Research Modeling the Desorption of Organic Contaminants from Long-Term Contaminated Soil Using Distributed Mass Transfer Rates TERESA B. CULVER,* STEPHEN P. HALLISEY, DIPAK SAHOO, JAMES J. DEITSCH, AND JAMES A. SMITH Department of Civil Engineering, University of Virginia, Charlottesville, Virginia 22903-2442
Simulation models for the fate and transport of groundwater contaminants are important tools for testing our understanding of transport phenomena at long-term contaminated sites and for designing remedial action plans. A finite difference formulation for contaminant transport including a distribution of contaminant mass-transfer rates between the water and soil is developed. Optimal model simulations using both log-normal and γ distributions of mass transfer rates are compared to the two-site equilibrium/ kinetic model. In all cases, optimal sorption parameters were determined by best fit to laboratory data. For desorption of trichloroethene from long-term contaminated soils, the distributed mass-transfer rate model provided significantly improved simulations of aqueous concentrations, as compared to the two-site model, for both batch and soil column experiments. However, use of an apparent partition coefficient demonstrated that the performance of the two-site model was very sensitive to the value of the partition coefficient, while the performances of the distributed models were robust over a wide range of partition coefficients. Desorption studies in continuous-flow stirred tank reactors with laboratory-contaminated soils demonstrated that as the length of the contamination period increases, the simulation capability of the two-site model decreases.
Introduction Simulation models for the fate and transport of groundwater contaminants are important tools for testing our understanding of transport phenomena and for designing remedial action plans. For organic contaminants, a key process to include in any transport model is sorption. Although most models have assumed linear equilibrium sorption, many laboratory and field observations cannot be accurately simulated using the assumption of linear equilibrium sorption (1). For reactive solutes, observed breakthrough curves may be asymmetrical with long tails (2), which may be difficult to reproduce assuming linear equilibrium sorption. In addition, long-term sorption nonequilibrium has been found under field conditions (3-5). Some researchers have characterized this nonequilibrium as a period of fast sorption kinetics * Corresponding author phone: 804-924-6375; fax: 804-982-2951; e-mail:
[email protected].
S0013-936X(96)00094-6 CCC: $14.00
1997 American Chemical Society
followed by a period of slow kinetics (6-8). Furthermore, there is growing evidence that desorption rates decrease as the length of exposure time increases (6, 8-10). From a management perspective, knowing both when a contaminant will arrive at a given location in an aquifer and how long the aquifer must be pumped to eliminate the contaminant are of extreme importance. Many groundwater contaminants are toxic at extremely low concentrations, making inaccurate description of tailing behavior costly and hazardous. This work focuses on simulation of the desorption of organic contaminants from long-term contaminated soil, which is especially important for groundwater remediation designs. Nonequilibrium sorption may govern systems with asymmetrical breakthrough curves and time-dependent sorption rates. At the particle scale, the governing processes of kinetic sorption and desorption are complex and poorly understood. A variety of processes, all related to the complexity and heterogeneity of natural particles, may interact to create the observed sorption and desorption rates (11). In addition to particle-scale processes, large-scale heterogeneity can create physical nonequilibrium, due to regions of mobile and immobile water, that may be observed in the transport of sorbing and nonsorbing solutes (1, 12). Although a variety of nonequilibrium sorption models have been formulated, they have been only partially successful in fitting experimental data, especially long-term desorption data (1, 11, 12). A commonly used nonequilibrium model is the two-site equilibrium/kinetic model, which may represent a variety of sorption mechanisms, including sorption rate limitations, first-order approximations of intrasorbent diffusion, and physical nonequilibrium (1, 12). The two-site equilibrium/kinetic model is intended to account for the experimental evidence of two different sorption or mass transfer rates. The following equations govern the sorption or mass transfer:
∂S1 ∂C ) fKD ∂t ∂t
(1)
∂S2 ) R[(1 - f)KDC - S2] ∂t
(2)
