Numerical Studies of the Impact of Spacer Geometry on Concentration

Jul 13, 2005 - The results suggest concentration polarization in spiral wound RO modules could be significantly alleviated by spacer optimization. Vie...
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Ind. Eng. Chem. Res. 2005, 44, 7638-7645

Numerical Studies of the Impact of Spacer Geometry on Concentration Polarization in Spiral Wound Membrane Modules Lianfa Song* and Shengwei Ma Division of Environmental Science & Engineering, Faculty of Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

The impact of the geometry of spacer filaments on concentration polarization and permeate flux in spiral wound reverse osmosis modules was studied using the 2-D streamline upwind Petrov/Galerkin finite element model. In this study, both salt concentrations and flow velocities on membrane surface are determined as results of numerical solution of the coupled governing equations for momentum and mass transfers, which is more appropriate for most applications of the spiral wound membrane modules. The patterns of concentration polarization along the spacer-filled membrane channel were first investigated respectively for the membranes attached to and opposite to the transverse filaments. The patterns were then analyzed with the hydraulic characteristics (e.g., reattachment point and recirculation) of flow in the membrane channel for the implications to the system performance. Finally, the effect of filament shape and size of the spacer on concentration polarization and system performance was simulated and discussed. The results suggest concentration polarization in spiral wound RO modules could be significantly alleviated by spacer optimization. 1. Introduction Concentration polarization has been identified since the 1960s as an important phenomenon that deteriorates the performance of the reverse osmosis (RO) systems by reducing the net driving pressure.1-4 Concentration polarization is also closely related to membrane fouling because foulant concentration may be significantly altered in the concentration polarization layer. Spacers are essential parts of the spiral wound RO modules that keep membranes apart to form the filtration channel, and therefore, a better understanding of the impact of spacers on concentration polarization is of primary importance for a better design or improvement of the spiral wound membrane module. It has been reported in the literature that the spacers can significantly alter the hydrodynamic conditions and mass transfer patterns in a membrane channel.5-13 Tien and Gill14 demonstrated that the alternatively placed impermeable sections and membrane sections could reduce concentration polarization and therefore increase membrane productivity noticeably. Kang and Chang15 and Kim et al.16 showed that a recirculation flow induced by spacers enhanced mass transfer in eletrodialysis systems. Recently, Schwinge et al.17 studied mass transfer enhancement for different filament configurations with a commercial CFD software (CFX). However, their work was limited by the uses of constant wall concentrations and impermeable walls, which were unrealistic to most spiral wound RO modules. Geraldes et al.19,20 simulated concentration polarization in nanofiltration (NF) spiral wound modules with fixed permeate velocity. Apparently, one important aspect of concentration polarization, the interaction between solute transport (concentration polarization) and momentum transfer (permeate velocity), could not be addressed. As indicated by Brian,21 the constant flux assumption * To whom correspondence should be addressed. E-mail: [email protected].

would lead to significant errors in concentration polarization simulations especially when wall concentration profiles are concerned. The purpose of this study was to investigate the effect of spacer filaments on concentration polarization in spiral wound RO modules with a more realistic mathematical description of the problem. Using the fully coupled streamline upwind Petrov/Galerkin (SUPG) finite element model developed earlier,1 both salt concentration profile (including wall concentrations) and the hydrodynamics (including permeate flux) can be simultaneously simulated in a RO membrane channel filled with spacer filaments. The impacts of different filament geometry, size, and mesh length (interval between filaments) on concentration polarization in the channel were simulated and discussed. Results from this study would provide a more realistic picture of concentration polarization in spiral wound RO modules so that they could be used to optimize spacer design. 2. Numerical Model The flow field and salt concentration profile in a membrane channel were fully described by the coupled continuity, Navier-Stokes, and convective diffusion equations:

∂u ∂v + )0 ∂x ∂y

(1)

( (

) )

∂2u ∂2u ∂u ∂u 1 ∂p ∂u +u +v )+ν 2+ 2 ∂t ∂x ∂y F ∂x ∂x ∂y ∂v ∂2v ∂2v ∂v ∂v 1 ∂p +u +v )+ν 2+ 2 ∂t ∂x ∂y F ∂y ∂x ∂y

(2)

∂2c ∂2c ∂c ∂c ∂c +u +v )D 2+ 2 ∂t ∂x ∂y ∂x ∂y

(3)

10.1021/ie048795w CCC: $30.25 © 2005 American Chemical Society Published on Web 07/13/2005

(

)

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Figure 1. Illustration of the spacer configuration.

