# Mass Transfer Principles in Column Percolation Tests: Initial Conditions and Tailing in Heterogeneous Materials

^{*}

## Abstract

**:**

^{−1}) and longitudinal dispersion leads to smaller initial concentrations than expected under equilibrium conditions. In order to elucidate the impact of different mass transfer mechanisms, film diffusion across an external aqueous boundary layer (first order kinetics, FD) and intraparticle pore diffusion (IPD) are considered. The results show that IPD results in slow desorption kinetics due to retarded transport within the tortuous intragranular pores. Non-linear sorption has not much of an effect if compared to ${K}_{d}$ values calculated for the appropriate concentration range (e.g., the initial equilibrium concentration). Sample heterogeneity in terms of grain size and different fractions of sorptive particles in the sample have a strong impact on leaching curves. A small fraction (<1%) of strongly sorbing particles (high ${K}_{d}$) carrying the contaminant may lead to very slow desorption rates (because of less surface area)—especially if mass release is limited by IPD—and thus non-equilibrium. In contrast, mixtures of less sorbing fine material (“labile” contamination with low ${K}_{d}$), with a small fraction of coarse particles carrying the contaminant leads to leaching close to or at equilibrium showing a step-wise concentration decline in the column effluent.

## 1. Introduction

## 2. Theory and Background

#### 2.1. Local Equilibrium: The Advection—Dispersion Equation

^{−1}], $n$ [-] and ${D}_{L}\text{}$($=\alpha v+{D}_{p})$ [L

^{2}T

^{−1}] denote the seepage velocity of the water, the intergranular porosity and the longitudinal dispersion coefficient. $\alpha $ [L], ${D}_{p}$ ($=$$n{D}_{aq}$) [L

^{2}T

^{−1}] and ${D}_{aq}$ [L

^{2}T

^{−1}] denote the dispersivity, the pore diffusion coefficient and the aqueous diffusion coefficient of the solute. $x$ [L] and $t$ [T] are the length of the column and time. ${\rho}_{b}\text{}$($=\left(1-n\right){\rho}_{s}$) [M L

^{−3}] is the dry bulk density of the packed bed in the column (porous media; ${\rho}_{s}$ is the solids density). For local equilibrium conditions the concentration in the solid phase $\left({C}_{s}\right)$ is in equilibrium with the solute concentration in water $\left({C}_{w}\right)$ and the distribution coefficient ${K}_{d}\text{}(={C}_{s}/{C}_{w})$ allowing for the calculation of the respective concentrations. Under these conditions, Equation (1) can be simplified as:

^{3}M

^{−1}] equals the liquid to solid ratio within the column, which in most cases is much smaller than in a batch leaching test (e.g., 0.25 L kg

^{−1}for a column test with a porosity of $n=$ 0.4 and a solid density of ${\rho}_{s}=$ 2.65 g cm

^{−3}, compared to e.g., 10 L kg

^{−1}in a batch test). Since leaching tests start for practical reasons with material packed more or less dry into the column, a uniform initial concentration is not necessarily achieved during the first flooding of the column. Initial conditions as assumed in Equation (4) (uniform concentration distribution), would only be achieved if the material is first mixed with water, equilibrated and then packed into the column, which is not practical. During the first flooding of the column, especially less sorbing solutes are displaced from the inlet and higher concentrations occur towards the outlet, as illustrated in Figure 1 (see also Appendix E). This may be accounted for by subtracting the distance of the solute displaced initially ($x/{R}_{d}$ with ${R}_{d}$ > 1) in Equation (4):

#### 2.2. Desorption Kinetics Limited by Film Diffusion

^{3}], ${m}_{d}$ [M] and $d$ [L] denotes the thickness of the external film, the volume of water, the dry mass of the solids in the column and the particle diameter, respectively. ${A}^{o}\text{}$($=6\text{}{m}_{d}$/(${V}_{w}\text{}{\rho}_{s}\text{}d$)) is the specific surface area of the particles per unit volume of water in the column [m

^{2}m

^{−3}$=$ m

^{−1}] (the term $6/{\rho}_{s}\text{}d$) represents the specific surface area of spherical particles per dry mass, e.g., in m

^{2}g

^{−1}). ${C}_{w}^{\prime}$ is the concentration at the solid-water interface where local equilibrium conditions apply (${C}_{w}^{\prime}={C}_{s}/{K}_{d}$). The external film thickness (${\delta}_{f}$) can be estimated from empirical Sherwood numbers ($Sh$) and the particle diameter ($d$):

#### 2.3. Desorption Limited by Intraparticle Pore Diffusion

^{2}T

^{−1}] the effective diffusion coefficient. ${C}_{w,intra}$ [M L

^{−3}] is the concentration of solute in the intra-granular pore water. $\epsilon $ [-] denotes the intraparticle porosity. $R$ [L] and ${\rho}_{p}$ [M L

^{−3}] ($={\rho}_{s}\left(1-\epsilon \right))$ denote the radius and bulk density of the particle (sphere).

