Environ. Sci. Techno/. 1995, 29, 1766-1772
Heterogeneous Sorption Processes in Subsurface Systems. 2. Diffusion Modeling Approaches JOSEPH A. PEDIT AND CASS T. MILLER* Department of Environmental Sciences and Engineering, CB 7400, 104 Rosenau Hall, University of North Carolina, Chapel Hill,North Carolina 27599-7400
The pore diffusion modeling approach is often used to model the sorption of organic chemicals by natural sorbents, implicitly assuming that the physical and sorptive properties of the sorbent are homogeneous. But application of such approaches to systems with significant variations in particle size and sorption equilibrium properties may be inappropriate; most natural systems have such variations. A general multiple-particle class pore diffusion model that accounts for variations in physical and sorptive properties was developed and used to interpret the results of a batch sorption rate study. Results indicated that multiple-particle class models provide a more accurate representation of long-term sorption rate data than traditional single-particle class approaches, the common approach of adding an instantaneous equilibrium fraction to single-particle class models leads to model parameters that are sensitive to the range over which data is collected, and tortuosity factors obtained with multiple-particle class modeling approaches were consistent with the range of values expected based upon mechanistic reasoning.
Introduction The models used to describe rate-limited sorption of organic chemicals by natural sorbents often assume the sorbent to be homogeneous with respect to its physical and sorptive properties. Application of these models to systems that are inherently heterogeneous often leads to poor agreement between experimental data and model fits. An instantaneous equilibrium fraction is sometimes included in sorption rate models in an effort to achieve better fits to experimental data (e.g.,refs 1-4), but the literature currently lacks an explanation as to what the instantaneously sorbed fraction represents for sorption studies conducted in batch systems. The objectives of this work were (1)to develop a general modeling approach to account for heterogeneity of sorption parameters expected for natural systems, (2) to compare sorption rate descriptions of experimental data obtained with models that account for heterogeneitywith traditional approaches, (3)to scrutinize the optimal parameter values for sensitivity to the data set used to estimate them, and (4) to compare effective tortuosity factors estimated from modeling results with the expected value based upon theory.
Background The variation of sorption equilibrium as a function of particle size within a bulk sample has been demonstrated in numerous studies (5-12). Some of these studies suggest that sorption equilibrium can vary by orders of magnitude within a bulk sample. For example, Karickhoff and coworkers (8) found variations of more than a factor of 400 in the distribution coefficients for the sorption of pyrene to different size fractions of a pond sediment, and Ball and Roberts (11) found variations of more than a factor of 30 in the distribution coefficients for the sorption of tetrachloroethene to different size fractions of a sandy aquifer material. Few studies have investigated the variation of sorption equilibrium as a function of particle mineralogy within a bulk sample. Ball and Roberts (11)separated the 0.0750.12-mm size fraction of a sandy aquifer material from Borden, ON, by magnetic properties and found distribution coefficients for the sorption of 1,2,4,5-tetrachlorobenzene to vary by more than a factor 7 . Similarly, Barber (22) separated the 0.125-0.250-mm size fraction of a sandy aquifer material from Cape Cod, MA, by magnetic properties and found distribution coefficients for the sorption of pentachlorobenzene to vary by more than a factor of 4. Weber and co-workers (13)conducted sorption equilibrium studies with 1,2,4-trichlorobenzene on quartz-like and shale-like fractions of a sandy aquifer material and found that sorption on the shale-like fraction was more than 20 times greater than that observed on the bulk material. Sorption on the quartz-like fraction was found to be negligible compared to sorption on the bulk material (14). These results suggest that sorption equilibrium variations within the sandy aquifer material studied by Weber and co-workers may span several orders of magnitude. These findings are particularly relevant in that the same sandy aquifer material that was used by Weber and co-workers was used in the sorption studies discussed in this paper. ' E-mail address: uncctmOgibbs.oit.unc.edu; Fax: 919-966-7911
1766 1 ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 29, NO. 7 , 1 9 9 5
0013-936W95/0929-1766$09.00/0
@ 1995 American Chemical Society
The sorption of organic chemicals by sediments, soils, and aquifer materials is assumed to be a diffusion-limited process (2,15).The most common approaches for modeling the diffusion process are by a fast-order mass transfer approximation or application of Ficks law for a well-defined geomeuy. Pedit and Miller (4) found that single-particle class models based on either approach underpredicted the rapid initial uptake observed in a batch sorption rate study. Miller and Weber (16) and Wu and Gschwend (17) have also observed the first-order mass transfer model to underpredict earlyuptakein batch sorption studies. Similar results have been noted using the surface diffusion model (16) and the pore diffusion model (2). Several investigatorshave found that single-particleclass model fits can be improved by inclusion of an instantaneously sorbed fraction (1-4). The bicontinuum model, which