Figure 2. Illustration of the computing domain.

The inlet of the membrane channel is described by boundary conditions of Dirichlet type (prescribed velocity and salt concentration):

u(x ) 0, y, t) ) U0(y, t) v(x ) 0,y, t) ) V0(y, t) c(x ) 0,y, t) ) C0(y, t) For the impermeable solid wall and the spacer/filaments in the membrane channel, no-slip, no-penetration conditions are imposed:

u)0 v)0 D

∂c )0 ∂n

At the membrane surface, the salt concentration and velocity cannot be specified a priori but defined with the following coupled equations:

vw ) A[∆p - ∆π(cw,cp)]

(4)

∂c D ) vw(cw - cp) ∂n

(5)

The outlet of the channel is described by the open boundary condition. A pressure boundary is not required because of the use of the penalty formulation of the Navier-Stokes equations. The model was numerically solved with the 2-D SUPG finite element method specially developed earlier1 for the study of concentration polarization in spiral wound RO modules. With this method, the momentum transfer

and solute transport are simultaneously calculated in a more realistic description of the spiral wound RO modules, where the wall concentrations and permeate flux are natural results of the numerical solution rather than inappropriately imposed conditions to the problem. For example, the effect of membrane permeability on the concentration profile on the membrane surface and in the channel can be well captured with this method. 3. Simulation Conditions and Assumptions Numerical simulations were carried out on supercomputers and PC clusters at the Supercomputing & Visualization Unit (SVU) of the National University of Singapore. Although different shapes of spacers are currently used in commercial spiral would modules, their function in promoting mixing and alleviating concentration polarization can be well represented by three regular types (cavity, zigzag, and submerged spacers).17,18 The computing domain employed in his study contains the cavity spacers only (Figures 1 and 2). However, the analysis and methodology are not difficult to be generalized to zigzag and submerged spacers. The channel height (1 mm) and mesh length (4.5 mm) used in this study fall in the common range of most commercial spiral wound RO modules. The channel length employed for numerical simulation in this study was 10 cm. Constant viscosity (ν ) 1.0 × 10-6 m2/s) of the solution was assumed, and the diffusivity was determined as sodium chloride in diluted solution (D ) 1.5 × 10-9 m2/ s). Complete (100%) salt rejection and linear dependency of osmotic pressure on salt concentration (∆π ) kcw, k

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Figure 3. Salt concentration (c/c0) profiles in a feed channel with 0.5 mm (in diameter) cylinder transverse filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

Figure 4. Wall concentration (cw/c0) profiles in an empty channel and a feed channel with 0.5 mm × 0.5 mm square bar transverse filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

Figure 5. Permeate flux profiles in an empty channel and a feed channel with 0.5 mm × 0.5 mm square bar transverse filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

) 75 kPa m3/kg) were used in the simulations. Fully developed parabolic flow profile was assumed in the inlet. No spacers were placed in the first 5 mm and the last 5 mm so that the entrance effect and exit effect due to the spacer filaments could be minimized. A successive refining meshing scheme toward the membrane surface that leads to “mesh-independent” solutions was used in all numerical simulations.1 4. Results and Discussion 4.1. Concentration Polarization Patterns in Spacer-Filled Channels. Because of salt accumulation and concentration polarization, both permeate flux decreases and wall concentrations increase monotonically in the cross-flow direction in an empty channel

(without spacers). In contrast, as shown in Figure 3, the monotonic trend of salt concentration profiles in a membrane channel is disrupted by the filaments. Wall concentrations are higher in the regions adjacent to the filaments but lower in the regions away from the filaments. The phenomenon implies that concentration polarization would be alleviated in some regions but aggravated in other regions by the existence of filaments in the membrane channel. The longitudinal profile of wall concentrations and distribution of permeate flux in a channel with multi square (0.5 mm × 0.5 mm) bar filaments are plotted with those in an empty channel respectively in Figures 4 and 5. Figure 4 shows there are periodic vibrations (variations) in wall concentrations on both membranes. The wall concentration on the membrane opposite to the

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Figure 6. Contour of x-component flow velocity in a feed channel with 0.5 mm (in diameter) cylinder transverse filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

Figure 7. Velocity field near the reattachment point in a feed channel with 0.5 mm (in diameter) cylinder transverse filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

