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Journal of Membrane Science 379 (2011) 488495

Contents lists available at ScienceDirect

Journal of Membrane Science

j ou rna l ho me page : www.e l sev i e r. com/ loca t e /memsc i

Computational uid dynamics simulations of ow and concentrationpolarization in forward osmosis membrane systems

M.F. Gruber a,b ,, C.J. Johnson c , C.Y. Tang d ,e , M.H. Jensen f , L. Yde c , C. Hlix-Nielsen a,g,a Aquaporin A/S, Ole Maales Vej 3, DK-2200 Copenhagen N, Denmarkb Nano-Science Center, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen , Denmarkc DHI Water & Environment, Agern Alle 5, DK-2970 Hrsholm,Denmarkd Singapore Membrane Technology Centre, Nanyang Technological University, Singapore 639798, Singaporee School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Singaporef Center for Models of Life, Niels Bohr Institute, Blegdamsvej 17,DK-2100 Copenhagen , Denmarkg The biomimetic membrane group DTU-Physics, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark

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Article history:Received 15 April 2011Received in revised form 10 June 2011Accepted 14 June 2011Available online 22 June 2011

Keywords:Forward osmosisComputational uid dynamics (CFD)Internal concentration polarizationExternal concentration polarizationDesalination

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Forward osmosis isan osmotically driven membrane separation process that relies on the utilization of a large osmotic pressure differential generated across a semi-permeable membrane. In recent years for-ward osmosis has shown great promise in the areas of wastewater treatment, seawater/brackish waterdesalination, and power generation. Previous analytical and experimental investigations have demon-strated how characteristics of typical asymmetric membranes, especially a porous support layer, inuencethe water ux performance in osmotically driven systems. In order to advance the understanding of membrane systems, models that can accurately encapsulate all signicant physical processes occurringinthe systems are required. The present study demonstrates a computational uid dynamics (CFD) modelcapable of simulating forward osmosis systems with asymmetric membranes. The model is inspired bypreviously published CFD models for pressure-driven systems and the general analytical theory for uxmodeling in asymmetric membranes. Simulations reveal a non-negligible external concentration polar-ization on the porous support, even when accounting for high cross-ow velocity and slip velocity atthe porous surface. Results conrm that the common assumption of insignicant external concentrationpolarization on the porous surface of asymmetric membranes used in current semi-analytical approachesmay not be generally valid in realistic systems under certain conditions; specically in systems withoutmass-transfer promoting spacers and low cross-ow velocities.

2011 Elsevier B.V. All rights reserved.

1. Introduction

During the last 40 years the separation of aqueous solutionsusing pressure-driven membrane systems have been intenselystudied both experimentally and theoretically. Over this time,many different semi-analytical models have been used to inves-tigate various features found in the pressure-driven systems: e.g.effects such as concentration polarization (CP) phenomena [1,2] ,changes in solute rejection [3] , the effect of wall slip velocity [4,5]and gravitational effects [6] have been studied.

Developing a generic model to encapsulate all aspects of membrane ltration in complex geometries can however not beaccomplished using an analytical model. During the last 15 years,

Corresponding author. Tel.: +45 31168385.

Corresponding author. Tel.: +45 27102076.E-mail addresses: [email protected] (M.F. Gruber),

[email protected] (C. Hlix-Nielsen).

computational uid dynamics (CFD) have become increasinglypopular formodelingcomplex owpatterns in membrane systemsbecause it provides a more robust approach capable of includ-ing many parameters. Many different CFD models dealing withpressure-driven membrane systems have been presented in liter-ature. Some models focus on the effects of variations in membraneproperties likesolute rejectionandsolutionproperties suchas den-sity, viscosity, and diffusivity [79] . Others are centralized aroundmass transfer optimizationby changingthe case geometry to mod-ulate an optimal ow, e.g. by using eddy-promoting spacers [9,10] .

