Assessing the Impact of Concentration-Dependent Fluid Properties on

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Ind. Eng. Chem. Res. 2008, 47, 1636-1649

Assessing the Impact of Concentration-Dependent Fluid Properties on Concentration Polarization in Crossflow Membrane Systems Shengwei Ma,*,† Stavros C. Kassinos,† and Despo Fatta Kassinos‡ Department of Mechanical and Manufacturing Engineering, UniVersity of Cyprus, 75 Kallipoleos St., P.O. Box 20537, Nicosia 1678, Cyprus, and Department of CiVil and EnVironmental Engineering, UniVersity of Cyprus

The effects of concentration-dependent fluid properties (viscosity, diffusivity, and osmotic-pressure coefficient) of sucrose-water solution on the permeate flux, wall concentration, Sherwood numbers, and velocity profiles in three different membrane systems were assessed using a 2D finite element model, which was developed by modifying the fluid property terms to incorporate the concentration dependency in a coupled concentration polarization model that was developed earlier. It was found that wall concentration, permeate velocity, and Sherwood numbers were significantly affected by the variations of the fluid properties. If any of the concentration dependencies of the fluid properties is neglected, wall concentrations would be either significantly overestimated or underestimated, whereas permeate velocity would always be overestimated. In different membrane systems, the relative importance of the effects of the concentration-dependent viscosity, diffusivity, and osmotic-pressure coefficient on wall concentration and permeate velocity may vary. These results suggest that the effects of concentration-dependent properties on concentration polarization would be strongly affected by the system operating conditions, especially the system value of ∆p/∆π0. It was also found that the errors in the solution of Berman for the velocity profile may be considerable when the variations of the fluid properties are considered under the simulation conditions. 1. Introduction In pressure-driven membrane systems, concentration polarization, which is an inherent phenomenon, may significantly affect the performance of the system.1-5 Several models that directly quantify the concentration polarization in reverse osmosis systems were proposed as early as 1960s.6-11 Because of the accumulation of the rejected solutes, concentration-dependent fluid properties, for example viscosity or diffusivity of the feed solution, may be significantly different in the concentration polarization layer from the values attained in the bulk.12,13 Furthermore, the flow and solute transport process in the concentration polarization layer is very likely to be affected by such variations of the fluid properties. Therefore, assessing the effects of concentration-dependent properties on concentration polarization and permeate flux would improve our understanding on the factors affecting concentration polarization in real systems. The impacts of the concentration-dependent viscosity and diffusivity on concentration polarization have been reported in various publications.14-18 For example, Doshi et al. assumed that the variation of viscosity in a sucrose solution was inversely proportional to that of diffusivity,17

[

]

1 - 0.114(c/c0) D µ0 ) ) D0 µ 1 - 0.114

2.5

(1)

and found as a result a very significant impact on wall concentrations. As stated in their article, the value of 0.114 in eq 1 is acceptable for the viscosity only; for the diffusivity this value should be 0.018, at least for sucrose solutions in the * To whom correspondence should be addressed. Tel.: +357 22894456. Fax: +357 22892254. E-mail: [email protected]. † Department of Mechanical and Manufacturing Engineering, University of Cyprus. ‡ Department of Civil and Environmental Engineering, University of Cyprus.

concentration range studied. Therefore, this assumption may result in an overestimation of the impacts of variable diffusivity on the mass transfer in the feed channel. In addition, the assumption that the product of µD remains constant is usually only acceptable in dilute solutions19 and may cause significant errors in concentration polarization layers where the concentrations are usually quite high. Gill et al. studied the impact of variable viscosity on wall concentrations.18 They found that the assumption of either variable or constant viscosity led to significant differences in the values of wall concentration obtained, especially when the inlet concentration was high. However, their work was based on constant permeate velocity, which may overestimate the blowup of wall concentration.20 Wiley and Fletcher simulated concentration polarization with the commercial computational fluid dynamics (CFD) software CFX.21 Several cases with variable viscosity and diffusivity were studied and compared with results obtained from other models. They concluded that using approximate solutions for the estimation of concentration polarization was highly questionable when the physical properties varied with the concentration. In addition, Geraldes et al. developed some numerical models for the flow and solute transport in membrane channels with concentration-dependent fluid properties.22-23 The impact of the variable physical properties on the mass transfer in the membrane channel was also reported in a few studies on limiting flux. For example, Shen and Probstein studied limiting flux in ultrafiltration of macromolecular solutions (bovine serum albumin solution) with concentrationdependent viscosity and diffusivity and found that a physicalproperty correction factor could be used to address the effects of variable properties.16 Aimar and Sanchez24 and Aimar and Field25 showed that the concentration-dependent fluid properties may result in the limiting flux without resorting to the assumption of gelling near the membrane surface. In previous research on concentration polarization, the osmotic pressure was usually assumed to be proportional to the concentration (i.e., the osmotic-pressure coefficient, k, is as-

10.1021/ie0713893 CCC: $40.75 © 2008 American Chemical Society Published on Web 01/23/2008

Ind. Eng. Chem. Res., Vol. 47, No. 5, 2008 1637

sumed constant). However, the nonlinear dependency of osmotic pressure on concentration has been reported for many solutions at relatively high concentrations.26 For example, for sucrosewater systems the dependency of osmotic pressure on concentration (up to 32% in w/w) can be fitted as (based on the data from Sourirajan27)

∆π ) 72.67c + 77.06c + 184.44c (× 10 Pa) 2

3

5

(2)

Therefore, in the widely used formula for variable permeate velocity,

Vw ) Vw0

(

)

cw 1-β c0

(3)

the parameter β, which can be expressed as β ) 1/(∆p/∆π0) 1 if pressure losses are negligible, should be concentration dependent and not be a constant, unless the nonlinear terms in expression 2 for the osmotic pressure are neglected. This suggests that the concentration-dependent osmotic-pressure coefficient might directly affect concentration polarization and permeate flux. However, this has rarely been reported in the literature. In addition, the impact of concentration-dependent properties on the axial variation of permeate velocity, which is the most essential parameter of system performance, is still not well understood. Therefore, until now the quantitative knowledge on the impact of concentration-dependent properties on concentration polarization and permeate flux was still very limited. The purpose of this research was to assess the role of the concentration-dependent viscosity, diffusivity, and osmoticpressure coefficient in concentration polarization and permeate flux in different membrane systems. A previously developed 2D fully coupled streamline upwind Petrov/Galerkin (SUPG) finite element model for concentration polarization in pressuredriven crossflow membrane systems28 was modified to incorporate concentration-dependent properties. The impacts of the concentration-dependent viscosity, diffusivity, and osmoticpressure coefficient of a sucrose solution on the axial variations and the averages of wall concentration and permeate velocity are discussed in this article. In addition, the effects of the concentration-dependent fluid properties on Sherwood numbers and velocity profiles are also assessed. The results of this article could improve our understanding of the role of variable fluid properties on concentration polarization and membrane system performance.

