Investigation of flow next to membrane walls

Journal of Membrane Science 264 (2005) 137–150

Investigation of flow next to membrane walls Margarita (Modek) Gimmelshtein, Raphael Semiat ∗ Rabin Desalination Laboratory, Grand Water Research Institute, Wolfson Department of Chemical Engineering, Technion – Israel Institute of Technology, Technion City, 32000 Haifa, Israel Received 10 October 2004; received in revised form 20 April 2005; accepted 23 April 2005 Available online 7 July 2005

Abstract The research reported here is based on a flat membrane demo-model demonstrating flow through the spacer between two membranes. The main objectives of the investigation were to measure velocity and estimate the magnitude of mixing index (MI) within a unit cell of the turbulence promoter net. The control of mixing intensity next to the membrane walls enables controlling parameters such as flux, product concentration and the local mass transfer coefficient. The particle image velocimetry (PIV) method was used for velocity measurements and MI estimations. The results show that flow direction changes occur near the spacer rods mainly due to the existence of an obstacle that the flow had to bypass. Significant mixing intensity can be found at higher flow rates than those used regularly in membrane processes. A simple, steady state two-dimensional flow analysis confirmed the measurements’ main findings. It was also shown that the spacer between the membranes may induce membrane fouling close to the filaments. © 2005 Elsevier B.V. All rights reserved. Keywords: Mixing index; Particle image velocimetry; Flow simulation

1. Introduction

1.1. Literature survey

As a mass transfer selective barrier, the membrane is affected by a severe problem known as concentrationpolarization. This is the accumulation of rejected species behind the membrane that could be suspended matter, large molecules, organic matter, bacteria and viruses, colloids, ionic salts, etc. Due to relatively high local concentration, this accumulation tends to increase osmotic pressure, reducing the flux through the membrane or causing solid precipitation on the membrane, which may simply clog it or sometimes cause severe damage. The concentration-polarization layer is very thin, down to an order of tens of nanometers. It is difficult to enter this sub-layer in order to mix its content. The only known way is to increase turbulence in the flow channel.

Flow fluctuations in the space between the membranes may improve mass transfer. Schock and Miquel [39] investigated the pressure drop in spiral-wound membranes in order to characterize mass transfer with flow conditions. Parvatiyar [32] proposed models for mass transfer dependence on turbulence. Brewster et al. [4], Chung et al. [8] and Kaur and Agarwal [21] proposed the use of Dean Vortices as a means for increasing mixing in membranes. Zimmerer and Kottke [48] injected ammonia as a reactive tracer in the flow between membranes in order to investigate mixing intensity and mass transfer based on flow and tracer parameters. Chang et al. [7] suggested introducing air bubbles during the operation of hollow-fiber membranes in order to clean the membranes and mix the boundary layer. Scot and Lobato [46] suggested using cross-corrugated membranes to improve mass transfer. Kaminski and Stawczyk [19] proposed using wavy membranes in order to increase rotational shear and consequently improve mass transfer. Neal et al. [31] used a video camera to investigate the influence of spacer

∗

Corresponding author. Tel.: +972 4 8292009; fax: +972 4 8295672. E-mail address: [email protected] (R. Semiat).

