Modeling, Simulation, and Experimental Validation for Aqueous

Jan 13, 2007 - Nibong Tebal, Penang, Malaysia. Detailed investigation on the influence of hydrodynamics, pressure, and transport properties toward a...
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Ind. Eng. Chem. Res. 2007, 46, 1316-1325

Modeling, Simulation, and Experimental Validation for Aqueous Solutions Flowing in Nanofiltration Membrane Channel A. L. Ahmad* and K. K. Lau School of Chemical Engineering, Engineering Campus, UniVersiti Sains Malaysia, Seri Ampangan, 14300 Nibong Tebal, Penang, Malaysia

Detailed investigation on the influence of hydrodynamics, pressure, and transport properties toward a concentration polarization phenomenon in a membrane channel is crucial for membrane module design purposes. Hence, a computational fluid dynamics (CFD) simulation and modeling approach has been conducted to reveal and anticipate the membrane transport properties and concentration polarization phenomenon. The current CFD model has been validated by experimental data, which exhibited excellent accordance between simulated and experimental values. The current study has also demonstrated that physical transport properties and boundary conditions should be modeled as variables in the solution of governing equations, since these parameters potentially affect the predicted values. 1. Introduction Comprehensive understanding on hydrodynamics and the concentration polarization phenomenon in the spacer-filled membrane channel is crucial to ensure that the concentration polarization phenomenon can be minimized at minimum energy consumption during desalination processes.1,2 To achieve this goal, studies for the computational fluid dynamics (CFD) simulation and modeling to predict hydrodynamics in the spacerfilled channel have been conducted intensively, parallel with the development of special numerical CFD codes and CFD simulation software. Li et al.3,4 conducted a 3-dimensional laminar CFD simulation for the spiral-wound membrane feed channel to evaluate mass transfer coefficients and power consumption for commercial net spacers. They investigated the relation between the Sherwood and power numbers using 3-D CFD transient models for low Reynolds flows in a nonwoven spacer-filled channel. The study revealed an optimum ratio for channel height over spacer length to yield a maximum mass transfer coefficient with moderate power consumption. A consequent study was also conducted experimentally to validate the optimum ratio.5 Modifications in the boundary condition in a CFD simulation can offer significant savings in computational time and costs. Hence, periodic boundary conditions in the streamwise direction were employed as an alternative to portray the CFD computational domain for a spacer-filled channel with a single spacer.6 Recently, Takaba and Nakao7 also carried out a CFD study on concentration polarization in the H2/CO separation membrane using commercial simulators. On the basis of the study, the importance of considering nonideal fluid dynamics in the design of a membrane module has been emphasized. Meanwhile, Geraldes et al.8 developed a special numerical code for 2-dimensional domain to yield the fluid flow profile and membrane permeation data. They applied this technique to reveal the hydrodynamics and the membrane transport mechanism in the empty membrane channel.9,10 A similar technique was also utilized by de Pinho et al.11 to determine and compare the intrinsic rejection coefficients obtained from the experiment and simulations. The modified governing equation in stream * To whom correspondence should be addressed. Tel.: +60 (4) 5941012. Fax: +60 (4) 5941013. E-mail: [email protected].