where S is the sorbed concentration (mass sorbed/mass soil) and subscripts 1 and 2 represent different sorption sites or regions, respectively; f is the fraction of sites in equilibrium; KD is the distribution coefficient (L3/M), C is the aqueous concentration (M/L3), R is a first-order mass-transfer rate coefficient (1/T), and t is the time dimension (T). The total sorbed concentration is the sum of S1 and S2. Typically both f and the mass-transfer rate, R, are fit to observed data (13). Due to localized soil heterogeneity and the complex structure of natural particles, the use of models with a single equilibrium partition coefficient (14-16) or a single masstransfer coefficient has been questioned (6, 9, 14, 15, 17, 18) for even a small soil sample. A simple solution to the problem of localized soil heterogeneity would be to extend the single kinetic site of the transport model into multiple kinetic compartments, each with a different mass-transfer rate. However, for each compartment added, the number of fitting parameters increases by two (the transfer rate and the associated likelihood of occurrence). Several approaches are
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possible for determining values for these additional parameters. A detailed soil analysis could be done, and the parameters could be based on some characteristic of the soil (14, 15, 19, 20). A second approach is to utilize a series approximation to the diffusion rates, which can be based on the geometry of the porous medium (14). The third approach, which is feasible without extensive characterization of the soil composition, is to assume that the variation in the masstransfer rates between kinetic compartments can be described by a probability density function. Two non-negative distributions, the γ distribution (6, 15, 21, 22) and the log-normal distribution (15), have been tested. Only two parameters are required to describe each of these distributions, thus they have the same degrees of freedom as the traditional two-site equilibrium/kinetic model. There are several reasons for employing the γ distribution. One can reasonably assume that the mass-transfer coefficient is in some way dependent upon pore size, and pore sizes in soils have been found to vary according to a γ distribution (23). Flexibility is another benefit of the γ distribution, since it can take on a wide variety of shapes (ref 24, p 151). The γ probability density function is given by
p(R) )
β-ηRη-1 R exp β Γ(η)
( )
(3)
where η (the shape parameter) and β (the scale parameter) are positive parameters, and Γ(η) is the γ function. Γ(η) is described by the equation:
Γ(η) )
∫
∞
0
x
η-1
exp(-x) dx
(4)
where x is a dummy variable of integration. Log-normal distributions have been widely used in subsurface applications. Both hydraulic conductivity and particle size distributions are typically considered to be lognormally distributed (25). Local soil diffusion rates have been related to particle size (14, 19), suggesting that a log-normal distribution would be a potential candidate to describe the mass-transfer rate distribution. The log-normal probability density function is given by
p(R) )
(
)
1 1 exp - 2(ln(R) - µ)2 2σ x2πσR
(5)
where µ and σ are distribution parameters. Connaughton et al. (6) utilized a γ distribution for the probability density function of the mass-transfer rates to model naphthalene desorption from long-term contaminated soils and laboratory-contaminated soils using batch experiments. They found that the γ-distributed rate model provided a far superior fit to the experimental data than the two-site equilibrium/kinetic model. Analysis of the fraction of the initial mass remaining versus time indicated a significant increase in resistance to desorption with time, which cannot be described by a single rate coefficient model that would predict the resistance to remain constant. A γ distribution was also used by Chen (21, 22) to describe the distribution of the time scales of mass transfer (the inverse of the masstransfer rate constant). For pesticide transport in soil columns, Chen (21) demonstrated that the γ model provided experimental data fits superior to those of the two-site equilibrium/kinetic model and the spherical diffusion model. To model batch desorption of diuron, Pedit and Miller (15) tested both the γ and log-normal probability distributions for the mass-transfer coefficient. These two-parameter probability distribution models were compared to a wide variety of other sorption formulations, including diffusion models based on multiple particle-size classes and probability distribution models where the equilibrium partition coefficient was correlated with the variation in the mass-transfer
