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Review of indoor emission source models. Part 2. Parameter estimation
Environmental Pollution 120 (2002) 551–564 www.elsevier.com/locate/envpol
Review of indoor emission source models. Part 2. Parameter estimation Zhishi Guo* US Environmental Protection Agency, Office of Research and Development, National Risk Management Research Laboratory, Mail Drop E-305-03, Research Triangle Park, NC 27711, USA Received 17 August 2001; accepted 9 March 2002
‘‘Capsule’’: Forty-eight methods for estimating the parameters in the indoor emission source models are compiled and reviewed. Abstract This review consists of two parts. Part 1 provides an overview of 52 indoor emission source models. Part 2—this paper—focuses on parameter estimation, a topic that is critical to modelers but has never been systematically discussed. A perfectly valid model may not be a useful one if some of its parameters are difficult to estimate in the absence of experimental data. This is true for both statistical and mass transfer models. Forty-eight methods are compiled and reviewed in this paper. Overall, developing methods for parameter estimation has fallen behind the development of models. Such imbalance is the main reason that many models have been left on the shelf since they were published. Published by Elsevier Science Ltd. Keywords: Parameter estimation; Model; Indoor air; Indoor source; Emission
1. Symbols and abbreviations
B0
The units of the symbols listed here are given in generic terms. If a parameter estimation method works only with a specific set of units, they are given in the text.
CT Cv
1.1. Symbols a a lumped parameter for computing Knudsen diffusivity (length2 time1 mass1/2 mol1/2), a0, a1, a2, a3, b0, b1, b2, b3, c 0, c 1, c2, c3 coefficients in methods P28 and P29 (dimensions vary) A, B, C, D, E coefficients in methods P16 to P21, P25 to P27 (dimensions vary)
d dm dw Da DKn DL Ds H k k1
k11 k12
* Tel.: +1-919-541-0185; fax: +1-919-541-2157. E-mail address: [email protected] (Z. Guo). 0269-7491/02/$ - see front matter Published by Elsevier Science Ltd. PII: S0269-7491(02)00188-4
proportionality coefficient in method P13 (mass1/2 pressure mol1/2 temperature3/2 time1) saturation concentration for TVOC (mass length3) saturation concentration for pure compound (mass length3) density of paint (mass length3) mean molecular diameter (length) diameter of water droplet (length) diffusivity in air (length2 time1) Knudsen diffusivity (length2 time1) diffusivity in water (length2 time1) diffusivity in solid (length2 time1) Henry’s constant (dimensionless) first-order decay rate constant (time1) first-order decay rate constant for wet-stage emissions with unit film thickness (length time1) first-order decay rate constant for wet-stage emissions from paint film (time1) first-order decay rate constant for dry-stage emissions from paint film (time1)
552
kD1
kg kL K KOL L m ma md m1 mi moth
mT Mr n n1, n2 P P0 Pi Pp PT q Re Sc t0.9 T u uc vA vAS vB vLB vd Wi WT y yi yoth yT Z
Z. Guo / Environmental Pollution 120 (2002) 551–564
first-order decay rate constant for dry-stage emissions with unit film thickness (length2 time1) gas-phase mass transfer coefficient (length time1) liquid-phase mass transfer coefficient (length time1) solid/air partition coefficient (dimensionless) overall liquid-phase mass transfer coefficient (length time1) characteristic length of a source (length) molecular weight (mass mol1) average molecular weight for air (mass mol1) mass of water droplet (mass) molecular weight for most abundant compound in solvent (mass mol1) molecular weight for compound i (mass mol1) average molecular weight for all VOCs in the solvent except the one in question (mass mol1) average molecular weight for TVOC (mass mol1) a parameter in methods P12 to P14, defined by Eq. 6 (mol mass1) carbon number index number (dimensionless) vapor pressure (pressure) atmospheric pressure (pressure) vapor pressure for compound i (pressure) partial pressure (pressure) total vapor pressure for TVOC (pressure) number of compounds quantified in a petroleum solvent Reynolds number (dimensionless) Schmidt number (dimensionless) 90% evaporation time (time) temperature (temperature), air velocity (length time1) velocity of water current (length time1) molar volume for air (length3 mol1) molar volume based on atomic and structural diffusion volume increments (length3 mol1) molar volume for the compound in question (length3 mol1) Le Bas molar volume (length3 mol1) velocity of water droplet (length time1) amount of compound i in paint film (mass) amount of TVOC in paint film (mass) VOC content in paint (fractional) content of compound i in paint (fractional) content of all VOCs in paint except the one in question (fractional) content of TVOC in paint (fractional) depth of water (length)
1.2. Greek proportionality constant in Eq. 15 (dimensionless), proportionality constant in method P48 (dimensionless) * proportionality constant in Eq. 15 (length1/3 time1/3), film thickness (length) viscosity of air (mass length1 time1) w viscosity of water (mass length1 time1) density of air (mass length3) surface tension of water (mass time2) AB characteristic length for diffusivity calculation used in method P13 (length) association factor for solvent (dimensionless), ! oscillation frequency for water droplet (time1) O collision integral used in method P13 (dimensionless) 1.3. Abbreviations IAQ indoor air quality PUF polyurethane foam SVOC semivolatile organic compound TVOC total volatile organic compound VOC volatile organic compound
2. Introduction Most indoor emission source models are given as a simple mathematical expression with one or more constants, known as model parameters. Loosely speaking, the mathematical expression determines the trend of the emission rate, and the parameters determine the magnitude. Since the emission source models are mainly used for predictions, the usefulness of a model is often judged by whether its parameters can be readily estimated in the absence of experimental data. Many reported model parameters are obtained by fitting a model to the experimental data. While this technique is useful in the early stage of model development, the results, in most cases, cannot be used for making predictions. Genuine predictive models are those whose parameters can be estimated independently. This paper compiles 48 published methods for estimating 15 model parameters, which represent only a fraction of the total number of parameters in the 52 models reviewed in Part 1. Some well-constructed models have not been widely used in exposure estimation simply because no methods are available to estimate some of the parameters. The methods reviewed here can be divided into two groups: relative and absolute methods. The former require the knowledge of a reference value for the same parameter, while the latter do not. Obviously, the absolute methods are much more useful than the relative ones.
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3. First-order decay rate constants
3.2. For petroleum-based indoor coating materials
3.1. Overview
Chang and Guo (1992) observed from chamber data that the decay rate constant for the fast, evaporative emissions (k11) is proportional to the vapor pressure of a compound (method P1), and that for the slow emissions from dry film (k12) is proportional to the Knudsen diffusivity (method P2), which is defined by Eq. 1 (Bennet and Myers, 1982).
The first-order decay models (see Table 1 in Part 1), especially models M1 to M3, are among the few indoor source models that are supported by a series of methods for parameter estimation. The simple first-order decay models contain two parameters: E0 and k for model M1; and M0 and k for model M2. The two models are equivalent because E0=M0k. For indoor coating materials, parameter M0 is usually determined from product formulation (e.g. VOC contents and product density) and application conditions (e.g. film thickness). Thus, the only remaining parameter to be estimated is the first-order decay rate constant. As shown in Tables 1 and 2, 11 methods are available for estimating k, k11, and k12. In general, the decay rate constants for evaporative emissions (k or k11) are a function of vapor pressure; that for the solid- phase diffusion-controlled emissions (k12) is a function of diffusivity in the solid or molecular weight. Dual first-order decay models (M3 and M4) have four parameters. In addition to estimating the decay rate constants, one has to determine how much of the solvent applied is available for evaporation and how much for the dry-stage emissions. Currently there are no methods available to estimate such partition.
a DKn ¼ pffiffiffiffi m
ð1Þ
Since the value of parameter a is often unknown, method P2 can be stated as: k12 is inversely proportional to the square root of the molecular weight. Both P1 and P2 are relative methods. Method P3, proposed by Evans (1994), is for estimating k for solvent emissions from coating materials based on the 90% drying time of the solvent (t0.9), which, in turn, is estimated by a method developed by Chinn (1981): log10 t0:9 ¼ 7:3698 0:95461og10 Cv
ð2Þ
where Cv
16040mP T
ð3Þ
Table 1 Methods for estimating the first-order decay rate constants for petroleum-based indoor coating materials Method ID
Formula
Description
Reference
Note
P1
k11 / P
VOC evaporation from wet film of wood stain
Chang and Guo (1992)
a
P2
k12 / DKn
VOC emissions from dry film of wood stain
Chang and Guo (1992)
a
P3
k¼
1n10 t0:9
TVOC evaporation from solvent-based paint; based on 90% drying time
Evans (1994)
b
P4
k¼
k g CT dyT
TVOC evaporation from petroleum-based coatings
Guo et al. (1999)
b,c
P5
k¼
kg Cv mT dyT m
VOC evaporation from petroleum-based coatings
Guo et al. (1999)
b,d
P6
k ¼ 2:95 109
VOC evaporation from alkyd paint
Koontz (2001)
b,e
P7
k¼
VOC evaporation from alkyd paint
Koontz (2001)
b,f
a b c d e f
P 0:27 2 r ¼ 0:86 m4:02 0:58
2 240Da P r ¼ 0:49 6:23 104 dyT
For model M3. For models M1 and M2. k (in h1); kg (in m/h); CT (in mg/m3); (in m); d (in g/m3); yT (in mg/g). k (in h1); kg (in m/h); Cv (in mg/m3); (in m); d (in g/m3); yT (in mg/g); mT and m (in g/mol). k (in h1); P (in mm Hg); m (in g mole1); (in mil). One mil=2.54105 m. k (in h1); Da in (m2/h); (P in mm Hg); d (in g/m3); yT (in mg/g).
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Table 2 Methods for estimating the first-order decay rate constants for water-based indoor coating materials Method ID P8 P9
Formula k1 k11 ¼ 2 k12 ¼ Ds 2
Description
Reference
Note
VOC and SVOC evaporation from wet film of latex paint
Clausen (1993)
a
VOC and SVOC emissions from dry film of latex paint
Clausen (1993)
a
P10a
k11 ¼ 233:25 P r2 ¼ 0:92
Glycol evaporation from latex paint
Wilkes et al. (1996)
a,b
P10b
k11 ¼ 264:7 P 4:961 r2 ¼ 0:96
Glycol evaporation from wet film of latex paint
Wilkes et al. (1996)
a,b
P11a
k12 ¼ 5:84 105 m r2 ¼ 0:96
Glycol emissions from dry film of latex paint
Wilkes et al. (1996)
a,c
P11b
k12 ¼ 7:08 105 m 0:002 r2 ¼ 0:995
Glycol emissions from dry film of latex paint
Wilkes et al. (1996)
a,c
a b c
For model M3. k11 (in h1); P (in mm Hg). k12 (in h1); m (in g mol1).
