PB87-180022
Compensating for Wall Effects in IAQ
(Indoor Air Quality) Chamber Tests by
Mathematical Modeling
Arkansas Univ., Fayetteville
Prepared for
Environmental Protection Agency
Research Triangle Park, NC
Apr 87
QL1 fepafeaiat of Commerce
TodHBMi Informatesi Service
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PB87-180022
EPA/600/D-87/111
April 1987
COMPENSATING FOR WALL EFFECTS IN IAQ CHAMBER TESTS
BY MATHEMATICAL MODELING
James E. Dunn
University of Arkansas, Department of Mathematical Sciences
Fayetteville, Arkansas 72701, USA
Bruce A. Tichenor
U.S. Environmental Protection Agency
Air and Bnergy Engineering Research Laboratory
Research Triangle Park, North Carolina 27711, USA
CR812305
EPA Project Officer
Bruce A. Tichenor
AIR AND ENERGY ENGINEERING RESEARCH LABORATORY
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
RESEARCH TRIANGLE PARK, NC 27711
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NOTICE
This document has been reviewed in accordance with
U.S. Environmental Protection Agency policy and
approved for publication. Mention of trade names
or commercial products does not constitute endorse-
ment or recommendation for use.
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INTRODUCTION
The purpose of this paper is to present a variety of extensions to a class
of mechanistic, time dependent models proposed by Dunn* which mathematically
decouple the true emission characteristics of an organic source from
"coloration" imposed by an emissions test chamber. Based on empirical evidence
of a longer than expected residue of emissions, Dunn's model incorporated a
tern for an adsorptive wall effect whose release rate was assumed to be equal
to that of the original source. The value of this approach is that it allowed
not only generation of a curve representing actual concentration in the cham-
ber over time, but produced an additional curve, obtained by setting the ad-
sorptive rate constant to zero, which represented the theoretical concentra-
tion in the chamber had there been no wall effect.
In keeping with the distinction proposed by Dunn, the models treated here
are thought to apply to thin film sources, as distinguished from deep sources
whose emission rates are diffusion limited. First, in line with the terminol-
ogy established by Renes et al?, we have generalized the specific concept of a
wall effect to the general notion of a sink effect, whether it be actual wall
or conduit adsorption, absorption by a gasket seal, or even accumulation in an
unventilated portion of the chamber. In a more sophisticated experiment in-
volving binary composite materials, one element might serve as a sink for
emissions from the other; e.g., a linoleum layer when testing emissions from
its adhesive. Our assumption differs slightly from that of Renes et al. in
that our sink has the capability of releasing adsorbed material; whereas, from
the form of the model shown, their sink is irreversible. Second, by including
an additional rate constant, we have relaxed the assumption that the release
rate from the sink is equal to that from the original source. It is very like-
ly that a different mechanism would be involved. Third, we have introduced
still another rate constant to reflect the potential effect of chamber concen-
tration on the emission rate of the source. Fourth, we have distinguished be-
tween thin film models which represent concentration for a decreasing source
(e.g., a swatch of floor wax or carpet glue on an inert carrier) and constant
source models (e.g., as might be obtained by use of permeation tubes or, in
our case, a cake of moth crystal).
Our development here is based.on an increasingly evident premise ths»t the
results of emission test chamber studies cannot be applied directly to a de-
termination of indoor air quality (IAQ). No one lives inside stainless steel >
walls. Since the adsorptive properties of typical indoor surfaces have now
been documented by Renes et al?, at least for N0£ and S0£, our tentative
approach is to start with an appropriate mathematical model, replace whatever
sink effects characterize the test chamber with those sink effects which
characterize a living/working environment, and then use the model to predict
the actual exposure level. Results are given of a reanalysis, based on these
new models, of the latex caulk data previously reported by Dunn1. Also treated
in some detail are the moth crystal emissons data which are discussed else-
where by Nelms et al? All emissions data were obtained in the 166 1, stain-
less steel test chamber described by Sanchez, et alt Its interior had been
electropolished after assembly; all conduits were either Teflon or Pyrex
glass.
GENERAL FORM OF THIN FILM MODELS
As visualized here, a test chamber operating system consists of four
(mathematical) compartments; namely, the source, the well-oixed contents of
the chamber, an exit, and a sink. Rate constants describe the flow among these
compartments, as illustrated by the following schematic:
2
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Sink(w)
kl k3 * 1 k4 k2
Source(A) Chamber(x-y-w) -* Bxit(y) ,
where
A = initial mass to be eaitted by the source,
x = x(t) = mass emitted to the chamber by tine t,
y = yet) = naas emitted froa the chamber by time t,
w = w(t) = mass in the sink at time t.