In recent years, forward osmosis (FO) has emerged as a popu-lar alternative to conventionalpressure-drivenmembrane systemssuchas microltration (MF), ultraltration(UF), nanoltration (NF)and reverse osmosis (RO). Unlike pressure-driven processes wherea hydraulic pressure is used to establish solvent ow through asemi-permeable membrane, FO uses a concentrated draw solutionand a dilute feed solution to generate solvent ow driven by anosmotic pressure difference across a semi-permeable membrane.The main advantages of FO compared to RO is the lack of a need for

0376-7388/$ seefront matter 2011 Elsevier B.V. All rights reserved.

doi: 10.1016/j.memsci.2011.06.022
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M.F. Gruberet al. / Journal of Membrane Science 379 (2011) 488495 489

hydraulicpressure, which makesFO potentially morecost-effective[11,12] . It has also been shown that FO has a lower propensity tomembrane fouling, possibly due to the lack of hydraulic pressure[1317] . Furthermore, the insertionof proteinswithspecicwater-or ion-transporting properties into biomimetic membranes showsgreat promise in FO applications [18] .

So far all studies found in the literature have observed farlower water uxes in FO experiments than expected based onbulkosmoticpressuredifferencesandmembrane water permeabil-ities. The reason for this is that membrane development over thepast four decades has been focused on pressure-driven processes,and as such current membranes typically utilize an asymmetricdesignwitha dense rejectionlayerand a thick poroussupport layerfor mechanical stability [19] . The porous support has a negligibleinuence when it comes to CP effects in RO, since only CP on thefeed-side of the membrane (i.e. the dense layer) is signicant withrespect to mass transfer [20] . In FO, however, CP effects are impor-tant on both sides of the membrane, and depending on how themembrane is positioned either dilutive or concentrative CP willoccur within the porous layer [21] . CP within the porous support isin a sense effectively decoupled from the external ow, and there-fore poses a reductionin effectiveosmotic driving force that cannotbe diminished using optimal ow conditions. CP within the porouslayer is generally termed internal concentration polarization (ICP)as opposed to the external concentration polarization (ECP) whichoccurs outside the membrane.

Several one-dimensional analytical and semi-analytical modelsderived from the work by Lee et al. [21] and Loeb et al. [22] havebeen proposed for determining the water ux through compos-ite membranes in FO systems, many of which correlate well withexperiments [15,21,2325] . In CFD simulations of RO systems, themembrane is often modeled as being symmetric: that is, as a sin-gle active separation layer with no porous support [9,10,26] . Thisis acceptable since the porous support has an insignicant effect inthis case. In the case of FO the effects of ICP must be included inthe model. To the best knowledge of the authors, no attempts havebeen made so far to fully simulate a FO system with an asymmetricmembrane using a CFD approach.

The objective of this paper is to extend the current semi-analytical models found in the literature used to describe ICP andwater ux in FO systems into an efcient CFD model. Using thisCFD model, we evaluate the semi-analytical approaches as well asinvestigate the effect of various major parameters such as cross-ow velocity, bulk osmotic pressure difference, and slip velocityon the CP proles and water uxes.

2. Model development

The basis for the model described in this paper is the opensource toolkit OpenFOAM (Open Field Operation And Manipula-

tion).OpenFOAMisaregisteredtrademarkofOpenCFDLimited,theproducer of the OpenFOAMsoftware. CFD packages have improvedsignicantly during the last decade and OpenFOAM provides aneasily extendable framework with an extensive range of featuresand robust solution algorithms. Custom implementation of themodel was performed in two steps: implementation of the solverfor the governing equations and implementation of proper bound-ary conditions on the membrane surface. The model developed forthe governing equations was validated by using the solver in a ROtest case, for which it was possible to reproduce previously pub-lished results in [9,26] . The boundary conditions were validatedby comparing obtained water uxes against published measure-ments; using values for membrane characteristics obtained in [27]for a chamber corresponding to that used in [27] , the boundary

conditions (with a no-slip tangential velocity) resulted in water

uxes within 10% of what was obtained in [27] . This discrepancyis easily explained based on discussions in Sections 3.2 and 3.3 aswell as various experimental factors, e.g. possibly peristaltic ow,membrane oscillations, etc.

2.1. Governing equations and membrane model

The ow in the membrane chamber is governed by equations

for conservation of mass and momentum as well as a convection-diffusion transport equation for the solute mass fraction. Thedeveloped model is capable of solving the governing equations inthree dimensions, but can also be used without modications fortwo-dimensional cases. The governing equationsused in the modelare:

t + ( U) =0 (1) Ut + ( U U) = [ ( U + U

T)] p + g (2) m A

t + ( Um A) ( D AB m A) =0 (3)

where symbol denitions can be found in the nomenclature. Theuid is assumed to be isothermal with the density being a functionof solute mass fraction only; i.e. the pressure dependence of thedensity is neglected. This is generallyknown as a weaklycompress-ible formulation of the governing equations, and has previouslybeen used successfully for studying pressure-driven membranesystems [9,26] . The viscosity and diffusion coefcient are allowedto be functions of the solute mass fraction. The ow is assumed tobe laminar, which is reasonable in most real membrane systems[10] .

