Volume 2 Number 4 Dennis R. Dinger 1 February 2004
An Update
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The topic in this issue covers fundamentals that I frequently must repeat as part of explanations concerning particle size distributions.
Some PSD Fundamentals
Milling Equations
There are a variety of milling equations that are supposed to characterize particle size distributions. Different people have their own pet equations that supposedly define all PSDs. I want to weigh in on this subject.
Andreasen Equation
Andreasen developed this equation for PACKING, not for milling. His papers were published around 1930, and his equation is as follows:
CPFT = 100(D/DL)n (1)
where CPFT = Cumulative Percent Finer Than; D = particle size; DL = largest particle size; and n = distribution modulus. He was trying to determine the PSD required for maximum packing and he selected this power law equation because he wanted an equation that would produce the similarity condition that he felt was required for perfect packing. Power law equations have that characteristic. In today's English, Andreasen's similarity condition defines fractal behavior. (Fractal geometry was defined in the 1960s.)
Gaudin-Schuhmann Equation
In the 1940s, Gaudin and Schuhmann published their milling equation which is identical to the Andreasen Equation (Eq. 1 above). They weren't studying packing of particles, but MILLING and BREAKAGE of particles. To my knowledge, their equation is the main milling equation (or at least one of the main milling equations) used among the mineral processing industries. There are other equations and other allegiances, but Gaudin & Schuhmann have a lot of support in that industry. And the mineral processing industries have done much, much more milling research than anyone in the ceramics industries -- so they know a lot about milling.
Normal and Log-Normal Distributions
There are some who recommend the use of normal or log-normal distributions for milling. These forms of distributions are useful for random processes. In my opinion, milling of minerals is not the random process required to produce normal and log-normal distributions. Chemical preparations of ultra-fine powders and polymers, however, do appear to be such random processes. Many times the goal of the preparation processes for these materials is a narrow monodispersion of particle sizes. The narrow particle size histograms produced by such processes typically resemble narrow, normal or log-normal distributions.
Recommendations
In my experience, the Gaudin & Schuhmann equation fits mill products rather well. Deviations from the ideal appear when the DL sizes become small. Then, instead of linear CPFT plots on log-log axes, the CPFT distributions tend to curve down towards zero at the finest particle sizes, rather than following the theoretical straight lines predicted by the equations. Jim Funk and I have explained the reason for these deviations from the linear: the mill product distributions curve away from the theoretical line, not because the milling changes away from the linear or first-order case, nor because there are too few or too many particles in the fine particle size classes, but because there is a finite smallest particle size in all milling processes, Ds. No powders are present below Ds. Milling parameters can certainly change and add curvature to the supposedly linear distributions. Too few or too many particles can certainly be produced in any of the fine particle size classes, but even when milling proceeds perfectly, the CPFT line will always curve towards zero because there are no particles in the distribution below some finest size, Ds.
The Gaudin & Schuhmann and Andreasen equations both assume infinitely small particles are possible. The fact that infinitely small particles are not possible always produces an effective smallest particle size for each mineral being milled. The fact that there are no particles finer than some Ds in each distribution produces the curvature away from the predicted linear behavior. Our equation handles this. We added a Ds to the Andreasen (& Gaudin & Schuhmann) equation:
CPFT = 100 (Dn - Dsn ) / (DLn - Dsn ) (2)
All of these equations (#1 & #2) have linear histograms when plotted on log-log axes with appropriately defined geometric size classifications (see later topic below). But as mentioned, random events always occur during milling, and the results tend to be linear histograms (consistent with the predictions of the equations) plus a coarse tail and a fine tail. The tails usually appear normal in nature.
When mill product distributions are extremely narrow (most everything resides in only one or two size classes), the linear portion of the histogram becomes narrow, and for all practical purposes, the linear portion effectively disappears. Such distributions resemble a narrow, non-linear distribution with only a coarse tail joined to a fine tail. If your goal is to produce very narrow distributions, they may appear normal because the linear portion is missing and the two tails together will resemble a normal form of distribution.
If your goal is to produce broad distributions which pack well, then you should use our equation (#2) and pay attention to the broad, linear central portion of the histogram. In most such cases, we would ignore any coarse or fine tails and pay most attention to the central portion of the histogram.
Similar to Andreasen, we developed Eq. 2 for PACKING, but it applies equally to MILLING. The histograms produced by both equations are identical except histograms from Eq. 1 continue to infinitely small particles, while histograms from Eq. 2 stop at a finest size, Ds, usually in the sub-micrometer range. How small a particle can still be recognized as a particular mineral species? Sub-Angstrom particles of minerals are impossible because atoms are in those size ranges. The smallest possible powder particle sizes must start in the nanometer ranges because quite a few of each of the appropriate atoms are required to produce any particular mineral. Log axes can continue to meaningless small sizes such as 10-100 or 10-1000000 meters (and beyond). An effective Ds is therefore required whenever using Eq. 1 -- and the appropriate equation to use is Eq. 2.
All of you who use Eq. 1 for milling should be using Eq. 2 instead.
Log-Log Axes
I recommend log-log axes for all PSD charts. MS Excel® can easily produce log axes and log-log charts. If you routinely use either of the two equations above, log-log axes linearize the results. CPFT curves and histograms of Eq. 1 are linear on log-log axes. Histograms of Eq. 2 are linear on log-log axes.
The other benefit to log-log axes is that they show details of the PSDs over broad ranges of size and percent.
Log-log axes are especially useful for histograms of distributions following either of the equations above. Since perfect distributions of Eq. 1 or Eq. 2 produce linear histograms on log-log axes, it becomes relatively easy to glance at a histogram on a log-log chart and quickly determine how the PSD should be altered to produce the desired results. For example, if you want to improve packing, you will want to change the PSD to produce the correct slope on the chart, and you will then also want to add or remove powder from various size classes to better produce the desired straight-line histogram. Yes, this can be done using other types of axes -- I just think it is easier for the average person to understand when log-log axes (and straight lines) are used.