It follows that the concentration in the chamber at time t is given by C(t) =
(x-y-w)/V, where V is the chamber volume, and I(tptg) = y(tg) - y(tj) is the
¦ass released froa the chamber in the tine interval (tptg). Clearly, y(t) =
1(0,t), assuming y(0) = 0. We postulate that, in a well nixed chamber, the
systea is described adequately by the following set of ordinary differential
equations:
^7 = kjg(x,t) - kg(x-y-w) ,
dt
(1)
^ = kg(x-y-w) ,
(2)
& = kg(x-y-w) - k4W .
(3)
Physically, the right-hand side of equation (1) defines the emission rate (BR)
of the source at any time t. It states that the rate of introduction of
material into the chamber is proportional to some function of time and the
amount already emitted, while being inhibited by increasing mass, or equiva-
lently concentration, in the vapor phase of the chamber. A precise definition
of the emission rate factor can then be given, namely
Emission rate factor(ERF) = lia — , (4)
t •+ •» dt
representing the steady state rate of mass introduction from the source into
the chamber atmosphere. Equation (2) states that the emission rate from the
chamber is proportional to the chamber concentration. Equation (3) states that
the rate at which material moves to the sink is proportional to the chamber
concentration, while the rate of removal from the sink is proportional to the
amount already in the sink. Both rate constants, k| and kg, are required to
describe the source. The rate of movement to the sink is described by the rate
constant kg, and the rate of release froa the sink is described by k^. Pre-
sumably, values of kg and k^ which characterize the test chamber differ dras-
tically from those which would be associated with an actual living/working .
environment. Assuming a constant air flow rate. F, through the chamber, then
kg = F/V, the number of air changes per unit tine. We assume that F is speci-
fied by the experimenter so that kg is a known rate constant.
The next section describes physical models which are obtained as solutions
of the above set of equations. Solutions with kg * 0 and kg * 0 are referred
to as full models. Solutions with kg ^ 0 but kg = 0 are sink models, since the
rate of emission froa the source is not then affected by increasing chamber
concentration. Conversely, solutions with kg = 0 and kg ft 0 are called vapor
pressure ( VP) models, and represent the absence of a sink. However, VP aodels
3
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Table I. Constant source models, showing the chamber concentration, C(t),
mass in the sink, w(t), and mass I(tpt2) emitted from the chamber
from tine tj to tine tg. l(k3) = 0 if kg = 0; otherwise L(kg) = 1.
Pull model (k3 * 0, kg 9f 0).
C(t) = k1k2[(k4-r2)2(l - e"r2t)/s2+L(k3)(k4-r3)2(l - e^VsgJ/V, (5)
w(t) = kjkgkgK^-rgXl - e ^J/sg + (k4-r3)(l - e 3 )/s3], (6)
I(tj,t2) = kjk2(t2—tj)/sj (7)
+ k1k2(k4-r2)[(k5-r2)(k3+k4-r2)-k3k5](e ^ - e *2 2)/(r2s2)
+ I>(kg)kjk2(k4_r3)[(kg—rg)(kg+k4—r3)~kgkg](e 3 ^ — e 3 2)/(r33g)•
r2.r3 = {k2+k3+k4+k5 ± [(k2+k3+k4+k5)2-4k4(k2+kg)]1/2}/2,
sj = k2 + kg,
8i = k2k5(k4"ri^2 + k2k3ri + Kk5~ri)(k3+k4~ri)~k3k5^2» tor i=2.3'
Sink model (k3 * 0, kg = 0).
C(t) = kjKr^Jd - e~rit)/rl - (r2-k4)(l - e'^Vr^/K^-r^V], (8)
w(t) = k^gtfl - e~rzt)/r2 - (1 - (9)
Ktptg) = kjtCr^-kg—k4) CrjCtg—tjJ-t- e ^ Z- e ^ 1]/r1 (10)
-(r2-kg-k4) [^(tg-tj) * e 2 2 - e 21 J/r2)/(rj-^),
rlfr2 = {k2+kg+k4 ± [(k2+kg+k4)2-4k2k4]1/2}/2 .
VP model (kg = 0, kg ft 0).
C(t) = kj[l - e"(k2+k5)t]/[(k24k5)V], (11)
Ktptg) = k^tjj-t^/^+kg) (12)
- k^re-^2^! - e"t2]/(k2+k5)2 .
Dilution model (kg =0, kg = 0).
C(t) - kjd - e~k2t)/(k2V) (13)
Ktptg) = kjt^ftg-tj) + e^2*2 - e"k2tl]/k2 . (14)
4
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play a dual role, as noted by Dunn1, in the sense that they are indistinguish-
able from sink models in which kg » kg and k4 = kjj i.e., where the rate con-
stant for removal from the sink is identical to that for the original source.
If both kg = 0 and kg = 0, then the result is called simply a dilation model.
