Measurement of Concentration Profiles by Holographic Interferometry andModelling in Unstirred Batch Reverse Osmosis

Julio Fernandez-Sempere, Francisco Ruiz-Bevia´ ,* Raquel Salcedo-Dı´az, and Pedro Garcı´a-Algado

Chemical Engineering Department, UniVersity of Alicante, Apartado 99, E-03080 Alicante, Spain

An optical method (real-time holographic interferometry) has been used to visualize concentration changesin the vicinity of the membrane surface during the dead-end reverse osmosis of salt solutions. Thisinterferometric technique is based on the fact that changes in refractive index, which are associated withchanges in concentration, can be visualized as interference fringes. Reverse osmosis experiments with NaCland Na2SO4 solutions with feed concentration in the range of 1-7 kg/m3 at a constant pressure of 600 kPahave been conducted. Interferograms obtained under different experimental conditions, as well as permeateflux and membrane rejection, are presented. Concentration profiles in the concentration polarization layerhave been determined from these interferograms and compared with those calculated using the mixedconvection-diffusion and osmotic pressure theory or Fick’s second law of diffusion, depending on whetherthe profiles correspond to the development or the disappearance of the layer. The reasonable agreement obtainedbetween experimental and calculated results seems to support the validity of the mathematical models proposedin the range of the experimental conditions studied.

1. Introduction

Experimental and theoretical studies about concentrationpolarization in ultrafiltration (UF) and reverse osmosis (RO)have been performed by many authors, as stated in a criticalreview by Sablani et al.1 During the mass-transfer processthrough a membrane, the permeate flux drives solute to themembrane. The buildup of rejected solute in the boundary layernear the membrane surface generates a concentration gradientand, as a consequence, a diffusive flow of solute back to thefeed bulk appears. The phenomenon is known as concentrationpolarization, and it is easier to study its properties throughmeasurements of the dissolved solute profiles in an unstirredbatch cell than in cross-flow processes, because, in cross-flowprocesses, the thickness of the boundary layer is limited by theflow parallel (especially if it is turbulent) to the membrane. Incross-flow processes, if steady state is attained, the convectivesolute flux to the membrane surface is balanced by the soluteflux though the membrane plus the diffusive and convectiveflow back to the bulk of the feed. The concentration profilenear the membrane is usually stable and the maximum concen-tration is not very high. However, in the case of membraneprocesses conducted in an unstirred batch cell or under dead-end conditions, a steady state is not easily attained, concentra-tions attained at the membrane surface (Cm) are very high, andthe thickness of the boundary layer (δ) grows continuously withtime (Figure 1). The process seems to reach a quasi-steady stateonly after a long period of time. When the concentration of thepermeate solution (Cp) tends toward the bulk concentration (Co),the convective solute flow to the membrane surface is balancedby the solute flux through the membrane and the diffusive flowback to the bulk solution, and no more accumulation of solutewill occur.

A good simulation model for the RO dead-end process mustpredict the evolution, with time, of concentration profiles in thepolarization layer, as well as the evolution of permeate flux. Thevalidation of the model proposed must be performed using experi-

mental studies. However, most authors only use permeate fluxexperimental data, so the validation of the model is not complete.

Experimental determination of the solute concentration pro-files in the polarization layer is a problem that has not yet beencompletely solved, because of the lack of an experimental tech-nique that allows a precise measurement of these profiles. Be-cause the solute accumulated at the membrane surface in RO isa salt, the variation of conductivity with concentration can beused to measure the concentration profile in the vicinity of themembrane. However, the probe used in this method can producea disturbance in the polarization layer. Chen et al.2 have com-pletely reviewed the so-called “noninvasive” methods, whichinvolve external signal generation and detection. Noninvasivetechniques include nonoptical (impedance spectroscopy, ultra-sonic reflectometry, etc.) and optical techniques. One of these

* To whom correspondence should be addressed. Tel:+34-965903547. Fax:+34-965903826. E-mail address: ruiz.bevia@ua.es.

Figure 1. Schematic concentration profiles at three different times in anunstirred batch cell.

7219Ind. Eng. Chem. Res.2006,45, 7219-7231

10.1021/ie060417z CCC: $33.50 © 2006 American Chemical SocietyPublished on Web 09/12/2006

optical techniques is interferometry, which has been used tovisualize the concentration polarization layer by several authors.3-6

In previous papers,7,8 holographic interferometry was usedto visualize the appearance, evolution with time, and disap-pearance of the concentration polarization layer during ultra-filtation of bovine serum albumin (BSA) and polyethylene glycol(PEG-2000). This technique, which has also been used to studydiffusion processes,9-11 allows a quantitative study of thepolarization phenomenon to be conducted, because each inter-ference fringe that is formed corresponds to a concentrationchange in the solution. In these previous papers, only theexperimental data of concentration profiles were presented andno theoretical model of simulation was used, because, duringultrafiltration, some additional processes occur, such as soluteadsorption or gel layer formation, which complicate the mod-elling of ultrafiltration.

In this paper, the same holographic interferometry techniqueis used to determine in situ and real-time concentration profilesduring the dead-end RO of salts, where the previously mentionedadditional processes are not expected. The aim of this study isto validate the theoretical model generally used to describe dead-end reverse osmosis processes using experimental data, not onlyof the permeate flux but also of the concentration profiles inthe polarization layer.

The concentration polarization phenomenon is a reversibleprocess; therefore, when the pressure is removed from thesystem, the polarization layer disappears. In this work, asimulation model for this disappearance is also proposed, andthe experimental and calculated results are compared.

