Environ. Sci. Technol. 1993, 27, 2341-2351
Surfactant-Enhanced Solubilization of Residual Dodecane in Soil Columns. 2. Mathematical Modeling Linda M. Abrlola,' Tlmothy J. Dekker, and Kurt D. Pennell
Department of Civil and Environmental Engineering, The University of Michigan, Ann Arbor, Michigan 48109-2125
Considerable effort is currently being directed toward the exploration and development of surfactant-enhanced methods for the recovery of entrapped nonaqueous-phase liquids (NAPLs) from aquifer systems (1-4). Although a number of laboratory and field studies have recently been conducted, relatively little attention has been focused on the mathematical modeling of such remediation strategies. Mathematical models could serve as useful tools in the interpretation of laboratory and field measurements. Models will be necessary for the quantitative integration and assessment of the various physical and chemical processes impacting surfactant-enhanced aquifer remediation (SEAR). Once validated, such models could prove invaluable in the prediction of SEAR performance and in the evaluation of alternative remediation schemes. In the petroleum industry, mathematical models have been applied for some time in the design and evaluation of surfactant-based technologies for enhanced oil recovery (e.g., refs 5-10). Such models have been developed to simulate the complex phase behavior of surfactant-oilaqueous mixtures, often in the presence of brine and cosolvents. Enhanced oil recovery process design has focused primarily on the mobilization of the oil through the formation of middle-phase microemulsions, which leads to ultralow interfacial tensions between the oleic and aqueous phases. Mathematical modeling of this mobilization process requires the solution of a highly nonlinear, coupled system of three-phase mass balance equations, one for each mixture component. These balance equations are typically predicated on the assumption of thermodynamic equilibrium between phases, that is, component
concentrations in contacting phases at the same spatial location are described by equilibrium partitioning relationships. In contrast to the concept of organic-phase mobilization, SEAR technologies have been directed primarily toward alternative NAPL recovery strategies which involve the promotion of micellar solubilization to enhance the performance of pump-and-treat remediation schemes. In this approach, surfactants are selected based upon their capacity to solubilize NAPLs with minimal mobilization of the entrapped organic phase (1). Mathematical modeling of enhanced solubilization is considerably more tractable than the modeling of the mobilization process. Coupled mass balance equations must still be solved for each system component, but these are simplified by assuming a stationary organic phase. In addition, complex phase behavior need not be modeled under many conditions. To date, very few papers have appeared in the environmental literature pertaining to the modeling of enhanced solubilization. In those studies which have been published (11,121,local equilibrium between the aqueous phase and the NAPL was assumed, consistent with the assumptions commonly employed in enhanced oil recovery simulation. The appropriateness of the local equilibrium assumption for predicting NAPL dissolution is the subject of current investigation. In the presence of entrapped NAPLs, concentrations of organic contaminants found in groundwater are typically well below their equilibrium solubilities (13-17). Several explanations have been proposed to account for these lower concentrations, including aqueousphase bypassing and dilution effects, rate-limited mass transfer, decreased solubilities due to the presence of mixtures, and aquifer heterogeneities (15,18-22). Recent laboratory column experiments (23-25) have investigated conditions under which rate-limited mass transfer may control dissolution. These investigations, coupled with theoretical modeling studies (21,26-28), suggest that significant deviations from local equilibrium may be encountered under typical pump-and-treat remediation conditions, especially in graded or heterogeneous aquifer materials and over large time scales. Data on the rates of surfactant-enhanced NAPL solubilization in porous media are scarce. A few studies, however, have considered the potential impact of nonequilibrium conditions on the mobilization of entrapped oil globules. In Lam e t al. (29),nonequilibrium effects were demonstrated to have an important impact on the critical capillary number under many laboratory conditions. Hirasaki (30) discusses other nonequilibrium phenomena which may influence oil mobilization. Results from the column experiments presented in part 1 of this paper (31) demonstrate that rate-limited, rather than instantaneous, solubilization of entrapped dodecane occurred in the sandhrfactant system studied. Nonequilibrium solubilization of residual NAPLs would clearly
0 1993 American Chemical Society
Envlron. Scl. Technol., Vol. 27, No. 12, 1993 2341
A mathematical model is developed to describe surfactantenhanced solubilization of nonaqueous-phase liquids (NAPLs) in porous media. The model incorporates aqueous-phase transport equations for organic and surfactant components as well as a mass balance for the organic phase. Rate-limited solubilization and surfactant sorption are represented by a linear driving force expression and a Langmuir isotherm, respectively. The model is implemented in a one-dimensional Galerkin finite element simulator which idealizes the entrapped residual organic as a collection of spherical globules. Soil column data for the solubilization of residual dodecane by an aqueous solution of polyoxyethylene (20) sorbitan monooleate are used to evaluate the conceptual model. Input parameters were obtained, where possible, from independent batch experiments. Calibrated model simulations exhibit good agreement with measured effluent concentrations, supporting the utility of the conceptual modeling approach. Sensitivity analyses explore the influence of surfactant concentration and flushing strategy on NAPL recovery. Introduction
0013-936X/93/0927-2341$04.00/0
have important practical implications for the application and design of SEAR schemes. Mass transfer limitations could potentially influence remediation duration, optimal pumping and injection strategies, and volume of surfactant solution required for aquifer restoration. For the accurate assessment of SEAR performance, interactions between the surfactant and aquifer matrix must also be addressed. Some recent studies have suggested that the sorption of nonionic surfactants to the solid phase may be significant in groundwater systems (32,331. Surfactant sorption was found to play a minor role in column observations of residual dodecane solubilization by polyoxyethylene (POE) (20) sorbitan monooleate reported in part 1(31). A more substantial sorption effect, however, would be anticipated for surfactants with shorter POE chain lengths or for matrix materials with higher organic carbon contents or larger specific surface areas. Previous models developed for enhanced solubilization (11,12) have neglected surfactant sorption. The primary objective of this paper is to develop and evaluate a conceptual modeling approach for the description of micellar solubilization of residual NAPLs in porous media. The model is designed to overcome the shortcomings of previously published models in applications to enhanced solubilization. Thus, based upon the foregoing discussion, the model incorporates nonequilibrium mass transfer processes and surfactant sorption as well as constituent mass balance equations. The model is implemented in a one-dimensionalnumerical simulator which is conceptually extendable to higher dimensions. Soil column data reported by Pennell e t al. (31)are employed to explore the ability of the model to simulate solubilization of entrapped dodecane by an aqueous solution of POE (20) sorbitan monooleate. Assessment of model performance in this application provides an evaluation of the conceptual modeling approach for description of SEAR. Additional model simulations illustrate the potential impact of rate-limited solubilization on the recovery of residual NAPLs. Although the calibrated model presented herein will not be generally applicable to systems with different physical or chemical properties, it is anticipated that the developed modeling approach will provide a useful framework for future modeling applications to SEAR and interpretation of experimental measurements of micellar solubilization. Model D e v e l o p m e n t
Mathematical Formulation. A general model of surfactant-enhanced solubilization must incorporate the transport of multiple chemical components (surfactant, water, organic) in a three-phase [organic (01, aqueous (w), solid (m)] system. The starting point for any mathematical formulation is a component balance equation (18):
i = 1, ..., n,
a = 0,w,m
where uf is the mass fraction of component i in phase a; e, is the volume fraction of the a phase; D & is the hydrodynamic dispersion tensor for component i in the a phase; qa is the Darcy velocity of the a phase, pa is the density of the a phase; Ego is the exchange of mass of
-
2342
Envlron. Scl. Technol., Vol. 27, No. 12, 1993
component i between the a and P phases; and n , is the number of components. Here the nonadvective flux has been assumed to be adequately represented by Fick's law. Component i has been assumed nonreactive, that is, there is no net production or loss of this component in the system. The mass exchange term in eq 1incorporates both sorption and liquid-liquid interphase mass transfer processes. Summing component balance equations over all species within a phase yields a phase balance equation:
a
-(E,P~)
at
+ V*(paqa) - - = Ea
a = 0,w ,m
where Ea =
E E;@.