PHYSICAL MODELS FOR EMISSIONS
Constant Source Models
Suppose we have a source which acts as if it were a constant emitter, at
least for a finite period of tine. A permeation tube exhibits this behavior; a
source such as Both crystal whose emission rate is limited by its surface area
also behaves this way to a good approximation. Thus A = •» effectively, and, if
we suppose that the potential for emissions is constant during this period,
then g(x,t) = 1 in equation (1). Table I shows the four different solutions of
equations (1) - (3) under these assumptions, given that the chamber
concentration is negligible at time zero. Clearly, these are related. The full
model reduces smoothly to either the sink model as kg •* 0 or the VP model as
kg -» 0. If both kg = 0 and kg = 0, then all models reduce to the simple dilu-
tion model.
The limiting forms of these models as t + » are given in Table II. Note
the intuitive result that the limiting steady state chamber concentration for
the sink model ic identical to that of the dilution model; i.e., independent
of the magnitude of any sink effect.
Table II. Steady state chamber concentration and mass in the sink,
corresponding to the constant source models listed in Table I.
Concentration, C("»)
Sink, w(>*>)
Full model* kjkgK^-^^/sg+ffy-rg^/SgJ/V kjkgkg[ (k4~r2)/s2+ (k4-r3)/s3]
Sink model kj/(k2V)» independent of kg & k4 ^1^3/
VP model kj/f(k2+kg)Vj 0
Dilution model kj/(k2V) 0
* r2» r3» °2> ^ Sg are as defined in Table I.
Table III. Theoretical emission rate factors for constant source models.
Full model* ERF = kjU - k2kg[(k4-r2)2/s2 + (k4-r3)2/s3]}
Sink & dilution models ERF = kj
VP model ERF = kjk^kg+kg)
* r2» r3» s2> Sg are as defined in Table I.
Theoretical forms of emission rate factors, as defined by equation (4).are
given in Table III. Again, there is nothing to distinguish between the sink
5
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and dilution model3. Note that the ERF for the VP model will always be leas
than that for the sink model to the extent that kg > 0.
The application of these principles is the central theme of this paper.
Figure 1 illustrates the forms taken by equations in Table I. Using 24.08 g of
noth crystal as the source, 1,4 dichlorobenzene concentrations were measured
in the test chamber, maintained at 35.6° C (a.d. = 0.10), 50.3 * relative
humidity (s.d. = 9.4), and k2 = 0.25 air changes per hour (ACH), and the
results are plotted as +*s. A least squares fit of the sink model, using SAS8
procedure NLIN, resulted in estimates kj = 259 mg/hr (s.e. = 8.32), kg = 0.128
hr~* (s.e. = 0.0196), and k^ = 0.0802 hr~* (s.e. = 0.0105). The solid line in
Figure 1 is a plot of the fitted model, and is our best estimate of the
actual, dynamic concentration in the chamber. The dashed line ( ), obtained
by setting kg = 0, represents our best assessment of the concentration which
would have resulted had there been no sink effect. The full model gave an
equally good fit (cf. Table V, test 10.1), but appeared to be overparameter—
ized by the presence of kg.
As a general principle: The proper way to remove a sink effect mathemati-
cally is to fit either the sink model or the full model, depending on whether
or not there is statistical evidence that kg = 0 {e.g., by examination of its
standard error or confidence interval), then setting kg = 0 in the resulting
fitied model.
Hote that the dashed line in Figure 1 is not the same as that which would
have resulted had the dilution model been fitted directly. In fact, the least
squares fit of the dilution model and that of the VP aodel were the same, kj =
211 mg/hour (s.e. = 8.26) and kg = 0 being estimated for the latter. The re-
sult is plotted in Figure 1 as the dot-dash ( ) line. It does not appear
to fit as well as the sink model, particularly the last three data points.
Using formulae from Table II, estimated steady state concentrations are 6230
mg/m for the sink model and 5080 mg/m^ for the VP-dilution model. If the sink
model is correct, then an unwary experimenter who felt that the last three
data points represented steady state would underestimate that value by about
15%.
From Table III, the emission rate factors are equal to the respective k^
estimates. These are also the emission rates, dx/dt, since kg is assumed to be
zero in the sink model and was estimated to be zero in the VP aodel.
Decreasing Source Models
Suppose that A < •». and that g(x,t) = (A - x) in equation (1). Physically,
this is an assumption that the source emission rate is proportional to the
amount remaining to be emitted, while being inhibited in proportion to the
chamber concentration. With equations (2) and (3) remaining unchanged, the
four possible models, obtained es solutions of equations (1) - (3) and corres-
ponding to those in Table I, are listed in Table IV. Again we have assumed
that the chamber concentration is negligible at tiae zero.
For all models listed in Table IV, both C(t) and w(t) decay to zero as t ¦»
«°. Also, since x = y = A in the limit, the emission rate factors are zero for
all Models. The implications of these properties will be discussed in the next
section in connection with examples of the analysis of mixed emissions from
latex caulk.
