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,daInstitutdeRadioprotectionetdeSûretéNucléaire(IRSN)–PSN-RES/SEREX/LE2M,Cadarachebât.327,13115StPaul-lez-Durance,France bInstitutdeRadioprotectionetdeSûretéNucléaire(IRSN)–PSN-RES/SAG/LEPC,Cadarachebât.700,13115StPaul-lez-Durance,France cUniversitédeToulouse–INPT–UPS–InstitutdeMécaniquedesFluidesdeToulouse(IMFT),AlléeCamilleSoula,F-31400Toulouse,France dCNRS–IMFT,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).
Nomenclature Latinletters
D beddiameter 93.96±0.04mm (mm) d sphericalparticlediameter (m) dhni numbermeandiameter
P
idiniP
ini (m) dhli lengthmeandiameter
P
id 2 iniP
inidi (m) dhsi surfacemeandiameterP
id 3 iniP
inid 2 i (m) dhvi volumemeandiameterP
id 4 iniP
inid 3 i (m) dSt Sauterdiameter 6VSpartpart (m)dS surfaceequivalentdiameter
Spart
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-sizedspher-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
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.
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
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
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.
(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
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)
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)
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
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=
6v
part1/3
; (23)
•Surfaceequivalentdiameter dS=
spart1/2
; (24)
•Specificsurfaceequivalentdiameter,orSauterdiameter dSt= 6Vpart Spart =6
v
part spart . (25) •ProductofSauterdiameterandanadimensionalsphericity coef-ficient,correspondingtotheratiobetweenthesurfaceofvolume equivalentsphereandtheactualsurfaceoftheparticle. dSt = 6v
part spart 1/3(6v
part)2/3 spart (26) 4.2.1. PermeabilityIdentificationof 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
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
idiniP
ini ; (34)•Lengthmeandiameter dhli=
P
id2iniP
idini ; (35)•Surfacemeandiameter dhsi=
P
id3iniP
id2ini ; (36)•Volumemeandiameter dhvi=
P
id4iniP
id 3 ini . (37) 4.3.1. PermeabilityIdentification 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
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 iniP
id2ini =6P
i d3 i 6 niP
id2ini =6P
iv
iniP
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
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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