Impact of Particle Size Distribution of Colloidal Particles on Contaminant Transport in Porous Media

applied sciences

Article Impact of Particle Size Distribution of Colloidal Particles on Contaminant Transport in Porous Media

Jongmuk Won 1 , Dongseop Lee 2, Khanh Pham 2, Hyobum Lee 2 and Hangseok Choi 2,*

1 Department of Civil and Environmental Engineering, University of Ulsan, Daehak-ro 93, Nam-gu, Ulsan 680-749, Korea; [email protected] 2 School of Civil, Environmental and Architectural Engineering, Anam-dong 5-ga, Korea University, Seoul 136-713, Korea; [email protected] (D.L.); [email protected] (K.P.); [email protected] (H.L.) * Correspondence: [email protected]; Tel.: +82-2-3290-3326; Fax: +82-2-928-7656

Received: 25 December 2018; Accepted: 27 February 2019; Published: 5 March 2019

Abstract: The presence of retained colloidal particles causes the retardation of contaminant transport when the contaminant is favorably adsorbed to colloidal particles. Although the particle size distribution affects the retention behavior of colloidal particles, the impact of particle size distribution on contaminant transport has not been reported to date. This study investigates the impact of the particle size distribution of the colloidal particles on contaminant transport through numerical simulation by representing the particle size distribution as a lognormal distribution function. In addition, the bed efﬁciency and contaminant saturation of simulated breakthrough curves were calculated, and a contaminant transport model with the Langmuir isotherm for the reaction between the contaminant–sand and contaminant–colloidal particle was introduced and validated with experimental data. The simulated breakthrough curves, bed efﬁciency, and contaminant saturation indicated that an increase in the mean and standard deviation of the particle size distribution causes the retardation of contaminant transport.

Keywords: particle size distribution; colloidal particle; contaminant transport; median particle size; standard deviation

1. Introduction It is now well-known that the transport and retention behavior of organic and inorganic colloidal particles (<10 µm) in porous media facilitates or retards contaminant transport when the contaminant is favorably adsorbed to colloidal particles [1–3]. Recent ﬁeld and laboratory studies demonstrated that even a small amount of colloidal particles lead to the facilitation and retardation of contaminants such as radionuclides (e.g., 137Cs and 239Pu) or heavy metals (Pb(II) and Ni(II)) [4–6]. Therefore, the presence of colloidal particles as well as the transport behavior of colloidal particles should be taken into account in order to properly predict the contaminant transport. This would minimize underestimation or overestimation of the travel distance of contaminants [7,8]. Relatively small sizes of colloidal particles and large hydrodynamic forces exerted upon them cause the particles to be transported rather than retained, leading to the facilitation of contaminant transport. In contrast, if a relatively large number of colloidal particles are retained and not mobile, contaminants are more likely to be retarded and transport more slowly compared to the pore ﬂuid. In this case, the larger amount of retained colloidal particles leads to the slower transport of contaminant because of more available adsorption sites in colloidal particles–sand media [9,10]. Therefore, an accurate estimation of the quantity of immobile colloidal particles cannot be overemphasized for the prediction of contaminant transport.

Appl. Sci. 2019, 9, 932; doi:10.3390/app9050932 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 932 2 of 12

In order to predict the amount of immobile colloidal particles, many preceding studies of colloid transport have documented the impact of size ratio between colloidal particles and sand, solution chemistry (ionic strength or pH), fluid drag force, and degree of saturation [11–15]. However, the monodisperse latex colloids used in most of these experimental studies cannot represent the polydisperse colloids in subsurface such as clay particles, commonly observed in nature, that contaminant (e.g., heavy metals and radionuclides) are favorably adsorbed. In addition, sand also can be polydisperse, showing a coefficient of uniformity of about 10 [16]. This implies that the particle size distributions of sand and colloidal particles should be considered for the accurate prediction of contaminant transport. The objective of this study was to numerically investigate the impact of the particle size distributions of sand and colloidal particles on contaminant transport. The breakthrough behavior of contaminant at the given mean and standard deviation of particle size distributions of sand and colloid particles was evaluated and discussed. In addition, the amount of immobile colloidal particles at given particles size distributions was evaluated using the model proposed by Won and Burns (2018) [17]. Consequently, the contaminant saturation and the bed efficiency of particle-retained sand medium were evaluated in order to represent simulated breakthrough curves quantitatively.

2. Mathematical Model

2.1. Colloidal Particle Transport in Saturated Porous Media Three main mechanisms can be taken into account for colloidal particle–sand interaction during the transport of particles in saturated porous media: straining, attachment, and detachment [12,18]. Straining is the retention mechanism of particles caused by a relatively large particle size compared to the size of pore space [19,20]. Attachment is another retention mechanism of particles induced by the net attraction energy between the colloidal particles and sand, typically explained by the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory [21,22]. The detachment mechanism represents the separation of retained colloidal particles from sand induced by fluid drag forces [23,24]. The one-dimensional continuum scale advection–dispersion-type governing equation with consideration of the above three main mechanisms can be expressed as:

∂(nC ) ∂(nC ) ∂ ∂C c = −q c + nD c − nk ψ C + ρ k S − nk ψ C , (1) ∂t ∂z ∂z c ∂z att att c b det att str str c