I B#a
Equation 2 may be incorporated into eq 1 to yield a conservative form of the component transport equation:
a@; capa-
dt
+ paqa-Vu; - w;Ea - = CEf@ - - - V.(capaD;;Vuq) @#a
(3)
Equations 2 and 3 are subject to the constraints: x u ;= 1
(4a)
I
Cc, = 1
(4b)
a
The Darcy velocity of the aqueous phase is customarily expressed by a modified form of Darcy's law: q a = - - *kkra (V Pa- pug) Fa
-
(5)
where k is the intrinsic permeability tensor; k,, is the relative permeability to the a phase; Pa is the a phase pressure; g is the gravity vector; and p a is the a-phase dynamic Gscosity. The system of eqs 2-5 can be simplified by considering the particular characteristics of the multiphase surfactant system to be modeled. In this work, micellar solubilization is presumed to be the sole organic recovery mechanism. Thus, the organic phase is assumed entrapped and immobile. Under these conditions, for a rigid porous medium, the organic phase mass balance eq 2 becomes: n-(sop") a = E" at
where n is the matrix porosity and so is the organic phase saturation (eo = ns,). Similarly, the solid-phase equation can be written as:
a
( 1 - n ) - (p") = Em (7) at The system of eqs 2-7 represents a general framework
for the modeling of surfactant-enhanced solubilization. To close the system, constitutive expressions must be developed to characterize the mass exchange terms. One approach, which obviates the need for explicit representation of these mass exchange terms, is to assume local thermodynamic equilibrium between phases. As discussed above, this is the approach which was previously employed to model surfactant-enhanced solubilization (11,12). Soil column solubilization data reported in part 1of this paper (31),however, exhibited rate-limited behavior. Thus, in
an effort to model this behavior, a more general expression of interphase mass exchange will be employed herein. Now consider application of the system of eqs 2-7 to the column experiments described in Pennell e t al. (31). These experiments involved the transport of three components (organic, surfactant, water) in a three-phase system. During the experiments, the aqueous-phase superficial or Darcy velocity was controlled. Thus, aqueous-phase flow (eq 2, a! = w ) need not be modeled, as long as the brief periods of flow transients during pumping rate adjustments are neglected. This is a very reasonable approximation for the short column lengths and permeable sand matrix used in the study. A one-dimensional balance equation for a component i [organic (0)or surfactant (s)] in the aqueous phase can be developed from eqs 1 and 3. First, from eq 1,an explicit representation for the exchange term EYw (mass transfer due to sorption) can be derived: (1 - n)
a
(p"w7)
=E
T
Here it has been assumed that the organic phase is nonwetting and, therefore, does not directly contact the solid phase. Thus, no mass exchange will occur between the organic and solid phases. Surface diffusion has also been neglected. Incorporating eq 8 into eq 3 yields the following aqueous-phase transport equation:
t=o,s
where Ci is the mass concentration of species i, Pb is the bulk density of the medium, (Pb (1 - n)pm),Qi is the sorbed mass fraction of species i (Qi a?),and E? is the exchange of species i between the aqueous phase and the entrapped organic phase. The hydrodynamic dispersion coefficient,Dhi, incorporates dispersion due to local velocity variation and Fickian diffusion. This coefficient can be expressed for the one-dimensional flow scenario as
where a ! is ~ the longitudinal dispersivity. The molecular diffusivity,Dli, of species i is modified by atortuosity factor, T , which accounts for the tortuous path traveled by diffusing solute molecules. In the formulation of eq 9, it has been assumed that the presence of the organic and surfactant species has a negligible effect on the density of the aqueous phase. This is a reasonable approximation for the experimental system, which had a maximum surfactant concentration of 43.2 g/L and a corresponding equilibrium solubility of 3500 mg/L for dodecane. Under these conditions, the density of the aqueous phase will vary by less than 0.2 % . In the preceding paper (31),the sorption of POE (20) sorbitan monooleate on Ottawa sand was shown to conform to a Langmuir isotherm: Qm
Q, = 1 + bC, where &,,represents the maximum sorption capacity and b is a constant equal to the rate of adsorption divided by the rate of desorption. Because dodecane was present as a separate phase liquid throughout the column and its
aqueous-phase solubility is exceedinglysmall, the sorption of this species was presumed to have negligible impact on the column effluent concentrations and Q,, was set equal to 0. The dominant interphase mass transfer term appearing in eqs 6 and 9 for the selected dodecane/surfactant system corresponds to the mass exchange of organic between the organic and aqueous phases. It was assumed that this mass exchange process can be adequately represented by a linear driving force model. Such models have traditionally been used in the chemical engineering literature to describe dissolution in packed beds (e.g., ref 34) and more recently have been successfully applied to modeling the dissolution of entrapped NAPLs in sandy soil or glass bead column experiments (23-25,35). The linear driving force model is derived from the assumption of diffusionlimited mass transfer through a stagnant boundary layer or "film" separating the two phases. Flux of solute between the phases in a direction normal to this interface can then be expressed by (36): where K'f is a film mass transfer coefficient for species i across the boundary layer and Ci and CIi are the concentrations of species i in the bulk phase and at the interface, respectively. The film transfer coefficient k'f is proportional to the diffusion coefficient of species i in the aqueous phase and is inversely proportional to the boundary-layer thickness. The specific interfacial area for mass transfer, a,, quantifies the contact area between the phases. Because the species concentration at the interface, CIi, is not typically a measurable parameter, the equilibrium saturation concentration (C,i) is often substituted, yielding a mass transfer expression of the form:
EY = kp,(C,i - Ci)
(12b)
