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Experimental investigation on single-phase pressure losses in nuclear debris beds: Identification of flow regimes and effective diameter

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To cite this version : Clavier, Rémi and Chikki, Nourdine and Fichot,

Florian and Quintard, Michel

Experimental investigation on

single-phase pressure losses in nuclear debris beds: Identification of flow

regimes and effective diameter

. (2015) Nuclear Engineering and

Design, vol. 292. pp. 222-236. ISSN 0029-5493

To link to this article : doi:

10.1016/j.nucengdes.2015.07.003

URL :

http://dx.doi.org/10.1016/j.nucengdes.2015.07.003

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Experimental

investigation

on

single-phase

pressure

losses

in

nuclear

debris

beds:

Identification

of

flow

regimes

and

effective

diameter

R.

Clavier

a,∗

,

N.

Chikhi

a,∗∗

,

F.

Fichot

b

,

M.

Quintard

c,d

aInstitutdeRadioprotectionetdeSûretéNucléaire(IRSN)PSN-RES/SEREX/LE2M,Cadarachebât.327,13115StPaul-lez-Durance,France bInstitutdeRadioprotectionetdeSûretéNucléaire(IRSN)PSN-RES/SAG/LEPC,Cadarachebât.700,13115StPaul-lez-Durance,France cUniversitédeToulouseINPTUPSInstitutdeMécaniquedesFluidesdeToulouse(IMFT),AlléeCamilleSoula,F-31400Toulouse,France dCNRSIMFT,F-31400Toulouse,France

h

i

g

h

l

i

g

h

t

s

Single-phasepressuredropsversusflowratesinparticlebedsaremeasured.

•Conditionsarerepresentativeoftherefloodingofanuclearfueldebrisbed.

•Darcy,weakinertial,stronginertialandweakturbulentregimesareobserved.

•ADarcy–Forchheimerlawisfoundtobeagoodapproximationinthisdomain.

•Apredictivecorrelationisderivedfromnewexperimentaldata.

a

b

s

t

r

a

c

t

Duringaseverenuclearpowerplantaccident,thedegradationofthereactorcorecanleadtotheformation

ofdebrisbeds.Themainaccidentmanagementprocedureconsistsininjectingwaterinsidethereactor

vessel.Nevertheless,largeuncertaintiesremainregardingthecoolabilityofsuchdebrisbeds.Motivated bythereductionoftheseuncertainties,experimentshavebeenconductedontheCALIDEfacilityinorder

toinvestigatesingle-phasepressurelossesinrepresentativedebrisbeds.Inthispaper,theseresults

arepresentedandanalyzedinordertoidentifyasimplesingle-phaseflowpressurelosscorrelationfor

debris-bed-likeparticlebedsinrefloodingconditions,whichcoverDarceantoWeaklyTurbulentflow

regimes.

Thefirstpartofthisworkisdedicatedtostudymacro-scalepressurelossesgeneratedby

debris-bed-likeparticlebeds,i.e.,highsphericity(>80%)particlebedswithrelativelysmallsizedispersion(from

1mmto10mm).ADarcy–Forchheimerlaw,involvingthesumofalineartermandaquadraticdeviation,

withrespecttofiltrationvelocity,hasbeenfoundtoberelevanttodescribethisbehaviorinDarcy,Strong

InertialandWeakTurbulentregimes.Ithasalsobeenobservedthat,inarestricteddomain(Re=15to

Re=30)betweenDarcyandWeakInertialregimes,deviationisbetterdescribedbyacubicterm,which

correspondstotheso-calledWeakInertialregime.

Thesecondpartofthisworkaimsatidentifyingexpressionsforcoefficientsoflinearandquadratic

termsinDarcy–Forchheimerlaw,inordertoobtainapredictivecorrelation.Inthecaseof

monodis-persebeds,andaccordingtotheErgunequation,theydependontheporosityofthemedium,empirical

constantsandthediameteroftheparticles.ApplicabilityoftheErgunequationfordebris-bed-likeparticle bedshasbeeninvestigatedbyassessingthepossibilitytoevaluateequivalentdiameters,i.e., character-isticlengthallowingcorrectpredictionsoflinearandquadratictermsbytheErgunequation.Ithasbeen

observedthattheSauterdiameterofparticlesallowsaveryprecisepredictionofthelinearterm,by

lessthan10%inmostcases,whilethequadratictermcanbepredictedusingtheproductoftheSauter

diameterandasphericitycoefficientasanequivalentdiameter,byabout15%.

∗ Correspondingauthor.Tel.:+33442199367. ∗∗ Correspondingauthor.Tel.:+33442199779.

E-mailaddresses:remi.clavier@irsn.fr(R.Clavier),nourdine.chikhi@irsn.fr(N.Chikhi).

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Nomenclature Latinletters

D beddiameter 93.96±0.04mm (mm) d sphericalparticlediameter (m) dhni numbermeandiameter

P

idini

P

ini (m) dhli lengthmeandiameter

P

id 2 ini

P

inidi (m) dhsi surfacemeandiameter

P

id 3 ini

P

inid 2 i (m) dhvi volumemeandiameter

P

id 4 ini

P

inid 3 i (m) dSt Sauterdiameter 6VSpartpart (m)

dS surfaceequivalentdiameter



S

part 



1/2

(m) dV volumeequivalentdiameter



6V part 



1/3

(m) di diameter of i-th sort of particlein a multi-sized

sphericalparticlebed (m) H Bedheight 499.0±1.6mm (mm) hK Ergunconstantforpermeability (–)

h Ergunconstantforpassability (–)

K permeability (m2)

mw massofwaternecessarytofillthebedup (kg)

ni numberofparticleofi-thsortinamulti-sized

spher-icalparticlebed (–) P pressure (Pa)

Q volumetricflow (m3/s)

Rep Reynoldsnumberinporousmedium (1−ε)UdSt (–)

Spart totalsurfaceofparticlesofabed (m2)

spart surfaceofasingleparticleofabed (m2)

si surfaceofi-thsortofparticleinamulti-sized

spher-icalparticlebed (m2)

Vpart totalvolumeofparticlesofabed (m3)

v

part volumeofasingleparticleofabed (m3)

v

i volumeofi-thsortofparticleinamulti-sized

spher-icalparticlebed (m3)

U filtrationvelocity Q

D2/4 (m/s) Greekletters

ε porosity (–)  passability (m)

 fluiddynamicviscosity (Pas)  fluiddensity (kg/m3)

w waterdensity (kg/m3)

sphericity 1/3(6Vpart)2/3 Spart (–)

1. Introduction

Inthecourseofaseverenuclearaccident,theheatupofthe coreaftercompleteorpartialdry-outcanleadtoacollapseoffuel assembliesandtotheformationofadebrisbed,whichismuch moredifficulttocooldownthantheintactfuelassemblies.This phenomenonhasbeenobservedintheTMI-2highlydamagedcore

in1979(Broughtonetal.,1989),andreproducedinmany

experi-mentalprograms:LOFT(HobbinsandMcPherson,1990),PHEBUS

(Repetto,1990),PBF(Pettietal.,1989).Removalofthedecayheat

fromthedebrisbedbyrefloodingisessentialforthemitigationof theaccident.However,thesuccessofthisoperationcanbe com-promizedbymanyfactors,suchasdecayheatpower,exothermal oxidationofZirconiumbysteam,oratooweakpermeabilityofthe

Fig.1.Schematicviewofflowstructureindebrisbedsduringreflooding.

bed,andcannotbepredictedonthebasisofcurrentknowledge andunderstanding.Thisimpliestostudyhowwaterpenetrates thisdegradedgeometry,whichcanbedescribedasahotporous medium.

Manyanalyticalexperimentalprogramshavebeen,andstillare dedicatedtothatquestion.Tutuetal.(1984)andGinsbergetal.

