Environ. Sci. Technol. 2002, 36, 2911-2918
In Situ Removal of 2-Chlorophenol from Unsaturated Soils by Ozonation MENGHAU SUNG AND CHIN PAO HUANG* Department of Civil and Environmental Engineering, University of Delaware, Newark, Delaware 19716
A mathematical model considering mass transfer process at the gas-liquid interface in soil ozonation was developed and validated with laboratory column experiments. Experimental data, specifically, concentration profiles of the organic contaminant and the ozone breakthrough curves, were obtained. In this model, the mass flux of ozone transferred from the gas phase into the liquid phase was described by the two-film theory incorporated with an enhancement factor approach as to account for chemical reactions. With the enhancement factor, the ozone gas transport in the experimental column can be described by an advection-dispersion-reaction equation with pseudosecond-order kinetics in the liquid film. This greatly simplifies the governing equations of the system. Results show that parameters such as degradation yield factor, diffusion coefficients, thickness of liquid film, ozone gas concentration, and gas-liquid interfacial area play an important role on the soil ozonation process. Using the scaled model, important universal dimensionless variables were obtained. The Stanton number (St) is the most important parameter in controlling the performance of system. When St approaches zero, the process is reaction-controlled. Conversely, when St is large, it is convection-controlled. Only when the system is convection-controlled (i.e., large St values) can an increase of ozone flow rate enhance the removal of soil contaminants such as 2-chlorophenol.
Introduction The in situ air-venting process has become the most commonly used soil remediation process (1, 2). However, the process is not totally feasible for low- or nonvolatile organic compounds regardless of soil permeability. Consequently, much recent effort has been made to modify the air-venting process with the use of a stronger reactive gas, such as ozone and hydrogen peroxide, to enhance the degradation of lowand nonvolatile organic contaminants. Several laboratory and field studies (3-5) have shown that this process is effective in removing contaminants such as polyaromatic hydrocarbons (PAHs), diesel fuels, and chlorinated solvents from soils with a removal efficiency of 40-98%. Most laboratory and field studies in soil ozonation investigate mainly the effects of major parameters such as ozone concentration, gas flow distribution, ozone gas residence time, soil type, precondition method, and moisture on the removal efficiency. Little attempt has been made to describe the process quantitatively. Hsu (5) developed a lumped kinetic expression to model the ozonation of PAHs in natural soils. The model approach does not separate the * Corresponding author phone: (302)831-8428; fax: (302)8313640; e-mail:
[email protected]. 10.1021/es010559e CCC: $22.00 Published on Web 05/29/2002
2002 American Chemical Society
effects of mass transfer and chemical reaction. Consequently, the model is rather system-specific and may not be applied universally. To properly describe the soil ozonation system, a predictive model considering all possible steps pertinent to the removal process is needed. Microscopically, the soil ozonation system can be treated as a gas-liquid mass transfer process coupled with chemical reactions. Unlike a homogeneous system, the liquid phase in the system is only a thin film covering soil particles. Because of the lack of a wellmixed liquid bulk, a diffusion process with chemical reactions needs to be formulated specifically. Prior to establishing governing equations for the soil ozonation process, the corresponding chemical kinetics must be understood. Such kinetic equations are believed to be specific to soil water conditions and can be obtained from the literature of pure aqueous ozonation. Detailed ozone decomposition mechanisms have been studied and reviewed extensively (6-8). Direct reactions were found to be dominant under acidic conditions or in the presence of radical scavengers. Second-order kinetic equations, first-order with respect to ozone and organic contaminant, respectively, have been applied in direct ozone reactions. Stoichiometric coefficients usually accompany with these equations to account for reactions with multiple elementary steps (9, 10). At high pH or in the presence of hydrogen peroxide, radical reactions prevail. Mathematical models describing these advanced oxidation processes (AOPs) have been developed by considering gas-liquid mass transfer and kinetics with series of elementary reactions in the aqueous phase (11, 12). The approach involving the enhancement factor (13) and two-film theory (14) is useful to study chemical mass transfer in relation to physical mass transfer. Additionally, the twofilm theory essentially assumes a steady-state process. If this criterion cannot be satisfied, a transient model (15) has to be considered. However, for some non-steady-state systems, the concept of pseudo-steady-state (PSS) has been applied (16). In essence, PSS neglects the transient behavior with respect to system characteristic times (17). Therefore, approximate solutions for rigorous non-steady diffusion problems can sometimes be made through the PSS approach. The objective of this study is to develop a simple mathematical expression for describing the soil ozonation process. The mathematical model combines ozone oxidation in stationary liquid phase, ozone gas-liquid interfacial mass transfer, and ozone gas-phase transport in the subsurface to simulate the transformation of organic contaminants in unsaturated soil zone exemplified by 2-chlorophenol (2-CP) in a one-dimensional column. Experiments were carried out in acidic soils. A pseudo-second-order kinetic equation with empirical degradation yield factor was assumed between ozone and 2-CP. Using the enhancement factor approach, the interfacial flux into the thin liquid film on the soil surface can be formulated. With the constitutive flux, governing equations were developed and verified experimentally.