The case considered in this paper is one in which the poroussupport of a composite membrane faces the draw solution and theactive separation layer of the membrane faces the feed solution:anorientation commonly known as the AL-FS orientation. The com-

posite membrane is modeled as a two-dimensional plane whichmeans that the thickness of the membrane is not resolved, see Fig.1. The fact that the membrane is modeled as a smoothplane meansthat potential effects caused by surface roughness are neglected.We believe that it is highly unlikely that roughness effects beyondthose represented by a slip velocity, especially on the micro-meterscale, will signicantly affect the observed results. Assuming zerohydraulic pressure applied over the membrane, the water ux Jwcan be written as [24] :

Jw = A( d,i f,m )n d (4)where A is the pure water permeability coefcient, d,i is theosmotic pressure between the porous support and the active layerof the membrane, f ,m is the osmotic pressure at the membrane

on the feed side, and n d is the unit normal vector on the porousboundary, i.e. the normal facing the draw solution, see Fig. 1. A isreadily measured in a pressure driven system with clean water onbothsides ofthe membrane; insucha system the water ux willbelinearly proportional to the applied pressure with A as the propor-tionality coefcient [19] . The model does not resolve the osmoticpressure d,i within themembrane, so additional informationaboutwhat happens in the porous support is required in order to calcu-late the water ux. The most widely accepted analytical model fordescribing the effectof theporous support layer on thewater uxisthat developed by Loeb et al., where a linear relationship betweensolute concentration and osmotic pressure is assumed [22] :

Jw =1K

ln B+ A d,m

B

+ | J

w | + A f,m

n d (5)

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Fig. 1. Schematic representation of the geometry considered in all simulations. The close-up illustrates how the grid is graded such that a very ne resolution is obtainedclose to themembrane. Themembrane is represented using a dashedline, reecting the fact that the thicknessof themembrane is notresolved, but ratherdescribed usingEq. (5) .

where B is the solute permeation coefcient and K is a parametertermed by Lee et al. [21] that describes how easily solute can dif-fuse into and out of the support layer. Eq. (5) provides the velocityboundary condition in the normal direction of the membrane. It isnoted that in thecaseswhere B A d ,m and B | Jw |Eq. (5) reducesto the following: Jw = A( d,m e|

Jw |K f,m )n d (6)By comparing Eq. (4) with Eq. (6) an ICP modulus is obtained:

d,i

d,m = e| Jw |K (7)

The ICP modulus in Eq. (7) is only strictly valid for perfect mem-branes (i.e. 100% solute rejection) and when assuming a linearrelationship between osmotic pressure and solute concentration.Nevertheless, the ICP modulus nicely illustrates howthecoefcientK inuences the ux equation to account for changes in concentra-tion across the porous support layer.

A no-slip velocityboundary condition in the tangential directionis often assumed at the membrane boundary in CFD simulations[9,26] , however this may not be accurate at potentially roughporous supports [5] . On porous surfaces where the no-slip bound-ary cannot be applied, the tangential slip velocity Uslip can bemodeled as being proportional to the shear rate U/nd at theboundary [4,5] :

Uslip =

Und

(8)

where is the slip coefcient, is the permeability (m 2 ) and ndis the direction towards the draw which is normal to the mem-brane.The slip coefcient often rangesfrom 0.5to 10 anddescribesthe ow hydrodynamics adjacent to and directly inside the poroussupport, such that it depends on surface characteristics such asroughness, pore size and structure. It has for instance been shownthat has a tendency to be higher for a densely packed porousmaterial than for a less densely packed material [28] . The perme-ability generally lies in the range of 10

11 m 2 to 10

15 m 2 for

most semi-pervious materials [29] . In this model it is assumed that and do not depend on permeation velocity and in general thatthey are simply membrane constants. The shear rate is obtaineddirectly from the velocity eld in the model and as such accountsfor potential ow effects caused by membrane permeation. Know-ing the value for the parameter / it is possible to calculate theslip velocity on the porous boundary explicitly in the CFD model.

