EPA/600/A-94/196
The Correction for Nonuniform Mixing in Indoor Microenvironments
David T. Mage1,2
Wayne R. Ott2
'World Health Organization
Geneva, Switzerland
2United states Environmental Protection Agency
Atmospheric Research and Exposure Assessment Laboratory
Human Exposure and Field Reserach Division
Research Triangle Park, NC 27711
ASTM Symposium
Methods for Characterizing Indoor Sources and Sinks
September 25 - 28, 1994
Washington DC
DISCLAIMER
The information in this document has been funded in part by
the United States Environmental Protection Agency. It has been
subjected to the Agency's peer and administrative review, and it
has been approved for publication as an EPA document. Mention of
trade names or commercial products does not constitute endorsement
or recommendation for use.
1
-------
Introduction
The modelling of the air pollution concentration distribution, C(x,y,z,t), in an
imperfectly mixed indoor setting of continuous air volume v m3 is a very complicated
process [v is defined here as the total volume minus the solid volume of objects within
it in cubic meters (m3)]. If the air is perfectly mixed then the concentration throughout
the chamber is the same at time t and is equal to the exit concentration Ce(t\ or
C(jc,y,z,t) = Ce (t) for all x, y, z. Then the mass of pollutant inside the chamber at
time t is given by v Ce{t). However, air velocities in rooms without forced-air
flow can be quite low [1], and concentrations can vary spatially within the mixing
volume.
The nonhomogeneities of concentration that exist can be influenced by the
placement of fans, internal stream lines connecting a closely spaced air inlet and air
return opening, internal barriers to air flow, thermal gradients in the room. etc. This
paper reviews the literature on how an empirical "mixing factor" has been introduced
into indoor air quality models to "correct" for lack of uniformity in concentrations and
mixing in indoor settings. We contend that this mixing factor is a simplistic
approximation of a complex phenomenon that may lead to invalid predictions of
indoor concentration, and recent advances in monitoring of indoor environments are
discussed that indicate an empirical approach that can be used to understand the
2
-------
phenomenon that have been observed. Recommendations for future experimental
studies are suggested that should help to clarify the prediction of the effects of
nonideal mixing on measurements of exposure to pollutants in indoor settings.
3
-------
The Perfectly Mixed Case
If the mixing in volume v is instantaneous and complete, the concentration
throughout v will be identical to the concentration Cc leaving v in the exit flows.
Figure 1 represents an idealized situation for a chamber representing a room in which
inlet air containing a tracer material at concentration C, enters with flow rate w(t)
m3/min and leaves with concentration Ce at the same flow rate w(f). If there is a
source in the room emitting pollutant at constant rate g(t) mg/min, and sinks in the
room remove the pollutant with a first-order rate constant k(t) min'1 we can model the
system using the mass balance, equation (1), by first setting the quantity entering
volume v over time interval T, minus the quantity leaving and removed, equal to the
change in concentration within v over the interval T, times the volume v.
T T
I (u C g)dt - | (u C - kvC)dt . v[ C( 7") - C(0)] (!)
0 0
where w = flow rate into volume v, [M/L']
C( = pollutant concentration in inlet flow [M/L3]
C, = pollutant concentration in exit flow [M/L']
C = pollutant concentration in volume v [M/L3]
4
-------
g = source strength [M/T]
k = first order rate constant for pollutant reaction [T1]
T = time since initiation of the process [T],
For the simple case where C = 0 at / = 0, Cc = C, and the variables C„ w, g and k are
held constant, the concentration in volume v will rise exponentially in the interval
0 < t < Thy the equation:
C-(C.. g/w)(l-e->> v/w - k v)].
The concentration in the room, ( '('/) = (' + g w, will decay exponentially by the
equation:
C(t) = C{T)e v
In an identical manner, the ideal system response to a 1 mg unit delta function
input (a pulse with infinite amplitude and zero width, and a unit area) will also be a
negative exponential decay from a maximum value of 1/v mg/'m3 with time constant
-------
{k + vv v) min"1.
Any arbitrary tracer input to a perfectly mixed indoor setting can be modelled
using the principle of superposition by a combination of the responses to a set of delta
functions and step functions (e.g. a source emitting 1 mg/min for 1 min can be treated
as the summation of two sources, one source emitting 1 mg/min for all time t > 0 and
a second source emitting -1 mg/min for all time t > 1 min.
In the literature, a mixing factor (m) where m < 1 often has been introduced to
account for positive deviations of observed concentrations from the spatial uniformity
assumed by Equation 1. Figure 2 shows the situation described in Figure 1, modified
by the use of a mixing factor (m), that has been used to describe the positive
deviations of observed values of C from the predictions of C by equations 2 and 3.
