Numerical Modeling of Concentration Polarization in Spacer-filled

Article pubs.acs.org/IECR

Numerical Modeling of Concentration Polarization in Spacer-filled Channel with Permeation across Reverse Osmosis Membrane Toru Ishigami and Hideto Matsuyama* Center for Membrane and Film Technology, Department of Chemical Science and Engineering, Kobe University, 1-1 Rokkodai, Nada, Kobe 657-8501, Japan ABSTRACT: This paper presents a numerical simulation method for reasonably describing concentration polarization in a spacer-filled channel of a spiral-wound reverse osmosis membrane module. The permeation across the membrane was modeled theoretically, based on nonequilibrium thermodynamics. We then simulated flow and mass transfer in a periodic unit model of the spacer-filled channel, with different Reynolds numbers, spacer separations, and angles between spacers, using the proposed method as a boundary condition for the reverse osmosis membrane. The results show that the concentration polarization and water flux distribution on the membrane surface can be reasonably well expressed by the simulation. The present numerical method is effective for modeling concentration polarization, and provides better descriptions of the flow and mass transfer characteristics in the spacer-filled channel than previous simulation methods.

1. INTRODUCTION Seawater desalination using reverse osmosis (RO) membranes has attracted attention as a key technology in solving global potable water shortages. The most common commercially available RO membrane modules are spiral-wound membrane (SWM) modules. SWM modules include spacer nets to keep the membrane layers apart, and these also enhance mass transfer in the feedwater channel, thereby suppressing the decrease in water permeability that occurs through concentration polarization on the membrane surface. However, closely separated mesh spacers within the channel cause a high pressure drop, therefore excess pumping power is needed to supply the seawater feed. Viable methods for reducing energy use and the cost of seawater desalination must therefore be developed, by reducing the hydrodynamic drag in the channels of RO elements. However, it is well-known that the relationship between mass transfer enhancement and pressure drop is usually a trade-off,1 therefore it is necessary to understand the flow and mass transfer characteristics in the channel and design appropriate spacer-filled channels to improve membrane system performances while minimizing energy losses. Computational fluid dynamics (CFD) techniques are effective for studying the flow field and mass transfer inside complicated and narrow channels. A number of two-dimensional CFD studies of spacer-filled channels have been carried out.2−6 Many researchers have used three-dimensional CFD to investigate the effects of the spacer arrangement, such as the spacer spacing,8,9 incident angle of the spacers against the mean flow direction,9−11 and mesh spacer geometry,10,12,13 on the flow and mass transfer characteristics in the feed channel.14 Recently, several research groups performed similar simulations for forward osmosis (FO)15−17 and pressure-retarded osmosis (PRO) systems.17 However, the results did not accurately describe the concentration polarization in the channel, because they ignored water and salt permeations through the membrane, and adopted an impermeable wall boundary condition for the membrane surface during simulation. To increase the applicability of CFD results to cases with high © 2015 American Chemical Society

permeation rates, a permeable wall boundary condition should be incorporated. Lau, and Ahmad et al. performed two- and three-dimensional simulations that considered water and solute permeations of the membrane.18,19 However, they assumed that the concentration boundary layer thickness was uniform on the membrane surface in calculating the permeate flux. The local concentration boundary layer thickness is directly affected by the complicated fluid field around the spacers and must not be uniform. To our knowledge, and according to the most recent review of numerical studies of unit models of SWM modules,14 appropriate boundary conditions on the membrane surface have not yet been implemented, because of the associated computational complexity. In this study, we constructed a numerical simulation technique for calculating the flow and solute concentration field in a unit model of a spacer-filled channel with membrane permeation, by incorporating a permeable wall boundary condition. As a demonstration of this method, we then used our new technique to investigate the effects of the seawater crossflow velocity, spacer separation, and angle between spacers on the pressure drop, concentration polarization, and permeation flux.