(1986)performedthefirstsmallscaleexperiments(18kgofdebris)

onrefloodingofsingle-sizesphericalparticlebeds,i.e., monodis-persebeds.ExperimentsofWangandDhir(1988),and,later,the SILFIDEprogrambyEdF(AtkhenandBerthoud,2006),allowedto reachlargerscales(70kgofdebris)andtosimulateresidualpower byaninductionheating.Duringthelate-2000s,more representa-tiveconditionswereinvestigatedduringexperimentsofDEBRIS

(Rashidetal.,2008),POMECO(LiandMa,2011a,b)andPRELUDE

(Repettoetal.,2011)programs,wherehighertemperatures,upto

1000◦C,wereinvestigated,aswellasmulti-dimensionaleffects,for

examplebyuseofalateralby-passtosimulatetheeffectof non-degradedassembliesintheouterpartsofthecore.Nowadays,the PEARLprogram(Chikhi,2014)aimsatinvestigatingtheeffectof highertemperatures,largerscalesandpressure.

Inthisframework,anewfacility,CALIDE,hasbeenbuiltatIRSN (Cadarache,France),inordertostudypressurelossesgeneratedin representativedebrisbedsinrefloodingconditions.Therelevance ofthisstudyliesinthefactthatpressurelossesconstituteakey parametergoverningwaterpenetrationinahotdebrisbed,and thatnoconsensusexistsoncorrelationsvalidforbedspackedwith coarseand/ormulti-sizedparticles(Ozahietal.,2008;LiandMa,

2011a,b;MacDonaldetal.,1991).Establishmentofaccurate

corre-lationsforpressurelossesinporousmediaarethereforenecessary forinterpretationofrefloodingexperimentsandtheirnumerical simulationwithsevereaccidentcodes.Bothsingleandtwo-phase flowcorrelationsareneeded,sinceboththeseconfigurationsoccur inahotparticlebedduringreflooding(seeFig.1):two-phaseflows occurnearthequenchingfront,whilesingle-phaseflows,steam orliquidwater,occurinupstreamanddownstreamparts.Inthis work,experimentalresultsobtainedwithsingle-phase flowsin theCALIDEfacilitywillbeanalyzedinordertoderivea simple macro-scalecorrelationforsingle-phasepressurelossesvalidated forporousmediarepresentativeofnucleardebrisbeds,andfor flowingconditionsrepresentativeofareflooding.

AdetaileddescriptionoftheCALIDEfacilityandofthe stud-iedparticlebedsareprovidedinSection 2.TheCALIDEfacilityis designedtoreproducethecharacteristicfiltrationvelocities dur-ingreflooding,whichbasicallycorrespondtoReynoldsnumbers

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rangingfromRe=15toRe=100inliquidparts,andtoRe≈1000in gasparts(Repettoetal.,2011),Reynoldsnumberbeingdefinedby: Re= dStU

(1−ε), (1)

whereandarethedensityanddynamicviscosityofthefluid, respectively,εistheporosityofthemedium,andUanddStarethe

characteristicspeedanddimension.Uisthefiltrationvelocity,or theDarcyvelocity,definedas:

U= Q

D2/4, (2)

whereQisthevolumetricflow,andDthediameterofthetest sec-tion.ThecharacteristicdimensiondStistheSauterdiameterofthe

particles,definedby: dSt=

6Vpart

Spart , (3)

whereVpartandSpartarethetotalvolumeandsurfaceofthe

parti-cles,respectively.

ManydefinitionsexistforReynoldsnumberinporousmedia. Theexpressionusedinthiswork(Eq.(1)),and inparticularthe 1/(1−ε)factor,isobtainedduringthederivationoftheErgun equa-tionfromtheKozenymodel(duPlessisandWoudberg,2008).It hasbeenusedinsimilarsituations(Ozahietal.,2008;LiandMa,

2011a,b).

Experimentalparticlebedsarechosentoberepresentativeof realdebrisbeds,which basicallyconsistincoarseconvex parti-clesofsizerangingfromafewhundredmicronstoapproximately 10mm,withaverageporosityof40%(Chikhietal.,2014).

Flowingconditionsduringrefloodingarebeyondthevalidity domainoftheDarcy’slaw,whichonlyholdsinthecreepingregime, uptoReynoldsnumberofafewunitstoadozen,dependingon themedium (Chauveteau and Thirriot,1967; Meiand Auriault,

1991;Lasseuxetal.,2011).Inamacroscopicallyisotropicmedium

throughwitchaone-dimensionalupwardflowoccurs,Darcy’slaw canbeexpressedas:

∂P

∂z

+g=



KU, (4)

where P is the pressure, g the gravity acceleration, and K the permeabilityof themedium.Whenthefluid velocityincreases, deformationof thestreamlinesoccurin largestpores,followed bylocalrecirculations,duetopore-scaleinertialeffects,then pro-gressivelyinsmallerandsmallerpores,asshownbyChauveteau

and Thirriot (1967). Similarly, for Reynolds numbers of a few

hundredsor thousands(DybbsandEdwards, ulent; Latifietal.,

1989; Kuwahara et al., 1998), pore-scale turbulence appears.

Macroscopically,itresultsindeviationstoDarcy’slaw,orso-called Forchheimereffects(Forchheimer,1901),whichareoften repre-sentedbythefollowingmathematicalform:

∂P

∂z

+g=

 KU+bU

n. (5)

Thevalueof exponentn depends onflowregime and pore-scalegeometricalstructure. Itsdeterminationfor CALIDEdebris bedsisthepurposeofSection 3.Instructureddistillation pack-ings,forexample,SoulaineandQuintard(2014)observedn=3/2, whileaquadraticdeviation(n=2)usuallyfitswellwith experimen-taldataforStrongInertialregimeindisorderedsphericalparticle

beds(Forchheimer,1901; Ergun,1952;MacDonaldet al.,1979;

Liand Ma,2011a,b), aswellas numericalsimulations (Lasseux

etal.,2008,2011).Furthermore,itcanbemathematicallyderived

fromNavier–Stokesequations,usinghomogenizationmethod(Mei

andAuriault,1991;WodiéandLevy,1991;Firdaoussetal.,1997),

that thefirst deviations to Darcy’slaw should fit witha cubic

law (n=3), which corresponds tothe so-called “Weak Inertial” regime. Althoughsupported bynumerical simulations (Lasseux

etal.,2011;YazdchiandLuding,2012),theoccurrenceofWeak

Inertialregimeindisorderedparticlebedshasneverbeen exper-imentallyobserved,toourknowledge, becauseofthevery low magnitudeofthecubicterm,whichisusuallylostinmeasurement uncertainties.Thisexplainswhyaquadraticlawisusually consid-eredasanaccuratedescriptionofdeviationstoDarcy’slawfrom DarcytoTurbulentregimesinthis classofmedia.Nevertheless, andgiventhehighprecisionoftheinstrumentationoftheCALIDE facility,theexistenceofWeakInertialregimeisworth investigat-ing,especiallyconsideringthelackofexperimentaldataconcerning thatquestionintheliterature.

Quantitativepredictionofpressurelossesnecessitates expres-sionsforcoefficientsoflinearandnon-linearterms.Inthiswork, expressionsforDarcy(linear)andStrongInertial(quadratic)terms willbedetermined.Mostoften,asitwillbediscussedlater,cubic deviationsdonotplayasignificantrole,giventheirmagnitudeand therangeofReynoldsnumbersencounteredinpractice.Forthat reason,noexpressionwillbesoughtforcoefficientofthecubic terminWeakInertialregime.

Inthecaseofmonodispersebeds,avalidatedmodelforpressure lossesistheErgun’slaw(Ergun,1952),widelyusedinchemicaland petroleumengineering(NemecandLevec,2005;ShabanandKhan,

1995): −

∂P

∂z

+g=  KU+  U 2, (6)

whereisthe“passability”ofthemedium,permeabilityand pass-abilitybeingcalculatedby:

K= ε 3d2 hK(1−ε)2 (7) = ε 3d h(1−ε), (8)

whereεistheporosityofthebed,dthediameteroftheparticles andhKandharetheErgunconstants.