∂u ∂V + )0 ∂x ∂y

∂u ∂u ∂u 1 ∂p +u +V )+ ∂t ∂x ∂y F ∂x ∂2u ∂2u ∂υ ∂u ∂υ ∂u + (5) υ 2+ 2 + ∂x ∂x ∂y ∂y ∂x ∂y

(

)

∂V ∂V ∂V 1 ∂p +u +V )+ ∂t ∂x ∂y F ∂y ∂υ ∂V ∂υ ∂V ∂2V ∂2V + (6) υ 2+ 2 + ∂x ∂x ∂y ∂y ∂x ∂y

( (

) )

∂ 2c ∂ 2c ∂c ∂c ∂c ∂D ∂c ∂D ∂c +u +V )D 2+ 2 + + (7) ∂t ∂x ∂y ∂x ∂x ∂y ∂y ∂x ∂y Compared with the governing equations for the original model,28 the last two terms in the right-hand side in eqs 5, 6, and 7 were added to address the effects of concentrationdependent viscosity (υ) and diffusivity (D). The initial condition can be arbitrary because the dynamic model is used to obtain a steady-state solution. In practice, however, good initial conditions, for example simulation results with similar conditions, may significantly shorten the CPU time required to achieve the steady-state solution. Dirichlet (first-type) boundary conditions (prescribed velocity and solute concentration) are imposed for the inlet of the channel:

u(x ) 0,y,t) ) u0(y,t); V(x ) 0,y,t) ) v0(y,t); c(x ) 0,y,t) ) c0(y,t) No-slip, no-penetration conditions are imposed for the impermeable solid wall and the spacer/filaments (if they are present):

u ) 0; V ) 0; D

∂c )0 ∂n

At the membrane surface, both the solute concentration and permeate velocity usually cannot be specified a priori but are defined with the coupled equations:

νw ) A(∆p - ∆π(cw,cp))

2. The Numerical Model Recently, a 2D SUPG finite element model (FEM) was developed to simulate concentration polarization in spacer-filled channels.28 It has been used to study the impact of filament geometry on concentration polarization, to simulate permeate flux enhancement by different spacers, and to assess the accuracy of wall-concentration estimations based on averaged permeate velocity.20,29,30 In this study, the model was modified to incorporate the concentration-dependent fluid properties (viscosity, diffusivity, and osmotic-pressure coefficient). Constant fluid density is used in this study because its variation is not significant for the sucrose solution in the concentration range of interest.27 Therefore, after defining the crossflow direction as the x axis (x ∈ [0,L]) and the channel-height direction as y (y ∈ [0,H]), the flow field and solute transport in a membrane channel can be described by the coupled 2D continuity, NavierStokes, and solute transport equations:

(4)

D

∂c ) νw(cw - cp) ∂n

(8) (9)

A penalty formulation is applied to the Navier-Stokes equations in this model; therefore, the pressure boundary condition is not required. The governing equations were solved numerically to obtain a steady-state solution with the 2D SUPG FEM. A four-node (2 × 2) Gauss quadrature was applied, except for the penalty terms, where a one-point quadrature was used to meet the Ladyzhenskaya-Babuska-Brezzi stability conditions. The implicit Euler method was applied for time integration except for convection, concentration-dependent viscosity and diffusivity, and boundary terms, which were treated explicitly. A steadystate solution was obtained when the partial derivatives of concentration and velocity with respect to time, ∂c/∂t, ∂u/∂t, and ∂V/∂t, were smaller than the preset tolerances. The details of

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Figure 1. Comparison of simulations results with other models11,17 (the problem was defined in ref 17).

the numerical methods and the impacts of meshing scheme on the accuracy were discussed elsewhere.28 3. Comparing with Other Models For constant viscosity and diffusivity, the model was compared with the experimental results and was thoroughly discussed elsewhere.28 In this article, simulation results of this SUPG FEM model are compared with those of several other models for the cases with the problems defined in these models (with variable viscosity and diffusivity). Figure 1 compares the simulation results of our model with Doshi’s rapidly varying boundary conditions (RVBC) model17 and Wiley’s CFD (CFX) model21 with identical boundary conditions as those defined in the article of Doshi et al.17 The problem was defined by eq 1 and ∆π0/∆p ) 0.135.17 Four different scenarios were simulated: constant viscosity and diffusivity, variable viscosity and constant diffusivity, constant viscosity and variable diffusivity, and variable viscosity and diffusivity. As shown in Figure 1, compared with the results of Doshi’s RVBC model, our model predicts consistently far lower wall concentrations in all four scenarios. As pointed out by Wiley and Fletcher,21 in the RVBC model, the transverse variations of viscosity and diffusivity were neglected. This leads to an underestimation of the solute flux from the membrane surface to the bulk. In addition, the rapidly varying boundary conditions assume a constant y component of the velocity. However, as shown in Figure 2, the y component of velocity decreases monotonically from the membrane surface to the bulk. This suggests that the assumption of a constant V may lead to an overestimation of the solute flux from the bulk to the membrane surface. Therefore, for these two reasons, the RVBC model is very likely to overestimate wall concentration considerably. Figure 1 also shows that compared with the RVBC model, our model predicts similar trends for the effects of the variable fluid properties on wall concentrations. However, for ζ (3((A∆p)3h)/ U0D02x) > 50, our simulations suggest that variable viscosity has similar effects on wall concentrations at variable diffusivity. This is different from the results of the RVBC model, probably due to the assumptions applied in the RVBC model. For constant viscosity and diffusivity, our results are almost identical to those of Wiley and Fletcher’s model21 as shown in

Figure 2. Profile of y-component velocity (variable diffusivity, the problem was defined in ref 17).

Figure 1. However, for variable viscosity or diffusivity, our model predicts noticeably higher wall concentrations. This may be related to different meshing schemes. In their studies, only about 60 cells along the channel (crossflow/axial direction) were used. However, in our study, more than 2000 elements in the crossflow direction were used to get the mesh-independent solutions. It was found that, to get a mesh-independent solution, for constant viscosity and diffusivity, only the meshing scheme in the channel-height direction is important as discussed previously;28 however, for variable diffusivity and viscosity, meshing schemes in both the crossflow and channel-height directions are sensitive. Insufficient meshing in the crossflow direction will underestimate wall concentrations. For example, Figure 3 compares the converged solutions obtained with identical conditions except that different crossflow meshing schemes were used. It shows that wall concentrations are significantly underestimated when there are only 1000 elements in the crossflow direction. This suggests that if the variations of viscosity or diffusivity in the crossflow direction are inadequately represented, the wall concentration would be very likely underestimated under simulation conditions. Figure 4 compares the simulation results of our SUPG FEM model with the solution of Gill et al.18 for the problem of the


Figure 3. Impact of meshing scheme on the accuracy of wall concentrations (the problem was defined in ref 17, with variable diffusivity and constant viscosity).

