Volume 2 Number 6 Dennis R. Dinger 1 April 2004
Short Courses to be offered again in June 2004
Feedback from participants in previous short courses has been excellent. Quite a few have been asking when these courses will again be offered. So I have made plans to offer them during the week of 21-25 June 2004. The Short Course Announcement has been distributed to all subscribers. Courses offered will be:
Course 1 -- 21-23 June 2004
Fine Particle Processing Using Predictive Process Control
(PPC) Principles
$1295.00 for the 3-day
course
Course 2 -- 24 June 2004
Performing Particle Calculations in
MS Excel®
Using the DRD Add-In Functions
$ 595.00 for the 1-day course
Course 3 -- 25 June 2004
Rheology for
Ceramists
$ 595.00 for the 1-day course
Discounts are available for a single attendee taking more than one course, for multiple attendees from the same company, and for early registration and payment. The Short Course Brochure and registration forms are available at the web site. Check there for full details.
• If you have never attended one of these courses, sign up now.
• Those of you who have attended in the past and found it to be a valuable course ... Help to spread the word! ... encourage your employees, bosses, business associates, engineers, technicians, and anyone else who could benefit from these courses to attend.
• If it has always been your intention to attend one of these courses, but you've been putting it off ... here's another chance for you to attend.
• If you attended one of these courses in the past and you want to refresh your understanding, here's another chance for you to attend.
• If you live outside the US (as many subscribers do), here's a chance for you to visit the US to attend some good courses.
June in Clemson is a beautiful time of the year. Summer will be in full swing. Golf courses and outdoor activities in this area abound. Come early! Bring the family! Spend the weekend, attend some courses, stay another weekend, and make it a fun, educational week in sunny SC.
All subscribers to the E-zine will received the complete Short Course Brochure as Supplement 2 of this E-zine. If someone forwarded this issue to you, and you did not receive the announcement, click here: Short Course Brochure. To anyone interested, the registration form and hotel information form are available at the web site and through direct links on the announcement.
Feel free to forward this issue of the E-zine and/or Supplement 2 to anyone who might be interested ... or simply point everyone to the Dinger Ceramics web site for all details.
Thanks.
An Update
As with all other issues of the E-zine, please forward this issue to any ceramists or materials engineers who might be interested. Or simply point friends and associates to the Dinger Ceramics web site.
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The two books, Rheology for Ceramists and Particle Calculations for Ceramists, can be purchased at the Books and Downloads page of the web-site. Quantity discounts are available on the paperback books. If interested, please contact me for details. Downloadable versions of each book are also available at the web-site.
The following topic is the result of an excellent suggestion from a reader.
Particle Size
Distribution Analysis
Using Normal and
Log-Normal Distributions
Some companies use normal and log-normal distributions to characterize their particle size distributions. The questions were asked, "How do we calculate particle size distribution parameters if we use these distributions? How do we perform the calculations if we have only the mean and standard deviation for each of our distributions?" We will discuss this topic briefly.
The DRD Add-In Functions (available at the web site) have the functions necessary to calculate a variety of powder properties from measured particle size distribution data. These functions and spreadsheets can be used with any type of particle size distribution -- as long as the distributions are divided into size classes defined by square root or fourth root of two size series (or any other complete, valid geometric size classification.) Any and all particle size distributions can be defined in this way, so this poses no problems to using normal or log-normal distributions.
Normal and Log-Normal Distributions
The equation for the frequency distribution of a normal distribution is:
Y = (1/(σ*√2π))*e-½(X-μ)^2/σ^2 (1)
where Y = probability (or frequency), X= deviation from the mean, μ = mean, σ = standard deviation. The integral of this function from 0 to X is the cumulative form of the normal distribution.
The equation for the histogram of a log-normal distribution is similar, but in this case Z = log X:
Y = (1/(σz*√2π))*e-½(Z-μz)^2/σz^2 (2)
where Y = probability (or frequency), Z= deviation from the mean of Z, μz = mean of Z, σz = standard deviation of Z. (T. Allen, Particle Size Measurement, 3rd Ed., Chapman and Hall, New York, pp. 133-139,1981.) The integral of this function from 0 to Z is the cumulative form of the log-normal distribution.