6
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Table IV, Decreasing source aodels, showing the cbasber concentration, C(t),
bub in tEe sink, w(t), and was* ICtj.tg) caitted from the cbasber
from ti* tj to tiae t2.
fWll Vtftl (k3 *0.ks* 0).
2 -r,t
C(t) = A 2 ¦irl(kl " ri,e /{DV)»
3 _r t
w(t) = A J ¦1[i| - (kj+kg+kg)^ + kjk^Je 1 /D, (16)
i=l
3 £
z(tj.t2) I H^Ti ~ ki*(* 4 2 ~ e 1 1i^D»
i=l
D = kjkgkgCrjtrl - r|) ~ r2(r| - rf) + r3(rf - r2)},
•l = frlr3 " kIk2 ~ Wrl + r2 " kl»-
sod rj^ r2, end r3 ere roots of
r3-(k1+k2+k3+k4+k5)r2+[k2(k1+k4)+k1(k3+k4>+k4k5]r-k1k2lf4 = 0,
Sink aodel (kg * 0, kg = 0).
C(t) = kjA{ (k^-kjje 1 /[ (rj-kj) (r2-kj) ]-(k2+k3-r2)e 1 /Krj-rgXrj-kj}] (18)
+ (ka+kg-rjje ^/[(rj-rg)(r2-kj)3}/V,
w(t) = A(k2-r1)(k2+k3-r2)fr1(l - e ^J-kjd - e ***)]/[rj(rg-rj)(rj-k x}] (19)
- A(k2-r2)(k2+k3-r1)[r2(l - e ^S-kjCl - e ^Sj/t^rg-rjHi^-kj)),
= Ak2(k4-kj)(e"kltl - e~klt2)/f(r1-k1)(r2-k1)] (20)
- Akik2(k2+k3-r2)(e "l*1 - e ritz)/[r1{r1-r2)(rj-k1)]
~ Ak1k2(k2+k3 r1)(e"rztl - e"rzt2)/[r2{rrr2)(r2-k1)],
r2,r3 = (k2+k3+k4 i I(k2+k3+k4)2 - 4k2k4]1/2}/2.
^ xxiel (k3 « 0, i5 P 0).
C(t) » k1A(e 2 - e 1 )/[(r1-r2)V], (21)
I(t],,t2) = Alr1(e"Tztl - e - r2(e Fjtl - m ritz)]/(r1-r2>, (22)
rl'r2 3 tkl+k2+k5 * I«cl+k2+k5>2 " «<2]1/2>/2.
Dilution sodel (kg = 0, k5 = 0).
C(t) = kjACe'^* - e^V^-k^?], (23)
I(tJtt2) = Afk^e"*2*1 - e"*2*2) - k^e"*1*1 - e*1*2)]/^-!^). (24)
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EVIDENCE FOR A CHAMBER SINE EFFECT
1,4 Dichlorobenzene Emissions from Moth Crystal
Experimental conditions for 13 emissions tests of moth crystal3 are given
in Table V. The sample size, N, refers to the number of separate determina-
tions of chamber concentration made during each test. The order shown is that
actually used in the laboratory. The four constant source models were fitted
to each data set using SAS procedure NIIN0, and the residual sums of squares
(SSE) are listed in Table V as a basis for choosing the most appropriate
model. C(t) forms were fitted since the data were obtained as virtually
instantaneous measurements of chamber concentration.
In most cases, the least squares fit of the full model converged to a
solution in which either kg = u or kg = 0 (or both). This is indicated in the
column, 'Full.* Where all parameter estimates were positive for the full
model, the residual SSE is listed explicitly. Even then, except for test 10.1,
huge standard errors associated with the estimates suggested that the full
model was overparameterized. Only for test 10.1 was SSE smaller than that
listed for either the sink or the VP model, and then only minimally. In this
case, kg was significantly greater than zero, but not kg, so that the sink
model seemed preferred. There appear to have been major sink effects for tests
3, 4, 10.1, and 12, and, to a lesser extent, for tests 6.1, 8.1, and 11,
though kg was not significantly different from zero in the sink model for the
latter cases. This is substantiated by the fact that Nelms et al? recovered
1B0 and 23 mg of 1,4 dichlorobenzene from the chamber walls at the end of
tests 10.1 and 11, respectively. The vapor pressure term, kg, was signifi-
cantly greater than zero for tests 7, 9, and 14, but not for tests 5.1, 10,
and 13. The simple dilution model did not seem to be adequate in any case.
Table V. Lack-of-fit (SSE) comparisons of all constant source models
applied to 1,4 dichlorobenzene emissions from moth crystal.
Test
Temp.