The solute ux Js through the membrane can be written asfollows assuming that solute separation occurs over the active sep-aration layer only [24] :

Js = B(C d,i C f,m )n d (9)Itis noted thatthe saltux is negativeto indicate thatit isoppo-

site ofthe water ow. Theconstant B is commonlydeterminedfrom

measurements of the salt separation coefcient R in a pressure-driven reverse osmosis experiment [19,27] . More specically, Bcanbe related to R as follows [21,15] :

B =1 R

R | Jw | (10)

R =1 C f,mC d,m

(11)

Given the denition of the salt separation coefcient in Eq. (11) ,it is apparent that R describes how the solute is separated acrossthe membrane; e.g. a value of 0 means that the membrane is fullypermeable and that the concentration on the feed and draw sideof the membrane will be the same. A value of 1 indicates that themembrane is completely impervious to a given solute. In a givencross-ow experiment, one can estimate R using Eq. (11) , and Eq.(10) can then be used to calculate the membrane constant B.

Back to Eq. (9) , if we assume a linear relationship between con-centration and osmotic pressure, i.e. = C , we can combine Eqs.(9) and (4) to obtain an expression for the solute ux: Js =

B

A Jw (12)

Eq. (12) expresses the solute ux Js as a function of experimen-tally determined parameters A, B and along with the water ux Jw which is calculated by solving Eq. (5) . Knowing the solute uxthrough the membrane, it is possible to write boundary conditionsfor the solute mass fraction on both sides of the asymmetric mem-brane because theconvective anddiffusiveuxesmustbe balancedwith the solute ux [21] :

m D ABm And

n d + mm A,m Jw = Js (13)

This provides a boundary condition for the solute mass fractionwhich must be satised at both sides of the membrane. In conclu-sion, with Eq. (5) f or the water ux through the membrane and Eq.(13) f or solute ux balance on the membrane, we have boundaryconditions available for the velocity eld and solute mass fraction

on both sides of the membrane.

2.2. Case geometry and parameters

Simulations were carried out in the simple cross-ow chambershown in Fig. 1. The chamber was inspired by the one used in [27]and the dimensions were 14cm 6mm. The membrane segmentwas 10cm and located in the middle of the chamber. The heightof both the feed and drawchannels were 3 mm. The computationalmesh consisted of 150cells perpendicular to themembrane in boththe feed and draw channel, graded such that the rst grid pointswere located within 5 m of the membrane in order to capture CPeffects [9] . Gridpoints were within 50 m of the upper and lowerchamber walls, which was found to be sufcient. 280 cells were

used along the length of the chamber.

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At the draw and feed inlet uniform initial solute mass fractionswere specied.On both inletsthe componentof the pressure gradi-entnormaltotheinletswassettozero;theerrorintroducedbysucha boundary condition on the inlet pressure is believed to be mini-mal, especially given the 2 cm inlet sections, which was conrmedthrough a range of verication tests. Fully developed velocity pro-les in the direction along the length of the channels were set atboth inlets as follows [9] :

U x =6U yh 1 yh (14)where x and y refer to the coordinate system seen in Fig. 1. Sincepressure is unimportantfor thehydrodynamics of the weaklycom-pressible formulation, a gauge pressure of zero was specied atboth outlets. For the solutemass fraction and velocityat theoutletswe set m A/n =0 and U n = 0 . On all non-membrane walls theno-slip velocity and zero gradient solute mass fraction boundaryconditions were applied; U = 0 and m A/n = 0, respectively. On themembrane the boundary conditions described in Section 2.1 wereimplemented for the velocity and solute mass fraction. The veloc-ity on the draw side of the membrane was corrected for densitychanges across the membrane as in [9] .