Figure 1 shows a normal distribution with mean = 0.0, and standard deviation = 1.0.
Figure 1. Normal Distribution
In this figure, with this particular mean and standard deviation, the X values are equivalent to the number of standard deviations of each X value from the mean. This shows that essentially all of the data points falls within ±4 standard deviations of the mean. 68.27% of all values fall within ±1 standard deviation, 95.45% fall within ±2 standard deviations, and 99.73% fall within ±3 standard deviations of the mean.
Figure 2 shows a log-normal distribution with mean = 1μm, and standard deviation = 10μm.
Figure 2. Log-Normal Distribution Shown On Semi-Log Axes
Whereas both X and Y axes in Figure 1 were linear axes, the X axis in Figure 2 is a log axis. The Y axis is similar to the Y axis in Figure 1. These two figures show the relationship between normal and log-normal distributions. When plotted on appropriate axes, the two types of distributions have identical shapes. But the log-normal distribution is actually quite different from the normal distribution.
Figure 3 shows the log-normal distribution of Figure 2 plotted using linear axes for both X and Y.
Figure 3. Log-Normal Distribution Shown on Linear Axes
The log-normal distribution may appear to be identical to a normal distribution (Figures 1 & 2) but it is actually skewed severely, as shown in Figure 3.
Before proceeding any further, I need to add a disclaimer: I'm not recommending that anyone should use normal or log-normal distributions. But if you do ... or if you must ... here's how to work with them.
MS Excel® Functions
There is a function in MS Excel® called NORMDIST. Its form, with arguments, is as follows:
NORMDIST ( X, Mean, Std Dev, Cumulative ).
All three charts above were created using this function. The first argument, X, is the particle size value of interest. The next two arguments are the mean and standard deviation values. The final argument, which is named Cumulative by the MS Excel® spreadsheet program, allows either the cumulative form of the normal distribution, or the frequency distribution form of the normal distribution (as shown in the three figures) to be calculated. This is a logical argument. If you enter the value true for this argument, the function calculates the cumulative distribution. If you enter the value false, the function calculates the frequency distribution.
The arguments for X, Mean, and Std Dev, may take the form of actual numbers, or they can be cell references which point to the spreadsheet cells which contain the X, mean, and standard deviation values. The Cumulative argument must be either the word true or the word false.
When the cells containing the NORMDIST functions are formatted for Numbers, all values returned from this function will be between 0.0 and 1.0. If percentages are desired, the cells containing the NORMDIST functions can be formatted to show percentages (using the Format menu, Cells, Number, and then select Percentage.) Alternatively, if normal numerical values for percentages between 0% and 100% are desired, the results from all NORMDIST functions can be multiplied by 100. This decision is ultimately up to the individual. Both methods produce identical answers.
Producing Distributions from Known Means and Standard Deviations
To produce normal distributions from known mean and standard deviation values, the function arguments should be filled as follows: X should be the particle size (μm) for which the distribution is to be calculated; the mean should be the measured particle mean (μm); and the standard deviation should be the measured standard deviation (μm).
For example, if you want the cumulative value of a normal distribution when X=20μm, the mean is 40μm, and the standard deviation is 20μm, the function to use in each cell is as follows:
=NORMDIST(X,40,20,true)
where X can be a numeric value or a cell address pointing to the cell containing the X particle size value.
To produce log-normal distributions from known mean and standard deviation values, the function arguments should be filled as follows: the X value should be the log of the particle size; the mean should be the log of the measured particle mean (μm), and the standard deviation should be the log of the measured standard deviation (μm).
