics . . . . . . . . . . . . . . . 43

Changes in fluid properties . . . . . . . . . . . . . . . . . . . . . 49

Cavitation noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Cavitation luminescence . . . . . . . . . . . . . . . . . . . . . . . 56

Appendix A1:List of references . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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CON

TEN

TS

Symbols and units

A free cross-sectional area at the restriction [mm2]

C bubble volume per volume unit [-]

cF sound velocity in the fluid [m/s]

CT torque coefficient [-]

dh hydraulic diameter [mm]

E modulus of elasticity [N/mm]

E(f) spectral energy [W]

f frequency [s–1]

FF critical pressure ratio factor [-]

FL pressure recovery factor [-]

G constant [Nm]

H pressure height [bar]

HV Vickers hardness [-]

i index of throttling step [-]

k exponent for the ratio of erosion velocity [-]

K compressibility [m2/N]

KC pressure ratio at incipient cavitation influenceon flow rate [-]

KR cavitation resistance [h/mm3]

Kv flow coefficient [m3/h]

LWi unweighted sound power (inside) [dB]

MT torque [Nm]

n number of stages, times or events [-]

N amount of substance [mol]

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p pressure [N/m2]

pC critical thermodynamic pressure [bar]

pC pressure in the bubble [N/m2]

pG partial pressure of the gas [N/m2]

pmin minimum pressure in the control valve [bar]

pN standard pressure [N/m2]

pV vapor pressure [N/m2],[bar]

p1 upstream pressure [bar]

p2 downstream pressure [bar]

Q flow rate [m3/h]

r distance between sound source and observer [m]

R nucleus or bubble radius [m]

RE final radius of cavitation bubble [m]

RG gas constant [N/mm2]

Rk critical nucleus radius [m]

Rmax maximum radius of cavitation bubble [mm]

Rm tensile strength [N/mm2]

Rp 0.2 yield point at 0.2 % expansion [N/mm2]

R0 initial radius of cavitation bubble [m]

s distance between the center of the bubbleto the boundary [mm]

t time [s]

Ti temperature in the bubble [K]

T ambient temperature [K]

U circumference of the restriction [mm]

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UR centripetal velocity [m/s]

UR ultimative resilience [N/mm]

V(t) rate of change in bubble volume [m/s]

v kinematic velocity [m/s]

W reference deformation [N/mm2]

xF pressure ratio [-]

xFZ valve-specific cavitation coefficient [-]

y load of the control valve in percent [%]

α capillary constant [N/m]

ε expansion [mm]

εB rupture expansion when yield point is equalto tensile stress [mm]

ηF acoustical efficiency factor [-]

σB rupture strength [N/mm2]

ϕ opening angle of the butterfly disc [°]

ν kinematic viscosity [mm2/s]

ρF fluid density [kg/m3]

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Introduction

Cavitation has been a familiar phenomenon for a long time particularly inshipping. In 1917, the British physicist Lord Rayleigh was asked to investi-gate what caused fast-rotating ship propellers to erode so quickly. He discov-ered that the effect of cavitation, already proved in experiments by Reynoldsin 1894, was the source of the problem. Despite numerous investigations intothe subject of cavitation in the years that followed, many of the accompany-ing effects have still not yet been completely explained. This is no wonderconsidering the complexity of the process involving the areas of acoustics,hydrodynamics, thermodynamics, optics, plasma physics and chemistry.

Cavitation can be caused in a fluid by energy input. For example, a laserbeam creates a plasma in liquids which causes the liquid to evaporate creat-ing a cavity. Ultrasonic waves can be used to induce complex high-fre-quency alternating compression and rarefaction phases in liquids whichcause cavitation. In this way, cavitation effects can be applied usefully forcleaning surfaces, for non-invasive operations in the field of medicine andfor breaking down agglomerates in the textile finishing industry. In sewagetreatment plants, cavitation is used to break down molecules and bacteriacell walls, break up pollutants and dissolve out minerals from organic mate-rial.

Furthermore, cavitation can arise in hydrodynamic flows when the pressuredrops. This effect is, however, regarded to be a destructive phenomenon forthe most part. In addition to pump rotors, control valves are particularly ex-posed to this problem since the static pressure at the vena contracta even atmoderate operating conditions can reach levels sufficient for cavitation tostart occurring in liquids.

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The consequences for a control valve as well as for the entire control processvary and are often destructive:

�Loud noise

�Strong vibrations in the affected sections of the plant

�Choked flow caused by vapor formation

�Change of fluid properties

�Erosion of valve components

�Destruction of the control valve

�Plant shutdown

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Cavitation

Cavitation shall be generally understood as the dynamic process of the for-mation and implosion of cavities in fluids.Cavitation occurs, for instance,when high flow velocities cause the local hydrostatic pressure to drop to acritical value which roughly corresponds to the vapor pressure of the fluid.This causes small bubbles filled with steam and gases to form. These bubblesfinally collapse when they reach the high-pressure areas as they are carriedalong by the liquid flow. In the final phase of bubble implosion, high pres-sure peaks are generated inside the bubbles and in their immediate sur-roundings. These pressure peaks lead to mechanical vibrations, noise andmaterial erosion of surfaces in walled areas. If cavitation is severe, the hy-draulic valve coefficients as well as the fluid properties change.

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Incipient cavitation

The term ‘cavitation’ is derived from the Latin verb cavitare meaning ‘to hol-low out’, thus referring to the formation of cavities.

To create cavities in fluids, the fluid must first be expanded and then rup-tured. Theoretically, fluids can absorb high tensile strengths or negativepressures. Ackeret [1] estimates the negative pressure required to cause cav-itation in pure water at 20 °C, based on the minimum of the van der Waalscurve, and receives a theoretically possible tensile strength of 104 bar whichcorresponds approximately to the reciprocal compressibility of water.

Inhomogeneities (disturbances) in the quasi-crystalline structure of water,however, reduce the possible tensile strengths by minimum one order ofmagnitude. In all probability, submicroscopic accumulations of steam or gasmolecules are created at these disturbances with the molecules being in anunstable equilibrium with the fluid. In the case of external tensile strengths(negative pressure), these nuclei can exceed a critical diameter and thengrow spontaneously as steam is formed.

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Tens

ilestr

engt

h

The tensile strength of

the medium is reduced

by the disturbance

1500p [bar]

1200

900

600

300

00 20 40 60 80 100 [oC]

T

Fig. 1: Theoretical tensile strength values for perfect water

By means of statistical examination, Becker and Döring [2] determined theprobability for a critical nucleus to occur in dependence of the temperature,resulting in the theoretical tensile strength values for perfect water shown inFig. 1.

If these values were actually reached for industrial fluids, cavitation in con-nection with hydraulic systems would not be a matter for discussion. Thehighest values known until now were derived by measuring centrifugal forceas a function of the temperature with extremely pure water. As shown inFig. 2, the highest value to be achieved was only at 280 bars [3].

Fig. 2 also shows one of the many anomalies of water, i.e. the strong reduc-tion of tensile strength near the freezing point which is caused by the forma-tion of water crystals.

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280

p [bar]

240

200

160

120

80

40

00 200 400 [oC]

T

Fig. 2: Tensile strength values derived from measurements for pure water

The discrepancy between the theoretically and experimentally (under idealconditions) determined strength values shows that the microscopic bubblesfilled with gas and steam (cavitation nuclei), which exist in the fluid andwhose existence in water can be explained according to the model byHarvey [4], are a decisive factor in the occurrence of cavitation.

