Permeate Flux in Ultrafiltration Membrane: A Review

]. Applied Membrane Science & Technology, Vol. 14, December 2011,9-20© Unlversiti Teknologi Malaysia

Permeate Flux in Ultrafiltration Membrane: A Review

A Beicha!", R. Zaarnouch'' & N. M. Sulatmarr'

'University of Jijel. Faculty of Sciences and Technology, Department of Process Engineering, PB. 98, Ouled

Aissa, 18000 Jijel, Algeria

2University of jijcl, Faculty of Sciences and Technology, Department of Mechanics, PB. 98, Ouled Aissa, 18000

Jijel, Algeria3University of Malaya, Chemical Engineering Department, 50603, Kuala Lumpur, Malaysia

ABSTRACT

Membrane processes exist for most of the fluid separations encountered in industry. The most Widely used is

membrane ultrafiltration, pressure driven process which is capable of separating particles in the approximate size

range of O. 00 I to 0.1 }Ill. The design of membrane separation processes, like all other processes, requires quantitative

expressions relating material properties to separation performance. The factors controlling the performance of

ultrafiltration are extensively reviewed. There have been a number of seminal approaches in this field. Most havebeen based on the rate limiting effects of the concentration polarization of the separated particles at the membrane

surface. Various rigorous, empirical and intuitive models exist, which have been critically assessed in terms of their

predictive capability and applicability. The decision as to which of the membrane filtration models is the mostcorrect in predicting permeation rates is a matter of difficulty and appears to depend on the nature of the dispersion

to separated.

Keywords: Ultrafiltration. mathematical modelling, permeate flux, limiting flux

1.0 INTRODUCTION

Separation and purification processes usingmembrane technology are gaining popularity inmany chemical and food processing as well as inwaste treating industries. The technology offersseveral advantages over and above the traditionaltechniques, including low energy requirement andlow temperature of operation. The ultrafiltrationprocess is found to be suitable for large-scaleoperations and numerous studies on itscommercial application to concentrate or purifysolutions and in the extraction of solvents havebeen reported in the literature,

Ultrafiltration is a pressure driven process forseparating particles in the approximate size rangeof 0.001 to 0.1 11m. Under a typical hydrostatic

"Correspondtng to: A. Beicha (email: [email protected])

pressure of 60 to 600 kPa applied across themembrane, the solvent is forced through it aspermeate, Solutes that are unable to pass throughare retained, concentrated and removed tangential to the membrane surface as retentate. Theflow ofbulk solution towards the membrane resultsin concentration polarization, with soluteconcentration at the membrane surface beinghigher than that in the bulk. The thickness ofthispolarized layer is significantly reduced by themagnitude of the cross-flow velocity of the feedsolution past the membrane surface, which shearsoff the layer. Depending on the macromolecules,high osmotic pressure could develop in thepolarized layer at the membrane-solution interfacewhich reduces the effective driving force for theflow of permeate. For high molecular weight solute,the effect of osmotic pressure is less important. Incertain cases, the solute wall concentration may

10 A. Beicha, R. Zaamouch & N. M. Sulaiman

reach its solubility limit and solute precipitationonto the membrane surface to form a gel or a gellayer can occur. Osmotic pressure, concentrationpolarization and gel layer formation are amongthe dominant resistances, which have been foundto control ultrafiltration and other membranefiltration processes, in addition to membranefouling by adsorption.

The development of quantitative predictivemodels is, therefore, of great importance for thesuccessful application of membrane separationprocesses in the process industries. The designand simulation of membrane separation processes,like all other processes, require quantitativeexpressions relating material properties toseparation performance. The physical theoriesgoverning the filtration models principally describethe effect of the concentration polarizationphenomena at the membrane surface. This articlepresents a review of the existing filtration modelsfor colloidal and fine particle dispersions withemphasis on their qualitative and quantitativepredictive capability, and their limitations.

