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Erosion Index for Particulates

Particulates disintegration, spontaneous or intentionally induced, is frequently the result of shattering and/or surface erosion. The dominance of one mechanism over the other can be characterized by an erosion index, , computed from the relative size and mass fraction of the fine and coarse particles at any given time. This Demonstration lets you select the fines' initial and momentary mass fractions, and the relative sizes of both the fine and coarse fractions, which are used to calculate and display the erosion index. If , then there is dominant erosion; if , there is dominant shattering.

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Snapshot 1: instant coffee agglomerates after 15 taps
Snapshot 2: instant coffee agglomerates after 120 taps
Snapshot 3: freeze-dried coffee agglomerates after 180 taps
Snapshot 4: freeze-dried coffee agglomerates after 260 taps
During their transport, handling, and processing, particulates tend to disintegrate by shattering or surface erosion. The result can be size reduction or size reduction accompanied by the formation of fines or dust. In shattering, the daughter particles' size is on the order of half that of the initial, whereas in erosion, the size of the produced fines is much smaller. Thus, in shattering, the particle size is reduced significantly and rapidly. In surface erosion, fines are produced almost exclusively, and therefore the coarse fraction's particle size is little affected. The relative contributions of these two mechanisms can be quantified by an erosion index, , which is defined as , where and are the initial and momentary mass fractions of the fines ( and ), is the dimensionless mode of the fines' size distribution, assumed to be constant, and and are the initial and momentary dimensionless modes of the coarse particles' size distribution [1]. If the fines or coarse fraction's size distribution is symmetric, the mode equals the mean particle size.
The formula for calculating is derived from the ASTM shattering index [2], amended to account for the fines formation and normalizing both fractions' particle sizes [1]. Notice that if the original mass contains no fines (i.e., ) and the size of the fines produced is very small relative to that of the coarse particles, then . Either way, implies dominant erosion and , dominant shattering. Thus, in processes where the mechanisms dominance is reversed, as in erosion turned into shattering (see snapshots 1 and 2), will drop from a value greater than one to smaller than one. The opposite will happen if shattering is replaced by erosion.
In this Demonstration, you can enter the values of , , , , and with sliders and the program will compute and display the corresponding numerically and graphically as a bar chart in 2D and as a red point on a 3D surface plot. You can choose the upper limit of the axis in the 2D bar chart with a setter bar for increased resolution.
References
[1] L. M. Popplewell and M. Peleg, "An 'Erosion Index' to Characterize Fines Production in Size Reduction Processes," Powder Technology, 58(2), 1989 pp.145–148. doi:10.1016/0032-5910(89)80027-0.
[2] Standard Test Method of Drop Shatter Test for Coal, ASTM D440 - 07(2012), ASTM International, West Conshohocken, PA, 2012. doi:10.1520/D0440-07R12.
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