0376-7388/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2005.04.033

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parameters on critical flux and membrane fouling by particle sedimentation. Light and Tran [27] and Da Costa et al. [9] investigated different spacer types in order to optimize mass transfer in RO and UF membranes. The same group, as well as Da Costa and Fane [10] and Da Costa et al. [11], continued to investigate parameters affecting mass transfer induced by the spacers’ geometry in the flow channel between the membranes. An important finding was that the increase in mass transfer is always followed by an increase in pressure drop along the membrane. Polyakov and Karelin [35] suggested that a highconcentration polarization layer exists close to the spacer filaments, where only diffusion controls the concentration. Kuhnel and Kottke [24] developed an ink chemisorption technique on a polyamid layer that may be used for membrane mass transfer investigation. Avlonitis et al. [2,3] theoretically and experimentally investigated the influence of geometries in spiral-wound membranes. Schwinge et al. [41] investigated a zigzag type spacer and found that the spacer significantly increased the mass transfer at the expense of increased pressure drop. Phattaranawik et al. [34] investigated 20 different spacers in perdistillation membranes, in comparison to a UF membrane. A theoretical analysis of the flow next to the membranes was carried out by many investigators. Kamiadakis et al. [18] investigated flow stability around an array of cylinders simulating spacer filaments. The transition from laminar stable flow to non-stable flow took place at Reynolds number (Re) ranging from 100 to 200. Above Re = 600, the flow was characterized by a large vortex flow behind the cylinders. Significant efforts were invested in simulating the flow between two membranes around the turbulence promoter spacer. Buffat [5], Rosen and Tragardh [38], Pellerin et al. [33], and Madireddi et al. [29] developed flow models simulating flow in the membrane channel by introducing laminar and turbulent models at low Re. Cao et al. [6] used the Fluent code to simulate the flow. Karode and Kumar [20] analyzed the flow at relatively higher velocities and showed that the flow is basically parallel to the spacer filaments. Li et al. [25,26] simulated the flow and mass transfer for different types of spacers and reported that vortices start to appear at Re > 150. Schwinge et al. [40,42–45] connected the flow simulation to the pressure drop in the cell. They checked many parameters related to this type of problem and presented some operative solutions for improved design. Geraldes et al. [12–14] simulated the flow around ladder-type spacers. Koutsou et al. [23] recently simulated the flow between membranes spaced by turbulence promoters. Kim et al. [22] simulated the flow around confined obstacles at high Re (approximately 3000). Pozrikidis [36] concentrated on the relative distances between cylinders representing the spacers. Lipnizki and Jonsson [28] attempted to simulate mass transfer along the membrane, with and without the spacer. Staudacher et al. [47] used Fluent to simulate mass transfer in gas separation and pervaporation.

In summarizing the efforts reported in the literature, it is clear that spacers may improve mass transfer through membranes. It is also clear that this improvement is involved with high pressure drop consuming high energy, so most membrane separations are performed at low Re, where the range of reported improved mass transfer is relatively low. This article deals with two directions for investigation: the first related to work carried out in respect to turbulence promotion close to the membrane wall, and the second dealing with the simulation of flow relative to turbulence promoters. The understanding of mass transfer behavior is essential for the optimization of membrane processes. However, a direct observation of flow phenomena, namely flow visualization, is essential for further understanding. One of these flow visualization techniques is particle image velocimetry (PIV). 1.2. Particle image velocimetry technique Using the PIV application, the fluid must be seeded with particles. Two light pulses illuminate these particles at short time differences. Laser light is usually formed into a thin light plane guided into the flow field. A rapid-frame transfer CCD captures two frames exposed by the laser light pulses. Both images are then divided into small interrogation areas for which a single velocity vector is calculated. Areas from both frames are checked for identical particles to calculate the displacements. Pixel calculations extract the velocity from the correlation, and a vector for this interrogation window is obtained. The controlled time difference and calculated particle displacement enable a calculation of the two-dimensional velocity map. More information about this technique may be obtained from Adrian [1], Raffel et al. [37], Melling [30], Huang et al. [17] and Grant [15]. Cross-correlation applied to two frames provides exact information about the direction of a particle’s movement, and the illumination pulses enable velocity calculations. With known pixel size and magnification (lens and distance), the information can be converted easily into a real velocity scale. A definition of a mixing index (MI) parameter, based on the local deviations from smooth flow may be given by:
(u − u) ¯ 2 + (v − v¯ )2 √ MI = (1) u¯ 2 + v¯ 2 where u and v represent the two-directional instantaneous velocities, and the bar represents the average directional velocity. It is important to mention that for the “home-made” equipment used in this work, no attempt was made to calculate time–average velocities. This is reasonable since the research aim was to determine if mixing exists in the flow field. The MI is replacing the more accurate time dependent turbulence intensity. Average velocities in this case represent averaging along the path line around a measurement point. The experimental setup is described in Section 2. Flow simulation is presented in Section 3, followed by results reported and discussed in Section 4.

