Measurement of Concentration Profiles by Holographic Interferometry

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

7219

Measurement of Concentration Profiles by Holographic Interferometry and Modelling in Unstirred Batch Reverse Osmosis Julio Ferna´ ndez-Sempere, Francisco Ruiz-Bevia´ ,* Raquel Salcedo-Dıáz, and Pedro Garcıá-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 changes in the vicinity of the membrane surface during the dead-end reverse osmosis of salt solutions. This interferometric technique is based on the fact that changes in refractive index, which are associated with changes in concentration, can be visualized as interference fringes. Reverse osmosis experiments with NaCl and Na2SO4 solutions with feed concentration in the range of 1-7 kg/m3 at a constant pressure of 600 kPa have been conducted. Interferograms obtained under different experimental conditions, as well as permeate flux and membrane rejection, are presented. Concentration profiles in the concentration polarization layer have been determined from these interferograms and compared with those calculated using the mixed convection-diffusion and osmotic pressure theory or Fick’s second law of diffusion, depending on whether the profiles correspond to the development or the disappearance of the layer. The reasonable agreement obtained between experimental and calculated results seems to support the validity of the mathematical models proposed in the range of the experimental conditions studied. 1. Introduction Experimental and theoretical studies about concentration polarization in ultrafiltration (UF) and reverse osmosis (RO) have been performed by many authors, as stated in a critical review by Sablani et al.1 During the mass-transfer process through a membrane, the permeate flux drives solute to the membrane. The buildup of rejected solute in the boundary layer near the membrane surface generates a concentration gradient and, as a consequence, a diffusive flow of solute back to the feed bulk appears. The phenomenon is known as concentration polarization, and it is easier to study its properties through measurements of the dissolved solute profiles in an unstirred batch cell than in cross-flow processes, because, in cross-flow processes, the thickness of the boundary layer is limited by the flow parallel (especially if it is turbulent) to the membrane. In cross-flow processes, if steady state is attained, the convective solute flux to the membrane surface is balanced by the solute flux though the membrane plus the diffusive and convective flow back to the bulk of the feed. The concentration profile near the membrane is usually stable and the maximum concentration is not very high. However, in the case of membrane processes conducted in an unstirred batch cell or under deadend conditions, a steady state is not easily attained, concentrations attained at the membrane surface (Cm) are very high, and the thickness of the boundary layer (δ) grows continuously with time (Figure 1). The process seems to reach a quasi-steady state only after a long period of time. When the concentration of the permeate solution (Cp) tends toward the bulk concentration (Co), the convective solute flow to the membrane surface is balanced by the solute flux through the membrane and the diffusive flow back to the bulk solution, and no more accumulation of solute will occur. A good simulation model for the RO dead-end process must predict the evolution, with time, of concentration profiles in the polarization layer, as well as the evolution of permeate flux. The validation of the model proposed must be performed using experi* To whom correspondence should be addressed. Tel: +34965903547. Fax: +34-965903826. E-mail address: [email protected].

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

mental studies. However, most authors only use permeate flux experimental data, so the validation of the model is not complete. Experimental determination of the solute concentration profiles in the polarization layer is a problem that has not yet been completely solved, because of the lack of an experimental technique that allows a precise measurement of these profiles. Because the solute accumulated at the membrane surface in RO is a salt, the variation of conductivity with concentration can be used to measure the concentration profile in the vicinity of the membrane. However, the probe used in this method can produce a disturbance in the polarization layer. Chen et al.2 have completely reviewed the so-called “noninvasive” methods, which involve external signal generation and detection. Noninvasive techniques include nonoptical (impedance spectroscopy, ultrasonic reflectometry, etc.) and optical techniques. One of these

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

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

optical techniques is interferometry, which has been used to visualize the concentration polarization layer by several authors.3-6 In previous papers,7,8 holographic interferometry was used to visualize the appearance, evolution with time, and disappearance of the concentration polarization layer during ultrafiltation of bovine serum albumin (BSA) and polyethylene glycol (PEG-2000). This technique, which has also been used to study diffusion processes,9-11 allows a quantitative study of the polarization phenomenon to be conducted, because each interference fringe that is formed corresponds to a concentration change in the solution. In these previous papers, only the experimental data of concentration profiles were presented and no theoretical model of simulation was used, because, during ultrafiltration, some additional processes occur, such as solute adsorption or gel layer formation, which complicate the modelling of ultrafiltration. In this paper, the same holographic interferometry technique is used to determine in situ and real-time concentration profiles during the dead-end RO of salts, where the previously mentioned additional processes are not expected. The aim of this study is to validate the theoretical model generally used to describe deadend reverse osmosis processes using experimental data, not only of the permeate flux but also of the concentration profiles in the polarization layer. The concentration polarization phenomenon is a reversible process; therefore, when the pressure is removed from the system, the polarization layer disappears. In this work, a simulation model for this disappearance is also proposed, and the experimental and calculated results are compared. 2. Theory Theoretical models that describe dead-end RO processes usually combine both the unsteady-state mass balance in the polarization layer and the osmotic pressure model. In the mass balance, a convection-diffusion mechanism is assumed:

Figure 2. Boundary conditions used for the mathematical model.