^{2}T

^{−1}] is the apparent diffusion coefficient, defined as:

#### 2.4. Set-Up of “Numerical” Column Tests

^{−1}. Lower ${K}_{d}$ values (<0.1 L kg

^{−1}) were not considered here (this would have resulted in very high initial aqueous concentrations). If the ${K}_{d}$ values become large (${K}_{d}$ > 100 L kg

^{−1}), then the differences between the pre-equilibrated case and the “first flooding” scenario vanish and effluent concentrations are constant over long time periods. The ${K}_{d}$ range chosen covers many frequent environmental contaminants, such as per- and polyfluoroalkyl substances (PFAS), chlorinated solvents, polycyclic aromatic hydrocarbons and some heavy metals.

## 3. Results and Discussion

#### 3.1. Impact of Initial Conditions on Leaching

^{−1}. Dispersion also reduces differences between the pre-equilibrated and the first flooding case. At high ${K}_{d}$ values, the maximum concentrations were still achieved but the tailings became smoother. With the decrease of the ${K}_{d}$ value, the concentration gradients at the inlet became steeper and the “back” dispersion fluxes towards the outlet increased as well. In extreme cases, the peak concentration at the column outlet was smaller than the maximum concentration expected (e.g., $\text{}{K}_{d}=$ 0.1 L kg

^{−1}). The effect of initial conditions on normalized concentrations looks like a phase shift (see Figure 5, 1st row). This would lead to an underestimation of ${K}_{d}$ values derived from the pre-equilibrium analytical solution (Equation (4)) if the conditions in the column after the first flooding are not appropriately considered. The lower the ${K}_{d}$, the earlier the cumulative leachate concentration reaches its maximum value (${m}_{cum,\text{}max}=1000\text{}\mathsf{\mu}$g kg

^{−1}). Dispersion shifts this point to later times (see Figure 5, 3rd row).

#### 3.2. Initial Conditions and Leaching with Mass Transfer Limitations

^{−1}, 1 L kg

^{−1}and 10 L kg

^{−1}(see Figure A2)). The length of the mass transfer zone for IPD is much longer than for FD, but differences decrease with increasing ${K}_{d}$ values. For ${K}_{d}$ values of 1 L kg

^{−1}and 10 L kg

^{−1}, the mass transfer zone lengths for IPD are much shorter than the column length (${X}_{col}$ = 30 cm), which indicates that the equilibrium concentration is achieved at the outlet of the column after the first flooding. For small ${K}_{d}$ values (e.g., ${K}_{d}=$ 0.1 L kg

^{−1}), the equilibrium concentrations are not achieved at the outlet if dispersion is considered (see Figure 6, lower panel) although the mass transfer zone length (${X}_{s,63.2\%}=10\text{}\mathrm{cm}$) is still shorter than the column length. This is because the “clean” water front is close to the column outlet and dispersion “dilutes” the steep concentration gradients (“back dispersion”). The deviations between FD and fast kinetics almost vanish when dispersion is considered, indicating that with film diffusion, equilibrium is almost achieved. The development of the concentration distribution for IPD is also illustrated in animated graphs provided in the Supplementary Material (SM).

#### 3.3. Nonlinear Sorption Isotherms

^{−1}when no dispersion is considered. The differences in the concentration distribution before percolation starts are moderate. Concentration profiles tend to be smoother with nonlinear sorption with a slightly lower maximum concentration at the column outlet for low to mid ${K}_{d}$ values if dispersion is considered (see SM, Figure S1). Differences become more obvious in the tailing part of the leaching curves. Freundlich exponents smaller than 1 result in a longer tailing as is expected. The effect of nonlinear sorption looks similar to the dispersion effect, in both cases the leaching curves show more tailing (see SM, Figure S2). Nonlinearity of sorption is notably less significant than kinetic limitations in the mass transfer mechanisms.

