Environ. Sci. Technol. 2000, 34, 2446-2452
Rate-Limited Sorption and Desorption of 1,2-Dichlorobenzene to a Natural Sand Soil Column TERESA B. CULVER,* ROBERTA A. BROWN,† AND JAMES A. SMITH Program of Interdisciplinary Research in Contaminant Hydrogeology (PIRCH), Department of Civil Engineering, University of Virginia, Charlottesville, Virginia 22904-4742
The rate of sorption of 1,2-dichlorobenzene (DCB) onto a natural sandy soil in water-saturated soil columns was measured. Desorption rates were also measured after allowing the solute-sorbent system to equilibrate during a no-flow period, ranging from 3 to 49 d. Two-site equilibrium/kinetic and Γ-distributed rate sorption models were then used to determine a single set of sorption parameters that could describe both the uptake and desorption of 1,2-DCB from these columns. Ninety-five percent confidence regions for the mass transfer rates and the two-site parameters were generated. Both the Γ-distribution model and the twosite model could describe uptake and desorption with a single set of parameters, although the Γ-distribution model was better at the 95% confidence level in three of five experiments. There was no statistically significant change in the sorption parameters between 3 and 49 d of noflow. Allowing the Γ-distributions of rates to differ between uptake and desorption improved the ability to fit the observed data, and for some experiments it could be concluded with 95% confidence that the rate of desorption is different than the rate of uptake.
Introduction Sorption is a key process impacting the subsurface fate and transport of an organic contaminant. However, the common assumption of equilibrium sorption cannot accurately describe many laboratory and field observations (1-6). Although there is growing awareness that sorption kinetics may be important, the kinetics of contaminant sorption is still relatively poorly understood (7). Three important issues that are yet to be resolved are as follows: (1) what is the most reasonable and accurate mathematical formulation to describe contaminant sorption, (2) is the rate of contaminant uptake the same as the rate of contaminant desorption, and (3) does the length of time that a soil is exposed to a contaminant impact the desorption kinetics. 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 (6-9). One formulation, the two-site equilibrium/kinetic model, may approximate a variety of transport mechanisms including physical nonequilibrium and intra* Corresponding author phone: (804)924-6375; fax: (804)982-2951; e-mail:
[email protected]. † Present address: Department of Geography and Environmental Engineering, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218. 2446
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ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 34, NO. 12, 2000
sorbent diffusion (6, 8). 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 ) feKD ∂t ∂t
(1)
∂S2 ) R[(1 - fe)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, fe is the fraction of sites in equilibrium, KD is the equilibrium 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 fe and the mass-transfer rate, R, are fit to observed data (10). Due to localized soil heterogeneity and the complex structure of natural particles, the use of models with a single mass-transfer coefficient has been questioned (4, 11-16) for even a small soil sample. One approach 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. Several approaches are possible for determining the parameter values for these multikinetic models, including basing the values on a detailed soil analysis (13, 14, 17, 18) or on a series approximation of the diffusion rates, which can be related to the geometry of the porous medium (13). A third approach, which is feasible without extensive characterization of the soil composition, is to assume that the variation in the mass-transfer rates between kinetic compartments can be described by a probability density function. Two nonnegative distributions, the Γ-distribution (11, 14, 19-25) and the log-normal distribution (14, 21, 26), 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. In the above mathematical formulations, the processes of contaminant uptake and desorption are typically not mathematically differentiated. However, researchers have observed a desorption hysteresis (7, 9, 25, 27, 28) suggesting that desorption may occur at a slower rate than sorption. Furthermore, the rate of desorption may not be constant. Desorption has been described as an initial immediate release of the contaminant followed by a much slower desorption of the chemical occurring over months or years (29-31). The rate of desorption may decrease as the period of contamination increases (11, 21). Variations in the rate of desorption over time may be a factor in the long-term sorption nonequilibrium that has been observed under field conditions (30-33). In this paper, the results and analyses of column experiments measuring the rates of uptake and desorption of 1,2dichlorobenzene (1,2-DCB) to and from Picatinny sand are presented. The objectives of this study are the following: to compare the ability of a two-site equilibrium/kinetic model and a distributed sorption model to describe the observed uptake and desorption in a soil column, to compare rates of uptake and desorption for a single solute-sorbent combination in a soil column, and to compare rates of desorption for 10.1021/es9906655 CCC: $19.00
