Model Simulations of Particle Aggregation Effect on Colloid Exchange

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Model Simulations of Particle Aggregation Effect on Colloid Exchange between Streams and Streambeds Trachu Areepitak and Jianhong Ren* Department of Environmental Engineering, Texas A&M University—Kingsville, Kingsville, Texas 78363, United States

bS Supporting Information ABSTRACT: Colloids found in natural streams have large reactive surface areas, which makes them significant absorbents and carriers for pollutants. Streamsubsurface exchange plays a critical role in regulating the transport of colloids and contaminants in natural streams. Previous process-based multiphase exchange models were developed without consideration of colloidcolloid interaction. However, many studies have indicated that aggregation is a significant process and needs to be considered in stream process analysis. Herein, a new colloid exchange model was developed by including particle aggregation in addition to colloid settling and filtration. Selfpreserving size distribution concepts and classical aggregation theory were employed to model the aggregation process. Model simulations indicate that under conditions of low filtration and high degree of particleparticle interaction, aggregation could either decrease or increase the amount of colloids retained in streambeds, depending on the initial particle size. Thus, two possible cases may occur including enhanced colloid deposition and facilitated colloid transport. Also, when the aggregation rate is high and filtration increases, more particles are retained by bed sediments due to filtration, and fewer are aggregated, which reduces the extent of aggregation effect on colloid deposition. The work presented here will contribute to a better understanding and prediction of colloid transport phenomena in natural streams.

1. INTRODUCTION Colloids such as river-borne particles (e.g., iron oxides), sediment colloids (e.g., kaolinite), and biocolloids (e.g., viruses and microorganisms) are commonly present in natural streams.1 They are particles with effective diameters less than 10 μm.1 Due to their small sizes, colloids have large specific surface area, which makes them important absorbents in natural waters. Therefore, colloids have a great potential to affect the transport of many pollutants in river systems.13 Hyporheic exchange, the exchange of solute between the stream and the shallow subsurface caused by streambed topography, is a very important process in regulating the transport of colloids, contaminants, and ecologically relevant substances in natural water systems.38 The earliest observation of hyporheic exchange can be dated back to Simons et al.9 and Einstein,10 who observed the penetration of fine particles into the bed sediments in alluvial channels or in laboratory flumes. The convective flow pattern in high and low pressure regions of the streambed induced by streamwater flowing over obstructions on the bed surface was systematically studied and termed advective pumping under stationary bedform conditions and turnover when the bedform moves.4,6,11,12 The fundamental knowledge of hyporheic exchange in regulating conservative solute, colloids, and reactive substances transport has advanced significantly during past decade.3,58 r 2011 American Chemical Society

Packman et al.7 modeled the colloid exchange in a finite-depth hyporheic zone by applying the advective pumping and residence time function approach of Elliot and Brooks.6 Colloid settling velocity was added in the solute velocity equations to simulate particle paths and colloid filtration was used to calculate the colloid removal in the streambed. In this model, both settling velocity and filtration coefficient are assumed constant throughout the transport and colloidcolloid interactions such as aggregation process were not considered.7 Aggregation, the process of producing large aggregates by collisions induced by inter particle motion,13 is an important process occurring among colloids. The presence and importance of aggregation have been documented in studies of natural streams, groundwater infiltration/transport, and water and wastewater treatment, especially in the nanotechnology field.1,1416 Researchers14,17,18 have found that aggregation causes rapid deposition of fine particles, suspended sediments, or flocs, in rivers and may change the sediment transport properties in terms of particle size and floc density. In groundwater, different types of particles exist and their concentrations may range from very dilute (below 1010 particles per liter) to high (up to 1015 particles Received: February 20, 2011 Accepted: May 18, 2011 Revised: May 15, 2011 Published: May 31, 2011 5614