RH
ACH
N
Full
Sink
VP
Dilution
(°C)
(*)
(=k2)
(SSE)
(SSE)
(SSE)
(SSE)
3
23.4
15.2
0.98
13
2410
2410(11)*
(=Dilution)
24015
4
23.2
14.9
0.50
14
16304
16304(12)*
(=Dilution)
89780
7
23.3
19.3
0.29
22
(=VP)
(=Dilution)
37144(20)*
50573
5.1
23.3
17.5
0.50
17
(=VP)
(=Dilution)
14282(15)*
17557
6.1
22.8
51.7
0.26
18
(=Dilution)
62938(15)*
(=Dilution)
68051
8.1
22.6
53.1
0.99
16
(=Dilution)
2583(13)*
(=Dilution)
6339
9
35.5
14.8
0.27
16
1462778
2384895
1462777(14)*
2385391
10
35.6
34.0
0.25
16
(=VP)
(=Dilution)
1639028(14)*
2200353
10.1
35.6
50.3
0.25
21
147658
147762(18)*
(=Dilution)
676431
11
35.4
18.1
0.99
18
96923
96923(16)*
(=Mlution)
108483
12
36.3
49.7
1.92
10
(=Sink)
17642(7)*
(^Dilution)
65913
13
35.4
54.8
0.49
13
(=VP)
549251
415228(11)*
577399
14
24.8
42.5
1.92
10
669
1738
669(8)*
1821
*Pref erred
model.
A general pattern is difficult to discern. These tests were dominated by
either a sink effect or a vapor pressure effect, but not by both simultan-
8
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eously. IVjo physical explanations are suggested. Table VI shows estimates of
steady state chamber concentrations and emission rate factors for the 13
tests, based on matching the preferred models indicated in Table V with their
corresponding computational formulae in Tables II and III. Within each temp-
erature octave, choice of the VP model over the sink model generally corres-
fonded to those tests with high chamber concentrations; i.e., tests 7, 5.1, 9,
0, and 13. This is consistent with a vapor pressure gradient theory proposed
by Nelms et al? The sink model was preferred if either the concentration was
relatively low (tests 3, 4, 8.1, 11, and 12), or the relative humidity was
high (tests 6.1 and 10.1). A finite adsorption capacity by the walls could
have accounted for a greater fraction of the emissions in the first case;
additional humidity might have conditioned the adsorptive sites by some mech-
anism in the second case. Test 14 was the exception to both rules, the VP
model being preferred even though the concentration was low and the humidity
high.
Table VI. Steady state concentrations and emission rate factors for 1,4 di-
chlorobenzene from moth crystal (s.e. = standard error).
Test
Sample
Model
!Steadv State
ERF/Sanmle
BRF/tf
ERF/cm2
Wt.
:Est.
s.e.
Est.
s.e.
Est.
s.e.
Est.
s.e.
«>
i (mg/nr)
(mg/hr)
(mg/hr/g)
(mg/hr/cm)
3
24.1
Sink
552
6.95
89.8
1.13
3.72
0.047
1.63
0.021
4
24.6
Sink
986
24.0
81.9
1.99
3.33
0.081
1.49
0.036
7
29.0
VP
(k5=0)
1230
(1390
14.5
56.5)
59.2
(66.8
0.70
2.72)
2.04
(2.03
0.024
0.094)
1.08
(1.21
0.013
0.049)
5.1
29.3
VP
(k5=0)
1060
(1160
11.4
49.4)
88.2
(96.4
0.94
4.10)
3.01 0.025
(3.29) 0.140)
1.60
(1.75
0.017
0.075)
6.1
29.1
Sink
1610
82.9
69.5
3.58
2.39
0.123
1.26
0.065
8.1
26.8
Sink
543
5.23
89.2
0.86
3.34
0.032
1.62
0.016
9
21.7
VP
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mathematically. For comparable tests 4 and 5.1, the net effect was to move
estimates of the BRF on a per gram basis into closer alignment while the
similarity between steady state estimates decreased. Both estimates for test
10 moved closer to those for test 10.1. Revised estimates for test 13 also
seem more in line with those for test 10.1.
In spite of the modest improvements noted above, we believe that both a
wall effect and a vapor pressure effect always were present in the chamber.
Analyzing the leachate from the chamber walls after every test might have
rovided a definitive answer, but this was not done. We suggest that the pre-
erred models indicated in Table V represent the dominant effects, depending
on the test conditions, and that the data were insufficient to support esti-
mation of the lesser effect in all cases except test 10.1. This test was the
basis of Figure 1.
In order to summarize the results of Table VI based on this conclusion, we
used Stepwise Regression9 to construct empirical models relating steady state
chamber concentration and BRF on a per gram basis to selected functions of
temperature (T), relative humidity (RH). and air exchange rate (ACH). The list
of functions made available included all products, all reciprocals, all loga-
rithms, and ratios of all pairs of T, RH, and ACH. The models selected are:
Steady State(mg/m3) = 225.1 -1585/ACH + 83.56 T/ACH, (R2=0.990) (25)
ERF(mg/hr/g) = -12.28 +0.6155 T + 32.38 ACH/RH, (R2=0.954) (26)
All coefficients were significant at the 0.05 level (or less) to be included.