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

Rm )

∂C ∂C ∂C ∂ D )J + ∂t ∂y ∂y ∂y

( )

(1)

Moreover, if the diffusion coefficient is considered to be a constant, because the variation of D with the concentration is very low, eq 1 becomes

∆P Jw

(7)

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

π)φ

RTC M

(8)

2

∂C ∂C ∂C )J +D 2 ∂t ∂y ∂y

(2)

using the following boundary conditions (see Figure 2):

at t ) 0, C ) Co

∀y

(3)

at y > δ, C ) Co

∀t

(4)

where δ is the polarization layer thickness

at t > 0 and y ) 0, JCm + D

(dCdy )

y)0

) JCp

(5)

Cm is the solute concentration at the membrane surface and Cp is the solute concentration in the permeate flux. The osmotic pressure model uses the equation

J)

∆P - ∆π Rm

(6)

where ∆P is the pressure applied, ∆π the osmotic pressure difference across the membrane (π(Cm) - π(Cp)), and Rm the

where φ is the number of ions, R the gas constant, T the absolute temperature, and M the molecular weight of the solute. The use of eq 8 is equivalent to assuming that, in an interval of dilute solutions, the osmotic pressure is a linear function of the concentration, which is the same hypothesis assumed by Wiley and Fletcher.12 The simultaneous solution of eqs 2-7 gives the concentration profile and the permeate flux at any time during the process. When the pressure ceases, only the diffusive movement of the solute from the polarization layer to the bulk solution occurs. Therefore, the constitutive equation in this case is

∂ ∂C ∂C ) D ∂t ∂y ∂y

( )

(9)

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

∂C ∂2C )D 2 ∂t ∂y which is Fick’s second law.

(10)

Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7221

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

3. Experimental Section 3.1. Experimental Setup. The optical setup (holographic interferometry system) was similar to that described in previous papers.7,8 The RO setup is shown in Figure 3. The pressure necessary to run the process was supplied by means of pressurized nitrogen. To avoid the appearance of bubbles in the solution, 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 requirements of the holographic interferometry technique. The cell was provided with two windows, thus allowing the membrane surface to be visualized. A detailed description of the cell used can be found in a previous paper.13 The active membrane dimensions are 10 cm × 1 cm, with an effective area of 10 cm2, which is a size that has been chosen to satisfy the interferometric requirements. The distance from the membrane surface to the top of the cell is 45 mm; thus, the volume of solution in the cell is large enough to guarantee that concentration changes inside the cell only would occur near the membrane. However, far from the membrane surface, the bulk concentration (Co) will remain unchanged during the process. The RO cell was horizontally placed on the optical table and was the common element between both RO and optical setups. 3.2. Materials. The membrane used was a thin film membrane (TFM-50, from Hydro Water S.L.). Suitable pieces for the size of the cell used were cut from the entire membrane. Each piece of membrane was used for several experiments, so after each experiment the cell was washed with distilled water until the salt was completely removed from the membrane surface. The membrane was considered to be clean when the permeate flux of water was recovered. The experiments were performed using solutions of two salts: NaCl and Na2SO4 (Panreac). Different feed concentrations (Co), in the range of 1-7 kg/m3, were used to study the effect of the solute and feed concentration on the polarization layer. Some physical properties of the solutes that have been used are shown in Table 1.12,14,15 3.3. Experimental Methodology. The experimental procedure is similar to that previously described.7,8

After the water flux was checked, the module was filled with the solution and correctly aligned with the optical setup. A hologram of the cell then was recorded on the holographic plate, to obtain the reference state for the interferometric study. After the hologram was obtained, pressure was applied to the system. All the experiments were conducted at a constant pressure of 600 kPa. As a consequence of the RO process, concentration (and, therefore, refraction index) changes occurred in the solution near the membrane surface. The comparison between the reference state recorded on the hologram and the evolution of the RO process causes the appearance of the interference fringes (interferogram). Each interference fringe corresponds to a certain concentration change in the solution and was different, depending on the salt used. The relationship between concentration and refraction index was measured using a refractometer (Leica, model AR600) at 25 °C, which was the operation temperature. The relations obtained for each salt appear in Table 1. The methodology to obtain concentration profiles quantitatively was the same as that used in previous papers.7,8 The interferograms were continuously recorded using a video camera that was connected to a personal computer (PC). The video-capture software that was used allowed 24 images to be captured each second; therefore, the quantity of interferograms registered during each experiment was very high. Data for weight and conductivity of the permeate solution were also continuously measured during the process, to obtain the permeate flux and concentration. The continuous measure of conductivity was performed using a continuous flux conductivity cell (Crison, model 5287) and a conductimeter (Crison, model GLP 32) that was connected to a PC. The conductivity cell was previously calibrated to obtain the relationship between conductivity and concentration for each solute used (see Table 1). The solute concentration of the permeate flux directly measured (Cp,m) was not actually the concentration of the solution passing through the membrane, because of the existence of a dead volume between the membrane and the conductivity cell (see Figure 3). Passing through this dead volume caused a delay in the conductivity measurement, which was dependent on the value of the flux (J), which also was time-dependent. Hence, a correction of the measured concentration value was made considering ideal plug-flow in the permeate collection pipe. The measured concentration (Cp,m) and the corrected concentration (Cp,c) were related by eq 11:

Cp,c(t) ) Cp,m(t + τ)

(11)

τ is the delay time of the permeate solution and was calculated as

τ)

Vd SJ(t)

(12)

where Vd is the dead volume (expressed in units of m3), S the membrane surface (given in units of m2), and J(t) the permeate flux (given in units of m3/(m2 s)). Because J was variable with time, τ was also variable, so this correction had to be applied to all experimental concentration data that were measured. Some authors16,17 had a similar problem with the measurements of the permeate concentration, but they assumed perfect mixing in the dead volume of the cell.