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rate. They found that the model with a log-normal distribution for the mass-transfer rate and a constant equilibrium partition coefficient provided the best fit of all two-parameter models tested and performed nearly as well as the best threeparameter models. In this research, a distributed mass-transfer rate model is developed and applied to desorption of trichloroethene (TCE) from soils that can be expected to exhibit slow mass transfer, namely, long-term field-contaminated soils and high organic content laboratory-contaminated soils. Desorption from long-term contaminated soils is modeled for both batch experiments, where only mass transfer into a static aqueous phase occurs, and for soil column experiments, where mass transfer, hydrodynamic dispersion, and advection all occur. To further explore the significance of the length of contamination on the desorption behavior, desorption from soils contaminated for different time periods is compared. For the laboratory contaminations, a high organic content soil is used. Since an organic contaminant would be expected to have high sorption to this soil, any differences in desorption that might occur over relatively short contamination periods would be easier to detect. Desorption was measured in continuous-flow stirred tank reactors (CFSTRs), which involve mass transfer into a flowing aqueous phase. The simulation capabilities of a γ distributed rate model, a log-normallydistributed rate model, and the two-site equilibrium/kinetic model are compared. Like the γ model described by Chen and Wagenet (22), our column model includes advection and dispersion, in addition to sorption processes. However, our model adheres to a conventional description of dispersion, as implemented in the advection-dispersion equation, instead of defining a random velocity field as done by Chen and Wagenet (22). Furthermore, Chen and Wagenet considered the variation in the time scale (1/R), as opposed to the variation in R directly. Our model also differs from Chen’s in that it provides an implicit finite difference solution to contaminant transport (26) rather than an analytical solution. The analytical solution offers the advantage of being capable of simulating a complete distribution of sorption sites, while the finite difference model is limited to a finite number of sorption sites. Deviation of the discrete approximation from the continuous distribution are explored in the following section. However, numerical models provide greater flexibility than analytical solutions, such as the ability to handle variability in flow rates, boundary conditions, and initial conditions. Furthermore, the initial sorption conditions do not need to be at equilibrium using the numerical model. This is an important capability, since field samples may exhibit long-term nonequilibrium.
Model Development and Verification The model developed in this study is a one-dimensional masstransfer rate model. Either a γ distribution or a log-normal distribution can be used to determine the mass-transfer rate coefficients for each site. Although the model is referred to as a multi-compartment model throughout this paper, the model does not infer anything about the mechanisms of mass transfer. The term “compartment” simply refers to a fraction of the soil that exhibits the same mass-transfer behavior. The partial differential equations defining the one-site first-order kinetic model are the starting point for the development of the multi-compartment model. These equations for the change in the sorbed concentration and the change in the aqueous concentration are described by eqs 6 and 7, respectively:
∂S ) R(KDC - S) ∂t
(6)
∂C RF ∂C ∂2C ) D 2 - v - (KDC - S) ∂t ∂x θ ∂x
(7)
where F is the soil bulk density (M/L3), θ is the volumetric water content (L3/L3), D is the hydrodynamic dispersion coefficient (L2/T), v is the average pore water velocity (L/T), and x is the spatial dimension (L). For incorporation into a finite difference numerical model, the continuous probability distribution of the mass-transfer rate must be discretized into a finite number of compartments, NK. In expanding to a multi-compartment kinetic model, each of these kinetic compartments is assumed to behave as the single site in the one-site model, except each compartment has a different characteristic mass-transfer rate. Furthermore, to reduce the number of variables, each compartment is assumed to occupy an equal fraction of the soil. This assumption does not seriously limit the ability of the model to approximate a continuous distribution of kinetic rate coefficients, as will be shown later. Given these assumptions for the discrete compartments, the change in the total sorbed concentration is given by the following equation:
)
∑ R (fK C - S ) k
D
k
(8)
k)1
where f is the fraction of sites in each compartment, Rk is the mass-transfer rate for compartment k, and Sk is the mass sorbed at compartment k with respect to the total mass of the soil. Similarly, extending eq 7 into a multi-compartment form of the advection-dispersion equation results in the following expression:
∂C dt
2
∂C
NK
∂C -v (fKDC - Sk) ∂x k)1 θ ∂x
)D
2
∑
FRk
Q )
dt
(Cin - C) -
Vw
value
β for γ distribution η for γ distribution v (cm/min) D (cm2/min) θ F (g/cm3) KD (cm3/g) length of domain (cm)
0.00225 0.4444 0.01 0.05 0.4 1.64 10.0 450
TABLE 2. Percent Deviation of γ-Distributed Results from Results Using NK ) 200 no. of compartments (NK)
% deviation from NK ) 200
100 50 25 20 10 1
0.000957 0.00324 0.0961 0.236 1.08 9.75
NK
Ms Rk
k)1
Vw
∑
(fKDC - Sk)
follows:
% error )
× 100 xSSD SSQ
(11)
where NP
SSD )
∑(C
m,n
- Cd,n)2
n)1
(9)
NP
SSQ )
Equations 8 and 9 are the governing equations for the column model developed in this work. CFSTR and the batch experiments have similar governing equations, which are essentially simplified forms of the full advection-dispersion equation given by eq 9. The general one-site kinetic mass-transfer model for the CFSTR (9, 27, 28) can be extended to the multi-site formulation, as follows:
dC
parameter
NK
∂S ∂t
TABLE 1. Input Values Used in Model Verification
(10)
where Q is the volumetric flow rate through the reactor (L3/ T), Cin is the inflow solute concentration (M/L3), Ms is the mass of the soil in the reactor (M), and Vw is the volume of the water in the reactor (L3). For no flow conditions, the first term on the right-hand side of eq 10 goes to zero, and the resultant equation is the governing equation for the batch experiments. Equation 8 describes the change in the sorbed phase for the CFSTR and batch systems. For each of the NK distributed compartments in eqs 8-10, a representative value for the mass-transfer rate coefficient must be determined. In this study, the median value of the mass-transfer rate coefficient within each compartment was utilized as the representative value. The median is a good approximation of the mean, and it can be obtained fairly easily. For each compartment, the median mass transfer rate is determined from the cumulative probability distributions using a bisection search method. The γ and log-normal cumulative probability functions are standard probability functions, which were evaluated using the IMSL Special Functions Library (29). The model fits observed data by determining the distribution parameters that minimize the percent error in the simulated concentrations. The percent error was defined as
∑(C
d,n)
2
n)1
In eq 11, SSD represents the sum of the squared deviations, SSQ represents the sum of the squared observed data points, NP is the total number of data points for comparison, Cm,n is the modeled concentration of the nth data point, and Cd,n is the corresponding observed data point. To verify the numerical model developed in this work, four steps were taken. First, the model was verified as a onesite kinetic model by setting the number of kinetic compartments to 1. Input parameters are listed in Table 1, and the simulation period was 1.14 years. On comparison to an analytical solution of the one-site kinetic model (13), the percent error in the estimates as defined by eq 11 was less than 1.5% when the Peclet number of the numerical approximation [(v∆x)/D] was less than 0.5 (26), indicating that the numerics of the basic transport model were performing well. In the second step, sensitivity of the solution to the number of discrete kinetic sorption compartments, NK, was tested using the parameter values given in Table 1. The following discussion is for the γ model; results using a log-normal distribution are similar. The percent deviation of each of the simulations from the 200 compartment simulation is displayed in Table 2. The percent error is defined in eq 11. In order to exactly represent a γ distribution, an infinite number of compartments must be used. But since the results do not significantly change for values of NK greater than 50, it is unnecessary to use more than 50 compartments, at least for the time scale shown. The 10- and 1-compartment runs deviate more substantially from the NK ) 200 results. Based upon Table 2, the number of kinetic sorption compartments in the future work was set to 25 as a reasonable numerical approximation of a continuous distribution (less than a 0.1% deviation).
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TABLE 3. System Parameters for All Experimentsa experiment
parameter
value
batch
initial sorbed TCE, µg/kg 1100 KD (cm3/g) 0.69 KDapp (cm3/g) 39.6 columns 1 and 2 initial sorbed TCE, µg/kg 1100 θ 0.38 1.61 F (g/cm3) KD (cm3/g) 0.69 KDapp (cm3/g) 39.6 D (cm2/h) 2.0 equilibration period (h) 2.0 length of domain (cm) 30 v (cm/h) 6.45 column 1 v (cm/h) 8.37 column 2 CFSTR 1-week contamination initial aq TCE (mg/L) 255 KD (cm3/g) 17.55 Q (mL/h) 2.16 CFSTR 4-week contamination initial aq TCE (mg/L) 245 KD (cm3/g) 17.55 Q (mL/h) 2.21 a
All input parameters were determined experimentally or by tracer
fits.
