A Correlation for the Collector Efficiency of Brownian Particles in

May 11, 2009 - BP America, Inc. Houston, Texas 77079, and Geography and Environmental Engineering, Johns Hopkins University, Baltimore, Maryland, ...
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Environ. Sci. Technol. 2009, 43, 4419–4424

A Correlation for the Collector Efficiency of Brownian Particles in Clean-Bed Filtration in Sphere Packings by a Lattice-Boltzmann Method W E I L O N G * ,† A N D M A R K U S H I L P E R T ‡ BP America, Inc. Houston, Texas 77079, and Geography and Environmental Engineering, Johns Hopkins University, Baltimore, Maryland, 21218

Received August 28, 2008. Revised manuscript received April 20, 2009. Accepted April 22, 2009.

cout ) cine-λ0L

(1)

where cout is the particle concentration at the filter outlet, cin is the particle concentration at the filter inlet, L is the filter length, and λ0 is the clean-bed filter coefficient. Colloid filtration theory allows for the quantification of λ0. Due to the complex geometry of filters, simple unit-cell models have been developed to explain λ0. These models represent a granular filter by a single collector, for which a single-collector efficiency η is determined. Because of the small Darcy velocities U used in water and wastewater treatment and also typical of natural groundwater flow, the models assume Stokes flow, i.e., they assume the flow to be laminar and inertial effects to be negligible. The singlecollector efficiency can be expressed as the product of the single-collector efficiency for the case where the collectorparticle interactions are favorable, η0, and the sticking efficiency R (5): η ) Rη0

In this paper, we develop a new correlation for the cleanbed filter coefficient (λ0) for Brownian particles, for which diffusion is the main deposition mechanism. The correlation is based on numerical Lattice-Boltzmann (LB) simulations in random packings of spheres of uniform diameter. We use LB methods to solve the Navier-Stokes equation for flow and then the advection-diffusion equation for particle transport. We determine a correlation for an “equivalent” single-collector diffusion efficiency, ηD, so that we can compare our predictions to “true” single-collector correlations stemming from unit-cell modeling approaches. We compared our new correlation to experiments on the filtration of latex particles. For small particle diameters, 50 nm < dp < 300 nm, when gravity and interception are negligible, our correlation for ηD predicts measurements better than unit-cell correlations, which overestimate ηD. The good agreement suggests that the representation of three-dimensional transport pathways in porous media plays an important role when modeling transport and deposition of Brownian particles. To model larger particles, for which gravity and interception are important too, we build a correlation for the overall singlecollector efficiency η0 by adding corresponding ηG and ηI terms from unit-cell correlations to our ηD model. The resulting correlation predicts experiments with latex particles of dp > 300 nm well.

Introduction During filtration, particles suspended in a liquid are removed from a feed stream, because the particles collide with and deposit on the surfaces of the collectors that make up the porous filter. Concern about the occurrence and transport of potentially harmful colloids or pathogenic microorganisms in natural subsurface systems highlights the need to extend our knowledge about the filtration of Brownian particles (1-3). Usage of deep-bed filtration in water and wastewater treatment also requires a mature understanding of the process of filtration. During the initial filtration period, a filter is largely devoid of deposited particles. Then the suspended particle concentration decreases exponentially with the filter depth during unidirectional deep bed filtration, as originally suggested by Iwasaki (4): * Corresponding author e-mail: [email protected]. † BP America, Inc. ‡ Johns Hopkins University. 10.1021/es8024275 CCC: $40.75

Published on Web 05/11/2009

 2009 American Chemical Society

(2)

Recent work (6, 7) has, however, suggested that η0 and R are not necessarily independent of each other. Yao et al. (8) developed a correlation for η0 (see Table 1) by representing a packing of spheres by an isolated single collector thereby neglecting the hydrodynamic effects of neighboring collectors. Yao et al. explain η0 in terms of three transport mechanisms: η0 ) ηD + ηI + ηG