Spherical cavitation nuclei are stable when the fluid pressure p acting on thebubble surface and the partial pressure 2α/ R resulting from the surface ten-sion are in equilibrium with the sum of the partial pressures inside the bub-ble, i.e. the vapor pressure pV and the pressure of the enclosed gas volumepG:

In this equation, R is the radius of the bubble and α is the capillary constant. Ifyou consider the volume change of the bubble to be isothermal, the followingcan be derived from the general gas law for spherical nuclei:

and with (1)

with G being proportional to the gas volume contained in the nucleus.

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p pR

pG V+ = +2α (1)

pN R T

R

GRG

G=⋅ ⋅

⋅ ⋅=

43

33

π(2)

Cavitation nuclei are

decisive for the occur-

rence of cavitation

p p GR RV− = −

3

2α (3)

The representation p – pV = f(R) with G as a parameter exemplifies incipientcavitation: (Fig. 3)

When parameters are fixed, at first the radius of the nucleus will increaseonly slowly as the static pressure decreases. A small change in pressure onlyresults in unlimited bubble growth, i.e. incipient cavitation, after a criticalpressure difference has been reached which is mathematically defined byd(p – pV)/dR = 0 and which is derived from:

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-0.10.001

0.01

0.1

1

10

Rk [mm]

0 0.1 [bar]

p – pVG=10-12 Nm

G=10-11 NmG=10-10 NmG=10-9 NmG=10-8 NmG=10-7 Nm

Fig. 3: Critical nucleus radius with varying gas concentrations

Unstablearea

( )p pGV krit− = −

⋅⋅4

3 32α α (4)

Large nuclei, i.e. bubbles with a large G value, start to cavitate first. Theirsudden growth accelerates the ambient fluid and reduces the pressure lo-cally. This pressure reduction causes the next smaller bubbles to collapse,which in turn build up a pressure field that causes even smaller bubbles tocollapse, and so forth.

As a result, the critical pressure at which cavitation stops is higher than thecritical pressure at incipient cavitation. Lehmann and Young [5] examinedthe phenomenon of cavitation hysteresis in depth and found that the end ofcavitation can be more easily reproduced than incipient cavitation. In partic-ular cases, it therefore depends on the state of the liquid, especially the tem-perature, spectrum of nuclei, the content of dissolved gases and the surfacetension, at which static pressure cavitation begins. Generally, this is just be-low the vapor pressure. In practice, it is impossible to determine a liquid’sspectrum of nuclei in advance for most applications. Therefore, it is commonpractice in control valve sizing to describe the critical state of the cavitationnuclei at incipient cavitation by means of the vapor pressure of the liquid.

The cavitation coefficient xFZ

In the case of less viscous liquid flows around streamlined bodies, the internalfriction compared to the pressure may be frequently neglected. The velocitydistribution of these types of flow can be calculated on the basis of the poten-tial theory if the flow conditions are known. The pressure distribution alongthe body contour is derived from Bernoulli’s equation so that a relationshipbetween the minimum pressure pmin and the critical pressure can be statedaccording to equation (4).

In case of stalling flows as they occur in control valves, the potential theorycannot be used to determine the minimum pressure. Instead, the cavitationcoefficient xFZ [6] has proven useful.

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Incipient cavitation

starts just below

vapor pressure

Stalling flows occur in

control valves

It is based on the assumption that, in a control valve, the ratio of the externalpressure difference (p1 − p2) to the internal pressure difference (p1 − pmin) forall cavitation-free operating states equals a valve-specific value xFZ (Fig. 4):

Since the minimum pressure occurs in one of the unsteady vortex coresdownstream of the restriction, it cannot be determined by direct measure-ment.

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p1

p2

pV

pmin

p1 p2

Fig. 4: Distribution of pressure in the valve

xp pp pFZ =

−−

1 2

1 min

(5)

Minimum pressure

occurs downstream of

the restriction

It is therefore assumed that the minimum pressure pmin equals the vapor pres-sure pV of the fluid when cavitation noise begins, thus determining the pres-sure ratio xFZ as a function of the valve load y by means of noisemeasurements [7] (Fig. 5).

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LWi [dB]

xFxFZ

[-]

Fig. 5: Determining the cavitation coefficient xFZ

xF = 0.4

xFZ = 0.3

xF = 0.6

Fig. 6: Formation of cavitation zones with various xF values

The cavitation coeffi-

cient xFZ is determined

by noise measurements

If a valve’s xFZ values are known over the entire travel range, it can be determin-ed in advance for all operating pressure ratios

whether cavitation effects are to be expected. In case of an operating pressureratio xF < xFZ, there is no danger of cavitation occurring; when xF ≥ xFZ, a statio-nary cavitation zone builds up whose expansion is roughly proportional to thedifference (xF – xFZ), (Fig. 6).

However, since the difference pV – pcrit according to equation (4) is not cov-ered by the operating pressure ratio xF, these relationships can strictly speak-ing only be applied to media which conform to the test medium waterregarding their nuclei spectrum, surface tension and viscosity. Oldenziel [8]clearly showed this by measuring the pressure ratio xF at incipient cavitationas a function of the gas content of water (Fig. 7).

The experimental xFZ values should therefore be rounded to full five hun-dredths to account for the accuracy limits of the process.

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The operating case

xF = xFZ indicates

incipient cavitation

xp

p pFV

=−∆

1

(6)

28

24

20

16

12

8

4

00.1 0.2 0.3 [–]

xF

lml

Fig. 7: Relation between the pressure ratio xF and the gas content

Con

tent

ofdi

ssol

ved

gase

s

Bubble implosion

Due to the pressure recovery inevitable in control valves, the bubbles that arefilled with steam and gases reach zones of higher pressure where they im-plode at high bubble wall velocities. Calculations on the kinetics of the bub-ble collapse were made as early as 1917 by Rayleigh [9]. While neglectingthe surface tension and viscosity, Rayleigh’s calculations provide the implo-sion pressure for a spherical evacuated single bubble

where cF and ρF are the sound velocity and the fluid density, UR is the centrip-etal velocity, p0 the pressure at indefinite distance, R0 the initial radius and RE

the final radius of the bubble.

Güth [10] additionally accounts for the gas content of the cavitation bubblesand, assuming an adiabatic compression (p V⋅ χ = const.), receives the fol-lowing equation for the maximum implosion pressure.

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Pressure recovery leads

to bubble implosion

p c U c pR

Ri F F R F FE

= ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ −

ρ ρ2

310

3

3

0(7)

p pppi N

G

Nmax ( )= ⋅ − ⋅

−

−−

χχ

χχ

1 1

11

(8)Maximum implosion

pressure of a single

bubble

Neglecting the thermal conduction which, however, should be very low dueto the short implosion time, for the occurring temperature

An evaluation of the equations as a function of the ratio partial pressure of thegas inside the bubble pG to standard pressure pN, which is a measure for thegas content of the bubble, supplies the following maximum values for T = 293 Kand χ = 1.4:

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Maximum occurring

temperature

pp

G

N

[-] pi max

Nmm 2

Ti max [K]

0.1 1.3 610

0.05 7.2 990

0.01 405 3140

0.005 2290 5160

Table 1: Maximum values (pressure, temperature) with various gas content

( )T Tppi

G

Nmax = ⋅ − ⋅

−χ

χ1 1 (9)

Due to the pressure gradients in the ambient fluid or the influence of rigid boun-daries, the cavitation bubbles generally deviate from the spherically symmetricshape. They implode forming a microjet, as high-speed films show and as indi-cated according to [11] in Fig. 8 for three typical cases. Plesset and Chapman[12] have analyzed the implosion process and found the proportionality

for the jet velocity

Lauterborn [13] used a rotating mirror camera with a frame rate of900,000 pictures per second to show the formation of a microjet and determin-ed maximum jet velocities between 50 and 100 m/s, which confirmed the valu-es that have been theoretically determined by several authors.

