Environmental Data Analysis BC ENV 3017
Indoor - Outdoor PM
Background
-
exposure to PM is associated with a number of health effects ranging from
asthma due to cockroach allergen particles to increased morbidity and mortality
from outdoor air pollutants
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particle size determines the residence time of the particle in the air
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PM < 2.5mm settle very slowly and also penetrate
deep into the lungs
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PM > 2.5mm settle quickly and exposure is predominantly
through ingestion from hand to mouth contact
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we are spending a significant amount of our time indoors and need to better
understand what controls indoor concentrations of particulate matter
-
the goal of the experiment below is to use temporal information to better
understand what the the primary sources and sinks are of indoor particulate
matter of different size fractions
-
the paper by long et al. (2000) shows some nice examples of indoor air
PM (Fig, Fig)
The experiment
-
particles indoor and outdoor were monitored for several days in summer
1999 in bedroom 1 of Steve Chillruds apartment. Steve is a researcher at
LDEO. He lives on the 5th floor of 720
W 181th St in Washington Heights.
-
Particles were collected outside and inside (bedroom
1, volume: 33 m3) using a particle counter and a pump (Fig).
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an optical particle counter was connected to a valve that switches between
outdoor and indoor sources of particles
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the line for sampling outdoor particles was passed through the window
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each sampling line was made out of metal and has the same length so that
any potential loss of particles in the tubing is the same for both cases
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the experiment was performed from 7/2/99 to 7/7/99
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in order to determine the air exchange rate between outside and inside,
SF6 was added to the room several times during the experiment
and the drop of its concentration was measured as a function of time
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our goal is to develop a conceptual and a mathematical model of the experiment
-
we can use these models to test hypotheses
Conceptual model
We are picturing the bedroom as a box in which the air is well mixed
(homogeneous). Exchange with the outside occurs through the windows or
possibly the doors. There may be sources or sinks or particles in the room
(deposition or re-suspension of particles). The following factors might
have an effect on the indoor PM concentration:
-
PM concentration outside
-
deposition rate inside (sinks)
-
resuspension rate inside (sources)
-
air exchange between outside and inside
-
air exchange between rooms in the apartment
Our goal is to construct a mathematical model that includes the relevant
processes above and allows us to predict the indoor PM time series. Unfortunately,
we do not have a good handle on some of the factors (in red) mentioned
above. We can measure the outdoor/indoor exchange rate using SF6
and can make an assumption about the deposition rate inside, and will ignore
all the other processes.
That means we envision two processes controlling the indoor PM concentrations:
-
exchange of particles between indoors and outdoors,
-
and deposition of particles on to surfaces.
Mathematical model
We are assuming that the change of PM concentration inside (Cin)
is proportional to the difference between outside and inside concentration
(Cout - Cin) and that the deposition rate is proportional
to the inside PM concentration (Cin).
dCin
----- = -l (Cin-Cout)-R
Cin
dt
-
l is the exchange coefficient, R the deposition
rate, both in hour-1 or day-1, in the same
unit.
-
l means how many times per unit time the volume
of the room is exchanged.
-
R is the fraction of indoor PM that is being deposited per unit time.
The same equation holds true for the SF6 experiment, except
that Cout is practically 0 and R is 0 as well:
dCSF6
----- = -l CSF6
dt
This differential equation has an analytical solution (a mathematical
function) while the other equation can only be solved numerically:
CSF6 (t) = CSF6(t=0) * exp (-lt)
-
1/l is the time after which CSF6
has dropped to 1/e (=1/2.718) of the initial value
-
ln(2)/l is the half-life of SF6 in
the room, or the time after which CSF6 has dropped to half
By fitting an exponential function to the measured SF6 data,
we can determine
l
and CSF6(t=0)
and use the same
l
to solve the differential
equation for PM.
The equation for PM can only be solved numerically:
(Cin (t+Dt) - Cin(t))
/ Dt = -l (Cin(t)
- Cout(t)) - R Cin(t)
or
Cin(t+Dt) = Cin(t)
+ Dt (-l (Cin(t)
- Cout(t)) -R Cin)
or
Cin(t+Dt) = Cin(t)
(1-Dt (l + R)) +
Cout(t) Dt l
We are basically calculating Cin at the end of the time step
Dt
from Cin at the beginning of the time step. Our best estimate
of Cout during the time step is actually the average of Cout(t)
and Cout(t+Dt). The final equation
then is:
Cin(t+Dt) = Cin(t)
(1-Dt (l + R)) +
0.5(Cout(t)+Cout(t+Dt))
Dt
l
In the lab, we will use this equation to calculate Cin as
a function of time and compare it to the measured data to find out if our
model does describe the experiment reasonably well.
