Desalination 346 (2014) 80–90

Contents lists available at ScienceDirect

Desalination

j ourna l homepage: www.e lsev ie r .com/ locate /desa l

Modeling of suspension fouling in nanofiltration

F. Faridirad a, Z. Zourmand a, N. Kasiri a,⁎, M. Kazemi Moghaddam b, T. Mohammadi b

a Computer Aided Process Engineering (CAPE) Lab, School of Chemical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran 1684613114, Iranb Research Centre for Membrane Separation Processes, School of Chemical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran 1684613114, Iran

H I G H L I G H T S

• Dissolution and sedimentation of solute particles during NF were considered.• Considering the fouling of solutes during NF of suspensions• Derive a new mathematical model which calculates permeate flux• Verifying the new model with experimental data• Comparing the performance of the new model with other existing models

⁎ Corresponding author.E-mail address: capepub@cape.iust.ac.ir (N. Kasiri).

http://dx.doi.org/10.1016/j.desal.2014.05.0140011-9164/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 23 March 2014Received in revised form 6 May 2014Accepted 7 May 2014Available online 2 June 2014

Keywords:Membrane processMathematical modelNanofiltrationDissolutionCross flow

Prediction of membrane process performance using experimental and mathematical models facilitatesoptimization of membrane processes and their operating conditions. On the other hand investigation ofnanofiltration process performance over time and estimation of the time for recovery are very essential in themembrane industry. In this work a mathematical model considering membrane resistance changes and alsodissolution of deposited particles in nanofiltration feed, for a cross flow nanofiltration system was developed.The validation of thismodelwith experimental data demonstrated a good agreement. At a constant concentrationof 0.2 g/L error of the presentmodels varied from 3% (at the pressure of 1 bar) up to about 3.8% (at the pressure of2.9 bar) while at a constant pressure of 2 bar error varied from 2.45% (at the concentration of 0.1 g/L) up to about4.2% (at the concentration of 0.4 g/L). A comparison between thismodel and three other previousmodels proveda better performance for the present model with an average error of 3.25%.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

Nanofiltration is a membrane process for liquid separation whichhas been extended recently [1]. Reverse osmosis is replaced bynanofiltration in many applications, due to the fact that NF requiresless energy comparatively. Going by its specifications, nanofiltrationlays between the ultrafiltrationmembrane and RO.Membrane filtrationprocesses, such as nanofiltration (NF) and reverse osmosis, play animportant role in the production of high quality reclaimed waterwhen small organic compounds, (e.g., pesticides, endocrine disruptorsand pharmaceutically active compounds) are to be removed frompolluted water [2]. The important application of NF is for watertreatment, waste water treatment and desalination. NF is a pressuredriven separation process which can separate particles based on theirsize and electrostatic interactions between them. For the unchargedmolecules, the dominant mechanism for separation is molecular sizewhile for the separation of ions with similar size, electrostatic forces

play a more important role [3]. NF membranes have pores of around 1nm [4], therefore they got higher water flux at lower pressurescompared to RO. The main problem in using NF technology is capacityreduction of industrial units caused by membrane fouling, particularlyduring inorganic separation [5,6], leading to membrane life reduction[7]. Inorganic fouling induced by concentration polarization and scaleformation is one of the major limitations of NF as in other membraneprocesses used in water treatment [6,8]. This necessitates foulingcontrol in membrane processing [9]. Suspension fouling is of particularinterest in themembrane field. Suspensions are defined as fine particleswhose characteristic size is roughly in the range of 1 nm–1 μm [10].In these processes flux decline is dependent on many factors such asmembrane characteristics & module geometry, feed condition, solutetype and also operating conditions [11]. Concentration polarizationrefers to the reversible accumulation of solute within a thin boundarylayer adjacent to the membrane surface. Membrane fouling can beirreversible with solute adsorption on or in the membrane pore walls,leading to complete or partial pore blocking [12]. Formation of cakelayer over the membrane surface can determine rejection propertiesof the system since the deposited layer will act as a “secondary”

