Problem Of The
Month
December 2000—Screen Size
Probability Plots From Pulverized Data



W.S. Tyler Sieve


Mesh 
Inches 
Millimeters 
Mesh 
Inches 
Millimeters 
20 
0.033 
0.84 
20 
0.033 
0.83 
30 
0.023 
0.59 
28 
0.023 
0.59 
40 
0.0165 
0.42 
35 
0.016 
0.42 
50 
0.0117 
0.30 
48 
0.0116 
0.30 
60 
0.0098 
0.25 
60 
0.0097 
0.25 
100 
0.0058 
0.149 
100 
0.0058 
0.15 
140 
0.0041 
0.105 
150 
0.0041 
0.10 
200 
0.0029 
0.074 
200 
0.0029 
0.074 
325 
0.0017 
0.044 
325 
0.0017 
0.043 
ASTME1 lists typical screen sizes as 80, 100, 120, 140,
170, 200, 230, 270, 325, 400.
The vibrating screens at usually stacked with coarse screens on top and fine screens on the bottom. Mixtures of crushed particles are place on the top screen and descend by gravity/vibrations. When the screening test is over, the product retained by a specified screen size is calculated and often reported as a percent of the total charge weight to the top screen.
Consider the following screen data for pulverized coal:
• 99.5% of the material will pass
through a 50 mesh screen,
• 96.5% of the material will pass through a 100 mesh
screen, and
• 80.0% will pass through a 200 mesh screen.
This is the same as saying
• 0.5% of the material is held on a 50 mesh screen,
• 3% of the material is held on a 100 mesh screen, and
• 16.5% of the material is held on a 200 mesh screen.
Probability plots represent cumulative quantities versus
screen size. The cumulative data from above is
• 0.5% will be 50 mesh screen or coarser,
• 3.5% (0.5% + 3% = 3.5%) will be 100 mesh or coarser,
and
• 20% (0.5% + 3% + 16.5% = 20%) of the material will
be 200 mesh or coarser.
Data input for WinSMITH Weibull (using an Xdatum and Ypercentage) requires
1) clicking on the icon for “Method/Input”
(bottom row, third from the left) and
2) selecting the icon for“Probit 2” (bottom row, seven from the left).
The data is correctly input as
(screen size)*(cumulative affected quantity)*(quantity
sampled).
The above coal example is shown below in column format for easily copying into
WinSMITH Weibull from this page:
50*0.5*100
100*3.5*100
200*20*100
Again, as a reminder, it’s
• screen size (the Xvalue), then
• cumulative quantity (cumulative percentage), and finally the
• total quantity (if you’re working in percentages, then the total quantity is 100%).
Or (screen size * cumulative quantity * total quantity) as show above in the data file. Please note the data must be in rank order by screen size.
Since the data is represented in % of total weight, the data is easy to input. Did you notice that 80% of the total cumulative information does not exist from the sample?
The graph for the three data points is shown below as a Weibull plot.
Solving for the mesh size, the Weibull equation is (size) =
eta*(ln(1/1F(size)))^(1/beta). For
example, if we wanted to know the particle size where 90% of the particles
would be of larger screen size, we would calculate (size) =
349.9*(ln(1/(10.9)))^(1/2.738) = 474.5 mesh size. The same calculation for where 0.1% of the
particles would be of larger screen size is 28.1 mesh size. You can use the
“Predict” icon in WinSMITH Weibull to give you the facts from the
Weibull plot.
The values shown in the plot give the characteristic mesh
size as 349.8 (eta), the shape factor 2.738 (beta) and the coefficient of
determination 0.999 (r^2) says this straight line explains ~99.9% of the
variation in the data, and for 100 percentage points 80 are suspended data
(n/s). How much uncertainty exist in the statistical values?—of
course with only three samples, you would expect large variances but some data
is better than no data and you can fit confidence limits to the trend line by
use of WinSMITH Weibull software.
When engineers have their results explained in straight
lines on a XY plot, they can see the curve fit and comprehend the data. For engineers the relationship is simple: No
cartoon, no comprehension.
Demonstration versions of WinSMITH Weibull are available at no cost by mouse click here.
The authentic WinSMITH Weibull data file for use with the demonstration version or authentic version is also available by mouse click here which can be imported into the demonstration software without randomizing the data (as occurs when you input data into the demo software). This COAL.W (2.3K) is a ZIP file COAL.ZIP so it very small (0.8K).
In case you’d like to see the probability density function (PDF), this plot was made from the WinSMITH Weibull plot using WinSMITH Visual software:
Notice how the plot has a slightly longer tail to the right. Also notice the variability in the typical central tendencies—this is why use of the Weibull characteristic value eta is so handy for describing the skewed metric with a single value.
You can download a copy of this December 2000 file as a PDF
file by clicking
here.
Refer to the caveats on the Problems Of The Month Page about the limitations of the solutions. Maybe you have a better idea on how to solve the problem. Maybe you’ll find where I screwedup the solution and you can point out my errors as you check my calculations. Email your comments, criticisms, and corrections to Paul Barringer. Return to top of page by clicking here.
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