Development and Application of a Dual-Impedance Radial Diffusion

Environ. Sci. Technol. 1997, 31, 2983-2990

Development and Application of a Dual-Impedance Radial Diffusion Model to Simulate the Partitioning of Semivolatile Organic Compounds in Combustion Aerosols MICHAEL R. STROMMEN AND RICHARD M. KAMENS* Department of Environmental Sciences and Engineering, The University of North Carolina at Chapel Hill, CB 7400, Rosenau Hall, Chapel Hill, North Carolina 27599-7400

The fate of semivolatile organic compounds in the atmosphere is largely dependent on their partitioning between the gas phase and sorption to particulate matter. Since real atmospheres have been shown to deviate significantly from gas-particle equilibrium under certain conditions, dynamic mass transfer models are needed to accurately predict partitioning. In this work, a dual-impedance radial diffusion model is presented that is able to simulate the partitioning of deuterated fluoranthene in wood and diesel soot atmospheres generated in a large outdoor Teflon film chamber. It is shown that the dual-impedance model produces significantly better fits to experimental results than a one-layer model and that surface mass transfer is not rate limiting in these systems. The sensitivity of optimized apparent diffusion coefficients to key input parameters is also explored. This work lays the foundation for incorporating dynamic gas-particle partitioning models into larger atmospheric models, such as urban airshed models. By conducting experiments under various conditions (e.g., temperature and humidity), values for apparent diffusivities as a function of compound, particle source, and atmospheric conditions may be developed. After including photochemical reactions, the model may be used to predict the fate of semivolatile organic compounds in real atmospheres.

Introduction The phase in which an organic compound exists in the atmosphere is greatly influenced by its vapor pressure. Nonvolatile compounds, such as large alkanes (C J 28), exist almost exclusively in and on particulate matter (i.e., in the “particle phase”), whereas highly volatile compounds, such as small alkanes (C j 18), remain mostly in the gas phase. However, semivolatile organic compounds (SOCs), which have ambient vapor pressures of approximately 10-3-10-7 Torr, demonstrate significant partitioning between the gas and particle phases. This group of compounds includes demonstrated and potentially toxic and carcinogenic compounds, such as polynuclear aromatic hydrocarbons (PAHs), polychlorinated biphenyls (PCBs), and chlorinated dibenzodioxins and dibenzofurans. In fact, approximately 50 of the 189 hazardous air pollutants listed in the 1990 Clean Air Amendments have the potential to partition significantly * Corresponding author phone: (919) 966-5452; fax: (919) 9667911; e-mail: [email protected].

S0013-936X(97)00079-5 CCC: $14.00

 1997 American Chemical Society

between the gas and particle phases under ambient conditions (1). Over the past 15 years, significant work has been conducted to develop theoretical atmospheric gas-particle equilibrium partitioning relationships (2-6). These efforts have substantially increased the understanding of gasparticle partitioning and have produced predictive capabilities that are generally within a factor of 3 (7, 8). As the control requirements for many hazardous SOCs become more stringent, a factor of 3 may no longer be considered a suitable predictive capability. Furthermore, if these predictions are to be used as inputs for other atmospheric chemistry models, then a factor of 3 at the input may be grossly amplified in the model output. Work by Rounds et al. (9) and Kamens et al. (10) suggests that under certain conditions (e.g., cold temperatures) many compounds in the atmosphere may deviate from gas-particle equilibrium. As such, in order to provide accurate predictions of the phase distribution of SOCs in the atmosphere, dynamic mass transfer models are required. The first attempt to simulate general atmospheric sorption kinetics with a dynamic radial diffusion model was made by Rounds and Pankow in 1990 (11). Their model particle was spherical and consisted of a nonporous, nonsorbing inner core surrounded by a uniformly porous, sorbing outer shell. Sorption within the micropores of the particle was assumed to be Langmuirian, reversible, and instantaneous. Surface diffusion was assumed to be negligible, so gas-phase pore diffusion was the predominant mechanism for intraparticle transport. Mass transfer across the external surface of the particle was assumed to be instantaneous. Rounds et al. (9) used this model to predict the desorption of SOCs from automobile exhaust particles collected on filters. They were able to fit the experimental results for several n-alkanes and PAHs by optimizing the gas-particle partition coefficient, Kp, and the effective intraparticle diffusion coefficient, Deff. While the optimized Kp values agreed well with theoretical and experimental results, the Deff values were approximately 6 orders of magnitude smaller than expected for sorption-retarded gas-phase diffusion. It was suggested that one possibility for this discrepancy is that intraparticle transport by diffusion occurs in a liquid-like organic phase rather than in the gas phase (9). In fact, there is mounting evidence that many types of combustion aerosols contain a liquid-like phase. This evidence includes the following findings: (a) the impaction of wood and diesel soot on a metal foil produces a viscous liquid slick that retains little or no particle integrity (12), (b) subcooled liquid vapor pressures yield better fits to Langmuirian partitioning models than solid vapor pressures (2), and (c) as much as 50-90% by weight of the carbon associated with many types of combustion aerosols is extractable (13). In response to these findings, Odum et al. (14) developed a model that was similar to the Rounds and Pankow model (9, 11) but included diffusion through a viscous, liquid-like organic layer rather than gas-phase diffusion in a porous matrix. This model also addressed mass transfer across the particle surface. The model was used to predict the particlephase uptake of deuterated pyrene on diesel soot in a chamber experiment by optimizing the diffusion coefficient and the surface mass transfer coefficient. The work by Odum et al. (14) demonstrated that a radial diffusion model in which diffusion occurs in a liquid-like organic layer may be used to simulate the partitioning of SOCs on diesel soot. In the present work, a model has been developed that advances the Odum model by (a) using a dual-impedance model to describe the particle, (b) including more detailed input parameters to accurately describe the aerosol system, and (c) implementing a more efficient and robust solver. The utility of this model