For example, if you want a log-normal cumulative distribution value for X=50μm, when the mean is 100μm, and the standard deviation is 10μm, the function to use in each cell is as follows:
=NORMDIST(Z, 2, 1, true)
where Z is either the numerical value of the log of X, or it can be a cell reference pointing to the cell that contains the log of X; the second argument (the mean) is 2, which is the log of 100 (the log of the measured value of the mean); and the third argument (the standard deviation) is 1, which is the log of 10 (the log of the measured value of the standard deviation.)
Particle Size Calculations
To perform all other particle size distribution calculations for normal distributions, set up your particle sizes in a 4th root of two series. Then calculate the cumulative distribution values (as percentages) for each particle size in the distribution. To produce the cumulative values, the fourth argument in the NORMDIST functions must be true. From the cumulative distribution results, histogram values can be calculated for each size class, and from the histogram values, all other values (such as surface areas, numbers of particles, and expected minimum porosities) can be calculated using the DRD Add-In Functions, or other equivalent calculations.
Don't use the NORMDIST function to calculate frequency values (thinking that they are histogram values. They are not!) The cumulative distribution is the integral of the frequency distribution. You can easily and accurately calculate cumulative distribution values using NORMDIST, and you can easily calculate histograms from the cumulative distribution values. (The histogram value for a size class is the difference between the cumulative values at the sizes that define the size class.) It is not easy to calculate cumulative distributions from NORMDIST frequency distribution results.
Trust me. The answer to this is long and it involves calculus (you know ... derivatives and integrals) which most engineers don't like. When working with particle size distributions, only use the NORMDIST function to calculate cumulative values; use the cumulative values to then calculate histogram values; and all will work fine.
To take this a little further, we'll look at individual cell formulas needed to produce a cumulative particle size distribution. This is a homework assignment for each of you who are interested in normal distributions. I'll walk you through it.
Place the mean value into cell B1 and the standard deviation into cell B2. Put the particle sizes (in a square root or fourth root of two series) into cells A10 through A30. Then, we'll put the cumulative distribution values into column B. Cell B10 should then contain:
=NORMDIST(A10,$B$1,$B$2,true)*100.
This will produce the cumulative distribution value in percent (number values from 0 to 100). Note that the second and third arguments should be fixed cells (using $ characters) so they won't change when this cell is dragged down the column. In this form, the contents of this cell can then be dragged down the column (from B10 to B30) to calculate all other cumulative particle size distribution values for this PSD. Column C can then be used to hold the calculated histogram values. Assuming particle sizes are from coarse at the top to fine at the bottom of column A, the formula for the histogram value for the first class size (in C10) should be:
=B10-B11
The contents in this cell (C10) can also be dragged down the column (from C10 to C29) to fill in all other histogram values. Since histograms are differences between cumulative values, histograms should always contain one less cell than the corresponding cumulative distributions.
If you are going to use this procedure for PSD calculations, pick a common distribution and play with the NORMDIST function in MS Excel® and make sure you understand all the details. Check the results with a calculator, if necessary. Then, use the successful test spreadsheets as the basis for your calculations on process data.
Conclusions
It's not necessary for anyone today to use the nasty formulas shown above. Just use NORMDIST in MS Excel®. It does all the work for you. It's quite easy to use to calculate cumulative distributions from measured means and standard deviations. Once cumulative distributions have been calculated, histograms and all other particle size distribution parameters can easily be calculated.
Miscellany
This topic turned out to be quite a bit easier to explain than I originally expected. The ease is due to the fact that MS Excel® contains a function to produce normal distributions. In preparation for this article, I started by searching my bookshelf for all of my statistics books. I found one. Then I found the Particle Size Measurement book by T. Allen which contained equations and discussion of normal and log-normal distributions. Finally, (and fortunately) I looked through the function list in MS Excel® and sure enough, there was the NORMDIST function. At that point I realized the calculations would be easy for anyone to perform.
Good suggestions are arriving for future column topics. This article was from one of them. Please continue to send your ideas. Thanks.
Until next time ...
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Copyright © 2004 Dennis R Dinger
103 Augusta Rd, Clemson, SC 29631 (864) 654-3155
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