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(30) Perrin, D.D.;Watt, A. E. Biochim. Biophys. Acta 1971, 230,96-104. (31) Movchan, V. V.;Tulyupa, F. M.; Baibarova, E. Ya. Russ. J . Inorg. Chem. 1979,24,889-891. (32) Butterworth, P.; Hillier, I. H.; Vaughan, D. J.; Tossell, J. A. J . Phys. Chem., in press. (33) Parkhurst, D. L.; Thorstenson, D. C.; Plummer, L. N. PHREEQE-A Computer Program for Geochemical Calculations; Water-Resour. Invest. (U.S. Geol. Surv. 1980, No. 80-96. (34) Nordstrom, D.K.; Plummer, N. L.; Langmuir, D.; Busenberg, E.; May, H. M.; Jones, F. B.; Parkhurst, D. L. In Chemical Modeling of Aqueous S y s t e m I& Melchior, D.
C.,Bassett, R. L., Eds.;ACS Symposium Series 416;American Chemical Society: Washington, DC, 1989;Chapter 31, pp 398413. (35) Haraldsson, C.; Westerlung, S. Mar. Chem. 1988, 23, 417-424. (36) Kremling, K. Mar. Chem. 1983,13,87-108. (37) Framson, P. E.;Leckie, J. 0. Environ. Sci. Technol. 1978, 12,465-469.
Received for review March 4, 1992. Accepted August 31,1992. This work was supported by the National Science Foundation, Grant EAR-8804200.
Convective Transport of Gases in Moist Porous Media: Effect of Absorption, Adsorption, and Diffusion in Soil Aggregates Benjamin J. McCoy"rtand Dennls E. Roistonr:
Department of Chemical Engineering and Department of Land, Air, and Water Resources, University of California, Davis, California 95616
rn The mass transport of volatile chemicals in porous
media is of increasing interest due to efforts to remediate contaminated soil and groundwater. The convective movement of gaseous chemical species in porous media is influenced by absorption (partitioning) of the species in water that wets the surfaces of the soil particles. Four different mathematical models based on chromatographic systems are herein proposed and compared to describe the convection, dispersion, and mass transfer of the species in the voids of the porous medium and into the water. Adsorption-desorption at the liquid-solid interface is included. Mean retention time and concentration band variance (first and second moments) are compared for four models. The major influence on transport is retardation and band broadening due to the total capacity of the porous medium for the volatile chemical (in voids, in water films, and adsorbed at the wateraolid interfaces). The relative influence of the different transport processes is assessed.
Introduction In a partially saturated soil (the vadose zone) the migration of gaseous chemical species can be retarded by the moist porous medium when the species are partitioned between the air in the voids and the water. This effect is of interest, for example, in soil venting or vacuum extraction of volatile contaminants (1,2). The water may fill pores external to or within soil aggregates or may exist as films covering particles or pore surfaces. Gases can dissolve into the water, increasing the capacity of the soil volume and hence the retardation of the gases. Moreover, adsorption of the dissolved gas species at the wateraolid .interfaces can substantially increase capacity. Diffusion of the dissolved gas species in the water is lo4times slower than in air, further affecting migration of the gas component by increased dispersion. Although the hydrodynamic-dispersion equation for a single-phase fluid in a porous medium has been exhaustively studied, the three-phase system representing a partially saturated porous medium has only recently been addressed. Rasmuson et al. (3) introduced a mathematical model for convective transport of gases that hydrolyze in water 'Department of Chemical Engineering. f Department of Land, Air, and Water Resources. 2488
Environ. Scl. Technol., Vol. 26, No. 12, 1992
(e.g., C02and SO,). Flow of gas was assumed to take place in gas-filled macropores, with absorption of gas into adjacent water-filled microporous aggregates. The absorption process included interphase mass transfer through a gas film resistance and rectilinear diffusion into the water-filled micropores. The linear partial differential equations were solved with a numerical integration technique. Breakthrough curves for a range of solubility and mass-transfer parameter values were presented. As an example of the type of result that can be obtained from this model, Rasmuson et al. (3) showed that SO2penetration into the partidy saturated soil is limited to a few centimeters after 1.5days. By comparison, relatively insoluble 0,penetrated much more deeply. Brusseau (4) recently introduced a model for transport of organic chemicals by gas convection in a heterogeneous porous medium. The porous medium incorporated immobile water in macro- and micropores, which were referred to as advective and nonadvective domains, respectively. One-dimensional velocity and longitudinal dispersion in the gas were assumed. The gas-phase concentration was considered to be governed by a rate equation with time and distance as the variables. Transport effects in an immobile liquid phase of constant volume were described by a second differential equation with time as the only variable. The model included a sorption