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for different particle types and under different conditions is demonstrated by simulating the partitioning of PAHs in diesel and wood soot atmospheres in chamber experiments. These two combustion sources represent a significant portion of the particulate matter in urban settings. For example, in an inventory of an 80 × 80 km area centered over downtown Los Angeles, Rogge (15) determined that over 23% of fine aerosol organic carbon resulted from wood and diesel combustion.

Experimental Section All experiments were conducted at the UNC smog chamber facility located near Pittsboro, NC (16). The 190 m3 chamber is a wooden A-frame covered with Teflon. The chamber leak rate, including sampling, was between 1 and 5% per hour, as measured by a sulfur hexafluoride tracer. Background aerosol concentration ranged from 0.010 to 0.030 mg/m3. All experiments were conducted at night to avoid photochemical reactions. Prior to addition of combustion soot, deuterated fluoranthene (Cambridge Isotope Laboratories, Andover, MA) was added to the chamber via a hot (∼200 °C) injector. Combustion soot was then added from either a 1980 Mercedes Benz 300SD diesel engine or a wood stove burning dry, aged yellow pine. Concentrations of both native and deuterated PAHs were approximately 2 orders of magnitude higher than background levels. Temperature and humidity were measured periodically using a temperature thermistor and a lightscattering dewpoint meter (EG&G Model 800, Waltham, MA). Particle size and number were measured using an electrical aerosol analyzer (Thermo Systems, Inc. Model 3030, Minneapolis, MN). Sampling. The duration of each experiment was between 5 and 6 h. Sampling was conducted at approximately 15min intervals for the first part of each experiment and then at approximately 30-min intervals for the remainder of the experiment. Samples were collected at a flow rate of approximately 20 L/min for 8-20 min using a three-piece sampling train consisting of (a) a 40-cm, 5-channel annular denuder with an inner diameter of 2.4 cm and 0.1 cm spaces (University Research Glassware, Carborro, NC), followed by (b) a preweighed, 47-mm, Teflon-impregnated glass fiber filter (T60A20, Pallflex Products Corp., Putnam, CT), followed by (c) a 24-cm, 5-channel annular denuder (10, 17). The walls of the denuders were coated directly with a fine XAD-4 resin (18). The denuders were extracted in the field and reused (10, 16, 17). Analytes collected by the top denuder were considered to be gas phase, and the sum of the mass collected on the filter and by the bottom denuder was considered to be particle phase. The efficiency of the top denuder in capturing gas-phase compounds was determined by sampling with two denuders in series and comparing the mass collected on each denuder. All denuders used in this study were >99% efficient for fluoranthene-d10. Immediately after sampling, an internal standard containing several deuterated PAHs was added to each denuder. Deuterated pyrene was used as an internal standard for quantification of fluoranthene-d10. The denuders were extracted three times with a 20:30:50 mixture of optima grade dichloromethane:acetone:hexane (Fisher Scientific, Fair Lawn, NJ). The extract was then placed in a round-bottom flask, sealed, covered with aluminum foil, transported to the laboratory, and stored at 0 °C. Within 2 days, the samples were rotary evaporated to 5 mL; further concentrated to 50 µL with clean, dry nitrogen; and stored at 0 °C until quantification. Filter samples were immediately removed from the sampling train, placed in glass jars with Teflon lids, and transported to the laboratory in a cooler at 0 °C. The filters were allowed to thermally equilibrate for 30 min before being reweighed on a three-place milligram microbalance. The deuterated PAH internal standard mixture was added, and the filters were Soxhlet extracted for 12-18 h using the 20:

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FIGURE 1. One-layer model consisting of a solid core surrounded by a liquid layer. 30:50 dichloromethane:acetone:hexane mixture. After extraction, filters were dried with clean, dry nitrogen and reweighed to determine the fraction of extractable particle mass. Extracts were concentrated by rotary evaporation to 5 mL and blown down with clean, dry nitrogen to approximately 50 µL. Extraction and workup efficiencies for various PAHs from a diesel particulate matter standard have been characterized by Kamens et al. (10). Analysis. All quantitative analyses were conducted on a Hewlett Packard (HP) 5890 Series II gas chromatograph (GC) equipped with an HP 5971A mass selective detector. Selected ion monitoring mode was used to obtain greater sensitivity. GC injections were performed on a split/splitless injector with an HP 7673 autoinjector. A J&W DB-5 fused silica column was used for all separations.

Theory and Model Development Modeling was based on the assumptions that advectivedispersive solute transport from the bulk air to the particle surface was instantaneous and that solute transport in the particles occurred only by diffusion. These assumptions should be valid since the aerosol was well mixed in the chamber and no external forces were acting to cause advection within the particles. Two separate particle models were implemented in this work. The first was conceptually the same as the Odum et al. model (14); a sphere composed of a solid, nonsorptive core surrounded by a layer containing liquid-like organic material (Figure 1). Diffusion of analyte in the radial direction in the organic layer is described by the spherical coordinate form of Fick’s second law:

[

]

∂C ∂2C 2 ∂C ) Da 2 + ∂t r ∂r ∂r

()

(1)

where C (ng/cm3 organic material) is the concentration of the analyte at distance r (cm) from the center of the particle and Da (cm2/s) is the apparent diffusion coefficient for the analyte in the material surrounding the core. The apparent diffusivity accounts for transport impedances caused by three factors: (a) free-liquid diffusion, (b) sorption to solid phases, and (c) tortuosity, which quantifies the additional diffusion distance resulting from restrictions in the diffusion pathway (19). Note that the apparent diffusion coefficient has the same meaning as the “effective diffusion coefficient” used by Rounds and Pankow (9, 11) and the “diffusion coefficient” used by Odum et al. (14). This terminology has been used here to maintain consistency with Weber and DiGiano (19).

At the surface of the particle, the boundary condition is (20)

-Da

∂C ) R(Cs - Co) ∂r

(2)

where Cs (ng/cm3 organic material) is the concentration of the analyte immediately below the surface of the particle, Co (ng/cm3 organic material) is the concentration of the analyte required to maintain equilibrium with the gas phase, and R (cm/s) is the surface mass transfer coefficient. The boundary condition at the interface of the particle core and the organic material is described by

∂C )0 ∂r

(3)

The model discretizes the particles into a series of concentric spheres (nodes) spaced at equal intervals. In this work, the smallest particles were discretized into approximately 50 nodes, and the largest were discretized into approximately 500 nodes. It was demonstrated that this was a sufficiently fine discretization to provide accurate results. The model then implements a finite-difference method of lines technique (21) to provide estimates for eq 1 at each node and eqs 2 and 3 at the boundaries. This produces a system of differential algebraic equations (DAEs) of the form

F(t, C(t), C′(t)) ) 0

(4)

This system of equations is solved using the DAE package DDASPK (21, 22), which uses a combination of backward differentiation formula methods and a direct solver for the resulting linear system of equations. DDASPK has the advantage of being both robust, since it implements backward differentiation formula methods, and efficient, since it implements variable time stepping and variable order of backward differentiation formulas. The model was validated against the analytical solution provided by Crank (20) for a monodisperse aerosol consisting of coreless particles. It produced sufficiently accurate results at absolute and relative local error tolerances of e10-2. All simulations for this work were conducted at absolute and relative tolerances of 10-3 to ensure accuracy. Optimization simulations were run on a Hewlett Packard K-400 (Model 9000/829) UNIX workstation, and individual simulations were run on a 90-MHz Pentium PC. Determination of Co. As can be seen from eq 2, when Cs * Co, a gradient is established at the surface of the particle that drives diffusion of the analyte into or out of the particle. Co may be derived from the equilibrium gas-particle partition coefficient (6-8):

Kp )

