Environ. Sci. Technol. 2002, 36, 1845-1850
Modeling of Polychlorinated Biphenyl Removal from Contaminated Soil Using Steam P I N G K U A N D I , * ,† DANIEL P. Y. CHANG,† AND HARRY A. DWYER‡ Department of Civil and Environmental Engineering and Department of Mechanical and Aeronautical Engineering, University of California, Davis, One Shields Road, Davis, California 95616
Microwave-generated steam technology shows promise as an effective remediation alternative for removal of polychlorinated biphenyls (PCBs) from contaminated soils, based on our laboratory-scale experiments. The overall process can be described by a nonisothermal, unsteady, coupled heat and multicomponent PCB mass-transport model in a multiphase, variably saturated, porous soil medium. In this paper, a multicomponent PCB mass-transport model is presented that assumes evaporation is an important removal mechanism and that is based on first-order mass transfer between the interface of PCB films and the bulk steam. The model was calibrated using the experimental data, and the calibrated model was verified by computational mass-balance checks and comparisons with laboratoryscale column experimental results. From a qualitative point of view, the calibrated model successfully simulated the transport of PCBs in variably saturated soil media. The calculated increase/decrease factors of physicochemical properties of PCBs as a function of temperature in the soil, water, and free phases were consistent with the model hypothesis of an evaporation process. The effects of masstransfer coefficients and initial PCB concentrations in the soils on PCB removal rates were investigated using the numerical code. It was determined that the PCB removal rates were sensitive to mass-transfer coefficients and initial PCB concentrations. Although the steam:soil mass ratios required to achieve a given percentage removal were lower for lower initial PCB soil concentrations, steam: soil mass ratios required to achieve a given unit mass removal were higher for lower initial PCB soil concentrations.
Introduction Removal of volatile and semivolatile organic contaminants from soils has been an important environmental and economic issue. Polychlorinated biphenyls (PCBs), a family of semivolatile compounds, have been recognized as priority soil/sediment contaminants at many Superfund sites (1). A variety of remediation technologies have been evaluated and documented by the U.S. EPA (1). Since PCBs have limited solubility in water and have very low vapor pressures at soil * Corresponding author present address: Stationary Source Division, California Air Resources Board, 1001 I St., Sacramento, CA 95814; phone: (916)327-5784; fax: (916)327-6251; e-mail:
[email protected]. † Department of Civil and Environmental Engineering. ‡ Department of Mechanical and Aeronautical Engineering. 10.1021/es010739o CCC: $22.00 Published on Web 03/14/2002
2002 American Chemical Society
ambient temperature, physical, chemical, and bioremediation technologies may not be very efficient under ambient conditions. In recent years, steam technologies for cleaning up volatile organic contaminated soils have received more attention (2-7). We have investigated the use of microwavegenerated steam (MGS) to remove PCBs from contaminated soils at the former Mare Island Naval Shipyard (MINSY) located in Vallejo, CA (8). In this process, liquid water is used to saturate, i.e., to wet, contaminated soil. The soil is placed in a microwave guide that is connected to a microwave generator. As the soil is irradiated by microwave energy, liquid water within the soil pores evaporates. Simultaneously, contaminant molecules are vaporized and/or solubilized due to elevated temperatures and removed via the generation and expansion of steam. The process can be repeated cyclically until a desired removal level of the contaminants in the soil is reached. Although the laboratory-scale experiments have shown that MGS is a feasible remediation alternative for removing PCBs from contaminated soils, the process principles and performance are not fully understood. We are unaware of any mathematical models available for describing PCB removal from contaminated soils by a MGS process. Existing models related to contaminant transport by steam-injection technologies are limited, and they can be classified into two types: equilibrium and nonequilibrium. Hunt et al. (9) developed a model to account for 1-D movement of a thermal front in a saturated, homogeneous soil system. Falta et al. (3, 4) developed a 3-D model to simulate movement of three