That is the concentration being measured is higher than predicted, as if all the air that
enters is not completely mixed with the entire mass of room air. Note that the fraction
of inlet air that is unmixed into the room (1 - m) must somehow get from the inlet to
the outlet intact without any mass transfer of pollutant between it and the rest of the
room air. We signify' this phenomenon by a streamline, represented by the
continuous line to the exit, which does not allow mass transfer. Then this flow
(1 - m) w must recombine with the flow (mw) at the outlet, to produce an exit
concentration (Cc) given by the mass balance
6
-------
Ce = m C -r (1 - m) C (
(4)
Literature Review of a Mixing Factor to Describe Nonideal Mixing
The first detailed discussion of nonideal mixing of air in a room was by Lidwell
and Lovelock in 1946 [2], They stated that "If the mixing of incoming air with the air
of the room is not sensibly complete (sic) the concentration of tracer substance will
not, in general, decay logarithmically nor will the rate of decay be the same in various
parts of the room." They state that if it is possible to fit a curve of the decay of a
tracer in the room by an equation of form C = C0 exp(-$ /), "the decay constant $
may be either greater or less than the ideal air exchange rate (w/v) according to the
disposition of the air circulation in the room relative to the point at which the
observations were made". They further state "This 'constant' may conveniently be
referred to as the equivalent ventilation at the point of measurement". Their use of
'constant' with single quotes is to emphasize that the measured value of 0
($ = m w/'v) is a function of position in the room [0(x,>,z)] so that m is also a
function of position, m{x.y,z), as w and v are constant.
The next discussion of nonideal mixing which first coined the term "mixing
factor", appears in a 1960 article by Brief [3], Brief states "This mixing factor will
depend on the vapor or gas toxicity, the uniformity of contaminant distribution within
7
-------
the room, location of the fans, construction of the room, and the population therein.
This factor (m) may vary from 1/3 to 1/10 and is used in conjunction with v. Vto yield
the effective number of air changes, m v/V\ We note that this article referred to a
specific application - determining the required number of air exchanges to take place
before entry can be safely made into a contaminated space (e.g. before a worker can
go into an emptied benzene storage tank for visual inspection and cleaning). Given
the dangers of explosion, toxicity and asphyxiation in such circumstances, the mixing
factor m = 1/10 is appropriately recommended as a safety factor that allows for
nonideal mixing, as in the corners of the room.
In 1963, Turk [4] introduced the mixing factor (m) "to account for the fact that
dilution of air is not instantaneous, and that concentration fall-off rates are actually
smaller than the ideal values ... Brief [3] suggests that m commonly ranges between
1/3 and 1/10" Unfortunately, Turk's statement does not recognize that m can be
greater or less than 1 depending on where in the room it is measured, and the cited
usage of m in the range of 1/3 to 1/10 is for the highly specialized purpose of handling
dangerous materials requiring safety factors for human activities [3],
In 1970, Constance [5] discussed the air exchange required to remove
contaminants from an enclosed air space. He stated "If m = 1, we have the
unattainable perfect mixing discussed previously. Actual mixing factor can vary from
8
-------
1/3 to l/l 0, depending on contaminant toxicity, uniformity of distribution in the space,
location of air inlets and outlets, enclosure geometry and population". Constance [5]
discussed the power cost of providing extra air flow when using too low a design
value of m and the competing need to use m < 1 as a safety factor.
Drivas, Simmonds and Shair in 1972 [6] reported actual measurements of m and
cited Constance [5] as the source of the phrase "mixing factor". They observed that
the idea is identical to that first suggested by Lidwell and Lovelock [2], They stated
"Values of m are normally estimated to be from 1/3 to 1/10", which includes the
"safety factor" based on toxicity that appears in Constance [5], They measured m
values in several rooms in the basement of building which were connected by a
common hallway. In their study, the entire basement was ventilated by a single air
supply distributed to all the rooms, and all the room air exhausted through a door into
the common hallway which was then exhausted by a single air outlet. An
instantaneous injection of SF6 into the basement air intake was used as a test signal.
Measurements of m were made in a room with a single inlet and outlet to the hall, and
a value of m = 0.9 was measured while using four large fans to simulate perfect
mixing. Repeating the experiment without the fans in operation, the value of m was
found to be 0.68 at the same measurement point in the center of the room. In the
same experiment in another room, a value of m = 0.3 was measured 3 ft from one
9
-------
comer of that room, which they note validated the range of 1/3 to 1/10 for m [3-5],
Such imperfect mixing in this case may be caused by a flow streamline from the air
intake in the ceiling directly to the hall through the door, which of course would
reduce the air flow to the far corner of the room where m was measured to be 0.3.
Simultaneous measurements made in the hallway at both ends showed variation by a
factor of 2 in the concentration, and the concentration decreased with time in a manner
not described by a single exponential term, as suggested by Lidwell and Lovelock [2]
for nonuniformly mixed rooms.
In 1978, Esmen [7] derived the mass balance equation for a conservative
pollutant in a room with m < I. His paper states "Let m be defined as the portion of
the [contaminant free] ventilation air flow (w) that is completely mixed with the room
air. Ln other words, we imagine that the m fraction of the air flowing in will be totally
replaced by the room air, and the remainder is totally unmixed. Under this condition
the mass balance is: v dCdt = -mwC".