2. NUMERICAL METHODS 2.1. Governing Equations. The flow and solute concentration field in a unit model of the spacer-filled channel were simulated to obtain the flow and mass transfer characteristics within the feedwater (retentate) channel. The governing equations are the three-dimensional continuity, momentum, and solute transport equations: ∇·(ρ u) = Sv Received: Revised: Accepted: Published: 1665

(1)

October 8, 2014 January 13, 2015 January 20, 2015 January 20, 2015 DOI: 10.1021/ie5039665 Ind. Eng. Chem. Res. 2015, 54, 1665−1674

Article

Industrial & Engineering Chemistry Research ∇·(ρ uu) = −∇p + ∇·(μ∇u)

(2)

∇·(ρ um) = ∇·(ρD∇m) + Ss

(3)

Jv mf − D∇mf = Jv mp

where D is the mass diffusivity. It should be noted that this equation was established under an assumption that the seawater density is constant. The present numerical model developed in this study considered the local seawater density as explained in Section 2.4. Thus, these models mismatched. However, the difference may hardly affected the mass transfer characteristics obtained by the theoretical model, because the change of the seawater density calculated by eq 21 is quite small. The following equation is obtained by integrating eq 9 with respect to mf across the boundary layer in the feed channel:

where u denotes the fluid velocity vector, p the pressure, ρ the fluid density, μ the fluid viscosity, and m the solute mass fraction. Although previous studies used not mass fraction but concentration, we use mass fraction because Fluent that we used as a CFD solver adopts not concentration but mass fraction. These equations are based on a steady and laminar flow field. The source terms Sv and Ss in the continuity and solute transport equations represent the fluid and solute mass changes, respectively, due to permeation, and are given in cells adjacent to the membrane surface. These source terms are explained in Section 2.2. 2.2. Permeation Model. In this study, to represent the permeate across the RO membrane, the fluid and solute mass changes adjacent to the membrane surface with permeation are given, and the flow and mass transfer inside the RO membrane are solved not numerically but theoretically, using the following equations (eqs 4 and 5). Based on nonequilibrium thermodynamics, the water flux, Jv, and solute permeation flux, Js, are given as follows20,21 Jv = Lp{pf − pp − (Δπf − Δπp)} Js = P(mf − mp) − (1 − σ )msJv

m w − mp mb − m p

k=

Ssi = −

ki = (5)

(11)

J

ln

v m w − mp mb − m p

(12)

n

km =

∑i k iA i

(13) W ·L Here, W and L are the lengths of the membrane surface over which the mass characteristics in the x- and y-directions, respectively, are analyzed, as described in Section 2.4. From the mean mass transfer coefficient, a correlation between the Sherwood number, Sh, and the Reynolds number in the feed channel, Re, can be obtained as follows

Js ρs A i (7)

ρ − ρw (1 − m) m

(10)

By substituting mw, mb, and mp calculated from the simulation, and Jv calculated from eq 4, into eq 12, the local mass transfer coefficient can be obtained. Lau et al.18,19 assumed a solute concentration boundary layer thickness corresponding to the distance between the membrane surface and the center of the adjacent cell. However, this assumption is not reasonable, because the boundary thickness has no relation to the numerical grid arrangement. The mean mass transfer coefficient is calculated as follows

where Vi is the cell volume, and Ai is the face area adjacent to the RO membrane. ρs is the pseudo solute density under the condition that salt is dissolved in the seawater and obtained from the weight-average of the seawater and water densities. ρs =

D

D δ

(4)

(6)

Vi

Jv δ

From the above equations, the local mass transfer coefficient can be expressed as

Jv ρ0 A i Vi

= exp

where δ is the boundary layer thickness and the subscript b indicates the bulk variable. The mass transfer coefficient k is given by

where Lp is the water permeability coefficient, P is the solute permeability coefficient, Δπ is the osmotic pressure difference, σ is the solute reflection coefficient, and ms is the average solute mass fraction. The subscripts f and p denote the feed and permeate sides, respectively. To obtain the local water and solute permeation fluxes using eqs 4 and 5, the local solute mass fraction on the membrane surface is needed. A numerically calculated solute mass fraction in the cell adjacent to the membrane surface is adopted. In this method, we describe the fluid and solute mass changes resulting from permeation using the source terms in the continuity [eq 1] and mass transport [eq 3] equations. The source terms are given by Jv and Js as follows Svi = −

(9)

Sh = b·Re a·Sc 0.25

(14)

Sh =

k mdh D

(15)

Re =

d h vρ μ

(16)

Sc =

μ ρD

(8)

Introducing these source terms into the governing equations allows the permeation across the RO membrane to be taken into consideration. 2.3. Flow and Mass Transfer Characteristics. Mass transfer characteristics of the spacer-filled channel were analyzed by a theoretical model by Kimura and Sourirajan,22 which has been used in a number of previous studies on RO membrane.23,24 The following equation is derived from the solute mass balance on the RO membrane surface:

(17) 1,25

where dh is the hydraulic diameter (dh = 2H) and v is the effective fluid velocity. The definition of the present Reynolds number, called the hydraulic Reynolds number,11 is that given in previous papers.1,25 The exponent of the Schmidt number, Sc, in eq 14 was set at 0.25, based on many previous studies,1,26−28 although researchers have used various exponents.9,29 In the near future, we will investigate the Schmidt 1666

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Figure 1. Computational domain and boundary conditions for spacer-filled channel.

not occur at the surface overlapped by the spacers. Varying the spacer spacing and the angle between spacers therefore changes the effective area of the membrane surface. For instance, on increasing the angle between the spacers from 0.5π to 0.667π, the effective area of the membrane surface decreases, because the spacer length across the domain increases. In a real SWM module, the spacers are touched to the membrane surface by winding membrane leaves, therefore this assumption, which was used in previous papers, is reasonable.11,36 Cartesian coordinates were defined from the origin set at the bottom-left corner of the inlet boundary. The mesh size was small enough to be independent of the numerical results. Mesh refinement was performed using grid adaption utility of Fluent to change the number of grid element in the computational domain so that results could be checked for grid independency. The scaled residuals difference of the streamwise velocity with grid variation were less than 1 × 10−8. For the boundary conditions, the inlet velocity was uniform and constant. The top and bottom faces parallel to the flow direction were assumed to be the RO membrane surface and an impermeable wall, respectively, for comparisons of the concentration polarization. A periodic boundary condition was used on the left and right faces. The computational conditions are tabulated in Table 1.

number dependence on the mass transfer characteristics, using the present method to accurately determine the exponent of the Schmidt number. Spacers may lead not only to mass transfer enhancement, but also to a high pressure drop. When discussing the power consumption for pumping seawater feed into a spacer-filled channel, Li et al. defined the specific power consumption (SPC) as follows30 SPC =

Δp v ΔL

(18)

where Δp/ΔL is the pressure gradient in the stream-wise direction. The dimensionless power number, Pn, is obtained by normalizing the SPC. Pn = SPC

ρ2 H μ3

(19)

2.4. Numerical Methods and Computational Conditions. The governing equations for the unit model of the spacer-filled channel were numerically solved using the commercial CFD software package Fluent 13.0.0, which uses the finite volume method with an unstructured grid system. The quadratic upstream interpolation for convective kinetics (QUICK)31 was applied for convective terms. For the velocity−pressure coupling, the semi-implicit method for pressure-linked equations (SIMPLE) algorithm32 was used. The calculation was continued until reaching a steady state. Figure 1 shows the computational domains for the simulation of the periodic unit model for the spacer-filled channel. The computed geometry is a zigzag spacer arrangement.33 This geometry was chosen because it has the most similarities with spacers used in current SWM modules, and was also found to perform better than the other geometries reported in a previous paper.34 This computational domain consists of a series of three units with periodic spacer geometries and is similar to those in many previous studies.9,11,25,30 Although the previous studies have used one periodic cell, the periodic boundary condition in streamwise direction can not be applied in the present simulation model because of the changes of the flow rate and the salt mass fraction with permeation. The geometrical characteristics of spacers include the ratio of the distance between spacers (W/d) and the angle between the spacers (θ). The computational results for the central unit were used to analyze the flow and mass transfer characteristics of the spacerfilled channel, because the other units are affected by the inlet and outlet boundaries. The spacer diameter, d, was 0.315 mm.35 It should be noted that the spacers overlap with the membrane surfaces, as shown in Figure 1, to avoid inferior mesh geometries around them. Permeation and mass transfer do

Table 1. Computational Conditions for Simulation of Spacer-filled Channel transmembrane pressure at outlet inlet solute mass fraction in feed channel solute mass fraction in permeate channel water permeability coefficient sodium reflection coefficient sodium permeability coefficient

5.0 MPa 0.035 0 1.08 × 10−11 m·Pa−1·s−1 1.00 6.9 × 10−8 m·s−1

To simplify this system, the fluid is assumed to be seawater, which is only a binary mixture of salt and water. The water permeability, solute reflection, and solute permeability coefficients were taken from a previous report.37 The user-defined function (UDF) was incorporated to model the permeate across the RO membrane in the spacer-filled channels. UDF is an optional function of FLUENT, programmed by the user, which can be dynamically linked with the solver. For the seawater physical properties, Miyake’s formula38 was used to calculate the osmotic pressure, and the density, viscosity, and mass diffusivity of the seawater were based on a previous report,39 as follows Osmotic pressure: c Δπ = (0.6955 + 0.00252T ) × 108 ρ (20) 1667