fluid phases (air, water, and organic) in a porous system. For ex situ steam treatment of soils, Lord (10) developed an empirical model to simulate the removal of volatile organic compounds from soils via vacuum-assisted steam-stripping. The model considered a homogeneous isotropic soil (described by a simple steam permeability coefficient), under conditions of steady-state steam pressure, velocity, and constant temperature. In summary, all models above assumed local chemical equilibrium between the liquid and the vapor phases for the contaminants, constant steam injection rate, and steady steam-front velocities. Such assumptions may not be suitable for practical situations. Yuan and Udell (5) developed a nonequilibrium model to simulate the distillation removal of a free NAPL and derived a mass-transfer coefficient. Brouwers (11) conducted experimental and theoretical studies for combined solvent and steam-stripping of contaminated soil and proposed a 1-D nonequilibrium model to describe the unsteady mass transfer among vapor, condensate, and soil phases in a column. Van der Ham and Brouwers (12) presented a 1-D model to address an unsteady process of steam-stripping for free NAPL removal from unsaturated soils. They accounted for nonequilibrium between the liquid and the vapor phases of the contaminants. Nevertheless, all nonequilibrium models above assumed isothermal operating conditions. The removal of PCBs from a soil by MGS is complicated, and it involves at least three processes: (i) heat transfer, (ii) water phase change and transport (liquid to steam phase), and (iii) PCB phase change and transport among soil, liquid water, and steam phases. These processes take place simultaneously and are strongly coupled. The overall goal of this research was to develop a description of the transport phenomena associated with the removal of PCBs from the soil in the MGS process. VOL. 36, NO. 8, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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hk,m ) c1vvvSc-1/3
(2)
where c1 is a constant and Sc is the Schmidt number (Sc ) µv/Dk,vFv). It can be seen that the value of hk,m is mainly determined by the packed bed hydrodynamics, which vary temporally and spatially in our system. (Note that the condition of Re < 100 is satisfied for our system based on our calculations.) An ad hoc assumption was made that the specific contact area is proportional to the volumetric fraction of the void space. That is, a ) c2v (with c2 as a constant). Thus, the mass-transfer parameter hk,ma in eq 1 can be described as
hk,ma ) ckv2vvSc-1/3
(3)
In eq 1, the mass fraction of the kth species at the liquid PCB interface, according to Raoult’s law, can be expressed as i
Ck,m ) FIGURE 1. Conceptual diagram for interphase mass transfer of PCBs.
xkPkvMWk PsteamMWsteam +
v
∂Ck,m ) ∇(vDk,v∇Ck,m) - ∇(vvvCk,m) + ∂t hk,ma(Ck,mi - Ck,m) (1)
where Ck,m and Ck,mi are the mass fraction concentrations of the kth species of PCBs in the bulk steam phase flow and at the liquid PCB interface, respectively; hk,m is the gas-phase mass-transfer coefficient of the kth species; and a is the specific contact area between the liquid PCB and steam phases. Before solving this set of differential equations, the masstransfer parameter hk,ma has to be specified. According to the studies of Kunii and Suzuki (17) and Nelson and Galloway (18), it is known that hk,m can be described by the following general equation in packed beds at the condition of Re < 100 (Re ) dpvvFv/µv, Reynolds number): 1846
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∑x P
k k
v
MWk
k)1
Model Development The development of a 1-D multiphase heat and masstransport model for a soil-water sample has been described in an earlier study (13). A brief description of that model is provided in Appendix A in the Supporting Information. In this study, we develop a 1-D multicomponent PCB masstransport model that is coupled with the nonisothermal and unsteady heat and water mass-transfer model in a multiphase variably saturated porous soil medium. We assume that the PCBs form a continuous oily type of liquid film along the water-steam and soil-steam interfaces after start-up of water evaporation by microwave heating (Figure 1). Such configurations have been observed with styrene polymer blob casts studied by Wilson et al. (14) and with NAPL volatilization in unsaturated porous media by Wilkins et al. (15). Rathfelder et al. (16) also applied such a conceptual model to study the interphase mass transfer during soil vapor extraction. Next, it is assumed that the liquid PCB-water interactions resulting after the start-up of water evaporation can be neglected. Mass transfer between the liquid PCB and steam phase then takes place through a boundary layer that is a part of the steam phase, the driving force being the difference in the vapor-phase PCB concentration at the liquid PCB interface and in the bulk steam flow. It is further assumed that the mass-transfer processes are approximated by a first-order kinetic expression and that the steam behaves as an ideal gas phase. Thus, the 1-D vaporphase transport of the kth PCB species via the steam in the soil column, which does not account for sorbed and dissolved phases, can be described by
