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Canadian Journal of Chemical Engineering, 82, April 2, pp. 343-348, 2004
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A new method for identifying osmotically limited and gel layer
controlled pressure independent flux in ultrafiltration
Zaidi, S. K.; Karode, S. K.; Kirpalani, D.; Kumar, A.
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*Authortowhomcorrespondencemaybeaddressed.E-mailaddress: ashwani.kumar@nrc.ca
U
ltrafiltrationisapressure-drivenmembraneseparationofchemical species from a solvent, usually water, using a semi-permeable membrane. The solution is brought into contact with the membrane under pressure that causes permeate to pass through the membranewhilethesoluteisretainedonthemembranesurface.The netresultisalayerofsolutiononthemembranesurfacewithsubstan-tiallyhighersoluteconcentrationthanthatofthebulksolution.Thisis usuallytermedasconcentrationpolarization.Concentrationpolarization has received considerable theoretical and experimental attention over last25years.Ithasbeenobservedthatwithanincreaseinfeedpressure, permeate flux first increases and then becomes more or less pressure independent.TrettinandDoshi(1981)concludedthatpressureindependentfiltra-tion was gel layer controlled. These authors ignored the existence of thepolarizationlayerandmodelledthefluxusingDarcy’slaw,setting the filtration resistance in the pressure dependent filtration regime to the membrane resistance for pure water. The filtration resistance for thepressureindependentregimewassettothesumofthemembrane resistance for water and an additional resistance termed as the resist-ance of the so-called gel layer. Wijmans et al. (1984) interpreted flux databasedontheosmoticpressuremodel(vanOersetal.,1992)and concluded that for macromolecules with molecular weight ranging from10-100kDa,osmoticpressurelimitationwasmorelikelythangel formation.Theseauthorsassumedthatfiltrationresistancewaswholly comprisedofthemembraneresistanceforpurewater.
Inultrafiltration(UF)ofmacromoleculesusingaretentivemembrane, the polarization ratio (defined as the ratio of the solute concentration at the membrane surface to that in the bulk, i.e. CCC /www CCC ) is expectedb tobehighduetothelowdiffusivityofthesolute.Insuchacase,the polarization layer could offer an additional filtration resistance. In this workweproposeamethodforanalysisofunsteadystatefiltrationdata to differentiate between osmotically limited and gel layer controlled pressure independent filtration regimes. Also, the proposed method couldbeusedtoestimatetheadditionalfiltrationresistanceofferedby thepolarizationlayerinmacromolecularUF.
ANewMethodforIdentifyingOsmoticallyLimited
andGelLayerControlledPressureIndependent
FluxinUltrafiltration
S.K.Zaidi
2
,S.K.Karode
1
,D.Kirpalani
1
andA.Kumar
1
*
1
InstituteforChemicalProcessandEnvironmentalTechnology,NationalResearchCouncilofCanada,
11
1200MontrealRoad,Ottawa,ONCanadaK1A0R6
22
DepartmentofChemicalEngineering,UniversityofOttawa,Ottawa,ONCanadaK1A9B4
An unequivocal determination of whether pressure independent flux regime is osmotically controlled orgellayerdominated,isstillopenfordiscussionin themembraneliterature.Thepresentworkreportsa methodthatcouldbeusedtoaddressthisissue.Itis shownthatanalysisofpoststeadystatetransientfiltra-tion data leads to clear demarcashownthatanalysisofpoststeadystatetransientfiltra-tion of osmotically limitedandgellayercontrolledfiltration.Themethod proposedinthisworkcanalsobeusedtoestimatethe additionalfiltrationresistanceofferedbythepolariza-tion layer to the permeate flow in macromolecular ultrafiltration and has been verified experimentally. It has also been shown that the polarization layer thickness is not sensitive to the feed pressure but variesasafunctionofthebulksoluteconcentration; higherthebulkconcentration,thickeristhepolariza-tionlayer.
