Abstract
A recent concept called the fragmentation-energy fan has been used to analyze drop weight testing (DWT) data and to obtain both the mathematical form of the breakage index equation, i.e., t10 versus impact energy and the parameter values needed for making an actual prediction with it. The fan is visualized by plotting the progeny size corresponding to a set of percent passing values versus scaled drop energy in log–log scale and fitting straight, i.e., linear fan lines with a common focal point to these data. The fan behavior lies inherent in the fact that the DWT sieving data closely follow the Swebrec distribution. A mathematical expression for t10 in closed form follows directly from a functional inversion, and this expression differs from the forms it has been given by the JKMRC. In most cases five fan lines suffice to provide a very accurate t10 equation. When applied to a suite of eight rocks, ores mostly, the coefficient of determination R^{2} for the equation lies in the range 0.97–0.99 or almost as high as when JKMRC’s size-dependent breakage model is used. To obtain such a high fidelity a generalization of the original linear fan concept to so-called double fans with piecewise linear rays is developed. The fragmentation-energy fan approach is more compact and general in that t_{n} for an arbitrary value of the reduction ratio n is obtained at the same time as t10 and with t_{n} the complete closed-form solution for mass passing Pleft( {x,D,E_{text{cs}} } right).
Highlights
BIE Breakage index equation cumulative distribution functions (CDFs) Cumulative distribution function DW Drop weight drop weight testing (DWT) Drop weight test or testing JK, JKMRC Julius Kruttschnitt Mineral Research Centre Swebrec Swedish Blasting Research Centre a0, a1 Constant in double fan, Eq 33a,b A Amplitude of BIE (%) b Undulation exponent in Swebrec function, Eq 9 bJK Rate parameter in BIE A ⋅ bJK Crushability parameter of rock
One key parameter extracted from the DWT is t10, the cumulative percentage passing 1∕10:th of the initial lump size and its dependence on Ecs . t10(Ecs ) is called a breakage index equation, and it is a measure of the high-energy impact breakage taking place in crushers and AG/SAG mills
The mathematical form of t10 Ec′s follows directly from a functional inversion and differs from the form it has been given by the JKMRC (Napier-Munn et al 1996; Shi and Kojovic 2007; or Shi 2016)
Summary
Laboratory drop weight testing (DWT) of lumps of rock is a way to characterize the breakage properties of ore and rock so that design and modeling of comminution circuits can be made with confidence (Napier-Munn et al 1996). In the JKMRC monograph (Napier-Munn et al 1996, Table 4.5) a testing matrix (combinations of test parameters) of 5 specimen sizes (13.2–16 mm up to 53–63 mm) × 3 energy levels that vary with the size in the interval 0.1–2.5 kWh/tonne is used for AG/SAG mill modeling. Banini tested eight different rocks, and table gives the testing matrix He finds not surprisingly that the t10 Esv curves depend on specimen size. Since the product A ⋅ bJK is used in the machine design, this seems to make Eq 5 unsuitable to use (Shi and Kojovic 2007) These authors chose to extend the size-independent model of Narayan and Whiten (1988) that is described in Napier-Munn et al (1996) by using an equation developed by Vogel and Peukert (2003). It will be discussed at the end of this article when our model has been presented
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