This derivation is not realistic because it assumes that the 'unmixed' fraction (1 -
m) moves through the room, from inlet to outlet, without any mixing at all as shown
in Figure 2. By conservation of mass (see Lidwell and Lovelock [2]), the
nonuniformity of mixing implies that there are points in the room where m > 1 and
m < 1. This practical situation could occur when a streamline connects a ceiling air
10
-------
inlet to a ceiling air outlet so that a portion of the inlet air bypasses the turbulent core
of the room where a concentration measurement might be made. Thus Esmen's
conceptual derivation [7] does not apply to real situations because the value of C in
the derivation cannot be uniform throughout the room unless m - 1. Esmen later
acknowledges this by stating that m can be a variable in the room, but claimed that it
depends on the source, giving an example of a person moving away from a smoker to
get away from the smell (e.g. to an area of lower concentration which is treated as an
area of higher m).
In 1980, Ishizu [8] pointed out that, given the derivation of the mixing factor (m)
by Brief [3] in which m is defined as the portion of the inlet air that is [apparently]
completely mixed with the room air, the pollutant generation rate within the room (g)
appears to be enhanced by the mixing factor as gim if m < 1.
In 1981, The NAS/NRC report Indoor Pollutants [9] reviewed the literature on
single-compartment models and the use of the mixing factor to account for the lower
dilution than predicted by a well mixed state. They concluded that even when a
compartment is not well mixed, the uncertamty of the flow patterns, and difficulty of
specifying source and sink patterns "usually does not justify the development of a
more sophisticated model".
In 1987, Nagda, Rector and Koontz [10] introduced the mixing factor into the mass
11
-------
balance equation, but with the definition "the ratio of the actual residence time of the
pollutant indoors to the residence time derived directly from the air exchange ratio".
We fell that this interpretation is incorrect because, if a pollutant generated indoors
does not disperse as rapidly as we might expect, its residence time indoors is greater
than expected, not less than expected, which would give an m value greater than 1.
However, they also give another expression for m as "the ratio of the 'exit stream'
concentration to the indoor concentration" which is valid if the inlet concentration is
equal to zero (C = 0) [See Equation 4], To our knowledge no such room
measurements are available in the literature as an exit stream concentration may not
be measureable if there can be multiple entry and exit points for air infiltration (e.g.
windows, doors, cracks, etc.).
In 1991 a National Research Council (NRC) report [11] summarized the usage of
the mixing factor as follows "Mixing of air within and between rooms vanes spatially
and temporally...so single-compartment and multicompartment mass-balance models
generally incorporate a mixing factor". As explained by Lidwell and Lovelock [2] a
single-compartment model is valid if and only if m = I, so any single-compartment
model incorporating a mixing factor violates a basic assumption. However, if a
single room is modelled by multiple compartments, with air exchange between them,
then that basic assumption is violated unless m = 1 in each compartment.
12
-------
Recent investigations have considered the spatial variation of concentration in
indoor settings which have incomplete mixing, and show that the concentration
distribution and it's modelling can be quite complex [12-20], For example,
Baughman, Gadgil and Nazaroff [20] report a study of a room in which a mixture of
SF6 and He was released as a neutrally buoyant mixture while heated to simulate the
smoke from a burning cigarette. Figure 3 shows the total measurements from some
of the 41 different measurement points in the room during the quiescent case
(w/v = 0.03 to 0.08 hr'1) in two duplicate test runs. Let the mean concentration in the
room of dimensions a, b, c, at any time t be defined as C,
If a "mixing factor" were applied to these experiments, those measurements above the
normalized C C = I line would be interpreted to have a value of m < 1 (C is greater
than that predicted by perfect mixing) and all those values below the line would be
interpreted to have a value of m > 1 ((.' is less than that predicted by perfect mixing).
In fact since C = tnC, and the integral of C dv v is by definition equal to C, the
a b c
C(.x .y.z.t)dx dydz
(5)
13
-------
integral of m dv'v must be equal to 1. As we stated earlier, and as stated by Lidwell
and Lovelock, nonuniform mixing requires that there must be points in the room
where the concentrations are higher and lower than the mixed mean concentration.
Figure 4 shows the data from a similar test, but with a 500 Watt heater creating
thermal eddies. The result is a reduction of time to achieve uniform mixing in the
same room, from approximately 100 minutes to 15 minutes. In both experiments
shown by Figures 3 and 4, more than a two order of magnitude difference in
concentration spatially persisted in the room for 30 minutes and 3 minutes
respectively.
14
-------
DISCUSSION
Our review of the literature about the way in which the spatial variability of
concentrations m indoor settings has been handled shows a rather simplistic reliance
on a single parameter -- the mixing factor m — to account for discrepancy between the
concentrations predicted by the mass balance equation and those actually observed.