Figure 4. Comparing simulations results with the solution of Gill et al. (the problem was defined in ref 18, sucrose solution).

ultrafiltration of sucrose with variable viscosity, whose dependency on concentration (in w/w) was described as:

µ ) 0.0009049 exp(0.03479c)

(10)

The permeate velocity and diffusivity were set as constant. Three cases with different initial sucrose concentrations (0.1, 1, and 5%) were simulated. Figure 4 shows that our model predicts very similar wall-concentration variation and blowup patterns for all three cases. However, our model predicts noticeably higher wall concentrations in most locations, especially in the far downstream regions near the outlet. This is very likely due to the two assumptions in the work of Gill et al.: (1) neglecting axial components of the diffusion of solute and momentum, (2) neglecting the transverse variations of viscosity and assuming that µ ) µ0. Both of these assumptions may result in considerable underestimations of wall concentrations. In downstream regions, as concentration polarization becomes stronger, both the axial diffusion and transverse variation of viscosity would be more significant. Therefore, these two assumptions may lead to a considerable underestimation of wall concentrations in downstream regions.

Another factor that may contribute to the differences is the semi-infinite assumption in the solution of Gill et al. In our simulations, channel height was set as 1 mm. In the regions near the inlet, the semi-infinite assumption used by Gill et al. is reasonable as the concentration polarization layer is thin. However, in downstream regions, the increased thickness of the concentration polarization layer may make this assumption less reliable. For example, for C0 ) 1%, at ζ ) 39.4, the thickness of the concentration polarization layer is about 0.03 mm; whereas at ζ ) 875.5, this thickness is about 0.1 mm, which is 20% of h. This suggests that the semi-infinite assumption may result in considerable differences in the estimation of the wall concentration between our model and the model of Gill et al. in downstream regions. These comparisons suggest that this modified SUPG FEM model is able to capture the effects of the concentrationdependent fluid properties reasonably. 4. Simulation Conditions and Assumptions The height H and length L of the feed channel used in the numerical simulations were set as 1 and 10 cm, respectively.

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Table 1. Membrane Properties and Operating Conditions A (10-11 m/s Pa) ∆p (105 Pa) c0 (in wt %) U0 (m/s)

system #1

system #2

system #3

2.5 20 0.2 0.1

2.7 20 2 0.1

2.7 20 4 0.1

5. Results and Discussion

Sucrose (C12H22O11)-water solution was used as the feed in all of the simulations. The dependency of the viscosity, diffusivity, and osmotic pressure (∆π ) kc) was obtained from the data from Sourirajan27 by a least-squares regression (up to 32% in w/w):

ν ) 0.901 + 1.089c + 11.362c2 (× 10-6m2/s)

(11)

D ) 0.522 - 0.387c + 0.521c2 (× 10-6 m2/s)

(12)

k ) 72.67 + 77.06c + 184.44c (× 10 Pa)

(13)

2

5

Note that these three properties are not fully independent. For example, diffusivity is related to osmotic pressure through a sedimentation coefficient due to the interparticle interactions.37 However, in this research, the experimental data of the viscosity, diffusivity, and osmotic-pressure coefficient, and the derived regression of eqs 11-13 were directly used in our simulations without considering such interactions. Three different crossflow membrane systems were studied, and the system variables used in simulations are summarized in Table 1. Simulations were conducted with eight different combinations of concentration-dependent fluid prosperities (eight cases): (1) constant υ, D, and k; (2) variable D, constant υ and k; (3) variable k, constant υ and D; (4) variable υ, constant D and k; (5) variable υ and D, constant k; (6) variable υ and k, constant D; (7) variable k and D, constant υ; and (8) variable υ, D, and k. Complete (100%) solute rejection (cp ) 0) was assumed in all cases. At the inlet, the averaged crossflow velocity was set to 0.1 m/s so that a reasonable recovery rate could be simulated with the given channel configurations. Because the inlet is the start of the feed channel with permeable walls, it is reasonable to assume that the inlet is preceded by impermeable walls. Therefore, a fully developed parabolic flow profile (without permeation), which is identical to eqs 5 and 6 in the article of Doshi et al.,17 was assumed on the inlet boundary (y ∈[0,2h]):

[ (y -h h) ]; ν ) 0

u0(y) ) 1.5U0 1 -

real situation to assess the possible errors and the role of variable properties on wall concentration, permeate flux, and Sherwood numbers.

2

0

As discussed in section 3, and in the previous article,28 the meshing scheme may have significant impacts on the accuracy, and coarse meshes may underestimate concentration polarization. In all of the simulations, a mesh-independent solution was obtained by refining the mesh until successive solutions ceased to vary with meshing schemes. For the range of conditions examined, we have found that, to obtain the mesh-independent solutions, more than 60 non-uniform elements were needed in the channel-height direction (with exponentially decreasing height toward the membrane surface); in the crossflow (axial) direction, 2000-3000 elements were needed. The total elements vary from 120 000 to 180 000. Because steady-state solutions were sought, the incremental time step was found to have a negligible impact on the results of the wall concentration when the local Courant number constraint is satisfied. In this article, we assume that the simulation results based on concentration-dependent υ, D, and k (case 8) represent the real situation. Results from other cases were compared with this

