Modeling of flat-sheet and spiral-wound nanofiltration configurations and its application in seawater nanofiltration

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1 Modelng of flat-sheet and spral-wound nanofltraton confguratons and ts applcaton n seawater nanofltraton The MIT Faculty has made ths artcle openly avalable. Please share how ths access benefts you. Your story matters. Ctaton As Publshed Publsher Roy, Yagnasen, Mostafa H. Sharqawy, and John H. Lenhard. Modelng of Flat-Sheet and Spral-Wound Nanofltraton Confguratons and Its Applcaton n Seawater Nanofltraton. Journal of Membrane Scence 493 (November 2015): Elsever Verson Author's fnal manuscrpt Accessed Mon Apr 16 16:13:03 EDT 2018 Ctable Lnk Terms of Use Creatve Commons Attrbuton-Noncommercal-Share Alke Detaled Terms

2 Ths pre-prnt corresponds to the followng artcle:y. Roy, M.H. Sharqawy, and J.H. Lenhard V, Modelng of Flat- Sheet and Spral-Wound Nanofltraton Confguratons and Its Applcaton n Seawater Nanofltraton, J. Membrane Sc., 493: , 1 Nov ' Modelng of Flat-Sheet and Spral-Wound Nanofltraton Confguratons and Its Applcaton n Seawater Nanofltraton Yagnasen Roy 1, Mostafa H. Sharqawy 2, John H. Lenhard V *, 1 1 Department of Mechancal Engneerng, Massachusetts Insttute of Technology, Cambrdge, MA , USA 2 Department of Mechancal Engneerng, Kng Fahd Unversty of Petroleum and Mnerals, Dhahran 31261, Saud Araba Abstract The Donnan Sterc Pore Model wth delectrc excluson (DSPM-DE) s mplemented over flatsheet and spral-wound leaves to develop a comprehensve model for nanofltraton modules. Ths model allows the user to gan nsght nto the physcs of the nanofltraton process by allowng one to adjust and nvestgate effects of membrane charge, pore radus, and other membrane characterstcs. The study shows how operatng condtons such as feed flow rate and pressure affect the recovery rato and solute rejecton across the membrane. A comparson s made between the results for the flat-sheet and spral-wound confguratons. The comparson showed that for the spral-wound leaf, the maxmum values of transmembrane pressure, flux and velocty occur at the feed entrance (near the permeate ext), and the lowest value of these quanttes are at the dametrcally opposte corner. Ths s n contrast to the flat-sheet leaf, where all the quanttes vary only n the feed flow drecton. However t s found that the extent of varaton of these quanttes along the permeate flow drecton n the spral-wound membrane s neglgbly small n most cases. Also, for dentcal geometres and operatng condtons, the flatsheet and spral-wound confguratons gve smlar results. Thus the computatonally expensve and complex spral-wound model can be replaced by the flat-sheet model for a varety of purposes. In addton, the model was utlzed to predct the performance of a seawater nanofltraton system whch has been valdated wth the data obtaned from a large-scale seawater desalnaton plant, thereby establshng a relable model for desalnaton usng nanofltraton. Keywords * Correspondng author Emal address: lenhard@mt.edu (John H. Lenhard V), Tel

3 Nanofltraton, Flat sheet, Spral wound, Seawater, Modelng 2

4 Nomenclature A k porosty of membrane C concentraton mol m -3 C membrane volumetrc charge densty mol m -3 X ds contact area between channel and membrane n each cell m 2 D solute dffusvty m 2 s -1 D H hydraulc dameter of feed channel m e 0 electronc charge ( C) C f frcton coeffcent F Faraday s constant C eq -1 G Gbbs free energy J v Van 't Hoff coeffcent I onc strength mol L -1 h channel heght m j solute flux mol m -2 s -1 J w solvent permeaton flux m s -1 k mass transfer coeffcent n feed channel m s -1 k B Boltzmann constant ( J K -1 ) J K -1 K hndrance factor l dstance along the feed channel m L membrane length m L mx mxng length of spacer m m! mass flow rate mol s -1 N A Avogadro number ( mol -1 ) mol -1 P pressure Pa Δ P net net drvng pressure Pa Δ P loss hydraulc pressure loss along feed channel Pa Q flow rate m 3 s -1 r Stokes radus of solute m r pore radus of membrane m pore R unversal gas constant J mol -1 K -1 T temperature K u Velocty m s -1 W membrane wdth m x dstance normal to membrane m Δ x membrane actve layer thckness m z valence of speces 3

5 Greek symbols γ actvty coeffcent ε absolute permttvty of vacuum ( Fm -1 ) F m -1 ε 0 delectrc constant of medum ζ potental gradent at feed-membrane nterface V m -1 η mxng effcency of spacer λ rato of solute Stokes radus to pore radus ν knematc vscosty m s -1 π osmotc pressure N m -2 ρ densty kg m -3 Φ sterc parttonng factor Φ Born solvaton factor for parttonng B ψ membrane potental V Subscrpts c Convectve d Dffusve D Donnan potental f feed bulk solute speces n Inlet m feed-membrane nterface out Outlet p permeate just outsde the membrane pore nsde pore w Solvent Bulk Dmensonless Parameters Pe Peclet number 2 Re Sc Reynolds number Schmdt number h f u D w u w D H ν ν D 4

