USE OF SMALL-CHAMBER DATA TO ESTIMATE AND MODEL 

CHEMICAL EMISSIONS FROM LATEX AND ALKYD PAINTS

IE-2812

Final Report

prepared for

Economics, Exposure and Technology Division

Office of Pollution Prevention and Toxics

U.S. Environmental Protection Agency

401 M Street, S.W.

Washington, DC 20460

under

EPA Contract No. 68-D3-0013

Work Assignment No. 3-19

prepared by

Charles R. Wilkes and Michael D. Koontz

GEOMET Technologies, Inc.

20251 Century Boulevard

Germantown, MD 20874

A Subsidiary of 

Versar, Inc.

6850 Versar Center

Springfield, VA 22151

February 4, 1997
ii

DISCLAIMER

Mention of the names of specific companies, organizations, or entities does not constitute 

an endorsement by EPA. This document does not represent official Agency Policy.

iv

TABLE OF CONTENTS

Executive Summary

Appendix A--Characterizing the Emissions of VOCs from Latex Paint

Appendix B--Detailed Explanation and Application of Technique C

Appendix C--Characterizing the Emissions of VOCs from Alkyd Paint

Appendix D--Model Algorithms

Appendix E--Exposure Scenarios
1

EXECUTIVE SUMMARY

Introduction

Estimation of human inhalation exposure in residential settings, due to use of consumer 

products or installation of building materials or furnishings, requires integration of time-varying 

pollutant concentrations with information on individuals’ locations and breathing rates. Often 

the concentration component must be estimated using an indoor air quality (IAQ) model. The 

accuracy of the estimate is enhanced by providing a faithful depiction of the time-varying 

airborne concentration which, in turn, relies on reasonable characterization of the emission 

profile for the chemical(s) of concern.

The main objective of the effort described in this report was to develop methods for 

estimating emission profiles for chemicals released from interior latex/alkyd paints, for two 

situations: (1) data are available from small-chamber experiments whereby airborne 

concentrations are measured in the chamber at selected points in time after the paint has been 

instantaneously applied to a substrate; and (2) only the weight fractions and physical/chemical 

properties (molecular weight and vapor pressure) of the paint constituents are known. Smallchamber data used for this analysis have been recently collected for latex paint applied to gypsum 

board and for alkyd paint applied to glass.

Latex Paint Emission Profiles

Appendix A describes the estimation methods that have been developed under this task 

for latex paint. These methods were applied to experimental data collected under sponsorship of 

the EPA Air and Energy Engineering Research Laboratory (AEERL) in Research Triangle Park, 

NC. In brief, these earlier experiments involved application of latex paint to a small piece of prepainted gypsum board. Following the paint application, the specimen was immediately inserted 

in a small stainless-steel chamber, and concentrations of four individual VOCs were measured at 

selected points in time over a period of several hundred hours. Bulk analysis of the paint sample 

was also performed to obtain an estimate of the respective weights of these VOCs.
2

The following three possible representations of the emission profile were considered:

1. An empirically fit single-exponential model to account for the (assumed) general 

decline in the emission rate over time;

2. An empirically fit double-exponential model that assumes two components of 

emissions decline--a “fast” (evaporation-dominated) component that primarily 

governs the early stage of emissions and a “slow” (diffusion-dominated) component 

that governs the later stage; and 

3. A mass-transfer representation developed by researchers at AEERL.

For estimation of the single- and double-exponential model parameters (an initial 

emission rate and a first-order rate of decline in the rate), nonlinear regression analysis was 

applied to the chamber data, taking advantage of the known chamber volume and the 

experimentally controlled air exchange rate for the chamber. Based on this exercise, the 

following conclusions were reached:

1. The single-exponential model does not fit the data well, most likely because it is 

insufficient to provide an empirical representation of the physical process as the 

paint dries.

2. The double-exponential model fits the data quite well in most cases, provided that 

the second (“slow”) exponential is estimated first, by fitting a single-exponential 

model to the data after the first 48 hours in the chamber. The first (“fast”) 

exponential can then be estimated using all the chamber data, by specifying a doubleexponential model with the second exponential’s parameters input as “knowns.”

3. Integration of the fitted double-exponential model to infinity provided an estimate of 

the total emitted mass of each VOC. Comparison of this emitted mass with the 

applied mass, as estimated from the bulk analysis, indicated that the emitted mass 

ranged from 20 to 35 percent for those compounds, averaging about 25 percent.

4. Integration of the respective exponentials indicated that the amount of emitted mass 

attributable to the first (“fast”) exponential was generally less than 10 percent. Thus, 

although this exponential dominates the early emission phase, the later phase of 

“slow” emissions accounts for most of the total VOC emissions.
3

The mass-transfer model developed by the AEERL researchers has been shown, in a 

separate report, to provide a good fit to measured air concentrations when painting a research 

house. However, this model has many parameters that are difficult to estimate and, thus, cannot 

readily be applied here. Based on the results of the estimation process described above, it is 

recommended that the double-exponential model be fit and used in cases where chamber data for 

painted gypsum board are available.

Chamber data for latex paint applied to gypsum board currently are limited to two

formulations. Consequently, methods were sought by which to estimate parameters of the 

double-exponential emission model for brands of latex paint for which only the weight fraction, 

molecular weight and vapor pressure of a VOC constituent are known. It was hypothesized that 

the “fast” decline would be evaporation-dominated and, hence, the rate term for this decline 

would be related to a chemical’s vapor pressure. Similarly, it was hypothesized that the rate term 

for the diffusion-dominated “slow” decline would be related to a chemical’s molecular weight. 

These hypotheses were indeed supported by the data -- regression through the origin indicated 

strong linear relationships (1) between the “fast” decline rate constant and vapor pressure (R2

= 

0.92), and (2) between the “slow” decline rate and molecular weight (R2 = 0.96).

These relationships, combined with estimates of the fraction of applied mass that is 

ultimately emitted (25 percent) and the relative contributions of the “fast” and “slow” 

components of the emission profile (10 percent and 90 percent, respectively), make it possible to 

estimate all parameters of the double-exponential emission model for a given VOC when 

knowledge is restricted to the total paint mass applied, the weight fraction of the VOC, and its 

molecular weight and vapor pressure. Further details are provided in Appendix A, and a detailed 

explanation of the parameter-fitting procedure used in Appendix A is presented in Appendix B.

Alkyd Paint Emission Profiles

Appendix C describes the estimation methods for alkyd paint, based on a separate set of 

experiments conducted at Research Triangle Park. These experiments were similar to those for 
4

latex paint, except that the painted substrate was glass in this case. Further EPA small-chamber

studies are in progress to examine the emission characteristics of alkyd paints on various 

substrates, including gypsum board. Based on preliminary results, there is some evidence 

indicating that emission profiles for alkyd paint applied to glass may not be much different than 

those for alkyd paint applied to gypsum board. Although the difference in emission behavior due 

to substrate effects is not well understood, it was felt that examination and estimation of the 

emission profile for a glass substrate would still be instructive.

Not surprisingly, it was found that a single-exponential (“fast”) model was adequate to 

describe the time-related emission behavior of VOCs escaping from alkyd paint applied to glass. 

Integration of the fitted emission functions to infinity indicated that, on average, close to 100 

percent of the applied VOCs were emitted from gloss paint, as compared to about 75 percent for 

semi-gloss paint and 50 percent for flat paint. Whether these differences have implications for 

different types of latex paint is unknown at this time.

Various empirical models were postulated and estimated for the relationship between the 

rate constant for exponential emissions decline and the molecular weight and/or vapor pressure 

of each VOC. It was found that a simple, linear regression of the rate of decline against vapor 

pressure fit the data as well as any other model, with R2 values of 0.84 for gloss paint, 0.82 for 

semi-gloss paint, and 0.87 for flat paint. The regression estimates were sufficiently different for 

the three types that a combined regression estimate (R2 = 0.64) could not be justified. Although 

the results have less applicability due to the glass substrate that was used, there is greater 

confidence in the regression estimates because they are based on 15 VOCs, as opposed to just 

four for latex paint.

The above relationships, described more fully in Appendix C, make it possible to predict 

the single-exponential profile for a VOC emitted from alkyd paint with knowledge restricted to 

total paint mass applied, the weight fraction of the VOC, and its vapor pressure. However, 

because the estimates are based on a glass substrate and are restricted to a single-exponential 
5

model, the resultant approximations are likely to over-estimate both the peak concentration and 

the total mass emitted for any given chemical.

Algorithms

Appendix D describes algorithms for implementing the results of the estimation 

procedures described above in indoor air quality models. A general equation is presented that 

represents painting as a progressive event whereby paint is applied to a surface at an essentially 

constant rate. The algorithm represents the source strength as a function of time for the 

combination of (1) continuous application of paint at a constant rate and (2) double-exponential 

emission profile for each instantaneously painted segment of gypsum board.

The parameters to be estimated for the general equation include (1) the total mass of the 

VOC of concern in the applied paint, (2) the total time required to apply the paint, (3) the rate 

constants for the double-exponential emission profile, and (4) the relative fraction of emitted 

mass attributed to the first (“fast”) exponential. The remainder of Appendix D provides specific 

guidance on estimating the rate constants and the relative fraction for latex paint and alkyd paint 

for cases (1) where chamber data are available, and (2) only the results of bulk analysis (VOC 

weight fraction) are known, together with the chemical’s molecular weight and vapor pressure. 

The parameters for total mass applied and application time are discussed in Appendix E, 

summarized below. It is currently planned that indoor-air modeling efforts for latex and alkyd 

paints will be implemented using MCCEM (Multi-Chamber Concentration and Exposure Model) 

software.

Exposure Scenarios

Appendix E discusses several important aspects of exposure scenarios that need to be 

considered in implementing the suggested emission algorithms. These aspects include (1) 

whether a single room or larger part of a building is painted, and (2) appropriate estimators for 

applied paint mass and application rate. The simplest approach is to restrict the model to a 

single-room painting event. A more complicated approach, which raises issues such as assigning 
6

air flows and their destinations, involves a scenario whereby a series of rooms are progressively 

painted. A suggested compromise is a flexible scenario whereby a single room is painted and 

connected to the rest of the building, but with the size of the room dependent on the size of the 

painting event. This approach is believed to minimize modeling complexities at little expected 

sacrifice in accuracy.

Estimates from a previous study indicate that 4-foot by 8-foot wallboard specimens can 

be painted (single coat) in about 10 minutes (i.e., 3.2 square feet painted per minute. ) The same 

study indicated that the nominal coverage of a gallon of latex paint is 400 square feet. These two 

estimates, taken together, imply that it would take about two hours to apply a gallon of latex 

paint, which would be sufficient to cover the walls and ceilings of a 10-foot by 9-foot by 8-foot 

room (volume of 720 cubic feet or 20 cubic meters). By extension, if we conceive of a 400 m3 

house as being fully painted with 16 such rooms being painted, then it would require 

approximately 16 gallons and 32 hours for one person to apply one coat throughout the house.

Recommendations

As efforts on this assignment progressed, several areas became apparent where improved 

information or theoretical developments would aid any future iterations of estimation procedures 

for latex or alkyd paints:

• For latex paints, more small-chamber experimental data for paint applied to gypsum 

board are needed so that (1) differences across flat, semi-gloss and gloss paints can be 

evaluated, and (2) a broader array of individual VOCs is available to support 

development of regression estimates.

• For alkyd paints, small-chamber data based on paint applied to gypsum board are 

being gathered in current EPA studies. These data should be used to examine the 

emission processes for VOCs from alkyd paints, and to determine if the relationships 

developed in Appendix C can be improved.

• A physical model is needed that accounts for the mass-transfer process yet requires 

only minimum parameter estimates such as VOC weight faction, vapor pressure and 

molecular weight, with possible addition of selected parameters specific to the 
7

painting event (e.g., paint application rate, indoor volume); such a model would 

initially treat the paint as a thin liquid film with suspended solid particles, evolving to 

a solid comprising a matrix of particles and void spaces. Some further specifics in 

this regard are provided in Section 5 (Discussion) of Appendix C.
A-i

APPENDIX A

Characterizing the Emissions of VOCs from Latex Paint
A-ii

Table of Contents

1. Task Description ................................................................................................................ A-1

2. Representation of Emissions of VOCs from Latex Paint................................................... A-1

2.1 Single Exponential Model ............................................................................................ A-1

2.2 Double Exponential Model........................................................................................... A-2

2.3 Semi-Empirical Mass Transfer Model.......................................................................... A-3

2.4 Discussion..................................................................................................................... A-4

3. Estimating the Total Mass Released ................................................................................. A-5

4. Estimation Techniques for the Parameters of the Single and Double Exponential .......... A-8

4.1 Approach A................................................................................................................. A-10

4.2 Approach B................................................................................................................. A-11

4.3 Approach C................................................................................................................. A-12

4.4 Application of the Three Approaches......................................................................... A-12
A-1

1. Task Description

The objective of the work presented in this appendix was to provide technical support for 

developing exposure assessment methodologies for latex wall paint. The following specific 

items were addressed:

1. Development of a methodology for representing the emission profile of VOCs 

from a surface painted with latex paint. Possible representations include: (1) an 

empirically fit single exponential based on chamber data; (2) an empirically fit 

double exponential based on chamber data; and (3) a semi-empirical mass-transfer 

representation based on the AEERL and Acurex studies.