81F. Faridirad et al. / Desalination 346 (2014) 80–90

membrane prior to the “real” membrane or support material [13].Fouling importance has initiated many efforts for modeling flux declineprofile during NF [14,15]. To improve fouling models many dynamicmodels have been derived [4,11,16]. Theoretical models are not capableof predicting the filtration flux precisely, due to the simplifying assump-tions required to enable their analytical solution [17]. Various fluxdecline mechanisms are studied which are generally governed by cakeor Gel layer [18–20], resistance in series and pore blocking [21,22].Ballet et al. studied experimentally the effect of feed pressure, ionicstrength, concentration, and pH on the retention of phosphate anions[23]. Li et al. investigated the effects of pressure, flow rate on flux, andretention in seawater desalination process [24]. Darvishmanesh et al.developed a new semi-empirical model based on the traditional solu-tion diffusion with imperfection model for solvent resistant NF [25].This model demonstrated a good prediction for the flux of solventthrough the membrane. Fadaei et al. developed a mass transfer modelto predict ion transport through the NF membrane to account for theconcentration polarization phenomenon and its influence on ion sepa-ration [26]. They used CFD techniques and successfully predicted thelocal concentration of ions; permeate flux, and rejection of ions in arectangular cross flow. In the field of CFD many other works havebeen done such as modeling the impacts of feed spacer geometry onNF processes by Guillen and Hoek [27] and CFD modeling of porousmembranes by Pak et.al [28]. Field et al. introduced the concept ofcritical flux for microfiltration, stating that there is a permeate fluxbelowwhich fouling is not observed. It was a very powerful optimizationtool for this kind of separation operation. It was possible then to identifya critical flux for UF and NF membranes [29].

The purpose of this research is to introduce a new fouling modelbased on cake formation to describe the flux decline caused by inorganicsolutes. Fouling basedmodels are appropriate for predicting thefiltrationand clean up time duration. Inmodel development the effect ofmany pa-rameters such as operating conditions, cake filtration characteristics,membrane permeability, cross flow velocity, specific cake resistanceand other experimental parameters on flux reduction has been investi-gated. The developed model was then validated against experimentaldata obtained from water–oil solution NF. Finally model performancewas compared with three other models developed by Lihan Huang [30]and Wu [31], as well as the cake or Gel Layer Filtration model [32]. Inthese models the fact of dissolution of solutes, transmission of particlesthrough the membrane, membrane resistance changes, and also the ef-fect of operating parameters such as cross flow velocity and temperaturewere neglected. Considering the solute dissolution,membrane resistancechanges and also the effect of operating conditions lead to a model withbetter performance and better agreement with experimental data.

2. Model development

In the NF process feed enters from one side of the cell, having passedover the membrane surface, leaves the cell from the other side withpermeate passing through the membrane and exiting from a secondexit of the chamber. Cross flow arrangement reduces solute particle

Table 1Estimated parameters of the developed model, obtained from GA, using 70% of experimental d

Pressure ρ α b

c = 0.2 g/L P = 1 bar 2300 1011 0.P = 2 bar 2400 1012

P = 2.4 bar 2450 1.4 × 1012

P = 2.9 bar 2500 1013

Concentration ρ α b

P = 2 bar c = 0.1 g/L 2350 1011 0.c = 0.2 g/L 2400 1.4 × 1011

c = 0.3 g/L 2400 1.6 × 1011

c = 0.4 g/L 2450 1.8 × 1011

accumulation on the membrane leading to reduced cake formationand fouling [33].Membrane fouling is related to crossflownanofiltrationvelocity with larger flow velocities causing more turbulence hence lessfouling [34].

One of the theories for explaining membrane fouling is cakefiltration model which was developed by Hoek et al. to evaluatereverse osmosis and nanofiltration processes [35,36]. According tothis model flux can be explained as:

J tð Þ ¼ ΔpRm þ Rcp þ Rc tð Þ : ð1Þ

In which J(t) is solvent (permeate) flux which is a function ofeffective pressure applied on the membrane surface (Δp), hydraulicmembrane resistance (Rm) and cake layer resistance (Rc) and also theresistance related to concentration polarization layer (Rcp). Althoughconcentration polarization layer effect is significant at the very earlystages of the separation process, this gradually fades away when poreblockage takes over and due to which concentration polarization hasbeen neglected in the model. Eq. (1) is therefore simplified to:

J tð Þ ¼ ΔpRm þ Rc tð Þ : ð2Þ

Cake layer resistance is calculated through Eq. (3) using specific cakeresistance (α).

Rc ¼ αMd ð3Þ

In Eq. (3),Md is accumulatedmass per surface unit. The constant α isrelated tomembrane porosity through Carman–Kozeny's equation [36]:

α ¼ 45μ 1−εð Þ2ρa2pε

3 : ð4Þ

Here μ is water (solvent) viscosity, ε is cake porosity, ρ is cakedensity and ap is particle radius. If temperature is fixed, viscositywould remain constant. Despite this, Arrhenius equation can be usedto take into consideration the effect of temperature on permeate flux.

μ ¼ μ0 � exp − ERT

� �ð5Þ

Pressure difference is calculated through Eq. (6) by using purewater permeate.

Δp ¼ J0Rm ð6Þ

In this model membrane fouling is modeled as a dynamic process.The model is based on the following assumptions:

1- Due to the smallmembrane length, a uniform cake layer of a uniformresistance will be formed through the membrane surface.

ata.

a M1 M2 M3

00027 3.42 × 109 0.03 0.00004 10000.07 0.00002 11000.1 0.00001 11800.1 0.000001 1200

a M1 M2 M3

000087 3.34 × 109 1000 0.000006 0.061070 0.00003 0.061100 0.00003 0.051130 0.00002 0.4

Fig. 1. Variation of present model parameters with pressure.

