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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