Which particle size distribution model to use?

TLDR version:

• To model the PSD of a SAG Feed use a Gaudin-Schuhmann model
• To model the PSD of a SAG Screen U/S use a Rosin-Rammler model
• To model the PSD of a Cyclone O/F use a Gaudin-Schuhmann model
• To model the PSD of a BM Discharge use a Rosin-Rammler model
• The PSD of a Cyclone U/F doesn't fit the models evaluated.

Particle size distributions are the bread-and-butter of comminution engineers. These are often interrogated to provide information like "what fraction of the material passes a particular size?", or "what is the size that 80% of the material passes?". One of the most common ways of doing these interrogations is just to pick two of the raw points that straddle the location of interest and do an interpolation (linear or otherwise) between those two points. This is generally fine, but moving to the "next level" means fitting a continuous & smooth model to the distribution.

The purpose of this Article is to look at sets of five particle size distributions from actual plant surveys, and observe which of three models demonstrates the best fit for each application.

The PSD models

Gaudin-Schuhmann model: This model roughly corresponds to a line plotted on a log-log graph.

• %P = (x/K)^m

Rosin-Rammler model: Also known as a Double-Weibull model, it roughly corresponds to a line plotted on a graph with a log X axis and double-log Y axis.

• %R = exp(-[x/D]^n)

Bond root-2 model: Fred Bond spent a lot of time examining PSD data from operating mines as part of the investigation that would lead to his Third Theory of Comminution, commonly called the Bond Work Index model. In the Third Theory, Bond says that the most common shape of a PSD is a Gaudin-Schuhmann model with slope m = 1/√2. One way to express this is to anchor at the 50th percentile of a PSD as follows:

• %P =0.5(x/P_50 )^(1/√2)

A less commonly used model is the Swebrec model, which was not included in this investigation. This model is, in my opinion, a rather brute-force fit that works for any size distribution. Since it fits anything, it doesn't give you any real insight into how "normal" your data is compared to other data sets. If all you want is to blindly interpolate data, then do consider the Swebrec model; but if you want to (for example) compare the slopes of primary crusher products across different projects, then use the Gaudin-Schuhmann model instead.

Nomenclature:

• x is an arbitrary particle size, measured in μm
• %R and %P are the cumulative percent retained or passing, respectively, expressed as a decimal (eg. 50% = 0.5, 25% = 0.25)
• P_50 is the size in μm that passes 50% of the material
• K, D, m, n are fitted constants.

Plant data

Summary of which PSD model works best for different process plant streams (click the table to download the full document):

Acknowledgements

• Munashe Kurisa did a lot of the regression number crunching on plant data.
• This work was derived in part from data donated by Centerra Gold and Newcrest to the SAG Conference Award Foundation for a student research program.
• This work was derived in part from data provided by MolyCop Adesur and several of my Clients.