Many power-based grinding models exist, and most operators are familiar with Fred Bond’s “third theory”. Bond’s model is most commonly used to describe primary and secondary grinding to product sizes above, for example, 100 µm. Operators sometimes use Bond’s equation to describe grinding in situations where it is not appropriate, such as fine grinding below 50 µm. Using an alternative model would be a better choice in this situation.

Bond’s equation is one in a large family of models. Related equations better suited to fine grinding include the “signature plot”, Von Rittinger’s model and Charles’ equation. These models have a similar form to Bond’s and can be fit to industrial regrind mills and laboratory tests using the simple regression tools in the charts of computer spreadsheets. Operators will find that fine grinding calculations are both easier and more accurate when using the alternative equations to fit their regrind milling surveys or when performing laboratory regrind mill scale-up for plant design.

None of these models are new: Charles’ equation was published in 1957 and Von Rittinger’s model was proposed in 1867. A quick refresh how to apply these equations can turn these oddities from the undergraduate curriculum into useful tools for plant optimization. They can be fit to any type of mill including stirred and tumbling ball mills.

</p>
<p>Building a case for fine grind modelling requires reviewing some theory before building a framework for doing fine grinding calculations. The actual calculation procedure recommended is easy and can be applied by plant metallurgists. Don’t be frightened by the theory; the end result is reasonably<br />
simple.</p>
<p>The observation that mineral breakage is related to the absorption of energy is usually attributed to Von Rittinger (1867). Lynch and Rowland (2005) describe that the ability to empirical test and calibrate energy models of breakage didn’t exist until the widespread adoption of electric motors in the mining industry during the 1930’s and 1949’s. A series of technical papers published by the Allis Chalmers company during the 1940’s demonstrate that they were attempting to do such a calibration using Von Rittinger’s model which is usually stated as: “the energy consumed in the size reduction is proportional to the area of new surface produced.”</p>
<p>The calibration effort ultimately failed, and in 1952 an employee of Allis Chalmers named Fred Bond proposed a different model which came to bear his name (Lynch & Rowland, 2005). Bond’s work index model was empirically fit to the wide variety of data collected over the previous decades and was particularly focused on the ball and rod mills that were in common use at the time (Bond, 1952).</p>
<p>Bond’s model wasn’t immediately accepted by the mineral dressing industry, and its applicability was debated for about a decade after publication. Alternative models appeared, such as the model by Charles (1957) where an ore-specific coefficient and exponent on the size term would be measured. The debate largely settled when Hukki (1962) published a reconciliation of the major competing models of the time and suggested that all models were valid, but each within a certain size range.</p>
<p>A crucial observation of Hukki is that Bond’s equation is generally only valid in the range of primary and secondary grinding (product sizes between 1 mm – 100 µm), and that different models apply below this size range. This paper will discuss how to use those alternative models in an industrial setting.</p>
<p>