As indicated above, use of a single factor m to characterize an indoor air pollution
episode - from emission of the pollutant until its eventual decay to negligible indoor
levels - cannot be justified, and more detailed treatment of indoor air pollution
episodes is required. Toward this end, we now propose a general conceptual
framework for indoor concentrations and pollutant exposures. Where possible, we
draw upon recent research that promises to provide important knowledge about the
spatial variability of pollutant concentrations in indoor settings. Finally, we suggest
criteria for situations in which the mass balance model can provide useful predictions
in indoor settings.
Concentration Fieid Model. Consider a rectangular volume (an empty room
or a chamber) and imagine a 3-dimensional coordinate system (x,y,z) with its origin
fixed in one of the lower corners of the room. Here .y and v denote the horizontal
distances from the corner and z denotes the vertical distance. Suppose a, b, and c
15
-------
denote the length, width, and height, respectively, of the room; then 0 < x <_ a. 0 <
y <_ b, and 0 < z <_ c. The concentration of an air pollutant at point (x,y\zj at any
instant of time t can be represented by:
C(x.y.z,t) (6)
We call this the "concentration field" in the room at time t. Next we are interested in
the exposure that occurs at some point {x(tj,y(l),z(l)} that is moving inside the space
inside the room (for example, the exposure of a point on the person's nose inside the
room as the person walks about the room), then the personal exposure £(t) is given by
substituting x(t), y(l), and z(l) into Equation 1 to give the exposure as a function of
time:
£(/) = C{.y(/).v(*U(0} (7)
The mean concentration in the room at time / will be given by the integral of the
concentration field over the entire room divided by the volume of the room v = abc,
as shown in equation 5. The variance of the concentration field at any time t can be
written as
16
-------
a b c
Var{C\t} = [C(x,y,zJ) - C{t)}2dxdydz
(8)
0 0 0
The variance of the concentration in the room is an indicator of the "amount of
variability" of the concentration across the room at any time t. The square root of the
variance is the standard deviation of the concentration across the room at any time t.
Because it is useful to normalize the amount of variability by dividing it by the level
of concentration present, another useful statistic is the the coefficient of variation
CV(t), which is the square root of the variance divided by the mean at time l, or:
Baughman, Gadgil, and Nazaroff [20] have studied the indoor concentration
field by measuring the decay of an instantaneously emitted tracer gas at 41 sampling
sites within a test room. For their experiments, they defuied the "characteristic mixing
time" as "the period required for the relative standard deviation of the tracer gas
concentrations measured at the 41 points to become less than 10%." Adopting the
CV(t) = ^Var(c 1'}
(9)
C(0
17
-------
Baughman, Gadgil, Nazaroff-criterion, the charactenstic mixing time tY is the time
required for the coefficient of variation (also called the relative standard deviation)
to be less than 0.10. That is,
CV(t^) < 0.1 for characteristic mixing
After the charactenstic mixing time tr t/he pollutant is sufficiently well-mixed in the
room that it makes little difference where the exposure point (for example, the
person's nose) is located in the room. Thus, an important question is how much of a
total indoor air pollution episode consists of this well-mixed state versus the poorly-
mixed state. If a large portion of an indoor episode consists of the well-mixed state
after time / then a 1-compartment mass balance model can be used to predict the
average exposure of a person in the room without the use of a mixing factor, because
a basic assumption of the mass balance equation — a uniform concentration field at
any time t > ty — has been met during most of this period.. However, for the period
i < t a multicompartment model and knowledge of a persons trajectory between
compartments is necessary to describe their exposure. .
Indoor Air Pollution Episode. An indoor air pollution "episode" -- that is,
a source emitting a pollutant inside a chamber, an automobile, or a room in the home
18
-------
— typically can be characterized by several different time periods (Figure 1). First,
there is the "source-on" period - the time period during which the pollutant source is
emitting - which in this figure lasts from time t = 0 until t = tn. Examples are
smoking a cigarette, burning toast, using a spray can indoors. A single cigarette
smoked in a room has been represented in exposure models as a continuously-emitting
point source usually lasting 6-11 minutes [21]. Some sources, like freshly painted
surfaces, can have very long "source-on" periods as they outgas volatile organic
compounds as the material cures and harden.. Chlordane emitted into a home from
the soil beneath the foundation is another example of a source with a long on-time.
Other sources, like cigarettes, have relatively short source-on periods, typically about
7 minutes.
As long as the source is emitting, there will be relatively high concentrations
very close to the source, and the concentration field across the room will be
nonuniform. Although mechanical mixing by fans in a chamber with a burning
cigarette can bring about some uniformity, concentrations always will be higher
within a few centimeters above the tip of a cigarette while it is burning, and we
cannot describe the concentration during this period as uniform over the entire mixing
volume. Therefore, Figure 5 characterizes the source-on period, or a-phase, as a
"poorly mixed" state, and concentrations will vary spatially across the room.