These functions are plotted as response surfaces in Figures 2-4, respect-
ively.
Each contour in Figure 2 indicates how ACH must be adjusted to compensate
for increasing temperature in order to maintain a constant chamber concentra-
tion. Clearly, the chamber concentration of dichlorobenzene builds up more
rapidly at high temperatures with decreasing ACH than at low temperatures.
These features seem to hold regardless of the humidity level, since no func-
tion of RH was selected for inclusion in equation (25) by the stepwise pro-
cedure.
The dependence of BRF on the reciprocal of RH resembles that of the Berge
equation". The effect of RH at T = 23° C is shown in Figure 3. Bach contour
reflects how the ACH must be adjusted to compensate for increasing RH in order
to maintain a constant ERF on a per gram basis. Increasing ACH will increase
the ERF more rapidly at low RH than at high RH. Increasing RH will offset the
effect of increasing ACH, particularly at high ACH values. These relationships
appear to hold over the temperature range used here, since T occurs only
linearly in equation (26). Based on the same equation, Figure 4 shows the ef-
fect of temperature variation at 5OX RH. Clearly, temperature was the major
determinant of ERF over the range of experimental conditions used in these
tests. The near vertical appearance of the contours suggests that ACH has only
a limited effect on ERF at any temperature setting.
Based on our repeated observation of the tendency of 1,4 dichlorobenzene
concentrations to rise smoothly to a steady state level, we strongly maintain
that moth crystal emission is a surface phenomenon. Hence, we would have pre-
ferred to express BRF on a per square centimeter surface area basis rather
than a per gram basis. However, we could not do this because individual sur-
p
face areas were not measured: only an average value of 55 cm was calculated
from the geometry of the moth crystal cakes. Since, the weight values shown in
Table VI reflect considerable variation in the actual sample sizes, Figures 3
and 4 most accurately represent ERF relationships, considering this limitation
of the data set.
10
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Mixed Emissions from Latex Caulk
In previous work, Dunn1 reported fits of the VP model to emissions from a
commercial latex caulk. However, the VP model was treated there as a model for
"wall effects," whose influence could be removed mathematically as we have
done here. Since then, we have derived the more general form of sink model
shown in Table IV, as well as its full model counterpart. Table VII compares
the residual sums of squares obtained from a least squares fit of each of the
four decreasing source models using SAS8 procedure NLIN. I(tj,t2> forms were
fitted since the data were obtained by accumulations over time intervals in
Tenax samplers.
Table VII. Lack-of-fit (SSS) comparisons of all decreasing source models
fitted to emissions from latex caulk at 23° G and 50V RH.
.
ACH = 0.
361. N
= 12
: ACH = 1
.85. N
= 9
.•Full
Sink
VP
Dilution
: Full
Sink
VP
Dilution
Total
18.2
20.9
37.1
72.1
282
282
412
479
organics
Methyl ethyl
0.430
0.583
0.582
0.603
0.583*
4.04
4.08
4.66
ketone
Cg alcohol
9.78
12.0
12.0
15.1
28.6
33.8
68.1
77.5
Butyl pro-
0.912*
2.66
2.66
2.92
4.63*
14.9
15.2
17.5
pionate
* Significantly better fit than all other models (a = 0.05).
Table VIII. Rate constant estimates for the decreasing source, full model
fitted to latex caulk emissions at 23° C and 50% RH.
ACH
A
kl
k3
k4
k5
/—s
It
to*
(hr-1)
(hr-1)
(hr-1)
(hr-1)
Total organics
0.361
604
0.746
0.440
0.167
5.55x10"®
1.85
856
0.891
1.79
0.0457
1.93xl0~3
Methyl ethyl ketone
0.361
84.8
2.62
49.1
4238
0.268
1.85
105
5.44
9.30
40.9
0.389
Cg alcohol
0.361
199
1.00
168
221
0.322
1.85
283
0.734
2.10
0.0881
2.66X10"3
Butyl propionate
0.361
82.2
1.07
794
1896
0.238
1.85
97.0
3.00
14.0
6.72
0.0145
There is statistically significant evidence that both a sink effect and a
vapor pressure effect were active for emission of butyl propionate at both ACH
levels, and for methyl ethyl ketone emission at high ACH. For methyl ethyl ke-
tone at low ACH, kg in the VP model was significantly greater than zero while
kg in the sink model :;as sot significant, even though both models fitted
11
-------
equally well. Therefore, we conclude that methyl ethyl ketone emission wa3
limited by the chamber concentration at both air exchange rates, but the wall
effect only became apparent at the high ACH. An explanation is suggested in
Table VIII which lists least squares estimates of the rate constants for the
full model. The rate constant exceeds kg by two orders of magnitude at low
ACH, but only by a factor of four at high ACH. At low ACH, the walls were able
to maintain equilibrium with a slowly decreasing chamber concentration, but
at high ACH, removal from the walls lagged behind. We believe that the same
argument accounts for the appearance of a wall effect for Cg alcohol at high
ACH, when none is apparent at low ACH. Again, a comparison of the kg and
estimates is informative.