5.1. The Effects of Concentration-Dependent Properties on Wall Concentration. Figure 5 compares wall-concentration profiles corresponding to different assumptions of the concentration-dependent properties in system #1 (Table 1). It shows that near the inlet, differences in wall concentrations are not significant. However, in the downstream, significant differences can be observed. Except for the case with variable υ and D (and constant k, case 5), in all other cases wall concentrations in this system are underestimated. As shown in Figure 5, the case with variable k (constant υ and D, case 4) predicts lower wall concentrations than that with constant υ, D, and k (case 1). Similarly, variable k and D results in significantly lower wall concentrations than constant k and variable D. This indicates that a concentration-dependent osmotic-pressure coefficient leads to lower wall concentrations compared with a constant osmotic-pressure coefficient. When the nonlinear term in eq 2 is included in the calculations, for any given concentrations, the osmotic pressure is larger than that for the case with only the linear term. This may effectively lower the permeate velocity according to eq 8. Hence, the convective solute transport toward the membrane surface is slowed down. Consequently, the wall concentration is lowered. Figure 5 also illustrates that both variable diffusivity and viscosity have positive impacts on wall concentration. Compared with constant properties, variable diffusivity and/or viscosity results in higher wall concentrations no matter whether variable osmotic-pressure coefficient is considered or not. This suggests that in this system (#1), both variable diffusivity and viscosity have stronger impacts on wall concentration than a variable osmotic-pressure coefficient does. Figure 5 also shows that the lines corresponding to variable D (variable or constant k) are far above those corresponding to variable υ. This suggests that the variable diffusivity has a significantly stronger impact on wall concentration than variable viscosity. However, when the variable viscosity and diffusivity are considered together, the combined effects are far more significant than a simple addition. For example, the averaged dimensionless wall concentration (1/ L‚c0 ∫L0 cwdx) corresponding to variable viscosity (and constant D and k, case 4) is about 3.6% higher than that corresponding to the constant properties; for variable diffusivity (and constant υ and k, case 2), this is about 23.6%, whereas for variable viscosity and diffusivity (and constant k, case 5), this number increases to 35.1%. Figure 5 also shows that the wall-concentration profiles cluster in two groups: one with variable diffusivity and the other with constant diffusivity. This suggests that, in this system, variable diffusivity is very likely the dominant factor for concentration polarization. Therefore, for this system, if the concentration dependency of diffusivity is ignored or underestimated in the simulations, significant underestimation of wall concentrations would be expected. Figure 6 shows wall-concentration profiles in system #3. It was found that, in this system, the impact of concentrationdependent properties exhibits a significantly different pattern than that in system #1. Wall-concentration profiles cluster in two groups based on a constant or variable osmotic-pressure coefficient, rather than on diffusivity. This suggests that, unlike that in system #1, the dominant factor for concentration polarization in this system is the osmotic-pressure coefficient.


Figure 5. Axial variation of wall concentrations (simulation conditions, A ) 2.5 × 10-11 m/s Pa; ∆p ) 20 × 105 Pa; c0 ) 0.2% sucrose; U0 ) 0.1 m/s).

Figure 6. Axial variation of wall concentrations (simulation conditions, A ) 2.7 × 10-11 m/s Pa; ∆p ) 20 × 105 Pa; c0 ) 4% sucrose; U0 ) 0.1 m/s).

If the effects of concentration-dependent osmotic-pressure coefficient are neglected, wall concentrations would be significantly overestimated: even a constant fluid properties assumption would lead to a noticeable overestimation of wall concentrations in this system. This suggests that omitting the effects of concentration-dependent properties does not necessarily lead to a conservative estimation of wall concentrations in certain membrane systems. The relative importance of the osmotic-pressure coefficient in this system is probably due to the value of ∆p/∆π0, which would be the upper limit of the concentration polarization modulus of the system if a constant osmotic-pressure coefficient is assumed. When this value is large (for example, in system #1 this value is about 138), the variations of ∆π in eq 8 have weaker effects on permeate velocity as compared to a system with a smaller value of ∆p/∆π0 (for example, in system #3 this value is about 6.6). Therefore, the concentration-dependent osmotic-pressure coefficient very likely has stronger influences on wall concentration in systems with a lower value of ∆p/ ∆π0.

Figure 6 also indicates that concentration-dependent viscosity has slightly stronger influences on wall concentration than concentration-dependent diffusivity does in this system. This trend is the opposite of that in system #1 as discussed earlier. This suggests that not only the expression of the concentration dependency but also the operating conditions of a membrane system determine the relative importance of viscosity and diffusivity on wall concentrations. Similar to the case in system #1, the combined effects of variable diffusivity and viscosity are obvious. For example, the averaged dimensionless wall concentrations corresponding to variable viscosity (constant D and k, case 4) and variable diffusivity (constant υ and k, case 2) are respectively about 4.5 and 3.5% higher compared with those corresponding to constant properties, whereas this number increases to more than 8% when both viscosity and diffusivity are concentration dependent (constant k, case 5). Figure 7 shows that, in system #2, the effects of concentration-dependent properties are noticeably different from those in systems #1 and #3. In this system, the clustering of the wallconcentration profiles is not as obvious as in the previous two

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Figure 7. Axial variation of wall concentrations (simulation conditions, A ) 2.7 × 10-11 m/s Pa; ∆p ) 20 × 105 Pa; c0 ) 2% sucrose; U0 ) 0.1 m/s).

systems. In the downstream (x/H > 35), the impact of the concentration-dependent osmotic-pressure coefficient seems dominant, and there the assumption of constant properties can lead to an overestimation of wall concentrations; this is similar to that in system #3, whereas near the inlet (x/H < 15), the effects of variable viscosity and diffusivity are significant and wall concentrations would be underestimated if constant properties are used. Figure 7 indicates that variable diffusivity has slightly stronger influences on wall concentrations than variable viscosity. However, like that in system #3, this difference is not significant. This suggests that in the low range of concentrations, the concentration-dependent diffusivity has stronger influences on wall concentrations than viscosity, and increasing the concentrations would very likely strengthen the impacts of concentrationdependent viscosity. This is probably related to the concentration dependency of viscosity and diffusivity for the sucrose solution. From eqs 11 and 12, the derivative of viscosity and diffusivity with respect to concentration can be written as

dV ) 1.089 + 11.362c dc

(14)

dD ) -0.387 + 0.521c dc

(15)

This indicates that the variation of viscosity is far greater than that of diffusivity, and therefore increasing concentrations would very likely increase the relative importance of variable viscosity. Note that in the concentration range investigated (i.e., less than 32% in w/w), dD/dc is always less 0. This suggests that the decrease of diffusivity would slow down when the concentration increases. In system #1, the averaged wall concentration (1/L ∫L0 cwdx) is about 7.04% (w/w; variable D, k and υ; case 8) and the influences of the concentration-dependent diffusivity on wall concentration are significantly stronger than those of viscosity; in system #3 on the other hand, this averaged wall concentration is about 17% (w/w) and correspondingly the variable viscosity has slightly stronger influences on wall concentrations than diffusivity. In system #2, this value is about 15.5% (w/w) and the relative importance of diffusivity and viscosity is intermediate between that in system 1 and 3.

