Analytical modeling of steady-state filtration process in an automatic self-cleaning filter

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To link to this article : DOI:10.1016/j.cherd.2015.04.030

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http://dx.doi.org/10.1016/j.cherd.2015.04.030

To cite this version : Meireles, Martine and Prat, Marc and Estachy,

Guillaume Analytical modeling of steady-state filtration process in an

automatic self-cleaning filter. (2015) Chemical Engineering Research

and Design, vol. 100. pp. 15-26. ISSN 0263-8762

Any correspondence concerning this service should be sent to the repository

Analytical

modeling

of

steady-state

filtration

process

in

an

automatic

self-cleaning

filter

M.

Meireles

a,b,∗

_,

_M.

_Prat

c,d

_,

_G.

_Estachy

e

a_{UniversitédeToulouse,INPT,UPS,LaboratoiredeGénieChimique,118RoutedeNarbonne,F-31062Toulouse,}

France

b_{CNRS,UMR5503,F-31062Toulouse,France}

c_{UniversitédeToulouse,INPT,UPS,InstitutdeMécaniquedesFluidesdeToulouse,AvenueCamilleSoula,}

F-31400Toulouse,France

d_{CNRS,UMR5502,F-31400Toulouse,France}

e_{AlfaLavalMoattiSAS,10RueduMaréchaldeLattredeTassigny,78997ElancourtCedex,France}

Keywords: Filtration Hydrodynamic Self-cleaning Multiscalemodeling

a

b

s

t

r

a

c

t

Anautomaticself-cleaningfilterisasemicontinuousmachineoperatedfortheremoval ofparticlesfromafluid.Technically,adistributordrivenbyahydraulicmotorrotatesat regularintervalstofeedtheslurrytotheinletchambersofNsegmentedelements hor-izontallystackedoverthedistributorandback-flushesthelastchamber.Inthisworka practicalcomputationally-affordablemodelhasbeendevelopedtodescribethedistribution offlowrateinthedifferentsectorsofadisc-typeelementandtodemonstratetheeffectof parameters,suchasbackflushingtimeorpollutionconcentration,accountingforparticle cloggingandperiodicoperationconditionsforperfectandimperfectefficiencyofthe clean-ingbyback-flushing.Thekeyresultsarethepredictionofthetimeofclogging,aswellasa numericaltooltooptimizecriticalback-flushingtimeandpollutionconcentration. ©2015TheInstitutionofChemicalEngineers.PublishedbyElsevierB.V.Allrightsreserved.

1.

Introduction

Fully automatic self-cleaning filters have been used for decades in applications wherever suspended solids need tobe removed from a pressurized water stream. They are mainlyusedinclosedloop industrialsystems,ondeep-sea oilplatformsandonlargeshipsforhydraulicsystems.Marine enginesoperateinmostsevereenvironments,runningnearly 24h a day and under extreme loads. It is very important that the engines and hydraulic systems operate efficiently andrequire littleto nodowntime.For automotiveengines, oilchangeisaroutinelytask,butformarineengines, oper-atorshavetofacethepotentialforanoilspill,thedifficulty todisposeoftheusedmotoroilandthecostassociatedwith

∗ _{Correspondingauthorat:UniversitédeToulouse,INPT,UPS,LaboratoiredeGénieChimique,118,routedeNarbonne,31062Toulouse,} France.Tel.:+33561558162.

E-mailaddress:meireles@chimie.ups-tlse.fr(M.Meireles).

thesetypesofchanges.Forthispurpose,compactautomatic self-cleaning filtershave been developedby several world-wide companiestobeoperatedonlargemarineenginesto removeparticlesfromlubeoilindieselengineofships.An interestinganddistinguishingfeatureofanautomaticfilter isthat the backflushingoccurs withoutinterruption ofthe flowdownstreamofthefilter.For thisendaspecificdesign is neededwhich allowsthe periodical rotation offiltration chambersinsidethe filterhousing. Suchadesignallowsto simultaneously feedsomechamberswhileothersare back-flushedtoremovetheaccumulatedparticlesfromthefilter. Thetechnicalschemeofacompactautomaticself-cleaning fil-terispresentedinFig.1.Itiscomposedofhorizontallystacked disc-typeelementsmountedoveradistributor.Eachdisc-type

Listofsymbols:

As screenareaAs

C concentrationofsolidparticles(gm−3)

dh screenholesize(m)

dp particleequivalentdiameter(m)

dmoy averageparticlesizecalculatedfromthe num-berniofparticlesofsizedipermloffluid(m) e(t) cakeheightattimet(m)

h sectorheight(m)

L sectorlength(m)

M+1 numberofsectionalchambersbydisc-type ele-ments

np total number ofparticles per unit volumeof apparentdiameterlargerthanthescreenhole sizedh

N numberofdisc-typeelements

no initialnumberofmeshholes n(t) numberofholesattimet

Qs volumetricliquidflowratethroughthesector (m3_s−1₎

Qt volumetricflowrateenteringthefilter(m3s−1) Ri internalradiiofthefiltercolumn(m)

Re externalradiiofthefiltercolumn(m) R filterspecificresistancetoviscousflow(m−1)

Rca cakeresistance(m−1)

S0 averagecross-sectionareaintheinletchamber

(m2)

tb backflushingtime(s) tc cleaningtime(s) u axialvelocity(ms−1)

v transversevelocity(ms−1)

u* _{adimensionalaxialvelocity}

v* _{adimensionaltransversevelocity}

Vw filtrationvelocityinthesector(ms−1) U0 averagevelocityintheinletchannel(ms−1)

x horizontalcoordinateofthesector(m)

y verticalcoordinateofthesector(m)

x* _{adimensionalhorizontalcoordinateofthe}

sec-tor

y* _{adimensionalverticalcoordinateofthesector}

˛b freeopeningfractionrightafterbackflushing  dynamicviscosity(Pas)

 density(kgm−3)

elementismadefromtheassemblyoftwoframes(Fig.2a).A disc-typeelementisdividedinto(M+1)sectionalsectorsby ribs(Fig.2b).ThenumberofsectorsM+1isalwaysevenandof theorderof8to12.AsoutlinedinFig.3,inasectormadefrom theassemblyoftwoframes,thereareaninletchamberand twooutletchambersthatwedenoteanupperchamberanda lowerchamber.Theinletandthetwooutletchambersare sep-aratedbythefiltermedia.Thischamberdesignthusprovides adouble-sidedfiltrationarea.Thefiltrationmediaisusuallya weave-wirescreen:asimpletwo-dimensionalsquare-weave withrectangularopenings.Byconventionreferringtothe tex-tileindustryporesoropeningsofscreenmediaaredescribed bythetermmesh.Thesizeofthe“mesh”alsocalledthe nom-inalfiltrationdegreeisanactualdimensionoftheshortest distanceacrosstheopening.Itmoreorless correspondsto thesmallestparticlesizewhichcanberemovedfromthefluid stream.