Figure 8. Comparison of local wall concentration (cw/c0) profiles (on the membrane attached to the transverse filaments) in feed channels with 0.5 mm (in diameter) cylindrical filaments and 0.393 mm × 0.5 mm (height) rectangular bar filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

transverse filaments shows smaller amplitude and is consistently lower than that in the empty membrane channel. The amplitude of wall concentration variation is much higher on the membrane attached to the transverse filaments, with the peak values higher and the valley values lower than those in the empty membrane channel. The patterns of wall concentration profiles are oppositely reflected in the distribution of the permeate fluxes, as shown in Figure 5. For the membrane opposite to the transverse filaments, the location of the minimum value of wall concentrations (or the maximum value of permeate flux) in each cycle is very close to the middle point of each transverse filaments. From Figure 4, it can be found the peak values of wall concentrations on the membrane attached to the transverse filaments occur at the joint points of the filaments to the membrane. This pattern of concentration polarization is caused by the periodic boundary layer disruption due to the recirculation flow formed in the regions between the filaments.1,17 Because tangential flow on the membrane surface is divided into upstream and downstream at a point (reattachment point) between two filaments, wall salt concentration and the boundary

layer thickness grow in two opposite directions from the reattachment point. The cavity effect of the filaments on the flow field and flow separation is better described by the contour graph in Figure 6. Some fluid deviates from the main stream over the two filaments and flows to membrane. The reattachment point occurs about 3 mm downstream of the upstream filaments in the graph. A recirculation region forms between the upstream filament and the reattachment point. The reversed flow in this recirculation region is mainly responsible for the special pattern of concentration polarization in the spacer-filled membrane channel as shown in Figure 4. The flow field around the reattachment point is enlarged in Figure 7. It can be found that the reattachment point is a division point where the flow breaks into upstream and downstream tangential flows along membrane surface. These tangential flows carry the retained salt toward filaments and form concentration peaks on both sides of the filaments (Figure 8). It should be pointed out, for the membrane attached to the transverse filaments, the minimum wall concentration does not concur exactly at the reattachment point but slightly upstream of it. As

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Figure 9. Velocity field near the small recirculation regions in a feed channel with 0.5 mm (in diameter) cylinder transverse filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

Figure 10. Comparison of local wall concentration (cw/c0) profiles (on the membrane opposite to the transverse filaments) in channels with 0.5 mm (in diameter) cylindrical filaments and 0.393 mm × 0.5 mm (height) rectangular bar filaments (simulation conditions: ∆p )800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

shown in Figure 8, the location of the minimum wall concentrations, point B, is actually on the upstream of the reattachment point, point A. A higher concentration on the reattachment point is caused by stagnant or extremely slow tangential flow at that point. Figure 8 also shows there is a small but noticeable secondary peak in the wall concentration profile near the downstream filaments (illustrated as point C in Figure 8). This is likely because of a small scale recirculation formed in this region as shown in Figure 9. The results from Schwinge et al.17,18 also showed there is a noticeable peak for wall shear stress in the similar location when lf ) 8hf, but it was not discussed in their papers. This small recirculation may disrupt the boundary layer and cause the boundary layer to grow in two directions in this narrow region. The recirculation actually enhances wall concentrations buildup in these recirculation regions as shown in Figure 9. This recirculation and the resulted secondary concentration peak are not desirable to improve system performance. It is also noted that such a secondary concentration peak does not occurs near the first filament. The secondary concentration peak does not appear if the neighboring filaments are sufficiently far so that they cannot “sense” each other or if the two filaments are close enough so that the recirculation region reaches the downstream filament to eliminate this boundary layer disruption. This phenomenon will be further discussed in section 4.3. 4.2. Impact of Filament Shape on Concentration Polarization. The effect of cylindrical and rectangular filaments on concentration polarization was simulated and compared in Figure 8, in which the wall concentrations of membranes attached to cylindrical filaments and rectangular bar filaments are presented with the broken line and solid line, respectively. The diameter

