Environ. Sci. Technol. 2003, 37, 3821-3827
Single-Layer Model To Predict the Source/Sink Behavior of Diffusion-Controlled Building Materials DEEPT KUMAR AND JOHN C. LITTLE* Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, 418 Durham Hall, Blacksburg, Virginia 24061-0246
Building materials may act as both sources of and sinks for volatile organic compounds (VOCs) in indoor air. A strategy to characterize the rate of absorption and desorption of VOCs by diffusion-controlled building materials is validated. A previously developed model that predicts mass transfer between a flat slab of material and the well-mixed air within a chamber or room is extended. The generalized model allows a nonuniform initial material-phase concentration and a transient influent gas-phase concentration to be simultaneously considered. An analytical solution to the more general model is developed. Experimental data are obtained by placing samples of vinyl flooring inside a small stainless steel chamber and exposing them to absorption/ desorption cycles of n-dodecane and phenol. Measured values for the material-air partition coefficient and the material-phase diffusion coefficient were obtained previously in a series of completely independent experiments. The a priori model predictions are in close agreement with the observed experimental data.
Introduction Considerable health and productivity gains stand to be realized by improving the indoor environment (1). One way to help achieve these gains is to reduce gas-phase emissions from the wide range of sources that are found indoors. Volatile organic compounds (VOCs) emanating from building materials are an important class of indoor air contaminants. Sources of VOCs include wood (2), sealants (3), carpets (4), adhesives (5), and vinyl flooring (6). Vinyl flooring (VF) is an example of a material for which the emission rate is controlled by the initial material-phase concentration (C0), the material-phase diffusion coefficient (D), and the material-air partition coefficient (K) (7). Other materials governed by the same phenomena include polyurethane foam (8), styrene-butadiene rubber carpet backing (9), and probably many other polymeric substances. These materials may also act as sinks for VOCs in the air, depending on the relative magnitudes of the gas- and material-phase concentrations (10). Little and Hodgson (11) proposed a strategy for characterizing both the source and sink behavior of such “diffusioncontrolled” materials. In the case of a source, the approach involves independently measuring C0, D, and K and then predicting the emission rate a priori. This strategy has recently * Corresponding author telephone: (540)231-8737; fax: (540)2317916; e-mail:
[email protected]. 10.1021/es026332g CCC: $25.00 Published on Web 07/23/2003
2003 American Chemical Society
been applied to predict the emission rate of three VOCs from VF (7). The very good model predictions suggest that relatively homogeneous diffusion-controlled building materials can be characterized in a way that is more direct than the traditional chamber study. Because D and K tend to correlate with molecular weight and vapor pressure (11, 13), relationships could be deduced for the typical diffusion-controlled materials found in buildings, perhaps based on different classes of organic compounds. This would greatly facilitate the characterization of diffusion-controlled sources because the identification of individual VOCs in the material phase and measurement of their initial concentrations (12) are all that is required. Once the VOCs have been identified and quantified (i.e., C0 has been determined), values for D and K could be obtained from appropriate correlation equations and used to predict the emission rates without further effort. This approach could be applied to predict both source and sink behavior for the entire class of indoor materials. Although the source model has been experimentally validated (7), the sink model has not. In addition, as originally formulated (9), the analytical solution to the source model requires that the initial material-phase concentration be uniform and that the inlet air stream contain no VOC. Recently, measurements of chemical concentrations within VF have revealed a nonuniform initial concentration profile for some compounds (7). In another study, the rate of emission of VOCs from a building material was found to increase with time (14), at least initially. An explanation for this somewhat unusual behavior could be an initial materialphase concentration profile that increases with depth. Also, if a material initially acts as a sink and then later emits the VOCs, a nonuniform material-phase concentration may need to be considered when predicting the emission rate during the desorption process. The concentration of VOCs in the air entering a room is also important in determining the overall system response. The influent concentration may be transient in nature as the original source emitting VOCs is frequently not constant (10). In this paper, the previously developed source (9) and sink (11) models are generalized to include a nonuniform initial material-phase concentration and a time-dependent function describing the gas-phase influent VOC concentration. This encompasses a range of more realistic conditions that may be encountered in indoor environments. A series of small-chamber experiments is conducted, and the data are used to more fully validate the overall modeling approach.