2. Evaluate the methodologies and select an appropriate methodology for use in 

exposure assessments.

3. Based on the chosen methodology, develop a representation of the emission 

strength as a function of time and application rate.

4. Develop a method for representing the emission profile based on chemical 

properties, for cases where no chamber data are available.

5. Specify an algorithm to simulate the emission rate as a function of time and mass 

applied for use in a mass-balance model.

2. Representation of Emissions of VOCs from Latex Paint

2.1 Single Exponential Model

In the past, the emission of VOCs has often been represented by single exponential. The 

single exponential can be empirically fit to chamber data, and the parameters estimated using 

a non-linear regression technique. The time-varying emission rate for the single exponential 

is given by the following equation:

S t E e

kt ( ) = −

0 (1)

where: S(t) = Source strength as a function of time (mass/time)

E0 = Initial emission rate (mass/time)

k = First-order rate constant (time-1)

t = Time.
A-2

The mass-balance equation for a chamber with a constant airflow rate and a single source is as 

follows:

V dC

dt = S(t) − QC (2)

where: C = Concentration in the chamber (mass/volume)

V = Volume of the chamber

Q = Airflow rate into and out of the chamber (volume/time).

Integrating Equation 2 with the source term defined by Equation 1 and assuming an initial 

concentration of zero gives the following equation:

C t

E

V Q

V

k

e e

kt

Q

V

t

( ) =

− 



 





− 













 − − 0 (3)

Using the chamber data and Equation 3, non-linear regression is used to obtain estimates of 

the initial emission rate, E0, and the first-order rate constant, k. 

The estimated parameters for the four VOCs measured in the AEERL study, test 7 (a chamber 

test applying Sherwin Williams 99 latex paint to pre-painted gypsum), and the resulting 

emission profiles are shown in Figures 1-4. The single exponential is shown to be reasonably 

proficient at representing the initial rise that takes place in the first 48 hours, but significantly 

under-predicts the long-term concentrations associated with later emissions.

2.2 Double Exponential Model

The double exponential uses one exponential to represent the “fast” emission rate that occurs 

primarily when the paint is in the wet phase of the drying process, and a second exponential is 

used to represent the “slow” emission rate that occurs during the dry phase. For this reason, 

the double exponential is expected to offer an improved fit to the concentrations observed 

after the initial 48 hours. The double exponential is given by the following equation:

S t E e E e

k t k t ( ) = + − −

01 02

1 2 (4)

where: S(t) = Source strength as a function of time (mass/time)

E01 = Initial emission rate for first exponential (mass/time)

E02 = Initial emission rate for second exponential (mass/time)

k1 = First-order rate constant for first exponential (time-1

)

k2 = First-order rate constant for second exponential (time-1)

t = Time.
A-3

Integrating Equation 2 with the source term defined by Equation 4 and assuming an initial 

concentration of zero gives the following equation:

C t

E

V Q

V

k

e e

E

V Q

V

k

e e

k t

Q

V t k t

Q

V t

( ) =

− 



 





− 



 



 +

− 



 





− 



 



 − − − − 01

1

02

2

1 2 (5)

Using the chamber data and Equation 5, non-linear regression is used to obtain estimates of 

the initial emission rates, E01 and E02, and the first-order rate constants, k1 and k2.

The estimated parameters for the four VOCs measured in the AEERL study, test 7, and the 

resulting emission profiles are shown in Figures 5-8. The double exponential provides an 

improved representation of both the initial rise that takes place in the first 48 hours and the 

later emissions. Note that three different estimation methods are shown in these figures; these 

methods are discussed later (see Section 3).

2.3 Semi-Empirical Mass Transfer Model

A mass transfer emission model (“Data Summary for the Latex Paint Assessment Program, 

II. Dynamic and Static Chamber Testing and Model Development,” June 9, 1995, Prepared by 

Acurex Environmental Corp) provides a semi-empirical method for estimating the emissions. 

The proposed model is as follows:

S t k C M

M C MM f M

t m vo v

vo

v

vo D

D ( ) = . 



 



 − 



 



 + − 



 



 



 

 1 

2

1 0 5 (6)

where: S(t) = Source strength as a function of time (mg/m2/hr);

km = Mass transfer coefficient (m/hr);

Cvo = Partial pressure for the VOC of interest (mg/m3

);

Mv = Mass of the VOC remaining in wet stage emissions (mg/m2);

Mvo = Total mass of the VOC in wet stage emissions (mg/m2

);

C = Air concentration for the VOC of interest (mg/m3);

fD1 = Diffusion constant 

= 0.632*Di

1/2 / λ ;

Di = Molecular diffusivity of compound i in solid or liquid phase 

(m2/hr);

λ = Thickness of the diffusion layer (m);

MD = Total mass of the VOC in dry stage emissions (mg/m2

);

t = time (hr).

The fit of the mass-transfer model to chamber data is shown for test 10 in Figure 5.3 in the 

Acurex report. In Figure 6.5 of that report, the model fit to a painting episode at the 

EPA/AEERL test house is shown. In both cases, the model provides a good fit to the data. 

However, this model has many parameters that are difficult to estimate because experimental 

data are required. In addition, the model has not been shown to represent other scenarios.
A-4

2.4 Discussion

Each of these representations of the emission of VOCs from paint as it dries offers advantages 

and disadvantages. The single exponential requires only two parameters, and therefore the 

estimation of the parameters can be accomplished with a relatively simple technique. The 

disadvantages include a limited ability to fit the expected shape of the emission profile, and 

the lack of representation of the physical process that occurs as the paint dries. The double 

exponential offers an improved ability to fit the expected shape of the emission profile at the 

expense of two additional parameters, but also does not represent the physical process. The 

semi-empirical mass-transfer model begins to use the underlying processes as a basis for the 

emission profile. However, the model only includes part of the processes to explain the 

variations in emission rates, and has many parameters that are difficult to estimate.

An ideal model could be expected to be based entirely on the physical and chemical processes 

occurring in the paint as it dries. The paint is probably best represented by a 

diffusion/evaporation process from media (the paint) that is continually changing its 

composition. The movement of the VOCs occurs at both interfaces: the interface with the 

room air and the interface with the substrate material. The diffusion process within the paint 

film is affected by the changing composition of the paint, primarily the changes in void spaces 

between the solid particles in the paint. As the paint dries, the volatile and liquid components 

leave the system and the grains coalesce, leaving smaller openings and more tortuous 

diffusion paths for the remaining components as the drying proceeds. This can be modeled in 

a similar manner as diffusion in porous media, where the reduction in void space is 

representative of the reduction in diffusivity.

However, for the scope of this task, the development of a truly representative physical model 

is not feasible. Of the current options, the semi-empirical mass transfer model is most 

representative of the physical processes. However, it requires estimation of a greater number 

of parameters without an obvious improvement in fit. Therefore, in light of the available 

information, the representation chosen for this task is either the single or double exponential 

model. An estimation technique for the parameters of both models is presented, and each 

technique is evaluated for ability to represent the emission profile of various chemicals from 

painted surfaces. 

These semi-empirical and empirical approximations share several limitations. The source 

strengths given by these representations are not adjusted for painting conditions such as 

temperature, relative humidity, and thickness of the applied paint coating. In addition, the 

effects of concentration in the adjoining room air and reversible/irreversible sinks within the 

room are not considered. 
A-5

3. Estimating the Total Mass Released

When estimating the exposure of an occupant, the most important quantity is the total amount 

of the VOC released from the paint (fraction of applied mass). To obtain a reasonable 

estimate of the total emission and to gain insights into the relationship between the parameters 

of the double exponential and the chemical properties of each VOC, double exponentials have 

been carefully fit to the AEERL chamber data from Test 7. Test 7 is the emission resulting 

from applying Sherwin William 99 paint (a flat latex paint) to pre-painted gypsum wallboard. 

The fitted double exponential can be integrated to yield an estimate of the total mass of the 

VOC released. Because it represents the most common “real-world” scenario, the chamber 

data for a pre-painted gypsum board substrate were used for all data analysis. For estimation 

of the parameters of a double exponential, three alternative fitting techniques were evaluated, 

as follows:

• Technique A: A single exponential is fit to the first 48 hours of data, then a second 

single exponential is fit to the difference between the data points and 

the first exponential. The two exponentials are summed for the 

double exponential model.

• Technique B: A single exponential is fit to the first 48 hours of data, then a second 

single exponential is fit to the data after 48 hours. The two 

exponentials are summed for the double exponential model.

• Technique C: A single exponential is fit to the data after 48 hours, then a second 

single exponential is fit to the difference between the data points and 

the first exponential. The two exponentials are summed for the 

double exponential model.

Each of these techniques was evaluated for the quality of the fit by visually evaluating the fit 

and by comparing the values of R2 . The resulting fit to the concentration data for each of the 

three techniques is shown in Figures 5-8. Technique C yielded the best overall fit to the data. 

This technique outperformed the other techniques primarily because it considered the tail of 

the data first; the resulting contribution of the exponential fitted to the tail was well under the 

data values for the first 48 hours, leaving adequate room to fit the other exponential to the 

difference. The other two techniques often over-predicted the peak. Technique C is described 

in greater detail in Appendix B, including examples of the technique applied to the data from 

Test 7.

Technique C utilizes the following two steps to estimate the parameters of the double 

exponential:

1. Use nonlinear regression to estimate the parameters of the single exponential 

that best fits the data after 48 hours. This exponential represents the second 
A-6

exponential in the double exponential (the “slow” emission). The equation 

representing the concentration in a chamber with a constant air flow and a 

single exponential source is:

C t

E

V Q

V

k

e e

k t

Q

V t

( ) =

− 



 





− 



 



 − − 02

2

2 (7)

where: C = Concentration in the chamber (mass/volume)

V = Volume of the chamber

Q = Air flow rate into and out of the chamber (volume/time).

E02 = Initial emission rate for the second exponential (mass/time)

k2 = First-order rate constant for the second exponential (time-1)

t = Time (t > 48 hours).

Each of the above parameters is known from the chamber test conditions 

except E02 and k2, which are estimated with nonlinear regression analysis. 

2. Use nonlinear regression to estimate the parameters for another single 

exponential, such that this exponential fits the difference between the data and 

the predicted concentrations from the exponential that best fits the data after 48 

hours. Operationally, this is accomplished by putting the parameters (E02 and 

k2 estimated in step 1 a above) for the second exponential into the equation as 

“knowns”. The equation representing the concentration in the chamber with a 

constant air flow and the double exponential source is:

C t

E

V Q

V

k

e e

E

V Q

V

k

e e k t

Q

V t k t

Q

V t

( ) =

− 



 





− 



 



 +

− 



 





− 



 



 − − − − 01

1

02

2

1 2 (8)

where: E01 = Initial emission rate for first exponential (mass/time)

k1 = First-order rate constant for first exponential (time-1

)

t = Time (entire time series).

Using the fitted parameters from step 1 leaves only two parameters to be 

estimated, E01 and k1.

Each of the fitted double exponentials is integrated to infinity to obtain an estimate of the total 

mass of each VOC released. Table 1 presents the estimated parameters and the estimated 

mass of VOC released from the paint using the fitted double-exponential source term. The 

first exponential is assumed to represent the short-term emissions, and the second is assumed 

to represent the long-term emissions.
A-7

Table 1. Summary of Parameters of the Double Exponential Fit of VOC data from Test 7 

using Technique C.

Compound E01

mg/hr

k1

1/hr

E02

mg/hr

k2

1/hr

Total Mass of

VOC Released

mg

Total Mass 

of VOC 

Applied

mg

Fraction of Applied 

Mass Released

mg/mg

1st 2nd Total 1st 2nd Total

Propylene Glycol 0.81498 49.678 0.005827 0.002987 0.0164 1.9508 1.967 8.86 0.0019 0.220 0.222

Ethylene Glycol 2.8118 1.3443 0.05483 0.002643 2.0916 20.746 22.838 91.68 0.0228 0.2263 0.249

Butoxyethoxyethanol 0.2530 0.1584 0.02246 0.009896 1.5968 2.2700 3.866 19.02 0.0840 0.1193 0.203

Texanol 1.4649 0.9441 0.2179 0.0131 1.5516 16.653 18.205 51.57 0.0301 0.3229 0.353

The mass applied, reported in Table 1, is taken from the bulk analysis reported by AEERL 

(Attachment D, “Phase I Data Summary, including QA/QC Data,” Table 3, August 16, 1994, 

Prepared by Acurex Environmental Corp.). The mass of VOCs released is estimated by 

integrating the fitted exponentials to infinity. Table 1 indicates that the total fraction of VOC 

leaving the paint is between 20% and 35% for these compounds, averaging close to 25%. In 

addition, the amount attributed to the first exponential is less than 10% for most compounds. 