82 F. Faridirad et al. / Desalination 346 (2014) 80–90

2- Solute particle sedimentation on the membrane surface is pro-portional to the volumetric filtrate rate passing through it [16].

3- Sedimentation and dissolution happen simultaneously.4- Fouling will occur when solute concentration is equal to the

saturation concentration. Before this, neither sedimentation nordissolution will take place.

5- Particles can pass through the pores and therefore there are somesolute particles in the permeate side. By passing the particlesthrough the membrane pores, some particles sediment withinthem, and therefore the diameter of pores reduces which causesmembrane resistance to be changed. Thus membrane resistance

Fig. 2. Variation of present model p

changes over time. As time passes and cake formation takes placeleading to cake ΔP build up passing flux diminishes leading tocesar of further particle sedimentation within membrane pores.

6- Cake layer resistance is time dependent.

As time passes membrane resistance to flow increases while rateof its increase over time or the increasing gradient is reduced. As thisbehavior could mathematically be represented by a logarithmicfunction, in the present work this is proposed to be presented bythe following function.

Rm tð Þ ¼ Rm 0ð Þ � a ln b� tð Þð Þ ð7Þ

arameters with concentration.

Fig. 3. Porosity changes of the cake layer with pressure in present model.

Fig. 4. Porosity changes of the cake layer with concentration in present model.

Fig. 5.Model and experimental data (fl

83F. Faridirad et al. / Desalination 346 (2014) 80–90

In Eq. (7), a and b are constants that could be determined usingexperimental data.

According to the previous assumptions solute sedimentation rate isdescribed as [19]:

Rdep ¼ M1Qp: ð8Þ

And solute dissolution rate is calculated through Eq. (9) [37].

Rdis ¼ M2Q c ð9Þ

Eq. (10) is used to describe particles passing through themembrane.

passed ¼ M3ð Þ2�

Rm tð Þð10Þ

In Eqs. (8), (9) and (10), M1, M2 and M3 are deposition, dissolutionand transmission constants respectively. Total accumulation of solute(M:) can be calculated through Eq. (11).

M¼ M1Qp−M2Q c−M3ð Þ2

�Rm tð Þ

¼ ρAdldt

ð11Þ

In Eq. (11), ρ is cake density, A is effective surface of themembrane and dl

dt is the cake layer growth rate. Permeate flux andrate, are determined through Eqs. (12) and (13) respectively.

J tð Þ ¼ J0 � Rm tð ÞRm tð Þ þ ρ� l tð Þ � α

ð12Þ

Qp tð Þ ¼ Q0

1þ k� l tð Þa ln b� tð Þð Þ

ð13Þ

Here l(t) is cakefiltration thickness and k ¼ Q0�ραΔP is themass transfer

resistance constant.

ux vs. time) at different pressures.

Fig. 6. Error variation of developed model versus pressure. Fig. 8. Error variation of developed model versus concentration.

84 F. Faridirad et al. / Desalination 346 (2014) 80–90

A mass balance for flows gives Eq. (14).

ρAdldt

¼ M1Q0

1þ kl tð Þa ln b� tð Þ

0BBB@

1CCCA−M2 Q− Q0

1þ kl tð Þa ln b� tð Þ

0BBB@

1CCCA− M3ð Þ2

Rm;0 a ln b� tð Þð Þ

ð14ÞTherefore the amount of cake layer thickness could be deter-

mined through solving Eq. (14) and the flux quantity will be calculatedafterwards.

3. Previous models

3.1. Lihan Huang model

In this model it is assumed that fouling is a dynamic process whichstarts with pore blocking and then continues with cake formation onthe membrane surface [38]. The initial permeate flux is calculatedthrough Eq. (15). In Eq. (15) V0, V1 and V2 are the first infiltratedvolumes and Δt is the time step between these measurements. In thismodel Eq. (16) is used to calculate flux. In Eq. (16) α is a function

Fig. 7. A comparison between developed model and e

of particle concentration and β is a constant which is determinedexperimentally [30].

J0 ¼ −3V0 þ 4V1−V2

2AΔtð15Þ

J ¼ 1A:d V−Vcð Þd t−tcð Þ ¼ ΔPαβ

μ1

V−Vcð Þ1α−1ð16Þ

3.2. Gel Layer Filtration model [32,39]

In this model it is assumed that the size of particles is thesame hence the porosity can be assumed constant. The transientflux is described by Eq. (17) [40]. In Eq. (17) solvent flux is a functionof TMP, Δp(=ΔP − Δπm), hydraulic resistance of the membrane,Rm(=ΔP/μ × Jw), and cake layer resistance. At constant pressurethe flux is calculated only through multiplying specific cake resistance(rc) and the mass of transient layer per surface unit (Mc), as it is shownin Eq. (18). Specific cake resistance is described through Carman–Kozeny's equation which is shown in Eq. (19). μ0 is solvent viscosity, ε is

xperimental data flux at different concentrations.