Eqs. (7) and (8) are empirical correlations established for monodispersebeds.InSection4,theirapplicabilityto debris-bed-like media will be investigated, by assessing the possibility to defineequivalentdiameters,i.e.,characteristicdimensions allow-ingcorrectpredictionsofpermeabilityandpassabilitytermswhen injectedinEqs.(7) or(8).Althoughphysicallymeaningless,this kindofapproachhasbeenfoundtoberelevant,froman empiri-calpointofview,forpermeabilitypredictionofthisclassofmedia

(Chikhietal.,2014).Inthiswork,differentequivalentdiameters

willbesoughtforpermeability(dK)andpassability(d),among

availablevaluesofliterature(Ozahietal.,2008;LiandMa,2011b). 2. CALIDE

2.1. Experimentalfacility

TheCALIDEfacility,asillustratedinFig.2,isanair/waterloopat roomtemperatureandpressure.Itsinstrumentationallows mea-surementofpressurelossesversusflow-ratesina93.96±0.04mm diametercylindricaltestsectionthatcontainsaparticlebed.Six radialholesuniformlydistributedalongthesectionallowpressure tappingatdifferentlevelsinside,downstreamandupstreamthe debrisbed.Astainlesssteelwiremeshisplacedatthebottomof thetestsectiontosupportthebed.Amarkisdrawnonthetest section499.0±1.6mmabovethebottomwiremeshandisusedto adjustthebedheight.

Particlesarechosentoberepresentativeofnuclearfueldebris, intermsofsizeandshape,andarepresentedinnextparagraph.

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Fig.2.SchematicrepresentationoftheCALIDEfacility.

Waterandgasflowratesaredeterminedsothatfiltrationvelocityis representativeofwaterorsteamfiltrationvelocityduringa reflood-ingofadebrisbed.AccordingtotheProbabilisticSafetyAnalysis

(Durinetal.,2013),refloodingofadebrisbedismostlikelytooccur

ataprimarypressurerangingfrom1to10bar.Forthispressure range,theamountofwaterthatcanbeprovidedbythesafety injec-tionsysteminaFrench1300MWePWRrangesbetween1300m3/h

and200m3/h,whichcorrespondstofiltrationvelocityofbetween

32mm/sand5mm/s(thediameterofthevesselis3.76m). Evalua-tionofflowingconditionsinthegasphaserequiresanassumption onsteamflowrate.InPRELUDEexperiments(Repettoetal.,2013), steamproductionsupto100g/shavebeenobservedfor reflood-ingvelocitiesof5mm/s,whichcorrespondstoasteamfiltration velocityof7.2m/s.

Thus, instrumentation allows measurements of air flows (up to 1000Nl/min±0.5%≡2.58m/s at 20◦C, 1atm), water

flows (up to 600kg/h±0.2%≡24mm/s), pressure drops (upto 200mbar±0.04%),absolutepressureandfluidtemperature.Range andaccuracyofeachsensorusedinCALIDEarelistedinTable1.Air flows,waterflowsandpressuredropsaremeasuredusing respec-tivelythreeairflow-meters-regulatingvalves(small,mediumand largerange),twowaterflow-meterscommandingtworegulated pumps(smallandlargerange)andtwodifferentialpressure sen-sors(smallandlargerange).Usingseveralsensorsforfluidflows andpressuredropswasfoundtobenecessaryinordertominimize measurementuncertaintiesatbothlowandhighflowrates.

Athermocoupleandanabsolutepressuresensormeasurethe temperatureinthetestsection,whichisconsideredasuniform, andtheabsolutepressureatthebottomofthedebrisbed,inorder todeterminethephysicalpropertiesofthefluid.Densityand vis-cosityofwateraredeterminedfromtabulatedvalues(Lide,1990) andtemperature.Theviscosityoftheairisdeducedfromthe tem-perature,usingtheSutherlandlaw(Lide,1990),anditsdensityis determinedfromtheperfectgaslaw:

= P−1P/2

rT , (9)

wherePistheabsolutepressure,1Pisthepressurelossinthe debrisbed,Ttheabsolutetemperatureandrthespecificgas con-stant,whichisequalto287forair.Densityiscalculatedforthe averagepressureinthedebrisbedP−1P/2toaccountforthegas compressibility(reporttoAppendixformoredetails).

2.2. Particlebeds

Anexhaustivestate-of-the-artondebrisbedgranulometrycan befoundinChikhietal.(2014).Inthispart,themainresultsofthat studyaresummarized.

Fuelpelletsnaturallycrackduringnormaloperation(Olander, 1990).Thesizeandshapeofdebriswereinvestigatedby exper-imental programs, like LOFT (Hobbins and McPherson, 1990), PHEBUS(Repetto,1990)orPBF(Pettietal.,1989)and examina-tionofthedebrisbedformedintheTMI-2damagedcore(Akers etal.,1986).FromthecrackingmodelofOguma(Oguma,1983) andliteraturereview,Coindreauetal.(2013)estimatedthatthe shapeofthefuelfragmentsshouldbesimilartoprismsandthat theirsizesshouldrangefrom2.4mmto3.2mmaslongasthefuel burn-upremainsunder40GWd/tU,andthatverythinparticlesare notproducedordonotremainintothebed(upto30mmparticles couldappearbeyondthisvalue,bythefragmentationofthehigh burn-upstructures).

ThesizeoffuelfragmentsinTMI-2corerangesfrom0.3mmto 4mm,anditsaveragevalueisoftheorderof2mm(Akersetal., 1986).ThisisconsistentwiththeresultsofLOFTandPBFprograms

(Chikhietal.,2014).Concerningcladdingparticles,itcanbe

cal-culated(Coindreauetal.,2013)thattheiraverageSauterdiameter rangesfrom1.1mmto1.7mm.

TheporosityoftheTMI-2debrisbedhasbeendeterminedby

Akersetal.(1986),andhasbeenfoundtorangefrom0.35to0.55.

Anaveragevalueof0.4isusuallyusedforsafetyanalysis,andfor moderatelyirradiatedfuel,forexamplebyBürgeretal.(2006).

Geometry of debris beds is different when created from re-solidificationofmoltenmaterialsfallinginliquidwater. Granu-lometryofthiskindofdebrisbedshasbeeninvestigatedbymany

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Table1

InstrumentationusedontheCALIDEfacility. (a)Flow-meters

Name Range Accuracy

Air flow

BRONKHORST® 0–10Nl/min 0.5%×Reading F-201-CV BRONKHORST® 0–200Nl/min F-202-AV +0.1%×Range BRONKHORST® 0–1000Nl/min F-203-AV Water flow BRONKHORST® 0–30kg/h 0.2%×Reading Cori-FlowM14 BRONKHORST® 0–600kg/h Cori-FlowM55 (b)Pressuresensors

Name Range Max.range(MR) Accuracy

Diff ROSEMOUNT®3051 −1–7mbar ±14.19mbar 0.045%×MR

ROSEMOUNT® −10–200mbar ±1246mbar 0.04%×Reading

3051 +0.023%×MR

Abs ROSEMOUNT® 0–2bar 55.2bar 0.025%×Range

3051

experimentalprogramsonsteamexplosions.Althoughthe aver-agesizeofparticleiscomparablewithin-coredebris,rangingfrom

1mm(Huhtiniemietal.,1997;HuhtiniemiandMagallon,1999)to

5mm(Spenceretal.,1994),thesizedistributionismuchwider.

Forexample,sizesvaryingfrom0.25 tomorethan10mmhave beenobservedintheFAROexperiments(MagallonandHuhtiniemi, 2001).Porosityofthiskindofbedhasbeenobservedtobehigher thanfueldebrisbeds,rangingfrom0.50toalmost0.70(Chikhietal.,

2014).ButMaandDinh(2010)pointedoutthatparticles,inthat

case,presentanimportantinternalporosity,whichleadsto over-estimatetheeffectiveporosity.Thus,thesevaluesshouldnotbe consideredasreferences.

Therefore,thesizeof thedebrisrangesfromafew hundred micronstoapproximately10mm.Porosityofdebrisbedsranges from35%to55%.

Asexplainedintheintroduction,particlebedsstudiedinCALIDE aredesignedtoberepresentativeofnucleardebrisbeds.Theyare presentedinTable2.Non-sphericalparticlebedsandmixturesof sphericalparticleshavebeenstudied.