On the basis of the analysis of the abovementioned three different cases, it is clear that wall concentrations are strongly affected by variable fluid properties. Compared with assuming constant properties, having variable viscosity and diffusivity results in higher wall concentrations, whereas a variable osmoticpressure coefficient leads to lower wall concentrations. Therefore, when variable diffusivity, viscosity, and osmotic-pressure coefficients are considered together in a real system, there may be a dominant factor for wall concentrations because of the opposing effects that the concentration-dependent osmotic pressure and viscosity/diffusivity have on wall concentrations. For systems with a high value of ∆p/∆π0, the concentrationdependent diffusivity may be the dominant factor for wall concentrations, whereas for systems with a low value of ∆p/ ∆π0, the osmotic-pressure coefficient could dominate. Therefore, the dominant factor for wall concentrations depends on both the operating conditions and the dependency of properties on solute concentrations. Note that the assumption of constant properties may result in a significant overestimation of wall concentrations in systems with a relatively low value of ∆p/ ∆π0. 5.2. Effects of Variable Properties on Permeate Velocity. Figure 8 shows the axial variation of dimensionless permeate velocity (Vw/Vw0) in system #1 with different assumptions concerning viscosity, diffusivity, and the osmotic-pressure coefficient. Compared with the case with concentration-dependent υ, D, and k (case 8), all other cases overestimate permeate velocity. Similar to the wall-concentration profiles shown in Figure 5, permeate velocity variation profiles also cluster in two groups based on variable or constant diffusivity. The overestimation would be more significant when constant diffusivity is assumed. In this system, concentration-dependent diffusivity is the dominant factor for concentration polarization; therefore, if constant diffusivity is assumed, wall concentration would be significantly underestimated and permeate velocity overestimated. Note that wall concentrations corresponding to a variable osmotic-pressure coefficient in this system are lower than those corresponding to constant properties as shown in Figure 5. However, Figure 8 shows the same trend for permeate velocity. In other words, compared with constant properties, a variable


Figure 8. Axial variation of permeate velocity (simulation conditions, A ) 2.5 × 10-11 m/s Pa; ∆p ) 20 × 105 Pa; c0 ) 0.2% sucrose; U0 ) 0.1 m/s).

Figure 9. Axial variation of permeate velocity in system #3 (simulation conditions: A)2.7 × 10-11 m/s Pa; ∆p)20 × 105 Pa; c0)4% sucrose; U0)0.1 m/s).

osmotic-pressure coefficient would result in both lower wall concentrations and lower permeate velocity in this system. When a concentration-dependent osmotic-pressure coefficient is applied, for any given concentrations, the osmotic pressure is higher than that with a constant osmotic-pressure coefficient. According to eq 8, this leads to a lower permeate velocity for the case with a variable osmotic-pressure coefficient. In return, lower permeate velocity slows down the solute transport from the bulk toward the membrane surface and therefore results in a lower wall concentration. The axial variation of permeate velocity in system #3 is shown in Figure 9. It indicates again that permeate velocity is overestimated if any of the concentration dependencies are neglected. This was also observed in system #2. The reason lies in the dependencies of the solution properties on the concentration used in this study. According to eqs 11 and 12, if the concentration dependency is neglected, for any given

concentrations, the diffusion coefficient would be overestimated and the viscosity underestimated. This results in lower wall concentrations (therefore higher permeate velocity) as mentioned above. The constant osmotic-pressure coefficient also tends to result in an overestimation of permeate velocity. Therefore, including concentration dependency for viscosity, diffusivity, or the osmotic-pressure coefficient would result in a lower permeate velocity. This suggests that permeate velocity would be always overestimated if any of the concentration dependencies are underestimated or neglected. In system #3, the osmotic-pressure coefficient is the dominant factor for wall concentrations. However, this is not clearly reflected on permeate velocity in Figure 9. For a variable osmotic-pressure coefficient, a lower permeate velocity is achieved from the balance of two competing processes: lower wall concentrations corresponding to lower permeate velocity leading to a lower osmotic pressure, and variable osmotic-

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Figure 10. Errors of averaged permeate velocity with different concentration dependency of fluid properties.

Figure 11. Errors of averaged wall concentrations with different concentration dependency of fluid properties.

pressure coefficient resulting in a higher osmotic pressure. Overall, in all of the systems under our simulation conditions, the second process dominates; that is, compared with a constant osmotic-pressure coefficient, the concentration-dependent k leads to higher osmotic pressures and lower wall concentrations. However, because of the first process involved, the influences of a concentration-dependent osmotic-pressure coefficient on permeate velocity are weakened. Therefore, this coefficient was not observed as a dominant factor for permeate velocity. However, the significant impact of this coefficient on permeate velocity in this system could be observed by comparing the cases with constant properties and a variable osmotic-pressure coefficient (and constant υ and D, case 3). As shown in Figures 9 and 10, in system #3, compared with constant properties, a variable osmotic-pressure coefficient results in a 10% lower averaged permeate velocity (1/L∫L0 Vwdx), whereas in system #1, this value is only about 2%.

Figure 10 compares the errors (overestimation) of the averaged permeate velocity if concentration-dependent properties are not sufficiently considered (assuming the real value would be obtained from the simulation results based on a variable viscosity, diffusivity, and osmotic-pressure coefficient). It was found that in all of the cases, errors in system #1 are the lowest. This is probably due to the high value of ∆p/∆π0. As mentioned above, when this value is high, permeate velocity is relatively less sensitive to concentration polarization, and therefore the overestimation of permeate velocity due to errors in the estimation of concentration polarization is very likely less significant. Note that the error patterns of averaged permeate velocity are significantly different from those of averaged wall concentrations. Comparing Figure 10 and Figure 11, it was found that an overestimation of permeate velocity does not necessarily imply an overestimation or underestimation of wall concentra-


Figure 12. Variation of local and average Sherwood numbers.

tions, and vice versa. It was also found that, although errors of permeate velocity in system #1 are the lowest in all seven cases, in five of the seven cases, averaged wall concentrations have the highest errors for this system. This suggests that the relationship between errors of averaged wall concentrations and permeate velocity is significantly related to the system conditions. 5.3. The Effects of Variable Fluid Properties on the Sherwood Number. The local mass transfer coefficient and Sherwood number were calculated from the simulation results:

K(x) )

Vw(x)cw(x) cw(x) - c0

Sh(x) )

2K(x)H Dw(x)

(16) (17)

The average Sherwood number

Sh(x) )

1 x

∫0x Sh(x*)dx*

(18)

was then calculated. The average Sherwood number was also calculated using the correlation developed by De and Bhattacharya31 and the simulated permeate velocity profile. For the cases with variable fluid properties, the average Schimdt number and average wall Peclet number were used:

Sc(x) )

1 x

∫1x Sc(x*)dx*

(19)

Pew(x) )

