Environ. Sci. Technol. 1996, 30, 770-778
Effective Gas-Phase Diffusion Coefficients in Soils at Varying Water Content Measured Using a One-Flow Sorbent-Based Technique STUART BATTERMAN,* IYER PADMANABHAM, AND PAUL MILNE Department of Environmental and Industrial Health, The University of Michigan, 109 Observatory Drive, Ann Arbor, Michigan 48109-2029
This paper focuses on techniques used to measure and predict effective gas-phase diffusion coefficients for volatile organic compounds in soils. Large differences found among the existing correlations for the effective diffusion coefficient, laboratory tests, and field experiments indicate the need for accurate measurements of diffusion parameters in soils. The theory, techniques, and experimental issues involved in laboratory measurements are summarized. A new one-flow sorbent-based laboratory experimental system is developed. The system maintains a constant concentration gradient across a soil column using a test gas flow at one side of the column and a highcapacity sorbent at the other. The diffusive flux and the effective diffusion coefficient are estimated using the difference between inlet and outlet concentrations. Mixing factors account for concentration gradients at column ends. A sequence of tests is used to quantify diffusion coefficients for trichloroethylene in laboratory-prepared soils at soil water contents from 0 to 80% of saturation (0-16% by weight). Results obtained are generally equivalent to measurements from a conventional two-flow experimental system. A curvilinear relationship is found between the airfilled porosity and the effective diffusion coefficient. Measured gas-phase diffusion coefficients at intermediate and high soil water contents are significantly larger than values estimated using literature correlations, although the correlations cover a large range. The new technique provides precision comparable to that of existing experimental techniques but offers greater convenience, flexibility, and control.
* Author to whom all correspondence should be addressed; telephone: (313)763-2417; fax: (313)763-9424; e-mail address:
[email protected].
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Introduction Gas-phase diffusion is an important transport mechanism for many contaminants in unsaturated soils, especially volatile organic compounds (VOCs) with low aqueous solubility, e.g., fuels, solvents, and degreasers. Gas-phase diffusion affects the mobility, lifetime, and fate of these contaminants in soils and often the selection of a remediation approach. Theoretical (1, 2) and experimental studies (3-10) suggest that effective gas-phase diffusion coefficients in porous media depend on macroscopic soil features, e.g., total porosity n, air-filled porosity θa, waterfilled porosity θw, pore geometry or tortousity τ, grain size, horizontal and vertical layering, desiccation cracking, and the interconnectively of air pathways; microscopic features, e.g., surface area, micropores, exposed or accessible organic and mineral material available for sorption; VOC characteristics, e.g., molecular size, Henry’s law constant, solubility, partition coefficient, and vapor density; and temperature and pressure. Because of the difficulty of representing or measuring the features affecting diffusion rates, simple correlations for the effective diffusion coefficient D based on laboratory experiments find widespread use. The correlations in the literature (discussed later) are based on porosity. Air and water-filled porosities, which range from 0 to ≈60%, may change in time and space due to infiltration, recharge, etc. Thus, a strong need exists to estimate diffusion properties as a function of water content (10). In addition, large differences exist between the empirical correlations, laboratory simulators (described below), and field study results (11-12). Diffusion rates are often overpredicted in partially saturated soils but underpredicted in other soils due to unaccounted features, e.g., heterogeneities and cracks. As a macroscopic property, D is determined as a spatial average over some volume. This averaging may lead to an apparent scale dependence that explains some differences between laboratory and field results, as has been observed in saturated media (13-16). A recent study of the spatial variability of D found that experimental variability and variability within tested soil units were large, and no meaningful differences were found between units of varying lithographies, contrary to expectations (10). This study focuses on laboratory measurements of diffusion coefficients in unsaturated soils. The theory, correlations, and experimental approaches are reviewed, after which a new measurement technique is described. Results are presented using the technique and a test soil over a wide range of water contents. Estimated diffusion coefficients are compared to results obtained using a conventional technique and the literature correlations. A critique of the experimental techniques concludes the paper.