Nonsphericalparticles(Table2)consistinthreekindsof cylin-dersandtwokindsofprisms.Theirdimensionsareoftheorderof fuelpelletsfragments.Foreachkindofparticle,geometrical charac-teristics(diameter,sidelength,height,surface,volume)havebeen determinedfromsamplesof10particles.Diametersandheights havebeen measuredwitha caliper, prismslengthand particle surfacehavebeendeterminedfrompictureanalysis,andparticle volumeshavebeendeterminedbythewaterdisplacementmethod, whichalsoallowedtomeasuretheparticledensities.Table3(a) summarizesporosityofeachbedforairandwaterexperiments.

Nominalsizesofsphericalparticlesrangefrom1.5mmto8mm. Foreachsize,actualdiametershavebeenmeasuredwithacaliper in4directionsforeachoneofa30particlesrepresentativesample. Theobtainedmeanvaluesandstandarddeviationsaresummarized

inTable2.Multi-sizedsphericalparticlebedshavebeen

consid-ered,inordertostudytheeffectofsizedistributiononpressure losses.Table3(b)summarizesthecompositionandporosityofeach mixture.

2.3. Determinationofporosity

Pressurelossisverysensitivetoporosity.Theadoptedmethod consistsindeterminingthevolumeoftheporesbymeasuringthe

massofwatermwthatisnecessarytofillitup.Thebedheightis

preciselyadjustedonareferencemarksituatedH=499.0±1.6mm abovethebottomsupportingwiremesh.KnowingthediameterD ofthetestsection,porositycanthenbedeterminedby:

ε= mw4 wD2H.

(10) PorositiesofeachbedaresummarizedinTable3(a)and(b).They rangefrom35%to40%,whichfitstheaverageporosityofdebris beds.

3. Identificationofsingle-phaseflowregimesinparticle beds

As explained in the introduction, Darcy’s lawonly holds in creepingregime,forReynoldsnumberslowerthanalimitvalue rangingfrom 1to 10 (Chauveteau and Thirriot, 1967;Mei and

Auriault,1991;Lasseuxetal., 2011).For higherReynolds

num-bers,modificationsinpore-scaleflowstructureleadtoadeviation toDarcy’slaw,orForchheimereffects.Innon-structuredparticle beds,thisdeviationisusuallyadmittedtobequadratic.However, ithasbeenshown,bytheoreticalconsiderationsandby numeri-calsimulations,thatthisisanapproximationandthatdeviation toDarcy’slawshouldbecubic(WodiéandLevy,1991;Firdaouss

etal.,1997;Lasseuxetal.,2011;YazdchiandLuding,2012).This

assertionhasneverbeenconfirmedbyexperimentalobservations, toourknowledge,indisorderedporousmedia.Inthissection,an investigationontheformofdeviationstoDarcy’slawisproposed forparticlebedsrepresentativeofnucleardebrisbeds.

3.1. Darcyregime

Darcyregimeischaracterizedbyalineardependencebetween pressurelossesandfiltrationvelocity:

∂P

∂z

+g=



KU, (11)

whereKisthepermeabilityofthemedium.Equation11is equiva-lentto:

U −∂P

∂z+g

(7)

Table2

Sphericalandnon-sphericalparticlesusedinCALIDEexperiments.Materialsareglassforsphericalparticlesandporcelainfornonsphericalparticles. Diameter(mm) Density(kg/m3) Spheres 1.5 1.574±0.031 2574.0±7.4 2 2.086±0.033 2568.0±7.4 3 2.940±0.044 2560.0±7.4 4 4.058±0.031 2560.0±7.4 8 7.877±0.116 2568.0±7.4

Diameter/side(mm) Height(mm) Surface(mm2) Volume(mm3) Density(kg/m3)

Cylinders 5×5 5.13±0.08 4.53±0.23 114.1±4.2 97.89±0.55 2572.0±7.4 5×8 4.86±0.08 7.39±0.37 150.3±6.7 140.46±0.51 3046.0±8.8 8×12 7.99±0.10 11.13±0.48 380±11 575.5±2.9 2568.0±7.4 Prisms 46×4 4.15±0.11 3.84±0.13 69.8±1.1 41.55±0.55 2568.0±7.4 ×6 6.11±0.15 5.87±0.19 156.8±4.1 144.2±1.0 2452.0±7.1 Table3

Characteristicsofparticlebeds.Differentbeds,althoughpackedwiththesameparticles,havebeenusedforairandwaterexperiments,whichexplainswhyporositycanbe slightlydifferentinairandwaterflows.

(a)Non-sphericalparticles

Cylinder Flowingfluid ε

5×5 AirWater 0.35250.3525±0.0041 ±0.0041 5×8 AirWater 0.39540.3843±0.0047 ±0.0045 8×12 AirWater 0.38550.3642±0.0053 ±0.0065

Prism Flowingfluid ε

4×4 AirWater 0.36460.3750±0.0041 ±0.0064 6×6 AirWater 0.36990.3666±0.0065 ±0.0065

(b)Mixturesofsphericalparticles

Mixturenumber 1.5mm 2mm 3mm 4mm 8mm Flowingfluid ε

(%w) (%w) (%w (%w) (%w) 1 68.81 – *– 21.05 10.14 Air 0.3592±0.0064 Water 0.3592±0.0064 2 59.48 – 28.28 12.24 – Air 0.3526±0.0039 Water 0.3646±0.0038 3 – 43.95 – 40.07 15.98 Air 0.3542±0.0049 Water 0.3578±0.0064 4 38.69 36.95 22.64 1.06 0.67 Air 0.3592±0.0064 Water 0.3587±0.0064

Thus,thelefthandsidetermofEq.(12)shouldbeconstantand equaltothepermeabilitywhenDarcyregimeoccurs.Thenitshould progressivelydecreasewheninertialeffectsappear.Fig.3showsits evolutioninthecaseof4×4mmprismsinairflow.Exceptforvery lowReynoldsnumbers,wheremeasurementuncertaintiescause animportantnoise,Darcyregimeisclearlyidentifieduptoalimit valueof2.5,approximately.

Precise determination of this limit is not obvious,since the magnitudeofForchheimereffectsaroundthispointisbelow mea-surementuncertainties. Anarbitrarycriterion is usedhere,and consistsinverificationthatthepermeability,identifiedby appli-cationofleastsquarefittingbetweenpressurelossesandfiltration velocity correspondingto diamond symbols in Fig. 3, and rep-resented by the continuous line, must be included within the uncertaintyrangeofallvaluesofexpression(12)inDarcyregime. Similarbehaviorsareobservedforallparticlebedsinthisstudy, andforairandwaterflows,andaresummarizedinFig.4(a)–(d).For reasonsofclarity,evolutionofexpression(12)isshowninDarcy regimeonly,uptoapparitionofForchheimereffects.Thus,Darcy’s lawconstitutesarigorousdescriptionofmacroscalebehaviorof

Fig.3. Evolutionofexpression(12)atlowReynoldsnumberfor4×4mmprismsin airflow.ItsvalueisconstantforRe<2.5.

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(a)

(b)

(d)

(c)

Fig.4.IdentificationofDarcyregimeforbedspackedwithnon-sphericalparticlesandmulti-sizedsphericalparticles.Foreachbed,thevalueofthepermeabilityidentified withleastsquaremethodisplottedasconstantcontinuouslines.c5×5means5mm×5mmcylinders,p4×4means4mm×4mmprisms,mix1meansmixturen◦1in

Table3(b),etc.

pressurelosses,exceptatverylowReynoldsnumberforcertain particlebeds,wherepermeabilityissignificantlysmaller.Thiskind of“pre-darcy”regime,whichisnotsupportedbyanytheoretical development,requiresfurtheranalysis.However,theauthorspoint outthatitoccursatsmallReynoldsnumbers,wherepressurelosses areveryweak(oftheorderof1Pa).Therefore,itcouldbecausedby anexperimentalerror,forexampleanunforeseensingularpressure drop,oradriftofthepressuresensor.

ThelimitReynoldsnumberforDarcy’sregime,inthese experi-ments,rangesbetween2(4×4mmprismsinairflow,Fig.4(a))and 15(8×12mmcylindersinwaterflow,Fig.4(b)),whichisconsistent withliterature(ChauveteauandThirriot,1967;MeiandAuriault,

1991;Lasseuxetal.,2011).