Cp CgTSP

(5)

where Cp and Cg (ng/m3 air) are the particle and gas-phase analyte concentrations at equilibrium, respectively, and TSP (mg/m3 air) is the concentration of total suspended particulate matter. Co can then be determined at each time step from

Co(t) )

Kp(t)Cg(t)Forg fom(t)

(6)

where Forg (mg/cm3) is the density of the organic material, and fom (mg of organic material/mg of particle) is the mass fraction of organic material in the particles. The values of Cg and fom can be readily determined experimentally (the value of fom is assumed to be equal to the fraction of extractable particle mass), and Forg can be estimated from published particle composition studies (15, 23). A semi-empirical approach was implemented to determine Kp values. The value

TABLE 1. Empirical Eq 8 Constants for Fluoranthene constant

wood soot

diesel soota

n aemp bemp cemp r2 SD log Kp estimate

27 -1.122 -0.566 -4.327 0.927 0.118

9 -1.426 0 -6.616 0.857 0.226

a Diesel soot experiments have not been conducted at a sufficient range of relative humidities to obtain a robust regression using eq 8. As such, a regression of the form log Kp ) aemp log p°L + cemp was used.

of Kp can be determined theoretically by (24)

Kp )

760fomRT

(7)

MWomγp°L 106

where R is the gas constant (8.2 × 10-5 m3‚atm/mol‚K), T (K) is the temperature, MWom (g/mol) is the number-average molecular weight of the organic material, γ is the mole fraction activity coefficient of the compound of interest in the organic material, and p°L (Torr) is the pure compound liquid vapor pressure (subcooled if necessary) of the analyte of interest at T. Note from eq 7 that the partition coefficient, Kp, for a given compound is a function of the compound’s vapor pressure, p°L, and the activity coefficient, γ. Vartiainen et al. (25) showed that both diesel and wood soot particles sorb a significant amount of water that varies as a function of relative humidity. Addition of water to a soot particle will change the polarity of the particle’s liquid-like material, thus changing the activity coefficient of a given compound in the liquid. As such, relative humidity (RH) may be used as a surrogate for γ in a semi-empirical expression of the form:

log Kp ) aemp log p°L + bemp log RH + cemp

(8)

Kamens et al. (10) demonstrated that PAHs produced from the combustion of wood and diesel fuel establish gas-particle partitioning equilibrium shortly after soot injection into a chamber (but may deviate from equilibrium over time). Additionally, Pankow (7) has shown that a plot of log Kp versus log p°L for a homologous series of compounds will be linear and have a slope of -1 for systems in gas-particle equilibrium. Accordingly, a database of equilibrium Kp values for several compounds in wood and diesel soot systems was compiled, and the empirical parameters, aemp, bemp, and cemp, in eq 8 were obtained by linear regression for each compound in each soot type. These constants for fluoranthene are shown in Table 1. Other Model Inputs. Model inputs that changed as a function of time, such as Cg, TSP, fom, T, RH, and the fraction of the total number of particles in each particle size range (i.e., bin, n), PFn, were fed into the model via an exponential expression of the form

X(t) ) X1f + (X1o - X1f)e-X1kt + X2f + (X2o - X2f)e-X2kt

(9)

where X is the parameter of interest and t is time after soot injection. Xab values (e.g., X1f, X1o, etc.) were optimized to fit the experimental data for the parameter X. This method produced excellent fits to the experimental data and thus provided an accurate and smooth representation of the experimental data in the model.

Results and Discussion A comparison of the experimental and simulation results for a diesel soot experiment conducted on 6/8/95 is shown in Figure 2 (one-layer model). A summary of the key model inputs is given in Table 2. The values of Da and R used in this


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TABLE 2. Summary of Key Model Inputs inputs

parameter bins (number of different particle sizes), n particle diameter, a (µm)

duration of chamber experiment (h) fluoranthene-d10 gas-phase concn, Cg (ng/m3 air)b total suspended particulates, TSP (mg/m3 air)b mass fraction of organic material in particles, fom

chamber temperature, T (K) % relative humidity, RH

wood (9/13/95) diesel (6/8/95)

Odum et al. (14).

b

6

6

electrical aerosol analysis (EAA) data from chamber experiments, particles sizes constituting >0.1% of the total number of particles were included 0.0422-0.422 0.0237 to 0.422 0.0237 to 0.422 EAA data from chamber experiments; changes in particle size distribution as a function of time are incorporated by introducing parameters for the fraction of particles in each bin, PFn, in eq 9 5.46 5.78 4.67 chamber experiments 1121-653

