Partition Moment Of Inertia: Techno-Economic Partition Curve Analysis with Applications to Coal Separations

"In coal and mineral processing, several numeric parameters are used to quantify the separation performance of unit operations. For density-based separations, the partition curve is used as the starting point for many stages of diagnostic analyses. The curve is often quantified by its midpoint slope sharpness, and this single value is subsequently used for technical evaluation, benchmarking, and comparison. While this analytical approach does depict misplacement by relative density class, it inherently prioritizes middling separation and does not consider process economics or the micro-price value of individual classes. An alternative parameter, the organic efficiency, does indirectly consider process economics, but it does not reveal information on the root cause of lost revenue or misplacement. In this paper, a new analytical approach, the partition moment of inertia, is introduced. This parameter reconciles the technical misplacement data derived from the partition curve with the economic data derived from micro-pricing. The result is a new analytical procedure that promises better use of performance data. IntroductionPartition curves are commonly used in the characterization, evaluation and simulation of various physical separators. In mineral processing, partition curve analysis is routinely applied to hydrocyclones, screens, density separators and other beneficiation technologies that are based on intensive physical properties. Graphically, the partition curve depicts the recovery of particles as a function of the physical property on which the separation is based, such as particle density, size or magnetic susceptibility. Given the probabilistic nature of many real separators, the partition curve often exhibits a characteristic S shape (Fig. 1), similar to a cumulative normal distribution. Other geometric features of the curve are also depicted in Fig. 1, including the high and low closeout, or bypass, values; the midpoint slope; and the separation cut point, defined as the property value with 50 percent recovery.Partition curve analysis was originally proposed to evaluate coal washability (Tromp, 1937). Given the limited computational resources at the time, the analytical methods relied heavily on graphical construction and interpretation of the curve. Owing to the simplicity of calculation and interpretation, many of these graphical approaches are still widely used for performance analysis today. The most common calculations are based on the slope of the partition curve at the midpoint. Increased slopes correspond to enhanced separation performance, as a perfect separation exhibits a stepwise transition from 0 to 1 at the desired cut point, that is, an infinitely large slope. One common formulation of this slope, the Ecart Probable, Ep, is derived by evaluating the slope between the 25 and 75 percent recovery points:"