function and vorticity was also utilized to link the membrane transport equation for the computation of fluid flow and membrane transport profile in the empty membrane channel.12 Besides, the membrane channel filled with ladder-type spacer was discretized in 2-D and laminar conditions to predict the concentration polarization by the employment of this CFD mathematical modeling method.13-15 CFD simulations for fluid flow through rectangular channels filled with several commercially available spacers for membrane modules were carried out by Karode and Kumar16 and Dendukuri et al.17 Excellent agreement was found between the experimentally determined dependence of the total drag coefficient on the Reynolds number and the CFD simulations in this work. The pressure drop through the channel was found to be governed by a loss of fluid momentum caused by an almost abrupt change in the direction of the velocity vectors across a thin transition plane corresponding to the plane of intersection of the spacer filaments. Meanwhile, Cao et al.18 simulated the velocity profile and turbulent kinetic energy distributions in the spacer-filled channel. The study proved that the existence of the spacer is plausible to improve the local shear stress on the membrane surface and to produce eddy activities for mass transfer enhancement and fouling reduction. Besides, Schwinge et al.19-21 employed commercial CFD simulation codes (CFX) to simulate the hydrodynamics and mass transfer phenomenon in the spacer-filled channel. They conducted studies on flow patterns and mass transfer enhancement in narrow spacer-filled channels with CFD. The flow patterns were examined for a single filament adjacent to the wall and centered in the channel and for three different spacer configurations, the cavity, zigzag, and submerged spacers, with variations in mesh length, filament diameter, and Reynolds number.19 Large recirculation regions were formed behind the filaments, and the flow around the filament increased the shear stress on the wall. The CFD model was also used to study the effects of Reynolds number, mesh length, and filament diameter on mass transfer enhancement for three spacer configurations, a cavity, a zigzag, and a submerged spacer.20 Efforts were also carried out to conduct the simulation of unsteady flow for the spacerfilled channel (Schwinge et al.21,22). The flow was computed for different filament configurations for channel Reynolds numbers up to 2000. For the filament diameter, channel height, and spacer mesh length examined, the transition to unsteady

10.1021/ie061065z CCC: $37.00 © 2007 American Chemical Society Published on Web 01/13/2007

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Figure 1. Cross-sectional drawing for membrane permeation test cell.

Figure 2. Schematic diagram for experimental setup.

Table 1. Experimental Permeation Test under Different Pressure (∆P) and Feed Reynolds Numbers (Re) operating pressure, ∆P (105 Pa)

feed Reynolds number, Re

Low Feed Reynolds Number Study 3 280 5 300 7 320 9 340 11 360

operating pressure, ∆P (105 Pa)

feed Reynolds number, Re

High Feed Reynolds Number Study 3 900 5 1150 7 1400 9 1650 11 1900

flow for multiple filament spacers occurs above Rech ) 400 for cavity, submerged, and zigzag spacers. CFD simulation and modeling for membrane systems that consider the permeation flux are found to be limited. Pellerin et al.23 carried out a numerical hydrodynamic simulation of the flow field inside a membrane module where the membrane surface is treated as a porous wall. The diffusion-convection equation was incorporated in a straightforward manner to model a membrane separation showing the solute concentration field under realistic prevailing hydrodynamic conditions. Meanwhile, Wiley and Fletcher24,25 conducted a fundamental study involving a membrane model that can be incorporated in CFD simulation codes. The authors studied the effect of fluid properties on the numerical estimation of flux through the membrane. They integrated the wall concentration value with the adjacent fluid hydrodynamics to suit the particular CFD simulation with the

commercial finite control volume simulator CFX 4. Besides, Ma et al.26 conducted a quantitative study on the effects of the spacer upon concentration polarization. A 2-D streamline upwind Petrov/Galerkin (SUPG) finite element model was developed by numerically solving the coupled convectiondiffusion equation and Navier-Stokes equations in the feed channel. This model was verified with published experimental data of permeate flux in an empty channel. Consequent studies using the SUPG finite element method have been carried out to investigate the impact of the geometry of spacer filaments on concentration polarization and permeate flux in spiral-wound reverse osmosis modules.27 The main objective for the current study is to employ commercial CFD code Fluent 6 for modeling and simulating the flow conditions and permeation properties in the empty narrow nanofiltration membrane channel. The Spiegler-Kedem model has been used to predict the permeate flux in the membrane channel. Several researchers have agreed that it is a challenge to integrate the permeation properties in the commercial CFD simulation codes.28 Previous work was conducted by integrating the permeation properties in the membrane wall concentration calculation with the absence of permeate flux as the boundary condition.29 Under the current study, all transport properties and permeation properties (including permeate flux) have been modeled as variable and incorporated in the solutions of governing equations. Simulated data are compared with experimental data to determine the validity of the CFD

