Moulik, S. P.

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indianJournal of ChemistryVol. 15A,January1977.pp.1-3ProposedSimpleMulticapillaryFlow Model to GenerateConstantPressure:ElectrofiltrationEquationS. P. MOULIKDepartmentof Chemistry,]adavpurUniversity,Calcutta700032Received15December1975;revised10May 1976;accepted1 June 1976With the help of a proposedsimpleparallelyarrangedmulticapillarysystemmodelforthe filter and filter-cakeequationsfor constantpressurefI.ltrationand electrofl.ltrationhavebeengeneratedon the basisof thefundamentalflowequationof Poiseuille.Theseareidenti-cal in formswith d'ArcyandSperryequationsfor normalfI.ltrationandtheequationof Moulikfor electrofiltration,suggestingthatthefundamentalbasisof theproposedmodelis real. Thepictureof thepermeability(reciprocalof resistance)of thefilter as well as thecakehasbeenmadephysicallymeaningful.LIQUID flow througha filter-bedof constantthicknessis knownto be governedby d'Arcylaw·. In the caseof a bedthat coptinuouslygrowsin thickness,as in filtration,modificationofd'Arcyequationmadeby Sperry2andCarman3hasbeenconsideredto be adequate.Tne lUlderlyingprinciple recognizestwo separateresistances,oneofferedhy the filter and the otherby the growingfilter-cake.The final equationhas been derivedfor liquid transportunder a gradientof constanthydrostaticpressure.Whenthefiltrationis effectedunderan appliedelectricfield4,electrokineticflowsmustappearin additionto the normalflow5,andmodificationof the equationof Sperry has beenmade accordinglyand has been tested takingmeasurementsin a laboratory-scaleapparatus5-7•In a very recentpapers,a complementaryanalysisof thebasicphenomenaoccurringin electrofiltrationhas been given applyingnonequilibriumthermo-dynamics.It is the purposeof the presentpaperto showthatpropositionof asimpleparallelyarrangedmodelmulticapillary system can generated'Arcy andSperry equationsas well as the constantpressureelectrofiltrationequationput forward earlier byMoulik5-1.The ProposedModelThe cross-sectionof the flow modelis showninFig. 1. It is consideredfor simplicity that theassemblyis a bunchof parallelyarrangedcapillariesof equaldimensions.In this arrangementin totalarea,A, formedby the placementof n capillaries,somemoreporesshoulddevelopdueto the circularshapeof the individual parentcapillaries. Thesearetheextravoids. A partof thewholeareawouldof coursebe thenon-voidregionswhichare formedby the participationof the ",aIlsof the capillaries.If thesevoidsareblockedor madedead,the bunchof thecapillaryarrangementwill representa matrixof uniformvoidandnon-voidregions. Permeabilityof sucha matrixwoulddependon the proportionsCapina"wou.Fig. 1- Cross-sectionof a simpleparallelyarrangedmulti-capillaryflowmodelof thevoidsand the non-voidspresentin the totalarea. If theextravoidsremainassuchor thematrixis consistedof capillariesof varyingdimensions,thecross-sectionof the matrix is non-unifonn inrespect of the void and non~voidregions. Amembranefilter canhavecloseresemblanceto sucha model.Recognizingthat the capillarylengthsare equalbut thediametersdifferent,thevolumerateof flow,V1/T, of a fluidofviscosity,'Yj, ata constantpressuredifferential,P, throughthe bOOChwould be givenby Eq. (1) proposedby Poiseuille.VI =P1t rnlrf+na'i+nat1+....] =P1t ~n.~ ...