&EPA
United States
Environmental Protection
Agency
Office of
Toxic Substances
Washington, D.C. 20460
EPA
April 1987
Toxic Substances
Methods for Assessing
Exposure to Chemical
Substances
Volume 12
Methods for Estimating the
Concentration of Chemical
Substances in Indoor Air
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EPA 560/5-85-016
APRIL 1987
METHODS FOR ASSESSING EXPOSURE
TO CHEMICAL SUBSTANCES
Volume 12
Methods for Estimating the Concentration of
Chemical Substances 1n Indoor A1r
Patricia D. Jennings, Clay E. Carpenter,
M. Slvarama Krlshnan
EPA Contract No. 68-02-4254
Project Officer
Elizabeth F. Bryan
Exposure Evaluation Division
Office of Toxic Substances
Washington, O.C. 20460
U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF PESTICIDES AND TOXIC SUBSTANCES
WASHINGTON, D.C. 20460
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DISCLAIMER
This document has been reviewed and approved for publication by the
Office of Toxic Substances, Office of Pesticides and Toxic Substances,
U.S. Environmental Protection Agency. The use of trade names or
commercial products does not constitute Agency endorsement or
recommendation for use.
\\-
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FOREWORD
This document 1s one of a series of volumes, developed for the U.S.
Environmental Protection Agency (EPA), Office of Toxic Substances (OTS),
that provides methods and Information useful for assessing exposure to
chemical substances. The methods described 1n these volumes have been
Identified by EPA-OTS as having utility 1n exposure assessments on
existing and new chemicals 1n the OTS program. These methods are not
necessarily the only methods used by OTS, because the state-of-the art 1n
exposure assessment 1s changing rapidly, as 1s the availability of
methods and tools. There 1s no single correct approach to performing an
exposure assessment, and the methods 1n these volumes are accordingly
discussed only as options to be considered, rather than as rigid
procedures.
Perhaps more Important than the optional methods presented 1n these
volumes 1s the general Information catalogued. These documents contain a
great deal of non-chemlcal-speelfic data which can be used for many types
of exposure assessments. This Information 1s presented along with the
methods 1n Individual volumes and appendices. As a set, these volumes
should be thought of as a catalog of Information useful 1n exposure
assessment, and not as a "how-to" cookbook on the subject.
The definition, background, and discussion of planning exposure
assessments are discussed 1n the Introductory volume of the series
(Volume 1). Each subsequent volume addresses only one general exposure
setting. Consult Volume 1 for guidance on the proper use and
Interrelations of the various volumes and on the planning and Integration
of an entire assessment.
The titles of the nine basic volumes are as follows:
Volume 1 Methods for Assessing Exposure to Chemical Substances
(EPA 560/5-85-001)
Volume 2 Methods for Assessing Exposure to Chemical Substances 1n the
Ambient Environment (EPA 560/5-85-002)
Volume 3 Methods for Assessing Exposure from Disposal of Chemical
Substances (EPA 560/5-85-003)
Volume 4 Methods for Enumerating and Characterizing Populations Exposed
to Chemical Substances (EPA 560/5-85-004)
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Volume 5 Methods for Assessing Exposure to Chemical Substances In
Drinking Water (EPA 560/5-85-005)
Volume 6 Methods for Assessing Occupational Exposure to Chemical
Substances (EPA 560/5-85-006)
Volume 7 Methods for Assessing Consumer Exposure to Chemical Substances
(EPA 560/5-85-007)
Volume 8 Methods for Assessing Environmental Pathways of Food
Contamination (EPA 560/5-85-008)
Volume 9 Methods for Assessing Exposure to Chemical Substances
Resulting from Transportation-Related Spills (EPA 560/5-85-009)
Because exposure assessment 1s a rapidly developing field, Its
methods and analytical tools are quite dynamic. EPA-OTS Intends to Issue
periodic supplements for Volumes 2 through 9 to describe significant
Improvements and updates for the existing Information, as well as adding
short monographs to the series on specific areas of Interest. The first
four of these monographs are as follows:
Volume 10 Methods for Estimating Uncertainties 1n Exposure Assessments
(EPA 560/5-85-014)
Volume 11 Methods for Estimating the Migration of Chemical Substances
from Solid Matrices (EPA 560/5-85-015)
Volume 12 Methods for Estimating the Concentration of Chemical
Substances 1n Indoor A1r (EPA 560/5-85-016)
Volume 13 Methods for Estimating Retention of Liquids on Hands
(EPA 560/5-85-017)
Elizabeth F. Bryan, Chief
Exposure Assessment Branch
Exposure Evaluation Division (TS-798)
Office of Toxic Substances
vi
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ACKNOWLEDGEMENTS
This report was prepared by Versar Inc. of Springfield, Virginia, for
the EPA Office of Toxic Substances, Exposure Evaluation Division,
Exposure Assessment Branch (EAB) under EPA Contract Nos. 68-01-6271,
68-02-3968, and 68-02-4254. The EPA-EA8 Task Manager for this task was
Karen A. Hammerstrotn; the EPA Program Managers were Michael A. Callahan
and Elizabeth F. Bryan. The support and guidance given by these, and
other EPA personnel, 1s gratefully acknowledged.
A number of Versar personnel have contributed to this task over the
period of performance, as listed below:
Program Management
Task Management
Principal Investigators
Technical Support
Editing
Secretarial/Clerical
- Gayaneh Contos
- Patricia Jennings
Sh1v Krlshnan
G1na D1xon
- Clay Carpenter
Sh1v Krlshnan
Patricia Jennings
- Bruce Woodcock
Thompson Chambers
- Juliet Crumrlne
- Shirley Harrison
Sue Elhusseln
Lynn Maxfleld
Kamml Johannsen
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TABLE Of CONTENTS
1 , INTRODUCTION
2, METHODS FOR ESTIMATING POLLUTANT GENERATION RATES
2.1 Simple Models - Gas Laws
2.2 Mass Transfer
2.2.1 Background
2.2.2 Diffusion Rate
2.3 Aerosol Formation
2.3.1 Background
2.3.2 Quantification of Aerosols
2.4 Migration
2.4.1 Background
2,4.2 Estimation of Migration ..
3. BEHAVIOR OF POLLUTANTS INDOORS ,
3.1 Ventilation
3.2 Participate Behavior
3.2.1 Participates 1n Indoor A1r
3.2.2 Physical Behavior of Partlculates
3.2.3 Adsorption of Vapor Phase Pollutants to
Partlculates
3.3 Chemical Decay
3.4 Calculation of Indoor Air Concentrations
3.4.1 Background
3.4.2 Model Derivations
4 . MODEL DEMONSTRATION
4.1 Scenario 1 - Continuous Release of Aerosols
4.2 Scenario 2 - Continuous Release from Films
4.3 Scenario 3 - Time-Dependent Release
Page No
1
3
3
6
7
7
22
22
22
24
24
24
27
27
28
30
30
35
38
39
41
44
57
57
60
62
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TABLE OF CONTENTS (continued)
Page No.
5. REFERENCES 69
APPENDIX A - Symbols Used 73
APPENDIX B - Derivation of Equations for Estimating
Concentrations of Chemical Substances In Indoor
A1r 79
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LIST OF TABLES
Page No.
Table 1. Commercial A1r Exchange Rates (Based on ANSI/ASHRAE
62-1981 Recommended Ventilation Rates and Assumed
Celling Heights) 29
Table 2. Summary of Data for Participate Hatter 31
Table 3. Particle Concentration and Size Distribution for
Overnight Measurements 33
Table 4. Coefficients of Rate Constants 1n Adsorption
Equations 37
Table 5. A1r Changes Occurring Under Average Conditions 1n
Residences Exclusive of A1r Provided for
Ventilation 43
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LIST OF FIGURES
Page No.
Figure 1. Theoretical settling velocities of fibers 34
Figure 2. Concentration estimation pathways ,. 40
XI 1
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1. INTRODUCTION
Exposure assessments have characteristically been hampered by a lack
of data on concentrations of chemical substances 1n Indoor air.
Monitoring data are generally not available, and the available estimation
techniques are few and unvalldated. The lack of understanding of the
behavior of pollutants Indoors has led Investigators to make simplifying
assumptions regarding the rates of pollutant generation and the fate of
chemical substances 1n the Indoor environment.
This report presents methods that take Into account the major
parameters affecting Indoor air pollutant concentrations. The equations
and data In this report are easily applied to a wide range of conditions.
Section 2 presents the methods for estimating pollutant generation
rates. Section 3 discusses the physical and chemical processes governing
the behavior of Indoor air pollutants and presents equations for the
calculation of pollutant concentrations. Three scenarios Illustrating
the use of the methods 1n this report are presented 1n Section 4.
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2.
METHODS FOR ESTIMATING POLLUTANT GENERATION RATES
The following subsections describe methods for estimating the rates
at which various types of pollutants are generated Indoors. The
techniques presented herein are based on the principles of chemistry and
thermodynamics, and are applicable to all types of substances regardless
of vapor pressure, particle size, or molecular weight. Section 2.1
discusses the gas laws governing the behavior of all gases, while Section
2.2 presents the theory and use of mass transfer to predict diffusion
Into air. Section 2.3 discusses the formation and release of aerosols,
and the final section (2.4) presents data on estimating the rate of
migration of additives from polymers.
2.1
Simple Models - fias Laws
Models describing mass transfer and concentration changes 1n an
Indoor environment are based on a number of simple gas laws. A general
knowledge of these gas laws 1s required to fully understand the more
complicated gas models. Consequently, this section presents a synopsis
of the following gas laws: (1) Dalton's law, (2) Raoult's law,
(3) Henry's law, (4) Graham's law, and (5) the Ideal gas law.
(1) Oalton's law. Dalton's law states that the total pressure
exerted by a mixture of gases 1s equal to the sum of the partial
pressures of the various gases. The partial pressure of a gas 1n a
mixture Is defined as the pressure the gas would exert 1f 1t were alone
1n a container (Slenko and Plane 1966).
In general, Oalton's law can be written:
ptotal « pl * P2 * P3 - • • •
where the subscripts denote the various gases occupying a given volume.
Therefore, this law Implies that for pollutants 1n an Indoor
environment, the total pressure 1s equal to the partial pressure of air
plus the partial pressures of the various pollutants. Assuming the total
pressure 1s equal to the standard atmospheric pressure, this can be
written as follows:
p , PA
p2
where
PA
PI
the partial pressure of air
the partial pressure of pollutant 1
the partial pressure of pollutant 2.
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(2) Raoult's law. Raoult's law states that the partial pressure of a
solvent over a liquid solution 1s proportional to the mole fraction of
the solvent 1n the solution, at equilibrium.
Raoult's law can be mathematically expressed as
where
PA° = vapor pressure of pure solvent
XA = mole fraction of solvent 1n the solution
PA » partial pressure of the solvent over the solution (Lyman
et al. 1982).
The above equation can be rewritten as
PA - (i - xB) PA°
or
P ° - P
_A A = X0
8
where
Xg = mole fraction of the solute.
This last relationship shows that the fractional lowering of the
vapor pressure of a solvent 1s equal to the mole fraction of the solute.
In practice, Raoult's law 1s an Idealization and 1s most representative
of the behavior of the solvent 1n dilute solutions.
(3) Henry's law. Henry's law states that at constant temperature,
the ratio of partial pressure of the solute to Its mole fraction 1n
solution 1s a constant (Slenko and Plane 1966 ).
Mathematically, Henry's law 1s expressed as follows:
H XB = PB
where
H = Henry's constant for the solute
g = partial pressure of the solute
B = mole fraction of the solute 1n solution,
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Henry's constant 1s a useful parameter 1n estimating the partial
pressure of a component. It can be computed from the relationship
H . PB° / SB
where
Pg° = vapor pressure of the pure solvent
SB = solubility of the solute (Lyman et al. 1982).
The estimation of solubility for binary systems requires a knowledge
of activity coefficients. However, measured activity coefficients are
available for only a relatively small number of chemical systems. It 1s
possible to estimate these coefficients using Information on the
molecular structures of the chemicals. This process also poses
considerable difficulty because the requisite parameters are available
for only a limited number of chemical functional groups, and can be
applied only to molecules with relatively simple structures (Lyman et al.
1982). These constraints Introduce a level of difficulty 1n the
estimation of solubility and Henry's constant for systems where water 1s
not the solvent. Henry's constant has been calculated for a number of
binary systems by Lyman et al. (1982),
(4) Graham's law of diffusion. This law can be stated as follows:
the rate of diffusion of a gas 1s observed to be Inversely proportional
to the square root of Its molecular weight. In other words, this law
quantifies the rate at which a gas diffuses to occupy a given volume.
This law can be written as follows:
Gr
D -
where
D = diffusion rate
Gr = Graham's constant
MW = molecular weight.
This law 1s useful 1n quantifying diffusion rates of pollutants In Indoor
air.
(5) Ideal gas law. The Ideal gas law combines a series of
experimentally-determined proportionality relationships among volume,
temperature, pressure, and the number of moles. This law 1s also called
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an equation of state, because 1t tells how, 1n going from one gaseous
state to another, the four variables (volume, pressure, temperature, and
the number of moles) change. This equation can be written as follows:
P V = n R T
where
P = pressure
V = volume
n = number of moles
T = temperature
R = gas constant.
The value of R varies depending on the units of the four variables.
For example, for the variables P 1n atmospheres, V 1n liters, n 1n moles,
and T 1n degrees Kelvin, R has a value of 0.082057
l1ter-atmospheres/moles-0K.
When using the Ideal gas law, 1t Is Important to know the conditions
under which gases deviate from Ideal behavior. Experimental evidence
Indicates that gases are generally Ideal (follow the Ideal gas law)
except under extreme conditions (for example, high pressure and low
temperature). Furthermore, some gases deviate from Ideal behavior
depending on such properties as molecular size and polarity, although
variations are generally very small. For example, the volume of four
common gases (hydrogen, nitrogen, oxygen, and carbon dioxide) was
measured at standard temperature and pressure (1 mole of gas at 1
atmosphere pressure and 25°C), and the largest deviation from Ideal
behavior was about 0.7 percent (Slenko and Plane 1966).
As part of determining pollutant behavior 1n an Indoor environment,
the Ideal gas law can be used to determine concentrations In the boundary
layer just above an area of release.
2.2 Mass Transfer
In general, mass transfer can be thought of as the migration of a
component 1n a mixture, either within the same phase or from phase to
phase, because of a displacement from physical equilibrium (Greenkorn and
Kessler 1972). Hass transfer operations account for the release and
movement of Internally generated Indoor pollutants. This Includes
evaporation of solvents from paints and varnishes, releases during
aerosol applications, and migration from solids.
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Mass transfer 1n an Indoor environment occurs by diffusion, where the
driving force 1s based on the phenomenon that systems not 1n equilibrium
win tend to move toward equilibrium (the second law of thermodynamics).
There are actually two types of diffusion processes: molecular diffusion
and eddy (convectlve) diffusion. In both cases, diffusion occurs as a
result of a concentration gradient; however, 1n eddy diffusion, the mass
transfer 1s greatly aided by the dynamic characteristics of the flow
(Welty et al. 1976).
2.2.1 Background
It 1s Important that the principles of the molecular diffusion
process be well understood before they are used to help estimate Indoor
air concentrations. The principles of diffusion are best seen by
examining two different enclosed gases separated by a glass barrier.
Within each enclosed area, the molecules move 1n a random zig-zag path
striking other molecules of the same gas and the walls of the container.
Once the barrier 1s lifted, molecules of both gases will also begin to
strike each other. Although 1t 1s not possible to state which way any
particular molecule will travel 1n a given Interval of time, a definite
number of molecules from each gas will travel to the region previously
occupied only by molecules of the other gas. Accordingly, an overall net
molecular transfer from a region of higher concentration to one of lower
concentration will occur, with each molecular species moving 1n the
direction of a negative concentration gradient. This molecular migration
will continue until equilibrium 1s reached.
2.2.2 Diffusion Rate
The rate or molar flux of diffusion (moles/tlme-area) 1s the
parameter needed to ultimately determine chemical concentrations 1n
Indoor air. This rate 1s essentially the average flow of the diffusing
molecules per unit area (during diffusion) per unit time. It depends not
only on the concentration gradient, but also on the characteristics of
the diffusing compounds and environmental parameters (temperature,
pressure, etc.).
