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Equivalent

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EquivalentEquiva

    Equivalent

Chinese~umalofChemicalEngineering,16(2)2l42l7(2008)

    ;EquivalentCakeFiltrationModel

    ;uTan(徐坦),ZHUQixin(朱企新),clmNXu(陈旭)andLIWenping(李文苹)

    ;SchoolofChemicalEngineeringandTechnology,TianjinUniversity,Tianjin300072,Chi

    na

    ;AgrilectricResearchCompanyLakeCharles,LA70615,USA ;AbstractCakefiltrationhasbeenwidelyusedinmanychemicalprocesseswithmorenon

    Newtonian.highlyvis.

    ;cousandcompressiblematerialsinvolved.Neithertraditionalnormodemfiltrationtheory

    canbeappliedinpractice

    ;“Equivalentcakefiltrationmodel”isarecentlydevelopedmathematicalmodeltodescribecakefiltrationforboth

    ;NewtohiallandnonNewtonianfluids.ineithersteadyorunsteadyfiltrationstages.Thismodelhastwostrengths:

    ;f11Itcanbeusedtodetermineequivalentcapillaryradiiandpredictfiltrafionqualitybasedo

    nthepropertiesof

    ;solid/liquidsystemandoperationparameters:and(2)tocalculatecakespecificresistancea

    nditsvariationswim

    ;timeatvariouscaketIlicknesslocations.

    ;Keywordscakefiltration,equivalentcakefiltrationmodel,specificcakeresistance,filtrate

    rate,filtratequality,

    ;equivalentcapillaryradii,nonNewtonianfiltration

    ;lINTRoDUCTIoN

    ;Cakefiltrationisoneofthemostcommonand

    ;importantunitoperationsinchemicalengineering

    ;process.Cakefiltrationmodelprovidesatheoretical ;basisforprocessdesign,equipmentselection,scale ;magnification,performanceprediction,andstructural ;optimization.TraditionaltheoriesbyKarmanand

    ;Rum[11focusedonaveragecakeresistance.However,

    ;thesetheoriesdidnotcoverthenonNewtonianslurry.

    ;ModernfiltrationtheorywasgivenbyTillerandShi

    ;ratof2,31,andRushton4formedthelocalspecific

    ;resistanceandcakecompressibilityconcepts,but

    ;theseweretoughforpracticaluse.Similarworkin

    ;cludedthecapillarynetmodelbyDullienandSOrbie

    ;[5,6],andcakegeometrystudybyXu[7]andLi[8].

    ;InrelationtothenonNewtonianfluidfiltration,

    ;themostacceptedmodel

;Shirato,basedonDarcy‟s

    ;wasformedbyKozickiand

    ;Law3asfollows:

    ;(=aac?l一一I=一一Il-dfA,dZ

    ;wheredv/dtistheflowrate(m?s),Aisthecakearea ;),nisnonNewtonianindex,isviscosity(Pa?s), ;isRuth‟sspecificresistance(m),csisthedrysol—

    ;idsmassperunitfiltratemass,apisthedifferential ;pressure(Pa),andZisthecakethickness(m). ;Equation(1)istheoreticallysound.However,the ;specificcakeresistanceisnotobtainedbecauseofthe ;complexityanddynamicvariationsofthefiltercake ;structure.Inaddition,itisdifficulttomeasureorcal

    ;culatetheviscosityofnonNewtonianfluid.

    ;Tsarticlefocusesonpredictingfiltrationqual

    ;ityandcalculatingcakespecificresistancebasedon ;operationalparameters.

    ;2EQUIVALENTMODEL

    ;2.1Generaldescription

    ;Thecakefiltrationistypicallyanunsteadyproc

    ;ess.enfiltrationproceedseffectivefilteringarea ;andrelatedflowratedecreasescontinuously.Besides, ;flowchannelsarepluggedgraduallyandnewones ;form.Moreover,averagespecificresistanceincreases, ;andpossibilitythatimpuritiesflowthroughthecake ;decline.Thesefeaturesaremathematicallyindescrib

    ;able,butcanbephysicallydescribedasfollows[9]: ;(1)Allcapillaryradiiinanequivalentcakere

    ;duceandtheirlengthsprolongsynchronouslywiththe ;filtrationprocess,sothecakeunsteadyfeaturesare ;easilydescribed,suchastheamendmentoffiltration ;andaugmentofresistance.