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vp p

jetV

F

≈−∞

ρ(10)

The jet velocity of the

microjet reaches

up to 100 m/s

Asymmetric bubble

implosion leads to the

formation of a microjet

Fig. 8: Diagram illustrating the collapse of a bubble

Bubble moving into aregion of higher pressure

Bubble collapsing near thewall of a rigid boundary

Hemispherical bubbleclinging to the wall of arigid boundary

The pressure created by a fluid jet impact on the wall of a rigid boundary can beeasily estimated if the fluid jet is assumed to be a elastic deformable solid.

The application of the principle of conservation momentum results in

For water (ρF ≈ 1000 kg/m3, cF ≈ 1500 m/s), pressure surges with amplitudesbetween 750 and 1500 N/mm² are reached with the jet velocities mentionedabove. Depending on the size of the bubble (radius R), the surge

lasts between several microseconds and several milliseconds. The effect of onesingle surge is limited to an area of only a few micrometers in diameter.

The damage to the surface is caused to a considerable extent by the impact ofthe striking fluid jet and by the shock wave of the imploding bubble. Most prob-ably, the high temperature in the imploding bubble itself is also a key factor.

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t Rcsurge

F

= 2 (12)

p c vccsurge F F jet

F F

W W

= ⋅ ⋅ ⋅⋅⋅

+ρρ

ρ1

and because w c cw F F⋅ >> ⋅ρ

p c vsurge F F jet≈ ⋅ ⋅ρ (11)

In his publication [14], Lauterborn showed that the collapsing bubble,which acquires a toroidal shape, becomes unstable and collapses at se-veral locations.

These centers of collapse are the starting point for shock waveswhich lead to the characteristic toroidal pattern of damage. The ex-tent of damage depends mainly on the dimensionless parameter fordistance γ = s/Rmax , where s stands for the distance between the cen-ter point of the bubble and the boundary at the maximum bubble radi-us Rmax. A correlation between liquid jet and pattern of damage arisesonly when γ < 0.7, if the bubble is already located at the solid bounda-ry before it collapses. In this case, the jet reaches unobstructed the sur-face and leaves an impression on the surface [14].

Knapp [15] showed for a stationary cavitation zone according to Fig. 9that only one out of 30,000 bubbles implodes near the wall, thus ha-ving a damaging effect. The number of high-energy surges per cm²reaches its maximum at the end of the cavitation zone (Fig. 9).

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80

[Surges/cm2]

60

40

20

00 1 2 3 [–]

II0

0 I0

Fig. 9: Number of high-energy surges in a stationary cavitation zone

Cavitation intensity

While the effect of a collapsing bubble can be very well described by theoret-ical models, the cavitation intensity of a cavitation zone can currently only bedescribed in terms of quality. The following parameters are significant:

�xFZ value of the control valve

�operating pressure ratio xF

�pressure difference p2 − pV

�geometry of the control valve downstream of the restriction

�gas contents of the fluid

� fluid viscosity

�surface tension of the fluid

� fluid density

The difference between the pressure in the bubble and the pressure at the siteof implosion acts as the driving force when the bubble collapses, see equa-tions (10) and (11). Since the pressure in the bubble is almost equal to the va-por pressure, and the ambient pressure at the site of implosion correspondsapproximately to the downstream pressure, the damaging effect of the bub-ble increases as the difference p2 – pV increases. If the difference betweendownstream pressure and vapor pressure is less than 2 bars, as often is thecase in heat supply systems, there is no significant material erosion whennon-corrosive media are used. The cavitation zone extends and the numberof cavitation bubbles grows as the difference between the pressure ratio xF

and the valve-specific coefficient xFZ for incipient cavitation increase. On theone hand, this leads to an increase in the damaging effect at first. On the ot-her hand, as the difference between xF and xFZ increases, the duration of thebubble growth phase increases as well. Consequently a larger amount of ga-ses dissolved in the fluid diffuses into the bubble.

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The gas diffusion

increases as the differ-

ence between xF and

xFZ increases

The diffused gas is transported with the bubble into the cavitation zone and isreleased when the bubble collapses. As a result, the compressibility K of thefluid increases in the area of the collapse, while the density ρF decreases. Thesound velocity drops as a consequence

The result according to equation (11) is a reduction in pressure surge ampli-tudes causing the eroding effect of the collapsing bubbles to decrease. Thecavitation wear under otherwise identical conditions is smaller with supersa-turated liquids than with undersaturated liquids due to the same reasons. Be-sides, an extreme undersaturation causes a drop in the critical pressure atwhich incipient cavitation occurs to values far below the vapor pressure. Thismeans the incipient cavitation first occurs when the pressure ratio is greaterthan xFZ and is less intense due to the lack of cavitation nuclei. The bubblesimplode all the more energetically, the larger the pressure gradient downst-ream of the restriction is. At the given operating conditions, the pressure gra-dient is determined by the geometry of the control valve. The gradient isparticularly large when the free jet hits the valve body wall close the restricti-on. The energy dissipation is low as turbulent mixing has just started in thiscase, and the pressures close to the wall can reach values similar to the up-stream pressure p1. The driving force for the bubble implosion is then appro-ximately proportional (p1 – pV) and not as described above (p2 – pV).

A reduction in viscosity causes, when all other conditions remain the same, anincrease in the number and size of the bubbles. Additionally, the kinematic im-pulse of the microjet at low liquid viscosity is greater than at high viscosity. Iflaminar flow conditions are assumed due to the small size for the microjet, thenthe jet velocity is inversely proportional to the viscosity.

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cKF

F

=⋅1

ρ(13)

Diffused gas increases

the compressibility of

the fluid

The eroding effect

decreases as the com-

pressibiliy increases

Turbulent mixing causes

the pressure to drop

Then the much quoted exponential correlation between the material erosion ∆mand jet velocity v follows

for the distinct influence of operating viscosity on the material erosion

The pressures occurring when the bubble implodes are proportional accord-ing to equation (7) to the root of the density. This is why the erosion rate isparticularly high when cavitation occurs in mercury or liquefied metals.

The surface tension or capillarity makes the pressure in the bubble increaseaccording to equation (1). Therefore, liquids with smaller surface tensionthan water cavitate at pressure ratios lower than xFZ. When the conditionsare otherwise the same, the size and number of cavitation bubbles increasesas the surface tension lessens, while the driving force during bubble implo-sion is reduced. Technical literature and other sources supplies deviating de-tails about the effect of surface tension on material erosion. It may, however,be assumed that the surface tension influences incipient cavitation consider-ably, but its effect on material erosion is small when the cavitation is pro-nounced.

A certain judgement, that experienced engineers are most likely to have, isneeded to assess cavitation erosion as the effects of the above described pa-rameters affect and overlap each other.