Thisstudyalsoallowedtodeterminethepermeabilitiesofthe testedparticlebeds,whichwillbeusedinnextparagraphsto deter-minethedeviationtoDarcy’slawathigherReynoldsnumbers,and inSection4toidentifyacorrelationforpermeabilityofthetested particlebeds.

3.2. Weakinertialregime

Asexplainedintheintroduction,firstpore-scaleinertialeffects shouldresultinacubicdeviationtoDarcy’slaw,whichcorresponds totheWeakInertialregime:

∂P

∂z

+g=

 KU+bU

3. (13)

TheWeakInertialregimehasbeentheoreticallyderivedfrom Navier–StokesequationsusingtheHomogenizationmethod(Mei

andAuriault,1991;WodiéandLevy,1991;Firdaoussetal.,1997)

andreproduced bynumericalsimulations(Lasseuxetal.,2011;

YazdchiandLuding,2012),butneverobservedexperimentallyin

disorderedparticlebeds.Indeed,experimentalobservationisvery difficult,becauseoftheverylowmagnitudeofthecubicdeviation, whichislostinmeasurementnoise.Nevertheless,andgiventhe highprecisionoftheinstrumentationoftheCALIDEfacility,the existenceofWeakInertialregimeintestedparticlebedsisworth investigating,especiallyconsideringthelackofexperimentaldata onthatquestion.

Eq.(13)isequivalenttoalineardependencebetweendeviation toDarcy’slawandReynoldsnumberpower3:

∂P

∂z

+g−  KU=bU 3 =b′Re3, (14)

or, equivalently, to a linear dependence between deviation to Darcy’slaw,normalizedbytheDarcyterm,andReynoldsnumber squared: −∂P ∂z+g−  KU  KU = bU 3  KU = bK U 2=b′′Re2. (15)

Fig.5isalog–logplotofnormalizeddeviationstoDarcy’slawfor 4×4mmprismswithReynoldsnumber.Deviationsemergefrom

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Fig.5. NormalizeddeviationstotheDarcy’slawversusReynoldsnumberinthecase of4×4mmprismsinwaterflow.

experimentalnoisebeyondRe>10,and fitwithapowerlawof exponent2.4.AchangeofbehaviorisobservedaroundRe=20,the slopeofthecorrelationdroppingfrom2.4to1.2beyondthatpoint, whichismuchclosertotheexpectedvalueincaseofquadratic deviationstoDarcy’slaw,whichis1.

Similarbehaviorhasbeenobservedforallparticlebeds,inair andwaterflow.ResultsaresummarizedinFig.6(a)–(d).First devia-tionstoDarcy’slawfitwithpowerlawsofexponentsrangingfrom

1.6to3.3andgloballycloseto2(expressionsaregiveninthelegend ofthegraphs).BeyondalimitvalueforReynoldsnumber,which dependsonthemediumbutrangesfrom5to50approximately, theslopeofthecorrelationdropstoavaluecloseto1.However, andforreasonsofclarity,dataarenotshownbeyondthatlimitin

Fig.6(a)–(d).

Thisshowsthat deviationstoDarcy’slawsignificantlydiffer fromthequadraticlawaslongasReynoldsnumberremainsbelow certainlimits,andthattheycanbedescribedbypowerlawsof exponentsgloballyoftheorderof2.Thissupportstheexistence oftheWeakInertialregimeindisorderedparticlebeds,andEq.

(13)seemsplausibletodescribethebehaviorofpressurelosses inthatregime.However,itshouldbepointedoutthat measure-mentuncertaintiesarequitelarge,becauseoftheweaknessofthe magnitudeoftheWeakInertialdeviations,andthatfurtherstudies shouldbeconductedonthatquestion.

3.3. Stronginertialregime

AquadraticdeviationtoDarcy’slawusuallyfitswellwith exper-imentaldataathighReynoldsnumbersindisorderedparticlebeds

(Forchheimer,1901;Ergun,1952;MacDonaldetal.,1979;Nemec

andLevec,2005;Ozahietal.,2008;LiandMa,2011a),which

cor-respondstotheStrongInertialregime,asopposedtotheWeak Inertialregime: −

∂P

∂z

+g=  KU+  U 2. (16)

(a)

(b)

(d)

(c)

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Fig.7.Evolutionofexpression(17)versusReynoldsnumberfor8×12mmcylinders inwaterflowbetweenWeakInertialregimeandmaximumflowvelocity.Itsvalue isconstantforRe>80.

Verificationoftheapplicabilityofthisequationtothebedtested inthis studywillbemadebyusingamethodanalogtotheone usedinDarcyregimeinSection3.1,whichconsistsinstudyingthe evolutionwithReynoldsnumberoftheexpression:

U2 −∂P ∂z+g−  KU , (17)

Fig.8. Schematicalevolutionofthebehaviorofpressurelossesinparticlebeds.

whichshouldbeconstantandequaltothepassabilityifEq.(16)

isverified.Fig.7presentstheevolutionofexpression(17)with ReynoldsnumberbetweenWeakInertialregimeandmaximum flowvelocity observedinthatstudy,for 8×12mm cylindersin waterflow.AconstantvalueisobservedforRe>80,whichmeans thatEq. (16)is verified,andsothat deviationstoDarcycanbe describedbyaquadraticlawinthatdomain.Ithasbeenshownthat thesedeviationsfitwithacubiclawinWeakInertialregime,which appliesbetweenDarcyregimeandRe<50inthatcase.Between Re=50andRe=80,neithercubicnorquadraticlawfitswith exper-imentaldata.Thiscorrespondstothe“transition”regimeinFig.7.

(a)

(b)

(d)

(c)

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Similarbehaviors,summarized in Fig.9(a)–(d), areobserved foreachparticlebed.Expression(17)isfoundtobealmost con-stantoveralargerangeofReynoldsnumber,whichmeansthata quadraticlawisrelevanttodescribedeviationstoDarcy’slawin theStrongInertialRegime.Sincenosignificantchangeofbehavior isobservedforReynoldsnumbersofseveralhundred,whereweak pore-scaleturbulence,atleast,shouldoccur,wemayconcludethat thevaliditydomainofthisapproximationcanbeextendedtoWeak Turbulentregimes.

3.4. Discussion

Atthispoint,pressurelossesgeneratedbynon-spherical parti-clebedsandmixturesofsphericalparticlebedsinsingle-phase airand waterflows havebeenmeasuredintheCALIDE experi-mental facility.Thesemeasurements allowedtoidentify3 flow regimes(Fig.8): Darcyregime,which ischaracterizedbya lin-earrelationshipbetweenpressurelossesandfiltrationvelocityor Reynoldsnumber,WeakInertialregime,characterizedbyacubic deviationtoDarcy’slaw,andStrongInertialregime,characterized byaquadraticdeviationtoDarcy’slaw.TransitionReynolds num-bersbetweentheseregimesdependonthemediumbutseemto be,fortheobservedparticlebeds,oftheorderofRe≈10for tran-sitionbetweenDarcyandWeakInertialregime,andoftheorder ofRe≈30fortransitionbetweenWeakInertialandStrongInertial regimes.Theseexperimentsdonotgiveinformationabout appari-tion of turbulence,but weak turbulence, at least,should occur forReynoldsnumbersofseveralhundred,afortioriforRe≈1000. However,nosignificant differencewasobserved in the macro-scalebehaviorofpressurelossesbetweenStrongInertialandWeak Turbulent regimes. Therefore, applicability of quadratic law to describedeviationstoDarcy’slawcanbeextendedtoWeak Tur-bulentregime,atleastforthetypeofmediumconsideredinthis analysis.

ItshouldalsobepointedoutthatdeviationstoDarcy’slawin WeakInertialregimesaresmallcomparedtotheDarcyterm(30% atmaximum,inFig.6(b)),andsometimesevensmallerthan mea-surementuncertainties.Thus,describingWeakInertialtermbya quadraticlaw,insteadofacubiclaw,doesnotleadtoanimportant erroronthetotalpressurelossprediction.Forthatreason,andfor simplicityofuseinpracticalapplicationsorcodes,quadratic devi-ationtoDarcy’slawwillbeconsideredasagoodapproximation, alsointheWeakInertialregime.