As discussed by Pennell et al. (31),the process of micellar solubilization is clearly more complex than that of organic dissolution in a pure aqueous phase. Thus, a model based upon simple boundary-layer diffusion may be inadequate to represent this process. Mechanistic models for micellar solubilization which have been proposed in the literature (37, 38) generally involve a series of steps which may include interface sorption, micellar dissociation/re-formation, micellar diffusion, and simple organic species diffusion (37, 38). Although one could infer from these mechanistic models that some diffusive process would be rate-limiting for micellar solubilization, the utility of the linear driving force model in this application can only be assessed through model comparisons with laboratory data. In the present study, the partitioning of surfactant into the organic phase was neglected ( E Y ) due to the relatively large hydrophilic-lipophilic balance (HLB = 15) of POE (20) sorbitan monooleate and the extreme hydrophobicity of dodecane. Similarly, mass transfer of water into the organic phase was considered negligible (E; = 0). The above assumptions result in Eo E EZw in eq 6. Equations 6 and 9 with their associated constitutive relations (eqs 10-12) and constraint (4b) form a closed system of equations in the unknowns C,, C,,,ands,. These partial differential equations are coupled through the explicit dependence of coefficients on saturation and the implicit dependence of equilibrium solubility on surfactant Environ. Scl. Technol., Voi. 27, No. 12. 1993 2343
Table I. Input Parameters Used for Numerical Simulations of Soil Column Experiments column ~~
parameter porosity NAPL saturation column length column diameter longitudinal dispersivity molecular diffusivity NAPL SA factor median blob radius dodecane water solubility dodecane density Langmuir parameters Langmuir parameters soil bulk density Ottawa sand parameter time step nodal spacing
symbol n so
1
0.324 0.2036 6.50 4.80 0.1540 1.75 X 0.8 0.044 3.7 x 104 766 1.327 1.22 x 10-3 1790 0.071 0.02-1.0 0.0878
concentration. The resulting system of equations is weakly nonlinear due to the use of the Langmuir equation and the implicit dependence of phase saturations on concentration. The coupling, nonlinearity, and temporal evolution of the coefficients preclude analytical solution of the system, even for the homogeneous conditions of the laboratory experiments. Numerical Solution. The mathematical model described above was implemented in a finite element simulator. Here the system of eqs 6 and 9 was discretized in space using the Galerkin finite element method. Linear basis and weighting functions were employed after reduction in the order of the dispersion term by application of Green's theorem. Spatially varying coefficients were treated as element-wise constants in the formulation. Time derivatives were discretized using a backwards finite difference operator. To simplify the solution algorithm, the resulting system of algebraic equations was decoupled and linearized by lagging the saturation- and concentration-dependent coefficients by one time step. Resulting equations were then solved sequentially using a Thomas algorithm. For all simulations reported herein, time step size was adjusted to maintain global cumulative mass balance errors below 0.5 % . As the NAPL is solubilized over time, the specific interfacial area across which mass transfer can occur (a,) will be expected to change, influencing the total phase mass exchange (eq 12b). A capability was incorporated in the numerical simulator to estimate this changing interfacial area so that long-term solubilization scenarios could be modeled. The approach is based upon an idealization of the entrapped organic configuration (geometry) similar to that presented by Powers et al. (21). Here the residual organic is presumed to be initially entrapped as a collection of spherical "blobs" of uniform size. This modeling approach has been demonstrated to calibrate well with NAPL dissolution data derived from column experiments with uniform sands (39). Within the simulator, the total number of blobs is computed at the beginning of each simulation as (21):
where the superscript e refers to an elemental value, r; is the average blob radius within a given element, s: is the 2344
Environ. Scl. Technol., Vol. 27, No. 12, 1993
2
3
0.326 0.1584 6.35 4.80 0.1385 1.75 X 0.8 0.044 3.7 x 10" 766 1.327 1.22 x 10-3 1787 0.071 0.02-1.0 0.0858
0.323 0.1970 6.43 4.80 0.1139 1.75 X 10-* 0.8 0.044 3.7 x 10" 766 1.327 1.22 x 10-3 1795 0.071 0.02-1.0 0.0809
units
cm cm cm cm cm mgicm3 mgicm3 wig Limg mg/cm3 cm
ref 31 31 31 31 31 43 24 24 45 45 31 31. 31 24
h
cm
saturation of organic within the element, and Axe is the length of the element. The number of blobs within each element is assumed constant over time until complete dissolution occurs. A t the end of each time step, a new average blob radius is determined for the element, based upon eq 13 and the updated value for organic-phase saturation. Specific interfacial area is then estimated from the following expression:
where f " is a factor which accounts for the proportion of the interfacial area in contact with mobile water. Here 3/ ri is the surface to volume ratio of an entrapped spherical globule. The developed numerical model was verified through comparisons with simulations of a NAPL dissolution model (21)and through comparisons with analytical solutions to the advection-dispersion-reaction equation (eq 1). For all simulations presented herein, heuristic convergence in space and time was also demonstrated. Parameter Evaluation. The experimental data reported by Pennell et al. (31) provide a means to estimate the parameters required by the numerical model. Sorption and tritiated water breakthrough studies yielded independent measures of a number of system coefficients including porosity, Langmuir sorption coefficients, initial saturations, and column dispersivities (31). The derived model input parameters are summarized in Table I. Batch solubilization data show that the equilibrium concentration of solubilized dodecane is a linear function of surfactant concentration (31). Experiments conducted for a surfactant concentration range of 20-160 g/L gave solubilized dodecane concentrations ranging from 1700 to 14 000 mg/L. These data indicate a solubility enhancement of 6-7 orders of magnitude for the surfactant concentrations studied. Based upon a least-,squaresfit to the slope of the batch solubilization data, the following expression for dodecane equilibrium solubility was incorporated into the numerical simulator: C,, = 3.7 X + 81.09C, (15) Here C,, is expressed in mg/L and C, is expressed in g/L.
