Kinetic Model for the Supercritical Extraction of Contaminants from Soil

Chapter 20

Kinetic Model for the Supercritical Extraction of Contaminants from Soil 1

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Ziyad G. Rahmé, Richard G. Zytner, and Warren H. Stiver School of Engineering, University of Guelph, Guelph, Ontario N1G 2W1, Canada

Supercritical carbon dioxide extraction is a novel technique for the remediation of soil contaminated with heavy organic chemicals. It has the potential to quickly and effectively remove the contaminants from soil, allowing for site redevelopment. Research on an analytical scale has proven its feasibility. However, there is limited information available on which to reasonably estimate the cost and timing involved in the application at a particular site. This research paper presents three kinetic models for the supercritical extraction of contaminants from soil. The models consider the following factors affecting the breakthrough curve: axial dispersion, particle-to-fluid mass transfer coefficient, intraparticle diffusion and the equilibrium partition coefficient. The first three of these factors will affect the shape of a breakthrough curve while the latter will affect the breakthrough time. The models are solved using a Crank-Nicolson finite difference scheme. The models have been applied to literature breakthrough data to determine parameter values necessary to fit the observations. As three values in the model affect breakthrough shape, many combinations successfully fit the experimental data. Comparing the parameters with independent correlations still leaves many combinations of plausible parameters that can fit the data. This points to a need for independent measures of the three kinetic parameters specifically for supercritical fluid - soil systems. Soil contamination is a significant problem and results from waste sites and chemical spillage, intentional applications, and atmospheric deposition. It involves a wide range of contaminants from dioxins to explosives, from mercury to radioactive materials. The contamination threatens our air, water and food quality. Complete clean-up will be a slow and expensive exercise. Cairney estimated cleanup costs of in the billions 1

Corresponding author

0097-6156/95/0608-0298$12.00/0 ©

1995 A m e r i c a n Chemical Society

Hutchenson and Foster; Innovations in Supercritical Fluids ACS Symposium Series; American Chemical Society: Washington, DC, 1995.

20.

RAHME ET AL.

Supercritical Extraction of ContaminantsfromSoil

of US dollars for each of the Netherlands, Germany and the US (7). A recent survey indicates that excavation followed by landfill remains the most popular 'solution' (2). Some argue that this is not much of a solution at all A number of alternative solutions are available or in various stages of development Established alternatives include soil vapour extraction, bioremediation and soil washing. However, given the diversity of contaminants and sites, additional innovative technology development remains necessary. Supercritical Fluid Extraction (SFE) is one of the innovative technologies under development. SFE is viewed as an on-site, ex-situ process in which the soil is excavated and transferred to a high pressure vessel for extraction of the contaminants from soil. SFE is part of a group of solvent extraction based approaches. However, it has a major benefit that solvent recovery is easily managed simply by depressurizing the fluid. Carbon dioxide is the fluid of choice owing to its benign character in the environment as a soil residue and as a result of its relatively low critical point. Laitinen et al. (3) in a recent review of SFE for soil cleaning indicated that no work beyond the pilot-scale has been undertaken. To assist in the development and scale up of SFE it is necessary to understand the thermodynamics and the dynamics of the extraction process. The availability of these parameters will assist in the assessment of the economic feasibility of SFE for full scale soil remediation. Akgerman and coworkers have made some significant progress on both the thermodynamics and kinetics using a laboratory system {4,5). In particular, Madras et al. (5) used a model incorporating three kinetic factors of axial dispersion, film mass transfer and intraparticle diffusion to describe the breakthrough curve from their packed soil columns. Each parameter was estimated from literature correlations based on systems other than the combination of supercritical fluids (SCFs) and soil. The agreement between their model predictions and experimental data is quite good. However, this agreement is not sufficient to validate the correlations used to predict the individual kinetic parameters. Valocchi (6) demonstrates that the shape of the breakthrough curve cannot distinguish between the three kinetic mechanisms, meaning that any one of the three mechanisms could be responsible for the observed shape of a breakthrough curve. Thus, a substantial error in one or more of the three parameters could exist and be compensated for by errors in the other parameters or be hidden if the parameter is not controlling in the laboratory setting. This has important implications for process scale up as each of the three mechanisms is affected differently by scale. The objectives of this paper are to use kinetic models of the extraction process coupled with experimental observations of breakthrough behaviour to explore the rate controlling mechanisms of the overall process. Three models will be utilized that include different assumptions regarding which mechanism(s) control the overall process. The fitted kinetic parameters will be compared to independent correlations to assess the physical likelihood of the fitted values. This will allow an assessment of which mechanism(s) contribute significantly to the observed data as well to assess the applicability of correlations for process scale up.