where t0.9 is in seconds, Cv in mg/m3, m in g/mol, P in torr, and T in K. Typically, this method is used for TVOC emissions if the solvent contains more than one compound. It may overestimate k for individual VOCs in a mixture. One should also be aware that the correlation (Eq. 2) was found with the 90% evaporation time being determined on paper filters with dry air flowing through (ASTM, 1977). Apparently, the experimental conditions (e.g. film thickness and ventilation rate) are far from realistic. Besides, this method ignores the effect of film thickness. Despite these limitations, it is very useful for rough estimation of k because all three parameters in Eq. 3 are easily obtained. Method P4 (Guo et al., 1999) is another method for estimating k for TVOC emissions from solvent-based indoor coatings. It is reduced from a mass transfer model (M10). The gas-phase mass transfer coefficient (kg) is estimated by method P39; the TVOC saturation concentration (CT) is converted from the total vapor pressure (PT), which is estimated by method P30. Other required parameters—film thickness (), paint density (d), and TVOC content in paint (yT)—are easy to determine. Due to the omission of the back pressure effect when this method was derived, it predicts a higher initial emission rate than its corresponding mass transfer model (M10). The total emittable amount remains the same, however. Fig. 1 compares the k values predicted by methods P3 and P4 for an imaginary solvent-based paint that has the following properties: TVOC content=25%, total vapor pressure=1.5 mm Hg, average molecular weight for TVOC=142 g/mol, and product density=1.3 g/cm3. It appears that the two methods are comparable only when the gas-phase mass transfer coefficient is around 5 m/h and the wet film thickness is between 60 and 80 mm. Three methods (P5–P7) are available for estimating the decay rate constant for individual VOCs. Method
P5 (Guo et al., 1999) is reduced from model M12. This method requires the average molecular weight for TVOC (mT), which can be approximated in two ways: if the user knows the contents of the major components in the solvent, use Eq. 4 (Guo et al., 1999); otherwise, use the molecular weight for the most abundant VOC in the solvent. mT ¼
q q X X yi = ðyi =mi Þ i¼1
ð4Þ
i¼1
Method P6 is for VOC emissions from alkyd paint, and is implemented in an IAQ simulation program for wall paint (Koontz, 2001). This correlation was established using the chamber data for about 20 VOCs commonly found in petroleum-based solvents. These compounds cover a vapor pressure range from 0.25 to 24 mm Hg (at room temperature) and a molecular weight range from 87 to 168 g/mol. The correlation is good. However, judging from the exponents in the formula (0.27 for vapor pressure and 4.02 for molecular weight), this method seems too sensitive to molecular weight and too insensitive to vapor pressure. Any extrapolation beyond the valid range for molecular weight should be avoided. Fig. 2 compares the k values predicted by methods P5 and P6 for evaporation of 1,2,3-trimethylbenzene from an imaginary alkyd paint, whose solvent has the same properties as described in the example for comparing methods P3 and P4. The two methods are close only when the gas-phase mass transfer coefficient is small (1 m/h). Method P7 (Koontz, 2001) was derived from method P5, simplified by setting several default values. The correlation for the alkyd paint data is not as good as method P6. Koontz (2001) did not implement P7 in his simulation program; it was used only for method comparison.
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Fig. 1. Comparison of the first-order decay rate constant for TVOC evaporation predicted by methods P3 and P4. Values in parentheses in the legend are gas-phase mass transfer coefficient in m/h.
3.3. For water-based indoor coating materials Methods P8 and P9 (Clausen, 1993) concern the dependence of the decay rate constants on the film thickness of the paint. Method P8—a relative method— requires knowledge of the decay rate constant with unity film thickness (k1). Method P9 states that, for a given paint/substrate combination, the decay rate constant for the dry emissions is proportional to the diffusivity of the compound in question and inversely proportional to the square of the film thickness. In the absence of data for the diffusivity (Ds), method P9 is used as a relative method (Eq. 5). k12 ¼
kD1 2
ð5Þ
where kD1=Ds(/2)2. Wilkes et al. (1996) found that the decay rate constant for the fast emissions from the wet film of latex paint (k11) is proportional to the vapor pressure of the compound (methods P10a and P10b), and that for the slow emissions from the dry film (k12) is proportional to the molecular weight of the compound (methods P11a and P11b). These correlations were developed based on the chamber data for four glycols in a latex paint. In both methods, adding a non-zero intercept (methods P10b and P11b) improves the correlation coefficient. These two methods work well for predicting the glycol emissions up to two weeks following paint application.
4. Diffusivity 4.1. Diffusivity in air Diffusivity in air is mainly used to estimate the gasphase or overall mass transfer coefficient. Three methods (P12–P14 in Table 3) are commonly used by IAQ
555
Fig. 2. Comparison of the first-order decay rate constant for 1,2,3trimethylbenzene evaporation predicted by methods P5 and P6. Values in parentheses in the legend are gas-phase mass transfer coefficient in m/h.
modelers. P12 and P13 are the two methods recommended by Lyman et al. (1990), who compiled and reviewed nine methods for estimating the diffusivity in air. Method P12 (Fuller et al., 1966)—known as the Fuller-Schettler-Giddings (FSG) method—is fairly accurate for non-polar compounds and less accurate for the polar compounds including glycols. Method P13 (Wilke and Lee, 1955)—known as the Wilke-Lee (WL) method—usually does a better job than the FSG method, but the computation is slightly more complex and requires knowledge of the boiling point. Both methods require calculations of the molar volume (vB) of the compound in question, which can be approximated by either the ‘‘atomic and structural diffusion volume increments (vAS)’’ or ‘‘Le Bas volumes (vLB)’’ (Tables 17–4 and 17–5, respectively, in Lyman et al., 1990). The former is simpler to use. For instance, the atomic diffusion volume increments for carbon and hydrogen are 16.5 and 1.98 cm3/mol, respectively; the structural diffusion volume increment for an aromatic ring is 20.2 cm3/mol. Thus, for decane (C10H22), vAS=1016.5+221.98=208.6 cm3/mol; and for benzene (C6H6), vAS=616.5+61.9820.2= 90.7 cm3/ mol. Le Bas volumes are additive-volume increments for estimating the molar volumes of liquids at the normal boiling points (Reid et al., 1977). This method takes into consideration more structural factors (e.g. bonding conditions and non-aromatic rings). However, in either case, no data are available for phosphorous-containing compounds. P14 is a much simpler method used by Koontz (2001). Molecular weight is the only information this method requires. The trade-off is its poor accuracy. As shown in Table 4, the results from this method are roughly 45% smaller than the WL method. This error will affect the accuracy of the gas mass transfer coefficient and, consequently, of the emission rate. In all the three methods, parameter Mr is calculated from Eq. 6.
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Table 3 Methods for estimating diffusivity in air and water Method ID
Formula pffiffiffiffiffiffiffi 103 T 1:75 Mr Da ¼ 2 1=3 P0 1=3 A þ B pffiffiffiffiffiffiffi B0 T 2=3 Mr Da ¼ 2 P0 AB O pffiffiffiffiffiffiffi Da ¼ m0:33 Mr
P12
P13 P14
Description
Reference
Note
Diffusivity in air (FSG method)
Fuller et al. (1966)
a
Diffusivity in air (WL method)
Wilke and Lee (1955)
b
Diffusivity in air
Koontz (2001)
c
Diffusivity in water
Hayduk and Laudie (1974)
d
5
P15 a b c d
DL ¼
13:26 10 0:589 1:14 w LB
Da (in cm2/s); T (in K); Mr (in mole/g); P0 (in atm); vA and vB (in cm3/mol). ˚ Da (in cm2/s); B0 =0.02170.0005 M1/2 r ; T (in K); Mr (in mol/g); P0 (in atm); AB (in A); is dimensionless. Da (in cm2/s); m (in g/mol); Mr (in mol/g). DL (in cm2/s); mw (in centipoise); vLB (in cm3/mol).
Table 4 Comparison of three methods for estimating the diffusivity in air at 298 K and 1 atm (units: cm2/s) Compound
n-decane m-chlorotoluene
Mr ¼
Formula
C10 H22 C7 H7 Cl
ma þ m ma m
Method P12
P13
P14
0.0583 0.0730
0.0714 0.0758
0.0397 0.0417
ð6Þ
4.2. Diffusivity in water Diffusivity in water is used to estimate the liquidphase and overall mass transfer coefficient. Lyman et al. (1990) compiled six available methods for estimating the diffusivity in water, and they recommended method P15 (Hayduk and Laudie, 1974). This method requires only two parameters: the viscosity for water (0.8904 centipoise at 25 C) and the Le Bas volume of the solute. 4.3. Diffusivity in solids All the mass transfer models for VOC emissions from building materials (i.e. models M23–M31 in Part 1) require knowledge of the diffusivity of the VOC in the material (Ds). Although development of methods for estimating this parameter has been in active study for a decade, most progress was made very recently (see Table 5). Method P16 (Berens and Hopfenberg, 1982) correlates the diffusivity to the mean molecular diameter (dm) of the penetrant molecule. The data used by the authors included three glassy materials (polyvinyl chloride, polystyrene, and polymethyl methacrylate) and more than 20 compounds. Since the figures of dm are not
available for some VOCs studied, the authors proposed three methods for estimating dm. The authors also tried to correlate Ds to the square of dm for some data. With the scattering of data, however, the authors could not decide which correlation is better. Zhao et al. (1999) found that, for a polyurathane foam (PUF), there was a linear correlation (R2=0.94) between diffusivity and vapor pressure on a log–log scale (method P17). The data used included water and eight aromatic hydrocarbons. The correlation becomes less satisfactory when this method is applied to the data reported by other authors. Both methods P18 (Bodalal et al., 2001) and P19 (Cox et al., 2001) correlate diffusivity to molecular weight. Mathematically, the two methods are identical (Table 5). This author deliberately put them in different forms to emphasize their difference. They differ in how to obtain the correlation constants (A and n1). Method P18 applies non-linear regression to data on normal scale, while method P19 uses linear regression with data on log–log scale. This difference makes the R2 values not directly comparable between the two methods. In general, method P18 gives more weight to the data points with large Ds values; method 19 is less discriminative against smaller Ds values. Table 6 summarizes the reported correlation constants for different material/ compound group combinations. Description of the test materials by Bodalal et al. (2001) was somewhat cursory. Further studies are needed to investigate the variations of these two parameters among the same type of materials. The weakness of methods P18 and P19 is that, for a given material, each compound class has its own correlation constants (e.g. A and n1), which cannot be used to predict the diffusivity of a chemical in a different compound class. To resolve this problem, this author suggests to express the diffusivity as a function of the molar volume of the VOC (methods P20 and P21). After all, it is the volume, not the weight, of the molecule, that really matters to solid-phase diffusion. Methods P20
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Z. Guo / Environmental Pollution 120 (2002) 551–564 Table 5 Methods for estimating diffusivity in solids Method ID Formula
Description
P16
1nDs ¼ A Bdm
Ds as a function of penetration diameter; for mixed compound classes Berens and Hopfengerg (1982)
P17
1nDs ¼ A B1nP
Ds as a function of vapor pressure; for mixed compound classes
Zhao et al. (1999)
P18
Ds ¼
Ds as a function of molecular weight; for a specific compound class
Bodalal et al (2001)
a
P19
1nDs ¼ 1nA n1 1nm Ds as a function of molecular weight; for a specific compound class
Cox et al. (2001)
a
P20
1nDs ¼ A B1n AS
Ds as a function of molar volume; for mixed compound classes
Guo (in preparation)
b
P21
1nDs ¼ C D1n LB
Ds as a function of Le Bas volume; for mixed compound classes
Guo (in preparation)
c
a b c
A mn1
Reference
Note
Ds (in m2/s); m (in g/mol). Ds (in m2/s); AS (in cm3/mol). Ds (in m2/s); LS (in cm3/mol).
Table 6 Correlation constants for diffusivity in solids (methods P18 and P19) Building material
Compound group
A
n1
R2
Note
Gypsum board Oriented strand board Particle board Particle board Plywood Plywood Vinyl tile Vinyl tile Vinyl flooring
Aromatics Alkanes Aromatics Aldehydes Aromatics Aldehydes Alkanes Aromatics Alkanes
4.481101 1.450107 1.692107 3.396108 1.337107 7.900104 2.104101 3.617102 5.006105
5.99 8.36 8.5 9.33 8.56 3.64 5.94 6.51 7.45
0.9816 0.9866 0.9618 0.8328 0.9969 0.8802 0.9864 0.9789 0.9973
a,b
a b c d e f
a,c a,b a,d a,b a,d a,c a,b e,f
Data source: Bodalal et al. (2001). Included benzene, toluene, ethylbenzene, propylbenzene, and butylbenzene. Included heptane, octane, nonane, decane, and undecane. Included pentanal, haxanal, heptanal, and octanal. Data source: Cox et al. (2001). Included decane, dodecane, tetradecane, and pentadecane.
and P21 differ in how the molar volume is estimated. Method P20 uses the atomic and structural diffusion volume increments (Fuller et al., 1966) while method P21 uses the Le Bas volumes (Reid et al., 1977). Table 7 shows the correlation constants for these two methods using the diffusivity data reported by Bodalal et al. (2001) and Cox et al. (2001). Overall, the correlations are satisfactory except for the vinyl flooring. A major advantage of methods P20 and P21 is that the constants (Table 7) are independent of compound classes. Of course, more data are needed to further evaluate these methods and to refine the correlation constants. The strength of the bonding force between the VOC and the substrate molecules may affect the diffusivity. It is well known that the glycol (polar compounds) emissions from painted gypsum board (polar substrate) are very slow (Chang et al., 1997). As a future research topic, this author suggests to investigate such effects by
experimental determination of enthalpy changes, from which the bounding force can be estimated.