Therefore, Eq.(16)is relevant in WeakInertial, Strong Iner-tialandWeakTurbulentregimes.Furthermore,sincethequadratic termbecomesnegligiblebeforetheDarcytermforsmallvelocities, itsvaliditycanobviouslybeextendedtotheDarcyregime.

Establishmentofapredictivecorrelationnowrequires expres-sions for permeability and passability, which range over more than one order of magnitude.This is theobjective of the next section.

4. Single-phasepressurelossescorrelationinparticlebeds

Inprevioussections,ithasbeenestablishedthatpressurelosses inthestudiedparticlebedscanbeaccuratelydescribedby: −

∂P

∂z

+g=  KU+  U 2, (18)

whereKandarecharacteristicsofthemediumcalledpermeability andpassability,respectively.Permeabilitiesandpassabilitieshave beenexperimentallydeterminedfornon-sphericalparticlebeds andmulti-sizedsphericalparticlebeds,andforairandwaterflows. ResultsaresummarizedinTable5,fornon-sphericalparticlebeds,

andinTable6,formulti-sizedsphericalparticlebeds.

AssaidintheIntroduction,expressions forpermeabilityand passability havebeenproposedbyErgun (1952),inthecase of monodispersebeds,i.e.,single-sizedsphericalparticlebeds: K= ε 3d2 hK(1−ε)2 (19) = ε 3d h(1−ε), (20) whereεistheporosityofthebed,dthediameteroftheparticlesand hKandhtwoempiricalparameterscalledtheErgunconstants.The

objectiveofthissectionistoinvestigatetheapplicabilityofthese expressionstothenon-monodispersebedsstudiedinthiswork,by assessingthepossibilitytodefineequivalentdiameters,definedas thedimensionthatallowsacorrectpredictionofpermeabilityor passabilitybyEqs.(19)and(20).Althoughphysicallymeaningless ifoneconsidertheupscalingofNavier–Stokesequations,thiskind ofapproachhasbeenfoundrelevant,fromanempiricalpointof view,topredictpermeabilityofdebris-bed-likemedia(Chikhietal., 2014).Itisproposedheretoextendthisconcepttonon-Darcyflows. Thus,permeabilitiesandpassabilitiesoftestedparticlebedswillbe expressedas: K= ε 3d2 K hK(1−ε)2 (21) = ε 3d  h(1−ε), (22)

wheredKanddaretheequivalentdiametersforpermeabilityand

passability,respectively.ItshouldbenotedthatdKanddmaybe

aprioridifferent. 4.1. Ergunconstants

Thereislittledispersiononvaluesof Ergunconstantsin lit-erature.Ergun(1952)recommendedhK=150andh=1.75,while

MacDonaldetal.(1979)preferredhK=180andh=1.8.Ozahietal.

(2008) observed hK=160and h=1.61for highReynolds flows

(800→8000).Thus,recommendedvaluesforhKandhmayvary

bymorethan20%.Furthermore,MacDonaldetal.(1979)pointed outthatsurfacerugosityofparticlesisofgreatinfluenceonh,

whichcouldrangefrom1.8forsmoothparticlestomorethan4for veryroughparticles.

Inordertoeliminatethissourceofuncertainty,specific exper-imentsonmonodisperse bedshave beenconductedinorder to determinethevaluesofhKandhforthesphericalparticlesused

inthatstudy(Table2).Thus,bestfittingvaluesforhKandhhave

beenidentifiedformonodispersebedspackedwith1.5mm,2mm, 3mm,4mmand8mmglassbeads,inairandwaterflows.Results aresummarizedinTable4.Ergunconstantsseemtodifferwhen identifiedwithairorwaterdata,whichcouldindicatethatthe pore-scaleflowstructureisdifferentinairandwaterflows.However, theidentifiedvaluesremainwithintherangeofliterature recom-mendations.Besides,theErgunlawisanempiricalcorrelation,and significantdispersionshavebeenobservedforhKandh.

Macdon-aldetal.(MacDonaldetal.,1979),forexample,observedthatthe

dispersiononhKcouldbegreaterthan50%.Sinceourdatapresent

aless-than-10%dispersion,wecanbeconfidentthatusingtheir meanvaluesisrelevantinthatstudy.

Mean value for hK is 181±17, which is in agreement with

MacDonaldetal.(1979),whilemeanvalueforh,of1.63±0.15,

isclosertorecommendationofOzahietal.(2008).Thesevalues willbeusedinnextparagraphstoidentifyequivalentdiameters fornon-monodisperseparticlebeds.

The authors point out that these values are specific to the particlesusedinthisstudy,i.e.,smoothparticles.Althoughthey

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Table4

Ergunconstantsbestfittingvaluesforeachparticlesize.Standarddeviations,whichcorrespondtotheuncertaintiesonthemeanvalues,arelessthan10%.

d(mm) hK h

Air Water Air Water

1.5 167.4±5.5 196.4±10.8 1.79±0.01 1.41±0.07 2 169.1±5.8 183.1±9.7 1.61±0.06 1.74±0.06 3 168.8±4.6 175.4±9.7 1.78±0.11 1.60±0.06 4 177.0±6.1 178.6±8.3 1.78±0.09 1.54±0.04 8 189.2±27.9 206.3±17.8 1.57±0.07 1.46±0.09 Mean 181±17 1.63±0.15

presentagoodagreementwithrecommendedvaluesinliterature, theyshouldbeusedcarefullywithotherclassesofparticles(very roughparticles,forexample),especiallyconsideringthedispersion reportedinMacDonaldetal.(1979).

4.2. Non-sphericalparticlesbeds

Fromvariousexistingdefinitionsinliterature(Ozahietal.,2008;

LiandMa,2011b)forequivalentdiameters,4havebeenidentified

aspotentiallyrelevantforthisstudy: •Volumeequivalentdiameter

dV=



6

v

part





1/3

; (23)

•Surfaceequivalentdiameter dS=



spart





1/2

; (24)

•Specificsurfaceequivalentdiameter,orSauterdiameter dSt= 6Vpart Spart =6

v

part spart . (25) •ProductofSauterdiameterandanadimensionalsphericity coef-ficient,correspondingtotheratiobetweenthesurfaceofvolume equivalentsphereandtheactualsurfaceoftheparticle. dSt = 6

v

part spart 1/3(6

v

part)2/3 spart (26) 4.2.1. Permeability

Identificationof equivalent diameterfor permeability, dK, is

doneby studyingthe differencebetween experimentaldata in DarcyregimeandpredictionofDarcy’slaw,wherepermeability iscalculatedwithEq.(19)usingdV,dS,dStordSt asthe

character-isticdimensiond.Thisdifferenceisnormalizedbytheexperimental pressureloss:



∂P

∂z

+g



(d) −



∂P

∂z

+g



exp



∂P

∂z

+g



exp = U K(d)− U Kexp U Kexp = 1 K(d)− 1 Kexp 1 Kexp . (27)

Expression(27)hasbeenevaluatedforeachoneofthe4 poten-tial equivalent diameters, for each one of the 5 non-spherical particlebedsconsidered,in airandwater flows,defining 4sets of10values(seeleftcolumnsTable5).Meanvalueandstandard deviationofthesesetshavebeendetermined.Inorderto repre-senttheseresultsinamorevisualway,Fig.10(a)showsGaussian

distributionfunctionsofcorrespondingmeanvaluesandstandard deviation.