Materials and Methods Materials. The soil used in this experiment was a play sand from Quikrete Co. (Columbus, OH). The chemical 2-CP was purchased from the Aldrich Co. (Milwaukee, WI). The sand was first sieved with U.S. standard sieve #20 with 0.084 cm opening and then dried overnight at 100 °C in an oven to eliminate the moisture before use. Column Experiments. A 2-CP solution (1.2 g/L) was prepared and mixed with a given amount of soil completely in a glass bottle to achieve a moisture content of 12 wt %. Before the experiment, three soil samples were taken VOL. 36, NO. 13, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Conceptual illustration of soil ozonation system. (a) Experimental column. (b) Distribution of water film on soil surface and uniform gas flow in the system. (c) Processes ocurring at individual sand particle. (d) Transient concentration distribution inside the liquid film. (e) Transient concentration distribution in liquid phase under Van Krevelen’s approximation. randomly to check for the uniformality of 2-CP distribution in soil. To analyze the residual 2-CP concentration in soil, a given amount of soil (ca. 3 g) was sampled and mixed with 3 mL of a H2SO4 (1 N) solution and 9 mL of hexane in a test tube. The soil sample in the test tube was then mixed on a shaker overnight to extract the 2-CP from the soil. After extraction, the hexane phase was analyzed by GC-ECD (HP5890) for 2-CP. The ozone gas concentrations in and out of the column were monitored by the iodometric method with potassium iodide (KI) solution (18). The soil sample was packed into a 17.6-cm long column (plexy glass) with a diameter of 3.125 cm. A total of 15 g of soil was packed evenly in the column at a length of every 1.5 cm until the full length was filled. The ozone generator is from Welsbach Co. (Philadelphia, PA). A pressurized air tank from Keen Co. (Wilmington, DE) was connected to the inlet of the generator to provide oxygen. For ozonation experiments, the soil-packed column was first ozonated for a predetermined period of time. Soil samples were taken from the column at different depths to analyze the remaining 2-CP concentrations. Three different 2912
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ozone concentrations, 2.4, 7.6, and 19.4 mg/L(g), at a flow rate of 100 mL/min were used. Each column was run for 10, 20, 30, and 40 min at each ozone concentration. Separate experiments of identical operational conditions were conducted as to obtain ozone breakthrough curves (BTCs). A total of 15 runs were completed in this study.