allows for an instantaneous equilibrium fraction and a rate-limited fraction described by a fast-order mass transfer expression,has been widely used for modeling rates of sorption in batch and column systems (1). Ball and Roberts (2) and Pignatello and co-workers (3) found excellent agreement between experimental data and a single-particleclass model that allows for an instantaneous equilibrium fraction and a rate-limited fraction governed by pore diffusion. Inclusion of an instantaneous equilibrium fraction is intended to account for deviations from an idealized model, such as those caused by particle size variations. The effects of particle size variations on diffusion into sphericalparticles are well documented (18, 19). Equilibrium is approached more slowly when larger than average particles are present than when all of the particles have the same size, and the initial uptake is more rapid when smaller than average particles are present. While particle size variations might explain part of the need to include an instantaneous equilibrium fraction for modeling sorption into materials with wide particle size variations, it does not explain why an instantaneous equilibrium fraction is sometimes needed to accurately model sorption into materials with narrow particle size variations (2). Another source for deviations from the idealized models might be equilibrium variations within a sample. The effects of these variations are easiest to understand within the context of the pore diffusionmodeling approach, which views the sorption process in terms of a solute diffusing into a particle through the intraparticle pores. The intraparticle diffusion process is retarded by sorption to the walls of the pores. Within this framework, sorption will be slower into particles with higher than average distribution coefficients,causing a slower approach to equilibrium than in the case where all of the particles have the same distribution coefficient. Conversely, sorption will be faster into particles with lower than average distribution coefficients, leading to a more rapid initial uptake than in the case where all of the particles have the same distribution coefficient.
Experimental Materials and Methods The first paper of this series (4) described the results of a set of batch sorption equilibrium and rate experiments for the herbicide diuron [3-(3,4-dichlorophenyl)1,l-dimethylureal on a sandy aquifer material (hereafter referred to as the Wagner material). The results reported in the previous study will be used here. A description of the materials and methods were given in the earlier work.
The physical properties of the Wagner material needed for modeling purposes are a solid-phase density of 2.673 & 0.006 (mean one standard deviation) g.cm+ and an intraparticle porosity of 0.012 f 0.004. The particle size distribution was presented in the earlier work. The sorption rate experiment was conducted in completelymixed batch reactors. The experiment had an initial bulk fluid-phasesolute concentration of 305OpgL-', a mass of solids of 5 g, and a volume of bulk fluid of 24.9 mL.
*
Modeling Methods Equilibrium Modeling. Sorption rate models require a description of the equilibrium distribution relationship of a solute between the fluid and solid phases. The Toth model (20) was used to describe the results of the equilibrium experiment. The Toth model is given by
where qe is the equilibrium solid-phase solute concentration, Ce is the equilibrium fluid-phasesolute concentration, QOis a capacity coefficient, b is an affinity parameter, and j3~ is a heterogeneity parameter. The Toth model was fit to the experimental data, the optimal parameter values being determined by nonlinear weighted least squares regression assuming a constant relative error (20). An analysis of the standard deviations of the experimentally determined solid-phasesolute concentrations as a function of initial bulk fluid-phase solute concentration revealed that the magnitude of the standard deviation varied directly with the magnitude of the solid-phasesolute concentration; hence, a constant relative error was assumed. The optimal parameter values (fstandard error) for the Toth model were QO= 0.49 f 0.13gg-', b = 5500 i 1400 mL.g-', and j3~ = 0.10 f 0.004. General Rate Model Formulation. A model was developed to simulate the results of the sorption rate study conducted in the completely mixed batch reactors. The model was formulated to allowfor multiple-particleclasses with different physical and sorptive properties. The model allows for an instantaneous equilibrium fraction and a ratelimited fraction governed by pore diffusion for each particle class. The model also accounts for first-order degradation reactions; this feature will be used in a future article of this series. A mass balance expression for the bulk fluid-phase solute concentration in a batch reactor is given by
where Ms is the mass of solids, Vis the volume of bulk fluid, i is the particle class index, np is the number of particle classes, is the mass fraction of particle class i, f: is the fraction of equilibrium-type sorption sites of particle class i, $,(a is the solid-phase solute concentration of particle class i at equilibrium with the bulk fluid-phase solute concentration, Cb@) is the bulk fluid-phase solute concentration, tis time, k;is the boundarylayer mass transfer coefficient of particle class i, r is radial distance, uiis the particle radius of particle class i, ea is the apparent phrticle
f6
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density of particle class i, C,(r,t) is the intraparticle fluidphase solute concentration of particle class i, A: is the equilibrium-type solid-phase first-order reaction rate coefficient for particle class i, and & is the bulk fluid-phase first-order reaction rate coefficient. For a sorption rate experiment, the initial condition for the bulk fluid-phase solute concentration is given by