2. Theory

Theoretical models that describe dead-end RO processesusually combine both the unsteady-state mass balance in thepolarization layer and the osmotic pressure model. In the massbalance, a convection-diffusion mechanism is assumed:

Moreover, if the diffusion coefficient is considered to be aconstant, because the variation ofD with the concentration isvery low, eq 1 becomes

using the following boundary conditions (see Figure 2):

whereδ is the polarization layer thickness

Cm is the solute concentration at the membrane surface andCp

is the solute concentration in the permeate flux.The osmotic pressure model uses the equation

where ∆P is the pressure applied,∆π the osmotic pressuredifference across the membrane (π(Cm) - π(Cp)), andRm the

membrane hydraulic resistance to the flux with pure water (Jw),which is calculated as

The osmotic pressure (π) is calculated using the van’t Hoffequation for ideal, diluted solutions:

whereφ is the number of ions,R the gas constant,T theabsolute temperature, andM the molecular weight of the solute.The use of eq 8 is equivalent to assuming that, in an interval ofdilute solutions, the osmotic pressure is a linear function of theconcentration, which is the same hypothesis assumed by Wileyand Fletcher.12

The simultaneous solution of eqs 2-7 gives the concentrationprofile and the permeate flux at any time during the process.

When the pressure ceases, only the diffusive movement ofthe solute from the polarization layer to the bulk solution occurs.Therefore, the constitutive equation in this case is

Assuming that the diffusion coefficient is a constant, eq 9becomes

which is Fick’s second law.

Figure 2. Boundary conditions used for the mathematical model.

Rm ) ∆PJw

(7)

π ) φRTCM

(8)

∂C∂t

) ∂

∂y(D ∂C∂y) (9)

∂C∂t

) D∂

2C

∂y2(10)

∂C∂t

) J∂C∂y

+ ∂

∂y (D ∂C∂y) (1)

∂C∂t

) J∂C∂y

+ D∂

2C

∂y2(2)

at t ) 0, C ) Co ∀ y (3)

aty > δ, C ) Co ∀ t (4)

at t > 0 andy ) 0, JCm + D(dCdy)y)0

) JCp (5)

J ) ∆P - ∆πRm

(6)

7220 Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006

3. Experimental Section

3.1. Experimental Setup.The optical setup (holographicinterferometry system) was similar to that described in previouspapers.7,8 The RO setup is shown in Figure 3. The pressurenecessary to run the process was supplied by means ofpressurized nitrogen. To avoid the appearance of bubbles in thesolution, the gas and the solution were separated by a damper(Hidracar S.A, model V007A05E1-AI).

A special RO cell was designed to adapt it to the require-ments of the holographic interferometry technique. The cellwas provided with two windows, thus allowing the membranesurface to be visualized. A detailed description of the cellused can be found in a previous paper.13 The active membranedimensions are 10 cm× 1 cm, with an effective area of 10cm2, which is a size that has been chosen to satisfy theinterferometric requirements. The distance from the membranesurface to the top of the cell is 45 mm; thus, the volumeof solution in the cell is large enough to guarantee thatconcentration changes inside the cell only would occur nearthe membrane. However, far from the membrane surface, thebulk concentration (Co) will remain unchanged during theprocess. The RO cell was horizontally placed on the opticaltable and was the common element between both RO and opticalsetups.

3.2. Materials. The membrane used was a thin film mem-brane (TFM-50, from Hydro Water S.L.). Suitable pieces forthe size of the cell used were cut from the entire membrane.Each piece of membrane was used for several experiments, soafter each experiment the cell was washed with distilled wateruntil the salt was completely removed from the membranesurface. The membrane was considered to be clean when thepermeate flux of water was recovered.

The experiments were performed using solutions of twosalts: NaCl and Na2SO4 (Panreac). Different feed concentrations(Co), in the range of 1-7 kg/m3, were used to study the effectof the solute and feed concentration on the polarization layer.

Some physical properties of the solutes that have been usedare shown in Table 1.12,14,15

3.3. Experimental Methodology.The experimental proce-dure is similar to that previously described.7,8

After the water flux was checked, the module was filled withthe solution and correctly aligned with the optical setup. Ahologram of the cell then was recorded on the holographic plate,to obtain the reference state for the interferometric study. Afterthe hologram was obtained, pressure was applied to the system.All the experiments were conducted at a constant pressure of600 kPa. As a consequence of the RO process, concentration(and, therefore, refraction index) changes occurred in thesolution near the membrane surface. The comparison betweenthe reference state recorded on the hologram and the evolutionof the RO process causes the appearance of the interferencefringes (interferogram). Each interference fringe corresponds toa certain concentration change in the solution and was different,depending on the salt used. The relationship between concentra-tion and refraction index was measured using a refractometer(Leica, model AR600) at 25°C, which was the operationtemperature. The relations obtained for each salt appear in Table1.

The methodology to obtain concentration profiles quantita-tively was the same as that used in previous papers.7,8

The interferograms were continuously recorded using a videocamera that was connected to a personal computer (PC). Thevideo-capture software that was used allowed 24 images to becaptured each second; therefore, the quantity of interferogramsregistered during each experiment was very high.

Data for weight and conductivity of the permeate solutionwere also continuously measured during the process, to obtainthe permeate flux and concentration. The continuous measureof conductivity was performed using a continuous flux con-ductivity cell (Crison, model 5287) and a conductimeter (Crison,model GLP 32) that was connected to a PC. The conductivitycell was previously calibrated to obtain the relationship betweenconductivity and concentration for each solute used (see Table1).

The solute concentration of the permeate flux directlymeasured (Cp,m) was not actually the concentration of thesolution passing through the membrane, because of the existenceof a dead volume between the membrane and the conductivitycell (see Figure 3).

Passing through this dead volume caused a delay in theconductivity measurement, which was dependent on the valueof the flux (J), which also was time-dependent. Hence, acorrection of the measured concentration value was madeconsidering ideal plug-flow in the permeate collection pipe.

The measured concentration (Cp,m) and the corrected con-centration (Cp,c) were related by eq 11:

τ is the delay time of the permeate solution and was calculatedas

whereVd is the dead volume (expressed in units of m3), S themembrane surface (given in units of m2), andJ(t) the permeateflux (given in units of m3/(m2 s)).