Fig. 9. The effect of time passage on cake thickness (a) and flux (b) with V ms

� �as parameter.

Fig. 10. The effect of time passage on cake thickness (a) and flux (b) with P (bar) as parameter.

85F. Faridirad et al. / Desalination 346 (2014) 80–90

cake layer porosity, dp is particle diameter and ρc is particle density.

J tð Þ ¼ ΔpRm þ Rc tð Þ ð17Þ

Rc tð Þ ¼ rcMc tð Þ ¼ 45μ0 1−εð Þρcd

2pε

3

" #Mc tð Þ ð18Þ

rc ¼45μ0 1−εð Þρcd

2pε

3

" #ð19Þ

3.3. Wu model

According to this model at the beginning of the process,flux decline is proportional to flux according to Eq. (20) [40].

Fig. 11. The effect of time passage on cake thickn

This model has two constants which are determined throughexperimental data.

dJdt

¼ −kP exp −k f tð Þ � J ð20Þ

4. Results and discussions

4.1. Parameter estimation

Experimental data for water–oil feed which has been obtained fromthe experimental work carried out by T. Mohammadi et al. [41] havebeen carried out at 4 different pressures and concentrations assuming

ess (a) and flux (b) withM1 as parameter.

Fig. 12. The variation of cake thickness (a) and flux (b) versus time with M2 as parameter.

86 F. Faridirad et al. / Desalination 346 (2014) 80–90

a constant temperature of 30 °C. The membrane made of polysulfonewas produced by DOW of Denmark. Of course the membranes whichwere used in this reference were ultrafiltration membranes, but dueto their pore size that is close enough to the range of nanofiltrationmembrane size, using the data obtained from this paper does notmake any significant problem in our work. The membrane surface was28.64 cm2 and the cross flow velocity was 1 cm/s. The concentratedflow which goes out from the cell and the permeated flow from themembrane are returned to the feed tank tomake the feed concentration

Fig. 13. The variation of cake thickness (a) and

Fig. 14. The variation of cake thickness (a) and

constant. The permeate fluxes were measured with an accuracyof 0.001 g.

Flow system was laminar according to the following calculations:

Re ¼ ρuDh

μDh ¼ 4rh

rh ¼ 4� 0:0030� 0:03352� 0:0030þ 0:0335ð Þ ¼ 0:0055

flux (b) versus time with M3 as parameter.

flux (b) versus time with α as parameter.

Fig. 15. A comparison of different model flux predictions at 4 different pressures (present model and previous models).

87F. Faridirad et al. / Desalination 346 (2014) 80–90

u ¼ 0:01ms

→ Re ¼ 220:

In the first part, experiments have been done at 1, 2, 2.4, and 2.9 barwith a constant concentration of 0.2 g/L. In the second part tests havebeen carried out at 2 bar and at different concentrations of 0.1, 0.2, 0.3and 0.4 g/L. Oil density was set equal to 890 kg/m3. The parametersof the model have been calculated from experimental data using theGenetic Algorithm method [42,43]. In the GA optimization method70% of the data are used for fitting the best values to these parameters,the other 30% for goodness of fit evaluation and validation, thusproviding the best fit given the data circumstances. The resultshave been tabulated in Table 1.

Figs. 1 and 2, show the parameters of themodel versus pressure andconcentration respectively. From Fig. 1 it is clear that by increasing theamount of pressure M1, M3 and α increase while M2 decreases. It isreasonable as higher pressures cause more particles to permeatewhich means larger M1 and M3, on the other hand this causes the cake

Fig. 16. A comparison of error at different pressures for different m

layer to be more resistant and therefore specific resistance to increase.But this increase leads to less particle dissolution and thus decreasesthe amount of M2. Concentration has had the same effect on modelparameters as well. By increasing the concentration a larger amountof solute finds a chance for passing through membrane pores andtherefore both sedimentation (M1) and transmission (M3) constantswill be increased as shown in Fig. 2. Also a larger amount of solutecausesmore resistant cake layer leading to a larger value for the specificcake resistance (α). The increase of specific cake resistance as a result ofthe concentration increase is demonstrated in Fig. 2 which also showsthat increasing the concentration decreases the dissolution constant asmore sedimentation takes place.

Porosity changes of the cake layer with pressure and concentrationare shown in Figs. 3 and 4. In reality by increasing the pressure thedriving force towards the membrane increases thus applying largerforce on the cake which causes it to become more compact reducingthe free volume within it. This phenomenon leads to less porositywhich is obvious from Fig. 3. Concentration has the same influence

odel flux predictions (present model and previous models).

Fig. 17. A comparison between different models at 4 different concentrations for flux prediction (present model and previous models).