∂S ρ c = nk ψ C − ρ k S + nk ψ C , (2) b ∂t att att c b det att str str c −3 where n (-) is the porosity, Cc (M L ) is the particle concentration in aqueous phases, t (T) is the time, q −1 2 (L T ) is the Darcy’s velocity, z (L) is the depth, Dc (L T) is the dispersion coefficient of particles, ρb is −3 −1 −1 −1 the bulk density of sand (M L ), and katt (T ), kdet (T ) and kstr (T ) are the first-order coefficients accounting for attachment, detachment, and straining, respectively. ψatt and ψstr are the dimensionless −1 functions for colloidal particle attachment and straining, respectively, and Sc (M M ) is the solid-phase −1 −1 concentration of retained colloidal particles (Sc = Satt + Sstr, where Satt (M M ) and Sstr (M M ) are the solid-phase concentrations of attached and strained colloidal particles). ψatt [25,26], ψstr [12], katt [27], and kstr [12] in Equation (1) can be expressed as functions of the radius of colloidal particle (rc) and the radius of sand (rs) (see details in [17]):

Satt ψatt(rc, rs) = 1 − , (3) Smax(rc, rs)

−β ds50 + z ψstr = , (4) ds50 Appl. Sci. 2019, 9, 932 3 of 12

1.42 rc kstr(rc, rs) = 4.495 , (5) rs

3(1 − θw) katt(rc, rs) = α(rc)η0(rc, rs)vs, (6) 2ds50 −1 where Smax (M M ) is the maximum attached mass of colloidal particles per unit mass of sand, ds50 (L) is the median size of sand, β (-) is a fitting parameter (β = 0.43, see [12]), α (-) is the attachment −1 efficiency, η0 (-) is the single-collector collision efficiency [28,29], and vs (L T ) is the average seepage velocity. Note that the Smax adopted in this paper was determined by the equation presented in [18], derived from the geometry of spherical colloidal particles and sand.

2.2. Colloidal-Particle-Associated Contaminant Transport in Saturated Porous Media Assuming the all pore space is accessible to colloidal particles (i.e., no dead pores for colloidal particle transport) and colloidal particles are irreversibly retained during contaminant transport (i.e., no detachment), the mass balance equation for the colloidal-particle-associated contaminant transport in saturated porous media can be expressed as [9,30]:

∂(nC) ∂S ∂S ∂S S ∂(nC) ∂ ∂C + ρ e + ρ k + ρ c ic = −q + nD , (7) ∂t ∂t ∂t ∂t ∂z ∂z ∂z where C (M L−3) is the dissolved contaminant concentration in aqueous phase, ρ (M L−3) is the bulk −1 density of sand medium, Se (M M ) is the concentration of contaminant sorbed instantaneously to the −1 sand, and Sk (M M ) is the concentration of contaminant sorbed kinetically to the sand. In addition, Sic (M M−1) is the contaminant sorbed by retained colloidal particles. For three of the time derivative terms on the left-hand side associated with the adsorption of contaminant, the application of the Langmuir isotherm model on the contaminant–sand adsorption and the contaminant-retained colloidal particles adsorption leads to [31]: ∂S ∂C ρ e = ρ f K q , (8) ∂t 1 max(1) ∂t ∂Sk qmax(1)K2C ρ = ρω1 (1 − f ) − Sk , (9) ∂t 1 + K2C ∂ScSic qmax(2)K2C ρ = ρω2 Sc − SicSc + kcρScSic, (10) ∂t 1 + K2C where f (-) is the fraction of instantaneously sorbed contaminant in the two-site adsorption model (f was −3 assumed to be 0.01 in this study), Ki (L M) is the distribution coefficient of dissolved contaminant to −1 the adsorbent (i = 1 for sand, i = 2 for colloidal particle), qmax(i) (M M ) is the adsorption capacity, ω −1 −1 (T ) is the first-order rate constant, and kc (T ) is the first-order reaction rate constant. The application of the Langmuir isotherm in Equations (8)–(10) keeps the amount of adsorbed contaminant less than qmax.

2.3. Sampling of Colloidal Particles and Sand The Latin hypercube sampling (LHCS) method ([32]) was applied for sampling colloidal particles and sand at given particle size distributions. This method provides an even distribution of sample colloidal particles throughout the particle size distribution curve. The particle size distributions are represented by lognormal distribution functions and the mathematical formulation of mth probability values in the LHCS method (Pm) and mth sampled rc (rc(m)) at the given lognormal distribution function F is given as: 1 m − 1 P = u + , (11) m N m N

−1 rc(m) = F (Pm), (12) Appl. Sci. 2019, 9, 932 4 of 12

where m is an integer between 1 and N, N is the sampling size, and um is the random number generated from 0 to 1 (0.5 was assumed in this paper for the reproducibility of sampling result). Using Equations (11) and (12) ensures the reﬂection of particle size distribution to the sampled rc at the given mean (µc) and standard deviation (σc), which will eventually affect the amount of retained colloidal particles by the calculated katt and kstr as presented in Equations (5) and (6), respectively.

