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VOL. 82, NO. 20 JOURNAL OF GEOPHYSICAL RESEARCH JULY 10, 1977

Sound Speed n Liquid-Gas Mixtures' Water-Air and Water-SteamSUSAN WERNER KIEFFER

Department f Geology,University f California,Los Angeles,California90024The soundspeedof a two-phase luid, such as a magma-gas,water-air, or water-steammixture, isdramaticallydifferent rom the soundspeedof either pure component. n numerous eologic ituationsthe soundspeedof such wo-phasesystemsmay be of interest: n the search or magma reservoirs,nseismic xplorationof geothermalareas, n predictionof P wave velocitydecreases rior to earthquakes,and in inversionof crustal and upper mantle seismic ecords.Probably most dramatically, luid flowcharacteristicsuring eruptionsof volcanoes nd geysers re stronglydependent n the soundspeedoferupting wo-phase or multiphase) luids. n this paper the soundspeeds f water, air, steam,water-airmixtures, and water-steammixtures are calculated. t is demonstrated hat sound speeds alculated romclassical cousticand fluid dynamicsanalyses greewith resultsobtained from finite amplitude vapor-ization wave' theory.To the extent hat air and steamare represented s perfectgaseswith an adiabaticexponent , independent f temperature, heir soundspeeds ary in a simple manner directly with thesquare oot of the absolute emperature. he soundspeedof pure liquid water is a complex unctionofpressure nd temperatureand is given here to 8 kbar, 900C. In pure water at all pressureshe soundspeedattains a maximum value near 100C and decreases t higher temperatures; t high pressureshedecreases continuous,but at pressures elow 1 kbar the soundspeed eaches minimum value in thevicinity of 500-600C,above which t again ncreases. he soundspeedof a water-air mixture dependson the pressure,he void or mass raction of air, the frequency f the soundwave,and, if surface ensioneffectsare included, on bubble radius. The admixture of small volume fractions of air causesa dramaticlowering f the sound peed y nearly3 ordersof magnitude. he sound peeds f the pure iquidandgasend-members re nearly ndependent f pressure, ut the soundspeedof a mixture s highlydependent npressure.Calculated values for water-air mixtures are in good agreementwith measuredvalues. Thesoundspeed n a single-componentwo-phasesystem, uchas a water-steammixture, depends n whetheror not equilibriumbetween he phaseson the saturationcurve s maintained.Heat and mass ransferwhich occur when equilibrium is maintainedcause he soundspeed o be much lower than under non-equilibrium conditions n which heat and mass ransfer are absent.The sound speed n a water-steammixture may be as low as I m s x

INTRODUCTIONThe presence f gas or vaporbubbles n a liquid dramati-cally reduces he soundspeed n the liquid [Mallock, 1910;

Karplus, 958,196 ; Barclay t al., 1969;Mc WilliamandDug-gins,1969].n particular,hesound peeds muchower n aliquid-gasmixture than in either the gas or the liquid com-ponents. For example, it is about 1440-1480 m s in waterand about 340 m s- in air, but in an air-water mixture falls toabout 20 m s [McWilliam and Duggins,1969]. Even verysmall concentrations f gas dramatically reduce the soundspeed:%by volume f air n waterowershevelocity y95%to 100 m s [McWilliam and Duggins,1969].This dramaticphenomenonoccursbecause he two-phasesystemhas thedensityf a liquidbut hecompressibilityf a gas. hesoundspeed s even ess n a water-steammixture than in a water-airmixture, as low as a few metersper second Barclayet al.,1969].The liquid-gasand liquid-vaporcasesdiffer because fmass ransfer and latent heat exchangeswhich may accom-pany passage f a soundwave in a liquid-vaporsystembutwhich are absent n a liquid-gassystem.There are numerous eologic ituations n which he soundspeedpropertiesof two-phase iquid-gas ystemsmay be im-portant.On the surfaceof the earth at ambientpressure ndtemperature he propagationof acousticsignals hroughbod-ies of water depends n air content Carstensennd Foldy,1947;Hsieh and Piesset,1961; Grouse nd Brown, 1964;LairdandKendig,952;MacPherson,957].White1976] assug-gested hat gas bubbleswhich come out of solution n connatewater during dilatation may causeP wave decreases rior toearthquakes. In geothermal areas, hot water and admixedgasesmay be present o fairly great depths n the crust. Re-

Copyright 1977 by the AmericanGeophysicalUnion.

centiy,hotspot' eismicechniquesave een evelopedolocate geothermal energy sources.A detailed model for thesoundspeed f water and magma,both of whichmay containvapor phases, s required or proper nversionof seismic atafor useof these echniques. olatile elements nd compounds(CO:, H:O, and S) are thought o be present n the lower crustand mantle in numerouspossible orms [Irving and Wyllie,1973]:as free vapors Wyllie and Huang, 1975;Eggler, 1976],as vapor dissolved n silicatemagma [Green, 1972; Eggler,1976], in crystallinecompounds Newtonand Sharp, 1975;Eggler, 1976], as interstitial mmiscible iquids [McGetchinetal., 1973], and as proxyingcomplexes r structurallyboundcomplexes. nowledge f the soundspeed f thesepossibleforms s required o test or their presence Y seismicmethods.However, perhaps the most dramatic manifestation of theeffectof soundspeed n liquid-gas ystemss the controlwhichthe soundspeed xertson fluid low duringvolcanic ndgeysereruptions. he role of two-phaselow n volcanic ruptions asbeendeveloped ualitatively y Bennett 1971, 1974].His con-siderations f flow processesn two-phase ystemsead todifferent conclusions about mechanisms of vesiculation andvolcanicash formation han are arrivedat from petrographicconsiderationsVerhoogen, 951;McBirney, 1963;McBirneyand Murase, 1970]. Models of eruption mechanicsof vol-canoes McGetchin,1974;Sanfordet al., 1975] and geysers[Kieffer,1975]showa strongdependencef the eruptionmani-festationson the sound speedof the fluid in the column. Forexample, he height o whicha geyser isesduringan eruptiondepends tronglyon the soundspeedof the fluid in the geyserconduit.Since he sound peed epends n the temperatureand conditionof the water in the conduit, t is possible o inferthe conditions of the geyser luid from observationsof itssurfacebehavior Kieffer, 1975, also manuscript n prepara-