88 F. Faridirad et al. / Desalination 346 (2014) 80–90

on porosity. By increasing the amount of solute, free space of the cakelayer will be decreased leading to less porosity as shown in Fig. 4.

4.2. Model validation and verification

Permeate flux variations with pressure as observed experimentallyand predicted by the model and their relative error are demonstratedin Figs. 5 and 6 respectively. This demonstrates good compatibility.

As is clear from Fig. 6 the model error increases slightly as thepressure increases. Error changes from 3% at P = 1 bar to about3.8% at P = 2.9 bar which is not significant. A comparison betweenexperimental data and model results has been done at differentconcentrations, which is shown in Fig. 7 again demonstratinga good agreement. Error variation at different concentrations isshown in Fig. 8.

As shown in Fig. 8, by increasing the concentration, error increasesfrom 2.45%, at a concentration of 0.1 g/L to 4.2% at a concentrationof 0.4 g/L.

Fig. 18. A comparison of error at different concentrations for different

4.3. Assessment of model parameters on membrane performance

The effect of different parameters on flux decline according tothe time is investigated in this section. The influence of parameterssuch as pressure, specific cake resistance, cross flow velocity, modelconstants and temperature on flux and fouling will be evaluated.

4.3.1. Cross flow velocityMore cross flow velocity causes more flow tension and therefore

reduces concentration polarization layer [44]. It also leads to moreturbulency in flowwhichprevents solute sedimentation andmembranefouling [5]. Cross flow velocity can be changed by feed flow rate. Fig. 9shows the effect of crossflow velocity on flux profile and cake thickness.In Fig. 9 crossflowvelocity changes from0.08m/s to 0.6m/swhile otherparameters remain constant. As can be seen, flux reduces over time(Fig. 9b) and at higher rates initially, while cake thickness increases(Fig. 9a) with both eventually becoming constant. The process of coursetends to a steady state status. By increasing the cross flow velocity, as it

model predictions of flux (present model and previous models).

89F. Faridirad et al. / Desalination 346 (2014) 80–90

is clear from Fig. 9, the final flux increases, while cake layer thicknessdecreases. It can also be seen from Fig. 9 that all the curves start fromthe same point. As can be seen the flux changes from 1:22� 10−5 m

s

� �,

when the cross flow velocity is 0.08 m/s, to 2:03� 10−5 ms

� �when the

velocity is 0.6 m/s. This fact can be explained as flow turbulency willincrease by increasing the cross flow velocity, which leads to less solutesedimentation on the membrane surface, smaller cake layer thicknessand therefore membrane fouling, which allows more solvent topermeate through the membrane and will increase the flux.

4.3.2. Transmembrane pressureTransmembrane pressure, TMP, is a pressure difference between

two sides of the membrane which has a direct effect on permeatedflux. Fig. 10 shows the effect of pressure increase on NF, when it ischanged from 20.7 kPa to 62.1 kPa. As the permeated flux is related totransmembrane pressure, the amount of solvent which passes throughthe membrane will be increased by increasing the pressure. This fact isdemonstrated in Fig. 10 where both permeated flux and cake layerthickness increase over time. As it can be seen in Fig. 10b eventuallyall systems yield to the same amount of final flux.

4.3.3. Sedimentation parameter (M1)Here the amount of solute which has been accumulated on the

membrane surface is equal to the difference between the amountwhich has been deposited and the amount which has been dissolvedin the feed solution, therefore the process is controlled by twoparameters M1 and M2 and it is important to evaluate the effect ofthese two parameters on the flux profile. Fig. 11 shows the effect ofsedimentation parameter on solvent flux (picture a) and cake thickness(picture b). As it can be seen from Fig. 11, by increasing the amountof M1, permeate flux decreases as the cake layer thickness increases.On the other hand larger sedimentation coefficient causes moresedimentation which causes less solvent to permeate through themembrane. It can be explained from Fig. 11 that the value of M1

does not affect the final flux but it has a great influence on theslope of flux diagram and increases the rate of flux reduction.

4.3.4. Dissolution parameter (M2)It was mentioned before thatM2 is the dissolution parameter of the

solutewhich has been deposited on themembrane surface. The effect ofthis parameter on the flux of permeated solvent (picture b) and cakelayer thickness (picture a) is shown in Fig. 12. Here M2 varies from anear zero value of 0.000001 to 0.0002. It can be deduced from Fig. 12that by increasingM2 the amount of dissolution solutes in feed solutionincreases and therefore the amount of sedimentation on membranesurface or cake layer thickness decreases. Sedimentation and alsofouling reduction, cause more solvent to permeate. This is apparentlyclear from Fig. 12 (picture b).