Development of Model A chamber with a slab of material acting as either a source or a sink is shown in Figure 1. The influent air contains a VOC at a concentration described by a known function of time, yin(t). Inter-phase mass transfer takes place between the air and the material depending on the relative concentrations in the gas and material phases. Diffusion within the slab is governed by
∂ 2C ∂C )D 2 ∂t ∂x
(1)
where C(x,t) is the material-phase concentration, D is the material-phase diffusion coefficient, x is distance from the base of the slab, and t is time. The initial condition is given by an arbitrary function describing the nonuniform materialphase concentration profile or
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and the qm values are the positive, nonzero roots of
qm tan(qmL) ) h - kqm2
(10)
and h and k are given by
h)
Q AKD
k)
V AK
(11)
A simple procedure for finding the roots of eq 10 is described elsewhere (7). A variation of this model can be obtained for the case where the airflow rate is reduced to zero. In this case, the solution is FIGURE 1. Simplified representation of VOC interactions in a room or chamber.
C(x,t) )
g(L) + kf (L)
The first boundary condition assumes that the flux through the base of the material is zero or
|
∂C )0 ∂x x)0
The second boundary condition is based on a mass balance about the chamber, expressed as
dy ∂C ) Qyin - AD - Qy dt ∂x
V
(4)
C|x)L
where K is the material-air partition coefficient. The second boundary condition thus becomes
V ∂C ∂C Q + AD + (C - Kyin) ) 0 K ∂t ∂x K
(6)
Note, however, that equilibrium does not necessarily exist between the air and the material beneath the surface layer. The condition that controls J, the inter-phase mass-transfer flux, is the direction and magnitude of the material-phase concentration gradient at the surface of the slab or
∂C | ∂x x)L
J|x)L ) -D
(7)
A nonzero gradient results in a flux in the appropriate direction. If the gradient is zero, there is no transfer of mass between the two phases. Using Laplace transforms, a solution to this system of equations is found. The inverse is obtained by the method of residues yielding
[
C(x,t) ) ∞
2
∑
m)1
qm2{hDR(t) cos(qmL) + kf (L) cos(qmL) + IC} cos(qmx) exp(-Dq2mt) cos2(qmL){L(h - kqm2)2 + qm2(k + L) + h}
]
(8)
where
R(t) )
∫ Ky t
0
in(τ)
exp(Dqm2τ) dτ IC )
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9
∫
L
0
+
[IC{kqn sin(qnL) - cos(qnL)} kf (L)] cos(qnx) exp(-Dq2nt)
kqnL sin(qnL) - (k + L) cos(qnL)
]
(12)
where
∫ f (x) dx L
0
IC )
∫
L
0
cos(qnx)f (x) dx (13)
and the qn values are the positive, nonzero roots of
(5)
y(t)
∑
n)1
g(L) )
where the air in the chamber is assumed to be completely mixed. Equilibrium is assumed to exist between the chamber air and the surface of the material slab, such that
K)
∞
2
(3)
[
k+L
cos(qmx)f (x) dx (9)
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 37, NO. 17, 2003
kqn cos(qnL) + sin(qnL) ) 0
(14)
The roots of eq 14 are also found using the procedure described previously (7). Details of the analytical solution and an executable version of the software are available from the authors upon request.