(3)

where ηD is the diffusion efficiency, ηI is the interception efficiency, and ηG is the sedimentation efficiency. Later, Rajagopalan and Tien (9) (RT) developed a correlation for η0 (see Table 1) by using a unit-cell approach based on the Happel sphere-in-cell model (10), which represents the flow field in a porous medium more realistically. Then Yoshimura (11) developed a correlation for η0 (see Table 1) that is based on the Kuwabara (12) unit cell, which has the same geometry as Happel’s model but uses different flow boundary conditions. Using Happel’s model, Tufenkji and Elimelech (13), (TE) then developed a correlation for η0 (see Table 1) which better describes how van der Waals forces, the lubrication effect due to the thin water film that forms when a particle approaches a collector surface (14), and anisotropic diffusion close to the collector surface (15) affect the deposition of Brownian particles. The TE model, however, does not account for the hydrodynamic effects of suspended particle rotation, as the RT model does. In all unit-cell approaches for modeling filtration, η is eventually related to λ0 by a filter coefficient-collector efficiency relationship, λ0(η), which depends on the particular single-collector model ((18), Sec. 6.1). For the models presented in Table 1, this relationship is λ0 )

31 - ε Rη0 2 dc

(4)

where dc is the diameter of the spherical collector, and ε is the porosity of the filter. Sphere packings have a much more complicated geometry than unit-cell models. Furthermore, it is difficult to estimate the error that occurs when η0 correlations derived from simple conceptual models are used to predict η0 for filters with a more complex pore geometry, such as random sphere packings. Indeed, the correlations listed in Table 1 do not provide such error estimates. VOL. 43, NO. 12, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 1. Some Unit-Cell Correlations for the Single-Collector Efficiency η0 (the Dimensionless Numbers Are Defined in Table 2) reference

single-collector efficiency η0

Yao (16) Yao et al. (8) Rajagopalan and Tien (9)a

-2/3 4.04N Pe + (3NR2)/2 + NG -2/3 4A1/3 + (3NR2)/2 + NG s N Pe -2/3 0.125 1.875 4A1/3 N + AsN Lo NR + s Pe -0.4 0.0038AsN 1.2 G NR 1/3 -2/3 1 - m 2(1 - m) m 4.0Kw N Pe + 1.5MKw NR N Lo + 1.5WKw2 - nN R2(1 - n)N nG - 0.081 -0.715 0.052 2.4A1/3 N Pe N vdw + s NR 1.11 0.053 0.55AsN R1.675N A0.125 + 0.22N R-0.24N G N vdw

Yoshimura (11)b Tufenkji and Elimelech (13) a

Note that a portion of the original RT correlation was corrected by Rajagopalan et al. (17). Tien and Ramarao (18) show the entire corrected correlation. Here, we show the Logan et al. version of the corrected RT correlation for use together with eq 4. b Refer to Yoshimura (11) for M, m, and n values.

algebraic equations, which can be chosen such that realistic physical behavior is mimicked. Recently, these methods were used to study filtration. Biggs et al. (27) used a LG method to study the effects of particle deposition on the flow field in two-dimensional porous media. Przekop et al. (28) used a LB method to study the fractal geometry of deposited particulate matter in fibrous filters. This paper will investigate the clean-bed filter coefficient of Brownian particles, λ0, in sphere packings under favorable chemical condition (R ) 1). Here we define Brownian particles as particles the diameter dp of which is so small that diffusion is the main deposition mechanism. Our objectives are to develop an LB method for filtration in sphere packings, and to derive a new correlation for λ0 in random packings of spheres with uniform size dc based on the LB method. Unlike unit-cell approaches, we do not simplify the geometry of the filter but simulate transport in entire sphere packings. To allow for easy comparison to diffusion efficiency ηD and single-collector efficiency η0 from unit-cell approaches, which are widely used in practice, we will derive correlations for an “equivalent/average” diffusion efficiency ηD and an “equivalent/average” single-collector efficiency η0, which is related to λ0 through eq 4.