(4)
NK
where xk is the mole fraction of kth species in the liquid PCBs, Pkv is the saturation vapor pressure of the kth species, MWk is the corresponding molecular weight, NK is the number of PCB species present in the system, and Psteam and MWsteam are the vapor pressure and the molecular weight of the steam, respectively. The vapor pressure, Pkv, of the kth species as a function of temperature is expressed by an equation of the following form (19):
log(Pkv) ) Ak -
Bk T
(5)
where Ak and Bk are the constants for the kth PCB species. Note that Pkv varies temporally and spatially within the system. In eq 1, the effective diffusivity of the kth species is a function of temperature and porosity of the water vapor phase, given by (20)
Dk,v ) Dk,vov2
T (298 )
1.5
(6)
where Dk,vo is the molecular diffusivity of the kth species at 25 °C. In our experimental design, all steam and PCB vapors exited the system from the top of the soil column. The bottom side of the column was sealed to have unidirectional flow of the steam and to prevent possible liquid water drainage from the soil by gravity and pressure forces. Therefore, the following boundary and initial conditions are applicable for solving eq 1:
Ck,m(Z,0) ) Ck,mo
(7)
∂Ck,m(Z,t) ) 0 at Z ) L ∂Z
(8)
Equations 1 and A1, A2, and A4, in Appendix A in the Supporting Information, constitute a set of partial differential equations with the applicable initial and boundary conditions. The set of the equations was solved using the finite differences method. A fully implicit time discretization scheme was used for the time domain, and a central difference scheme was used for the space domain. The detailed solution procedures are provided in Appendix B in the Supporting Information.
TABLE 1. Conditions Used in the Experiments and Model Predictions parameter
value
microwave power soil bed depth soil sample weight initial water weight initial PCB concn in soil pentachloro hexachloro heptachloro octachloro nonachloro total PCBs
TABLE 2. Physical, Thermodynamic, and Dielectric Properties Used as Model Inputs
unit
symbol
200 4 15 5
W cm g g
34 448 363 201 21 1067
mg/kg of soil mg/kg of soil mg/kg of soil mg/kg of soil mg/kg of soil mg/kg of soil
Mass-balance checks were calculated as a measure of the program code’s ability to correctly solve the heat and the PCB mass-transport equations. A global mass-balance error is calculated for each PCB species using
{ | |} NN
∑M
% mass error(t) ) 100 1 -
Cps Cpl Cpv ks kl kv keff s l v Fs Fl Fv Ft ′ ′′ ∆hevap
NN
0
-
k,i
i)1
∑M
t
k,i
i)1
NT NN
∑∑F
a
value or expression
unit
ref
880 (soil) 4216 1842 1.3 (soil) 0.653 0.027 sks + lkl + vkv 0.4 (soil) l(t, T)b v(t, T)b 2100 (soil) taken from steam table Fv ) 3.3211-0.07104T + 0.0004388T 2 c sFs + lFl + vFv ′ ) 85.2-0.3358T c ′′ ) 320T-1.03 c ∆hevap ) 2473.6-1.7097T 0.00457T 2 c
kg-1 K-1
21 21 21 22 21 21 a a a a a 23 23
This work.
b
J J kg-1 K-1 J kg-1 K-1 W m-1 K-1 W m-1 K-1 W m-1 K-1 W m-1 K-1
kg m-3 kg m-3 kg m-3 kg m-3
a 24 24 23
kJ kg-1
t, time; T, temperature. c T in °C.
(9)
k,i,j∆tj
j)1 i)1
where Mk,i0 and Mk,it are the kth species masses in grid cell “i” of the soil column at time zero and time t, respectively; Fk,i,j is the removal rate of the kth species in grid cell “i” and at time step “j”; NT is the number of the total time steps; and NN is the number of the grid cells in the soil column. To quantitatively evaluate the magnitude of deviation of PCB removal model predictions from the experimental data, we define the percent error as follows:
() ( ) 1/2
% error ) 100
SSD SSE
1/2
NP
∑(C
) 100
m,n
- Ce,n)2
n)1
NP
∑(C
(10)
2
e,n)
n)1
In eq 10, SSD represents the sum of the squared deviations, SSE represents the sum of the squared experimental data points, Cm,n is the modeled concentration of the nth data point, Ce,n is the corresponding experimental data point, and NP is the total number of data points for comparison.