The scope of this paper is limited to a theoretical and numericalinvestigation of FO systems andas such allnecessary model param-eters were inspired by previous literature in the eld, rather thanbeing obtained from our own experiments. Empirical expressionsforthe physical properties of a NaCl solution at 25 C were obtainedfrom [30] :

=805 .1 105 mA (15)

=0.89 10 3 (1 +1 .63 mA) (16)

D AB =max(1 .61 10 9 (1 14 mA), 1.45 10

9 ) (17)

=997 .1 +694 mA (18)where is the osmotic pressure (Pa), is the viscosity (Pa s), D ABis the diffusion coefcient of the solute (m 2 s1 ), and is the uid

density (kgm 3). Using Eqs. (15) and (18) we estimate the propor-tionality factor in Eq. (12) as:

=805 102 Pa m 3 kg1 (19)

The expressions (15) (19) are valid for NaCl mass fractions up to0.09, which corresponds to a NaCl concentration of approximately1.6M [30] . Inspired by FO experiments performedby Yip et al. [27] ,membrane characteristics were set as follows:

A=1 10 12 m(sPa) 1 (20a)

B =1 10 7 m s 1 (20b)

K =0.5 s m 1 (20c)

For the slip-boundary condition on the porous support, the slip

coefcient andtheporosity mustbe set.Basedon investigationsin [31] the slip coefcient was set to 5. The porosity was var-ied between the general values of 10 5 m 2 to 10 15 m 2 , i.e. valuesranging from highly pervious materials to more semi-permeablematerials [29] . It is noted that the value for and the values speci-ed in Eqs. (20a) (20c) are merely inspired by the values found in[31] and [27] , i.e. it is not attemptedto reproduce the experimentalresults presented in [27] .

2.3. Numerical procedure

The simulations were performed using the OpenFOAM library,version 1.7. A newsolver based on theweakly compressibleformu-lation describedin Section 2.1 w as implemented usingthe supplied

solvers twoLiquidMixingFoam and rhoPisoFoam as templates. A

PISO algorithm was used for treating the inter-equation pressure-velocity coupling of the governing Eqs. (1) and (2) f or mass andmomentumconservation [32] . Twoiterations in the PISO loop werefound to be sufcient to achieve stable simulations for all cross-ow velocities. In order to obtain temporal accuracy of the solutemass fraction, Eq. (3) w hich describes the convection and diffusionof solute, was solved within the PISO loop. Membrane boundaryconditions were explicitly implemented using eld values fromprevious time steps. The water ux equation, Eq. (5) , was solvedfor Jw at each point on the membrane using Ridders method forroot-nding [33] .

In this work the Reynolds numbers for the majority of thesimulations were around 300. It has been argued that isotropicturbulence models only are relevant to membrane chambers forReynolds numbers above 30,000 [10] . We therefore believe thatthelaminar model developed here is capable of describing thegen-eral owdynamics investigatedin this work. TheReynoldsnumberis proportional to the cross-ow velocity and reaches a value of 3000for a cross-ow velocityof 1 m s 1 in the present system. Onesimulation was carried out in this work with a cross-ow veloc-ity of 1 m s 1 . It is believed that the ow will enter a transitionalregime above a Reynolds number of about 2000, which may workto decrease ECPat the membrane because of increasedmixing [10] .Inclusionof turbulence in themodel is howeverbeyondthe scopeof this work, and it is assumed that results obtained with the presentmodel at Reynolds numbers of 3000 are sufciently accurate.

3. Results and discussions

All simulations were run until steady-state solutions with con-stant water and solute uxes were obtained. The exact time foreach simulation to reach steady-state was dependent on the owconditions; for the lowest cross-ow velocities it took about anhour of simulation for the ow to reach steady-state and for thehighest cross-ow velocities the simulation had reached steady-state after less than a minute of simulated ow. It was conrmed

that the model ensured overall mass balance and that the soluteux balance equation was satised at both sides of the membraneboundary during the simulations. Simulations were run on a per-sonal computer with a quad core processor (Intel Q9450) and ittook approximately 1 h for each simulation to reach steady-state.

Using the total amount of solute at the draw side of the mem-brane as the integralfunction in ananalysisof theGrid ConvergenceIndex (GCI) [10,34] , it was found that the GCI for the coarse gridcompared to that of a ner grid with twice the amount of cells inboth directions was below 0.1%. This conrms that grid indepen-dence was achieved.

3.1. Mass fraction proles vs. cross-ow velocity

The ultimate goal of implementing CFD models that describemembrane ows is to have models in which various ow parame-ters can readily be optimized to promote higher water ux acrossthe membrane. A signicant parameter to be investigated in thisregard is the average cross-ow velocity in the membrane cham-ber. It is generally known that higher cross-ow velocities work toreduce the severity of ECP. To show this effect, solute mass fractionproles are presented in Fig. 2(a) for mean inlet velocities of 0.001,0.01, 0.1, and 1m s 1 . All mass fraction proles in this paper arerecorded along the y axis at the center of the membrane, refer toFig. 1. At each time-step in the simulation, the total water uxesacross the membrane were recorded and the steady-state resultsare presented in Fig. 2(b). As expected it is evident that increasedcross-ow velocities reduce ECP at the membrane, which in turn

provides for higher water permeation rates.