Simulation of Filtration in Sphere Packings

FIGURE 1. Illustration of a column experiment to measure η0. The effluent concentration c approaches the quasi steady-state value cout, as particles are injected at a concentration cin. Hence, theoretical approaches have been developed that model a large number of collectors. Gal et al. (20), for example, represented a granular filter by a regular sphere packing and predicted the total filtration efficiency for aerosol filtration, where particle diffusion was assumed to be negligible. Later, Cushing and Lawler (21) used the same packing structure to investigate aqueous depth filtration. In this study, they derived a correlation for an “equivalent” single-collector efficiency η; however, their modeling approach did not account for Brownian diffusion. Also pore-network modeling approaches (22, 23) have been employed to investigate depth filtration. Simulations in both regular sphere packings and the aforementioned pore networks, however, do not accurately capture the complex flow and particle transport patterns that occur in random granular filters. Lattice-Boltzmann (LB) methods (24), as well as the preceding Lattice-gas (LG) methods (25), are numerical techniques that are particularly well suited to model a broad range of transport phenomena in complex pore geometries at the pore scale (26, 24). LB and LG methods represent fluid and solute particles by quasi-particles that move with a discrete set of velocities along a numerical lattice, at the nodes of which they collide. The collision rules are given by simple

Lattice-Boltzmann Modeling Approach. We implemented an LB method (29) to solve the Navier-Stokes equation. The pore-scale velocity b v obeys the no-slip boundary condition, b v ) 0, at the collector surfaces. The Reynolds numbers NRe of the flows that we simulated were much smaller than 1, because water filtration is typically performed under conditions of Stokes flow (18). The simulated pore-scale velocity field b v is then used to solve the advection-diffusion equation for particle transport (30),

( )

b ∂c b ·(v b ·(D∇ b c) - ∇ b · DF +∇ bc) ) ∇ c ∂t kT

where k is the Boltzmann constant, T is absolute temperature, b F accounts for external forces acting on the particles (e.g., gravity and DLVO forces), and D is the particle diffusion coefficient. The latter can be estimated from the EinsteinStokes equation, D ) kT/(3Fπf νdp), where ν is kinematic viscosity, Ff is fluid density, and dp is the diameter of the spherical particles. The condition R ) 1 can be accounted for by the perfect-sink boundary condition, c ) 0 on the collector surfaces if dp , dc. To solve eq 5 we generalized an LB method (31) for solving the advection-diffusion equation to account for an external force due to gravity that acts on the particles. Supporting Information A contains more details of our LB method. As shown in Supporting Information B, we validated our LB method for filtration by comparing ηD from simulations to an analytical solution (32, 33) for Happel’s sphere-in-cell model. Numerical Simulation Setup. Our numerical experiments used to determine λ0 in sphere packings mimic real column experiments, in which a liquid containing suspended particles

TABLE 2. Summary of Dimensionless Parameters Used in Table 1 (Variables are Defined in the Text)

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parameter

definition

explanation

NR NPe NLo NG NA Nvdw As Kw

dp/dc Udc/D 4A/(9Fπfνd2pU) d2p(Fp - Ff)g/(18FfνU) A/(3Fπfνd2pU) A/kT 2(1 - p5h)/(2 - 3ph + 3p5h - 2p6h) 5(1 - p3h)/(5 - 9ph + 5p3h - p6h)

interception number Peclet number London number, where A is Hamaker constant gravitation number attraction number van der Waals number parameter for Happel’s sphere-in-cell model parameter for Kuwabara model

ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 43, NO. 12, 2009

(5)

FIGURE 2. Particle concentration during filtration in a packing of spheres at selected times. The color represents the particle concentration and the black represents the solid phase. Simulation parameters: U ) 0.35 µm/s, grain diameter dc ) 218 µm, ε ) 0.358, L ) 1.07 mm, dp ) 1 µm. with concentration cin is injected into a column (see Figure 1). In such experiments, one seeks to establish conditions such that R ) 1, for example, by using oppositely charged particles and collectors at a high ionic strength (2). The effluent concentration of particles in clean-bed filtration is then continuously measured, and η0 can be calculated from eqs 1 and 4: η0 ) -