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Table 1. Physical Properties of the Solutes Useda Expression property

NaCl

Na2SO4

diffusion coefficient, D (m2/s) density, F (kg/m3) refractive index, n conductivity, µ (µS) osmotic pressure, Π (atm)

max (1.61 × 10-9(1-14) × m), 1.45 × 10-9) 7.24C + 997.1 1.76 × 10-4C + 1.33299 1799.2C + 2.29 0.835C

-3.9 × 10-12C + 1.16 × 10-9 9.80C + 997.1 1.54 × 10-4C + 1.33299 1227.3C + 2.29 0.516C

a

reference(s) 12, 15 14 b b

van’t Hoff equation

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

Table 2. Experimental Conditions Co (kg/m3)

Jw (× 106 m3/(m2 s))

I II III IV V

1 2 3.5 5 7

5.61 5.78 5.71 5.42 5.69

VI VII VIII IX X

1 2 3.5 5 7

6.55 6.61 6.61 6.39 6.47

experiment

Rm (× 10-11 Pa s/m2)

area observed (mm × mm)

1.08 1.05 1.06 1.12 1.07

3.40 × 9.40 3.40 × 9.40 3.40 × 9.40 3.40 × 9.40 3.40 × 9.40

0.93 0.91 0.92 0.95 0.94

2.95 × 5.75 2.95 × 5.75 2.95 × 5.75 2.95 × 5.75 2.95 × 5.75

NaCl

Na2SO4

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

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

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


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

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

Ro ) 1 -

Cp,c Co

(13)

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

R)1-

Cp,c Cm

(14)

Although eq 13 actually applies to well-mixed bulk solutions, in unstirred batch RO, the value of Ro can be useful to indicate how similar Co and Cp are and, therefore, how far the process is from the steady state. After ∼2 h, 30 min, the pressure was removed and the RO process ended. From that moment, the concentration polarization layer started to disappear. During the process of disappearance of the polarization layer, interferograms were also recorded until almost all the interference fringes disappeared. The procedure to obtain concentration profiles quantitatively was the same as that used during the RO process. 4. Results and Discussion 4.1. Visualization of the Polarized Layer. By means of holographic interferometry, it has been possible to follow, in real time, the appearance, evolution, and disappearance of the concentration polarization layer during dead-end RO experiments. Two groups of dead-end RO experiments were performed. A different salt (NaCl and Na2SO4) was used in each one, varying the feed concentration between 1 kg/m3 and 7 kg/ m3. In all the experiments, the transmembrane pressure was kept

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

constant at 600 kPa. Very similar behaviors were observed with both solutes. Experimental conditions for each one of the salts studied are shown in Table 2, where Jw is the volumetric permeate flux with pure water before the beginning of the RO experiment and Rm is the membrane hydraulic resistance to the flux with pure water, which is calculated with eq 7. The area observed indicates the true dimensions of the rectangular zone visualized by means of the optical system and is dependent on the magnification needed in each case, which is obtained using a lens to focus the image on the camera. Reproducibility of the behavior observed was confirmed by repeating the RO runs under the same conditions (solute and initial feed concentration). A few minutes after the RO process started, some interferometric fringes near the membrane surface appeared. The amount of fringes continued to increase throughout the process, thus indicating that the concentration at the membrane surface (Cm) was increasing, as well as the thickness of the boundary layer (δ). The rate of appearance of new interference fringes was high during the first few minutes of the experiment; but, later, it decreased and Cm seemed to tend toward a constant value. However, the thickness of the polarization layer (δ) continually increased. This behavior was already predicted by the mathematical models for unstirred dead-end processes used by other authors.18,19 Interferograms at 5, 35, 90 and 120 min, corresponding to two experiments with different initial concentration of Na2SO4 (2 kg/m3 and 5 kg/m3), are shown in Figures 4 and 5. Because each experiment was recorded in a continuous way, many interferograms were available. Interferograms in these figures have been selected, as an example, to illustrate the evolution of the system. In each interferogram, a horizontal line and an auxiliary scale have been drawn to show where the membrane surface initially was, as well as the magnification used in the experiment.

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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.)

The behavior in both experiments was very similar. As can be seen in Figures 4 and 5, after 5 min, some interference fringes had already appeared, thus meaning that a concentration gradient existed from the beginning of the process. New fringes were continuously appearing. The number of fringes at 35 min doubled at 5 min. With increased time, the rate of appearance of new fringes became slower. At 120 min, there were only 1 or 2 more fringes than those observed at 90 min and, as a consequence, Cm was very similar in both cases. This behavior indicates that most of the polarization layer was developed during, approximately, the first hour of the process.

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.)

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


Figure 11. Experimental and calculated concentration profiles at different times: (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. (Figure legend: (O) 5 min, (4) 35 min, (/) 90 min, and (]) 120 min; solid line represents the model calculation.)

Figure 12. Experimental and calculated concentration profiles at different times: (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) 35 min, (/) 90 min, and (]) 120 min; solid line represents the model calculation.)

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Figure 13. Experimental and calculated concentration profiles at different times 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 ) 3.5 kg/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 different times 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) 15 min, (/) 30 min, and (]) 60 min; solid line represents the model calculation.)

Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7227 Table A1. NaCl Concentration Profiles y (mm) interferogram fringe order number 1 2 3 4 5 6 7 8 9 10 11 12

at 5 min

at 35 min

at 90 min

Experiment I: Co ) 1 kg/m3 0.99 4.69 7.51 0.33 1.93 3.27 0.11 1.29 2.17 0.92 1.63 0.67 1.26 0.49 1.00 0.35 0.77 0.19 0.65 0.09 0.47 0.33 0.19

y (mm) at 120 min

C (kg/m3)

interferogram fringe order number

8.31 4.23 2.77 2.12 1.67 1.34 1.08 0.87 0.67 0.49 0.35 0.19

1.36 1.72 2.08 2.44 2.80 3.16 3.52 3.88 4.24 4.60 4.95 5.31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 2 3 4 5 6 7 8 9 10 11 12 13

1 2 3 4 5 6 7 8 9 10 11 12 13

Experiment III: Co ) 3.5 kg/m3 0.81 4.35 5.70 0.49 2.79 3.97 0.31 2.08 3.10 0.17 1.64 2.49 1.28 2.08 1.00 1.69 0.77 1.38 0.58 1.09 0.40 0.89 0.22 0.66 0.05 0.48 0.28 0.11

5.90 4.44 3.51 2.88 2.37 1.93 1.54 1.26 0.91 0.62 0.39 0.19 0.11

3.86 4.22 4.58 4.94 5.30 5.66 6.02 6.38 6.74 7.10 7.45 7.81 8.17

1 2 3 4 5 6 7 8 9 10 11

Experiment V: Co ) 7 kg/m3 0.87 3.12 4.78 0.43 2.25 3.56 0.23 1.73 2.81 0.11 1.13 2.21 0.76 1.83 0.42 1.39 0.22 1.01 0.09 0.69 0.46 0.27 0.12

5.42 4.15 3.30 2.63 2.15 1.69 1.30 0.96 0.66 0.42 0.19

7.36 7.72 8.08 8.44 8.80 9.16 9.52 9.88 10.24 10.60 10.95

other hand, if the membrane is not completely selective, the permeate concentration (Cp) will increase as the transmembrane concentration (∆C) does. Finally, the diffusive flow toward the solution bulk will also increase as the concentration difference (Cm - Co) does. In these circumstances, if the permeate concentration (Cp) equals the bulk concentration (Co), the convective solute flow to the membrane surface will be balanced by the solute flux through the membrane and the diffusive flow back to the bulk solution, and no more accumulation of solute in the vicinity of the membrane will occur. This tendency to reach the stationary state, also noted by Nicolas et al.,17 can be observed when permeate flux and membrane rejection are studied (see Section 4.20) because, in some experiments, the permeate flux stabilizes and does not continuously decrease, and also because the observed retention (Ro) tends toward zero (Cp ≈ Co). After the pressure ceased and the RO process ended, the fringe pattern changed. The interference fringes became wider and more separated, slowly disappearing. Interferograms in Figures 6 and 7 correspond to 2, 15, 30, and 60 min after the end of the RO process, for the two previously mentioned experiments (Na2SO4; Co ) 2 and 5 kg/m3). As can be observed in both figures, the number of fringes decreased with time and the remaining fringes were wider and more separated between themselves, as well as from the membrane. This behavior indicates that the concentration at the membrane surface (Cm)

at 5 min

at 35 min

at 90 min

Experiment II: Co ) 2 kg/m3 2.34 2.37 4.49 0.67 1.75 3.23 0.34 1.44 2.53 0.20 1.09 2.12 0.86 1.68 0.68 1.38 0.49 1.13 0.35 0.92 0.21 0.69 0.12 0.52 0.33 0.17 0.09 Experiment IV: Co ) 5 kg/m3 1.47 3.95 5.26 0.82 2.68 4.18 0.49 2.06 3.29 0.28 1.58 2.66 0.12 1.23 2.16 0.97 1.79 0.70 1.40 0.49 1.07 0.29 0.78 0.16 0.54 0.34 0.14

at 120 min

C (kg/m3)

5.13 3.73 2.95 2.40 1.92 1.63 1.33 1.09 0.86 0.65 0.50 0.34 0.20 0.12

2.36 2.72 3.08 3.44 3.80 4.16 4.52 4.88 5.24 5.60 5.95 6.31 6.67 7.03

6.91 5.00 3.93 3.17 2.58 2.17 1.75 1.38 1.06 0.73 0.52 0.32 0.11

5.36 5.72 6.08 6.44 6.80 7.16 7.52 7.88 8.24 8.60 8.95 9.31 9.67

decreases and the thickness of the boundary layer (δ) increases with time. Therefore, the slope of the concentration profiles gets smoother with time, until the concentration gradient completely disappears. The disappearance of the polarization layer is a very slow process (for instance, 60 min after the end of the RO process, some interference fringes were still visible). This evolution of the interference fringes is a consequence of the diffusive solute flow from the membrane surface to the bulk solution, because of the concentration gradient between them that is generated during the RO process. This proves that concentration polarization is a reversible process. 4.2. Permeate Flux and Membrane Rejection. Data regarding permeate weight and concentration were continuously measured for each experiment, to obtain the permeate flux (as the curve derived from the weight) and the membrane rejection (see eqs 13 and 14) and the effect of initial concentration on both parameters was studied. Both parameters are indicative of the membrane performance and, therefore, of the effectiveness of the process. Both varied with time, because of the fact that dead-end RO is an unsteady-state process. As a consequence of the concentration polarization, a decrease in the permeate flux was observed during the process. As concentration increased in the vicinity of the membrane, the osmotic pressure of the solution also did and, hence, the driving force (∆P - ∆π) decreased. Other authors who have been

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Table A2. Na2SO4 Concentration Profiles y (mm) interferogram fringe order number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

at 5 min

at 35 min

at 90 min

Experiment VI: Co ) 1 kg/m3 0.62 1.97 4.61 0.27 1.24 2.30 0.15 0.93 1.76 0.05 0.74 1.41 0.61 1.20 0.51 1.03 0.42 0.89 0.33 0.78 0.28 0.68 0.22 0.57 0.16 0.50 0.10 0.43 0.05 0.36 0.30 0.23 0.19 0.13 0.06