The above correlation assumes an organic concentration intercept at the aqueous solubility of dodecane, in the absence of surfactant. This intercept has little effect on model predictions, since the second term in eq 15 is typically orders of magnitude larger than the first. A linear dependence of solubility on surfactant concentration is typically reported for surfactant concentrations in excess of the critical micelle concentration (CMC), the concentration at which surfactant micelles begin to form. The CMC for POE (20) sorbitan monooleate is reported as 13 mg/L, or about 0.001% on a volume basis (40).Thus, for the range of surfactant concentrations studied, potential dodecane solubility enhancements below the CMC were neglected. Because no independent information was available on the kinetics of dodecane solubilization, results of the column solubilization experiments were used to develop a phenomenological expression for the mass transfer coefficient appearing in eq 12b. Column data provided a means for estimating alumped or “effective”mass transfer coefficient (keff = kfa,), which incorporates both entrapment geometry and diffusion limitations. Here it was implicitly assumed that the specific interfacial area did not change appreciably over the course of a column experiment, since only approximately 10% of the entrapped dodecane was solubilized. On the basis of the column effluent data for various flushing velocities and flow interruption period durations, the following functional form was hypothesized for keff:
ke, = aqb + 6, where a and b are empirical constants and 6, is the value of k,ff under conditions of no flow. Column effluent data following periods of flow interruption were used to estimate 6,, in eq 16. Conceptually, the above expression may be thought of as a representation of a quasi-steady-state diffusion-limited mass transfer process, which is enhanced by the advective flux of the bulk fluid past the interface. Similar phenomenological expressions exhibiting a power law dependence on velocity have been employed in the description of simple dissolution in packed bed systems (21). The functional form of eq 16has also been employed to describe the larger scale dissolution of an entrapped NAPL “pancake” in the capillary fringe zone of an aquifer (41). Now consider eq 9, written for the organic species. Under no-flow conditions and neglecting any bulk-phase transport due to molecular diffusion,this equation becomes dC0 ns, -- = &,(c,,- C,) at Further, if one neglects any variations in the initial concentrations along the length of the column, eq 17 may be solved to develop an expression for 6,:
where t represents the duration of the flow interruption and Ci is the initial concentration. In accordance with eq 18, flow interruption data reported by Pennell et al. (31) are plotted in Figure 1, using values for Ci based upon steady-state effluent concentrations prior to flow interruption. Linear regression procedures were utilized to obtain the value of A,. The value of 6, was found to be
1.2
r
1
L
A
data
-y=0.0109
x (rA2=.959)
0.8
0.6
0.4
0.2
0
0
40
20
60
100
80
I (hre)
FlgFre 1. Linear fit of the flow interruption data to obtain the values of ko used in model simulations.
-3.2 -3’0
3
-3.4
I
0
y
I
-3.8W7
+ 0.1021% R”2
I
0.532
-3.8 -4.0 0.5
1.5
Wq)
Figure 2. Log-log plot of effective mass transfer coefficients versus Darcy velocity used to obtain estimates of parameters a and b in eq 16.
insensitive to variations in the initial dodecane concentration, Ci. Figure 1also shows an alternative interpretation of the data. Here separate fits of the long- and short-term data produce a higher quality of fit for the short-term data. Possible explanations for the change in apparent rate of solubilization as the equilibrium concentration is approached are offered by Pennell et al. (31). The changing rate may reflect a reduction in the rate of organic incorporation within the micelle or a reduction in the micelle diffusivity as the micelles become swollen with dodecane. The column data may also be used to estimate the parameters a and b in eq 16. Under steady-state conditions, eq 9 for organic species transport may be written as
Here, as discussed previously, it is assumed that the saturation profile did not change appreciably during the course of each experiment. An analytical solution to the above equation for a semi-infinite column and subject to a first-type boundary condition at the upstream end was employed to determine values of k,ff at each flow velocity (42).In Figure 2, these values of keff are combined with the average &, value and plotted against Darcy velocity. The linear regression for this log-log plot provided Environ. Scl. Technol., Vol. 27. No. 12, 1993
2345
~
~~
~~~~
Table 11. Values of Empirical Constants Derived from Experimental Data for Effective Mass Transfer Coefficient of Form: ken = aqb + k,, (&, l/h, q, cm/h) empirical constant a
A,
b
all data 0.0250 f 0.0033 0.192 f 0.099 0.0109 f 0.0010 short-term 0.0207 f 0.0032 0.224 f 0.115 0.0153 f 0.0018 interruption long-term 0.0252 f 0.0034 0.191 f 0.098 0.0107 f 0.0014 interruption
estimates for the parameters a and b. Although the measured data exhibit a definite, reproducible trend with velocity ( 3 0 , the regression plot indicates that the mass transfer rate is only weakly correlated to Darcy velocity. The range of computed mass transfer coefficients is quite small over the 2 order of magnitude veIocity variation. Use of an expression of the form of eq 16 but written in terms of interstitial velocity did not improve the correlation. A statistical software package (SAS, v.6.08,1992) was employed to test the hypothesis of zero slope (no velocity correlation) for the data presented in Figure 2. This analysis produced a p-value of 0.0009 for the zero slope hypothesis, indicating that a non-zero slope (velocity correlation) can be assumed with a high degree of certainty. Table I1 summarizes the best fit values of the mass transfer parameters derived from the data and their associated 95% confidence intervals. The phenomenological model represented by eq 16 can be recast in terms of dimensionless quantities for comparisons with previous investigations. Powers et al. (24) developed the following correlation for rate-limited dissolution of entrapped NAPL in terms of a modified Sherwood number:
*'
Sh' = 5 7 . 7 R e o . 6 1 d ~ ~ 3 ~ (20d Here the modified Sherwood number is a dimensionless parameter relating interphase mass transport resistance to molecular transport resistance [Sh' = k,ff (dEolDdl where d50 is the median grain size and DI is the free liquid diffusivity. Re is the Reynold's number, here defined in terms of the superficial velocity (Re = d50qpw/pW).Vi is the uniformity index of the porous medium. In eq 20a, d50 must be specified in cm. The above correlation was developed and verified for the dissolution of styrene and trichloroethylene in a number of sandy media, including Ottawa sand. From Table I, the substitution of grain size data for Ottawa sand (Vi= 1.21) into eq 20a yields the following expression: Sh' = 11.48Re0.61 @Ob) The correlation shown in eq 20b can be compared to that developed for the surfactant-enhanced solubilization of dodecane in this work. In dimensionless form, this expression may be written as Sh' = 0.0049Re0.'92+ 0.00315 (21) The coefficient of the Reynold's number term in eq 21 is more than 3 orders of magnitude smaller than that appearing in eq 20b. This indicates that the departure from local equilibrium conditions is considerably more pronounced for the micellar solubilization of dodecane than for simple dissolution of an entrapped NAPL. It is anticipated that interphase mass transfer will be impacted to some degree by compound free liquid diffu2346
Envlron. Scl. Technol., Vol. 27, No. 12, 1993
sivity. Published mass transfer correlation expressions for dissolution in packed beds typically include a dependence on the Schmidt number (S, = pw/D1pw).Exponents of the Schmidt number in these correlations are within the range of 0.3-0.5 (21). The Schmidt number of a dodecanelwater system differs from that of the styrene/ water system employed in the development of eq 20a by less than a factor of 2 (dodecanetwater = 2057; styrene/ water = 1250) (43). Although the presence of the surfactant might reduce the diffusivity of the dodecane in solution, it is clear that the enhanced departure from equilibrium exhibited by the dodecanelsurfactant system cannot be explained solely by differences in system diffusivities. Also observe that the velocity dependence of the rate of mass transfer, as indicated by the exponent of the Reynold's number, differs markedly in the two correlations. This suggests a significantly reduced dependence of mass transfer on system hydrodynamics for surfactant-enhanced solubilization. Taken in concert, the above observations point to fundamentally different mechanisms governing micellar solubilization in aqueous surfactant systems in comparison with simple NAPL dissolution. Modeling Results and Discussion
Model Simulations. Following parameter evaluation, model simulations were conducted to assess the applicability of the conceptual model to describe the three column experiments reported by Pennell et al. (31). Model parameters for these simulations are summarized in Tables I and 11. Third-type boundary conditions (constant total flux) for the organic and surfactant concentrations were implemented at the upstream boundary, and second-type conditions (zero dispersive flux) were implemented at the downstream end of the column. The initial dodecane concentration was set at its aqueous solubility (3.7 X 10-3 mg/L) throughout the column. Flow interruption periods were simulated by employing a time-varying value of q and zero flux boundary conditions. The coefficients appearing in the mass transfer correlation (eq 16) were assumed independent of time. The effective mass transfer coefficient keff, as given in eq 12b, is the product of a mass transfer coefficient, kf, and a specific interfacial area a,. As discussed above, a correlation for k,ff was developed from experimental column effluent data. Recall that, in an effort to incorporate the changes in available interfacial area as the organic solubilizes over time, the simulator requires an estimate of the original interfacial area and average entrapped globule size (see eqs 13 and 14). The estimate of median blob size for the dodecane10ttawa sand system given in Table I was derived from measurements of styrene entrapment in this same Ottawa sand (24, 39). In that study, styrene was emplaced in a manner similar to that employed by Pennell et al. (31) to entrap dodecane, and the styrene was subsequently polymerized to evaluate its distribution. The available surface area factor, f e, appearing in Table I was obtained by model calibration to transient styrene dissolution data in the same Ottawa sand (39). It is anticipated that the similarities between the densities of styrene and dodecane and their interfacial tensions with water would tend to create similar entrapment geometries in the two systems. Thus, these estimates of interfacial area were considered to be adequate for use in the limited model sensitivity analyses presented below.
Duration of flow Interruption:
a
2500
g 5 2s
2000
-Model: one KO lit
_ _ 1500
8
g
-.data
A
7
too0
d 500
Table 111. Comparison of Experimental Measurements and Model Predictions of Dodecane Saturations (so) before and after Surfactant Flushing
16.5 his
Model: two KO fits
1
1 -
crnlhr
I
0
3 100
0
200
300
400
500
600
1000
Flushing Volume (mL)
3500
A
_ _
48.2 hrs
Emdata ode^: one KO in
Model: two
KO
fits
22.2 hrs
2500
5E
'f
2000
Duration of flow interrupt1on:e 7.1 hrs
0
g
'
A
1500
1000
500 0 400
200
0
600
Flushing Volume (mL) 88.6 hf3
n
n
5 g
'
2000
0
:
E
1500
500 crnlhr
0 0
100
200
300
400
500
600
700
Flushing Volume (mL)
Flgure 3. Comparlson of measured and simulated effluent dodecane concentrations for (a) column 1, (b) column 2, and (c) column 3.
Figure 3a-c presents comparisons of simulated and measured organic effluent concentrations for the three column experiments. Inspection of these figures reveals that the column data are generally well-modeled by the simulator. The shape and timing of initial organic breakthrough curves are reproduced quite accurately, as are the effluent concentration variations with changes in superficial velocity. The slight retardation in the breakthrough of dodecane is attributed to the sorption of the surfactant. Here, the use of batch-determined Langmuir sorption parameters yielded good predictions of the surfactant sorption behavior and corresponding dodecane breakthrough.