299

300

INNOVATIONS IN SUPERCRITICAL FLUIDS

Model Development To model the process, the following initial assumptions were made: the system operation is isothermal and isobaric; C 0 flow is at a constant velocity; the contaminant is initially sorbed homogeneously throughout the packed bed; initially, the solvent rapidly fills the intraparticle pores; equilibrium adsorption isotherm is linear; soil consists of single sized spheres that are uniformly distributed, and have a constant porosity; and, radial concentration gradients within the vessel are negligible, however, radial gradients within soil aggregates or particles are possible. The models are all based on a differential mass balance describing transport of contaminant through a packed bed. The general form of the balance is given by Equation 1. 2

dC

„ dC

dC

2

» r

(i \

where N is a sink/source term. Equation 1 is subject to the following three boundary conditions. p

Λίζ=0,^-ς.„) = ^ | ι

(2)

Aiz=L, ^ £ = 0

(3)

dz

At ί=0, C=C

to

(4)

The particles making up the soil are viewed as porous spherical aggregates of finer grains in which chemicals are either sorbed to the soil or freely dissolved in the intraparticle pore fluid. The desorption path of the contaminant follows four steps. First, the extract must be released from the sorbed state to the carbon dioxide. Second, the contaminant diffuses through the internal pores of the soil aggregates to the aggregate surface. Then, the contaminant diffuses through the external supercritical fluid film at the soil interface. Finally, the contaminant moves with the bulk fluid through the interparticle pores of the soil column. Based on this desorption path, three models are presented. The models differ in which of the steps is considered to be controlling or significantly contributing to control of the overall rate of the extraction process. The first model considers the axial dispersion associated with the final step to be the only controlling factor and assumes that the internal and film diffusion is rapid. The second model includes both axial dispersion and film mass transfer as the controlling steps in the desorption process. Finally, the third model considers axial dispersion, film mass transfer and internal diffusion in the overall desorption process. For all three models the first


20. RAHME ET AL.

Supercritical Extraction of Contaminants from301 Soil

desorption step of the contaminant leaving the sorbed state is considered to be rapid and not rate limiting. For each of the three models only the sink/source term in Equation 1 is different. Local Equilibrium with Axial Dispersion - Model Type I. The first type of the model assumes local equilibrium is valid. For this case, the concentration in the soil will be proportional to the concentration in the bulk supercritical fluid at all times and locations. The proportional constant will be the equilibria partition coefficient, K . The N term in Equation 1 is replaced as follows, a

p

(1-ε)

„ dC

Therefore, substituting into Equation 1 and isolating the concentration-time derivative results in 9C _ 2

d t

1• ϋ ^ *

dC

β

Ρ ,

The only rate parameter controlling the desorption profile for the Type I model is axial dispersion. Film Mass Transfer with Axial Dispersion - Model Type II. The Type Π model incorporates a kinetic submodel to describe mass transfer through the film on the SCF side of the aggregate-fluid interface. In this case, the mass exchange rate with the packing is proportional to the driving force for transfer and is given by Equation 7. 3* d-e) w

ËR

C-

(7)

In addition to the axial dispersion coefficient, a film mass transfer coefficient also controls the desorption profile. Internal Soil Resistance with Axial Dispersion and Film Mass Transfer - Model Type III. The Type III model utilizes a kinetic submodel which has three steps controlling the desorption process. The model extends the Type II model to include a concentration gradient as a function of radial position in the aggregate as part of the rate limiting process. For this case, the expression for Ν is given by,


302

INNOVATIONS I N SUPERCRITICAL FLUIDS

In this model the packed bed equation must be coupled with an equation describing the profile within the soil aggregates. This latter equation is based on a differential mass balance of solute in the intraparticle pore region and is given as ( 'a

dt

C

e

η + ρ *Γ (1-η) 2 5

β

TrOEl + r^ " dr

Γ

3 r

(9)

2

where

D =D l e

(10)

m

The additional boundary conditions for Equation 9 are: Att=0,C =C a

« « a

ao

(11)

K

Atr = 0 , ^ 1 = 0

(12)

3/·

Atr=R,D^El=k (C-C ) m

a

(13)