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In Fig. 2(a ), signicant ECP is observed at the draw side of themembrane for cross-ow velocities

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Fig. 3. Solute mass fraction proles away from membrane at thedraw side for dif-ferent / values.The mean cross-ow velocity forall simulations was 0.1m s 1 .The values seen in thelegend represent / .

as [24] :

d,m

d,b = e| Jw |/k d (22)

where kd is a mass transfer coefcient. CombiningEqs. (7), (21) and(22) we obtain:

d,i = d,b e| Jw |(F +1/k d ) = d,b e|

Jw |K (23)

From this we see that what is usually denedas the soluteresis-tivity K in theporous support is actually a combination of theactualsolute resistivity in the porous layer F and a mass transfer coef-cient kd :

K =F +1kd

(24)

In accordance with the model developed by Lee et al., the soluteresistivity can be written as [21] :

K =S

D AB (25)

where S is a structural parameter for themembranewhich dependson its thickness, tortuosity andporosityonly. The structural param-eter S is usually calculated using K in place of F in Eq. (25) , and K isin turn estimated by tting a water ux equation to experimentalresults forthewaterux.Since K depends onthemass transfercoef-cient itin turn also depends onthe owconditions,so toobtainthe

Fig. 4. Using a geometric relation for kd as described in [36] , this gure depicts ananalytical expression for K as a function of mean cross-ow velocity using Eq. (24) .Assuming negligibleECP, i.e. f ,m = f ,b and d ,m = d,b , the mass-transfercoefcientsK as determined from the numerical results presented in Fig. 2 are also depicted inthe gure. The dotted line represents the solute resistivity to diffusion within the

porous support layer F .

Fig.5. Simulations performedat differentbulk draw/feed solute concentrations. (a)Zero feed solute concentration with increasing draw solute concentration. (b) Con-stant draw solute mass fraction of 0.09 with increasing feed solute concentration.(c) Water permeation uxes corresponding to thesimulationsshown in (a)and (b).

actual structural parameter S which should only depend on mem-brane characteristics, one must consider the ow conditions. It isrecommended that ow conditions which minimize ECP are uti-lized in order to ensure a large kd and thereby a value of K whichapproaches F , thus making the calculated S value independent of ow conditions.

Based on correlations for the geometry used here, the masstransfer coefcient kd can be calculated as a function of ow rateusing lm-theory models [36] . Using the relation for kd found in[36] we depict the value of K as a function of cross-ow veloc-ity using Eq. (24) in Fig. 4. From the gure it is clear that for K to

attaina value close to F the cross-owvelocitymust be higher than

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0.1ms 1 . It is notedthat turbulence-promoting spacers,morecom-plex case geometries and slip velocity at the boundary may workto enhance mass transfer and thereby reduce K at a given meancross-ow velocity [4,10,35] . Naturally, the coefcient F could beestimated from a given experiment simply by using Eq. (24) andexpressions for kd obtained with lm-theory or by some othermeans. Such simple expressions for the mass transfer coefcientskd are however only available for the simplest of geometries, and

even for the geometry investigated here some discrepancy is seenbetween the analytical expression and the results obtained in theCFD model, see Fig. 4. The advantage of using a CFD model is thatthe ECP proles are fully resolved directly in the model.

3.4. Osmotic pressure dependence

To study how the osmotic pressure difference across the mem-brane inuences the solute mass fraction proles andwater uxes,a seriesof 9 simulations were performedwithdrawinlet m A-valuesranging from 0.01 to 0.08, keeping the feed inlet m A = 0. Resultingmass fraction proles are presented in Fig. 5(a ). Another series of 8simulations were run with the draw inlet m A kept at 0.09 and withfeed inlet m A-values ranging from 0.01 to 0.09. Solute mass frac-tion proles areshown in Fig. 5(b).The wateruxes recorded forallsimulations are presented in Fig. 5(c). A no-slip velocity boundaryconditionwas usedin allthe simulations,see discussion Section 3.2 .