Unedéterminationclairedesilerégimeindépendant de flux de pression est osmotiquement contrôlé ou la couche de gel dominée est encore ouverte pour la discussion dans la littérature de membrane. Le travail actuel indique une méthode qui pourrait être employéepourabordercettequestion.Onluimontre que l’analyse des données passagères de filtration d’étatd’équilibreultérieurmèneàladélimitationclaire de la filtration contrôlée par couche osmotiquement limitée et filtration contôlée par une couche de gel. Laméthodeproposéedanscetravailpeutégalement êtreemployéepourestimerlarésistancesupplémen-taire de filtration offerte par la couche de polarisa-tion à l’écoulement perméable dans l’ultrafiltrapolarisa-tion macromoléculaire.Laméthodeproposéeaétévérifiée expérimentalement. Il y a aussi eu démonstration que l’épaisseur de couche de polarisation n’est pas sensible à la pression d’alimentation mais change en fonction de la concentration en bloc de corps dissous;plusélevéeestlaconcentrationenbloc,plus profondémentestlacouchedepolarisation.
Keywords: membranes,osmoticpressure,gellayer,
Theory
Permeation of pure solvent through membranes is usually modelledusingDarcy’slawwherethesolventfluxisdirectly proportionaltotheeffectivepressuredifferenceandinversely proportional to the filtration resistance. The constant of proportionalityistheinverseofsolventviscosity(vanOerset al.,1992).Mathematically,thiscanbewrittenas: J P R w m = ∆ µ (1)
where,Jwwwisthefluxofpuresolvent,Rm isthemembranefiltra-tionresistanceandµissolventviscosity.
In the case of UF with a solute that is retained by the membrane,thedrivingforceandtheflowresistancewouldbe modifiedduetotheoccurrenceofconcentrationpolarization or gel layer formation. Permeate flux for such cases can be obtainedusingthemodelreportedinliterature(Choeetal., 1986;vanOersetal.,1992): J P R s f = ∆PP−− π∆ µ (2)
where,Jsssisthesoluteflux,∆Pistheappliedpressure,PP Rfffisthe total filtration resistance and∆π the osmotic pressure differ-ence across the membrane. Equation (2) has been used to modelultrafiltrationsincefirstproposedbyGoldsmith(1971). Subsequently, several authors (Bhattaharjee et al.,1992; Elimelechetal.,1998)haveshowntheequivalenceofosmotic pressure and gel layer controlled concentration polarization models.
Differentiating Equation (2) with respect to time we get followingequation: dJ dt R d P dt d dt P R dR dt s f f f = − = − = − = − = − = − − 1 2 µ π π µ ∆ ∆ d P d P dd ππ ∆PPPPPP−−−−∆ππ (3)
The second term on the right-hand side of the above equation was neglected as it was found to be too small to makeanysignificantcontributionstothecalculations,e.g.on theright-handsidethesecondtermwasfoundtobe0.1-2.0% ofthefirstterminthepresentstudy.Forsubsequentcalcula-tionsEquation(4)wasused. dJ dt R d P dt d dt s f = − = − = − = − 1 µ π ∆ ∆ d P d P dd (4)
By analyzing post steady state transient filtration data obtained by reducing the driving pressure as a function of time,theactualfiltrationresistancecanbecalculatedfromthe slope of the graph obtained by plotting dJs/dt versus d/dt versus d/dt versus d P/dt.∆ Theslopeoftheresultinggraphwouldbelinearifd∆π/dtwas equaltozeroorthedifferencebetweenddd P/∆PP dtanddtdt ddd /dtwas∆π constant.Moreover,sincetheshapeofthefluxversuspressure curve is non-linear (linear at low pressure and tapering off at high pressure) the only case where the variation of dJs/dt and ddd P/∆PP dt would yield a linear slope would be at adtdt ddd∆π/dt valueofzero.Inthecaseofgellayercontrolledfiltration,the soluteconcentrationatthesurfaceofthemembraneremains
constant (time independent) at the gel concentration. This wouldproducealinearslopeforthedJs/dtversus/dt/dt dP/dPdP dtgraph.dtdt Inthecaseofosmoticallylimitedpressureindependentfiltra-tion,thesoluteconcentrationatthesurfaceofthemembrane wouldbeafunctionoftheappliedpressure.Therefore,after achievingasteadystateintheexperiment,areductioninthe drivingpressurewithtimewouldcauseachangeinthesolute concentration at the surface of the membrane with time, resulting in a non-linear graph between dJs/dt versus/dt/dt dP/PP dt. Usingthesimpletechniquedescribedabove,itispossibleto unequivocally determine whether the pressure independent filtrationregimeisaresultofagellayercontrolledfiltrationor anosmoticallylimitedfiltration.