3910-1706

397-260

top denuder samples from chamber experiments

1.05-0.66

1.05-0.43

0.93-0.69

mass on filters from chamber experiments

0.90-0.92

0.55-0.38

0.55-0.38

296.5-294.5 56.4-73.1

295.7-291.0 100 (const)

297.5-295.6 94.0 (const)

840

840

2000

2000

comparison of particle mass before vs after Soxhlet extraction; this was not measured in the 7/1/93 diesel experiment, so it was assumed to be the same as the 6/8/95 diesel experiment temperature probe in chamber dewpoint meter; no data were obtained for the 6/8/95 experiment; since it rained periodically throughout the experiment, RH was estimated as 100% estimated from particle-composition data of Rogge et al. (15, 23) Rounds and Pankow (11)

These values represent the data range. The model inputs are actually the optimized Xab values shown in eq 9.

FIGURE 2. Comparison of particle-phase fluoranthene-d10 concentrations obtained by one-layer and dual-impedance models versus experimental results for a chamber experiment with diesel soot (6/8/95). For the one-layer model: Da ) 9.3 × 10-16 cm2/s, r ) 1 × 10-6 cm/s, and the sum of squared residuals (SSR) ) 8.7 × 105. For the dual-impedance model: Da,1 g 1 × 10-13 cm2/s, Da,2 ) 4.0 × 10-17 cm2/s, r ) 1 × 10-6 cm/s, Ep,2 ) 0.590, and SSR ) 1.9 × 105. The “equilibrium concentration” is the concentration obtained when C ) Co throughout the organic material of the particles (eqs 1 and 2). simulation were optimized using a Levenberg-Marquardt routine (26). Note that the surface mass transfer coefficient, R, was set at a high value (1 × 10-6 cm/s). This was done because parameter optimizations indicated that the system was limited by the value of Da rather than the value of R (i.e., diffusion limited). This was the case for all simulations conducted in this work and, as discussed later, is supported by recent work of Kamens and Coe (27). As can be seen in Figure 2, a poor fit of the experimental data was obtained using the one-layer model. The experimental particle-phase concentration increased rapidly shortly after soot injection and then remained relatively stable for the remainder of the experiment. To fit the earlier experimental points in Figure 2 with the one-layer model, a large apparent diffusion

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5

density of organic material in 1190 particle, Forg (mg/cm3) 3 density of particle core, Fcore (mg/cm ) 2000 a

diesel (7/1/93)a


FIGURE 3. Dual-impedance model consisting of an inner layer of discrete solids surrounded by an outer layer primarily composed of liquid-like organic material. coefficient, Da, would be needed. However, this large Da would cause the simulated particle-phase concentration to approach the equilibrium concentration too rapidly, and thus, the latter points would be “overshot”. As a result, the model optimized on a lower value of Da, which caused it to “undershoot” the initial points. It is clear from these results that a model is needed that allows rapid initial migration into the particle and slower subsequent migration. Accordingly, a dual-impedance model was developed in which the conceptual particle consists of two layers: an outer layer primarily composed of liquid-like organic material that covers an inner spherical layer composed of many discrete, solid, and impenetrable masses submerged in the liquid-like organic material (Figure 3). The rate of diffusion in each layer is quantified by an apparent diffusion coefficient, which accounts for liquid diffusion, adsorption,

TABLE 3. Sensitivity Tests for Dual-Impedance Model Using Fluoranthene-d10 Data from 9/13/95 Wood Soot Experiment optimized variables

value

Da,1 (cm2/s)

Da,2 (cm2/s)

SSR

cemp ) -4.445 (-23.8% change in Kp) cemp ) -4.386 (-12.7% change in Kp) cemp ) -4.327 (0% change in Kp) cemp ) -4.268 (+14.6% change in Kp) cemp ) -4.209 (+31.2% change in Kp) 1 - p,2 ) 0.120 (-23.8%) 1 - p,2 ) 0.138 (-12.7%) 1 - p,2 ) 0.158 (+0%) 1 - p,2 ) 0.181 (+14.6%) 1 - p,2 ) 0.207 (+31.2%) Forg ) 907 mg/cm3 (-23.8%) Forg ) 1039 mg/cm3 (-12.7%) Forg ) 1190 mg/cm3 (+0%) Forg ) 1364 mg/cm3 (+14.6%) Forg ) 1561 mg/cm3 (+31.2%)

2.9 × 4.9 × 10-15 4.3 × 10-15 1.5 × 10-15 1.5 × 10-15 6.0 × 10-15 5.0 × 10-15 4.3 × 10-15 3.7 × 10-15 3.3 × 10-15 4.3 × 10-15 3.9 × 10-15 4.3 × 10-15 4.1 × 10-15 4.5 × 10-15