Although the double exponential source term offers a reasonable fit to the concentration 

profiles, the ultimate fraction of each VOC that is being emitted is only an estimate.

Technique C was then applied to the chamber data from Test 6. Test 6 is the emissions 

resulting from applying Sherwin William 1629 paint (a flat latex paint), hereafter referred to 

as the RTI paint, to pre-painted gypsum wallboard. The bulk analysis for the RTI paint, 

whose results are shown in Table 2, was conducted separately by RTI and by AEERL. It is 

noteworthy that the values reported by RTI are considerably lower than those reported by 

AEERL, for the four compounds analyzed in this appendix. For consistency, this analysis 

used the values reported by AEERL for both Tests 7 and 6. The parameters for the resulting 

fits are shown in Table 3. The estimated fraction released is less for Test 6, ranging from 10 

to 30%, and averaging close to 15%. However, if the RTI bulk analysis (shown in Table 2) 

were used, the average would be close to 25%.

Table 2. Bulk Analysis for RTI Paint (After AEERL, Attachment D, “Phase I Data 

Summary, including QA/QC Data,” Table 4, August 16, 1994, Prepared by Acurex 

Environmental Corp.)

By AEERL 

(mg/g)

By RTI

(mg/g)

Propylene Glycol 3.27 1.64 

Ethylene Glycol 44.5 29.0

Butoxyethoxyethanol 1.83 1.54 

Texanol 10.2 5.13 
A-8

Table 3. Summary of Parameters of the Double Exponential Fit of VOC data from Test 6.

Compound E01

mg/hr

k1

1/hr

E02

mg/hr

k2

1/hr

Total Mass of

VOC Released

mg

Total Mass 

of VOC 

Applied

mg

Fraction of Applied 

Mass Released

mg/mg

1st 2nd Total 1st 2nd Total

Propylene Glycol 1.0072 81.218 0.01468 0.01251 0.0124 1.1738 1.1862 11.61 0.0011 0.101 0.1022

Ethylene Glycol 0.9271 0.4223 0.05125 0.003069 2.1953 16.701 18.896 157.98 0.0139 0.1057 0.1196

Butoxyethoxyethanol 0.1219 0.4656 0.00879 0.01727 0.2618 0.5088 0.7707 6.50 0.0403 0.0783 0.1186

Texanol 0.8869 0.4676 0.1226 0.0150 1.8968 8.1664 10.063 36.21 0.0524 0.2255 0.2779

4. Estimation Techniques for the Parameters of the Single and Double Exponential

The parameters of the fitted double exponentials shown in Table 1 (E0 and k) were evaluated 

for correlation with chemical properties. The vapor pressures and molecular weights used for 

this investigation are shown in Table 4. The vapor pressures were provided by OPPT.

Table 4. Chemical Properties used for Correlation Analysis.

Compound

Molecular Weight 

(gm/mol)

Vapor Pressure 

(Torr)

Propylene Glycol 76.1 0.2

Ethylene Glycol 62.07 0.05

Butoxyethoxyethanol 162.2 0.02

Texanol 216 0.00189

Initially, the correlation between Chinn’s k (represented as evaporation time, Chinn, K.S.K., 

“A Simple Method for Predicting Chemical Agent Evaporation”, Technical Report, U.S. 

Department of Defense, Defense Technical Information Center, Cameron Station, Alexandria, 

VA, 1981) and k1 was examined, hypothesizing that Chinn’s k could be used as a predictor for 

k1. This was thought to be a reasonable approach, since Chinn’s k was developed based on 

evaporation of VOCs from a pure film, and since k1 is primarily representing the “fast” 

emissions immediately following application of the paint. However, in the analysis of the 

alkyd paints (presented in Appendix C), it was found that the correlation between the vapor 

pressure and k1 was stronger than the correlation for Chinn’s k, and this analysis was 

re-visited.

The analysis revealed a correlation between the values of k1 and the vapor pressure of the 

VOC. The correlation between the fitted values of k1 from Test 7 and the vapor pressures of 

the four VOCs are shown in Figure 9. Regression through the means of the independent and 

dependent variables produced a relationship with a negative intercept, resulting in a negative 

value for k1 in cases where the vapor pressure is less than approximately 0.02 torr. This result 
A-9

would be unacceptable, and therefore the analysis was repeated, forcing it through the origin, 

as shown in Figure 9. 

Since, in the initial phases of the paint drying process, the VOCs are leaving the paint 

primarily through evaporation, it is reasonable for the initial rate constant to be correlated with 

the VOC’s vapor pressure. The number of data points is small, and therefore this relationship 

cannot be used with great confidence. But, without further chamber tests for additional 

compounds, this relationship provides a simple predictor for k1.

The analysis also revealed a correlation between the values of k2 and the molecular weight. 

The correlation between the fitted values of k2 from Test 7 and the molecular weights of the 

four VOCs are as shown in Figure 10. Regression through the means of the independent and 

dependent variables again produced a relationship with a negative intercept. The analysis 

forcing a zero intercept is also shown in Figure 10. Although the number of data points is 

admittedly small, a linear relationship is apparent.

The correlation of k2 with molecular weight can be explained by diffusion theory. The 

long-term flux of the VOC from the paint is primarily a diffusion process, and the diffusivity 

of a compound is highly correlated with its molecular weight. 

0

10

20

30

40

50

60

70

0 0.05 0.1 0.15 0.2 0.25 0.3

Vapor Pressure (torr)

k 1 (hr-1 )

k 1 = (233.25)*VP

R2

 = 0.92

k 1 = (264.7)*VP - 4.961

R2

 = 0.96

Figure 9. Correlation between Vapor Pressure and k1 for Latex Paint.
A-10

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 50 100 150 200 250

Molecular Weight (g/mol)

k 2 (hr-1 )

k 2 = (7.08E-5)*MW -0.002

R2

 = 0.995

k 2 = (5.8385E-5)*MW

R2

 = 0.96

Figure 10. Correlation between Molecular Weight and k2 for Latex Paint.

An analysis of the total fraction of the VOC emitted allows an estimation of the initial 

emission rate, E0. The estimated mass of VOC released by the paint using the fitted double 

exponential source term, shown in Table 1, is used as a guideline for the estimating the 

fraction of applied mass that is ultimately emitted. The table indicates that the total fraction of 

VOC leaving the paint is between 20% and 35% for these compounds, and for most 

compounds, less than 10% of this is attributed to the first (“fast”) exponential. This suggests 

that, in the absence of a better model, an empirical approach can be developed to yield a 

reasonable estimate of the emission profile. Two suggested approaches, labeled Approach A 

and Approach B, are presented below. In addition, Approach C suggests a parameter 

estimation technique for paints where chamber data are available.

4.1 Approach A

Approach A relies on the results of the above regressions to estimate a double exponential 

emission term. The following steps are undertaken to estimate the parameters of the two 

exponentials:

1. Assume 25% (or another chosen percentage) of the total mass is released. Allocate 

10% of the amount released to the first exponential and the remaining 90% to the 

second exponential. 
A-11

2. Calculate k1 from empirical relationship between k1 and vapor pressure:

k1 (hr ) ( ) (Vapor Pressure(torr)) 1 23325 − = . ∗

3. Calculate k2 from empirical relationship between k2 and molecular weight

k2 (hr ) ( ) ( Molecular Weight) 1 5 5839 10 − − = . × ∗

4. Calculate E01 and E02 such that the correct mass of the VOC is released for each 

exponential, as follows:

Total Mass of VOC Released = ( Mass of Paint Applied) ∗ (VOC Weight Fraction) ∗ (0.25)

E (Total Mass of VOC Released) k 01 1 = 0.10 ∗ ∗

E (Total Mass of VOC Released) k 02 2 = 0.90 ∗ ∗

4.2 Approach B

Approach B uses Chinn’s estimate of the first order release rate and the estimate of the 

fraction of the VOC released to estimate a single exponential emission term. The following 

steps are undertaken to estimate the parameters of the exponential:

1. Calculate k1 using Chinn’s approximation:

( ) ( ) ( ) Volatility mg m Molecular Weight g mol Vapor Pressure mm Hg

Temperature K

o ( / ) ( / ) ( )

( )

3 16036 = ∗ ∗

( ( )) Evaporation Time seconds

Volatility

( )

. . log

=

− ∗

10

7 3698 0 9546 10

( )

k hr hr

Evaporation Time

− 



 



 =





 





1

3600 01 sec ln .

(sec)

2. Assume 25% (or another chosen percentage) of the total mass is released. 

Total Mass of VOC Released = ( Mass of Paint Applied) ∗ (VOC Weight Fraction) ∗ (0.25)

3. Use the value of k, estimated above, to calculate the initial emission rate:

E (Total Mass of VOC Released) k 0 = ∗
A-12

4.3 Approach C

Approach C is used when chamber data are available. Approach C uses the parameters of the 

double exponentials fitted to the chamber data to represent the emission rate of the VOCs. 

The double exponentials are fitted to the data using the following technique:

• A single exponential is fit to the data after 48 hours and the resulting parameters 

are used to represent the second “slow” exponential. Then another single 

exponential is fit to the difference between the data points and the “slow” 

exponential. The resulting parameters are used to represent the first “fast” 

exponential. The two exponentials are summed for the double exponential model.

4.4 Application of the Three Approaches

The above approaches have been applied to the AEERL chamber data for Sherwin Williams 

99 paint, upon which the parameter estimates are based, and their resulting fits are shown in 

Figures 11-14. In addition, each approach has been applied to chamber data from another 

paint, referred to as the RTI paint (Sherwin Williams 1629 latex paint, but with a different 

composition). The comparisons between the chamber data and the predicted concentrations 

using each approach are shown in Figures 15-18. Visual inspection of the resulting fits to the 

data suggests that Approach C should be used whenever chamber data are available.

Each of these approaches produces the same mass of VOC emissions, by design. The double 

exponential provides a better fit to the data, and should be used if long-term exposure is of 

interest. However, if only worst-case, short-term exposure estimates are of interest, the single 

exponential may be sufficient. 

Approaches A and B offer an extremely simple technique for approximating the emission rate 

of VOCs from paint. Each approach requires the knowledge of the mass fraction of the VOC 

of interest in the paint, and the molecular weight and vapor pressure of the VOC. Given this 

information, either approach can be implemented to estimate the emission profile. When 

chamber data are not available, Approach A provides a reasonable approximation of the 

emission profile. Although the initial peak is often over- or under-predicted, the fit at the tail 

appears to be very good and the overall fit is adequate.
A-13

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Propylene Glycol

Single Exponential Fit

The single exponential is fit using the first 48 hours of data to a single exponential such 

that: E0 = 0.009124 mg/hr; k = 0.030486 hr-1

(mg/m3

)

Figure 1. Single Exponential Fit of Propylene Glycol Data from Test 7.
A-14

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

5

10

15

20

25

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Ethylene Glycol

Single Exponential Fit

The single exponential is fit using the first 48 hours of data to a single exponential such 

that: E0= 2.44047 mg/hr; k = 1.03626 hr-1

(mg/m3)

Figure 2. Single Exponential Fit of Ethylene Glycol Data from Test 7.
A-15

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

1

2

3

4

5

6

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Butoxyethoxyethanol

Single Exponential Fit

The single exponential is fit using the first 48 hours of data to a single exponential such 

that: E0 = 0.243392 mg/hr; k = 0.096014 hr-1

(mg/m3

)

Figure 3. Single Exponential Fit of Butoxyethoxyethanol Data from Test 7.
A-16

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

2

4

6

8

10

12

14

16

18

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Texanol

Single Exponential Fit

The single exponential is fit using the first 48 hours of data to a single exponential such 

that: E0 = 0.606373 mg/hr; k = 0.053051 hr-1

(mg/m3

)

Figure 4. Single Exponential Fit of Texanol Data from Test 7.
A-17

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Propylene Glycol (mg/m3)

Double Exponential Fit B

Double Exponential Fit A

Double Exponential Fit C

Double Exponential A is fit by first fitting a single exponential to first 48 hours, and then fitting a second exponential to 

the difference between the data points and the first single exponential. The results are summed for the fit below such that: 