Paper number 7B0034. 2895


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2896 KIEFFER:SOUND SPEED N LIQUID-GAS MIXTURESi i i i i ' i i i

240 - -CRITICAL220 -200 -180 -

- 0-'-' 40

t2o-) lOO- -Ii o-

60-40- -

O 100 200 00 400Temperature (C)Fig. 1. Pressure nd temperature f water at saturation.

,

.

tion,1977].t is apparenthatmuchemainso be earnedabout the relationshipbetween low during eruptionsand thepropertiesof material during the flow. The calculations res-ented here of sound speed n a two-phase medium representthe initial phaseof an attempt o developa quantitativemodelfor volcaniceruption phenomenaand to relate the thermody-namic parametersof water-steamand magma-gassystemsothe flow parametersobservedduring an eruption.Althoughonly a few theoriesor sound peedn liquid-gasmixtures exist, they are more abundant than the extremelysparseexperimentaldata. Much of the theory was developedin studies of containment of water-moderated nuclear reac-tors. Itis the purposeof this paper o reviewand synthesizeheconceptsequired or calculations f the sound peedn liquid-gas and liquid-vapor mixtures and to examine the effect ofvolatile content on the sound speed or water-air and water-steam systems.This work is being extended to magma-gasmixtures.

EQUATIONSOF STATEAND SOUND SPEEDSOFWATER, AIR, AND STEAM

A simplifiedphasediagramof water and its saturation urveareshownn Figure .Becausef hegreat olumeifferencesbetweenhe iquid nd apor fiasesnd ecausef he riticalphenomenont is difficult to express he equationof stateofwater in analytical form over a wide range of pressures ndtemperaturese.g., seediscussionsy Burnham t al. [1969]and Helgesonand Kirkham [1974]). Becauseof thesecom-plexities the sound speed of water and water-gasmixturescannot be calculated t all pressures nd temperaturesrom asingleequationof state.Therefore n this paper severaldiffer-ent expressionsor the equationof stateof the water, air, andsteam components re used, each of which is an appropriateand relatively simpleexpressionor the pressure nd temper-ature conditions for which it is used.

1. For liquid water at 25C an adiabaticequationof stateis used:

= PA exp [(P - PA)/K] (1)wherep is the densityof the liqud phase,PA s the densityofthe liquid phase n a referencestate 1.0 gcm - at 1 bar), P isthe pressure,Pn is a referencepressure 1 bar), and K is the

bulkmodulusfwater2.2X 106 ars).he oundpeedappropriate o this expressionsc = (K/o)'/' (2)

2. ForWater n hesaturationurve nd o pressuresf 8kbar and temperaturesbetween 25 and 900C, internallyconsistentabulatedataof HelgesonndKirkham1974] reused. he tabulatedsothermalornpressibilitiesr werecon-vertedo adiabaticompressibilitiess, and hesoundpeedwascalculatedrom c = (Ksp)/'.3. For water on the saturationcurve an equationof stateof the form P(T) is requiredby the theory developedn thispaper. woexpressionsreusedKeenantal., 1969; elgesonand Kirkham, 1974, p. 1102]:Ps,tPcxp0-Sr(tct) %x(0.650.01t)-x (3a)In this quaV_on,c s he riticalressure220.38ars), isthe temperature in degrees,Kelvin), is 1000/T, t is thetemperature n degreesCelsius, c is the critical tempera-ture (374.136C), nd the Ft are coeffiCien{ssed:F =-741.9242, F. = -29.7210, Fa = - 11.5529, F = -0.8686, F5= 0.1094, F6 = 0.4400, F7 = 0.2521, and Fa = 0.0522. Thesecond,and simpler, assumption s that

Paat - A Ta exp (- AH/R T) (3b)(A common assumption s that a = 0.) This simpler orm iscommonlyused or vaporization urves nd is directly elatedto the Clausius-Clapeyron quation with the simplifyingas-sumptions hat the gas phase obeys he perfect gas aw, thevolume f thegasphases muchgreaterhan hat of the iquidphase,and the heat of vaporization s constant.For water thecoefficientA is 2.6 X 10 dyn cm ', and the ratio AH/R is4.6 X 10a K.