dx.doi.org/10.1021/es200586v | Environ. Sci. Technol. 2011, 45, 5614–5621

Environmental Science & Technology

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travel through and out of the streambed. Thus, five types of particle phases are considered in the new model including mobile primary colloids (colloids without aggregation), mobile aggregates, stationary bed sediments, primary colloids attached to bed sediments, and aggregates attached to bed sediments (Figure 1). Advective Pumping. Stream flow over bedforms may produce a dynamic head variation at the bed surface, which pumps flow into and out of the streambed. This mass transfer between the stream and the subsurface was described using advective pumping theory (see Text S1 in the Supporting Information (SI)).6 Colloid Settling. Since pore water velocity during hyporheic exchange is usually very slow, Stoke’s law can be used to model the settling velocity of spherical particles.7 The colloid path can then be calculated by the following:7 u ¼ kKhm cosðkxÞ½tanhðkdb ÞsinhðkyÞ þ coshðkyÞ þ uu ð2Þ Figure 1. Processes and particle phases considered in the new colloid exchange model. 2,19

per liter). These particles can be initially very small, but may grow and build up agglomerates when the hydrochemical subsurface environment changes such as during artificial groundwater recharge, when ionic strength is increased or ion balance is shifted from one dominated by Naþ to one dominated by Ca2þ.2,20,21 When aggregates formed are large enough, they may be immobilized; however, if the aggregates formed are sufficiently small with respect to the pore diameters, they may still be transported.2,19,21 Thus, it is important to consider aggregation in colloid transport models and fundamentally understand its effect on colloid transport, especially in the hyporheic zone. In this work, a colloid exchange model in consideration of aggregation, advective pumping induced by a stationary dune shaped bedform, particle filtration, and settling was developed. The effect of aggregation on colloid exchange under different physicochemical conditions was evaluated by conducting various model simulations.

2. THEORY In natural streams, when the in-stream concentration variations in the longitudinal direction are not important, the advection dispersion transport equation modified by Bencala and Walter22 can be written as follows: DC ¼ BðtÞ Dt

v ¼ kKhm sinðkxÞ½tanhðkdb ÞcoshðkyÞ þ sinhðkyÞ vs θ ð3Þ where vs is the Stokes settling velocity and θ is the sediments porosity. All other parameters are defined in SI Text S1. Colloid Aggregation. The classical aggregation theory proposed by Smoluchowski13 assumes a second order aggregation rate process, steady-state conditions, and irreversible aggregation (no break-up after aggregation). For aggregation occurring in solution containing initially identical spherical particles, the change of total number concentration of all particles is given as1,13 dnT ¼ ka n2T dt

where nT is the total number concentration, and ka is the aggregation rate. Self-Preserving Size Distribution. Friedlander23 discovered that after a certain period of time, particle size distribution (PSD) of aerosols reaches a self-preserving condition and is independent of its initial condition, that is, PSD only changes in scale, but keeps the same distribution shape. Thus, one can obtain self-preserving size distributions at different times by applying a similarity transformation. This similarity transformation approach can be used to predict the changes of PSD in coagulation processes.23,24 The similarity transformation proposed by Friedlander23 is nðvp , tÞ ¼

nT 4=3 ψðηÞ φ1=3

1=3 nT η¼ dp φ

ð1Þ

where C is the in-stream concentration, t is time, and B(t) is the sinks and sources term caused by processes such as hyporheic exchange. For colloids, the processes that contribute to term B(t) include hydraulic exchange and nonlinear processes such as colloid colloid interaction and colloidsediment interaction. The specific processes considered here include advective pumping, colloid settling, filtration, and aggregation. When colloids are discharged into a stream, they may travel in, through and out of the streambed via advective pumping (Figure 1). During this transport, colloids may collide with other colloids and produce large aggregates. Their travel paths may deviate from the solute paths due to settling. Meanwhile, they may be removed by sediments due to filtration. The deposited colloids/aggregates will remain in the bed while mobile colloids/aggregates may continue to

ð4Þ

ð5Þ

ð6Þ

where n(vp,t) is the number concentration for each size class, η is the dimensionless volume function, ψ is the similarity function, dp is the effective particle diameter, φ is the volume fraction, and vp is the volume of spherical particle, vp = π d3p/6. The applicability of this self-preserving size distribution theory to hydrosol aggregation in Brownian motion and laminar shear flow was proved both experimentally and theoretically by Swift and Friedlander and Suzuku et al.25,26 The theory was also found mathematically consistent with the Smoluchowski’s theory of aggregation although it was not derived from the Smoluchowski theory.25 Thus, this theory was applied here to model the size distribution evolution of aquatic colloids during aggregation. To estimate the time-lag tsp, the time required to reach the selfpreserving particle size distribution, Friedlander proposed the 5615

