Evaporation condensation-induced bubble motion after temperature gradient set-up

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Evaporation condensation-induced bubble motion after

temperature gradient set-up

Vadim S. Nikolayev, Yves Garrabos, Carole Lecoutre, Guillaume Pichavant,

Denis Chatain, Daniel Beysens

To cite this version:

Vadim S. Nikolayev, Yves Garrabos, Carole Lecoutre, Guillaume Pichavant, Denis Chatain, et al..

Evaporation condensation-induced bubble motion after temperature gradient set-up. Comptes Rendus

Mécanique, Elsevier Masson, 2017, 345, pp.35-46. �10.1016/j.crme.2016.10.002�. �cea-01485391�

Comptes

Rendus

Mecanique

www.sciencedirect.com

Basic and applied researches in microgravity/Recherches fondamentales et appliquées en microgravité

Evaporation

condensation-induced

bubble

motion

after

temperature

gradient

set-up

Vadim

S. Nikolayev

a

,

∗

,

Yves Garrabos

b

,

Carole Lecoutre

b

,

Guillaume Pichavant

c

,

Denis Chatain

c

,

Daniel Beysens

d

,

e

a_{Servicedephysiquedel’étatcondensé,CEA,CNRS,UniversitéParis-Saclay,CEASaclay,91191Gif-sur-Yvettecedex,France} b_{CNRS,UniversitédeBordeaux,ICMCB,UPR9048,33600Pessac,France}

c_{UniversitéGrenobleAlpes,CEAINAC–SBT,38000Grenoble,France} d_{Univ.GrenobleAlpes,CEAINAC–SBT,38000Grenoble,France} e_{ESEME,PMMH–ESPCI,10,rueVauquelin,75231Pariscedex5,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received26February2016 Accepted21April2016 Availableonline31October2016

Keywords:

Bubblemotion Thermalgradient Criticalphenomena Pistoneffect

Thermocapillary(Marangoni)motionofagasbubble(oraliquiddrop)underatemperature gradientcanhardlybepresentinaone-componentfluid.Indeed,insuchapuresystem, the vapor–liquid interface is always isothermal (at saturation temperature). However, evaporation onthe hot side and condensationonthe coldside can occur and displace the bubble.Wehave observedsuchaphenomenon intwodifferent fluidssubmitted to atemperaturegradientunderreducedgravity:hydrogenundermagneticcompensationof gravity inthe HYLDEfacility atCEA-Grenobleand water inthe DECLIC facilityonboard the ISS. The experiments and the subsequentanalysis are performedin the vicinity of the vapor–liquid critical point to benefit from critical universality. In order to better understand the phenomena, a1D numerical simulation has been performed. After the temperature gradient is imposed, two regimes can be evidenced. At early times, the temperaturesinthebubbleand thesurroundingliquidbecomedifferentthankstotheir different compressibility and the “piston effect” mechanism, i.e. the fast adiabatic bulk thermalization inducedbytheexpansionofthethermal boundarylayers.The difference in local temperature gradients at the vapor–liquid interface results in an unbalanced evaporation/condensation phenomenon that makes the shape of the bubble vary and provokeits motion.Atlongtimes, asteadytemperaturegradient progressively formsin theliquid(butnotinthebubble)andinducessteadybubblemotiontowardsthehotend. Weevaluatethebubblevelocityandcomparewithexistingtheories.

©2016Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Classically,whenabubbleofgas(radiusR) isimmersedina liquidandsubjectedtoatemperaturegradient,abubble driftalong thegradientisobservedwhen thegravityeffectsare negligible.Thismotionisclassically attributedto a ther-mocapillary(Marangoni)convection,thetemperaturegradientinducingasurfacetensiongradientthatdrivestheflow.The bubblevelocityinasteadygradientisgivenbytheexpression[1]

*

Correspondingauthor.

E-mailaddress:vadim.nikolayev@cea.fr(V.S. Nikolayev).

http://dx.doi.org/10.1016/j.crme.2016.10.002

1631-0721/©2016Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).



vM

= −

2 2

η

L

+

3

η

G d

σ

dT R 2

+ λ

G

/λ

L

∇

T (1)

Here T is temperature,

∇

T is the temperature gradient,

σ

is surface tension,

η

L and

η

G are the liquid and gas shear

viscosities and

λ

Land

λ

G aretheliquidandgasthermalconductivities,respectively.Dependingonthesignofd

σ

/

dT ,the

gasbubblewillmoveparallelorantiparalleltothethermalgradient.Whenbothliquidandgasarethesamesubstance,the gascorrespondstothepurevaporinequilibriumwithitsliquid.Thevapor–liquidinterfaceisatthesaturationtemperature. Anyinterfacetemperaturechangethenleadstoevaporationorcondensationandwillthusbeimmediatelycounterbalanced by thelatent heateffect[2].The thermocapillarymotionis thushardly possible.However, anotherreasonforthebubble motioncanexist.

Asimple1D modelwherethegradientisdirectedalong thez direction showsthat evaporation,whichaddsthevapor tothehotsideofthebubble,andcondensation,whichremovesitfromthecoldside,correspondstoabubbledriftwithan (apparent) velocity vDequaltotheevaporation(condensation)interface velocity.The rateofevaporationdm

/

dt,wherem

ismass,t istime, S istheinterfaceareaperpendiculartoz andL isthelatentheatcanbeexpressedas dm

dt

=

L vDS

ρ

L

= λ

LS dT

dz

.