4.3.5. Transmission parameter (M3)This parameter denotes particle transmission. The effect of this

term is shown in Fig. 13. It is clear that by increasing the amountof transmission parameter, more solution permeates through themembrane therefore permeate flux will be increased (picture a) whilecake layer thickness (picture b) will be decreased. As it is obvious allthe diagrams in both pictures a and b start from the same point butterminate at different values of flux and cake thickness.

4.3.6. Specific cake resistance (α)This parameter expresses the amount of cake layer resistancewhich

is formed on the membrane surface. Therefore it is obvious that byincreasing the amount of this parameter its effect on solvent fluxis more, allowing a lesser amount of solvent to pass through themembrane leading to a decrease in the amount of cake layer thickness.In Fig. 14 the effect of specific cake resistance on permeated flux(picture b) and cake layer thickness (picture a) is shown. Here specific

cake resistance was changed from 50� 1011 1�h to 12� 1013 1

�h .

As it is shown in Fig. 14 by increasing α the amount of permeated fluxis decreased leading to a shortening of the transient period. By increas-ing the amount of α, cake layer resistance increases leading to lesssolvent permeation. Another point is that by increasing α permeatedflux reaches to its constant amount in a shorter time. Also from Fig. 14it is clear that specific cake resistant does not affect the initial and finalflux and in the long run all diagrams become close together and theirfinal flux is similar. The cake layer thickness reduction is shown inFig. 14 (picture a). Other than the influence on flux, by increasing theamount of specific cake resistance the slope of the diagrams is increasedwhich causes the system to take longer to become a steady state.

4.4. The comparison of present model against previous models

The developed model has an acceptable performance with anaverage error of 3.2%. This model can predict the behavior of oil–watersystems in an appropriate way which is due to the realistic setof assumptions the model is based on for the oil–water system.A comparison between different models at various pressures hasbeen carried out with the results shown in Fig. 15.

According to the assumptions of the previous models and thepresented model it is apparently clear that this model performs betterin flux prediction. Lihan Huang model had an acceptable performancein predicting experimental data related to oil–water mixtures. Thisrelatively good performance is due to the acceptable assumptions thatfouling is a dynamic process starting by pore blockage and continuingby cake formation. Pore blocking is a natural phenomenonwhich occursat the beginning of oil–water separation, because of the smallmoleculescapable of passing through the membrane pores. This little deviationincreases by increasing the pressure. This can be justified by increasingthe pressure cake layer resistance increases resulting to reduced fluxwhile at the same time driving force increases leading to permeationof more water. In reality the second feature was dominant hencethe model under predicts the flux. This difference can also be due toneglecting the dissolution effect of the particles which causes the cakelayer resistance to be reduced. This difference increases by increasingthe driving force. Gel Layer Filtration and Wu models do not considermembrane fouling and pore blocking which causes these models toevaluate a lower resistance against permeate flow leading to the overprediction of permeate flux compared to the experimental data. Theamount of error and its changes versus pressure for different modelsis shown in Fig. 16. It is clear that the developed model has the smallesterror compared to the others. Increased pressure only increasesthe error slightly. After this model, Lihan Huang showed the bestperformance compared to other previously developed models.

A comparison among models and experimental data has been doneat different concentrations, as shown in Fig. 17. As in the previoussection, it can be seen from Fig. 18 that the error related to the presentmodel is the least compared to other models. By increasing theconcentration a little raise in error was detected.

5. Conclusions

Modeling is an appropriate tool for predicting processes behavior innanofiltration of suspensions. Here a mathematical model was extend-ed for predicting and describing the behavior of a nanofiltration systemand fouling. In this work available suitable models were analyzed andcompared to the experimental data and also with the presently devel-oped model. The comparison between model results and experimentaldata showed that the model assumptions play an important role inthe final relations and therefore prediction of experimental data. Asmentioned before, due to the conditions of the evaluated system andthe fact that membrane resistance changes are accompanied with cakelayer formation during filtration, the assumption of current model isclose enough to the behavior of oil–water system enabling the model

90 F. Faridirad et al. / Desalination 346 (2014) 80–90

to predict the experimental data better than the three other modelsevaluated here. The average error of developed model is approximately3.2% compared to the average errors of Lihan Huang, Wu and Gel LayerFiltration models which are 6%, 8.2% and 9.6% respectively. This in-creases only slightly as pressure and concentration increase demon-strating the acceptable performance of the model at higher pressuresand concentrations of up to 2.9 bar and 0.4 g/L respectively.