Results and Discussion The experimental procedures and results of PCB removal from the contaminated soils by the MGS have been addressed in detail elsewhere (8). Table 1 summarizes the experimental conditions. Table 2 lists the pertinent physical, thermodynamic, and dielectric properties of the soil, water, and steam used in the model inputs. Model Predictions versus Experimental Results. To simulate the PCB removal rates using eq 1, a value of ck in eq 3 for the kth PCB species was calibrated to the experimental data. An optimal value of ck was determined in an iterative manner to minimize the error between the experimental data and the model simulations. Then simulations of the model were carried out with the optimal ck value, and the results were compared with the experimental data. Figure 2 shows comparisons of the experimental data with the PCB removal model simulations. Here the “time” scale is implicitly represented as cumulative steam/soil mass ratio, and a unit ratio is equivalent to a treatment time of 7.5 min for this
FIGURE 2. PCB model simulations vs experimental results (initial total PCB concentration ) 1067 ppm).
TABLE 3. Percent Errors of the PCB Removal Model PCB species
% error defined by eq 10
pentachlorobiphenyl hexachlorobiphenyl heptachlorobiphenyl octachlorobiphenyl nonachlorobiphenyl overall
5.4 3.3 7.7 15.4 23.3 6.0
experimental data set. Note that in our simulations, soil mass was fixed and rewetted with water repeatedly for different cycles. The total steam employed at the end of the Nth cycle was the cumulative sum of the previous N cycles’ water addition. For PCBs, mass and mole fraction remaining in the ith cell of the soil column at the end of the Nth cycle served as the initial mass and mole fraction of PCBs in the ith cell for cycle N+1, and so on. Thus an increasing number of cycles resulted in a decrease in the amount of PCB mass remaining in the soil or an increased PCB removal. It can be seen from Figure 2 that the PCB removal model provided a reasonably good presentation of the experimental data with an overall error of 6.0% (Table 3), although it did not mimic the onsets for octachlorobiphenyl and nonachlorobiphenyl removal very well. Both the experimental and the predicted VOL. 36, NO. 8, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 4. Cumulative Mass-Balance Errors in the Simulations of the Experimental Results (15 Water Addition Cycles) PCB species
% mass error defined by eq 9
pentachlorobiphenyl hexachlorobiphenyls heptachlorobiphenyl octachlorobiphenyl nonachlorobiphenyl
0.2 2.0 1.1 1.2 0.4
results showed that the more volatile PCB species were removed ahead of successively less volatile species. This trend is consistent with physicochemical properties of PCB species, especially vapor pressures. According to Raoult’s law, for the mixed PCB species system, the liquid composition would change continuously toward heavier fractions, thus increasing the mole fractions of the heavier fractions remaining in the liquid phase. Nevertheless, the uncertainty of octachlorobiphenyl and nonachlorobiphenyl property estimates could be a reason the model did not capture the onset portions of the two removal curves very well. Table 4 lists the results of numerical PCB mass-balance calculations for simulations of the PCB removal model. For all PCB species, cumulative mass-balance errors were equal to or less than 2%. On the basis of the PCB mass-balance checks and reasonably good comparisons of the PCB model simulations with the experimental data given the uncertainties in the physicochemical parameter estimates, we are confident that the coupled heat and PCB mass-transport equations were solved correctly by the numerical methods used. PCB Removal Mechanisms. As assumed implicitly in the Model Development section, evaporation was as an important mechanism for PCB removal from the bulk soils. To justify this hypothesis, the increase/decrease factors of physicochemical properties of PCBs from the ambient to averaged steam temperatures are calculated first. For the sake of simplicity, assuming that the average temperature of the system during PCB removal by MGS was 120 °C, the calculated increase/decrease factors for vapor pressure (P), solubility (S), Henry’s constant (H), soil-water distribution coefficient (Kd), and diffusivity (D) are given in Table 5 (details presented in Appendix C in the Supporting Information). One can see that the effect of temperature on the vapor pressures was much stronger than all other properties. As the temperature increased from 20 to 120 °C, the vapor pressures of PCBs were increased by factors of 8400, 5900, 16 000, 25 000, and 110 000 for penta-, hexa-, hepta-, octa-, and nona-PCBs, respectively. To achieve the experimentally determined PCB removal efficiency, the calculated steam requirements based on temperature change of the solubility data would be much higher than those calculated based on the vapor pressure data, and the latter were of the same magnitudes as the experimental results (data not shown). This implies that evaporation would be an important control mechanism for PCB removal from the soils by the MGS.