The rate of molecular movement for a binary solution has
traditionally been expressed by Pick's law:
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where
JB = flux or rate (1n a moving coordinate system)
DAB = diffusion coefficient
dCg/dz = concentration gradient.
The diffusion coefficient contains the parameters, other than the
concentration gradient, that affect the diffusion rate. If possible, the
diffusion coefficient should be experimentally determined. However, 1n
the absence of such data, the Fuller, Schettler, and Biddings (FS6)
method or the W1lke and Lee (WL) method can be used to estimate the
diffusion coefficients (Lyman et al. 1982). The FSG method 1s reportedly
applicable to nonpolar gases at low to moderate temperatures, and the WL
method 1s applicable to a wide range of compounds over a fairly wide
temperature range. The WL method 1s presented here because the average
number of errors obtained with this method Is considerably less than that
obtained with the FSG method.
(o.0021? - 0.00050 V1/MWA *• 1/MWg JT3/2 Vl/MWA * 1/NW 2 _
D „ - -_-
P°A8 Q
where
DAB = d1ffus1v1ty, cm2/sec
T = ambient temperature, °K
MW^, MWg = molecular weight of air and pollutant,
respectively
P = absolute pressure, atm „
0AR = characteristic length of molecule, A (Angstrom
AB units)
n = collision Integral
The collision Integral, Q, 1s a function of the molecular energy of
attraction, c, and the Boltzmann Constant, kB, as given below:
e
(T*)b * exp (T*d) * exp (T*f) * exp (T*h) (2~3)
where
the values a-h are as given 1n Chapter 17, Lyman et al. (1982),
and
T
(2-4)
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Values of e/kg are given 1n Treybal (1968) for some of the more
common gases. Values for other gases can be approximated using the
following formula:
e/kr
1.15Tt
(Lyman et al. 1982)
(2-5)
where
Tj, = normal boiling point, °K.
The characteristic length,
following relationship:
, can be calculated using the
JAB
(2-6)
where
o, = 3.711 A (angstrom units)
= 1.18 V'B1/3
V'g = molar volume of the pollutant at Its normal boiling point
(cm3/mole). (See Treybal (1968) or Lyman et al, (1982) for
atomic and molecular volumes.)
There are numerous solutions to the partial differential equation
presented 1n Pick's law depending on the parameters of the particular scenario
being examined. For pollutants released 1n an Indoor air environment, It 1s
assumed that the pollutant moves through the stagnant (or very slowly moving)
Indoor air. However, before solving Pick's law based on this scenario, one
must convert the moving coordinate system (Jg 1n Pick's law) to a stationary
one. In a binary system subscript A denotes the Indoor air and subscript B
denotes the pollutant. Pick's law with a stationary coordinate system can be
described as follows:
and
NA -
NB = JB * N YB
(2-7)
|2-8)
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where
NA and NB = flux with respect to a stationary coordinate
system
JA and JB - flux with respect to a moving coordinate system
N YA and N YB = the additional apparent flux from changing from a
moving to a stationary coordinate system; YA and YB
refer to the mole fraction of air and of the pollutant,
respectively.
Furthermore, the following equation describes the total molar flux with
respect to stationary coordinates:
N = NA * NB (2-9)
where
N = total molar flux
NA = molar flux of the Indoor air
NB = molar flux of the pollutant.
However, for the Indoor air scenario described above, the Indoor air,
component A of the binary system, 1s essentially stagnant. Therefore,
the molar flux (movement) of component A 1s zero or NA » 0.
Consequently,
N = NB where NB = JB * N YB
Using the relationship
Cm = CA + CB = cm YA + cm YB
where
Cm = molar density of the mixture
CA»CB - "»olar densities of A and B
YA,YB = mole fractions of A and B 1n the vapor,
and substituting Equation (2-1) for Jg» NB becomes:
NB>Z= -VAB- * NBYB ^2-11>
where
= molar flux of pollutant B across any plane of
constant z.
10
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By rearranging, Integrating, and assuming z varies from 0 to L and
YB varies from
Kessler 1972):
to Yg£, Equation (2-11) becomes (Greenkorn and
'8,2
-CmDAB in 1 - YB1
1 _
(2-12)
B2
By rearranging Equation (2-12), substituting Y/^ for (1 - Yg), and
by using the definition of log mean. Equation (2-12) can be rewritten as
follows (Greenkorn and Kessler 1972):
-Cm°AB
'B.Z
B2
- Y
B1
(2-13)
where
(VA)lm = YA2 - YA1 .
in (VA2/YA1)
For gases at low to moderate pressures, the partial pressures of the
two components can be substituted for mole fraction (since 1n a binary
system, mole fraction 1n the vapor 1s equal to P^/P). Therefore, the
final flux equation becomes:
-PD
AB
RTlm
B2
- P
B1
(2-14)
where
- P
A1
in (PA2/PA])
(2-15)
Consequently, by using Equations (2-2) and (2-14), the molar flux, N,
can be calculated. The units of molar flux are moles/cm^-sec. If the
surface area from which the pollutant 1s released and the molecular
weight of the pollutant are known, a continuous evaporation rate. G, can
be determined. For a scenario 1n which the release rate 1s continuous,
the film containing the pollutant 1s Instantaneously applied to the
surface. The amount of pollutant remaining on the surface 1s assumed to
be the same for each point on the surface at any given point In time.
11
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For a scenario 1n which the release rate 1s time-dependent, the rate
at which the film containing the pollutant Is applied affects the total
mass of pollutant available for release from a given point on the surface
at any point In time. The amount of pollutant remaining on the surface
differs from point to point on the surface at any given point 1n time.
If the surface area onto which the film 1s applied, the molecular weight
of the pollutant volatilizing from the film, and the rate at which the
film 1s applied to the surface are taken Into account, a time-dependent
release rate, G^AR, can be determined.
The next step 1s to estimate the Indoor air concentration based on
the release rate. This 1s based on numerous factors, such as ventilation
rates, mixing factors, and sorptlon properties. Section 3 describes the
behavior of Indoor pollutants based on these and other factors.
Section 4 describes the calculation of the various parameters leading to
the estimation of Indoor pollutant concentrations.
One needs to compute the partial pressures and diffusion coefficient
before the molar flux can be estimated. The computation of molar flux 1s
one route leading to the estimation of a continuous or time-dependent
release rate. Other methods to estimate release rates are presented 1n
Section 3.4.1. The various steps Involved 1n the calculation of partial
pressures, diffusion coefficient, molar flux, and evaporation rate/mass
flux for diffusion 1n a two-component system are discussed below. Sample
calculations are provided for the estimation of parameters for the
diffusion of 2-methoxyethanol Into air.
Sample Calculation of Partial Pressures
In computing the partial pressures, one has to consider the two
Interfaces Involved In the diffusion process. Diffusion takes place
essentially through a gas film as a result of Its concentration
gradient. If the pollutant and air are considered as the two components
of a system, the Interfaces could be defined as (1) the Interface between
the liquid film on the surface and the gas film and (2) Interface between
the gas film and the main air stream (this numbering convention for the
Interfaces will be followed throughout this report).
The partial pressures can be calculated using Dalton's law as
discussed 1n Section 2.1.
Basic Steps
(1) Determine the partial pressure of the pollutant, Pg1 , at the
liquid film-gas film Interface. The partial pressure of the
pollutant, 1n atmospheres. Is obtained by dividing the vapor
pressure (VP), In mm Hg. by the atmospheric pressure ( P) 1n
mm Hg.
12
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(2) Determine the partial pressure of air, P/^-j, at the liquid
film-gas film Interface. Use the partial pressure of the
pollutant at this Interface and apply Oalton's Law.
(3) Determine the partial pressure of the pollutant, Pg2, at the
Interface of the gas film-main air stream. Assume that the
partial pressure of air at this Interface, P/^t 1s equal to
the atmospheric pressure (P). Use the partial pressure of air
at this Interface and apply Dalton's law.
Example 1:
Calculate the partial pressure at the two Interfaces for the
evaporation of 2-methoxyethanol (2-ME) Into air at 25°C (298°K) and 1
atmosphere. The vapor pressure of 2-ME at 25°C 1s 6.2 mm Hg.
(1) Partial pressure of the pollutant, Pg^, at the liquid film-gas
film Interface.
VP
PBI • p-
6.2 mm Hg
760 mm Hg
= 0.008 atm.
(2) Partial pressure of air, P^i, at the liquid film-gas film
Interface.
PAI - i - PBI
= 1 - 0.008
= 0.992 atm.
(3) Partial pressure of the pollutant, Pg2, at the Interface of
the gas film-main air stream.
PA2 = 1 atm
From Dalton's law,
PB2 = 1 - PA2
= 1-1
= 0 atm.
13
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Sample Calculation of Diffusion Coefficient
A step-by-step procedure for calculating the diffusion coefficient
for the diffusion of a pollutant through air 1s given below.
Basic Steps
(1) Obtain Wig, the molecular weight of pollutant B, and Its
boiling point (°K) from the literature,
(2) Use the following parameters for air: (Lyman et al. 1982)
MWA = 28.97 g/mole
VA = 20.1 cm3/mole
(e/ka)A = 78.6 °K
OA = 3.711 X
(3) Obtain Vg, the molar volume for the pollutant, from the
literature, or estimate using Lyman et al. (1982).
(4) Calculate a$ using Vg In Equation (2-6).
(5) Calculate aAg as (0A +
(6) Use Equation (2-5) to determine (e/kg)g for the
pollutant using Its boiling point, Tjj.
(7) Calculate T* at the desired temperature, using Equation (2-4)
(8) Use the constants
a = 1.06 c = 0.193 e = 1.04 g = 1.76
b = 0.156 d = 0.476 f = 1.53 h = 3.89
and T* to compute the value of the collision Integral, a,
using Equation (2-3). ~
(9) Calculate the value of -Jl/MWA * 1/MWB.
(10) Calculate DAB, the diffusion coefficient, from Equation (2-2)
using the known values of P and T and the values of other
variables as derived above.
14
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Example 2:
Calculate the coefficient DAg for the diffusion of 2-methoxyethanol
through air using the WHke-Lee method, at 25°C (298°K) and 1 atmosphere.
(1) From the literature,
MWB = Molecular weight of 2-ME = 76.
Tb = Normal boiling point of 2-ME = 397.5°K.
(2) For air
MWA > Molecular weight of air = 28.97.
VA = Molar volume of air = 20.1 cm3/mole.
(c/kB)A - ?8.6 °K.
OA = 3.711 A.
(3) Vg = molar volume of 2-ME at normal boiling point
= 90.5 cm3/mole (Versar 1983).
(4) Using Equation (2-6),
0B = 1.18 VB 1/3
= 1.18 (90.5)1/3
= 5.298 A.
(5) aAB = (0A + crB)/2
= (3.711 + 5.298)/2 = 4.5045 A.
(6) Using Equation (2-5),
(e/kB)B . 1.15 Tb
• 1.15 x 397.5
= 457.1°K.
T* =
=•^457.1x78.6
= 189.55°K
T
298 = 1.512
189.55
15
-------
(8) Using Equation (2-3),
1.06 + 0.193 * 1.04
0.156 exp (1.572 x 0.476) exp (1.572 x 1.53)
1.76
exp(1.572 x 3.89)
1.177
(9) V1/MWB + 1/MWA - V1/76 * 1/28.97 - 0.2183 .
(10) Using the above variables 1n the WHke-Lee equation,
T = temperature - 298°K
P * pressure = 1 atm,
10.00217 - 0.0005 ..
DAD -
Au
/i/»A.i/»,)T"*y
" °AB * a
1/liU ^ "1 /ULJ
/nW. T 1 /ITWr,
A B
(0.00217 - 0.0005 x 0.2183) 2981'5 x 0.2183
1 x (4.5045)2 X 1.177
= 0.0969 cm2/sec .
Sample Calculation of Molar Flux
The necessary parameters to compute the molar flux of a two-component
system are (1) the partial pressures of the system and (2) the
coefficient for the diffusion of one component through the other. These
parameters could be estimated using the techniques discussed before In
this section. A step-by-step procedure for the computation of the molar
flux follows:
Basic Steps:
(1) Compute the coefficient, D^g, for the diffusion of the
contaminant through air, using the W1lke-Lee method
(Equation 2-2).
16
-------
(2) Calculate the partial pressures at the two prevalent
Interfaces: (1) liquid film-gas film Interface, and (2) gas
film-main air stream Interface.
(3) Use Equation (2-15) to calculate the log mean of air partial
pressures
(4) Obtain the value of the gas constant, R, from the literature,
and assume the boundary layer thickness, L.
(5) Calculate the molar flux, Ng>z using the known values of P, T,
and R, and the derived values of OAB« (PA)lm» ano< partial
pressures.
Example 3:
Calculate the molar flux for the molecular diffusion of
2-methoxyethanol through air, at 25°C (298°K) and 1 atmosphere.
(1) From Example #2,
DAB = Affusion coefficient = 0.0969 cm2/sec.
(2) From Example #1 ,
PBI = Partial pressure of 2-HE at Interface of liquid and
boundary layer
= 0.008 atm.
Pg2 = Partial pressure of 2-ME at Interface of boundary
layer and main air stream
= 0 atm.
PAI = Partial pressure of air at Interface of liquid and
boundary layer
= 0.992 atm.
?A2 = Partial pressure of air at Interface of boundary
layer and main air stream
= 1 atm.
(3) Using Equation (2-15),
P - P
. A2 A1
1m in (PA2/PA,)
17
-------
1 - 0.992
s ln (1/0.992)
= 0.996 atm.
(4) R « Gas constant » 82.05 atm-cm3/ mole-°K
Assume L =• Gas film thickness = 2.54 cm.
(5) From Equation (2-14), at
T = Temperature = 298°K
P « Pressure = 1 atm,
- Px
N
B'Z L x R x T x (PJ1m
A im
- 1 x .0969 x (0 - 0.008)
= 2 54 x 82 Q5 x 298 x 0 9g6
—8 2
= 1.25 x 10 mole/cm -sec.
Sample Calculation of Continuous/Time-Dependent Release Rate
For those circumstances 1n which a continuous rate of release 1s
assumed, the evaporation rate of the pollutant, 1n grams/hour, can be
calculated from (1) the molar flux of the pollutant; (2) the surface area
covered by the liquid film from which the pollutant evaporates; and
(3) the molecular weight of the pollutant. For those circumstances 1n
which a time-dependent rate of release 1s assumed, the mass flux, G^,
1n grams/cm2-hr, can be calculated from the molar flux of the pollutant
and the molecular weight of the pollutant. The time-dependent release
rate, G^A^, 1s the product of the mass flux, G^, and the rate of
application of the liquid film to the surface, A^, 1n cm2/hr. The
units for the time-dependent release rate are grams/hours^.
It must be noted that the method for estimating the rate of release
of a chemical substance from a liquid film assumes that the film from
which the chemical substance 1s being released behaves as an Ideal
solution. To behave as an Ideal solution, the film must consist of only
the pure chemical substance or a combination of chemical substances that
have similar molecular sizes and structures. Most liquid films, however,
are a mixture of a number of different solvents, resins, and pigments;
18
-------
because they have a vast range of molecular sizes and structures, they do
not behave as Ideal solutions. In a non-Ideal mixture, an Individual
volatile chemical substance will exhibit a rate of evaporation different
from that which 1t exhibits 1n the pure state.
To adjust the release rate of the pure chemical to account for
non-Ideal behavior as a result of Interactions among mixture components
and to account for the effects of dilution of the mixture in water or
other solvents, the release rate of the pure chemical must be multiplied
by a correction factor. The correction factor can be, 1n order of
preference, the activity coefficient, the mole fraction, or the weight
fraction of the chemical 1n the mixture from which 1t 1s evaporating.
Methods for estimating activity coefficients for two-component
systems are presented 1n the Handbook of Chemical Property Estimation
Methods (Lyman et al. 1982). Although the activity coefficient 1s the
most accurate correction factor, the detailed data required regarding
components of the mixture and the length of time required to complete the
calculations are major disadvantages. The weight fraction of each
component of the mixture must be known. In addition, the activity
coefficient for each component with each of the other components of the
mixture must be obtained.