1 x

∫0x Pew(x*)dx*

(20)

where Sc(x) and Pew(x) are the local Schimdt number and the wall Peclet number based on the local wall concentrations and local permeate velocity. Figure 12 compares the calculated Sherwood numbers in three systems. It was found that the Sherwood number would be overestimated if the variations of fluid properties are neglected. For example, as shown in part a of Figure 12, in system #1, at x/H ) 100, the average Sherwood number would be overestimated by 9% if the dependency of the fluid properties on concentration is omitted under the simulation conditions. Increasing the initial sucrose concentration

to 2% (w/w, system #2) increases this overestimation. As shown in part b of Figure 12, at the same location (x/H ) 100) in system #2 this overestimation of the Sherwood number is about 21.1%. However, further increasing the initial concentration to 4% (w/w, system #3) does not result in a significant increase in this overestimation. This is very likely related to the maximum wall concentrations. In our simulations, the applied pressure was set as 20 × 105 Pa. This suggests that when the variation of the fluid properties is considered, the maximum possible value of wall concentration, which corresponds to Vw ) 0 according to eq 8, is about 20% (w/w). As shown in Figure 7, in system #2, for the case with variable fluid properties, the maximum wall concentration is about 17.4%, which is about 87% of the maximum possible value. In system #3 (Figure 6), the maximum wall concentration is about 18.3% (w/w). This suggests that the variable fluid properties have similar effects on the mass transfer coefficients in these two systems, and hence the differences of the errors due to the omission of the variable fluid properties are not large. Figure 12 also shows the average Sherwood number calculated with the correlation of De and Bhattacharya.31 Note that this correlation is based on the assumptions of constant wall concentrations and x1/3Vw ) constant. In our calculations, the local wall concentrations and permeate velocity from the simulation results were used to get the local Schimdt number and the local wall Peclet number. It was found that for the cases with constant fluid properties, this correlation underestimates the Sherwood numbers noticeably. For the cases with variable fluid properties, in systems #2 and #3, this correlation overestimates Sherwood numbers, whereas in system #1, the average Sherwood numbers are underestimated for all lengths. However, for most lengths (x/H), the differences between the average Sherwood numbers calculated with this correlation (using the simulation results) and the Sherwood numbers calculated directly from the simulation results are less than 15%. These suggest that this correlation may provide a reasonable estimation of the mass transfer rate. De and Bhattacharya (1999) introduced a correction factor (Sc0/Scw)n to represent the impacts of the variable fluid properties.32 However, as shown in Figure 13, the relationship between ln(Sh/Shc and ln(Sc0/Scw) is obviously nonlinear. This implies that this correction factor is not suitable for the three systems investigated in this article.

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Figure 13. Effects of Schmidt numbers on Sherwood numbers.

Figure 14. Variation of local and average Sherwood numbers based on the stagnant film model.

In membrane system analysis, a stagnant film model approximation is often used. In this model, the mass transfer coefficient is defined as (assuming 100% rejection)33

K2(x) )

Vw(x) ln(cw(x)/c0)

(21)

Then local Sherwood number, Sh2(x), and average Sherwood number, Sh2(x), can be calculated with eqs 17 and 18, respectively. Schock and Miquel (1987) reported that the experimental data fit the Leveque correlation fairly well:34

Sh2(x) ) 1.85(Re‚Sc2H)1/3x-(1/β)

(22)

Figure 14 compares the local and average Sherwood numbers in three systems with constant and variable fluid properties. It was found that the Sherwood number based on the stagnant film model is consistently lower than the standard Sherwood number (Figure 12) because ln(cw/c0) is always larger than 1 (c0/cw). This suggests that it may be inappropriate to compare the Sherwood number based on the stagnant film model with the correlations developed for the standard Sherwood numbers. Figure 14 also shows that, compared with the cases with variable fluid properties, the mass transfer is always overestimated in

all of the three systems when the concentration-dependent properties are neglected. Figure 14 also indicates that the Leveque correlation is unable to represent the effects of fluid properties on mass transfer in membrane systems. In system #1, the effects of the concentration-dependent fluid properties are not clearly represented by the Leveque correlation. Moreover, in all three systems, the Leveque correlation predicts counterintuitive effects of the variable fluid properties: the Sherwood number would increase noticeably when the concentration-dependent fluid properties are considered. For example, in system #2, at x/H ) 100, for the case with constant fluid properties the Leveque correlation predicts a Sherwood number of around 35.82, which is about 16.4% lower than that for the case with variable fluid properties (calculated with the Leveque correlation). This is probably related to the mass transfer coefficient definition (eq 21) in the stagnant film model. Interestingly, for systems #2 and #3, the Leveque correlation gives smaller errors for the cases with variable fluid properties than for those with constant properties. For example, as shown in part b of Figure 14 at x/H ) 100, in system #2, the error of the average Sherwood number estimated by the Leveque correlation is about 7.2% for the case with variable fluid properties, whereas for the case with constant properties, the error is about 30%. In system #3, at the same


Figure 15. Effects of variable fluid properties on the profile of y-component velocity (V).

location, the errors are 12.8% and 19.9% for variable and constant fluid properties, respectively. In system #1, as shown in part a of Figure 14, the Leveque correlation predicts the average Sherwood numbers with significant errors. For the case with constant and variable fluid properties, the errors are more than 30 and 50% respectively for most of the lengths. The relative high concentration polarization modulus (cw/c0) may result in the invalidity of the stagnant film model in this system. This is consistent with Zydney’s findings.35 Zydney (1997) theoretically analyzed the validity of the stagnant film model for membrane separation systems and found that the stagnant film model would be invalid if the concentration polarization modulus is larger than 17.35 In system #1, as shown in Figure 5, the concentration polarization modulus is far larger than 20 once x/H > 25. This very likely invalidates the stagnant film model assumption, and therefore the Leveque correlation in this system. Comparing with the Leveque correlation, the correlation developed by De and Bhattacharya gives a better approximation on average Sherwood numbers in all three systems, especially in system #1 (with a high concentration polarization modulus). However, it should be pointed out that, as mentioned earlier, the error in the correlation of De and Bhattacharya is considerable possibly due to the assumptions of constant wall concentrations and x1/3Vw ) constant. 5.4. The Effects of Variable Fluid Properties on the Velocity Profile. The variable fluid properties would not only affect the solute transport but also the flow field. The variations of viscosity directly affect the velocity profile. Moreover, the flow field is affected by the variable diffusivity and osmoticpressure coefficient through the coupling of solute transport and momentum transfer (eqs 8 and 9). However, the effects of the concentration-dependent fluid properties on the velocity profile in membrane systems have been rarely reported. The approximate velocity solution of Berman36 has been widely used in previous research. However, this solution is based on constant fluid properties, and therefore it may result in some errors for the cases with concentration-dependent fluid properties. So, our simulation results of the velocity profile were compared with this solution for both constant and variable fluid properties. It was found that, for the cases with constant fluid properties, the solution of Berman matches the simulation results very well