( )

cout dc 2 ln 3 (1 - ε)L cin

(6)

where cout is the effluent concentration at quasi steady-state. For real experiments, we use the term “quasi steady-state” to make clear that the filter gradually starts fouling, even during clean-bed filtration. Our numerical model, which does not simulate fouling, may approximate, however, steadystate conditions. Figure 2 illustrates a numerical experiment. We first generate a packing of nonoverlapping spheres using a numerical code developed by Yang et al. (34). Such packings represent real porous media quite well (35, 36) and can match a prescribed porosity and sphere-size distribution. We apply the LB method to simulate the fluid velocity field b v, which is then used in the transport simulation. Initially no suspended particles are present in the domain. We apply the Dirichlet boundary condition at the inlet, c(x,y,z ) 0,t) ) cin, and the Neumann boundary condition at the outlet, ∂c(x,y,z ) L,t)/∂z ) 0. The perfect-sink boundary condition is implemented at the collector surfaces to model R ) 1 (13). At steady-state, the concentration at the outlet, cout, is retrieved from the simulations, and a collector efficiency η0 is calculated from eq 6. Note that our collector efficiency is an average/equivalent one, because η0 is derived from numerical simulations in packings that contain a large number of collectors. This differs from unit-cell approaches that consider filtration in a domain containing exactly one collector. Effects of Packing Randomness on η0. We can simulate only a small portion of a column that would typically be used in real filtration experiments due to the computational cost. Since random sphere packings (even if grain size is uniform) are heterogeneous and random at the grain scale, a numerical domain that is too small could result in a simulated η0 value, the stochastic error of which is too big to draw meaningful conclusions. To quantify the effects of packing randomness on η0, we created five statistically identical sphere packings with a porosity ε ) 0.358 that contain about 700 spherical collector particles. The size of the cubic domain was 0.5 mm (discretized by 1003 voxels). We then simulated filtration in the five different domains and calculated η0 as described before. Figure 3 shows how η0 varies in the different domains for selected dp values. For

FIGURE 3. Packing randomness effect on η0. The symbols stand for the simulation results in five statistically identical sphere packings. The solid line and error bars represent the means and 90% confidence intervals, respectively. Simulation parameters: U ) 0.368 µm/s, dc ) 0.06 mm, ε ) 0.368, L ) 2 mm, and ∆x ) 5 µm. a given dp and at a 90% confidence level, the confidence j 0(100 ( 15)% where η j 0 is the mean value. interval of η0 is η A New Correlation for ηD. To establish the need for an improved correlation for ηD let us compare predictions of the RT and TE correlations to numerical simulations, for which the effects of gravity and interception on deposition are negligible. We generated a packing of spheres of uniform size and then modeled fluid flow and particle transport in the packing. Figure 4a shows that the η0 ≈ ηD values calculated from the simulations are lesser than the ones predicted from the RT and TE correlations, especially for small particle diameters (dp). As dp increases, the difference between the numerical experiments and the various correlations decreases. Hence in this specific example, the RT and TE correlations predict ηD ≈ η0 less accurately, particularly for smaller particle sizes dp. We performed a set of numerical breakthrough experiments (85 all together) in 10 uniform sphere packings that differed in porosity ε to derive a correlation for ηD. For each packing, we varied the flow velocity U and particle size dp. Table 3 presents the parameters used in the simulations. To allow for comparison to single-collector efficiencies stemming from unit-cell approaches, we determined an “equivalent” single-collector efficiency η0 from eq 6. We assumed the following functional form for this equivalent single-collector efficiency: a2 a3 NR η0 ≈ ηD(NPe, NR, ε) ) a1f (ε)NPe