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Ind. Eng. Chem. Res., Vol. 46, No. 4, 2007 Table 2. Transport Properties for Different Types of Solutes properties

values a

osmotic pressure, π (105 Pa) density, F (kg/m3) viscosity, µ (10-3 Pa‚s) diffusion coefficient, DAB (10-9 m2/s) osmotic pressure, π (105 Pa) density, F (kg/m3) viscosity, µ (10-3 Pa‚s) diffusion coefficient, DAB (10-9 m2/s)

Figure 3. Part of the computational domain.

osmotic pressure, π (105 Pa) density, F (kg/m3) viscosity, µ (10-3 Pa‚s) diffusion coefficient, DAB (10-9 m2/s) a

Curve-fitting values.32

b

MgSO4 4.97mA(49.55 - 136.7mA + 1207mA2 ) 997 + 1036mA 0.8972(1 + 3.713mA + 26.0614mA2) 0.849(1 - 0.41873mAw0.0872) (for mA < 0.06) Na2SO4b 337.8mA0.95 997.1 + 909mA 0.89(1 + 3.52mA) 1.23(1 - 0.76mA0.4 ) Sucroseb 72.18mA(1 + 0.94mA + 2.93mA2) 997.1 + 416.7mA 0.89(1 + 1.31mA + 16.83mA2) 0.52(1 - 1.33mA) Reference values.9

Figure 4. Definition for δc used in modified film theory.29

simulation/modeling approaches. Solution transport properties, the concentration polarization phenomenon, and membrane intrinsic properties are evaluated under the current study. Figure 5. Calculation loop for the predicted values.

2. Experimental Method 2.1. Materials. Commercial flat-sheet NF membranes NF90 (membrane samples for FilmTec Corp.) were used in the current study. Membrane samples were soaked in deionized water overnight before they were used. MgSO4 solution (0.5% w/w), Na2SO4 solution (0.2% w/w), and sucrose solution (0.2% w/w) were prepared from the anhydrous MgSO4 powder (SigmaAldrich), anhydrous Na2SO4 powder (Merck), and sucrose powder (Merck). Deionized water (resistivity < 18.2 (M Ohm)/ cm and conductivity < 0.1 µS/cm) were used in the pure water permeability test and preparation of aqueous solutions. 2.2. Permeation Test Cell and Experimental Setup. Figure 1 demonstrates the cross-sectional drawing for the NF membrane permeation test cell used in the current study. This permeation test cell consisted of two detached parts. The total active permeation area of the test cell was 0.025 m (w) × 0.255 m (l). The height of the narrow membrane channel (slit between two detached parts) was 1 mm. There were five four-permeate collectors that were located below the porous support. The NF membrane was placed on a nonwoven polyester (Ahlstrom Inc.) paper, which was supported by the stainless steel porous material

as shown in the Figure 1. This cell was installed in the experimental setup as shown in Figure 2. 2.3. Experimental Procedures. Before conducting the permeation test, the soaked NF membrane was compressed under 15 bar of pressure for 8 h. The pure-water permeability test was conducted with deionized water for pressures of 3 × 105, 5 × 105, 7 × 105, 9 × 105, and 11 × 105 Pa, respectively, under feed Reynolds number (Re) 300 at a temperature of 25 ( 0.5 °C. The pure water permeability for the used NF membranes was determined and has an average value of 4.72 × 10-11 m/(Pa‚s). The permeation tests with the different aqueous solutions (refer to Section 2.1) were performed for pressures of 3 × 105, 5 × 105, 7 × 105, 9 × 105, and 11 × 105 Pa, respectively, under two different ranges of feed Reynolds numbers, as listed in Table 1, at temperature 25 ( 0.5 °C. Samples were collected in terms of permeation flux for every interval of 2 h. The collected samples were analyzed by conductivity meter (MgSO4 and Na2SO4) and spectrophotometer (sucrose) for the determination of permeate concentration.

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Figure 6. Comparison between simulated and experimental data for MgSO4 solution (0.5% w/w) (a) under low feed Reynolds number (Re ) 280360) and (b) under high feed Reynolds number (Re ) 900-1900).

Figure 7. Comparison between simulated and experimental data for sucrose solution (0.2% w/w) (a) under low feed Reynolds number (Re ) 280360) and (b) under high feed Reynolds number (Re ) 900-1900).