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Environmental Science & Technology following equation:23

tsp ¼ 13

2kb Tn0 3μ

1

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ð7Þ

for initially monodisperse particles, where kb is the Boltzmann’s constant (kb = 1.3805 1023 J/K), n0 is the initial number concentration, T is the temperature, and μ is the dynamic viscosity. Equation 7 was developed for systems with no potential energy barrier;27 however, this assumption might not always be valid for aquatic systems where varying water chemical conditions such as ionic strength often exist and can impose energy barriers between particles, which may lead to underestimation of the time-lag.26 In such cases, the time-lag for aquatic colloids can be estimated using laboratory experiments following the approach of Swift and Friedlander.25 Colloid Filtration. The mobile colloid concentration after a travel path of L in the streambed was calculated by7,28 C ¼ eλf L C0

ð8Þ

where C is the effluent colloid concentration, C0 is the influent colloid concentration, and λf is the filtration coefficient and calculated by28,29 λf ¼

3 ð1 θÞ Rf η0 2 dc

ð9Þ

where dc is the sediment collector diameter, Rf is the collision efficiency, and η0 is the single-collector contact efficiency. The single-collector contact efficiency (η0) can be calculated by29 0:715 0:052 η0 ¼ 2:4AS NR0:081 NPe NvdW 1=3

0:125 0:125 þ 0:55As NR1:675 NPe NvdW

1:11 0:053 1:11 þ 0:475NR1:35 NPe NvdW Ngr

ð10Þ

where As is a porosity-dependent parameter and given as As = (2(1γ5))/(23γ þ 3γ5 2γ6), γ = 1 θ, NR is the aspect ratio, particle diameter over collector diameter (NR = dp/dc), NPe is the Peclet number (NPe = wdc/Di), NvdW is the van der Waals number (NvdW = A/kbT), w is the velocity of fluid in porous media, Di is the diffusion coefficient, and A is the Hamaker constant. Mass Transfer between Streams and Streambeds. To quantify the mass transfer between streams and streambeds, residence time function, R(t, t0, x0), the fraction of tracers that entered bed at time t0 and position x0 which remained in the bed at a later time t, is used.6 The flux weighted average residence time function R is given as follows:6,7 Z 1 λ=2 vðxÞRðτ, xÞdx λ RðτÞ ¼ 0 Z λ=2 ð11Þ 1 vðxÞdx λ 0 where τ is the time that has elapsed since particles entered the bed, τ = t t0. To obtain the accumulated mass transfer or depth of penetration M(t) in the streambed, R is integrated as follows:6,7 Z q t MðtÞ ¼ RðτÞCðt τÞdτ ð12Þ θ 0 where C* is the dimensionless in-stream concentration (C/C0), C is the in-stream colloid concentration, C0 is initial in-stream colloid

concentration, and q is the average flux of tracers entering the bed surface. For a closed system such as a laboratory recirculating flume, correlation between the depth of penetration M(t) and in-stream concentration is given by30 C=C0 ¼

