Characterisation and prediction of separation performance of nanofiltration membranes

Transcription

1 ELSEVIER Journal of Membrane Science 112 (1996) journal of MEMBRANE SCIENCE Characterisation and prediction of separation performance of nanofiltration membranes W. Richard Bowen *, Hilmi Mukhtar l Biochemical Engineering Group, Department of Chemical Engineering, University of Wales, Swansea SA2 8PP, United Kingdom Received 18 July 1995; revised 6 November 1995; accepted 7 November 1995 Abstract The rejection of single electrolytes at six nanofiltration membranes has been experimentally studied. The membranes were chosen to cover the range from ultrafiltration to reverse osmosis and to represent a diversity of polymers used for membrane fabrication. Such experimental data has been interpreted using a model based on the extended Nernst-Planck equation. The model accounts for the hindered nature of transport in the membranes. Such an interpretation allows a characterisation of the membranes in terms of two parameters, an effective membrane thickness and an effective membrane charge density. The latter may be related to the solution ionic content by means of an isotherm. The method presented also allows an estimation of the effective pore size of the membrane. A knowledge of the effective membrane thickness, effective charge density and effective pore size allows use of the model to predict the separation of mixtures of electrolytes at a membrane. Very good agreement between such a prediction and experimental data has been obtained. Keywords: Nanofiltration; Nernst-Planck; Rejection; Pore-size; Process prediction 1. Introduction simple ions is influenced by the inherent charge of the membranes [2]. The technical potential of One of the most important recent advances in nanofiltration lies between ultrafiltration and reverse membrane technology has been the development of osmosis [3]. Major applications of nanofiltration innanofiltration membranes. These membranes have clude the removal of salts in water treatment and the received their name as they have a molecular weight fractionation of salts and small molecules in a numcut-off for uncharged molecules corresponding to ber of industries. A great advantage of nanofiltration pores of about one nanometre in diameter [1]. How- is that the operating pressures can be much lower ever, rejection of charged molecules and especially than those required for reverse osmosis. For all membrane processes, the optimisation of * Corresponding author, existing equipment and the widening of the range of l Current address: Department of Chemical Engineering, Uni- application would be greatly facilitated if quantitaversiti Sains Malaysia, Perak Branch Campus, Tronoh, tive methods for predicting process performance were Perak, Malaysia. available. Ideally, such methods would use physical /96/$ Elsevier Science B.V. All rights reserved SSDI (95)

2 264 W.R. Bowen, H. Mukhtar / Journal of Membrane Science 112 (1996) property data of a process stream and a membrane in membrane charge with such small effective pore conjunction with a fundamental mathematical de- size. Also, although the effective membrane thickscription of the process so that the conditions giving ness is related to the physical thickness of the active the required separation could be predicted. Such an layer of the membrane its calculation requires an approach is especially challenging for the case of assumption about the porosity of the membrane, nanofiltration as the length scales determining sepa- which cannot be determined directly for nanometre ration are only about one order of magnitude greater dimension pores. The best way to obtain these pathan atomic dimensions. This is a scale at which rameters is by measurement of the rejection of single macroscopic descriptions of hydrodynamics and in- salt solutions. When such characterisation has been teractions are beginning to break down. It may be carried out it is possible to predict the separation of asked, for example, as to whether it is sensible to mixtures of electrolytes at the membrane. describe the transport channels in such membranes The present paper develops such an approach in a as being pores - though recent atomic force micro- number of ways to increase the predictive power. scope studies indicate that, at least for some mem- The most important developments are: branes, pores may exist [4]. (a) A hybrid model has been formulated [13] in A practically useful theoretical model can provide which the extended Nernst-Planck equation was quantitative process prediction from accessible phys- solved as if the membrane were homogeneous and ical property