(7)

where a1, a2, and a3 are constants which need to be determined and f(ε) is a function that depends on porosity. Note that we assume the effects of interception and gravity to be negligible for the small particle sizes dp investigated; VOL. 43, NO. 12, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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deposition of Brownian particles onto collectors. In a statistical analysis, we showed that we can indeed drop ηI and ηG in eq 7 as well as the potential predictor variables NG and NR. The adjusted R2 value for the correlation without the ηI and ηG terms is 0.87. When we accounted for ηI, ηG, NG, and NR, we obtained R2 ) 0.86. The lower value justifies our approach. Building a Correlation for η0. Yao et al. (8) suggested in eq 3 that the collector efficiencies by the three transport mechanisms are additive. Also employing this assumption, we use the interception and sedimentation efficiencies from unit-cell correlations to develop a correlation for the overall collector efficiency η0. Using the ηI and ηG terms from the TE correlation (see Table 1), for example, yields η0 ) ηD + ηI + ηG ) (1 - ε)3 -0.65(0.023 0.19(0.03 NPe NR + ε2 0.053 0.55AsNR1.675NA0.125 + 0.22NR-0.24NG1.11Nvdw (9)

(15.56 ( 0.21)

FIGURE 4. (a) Comparison of η0 among the RT and TE correlations, our new correlation, and numerical experiments in a packing of spheres of uniform size. Simulation parameters: U ) 1.29 µm/s, dc ) 0.04 mm, ε ) 0.385, and L ) 0.2 mm. (b) Comparison between the η0 ≈ ηD values from all of our numerical breakthrough experiments and the correlation that has been derived from these experiments. The symbols stand for the different packings used in the simulations.

TABLE 3. Summary of Parameters Used in Numerical Simulations (Values of T, A, ν, and Gp Adopted from Tufenkji and Elimelech (13)) parameter

range

dp ε U T k A ν dc Fp Ff NRe

0.1-1 µm 0.30-0.42 0.1-10 µm/s 298 K 1.38 × 10- 23 J/K 1.0 × 10- 20 J 0.8 × 10- 6 m2/s 20-450 µm 1.05 g/cm3 1.0 g/cm3 10- 5 to 10- 3

hence, the ηI and ηG terms are not present. The power law form is consistent with the single-collector efficiency models listed in Table 1. Also note that our correlation does not depend on Nvdw, because we do not account for the lubrication effect and van der Waals forces. We used f(ε) ) (1 - ε)3/ε2 from the Carman-Kozeny relation (37), because it describes the numerical breakthrough experiments better than f(ε) ) As1/3, which is used in the RT and TE correlations. Moreover, there is no theoretical justification for using Happel’s As function in random sphere packings. To derive the correlation for ηD, we performed a multivariate regression analysis on the λ0 values inferred from the set of numerical column experiments. We obtained ηD ) (15.56 ( 0.21)

(1 - ε)3 -0.65(0.023 0.19(0.03 NPe NR ε2

(8)

0.46 , whereas it is Note that our correlation gives ηD ∼ dp 0.6 0.80 in the RT and ηD ∼ din the TE correlation. ηD ∼ dp p We attribute this difference to the effects that neighboring collectors have on the hydrodynamics and hence the