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Industrial & Engineering Chemistry Research Density: ρ = 498.4n +

2.484 × 105n2 + 752.4nc

n = 1.0069 − 2.7507 × 10−4T

(21)

Viscosity: ⎛ 1.965 × 103 ⎞ μ = 1.234 × 10−6exp⎜2.12 × 10−3c + ⎟ 273.15 + T ⎠ ⎝ (22)

Mass diffusivity: ⎛ 2.513 × 103 ⎞ D = 6.725 × 10−6exp⎜0.1546 × 10−3c − ⎟ 273.15 + T ⎠ ⎝

Figure 2. Pressure contours for different Reynolds numbers at x = 0.25W, W/d = 4.76, and θ = 0.417π: (a) Re = 4.63; (b) Re = 23.4; (c) Re = 35.3; (d) Re = 47.0. The range of the contour legend is adjusted in each case.

(23)

where c is the salt concentration and calculated as follows c = mρ (24) The procedures for installation of permeation across the RO membrane in the simulation are as follows (1) the flow and solute mass fraction fields are updated by numerical computation; (2) the local water and solute mass fluxes are calculated by substituting the solute mass fraction and pressure on a cell adjacent to the membrane surface obtained from step 1 into eqs 4 and 5; (3) the source terms of eqs 1 and 3 are calculated using eqs 6 and 7, respectively. These steps were used for every iteration of the simulation.

3. RESULTS AND DISCUSSION As a demonstration, we used the proposed method to investigate the effects of the Reynolds number, spacer spacing, and angle between the spacers on the flow and mass transfer characteristics in the channel. 3.1. Effects of Reynolds Number. The effects of the Reynolds number on the fluid and mass transfer characteristics in the channel, and on the water permeability, were investigated. W/d was set at 4.76.35 Re was changed in four steps, i.e., 4.63, 23.4, 35.3, and 47.0, by varying the inlet uniform velocity. It should be noted that the Reynolds number is much smaller than those in previous studies.8−11 This Reynolds number is set under the assumption that the permeate water and recovery rate are 2.3 m3·d−1 and 50% of each membrane leaf, respectively, which are the usual values in current RO systems using SWM modules. Figure 2 shows the pressure contours in the cross section at x = 0.25W. The pressure sharply decreases around the spacers in all cases. This is because there is significant shear stress, because of the higher fluid velocity caused by the small cross-sectional area due to the presence of spacers. In addition, the absolute value of the pressure drop increases with increasing Re. In contrast, the distribution profiles are similar. These results show that the shear stress around the spacer strongly contributes to the pressure drop under the present conditions. Figures 3 and 4 show the solute mass fraction distributions and velocity vectors, respectively, for different Re values in the cross section at x = 0.25W. From Figure 3, it can be seen that the solute mass fraction becomes high on the membrane surface in all cases, indicating that concentration polarization is successfully described by the simulation. In addition, a higher mass fraction region is observed near the spacers adjacent to the membrane surface, because the fluid cannot fully penetrate into the regions where spacers are attached to the membrane

Figure 3. Contours of solute mass fraction on membrane surface for different Reynolds numbers at x = 0.25W, W/d = 4.76, and θ = 0.417π: (a) Re = 4.63; (b) Re = 23.4; (c) Re = 35.3; (d) Re = 47.0.

Figure 4. Velocity vectors for different Reynolds numbers at x = 0.25W, W/d = 4.76, and θ = 0.417π: (a) Re = 4.63; (b) Re = 23.4; (c) Re = 35.3; (d) Re = 47.0. The range of the contour legend is adjusted in each case.

surface and the fluid velocity decreases, as shown in Figure 4. Consequently, the mass transfer becomes low in this region. However, the concentration boundary thickness in the central region between spacers is reduced, and the solute mass fraction on the membrane surface becomes lower in all cases. This is because the fluid flows with high velocity toward the membrane surface, as a result of spacers adjacent to the impermeable wall. The overall concentration boundary layer thickness is reduced, and the surface mass fraction decreases with increasing Re, indicating that mass transfer is enhanced. Previous studies11,36 1668