data. Such a model should be physi- non-porous, but hindrance factors for diffusion and catly realistic and require a minimum number of convection were included to allow for the transport (plausible) assumptions. It is also an advantage if of ions in the membrane taking place within a consuch a model does not involve mathematics which is fined space. The values of the hindrance factors, tedious, complicated or difficult to follow. Models which depend on the ionic radius/pore radius ratio, with the greater predictive capability are to be pre- were calculated using a numerical technique [14]. ferred. Of the various possible approaches to the (b) Optimisation of the hindrance factors allows description of nanofiltration, the use of the extended calculation of the effective pore diameter of the Nernst-Planck equation [5,6] best fulfils these re- membrane and further improves the accuracy of quirements. Such an approach describes the transport prediction of the rejection of mixtures. of ions across the membrane in terms of diffusion (c) The effective membrane charge density was and migration terms, as a result of concentration and related to the solution ionic content in terms of a electrical potential gradients, as well as convection Freundlich isotherm. Expressing the solution propercaused by the pressure difference across the mem- ties in terms of charge (equivalents/m 3) can result brane. When coupled with a suitable description of in a single isotherm for both monovalent and divathe partitioning of solutes between solution and lent electrolytes. membrane, for example a Donnan expression for Using this approach, six nanofiltration membranes ions, the extended Nernst-Planck equation provides chosen to cover the range from ultrafiltration to a complete description of the process. Such an ap- reverse osmosis and to represent a diversity of polyproach has previously found application in the de- mers used for membrane fabrication have been charscription of "reverse osmosis" [7-9] and in a re- acterised, using the model and data for single elecstricted forms to nanofiltration [10,11]. trolyte solutions, in terms of effective membrane The use of the extended Nemst-Planck equation thickness and effective charge density. Then, taking as a predictive method requires determination of at one membrane as an example, the effective pore least two parameters, the effective membrane charge radius is estimated. This leads to an optimisation of and the effective membrane thickness [7-9]. A1- the parameters and, through use of an isotherm, though a number of independent methods are in allows prediction of the separation of mixtures of principle applicable [12], such methods are in prac- electrolytes. To illustrate the accuracy of this aptice not suitable for nanofiltration membranes. Thus, proach, such a prediction is compared to experimenneither electrokinetic measurements nor surface ph tal data for mixed NaC1/Na2SO 4 solutions under titration can be used to unequivocally determine the conditions where negative rejection (a surprising

3 W.R. Bowen, H. Mukhtar /Journal of Membrane Science 112 (1996) phenomenon) is predicted and observed. Very good Non-idealities of properties inside the membrane quantitative agreement between theory and experi- as a result of coupling between ions and ions, or ions ment is obtained, and membrane, are assumed to be accounted for inside this effective charge density. There is also a requirement for electroneutrality in 2. Theoretical background the solution, 2.1. Extended Nernst-Planck equation L zici = 0 (6) i=1 The extended Nernst-Planck equation may be written, and for no overall electrical current passing through the membrane, dci zicidi'pfd~b+kiccij v (1) I = LFziji=O (7) Ji = -Di,p dx RT dx ' i= l where Ji is the flux of ion i and the terms on the Donnan equilibrium is assumed to apply at the right hand side represent transport due to diffusion, interfaces between the membrane and the external electric field gradient and convection respectively solutions (feed and permeate), (all symb ls are defined in the List f symb ls)" DLp ( c ~ ) ( z ~ F ) is the diffusivity of the ion i in the membrane = exp ---~TAOD (for x= 0 and x= Ax) (hindered diffusivity) and K~. is the hindrance factor for convection inside the membrane. The calcula- (8) tion of Di. p and Ki, c will be considered later. For the hybrid model developed here j~ and Jv are fluxes based on membrane area (in a way implicitly setting the fractional porosity of the membrane (P) to 1.0 for the purposes of calculation), where activity coefficients have not been included as they are assumed to be accounted for inside the membrane by the effective charge density (X) and only solutions dilute enough for 3, ~ 1 are considered in the present work. In using such a Donnan The concentration gradient can be obtained by type expression it is also implied that, for the simple rearranging Eq. (1), ions to be considered in the present work, steric dc~ Jv z~c~ d~b effects on entrance to the membrane are not signifi-. dx. Di,. p (Ki'cCi.. Ci'p) RT F-- dx (2) cant. Eqs. (2) and (4) may then be solved with the where the flux of ion j~ is expressed as, boundary conditions, Ji = JvCi.p (3) x -~. 