BecauseJ was variable with time,τ was also variable,so this correction had to be applied to all experimentalconcentration data that were measured. Some authors16,17 hada similar problem with the measurements of the permeateconcentration, but they assumed perfect mixing in the deadvolume of the cell.

Figure 3. Schematic diagram of the reverse osmosis (RO) system.Legend: (1) nitrogen cylinder, (2) pressure-control valve, (3a,b) precisionregulation valves, (4) pressure gauges, (5) damper, (6) RO module, (7) inletand outlet valves, (8) feed and water tanks, (9) pump, (10) conductivityprobe, (11) permeate collector vessel and balance, (12) conductimeter, and(13) computer.

Cp,c(t) ) Cp,m(t + τ) (11)

τ )Vd

SJ(t)(12)

Ind. Eng. Chem. Res., Vol. 45, No. 21, 20067221

To obtain weight data from the beginning of the process, thepermeate pipe was filled with pure water. However, the diffusionprocess inside the pipe was assumed to be negligible, becauseof the small value of the solute diffusion coefficients.

Figure 4. Interferograms belonging to experiment VII (Na2SO4: Co ) 2kg/m3) at 5, 35, 90, and 120 min. The membrane position in eachinterferogram is indicated by means of an additional longer line that extendson each side of the interferogram.

Figure 5. Interferograms belonging to experiment IX (Na2SO4: Co ) 5kg/m3) at 5, 35, 90, and 120 min.

Table 1. Physical Properties of the Solutes Useda

Expression

property NaCl Na2SO4 reference(s)

diffusion coefficient,D (m2/s) max (1.61× 10-9(1-14)× m), 1.45× 10-9) -3.9× 10-12C + 1.16× 10-9 12, 15density,F (kg/m3) 7.24C + 997.1 9.80C + 997.1 14refractive index,n 1.76× 10-4C + 1.33299 1.54× 10-4C + 1.33299 b

conductivity,µ (µS) 1799.2C + 2.29 1227.3C + 2.29 b

osmotic pressure,Π (atm) 0.835C 0.516C van’t Hoff equation

a Parameter legend:m, mass fraction;C, concentration (expressed in units of kg/m3). b Experimentally measured.

Table 2. Experimental Conditions

experimentCo

(kg/m3)Jw

(× 106 m3/(m2 s))Rm

(× 10-11Pa s/m2)area observed(mm× mm)

NaClI 1 5.61 1.08 3.40× 9.40II 2 5.78 1.05 3.40× 9.40III 3.5 5.71 1.06 3.40× 9.40IV 5 5.42 1.12 3.40× 9.40V 7 5.69 1.07 3.40× 9.40

Na2SO4

VI 1 6.55 0.93 2.95× 5.75VII 2 6.61 0.91 2.95× 5.75VIII 3.5 6.61 0.92 2.95× 5.75IX 5 6.39 0.95 2.95× 5.75X 7 6.47 0.94 2.95× 5.75

7222 Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006

The corrected permeate concentration (Cp,c) was used to simu-late the process and to calculate the observed rejection (Ro),using eq 13:

The intrinsic rejection (R) was calculated using eq 14:

Although eq 13 actually applies to well-mixed bulk solutions,in unstirred batch RO, the value ofRo can be useful to indicatehow similarCo andCp are and, therefore, how far the processis from the steady state.

After ∼2 h, 30 min, the pressure was removed and the ROprocess ended. From that moment, the concentration polarizationlayer started to disappear. During the process of disappearanceof the polarization layer, interferograms were also recorded untilalmost all the interference fringes disappeared. The procedureto obtain concentration profiles quantitatively was the same asthat used during the RO process.

4. Results and Discussion

4.1. Visualization of the Polarized Layer.By means ofholographic interferometry, it has been possible to follow, inreal time, the appearance, evolution, and disappearance of theconcentration polarization layer during dead-end RO experi-ments. Two groups of dead-end RO experiments were per-formed. A different salt (NaCl and Na2SO4) was used in eachone, varying the feed concentration between 1 kg/m3 and 7 kg/m3. In all the experiments, the transmembrane pressure was kept

constant at 600 kPa. Very similar behaviors were observed withboth solutes. Experimental conditions for each one of the saltsstudied are shown in Table 2, whereJw is the volumetricpermeate flux with pure water before the beginning of the ROexperiment andRm is the membrane hydraulic resistance to theflux with pure water, which is calculated with eq 7. The areaobserved indicates the true dimensions of the rectangular zonevisualized by means of the optical system and is dependent onthe magnification needed in each case, which is obtained usinga lens to focus the image on the camera. Reproducibility of thebehavior observed was confirmed by repeating the RO runsunder the same conditions (solute and initial feed concentration).

A few minutes after the RO process started, some interferomet-ric fringes near the membrane surface appeared. The amount offringes continued to increase throughout the process, thus indi-cating that the concentration at the membrane surface (Cm) wasincreasing, as well as the thickness of the boundary layer (δ).The rate of appearance of new interference fringes was highduring the first few minutes of the experiment; but, later, it de-creased andCm seemed to tend toward a constant value. How-ever, the thickness of the polarization layer (δ) continually in-creased. This behavior was already predicted by the mathematicalmodels for unstirred dead-end processes used by other authors.18,19

Interferograms at 5, 35, 90 and 120 min, corresponding to twoexperiments with different initial concentration of Na2SO4 (2kg/m3 and 5 kg/m3), are shown in Figures 4 and 5. Because eachexperiment was recorded in a continuous way, many interfero-grams were available. Interferograms in these figures have beenselected, as an example, to illustrate the evolution of the system.In each interferogram, a horizontal line and an auxiliary scalehave been drawn to show where the membrane surface initiallywas, as well as the magnification used in the experiment.

Figure 6. Interferograms belonging to experiment VII (Na2SO4: Co ) 2kg/m3) at 5, 15, 30, and 60 min after the pressure ceased. Figure 7. Interferograms belonging to experiment IX (Na2SO4: Co ) 5

kg/m3) at 5, 15, 30, and 60 min after the pressure ceased.