Even though the sink model fitted the total organics data almost as well
as the full model, the fact that individual compounds exhibited both sink and
vapor pressure effects suggests that the latter should be used. Both kg in the
sink model and kg in the VP model were significantly greater than zero at both
ACH levels when these models were fitted separately to the data. Unfortunate-
ly, estimating five parameters in the full model based on a limited number of
data points often led to highly correlated estimates; so much so, in fact,
that the information matrix was nearly singular, and unreasonably large stand-
ard errors were produced. This is why standard errors are not listed an Table
VIII. Hence, in future work with these models, we suggest fitting the simpler
models first, then making comparisons to the fit of the full model using the
SSS criterion as we have done.
Estimates of A in Table VIII are expressed as micrograms of emittable
material per gram of source. He use this fact to illustrate the application of
these models in adjusting out any generic eir.lt effect. Suppose that 5 g of
latex caulk were introduced into the chamber, maintained at 23° C and 50* RH,
with 1.85 ACH, The appropriate full model for the chamber concentration of
butyl propionate is equation (15) with A = 5 x 97.0 = 485 jig/sample, and rate
constants taken directly from Table VIII. Mass [= C(t) x 0.166] xn the vapor
fhase of the chamber rather than concentration has been plotted as a solid
ine in Figure 5. The theoretical mass, had there been no sink effect, is
plotted as the dot-dash line ( ), and was obtained by setting kg = u in
equation (15). Thus, the transient sink effect of the walls would be to reduce
peak concentration by about 50*. The mass on the walls, obtained from equation
(16)j is plotted as the dashed line (—). Note that its response in the cham-
ber is somewhat delayed compared to that of the vapor phase curves. Note also
that at its peak, the wall effect accounted for nearly twice the mass of that
in the vapor phase. This is the most extreme case of those listed in Table
VIII, but clearly it contradicts the notion that a state-of-the-art emissions
test system automatically will be non-adsorptive. In order to complete a dis-
cription of the butyl propionate emissions, its chamber-based emission rate
function, dx/dt = kj[A - VC(t) - 1(0,t) -wj - kgVC(t), as defined by equation
(1), is plotted as the solid line in Figure 6. The theoretical rate function,
had there been no wall effect and obtained by setting kg = 0 in dx/dt, is
plotted as the dashed ( ) line. Again the difference is striking. The
material coming off the walls was able to contribute to the vapor pressure ef-
fect, thereby suppressing source emissions and leading to a relatively sus-
tained emission rate. This effect only emerged after the intial "explosion"
of butyl propionate essentially had been removed from the chamber; i.e., after
about 3-1/2 hours. Both emission rate functions are tending toward zero, as
predicted by theory.
SUMMARY AND CONCLUSIONS
In spite of meticulous care in chamber design, construction, and
operation, we have presented evidence that a state-of-the-art emissions test
chamber can act as a transient sink. At a low air exchange rate, particularly
at low humidity, the expression of this effect may be obscured by the other
12
-------
laain effect in the chamber; namely, that of increased concentration as it af-
fects the emission rate of the source. We have termed this the vapor pressure
(VP) effect. He feel that a major contribution of this paper has been to pre-
sent a class of mathematical models which accounts for both phenomena. The
source emission rate (BR) as a function of time and the steady state, emission
rate factor (ERF) are given precise definitions as a result. Rather trivially,
the effect or a chamber sink is adjusted out simply by setting to zero the
rate constant which governs sink adsorption/absorption. On a more ambitious
scale, these results provide a mathematical framework in which the sink con-
stants for the test chamber can be replaced by those of an actual living/
working environment, the concentration in that environment then being esti-
mated from the appropriate model. Clearly, this can answer "what if" questions
concerning untested combinations of sources and sinks. It also adds motivation
to the use of controlled test chambers in order to ascertain the sink proper-
ties of various indoor building materials.
ACKNOWLEDGEMENT
This research was accomplished with partial support from the U.S. Envir-
onmental Protection Agency under Contract No. CR-812305-01-0.
REFERENCES
1. J. E. Dunn, "Models and statistical methods for gaseous emission testing of
finite sources in well-mixed chambers," Atmos. Bnv. 21(2): 425-430 (1987).
2. S. Renes, B. P. Leaderer, L. Schaap, H. Verstraelen, and T. Tosun. "An
evaluation of sink terms in removing NO2 and S02 from indoor air," CLIMA
2000Ax 221-226 (1985).
3. L. H. Nelms, M. A. Mason, and B. A. Tichenor, "Determination of emission
rates and concentration levels of pr-dichlorobenzene from moth repellent,"
Proc. 80th Annual Meeting of the Air Pollution Control Association, paper
no. 87-83.6 (1987).
