Performance decline due to cake enhanced concentration polarization in cross flow membrane filtration: A transient electrokinetic model and experimental observation

Performance decline due to cake enhanced concentration polarization in

cross flow membrane filtration: A transient electrokinetic model and

experimental observation

Md. Abdullaha-Al-Mamun, Mohtada Sadrzadeh, Subir Bhattacharjee

Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2G8

October 31, 2012

Abstract

Fouling of nanofiltration membranes by multiple rejected components adversely increases the op-erating pressure, decreases the performance and requires cleaning. Considerable attention has beendevoted toward understanding the fouling phenomena and developing models. However, the models aremostly empirical and fail to establish mechanistic link between the different phenomena. In this con-text, a transient model has been developed to predict the performance decline due to colloidal fouling ofnanofiltration membranes. The model combines the transient growth of cake layer and cake enhancedconcentration polarization (CECP) phenomenon to predict the permeate flux and observed rejectiondecline during cross flow membrane filtration. The cake layer has been considered as a swarm of non-interacting, incompressible, spherical, charged particles and represented using the Kuwabara cell model.The model also provides fundamental insight to the development of streaming potential and electroos-motic back flow, due to transport of ions around the charged colloidal particles of the cake layer, in lightof classical Levine-Neale model of electrophoresis in concentrated dispersions. The CECP phenomenahas been accounted by considering the hindered back diffusion of ions through the tortuous path of thecake layer. These phenomena are then combined with the film theory of cross flow filtration for pred-icating the permeate flux and observed rejection. In order to validate the developed model fouling ofnanofiltration membrane by colloidal silica particles were carried out and the results were compared withthe model predictions. Statistical analysis reveals the model is in good agreement with experimentalresults with only one adjustable parameter, cake porosity.

Keywords: Membrane Filtration, Nanofiltration, Cake Enhanced Concentration Polarization, FilmTheory, Kuwabara Cell Model, Levine-Neale Electrophoretic Mobility, Electroosmotic Back Flow

1 Introduction

Over the past decades, several models have been postulated to elucidate the mechanism of permeateflux and observed rejection decline due to colloidal fouling of salt rejecting membrane processes. Hoek andElimelech [1] successfully explained the performance decline mechanism for nanofiltration (NF) and reverseosmosis (RO) processes, and reported the flux decline due to cake enhanced osmotic pressure (CEOP) orcake enhanced concentration polarization (CECP). They explained the fouling mechanism as hinderedback-diffusion of salt ions within colloidal cake layers resulting in an increase of concentration polarization(CP) as well as the transmembrane osmotic pressure (TMOP). Later, Wang and Tarabara [2] proposedanother approach to determine the contribution of CP and cake resistance on performance decline based onsalt permeability constant measurement. These approaches, quantify the relative contribution of CP andcake resistance during constant pressure filtration experiments. However, none of these models consideredthe coupled effect of transient growth of cake layer and CP phenomenon to establish a mechanistic model

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of performance decline independent of experimental results. Besides, these models used Kozeny-Carmanequation to determine the cake resistance instead of cell model of swarm of particles which restricts theapplicability for fixed cake structures only.

Earlier, CECP models also neglected the zeta potential effect of charged colloidal particles. Fluidflow and charge transport through the colloidal cake layer are interrelated due to the formation of electricdouble layer (EDL) around the particles. Transport of electrolyte through the charged packed bed developselectroviscous resistance [3, 4]. Recently, De and Bhattacharjee [5] proposed an electrokinetic model forflux decline during the filtration of charged colloids and electrolyte. The model assumed the cake layeras a bundle of charged cylindrical nano-pores and considered the electroviscous effect created by them.This model does not provide mechanistic links between the transient growth of cake layer and CP of saltrejecting membranes. Besides, the electroosmotic flow through a bed of stationary colloidal particles getaffected by the surrounding particles because the presence of neighboring particles affects the fluid velocityfield and ion distribution around each particle [6]. It is therefore important to know how the concentrationof cake layer affects the electroosmotic flow of electrolyte. This problem can be solved by applying the cellmodel approach to the electroosmotic flow of electrolyte.

In this study, a mechanistic model is presented for transient growth of cake layer and CECP consideringthe cell model of cake structure and electroosmotic back flow (EBF) across the cake layer. To validate themodel with experimental results, cross flow NF was conducted with silica particles and NaCl solution overa range of operating conditions.