(1)T 81)L " 8'YjL L., ••wheren,refersto the numberof ca.pillariesof typei having radius" and length(L).Now in a total area,A, of the cross-sectionof amulticapillarybunchif therearen total capillaries?f types,1, 3,2, ...., thevoid areaperunit faceareaIS- 2_ 1t~n", (2)~- ~ ...~maybe definedas thepermeabilityor the trans-porting factor of the capillary assembly,andn =nl+n2+na+ =:En,.1INDIAN J. CHEM., VOL. 15A, JANUARY 1977Taki/ng Eq. (2) into consideration, Eq. (1) canbe wr ltten asEq. (4) is the d'Arcy flow equationI for the flowof a c eanliquid through a filter-bed. The permea.bility constant is then the reciprocalof Ri.Let us now consider two individual bunches ofmulti apillary systems,type-l and type-2, throughwhic a liquid flows at a constantpressure,P. Thevol e rates of flow will dependon the pressureas'Well the dimensionsof the capillary systemsandshoul follow equation (4). If these two bunchesare n w joined in series, and the overall pressureis ma ntained again at P, then the overall volumerate f flow will be... (16)...(15),..(13)I- ='lJR:1+'lJL2OC (11)V12 PA 2PA ,,,where_ 16(:Enl~)= 16A (12)oc - --4 4 ...~2 :Enjrj rtr.njrj(For capillaries of uniform dimensions, oc =16Atrtnr4).Eq. (11) is identical in form with the equationof Sperry2put forward for filtration of a suspensionthat can form a filter-cake. The resistancesof thefilter and the cake can be then modelled with theresistancesof the multicapillary systemsconsideredabove.Theoryof FiltrationNormal filtration - Let us consider that amembrane filter is equivalent to a bunch oJ tinycapillaries arranged in parallel. In a filtratione,xperiment,suspendedparticles would be depositedon this filter forming a cake. With the progressof filtration, this growing cake can be imagined tobe equivalent to continuous coupling in series of asecondbunch of capillaries with the parent matrix.If the deposition is madeslow and smooth, then anuniform cake-matrix is possible to be formed, whichmay impart a copstant specific cake resistance3,oc. For such a systemEq. (11) is applicable. Takingv as the volumeof the cake formed per unit volumeof the filtrate, L2 of Eq. (11) can be eliminated2,3by L2 =vV12/A.T 'lJRi 'f)cx.vV12V-=PA+ 2PA212This is the exact filtration equation proposedbySperry2 and Carman3.Electrofiltration- The increased flow of anaqueoUs electrolyte solution at constant pressurethrough a filter under an applied electric field wasconsideredearlier5to be a phenomenonof decreasedfilter resistancecausedby the contribution of filter-poreelectroosmosis9. Under conditiODsof no surfaceConductance9-12,the new resistance,R;, was equatedwith the original resistance,R1,by a relationR; ~(1+~) ...(1'1)where K is a constant, ~1is the electrokineticpotential of the double layer existing at the solid-liquid interface inside the filter-pores, E is theelectric field and VI is the volume flowed throughthe filter (the filter has beep consideredto be thebunch of the multicapillary system, type-I) at zerovoltage. The volume rate of flow is then given byVi PA PA(Vl+K~lE)T ='lJR~= 'lJR~VlA similar equation holds also separately for thefilter-cake.V~ PA PA(V2+K~2E)- =-" =------T 'lJR2 'lJR~V2In this equation all terms have the same signi.ficance as those appearing in Eq. (15).... (9)... (8)... (7)... (3)... (10)T.'