Experimental
Materials
TheUFmembraneusedinthisworkhadanominalMWCOof 6kDaandwaspreparedinthelaboratoryona1079backing (Tyvek,SuppliedbyDu-Pont).Membranecastingformulation contained25%ofpolysulfone,(Radel-RAmoco,U.S.)and21% polyvinylpyrrolidone (PVP) (Sigma, U.S.) in N-methyl-2-pyrrolidinone (NMP) (Anachemia, U.S.). The membrane was prepared by the phase inversion method, details of that are givenelsewhere(Dal-Cinetal.,1994).Throughoutthisinvesti-gation, Polyethylene Glycol (PEG) (Fluka chemie AG, Switzerland) with 35 kDa molecular weight was used as a standard solute. Concentration of PEG in the feed as well as permeatestreamwasmeasuredonthebasisofTotalOrganic Carbon (TOC) content. TOC content was estimated using a TOCAnalyzer(ShimadzuCorporation,Japan).Feedconcentra-tionwasvariedfrom0.2to5kg/m3.Apparatus
Figure1showsaschematicdiagramoftheexperimentalsetup. AsshowninFigure1,alltheexperimentswereperformedon acommercialultra-filtrationstirredcellunit(CellModelNo.: 8050;Amicon,U.S.).Thespecificationsofthestirredcellunit arelistedbelow: Areaofmembrane: 3.2x10-3m2 Diameterofcell: 0.063m Operatingpressure: 517kPa(75psi) Feedvolume: 1.6x10-4m3(160mL)For each experimental run, the feed chamber was purged withnitrogen.Themembranewassetupatthebaseofthe feed chamber and permeate was collected at the bottom of
Figure
the cell unit through a 1.5mm diameter tube. Membrane flux was measured by recording the change in weight of a standardbeakerovertime.Inordertodeterminethetransient flux through the membrane, permeate weight was recorded continuouslyusingastandardweighingbalance(SARTORIUS ModelNo.:BP221S).Theserialportcommunicationprogram (RS-232) from the weighing balance was developed using LabVIEW software. A static pressure transducer (Ashcroft, Model No.: K5) was installed on the feed side of the cell. The analog output of a static pressure transducer was digitized using a National Instrument PCI 6023E A/D card andaCB-68LPconnectingblock.Adataacquisitionsystemin LabVIEWwasdevelopedtosynchronizethepressureandmass measurements.Therateofchangeinfluxandpressurewere also plotted simultaneously in real-time with the developed softwaretomonitorthetransientresponse.
ExperimentalProcedure
1. Initially Pure Water Permeation (PWP) was measured for a new membrane. The mass of permeated solvent was recorded every two seconds through the data acquisition softwareLabVIEW.Anerroroflessthan0.1%inmeasuring solventfluxwasobserved. 2.Thefiltrationresistanceforpurewaterwascalculatedusing Equation(1). 3.Foragivenfeedsolution,fluxversustimedatawascollected ataconstantfeedpressureuntilasteadystatewasreached, i.e.whenthefluxvaluesbecametimeindependent.Oncea steadystatewasachieved,thebypassvalve(item9inFigure 1)wasopenedacracktoletthefeedpressuredecayasa functionoftime. 4.Theunsteadystatefiltrationdatawascontinuouslyrecorded usingadataacquisitionsystem.
