Accumulation
Part 2: Summations: Size
Distribution of Particles
In your worksheet you will
find data on the atmosphere over Pasadena, California, in August and September
of 1969. These data provide information about the size distribution of
particles (all in the PM10 range).
Suppose we let p
denote diameters of particles, and we let N(p) be a function whose
value at p is the number of particles of size less than or equal
to p in a cubic centimeter (cc) of air. N is called the size
distribution function, and its derivative dN/dp is called the
size density function. We have no way to measure N directly,
but we can measure dN/dp with reasonable accuracy in the following
way: By using filters for particles of sizes p and p + Delta-p
(the second one first), we can measure the number Delta-N of particles
between these two sizes. If the two sizes are close together, then Delta-N
/ Delta-p is approximately the value of dN/dp at the particular
size p. These approximate rate-of-change numbers are tabulated in
your worksheet.
- Plot the data for dN/dP
as a function of p. What do you see?
- It's very difficult to
get a good picture of data that vary as these do -- over nine orders of
magnitude. Experiment with shorter intervals on the p-axis to see
if you can get a sense of how the data vary.
- Another way to plot data
with such a large range of values is with a semilog graph. Do it. Now do
you get a sense of how the size density function varies?
- Zipf's Rank-Size Law,
originally formulated as a principle in linguistics, asserts that things
of widely varying sizes tend to be distributed like a decaying exponential
function. (Reference: Herbert A. Simon, "The Sizes of
Things," in Statistics: A Guide to the Unknown, edited by Judith
Tanur, Wadsworth & Brooks/Cole, 1985.) Are sizes of particles
in the atmosphere distributed according to Zipf's law? Explain.
- In each size interval [p_{1},p_{2}]
we can count the number of particles: Delta-N = (Delta-N/Delta-p)xDelta-p.
This is another instance of the familiar principle rise = rate times
run. How many particles are there of size between 0.00875 and 0.0125
microns in a cc of air? Of size between 0.07 and 0.09 microns?
- If we want to count all
the particles (of the sizes for which we have data) in a cc of air, we
could add up all the rises for all the size intervals. That would be very
tedious if you had to do it for one interval at a time. Follow the instructions
in your worksheet to carry out this calculation.
- How many of the particles
counted in the preceding step have diameters less than 2.5 microns? [A
small change in the command you used in step 6 will answer this.] What
percentage of the particles have diameters less than 2.5 microns
(i.e., would be subject to the PM2.5 standard)?
Send comments to the
authors <modules at math.duke.edu>
Last modified: November
18, 1997