0, C i = Ci. f and x = Ax, C i = C~,p (9) where Ci, p is the concentration of ion i in the where C~.f and Ci, p are the concentrations of ion i at permeate. The potential gradient is given by, the feed side of the membrane and the permeate side L z~jv of the membrane, respectively (at the interfaces). do -n-'--(ki,c Ci- Ci,p) The real rejection of ion i is then given by, i=l Ui,p" - - = C i.p dx F ~ z 2ci (4) R = 1 - Ci (10) RT i=1 In deriving Eq. (4) it has been assumed that the effective membrane charge density (X) is constant throughout the membrane and is given as, The equations were solved numerically using the Runge-Kutta-Gill method Calculation of hindrance factors ~_azic i = --X (for 0 < x > Ax) (5) Solutes moving in free solution experience a drag i= 1 force exerted by the solvent. When solutes move in

4 266 W.R. Bowen, H. Mukhtar / Journal of Membrane Science 112 (1996) confined spaces, such as membrane pores, the drag is Table 1 modified and the transport may be considered to be Ionic properties and hindrance factors (for rp = l rim) hindered. Theoretical descriptions for hindered trans- Ionic D i= l09 r s 109 )t Ki,d Kic D i X 109 port in pores has been extensively reviewed [15]. The type (m 2 s J) (m) ' (m'~ s- t) present work will make use of recent finite element Na calculations for the movement of a spherical solute El along the centreline of a pore [14]. The results of the soj calculations may be expressed in terms of hindrance factors, for diffusion, Di,p It should be noted in Fig. 1 that the correction for Ki, d - (11) hindered diffusion is substantially greater than the Di,~ correction for hindered convection. and for convection, Calculation of the hindrance factors requires a us knowledge of the ionic radii and the pore radii. Ionic Ki, c =- (12) radii calculated from ionic mobilities [16] are most ux appropriate here as it is the transport properties of where both hindrance factors are functions of the the ions which are under consideration. At first in the ionic radius/pore radius ratio (A). These hindrance present work it will be assumed that the pores have a factors are used to calculate the transport terms in radius of 1 rim. Data and calculations for the ions to Eq. (1). For 0 < A < 0.4, the results of the numerical be used in experimental work are shown in Table 1. calculations are well-expressed by linear relation- For the first initial comparison purposes with NaCI ships, as shown in Fig. 1. Hence, in the present work solutions, the assumption of a given value for the the hindrance factors for any ion i have been calcu- pore radius is not critical. This is illustrated by the lated using, data of Table 2. The first two blocks of rows of calculations are for a monovalent electrolyte with the Ki. ~ = A (13) anions and cations having equal radii. It may be seen Ki, c =-0.301A (14) that varying Ki, d but keeping the ratio Ki.d/AX constant results in identical predictions for rejection. Allowing for the different radii of the ions (third block) does not change the predicted rejection sub- 1.2 Table ~ Compensating effect of Ki. d and Ax if the ratio Ki, d/ax is kept constant. Sets of data in first two rows are for a 1 : 1 electrolyte,, 0.8 [ ~ with identical sizes of anions and cations. Last row is for NaCI ~' with the mean value of Ki, d (shown) being used to calculate Ax 0.6 (% = 1 nm). X/C b = 10 for all calculations ~-" 0.4 Jv x 106 Ax X 106 Ki, d Ki, d lax R (m s- 1) (m) (m-') (%) 0.2 I i i i i ~, Fig. 1. Hindrance factors as a function of ionic radius/pore radius ratio: ([3) hindrance factor for diffusion; (A) hindrance factor for convection. Results of numerical calculations using the method of [14].