6 1. Introducton Nanofltraton (NF) s a pressure drven membrane-based water purfcaton process wth performance between that of reverse osmoss (RO) and ultrafltraton (UF) [1], [2]. The nterplay of three excluson mechansms, the sterc effects, Donnan excluson effects and delectrc effects allow a great degree of varablty n membrane selectvty [2], [3], [4], [5]. In general, nanofltraton shows hgh rejecton of dvalent and multvalent ons [6], [7], [8]. It created a revoluton n the world of separaton technology, prevously domnated by RO, due to ts hgh water permeablty and hence lower energy consumpton n addton to ts on selectvty [2], [9]. In ts early days, nanofltraton was utlzed predomnantly n the dary and chemcal ndustres applcatons [9]. In more recent years, t has been used n a varety of applcatons such as desalnaton [2] [8], wastewater treatment [10], dafltraton [11], petroleum fractonaton [12], and treatment of mnng water [13]. The Donnan Sterc Pore Model wth delectrc excluson (DSPM-DE) s a comprehensve model of the mechansm of nanofltraton. Ths model solves the Extended Nernst Planck equaton (ENP) for each solute speces through the membrane and uses boundary condtons at the membrane surfaces to account for the Donnan excluson, delectrc excluson, and sterc excluson effects. It s an mprovement upon the orgnal Donnan Sterc Pore Model (DSPM) [1] [3] [11] [14], as t explans the mechansm of delectrc excluson, whch s vtal for the correct predcton of the rejecton of multvalent ons by the nanofltraton membrane. In the current work, the delectrc excluson mechansm based on the Born effect s consdered. The Born effect accounts for the energy barrer for solvaton nsde the pores and hence decreased delectrc constant of the solvent [3] [15] [16]. Accordng to the work of Bowen et al [3], ths mechansm 5

7 of delectrc excluson s domnant over the other effect used to explan delectrc excluson, nvolvng mage charges that develop at the nterface of the bulk soluton and membrane (as descrbed by Bandn et al [5]), for most nanofltraton condtons. Ths s explaned by the fact that the small pores n nanofltraton membranes cause the value of the delectrc constant of the solvent nsde the membrane to approach that of the membrane tself and moreover, the mage charges are screened n electrolyte solutons due to the formaton of electrcal double layers [3]. The DSPM-DE model usng the Born effect for delectrc excluson has been well valdated wth lab-scale experments [4]. Geraldes et al. [4] ntroduced the software 'Nanofltran' that solves the dscretzed and lnearzed ENP equatons. Nanofltran s a robust and comprehensve software that consders the non-dealty of solutons and the concentraton polarzaton effect at the feed-sde of the membrane. However, t models a 'small patch' of membrane and does not account for the streamwse dstrbuton of varous quanttes, namely flow parameters such as cross-flow veloctes, solute concentratons, and transmembrane flux as well as solute rejecton profles along a large membrane leaf. Thus 'Nanofltran' cannot be used to descrbe large membranes that are used n large-scale nanofltraton unts. Htherto, to the best knowledge of the authors, a comprehensve model of a spral-wound module of nanofltraton that accounts for the detaled mechansm of nanofltraton has not been ntroduced. Schwnge et al. [17] showed a detaled analyss of spral wound membranes and the spatal dstrbuton of quanttes such as transmembrane flux, transmembrane pressure dfference, feed concentraton, and crossflow velocty along the membrane. Ths study, however, s general and can be appled to reverse osmoss, nanofltraton, ultrafltraton, or mcrofltraton membranes. A complete study of nanofltraton membranes demands attenton not only to the 6

8 general features of the membrane, but also to ts unque separaton capablty and mechansm. A comprehensve study of nanofltraton mechansm nvolves a combnaton of the dffusve transport, electro-mgraton and convectve transport through narrow pores, therefore requrng use of the Extended Nernst-Planck equaton, modfed by the hndered transport theory [3], [4], [5]. The NF model ntroduced n the present work s based on the DSPM-DE model, appled over a flat-sheet and spral-wound leaf. The results from the ndvdual leaves can be easly treated as f n a parallel connecton to depct a spral-wound element, whch may n turn be put nto a tran of spral-wound elements that exst n seres wthn a pressure vessel. The user can make use of several degrees of freedom n the defnton of the membrane, namely the membrane pore radus, membrane effectve thckness, membrane charge, pore delectrc constant and membrane dmensons. It s also possble to test the behavor of ndvdual leaves or an ndvdual element for dfferent feed flow rates, compostons, and transmembrane pressures. Varous feed water propertes, such as ph levels and temperature can be ncorporated nto the model by characterzng the membrane and subsequently usng these parameters n the model [14]. Another mportant am of ths work s to provde results for each consttuent on of seawater from ts nanofltraton modelng. Most commonly, seawater s modeled by a sodum-chlorde soluton at a concentraton smlar to that of seawater [18]. Whle ths s a reasonable approxmaton for seawater [19] [20], t does not gve any nformaton about the permeate concentratons of the many ndvdual ons n seawater. Thus, t fals to provde essental nformaton regardng concentraton of scale-causng ons such as magnesum, calcum, sulphate and carbonate ons that enter thermal desalnaton processes for whch nanofltraton s used as a pretreatment [21]. 7

9 The use of nanofltraton as a pretreatment stage n thermal desalnaton processes, namely, Mult-Stage Flash (MSF) and Mult-Effect-Dstllaton (MED) seawater desalnaton plants, n order to ncrease the top brne temperature (TBT), has been a subject of nterest and study by several researchers [13], [21], [22], [23]. Nanofltraton effcently removes scale-causng ons such as calcum, magnesum, sulphate, and carbonate ons and hence adds potental to ncrease the top brne temperature (TBT) n an MSF or MED plant. In reference [21], the Salne Water Converson Corporaton, Research and Development Center (SWCC-RDC) demonstrated that the addton of a nanofltraton unt as pretreatment to MSF was found to be successful n the removal of turbdty, resdual bactera, and seawater total dssolved solds (TDS). Moreover, snce t resulted n lower concentratons of the scale formng consttuents, the TBT could be ncreased up to 160oC [21]. Consequently, t reduced the thermal energy nput and decreased the antscalant addtves, as evdent from expermental results of a plot plant. Several expermental efforts have been made on nanofltraton of seawater, both at lab scale as well as n desalnaton plants [23], [24], [25]. However, the am of ths work s to provde a useful model to reduce the number of experments requred for such studes. In summary, ths work ams at ntroducng a comprehensve model for flat-sheet and spralwound nanofltraton membranes and evaluates ther performance for the seawater desalnaton applcaton. A model s ntroduced for analyzng commercally used nanofltraton elements that allows the user to understand the mechansm of fltraton and provdes the flexblty to smulate a wde range of membrane types by adjustng the varous key parameters that characterze the membrane. Further, a detaled analyss of seawater nanofltraton usng ths model s descrbed. 8