3. CFD Methodology 3.1. Simulation Method. The hydrodynamics of the fluid flow condition can be discretized by the solution of the governing equations. The governing eqs (1-4) were used to solve the incompressible steady-state laminar flow for 2-dimensional membrane channel domain with the consideration of solute concentration.

Continuity equation: ∂(Fu) ∂(FV) + )0 ∂x ∂y Navier-Stokes equation in x-direction:

(1)

∂(Fuu) ∂(FVu) ∂u ∂ ∂u ∂V ∂P ∂ µ + µ + )+2 + ∂x ∂y ∂x ∂x ∂x ∂y ∂y ∂x 2 ∂ ∂u ∂V + µ (2) 3 ∂x ∂x ∂y Navier-Stokes equation in y-direction:

( )

[( [(

)] )]

∂(FuV) ∂(FVV) ∂P ∂ ∂V + )+2 µ + ∂x ∂y ∂y ∂y ∂y ∂u ∂V ∂u ∂V ∂ 2 ∂ + + µ µ ∂x ∂y ∂x 3 ∂y ∂x ∂y Solute conservation equation:

( ) )] [ (

[(

(

) (

)] (3)

)

∂mA ∂mA ∂(FumA) ∂(FVmA) ∂ ∂ + ) FDAB + FDAB ∂x ∂y ∂x ∂x ∂y ∂y (4) Fluent v6 was used as the CFD simulation package in this work to simulate the flow condition in this narrow membrane channel. Discretization of the governing equations was carried out based on control volume technique. The discrete velocities

Figure 8. Comparison between simulated and experimental data for Na2SO4 solution (0.2% w/w) (a) under low feed Reynolds number (Re ) 280360) and (b) under high feed Reynolds number (Re ) 900-1900).

and pressures were stored by a nonstaggered system that consists of cells and faces. These values were stored in the cells’ center.

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Table 3. Schmidt Number and Osmotic Pressure for Different Solutions at Feed Concentration solution

Schmidt number (Sc ) µ/FDAB)

osmotic pressure, π (105 Pa)

MgSO4 Na2SO4 sucrose

1460 779 1782

1.215 0.922 0.145

Face values for velocity and concentration would be evaluated through interpolation using the second-order upwind scheme. The velocity and pressure parameters would be linked and solved by SIMPLE algorithm and accelerated by algebraic multigrid solver (AMG). Because of the laminar conditions in the membrane channel, steady-state simulation was used. High convergence criterion was set for velocity and concentration (0.000 01%) to offer sufficient iteration for complete convergence between the boundary grid and the interior mesh grid. The simulation domain (drawn by Gambit 2.1) was identical with the active experimental permeation area, which was 255 mm in length and 1 mm in height. In order to eliminate the effect of mesh quality and size on the results, initial simulations were carried out with four mesh resolutions (l × h), 250 × 45, 500 × 65, 750 × 85 , and 1000 × 105. It was found that, when the resolution was increased above 500 × 65, it had little effect on the solution in terms of velocity and concentration. Therefore, the computational grid was constructed with 500 × 65 cells. Approximately 70% of the generated nodes grids were allocated at the near-membrane region because of the existence of a high concentration profile adjacent to the membrane interface. The computational domain was shown in Figure 3. 3.2. Modified Boundary Condition and Transport Properties. The boundary conditions that describe the current simulated computational domain are represented by eqs 5-8. A plug flow velocity inlet was applied in the entrance of the membrane channel. The top part of the channel was treated as an impermeable wall. Fluid flowing out of the channel was assumed to exist in a fully developed condition where all changes for the flow parameters were equal to zero. The no-slip condition (where u ) 0) was fixed for the wall and membrane interface.