 E01 = 0.009124 mg/hr k1 = 0.030486 hr-1

 E02 = 0.001134 mg/hr k2 = -0.002433 hr-1

Double Exponential C is fit by first fitting a single exponential data after 48 hours, and then fitting a second 

exponential to the difference between the data points and the fitted exponential. The results are summed 

for the fit below such that: 

 E01 = 0.814975 mg/hr k1 = 49.6775 hr-1

 E02 = 0.005827 mg/hr k2 = 0.002987 hr-1

Double Exponential B is fit by first fitting a single exponential to first 48 hours, and then fitting a second exponential to 

the data after 48 hours. The results are summed for the fit below such that: 

 E01 = 0.009124 mg/hr k1 = 0.030486 hr-1

 E02 = 0.005959 mg/hr k2 = 0.003077 hr -1

Figure 5. Double Exponential Fit of Propylene Glycol Data from Test 7.
A-18

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

5

10

15

20

25

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Ethylene Glycol (mg/m3) Double Exponential Fit A

Double Exponential Fit B Double Exponential Fit C

Double Exponential A is fit by first fitting a single exponential to first 48 hours, and then fitting a second 

exponential to the difference between the data points and the first single exponential. The results are summed 

for the fit below such that: 

 E01 = 0.244047 mg/hr k1 = 1.03626 hr-1

 E02 = 0.06000 mg/hr k2 = 0.003168 hr-1

Double Exponential C is fit by first fitting a single exponential data after 48 hours, and then fitting a second 

exponential to the difference between the data points and the fitted exponential. The results are summed for 

the fit below such that: 

 E01 = 126.079 mg/hr k1 = 73.3729 hr-1

 E02 = 0.054833 mg/hr k2 = 0.002643 hr-1

Double Exponential B is fit by first fitting a single exponential to first 48 hours, and then fitting a second 

exponential to the data after 48 hours. The results are summed for the fit below such that: 

 E01 = 0.244047 mg/hr k1 = 1.03626 hr-1

 E02 = 0.061181 mg/hr k2 = 0.00313 -1

Figure 6. Double Exponential Fit of Ethylene Glycol Data from Test 7.
A-19

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

1

2

3

4

5

6

7

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Butoxyethoxyethanol (mg/m3) Double Exponential Fit A

Double Exponential Fit B Double Exponential Fit C

Double Exponential A is fit by first fitting a single exponential to first 48 hours, and then 

fitting a second exponential to the difference between the data points and the first single 

exponential. The results are summed for the fit below such that: 

 E01 = 0.243392 mg/hr k1 = 0.09014 hr-1

 E02 = 0.005387 mg/hr k2 = 0.001103 hr-1

Double Exponential C is fit by first fitting a single exponential data after 48 hours, and then 

fitting a second exponential to the difference between the data points and the fitted exponential. 

The results are summed for the fit below such that: 

 E01 = 0.252980 mg/hr k1 = 0.158431 hr-1

 E02 = 0.022464 mg/hr k2 = 0.009896 hr-1

Double Exponential B is fit by first fitting a single exponential to first 48 hours, and then fitting a second 

exponential to the data after 48 hours. The results are summed for the fit below such that: E01 = 

0.243392 mg/hr k1 = 0.09014 hr-1

 E02 = 0.018773 mg/hr k2 = 0.008486 -1

Figure 7. Double Exponential Fit of Butoxyethoxyethanol Data from Test 7.
A-20

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

5

10

15

20

25

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION

Texanol (mg/m3) Double Exponential Fit A

Double Exponential Fit C Double Exponential Fit B

Double Exponential A is fit by first fitting a single exponential to first 48 hours, and then fitting a second 

exponential to the difference between the data points and the first single exponential. The results are summed 

for the fit below such that: 

 E01 = 0.606373 mg/hr k1 = 0.053051 hr-1

 E02 = 0.015984 mg/hr k2 = -0.00033 hr-1

Double Exponential C is fit by first fitting a single exponential data after 48 hours, and then fitting a 

second exponential to the difference between the data points and the fitted exponential. The results 

are summed for the fit below such that: 

 E01 = 1.46487 mg/hr k1 = 0.944103 hr-1

 E02 = 0.217927 mg/hr k2 = 0.013086 hr-1

Double Exponential B is fit by first fitting a single exponential to first 48 hours, and then fitting a second 

exponential to the data after 48 hours. The results are summed for the fit below such that: 

 E01 = 0.606373 mg/hr k1 = 0.053051 hr-1

 E02 = 0.201558 mg/hr k2 = 0.012332 hr -1

Figure 8. Double Exponential Fit of Texanol Data from Test 7.
A-21

TEST 7: SW 99 PAINT APPLIED TO PAINTED GYPSUM

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Propylene Glycol 

Approach A: Double exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

TEST 7: SW 99 PAINT APPLIED TO PAINTED GYPSUM

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION (mg/m3

)

Propylene Glycol 

Approach A: Double exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

Figure 11. Comparison of the Estimated Chamber Concentrations by the Three 

Approaches to the Propylene Glycol Data from Test 7.
A-22

TEST 7: SW 99 PAINT APPLIED TO PAINTED GYPSUM

0

5

10

15

20

25

30

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Ethylene Glycol

Approach A: Double exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

TEST 7: SW 99 PAINT APPLIED TO PAINTED GYPSUM

0

5

10

15

20

25

30

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION (mg/m3

)

Ethylene Glycol

Approach A: Double exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

Figure 12. Comparison of the Estimated Chamber Concentrations by the Three 

Approaches to the Ethylene Glycol Data from Test 7.
A-23

TEST 7: SW 99 PAINT APPLIED TO PAINTED GYPSUM

0

1

2

3

4

5

6

7

8

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Butoxyethoxyethanol

Approach A: Double exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

TEST 7: SW 99 PAINT APPLIED TO PAINTED GYPSUM

0

1

2

3

4

5

6

7

8

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION (mg/m3

)

Butoxyethoxyethanol

Approach A: Double exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

Figure 13. Comparison of the Estimated Chamber Concentrations by the Three 

Approaches to the Butoxyethoxyethanol Data from Test 7.
A-24

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

2

4

6

8

10

12

14

16

18

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Texanol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach C: Fitted Double Exponential

Approach B: Single Exponential based on Chinn's k

TEST 7: PAINT APPLIED TO PAINTED GYPSUM

0

2

4

6

8

10

12

14

16

18

0 50 100 150 200 250 300 350

TIME (HOURS)

CONCENTRATION (mg/m3

)

Texanol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach C: Fitted Double Exponential

Approach B: Single Exponential based on Chinn's k

Figure 14. Comparison of the Estimated Chamber Concentrations by the Two 

Approaches to the Texanol Data from Test 7.
A-25

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Propylene Glycol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375

TIME (HOURS)

CONCENTRATION (mg/m3

)

Propylene Glycol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

Figure 15. Comparison of Estimated Emission by the Three Approaches to the 

Propylene Glycol Data from Test 6.
A-26

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

5

10

15

20

25

30

35

40

45

50

55

60

65

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Ethylene Glycol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

5

10

15

20

25

30

35

40

45

50

55

60

65

0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375

TIME (HOURS)

CONCENTRATION (mg/m3

)

Ethylene Glycol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

Figure 16. Comparison of the Estimated Chamber Concentrations by the Three 

Approaches to the Ethylene Glycol Data from Test 6.
A-27

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

0.5

1

1.5

2

2.5

3

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Butoxyethoxyethanol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

0.5

1

1.5

2

2.5

3

0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375

TIME (HOURS)

CONCENTRATION (mg/m3

)

Butoxyethoxyethanol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

Figure 17. Comparison of the Estimated Chamber Concentrations by the Two 

Approaches to the Butoxyethoxyethanol Data from Test 6.
A-28

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 4 8 12 16 20 24

TIME (HOURS)

CONCENTRATION (mg/m3

)

Texanol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

TEST 6: RTI PAINT APPLIED TO PAINTED GYPSUM

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375

TIME (HOURS)

CONCENTRATION (mg/m3

)

Texanol

Approach A: Double Exponential based on Vapor Pressure and Molecular Weight

Approach B: Single Exponential based on Chinn's k

Approach C: Fitted Double Exponential

Figure 18. Comparison of the Estimated Chamber Concentrations by the Three 

Approaches to the Texanol Data from Test 6.
B-i

APPENDIX B

Detailed Explanation and Application of Technique C
B-ii

Table of Contents

1. Background ..............................................................................................................................B-1

2. Fitting the Double Exponential using STATISTICA...............................................................B-2

3. Fitting the Double Exponential using SPSS ............................................................................B-5

4. Fitting the Double Exponential using R0k...............................................................................B-6
B-1

1. Background

The technique used to estimate the parameters to the double exponentials, presented in Appendix 

A as Technique C, is based on separately fitting two single exponentials to characterize the major 

effects of the initial (“fast”)and long-term (“slow”) emissions represented by each exponential. 

The parameters to either of the single exponentials can be estimated by using one of several 

available statistical software packages that has nonlinear parameter estimation capabilities. The 

parameters presented in this report were estimated using either STATISTICA Version 5.0 

(StatSoft, Inc., 2325 East 13th Street, Tulsa, OK, 74104, Phone: 918/583-4149, Copyright 1995) 

or SPSS/PC+ Version 2.0 (SPSS Inc., 444 N. Michigan Avenue, Chicago, IL 60611, Phone: 

312/329-2400, Copyright 1988). Other statistical analysis software packages, such as SAS (SAS 

Institute Inc., SAS Campus Drive, Cary, NC 27513-2414, phone: 919/677-8000 ), SPSS for 

Windows, and Systat (owned by SPSS Inc.), have similar capabilities and also could have been 

used.

The single exponential, as given by equation 1 in Appendix A, is often used to represent a 

contaminant emission rate from a source that has a finite mass of the contaminant, where the 

emission rate declines with the fraction released and, therefore, with time. The parameters of the 

single exponential, E0 and k, affect the shape, including the peak emission rate, the rate of 

decline, and the ultimate mass released by the exponential. The parameter, E0, represents the 

initial emission rate of the exponential, and the rate parameter, k, is related to the half-life, and 

determines the rate at which the emissions approach zero. The half-life (the time required to 

release half of the ultimate emissions) of the emission profile is:

Half Life = − = ln(0.5) 0.6931

k k (1)

Larger values of k indicate that the mass is being released at a faster rate. A single exponential 

can be integrated from the time of application to infinity to calculate the amount of mass that is 

released. The resultant mass is given by:

Mass Released = =

=

∞

−

∫ E e

E

k t

kt

0

0

0 (2)

Given the above solution, there are infinite ratios of E0/k that yield the same release of mass. 

However, the shape of the resultant profile changes with the magnitudes of the parameters. 

When evaluating alternative possible parameter sets, the most important feature to consider is the 

quantity of mass released. 

Consider two examples; in the first case, E0 = 1 and k = 0.1, and in the second case E0 = 10 and 

k = 1. Since the ratio E0/k is the same for both cases, the mass released is the same. But, the 

second case has an initial emission rate a factor of ten larger and a half-life a factor of ten smaller 

as compared to the first case. This indicates that the emissions represented by the second case 
B-2

are initially much higher, and approach zero much sooner that the emissions represented by the 

first case.

2. Fitting the Double Exponential using STATISTICA

STATISTICA was used to estimate the parameters as described in Appendix A. The following 

steps can be performed to estimate the parameters of the double exponential:

1. Enter the data after 48 hours into the STATISTICA data management spreadsheet, 

as shown in Figure 1.

2. Use nonlinear regression to estimate the parameters of the single exponential that 

best fits the data after 48 hours. This exponential represents the second 

exponential in the double exponential (the “slow” emission). The equation 

representing the concentration in a chamber with a constant air flow and a single 

exponential source is:

C t

E

V Q

V

k

e e

k t

Q

V t

( ) =

− 



 





− 



 



 − − 02

2

2 (3)

where: C = Concentration in the chamber (mass/volume)

V = Volume of the chamber

Q = Air flow rate into and out of the chamber (volume/time).

Figure 1. STATISTICA Data Entry Screen with the Data for Propylene Glycol after 48 

Hours from Test 7.
B-3

E02 = Initial emission rate for the second exponential (mass/time)

k2 = First-order rate constant for the second exponential (time-1

)

t = Time (t > 48 hours).

Equation 3 is entered into the nonlinear parameter estimation module within 

STATISTICA, by specifying “User-specified regression” and opening the 

“Function to be estimated & loss function” window, as shown in Figure 2.

The regression analysis is then conducted, taking care to ensure that the starting 

values entered for the two parameters are estimated appropriately. From the 

completed analysis shown in Appendix A, E02 can be expected to be on the order 

of 0.1 or smaller, and k2 can be expect to be on the order of 0.01 or smaller. In 

addition, the time step must be specified to be sufficiently small to avoid 

instability problems (on the order of 0.01).