(2)

Here

ρ

Land

ρ

V aretheliquidandthevapordensity,respectively.Thisgivestheinterfacevelocity

vD

=

λ

L

ρ

LL

dT

dz

.

(3)

Asimilarreasoningappliedtoa3Dmodel(seeAppendix A)resultsinafactor3,

vD

=

3

λ

L

ρ

LL

dT

dz

.

(4)

In contrast to thermocapillary migration, the bubblealways movesin the direction of the temperaturegradient, with a constant speedindependentofitsradius.ThisapproachwasfollowedbyMoketal.[3]whenanalyzingtheir experiments inhydrogenH2.Thethermalgradientwasusedtheretocompensatebuoyancy.

AfurtherstudywasperformedbyOnukiandKanatani[4]intheframeworkofadynamicvanderWaalstheorystarting withentropyandenergyfunctionalwithgradientcontributions[5].Theresultanthydrodynamicequationscontainthestress arising from the densitygradient. It provides a general scheme oftwo-phase hydrodynamics involving the vapor–liquid transitionatnon-uniformtemperature.Accountingalsoforevaporationandcondensation,thevaporbubblevelocityvDwas

foundtobe vD

=

ˆ

η

+



1

+ ˆ

η

/

2



ρ

ˆ



1

/

3

+ ˆ

η

/

2



ρ

VL

λ

L

∇

T (5)

where

η

ˆ

=

η

V

/

η

L and

ρ

ˆ

=

ρ

V

/

ρ

L. Far from Tc (

η

ˆ

→

0), this expression reduces to Eq.(4), which neglects however the

hydrodynamic flow induced by the phase change. Note that close to Tc, where

η

ˆ

∼

1 Eq. (5) yields exactly the same

expression as

ρ

ˆ

∼

1 in numeratorand

ρ

V

∼

ρ

L in denominator.This showsthat the hydrodynamic flows caused by the

phasetransitionhaveonlyasmallimpactonthisphenomenon.Theywillbeneglectedinthetheoreticalpartofthepresent article.Theflowcanhoweverbeimportantinconstrainedgeometries(wherethemoving bubblesizeiscomparabletothe vesselsize),asitwillbediscussedlateron.

In thispaper, we report preliminary experiments performed(i) witha H2 vapor bubble inliquid H2 under magnetic

compensationofgravitynearitscriticalpointand(ii)inliquid–vaporwateratsaturationverynearitscriticalpointunder weightlessness.Thedataareanalyzedintheframeworkofa1Dmodelthatignoreshydrodynamiceffects,butcapturesthe maincharacteristicsoftheproblem,includingphasechange,releaseoflatentheatandcompressibility(pistoneffect).A re-alistic temperaturedistributioninthefluidiscalculated.Thebubbledisplacementcausedbytheevaporation/condensation processisevaluated,inparticularatshorttimesaftertemperaturehasbeenchangedattheboundary,andlatetimeswhen asteadygradienthastakenplace.

2. Experiment

2.1. Hydrogenundermagneticcompensationofgravity

Gravity forces can be compensatedby magnetic forcesthat are thestrongest nearthe endof a solenoid. The HYLDE (HYdrogenLevitationDEvice)facilityhasbeensetupatCEA–Grenobletoworkwithhydrogen.Detailscanbefoundin[6]. It can be shown[7]that a perfecthomogeneous acceleration field cannot be obtainedinthe whole volume.In practice, a zerovalueofeffectivegravityg∗isachievedatoneorseveralpointsinspaceandcanbemadeassmallasneededwithin a finite volume by configuring themagnetic field [8].When the condition g∗

=

0 is achievedat some point, thespatial distribution of g∗ is called “residual gravity”. The residual gravity is directed upward (downward) atthe upper(lower) part of the cell,corresponding to bubbleattraction to the cell center (i.e.to the stable positionfor bubble levitation at

Fig. 1. The HYLDE cell. (a) Photo. (b) Observation of the bubble. The external diameter of the cylinder (4 mm) gives the scale.

Fig. 2. Experimental cell for experiments with near critical pure water, in the High-Temperature Insert (HTI) inside the DECLIC facility.

equilibriuminthevertical direction).Theradial variationof g∗ correspondsto vaporrepulsionfromthe coilaxis.Thisis a severelimitation to studythelong-termmotionofbubbles, asthelatterwill alwaysbe trapped insome points ofthe cell.However, some qualitativeobservations canbe made.Preliminaryexperiments havebeenperformedinacylinder of

L

=

16

.

5 mm inlength, 3 mm ininnerdiameter, and4 mm inouter diameter,madeof polymethylmethacrylate(PMMA, denoted hereafter by the subscript S) (Fig. 1). The cylinder axiscoincides withthe magnetic coil axis. PMMA has been chosenbecauseitistransparentandhasalowthermalconductivity(

λ

S

=

0

.

125 W

·

m−1

·

K−1 at33 K)andthespecificheat

ofthewallsislow(CS

=

180 J

·

kg−1

·

K−1at33 K).Withamassdensityof1

.

15

·

103 kg

·

m−3,onegetsathermaldiffusivity

DS

=

6

.