The detailed operating principle of an automatic self-cleaning filter is described below. The system is fed with the slurry from the filterhousing tothe inletchambersof

N segmented disc-type elements horizontally stacked over thedistributor.Foreachelement,theMsectorsarefedfrom thegapbetweenthedistributorandtheinternalsideofthe disc-typeelement(seeFig.2a).Theslurryentersthesector through the inlet chamberentrance. Uponfiltration, parti-clesbuilduponthesurfaceofthetwofiltermedialocated attheupper/lowersideoftheinletchamber,and theclean fluidleavesthesectorthroughtheupper/loweroutlet cham-bers.Particleaccumulationcausesanincreasedpressuredrop through the sectors and therefore through each disc-type element.Theliquidflowratethroughthesectorsofa disc-typeelementismaintainedataconstantvaluebymeansof adisplacement pump.Thedistributor whichisdrivenbya hydraulicmotoronthetopofthefilterhousingcontinuously rotatestosimultaneouslyfeedMsectorsandbackflushwith someamountoffiltratethe(M+1)sector,flushingthe accu-mulatedparticlesfromthesurfaceofthemedia(seeFig.2a). Inthisway,allsectorsarebackflushedonceperfullrotationof thedistributorandsolidswhicharecollectedduringthe filtra-tionperiodcanberemovedbybackflushingafterafullrotation iscompleted.Backflushingiscommoninfiltrationsystems, e.g.(Andoetal.,2012;Bacchinetal.,2011;Benmachouetal., 2003;Berman,1953;DuignanandNash,2012)tonameonlya few,toclean-upthefiltersurfaceperiodically.Theinteresting and distinguishingfeatureofanautomaticfilteristhatthe backflushingoccurswithoutinterruptionofthefiltration pro-cess.Backflushingtime(whichcorrespondstoagivenrotating speed)isanoperatingparameteroftheorderof10to60s.It canbespecifiedand optimizedinordertolimit accumula-tioninthefiltrationchamberswhichdependsamongothers onfiltrationmediacharacteristicsandpropertiesoftheslurry suchassizedistributionandshapeoftheparticles.Evenfor agivenapplicationlikelubeoilindieselengineofships,fluid propertiesmayvaryfromoneenginetoanotherandevenwith operatingtimeforagivenengine.Thisisthereasonwhy,when optimizingthedesign(numberandgeometricaldimensionsof elements)ortheoperatingconditions,suchasthe backflush-ingtime,engineeringcompaniesstronglyrelyonstandardized qualificationtestsusuallycarriedoutwithstandardizedfine Arizonasandwithawell-characterizedsizedistribution.Such standardizedqualificationtestsalsoallowcomparing auto-maticfiltersfromdifferentcompanies.Aqualificationtestofa filterprovideswiththevariationofpressuredropasafunction ofthetimeoffiltrationforaknownconcentrationofparticles introducedinthefilteratagivenvolumetricrate.Thisallows qualifyingafilteroraprocedureregardingthetimeneeded toreachamaximumpressuredropwhichisnormallysetto around2×105_{Pa,alsocalledtimeforclogging.Thesetestsare}

extensivelyusedtocomparedifferentdesignsandoptimize operatingparametersbuttheyarequitecostly.Thereforefor mostengineeringcompanies,ageneralobjectiveistodevelop aCPU-friendlyandreliablenumericaltoolinordertoreduce thetimeofdesign,thenumberofexperimentaltrials,aswell astooptimizeoperatingconditions.

Inthisworkapracticalcomputationallyaffordablemodel has been developed to describe the distribution of flow rate in the differentsectors ofadisc-type element and to demonstratetheeffectofparameterssettings,suchas back-flushingtime orpollution concentration.Theaimisrather todemonstratetheeffectofparametersettingsthan repre-sent realcases. Aswillbeseenfurtherinthe paper,some

Fig.1–Technicalschemeofanautomaticfilterwithdisk-typeelementsstackedoveradistributortoformthefilteringunit.

Fig.2–Left:schemeofthedisk-typechambercomposedoftheassemblyoftwoframes.EachframeisdividedinM+1

sectionalchambersbyribs.Right:schemeoftheoperatingprinciple:Adistributordrivenbyahydraulicmotorrotatesat

regularintervalstofeedtheslurryinMchambersandbackflushthelastchamber.

Fig.3–Left:designofsectorgeometry,right:schematicrepresentationofhalfandinletandhalfandoutletchambers.Lis

thelengthofthesectorandhthehalfwidth,xandyarethecoordinates.Notethatonlyhalfofinletchamber(outlet

chamberrespectively)isshown.Henceeachinletchamber(outletchamberrespectively)inthesystemisborderedbytwo

filtrationscreens.Theplanelimitedbythedashedlineistheplanecorrespondingapproximatelytothe2Dmodel.

geometricaldataaretakenfromthedesignofarealAlfaMoatti filterforthesolepurposeofmodeldevelopment(seeTable1

forinstance).Theseinputscaneasilybereplacedwith alterna-tivedatawithoutaffectingthegeneralityofthemethodology developedhere.Alsoacomparisonbetweenthe calculation oftimeforcloggingandexperimentaltestofanautomatic self-cleaningfilterwillbeprovided.Theproblemrequires a multi-scaleapproachfromthefiltermediamicro-scaleupto themacro-scale oftheautomaticself-cleaning filter,which

impliestheconsiderationoftwointermediatescales (cham-bersofasectoranddisc-typeelement).Itshouldalsoaccount fortheperiodicityoffiltrationandbackflushingintheM+1 sectorsofa disc-typeelement duetothe continuous rota-tionofthefilter,notingthatafteronefillrotation,filtration occursatasteadystate.Theobjectiveisnotadetailed mod-elingbut asimplified CPUfriendlyapproachuseful togain insightsintotheinfluenceofmainparameters.Forthisend, weshalluseasimplifiedmodeloftheflow(intheabsenceof

Table1–GeometricaldataforAlfa-LavalMoattiFilter150andphysicalcharacteristicsofAlla-LavalA03screenmedia.

Classsize(␮m) >6 >8 >10 >14 >17 >4 >20 >22

Numberofparticlesfor20ml 10,4131 46,452 124,845 8424 4058 28,736 2058 1391

Classsize(␮m) >28 >30 >35 >40 >45 >25 >50 >70

particles)derivedfromcrosssectionalaveragingoftheNavier Stokesequationsinasectorusingsimilaritysolutions(Section

2).CloggingscenariiarestudiedinSection3basedonstandard modelsandexperimentalevolutionofpressuredrop.In Sec-tion4,acompletemodelforstationaryregimeisdeveloped andusedforstudyingtheinfluenceofbackflushingtimeand pollutionconcentrationontheflowdistributioninthe differ-entsectorsandtocalculatethepressuredropincreaseofa filter.

2.