Ro ) 1 -Cp,c

Co(13)

R ) 1 -Cp,c

Cm(14)

Ind. Eng. Chem. Res., Vol. 45, No. 21, 20067223

The behavior in both experiments was very similar. As canbe seen in Figures 4 and 5, after 5 min, some interference fringeshad already appeared, thus meaning that a concentration gradientexisted from the beginning of the process.

New fringes were continuously appearing. The number offringes at 35 min doubled at 5 min. With increased time, therate of appearance of new fringes became slower. At 120 min,there were only 1 or 2 more fringes than those observed at 90min and, as a consequence,Cm was very similar in both cases.This behavior indicates that most of the polarization layer wasdeveloped during, approximately, the first hour of the process.

Cm seemed to slowly tend toward a constant value; however,fringes appeared further and further away from the membranesurface, which indicates that the thickness of the boundary layer(δ) was increasing and, therefore, unstirred dead-end RO is anunsteady-state process. However, with time, the dead-end processcould reach a stationary state, as a consequence of the greatincrease of the concentration at the membrane surface (Cm).First, the osmotic pressure in the membrane surface increasesas a consequence of a higher concentration and, therefore, theeffective driven force (∆P - ∆Π) could become zero. On the

Figure 8. Evolution of the experimental and calculated dimensionless flux (J/Jw): (a) NaCl and (b) Na2SO4. (Figure legend: (/) 1 kg/m3, (4) 2 kg/m3, (0)3.5 kg/m3, (b) 5 kg/m3, and (×) 7 kg/m3; the solid line represents the model calculation.)

Figure 9. Evolution of the rejection observed: (a) NaCl and (b) Na2SO4.(Figure legend: (/) 1 kg/m3, (4) 2 kg/m3, (0) 3.5 kg/m3, (b) 5 kg/m3, and(×) 7 kg/m3.) Figure 10. Evolution of the intrinsic rejection: (a) NaCl and (b) Na2SO4.

(Figure legend: (/) 1 kg/m3, (4) 2 kg/m3, (0) 3.5 kg/m3, (b) 5 kg/m3, and(×) 7 kg/m3.)

7224 Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006

Figure 11. Experimental and calculated concentration profiles at differenttimes: (a) experiment I, NaCl,Co ) 1 kg/m3; (b) experiment II, NaCl,Co

) 2 kg/m3; (c) experiment III, NaCl,Co ) 3.5 kg/m3; (d) experiment IV,NaCl, Co ) 5 kg/m3; and (e) experiment V, NaCl,Co ) 7 kg/m3. (Figurelegend: (O) 5 min, (4) 35 min, (/) 90 min, and (]) 120 min; solid linerepresents the model calculation.)

Figure 12. Experimental and calculated concentration profiles at differenttimes: (a) experiment VI, Na2SO4, Co ) 1 kg/m3; (b) experiment VII, Na2-SO4, Co ) 2 kg/m3; (c) experiment VIII, Na2SO4, Co ) 3.5 kg/m3; (d)experiment IX, Na2SO4, Co ) 5 kg/m3; and (e) experiment X, Na2SO4, Co

) 7 kg/m3. (Figure legend: (O) 5 min, (4) 35 min, (/) 90 min, and (])120 min; solid line represents the model calculation.)

Ind. Eng. Chem. Res., Vol. 45, No. 21, 20067225

Figure 13. Experimental and calculated concentration profiles at differenttimes after the pressure ceased: (a) experiment I, NaCl,Co ) 1 kg/m3; (b)experiment II, NaCl,Co ) 2 kg/m3; (c) experiment III, NaCl,Co ) 3.5kg/m3; (d) experiment IV, NaCl,Co ) 5 kg/m3; and (e) experiment V, NaCl,Co ) 7 kg/m3. (Figure legend: (O) 5 min, (4) 15 min, (/) 30 min, and (])60 min; solid line represents the model calculation.)

Figure 14. Experimental and calculated concentration profiles at differenttimes after the pressure ceased: (a) experiment VI, Na2SO4, Co ) 1 kg/m3;(b) experiment VII, Na2SO4, Co ) 2 kg/m3; (c) experiment VIII, Na2SO4,Co ) 3.5 kg/m3; (d) experiment IX, Na2SO4, Co ) 5 kg/m3; and (e)experiment X, Na2SO4, Co ) 7 kg/m3. (Figure legend: (O) 5 min, (4) 15min, (/) 30 min, and (]) 60 min; solid line represents the model calculation.)

7226 Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006

other hand, if the membrane is not completely selective, thepermeate concentration (Cp) will increase as the transmembraneconcentration (∆C) does. Finally, the diffusive flow toward thesolution bulk will also increase as the concentration difference(Cm - Co) does. In these circumstances, if the permeate concen-tration (Cp) equals the bulk concentration (Co), the convectivesolute flow to the membrane surface will be balanced by thesolute flux through the membrane and the diffusive flow backto the bulk solution, and no more accumulation of solute in thevicinity of the membrane will occur. This tendency to reach thestationary state, also noted by Nicolas et al.,17 can be observedwhen permeate flux and membrane rejection are studied (seeSection 4.20) because, in some experiments, the permeate fluxstabilizes and does not continuously decrease, and also becausethe observed retention (Ro) tends toward zero (Cp ≈ Co).

After the pressure ceased and the RO process ended, thefringe pattern changed. The interference fringes became widerand more separated, slowly disappearing. Interferograms inFigures 6 and 7 correspond to 2, 15, 30, and 60 min after theend of the RO process, for the two previously mentionedexperiments (Na2SO4; Co ) 2 and 5 kg/m3). As can be observedin both figures, the number of fringes decreased with time andthe remaining fringes were wider and more separated betweenthemselves, as well as from the membrane. This behaviorindicates that the concentration at the membrane surface (Cm)

decreases and the thickness of the boundary layer (δ) increaseswith time. Therefore, the slope of the concentration profiles getssmoother with time, until the concentration gradient completelydisappears. The disappearance of the polarization layer is a veryslow process (for instance, 60 min after the end of the ROprocess, some interference fringes were still visible). Thisevolution of the interference fringes is a consequence of thediffusive solute flow from the membrane surface to the bulksolution, because of the concentration gradient between themthat is generated during the RO process. This proves thatconcentration polarization is a reversible process.