5. Solid/air partition coefficient Solid/air partition coefficient (K ) is a parameter required by models M25–M31. As shown in Table 8, both methods P22 (Zhao et al., 1999; Cox et al., 2001) and P23 (Bodalal et al., 2001) correlate K to the vapor pressure of the VOC. The two methods differ slightly in how the constants are determined (see Section 4.3). Table 9 lists the values for B and n2 reported by Bodalal et al. (2001), Cox et al. (2001), Zhao et al. (1999). Note that each building material has its own constants. In an effort to develop a correlation independent of compound classes and material types, this author found that there exists a fair correlation (Fig. 3) between
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Table 7 Correlation constants for methods P20 and P11 Building material
Method P20
Gypsum board Oriented strand board Particle board Plywood Vinyl tile Vinyl flooring a b c d e f g
Method P21
Note
A
B
R2
C
D
R2
1.536 19.73 10.59 0.6787 3.546 24.15
4.674 8.401 6.970 4.720 4.286 0.9349
0.997 0.973 0.785 0.815 0.917 0.689
1.652 21.45 14.09 0.0644 4.445 23.58
4.589 8.533 7.575 4.780 4.043 1.023
0.997 0.974 0.864 0.847 0.895 0.665
a,b a,c a,d a,d a,e f,g
Diffisivity data from Bodalal et al. (2001). Based on a single class of compounds (5 aromatics). Based on a single class of compounds (5 alkanes). Based on 3 classes of compounds (5 aromatics, 4 aldehydes, and limonene). Based on 2 classes of compounds (5 alkanes and 5 aromatics). Diffisivity data from Cox et al. (2001). Based on 5 classes of compounds (4 alkanes, water, butanol, toluene, and phenol).
Table 8 Methods for estimating solid/air partition coefficients Method ID
Formula
Description
Reference
Note
P22
1nK ¼ 1nB n2 1nP
K as a function of vapor pressure; for a specific material and mixed compound classed
Zhao et al. (1999)
a
P23
K¼
K as a function of vapor pressure; for a specific material and compound class
Bodalal et al. (2001)
a
P24
1nK ¼ 8:76 0:7851nP
K as a function of vapor pressure; for all materials and compound classes
Guo (in preparation)
a
a
B P n2
K is dimensionless; P (in mm Hg).
Table 9 Correlation constants for methods P22 and P23 Building material Gypsum board Oriented strand board Particle board Particle board Plywood Plywood Vinyl tile Vinyl tile Vinyl flooring PUF a b c d e f g h
Compound group Aromatics Alkanes Aromatics Aldehydes Aromatics Aldehydes Alkanes Aromatics Alkanes Aromatics
B 4
1.06910 1.073104 1.305104 1.037104 1.108104 6.581103 2.032104 1.111104 2.566103 1.238103
n2
R2
Note
0.772 0.886 0.892 0.51 0.944 0.644 1.051 0.863 0.772 0.921
0.9853 0.9895 0.9834 0.9466 0.9801 0.9412 0.9999 0.9933 0.9986 0.9624
a,b
Source: Bodalal et al. (2001). Compounds included benzene, toluene, ethylbenzene, propylbenzene, and butylbenzene. Compounds included heptane, octane, nonane, decane, and undecane. Compounds included pentanal, haxanal. heptanal, and octanal. Source: Cox et al. (2001). Compounds included decane, dodecane, tetradecane, and pentadecane. Source: Zhao et al. (1999). Compounds included: benezone, toluene, p-xylene, ethylbenzene, chlorobenzene, 1,2,4-trimethylbenzene, styrene, and naphthalene.
a,c a,b a,d a,b a,d a,c a,b e,f g,h
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559
ture range. Two methods (P28 and P29) developed by Makar (2000) allow the user to estimate the vapor pressure for an organic compound by knowing (1) its compound class, (2) the carbon number in the molecule, and (3) the temperature of interest. In doing so, Makar divided the organic compounds into 39 classes (e.g. nalcohols, n-alkanes, and polyaromatics) and established two correlations for each compound class based on available experimental data: Eq. 7 for method P28 and Eq. 8 for P29. log10 P ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 Fig. 3. Comparison of gas-phase mass transfer coefficient for n-decane estimated with six absolute methods, assuming that the air velocity is 10 cm/s and the source area is 1 m2.
partition coefficients and vapor pressures for all 56 material/compound combinations reported by Bodalal et al. (2001), Cox et al. (2001), and Zhao et al. (1999). Thus, method P24 can be used to roughly estimate the partition coefficients for all material/compound combinations.
6. Vapor pressure, partial pressure, and total vapor pressure 6.1. Vapor pressure as a function of temperature Vapor pressure, partial pressure and total vapor pressure are key parameters in modeling VOC emissions from liquids and solids. Commonly used methods for estimating these parameters are listed in Table 10. Vapor pressure is an important parameter in many indoor source models and parameter estimation methods. Many compounds have experimentally determined vapor pressure, which can be found in the literature. Frequently cited references for vapor pressure include Yaws (1994), Boublik et al. (1984), and Schlessinger (1972). Since vapor pressure is strongly affected by temperature, experimental data are often presented as a certain temperature function, whose coefficients are given in a large table. Methods P25 (Schlessinger, 1972), P26 (Boublik et al., 1984), and P27 (Yaws, 1994) are three commonly used temperature functions. One should be aware that the coefficients in these three methods (e.g. A and B in P25) are valid only in a certain temperature range. Extrapolation beyond the valid range is not recommended. 6.2. Estimation of vapor pressure in the absence of experimental data Not all compounds have experimentally determined vapor pressure. For some compounds, the experimental data in the literature may not cover the room tempera-
þ n b0 þ b1 T þ b2 T 2 þ b 3 T 3
ð7Þ
log10 P ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 þ n b0 þ b1 T þ b2 T 2 þ b 3 T 3 þ n2 c 0 þ c 1 T þ c 2 T 2 þ c 3 T 3
ð8Þ
where P is in mm Hg and Tin K. The coefficients in Eqs. 7 and 8 are given in two large tables. In general, the correlations are good. These two methods are very useful in the absence of experimental data at room temperature. It is the user’s responsibility, however, to determine the class to which a given compound belongs. In other words, the user must know the molecular structure of the compound. This author prefers method P29 over P28 because its coefficient table contains the valid range for the carbon number. 6.3. Total vapor pressure The total vapor pressure for TVOC is the sum of the partial pressures for all compounds in the solvent mixture. It is required by models M10–M12 and method P4. For a petroleum- based solvent, which may contain hundreds of compounds, summing up all partial pressures is difficult. Guo et al. (1999) proposed to approximate the total vapor pressure from the imaginary solvent consisting of the known major components in the mixture (method P30). Knowledge of the contents of about a dozen major components in the solvent is usually sufficient for a reasonable estimate of the total vapor pressure. 6.4. Partial pressure Partial pressure is needed to predict the emissions of individual VOCs from products that contain a solvent mixture. For ideal and non-ideal solutions, the partial pressure is determined by Eqs. 9 and 10, respectively. Pp ¼ xL P
ð9Þ
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Table 10 Methods for estimating vapor pressure, partial pressure, and total vapor pressure Method ID
Formula
Description
Reference
Note
P25
log10 P ¼
Vapor pressure as a function of temperature
Schlessinger (1972)
a
P26
log10 P ¼ A
B TþC
Vapor pressure as a function of temperature
Boublik et al. (1984)
a
P27
log10 P ¼ A þ
B þ Clog10 T þ DT þ ET 2 T
Vapor pressure as a function of temperature
Yaws (1994)
a
P28
See text
Vapor pressure by compound class
Makar (2000)
P29
See text
Vapor pressure by compound class
Makar (2000)
P30
PT ¼
Pq Pi yi Xq yi = i¼1 i¼1 mi mi
Total vapor pressure for TVOCS
Guo et al. (1999)
P31
Pp ¼
Wi mT Pi mi WT
Partial pressure for individual vocs in paint film
Guo et al. (1999)
P32
Pp ¼
Wi m1 Pi mi WT
Partial pressure for individual vocs in paint film
Guo (2000)
P33
Pp ¼
yi =mi Pi yi =mi þ yoth =moth
Partial pressure for individual vocs in paint film
van Veen et al. (1999)
a
0:218A þB T
P (in mm Hg); T (in K).
Pp ¼ xL P
ð10Þ
Methods for estimating the activity coefficient have been developed. A good introduction is given by Lyman et al. (1990). Eq. 10 has not been widely used by IAQ modelers, however. P31 and P32 (Guo et al., 1999) are two approximate methods for estimating the partial pressure for an individual VOC in petroleum-based solvents. Both methods are based on the contents of the major components, and they differ only in how the average molecular weight is calculated. Method P31 uses Eq. 4 to estimate the average molecular weight for TVOC. In method P32, it is represented by the molecular weight of the most abundant component in the solvent. Method P33 (van Veen et al., 1999) treats the solvent as a bicomponent system, the compound in question and the rest of compounds. The authors did not specify how to calculate the average molecular weight (moth), however. All three methods were developed for petroleum solvent-based paint. Methods P31 and P32 have also been applied to solvent spills (Guo, 2000).
7. Mass transfer coefficients 7.1. Gas-phase mass transfer coefficient Seven methods have been used to estimate the gasphase mass transfer coefficient in the indoor environ-
ment (Table 11). P34 (Bennett and Myers, 1962) is the conventional method favored by chemical engineers. This method is based on correlations between three dimensionless numbers: Sherwood number (Sh), Schmidt number (Sc) and Reynolds number (Re), with the latter two being defined by Eqs. 11 and 12, respectively. Lu Sc ¼ Da Re ¼
ð11Þ ð12Þ
A series of correlations have been developed between Sh, Sc and Re (Bennett and Myers, 1962; White, 1991). For example, Eq. 13 applies to laminar flow conditions. Sh ¼ 0:664Sc1=3 Re1=2
ð13Þ
While P34 remains the method of choice, several simpler models have been used by IAQ modelers to estimate the gas-phase mass transfer coefficient. Methods P35 (Higbie, 1945), P36 (Mackay and Matsugu, 1973), P37 (Geankoplis, 1993), and P38 (Jayjock, 1994) were borrowed from studies for the ambient environment, while P39 (Sparks et al., 1996) and P40 (Haghighat and Zhang, 1998) were developed exclusively for the indoor environment. Method P36 consists of two equations. The one in Table 11 is for Reynolds numbers smaller than 15 000—the condition most likely to be found in the indoor environment. Fig. 4 compares the mass
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Z. Guo / Environmental Pollution 120 (2002) 551–564 Table 11 Methods for estimating of the gas-phase mass transfer coefficient Method ID Formula kg L Da
Description
Reference
Note
Conventional method based on Sherwood number
Bennett and Myers (1962), Sparks et al. (1996)
a
Used by model M34
Higbie (1945), Reinke and Brosseau (1997)
a
P34
Sh ¼
P35
kg ¼ 2
P36
kg ¼ 17:35u0:78 L 0:11 Sc0:67 Used by model M35
Mackay and Matsugu (1973), Reinke and Brosseau (1997)
a,b
P37
kg ¼ 0:664Re1=2 uSc2=3
For solvent spill; model not included in Part 1
Geankoplis (1993), Reinke and Brosseau (1997)
a
P38
kg ¼ 0:5
Used by model M32
Jayjock (1994), van Veen et al. (1999)
c
P39
kg ¼ 0:33Da L 3
P40
See text
a b c d
rffiffiffiffiffiffiffiffiffi Da u L
rffiffiffiffiffi 3 18 m 1
23 u
Used by models M10, M11, M17, M37, and M38 Sparks et al. (1996) See text
d
Haghighat and Zhang (1998)
The first reference is for the original method development and the second for indoor application. kg and u (in m/s); L in (m); Sc is dimensionless. kg (in m/min); m (in g/mol). kg (in m/h); Da (in m2/h); L (in m); u (in m/h); (in g/m3); (in g/m/h).