19
-------
Furtaw [22] has found that concentrations measured in a chamber very close to the
source (less than I meter) can be very high during the time period that the source is
emitting. When close to the source, it appears that small "pockets" of high
concentrations occur as a random process, causing the concentration values to be
significantly higher than predicted by the mass balance equation.
At time t > la in the conceptual model, the source stops emitting (for example,
the cigarette smoking ends or toast stops being toasted), and the "source off' time
period begins. In Figure 5, we characterize the source-off period as consisting of two
time periods, a transition period called the p-period lasting for time duration tB and a
well-mixed phase called the y-period lasting for time duration The y-period can
be characterized as meeting the Baughman-Gadgil-Nazaroff criterion discussed
above). That is, the spatial coefficient of variation obtained from Equation 10 for this
pollutant and this source during the y-period over the entire volume of the room is less
than 0.1, or Cr< 0.1.
During the poorly mixed a- and p-periods, the exact position of the receptor
point xftJ.yOj.zftJ is important, because different points in the room experience
different concentrations During the y-penod, the position of the receptor point is not
very important, because the normalized standard deviation of the exposure at any
instant of time throughout the room is less than 0.10. (Problems with surface effects —
20
-------
sinks that absorb and then release some pollutants near surfaces ~ have been ignored
to simplify our analysis.)
To compute the average concentration at any point (x,y,z) over the entire indoor
air pollution episode, we form the weighted average of the three time periods:
C- '> Cp. ^ cv (10)
/ + tR t t t + t a + t P t - t* + t Y
a p Y apv a p y
We shall define an indoor air pollution episode as an event lastmg until the
concentration becomes negligible (for example, less than 1% of the value attained at
the end of the source-on period), assuming that no additional pollutant enters the
room during this period. Thus, because exponential decay implies that the
concentration will approach but never actually reach zero, we assume that the
concentration is approximately zero at time / > 7 at the end of the episode.
Recent experiments performed at Stanford University using real-time
instruments operating in a home with a smoker present suggesst that the a-period and
P-period both are relatively brief compared with the y-period. For a cigarette, for
example, the a-period ~ the time during which the cigarette is being smoked —
21
-------
typically is about 7 minutes. The (3-period -- the time it takes once the source has
ended to reach characteristic or uniform mixing ~ also is relatively brief because of
the rapidity with which the air in a room disperses and mixes the pollutant by
advection. As noted by Baughman, Gadgil, and Nazaroff [20] "...the transport of a
pollutant over a significant distance is driven by advection since the diffusion rate is
relatively small by comparison." In still air, they calculate a typical characteristic time
of 450,000 seconds or more than 5 days for a pollutant to migrate 3 meters. They point
out that "...from the common experience of how rapidly odors and smoke spread
indoors, it is apparent that the characteristic time for contaminant dispersion is more
on the order of minutes, not days." Our own experiments agree with their
conclusions, and we have found that pollutant migration across a large living room ,
or even from a bedroom to an adjacent living from, takes only a few minutes if the
door between the two adjacent rooms is open.
Baughman, Gadgil, and Nazaroff [20] have measured the concentrations of a
tracer gas SF6 at 41 points in a 31 m3 test room with a very low air exchange rate (0.03
to 0.08 ach). Thus their residence time ranged from about t =12.5 hours to 33.3
hours. They used an instantaneous source; their experiment has virtually no source-on
period (that is, tK ~ 0). They define the characteristic mixing time as the time elapsed
until the relative standard deviation becomes less than 0.10, which corresponds to the
22
-------
P-period in Figure 5. For two "quiescent" experiments with no internal sources in the
room, they found that l[S = 80 minutes in the first experiment and fp= 100 minutes.
When a 500-w heater was added to the room, two experiments gave /(t= 7 minutes and
tp = 13 minutes. Finally, with incoming solar radiation, = 7 min and lp = 10
minutes. If we compare these value with the residence time x (= 1 /cj>) of air in their
experimental chamber, we find that t ranges from 12.5 hours to 33.3 hours, and the
ratio of the transition tune to the residence time, or /p/r, ranges from 0.05 to 0.10.
Thus, once the source turns off, the period in which the pollutant is poorly mixed is
only about 5-10% of the residence time. However, comparing the transition period
with the residence time is not really useful, because the indoor concentration is only
Me = 37% of its starting value after t tune units have elapsed. Instead, we arbitrarily
choose the endpoint of the indoor air pollution episode as the time at which the
theoretical exponential decay reaches 1 % of its initial value the time when the source
turns off. If we set the exponential decay function equal to 0 .01, we find the resulting
time as:
Ca ' ' 1
C(/R - t ) --—-
2-—{— - e 1 0.01, or - ty -ln(0.01) = 4.6It (11)
^ O
Using this equation, we see that the p-period in the experiments of Baughman,
23
-------
Gadgil, and Nazaroff [20] gives a p-period time of t(l that typically is less than 5% of
the total source-off time, as can be seen from Table 1. The minimum and maximum
values in this table are obtained by substituting the minimum and maximum air
residence times (t ranging from 12.5 to 33.3 hours) into the above equation. When
viewed from this perspective, the portion of the source-off period in which the room
is poorly mixed ranges between 0.1% and 2.9% of the total source-off time. The
proportion is greatest (0.9% to 2.9%) when the room is in its quiescent condition and
relatively free of mechanical or thermal agitation.