#### 3.4. Impact of Heterogeneous Sample Composition

^{−}

^{1}, respectively). A small fraction of strong sorbents showed lower desorption rates compared to a large fraction of the weak sorbents. For this “exotic” case where only 1% of the particles carries all the contamination, initial nonequilibrium and long tailing was observed. This effect was very pronounced for intraparticle pore diffusion; the concentrations initially started on a plateau (“like equilibrium”), but then rapidly declined and showed a pronounced tailing and decrease with the square root of time (or LS). It may be noted, that longitudinal dispersion becomes less relevant if non-equilibrium conditions prevail at high ${K}_{d}$ values (see Figures S3 and S4 in SM). If such pronounced initial nonequilibrium is observed, then extended periods of time would be needed to equilibrate the water in the column with the solids (e.g., a manifold of the contact time of 5 h).

^{−1}). While the shapes of all leaching curves are very similar, their locations are shifted in time according to the different ${K}_{d}$ values by a factor of 10. If intraparticle pore diffusion is considered, tailing is observed if coarse particles predominate. This applies to both, the development of initial conditions in the column and leaching. If fine particles predominate, the leaching is close to equilibrium at early times; at later times, tailing is observed with the typical square root of time behavior. Considering the dispersion effect, non-equilibrium concentrations can be seen at the column effluent after first flooding especially at low ${K}_{d}$ values (${K}_{d}=$0.1 L kg

^{−1}). Initial non-equilibrium conditions become more salient for intraparticle pore diffusion if coarse particles predominate (see Figures S5 and S6 in SM).

^{−}

^{1}) and 90% of coarse particles with high sorption capacity (${K}_{d}=$ 100 L kg

^{−}

^{1}) is compared with two extreme cases where a hypothetical sample only contains pure fine particles with low sorption capacity and another hypothetical sample contains pure coarse particles with high sorption capacity. Figure 11 shows the initial concentration distribution for these three compositions after the first flooding period as well as the corresponding leaching curves. Sorption equilibrium is achieved rapidly if the sample consists of only fine particles with a small ${K}_{d}$ or only coarse particles with a high ${K}_{d}$. Pure coarse material with a high ${K}_{d}$ shows a low equilibrium concentration (${C}_{w,eq}={C}_{s,ini}/{K}_{d}=$ 1000 $\mathsf{\mu}$g kg

^{−}

^{1}$/$100 L kg

^{−}

^{1}$=$ 10 $\mathsf{\mu}$g L

^{−}

^{1}) while pure fine material with a low ${K}_{d}$ presents a much higher equilibrium concentration (${C}_{w,eq}={C}_{s,ini}/{K}_{d}=$ 1000 $\mathsf{\mu}$g kg

^{−}

^{1}/10 L kg

^{−}

^{1}= 100 $\mathsf{\mu}$g L

^{−}

^{1}) after a short flow distance. Interestingly, the mixed case where 10% of the column is fine material caused a high concentration which would be sorbed by the coarse materials leading to a slightly higher plateau concentration compared to pure coarse materials. The pollutants were redistributed between fine and coarse materials during the first flooding of the column. The concentration increase towards the outlet of the column in the mixed case is due to fast desorption from the fine material followed by slow sorption by the coarse material. The redistribution is almost complete at the inlet of the column because of the long residence time (${t}_{c}\text{}$= 5 h). Since the fine particles make up only to 10% of the total mass, they are already depleted in contaminant concentrations inside the column and in equilibrium with the coarse particles (reflecting both extreme cases). The front of the high concentration caused by the fine particles is already close to the outlet, while the rest is in equilibrium with the 90% coarse particle fraction.

## 4. Summary and Conclusions

^{−1}) in column leaching tests. Two different scenarios for the establishment of the initial conditions before the start of the leaching phase were considered: a fully pre-equilibrated column and a more realistic scenario where a column is flooded with water from the bottom. In order to highlight the effect of mass transfer limitations, two mechanisms are compared: film diffusion and intra-particle diffusion. Cases without and with dispersion illustrate how dispersive mixing may mask diffusion limited mass transfer. Furthermore, we looked into the impact of heterogeneous sample compositions in terms of reactive particle fractions and particle sizes. Since possible parameter combinations amount to almost infinite numbers, we have limited our analysis to just a few exemplary cases that illustrate the role of individual material properties. These few cases already show that virtually any leaching behavior can be produced with highly heterogeneous samples (depending on the mixing of different materials). The most important conclusions are:

**Initial conditions**have a significant impact on leaching at low ${K}_{d}$ values (${K}_{d}\text{}$< 1 L kg

^{−1}). With increasing ${K}_{d},$ the differences between pre-equilibrium and non-equilibrium conditions gradually vanish for ${K}_{d}\text{}$> 10 L kg

^{−1}(see Figure 5). Compounds with very low ${K}_{d}$ (“salts”) would reach extremely high concentrations (${K}_{d}\text{}$<< 1 L kg

^{−1}) at the column outlet (see Figure 4) potentially leading to enhanced dispersion due to density fingering. The ${K}_{d}$ values derived from retardation factors (${R}_{d}$ in Equation (4)) would be underestimated if the conditions in the column after the first flooding are not appropriately considered, due to a “phase shift” in normalized concentrations curves (${C}_{w}/{C}_{w,eq}$ vs. LS).

**Dispersion**generally causes “smoothing” of concentrations gradients in the column and tends to “mask” film and intraparticle diffusion characteristics due to enhanced “mixing” of the solute within the column. It may lead to smaller initial concentrations at the column outlet after the first flooding period than expected for equilibrium; this is pronounced especially at low ${K}_{d}$ values (see Figure 7 and Figure S2), which may be interpreted as non-equilibrium, but is just a consequence of dilution by dispersive mixing.

**Intraparticle pore diffusion**(IPD) generally shows slower desorption kinetics than

**film diffusion**(FD) through an aqueous boundary layer. This is due to the much smaller effective diffusion coefficient in the intraparticle pores and the large diffusion distance that develops inside the particle over time, resulting in the typical square root of time decrease of concentrations (a slope of 1/2 is observed in log-log plots of leaching curves, see Figure 7, Figure 9 and Figure 10). IPD is more sensitive to the variation of particle sizes than FD (see Figure 10). Mass transfer limitations in an aqueous boundary layer commonly exists for surface adsorbed compounds and easily soluble solids (“salts”). Elements such as heavy metals, which are slowly released from the solid phase, would require much lower solid state diffusion coefficients; if reaction fronts propagate into the particle releasing metals, intraparticle (solid) diffusion models apply again (shrinking core), which are very similar to the IPD approach used here.

**Non-linear sorption**has little influence on the leaching test results if the “right” effective ${K}_{d}$ value is calculated for the proper concentration range (since for the nonlinear sorption the ${K}_{d}$ depends on the concentration, large deviations may occur if just the ${K}_{fr}$ is determined far away of the sample’s concentration is used as “${K}_{d}$”); nevertheless, as concentrations decrease nonlinear sorption causes more tailing (see Figure 8).

**Heterogeneous samples**with only a small fraction of strongly sorbing particles lead to much slower desorption rates (because of less surface area), especially if mass release is limited by intraparticle pore diffusion (see Figure 9). In extreme cases (just 1% of the material is contaminated at ${K}_{d}=100\times {K}_{d,av}$), leaching may start at a plateau (suggesting equilibrium), but far below equilibrium concentrations (${C}_{w,peak}\ll {C}_{w,eq}$) and concentrations later decrease further; The ${K}_{d}$ values derived from the initial aqueous concentration (${K}_{d}={C}_{s,ini}/{C}_{w,peak}$) would be overestimated while the ${K}_{d}$ values calculated from retardation factors would be underestimated.

^{−}

^{1}) and large amounts of coarse particles with high sorption capacity (${K}_{d}=$ 100 L kg

^{−}

^{1}), exhibit the respective characteristics of each of the individual components in different time periods (see Figure 11). Small amounts of fine particles with low sorption capacity dominate short term behavior of the mixtures and lead to a peak effluent concentration (${C}_{w,peak}$) which approaches the equilibrium concentration expected for fine particles (see Figure 11). Since the mass fraction of fine particles is small (10%), the leachate concentrations drop rapidly and reach slightly higher equilibrium levels of 100% pure coarse particles due to the redistribution of pollutants between fine and coarse particles. Ten percent of fine particles with low sorption capacity causes a high equilibrium concentration which are sorbed by the coarse particles with high sorption capacity. ${K}_{d}$ values derived from the initial aqueous concentration (${K}_{d}={C}_{s,ini}/{C}_{w,peak}$) would be underestimated, while ${K}_{d}$ values derived from the following plateau concentration would be overestimated. Cumulative mass release, however, is often quite insensitive to mass transfer mechanisms (FD or IPD) especially for LS < 5 (see Figure 11).