where c b o is the initial bulk fluid-phase solute concentration. The pore diffusion model is usually formulated assuming that the transfer of solute between the intraparticle fluid and solid phases is rapid relative to the intraparticle pore diffusion processes. Under this assumption, the sorbentphase mass balance equation for each particle class is
ObibC b(r,t) - e$.iq:(r,t) (4)
where 0; is the intraparticle porosity of particle class i, q:(r,t)is the intraparticle solid-phase solute concentration of particle class i, Dk is the pore diffusion coefficient for particle class i, 1',is the intraparticle fluid-phase first-order reaction rate coefficient for particle class i, and 1: is the intraparticle solid-phase first-order reaction rate coefficient for particle class i. In theory, an upper bound for the pore diffusion coefficient of a solute is its bulk fluid-phase diffusion coefficient The relationship between the diffusion coefficients is given by
(a).
(5) where K : is the constrictivity factor for particle class i, and X I is the tortuosity factor for particle class i (2, 21). The constrictivity factor represents the fractional reduction in diffusivity that occurs when the solute and intraparticle pore radius are of comparable magnitude (22);the tortuosity factor accounts for discrepancies between the actual diffusional path length and an assumed straight diffusional path length within a particle. The effects of constrictivity and tortuosity are sometimes lumped into an effective tortuosity factor, x:, which is given by (21)
The effective tortuosity factor for a particle class can be estimated as the ratio of a to D;. It is important to note that estimates of DL and 2: from experimental data represent an average within and among all of the particles that comprise a particle class. The boundary conditions for each particle class are given by
aCb(r = 0,t) =O ar
ns
(7)
If the sorbent is initially free of solute, then the appropriate initial condition for each particle class is 1768
The governing system of equations outlined above were solved numerically. A brief description of the numerical methods will be given here; details of the numerical methods and model validation are discussed elsewhere (23). The method of lines (24) was used to reduce the governing system of partial differential equations to a system of ordinary differential equations. The Bubnov-Galerkin finite element method (25) was used to approximate the derivatives of the partial differential equations. Third-order Lagrange polynomials were used for the basis functions. The resulting system of ordinary differential equations was integrated in time by the backward differentiation formula method (24). Four third-order elements (13 nodes) per particle class were sufficientto achieve a converged solution for the sampling times used in the sorption rate study (from 2 to 3355 h) and the range of parameter values estimated from the data. The general model requires specification of a boundary layer mass transfer coefficient for each particle class. In the first paper of this series (4),it was assumed that sample mixingwas sufficientsuch that boundqlayer mass transfer was rapid relative to intraparticle diffusion. This assumption, justified by Miller (26) and Ball (23, was used in all of the modeling approaches discussed below. The boundary layer mass transfer coefficient for each particle class was set to lo3 cmh-', which was large enough to make boundary layer mass transfer resistance negligible. Submodels of the General Rate Model. The results of the sorption rate experiment were modeled by several submodels of the general rate model. Table 1 summarizes the key assumptions of the submodels investigated. The single-particle class pore diffusion model (hereafter referred to as the SP-P model) is the simplest submodel. It assumes the physical and sorptive properties of the sorbent to be homogeneous. This model was fit to the sorption rate experiment in the first paper of this series (4) and will be referenced in this work as the base case against which the heterogeneous modeling approaches were compared. In the earlier work, optimal parameter values for several sorption rate models were determined by nonlinear least squares regression assuming a constant absolute error in the sorption rate data (20). An analysis of the standard deviations of the experimentally determined bulk fluidphase solute concentrations as a function of time showed no trends; hence, a constant absolute error was assumed. All of the model fits in this paper were also determined under the assumption of a constant absolute error in the data. A representative particle radius was needed for the single-particle size submodels in order to estimate a pore diffusion coefficient. The Sauter mean radius (28) was chosen as the representative radius for the particle size distribution of the sandy aquifer material. The Sauter mean radius (asm)is given by
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 29. NO. 7 , 1995
where n, is the number of size fractions being pooled,
TABLE 1
Summary of Sorption Rate Models particle size variations
model abbrev
equilibrium variations
SP-P
1 division fm= 1.000, a = 0.198 rnm &t
SP-EP
1 division fm= 1.000, a = 0.198 m m
SP-LN(&) fm=
MP-LN(&)
1 division 1.000,~=0.198 rnm
3 divisions = 0.337, a = 0.545 rnm fm= 0.448, a = 0.202 m m fm= 0.176, a = 0.087 rnm fm