Experiment VIII: Co ) 3.5 kg/m3 1.19 2.96 4.65 0.68 2.18 3.53 0.47 1.77 2.94 0.34 1.49 2.54 0.25 1.28 2.23 0.18 1.09 2.00 0.12 0.95 1.77 0.06 0.82 1.57 0.70 1.40 0.60 1.23 0.52 1.11 0.43 0.96 0.36 0.86 0.29 0.74 0.23 0.63 0.16 0.54 0.10 0.44 0.05 0.35 0.26 0.18 0.12 0.06 Experiment X: Co ) 7 kg/m3 1.08 3.01 4.89 0.71 2.35 3.97 0.50 1.97 3.35 0.38 1.68 2.86 0.28 1.42 2.50 0.18 1.21 2.20 0.11 1.04 1.94 0.05 0.86 1.71 0.73 1.50 0.60 1.31 0.48 1.11 0.36 0.95 0.25 0.80 0.15 0.63 0.06 0.50 0.38 0.26 0.15

y (mm) at 120 min

C (kg/m3)


5.72 3.00 2.14 1.73 1.44 1.25 1.08 0.95 0.83 0.73 0.64 0.55 0.49 0.40 0.34 0.29 0.22 0.16 0.11 0.05

1.41 1.82 2.23 2.64 3.05 3.46 3.87 4.28 4.70 5.11 5.52 5.93 6.34 6.75 7.16 7.57 7.98 8.39 8.80 9.21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

5.42 4.10 3.38 2.91 2.56 2.29 2.02 1.83 1.65 1.47 1.31 1.17 1.03 0.89 0.78 0.67 0.56 0.48 0.38 0.30 0.20 0.14 0.05

3.91 4.32 4.73 5.14 5.55 5.96 6.37 6.78 7.20 7.61 8.02 8.43 8.84 9.25 9.66 10.07 10.48 10.89 11.30 11.71 12.12 12.53 12.94

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

5.70 4.64 3.88 3.35 2.91 2.54 2.24 2.01 1.76 1.54 1.33 1.15 0.95 0.80 0.65 0.50 0.36 0.25 0.15

7.41 7.82 8.23 8.64 9.05 9.46 9.87 10.28 10.70 11.11 11.52 11.93 12.34 12.75 13.16 13.57 13.98 14.39 14.80

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

at 5 min

at 35 min

at 90 min

Experiment VII: Co ) 2 kg/m3 0.95 2.48 3.53 0.50 1.69 2.64 0.33 1.33 2.20 0.20 1.13 1.86 0.14 0.95 1.61 0.08 0.83 1.45 0.71 1.26 0.61 1.13 0.52 1.01 0.44 0.89 0.38 0.78 0.31 0.69 0.26 0.60 0.19 0.53 0.14 0.45 0.08 0.38 0.31 0.25 0.18 0.11 0.06 Experiment IX: Co ) 5 kg/m3 0.81 2.68 4.94 0.57 2.10 3.80 0.44 1.77 3.16 0.32 1.52 2.73 0.25 1.31 2.38 0.18 1.14 2.10 0.12 0.99 1.84 0.06 0.86 1.64 0.74 1.45 0.63 1.29 0.52 1.13 0.43 0.98 0.35 0.85 0.27 0.73 0.20 0.61 0.14 0.50 0.07 0.40 0.29 0.21 0.12

at 120 min

C (kg/m3)

3.98 3.06 2.52 2.14 1.87 1.66 1.48 1.31 1.15 1.04 0.93 0.82 0.72 0.64 0.55 0.47 0.39 0.32 0.25 0.18 0.13

2.41 2.82 3.23 3.64 4.05 4.46 4.87 5.28 5.70 6.11 6.52 6.93 7.34 7.75 8.16 8.57 8.98 9.39 9.80 10.21 10.62

5.68 4.48 3.68 3.19 2.78 2.45 2.17 1.95 1.73 1.55 1.37 1.20 1.06 0.92 0.79 0.67 0.56 0.43 0.34 0.25 0.16 0.06

5.41 5.82 6.23 6.64 7.05 7.46 7.87 8.28 8.70 9.11 9.52 9.93 10.34 10.75 11.16 11.57 11.98 12.39 12.80 13.21 13.62 14.03

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

Ind. Eng. Chem. Res., Vol. 45, No. 21, 2006 7229 Table A3. NaCl Concentration Profiles during the Disappearance of the Polarized Layer y (mm) interferogram fringe order number

at 5 min

at 35 min

y (mm)

at 90 min

1 2 3 4 5 6 7 8 9

Experiment I: Co ) 1 kg/m3 4.00 4.50 5.24 2.76 3.38 3.79 2.19 2.60 2.71 1.82 1.99 1.68 1.50 1.37 0.37 1.21 0.55 0.97 0.63 0.24

1 2 3 4 5 6 7 8 9 10

Experiment III: Co ) 3.5 kg/m3 6.09 6.43 7.12 4.35 4.79 5.35 3.41 3.87 4.25 2.77 3.13 3.21 2.31 2.49 2.26 1.90 1.78 0.86 1.58 0.90 1.24 0.88 0.46

1 2 3 4 5 6 7 8 9 10 11

Experiment V: Co ) 7 kg/m3 8.10 7.83 8.13 6.17 6.13 6.45 4.96 4.98 5.17 4.05 4.09 4.19 3.27 3.23 3.13 2.65 2.48 1.83 2.11 1.57 0.85 1.68 0.56 0.14 1.15 0.68 0.09

at 120 min

C (kg/m3)