## Supplementary Materials

^{−}

^{5}m s

^{−}

^{1}, $\alpha $/x = 0 or 0.1, ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−}

^{1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−}

^{9}m

^{2}s

^{−}

^{1}, $\epsilon $ = 0.05, ${d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m, Figure S2. Normalized concentrations (${C}_{w}/{C}_{w,eq}$) as well as cumulative concentrations (${m}_{cum}$) in the column effluent vs. time (expressed as liquid to solid ratio: LS) for different initial conditions depicted in Figure S1; solid lines: linear sorption; dashed lines: nonlinear sorption. Left column: without dispersion; right column: with dispersion, Figure S3. Initial concentration distribution in the column after the first flooding (up-flow) for different bi-modal compositions of sorbing and non-sorbing particles; left column: homogeneous case with average ${K}_{d}$ (=${K}_{d,av}\text{}$= 1 L kg

^{−}

^{1}); mid column: only 10% of the particles carry the contaminant at ${K}_{d}\text{}$=$\text{}10\times {K}_{d,av}$; right column: only 1% of the particles carry the contaminant at ${K}_{d}$ = $100\times {K}_{d,av}$; the average ${K}_{d,av}$ of the entire material is the same for all compositions; solid lines: film diffusion case, dashed lines: intraparticle diffusion case. Top panel: without dispersion; bottom panel: with dispersion; n = 0.45, v = 1.67 × 10

^{−}

^{5}m s

^{−}

^{1}, $\alpha $/x = 0 or 0.1, ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−}

^{1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−}

^{9}m

^{2}s

^{−}

^{1}, $\epsilon $ = 0.05, ${d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m, Figure S4. Normalized concentrations (${C}_{w}/{C}_{w,eq}$) as well as cumulative concentrations (${m}_{cum}$) in the column effluent vs. time (expressed as liquid to solid ratio: LS) for different combinations of sorbing particles and distribution coefficients (initial conditions depicted in Figure S3); left: without dispersion; right: with dispersion; solid lines: film diffusion cases, dashed lines: intraparticle diffusion cases, Figure S5. Initial concentration distribution in the column after the first flooding (up-flow) for two different bi-modal grain size distributions of fine and coarse particles; solid lines: fine particle mass fraction 10%; dashed lines: fine particle mass fraction 90%. (n = 0.45, v = 1.67 × 10

^{−}

^{5}m s

^{−}

^{1}, $\alpha $/x = 0 or 0.1, ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−}

^{1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−}

^{9}m

^{2}s

^{−}

^{1}, $\epsilon $ = 0.05,$\text{}{d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m, ${d}_{p,fine}$ = 63 $\mathsf{\mu}$m); top panel: without dispersion; bottom panel: with dispersion, Figure S6. Influence of different grain size fractions and distribution coefficients on normalized concentrations (${C}_{w}/{C}_{w,eq}$) as well as cumulative concentrations (${m}_{cum}$) in the column effluent vs. time (expressed as liquid to solid ratio LS); left: without dispersion; right: with dispersion; solid lines: fine particle mass fraction 10%; dashed lines: fine particle mass fraction 90%; kinetic parameters are the same as Figure S5, Figure S7. Initial concentration distribution in the column after the first flooding (up-flow) for different bi-modal material compositions of fine particles with low sorption capacity (${K}_{d}=$ 10 L kg

^{−1}) and coarse particles with high sorption capacity; left: 100% coarse particles (${K}_{d}=$ 100 L kg

^{−1}); middle: mixed sample with 10% fine particles; right: 100% fine particles; solid lines: film diffusion (FD), dashed lines: intraparticle diffusion cases (IPD); n = 0.45, v = 1.67 × 10

^{−}

^{5}m s

^{−}

^{1}, $\alpha $/x = 0 or 0.1, ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−}

^{1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−}

^{9}m

^{2}s

^{−}

^{1}, $\epsilon $ = 0.05,$\text{}{d}_{p,coarse}\text{}$= 2000$\text{}\mathsf{\mu}$m, ${d}_{p,fine}$ = 63$\text{}\mathsf{\mu}$m; top panel: without dispersion; bottom panel: with dispersion, Figure S8. Leachate concentrations (${C}_{w}$) as well as cumulative concentrations (${m}_{cum}$) in the column effluent vs. time (expressed as liquid to solid ratio: LS) for different combinations of fine particles with low sorption capacity (${K}_{d}=$ 10 L kg

^{−1}) and coarse particles with high sorption capacity (${K}_{d}=$ 100 L kg

^{−1}); left: without dispersion; right: with dispersion; solid lines: film diffusion cases, dashed lines: intraparticle diffusion cases; kinetic parameters are the same as Figure S7.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Empirical Relationships for the Estimation of Sherwood Numbers

^{−}

^{1}T

^{−}

^{1}] denotes the dynamic viscosity of the fluid. ${\rho}_{L}$ [M L

^{−3}] is the density of the fluid and $v$ [L T

^{−1}] denotes the flow velocity.