A simpler approach for obtaining the correction factor 1s to use the
mole fraction of the chemical In the mixture as a substitute for the
activity coefficient. The mole fraction, although less accurate than the
activity coefficient as a correction factor, 1s easier to calculate. To
calculate the mole fraction, one must know the weight fraction and the
molecular weight of each component of the mixture.
In the absence of data on the weight fraction and/or molecular weight
of each component of a mixture, 1t 1s suggested that the weight fraction
of the chemical substance In the mixture be used as a correction factor.
Use of the weight fraction as a correction factor provides the lowest
degree of accuracy of the three variables suggested. The major
advantage, however, Is the ease with which a value for correction factor
can be obtained.
A step-by-step procedure for calculating a continuous or
time-dependent release rate follows:
19
-------
Basic Steps:
(1) Compute the molar flux of the pollutant, 1n moles/cm^-sec.
(2) Determine the surface area that 1s covered with liquid film, 1n
(3) If computation of a time-dependent release rate 1s desired, 1n
addition to the Information 1n Steps (1) and (2), the rate of
application of liquid film to the surface, 1n cm^/hour, must
be determined.
Example 4:
Calculate the continuous and time-dependent release rate for
2-methoxyethanol volatilizing from varnish applied to a table. The room
temperature 1s assumed to be 25°C (298°K), and the atmospheric pressure
1s assumed to be 1 atmosphere.
(1) From Example #3,
N = molar flux = 1.25 x TO'8 moles/cm2-sec.
(2) The following equation 1s used to calculate a continuous release
rate:
G = N x SA x MW x 3600
where
G = release rate of chemical substance (g/hr)
N = molar flux of pure chemical substance
(mole/cm^-sec)
SA = surface area covered by liquid film (cm^); the
surface area 1s 25,400 cm2.
The value, 3600, 1s used to convert from seconds to hours,
Consequently, the continuous release rate 1s obtained from
„ mole _ g sec
G ' K25 x 1(H3 c^c" x 25'400 cm2 x 76 m0le x 360° "^
= 86.9 grams/hour.
The composition of the varnish 1s not known. It 1s known,
however, that the weight fraction of 2-methoxyethanol 1n the
varnish Is 0.02. The weight fraction 1s used as the correction
factor to adjust the release rate of the pure chemical to
account for non-Ideal behavior as a result of Interactions
20
-------
among mixture components. If the continuous release rate, G, 1s
multiplied by the correction factor, a value of 1.74 grams/hour
1s calculated for G.
(3) The following equation 1s used to calculate a time-dependent
release rate:
GNAR = N x HW x 3600 x (SA/ta)
where
= time-dependent release rate
ta = time required to apply the lquid film to the
table surface, 1.5 hours,
and all other parameters and values are as discussed earlier In
this example. It must be noted that GNA^ 1s the product of
GN and AR. The parameter, 6^, 1s calculated using the
following expression:
GN = N x m x 3600
The parameter, AR, 1s calculated as follows:
AR = SA/ta
In this example, the time-dependent release rate 1s obtained as
follows:
GNAR . 1.25 x 10-8 JtMUL x 76 _J__ x 36QO _J»ec x 25,400 cm?
cm'-see g-mole hour 1.5 hours
= 57.9 g/hour2.
Although the composition of varnish 1s not known, the weight
fraction of 2-methoxyethanol 1n the varnish 1s known to be
0.02. The weight fraction 1s used as the correction factor to
adjust the release rate of the pure chemical to account for
non-ideal behavior as a result of Interactions among mixture
components. Upon multiplying the time-dependent release rate,
G^AR, by the correction factor, a value of 1.16 grams/hour
1s calculated for
21
-------
2.3 Aerosol Formation
2.3.1 Background
This section addresses the generation of aerosols from liquid
chemicals as a source of Indoor air pollution. An aerosol can be
composed of almost any type of chemical and 1s typically characterized by
constituent particles (either solid or liquid) with a size ranging from
the molecular level to 0.5 mm (Meyer 1983). Aerosol particles within the
range of 0.1 to 1.5 microns (called resplrable particles, or RSP) are
most significant because they are known to enter the respiratory tract
and penetrate deep Into the lung The lifetime of the aerosol will also
be a function of partlculate size; this 1s discussed later 1n Section 3.2.
The phenomenon of aerosollzatlon Is related to the expenditure of
energy for the agitation or propulsion of a liquid 1n an Indoor
environment. Chemical or physical (phase) changes of the aerosol are not
considered here. An Indoor liquid aerosol may be formed either
Intentionally or unintentionally. Examples of each category Include the
Intentional spraying of a disinfectant 1n the home and the unintentional
aerosol resulting from the mixing or pouring of a liquid chemical 1n an
enclosed Industrial facility.
2.3.2 Quantification of Aerosols
Methods for estimating generation rates of, and ultimately pollutant
concentrations resulting from, liquid aerosollzatlon 1n an Indoor
environment have not been developed. An extensive search of the
literature and consultation with authorities 1n the field of aerosol
research concluded that no generic data (other than that presented below)
or estimation techniques or models are available for this type of
analysis.
For the purpose of estimating the concentration of a liquid chemical
delivered to the Indoor environment as an aerosol, a percent of the total
liquid mass released In the form of an aerosol can be estimated for both
Intentional and unintentional aerosollzatlon. By assuming a volume of
air Into which the aerosol 1s released, this would yield an estimated
atmospheric concentration 1n mass per volume (e.g., mg/m^).
(1) Intentional release. The objective of an aerosol spray can or
any other type of pneumatic device 1s to deliver one hundred percent of
the contained product In the form of an aerosol via air pressure and/or
propellant. Intentional releases can consist of short-term releases of
product (e.g., a few seconds) from an aerosol container, longer-term
22
-------
releases of product from an aerosol container (e.g., from several seconds
to a few minutes), or multiple discharges of product from aerosol
containers 1n which each discharge occurs within a short time of the
other. A short-term release can be characterized as Instantaneous.
Longer-term releases and multiple discharges occurring within short times
of each other can be characterized as continuous releases. Any type of
release, however, requires some amount of time to occur and, by
definition, cannot be truly Instantaneous. For Increased accuracy, 1t 1s
suggested that all releases, even those that can be characterized as
Instantaneous, be characterized as continuous releases. The following
equation 1s used to estimate the release of a chemical substance from a
pressurized aerosol container.
MF x H x 0V
DO
(2-16)
where
WF
G
M
0V
DD =
weight fraction of chemical substance 1n product (unltless)
release rate of chemical substance (mass/hr)
mass of product discharged during active use (units of mass)
fraction of product that 1s overspray and does not contact
Intended target (unltless)
duration of active discharge of the pressurized aerosol product
(hours).
General guidelines for estimating the parameters of Equation (2-16) are
presented 1n Volume 7 of Hethods for Assessing Exposure to Chemical
Substances (Jennings et al. 1987).
If the product 1s designed to be released Into air rather than
directed at a surface, the entire mass of product released can be
considered overspray, and the value for 0V can be set equal to one.
Products of this type Include aerosol air fresheners, pesticide space
fumlgants, and solid room deodorizers.
(2) Unintentional release. Data available regarding unintentional
releases of aerosols are limited to airborne releases of partlculate
radioactive materials. In a study by Sutter et al. (1982), a range of
volumes of radioactive liquids and a range of masses of radioactive
powders were spilled from heights of 1 and 3 meters. The volume of
radioactive liquid spilled ranged from 125 to 1,000 cubic centimeters.
Based on experimental results reported by Sutter et al., an average
23
-------
weight fraction of 0.00003 of "spilled" liquids can be expected to become
airborne In static air when spilled from a height of one meter onto the
floor of a room-sized enclosure (HammerStrom et al. 1985). The mass of
radioactive powder spilled ranged from 25 to 1,000 grams. Based on
experimental results reported by Sutter et al., an average weight
fraction of 0.00019 of a "spilled" powder can be expected to become
airborne 1n static air when spilled from a height of one meter onto the
floor of a room-sized enclosure (Hammerstrom et al. 1985).
2.4 Migration
2.4.1 Background
Plastic and elastomerlc materials may contain relatively low
molecular weight substances that may leave the polymer matrix and enter
an external phase. The widespread use of plastics and elastomers 1n
products found 1n homes, offices, transportation vehicles, and commercial
and Industrial buildings Implies that persons are exposed to these
migrant substances 1n Indoor air. To quantify the extent of that
exposure, one must understand the forces governing migration,
Schwope et al. (1985), describe the mechanism for migration as one of
diffusion of migrant through the polymer to Its surface, transfer across
the polymer/external phase Interface, and assimilation Into the bulk of
the external phase. The migrant may be an additive (plastldzer,
stabilizers, etc.), unreacted catalyst, unreacted monomer, or low
molecular weight polymer (I.e., ollgomer). The factors that Influence
the rate of migration are numerous and are beyond the scope of this
report. However, Schwope et al. provide a comprehensive analysis of this
complex phenomenon.
2.4.2 Estimation of Migration
Schwope et al. (1985) present a model based on a polymer having the
following properties:
* It 1s a flat sheet with no edges.
• The migrant 1s Initially homogeneously distributed through the
polymer.
• The migrant of concern 1s unaffected by any other migration that
may be occurring.
• The external phase does not penetrate the polymer.
24
-------
* The diffusion coefficient of the migrant 1s a function only of
temperature (and not of position or time).
• Pick's laws apply (migration 1s proportional to the concentration
gradient between the polymer surface and Interior).
» The external phase may be either Infinite or finite.
The model developed by Schwope et al. can be used to estimate the
fraction, F, of migrant that has been released from the polymer at time,
t. The fraction of migrant released at time, t, can then be multiplied
by the original concentration In the polymer and by the polymer volume to
determine the total mass of migrant released as follows:
Ht = FCSOVS (2-17)
where
M-t = mass of migrant released (grams)
F = fraction of migrant released
Cso = original concentration of migrant 1n polymer (g/cnfl)
Vs = volume of polymer (cm3).
The mass, M^, should be divided by the time to obtain the average
release rate, G, 1n grams per hour for the period of release of the
migrant from the polymer.
25
-------
3.
BEHAVIOR OF POLLUTANTS INDOORS
Once Introduced Into the Indoor atmosphere (by processes such as
those discussed 1n the previous section), pollutants may undergo a number
of transport and transformation phenomena. The major determinants of a
pollutant's fate are described 1n subsequent subsections; they are:
• Removal by ventilation.
• Interactions between gaseous or vapor stages of chemicals and
partlculates and surfaces.
• Settling and resuspenslon of partlculates (and pollutants sorbed
to them).
3.1
Removal by chemical decay processes.
Ventilation
Ventilation can be defined 1n two ways: (1) a system providing fresh
outdoor air and (2) circulation of air within a building (Meyer 1983).
This 1s an Important distinction since the circulated air may be
contaminated. Heyer (1983) estimated that heat and air conditioning
systems redrculate between 80 percent and 100 percent of the air. Fresh
air will, however, still be entering the building through leaks, cracks,
etc. It can therefore be assumed that 1n most enclosed buildings, the
ventilation air 1s a combination of redrculated and fresh air. The
exact percentage of each will vary among buildings depending on such
factors as the quality of Insulation, number of leaks and cracks, and
existence of storm windows.
A1r reclrculatlon may complicate some Indoor air modeling; however,
1t does not apply to some scenarios. For example, when a contaminant 1s
released 1n only one room of a building and the length of exposure 1s
brief, 1t can be assumed the air coming from other rooms 1s
contaminant-free. However, for a long period of exposure, this will not
apply since the air 1n other rooms will have become contaminated.
The ventilation rate 1s Important for calculation of concentrations
of pollutants 1n Indoor air. This 1s usually expressed 1n terms of an
air exchange rate, which 1s defined as the rate at which Indoor air 1s
replaced by outdoor air. However, 1f exposure 1n only one room 1s of
concern, and the Incoming air from other rooms 1s contaminant-free, this
air would also be Included 1n the air exchange rate. Therefore, an air
exchange rate of one per hour Implies that the air In a given room or
building 1s completely exchanged with uncontamlnated air 1n one hour.
21
-------
Typical air exchange (fresh air Infiltration) rates 1n residences
vary from under 0.5 per hour to over 4 per hour. One study (McNall 1981)
measured air exchange rates 1n homes 1n 14 cities throughout the United
States and found a mean air exchange rate of 0.86 per hour. Another
study measured homes 1n Colorado and found a mean air exchange rate of
0.82 per hour (Heyer 1983).
In most Indoor air models, the air exchange rate Is expressed 1n
terms of the amount of air Infiltration (ft^/hr) per room or building
volume (ft-*), if one or both of these variables are unknown, the
average air exchange rates Identified above can be used for buildings.
However, for specific rooms, the air exchange rate may be slightly higher
since uncontamlnated air from other rooms may also be Included 1n the
amount of Infiltration air.
Table 1 lists air exchange rates based on ANSI/ASHRAE recommended
ventilation rates and assumed celling heights. Ventilation rates are
expressed as cubic feet per minute (cfm) per person In the room; the
first column of the table lists the number of persons expected In various
commercial establishments. These two values can be used to calculate an
air exchange rate (number of room air changes per hour) that can be used
1n models to estimate concentrations of pollutants Indoors. A celling
height was assumed for each type of commercial area to enable calculation
of room volume. The last column of Table 1 Indicates that air exchange
rates computed for the commercial areas, where smoking 1s not permitted,
typically fall within a range of 0.5 to 5 per hour.
3.2 Participate Behavior
Much of the recent research Into Indoor air pollution has centered on
aerosols, fibers, dusts, and other partlculates. Hollowell et al.
(undated) state that early researchers assumed that Indoor pollution
arises solely from outdoor sources and enters buildings through
ventilation air. This has been shown to be true 1n some cases, yet
Investigators (Moschandreas et al. 1979; Lefcoe and Inculet 1975) have
also Identified Indoor sources of pollution. The role of partlculates
must be accounted for 1n any comprehensive assessment of exposure to
chemicals 1n Indoor air, whether the pollutant of Interest 1s gaseous or
partlculate, of Indoor or outdoor origin. Gaseous or vapor-phase Indoor
air contaminants may adsorb to partlculates, such as household dust or
aerosols; the fraction adsorbed thereafter behaves as a partlculate and
1s subject to settling, resuspenslon, removal by air filtration, etc.
The following subsections discuss the characteristics of partlculate
matter 1n Indoor air, the tendency of vapor-phase pollutants to adsorb to
partlculates, and the physical processes governing parttculate behavior.
28
-------
Table 1. Counercial Air Exchange Rates
(Based on ANSI/ASHRAE 62-1981 Recommended
Ventilation Rates and Assumed Ceiling Heights)
Estimated ASHRAE 62-1981 Air exchange rates
persons per Ceiling* ventilation rates per hour
1000 ft2 height Non-
Comnercial area floor area Smoking smoking Smoking Non-smoking
Shoe repair shops (combined workrooms/
trade areas)
Theatres
Auditoriums in motion picture theatres.
lecture, concert and opera halls
Stages, TV and movie studios
Ballrooms (public)
Bowling alleys (seating area)
Gymnasiums and arenas
Playing floors -minimal or no seating
Locker rooms
Spectator areas
Game rooms (e.g., cards and billiards)
Sni timing pools - spectators' area
Ice skating, curling, and roller rinks
Transportation
Waiting rooms
Ticket and baggage areas, corridors, and
gate areas
Concourses
Offices
General office space
Conference rooms
Waiting rooms
Educational facilities
Classrooms
Laboratories
Libraries
10
150
70
100
70
70
20
150
70
25
30
150
150
150
7
50
60
50
30
20
ft
10
30
30
35
15
30
10
20
10
35
25
15
15
15
10
10
10
10
15
15
cfm/person
15
35
—
35
35
__
35
35
35
35
—
35
35
35
20
35
35
25
—
—
cfm/person
10
7
10
7
7
20
15
7
7
7
20
7
7
7
5
7
7
5
10
5
0.9
10.5
—
6
9.8
—
4.2
15.8
14.7
1.5
--
21
21
21
8.4
10.5
12.6
7.5
—
—
0.6
2.1
1.4
1.2
2.0
2.8
1.8
3.2
2.9
0.3
1.4
4.2
4.2
4.2
0.2
2.1
2.5
1.5
1.2
0.4
Source: ASHRAE (1981a).
*Values for ceiling height are based on professional judgement.