transfer coefficient for n-decane calculated from methods P34 to P39. Overall, the difference between these methods is fairly large. In particular, the two simpler methods (P35 and P38) are significantly different from the other four methods. Further evaluation is needed to determine which methods perform better under realistic indoor conditions. Developed by Haghighat and Yang (1998), P40 is a more refined method than all the methods discussed earlier. It takes into consideration such factors as surface roughness, which are ignored by other methods. This author was unable to compare P40 with other methods because of insufficient information in the original paper. A potential weakness of this method is that one of its coefficients may have to be determined experimentally. More work is needed to make this method practically useful. 7.2. Liquid-phase mass transfer coefficient The liquid-phase mass transfer coefficient can be used alone to predict the VOC emissions from water and aqueous solutions if the Henry’s constant is fairly large. It is also required to calculate the overall mass transfer coefficient. Method P41 (Little, 1992a; Moya et al., 1999) in Table 12 is a relative method. The value of n2 varies from 0.5 to 1 and a value of 0.67 is often applied. P42 (Lyman et al., 1990) and P43 (Southworth, 1979) are two absolute methods borrowed from the studies in the ambient environment. Their usefulness in the indoor environment has not been evaluated. There is an
Fig. 4. Correlation between solid/air partition coefficients and vapor pressure for all material/compound combinations (n=56, R2=0.734).
obvious flaw in method P43: it becomes zero if the water body is still (uc=0). Thus, this method should be used with caution. Methods P44 (Angelo et al., 1966), P45 (Handlos and Baron, 1957), and P46 (Ruckenstein, 1967) are used for estimating the liquid-phase mass transfer coefficient for VOC emissions from water drops. Giardino et al. (1992) applied these methods to domestic shower spray and tested them against the experimental data for volatilization of trichloroethylene. The authors found that methods P44 and P45 fit the data well and that method P44 is more convenient to use because it does not require the velocity of water drops. The oscillation frequency of water drop (!) in method P44 is calculated from Eq. 14.
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Table 12 Methods for estimating of the liquid-phase mass transfer coefficient Method ID
Formula
Description
Reference
P41
kL / DLn2
kL As a function of diffusivity
Little (1992a), Moya et al. (1999)
P42
kL ¼ 20
For VOC from ambient water bodies
Lyman (1990)
a
P43
kL ¼ 23:51
For VOC from ambient water bodies
Southworth (1979)
b
For VOC from water droplets
Angelo et al. (1966), Giardino et al. (1992)
c,d
For VOC from water droplets
Handlos and Baron (1957), Giardino et al. (1992)
c,e
For VOC from water droplets
Ruckenstein (1967), Giardino et al. (1992)
c,f
rffiffiffiffiffi 44 m
u0:969 c Z 0:673 rffiffiffiffiffiffiffiffiffiffi !DL kL ¼ 2
P44
0:00375 d 1 þ w = rffiffiffiffiffiffiffiffiffiffiffi d DL kL ¼ 2 dw
P45
kL ¼
P46 a b c d e f
rffiffiffiffiffi 32 m
Note
kL (in cm/h); m (in g/mol). kL (in cm/h); uc (in m/s); Z (in m); m (in g/mol). The first reference is for the original method development and the second for indoor application. kL (in cm/s); ! (in s1); DL (in cm2/s). kL (in cm/s); d (in cm/s); w and (in cm2/s). kL (in cm/s); d (in cm/s); DL (in cm2/s); dw (in cm).
Table 13 Methods for estimating of the overall mass transfer coefficient Method ID
Formula
P47
1 DLi KOLi DLr
P48
KOL
a b
n2
1 1 Dai þ kLr Hi kgr Dar !1 2:5 1 ¼ þ DL2=3 D2=3 a H ¼
n1
DLi DLr
Description
Reference
Note
A relative method for estimating KOL
Little (1992a)
a
McKone method for estimating KOL
McKone (1987)
b
n2
Subscripts i and r represent the compound of interest and the reference compound, respectively. KOL (in m/s); and H are dimensionless; DL and Da (in m2/s).
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8=ð3md Þ !¼ 2
ð14Þ
P48 (Eq. 15) and suggested that should be approximately 70. He pointed out that method P48, in which is set to 1, may underestimate the effect of gas-phase resistance.
7.3. Overall mass transfer coefficient The overall mass transfer coefficient is given as either the gas- or liquid-phase mass transfer coefficient and is used in models for emissions from tap water and waterbased products [see Eq. (15) in Part 1]. Due to the lack of experimental data, accurate prediction of this parameter has been difficult. P47 (Little, 1992a) in Table 13 is a fairly complex relative method, in which the two indices (n1 and n2) may vary. P48 (McKone, 1987) is an approximate, absolute method based on data for radon emissions from tap water that are adjusted to VOCs. Little (1992b) proposed an improved version of method
KOL ¼
2:5 DL2=3
þ
Da2=3 H
!1 ð15Þ
In both method P48 and Eq. 15, the proportionality constants ( and *) are independent of the compounds in question, but depend on hydrodynamic conditions. This method is fairly simple, but need more experimental data to determine the values of the proportionality constants under different physical conditions.
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8. Parameters for fugacity models Fugacity models (see Table 9 in Part 1) utilize a special set of parameters that are not used by other indoor source models. Examples of such parameters are: fugacity capacity, transfer parameter, and time-varying compartment volume for building materials. The reader is referred to the original papers (see references in Part 1) for more details.
9. Conclusion and discussion Forty-eight methods for parameter estimation were compiled and reviewed. Eleven of them are for estimating the decay rate constants in the first-order decay models. The other 37 methods are for estimating diffusivity, partition coefficient, vapor pressure/volatility, and mass transfer coefficient. Methods for estimating diffusivity in air and water are well established. It is more difficult to estimate diffusivity in solids because it is dependent on the properties of the material. Progress has recently been made to estimate this parameter. Several methods have also been proposed to estimate the solid/air partition coefficient, which is required by seven mass transfer models for building materials (M25–M31). Vapor pressure is usually found in the literature, but the data may not be valid at room temperature. In the absence of experimental data, methods P28 and P29 are useful. Seven absolute methods are available for estimating the gas-phase mass transfer coefficient. However, significant difference exists between them. The two simplest methods (P35 and P38) seem less accurate than the rest and should be avoided. Further evaluation of these methods is needed. Methods for estimating the liquid-phase mass transfer coefficient have not been adequately evaluated. Currently there are no reliable methods for estimating the overall mass transfer coefficient, which is a critical parameter for VOC emissions from water and aqueous solutions. It is a common misconception to think that parameter estimation is a problem only for statistical models. In fact, many mass transfer models contain one or more parameters that are difficult to estimate. Overall, developing of methods for estimating the model parameters has not progressed to the extent that IAQ modelers would like to see. This is evidenced by the fact that many models discussed in Part 1 are not supported by any parameter estimation methods. Current practice in developing indoor source models often involves two steps: model development and validation. While these steps are essential, a third step—finding ways to estimate the model parameters—should follow. Otherwise, the usefulness of the model will be limited.
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References Angelo, J.B., Lightfoot, E.N., Howard, D.W., 1966. Mass and heat transfer from drops in liquid–liquid extraction. American Institute of Chemical Engineers Journal 12, 751–760. ASTM, 1977. Standard methods for evaporation rates of volatile liquids. In: Annual Book of ASTM Standards, Part 27. American Society of Testing and Materials, Philadelphia, PA, pp. 716–726. Bennett, C.O., Myers, J.E., 1982. Momentum, Heat, and Mass Transfer, third ed. McGraw-Hill, New York, NY. Berens, A.R., Hopfenburg, H.B., 1982. Diffusion of organic vapors at low concentrations in glassy PVC, Polystyrene, and PMMA. Journal of Membrane Science 10, 283–303. Bodalal, A., Zhang, J.S., Plett, E.G., Shaw, C.Y., 2001. Correlations between the internal diffusion and equilibrium partition coefficients of volatile organic compounds (VOCs) in building materials and the VOC properties. ASHRAE Transactions 107, 789–800. Boublik, T., Fried, V., Ha´la, E., 1984. The Vapor Pressures of Pure Substances, second ed. Elsevier, New York, NY. Chang, J.C.S., Guo, Z., 1992. Characterization of organic emissions from a wood finishing product—wood stain. Indoor Air 2, 146– 153. Chang, J.C.S., Tichenor, B.A., Guo, Z., Krebs, K., 1997. Substrate effects on VOC emissions from latex paint. Indoor Air 7, 241–247. Chinn, K.S.K., 1981. A Simple Method for Predicting Chemical Agent Evaporation. US Army Dugway Proving Ground, Dugway, UT (DPG-TR-401). Clausen, P.A., 1993. Emission of volatile and semivolatile organic compounds from waterborne paints—the effect of the film thickness. Indoor Air 3, 269–275. Cox, S.S., Zhao, D., Little, J.C., 2001. Measuring partition and diffusion coefficients for volatile organic compounds in vinyl flooring. Atmospheric Environment 35, 3823–3830. Evans, W.C., 1994. Development of continuous-application source terms and analytical solution for one- and two-compartment systems. In: Tichenor, B.A. (Ed.), Characterizing Sources of Indoor Air Pollution and Related Sink Effects, ASTM STP 1287. American Society of Testing and Materials, Philadelphia, pp. 279–293. Fuller, E.N., Schettler, P.D., Giddings, J.C., 1966. A new method for prediction of binary gas-phase diffusion coefficients. Industrial and Engineering Chemistry 58, 19–27. (Cited by Lyman et al., 1990). Geankoplis, C.J., 1993. Transport Processes and Unit Operations, third ed. PTR Prentice Hall, Englewood, Cliffs, NJ. Giardino, N.J., Esmen, N.A., Andelman, J.B., 1992. Modeling volatilization of trichloroethylene from a domestic shower spray: the role of drop-size distribution. Environmental Science and Technology 26, 1602–1606. Guo, Z., Chang, J.C.S., Sparks, L.E., Fortmann, R.C., 1999. Estimation of the rate of VOC emissions from solvent-based indoor coating materials based on product formulation. Atmospheric Environment 33, 1205–1215. Guo, Z., 2000. Simulation Tool Kit for Indoor Air Quality and Inhalation Exposure (IAQX) Version 1.0 User’s Guide. U.S. Environmental Protection Agency, National Risk Management Research Laboratory, Research Triangle Park, NC. (Publication No. EPA600/R-00–094 NTIS PB2001–101221). Guo, Z. Review of indoor emission source models—Part 1. Overview (in preparation). Handlos, A.E., Baron, T., 1957. Generalization of the penetration theory for surface stretch: application to forming and oscillating drops. American Institute of Chemical Engineers Journal 3, 127–136. Hayduk, W., Laudie, H., 1974. Prediction of diffusion coefficient for non-electrolysis in dilute aqueous solution. American Institute of Chemical Engineers Journal 20, 611–615. (Cited by Lyman et al., 1990).