Experimental Approach Small-chamber experiments were conducted to generate data for model validation. Samples of VF were placed inside the chamber and subjected to absoption/desorption cycles using both constant and transient influent concentrations. During the sorption period, the material-phase concentration within the VF increased. The experiment was terminated before steady state was achieved so that a nonuniform materialphase concentration was established. The model-predicted final concentration profile created during the absorption period served as the initial condition for the subsequent desorption period. Phenol, an aromatic compound, and n-dodecane, an aliphatic compound, were selected as representative VOCs for the experiments because these two compounds are found in the type of VF used and the K and D values were measured previously (13). A gravimetrically calibrated diffusion oven was used to establish a range of gas-phase concentrations for each of the VOCs. Increasing the temperature inside the oven increased the emission rate from the diffusion tube and hence the resulting gas-phase concentration. The accuracy of the photoacoustic gas analyzer used to measure VOC concentrations in the effluent from the chamber was checked against the series of standard gas-phase concentrations generated by the diffusion oven. Calibration of Diffusion Oven. A diffusion tube containing the VOC in liquid form was placed inside the diffusion oven and maintained at a constant temperature. Clean dry air was passed through the oven at a known and controlled flow rate. The VOC diffused out of the tube at a constant rate and mixed with the air. The diffusion tube was removed and weighed to determine the initial mass of VOC inside the tube. After replacing the tube, air was passed through the oven for
FIGURE 2. Microbalance apparatus used to measure D and K. an additional time period, and a second mass reading was obtained. The difference in the mass of the tube between the two readings gave the mass flow rate into the air. The gasphase VOC concentration was calculated using the VOC emission rate and the airflow rate. The procedure was repeated at different temperatures, thereby establishing a range of known gas-phase concentrations. Calibration of Gas Analyzer. The photoacoustic gas analyzer (model 1312, California Analytical) used to measure the gas-phase concentration of the two VOCs was calibrated. A zero calibration against a null gas and a humidity interference calibration were done by the manufacturer. A compound-specific two-point calibration was performed in the laboratory for each of the VOCs. The two concentration values used in the calibration were selected based on the range over which the instrument needed to measure the concentration in the chamber experiment. After the twopoint calibration was completed, the accuracy was confirmed by comparison to known gas-phase VOC concentrations established using the gravimetrically calibrated diffusion oven. The instrument was found to provide a linear response and to be accurate for n-dodecane concentrations above 1 mg m-3 and for phenol concentrations above 2 mg m-3. For concentrations below these levels, the gas analyzer is unable to accurately determine the gas-phase concentration. The calibration and use of the gas analyzer required a gas flow rate of at least 0.3 L min-1 as this was the minimum flushing rate through the measurement cell of the instrument. This flow rate determined the maximum concentration of VOC possible for a given diffusion oven temperature. Determination of K and D. The method used to measure the diffusion and partition coefficient is related to similar methods to measure analogous properties in other materials (15, 16). The method used to measure D and K in vinyl flooring (13) is briefly reviewed. A thin slice of the VF was obtained using a microtome and suspended in a microbalance as shown in Figure 2. The original VF material was about 2 mm thick while the thin slice was about 0.1 mm thick. This 20fold decrease in the diffusion path length decreases the time taken to reach equilibrium by a factor of about 400. The thin VF slice was subjected to absorption/desorption cycles with K determined from the interim equilibrium
condition and D established from the kinetic data. The initial mass of the slice was recorded. A constant flow of air with a known gas-phase concentration was passed through the microbalance. The VOC transferred from the air into the VF, resulting in an increase in the mass measured by the microbalance. The process continued until equilibrium was reached. The increased mass of the VF slice was due to the presence of a uniform concentration of the VOC. The absorbed mass and hence the concentration within the VF was known. The ratio of the material-phase concentration to the applied gas-phase concentration gave the partition coefficient, K, for the specific VOC/VF system. After equilibrium was reached, the influent stream was switched to VOC-free air, and data were collected during the desorption period. The diffusion coefficient, D, was fitted to the kinetic data obtained during both absorption and desorption periods using
Mt M∞
∞
)1-
[
8
∑ (2n + 1) π
n)0
2 2
{
exp -
}]
D(2n + 1)2π2t 4L2
(15)
where Mt is the total mass of VOC that has either entered or left the slice in time t and M∞ is the corresponding quantity at equilibrium. The experiments to determine K and D were conducted at a temperature of 25.6 ( 0.3 °C (13). Chamber Experiments. The arrangement for the chamber experiments is shown in Figure 3. Clean air from a gas cylinder was passed through the diffusion oven. The diffusion oven provided a constant emission rate of VOC to the air. To create conditions under which the VF inside the chamber would absorb VOCs, the airstream was directed through the diffusion oven. When the VF was later required to desorb the VOCs, the diffusion oven was bypassed. An empty chamber was used to create an influent stream with a transient gas-phase concentration. The second chamber housed the samples of vinyl flooring that acted as either sink or source. The photoacoustic gas analyzer was used to directly measure the concentration of VOC in the air leaving the chamber. The chamber was maintained at a temperature of 24 ( 1 °C. The VF sample used in the chamber experiments was “conditioned” in the open air in the laboratory for a period VOL. 37, NO. 17, 2003 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 3. Experimental apparatus showing airflow arrangement, diffusion oven, two small chambers (each approximately 7 L in volume) and gas analyzer.