Flow

characteristics

in

the

filtration

chambers

Webegintheanalysisbyassumingthatthemaincontribution tothe pressuredropisduetotheflowwithinthefiltration chambers.Allother contributionssuchaspressuredrop in thedistributororinthefilterhousingareneglected. Consis-tentlywiththisassumption,wefurtherassumethattheflow througheachdisc-typeelementisthesamewhateverthe posi-tionofthe elementinthe stack.Theseassumptionsimply that,

Pf(t)≈Pei(t) 1≤i≤N (1)

whereNisthenumberofdisc-typeelement,Pf(t)andPei(t) arerespectively,thepressuredropsacrossthefiltercomposed ofNelementsandelementi,respectively.

Inthis section, westudy the characteristics of theflow withintheinletandupper(orlower)outletchambersofoneof theM+1sectorsofadisc-element.Essentially,weshowthat thefiltrationvelocityintheinlet/outletchamberscanbe con-sideredasuniformalongthechamberlongitudinalaxisforthe typicalconditionsandgeometricaldimensionsencountered inanautomaticfilter.Theanalysisisbasedona1Dflowmodel, whichhasbeenusedinpreviousfiltrationstudiesforlowto moderatefiltrationvelocities.Themainfeaturesofthemodel arerecalledherebutmoredetailscanbefoundinreferences (GanandAllen,1999;Koltuniewiczetal.,2004).

Duetosymmetry, weconsiderhalf ofasector(in the y plane),comprisinghalfoftheinletchamberand theupper chamber of height h (see Fig. 3). Note that h in Fig. 3

whichcorresponds to thehalf-heightof theinlet chamber isalsotheheightofthe upper/lower outletchambers.The screen thickness of the order of 0.5 micron is considered asnegligible comparedwith h. Typical geometrical dimen-sionsforflowchambersare7mmforh,and65mmforthe lengthL.

Therearethreemainfeaturesoftheinletandoutlet cham-bers:(i)theratiooftheheighttochamberlengthh/Lissmaller than1,(ii)thecross-sectionalareaincreasesinthex-direction,

(iii)constrictioneffectsareexpectedattheentrance(forthe inletchamber)andattheexit(foroutletchambers)sincethe fluidentrance(exitrespectively)isthroughafractionofthe sectionoftheinletchamber(oftheoutletchamber respec-tively). However,due tothe relativesmall aspect ratio h/L,

theazimuthaldeformationofstreamlinescanbeassumed tooccuronlyoverasmallregionofthechambers.Asaresult, ourapproachisbasedontheconsiderationofthesimplified 2Dgeometry consideringthe y-axis symmetrysketched in

Fig.3wherey=0atthecenteroftheinletchannelandy=h

atthesurfaceofthefiltermedia.Theflowisdescribedusing theNavier–Stokesequationsexpressedforthis2Dgeometry

with the mass balance and momentum balance equations expressedindimensionlessformas:

∂u∗ ∂x∗ + ∂v∗ ∂y∗ =0 (2)



u∗∂u∗ ∂x∗+v∗ ∂u∗ ∂y∗



=−∂p_∂x∗_∗+_Re1 0



∂2u∗ ∂2_x∗+ ∂2u∗ ∂2_y∗



(3)



u∗∂v∗ ∂x∗+v ∗∂v∗ ∂y∗



=−∂p∗ ∂y∗+ 1 Re0



∂2_v∗ ∂2_x∗+ ∂2_v∗ ∂2_y∗



(4) wherex∗=x h,y∗= y h,u∗= Uu0,v ∗₌ v U0,p ∗₌ p  U2 0 U0isthe

aver-agevelocityintheinletchannelandRe0=U_0h;

andarethefluiddensityanddynamicviscosity, respec-tively.

The above equations haveto besolved in conjunctions withtheboundaryconditionsatthesurfaceofthefiltermedia (y=h),

u∗(x∗,1)=0; v∗(x∗,1)=Vw

U0

(5) Intheaboveequation,Vwistheaveragevelocitythrough thefiltermedia.Duetomassconservationforasteadyflow,

U0isalsotheaveragevelocityattheexitofoutletchanneland

isestimatedas: U0=Qs

S0 (6)

Intheaboveequation,Qsisthevolumetricflowrate enter-ing a sectorandS0 is theaverage cross-sectionarea given

by

S0=

2



Ri+Re−R2 i



M+1 h (7)

RiandRearetheinternalandexternalradiiofthedisc-type element,respectively.

Under the following assumptions: (i) laminar flow, (ii) quasi-uniform filtration velocity along the channel v(x, y)=v(y),and(iii)Vwy <<U0, ananalyticalsolution tothis

systemcanbeproposedprovidedthatRew= V_wh isofthe

orderof1,(Lietal.,1998).

Accordingtothissolutionthelocalvelocityfield(u*_,v*_)is

givenby: u∗(x∗,y∗) U0 = 3 2 u∗_ y∗ U0



1−y∗2

 

1−Rew 420



2−7y∗2−7y∗4



+o



Re2_w



(8) v∗(y∗) U0 = 1 2 Rew Re0y ∗_(3−y∗2_)+o



_Re2 w



(9) u∗_

y∗isthedimensionlessaverageaxialvelocitydefinedby

u∗



y∗= 1 h h



0 u∗(x∗,y∗)dy∗ (10)

BysubstitutingEqs.(8)and(9)intotheconservative equa-tionsandapplyingtheaveragingoperatordefinedbyEq.(10),

oneobtains,(seeGanandAllen,1999forthedetails),the fol-lowinglocalmassbalanceandmomentumbalanceequations,

du∗y∗ dx∗ + Rew Re0 = 0 (11) 81 70 du∗2 y∗ dx∗ + 3 Re0

u∗



y∗− 1 Re0 du∗2 y∗ dx∗2 + d

p∗



y∗ dx∗ =0 (12) Eqs.(11)and(12)areusedtodescribetheflowinboththe inletand upperoutletchambers.To couplethesystems of Eqs.(11)and(12) relativetoeachchamber, weexpressthe localvelocityVwthroughthefiltermediaintermsofthelocal pressuredifferenceaccordingtoalinearrelation:

Vw=p1−p2

R (13)

wheresubscripts1and2refertoinletandupperoutlet chan-nelseparatedbythefiltermedia;Risthehydraulicresistance ofanytypeoffiltrationmedia.

