Navigating Particle Size Distributions: A Guide to Choosing the Right Model for Your Mineral Processing Operations

Particle size distribution (PSD) is a critical parameter in mineral processing as it determines the efficiency of various separation and classification processes. In addition, PSD can also serve as a direct performance indicator in certain mineral processing operations, where the PSD of the final product is directly linked to its quality. Therefore, accurate determination and control of PSD is crucial for optimizing both the efficiency and quality of mineral processing operations. In this article, we will discuss the most commonly used models for PSD, the Gaudin Schumann and Rosin Rammler models, and how to determine which model to use based on curve fitting, R-squared value, and mean squared error (MSE).

While sieve analysis is a common method for determining PSD in mineral processing, the resulting data may not always be easily interpretable or usable. Often, the relationship between size and mass percent is not linear, requiring interpolation of the data to obtain specific screen sizes. To make this process more accurate, empirical relationships such as cumulative passing or retaining percentages are calculated from the sieve analysis data. However, different mineral processing operations may require different PSD models based on their specific characteristics and requirements. Therefore, different models have been developed over time to fit different datasets and provide more accurate PSD predictions. The most widely known and used models for PSD are the Gaudin Schumann and Rosin Rammler models, which we will discuss further in the following sections.

Gaudin Schuhmann Model

When dealing with skewed particle size distributions, the traditional linear plot of size against mass percent may not be appropriate. In such cases, a more useful plot can be obtained by taking the logarithm of both the cumulative undersize and the screen aperture, and then plotting these values against each other on a linear axes or directly plotting original values on a log–log paper. This type of plot is commonly known as a Gaudin-Schuhmann plot and can often result in a straight line, providing a linear relationship between the two variables. The GGS model can be expressed by the following equation, where x represents the particle size and P denotes the cumulative percentage of particles passing through the sieve.

Rosin Rammler Model

For finely ground particles, such as those produced in tumbling mills, an alternative plot can be used to analyze particle size distribution. This plot involves taking the natural logarithm of either ln[ln(100/(100-P))] or ln[ln(100/R)], where P is the cumulative % passing and R is the cumulative % retained, and plotting against the ln of sieve size. These types of plots are commonly referred to as Rosin-Rammler plots. The double log scale expands the fine and coarse ends of the size range (<25% and >75%) and compresses the mid-range (30-60%). In most operating plant conditions, approximations in particle size computation and estimation are sufficient for most purposes. The Rosin–Rammler (or Weibull) distribution is expressed as;

A straight line can be obtained when plotting ln[ln(100/R)] versus ln(x). The slope of the straight line and the intercept at R = 36.8 can be used to obtain the parameters of the Rosin-Rammler distribution, m and x63.2, respectively. These parameters completely describe the size distribution. To simplify the calculation of the double log, special graph paper, known as Rosin Rammler/Weibull paper, is available, which allows for the direct plotting of cumulative % retained (or cumulative % passing) values on the Y-axis. A line is included on the graph paper at a cumulative % retained value of 36.8 to facilitate the estimation of the parameter x63.2.

Determining which Model to Use

To determine which model to use, we need to perform curve fitting to determine the best-fit parameters for each model. Curve fitting involves adjusting the model parameters to minimize the difference between the measured and calculated PSD values. We can use Excel Solver, trendline option over Excel chart or Linest function in Excel to determine model parameters.

We can compare the goodness of fit for each model by calculating the R-squared value and the mean squared error (MSE). R-squared is a statistical measure that represents the proportion of the variance in the dependent variable that is explained by the independent variable(s). The closer the R-squared value is to 1, the better the model fits the data. MSE, on the other hand, measures the average of the squared differences between the measured and calculated values. A lower MSE value indicates a better fit.

Step-by-Step Tutorial

Let's walk through a step-by-step tutorial to determine the PSD using the Gaudin Schumann and Rosin Rammler models. Suppose we have the following sieve analysis data:

We can calculate the required data to linearize the size distribution according to different models as follows.

The GGS model parameters can be found by plotting Ln(x) versus Ln(P) values on the millimetric axis as follow. The slope of the curve will give the parameter m, and the x_max parameter can be calculated by rearranging the constant term in the model.

To determine the Rosin Rammler model parameters, plot the Ln(x) versus Ln[Ln(100/R)] values. Then, find the m and x63.2 parameters using a similar method as before.

The calculated model parameters are summarized in the following table;

If you want to see the Gaudin Schuhmann distribution on the log-log axis and the Rosin Rammler distribution on the Weibull graph paper, here is the results;

Here is the summary of the measured and calculated cumulative passing percentages and their graphical representations:

Based on the results, it is more appropriate to choose the Gaudin Schuhmann function as it has a higher R-squared value and a lower MSE value. By using this model, the size distribution can be accurately described, and various inferences or simulations can be made.

Conclusion

Accurate determination and control of particle size distribution (PSD) is crucial for optimizing the efficiency and quality of mineral processing operations. While sieve analysis is a common method for determining PSD, it may not always be easily interpretable or usable. Therefore, different PSD models, such as the Gaudin Schuhmann and Rosin Rammler models, have been developed to fit different datasets and provide more accurate predictions. By understanding the characteristics and requirements of different mineral processing operations, one can determine which model to use based on curve fitting, R-squared value, and mean squared error (MSE) analysis. Implementing the correct PSD model can lead to improved process efficiency, product quality, and overall profitability in mineral processing.