29
-------
3.2.1 Participates 1n Indoor A1r
Moschandreas et al. (1981) performed the most recent, comprehensive
assessment of Indoor air pollution Identified during a thorough
literature review. They studied Indoor-outdoor air quality relationships
of NO, N02, S02, C02, metals, benzo(a)pyrene (B(a)P), total
suspended partlculates (TSP), and resplrable suspended partlculates
(RSP). A summary of their Indoor and outdoor partlculate matter data 1s
presented 1n Table 2. Though outdoor data are limited, 1t 1s apparent
that Indoor air has consistently higher concentrations of TSP, RSP,
504, and Iron and often exceeds outdoor air levels of other metals and
B(a)P. The type of heat used 1n homes can Impact the level of
partlculates; combustion processes associated with gas heat result 1n
higher levels of TSP and RSP.
Temperature and humidity have a profound effect on the
characteristics and behavior of Indoor air partlculates. These
parameters affect the rate of formation and evaporation of liquid
aerosols (and thus their conversion to dry partlculates) (Meyer 1983).
Temperature may affect the mixing or stratification of Indoor air and
thus affect the movement of partlculates.
Activities within the home are believed to have the most profound
effect on partlculate concentrations. Cooking, smoking, children
playing, and house cleaning either generate or resuspend partlculates
(Lefcoe and Inculet 1975). The World Health Organization (WHO 1979)
focused on smoking and cooking (especially with gas) as culprits for
generating aerosols and dry partlculates 1n homes. In offices, smoking
1s probably responsible for partlculates and trace metals observed.
3.2.2 Physical Behavior of Partlculates
Partlculates 1n air are not stable under chemical equilibrium
conditions; settling, separation, and recombination are expected to
occur, especially 1n static air. As a result, partlculate accumulation
occurs 1n buildings 1n aerodynamlcally favorable places. Indoor air
disturbances stir up settled dust and fibers and resuspend them
(Meyer 1983).
Sansone and Sleln (1977/78) reviewed the pertinent literature on
resuspenslon of Indoor surface contamination. They express the
relationship between surface and Indoor contamination as:
KD Aconc
R = S
cone
30
-------
Table 2. Summary of Data for Paniculate Matter
1SH
(mg/m3)
Sites
Residence -
electric heat
1
2
3
4
Hos i dem:e -
gas heat
1
2
3
4
5
6
;
8
Outdoor
1
2
Hax
24 Hr
Aver
89.1
142.3
202.5
202.6
283.6
449.7
349.3
348.6
342.6
256.5
2(3.8
233.0
56.9
129.6
Hean
24 Hr
Aver
47.9
54.1
35.6
61 2
5;. 7
109.9
112.8
110.9
78.8
90.;
14;. 2
42 6
29.;
36.5
RSP*
(mg/m3)
Max
24 -Hr
Aver
45.8
2;. 6
118.9
72.5
67.6
120.1
160.0
61.5
70.4
107.3
149.1
91.4
38.5
53.0
Hean
24-Hr
Aver
23.0
19.4
27. 5
40.1
22.9
53.0
63.6
42.0
36.9
49 3
99.9
30.3
16.8
20.2
(mg/m3)
Max
24-Hr
Aver
9.4
7.1
20.9
12.1
21.7
13.1
16.4
5.8
8.7
4.1
56
14.4
8.0
7.0
Hean
24 -Hr
Aver
5.9
3.0
5.2
5.6
9.5
5.4
5.3
3.1
3.1
2.1
2.6
6.1
3.0
2.5
M03
(mg/m)3
Hax
24 -Hr
Aver
1.3
0.8
1.4
0.8
1.7
6.8
4.0
2.1
3.2
5.6
7.6
5.2
0.6
1.1
Hean
24 -Hr
Aver
0.8
0.5
0.5
0.4
0.5
1.7
1.9
1.1
1.1
1.9
2.8
0.8
0.3
0.5
B(a)P
(ng/m3)
Hax
24 -Hr
Aver
i.;
2.8
3.3
8.0
—
0.9
11.4
1.8
1.6
1.1
6.7
0.6
1.3
2.2
Hean
24 -Hr
Aver
1.0
1.1
1.0
4.1
...
0.4
1.5
0.9
0.7
0.4
1.2
0.3
0.5
1.0
Pb
(mg/g
Hax
24 -Hr
Aver
0.98
0.94
1.00
1.14
0.39
0.32
0.%
1.88
0.40
0 28
1.35
2.11
0.66
1.81
Fe
ii3) (ug/m3)
Hean
24-Hr
Aver
0.62
0.43
0.36
0.50
0.19
0.22
0.30
0.52
0.21
0.13
0.27
0.40
0.30
0.58
Hax
24-Hr
Aver
0.41
0.85
0.36
0.34
0.12
0.86
0.82
0.45
0.48
1.09
0.73
0.49
0.19
0.43
Hean
24-Hr
Aver
0.2S
0.44
0.19
0.16
0.03
0.52
0.26
0.44
0.26
0.43
0.35
0.20
0.08
0.18
Hn
(ug/m3)
Hax
24 -«r
Aver
1.41
0.17
0.02
0.03
0.46
0.03
0.02
2.68
0.01
0.02
0.05
0.02
0.01
0.04
Hean
24-«r
Aver
0.88
0.03
0.01
0.02
0.06
0.02
0.01
1.44
0.01
0.01
0.02
0.01
0.01
0.02
V
(ng/m3)
Hax
24 -Hr
Aver
41.8
29.8
18.4
25. 4
43.2
17.8
42.1
28.4
54.0
18.9
39.5
10/.5
22.3
65.6
Hean
24 -Hr
Aver
25.1
19.0
12.3
11.1
H.4
11.1
23.0
16.3
34.6
12.1
11. 1
43.1
12.0
30.3
As
(ng/m)3
Hax
24 -Hr
Aver
6.4
5.9
2.7
3.5
4.4
4.7
3.2
4.3
4.4
5.1
5.8
2.5
1.1
10.3
Hean
24 -Hr
Aver
1.9
2.7
1.2
1.3
1.2
2.5
14
2.3
2.0
3.0
3.5
0.9
0.5
2.5
Cd
(ng/m1)
Hax
24-Hr
Aver
3.0
4 ;
6.1
3.6
3.8
12.8
3.7
5.6
4.9
1 1
; 6
3.2
44
10.9
Hean
24 -Hr
Aver
1.2
1.3
20
?.o
1 1
3. 1
2.5
2.0
2.5
3.6
53
1. 1
15
2.2
«RSI' Indoor average based on one location (zone) only: living roan.
Source: Hoschandreas et al. (1981).
-------
where
KR = the resuspenslon factor 1n m~^, cnr^, or
Aconc = the air concentration of the substance
Sconc = the surface concentration of the substance.
A resuspenslon factor of KR = 10~*m~l Indicates that one mass unit
of surface contamination per square meter would yield 10-* mass units
of contamination per cubic meter. Average values of KR reported by
Sansone and Sleln range from 2 x 10-fyiH 1n quiescent conditions to
4 x IQ-2™-1 when a contaminated floor 1s being swept. Other
Investigators have reported KR values generally one or two orders of
magnitude higher than those. The variability In resuspenslon factors 1s
attributed by the authors to the nature of the contaminant, the nature of
the surface, and the area of the room 1n which the airborne
concentrations were measured. Most of the resuspenslon data reported
pertain to radioactive dusts and aerosols.
Particle size, shape, and density determine the aerodynamic behavior
of particles and the likelihood that suspended matter 1s Inhaled (Sansone
and Sleln 1977/78). Though several means are available for
characterizing aerosols 1n an Indoor aerosol environment, relatively
little work on the sizes of Indoor air partlculates has been performed.
A simple measurement technique 1s to determine the total mass of
partlculate matter per unit volume of air; this 1s known as the
"partlculate loading." A more sophisticated technique 1s the "clean
room" classification, which recognizes that large numbers of small
particles may have a marked effect on product performance while
contributing Uttle mass to the partlculate loading environment.
"Clean-room11 ratings are determined by a total count of the number of
particles per unit volume within a given range of sizes. Lum and Graedel
(1973) studied the size spectra of Indoor aerosols; their results are
presented 1n Table 3. These were, however, overnight measurements;
Lefcoe and Inculet (1975) pointed out that daytime activities often
result 1n the resuspenslon of the larger particles, which would not be
seen at night.
The size and shape of the partlculate affect settling by dictating
the extent to which gravity pulls the particle. Sawyer and Spooner
(1978) demonstrated the effect of asbestos fiber diameter on settling
velocities; their results, 1n Figure 1, show that the effect of fiber
size Is logarithmic. The logarithmic relationship Indicates that the
settling velocity 1s only weakly dependent upon fiber length but strongly
dependent upon fiber diameter. The parameter, pi, 1n Figure 1 1s the
fiber density. The parameter, n, 1n this figure, 1s the coefficient of
gas viscosity. The parameter, g, 1s the acceleration due to gravity.
32
-------
Table 3. Particle Concentration and Size Distribution
for Overnight Measurements.
Time
July 1971
1700
2030
2300
0400
1000
7 July 1971
1700
2023
2300
0400
1000
Total
Counts Ft"**
(0.25-2,5
1,373.000
1,347,000
379,000
309,000
1,442,000
536,000
140,000
592,000
106,000
689,000
Percent of Total Number
0.20-0.25
(pm)
40.1
39.9
59.4
62.7
38.9
55.7
67.6
43.9
67.5
51.1
0,25-0.35
(wm)
53.2
53.5
38.9
36.2
54.2
42.6
31.0
51.1
30.7
45.4
0.35-0.50 0.50-0.75
(yfn)
m
6.4
6.4
1.5
1.0
6.7
1.6
1.1
4.8
1.5
3.3
(ym)
0.14
0.13
0.13
0.06
0.15
0.09
0.17
0.17
0.28
0.13
0.75-1.50
(pm)
0.007
0.005
0.025
0.01
0.01
0.003
0.13
0.01
0.01
0.006
*This number is equivalent to the "clean room" rating.
Source: Lum and Graedel (1973).
33
-------
101
u
u
>
£
U
C
ui
O
z
ui
10
,-1
10
-2
10
-3
10'
—i 1—(—r M ui i
pi« 2.6 a/em
tj* 1,S* 10~4gem~1 we."1
g * 981 em we."
— — FIBER AXIS VERTICAL
—— FIBER AXIS HORIZONTAL
10
FIBER LENGTH,,
100
Source: Sawyer and Spooner (1978).
500
Figure 1. Theoretical settling velocities of fibers.
34
-------
Aerosols have been shown, under certain conditions found Indoors, to
coagulate and form larger particles that therefore settle faster (Meyer
1983). The conditions that favor the formation of aerosol aggregates are
high humidity and large temperature gradients within the air space
(Meyer 1983, Strey and Wagner 1982).
3.2.3 Adsorption of Vapor Phase Pollutants to Partlculates
The simplest methods for estimating gaseous concentrations of
pollutants, as presented 1n Section 2.1, depend on the assumption that
the concentration of a chemical at equilibrium cannot exceed that which
would be predicted by the Ideal gas law. If, however, gaseous
constltutents can adsorb to airborne partlculates, the concentration of
that constituent may Indeed exceed the equilibrium concentration as a
vapor.
Few researchers have addressed this phenomenon; those who have
studied adsorption have used benzo(a)pyrene as the subject (Butler and
Crossley 1979, Natush and Tomklns 1978). Natush and Tomklns presented
data on adsorption of B(a)P to fly ash 1n the stacks of coal-fired power
plants. They discussed the theoretical basis for that adsorption and
concluded that the theory was applicable to any vapor species and any
partlculate adsorbent.
Natush and Tomklns (1978) present the mathematical basis for the
theory as follows:
"ad
HC*Sur
where
p
HCp = the polycycllc aromatic hydrocarbon (e.g., B(a)P)
Sur = the surface of the partlculate
HCads = tne adsorbed entity
ka(]S and kdes = adsorption and desorptlon rate constants
(sec-1).
The rate constant for desorptlon 1s determined by the activation energy
barrier:
k T exp[-Ed/RT]
k, = -r- (3-2)
des h
35
-------
where
kg « Boltzmann's constant = 1.38044 x 1Q-16 erg/°K
h = Planck's constant = 6.6252 x 10~27 erg-sec
T = temperature (°K)
R = the gas constant = 1.9872 cal/mole-°K
E = kadsna ' kdes "ads ' kads nT '
36
-------
Table 4. Coefficients of Rate Constants
in Adsorption Equations
Coefficient Value Comments
c KT4 to 1 Natush and Tonkins (1978)
(unitless) cite Roberts and Vanderslice
(1967)
HWa(j chemical specific
np See Table 3-3
dp See Table 3-4
Ea 10 Kcal/mole These data are specific to
Ed 30-40 Kcal/mole PAHs and fly ash; Natush and
Tonkins (1978) cite Hayward
and Trapnell (1964).
6 0.5 Assumed; equation relatively
insensitive to this parameter
37
-------
where
nads = the number of vapor molecules adsorbed
nj = the total number of PAH molecules per unit volume, and
"T = na * "ads-
Equation (3-4) can be used to establish the mole fraction, Xv, of
all PAH molecules that are adsorbed under steady-state conditions. Thus,
the derivative of the net rate of adsorption 1s equal to zero and
T ads des
Integration of Equation (3-4) provides an expression for the time, t-|/q,
required to reduce the concentration of vapor species to a fraction 1/q of
Its Initial value. Table 4 presents values for the coefficients defining
and kdes-
Since partlculate surface area 1s a large determinant of the degree
of adsorption, 1t would appear that vapors will preferentially adsorb to
smaller particles (with higher surface area-to-volume ratios). Butler and
Crossley (1979), again using B(a)P as an example, prove that to be the
case. They demonstrated that less than 70 percent of the partlculate
matter they measured was less than 3j*m 1n diameter, while 90 percent of
the total B(a)P was associated with particles less than or equal to
3ym. They also point out that these smaller particles are more likely
to be resplrable.
3.3 Chemical Decay
Airborne pollutants will be removed from the Indoor environment via
atmospheric degradation processes such as oxidation, hydrolysis, and
photolysis. Also, It has been shown (Sutton et al. 1976, Meyer 1983) that
highly reactive airborne pollutants degenerate rapidly when 1n contact
with typical living space surfaces and materials. The Importance of these
removal (or fate) processes to the Indoor pollutant concentration over
time depends on the chemical nature of the pollutant, the generation and
Infiltration rates of the Indoor pollutant, and the air change rate within
the Indoor environment. Clearly, 1f the rates of the removal processes
for a particular pollutant are considerably slower than the air exchange
rate, ventilation will prevail as the primary process of pollutant removal
from the Indoor environment. To Illustrate this point, the atmospheric
residence times for dlchloromethane and trlchloroethene are 15 to 33 weeks
and about 4 days, respectively (Versar 1980); the typical air exchange
rates for modern buildings range from 0.4 to 10 air changes per hour
(Meyer 1983).
38
-------
Unfortunately, very little research has been done 1n the area of
comparing outdoor atmospheric decomposition with Indoor atmospheric
decomposition processes even though differences 1n conditions have been
well characterized (I.e., air flow, dispersion and mixing, temperature,
and humidity). There are no estimating methods for predicting rates of
decay reactions Indoors,
Atmospheric residence times and half-lives are available for 49 of
the 129 priority pollutants 1n the Non-Aquatic Fate Studies (Versar 1980)
performed for the EPA. An estimation technique for calculating these
values for other chemical pollutants 1s available 1n the Handbook of
Chemical Property Estimation Methods (Lyman et al. 1982).
Oxidation, photolysis, and hydrolysis reaction rates are tabulated
1n two sources, Versar (1980) and Data Sheets for Atmospheric Reactions
(Hampson 1980, prepared by the National Bureau of Standards). Data 1n
both sources are limited 1n scope and reflect values for gas phase
chemical reactions 1n the outdoor atmosphere. Photolysis 1s not
considered a significant degradation process 1n Indoor air because of the
absence of ultraviolet light. Oxidation would be the process of primary
concern 1f rates are available for a particular chemical pollutant.
Reaction on and sorptlon to Indoor material surfaces can be
significant removal processes because of the higher surface-to-volume
ratio Indoors. Highly reactive chemicals and organic adds adhere
tenaciously to surfaces and are most likely to be appreciably affected by
this process (Meyer 1983).