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2. Experimental setup The experimental system is described schematically in Fig. 1. A flat membrane cell made of aluminum equipped with 1.3 cm diameter transparent Plexiglas windows allowing light transfer was built. The inner part of the cell is comprised of a narrow membrane box (internal size 190, 40 and 1 mm, respectively in the x, y and z directions) containing a turbulence/miximg promoter spacer. The system also includes a 20 L feed vessel, a centrifugal pump, a variable area flow meter, piping and valves. The spacer had a flexible rhombus-like structure made of four 2.4 mm wires of 0.37 mm average diameter arranged so that the longer diagonal is about 3.5 mm and the shorter about 2.5 mm. Spacer thickness was measured at approximately 0.8 mm. A water solution containing fresh seeding of yeast

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particles of normal size of approximately 5 ␮m was used in each experiment. The PIV system included a pixel-fly double-shutter HiRes 1360(H) × 1024(V) CCD camera, a VZM”1000 2.5× to 10× lens and a red LED-based light system operated by a homemade pulse generator. An image processing frame-grabber was installed in a PC computer, also equipped with the Matlab image-processing program. The pulse generator was used for camera trigger control and was synchronized with the light source. The camera worked in the double-shutter mode, which enables capturing two separated full-frame images within a short inter-framing time. The exact timing between the images was controlled by the time between two illuminations of the LED, and varied between 2–10 ␮s, depending on flow rate. Since the LED illuminated the entire flow volume, the location of the investigated x–y plane was obtained by controlling the camera movement, based on the narrow focal depth of the camera-lens system. The PIV measurements were performed based on the principle described earlier. Image capture included two images that were cross-correlated using the Matlab program. An appropriate interrogation area (32 × 32 pixels) was used to calculate two-directional velocities based on the timing parameters of the pulse generator. Seeding density was between 12–15 particles for an interrogation area of 32 × 32 pixels. Error correction algorithms for the velocity calculations in this work were based on global filter, local median filter and vector interpolation techniques. The calculated standard deviation based error was 4–6%. The computer program used was obtained from the Technion’s Department of Mechanical Engineering (Gurka [16]). A text file with values of flow velocities in the x and y directions was obtained as a result of these actions. Parameters and work conditions during the experiments: • Velocity range between 0.06 and 1.3 m/s. • Re calculated between 240 and 4000.

Fig. 1. Experimental system: (a) schematic representation of the experimental system; (b) membranes flow chamber.

It is important to mention here that no membranes were used in the experiments. The chamber walls simulated membranes with no permeation through the walls. The permeation expected in the working conditions for a real RO membrane is in the order of 0.02% of the total flow rate, justifying neglecting permeation for the experiments. For UF membranes where high permeation exists, the situation is different. Permeation causes concentration polarization that affects the flow behavior next to the membrane wall. The associated effects were not considered here. The operational procedure included cleaning the system with a dilute solution of NaClO3 and filling the vessel with the water-containing particles. Several grams of yeast were needed for 10 L of water. The flow was adjusted to a chosen flow rate using the pump and flow meter. When proper flow was established and particles were seen through the camera on the computer screen, a capture was made of two images in the double-shutter camera mode and the frame-grabber

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connecting the CCD camera to the PC computer. The image processing stage followed. The last step was the image processing using the Matlab program for the velocity and the calculation of velocity derivatives.

3. Flow simulation

Fig. 2. Schematic description of the simulation model.

A two-dimensional steady state flow model was built using Femlab software (Matlab group) that computes the velocity components u = (u, w) in the x and z directions, and pressure p for the fluid in the region, defined by the system geometry. The x direction is the same x direction of the PIV measurements, yet the z direction is the direction perpendicular to the membrane plane, x–y. The simulation system was used in order to add the third dimension that is not included in the experiments and to make some estimates for parameters associated with the flow around the spacers. Since the spacers exhibit three-dimensional structure, located in the flow, the two-dimensional approach can give some illumination on this type of flow without the large efforts needed for solving the three-dimensional real case. Fig. 2 presents the flow configuration chosen for the flow simulation. The circles represent the filaments of the spacer between the two lines that represent the membranes. The PDE (partial differential equations) model for this application uses the steady incompressible Navier–Stokes equations followed by the continuity

equation: −η∇ 2 u + ρ(u · ∇)u + ∇p = F ,

∇ ·u=0

(2)

where η is the dynamic viscosity, ρ the density and F represents the force field. As can be seen in Fig. 2, the boundary conditions chosen where non-slip conditions at the membrane boundaries and the model was solved with an initial parabolic flow profile entering between two parallel plates: ¯ u = 4us(1 − s)

(3)

where s is the non-dimensional distance between the plates. The laminar profile assumption is due to a low Re, which is defined as: Re =

¯ 2ul η

The length l is the distance between the two membranes.