    ;(2)Cakeagingfeatures,likethedepositionofthe ;impurityandtheshrinkingoftheeffectivearea,are ;„„equivalent‟‟toreductionincapillaryradiusandthe

    ;increaseinlengm.Differenttimeanddifferentradii ;correspondstodifferentflowratesandspecificresis

    ;tances.respectively.Whereasdifferentcakelocations ;withthesameequivalentradiuscorrespondtodiffer

    ;entspecificresistancesasthecakegrowstoacertain ;thickness.Themacroscopicequivalentisofmoreim- ;portancewhencomparedtomicroscopicsimilarity,in ;genera1.

    ;Equivalentcakefiltrationmodelassumesthatthe

    ;liquiduniformlyflowsthroughabundleofcapillaries ;underthepressuredropap,andtheflowratethrough ;thecapillariesisthesRffleasthatthrougharealfilter ;cake.Thus,thecapillariesbundleiscalled”the

    ;equivalentcake”,andradiiofthecapillariesarecalled

    ;“equivalentradii”.

    ;2.2Basicassumptions

    ;f1)Theamountofthecapillariesthatconstitute

    .withoutfluctuation,andthereisnocap ;thecakeis?

    ;illarycloggingorpluggingduringthefiltration

    ;cyclicalprocess.

    ;(2)Allcapillariesarecylindricalandstraight ;withthesameradiusR(f),whichreducesasfiltration ;proceeds.Moreover,flowrateq(t)througheachcap

    ;illaryisinvariable.

    ;(3)Thelengthofeachcapillaryl(t)isequaltothe ;Received20070911.accepted20080220.

    ;Towhomcorrespondenceshouldbeaddressed.E-mail:qxzhu@eyou.com

    ;

    ;Chin.J.Chem.Eng.,Vo1.16,No.2,April2008215 ;cakethickness,whichvariesasfiltrationproceeds. ;(4)Flowrateoftheliquidflowingthroughthe ;simulatedfiltercakeissimilartothatofactualfilter ;cakewhenpressuredrop?pisset.

    ;Duringarealfiltrationprocess.filtrateflow ;channelsgraduallydwindlewithcontinuousreduction ;inflowrateandincreaseinresistance0c(.Corre

    ;spondingly,inanequivalentcake,thedecreaseincap

    ;illaryradii(,theincreaseincakespecificresistance ;0c(andthedecreaseinflltrateratetakesplacewim ;thefiltrationinprogress.ThereductionofR(t1during ;theprocessprovidesakeYtosolveunsteadyanddy

    ;namicfiltrationproblem.

    ;3NoN.NEWTONIANCAPILLARYFLoW

    ;NavierStokesequation(NS)101isusedtoform

    ;capillaryflowequations,withviscosityasaconstant ;forNewtonianflowbutasavariablefornonNewtonian

    ;flow.AnonNewtoniancapillaryflowmodel(Fig.11 ;withfollowingassumptionsisprovided:

    ;(1)Roundandstraightcapillarywith?=PlP2

    ;betweentwoends;

    ;(2)ThevalueoftIleReynoldsnumberislow,and ;laminarflowonlyexistsindirectionzwithnorotation ;around0andhowever,alltheinertiaandweight

;forcesareignored;

    ;(3)Ruidsareincompressible,andVV=0,where ;isflowspeedtensor;

    ;f4)Theflowissteady;

    ;(5)NonNewtonianflowisgovernedbyOst

    deWaele‟spowerlaw7: ;wald

    ;=M(2)

    ;whereMisviscositycoefficient,2-isshearstress(Pa),

    ;andisshearrate(s-i).

    ;r,,//

    ;

    ;_)…………….0/,

    ;Figure1Thecapillaryflowmodel

    ;InFig.1,allvelocitycomponentsarezeroexcept ;indirectionz,i.e.,equationbecomes=0,=0.Thus,theNS

    ;dVz

    ;=.ltV.V+pg

    ;z

    ;D

    ;d

    ;p

    ;z

    ;-

    ;,(3)

    ;wherepismassperunitvolume(kg.m.).

    ;InnonNewtonianflow,isnotaconstantinthe ;Laplacian.Incorporatingtheassumptions,Eq.(3)can ;berewrittenas

    ;=

    ;Dr(,.Or],,.

    ;where,.isvariableradius,Vzisthevelocitycompo

    ;nentindirectionz,and/bz=Ap/listhepressure ;gradient.