Table 1. Ratio of Transition Time (/„) to Residence Time (t ) and to Source-Off Time
Op ~1 ) Expressed as a Percentage, Based on the Experiments by Baughman, Gadgil, and
Nazaroff [20]
No
Experiment
Type
Transition
Time
(hours)
Transition
Residence,
(minimum)
(%)
Transition-
Residence,
' p mm
(maximum)
(%)
Transition -
Source-Off
V'r. + O
(minimum)
(%)
Transition +
Source-Off
f,'/('/, ¦ 'J
(maximum)
(%)
I
Quiescent
1 67
13
5
2.9
1.1
2
Quiescent
133
11
4
2.3
0 9
3
500-VV Heater
0.25
2
1
0.4
0 2
4
500-W Heater
0.22
2
1
0.4
0 1
5
Solar
Radiation
0.12
1
0.4
0.2
0.2
6
Solar
Radiation
0.17
1
1
0.3
0.1
24
-------
As soon as as a heater or solar radiation is added, the poorly mixed period tp
becomes almost negligible (0.1-0.4%) of the overall source-off period. Thus, if the
average human exposure is computed for the entire indoor air episode using
Equation 10, then the proportion contributed to this average when the pollutant is
poorly mixed in the source-off period is likely to be very small or negligible, based
on these experiments. In other experiments that we have performed with a
cigarette smoked in a real home, ip has been so small that we were unable to see it,
relative to the duration of episode. The reason for the relatively small poorly
mixed period appears to be the rapidness with which the pollutant mixes throught
the room of a home ~ typically a few minutes ~ combined with the relatively low
air exchange rates of homes, typically about 1 ach (see, for example, Pandian, et
al., [23] Thus, the experiments suggest that the pollutant exposures to some
sources - such as the cigarette ~ can be modeled well using the well-mixed
assumption in the mass balance equation, since the source-on time (/J is relatively
small (about 7 minutes), the poorly mixed source-off time period (/p) is only a few
percent of the total source-off period, which can last for several hours until the
episode ends. Of course, this hypothesis needs more testing in future experiments
in real homes, but the evidence now suggest that modeling environmental tobacco
25
-------
tobacco smoke (ETS) exposures in the home will be reasonably accurate if the
mixing factor m is eliminated entirely from the equation. This elimination of the
mixing factor is equivalent to setting m = I. This approach was found to work well
in experiments on cigarette smoking activity and exposures in a moving
automobile and a chamber [21],
However, as we have shown in this paper, use of a uniformly mixed
assumption is valid only if the a-period and p-period are small relative to the y-
period. This result may not be valid for some other common indoor sources of air
pollution, such as a carpet that continuously erruts a pollutant and therefore has a
large a-period relative to the indoor air residence period. The concepts presented
in this paper show the need for more research on events occuring in the a-period
during which the source is on and for experiments conducted in realistic settings in
which people actually experience exposures from indoor sources.
26
-------
Summary and Conclusion
Our analysis of the literature concerning the correction for nonuniform mixing
by use of a mixing factor, m < 1, shows that this concept is not appropriate on
theoretical grounds. It was derived originally as a safety factor to account for the
existence of poorly ventilated compartments in a multicompartment setting.
However, if such deviations from uniform mixing occur then a multicompartment
model with m = 1 in each compartment could be used. Certainly, if a nonventilated
compartment, such as a hold of a ship, is to be entered by a worker, a safety factor
is warrented to estimate how long to wait before entering, but it could not be used
to estimate that worker's exposure.
Development of our framework eliminates the perceived need for the use of a
"mixing factor m", and indeed, we have found it desireable to discard this concept
altogether. We show that in many cases, where air exchange is typical of that in
homes and offices, the concentration between multicompartments equilibrates
quickly to meet the criterion suggested by Baughman et al. [20], Even in the first
phase during the source on period, the concentrations fluctuate rapidly about the
mean or expected value of uniform mixing. Therefore a person moving about the
room may experience these fluctuations, and the average exposure dunng this
brief interval might even be represented by the expected exposure with m = 1.
27
-------
Thus, we hope that indoor and exposure modelling research of the
fiiture will look more precisely at the processes that occur in mixing
volumes, using our proposed framework, and will attempt to account for
differences between observed and predicted concentrations without using a
gross single parameter such as m. As we have argued, uniform mixing
requires m = 1, all other cases require that m vary across the volume being
modelled by mass balance considerations. If there is a significant
concentration gradient, then a multicompartment model with m = I in each
compartment should be used. Therefore, we do not feel that a single
parameter m unequal to 1 is useful for indoor air and exposure modelling
applications when mixing is not uniform.