2,0 FACTORS LIMITING PERMEATEFLUX IN ULTRAFILTRATION

During an actual separation, the membraneperformance can change drastically with time, anda typical flux-time behaviour observed which is adecrease in flux through the membrane over time.This flux decline behaviour is mainly due tomembrane fouling (deposition of solute at themembrane surface and in pores or phase

Concentration polarization (Rop )

Retentate

Membrane (Rm)

Permeate _+"-_Adsorption (Rods)

boundary). The concentration of the accumulatedsolute molecules over the membrane surface maybecome so high and exceeds its solubility limit. Inthis case a gellayer can be formed on the membranewhich in tum exerts the gel layer resistance R{fThis phenomenon mainly happens when thesolution contains proteins. It is possible for somesolutes to penetrate into the membrane and blockthe pores thus leading to the pore-blockingresistance Rpb' Adsorption can also take place onthe membrane surface as well as within the poresthemselves. Figure 1 provides a schematicrepresentation of the various resistances that canoccur.

The extent of these phenomena is stronglydependent on the type of membrane process andfeed solution employed [1-5]. Even for a givensolution, fouling will depend on physical andchemical parameters such as concentration,temperature, pH, ionic strength and specificinteractions such as hydrogen bonding and dipole­dipole interaction.

Several parameters may affect the outcome of adeposition process during UFo Persson and Nilsson[6] measured the amount of whey proteindeposited, indirectly as deposit resistance, afterstatic protein exposure. They found that the depositresistance depended on the membrane pore size;the larger the nominal pore size, the less the depositresistance. Recent experimental investigations [7]suggest that interparticle and especiallyelectrostatic interactions which are important inthe colloidal size range of 5-500 nm play animportant role. It has been found that thepermeability of layers of proteins formed at

Pore-blocking (Rpb)

Figure 1 Overview of various resistances to mass transport

Permeate Flux in Ultrafiltration Membrane: A Review 11

membranesurfaces during ultrafiltrationdependson solution conditions [8]. Wakeman and Tarleton[9] have found that varying the plI. and hence thezeta-potential, of anatase dispersions has a five­fold effect on llltration rate, Heinemann et a], [10]studied the effect of pH and ionic strength on therejection of whey protein and found that therejectionincreasedwith these two parameters. Theincreased in rejection was due to electrostaticinteraction between solute and membrane.

Fordham and Ladva [11] have studied the cross­flow filtration of bentonite suspensions bymeasuring the filtrate flux with time and postmortem measurements of the steady-state gelthickness. The steady-state flux was observed toincrease with increasing cross-flow velocitywhereas the gel thickness decreased, indicatingthat higher shear rates reduced the gel height, andthus lowered the resistance to fluid out flow.Measurements of the filtrate flux, showed that afteran initial period of flux decline, a steady-state fluxwas reached. In general, this steady-state value ISfound to increase with increasing applied cross­flow velocity. However, in a few cases e.g. Fischerand Raasch [12], Lu and Ju [13] and Wakemanand Tarleton [9], reduced values have also beenreported which are explained in terms of theselective deposition of fine partides into the filtergel.

Hydrophobicity of particles may affect theirdeposition on the membrane surface. Perssonet a], [14] have studied the fouling behaviour ofsilica on fourdifferentmicro filtration membranes.They observed that hydrophobic particlesaggregated more and formed a less dense gel layeron the membrane than hydrophilic partides did.They found that heavy fouling on membranesoccurred with mixtures of protein with hydrophilicparticles but not with hydrophobic partides andprotein mixture.

There is considerable experimental evidencethat protein adsorption within the pores ofultrafiltration reducesthe effectivemembrane poresize and therefore alter the membrane transportproperties [15-18]. The adsorption phenomenondepends on the extent and strength of theinteraction governed by the physical and chemicalproperties of the protein, adsorbent surface andsolvent (e.g., pH and ionic strength). Norde et a],

[19] found that protein adsorption is stronglyaffected by electrostatic interactions, withmaximum adsorption attained near the proteinisoelectric point, i.e. at the point at which theprotein is electri.cally neutral. The adsorptiondependency on plI is largely determined by therelative contribution of intramolecularhydrophobic interaction to the stabilization of theproteinstructurein solution [19]. Bowen and Gan[20] studied bovine serum albumin (BSA)adsorptionon polyvinylidene fluoridemembranes,and found that the adsorption was rapid initially,reaching equilibrium after 30 min to 3 hr. Theadsorption isotherms indicated two differentadsorption sites, one of high affinity and the otherof low affinity. At pH ~ 7, they did not find anyadsorption at all. Exposure of track etchedpolycarbonate or mica membranes to BSA leadsquickly to an irreversible decrease in pore radiusroughly comparable to that expected for anadsorbed monolayer [17, 18, 21], Schultz et a1.[22] found little or no change in pore radius withdextran solution but substantial changes withsolutions ofvarious proteins. Matthtasson [23]studied the adsorption ofBSA in cellulose acetate,polysulphone and polyamide membranes, Theyfound that adsorption was greatest On thehydrophobic polysulphone membranes and wasleast on the hydrophilic cellulose acetatemembranes. Protein adsorption increases withincreasing bulk protein concentration andsubstantially reduces the membrane hydraulicpermeability,