FIGURE 3. Effect of mass-transfer coefficients on total PCB removal. An additional consideration is that the distribution of initial PCBs in the soil, water, and free phase (8) indicated that only one-third of PCBs were sorbed onto the soil particles and another two-thirds were in free phase that were probably loosely deposited on the soil particle surfaces and/or within the soil pores. This again supports the hypothesis that evaporation is an important mechanism in removing most, i.e., about two-thirds, of the PCBs. Nevertheless, the experimentally determined PCB removals were higher by 5-15%, depending on the PCB species, than those predicted by the model for the tail portions of the curves (see Figure 2). Other effects, such as increased solubility of PCBs into the heated aqueous phase, may have also been mechanisms for PCB removal. In addition, in the microwave heating process, since heating is generated internally and selectively, the internal volumetric heating gives rise to locally higher internal temperature, which could result in the evolution and formation of multiple gas bubbles of varying size at nucleation sites in the soil matrix. The formation of gas bubbles may produce steam and/or PCB vapor-filled pores or even small channels that enhance advection of steam and evaporation of PCB molecules within the soil. Thus, we speculate that rapid expansion and evaporation of water within the soil pores could be an additional mechanism for enhanced PCB removal. Sensitivity Analysis. A sensitivity analysis was conducted to test the PCB removal model’s response to changes of the model inputs. The initial distribution of PCB species in the soil was assumed to be the same in all cases. The effect of changing hk,ma values on mass-transfer rates of PCBs is presented in Figure 3. In this sensitivity run, the base case was that values of hk,ma were those calibrated from the experimental data. As the hk,ma values were halved, PCB removal rates were reduced. In contrast, higher hk,ma values resulted in faster PCB removal rates. To achieve an overall PCB removal of 90%, doubling hk,ma required a steam/soil mass ratio of 3.9, which was 75% of the ratio for the base case. Figure 4 shows the effect of the initial PCB concentrations on PCB removal rates. The rates of PCB removal with lower initial PCB concentrations were higher than those with
TABLE 5. Calculated Increase/Decrease Factors for Physicochemical Properties of PCBs PCB species
vapor pressure increase factor (P120/P25)
solubility increase factor (S120/S25)
Henry’s constant increase factor (H120/H25)
partitioning coeff decrease factor (Kd,25/Kd,120)
diffusivity increase factor (D120/D25)
penta hexa hepta octa nona
8 378 5 893 16 249 24 624 106 766
21 77 98 125 115
303 58 126 149 704
23 86 110 141 129
1.5 1.5 1.5 1.5 1.5
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(13), microwave energy distribution along the height of the soil sample was assumed to decay exponentially within a given volume element. Thus, nonuniform profiles of “steam generation rate” and temperature formed. Consequently, the largest rates of water vapor and PCB evaporation occurred in the top portion of the soil sample. As the top of the soil sample dried out, microwave energy penetrated into the deeper parts of the soil sample. Finally, the nonuniformity formed at the beginning tended to level off.
Acknowledgments
FIGURE 4. Effect of initial PCB concentrations on total PCB removal.
TABLE 6. Calculated Steam Requirements (g) for Removal of 1 mg of PCB Speciesa cases
pentaPCB
hexaPCB
heptaPCB
octaPCB
nonaPCB
case 1 (533 ppm) base case (1067 ppm) case 3 (1600 ppm)
65 60 58
9 8 8
25 17 14
55 41 37
520 388 290
This publication was made possible by Grant IEHS 5 P42 ES04699 from the National Institute of Environmental Health Sciences, NIH with funding provided by the U.S. EPA. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the NIEHS, NIH, or U.S. EPA. The authors thank Dr. Dan Jones and Mr. Dale Uyeminami for their help in the laboratory GC/MS analyses. The authors also are grateful to two anonymous reviewers for their constructive comments.
