Mineral liberation and the batch communition equation

The batch comminution equation for multi-component mineral systems is a multidimensional integrodifferential equation ∂P(g,D;t)∂t=∫R′∫S(g′,D′)B(g,D|g′,D′)p(g′,D′;t)dg′dD′ that cannot be solved analytically. The phase space (g, D) was discretized and the equation formulated as dpijdt=−Sijpij+∑i=1j−1∑k=Kl∗Kl∗∗Sklbijklpkl Solutions to this discretized form of the equation are easy to generate provided that models for the selection function Sy and the multi-component breakage function bykl are known. ral solution to the batch comminution equation that does not rely on the random fracture assumption is derived in this paper and is given by pij=∑l=1j∑k=112αijkle−Sklt with the coefficients given by αijkl=0 ifi≠mαijij-pij(0)−∑l=1j∑k=112αijklαijmn=∑l=1j∑k=112SklbijklαklmnSij−Smn olution includes the following non-random fracture processes: selective phase breakage, differential breakage, preferential breakage, liberation by detachment and boundary-region fracture. This solution was compared with experimental data obtained from batch tests on two ores. The model was found to be reliable and the derived Andrews-Mika diagram is shown. rived models for the selection function and the Andrews-Mika diagram can be used to simulate the effects of the processes that produce non random fracture, and thus to assess their relative importance, and also to simulate the behavior of the ore in continuous milling and concentrating circuits. umption of random fracture provides an important simplification because, under this assumption, the selection function is independent of the particle composition. The random fracture assumption has been used by researchers who developed models for the prediction of mineral liberation by comminution. The solution of the batch comminution equation under this assumption was found to be pij(t)=∑l=1jAijle−Slt with the coefficients given by Aijjj=pij(0)−∑i=1j−1Aijl and Aijn∑l=nj−1Sl∑k=112bijklAklnSj−Sn