4.2. Permeate Flux and Membrane Rejection.Data regard-ing permeate weight and concentration were continuouslymeasured for each experiment, to obtain the permeate flux (asthe curve derived from the weight) and the membrane rejection(see eqs 13 and 14) and the effect of initial concentration onboth parameters was studied. Both parameters are indicative ofthe membrane performance and, therefore, of the effectivenessof the process. Both varied with time, because of the fact thatdead-end RO is an unsteady-state process.

As a consequence of the concentration polarization, a decreasein the permeate flux was observed during the process. Asconcentration increased in the vicinity of the membrane, theosmotic pressure of the solution also did and, hence, the drivingforce (∆P - ∆π) decreased. Other authors who have been

Table A1. NaCl Concentration Profiles

y (mm) y (mm)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

Experiment I:Co ) 1 kg/m3 Experiment II:Co ) 2 kg/m3

1 0.99 4.69 7.51 8.31 1.36 1 2.34 2.37 4.49 5.13 2.362 0.33 1.93 3.27 4.23 1.72 2 0.67 1.75 3.23 3.73 2.723 0.11 1.29 2.17 2.77 2.08 3 0.34 1.44 2.53 2.95 3.084 0.92 1.63 2.12 2.44 4 0.20 1.09 2.12 2.40 3.445 0.67 1.26 1.67 2.80 5 0.86 1.68 1.92 3.806 0.49 1.00 1.34 3.16 6 0.68 1.38 1.63 4.167 0.35 0.77 1.08 3.52 7 0.49 1.13 1.33 4.528 0.19 0.65 0.87 3.88 8 0.35 0.92 1.09 4.889 0.09 0.47 0.67 4.24 9 0.21 0.69 0.86 5.24

10 0.33 0.49 4.60 10 0.12 0.52 0.65 5.6011 0.19 0.35 4.95 11 0.33 0.50 5.9512 0.19 5.31 12 0.17 0.34 6.31

13 0.09 0.20 6.6714 0.12 7.03

Experiment III:Co ) 3.5 kg/m3 Experiment IV:Co ) 5 kg/m3

1 0.81 4.35 5.70 5.90 3.86 1 1.47 3.95 5.26 6.91 5.362 0.49 2.79 3.97 4.44 4.22 2 0.82 2.68 4.18 5.00 5.723 0.31 2.08 3.10 3.51 4.58 3 0.49 2.06 3.29 3.93 6.084 0.17 1.64 2.49 2.88 4.94 4 0.28 1.58 2.66 3.17 6.445 1.28 2.08 2.37 5.30 5 0.12 1.23 2.16 2.58 6.806 1.00 1.69 1.93 5.66 6 0.97 1.79 2.17 7.167 0.77 1.38 1.54 6.02 7 0.70 1.40 1.75 7.528 0.58 1.09 1.26 6.38 8 0.49 1.07 1.38 7.889 0.40 0.89 0.91 6.74 9 0.29 0.78 1.06 8.24

10 0.22 0.66 0.62 7.10 10 0.16 0.54 0.73 8.6011 0.05 0.48 0.39 7.45 11 0.34 0.52 8.9512 0.28 0.19 7.81 12 0.14 0.32 9.3113 0.11 0.11 8.17 13 0.11 9.67

Experiment V:Co ) 7 kg/m3

1 0.87 3.12 4.78 5.42 7.362 0.43 2.25 3.56 4.15 7.723 0.23 1.73 2.81 3.30 8.084 0.11 1.13 2.21 2.63 8.445 0.76 1.83 2.15 8.806 0.42 1.39 1.69 9.167 0.22 1.01 1.30 9.528 0.09 0.69 0.96 9.889 0.46 0.66 10.24

10 0.27 0.42 10.6011 0.12 0.19 10.95

Ind. Eng. Chem. Res., Vol. 45, No. 21, 20067227

working with unstirred batch cells have observed similarbehavior.16,20 During the first few minutes of the process, thepermeate flux underwent a great reduction, whereas during therest of the process, a smaller reduction occurred. Figure 8 showsthe evolution of the nondimensional flux (J/Jw) with time. Data

regarding permeate flux with pure water (Jw) are presented inTable 2. It can be observed that, because of the greater osmoticpressure, the higher the feed concentration, the higher thereduction of the permeate flux, with respect to pure water flux(Jw). The initial concentration had a great influence on the

Table A2. Na2SO4 Concentration Profiles

y (mm) y (mm)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

Experiment VI:Co ) 1 kg/m3 Experiment VII:Co ) 2 kg/m3

1 0.62 1.97 4.61 5.72 1.41 1 0.95 2.48 3.53 3.98 2.412 0.27 1.24 2.30 3.00 1.82 2 0.50 1.69 2.64 3.06 2.823 0.15 0.93 1.76 2.14 2.23 3 0.33 1.33 2.20 2.52 3.234 0.05 0.74 1.41 1.73 2.64 4 0.20 1.13 1.86 2.14 3.645 0.61 1.20 1.44 3.05 5 0.14 0.95 1.61 1.87 4.056 0.51 1.03 1.25 3.46 6 0.08 0.83 1.45 1.66 4.467 0.42 0.89 1.08 3.87 7 0.71 1.26 1.48 4.878 0.33 0.78 0.95 4.28 8 0.61 1.13 1.31 5.289 0.28 0.68 0.83 4.70 9 0.52 1.01 1.15 5.70