Ultrafiltrationmembraneshave an asymmetricstructure whereby the hydrodynamic resistanceis mainly determined within a small layer of thetotal membrane thickness [1], Robertson andZydney [24] have studied the BSA adsorption inthe NOVA and OMEGA polyethersulphonemembranes. They have found that even thoughadsorption in the skin of an asymmetric UFmembranes is only a small fraction of the total (skin,substructure, and matrix adsorption), it is this skinadsorption that has such profound effects on themembrane hydraulic permeability as well asmembrane transport properties in general.

The effect of hydrodynamic conditions on theadsorption of macromolecules onto the surfacewere reported by some workers, Fuller and Lee

12 A. Beicha, R. Zaarnouch & N. M. Sulalman

Using these boundary conditions, Eq. (1) isintegrated over the boundary layer of thickness8 over which the concentration varies, to give:

(1)

(2)

(3)

Y=O---7C=Cb }

Y = 8 ---7 C = Cg

(c -C J DJ = kin g P. k = ~Cb -Cp ' 8'

dCJxC =JxC-D-

P dy

In the conventional gel-polarization model, theconcentration of the gel layer, CfJ' is assumedconstant, Thus, boundary conditions at steady stateafter gel layer formation are:

are conveyed by permeate flux to the membranesurface, and a portion of them permeate throughthe membrane, but the rest ofthem are rejected bythe membrane and diffuse back into the bulksolution. At steady state, the quantity of solutesconveyed to the membrane is equal to the sum ofthose that permeate through the membrane andthat which diffuse back.

where J is the flux through the membrane, Cpthe permeate concentration, Cb the bulk streamconcentration, Cg the gel concentration at themembrane surface, D the diffusivity coefficient,and k the mass transfer coefficient. When theconcentration of permeate tends to zero), Eq. (3)becomes:

J = kIn ( ~: J (4)

Equation (4) shows that no solutes are able topass through the membrane and the flux of solvent(water) is dependent only on the characteristicsof D, Cg and the boundary layer thickness 8.Equation (4) is ordinarily used for the analysis ofultrafiltration fluxes. Mass transfer coefficient inlaminar flow can be evaluated by using thefollowing Leveque solution applicable for all thin­channel lengths [30]:

3.1 Gel-polarization Film Model

3.0 ULTRAFILTRATION MODELS

[25] reported that hydrodynamic forces candramatically increase the rate of desorption inpolymer systems which are otherwise irreversiblyadsorbed under no flow conditions. At the highestvelocity gradient and for the higher molecularweights, the film thickness was observed todecrease. Using the membrane hydraulicpermeability measurements before and afterprotein adsorption together with proteinadsorption experimental data. Robertson andZydney [24] indicated that protein adsorptionshould have substantially different effects onmembrane transport characteristics, dependingupon the pore size of the membrane. For the largerpore membranes, the pore size is reduced by thesize of monolayer adsorption (size of an adsorbedalbumin molecule) and for smaller poremembranes, the pores will be blocked partially orcompletely.

The solvent always plays a double role, affectingboth lateral interaction between the adsorbatemolecules and determining the effectiveinteraction between the surface and the adsorbate.This means that they adsorb strongly from somesolvents, and not from others [26]. The flux declinealso depends on temperature and the viscous flowinside the pores. An increase in the temperatureby one Kelvin degree decreases the viscosity by2-3%, such that the flux would in that case, increasecorrespondingly [27]. A salt addition increasedthe water permeability by 2-6%, which wasexplained by electroviscous effects such as reducedstreaming potentials and less interference fromoverlapping electrical double layers [28]. The poorseparation efficiency may also be due to wide poresize distribution in the membrane. It may alsoinvolve concentration polarization, resistanceassociated with protein deposition, adsorption,and pore plugging, or a combination of thesefactors [1].