rate expression based on a rate constant and an equilibrium sorption coefficient. Combined transport resistances in the gas and liquid phases were described by a mass-transfer coefficient, which lumped together the diffusion and film mass-transfer processes. The model, which was solved numerically, agreed with data in the literature when nonequilibrium effects were incorporated. Gierke et al. (5) modeled the movement of a volatile organic chemical in columns of unsaturated soil by including advection in both air and water. The complete model included dispersion in air and water, *water mass transfer, diffusion in immobile water, mass transfer between mobile and immobile water, and sorption at the water-soil interface. The coupled partial differential equations were solved by an orthogonal collocation technique and applied to experimental data for the movement of trichloroethene in sand and in clay aggregates. Smith (6, 7)considered aeration of water-logged aggregates in the absence of adsorption. Slow diffusion within the water-filled pores of the aggregate was contrasted to
0013-936X/92/0926-2468$03.00/0
0 1992 American Chemical Society
mass transfer in the interparticle voids. The behavior of flowing gases in partially saturated soil is analogous to three-phase chromatography, for example, gas-liquid partition chromatography, where the solid matrix supports a layer of liquid into which the gaseous species dissolve and diffuse (8). As they are convected through the voids of the porous medium, the gaseous species experience hydrodynamic dispersion and mass transfer to the wetted surfaces. The degree of partitioning of different chemical species in the liquid film affects the chromatographic retention time, as well as the band broadening (dispersion) due to diffusional mass-transfer processes. We hypothesize that the mathematical models for such chromatographic processes can be productively applied to understand convective transport in partially saturated soils. In addition to the partitioning of the gas species between the gas and liquid, we account for adsorption of the dissolved species at the liquid-solid interface. Leij and Dane (9) have also applied temporal moments to study solute transport in a single fluid in soil. The benefit to be gained from the simulation of soils by these chromatography models depends on the nature of the specific issues that are addressed. As with any mathematical model of physical processes, certain detailed quantitative features will be approximately depicted, due to the nature of the assumptions that restrict the models. Several general features, however, can be suitably represented by the present model. For example, a motivating question for this work was, when can local equilibrium be assumed for migration of contaminants in moist soil? A nonequilibrium effect that one would like to ignore is the diffusion of the contaminant in the water layer on the surfaces of the soil particles. The magnitude of such diffusion relative to other transport processes that occur in porous media, e.g., the mass transfer through the boundary layer surrounding each particle, can be quantified by the chromatographic models. Chromatographic processes are conveniently described by temporal moments of responses to pulse inputs to the column. The first normalized moment, p1, for a concentration pulse migrating in the porous medium represents the mean retention time, which is independent of diffusive resistances for reversible kinetics. The first moment provides the penetration time for the component to reach a given distance into the porous medium. The normalized second central moment, p 2 (the variance), represents the spreading of the pulse due to dispersive mass-transfer and rate processes. Thus, the second moment contains the information needed for estimating the influence of nonequilibrium effects. For an arbitrary input in a linear system, the response is simply an integration (superposition) of the input function with the impulse response (IO). A breakthrough curve is the time integral over the impulse response multiplied by a step-function input. A distinct advantage of the moment approach is that the algebraic moment expressions can be evaluated easily for chosen values of the parameters. A potential disadvantage is that chromatographic moment theories are typically restricted to linear systems.
Mathematical Models Four models of the wetted solid matrix are proposed and compared (Figure 1): (A) capillary tubes, (B) nonporous spheres, (C)porous spheres with uniform-thickness film covering the pore surfaces, and (D)porous spheres with liquid-filled pores. In the first three models, all surfaces accessible to the gas phase are assumed to be covered with a liquid film. These models are simplified by the assumption of uniform
I
A. Capillary Tubes
I B.
C. Porous Particles
I
I
I
I
Nonporous Particles
D. Porous Particles With
Water-Filled Pores
I
Flgwe 1. Schematic drawings of lour models of partially saturated sol1 aggregates. The radli of cyllndrlcal tubes and spherlcal partlcles are R ; the thickness of the liquid film In A-C Is 6.