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Figure 4a shows that the new correlation eq 9 agrees better with numerical column breakthrough experiments than the RT and TE correlations. Note that the results from the numerical experiments are presented together with a 15% error bar that represents the uncertainty in ηD due to the randomness of the pore structure. Figure 4b summarizes the results from the 85 numerical column breakthrough experiments that we performed for different values of ε, U, dc, and dp. The η0 ≈ ηD values from the numerical experiments agree well with the correlation 9, indicating the appropriateness of the power law form for ηD that we used in eq 8. If we increase the domain size of the simulations used to determine the correlation for η0, the mean value of η0 for statistically identical representations of a porous medium can be expected to remain about constant; however, the 15% variation due to packing randomness would decrease. Permeability in random sphere packings, for example, exhibits such a size dependence (35). A smaller variation in the simulated η0 values would result in a higher R2 value for the correlation. Comparison of Our New Correlations for ηD and η0 to Experiments. We compared eqs 8 and 9 for ηD and η0 as well as the RT, Yoshimura, and TE correlations with experimental data chosen from publications on the filtration of uniform Brownian latex particles in uniform sphere packings (8, 2, 3). These papers investigate the dependence of η0 on particle size dp and provide all the parameters needed in the various correlations. Figure 5 shows that overall our correlations for ηD and η0 agree better with the experiments than the RT, Yoshimura, and TE correlations. For small particle sizes, dp < 300 nm, the experimental data can solely be described by our correlation for ηD. Gravity and interception have negligible impact on particle deposition. This is consistent with the correlations for η0 shown in Table 1, which predict that ηD . ηG and ηD . ηI as dpf0. Therefore, our ηD is nearly indistinguishable from our η0 for dp < 300 nm. To evaluate whether our correlation for ηD explains the (independently obtained) experimental data, we estimated the uncertainty in the prediction from the correlation (38). To that end, we determined the 95% confidence interval for the mean response [see ch 12 in (39)]. The error bars in Figure 5 show this estimation error. All error bars overlap with the experimental values. Hence, the correlation predicts the experimental data at a 95% confidence level. Note that we could even decrease the confidence level by accounting for the experimental error; however, the latter data were not available. For larger particle sizes, dp > 300 nm, the experimental data are better described by our correlation for η0, because gravity

FIGURE 5. Comparison of η0 among the RT, Yoshimura, TE correlations, our new correlation, and experimental data on the filtration of latex particles by (a) U ) 0.136 cm/s, dc ) 397 µm, ε ) 0.36, and L ) 14 cm, (b) U ) 0.136 cm/s, dc ) 200 µm, ε ) 0.4, and L ) 20 cm, and (c) U ) 0.377 cm/s, dc ) 200 µm, ε ) 0.4, and L ) 20 cm. and interception become more important. For the dp ) 750 nm particles, ηG and ηI make up about 25% of the total η0. For increasing dp values, our correlation for η0 will eventually converge to the predictions from single-collector correlations, because our η0 uses their ηG and ηI terms which then outweigh ηD.

Discussion We used an LB method to simulate particle transport during filtration. As opposed to unit-cell approaches, our LB method predicts filtration in packings of spheres. By performing a set of numerical breakthrough experiments, we derived the correlation for the diffusion efficiency ηD given by eq 8. An overall correlation for η0 can be obtained by employing the ηI and ηG terms from unit-cell correlations. We compared our new correlations for ηD and η0 first to experimental data on the filtration of latex particles in packings of spheres of uniform size, and second to η0 from single-collector correlations. Figure 5 shows that for the smaller particle sizes, dp < 300 nm, for which gravity and interception are negligible, our correlation for ηD predicts measured values better than unit-cell correlations. The better agreement suggests that it is important to model the 3D pore geometry and the transport pathways therein accurately when deriving a correlation for ηD. Our correlation for η0 also predicts measurements well for larger sizes of latex particles, dp > 300 nm, for which ηG and ηI are not negligible but still smaller than ηD. Comparison to more experimental data is necessary to judge whether our correlation for η0 yields better predictions than unit-cell correlations in that transport regime. The difference between our model and