ResultsandDiscussions
Figure 2 shows the permeate flux as a function of time for three different (constant) feed pressures (135, 270 and 405 kPa) for a bulk feed concentration of 1 kg/m3. As expected,
thereisasharpdropinthepermeatefluxfromtheinitialvalue (at time t = 0, corresponding to the flux of pure water) for shortfiltrationtimes.Atlongerfiltrationtimes,thefluxdecline graduallyreducesandeventuallyattainssteadystate.Itshould benotedthatduringthetimerequiredtoattainsteadystate
thechangeinthebulkfeedconcentrationwasnegligible.Only about 20–25 mL of the initial feed volume of 150 mL was collected as the cumulative permeate during these experi-ments.
Table1showsthesteadystatemembranerejection(R)based onthebulkfeedconcentration(R=1-CCC /Cp CC wherebbb C andCCppp CCCb are respectively the solute concentration in permeate andpp pp feed)fortheentireconcentrationandpressurerangesstudied inthiswork.Ascanbeseenfromthistable,steadystatesolute rejection was always greater than 97%. The membrane can therefore be treated as being completely retentive to the solute. As a consequence, the osmotic pressure of permeate canbeneglectedandtheeffectiveosmoticpressuredifference acrossthemembrane(∆π)canbesetequaltoπw. Figure3showsthesteadystatefluxforasoluteconcentra-tionof0.2,1and5kg/m3asafunctionoffeedpressure.Itis clearthatpermeatefluxbecomesindependentofpressurefor afeedconcentrationof5kg/m3.Forafeedconcentrationof 1kg/m3,permeatefluxisonlyslightlypressuredependent.The pressurerangewasrestrictedto400kPaduetothelimitations of the test cell. After steady state was reached at a constant feed pressure, the bypass valve (see Figure 1) was opened a crack.Thisresultedinadecayofthefeedpressure(pressure reduction)asafunctionoftime.Theunsteadypermeateflux and feed pressure was recorded automatically using a data acquisitionsystemasdiscussedintheexperimentalsection.
Figure4showsthevariationoftheunsteadystatepermeate flux as a function of reducing feed pressure achieved by opening the bypass valve a crack once steady state had
Figure2.Variationofpermeatefluxasafunctionoftimeforvarious valuesoffeedpressure(bulkconcentrationofPEG=1kg/m valuesoffeedpressure(bulkconcentrationofPEG=1kg/m3).). Table1. SteadystatePEGrejectionfortheentirepressureand concentrationrangestudiedinthiswork. ∆P,(kPa)PP CCC ,(kg/mb 3) 0.2 1 5 135 99.7 99.6 99.5 270 99.1 98.5 98.3 405 97.9 97.5 97.2 Figure3.Steadystatepermeatefluxasafunctionoffeedpressurefor differentfeedconcentrationsofPEG.
beenreachedforafeedconcentrationof0.2and1kg/m3.A
similarplotfor5kg/m3isshowninFigure5.Thesefiguresare
remarkablebecausetheyclearlyshowthecontributionofthe polarizationlayertotheoverallfiltrationresistance.Itshould be noted that as the pressure decays from a higher initial value,thetransientfluxatanygivenpressureisslightlylower thanthefluxatthesamepressurewhenthepressuredecayis initiatedfromalowerinitialvalue.ForexampleinFigure4,if astraightlineisdrawnverticallyat100kPa,thefluxresulting fromapressuredecaystartingfrom405kPaisslightlylower thanthatresultingfromapressuredecaystartingat270kPa, whichinturnislowerthanthefluxat100kParesultingfroma pressuredecayinitiatedat135kPa.Thisobservationcouldbe explainedasfollows:Inthecasewheresteadystateisreached atahigherpressure,theamountofsoluteaccumulatedinthe polarizationlayerislarger.Thishasresultedinhigherfiltration resistancetothesolvent(permeate)flow.Consequently,when thefeedpressureisrapidlyreducedfromahighinitialvalue, thepolarizationlayeroffersahigherresistancetothepermeate flow,resultinginaslightlylowerfluxthanifthepressuredecay wasinitiatedfromalowerinitialvalue. Furthermore,Figure5showsthatirrespectiveoftheinitial valueofthepressure,oncethebypassvalveiscrackedopen
to reduce the feed pressure, the balance registers a loss in weightoncethefeedpressuredropstolessthanabout90kPa. Theonlyreasonthebalancecouldregisteralossinweight(a negativeflux)isbecausethefeedpressuredropstoavalueless thantheosmoticpressurecorrespondingtotheaccumulated solute at the membrane surface. This is indeed remarkable. Tothebestofourknowledge,thisisthefirsttimethatsuch a phenomenon of negative flux has been demonstrated experimentally.