of the cylindrical filaments is 0.5 mm. The height and width (longitudinal direction) of the rectangular bar filaments are 0.5 and 0.393 mm so that the cross-sectional area is identical with that of the cylindrical filaments. The distance between two neighboring filaments is set as 4.5 mm. Simulations show that the overall oscillatory concentration profiles are similar for both filament types. Figure 8 shows the wall concentrations are generally higher for cylindrical filaments with a much higher peak value than those for rectangular bar filaments. It reflects that the rectangular bar filaments can create stronger recirculation flows than the cylindrical filaments. The higher wall concentration near the contact point of cylindrical filaments with membrane can be attributed to the larger stagnant cavity formed adjacent to the contact point. Simulation results show that the average permeate flux of a 10 cm membrane channel with cylindrical filaments is about 5.4% lower than that with rectangular bar filaments under otherwise identical conditions, which has about 8% smaller membrane surface than that with cylindrical filaments. Figure 10 compares wall concentrations on the membrane opposite to the transverse filaments. Similarly, the wall concentrations with rectangular bar filaments are slightly yet consistently lower than that with cylindrical filaments. This shows the square bar filaments are slightly more effective in reducing concentration polarization than the cylindrical filaments. The effect of the filament shape on permeate flux is relatively weaker on the membrane opposite to the transverse filaments, and the differences in permeate flux (averaged in 10 cm long channels) are less than 2% under the simulation conditions. It should be pointed out the slightly higher permeate flux with rectangular filaments is obtained at the cost of slightly higher pressure drop in the channel than

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Figure 11. Comparison of local wall concentration (cw/c0) profiles (on the membrane opposite to the transverse filaments) in channels with square bar filaments with different filament sizes (0.25, 0.5, and 0.75 mm in height and 0.5 mm in width) (simulation conditions: ∆p ) 225 psi; c0 ) 2000 mg/L; A ) 1.59 × 10-11 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

Figure 12. Comparison of local wall concentration (cw/c0) profiles (on the membrane attached to the transverse filaments) in channels with square bar filaments with different filament sizes (0.25, 0.5, and 0.75 mm in height and 0.5 mm in width) (simulation conditions: ∆p )800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

cylindrical filaments. In the 10 cm long channel with 20 cylindrical filaments, the pressure drop is about 450 Pa (cross-flow velocity is 0.1m/s), which is about 15% lower than that with the same number of rectangular bar filaments. 4.3. Impact of Filament Size on Concentration Polarization. To simulate the effect of filament size on concentration polarization, the height of rectangular bat filaments varied from 0.25 to 0.75 mm. The width of these filaments was set as 0.5 mm so that the longitudinal distribution of wall concentration can be compared on all locations. Figure 11 compares wall concentration on the membrane opposite to the transverse filaments. It can be seen that the concentration on the membrane surface is significantly affected by the size of filaments. Compared with the wall concentration in the empty channel, the 0.25 and 0.75 mm filaments reduce the averaged wall concentrations about 7% and 31%, respectively. Concentration polarization alleviation on the membrane opposite to the transverse filaments is mainly achieved by the periodical acceleration and deceleration of the cross-flow. A larger filament makes the passage with the opposite membrane smaller, and therefore, greater acceleration and deceleration of the fluids causes forced entering and leaving of this narrow region.

The impact of filament size on wall concentration on the membrane attached to transverse filaments is shown in Figure 12, which is more complex than that on the membrane opposite to the transverse filaments. The minimum wall concentration occurs close to the upstream filament for the 0.25 mm filament, which indicates the filaments unable to cause a strong recirculation flow between the filaments. On the contrary, the minimum wall concentration approaches downstream for the 0.75 mm filament, which means an extended strong recirculation region is induced by the filament. The effect of the 0.50 mm filament on concentration polarization is moderate, and the minimum wall concentration occurs in the middle of two neighboring filaments. As mentioned earlier, concentration boundary layer grows in two opposite directions, and therefore, there are two peaks in the wall concentration profile by the upstream and downstream filaments. As can be seen from Figure 12, the peak values are strongly affected by the filament sizes. This is because the recirculation flow pattern varies with filament sizes. The length for the concentration boundary layer to develop in the reversed flow region of 0.25 mm filament is shorter than that of the large filaments, which leads to lower peak wall concentrations near the upstream filaments. The

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Figure 13. Contour of x-component flow velocity in a feed channel with 0.75 mm (in diameter) cylinder transverse filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 4.5 mm).

Figure 14. Local wall concentration (cw/c0) profiles (on the membrane attached to the transverse filaments) in feed channels with 0.5 mm square bar filaments (simulation conditions: ∆p ) 800 psi; c0 ) 32 000 mg/L; A ) 7.3 × 10-12 m/s Pa; u0 ) 0.1 m/s; h ) 1 mm; lf ) 2.5 mm).