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∆∆

mm

vv

kk

1

2

1

2

1 8=

= ... (14)

∆∆

mm

k

1

2

2

1

=

νν

(15)

The erosion rate becomes

more rapid as medium

density increases

Cavitation erosion

The material surface, depending on its structure, is deformed, loosened andeventually eroded in particles in various ways due to the frequent strain fromthe pressure waves created by the microjet occurring when the bubble col-lapses. Fig. 10 shows how cavitation erosion is generally subdivided intothree areas.

� In area I, termed the incubation period, a loss in weight is not yet measura-ble.

�Area II is characterized by an almost constant erosion rate. Areas anddepths of the material erosion increase with time.

� In area III, the surface that is already strongly fractured reduces the proba-bility for an implosion close to the surface as it acts as a kind of protectivecushion; the material erodes at a much slower rate.

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I

II

III

Fig. 10: The course of development of cavitation erosion

Loss

inw

eigh

t

Time

Cavitation erosion is

subdivided into three

areas

During the incubation period, the surface of ductile metals first only under-goes elastic deformation and then plastic deformation. Dents and bulgesarise. Their number increases as time passes. After the deformability of thematerial is exceeded at the end of the incubation period, fractures occur andindividual particles break off.

The deformation phase does not take place to a great extent with brittle met-als due to the high density of dislocation obstacles. As a consequence, inter-nal tensions form that exceed the material strength at the end of theincubation phase. Fractures and ruptures occur that cause a monotonouslyincreasing weight loss. In cast iron, the graphite phase is eroded when thecavitation strain starts, meaning an incubation period can only be mentionedin conjunction with pearlite or ferrite erosion.

After the graphite nodules have been eliminated from spheroidal cast iron,the soft ferrite phase flows into the emptied troughs. The pearlite acts as asupporting frame and slows down surface deformation. The end of the incu-bation period is reached when the ferrite breaks off at the trough edges afterstrong deformation and the pearlite material areas break away due to mate-rial fatigue.

In cast iron with lamellar graphite, the ferrite is eroded after the incubationtime elapses without much plastic deformation because after the graphite hasbeen removed, the contact between the ferrite blocks remains restricted to afew metal bridges which cannot withstand the constant strain.

In ceramic material, microfractures start to appear right from the beginningof the strain without undergoing any noticeable plastic deformation before-hand. As the strain continues, the density of fractures continuously grows. Af-ter the incubation time has elapsed, the fractures have spread and joinedeach other and breakages occur. The incubation in plastics is similar to duc-tile metals where plastic deformation, formation of cracks and spread ofcracks occurs. In crystalline plastics, these structure faults are formed by dis-location, in amorphous plastics by the breaking up of atomic bonds.

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In all materials, surface roughness in the area where the bubbles implodeleads to a noticeable increase in material erosion since the machiningnotches help the bubbles to implode energetically and the ridges of theroughly machined surfaces resist the imploding bubbles less than even sur-faces would. The influence of the surface roughness must though be observedin conjunction with the microhardness achieved by machining. According to[16], an austenitic steel (EN material no. 1.4919, corresponding toAISI 316 H) with a milled surface has a microhardness HV0.05 of 430,whereas it has just 246 with an electrolytically polished surface.

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Cavitation corrosion

Apart from mechanical strain, corrosive and stress corrosive influences de-termine the erosion rate of the material. The combination of cavitation erosi-on and corrosion where the aggressive components can intensify each otheris termed cavitation corrosion. The effect of impacting liquid jets intensifiesthe corrosive attack as the forming top and passivation layers are immedi-ately worn away, causing the high initial corrosion rate typical of bright me-tal surfaces to be kept as long as the strain lasts.

A beneficial factor for the corrosion process is also the free oxygen which thecavitation bubbles absorb by diffusion, also from undersaturated liquids,while they are growing and which they then release in the cavitation areawhen they implode.

Additionally, the ions of a corrosive medium are likely to interact with thecrystallographic slip steps and fractures of areas that have undergone plasticdeformation due to liquid pressure waves. This leads to an accelerated mate-rial destruction due to intensified fracture formation or due to fracturespreading.

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Cavitation resistance

Magnetostrictive transducers (Fig. 12) as well as cavitation chambers(Fig. 11), electromagnetic, piezoelectric and ultrasonic vibrating instrumentsare mostly used in laboratories to investigate cavitation resistance of materi-als. In the magnetostrictive transducer, nickel laminations are excited to vi-brate by high-frequency alternating currents.

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Fig. 11: Cavitation chamber

Fig. 12: Magnetostrictive transducer

Heating/cooling

Field coil

Sample

Sonotrode

Sample

These vibrations are transferred to a coupled sonotrode, which is designed tovibrate in response at the predetermined frequency. The material samplesare attached either to the free end of the rod and/or face it at a defined dis-tance.

The liquid surrounding the free end of the rod cannot follow the high-fre-quency rod oscillations due to its mass inertia, causing liquid cavities in theform of small bubbles to be formed which erode the material when they im-plode. The test set ups allow cavitation resistance of various materials to beinvestigated under controlled conditions, making it possible to specify a cor-relation between the mechanical material characteristics and the erosionrate.

According to the investigations of R. Garcia and F.G. Hammitt [17], the cavi-tation resistance KR is proportional to the deformation energy (introduced byHobbs) up to the point of fracture UR (ultimative resilience – Fig. 13).

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σB

σ

εBε

UR= .σB.εB

ε

2

=

1

.σB2

21

σS

[N/mm2]

[mm]

Fig. 13: Material property UR

K URR ≈ (16)

Cavitation resistance KR

UR is a combined material property that describes to some extent the energywhich could be stored elastically in a material per unit of volume if the yieldpoint could be raised to the level of tensile stress.

Berger [16] describes the cavitation resistance with the relation

In the equation, E stands for elasticity module, HV for Vickers hardness, Rm

for tensile strength and Rp 0.2 for the top yield point or the strength at 0.2%elasticity. The good match of the values calculated according to the equa-tion (17) with values determined in a cavitation chamber (nozzle sample)can be seen in Fig. 14.

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K R W HVE RR

m

p

≈ ⋅ ⋅⋅

1071 0 125 1971

0 5620 2

0 618

. . .

..

.(17)

14

KRCK 45

12

measuredcalculated

material

10

8

6 GG

G 4

0G

GG

40

(aus

teni

tic)

C10

St 5

0-2

Ck

45C

45 te

mpe

red

90 M

n V

8X

22 C

r N

i 17

X 90

Cr

Mo

V 1

8X

5 C

r N

i 18

934

Cr

Ni M

o 6

16 M

n C

r 5

42 M

n C

r 4

42 M

n C

r 4

tem

pere

dSp

ecia

l bra

ssC

u A

I 10

Ni

AI C

u M

g 1

AI Z

n M

g

4

2

0

KR

Fig. 14: Cavitation resistance

The equations (16) and (17) are only suitable for assessing metals with suffi-cient elasticity which allow the yield point and deformation energy to be de-termined in a tensile test.

The applicability of the equations (16) and (17), that were derived from testswith clear water and hydraulic fluid, is still restricted to cases using less cor-rosive fluids. If cavitation corrosion as described in the section on cavitationcorrosion is expected, it is recommendable upon selecting the material topay attention primarily to the good corrosion resistance and then to take thecavitation resistance into consideration.