in all three systems. In addition, for the cases with constant fluid properties, the profile of V/Vw does not have noticeable variations in the three systems. This is due to the relatively low permeate velocity in all three systems. In the solution of Berman (eq 33 in ref 36), the second term is negligible, and hence the value of V/Vw is a function of y only when the permeate velocity is on the order of 10-5m/s. These results indicate that the solution of Berman is a very accurate approximation of the velocity profile in membrane systems when the variation of the fluid properties is negligible. However, as shown in Figure 15, for the cases with variable fluid properties, the profile of V differs from the solution of Berman significantly. In all three systems, the value of V/Vw is smaller than that predicated by the solution of Berman in all locations except at the membrane surface (V/Vw ) 1) and at the symmetric plane (y ) h, V/Vw ) 0). Moreover, this difference is dependent on both the location and the system parameters. For example, at x/H ) 100 and y ) 0.5 h (or 1.5 h), in system #3 the value of V/Vw is 0.35, which is about 51% of the value estimated by the solution of Berman, whereas in the same location in systems #1 and #2, this value is about 78.7 and 67% of that estimated by Berman, respectively. This suggests that the concentration-dependent fluid properties significantly affect the velocity profile, and therefore in these cases the solution of Berman may have relatively large errors for the y-component velocity. The profile of the x-component velocity (u) in three systems, together with the solution of Berman and the Poiseuille parabola, are shown in Figure 16. It was found that, for the cases with variable fluid properties, there are noticeable discrepancies between the solution of Berman and our simulation results. Near the symmetric plane (i.e., center of the feed channel in the y direction), the solution of Berman overestimates the degree of deviation (flattening) from the Poiseuille parabola, whereas near the membrane surface noticeable underestimation is present. This overestimation near the center of the channel and the underestimation near the membrane surface are more pronounced in systems #2 and #3 than those in system #1. This suggests that increasing the variations of the fluid properties may result in larger errors in the solution of Berman for u.

1648


Figure 16. Effects of variable fluid properties on the profile of x-component velocity (u).

6. Conclusions In this article, a 2D numerical model was developed to simulate concentration polarization with concentration-dependent viscosity, diffusivity, and the osmotic-pressure coefficient. By comparing the simulations results with other approximation models and a CFD model, it was found that the model is able to reliably simulate the effects of concentration-dependent fluid properties on concentration polarization. It was found that concentration-dependent solution properties have significant impacts on concentration polarization. Compared with the constant properties, variable viscosity and diffusivity result in higher wall concentrations, whereas a variable osmotic-pressure coefficient leads to lower wall concentrations. In different systems, there could be different dominant factors for wall concentrations. The value of ∆p/∆π0 may be an indicator of the relative importance of viscosity, diffusivity, and osmotic pressure in a given system under the simulation conditions. In systems with a high value of ∆p/∆π0, concentration-dependent diffusivity may play a dominant role in concentration polarization, and neglecting the concentration dependency of diffusivity would result in a significant underestimation of wall concentrations. In systems with lower values of ∆p/∆π0, the effects of the concentration-dependent osmoticpressure coefficient may dominate and viscosity may have slightly stronger impacts on wall concentrations than diffusivity. In these systems, constant fluid properties may result in significant overestimation of wall concentrations. These findings suggest that the relative importance of viscosity, diffusivity, and the osmotic-pressure coefficient on wall concentrations is not only determined by the expression of the concentration dependency but also by operating conditions. It was also found that permeate velocity would be noticeably overestimated if the dependency of any fluid properties (viscosity, diffusivity, and osmotic-pressure coefficient) on concentration is neglected. In systems with a high value of ∆p/∆π0, diffusivity tends to have stronger influences on permeate velocity than viscosity and the osmotic-pressure coefficient, whereas in systems with a low value of ∆p/∆π0, a dominant impact of the osmotic-pressure coefficient was not observed. Although averaged permeate velocity would be always overestimated if the dependency of solution properties on

concentration is neglected, averaged wall concentrations could be overestimated or underestimated. This indicates that an overestimation of permeate velocity does not necessarily imply an overestimation or underestimation of wall concentrations and vice versa. The concentration-dependent fluid properties also affect both of the local and average Sherwood numbers. Compared with the Leveque correlation, the correlation developed by De and Bhattacharya gives a better approximation on average Sherwood numbers, especially for systems with a high concentration polarization modulus. However, the errors of both correlations are considerable. It was also found that the velocity profile is noticeably affected by the concentration-dependent fluid properties, and the solution of Berman may have considerable errors, especially for the profile of y-component velocity, when the variations of the fluid properties are significant. We readily acknowledge that our simulation is based on a sucrose-water solution, and therefore the results may be inapplicable for other macromolecules. In addition, because of the limitations of the computing resources, a relatively low crossflow velocity (U0 ) 0.1 m/s, Re ≈ 200) was deliberately used to simulate a reasonable recovery rate in the systems. Therefore, for membrane systems operated under a turbulent flow regime, the results may be not applicable. List of Symbols A ) membrane permeability (m/s Pa) c ) solute concentration (w/w) D ) diffusivity (m2/s) H ) channel height (m) h ) H/2 (m) k ) osmotic-pressure coefficient (Pa/wt %) K ) mass transfer coefficient K2 ) mass transfer coefficient based on stagnant film model L ) channel length (m) n ) normal direction p ) pressure (Pa) ∆p ) applied pressure (Pa) Pew ) wall Peclet number (Vw‚2H/Dw) Re ) Reynolds number Sc ) Schimdt number


