The log-normal curve offers its own perspective on particle distribution.
The purpose of this article is to present an alternative method to a linear segmented granulation curve often used to describe the particle size distribution of mill stocks.
Having access to the granulation profile of mill stocks is essential in mill design and operation. An established planned granulation profile allows the miller to identify the planned material flow needed to size material handling and processing equipment properly.
Mill operators can monitor changes or shifts in granulation curves that may indicate incorrect equipment settings or wear, in addition to unplanned changes in the mill mix. The granulation curve is a useful tool in establishing mill balance whether from a design or operational perspective.
Traditionally, a linear segmented granulation curve is used to describe the granulation profile of a stock in the mill flow or, in some cases, mill products such as wheat bran or whole wheat flour, where particle size differences may impact a consumer product significantly.
Granulation Curve Example
An example of a granulation curve for ground second break stock taken under the rollstand in a hard red winter wheat mill is provided in Figure 1 below. The points shown are the average of four samples taken from under the second break rollstand.
In addition to second break stock prior to the roll, as well as samples from fourth break, coarse sizing, fine sizing, first middling, and third middling prior to and after grinding operations were evaluated. The samples were screened on a Ro-Tap sifter and U.S. standard sieves to determine the size distribution. The entire sample taken from the mill was used for the sieve test to eliminate the effect of segregation in the sample bag.
The screen test was conducted for two minutes with the Ro-Tap hammer striking the top of the sieve stack. The screen stack employed was selected to minimize the amount of material over the top screen and below the bottom screen. The material remaining on each screen was weighed and the data recorded with the average values for ground second break stock shown in Table 1 on p. 33, in addition to the natural logarithm of the screen size.
Evaluating a potential change in sifter clothing, one must identify the slope of the line between two known points inclusive of the size change proposed to interpolate the change in stock mass flow. The process is simple linear interpolation
The segments between known screen test points are not appreciably different in this example but can be quite exaggerated when looking at a variety of mill stocks. Evaluating a potential change in sifter clothing, one must identify the slope of the line between two known points inclusive of the size change proposed to interpolate the change in stock mass flow. The process is simple linear interpolation.
Log-Normal Curve
Engineers working in the feed and milling industries often represent the granulation of mill stocks using the log-normal curve. A log-normal plot using the same data can be used to produce a single linear line to represent the granulation profile of the same product as shown in Figure 2 p. 34. Instead of the natural logarithm to the base 10 as implied by the abbreviation “log,” the natural logarithm is employed with the base, a fixed irrational number approximately equal to 2.718281828459.
The scale for both axes is logarithmic, using the natural logarithm designated as Ln. The logarithm of the screen opening is shown on the vertical axis, while the cumulative percent is shown on the horizontal axis. The linear equation shown on the graph for ground second break stock has a significantly strong R2 value of 0.9949. The labels identify the X/Y pair for the screen size test point identified. Using the equation, one can plug in a desired X or Y value to calculate the missing variable. As with the traditional linear approach, the difference between cumulative material passing through two different screens will identify the present of material between the two screens of interest.
Probit transformations allow one to conduct a regression analysis to determine if the points fell on a straight line using the logarithmic scale.
Analysis of the data is made to determine if the particle size distribution is normal, allowing for development of a useful linear relationship. The cumulative percent through a particular screen is transformed to a probability unit called “Probit.” The Excel statistical function, NORMINV(probability, mean, standard deviation), calculates the Probit value using the cumulative proportion through the mean equal to zero and the standard deviation equal to one for material on each screen by returning the inverse of the normal cumulative distribution function. The natural logarithm of the screen size opening in microns was used to calculate Ln (microns).
Probit transformations allow one to conduct a regression c analysis to determine if the points fell on a straight line using the logarithmic scale. Statistical analysis and residual plots indicate various mill stock particle sizes have log-normal distribution, and the regression equations could be used to determine the mass median diameter (MMD), and the geometric standard deviation (GSD).
Note the cumulative percentage of material held on the pan has not been included in this analysis, because the exact cutoff and size and mass contribution between the smallest screen and the pan is unknown. An assumption of near zero size and near unity mass above this size creates limits on the ability to fit a straight line and places too much weight on an area of particle size for which there is little interest. Generally, the particle size distribution of flour fewer than 150 microns is not useful from a practical operational milling or engineering standpoint.
The typical R2 value for each stock evaluated is high, indicating the model explained a high proportion of the observed variation. The regression equation can be used to calculate MMD = d50 and GSD. The key information in this report is the value of the F test statistic and the P value. For all tests, the F statistic was quite large, and the P value quite low, indicating the model fit the data well.
Regression equations at the 50% and 84% levels are used to determine MMD and GSD. The Probit for 50% is zero, making the MMD or d50 equal to EXP (intercept). The MMD data for the various stocks followed expected trends, with stock samples taken below the roll having smaller MMDs than samples taken above the roll. This makes sense, because the rolls are grinding the material thereby reducing the particle size. The Probit for 84% is 0.99 and was used to calculate d84. The ratio of d84/d50 is the GSD. These data are summarized for the various stock samples evaluated in Table 2 below for the stocks identified.
Conclusions
Ground second break stock, in addition to second break stock prior to the roll, as well as samples from fourth break, coarse sizing, fine sizing, first middling, and third middling prior to and after grinding operations were shown to have log-normal distribution.
The log-normal distribution and the linear regression developed using this approach may be easier to use and more accurate in developing a mass balance model of mill flow. The log-normal distribution may be an improvement over the linear granulation curve and appears to be adequate for milling purposes. Effort should be made to integrate this analysis technique in mill flow development and management.
So far, there is no mathematical granulation profile method that reflects quality characteristics of various size particles. We still need millers with a keen eye for detail and sound judgment.
While graphical analysis and mathematical models for particle size distributions are useful, they still do not tell us about the characteristics of the product. In milling, we must concern ourselves with both the quantity and mass flow rate of material, as well as the quality of the material. So far, there is no mathematical granulation profile method that reflects quality characteristics of various size particles. We still need millers with a keen eye for detail and sound judgment.
Dr. Jeff Gwirtz is CEO of JAG Services, Inc., an international consulting company in Lawrence, KS; 785-341-2371; jeff@jagsi.com. He also is adjunct professor in the Department of Grain Science and Industry at Kansas State University, Manhattan.