9.3 × 3.1 × 10-15 1.4 × 10-16 1.2 × 10-18 4.5 × 10-19 3.1 × 10-16 2.0 × 10-16 1.4 × 10-16 9.2 × 10-17 5.2 × 10-17 7.7 × 10-17 1.1 × 10-16 1.4 × 10-16 1.7 × 10-16 2.1 × 10-16

1.8 × 104 2.0 × 103 3.5 × 103 2.4 × 103 2.3 × 103 4.8 × 103 4.0 × 103 3.5 × 103 3.0 × 103 2.7 × 103 3.1 × 103 3.4 × 103 3.5 × 103 3.9 × 103 4.4 × 103

parameter equilibrium gas-particle partition coefficient,

Kpa

fraction of layer 2 composed of solid material, 1 - p,2b

density of organic material in particles, Forg

10-14

10-15

a The value of K was modified by changing c p emp (eq 8). The listed values were chosen, because they represent -1σ, 1/2σ, +0σ, +1/2σ, and +1σ in the estimate of log Kp in Table 1. b 1 - p,2 ) 0.158 was obtained by parameter optimization (see Figure 5).

and tortuosity (19). Diffusion in the outer layer is more rapid since impedances due to adsorption and tortuosity are less significant. This model is physically more consistent with scanning electron micrographs of combustion particles, which show an aggregation of individual microspheres surrounded by and interspersed with what appears to be liquid. The dual-impedance model is also supported by combustion research, which indicates that combustion particles are formed by the coagulation of “crystallites” in the range of 20-30 Å in the flame zone (28). These particles then continue to grow by surface growth, coagulation into chains and clusters, and condensation of organic compounds as they cool. This type of model has also been used to simulate the uptake of analytes by soil particles. In these models, termed dual intra-aggregate porosity models, two diffusion coefficients are used to differentiate the transport rate in regions of the particle having different pore sizes (29). The dual-impedance model uses the same numerical technique as the one-layer model to solve eqs 1-3. However, an additional boundary condition is needed to quantify mass transfer at the interface of layers 1 and 2:

∂C1 ∂C2 Da,1 ) Da,2 ∂r ∂r

(10)

where C1 is the concentration (ng/cm3 organic material) adjacent to the layer 1/layer 2 interface on the layer 1 side, and C2 is the concentration adjacent to the interface on the layer 2 side. Simulation results from the revised model are shown by the dual-impedance model line in Figure 2. Note that a significantly better fit is obtained as compared to the onelayer model. A similar observation is made when modeling the pyrene-d10/diesel soot data of Odum et al. (14) (Figure 4). The dual-impedance model introduces two additional paraameters to the one-layer model: (1) a second apparent diffusion coefficient, Da,2, and (2) the porosity of the inner layer, p,2, which, in turn, dictates the thickness of the layers. The values of Da,1, Da,2, and p,2 used in Figures 2 and 4 were obtained by parameter optimization. Since the fraction of organic material in the particle, fom, changes as a function of time, either p,2 or the thickness of the layers must also change. In this work, the thickness of the layers was kept constant, and p,2 was allowed to change over time according to

(

)

Fpart(t)fom(t) -1 Forg p,2(t) ) 1 + b3 a

()

(11)

FIGURE 4. Comparison of particle-phase pyrene-d10 concentrations obtained by one-layer and dual-impedance models versus experimental results for a chamber experiment with diesel soot conducted by Odum et al. (14) on 7/1/93. For the one-layer model: Da ) 1.2 × 10-16 cm2/s, r ) 1 × 10-6 cm/s, and SSR ) 5.7 × 103. For the dual-impedance model: Da,1 g 1 × 10-15 cm2/s, Da,2 ) 2.9 × 10-17 cm2/s, r ) 1 × 10-6 cm/s, Ep,2 ) 0.706, and SSR ) 1.2 × 103. where Fpart and Forg are the densities of the particles and the organic material, respectively, b is the distance between the center of the particle and the interface between layer 1 and layer 2 (Figure 3), a is the radius of the particle, and p,2 is the porosity of layer 2. This indicates that the inner layer would become more tightly “packed” as the particle ages or “dries” (i.e., as fom decreases). This method gave better fits than those obtained when holding p,2 constant and varying b/a. The results for fluoranthene-d10 in a wood soot experiment are shown in Figure 5. Note that the difference in the quality of fit between the one-layer and dual-impedance models is not as significant as that of the diesel soot experiment shown in Figure 2. This is because wood soot particles are more liquid-like in nature than diesel soot particles as shown by the higher fom values for wood soot in Table 2. Accordingly, there is less core material, and the magnitude of impedances in layer 2 is not significantly different than that in layer 1. Modeling of phenanthrene-d10 data for this experiment showed a similar tendency. Sensitivity Analyses. The sensitivity of the optimized apparent diffusion coefficients, Da,1 and Da,2, to the following key input parameters was explored using the dual-impedance model: (a) the equilibrium partition coefficient, Kp (b) the fraction of layer 2 composed of solid material, 1 - p,2, and (c) the density of the organic material in the particles, Forg. These results are shown in Figure 6a-c and Table 3 for fluoranthene-d10 data from the 9/13/95 wood soot experiment.