The rate parameters of concern are the axial dispersion coefficient, particle-tofluid mass transfer coefficient and the effective diffusion coefficient. However, since the molecular diffusion coefficient and intraparticle porosity are fixed, the effective diffusion coefficient uncertainty is entirely in the tortuosity. Model solution. All three models were solved using a Crank-Nicolson finite difference scheme (central difference in time and space). When the grid Peclet number exceeds two, the central difference scheme may diverge and lead to inaccurate solutions. Therefore, the convection terms in Equations 1 were discretized using the exponential scheme (7). In all of the numerical solutions presented here, mass closure was 100 ± 1%. For each of the models a single fitting parameter was used to fit the breakthrough front of the desorption profile. The region of the breakthrough curve used for the fitting corresponds to the experimental data between approximately 84% and 16% of the initial concentration, which represents the mean time plus one standard deviation. The spreading in this region of the breakthrough curve is


20.

RAHMEETAL.

Supercritical Extraction of Contaminants from Soil

303

dependent on the contribution of axial dispersion, film mass transfer and intraparticle diffusion. The best fit for a given parameter was determined by producing the minimum mean squared error between the model and the experimental data. Parameter Estimation Two types of parameters have been estimated. One type is referred to as FIXED parameters based on the known operating conditions of the experiments and reliably estimated values of density, viscosity and partition coefficients. The second type is referred to as VARIABLE parameters which are the parameters associated with the kinetics of the extraction process. Experimental results from two different sources are used in this paper. The first set of experimental data is provided in Erkey et al. (4) and will be referred as the Texas data. The second set of data is given in Kothandaraman et al. (8) and will be referred to as the Rutgers data. The Texas data contains desorption runs for two contaminants, naphthalene and phenanthrene, at three temperatures. The Rutgers data contains runs for phenanthrene and anthracene at one temperature. Table I summarizes the properties of the packed beds for these data sets. Fixed Parameters. Table II summarizes the operating conditions used in the Rutgers and Texas experiments and calculated values of the fixed parameters. The molecular diffusivities in supercritical carbon dioxide are derived from the following correlation (9). r0.75

D

m

= κ*10~

5

μ*10

(14)

6

For naphthalene a value of 19.49 for κ gave a average deviation with experimental data of ±5.2%. The diffusivities for phenanthrene and anthracene can be determined from the value for naphthalene based on the ratio of the molar volumes raised to the 0.6 power (10). Thus, κ is 16.24 and 16.86 for phenanthrene and anthracene, respectively. For the Texas data, the partition coefficients, or the adsorption equilibrium constants, were determined from the adsorption isotherms found in Erkey et al. (4). For the Rutgers data, Andrews et al (77) reported non-linear adsorption isotherms for phenanthrene and anthracene. However, the models utilized in this work required a linear approximation of the adsorption constant, which was made using a moment analysis. A linear isotherm assumption was incorporated in the models of this paper based on the well established linear behaviour of hydrophobic organics in soil-water systems (72). The partition coefficients used for both the Texas and the Rutgers soil are presented in Table II. Variable parameters. The rate parameters of concern are the axial dispersion coefficient (D ), the film mass transfer coefficient (k, ) and the tortuosity (τ). They are calculated using the correlations presented below. ax

n



Rutgers 170

170 0.024

0.024

1.48

0.99

0.74

1.61

1.05

0.74

B e n d e r E q u a t i o n o f State (13)

318

Anthracene

100

328 318

100

318

Phenanthrene

100

100

328 308

100

318

Phenanthrene

100

308

72.8

velocity

(atm)

(Κ) (cm/s)

Interstitial

Ρ

Τ

Texas

Naphthalene

0.505

0.50

(v/v)

200*

230 0.018*

0.12

(v/v)

Porosity

Diameter (um)

Intraaggregate

Aggregate Particle

2.65

2.52

3

Density 5

6.97

6.97

D m 4

0.94

1.08

1.88

1.40

0.97

2.26

1.68

1.17

cm /s)

2

(10"

0.3

0.3

22

15

10

24

16

10

Re

9.4

8.2

4.2

5.3

8.8

3.5

4.5

7.3

Sc

K

6.0

3.5

16.4

3.06

1.24

2.26

0.46

(14)

fluid/

0.17

cm3

a
18

Internal Soil Resistance. The internal soil resistance is controlled by the effective diffusion coefficient within the soil aggregates. However, if the molecular diffusion coefficient and the aggregate porosity are known then the remaining variable parameter is the aggregate tortuosity. Chang (20) mathematically derived the following relationship between tortuosity and porosity, τ = 2-η