Fig. 3. Two successive flow images of yeast particles in water in a unit cell.

Fig. 4. Velocity map: (a) superimposed on flow picture; (b) without flow picture.

(4)

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Fig. 5. Velocity maps in a unit cell of the spacer: (a) u velocity component; (b) v velocity component.

Fig. 6. Effect of interrogation areas: (a) 16 pixels; (b) 32 pixels; (c) comparing average velocities for the two interrogation areas.

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Fig. 7. Mixing index map and velocity field in a spacer unit cell.

The exit from the flow section was based on constant static pressure. Non-slip conditions for other boundaries were assumed as the boundary conditions. Since the model solves for continuous stable flow, the general expression for turbulence intensity or the defined MI cannot be evaluated from the results. Instead, the vorticity ω was chosen to represent the velocities gradients, or the tendency to form vortices: ω=

∂w ∂u + ∂x ∂z

(5)

The solution grid consisted of 666 nodes. The convergence criterion was chosen as 1 × 10−6 .

Fig. 8. Velocity profile between two membranes-comparison between two unit cells, similar velocity.

Fig. 9. Average velocity measured, u at the cell center, along y-axis, at two levels between the membranes.

4. Results and discussion 4.1. Flow velocity measurement Fig. 3 shows the raw results in the form of two successive pictures generated by the CCD camera. The two raw images were transferred to a Matlab processing program. The picture quality was good enough for the measurements. The results are shown in Fig. 4a with the velocity field superimposed on the particles in the flow picture, and in Fig. 4b where the velocity field without the particles is shown.

Fig. 10. The influence of flowrate, expressed as Reynolds number on the velocity profile along the y-axis.

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In observing Fig. 4, it is clear that the velocity is basically found in the expected flow direction, with a minor component in the direction perpendicular to the flow within the twodimensional flow area. Close to the spacer rods, the velocity is almost zero, as can be seen in Fig. 4b by the size of the velocity vectors, which are reduced to dots or disappear completely. The reason for this is the required change in velocity in order to bypass the obstacle—the spacer wire. The darkness near the filament does not impede detection of the velocity component. The third dimensional velocity component cannot be detected with the current setup of the optical system. The velocity above and below the filament must be higher in order

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to maintain the mass balance, yet only a low velocity in the horizontal plane was detected at the focal region level for the picture presented. Fig. 5 presents the velocity field detected by the optical system, represented as velocity components in the main (x) direction and the perpendicular (y) direction. The components in the y direction appear to be significantly smaller than the components in the x direction. This is a first sign of low-order mixing in the flow. The velocity field shows what is difficult to see from the velocity vectors—the local change in velocity led to the ability to calculate spatial velocity changes.

Fig. 11. Field velocity and mixing index as function of average velocity: (a) average velocity −0.06 m/s, Re = 180; (b) 0.17 m/s, Re = 420; (c) 0.3 m/s, Re = 900; (d) 0.68 m/s, Re = 2050; (e) 0.97 m/s, Re = 5000.

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Fig. 11. (Continued ).

Fig. 6 shows the precision of the calculations achieved using the technique. Velocity vectors represented in the previous figure are calculated by interrogation areas made of a matrix of 32 × 32 and 16 × 16 pixels. Too small interrogation areas loose accuracy; too large areas tend to average the velocity while filtering out the flow fluctuations. The comparison made in Fig. 6 shows no significant differences and hence represents good precision of the velocities calculated. Fig. 7 presents a MI map calculated with Eq. (1). The MI in the main area of the cell center is low, ranging from 0.0 to 0.2 of the average velocity, and increases to 1 close to the cell wires due to flow direction changes. This is not real mixing, rather a product of the calculation method used here. Fig. 8 presents the velocity profile between the two walls, representing the two membranes, in the center of the unit cell. This velocity profile may be obtained by moving the microscopic lens with the camera in the z direction. The length of the focal region is estimated at 20 ␮m based on the optics used. The figure contains two similar profiles at the same average velocity taken in two different measurement sessions. The parabolic shape profile shown represents a basically laminar flow. The difference between the two profiles shows the nature of the flow that takes place in a variable and flexible measurement cell due to the nature of the spacer grid.