£1'rjRi+'lJR~V121PA PAandP PRI+R2 -YP TP-+-VI V2Su a situation prevails in practical electrofil-tratio experimepts4-6. The physical significanceof E . (7) is that the resistancetermsRI and R2 aredime sional quantities, and for the same bunches,their magnitudes are independent of the appliedpress re and are given by the ratios between thepress re and the volume rates of flow.Th inverse of Eq. (7) isI... !+TVI2 VI V2Th s equation means that working at a constantpress re helps to eliminate it in the equation andthe i verse of the total flow is just the linear com-bina 'on of the inverse of the individual flows.Th detailed form of Eq. (8) is8'lJLI (:En.r~) 8'lJL2(:Enil~)V12 }PA~' :Enirt +PA~2 :Enjr'In Eq. (9), i and j terms are used to differentiatebetw.en the two typesof the multicapillary systems.W h reference to Eq. (4) we can also writeVI = A~(:En.r~)T 8'lJL :En.,.-orVI P (T = R~=Rl ... 4)'Wherethe total system resistance,Rl =¢(:En.r1) ...(5)A :En.r..and its conventional (partial) resistance,R~=t(:En.~) ...(6)~. :E .r.I{Whe~ t.hedimensionsof the capillaries are equal,Rl = 'lJL/~Ar2,ard Rl =8D/~r2).2I I I'" 11111 I-~------~----MOULIK: ELECTROFILTRATION EQUATIONThe rateof flow for thecombinedsystemis thenT T T 7jR;Vl-=-,+-,=---~-Vl2 Vl V2 PA(Vl+K~lE)+ [2~CL~2(V :~~2E)] V~2...(17)It has beenshownearlier5-7that at the criticalvoltage,Ec" theparticleflowdueto electrophoresiscanbemadeequalto thefluidflow makingthenettransferof the suspendedparticleszero. Then atany voltage,E <Ec" the resultantflow towardsthe filter isV~2Vf =T -!LEA ...(18)where !L is the electrophoreticmobility of thesuspendedparticles. Eq. (18)will guidethe for-mationof thecake,andat E =Em VI =0,yieldingVI =!J-Ec,A-!LEA ...(19)The appropriatethicknessof the cake,L~,inelectrofilt.ration6is then, (VI ) (Ecr-E)L2 =L2 ~.JT =L2 ;: ..,(20)Introductionof thiscorrectionmodifiesEq. (17)toT 7jR;V1-, =Vl2 PA(Vl+K~lE)[( 7jCLV )(Ecr-E)( V2 )]'+ 2PA2 ~ V2+K~2E Vl2 ...(21)At the early stages of filtration the factor(V2fV2+K~lE),may be nearly equal to unityl-f.This may also be the case for systemswhereV2~~2E. Eq. (21)is thenT 7jR;VJ [( 7jCLV)(Ec,-E)] V'V;2=PA(Vl+K~lE)+ ZPA2 ~ 12 4...(22)Eq. (22)is e~actlythe electrofiltrationequationdeducedearlierby Moulik6•References1. D'ARCY, H. P. G., Les fontainespubliquede la ville de-dijon (Victor Dalmont, Paris), 1856,113.2. SPERRY, D. R., Chem.metall.Engng,15(1916),198; 17'(1917),161.3. CARMAN,P. C., TYans. Inst. chem.Engys, London, 15·(1937),150.4. MOULIK, S. P. & BIER, M., Pyoceedingsof the thiytIAmeyicanwateyyesouycesconfeyence,Ser. No.3, 1967.524.5. MOULIK, S. P., COOPER,F. C. &- BIER, M., J. ColloillInterface5ci., 24 (1967),427.6. MOULIK, S. P., Enviyonmental5ci. & Tech., 5 (1971),771.7. MOULIK, S. P., Kolloid Zeit und Zeit Polymeye,254,(1976),39.8. MOULIK, S. P., J. Colloid Interface 5ci., 53 (1975)447..9. ADAMSON,A. W., Physical chemistyyof sUYfaces(Inter-science,New York), 1960,193.10. KRUYT, H. R., Colloid science,Vol. I (Elsevier,An:ister-dam), 1952,76.11. RUTGERS,J., TYans.Fayaday50c., 36 (1940),69.12. GHOSH,B. N., BANDOPADHYAY,S. & MOULIK, S. P.,Pyoc. Natn Inst. 5ci., India, 27 (1961),193.13. GHOSH,B. N., MOULIK, S. P. & SENGUPTA,S. K., J.electroanal.Chem.,9 (1965),372.3Ii....--1

Proposed Simple Multicapillary Flow Model to Generate Constant Pressure: Electrofiltration Equation

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Proposed Simple Multicapillary Flow Model to Generate Constant Pressure: Electrofiltration Equation

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