10 0.22 0.57 0.73 5.11 10 0.44 0.89 1.04 6.1111 0.16 0.50 0.64 5.52 11 0.38 0.78 0.93 6.5212 0.10 0.43 0.55 5.93 12 0.31 0.69 0.82 6.9313 0.05 0.36 0.49 6.34 13 0.26 0.60 0.72 7.3414 0.30 0.40 6.75 14 0.19 0.53 0.64 7.7515 0.23 0.34 7.16 15 0.14 0.45 0.55 8.1616 0.19 0.29 7.57 16 0.08 0.38 0.47 8.5717 0.13 0.22 7.98 17 0.31 0.39 8.9818 0.06 0.16 8.39 18 0.25 0.32 9.3919 0.11 8.80 19 0.18 0.25 9.8020 0.05 9.21 20 0.11 0.18 10.21

21 0.06 0.13 10.62

Experiment VIII:Co ) 3.5 kg/m3 Experiment IX: Co ) 5 kg/m3

1 1.19 2.96 4.65 5.42 3.91 1 0.81 2.68 4.94 5.68 5.412 0.68 2.18 3.53 4.10 4.32 2 0.57 2.10 3.80 4.48 5.823 0.47 1.77 2.94 3.38 4.73 3 0.44 1.77 3.16 3.68 6.234 0.34 1.49 2.54 2.91 5.14 4 0.32 1.52 2.73 3.19 6.645 0.25 1.28 2.23 2.56 5.55 5 0.25 1.31 2.38 2.78 7.056 0.18 1.09 2.00 2.29 5.96 6 0.18 1.14 2.10 2.45 7.467 0.12 0.95 1.77 2.02 6.37 7 0.12 0.99 1.84 2.17 7.878 0.06 0.82 1.57 1.83 6.78 8 0.06 0.86 1.64 1.95 8.289 0.70 1.40 1.65 7.20 9 0.74 1.45 1.73 8.70

10 0.60 1.23 1.47 7.61 10 0.63 1.29 1.55 9.1111 0.52 1.11 1.31 8.02 11 0.52 1.13 1.37 9.5212 0.43 0.96 1.17 8.43 12 0.43 0.98 1.20 9.9313 0.36 0.86 1.03 8.84 13 0.35 0.85 1.06 10.3414 0.29 0.74 0.89 9.25 14 0.27 0.73 0.92 10.7515 0.23 0.63 0.78 9.66 15 0.20 0.61 0.79 11.1616 0.16 0.54 0.67 10.07 16 0.14 0.50 0.67 11.5717 0.10 0.44 0.56 10.48 17 0.07 0.40 0.56 11.9818 0.05 0.35 0.48 10.89 18 0.29 0.43 12.3919 0.26 0.38 11.30 19 0.21 0.34 12.8020 0.18 0.30 11.71 20 0.12 0.25 13.2121 0.12 0.20 12.12 21 0.16 13.6222 0.06 0.14 12.53 22 0.06 14.03

0.05 12.94

Experiment X:Co ) 7 kg/m3

1 1.08 3.01 4.89 5.70 7.412 0.71 2.35 3.97 4.64 7.823 0.50 1.97 3.35 3.88 8.234 0.38 1.68 2.86 3.35 8.645 0.28 1.42 2.50 2.91 9.056 0.18 1.21 2.20 2.54 9.467 0.11 1.04 1.94 2.24 9.878 0.05 0.86 1.71 2.01 10.289 0.73 1.50 1.76 10.70

10 0.60 1.31 1.54 11.1111 0.48 1.11 1.33 11.5212 0.36 0.95 1.15 11.9313 0.25 0.80 0.95 12.3414 0.15 0.63 0.80 12.7515 0.06 0.50 0.65 13.1616 0.38 0.50 13.5717 0.26 0.36 13.9818 0.15 0.25 14.3919 0.15 14.80

7228 Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006

permeate flux. However, the evolution of permeate flux withtime was very similar from one experiment to another: afterthe initial reduction, the slope of the curves was very small andsimilar in all the experiments.

Membrane-observed rejection (Ro) and intrinsic rejection (R)were calculated with eqs 13 and 14.R is an interesting valuewhen the transport mechanism through the membrane is studied.Few reliable data ofRexist in the literature, because of the lackof an experimental technique that allows a precise measurementof Cm. Usually, an indirect approach to determineR is made bysolving a simplified transport equation in the polarization layer.17

The use of this simplification assumes a steady-state situationfor stirred batch cells or cross-flow cells. Therefore, this formulais not correct for unstirred dead-end RO processes.

Using holographic interferometry, it has been possible to mea-sureCm, which corresponds to the nearest fringe to the mem-brane surface. The measurement ofCm during each experimenthas allowed a more-accurate value ofR to be calculated.

The initial concentration also had a great influence on themembrane rejection, as can be observed in Figures 9 and 10. Forthe two solutes studied, the higher the feed concentration, thehigher the rejection observed. During the entire process,Ro under-went a continuous reduction, reaching very low values in someof the experiments that were performed. For example, in experi-ment I (NaCl,Co ) 1 kg/m3), over long periods,Ro tends towardzero (Figure 9). This reduction ofRo indicates an increase in

permeate concentration (see eq 13). As previously mentioned,whenCp ) Co, the process will reach a stationary-state. In thatcase, the value ofRo will be zero. The shape of theRo curvesindicates that the evolution of dead-end RO processes tendstoward the previously mentioned steady-state, which is attainedearlier in processes that have been performed with lowconcentration solutions.

On the other hand, the intrinsic rejection is dependent directlyon the maximum concentration at the membrane surface (seeeq 14). The higher the value ofCm, the more solute that is ableto pass through the membrane and therefore, the lower the valueof R. In experiments where low concentration solutions wereused, intrinsic rejection became almost stable after∼1 h ofexperiment (see Figure 10). This behavior reinforces the theorythat the more diluted the feed solution, the faster the approachto a steady-state condition.