In the case of ultrafiltration, the solutes aremacromolecules or colloids which tend to formthe gel layer on the membrane surface [29] . Solutes

(d )0.33 d

Sh = 1.62 ReSc l for 100 < ReSc l < 5000

(5)

Permeate Flux in Ultrafiltration Membrane: A Review 13

where Sh is the Sherwood number, Re theReynolds number, Sc the Schmidt number, dhthe equivalent hydraulic diameter and L thechannel length. Hence from Eq. (5), we obtain themass transfer coefficient:

ultrafiltration system, with the coefficient ofdiffusion calculated at the gel concentration, CgoProbstein et a1. [38] used this expression ofpermeate flux in determining the diffusivity andgel concentration in macromolecular solutions.Nakao et a1. [39] have analyzed many ultraftltrateflux data based on this treatment of the gelpolarization model. In their analyses, they foundthat Cg determined by the extrapolation onlogarithm concentration axis, depended mostdefinitely on the type of apparatus or modules andthe experimental conditions, They discovered thata solution, whose concentration was made equalto Cgsometimes had fluidity, in sharp contrast to anon-fluid gel-like state, and that the ultrafiltrationflux did not become zero when this solution wasused as a feed,They were able to measure Cgdirectlyand found a relationship between Rg and Cg of1.7 power. regardless of the kind of solute used.They were of the opinion that Cg has no physicalsignificance.

In orderto enhance the accuracy of predictionof limiting flux during filtration of particulatesuspensions, a number of mechanisms other thenBrownian back diffusion have been proposed.Zydney and Colton [40J proposed thatthe steady­state permeate flux for cross-flow microfiltrationcould be predicted by the classical gel-polarizationmodel (Eq. (4), In place ofthe Stokes-Einsteinparticle diffusivity, Do ~ KT16J!/la, they used a"shear enhanced" particle diffusivity based uponthe experiments of Eckstein et a1. [41]. A constantdiffusion coefficient of D ~ 0,03a'y, where a isparticle radius, was used. Due to the approximateexpression of shear-diffusivity, and invalidity ofLeveque solution for mass transfer (Eq, (6») indilute solution [42], the expression predictssteady-state permeate fluxes that are an order ofmagnitude lower than the measured fluxes [35].

Among the arguments against the gelpolarization film model are, the gel polarizationmodel itself does not specify what mass transfercoefficient, k, should be used, the inability toaccount for the anomalous low and high flux incolloidal suspensions, and flux dependence on thefeed velocity varies with the type of solute and themembrane system [33],

There are other models which proposed toexplain the gel formation on the membranesurface

(6)

(7)

(8)

(, JO

'33k ~ 0,816 ZD 2

( J0.5 DO 66

k ~ 0.664 i V 0 17

where it is the fluid shear rate at the membranesurface. For higher power dependence of flux onfeed velocity than that indicated by Eq. (5), thecorrelation of Grober et a1. [31] can be usedinstead. This correction is applicable for velocityand concentration profiles both developing downthe full channel length.

Sh ~ 0,664(Re? JSScO,33

where uzis the feed velocity and n the kinematicviscosity. Using values calculated from Eqs. (6)or (8) and the value of CIf' the concentration atwhich the flux drops to zero, Eqs, (4) predictssteady-permeate flux less satisfactorily comparedwith experimental flux [32, 33], Porter [33]explained that the discrepancy between thepredicted values given by Eq. (4) and the data isdue in fact to the back-diffusion of particles awayfrom the membrane which is supplemented by alateral migration of particles due to inertial lift orthe so called "tubular-pinch effect". Unfortunately,it was found from hydrodynamic calculations,that the inertial lift velocity is often less than thepermeate velocity in typical cross-flow micro­filtration systems [34, 35], On the other hand,Altena and Belfor [36] showed that for solutionsof smaller particles, the permeation drag forcedominates compared to inertial force. Shen andProbstein [37] suggest that the discrepancybetween Eq. (4) and experimental data is due tothe dependence of concentration on diffusion andviscosity, They incorporated the dependence ofconcentration on D in the steady-stateconcentration diffusion equation and arrived atan equation similar to Blatt's equation for masstransfer (Eq. (6» in parallel channel iaminar

