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.

1. Plot the data for dN/dP as a function of p. What do you see?
2. 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.
3. 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?
4. 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.
5. In each size interval [p₁,p₂] 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?
6. 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.
7. 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)?

Last modified: November 18, 1997