M E 433
Professor John M. Cimbala
Lecture 38
Today, we will:
•
Final Topic of the Semester: Discuss particle statistics: grouped data vs. listed data,
histograms, PDFs, and cumulative distribution functions
•
Analyze in detail a sample of polluted air from Hinds’ air pollution book. See class
handout, and also the posted Excel spreadsheet on the course website.
Example: Converting from mass distribution to number distribution
Given: A cascade impactor is used to sample air in a city during a temperature inversion.
Tray number 5 of the impactor weighs 13.20 mg clean. After the sample is taken, the same
tray (with collected particles) weighs 13.32 mg. This tray (Tray 5) has been calibrated to
collect particles between 4 and 6 microns. Assume unit density spherical particles (ρp = 1000
kg/m3), and that the air is at STP, ρ = 1.184 kg/m3, µ = 1.849 × 10-5 kg/(m s).
To do: Estimate the number of particles collected on Tray 5. Hint: We approximate all the
particles on Tray 5 as having the middle diameter in its range, i.e., 5 microns (half-way
between 4 and 6 microns). Give your answer in millions of particles to 3 significant digits.
Solution:
The data shown below are from the class handout, so you can follow along.
Minimum
diameter in
class j
Maximum
diameter in
class j
Number of
particles in class j
(see Plot 1)
Width of class j:
∆Dp,j = Dp,max,j – Dp,min,j
Middle diameter in class j:
Dp,j = (Dp,min,j + Dp,max,j) / 2
Probability in class j:
f(Dp,j)*∆Dp,j
also = nj/nt
Fraction of particles in class j
divided by class width:
nj/(∆Dp,jnt)
(see Plot 3)
Number of particles in class j
divided by class width:
nj/∆Dp,j
(see Plot 2)
90
200
180
160
140
120
n j 100
80
60
40
20
0
Plot 1 - Simple
histogram
Plot 2 - Modified
histogram
count per class width
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
0
10
20
50
0.09
0.09
0.08
0.08
Plot 3 - Histogram of
fraction per class width
0.07
Plot 4 - Probability
density function (PDF),
linear scale
0.07
0.06
0.05
0.04
0.03
0.06
0.05
0.04
0.03
0.02
0.02
0.01
0.01
0
0.00
0
10
20
30
40
50
0
10
20
Dp
F (Dp ,j ) = cumulative dist. function
0.08
Plot 5 - PDF,
log scale
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
1
D p ,j
30
40
50
D p ,j
0.09
PDF = f (Dp ,j )
40
Dp
PDF = f (Dp ,j )
f (Dp ,j ) = fraction per class width
Dp
30
10
on a log scale
100
1.2
1
0.8
0.6
0.4
Plot 6 - Cumulative
distribution function
0.2
0
0
10
20
30
D p ,max,j
40
50
How to calculate the final column on the class handout, the cumulative distribution function:
Skipped
some
columns
.
.
.
...
Plot the cumulative distribution
function against maximum
diameter in class j, not against
minimum diameter!
Cumulative
distribution
function