04

·

10−7 m2

·

s−1,corresponding toacharacteristicdiffusiontimetS

=

L2

/

DS



400 s.ThePMMAcylinderissealed

withstainless steelringstotwo parallelelectrolyticcopper (thermalconductivity at33Kis

≈

1130 W

·

m−1

_·

_K−1_{) blocks.}

Theyareinthermalcontactwiththeheliumbathbythermalconductors.Stainlesssteelisathermalinsulatorattheselow temperatures(thermal conductivityis 3

.

37 W

·

m−1

·

K−1). The uppercopper block iscalled“head” (H).Theother block is the“base”(B); itiskept ataconstanttemperature TB withcontrolaccuracy

±

0

.

3 mK at 33 K;theworkingtemperature

rangeis15–40 K. Thecell canbe filledwithpurepressurizedH2 throughacapillary.This capillaryisclosed bya H2 ice

plug(theH2solidificationtemperatureis14 K),whoseformationisprovidedlocallybyathermalconductorincontactwith

theheliumbath.Thecellisobservedby coherent(parallel)light transmission(Fig. 1b).The cellisfilledattheH2 critical

density.

Theprocedureconsistsinhavinginitiallybase,head,fluidandcellwallsatthesametemperature.Thenthebaseand/or theheadtemperatureischangedbyagivenamount.Undergravitationcompensation,weakthermalflowscanappearwhen theheadorthebasisisheatedduetotheremaininggravitydirectedtowardsthecellcenter.Whentheheadorthebasisis cooled,suchflowsarenotpresent.

2.2. DECLICexperiment

Webriefly presentheretheoptical cellforexperimentalobservationsofcriticalphenomenaathighpressureandhigh temperature, using the HTI (High-TemperatureInsert) module of theDECLIC instrument. DECLIC is theCNES–NASA joint projectonboardtheInternationalSpaceStationsince2009[9].ApictureoftheopticalcellisgiveninFig. 2,whiledetails

Fig. 3. (a)Side-view(withrespectto

Fig. 2

)crosssectionoftheHTIcellshowingtheopticalchannelforlighttransmissionobservationanddetailsofthe mechanicalassembly.(b)Front-viewcrosssectionshowingtheopticalchannelfor90◦lightscatteringmeasurementsthroughthediamondwindow,and detailsofthePt100temperaturesensor(DOCs).

can befound in[10].High-resolutionandhigh-speedopticaldiagnosticscan besynchronizedwithtemperature measure-ments andadjustedtotheselectedmonitoringrateofthethermalpulsesproducedbyone heatingactuator.Temperature regulationisensuredtothesamplecellunitthatcontainsthehigh-pressurecell.Temperatureismaintainedconstantwithin 0.5mKaroundthecriticalpointofwater(

≈

375◦C)andgradientsoftheorderof0

.

1 K

·

cm−1_{canbecreated.}

Thecelldesignsatisfiesseveralscientificandsafetyrequirements.Thecellisintendedtostudywaterinthevicinityof its criticalpoint usingtheoptical diagnosticsofthe DECLICinstrument. The diagnosticsmethodsinclude(coherent)light transmission,gridshadowgraphyofthecompletecellvolume,theturbiditymeasurementsby laserlightattenuation,static small angle light diffusion measurements, and90◦ laser light scattering. For compatibility withthe safety requirements of NASA on boardthe ISS, itcan operate up to 405◦C and33MPa, andsatisfies theleak-before-burst safetyconstraint. Boththecellbodyandthetransparentmaterialsneedtoberesistanttocorrosionathightemperature,especiallycorrosion withinsupercriticalwaterandaqueousmedia.

The transmissionobservationof thefluid volumecan be madethrougha 8-mmobservable diameter.In addition,90◦ light scatteringis measured by aphotodiode through a1.6-mmdiameter. SeveralPt sensors(resistance 25



or 100



) locatedinthecellbodyanditssamplecellhousing(SCH)areusedtomonitortemperatureand/ortogenerateaheatflux closetothefluid.

Fig. 3a shows the cell and the cross-section of the fully assembled sample cell whose typical external dimensions are 62 mm in length, 55 mm in height, and72 mm in depth. The cell body andtransmission flanges are made of In-conel 718 (a nickel-based alloy manufactured by Aubert & Duval, France). The optical windows chosen for transmission observation are madeofsapphire (18 mm indiameter,9 mm inthickness), whilethe one for90◦ light scattering mea-surements is made of synthetic bulk poly-crystalline diamond (5.5 mm in diameter, 1.2mm inthickness). The thermal differentialdilatationbetweendifferentmaterialsisaccountedforbyusinganappropriatenumberofelasticwashersmade of Inconel 718 (from Ressorts Masselin, France). Afterfilling at a near-criticaldensity of water (quality Ultrex II, Ultra-pure Reagent, from J.T. Baker, USA), the cell is closed by a “blind window” made of Inconel 718 (5.5 mm in diameter – 2.5 mm in thickness), using a similar design as the 90◦ light scattering part. The tightening is achieved using gold sealings.

Fig. 4. EvolutionofH2bubbleupperandlowerpoints(zuandzl,respectively),bubblemasscenterpositionszg(leftordinateaxis)andbubblediameter D=zu−zl(rightordinateaxis).Initially,thebaseandtheheadtemperaturesareatT0=32.0 K,thenthebaseiscooledtoTB=31.8 K.