(19)


306 Results and Discussion

Figures 1 to 3 indicate the degree of fit between the three model types and the observed experimental data. Each figure is representative of the fit possible by each model for all of the data. Although the fit was based on a least squares error for data from 16 and 84% of the initial concentration, the resulting fit typically matched from the beginning down to approximately 25% of the initial level. Beyond this point, in all but one case (Texas, naphthalene data at 35°C) the data began to tail in relation to the model. If the results from the Type I, Π and ΠΙ models for a given run were to be plotted on the same figure, the three fitted curves would be virtually superimposed. This was observed in almost all cases. This observation is consistent with the work of Valocchi (6) in which several different models could fit the a breakthrough curve in the same way. In an attempt to determine the most plausible model and to isolate the controlling mechanism behind the process, the curve fitted values were analyzed to assess whether they are physically plausible. The fitted values from each of the models are summarized in Table III. In addition, the values determined by the correlations are included for comparison. Axial Dispersion - Model Type I. The axial dispersion parameters fitted to the data based on the model are approximately 4-20 times greater than those calculated from the Tan-Liou correlation and 7.5-55 times greater than those based on Freeze and Cherry's soil-water data. To decide whether this is sufficient evidence to reject the premise of model Type I that axial dispersion is the only controlling rate parameter, it is necessary to consider the certainty and applicability of the correlations. The Tan and Liou (75) experiments were performed with spherical glass beads. On the other hand, soil particles can have shape factors ranging from 0.43 to 0.95 (27). This difference in sphericity can have a significant affect on axial dispersion. For water flow in soil, changingfromglass spheres to like-grained sand has resulted in an increase of axial dispersion by between 35 and 700% (22). As well, the bed used in the Tan and Liou (75) experiments were uniform diameter glass beads. While a mean particle diameter is used for a packed bed of soil, in reality a substantial distribution of particle sizes exists. This can result in a substantially higher axial dispersion coefficients (22-24). The Freeze and Cherry axial dispersion depends on the value of a. This value could be as much as 100 times larger than what was used in Table III which would directly increase the calculated value of the axial dispersion. It is uncertain if the axial dispersion of water and SCFs would be similar in the same system. Caution should be taken in applying data from the soil-water domain. It is likely that soil-water experimental measurements of axial dispersion did not correct or account for intraparticle diffusion in their analysis. This could explain the typically higher axial dispersion coefficients in soil-water systems. Overall, the fitted axial dispersion values exceed the available calculated values; however, once the uncertainty in the calculated values is reviewed the fitted values must be considered plausible.


20.

RAHMEETAL.


0.025

t(s)

Figure 1 : Illustration of Fit for Model Type I

Figure 2: Illustration of Fit for Model Type II


30

308


0.005

0

100

200

300

400

500

600

700

t(s)

Figure 3: Illustration of Fit for Model Type III

Film Mass Transfer - Model Type II. For the Type II model, the computed mass transfer coefficients are, with the exception of the Texas naphthalene 55°C., approximately one order of magnitude lower than those predicted by the correlation. For this calculation the axial dispersion was assumed to be that calculated from Equation 15. Once again, in order to determine the plausibility of the computed values of kj^, it is useful to examine the conditions for which the correlations are valid. The Lim et al. (17) experiments were conducted for Reynolds numbers ranging from 4 to 135 and Schmidt numbers ranging from 2 to 11. An examination of Table Π indicates the Re and Sc numbers for Texas runs fall within this range. For the Wakao-Kaguei relationship, the lumped parameter S c ^ R e for the Texas data once again falls within the range of validity of the correlation. As well, both correlations agree fairly closely, indicating the predicted mass transfer coefficient are within the right range. All these factors suggest that the fitted mass transfer coefficients are not plausible, assuming the Tan and Liou correlation for calculating axial dispersion is valid. However, neither correlation is derived from observations involving soils or SCF/soil systems. A second approach to fitting the data with the Type II model was to assume the mass transfer correlation of Lim et al. (17) was valid and fit the curve by adjusting the value of the axial dispersion. This resulted in nearly identical axial dispersion values as in the Type I model and presented in Table III. Then, if the Lim correlated values are accurate, this would indicate that the particle-to-fluid mass transfer limitation has a negligible effect on the breakthrough curve and thus this limitation itself is negligible. 0,6



Rutgers

Naphthalene

Texas

3.0

0.72/1.4

0.03

* based o n a?=0.02 c m

6.1/55

0.0018/0.0002

0.04

0.04A).02

45

0.05/0.02

0.79/1.6

0.04

20/23

0.023/0.02

0.45

55

Anthracene

2.3

0.08/0.09

12./11.