AccordingtoLinetal.(2009),thepressure dropacrossa meshtypefiltrationmediacanbeexpressedas

10 Resc1.14 = 2P (Vw/ (1−Am))2 1 Am with Resc= (Vw/(1−Am)dh  (14)

whereP=p1−p2anddh isthesizeofthemeshandAm is theopensurfaceofthe screenmediaorthesumofallthe areasoftheopeningsthroughwhichthefluidcanpass.On thegrounds,thatReynoldsnumber,Resc,issmallcompared

withone,onecansimplifyEq.(14)as: 10 Resc ≈ 2P (Vw/ (1−Am))2 1 Am (15)

Now,combiningEqs.(13)and(15),givesusasimplerelation forthehydraulicresistanceofascreenfiltrationmedia: R= 5Am

(1−Am)dh

(16) TogetherwithEq.(13),Eqs.(11)and(12)governingtheflow intheinletandoutletchannelsaretherefore,Inletchannel (subscript1referstoinletchannel)

d

u∗₁



y dx∗ +Re0



p∗₁−p∗₂



hR =0 (17) 81 70 d

u∗₁



2 y∗ dx∗ + 3 Re0

u∗1



y− 1 Re0 d

u∗₁



2 y∗ dx∗2 + d

p∗₁



y dx∗ =0 (18) Outletchannel(subscript2referstooutletchannel) d

u∗₂



y dx∗ +Re0



p∗₁−p∗₂



hR =0 (19) 81 70 d

u∗₂



2∗ y dx∗ + 3 Re0

u∗2



y− 1 Re0 d

u∗₂



2 y∗ dx∗2 + d

p∗₂



y∗ dx∗ =0 (20)

Thesystem ofEqs.(17)–(20)issolvedcombining afinite differencemethodofdiscretizationwithaNewton–Raphson method (to deal with the non-linearities and the coupled

Fig.4–Filtrationvelocityalongthefilterscreen:fromtopto

bottom:filtrationvelocitycalculatedbytheanalyticalmodel

solution,filtrationvelocityfrom3DCFDsimulationinthe

planelimitedbythedashedlineinFig.3.

natureofthesystem).DetailscanbefoundinOxarangoetal. (2004). Asensitivitystudy ofthenumber ofgridpointshas beencarriedout,consideringupto2500equallyspacedgrid points.Thisstudyindicatedthatagridof200upto300points was agood trade-off between accuracyand computational time.Typically,thetimeforcomputingtheflowusingthis2D modelisoftheorderofonetotwosecondsonastandardLinux PC.Thistimeistobecomparedwiththetimeof5minneeded foradirectcomputationusingtheCFDcommercialsoftware onthesameplatform,withoutmentioningthetimeneeded forpreparingtheCFDcomputation(gridconstruction,etc.).

Toinvestigatetheflowcharacteristicsinthechambersof asectorwehaveusedgeometricalandoperatingdataofan existingautomaticself-cleaningfilterdesignedbyAlfa-Moatti. ThegeometricaldataofAlfa-Moatti150filterarereportedin

Table1.Fortheflowsimulationtheinputsare:anaverageinlet velocityU0 equalto0.2488ms−1,an averagefiltration Vw equalto0.0268ms−1,anhydraulic resistanceofthe screen mediaRequalto47.6×105_m−1_{asgivenfora25␮mabsolute}

meshsizeA03filtrationmedia(AlfaLavalA03filter).

Forthesesetofparameters,theReynoldsnumberRe0inthe

filtrationchambersisequalto58.5forlubeoil(=850kgm−3, =2.55×10−2Pasat25◦C)whichcorresponds toalaminar flowwhereastheReynoldsnumberRewoftheflowthroughthe screenisequalto6.Thisdoesnotentirelymeetthe require-mentRewoftheorderof1butissufficientlysmalltoobtain reasonablygoodresults.

Inordertovalidatethe2D-simplifiedmodel,comparisons withthreedimensionalCFDcomputationshavebeencarrying outusingcommercialsoftware(Fluent,ANSYS).Onthewhole theagreementbetweenourmodelandtheCFDdirect simula-tionisverygood.Arepresentativeexampleoftheresults,Fig.4

showsratioofthelocalvelocityVwtotheaverageVwalong thedimensionlesssectorlength.Our2Dmodelpredictsa uni-formvelocity,atrendalsopredictedby3DCFDsimulations. Themostnoticeabledifferenceoccurattheentranceofthe sectorwheretheflowisnotyetfullyestablishedandbottom ofthesectorwhereisthefiltrationvelocityisonly25%ofthe averagevelocity,whereasinthemainpartofthesector,the localvelocitypredictedbythe 3Dsimulations issomewhat lower by20%,which isleftunexplained atthis stage.Note thatthefiltrationvelocityalongthesectorlengthx,isakey elementintheperspectiveofcloggingsimulations.Fromthis

comparisonwevalidatetheassumptionofauniformfiltration velocitythroughthemediaalongthesectorlength.

Thepressuredropthroughthefiltermediacanalsobe esti-matedwithour2Dmodelaswellasthepressuredropinthe inletandoutletchambers.For thespecifiedconditions,the pressuredrop throughthe filtermediaisnumerically eval-uatedat3372Pa (0.03372bar).For comparison,thepressure dropintheinletand outletchannelsare respectively48Pa and72Pa.Akeyresultisthatthepressuredropthroughthe filtrationchambersisdominatedbythepressuredropthrough thefiltrationmediaandnotbythehydrodynamicthroughthe chambers.

Alsoasdiscussedinpreviousworks,(e.g.GanandAllen, 1999; Koltuniewicz et al., 2004; Oxarango et al., 2004), it isimportanttorealizethat aimingatoptimizingoperating conditions or filter design, this simplified model is much easy to use for a parametric study than 3D CFD simula-tions that are less affordable in terms of computational time.

3.

Clogging

the

filtration

media

The development of a model for the clogging of filtration mediabeginswiththeconsideration oftwoclogging mech-anisms: surface blocking of media openings by particles, building ofacakeon the topofthe media. Because filtra-tionmediahaveameshgeometrycharacterizedbyopenings of constant section in the media thickness direction, we assumethatdepthblockingisnegligible,adifferentcasefrom non-wovenfibrous media.Depthblockingmay occurwhen particleswithsizesmallerthanopeningsizeare caughton theinnerwallsoftheopeningsduetospecificadsorptionor byflowconstrictions.

Experiments including visualizations of particle capture are desirable for a better understanding ofcapture mech-anisms.Suchapproaches are currentlyunder development formodelgridsandmonodisperseparticles,(e.g. Psochand Schiewer,2006;Rebaietal.,2010).Analternativeconsistsin carryingoutfiltrationteststomeasurethedynamicsof pres-sure increase during a filtration and to compare with the prediction ofcloggingmodels representative for the afore-mentionedmechanisms.Wethuscarriedoutfiltrationstests withastandardizeddispersion,referredaswithArizonatest finesand(READEadvancedmaterialsUSA),commonlyused fortestsinhydraulicfluidforinhydraulic,automotive,and aerospaceapplications.