3.4 Calculation of Indoor A1r Concentrations
Indoor air pollutant concentrations can be measured by monitoring
studies or estimated by mathematical modeling techniques. Obviously,
monitoring Is preferred for developing accurate data; however, because of
Its relatively high cost and application to only one set of conditions,
1t 1s not always feasible. Consequently, mathematical models have become
the more common method for estimating Indoor air concentrations.
An estimation technique flow diagram 1s presented as Figure 2. One
of the three concentration pathways Indicated 1n the diagram could be
selected depending on (1) data availability, (2) desired computational
difficulty, and (3) preferred confidence level of the estimations.
Pathways 1 and 2 are fairly simple to use and are based on the gas laws
presented 1n Section 2. Pathway 3 Is comparatively difficult and deals
with (1) Instantaneous, (2) continuous, and (3) time-dependent
contaminant releases. This pathway requires comprehensive computations
and uses mathematical models for estimating concentrations.
39
-------
Setecl Preferred Pathway
'
i
1
Vapor Pressure
F
'
Model*
Con tarn In am
'
Mole Fraction
1 '- ! 1
Pathway 1 Pathway 3 Pathway 2
t
' Requisit* Osta: Vapor Pressure
• Easy Computations
• Low Confidence Level
1
i i
• Uae Literature, Estimate.
or Assume Vapor Pressure
of Conlamkianl
• Compute Partial Pressure*
(Section 2.2}
• Calculate Concentration
ualng Meal Gaa Law
(Section 2.1)
1
r
• Requisite Data; Mole Fraction
of Contaminant
• Eaay Computation*
• Low Confidence Level
1
Estimate or Assume
• Room Volume
• Air Exchange Rate
• Mlilng Factor
(Section 3.4.1)
F
• Compute Partial Pressure*
using Raoult's Law
(Section 2.1)
* Calculate Concentration
mktg Ideal Gas Law
(Section 2.1)
Instantaneous Release
-fa
O
No LbnHalton on Exposure Time
Initial Concentration Required
Moderately Difficult Computations
Low/Medium Confidence Level
Eallntate Initial Concentration
Bleed on Total Release
(Section 3.4.2)
Concentration Esthnatea
U*e Equation* (1-7) Through
(3-11) to Eattmaie Concentration
Conllnuou* Release
t
No Limitation on Esposute TtilM
Release Rate) Required
Difficult Computations
Low/Medium Confidence Level
Release Rale
Calculate Partial Pressures
Calculate Diffusion Coefficient
Calculate Molar Fkix
Uae Molar Ftui and Surface Area
to Compute Release Rate
(Section 2.2)
Concentration Estimates
Till the End of Release, use
Equatlona (3-13) Through (3-18) to
Estimate Concentration
Alter Ike End of Release, use
Equatlona (3-11) Through! (3-14)
Compute Weighted Average to
Find Final Average Concentration
(Section 3.4.2)
Thn* Dependent RelMM
*
II*** Fkn and Application
Rate Required
Difficult Cornputattons
HedkimfHIgh Confidence Level
Release Paremelere
CakuUte Partial Preesure*
Calculate DHfualon Coefficient
Calculate Mater/Mass Flu*
Eetimale/Aeeuine Contaminant
Application Rat* (Section 2J)
Concentration EaUmale*
Uae Equatlona (3-1*) Tkrougk
(3-27) to EsUmal* Concentration
(Section 14 J)
Figure 2. Concentrfltion estimation pathways.
-------
This section deals exclusively with the models used to estimate
Indoor air concentrations 1n the Computerized Consumer Exposure Models
(CCEM). These models were developed for the Exposure Evaluation Division
of the Office of Toxic Substances of the U.S. Environmental Protection
Agency. Background Information on the model Input parameters 1s
presented followed by model derivations for three different types of
release scenarios.
3.4.1 Background
The concentration of pollutants 1n an Indoor environment depends on
numerous factors. Those Include the release rates, ventilation rates.
the extent of nonldeal Indoor air mixing, room volume, the length of
exposure, pollutant decay, pollutant sorptlon onto partlculates, and
chemical reactions that affect concentrations. Most of these factors are
considered 1n estimating Indoor air concentration; some, however, are
not, because of their limited or unknown effect, the wide variation among
contaminants, and the complexity they add to the estimation.
General Information and assumptions on the factors that have the
major effect on Indoor air concentration are presented below:
Release rates - Release rates from continuous and time-dependent
operations are assumed constant. A technique to derive release rates
from such operations was presented In Section 2.2. Simplified methods
for estimating release rates can also be used, such as: (1) assuming an
Instantaneous release, where 100 percent of the contaminant 1s assumed to
be released 1n an Instant and (2) assuming release of 100 percent of the
material over a given period of time, e.g., one year. Release rates are
usually expressed 1n grams per second. A constant release rate for a
continuous operation assumes that the amount of pollutant released from
the surface 1s the same from each point on the surface at any given point
In time. A constant release rate for a time-dependent operation assumes
that the amount of pollutant available for release from the surface
differs from point to point on the surface at any given point 1n time.
An example of a continuous operation 1s the volatilization of a chemical
substance from a bucket of liquid or from a surface to which a film has
been rapidly applied. For a continuous release, Instantaneous
application 1s assumed.
An example of a time-dependent operation 1s the volatilization of a
chemical substance from paint on the surface of walls 1n a room. Since a
few hours are required to apply paint to the walls, the rate at which the
paint 1s applied to the surface must be taken Into account. A chemical
1n paint applied to the walls during the first few minutes of the
painting session has had a few hours to evaporate by the time the last
41
-------
Table 5. Air Changes Occurring Under Average Conditions in Residences
Exclusive of Air Provided for Ventilation*
Kind of room No. air changes/hour (ACH)
Rooms with no windows or exterior doors 0.5
Rooms with windows or exterior doors on one side 1
Rooms with windows or exterior doors on two sides 1.5
Rooms with windows or exterior doors on three sides 2
Entrance halls 2
aFor rooms with weatherstripped windows or with storm sash, use two-thirds of
these values.
Source: American Society of Heating, Refrigerating, and Air-Conditioning Engineers,
Inc. (1981b).
43
-------
bit of paint has been applied. The chemical 1n the last bit of paint
applied, however, has only begun to volatilize from the surface. The
rate of application of the paint to the surface has an effect on the
amount of chemical released to the air at any given point 1n time.
Ventilation flow rates - Ventilation rates are discussed 1n detail 1n
Section 3.1. For modeling purposes, ventilation rates are assumed to be
constant, I.e., a constant, unchanging flow of air for a given
ventilation rate. Furthermore, the Incoming air 1s assumed to be
uncontamlnated. Ventilation rates are generally given 1n cubic feet per
hour, or air exchanges per hour. Typical air exchange rates 1n
residences are presented 1n Table 5.
Indoor air mixing - Indoor air mixing describes how well the dilution
ventilation air mixes with the contaminated room air. If the room air 1s
not well mixed, pockets of poorly mixed air may be found 1n the room.
Poor mixing can greatly affect contaminant concentration since the time
for a complete air exchange to occur 1n actuality will be much greater
than the required theoretical time for an air exchange. To compensate
for this effect, a mixing factor, m, has been developed. Mathematically,
m 1s defined as the ratio of time for a theoretical air change to occur
to the time observed experimentally. A mixing factor of 1 means that the
room has Ideal, perfect mixing. Actual values of m range from 1/3 to
1/10, depending mostly on the air supply system and room size
(Clement 1981). The mixing factor 1s unltless.
Room volume - Rooms are assumed to be rectangular and enclosed. Room
volume 1s given 1n cubic meters.
Elapsed time - This 1s the time period during which the concentration or
exposure 1s of concern. The concentration 1s constantly changing with
time, and thus the length of exposure must be defined 1n order to
estimate average exposure levels. Time 1s expressed 1n hours.
These are the only parameters considered 1n the core models. As
previously stated, other parameters (pollutant decay, pollutant sorptlon
onto partlculates and surfaces, and chemical reactions) affect
concentration, vary considerably among pollutants, and, depending on the
assumptions used, may greatly complicate the concentration model.
However, 1f a function describing one of these parameters 1s known for a
given pollutant, 1t may be possible to Incorporate 1t Into the
concentration model.
42
-------
and Integrating
C(t)dt
ftu
= V C
Jtn
klt
*lldt
(3-8)
where
t0 = time at which Instantaneous release occurs (hours)
tu = any point 1n time after Instantaneous release occurs
(hours).
Noting that
C(t)dt
(tu - t0)
One can write Equation (3-9) as:
to
(3-10)
ave =
tu - t0
substituting m(Q/V) for
ave '
-C0V\e-»(0/V)t |u
lJ~' I to
t - t
It must be noted that the time at the beginning of exposure, tjj,
and the time at the end of exposure, te, can be specified at any time
after release begins. The only stipulation 1s that the time at the
beginning of exposure must occur before the time at the end of exposure.
The average concentration during exposure 1s calculated using
Equation (3-11). The variable, tb, 1s
variable, te, 1s substituted for tu 1n Equation (3-11).
substituted for t0 and the
Continuous Release - Continuous release scenarios are characterized by a
chemical substance that 1s released at a constant rate until the period
of exposure ends or the source ceases to emit the chemical substance,
whichever comes first. If the exposure continues after the source ceases
to emit the chemical substance. Equation (3-7) for calculating room air
concentrations during Instantaneous releases is used. The value for C0
used 1n Equation (3-7) would be the value of C at the time at which the
45
-------
3.4.2 Model Derivations
Instantaneous Release - The mass balance for this scenario 1s given as
fol lows:
r = -CQ (Porter 1983) (3-5)
where
V = room volume
C = contaminant concentration
Q = ventilation air flow.
Equation (3-5) assumes Ideal mixing. If nonldeal mixing 1s assumed,
Equation (3-5) becomes:
dC
V— = -mCQ . (3-6)
Assuming the Initial concentration 1s C0, the solution to
Equation (3-6) 1s
C = C0 e-n (3-7)
where
C = contaminant concentration at time t (g/m3)
C0 = the Initial contaminant concentration (g/m3)
m = mixing factor (unltless)
Q = ventilation air flow (m3/hr)
V = room volume (m3)
t = time (hours).
Equation (3-7) provides an exponential relationship between time and
contaminant concentration. Therefore, for any given time, t, the
concentration can be estimated.
The average concentration following an Instantaneous release can be
estimated by Integrating Equation (3-7) from the time of Instantaneous
release to any point 1n time after the Instantaneous release occurs. The
expression, mQ/V, 1n Equation (3-7) can be set equal to k-| , which 1s a
constant.
Therefore, rewriting Equation (3-7)
44
-------
r -"i* i
C = G/mQ [1 -e J. (3-15)
Integrating between the limits t0 and tg,
t ft r -k t
0
where
ft r -k t I
C(t)dt » G/mQ )t M - « dt|
tp = time at the beginning of release (hours)
tg = time at the end of release (hours).
Since G, Q, and Iq are constants. Equation (3-16) becomes:
(6/mQ)t * (G/mQk.) e"klt 1*9 ; (3-17)
I |tn
tg - t0
substituting (mQ/V) for K^ ,
(G/mQ)t * (GV/m2Q2) e-m(Q/V)t £ . (3-18)
______
The room air concentration at the time at which the chemical
substance 1s no longer released, C^g, 1s calculated by substituting
tg Into Equation (3-13). The parameter, C0 1s assumed to equal zero
1n Equation (3-13). For continuous releases, the parameter, tg, 1s
calculated using the following equation:
tg = •: (3-19)
where
M = mass of chemical substance released from aerosol product, or mass
applied, sprayed, or spilled onto a surface
G = release rate of the chemical substance calculated using the
appropriate method (see Section 2) (mass/hour).
Equation (3-20), a variation of Equation (3-7), 1s used to calculate
the room air concentration at any time, t, after release of chemical
substance from the source has ceased.
47
-------
chemical ceased to be emitted. Examples of scenarios 1n which the rate
of release 1s continuous Include: (1) single releases of chemical
substances from pressurized aerosol products for time periods of more
than a few seconds; (2) multiple discharges of chemical substances from
pressurized aerosol products 1n which each discharge occurs within a
short time of the other; (3) releases of chemical substances from films
formed when products are spilled or sprayed Instantaneously onto
surfaces; and (4) releases of migrants from solid matrices or bulk
liquids.
The mass balance for this scenario, assuming nonldeal mixing, 1s
given as follows:
df
= 6 "
(Porter 1983)
(3-12)
where
V = room volume (m^)
C = contaminant concentration (g/m3)
G = release rate (g/hr)
m = mixing factor (unltless)
0 = ventilation flow (m-Vhr).
After rearranging and Integrating, the solution to Equation (3-12)
becomes (Clement 1981):
C = G/mQ + (CQ - G/mQ) e
-m(Q/¥)t
(3-13)
where
C0 = the Initial concentration
t = time.
If the contaminant concentration 1n the room Is zero as release G
begins, Equation (3-13) becomes:
c . G/mq T - e-m(Q/v)t
Equation (3-13) 1s only valid for the duration of the release.
(3-14)
To determine the average concentration for any time period while
release G 1s occurring, Equation (3-14) needs to be Integrated. To
accomplish this, (mQ/V) 1n Equation (3-14) Is set equal to k-j. Thus,
Equation (3-14) becomes:
46
-------
• Case 2: The time at the beginning of exposure 1s before the time
at the end of release, and the time at the end of
exposure 1s after the time at the end of release. In
this case, the average concentration during exposure 1s
calculated using Equations (3-18), (3-21), and (3-22).
The variable, t^, 1s substituted for t0 In Equation
(3-18). The variable, te, 1s substituted for tu 1n
Equation (3-21). The average concentration during
release obtained from Equation (3-18) and the average
concentration after release obtained from Equation
(3-21) are used 1n Equation (3-22) to obtain the average
concentration during exposure. In addition, t^ 1s
substituted for t0, and te 1s substituted for tu
1n Equation (3-22).
* Case 3: The time at the beginning of exposure and the time at
the end of exposure are after the time at the end of
release. In this case, the average concentration during
exposure 1s calculated using Equation (3-21). The
variable, t(j, 1s substituted for tg, and the
variable, te, 1s substituted for tu 1n Equation
(3-21).
Time-Dependent Release - The following equations are applicable for
estimating concentrations that occur when the rate of release of a
chemical substance 1s time-dependent. These equations are applicable to
scenarios 1n which a coating or film containing the chemical substance
for which exposure 1s being assessed Is applied to a surface and the time
required for application 1s more than a few minutes. To use this method,
the assessor must first ascertain whether the time for evaporation of the
chemical substance from the film, once 1t 1s applied to the surface
(tg), 1s less than, greater than, or equal to the time required to
apply the film to the surface (ta). The application time, ta, may be
measured, estimated, or calculated by dividing the surface area covered,
SA, by the application rate, AR. The following expression 1s used to
estimate tg for a time-dependent release.
tg = M/ta/GNAR (3-23)
where H 1s the total mass of chemical substance 1n the film applied
to the surface and G^A^ Is calculated using the method cited 1n
Section 2.2.2, Example 4.
This method for estimating tg assumes that the release rate per
unit area (the mas.s flux, GN) 1s constant over time because of the
effect that drying of the coating or film may have on the release rate
and because some chemical substances do not evaporate completely from the
49
-------
C .
(3-20)
Equation (3-21) 1s used to calculate the average concentration from the
time the release of chemical substance has ceased until any point 1n time
after the time at which release of the chemical ceases.
-ra(Q/V)(t-t )
ave =
tu -
(3-21)
where
tu = any point 1n time after the time at which release of
the chemical ceases (hours).