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FIGURE 5. Comparison of particle-phase fluoranthene-d10 concentrations obtained by one-layer and dual-impedance models versus experimental results for a chamber experiment with wood soot (9/13/95). For the one-layer model: Da ) 2.1 × 10-15 cm2/s, r ) 1 × 10-6 cm/s, and SSR ) 1.2 × 104. For the dual-impedance model: Da,1 ) 4.3 × 10-15 cm2/s, Da,2 ) 1.4 × 10-16 cm2/s, r ) 1 × 10-6 cm/s, Ep,2 ) 0.842, and SSR ) 3.5 × 103. The listed percent changes in Kp were chosen because they represent a standard deviation of -1σ, -1/2σ, +0σ, +1/2σ, and +1σ in the estimate of log Kp in Table 1. These percentages were also used for the 1 - p,2 and Forg sensitivity tests because they represent reasonable levels of uncertainty in these parameters and they allow in intercomparison of the sensitivity of the results to each of the three parameters. The fraction of layer 2 composed of solid material, 1 - p,2, was used instead of simply the porosity, p,2, because the largest change (31.2%) would have raised the value of p,2 over 1, which is physically impossible. Note from the sum of squared residuals (SSR) values in Table 3 that excellent fits of the experimental data were obtained for each simulation (these fits are similar to that shown for the dual-impedance model in Figure 5). As shown in Figure 6a, the optimized Da,1 and Da,2 values decrease as Kp increases. This is a result of higher equilibrium particle-phase concentration at the surface of the particle, Co, at higher Kp values (eq 6). As a result, there is a higher concentration gradient driving diffusion (eq 2), and smaller apparent diffusion coefficients are needed to fit the experimental data. Similar results were obtained for the 1 - p,2 sensitivity tests (Figure 6b). In eq 11, the quantity Fpartfom/Forg represents the ratio of organic material volume to particle volume, which is always e1. Accordingly, increasing the value of the quantity 1 - p,2 results in a smaller b/a value. Given that the particle radius, a, is constant in each bin (taken from electrical aerosol analyzer measurements), the volume of layer 1 increases and layer 2 decreases as the value of 1 - p,2 is increased. Since diffusion is significantly more rapid in layer 1, diffusion in layer 2 is primarily rate limiting except at the beginning of the experiment. Accordingly, a smaller Da,2 is needed to fit the experimental data, while Da,1 is nearly constant as 1 - p,2 changes. The Forg sensitivity results (Figure 6c) show the opposite tendency in Da,2, since b/a increases as Forg increases (eq 11). The results shown in Figure 6a-c indicate that the optimized Da values include error resulting from uncertainty in the input parameters. In particular, the results indicate that the optimized value of Da,2 is very sensitive to changes in each of the three parameters, with Kp being the most sensitive parameter. As such, it is valuable to understand the effect that these optimized apparent diffusivities have on the results of individual simulations. Figure 7 compares the results of four simulations of fluoranthene-d10 in the 9/13/95 wood soot experiment. The Da values for these simulations

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FIGURE 6. Results of sensitivity tests using fluoranthene-d10 data from the 9/13/95 wood soot experiment. Da values (in cm2/s) were obtained by optimization. were taken from (a) the optimization simulation using the Kp values obtained from the parameters in Table 1 (i.e., 0% change in Kp) and (b) the optimization simulation using 23.8% smaller Kp values (i.e., -1σ of Kp estimate in Table 1). These results indicate that the increased Da values result in an error of approximately 61% at 0.40 h1/2 (0.16 h), which is primarily due to the increase in the Da,1 value, and an error of approximately 9.9% at 2.3 h1/2 (5.3 h), which is primarily due to the increase in the Da,2 value. Simulations comparing the effects of the Da values obtained from optimization simulations using the “0% change” Kp values versus those obtained using 31.2% larger Kp values show a similar magnitude of error (-27% error at t ) 0.40 h1/2 and -27% error at t ) 2.3 h1/2). These results indicate that the errors in the optimized apparent diffusivities caused by uncertainty in input parameters for optimization simulations can result in moderate errors in the results of individual simulations. Significant Observations. Several significant observations may be drawn from these results. First, a dual-impedance model gives significantly better reproductions of non-

expanded to include photochemical reactions, and finally (c) incorporated into larger atmospheric models, such as urban airshed models, to predict the fate of SOCs in real atmospheres.

