Equivalent
Chinese~umalofChemicalEngineering,16(2)2l4，2l7(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
;Newtohiallandnon—Newtonianfluids.ineithersteadyorunsteadyfiltrationstages.Thismodelhastwostrengths:
;f11Itcanbeusedtodetermineequivalentcapillaryradiiandpredictfiltrafionqualitybasedo
nthepropertiesof
;solid/liquidsystemandoperationparameters:and(2)tocalculatecakespecificresistancea
nditsvariationswim
;timeatvariouscaketIlicknesslocations.
;Keywordscakefiltration,equivalentcakefiltrationmodel,specificcakeresistance,filtrate
rate,filtratequality,
;equivalentcapillaryradii,non—Newtonianfiltration
;lINTRoDUCTIoN
;Cakefiltrationisoneofthemostcommonand
;importantunitoperationsinchemicalengineering
;process.Cakefiltrationmodelprovidesatheoretical ;basisforprocessdesign,equipmentselection,scale ;magnification,performanceprediction,andstructural ;optimization.TraditionaltheoriesbyKarmanand
;Rum[11focusedonaveragecakeresistance.However,
;thesetheoriesdidnotcoverthenon—Newtonianslurry.
;ModernfiltrationtheorywasgivenbyTillerandShi—
;ratof2,31,andRushton【4】formedthelocalspecific
;resistanceandcakecompressibilityconcepts,but
;theseweretoughforpracticaluse.Similarworkin—
;cludedthecapillarynetmodelbyDullienandSOrbie
;[5,6],andcakegeometrystudybyXu[7]andLi[8].
;Inrelationtothenon—Newtonianfluidfiltration,
;themostacceptedmodel
;Shirato,basedonDarcy‟s
;wasformedbyKozickiand
;Law【3】asfollows:
;(=aac?l一一I=一一Il-dfA,dZ
;wheredv/dtistheflowrate(m?s),Aisthecakearea ;佃),nisnon—Newtonianindex,isviscosity(Pa?s), ;isRuth‟sspecificresistance(m),csisthedrysol—
;idsmassperunitfiltratemass,apisthedifferential ;pressure(Pa),andZisthecakethickness(m). ;Equation(1)istheoreticallysound.However,the ;specificcakeresistanceisnotobtainedbecauseofthe ;complexityanddynamicvariationsofthefiltercake ;structure.Inaddition,itisdifficulttomeasureorcal—
;culatetheviscosityofnon—Newtonianfluid.
;Tsarticlefocusesonpredictingfiltrationqual—
;ityandcalculatingcakespecificresistancebasedon ;operationalparameters.
;2EQUIVALENTMODEL
;2.1Generaldescription
;Thecakefiltrationistypicallyanunsteadyproc—
;ess.饧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 ;Received2007—09—11.accepted2008—02—20.
;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
;Navier—Stokesequation(N—S)『101isusedtoform
;capillaryflowequations,withviscosityasaconstant ;forNewtonianflowbutasavariablefornon—Newtonian
;flow.Anon—Newtoniancapillaryflowmodel(Fig.11 ;withfollowingassumptionsisprovided:
;(1)Roundandstraightcapillarywith?=Pl—P2
;betweentwoends;
;(2)ThevalueoftIleReynoldsnumberislow,and ;laminarflowonlyexistsindirectionzwithnorotation ;around0andhowever,alltheinertiaandweight
;forcesareignored;
;(3)Ruidsareincompressible,andVV=0,where ;isflowspeedtensor;
;f4)Theflowissteady;
;(5)NonNewtonianflowisgovernedbyOst
deWaele‟spowerlaw【7】: ;wald—
;=M(2)
;whereMisviscositycoefficient,2-isshearstress(Pa),
;andisshearrate(s-i).
;r曼,,//
;一
;_)…………….一0/,
;Figure1Thecapillaryflowmodel
;InFig.1,allvelocitycomponentsarezeroexcept ;indirectionz,i.e.,equationbecomes=0,=0.Thus,theN—S
;dVz
;=.ltV.V+pg
;z
;D
;d
;p
;z
;-
;,(3)
;wherepismassperunitvolume(kg.m一.).
;Innon—Newtonianflow,isnotaconstantinthe ;Laplacian.Incorporatingtheassumptions,Eq.(3)can ;berewrittenas
;=旦
;Dr(,.Or],,.
;where,.isvariableradius,Vzisthevelocitycompo—
;nentindirectionz,and/bz=Ap/listhepressure ;gradient.