## Appendix B. Film Diffusion Coupled to Advective-Dispersive Transport

**Figure A1.**Discretization of the column into N parts (

**a**). Representation of the solid phase as a composition of grains having different sizes and properties, each discretized by a number of L shells (

**b**).

^{−}

^{3}], ${C}_{w,j-1}$ [M L

^{−}

^{3}] and ${C}_{w,j+1}$ [M L

^{−}

^{3}] denote the solute concentration in the water phase in volume $j$, $j-1$ and $j+1$, respectively. ${C}_{s,j}$ [M M

^{−}

^{1}] denotes the solute concentration in the solid phase in volume $j$.

^{−}

^{15}).

## Appendix C. Intraparticle Pore Diffusion Coupled to Advective-Dispersive Transport

^{−2}T

^{−1}] denotes the solute flux density into the external water phase. R and ${N}_{p}$ denote the radius and the total number of the spherical particles. The latter can be calculated by:

^{−15}).

## Appendix D. Length of the Mass Transfer Zone (${X}_{s}$) for the First Order Analytical Solution

#### Appendix D.1. Analytical Solution Based on the Film Diffusion Model

#### Appendix D.2. Analytical Solution Based on the Intraparticle Pore Diffusion Model

^{2}) and the mean square displacement ${\delta}_{p}$ (${\delta}_{p}=\sqrt{\pi {D}_{a}\text{}{t}_{c}})$ representing the diffusion distance, which grows with the square root of contact time between particles and water (${t}_{c}$) at early times, leads to:

#### Appendix D.3. Comparison of Analytical and Numerical Solution and Estimation of Mass Transfer Zone Length (${X}_{s}$)

^{−1}, 1 L kg

^{−1}and 10 L kg

^{−1}, respectively. The deviations between FD and IPD gradually vanish with increasing ${K}_{d}$ values. If the initial concentration in the column leachate is close to equilibrium, it may be used for the determination of ${K}_{d}$ (${K}_{d}={C}_{s,ini}/{C}_{w,peak}$); ${K}_{d}$ is overestimated if the initial effluent concentration does not reach equilibrium (${C}_{w,peak}<{C}_{w,eq})$. The length of the mass transfer zone (Equations (A28) and (A32)) may be used to assess equilibrium at the beginning of the column test.

**Figure A2.**Concentration increase in a water parcel (${C}_{w,peak}$) in the column during the first flooding (up-flow); solid lines: film diffusion; dashed lines: intraparticle diffusion; comparison between analytical (ana.) and numerical (num.) solutions. n = 0.45, v = 1.67 × 10

^{−}

^{5}m s

^{−}

^{1}, $\alpha $ /x = 0 (no dispersion), ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−}

^{1}, $t$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−}

^{9}m

^{2}s

^{−}

^{1}, $\epsilon $ = 0.05, ${d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m.

## Appendix E. Comparison of Analytical and Numerical Solution (Code Verification)

**Figure A3.**Concentration vs. distance in the up-flow column test after the first flooding of the column (initial condition). Comparison of the analytical solution (Equation (6), dashed lines) and numerical solution (solid lines); n = 0.45, v = 1.67 × 10

^{−5}m s

^{−1}, $\alpha $ = 0 (no dispersion), ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$ g kg

^{−1}, ${t}_{c}$ = 5 h, ${d}_{p,fine}$ = 63 $\mathsf{\mu}$ m.

**Figure A4.**Normalized and absolute concentration (${C}_{w}/{C}_{w,eq},{C}_{w}$) as well as cumulative concentration (${m}_{cum}$ ) in the column effluent vs. time (expressed as liquid to solid ratio: LS) for initial conditions shown in Figure A3; comparison of analytical solution (Equation (6), dashed lines) and numerical solution (solid lines).