Acknowledgments This work was supported by grants from the National Science Foundation (ATM 940848, Dr. Sherry O. Farwell, Project Officer) and the North Carolina Super Computing Center (Kenneth Galluppi, Project Officer). The authors would also like to thank Professor Cass T. Miller for his insight and the use of his computing resources and Keri Leach, Myoseon Jang, Heejeong Latimer, Jianbo Zhang, and Jianxin Hu for assisting with chamber experiments.

FIGURE 7. Sensitivity of individual simulation results to the apparent diffusion coefficients obtained by optimization with Kp ) Kp - 0.238 Kp. Da values are in cm2/s. equilibrium gas-particle partitioning than does a simple, onelayer model for diesel soot aerosols. For wood soot, the dualimpedance gives slightly better results, but the one-layer model results are still reasonable. Second, surface mass transfer is not rate limiting in these systems. In each of the simulations, the surface mass transfer coefficient, R, optimized to a large value (i.e., R optimized to a value such that increasing its value did not change the simulation results, but decreasing its value resulted in a poorer fit of the experimental data). Accordingly, a large value (R ) 1 × 10-6 cm/s) was used for all simulations. This observation is supported by a recent study (27) in which diesel soot was passed through a large gas-phase stripping device to remove gas-phase compounds from the aerosol. This caused fluoranthene to rapidly off gas from the particles in an attempt to re-establish equilibrium. It was noted that approximately 50% of the particle-phase fluoranthene was off gassed when the particles were resident in the stripping device for approximately 10 s. The rapid rate of off gassing implies that surface mass transfer is rapid. The observation of rapid mass transfer across the particle surface is also consistent with observations of the uptake of solutes by soil particles in subsurface environments, in which intraaggregate diffusion rather than surface mass transfer is usually determined to be rate limiting (29). Third, conclusions can be drawn regarding the relative importance of the three factors impeding transport of analyte: (a) free-liquid diffusion, (b) adsorption to solid surfaces, and (c) restrictions in the diffusion pathway (i.e., tortuosity). The diesel soot simulation results indicate that the value of Da,1 is g 1 × 10-13 cm2/s (Figure 2). Since free-liquid diffusivities are expected to be in the range of 10-5-10-7 cm2/s (19), the optimized value of Da,1 does not preclude the possibility that adsorption and/or tortuosity impedes mass transport in the outer layer of the diesel soot particles. However, these results indicate that transport in the outer layer is not rate limiting. The inner layer apparent diffusivity is significantly smaller (Da,2 ) 4.0 × 10-17 cm2/s), indicating that the adsorption and/or tortuosity are more significant in the inner layer and that transport in this layer is rate limiting. Alternately, diffusion in both layers of wood soot particles is rate limiting (Da,1 ) 4.3 × 10-15 cm2/s and Da,2 ) 1.4 × 10-16 cm2/s in Figure 5). This indicates that adsorption and/or tortuosity are impeding mass transport in each layer. By conducting several experiments using various combustion sources under a range of atmospheric conditions (e.g., temperature and humidity), optimized apparent diffusivities may be determined for both the inner and outer particle layers. In this manner, values for Da,1 and Da,2 as a function of atmospheric conditions, soot type, and analyte may be determined. The model may then be (a) verified in atmospheres containing mixtures of particle types, (b)

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(28) VanDell, R. D.; Mahle, N. H. The role of carbon particulate surface area on the products of incomplete combustion (PICs) emission. In Emissions from Combustion Processes: Origin, Measurement, Control; Clement, R., Kagel, R., Eds.; Lewis Publishers: Boca Raton, FL, 1990. (29) Brusseau, M. L.; Rao, P. S. C. Crit. Rev. Environ. Control. 1989, 19, 33-99.

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Received for review January 29, 1997. Revised manuscript received June 11, 1997. Accepted June 24, 1997.X ES970079G X

Abstract published in Advance ACS Abstracts, August 1, 1997.

Development and Application of a Dual-Impedance Radial Diffusion

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