3.1. Surfaceblockingmode

In this model, the fluid issupposed to containsuspended sphericalsolidparticleswithasizedistributionintherange [dpl−dpu].ForArizonatestfinesand,thesizedistributionas

wellasthenumberofparticlesperunitvolumeisestimated for16classesbetween4␮mand70␮masreportedinTable2. Thescreenmediahascharacteristicopeningsizeourmesh sizedh.Weshallassumeamechanisticcriterionforparticle blocking:onlytheparticles withequivalentdiameterlarger thantheopeningsizedh,arecaught.Moreover,weconsider eachparticlecoversandcompletelyclogstheopening. Aggre-gationofparticlesovertimeisnotconsideredinthismodel. Theinitialnumberofopeningsnoiscalculatedfrom geometri-calscreenareaforonesectorAs,openingsizedhandhydraulic porosity␧ofthescreenbyEq.(23),notingthatAmtheopen

1 10 100 1000 10000 6000 5000 4000 3000 2000 1000 0 Pressure drop (Pa) Time ( s)

> 25 microns Experimental data > 28 microns

Fig.5–Variationsofpressuredropversusfiltrationtime:

curvesfromtoptobottom:numericalresultsassuming

cloggingbyparticlesofminimalsize28microns,datafrom

experimentaltests,numericalresultsassumingcloggingby

particlesofminimalsize25microns.

surfaceofthescreenmediaisequaltotheproductofAsand dh:

n0= Asε

dh2

(21)

If np isthe totalnumberofparticles perunitvolumeof fluidofdiameterlargerthanthemeshsizedh,thatcontribute toblockthescreen,thenumbern(t)ofoutstandingopenings attimetisgivenby:

n(t)=no−Qsnpt (22)

whereQsisthevolumetricliquidflowrateinthesector. Theincreaseofthepressuredropduetotheprogressive cloggingofopeningsbyparticlesisthencalculatedfromthe followingequationwhichreferstochangeinthenumberof openingswithtime:

P=Rn0Qs

Asn (t)

(23)

InsertingEq.(22)intoEq.(23)yieldstheevolutionof pres-suredropforasectorasafunctionoftimeforthegridclogging model, P = Rn0Qs As



n0−Q_As_snpt



(24)

Wehavecomputedtheevolutionofpressuredropversus timeusingEq.(24)assumingcloggingofthemeshbyparticles of(i)sizeequalorlargertothemeshsize(largerthan25␮m) (ii)largerthanthemeshsize(largerthan28␮m).

The volumetric liquid flow rate Qs is equal to 1.09×10−5m3_s−1 _{and the fluid viscosity is 0.015Pas for}

atemperatureof40◦C.Fig.6showstheevaluationofpressure drop P acrossasectorasafunctionoftime during stan-dardizedfiltrationexperimentsusingthe Arizonafinesand dispersiontforavolumetricflowrateto1.09×10−5m3_s−1_.

As can be seen from Fig. 5, the pressure drop evolu-tion collapsesreasonablywellontoasinglecurve withthe values predicted by the grid clogging model for a criteria based on size equal or larger to the mesh size. The results show an evolution in two main periods, namely a period ofvery moderate increasewherethe particles are progres-sively trappedatthe surfaceofthe openingsfollowedbya

Table2–Sizedistributionofsolidparticlesenteringthefilterexpressedintermsofnumberofparticlesper20mloffluid

for16classesbetween4␮mand70␮m.Totalmassconcentrationofparticlesis3mgl−1.

Internal disk-type diameter(m) Externaldisk typediameter (m) Numberof sectorsby element Numberof elementsin thefilter Chamberhalf height(m) Chamber length(m) Hydraulic resistanceof screen(m−1) Absolute meshsize (␮m) 0.078 0.150 8 16 0.007 0.065 47.6×105 ₂₅

periodof sharp increase in pressure drop. When the time ofoperationexceedsacritical timeof4500s, experimental datashowthatthepressuredropdivergestowardvaluesof about400Pa.Beforethatcriticaltime,constantlowpressure dropofabout80to90Pahasbeenrecordedinthetest experi-ment.Wecannotethatthemodelfitsquitewelltheclogging timeexperiencedinthe testwhentakinginto accountthe totalnumberparticleswithnominalsizelargerthan25␮m entering the filter. Wehave plottedin Fig. 6, the variation ofpressuredropasafunctionofthefractionofopenholes usingEq.(22)tocomputethefractionofopen holesn(t)/no versustime.Wenotethatthesharpincreaseinpressuredrop iscorrelated toa reduced fraction ofopen holes. Wemay assessthat whenthefractionofthe remainingopen holes islower than a critical fractionof10% then pressure drop diverges.

3.2. Cakefiltrationmode

Inthismodel,weassumethatthefiltermediaisfouledby alayerofparticlesbeforealltheholesarephysicallyclogged. Thepressuredropwillthenbepreferentiallyrelatedtotherate ofcakegrowthdescribedbyawell-establishedcake-filtration model(Tingetal.,2006).

Thepressuredropiscomputedas,

P=Qs(R∗+Rca) (25)

whereisthefluidviscosity,Qsistheflowrateinthesector, R*_{istheresistanceofthemeshsheetandR}

catheresistance

ofthecake,whichisexpressedas,

Rca=e(t)

kca

(26)

wheree(t)isthecakethicknessattimetoffiltration.

0 500 1000 1500 2000 2500 5% 17% 29% 41% 53% 65% 76% 88% 100% Pressure dr op (P a)

Fracon of open holes (%)

10% = critical fraction

Fig.6–Pressuredropthroughthescreen(numerical

results)asafunctionofthefractionoffreeopeningsatthe

screensurface.

kcaisthecakepermeability,whichisexpressedbythe

clas-sicalKozeny–Carmanequation,

kca=

ε3 cad2pmoy

180 (1−εca)2

(27)

whereεcaisthecakeporositytakenasequalto0.36;dpmoyis the averageparticlesizecalculatedfrom thenumbernpi of

particlesofsizedpiperunitvolumeoffluid.

dpmoy=

i npid3pi

i npid2pi (28)

Theevolutionofcakethicknesse(t)asafunctiontimecan bededucedfromthevolumeofparticlescapturedattimet

andisgivenby:

e(t)= Qs

i npid3_pit 6(1−εca)As (29)

InsertingEqs.(26)and(29)intoEq.(25)givestheexpression forthepressuredropforcakefiltrationmodel

P=Qs

⎡

⎢

⎢

⎣

R∗+

⎛

⎜

⎜

⎝

180Qs(1−εca)

i npid3pi 6Asε3cad2pmoy t

⎞

⎟

⎟

⎠

⎤

⎥

⎥

⎦

(30)

Fig.7presentsthevariationsofpressuredropversustime obtainedwiththecakemodel.Eq.(30)predictsalinear varia-tionofpressuredropwithtime,averydifferentfeaturefrom experimental data curve. Moreover the time necessary for clogging isvery small: afew seconds compared to experi-mentaltime4000s.Thepressuredroppredictedbythecake

0 2000 4000 6000 8000 5500 5000 4500 4000 3500 3000 2500

Pressure Drop (Pa)

Time (s)

Fig.7–Variationsofpressuredropthoughversusfiltration

timeassumingcakegrowthatthesurfaceofthescreen:

curvewithblacksquaresrepresentsnumericalresultsfrom

cakefiltrationmodel,curvewithemptytriangles

modelafterafiltrationtimeof4500sis42timeslargerthan theexperimentalvalue.