Theory The following assumes that the air phase is at atmospheric pressure; the porous media is homogeneous, isotropic, and isothermal; and the diffusing VOC is inert (does not sorb or dissolve). Then, the effective gas-phase diffusion coefficient D (cm2/s) is
D ) θgτDg
0013-936X/96/0930-0770$12.00/0
(1)
1996 American Chemical Society
where θg is the effective gas-phase area for diffusion, τ is the air-phase tortuosity accounting for the “tortuous” diffusion path in porous media, and Dg is the “free-air” diffusion coefficient of the specific vapor (cm2/s). Conceptually, τ is the ratio of the straight path length to the tortuous path length. θg and τ are media properties, and Dg is assumed to be independent of concentration. Note that this formulation has not been presented consistently in the literature. Sometimes τ has been defined to include θg, and τ-1 has also been used for τ. Equation 1 is advantageous, however, as effects of media and water content are separated. Free-air diffusion coefficients Dg are available for many vapors in air at 25 °C (17) and may be adjusted to temperature T (K) by (18):
DT ) Dg(T/298)1.5
(2)
Diurnal temperature changes in the upper (≈0.3 m) soil horizon and seasonal temperature changes through the upper (≈10 m) soil layers may affect D, although shortterm effects are usually small (19). If neat or residual VOC is present, a change in temperature will affect the vapor pressure and the concentration gradient, potentially causing large differences in diffusive fluxes.
Measurement Approaches Steady-State Methods. Several steady- and unsteady-state methods have been used to measure D. The methods vary in the manner that boundary conditions are maintained and diffusive fluxes are derived. The following assumes that pressure, diffusivity, water content, and temperature are constant and uniform throughout the column. The steady-state approach gives the contaminant flux F (µg/s) through a plane in the soil using Fick’s first law:
F ) -AD dC/dz
(3)
where A is the cross-sectional area of the column (cm2), D is the effective diffusion coefficient (cm2/s), and dC/dz is the concentration gradient (µg/cm2). With a uniform gradient, dC/dz ) ∆C/L, where ∆C is the concentration difference between opposite faces of the soil column, and L is the column length. If sorption and dissolution processes are negligible, the approximate time to steady-state conditions tss is L2/(6D) (19). Most steady-state studies (8, 9, 20-23) have used a liquid VOC reservoir, and vapor equilibrium is assumed to maintain the high concentration boundary. Early studies left the top end of the column open to the atmosphere and estimated F by weight loss. Most later studies used an air flow (Figure 1A) and estimated F as C4Q4 (assuming C3 ) 0). The reservoir technique is suited to compounds that are liquids at laboratory temperatures. Concentrations can only be controlled by changing the temperature. This technique may suffer from liquid-to-vapor mass transfer limitations, unknown or unaccounted for concentration gradients above the reservoir, vapor condensation on the media, and sensitivity to temperature changes at the reservoir. Other studies (24-26) have maintained nearconstant concentrations using gas flows at both ends of the soil column (Figure 1B). This technique is prone to pressure imbalances that produce advective flows. Diffusion coefficients have also been derived using the weight gain of a sample (not soil) backed by a sorbent suspended
FIGURE 1. Schematics of four experimental techniques. Steadystate techniques: (A) liquid reservoir, (B) two-flow, and (C) proposed one-flow sorbent-based. Unsteady-state techniques: (D) concentration profiling using reservoir. C1-C4 (µg/cm3) and Q1-Q4 (cm3/s) denote concentrations and flow rates, respectively; subscripts 1 and 2 denote high concentration side; subscripts 3 and 4 denote low concentration side.
in a cell containing the test gas, (27) similar to that depicted in Figure 1C. Unsteady-State Methods. Unsteady-state methods use the one-dimension diffusion equation:
θg ∂C/∂t ) ∂/∂z (D ∂C/∂z)
(4)
and estimate D by fitting either the concentration profile along the soil column, using measurements at several locations (Figure 1D) (10, 28, 29), or “breakthrough curves”, using sequential measurements of the flux diffusing through the column, to a solution of eq 4. Unsteady-state results may be difficult to interpret given sorption on soil particle surfaces, intraparticle (micropore) diffusion, dissolution in soil water, and other processes that affect transient behavior. Some processes depend upon the specific soil-contaminant combination, e.g., sorption isotherms are influenced by temperature, water content, soil organic matter content, etc. Because these effects cannot be accurately predicted, most estimates of D rely on steady-state techniques that avoid these complications. (Unsteady-state tests can also estimate sorption parameters, and tests using inert tracer gases, e.g., SF6, can measure tortuosity.) One-Flow Sorbent-Based Technique. The proposed measurement technique (Figure 1C) maintains the high concentration side of the soil column near desired concentration C1 by flow Q1. The low concentration side is kept at near zero concentration without flows using a sorbent. At steady-state, the diffusive flux through the soil is the difference between incoming and outgoing fluxes:
F ) (C1 - C2)Q2
(5)
assuming a leak-free system, thus Q1 ) Q2, and a perfect sorbent, thus C3 ) 0. Figure 2 displays steady-state concentration differences predicted for this system at several flow rates (based on a model described later). The
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FIGURE 2. Predicted concentration differences (as ratio C2/C1) for the one-flow, sorbent-based system. Column length L ) 5 cm and mixing factor r ) 0.5, except for * where r ) 1.