FIGURE 4. Check for mixing and sink-effect within the chamber using n-dodecane. of 2 yr to provide sufficient time for desorption of any residual phenol and dodecane initially present. Although the edges of the VF samples were not sealed in these experiments, in no case did the surface area of the edges exceed 15% of the total exposed surface area, and in most cases, the edge surface was less than 10% of the total surface. The model was used to check the potential impact of this edge effect on the chamber experiments. For the absorption of phenol and under the conditions used in the chamber experiments, a 15% increase in surface area of the VF sample resulted in an average decrease in predicted gas-phase chamber concentrations of only 4%. A check on mixing within the chamber as well as the potential sink-effect of the chamber walls was obtained using n-dodecane. A single chamber without any VF was subjected to a step-up and then a step-down in the influent concentration. The experimental data are compared to the theoretical well-mixed case in Figure 4. The results show that the conditions inside the chamber are very close to being wellmixed. The fact that the chamber concentration took slightly longer to reach steady-state than expected in both step-up and step-down experiments suggests that a small sink-effect may exist due to interactions with the stainless steel chamber surfaces. Although interactions with the chamber walls are expected to be somewhat more pronounced for phenol (7), the modest sink effect is not expected to significantly impact the sorption/desorption experiments conducted with VF present in the chamber. 3824
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The usefulness of the general single-layer model is realized when the initial material-phase concentration profile and the influent gas-phase concentration vary. During absorption the VF acts as a sink. A nonuniform material-phase concentration was created by stopping the absorption period before steady-state was reached. The final concentration in the VF (as predicted by the model) became the initial condition for the subsequent desorption period. For a constant inlet concentration (yc), VOC-laden air at a constant flow rate was passed through the chamber. To create a transient gas-phase influent concentration, two chambers were used. The first one did not have VF present. The outlet concentration from this chamber was time-dependent until steady state was reached. The stream coming out of the first chamber was used as the influent to the second chamber, which contained the VF. In this way, a time-dependent influent gas-phase concentration was generated, with the transient concentration given by
[
( QtV)]
yin(t) ) yc 1 - exp -
(16)
The gas-phase effluent concentration and the evolving material-phase concentration profiles were predicted by eq 4 with yin(t) given by eq 16 and f (x) ) 0. During desorption, the VF acted as a source and the VOC supplied to the chamber was stopped. When a variable inlet concentration was desired, a clean stream of air was first passed through the empty chamber and then into the second chamber. The influent concentration in this case was given by
( QtV)
yin(t) ) yc exp -
(17)
The predicted gas-phase concentration was based on the final material-phase concentration from the sorption period serving as f (x) with the imposed effect of yin(t) given by eq 17. For the case of a constant influent concentration, a stream of clean dry air was passed directly through the chamber (yin(t) ) 0). A final consideration in the chamber experiments was the orientation of the VF. Previous experimental work (7) revealed a difference in the emissions rate from new VF when placed either top-up or bottom-up. The VF in the top-up orientation released VOCs at a slightly lower rate when compared to the rate for the bottom-up orientation. These findings were consistent with the VF material-phase concentration profiles measured at the completion of the
FIGURE 5. Absorption and desorption of n-dodecane starting with initial material-phase concentration of zero.
TABLE 1. Parameter Values Used for Model Predictions parameter
value
volume of chamber, V thickness of VF, L area of VF, A diffusion coefficient (n-dodecane) partition coefficient (n-dodecane) diffusion coefficient (phenol) partition coefficient (phenol)
0.0069 m3 0.0019 m 0.0128 m2 3.4 × 10-13 m2 s-1 17 000 1.2 × 10-13 m2 s-1 120 000
chamber tests. Experiments were therefore conducted in this study to see whether a similar effect would be observed during the absorption/desorption tests.
FIGURE 6. Absorption and desorption of phenol starting with initial material-phase concentration of zero. The predicted material-phase concentration during absorption is also shown.