0.75 mm filaments create recirculation in the whole region between the two neighboring filaments and result in the highest peak values near the upstream filament, the lowest peak values near the downstream filaments, and the lowest average salt concentration in the whole interval. The peak values in wall concentration near the downstream filament do not show significant difference for the 0.25 and 0.5 mm filaments. This is probably related to the small secondary peak in wall concentration profile of the 0.5 mm filament. The contour of flow filed for the case of 0.75 mm filaments is shown in Figure 13. It can be seen that the recirculation regions actually reach the downstream filament, which eliminates the possibility of the small recirculation in the front of the downstream filaments. To verify the concept, simulations are carried out for the 0.5 mm filaments with a reduced mesh length of 2.5 mm, and the results are shown in Figure 14. It is found the secondary peaks truly disappear and the peak concentration near the downstream filaments is much lower than that near the upstream filaments. It is interesting to note that the small recirculation does not exist for the first filament. This observation indicates the occurrence of the small recirculation may be suppressed by the fully developed bulk flow. This is also very likely the reason for the cases with 0.25 mm filaments. The disturbance on the flow by the 0.25 mm is so small that it can restore to the fully developed state before it encounters the next filament. Generally, if the mesh length is sufficiently long so that the downstream filament cannot “sense” the upstream filament or if the filaments are close enough so that the recirculation region reaches the downstream filament, the secondary peak would not appear. Because very long or short mesh length will either reduce the ability for concentration polarization alleviation or increase pressure loss, the proper mesh length has to be determined on the basis of trade off between flux improvement and energy

pressure loss (system optimization). Further studies are needed on the importance of the secondary concentration peak on the overall performance of the RO modules. It should be pointed out that large filaments also cause higher pressure drops than smaller ones. Under the simulation conditions, the pressure drop in a 10 cm long channel with 20 rectangular bar filaments of 0.5 mm (width) × 0.75 mm (height) is about 3200 Pa, which is about 6 and 16 times that in the same channel with same number of square bar filaments of 0.5 mm × 0.5 mm and 0.5 mm × 0.25 mm, respectively. Greater restriction of the channel causes more drastic change of the flow velocity and greater energy dissipation. This finding suggests that pressure drop along the membrane channel may become a concern when large filaments are used to alleviate concentration polarization in spiral wound RO modules. 5. Conclusions Concentration polarization is viewed as one of the most important factors that affect fouling and performance of membrane systems. However, most of the previous studies on concentration polarization in the spacer-filled membrane channel were conducted with either prespecified salt concentrations or flow velocities on membrane surfaces or both. In fact, the salt concentration and flow velocities on the membrane surfaces in most applications are not a prior knowledge. On the contrary, the determination of these variables, which are major indicators of the system performance, is one of the motivations to study concentration polarization. In this study, both salt concentration profiles and flow velocities on the membranes are sought as results of interaction between momentum and mass transfers in the membrane channel, a more realistic reflection of the real applications.

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It was found that the recursive pattern of the spacers causes periodical variations on both wall concentration and permeate flux in the membrane channel. The variations on the membrane attached to the transverse filaments are more drastic than those on the membrane opposite to the transverse filaments. Although there are peak wall concentrations around the filaments, the overall effect of the spacers is to alleviate concentration polarization and to increase permeate flux. Filament shape has limited impact on concentration polarization for the membrane opposite to transverse filaments. For the membrane attached to the filaments, the cylindrical filaments usually result in noticeably lower permeate flux compared to square bar filaments with identical voids under the same simulation conditions. Filament size (height) has significant impact on concentration polarization and permeate flux. The larger filaments can alleviate concentration polarization and increase permeate flux more effectively even though peak wall concentration near the filaments is higher. However, the improvement of system performance by the larger filaments is obtained at the cost of increased pressure drop along the membrane channel. The effect of the spacer on alleviating concentration polarization can be deteriorated by the occurrence of a small secondary recirculation in the front of filaments. A proper choice of the interval between two neighboring filaments of given height can eliminate the secondary recirculation. The results suggest the size and interval of filaments might be the two important parameters to optimize spacer design for the membrane channel configuration considered in this study. List of Symbols A ) membrane permeability (m/s Pa) c ) salt concentration (kg/m3) D ) diffusivity (m2/s) h ) channel height (m) K ) osmotic pressure constant (Pa m3/kg) L ) channel length (m) lf ) mesh length p ) pressure (Pa) ∆p ) applied pressure (Pa) Pe ) Peclet number ()hu0/D, H rather than hydraulic diameter is used in this paper) Re ) Reynolds number ()hu0/ν, H rather than hydraulic diameter is used in this paper) t ) time (s) u ) axial velocity (x direction) (m/s) v ) lateral velocity (y direction) (m/s) x ) axial coordinate (cross-flow direction) (m) y ) lateral coordinate (channel height direction) (m) Greek Letters ν ) viscosity (m2/s) F ) density (kg/m3) π ) osmotic pressure (Pa)