Fig. 9 shows the velocity profile at two different heights above the lower membrane along the y-axis, with repetition for the same flow rate. The profile changes from being almost flat at the lower point to parabolic at the center between the two walls. The velocity at the two heights is in good agreement with the profile presented in the previous figure. The profile shows that the velocity reaches a maximum at the center of the flow cell and a minimum close to the intersection points of the wires. This represents a three-dimensional velocity profile within a unit cell. The variations between the repeated profiles represent the different behaviors in the different experiments. Fig. 10 shows a similar velocity profile, except that the parameter changed in the figure is the flow rate along the membrane expressed as the Re. As expected, the profiles are close to parabolic shape, being flatter for low Re and sharper for high flow rates. Disturbances at the center for high Re are due to increased flow fluctuations with increased flow rate. This may be better seen in Fig. 11. The figures represent the flow in the major part of the cell for different flow rates and the calculated MI for the different experiments. The MI increased significantly with flow rate, as expressed by the darker areas in the picture. Higher MI is always shown, even at low flow rates, close to the wires due to the velocity changes there. However,

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except for Fig. 11a where the flow rate is in the range of common RO operation, the increased MI due to flow rate also exerts an increase in pressure drop across the membrane. Too high-pressure drop is undesirable due to energy dissipation, therefore it is important to optimize pressure drop against flow rate with relatively high mixing or turbulence in order to improve operational conditions. The results shown in Fig. 11 can explain the literature findings reported above whereby mass transfer is increased with an increase in pressure drop. This may be explained by the increase in mixing intensity in the gap between the membranes.

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While Fig. 11 represents the flow inside a unit cell, it was interesting to observe what happens close to the spacer rods. Fig. 12 presents the results around and inside a unit-spacer cell. The figure is a superposition of four different pictures taken at the four quarters of the unit cell and combined together to form a complete picture. The reason for this was the need to maintain the accuracy gained within the limitation of the optical system—no single picture could be taken of the entire area. It is clear from the picture that the turbulent behavior after passing the tracer rods is similar to that inside the cell. Some changes may be seen in the

Fig. 12. Flow in and around a unit cell—superposition of four picture: (a) velocity map superimposed on cell pictures; (b) four mixing index maps.

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different quarters since the rods are not located on the same plane, therefore changes in the MI may be detected. In the lower left quarter, the rod is elevated in comparison to the other three quarters, which is revealed by the intensity of the velocity vectors shown on the rods. 4.2. Simulation results The simulation describing the flow in a cell is presented in Figs. 13 and 14. Re number for the calculations was 1000. The flow in the unit cell is three-dimensional, however, for the sake of simplicity, based on the experimental results it was decided to settle for only two-directional flow, as described above. Figs. 13 and 14 present the flow in different slices through the unit cell. The height in the figures represents the distance between the membranes, while the horizontal distance, x, represents the path along the flow direction. The

circles represent the spacer filaments located at different distances, as expected in the y direction. The simulation presents the steady-state conditions, since no attempt was made in this work to measure time variations. Fig. 13 shows the calculated velocity vectors as white arrows; the background, shown in different gray levels, represents the velocity in the figures on the left and the vorticity in the figures on the right. It may be seen that the velocity at the center of the slit shows a parabolic profile, as demonstrated experimentally. The bypass flow along the filaments is seen very well in the figures. The highest velocities are observed, as expected, above and below the filaments. The vorticity is seen to be very low throughout the flow area but has higher values above and below the tracer filaments. This agrees with the previously reported experimental findings. Fig. 14 presents a similar picture for different ratios of the distance between the membranes and the diameter of the

Fig. 13. Velocity and vorticity fields: (a) velocity, distance between cylinders centers is 2 mm (Re = 1000); (b) vorticity for (a); (c) velocity, distance between cylinders centers is 1.5 mm; (d) vorticity for (c); (e) velocity, distance between cylinders centers is 0.5 mm; (f) vorticity for (e).