For x ) 0 and 0 < y < h, u ) uf; V ) 0; mA ) mAf

(5)

For x ) l and 0 < y < h, ∂mA ∂u ∂V ) 0; ) 0; )0 ∂x ∂x ∂x

(6)

For y ) 0 and 0 < x < l, u ) 0; V ) -Jv ) -Lp{∆P - σ[π (mAw) - π(R′mAw)]} mA ) mAw ) exp

( )

Jvδc 1 ‚ ‚m DAB Jvδc Ac R + R′ exp DAB

( )

(7)

Table 4. Curve-Fitted Spiegler-Kedem Parameters at Feed Concentration

For y ) h and 0 < x < l, u ) 0; V ) 0;

∂mA )0 ∂y

Figure 9. Curve-fitting of Spiegler-Kedem parameters for various types of solutions: (a) curve-fitting for σ and Ps with true rejection, R, and permeate flux, Jv (MgSO4 ) 0.5% w/w); (b) curve-fitting for σ and Ps with true rejection, R, and permeate flux, Jv (sucrose ) 0.2% w/w); and (c) curve-fitting for σ and Ps with true rejection, R, and permeate flux, Jv (Na2SO4 ) 0.2% w/w).

(8)

A user-defined function (UDF) was incorporated in the membrane boundary condition to model the concentration profile for the membrane interface. Film theory (in the form of a partial differential equation) was applied as the membrane interface boundary condition. The partial differential equation was

solution

σ

Ps

Rregression2

MgSO4 Na2SO4 sucrose

0.9627 0.9945 0.9994

4.3359E-7 1.7526E-8 1.0881E-7

0.9445 0.9781 0.9919

changed into analytical form (refer to eq 7) because of the limitation of the simulator in handling a partial differential equation as a boundary condition. Since mAw was one of the boundary conditions for the simulation and this boundary was affected by hydrodynamics and transmembrane pressure, to

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concentration at temperature 25 ( 0.5 °C. Transport physical properties for these aqueous solutions were listed in Table 2.

Spiegler-Kedem Model Jv ) Lp{∆P - σ[π(mAw) - π(mAp)]} Ns ) ω∆π + (1 - σ)JvC h Given mAp ) mAw(1 - R) and R′ ) 1 - R,

Jv ) Lp{∆P - σ[π(mAw) - π(R′mAw)]}

(9)

the reflection coefficient, σ, can be determined by curve-fitting of eq 10. Refer to the Spiegler-Kedem model30 for the derivation of eq 10.

(1 - σ) (1 - σ)Jv 1 - σ exp Ps

(

R)1-

)

(10)

Modified Film Theory

( )

∂mA + JvmA ) JvmAp ∂y

DAB

(11)

Integrating eq 11 under these limits, y ) 0 w mA ) mAw, y ) δc w mA ) mAc, yields

ln

(

)

Jvδc mAw - mAp ) mAc - mAp DAB

(12)

( )

Jvδc RmAw ) exp mAc - R′mAw DAB mAw ) exp

Figure 10. Membrane wall Schmidt number, Scw, for various solutions: (a) membrane wall Schmidt number, Scw, for MgSO4 (0.5% w/w); (b) membrane wall Schmidt number, Scw, for sucrose (0.2% w/w); and (c) membrane wall Schmidt number, Scw, for Na2SO4 (0.2% w/w).

relate the mAw with these properties, a relationship between mAw and the wall adjacent cell mass fraction, mAc, was constructed (refer to eq 7). To construct this relationship, modified film theory was built from the permeation flux predicted by the phenomenological equation (Spiegler-Kedem model30). To ensure this modified film theory was valid to be used, the distance between the wall and the wall adjacent cell, δc, must be very small (in this simulation, δc ) 2.5 × 10-7 m). This was vital to ensure that this point fell inside the laminar region, which is the prerequisite in the film theory. Refer to Figure 4 for the definition of δc used in the modified film theory.31 Physical transport properties for various aqueous solutions were modeled as variables that depend on the aqueous solution

( )

Jvδc 1 ‚ ‚m DAB Jvδc Ac R + R′ exp DAB

( )