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Higbie, R., 1945. The rate of absorption of a pure gas into a still liquid during short periods of exposure. Transactions of American Institute of Chemical Engineers 31, 365–389. (Cited by Reinke and Brosseau, 1997). Haghighat, F., Zhang, Y., 1998. Modeling of emission of volatile organic compounds from building materials—estimation of gasphase mass transfer coefficient. Building and Environment 34, 377–389. Jayjock, M.A., 1994. Back pressure modeling of indoor air concentrations from volatilizing sources. American Industrial Hygiene Association Journal 55, 230–235. Koontz, M., 2001. Wall Paint Exposure Model (WPEM) Version 3.2 User’s Guide, developed by Geomet Technologies, Inc., Germantown, MD, for US EPA Office of Pollution Prevention and Toxics, Washington, DC, and National Paint and Coatings Association, Washington, DC. http://www.epa.gov/opptintr/exposure/ docs/wpem.htm (accessed on 20 March 2001). Little, J.C., 1992a. Applying the two-resistance theory to contaminant volatilization in showers. Environmental Science and Technology 26, 1341–1349. Little, J.C., 1992b. Comment on ‘‘Human exposure to volatile organic compounds in household tap water: the indoor inhalation pathway.’’. Environmental Science and Technology 26, 836–837. Lyman, W.L., Reehl, W.F., Rosenblatt, D.H., 1990. Handbook of Chemical Property Estimation Methods: Environmental Behavior of Organic Compounds. American Chemical Society, Washington, DC. Mackay, D., Matsugu, R.S., 1973. Evaporation rates of liquid hydrocarbon spills on land and water. Canadian Journal of Chemical Engineering 51, 434–439. Makar, P.A., 2000. The estimation of organic gas vapor pressure. Atmospheric Environment 35, 961–974. McKone, T.E., 1987. Human exposure to volatile organic compounds in household tap water: the indoor inhalation pathway. Environmental Science and Technology 21, 1194–1201. Moya, J., Howard-Reed, C., Corsi, R.L., 1999. Volatilization of chemicals from tap water to indoor air from contaminated water used for showering. Environmental Science and Technology 33, 2321–2327.
Reid, R.C., Prausnitz, J.M., Sherwood, T.K., 1977. The Properties of Gases and Liquids, third ed. McGraw-Hill, New York, NY. Reinke, P.H., Brosseau, L.M., 1997. Development of a model to predict air contaminant concentrations following indoor spills of volatile liquids. Annals of Occupational Hygiene 41, 415–435. Ruckenstein, E., 1967. Mass transfer between a single drop and a continuous phase. International Journal of Heat and Mass Transfer 10, 1785–1792. Schlessinger, G.G., 1973. Vapor pressure, critical temperature, and critical pressures of organic compounds. In: Weast, R.C. (Ed.), Handbook of Chemistry and Physics, fifty-third ed. The Chemical Rubber Co, Cleveland, OH, pp. D151–D170. Southworth, G.R., 1979. The role of volatilization in removing polyaromatic hydrocarbons from aquatic environments. Bulletin of Environmental Contamination and Toxicology 21, 507–514. (Cited by Lyman et al., 1990). Sparks, L., Tichenor, B., Chang, J., Guo, Z., 1996. Gas-phase mass transfer model for predicting volatile organic compound (VOC) emission rates from indoor pollutant sources. Indoor Air 6, 31–40. van Veen, M.P., Fortezza, F., Bloemen, H.J.Th., Kliest, J.J., 1999. Indoor air exposure to volatile compounds emitted by paints: experiment and model. Journal of Exposure Analysis and Environmental Epidemiology 9, 569–574. White, F.M., 1991. Heat and Mass Transfer. Addison-Wesley, New York, NY, USA. Wilke, C.R., Lee, C.Y., 1955. Estimation of diffusion coefficients for gases and vapors. Industrial Engineering Chemistry 47, 1253–1257. (Cited by Lyman et al., 1990). Wilkes, C., Koontz, M., Rayn, M., Cinalli, C., 1996. Estimation of emission profiles from interior latex paints. In: Indoor Air ‘96, Proceedings of the Seventh International Conference of Indoor Air Quality and Climate, vol. 2. SEEC Ishibashi, Japan, pp. 55–60. Yaws, C.L., 1994. Handbook of Vapor Pressure, 3 vols. Gulf Publishing, Houston, TX. Zhao, D.Y., Cox, S.S., Little, J.C., 1999. Source/sink characterization of diffusion-controlled building materials. In: Indoor Air 99: Proceedings of the Eighth International Conference on Indoor Air Quality and Climate, vol. 1. pp. 408–413.
Review of indoor emission source models. Part 2. Parameter estimation Zhishi Guo* US Environmental Protection Agency, Office of Research and Development, National Risk Management Research Laboratory, Mail Drop E-305-03, Research Triangle Park, NC 27711, USA Received 17 August 2001; accepted 9 March 2002
‘‘Capsule’’: Forty-eight methods for estimating the parameters in the indoor emission source models are compiled and reviewed. Abstract This review consists of two parts. Part 1 provides an overview of 52 indoor emission source models. Part 2—this paper—focuses on parameter estimation, a topic that is critical to modelers but has never been systematically discussed. A perfectly valid model may not be a useful one if some of its parameters are difficult to estimate in the absence of experimental data. This is true for both statistical and mass transfer models. Forty-eight methods are compiled and reviewed in this paper. Overall, developing methods for parameter estimation has fallen behind the development of models. Such imbalance is the main reason that many models have been left on the shelf since they were published. Published by Elsevier Science Ltd. Keywords: Parameter estimation; Model; Indoor air; Indoor source; Emission
1. Symbols and abbreviations
B0
The units of the symbols listed here are given in generic terms. If a parameter estimation method works only with a specific set of units, they are given in the text.
CT Cv
1.1. Symbols a a lumped parameter for computing Knudsen diffusivity (length2 time1 mass1/2 mol1/2), a0, a1, a2, a3, b0, b1, b2, b3, c 0, c 1, c2, c3 coefficients in methods P28 and P29 (dimensions vary) A, B, C, D, E coefficients in methods P16 to P21, P25 to P27 (dimensions vary)
d dm dw Da DKn DL Ds H k k1
k11 k12
* Tel.: +1-919-541-0185; fax: +1-919-541-2157. E-mail address: [email protected] (Z. Guo). 0269-7491/02/$ - see front matter Published by Elsevier Science Ltd. PII: S0269-7491(02)00188-4
proportionality coefficient in method P13 (mass1/2 pressure mol1/2 temperature3/2 time1) saturation concentration for TVOC (mass length3) saturation concentration for pure compound (mass length3) density of paint (mass length3) mean molecular diameter (length) diameter of water droplet (length) diffusivity in air (length2 time1) Knudsen diffusivity (length2 time1) diffusivity in water (length2 time1) diffusivity in solid (length2 time1) Henry’s constant (dimensionless) first-order decay rate constant (time1) first-order decay rate constant for wet-stage emissions with unit film thickness (length time1) first-order decay rate constant for wet-stage emissions from paint film (time1) first-order decay rate constant for dry-stage emissions from paint film (time1)
552
kD1
kg kL K KOL L m ma md m1 mi moth
mT Mr n n1, n2 P P0 Pi Pp PT q Re Sc t0.9 T u uc vA vAS vB vLB vd Wi WT y yi yoth yT Z
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first-order decay rate constant for dry-stage emissions with unit film thickness (length2 time1) gas-phase mass transfer coefficient (length time1) liquid-phase mass transfer coefficient (length time1) solid/air partition coefficient (dimensionless) overall liquid-phase mass transfer coefficient (length time1) characteristic length of a source (length) molecular weight (mass mol1) average molecular weight for air (mass mol1) mass of water droplet (mass) molecular weight for most abundant compound in solvent (mass mol1) molecular weight for compound i (mass mol1) average molecular weight for all VOCs in the solvent except the one in question (mass mol1) average molecular weight for TVOC (mass mol1) a parameter in methods P12 to P14, defined by Eq. 6 (mol mass1) carbon number index number (dimensionless) vapor pressure (pressure) atmospheric pressure (pressure) vapor pressure for compound i (pressure) partial pressure (pressure) total vapor pressure for TVOC (pressure) number of compounds quantified in a petroleum solvent Reynolds number (dimensionless) Schmidt number (dimensionless) 90% evaporation time (time) temperature (temperature), air velocity (length time1) velocity of water current (length time1) molar volume for air (length3 mol1) molar volume based on atomic and structural diffusion volume increments (length3 mol1) molar volume for the compound in question (length3 mol1) Le Bas molar volume (length3 mol1) velocity of water droplet (length time1) amount of compound i in paint film (mass) amount of TVOC in paint film (mass) VOC content in paint (fractional) content of compound i in paint (fractional) content of all VOCs in paint except the one in question (fractional) content of TVOC in paint (fractional) depth of water (length)
1.2. Greek proportionality constant in Eq. 15 (dimensionless), proportionality constant in method P48 (dimensionless) * proportionality constant in Eq. 15 (length1/3 time1/3), film thickness (length) viscosity of air (mass length1 time1) w viscosity of water (mass length1 time1) density of air (mass length3) surface tension of water (mass time2) AB characteristic length for diffusivity calculation used in method P13 (length) association factor for solvent (dimensionless), ! oscillation frequency for water droplet (time1) O collision integral used in method P13 (dimensionless) 1.3. Abbreviations IAQ indoor air quality PUF polyurethane foam SVOC semivolatile organic compound TVOC total volatile organic compound VOC volatile organic compound
2. Introduction Most indoor emission source models are given as a simple mathematical expression with one or more constants, known as model parameters. Loosely speaking, the mathematical expression determines the trend of the emission rate, and the parameters determine the magnitude. Since the emission source models are mainly used for predictions, the usefulness of a model is often judged by whether its parameters can be readily estimated in the absence of experimental data. Many reported model parameters are obtained by fitting a model to the experimental data. While this technique is useful in the early stage of model development, the results, in most cases, cannot be used for making predictions. Genuine predictive models are those whose parameters can be estimated independently. This paper compiles 48 published methods for estimating 15 model parameters, which represent only a fraction of the total number of parameters in the 52 models reviewed in Part 1. Some well-constructed models have not been widely used in exposure estimation simply because no methods are available to estimate some of the parameters. The methods reviewed here can be divided into two groups: relative and absolute methods. The former require the knowledge of a reference value for the same parameter, while the latter do not. Obviously, the absolute methods are much more useful than the relative ones.
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3. First-order decay rate constants
3.2. For petroleum-based indoor coating materials
3.1. Overview
Chang and Guo (1992) observed from chamber data that the decay rate constant for the fast, evaporative emissions (k11) is proportional to the vapor pressure of a compound (method P1), and that for the slow emissions from dry film (k12) is proportional to the Knudsen diffusivity (method P2), which is defined by Eq. 1 (Bennet and Myers, 1982).