ratio of outlet to inlet concentrations, C2/C1, increases with higher D and lower flow rates. Either steady- or unsteadystate methods can be used to estimate D from this ratio. Note that D might also be measured by determining the VOC mass collected on the sorbent. This is not recommended, however, as both unsteady- and steady-state diffusive fluxes load the sorbent, and these portions cannot be separated. (In long-term tests, however, the unsteadystate portion of the flux might be neglected.) The proposed technique appears applicable to a broad range of diffusion coefficients and has two major advantages: the single-flow path eliminates pressure gradients, and vapor concentrations are easily controlled and monitored. To our knowledge, this technique has not been previously used, although it resembles an approach used to estimate D in polymer sheets by the disappearance of a solute in a bath in which the sheet is immersed (19). Such techniques, however, are limited to unsteady-state approaches. It also resembles a sorbent-based system that uses weight gain to derive fluxes (27), a technique with limited applicability to soil since water and VOC mass changes cannot be separated, and small mass changes at low concentrations cannot be detected.
Experimental Methods Measurements of D were obtained using one- and and twoflow experimental systems that shared many features and procedures. The convention two-flow system is described first. Two-Flow System. Soil columns were constructed of borosilicate glass with Teflon end caps. Internal dimensions were 5 cm diameter and 13 cm length. Each end cap had three 0.635-cm diameter ports, two for flushing the chamber space, and a third to measure differential pressure across the column using a Magnehelic gauge with a readability of 0.05 mm H2O (Dwyer Instruments Corp., Michigan City, IN). A rigid stainless steel screen in the bottom end cap supported the soil. Columns were clamped in a vertical position and packed with media to a height of ≈8 cm. VOC vapor in humidified air (C1, Q1) was introduced at the top of the column and controlled by precision needle valves.
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The top outlet port was connected to a second needle valve and vacuum, drawing the vapor (C2, Q2) from the soil column. A third needle valve at the bottom inlet port controlled air drawn from the laboratory (C3, Q3). Bottom outlet flows (C4, Q4) were controlled by a fourth needle valve connected to the vacuum. Exiting flows Q2 and Q4 passed through a carbon trap, a vacuum regulator, and a rotary vane vacuum pump. This balanced design minimized flow and pressure deviations that might occur over the test duration. Each flow line was equipped with a septum port, allowing gas withdrawals for concentration measurements. Teflon and stainless steel tubing and fittings were used throughout. All experiments used trichloroethylene (TCE), a common and toxic VOC. A test gas containing water and TCE vapor at the desired concentrations was prepared by blending three of four flows. A four-way valve allows the solvent vapor to be switched on or off without disturbing the total flow rate. Separate bubblers were used to achieve vapor concentrations near water and TCE saturation. Typical flow rates were 0.3, 0.5, and 0.2 L/min for dry dilution air, water-saturated air, and TCE-saturated air, respectively. Each flow was controlled by a calibrated rotameter and pressure regulator. The test gas generation system is housed in a temperature-controlled enclosure set at 30 °C. At the laboratory temperature (22 °C), this yielded a TCE concentration of 140 µg/cm3 and a relative humidity of 80%. (Actual concentrations were measured in each experiment.) Only a small portion of the test gas flow was utilized, and the remainder was vented through large diameter tubing to ensure that Q1 was at atmospheric pressure. All experiments used the fine sand (300-600 µm diameter) portion of a clean soil. A graded soil was preferred for consistency, uniformity in packing, and higher experimental precisions. This soil has a particle density of 2.6 g/cm3 and an organic content of 0.014% (23). The soil was brought to the desired water content by thoroughly mixing appropriate amounts of