2.4. Numerical Modeling The equation for colloidal particle transport presented in Equation (1) was solved ﬁrst at the inlet concentration of 5 g L−1 for 30 pore volumes (PVs, 1 PV represents the volume of colloidal particle suspension equal to the pore volume of sand medium) of injection. The obtained retention proﬁle of colloidal particles (Sc in Equation (2)) was applied to the initial retained colloidal particles for solving the contaminant transport equation (Equation (7)). An inlet contaminant concentration of 5 mg L−1 was applied to the boundary condition for 30 PVs of injection. All parameters used in simulations were identical, except the mean and the standard deviation of the particle size distribution of colloidal particles. The backward Euler method was used for discretizing the governing equations, and the Picard iteration [33,34] was implemented in the model for the convergence in each time step. The dimension of the sand medium domain in all the numerical simulations was set to 30.48 cm (=1 foot) long, except the model validation (the dimension of domain = 4.61 cm). All parameters adopted in the numerical simulation are summarized in Table1.

Table 1. Parameters adopted in numerical simulation.

Parameters Value Note and Related Reference n 0.365 Updated every time step. Decreased over time as colloidal particles retained

N 1 or 100 N = 100 when σc > 0 −3 2 −1 Dc (10 cm s ) 6.1–6.4 Calculated using equations proposed in [35], mean value was used if N > 1 D (10−3 cm2 s−1) 2.9 Experimentally obtained using non-reactive tracer (bromide) q (cm s−1) 0.0123 - −3 ρb (g cm ) 1.68 The speciﬁc gravity of sand was assumed 2.65

ds50 (cm) 0.072 Typical median size of coarse sand µ µc 0.1–2 Typical median size of clay colloidal particles (rc50 = e )

σc −2.3 to 0.69 Equivalent to the coefﬁcient of uniformity ranged from 0 to 10 f (-) 0.01 Assumed value ([36]) −3 −1 K1, K2 (10 L mg ) 2.202, 6.102 Obtained by batch experiment ([9]) −1 −3 qmax(1), qmax(2) (mg g ) 1.081 × 10 , 2.854 Obtained by batch experiment ([9]) −3 ω1 1.859 × 10 Obtained by batch experiment ([9])

2.5. Bed Efﬁciency and Contaminant Saturation The evaluation of breakthrough curves of contaminant allows the qualitative description of contaminant transport at varied µc and σc. However, obtaining the bed efﬁciency (β) and the contaminant saturation (θ) from the evaluated breakthrough curves provides the quantitative description of breakthrough curves in each condition. The β value represents the fraction of adsorbed contaminant during its transport:

 PV  f ! λ C C β   = 1 −  ∑ 0.5λ + /PVf , (13) j=1 C0 j C0 j+1 where PVf (-) is the ﬁnal elapsed pore volume (PVf = 30 in this work), λ (-) is the pore volume interval, −3 and j is an order of normalized concentration starting from 0 PV (j = PVf/λ + 1), and C0 (M L ) is Appl. Sci. 2019, 9, 932 5 of 12

the inlet concentration of contaminant. On the other hand, θ quantiﬁes the saturation of bed at PVf (i.e., the fraction of occupied adsorption sites in the bed), expressed as:

N θ = (CiVPVβPVf)/( ∑ qmax(k)Wk), (14) k=1

3 where VPV (L ) is the volume of 1 pore volume calculated by the porosity of the bed, N is the total −1 number of materials (N = 2 in this study), qmax(k) (M M ) is the maximum adsorption capacity of contaminant, and Wk (M) is the weight of the kth material in the bed.

3. Results and Discussion

3.1. Model Validation While the equations for contaminant adsorption to sand adopted in this paper (Equations (8) and (9)) have been reportedAppl. Sci. 2019 and, 9, x used FOR PEER in REVIEW the literature [31,37], the equation describing the adsorption5 of 12 mechanism of contaminant to retained colloidal particles presented in Equation (10) has not been reported. Therefore, the model presented3. Results and in the Discussion previous sections was validated by comparing with laboratory experimental data as follows.3.1. Model Validation Two casesWhile of the the experimentalequations for contaminant data presented adsorption to insand [9 adopted] were in selectedthis paper (Equations to validate (8) the model documentedand in (9)) the have previous been reported sections and used and in the to literat obtainure ω[31,37],2 and the kequationc, which describing describe the adsorption the adsorption rate of contaminantmechanism to retained of contaminant colloidal to retained particles. colloidal A columnparticles presented experiment in Equation was (10) performed has not been to investigate reported. Therefore, the model presented in the previous sections was validated by comparing with the transportlaboratory of Pb(II) experimental under the data presence as follows. of retained kaolinite colloidal particles. The column with 5.08 cm diameterTwo and cases 4.61 of the cm experimental height was data designed presented in and [9] uniformlywere selected packedto validate with the model 1%–5% kaolinite content. Afterdocumented equilibrating in the previous the system sections usingand to obtain the background ω2 and kc, which solution describe the without adsorption Pb(II), rate of 5 mg L−1 of contaminant to retained colloidal particles. A column experiment was performed to investigate the Pb(II) solutiontransport was injectedof Pb(II) under for the 100 presence PVs. Conditionsof retained kaolinite of the colloidal selected particles. experiment The column caseswith 5.08 for the model validation incm this diameter paper and were 4.61 cm 1% height and was 5% designed of kaolinite and uniformly content packed at 6 with mL 1%–5% min− kaolinite1 of flow content. rate and pH ~6. −1 Detailed informationAfter equilibrating on the the experimental system using the procedure background is solution documented without inPb(II), [9]. 5 mg L of Pb(II) solution was injected for 100 PVs. Conditions of the selected experiment cases for the model The trust-regionvalidation in algorithmthis paper were was 1% usedand 5% to of obtainkaolinite thecontent best at 6 curve-fittedmL min−1 of flow breakthrough rate and pH ~6. curves and ω2 and kc atDetailed given observedinformation on breakthrough the experimentalcurves procedure (the is documented step and in function [9]. tolerance for the stopping criteria was set asThe 10 trust-region−6). As seen algorithm in Figure was used1, theto obtain proposed the best modelcurve-fitted in thisbreakthrough paper simulated curves and ω the2 observed and kc at given observed breakthrough curves (the step and function tolerance for the stopping breakthroughcriteria curves was accurately.set as 10−6). As Evenseen in thoughFigure 1, the the proposed back-calculated model in this paper coefficients simulated in the Figure observed1 may be able −1 to exclusivelybreakthrough describe the curves Pb(II) accurately. transport Even though under the the back presence-calculated of coefficients kaolinite, in Figure the values 1 may be of ableω 2 = 0.016 s to exclusively−1 describe the Pb(II) transport under the presence of kaolinite, the values of ω2 = 0.016 s−1 and kc = 0.0152 s were used in further simulation in order to investigate the impact of µc and σc of and kc = 0.0152 s−1 were used in further simulation in order to investigate the impact of µc and σc of particle size distributionparticle size distribution on contaminant on contaminant transport transport at given given ω2 ωand2 andkc. kc.