Results and Discussion Table 1 provides the model parameter values used to predict the gas-phase chamber concentrations for the various experimental scenarios while Table 2 provides details of the series of experiments conducted. Top-up and bottom-up experiments were performed over a single absorption/ desorption cycle using either n-dodecane or phenol. A transient influent gas-phase concentration was investigated for phenol alone. In general, phenol induces a stronger response than dodecane because the material-air partition coefficient is higher. In Figures 5-7, the observed gas-phase concentration profiles are compared to those predicted by the model. In Figure 6, the evolution of the predicted materialphase concentration profile is shown for the absorption period. The final concentration profile at 1200 min is used as the initial condition for the desorption predictions. As shown in Table 2 and in Figures 5-7, the model predictions are in close agreement with the experimental results. Correlation coefficients varying between 0.71 and 0.98 were obtained for the entire series of tests. All the
FIGURE 7. Absorption and desorption of phenol starting with initial material-phase concentration of zero. A transient influent gas-phase concentration is used during absorption and desorption. predicted results are based on values of K and D that were determined in a completely independent series of experi-
TABLE 2. Experimental Conditions and Goodness of Fit for Model Predictions Compared to Experimental Dataa
a
compound
period
Q (m3 s-1)
yin(t) (mg m-3)
data points
R2
n-dodecane n-dodecane n-dodecane n-dodecane phenol phenol phenol phenol phenol phenol
absorption desorption absorption desorption absorption desorption absorption desorption absorption desorption
6.72 × 10-6 6.72 × 10-6 6.72 × 10-6 6.72 × 10-6 7.40 × 10-6 7.40 × 10-6 7.40 × 10-6 7.40 × 10-6 6.42 × 10-6 6.42 × 10-6
11.4 0 11.4 0 20.8 0 20.8 0 24 × (1 - exp(-0.0009t)) 24 × exp(-0.0009t)
161 161 161 161 81 98 81 98 282 109
0.98 0.71 0.97 0.79 0.91 0.94 0.87 0.95 0.88 0.94
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ments. The good match between the model and the observed data generally confirms the reliability of the approach for predicting the source/sink behavior of diffusion-controlled materials. The data shown in Figures 5 and 6 suggest that there is no significant difference between the top-up and the bottomup orientations. While it is not clear why this is the case, it may be the result of the 2-yr conditioning period in the laboratory. If some form of sealer was originally applied to create a more impermeable surface layer (7), this substance may have slowly degraded during the ensuing years. This study has provided further evidence in support of the proposed source/sink characterization strategy for diffusion controlled materials. There are, however, several model assumptions that need to be considered. The parameters K and D are assumed to be independent of concentration. Cox et al. (13) showed that this ideal “Fickian” behavior holds for a wide range of VOCs in vinyl flooring. In the experiments to measure K and D (13), the gas-phase concentrations were in the milligrams per cubic meter range. The K and D values measured at these relatively high concentrations were then successfully used to predict emissions from VF in a chamber with gas-phase concentrations in the micrograms per cubic meter range (7). This suggests, as would be theoretically expected, that the values for K and D remain independent of concentration at lower concentrations. The validity of the overall modeling approach is extended in the current series of experiments where the model successfully predicts gasphase concentrations in the milligrams per cubic meter range. A further implication of the ideal concentration-independent behavior is that the VOCs would not be expected to interact with one another if more than one VOC were being simultaneously absorbed or desorbed. Although only limited data were collected (13), this has been experimentally verified during the simultaneous sorption of two compounds, phenol and n-dodecane, in VF. Schwope et al. (17) suggested that the assumption of concentration independence is generally valid for polymeric materials where the material-phase concentration is below 1% by mass. If the emission rate from a diffusion-controlled source is to be predicted, the initial material-phase concentration must be known. Although it has been shown that the concentration profile of several VOCs in recently manufactured VF is initially uniform (7), this may not always be the case, especially if the material is not new or if the VOC has been introduced into the material in a nonuniform fashion. Thus, it may be necessary to measure the initial material-phase concentration profile. A method for doing this in vinyl flooring is available (7, 12), and a similar approach should work for most other polymeric materials. The resistance to mass transfer in the boundary layer between the bulk chamber air and the material surface is assumed to be negligible. For diffusion through a 1-cm layer of stagnant air and a gas-phase diffusion coefficient of 1 × 10-5 m2 s-1, a characteristic time of 5 s is obtained (10). Thus, the time taken to establish equilibrium at the material surface is fast relative to the time scale over which the gas-phase chamber concentration changes. A comprehensive analysis of external gas-phase resistance to mass transfer is provided by Haghighat and Zhang (18). Xu and Zhang (19) have developed a convenient analytical solution to a similar model that incorporates gas-phase resistance, although the solution is constrained to a uniform initial material-phase concentration and constant influent gas-phase concentration. Although the new analytical solutions to the single-layer model are more generally applicable than the previous versions, the primary purpose of this research was to further validate the overall source/sink characterization strategy. Clearly, there will be other situations where these particular solutions do not apply. With the increasing availability of 3826
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powerful numerical software, the same governing phenomena could be included in more complex scenarios.