10 2. Mathematcal Model The model presented n ths work s an ntegrated verson of the prevously developed DSPM-DE model [4]. In ths work, the elemental equatons of that model are 'threaded together' to smulate the transport over a large membrane leaf wth locally varyng condtons. The large membrane leaf s dvded nto cells and the DSPM-DE model equatons are appled by movng from one cell to another whle accountng for the mass conservaton of each solute speces and of the solvent. In addton, the hydraulc pressure losses along the feed flow drecton n the feed channel are consdered. Fgure 1a and 1b are schematc dagrams of the flat-sheet membrane leaf and the spral-wound membrane leaf confguratons respectvely. The two confguratons dffer by the flow arrangement. Fgure 1a shows the flat-sheet membrane confguraton wth the feed and permeate flows n ther respectve channels, flowng parallel to one-another. The membrane leaf has a wdth W and a length L along the feed flow drecton. As shown, the cells have a wdth equal to that of the membrane leaf and they splt the length of the membrane nto several segments. Fgure 1b shows an unwound spral-wound membrane leaf. In ths confguraton, the feed and permeate flow perpendcular to each-other n ther respectve channels. Therefore, n order to capture the varaton of the flow parameters and the rejecton performance of the membrane n both the longtudnal and transverse drectons, the cells are square elements that splt both the wdth and length of the membrane nto segments. Grd ndependence studes of the present work have showed that beyond 100 cells (n the feed flow drecton for the flat-sheet case and 100 cells each n the feed and permeate flow drectons n the spral-wound case), the computatonal results vary by less than 1%. Therefore, 100 cells were taken for all cases n the present work. 9

11 (a) (b) Fg. 1 Schematc dagrams of (a) flat-sheet membrane, and (b) unwound spral-wound leaf. 10

12 2.1. Governng Equatons The Extended Nernst-Planck equaton (ENP) descrbes the transfer of ons under the nfluence of concentraton gradent, electrc feld, and nerta forces. For each solute '' transferrng through the membrane pores, the ENP equaton s gven by Eq. (1). dc z C dψ dx, pore, pore, pore j, pore = D, pore F + K, cc, pore dx RT D J w (1) where j, pore s the flux of the speces '' nsde a pore, the frst term on the rght represents the transport due to dffuson (concentraton gradent), the second term represents the transport due to electrc feld (potental gradent), and the last term represents the transport due to convectve forces. Due to the extremely small pore szes n nanofltraton membranes, the transport of the solute s hndered. Thus, the ENP has been modfed by the hndered transport theory [26] [27] through ntroducton of the coeffcents K c K, d,, whch gve a measure of the lag of a sphercal solute movng nsde a cylndrcal pore and the enhanced drag experenced by the solute respectvely. Both these coeffcents are functons of the rato of solute radus to pore radus, λ [3]. The dffusvty (dffuson coeffcent) of the solute nsde the pore s related to the dffusvty of the solute n the bulk soluton as gven by Eq. (2): D, pore = K, d D, (2) For λ 0. 95, [4] 11

13 K, d (9 / 8) λ lnλ λ λ λ λ λ λ λ = Φ (3) and for λ > 0. 95, [4] K, d 5 / 2 1 λ (4) = λ For convecton, the hndrance factor s [4] K, c λ 1.907λ 0.834λ (5) λ 0.741λ = 2 The equlbrum boundary condton at the membrane-feed soluton nterface due to the combnaton of the sterc, Donnan and delectrc effects s gven by γ, pore γ, m C C, pore, m z F = ΦΦ B exp Δ D, m RT ψ n (6) It s to be noted that C, n equaton (6) s the solute concentraton just wthn the pore pore 'entrance'. Ths s mportant because the solute concentraton vares along the pore. C, s the m feed concentraton at the membrane-feed soluton nterface. γ, pore,γ, m are solute actvty coeffcents just wthn the pore entrance and at the membrane and feed soluton nterface respectvely; Φ, Φ are the sterc parttonng factor and solvaton energy contrbuton to B parttonng respectvely, and Δ ψ D, m s the Donnan potental on the feed sde, whch s the 12

14 potental dfference between the pont just wthn the pore entrance and the soluton at the feedmembrane nterface [4]. Furthermore, the sterc parttonng factor s gven by [4] Φ 2 r = ( 1 λ ), where λ = (7) r pore The Born solvaton energy contrbuton to parttonng s gven by [4] ΔG Φ = B (8) k BT where ΔG s the Gbbs free energy of solvaton gven by [4] ΔG = z 2 e πε r 1 ε pore 1 ε f (9) where e0 s the electronc charge, ε 0 s the absolute permttvty of vacuum, ε pore, ε f are the relatve permttvty of the solvent wthn the pore and n the bulk feed soluton (taken equal to the delectrc constant of pure water) respectvely and r s the Stokes radus of the solute. Smlarly, the equlbrum boundary condton at the membrane-permeate soluton nterface s gven by γ, pore γ, p C C, pore, p z F = Φ Φ B exp Δψ D, p RT out (10) 13

15 In Eq. (10), C, pore s the concentraton at the ext of the pore, just wthn the membrane and C, p s the concentraton n the permeate soluton just outsde the membrane [4]. Δ ψ D, p s the Donnan potental dfference between the pont just wthn the pore ext and the soluton at the permeate-membrane nterface. The actvty coeffcents are calculated by the Daves equaton gven by [4]. 1/ 2 2 I ln( γ ) = Az 0. 3I 1/ 2 (11) 1+ I where I s the onc strength gven by I 2 = 0.5 z C and A s a temperature-dependent parameter gven by A 3 1/ 2 e0 N A 1/ 2 3 ln( 10) 4π 2 ( ε k = 3 3 1/ T ) 2 B (12) (13) where k s the Boltzmann constant, ε s the permttvty of the medum (the value wthn the B pore or n the bulk feed/permeate), N s the Avogadro number. A The ENP equaton (gven by Eq. (1)) and the relevant boundary condtons (gven by Eq. (6) and Eq. (10)) are solved numercally for each solute n each cell. Equatons (6) and (10) state that the concentraton just wthn the membrane versus that at the contact surface of the membrane and the feed/permeate soluton s governed by the sterc, Donnan, and delectrc excluson effects. It s assumed that the membrane element s workng under steady state condton and both solute and solvent mass flow rate are conserved n travellng from one cell to the next. Snce at steady state, the molar flux of the solute s ndependent of ts poston nsde the pore, the followng relaton s vald [4]: 14