1 division given by experimentally determined bulk value 1 division & given by experimentally determined bulk value 13 divisions &treated as a lognormally distributed random variable 13 divisions & treated as a lognormally distributed random variable
f Q is the mass fraction contained between adjoining sieves, and a' is the geometric mean radius of the material contained between adjoining sieves. A Sauter mean radius of 0.198 mm was calculated from the results of the entire sieve analysis. A single-particleclass submodel of the generalrate model that allows for an instantaneous equilibrium fraction and a rate-limitedfraction governed by pore diffusion (hereafter referred to as the SP-EP model) was also fit to the sorption rate experiment in the first paper of this series (4). The investigation described here will demonstrate how the SPEP model's optimal parameter values change as early data is removed from the data set. If the instantaneous equilibrium fraction represents a real physical phenomenon, then its magnitude should be independent of the quantity of data collected during the earlyportion of a batch sorption rate experiment. The duplicate samples collected at the earliest time (2 h) were removed from the data set, after which the instantaneous equilibrium fraction and the pore diffusion coefficient were estimated for the truncated data set. The duplicate samples collected at the next earliest time (6 h) were then removed from the truncated data set, and the rate model parameters were estimated again. This procedure was repeated until all data collected before 837 h was removed from the original data set. As discussed earlier, inclusion of an instantaneous equilibrium fraction in the SP-EP model is intended to account for deviations from the idealized SP-P model. If the pore diffusion model accuratelydescribes sorption rates for an isolated particle, then variations in physical and sorptiveproperties within a bulk sample would be expected to cause deviations from the idealized SP-P model. A submodel of the general rate model was developed based on the premise that the need to use an instantaneous equilibrium fraction in batch sorption rate modeling is primarily a reflection of equilibrium heterogeneities (Le., variations of the equilibrium distribution relationship from particle to particle). With this premise in mind, the submodel was based on the following assumptions: (1)The Toth model describes equilibrium for all particle classes. (2) All particle classes exhibit the same equilibrium nonlinearity. For the Toth model, this assumption implies that b and BT are constants across all particle classes. (3)The equilibrium capacity coefficientis alognormally distributed random variable, the random variable being designated by Qo.
n,
key model assumptions
1 all particles have identical physical and sorptive properties 1 sorption rates within the single-particle class occur via an instantaneous equilibrium fraction and a rate-limited fraction 13 assumes effects of particle size variations on sorption rates are small compared to equilibrium variations 39 accounts for particle size variations and equilibrium variations
(4) The pore diffusion model provides an accurate description of the rate of sorption for each particle class. (5) There is no correlation between the equilibrium capacity coefficient and particle radius. (6)All particle classes have the same solid-phase density and intraparticle porosity as the bulk material. These assumptions make it possible to formulate a model with relatively few parameters. Justificationfor the second assumption was discussed in the first paper of this series (4). There have been numerous investigations on the variation of sorption equilibrium as a function of particle size within a bulk sample (5-121, but no definitive conclusion as to a universal correlation structure can be gleaned from these studies. The fifth assumption was invoked because of these mixed results. The mass-fraction probability density function for a lognormally distributed equilibrium capacity coefficient is given by
4
where ,UQand are the mean and variance of the logtransformed random variable Qo. The mean and variance of ln(Qo)are restricted because the expected value of QO must equal the experimentally determined bulk equilibrium capacity coefficient. This restriction is given by (29) (12)
where E(Q0) is the expected value of Qo. The continuous distribution was reduced to a discrete distribution such that the expectedvalue of Qoof the discrete distribution was the same as that of the continuous distribution. This is the same approach that was used by Pedit (23)for distributed first-order mass transfer models; the reader is referred to the earlier work for details. The continuous distribution was discretized such that each fraction accounted for 10% of the overall sorption at equilibrium (Le., the product fkQi equaled 10% of E(Qo) for each particle class). Finer discretizationswere used in the upper 5% tail and lower 5% tail of the distribution to better capture extreme events. For example, the lower tail was discretized such that the first and second fractions accounted for 1% and 4% of the overall sorption at equilibrium, respectively. The upper 5%tail was discretized in a fashion complementary to that used on the lower 5% tail. The 13 mass fractions and corresponding equilibrium VOL. 29, NO. 7, 1995 /ENVIRONMENTAL SCIENCE &TECHNOLOGY m.1769
TABLE 2
Summary of Optimal Rate Model Parameters model'
RMSEb
effective tortuosity
paramateP
value f standard error
SP-.P
0.045
730
DP
2.8 x
SP--EP
0.033
990
fe
2.7 x 2.9 x 2.1 x 10-5 i 7.9 x 5.6 x O ' 0 1 jI 2.3 x 1.9 x i 1.4 x 5.2 x loAo i 8.8 x 2.0 x 5.3 x
DP
S P - L N(00)
0.024
M P- LN(Q0)
0.025
110
a', DP
20
100
D!J a Model abbreviations are defined in the text. parameters are defined in the text.