Sh ) Sherwood number Sh2 ) Sherwood number based on the stagnant film model t ) time (sec) U0 ) 1/H ∫H0 u0dy (m/s) u ) axial velocity in the x direction (m/s) V ) lateral velocity in the y direction (m/s) x ) axial coordinate in the crossflow direction (m) y ) lateral coordinate in the channel-height direction (m) Greek Letters υ ) kinematic viscosity (m2/s) F ) density (kg/m3) ∆π ) osmotic pressure (Pa) µ ) dynamic viscosity (Pa s) ζ ) 3((A∆p)3h)/U0D20x (in Figure 4, A∆p ) Vw) Subscripts 0 ) at x ) 0 p ) permeate w ) membrane surface Acknowledgment This work has been performed under the UCY-CompSci project, a Marie Curie Transfer of Knowledge (TOK-DEV) grant (contract no. MTKD-CT-2004-014199), and under the SBMEuroFlows project, a Marie Curie IRG grant (contract no. MIRG-CT-2004-511097); both funded by the CEC under the Sixth Framework Program. Literature Cited (1) Matthiasson, E.; Sivik, E. Concentration Polarization and Fouling. Desalination 1980, 35, 59. (2) Belfort, G. Synthetic Membrane Processes: Fundamentals and Water Applications; Academic Press: Orlando, FL, 1984. (3) Jonsson, G.; Boesen, C. E. Concentration Polarization in a Reverse Osmosis Test Cell. Desalination 1977, 21, 1. (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) Song, L.; Elimelech, M. Theory of Concentration Polarization in Crossflow Filtration. J. Chem. Soc., Faraday Trans. 1995, 91, 3389. (6) Dresner, L. Boundary Layer Buildup in the Demineralization of Salt Water by ReVerse Osmosis, special report ORNL-3621; Oak Ridge National Laboratory: Oak Ridge, TN, 1964. (7) Sherwood, T. K.; Brian, P. L. T.; Fisher, R. E.; Dresner, L. Salt Concentration at Phase Boundaries in Desalination by Reverse Osmosis. Ind. Eng. Chem. Fundam. 1965, 4, 113. (8) Brian, P. L. T. Concentration Polarization in Reverse Osmosis Desalination with Variable Flux and Incomplete Salt Rejection. Ind. Eng. Chem. Fundam. 1965, 4, 439. (9) Gill, W. N.; Tien, C.; Zeh, D. W. Analysis of Continuous Reverse Osmosis Systems for Desalination. Int. J. Heat Mass Transfer 1966, 9, 907. (10) Gill, W. N.; Tien, C.; Zeh, D. W. Concentration Polarization Effects in a Reverse Osmosis System. Ind. Eng. Chem. Fundam. 1966, 5, 637. (11) Srinivasan, S.; Tien, C.; Gill, W. N. Simultaneous Development of Velocity and Concentration Profiles in Reverse Osmosis Systems. Chem. Eng. Sci. 1967, 22, 417. (12) Placidi, M.; Cannistraro, S. A Dynamic Light Scattering Study on Mutual Diffusion Coefficient of BSA in Concentrated Aqueous Solutions. Europhys. Lett. 1998, 43, 476. (13) Vand, V. Viscosity of Solutions and Suspensions. III: Theoretical Interpretation of Viscosity of Sucrose Solutions. J. Phys. Colloid Chem. 1948, 63, 314.

(14) Goldsmith, R. L. Macromolecular Ultrafiltration with Microporous Membranes. Ind. Eng. Chem. Fundam. 1971, 10, 113. (15) Kozinski, A. A.; Lightfoot, E. N. Proteins Ultrafiltration: A General Example of Boundary Layer Filtration. AIChE J. 1972, 18, 1030. (16) Shen, J. J. S; Probstein, R. F. Ultrafilatration of Macromolecular Solutions. Ind. Eng. Chem. Fundam. 1977, 16, 459. (17) Doshi, M. R.; Dewan, A. K.; Gill, W. N. The Effect of Concentration Dependent Viscosity and Diffusivity on Concentration Polarization in Reverse Osmosis Systems. AIChE Symp. Ser. 1971, 68, 323. (18) Gill, W. N.; Wiley, D. E.; Fell, C. J. D.; Fane, A. G. Effect of Viscosity on Concentration Polarization in Ultrafiltration. AIChE J. 1988, 34, 1563. (19) Wilke, C. R.; Chang, P. Correlation of Diffusion Coefficients in Dilute Solutions. AIChE J. 1955, 1, 264. (20) Ma, S.; Kassinos, S.; Fatta, D. Assessing the Accuracy of Wall Concentration Estimation Based on Averaged Permeate Velocity in SpacerFilled RO Membrane Systems. Ind. Eng. Chem. Res. 2006, 45, 8134. (21) Wiley, D. E.; Fletcher, D. F. Techniques for Computational Fluid Dynamics Modeling of Flow in Membrane Channels. J. Membr. Sci. 2003, 211, 127. (22) Geraldes, V.; Semiao, V.; de Pinho, M. N. Flow and Mass Transfer Modelling of Nanofiltration. J. Membr. Sci. 2001, 191, 109. (23) Geraldes, V.; Semiao, V.; de Pinho, M. N. The Effect on Mass Transfer of Momentum and Concentration Boundary Layers at the Entrance Region of a Slit with a Nanofiltration Membrane Wall. Chem. Eng. Sci. 2002, 57, 735. (24) Aimar, P.; Sanchez, V. A Novel Approach to Limiting Phenomena during Ultrafiltration of Macromolecules, Ind. Eng. Chem. Fundam. 1986, 25, 789. (25) Aimar, P.; Field, R. Limiting Flux in Membrane Separations: A Model Based on the Viscosity Dependency of the Mass Transfer Coefficient. Chem. Eng. Sci. 1992, 47, 579. (26) Frazer, J. C. W.; Myrick, R. T. The Osmotic Pressure of Sucrose Solutions at 30°. J. Am. Chem. Soc. 1916, 38, 1907. (27) Sourirajan, S. ReVerse Osmosis; Logos Press: London, 1970. (28) Ma, S.; Song, L.; Ong, S. L.; Ng, W. J. A 2D Streamline Upwind Petrov/Galerkin Finite Element Model for Concentration Polarization in Spiral Wound Reverse Osmosis Modules. J. Membr. Sci. 2004, 244, 129. (29) Ma, S.; Song, L. Numerical Study on Permeate Flux Enhancement by Spacers in a Crossflow Reverse Osmosis Channel. J. Membr. Sci. 2006, 284, 102. (30) Song, L.; Ma, S. Numerical Studies of the Impact of Spacer Geometry on Concentration Polarization in Spiral Wound Membrane Modules. Ind. Eng. Chem. Res. 2005, 44, 7638. (31) De, S.; Bhattacharya, P. K. Predication of Mass Transfer Coefficient with Suction in the Applications of Reverse Osmosis and Ultrafiltration. J. Membr. Sci. 1997, 128, 119. (32) De, S.; Bhattacharya, P. K. Mass Transfer Coefficient with Suction Including Property Variations in Applications of Cross-Flow Ultrafiltration. Sep. Purif. Technol. 1999, 16, 61. (33) Michaels, A. S. New Separation Technique for the CPI. Chem. Eng. Prog. 1968, 64, 31. (34) Schock, G.; Miquel, A. Mass Transfer and Pressure Loss in Spiral Wound Modules. Desalination 1987, 64, 339. (35) Zydney, A. L. Stagnant Film Model for Concentration Polarization in Membrane Systems. J. Membr. Sci. 1997, 130, 275. (36) Berman, A. S. Laminar Flow in Channels with Porous Walls. J. Appl. Phys. 1953, 24, 1232. (37) Bhattacharjee, S.; Kim, A. S.; Elemelech, M. Concentration Polarization of Interacting Solute Particles in Crossflow Membrane Filtration. J. Colloid Interface Sci. 1999, 212, 81.

ReceiVed for reView October 16, 2007 ReVised manuscript receiVed November 27, 2007 Accepted November 30, 2007 IE0713893

Assessing the Impact of Concentration-Dependent Fluid Properties on

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