ItcanbeseeninFig.10(a)thattheSauterdiameterdStistheone

whichallowsthebestprediction,onaverage,ofpressurelossesin Darcyregime,whiledSt leadstounderestimatethem,by24.5%,

anddVanddStooverestimatethem,by34.2%and55.3%,

respec-tively.Furthermore,predictionsofpressurelossesusingdSt and

dStpresentthelowestdispersions,whichmeansthatdStallowsa

correctpredictionnotonlyonaveragebutalsoforeachtestedbed individually.Indeed,itcanbeseeninTable5thatthisprediction alwayspresentsaless-than-16%differencewithexperimentaldata, whichisverysmallforsuchanexperimentalcorrelation.Therefore, theSauterdiameterdStconstitutesagoodequivalentdiameterfor

permeabilityofnon-sphericalparticlebeds. 4.2.2. Passability

Similarprocedureisusedtodeterminetheequivalent diame-terforpassabilityd.Itisdonebystudyingthedifferencebetween

experimentaldeviationstoDarcy’slawandpredictionusingEq.

(20),wherecharacteristicdimensionisdV,dS,dStordSt ,

normal-izedbyexperimentaldeviationstoDarcy’slaw:



∂P

∂z

+g− U K



(d) −



∂P

∂z

+g− U K



exp



∂P

∂z

+g− U K



exp = U2 (d)− U2 exp U2 exp = 1 (d)− 1 exp 1 exp . (28)

Meanvalues and standard deviations of expression(28),for eachpotentialequivalentdiameter(dV,dS,dStanddSt ),havebeen

determined(seerightcolumnsofTable5),andarerepresentedin

Fig.10(b)byGaussiandistributionfunctions.ContrarytoFig.10(a), bestpredictionofdeviationtoDarcy’slawisobtainedwiththe productofSauterdiameterandsphericitycoefficientdSt ,which

leadstoa9.0%overestimationofdeviationtoDarcy’slaw,on aver-age,whiledSt,dVanddSleadtooverestimateitby26.2%,46.1%and

57.3%,respectively.Therefore,theproductofSauterdiameterand sphericityconstitutesagoodchoicefortheequivalentdiameterfor passabilityofnon-sphericalparticlebeds.

However,itshouldbepointedoutherethatpredictionof devia-tionstoDarcy’slawusingthisdiameterisnotcorrectforalltested beds(a23.2%differenceisobservedfor5×8mmcylindersinwater flow),whichsuggeststhatthis correlationwouldturnincorrect iftheDarcytermwassmallcomparedtothequadraticone,and, therefore,thatitpresentsalimitedvaliditydomain.Nevertheless, andlookingatexperimentaldataobtainedinthepresentstudy, theuseofthiscorrelationcandefinitelyberecommendedwithin

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Table5

Experimentalpermeabilitiesandpassabilitiesofnon-sphericalparticlebeds.

Particle Kexp 1 K(d)−Kexp1 1 Kexp (%) exp 1 (d)−exp1 1 exp (%) (×10−9m2) d V dS dSt dSt (×10−5m) dV dS dSt dSt c5×5 AirWater 15.0016.550±0.074 13.8 26.5 −7.8 −25.3 16.90±0.99 40.5 48.0 26.4 13.8 ±0.22 24.2 38.0 0.6 −18.5 17.74±0.15 32.5 39.6 19.3 7.3 c5×8 AirWater 17.2427.21±0.11 41.2 62.5 6.7 −19.4 30.30±0.67 32.5 42.1 15.2 0.1 ±0.19 48.6 70.9 12.3 −15.1 17.41±0.14 63.0 74.9 41.7 23.2 c8×12 AirWater 47.1566.00±0.49 35.1 53.6 4.5 −19.2 46.81±0.68 26.2 34.6 11.0 −2.3 ±0.34 48.9 69.3 15.2 −10.9 37.75±0.35 27.5 36.0 12.2 −1.4 p4×4 AirWater 9.6310.054±0.040 21.7 46.5 −16.0 −42.1 12.44±0.77 61.9 77.6 34.5 11.7 ±0.18 42.8 71.9 −1.5 −32.1 13.11±0.12 69.8 86.3 41.1 17.2 p6×6 Air 24.78±0.16 20.2 41.7 −13.5 −37.8 20.9±1.0 53.7 66.9 30.3 10.5 Water 19.70±0.27 45.7 71.8 4.8 −24.6 20.29±0.18 53.2 66.4 30.0 10.2 Mean 34.2 55.3 0.5 −24.5 46.1 57.3 26.2 9.0 Standarddeviation 13.1 16.4 10.4 10.0 16.2 19.3 11.3 8.3

(a)

(b)

Fig.10. Identificationofequivalentdiameterfornon-sphericalparticlebeds.

theinvestigated rangeof Reynolds numbers,i.e., fromRe=0to Re=1500: −

∂P

∂z

+g= hK(1−ε)2 ε3dSt U+ h(1−ε) ε3dSt U 2, (29) where: dSt= 6Vpart Spart (30) = 1/3(6V part)2/3 Spart (31) hK=181 (32) h=1.63. (33)

4.3. Multi-sizedsphericalsphericalparticles

Theobjectiveofthisparagraphistodeterminemean diame-tersforthestudiedmulti-sizedsphericalparticlebeds.Frommany definitionsof meandiametersexisting intheliterature,4seem particularlyinteresting:

•Numbermeandiameter dhni=

P

idini

P

ini ; (34)

•Lengthmeandiameter dhli=

P

id2ini

P

idini ; (35)

•Surfacemeandiameter dhsi=

P

id3ini

P

id2ini ; (36)

•Volumemeandiameter dhvi=

P

id4ini

P

id 3 ini . (37) 4.3.1. Permeability

Identification of equivalent diameters for permeability, dK,

and passability, d,is doneby a similarmethod tothecase of

non-sphericalparticlebeds.Expressions(27)and(28)havebeen evaluatedforeachoneofthe4potentialequivalentdiameters,for eachoneofthe4non-sphericalparticlebedsconsidered,inairand waterflows(seeTable6).Fig.11(a)summarizesresultsfor perme-ability,andshowsthattheSurfacemeandiameter,dhsi,allowsthe

bestpredictionofpressurelossesinDarcyregime,sinceitleads toslightlyoverestimateitbylessthan4.6%onaverage,whilethe Volumemeandiameterdhvileadstounderestimateitby26.8%,and

LengthmeandiameterdhliandNumbermeandiameterdhnileadsto

overestimateitrespectivelyby32.2%and51.0%.Furthermore, pre-dictionsofpressurelossesinDarcyregimecorrespondingtodhsi

(14)

Table6

Experimentalpermeabilitiesandpassabilitiesofmulti-sizedsphericalparticlebeds.

Mixture Kexp 1 K(d)−Kexp1 1 Kexp (%) exp 1 (d)−exp1 1 exp (%) (×10−9m2) d

hni dhli dhsi dhvi (×10−5m) dhni dhli dhsi dhvi

1 Air 2.499±0.083 50.3 34.5 0.2 −46.9 8.87±0.15 22.2 15.6 −0.3 −27.3 Water 2.402±0.028 44.4 29.2 −3.8 −48.9 7.54±0.33 3.9 −1.7 −15.2 −38.2 2 Air 2.512±0.012 50.2 32.7 8.9 −16.6 7.532±0.075 6.4 0.0 −9.4 −20.7 Water 2.735±0.038 63.5 44.5 18.5 −9.2 8.23±0.32 16.3 9.3 −1.0 −13.4 3 air 5.837±0.029 80.5 48.8 7.8 −31.6 13.48±0.13 36.0 23.5 5.1 −16.2 Water 5.50±0.11 69.9 40.1 1.5 −35.6 13.55±0.35 36.7 24.1 5.7 −15.8 4 Air 2.723±0.013 34.7 23.4 9.9 −5.8 7.724±0.088 −3.2 −7.4 −12.6 −19.1 Water 2.315±0.027 14.5 4.9 −6.5 −19.9 9.10±0.37 14.0 9.1 3.0 −4.7 Mean 51.0 32.2 4.6 −26.8 16.5 9.1 −3.1 −19.4 Standarddeviation 20.8 13.8 8.2 16.4 14.5 11.6 8.2 10.0

(a)

(b)

Fig.11.Identificationofequivalentdiameterformulti-sized-sphericalparticlebeds.

allowsacorrectpredictionnotonlyonaveragebutalsoforeach testedbedindividually.Thus,itcanbeseeninTable6thatthis predictionalwayspresentsaless-than-10%differencewith exper-imentaldata,exceptformixture2inwaterflow,forwhichthis differenceis18.5%,whichremainsreasonable.Therefore,the equiv-alentdiameterforpermeability ofmulti-sizedsphericalparticle bedsistheSurfacemeandiameterdhsi.