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Figure 5. Contours of mass transfer coefficient on membrane surface for different Reynolds numbers, W/d = 4.76, and θ = 0.417π: (a) Re = 4.63; (b) Re = 23.4; (c) Re = 35.3; (d) Re = 47.0. The range of the contour legend is adjusted in each case.

reported that the inertial flow around spacers also affected the flow and mass transfer characteristics. However, the fluid hardly separates from the spacer surfaces as seen in Figure 4, because the Reynolds number of this study is much smaller than those of the previous studies. The inertial effects are not so significant in this study. Figure 5 shows the mass transfer coefficient distribution on the membrane surface for different Re values. The mass transfer coefficients are high near the inlet in all cases. However, it should be noted that this is simply because a uniform velocity was specified for the inlet boundary condition. The mass transfer coefficient becomes low near the spacers, adjacent to the membrane surface, and is high above the spacers, adjacent to the impermeable wall. This mass fraction distribution is affected by the nonuniform fluid velocity profile caused by the installation of spacers. Furthermore, the mass transfer coefficient decreases slightly in the stream-wise direction, because the flow rate decreases in that direction because of permeation. The mass transfer coefficient decreases overall with decreasing Re. In addition, the mass transfer coefficient significantly decreases in the stream-wise direction. Again, this is because of the decrease in flow rate in the stream-wise direction as permeation becomes large with decreasing Reynolds number. Figure 6 shows the relationship between Sh and Re. As can be seen, Sh increases with increasing Re. This behavior is consistent with the calculation results for the mass fraction and mass transfer coefficient distributions discussed above. Several researchers have reported the correlation between the Sherwood number and the Reynolds number. In FimbresWeihs and Wiley’s paper,11 when the Reynolds number is 50, the Sherwood number is in the range 20−30 for zigzag spacers with different arrangements. In a correlation reported by Koutsou et al., the exponent of the Reynolds number is in the range 0.57−0.87 for different Schmidt numbers and spacer arrangements.9 These values differ from the results of the present simulation, but are comparable. These studies and the present study differ in many aspects such as the permeation

Figure 6. Relationship between Sherwood and Reynolds numbers, at W/d = 4.76 and θ = 0.417π.

modeling and the region of the Reynolds number analyzed. A unified numerical framework is expected to be necessary. Figure 7 shows the contours of the water flux distributions on the membrane surface. It was found that the water flux was lower near the spacers, adjacent to the membrane surface, and higher above the spacers, adjacent to the impermeable wall. In addition, the water flux decreased in the stream-wise direction because the solute mass fraction increases as a result of permeation. This distribution profile is similar to that for the mass transfer coefficient, shown in Figure 5. The water flux becomes high in the region where the mass transfer is enhanced and the solute mass fraction is low, because the water flux is significantly affected by the osmotic pressure, as described by eq 4. Based on these investigations, the local water flux is reasonably correlated with the local mass transfer. 1669

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Figure 7. Contours of water flux for different Reynolds numbers, W/d = 4.76, and θ = 0.417π: (a) Re = 4.63; (b) Re = 23.4; (c) Re = 35.3; (d) Re = 47.0. The range of the contour legends is adjusted in each case.

Figure 8. Contours of mass transfer coefficient on the membrane surface for different W/d, Re = 47.0, and θ = 0.417π: (a) W/d = 4.76; (b) W/d = 9.52; (c) W/d = 14.3; (d) W/d = 19.0. The range of the contour legends is adjusted in each case.

mass fraction field on the membrane surface is strongly affected by the mass transfer characteristics. The mass transfer coefficient is high at locations above the spacers, adjacent to the impermeable wall, and low near the spacers, adjacent to the membrane surface, in all cases. In addition, the overall mass transfer coefficient increases with increasing W/d. In the case of W/d = 4.76, the distribution profile is significantly different from those in other cases, and the high mass transfer coefficient and low solute mass fraction areas are wider than those in other cases. The fluid velocity distribution in the channel was investigated. Figure 10 shows the cross-sectional velocity vectors at x = 0.25W. A high-velocity region can be observed