4. For steaman ideal gasequationof state s used:PV* = const = Gst (4)

where Gst = ToRo/Mpo-, To is a reference emperature(100C at I bar), V is the volume, Ro s the gas constant 8.32X 107ergsC mol-X), M is the molecularweight 18.02 forsteam), o is the density n the referencestate (0.96 g cm-a),and3' s the sehtropicxponent1.31 or steam). or an dealgas,

c = (7RT) '/' (5)where R = Ro/M.5. For air an ideal gas equation of state s used:

PV* = const = Gtr (6)where Gtr = ToRo/Mpo M, M = 28.98 for air, and 3' = 1.40for an adiabatic process.To the extent that air and steam are represented s perfectgases ith 3' independentf temperature,he sound peedvariesdirectlywith the square oot of the absolute emperature(Figure ). Thesound peedf a perfect as s ndependentfpressure or a fixed emperature.Measureddata for'air showaglightdependence n pressure Hilsenrathet al., 1955],but thisdependence an be ignoredhere. Becausehe specific ascon-stant R depends n the molecularweight, the soundspeedofsteam s higher than that of air (c : M-/2).The behavior of the sound speedof water is much morecomplex and depends n both pressure nd temperature Fig-ure 2). Values of soundspeedcalculated rom the staticcom-pressibilitydata increase ystematically ith pressure ut be-


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KIEFFER:SOUND SPEED N LIQUID-GAS MIXTURES 2897

cause of the anomalous compressibilityare a complicatedfunction of temperature.The high-pressure nd high-temper-ature values or water obtained rom the staticcompressibilitydata are systematically -10% lower than thoseobtainedex-perimentally rom dynamicultrasonic xperimentsSmithandLawson, 1954] but show the same trend. The sound speedinitially increases ith temperaturen the range25-100C.Amaximum value is attained near 100C, and with further tem-perature ncreasehe soundspeed ecreasesapidly. The max-imum in soundspeed s alwaysattained at temperatures ome-what larger than the minimum in compressibility.CompareFigure 2 of this work with Figure 7 of Helgeson nd Kirkham[1974].) Sound speeds btained rom the compressibility atashow hat the temperatureat which the maximum s attainedincreases ith increasing ressure. he direction n which hismaximum moveswhen temperature s increased as been hesubjectof controversy. xperimental esults re contradictory.The mostrecent, and apparentlymost reliable,data of Smithand Lawson [ 1954] show the same rend as the static data, butearlier data [Holton, 1951] show the opposite rend: the tem-perature at which the maximum is attained decreases ithincreasingpressure.Besides hese low-temperature nflection points there arehigh-temperaturenflectionsn the velocities t 0.5 and 1 kbar.The minima in the 0.5- and 1-kbar sound velocity curvescorrespond o minima in 0.5- and 1-kbar compressibilitycurvesof Helgeson nd Kirkham [ 1974,p. 1112].There do notappear o be any experimental igh-pressureigh-temperatureultrasonicdata to comparewith the calculatedcurves.

SOUND SPEED IN A Two-COMPONENT SYSTEMIn this section he word 'gas' refers o a gasphaseof differ-ent composition han the liquid phase. The most obvious

geologic ystemsnvolvingwater-gasmixturesare at low pres-sure and low temperature, e.g., surfacewater systemswithadmixed air bubbles or shallow crustal water with admixedgas. In order to avoid tedious algebraic complexities asso-ciated with the water equation of state) which do not providemuch additional insight, the general propertiesof the soundspeed n suchsystems re illustratedby considering water-airsystemat 25C and pressures rom 1 to 500 bars.The first quantitative analysisof the effectof gasbubblesonthe soundspeed n a liquid waspublished y Mallock [1910] nan investigation f the dampingof soundby frothy liquids.Amore elegant calculation has been given by McWilliam andDuggins [1969], and their treatment is followed here. Theassumptionsnherent n their model are as follows. 1) The li-quid and gas phases re in equilibrium, and there is negligiblemass ransfer between he phasesowing to gas becomingdis-solvedor liquefied; 2) There is no slip between he liquid andgas phases; 3) The wavelength of the sound wave is muchlarger than the average dimension of nonuniformity of themixture;and (4) The gas s compressiblend obeys he perfectgas aw (equation 6)), and the liquid is elasticwith a constantbulk modulus.

Two cases are considered:1. The gas bubblesare sufficiently arge that surface en-sion effectscan be neglected.2. The gas bubbles are small, so that surface tension isappreciable. t is assumedhat the bubblesare spherical nd of

uniform sizeand that classical urface ension heory applies.Although case 1 is a limiting case of 2, it is instructive oconsider t first and separately n order to illustrate the depen-denceof sound speedon pressureand vapor fraction.Case 1: surface ension ffectsneglected. In the absence f

2800] ,,..,9.4 Kb26-.,...7..37b 5.54 Kb2200 - -

2000-t800 -..,,'1704 Kb/?'1 par,.ooI,ooo-

600

% Temperature (=C)Fig. 2. Sound speed of the pure phasesdiscussedn this paper,water, steam, and air. Heavy curves for water were calculated fromcompressibility nd volume data tabulated by Helgeson nd Kirkham[1974]. Isothermal compressibilities ere converted o adiabaticcom-pressibilities y using their tabulated thermal expansionand specificheat data. Dotted curves of measuredsound speedsof water [Smithand Lawson, 1954] are shown for comparison. Data for air are fromHilsenrathet al. [1955], and data for steamare from Hodgroanet al.[1958, p. 2320].

surface ension i.e., with the assumption hat the bubbleshaveradii greater han 10 4 cm) the pressurewithin the gas bub-blesPc in a liquid is the sameas he pressuren the liquid PL:Pc = PL ----P (7)