If average concentrations are desired for a time starting before release
ceases and ending after release of the chemical substance from the source
has ceased, the average concentration during this period can be estimated
by calculating a time-weighted average of the average concentration
during release and the average concentration after release has ceased:
Cave (during period that starts before release ceases, but ends
after release ceases) =
t -t
1C x -*—*
ave (during release) tu-t0
t -t
I
C x -—
ave (after release) t -t
(3-22)
In estimating average concentrations during exposure, one should note
that the time at the beginning of exposure and the time at the end of
exposure can be specified at any time after release begins. The only
stipulation 1s that the time at the beginning of exposure must occur
before the time at the end of exposure. Possible cases for the time at
the beginning of exposure and the time at the end of exposure Include:
• Case 1: The time at the beginning of exposure and the time at
the end of exposure are before the time at the end of
release. In this case, the average concentration during
exposure 1s calculated using Equation (3-18). The
variable,
variable,
(3-18).
is substituted for tn, and the
1s substituted for
1n Equation
48
-------
(!) From t0 to t]:
C =
Vk
f 1
r •
GNAR t2 -
Vk I 2
ave
- *
(2) From t] < t <
Vk
GNAR
Vk
X2
ave
(3) From t2 < t < tr
C = -e
Vk k
(3-24)
(3-25)
(3-26)
(3-27)
- t
>28)
vk k
t
(3-29)
ave
where
t2
C, = concentration at the time, t2, calculated using
2 Equation (3-26).
-------
film or coating before the coating dries (Newman et al. 1975; Newman and
Nunn 1975), the assumption of constant flux until the entire mass of
chemical substance 1s released may not be true. It 1s recommended that
the value of tg obtained using Equation (3-23) be used only to estimate
concentrations during exposure up to the time reported for the specific
coating to form a dry film, 1f this time 1s known.
Four equations are used to calculate concentrations 1n Indoor air
resulting from time-dependent releases of chemical substances. These
equations mathematically describe four physical situations that occur at
specific Intervals that characterize a time-dependent release. Based on
letting t] equal ta or tg, whichever 1s smaller, and on letting
t2 equal ta or tg, whichever 1s larger, the following physical
situation during each of the four Intervals occurs:
(1) t0 < t < t-j: Mass of chemical substance released 1s Increasing,
and concentration at any time, t, 1s Increasing during
this Interval. The mass of chemical substance released
1s Increasing because of the additional mass being
applied to the surface.
(2) t] < t < t2' The mass of chemical substance released remains
constant, and the concentration 1s Increasing. The
additional mass applied to the surface 1s balanced
equally by the portion of the surface from which the
chemical has already evaporated.
(3) t2 < t < tr: The variable tr denotes the time at which the very
last bit of chemical substance has been released from
the surface. During this Interval, the film 1s no
longer being applied but chemical substance 1s still
being released. Therefore, the mass of chemical
substance released 1s decreasing. Whether the
concentration at any time, t, during this Interval 1s
Increasing or decreasing 1s determined by the air
exchange rate.
(4) tr < t < tu: The mass released 1s zero. The concentration 1s
decreasing with time as ventilation air flows out
of the room.
The following equations are used to determine the concentration of
chemical substance at any time, t, during each Interval and the average
concentration during each Interval. The average concentration during the
exposure period 1s determined by calculating a time-weighted average
based on the average concentration calculated during each interval.
50
-------
• Case 3: The time at the beginning of exposure 1s less than or
equal to t-|, and the time at the end of exposure 1s
greater than t2, but less than or equal to tr. In
this case, the average concentration during exposure 1s
the weighted average of average concentrations calculated
using Equations (3-25), (3-27), and (3-29). The
variable, tb, 1s substituted for t0 1n Equation
(3-25). The variable, te, 1s substituted for tr 1n
Equation (3-29).
• Case 4: The time at the beginning of exposure 1s less than or
equal to t-|, and the time at the end of exposure 1s
greater than tr. In this case, the average
concentration during exposure 1s the weighted average of
average concentrations calculated using Equations (3-25),
(3-27), (3-29), and (3-31). The variable, tb, 1s
substituted for t0 1n Equation (3-25). The variable,
te, 1s substituted for tu 1n Equation (3-31).
• Case 5: The time at the beginning of exposure and the time at the
end of exposure are greater than t-j, but less than or
equal to t2- In this case, the average concentration
during exposure 1s calculated using Equation (3-27). The
variable, tb, 1s substituted for t-| and the variable,
te, 1s substituted for t2 1n Equation (3-27).
• Case 6: The time at the beginning of exposure 1s greater than
t-|, but less than or equal to t2. The time at the
end of exposure 1s greater than t2, but less than or
equal to tr. In this case, the average concentration
during exposure 1s the weighted average of average
concentrations calculated using Equations (3-27) and
(3-29). The variable, tb, 1s substituted for t-| 1n
Equation (3-27). The variable, te, 1s substituted for
tr 1n Equation (3-29).
• Case 7: The time at the beginning of exposure 1s greater than
t-j, but less than or equal to t2. The time at the
end of exposure 1s greater than tr. In this case, the
average concentration during exposure 1s the weighted
average of average concentrations calculated using
Equations (3-27). (3-29), and (3-31). The variable.
tb, 1s substituted for t] 1n Equation (3-27). The
variable, te, 1s substituted for tu 1n Equation
(3-31).
53
-------
(4) From tr < t < tu
C . C+ |e-*-V I (3-30)
t
Cave ' '
f ..-Mt-y 1 *u
rl k2 J *r
C, _e-*^-V u (3-31)
where
C = concentration at the time at which release ends calculated
r using Equation (3-29) (mass/volume)
It must be noted that the time at the beginning of exposure and the
time at the end of exposure can be specified at any time after release
begins. The only requirement 1s that the time at the beginning of
exposure must occur before the time at the end of exposure. To delineate
the possible cases that can occur, the convention that t] corresponds
to ta or tg, whichever 1s smaller, and that t2 corresponds to ta
or tg, whichever 1s larger, 1s used. Possible cases for the time at
the beginning of exposure and the time at the end of exposure 1n relation
to t^, t£, tr, and tu Include:
• Case 1: The time at the beginning of exposure and the time at the
end of exposure are less than or equal to t-j. In this
case, the average concentration during exposure 1s
calculated using Equation (3-25). The variable, t^, 1s
substituted for t0 and the variable, te, 1s
substituted for t] 1n Equation (3-25).
• Case 2: The time at the beginning of exposure 1s less than or
equal to t-j and the time at the end of exposure 1s
greater than t], but less than or equal to t2- In
this case, the average concentration during exposure 1s
the weighted average of average concentrations calculated
using Equations (3-25) and (3-27). The variable, t^,
Is substituted for t0 1n Equation (3-25). The
variable, te, 1s substituted for t2 1n Equation
(3-27).
52
-------
Under some circumstances, the concentrations predicted using these
methods may exceed the saturation concentration of the chemical substance
under the environmental conditions for which 1t 1s being modeled. Such
circumstances are most likely to be encountered 1n scenarios 1n which a
large mass of a chemical substance with a relatively low vapor pressure
1s available for release from surfaces to which a film containing the
chemical substance 1s applied.
A reason for predicted concentrations exceeding the saturation
concentration Is the underlying assumption of the methods presented 1n
this section that release occurs at a constant rate. Release at a
constant rate can only be assumed 1f the release rate 1s slow enough that
the concentration of the chemical substance 1n the room never reaches a
level that 1s high enough to have an appreciable effect on the rate of
release. If the release rate of the chemical to pure air 1s fast enough,
the release rate will begin to decrease as concentrations become higher
and approach equilibrium.
In the event that the average concentration predicted using the
equations described previously exceeds the saturation concentration of
the chemical substance 1n any of the four Intervals during exposure, 1t
1s suggested that the assessor substitute the saturation concentration
for the value of the average concentration during that Interval. The
average concentration during the exposure period 1s then determined by
calculating a time-weighted average from the average concentration used
during each applicable Interval.
55
-------
• Case 8: The time at the beginning of exposure and the time at the
end of exposure 1s greater than t2, but less than or
equal to tr. In this case, the average concentration
during exposure 1s calculated using Equation (3-29). The
variable, tb, 1s substituted for t2 and the variable,
te, 1s substituted for tr 1n Equation (3-29). The
concentration at the time at the beginning of exposure,
Ctb, 1s substituted for Ct2 1n Equation (3-29).
Ct|j 1s calculated by substituting tb for t 1n
Equation (3-26).
* Case 9: The time at the beginning of exposure 1s greater than
t2, but less than or equal to tr. The time at the
end of exposure 1s greater than tr. In this case, the
average concentration during exposure 1s the weighted
average of average concentrations calculated using
Equations (3-29) and (3-31). The variable, tb, 1s
substituted for t2 1n Equation (3-29). The variable,
te, 1s substituted for tu 1n Equation (3-31). The
concentration at the time at the beginning of exposure,
Ctb, 1s substituted for Ct2 1n Equation (3-29).
Ctb ^s calculated by substituting tb for t 1n
Equation (3-26).
« Case 10: The time at the beginning of exposure and the time at the
end of exposure are greater than tr. In this case, the
average concentration during exposure 1s calculated using
Equation (3-31). The variable, tb, 1s substituted for
tr and the variable, te, 1s substituted for tu 1n
Equation (3-31). The concentration at the time at the
beginning of exposure, Ctb, 1s substituted for Ctr 1n
Equation (3-31). Ctr 1s calculated by substituting
tb for t 1n Equation (3-28).
It must be noted that the physical situations described for the
Intervals comprising these two cases are generally applicable under
environmental conditions considered most likely to occur Indoors. If,
however, the ventilation air flow rate 1s sufficiently high to offset the
rate at which the chemical substance 1s released, the physical situations
that occur during each Interval will vary from those previously
described. Additional data are required to determine the combination of
release rate of chemical substance and air exchange rate that would cause
the actual physical situation to deviate. The derivations of Equations
(3-24) through (3-31) are presented In Appendix B.
54
-------
4. MODEL DEMONSTRATION
Three exposure scenarios were selected to demonstrate how the
equations presented 1n Section 3 are used to estimate average
concentrations of chemical substances 1n Indoor air. One scenario was
selected to demonstrate calculation of average concentrations during
exposure for each of three generic releases. These Include continuous
releases from aerosol containers, continuous releases from spilled
liquids, and time-dependent releases from films or coatings applied to
surfaces.
Scenario 1 demonstrates how the equations presented 1n Section 3 can
be used to estimate average concentrations 1n Indoor air as a result of a
continuous release of a chemical substance from an aerosol spray.
Scenario 2 demonstrates estimation of average concentrations 1n Indoor
air as a result of a continuous release of a chemical substance from a
spilled liquid. Estimation of average concentrations 1n Indoor air as a
result of a time-dependent release 1s demonstrated 1n Scenario 3.
To estimate Inhalation exposure, calculations of average
concentrations during exposure do not have to be performed by hand. The
Computerized Consumer Exposure Model (CCEM) can be used to calculate
average concentrations and Inhalation exposure resulting from each of the
three generic releases.
4.1 Scenario 1 - Continuous Release of Aerosols
The entire contents of a 13-ounce aerosol spray can are discharged at
a constant rate over a period of 16.5 minutes 1n a kitchen of typical
size. The contents of the can consist of several chemical substances,
Including toluene. The room has a window on one side. It 1s assumed
that mechanical ventilation does not occur during the period of
exposure. Therefore, exchange of air 1s assumed to occur only by
Infiltration of air through cracks around the window frame. The air 1n
the room 1s assumed to be Ideally mixed. The Individual that dispenses
the product from the aerosol container Is assumed to remain 1n the room
for one-half hour after discharge of product from the aerosol container
begins.
Parameters required to estimate concentrations and values used for
these parameters are as follows:
tg = duration of discharge of the pressurized aerosol product, 1n
hours. The value assumed for tg 1n this scenario 1s 16.5
minutes, or 0.275 hours.
57
-------
/ S\ t / GV \ It9
I ~~~ I 4. j —5j j g-m(Q/V)t I
~ ' mQ ' ' m Q tg
ave " tg - to
Substituting the values for this scenario 1n Equation (3-18) yields
C
1
ave
(.275-0)
(1)(20)/
(107345)
d)(20)
>75) Jimmmm t
\ (1) (20) /
.\(0) /(107345)(20)\ e-l(20/20)(0)1
/ \ (1)2(20)2 / j
-1(20/20)(.275)
or 674.76 mg/nfl.
Equation (3-21) 1s used to estimate the average concentration 1n
Indoor air from the time at the end of discharge until the time at the
end of exposure. To use equation (3-21), the concentration at the point
1n time at which discharge ends, C^g, must be determined. Equation
(3-14) 1s used to estimate the concentration at the time when discharge
ends.
Ctg = G/rnqfl - e-to(W)t* 1 (3-14)
Substituting the values for this scenario Into equation (3-14) yields
Ctg - 107345/<1)(20) [l - e-H20/20). 275 J ,
or 1290.44 mg/m3.
Equation (3-21) 1s
/-C. V\ t
\-^He-
''ave - tu - tg
u
tg
Substituting the value for Ctg and the values for other parameters 1n
this scenario Into equation (3-21) yields
1 j r/(-129Q.44)(20)\ !,.,n/.,nw c oicx
Cave • (.5-. 275) j |_\ ! e-1 (20/20)( .5-.275)
f/C-1290.
L\ (i)
e-l(20/20)(.275-.275)
\ \ • / \ «•«/ / \
or 1155.57 mg/m3.
59
-------
M = mass of product discharged, 1n grams. Multiplying 13 ounces by
28.5 grains per ounce yields a value of 369 grams for M. This
parameter 1s used to estimate G.
WF « weight fraction of chemical substance 1n the product. It 1s
assumed that the weight fraction of toluene 1n the product 1s
0.20. This parameter 1s used to estimate G.
0V = fraction of product that 1s overspray and does not contact the
Intended target. For conservative estimates of concentrations
of toluene 1n air, the maximum value for overspray of spray
products 1s assumed. According to Jennings et al. (1987), the
maximum overspray fraction 1s 0.40. This parameter 1s used to
estimate G.
G = release rate of the chemical substance, 1n mg/hour. The
parameter G 1s calculated from (M x WF x 0V x l,000)/tg. The
quantity, 1,000, 1s used to convert M from grams to milligrams.
A value of 107,345 mg/hour 1s estimated for this scenario.
V = room volume, 1n cubic meters. According to Jennings et al.
(1987), the volume of a kitchen 1n a typical house 1s 20 cubic
meters.
m = mixing factor. Since Ideal mixing 1s assumed, a value of 1.0 1s
used for mixing factor.
Q = ventilation flow rate, 1n cubic meters per hour. This parameter
1s the product of the room volume and the air exchange rate
(A). Since air exchange results only from Infiltration and the
room has one window, a value of 1.0 for number of room air
changes per hour as reported 1n Jennings et al. (1987) 1s used.
Upon multiplying A by V, a value of 20 m^/hour 1s obtained.
tu = any point 1n time greater than the time at which release of
the chemical ceases, 1n hours. In this scenario, the time at
the end of exposure, te, 1s set equal to tu. A value of 0.5
hours 1s used.
t0 = time at the beginning of release, 1n hours. For this
scenario, the time at the beginning of release 1s the same as
the time at the beginning of exposure, t^.
Equation (3-18) 1s used to estimate the average concentration 1n
Indoor air during the period while the contents of the aerosol container
are being discharged.
58
-------
G = release rate of the chemical substance from the surface, In
g/hour. The calculations used to obtain G are presented 1n
Section 2. The value of G assumed for this scenario 1s 1.74
grams/hours, or 1740 milligrams per hour. This release rate was
obtained by multiplying the release rate calculated for pure
2-methoxyethanol by a correction factor of 0.02. For the
purpose of estimating concentrations during exposure, 1n
mg/m^, the value for G 1s converted from grams/hour to
milligrams/hour.
M = mass of chemical substance available for release, 1n grams.
This value 1s obtained by multiplying the mass of product
available for release (I.e., the mass of product spilled) by the
weight fraction of chemical substance 1n the product (I.e.,
100 g x .02 = 2 grams). This parameter 1s used to estimate G.
tg = time for evaporation of the chemical substance, 1n hours. This
value was obtained by dividing mass of chemical substance
available for release by the release rate of the chemical
substance. The value assumed for tg 1n this scenario 1s 1.15
hours.
V = room volume, 1n cubic meters. According to Jennings et al.
(1987), the volume of a kitchen 1n a typical house 1s 20 cubic
meters.
m = mixing factor. Since Ideal mixing 1s assumed, a value of 1.0 1s
used for the mixing factor.