The transient data presented in Figure 5 can be further analyzedbyplottingthetimederivativeoftheunsteadyflux (dJs/dt)versusthetimederivativeofthefeedpressure(dtdt dP/dPdP dt).dtdt ThisdataisshowninFigure6.Itisobviousthatthereisavery good linear correlation for each value of starting pressure. Using the slope values of these straight lines the filtration resistance (Rfff) for each of the three starting pressures, 135, 270and405kPawas1.84,2.95and4.30x1014m-1
,respec-tively. The resistance for pure water was measured to be 3.75 x 1013m-1. Utilizing this estimated filtration resistance and
Equation(2)theosmoticpressure(πwww )ofthesoluteaccumu-latedatthemembranesurfacewascalculatedasafunctionof theunsteadypressure.Theosmoticpressureatthemembrane surfaceisfoundtobeindependentofthefeedpressureandit clearlyindicatesthegellayerdominatedfiltration.Considering thattheosmoticpressureisindependentofappliedpressure and is only dependent on solute concentration, increased resistanceatahigherappliedpressurewasduetoanincrease in gel layer thickness on the membrane surface. Further, it shouldbenotedthatusingEquation(2),theosmoticpressure ofthePEGgelwasfoundtobeabout87±3kPa.Thisvalueis in very good agreement with the experimental evidence of theonsetofthenegativefluxwhenthefeedpressuredropsto valuesbelow90kPaasshowninFigure5.
A similar analysis of the data presented in Figure 4 results in Figure 7 (for a PEG concentration of 1 kg/m3). As can
be seen, the graph between dJs/dt versus/dt/dt dP/dt is no longer linear,whichclearlyindicatesthatthesoluteconcentration(or osmoticpressure)atthemembranesurfacevariesasafunction ofthefeedpressure.Thefiltrationresistanceatdifferentfeed pressurescanbecalculatedbydrawingtangentstothecurves inFigure8.Usingtheseresistancevaluesateachfeedpressure, theosmoticpressureofthesoluteatthemembranesurface(or πwww)canbecalculatedusingEquation(2).Theresultsareshown inFigure8.Itisclearfromthisfigurethatπwwwisafunctionof thefeedpressureandisalwayslessthan90kPa,indicatingan Figure4.Poststeadystatetransientvariationofpermeatefluxasa functionofreducingfeedpressureforthreedifferentinitialpressures (bulkconcentrationofPEG=0.2and1kg/m (bulkconcentrationofPEG=0.2and1kg/m3).). Figure5.Poststeadystatetransientvariationofpermeatefluxasa functionofreducingfeedpressureforthreedifferentinitialpressures (bulkconcentrationofPEG=5kg/m (bulkconcentrationofPEG=5kg/m3).).
almostosmoticallylimitedpermeateflow.However,athigher applied pressures the contributions of the gel layer towards controllingpermeateflowwillplayaprominentrole.
Itisinstructivetocalculatethesteadystatesoluteconcen-trationprofileinthepolarizationlayer.Acorrelationbetween the solute osmotic pressure and its concentration correla-tion has been provided by Bhattacharjee et al. (1996). They demonstratedthatFlory’sequationforthechangeinchemical potential (the osmotic pressure) of PEG, as a function of volume fraction was an accurate correlation for osmotic pressureandconcentrationofPEG.Flory’sequationrelatesthe osmoticpressurewithasolutevolumefractionas π γ π γ γ χ γ π γ π γ −− γγ ++χχ π γ π γ π γ π γ π γ π γ π γ π γ RT V n π γ π γ π γ π γ π γ π γ π γ π γ π γ π γ π γ π γ π VV γpp −− nn γγγγγγpp++χχχχχχ p 1 V n V 1 1 n 12 π γ π −−γ ++ π γ π γ π γ π ln( −−−−γpp ++++ 1 pp π γ π γ π γ π γ π γ π VVVV 11−−−−−−−−−−−γ ) (pppppppp +++++++++++ 11−−−−−− nnnn) pppppppp++++++ (5)
where,γγγ =ppp c/ccρppp is the volume fraction of polymer,c is thecc polymerconcentration,pp pp ρpppisthepolymerdensityandnisthe
numberofmonomersubunitsinthepolymerchain.TheFlory-pp Huggins interaction parameterχ12is a linear function of the
solute concentration for linear chain polymers, and the concentrationdependenceisbestexpressedas
lnχ12 = +p qc (5a)
where p andpp q are constants having values of –0.604 andqq 1.7452respectively.