4.3. Concentration Profiles. Interference fringes that ap-peared in the interferograms were the result of a refractive indexgradient in the vicinity of the membrane that were due to aconcentration gradient. Because a relationship exists betweenthe refractive index and concentration (Table 1) in aqueoussolutions of salts, it was possible to obtain the concentrationprofile from the interference fringes, using the methodologyexplained in a previous paper.7 The fringes farthest from themembrane surface were broader and it was more difficult toassign their position precisely, because of the difficulty involvedto ascertain exactly where the maximum or the minimum oflight intensity was. The evolution of the concentration profilesof the experiments shown in Table 2 can be observed in Figures11 and 12. As can be observed, the development of the

Table A3. NaCl Concentration Profiles during the Disappearance of the Polarized Layer

y (mm) y (mm)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

Experiment I:Co ) 1 kg/m3 Experiment II:Co ) 2 kg/m3

1 4.00 4.50 5.24 6.24 1.36 1 4.66 5.19 5.90 6.53 2.362 2.76 3.38 3.79 4.14 1.72 2 3.53 4.13 4.57 4.97 2.723 2.19 2.60 2.71 2.35 2.08 3 2.82 3.31 3.57 3.35 3.084 1.82 1.99 1.68 2.44 4 2.33 2.66 2.64 1.20 3.445 1.50 1.37 0.37 2.80 5 1.93 2.04 1.58 3.806 1.21 0.55 3.16 6 1.62 1.40 0.66 4.167 0.97 3.52 7 1.35 0.64 4.528 0.63 3.88 8 1.04 4.889 0.24 4.24 9 0.75 5.24

10 0.16 5.60

Experiment III:Co ) 3.5 kg/m3 Experiment IV:Co ) 5 kg/m3

1 6.09 6.43 7.12 8.54 3.86 1 8.63 8.46 8.82 5.362 4.35 4.79 5.35 6.21 4.22 2 5.97 5.99 6.22 6.59 5.723 3.41 3.87 4.25 4.65 4.58 3 4.56 4.75 4.92 4.91 6.084 2.77 3.13 3.21 3.15 4.94 4 3.62 3.84 3.85 3.42 6.445 2.31 2.49 2.26 0.90 5.30 5 2.95 3.06 2.84 0.65 6.806 1.90 1.78 0.86 5.66 6 2.37 2.35 1.64 7.167 1.58 0.90 6.02 7 1.93 1.54 0.77 7.528 1.24 6.38 8 1.59 0.62 7.889 0.88 6.74 9 1.12 8.24

10 0.46 7.10 10 0.70 8.6011 0.15 8.95

Experiment V:Co ) 7 kg/m3

1 8.10 7.83 8.13 8.45 7.362 6.17 6.13 6.45 6.75 7.723 4.96 4.98 5.17 5.21 8.084 4.05 4.09 4.19 3.95 8.445 3.27 3.23 3.13 1.86 8.806 2.65 2.48 1.83 0.78 9.167 2.11 1.57 0.85 9.528 1.68 0.56 0.14 9.889 1.15 10.24

10 0.68 10.6011 0.09 10.95

J ) Dδ

ln(Cm - Cp

Co - Cp) (15)

Ind. Eng. Chem. Res., Vol. 45, No. 21, 20067229

polarization layer mainly occurred during the first few minutesof the process, whereas, at longer time periods, the variation ofthe concentration profile was very slight. Despite the continuousgrowing of the boundary layer, its maximum thickness, in allthe experiments, was no more than 7-8 mm, even at longperiods. The bulk concentration is considered to be constant,because of the great volume of the solution inside the cell (height) 45 mm), in comparison to the small thickness of the boundarylayer. Moreover, at greater distances from the membrane surface,no interference fringes were observed, which means that thebulk solution had maintained its initial concentration (Co).

The same procedure can be used to obtain concentration pro-files during the disappearance of the fringes. The evolution ofthe concentration profiles of the experiments shown in Table 2can be observed in Figures 13 and 14. It can be seen that theslope of the concentration profile becomes smoother with time,as a consequence of the decrease in the solution concentration

at the membrane surface and the increase of the thickness ofthe boundary layer.

4.4. Application of a Mathematical Model to the Experimen-tal Data. Using the same operation conditions, all the experi-ments conducted were also simulated by the mathematical modelexplained previously for dead-end RO processes (unsteady state).Using the boundary conditions given in eqs 3, 4, and 5, eqs 2,6, and 7 were simultaneously solved by a numerical method. Bymeans of this set of equations, the concentration profiles (C(y,t))and permeate flux (J) were calculated at any time during theRO process. To model the process, the following data were need-ed: feed solution concentration (Co), transmembrane pressure(∆P), diffusion coefficient of the solute used (D), permeate fluxwith pure water (Jw), and permeate concentration (Cp,c). Jw andCp,c were experimentally measured. In eq 4, to initiate the calcu-lation process, a high value ofδ was assumed, where the soluteconcentration was not expected to change. After the calculation

Table A4. Na2SO4 Concentration Profiles during the Disappearance of the Polarized Layer

y (mm) y (mm)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

interferogram fringeorder number

at5 min

at35 min

at90 min

at120 min

C(kg/m3)

Experiment VI:Co ) 1 kg/m3 Experiment VII: Co ) 2 kg/m3

1 5.24 5.62 1.41 1 4.27 4.79 5.47 2.412 2.86 3.86 4.68 5.36 1.82 2 3.29 3.96 4.49 5.16 2.823 2.28 3.12 3.76 3.98 2.23 3 2.75 3.40 3.80 4.29 3.234 1.94 2.62 3.06 2.93 2.64 4 2.37 2.96 3.31 3.55 3.645 1.70 2.24 2.46 1.84 3.05 5 2.10 2.60 2.82 2.76 4.056 1.50 1.91 1.89 0.35 3.46 6 1.91 2.29 2.36 1.93 4.467 1.34 1.53 1.21 3.87 7 1.70 1.96 1.84 0.39 4.878 1.19 1.15 0.28 4.28 8 1.54 1.64 1.34 5.289 1.06 0.66 4.70 9 1.39 1.31 0.43 5.70