28
-------
1. Matthews, T.G., Thompson, C. V., Wilson, D. L., Hawthorne, A, R., and
Mage,D. T., Air Velocities Inside Domestic Environments: An Important
Parameter in the Study of Indoor Air Quality and Climate. Environment
International Vol. 15, 1989, pp. 545-550.
2. Lidwell, O. L. and Lovelock, J. E., "Some Methods of Measuring
Ventilation," Journal of Hygiene (Cambridge), Vol. 44, 1946, pp. 326-332.
3. Brief, R. S,, "Simple Way to Determine Air Contaminants," Air
Engineering, Vol. 2, 1960, pp. 39-41.
4. Turk, A., "Measurements of Odorous Vapors in Test Chambers:
Theoretical," ASHRAE Journal, Vol. 5, 1963, pp. 55-58.
5. Constance, J. D., "Mixing Factor is Guide to Ventilation", Power, Vol. 114,
1970. pp. 56-57.
6. Drivas, P. J., Simmonds, P. G. and Shair, F. H., "Experimental
Characterization of Ventilation Systems in Buildings", Environmental
Science & Technology, Vol. 6, 1972, pp. 609-614.
7. Esmen, N. A., "Characterization of Contaminant Concentrations in
Enclosed Spaces", Environmental Science & Technology, Vol. 12, 1978,
pp. 337 -339.
8. Ishizu, Y., "General Equation for the Distribution of Indoor Pollution",
Environmental Science & Technology, Vol. 14, 1980. pp. 1254-1257, and
correction in Vol. 14, 1980, p. 1460.
9. National Academy of Sciences, National Research Council, Indoor
Pollutants. National Academy Press, Washington, 1981, pp. VI-24 - VI-28.
10. Nagda, N. L.. Rector, H. E., and Koontz, M. D., Guidelines for Monitoring
Indoor Air Quality. Hemisphere Publishing Corporation, New York, 1987,
p. 26
11 National Academy of Sciences, National Research Council, Human
Exposure Assessment for Airborne Pollutants - Advances and
Opportunities. National Academy Press, Washington, 1991, p. 192.
12. Koganei. M, Holbrook, G. T., Olesen, B. W., and Woods, J. E,, "Modeling the
Thermal and Indoor Air Quality Performance of the Vertical Displacement
Ventilation Systems".Proceedings of Indoor Air ' 93 , Vol. 5, 1993, pp.
241-246.
13 Davidson, L . "Ventilation by displacement in a Three-Dimensional Room -
A Numerical Survey", Building and Environment, 1989, Vol. 24, pp. 363-
372.
29
-------
14. Sherman, M. H., "On the estimation of Multizone Ventilation Rates from
Tracer Gas Measurements", Building and Environment, 1989, Vol. 24, pp.
355-362.
15. Sherman, M. H., "Uncertainty in Air Flow Calculations Using Tracer Gas
Measurements", Building and Environment, 1989, Vol 24, pp. 347 - 354.
16. Axley, J. W., "Multi-Zone Dispersal Analysis by Element Assembly",
Building and Environment, 1989, Vol. 24, pp 113-130.
17. Sherman, M. H., "Tracer-Gas Techniques For Measuring Ventilation in a
Single-Zone", Building and Environment, 1990, Vol. 25, pp. 365-374.
18. Mokhtrazdeh-Dehghan, M. R., El Telbany M., M., M., and Reynolds, A. J.,
"Transfer Rates in Single-Sided Ventilation", Building and Ventilation,
1990, Vol. 25, pp. 155-161.
19 Yost, M. G., Gadgil, A. J., Drescher, A. C., Zhou, Y., Simonds, M. A.,
Levine, S. P., NazarofF, W., W., and Saisan, P. A., "Imaging Indoor Tracer-Gas
Concentration with Computed Tomography: Experimental Results with a
Remote Sensing FTIR System", American Industrial Hygiene Association
Journal, 1994, Vol. 55, pp. 395-402.
20. Baughman, A. V., Gadgil, A. J., and Nazaroff, W. W., "Mixing of a Point
Source Pollutant by Natural Convection Flow Within a Room", Indoor
Air, J994, Vol. 4, pp. 114-122.
21. Ott, W. R., Langan, L., and Switzer, P., "A Time Series Model for Cigarette
Smoking Activity Patterns: Model Validation for Carbon Monoxide and
Respirable Particles in a Chamber and an Automobile," Journal of
Exposure Analysis and Environmental Epidemiology, 1992, Vol. 2
Supplement 2, pp. 175-200.
22. Furtaw, E,, personal communication, U S E P A. and University of Nevada,
Las Vegas, NV, 1993.
23. Pandian, M.D., Ott, W. R., and Behar, J. V., "Residential Air Exchange Rates
for Use in Indoor Air and Exposure Modeling Studies," Journal of
Exposure Analysis and Environmental Epidemiology>, 1993, Vol. 3, pp.