crn2.h-l
lo-' crn2.h-l lo-' crn2.h-' crn2.h-'
RMSE, root mean square error in units of relative bulk fluid-phase solute concentration. Model
capacity coefficients generated by this procedure were used as part of the input to the general rate model described earlier. The model was fit to the batch sorption rate experiment under the assumption that particle size variations were not important compared to equilibriumvariations. This singleparticle size lognormally distributed capacity coefficient model will be referred to as the SP-LN(Q0) model. When the boundary layer mass transfer resistance is ignored, as it is here, the SP-LN(Q0) model has only two adjustable parameters: the pore diffusion coefficient and the mean or variance of ln(Q0). This model has the same number of adjustable sorption rate parameters as the first-order masstransfer-based bicontinuum model and the SP-EP model. The SP-LN(Qo) model was also fit to truncated data sets to investigate how the optimal parameter values change due to the lack of early data. The last submodel considered here accounts for particle size variations as well as equilibrium variations. This multiple-particle size lognormally distributed capacity Coefficient model. will. be referred to as the MP-LN(Q0) model. This model was fit to the batch sorption data set to determine if the effects of particle size variations on sorption rates were significant compared to the effects of equilibrium variations for the sorbent studied. Equilibrium variations were handled in the same manner as for the SP-LN(Qo) model. To make the problem numerically tractable, the particle size distribution observed in the sieve analysis were represented by three particle sizes. The Sauter mean radiiwere calculated for the mass fractions contained on the three largest sieves (0.71-2.00 mm),the three middle sieves (0.25-0.71 mm),and the four smallest sieves ('0.090.25 mm). The calculated Sauter mean radii and corresponding mass fractions are listed in Table 1.
Results and Discussion The results of the sorption rate experiment and the model fits for the SP-EP, SP-LN(Qo), and MP-LN(Q0) models are shown in Figure 1. The root mean square error (RMSE) and optimal rate model parameters for all of the models discussed in the previous section are summarized in Table 2. The SP-P model fit (notshown in Figure 1)was discussed in the first paper of this series (4). The SP-EP model fit overpredicts the extent of sorption observed in the experiment before 12 h and overpredicts the approach to equilibrium observed in the experiment at times greater than 1500 h. The SP-LN(Q0) and MP-LN(Q0) models provide a better fit to the data (as indicated by the lower RMSE value) than the SP-EP model. The SP-LN(Q0) and 1770
1.7 x
ENVIRONMENTAL SCIENCE &TECHNOLOGY / VOL. 29, NO 7 , 1995
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'
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i
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FIGURE 1. Experimental data and model fits for diuron sorption on the Wagner aquifer sand.
MP-LN(Q0) model fits slightly underpredict the extent of sorption before 6 h but provide an accurate description of the experimental results after 6 h. Model Sensitivity to Lack of Early Data. The results of the study, which was intended to demonstrate the sensitivity of the SP-EP model's optimal parameter values to the lack of early data, are shown in Figure 2a. The figure shows how the optimal parameter values change as a function of the time of the first data points used in the optimization. The optimal parameter values are scaled to the optimal parameter values obtained using the complete data set, which werefe = 0.027 k 0.0029 and D, = 2.1 x lo-' f 7.9 x lo-' cmZ.h-lI The optimal instantaneous equilibrium fraction steadily increased and the optimal pore diffusion coefficient steadily decreased as the early data were removed from the data set. The optimal instantaneous equilibrium fraction increased to 0.25 f 0.053, and the optimal pore f 1.9 x diffusion coefficient decreased to 5.0 x cm2W1after all the data prior to 837 h had been removed from the data set. A similar study was done for the firstorder mass-transfer-based bicontinuum model. The optimal model parameters showed the same behavior as the SP-EP model (results not shown). The optimal instantaneous equilibrium fraction was 0.098 i 0.0091 when all of the data was used and steadily increased to 0.35 f 0.022 when all the data prior to 837 h had been removed from the data set.