5 W.R. Bowen, H. Mukhtar /Journal of Membrane Science 112 (1996) NaC1/Na~SO, the experimental work of the present paper. The loo~ mass transfer coefficient in such a cell may in gen- 80- s ~,,11,,~'~"... eral be written as, 6o- Ush = 9(URe) Y( Use) 33 (16) 5"2 40- ~',~ D where y=0.567, or a value very close [17-19]. 20- " ~ ~ ~ D Thus the mass transfer coefficient can be written,.":" Ion ryp~s k = k'09 0"567 (17) o-.-ii. m Na" -20- '~... '::"... c~ When taken in conjunction with the concentration...." so,' polarisation model for membrane filtration [20] this i... i... i... leads to [21], 10" ~ JvAx.101O/(m2s -') In Robs =ln m ~ (18) Fig. 2. Effect of correction for hindered diffusivity on real rejection for a mixed NaC1/Na~SO 4 solution as a function of nor- where m = 1/k'. Hence, the real rejection of a malised volume flux. Mole fraction NaCI = 0.5, X/C b = 10, membrane may be determined from the experimen-,:ix = m, r v = 1.0 nm. HD, calculation using hindered tally observed rejection by extrapolation to infinite w diffusivity. BD, calculation using bulk diffusivity. {y-axis intercept on plotting ln[(1 -- Robs)/Rob s] stantially if Ki,d/AX (based on the mean value of against (Jv/tO '567)}. K~, d) remains constant. That is, there is a compensation between the pore radius (rp) and the membrane thickness (Ax) in the calculations. (Changes in Ki, c 3. Materials and methods are much less important than changes in K~,o.) However, for ions of very different radii or for mixtures Six different nanofiltration membranes have been of ions the effects of allowing for hindered diffusiv- investigated. Four were obtained from Hoechst Sepaity become more significant. This is illustrated for a ration Products (Germany), PES5, PES10, CA20 and mixture of NaCI and Na2SO 4 (used for experimental CA50. PES and CA refer to polyethersulfone and work reported in Section 4.3) in Fig. 2. Hence, in cellulose acetate, respectively. Two were obtained using the mathematical model predictively it is im- from DDS Filtration Advanced Membrane Technolportant to evaluate the effective pore radius, as will ogy (Denmark), ETNA 01A and HC50. ETNA is a be shown later, coated hydrophilic membrane and HC50 has a polyamide separation layer on a polysulfone support Observed and real rejection Experiments were carried out in a stirred Amicon ultrafiltration cell (Model 8200) of 200 ml capacity. The model presented in Section 2.1 allows calcu- Stirring speed was varied using a purpose built digilation of the real rejection of the membrane. How- tally controlled magnetic stirrer. The cell was presever, in most practical cases only the observed rejec- surised with oxygen free nitrogen gas with the prestion may be experimentally measured, and this is sure determined accurately by means of a transducer. dependent upon the concentration polarisation occur- The amount of permeate collected was determined ring at the membrane surface. The observed rejection using a balance linked to a computer. No more than may be defined as, 20 ml of permeate was collected in any experiment. COp All calculations were based on mean feed and per- Rob s = (15) meate concentrations. The permeate flux was calcu- C i,f lated from the slope of weight versus time using a where the subscripts f and p refer to the feed and linear regression. In carrying out experiments the permeate respectively. A stirred cell has been used in stirring speed ( rpm) and pressure (

6 268 W.R. Bowen, H. Mukhtar /Journal of Membrane Science 112 (1996) kn m -2) were first increased and then decreased, with excellent reproducibility. Conductivity and ph J / ~ of solutions were accurately determined using digital probes calibrated on a daily basis. In mixtures of ~ 2.5 electrolytes, a chloride ion selective electrode used ~ with a double junction reference electrode (Russell ~ // / ' models and , recalibrated every 2 h? 2.0 and using 1.0 M NaNO 3 as an ionic strength adjuster) were used in conjunction with the conductivity measurements in analysing feed and permeate "~ solutions. High purity NaCI and Na2SO 4 were used Solutions were prepared with water produced by ~ ~ ~ ~ reverse osmosis, ion-exchange, carbon adsorption and then microfiltration. J~/o,) 0567 x 106/(m S 433) 4. Results and discussion Fig. 4. Observed rejection analysed using Eq. (18). PES5 membrane, 10-3 M NaCI solution, Ap = 400 kn m -z Membrane characterisation using single elec- membranes cover much of the nanofiltration range, trolyte solutions from fluxes similar to those typical of ultrafiltration to those typical of reverse osmosis. The high fluxes The hydraulic properties of the nanofiltration of some of the membranes at the relatively low membranes studied were determined by measure- pressures used confirm the potential of nanofiltration ment of the water flux as a function of applied as an energy saving process. pressure. Data for the six membranes is shown in To allow comparative characterisation of the Fig. 3. In all cases the water flux was a linear membranes in terms of X and Ax, the rejection of function of the applied pressure. The permeability of 10-3 M NaCI was measured at all six membranes as the membranes decreased in the sequence ETNA01A a function of stirring speed at different applied pres- > PES5 > PES10 > CA20 > HC50 > CA50. The sures. The observed rejections (Rob s) were used to magnitudes of the water fluxes show that the chosen calculate the real rejection (R) using the analysis of Section 2.3. An example of this analysis is given in Fig. 4 which shows that Eq. (18) describes the data i,,m well. Resulting real rejections at different applied 150- I--~-ErNAOIA I Membrane Types / / pressures are shown in Fig. 5. For each membrane i..o. yes5,aa..