Mathematical Modeling Ozone Gas Transport. Figure 1 shows the microscopic view of the soil-water system during ozonation. The ozone gas transport in the column along the z variation, or the longitudinal direction, is described by the following equation:
∂CAg ∂2CAg ∂CAg 1 A N ) EAz 2 - vg ∂t ∂z nSa V A,int ∂z
(1)
where CAg is the average ozone gas concentration, t is the reaction time, EAz is the gas-phase dispersion coefficient, z is the column distance from the gas entrance, vg is the gas velocity, n is the soil porosity, Sa is the degree of saturation
of gas phase, AV is the specific gas-liquid interfacial area, and NA,int is the interfacial ozone flux. Equation 1 is subject to an initial condition of zero ozone gas concentration throughout the column (i.e., CAg ) 0 at t ) 0) and two boundary conditions (i.e., CAg ) C0 at z ) 0; a constant entrance concentration, C0, and a zero-flux equation at z ) L, the exit, that is, EAz(∂CAg/∂z) ) 0). Parameters Estimation. The dispersive effect of ozone gas molecules in the soil system is due to diffusion through tortuous air-filled pores. Thus a free-air diffusion coefficient (Dg) can be corrected for the tortuosity (τa) of the air-filled pores to obtain the field gas dispersion coefficient, that is, EAz ) Dgτa-1. In this work, the optimized Gilliland-type equation (19), suitable for simple system at low pressure, is used to calculate the diffusivity of ozone gas (Dg) in the air. Many correlations are available for determining the tortuosity of the air-filled pores. A relationship adapted from Millington (20) was used in this work: τa ) n2[n(1 - Sw)]-7/3, where Sw ) 1 - Sa. The diffusion coefficients in liquid phase for ozone and 2-CP were calculated by the equation of Wilke and Chang (21). Constitutive Interfacial Flux. Several assumptions must be made prior to seeking solution for NA,int. First, gas-phase resistance of ozone can be neglected; therefore, ozone concentration at the interface can be obtained from CAg/H, where H is the Henry’s constant of ozone. Second, it is assumed that the organic contaminant concentration in the liquid film is in large excess as compared to the ozone concentration. Consequently, the concentration changes of organic contaminant inside the film can be considered as PSS. Third, the reaction kinetics between ozone and the organic compound in the liquid phase follow a pseudosecond-order expression which encompasses all possible reactions such as direct, decomposition, catalytic, and side. The diffusion process coupled with chemical reaction inside the liquid film can be described by the following equations with respect to ozone and the organic compound:
DAL
DBL
d2CAL dy2
d2CBL dy2
) kCALCBL
(2)
) νkCALCBL
(3)
where DAL and DBL are the liquid-phase molecular diffusivity of ozone and 2-CP, respectively; y is the distance in the liquid film measuring from the gas-liquid interface; CAL and CBL are the concentrations of ozone and 2-CP in the liquid film, respectively; k is the pseudo-second-order rate constant; and ν is the degradation yield factor defined as moles of 2-CP degraded per mole of ozone consumed. Equation 2 is subject to a constant ozone concentration (i.e., CAi ) CAg/H at y ) 0) and a zero-flux boundary condition at y ) λ, where λ is the thickness of the liquid film on soil particle surface. The boundary conditions for CBL are dCBL/dy ) 0 at y ) 0 and CBL ) CBb at y ) λ, where CBb is the 2-CP liquid-phase concentration at the liquid-solid boundary. An approximate analytical solution of eqs 2 and 3 was obtained by Van Krevelen (22) under the assumptions that CBL remains constant within a certain region of λR, adjacent to the gas-liquid interface, and that CAL equals zero outside that region. Figure 1 (parts d and e) shows the concentration profiles in the liquid film of both numerical and Van Krevelen’s approximate solutions. The approximate solution is often expressed in terms of the enhancement factor, which is defined as the chemical mass transfer rate to pure physical mass transfer rate (13). An explicit enhancement factor equation used in this study is expressed as (23)
FA ) 1 +
[
( )]
γ-1 1 DBL CBb 1 - exp ν DAL CAi 1 DBL CBb ν DAL CAi
(4)
where γ is the Hatta number and is defined as γ ) xkCBbDAL/KL, where KL ) DAL/λ from the two-film theory. The reduced distance λR now becomes λR ) λ/FA. After obtaining the profiles of CAL and CBL, the CBb value at the next time step is determined by the volume-average concentration of CBL across the film when conducting a transient computation. The accuracy of this scheme is high when CBL does not change significantly inside the film. Given the enhancement factor equation, NA,int can now be found as NA,int ) (FAKLCAg)/H. By further defining a new parameter, kp as
kp )
AVFAKL nSaH
(5)
eq 1 now becomes
∂CAg ∂2CAg ∂CAg ) EAz 2 - vg - kpCAg ∂t ∂z ∂z
(6)
Similarly, the mass balance equation for 2-CP in the liquid film at the soil surface can be written as
Sa ∂P 1 )(νA N ) ) -kp νCAg, at P ) P0 as t ) 0 ∂t nSw V A,int Sw (7) where P is the average liquid-phase 2-CP concentration obtained by integrating CBL over the liquid-phase thickness, λ. P0 is the initial concentration of 2-CP. Because 2-CP in the liquid phase is relatively stationary as compared to ozone, both dispersive and advective transport forces are negligible. Dimensionless Governing Equations. By introducing the following dimensionless variables, π ) (CAg/C0) (liquid-phase ozone concentration), φ ) (P/P0) (liquid phase 2-CP concentration), ζ ) (z/L) (column length), τ ) (vgt/L) (time), and R ) (νSaC0/SwP0) (degradation yield), where L is the total length of the column, eqs 6 and 7 can be simplified further
∂π 1 ∂2π ∂π ) - Stπ ∂τ Pe ∂ζ2 ∂ζ
(8)
∂φ ) -RStπ ∂τ
(9)
where Pe is the Pe´clet number, Pe ) (vgL/EAz), and St is the Stanton number, St ) (kpL/vg). The scaled dimensionless governing equations are subject to the following initial and boundary conditions, that is, τ ) 0, π ) 0; ζ ) 0, π ) 1; ζ ) 1, ∂π/∂ζ ) 0; and τ ) 0, φ ) 1. From eqs 8 and 9, it is clear that the most important parameters affecting ozone gas concentration in the soil column are Pe and St, whereas Pe, St, and R are the most important parameters governing the removal of 2-CP in soils.