d0 d0 þ MðtÞθ

ð13Þ

where d0 is the effective stream depth (total volume of water in the stream per unit bed surface area). The flux-weighted average residence time function, R, is found numerically using particle tracking. In this method, the colloid suspension is represented as a large number of discrete tracer particles, which are tracked as they propagate into, through, and out of the bed (Figure 1). The solute paths and colloid paths, which define the streamtubes in the streambed, are calculated using interstitial velocities (eqs 2 and 3). Thus, the settling of colloids/aggregates is included in eq 3 for calculating the colloid paths. For each streamtube (see Figure 1), the total number concentrations of particles after aggregation are calculated at the first time step using eq 4 based on the initial total number concentrations. These calculated total number concentrations are then used to obtain the particle size distribution (PSD) at that time step using eqs 5 and 6. The filtration coefficient and singlecollector contact efficiency are calculated using eqs 9 and 10 based on the new PSD. The amount of colloids and aggregates that are attached to bed sediments are then obtained using eq 8. The amount of colloids/aggregates that are not retained in the streambed are transported to the next time step and a new corresponding total number concentration nT is calculated and used as the new initial total number concentration for the aggregation calculation in the second time step (see Figure 1). Similar calculations conducted in the first time step are carried out in the second time step and such calculations are continued for the following time steps until the particle suspensions are transported out of the streamtube. Thus, the codependence of filtration and aggregation is considered in the new model. The residence time function of the colloids is obtained based on the distribution of tracer particles between mobile and immobile phases at each time step. Flux-weighted residence time function R is then calculated using eq 11. The accumulated mass transfer and in-stream particle concentration are obtained by solving eqs 12 and 13 simultaneously. To prevent numerical dispersion, sufficiently small time step (dt) is used following the approach of Ren and Packman,3 dt = 0.01/[3 þ log10(uu þ 0.01)]3. Model Input Parameters. Model input parameters include (i) stream parameters such as average stream velocity, underflow velocity, stream depth, effective stream depth, water density, dynamic viscosity, and temperature; (ii) Streambed and bedform parameters including bed depth, sediment size, average bedform height, hydraulic conductivity, bedform wavelength, and sediment porosity; and (iii) colloidal process parameters such as aggregation rate, primary particle size for each size class, initial number concentration for each size class, density of particle, stable time tst (the time when the aggregation process ceases), and filtration collision efficiency (Table 1). All input parameters are readily obtainable through direct measurements, estimation by conducting simple independent experiments, or from current literature (see details on example model input parameters estimation in SI Text S2). Since filtration and aggregation are modeled using two separate physicochemical parameters (collision 5616



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Table 1. Model Input Parameters Used in the New Colloid Exchange Model input parameters

symbol

units

Stream Parameters average stream velocity

U

cm/s

underflow velocity stream depth

uu d

cm/min cm

effective stream depth

d’

cm

density of water

Fw

kg/m3

dynamic viscosity

μ

kg/ms

temperature

T

K

Streambed and Bedform Parameters bed depth

db

cm

sediment size

dc

cm

average bedform height hydraulic conductivity

H K

cm cm/s

bedform wavelength

λ

cm

sediment porosity

θ

dimensionless

ka

m3/s

Colloidal Process Parameters aggregation rate Boltzmann constant

kb

J/K

density of particle

Fp

kg/m3

gravity initial number concentration for each size class

g n0

m/s2 number of

primary particle size for each size class

dp0

μm

stable time (the time when ka = 0)

tst

hour

filtration collision efficiency

Rf

dimensionless

particles/L

efficiency Rf and aggregation rate ka, respectively), the model can be used to simulate a system that has different physicochemical interactions among colloids (for aggregation) and between colloid and bed sediment (for filtration).

3. RESULTS AND DISCUSSION The first set of simulations conducted were to verify the new model by simulating the flume experiment results presented in Ren and Packman8,31 for the case of no aggregation. The simulation results were also compared with those obtained using the existing colloid exchange model developed by Packman et al.7 The second set of simulations demonstrated the effect of aggregation on modifying the colloid paths in the streambed and colloid exchange between streams and streambeds under various physicochemical conditions occurring in natural streams. The simulation results are presented using dimensionless in-stream colloid concentration (C*= C/C0) and dimensionless time (t*/θ), where t*/θ = k2Khmt/θ. Simulations of Colloid Exchange in the Absence of Aggregation. SI Figure S1 shows the simulations of exchange of

silica, clay, and hematite8,31 with a clean silica sand bed and the comparison with those obtained from Packman’s model.7 These flume experiments8,31 were designed to minimize aggregation by using: low ionic strength conditions (10 mM NaCl), very stable silica colloids obtained from Nissan Chemical Industries, Ltd. (Tokyo), and low particle number concentrations for reactive colloids such as clay and hematite (See SI Figure S1). Thus, aggregation was not expected in these experiments.8,31Parameters