As givenby (1) and (6), the densities f the liquid o. and gasoc, respectively, re given byp. = P.Aexp [(P- PA)/K] (8)

andPo = (P/Ga,r) x/'r (9)

The density p of a two-phase mixture which has gas massfraction r/ = Mc/M. is given mplicitly by :(1 + rt)/p - (rt/pc) + (1/pL) (10)

The void fraction x is simply related to the mass raction r/throughx = (1 + pc/rlpr) - (11)

Substitution for p. and pc in terms of P gives the adiabaticequationof stateof the mixture:110Lair --exp A - (12)p = (1+ TI)pL p/'r K

from which he soundspeedc = (dP/dp)X/2sC = TIPLA + exp K [(1 + r/)p

TIPLAair 1 PA- -'yp('r+xl/,exp ' K (13)


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2898 KIEFFER: OUNDSPEEDN LIQUID-GASMIXTURES

14OOi i i i i

ADIABATIC SOUND SPEED1zoo

Tc.)00 800

1 60003 00

00 Q1 0.2 03 Q4 0.5 0.6 0.7 0.8 0.9PUREIQUIDX (Void Fraction,Air)

ISOTHERMALOUND PEED

i".

10 0 0.1 Q2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0PURE ASPUREIQUID X (Void Fraction, Air) PURE AS

Fig. 3. Calculateddependence f (a) adiabaticand (b) isothermal oundspeed f water-airmixtureon volumecontentofgasand on pressure. urface ension s neglected.

Values of the sound speed according o this model areshown n Figures 3 and 4, where the sound speed s plottedagainstvoid fraction and mass raction, respectively. onsiderfirst only Figure 3a, which shows he dependence f the adia-batic sound speed c = (PP/Pp)s 1/' n a void fraction. Themost dramatic effect of the admixture of small-volume frac-tions of air is the lowering of the sound speed;e.g., at 1-barpressurewith one part in 104by volume of air the soundspeedis lowered from that of pure water, 1470 ms-', to 900 ms-X.The decrease s most dramatic at low pressures.At 1-barpressure minimum value of the soundvelocity s attained atvoid fraction x = 0.5. At higher pressures he minimum isshifted to larger values of x, and at 500-bar pressure heminimum vanishes,and the sound speeddecreasesmonoto-nically to that of pure air. At very large values of the voidfraction (x > 0.9) the soundspeedapproacheshat of the gas.The soundspeedof the mixture is highly pressure ependent,whereas he soundspeeds f the pure end-members re inde-pendentof pressure.The variation of soundspeedwith mass raction s shown nFigure 4 on a semilogarithmic raph to emphasize he magni-tude of the effect for small mass fractions of air. Consider firstthe dotted curves,which show he adiabaticsoundspeed.At 1-bar pressure,1 ppm air causes he soundspeed o drop to onequarter of the value for pure water. At higherpressures,argermass ractionsof air are required o cause he samedecrease.The minimum value attained at 1 bar is 24 m s x.

Measured data [Karplus, 1958] are compared with calcu-lated values n Figure 5. At mass raction r/ = 10 5 the soundspeed s approximately100 m s x, and at mass ractionr/ =10 3 it decreaseso 24 m s x. There is a definitedependence fsound speedon frequency considered elow), but in generalthe data confirm the theoreticalpredictionof a large decreasewith admixture of gas nto the liquid.The propagation of a sound wave through a two-phaseliquid-gasmixture may be an isothermal ather than an adia-batic process.Heat may be conducted between the gas andliquid phases, nd because f the large heat capacityof water

the temperature variation within the water may be smallenough o be ignored [Karplus, 1958]. The soundspeedof themixture may therefore depend on the frequencyor, equiva-lently, the period of the sound waves. If the period of thewaves is less than the time in which heat can be conductedfrom the gas bubbles into or away from the surroundingliquid, the compressions nd rarefactionsaccompanyinghepassage f the sound wave will be adiabatic,and the behaviorof hegaswillbegiven y headiabaticolytropicxponent?= 1.4 for air). If, however, the period of the soundwaves sgreater than the time required for conductiveheat flow fromthe bubbles nto the liquid, the processwill be nearly sother-mal, and the behaviorof the gaswill be givenby the isothermal

1400

..1000l. 00C')D0020O

0-8 107 106 105 104 I0 3 I0-2 101 10 10 (Mass Fraation, Air)Fig. 4. Calculateddependence f soundspeedof water-airmixtureon mass raction of gas and on pressure,plotted on semilogarithmicpaper to show the effect of small volume fractions of air. Surfacetension s neglected. olid curves ndicate sothermal oundspeed,anddotted curves ndicateadiabaticsoundspeed. nset regionat bottom isshownenlarged n Figure 5.