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Avoiding cavitation

As already explained on page 14, cavitation is avoided when the pressureratio xF at the control valve is smaller than the corresponding xFZ value for alloperating cases. If the operating pressure ratio xF is kept small by a cleverplant layout, the selection of the control valve decides whether the ratioxF < xFZ is fulfilled and a cavitation-free flow can be guaranteed.

The xFZ value, which can theoretically be 1 (no pressure recovery) is deter-mined considerably by the Carnot impact loss which is a function of the ope-ning ratios (nominal size cross-section to restriction cross-section). Theimpact loss and the xFZ value with it, increases as the opening ratio increases,whereas the Kv value drops as the opening ratio increases, which can also beexpressed by Kv Control valve/Kvs Ball valve.

The xFZ value range of various types of control valves is shown in Fig. 15 as afunction of the Kv value of the valve related to the Kvs value of a completelyopen ball valve in the same nominal size.

The bandwidth of the xFZ value range is expressed to a great extent by the hy-draulic diameter

the shape of the valve plug and seat and the number of pressure reductionstages.

For the free cross-section A = DN2⋅ π/4 of a non-reduced ball valve, the cir-cumference U = DN · πand thus the hydraulic diameter is equal to the nomi-nal size DN.

Another expression for the hydraulic diameter is the valve style modifier Fd

used in the IEC 60534-2-1 standard [25].

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The xFZ value increases

as the opening ratio

increases

dAUh = ⋅4 (18)

In a linear valve with a parabolic plug, the free cross-section is ring-shapedat small loads and the hydraulic radius is just a fraction of the nominal diam-eter DN. This results in a large surface area of the free jet downstream of therestriction which leads to an intensive impulse exchange with the surround-ing medium and causes high pressure losses.

As can be seen in Fig. 15, ball valves only allow the medium to be controlledwithout cavitation at small pressure ratios. Butterfly valves and rotary plugvalves are slightly better, whereas linear valves allow control without anycavitation even at high pressure ratios when the plug is designed accord-ingly.

Control valves which can be fitted with anti-cavitation trims (Fig. 16) for re-ducing cavitation and multi-stage axial plugs (Fig. 17) should be given aspecial mention here.

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1

xFZ [-]

[-]

Kv Control valve

Kvs Ball valve

00 1

dh < DNdh = DN

Linear valves

Rotary valve/ Butterfly valve

Ball Valves

Fig. 15: Control valve operating range without cavitation

Linear valves allow

control without any

cavitation even at

high pressure ratios

The system illustrated in Fig. 16 is a seat and plug trim specially develo-ped for cases where cavitation occurs and designed to be fitted in existingvalves. The trim has been optimized in a series of flow simulations and in-tensive tests. The plug is double guided in the body to prevent mechanicalvibrations, and the plug contours have been designed for better flow cha-racteristics. The seat diameter has not been reduced to keep the hydraulicdiameter as small as possible. In combination with the special shape ofthe plug and seat, this is particularly effective. Additionally, a maximumof four attenuation plates can be integrated into the seat to additionallyincrease the xFZ value at high valve loads [18].

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Fig. 16: AC trim system (SAMSON)

Fig. 17: Multi-stage axial plug (SAMSON Type 3255 Valve)

For an initial rough assessment, it is possible to assume that the xFZi values ofeach individual stage are the same in a valve with a multi-stage plug. Then,the xFZ value is obtained for an n-stage control valve according to

and for the Kv value ratio of each stage

The Kv value of an n-stage control valve arises from the Kvi values of eachindividual stage according to

The number of stages n required to manage an operating pressure ratio xF

free of cavitation is obtained when the xFZi value is known from

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( )x xFZ FZin= − −1 1 (19)

KK

xvi

viFZi

+

= −1

1 (20)

K

K K K

v

v v vn

=+ + +

11 1 1

12

22 2

...(21)

nxx

F

FZi

=−−

lg( )lg( )

11

(22)

The relationships are easier to understand when looking at Fig. 18. It can beseen that a five-stage valve whose individual stages each have an xFZi valueof 0.3 reaches an xFZ value of > 0.8, its Kv value is however just 30% of theKvn value of the last stage. The Kvn value corresponds approximately to theKvs value possible in a one-stage valve of the same nominal size. Besides theaxial stage plug, the radial stage plugs have proven themselves well in prac-tice (Fig. 19, left).

The xFZ value of the perforated plug (Fig. 19, center) is determined by theopening ratio and the hydraulic radius of the largest hole. The division ofholes should be kept to at least three hole diameters to avoid the free jets fromjoining together before the impulse exchange with the surrounding mediumhas finished. Just relatively small Kvs values (or large opening ratios) are pos-sible especially with equal percentage characteristics. But they lead to highxFZ values as shown in Fig. 15. Valves with perforated cages (Fig. 19, right)function similarly to valves with perforated plugs.

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0.10.1

0.2

0.4

0.6

0.8

1.0

Kv/Kvn

0.2 0.3 0.4

n=5

n=4

n=3

n=2

n=1

xFZi = 0.1

0.5 0.6 0.7 0.8 0.9 1.0 xFZ

0.2 0.3 0.4 0.5 0.6

Fig. 18: xFZ values of multi-stage valves

A special design (deviations, restriction cascades) of the ducts allows thecountermeasures which increase the xFZ value, for example, high openingratio, small hydraulic radius and multi-stage throttling, to be combined inone valve. High xFZ values (i.e. low pressure recovery), on the one hand, re-quire large opening ratios, and thus relatively small Kv values. Large Kv val-ues related to the nominal size, on the other hand, always lead to relativelysmall xFZ values. The xFZ values are listed in the data sheets of control valvemanufacturers. Specifications that are outside of the value range shown inFig. 15 should be viewed critically though.

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Fig. 19: Plug designs

Operation with cavitation

Since xFZ values necessary for cavitation-free operation cannot always beachieved, or the control valves are too susceptible to dirt or too expensi-ve, making their use uneconomical, valves are often operated in condi-tions where cavitation occurs. A troublefree operation with a satisfactoryservice life can be guaranteed though if the effects of cavitation are takeninto consideration on sizing and selecting the valve. To ensure that thevalve body walls are outside the area at risk of erosion, a larger valveshould be selected than required by the valve sizing simply based on theseat diameter. To achieve this, the nominal diameter should be selected toensure that a certain outlet velocity is not exceeded. It can then be assu-med that the cavitation zones will not reach up to the valve walls, and thatthe bubbles implode without any harm and with little effect (apart fromnoise).The following table helps to show whether erosion damage due to cavita-tion is to be expected. This is the case when xF > xFcrit,cav as well as whenp1-p2 > ∆pcrit,cav [18]. Additionally, a nominal valve size should be selec-ted to ensure that the outlet velocity does not exceed 4 m/s.

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Valve designxFcrit,cav[-]

∆pcrit,cav[bar]

Single-stage linear valves 0,7 15

Single-stage linear valves withstellited or hardened trim

0,7 25

3-stage linear valves 1,0 100

5-stage linear valves 1,0 200

Rotary plug valves 0,4 10

Butterfly and ball valves 0,25 5

Table 2: Limit values for preventing cavitation erosion

The sealing slope and plug surface of parabolic plugs are particularly atrisk from erosion (Fig. 20).The area around the sealing edge should begiven a Stellite facing. A hard facing of the whole surface is recommen-ded when the cavitation intensity increases.