The first-order decay models (see Table 1 in Part 1), especially models M1 to M3, are among the few indoor source models that are supported by a series of methods for parameter estimation. The simple first-order decay models contain two parameters: E0 and k for model M1; and M0 and k for model M2. The two models are equivalent because E0=M0k. For indoor coating materials, parameter M0 is usually determined from product formulation (e.g. VOC contents and product density) and application conditions (e.g. film thickness). Thus, the only remaining parameter to be estimated is the first-order decay rate constant. As shown in Tables 1 and 2, 11 methods are available for estimating k, k11, and k12. In general, the decay rate constants for evaporative emissions (k or k11) are a function of vapor pressure; that for the solid- phase diffusion-controlled emissions (k12) is a function of diffusivity in the solid or molecular weight. Dual first-order decay models (M3 and M4) have four parameters. In addition to estimating the decay rate constants, one has to determine how much of the solvent applied is available for evaporation and how much for the dry-stage emissions. Currently there are no methods available to estimate such partition.
a DKn ¼ pffiffiffiffi m
ð1Þ
Since the value of parameter a is often unknown, method P2 can be stated as: k12 is inversely proportional to the square root of the molecular weight. Both P1 and P2 are relative methods. Method P3, proposed by Evans (1994), is for estimating k for solvent emissions from coating materials based on the 90% drying time of the solvent (t0.9), which, in turn, is estimated by a method developed by Chinn (1981): log10 t0:9 ¼ 7:3698 0:95461og10 Cv
ð2Þ
where Cv
16040mP T
ð3Þ
Table 1 Methods for estimating the first-order decay rate constants for petroleum-based indoor coating materials Method ID
Formula
Description
Reference
Note
P1
k11 / P
VOC evaporation from wet film of wood stain
Chang and Guo (1992)
a
P2
k12 / DKn
VOC emissions from dry film of wood stain
Chang and Guo (1992)
a
P3
k¼
1n10 t0:9
TVOC evaporation from solvent-based paint; based on 90% drying time
Evans (1994)
b
P4
k¼
k g CT dyT
TVOC evaporation from petroleum-based coatings
Guo et al. (1999)
b,c
P5
k¼
kg Cv mT dyT m
VOC evaporation from petroleum-based coatings
Guo et al. (1999)
b,d
P6
k ¼ 2:95 109
VOC evaporation from alkyd paint
Koontz (2001)
b,e
P7
k¼
VOC evaporation from alkyd paint
Koontz (2001)
b,f
a b c d e f
P 0:27 2 r ¼ 0:86 m4:02 0:58
2 240Da P r ¼ 0:49 6:23 104 dyT
For model M3. For models M1 and M2. k (in h1); kg (in m/h); CT (in mg/m3); (in m); d (in g/m3); yT (in mg/g). k (in h1); kg (in m/h); Cv (in mg/m3); (in m); d (in g/m3); yT (in mg/g); mT and m (in g/mol). k (in h1); P (in mm Hg); m (in g mole1); (in mil). One mil=2.54105 m. k (in h1); Da in (m2/h); (P in mm Hg); d (in g/m3); yT (in mg/g).
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Table 2 Methods for estimating the first-order decay rate constants for water-based indoor coating materials Method ID P8 P9
Formula k1 k11 ¼ 2 k12 ¼ Ds 2
Description
Reference
Note
VOC and SVOC evaporation from wet film of latex paint
Clausen (1993)
a
VOC and SVOC emissions from dry film of latex paint
Clausen (1993)
a
P10a
k11 ¼ 233:25 P r2 ¼ 0:92
Glycol evaporation from latex paint
Wilkes et al. (1996)
a,b
P10b
k11 ¼ 264:7 P 4:961 r2 ¼ 0:96
Glycol evaporation from wet film of latex paint
Wilkes et al. (1996)
a,b
P11a
k12 ¼ 5:84 105 m r2 ¼ 0:96
Glycol emissions from dry film of latex paint
Wilkes et al. (1996)
a,c
P11b
k12 ¼ 7:08 105 m 0:002 r2 ¼ 0:995
Glycol emissions from dry film of latex paint
Wilkes et al. (1996)
a,c
a b c
For model M3. k11 (in h1); P (in mm Hg). k12 (in h1); m (in g mol1).
where t0.9 is in seconds, Cv in mg/m3, m in g/mol, P in torr, and T in K. Typically, this method is used for TVOC emissions if the solvent contains more than one compound. It may overestimate k for individual VOCs in a mixture. One should also be aware that the correlation (Eq. 2) was found with the 90% evaporation time being determined on paper filters with dry air flowing through (ASTM, 1977). Apparently, the experimental conditions (e.g. film thickness and ventilation rate) are far from realistic. Besides, this method ignores the effect of film thickness. Despite these limitations, it is very useful for rough estimation of k because all three parameters in Eq. 3 are easily obtained. Method P4 (Guo et al., 1999) is another method for estimating k for TVOC emissions from solvent-based indoor coatings. It is reduced from a mass transfer model (M10). The gas-phase mass transfer coefficient (kg) is estimated by method P39; the TVOC saturation concentration (CT) is converted from the total vapor pressure (PT), which is estimated by method P30. Other required parameters—film thickness (), paint density (d), and TVOC content in paint (yT)—are easy to determine. Due to the omission of the back pressure effect when this method was derived, it predicts a higher initial emission rate than its corresponding mass transfer model (M10). The total emittable amount remains the same, however. Fig. 1 compares the k values predicted by methods P3 and P4 for an imaginary solvent-based paint that has the following properties: TVOC content=25%, total vapor pressure=1.5 mm Hg, average molecular weight for TVOC=142 g/mol, and product density=1.3 g/cm3. It appears that the two methods are comparable only when the gas-phase mass transfer coefficient is around 5 m/h and the wet film thickness is between 60 and 80 mm. Three methods (P5–P7) are available for estimating the decay rate constant for individual VOCs. Method
P5 (Guo et al., 1999) is reduced from model M12. This method requires the average molecular weight for TVOC (mT), which can be approximated in two ways: if the user knows the contents of the major components in the solvent, use Eq. 4 (Guo et al., 1999); otherwise, use the molecular weight for the most abundant VOC in the solvent. mT ¼
q q X X yi = ðyi =mi Þ i¼1
ð4Þ
i¼1
Method P6 is for VOC emissions from alkyd paint, and is implemented in an IAQ simulation program for wall paint (Koontz, 2001). This correlation was established using the chamber data for about 20 VOCs commonly found in petroleum-based solvents. These compounds cover a vapor pressure range from 0.25 to 24 mm Hg (at room temperature) and a molecular weight range from 87 to 168 g/mol. The correlation is good. However, judging from the exponents in the formula (0.27 for vapor pressure and 4.02 for molecular weight), this method seems too sensitive to molecular weight and too insensitive to vapor pressure. Any extrapolation beyond the valid range for molecular weight should be avoided. Fig. 2 compares the k values predicted by methods P5 and P6 for evaporation of 1,2,3-trimethylbenzene from an imaginary alkyd paint, whose solvent has the same properties as described in the example for comparing methods P3 and P4. The two methods are close only when the gas-phase mass transfer coefficient is small (1 m/h). Method P7 (Koontz, 2001) was derived from method P5, simplified by setting several default values. The correlation for the alkyd paint data is not as good as method P6. Koontz (2001) did not implement P7 in his simulation program; it was used only for method comparison.
Z. Guo / Environmental Pollution 120 (2002) 551–564
Fig. 1. Comparison of the first-order decay rate constant for TVOC evaporation predicted by methods P3 and P4. Values in parentheses in the legend are gas-phase mass transfer coefficient in m/h.
3.3. For water-based indoor coating materials Methods P8 and P9 (Clausen, 1993) concern the dependence of the decay rate constants on the film thickness of the paint. Method P8—a relative method— requires knowledge of the decay rate constant with unity film thickness (k1). Method P9 states that, for a given paint/substrate combination, the decay rate constant for the dry emissions is proportional to the diffusivity of the compound in question and inversely proportional to the square of the film thickness. In the absence of data for the diffusivity (Ds), method P9 is used as a relative method (Eq. 5). k12 ¼
kD1 2
ð5Þ
where kD1=Ds(/2)2. Wilkes et al. (1996) found that the decay rate constant for the fast emissions from the wet film of latex paint (k11) is proportional to the vapor pressure of the compound (methods P10a and P10b), and that for the slow emissions from the dry film (k12) is proportional to the molecular weight of the compound (methods P11a and P11b). These correlations were developed based on the chamber data for four glycols in a latex paint. In both methods, adding a non-zero intercept (methods P10b and P11b) improves the correlation coefficient. These two methods work well for predicting the glycol emissions up to two weeks following paint application.
4. Diffusivity 4.1. Diffusivity in air Diffusivity in air is mainly used to estimate the gasphase or overall mass transfer coefficient. Three methods (P12–P14 in Table 3) are commonly used by IAQ
555
Fig. 2. Comparison of the first-order decay rate constant for 1,2,3trimethylbenzene evaporation predicted by methods P5 and P6. Values in parentheses in the legend are gas-phase mass transfer coefficient in m/h.
modelers. P12 and P13 are the two methods recommended by Lyman et al. (1990), who compiled and reviewed nine methods for estimating the diffusivity in air. Method P12 (Fuller et al., 1966)—known as the Fuller-Schettler-Giddings (FSG) method—is fairly accurate for non-polar compounds and less accurate for the polar compounds including glycols. Method P13 (Wilke and Lee, 1955)—known as the Wilke-Lee (WL) method—usually does a better job than the FSG method, but the computation is slightly more complex and requires knowledge of the boiling point. Both methods require calculations of the molar volume (vB) of the compound in question, which can be approximated by either the ‘‘atomic and structural diffusion volume increments (vAS)’’ or ‘‘Le Bas volumes (vLB)’’ (Tables 17–4 and 17–5, respectively, in Lyman et al., 1990). The former is simpler to use. For instance, the atomic diffusion volume increments for carbon and hydrogen are 16.5 and 1.98 cm3/mol, respectively; the structural diffusion volume increment for an aromatic ring is 20.2 cm3/mol. Thus, for decane (C10H22), vAS=1016.5+221.98=208.6 cm3/mol; and for benzene (C6H6), vAS=616.5+61.9820.2= 90.7 cm3/ mol. Le Bas volumes are additive-volume increments for estimating the molar volumes of liquids at the normal boiling points (Reid et al., 1977). This method takes into consideration more structural factors (e.g. bonding conditions and non-aromatic rings). However, in either case, no data are available for phosphorous-containing compounds. P14 is a much simpler method used by Koontz (2001). Molecular weight is the only information this method requires. The trade-off is its poor accuracy. As shown in Table 4, the results from this method are roughly 45% smaller than the WL method. This error will affect the accuracy of the gas mass transfer coefficient and, consequently, of the emission rate. In all the three methods, parameter Mr is calculated from Eq. 6.
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Table 3 Methods for estimating diffusivity in air and water Method ID
Formula pffiffiffiffiffiffiffi 103 T 1:75 Mr Da ¼ 2 1=3 P0 1=3 A þ B pffiffiffiffiffiffiffi B0 T 2=3 Mr Da ¼ 2 P0 AB O pffiffiffiffiffiffiffi Da ¼ m0:33 Mr
P12
P13 P14
Description
Reference
Note
Diffusivity in air (FSG method)
Fuller et al. (1966)
a
Diffusivity in air (WL method)
Wilke and Lee (1955)
b
Diffusivity in air
Koontz (2001)
c
Diffusivity in water
Hayduk and Laudie (1974)
d
5
P15 a b c d
DL ¼
13:26 10 0:589 1:14 w LB
Da (in cm2/s); T (in K); Mr (in mole/g); P0 (in atm); vA and vB (in cm3/mol). ˚ Da (in cm2/s); B0 =0.02170.0005 M1/2 r ; T (in K); Mr (in mol/g); P0 (in atm); AB (in A); is dimensionless. Da (in cm2/s); m (in g/mol); Mr (in mol/g). DL (in cm2/s); mw (in centipoise); vLB (in cm3/mol).