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**Figure 1.**Initial concentration distribution in a column of length $x$ for the “pre-equilibrated” case (dashed line) and after the first flooding of an initially dry column from the bottom (solid line); no dispersion, ${R}_{d}=2$, after Grathwohl and Susset, 2009 [21].

**Figure 2.**Scheme of mass transfer limited by film diffusion during the first flooding with fixed concentration at the interface (because the infiltrating water is always contacting fresh material as it advances).

**Figure 4.**Initial aqueous concentration distributions in the column after the first flooding (solid lines) if mass transfer is controlled by film diffusion for three different distribution coefficients, ${K}_{d}$, compared to the pre-equilibrated case (dashed lines).

**Top panel**: without dispersion;

**bottom panel**: with dispersion; n = 0.45, v = 1.67 × 10

^{−5}m s

^{−1}, $\alpha $/x = 0 or 0.1, ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−1}, ${t}_{c}$ = 5 h, ${d}_{p,fine}$ = 63$\text{}\mathsf{\mu}$m.

**Figure 5.**Normalized and absolute concentration (${C}_{w}/{C}_{w,eq},{C}_{w}$) as well as cumulative concentration (${m}_{cum}$ ) in the column effluent vs. time (expressed as liquid to solid ratio: LS) for the initial conditions (depicted in Figure 4) established after the first flooding of the column (solid lines) compared to the pre-equilibrated case (dashed lines).

**Left column**: without dispersion;

**right column**: with dispersion.

**Figure 6.**Initial aqueous concentration distributions in the column after the first flooding depending on the mass transfer limitation; dotted lines: film diffusion (FD), dashed lines: intraparticle diffusion (IPD); solid lines: fast kinetics (equilibrium, fine particles).

**Top panel**: without dispersion;

**bottom panel**: with dispersion; n = 0.45, v = 1.67 × 10

^{−}

^{5}m s

^{−}

^{1}, $\alpha $/x = 0 or 0.1, ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−}

^{1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−}

^{9}m

^{2}s

^{−}

^{1}, $\epsilon $ = 0.05, ${d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m, ${d}_{p,fine}$ = 63 $\mathsf{\mu}$m.

**Figure 7.**Normalized concentrations (${C}_{w}/{C}_{w,eq}$) as well as cumulative concentrations (${m}_{cum}$ ) in the column effluent vs. time (expressed as liquid to solid ratio: LS) for different mass-transfer processes, given the initial conditions depicted in Figure 6.

**Left column**: without dispersion;

**Right column**: with dispersion.

**Figure 8.**Influence of sorption non-linearity: initial aqueous concentration distribution in the column after the first flooding (

**left graph**) and column effluent concentration (normalized:

**mid graph**, cumulative:

**right graph**) vs. time (expressed as liquid to solid ratio: LS); solid lines: linear sorption (${K}_{d}$ = 1 L kg

^{−1}); dashed lines: non-linear sorption case (Freundlich coefficient ${K}_{fr}$ = 7.94, exponent $1/n$ = 0.7); n = 0.45, v = 1.67 × 10

^{−5}m s

^{−1}, $\alpha $/x = 0 or 0.1, ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−9}m

^{2}s

^{−1}, $\epsilon $ = 0.05, ${d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m.

**Figure 9.**Behavior of bi-modal material compositions of sorbing and non-sorbing particles: initial concentration distribution in the column after the first flooding (

**top panel**) and column effluent concentration (normalized:

**mid panel**, cumulative:

**bottom panel**) vs. time (expressed as liquid to solid ratio: LS). Left column: homogeneous case with average ${K}_{d}\text{}$(= ${K}_{d,av}\text{}$ = 1 L kg

^{−1}); mid column: only 10% of the particles carry the contaminant at ${K}_{d}\text{}$ =$\text{}10\times {K}_{d,av}$; right column: only 1% of the particles carry the contaminant at ${K}_{d}$ = $100\times {K}_{d,av}$; the average ${K}_{d,av}$ of the entire material is the same for all compositions; solid lines: film diffusion cases, dashed lines: intraparticle diffusion case; n = 0.45, v = 1.67 × 10

^{−5}m s

^{−1}, $\alpha $ = 0 (no dispersion), ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−9}m

^{2}s

^{−1}, $\epsilon $ = 0.05,$\text{}{d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m.