6.24 4.14 2.35

1.36 1.72 2.08 2.44 2.80 3.16 3.52 3.88 4.24

1 2 3 4 5 6 7 8 9 10

Experiment II: Co ) 2 kg/m3 4.66 5.19 5.90 3.53 4.13 4.57 2.82 3.31 3.57 2.33 2.66 2.64 1.93 2.04 1.58 1.62 1.40 0.66 1.35 0.64 1.04 0.75 0.16

8.54 6.21 4.65 3.15 0.90

3.86 4.22 4.58 4.94 5.30 5.66 6.02 6.38 6.74 7.10

1 2 3 4 5 6 7 8 9 10 11

Experiment IV: Co ) 5 kg/m3 8.63 8.46 5.97 5.99 6.22 4.56 4.75 4.92 3.62 3.84 3.85 2.95 3.06 2.84 2.37 2.35 1.64 1.93 1.54 0.77 1.59 0.62 1.12 0.70 0.15

8.45 6.75 5.21 3.95 1.86 0.78

7.36 7.72 8.08 8.44 8.80 9.16 9.52 9.88 10.24 10.60 10.95

permeate flux. However, the evolution of permeate flux with time was very similar from one experiment to another: after the initial reduction, the slope of the curves was very small and similar in all the experiments. Membrane-observed rejection (Ro) and intrinsic rejection (R) were calculated with eqs 13 and 14. R is an interesting value when the transport mechanism through the membrane is studied. Few reliable data of R exist in the literature, because of the lack of an experimental technique that allows a precise measurement of Cm. Usually, an indirect approach to determine R is made by solving a simplified transport equation in the polarization layer.17

J)

(

)

D C m - Cp ln δ Co - C p

(15)

The use of this simplification assumes a steady-state situation for stirred batch cells or cross-flow cells. Therefore, this formula is not correct for unstirred dead-end RO processes. Using holographic interferometry, it has been possible to measure Cm, which corresponds to the nearest fringe to the membrane surface. The measurement of Cm during each experiment has allowed a more-accurate value of R to be calculated. The initial concentration also had a great influence on the membrane rejection, as can be observed in Figures 9 and 10. For the two solutes studied, the higher the feed concentration, the higher the rejection observed. During the entire process, Ro underwent a continuous reduction, reaching very low values in some of the experiments that were performed. For example, in experiment I (NaCl, Co ) 1 kg/m3), over long periods, Ro tends toward zero (Figure 9). This reduction of Ro indicates an increase in

at 5 min

at 35 min

at 90 min

at 120 min

C (kg/m3)

6.53 4.97 3.35 1.20

2.36 2.72 3.08 3.44 3.80 4.16 4.52 4.88 5.24 5.60

8.82 6.59 4.91 3.42 0.65

5.36 5.72 6.08 6.44 6.80 7.16 7.52 7.88 8.24 8.60 8.95

permeate concentration (see eq 13). As previously mentioned, when Cp ) Co, the process will reach a stationary-state. In that case, the value of Ro will be zero. The shape of the Ro curves indicates that the evolution of dead-end RO processes tends toward the previously mentioned steady-state, which is attained earlier in processes that have been performed with low concentration solutions. On the other hand, the intrinsic rejection is dependent directly on the maximum concentration at the membrane surface (see eq 14). The higher the value of Cm, the more solute that is able to pass through the membrane and therefore, the lower the value of R. In experiments where low concentration solutions were used, intrinsic rejection became almost stable after ∼1 h of experiment (see Figure 10). This behavior reinforces the theory that the more diluted the feed solution, the faster the approach to a steady-state condition. 4.3. Concentration Profiles. Interference fringes that appeared in the interferograms were the result of a refractive index gradient in the vicinity of the membrane that were due to a concentration gradient. Because a relationship exists between the refractive index and concentration (Table 1) in aqueous solutions of salts, it was possible to obtain the concentration profile from the interference fringes, using the methodology explained in a previous paper.7 The fringes farthest from the membrane surface were broader and it was more difficult to assign their position precisely, because of the difficulty involved to ascertain exactly where the maximum or the minimum of light intensity was. The evolution of the concentration profiles of the experiments shown in Table 2 can be observed in Figures 11 and 12. As can be observed, the development of the

7230


Table A4. Na2SO4 Concentration Profiles during the Disappearance of the Polarized Layer y (mm) interferogram fringe order number

at 5 min

at 35 min

at 90 min

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Experiment VI: Co ) 1 kg/m3 5.24 5.62 2.86 3.86 4.68 2.28 3.12 3.76 1.94 2.62 3.06 1.70 2.24 2.46 1.50 1.91 1.89 1.34 1.53 1.21 1.19 1.15 0.28 1.06 0.66 0.94 0.78 0.63 0.47 0.30 0.08

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Experiment VIII: Co ) 3.5 kg/m3 5.29 5.21 4.12 4.36 4.77 3.48 3.83 4.14 3.05 3.37 3.62 2.72 3.01 3.14 2.44 2.67 2.68 2.20 2.35 2.26 1.99 2.03 1.80 1.80 1.72 1.29 1.61 1.42 0.57 1.45 1.02 1.29 0.59 1.14 0.09 0.92 0.69 0.47 0.23

y (mm) at 120 min

5.36 3.98 2.93 1.84 0.35

5.39 4.60 3.87 3.18 2.48 1.72 0.54

C (kg/m3)