To compare our numerical model to the only analytical solution of the γ-distributed sorption transport model, including the impact of advection and dispersion (21, 22), we had to convert a random velocity field to an equivalent dispersion coefficient and average porewater velocity. Chen (21) provided the equations for the conversion (p 127). Furthermore, our model had to be written in terms of the distribution of the time scale of mass transfer (1/R). Using our 25-compartment numerical model and the γ sorption parameters determined in the earlier study, we were able to reproduce a column transport simulation of Chen (ref 21, p 70). Thus, we concluded our discrete kinetic simulation model was a good approximation of the continuous model. As a final confirmation, the parameter fitting routine was compared to the nonlinear regression model of Parker and van Genuchten (13). The best two-site parameters were determined by fitting the observed aqueous concentrations for column 1 (see Figure 2). The system parameters listed in Table 3 were used with the following variations required to satisfy the assumptions of the analytical transport model (13). To satisfy the analytical model steady-flow conditions, the preliminary equilibration period with no flow was not modeled, and it was assumed that the sorption was in equilibrium at the start of the flow period. With the above assumptions, which do not apply to our actual system, it was necessary to assume a lower total initial concentration to reasonably approximate the observed aqueous concentrations. For this hypothetical test, we assumed that the initial aqueous concentration was 300 µg/L with a corresponding equilibrium initial sorbed concentration of 300 µg/kg. Under these conditions, the Parker and van Genuchten model determined the best-fit parameters for the two-site equilibrium/kinetic sorption model to be f ) 0.415 (fraction of equilibrium sites) and R ) 0.00443/h (mass-transfer rate). The corresponding parameters for our routine were f ) 0.407 and R ) 0.00442/h, showing an excellent selection in the optimal parameters. The slight difference is likely caused by the small degree of numerical error in the finite difference simulations when compared to the analytical simulation used in the Parker and van Genuchten model, although the deviation is well within normal experimental accuracy.
Materials and Methods The soil utilized in the batch, CFSTR, and column experiments was collected at Picatinny Arsenal, NJ. Because of improper disposal of trichloroethene (TCE) at the Arsenal between 1950 and 1985, the unconfined, sand-and-gravel aquifer has been
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contaminated for more than 10 years. A detailed description of the field site’s hydrogeology and groundwater contamination is given by Imbrigiotta et al. (30). For the batch and column experiments, aquifer material from different sites, including a peat layer, in the TCE-contaminated zone of the Picatinny site was composited and used to represent longterm contaminated soil. Prior to being used in either batch or column experiments, the soil was partially dried for 24 h at 20 °C and passed through a 1-mm sieve. For the CFSTR experiments, the soil samples were collected from a peat layer at the site. The peat soil was dried for 48 h at 105 °C to remove any sorbed TCE. Subsequent extraction of the soil sample and analysis of the extract for TCE indicated that the remaining soil TCE concentration was less than 1% of the sorbed concentrations used in laboratory experiments. Additional details of the soil preparation procedure are given elsewhere (9). The organic carbon content of the soil samples was quantified by Huffman Laboratories (Golden, CO) and determined to be 1.04% for the field-contaminated soil used in the batch and column experiments and 24% for the laboratory-contaminated peat soil used in the CFSTR experiment. A mixture of a 500-µCi sample of [14C]TCE (specific activity equal to 6.2 mCi/mmol) obtained from Sigma Chemical Co. and non-radioactive TCE was used to contaminate soils for the CFSTR experiments. The resultant chemical and radiochemical purity of the radioisotope was greater than 98% (9). The initial concentration of TCE in the partially air-dried field-contaminated soil was measured using a hot methanol extraction (4, 31). Koller et al. (32) observed that air-drying of contaminated soil from the Picatinny Arsenal had a negligible effect on the measurement of sorbed TCE concentrations. A 15-mL glass centrifuge tube was filled with 6 g of soil and 11 mL of methanol and incubated at 70 °C for 48 h with intermittent shaking. The soil and methanol were separated by centrifugation for 30 min at 1000g. The supernatant methanol was mixed with water and hexane, and the TCE concentration in the hexane extract was quantified with a Perkin Elmer Autosystem gas chromatograph with an electron-capture detector. Equilibrium distribution coefficients (KD) for TCE sorption to the soils were determined by a conventional batch uptake methodology, which is described in detail elsewhere (9). Equilibrium was assumed after 72 h of uptake for the soil used in the batch and column experiments and after 48 h for the high organic content soil used in the CFSTR experiments. Prior to use in the equilibrium experiments, the TCE concentration of the field-contaminated soil was reduced to below the detection level by heating the soil sample for 48 h at 105 °C. In addition, for