0.9 R2 = 0.97 0.8

0.7

0.6 0 0.5 C / C C / 0.4

0.3 R2 = 0.97 0.2 Sc = 1% Sc = 5% 0.1 Model Simulation 0 0 20406080100120 PV

Figure 1. Observed and modeled breakthrough curves of Pb(II) under the presence of retained kaolinite Figure 1. Observed and modeled breakthrough curves of Pb(II) under the presence of retained 2 colloidal particleskaolinite at 1% colloidal and 5%. particles Back-calculated at 1% and 5%. Back-calculatedω2 and kc are ω also2 andpresented kc are also presented with R withvalues. R2 values. PV: pore volume. PV: pore volume.

3.2. Impact of Median Colloidal Particle Size (rc) Figures 2 and 3 illustrate the contaminant breakthrough curves and the retention profile of colloidal particles at rc = 0.1, 0.5, 1, and 2 µm. While Figure 2 shows the concentration of contaminant Appl. Sci. 2019, 9, 932 6 of 12

3.2. Impact of Median Colloidal Particle Size (rc) Figures2 and3 illustrate the contaminant breakthrough curves and the retention profile of colloidal particles at rc = 0.1, 0.5, 1, and 2 µm. While Figure2 shows the concentration of contaminant at the outlet of the sand medium during the injection, Figure3 presents the amount of retained particles with depth afterAppl. Sci. injecting 2019, 9, x FOR 30 PEER pore REVIEW volumes of particle suspension. As seen in Figure6 2of, 12 the transport Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 12 of contaminant was retarded when rc increased because of a more significant amount of retained at the outlet of the sand medium during the injection, Figure 3 presents the amount of retained colloidal particlesatparticles the outlet at with higher of depth the sandrafterc as mediuminjecting illustrated during30 pore in thevolumes Figure injection, of3 particle. Figure The highersuspension. 3 presents value theAs seenamount of inrc Figure ledof retained to 2, greaterthe katt and particles with depth after injecting 30 pore volumes of particle suspension. As seen in Figure 2, the kstr in Equationtransport (1) as of presentedcontaminant was in Figure retarded4 when(except rc increasedrc < 0.4 becauseµm becauseof a more significant of large ηamount0 value of at relatively transportretained colloidal of contaminant particles was at higher retarded rc as when illustrated rc increased in Figure because 3. The of higher a more value significant of rc led amount to greater of low rc, see details in [29]), which resulted in a larger amount of retained colloidal particles. It can be retainedkatt and k colloidalstr in Equation particles (1) asat higherpresented rc as in illustrated Figure 4 in(except Figure rc 3.< The0.4 µm higher because value of of large rc led η to0 value greater at anticipated thatkrelativelyatt and increases kstr low in Equation rc, see in details the (1) asnumber in presented[29]), which of inretained resultedFigure 4 in (except colloidala larger rc amount< 0.4 particles µm of because retained would of colloidal largecorrespond η0 particles. value at to slower the contaminantrelativelyIt can be transport anticipated low rc, see at detailsthat the increases givenin [29]), in which adsorption the number resulted of inrate. retained a larger The colloidal amount results particlesof retained presented would colloidal correspond in particles. Figure to2 imply that It can be anticipated that increases in the number of retained colloidal particles would correspond to the size of theslower colloidal the contaminant particles transport should at the begiven taken adsorp intotion rate. account The results as presented one of thein Figure critical 2 imply parameters in slowerthat the the size contaminant of the colloidal transport particles at the should given be adsorp takention into rate. account The resultsas one presentedof the critical in Figure parameters 2 imply in contaminantthatcontaminant transport the size of throughtransport the colloidal through porous particles porous media. should media. Thatbe taken That is, into a is, slight account a slight increase as oneincrease of the in incriticalrc rcanc can parameters retard retard thethe in contaminant contaminant transport through porous media. That is, a slight increase in rc can retard the transport significantly.contaminant transport The retardation significantly. of The transport retardation would of transport be more would substantialbe more substantial if ω 2if andω2 andk c were larger contaminant transport significantly. The retardation of transport would be more substantial if ω2 and than the valueskc were considered larger than the in values this paper. considered in this paper. kc were larger than the values considered in this paper. 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0 0.6