;Accordingtothedefinitionofviscosity,oneob—
;tains
;/1=r/and=一DVz/a,.,(5)
;whereVzdecreaseswithincreasein,.,thus,thereisa ;minussignintheequation.Becausefiltrationflow ;alwayslocatesinlowshearstressarea,theremustb. ;e
;r0,whichcorrespondstoapointwhere=1s1, ;orr0==.Therefore,
;=
;(]?:(]”.,=一,c6
;SubstitutingEqs.(5)and(6)intoEq.(4),thenobtains ;D(r/102~n)一
;,.卸
;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)canbeappliedtonon—Newtonian
;flowwhenncorrespondstothepointwhere:1s1. ;4EQUIVALENTFLOWRATE
;Onthebasisoftheassumptionofequivalentcake ;filtrationmodel,theflowrateofliquidflowing ;throughthecapillariesisthesameasthatoftheactual
;filtercakeunderthesamepressuredrop卸atdiffer—
;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.=一l—l?NIrl+L)J
;=
;3n1[卜+l2z(f)I，
;where(istheequivalentradiiofcaoillarytubes. ;andA(f)isthecrosssectionareaofeachcapillary ;tube.IfthetotalfiltrationareaisA,thentheporosity ;ofequivalentcakeisequaltothatofrealcakeandone ;obtains
;f)or舭)=枷).(10)
;SubstitutingEq.(10)intoEq.(9)yields ;,,??_,\
;
;216Chin.J.Chem.Eng.,Vo1.16,No.2,April2008 ;Nq(t)-3
;n+1[]n+lR0)一,
;wherea(t)isporosityofcake,andcanbedetermined ;byexperiments.Eq.(11)canberewrittenas
;肌
;_[][Duringanunsteadyfiltrationprocess,theflow ;ratecontinuouslydecreases,whichcorrespondstothe ;decreasein尺(.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尺(f).
;Thoughequivalentcakefiltrationmodelassumes ;tllatthecapillaryisroundandstraight.buttheactual ;situationisdifferent.Poresize,shapeandlengthare ;variable,andinletBagleyphenomenonandoutlet ;streamlinediflusion【6,10】arenotconsidered.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)一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)=——__=一.(17)
;(f)尺(f).Cs
;Equations(15)and(17)provideamethodto ;calculatelocalandaveragecakespecifcresistance ;whenporositys(canbemeasured,andsometimes ;obtainedbyliquidmaterialbalanceequation.Fromthe ;twoequationsabove,itcanbeconcludedthat ;(1)ForNewtonianfluid,thespecificcakeresis ;tanceisonlyrelatedtoparticledepositionandcake ;structureparameterssuchass(f)and尺(,anddoes
;notrelatetofluidproperties.西ereasin
;non—Newtonianfluid,bothcakestructureparameters ;andfluidfactornareinvolved;thisisconsideredthe ;maindifferencebetweenthetwofluids.
;(2)ForbothNewtonianandnon—Newtonian
;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?口of0.4MPa,withdopeviscosity
;of62Pa-sat50.Candn=0.60.Liquidincakecanbe ;usedasameasureofporositybasedonmaterialbalance. ;Table1Calculationandtestresults
;Note:Plateandframepresseswereusedwithafiltrationarea ;0f50meach,amaximalcakethicknessof14mm.andadif- ;ferentialpressureof0.4MPa.Particularr50infiltra~usedto ;characterizefiltratepurityisanalyzedbylaserparticlesize ;analyzer(HRI,D一150JA,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)Both山e尺(andparticularradiiinfiltrate
;rimpurityradii)decreasesynchronouslyasthefiltra—
;tionprocessproceeds.
;r3)Rfare2.8—3.3timeslargerthanfiltrate,par—
;ticularr50,although尺(f)andrs0areabsolutelytwo
;differentissuesfromdifferentsources.Theerrormay ;becausedbythehypothesesmadeforthemodeland ;complexityofcakefiltration.Comparedwithother ;models,theerrorisnottoobad.
;r4)Itiseasiertousecakemoistureinsteadof ;porosity,butmaycauseproblemsbecauseofanidle ;voidexistinginsidethecake.尺ff)maybeunderesti—
;matedandresistanceoverrated,althoughtheerror ;seemstobeacceptable.
;2.1O
;2.O5
;皇2.OO
;趸1.95
;1.9O
;1.85
;莹
;.
;老
;葛
;?
;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). ;尺rcouldbeusedasanindicatoroffiltrationquality, ;and0crcouldbeusedasapredictionoffiltraterate ;forbothNewtonianandnon—Newtonianfluids.Un—
;steadyfeaturesofcakefiltrationarefinitelydescribed. ;NOMENCLATURE
;cakefiltrationarea,m
;crosssectionareaofeachcapillarytubeattimet,m ;drysolidsmassperunitfiltratemass
;flowrate,m.s一
;cakethickness,m
;cakethicknessattimeLm
;viscositycoefficient
;amountoftotalcapillarytubes
;non.Newtoniallindex
;differentialpressure,MPa
;flowratethroughtherealcake,m3.s，l
;flowrateofasinglec~illaryattimem31s
;equivalentradiiofcapillarytubesattimet,m ;variablera~us,m
;time
;flowspeedtensor,m-s
;thecomponentindirectionz.m?s_.
;flowdirection
;Ruth‟sspecificresistance,m
;shearrate,s一1
;porosityoffiltercakesattimet ;viscosity,Pa?s
;aspecialviscositypointwhen=一1,Pa?s
;weightperunitvolume,kg-m一
;shearstress,Pa
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