Themain volume(280 mm3)of thefluidsample observedby transmissionisa cylinder(8mm indiameter,5mmin thickness),asshowninFig. 3b.Thedeadvolumesarelimitedtothesmallscatteringandfillingchannels(1.6mmand1 mm indiameter,respectively,7mminlength).Whenfilledatthecriticaldensityofwater(

ρ

c

=

322 kg

·

m−3),themassofwater

is

≈

90 mg.

Thecellisintegratedwithinanickeledcopperalloyhousing(labeledSCHforSampleCellHousing)toformtheHTI Sam-pleCellUnit(SCU).The twoPt25sensors usedfortemperatureregulation(SCUr)andtemperaturemeasurements (SCUm) arelocatedinsidethecopperalloy housing,whilethePt100sensorusedfortemperaturemeasurements(DOCs)insidethe cellbodyislocatedinachannelinfrontofthatusedfor90◦lightscattering(seeFig. 3fordetails).Temperatureregulation (within

±

1 mK)isprovidedwithtwoPeltierelementslocatedsymmetricallyaboveandbelowtheSCU.

BecauseofadisbalanceofthesePeltierelements,atemperaturegradientwas detectedalong theSCUaxis.Inorderto compensatethiseffect,an additionaltemperaturemonitoringprocedurehasbeenimplemented[10].Basedonthecontrol ofthegradientbetweenthetwoPtsensorslocatedonbothpartsoftheHTIcellinsidetheSCU,itallowedthetemperature gradientthroughthecelltobeminimizedinnominalsituations.Thisdisbalancehasalsobeenusedtoimposeacontrolled temperature gradient perpendicular to the cell optical axis. This gradient could be varied by amplitude andby sign. Its typicalamplitudewas 1.6 K/mandthetimetogofromonesteadygradientvaluetoanotheronewas about3 h.Wehave takenadvantageofittocarryouttheexperimentsonthebubbledrift.

Prior to anymeasurement, the criticaltemperaturehas been determined by visual observationof the onsetof phase separation,resultinginthecompletedarkeningofthecell(criticalopalescence).Thefoundvalue Tc

= (

373

.

995

±

0

.

001

)

◦C

is in agreement with the preliminary determination at our ground-based laboratory. It is at present the most accurate determinationofwatercriticaltemperature[11].

3. Measurementresults

3.1. Hydrogen

Intheadoptedmodusoperandi,asteadytemperaturegradientisnotestablishedimmediatelyaftertheheadand/orthe basetemperaturechange.Initially, diffusivethermallayers develop insidethe fluidnearboth theheadandthebase.The bulktemperaturealsochanges,duetothepressurechangeinthebulkfluid.Thetemperaturechangeislargerinthevapor bubblethanintheliquidbecausethederivative

(∂

T

/∂

p

)

ρ islargerinthevapor phasethanintheliquidphase [12].The

time scaleofthispressure riseis quitesmall(“pistoneffect” [13]). This“piston effect”isall themorepronounced when thesampleiscloserto Tc.Thusatemperaturegradientcanimmediatelyformneartheliquid–vaporinterfaceandcausesa

phasechangethatresultsinthemodificationofthebubblediameterandalsoitsmotionifthegradientsaroundthebubble arenotfullysymmetric.InFig. 4isreportedthebubblerelaxationat

(

T0

−

Tc

)/

Tc

=

0

.

970

,

∇

T

=

0

.

129 K/cm;thepressure

drops,thebubblediameterincreases,butthemeandisplacementofthecenterofmass,zg

= (

zu

+

zl

)/

2 (zu andzl arethe

upperandlowerpointsofthebubble,seeFig. 1b),remainsnegligible,inagreementwiththeformationaroundthebubble ofpiston-effect-inducedsymmetrictemperaturegradients.

Fig. 5. Patternofwaterbubbles∼10,800 s≈3 h afterimposingaverticaltemperaturegradientof(a)1.7 K/mand(b)−1.7 K/matTc−200 mK.Themean

displacementis≈1 mm towardsthehotend.Thepatternslookdifferentbecausecoalescenceoccursduringthebubbledrift.

3.2. Water

The results onwaterare concerned withtemperature gradients intwo opposite directionsalong the SCUaxis witha 1.7 K/m temperaturegradient(Fig. 5).Notethatthisisthegradientimposedtothecellexterior.Ithasbeenobservedthat vaporbubblesalwaysmovetowardsthehotterside,asexpected,withavelocityofabout0.37mm/h(1

.

02

·

10−7_m

_·

_s−1_).

It is worth comparing the drift velocity value (4) withthe experimental results.For water at Tc

−

200 mK under a

temperature gradient of

≈

1

.

7 K

·

m−1, where L

≈

150 kJ

·

kg−1,

λ

L

≈

0

.

94 W

·

m−1

·

K−1, and

ρ

L

≈

367 kg

·

m−3, one

ob-tains v

≈

0

.

29 mm

/

h, which compares well withthe above observation of the bubble drift of 0.37 mm/h. In addition, the displacement doesnot depend on the bubblesize, which isin agreement withthe underlyingprocess of condensa-tion/evaporation-induceddisplacement.