(7)

where Jv ) Lp{∆P - σ[π(mAw) - π(R′mAw)]} 3.3. Numerical Model Validation. Current CFD models were used to predict permeation flux (Jv), reflection coefficient (σ), membrane wall concentration (mAw) (or true rejection), and membrane wall transport properties (viscosity and diffusion coefficient). To ensure the data predicted by the model were valid, permeate flux would be recalculated by employing the CFD simulation. The recalculated permeate flux was compared with the experimental values to confirm the validity of the computed values. With the experimental permeate concentration (mAp) and water permeability (LP), the computation of the permeate flux (Jv) started with the guess of true rejection [Rguess ) ((mAw - mAp)/mAw)] and reflection coefficient (σ). By solving the governing equations with all the boundary conditions (including the modified boundary condition), permeate flux (Jv) and membrane wall concentration (mAw) could be computed. The reflection coefficient (σ) and the true rejection [R ) ((mAw - mAp)/mAw)] could be revised by substituting the guessed values with these calculated values. This calculation loop would be carried out until the error between Rguess & Rcomputed and σguess & σcurve-fitted were 0.94. 4.3. Membrane Wall Physical Transport Properties. To give a better description for the membrane wall transport properties, Figure 10 demonstrates the evolution of Schmidt Number (Scw ) µ/FDAB) under different pressures for various types of solutions. For instance, Schmidt number is one of the indicators for concentration polarization development on the membrane wall.10 On the basis of Figure 10, the membrane wall Schmidt Number, Scw, increases along the membrane channel and demonstrates a higher magnitude under higher transmembrane pressure. This phenomenon signifies that hydrodynamics and pressure tend to affect the membrane wall Schmidt number, which has a crucial role in influencing the concentration polarization phenomenon in the membrane channel. On the basis of the results, the changes of Scw (along the membrane channel) are highest for sucrose followed by MgSO4 and Na2SO4. Therefore, sucrose solution is expected to exhibit a higher

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Figure 14. Evolution of predicted permeate flux along the membrane channel for MgSO4 under low feed Reynolds number.

Figure 13. Evolution of concentration polarization factor, Γ, along the membrane channel for Na2SO4 (0.2%) (a) under low feed Reynolds number and (b) under high feed Reynolds number.

Figure 15. Evolution of predicted permeate flux along the membrane channel for sucrose under low feed Reynolds number.

concentration polarization effect followed by those for MgSO4 and Na2SO4 solutions (refer to Section 4.4). On the basis of this study, the changes of transport properties in the membrane channel, especially on the membrane wall, are significant and cannot be neglected. Under constant temperature, these properties should be modeled as variables to concentration, which is directly affected by pressure and hydrodynamics in the membrane channel. 4.4. Concentration Polarization Factor. Figures 11-13 depict the evolution of concentration polarization factor (Γ ) (CAw - CAf)/CAf) for various types of solutions along the membrane channel. On the basis of the results, the increment of Γ for sucrose solution is found to be the highest followed by those for MgSO4 and Na2SO4. This occurrence is attributed to its highest Schmidt feed number and high rejection, as shown in Table 3 and Figure 9. The increment of Γ along the membrane channel is due to reduction of wall shear stress, as explained in previous work.29 Under higher feed Reynolds number, the flow in the membrane channel tends to generate higher wall shear stresses on the membrane wall. This phenomenon tends to produce a higher scouring effect on the membrane wall, which subsequently reduces the membrane wall concentration.30 Comparing among different types of solutions, sucrose tends to yield the highest degree of concentration factor reduction under higher Reynolds number followed by Na2SO4 and MgSO4 (Figures 11-13). This occurrence signifies that Γ for an aqueous solution

that exhibits higher Schmidt number and better rejection tends to be influenced by hydrodynamics. On the basis of the results, higher transmembrane pressure produces higher Γ. The increment in applied pressure tends to generate a higher permeate flux and reject more solute from the membrane. This phenomenon will increase the membrane wall concentration and Schmidt number (Scw), which consequently promote the formation of a higher concentration polarization factor. 4.5. Constant and Varying Permeate Flux. Figures 1416 depict the evolution of permeate flux predicted using CFD simulation under different operating conditions for various types of solutions. The average values of these predicted data have been compared with the experimental data in Section 4.1 (Figures 6-8). On the basis of the figures, permeate fluxes are found to be reduced along the membrane channel. The reduction slope of permeate flux along the membrane channel was found to be higher at higher pressure. Permeate flux was employed as one of the boundary conditions in the CFD solution of governing equations (refer to Section 3.2). This value will change according to pressure and hydrodynamics. Hence, modeling of Jv as a constant is inappropriate, since this boundary condition value will affect the CFD calculations. Figure 17a demonstrates the comparison between the membrane wall concentrations predicted by assuming constant and variation of permeation flux. If Jv is modeled as a constant, the Jv at the entrance region tends to be underpredicted (refer to Figure 17b). On the basis of eq 7, a