14 A. Belcha, R. Zaamouch & N. M. Sulaiman

(9)

based on concentration polarization. Le andHowell [43] developed a model for ultrafiltrationto predict steady-state permeate fiux which isdescribed as the pore-blocking model. Theyhave shown that their model can replace theclassical gel-polarization model. Their modelpostulated that the limiting concentrationoccurring at the membrane surface as TMP israised, is a result of the interaction between thesolute and the surface. Their model again needsthe mass transfer coefficient, k, as was in the caseof the previous gel-concentration model, limitingconcentration and another parameter relating thepore size distribution of membrane to soluteparticle size.

Trettin and Doshi [42] integrated theconcentration diffusion equation in the case of adead-end system by assuming a concentrationprofile in the boundary layer and that the solutegel concentration is reached instantaneously at themembrane surface.

Bowen and Jenner [44J have deveioped arigorous dynamic mathematical model forpredicting the rate of uitraflitration in dead-endsystem. The model is based on Darcy's equationand the gel formed was compressibie. The locals-pecific resistances are calculated frominterparticle interaction approach. The model isonly valid for charged colloidal dispersions.

Davis and Leighton [34] have presented atheory which describes the transport of aconcentrated layer of particles alonga porous wallunder laminar flow conditions. A shear inducedhydrodynamic diffusion mechanism was proposedto describe the lateral migration of particles awayfrom the porous wall as the layer is sheared. Atsteady-state, particle diffusion within the iayer isbalanced by the connective flux of particles towardthe porous wali due to the fluid into the wall. Theirmodel predicts the nonlinear veiocity andconcentration profiles within the sheared particlelayer, as well as the layer thickness and the wallconcentration.

Romero and Davis [45] have proposed atheoretical model of crossflow microfiltrationwhich includes the time dependent decltne ofpermeate flux due to particle iayer build up. Themodei is based on the shear induced hydrodynamicdiffusion mechanism of particle motion within a

concentrated flowing layer near the membranesurface, balancing the consecutively driven flux ofparticles toward the membrane surface.

Another genre of models exists that may bedescribed as particle adhesion models, e.g. thecriticai flow modei ofRautenbach and Schock [46],which assumes that gel deposition ceases whenthe feed velocity exceeds some multiple of theflitrate flux, the constant of proportionality beingdetermined empirically for a given suspension.

3.2 Concentration Polarization-osmoticPressure Model

These models consider the flux as being limitedby the high osmotic pressure arising in theconcentration-polarized layer close to themembrane surface, with no gel proposed. Thewater flux through a membrane of constantpermeability is reported by Merten [47]:

]~ _!':E~_I1"--fiRm

where Rm is the membrane resistance, II thesolvent viscosity and Mr ~ ,,(cm ) - ,,(c;! with theconcentrations em and cp at the membrane surfaceand in the permeate, respectively. The osmoticpressure n is often represented in terms of apolynomial

" ~ 81 C + 8 2C' + 8 3C' (10)

where 81 is the coefficient in van't Hoffs law forinfinitely dilute solutions and 82' 83 represent thenon-ideality of the solution.

Goldsmith [48] used Eq. (3) together withEq. (9) in his analysis, and showed that flux islimited by mass transfer conditions on the feedsolution of macromolecules side of the membrane(concentration polarization). Although based onmolecular weight considerations, the osmoticpressures of macromolecule solutions wouldappear to be insignificant.

Brian [49J integrated numerically theconvective-diffusion equation at steady-statecondition with osmotic pressure controlledpermeate flux (Eq. (9». He incorporated the axiaiand normal velocities expressions developed firstlyby Berman [50] for the Newtonian flow field withnormal velocity uniform along the channel iength.

Permeate Flux in Ultrafiltration Membrane: A Review 15

By taking the same profiles of velocities used byBrian and assuming a concentration profile,Leungand Probstein 151] proceeded to integrate thesteady-state convection-diffusion equation overthe boundary layer. They have compared theirintegral solution to the finite difference solutionused by Brian and found agreement.