~ the real rejection increases as the applied pressure I -'" vest0 /... '~.--~-CA20 /... (and hence volume flux) increases, approaching a I-*- [ -. c,,s0.c50 /" W"... o- limiting value at the higher pressures. Comparison of ~'E100- the rejection at the different membranes shows that it... decreases in the sequence CA50 > HC50 > PES10 =... A ~ ETNA01A > PES5 ~ CA20, which is to a certain 50-.* extent, though not exactly, the reverse of the se ~ _ - quence for water flux. ~r.,-~,.-,,~-'=---:-=~*'" - "* "- -- ~: -- -" ~ It is data such as that in Fig. 5 that allows the first 0 ~ ~ J ~ ~ type of characterisation of these membranes by ap plying the model described in Section 2 to calculate AP /(kn m -~) X and Ax. This is achieved by fitting the experimen- Fig. 3. Water flux at six nanofiltration membranes as a function of applied pressure, tal data to the results of the theoretical calculations as shown in Fig. 6 for the example of PES5 (this

7 W.R. Bowen, H. Mukhtar / Journal of Membrane Science 112 (1996) Table 3 M,,,~-~r~,, Parameters for six nanofiltration membranes by analysis of rejec- 80-.~-- ~rn^olt,.o-r~s5 tion in 10-3 M NaCI -*.- w..s,0 Membrane /Ix X Error -~ - CA ~- Mcs0,- --*"" type (m) (equivalents m - 3 ) (%) ,- CA~0 m m ~.- *" ""._.-~,--'~'--'" ETNA01A 1.60X ~",," ~" ~'~" PES5 2.00X 10 -~ "] ~,, s~,,,,~ ~ PESI s S S ], / ~.,~k''--..~-.~ "''' -:~... CA ).( s S [, ~ _~,~.,..~.,..-,'~" "~ - HC '[ //'" _~..""~ CA ,f...'"... f,f " 0~,, I ' I ' t I ' I AP /(kn m 2) Fig. 5. Real rejection for a 10-3 M NaC1 solution as a function of applied pressure for six nanofiltration membranes. membrane will be used as an example in further developments in the paper). Fig. 6 shows that very good agreement between experiment and calculation may be achieved. The fit between experiment and theory is quite sensitive to the values of X and Ax as shown by the curves which result if the best fit values of X and Ax are varied by + 10%. Such variation leads to a substantially worse match be- tween experiment and theory. The values of the ~ resulting parameters are given in Table 3. It may be seen that for this group of membranes there is an order of magnitude range of /Ix with the membranes forming two groupings, those with Ax~ 10-5 (ETNA01A, PES5, PES10) and those with Ax~ 10-4 (CA20, HC50, CA50). It should be remembered that Ax is an effective membrane thickness and includes variation of porosity (it is assumed in the hybrid model that P = 1.0) and also reflects the choice of pore radius. The variation in the value of X for this group of membranes covers a factor of ~ 3. Some interesting conclusions regarding the way in which separation is achieved at the membranes may be drawn from the data of Table 3. For example, for the cellulosic membranes, the greater rejec- tion obtained with CA50 compared to CA20 is due to an increase in both Ax (a factor of ~ 1.8) and 40- IX[ (a factor of ~ 3.5). However, for the polyether- :.,.-.-.'.:.:. sulfone membranes the greater rejection obtained ~ ~ with PES10 compared to PES5 is mostly due to the 30- ~," increase in I xi (a factor of ~ 2.3) as the values of 20- Ax for these membranes are very close. ":'"" " / The rejection at these nanofiltration membranes,.-..~,~" s. o [ Pes5 also depends on the electrolyte concentration, as ~'" I ~ B~tN~,~ F~ shown in Fig. 7 for single salt solutions of NaC1 and 10- ~/~",.."Z ~' o E~r~,, Na2SO 4 at the PES5 membrane. Applying the analy- I "~ N.,~., + 10% ~ ~" [---s,,~,~ x sis of Section 2, with the assumption that Ax is rely aar," I- Numerical-10% *x i --. I~ummcal- x independent of salt concentration, allows evaluation 0 - ~ of X at the different concentrations. It was found that I xi increases with salt concentration indicating J, x 106 /(m3/m2s) that co-ion adsorption occurs at the membrane. In effect, this adsorption increases the ability of the Fig. 6. Experimental and numerically calculated real rejection as a function of volume flux for a 10-3 M NaCI solution at a PES5 membrane. Lines shown are best fit and also + 10% of the optimised values of X and Ax. membrane to reject salts at high concentration. If the bulk salt concentration is expressed in terms of equivalents/m 3 (Cb), then the dependence of X on