Results and Discussion Preliminary Assumption. It is necessary to compare various system characteristic times (17) in order to justify the use of PSS. The specific nature of the holdup and the throughput vary according to the process being considered. During the soil ozonation process, the gas-phase ozone concentration (CAg) changes as a function of time and distance along the column. That change will impose time-dependent boundary conditions at the liquid film surface. We want to know how the concentration change at the boundary affects the VOL. 36, NO. 13, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. Experimental and model fitting results of residual 2-CP profiles at various experimental conditions: (a) f ) 0.1, O3,o ) 2.4; (b) f ) 0.1, O3,o ) 7.6; (c) f ) 0.1, O3,o ) 19.4 (f ) ozone flow rate (L/min), O3,o ) ozone gas concentration (mg/L)). concentration profile inside the film. A fast response can justify the use of a PSS assumption (17). Under the aforesaid experimental conditions, a typical characteristic time for ozone (tO3) concentration change across the entire column is about 300 s, which was estimated empirically from the breakthrough curves by observing the time when ozone first appeared at the column outlet or when τ ) 10 (tO3 ∼ τL/vg ) (10 × 17.6 cm)/(0.575 cm/s) ) 306 s), because the characteristic time of ozone diffusion in the liquid film (tD) is about 3.2 s (note that tD ∼ λ2/DAL ) (0.008 cm)2/(2 × 10-5 cm2/s) ) 3.2 s). Moreover, the characteristic time of pseudosecond-order reaction (trxn) of ozone is approximately 0.11 s (note that trxn ∼ 1/(kCBL0) ) 1/(103 M-1 s-1)(9.4 × 10-3 M) ) 0.11 s, where CBL0 is the initial concentration of organic compound inside the liquid film), because the response time of liquid film system being modeled is much shorter than the time required for significant changes in the boundary condition. Therefore, the ozone concentration change inside the liquid film can be treated as PSS with respect to the ozone concentration change in the gas phase. This implied that the system in the liquid phase would respond almost immediately to changes in the boundary conditions. The Hatta number in this system is 5.5 (γ ) xkCBbDAL/KL )
x(103M-1s-1)(9.36×10-3M)(2×10-5cm2/s)/(2.5
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cm/s) ) 5.5). This implies that the rate of chemical reaction is faster than diffusion; therefore, diffusion is the rate-limiting step inside the liquid film. Under this condition, the concentration change of 2-CP across the liquid film is small because 2-CP concentration is much greater than the corresponding ozone concentration. As a result, the use of an average value of 2-CP in transient computation is acceptable. Degradation Yield Factor and Model Fitting. Experimental data obtained from this study are the residual 2-CP concentration profiles along the column and ozone BTCs, as shown in Figures 2 and 3 (in open points.) Note that the dimensionless ozone concentration, R0, is the empirical yield factor at time zero, which can be obtained by assigning a unity ν value to R. As a result, the ozone gas concentration of 2.4, 7.6, and 19.4 mg/L(g) will have R0 values of 9.3 × 10-3, 1.4 × 10-2, and 3.4 × 10-2, respectively. The area above each 2-CP profile curve in Figure 2 represents the fraction of 2-CP degraded, and the area above each ozone BTC in Figure 3 during a specific time interval represents ozone consumption in that period. Therefore, ν values can be calculated using these data. Open circles in Figure 4 are results of ν as a function of (C0/H)/P0, which is the initial contact ratio between ozone and 2-CP in liquid phase. From Figure 4, ν
TABLE 1. In Situ Soil Ozonation System Parameters of Soil, 2-CP, and Ozone parameter
FIGURE 3. Experimental and model fitting results of ozone BTCs under various experimental conditions.
