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Tutorial 1

This document contains data on the size distribution of particles from 0-16 microns in size. It includes the number of particles, average size, mass fractions, and cumulative distributions. The data fits a lognormal distribution curve with a geometric mean of 3.8 microns. The median size is calculated as 6.5 microns, the mode is 6 microns, and the mean size is 6.65 microns based on the mass distribution. Charts of the mass fraction and cumulative distributions are included to determine the median, mode and mean.

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Aakriti Bhandari
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71 views4 pages

Tutorial 1

This document contains data on the size distribution of particles from 0-16 microns in size. It includes the number of particles, average size, mass fractions, and cumulative distributions. The data fits a lognormal distribution curve with a geometric mean of 3.8 microns. The median size is calculated as 6.5 microns, the mode is 6 microns, and the mean size is 6.65 microns based on the mass distribution. Charts of the mass fraction and cumulative distributions are included to determine the median, mode and mean.

Uploaded by

Aakriti Bhandari
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as DOCX, PDF, TXT or read online on Scribd
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1.2 b.

From

To

0.00
2.00
2.52
3.18
4.00
5.04
6.35
8.00
10.10
12.70

2.00
2.52
3.18
4.00
5.04
6.35
8.00
10.10
12.70
16.00
Sum =
Assuming a Sphere,

Average
number(ni) nixi
nixi^2
nixi^3
(xi)
1.00
2738.00
2738.00
2738.00
2738.00
2.26
5108.00
11544.08
26089.62
58962.54
2.85
5396.00
15378.60
43829.01
124912.68
3.59
5510.00
19780.90
71013.43
254938.22
4.52
5109.00
23092.68 104378.91
471792.69
5.70
5151.00
29334.95 167062.51
951421.00
7.18
3869.00
27760.08 199178.54 1429106.01
9.05
1799.00
16280.95 147342.60 1333450.51
11.40
329.00
3750.60
42756.84
487427.98
14.35
24.00
344.40
4942.14
70919.71
61.89
35033.00 150005.23 809331.60 5185669.34

Calculating the dSn , dVn and dSvn .


Number Basis,

dSn =

DVn =

dSVn =

From
0.00
2.00
2.52
3.18
4.00
5.04
6.35
8.00
10.10
12.70

To
2.00
2.52
3.18
4.00
5.04
6.35
8.00
10.10
12.70
16.00

Average
number(ni) nixi^4
nixi^5
nixi^6
(xi)
1.00
2738.00
2738.00
2738.00
2738.00
2.26
5108.00
133255.35
301157.08
680615.01
2.85
5396.00
356001.13
1014603.23
2891619.21
3.59
5510.00
915228.20
3285669.24
11795552.57
4.52
5109.00
2132502.96
9638913.36
43567888.40
5.70
5151.00
5418342.62
30857461.23 175733241.68
7.18
3869.00
10253835.63
73571270.64 527873866.84
9.05
1799.00
12067727.09 109212930.18 988377018.13
11.40
329.00
5556678.93
63346139.76 722145993.27
14.35
24.00
1017697.82
14603963.78 209566880.19
37854007.73 305834846.50 2682635413.31

Derivation of the weight basis formula is included in the other page.


Weight Basis,
dSw =

DVw =

dSVw =

1.2.c
Showing that the size distribution follows the log-normal Distribution:
From(m)

0.00
2.00
2.52
3.18
4.00
5.04
6.35
8.00
10.10
12.70

To (m)

2.00
2.52
3.18
4.00
5.04
6.35
8.00
10.10
12.70
16.00

Average
1.00
2.26
2.85
3.59
4.52
5.70
7.18
9.05
11.40
14.35

Number

2738.00
5108.00
5396.00
5510.00
5109.00
5151.00
3869.00
1799.00
329.00
24.00

fn(x)

0.07815
0.14581
0.15403
0.15728
0.14583
0.14703
0.11044
0.05135
0.00939
0.00069

Cumul.Under

0.07815
0.22396
0.37799
0.53527
0.68110
0.82813
0.93857
0.98992
0.99931
1.00000

Cumul.Over

1.00
0.92
0.78
0.62
0.46
0.32
0.17
0.06
0.01
0.00

Particle Size (um)

Arithmetic Normal Dis


18.00
16.00
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00

Series1

Cumulative Undersize

As the graph produced from the Logarithmic Normal Distribution is linear for Particle size vs
Cumulative undersize, the data given follows the log normal distribution.
Calculating the Geometric Mean using the graph from the previous page
D50 = 3.80
D16 = 2.25
D84 = 6.4
=

or, =

= 1.68
=

= 1.68

Therefore, = 1.68
And
Dgn = 3.80
Calculating
Dan, dSn , dVn and dSvn
(

( ( ))

( ( ))

( ( ))

( ( ))

1.4 a
Calculating the median, mode and mean of the mass distribution
From

To

0
1
3
5
7
9
11
13

frequency(ni) Average(xi)

1
3
5
7
9
11
13
0
sum =

0
5
8
12
8
6
4
0
43

0.5
2
4
6
8
10
12
6.5

Mass
fraction
fn(x)

cumulative
under

0
0.116279
0.186047
0.27907
0.186047
0.139535
0.093023
0

0
0.11627907
0.302325581
0.581395349
0.76744186
0.906976744
1
1

cumulative over

fn*xi

1
1
0.88372093
0.697674419
0.418604651
0.23255814
0.093023256
0
Mean =

0
0.232558
0.744186
1.674419
1.488372
1.395349
1.116279
0
6.651163

To calculate mode
0.3

mass fraction, fn

0.25
0.2
0.15
0.1
0.05
0
0

10

12

14

Size (um)

From the graph above, the highest mass fraction occurs when the size is 6

Cumulative

To calculate the median


1.2
1
0.8
0.6
0.4
0.2
0

Undersize
Oversize
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Size(micron)

The above plot of cumulative mass fractions vs size gives a d50 = 6.5m

. So, the median is

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