ME 405
Fall 2006
Professor John M. Cimbala Lecture 34
11/27/2006
Today, we will:
•
•
Discuss Particle Statistics in Section 8.2
Discuss an example particle size distribution in detail – see class handout
The data shown below are from the class handout, so you can follow along.
j = class
(bin
number)
D p ,min,j
(lower
limit)
D p ,max,j
(upper
limit)
D p,j
(middle
value)
ΔD p ,j =
class
width
nj =
frequency
(count per
class)
1
2
3
4
5
6
7
8
9
10
11
1
4
6
8
9
10
14
16
20
35
50
4
6
8
9
10
14
16
20
35
50
100
2.5
5
7
8.5
9.5
12
15
18
27.5
42.5
75
3
2
2
1
1
4
2
4
15
15
50
104
160
161
75
67
186
61
79
90
17
0
Minimum
diameter in
class j
Maximum
diameter in
class j
f (D p ,j ) = probability in this
n j /ΔD p ,j =
class =
count per n j /(ΔD p ,j n t )=
f
(D
)*ΔD
class width
p,j =
p ,j
fraction per
n j /n t
class width
Number of
particles in class j
(see Plot 1)
Width of class j:
ΔDp,j = Dp,max,j – Dp,min,j
Middle diameter in class j:
Dp,j = (Dp,min,j + Dp,max,j) / 2
34.667
80.000
80.500
75.000
67.000
46.500
30.500
19.750
6.000
1.133
0.000
0.0347
0.0800
0.0805
0.0750
0.0670
0.0465
0.0305
0.0198
0.0060
0.0011
0.0000
0.104
0.16
0.161
0.075
0.067
0.186
0.061
0.079
0.09
0.017
0
Probability in class j
Fraction of particles in class j
divided by class width:
nj/(ΔDp,jnt)
(see Plot 3)
Number of particles in class j
divided by class width:
nj/ΔDp,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
40
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
30
40
50
D p ,j
0.09
PDF = f (Dp ,j )
30
Dp
PDF = f (Dp ,j )
f (Dp ,j ) = fraction per class width
Dp
10
D p ,j 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