0 0.5

C /C 0.5 0.4 C /C 0.4 0.3 No particle r = 0.1 µm 0.3 Noc particle 0.2 rc = 0.10.5 µm 0.2 rc = 0.51 µm µm 0.1 c rc = 12 µm 0.1 0 rc = 2 µm 0 0 5 10 15 20 25 30 35 0 5 10 15PV 20 25 30 35 PV Figure 2. Simulated breakthrough curves of the contaminant at inlet colloidal particle sizes ranging Figure 2. SimulatedFigurefrom 0.1 2. to Simulated breakthrough2 µm. breakthrough curves curves of of the the contaminantcontaminant at inlet at inletcolloidal colloidal particle sizes particle ranging sizes ranging from 0.1 to 2fromµm. 0.1 to 2 µm.

Retained particle (g particle / g sand) 0.0001Retained 0.001 particle 0.01 (g particle / g sand) 0.1 1 0.00010 0.001 0.01 0.1 1 0 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.6 0.6 0.7 Normalizeddepth(-) 0.7 rc = 0.1 µm

Normalizeddepth(-) 0.8 rc = 0.10.5 µm 0.8 r = 1 µm 0.9 rc = 0.5 µm r = 12 µm 0.9 c 1 rc = 2 µm 1 Figure 3. Simulated retention proﬁles of colloidal particles at in inlet colloidal particle sizes ranging Figure 3. Simulated retention profiles of colloidal particles at in inlet colloidal particle sizes ranging from 0.1 to 2Figurefromµm. 0.1 3. to Simulated 2 µm. retention profiles of colloidal particles at in inlet colloidal particle sizes ranging from 0.1 to 2 µm. Appl. Sci. 2019, 9, 932 7 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 12

35 3

rs = 360 µm 30 2.5

25 k

2 str ×10

(/sec) 20 3 1.5 3 (/sec)

×10 15 att

k 1 10 katt 5 0.5 kstr

0 0 0 0.5 1 1.5 2 2.5

rc (µm)

Figure 4. Calculated katt and kstr using Equations (5) and (6) at varied rc (0.1 µm interval) and rs = 360 µm. Figure 4. Calculated katt and kstr using Equations (5) and (6) at varied rc (0.1 µm interval) and 3.3. Impact of ther Standards = 360 µm. Deviation of the Particle Size Distribution of Colloidal Particles The median3.3. Impact particle of the Standard radii ofDeviation 0.18 andof the Pa 0.55rticleµ Sizem Distribution were selected, of Colloidal which Particles correspond to the median particle diameterThe of median kaolinite particle SA1 radii and of 0.18 kaolinite and 0.55 RP2µm were in [selected,38], in which order correspond to investigate to the median the impact of the particle diameter of kaolinite SA1 and kaolinite RP2 in [38], in order to investigate the impact of the standard deviationstandard deviation of particle of particle size size distribution. distribution. Figure Figure 5 illustrates5 illustrates simulated simulated breakthrough breakthrough curves of curves of contaminantscontaminants and the and retention the retention profiles profiles of particles particles at varied at varied σc and µσcc = and−1.72 µ(correspondingc = −1.72 (corresponding to the to the median particlemedian particle size =size 0.18 = 0.18µm). µm). It It isis clearly observed observed that the that increase the increasein the standard in the deviation standard of deviation particle size distribution (i.e., an increase in the coefficient of uniformity) retarded the transport of of particle sizecontaminant distribution (Figure (i.e.,5a). This an is increase attributed in to thethe larger coefficient amount of retained uniformity) particles retarded at higher σ thec transport of contaminant(Figure (Figure 5b), which5a). is caused This isby attributedlarger values of to sample the largerrc at higher amount σc (Figure of 6).retained As presented particles in [17], at higher in the identical median size, the amount of retained particles increased as σc increased. In other words, σc (Figure5b), which is caused by larger values of sample rc at higher σc (Figure6). As presented no consideration of σc in colloidal particle transport in porous media would lead to a significant in [17], in theunderestimation identical median of the amount size, the of retained amount particles. of retained The results particles obtained increased in Figure 5 indicate as σc increased. that σc In other words, no considerationcan be a critical of factorσc in not colloidal only for particlethe colloidal transport particle transport in porous but mediaalso for wouldthe contaminant lead to a significant underestimationtransport of when theamount the contaminants of retained are favorably particles. adsorbed The to retained results colloidal obtained particles. in Figure 5 indicate that The impact of σc became more significant when µc increased, as demonstrated in Figure 7. At µc σc can be a critical= −0.6 (corresponding factor not toonly the median for theparticle colloidal size of 0.55 particle µm), the retardation transport of contaminant but also for transport the contaminant transport whenwas more the contaminantssignificant compared are to favorably the case of µ adsorbedc = −1.72, as illustrated to retained in Figure colloidal 7a. This result particles. may be explained mainly by the fact that an increase in the amount of strained particles due to increasing σc The impact of σc became more significant when µc increased, as demonstrated in Figure7. At µc is greater at larger µc (showing a more exponential retention profile according to σc at µc = −0.6 (Figure = −0.6 (corresponding7b) than at µc to= − the1.72 (Figure median 5b). particle The significant size ofincrease 0.55 µinm), retained the retardationparticles at the oftop contaminant layer of the transport was more significantsand medium compared corresponding to theto the case increase of µofc σ=c provided−1.72, a as large illustrated number of adsorption in Figure sites7a. at This the result may be explainedtop mainly layer, which by the resulted fact in that the ansignific increaseant retardation in the of amount contaminant of strainedtransport. particles due to increasing σc is greater at larger µc (showing a more exponential retention profile according to σc at µc = −0.6 (Figure7b) than at µc = −1.72 (Figure5b). The significant increase in retained particles at the top layer of the sand medium corresponding to the increase of σc provided a large number of adsorption sites at the top layer, which resulted in the significant retardation of contaminant transport. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 12 Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 12 Appl. Sci.1 2019, 9, 932 8 of 12 Retained particle (g particle / g sand) Appl. Sci.0.9 2019, 9, x FOR PEER REVIEW (a) 0.0001 0.001 0.018 of 0.1 12 1 0 0.8 Retained particle (g particle / g sand) 0.9 (a) 0.0001 0.001 0.01(b) 0.1 1 0.1 0.7 0 Retained particle (g particle / g sand) 0.8 (a) 0.2 0.90.6 0.10.0001 0.001 0.01(b) 0.1 0.70 0.30 0.80.5 0.2 0.6 (b) C /C µc=-1.72 0.1