4. Theory

The experimental dataare very difficulttobe compared withthetheoretical expression(4) becausetheexperimental local valueofthe temperaturegradientisunknown;one knowsonlythetemperaturedifference ofthecellends andthe cell length to calculatethe average value ofthe gradient. Inthe vicinity ofthe criticalpoint, thermal diffusionbecomes very slow, so that the stationarytemperature distribution is long to attain. In the near-critical region, the piston effect influences stronglythetemperaturevariationinthecell.Whilethephysicaloriginofthepiston effectiswellunderstood, thecalculationsrequiredtorepresentrealisticexperimentalconditionsaredifficulttocarryout.Thebehaviorofnear-critical fluids isindeedcomplicatedby thehighly non-linear equationsofstate (EOS) usedto describethem andbecauseofthe presenceofthevapor–liquidinterface,thepositionofwhichisapriori unknownandshouldbedetermined.Threetheoretical approacheshavebeensuggestedforconfinednear-criticalfluidsintheabsenceofconvection.Thefirstisthehydrodynamic approach[13,14],whichhasbeenappliedtoourknowledgeonlytosingle-phasefluids.Thesecond isthediffuseinterface modelapproach[5];onlysmallsizesystemscanbeanalyzedwithit.Withinthethird(“thermodynamic”)approach[15,16], thepistoneffectistakenintoaccountbyasupplementaryterm

g

(

T

)

=



1

−

CV CP

 

∂

T

∂

P



ρ dP dt (6)

introducedintheheatconductionequationasfollows,

∂

T

∂

t

=

1

ρ

CP

∇ · (λ∇

T

)

+

g

(

T

)

(7)

where T isthelocalfluidtemperature, CV

(

CP

)

,thespecificheat atconstant volume(pressure).Notethat thetermg

(

T

)

isimportantnearthecriticalpointwhereCP

CV.Eq.(7)isequivalenttotheequationsofthehydrodynamicapproachif thetimederivativesarereplacedbytheconvectivederivatives.

The bulk fluid motion isneglected and thepressure P ,assumedto be homogeneous, is onlya function oftime. The pressureisdeterminedfromthefluidmassconservationandcomputedviathenonlinearexpression[15]:

dP dt

= −



V

(∂

ρ

_

/∂

T

)

P

∂

T

/∂

t dV V

ρχ

TdV (8) where

χ

T

=

ρ

−1

(∂

ρ

/∂

P

)

T isthe isothermalcompressibilityand V the volumeof thefluid sample. Theresolution of(8) requiresaniterativeprocedureforeachtimestep.ItconsistsincalculatingthetemperaturewithEqs.(6)–(7)forsometrial valueof P .Theotherthermodynamicparameters(

ρ

,

χ

T

,

. . .

)aredeterminingwiththeEOS

a1Dcalculationforsingle-component(supercritical)fluids.Thetwo-phase casehasbeensolvedwithinasimilarapproach in1D[19,20].Theinterfacepositionwascalculatedfromtheheatbalanceatthevapor–liquidinterface.

Such a methodhas proved to be efficient in 1D. However, its extension to higher dimensionswould require a large computational effort.The computer resources wouldrisesteeply becausethe thermodynamic variableswouldhaveto be evaluatedateachgridpointofthecomputationaldomainbymeansofaniterativeprocedure.Thelatterisnecessarybecause theproblemisnonlinear.Theproblemisstillmoredifficultfortwo-phasefluidsbecauseoftheapriori unknowninterface position.

In [21], some of usproposed an approximate but more computationally efficient method for the calculation of heat andmasstransferinsingle-phasefluids. Itwascalled“fastcalculationmethod”.Ithasbeenshownthat, comparedtothe rigoroushydrodynamicapproach,thefastmethodprovidesasufficientlygoodaccuracy.Inthepresentwork,itisgeneralized tothetwo-phasefluidcase.Itisthenappliedin1Danditsresultsarecomparedtotheexperimentaldata.

4.1. Fastcalculationmethod

The method is based on the energy equation (7) with the initial condition of homogeneous temperature T0 inside

the fluid.It will be solved withtheboundary element numericalmethod (BEM)[21],which usesonly thevalues ofthe variablesatthedomainboundariesasunknowns.Forthisreason,itisfarmoreefficientthanthefinite-differencemethod usedpreviouslyfortwo-phase calculations[19,20].However,thisBEMadvantagewouldbelostifthepressureequationin the form(8) were employed. Eq.(8) requiresthe knowledge ofthe variables atthe internal domain points;they would need tobe calculated anywayateach iteration fromthe boundaryvalues (ina separate calculation). A differentformof thepressure equation that usesonly theboundary valuesofthevariables thus needsto be derived.The thermodynamic approachwillbeusedforitsderivation.Wesummarizefirstthesingle-phasecase[21].