column
initial so
final so (exp)
final so (model)
1 2 3
20.36 15.84 19.70
19.68 14.12 17.64
19.72 14.17 17.79
The timing and shape of flow interruption peaks are also well-reproduced by the simulator. A steep rise in dodecane concentration occurs when the surfactant solution residing within the column during flow interruption is displaced upon recommencement of flow. The concentration peaks are followed by aslightly less rapid decline to steady-state concentration levels which are dependent upon the aqueous-phase flow velocity. Experimental data and model predictions indicate that the flushing volume required to reach preflow interruption effluent concentrations is on the order of 1.5-2.0 pore vol. Examination of the maximum peak values, however, reveals that these are underpredicted by the model, particularly for the flow interrupt periods of shorter duration. Use of two different k,ff correlation expressions (see Table II), based upon the data fit shown in Figure 1 for the short- and long-duration interruption periods, produced a better model fit to the data, as shown in Figure 3. Here the correlation with the larger ho value was employed for flow interruption periods of duration less than 40 h. A more sophisticated approach to modeling the apparent time variation of h,, might involve specifying h,, as a function of organic concentration. However, further investigation of the kinetics of solubilization would be required to develop the data for such a model. The experimental data and model predictions shown in Figure 3 can also be compared with respect to the total amount of NAPL dissolved over the course of each experiment. Weight measurements of the soil column before surfactant flushing and calculations of the area under the effluent curves permitted estimation of the mass of dodecane removed in each experiment (31). Changes in dodecane saturation computed by this approach can be compared with simulator predictions, expressed as a volume average of final dodecane saturations throughout the domain. Experimentally determined dodecane saturations are compared with simulator predictions in Table I11for each column experiment. Model simulations agree extremely well with experimental estimates of final saturation for all three column experiments. Sensitivity Analysis. In the preceding section, column simulations were based upon actual laboratory column conditions in an effort to evaluate the applicability of the modeling approach. These simulation results suggest that the developed conceptual model can adequately represent system behavior, with the calibration of the effective mass transfer coefficient. In this section, the calibrated numerical simulator is employed to explore model predictions for a wider range of experimentalconditions. The potential impact of applied surfactant concentration and column flushing procedure on dodecane recovery are examined. The potential effect of influent surfactant concentration on surfactant and organic breakthrough is illustrated in Figure 4a,b. Here simulations were conducted using the column parameters from experiment 3, with an initial so = 0.17, for a range of surfactant concentrations (0.5-6% Envlron. Sci. Technol., Vol. 27, No. 12, 1993 2347
0.2 .~ .
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Figure 4. Effect of influent surfactant concentration on (a) surfactant and (b) dodecane breakthrough curves.
by volume). Recall that this range of concentrations falls within that employed in batch solubilization studies with the dodecane/surfactant system (31). A flow rate of 0.5 mL/min was maintained in all simulations. Surfactant breakthrough curves are plotted in Figure 4a. Note that curves shift to the left with increasing surfactant concentration, indicating a decrease in the apparent surfactant retardation. As the influent concentration of surfactant (C,J increases, the pore volume breakthrough point corresponding to a relative surfactant concentration (Cs/ C,i) of 0.5 approaches a value of 1.0,which is characteristic of unretarded solute transport. This decrease in apparent retardation can be explained by the properties of the Langmuir isotherm. The quantity of sorbed surfactant is limited and constitutes a smaller fraction of the total surfactant mass as the surfactant concentration increases. Similar trends in surfactant breakthrough curves have been observed by Abdul and Gibson (32). Figure 4a also illustrates that predicted concentration curves are characterized by a sharp initial rise followed by a more dispersed approach to a maximum value. This "selfsharpening" or reduction in apparent dispersion is characteristic of breakthrough curves influenced by Langmuir sorption. The invading surfactant mass comprising the leading edge of the solute front will be most strongly sorbed, thereby reducing the degree of spreading in the initial portion of the breakthrough curve. In Figure 4b, the corresponding breakthrough curves for solubilized dodecane are plotted. Dodecane breakthrough mimics the behavior exhibited by the surfactant with decreased apparent dispersion and earlier breakthrough at higher surfactant concentrations. Higher surfactant levels clearly produce higher solubilized dode2348
Envlron. Scl. Technol., Vol. 27, No. 12, 1993
cane concentrations, consistent with the batch solubilization studies. Note that these simulations are based upon the assumption that solubilization kinetics will not be influenced by the concentration of surfactant over the range 0.5-6%. Limited kinetic solubilization data for similar organic/surfactant systems (37, 381, however, suggest that rate of solubilization may increase with surfactant concentration. Thus, it is quite possible that differences among the maximum effluent concentrations of dodecane shown in Figure 4b would be magnified under actual experimental conditions. For surfactant concentrations less than -4%, the value for which the mass transfer correlation parameters were calibrated, dodecane may be solubilized to a smaller extent than predicted by the model. The opposite effect is likely to be observed for surfactant concentrations greater than 4 % . The concentration breakthrough and dispersion trends noted above would not be expected to change, since they are governed by surfactant sorption behavior. The developed mathematical simulator can also be employed to explore the potential performance of various column flushing strategies. I t should be noted that for the simulations presented below, effective mass transfer rates are influenced substantially by the interfacial area algorithm summarized by eqs 13 and 14. Because the solubilization experiments presented in Pennell et al. (31) were conducted over a relatively small range of saturations, these data are inadequate to evaluate the ability of this interfacial area algorithm to capture large time, transient solubilization behavior in the dodecane/surfactant system. As discussed above, however, the utility of this algorithm has been demonstrated in the modeling of entrapped organic dissolution in similar media (39). Incorporation of the algorithm in the SEAR simulator permits qualitative assessment of the performance of alternative flushing strategies. The potential influence of pumping rate on the total aqueous phase volume required to flush the column is illustrated in Figure 5. Column parameters and initial conditions are again taken from experiment 3, with an initial so = 0.17. Here cleanup is operationally defined as the time at which the dodecane saturation in the last element reaches a value of Figure 5 depicts average organic saturation versus flushing volume as a function of flow rate. As anticipated, due to mass transfer limitations, model predictions indicate that greater volumes of water would be required to remove the entrapped dodecane at
a
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500
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Figure8. Performanceof column flushing strategies: (a) total treatment time versus flushing volume, and (b) effluent concentration evolution with flushing volume.