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KIEFFER: OUNDSPEEDN LIQUID-GASMIXTURES 2899

160 I I IIIIIII I I I11111- Data f Karplus1958)140-7 "1000 Hz20-XI00 o Extrapolatedo zerorequency- Averageubbleiametereo- ADIABATIC : 0.'mm - - SOUNDAVE //I ISOTHERMAL'S%WVE

Ol I I I IIIIII I I I IIIIII I I I IIIIII I I{0-5 0-4 0-3 O-Z (MassFraction, ir)Fig. 5. Comparisonof calculated oundspeedswith data of Kar1us[1958].

polytropic exponent 7 = 1 for air). Correspondingly, heisothermal bulk modulus for the liquid should be used.The isothermal sound speed of a two-phase mixture de-creaseswith the admixture of small amounts of air, as does theadiabatic sound speed Figure 3b and Figure 4; solid curves).However, the sound speedof air compressedsothermally sless than that of air compressed diabatically by the factor7/', so that isothermalsound speeds f a two-phasemixtureare lower than adiabatic sound speeds.The assumptionofisothermalpropagation must break down at large void frac-tions of air, sincepropagation n pure air must be adiabatic.For this reason he isothermal sound speedcurvesshown nFigure 3b are dashed or large void fractions.The frequencyat which thermal conductivityeffectsmaybecome significantcan be estimatedas follows. A character-istic time constant or thermal coolingof small spheres s t =0.05D'/K, where D is the diameter of the sphereand K is itsthermal diffusivity (seeKarplus, 1961, p. H-1 or CarslawandJaeger, 1959, chapter 9). For a 0.1-mm sphere of air withdiffusivity0.187 cm ' s z [Carslawand Jaeger, 1959,p. 497] thecharacteristic ime is 2.5 X l0 -5 s. The reciprocalof this timeconstant ivesa valueof frequency elowwhich he isothermalapproximation should be approachedand above which theadiabatic approximation should be approached.For air bub-bles of diameter 0.1 mm the characteristic requency or con-ductive heat transfer is 40,000 Hz. Karplus [1958, p. l l] ob-tained experimental values of sound speeds in water-airmixtures which had bubbles of this diameter. The data weremeasured t frequenciesrom 250 to 1750 Hz and the bubblemass fractions from 1.2 X 10 -5 to 2.4 X l0 -8. All of the datawere obtainedat frequenciesow in comparisono the charac-teristic requencyof 40,000 Hz. The lowest requencydata lienear the calculated sothermalcurve Figure 5), supporting heidea that low-frequency oundpropagation n a water-air sys-tem is an isothermalprocess. here is an increase n the soundspeedwith increasing requency,as would be expected romthe above discussion, ut the adiabatic curve is approached(and even exceeded y a few data points) at frequenciesmuchless han the calculatedvalue of 40,000 Hz. Karplus himselfattributed he frequency ependenceo the presence f a deter-gent n the experimentalmixture,but his experiments esignedto test this hypothesiswere inconclusive. hus although the

frequencydependence f the soundspeeddue to this effecthasnot been experimentallyproven, it is important to keep inmind that at low frequencies he sound speedof a water-airmixture appears o be isothermaland that the soundspeedof atwo-phasemixture is sensitiveo frequency.Entirely differentresultsshould be expected n laboratory experimentsat, say, 1MHz and field experimentsat, say, 1-10 Hz.The variation of sound speedwith void fraction is given inFigures3a and 3b because oid fraction s the natural variableof mixing theory and previous investigatorsgenerally havefollowed this convention. For many engineeringpurposes t ispreferable o specifymass raction as a measureof gascontent,rather than void fraction, because mass fraction is invariantwith surface tension and pressure McWilliam and Duggins,1969, p. 105]. There are many geologicsituations n which itis also preferable o specifymass raction. One suchsituationis the passage f a gas-liquidmixture up a geyserconduit or avolcanic neck. During the expansionof the mixture the voidfraction may vary greatly as the fluid rises, but the massfraction will not if gas s neither condensing or dissolving. nthe remainingcalculations,mass raction will be specified.In the following section the effect of small bubbles whichhave appreciablesurface ension s examined.M cWilliam andDugginsdeveloped his model for the specific aseof isother-mal compression.Generalization to the caseof adiabatic com-pression equiresconsiderablealgebraic complexity which isnot warranted at this time becauseof the paucity of experi-mental data.Case2: surface ension ffects onsidered. When the surfacetensionbecomessignificant, he pressure n a gas bubble ex-ceeds he pressure n the surrounding iquid:

Pa: Pt, + (2c/r) (14)where r is the bubble radius. For the case of isothermal com-pressionof the gas bubbles the same procedure used in theprevious sectiongives the following sound velocity [McWil-liarn and Duggins, 1969, p. 104]: '

O.Ol=z_

0.001 ' ''"'' ' ' '"' flo-e lO-? io-6 Io-5Bubble Radius (cm)Fig. 6. Density of air in bubbles n water mixture as a function ofbubble radius and pressure.


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2900 KIEFFER: OUNDSPEEDN LIQUID-GASMIXTURESIII1I I II1111I IIIIII l llllll I [111111l Ill

-

To OOO ooo

:2OO

oo - oo 8 = 0_0-8 O-SO-e O-Z 0-6 0-5 0-4 O-S O-Z 0-6 0-5 10-4

400 400( -- 200 - - 200

-50060200

_ --000 000 800 800 600 600 400 400

_ _ooo10-8 1 bar -. ,,,,,I 1111111 I I IIIIIII i I II10-7 10-6 I0-5 10-4Bubble Radius (cm) ]llllllu [lliiin .l. liltlill i tiiillU l iit.l 010-5 10-8 107 10-6 10-5 10-4 I0-5Bubble Rodius (cm)Fig. 7. Calculated ound peed f water-airmixture sa function f bubble adius ndpressureor fourmassractions,r/ = lO , lO ', lO 3, and lO 4

c= 3:0-b2,/ (]5)in which

= 2a/r(1+r/) 2_..ap7 - p:rlG r

b = 32mlaa/3m = 71rig

0 ! - exp(PA- PL)/K][(2a/r)PL]KpArIG(16)

X =(0- 1)/rigA valueof 72.2dyn cm- xwasused or the surfaceension f anair-water interface.