Q = ventilation flow rate, 1n cubic meters per hour. This parameter
1s the product of the room volume, V, and the air exchange rate,
A. Since air exchange results only from Infiltration and the
room has one window, a value of 1.0 for number of room air
changes per hour, as reported 1n Jennings et al. (1987), 1s
assumed. Upon multiplying A and V, a value of 20 mVhour 1s
obtained.
te = time at the end of exposure, 1n hours. A value of one hour 1s
assumed 1n this scenario.
t0 = time at the beginning of release, 1n hours. For this
scenario, the time at the beginning of release 1s the same as
the time at the beginning of exposure, t^.
61
-------
Equation (3-22) 1s used to estimate the average concentration of
toluene 1n Indoor air for the entire period of exposure
Caye(total) .
[Cave (dur1"9 release) x\(tu - t0)
/(t - t )
Cavfi (after release) x (^ _ ^
(3-22)
Substituting the values for parameters 1n this scenario Into equation
(3-22) yields
Cave(total) .
or 891.12 mg/m3.
.5 - .275)^
(-5-0)
It must be noted that the average concentration estimated for this
scenario accounts only for levels of toluene released directly from the
aerosol container to air. Concentrations of toluene resulting from
volatilization from surfaces contacted by aerosol spray are not Included
1n this estimate, A time-dependent release must be used to estimate
concentrations of chemicals 1n air due to volatilization from films
applied to surfaces.
4.2
Scenario 2 - Continuous Release from Films
One hundred grams of a liquid containing two percent 2-methoxyethanol
are spilled onto a tile floor of a kitchen. The spilled liquid 1s
distributed evenly over a 27.35 square foot (I.e., 25,400 cm?) area.
The room 1n which the liquid 1s spilled has a window on one side. It 1s
assumed that mechanical ventilation does not occur during the period of
exposure. Therefore, exchange of air 1s assumed to occur only by
Infiltration of air through cracks around the window frame. The air 1n
the room 1s assumed to be Ideally mixed. The Individual that spills the
product 1s assumed to remain 1n the room for one hour after the spill has
occurred.
Parameters required to estimate concentrations and values used for
these parameters are as follows:
60
-------
Parameters required to estimate concentrations and values used for
these parameters are as follows:
ta = time for application of paint remover to the surface, 1n
hours. In this scenario, ta 1s ~1.2 hours.
GN = mass flux of pure methylene chloride, 1n grams/cm^-hour.
The mass flux of pure methylene chloride has been calculated
using methods described 1n Section 2. The value for mass flux
of pure methylene chloride, based on these calculations, 1s
0.4173 g/cm2-hr.
AR = rate of application of paint remover to the surface, 1n
cm2/hr. The surface area covered with paint remover was
converted from square feet to square centlmers by multiplying
by 929.03. The resulting value was divided by the application
time (I.e., about 1.2 hours). The value for AR obtained for
this scenario was 301,935 cm^/hr.
= time-dependent release rate of methylene chloride, 1n
grams/hour^. The value for G^AR 1s the product of G^
and AR. The resulting value 1s 1.26 x 10$ grams/hour^.
For the purpose of estimating concentrations during exposure,
1n mg/m^, the value for G^AR 1s converted from
grams/hour? to milligrams/hour^.
tg = time for evaporation of methylene chloride from the surface,
1n hours. This value 1s calculated from M/ta/GjjAR. The
value for tg 1n this scenario 1s 8.07 x 10-* hours (I.e.,
about three seconds).
tr = time for all methylene chloride applied to be released from
the surface, In hours. The parameter, tr, 1s calculated by
summing ta and tg. The value for tr used In this scenario
1s about 1.207 hours.
te = time at the end of exposure, 1n hours. A value for te
equal to the application time 1s assumed for one Individual
(I.e., 1.206 hours). A value for te of 1.5 hours 1s assumed
for the other Individual.
t0 = time at the beginning of release, 1n hours.
63
-------
It must be noted that the time at the end of exposure, te, 1s less
than the evaporation time, tg. This situation corresponds to that
described 1n Section 3 for Case 1 of a continuous release. Equation
(3-18) 1s used to estimate the average concentration of 2-methoxyethanol
during the period of exposure. It must be noted that the time at the end
of exposure, te, 1s less than the evaporation time, tg. The variable,
te, 1s substituted for tg 1n Equation (3-18).
mQ / \ m Q / I t0
"ave " te - t0
Substituting the values for this scenario 1n equation (3-18), yields
1
Cave = (1.0-0)
or 32 mg/m^.
4.3 Scenario 3 - Time-Dependent Release
Paint remover 1s applied to the walls and celling of a dining room of
typical size. The paint remover 1s a commercial grade consisting of pure
methylene chloride. The area of the walls and celling covered with paint
remover 1s 392 square feet. The room 1n which the paint remover 1s
applied has a window on one side. It 1s assumed that mechanical
ventilation does not occur during the period of exposure. Therefore,
exchange of air 1s assumed to occur only by Infiltration of air through
cracks around the window frame. The air 1n the room 1s assumed to be
Ideally mixed. The mass of paint remover used 1s measured directly and
1s determined to be 122.65 grams. Two Individuals are 1n the room. One
Individual enters the room about 15 minutes after application of paint
remover begins and leaves the room Immediately upon completing
application of remover (I.e., about 1.2 hours after application begins).
The other Individual remains 1n the room for 1.5 hours after application
of remover begins. Average concentrations during exposure are estlmateed
for each Individual.
62
-------
Substituting the values for this scenario 1n equation (3-25) yields
1.26 x 1C
8
'ave (8.07 x 10"4 - 0)(20)(1)
(8.P7 x 1P~4)2 (8.P7 x IP"4 e"1(8'07 x
,2
(1)
(1)'
(0)
-d)(0)
L 2 (1) (!)< .
or 7.807 mg/m3.
Equation (3-27) 1s used to estimate the average concentration of
methylene chloride 1n air for the Interval of time from tg to ta.
i A [tTt * 1 i -k(t-t!) -kt\ 1
~2\ 6 ' 6 /
vk L k v ; J
ave t2 - ti
Substituting the values for this scenario 1n equation (3-27) yields
1.26 x IP8
(3-27)
"4
(8.07 x 10-4)(1.20615
(1.20615 - 8.07 x 10 ")(2P)(1) '
.20615 - 8.07 x IP-4) _ e-(l)(l.20615))
|~,n «-, .. ,n-dv? . J /.-(1)(8.07 x IP'4 - 8.07
L'
_ e-(l)(8.07 x IP'4)
or 2130.98 mg/m3.
Equation (3-29) 1s used to estimate the average concentration of
methylene chloride 1n air for the Interval of time from ta to tr.
The value of 3561.573 mg/m3 for Ct2 was obtained by substituting t2
for t 1n equation (3-36).
65
-------
tt, = time at the beginning of exposure, hours. For the
Individual that remains 1n the room from the start of
application of remover until after completion, tjj 1s equal
to t0, or zero. For the Individual that enters the room
after application has begun t0 equals 0.25 hours (I.e., 15
minutes).
V = room volume, 1n cubic meters. According to Jennings et al.
(1987), the volume of a dining room 1n a typical house 1s 20
cubic meters. This parameter 1s used to estimate k-j.
m = mixing factor. Since Ideal mixing 1s assumed, a value of 1.0
1s used for the mixing factor. This parameter 1s used to
estimate k^.
Q = ventilation flow rate, 1n cubic meters per hour. This
parameter 1s the product of the room volume, V, and the air
exchange rate, A. Since air exchange results only from
Infiltration and the room has one window, a value of 1.0 for
number of room air changes per hour, as reported In Jennings
et al. (1987), 1s assumed. Upon multiplying A by V, a value
of 20 nr per hour 1s obtained.
k^ = a constant obtained by dividing the product of the
ventilation flow rate. A, and mixing factor, m, by the room
volume, V. In this scenario, k] 1s always equal to 1.0.
To estimate average concentrations during exposure for the Individual
present 1n the room from the time paint remover Is applied until 90
minutes later, equations (3-25), (3-27), (3-29) and (3-31) are used. In
this scenario, tg < ta. Therefore, the value for tg Is substituted for
t], and the value for ta 1s substituted for t2- The variable,
te, 1s substituted for tu 1n equation (3-31).
Equation (3-25) 1s used to estimate the average concentration of
methylene chloride 1n air from the time at the beginning of exposure,
tfj, until the time required for methylene chloride applied at the very
beginning of exposure to evaporate, tg.
C
GNAR j"t2 t e-kt
Vk L 2 k k2
ave t] - t0
(3-25)
64
-------
'ave(TOTAL)
rc x((t> - V) i
L ave(to-ti). x\(te - t0)/J
/ (t - t )v
Cave(t.-t ) x\(t -t ) J
* Cave(t1-t2) X
r e o
Substituting the values for this scenario yields
Cave(tg-tr) x
(te - t0)/
W1)]
'ave(TOTAL)
.20615- 8.07
1.5 - 0
,536.765
3085.785 x
.206558)\
- 0)
or 2317 mg/m3.
To estimate average concentrations during exposure for the Individual
who enters the room about 15 minutes after application of paint remover
begins and leaves the room Immediately upon completing application of
remover, equation (3-27) alone 1s used. The variable, tjj, 1s
substituted for t] and the variable, te, 1s substituted for t2 1n
equation (3-27). Substituting the values for this scenario 1n equation
(3-27) yields
1.26 x 10
8
ave
(1.20615-.15)(20)(1)
.25)(1.20615) *
(1)'
.20615-.25) _ e-(l.20615)
(.25X.25)
.25-.25) _ e-(l)(.25)
or 2530 mg/m3.
67
-------
ave
tr - t?
Substituting the values for this scenario 1n equation (3-29) yields
.20615) "
1
'ave " (1.206558-1.20615)
ne-(1)(1.206558-1.
0 ]
. 26
(3-29)
-20&558-1.2061f
- 3561
5731 Jl.26 X
J L(20)(1
,8
j2- (1.206558)f } * 1.206558 - T-20*558
(20)(1)
or 3537 mg/m3.
|_ 1.26 x 10 ^1 ^ 1 206558 - 1 .20615J - 3561 .573
\1 1
/J J
1 n 1.20615
(1-20615)*1.206558- 2
Equation (3-31) 1s used to estimate the averge concentration of
methylene chloride 1n air for the Interval of time from tr to the time
at the end of exposure, te. The value of 3560.646 mg/m3 for C. was
obtained by substituting tr for t 1n equation (3-28). r
^1 ».
! 1 tr
"ave ~ te - tr
Substituting the values for this scenario 1n equation (3-31) yields
)-l. 206558)"]
(3-31)
3560.646
[-(1.5-1.206558)1 f -(1.206558-1.;
^e _ ^e
(i)2 J L (i)2
ave
or 3086 mg/m3.
1.5 - 1.206558
The average concentration of methylene chloride 1n air during the
exposure period 1s a weighted average of the average concentrations for
each of the four Intervals.
66
-------
5. REFERENCES
ASHRAE. 1981a. American Society of Heating, Refrigerating and
A1r-Cond1t1on1ng Engineers. ASHRAE handbook 1981 fundamentals. Atlanta,
GA: American Society of Heating, Refrigerating and A1r-Cond1t1on1ng
Engineers, Inc.
ASHRAE. 1981b. American Society of Heating, Refrigerating and
A1r-Cond1t1on1ng Engineers. Ventilation for acceptable Indoor air
quality. ASHRAE 62-1981. Atlanta, GA: American Society of Heating,
Refrigerating and A1r-Cond1t1on1ng Engineers, Inc.
Butler JO, Crossley P. 1979. An appraisal of relative airborne
sub-urban concentrations of polycycllc aromatic hydrocarbons monitored
Indoors and outdoors. Sd. Total Environ. 11: 53-58.
Clement Associates Inc. 1981. Mathematical models for estimating
workplace concentration levels: a literature review. Washington, DC:
U.S. Environmental Protection Agency.
Greenkorn RA, Kessler DP. 1972. Transfer operations. New York, NY:
McGraw-Hill Book Co.
Hammerstrom K, Schweer LG, Adklns L, et al. 1985. Exposure assessment
for polychlorlnated blphenyls (PCBs): Incidental production, recycling,
and selected authorized uses. Final report. Washington, DC: Office of
Toxic Substances, U.S. Envlornmental Protection Agency. EPA Contract No.
68-01-6271.
Hampson RF. 1980. Chemical kinetic and photochemical data sheets for
atmospheric reactions. Washington, DC: U.S. Department of
Transportation, Office of Environment and Energy.
Hayward D, Trapness BMW. 1964. Chem1sorpt1on. Butterworths, London (as
cited by Natush and Tomklns).
Hollowell CD, BudnHz RJ, Traynor GW. Undated. Combustion-generated
Indoor air pollution. Fourth International Clean A1r Congress, Japanese
Union of A1r Pollution Prevention Associations, pp. 684-687.
Jennings PD, Hammerstrom KA, Adklns LC, et al. 1987. Methods for
assessing exposure to chemical substances. Volume 7. Methods for
assessing consumer exposure to chemical substances. Washington, DC:
Office of Toxic Substances, U.S. Environmental Protection Agency. EPA
560/5-85-007.
Lefcoe NM, Inculet II. 1975. Partlculates 1n domestic premises II.
Ambient levels and Indoor-outdoor relationships. Arch. Environ. Health
30: 565-570.
69
-------
Schwope AD, Lyman WJ, Reid RC. 1985. Methods for assessing exposure to
chemical substances. Volume 11. Methods for estimating the migration of
chemical substances from solid matrices. Washington, DC: U.S.
Environmental Protection Agency. EPA 560/5-85-015,
Slenko MJ, Plane RA. 1966. Chemistry: principles and properties. New
York, NY: McGraw-Hill Book Co.
Strey R, Wagner PE. 1982. Measurements of homogeneous nucleatlon rates
for the homologous series of alcohols and comparison with the classical
nucleatlon theory. Jour. Aerosol Sc1., 162-164.
Sutter DJ, Nodolf KM, Maklno KK. 1976. Predicting ozone concentrations
1n residential structures. ASHRAE Journal. September 1976. pp. 21-26.
Sutter SL, Johnston JW, M1sh1ma J. 1982. Investigation of
accident-generated aerosols: releases from free fall spills. Am. Ind.
Hyg. Assoc. J. 43{7):540-544.
Treybal RE. 1968. Mass-transfer operations. New York, NY: McGraw-Hill
Book Co.
Versar. 1983. Prediction of human exposure to cellosolves. Draft
report. Washington, DC: U.S. Environmental Protection Agency.
Versar. 1980. Non-aquatic fate studies. Draft reports. Washington,
DC: U.S. Environmental Protection Agency, Office of Water Regulations
and standards. Contract No. 68-01-3852.
WHO. 1979. World Health Organization. Health aspects related to Indoor
air quality. Copenhagen: WHO Regional Office for Europe.
Welty JR, Wicks CE, Wilson, RE. 1976. Fundamentals of momentum, heat
and mass transfer. New York, NY: John WHey & Sons.
71
-------
Lum RM, Graedel TE. 1973. Measurements and models of Indoor aerosol
size spectra. Atmos. Environ. 7: 827-842.
Lyman WJ, Reehl WF, Rosenblatt DH. 1982. Handbook of chemical property
estimation methods. New York: McGraw-Hill Book Co.
McNall PE. 1981. Building ventilation measurements, predictions, and
standards. Bull. N.Y. Acad. Med. 57(10): 1027-1042.
Meyer B. 1983. Indoor air quality. Reading MA: Addlson-Wesley
Publishing Co., Inc.
Moschandreas DJ, Winchester JW, Nelson JW, Burton RM. 1979. Fine
particle residential Indoor air pollution. Atmos. Environ. 13:
1413-1418.
Moschandreas DJ, Zabransky J, Pelton OJ. 1981. Geomet, Inc. Comparison
of Indoor and outdoor air quality. Final report. Palo Alto, CA:
Electric Power Research Institute. EPRI-EA-1733.
Natush PFS, Tomklns BA. 1978. Theoretical consideration of the
adsorption of polynuclear aromatic hydrocarbon vapor onto fly ash 1n a
coal-fixed power plant. Cardnogenesls 3: 145-153.
Newmann DJ, Nunn CJ, Oliver JK. 1975. Release of Individual solvents
and binary solvent blends from thermoplastic coatings. JPT. 47: 70-78.
Newmann DJ, Nunn CJ. 1975. Solvent retention 1n organic coatings.
Progress 1n Organic Coatings 3: 221-243.
Porter WK. 1983. Division of Health Sciences Laboratories, U.S.