Nomenclature A
cross-sectional area of the soil column (m2)
a
specific surface contact area (m2 m-3)
ai
coefficient used in Thomas algorithm (dimensionless)
Ak
constant used in the vapor pressure equation (dimensionless)
bi
coefficient used in Thomas algorithm (dimensionless)
Bk
constant used in the vapor pressure equation (dimensionless)
ci
coefficient used in Thomas algorithm (dimensionless)
Ck
constant used in the solubility equation (dimensionless)
Ck,m
mass-fraction concentration of the kth PCB species in the bulk steam phase (kg kg-1)
Ck,mi
mass-fraction concentration of the kth PCB species at the liquid PCB interface (kg kg-1)
Cp
specific heat (J kg-1 K-1)
di
coefficient used in Thomas algorithm (dimensionless)
Dk,v
vapor-phase diffusion coefficient of the kth PCB species (m2 s-1)
dp
particle diameter (m)
Fk,i,j
removal rate of the kth species in grid cell “i” and time step “j” (kg s-1)
H
Henry’s law constant (dimensionless)
hk,m
gas-phase mass-transfer coefficient of the kth PCB species (m s-1)
Hk,s
enthalpy of dissolution of the kth species (J mol-1)
∆hevap
latent heat of evaporation (kJ kg-1)
k
thermal conductivity (W m-1 K-1)
keff
effective thermal conductivity (W m-1 K-1)
Kk,d
partitioning coefficient for the kth PCB species (m3 kg-1)
L
length of the packed column (m)
a
Assume that (i) PCB species compositions were the same for the three cases and (ii) PCB species were independently removed by 90%.
FIGURE 5. PCB removal profile along soil column depth. higher initial PCB concentrations. To achieve the same proportionate reduction of 90%, the steam requirements were 50, 75, and 115 g (in liquid phase), corresponding to 10, 15, and 23 water addition cycles, for PCB concentrations of 533, 1067 (base case), and 1600 ppm, respectively. On the other hand, to remove 1 mg of PCBs, the steam requirements were 6.0, 5.3, and 5.3 g for the three cases, respectively. This indicates that the steam required to remove a given mass of PCBs would be the same or less when PCBs in the soil are more concentrated. Another interesting finding from the modeling simulations was that the effects of initial PCB concentrations on steam requirements were different for different PCB species. For an initial PCB concentration of 533 ppm, the ratio of steam required to remove 1 mg of nona-PCB as compared to that required to remove 1 mg of penta-PCB based on a removal of 90% was about 8.0 (see Table 6). As the initial PCB concentration in the soil increased to 1067 or to 1600 ppm, the ratio became 6.6 or 5.0. The vertical profile of PCB removal within the soil column predicted by the model indicated that for a given time, or steam/soil mass ratio, PCB removals were not uniform along the soil column depth (Figure 5). This may be attributable to microwave heating characteristics. As discussed elsewhere
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m ˘
steam generation rate (kg m-3 s-1)
s
solid phase
Mk,i
kth species mass in grid cell “i” of the soil column (kg)
t
total
v
vapor phase
molecular weight (dimensionless)
w
water
MW
Glossary
Supporting Information Available
NK
number of PCB species
NN
number of grid cell in the soil column
NP
number of data point
Appendix A, heat and water mass-transport model; Appendix B, solution procedures; Appendix C, calculations of PCB physicochemical properties. This material is available free of charge via the Internet at http://pubs.acs.org.