**Figure 10.**Behavior of the bi-modal material compositions of fine and coarse particles: initial concentration distribution in the column after the first flooding (

**top panel**) and column effluent concentration (normalized:

**mid panel**, cumulative:

**bottom panel**) vs. time (expressed as liquid to solid ratio: LS); solid lines: fine particle mass fraction 10%; dashed lines: fine particle mass fraction 90%. (n = 0.45, v = 1.67 × 10

^{−5}m s

^{−1}, $\alpha $ = 0 (no dispersion), ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−9}m

^{2}s

^{−1}, $\epsilon $ = 0.05,$\text{}{d}_{p,coarse}$ = 2000 $\mathsf{\mu}$m, ${d}_{p,fine}$ = 63 $\mathsf{\mu}$m).

**Figure 11.**Behavior of bi-modal material compositions of fine particles with low sorption capacity (${K}_{d}=$10 L kg

^{−}

^{1}) and coarse particles with high sorption capacity (${K}_{d}=$ 100 L kg

^{−}

^{1}): initial concentration distributions in the column after the first flooding (

**top panel**) and the column effluent concentration (normalized:

**mid panel**, cumulative:

**bottom panel**) vs. time (expressed as liquid to solid ratio: LS). Left column: 100% coarse particles; mid column: mixed sample with 10% fine particles; right column: 100% fine particles; n = 0.45, v = 1.67 × 10

^{−5}m s

^{−1}, $\alpha $ = 0 (no dispersion), ${C}_{s,ini}$ = 1000 $\mathsf{\mu}$g kg

^{−1}, ${t}_{c}$ = 5 h, ${D}_{aq}$ = 1 × 10

^{−9}m

^{2}s

^{−1}, $\epsilon $ = 0.05,$\text{}{d}_{p,coarse}\text{}$ = 2000$\text{}\mathsf{\mu}$m, ${d}_{p,fine}$ = 63$\text{}\mathsf{\mu}$m.

Property | Symbol (Unit) | Reference and [Alternative Values] |
---|---|---|

Net column length | ${X}_{col}$ (cm) | 30 |

Inner column diameter | ${D}_{c}$ (cm) | 5.46 |

Total volume of column | ${V}_{tot}\text{}\left(\mathrm{L}\right)$ | 0.70 |

Dry solid density | ${\rho}_{s}$ (kg L^{−1}) | 2.60 |

Inter-granular porosity | n (-) | 0.45 |

Intraparticle porosity | $\epsilon $ (-) | 0.05 |

Solid mass in column | ${m}_{d}$ (kg) | 1 |

Liquid to solid ratio in column | $L{S}_{col}\text{}$(L kg^{−1}) | 0.31 |

Initial concentration in solid phase | ${C}_{s,ini}$ (µg kg^{−1}) | 1000 |

Contact time | ${t}_{c}$ (h) | 5 |

Dispersivity | $\alpha $ (m) | [0, 0.03] |

Water flow velocity | v (m s^{−1}) | 1.67 × 10^{−5} |

Aqueous diffusion coefficient | ${D}_{aq}$ (m^{2} s^{−1}) | 1 × 10^{−9} |

Particle diameters | $d$ (µm) | [63, 2000] |

Distribution coefficients | ${K}_{d}$ (L kg^{−1}) | [0.1, 1, 10, 100] |

Freundlich coefficients | ${K}_{fr}$ (µg kg^{−}^{1}:(µg L^{−}^{1})^{1/n}) | [1.58, 7.94, 39.81] |

Freundlich exponent | $1/n$ | 0.7 |

Sherwood number | $Sh=2+0.1P{e}^{1/2}$ (-) | [2.1, 2.6] |

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**MDPI and ACS Style**

Liu, B.; Finkel, M.; Grathwohl, P.
Mass Transfer Principles in Column Percolation Tests: Initial Conditions and Tailing in Heterogeneous Materials. *Materials* **2021**, *14*, 4708.
https://doi.org/10.3390/ma14164708

**AMA Style**

Liu B, Finkel M, Grathwohl P.
Mass Transfer Principles in Column Percolation Tests: Initial Conditions and Tailing in Heterogeneous Materials. *Materials*. 2021; 14(16):4708.
https://doi.org/10.3390/ma14164708

**Chicago/Turabian Style**

Liu, Binlong, Michael Finkel, and Peter Grathwohl.
2021. "Mass Transfer Principles in Column Percolation Tests: Initial Conditions and Tailing in Heterogeneous Materials" *Materials* 14, no. 16: 4708.
https://doi.org/10.3390/ma14164708