3. Enter the data for the entire time series into the STATISTICA data management 

spreadsheet, as shown in Figure 3.

4. Use nonlinear regression to estimate the parameters for the other (“fast”) single 

Figure 2. Statistica Entry Screen for “User-specified regression” Equation with Equation 

for Single Exponential to be Fitted to the Propylene Glycol Data after 48 Hours.
B-4

exponential, such that this exponential fits the difference between the data and the 

predicted concentrations from the exponential that best fits the data after 48 hours. 

Operationally, this is accomplished by putting the parameters (E02 and k2 

estimated in step 2 above) for the second exponential into the equation as 

“knowns”. The equation representing the concentration in the chamber with a 

constant air flow and the double exponential source is:

C t

E

V Q

V

k

e e

E

V Q

V

k

e e k t

Q

V t k t

Q

V t

( ) =

− 



 





− 



 



 +

− 



 





− 



 



 − − − − 01

1

02

2

1 2 (4)

where: E01 = Initial emission rate for first exponential (mass/time)

k1 = First-order rate constant for first exponential (time-1

)

t = Time (entire time series).

Equation 4 is entered into the nonlinear parameter estimation module within 

STATISTICA, specifying by “User-specified regression” and opening the 

“Function to be estimated & loss function” window, as shown in Figure 4.

Figure 3. STATISTICA Data Entry Screen with the Data for Propylene Glycol from Test 7.
B-5

The regression analysis is then conducted, taking care to ensure that the starting 

values entered for the two parameters are estimated appropriately. From the 

completed analysis shown in Appendix A, both E01 and k1 can be expected to be 

on the order of 1 or larger. In addition, the time step must be specified to be 

sufficiently small to avoid instability problems (on the order of 0.01).

3. Fitting the Double Exponential using SPSS

The parameter estimation is conducted functionally in same manner with SPSS as described 

above for STATISTICA. SPSS uses either a command line interface or a batch file of coded 

instructions to accomplish this task. A sample batch file to SPSS is shown in Figure 5 for the 

case of fitting the first exponential after the parameters to the second exponential have been 

estimated. As shown in Figure 5, E01 and k1 are input as variables to be estimated, whereas E02

(0.005827) and k2 (0.002987) are entered as knowns. The data for the analysis is a fixed-column 

file, with the first 8 columns containing the number representing the current time, and columns 9 

through 16 containing the number representing the air concentration for the compound at the 

current time. An example of the data file is shown in Figure 6.

Figure 4. STATISTICA Entry Screen for “User-specified regression” Equation with 

Equation for the Double Exponential to be Fitted to the Propylene Glycol Data 

from Test 7.
B-6

Figure 5. SPSS Batch File of Coded Instructions for Nonlinear Regression Analysis with 

Equation for the Double Exponential to be Fitted to the Propylene Glycol Data 

from Test 7.

Figure 6. SPSS Fixed-Column Input File with the Propylene Glycol Data from Test 7.

4. Fitting the Double Exponential using R0k

The parameter estimation needs to be conducted in a slightly different fashion in R0k because the 

program does not allow the user to enter a user-defined equation. The program’s input screen, 

shown in Figure 7, requires the user to input values for the chamber volume, the chamber air 

exchange rate, the exposed surface area of the source, an estimated peak concentration, and the 

estimated time at which the peak occurs. R0k requires that the effect of the fitted second 

(“slow”) exponential be removed from the data prior to fitting the first (“fast”) exponential. This 

SET EJECT=ON SCREEN=OFF DISK='c:TEST7.OUT' WIDTH=132.

TITLE NONLIN REG FOR CHAMBER PAINT CONCS.

DATA LIST FILE = 'C:TEST7.PRN'

 /T 1-8 C1 9-16.

MODEL PROGRAM E01=10 k1=5.

COMPUTE PRED=(E01/(0.053*(0.535-k1)))*(EXP(0-k1*T)-EXP(0-0.535*T)) + 

(0.005827/(0.053*(0.535-0.002987)))*(EXP(0-0.002987*T)-EXP(0-0.535*T)).

NLR C1 WITH T.

FINISH.

 0 0.01

 0.48 0.31

 1.02 0.24

 2.13 0.22

 3.05 0.24

 3.08 0.22

 5.53 0.23

 7.03 0.23

 10.8 0.22

 24.78 0.11

 27.8 0.12

 31.4 0.14

 31.5 0.14

 48.05 0.17

 55.25 0.14

 75.45 0.15

 97.92 0.17

 170.48 0.17

 221.13 0.07

 221.22 0.21

 268.05 0.12

 343.1 0.01

 343.3 0.02
B-7

is mathematically equivalent to constructing the double exponential equation with the parameters 

to one entered as “known”, as shown in the previous two examples. R0k also has several 

limitations that are important to note. The first limitation is that the value of “R0” estimated by 

program appears to always be reported as an integer. This results in initial emission rates of less

than 1 being reported as zeros. Since nearly all the initial emission rates reported here are less 

than 1, this limitation must be addressed to utilize R0k. The actual emission rate in R0k is the 

product of the “Source Area” multipled by “R0”. Therefore, the “Source Area” can be reduced 

as necessary to increase the value of R0 to an integer. 

The program is also sensitive to the initial inputs for the “Estimated peak conc” and the 

“Estimated time to peak”. When estimating the parameters to the second exponential, the peak 

in the exponential occurs later than the peak in the data set, because the second exponential is not 

attempting to fit the peak in the data, but rather the long-term emissions at the tail of the data. 

Therefore, the peak concentration for the second exponential can be expected to be lower and 

occur later than the peak in the actual data.

R0k can be used to estimate the parameters to the double exponential using the following steps:

1. Enter the data after 48 hours into the R0k spreadsheet, as shown in Figure 7. In 

addition enter the actual chamber volume, chamber air exchange rate, source area 

Figure 7. R0k Data Entry Screen with the Data for Propylene Glycol after 48 Hours from 

Test 7.
B-8

and estimates for the “Estimated peak conc” and the “Estimated time for peak”. 

As a rule-of-thumb, the “Estimated peak conc” can be entered as the largest 

concentration value in the data set, and the “Estimated time for peak” can be 

entered as approximately 12 hours after the occurrence of the peak. 

2. The regression analysis is then conducted, evaluating the estimated parameters on 

a case-by-case basis for reasonable ranges. The value entered for the source area 

should be adjusted downward from the actual value, until the appropriate number 

of significant digits is achieved. From the completed analysis shown in Appendix 

A, E02 can be expected to be on the order of 0.1 or smaller, and k2 can be expected 

to be on the order of 0.01 or smaller. The value of E02 is the multiple of the value 

entered for the source area and the predicted value for “R0.”

3. After the parameters to the second exponential are estimated, subtract out the 

effect of the second exponential from the data, as shown in Figure 8. Enter the 

resultant data (Figure 8, column F) for the entire time series into the R0k 

spreadsheet, as shown in Figure 9.

Figure 8. Excel Spreadsheet Used to Manipulate the Data for Entry into R0k.
B-9

The second regression analysis is then conducted, evaluating the estimated 

parameters on a case by case basis for reasonable ranges. The value entered for 

the source area should be adjusted downward from the actual value, until the 

appropriate number of significant digits is achieved. From the completed analysis 

in Appendix A, both E01 and k1 can be expected to be on the order of 1 or larger. 

The value of E01 is the multiple of the value entered for the source area and the 

predicted value for “R0.”

Figure 9. R0k Data Entry Screen for the “fast” Exponential with the Residuals for 

Propylene Glycol from Test 7.
C-i

APPENDIX C

Characterizing the Emissions of VOCs from Alkyd Paint
C-ii

Table of Contents

1. Task Description ......................................................................................................................C-1

2. Representation of Emissions of VOCs from Alkyd Paint........................................................C-1

3. Estimating the Total Mass Released .......................................................................................C-2

4. Parameter Estimation..............................................................................................................C-6

5. Discussion...............................................................................................................................C-9
C-1

1. Task Description

The objective of the work presented in this appendix was to provide technical support for 

developing exposure assessment methodologies for alkyd wall paint. The following specific 

items were addressed:

1. Development of a methodology for representing the emission profile of VOCs from a 

surface painted with alkyd paint. Possible representations include: (1) an empirically 

fit single exponential based on chamber data; and (2) an empirically fit double 

exponential based on chamber data.

2. Evaluate the methodologies and select the appropriate methodology for use in 

exposure assessments.

3. Based on the chosen methodology, develop a representation of the emission strength 

as a function of time and application rate.

4. Develop a method for representing the emission profile based on chemical properties, 

for cases where no chamber data are available.

5. Specify the algorithm to simulate the emission rate as a function of time and mass 

applied for use in a mass-balance model.

2. Representation of Emissions of VOCs from Alkyd Paint

The RTI data for VOC emissions from alkyd paint applied to glass comprise 5 tests, with each 

duplicated. Several of the 5 test cases are for specific circumstances not applicable to this work, 

leaving 3 tests, each with a duplicate. The three tests are identical except for the type of paint 

used, described as tests 5 and 6 for gloss paint, 11 and 12 for flat paint, and 13 and 14 for semigloss paint. During each test, the chamber concentration was measured for 15 VOCs at eight 

time intervals, ranging from approximately 1 hour to approximately 24 hours after the paint was 

applied. A single exponential was found to adequately represent the emission rates of the VOCs 

from the drying alkyd paint on glass. As additional information pertaining to alkyd paint applied 

to wallboard is obtained from ongoing EPA small-chamber studies, this finding should be 

re-evaluated. 

The emission rate for the single exponential is given by the following equation:

S t E e

kt ( ) = −

0 (1)

where: S(t) = Source strength as a function of time (mass/time)

E0 = Initial emission rate (mass/time)

k = First-order rate constant (time-1)

t = Time.
C-2

The mass-balance equation for a chamber with a constant airflow rate and a single source is as 

follows:

V dC

dt = S(t) − QC (2)

where: C = Concentration in the chamber (mass/volume)

V = Volume of the chamber

Q = Air flow rate into and out of the chamber (volume/time).

Integrating Equation 2 with the source term defined by Equation 1 and assuming an initial 

concentration of zero gives the following equation:

C t

E

V Q

V

k

e e

kt

Q

V

t

( ) =

− 



 





− 













 − − 0 (3)

3. Estimating the Total Mass Released

When estimating exposure of an occupant, the most important quantity is the total amount of the 

VOC released from the paint (fraction of applied mass). To obtain a reasonable estimate of the 

total emission and to gain insights into the relationship between the parameters of the single 

exponential and the chemical properties of each VOC, a single exponential has been fit to the 

chamber data. A single exponential was used to represent the emission profile because of the 

relatively few data points and the faster emission profile, probably due to the glass substrate. 

There is some uncertainty as to whether the tail is well represented, but without more data the fit 

cannot be objectively evaluated.

Using the chamber data and Equation 3, non-linear regression was used to obtain estimates of the 

initial emission rate, E0, and the first-order rate constant, k. The fits of the single exponentials to 

the measured concentrations for RTI test 11 are in shown Figure 1. The parameters of the fitted 

single exponentials are shown in Tables 1, 2, and 3, for each of the RTI tests. The fitted single 

exponentials were integrated to infinity (time) for each VOC and test to estimate the total mass of 

VOC emitted for the conditions of each chamber test. The estimated emitted mass is shown in 

Tables 1, 2, and 3 along with the estimated fraction of the applied mass emitted during each test. 

The estimated fraction emitted is lowest for flat paint and highest for gloss paint in nearly all 

cases. In addition, for each paint type, the fraction emitted is higher for thinner paint films 

(lower applied unit mass).
C-3

Table 1. Comparison of the Parameters of the Fitted Single Exponential to the Chamber Concentrations for Tests 5 and 6. 

Paint Name: Glidden 4550; Paint Type: Alkyd; Gloss Type: Gloss; Paint Color Name: Hyacinth.