the field-contaminated soil, an apparent KDapp was defined as the ratio of the final sorbed concentration to the final aqueous concentration for the batch desorption experiment. Each batch reactor consists of 6 g of field-contaminated soil and 11 mL of deionized, organic-free water combined in a 15-mL glass centrifuge tube with a Teflon-lined septum cap. A set of these batch reactors were shaken at 20 °C on a Boekel Orbitron rotator (Model 260200). Duplicate batch reactors were sacrificed over time for analysis of the aqueousphase TCE concentration. The soil and water phases were separated by centrifugation with a DuPont Sorvall RT 6000B centrifuge at 1000g for 30 min. Five milliliters of the supernatant was transferred from the batch reactor to a 7-mL glass vial containing 1 mL of hexane. Following mixing and phase separation, the concentration of TCE in the hexane extract was quantified by gas chromatography with an electron-capture detector. A separate TCE recovery analysis was completed using a set of centrifuge tubes containing 15 mL of water and TCE but no soil to determine if losses by volatilization or sorption to glassware was significant. These tubes were shaken and centrifuged in a manner analogous
to tubes containing contaminated soil, and the aqueous-phase TCE concentration was quantified for different incubation times. The average recovery of TCE from these tubes was 94.8 ( 5.9%. The procedure used for column experiments is similar to the procedure described by Koller et al. (8). Duplicate glass columns (30 cm length × 2.5 cm i.d.) with Teflon end fittings and stainless steel screens were filled with partially air-dried field-contaminated soil. A Manostat peristaltic cassette pump was used to pump deionized, organic-free water through the columns. To wet the columns, water was allowed to fill the column from the bottom. Once the water filled the column, flow was stopped, and the column was allowed to equilibrate for 2 h. After the equilibration period, water was pumped through the column at a constant rate. Effluent from each column passed through a stainless steel syringe needle into a 5-mL glass sampling tube (7 cm length × 1.5 cm i.d.) sealed at each end with Supelco mininert valves. For analysis of the effluent concentration, the sampling tube was periodically disconnected from the flow path, and TCE in the 4-mL water sample was extracted into 1 mL of hexane. The TCE concentration in the hexane extract was quantified by gas chromatography with an electron-capture detector. At the conclusion of each desorption experiment, a Br- tracer test was conducted for each soil column to quantify the average porewater velocity (v) and the coefficient of hydrodynamic dispersion (D). Br- at a concentration of 800 mg/L was added to the inflow water for each column. The Br- in the effluent was quantified over time by analysis with a specific-ion electrode. The average pore water velocity and the coefficient of hydrodynamic dispersion were determined by adjusting v and D until an optimum fit to the tracer data was obtained. A continuous-flow stirred tank reactor (CFSTR) apparatus was used to study the desorption kinetics of TCE from the laboratory-contaminated peat soil to water. Cylindrical glass reactors (1.0 cm i.d. × 15 cm) with Teflon end fittings and stainless steel screens were used for the CFSTRs. Each CFSTR contained 5 g of soil and approximately 10 mL of deionized, organic-free water. The CFSTRs were contaminated with [14C]TCE. A specific volume of [14C]TCE was injected into each of the CFSTRs, such that an approximate equilibrium aqueous TCE concentration of 250 mg/L would result. The CFSTRs were then continuously shaken at 150 rpm for either a 1- or 4-week equilibration period. Following the equilibration period, deionized, organic-free water was pumped through the shaking CFSTR at 2 mL/h. The effluent from the CFSTRs was collected in 7-mL scintillation vials filled with 5 mL of scintillation cocktail and analyzed for TCE concentration on a Packard 1900CA liquid-scintillation counter. Experiments for each equilibration period were performed in duplicate, and the data in the results section are the averages of the duplicate CFSTRs. A complete description of the CFSTR apparatus and experimental procedure is given by Deitsch and Smith (9). Contaminant concentrations for each experimental system were simulated using three different sorption models: the two-site equilibrium/kinetic model, the γ-distributed masstransfer model, and the log-normally-distributed masstransfer model. All system input parameters are listed in Table 3, including the measured values of the initial sorbed concentration and the tracer-based dispersion coefficient. For each model, the two parameters that provided the minimum percent error in the simulated concentrations (eq 11) were determined. For the CFSTR experiments, the system was assumed to have reached equilibrium after the equilibration period. For the column experiments, there was no inflow during the preliminary equilibration period. Once flow began, the upper boundary condition was a Cauchy condition with zero flux (no TCE in influent water), and the lower boundary condition was a Neumann condition with the concentration gradient set to zero. Due to the inability to
FIGURE 1. Measured aqueous TCE concentrations and optimal model simulations for batch desorption experiments. The distributed curve is for both the γ and log-normal simulations that overlap.