16 j, pore = C, p J w (14) wherec, s the permeate concentraton just outsde the membrane at the permeate sde [4]. p The mass balance of each solute speces '' n the feed channel s gven by Eq. (15). dm! = C J ds (15), f, p w where m, s the mass of the solute '' n the feed sde of a cell. Smlarly, the mass balance of f each solute speces '' n the permeate channel s gven by Eq. (16). dm! C J ds (16), p =, p w On the other hand, the solvent mass balance on the feed sde s gven by Eq. (17). dq f = J ds (17) w Smlarly, the solvent mass balance on permeate sde s gven by Eq. (18). dq p = J wds (18) The DSPM-DE model equatons descrbed above (Eq. 1 to 13) are dscretzed as shown n reference [4] and solved numercally usng MATLAB (verson R2013b). Equatons are dscretzed by the forward dfferences method. Alongsde solvng the model equatons and gettng the velocty and concentraton felds, the hydraulc pressure loss along the feed flow drecton s determned by the frcton factor. The correlaton for the frcton coeffcent n the 15

17 feed channel of FlmTec membrane element s taken from [28] whch was ftted wth respect to expermental data f = (19) 0.3 Re Accordngly, the pressure drop along the feed channel n the feed flow drecton s gven by Eq. (20). f l 2 Δ Ploss = ρ wuw (20) 2 D H where l s the length along the feed channel n the feed flow drecton, of flow at that locaton, and uws the bulk velocty D s the hydraulc dameter of the feed channel. In a sngle leaf, the H permeate flow rate s low compared to the feed flow rate even at hgh recovery ratos. Consequently, the permeate Reynolds number s also low and there s no sgnfcant hydraulc pressure loss n the permeate channel. Therefore, the hydraulc pressure drop n the permeate channel was not ncluded and the permeate channel was consdered to be unformly at atmospherc pressure. For the mass transfer coeffcent, the expresson gven by [29] and [30] for spral wound membranes (whch ncludes the effect of spacers) was used as gven by Eq. (21) k η = η 1/ 2 D h, f Sc 1/ 6 Pe h f L mx 1/ 2 (21) 16

18 where: η s the mxng effcency of spacer; L mx s the mxng length of the spacer; h f s the feed channel heght; Pe s the Peclet number n the channel gven by Pe = 2h u D f, w ; and Sc s the Schmdt number for each solute speces, gven by Sc = ν. D, Concentraton polarzaton on the feed sde s consdered by applyng a mass balance at the nterface between the feed soluton and the membrane, as gven by Eq. (22) [4]. However, the permeate sde concentraton polarzaton s neglected, whch s a reasonable assumpton for pressure-drven membrane processes such as nanofltraton and reverse osmoss [31], [32], [33]. j = k ( C, m C, f ) + J wc, m zc, md, F ξ RT (22) where ξ s the electrcal potental gradent at the feed-membrane nterface n the contnuum phase, just outsde the electrcal double layer [4]. The transmembrane osmotc pressure s calculated by the Van 't Hoff equaton n any cell Δπ = RT C C ) (23) v ( m p where Cms the salt concentraton at the feed-membrane nterface, v s the Van 't Hoff coeffcent, R s the unversal gas constant, and T s the absolute temperature. Fnally, the transmembrane solvent flux s calculated from Eq. (24) as shown below: 17

19 J w = ΔP net 2 2 rpore rpore = (( Pf Pp ) Δπ ) Δx Δx 8νρ w 8νρw Ak Ak (24) For our smulatons, the NF270 membrane manufactured by Dow and FlmTec was consdered. Several authors have nvestgated ths membrane and reported expermental results for the rejecton rato at dfferent fluxes. By fttng the expermental data to the DSPM-DE model, t s found that the membrane has an average pore radus of 0.43 nm and an actve layer thckness to porosty rato of (Δx/A k ) about 1µm [15], [34]. In addton, the pore delectrc constant s 42.2 from fttng wth experments wth sodum-chlorde [15]. Wth the excepton of the fttng from magnesum-sulphate expermental data, the NF 270 membrane s found to have a pore delectrc constant close to 40 after fttng wth several other solutes [15]. In fact, t s seen that for membrane characterzaton purposes, among the four parameters requred to characterze a nanofltraton membrane, namely pore radus ( r pore ), rato of the membrane actve layer thckness to porosty ( Δ x / Ak ), pore delectrc constant ( ε pore ), and membrane volumetrc charge densty ( C ), the frst three parameters can be assumed unque for X a gven membrane wthout much error. These three parameters do not change wth the solute concentraton n the feed, soluton ph or the nature of the solute [14], [15]. However, when any of these parameters are ftted wth respect to data from dfferent solutes, ther ftted values may vary slghtly [3], [14], [15]. These values are numercally very close and therefore, an average value s usually taken [15]. 18