The densityof the air dependson the bubble size as well ason the pressure, s is shown n Figure 6. The gas density sinsensitiveo radius f the bubbles re argeand becomes qualto the valueobtained n the absence f surface ension. he gasdensity ncreases ecauseof surface ension as the bubble sizedecreases.t radiiof !-5 X l0 7 cm 10-50 ) thegasdensitywould become as large as the liquid density;presumablyatsmaller radii, bubbles are unstable.When surface ension s taken into account, both pressureand bubblesizeaffect he soundspeedof a two-phasemixture,as s shown n Figure 7: Consider irst the case = 10- , shownin the upper left of this figure. At 1-bar pressure he soundspeed s close o that of the pure liquid if the bubble radius ssmaller han about3 X l0 8 cm (3 /[). The sound peeddecreases apidly in the range 3 X 10 8 to about l0 -6 cm. It isnearly constant, equal to the value obtained in the model inwhich surface ension s ignored, or bubbleradii greater hanabout l0 -6 cm. With increasing ressure he decreasen soundspeed occursover approximately he same range of bubblesizes,but the magnitudeof the drop is smaller.Three distinct regionsof behavior are recognizable:1) a


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KIEFFER: OUND PEEDN LIQUID-GAS IXTURES 2901400

:350 -

300 -0(1)250200.'150-t00

50

o to

i i i i i i i i iSUBCOOLEDSUPERHEATEDWAT TEAMA E

t 2 :3 4 5 6 7 8 9Entropy (Joules g-1 OK-l)Fig. 8. Temperature-entropy iagram or H.O (data from Keenanet al. [1969]). The thermodynamic athswhich llustrate he effectofadiabatic compressionand decompression n the mixture [afterDavies, 1965] are discussedn the text. Vertical scale of paths isexaggerated. hin linesextendingnto the singlephase egions rom A,F, and I and from E, H, and M are constantpressure ines.

region at small bubble radii where the soundspeed s approxi-mately independent of both bubble size and pressure, 2) aregion at intermediatebubble radii where the sound speeddependssensitivelyon both pressureand bubble radius, and(3) a region at larger bubble radii where the sound speed sindependentof bubble radius but sensitive o pressure.For smaller mass fractions (r = 10 2, 10 3, and l0 -4 inFigure 7) these regions shift in a systematic manner. Theregion at small bubble radius where the sound speed s inde-pendent of both pressureand bubble size is enlarged.Theregionwhere otheffectsre mportantmoves rogressivelyolargerbubble adii Withdecreasing ass raction.Finally, hemagnitudeof the soundspeeddrop decreases ith decreasingmass fraction.

SOUNDSPEEDOF SINGLE-COMPONENTwO-PHASESYSTEMGeneralconsiderations. Calculation of the soundspeedof atwo-componentwo-phase ystems an easyproceduref adia-batic equationof statedata are available,because ressure ndtemperature aybeconsideredo be ndependentariablesnsuch syste . Calculationf thesound peedn a one-com-ponent two-phasesystem s a more difficult matter becausepressure nd temperature re not independent ariablesbutare relatedbY the Gibbs equation or equilibriumbetween hephases..The physicalprocesses hichoccurduringpropagation f a

sound wave through a two-phasemixture are much morecomplexhan for a single-phaseixture e.g.,Davies,1965].Consider the temperature-entropy TS) diagram of watershown n Figure 8. A mixture of saturatedwater and steam srepresented s point G, where he chord ratio FG/FH is themass fraction r of steam in the mixture. Isentropic pressurechanges, uch as the compressionsnd rarefactionswhichoccurduring propagationof a soundwave,are represented ymovementup and down the constantentropy line CGK. Ifsteamand water remain n thermal equilibriumon the satura-tion line, there must be mass ransferbetween he phases, incethe fraction of steam in the mixture changes BC/BD FG/FH IK/IM). This requires hat condensation r evapo-ration take place.

An isentropic pressure ncrease from P to P + /xp corre-sponds o movementof the mixture from G to C in the temper-ature-entropy diagram of Figure 8. The pressure ncrease nthe water phase corresponds o movement rom F to A' as aresult, the water phase becomessubcooled.The pressure n-crease n the steamphasecorrespondso movement rom H tOE as the steambecomes uperheated. he induced emperaturedifference between the steam and the water leads to heattransfer rom the superheated team o the subcooledwater. fthe original compositionof the mixture G lies to the right ofthe peakof the two-phaseoop, as s shown n Figure8, somewater will be vaporized, and the mass raction of steam n themixture will increaseduring adiabaticcompressionBC/BD >FG/FH). (This phenomenon occurs during meteorologicalconditionswhen saturatedair is adiabaticallycompressed ur-ing descentand a decrease n relative humidity is induced.) Ifthe original composition ies to the left of the top of the two-phase oop (assumed o be symmetric),somesteam will con-dense, and the mass fraction of steam in the mixture willdecrease uring adiabaticcompression. hus by heat and masstransfer both the water and the steam are restored to thesaturation ine, the water by the path A to B, and the steambythe path E to D.An isentropicpressure eduction from P to P - /xp corre-sponds o movement from G to K in the TS diagram. Thepressure ecreasen the water phasecorrespondso movementfromF tR; asa result,hewater ecomesupe{heatedboveits saturation temperature,point I (IR is a continuation of aconstant pressure ine from the water region). The pressuredecrease n the steam phase alone corresponds o movementfrom H to N in the TS diagram,and the steambecomessubcooled r supersaturatedwith respect o its saturation em-perature, point M (MN is a continuation of a constant pres-sure line from the superheated egion). If the original compo-sition of the mixture G lies to the right of the peak of thetwo-phase oop, some vapor will condense o form a mixtureof saturatedwater and steam point L), and thus the subcooledsteammaymove oward hestable tateM. Themassractlonof steam n sucha mixture will decreaseIK/iM < FG/FH) (adirect application of this phenomenon s found in the Wilson9loudchamber). f the originalcompositionies o the left ofthe two-phase oop (assumed o be symmetric),cavitation, andsome vaporization, of the subcooled iquid will be the domi-nant process,and the mass fraction of steam in the mixturewill increase. Cavitation of the water creates a local mixturewith the compositionof point J, and the superheatedwater atpoint R thus moves toward the stable state I, a mixture ofwater and steam.