TABLE 4. Minimum Percent Error and Respective Distribution Parameters for Each Experiment Given Each Desorption Modela experiment batch
model
two-site γ log-normal column 1 two-site γ log-normal column 2 two-site γ log-normal CFSTR (4 week) two-site γ log-normal CFSTR (1 week) two-site gamma log-normal a
% error
parameter 1
parameter 2
17.68 13.33 13.34 45.71 18.04 19.01 58.20 30.51 31.24 11.12 9.95 8.77 8.36 14.44 13.38
R ) 0.000077 β ) 0.0306 µ ) -18.12 R ) 0.000318 β ) 0.0162 µ ) -10.13 R ) 0.000296 β ) 0.0469 µ ) -11.01 R ) 0.00725 β ) 0.17 µ ) -5.4 R ) 0.0064 β ) 1.39 µ ) -4.07
f ) 0.0064 η ) 0.0154 σ ) 6.32 f ) 0.0100 η ) 0.0731 σ ) 2.97 f ) 0.044 η ) 0.0434 σ ) 3.68 f ) 0.90 η ) 0.211 σ ) 2.57 f ) 0.68 η ) 0.173 σ ) 3.32
For batch and column studies, the KD was 0.69 cm3/g.
reach sorption equilibrium with field-contaminated soils without an extremely long equilibration period, the numerical model simulated both the no-flow equilibration period and the period of flow. For the column experiments, the initial conditions given in Table 3 are for the start of the equilibration period. Thus, for the column experiments, it was not assumed that sorption equilibrium had been reached after the short equilibration period.
Results The batch experimental data was modeled using each desorption model. Based on the uptake KD, both the distributed models provided an excellent fit of the data (Figure 1), while the equilibrium/kinetic two-site model was unable to capture the general behavior of the TCE desorption from the long-term field-contaminated soil. The best fit of the two-site model had a significantly greater percent error than the best fits of the distributed parameter models (Table 4). However, if the apparent desorption KDapp is used, which was based on the final concentrations of the batch desorption experiment and thus was not an independent experiment, the performance of the two-site model improves dramatically (Figure 1), and the percent error drops to 9.76% (Table 5). For the distributed model, the overall fit is not significantly affected, with only a 0.4% increase in the percent error (Table 5) and the simulated concentrations (not shown) fall between the curves for the distributed models using KD and for the two-site model using KDapp.
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TABLE 5. Minimum Percent Error and Respective Distribution Parameters for Each Experiment with Field-Contaminated Soil using KDapp (39.7 cm3/g) experiment batch column 1 column 2
model
% error
parameter 1
parameter 2
two-site γ log-normal two-site γ log-normal two-site γ log-normal
9.76 13.74 13.74 36.24 19.54 19.88 21.48 30.13 26.41
R ) 0.000243 β ) 6.3 × 10-9 µ ) -8.291 R ) 0.000285 β ) 0.0492 µ ) -10.63 R ) 0.000169 β ) 0.4879 µ ) -17.22
f ) 0.0038 η ) 39745 σ ) 0.00712 f ) 0.034 η ) 0.0541 σ ) 3.76 f ) 0.049 η ) 0.0230 σ ) 8.66
FIGURE 4. Optimal mass-transfer distributions to describe desorption from long-term contaminated soils under batch and column conditions using KD, where (a) compares the γ-distributed rates and (b) compares the log-normally-distributed rates. r j is the mean of the set of discrete rk. Values of rk that have little contribution to desorption over the entire period of the experiment (rk × length of experiment < 0.01) are not shown. FIGURE 2. Measured effluent aqueous TCE concentrations and optimal model simulations for column experiment 1.
FIGURE 3. Measured effluent aqueous TCE concentrations and optimal model simulations for column experiment 2. The poor performance of the two-site formulation was most dramatic for the two column experiments. Based on KD, the percent error for the equilibrium/kinetic two-site model was approximately double the percent errors achieved by the distributed models (Table 4). The distributed models captured the observed TCE desorption behavior more accurately, especially at extended time periods (Figures 2 and 3). With less than 25% of the initial sorbed mass desorbed, from the field-contaminated soils, the column experiments were terminated, due to aqueous non-detectability (