Acknowledgments Financial support for this research was provided by the National Science Foundation (NSF) through an NSF CAREER Award (BES 9624488) and an NSF PATH Award (CMS 0122165). We thank Steve Cox for his valuable help with the experimental work and Al Hodgson for his insightful review of the manuscript.
Notation A
surface area of building material (m2)
C(x,t) concentration of VOC in material phase (mg m-3) C0
uniform initial concentration of VOC in material phase (mg m-3)
D
material-phase diffusion coefficient (m2 s-1)
f (x)
nonuniform initial concentration of VOC in material phase (mg m-3)
J(x,t)
inter-phase mass-transfer flux (mg m-2 s-1)
K
material-air partition coefficient (dimensionless)
L
thickness of building material (m)
Q
volumetric airflow rate through chamber (m3 s-1)
t
time (s)
V
volume of air in chamber (m3)
x
linear distance (m)
y(t)
gas-phase concentration of VOC inside the chamber (mg m-3)
yin(t)
gas-phase concentration of VOC entering the chamber (mg m-3)
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(11) Little, J. C.; Hodgson, A. T. In A Strategy for Characterizing Homogeneous, Diffusion-Controlled, Indoor Sources and Sinks; Tichenor, B. A., Ed.; ASTM STP 1287; American Society for Testing and Materials: Philadelphia, 1996; pp 294-304. (12) Cox, S. S.; Hodgson, A. T.; Little, J. C. Measuring concentrations of volatile organic compounds in vinyl flooring. J. Air Waste Manage. Assoc. 2001, 51, 1195-1201. (13) Cox, S. S.; Zhao, D. Y.; Little, J. C. Measuring partition and diffusion coefficients of volatile organic compounds in vinyl flooring. Atmos. Environ. 2001, 35, 3823-3830. (14) Hodgson, A. T.; Beal, D.; Chandra, S. Concentrations and sources of formaldehyde and volatile organic compounds in four new manufactured houses. In Proceedings of the 8th International Conference on Indoor Air Quality and Climate, Edinburgh, Scotland, August 8-13; Construction Research Communications Ltd.: London, 1999; Vol. 4, pp 119-124. (15) Lin, T. F.; Little, J. C.; Nazaroff, W. Transport and sorption of VOCs and water vapor within dry soil grains. Environ. Sci. Technol. 1994, 28, 322-330.
(16) Lin, T. F.; Little, J. C.; Nazaroff, W. Transport and sorption of organic gases in activated carbon. J. Environ. Eng. ASCE 1996, 122, 169-175. (17) Schwope, A. D.; Lyman, W. J.; Reid, R. C. Methods for Assessing Exposure to Chemical Substances; EPA Report Vol. 11, U.S. Environmental Protection Agency, Office of Toxic Substances: Washington, DC, 1989; EPA 560/5-85-015. (18) Haghighat, F.; Zhang, Y. Modeling of emissions of volatile organic compounds from building materialssestimation of gas-phase mass transfer coefficient. Build. Environ. 1999, 34, 377-389. (19) Xu, Y.; Zhang, Y. An improved mass transfer based model for analyzing VOC emissions from building materials. Atmos. Environ. 2003, 37, 2497-2505.
Received for review November 14, 2002. Revised manuscript received May 21, 2003. Accepted June 6, 2003. ES026332G
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