Literature Cited (1) Ma, S.; Song, L.; Ong S. L.; Ng, W. J. A 2-D streamline upwind Petrov/Galerkin finite element model for concentration polarization in spiral wound reverse osmosis modules. J. Membr. Sci. 2004, 244, 129. (2) Matthiasson, E.; Sivik, E. Concentration polarization and fouling. Desalination 1980, 35, 59. (3) Belfort, G. Synthetic Membrane Processes: Fundamentals and Water Applications; Academic Press: Orlando, FL, 1984. (4) 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. (5) Winograd, Y.; Solan, A.; Toren, M. Mass transfer in narrow channels in the presence of turbulence promoters. Desalination 1973, 13, 171. (6) Chiolle, A.; Gianotti, G.; Gramondo, M.; Parrini, G. Mathematical model of reverse osmosis in parallel wall channels with turbulence promoting nets. Desalination 1978, 26, 3. (7) Miyoshi, H.; Fukuumoto, T.; Kataoka, T. A consideration of flow distribution in an ion exchange compartment with spacer. Desalination 1982, 42, 47. (8) Focke, W. W.; Nuijens, P. G. J. M. Velocity profile caused by a high porosity spacer between parallel plates (membrane). Desalination 1984, 49, 243. (9) Schock, G.; Miquel, A. Mass transfer and pressure loss in spiral wound modules. Desalination 1987, 64, 339. (10) Da Costa, A. R.; Fane, A. G. Net-type spacers: effects of configuration on fluid flow path and ultrafiltration flux. Ind. Eng. Chem. Res. 1994, 33, 1845. (11) Van Gauwbergen, D.; Baeyens, J. Macroscopic fluid flow conditions in spiral-wound membrane elements. Desalination 1997, 110, 287. (12) Cao, Z.; Wiley: D. E.; Fane, A. G. CFD simulations of nettype turbulence promoters in a narrow channel. J. Membr. Sci. 2001, 185, 157. (13) Karode, S. K.; Kumar, A. Flow visualization through spacer filled channels by computational fluid dynamics I. pressure drop and shear rate calculations for flat sheet geometry. J. Membr. Sci. 2001, 193, 69. (14) Tian, C.; Gill, W. N. The relaxation of concentration polarization in a reverse osmosis desalination system. AIChE J. 1966, 12, 722. (15) Kang, I. S.; Chang, H. N. The effects of turbulence promoters on mass transfer-numerical analysis and flow visualization. Int. J. Heat Mass Transfer 1982, 25, 1167. (16) Kim, D. H.; Kim, I. H.; Chang, H. N. Experimental study of mass transfer around a turbulence promoter by the limiting current method. Int. J. Heat Mass Transfer 1983, 26, 1007. (17) Schwinge, J.; Wiley: D. E.; Fletcher, D. F. Simulation of the Flow around Spacer Filaments between Narrow Channel Walls. 1. Hydrodynamics. Ind. Eng. Chem. Res. 2002, 41, 2977. (18) Schwinge, J.; Wiley: D. E.; Fletcher, D. F. Simulation of the Flow around Spacer Filaments between Narrow Channel Walls. 2. Mass Transfer Enhancement. Ind. Eng. Chem. Res. 2002, 41, 4879. (19) Geraldes, V.; Semiao, V.; Pinho, M. N. de. The effect of the ladder-type spacers configurations in NF spiral wound modules on concentration boundary layers disruption. Desalination 2002, 146, 187. (20) Geraldes V.; Semiao, V.; Pinho, M. N. de. Concentration polarization and flow structure within nanofiltration spiral-wound modules with ladder-type spacers. Comput. Struct. 2004, 82, 1561. (21) Brian, P. L. T. Concentration polarization in reverse osmosis desalination with variable flux and incomplete salt rejection. Ind. Eng. Chem. Fundam. 1965, 4, 439.

Subscripts 0 ) at x ) 0 p ) permeate w ) wall or membrane surface f ) filament

Received for review December 13, 2004 Revised manuscript received June 9, 2005 Accepted June 15, 2005 IE048795W