4.1.1. Single-phasefluid

The totalamountof heat

δ

Q given tothe fluidduring the time

δ

t canbe calculatedby integrationover theexternal boundary S ofthefluid,

δ

Q

= δ

t

S

jndS (10)

where jn istheheatfluxdirectedinwardstothefluiddomain.Accordingtoageneralthermodynamicexpression,

δ

Q

=

V

CV

ρ



∂

T

∂

P



ρ

δ

P

−

CP

α

P

δ

ρ



dV (11)

wheretheintegrationisperformedoverthefluidvolume V and

α

P

= −

1

ρ



∂

ρ

∂

T



P (12) istheisobaricthermalexpansioncoefficient.Byapplyingtheintegralmeanvaluetheoremto(11),onefinds

δ

Q

=

V Ca_V

ρ

a



∂

T

∂

P



a ρ

δ

P (13)

wherethesuperscript‘a’meansthevaluecalculatedfortheaveragedensity

ρ

a_{andthespatiallyhomogeneouspressureP .}

Thesecondtermof(11)disappearsaftertheaveragingbecausetheaveragedensityisconstant(thefluidcellisclosed)and

δ

ρ

a

_≡

_0.

Thesingle-phaseversionofthepressureequationisthus dP dt

=



∂

P

∂

T



a ρ 1 V

ρ

a_Ca V

S jndS (14)

TheadvantageofthisformwithrespecttoEq.(8)isthatitusesonlytheboundaryandspatiallyaveragedvalues. Thesameideaofthevolumeaveragingisappliedtotheenergyequationwheretheterm(6)isreplacedby

ga

(

t

)

=



1

−

C a V Ca_P

 

∂

T

∂

P



a ρ dP dt (15)

andmaterialparametersarereplacedbytheir averagedvalues,whichthusbecomeindependentofthespatialvariableand dependonlyontime.TheaveragingpermitstosimplifyEq.(7)byintroducingavariable

ψ (



x

,

t

)

=

T

(

x



,

t

)

−

Ea

(

P

)

(16) with



x thepositionvectorand

Ea

(

P

)

=

P

P0



1

−

C a V Ca_P

 

∂

T

∂

P



a ρ dP

+

T0 (17)

where P0 istheinitialpressurerelatedtotheinitialtemperatureT0throughtheEOS.Thereasonoftheintroductionof

ψ

isthatitobeysasimplerequation

∂ψ

∂

t

=

D

a

_∇

2

_ψ

₍₁₈₎

withthetrivialinitialcondition.ThethermaldiffusioncoefficientDa

_{= λ}

a

_/

_ρ

a_Ca

P dependson P only.Thisallowsthetimet tobereplacedbyanewindependentvariable

τ

definedbytheequation

d

τ

dt

=

D

a

₍

_P

₎

₍₁₉₎

whose initial condition can be imposed as

τ

|

t=0

=

0.Since P is a function of t only, this initial value problemis fully

defined.ThesubstitutionofEq.(19)intoEq.(18)resultsinthelineardiffusionproblem

∂ψ

∂

τ

= ∇

2

_ψ

ψ

|

τ=0

=

0

(20) ItcanbesolvedwithBEM.Acomparisonwiththerigoroushydrodynamicapproach[21]showsthatthefastmethodresults inaccuratecalculationsinspiteofthesimplifyingassumptions.

4.1.2. Two-phasefluid

Asimilarapproachcanbeusedforthetwo-phasecase.Forthesakeofgenerality,weperformthederivationsforacase where N domainsofone(liquidorvapor)phasedenotedbythesuperscriptk

=

1

. . .

N aresurroundedbyanother(vapor or liquid,respectively) phase denoted by k

=

N

+

1.The volume fractions corresponding to each of thedomains can be introducedvia

φ

k

=

Vk

/

V (21)

Thefluidisassumedtobeconfinedinaclosedcellsothatboththetotalvolumeandthetotalmassareconserved, N+1



k=1

φ

k

=

1 (22) N+1



k=1

φ

k

ρ

k

₌

_ρ

a ₍₂₃₎

while boththeaveragedensity

ρ

k _{andthevolumefraction}

_φ

k _{ofeach domainmaychangeintime dueto phasechange.} Byproceedingsimilarlytothesingle-phasecase, onecanapply theintegralmeanvaluetheoremto(11),writtenforeach domain,

δ

Qk

=

V

φ

k

Ck_V

ρ

k



∂

T

∂

P



k ρ

δ

P

−

C k P

α

k P

δ

ρ

k



(24) where the pressureis thesame (because ofits spatial homogeneity) forall domains. Theupper indexk means that the variableiscalculatedforthethermodynamicstatedefinedby

(

ρ

k

_,

_P

₎

_{,i.e.theyarespatiallyaveraged.Unlikethesingle-phase} case(cf.Eq.(13)),thesecondtermdoesnotdisappearinEq.(24)because

δ

ρ

k

₌

_{0.Similarlyto(10),the}

_δ

_Qk_valuescanbe obtainedbyintegrationovertheboundarySk_{ofthek-thdomain,}

δ

Qk

= δ

t

Sk

jk_ndS (25)

where jkn istheheatfluxalongthevectornormalto Skanddirectedinwardstothek-thdomain,k

=

1

. . .

N

+

1.

Apartoftheheatsuppliedtothefluidisusedforevaporationorcondensationateachoftheinterfaces.Asinprevious works[19,20],itisassumedthatthephasetransitiondoesnotoccurinsidethedomainssothattheliquidcanbeoverheated

δ

Q_ik

= −δ

t

Sk

(

jk_n

+

j_nN+1

)

dS (26)

isconsumedatthek-thinternalvapor–liquidinterface,sothat

δ

Q_ik

= ∓

V L

δ(

ρ

k

φ

k

)

(27)

where

L

isthelatentheat.Theuppersigncorrespondstoacasewherethedomainsk

=

1

. . .