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Fig. 14. Comparison between spacer cylinders thickness (Re = 1000): (a) 0.3 mm thickness, velocity map; (b) 0.3 mm thickness, vorticity map; (c) 0.6 mm thickness, velocity map; (d) 0.6 mm thickness, vorticity map.

spacer filaments. Smaller diameter filaments do not differ significantly but wider filaments show higher vorticity below and above the filaments, as should be expected due to the lower gaps between the membranes and the filaments. A comparison between the experimental results shown above and the simulation calculations are shown for two average velocities in the slit in Fig. 15. Simulation was performed when the inlet average velocity was 0.35 m/s (Re number around 1000) and 1 m/s (Re = 2860). The comparison made at the center of the cell along the x-axis presents satisfactory agreement. The spacer between membranes is expected to increase mixing or turbulence and hence to reduce fouling. Fig. 16 shows a fouled membrane with small particles of rust. The membrane shown is part of an industrial 8 in. membrane that deteriorated and was checked in our laboratory. The left picture shows the fouled membrane with the spacer and the right picture shows the membrane without the spacer. The rust accumulates on the membrane very close to the spacer filaments, giving an accurate replica of the spacer, as can be seen on the right picture. The entire area of the membrane was fouled in the same way, which tells us that fouling was conducted by very small particles that followed the flow throughout the membrane. All membranes in the module were fouled at the same way. It seems that some high fouling exists at the junction points where the velocity might be low, at the points the strands touch the membrane, however fouling exist also along the strands in both directions, where the velocity is higher. Some fouling could be made where the flow

close to the spacer is low, causing calm precipitation. No fouling, however, exists inside the cell on the membrane where the flow parallel to the membrane is zero and the supersaturation is increased by concentration polarization. Yet the iron oxide lines are describing the exact replica of spacer, on both sides,

Fig. 15. Comparing experimental velocity profile with the simulation of flow between two membranes and around the cylinders.

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Fig. 16. Pictures of used membrane: (a) with spacer; (b) without spacer.

even at places where high velocities exist around the spacer. It may be concluded that the spacer could promote fouling under some conditions, fouling that should be avoided with proper pretreatment. The explanation for the fouling at the high velocity regions is shown in the simulation, as expressed in Fig. 17. The flow must bypass the spacer rods, reducing the flow area, thus increasing the velocity there. As may be

seen in the picture, the white arrows represent the velocity vectors. The different gray levels represent the x component in the upper figure and the y component in the lower one. Clearly, the high velocity components exist next to the membrane walls. Some of the small rust particles cannot follow the flow exactly so they tend to precipitate at the membranes due to these components. Hence, the spacer enhances in dif-

Fig. 17. Flow simulation around two cylinders: (a) gray levels represent u; (b) gray levels represent v.

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ferent mechanisms particulate fouling that should be avoided with proper pretreatment.

5. Conclusions The flow in a unit cell of a spacer between two membranes was investigated experimentally using the PIV technique and theoretically using Femlab. Good agreement was found between the two techniques, which may explain phenomena associated with this type of flow. The PIV method was used to map the two-directional velocity at various flow parameters at different levels between the two walls. No attempt was made to take time-based measurements, however, the flow was found to be basically laminar with only small fluctuations. At low Re (up to 1200), the calculated MI was found to be very low at the center yet increased towards the spacer filaments. This increased MI is not real mixing; it was detected as a result of the change in direction generated by the obstacle in flow direction. Significant MI may be found elsewhere at significantly increased Re. The high range is generally not used in field operations due to the high shear rate associated with increased pressure drop and loss of energy. The theoretical two-dimensional steady-state analysis exhibits similar results in terms of laminar flow and vorticity, and the tendency to form vortices. Understanding flow behavior could lead to an understanding of flow phenomena, as demonstrated by a sample of membrane fouling. In some cases at least, the spacer can apparently increase fouling instead of reducing it. The technique developed here, after some improvements are incorporated, may be used to detect the real turbulence intensity in this type of flow and help in the design of improved spacers.

Acknowledgements The authors wish to thank the infrastructure project of the Ministry of Science and Technology for support of this work. Technion’s Grand Water Research Institute appreciates the partial support provided for this study and thanks to Malcolm and Lyn Chaikin for providing the infrastructure of the Membranes Laboratory. This work forms part of the M.Sc. of M.M.G.

Nomenclature F MI p Re s

force field (N) mixing intensities pressure (Pa) Reynolds number non-dimensional distance

t u u¯ U v v¯ x

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time between two images (s) velocity in x direction (m/s) average velocity in x direction (m/s) average velocity across the channel (m/s) velocity in y direction (m/s) average velocity in y direction (m/s) distance in x direction (pixel)

Greek letters µ viscosity (Pa s) ω vorticity (s−1 ) ρ density (kg/m3 ) η dynamic viscosity (m2 /s)

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