films of thickness 6. It is further assumed that the film is very thin, so that the diffusional process can be considered one-dimensional in rectangular coordinates. The parameters that govern the partitioning phenomena are the wetted surface area/volume ratio, A,, the liquid film thickness, 6, the “dimensionless” Henry’s law (partition) coefficient, KH,and the diffusion coefficient for the species dissolved in the liquid, DL. In the fourth model, all the pores in the spherical aggregate are filled with water, and reversible adsorption occurs at the liquid-solid interface. Thus, while the diffusion processes of models A-C are restricted to thin films of water, for model D there is no such limitation on water content. As all solid surfaces are covered with water, we do not consider the case when adsorption occurs on dry regions of the surfaces. This case can be treated by appropriate factors multiplying the terms for the partitioning effects; see, for example, Alkharasani and McCoy (II), who postulated that dry regions can adsorb the migrating chemical species. Ong and Lion (12) recently presented experimental evidence that the moisture content has a prominent effect on partitioning. For very low amounts of liquid, the partitioning first decreased with increased moisture due to leas than monolayer coverage by water. The partitioning attained a minimum and then increased with moisture content due to Henry’s law solubility. In the present model, we assume that the film is always thick enough (greater than five molecular layers according to Ong and Lion) that Henry’s law solubility is valid. The governing differential equations for the four models, shown schematically in Figure 1,are provided in Table I. Their respective moment expressions are given in Table 11. The governing equations for the interparticle void concentration, c(z,t), in all four models include similar terms for transient accumulation, convection, hydrodynamic dispersion, and mass transfer. The differential equations provide expressions for the moments by means of a straightforward, but algebraically complex, procedure. Laplace transformation is applied to the partial differential equations, and the resulting ordinary differential equations are solved for
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Table I. Governing Differential Equations with Boundary and Initial Conditions for (A) Capillary Tubes, (B) Nonporous Particles, (C) Porous Particles, and (D) Porous Particles with Water-Filled PoresD
A. Capillary Tube with Water Film Covering Inner Surface
in the pores
in the voids ac
- + vE at
a2c
az
= Do - - A,k,(c - c,) a22
in the liquid film
in the liquid film
BC
k,(c - c,) = -DL
8CL aY
@Y = 6
where c, = KHcL (y = 6) B. Nonporous Particles with Water Film Covering Surface in the voids
and
in the liauid film 1'CL -=DLat
D. Porous Particles with Water-Filled Pores in the voids
a2CL
ad
BC
@Y = 0
in the pores
C. Porous Particles with Water Film Covering Pore Surfaces in the voids t
"The four models all have the boundary conditions (BC) c(z=O,
t ) = c o ( t ) and c(z=-,t)
The temporal moments are obtained as limits of derivatives of the Laplace-transformed concentration, c
m,(z) = Jmc(z,t)tn dt = (-l), limp+ d"Z/dp" 0
(2)
These computationalmanipulations, extremely tedious and time-consumingif done by hand, were performed with the computer algebra system, Mathematica (13). The normalized first moment, the mean retention time for a concentration pulse entering the system at time t = 0, is Pl
= m/mo
(3)
The normalized second central moment, or variance, is p2
= L a c @- pl)' dt = m2/m0- p12
(4)
and is a measure of dispersive broadening of a concentration pulse entering at t = 0. The similar forms of the differential equations lead to moment expressions that have a similar structure (Table 11). Each model allows for equilibrium adsorption of the dissolved species at the water-solid interface. The adsorption is assumed linear, with adsorption equilibrium coefficient Kd. The component is partitioned between the gas and the liquid according to the dimensionless Henry's coefficient KH,the ratio of the equilibrium concentration Environ. Sci. Technol., Vol. 26, No. 12, 1992
dCL DLj
BC
ac ac a2C 3 + YO= Do- - (1 - e ) ~ k , ( c- c,(R)) at az az2
0, ci(z,t=O) = 0, etc.
2470
BC
= k,(c - KHcL) dr aCL
-ar- - 0
@r = R
@r=O
= 0 and the zero initial conditions (IC) c(z,t=O) =
in the gas to the concentration in the liquid. The velocity of gas in the voids is u = uo/e, where the superficial velocity uo is the volumetric flow rate divided by the sample cross-sectional area and e is the air-filled void fraction (excludingintraparticle porosity). The diffusion coefficient for the dissolved component in the stagnant liquid film is DL.Differences in the moment expressions are due to geometrical differences in the models (Figure 1). A. Capillary Tubes. The gaseous channels (voids) through the soil are considered to be tubes of radius R with the inner surface covered with a liquid film of thickness 6,where we will take 6 > 1the component absorption in the liquid is negligible, and the models reach different asymptotes because model C has the additional capacity of the empty pores relative to model D with liquid-occluded pores. The influence of the magnitude of adsorption at the water-solid surface is to increase the moments by increasing the soil capacity for the component. For large values of Kd (Kd >> 1)and Small 2474
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Flgure 0. Effect of adsorption constant, Kd, on ratio of particle mass-transferresistance to hydrodynamic dispersion (porous-particle model C, wetted-pore surfaces, and model D, liquid-filled pores) for parameters of Table I I I and v = 0.1 cm/s.
values of K H ( K H