N areliquidandthedomain

N

+

1 isthevapor;thelowersignreferstotheoppositesituation.BycombiningEqs.(26)and(27),oneobtains

±

Ld

(

ρ

k

_φ

k

₎

dt

=

1 V

Sk

(

jk_n

+

j_nN+1

)

dS (28)

wherek

=

1

. . .

N.Similarly,Eq.(24)combinedwith(25)resultsin

φ

k

Ck_V

ρ

k



∂

T

∂

P



k ρ dP dt

−

Ck_P

α

k_P d

ρ

k dt



=

1 V

Sk jk_ndS (29)

wherek

=

1

. . .

N

+

1.Weobtainasetof2N

+

1 ordinarydifferentialequations(ODE):N equations(28),andN

+

1 equations

(29).Twoconservationlaws((22), (23))canbeusedtoeliminate

φ

N+1_and

_ρ

N+1 _{variablesfromthelastofequations(29).}

Thissetof2N

+

1 equationsislinearwithrespecttothe2N

+

1 timederivativesof

φ

k,

ρ

k_{,andP .Thesetcanthusbeeasily} solvedwithrespecttothederivatives,eithernumericallyoranalytically.Theresultingsetof2N

+

1 ODEinthecanonicform canbesolvedateachiterationstepbyanynumericalmethod,likethatofRunge–Kutta.

Theequationsforthereducedtemperature

ψ

k inthek-thdomainare similartoEqs.(16)–(20),wherethe superscript aneedtobereplacedbyk;notethat

τ

k_{willalsobedifferentforeach domain.Inadditiontotheboundaryconditionsat} theexternalsurfaceofthefluid,extraconditionsare necessaryatthevapor–liquidinterfaces.Thepressurefixestherethe temperaturetobeequaltothesaturationtemperatureTsat

=

Tsat

(

P

)

.AttheinterfacesSk,theconditionsfor

ψ

read

ψ

N+1

+

EN+1

= ψ

k

+

Ek

=

Tsat

(

P

)

(30)

wherek

=

1

. . .

N. 4.2. Problemstatement

Usually,the “temperaturestep” boundarycondition isapplied for1D problems.Thisheating process correspondsto a fluidcell,initiallyatuniformtemperature,whichissubmittedtosuddentemperatureincrease



T atoneofitsboundaries, whiletheotheriskeptattheinitialtemperatureT0.Thisheatingconditionisphysicallyunrealisticbecausetheinitialvalue

fortheheatfluxattheheatedboundaryisinfinite.Inthiswork,weuseinsteadthegradualtemperatureincrease,

TH

=

T0

+ 

T

[

1

−

exp

(

−

t

/

τ

r

)

]

(31)

where

τ

risthetransitiontime.

4.3. 1Dnumericalresults

Wereportthe1DsimulationofthebehaviorofaH2 bubbleofsizeD

=

2

.

1 mm initiallylocatedinthemiddleofthecell.

Thetheoryisquitegeneralandcanbeappliedtobothliquiddropsandvaporbubbles.Forpracticalreasons(thecontainer walls are in generalwetted by liquid), bubble motion inside the liquid is often considered. The cell is initially at T0

=

0

.

875Tc (28.875 K).Attimet

=

0,thetemperatureoftheheadlocatedat z

=

16

.

5 mm ischangedtoT0

+ 

T

=

0

.

945Tc

(31.185 K),withtypicaltime

τ

r

=

1 s,whilethebasetemperatureTB

≡

T

(

z

=

0

)

=

T0.Twotimeregionscanbeevidenced

withrespecttothethermaldiffusiontime,tD

=

L2

/

Da

≈

450 s.The1D simulationseemstobesufficientforthefollowing

reasons.ThelateralPMMAcellwallshavea smallthermalconductivitysimilartothatoftheliquidandofthevapor. The exteriorofthecellisundervacuumandtheradiationheatlossesarelowbecauseoflowtemperatures;theexteriorofthe cellisthusadiabatic,andradialthermalgradientsarenotexpected.

4.3.1. Shorttimes

WhenTHisraised,thetemperatureprofileissoonmodifiedasseeninFig. 6a.First,ahotthermalboundarylayer

devel-opsatthehot endwhile,thanks tothepistoneffect,thebulktemperatureandsamplepressureincrease homogeneously. Thistemperatureincreaseismuchmorepronouncedinthevaporphasebecausethetermg ismuchlargerthere[12].This createshotboundarylayersclosetothebubbleinterface.Theweakrecondensationprocessoccursandthebubblevolume

Fig. 6. Simulatedevolutionofthespatialtemperaturedistributioninthe1Dcell.Thecurvelegendsshowthecorrespondingtimesandpositionsofthe lowerandupperbubbleends(inmm).(a)Shorttimes(semi-logplot).(b)Longtimes.

Fig. 7. (a)Simulatedevolutionoftheinterfacetemperatureandthelocationofinterfaceszl,zu andzg= (zl+zu)/2.(b)Correspondingvelocitydzg/dt

(interruptedcurve)ascomparedtothevelocityvD(fullcurve)calculatedwithEq.(3).

decreases.Thedynamicsofthecenterofmassofbubbleisdeterminedbytherecondensationprocess,i.e.bytheasymmetry ofthetemperaturedistributionaroundthebubble.Thebubbleslightlydisplacestothecoldendofthecell(Fig. 7).

