Liberation 1, 2, 3: Theoretical analysis of the effect of space dimension on mineral liberation by size reduction☆

Abstract A derivation is made, for space dimensionality of 1, 2 and 3, of mineral liberation by size reduction. The mathematical solution to the problem of mineral liberation prediction is derived through the use of mathematical morphology and integral geometry methods. The solution is worked out for real particles produced by fragmentation of ores, and for ore textures which correspond to Boolean schemes with primary Poisson polyhedra, and Poisson polyhedral textures. These textures have characteristics which are often encountered in ores. For a sparse phase it is demonstrated that the degrees of liberation in 1, 2 and 3 dimensions are related by a simple relationship: L d+i (D) = ( L d ( D )) 2 i where L d ( D ) is the degree of liberation at screen size D measured in d dimensions. The degrees of liberation follow an exponential decay: l d ( D )= exp [−0.693( D D 50,d )] with D 50,d+i = D 50,d /2i