1.0

19/54

0.028/0.01

0.54

45

3.9/35

0.07/0.08

8.2/7.2

0.6

10/53

0.037/0.007

0.37

35

0.0018/0.0002

0.07/0.09

5.4/4.7

0.4

6/7.5

0.025/0.02

0.15

55

0.007

0.8/0.93

15./13.

12.

18/52

0.029/0.01

0.52

45

45

3.0

O.O3/0.O4

9.4/8.5

0.8

1.2

0.8

n/a

2.4

0.05/0.06

6.0/5.3

0.3 0.3

Fitted

3.5/19

Eq'n

F i t t e d to

Ratio

E q ' n 17/18

2

( Ι Ο " cm/s)

Fitted

k,,,

0.037/0.007

Eq'n

E q ' n 15/16*

Fitted to

Ratio

0.13

Fitted

2

E>ax ( c m / s )

35

(°C)

Temp

Phenanthrene

Phenanthrene

Chemical

Data

τ

2.0

2.0

1.9

1.9

1.9

1.9

1.9

1.9

E q ' n 19

1.5

1.2

0.42

0.63

0.42

-

1.6

1.3

Ratio

M o d e l T y p e III Internal R e s i s t a n c e

M o d e l T y p e II F i l m M a s s Transfer

Model Type I A x i a l Dispersion

Table III: Model Results

310


Internal Resistance - Model Type III. For the Type ITT model the axial dispersion from Equation 15 and the mass transfer coefficient from Equation 17 were assumed to apply. The value of the tortuosity in Equation 10 was used as the fitting parameter. The ratio of the fitted tortuosities to the Chang (20) values range from 0.4 to 1.6. This apparent agreement is encouraging. However, it should not be viewed as validation of the model and the independent correlations. To fully validate the model and correlations further studies are required including independent measures of the parameters for SC-C0 /soil systems and measures of breakthrough curves for a wider range of conditions and soils. It is necessary to recognize that the requirement to assume values for the particle diameter and the intraparticle porosity for the Rutgers data directly affects the fitted tortuosity. Other possible values for the diameter and porosity could result in the Rutgers tortuosity falling outside the 'correct' range. For example, if a particle diameter of 700 pm is assumed, the tortuosity for anthracene becomes 0.16. In essence because of the unknown values for the Rutgers experiments, the tortuosities derived from the Type III model can not be fairly evaluated. Madras et al. (5) use the same system of equations, but a different solution technique, to model the Texas data. Their results are comparable to the results presented in this paper. The differences can be attributed to the use of different property estimation techniques and rate parameter correlations. Their model as well cannot satisfactorily predict the desorption tail for the extraction of contaminants from soil. 2

Tailing. Each model fails to predict the desorption tail, which is a major shortcoming. In the remediation of contaminated soils, the elution tail can have a large impact on economics. For the desorption of contaminants from activated carbon, for example, it is sufficient to regenerate the bed to within 5% of the initial concentration. The desorption tail in this case would not have to be considered. However, due to the low clean-up levels required in environmental applications, soil often needs to be remediated to under 0.1% of original levels. Therefore, the tail can have a large impact on the desorption process and needs to be understood. Several mechanisms have been proposed to describe excessive tailing of the breakthrough curve, including: nonequilibrium, nonlinearity of the adsorption isotherm, nonsingularity, transformation reactions and aggregate size distribution (25,26). Further modelling research is necessary. Summary Three models incorporating axial dispersion, film mass transfer and intraaggregate diffusivity were applied to literature breakthrough data for the supercritical extraction of contaminants from soil. All three models fit the breakthrough front successfully. Kinetic parameters were fitted and compared to independent correlations. The greatest consistency between the fitted parameters and the existing correlations occurred when the full Type III model was utilized. This should not be interpreted as validation of each of the existing correlations for two specific reasons. One, for the film mass transfer component the correlations suggest that it is the least contributing step. Thus, a substantial error in this correlation could exist and go


20.