Table 4 Comparison of three methods for estimating the diffusivity in air at 298 K and 1 atm (units: cm2/s) Compound
n-decane m-chlorotoluene
Mr ¼
Formula
C10 H22 C7 H7 Cl
ma þ m ma m
Method P12
P13
P14
0.0583 0.0730
0.0714 0.0758
0.0397 0.0417
ð6Þ
4.2. Diffusivity in water Diffusivity in water is used to estimate the liquidphase and overall mass transfer coefficient. Lyman et al. (1990) compiled six available methods for estimating the diffusivity in water, and they recommended method P15 (Hayduk and Laudie, 1974). This method requires only two parameters: the viscosity for water (0.8904 centipoise at 25 C) and the Le Bas volume of the solute. 4.3. Diffusivity in solids All the mass transfer models for VOC emissions from building materials (i.e. models M23–M31 in Part 1) require knowledge of the diffusivity of the VOC in the material (Ds). Although development of methods for estimating this parameter has been in active study for a decade, most progress was made very recently (see Table 5). Method P16 (Berens and Hopfenberg, 1982) correlates the diffusivity to the mean molecular diameter (dm) of the penetrant molecule. The data used by the authors included three glassy materials (polyvinyl chloride, polystyrene, and polymethyl methacrylate) and more than 20 compounds. Since the figures of dm are not
available for some VOCs studied, the authors proposed three methods for estimating dm. The authors also tried to correlate Ds to the square of dm for some data. With the scattering of data, however, the authors could not decide which correlation is better. Zhao et al. (1999) found that, for a polyurathane foam (PUF), there was a linear correlation (R2=0.94) between diffusivity and vapor pressure on a log–log scale (method P17). The data used included water and eight aromatic hydrocarbons. The correlation becomes less satisfactory when this method is applied to the data reported by other authors. Both methods P18 (Bodalal et al., 2001) and P19 (Cox et al., 2001) correlate diffusivity to molecular weight. Mathematically, the two methods are identical (Table 5). This author deliberately put them in different forms to emphasize their difference. They differ in how to obtain the correlation constants (A and n1). Method P18 applies non-linear regression to data on normal scale, while method P19 uses linear regression with data on log–log scale. This difference makes the R2 values not directly comparable between the two methods. In general, method P18 gives more weight to the data points with large Ds values; method 19 is less discriminative against smaller Ds values. Table 6 summarizes the reported correlation constants for different material/ compound group combinations. Description of the test materials by Bodalal et al. (2001) was somewhat cursory. Further studies are needed to investigate the variations of these two parameters among the same type of materials. The weakness of methods P18 and P19 is that, for a given material, each compound class has its own correlation constants (e.g. A and n1), which cannot be used to predict the diffusivity of a chemical in a different compound class. To resolve this problem, this author suggests to express the diffusivity as a function of the molar volume of the VOC (methods P20 and P21). After all, it is the volume, not the weight, of the molecule, that really matters to solid-phase diffusion. Methods P20
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Z. Guo / Environmental Pollution 120 (2002) 551–564 Table 5 Methods for estimating diffusivity in solids Method ID Formula
Description
P16
1nDs ¼ A Bdm
Ds as a function of penetration diameter; for mixed compound classes Berens and Hopfengerg (1982)
P17
1nDs ¼ A B1nP
Ds as a function of vapor pressure; for mixed compound classes
Zhao et al. (1999)
P18
Ds ¼
Ds as a function of molecular weight; for a specific compound class
Bodalal et al (2001)
a
P19
1nDs ¼ 1nA n1 1nm Ds as a function of molecular weight; for a specific compound class
Cox et al. (2001)
a
P20
1nDs ¼ A B1n AS
Ds as a function of molar volume; for mixed compound classes
Guo (in preparation)
b
P21
1nDs ¼ C D1n LB
Ds as a function of Le Bas volume; for mixed compound classes
Guo (in preparation)
c
a b c
A mn1
Reference
Note
Ds (in m2/s); m (in g/mol). Ds (in m2/s); AS (in cm3/mol). Ds (in m2/s); LS (in cm3/mol).
Table 6 Correlation constants for diffusivity in solids (methods P18 and P19) Building material
Compound group
A
n1
R2
Note
Gypsum board Oriented strand board Particle board Particle board Plywood Plywood Vinyl tile Vinyl tile Vinyl flooring
Aromatics Alkanes Aromatics Aldehydes Aromatics Aldehydes Alkanes Aromatics Alkanes
4.481101 1.450107 1.692107 3.396108 1.337107 7.900104 2.104101 3.617102 5.006105
5.99 8.36 8.5 9.33 8.56 3.64 5.94 6.51 7.45
0.9816 0.9866 0.9618 0.8328 0.9969 0.8802 0.9864 0.9789 0.9973
a,b
a b c d e f
a,c a,b a,d a,b a,d a,c a,b e,f
Data source: Bodalal et al. (2001). Included benzene, toluene, ethylbenzene, propylbenzene, and butylbenzene. Included heptane, octane, nonane, decane, and undecane. Included pentanal, haxanal, heptanal, and octanal. Data source: Cox et al. (2001). Included decane, dodecane, tetradecane, and pentadecane.
and P21 differ in how the molar volume is estimated. Method P20 uses the atomic and structural diffusion volume increments (Fuller et al., 1966) while method P21 uses the Le Bas volumes (Reid et al., 1977). Table 7 shows the correlation constants for these two methods using the diffusivity data reported by Bodalal et al. (2001) and Cox et al. (2001). Overall, the correlations are satisfactory except for the vinyl flooring. A major advantage of methods P20 and P21 is that the constants (Table 7) are independent of compound classes. Of course, more data are needed to further evaluate these methods and to refine the correlation constants. The strength of the bonding force between the VOC and the substrate molecules may affect the diffusivity. It is well known that the glycol (polar compounds) emissions from painted gypsum board (polar substrate) are very slow (Chang et al., 1997). As a future research topic, this author suggests to investigate such effects by
experimental determination of enthalpy changes, from which the bounding force can be estimated.
5. Solid/air partition coefficient Solid/air partition coefficient (K ) is a parameter required by models M25–M31. As shown in Table 8, both methods P22 (Zhao et al., 1999; Cox et al., 2001) and P23 (Bodalal et al., 2001) correlate K to the vapor pressure of the VOC. The two methods differ slightly in how the constants are determined (see Section 4.3). Table 9 lists the values for B and n2 reported by Bodalal et al. (2001), Cox et al. (2001), Zhao et al. (1999). Note that each building material has its own constants. In an effort to develop a correlation independent of compound classes and material types, this author found that there exists a fair correlation (Fig. 3) between
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Z. Guo / Environmental Pollution 120 (2002) 551–564
Table 7 Correlation constants for methods P20 and P11 Building material
Method P20
Gypsum board Oriented strand board Particle board Plywood Vinyl tile Vinyl flooring a b c d e f g
Method P21
Note
A
B
R2
C
D
R2
1.536 19.73 10.59 0.6787 3.546 24.15
4.674 8.401 6.970 4.720 4.286 0.9349
0.997 0.973 0.785 0.815 0.917 0.689
1.652 21.45 14.09 0.0644 4.445 23.58
4.589 8.533 7.575 4.780 4.043 1.023
0.997 0.974 0.864 0.847 0.895 0.665
a,b a,c a,d a,d a,e f,g
Diffisivity data from Bodalal et al. (2001). Based on a single class of compounds (5 aromatics). Based on a single class of compounds (5 alkanes). Based on 3 classes of compounds (5 aromatics, 4 aldehydes, and limonene). Based on 2 classes of compounds (5 alkanes and 5 aromatics). Diffisivity data from Cox et al. (2001). Based on 5 classes of compounds (4 alkanes, water, butanol, toluene, and phenol).
Table 8 Methods for estimating solid/air partition coefficients Method ID
Formula
Description
Reference
Note
P22
1nK ¼ 1nB n2 1nP
K as a function of vapor pressure; for a specific material and mixed compound classed
Zhao et al. (1999)
a
P23
K¼
K as a function of vapor pressure; for a specific material and compound class
Bodalal et al. (2001)
a
P24
1nK ¼ 8:76 0:7851nP
K as a function of vapor pressure; for all materials and compound classes
Guo (in preparation)
a
a
B P n2
K is dimensionless; P (in mm Hg).
Table 9 Correlation constants for methods P22 and P23 Building material Gypsum board Oriented strand board Particle board Particle board Plywood Plywood Vinyl tile Vinyl tile Vinyl flooring PUF a b c d e f g h
Compound group Aromatics Alkanes Aromatics Aldehydes Aromatics Aldehydes Alkanes Aromatics Alkanes Aromatics
B 4
1.06910 1.073104 1.305104 1.037104 1.108104 6.581103 2.032104 1.111104 2.566103 1.238103
n2
R2
Note
0.772 0.886 0.892 0.51 0.944 0.644 1.051 0.863 0.772 0.921
0.9853 0.9895 0.9834 0.9466 0.9801 0.9412 0.9999 0.9933 0.9986 0.9624
a,b
Source: Bodalal et al. (2001). Compounds included benzene, toluene, ethylbenzene, propylbenzene, and butylbenzene. Compounds included heptane, octane, nonane, decane, and undecane. Compounds included pentanal, haxanal. heptanal, and octanal. Source: Cox et al. (2001). Compounds included decane, dodecane, tetradecane, and pentadecane. Source: Zhao et al. (1999). Compounds included: benezone, toluene, p-xylene, ethylbenzene, chlorobenzene, 1,2,4-trimethylbenzene, styrene, and naphthalene.
a,c a,b a,d a,b a,d a,c a,b e,f g,h
Z. Guo / Environmental Pollution 120 (2002) 551–564
559
ture range. Two methods (P28 and P29) developed by Makar (2000) allow the user to estimate the vapor pressure for an organic compound by knowing (1) its compound class, (2) the carbon number in the molecule, and (3) the temperature of interest. In doing so, Makar divided the organic compounds into 39 classes (e.g. nalcohols, n-alkanes, and polyaromatics) and established two correlations for each compound class based on available experimental data: Eq. 7 for method P28 and Eq. 8 for P29. log10 P ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 Fig. 3. Comparison of gas-phase mass transfer coefficient for n-decane estimated with six absolute methods, assuming that the air velocity is 10 cm/s and the source area is 1 m2.
partition coefficients and vapor pressures for all 56 material/compound combinations reported by Bodalal et al. (2001), Cox et al. (2001), and Zhao et al. (1999). Thus, method P24 can be used to roughly estimate the partition coefficients for all material/compound combinations.
6. Vapor pressure, partial pressure, and total vapor pressure 6.1. Vapor pressure as a function of temperature Vapor pressure, partial pressure and total vapor pressure are key parameters in modeling VOC emissions from liquids and solids. Commonly used methods for estimating these parameters are listed in Table 10. Vapor pressure is an important parameter in many indoor source models and parameter estimation methods. Many compounds have experimentally determined vapor pressure, which can be found in the literature. Frequently cited references for vapor pressure include Yaws (1994), Boublik et al. (1984), and Schlessinger (1972). Since vapor pressure is strongly affected by temperature, experimental data are often presented as a certain temperature function, whose coefficients are given in a large table. Methods P25 (Schlessinger, 1972), P26 (Boublik et al., 1984), and P27 (Yaws, 1994) are three commonly used temperature functions. One should be aware that the coefficients in these three methods (e.g. A and B in P25) are valid only in a certain temperature range. Extrapolation beyond the valid range is not recommended. 6.2. Estimation of vapor pressure in the absence of experimental data Not all compounds have experimentally determined vapor pressure. For some compounds, the experimental data in the literature may not cover the room tempera-
þ n b0 þ b1 T þ b2 T 2 þ b 3 T 3
ð7Þ
log10 P ¼ a0 þ a1 T þ a2 T 2 þ a3 T 3 þ n b0 þ b1 T þ b2 T 2 þ b 3 T 3 þ n2 c 0 þ c 1 T þ c 2 T 2 þ c 3 T 3
ð8Þ
where P is in mm Hg and Tin K. The coefficients in Eqs. 7 and 8 are given in two large tables. In general, the correlations are good. These two methods are very useful in the absence of experimental data at room temperature. It is the user’s responsibility, however, to determine the class to which a given compound belongs. In other words, the user must know the molecular structure of the compound. This author prefers method P29 over P28 because its coefficient table contains the valid range for the carbon number. 6.3. Total vapor pressure The total vapor pressure for TVOC is the sum of the partial pressures for all compounds in the solvent mixture. It is required by models M10–M12 and method P4. For a petroleum- based solvent, which may contain hundreds of compounds, summing up all partial pressures is difficult. Guo et al. (1999) proposed to approximate the total vapor pressure from the imaginary solvent consisting of the known major components in the mixture (method P30). Knowledge of the contents of about a dozen major components in the solvent is usually sufficient for a reasonable estimate of the total vapor pressure. 6.4. Partial pressure Partial pressure is needed to predict the emissions of individual VOCs from products that contain a solvent mixture. For ideal and non-ideal solutions, the partial pressure is determined by Eqs. 9 and 10, respectively. Pp ¼ xL P
ð9Þ
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Z. Guo / Environmental Pollution 120 (2002) 551–564
Table 10 Methods for estimating vapor pressure, partial pressure, and total vapor pressure Method ID
Formula
Description
Reference
Note
P25
log10 P ¼
Vapor pressure as a function of temperature
Schlessinger (1972)
a
P26
log10 P ¼ A
B TþC
Vapor pressure as a function of temperature
Boublik et al. (1984)
a
P27
log10 P ¼ A þ
B þ Clog10 T þ DT þ ET 2 T
Vapor pressure as a function of temperature
Yaws (1994)
a
P28
See text
Vapor pressure by compound class
Makar (2000)
P29
See text
Vapor pressure by compound class
Makar (2000)
P30
PT ¼
Pq Pi yi Xq yi = i¼1 i¼1 mi mi
Total vapor pressure for TVOCS
Guo et al. (1999)
P31
Pp ¼
Wi mT Pi mi WT
Partial pressure for individual vocs in paint film
Guo et al. (1999)
P32
Pp ¼
Wi m1 Pi mi WT
Partial pressure for individual vocs in paint film
Guo (2000)
P33
Pp ¼
yi =mi Pi yi =mi þ yoth =moth
Partial pressure for individual vocs in paint film
van Veen et al. (1999)
a
0:218A þB T
P (in mm Hg); T (in K).