3.3. Choiceofacloggingmodel

Basedonthecomparisonofthepredictionsofthetwo mod-elswithexperimentaldata,theconclusionisclearlythatthe mostrelevantmodelisthegridcloggingmodel.Naturally,this modelisbasedonsimplificationslikeparticlesformfactorand doesnottakeintoaccountthepossibleoccultationofseveral poresbyasingleparticlenottheoccurrenceofparticle aggre-gation.Notethatwearemainlyinterestedinthepredictionof pressuredrop,akeyelementinthemulti-scalemodelingwe aimtodevelop.Inthatrespect,webelievethatthesimplified gridcloggingmodelissufficientforthisissue.

4.

Modeling

of

stationary

regime

for

an

automatic

self-cleaning

filter

This section reports on the model we have developed to computethepressuredropatthescaleoftheautomatic self-cleaningfilter.

Oneimportantparameterwhenoperatingsuchan auto-maticfilteristhebackflushingtimeTheback-flushingtimeis thetimebetweentwoconsecutiveshiftsofthefluid distribu-tor.Atanytime,therearealwaysMsectorsinfiltrationmode and1sectorinback-flushingmode.Weshallassumethatthe timerequiredforashiftofthefluiddistributorisverysmall comparedwiththeback-flushingtime.

Inthismodeltheimpactofcloggingonthepressuredropis describedbythegridcloggingmodelwhereastheefficiencyof theperiodicalbackflushingisdescribedthroughaparameter denoted˛bwhichrepresentsthefractionoffreeopeningsjust afterbackflushingperiod.

4.1. Modeldevelopment

Theprimaryobjectiveofthemodelistocomputetheevolution ofpressure dropP(t)asafunctionofthe totalvolumetric rateenteringthefilterQf,thenumberofstackedelementsN, thenumberofsectorsM+1,theparticlesizedistribution(or pollutionprofile)enteringthefilter,backflushingtimetb,mesh sizedhandscreenareaforonesectorAs.Asmentionedbefore, thebackflushingtimeisthetimebetweentwoconsecutive shiftsofthefluiddistributor.Whenthefilterbeginstooperate orwhenthereisachangeintheoperationconditions,there existsatransientperiodbeforereachingastationnaryregime, wherethepressure dropacrossany particularsectorvaries periodically.Inthisworkwefocusonthestationnaryregime. Onlydimensionalvariablesareusedinthissection.

WerecallthatweassumeM+1sectorswithMthenumber ofsectorsinfiltrationmode.Notethatowingtothedesignof disktypechamber,Mhastobeanoddnumber.Thepressure dropisassumedtobethesamethrougheachelement.Thus P(t)≈Pf(t)≈Pei(t) 1≤i≤N (31)

wherePf(t)andPei(t)arethepressuredropacrossthewhole filterandelementirespectively.Supposewestartacycleat timet=0,justafterashiftofdistributorandwewantto com-putetheevolutionofP(t)fromt=0tot=tb,i.e.justbeforea newshiftofdistributor(werecallthattbisthebackflushing time).Weassumethatsector#1hasjustbeenbackflushedat

t=0.ThussectorMisthedirtiestatt=0.

Let˛j0bethefractionoffreeopeningsinthescreenofsector jatt=0(1≤j≤M).

Webegintheanalysisbyexpressingthatthesumofflow ratesthrougheachsectorinfiltrationmodeshouldbeequal toQfthetotalvolumetricflowrateenteringthefilter.

Qf = M

j=1

Qsj (32)

whereQsjistheflowrateinsectorj.

Letno bethenumberofopeningsofthemesh,and sup-posethattherearenowonlyntopeningsleft,theothersbeing blockedbyparticles.Theaveragefiltrationvelocityseenbythe freeopeningisthengivensimplyby

Vw(t)=Vw0/˛ (33)

whereVw0istheaveragefiltrationvelocitywhenthemeshis

cleanand˛=nt/no(˛isthefractionoffreeopeningsattimet). Asaresult,thepressuredropthroughthesectorbecomes P=˛−1RVw0=˛−1R_2AQs

s (34)

Thefactor2inEq.(34)comesfromthefactthattherearean upperandalowerfiltermedia,eachofareaAs,ineachsector.

Fromthis,Eq.(32)gives

Qf = P(t)As R M

j=1 ˛j= P(t)As R∗ ˝(t) (35) ˝(t)= M

j=1 ˛j (36)

Weexpresstheevolutionof˛jforeachscreenofeachsector as d˛j dt =− np n0 Qsj 2 =− npAs n0R˛j P with ˛j=˛j0 at t=0(37)

where we recall that np is the number ofparticles ofsize greater thanthe screenopeningsizedh perunit volumeof fluid.SumminguptheMEq.(37)andusingEq.(36)leadto

d˝(t) dt =−

np

2n0Qf

(38) whichcanbereadilyintegratedtogive

˝(t)=˝0− npQf 2n0 t (39) with˝0= M

j=1 ˛j0.

ThenwededucefromEq.(34)anexpressionforthe varia-tionofpressuredropwithtimeofoperation

P(t)= RQ_2Af s 1



˝0− npQf 2n0 t



(40)

Inthefollowingsection,anexpressionforthecalculation of˝0isestablished

FirstwecombineEqs.(37)and(40)toyield,Eq.(41)

1 ˛j d˛j dt =− npQf 2n0 1



˝0− npQf 2n0 t



(41)

whichcanbeintegratedtogive:

˛j=˛j0



1−_2˝npQf 0n0t



(42)

Finally,weexpressthattheinitialfreeopeningfraction˛j0 ofsectorjisequaltothefreeopeningfractionineachscreen ofsectorj−1attimetb(stationnaryregime),

˛j0=˛j−1(tb) j=2,M (43)

withtheadditionalconditionthat

˛10=˛b (44)

whereasmentionedbefore,˛b isthe freeopeningfraction rightafterbackflushing.

CombiningEqs(42)and(43):

˛j0=˛b



1− npQf 2˝0n0tb



j−1 j=1,M (45)

fromwhichweget

˝0=˛b M

j=1



1− npQf 2˝0n0tb



j−1 (46)

Solvingfor˝0yields

˝0= npQftb 2n0 1



1−



1−npQftb 2˛bn0



1/M



(47)

whichcanbeusedtogetherwithEq.(40)tocalculatethe pres-suredropwheretheuniqueparameteris˛bthefreeopening fractionrightafterbackflushing.

Fortheparameter˛b,wecandistinguishtwosituations: - perfectcleaning,whichcorrespondsto˛b=1.Inthiscaseat

theendofbackflushingperiod,alltheparticleshavebeen removedandthenumberofaperturesleftopenisequalto theinitialone.

- imperfect(partial)cleaning.Inthiscase,weshallassume thatthebackflushingtimetbislessthanthetimeneeded toremovealltheparticlesfromthemeshopenings.Inthis casewedevelopedanapproach(seeAnnexI)toestimate ˛b(tb)thefractionofopeningattheendofthebackflushing period.