Theshort-timebehavioroftheH2 bubble(motionandbubbleshrinkingimmediatelyafterheating)canthusbe

satisfac-torilyexplainedbytheshort-timetransientbehavior.

4.3.2. Longtimes

Astimegoeson,thehotboundarylayerdevelopedatthehotendattainsthebubble.Themasstransferinversesthesign andtheevaporationbeginsatthehotendofthebubble,whilecondensationcontinuesatitscoldend. Asthetemperature profile developstowardsa linearprofile,both interfacesstartto movewiththesamevelocity inthesame(hot)direction, resulting inaconstant bubblesize. Fig. 6bandFig. 7a showthisbehavior, wherea constantshape andconstantvelocity are observedfortimeslonger thanthediffusion time(

≈

450 s).Itis interestingto comparethevalue ofthisasymptotic velocity with that obtainedfromEq. (3), wherethe value of thegradient is takenasan average of thegradients at the bubbleinterface.ThecomparisonneedstobemadewithEq.(3),whichissuitableforthe1Dcase,ratherthanwithEq.(4), derivedforthe3Dcase(seeAppendix A).Theagreementissatisfactoryatlongtimeswherethepistoneffectisnegligible.

Onlythecaseofthenon-symmetricinitialtemperaturechangeissimulatedhere.Inthecaseofthesymmetric temper-aturechange,thepistoneffectisnotexpectedtobepronounced,andthebubblewillnotmoveorchangeitssizeuntilthe diffusiveboundarylayerattainsit.Inotherwords,thebubbleisexpectedtoremainstillandunchangedatshorttimescale; itsbehaviorisexpectedtobesimilartotheabove-simulatedcaseatlongtimes.

A gas bubbleimmersed in a liquid submitted to a temperaturegradient is usually assumedto move because of the interfacial tensionvariationalong itsinterface. Whenthe gasisthepure vapor ofthesamefluid,the interfacehasto be isothermalatthesaturationtemperatureandthermocapillarymotionshouldnotoccur.However,evaporationonthehotside andcondensationonthecoldsidecanoccurandindeeddisplacethebubble.Experimentsperformedunderweightlessness to cancel the buoyancyeffects with waternear its critical point (

≈

647 K) have confirmedthis behavior. Another series of experiments with hydrogen nearits critical point (

≈

33 K) under magnetic compensation of gravity shows transient behavior.Thelatterinvolvescomplicatedinterplayofpressurechangeduetopistoneffect,releaseoflatentheatandthermal diffusion,andcanbewellunderstoodby1Dnumericalsimulations.Thatmethodofsimulations,whichhasbeenpreviously validatedforthesupercritical(singlephase)fluids,hasbeenextendedhereforthetwo-phasefluidcase.Comparedwiththe existingmethods,thepresentmethodrequiresmuchsmallercomputationresourcesandisthussuitableforsimulationsin higherdimensionsintheabsenceofgravity.

Acknowledgements

ThisworkwassupportedinpartbyCNESMFAprogram.WearegratefultoC.Marietteforhelpfulcommentsandthank CNESandNASAforhavingprovidedustheaccesstotheInternationalSpaceStation.

Appendix A. 3Dbubbledriftinaconstanttemperaturegradientmodel

ConsiderasphericalbubbleofradiusR inanexternallyimposedconstanttemperaturegradientdT

/

dz.Thebubble inter-facewiththeimposedsaturationtemperature(withrespecttowhichT willbe given)willperturbthelineartemperature field.Thesolutiontotheoverallfieldisgivenin[22](problem1ofSec. 3)forthecasewhenthebubblecenteris momen-tarilysituatedatz

=

0 point,wheretheunperturbedvalueofthetemperatureT

=

0 (thisassumptiondoesnotleadtothe lossofgenerality).Inthesphericalreference(z

=

r cos

θ

)

T

=

rdT

dzcos

θ (

1

−

R

3

_/

_r3

₎

_(A.1)

Thelocalvelocityofthebubblesurfaceduetoevaporation–condensationisfoundfromtheheatflux jn,

vn

(θ )

=

jn

ρ

LL

=

λ

L

ρ

LL dT dr

r=R

=

3

λ

L

ρ

LL dT dzcos

θ

(A.2)

Sincethepositionofthemasscenterofthebubbleisdefinedby

zg

=

1 Vb

Vb z dV

whereVb

=

4

π

R3

/

3 isthebubblevolume,thebubblevelocityis

vD

=

dzg dt

=

1 Vb d dt

Vb z dV

−

1 Vb dVb dt zg

Thesecondtermdisappearssince zg

=

0 accordingtoourinitialassumption. ByusingtheReynoldstransporttheorem,oneobtains

vD

=

1 Vb

Sb z vndS (A.3)

where vn fromEq.(A.2) hasto be used.The integration overthe bubblesurface Sb can be reducedto that over

θ

that

variesfrom0to

π

,dS

=

2

π

R2sin

θ

d

θ

.Bycarryingouttheintegration,onerecoversEq.(4).

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