    ;Accordingtothedefinitionofviscosity,oneob

    ;tains

    ;/1=r/and=DVz/a,.,(5)

    ;whereVzdecreaseswithincreasein,.,thus,thereisa ;minussignintheequation.Becausefiltrationflow ;alwayslocatesinlowshearstressarea,theremustb. ;e

    ;r0,whichcorrespondstoapointwhere=1s1, ;orr0==.Therefore,

    ;=

    ;(]?:(]”.,=,c6

    ;SubstitutingEqs.(5)and(6)intoEq.(4),thenobtains ;D(r/102~n)

    ;,.

    ;a,.f

    ;Theaboveequationcanbeintegratedunderthe ;followingboundaryconditions:

    ;(1)Atthecenterofthecapillary,.=0,and ;DV/=,,bt0

    ;(2)Attheinternalsurfaceofthecapillary,.=R, ;andVz=0.

    ;FromtIledefinitionofcapillaryflowrate. ;qJ:Vz(r)‟21tr‟dr,onegets

    ;nf1/n

    ;nR0

    ;Equationf7)canbeappliedtononNewtonian

    ;flowwhenncorrespondstothepointwhere:1s1. ;4EQUIVALENTFLOWRATE

    ;Onthebasisoftheassumptionofequivalentcake ;filtrationmodel,theflowrateofliquidflowing ;throughthecapillariesisthesameasthatoftheactual

     ;filtercakeunderthesamepressuredropatdiffer

    ;enttime:

    ;Q(f)=Nq(t),(8)

    ;whereQ(t)isflowratethroughtherealcake,which ;canbedeterminedbyexperiments;Nisthenumberof ;totalcapillarytubes,andq(t)istherateflowing ;throughasinglecapillary.

    ;SubstitutingEq.(7)intoEq,(8),onegets ;Q(t)=Nq(t)

    ;=

    ;3n1{2ltol(t.=ll?NIrl+L)J

    ;=

    ;3n1[+l2z(f)I

    ;where(istheequivalentradiiofcaoillarytubes. ;andA(f)isthecrosssectionareaofeachcapillary ;tube.IfthetotalfiltrationareaisA,thentheporosity ;ofequivalentcakeisequaltothatofrealcakeandone ;obtains

    ;f)or)=).(10)

    ;SubstitutingEq.(10)intoEq.(9)yields ;,,??_,\

    ;

    ;216Chin.J.Chem.Eng.,Vo1.16,No.2,April2008 ;Nq(t)-3

;n+1[]n+lR0),

    ;wherea(t)isporosityofcake,andcanbedetermined ;byexperiments.Eq.(11)canberewrittenas

    ;

    ;_[][Duringanunsteadyfiltrationprocess,theflow ;ratecontinuouslydecreases,whichcorrespondstothe ;decreasein(.Eq.(7)showssteadyflow,andthe

    ;deductionofEqs.(11)and(12)basedonEq.(7)by ;introducingtimetprovidesawaytodescribetheun

    ;steadyfiltrateflow.Duringfiltrationprocess,theR(f) ;decreasesasa(f)decays.whichisrelatedtothede

    ;creaseintheactualcakeporesizeduringarealfiltra

    ;tionprocess.Furthermore,Eq.(10)indicatesthat ;(1)Whenotherconditionsarefixed;thelower/20 ;istheR(alsodecreases.WlavstodecreaseR(f)in

    ;eludetemperaturecontrol,solventcomponentsopti

    ;mization,andapplicationofreagentstoincreasesolu

    ;bility.

    ;(2)Whenotherconditionsarefixed;thelower ;thefiltrationpressureisthehigheristhefiltratequality. ;(3)Therelationbetweenporosity,Ap,andflow ;rateonfiltratequalityisquitecomplicated.Eq.f1)can ;assistintheselectionofoptimizedoperatingparame

    ;terstoachievethebestfi1tratequality.

    ;(4)Porositys(f)dependsonparticle‟sphysical

    ;andchemicalpropertiesinsuspensions,andtheman

    ;neroftheparticledeposition.Furthermore,s(isre

    ;latedto(f).