the unit-cell correlation used to augment our ηD correlation by the ηG and ηI terms vanishes as dp further increases, because then the ηD term looses significance. Future research should, however, develop correlations for ηG and ηI that are based on simulations in sphere packings instead of unit-cell considerations. The critical particle size, below which ηG and ηI are negligible, depends on a variety of factors including Darcy velocity U and particle density Fp. Many manufactured nanoparticles, e.g. fullerenes, possess a larger density than latex particles. The unitcell correlations in Table 1 show that the critical particle size is smaller for fullerenes than for latex particles. We also provided confidence intervals for the predictions of our correlation for ηD. The experimental values lie within these confidence intervals. As such, we can say that our new correlation predicts the experimental data investigated. Such a conclusion would be much harder to draw when comparing unit-cell correlations for ηD to filtration experiments in sphere packings, because the significant simplification of the pore geometry and flow field introduces a systematic error in ηD that is hard to quantify. For our correlation for η0, which needs to be used for larger particle sizes (dp) when gravity and interception are important, we were, however, not able to provide confidence intervals for η0, because our correlation equation uses ηI and ηG terms from unit-cell correlations. For nanoparticle filtration and virus transport, where particle size dp is on the order of nanometers, the collector efficiency η0 from our correlation and unit-cell correlations differ by up to 100% as can be seen from Figures 4a and 5. Comparison to experimental data (see Figure 5) also suggests that unit-cell correlations systematically overestimate the filter efficiency for nanoparticles; the particles may travel further than expected. Tufenkji and Elimelech (13) (not Rajagopalan and Tien (9)) account for the lubrication effect and anisotropic dispersion when deriving a correlation for ηD. Our simulations, however, do not account for these effects. Both of these effects are represented by Nvdw in the TE correlation. By comparing the small difference between the predictions from the RT and TE correlations to the large difference between our numerical experiments in sphere packings and the RT and TE correlations (e.g., see Figure 4a), one can conclude that the accuracy with which flow and transport in a sphere packing is modeled affects η0 more than accounting or not accounting for the lubrication effect and anisotropic diffusion. This view is supported by the relatively small role of Nvdw in the TE correlation (due to the small exponent of Nvdw, 0.052). Moreover, the closer agreement between representative experimental data (see Figure 5) and our new correlation suggests that realistically representing the 3D transport pathways in sphere packings might be more important than accounting for the lubrication effect and anisotropic diffusion. It is possible, however, that for nonlatex particles, which may possess a significantly reduced Hamaker constant A, DLVO forces and thus Nvdw could play a greater role. Like the TE correlation, our numerical model does not account for suspended-particle rotation and its effect on ηD. This might be a sensible simplification, because this effect decreases with decreasing particle size (18). It would be difficult to account for this effect in an LB method, for example, by using the approach developed by Nguyen and Ladd (40), because their approach would require a spatial discretization smaller than the suspended-particle size. This in turn would limit the size of the computational domain to only a few collector particles, which would not adequately represent the geometry and topology of a random sphere packing. Even though our new correlation agrees well with experimental data obtained for filtration velocities on the order of cm/s, the velocities in our numerical simulations are on the order of µm/s because of numerical instability VOL. 43, NO. 12, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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that occurs for higher Peclet numbers NPe. Nonetheless, we were able to predict ηD for the various experiments with our new correlation. For that purpose, we used the correlation in NPe regimes which have not been accessed in the numerical experiments used to derive our correlation; however, we estimated confidence intervals for the predicted ηD values. It would be good to replace our single-relaxation time LB method by a multiple-relaxation time method (41), which would allow the simulations of higher filtration velocities. Such simulations would in turn allow to reduce the confidence intervals of predicted ηD values. An improved correlation for η0 can be widely used in engineered and natural systems, as elaborated by Tufenkji and Elimelech (13). In deep-bed filtration under unfavorable chemical conditions where R < 1, column experiments can be used to determine R ) -2dc 1n(cout/cin)/(3(1–ε)Lηo) if η0 is estimated from a correlation (8). Our new correlation for η0 can thus be used to estimate R more accurately. Both the filter coefficient λ and the kinetic attachment rate kd ) 3(1–ε)Rηo|v|/ (2dc) can also be estimated more accurately from an improved correlation for η0.

Acknowledgments We thank Dr. Haiou Huang and Professor Charles R. O’Melia for stimulating discussions, and Professor Hugh Ellis for supercomputing resources at Johns Hopkins University.

Supporting Information Available Details of our Lattice-Boltzmann method and validation of our model. This material is available free of charge via the Internet at http://pubs.acs.org.

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