soil column length, cm soil column diameter, cm specific gravity of soils average soil particle size, mm water film thickness (λ), mm specific gravity of soils porosity in column (n) liquid degree of saturation (Sw) gas degree of saturation (Sa) air entry value for sand (h0), cm‚H2O porosity index (β) pH of aqueous 2-CP solution pKa value of 2-CP aqueous 2-CP concentration (P0), g/L second-order reaction rate constant (k), (mol/L)-1 s-1 aqueous diffusion coefficient of ozone (DA), cm2/s aqueous diffusion coefficient of 2-CP (DB), cm2/s gas diffusion coefficient of ozone (Dg), cm2/s gas dispersion coefficient of ozone (Eg), cm2/s tortuorsity of gas in soil column (τa) dimensionless Henry’s constant of 2-CP dimensionless Henry’s constant of ozone (H) ozone gas velocity (vg), cm/s specific gas-liquid interfacial area in model (cm-1)
value 17.60 3.18 2.6 0.84 0.08a 2.6 0.51a 0.34a 0.66a 2b 2b 3 ( 0.05 8.3 1.2 103c 2.0 × 10-5d 0.9 × 10-5d 0.42d 0.12d 3.3e 10-3.74f 3.85f 0.575g 20h
a From theoretical calculations. b From Cihacek (1979). c From Hoigne ´ (1983). d From Wilke and Chang (1955). e From Millington (1959). f From g Chemical Engineers’ Handbook (Perry 1976). Vg at Pe value of 100. h From model fitting.
FIGURE 4. Plot of the degradation yeild factor (ν) as a function of contact ratio (C0/H)/P0 at various ozone concentrations. decreases gradually as the contact ratio increases. This implies that the degree of oxidation is dependent on contact ratio. The temporal and spatial yield factor, ν(z,t), can be described by the following power equation: ν(z,t) ) 0.0802(CAi/P)-0.3063, where (CAi/P) is the instantaneous contact ratio between ozone and 2-CP. In fitting the column profiles and ozone BTCs, various power equations within the range of two lines in Figure 4 were used. Moreover, The second-order reaction rate constant (k) used in the model was estimated from Hoigne´’s (24). In a final manner, the AV value is used as an adjustable parameter in fitting the model to experimental data of column profiles. Column Profiles and Breakthrough Curve. Residual 2-CP at the soil surface in the column and ozone BTCs can be obtained by solving eqs 8 and 9. Figures 2 and 3 show the best-fitted results (see various lines in the figures) from model calculation. Results clearly indicate that the observed data agree well with model simulation over a wide range of time. Inspection of eq 8 reveals that ozone concentration appears to be independent of Pe. This is easily realized because all dimensionless quantities have values ranging from 0 to 1, and a Pe value greater than 100 would diminish the dispersive term in eq 8. This leads π to depend entirely on St and R. Under the same kp value as in this case, the
effects of ozone concentration on St are not significant. Therefore, the dependence falls on R. From Figure 3, the functional dependence of BTCs on R0 is evident. At a large R0 value of 3.4 × 10-2, 2-CP is removed faster than at a small R0 value of 9.3 × 10-3. Therefore, ozone breaks through faster at higher R0 values. Gas-Liquid Interfacial Area. Some theoretical and empirical correlations for estimating AV have been proposed. The following two equations are from Miller (25) and Kim (26), respectively.