Bowen et el. [52] have followed the sameapproach taken by Leung and Probstein [51] forpredicting the rate of cross-flow membraneultrafiltration in a rectangular channel with oneporous wall. However they focused on a detaileddescription of the dependence of both osmoticpressure and gradient diffusion coefficient onconcentration and physicochemical parametersdeveloped originally to describe the dead-endultrafiltration of colloids [44j.

Vilker et aJ. [53] solved the unsteady convectivediffusion equation in the case of unstirred cellsystem, together with osmotic pressure-typeboundary condition at the membrane surface(Eq. (9». Regular perturbation theory was used todescribe concentration polarization duringultrafiltration of albumin solutions for the case ofa highly rejecting membrane. They showed thatthe flux diminishes like the inverse square root oftime.

3.3 Gel Growth Rate Models

Many different models have been proposed topredict permeate flux during ultrafiltration andmicrofiltration, The gel layer and membrane maybe considered as two resistances in series, and thepermeate flux up: can be calculated from theexpression:

u = ~E-__ (11)P Rm + Rg

A gel layer is formed by deposition on themembrane surface and affects the flux as aresistance in series with the membrane resistance.The resistance ofthe deposit, Rgo may be expressedas follows:

(12)

where Rg is the specific resistance of deposit, and1the gel layer thickness. The gel growth rate isgiven by the static gel filtration theory:

dl cb-- = ------ u (13)dt Eg-Cb p

where Eb is the solidosity of bulk stream, cg thesolidosity of gel formed, up the permeate flux,and t the time. This law was established todescribe the dynamic ultrafiltration in dead­ended system. The theory assumes that solidosityof solution over gel is constant and equal to bulksolidosity,cb' and that the solidosity of gelformed, Ego is also constant.

Equation (13) indicates that the gel thicknesswill grow continuously with time resulting in adecrease of permeate flux. Thus the flux in dead­ended systems is so small as to be virtually nonexistent. However in cross-flow filtration thesituation is different, Porter [33] and others havereported an increase in the permeation nux withincreasing tangential shear, indicating that a highshear rate is effective in reducing the gel layerthickness. Some researchers have hypothesizedthat the gel layer accumulates only until thehydrodynamic shear exerted by the flow ofsuspension causes the gel to flow tangentiallyalong the membrane surface at a rate. whichbalances the deposition of particles.

In order to account for the effectof feed velocityon deposition of particles on gel layer, the so calledterm "function probability" must be incorporatedin expression (13), that can be rewritten as follows:

dl Ebup-- = ------ y (14)dt Cg-Eb

where, y is the fraction of particles transportedto membrane surface which achieved deposition.This function, Yo decreases with time until it reachesthe zero value and hence the steady-state is reached.By defining the forces acting on spherical particletransported to the vicinity of the gel suspensioninterface with a protrusion of finite height, alongthe direction of the main flow and along thedirection of permeation, and by application of'principal of moments, Stamatakis and Chi [54]obtained the condition for which the sphericalparticle remains static and finally arrived at oneexpression of y. The investigators were able topredict the dynamic permeate flux data of Murkesand Carlson [55] on the cross-flow filtration offinely dispersed kaolin at a bulk soudosity of

16 A. Beicha, R. Zaamouch & N. M. Sulaiman

(16)

0.0033. The unknown gel solidosity andpermeability were obtained by curve fitting. Bydirect experimental observation of particlesdeposition on the gelsurface, Makley and Sherman[56] defined the function, y, by measuring theangle between the impacting particle trajectoriesand the gel surface at which rolling commences.As expected, they also reported the thinning of thegel layerat high cross-flowvelocity, but surprisinglyobserved a continuous decrease in permeate fluxwith increasing velocity, They indicated thatthinner gels produced at higher cross-flowvelocities offer more hydraulic resistance thanthicker onesformed at lower velocities and relatedthe phenomenon to the packing density of particlesin the gel layer. The model contained an adjustableparameter R' (ratio of gel resistance formed at highcross-flowvelocitywith negligibleflux to that withno cross-flowvelocity applied) which was fitted tothe data. Noticeable variations between predictedand experimental fluxes were observed especiallyat long operating time.