4. D. C. Sanchez, M. A. Mason, and C. L. Norris, "Methods and results of
characterization of organic emissions from indoor materials," Ataos. Env.
21(2): 337-345 (1987),
5. SAS Institute. Inc., SAS User's Guide: Statistics, Version 5 Edition,
Cary, N.C. (1985).
6. T. Godish and J. Rouch, "An assessment of the Berge equation applied to
formaldehyde measurements under controlled conditions of temperature and
humidity in a mobile home," J, Air Pollution Control Asso. 35(11): 1186-
1187 (1985).
13
-------
HOURS
SINK . SINK(K3=0) , DILUTION
FIGURE 1. 1,4 DICHLOROBENZENE FROM MOTH
CRYSTAL. T ¦ 36 C, RH«50*, ACH = 0.25.
TEMPERATURE 0
MG/H"3 500 750 1000 1500
¦ » 2000 4 3000 4000 5000
FIGURE 2. STEADY STATE CONCENTRATION OF
1,4 DICHLOROBENZENE FROM MOTH CRYSTAL.
14
-------
* RELATIVE HUMIDITY
2.5 3.0 3.5
4.5 t 5.0 // 5.5
4.0
6.0
FIGURE 3. EMISSION RATE FACTOR FOR 1,4
DICHLOROBENZENE FROM MOTH CRYSTAL AT 23 C.
HG/HR/G
- 3
-• 7
TEMPERATURE C
4
8 -rf
5
9
6
10
FIGURE 4. EMISSION RATE FACTOR FOR 1.4 DICHLORO-
BENZENE FROM MOTH CRYSTAL AT 50* RH.
15
-------
HOURS
CHAMBER , CHAMBER(K3=0) . WALL
FIGURE 5. BUTYL PROPIONATE EMISSION FROM
CAULK AT 23 C, 50* RH. AND 1.85 ACH.
HOURS
ER , ER(K3=0)
FIGURE 6. BUTYL PROPIONATE EMISSION RATE
FOR CAULK AT 23 C, 50* RH, AND 1.85 ACH.
16
-------
TECHNICAL REPORT DATA
(Pleat read hamtctlom on the reverie be/are completing!
1. REPORT NO. 2.
FPA/600/n-R7/ni
3mrisrwmN%2i*s
4. title and subtitle
Compensating for Wall Effects in LAQ Chamber Tests
by Mathematical Modeling
5. REPORT DATE
April 1987
6, PERFORMING ORGANIZATION CODE
7. AUTHORISI
James E. Dunn (University of Arkansas) and
Bruce A. Tichenor (EPA/AEERL)
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING OROANIZATION NAME AND ADDRESS
University of Arkansas
Department of Mathematical Sciences
Fayetteville. Arkansas 72701
ID. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO,
CR812305
13, SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Air and Energy Engineering Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Published Paper; 6/86 - 2/87
14. SPONSORING AGENCY CODE
EPA/600/13
15 supplementary NOTES AEERL project officer is Bruce A. Tichenor. Mail Drop 54, 919/
541t2991.
V
16 ABSTRACT
The paper presents mechanistic mathematical models that account for two
phenomena: (1) interior surfaces of a state-of-the-art emissions test chamber acting
as a transient sink for organic emissions; and (2) the effect of increasing chamber
concentration on the emission rate of the source. A key point is that the effect of the
chamber sink can be adjusted out simply by first fitting-the^appropriate model, then
setting to zero the rate constant which governs sink adsorption/ab'sorption.]/As a con-
sequence of this mathematical development, a source emission rate as a function of
time and a steady state emission rate factor are given precise definitions. Applica-
tions involve modeling 1,4 dichlorobenzene emission from moth crystal, and mixed
emissions from latex caulk. J[n the first case, at a low air exchange rate and low
humidity, the repressive effect of increasing vapor pressure tends to overshadow
the sink effect. Increased humidity tends to offset the increase in emission rate
which otherwise would occur with increased air exchange. Temperature is the prin-
cipal determinant of the steady! state emission rate. For the latex caulk, the effect
of a sink is to retard the apparent emission rate but lengthen the period of emissions.
\
17. KEY WORDS AND DOCUMENT ANALYSIS
», DESCRIPTORS
b.lOENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
Pollution Air
Mathematical Sorption
Models Insecticides
Walls Latex
Test Chambers Caulking
Emission
Pollution Control
Stationary Sources
Wall Effects
Indoor Air Quality
Moth Crystal
13B 14A
07D
12A 06F
13 M 11J
14B 13J
14G
IS. DISTRIBUTION STATEMENT
Release to Public
19. SECURITY CLASS (THil Rtport) "
Unclassified
21. NO. OF PAGES
18
20. SECURITY CLASS (ThU pate/
Unclassified
22. PRICE
EPA r*tm 2220*1 (1-79)
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