used in the simulations are presented in SI Table S1. Simulations for both residence time functions and in-stream colloid concentrations obtained using the new model and Packman’s model match very well for different colloids and flume conditions. The new model also accurately simulates all experimental data. Effects of Aggregation on Colloid Exchange. In natural aquatic systems, properties of particles (such as particle number concentration and chemical composition) and water chemistry (such as pH and ionic strength) are all important factors responsible for the occurrence of aggregation.1 For reactive colloids, such as kaolinite clay and hematite, when the particle number concentrations and background ionic strength conditions are high enough, significant aggregation will occur. In addition, slow fluid shear could also help promote aggregation.1,15 The new model described here is developed to simulate the particle deposition in such aquatic systems that favor particle aggregation. To demonstrate the diverse effects that aggregation can have on colloid exchange, a wide range of simulation conditions were selected (see SI Table S2). These inputs represent a range of physicochemical conditions one usually sees in either natural systems or laboratory experiments1,8,15,31,32 (see details in SI Text S3). SI Figure S2 shows that aggregation substantially modified the colloid paths in the streambed by causing deviation from the solute paths in comparison to the colloid paths for the case of no aggregation. Figure 2 shows two possible effects of aggregation on colloid exchange including enhanced colloid deposition (case I) and facilitated colloid transport (case II). For case I, increase in ka values cause more colloids to be retained in the streambed (Figure 2A and C). Thus, the normalized in-stream colloid concentration curves shift downward. For case II, increase in ka values from 0 to 7.0 1020 m3/s decreases the amount of colloids retained in the streambed (Figure 2B and D although they are hard to be distinguished in Figure 2D). Thus, the normalized instream colloid concentration curves shift upward indicating more colloids remaining in the streamwater. Further increase in ka values results in enhanced colloid deposition in Figure 2B and D. The observation of the two possible cases can be explained by the relationship between filtration coefficient (λf) and colloid sizes (eqs 810). As presented by Tufenkji and Elimelech,29 filtration coefficients decay to a minimum point at a certain particle size and then rise after that. Under the simulation conditions used here, the minimum filtration occurs when the colloid size is 0.48 μm (see SI Figure S3). Thus, when the primary particle size is close to or bigger than 0.48 μm (Figure 2A and C), increase in particle size caused by aggregation results in additional colloid deposition due to the increase in filtration coefficient (enhanced colloid deposition). When the aggregate size becomes larger than 2 μm, settling becomes significant.1 However, when the primary colloid size is significantly smaller than 0.48 μm (Figure 2B and D), and if the aggregate size never exceeds 0.48 μm during aggregation, the resulting filtration coefficient decreases, causing less retention of colloids/aggregates in the streambed and more remaining in the streamwater (facilitated colloid transport). Comparing the results of low filtration (Figure 2A and B) with high filtration (Figure 2C and D) for the same ka values, effect of aggregation is more pronounced for the case of the low filtration condition than that for the high filtration condition. This is because when filtration is low, colloid deposition is mainly dominated by aggregation. When Rf increases (Figure 2C and D), more colloids tend to attach to bed sediments and leave smaller amount of colloids to be aggregated, which reduces the effect of aggregation. The interplay between particle filtration 5617



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Figure 2. Effects of aggregation on colloid exchange: (A) dp0 = 0.34 μm, low filtration (Rf = 0.03); (B) dp0 = 0.05 μm and Rf = 0.03; (C) dp0 = 0.34 μm, high filtration (Rf = 0.85); and (D) dp0 = 0.05 μm, and Rf = 0.85. Stable time, tst, used for all simulations is 104 h or t*/θ = 632.

and aggregation and its effect on colloid deposition is further demonstrated in Figure 3. Conventionally, it is expected that filtration causes additional colloid trapping and exchange in the streambed.7 This is consistent with what is shown in Figure 3A where ka is relatively low and colloid deposition is dominated by colloid filtration. However, when the aggregation rate is high (Figure 3B), increase in filtration causes a smaller amount of colloids to be retained in the streambed. This result further shows that when filtration is low and particleparticle interaction is high, most particles are aggregated rapidly and produce aggregates larger than 2 μm, which causes colloid/aggregate settling dominated colloid deposition in the streambed. However, when filtration increases, the extent of aggregate formation and the effect of aggregation on colloid deposition are reduced due to the high amount of particle removal by the bed sediments via filtration. Thus, increase in filtration causes less net exchange of particles in the streambed under high ka conditions (Figure 3B). In natural streams, reactive colloids could be strongly attached to other colloids, which results in high aggregation conditions. In such cases, generalized assumptions based on common knowledge that an increase in filtration enhances the deposition of colloids in the streambed might not always be valid. Thus, the interplay between particle filtration and aggregation discussed here has important implications on colloid transport in natural streams. The effect of primary particle size on colloid transport was evaluated by applying the model to simulate the exchange of four types of colloids under different aggregation and filtration conditions (Figure 4). When ka = 0, that is, the absence of aggregation, colloids of primary particle size of 1.2 μm are retained most in the bed, followed by 0.10 μm and 0.34 or 0.45 μm colloids. In the presence of aggregation, colloids of primary particle size of 1.2 μm are retained most in the bed, followed by 0.10 and 0.34 μm