TEST 5

Mass Applied Paint =9.50 grams

TEST 6

Mass Applied Paint =8.78 grams

Fitted 

Parameters

Mass 

VOC 

Emitted

Weight 

Fraction

Mass 

VOC 

Applied

Fraction 

of VOC 

Emitted

Fitted 

Parameters

Mass 

VOC 

Emitted

Weight 

Fraction

Mass 

VOC 

Applied

Fraction 

of VOC 

Emitted

Eo k R2 (Eo/k) mg VOC Emitted Eo k R2 (Eo/k) mg VOC Emitted

COMPOUND (mg/

h)

(h-1) (mg) g Paint (mg) Applied (mg/

h)

(h-1) (mg) g Paint (mg) Applied

m,p-Xylene 16.580 0.779 0.99 21.28 3.58 34.01 0.626 8.391 0.474 0.92 17.70 3.58 31.43 0.563

n-Nonane 30.692 0.408 0.99 75.15 9.82 93.29 0.806 21.566 0.285 0.84 75.75 9.82 86.22 0.879

o-Xylene 5.175 0.673 0.99 7.69 1.06 10.07 0.764 2.680 0.402 0.92 6.67 1.06 9.31 0.716

Propylcyclohexane 10.388 0.340 0.96 30.59 4.19 39.81 0.768 7.741 0.272 0.90 28.42 4.19 36.79 0.772

3- & 4-Ethyltoluene 5.868 0.308 0.99 19.07 1.82 17.29 1.103 4.099 0.222 0.96 18.49 1.82 15.98 1.157

1,3,5-Trimethylbenzene 1.946 0.249 0.99 7.82 0.79 7.49 1.045 1.447 0.188 0.95 7.71 0.79 6.92 1.114

n-Decane 47.763 0.210 0.97 227.22 18.00 171.00 1.329 38.380 0.168 0.91 228.49 18.00 158.04 1.446

2-Ethyltoluene 1.812 0.314 0.97 5.77 0.62 5.86 0.984 1.255 0.232 0.93 5.41 0.62 5.42 0.999

1,2,4-Trimethylbenzene 6.142 0.240 0.99 25.55 2.76 26.22 0.975 4.420 0.173 0.95 25.59 2.76 24.23 1.056

1,2,3-Trimethylbenzene 1.646 0.178 0.99 9.27 0.84 7.97 1.163 1.182 0.127 0.97 9.33 0.84 7.37 1.266

2-Methyldecane 2.719 0.127 0.94 21.42 1.98 18.81 1.139 2.091 0.103 0.84 20.29 1.98 17.38 1.167

transDecahydronaphthalene4.367 0.177 0.98 24.67 2.10 19.95 1.236 3.340 0.135 0.89 24.71 2.10 18.44 1.340

n-Undecane 8.294 0.080 0.93 104.10 9.11 86.55 1.203 6.562 0.056 0.85 116.24 9.11 79.99 1.453

Pentylcyclohexane 0.721 0.041 0.83 17.51 0.52 4.92 3.559 0.463 0.026 0.63 17.65 0.52 4.55 3.881

n-Dodecane 0.777 -0.007 0.92 2.60 24.70 0.000 0.639 -0.018 0.78 2.60 22.83 0.000

TVOC 669.65 0.252 0.99 2652.67 284.00 2698.00 0.983 468.57 0.181 0.96 2590.05 284.00 2493.52 1.039
C-4

Table 2. Comparison of the Parameters of the Fitted Single Exponential to the Chamber Concentrations for Tests 11 and 12. 

Paint Name: Glidden 5700; Paint Type: Alkyd; Gloss Type: Flat; Paint Color Name: Chim Cham.

TEST 11

Mass Applied Paint =13.53 grams

TEST 12

Mass Applied Paint =14.83 grams

Fitted 

Parameters

Mass 

VOC 

Emitted

Weight 

Fraction

Mass 

VOC 

Applied

Fraction 

of VOC 

Emitted

Fitted 

Parameters

Mass 

VOC 

Emitted

Weight 

Fraction

Mass 

VOC 

Applied

Fraction 

of VOC 

Emitted

Eo k R2 (Eo/k) mg VOC Emitted Eo k R2 (Eo/k) mg VOC Emitted

COMPOUND (mg/

h)

(h-1) (mg) g Paint (mg) Applied (mg/

h)

(h-1) (mg) g Paint (mg) Applied

m,p-Xylene 12.294 1.278 0.97 9.62 1.56 21.11 0.456 10.692 0.955 0.99 11.19 1.56 23.13 0.484

n-Nonane 4.257 0.683 0.93 6.23 0.68 9.13 0.682 3.345 0.543 0.96 6.16 0.68 10.01 0.615

o-Xylene 3.011 1.192 0.97 2.53 0.37 4.97 0.509 2.633 0.944 0.95 2.79 0.37 5.44 0.512

Propylcyclohexane 1.141 0.713 0.93 1.60 0.20 2.65 0.604 0.799 0.474 0.93 1.69 0.20 2.91 0.581

3- & 4-Ethyltoluene 0.820 0.585 0.94 1.40 ND -- -- 0.653 0.443 0.89 1.47 ND -- --

1,3,5-Trimethylbenzene 0.255 0.430 0.93 0.59 ND -- -- 0.185 0.282 0.88 0.66 ND -- --

n-Decane 10.816 0.324 0.93 33.36 4.51 61.02 0.547 7.594 0.245 0.89 31.05 4.51 66.88 0.464

2-Ethyltoluene 0.250 0.493 0.93 0.51 ND -- -- 0.194 0.349 0.89 0.55 ND -- --

1,2,4-Trimethylbenzene 0.703 0.336 0.93 2.10 ND -- -- 0.538 0.267 0.84 2.01 ND -- --

1,2,3-Trimethylbenzene 0.195 0.260 0.91 0.75 ND -- -- 0.153 0.204 0.84 0.75 ND -- --

2-Methyldecane 4.016 0.212 0.89 18.92 3.24 43.84 0.431 2.900 0.192 0.83 15.13 3.24 48.05 0.315

transDecahydronaphthalene6.076 0.238 0.95 25.57 3.24 43.84 0.583 4.882 0.230 0.85 21.27 3.24 48.05 0.443

n-Undecane 17.572 0.235 0.91 74.62 19.10 258.42 0.289 15.161 0.268 0.94 56.56 19.10 283.25 0.200

Pentylcyclohexane 2.601 0.188 0.85 13.85 3.01 40.73 0.340 1.786 0.148 0.86 12.10 3.01 44.64 0.271

n-Dodecane 2.716 0.074 0.69 36.59 11.50 155.60 0.235 1.982 0.075 0.68 26.44 11.50 170.55 0.155

TVOC 297.42 0.187 0.86 1593.42 179.00 2421.87 0.658 198.77 0.166 0.76 1197.42 179.00 2654.57 0.451
C-5

Table 3. Comparison of the Parameters of the Fitted Single Exponential to the Chamber Concentrations for Tests 13 and 14.

Paint Name: Glidden UH8000; Paint Type: Alkyd; Gloss Type: Semi-Gloss; Paint Color Name: Seafoam.

TEST 13

Mass Applied Paint =11.19 grams

TEST 14

Mass Applied Paint =12.23 grams

Fitted 

Parameters

Mass 

VOC 

Emitted

Weight 

Fraction

Mass 

VOC 

Applied

Fraction 

of VOC 

Emitted

Fitted 

Parameters

Mass 

VOC 

Emitted

Weight 

Fraction

Mass 

VOC 

Applied

Fraction 

of VOC 

Emitted

Eo k R2 (Eo/k) mg VOC Emitted Eo k R2 (Eo/k) mg VOC Emitted

COMPOUND (mg/

h)

(h-1) (mg) g Paint (mg) Applied (mg/

h)

(h-1) (mg) g Paint (mg) Applied

m,p-Xylene 12.673 0.588 0.99 21.54 3.97 44.42 0.485 12.455 0.494 0.99 25.23 3.97 48.55 0.520

n-Nonane 13.905 0.370 0.97 37.62 5.32 59.53 0.632 13.178 0.294 0.99 44.85 5.32 65.06 0.689

o-Xylene 3.600 0.510 0.99 7.05 1.08 12.09 0.584 3.837 0.495 0.98 7.75 1.08 13.21 0.587

Propylcyclohexane 3.336 0.329 0.98 10.13 1.45 16.23 0.625 3.075 0.281 0.98 10.96 1.45 17.73 0.618

3- & 4-Ethyltoluene 2.762 0.223 0.98 12.41 1.30 14.55 0.853 2.430 0.178 0.97 13.67 1.30 15.90 0.860

1,3,5-Trimethylbenzene 0.941 0.205 0.98 4.58 0.52 5.82 0.788 0.781 0.158 0.96 4.94 0.52 6.36 0.777

n-Decane 13.785 0.239 0.95 57.63 14.10 157.78 0.365 15.039 0.243 0.96 61.94 14.10 172.44 0.359

2-Ethyltoluene 0.922 0.244 0.97 3.78 0.48 5.40 0.700 0.811 0.208 0.97 3.91 0.48 5.91 0.661

1,2,4-Trimethylbenzene 2.619 0.182 0.97 14.41 1.81 20.25 0.712 2.263 0.153 0.94 14.83 1.81 22.14 0.670

1,2,3-Trimethylbenzene 0.733 0.124 0.98 5.91 0.64 7.11 0.831 0.576 0.084 0.95 6.82 0.64 7.77 0.879

2-Methyldecane 1.557 0.077 0.83 20.18 2.63 29.43 0.686 1.329 0.064 0.86 20.77 2.63 32.16 0.646

transDecahydronaphthalene2.450 0.115 0.94 21.32 2.58 28.87 0.739 2.039 0.085 0.91 24.02 2.58 31.55 0.761

n-Undecane 8.344 0.191 0.89 43.65 16.20 181.28 0.241 8.287 0.200 0.93 41.48 16.20 198.13 0.209

Pentylcyclohexane 0.670 0.038 0.87 17.58 2.06 23.05 0.763 0.603 0.039 0.85 15.48 2.06 25.19 0.614

n-Dodecane 0.874 -0.0005 0.90 8.10 90.64 0.000 0.776 0.005 0.94 143.97 8.10 99.06 1.453

TVOC 214.71 0.157 0.92 1369.50 207.00 2316.33 0.591 201.23 0.178 0.93 1131.97 207.00 2531.61 0.447
C-6

4. Parameter Estimation

The fitted rate constant k for the single exponential is likely to be correlated with the chemical 

properties of the VOCs, with the largest correlation being with vapor pressure, since VOC 

emission due to initial paint drying is primarily an evaporative process. The relationship between 

the chemical properties and k was evaluated using both linear and non-linear regression analysis. 

The chemical properties are given in Table 4. Several alternative transformation models were 

examined including log transformations and multiplicative combinations of chemical properties 

similar to Chinn’s model (Chinn, K.S.K., “A Simple Method for Predicting Chemical Agent 

Evaporation”, Technical Report, U.S. Department of Defense, Defense Technical Information 

Center, Cameron Station, Alexandria, VA, 1981). Several models provided useful 

representations of the relationship between k and the chemical properties, as follows (see Table 5 

for parameter estimates):

Model 1: Predicted k = A + B * MW + C *VP

Model 2: Predicted k = A + B * MW *VP

Model 3: Predicted k = A + B *Chinn's k

Model 4: Predicted k A B MW C VP D = + * + *

Model 5: Predicted k = A + B * VP (Recommended)

Table 4. Chemical Properties used in the Regression Analysis.

COMPOUND MW VP

m,p-Xylene 106.17 8.29

n-Nonane 128.26 3.22

o-Xylene 106.17 6.61

Propylcyclohexane 126.24 4.192

3- & 4-Ethyltoluene 120.19 1.36

1,3,5-Trimethylbenzene 120.2 1.36

n-Decane 142.28 2.7

2-Ethyltoluene 120.19 1.36

1,2,4-Trimethylbenzene 120.2 2.1

1,2,3-Trimethylbenzene 120.2 1.36

2-Methyldecane 156.31 --

trans-Decahydronaphthalene 138.25 1.23

n-Undecane 156.31 1

Pentylcyclohexane 154.3 0.5

n-Dodecane 170.34 0.2
C-7

Table 5. Parameters Estimated by Regression Analysis for Models 1 through 5.

Tests 5 & 6 Tests 11 & 12 Tests 13 & 14 All Tests

(Gloss) (Flat) (Semi-Gloss) Combined

Model 1

A (hr-1) 0.5957 0.6518 0.2456 0.4977

B (mol g-1 hr-1) -0.0370 -0.00365 -0.00125 -0.00286

C (Torr-1 hr-1) 0.0504 0.1065 0.0558 0.07089

R2 0.85 0.86 0.88 0.65

Model 2

A (hr-1) 0.0427 0.0914 0.04436 0.05949

B (mol g-1 Torr-1 hr-1) 0.00065 0.00116 0.00059 0.000802

R2 0.73 0.80 0.87 0.60

Model 3

A (hr-1) -0.0076 0.00134 0.00779 0.000497

B (dimensionless) 0.0546 0.09769 0.0476 0.06666

R2 0.86 0.93 0.94 0.69

Model 4

A (hr-1) 1.1237 1.3782 0.04599 0.9080

B(mol0.5 g-0.5 hr-1) -0.0882 -0.1004 -0.00783 -0.0685

C (Torr-D hr-1) 0.0470 0.0606 0.5790 0.0688

D (dimensionless) 1.0227 1.2388 0.1707 1.0084

R2 0.85 0.87 0.90 0.65

Model 5

A (hr-1) 0.0628 0.1259 0.0660 0.08491

B (Torr-1 hr-1) 0.07007 0.1259 0.0624 0.08610

R2 0.77 0.84 0.87 0.62

Based on the above analysis, the various models provided approximately equal representations of 

the relationship between k and the chemical properties. Model 5 was selected as the most useful 

because the predicted k is based on only one chemical property, the vapor pressure. 