sand and water in a beaker. Water contents of 0, 20, 40, 60, and 80% of saturation were used, corresponding to 0, 4.6, 8.7, 12.5, and 16.0% water by weight. Moist sand was carefully packed inside the column and tapped down every 1 cm to obtain a uniform porosity. Steps to minimize any evaporation that might alter the water content included covering beakers and columns, using cold water, and completing the process as rapidly as possible. The length and weight of the packed soil in the column were measured, in part to confirm the soil water content. The total porosity in the packed columns was 0.383 ( 0.050 (N ) 5). After packing and assembling the column, the experimental system was capped, checked for leaks, and allowed to equilibrate to laboratory temperature. Humidified air flows of ≈60 mL/min were measured using manual or electronic bubble meters. After flows were balanced and zero pressure differential was obtained, TCE vapor was introduced. Periodically, 50 or 100 µL gas samples were withdrawn using a gas-tight syringe for quantitation of TCE. Two replicate samples from each port were taken every few hours until steady-state conditions were reached. Final samples were taken overnight (≈20 h after the start time), and flow rates were rechecked. In several experiments, the column orientation was reversed so that TCE vapor entered at the column bottom. In the analysis of the two-flow system, flows are assumed to be balanced so that Q1 ) Q2 and Q3 ) Q4. Substituting
effective concentrations at the sweep and purge faces of the column into eq 3 leads to
F ) (Csweep - Cpurge)AD/L
(6)
Effective concentrations are estimated using mixing factors R and β:
Csweep ) (1 - R)C1 + RC2
(7)
Cpurge ) (1 - β)C3 + βC4
(8)
A mixing factor of 0.5 averages inlet and outlet concentrations and may be used if strong concentration gradients exist within the end cap. A mixing factor of 1 uses the exiting concentration and represents complete mixing. Flow and concentration distributions in the column end caps were not measured. The estimated Reynolds number in the end caps for most experiments is ≈500. Turbulence and mixing in the end caps would be promoted by the rough face of the soil, the shape of the chamber, and the bends in the flow path. This suggests transitional or somewhat turbulent flow, supporting the fully mixed assumption. (Results shown later are also consistent with this assumption.) Substituting eqs 7 and 8 into eq 6 yields
F ) [(1 - R)C1 + RC2 - (1 - β)C3 - βC4]AD/L (9) The steady-state flux balance is
C1Q1 + C3Q3 ) C2Q2 + C4Q4
(10)
Purge flows use clean air so C3 ) 0, thus the flux through the soil is F ) C4Q4, and exiting sweep and purge flow concentrations are C2 ) C1 - F/Q2 and C4 ) F/Q4. From this, the steady-state flux though the column is
F ) C4Q4 ) C1/(L/AD + R/Q2 + β/Q4)
(11)
Note that as flows Q2 and Q4 increase, eq 11 is reduced to eq 3, and mixing corrections become unnecessary. Long columns giving small fluxes also will diminish the effects of mixing as C1 ≈ C2 and C3 ≈ C4 ≈ 0. The estimated effective diffusion coefficient D* is derived from eq 11 with measurements of concentrations C1 and C4 and flows Q2 and Q4:
D* ) L/{A[C1/(C4Q4) - R/Q2 - β/Q4]}
(12)
or from eq 9 if an additional measurement of C2 is available:
D* ) LC4Q4/[A{(1 - R)C1 + RC2 - βC4}]
(13)
Results following use eq 13 and the fully mixed assumption. It should be noted that D* in some of the literature is not based on effective concentrations given by eqs 12 and 13, although biases appear to be minor. One-Flow System. The experimental system used the same column and end caps; however, an insert beneath the screen supporting the soil provided a 0.5-cm air gap above a 1 cm thick sorbent bed, and the soil length was shorter (4-5.5 cm). The air gap minimized migration of water from the soil to the sorbent. (This occurred in preliminary experiments at high soil water contents where it led to C2/C1 ratios that continued to increase as water drained through the soil and screen and onto the sorbent where it decreased its effectiveness. This was eliminated by the air gap.) In most cases, 4 g of 40-60 mesh of activated
FIGURE 3. Details of the one-flow sorbent-based experimental system. Two of three columns used are depicted.