4.3.2. Passability

Similarobservationscanbemadeforpassabilityresults.Mean valuesandstandarddeviationsofexpression(28),foreach poten-tialequivalentdiameter,havebeendetermined(seerightcolumns

ofTable6),andarerepresentedinFig.11(b)byGaussianlaws.It

canbeseenthatSurfacemeandiameterallowsthebestprediction ofdeviationtoDarcy’slaw.Indeed,itleadstoaslight underesti-mation,by3.1%,whiledhvileadstounderestimateitby19.4%,and

dhlianddhnileadtoanoverestimationby9.1%and16.5%,

respec-tively.Furthermore,predictionsofdeviationstoDarcy’slawusing thisdiameterpresentthesmallestdispersion.Therefore,the equiv-alentdiameterforpassabilityofmulti-sizedsphericalparticlebeds istheSurfacemeandiameter.

Thus,ithasbeenshownatthispointthatpressurelosses gen-eratedbynon-spherical particlebedsand multi-sizedspherical particlebedscanbeaccuratelypredictedbyErgun’slawusingan “equivalentdiameter”approachforpermeabilityandpassability. Ithasbeenobservedthatequivalentdiametersfor permeability andpassabilityofnon-sphericalparticlebedsaretheSauter diam-eteroftheparticlesandtheproductoftheSauterdiameterandthe sphericitycoefficient,respectively.Thesamestudy,onmulti-sized

sphericalparticlebeds,showedthatequivalentdiametersfor per-meabilityandpassabilityofthiskindofparticlebedarebothequal tothesurfacemeandiameter.Itisoffundamentalinteresthereto remarkthattheseobservationsareconsistent:indeed,thesurface meandiameterofamulti-sizedsphericalparticlebedisequaltoits Sauterdiameter: dhsi=

P

id 3 ini

P

id2ini =6

P

i d3 i 6 ni

P

id2ini =6

P

i

v

ini

P

isini =6Vpart Spart =dSt, (38)

andthesphericitycoefficientis1forsphericalparticles.

Therefore, applicabilityofEqs. (29)–(33)can beextendedto alltestedparticlebeds,i.e.,bothnon-sphericalparticlebedsand multi-sizedsphericalparticlebeds,andinDarcytoWeakTurbulent regime.

5. Conclusions

Motivated by uncertainty reduction in nuclear debris beds coolability, experiments have been conducted on the CALIDE facilityinordertoinvestigatesinglephasepressurelossesin rep-resentativedebrisbeds.

Inthispaper,experimentalresultsobtainedontheCALIDE facil-ityhavebeenpresentedandanalyzedinordertoidentifyasimple single-phaseflowpressurelosscorrelationfordebris-bed-like par-ticlebedscoveringvariousrefloodingflowconditions.

Onthebasisofcurrentliterature,Reynoldsnumberin single-phasepartsofanucleardebrisbedwasestimatedtorangefrom Re=15tomorethanRe=1000,anddebrisbedgranulometrywas

(15)

estimatedbetweenafewhundredsmicronsto10mm,withconvex highsphericityshapesandporosityoftheorderof40%.

ExperimentaldataobtainedontheCALIDEfacilityhavebeen interpreted to determinethe macro-scale behavior of pressure lossesinparticlebeds.ItwasobservedthataDarcy–Forchheimer law,involvingthesumofalineartermandaquadraticterm,with respecttofluid velocity,wasrelevant todescribethis behavior inDarcy,InertialandWeakTurbulentregimes.It hasalsobeen observedthat,inarestricteddomainbetweenDarcyandInertial regimes,deviationtoDarcy’slawisbetterdescribedbya cubic term.Thiscorrespondstotheso-calledWeakInertialregime,which canbetheoreticallyderivedfrompore-scaleNavier-Stokes equa-tions,andreproducedinnumericalsimulations,buthasneverbeen observedexperimentally,toourknowledge,ondisorderedporous media.Nevertheless,itwaspointedoutthatWeakInertial devi-ationisverysmallcomparedtoDarcyterm,and,therefore,that Darcy–ForchheimerlawremainsrelevantevenintheWeakInertial regime.

Darcy–Forchheimerlaw allowingaqualitative descriptionof pressurelossesonly,itwasnecessarytodetermineexpressionsfor coefficientsoflinearandquadratictermsinordertoobtaina pre-dictivecorrelation.ApplicabilityoftheErgun’slaw,whichisvalid formonodisperseparticlebedsonly,wasinvestigatedbyassessing thepossibilitytodefineequivalentdiametersforthestudiedbeds. Thisapproachhasbeenfoundtoberelevantforboth permeabil-ityandpassabilityprediction.Itwasobservedthatpermeabilitiesof alltestedbeds,i.e.,non-sphericalandmulti-sizedsphericalparticle beds,couldbepreciselypredicted,bylessthan10%inmostcases, byErgunexpressionusingtheSauterdiameter,whiletheproduct oftheSauterdiameterandthesphericitycoefficient allowedfor anaccuratepredictionofpassabilities,byabout15%.

Therefore,thefollowingcorrelationcanberecommendedfor calculationofsingle-phasepressurelossesinnucleardebrisbeds duringreflooding,forReynoldsnumbersrangingfrom0to1500:

∂P

∂z

+g= 181(1−ε)2 ε3d2 St U+1.63(1−ε) ε3d St U2, (39)

AppendixA. Effectofgascompressibility

Inairflows,thedensityisafunctionofpressure.Theperfectgas hypothesisstates:

= P

rT, (40)

wherePistheabsolutepressure,Ttheabsolutetemperatureand rthespecificgasconstant,whichisequalto287forair.Thus,the pressurelossesalongthetestsectionarecausingdropsofthegas densityandvelocity.Whencompressibilityeffectsaretakeninto account,theErgunequationcanbere-writtenas:

−dP dz =  K ˙m (P)D2/4+ (P) 



˙m (P)D2/4



2 (41) ⇔−(P)dP=

"

 K ˙m D2/4+ 1 



˙m D2/4



2

#

dz (42) ⇔−P rTdP=

"

 K ˙m D2/4+ 1 



˙m D2/4



2

#

dz, (43)

where ˙misthemassflowrate,whichisconstantalongthetest section.Byintroducingtheperfectgashypothesis,weget:

IntegrationofEq.(43)overthecolumnheightleadsto:

Z

P2 P1 P rTdP=

Z

z2 z1

"

 K ˙m D2/4+ 1 



˙m D2/4



2

#

dz (44) ⇒− 1 2rT P 2 2−P12



|

{z

}

(P2−P1)(P1+P2) =H

"

 K ˙m D2/4+ 1 



˙m D2/4



2

#

(45) ⇔−1P P1+P2 2rT

|

{z

}

 P1+P22



=¯ =H

"

 K ˙m D2/4+ 1 



˙m D2/4



2

#

(46) ⇔−1P H =  K ˙m ¯D2/4

|

{z

}

U( ¯) + ¯ 



˙m ¯D2/4



2

|

{z

}

U( ¯)2 (47)

Therefore,compressibilityeffectsaretakenintoaccountby cal-culatingthegasdensityattheaveragepressureP1+P2

2 .Theauthors

wouldliketopointoutthatthiscorrectionhasaveryweakimpact inthiswork,sincethepressurelossmagnitudeisofafewmillibars atmaximum,whichisvery smallcompared totheatmospheric pressure.However,compressibilityeffectswouldplayasignificant roleifpressurelosseswerenotsmallcompared totheabsolute pressureofthesystem(highercolumns,greatervelocities...) References

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Figure

Fig. 1. Schematic view of flow structure in debris beds during reflooding.
Fig. 2. Schematic representation of the CALIDE facility.
Fig. 3. Evolution of expression (12) at low Reynolds number for 4 × 4 mm prisms in air flow
Fig. 4. Identification of Darcy regime for beds packed with non-spherical particles and multi-sized spherical particles
+5

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