The common assumption used amonginvestigators to model the process was a constantsolidosity of suspension above the gel layer,which was taken to be equal to that of the bulkvalue. In addition to the feed velocity, the soluteconcentration in the feed solution would also affectthe extent of gel layer formation, a parameter thatwas not considered in most of the models citedearlier. Furthermore, using the bulk solidosityvalue to characterize the solution just above thegel layer may only be valid for dilute systems. Forconcentrated solutions, the development ofconcentration polarization would result in thesolidosity value at the interface to be much higherthan the bulk soIidosity. Sulaiman et a1. [57] andBeicha et a1. [58] developed a model which couplesthe formation of a gel layer on the membranesurface and the presences of a polarized layerabove the gel. The model was compared withexperimental permeate fluxes obtained from theultrafiltration of polyethyleneglycol (PEG) usingpolyethersulfone membrane (9000 MWCO). Themodel gave an excellent prediction ofthe permeatefluxes. However, for higher transmembranepressures the model over predicts the permeatefluxes. The same concept employed by Beichaet a1. [58] and Zaamouche et a1. [59] was used to

predict limiting flux when the permeate fluxbecomes independent of pressure during tubularultrafiltration. By combining the effects of theeffect of the feed velocity and the bulkconcentration, Beicha and Zaamouche [60]showed that the limiting permeate flux can beexpressed as:

(15)

where £Sf is the steady-state value of the averagesolidosity in the polarized layer, and Uo the feedvelocity. The constant A depends on the solute­membrane system. The limiting flux was found tobe proportional to the square root of the feedvelocity. For bulk concentrations near the criticalconcentration, the concentration polarization hasno effect on limiting flux. Its effect is morepronounced when increasing bulk concentrationbeyond the critical concentration at which the twostraight lines on the plot of permeate flux at steady­state versus In (bulk concentration) crossed eachother [60].

3.4 Adsorption Models

Pore restriction models assume that the fluxdecrease is due to the pore becoming narrowerbecause of adsorption, e.g.:

J1 (ra -61)4fa = ---;~C-

where J1 is the pure water flux of fouledmembrane, Jo the pure water flux of cleanedmembrane, IO the pore radius of cleanedmembrane, and 6r decrease in radius. Equation(16) which is based on the Hagen-Poiseuilleequation, indicates that the radius decrease due toadsorbed solute, is assumed to be smeared outequally throughout the pore length and that thepore radius is assumed to be sharply defined.

Simulations have been gradually adopted invirtually any branch of science, due to the gain intime and money they offer compared with practicalwork. Simulation is especially useful when oneconsiders mechanisms which are kinetically ratherthan thermodynamically limited.

An alternative way to study the adsorption­polarization interplay in ultrafiltration is presented

Permeate Flux in Ultrafiltration Membrane: A Review 17

by Gekas et al. [61]. They incorporated the effectsof membrane fouling due to the adsorption ofsolute to the generalized diffusion equation forthe polarized boundary layer in a dead-endultrafiltration processusing platemembranes. Thisfouling mechanism, which is time dependent, isparticularly applicable to the ultrafiltration ofproteins.Using the model, these investigatorswereable to assess individually as well as collectivelythe roles of concentration polarization andadsorption on the profiles of permeate flux. Theeffectsof cross-flowvelocity were only investigatedindirectly by varying the values of mass transfercoefficient.

Flora [62] simulated the flux decline due tosurface fouling and found good agreement withfouling experiments performed in an unstirrcdultrafiltration cell. Doshi [63] has developed amodel for the interplay of adsorption andpolarization in an unstirred batch cell. His aimwas to identify factors limiting flux in theultrafiltrationof macromolecules.

4.0 CURRENT & FUTUREDEVELOPMENTS

The decision as to which of the filtration modelsis the most correct in predicting flux values is amatter of difficulty and a certain amount ofcontroversy.Eachmodel appears to be consistentwith selected experimental data. A large majorityof modeling works on ultrafiltration has been bymodels 'based on the concentration-diffusionequation. A good model will lead towards belterprediction and optimization of ultrafiltrationmembrane processes. The resistance modelshave the advantages of including a description ofnon-Newtonian polarized layers. A theoreticalbasis has to be established for the governing modelequations to enable furtherdevelopments.

ACKNOWLEDGEMENT

The authors are grateful for the financialsupport from ministry for higher education andscientific research through national programs ofresearch.

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