colloids in Figure 4A and C and by 0.45 and 0.10 μm colloids in Figure 4B. The difference observed in this trend is caused by the different effect of aggregation on each size of colloids. For example, under the aggregation and filtration conditions used here, all colloid deposition in the streambed was enhanced due to the presence of aggregation (case I) in Figure 4A; however, the 0.10 μm colloids in Figure 4B and C are retained less in the bed (case II) compared to the case of no aggregation. The facilitation of the mobility of the 0.10 μm colloids due to aggregation is so pronounced in Figure 4B that it results in a shift in the order of deposition for the particles simulated compared to the case of no aggregation. The effect of primary particle size on colloid deposition can also be explained using the filtration-size and settling-size relationships mentioned above. Since the calculated filtration is lower for the colloids with primary size of 0.34 or 0.45 μm than that for 0.10 μm (see SI Figure S3), the 0.10 μm colloids deposit more in the streambed when ka = 0 m3/s. The 1.2 μm colloids are subject to the highest settling and filtration, thus they deposit most in the streambed. When ka = 5.4 1017 m3/s (Figure 4A), due to the combined effect of aggregation, filtration, and settling and the difference in the particle number concentrations used, the aggregates developed from 0.34 μm particles are smaller than those developed from 0.10 μm colloids, thus the 0.34 μm particles/aggregates are subjected to lower filtration and settling than the 0.10 μm particles. The 1.2 μm colloids aggregate to the largest size, therefore they are subject to the highest filtration and settling, depositing most in the bed. The same explanation can also be used to account for the order of colloid/aggregate deposition observed in Figure 4B and C. The stable time, tst, defines the time it takes for the aggregation to reach equilibrium, that is, aggregation process stops and ka = 0. 5618



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Figure 3. Effect of filtration on colloid exchange (A) for ka = 2.81 1018 m3/s and tst = 0.5 h (t*/θ = 3.0); and (B) ka = 5.0 1013 m3/s and tst = 104 h (t*/θ = 632). dp = 0.25 μm for both plots.

Figure 4. Effect of primary particle size on colloid exchange. Dashed lines are for ka = 0 m3/s, solid lines for ka values specified in each plot, black lines for 0.10 μm colloids, red lines for 0.34 μm colloids, orange lines for 0.45 μm and blue lines for 1.2 μm colloids. The mass concentration used for all particles is 100 mg/L and the corresponding number concentrations are 8.1 1013, 1.9 1012, 7.8 1011, and 4.0 1010 #/L for 0.10, 0.34, 0.45, and 1.2 μm colloids, respectively.

The stable time depends on physicochemical conditions such as ionic strength and colloid type and concentration. For example, our laboratory studies show that tst for the aggregation of 100 mg/ L kaolinite clay colloids (0.45 μm) in 300 mM NaCl solution is about 40 min. Figure 5 indicates that when the aggregation process continues, that is, the stable time increases, the colloid deposition for case I is enhanced and the colloid mobility for case II is facilitated to a greater extent. As a result, the normalized in-stream concentration curves shift downward in Figure 5A and upward in Figure 5B as tst values increase. Thus, stable time tst considerably affects the colloid exchange between streams and streambeds. The evolution of particle size distribution in the streambed over time corresponding to the colloid exchange simulated in

Figure 5A and B is shown in SI Figure S4. Particles develop to a larger size as stable time increases. They rapidly grow in the first hour, but the growth slows down during later hours because Smoluchowski equation describes exponential decay of the aggregation process13 and also the total number concentrations of colloids/aggregates decrease over time due to the removal by the sediments during pore water transport. At 104 h, the particle size distribution becomes relatively large since more aggregates are formed during the transport process. Since this new model did not consider disaggregation process, the simulated particle size always increases (SI Figure S4) even though the total number concentration decreases during the pore water transport. Also, a weight average particle size, that is, the summation of product of 5619