Consumer Product Safety Commission. Briefing paper on n-hexane:
evaluation of consumer exposure and health risk. Inter-agency letter to
Stephen Nacht, Exposure Evaluation Division, U.S. Environmental
Protection Agency.
Roberts RW, Vandersllce TA. 1967. Ultrahlgh vacuum and Its
applications. Englewood Cliffs, NJ: Prentice-Hall (as cited by Natush
and Tomklns).
Sansone EB, Sleln MW. 1977/78. Red1spers1on of Indoor surface
contamination: a review. Journ. Hazardous Materials 2: 347-361.
Sawyer RN, Spooner CM. 1978. Sprayed asbestos-containing materials 1n
buildings: a guidance document. Research Triangle Park, NC: U.S.
Environmental Protection Agency. EPA-450/2-78-014.
70
-------
APPENDIX A
Symbols Used
73
-------
APPENDIX A (continued)
Symbols Used
Gr = Graham's constant.
GN = Mass flux of contaminant (g/cm2 hr).
GNAR = Time-dependent release rate (g/hr2).
h = Planck's constant = 6.6252 x 10~27 erg-sec.
H = Henry's constant.
HCa(js = Adsorbed entity (e.g., hydrocarbon).
HCp = Polycycllc aromatic hydrocarbon (e.g., B(a)P).
JA,jg = Flux of A and B with respect to a moving coordinate system.
k-| = Constant 1n concentration models.
= m Q/V
= Adsorption and desorptlon rate constants 1n Eq. (3-1).
kB = Boltzmann constant = 1.38044 x 1(H6 erg/°K.
KR = Resuspenslon factor.
L = Gas film thickness 1n flux equation.
m = Mixing factor.
M = Mass of product.
W = Molecular weight.
Mt = Migration rate (g/cm2)-
n = Number of moles.
na(js = Number of vapor molecules adsorbed 1n Eq. (3-4).
np = Number of particles/unit volume gas 1n Eq. (3-3).
75
-------
APPENDIX A
Symbols Used
a through h = Constants 1n collision Integral equation.
Aconc = A1r concentration of a substance.
AR = Contaminant application rate (m^/hr).
c = Sticking coefficient 1n Eq. (3-3).
C = Contaminant concentration (g/nfl) .
C0 = Initial concentration of contaminant (g/tn^).
Cave = Average concentration of contaminant (g/nfl) .
densities of conponents A, B and the mixture
(moles/cm3) .
Cso = Original concentration of migrant 1n polymer (q/citfl)
dp = Diameter of particles 1n Eq. (3-3).
D = Diffusion rate (cm2/sec).
D^i = Diffusion coefficient for diffusion of B through A.
DO = Duration of discharge (hours).
e = The base of the natural system of logarithms.
* 2.71828.
F = Fraction of migrant released from a polymer.
£a = Activation energy for adsorption 1n Eq. (3-3).
E(j = Activation energy for desorptlon 1n Eq. (3-2).
g = Acceleration due to gravity (981 cm/sec2).
G = Contaminant release rate (g/hr).
74
-------
APPENDIX A (continued)
Symbols Used
te = Time at the end of exposure.
tg = Evaporation time.
t0 = Time at the beginning of release of chemical.
tr = Time for release of chemical to cease (ta + tg).
tu = Any time greater than the time at which release of the
chemical ceases.
t] = ta or tg, whichever 1s smaller.
t2 = ta or tg, whichever 1s larger.
T = Temperature, °K.
T* = D1mens1onless temperature.
Tb = Normal boiling point (°K).
V = Volume (m3).
V/\ = Molar volume of air (cm3/mole)
V'g = Molar volume of B (cm3/mole) at normal boiling point.
Vs = Volume of polymer (cm3).
WF = Weight fraction of chemical substance 1n product.
VP = Vapor pressure.
X/\,Xg = Mole fraction of components A and B 1n liquid.
Xv = Mole fraction of adsorbed material 1n Eq. (3-4).
Y/y, YB = Mole fraction of components A and B 1n vapor.
z = Plane area.
77
-------
APPENDIX A (continued)
Symbols Used
nj = Total number of vapor molecules 1n eq. (3-4).
N = Total molar flux with respect to a stationary coordinate
system.
NA.NB = Molar flux of A and B with respect to a stationary
coordinate system.
Ng j = Molar flux of B across any plane of constant z.
0V = Fraction of product that 1s overspray and does not contact
Intended target.
P = Pressure (atm).
Ptotal = Total pressure (atm).
h.p2.P3.
PA.PB = Partial pressures of components 1,2,3,A, and B (atm).
PA°«')BO = VaP°r pressure of pure components A and B (atm).
Q = Ventilation air flow (m3/hr).
R = Gas constant = 1.987 cal/mole-°K
= 82.054 cm3-atm/mole-°K
SB = Solubility of component B.
Sconc = Concentration of a substance on a surface.
Sur = Surface of a particle.
SA = Surface area from which chemical evaporates.
t = Time (hrs).
ta = Time to apply film or coating to a surface.
t[3 = Time at the beginning of exposure.
76
-------
APPENDIX B
Derivation of Equations for Estimating
Concentrations of Chemical Substances
1n Indoor A1r
79
-------
APPENDIX A (continued)
Symbols Used
Greek
c = Energy of molecular Interaction.
vim = Micrometers.
ft = Collision Integral.
p, pL = Density, (mass/volume).
°A«0B'aAB = Characteristic length of A, B, and AB
1n WHke-Lee equation.
e = Fraction of adsorption sites occupied 1n Eq. (3.3).
n = Coefficient of gas viscosity, grams/cm/sec.
* = The ratio of the circumference to the diameter of a circle.
s 3.14159.
78
-------
and substituting k-) and k2 as appropriate Into Equation (B-2), the
resulting expression 1s
-r^ = k2t - kiC. (B-3)
By letting
z = k2t - k-|C, (B-4)
and by differentiating with respect to time, the resulting expression 1s
dz , dC /D c.
dt - k2 - kl dt- (B-5)
By rearranging Equation (B-5), Equation (B-6) 1s
dt ' k^\df - k
When Equation (B-6) and Equation (B-4) are substituted Into Equation
(B-3), the resulting expression 1s
J~(TT - k2 I = z. (B-7)
When Equation (B-7) 1s multiplied by k-| and rearranged, the resulting
expression 1s
-rf- = -k]z * kg. (B-8)
Equation (B-8) can be rearranged to
dz
= dt. (B-9)
Multiplying Equation (B-9) by k2 yields
dz
1 - (k!/k2)z
When Equation (B-10) 1s Integrated, the resulting expression is
",).
_ Inll - — z)= k2t * C*. (B-ll)
31
-------
APPENDIX B
The derivation of equations for estimating concentrations of chemical
substances 1n Indoor air at any point 1n time, t, as a result of a
time-dependent release are presented for four Intervals. These are:
(1)
(2)
(3)
(4)
0
t2
t
< t
< t
< t
> tr
< ti;
< t2;
< tr;
and
Ths parameters, t^ and t2, can represent the time to apply the liquid
film to a surface from which a chemical substance volatilizes (ta) or
the time required for a chemical substance to evaporate from a liquid
film once 1t has been applied to a surface (tg). If ta 1s smaller
than tg, then t-\ equals ta and t2 equals tg. If tg 1s
smaller than ta, then ti equals tg and t2 equals ta. The
parameter, tr, 1s the sum of t] and t2.
(1) For t < t-\
The mass balance equation and the physical Interpretation of each
group of terms 1s
V ^ - GNAR t - mQC. (B-1)
(The net mass of } (The mass of chemical } ( The mass of chemical)
-
(in air 1n the room) (to air 1n the room ) (air leaving the room)
Upon dividing Equation (B-1) by V, Equation (B-1) becomes
dC GwAo mQ
_ = JLH t - _ C. (B-2)
dt V V
By letting
k] = and k2 =
V
30
-------
where
tjj = time at the beginning of the desired time Interval
te « time at the end of the desired time Interval.
(2) For t-i < t < t2
The mass balance equation and the physical Interpretation of each
group of terms 1s
V JJ-JT = GNARtl - mQC. (B-19)
fThe net mass of \ {The mass of chemical^ (The mass of chemical \
= - .
(in air 1n the room; (to air 1n the room ; (air leaving the room;
If Equation (B-19) 1s divided by V, Equation (B-19) becomes
dC
df •—¥--—• (B-20)
By letting
m() S^R*!
k2 = ~~w and k2 = r/
and substituting k-j and k2t as appropriate Into Equation (B-20), one
obtains the resulting expression of
H = k2 - k]C. (B-21)
Rearranging Equation (B-21) yields the resulting expression
dC
k2 - kiC
which, when multiplied by k2, becomes
dC
= dt (B-22)
Integrating Equation (B-23) yields the following expression
-k2
(B"24)
83
-------
When C = 0 at t » 0, this Implies that z = 0 at t = 0 and that C* = 0.
Setting C* = 0 and dividing Equation (B-ll) by (-k2/k-|) yields
/ k \
In II - _L * I = -kit. (B-12)
\ *2 ;
Taking the antllog of Equation (B-12) yields
1 -
(B-13)
Substituting the expression from Equation (B-4) for z Into Equation
(B-13) and rearranging yields
-kit
By solving for C, the resulting expression 1s
C . ^
^ - k2t]
/ J
which, upon rearranging, 1s
Substituting SNAR/V for k2 yields
t
(B-14)
(B-15)
(B-16)
(B-17)
The Equation to calculate the average concentration for any Interval of
time where the time at the end of exposure 1s less than or equal to t-j
1s obtained by Integrating Equation (B-17) with respect to time and
dividing the resulting equation by the length of the exposure Interval.
The resulting expression 1s
-ave -
QNAR
kiV(tb - te)
> _ t _ e-k1f
2
2 ki ki
rit = tb
Jt = t,
(B-18)
82
-------
or
C* «
kit]
(B-32)
Substituting the expression for C* 1n Equation (B-32) for C* 1n Equation
(B-26) yields
1
u = M
•"• «i
Multiplying Equation (B-33) by -1 and rearranging yields
1
- e
")
Substituting 6NARt-i/V for k2 1n Equation (B-34) yields
ki(t-|-t) -k]t^
- e
or
- e
Simplifying Equation (B-36) yields
C =
K]V
Rearranging Equation (B-37) yields
GNAR f «
C = -r-
(B-33)
(B-34)
(B-35)
(B-36)
(B-37)
(B-38)
The equation to calculate the average concentration for any Interval of
time less where te 1s less than or equal to t2 and t^ 1s greater
than or equal to t] 1s obtained by Integrating Equation (B-38) with
respect to time and dividing the resulting equation by the length of the
exposure Interval. The resulting expression 1s
85
-------
Solving for C by dividing by -k2/k-| yields
•(-SO
"l
Inll - j— C 1= -kit + C*. (B-25)
Taking the antllog of Equation (B-25) yields
1 - j^ C = C* e . (B-26)
To determine C*t substitute the expression for C 1n Equation (B-17) for
the parameter, C, 1n Equation (B-26). Substitute t-| for the parameter,
t, 1n Equation (B-17). The resulting expression after these
substitutions are made 1s
GNAR / 1 e
tl - U7 * -T7~/!= c*e • (B-27)
GNAR/klv 1n Equation (B-27) can be simplified 1f one multiplies by
t-|/t-|. The parameter k2 can be substituted for GflAflt]/V,
after which Equation (B-28) 1s obtained:
i e-'i'ni
kT + ~h—/J-
kll *2
(B-28)
Equation (B-28) can be simplified to Equation (B-29) by cancelling like
terms:
b _ 4-_
-kit]
Multiplying Equation (B-29) by e ] 1 and cancelling like terms yields
- 1 + , ] - ^-——I = C*. (B-30)
kit] k]ti / '
Equation (B-30) can be further simplified to Equation (B-31).
34
-------
If Equations (B-43) and (B-45) are substituted Into Equation (B-42), the
resulting expression 1s
Multiplying Equation (B-46) by -k] yields
jjf * "2 - -*iz. (B-47)
Rearranging Equation (B-47) yields
gf = -(k2 + kiz). (B-48)
Rearranging Equation (B-48) yields
r ^r— = -dt. (B-49)
k2 * k]Z v '
Integrating Equation (B-49) with respect to
J- In (k2 * kiz) « -t + B". (B-50)
Multiplying Equation (B-50) by k-| yields
In (k2 + M) = -k]t + B1. (B-51)
If one takes the antllog of Equation (B-51), the resulting expression 1s
k2 * ^i = Be"klt. (B-52)
To determine B 1n Equation (B-52), substitute the expression for z from
Equation (B-43) In Equation (B-52). When t equals t£, C 1s equal to
Ct2. Therefore, substitute t2 for t and Ct2 for C to determine B.
Ct2 1s calculated by substituting t2 for t 1n Equation (B-37). The
resulting expressing 1s
k2 * k! k2 (tr-t2) - k1Ct2| = Be"klt2. (B-53)
Rearranging Equation (B-53) yields
37
-------
*ave =
t]t *
t-t,
t-tb
(B-39)
(3) For t2 < t < t
The mass balance equation and the physical Interpretation of each
group of terms 1s
dC
v =
(tr-t) - moc.
(B-40)
I The net mass of | (Jt\e mass of chemical^ ^The mass of chemical \
=< substance released V-.
(in air 1n the room) (to aVr 1n the room j (air leaving the room I
Dividing Equation (B-40) by V yields
dC GNAR
dt = V
mQ
(tr-t) - rr C.
(B-41)
By letting
and
GNAR
and by substituting k-] and k2, as appropriate, Into Equation (B-41),
one obtains
H = k2(tr-t) - kiC. (B-42)
If we let
z = k2(tr-t) - k^t (B-43)
and differentiate with respect to time, the resulting expression 1s
d z . , dC
Upon being rearranged, Equation (B-44) becomes
kTldf *»
dC
df
(B-44)
(B-45)
06
-------
Substituting
for k2 1n Equation (B-61) and rearranging yields
(tr-t2) - Ct2
-Mt-t2)
= -e
GNAR GNAR
11 p 4- (
GNAf
-kl2V
fcr-t)
i GNAR
M
'
(B-62)
Is calculated using Equation (B-38) by setting t equal to t2- The
equation to calculate the average concentration for any Interval of time where
t 1s less than or equal
to tr and
Is greater than or equal to
1s derived by Integrating Equation (B-62) with respect to time and dividing
the resulting equation by the length of the exposure Interval. The resulting
expression 1s
1
3Ve = (te-tb)
GNAR GNAR
.
+ (
-ki(t-t2)
e
*1
™"
tr-i )
_kl2V k!V C
"GNAR
k^V
t=te
•
t=tb
t
Equation (3-7) with Ctr substituted for C0 and with (t-tr)
substituted for t 1s used to calculate the concentration at any time after
tr. Ctr 1s calculated by substituting tr for t 1n Equation (B-62). As
the time during this Interval Increases, the concentration decreases
exponentially as air containing the chemical flows out of the room.
C = Ct
re
-m(Q/V)
-------
(tr-t2) - h2 Ct2]-
(B-54)
Substituting the expression for z from Equation (B-43) Into Equation
(B-52) yields
k2 * k-|j"k2 (tr-t) - k-|Cj = Be~klt. (B-55)
Multiplying the expression 1n brackets 1n Equation (B-55) by k^ yields
k2 + k-|k2 (tr-t) - k-|2C = Be~klt.
Rearranging Equation (B-56) yields
(B-56)
= Be~klt - k2 -
(tr-t).
(B-57)
(B-58)
Substituting the expression for B from Equation (B-54) Into
Equation (B-57) yields
, -k]t ( k-,t2r , -, )
-k^C = e je [k2 + k-|k2 (tr-t2) - k-|2Ct2 V
-Fk2 * k!k2 (tr-t)l.
Simplifying Equation (B-58) results 1n
-k!2C = e("klt * klt2)[k2 * kTk2 (tr-t2) - kf Ct2]
-[k2 * k!k2 (tr-t)]. (B-59)
Simplifying Equation (B-59) results 1n
2 _k1(t-t2)|"
-[k2 * k!k2 (tr-t)].
Dividing Equation (B-60) by -k-|2 yields
-k!(t-t2)
(tr-t2) -
(B-60)
C = e
(tr-t2)
k2
(tr-t)
(B-61)
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