The latex paint emissions from wallboard, presented in Appendix A, were best described by a 

double exponential, with the initial rise primarily represented by the first exponential and the tail 

represented primarily by the second exponential. We found that the fitted rate parameters to the 

exponentials, k1 and k2, were correlated to varying degrees with the chemical properties. The 

fitted k to the second (“slow”) exponential, k2, was found to be highly correlated with molecular 

weight, indicating that a diffusive process is occurring. Although there were only four fitted k 

and molecular weight pairs upon which to base this correlation, this is what is expected. The 

fitted k for the first (“fast”) exponential, k1 , was found to be loosely correlated to Chinn’s k 

(based on vapor pressure and molecular weight).

The results from the latex paint analysis indicating that the rate constant for the second 

exponential is a function of the molecular weight, together with this analysis for alkyd paint, 

suggests that it is likely that both processes should indeed be represented by a double 

exponential. The first exponential represents the “fast” emissions taking place when diffusion 
C-8

through the boundary layer is the limiting mass-transfer mechanism. The second exponential 

represents the “slower” emissions, occurring when diffusion through the liquid/solid drying paint 

film is the limiting mass-transfer mechanism. The rate of diffusion through the boundary layer is 

controlled by the concentration gradient across the layer and therefore primarily by the vapor 

pressure of the compound. 

The data used for analysis of alkyd paint emissions are for a glass substrate, and were recorded 

for only 24 hours. The emissions from a painted glass substrate may not be representative of 

emissions from painted gypsum. The lack of data after 24 hours prevents a meaningful analysis 

of the tail (“slow” emissions). The value of k1 (“fast” emissions) for any given VOC appears to 

be quite different across gloss types, as can be seen in Tables 1, 2, and 3. Therefore, it is 

recommended that the data gathered in the current EPA small-chamber studies of emissions from 

alkyd paint on gypsum board be used to evaluate the parameter estimation results presented in 

this report. 

Although only one alkyd paint formulation was examined for each gloss type, the gloss type 

appears to have a distinct impact on the emission profile. For all reported VOCs, the value of the 

fitted rate parameter, k, for the flat paint is larger than for the semi-gloss and gloss paints. There 

is no clear or consistent difference between the fitted values of k for gloss and semi-gloss. An 

analysis of the estimated fraction emitted indicates that, in general, the fraction emitted is largest 

for gloss paint and smallest for flat paint. Based on this analysis of only three alkyd paints, this 

suggests that a single algorithm cannot be applied to all paint gloss types.

A conclusive set of algorithms cannot be specified based on the limited number of paints and 

substrates analyzed. However, based on this analysis, preliminary estimates of the emission 

profiles have been developed, for use until more conclusive evidence is available. The suggested 

techniques given below differentiate between gloss type in specifying the fraction emitted, and 

differentiate for the chemical in specifying the rate constant, k. The fraction emitted for each 

gloss type is specified based on an analyses of the results contained in Tables 1, 2, and 3. The 

median fraction emitted is approximately 0.5, 0.7, and 1.0 for the flat, semi-gloss, and gloss 

paints, respectively. The techniques presented below are based on this observation.

These proposed techniques represent the emission profile for alkyd paint as a single exponential 

as given by equation 3. The parameters E0 and k are estimated as follows:

A. Gloss Type: Flat

1. Set fraction emitted to 50%

2. Estimate k according to model 5, presented in Table 4, as follows:

k (hr-1) = 0.066 + 0.129 * (Vapor Pressure (Torr))

3. Estimate E0 as follows:

E0 = (fraction emitted) * (mass applied) * k

= 0.5 * (mass applied) * k
C-9

B. Gloss Type: Semi-Gloss

1. Set fraction emitted to 75%

2. Estimate k according to model 5, presented in Table 4, as follows:

k (hr-1

) = 0.046 + 0.061 * (Vapor Pressure (Torr))

3. Estimate E0 as follows:

E0 = (fraction emitted) * (mass applied) * k

= 0.75 * (mass applied) * k

C. Gloss Type: Gloss

1. Set fraction emitted to 100%

2. Estimate k according to model 5, presented in Table 4, as follows:

k (hr-1) = 0.024 + 0.074 * (Vapor Pressure (Torr))

3. Estimate E0 as follows:

E0 = (fraction emitted) * (mass applied) * k

= 1.0 * (mass applied) * k

5. Discussion

When estimating exposure of an occupant, the most important quantity is the total mass of the 

compound emitted. The analyses and techniques presented above provide a method of 

estimating the mass released and the general shape of the emission profiles, based on the limited 

available information. These techniques are intended to be used only when no other information 

is available, and should be replaced by better estimates whenever possible.

The above results reinforce the conclusion that an empirical representation may be sufficient for 

initial exposure assessments, but will not adequately account for all the variability across paints 

and substrates. This suggests the need to develop a physically based model to represent to masstransfer processes. A physical model needs to account for the paint initially as a thin liquid film 

with suspended solid particles, evolving to a solid made up of a matrix of particles and void 

spaces. On the wallboard side of the paint film, the liquid portion of the paint is drawn into the 

paper backing primarily due to capillary forces, and the VOCs diffuse into the porous paper and 

gypsum. As the paint film dries, the particles move closer together, eventually becoming nearly 

impervious to diffusion, and trapping the remaining VOCs in the wallboard.

In absence of a physically based model at present, the best recourse is to use the empirically 

derived estimates given above, recognizing their limitations. Perhaps the greatest limitation is 

that the estimates are based on alkyd paint applied to a glass substrate; however, a preliminary 

analysis of data gathered in recent chamber studies appears to provide evidence that a single 

exponential may adequately represent the emission profile. Further analysis of the data gathered 

in current EPA small-chamber studies is required to validate and, if necessary, refine the 

estimation methods and results presented above.
0

10

20

30

40

50

60

70

80

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

m,p-Xylene

FITTED SINGLE EXPONENTIAL

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

n-Nonane

FITTED SINGLE EXPONENTIAL

0

1

2

3

4

5

6

7

8

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

3- & 4-Ethyltoluene

FITTED SINGLE EXPONENTIAL

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

1,3,5-Trimethylbenzene

FITTED SINGLE EXPONENTIAL

Figure 1. Comparison of the Chamber Concentrations to the Fitted Single Exponential for Test 11.
0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

1,2,4-Trimethylbenzene

FITTED SINGLE EXPONENTIAL

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

1,2,3-Trimethylbenzene

FITTED SINGLE EXPONENTIAL

0

50

100

150

200

250

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

n-Undecane

FITTED SINGLE EXPONENTIAL

0

5

10

15

20

25

30

35

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

Pentylcyclohexane

FITTED SINGLE EXPONENTIAL

Figure 1 (CONTINUED). Comparison of the Chamber Concentrations to the Fitted Single Exponential for Test 11.
0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

o-Xylene

FITTED SINGLE EXPONENTIAL

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

Propylcyclohexane

FITTED SINGLE EXPONENTIAL

0.0

20.0

40.0

60.0

80.0

100.0

120.0

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

n-Decane

FITTED SINGLE EXPONENTIAL

0

0.5

1

1.5

2

2.5

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

2-Ethyltoluene

FITTED SINGLE EXPONENTIAL

Figure 1 (CONTINUED). Comparison of the Chamber Concentrations to the Fitted Single Exponential for Test 11.
0

10

20

30

40

50

60

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

2-Methyldecane

FITTED SINGLE EXPONENTIAL

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

trans-Decahydronaphthalene

FITTED SINGLE EXPONENTIAL

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

n-Dodecane

FITTED SINGLE EXPONENTIAL

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25

Time (hours)

Concentration (mg/m3)

TVOC

FITTED SINGLE EXPONENTIAL

Figure 1 (CONTINUED). Comparison of the Chamber Concentrations to the Fitted Single Exponential for Test 11.
D-i

APPENDIX D

Model Algorithms
D-ii

Table of Contents

1. General Equation for Representing Emissions from Paint during Continuous Application... D-1

2. Estimation of Equation Parameters for Latex Paint................................................................ D-2

2.1 Method 1: Estimation of Equation Parameters for Latex Paint Using Appropriate 

Substrate Chamber Data................................................................................................... D-2

2.2 Method 2: Estimation of Equation Parameters for Latex Paint Using Glass or 

Stainless Steel Substrate Chamber Data........................................................................... D-4

2.3 Method 3: Estimation of Equation Parameters for Latex Paint Using Bulk Analysis...... D-5

3. Estimation of Equation Parameters for Alkyd Paint............................................................... D-6

3.1 Method 1: Estimation of Equation Parameters for Alkyd Paint Using Glass or 

Stainless Steel Substrate Chamber Data........................................................................... D-6

3.2 Method 2: Estimation of Equation Parameters for Alkyd Paint Using Bulk Analysis..... D-7
D-1

1. General Equation for Representing Emissions from Paint during Continuous Application

Modeling indoor air concentrations of the VOCs with an ultimate objective of estimating 

inhalation exposure to the VOCs requires that the painting event be represented as a 

progressive event, where paint is being applied to a surface at an essentially constant rate. 

The application of the double-exponential source term to an ongoing event was presented 

in a recent paper by Evans (WC Evans, “Development of Continuous-Application Source 

Terms and Two-Compartment Analytical Solutions,” ASTM Symposium on Methods for 

Characterizing Indoor Sources and Sinks, September 25-28, 1994, Washington, DC). 

This algorithm can be used in a predictive model to represent the source strength as a 

function of time for the combination of (1) continuous application of paint at an 

essentially constant rate, and (2) double exponential emission profile.

Representation of the time-varying source strength for a double exponential during a 

continuous painting event is given by the following equation:1

( ) [ ( ) ( )( )]

( )

( ) ( ) ( )

[ ( )] ( )

S t

M

t

f e f e

M

t

f e f e H t t

a

k t k t

a

k t t k t t

a

a a

= − + − −

− − + − − −

− −

− − − −

1 1 1

1 1 1

1 2

1 2

(1)

where: S(t) = Source strength (mg/hr)

M = Total mass to be emitted (mg)

ta = Paint application time (hours)

f = Fraction of mass emitted from the first exponential (unitless)

k1 = First-order rate constant for first exponential (hours-1

)

k2 = First-order rate constant for second exponential (hours-1

)

t = Time (hours)

and: ( ) ( )

( ) H t t if t t

if t t a

a

a

− = − ≤

− >







0 0

1 0

;

;

(2)

In addition, an attribute specific to the modeled painting event needs to be 

designated:

MEV = Mass of paint applied during exposure event (mg)

 1 WC Evans, “Development of Continuous-Application Source Terms and 

Two-Compartment Analytical Solutions”, ASTM Symposium on Methods for 

Characterizing Indoor Sources and Sinks, September 25-28, 1994, Washington, DC
D-2

One gallon of paint has an approximate mass of 4700 grams and takes about 2 hours to 

apply to gypsum board. This quantity is sufficient to paint a room with approximately 

400 square feet of surface area (e.g., walls and ceilings of a room with length of 10 feet, 

width of 9 feet and height of 8 feet, with a resultant volume of 720 cubic feet or 20 cubic 

meters; See Appendix F for details). The estimation of the remaining parameters to 

Equation 1 for latex and alkyd paints, given various combinations of information about 

the paint, is presented below. 

2. Estimation of Equation Parameters for Latex Paint

The estimation of the parameters to Equation 1 for latex paints uses one of three 

alternative techniques. If chamber data are available for the correct substrate (gypsum 

board), a curve-fitting technique for a double exponential is recommended, with the 

resulting fitted parameters applied to Equation 1. If chamber data are available for the 

glass or stainless steel substrate, a single exponential based upon a curve-fitting technique 

and 25% mass released is recommended, with the resulting fitted parameters applied to 

Equation 1. If no chamber data are available, a bulk analysis of the paint is recommended 

along with parameter estimation based on the analysis presented in Appendix A. Each of 

these parameter estimation techniques is discussed below.

2.1 Method 1: Estimation of Equation Parameters for Latex Paint Using 

Appropriate Substrate Chamber Data

This technique is recommended whenever chamber data are available for painted gypsum 

board. The chamber test should use the paint and substrate for which the emission profile 

is being estimated.

User Inputs

1. Concentration vs. time profiles from chamber tests for each VOC of interest in the 

paint.