4.2. Capabilitiesofthemodel

4.2.1. Influenceofbackflushingtimeandpollution concentrationparameters

Toillustrateitscapabilities,themodelhasbeenusedtostudy theinfluenceofbackflushingtimeon Qsj, theflowrate for

Fig.8–Specificloaddistributionrate(m3_s−1_)bysector

versusbackflushingperiod(s).Thenumberof

particlescm−3greaterthan25␮mindiameteris

60particlescm−3.Thereferenceflowrateis

Q_sref=1.1×10−5m3s−1.

each sectorofanelement.For thesecalculations, weused thesamepollutionprofiledataasusedbefore,aswellasthe samegeometricalandoperatingdata.Accordingtotheabove considerations,thereducedflowrateineachsectorisgiven by, Qsj Qf = ˛b ˝0



1− npQf 2˝0n0tb



j−1 (48)

Wecalculatedthevaluesofthereducedflowratesinthe

M+1=8sectorsofanelementwiththeassumptionofaperfect cleaning.

ResultsarereportedinFig.8forincreasingbackflushing time.Thetopcurveshowstheflowrateinsector#1,nextcurve belowtheflowrateinsector#2andsoon.Asanexample,we seethatfortboftheorderorlargerthan10s,because sec-tor#7clogsduringthebackflushingperiod,theflowrateis reducedcomparedwithaverageentranceflowrate,whereas theflowrateonsector#1(justaftertheshiftofdistributor) islarger.Forabackflushingtimeof20swecanestimatethat theflowrateinsector#1is10%morethaninsector#7.Of coursethedistributionoftheflowoverthedifferentsectors oftheelementisadirectconsequenceofthedistributionof thepressuredropinthedifferentsectors:insector#1, clog-ginghasbeenremovedsothepressuredropislow,whereas insector#7,pressuredropprogressivelyincreasesduringthe fullrotationofthedistributor.

Wealsocalculated the valuesofthe reducedflow rates in the M+1sectorsas afunction ofthe pollution concen-tration.Theconcentrationofsolidparticlesperunitvolume is expected togreatly influencethe pressure drop ineach sector.ResultsarereportedinFig.9forincreasingpollution concentration. The influence of particle concentration is pretty much the same as for increasing the backflushing time. Again the distribution of the flow over the different sectorsisrelatedtothepressuredropineachsectoradirect consequenceofthenumberofparticlesperunitofvolumetric

Fig.9–Specificloaddistribution(flowrate)foreachsector

asafunctionoftheparticleconcentration(numberof

particlescm−3greaterthan25␮mindiameter;thepollution

profileissimilarforeachconcentration).Thebackflushing

periodis15s.Thereferenceflowrateis

Q_sref=1.1×10−5m3_s−1_.

flowrate(concentration)caughtatthesurfaceofthescreen media. More simulation would be needed to determine critical backflushing time or critical particle concentration correspondingtoaseverecloggingofsector#7,sector#6,and sector#5,thatwouldcause adrasticreductionofflowrate intheseparticularsectors.Thesecriticalparametersmustbe avoidedbecausetheycorrespondtosituationsforwhichonly a fractionof the filteractually deliver a downstreamflow. Sincethepurposeofthisworkisnotarealisticoptimization

4000 3000 2000 5000 6000 t (s) 0 2 4 6 8 10 12 Δ P / Experimental data d_p >28 μm d_p >25 μm 10 2 (Pa)

Fig.10–Pressuredropinthefilterasafunctionof

operationtime:experimentaldata(blacksolidcircles),

numericalresultsassumingparticlesgreaterthanorequal

to25␮marecaught(emptysquares),numericalresults

assumingparticlesgreaterthanorequalto28␮mare

caught(emptytriangles).

ofaparticularfilterbutanassessmentofmodelcapabilities, wedidnotinvestigatelargerangeofparametersettings. 4.2.2. Pressuredropthroughthecompletefilter

Onecapabilityofthemodelistocalculatethepressuredrop atthescaleofone-disctypeelement.Thenusingina sim-ple analytical up scaling procedure andassuming that the flowthrougheachdisc-typeelementisthesamewhateverthe positionoftheelementinthestack,onecanestimatethe evo-lutionofpressuredropatthescaleofthefiltercomposedof Ndisc-typeelements.Fig.10comparesthecalculationsusing geometricaldatafromTable1aswellasdatefortheparticle sizedistributionandoperatingconditionsusedinthe previ-oussectionexceptforback-flushingtime.Thenumericaland experimentaldataarecomparedforaback-flushingtimeof3s correspondingtoanimperfectcleaning(˛bisalmostequalto zero).Thissettingwaschoseninordertobeabletomeasurean increasedpressuredropinanhourrange.Realisticconditions withbackflushingtimeoftheorderof20swouldgivean oper-atingrangeoftheorderofseveralweeks.Twocaseshavebeen consideredasfarasparticlefiltrationdegreeisconcerned.The solidlinewithemptysquarescorrespondstoresultsobtained assumingthanparticlesofsizegreaterthanorequalto25␮m arecaughtbythescreen.Thesolidlinewithemptytriangles correspondstotheresultsobtainedassumingthatthe parti-clesofsizegreaterthanorequalto28␮marecaughtbythe screen.

Ascanbeseen,thedifferenceintimeofclogginghighlights theimportanceofthemodelingofparticlecapture(specifically forparticles closetothecritical size).Thegoodqualitative agreementwiththeexperimentaldatahoweverindicatesthat themainfeaturesofcloggingatelementscalearecorrectly takenintoaccountinthemodelatthefilterscale.

5.

Conclusions

Inthispaperwepresentedananalyticalmodelforstationnary (periodic)operatingconditionsoffiltrationandbackflushing inanautomaticfilter.Thismodelaccountsformulti-scaleand multi-physicsconsiderations.Weshowedthat thefiltration velocityinasectionofdiscelement,hereindefinedasa sec-tor,canbeconsideredasuniformforthetypicalconditions encounteredinliquidfiltrationandthatthe1Dmodeling cor-rectlypredictsthefiltrationvelocity.Wealsoconcludedthat themostrelevantmodelisascreencloggingmodelforwhich the increaseinpressuredrop isrelated tothe reductionof meshfreeopenings.Wealsodevelopedamodeltoestimate thetimeneededforasectortobecorrectlybackflushedand definedaskeyparameterthefreeopeningfractionofasector afteragivenbackflushingtime.

Themodelwasthenusedinasimpleanalyticalupscaling procedureallowingtoestimatetheevolutionofpressuredrop atthescaleofafilterelementanddeducethetimeof clog-ging.Cloggingscenarioasafunctionofparticleconcentration andbackflushingtimewerealsoinvestigated.Akeyresultis thatthelongerthebackflushingtime,thelargerthedifference betweentheflowratesineachsectorinthefilter.

The modelisbased on several simplifyingassumptions allowing onetodevelopquasi-analyticalsolutions. For this reasonitgreatlyfacilitatesparametricstudiesasillustratedin thiswork.Inadditiontopermithighlightingthemainfeatures of the automaticfilter under normaloperation conditions, themodelcanalsoserveasabasisforthedevelopmentof moresophisticatedmodels.Henceitopensuptheroutefor

modelsaimingatoptimizingtheoperationoftheautomatic filterand/oroptimizingitsdesign.