8 270 W.R. Bowen, H. Mukhtar /Journal of Membrane Science 112 (1996) sity of some types of membrane depends on the feed 10o- ~-,,.. ~olu~o, Ty~ concentration [8]. The analysis in terms of an ~" "~-, isotherm is very useful for prediction of rejection ~" %. -~- Nat1 80- ~ '~ --~- N~o. expected with mixtures of electrolytes.. \ 60- \, \\ 4.2. Determination of effective pore radius 40- ~, \ The incorporation of a description of the hindered o,, 0 x nature of ionic transport in the extended Nernst , Planck equation provides the possibility of character- ~-. ising the effective pore radius of the membranes. 0 ~ 1 ~ This leads to a three parameter characterisation of " 10' 10' 10 ~ the membranes in terms of X, Ax and rp. Obtaining three parameters from rejection data is in general Col(tool m-3) subject to substantially greater uncertainty than ob- Fig. 7. Real rejection of NaCI and Na2SO4 as single electrolytes taining two parameters, especially due to the interacat a PES5 membrane as a function of bulk electrolyte concentra- tion between parameters of the type discussed in tion. (Jr = m s- ~). Section 2.2. However, several approaches are possible and the simplest will be used here, taking the C b for both types of single salt solutions lies on a example of the PES5 membrane. single isotherm of the type, Taking the values of X and Ax given in Table 3 logl0lxi = s log 0C b -t-q (19) as a starting point, it is possible to calculate the effect of variation of rp on the rejection of single which is Freundlich in form. Fig. 8 shows the data salt solutions of NaC1 and Na2SO 4 at the PES5 plotted in this way. For the case of PES5 it was membrane. Fig. 9a shows that for NaC1 there is only found that s = and q = Similar rela- a small variation of the predicted rejection on rp. tionships were found for the other two membranes However, for Na2SO 4 the dependence of the preinvestigated in this way, CA20 (s = 0.499, q = dicted rejection on rp is much greater as shown in 0.859) and HC50 (s = 0.875, q = 0.999). Other au- Fig. 9b due to the larger ionic radius of the sulfate thors have also found that the effective charge den- ion. In these figures the curve for rp = 1.0 corresponds to the initial assumption in the present paper 2.0 and the curve for rp = oc corresponds to the calcula-,, tions of previous workers [8]. Hence, using the values of X and /Ix of Table 3 and using the isotherm ~ Best Fit ] of Eq. (19) it is possible to analyse rejection data for ~./-"1.5- o NaCI J Na2SO 4 solutions to obtain a value for the effective,, N~,so, I pore radius of the membrane. Experimental data, ~ together with the best fit line, as well as lines for ;~1.0- Z ~ other values of rp, are shown in Fig. 10. The best fit,-..g e~ value obtained in this way is rp = 0.72 rim. Using this value it is possible to reanalyse data for NaCI 0.5- solutions such as that in Fig. 6. This leads to small changes in the characteristic parameters (/Ix= 1.7 J ~ 10-5, s = 0.508, q = 0.874), but iteration through -2-1 o 1 2 the calculation procedure does not significantly log~0[ C~ /(eq m -3) ] change the calculated value of rp. Fig. 8. Effective charge density at a PES5 membrane as a function of bulk salt concentration plotted according to Eq. (19). This method of obtaining a value for the effective pore size of a nanofiltration membrane through anal-

9 NaC1 W.R. Bowen, H. Mukhtar /Journal of Membrane Science 112 (1996) Rejection of mixed NaCl / NaSO 4 electrolytes 80- r,=0.5.m Having obtained values for the physical memr,=l.0nm brane parameters Ax and rp as well as values of s r, = 2.0 nm 60- T=o. and q to allow calculation of X it is possible to &. predict the rejection of mixtures of electrolytes at the PES5 membrane. The usefulness of this approach 40- will be illustrated with NaC1/Na2SO 4 solutions un- S der conditions where the rather surprising phe- 20- nomenon of negative rejection is predicted and observed. 0,,,,,,,, Experimental data and theoretical predictions are shown in Fig. 11 for NaC1/Na2SO 4 molar ratios of Jv x 106/(m3/m2s) 0.2, 0.4, 0.6 and 0.8. (It should be noted that the effective membrane charge is varying with these 100 conditions as the isotherm varies with total concenb tration expressed in equivalents/m3.) Very good agreement is obtained between prediction and exper- 80- iment. In particular, the highly negative rejection of ~ chloride ions at low molar ratio and low flux is well 60- / ~ -..'"