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Figure 5. (A) Effect of stable time on colloid exchange for case I: enhanced colloid deposition; and (B) Effect of stable time on colloid exchange for case II: facilitated colloid transport. Rf = 0.03 for all plots. The equivalent dimensionless tst used for 0, 0.1, 0.5, 1, 104 h are 0, 0.6, 3.0, 6.1, and 632, respectively.

particle size and number concentration of each size class divided by the total number concentration (dp = ∑(dpini)/nT), is used at each time step to calculate the filtration coefficient using eqs 9 and 10 and settling velocity that is used in eq 3 in the new model. Thus, the removal of each specific particle size class is not considered at each time step. Therefore, the new model will primarily provide reasonable simulations for pseudomonodisperse particles having a unimodal size distribution. When the primary particle size distribution is highly polydisperse or multimodal, the application of this model should be carefully conducted and a simulation procedure that involves consideration of the filtration of each particle or aggregate size class as described in ref 33 needs to be followed. The model applied the self-preserving size distribution concept to calculate the particle size evolution at different times caused by aggregation. Thus, to apply the model, the self-preserving size distribution condition needs to be met, that is, the simulation time should be much longer than the time to reach the self-preserving distribution or the time-lag tsp. Therefore, it is important to know the time-lag before applying the model. Based on the current literature25 and results obtained in the authors’ lab for aggregation of 100 mg/L kaolinite clay colloids (0.45 μm) under ionic strength conditions of 100, 300, and 700 mM NaCl, the time-lag is generally found in the order of seconds to a few minutes. Thus, this model should be applicable in most natural and laboratory systems. This work is the first attempt to incorporate aggregation into process-based stream-subsurface exchange model. Due to the complexity of this kind of process-based models, the authors decided to incorporate one additional process at a time into the existing colloid exchange model that has already been verified using various laboratory flume experiments. Thus, the new model did not consider straining, a potentially important particle deposition mechanism. Further model verification is currently carried out using flume experiments conducted at favorable aggregation conditions. Model refinement will be carried out to include straining if needed in future work. In natural streams, colloids can come from many sources such as surface runoff, sediment resuspension and metal precipitation.1 Reactive colloids can form flocs which influence the particle settling and filtration. The new model illustrates an effective approach to incorporate aggregation into transport models and provides a mechanistic level understanding of the effect of aggregation on colloid transport in streams. The combined effect of aggregation, filtration, and settling on particle deposition demonstrated here can be very useful in interpreting field scale particle transport results such as those presented by Karwan and Saiers34 (see SI Text S4).

Natural water systems are very complex and include many nonideal conditions such as heterogeneous streambeds and modification of surface chemical conditions of colloids under different experimental conditions. While the new model only considers homogeneous streambeds and a few processes governing colloid transport, consideration of additional processes and nonideal conditions can be included in future model development. Although the new model is still quite limited in terms of application in natural water systems, it does provide us an opportunity to closely examine the effect of aggregation, an important process in natural waters, on colloid transport.

’ ASSOCIATED CONTENT

bS

Supporting Information. Texts S1S5, Tables S1S2, and Figures S1S4. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*CPhone: 361-593-2798; fax: 361-593-2069; e-mail: jianhong. [email protected].

’ ACKNOWLEDGMENT This work was supported by the National Science Foundation CAREER award CBET-0449014 to J.R. and a Texas A&M University—Kingsville Graduate Assistantship to T.A. We thank Andrew G. Smith for reviewing the manuscript before submission. ’ REFERENCES (1) Stumm, W.; Morgan, J. J. Aquatic Chemistry; Wiley-Interscience: New York, 1996. (2) McCarthy, J. F.; Zachara, J. M. Subsurface transport of contaminants. Environ. Sci. Technol. 1989, 23 (5), 496–502. (3) Ren, J.; Packman, A. I. Modeling of simultaneous exchange of colloids and sorbing contaminants between streams and streambeds. Environ. Sci. Technol. 2004a, 38 (10), 2901–2911. (4) Thibodeaux, L. J.; Boyle, J. D. Bedform-generated convective transport in bottom sediment. Nature 1987, 325 (6102), 341–343. (5) Eylers, H.; Brooks, N. H.; Morgan, J. J. Transport of adsorbing metals from stream water to a stationary sand-bed in a laboratory flume. Mar. Freshwater Res. 1995, 46, 209–214. 5620


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Model Simulations of Particle Aggregation Effect on Colloid Exchange

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