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5. Conclusion

Figure 16. Evolution of predicted permeate flux along the membrane channel for Na2SO4 under low feed Reynolds number.

The CFD simulation and modeling approach has been conducted to reveal and anticipate the membrane transport properties and concentration polarization phenomenon. This model has been validated by experimental data, which exhibited excellent accordance between simulated and experimental values. On the basis of the current study, several findings can be concluded as follow: • The current CFD model can precisely anticipate the permeate flux/membrane wall concentration and the transport properties in the membrane channel. • The validation of the membrane transport model is recommended to be conducted under different ranges of feed Reynolds number (high and low ranges) using an aqueous solution with high Schmidt number and osmotic pressure, since, under this condition, the permeate flux is more “sensitive” to the variation of hydrodynamics. • All physical transport properties and boundary conditions (i.e., permeate flux) should be modeled as variables in the solution of governing equations since these parameters potentially affect the predicted value (i.e., membrane wall concentration). Acknowledgment This work was funded by Grant IRPA EA by Ministry of Science, Technology and Innovation, Malaysia. The authors would like to show appreciation to NSF MOSTI for providing financial support to K.K. Lau. The authors would also like to thank the School of Mechanical Engineering, Universiti Sains Malaysia, for providing computing software FLUENT v6.0.12. Nomenclature

Figure 17. Effect of constant and varying permeate flux on membrane wall concentration polarization prediction: (a) comparison of membrane wall concentration predicted by constant and variation of permeate flux for MgSO4 solution under low feed Reynolds number and (b) comparison of predicted constant and varying flux along the membrane channel for MgSO4 under low feed Reynolds number.

smaller value of Jv will produce lower membrane wall concentration compared to the actual condition. At the channel exit region, the overpredicted Jv (Figure 17b) tends to generate a higher membrane wall concentration (Figure 17a). By employing constant Jv as a boundary condition, the error of prediction for the membrane wall concentration becomes more obvious at higher pressure, since, referring to Figure 17b, the reduction slope of the permeate flux was found to be higher at higher pressure.

CA ) concentration (kg/m3) C h ) average concentration (kg/m3) DAB ) binary mass diffusion coefficient (m2/s) h ) narrow membrane channel height (m) H ) diffusive hindrance factor Jv ) permeate volume flux (m/s) Jw ) pure water flux (m/s) l ) narrow membrane channel length (m) Lp ) hydraulic permeability (m/(Pa·s)) mA ) solute mass fraction ((kg of solute)/(kg of solution)) Ns ) solute flux (kg/m2·s) Ps ) overall solute permeability(m/(Pa‚s)) R ) true rejection R′ ) 1 - R Re ) feed Reynolds number (Re ) Fufh/µ) Sc ) Schmidt number (Sc ) µ/FDAB) u ) velocity in x-direction (m/s) V ) velocity in y-direction (m/s) w ) width (m) x ) x-coordinate y ) y-coordinate Greek Letters σ ) reflection coefficient Γ ) concentration polarization factor, defined by Γ ) (CAW CAf)/CAf F ) density (kg/m3) µ ) viscosity (kg/m‚s) ∆P ) transmembrane pressure (Pa) δc ) distance between wall and wall adjacent cell (m) ω ) solute permeability (kg/(N‚s)) π ) osmotic pressure (Pa)

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ReceiVed for reView August 14, 2006 ReVised manuscript receiVed December 6, 2006 Accepted December 12, 2006 IE061065Z