Pp ¼ xL P
ð10Þ
Methods for estimating the activity coefficient have been developed. A good introduction is given by Lyman et al. (1990). Eq. 10 has not been widely used by IAQ modelers, however. P31 and P32 (Guo et al., 1999) are two approximate methods for estimating the partial pressure for an individual VOC in petroleum-based solvents. Both methods are based on the contents of the major components, and they differ only in how the average molecular weight is calculated. Method P31 uses Eq. 4 to estimate the average molecular weight for TVOC. In method P32, it is represented by the molecular weight of the most abundant component in the solvent. Method P33 (van Veen et al., 1999) treats the solvent as a bicomponent system, the compound in question and the rest of compounds. The authors did not specify how to calculate the average molecular weight (moth), however. All three methods were developed for petroleum solvent-based paint. Methods P31 and P32 have also been applied to solvent spills (Guo, 2000).
7. Mass transfer coefficients 7.1. Gas-phase mass transfer coefficient Seven methods have been used to estimate the gasphase mass transfer coefficient in the indoor environ-
ment (Table 11). P34 (Bennett and Myers, 1962) is the conventional method favored by chemical engineers. This method is based on correlations between three dimensionless numbers: Sherwood number (Sh), Schmidt number (Sc) and Reynolds number (Re), with the latter two being defined by Eqs. 11 and 12, respectively. Lu Sc ¼ Da Re ¼
ð11Þ ð12Þ
A series of correlations have been developed between Sh, Sc and Re (Bennett and Myers, 1962; White, 1991). For example, Eq. 13 applies to laminar flow conditions. Sh ¼ 0:664Sc1=3 Re1=2
ð13Þ
While P34 remains the method of choice, several simpler models have been used by IAQ modelers to estimate the gas-phase mass transfer coefficient. Methods P35 (Higbie, 1945), P36 (Mackay and Matsugu, 1973), P37 (Geankoplis, 1993), and P38 (Jayjock, 1994) were borrowed from studies for the ambient environment, while P39 (Sparks et al., 1996) and P40 (Haghighat and Zhang, 1998) were developed exclusively for the indoor environment. Method P36 consists of two equations. The one in Table 11 is for Reynolds numbers smaller than 15 000—the condition most likely to be found in the indoor environment. Fig. 4 compares the mass
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Z. Guo / Environmental Pollution 120 (2002) 551–564 Table 11 Methods for estimating of the gas-phase mass transfer coefficient Method ID Formula kg L Da
Description
Reference
Note
Conventional method based on Sherwood number
Bennett and Myers (1962), Sparks et al. (1996)
a
Used by model M34
Higbie (1945), Reinke and Brosseau (1997)
a
P34
Sh ¼
P35
kg ¼ 2
P36
kg ¼ 17:35u0:78 L 0:11 Sc0:67 Used by model M35
Mackay and Matsugu (1973), Reinke and Brosseau (1997)
a,b
P37
kg ¼ 0:664Re1=2 uSc2=3
For solvent spill; model not included in Part 1
Geankoplis (1993), Reinke and Brosseau (1997)
a
P38
kg ¼ 0:5
Used by model M32
Jayjock (1994), van Veen et al. (1999)
c
P39
kg ¼ 0:33Da L 3
P40
See text
a b c d
rffiffiffiffiffiffiffiffiffi Da u L
rffiffiffiffiffi 3 18 m 1
23 u
Used by models M10, M11, M17, M37, and M38 Sparks et al. (1996) See text
d
Haghighat and Zhang (1998)
The first reference is for the original method development and the second for indoor application. kg and u (in m/s); L in (m); Sc is dimensionless. kg (in m/min); m (in g/mol). kg (in m/h); Da (in m2/h); L (in m); u (in m/h); (in g/m3); (in g/m/h).
transfer coefficient for n-decane calculated from methods P34 to P39. Overall, the difference between these methods is fairly large. In particular, the two simpler methods (P35 and P38) are significantly different from the other four methods. Further evaluation is needed to determine which methods perform better under realistic indoor conditions. Developed by Haghighat and Yang (1998), P40 is a more refined method than all the methods discussed earlier. It takes into consideration such factors as surface roughness, which are ignored by other methods. This author was unable to compare P40 with other methods because of insufficient information in the original paper. A potential weakness of this method is that one of its coefficients may have to be determined experimentally. More work is needed to make this method practically useful. 7.2. Liquid-phase mass transfer coefficient The liquid-phase mass transfer coefficient can be used alone to predict the VOC emissions from water and aqueous solutions if the Henry’s constant is fairly large. It is also required to calculate the overall mass transfer coefficient. Method P41 (Little, 1992a; Moya et al., 1999) in Table 12 is a relative method. The value of n2 varies from 0.5 to 1 and a value of 0.67 is often applied. P42 (Lyman et al., 1990) and P43 (Southworth, 1979) are two absolute methods borrowed from the studies in the ambient environment. Their usefulness in the indoor environment has not been evaluated. There is an
Fig. 4. Correlation between solid/air partition coefficients and vapor pressure for all material/compound combinations (n=56, R2=0.734).
obvious flaw in method P43: it becomes zero if the water body is still (uc=0). Thus, this method should be used with caution. Methods P44 (Angelo et al., 1966), P45 (Handlos and Baron, 1957), and P46 (Ruckenstein, 1967) are used for estimating the liquid-phase mass transfer coefficient for VOC emissions from water drops. Giardino et al. (1992) applied these methods to domestic shower spray and tested them against the experimental data for volatilization of trichloroethylene. The authors found that methods P44 and P45 fit the data well and that method P44 is more convenient to use because it does not require the velocity of water drops. The oscillation frequency of water drop (!) in method P44 is calculated from Eq. 14.
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Table 12 Methods for estimating of the liquid-phase mass transfer coefficient Method ID
Formula
Description
Reference
P41
kL / DLn2
kL As a function of diffusivity
Little (1992a), Moya et al. (1999)
P42
kL ¼ 20
For VOC from ambient water bodies
Lyman (1990)
a
P43
kL ¼ 23:51
For VOC from ambient water bodies
Southworth (1979)
b
For VOC from water droplets
Angelo et al. (1966), Giardino et al. (1992)
c,d
For VOC from water droplets
Handlos and Baron (1957), Giardino et al. (1992)
c,e
For VOC from water droplets
Ruckenstein (1967), Giardino et al. (1992)
c,f
rffiffiffiffiffi 44 m
u0:969 c Z 0:673 rffiffiffiffiffiffiffiffiffiffi !DL kL ¼ 2
P44
0:00375 d 1 þ w = rffiffiffiffiffiffiffiffiffiffiffi d DL kL ¼ 2 dw
P45
kL ¼
P46 a b c d e f
rffiffiffiffiffi 32 m
Note
kL (in cm/h); m (in g/mol). kL (in cm/h); uc (in m/s); Z (in m); m (in g/mol). The first reference is for the original method development and the second for indoor application. kL (in cm/s); ! (in s1); DL (in cm2/s). kL (in cm/s); d (in cm/s); w and (in cm2/s). kL (in cm/s); d (in cm/s); DL (in cm2/s); dw (in cm).
Table 13 Methods for estimating of the overall mass transfer coefficient Method ID
Formula
P47
1 DLi KOLi DLr
P48
KOL
a b
n2
1 1 Dai þ kLr Hi kgr Dar !1 2:5 1 ¼ þ DL2=3 D2=3 a H ¼
n1
DLi DLr
Description
Reference
Note
A relative method for estimating KOL
Little (1992a)
a
McKone method for estimating KOL
McKone (1987)
b
n2
Subscripts i and r represent the compound of interest and the reference compound, respectively. KOL (in m/s); and H are dimensionless; DL and Da (in m2/s).
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8=ð3md Þ !¼ 2
ð14Þ
P48 (Eq. 15) and suggested that should be approximately 70. He pointed out that method P48, in which is set to 1, may underestimate the effect of gas-phase resistance.
7.3. Overall mass transfer coefficient The overall mass transfer coefficient is given as either the gas- or liquid-phase mass transfer coefficient and is used in models for emissions from tap water and waterbased products [see Eq. (15) in Part 1]. Due to the lack of experimental data, accurate prediction of this parameter has been difficult. P47 (Little, 1992a) in Table 13 is a fairly complex relative method, in which the two indices (n1 and n2) may vary. P48 (McKone, 1987) is an approximate, absolute method based on data for radon emissions from tap water that are adjusted to VOCs. Little (1992b) proposed an improved version of method
KOL ¼
2:5 DL2=3
þ
Da2=3 H
!1 ð15Þ
In both method P48 and Eq. 15, the proportionality constants ( and *) are independent of the compounds in question, but depend on hydrodynamic conditions. This method is fairly simple, but need more experimental data to determine the values of the proportionality constants under different physical conditions.
Z. Guo / Environmental Pollution 120 (2002) 551–564
8. Parameters for fugacity models Fugacity models (see Table 9 in Part 1) utilize a special set of parameters that are not used by other indoor source models. Examples of such parameters are: fugacity capacity, transfer parameter, and time-varying compartment volume for building materials. The reader is referred to the original papers (see references in Part 1) for more details.
9. Conclusion and discussion Forty-eight methods for parameter estimation were compiled and reviewed. Eleven of them are for estimating the decay rate constants in the first-order decay models. The other 37 methods are for estimating diffusivity, partition coefficient, vapor pressure/volatility, and mass transfer coefficient. Methods for estimating diffusivity in air and water are well established. It is more difficult to estimate diffusivity in solids because it is dependent on the properties of the material. Progress has recently been made to estimate this parameter. Several methods have also been proposed to estimate the solid/air partition coefficient, which is required by seven mass transfer models for building materials (M25–M31). Vapor pressure is usually found in the literature, but the data may not be valid at room temperature. In the absence of experimental data, methods P28 and P29 are useful. Seven absolute methods are available for estimating the gas-phase mass transfer coefficient. However, significant difference exists between them. The two simplest methods (P35 and P38) seem less accurate than the rest and should be avoided. Further evaluation of these methods is needed. Methods for estimating the liquid-phase mass transfer coefficient have not been adequately evaluated. Currently there are no reliable methods for estimating the overall mass transfer coefficient, which is a critical parameter for VOC emissions from water and aqueous solutions. It is a common misconception to think that parameter estimation is a problem only for statistical models. In fact, many mass transfer models contain one or more parameters that are difficult to estimate. Overall, developing of methods for estimating the model parameters has not progressed to the extent that IAQ modelers would like to see. This is evidenced by the fact that many models discussed in Part 1 are not supported by any parameter estimation methods. Current practice in developing indoor source models often involves two steps: model development and validation. While these steps are essential, a third step—finding ways to estimate the model parameters—should follow. Otherwise, the usefulness of the model will be limited.
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