As the bulk draw concentration is increased in Fig. 5(c), a non-linear increase in water ux was observed as expected from Eq.(5) . From Fig. 5(a ), it is seen that increasing the draw concentrationresults in increasedECP, whichis to be expected from the ECPmod-ulus inEq. (22) . Upon addition of soluteto thefeed,the waterux isdecreased by the combined effect of increased ECP on the feed sideanddecreased bulkdrivingforceacross themembrane. InFig. 5(b ) itis seen how when the feed concentration is increased initially, ECPon the feed side becomes more signicant, however as the feedconcentration approaches the draw concentration, the decreasedwater ux results in lower ECP on bothsides of the membrane. Thelower water ux additionally decreases the severity of ICP and all

the effects taken together are what cause the hysteresis observedfor the water ux in Fig. 5(c), which corresponds to what has pre-viously been observed experimentally for FO membranes [24] .

4. Concluding remarks

We have developed a generalized computational model formembrane systems and used it to investigate water ux and con-centration polarization in FO systems with composite membranes.The model was developed within an open source CFD frameworkwith a custom built weakly compressible transient solver andexplicit boundary conditions for the velocity and solute mass frac-tion on the membrane. The code was validated by comparing theresults to previously published water and solute uxes, and veri-

ed by the existence of overall mass balance in the chamber andux balance at the membrane boundary.

The model was used to reveal the existence of external dilutiveconcentration polarizationon the draw side of an FO experiment inwhich an asymmetric membranewas faced withits porous supportagainst the draw side. The external concentration polarization onthe porous support is not normally considered in semi-analyticalapproaches, since for realistic applications it can be assumed thatthe presenceof eddy-promoting spacers andhighcross-ow veloc-ities will largely negate the external polarization. In cases wheresuch optimized conditions cannot be assumed, our results showhow CFD simulations can be used to reveal and quantify externalconcentration polarization.

It was demonstrated that a tangential slip-velocity could be

included in the CFD model in order to parameterize and investi-

gate the effects of complex ow dynamics near the membrane.The results show that a tangential ow along the membraneresult in decreasedexternalpolarization andthereforeenhance thewater ux, however not to such a degree that external polariza-tion becomes negligible. Investigations of the inuence of osmoticpressure difference across the membrane revealed additionalinformation about how different feed and draw concentrationscontribute differently to the effective osmotic pressure across theactive layer of the membrane.

The main advantage of the presented CFD model over tradi-tional semi-analytical approaches is itscapabilityof simultaneouslyresolving the effects of parameters such as slip velocity, externaland internal concentration polarization, physical uid propertiesand cross-ow velocity in a membrane chamber. The model is eas-ily extended to three-dimensionalgeometries andcanfurthermorereadily be usedto studythe effect of mass-transfer promoting spac-ers, chamber geometry and chamber size, albeit at a signicantextra computational cost.

Acknowledgements

C. Hlix-Nielsen, L. Yde and C.Y. Tang were supported by the

Environment & Water Industry Programme Ofce of Singapore(EWI) through a collaboration project #MEWR 651/06/169. L. Ydeand C.J. Johnson were also supportedby the DanishScience of Tech-nology and Innovation.

Nomenclature Nomenclature

Symbol A pure water permeability (m(s Pa) 1 ) slip coefcientB solute permeation coefcient (ms 1 )C solute concentration (kgm 3 )D AB solute diffusion coefcient (m 2 s1 )g gravitational acceleration (m s 2 )h height of chamber (m) Js solute ux (kg (m 2 s)1 ) Jw water permeation ux (ms 1 )K diffusion resistivity (sm 1 )kd mass transfer coefcient (ms 1 )

permeability (m 2 )m A solute mass fraction (kgkg 1 )n surface normal vectorn surface normal direction (m)

viscosity of uid (Pa s) p pressure (Pa)

osmotic pressure (Pa)R solute separation coefcient

Re Reynolds numberuid denisty (kg m 3 )

S structural parameter (m)U uid velocity vector (ms 1 )U mean cross-ow velocity (m s 1 )Uslip surface slip velocity (ms 1 ) x membrane tangential axis (m) y membrane normal axis (m)

Indexd draw-side of membrane f feed-side of membranei interface between active layer and supportm at the membrane surface

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