R A H M E ET AL.

311


undetected by the experimental observations. Two, since the other two parameters are both contributing significantly, it is only the combination of the two parameters that has been validated. Consequently, given the apparent uncertainty in the available correlations, it is possible that significant errors prevailing in one parameter may be compensated for by errors in the other parameter (s). As each of the three parameters are affected differently by the scale of the system, it is not sufficient to have only validated the combination of two parameters if the end goal is to use all three parameters for process scale up. The uncertainty associated with all of the independent correlations suggests direct measures of each individual parameter must be undertaken specifically for supercritical fluid - soil systems. Nomenclature C (C ) C (C ) C d t0

a

in

p

ao

3

= concentration of solute in bulk fluid (at time=0), mol/L fluid = concentration of solute in intraparticle pores (at time=0), mol/L fluid = inlet concentration, mol/L fluid = particle diameter, L = molecular diffusivity, L /T 3

3

z


312


Acknowledgments This work has been funded by the Natural Sciences and Engineering Research Council of Canada and the Groundwater and Soil Remediation Program. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Cairney, T. in Contaminated Land: Problems and Solutions, Cairney, T., Ed., Chapman & Hall, 1993, p1-7. Parker, S. Soils, 1993, March Issue, p41. Laitinen, Α.; Michaux, Α.; Aaltonen, Ο. Environ. Technol., 1994, 15, 715-727. Erkey, C.; Madras, G.; Orejuela, M.; Akgerman, A. Environ. Sci. Technol., 1993, 27, 1225-1231. Madras, G.; Thibaud, C.; Erkey, C.; Akgerman, A. AIChE J., 1994, 40, 777785. Valocchi, A.J. Water Res. Res., 1985, 21, 808-820. Patankar, S.V. Numerical Heat Transfer and Fluid Flow, McGraw-Hill Inc., New York, 1980. Kothandaraman, S.; Ahlert, R.C.; Venkataramani, E.S.; Andrews, A.T. Environ. Prog., 1992, 11, 220-222. Lamb, D.M.; Adamy, K.W.; Jonas, J. J. Phys. Chem., 1989, 93, 5002-5005. Sassiat, P.R.; Mourier, P.; Caude, M.H.; Rosset, R.H. Anal. Chem., 1987, 59, 1164-1170. Andrews, A.T.; Ahlert, R.C.; Kosson, D.S. Environ. Prog., 1990, 9, 204-210. Karickhoff, S.W. Chemosphere, 1981, 10, 833-846. Knaff, G.; Schlunder, E.U. 1987 Chem. Eng. Processing, 21:151-162. Reid, R.C.; Prausnitz, J.M.; Poling B.E. 1987 The Properties of Gases and Liquids, 4 edition, McGraw-Hill Inc., NY, p388-490. Tan, C.-S.; Liou, D.-C. Ind. Eng. Chem. Res., 1989, 28, 1246-1250. Freeze, R.A.; Cherry, J.A. Groundwater, Prentice-Hall Inc., Englewood Cliffs, NJ, 1979, p430. Lim, G.-B.; Holder, G.D.; Shah, Y.T. J. Supercritical Fluids, 1990, 3, 186-197. Pongsiri, N.; Viswanath, D.S. Ind. Eng. Chem. Res., 1989, 28, 1918-1921. Wakao, N.; Kaguei, S. Heat and Mass Transfer in Packed Beds, Gordon and Breach, NY, 1982, p156-157. Chang, H.C. Chem. Eng. Communication, 1982, 15, 83-91. Chemical Engineers Handbook,6 Ed., Perry, R.H.; Green, D. Eds.; McGrawHill Inc., NY, 1984, p5-54. Klotz, D.; Moser, H. Isotope Techniques in Groundwater Hydrology, International Atomic Energy Agency, Vienna, 1974, Vol 2, pp 341-355. Han, N.W.; Bhakta, J.; Carbonell, R.G. AIChE J., 1985, 31, 277-288. Rasmusson, A. Chem. Eng. Sci., 1985, 40, 621-629. Brusseau, M.L.; Rao, P.S.C. Crit. Rev. Environ. Cont., 1989, 19, 33-99. Fong, F.K.; Mulkey, L.A. Water Res. Res., 1990, 26, 1291-1303. th

15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

th

RECEIVED June 20,

1995


Kinetic Model for the Supercritical Extraction of Contaminants from Soil

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