AV ) 13.6h0(θ1-β/θ0-β) (Miller)
(10)
AV ) - 64.7Sw + 64.7 (Kim)
(11)
where h0 is the air entry value, θ is the volumetric water content in soil, θ0 is the volumetric fraction of pore space, β is the porosity index. Typical values of h0 and β in eq 10 were obtained from the literature (27, 28) and are listed in Table 1. After substituting parameters into eqs 10 and 11, AV values predicted from Miller’s and Kim’s equations are 41.2 and 43 cm2/cm3, respectively. The fitted value in our soil ozonation system is about 20 cm2/cm3 (see Table 1), which is of the same order of magnitude as those predicted previously. Intuitively, if the results predicted by Miller and Kim were appropriate, it means that approximately 50% of the available gas-liquid interfacial area is in contact with ozone gas in the process. Stanton Number. The kp term has a unit of s-1 and actually represents the rate of interfacial chemical absorption (or VOL. 36, NO. 13, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 5. Plot Stanton number (St) as a function of Pe´ clet number (Pe) for runs with the same r0 of 0.00927 at the column inlst (ζ ) 0). termed “surface reaction” hereafter) of ozone from the gas into the liquid phase. And, vg/L represents the convective rate of ozone mass transfer in the gas. As a result, St stands for the relative rate of surface reaction to convection. When St . 0, the surface reaction rate is much faster than convection. Therefore, the process is convective mass transfer controlled. On the contrary, the process is surface reaction controlled when St ∼ 0. Figure 5 shows St as a function of Pe at the inlet of the column (ζ ) 0) under a constant initial R0 of 9.27 × 10-3. At the beginning of the operation (τ ) 0), St values decrease as Pe increases. This implies that it is more surface-reaction-controlled at higher Pe values. As reaction proceeds or as τ increases in Figure 5, surfacereaction-controlled process dominates since St is approaching zero. Under a convection-controlled condition (St > 0), the increase of ozone flow rate or Pe can increase the removal efficiency. However, this increase is minimum in this case because the process quickly approaches the state of surface reaction control (St f 0) as Pe increases. Therefore, in the transition region between convection and surface reaction control, the effect of Pe is insignificant. Consequently, it can be inferred that increasing Pe only under a large St value can increase the 2-CP removal efficiency. Parameters Controlling Ozone Gas Transport and 2-CP Removal. To evaluate the effect of pertinent parameters on ozone gas transport and 2-CP removal, a special case was considered. At the beginning phase of the operation when the ozone concentration is low, the ozone gas concentration applied is much smaller than 2-CP concentration in the soil liquid film. Under this condition, there is a short period during which 2-CP concentration remains nearly constant along the column. During this period, the system can be at its steady-state so that there is no time-dependence in the ozone gas transport (eq 8). In addition, the effect of gas dispersion can be neglected under a large Pe´clet number (Pe . 1). Therefore, eq 8, which describes ozone gas transport in terms of first-order decay, is reduced to an ordinary differential equation (ODE) as given here
0)-
dπ - Stπ dζ
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Inspection of eq 13 reveals that π f 1 as St f 0. This means that the surface reaction is controlling and that the reactant concentration is nearly uniform throughout the column. At the other extreme, as St f ∞, the process is controlled entirely by convective mass transfer and π f 0. That is, the concentration along the column approaches zero. Concentration profiles along the column at several St values are shown in Figure 6. The transition from surface reaction to convection control as St increases is evident. In the previous analysis, it is stated that Pe can only be adjusted to reduce St in order to increase the 2-CP removal efficiency under an initially large St value. This point is further enforced from the ozone gas transport profile. From Figure 6, it is seen that at large St values ozone reacts extremely fast so that the ozone gas only exists in region close to the inlet of the column. Reducing St by increasing Pe under this condition is actually extending the utilization zone further into the column and, therefore, increases the contaminant removal efficiency. However, when St < 1, ozone gas is adequately supplied as its concentration is already well-distributed along the column. Consequently, convective mass transfer is no longer the limiting factor in the process; therefore, in this case, the ozone utilization zone cannot be extended further, even when St is reduced. In addition to ozone gas transport, 2-CP removal can be analyzed in a similar way. This is done by considering the aforesaid steady-state ozone gas transport while assuming that the 2-CP concentration (φ) changes slightly. To estimate this change in the column, we have to evaluate eq 9 at the inlet of the column (ζ ) 0). By neglecting the distance dependence and assuming that R does not change (i.e., R ) R0) eq 9 reduces to the following ODE:
dφ ) -R0Stπ dτ
φ ) 1 - R0Stπτ ) 1 - κπτ
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 36, NO. 13, 2002
(15)
Here, the new dimensionless parameter κ is the product of R0 and St. Examination of κ reveals that
κ)
(13)
(14)
Equation 14 has the following solution:
(12)
The solution to eq 12 is simply an exponential equation
π ) e-Stζ
FIGURE 6. Plot of the dimensionless ozone gas concentration (π) as a function of dimensionless distance (ζ) along the column under a steady state condition at various Stanton numbers (St ).