Acknowledgements

The authors are grateful to P. Schmitz forinteresting dis-cussionsand to A. Ramadane forcarrying out preliminary simulations.

Appendix

A.

Annex

I:

Modeling

for

the

efficiency

of

the

backflushing

Theprimaryobjectiveofthesimplisticmodelistocalculate thefractionofaperturesleftopenafterabackflushingperiod ˛b(tb) depending on specified backflushing time and flow rate.

Thebackflushingmodelisbasedontheassumptionthat only a discrete number of particles block the filter media openings.This isofcourse consistentwith the conclusion onthecloggingmodeloftheprevioussection.Herewe sup-posethat np blocking particlesare evenlydistributed along the filtermedia. Weconsider again a 2Ddomain identical totheoneconsideredinSection2.Theobjectiveisto com-putetheexittimeforeachparticle.Theexittimeisthetime neededforaparticletotravelfromitsinitialpositiononthe filtermediatotheentranceoftheinletchamber.Thecleaning timetccorrespondsthereforetotheexittimeoftheparticle locatedinitiallythefarthestfromtheentranceofthe cham-bersection.Whenthebackflushingtimetbisgreaterthanor equaltothecleaningtime,thebackflushingoperationmay beconsidered as perfectsince the filtermediais perfectly cleanattheendofbackflushing.Onthecontrary,ifthe back-flushingtimeissmallerthanthecleaningtime,afractionof particles does notexit the inletchannel beforethe end of backflushing.

InaccordancewiththesimulationpresentedinSection2, theparticlesare supposedtofollow thestreamlines.If we knowthevelocityfield(u,v)intheinletchamberduring back-flushing,thenthetrajectoryofaparticleisgivenbysolvingthe equationsstatingthatStokesnumberismuchsmallerthan1 (St1,noslipcondition)

dxp

dt =u; dyp

dt =v (A-1)

where(xp,yp)arethedimensionlesscoordinatesoftheinitial positionofparticle.

WecheckedthenoslipconditionsSt1,assumingan aver-ageparticlesizeof4␮m,aparticledensityof2500kgm−3,a fluidviscosityof0.0255Pas,anaveragefluidvelocityuinthe chamberof0.25ms−1andacharacteristiclength correspond-ingtoL,thelengthofachamber.ThevalueofStokesnumber definedaspd

2 pU

L isequalto5.10−6.

Eq. (A-1) are solved using a fourth order Runge Kutta method. This gives the position of particle (xp(t), yp(t)) in the inlet chamber asa function of time. The particle exit timeisobtainedwhenxp=0(entranceoftheinletchamber). Since the results reported in Section 2 indicate that the localvelocity is quasi-uniform along the filtermedia (this also holds during backflushing), the velocity field can be

0,4 0,2 0 0,6 0,8 1 x_p0 / L 0 0,5 1 1,5 2

Particle exit time (s)

Fig.A.1–(a)Examplesofparticletrajectoriesintheinlet

channelduringback-flushing.Notethaty=0corresponds

tothescreenandy=1toinletchannelcenterinthefigure.

Thearrowindicatesthemaindirectionoftheflowin

directionofchannelentrance,(b)particleexittimeasa

functionofparticleinitialpositionx_p0onfiltercloth.

determinedanalyticallyasbeforeusingthesolutionproposed inKoltuniewiczetal.(2004).Thissolutionreads,

u(x,y) Uo = 3 2 uy Uo



1−y2

 

₁₋Rew 420



2−7y2_−7y4



_+O



_Re2 w



v(x,y) Uo = 1 2 Rew Re0y (3−y2₎₊Re2w Re0



₁ 280



−2y+3y3_−y7



_+O



_Re3 w



(A-2) Werecallthattheoriginofthecoordinatesystemisatthe centerofthechannelhere(hencethefiltermediaisaty=1 andthecenterofchamberisaty=0).

Theaveragevelocityu asafunctionofxisdeducedfroma simplemassbalance,whichgivesalinearvariationsincethe localfiltrationvelocityisassumedtobeuniform:

u =U0



1−hx L



(A-3) Here as before U0 is the average velocity in the inlet

chambersection(U0isnegativeduringback-flushing).Foran

illustrationofthemethodology,wechoosetheaverage veloc-ityequalto0.2488ms−1andtheaveragefiltrationvelocityVw isequalto0.0268ms−1.Theseinputscaneasilybereplaced withalternativedata.

Fig.A.1ashowsexamplesofparticletrajectoriesintheinlet chamberforthe2Dgeometry,flowconditionsandmesh char-acteristics detailed inSection 2. Thechannel being 65mm long,it gives2600possibleparticle positionalong thefilter cloth(L/dh=2600fordh=25␮m).Fig.A.1balsoshowsthe par-ticleexittimeasafunctionofparticleinitialpositiononthe filtercloth.Theexittimeofthefarthestparticleisfoundto be1.6sfortheseconditions.However,mostofparticlesexit the channelin lessthan 0.7s.This isfurtherillustratedin

Fig. A.2which showstheevolutionofthenumber of parti-clesmp presentwithinthechannelduringback-flushingas a functionoftime. Wehave assumedhere that one parti-clewaspresentoneachmeshopeningwhenback-flushing starts(thusleadingtomp0=2600particlesinitially).Onecan

Fig.A.2–Evolutionofthenumberofparticlespresent

withinthechannelduringbackflushingasafunctionof

time.

distinguishtwophases:afirstphaseduringwhichabout75% oftheparticlesexitthedomain,thistakesabout0.3sanda significantlylongertailduringwhichtheremainingparticles exit.

Asillustrated inFig.A.2,wecandeducefromthis back-flushingmodelafunctionalrelationshipoftheform:

1−˛b(tb)

1−˛b(0) =f(tb/tc

)=a4(tb/tc)4+a3(tb/tc)3+a2(tb/tc)2

+a1(tb/tc)+a0 (A-4)

wheretbisthetimeofbackflushandtcthetimeneeded toremoveallthe particles blockedonthe opening(perfect cleaning)isequalfortheflowconditionsconsideredhereto 1.6sasshownonFig.A.1b.Theterm1−˛b(0)inrelation

(A-4)representsthefractionofopeningsoccupiedbyparticlesat thebeginningofbackflushingwhereas 1−˛b(tb)

1−˛b(0) isthefraction ofparticlesstillpresentattheendofback-flushing.

Foragivenflowrateenteringthesectorduringtheback flushingperiod,thebackflushingefficiencycanbeestimated dependingontwosituations:

- fullcleaning,whichcanbeexpectedwhentc<tb.Therefore ˛b= 1 is this case (mesh completely clean after back-flushing)

- imperfect(partial)cleaningwhentc>tb.Inthiscase,˛bcan beestimatedfromthefunctionalformofrelation(A-4).

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