~;[~ i ~. _ i i i ' ~ '. f.' S "i'~'-'ii~,n~, described, as is the decreasingly negative rejection at -'~ low flux as the molar ratio is increased. Good agree "" "" "" ment between the predicted and experimental rejecs tions of sulfate and sodium ions is also obtained. The i --,,=05.m use of the calculated value for the effective pore..',-,s... r,= 1.0nm...;-..-'" -... :.o m radius was very important in obtaining the very good '~" -- ~=~ agreement shown. The quantitative agreement ob- 0 I I I I I I I Jv x 106/(m3/m~s) Fig. 9. (a) Predicted rejection of a PES5 membrane in 10-3 M 100 NaCI solution as a function of volume flux at various values of effective pore radius. (b) Predicted rejection of a PES5 membrane 80-.Q.... O... in 10-3 M Na2SO 4 solution as a function of volume flux at o various values of effective pore radius.."'" o~'".---'"~ ~..-"... ss..i. ysis of transport properties has analogies to obtaining ~.[ :,.-"~,,". f..f P~5 I.'.." estimates of the effective pore size of ultrafiltration 40-..:../ ~r F =0.50nml membranes through solute challenge [22]. The value ~,,../'... =0.72rim] ::,,./ r, = 1.0o nm obtained is in the range expected. It should be noted 20-,,./ --.. r,=~* I that the identification of an effective pore size does ;;'i o Experiment I not necessarily imply the existence of geometrically o,,, well-defined pores in nanofiltration membranes. A more general interpretation would be that the hin- J, x 106 /(m3/m2s) drance to transport for the passage of ions through the polymer structure of such a membrane is equiva- Fig. 10. Experimental and predicted rejection of a PES5 membrane in M Na2SO 4 solution as a function of volume lent to that for passage through pores of such an flux as various values of effective pore radius: Ax = m. effective size. s = 0.521, q = I

10 272 W.R. Bowen, H. Mukhtar /Journal of Membrane Science 112 (1996) t I.. ' 1Y I,o V ~... I ~ ' 0 t : ~ ~ ; I~,0-,.o Io...=,. II...,o..=.,c,,,.",,. Numerical CI." - - Numerical SO, ~ J i..., :... "<... " e- ~ I ~ ~...=,so: I[ I I I I ~lO an L x 10 6 /(m~/m~s) L x 10 ~ /(rn~/m~s) 100- G A A A _A... ~_=_ 100- d.~_.~. _.~ ~-''''" fo A f [1"1 I 50-/~.~--"~~ o 50- []... i50 ~ Numerical CI.... Nur~nc.al el" Numerical SO, ~ Nutaefical SO, ~ [] E0cpe*irnemSa" [] ExperirmmtNa" o I I ' I [ I I I I I I I I I [ I I Jv x 106 /(m3/m2s) Jv X 106 /(m3/m2s) Fig. 11. Experimental and predicted rejections of a PES5 membrane in mixtures of 10-3 M NaCI and Na2SO 4 solutions at molar ratios NaCI:NazSO 4 of (a) 0.2, (b) 0.4, (c) 0.6, (d) 0.8; Ax = m, rp = 0.72 nm, s = 0.508, q = tained on assuming rp = 1.0 nm or rp = ~ [8] is membranes allows a preliminary characterisation of significantly less good. membranes for given electrolyte conditions in terms of two parameters, an effective membrane charge (X) and an effective membrane thickness (Ax). 5. Conclusions There was a substantial variation in the values of these parameters for the six membranes studied. The Use of the extended Nernst-Planck equation in effective membrane charge depends on the bulk coconjunction with a Donnan condition at the mem- ion concentration in a way that may be described by brane/solution interfaces is a useful means of a single isotherm (involving two parameters, s and analysing and predicting the performance of nanofil- q) for both a monovalent and divalent co-ion if tration membranes. Formulation of a hybrid model solution concentration is described in equivalent/m 3. allows account to be taken of the hindered nature of A more detailed analysis for one of the memtransport in the membranes. Making a reasonable branes studied (PES5) allowed a value to be obtained assumption about the effective pore radius of the for the effective pore radius by analysis of single salt

11 W.R. Bowen, H. Mukhtar / Journal of Membrane Science 112 (1996) solutions of NaCI and Na2SO 4. The value obtained s isotherm parameter [defined by Eq. (19)] corresponded well to expectations. Characterisation T absolute temperature (K) of the PES5 membrane in terms of Ax and rp, as u s solute velocity (m s -~) well as values of s and q to allow calculation of X, u x average solvent velocity (m s -~ ) allowed prediction of the rejection of mixtures of x distance normal to membrane (m) electrolytes at the membrane. Very good agreement ax effective membrane thickness (m) was found between prediction and experiment. X effective membrane charge (equivalents m -3) The generality and relative ease of application of z~ valence of ion the extended Nemst-Planck approach to nanofiltra- q~ constant in Eq. (16) tion suggests that further attention should be given to y activity coefficient its use as a means of process prediction in the A ionic radius/pore radius ratio