kpL νSaC0 vg SwP0
(16)
where κ is actually the ratio of the characteristic time of convection (L/vg) to the characteristic time of 2-CP reaction
ν
degradation yield factor between ozone and 2-CP
θ
volumetric water content in soil matrix
θ0
volumetric fraction of pore space
τa
tortuosity
θm
volumetric fraction of a monolayer of H2O
AV
specific gas-liquid interfacial area (cm2/cm3)
C0
ozone gas-phase concentration at the column inlet (mol/cm3)
CAg
average ozone gas concentration in the column (mol/cm3)
CAi
ozone concentration at the gas-liquid interface (mol/cm3)
CAL
ozone liquid concentration in the film (mol/cm3)
CBb
2-CP liquid-phase concentration at the liquidsolid boundary (mol/cm3)
CBL
2-CP liquid-phase concentration in the film (mol/cm3)
DAL
liquid-phase diffusion coefficient of ozone (cm2/s)
DBL
liquid-phase diffusion coefficient of 2-CP (cm2/s)
Dg
molecular diffusion coefficient of ozone in gas phase (cm2/s)
EAz
gas-phase dispersion coefficient (cm2/s)
FA
enhancement factor
H
dimensionless Henry’s law constant
h0
air entry value (cm‚H2O)
k
second-order reaction rate constant between ozone and 2-CP (cm3/(mol‚s))
KG
gas-phase mass transfer coefficient (cm/s)
Acknowledgments
KL
mass transfer coefficient (cm/s)
This work was supported partially by a DOE Grant No. DE-FG07-96ER14716. The views expressed here are those of the authors and do not imply endorsement of the funding agency. We thank Professor Abraham Lenhoff of the Department of Chemical Engineering at the University of Delaware for many valuable suggestions and comments on the modeling. We also thank our unanimous reviewers who have made excellent suggestions on data interpretation and organization.
kp
lumped coefficient defined as kp ) AVFAKL/nSaH (1/s)
L
length of the column (cm)
n
porosity of soil matrix in the column
NA,int
interfacial flux of ozone from gas phase to liquid phase (mol/(cm2‚s))
P
liquid phase 2-CP concentration in soil liquid (mol/cm3)
P0
initial 2-CP concentration in the liquid phase (mol/cm3)
FIGURE 7. Plot of the dimensionless 2-CP concentration (φ) as a function of dimensionless time (τ) at a certain point in the column where dimensionless ozone gas concentration (π) equals 0.8. (SwP0/νkpSaC0). Thus, κ is a measurement of the intrinsic rate of 2-CP reaction to the convection rate of ozone gas. It is similar to the Stanton number mentioned previously; however, the reaction rate is related to 2-CP but not to ozone. As can be seen, κ involves an additional term R0, which accounts for the overall mass of 2-CP removed as a result of ozone mass transferred across the interface. Figure 7 shows the plot of φ under various StR0 (or κ) values at the column inlet when π ) 0.8. From Figure 7, as κ f 0, φ is near unity because the surface reaction rate is small relative to convection. Also, φ decreases as κ increases, meaning that the 2-CP reaction rate increases relative to rate of convection. It should be noted that κ could be increased not only by raising the St value, as mentioned before, but also by raising the R0 value. This, of course, is achieved by increasing the ozone gas concentration.
Glossary R0
dimensionless degradation yield factor at time zero
Pe
Pe´lect number, VgL/EAz
R
dimensionless degradation yield
Sa
gas degree of saturation
γ
Hatta number, γ )
xkCBbDAL/KL
Sw
liquid degree of saturation
ζ
dimensionless distance along the column
St
Stanton number
π
dimensionless ozone gas-phase concentration
t
time (s)
φ
dimensionless 2-CP concentration
vg
average gas velocity in the column (cm/s)
κ
dimensionless quantity defined as R0St
yG
the film thickness in the gas phase (cm)
λ
total liquid thickness on soil surface (cm)
z
distance along the column (cm)
τ
dimensionless time
β
porosity index
γ
Hatta number
Literature Cited (1) Fischer, U.; Schulin, R.; Keller, M.; Stauffer, F. Water Resour. Res. 1996, 32, 3413-3427. VOL. 36, NO. 13, 2002 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Received for review January 22, 2001. Revised manuscript received March 26, 2002. Accepted March 27, 2002. ES010559E