20 The remanng parameter, the membrane volumetrc charge densty ( C ), depends on the solute and solvent nature, the solute concentraton, and the ph of the soluton [3], [15], [16]. Therefore, ths parameter must be carefully determned for each case nvestgated. For the present work, the values of pore radus, actve layer thckness to porosty rato, and the pore delectrc constant for NF270 as mentoned prevously were taken, and an effectve membrane volumetrc charge densty was ftted to data taken from [35] (see Table 1.2 n ths reference). For ths, the flow parameters n the present model were adjusted smlar to those n [35] and the rejecton rato and recovery rato from the model are then matched (wth those measured n [35]) by adjustng the membrane charge n the model. For nstance, for an nlet feed concentraton of 2000 ppm sodum-chlorde, n order to acheve a recovery rato of about 10% and mean rejecton of sodum-chlorde of 80%, an effectve membrane charge densty of X C =-45mol/m 3 was ftted. X Comparng ths value wth values ftted by other researchers for FlmTec membranes, t was found that ths value s wthn reasonable lmts for the DSPM-DE model [16]. Therefore, a unform average membrane charge across the entre membrane s assumed n our model usng the calbraton step dscussed above. It s mportant to note that n reference [16], the varaton of the membrane charge densty wth solute concentraton s nvestgated for NF250 and NF300, showng that the membrane charge densty ncreases lnearly and monotoncally wth the concentraton of sodum-chlorde and consequently, the rejecton rato ncreases monotoncally. Ths s verfed by [3] where they show the same trend for Desal-DK membrane for both sodumchlorde and magnesum chlorde solutons. Further, from our smulatons, t s observed that wth ncreasng membrane charge, each of the quanttes such as retentate concentraton, permeate concentraton, rejecton rato, recovery rato, transmembrane flux and feed flow rates ether ncrease or decrease monotoncally. Therefore, f varaton of membrane charge across the 19

21 membrane were ncluded n the smulaton, t would fne-tune the results for each of these quanttes, but would not affect the trends observed n the study. The values used to characterze the NF-270 membrane for ths case are gven n Table 1. Table 1 Values of membrane characterzaton parameters Parameter Value r 0.43 nm pore Δ x / 1 µm A k ε 42.2 pore C -45 mol/m 3 X For the spral-wound membrane confguraton, the conservaton equatons are modfed to allow for varaton of varous flow parameters and concentraton profles both n the drecton of flow of feed as well as n the perpendcular drecton, due to the cross-flow of the permeate stream. Accordng to [36], for a spral-wound leaf, the heght of the feed channel s very small whch allows the curvature of the channel to be gnored. Thus, the feed channel n a spralwound leaf can be modeled as a thn rectangular duct wth a heght range of mm. In the present model, the feed channel heght was fxed at 0.7 mm, the permeate channel heght s of 0.3 mm, and each leaf has a dmenson of 1m 1m, whch are commonly used values n commercal spral-wound membranes [28]. For the flat-sheet membrane, we assumed the same dmensons as for the spral-wound module n order to make ther comparson easer. Snce the am of our study s to nvestgate the effect of dfferent flow parameters on the nanofltraton performance n the two confguratons, t s necessary for the two modules to be smlar n structure, thereby allowng us to study the dfference n performance due to ther dfferent flow confguratons. 20

22 2.2 Model Valdaton Referrng to the schematc dagram shown n Fg. 1, valdaton of our model s performed by comparng the performance results at varous operatng lmts. For nstance, when the wdth and length of the leaf are reduced to a few centmeters, the model results from both the flat-sheet and spral-wound confguratons were compared wth the expermental measurements conducted at the lab scale usng test cells. In ths manner, the large membrane leaf was geometrcally reduced to a 'small patch' of membrane. At ths lmt, there s neglgble varaton of quanttes such as feed concentraton and rejecton rato along the length and wdth of the membrane. Excellent agreement wth the expermental data presented by [15] s obtaned. In ths reference, experments are performed usng a cross-flow test cell manufactured by GE Osmoncs, usng NF270 and NF99HF membranes at ther respectve soelectrc ponts (when membrane charge s effectvely zero). The comparson of the smulaton results at ths lmt and the expermental data s shown n Fg. 2. In ths lmt, snce the flat-sheet and spral-wound modules gve very smlar results, only one set of smulaton data s presented for valdaton. 21

23 Fg. 2 Comparson between model results and lab-scale experments for strred-cell szed membranes [15] Another valdaton was conducted wth respect to the data provded n Dow techncal manual whch descrbes the expermental performance of NF270 membrane under standard test condtons. For the gven set of nput condtons, the smulaton results of the NF270 membrane from the present model are compared wth the data provded by Dow [35] usng the membrane characterstcs shown n Table 1. Our model predcts a recovery rato of 10% and a mean rejecton rato of 80% for a feed soluton of 2000 ppm sodum-chlorde as tested and reported n the Dow s manual [35]. These values are n exact agreement wth the expermental values reported n Dow manual for the recovery rato and rejecton rato respectvely. Furthermore, from our model, the characterstc features of the spral-wound membranes can be observed. These features are n good agreement wth the observatons found from the detaled 22

24 modelng study of spral-wound leaves presented n [17]. Fgure 3a, 3b, and 3c show the varatons of the trans-membrane pressure (TMP), trans-membrane flux, and feed concentraton on the membrane surface respectvely. Ths smulaton s conducted at a feed pressure of 1Mpa, feed flow rate of 144 L/h, and an nlet sodum-chlorde feed concentraton of 2000ppm. It s shown that the maxmum values of trans-membrane pressure, trans-membrane flux, and velocty occur at the feed entrance (near the permeate ext sde), where the salt concentraton s the lowest [17]. Ths trend s the most promnent for the trans-membrane pressure and to a lesser extent for trans-membrane flux as shown n [17]. In addton, at the dagonally opposte corner of the membrane, at the feed ext (near the permeate entrance) the trans-membrane pressure, transmembrane flux, and velocty show mnmum values where the feed salt concentraton s the hghest. Fgure 3 llustrates the mportant characterstc trats of a spral-wound membrane usng our model. Exact values of concentraton and other quanttes at dfferent ponts on the membrane surface for a nanofltraton spral-wound membrane were not found n lterature whch ndcates the mportance of the present model. 23

25 (a) (b) (c) Fg. 3 Varaton of the trans-membrane pressure, trans-membrane flux, and feed concentraton over the spral-wound membrane leaf at feed flow rate of 144 L/h and nlet feed pressure of 1 MPa (a) the trans-membrane pressure varaton, (b) the trans-membrane flux varaton, (c) feed concentraton. 24