1.41 1.82 2.23 2.64 3.05 3.46 3.87 4.28 4.70 5.11 5.52 5.93 6.34 6.75 7.16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Experiment VII: Co ) 2 kg/m3 4.27 4.79 5.47 3.29 3.96 4.49 2.75 3.40 3.80 2.37 2.96 3.31 2.10 2.60 2.82 1.91 2.29 2.36 1.70 1.96 1.84 1.54 1.64 1.34 1.39 1.31 0.43 1.24 0.91 1.11 0.37 0.98 0.81 0.62 0.43 0.23

3.91 4.32 4.73 5.14 5.55 5.96 6.37 6.78 7.20 7.61 8.02 8.43 8.84 9.25 9.66 10.07 10.48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Experiment IX: Co ) 5 kg/m3 5.20 5.27 5.72 4.32 4.53 4.93 3.70 3.97 4.29 3.22 3.50 3.77 2.84 3.11 3.30 2.57 2.76 2.83 2.29 2.42 2.35 2.06 2.09 1.82 1.85 1.71 0.87 1.66 1.29 0.09 1.46 0.67 1.27 1.04 0.79 0.53 0.23

at 5 min

at 35 min

at 90 min

at 120 min

5.16 4.29 3.55 2.76 1.93 0.39

5.501 4.689 4.011 3.293 2.549 1.175

C (kg/m3) 2.41 2.82 3.23 3.64 4.05 4.46 4.87 5.28 5.70 6.11 6.52 6.93 7.34 7.75 8.16 8.57 5.41 5.82 6.23 6.64 7.05 7.46 7.87 8.28 8.70 9.11 9.52 9.93 10.34 10.75 11.16 11.57

Experiment X: Co ) 7 kg/m3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

4.89 4.13 3.62 3.17 2.79 2.49 2.20 1.96 1.71 1.48 1.22 0.90 0.53 0.14

5.23 4.48 3.93 3.41 3.01 2.60 2.19 1.78 1.33 0.62

5.47 4.71 4.08 3.50 2.94 2.39 1.76 0.75

5.21 4.38 3.55 2.67 1.45

7.41 7.82 8.23 8.64 9.05 9.46 9.87 10.28 10.70 11.11 11.52 11.93 12.34 12.75 13.16

polarization layer mainly occurred during the first few minutes of the process, whereas, at longer time periods, the variation of the concentration profile was very slight. Despite the continuous growing of the boundary layer, its maximum thickness, in all the experiments, was no more than 7-8 mm, even at long periods. 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 boundary layer. Moreover, at greater distances from the membrane surface, no interference fringes were observed, which means that the bulk solution had maintained its initial concentration (Co). The same procedure can be used to obtain concentration profiles during the disappearance of the fringes. The evolution of the concentration profiles of the experiments shown in Table 2 can be observed in Figures 13 and 14. It can be seen that the slope 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 of the boundary layer. 4.4. Application of a Mathematical Model to the Experimental Data. Using the same operation conditions, all the experiments conducted were also simulated by the mathematical model explained 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. By means of this set of equations, the concentration profiles (C(y,t)) and permeate flux (J) were calculated at any time during the RO process. To model the process, the following data were needed: feed solution concentration (Co), transmembrane pressure (∆P), diffusion coefficient of the solute used (D), permeate flux with pure water (Jw), and permeate concentration (Cp,c). Jw and Cp,c were experimentally measured. In eq 4, to initiate the calculation process, a high value of δ was assumed, where the solute concentration was not expected to change. After the calculation


program was run, it was checked that the value of δ that was calculated was lower than the value of δ that was assumed. Concentration profiles and permeate flux calculated for each experiment were compared with those experimentally obtained to validate the model that was proposed. As can be observed in Figures 8a and 8b, a reasonable agreement between experimental and calculated permeate fluxes is obtained, better at larger times than at short times. In Figures 11 and 12, the experimental and calculated concentration profiles are very similar, better as the time and initial concentration increases, thus indicating that the combination of the convection-diffusion mechanism and the osmotic pressure theory could be quite an adequate model to describe, in the range of the experimental conditions studied, RO processes operating under unstirred batch conditions. The polarization layer disappeared as a consequence of the diffusive movement from the membrane surface to the bulk solution of the solute accumulated during the RO process, caused by the concentration difference between both places. After the convective solute flow due to the pressure applied ceased, diffusion was the only mass-transfer mechanism that was involved. To calculate concentration profiles during the disappearance of the polarized layer, eq 10 was solved numerically, using the first concentration profile obtained by holographic interferometry (60 s after the RO process finished) as the initial condition. Experimental and calculated concentration profiles are shown in Figures 13 and 14. In both figures, the slope of the concentration profile becomes smoother with time and, therefore, the change in concentration is less. This small change in concentration explains the intersection of the profiles at different times. In the same way as in the RO process, experimental and simulated results were rather concordant (see Figures 13 and 14); thus, the diffusion theory describes quite precisely the process of the disappearance of the polarization layer. 5. Conclusions The experimental results obtained proved that real-time holographic interferometry is a useful noninvasive technique to study the concentration polarization phenomenon during reverse osmosis (RO) processes. It has been used to follow, as interference fringes, the appearance, the evolution with time, and the disappearance of the polarization layer in the dead-end RO of two salts (NaCl and Na2SO4). Concentration profiles and permeate flux have been determined. While permeate flux undergoes a reduction with time, the concentration at the membrane surface (Cm), as well as the thickness of the boundary layer (δ), were continuously increasing. A mathematical model has been proposed for the simulation of dead-end RO processes. The model, which is a combination of the convection-diffusion mechanism and the osmotic pressure theory, has been validated, under the experimental conditions used, by a reasonable agreement between experimental and calculated results (concentration profiles and permeate flux). Concentration profiles, after the pressure ceased, have been experimentally obtained and it has been possible to study the disappearance of the polarization concentration layer, concluding that the polarization phenomenon is a reversible process. A mathematical model, based on a diffusive mechanism, has been proposed. Agreement between the experimental and calculated concentration profiles validates the model. Acknowledgment This research was sponsored by the Plan Nacional de I+D+I BQU2000-0456 (Ministerio de Educacioń y Cultura) and by the Ajuda per a Grups de I+D+I de la Consellerıá d′Empresa,