2 Mathematical model

During filtration of electrolyte solution and charged particles above the critical flux of particle deposition, astationary cake layer forms at the membrane surface. The cake layer resistance consists of the hydrodynamicresistance of the cake layer and electroviscous resistance due to the charged colloids. Also, the totalresistance to permeate flow comprises membrane resistance (Rm) and total cake layer resistance (Rc). SinceNF membrane also reject salt, the developed TMOP (πm) also opposes the applied pressure. Therefore,the effective driving force for permeation is the difference between the operating pressure and TMOP [7].

Colloid

EDL

Eα U

(b)

y=L+δ

Ci,f

y

C

y = δ

(a)

Ci,δ

Ci,m

Ci,p

∆P

Βulk

∆Pc/µrcL

uw

θ

b

a∆Pc

Figure 1 – a: Transport of electrolyte through CP Layer, cake layer and membrane b: Levine-Neale elec-trophoretic model based on Kuwabara Cell model

The CP layer and cake layer in cross flow membrane filtration is illustrated in Fig. 1a. The retainedsolutes at the membrane surface form cake and CP layers. To elucidate the transport of solvent and

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electrolyte through the interstices of cake layer, consider a single colloidal particle within the cake illustratedin Fig. 1b. According to the figure, a colloidal particle of hydrodynamic radius a (m) and zeta potentialψp (mV ) is considered stationary in a spherical coordinate. The colloidal cake structure is representedusing the Kuwabara cell model. The cell model defines a spherical fluid shell of radius b which is related

to cake volume fraction (ϕc) as, b = a/ϕ1/3c .

The permeate flux due to the hydrodynamic Stokes drag of the cake layer can be written based on theresistance in series and Kuwabara cell models as follows:

vw =∆Pc

µrcL(1)

here, ∆Pc is the pressure drop across the cake layer, L is the cake layer thickness, rc = 9ϕcAK

2a2is the specific

cake layer resistance, µ is fluid viscosity, and AK is Kuwabara correction factor.To calculate the electroosmotic flow of the salt solution pass through the stationary cake layer, we

use the Levine-Neale electrophoretic model. According to the Levine and Neale [8] cell model, the elec-trophoretic mobility is given by:

η =U

E∞=

(εψp

µ

)g2(κa, ϕc) (2)

here, ε is electrical permittivity, U is the mean relative velocity of the interstitial fluid with respect to thestationary colloidal particle at the cell boundary, E∞ is the electric field induced by the streaming potentialdeveloped across the cake layer, g2(κa, ϕc) accounts for the presence of neighboring particles, and κ is theinverse Debye length.

The expression of electric field (E∞) developed by the induced streaming potential at zero currentcondition, is written based on the Onsager principle of reciprocity and Debye-Huckel approximation(zeψp/kBT ≤ 1) as follows [6]:

E∞ =∆Pc

L

(εψp

µσ∞

)(1− 2 I1(κa)

κa I0(κa)

)(3)

Here, I0(κa) and I1(κa) are zeroth and first order modified Bessel function of the first kind, respectively.The expression for bulk electroosmotic flux across the cake layer can be written from interstitial fluidvelocity using the continuity assumption as follows:

uw = U(1− ϕc) =∆Pc

Lσ∞

(εψp

µ

)2

(1− ϕc)g2(κa, ϕc)

(1− 2 I1(κa)

κa I0(κa)

)(4)

Therefore, the total flux across the cake layer at no current condition due to effective cake pressure∆Pc and reverse electroosmotic flow, is given as follows:

vw =∆Pc

µrcL− uw =

∆Pcg∗

µrcL(5)

The parameter g∗ accounts for electroosmotic effect in swarm of charged colloidal particles and expressedas follows:

g∗ = 1− 9ϕc(1− ϕc)AKβg2(κa, ϕc)

(1− 2 I1(κa)

κa I0(κa)

)(6)

This parameter, g∗, is accounted as the electroviscous resistance, Rc = rcL/g∗. The parameter β

quantifies the effect of the colloid zeta potential (ψp) and solution conductivity (σ∞) on EBF, and expressedas:

β =(εψp)

2

2a2µσ∞(7)

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For 1:1 electrolyte concentration, the TMOP is expressed by the van’t Hoff equation [7],

∆πm = 2<T (Ci,m − Ci,p) (8)

The expression of Ci,m, derived by integrating the mass balance of electrolytes within the CP and cakelayers with appropriate boundary conditions, is:

Ci,m

Ci,f=

exp [vwδ/Di + vwL/D∗i ]