V-port plugs are exposed less to cavitation attacks than parabolic plugsunder the same operating conditions. Due to the diverted jet (Fig. 21), thecavitation zones do not stick to the plug surface, meaning the cavitationbubbles implode almost without any effect.

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Fig. 20: Areas of a parabolic plug at risk from erosion

Fig. 21: Diverted jet occurring with a V-port plug

The diverted jet pre-

vents the V-port plug

from being eroded

The sealing slope and

plug surface of parabo-

lic plugs are particulary

at risk from erosion

The valve seat is not exposed to the cavitation attack when the mediumflows against the closing direction. However, if the medium flows in thereverse direction to protect the valve body walls and plug facing (risk ofpressure surge waves), the seat and plug surface are then particularly atrisk from erosion. A satisfactory service life can only be achieved in thiscase by using highly resistant materials.

The perforated plug (Fig. 22) is better suited for medium flow in the reverse di-rection. A steep pressure gradient that makes the cavitation bubbles implodeforms in the center of the plug, i.e. in sufficient distance away from the surfacedue to the collision of the partial flows. However, the cavitation zone moves outof the plug when the differential pressure rises, causing the body floor toerode. The above mentioned pressure ratios and differential pressures shouldonly be used as a reference for identifying the cavitation intensity. The limitscan be considerably reduced as a result of cavitation corrosion, especiallywhen corrosive fluids are used.

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A rising differential

pressure can cause the

body floor to erode

A medium flowing in

the reverse direction

increases the risk of

erosion

Fig. 22: Cavitation zone with a perforated plug

Influence on the hydrauliccharacteristics

The continuity of the fluid phase is interrupted when cavitation occurs and thedynamic interaction between flow and its restriction is affected. Additionally,cavitation makes the compressibility of the fluid increase locally and in thisway reduces the sound velocity (equation 13). The density of the fluid, too, isdrastically reduced by the bubble volume in the area of the restriction as thepressure ratio rises.

These effects limit the flow rate in control valves (choked flow) if a certain dif-ferential pressure (∆pmax) is exceeded as shown in Fig. 23.

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Cavitation changes the

density and compressi-

bility of a medium

xFZ

KC

FL

∆p

Q

∆pmax

Fig. 23: The valve-specific factor FL

The critical pressure pDK is reached at the vena contracta at the differentialpressure ∆pmax, which is below the vapor pressure according to measure-ments by Stiles [19] (Fig. 24).

Stiles found the relationship shown in Fig. 25 for the ratio between criticalpressure at the vena contracta and the vapor pressure

from measurements using Frigen 12, which can also be approached usingthe following equation:

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p1

p2

pV

pDK

∆pDK ∆pmax

Fig. 24: The critical pressure pDK

The critical pressure is

below vapor pressure

FppF

DK

V

= (23)

FppF

V

C

= −0 96 0 28. . (24)

Baumann [20] introduced the valve-specific factor FL for the ratio bet-ween the smallest differential pressure across the valve at which the cho-ked flow starts, ∆pmax and the critical differential pressure at the venacontracta, ∆pDK.

FL value is the pressure recovery factor. It is determined from a flow rate measu-rement as in Fig. 23. In this measurement, the upstream pressure is kept cons-tant and the downstream pressure is reduced until choked flow starts. A detaileddescription of the procedure is specified in IEC 60534, Part 3.

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FppL

DK

2 =∆∆

max (25)

0.600 0.400.20

0.6

0.7

0.8

0.9

1.0

1.000.80

FF

pVpC

Fig. 25: FF factor used to determine the critical pressure ratio

FL: the factor for

pressure recovery

The maximum value for differential pressure at which the flow rate isachieved due to cavitation is obtained with the FL value and equation (23).

The first deviation in flow rate from the general measurement equation

is indicated by the pressure ratio.

The Kc value is often referred to as the incipient cavitation index. Yet, inci-pient cavitation already occurs, as shown in Fig. 23, at the much smallerpressure ratio xFZ. If both coefficients are confused with another, it causes se-rious mistakes in the assessment of the pressure recovery, the cavitation in-tensity and noise emission.

The distribution of pressure on the surface of the closure member is alsochanged by the change in flow pattern due to cavitation.

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( )∆p F p F pL F Vmax = − ⋅21 (26)

Kp

p pCV

=−∆

1

(28)

Q K pv= ⋅ ⋅∆ ρρ0

(27)

Therefore, the hydraulic torque in butterfly valves, which is calculated withcavitation-free flow according to

and works in the closing direction, struggles against a limit value as the pres-sure ratio grows. In equation (29), D is the disc diameter, CT the torque coef-ficient dependent on the opening angle ϕ, and H the static differentialpressure p1 – p2 across the butterfly valve, increased by the impact pressureof the flow velocity v:

In Fig. 26, the torque for an opening angle ϕ of 60° is shown as a function ofthe pressure ratio xF at constant upstream pressure.

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Hydraulic torque with

cavitation-free flow

600

1000 0.5

FL²

1.0

xF

MT [Nm]

[-]

Fig. 26: Course of torque in a butterfly valve

H p F= +ρ

ν2

2 (30)

M C D HT T= ⋅ ⋅3 (29)

The torque reaches its maximum at approximately the same differential pres-sure at which also the choked flow occurs (equation 26). As the pressure ra-tio (or ∆p) further increases, the torque drops slightly at first and then reachesa limit value irrespective of the differential pressure.

When the limit value is reached, the cavitation zone stretches over the wholedownstream pressure side, causing the pressure distribution on the inlet sideto be approximately constant with the vapor pressure pV irrelevant of howthe pressure ratio increases.

The flow forces around the plug change due to cavitation even in linearvalves. However, this change in force is insignificant, except in self-operatedregulators where the plug forces must be carefully balanced out by means ofpressure balancing.

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Torque reaches its

maximum at

choked flow

Changes in fluid properties

Not just the surface of the hydraulic components, but even the fluid itself, isexposed to extreme loads when gas-filled cavitation bubbles implode.

For example, when bubbles implode in hydraulic fluids, temperatures canoccur that are sufficient under certain circumstances to ignite bubbles con-taining air and oil vapor. This process, which is called the microdiesel effect,leads to an accelerated ageing of hydraulic fluids. Cavitation increases thefree gas content in the fluid: parts of the gas dissolved in the fluid diffuse intothe cavitation bubbles during the growth phase.

The gas parts are released when the bubbles implode and locally increasethe compressibility of the fluid. This is accompanied by a reduction in soundvelocity, meaning pressure surge and sound propagation calculations inpipelines where cavitation occurs is made more difficult. Fig. 27 shows thesound velocity in water as a function of the bubble concentration accordingto [21].

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The microdiesel effect

leads to an accelerated

aging of hydraulic

fluids

104

[m/s]

[%]

103

102

101

0.0001 0.1 100

1080m/s1500m/s

p = 10 barp = 1 barp = 0.1 bar

340m/s

108m/s

cF

Fig. 27: Sound velocity as a function of the bubble concentration

If oxygen is also released by the cavitation process, the oxidizing effect of thefluid increases. The contamination of the medium caused by cavitation-in-duced material erosion should not be underestimated in closed circuits.