2000 American Chemical Society Published on Web 04/29/2000
the same solute-sorbent combination given different length periods of no-flow. Previous studies, which have indicated that the rates of uptake and desorption of organics onto natural soils and organobentonites may differ, have been performed in batch systems (9, 25, 27, 28, 34). Mass-transfer rates in batch systems, which are generally shaken and exhibit no large-scale soil structure, may not be representative of rates measured in a bulk soil (21). Modeling Approach. The numerical model developed and presented in ref 21 was utilized for the analysis of the uptake and desorption experiments. This one-dimensional finite difference model can simulate advection-dispersion with two-site equilibrium/kinetic sorption or with a distribution of mass transfer rates. For the distribution of mass transfer rates, this work assumed a gamma (Γ) probability density function. In the numerical model, the continuous Γ-distribution of rates is approximated by NK discrete kinetic sorption sites, each representing the same fraction of sorption sites but having different rates of sorption (21, 25). The model (21) can also fit experimental data by determining the set of fitting parameters, θ, that will minimize the percent error. In this work, θ is equal to (R j ,σ), the mean and standard deviation of the distribution of mass transfer rates. The best fit parameters, θˆ , minimize the percent error, which is defined as
E(θˆ ) )
∂S
NK
)
∂t ∂C ∂t
)D
∂2 C ∂x
2
x
(3)
NP
∑(C
m,j
NP
∑(C
2
d,j)
j)1
SSD is the sum of squared deviations and SSQ is the sum of squared experimental data points evaluated at θˆ . NP represents the total number of experimental data points, Cm,j is the modeled concentration of the jth data point, and Cd,j is the corresponding experimental data point. The percent error is minimized using a constrained quasi-Newton algorithm (35). The numerical model developed in Culver et al. (21) was extended in two ways for this study. The boundary conditions were modified to allow for equilibration periods with no flow into or out of the column. A routine to generate confidence intervals around the fitted parameters was also added. Draper and Smith (36) defined the confidence interval for the parameters of a nonlinear regression as all sets of parameters that satisfy the following relationship
x1+ (NPm- m)F(m,NP - m,ω) ) E
∂C
-v
∂x
D
m,ω
(4)
where F is the standard F-statistic, m is the number of fitting parameters, and ω is the expected confidence level (e.g., 95% confidence). Em,min is the minimum or best-fit error and Em,ω is the percent error at the boundary of the ω-confidence interval when m fitting parameters are used. If the regression model is linear, eq 4 will provide exact confidence intervals. However, for nonlinear models, such as the distributedsorption advection-dispersion equation, the shape of the contour will be correct, but the level of confidence, ω, will be approximate (36). Equation 4 directly incorporates the impact of differing numbers of data points or fitting
(5)
k
NK FR
-
∑ n (fK C - S ) k
D
(6)
k
k)1
where S is the total mass sorbed, f is the fraction of sites in each discrete kinetic compartment, Rk is the mass-transfer rate for compartment k (1/T), and Sk is the mass sorbed at compartment k with respect to the total mass of the soil. Furthermore, F is the soil bulk density (M/L3), n 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). The number of sorption sites was set to 25, which has been found to be sufficient to approximate the continuous distribution (37). The initial and influent conditions are
vC ) vinCo
for t ) 0
for x ) 0,
(7)
0 < t e tup
(8)
for tup e t e tdes
v ) vin, Co ) 0
- Cd,j)2
j)1
SSQ )
k
k)1
v)0 SSD )
∑R [fK C - S ]
C ) 0, S ) 0
SSD *100 SSQ
where
E(θ) e Em,min
parameters. This approach to confidence intervals has been utilized to characterize the uncertainty in batch desorption/ sorption experiments (25, 27, 28). The governing equations for contaminant transport with distributed sorption through the columns are
(9)
for tdes