0 0.4 0.70.4 0.3 0.5 σc=0 0.2 0.6 σ =0.1 0.5 µc=-1.72 C /C 0.3 µcc=-1.72

0 0.4 0.4 σc=0.5 0.3 σ =0 0.5 σc=0 0.6 c 0.2 σc=1.0 0.5 µσc=-1.72c=0.1

C /C σ =0.1 0.3 µcc=-1.72 0.4 σc=1.2 0.7 σc=0.5 0.4 σc=0.5 σc=0 0.1 σc=0 (-) depth Normalized 0.6 σc=1.5 σc=1.0 0.2 σc=1.0 0.5 µσc=-1.72c=0.1 σc=0.1 0.8 0.3 σc=1.2 0 σc=1.2 0.7 σc=0.5 σc=0.5 σc=0 0.1 (-) depth Normalized 0.6 σc=1.5 0.2 0 5 10 15 20σc=1.5 25 30 0.9 σc=1.0 σc=1.0 0.8 σc=0.1 0 PV σc=1.2 σc=1.2 0.71 σc=0.5

0.1 (-) depth Normalized 0 5 10 15 20σ =1.5 25 30 0.9 σ c=1.0=1.5 c 0.8 c PV σ =1.2 0 1 c Figure0 5. 5(a) Simulated 10 breakthrough 15 20 cu 25rves of 30 contaminant 0.9 and (b) the retention profiles of particlesσc=1.5 PV 1 Figureat µc = −5.1.72 (a) Simulatedand σc = 0, breakthrough0.1, 0.2, 1.0, 1.2, cu andrves 1.5. of contaminant and (b) the retention profiles of particles FigureatFigure µc = −5. 5.1.72 ((aa)) andSimulated Simulated σc = 0, 0.1,breakthrough breakthrough 0.2, 1.0, 1.2, cu curves andrves 1.5. of of contaminant contaminant and and ( (bb)) the the retention retention profiles proﬁles of of particles particles at µc = −1.72 and σc = 0, 0.1, 0.2, 1.0, 1.2, and 1.5. at µc = −1.72 and100 σc = 0, 0.1, 0.2, 1.0, 1.2, and 1.5. 10090 1009080 908070 807060 706050 µc=-1.72 6050 σ =0 40 µc=-1.72c σ =0.1 50 σ c=0 4030 µσcc=-1.72=0.5 σ c=0.1 Percentagefiner than σσc=0=1.0 403020 σcc=0.5 σσc=0.1=1.2 Percentagefiner than σcc=1.0 3020 σσc=0.5=1.5 10 σcc=1.2

Percentagefiner than c σc=1.0 20100 σc=1.5 σc=1.2 0.001 0.01 0.1 1 10 100 σc=1.5 0.001 0.01rc 0.1(µm) 1 10 0 0.001 0.01rc 0.1(µm) 1 10