In general,condensation nd evaporationcannot ake placeinstantaneously, ince he transportationof heat and masscanonly occur at a finite speed.The time lag in flashingwater tosteamor condensing team o water s important n determin-ing the degree of equilibrium obtained in the sound wave.Since ondensationnd evaporation enerallyprOceedt dif-ferent rates, t should be expected hat compression nd rare-factionwavesbehav differently.Experiments aveconfirmedthat finite amplitude rarefaction waves in water-steam mix-tures have lower velocities han compressionwaves becauserarefactionwaves end to maintaincontinUOushermalequi-librium [Barclay et al., 1969].In summary, a mixture of liquid and its vapor may respondto pressuredisturbancesby (1) nonequilibrium response inwhich there is no mass transfer between he phases; .e., theliquid and the vapor are independently sentropic,and both


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2902 KIEFFER: OUND PEEDN LiQUID-GASMIXTURES10,000

1,000'o 100

; {-:D WATER-STEAM: PHASES0 NOTNTHERMODYNAMIC('/') -- EQUILIBRIUM)_ I IIIIllIIIIIIIII IIIIIII IIIIIIIIIIIIII5 iCF4 I0- 10-2 I0- 10- I- 200 bars (36GC)

50 ars2G4C)/() OE 0 arsA80C)'o7 MassFraction, team)

Fig. 9. Calculated sound speedof water-steammixture (a) not inequilibrium and (b) in equilibrium as a function of mass raction andpressure.

givingOV e,s - (19)

In the limiting caseof P O, dx/dP becomes s - Vt. Thesystem erivative0V/0P)s,,,can be writtenas the sumofcomponent erivatives1 - )(OVt/P) + (PVs/PP). Hence(18) becomes

(o.) (20))The derivative 0rt/0P)s can be determined rom the first lawof thermodynamics:

bQ = bE + PiJV (21)For an adiabatic change, bQ = 0. The differential nternalenergy in this equation is more meaningfully expressedntermsof the enthalpyof reaction

bH = bE+ PbV + VbPso the first law becomes

depart from the saturation ine) and (2) equilibrium esponse(in which there is mass ransfer between he phasesand theliquid and vapor remain on the saturation ine). Real systemswill probably show behavior between hese wo extremes.Case 1: nonequilibriumesponse. If there is no mass rans-fer between he liquid and vapor phases,he soundspeedwillbe given by the theory of the previoussection.Surface ensioneffects houldnot be important or a liquid-vapor ystem nthe saturationcurve, and the sound speed s therefore nde-pendentof bubblesize.The adiabaticsoundspeed n this caseis givenby (13). This wouldmostnearlybe the case or high-frequencywaves; he case of low-frequency onequilibriumpropagation, hichwouldbe morenearly sothermal,s notconsideredhere becausemass transfer effectsmay becomeimportant (case2, discussed elow).The sound speed of a water-steamsystem n which thephasesare not in thermodynamicequilibrium is shown inFigure 9a. At 1-bar pressure 100C) the sound speedde-creases teadilywith the addition of smallamountsof steam o24 m s at rt = 10 8 and then increases o 446 m s s, the soundspeedof pure steamat 100C.At higherpressureshe decreaseis less pronounced,and the minimum value is attained athigher mass ractionsof steam.Case : equilibriumesponse.Equilibrium etweenhe iq-uid and vapor phases can be maintained if mass transferbetween he liquid and vapor phases ccurs n a time short ncomparisono the acousticwaveperiod.The soundspeed anbe calculated rom the equationof stateof the system n theform V(P, S, x) [Davies, 1965].A differentialchange n vol-ume of the two-phasesystem s given by

For adiabaticpropagationof sound,dS = 0:

S, P,S(18)

bH- VbP = 0For the liquid-vapormixture,

H =(1 -rt)Ht +rtHs = Ht +rtLwhere L is the latent heat of transformation. For this,

bH = bHt + nbL + Lbrtand substitutingnto (22) gives

bHt + nbL + Lbn - VbP = 0Rearranging terms gives

L&/ = VbP - bHt -o

s s=L LFinally, then, the soundspeedobtained from

c: = - (dV/dP )s

-- C2

(22)

(23)

(24):,

(25)

(26a)

(26b)

(27)