407-416.
30
-------
SOURCE INPUT
Concentration C^(t)
OaO
C(t)
OUTPUT
W,
'out
Concentration Cp(t)
VOLUME v
Figure 1. Air mixing chamber with source input port flow rate win at
concentration (t) and output port with flow rate wout al
concentration Ce (t).
CO
-------
SOURCE INPUT
w
Concentration C^(t)
OUTPUT
rn uu
Concentration C£(t)
C(i)
VOLUME v
Figure 2. Air mixing chamber with source input flow rate w
at concentration C^(t) and output port with flow rate
uj at concentration C£(t) - mC(t) + (1 -m)C^(t).
Usage of the mixing factor m implies that a fraction
(/-rn) bypasses the chamber without mixing in it.
CO
-------
10
c.
o
flj
l~
c
OJ
u
c
o
U
\0
U-
in
"S
~f0
E
o
z
.1
.01
:!j!j,
f X f
!.:i:
Mo.
TEST 2
1 i *
b o 1 0
Ml
i
1
1
| I
i *
¦ J * 5
:fi oIk
~ 0 '
~ X 4
r* 1 1
iK
nnn.
i
i
1
1
1 1
«i
\ * n
;
\
* O °
! * ° '
8
o :
o
* X o
A
i O
4 1
i
(———
•
20
40 60
Time (min)
80
100
120
Fig- 3 Scattergram of normalized tracer gas concentration versus lime under nearly
isothermal conditions. The tracer gas was released at time = 0 minutes. The
symbols represent different sampling zones: A => core at 76 cm height; O => core
at 160 cm height; V =» core at 206 cm height; X => comers; and + =* surfaces.
t-i
UJ
-------
10
c
o
To
u.
C
u
c
o
U
LL.
1/5
T3
OJ
N
m
O
Z
.1
.01
! x
1 |
1 1
i i
¦ ¦
,—1
TES
V-
T 4)
! I
: i
(
+
!!
i A A
A
i
i
! 1
¦ i
i ¦
«p
w
' *
~ 0
X
~
~
¦
10 15 20 25
Time (min)
30
35
40
Scattergram of normalized tracer gas concentration versus time with 500-W heater
operating. See caption to Fig. 3 for symbol legend.
-c
-------
Source
Source
On
Off
t(X
h
1 ~
Not
Detectable
Poorly Mixed
Well Mixed*
Episode (Concentration Detectable)
0
Time
Not
Detectable
* Baughman-Gadgil-Nazaroff criterion
Figure 5. Schematic representation of three concentration phases
of a pollutant emitted into a room. Concentration is
not detectable after ty.
-------
TECHNICAL REPORT DATA
1. REPORT NO.
EPA/600/A-94/196
i.
3.RJ
4. TITLE AND SUBTITLE
The correction for nonuniform mixing in indoor
microenvironments
5.REPORT DATE
6.PERFORMING ORGANIZATION CODB
7. AUTHOR(S)
David T
8.PERFORMING ORGANIZATION REPORT
NO.
nd Wayne R. Ott
9. PERFORMING ORGANIZATION NAME AND ADDRESS
USEPA/AREAL/HEFRD (MD-56)
RTP, NC 27711
10.PROG RAM ELEMENT NO.
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
13.TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
Work was begun by Dr. Mage while employed with WHO and continued after return to
USEPA
16. ABSTRACT
The modelling of the indoor concentration distribution produced by sources
and Binks of pollutants is complicated by nonuniform mixing within the indoor
settings. Two common approaches to predicting the concentration distribution are
to either treat the indoor volume as containing multiple compartments with uniform
mixing within each, or to treat the entire indoor volume as a single uniformly
mixed compartment with an empirical mixing factor m that iB introduced to correct
for nonuniform mixing. We review the literature on m and Bhow that thiB empirical
approach violates a basic principle of conservation of mass.
^ we propose a new conceptual mod¥"l~for-'eh"e~caso"f"~a source of pollution in an
indoor setting^by defining the source operating conditions within three periods,
tof t, and t7, where t„ is the time while the source is emitting, iB the time
after the source^stops emitting, but while the concentration distribution is
nonuniform in the- indoor setting, and ty is the time from the point where the
indoor concentration becomes uniform until it becomes nondetectable above the
background value. We define the state of uniform concentration as when the
coefficient of variation of concentration (standard deviation/mean) throughout the
volume becomes less than 0.1. We show, that with this definition, the assumption
of uniform mixing for the entire volume will not lead to serious errors in
predictions of exposures if a subject is moving about in the indoor setting and ty
» t.+ tn-
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.IDENTIFIERS/ OPEN ENDED
TERMS
c. COS ATI
18. DISTRIBUTION STATEMENT
19. SECURITY CLASS (This Report)
21.NO. OF PAGES
J£.
20. SECURITY CLASS (This Page)
22. PRICE
------- |