3.2. Effects of Spacer Spacing. The effects of the spacer spacing on the flow and mass transfer characteristics, and on the filtration performance, were investigated. The spacer spacing was changed by varying W/d in four steps: 4.76, 9.52, 14.3, and 19.0. Re and θ were set at 47.0 and 0.417π, respectively. Figures 8 and 9 show the mass transfer coefficient and solute mass fraction distributions, respectively, on the membrane surface, for different W/d values. The mass transfer coefficient and mass fraction are high, because a uniform velocity is specified for the inlet boundary condition. From these figures, it can be seen that the distribution profiles of the mass transfer coefficient and the mass fraction are similar, showing that the 1670

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Figure 9. Contours of solute mass fraction on membrane surface for different W/d, Re = 47.0, and θ = 0.417π: (a) W/d = 4.76; (b) W/d = 9.52; (c) W/d = 14.3; (d) W/d = 19.0.

above the spacers, adjacent to the impermeable wall, as a result of the decreased cross-sectional area. Furthermore, the main flow separates from the surfaces of the spacers adjacent to the membrane surface and attaches posteriorly to the membrane surface. Consequently, the fluid velocity is low behind the spacers, and a high velocity toward the membrane surface is observed. The positions of high velocity correspond to those of high mass transfer, as shown in Figure 8. However, the area occupied by the high velocity region decreases with decreasing W/d. The high mass transfer area therefore decreases, and the overall mass transfer coefficient decreases, with increasing W/d. Figure 11 shows the water flux contours on the membrane surface. The water flux distribution profiles are quite similar to

Figure 10. Velocity vectors for different W/d at x = 0.25W, Re = 47.0, and θ = 0.417π: (a) W/d = 4.76; (b) W/d = 9.52; (c) W/d = 14.3; (d) W/d = 19.0.

Figure 11. Contours of water flux for different W/d at x = 0.25W, Re = 47.0, and θ = 0.417π: (a) W/d = 4.76; (b) W/d = 9.52; (c) W/d = 14.3; (d) W/d = 19.0. The range of the contour legend is adjusted in each case. 1671

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Industrial & Engineering Chemistry Research those for the mass transfer coefficient and the solute mass fraction, shown in Figure 8, because the osmotic pressure increases in regions where the solute mass fraction is high because of low mass transfer. These investigations show that to predict the local water flux, it is necessary to estimate the local mass transfer coefficient on the membrane surface accurately. 3. 3 Effects of Angle between Spacers. The effects of the angle between the spacers (θ) on the mass transfer characteristics and the filtration performance were investigated. The angle was changed in three steps: 0.417π, 0.5π, and 0.590π. Re and W/d were set at 47.0 and 4.76, respectively. Figure 12 shows the contours of the solute mass fraction on the membrane surface for different θ values. In all cases, the

Figure 13. Contours of water flux on membrane surface for different θ, Re = 47.0, and W/d = 4.76: (a) θ = 0.417π; (b) θ = 0.5π; (c) θ = 0.590π.

3.4. Summary of Spacer Performance. Finally, to investigate the trade-off between the mass transfer and pressure drop, the performances of spacers with different spacings and angles were compared, based on the Sherwood and power numbers, as shown in Figure 14. The spacer performances are better when the plots are located at the left and upper regions in these figures, representing a lower pressure drop and higher mass transfer. As shown in Figure 14a, Sh increases with increasing Pn. Pn is the normalized value of the pressure drop in the channel, which is correlated with Re. The increase in Sh is therefore caused by the increase in Re, as shown in Figure 6. The plots shift to the left and upper regions with decreasing W/d, indicating that the spacer performance improves slightly. The bulk fluid has a higher velocity sweep on the membrane surface, because of the presence of spacers. This effect leads directly to mass transfer enhancement. As shown in Figure 8, the area ratio for a high mass transfer coefficient decreases with increasing W/d. However, a decrease in W/d reduces the effective area of the membrane surface, because the area where the spacers are attached to the membrane surface per unit area increases. Consequently, the enhancement of the spacer performance is lower than expected. The effect of the angle between spacers is shown in Figure 14b. The plots shift to the upper left region in the figure with decreasing angle, explaining the better spacer performance. As shown in Figures 12 and 13, the mass transfer characteristics do not differ significantly. However, decreasing the angle reduces the effective area of the membrane surface, therefore the spacer performance is better in the case of a low angle. Koutsou et al. also reported that the Sherwood number when the angle between the spacers was 90° was higher than

Figure 12. Contours of solute mass fraction on membrane surface for different θ, Re = 47.0, and W/d = 4.76: (a) θ = 0.417π; (b) θ = 0.5π; (c) θ = 0.590π.