The results of the study, which was intended to demonstrate the sensitivity of the SP-LN(Q0) model’s optimal parametervalues to the lackof earlydata, are shown in Figure 2b. The optimal parameter values are scaled to the optimal parameter values obtained using the complete data set, which were = 5.6 f 0.23 and Dp = 1.9 x f 1.4 x cmah-l. The optimal value varied by less than 6%, and the optimal D,value varied by less than 14% as data prior to 239 h was removed from the data set. The variations in the optimal parameter values for the SPLN(Qo)model are small compared to the variations in the optimal parameter values for the SP-EP model during the same time frame. The optimalf,value for the SP-EP model increased by 390%,and the optimal D,, value for the SP-EP model decreased by 40% as data prior to 239 h were removed from the data set. However, the optimal Dpvalue for the SP-LN(Q0) model varied by more than 2 orders of magnitude as additional data was removed from the data set. Multiple-Particle Class Approaches. The SP-LN(Q0) model is identical to the SP-P model when approaches zero. If the pore diffusion model accurately describes sorption rates for an isolated particle, then the magnitude of ought to reflect the magnitude of the variations of the equilibrium capacity coefficient within a bulk sorbent sample. However, the SP-LN(Q0) model does not explicitly account for particle size variations, so the magnitude of is Muenced by particle size variations as well as equilibrium capacity coefficient variations. The MPLN(Qo)model does account for particle size variations, so the magnitude of would be expected to only reflect the magnitude of the variations of the equilibrium capacity coefficient. The optimal values of were 5.6 f 0.23 and 5.2 f 0.088 for the SP-LN(Q0) and MP-LN(Q0) models, respectively. The MP-LN(Q0) model yields alower estimate because it accounts for particle size variations. of However, recall that these variances are for the natural logtransformed equilibrium capacity coefficient. As such, these variances represent variations over several orders of magnitude. The small decrease in between the SPLN(Q0) and MP-LN(Q0) approaches suggests that equilibrium variations within the Wagner material may be more important than particle size variations for diffusion-based modeling approaches. Further evidence for this conclusion can be found by noting that the optimal SP-LN(Q0) and MP-LN(Qo) model fits (Figure 1) are nearly identical, indicating that incorporating particle size effects into the model did little to change the model fit. One of the assumptions of the SP-LN(Q0) and MPLN(Qo) modeling approaches was that there was no correlation between the equilibrium capacity coefficient and particle radius. There have been numerous investigations on the variation of sorption equilibrium as a function of particle size within a bulk sample (5-12). No definitive conclusion as to a universal correlation structure can be ascertained from these studies. If the Wagner material had a positive correlation structure between the equilibrium capacity coefficient and particle radius, then the MPLN(Q0) modeling approach would yield artificially high estimates. Small particle radius and low equilibrium capacity both result in a rapid approach to equilibrium within the retarded intraparticle diffusion modeling framework. Conversely, large particle radius and high equilibrium capacity both result in a slow approach to equilibrium.
4
t+ I
1E-1
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FIGURE 2. Optimal parameter values for the SP-EP model (top figure) and the SP-LN(Q) modal (bottom figure) as a function of the time of the first data points. The results of the sensitivitystudies with the SP-EP and the first-order mass-transfer-based bicontinuum models demonstrate that the optimal parameter values for models that invoke an instantaneous equilibrium fraction are dependent on the amount of data collected at early times when the fractional approach to equilibrium is small. Such an effect is seen in the batch sorption rate experiments of Ball and Roberts ( Z ) , who applied the SP-EP model to the results of sorption rate experimentswith tetrachloroethene (PCE) and 1,2,4,5-tetrachlorobenzene on various size fractions of a sandy aquifer material from Borden, ON. Their optimal instantaneous equilibrium fractions ranged from zero for the sorption of PCE on the largest (0.85-1.70mm) particle size fraction to 0.31 for sorption of PCE on the smallest (