Figure 6. Particle size distributions for colloidalrc (µm)particles (µc = −1.72) used in the simulation. Figure 6. Particle size distributions for colloidal particles (µc = 1.72) used in the simulation. Figure 6. Particle size distributions for colloidal particles (µc = −1.72) used in the simulation. 1 Retained particle (g particle / g sand) 0.9 Figure(a) 6. Particle size distributions for colloidal particles0.001 (µc = −1.72) used in 0.01 the simulation. 0.1 1 0 0.8 Retained particle (g particle / g sand) (a) 0.9 0.10.001(b) 0.01 0.1 0.71 0 0.8 Retained particle (g particle / g sand) (a) 0.2 0.90.6 0.10.001(b) 0.01 0.1 0.70 0.30 0.80.5 0.2 µc=-0.60 (b) C /C 0.6 0.1 0 0.7 0.4 0.4 σc=0 0.3 0.5 σ =0.1 0.2 0.6 µcc=-0.60 0.5 µc=-0.60 C /C 0.3

0 0.4 σσc=0=0.5 0.4 c 0.30.6 σc=0 0.5 σc=1.0 0.2 µσcc=-0.60=0.1 0.5 µσc=-0.60c=0.1 C /C 0.3 σc=1.2 0.4 σc=0.5 0.7 σc=0.5 0.4 σc=0 σc=0 0.1 σσc=1.0=1.5 (-) depth Normalized 0.6 0.2 σ c=0.1 σc=1.0 c 0.50.8 µσc=-0.60c=0.1 0.3 σc=1.2 σc=1.2 0 σc=0.5 0.7 σc=0.5 σc=0 0.1 (-) depth Normalized σ c=1.0=1.5 0.6 σc=1.5 0.2 0 5 10 15 20c 25 30 35 0.9 σc=1.0 0.8 σc=0.1 0 PV σc=1.2 σc=1.2 0.71 σc=0.5

0.1 σ =1.5 (-) depth Normalized 0 5 10 15 20c 25 30 35 0.9 σ c=1.0=1.5 0.8 c PV σ =1.2 0 1 c Figure0 7. 5 (a) Simulated 10 15 breakthrough 20 25 curvescu 30rves of 35 contaminant 0.9 and (b) the retention profilesprofiles of particlesσc=1.5 PV Figureat µcc = −7.−0.60 0.60(a) Simulatedand and σσc =c =0, 0,breakthrough0.1, 0.1, 0.2, 0.2, 1.0, 1.0, 1.2,1.2, cu andrves and 1.5. of 1.5. contaminant 1 and (b) the retention profiles of particles 3.4. BedFigureat µ Efficiencyc = −7.0.60 (a) andSimulated and σc Contaminant= 0, 0.1,breakthrough 0.2, 1.0, Saturation 1.2, cu andrves 1.5. of contaminant and (b) the retention profiles of particles 3.4. Bed Efficiency and Contaminant Saturation at µc = −0.60 and σc = 0, 0.1, 0.2, 1.0, 1.2, and 1.5. β θ 3.4. BedFigures Efficiency8 8 andand and9 9 illustrate illustrateContaminant thethe calculatedSaturation β andand θ usingusing Equations Equations (13) (13) and and (14) (14)for Figures for Figure 2, 5a,2, Figure5a, and Figure7b. The value of β almost linearly increased as rc increased because of the 3.4.and Bed Figures7b. Efficiency The 8 valueand and 9 ofillustrateContaminant β almost the Saturationlinearlycalculated increased β and θ usingas rc increasedEquations because(13) and of(14) the for retardation Figures 2, 5a, of retardation of contaminant transport, while θ decreased as rc increased, caused by the increase in andcontaminant Figures7b. The 8 transport,value and 9 ofillustrate βwhile almost θthe decreased linearlycalculated asincreased βrc andincreased, θ usingas r ccaused increasedEquations by the because(13) increase and of(14) in the retainedfor retardation Figures colloidal 2, 5a,of retained colloidal particles. Forty percent of available adsorption sites (θ ~ 0.6) in the bed at rc = 2 µm contaminantparticles. Forty transport, percent while of available θ decreased adsorption as rc increased, sites (θ ~ caused 0.6) in bythe the bed increase at rc = in2 µmretained after colloidal30 PV of andafter 7b. 30 PVThe of value contaminant of β almost injection linearly implies increased that the increaseas rc increased in adsorption because sites of affects the retardation the long-term of contaminantparticles. Forty transport, percent while of available θ decreased adsorption as rc increased, sites (θ ~ caused 0.6) in bythe the bed increase at rc = in2 µm retained after colloidal30 PV of particles. Forty percent of available adsorption sites (θ ~ 0.6) in the bed at rc = 2 µm after 30 PV of Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 12 contaminantAppl. Sci. Sci. 20192019,, 99 ,,injection x 932 FOR PEER implies REVIEW that the increase in adsorption sites affects the long-term transport9 9of of 12 12 behavior of contaminants as well as the retardation of transport. In addition, the evaluation of β providescontaminant the thresholdinjection implies rc at which that the the contaminant increase in adsorptionis not detected sites at affects the outlet the long-termafter the given transport PVs. transport behavior of contaminants as well as the retardation of transport. In addition, the evaluation Forbehavior instance, of contaminants the extrapolation as well of β valuesas the presentedretardatio nin ofFigure transport. 8 indicates In addition, that the thecontaminant evaluation would of β of β provides the threshold rc at which the contaminant is not detected at the outlet after the given notprovides be detected the threshold during 30rc atPV which of injection the contaminant at rc > 2.3 µm is not(i.e., detected β = 1 at ratc ~ the 2.3 outletµm). after the given PVs. PVs. For instance, the extrapolation of β values presented in Figure8 indicates that the contaminant For instance,In contrast, the theextrapolation increase in of σ cβ led values to an presented increase inin βFigure and θ 8 at in µdicatesc = −1.72 that and the − 0.6.contaminant However, would while would not be detected during 30 PV of injection at rc > 2.3 µm (i.e., β = 1 at rc ~ 2.3 µm). βnot increased be detected significantly during 30 as PV σc ofincreased, injection a at slight rc > 2.3 increase µm (i.e., in βθ =was 1 at observed rc ~ 2.3 µm). at µ c = −0.6. This implies In contrast, the increase in σc led to an increase in β and θ at µc = −1.72 and −0.6. However, the exponentialIn contrast, increasethe increase in the in σadsorptionc led to an increasecapacity in of β the and bed θ at as µ cσ =c −increases1.72 and −when0.6. However, µc is relatively while while β increased significantly as σc increased, a slight increase in θ was observed at µc = −0.6. high.β increased In other significantly words, substantial as σc increased, adsorpti a onslight sites increase were still in θavailable was observed at high at σµc c after= −0.6. the This significant implies This implies the exponential increase in the adsorption capacity of the bed as σc increases when µc retardationthe exponential of contaminant increase in thetransport. adsorption As statedcapacity above, of the the bed calculation as σc increases of β whenand θ µ cprovides is relatively the is relatively high. In other words, substantial adsorption sites were still available at high σc after the quantitativehigh. In other description words, substantial of observed/simulated adsorption sites breakt werehrough still available curves as at well high as σ cthe after prediction the significant of the significant retardation of contaminant transport. As stated above, the calculation of β and θ provides breakthroughretardation of point.contaminant transport. As stated above, the calculation of β and θ provides the quantitativethe quantitative description description of observed/simulated of observed/simulated breakt breakthroughhrough curves curves as well as wellas the as prediction the prediction of the of breakthroughthe breakthrough point. point.1 0.9 1 0.8 0.9 0.7 0.8 0.6 0.7 0.5 0.6 βor θ βor 0.4 0.5