Rr + (1−Rr) exp [vwδ/Di + vwL/D∗i ](9)

here, Rr = 1 − Ci,p/Ci,m is the real rejection of the membrane. Hindered diffusivity of electrolyte iscalculated from the tortuosity (ς) of the cake layer as D∗i = Diες

−1 where, ς = [1− ln(ε2)] [9].Transient growth of cake layer is obtained by integrating the mass balance equation of colloids in the

cake layer with appropriate boundary conditions [10]:

ρpdL

dt= vw

Cp,m − Cp,f exp(vw/kp)

1− exp(vw/kp)(10)

Due to difficulty of determining the colloid mass transfer coefficient kp at specific length of the slitchannel and as the cake layer thickness is usually very small compared to the hydrodynamic diameter ofthe channel, constant length average kp for laminar flow in channel has been considered using Levequeexpression [11]. The thickness of the cake layer can be known by integrating the Eq. 10 over the time.The cake layer mass is obtained using Mc = LρpϕcAm where, Am is the effective membrane area and ρp isparticle density. The permeate flux for cake filtration can be written using the modified Darcy‚s law [12]:

vw =∆P −∆πmµ(Rm +Rc)

(11)

where, ∆P is applied pressure and Rm is membrane resistance. The Eqs. 8, 10 and 11 are solved simul-taneously to obtain the permeate flux and observed rejection. The observed rejection (Ro) can be writtenas follows:

Ri,o = 1− Ci,p

Ci,f(12)

Overall pressure drop during operation is the summation of trans-cake hydraulic pressure, cake-enhancedTMOP and transmembrane pressure. Therefore,

∆P = ∆Pc + ∆Pm + ∆πm (13)

where, ∆Pm is the pressure drop across the membrane.

3 Results and discussion

The resulting set of differential-algebraic equations are solved using a coupled differential algebraic equationsolver (DASSL) in Fortran. Colloidal fouling experiments in presence of NaCl solution were conducted forNF90 (Dow FilmTec) membrane to compare with the electrokinetic model. The transient permeate flux,observed rejection, and deposited mass were recorded as a function of time for 5 hrs. Simultaneously,the embedded parameters, namely, TMOP and cake porosity were calculated following the Hoek andElimelech [1] procedure. The experiments were conducted at constant temperature (25℃) and pH (7.0),which allowed fixed zeta potential (-35 mV ) for a given particle (100 nm Snowtex ZL, Nissan Chemical).In all plots, permeate flux has been presented as normalized flux (vw/vi) and the time of silica additionis considered as zero. The steady state salt water flux (vi) and observed rejection (Ro) was obtained bysolving Eqs. 8, 9, and 11 in absence of cake layer (i.e., L = 0 and Rc = 0). The pressure drops are

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Table 1 – Experimental conditions for fouling experiment at 25± 1℃ and pH=7.0

Experiment Operating ParametersNo. ∆P (kPa) Ci,f (mM) Cp,f (ppm) uc (m/s) 2a (nm)1 965 10 300 0.1 1002 689 10 300 0.1 1003 1033.5 20 300 0.1 1004 965 10 500 0.1 1005 965 10 500 0.2 100

presented in normalized form namely normalized trans-membrane pressure (∆Pm∗ = ∆Pm/∆P ), trans-

cake hydrodynamic pressure (∆Pc∗ = ∆Pc/∆P ) and cake-enhanced TMOP (∆πm

∗ = ∆πm/∆P ). Theexperimental results, raw data and calculated parameters, are shown as symbol in the graphs. The modelpredictions (lines) are superimposed on experimental results to compare with each other. The averageporosity (ε = 1 − ϕc) obtained from each experiment was considered as the only adjustable parameter inthe model. For the silica colloidal particles bulk concentration of 0.2 kg/m3, the cake concentration wasreported to be about 200 kg/m3 [13] which was considered throughout the analysis.

3.1 Comparison of model and experiment

Five experiments were conducted varying the salt concentration, cross flow velocity, operating pressure, andsilica concentration. Each parameter varied at two levels, and the results obtained from the experimentsafter 2 hrs and 4 hrs were used to demonstrate the goodness of fit of the model. Four responses namely fluxdecline, observed rejection, deposited mass, and normalized CEOP were investigated using the statisticalparameters including mean squared error (MSE), root mean squared error (RMSE), correlation coefficient(R), and coefficient of determination (R2). The statistical results for the responses are listed in Table 2.As can be seen, the model predictions are in good agreement with experimental results with RMSE valueof less than 0.1 and correlation coefficient of greater than 0.9.