26 3. Results and Dscusson 3.1 Analyss of flat-sheet module In ths secton, parametrc studes are conducted n order to understand the operaton of a nanofltraton module and the effect of dfferent flow parameters on ts performance. The results presented here are for a 2000 ppm soluton of sodum-chlorde at 25 o C. For the parametrc study, a membrane charge of mol/m 3 and a pore delectrc constant of 40.4 are consdered. These values dffer from those s Table 2.1 whch provdes the parameters used n valdaton of the model usng the NF270 membrane. However, as detaled n secton 2.1, the charge of a gven membrane s a functon of feed operatng condtons and so a charge of mol/m 3 corresponds to, for example, a feed ph closer to the so-electrc pont. Furthemore, t s mentoned n secton 2.1 that the NF270 membrane has a pore delectrc constant ~40, as was ftted n reference [15]. Frstly, a sngle leaf of a flat-sheet membrane as shown n Fg. 1a s consdered. The varaton of feed and permeate Reynolds numbers along the membrane n the drecton of feed flow are fundamental n explanng several other trends, so they are nvestgated frst. The feed Reynolds number decreases along the membrane, snce the average bulk flow velocty decreases along the membrane. Ths results from the decrease n the feed flow volume due to permeaton of soluton to the permeate sde through the membrane. At hgher feed flow rates, the feed Reynolds number s greater, as there s greater flow through fxed channel dmensons, causng average bulk flow velocty to be hgher. The permeate Reynolds number ncreases along the membrane, due to the ncrease n permeate flow rate as a result of the flux enterng through the membrane. Accordng to Vtor et al. [36] for rectangular channels, the transton between lamnar and turbulent flow occurs at a Reynolds number between 150 and

27 n the presence of spacers. In our work, operaton over a large range of feed Reynolds number s shown. For nstance, at the mnmum feed flow rate of 60 L/h per leaf, the feed Reynolds number at the nlet s 50, whle for the maxmum feed flowrate of 1000 L/h per leaf, the Reynolds number at the nlet s 600. Fgure 4 shows the varaton of the feed pressure along the membrane length. The feed pressure decreases along the membrane n the feed flow drecton due to hydraulc losses. At hgher flow rates and hence hgher feed Reynolds numbers, the hydraulc losses are greater due to greater average velocty n the feed channel. Thus, the pressure varaton lnes slope down at greater angles for greater flow rates. On the other hand, the permeate Reynolds number s maxmum at the permeate ext, snce the flux permeated through the entre membrane adds up at that pont. However, the maxmum value of permeate Reynolds number does not exceed 20 at any of the feed flow rates nvestgated. Thus, due to the very low permeate Reynolds numbers, the hydraulc losses are nsgnfcant and permeate hydraulc pressure remans essentally unform along the flow drecton. Therefore, the trans-membrane hydraulc pressure (TMP) s essentally a sole functon of the feed pressure. 26

28 Feed pressure [kpa] L/h 100 L/h 600 L/h 1000 L/h Dstance along feed channel length [m] Fg. 4 Feed pressure varaton along the feed flow drecton at 480 kpa nlet feed pressure and dfferent feed flow rates. The varaton of the solute mass transfer coeffcent n the feed channel wth respect to feed flow rate s essental n descrbng the concentraton polarzaton at dfferent flow rates. Its value s proportonal to the feed Reynolds number values, and thus at hgher Reynolds numbers, the concentraton boundary layer s thnner, resultng n lower concentraton polarzaton. Fgure 5a and 5b show the varatons of the bulk feed concentraton at dfferent feed flow rates and feed nlet pressures respectvely. Fgure 6 shows the concentraton at the membrane surface at dfferent feed flow rates. It s notced n Fgs 5 and 6 that at the lowest feed flow rates there s a steep ncrease of the feed concentraton and the concentraton at the membrane surface n the flow drecton. Ths s because the Reynolds number decreases to a small value along the feed channel as a result of a low mass flow rate, whch also leads to a low mass transfer coeffcent, so there s greater concentraton polarzaton. Snce at hgher values of Reynolds number, there s 27

29 the combned effect of the hgher bulk solute mass flow rate n the feed channel together wth the ncreased transport from the membrane to the bulk of the feed (lower concentraton polarzaton), very lttle solute enters the permeate channels. Ths s evdent from Fg. 7, whch clearly shows that the permeate concentraton decreases at hgher feed flow rates. Due to the nverse argument, the permeate concentraton ncreases along the membrane due to the decreasng feed Reynolds numbers Feed bulk concentraton [ppm] L/h 100 L/h 600 L/h 1000 L/h 2000 (a) Dstance along feed channel length [m] 28

30 kpa 750 kpa 1000 kpa Feed bulk concentraton [ppm] Dstance along feed channel length [m] (b) Fg. 5 Varaton of feed bulk concentraton along feed flow drecton at (a) 480 kpa nlet feed pressure and dfferent feed flow rates, and (b) 100 L/h flow rate and dfferent feed nlet pressures Membrane concentraton [ppm] L/h 100 L/h 600 L/h 1000 L/h Dstance along feed channel length [m] Fg. 6 Varaton of the feed concentraton at membrane surface on feed sde along the feed flow drecton at 480 kpa nlet feed pressure and dfferent feed flow rates. 29

31 1500 Permeate concentraton [ppm] L/h 100 L/h 600 L/h 1000 L/h Dstance along feed channel length [m] Fg. 7 Varaton of permeate concentraton along feed flow drecton at 480 kpa nlet feed pressure and dfferent feed flow rates. The trans-membrane osmotc pressure, as gven by Eq. (23) follows the trend of the membrane concentraton, snce the varaton of permeate concentraton along the channel s relatvely small. For a gven feed pressure, the net drvng pressure, defned n Eq. (24) as ΔP net = (( P P ) Δπ ), frst ncreases wth the flow rate and then decreases (see Fg. 8a). Ths f p s because ntally, at lower feed flow rates, the feed pressure domnates over the transmembrane osmotc pressure, resultng n hgh net drvng pressure. However, wth ncreasng feed flow rates, the ncreased hydraulc losses cause the feed pressure to decrease rapdly along the feed flow drecton. Thus, the effect of the trans-membrane osmotc pressure s more promnent at hgher feed flow rates and the net drvng force s decreased. Further apprecaton of ths trend of varaton of net drvng pressure can be obtaned by observng ts varaton at any 30