    ;Thoughequivalentcakefiltrationmodelassumes ;tllatthecapillaryisroundandstraight.buttheactual ;situationisdifferent.Poresize,shapeandlengthare ;variable,andinletBagleyphenomenonandoutlet ;streamlinediflusion6,10arenotconsidered.How

    ;ever,themode1andcalculationofR(t)canstillserve ;asaneffectivetooltopredictfiltrationflowrateand ;qualityvariationduringthedynamicfiltrationprocess. ;5CAKESPECIFICRESiSTANCECALCULA

    ;TIoN

    ;Withthesamepressuredrop,filtrateflowrateof ;theequivalentcakeisequivalenttothatofanactual ;one,thus,theyshouldsharethesamecakespecific ;resistancebasedonEq.(11),whichcanbesimplified ;tothefollowingform:

    ;(l().CombiningEq.(13)withEq.(1)andDarcy‟slaw,

;cakepermeabilityKcanbedeductedas

    ;:

    ;fR.(14)23,z+1/

    ;Substituting=(KCs)intoEq.(13),gets

    ;_()2R(15)

    ;SubstitutionofEq.(15)intoEq.(13)yields

    : ;f,垒丫

    ;A~xCs

    ;?

    ;,

    ;(16)

    ;FormationofEq.(16)isbasicallyidenticaltoEq. ;(1)exceptthatviscosity/-toisreplacedby/.tinEq.(1). ;/.t0isobtainedwithaaccuraterheometrictestofthe ;slurry.Onthecontrary,itisalmostimpossibletoca1 ;culate/.tinEq.(1).Theintroductionofp0makesEq. ;(1)effectiveinapplication.

    ;ForNewtonianfluid,whenn=1,and=,

    ;Eq.(15)canbesimplifiedto

    ;R

    ;(f)=——__=.(17)

    ;(f)(f).Cs

    ;Equations(15)and(17)provideamethodto ;calculatelocalandaveragecakespecifcresistance ;whenporositys(canbemeasured,andsometimes ;obtainedbyliquidmaterialbalanceequation.Fromthe ;twoequationsabove,itcanbeconcludedthat ;(1)ForNewtonianfluid,thespecificcakeresis ;tanceisonlyrelatedtoparticledepositionandcake ;structureparameterssuchass(f)and(,anddoes

    ;notrelatetofluidproperties.西ereasin

    ;nonNewtonianfluid,bothcakestructureparameters ;andfluidfactornareinvolved;thisisconsideredthe ;maindifferencebetweenthetwofluids.

    ;(2)ForbothNewtonianandnonNewtonian

    ;slurry,a(t)increasesasR(t)deceases.

    ;(3)Specificcakeresistance(f)risesgreatlyasn ;becomeslargercontinuously.whentheslurrychanges ;fromshearthinningtoshearthickening.

    ;6EXAMPLE

    ;Table1givesthetestedfiltrationdata.suchasfil

    ;traterate,cakemoisturecontent,particularr50infil

    ;trate,andcalculateddata,suchasR(t)andspecific ;cakeresistancebasedonEqs.(12)and05).Fig.2

    ;showscorrelationsofR(f),r50,andpresslife.Thetest ;isunderaconstant?of0.4MPa,withdopeviscosity

    ;of62Pa-sat50.Candn=0.60.Liquidincakecanbe ;usedasameasureofporositybasedonmaterialbalance. ;Table1Calculationandtestresults

    ;Note:Plateandframepresseswereusedwithafiltrationarea ;0f50meach,amaximalcakethicknessof14mm.andadif- ;ferentialpressureof0.4MPa.Particularr50infiltra~usedto ;characterizefiltratepurityisanalyzedbylaserparticlesize ;analyzer(HRI,D150JA,mAC/R0YC0).Presslifewasiust ;54dayslong.

    ;

    ;Chin.J.Chem.Eng.,Vo1.16,No.2,April2008217 ;AcomparisonisshowninTlable2beforeandaf- ;tercorrectiveactionsaccordingtoEq.(12),including ;1argerfiltrationarea,lessfiltraterecyclerate,and ;higherdopetemperature.Extrusionstabilityisusedas ;animportantparameteroffiltrationquality.From1Ia_ ;bles1,2andFig.2,wehavethefollowingdiscussions”

    ;(1)Specificcakeresistancesgoupsharpl~

    ;whereasflowratedecreaseswithtimeasthecakeages. ;r2)Bothe(andparticularradiiinfiltrate

    ;rimpurityradii)decreasesynchronouslyasthefiltra

    ;tionprocessproceeds.