application of such processes, v kinematic viscosity (m 2 s- 1) 6o stirring speed (s- 1 ) qj electric potential (V) 6. List of symbols Aq~ n Donnan potential (V) c~ concentration in membrane (mol m-3) C~ concentration in solution (mol m -3) Acknowledgements Ci. f concentration on feed side of membrane (mol m -3) Hoechst Separation Products (Germany) are C~,p concentration in permeate (mol m -3) thanked for the donation of membranes for this C b concentration in solution (equivalent m -3) work. The Malaysian Public Services Department C bulk solution concentration (in absence of and Universiti Sains Malaysia are thanked for proconcentration gradient) (mol m-3) viding support to Hilmi Mukhtar. Oi, p hindered diffusivity (m 2 s- i) Di, ~ bulk diffusivity (m 2 s -I) F Faraday constant (C mol ~ ) References I current density (A m -2) j~ ion flux (based on membrane area) (mol m -2 [1] P. Eriksson, Nanofiltration extends the range of membrane S- I) ultrafiltration, Environ. Prog., 7 (1988) Jv volume flux (based on membrane area) (m [2] R. Rautenbach and A. Groschl, Separation potential of nanofiltration membranes, Desalination, 77 (1990) s-l) [3] L.P. Raman, M. Cheryan and N. Rajagopalan, Consider k mass transfer constant (m s- 1 ) nanofiltration for membrane separations, Chem. Eng. Prog., k' mass transfer constant defined by Eq. (17) 9o (1994) Ki. c hindrance factor for convection [4] W.R. Bowen, R.W. Lovitt, N. Hilal and P.M. Williams, Ki, a hindrance factor for diffusion Atomic force microscope studies of synthetic membranes, in W.R Bowen, R.W. Field and J.A. Howell (Eds.), Proc. m constant in Eq. (18) Euromembrane'95, Vol. 1, ESMST, Bath, 1995, pp NRe Reynolds number (ojr2/n) [5] R. Schogl, Membrane permeation in systems far from equi- Nsc Schmidt number (v/d) librium, Ber. Bunseng. Physik. Chem., 70 (1966) NSh Sherwood number (kr/d) [6] L. Dresner, Some remarks on the integration of the extended P fractional membrane porosity Nemst-Planck equations in the hyperfiltration of multicomponent solutions, Desalination, l0 (1972) q isotherm parameter [defined by Eq. (19)] [7] T. Tsuru, S. Nakao and S. Kimura, Calculation of ion r radius of filtration cell (m) rejection by extended Nernst-Planck equation with charged rp effective pore radius (m) reverse osmosis membranes for single and mixed electrolyte r s ionic radius in solution (m) solutions, J. Chem. Eng. Jpn., 24 (1991) R real rejection [defined by Eq. (10)] [8] T. Tsuru, M. Urairi, S. Nakao and S. Kimura, Reverse osmosis of single and mixed electrolytes with charged mem- R gas constant (J mol-1 K-l) branes: experiment and analysis, J. Chem. Eng. Jpn., 24 Rob s observed rejection [defined by Eq. (15)] (1991)

12 274 W.R. Bowen, H. Mukhtar /Journal of Membrane Science 112 (1996) [9] T. Tsuru, M. Urairi, S. Nakao and S. Kimura, Negative [16] P.W. Atkins, Physical Chemistry, 4th edn., OUP, Oxford, rejection of anions in the loose reverse osmosis separation of mono- and divalent ion mixtures, 1ChemE Symp. Ser., 125 [17] K.A. Smith, C.K. Colton, E.W. Merrill and L.B. Evans, (1991) Convective transport in a batch dialyser: determination of [10] D.W. Nielsen and G. Jonsson, Bulk-phase criteria for nega- true membrane permeability from a single measurement, rive ion rejection in nanofiltration of multicomponent salt AIChE Symp. Ser., 64 (1968) solutions, Sep. Sci. Technol., 29 (1991) [18] D.M. Malone and J.L. Anderson, Diffusional boundary-layer [11] J.M.K Timmer, H.C. van der Horst and T. Robbertsen, resistance for membranes with low porosity, AIChE J., 23 Transport of lactic acid through reverse osmosis and nanofil- (1977) tration membranes, J. Membrane Sci., 85 (1993) [19] W.S. Opong and A.L. Zydney, Diffusive and convective [12] W.R. Bowen, Electrochemical aspects of microfiltration and protein transport through asymmetric membranes, AIChE J., ultrafiltration, in Membranes in bioprocessing, J.A. Howell, 37 (1991) V. Sanchez and R.W. Field (Eds.), Blackie, London, [20] A.S. Michaels, New technique for CPI, Chem. Eng. Prog., 64 [13] W.M. Deen, B. Savat and J.M. Jamieson, Theoretical model (1968) for glomerular filtration of charged solutes, Am. J. Physiol., [21] S. Nakao and S. Kimura, Analysis of solutes rejection in 38 (1980) ultrafiltration, J. Chem. Eng. Jpn., 14 (1981) [14] W.R. Bowen and A.O. Sharif, Transport through microfiltra- [22] M. Mulder, Basic Principles of Membrane Technology, tion membranes - particle hydrodynamics and flux reduc- Kluwer, Dordrecht, tion, J. Colloid Interface Sci., 168 (1994) [15] W.M. Deen, Hindered transport of large molecules in liquidfilled pores, AIChE J., 33 (1987)

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