solute mass fraction on the membrane surface is higher near the spacer and lower above the spacer. This is because the mass transfer characteristics are affected by the fluid flow around the spacers, as described above. In addition, the distribution profiles do not differ markedly, indicating that varying the angles between the spacers does not significantly affect the mass transfer characteristics. The contours of the permeation fluxes are shown in Figure 13. In all cases, the permeation fluxes are high and low at locations where the solute mass fractions are low and high, respectively. Furthermore, the distribution profiles are quite similar. Lau et al. investigated the effect of the angle on the concentration polarization.19 The concentration factors in the cases θ = 0.5π and 0.667π did not differ markedly, which is the same trend as for the present results. Our results show that the angle between spacers does not significantly affect the mass transfer characteristics and the permeation performance. 1672

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Figure 14. Relationship between Sherwood number and Power number: effects of (a) spacer spacing and (b) angle between spacers.

that for 120°.9 This trend is consistent with the present results. It is important to note that the spacer performance is affected not only by the flow and mass transfer in the channel but also by the effective area of the membrane surface under the present conditions. We must therefore accurately estimate the area where the spacers are attached to the membrane surface.

appreciation also goes to the Mega-ton Water System project led by Dr. Kurihara, Toray Industries, Inc.

■

4. CONCLUSIONS In this study, we constructed a numerical simulation method for reasonably describing concentration polarization in the spacer-filled channel of an SWM module. The permeation rate across the RO membrane was theoretically modeled. As a demonstration of the present simulation model, we then carried out numerical simulations of the fluid flow and mass transfer in a spacer-filled channel, using the model as a boundary condition for the RO membrane. To our knowledge, this is the first time that a reasonable permeable wall boundary condition has been introduced into a three-dimensional numerical simulation. The effects of the spacer spacing and the angle between spacers on the mass transfer characteristics in the channel and the water flux were investigated. The present model is useful for understanding the local solute concentration, water flux, and mass transfer coefficient distributions on the membrane surface. Although the simulation results have not been validated, we intend to compare them with experimental results in the future. It is expected that the present method can be applied not only to RO systems, but also to FO and PRO systems. We believe that the present simulation method can be a powerful tool for optimizing the spacer arrangement in spacer-filled channels and for predicting the product flow rate from SWM modules in RO, FO, and PRO processes.

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AUTHOR INFORMATION

Corresponding Author

NOMENCLATURE A = face area adjacent to the reverse osmosis membrane, m2 c = solute concentration, kg·m−3 d = spacer diameter, m dh = hydraulic diameter, m D = mass diffusivity, m2·s−1 H = channel height, m J = permeation flux, m·s−1 k = mass transfer coefficient, m·s−1 km = mean mass transfer coefficient, m·s−1 L = channel length in y-direction, m Lp = water permeability coefficient, m·Pa−1·s−1 m = solute mass fraction p = pressure, Pa P = solute permeability coefficient, m·s−1 Pn = power number Re = Reynolds number Sv = source term of continuity equation, kg·m−3·s−1 Ss = source term of transport equation, s−1 Sc = Schmidt number Sh = Sherwood number SPC = specific power consumption, Pa·s−1 t = time, s T = temperature, °C u = fluid velocity vector, m·s−1 v = mean fluid velocity, m·s−1 V = cell volume, m3 W = channel length in x-direction, m x = x-coordinate, m y = y-coordinate, m z = z-coordinate, m

Greek Letters

*H. Matsuyama. E-mail: [email protected]. Tel.: +81-78803-6180. Fax: +81-78-803-6180.

δ = boundary layer thickness, m Δp/ΔL = pressure gradient in stream-wise direction, Pa·m−1 Δπ = osmotic pressure difference, Pa μ = fluid viscosity, Pa·s ρ = fluid density, kg·m−3 σ = solute reflection coefficient

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was funded by the Japan Society for the Promotion of Science (JSPS) through the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program) ”, initiated by the Council for Science and Technology Policy (CSTP). The author’s deepest

Subscripts

f = feed side p = permeate side s = solute v = fluid 1673

DOI: 10.1021/ie5039665 Ind. Eng. Chem. Res. 2015, 54, 1665−1674

Article

Industrial & Engineering Chemistry Research

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i = cell index b = bulk w = membrane surface

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REFERENCES

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DOI: 10.1021/ie5039665 Ind. Eng. Chem. Res. 2015, 54, 1665−1674