Symbols

J(t) permeation flux (m/s)J0 initial permeation flux (m/s)Δp pressure difference between two sides of themembrane (Pa)Rm membrane resistance (kg/s·m2)Rcp polarization layer resistance (kg/s·m2)Rc(t) cake layer resistance (kg/s·m2)μ solution viscosity (Pa·s)μ0 solvent viscosity (Pa·s)α specific cake resistance (1/s)Md accumulated mass surface unit (kg/m2)Rm (0) distinct membrane resistance (kg/m2)a membrane resistance parameter (1/s)b membrane resistance parameter (s)t nanofiltration time (s)Rdep deposition rate of solute (kg/s)Rdis dissolution of solute (kg/s)Q0 feed intel rate (m3/s)Qc cake formation rate (m3/s)Qp permeate rate (m3/s)M1 deposition constant (kg/m3)M2 dissolution constant (kg/m3)M3 passed constant (kg/m·s)l cake layer thickness (m)ρ solution density (kg/m3)ρc cake layer density (kg/m3)A membrane surface area (m2)M

:mass accumulation on the membrane surface (kg/s)

k mass transfer resistance constant (1/s·m3)v0 initial permeate volume (m3)α Lihan Huang constant (1)ε cake layer porosity (1)dp particle diameter (m)

References

[1] W.R. Bowen, J.S. Welfoot, Modelling the performance of membrane nanofiltration—critical assessment and model development, Chem. Eng. Sci. 57 (2002) 1121–1137.

[2] E.-E. Chang, Y.-C. Chang, C.-H. Liang, C.-P. Huang, P.-C. Chiang, Identifying therejection mechanism for nanofiltration membranes fouled by humic acid andcalcium ions exemplified by acetaminophen, sulfamethoxazole, and triclosan, J.Hazard. Mater. 221 (2012) 19–27.

[3] A.E. Childress, M. Elimelech, Relating nanofiltration membrane performance tomembrane charge (electrokinetic) characteristics, Environ. Sci. Technol. 34 (2000)3710–3716.

[4] B. Tansel, W. Bao, I. Tansel, Characterization of fouling kinetics in ultrafiltrationsystems by resistances in series model, Desalination 129 (2000) 7–14.

[5] S. Bhattacharjee, G.M. Johnston, A model of membrane fouling by salt precipitationfrom multicomponent ionic mixtures in crossflow nanofiltration, Environ. Eng. Sci.19 (2002) 399–412.

[6] C.-J. Lin, P. Rao, S. Shirazi, Effect of operating parameters on permeate flux declinecaused by cake formation — a model study, Desalination 171 (2005) 95–105.

[7] A. Seidel, M. Elimelech, Coupling between chemical and physical interactions innatural organic matter (NOM) fouling of nanofiltration membranes: implicationsfor fouling control, J. Membr. Sci. 203 (2002) 245–255.

[8] S. Mattaraj, C. Jarusutthirak, C. Charoensuk, R. Jiraratananon, A combinedpore blockage, osmotic pressure, and cake filtration model for crossflownanofiltration of natural organic matter and inorganic salts, Desalination 274(2011) 182–191.

[9] Y.A. Le Gouellec, M. Elimelech, Calcium sulfate (gypsum) scaling in nanofiltration ofagricultural drainage water, J. Membr. Sci. 205 (2002) 279–291.

[10] C.Y. Tang, T. Chong, A.G. Fane, Colloidal interactions and fouling of NF and ROmembranes: a review, Adv. Colloid Interf. Sci. 164 (2011) 126–143.

[11] H.F. Shaalan, Development of fouling control strategies pertinent to nanofiltrationmembranes, Desalination 153 (2003) 125–131.

[12] B. Sarkar, S. De, A combined complete pore blocking and cake filtration model forsteady‐state electric field‐assisted ultrafiltration, AICHE J. 58 (2012) 1435–1446.

[13] M.E. Ersahin, H. Ozgun, R.K. Dereli, I. Ozturk, K. Roest, J.B. van Lier, A review ondynamic membrane filtration: materials, applications and future perspectives,Bioresour. Technol. 122 (2012) 196–206.

[14] E. Fountoukidis, Z. Maroulis, D. Marinos-Kouris, Crystallization of calcium sulfate onreverse osmosis membranes, Desalination 79 (1990) 47–63.

[15] L. Song, M. Elimelech, Theory of concentration polarization in crossflow filtration, J.Chem. Soc. Faraday Trans. 91 (1995) 3389–3398.

[16] H.R. Rabie, P. Côté, N. Adams, A method for assessing membrane fouling in pilot-and full-scale systems, Desalination 141 (2001) 237–243.

[17] S. Ghandehari, M.M. Montazer-Rahmati, M. Asghari, A comparison between semi-theoretical and empirical modeling of cross-flow microfiltration using ANN,Desalination 277 (2011) 348–355.

[18] P. Rai, G. Majumdar, S. DasGupta, S. De, Modeling of permeate flux of syntheticfruit juice and mosambi juice (Citrus sinensis (L.) Osbeck) in stirred continuousultrafiltration, LWT-Food Sci. Technol. 40 (2007) 1765–1773.

[19] B. Sarkar, S. Pal, T.B. Ghosh, S. De, S. DasGupta, A study of electric field enhancedultrafiltration of synthetic fruit juice and optical quantification of gel deposition, J.Membr. Sci. 311 (2008) 112–120.