charcoal was used as a sorbent. Each experiment used fresh sorbent. The same media and packing procedure described earlier were used. TCE vapor in humidified air was introduced into three columns allowing three simultaneous replicates (Figure 3). Flow rates from 3 to 26 cm3/min were tested; however, flows were ≈10 cm3/min in most cases. Gas samples of 100 or 200 µL were withdrawn from incoming and exiting lines to determine C1 and C2. Three vapor samples were taken and analyzed at each port until steady-state conditions were reached (typically 5-32 h after startup). Flows were rechecked at the end of the experiment. The analysis of the one-flow system uses eq 9 but assumes C3 ) C4 ) 0 with an efficient sorbent. The steadystate flux F through the column becomes
F ) (C1 - C2)Q2 ) AD[(1 - R)C1 + RC2]/L
(14)
from which D is estimated as
D* ) (C1 - C2)Q2L/[A[(1 - R)C1 + RC2)]
(15)
This approach should be valid if the diffusive flux through the soil is between about 5% and 30% of incoming flux C1Q1, i.e., 0.7 e C2/C1 e 0.95. With smaller C2/C1 ratios, much of the incoming flux enters the soil, C2 approaches zero, and strong nonlinearities in concentrations and fluxes at the column face complicate the analysis. At the extreme, C2 will remain near zero for all soils with large D, and D* will be unmeasurable. On the other hand, if C2/C1 approaches one, the concentration difference C1 - C2 may not be measured accurately. Fortunately, the flow rate and system geometry can be selected to optimize system performance. Figure 2 shows the sensitivity of the system to incompletely (β ) 0.5) and fully (β ) 1) mixed conditions with a typical flow rate of Q2 ) 10 cm3/min. Estimated D’s are similar until C2/C1 falls below 0.7 (D > 0.015 cm2/s). The figure also shows that a higher flow rate (20 cm3/min) will maintain C2/C1 above 0.7 for D up to 0.03 cm2/s, about the largest value anticipated in soils for VOCs. (Light gases such as CH4 have higher diffusion coefficients.) For C2/C1 ) 0.7, the two mixing assumptions yield D*’s that differ by 15%, thus this value is suggested as a lower bound. Oneflow system estimates are calculated using eq 15 and the fully mixed assumption. TCE Analysis. Gas samples were analyzed by direct injection onto a 1/8 in. × 6 ft. packed column (Supelcowax 5% GP SP 1200, Supelco Inc., Bellefonte PA) of a Varian
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TABLE 1
Conditions and Results of Diffusion Experiments Using the Two-Flow System flow rate (mL/min)
concn C1 (µg/mL)
concn C2 (µg/mL)
concn C4 (µg/mL)
diffusion coeff (cm2/s)
water content (%)
run no.
Q2
Q4
av
SD
av
SD
av
SD
av
0
1 2 3 4 av 5 6 av 7 8 av 9 10 av 11 12 av
64 60 63 54
48 57 56 63
117 143 127 142
16.3 3.5 3.2 3.6
127 141 124 126
3.5 0.7 3.1 3.2
8.1 8.1 6.2 7.3
0.2 0.2 0.2 0.2
62 61
45 48
128 124
0.7 2.1
126 129
7.1 4.2
3.3 5.4
0.1 0.1
60 61
52 53
137 130
2.8 4.9
142 135
3.5 1.4
1.9 2.8
0.1 0.1
61 60
60 63
144 142
4.9 2.8
136 146
0.7 2.1
1.5 1.3
0.1 0.1
62 61
61 62
129 143
0.7 7.1
131 145
0.7 0.7
0.9 0.9
0.1 0.1
0.0334 0.0291 0.0245 0.0295 0.0291 0.0115 0.0197 0.0156 0.0056 0.0089 0.0072 0.0045 0.0036 0.0041 0.0028 0.0026 0.0027
20 40 60 80
3700 (Mountain View, CA) gas chromatograph (GC) equipped with a flame ionization detector. The GC’s electrometer output was stored using a PC data acquisition system. Peak areas, corrected for baseline, were computed and converted to concentrations and fluxes using measured flow rates. Multipoint calibrations were performed daily using TCE standards prepared in air. Typical linearities exceeded 0.99 (as R2), and method detection limits for TCE were