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Cavitation noise

The pressure peaks induced by the bubble implosions cause material erosionas well as a loud typical noise. The theoretical approaches to explain hownoise develops are based on individual bubbles which implode concentri-cally in an infinitely expanded fluid and without affecting each other. Thecavitation bubble can then be regarded as an isotropic radiator of zero or-der (monopole source) which creates in the far field the sound pressure

In the equation, V(t) is the rate of change in bubble volume which can be de-termined approximately using the Rayleigh-Plesset differential equation, andr is the distance between the sound source and the point of observation. Thespectral distribution of the energy E(f) of the noise created by a bubble is ob-tained from the squared Fourier transform of the pressure over time in the farfield

while neglecting the retarded time trcf

−

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Cavitation bubbles act

as an isotropic radiator

of zero order

p r tr

ddt

V t t rc

F

F

( , ) ( )= ⋅ ⋅ ⋅ −

ρπ4

2

2(31)

E f p t e dtift( ) ( )= ⋅

−

−∞

+∞

∫ 2

2

π (32)

( )( )E fr

f V t e dtift( ) = ⋅

⋅ ⋅ ⋅ −

−∞

+∞

∫ρ π π

24 2

2

(33)

The spectral energy rises at first with the frequency f raised to the power offour and reaches a maximum at a frequency whose the inverse ratio is ap-proximately the same as the collapsing time of the cavitation bubble. As aconsequence of the collapsing time being directly proportional to the bubbleradius, the maximum noise is shifted to lower frequencies as the bubble ra-dius increases. At frequencies above the maximum, the spectral energy de-creases at f –2/5.

If the bubble collapses are observed as random events not connected withone another, whose frequency follows a Poisson distribution, it is possible toderive the total spectral sound energy Etotal(f) from the spectral energy of indi-vidual events E(f) and a mean number of bubble collapses per unit of time n.

These theoretical considerations made for individual bubbles apply approxi-mately when cavitation just starts. As the cavitation progresses, distinctivecavitation zones form in which the effects of each individual bubble overlapeach other, and the contribution of each bubble to the entire noise dependson the history of the neighboring bubble.

More extensive investigations therefore observe the cavitating fluid as aquasi-continuum of density

In the equation, C is the total volume of all bubbles per volume unit. Lyamshev[22] discovered on this basis that the sound intensity of a cavitation zoneequals the flow velocity raised to the power of four, or is proportional to thesquare of the differential pressure.

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E f n E ftotal( ) ( )= ⋅ (34)

Observations of individ-

ual bubbles only apply

to incipient cavitation

ρρ

=+

F

C( )1(35)

The theoretical approaches were extended and refined by various authors.Despite this, no one has yet successfully described the exceptionally complexcorrelations in the cavitation zone downstream of a control valve for variousmedia and pressure ratios.

Therefore, the currently valid equations as per the German directiveVDMA 24422 to predict noise emission of cavitating flows of control valvesare not based on hydrodynamic and thermodynamic models, but instead de-scribe the course of internal sound power level on the basis of measurementsrelated to xF, z, ρF and ∆p.

In this case, neither the influence of surface tension nor viscosity nor the influ-ence of the gas contents are taken into account. The influence of the densityand the differential pressure important for the course of the bubble implosion

are also only reproduced inexactly in the empirical VDMA equations which arebased mainly on measurements with cold water (pV ≈ 0 bar, ρF ≈1000 kg/m3).The VDMA calculation methods [23] introduced in 1979 and improved in1989 are, however, well proven and recognized worldwide.

Note: The IEC 60534-8-4 international standard is currently being revised.The main features of the new standard are shown in [25].

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The equations accor-

ding to VDMA are ba-

sed on empirical data

p px

xpV

F

F2

1− =

−⋅ ∆ (36)

When cavitation (xF >> xFZ) occurs, the acoustic power emitted in the octaveband range of 0.5 to 8 kHz in the pipeline is obtained according to

The first row of the equation (37) represents the sound power level which iscaused by the turbulent flow noise. The second row serves to calculate thesound power component which arises when the flow noise caused by the tur-bulence overlaps the cavitation noise resulting from the temporal static bub-ble collapse. Fig. 28 shows the course of the standardized sound power forstandard valves with parabolic plugs (∆LF = 0) in relation to the operatingpressure ratio with xFZ as a parameter. When xF > xFZ applies, the noiseemission rises steeply, reaches a maximum and drops back to the sound po-wer level caused by the turbulence when xF = 1.

The drop in sound power level at high pressure ratios is based, on the onehand, on the compressibility of the fluid that increases with xF (see section onavoiding cavitation) and on the reduction of the driving force p2 – pV whenthe bubble implodes (equation 10). Cavitation behavior deviating from thecalculation can be taken into consideration by the valve manufacturer byspecifying a valve-specific correction coefficient ∆LF = f (xF, y).

The spectral distribution of the internal sound power level depends on the de-sign, pressure ratios, load and xFZ value of the valve. According toVDMA 24422, the spectral distribution can be reproduced for practicalapplication in the octave band range 0.5 to 8 kHz irrelevant of the operatingcondition using a noise spectrum that drops 3 dB per octave.

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LW p

FWiF

F L

= + ⋅⋅ ⋅

⋅134 4 10

2. lg

∆ ηρ

( )Lwi

x xx

xx

LFZ F

Fx

F

FZFFZ

− ⋅⋅ −

⋅−

−+120

1 10011

0 0625 0 8. .

lg.

∆

with p F p F pL F V≤ − ⋅21( ) (37)

Several measurements performed by SAMSON resulted, however, in a soundpower spectrum as shown in Fig. 29. At incipient cavitation (xF = xFZ), the soundpower level radiated in the individual octave bands is approximately the same;the low frequencies dominate as the pressure ratio rises.

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. ∆p

ρ F. F

L2L W

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lg–1

34.4

15

10

5

00 0.5

z y=0.

1

z y=0.

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z y=0.

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71

xF

Fig. 28: Sound power for standard valves with parabolic plug (∆LF = 0)

L Wi –

LW

i(fm

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–40

–20

01

xF

8 kHz

8 kHz

4 kHz

2 kHz1 kHz0.5 kHz

4 kHz

2 kHz

1 kHz0.5 kHz

Fig. 29: Measured sound power spectrum (see also [25])

The low frequencies

dominate as the

pressure ratio rises

Cavitation luminescence

To conclude with, another cavitation phenomenon should be introducedwhich is insignificant in conjunction with control valves, but is very suitable todemonstrate again the extreme, mostly uninvestigated conditions that existwhen bubbles implode: cavitation luminescence. In the cavitation zone, smallflashes of light can be seen under certain conditions that have a spectral dis-tribution from near infrared right up to the ultraviolet range. Several authorstrace the light emission back to the photochemical recombination of gas mol-ecules that have thermally dissociated when the bubbles imploded. Othersbelieve that the flashes of light occur when the free ions recombine whicharise on bubble implosion due to the mechanical load of the molecules. Afurther theory postulates aspherical bubbles which are surrounded by alayer of dipoles. The light appearances arise when the dipoles discharge onbubble implosion. Finally, there is the theory termed “hot spots” that assumesthe gas compression linked with the bubble implosion leads to such ex-tremely high temperatures that the gas starts to glow.