10 0.94 5.11 10 1.24 0.91 6.1111 0.78 5.52 11 1.11 0.37 6.5212 0.63 5.93 12 0.98 6.9313 0.47 6.34 13 0.81 7.3414 0.30 6.75 14 0.62 7.7515 0.08 7.16 15 0.43 8.16

16 0.23 8.57

Experiment VIII:Co ) 3.5 kg/m3 Experiment IX:Co ) 5 kg/m3

1 5.29 5.21 3.91 1 5.20 5.27 5.72 5.412 4.12 4.36 4.77 5.39 4.32 2 4.32 4.53 4.93 5.501 5.823 3.48 3.83 4.14 4.60 4.73 3 3.70 3.97 4.29 4.689 6.234 3.05 3.37 3.62 3.87 5.14 4 3.22 3.50 3.77 4.011 6.645 2.72 3.01 3.14 3.18 5.55 5 2.84 3.11 3.30 3.293 7.056 2.44 2.67 2.68 2.48 5.96 6 2.57 2.76 2.83 2.549 7.467 2.20 2.35 2.26 1.72 6.37 7 2.29 2.42 2.35 1.175 7.878 1.99 2.03 1.80 0.54 6.78 8 2.06 2.09 1.82 8.289 1.80 1.72 1.29 7.20 9 1.85 1.71 0.87 8.70

10 1.61 1.42 0.57 7.61 10 1.66 1.29 0.09 9.1111 1.45 1.02 8.02 11 1.46 0.67 9.5212 1.29 0.59 8.43 12 1.27 9.9313 1.14 0.09 8.84 13 1.04 10.3414 0.92 9.25 14 0.79 10.7515 0.69 9.66 15 0.53 11.1616 0.47 10.07 16 0.23 11.5717 0.23 10.48

Experiment X:Co ) 7 kg/m3

1 7.412 4.89 5.23 5.47 7.823 4.13 4.48 4.71 5.21 8.234 3.62 3.93 4.08 4.38 8.645 3.17 3.41 3.50 3.55 9.056 2.79 3.01 2.94 2.67 9.467 2.49 2.60 2.39 1.45 9.878 2.20 2.19 1.76 10.289 1.96 1.78 0.75 10.70

10 1.71 1.33 11.1111 1.48 0.62 11.5212 1.22 11.9313 0.90 12.3414 0.53 12.7515 0.14 13.16

7230 Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006

program was run, it was checked that the value ofδ that wascalculated was lower than the value ofδ that was assumed.Concentration profiles and permeate flux calculated for eachexperiment were compared with those experimentally obtainedto validate the model that was proposed. As can be observed inFigures 8a and 8b, a reasonable agreement between experimentaland calculated permeate fluxes is obtained, better at larger timesthan at short times. In Figures 11 and 12, the experimental andcalculated concentration profiles are very similar, better as thetime and initial concentration increases, thus indicating that thecombination of the convection-diffusion mechanism and theosmotic pressure theory could be quite an adequate model todescribe, in the range of the experimental conditions studied,RO processes operating under unstirred batch conditions.

The polarization layer disappeared as a consequence of thediffusive movement from the membrane surface to the bulk solu-tion of the solute accumulated during the RO process, caused bytheconcentrationdifferencebetweenbothplaces.After theconvec-tive solute flow due to the pressure applied ceased, diffusionwas the only mass-transfer mechanism that was involved.

To calculate concentration profiles during the disappearanceof the polarized layer, eq 10 was solved numerically, using thefirst concentration profile obtained by holographic interferometry(60 s after the RO process finished) as the initial condition.Experimental and calculated concentration profiles are shownin Figures 13 and 14. In both figures, the slope of theconcentration profile becomes smoother with time and, therefore,the change in concentration is less. This small change inconcentration explains the intersection of the profiles at differenttimes. In the same way as in the RO process, experimental andsimulated results were rather concordant (see Figures 13 and14); thus, the diffusion theory describes quite precisely theprocess of the disappearance of the polarization layer.

5. Conclusions

The experimental results obtained proved that real-timeholographic interferometry is a useful noninvasive techniqueto study the concentration polarization phenomenon duringreverse osmosis (RO) processes. It has been used to follow, asinterference fringes, the appearance, the evolution with time,and the disappearance of the polarization layer in the dead-endRO of two salts (NaCl and Na2SO4).

Concentration profiles and permeate flux have been determined.While permeate flux undergoes a reduction with time, theconcentration at the membrane surface (Cm), as well as the thick-ness of the boundary layer (δ), were continuously increasing.

A mathematical model has been proposed for the simulationof dead-end RO processes. The model, which is a combinationof the convection-diffusion mechanism and the osmotic pres-sure theory, has been validated, under the experimental condi-tions used, by a reasonable agreement between experimentaland calculated results (concentration profiles and permeate flux).

Concentration profiles, after the pressure ceased, have beenexperimentally obtained and it has been possible to study thedisappearance of the polarization concentration layer, concludingthat the polarization phenomenon is a reversible process. Amathematical model, based on a diffusive mechanism, has beenproposed. Agreement between the experimental and calculatedconcentration profiles validates the model.

Acknowledgment

This research was sponsored by the Plan Nacional de I+D+IBQU2000-0456 (Ministerio de Educacio´n y Cultura) and bythe Ajuda per a Grups de I+D+I de la Consellerı´a d′Empresa,

Universitat i Ciencia (Generalitat Valenciana). The authors aregrateful to Dr. Jose´ A. Caballero-Suarez and Dr. Vicente Gomis-Yagues for their assistance in the application of the model.

Appendix

The number of interference fringes, the distance (y) to themembrane for each fringe and its concentration (C) for someselected times of the experiments presented in Table 2 are shownin Tables A1 and A2. Same information corresponding to thedisappearance of the polarized layer is shown in Tables A3and A4.

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ReceiVed for reView April 3, 2006ReVised manuscript receiVed July 12, 2006

AcceptedAugust 1, 2006

IE060417Z

Ind. Eng. Chem. Res., Vol. 45, No. 21, 20067231