higher velocities. The theoretical lower bound of the required aqueous-phase volume occurs when concentrations are presumed at equilibrium solubility. This lower bound volume is indicated by the local equilibrium curve in Figure 5. For lower flow rates, however, cleanup times will be substantially increased. Thus, a distinct relationship exists between treatment time and volume. This relationship is illustrated in Figure 6a, where each point on the solid curve represents a specific constant flushing flow rate. In this figure, the local equilibrium minimal volume prediction is given by the vertical line. The constant flow rate predictions can be seen to be asymptotic to this Iine. Inspection of Figure 6a reveals that, beyond a certain flow rate (approximately 0.03 mL/min), an increase in flushing rate yields only marginal improvement in treatment time, while greatly impacting the required volume of surfactant solution. Estimation of the total flushing volume will be critical to an evaluation of the overall cost of aremediation process. The flushing volume represents the volume of contaminant solution requiring treatment and relates directly to the quantity of surfactant required to achieve restoration. Flow interruption was also explored as an alternate method of column cleanup. In this approach, higher aqueous-phase concentrations are achieved by alternating periods of flow with periods of flow interruption, during which the organic and aqueous phases are permitted to approach equilibrium. A flow interruption or pulse pumping scheme was simulated by modeling initial injection of surfactant solution at a rate of 1mL/min for 1.5 pore vol. Flow through the column was then halted,
allowing partial equilibration between the aqueous and organic phases for a fixed time period. Flow was then resumed at a rate of 1mL/min until 1.5pore vol was again displaced from the column. This alternating sequence of flushing and flow interruption was repeated until complete removal of the organic was achieved. The effectiveness of pulse pumping for column remediation is illustrated in Figure 6a. Here total treatment time and volume of surfactant solution are plotted as a function of flow interruption duration. When the specified duration is zero, the scheme is effectively operating at a constant flow rate, and the pulse pumping curve intersects the constant flow curve at the 1 mL/min point. As flow interruption duration increases, the required flushing volume decreases, approaching the local equilibrium limit. Thus, the pulse pumping approach produces a substantial reduction in flushing volume over the corresponding continuous pumping scheme at the 1.0 mL/min flow rate. It is clear from Figure 6a, however, that the pulse pumping curve never passes below the constant flow curve, indicating that no pulse pumping configuration is superior to all constant flow strategies in both cleanup time and flushing volume. For example, consider time and volume requirements for the 50-h flow interrupt scheme. An equally effective constant flow approach, with respect to volume, would be to pump at a rate of 0.03 mL/min. The total treatment time for this constant flow scheme, however, would be approximately 50% less than that of the pulse pumping alternative. The effluent dodecane concentration profiles for each of these treatment alternatives are plotted in Figure 6b. As anticipated, the figure reveals that the flow interrupt strategy substantially increasesthe aqueous-phase dodecane concentration above that produced by the constant flow (1mL/min) scheme. Maximum effluent concentration values for the pulse pumping scenario also exceed those produced by aconstant flow strategy with a substantially lower flow rate (0.032 mL/min). When dodecane concentrations are averaged over pore volumes, however, the volume-averaged concentration distributions of the latter two schemes are quite similar. This example demonstrates that evaluation of alternative column flushing strategies must include consideration of the trade-offs between aqueous-phase volume, flushing time, and pumping time. The developed simulator can provide a useful tool for this type of assessment.
Conclusions In this paper, a mathematical model was developed to describe the surfactant-enhanced recovery, via micellar solubilization, of an entrapped organic liquid in a porous medium. On the basis of available experimental observations, the conceptual model was designed to incorporate rate-limited interphase mass transfer and surfactant sorption. A linear driving force expression was employed to model solubilization, where the effective mass transfer coefficient was a function of entrapped organic geometry. The mathematical model was implemented in a onedimensional simulator. Model simulations of a series of column experiments, described in part 1 (321, were conducted to evaluate the applicability of the conceptual modeling approach. A Langmuir sorption isotherm, with parameters derived from batch measurements, was used in the simulator to predict surfactant sorption behavior. The effective mass Envlron. Scl. Technol., Vol. 27, No. 12, 1993 2349
transfer coefficient correlation employed in the simulator was derived from the column data. Good agreement between calibrated model simulations and experimental measurements supports the utility of the conceptual modeling approach. Further experimental investigations will be required to refine our understanding of the kinetics of micellar solubilization. Such investigations could lead to the development of a more generally applicable expression for the effective mass transfer coefficient. Model sensitivity analyses for the investigated dodecane/ surfactant system illustrated the characteristic behavior of Langmuir sorption and the potential impact of surfactant concentration on organic recovery. The influence of column flushing strategy was also explored. Examples highlighted the need for evaluating the trade-off between total flushing time and volume of surfactant solution required for remediation. Flow interruption schemes were shown to reduce required flushing volume at the expense of total treatment time and were found to produce dodecane recovery performance comparable to that exhibited by continuous pumping schemes at lower flow rates. This paper has focused on the influence of interphase mass transfer limitations on the remediation of a laboratory scale, homogeneous, one-dimensional system. Experimental measurements (31) and model simulations demonstrate that mass transfer limitations can have a large impact on mass recovery in these relatively simple systems. Applications of SEAR technologies to field remediation scenarios will necessarily involve many additional complexities, including dimensionality, variability of NAPL distributions, and media heterogeneities,which will further complicate scheme design and may create additional mass transfer limitations. The mathematical model presented herein is intended to provide a framework and foundation for future modeling applications to these more complex systems. Acknowledgments
