Discussion - Mathematical Model of Grinding at Different Conditions in Ball Mills – Transactions SME/AIME, Vol. 252, No. 4, December 1972, pp. 452-457 – Olsen, T. O. and Krogh, S. R.

John Philip Zannaras (Registered Professional Engineer, Congress, Ariz.)- The authors show in their [Fig. 4] the cumulative results of grinding with two sizes of bails, one 30 mm and the other 20 mm. The Gaudin-Meloy probability for breakage for a single fall used by the authors in their paper is given by their [Eq. 6]. Another form of [Eq. 6] is given by Gaudin-Meloy in their paper as follows: [ ] From the definition of A given by Gaudin-Meloy it is obvious that A30 is greater than A., it therefore follows from the foregoing equation that according to the Gaudin-Meloy formula the probability of breakage for a single fall of all the balls in the 20-mm experiment was more than 2.25 times the probability of breakage in a single fall of all the 30-mm balls which is contradicted by the authors' experimental evidence in [Fig. 4] in which is shown that much more grinding took place with the 30-mm balls. In fact the authors present convincing evidence that the Gaudin-Meloy work is erroneous because their work is based only on a geometric configuration and on ambiguous and uncertain definition of A and disregards the fact that the distribution of fracture depends on other important and indispensable factors including the intensity of stress caused on the feed grain by the balls. That there is an immense and chaotic variation of stresses caused on the feed grains by the balls is shown by a formula, derived by this writer and published in 1955. The derivation of this formula is given in the Appendix of this discussion; it contains the variables in the operation of ball mills and is based on the science of mechanics-not on probabilities. [ ] where T is the unit stress in pounds per square inch produced by the moving load, h is the height in feet through which P has fallen, E is the modulus of elasticity of the rock, P is weight of one ball, and L is the size of the grain (assumed to be a cube of side L). It will be shown by this writer's formula that when a 30-mm ball strikes any size of feed grain it will cause 1.837 times higher stress than the 20-mm ball when striking the same size grain. It will be shown that when the 30-mm ball or the 20-mm ball strikes a feed grain of 0.1 mm it will cause 90 times more stress on the grain than when it strikes a 2-mm grain.