[20] S. De, P. Bhattacharya, Modeling of ultrafiltration process for a two-componentaqueous solution of low and high (gel-forming) molecular weight solutes, J.Membr. Sci. 136 (1997) 57–69.

[21] S. Nataraj, R. Schomäcker,M. Kraume, I.Mishra, A. Drews, Analyses of polysaccharidefouling mechanisms during crossflow membrane filtration, J. Membr. Sci. 308(2008) 152–161.

[22] Ö. ÇEtinkaya, V. GÖKmen, Assessment of an exponential model for ultrafiltration ofapple juice, J. Food Process Eng. 29 (2006) 508–518.

[23] G.T. Ballet, A. Hafiane, M. Dhahbi, Influence of operating conditions on the retentionof phosphate in water by nanofiltration, J. Membr. Sci. 290 (2007) 164–172.

[24] X.-m. LI, D. WANG, T. CHAI, B.-w. SU, C.-j. GAO, Study on seawater softening bynanofiltration membrane, J. Chem. Eng. Chin. Univ. 4 (2009) 008.

[25] S. Darvishmanesh, A. Buekenhoudt, J. Degrève, B. Van der Bruggen, General modelfor prediction of solvent permeation through organic and inorganic solvent resistantnanofiltration membranes, J. Membr. Sci. 334 (2009) 43–49.

[26] F. Fadaei, S. Shirazian, S.N. Ashrafizadeh, Mass transfer modeling of ion transportthrough nanoporous media, Desalination 281 (2011) 325–333.

[27] G. Guillen, E. Hoek, Modeling the impacts of feed spacer geometry on reverseosmosis and nanofiltration processes, Chem. Eng. J. 149 (2009) 221–231.

[28] A. Pak, T. Mohammadi, S. Hosseinalipour, V. Allahdini, CFD modeling of porousmembranes, Desalination 222 (2008) 482–488.

[29] M. Iaquinta, M. Stoller, C. Merli, Optimization of a nanofiltration membraneprocess for tomato industry wastewater effluent treatment, Desalination 245(2009) 314–320.

[30] L. Huang, M.T. Morrissey, Fouling of membranes during microfiltration of surimiwash water: roles of pore blocking and surface cake formation, J. Membr. Sci. 144(1998) 113–123.

[31] D. Wu, J. Howell, N. Turner, New method for modelling the time-dependence ofpermeation flux in ultrafiltration, Food Bioprod. Process. 69 (1991) 77–82.

[32] P. Hermans, H. Bredée, Principles of the mathematical treatment of constant-pressure filtration, J. Soc. Chem. Ind. 55 (1936) 1–4.

[33] W.R. Bowen, A. Mongruel, P.M. Williams, Prediction of the rate of cross-flowmembrane ultrafiltration: a colloidal interaction approach, Chem. Eng. Sci. 51(1996) 4321–4333.

[34] R. Sheikholeslami, Foulingmitigation inmembrane processes: report on a workshopheld January 26–29, 1999, Technion—Israel Institute of Technology, Haifa, Israel,Desalination 123 (1999) 45–53.

[35] R.H. Davis, Modeling of fouling of crossflow microfiltration membranes, Sep. Purif.methods 21 (1992) 75–126.

[36] E.M. Hoek, A.S. Kim, M. Elimelech, Influence of crossflow membrane filter geometryand shear rate on colloidal fouling in reverse osmosis and nanofiltration separations,Environ. Eng. Sci. 19 (2002) 357–372.

[37] R. Bian, K. Yamamoto, Y. Watanabe, The effect of shear rate on controlling the con-centration polarization and membrane fouling, Desalination 131 (2000) 225–236.

[38] R.S. Faibish, M. Elimelech, Y. Cohen, Effect of interparticle electrostatic doublelayer interactions on permeate flux decline in crossflow membrane filtrationof colloidal suspensions: an experimental investigation, J. Colloid Interface Sci.204 (1998) 77–86.

[39] V. Gonsalves, A critical investigation on the viscose filtration process, Recueil desTravaux Chimiques des Pays-Bas 69 (1950) 873–903.

[40] T. Arnot, R. Field, A. Koltuniewicz, Cross-flow and dead-end microfiltration of oily-water emulsions: part II. Mechanisms and modelling of flux decline, J. Membr. Sci.169 (2000) 1–15.

[41] T. Mohammadi, A. Kohpeyma, M. Sadrzadeh, Mathematical modeling of flux declinein ultrafiltration, Desalination 184 (2005) 367–375.

[42] R. Poli, J. Koza, Genetic Programming, Springer, 2014.[43] D.E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning,

Addison-Wesley Reading Menlo Park, 1989.[44] M.A. Razavi, A. Mortazavi, M. Mousavi, Dynamic modelling of milk ultrafiltration by

artificial neural network, J. Membr. Sci. 220 (2003) 47–58.