Histograms vs Cumulative Distributions
The information in a histogram is also present in a CPFT plot, and vice versa. Why does it matter which form is used? It matters because all PSD information is masked in a CPFT presentation. It is much simpler to understand a histogram than to decipher the exact same data plotted in CPFT form. For example, adding powder in a single size class to a histogram produces a peak in the histogram at that size class. It doesn't affect any of the amounts in any other size classes.
Adding powder to a single size class in a CPFT plot, however, affects the values at that size and all greater sizes. If you don't believe this, take any two CPFT distributions and try to predict the PSD of a mixture of 50% of each. Like I said -- sure, it can be done, but it's much easier to do using histograms.
Relationship Between Andreasen Equation and Histograms
The slopes of the CPFT curve and the histogram of the Andreasen Equation (#1 above), as well as the slope of the histogram of our equation (Eq. #2 above), are identical for constant distribution moduli, n. If you want to see different distribution moduli on a chart, the easiest way to plot the several lines with different distribution moduli is to plot several Andreasen CPFT curves (Eq. 1) with the desired moduli. Even if you're working with histograms, it's still easier to plot some Andreasen CPFT distributions to define the various distribution moduli, n, values. For each Andreasen distribution, plot one point at the DL and the second point at the smallest particle size on the chart. MS Excel® will connect the two points with a straight line. The slopes of the lines created in this way are identical to the slopes of the corresponding histograms.
If you need to add to the chart a movable line with a particular slope, n, (that is, with a particular distribution modulus) plot the Andreasen CPFT distribution with that slope. It doesn't matter which DL you chose, or even the nature of the size series used. The Andreasen line will quickly show the desired slope. Within MS Excel®, then use the drawing functions to create a straight line on top of the Andreasen distribution line. Then, click on the hand-drawn line, and you can move it around the chart as desired.
When you divide an Andreasen CPFT distribution into size classes to produce a histogram, as long as you use a proper geometric series of size classes (see next section) for the histogram, the slope across the top of the histogram will be identical with the slope of the original Andreasen CPFT distribution.
Geometric Series of Size Classes
An arithmetic series is one in which the difference between any two adjacent values is a constant. For example, 1, 3, 5, 7, ... is an arithmetic series. The difference between any two adjacent sizes in this series is 2. A geometric series is one in which the ratio between any two adjacent values is a constant. For example, 1, 2, 4, 8, 16, 32, ... is a geometric series. The ratio between any two adjacent sizes in this series is 2. One could also say that each size in this series is twice the previous size.
Multiplying by a constant value is the same as adding a constant log to the log of the previous value. For example, log 1 + log 2 = log 2; log 2 + log 2 = log 4; log 4 + log 2 = log 8; etc. Since log axes are based on logarithms, all proper geometric series will produce constant class sizes on log axes.
Therefore, when using log axes, ALWAYS use size classes chosen to fit a geometric series. It doesn't matter which geometric series you use -- the only requirement is that it MUST be a geometric series. Standard sieves fit a 4th root of 2 geometric series. Taking every other sieve out of the series produces a square root of 2 geometric series. Taking every fourth sieve out of the series produces a factor of two series. Etc.
So for particle size analyses, it is appropriate to use a 4th root of 2, a square root of 2, or a factor of 2 series to define the sub-sieve size classes. For automatic analyses, 4th root of 2 series are best. Even if particles in your production body are not coarse enough to be covered by sieves, it is still advisable to use a geometric series that continues the sieve size geometric classification.
When we first started using automatic particle size analyzers back in the late 1970s and we wanted to merge the data with sieve analyses, we frequently had to contend with sieve sizes such as 150μm, 75μm, 37μm, and automatic analysis results such as 40μm, 30μm, 20μm, 10μm, 9μm, 8μm, 7μm, ... Today's analyzers usually allow you to specify the size classes you want to use. There is no excuse any more to not be using sizes forming a proper geometric series. In the example just mentioned, the continuation of the 150μm, 75μm, 37μm, sieve series should be 18.5μm, 9.25μm, 4.63μm,2.31μm, 1.16μm, and 0.578μm. When the automatic analyzer allows you to specify the class sizes, you simply enter those values to the level of precision possible.
Also, when using a geometric series of class sizes, don't skip any classes! Sure, you can leave a broken screen out of a sieve stack and still perform the sieve analysis. But remember that the next smaller sieve (relative to the one that is missing) will contain approximately twice as much as all other screens because it now contains the powder that would have been on the missing screen, as well as the powder it would normally have collected. The same is true for missing size classes in automatic particle size analyses.
If the original PSD data is a detailed, complete CPFT curve, the histogram amounts in any size class can be calculated from it. Again, use all values in the appropriate geometric series. The interpolate function in the DRD Add-In Function set performs this calculation within MS Excel®.
Why is it important to use proper geometric series for size classes? In most cases, PSD histograms in MS Excel® or in other spreadsheet software are usually plotted using line charts (rather than the MS Excel® bar charts, which were designed for business users.) Histogram bars can be plotted using line charts (see for example the histogram line in Figure 1, of the E-zine, Vol 1, No 6) but they are much more difficult to successfully produce. Instead, they are frequently simplified by replacing each bar with a single point from the center-top of the bar. When that is done, the histogram will be shown as a line on the chart (without clear size class markings). If random size classes are missing from such charts, the persons reading the charts will have no idea that size classes are missing and data will be interpreted incorrectly. Results will then be disastrous.
Miscellany
I'm still looking for more suggested topics for future columns.
Until next time ...
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