Universitat i Cie`ncia (Generalitat Valenciana). The authors are grateful to Dr. Jose´ A. Caballero-Suarez and Dr. Vicente GomisYaguës for their assistance in the application of the model. Appendix The number of interference fringes, the distance (y) to the membrane for each fringe and its concentration (C) for some selected times of the experiments presented in Table 2 are shown in Tables A1 and A2. Same information corresponding to the disappearance of the polarized layer is shown in Tables A3 and A4. Literature Cited (1) Sablani, S. S.; Goosen, M. F. A.; Al-Belushi, R.; Wilf, M. Concentration polarization in ultrafiltration and reverse osmosis: a critical review Desalination 2001, 141, 269-289. (2) Chen, V.; Li, H.; Fane, A. G. Noninvasive observation of synthetic membrane processessa review of methods. J. Membr. Sci. 2004, 241, 2344. (3) Johnson, A. R. Experimental investigation of polarization effects in reverse osmosis. AIChE J. 1974, 20, 966-974. (4) Clifton, M.; Sanchez, V. Holographic interferometry applied to the measurement of boundary layers in electrodialysis and ultrafiltration. SPIE, Opt. Photonics Appl. Med. 1979, 211, 111-115. (5) Mahlab, D. M.; Yosef, N. B.; Belfort, G. Concentration polarization profile for dissolved species in unstirred batch hyperfiltration (reverse osmosis)sII. transient case. Desalination 1978, 24, 297-303. (6) Mahlab, D.; Yosef, N. B.; Belfort, G. Interferometric measurement of concentration polarization profile for dissolved species in unstirred batch hyperfiltration (reverse osmosis). Cem. Eng. Commun. 1980, 6, 225-243. (7) Fernańdez Torres, M. J.; Ruiz-Bevia´, F.; Fernańdez-Sempere, J.; Lo´pez-Leiva, M. Visualization of the UF Polarizad Layer by Holographic Interferometry. AIChE J. 1998, 44, 1765-1776. (8) Fernańdez-Sempere, J.; Ruiz-Bevia´, F.; Salcedo-Dıáz, R. Measurements by hologrphic interferometry of concentration profiles in dead-end ultrafiltration of poly(ethylene glycol) solutions. J. Membr. Sci. 2004, 229, 187-197. (9) Ruiz-Bevia´, F.; Fernańdez-Sempere, J.; Colom-Valiente, J. Diffusivity Measurement in Calcium Alginate Gel by Holographic Interferometry. AIChE J. 1989, 35 (11), 1895-1898. (10) Kosar, T. F.; Phillips, R. J. Measurement of protein diffusion in dextran solutions by holographic interferometry. AIChE J. 1995, 41 (3), 701-711. (11) Fernańdez-Sempere, J.; Ruiz-Bevia´, F.; Colom-Valiente, J.; Ma´sPe´rez, F. Determination of Diffusion Coefficients of Glycols, J. Chem. Eng. Data 1996, 41, 47-48. (12) Wiley, D.; Fletcher, D. A computational fluids dynamics study of buoyancy effects in reverse osmosis. J. Membr. Sci. 2004, 245, 175-181. (13) Fernańdez-Sempere, J.; Ruiz-Bevia´, F.; Salcedo-Dıáz, R.; GarcıáAlgado, P. Equipo experimental para visualizar la formacioń de la capa de polarizacioń durante el proceso de o´smosis inversa. Ing. Quı´m. 2006, 438, 147-154. (14) Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’ Handbook; 7th Edition; McGraw-Hill: New York, 1997. (15) Sourirajan, S. ReVerse Osmosis; Academic Press: New York, 1970. (16) Balannec, B.; Nicolas, S.; Bariou, B. Experimental study and modelization of reverse osmosis with salt solutes in an unstirred batch cell. Desalination 1999, 122, 43-51. (17) Nicolas, S.; Balannec, B.; Beline, F.; Bariou, B. Ultrafiltration and reverse osmosis of small noncharged molecules: a comparison study of rejection in a stirred and an unstirred batch cell. J. Membr. Sci. 2000, 164, 141-155. (18) Van den Berg, G. B.; Smolders, C. A. Flux decline in ultrafiltration processes. Desaliation 1990, 77, 101-133. (19) Bhattacharjee, S.; Bhattacharya, P. K. Flux decline behaviour with low molecular weight solutes during ultrafiltrarion in an unstirred batch cell. J. Membr. Sci. 1992, 72, 149-161. (20) Boulanouar, I.; Nicolas, S. Bariou, B. Ultrafiltration and reverse osmosis in unstirred batch cell of charged solutes (protein, salts) with total retention. Desalination 1996, 104, 83-93.

ReceiVed for reView April 3, 2006 ReVised manuscript receiVed July 12, 2006 Accepted August 1, 2006 IE060417Z

Measurement of Concentration Profiles by Holographic Interferometry

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