32 fxed pont of the membrane wth respect to flow rate. The net drvng pressure at the md-pont of the membrane at dfferent flow rates s shown n Fg. 8b. It s to be noted that the flow rate at whch the net drvng pressure s maxmum wll be dfferent f a dfferent nlet feed concentraton s consdered or dfferent membrane propertes are consdered but the trend of varaton wth flow rate wll be smlar. Snce the trans-membrane flux s drectly dependent on the drvng force, the varaton of the trans-membrane flux s exactly smlar to the drvng force as clearly llustrated n Fg Drvng pressure (ΔP-Δπ) [kpa] L/h 100 L/h 600 L/h 1000 L/h Dstance along feed channel length [m] (a) 31

33 (b) Net drvng pressure (Δ P-Δ π) [kpa] Feed flow rate [L/h] Fg. 8 Drvng pressure at 480 kpa nlet feed pressure and dfferent flow rates (a) varaton along the membrane length, (b) at mdpont of the feed channel 32

34 L/h 100 L/h 600 L/h 1000 L/h Transmembrane Flux [L/m 2 -h] Dstance along feed channel length [m] Fg. 9 Varaton of transmembrane flux along feed flow drecton at 480 kpa nlet feed pressure and dfferent feed flow rates. Fgure 10 shows the varaton of the overall rejecton rato of the membrane wth feed pressure and feed flow rate. It s llustrated that the rejecton rato ncreases monotoncally wth the feed pressure. Ths s because the drvng force s hgher, causng solvent permeaton to be hgher, therefore 'leavng behnd' the solute ons. On the other hand, the rejecton rato ncreases wth the flow rate but reaches an asymptotc value. Ths s because when ncreasng the flow rate, ntally the drvng force and hence the solvent permeaton ncreases but as the flow rate s further ncreased, the drvng force s decreased (due to the role of hydraulc pressure losses), causng lower solvent flux and hence decreased rejecton. Fgure 11 shows the varaton of the net recovery rato wth flow rate, at dfferent feed pressures. As shown n ths fgure, at hgher flow rates, the recovery rato decreases. Ths s because the net drvng pressure decreases, 33

35 causng a decrease n the trans-membrane flux. Snce the permeate flow s created by the flux comng n from the feed sde, the decreased trans-membrane flux mples a reduced recovery rato. Smlarly, the recovery rato ncreases wth the feed pressure because the drvng force for permeaton ncreases causng a greater trans-membrane flux and hence greater recovery Rejecton rato kpa 750 kpa 1000 kpa Feed flow rate [L/h] Fg. 10 Rejecton rato for NaCl at dfferent feed flow rates and nlet feed pressures. 34

36 kpa 750 kpa 1000 kpa 0.7 Recovery rato Feed flow rate [L/h] Fg. 11 Recovery rato at dfferent feed flow rates and nlet feed pressures. 3.2 Analyss of spral-wound module The operaton of the spral-wound membrane can be explaned smlarly to that of the flatplate confguraton. The trends of varaton of quanttes such as the Reynolds numbers, mass transfer coeffcents, feed concentratons, rejecton ratos and recovery rato wth respect to feed flow rate and feed pressure are smlar to the flat-sheet case. The key dfference s that n the spral-wound module, these quanttes also vary n the permeate-flow drecton, perpendcular to the feed flow. To compare the performance of the spral-wound membrane wth the flat-sheet, Fg. 12 shows surface plots of the trans-membrane hydraulc pressure (TMP), trans-membrane flux and rejecton rato n the two confguratons under smlar operatng condtons (.e. flow 35

37 rate of 250 L/h, feed nlet pressure of 480 kpa, and nlet feed concentraton of 2000 ppm sodumchlorde). The spral-wound membrane shows the maxmum and mnmum values of the TMP and flux at opposte corners of the membrane. The range of the three quanttes plotted s smlar n both the flat-sheet and spral-wound membranes. Therefore, for dentcal leaf geometry and dentcal flow condtons, the flat-sheet and spral-wound confguratons gve smlar results. Snce the computatonal model for the flat-sheet membrane s computatonally less tme consumng and less complex compared wth the spral-wound confguraton, t would be advantageous to use the flat-sheet confguraton model nstead of the spral-wound one wthout losng sgnfcant nformaton. (a) (b) 36

38 (c) Fg. 12 Comparson of values for spral-wound membrane and flat-sheet membrane at 250 L/h feed flow rate and 480 kpa feed pressure: (a) trans-membrane pressure (TMP), (b) trans-membrane flux, and (c) rejecton rato. To nvestgate ths further, t s necessary to check f under dfferent flow condtons, the varaton of quanttes n the permeate flow drecton s sgnfcant. Four flow rates spread over a wde range, 1805 L/h, 361L/h, 110 L/h and 50L/h were nvestgated at dfferent values of feed nlet pressures (480 kpa, 750 kpa and 1000kPa) and 2000 ppm sodum-chlorde soluton was consdered. It was observed that the varaton of the feed flow rate, net drvng pressure, transmembrane flux, rejecton rato, and feed concentraton n the drecton of permeate flow (percentage varaton from begnnng of permeate flow to the permeate ext) s less than 10 % n all cases and less than 5% n most cases. The only quantty that shows marked varaton along the permeate flow drecton s the permeate Reynolds number whch has a maxmum value of about 20 at the feed entrance, near the permeate ext. It s close to zero along the edge of the leaf where the permeate flow begns. 37