Equation (5) can be used to calculate CCC (the solute wallwww concentration) from the estimatedπwww values from
experi-mentaldata.UsingthisvalueofCCC andthesteadystateflux,www Js, the solute concentration profile as a function of distance perpendicular to the membrane surface can be estimated usingthefilmtheory(Zydney,1997;vandenBergetal.,1989; BhattacharjeeandDatta,2001)andthefollowingequation: C C J x D x b C C C =C + w b J xJ xs C C Cxx Cbb C C C C C C C C (CCCCCCCCCCCCCCCCCww CCCCCC exp(CCCCCCCCCCCbb) − ) (6) whereDisdiffusivityandiscalculatedusingliteraturecorrela-DD tion (Bhattacharjee and Datta, 2001). The boundary layer thickness(δ)canthenbeestimatedasthatvalueofxatwhichxx the solute concentration becomes almost equal to the bulk soluteconcentration.
Figure9showsthesteadystatesoluteconcentrationprofile within the polarization layer for three different steady feed
Figure
Figure7.AplotofAplotofdJdJdJ /dtversusssdtdt ddd P/∆PP dtfordatapresentedinFigure4.dtdtfordatapresentedinFigure4.
Figure8.PEGosmoticpressureatthemembranesurface(πwww)fora
PEGbulkconcentrationof1kg/m3calculatedasdetailedinthetext. Figure10.Calculatedthicknessofthepolarizationlayer(functionofbulkPEGconcentrations. δ)asa
Figure9.VariationofsteadystatePEGconcentrationasafunctionof
distancefromthemembranesurfaceforthreedifferentfeedpressures (bulkPEGconcentration=5kg/m3).
pressuresandabulkfeedconcentrationof5kg/m3.Fromthis
figure,itcanbeseenthatforagivenbulkfeedconcentration, theextentofthepolarizationlayerisindependentofthefeed pressure. The extent of the polarization layer for each value ofbulkfeedconcentrationisplottedinFigure10.Thisclearly shows that the extent of the polarization layer is a function of the bulk solute concentration and that the makeup of the polarization layer could be such as to provide significant hydraulicresistancecomparedtothemembraneresistance.It is therefore necessary to account for the additional filtration resistanceofthepolarizationlayerwhencalculatingtheoverall filtrationresistance.Applicationofthisanalysiswillbelimited tothosegellayersthatarenotsufficientlydiffusedandoffer significantresistancetoflowofsolvent.
Conclusions
Inthisworkwepresentamethodtounequivocallydetermine whetherthepressureindependentfiltrationregimeisosmoti-callylimitedorgellayercontrolled.Thisisdonebytheanalysis ofunsteadystatefiltrationdatameasuredafterasteadystate hadbeenreachedatconstantfeedpressure.The method of analysis has been validated by the experi-mentalobservationofanegativefluxinducedasaresultofgel layercontrolledfiltrationwhenthefeedpressureisreducedto avaluelessthantheosmoticpressureofthegel.
Itwasshownthatusingthemethodproposedinthiswork, it is possible to estimate the filtration resistance offered by the polarization layer in UF of macromolecular solutions. It is shown that the thickness of the polarization layer in such situationscanbesignificantlylarge.Further,itwasshownthat thethicknessofthepolarizationlayerisafunctionofthebulk soluteconcentrationandahigherbulkconcentrationledtoa thickerpolarizationlayer.