2. MCH = Mass of paint applied during chamber test (mg)

3. MEV = Mass of paint applied during exposure event (mg)

4. ta = Duration of event (hours)

Step 1: Fit Double Exponential to Chamber Data

i. Use nonlinear regression to estimate the parameters of the single exponential that best 

fits the data after 48 hours. This exponential represents the second exponential in the 

double exponential (the “slow” emission). The equation representing the 

concentration in a chamber with a constant air flow and a single exponential source is:

C t

E

V Q

V

k

e e

k t

Q

V t

( ) =

− 



 





− 



 



 − − 02

2

2 (3)
D-3

where: C = Concentration in the chamber (mass/volume)

V = Volume of the chamber

Q = Air flow rate into and out of the chamber (volume/time).

E02 = Initial emission rate for the second exponential (mass/time)

k2 = First-order rate constant for the second exponential (time-1)

t = Time (t > 48 hours).

Each of the above parameters is known from the chamber test conditions except 

E02 and k2, which are estimated with nonlinear regression analysis. 

ii. Use nonlinear regression to estimate the parameters for another single exponential, 

such that this exponential fits the difference between the data and the predicted 

concentrations from the exponential that best fits the data after 48 hours. The 

equation representing the concentration in the chamber with a constant air flow and 

the double exponential source is:

C t

E

V Q

V

k

e e

E

V Q

V

k

e e

k t

Q

V t k t

Q

V t

( ) =

− 



 





− 



 



 +

− 



 





− 



 



 − − − − 01

1

02

2

1 2 (4)

where: E01 = Initial emission rate for first exponential (mass/time)

k1 = First-order rate constant for first exponential (time-1)

Operationally, this can be accomplished by using all the data and inputting the 

fitted parameters E02 and k2 from step i as “knowns,” leaving only two parameters 

to be estimated, E01 and k1.

Step 2: Estimate Parameters to Equation 1

M E

k

E

k

M

M

EV

CH

= +





 



 01

1

02

2

(5)

f E

k E

k

E

k

=

∗ +





 





01

1

01

1

02

2

(6)

k1 = fitted parameter from step 1ii

k2 = fitted parameter from step 1i

Step 3: Calculate emission profile

Using the parameters calculated above, implement Equation 1 to give the emission rate as 

a function of time. These algorithms are intended for inclusion in a mass-balance model, 
D-4

to be used to simulate source strength, indoor-air concentrations and, ultimately, exposure 

scenarios. 

2.2 Method 2: Estimation of Equation Parameters for Latex Paint Using Glass or 

Stainless Steel Substrate Chamber Data

This technique is recommended whenever chamber data are available, but the data are 

from tests on a glass or stainless steel substrate. The chamber test should use the paint for

which the emission profile is being estimated.

User Inputs

1. Concentration vs. time profiles from chamber tests for each VOC of interest in the 

paint

2. Weight fraction for each VOC in the paint (mg VOC/mg paint)

3. MEV = Mass of paint applied during exposure event (mg)

4. ta = Duration of event (hours)

Step 1: Fit Single Exponential to Chamber Data

Use nonlinear regression to estimate the parameters of the single exponential that best fits 

all the data. The equation representing the concentration in a chamber with a constant air 

flow and a single exponential source is:

C t

E

V Q

V

k

e e

k t

Q

V t

( ) =

− 



 





− 













 − − 01

1

1 (7)

where: C = Concentration in the chamber (mass/volume)

V = Volume of the chamber

Q = Air flow rate into and out of the chamber (volume/time).

E01 = Initial emission rate for the exponential (mass/time)

k1 = First-order rate constant for the exponential (time-1

)

t = Time (hours).

Each of the above parameters is known from the chamber test conditions except 

E01 and k1, which are estimated with nonlinear regression analysis. 

Step 2: Calculate parameters to equation 1 as a single exponential, using the fitted value 

for k1

1. Calculate total mass to be emitted

M = 0.25 * (Weight fraction of VOC) * MEV (8)

2. f = 1.0

3. k1 = fitted parameter from step 1

4. k2 = 0
D-5

Step 3: Calculate emission profile

Using the parameters calculated above, implement Equation 1 to give the emission rate as 

a function of time.

2.3 Method 3: Estimation of Equation Parameters for Latex Paint Using Bulk 

Analysis 

This technique is recommended when chamber data are not available. The bulk analysis 

and chemical properties of the VOC are used to estimate the emission profile.

User Inputs

1. Weight fraction for each VOC in the paint (mg VOC/mg paint)

2. Molecular weight of each VOC in the paint (g VOC/mol VOC)

3. Vapor pressure of each VOC in the paint (mm Hg)

4. MEV = Mass of paint applied during event (mg)

5. ta = Duration of event (hours)

Step 1: Calculate k1 from relationship with Vapor Pressure

k1 (hr ) ( ) (Vapor Pressure) 1 23325 − = . ∗ (9)

Step 2: Calculate k2 from relationship with Molecular Weight

k2 (hr ) ( ) ( Molecular Weight) 1 5 584 10 − − = . × * (10)

Step 3: Calculate parameters to equation 1

1. Calculate total mass to be emitted

M = 0.25 * (Weight fraction of VOC) * MEV 

2. f = 0.1

3. k1 = fitted parameter from step 1

4. k2 = fitted parameter from step 2

Step 4: Calculate emission profile

Using the parameters calculated above, implement Equation 1 to give the emission rate as 

a function of time. These algorithms are intended for inclusion in a mass-balance model, 

to be used to simulate source strength, indoor-air concentrations and, ultimately, exposure 

scenarios.
D-6

3. Estimation of Equation Parameters for Alkyd Paint

The estimation of the parameters to Equation 1 for alkyd paints use one of two alternative 

techniques. If chamber data are available for the glass or stainless steel substrate, a single 

exponential based upon a curve-fitting technique is recommended, with the resulting 

fitted parameters applied to Equation 1. If no chamber data are available, a bulk analysis 

of the paint is recommended along with parameter estimation based on the analysis 

presented in Appendix C. Each of the parameter estimation techniques is discussed 

below. No chamber data for alkyd paint on gypsum substrate were available;

consequently, no recommendation is made for use of chamber data on the correct 

substrate. Further EPA small-chamber studies are in progress to examine the emission 

characteristics of alkyd paint applied to a gypsum board substrate. Based on preliminary 

results, there is some evidence indicating that emission profiles for alkyd paint applied to 

glass may not be much different than those for alkyd paint applied to gypsum board. 

When these data become available, techniques for estimating emission profiles can be 

developed and recommendations can be made. 

3.1 Method 1: Estimation of Equation Parameters for Alkyd Paint Using Glass or 

Stainless Steel Substrate Chamber Data

This technique is recommended whenever chamber data are available, but the data are 

from tests on a glass or stainless steel substrate. The chamber test should use the paint for

which the emission profile is being estimated.

User Inputs

1. Concentration vs. time profiles from chamber tests for each VOC of interest in the 

paint.

2. Weight fraction for each VOC in the paint (mg VOC/mg paint)

3. MCH = Mass of paint applied during chamber test (mg)

4. MEV = Mass of paint applied during exposure event (mg)

5. ta = Duration of event (hours)

Step 1: Fit Single Exponential to Chamber Data

Use nonlinear regression to estimate the parameters to the single exponential that best fits 

all the data. The equation representing the concentration in a chamber with a constant air 

flow and a single exponential source is:

C t

E

V Q

V

k

e e

k t

Q

V t

( ) =

− 



 





− 













 − − 01

1

1 (11)

where: C = Concentration in the chamber (mass/volume)
D-7

V = Volume of the chamber

Q = Air flow rate into and out of the chamber (volume/time).

E01 = Initial emission rate for the exponential (mass/time)

k1 = First-order rate constant for the exponential (time-1

)

t = Time (t > 48 hours).

Each of the above parameters are known from the chamber test conditions except E01 

and k1, which are estimated with nonlinear regression analysis. 

Step 2: Calculate parameters to equation 1 as a single exponential using the fitted 

parameters for E01 and k1

1. Calculate total mass to be emitted

M E

k

M

M

EV

CH

= 



 



 01

1

2. f = 1.0

3. k1 = fitted parameter from step 1

4. k2 = 0

Step 3: Calculate emission profile

Using the parameters calculated above, implement Equation 1 to give the emission rate as 

a function of time.

3.2 Method 2: Estimation of Equation Parameters for Alkyd Paint Using Bulk 

Analysis 

This technique is recommended when chamber data are not available. The bulk analysis 

and chemical properties of the VOC are used to estimate a single exponential 

representation of the emission profile.

User Inputs

1. Weight fraction for each VOC in the paint (mg VOC/mg paint)

2. Molecular weight of each VOC in the paint (g VOC/mol VOC)

3. Vapor pressure of each VOC in the paint (mm Hg)

4. MEV = Mass of paint applied during event (mg)

5. ta = Duration of event (hours)

Step 1: Calculate k1 from relationship with Vapor Pressure

For gloss type alkyd paint:

k1 (hr ) ( ) (Vapor Pressure ) 1 0 024 0 074 − = . + . ∗ (12)
D-8

For semi-gloss type alkyd paint:

k1 (hr ) ( ) (Vapor Pressure ) 1 0 046 0 061 − = . + . ∗ (13)

For flat type alkyd paint:

k1 (hr ) ( ) (Vapor Pressure ) 1 0 066 0129 − = . + . ∗ (14)

Step 2: Calculate parameters to equation 1 as a single exponential using the fitted value 

for k1

1. Calculate total mass to be emitted

For gloss type alkyd paint:

M = 1.0 * (Weight fraction of VOC) * MEV (15)

For semi-gloss type alkyd paint:

M = 0.75 * (Weight fraction of VOC) * MEV (16)

For flat type alkyd paint:

M = 0.50 * (Weight fraction of VOC) * MEV (17)

2. f = 1.0

3. k1 = fitted parameter from step 1

4. k2 = 0

Step 3: Calculate emission profile

Using the parameters calculated above, implement Equation 1 to give the emission rate as 

a function of time.
E-i

APPENDIX E

Exposure Scenarios
E-1

The emission algorithms presented in Appendices A and C can be applied to any of 

several approaches for representing exposure scenarios. An approach where the scenario 

consists of a single room being painted and connected to the rest of the house (i.e., a twochamber model) would be the simplest approach. This approach would yield a 

reasonable estimate of consumer exposure to a single-room painting event. However, 

such a scenario would not be of use in the cases of occupational exposure, particularly in 

the case where a painter spends the entire workday in one house.

A more complicated approach could involve a scenario where a series of chambers, each 

the size of normal rooms in a house, are used, and the painting event moves to the next 

chamber after the proper amount of mass is applied in the current chamber. This 

approach introduces a variety of other issues, including assigning air flows, destinations 

of air flows, and room volumes.

A suggested compromise is a flexible scenario whereby multiple painted rooms are 

represented as a single room that connected to the rest of the house, with the size of the 

“painted room” dependent on the amount of surface area painted. The size of the 

“painted room” would comprise a volume equal to the summed volume of all painted 

rooms, and all painting would occur within this summed volume. This flexible, twochamber model would collapse to a single chamber for the scenario where the entire 

house interior is painted. Such an approach would be adaptable to each of the above

exposure scenarios without undue complexity. An important point to consider in using 

this approach is how the room volume impacts the air concentrations in a model based on 

a well-mixed assumption. It is likely to under-predict the peak concentrations that could 

be experienced in any one room. However, it is also likely to effectively over-predict the 

starting concentration in a room where painting has just begun. It is unclear what the net 

effect would be, but it is unlikely to have a large impact.

Other important considerations for the exposure scenario include the amount of paint 

applied and the duration of the painting event, which are related through the application 

rate. In a study by the Eastman Kodak Company (D. Guest, P.J. Dersinger and J.G. 

Winter, 1985, “Estimation of the Atmospheric Concentration of Diethylene Glycol 

Monobutyl Ether Acetate Resulting from the Application of Latex Paint”), eight 4-foot by 

8-foot gypsum board specimens were painted in 80 minutes (10 minutes per specimen), 

covering a total of 256 square feet. The same report indicated the nominal coverage rate 

of a gallon of paint as 400 square feet. 

By extension of the above application rate (3.2 square feet per minute), it takes 

approximately 2 hours to apply one gallon of paint to 400 square feet of surface area. 

Such a surface area corresponds, for example, to the walls and ceiling of a room with 

length of 10 feet, width of 9 feet and height of 8 feet, with a total volume of 720 cubic 

feet or about 20 cubic meters. If we conceive of a house with 400 cubic meters volume as 

being “fully painted” with 16 such rooms being painted, then it would require 16 gallons 

and 32 hours to paint the entire house. With a two-person painting team, the application 

time would be halved.