NT
number of total time steps for simulation
P
microwave power (W) v
Pk
saturation vapor pressure of the kth species (Pa)
Psteam
steam pressure (Pa)
Qabs
rate of microwave energy absorbed per unit volume (W m-3 s-1)
R
gas constant (J mol-1 K-1 or Pa m3 mol-1 K-1)
Re
Reynolds number (dimensionless)
S
solubility (kg m-3)
Sc
Schmidt number (dimensionless)
t
time (s)
T
temperature (K)
v
velocity (m s-1)
xk
mole fraction of the kth species in the liquid PCBs (dimensionless)
Z
distance (m)
Greek Letters R
attenuation factor (m-1)
volume fraction (dimensionless)
θ
liquid saturation (dimensionless)
F
density (kg m-3)
δ
tan-1 (′′/′) (dimensionless)
λ
wavelength (m)
µ
viscosity (kg m-1 s-1)
Superscripts and Subscripts
Literature Cited (1) Technology Alternatives for the Remediation of PCB-Contaminated Soils and Sediments; U.S. Environmental Protection Agency: Washington, DC, 1993; EPA/540/S-93-506. (2) Udell, K. S.; Steward, L. D. In Subsurface Contamination by Immiscible Fluids; Weyer, K. U., Ed.; Rotterdam: The Netherlands, 1990; pp 327-335. (3) Falta, R. W.; Pruess, K.; Javandel, I.; Witherspoon, P. A. Water Resour. Res. 1992, 28, 443-449. (4) Falta, R. W.; Pruess, K.; Javandel, I.; Witherspoon, P. A. Water Resour. Res. 1992, 28, 451-465. (5) Yuan, Z. G.; Udell, K. S. Int. J. Heat Mass Transfer 1993, 38, 1965-1976. (6) Itamura, M. T.; Udell, K. S. Proceedings of the ASME Heat Transfer and Fluid Engineering Division; 1995; pp 651-660. (7) Imhoff, P. T.; Frizzell, A.; Miller, C. T. Environ. Sci. Technol. 1997, 31, 1615-1621. (8) Di, P.; Chang, D. P. Y. J. Air Waste Manage. Assoc. 2001, 51, 174-180. (9) Hunt, J. R.; Sitar, N.; Udell, K. S. Water Resour. Res. 1988, 24, 1247-1258. (10) Lord, A. E., Jr. Geotech. Test. J. 1995, 18, 32-40. (11) Brouwers, H. J. H. J. Hazard. Mater. 1996, 50, 47-64. (12) Van der Ham, A. G. J.; Brouwers, H. J. H. Water Resour. Res. 1998, 34, 47-54. (13) Di, P.; Chang, D. P. Y.; Dwyer, H. A. J. Environ. Eng. 2000, 126, 1108-1115. (14) Wilson, J. L.; Conrad, S. H.; Hagan, E.; Mason, W. R.; Peplinski, W. Proceedings of Organic Chemical and Petroleum Hydrocarbons in Groundwater; National Water Well Association: Houston, TX, 1988; pp 107-132. (15) Wilkins, M. D.; Abriola, L. M.; Pennell, K. D. Water Resour. Res. 1995, 31, 2159-2172. (16) Rathfelder, K.; Yeh, W. W.-G.; Macky, D. J. Contam. Hydrol. 1991, 8, 263-297. (17) Kunii, D.; Suzuki, M. Int. J. Heat Mass Transfer 1966, 10, 845852. (18) Nelson, P. A.; Galloway, T. R. Chem. Eng. Sci. 1975, 30, 1-6. (19) Burkhard, L. P.; Armstrong, D. E.; Andren, A. W. J. Chem. Eng. Data 1984, 29, 248-250. (20) Lighty, J. S.; Silcox, G. D.; Pershing, D. W. Environ. Sci. Technol. 1990, 24, 750-757. (21) Perry, R. H.; Chilton, C. H.Chemical Engineer’s Handbook, 5th ed.; McGraw-Hill: New York, 1973. (22) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley: New York, 1960. (23) Moran, M. J.; Shapiro, H. N. Fundamentals of Engineering Thermodynamics, 2nd ed.; John Wiley & Sons Inc.: New York, 1992. (24) Von Hippel, A. R. Dielectrics and Waves; MIT Press: Cambridge, MA, 1954.
abs
absorption
sat
saturation
eff
effective
evap
evaporation
i
general index for vector
l
liquid phase
max
maximum value
n
node number
Received for review March 14, 2001. Revised manuscript received January 16, 2002. Accepted January 21, 2002.
o
initial condition
ES010739O
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