Figure 2 – a) Normalized permeate flux (vw/vi), Observed rejection (Ro) and Deposited cake mass (Mc) onNF90 membrane. Fouling experiment conducted at 965 kPa, cross flow velocity 0.1 m/s, 10 mM NaCl solution,300 ppm of Silica, temperature of 25℃ and pH 7.0, b) Comparison of ∆Pm

∗, ∆Pc∗ and ∆πm

∗.

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Figure 3 – Effect of a) salt concentration and b) cross flow velocity on water flux, mass deposition, and CECP.

Fig. 2 shows the comparison of model predictions and experimental results for experiment 1 (Table1). The permeate flux and observed salt rejection declined over the filtration time with the increase ofcake layer thickness. Contribution of cake layer resistance on pressure drop is insignificant compared toTMOP validating the primary mechanism of flux decline as CECP. The model captures the coupled effectof cake layer growth and CECP phenomena which allows accurate prediction of experimental results. Theexperimental average porosity, 0.48, was used for the model prediction as well.

Table 2 – Statistical criteria for validation of the developed model

ResponseCriterion Flux Observed salt Deposited Normalized

decline, rejection, silica mass, CEOP,(1-vw/vi) Ro Mc ∆π∗m

MSEa 1.11× 10−3 1.85× 10−3 9.34× 10−3 8.45× 10−4

RMSEa 0.033 0.043 0.096 0.029R 0.959 0.952 0.901 0.965R2 0.92 0.906 0.813 0.931

a The MSE has the unit of squared of response unit while RMSEhas the same unit as response

3.2 Effect of salt concentration and cross flow velocity

The effect of salt feed concentration and cross flow velocity on water flux, mass deposition, and CECPare shown in Fig. 3a and 3b, respectively. To study the effect of salt concentration on cake structure andCECP, two experiments (1 and 3 in Table 1) were conducted at same initial flux condition. The permeateflux decline is higher for 20 mM NaCl concentration even though the mass deposition is less. This is dueto denser structure of cake layer and enhanced CP at higher salt concentration. Average experimentalporosity for 20 mM was less, 0.45, as compared to 0.48 for 10 mM . Experimental results for two crossflow velocities (4 and 5 in Table 1) are shown in Fig. 3b. Flux decline is higher for increased cross velocity

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condition which seems to be counter intuitive. The result can be attributed to the fact of less initialTMOP and higher initial salt flux resulting in excessive mass deposition and increase in TMOP. Hence,the coupling between cake layer growth and CECP is significantly important for the development of amechanistic model which has been accounted in the present study.

3.3 Electroosmotic back flow

The effect of EBF is captured in parameter g∗ which quantifies the increase in cake pressure drop due toEBF in terms of pressure drop ratio (∆P ∗c /∆P

∗c,h). This is the ratio of total pressure drop including EBF

to the pressure drop due only to hydrodynamic resistance of cake layer. Fig. 4 shows the effect of zetapotential (ψp) and cake volume fraction (ϕc) on pressure drop ratio as a function of κa. The ratio exhibitsa maximum value with respect to a smaller κa value for constant ϕc and it increases with the increase of ψp.However, the maximum value shifts to larger κa values and increases with the increase of ϕc for constantψp. This indicates that at higher ϕc values the EBF becomes important at larger κa values. Therefore,the EBF is more important in presence of denser cake layer having higher zeta potential for 2< κa <15.In case of denser cake layer the outer cell envelop, b, becomes comparable with the EDL thickness. At thiscondition, fluid flow modifies the ion distribution around the particles of cake layer and creates the EBF.

Figure 4 – Effect of a) zeta potential (ψp) and b) volume fraction (ϕc) on pressure drop ratio (∆P ∗c /∆P

∗c,h) as

a function of κa.

4 Conclusion

A mechanistic model has been developed for investigating the coupled effect of transient cake layer growthand CECP on performance decline of salt rejecting membranes. The model also captures the effect of cakevolume fraction on EBF in light of Levine-Neale cell model of electrophoresis. In presence of denser andcharged cake layer during low ionic strength operation the effect of EBF become important. Simultaneously,experimental study was conducted to investigate the effect of operating parameters and goodness of fit of themodel. The model can predict the experimental results with reasonable accuracy with only one adjustableparameter, cake volume fraction. The RMSE values were lower than 0.1 and correlation coefficients werehigher than 0.9.

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