βor θ βor 0.3 N=1 0.4 0.2 β 0.3 N=1 θ 0.1 0.2 β 0 θ 0.1 00.511.522.5

0 rc (µm) 00.511.522.5

rc (µm) Figure 8. Calculated β and θ for Figure 2. Figure 8. Calculated β and θ for Figure2. Figure 8. Calculated β and θ for Figure 2. 1 β(μ = -1.72) 0.9 c 1 θ(μc = -1.72) 0.8 β(μc = -0.6)-1.72) 0.9 c θ(μc = -0.6)-1.72) 0.7 0.8 β(μc = -0.6) 0.6 θ(μc = -0.6) 0.7 0.5 0.6 βor θ βor 0.4 0.5

βor θ βor 0.3 0.4 0.2 0.3 0.1 0.2 N=100 0 0.1 0 0.5 1 1.5N=100 2

0 σc (µm) 0 0.5 1 1.5 2 Figure 9. Calculated β and θ for Figures5a and7a. σc (µm) Figure 9. Calculated β and θ for Figures 5a and 7a.

Figure 9. Calculated β and θ for Figures 5a and 7a. Appl. Sci. 2019, 9, 932 10 of 12

4. Conclusions This work investigated the impact of particle size distribution for colloidal particles on contaminant transport by performing a numerical simulation. The colloidal-particle-associated contaminant transport model along with the Langmuir adsorption model for the reaction between contaminant–sand and contaminant–colloidal particles was introduced, and a parametric study was performed at varied mean and standard deviation of particle size distribution. Based on the observations from simulated breakthrough curves, the key ﬁndings can be summarized as follows:

1. The experimental breakthrough curves of Pb(II) under the presence of retained kaolinite particles were described well by the colloidal-particle-associated contaminant transport model introduced in this paper. This implies that the model can be used for the prediction of transport of heavy metals under the presence of immobile kaolinite particles or other particles that favorably adsorb the heavy metals. 2. As the median size of the colloidal particles and the standard deviation increased, the transport of contaminants was retarded. The impact of a standard deviation on the retardation of transport became more signiﬁcant as the median size of the colloidal particles increased. This is attributed to the variation in the amount of retained colloidal particles by the median size and the standard deviation of particle size distribution. Therefore, the particle size distribution of particles should be taken into account in the prediction of contaminant transport. 3. A quantitative description of simulated breakthrough curves from the calculated β and θ provides the remaining available adsorption sites of the bed and the prediction of the breakthrough point at a given condition. 4. Further investigation is needed for the impact of particle size distribution on contaminant transport when particle and contaminant transport simultaneously. In addition, the unsaturated ﬂow can also be considered in the modeling work to predict the contaminant transport from the surface to groundwater.

Author Contributions: Writing draft paper, J.W.; Investigation and discussion, J.W., D.L, K.P, H.L; Visualization, H.L; Supervision and editing the manuscript, H.C. Acknowledgments: The authors appreciate the support partially by the Korea Agency for Infrastructure Technology Advancement under the Ministry of Land, Infrastructure and Transport of the Korean government (No. 18SCIP-B108153-04). The authors also wish to thank the anonymous reviewers for their careful review of the manuscript and valuable comments. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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