Funding for this research was provided by the Office of Research and Development, U.S.Environmental Protection Agency under Grant R815750 to the Great Lakes and Mid-Atlantic Hazardous Substance Research Center. Partial funding of the research activities of the Center is provided by the State of Michigan, Department of Natural Resources and Department of Energy. Additional funding was provided by the Ford Motor Co. and by the US. Environmental Protection Agency through Cooperative Agreement CR-818647 with R. S. Kerr Environmental Research Laboratory. The research described in this article has not been subject to Agency review and, therefore, does not necessarily reflect the views of the Agency, and no official endorsement should be inferred. Literature Cited (1) Fountain, J. C.; Klimek, A.; Beikirch, M. G. J . Hazard. Mater. 1991, 28, 295. (2) Abdul, A. S.;Gibson, T. L.; Rai, D. N. Ground Water 1990, 28, 1920. (3) Vigon, B. W.; Rubin, A. J. Res. J . Water Pollut. Control Fed. 1989, 61, 1233. (4) Nash, J. H. Field Studies of In-Situ Soil Washing; U.S. Environmental Protection Agency: Cincinnati, OH, 1987; EPA/600/2-87/110. 2350
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(5) Camilleri, D.; Fil, A.; Pope, G. A.; Rouse, B. A.; Sepehrnoori, K. SPE Reservoir Eng. 1987,2, 433. (6) Scott,T.; Sharpe, S. R.;Sorbie, K. S.;Clifford, P. J.;Roberh, L. J.; Foulser, R. W. S.; Oakes, J. A. In Proceedings of the Ninth SPE Symposium on Reservoir Simulation; San Antonio, T X 1987, p 367. (7) Bang, H. W.; Caudle, B. H. SPEJ, Soc. Pet. Eng. J . 1984, 24, 617. (8) Thomas, C. P.; Fleming, P. D., 111; Winter, W. K. SPEJ, SOC.Pet. Eng. J . 1984, 24, 606. (9) Saad, N. Ph.D. Dissertation, The University of Texas a t Austin, 1989. (10) Pope, G. A+;Baviere, M. In Basic Concepts in Enhanced Oil Recovery Processes; Baviere, M., Ed.; Elsevier: New York, 1991; pp 89-122. (11) Wilson, D. J. Sep. Sei. Technol. 1989, 24, 863. (12) Wilson, D. J.; Clarke, A. N. Sep. Sci. Technol. 1991, 26, 1177. Roberts, P. V.; Cherry, J. A. Environ. Sci. (13) Mackay, D. M.; Technol. 1985, 19, 384. (14) Schwille, F. Dense Chlorinated Solvents in Porous and Fractured M e d i a ; Pankow, J. F., Translator; Lewis Publishers: Chelsea, MI 1988; Chapter 7. (15) Feenstra, S.;Coburn, J. Calif. WPCF Bull. 1986, 23, 26. (16) Mercer, J. W.; Cohen, R. M. J . Contam. Hydrol. 1990, 6 , 107. (17) Patrick, G. C.;Burgess, A. S. Insubsurface Contamination by Immiscible Fluids;Weyer, K. U., Ed.; Balkema: Brookfiled, VT, 1992; pp 489-501. (18) Abriola, L. M. Environ. Health Perspect. 1989, 83,117. (19) Razakarisoa, 0.;Rasolofoniaina, J. D.; Muntzer, P.; Zilliox, L. In Contaminant Transport in Groundwater; Kobus, H. E., Kinzelbach, W., Eds.; BalkemaPress: Rotterdam, 1989, pp 405-412. (20) Mackay, D. M.; Cherry. J. A. Environ. Sci. Technol. 1989, 23, 630. (21) Powers, S.E.; Loureiro, C. 0.;Abriola, L. M.; Weber, W. J., Jr. Water Resour. Res. 1991,27,463. (22) Wilson, J. L.; Conrad, S. H.; Mason, W. R.; Peplinski, W.; Hagan, E. Laboratory Investigations of Residual Liquid Organics from Spills, Leaks and Disposal of Hazardous Wastes in Groundwater; U. S. Environmental Protection Agency: Ada, OK, 1990; EPA/600/6-90/004. (23) Geller, J. T. Ph.D. Dissertation, University of California a t Berkeley, 1990. (24) Powers, S.E.; Abriola, L. M.; Weber, W. J.,Jr. WaterResour. Res. 1992, 28, 2691. (25) Imhoff, P. T.; Jaffe, P. R.; Pinder, G. F. In Proceedings of the American Society of Civil Engineers, Specialty Conference; American Society of Civil Engineers: Arlington, VA, 1990; pp 290-298. (26) Hunt, J. R.; Sitar, N.; Udell, K. S. Water Resour. Res. 1988, 24, 1247. (27) Dorgarten, H. W.; Tsang, C. F. In Subsurface Contamination by Immiscible Fluids;Weyer, K. U., Ed.; Balkema: Brookfiled, VT, 1992; pp 149-158. (28) Robinson, G.C.; Bedient, P. B.In Proceedings of Petroleum Hydrocarbons and Organic Chemicals in Groundwater: Prevention, Detection and Restoration; National Water Well Association: Dublin, OH, 1991; pp 531-540. (29) Lam, A. C.; Schechter, R. S.; Wade, W. H. SPEJ, Soc. Pet. Eng. J . 1983, 23, 781. (30) Hirasaki, G. J. In Proceedings of the SPEIDOE Enhanced Oil Recovery Symposium, Tulsa, OK; Society of Petroleum Engineers: Dallas, TX, 1980; pp 323-343. (31) Pennell, K. D.; Abriola, L. M.; Weber, W. J., Jr. Environ. Sci. Technol., preceding paper in this issue. (32) Abdul, A. S.;Gibson, T. L. Environ. Sei. Technol. 1991,25, 665. (33) Palmer, C.; Sabatini, D. A.; Harwell, J. H. In Transport and Remediation of Subsurface Contaminants: Collodial, Znterfacial, and Surfactant Phenomena; Sabatini,D. A., Knox, R. C., Eds.; ACS Symposium Series 491; American Chemical Society: Washington, DC, 1992; pp 169-181.
(34) Wilson, E. J.; Geankoplis, C. J. Ind. Eng. Chem. Fundam. 1966,5, 9. (35) Miller, C. T.; Pokier-McNeill, M. M.; Mayer, A. S. Water Resour. Res. 1990, 26, 2783. (36) Weber, W. J., Jr. Physicochemical Processes for Water Quality Control;John Wiley & Sons: New York, 1972; pp 510-512. (37) Caroll, B. J. J. Colloid Interface Sci. 1981, 79, 126. (38) Caroll, B.J.; O'Rourke, B. G. C.; Ward, A. J. I. J. Pharm. Pharmacol. 1982, 34, 287. (39) Powers, S.E. Ph.D. Dissertation, The University of Michigan, 1993. (40) Becher, P. In Nonionic Surfactants; Schick, M. J., Ed.; Surfactant Science Series; MarcelDekker: New York, 1967; Vol. 1,pp 478-515. (41) Pfannkuch, H. 0.In Proceedings of the NWWAIAPZ Conference on Petroleum and Organic Chemicals in Groundwater: Prevention, Detection, and Restoration;
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National Water Well Association: Dublin, OH, 1984; pp 111-129. Carslaw, H.S.;Jaeger, J. C. Conduction of Heat in Solids; Oxford University Press: New York, 1959; Chapter 4. Lyman, W. J.; Reehl, W. F.; Rosenblatt, D. H. Handbook of Chemical Property Estimation Methods; McGraw-Hill: New York, 1982; Chapter 17. Hayduk, W.; Laudie, H. AZChE J. 1974, 20, 611. Vershueren, K. Handbook of Environmental Data on Organic Chemicals;Van Nostrand Reinhold: New York, 1977; pp 289-290.
Received for review November 16, 1992. Revised manuscript received July 7, 1993. Accepted July 16, 1993.' Abstract published in Advance ACS Abstracts, September 15, 1993. @
Envlron. Scl. Technol., Vol. 27, No. 12, 1993 23S1