The four derivativesequired n this equation an be obtainedfrom tablesof thermodynamic ata if the data exist or theliquid-gas ystemsn question. or this study,data wereob-tained rom Keenan t al. [1969].The sound speedof a water-steammixture in which ther-modynamic quilibriumbetween he phasess maintainedsshown n Figure9b. The behavior s quite different rom thenonequiiibriumasehownnFigurea.At 1-bar ressurehevelocitydropsdiscontinuouslyo 1 m s with the additionofa finite amount of steam an'd remains at this low value for massfractions t less han 10 4. With the formationof larger

(28)


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KIEFFER:SOUNDSPEED N LIQUID-GASMIXTURES 2903amounts of steam in the mixture the velocity rises slowlytoward, but does not attain, the value for the pure steamphase. At higher pressureshe velocity decrease s less dra-matic but is maintained to higher mass fractions. The mostpronouncedeffectof equilibrium behavior n contrast o non-equilibrium behavior s the magnitudeof the decrease: t 1-barpressuret drops o approximately m s 1, much ower hanthe minimumvelocityof 24 m s obtained or the nonequilib-rium case.

The soundspeedof a two-phase iquid-vapormixture doesnot smoothlyapproach he component elocities t r/= 0 andr/ -- 1, as may be seenby writing (28) in the form [Davies,1965, p. 5]:

(29)The term in bracketsrepresentshe differencebetween henonequilibrium aseand the equilibriumcase.The nonequi-librium casecorrectlyconvergeso the liquid and gas soundvelocitiesat r/ = 0 and r/ = 1. Since he term in bracketsdoesnot reduce to zero at n = 0 and rt = 1, there are mathematicaldiscontinuitiesith the formation f an infinitesimallymallamount of vapor in liquid or droplets in vapor.Approximationso (29) canbe made or verysmall 7


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2904 KIEFFER: OUNDSPEEDN LIQUID-GASMIXTURESHolton, G., Ultrasonicpropagation n liquidsunder high pressure:Velocitymeasurements'onater, . Appl.Phys., 2, 1407, 951.Hsieh,D. Y., and M. S. Piesset, n the propagation f sound n aliquidcontainingasbubbles, hys. luids, , 970-975,1961.Irving,A. J., andP. J. Wyllie,Melting elationshipsn CaO-CO2ndMgO-CO2o 36 kilobarswith commentsn CO in the mantle,Earth Planet. Sci. Lett., 20, 220-225, 1973.Karplus,H. B., The velocity f soundn a liquidcontainingas

bubbles, es.Develop. ep.C00-248, 2 pp., AtomicEnergy om-mission,Chicago, II., 1958.Karplus, . B., Propagationf pressureavesn a mixture f waterandsteam, es.Develop. ep.ARF-4132-12, 8pp.,AtomicEnergyCommission,Chicago, II1., 1961.Keenan,J. H., F. G. Keyes,P. G. Hill, and J. G. Moore, SteamTables,1962pp., JohnWiley, New York, 1969.Kieffer, .W., Geysers: hydrodynamicodelor heearly tagesferuption, eol. oc.Amer.Abstr.Programs,, 1146, 975.Laird, D. T., and P.M. Kendig, Attenuationof sound n watercontainingir bubbles,. Acoust. oc.Amet.,24, 29-32,1952.Landau,L. D.,andE. M. Lifshitz,FluidMechanics,ranslatedromRussian y J. Sykes ndW. Reid,pp. 248-249, ergamon,ewYork, 1959.MacPherson,. D., The effect f gasbubbles n sound ropagationnwater,Proc.Phys.Soc.London, ect.B, 70, 85-92, 1957.Mallock,A., The damping f sound y frothy luids,Proc.Roy.Soc.London, Set. A, 84, 391, 1910.McBirney, . R., Factors overninghe nature f submarineol-canism,Bull. Volcanol.,26, 455, 1963.McBirney,. R., andT. Murase, actorsoverninghe ormationfpyroclasticocks,Bull. Volcanol.,4, 372,1970.

M cGetchin, . R., Solidearthgeosciencesesearchctivities t LASL,Rep. A-5956-PR,. 59,LosAlamosci. ab.,N. Mex., 974.McGetchin,T. R., Y. S. Nikhanj, and A. A. Chodos,Carbonatite-kimberlite elationsn the CaneValley diatreme,SanJuanCounty,Utah, J. Geophys.es.,78, 1854-1869, 973.McWilliam,., andR. K. Duggins,peedfsoundnbubblyiquids,Proc. nst. Mech. Eng., 184(3C), 102, 1969.Newton,R. C., andW. E. Sharp,Stability f forsterite CO and tsbearing n he oleof CO n themantle, arthPlanet. ci.Lett.,26,239-244, 1975.Sanford,M. T., E. M. Jones, ndT. McGetchin,Hydrodynamicsfcaldera-formingruptions, eol.Soc.Amer.Abstr.Programs,,1257, 1975.Smith, A. H., and A. W. Lawson,The velocityof sound n wateras afunction f temperaturend pressure,. Chem.Phys., 2, 1351,1954.Verhoogen,., Mechanicsf ash ormation, mer. . Sci.,249,729,1951.White, J. E., Role of dissolvedas n dilatancy-diffusionodel ab-stract),Eos Trans.AGU, 57, 1010,1976.Wyllie,P. J., and W. L. Huang, nfluence f mantleCO in thegenerationf carbonatitesndkimberlites,ature, 57,297-299,1975.

(ReceivedJune 23, 1976;revised December 20, 1976;accepted ecember27, 1976.)

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