39 Therefore, t can be concluded that for smlar geometrc specfcatons, the computatonally less ntensve flat-sheet model ntroduced n ths work can be used to predct the performance of the spral-wound membrane. Ths observaton s especally helpful for the nvestgaton of large scale-systems where seres of spral-wound membrane elements are used. Snce n a membrane element the spral-wound leaves are n parallel, the rejecton rato and recovery rato for the entre element s the same as that of the ndvdual leaf. In order to model a seres of elements, the modelng can be smply made so that the ext feed flow rate and pressure wll be the nlet values for the next element. 3.3 Seawater Nanofltraton The model of nanofltraton ntroduced n ths work s now appled to seawater and t s valdated wth respect to a large scale desalnaton system, the Umm Lujj NF-SWRO plant owned by the Salne Water Converson Corporaton, Research and Development Center (SWCC- RDC) [37], [38]. Gulf seawater concentraton [23], as shown n Table 2, s used as the ntal feed soluton. The setup of the Umm Lujj NF unt descrbed n the references s modeled, and t s attempted to match the overall recovery rato and rejecton rato for each ndvdual on when the flow condtons as specfed n [37] are appled. The nanofltraton unt of the desalnaton plant descrbed n references [37] and [38], conssts of several pressure vessels n parallel. Each vessel conssts of sx spral-wound elements of the DK8040F membrane manufactured by GE- Osmoncs, n seres. Snce the pressure vessels operate n parallel, the rejecton rato and recovery rato of the entre NF unt s represented by the values obtaned for a sngle vessel. In order to model a pressure vessel, sx elements are modeled such that the ext feed pressure and 38

40 flow rate from each element s the nlet for the next element. The nlet feed pressure and nlet feed flow rates are taken as 30 bar and 13.3 m 3 /h respectvely, whch are the values gven for each vessel n [37]. The recovery rato n each vessel n reference [37] was found to be 65%. Table 2 Valdaton of solute rejecton ratos from model for sea-water nanofltraton n a large scale desalnaton plant Ion Concentraton n seawater (ppm) Rejecton rato [37] Rejecton rato from present model % Devaton Ca Mg SO Cl HCO Snce the NF unt descrbed n the Umm Lujj plant [37] [38] uses DK8040F membrane whch falls under the broad category of Desal-DK membranes manufactured by GE-Osmoncs, t s necessary to obtan the membrane characterstcs whch gve a good ft wth the DSPM-DE model. In ths smulaton, the results from the characterzaton of Desal-DK membranes gven n references [1], [3]are used. They found the pore radus to be 0.45 nm and the actve layer thckness to porosty rato ( Δ x / Ak ) to be 3µm (when characterzed by glucose). The pore delectrc constant s found to be 38. However, n order to obtan good correspondence wth the data n [37], values of pore delectrc constant and membrane charge densty are set to ε = 56.5 and C = 80 mol/m 3. The devaton of the pore delectrc constant from the value pore X n lterature may be due to the fact that n the references, the fttng s done wth respect to the rejecton data of a sngle solute such as sodum-chlorde, whereas n seawater there s a mxture 39

41 of ons, each of whch has ts own unque behavor wth the membrane. The nteracton of dfferent solutes wth the same membrane can ndeed be very dfferent. For example, n [15] (Table 4), t s found that the delectrc constant of the membranes NF99HF and NF270 s around 40 from characterzaton wth respect to sodum-chlorde, potassum-chlorde, and sodumsulphate, whle magnesum-sulphate, gves a value around 75 for NF99HF and 65 for NF270, thereby ndcatng that ths salt has a unque chemstry wth the membrane [15]. In addton, n [16] (Table 2), t s seen that characterzaton of volumetrc charge densty for the both the membranes NF300 and NF250 wth respect to sodum-chlorde gve negatve values, whle fttng wth respect to magnesum-chlorde gves a postve membrane charge densty. Snce seawater contans not only sodum, chlorde, sulphate, and magnesum ons, but also several other ons, t s expected that the fnal values of the pore delectrc constant and membrane charge to be an 'average' value that represents the nteracton of all these ons wth the membrane. Furthermore, the membrane charge densty changes wth solute concentraton and therefore wll vary from element to element n the seres. However, n order to fnd a relaton for the varaton of membrane charge densty as a functon of feed concentraton (by the method demonstrated n reference [16]) for seawater, a few experments need to be performed, whch s beyond the scope of our research thus far. As mentoned prevously, a smple correlaton between solute concentraton and membrane charge densty can be obtaned from expermental analyss. However, for seawater the stuaton s made more complex by the fact that t contans a varety of solute ons and the type of correlaton vares from solute to solute. For example, n [39], the charge-concentraton correlaton for calcum-chlorde s found to be almost parabolc 40

42 whle that for sodum-chlorde s lnear. Therefore, to smplfy the stuaton, the same value of membrane charge densty for all the sx elements s taken for the smulaton. As mentoned earler, even though a charge densty for all the elements n seres s consdered, whch s not completely rgorous, the present model s able to predct the relatve values of rejecton of each on wth respect to the others correctly; and, for the flow parameters used n the Umm Lujj plant, the model s able to correctly predct the recovery rato. Further expermentaton to correlate seawater concentraton wth membrane charge wll fne-tune the values of rejecton rato of each on and mprove the agreement wth expermental values. As observed n the Table 2, there s a good agreement of values and trends wth the reference. A recovery rato of 65.41% for each vessel s obtaned by the smulaton, whch has an error of only 0.63% wth respect to the reference. Thus our model can be used wth confdence n the modelng of seawater nanofltraton. 4. Conclusons In ths work comprehensve models for large-scale nanofltraton usng the flat-sheet and spral-wound confguratons are developed by extendng the DSPM-DE model over a membrane leaf. The models for the ndvdual leaves can be easly extended to put them n seres or parallel, n order to smulate membrane modules and trans of modules, as one would fnd n desalnaton plants. The effects of flow parameters such as feed pressure and flow rate on the solute rejecton and recovery rato of the membrane have been nvestgated. The varaton of other quanttes such as feed concentraton, permeate concentraton and trans-membrane flux over the leaf have also been presented. These studes have shown that the rejecton and recovery ratos of the NF 41