    ;r3)Rfare2.83.3timeslargerthanfiltrate,par

    ;ticularr50,although(f)andrs0areabsolutelytwo

    ;differentissuesfromdifferentsources.Theerrormay ;becausedbythehypothesesmadeforthemodeland ;complexityofcakefiltration.Comparedwithother ;models,theerrorisnottoobad.

    ;r4)Itiseasiertousecakemoistureinsteadof ;porosity,butmaycauseproblemsbecauseofanidle ;voidexistinginsidethecake.ff)maybeunderesti

    ;matedandresistanceoverrated,althoughtheerror ;seemstobeacceptable.

    ;2.1O

    ;2.O5

    ;2.OO

    ;1.95

    ;1.9O

    ;1.85

    ;

    ;.

    ;

;

    ;?

    ;presslife/d

    ;Figure2Equivalentradiiandaverageimpuritysize ;Table2Filtrationqualityimprovement

    ;Filtrationisaverycomplicatedprocess.inwhich ;bothsolidsandliquidmove,filterchannelsdeform, ;filtrationeffectchange.andhugediflficultiesbegin ;whenonetriestomodelit.Thisiswhyaprecise ;mathematicaldescriptionforcakefiltrationdoesnot ;existtilldate.Therefore,themodelfocusesonmac

    ;roscopicequivalenceinfiltrationprocessandnotmi

    ;croscopicsimilarity.Nevertheless,relativelystrong ;hypothesesmadeinthisarticleisnotanobstaclein ;termsoftheequivalenceofflowrateasexpressedin ;Eq.(8),althoughfurtherworkneedstobedone. ;7CONCLUSIONS

    ;Equivalentconceptisusedtomodelcakefiltra

    ;tion.Rrandcakeresistancescanbeeasilycalculated ;basedonfiltrationparametersbyEqs.(12)and(15). ;rcouldbeusedasanindicatoroffiltrationquality, ;and0crcouldbeusedasapredictionoffiltraterate ;forbothNewtonianandnonNewtonianfluids.Un

    ;steadyfeaturesofcakefiltrationarefinitelydescribed. ;NOMENCLATURE

    ;cakefiltrationarea,m

    ;crosssectionareaofeachcapillarytubeattimet,m ;drysolidsmassperunitfiltratemass

    ;flowrate,m.s

    ;cakethickness,m

    ;cakethicknessattimeLm

    ;viscositycoefficient

    ;amountoftotalcapillarytubes

    ;non.Newtoniallindex

    ;differentialpressure,MPa

    ;flowratethroughtherealcake,m3.sl

    ;flowrateofasinglec~illaryattimem31s

    ;equivalentradiiofcapillarytubesattimet,m ;variablera~us,m

    ;time

    ;flowspeedtensor,m-s

    ;thecomponentindirectionz.m?s_.

    ;flowdirection

    ;Ruth‟sspecificresistance,m

;shearrate,s1

    ;porosityoffiltercakesattimet ;viscosity,Pa?s

    ;aspecialviscositypointwhen=1,Pa?s

    ;weightperunitvolume,kg-m

    ;shearstress,Pa

    ;REFERENCES

    ;1Ruth.B.E,”Correlatingfiltrationtion”,

    ;MemoirsoftheFacultyofEngineering,NagoyaUniversity,37(1),

    ;3888f1985).

    ;4Rushton,A.,SolidLiquidFiltrationandSeparationTechnology,2nd

    ;edition.WEY_VCH,London,491(2000).

    ;5DuUien,EA.L.,”Newnetworkpermeabilitymodelofporousmedia‟‟,

    ;AIchEj.21,299-307(1975).

    ;6Sorbic,K.S.,Cliff0rd,P.J.,Jones,E.R.,‟herheologyofpseudo ;plasticfluidsinporousmediausingnetworkmodeling”.J.Colloid

    ;InteSci.,130t.508534(1989).

    ;7xu,X.Y”Fractalstudyoncakefiltration‟‟,J.MetafMlnP,207(9),

    ;4244(1993).

    ;8Li,W.E,‟1rheories&ApplicaftonsCeramicmembraneforcrossflow

    ;micro~tration‟‟,Ph.D.Thesis,TianjinUniversity,Tianjin(1998).

    ;9X|u.T..‟‟F1lndationofequivalentcakefiltrationmodelanditsappli.

    ;cationinnon.Newtonianslurry‟‟.Ph.D.esis.TianjinUniversity,

    ;Tianjin(2006).

    ;10Wters,K.,Rheometry:IndustrialApplications,ResearchStudies

    ;Press.?

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