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Appendix A1:List of references

[1] Ackeret, J.: Experimentelle und theoretische Untersuchungen überHohlraumbildung im Wasser, Techn. Mechanik undThermodynamik, Band 1, Nr. 1, Berlin, Januar 1930

[2] Becker, R.; Döring, W.: Kinetische Behandlung in übersättigtenDämpfen, in Volmer, M.: Kinetik der Phasenbildung, VerlagSteinkopf, Leipzig 1939, pages 156–165

[3] Briggs, L. J.: The Limiting Negative Pressure of Water, Journal ofApplied Physics, Vol. 21, July 1950, pages 721–722

[4] Harvey, E. N.; McElroy, W. D.; Whitely, A.H.: On Cavity Forma-tion in Water, Journal of Applied Physics, Vol. 18, Feb. 1947,pages 162–172

[5] Lehmann, A. F.; Young J.O.: Experimental Investigations of Incipi-ent and Desinent Cavitation, Journal of Basic Engineering, June1964, pages 275–284

[6] Jeschke, N. und Gruner, K.: Geräuschverhalten von Stellventilen,Regelungstechnische Praxis, Mai 1975

[7] DIN 45635: Geräuschmessungen an Maschinen, Blatt 50:Luftschallemission, Hüllflächenverfahren, Armaturen, August 1987

[8] Oldenziel, D. M.: Bubble Cavitation in Relation to Liquid Qual-ity, Delft Hydraulics Laboratory, Publication No. 21, May 1979,pages 224–240

[9] Rayleigh, L.: On the Pressure Developed in a Liquid During the Collapseof a Spherical Cavity, Phil. Marg., Vol. 34, 1917, pages 94–98 APP

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[10] Güth, W.: Zur Entstehung der Stoßwellen bei der Kavitation,Acustica, Vol. 6, 1956, pages 526–531

[11] Knapp, R.T.; Daily, J.; Hammit F.: Cavitation, McGraw-Hill BookCompany, New York, 1970, pages 343–347

[12] Plesset, M. S.; Chapman R. S.: Collapse of an Initially SphericalVapour Cavity in the Neighbourhood of a Solid Boundary, Jour-nal of Fluid Mechanics, Vol. 47, Part 2, pages 283–290

[13] Lauterborn, W.: Kavitation durch Laserlicht, Acustica, Vol. 31,1974, pages 51–78

[14] Lauterborn, W.: Jetbildung und Erosion, VDI TechnologiezentrumPhysikalische Technologien, Technologiefrüherkennung Band 32,Von der Kavitation zur Sonotechnologie, 2000, pages 35–36

[15] Knapp, R.T.: Recent Investigations of Cavitation and CavitationDamage, Trans. ASME, 77, 1955, pages 1045–1054

[16] Berger, J.: Kavitationserosion und Maßnahmen zu ihrer Vermei-dung in Hydraulikanlagen für HFA-Flüssigkeiten, Dissertation,TH Aachen, 1983

[17] Garcia, R.; Hammitt, F.G.: Cavitation Damage and Correlationswith Material and Fluid Properties, Journal of Basic Engineering,December 1967, pages 753–763

[18] Kiesbauer, J: Stellventile bei kritischen Prozessbedingungen inRaffinerien (Control valves for critical applications in refineries),Sonderdruck „Industriearmaturen”, 2001, SAMSON AG, pages4–5

[19] Stiles, G. F.: Sizing Control Valves for Choked Conditions Due toCavitation or Flashing, ISA Handbook of Control Valves, InstrumentSociety of America, 1976A

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[20] Baumann, H. D.: The Introduction of a Critical Flow Factor for ValveSizing, Paper presented at ISA Annual Conference, October 1962

[21] Wijngaarden, L.: Sound and Shock Waves in Bubbly Liquids, inLauterborn, W.: Cavitation and Inhomogenities in UnderwaterAcoustics, Springer Series in Electrophysics 4, Springer Verlag,Berlin Heidelberg, 1980, pages 127–140

[22] Lyamshev, L. M.: On the Theory of Hydrodynamic Cavitation Noise;Soviet Physics-Acoustics, Vol. 14, No. 4, April-June 1970

[23] VDMA-Einheitsblatt 24422, Richtlinien für die Geräuschberechnungbei Regel- und Absperrarmaturen, Beuth Verlag, Berlin, Januar 1989

[24] Vnucec, D., Kiesbauer, J.: Übersicht über die Genauigkeit vonSchallberechnungsnormen b. Stellgeräten (Evaluating the calcula-tion accuracy provided by the relevant noise prediction standardsfor control valves), Oldenbourg Verlag, München, Germany,Heft 7, 2003

[25] Kiesbauer, J., Baumann, H.D.: “News in the prediction of hydro-dynamic noise of control valves“, Industriearmaturen , Vulkanver-lag, Essen, Germany, Heft 3,2002

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Figures

Fig. 1: Theoretical tensile strength values for perfect water . . . . . . . 10

Fig. 2: Tensile strength values derived from measurements for pure water 11

Fig. 3: Critical nucleus radius with varying gas concentrations . . . . . 13

Fig. 4: Distribution of pressure in the valve . . . . . . . . . . . . . . . 15

Fig. 5: Determining the cavitation coefficient xFZ . . . . . . . . . . . . . . . . 16

Fig. 6: Formation of cavitation zones with various xF values. . . . . . . . . . 16

Fig. 7: Relation between the pressure ratio xF and the gas content . . . . 17

Fig. 8: Diagram illustrating the collapse of a bubble . . . . . . . . . . 20

Fig. 9: Number of high-energy surges in a stationary cavitation zone . . 22

Fig. 10: The course of development of cavitation erosion . . . . . . . . 26

Fig. 11: Cavitation chamber . . . . . . . . . . . . . . . . . . . . . 30

Fig. 12: Magnetostrictive transducer . . . . . . . . . . . . . . . . . 30

Fig. 13: Material property UR . . . . . . . . . . . . . . . . . . . . 31

Fig. 14: Cavitation resistance. . . . . . . . . . . . . . . . . . . . . 32

Fig. 15: Control valve operating range without cavitation. . . . . . . . 35

Fig. 16: AC trim system (SAMSON). . . . . . . . . . . . . . . . . . 36

Fig. 17: Multi-stage axial plug (SAMSON Type 3255 Valve) . . . . . . 36

Fig. 18: xFZ values of multi-stage valves . . . . . . . . . . . . . . . . 38

Fig. 19: Plug designs. . . . . . . . . . . . . . . . . . . . . . . . . 39

Fig. 20: Areas of a parabolic plug at risk from erosion . . . . . . . . . 40

Fig. 21: Diverted jet occurring with a V-port plug. . . . . . . . . . . . 41

Fig. 22: Cavitation zone with a perforated plug . . . . . . . . . . . . 42

Fig. 23: The valve-specific factor FL . . . . . . . . . . . . . . . . . . 43FIG

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Fig. 24: The critical pressure pDK . . . . . . . . . . . . . . . . . . . 44

Fig. 25: FF factor used to determine the critical pressure ratio . . . . . . 45

Fig. 26: Course of torque in a butterfly valve . . . . . . . . . . . . . . 47

Fig. 27: Sound velocity as a function of the bubble concentration . . . . 49

Fig. 28: Sound power for standard valves with parabolic plug (∆LF = 0) . 55

Fig. 29: Measured sound power spectrum . . . . . . . . . . . . . . . 55

Table 1: Maximum values with various gas content . . . . . . . . . . . 19

Table 2: Limit values for preventing cavitation erosion . . . . . . . . . 40

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Cavitation in Control Valves

Engineering

Transcript of Cavitation in Control Valves