A matrix analysis of processes involving particle assemblies

Abstract

The processes analyzed in this paper are the size reduction and size classification of particle assemblies. Particle size distributions are described by vectors, and alterations to size distributions during breakage processes are described by matrices multiplying the vectors. The matrix approximation gives an adequate representation of the processes studied, and the manipulation of the matrices is easy and flexible. The breakage of a particle assembly is thought of as two processes. In the first, the machine breaking the particles is said to select for breakage a proportion of the particles, and the remaining particles are unbroken. To discover a function or matrix which describes the process of selection is to understand how the machine operates. In the second process, the particles selected are broken in a regular way; the proportions of particles of each size formed by the breakage are described by a breakage function or a breakage matrix. The analysis of breakage is in this way given convenient mathematical form. These matrices depend on the characteristics of the machine and on the nature of the particle assembly. After breaking the particles, crushing and grinding machines frequently pass the product assemblies to a classifier from which the larger particles are returned, mixed with fresh material, to the grinding zone. The analysis is extended to the description of such circuits. The experimental work reported concerns the breaking of coal particles in a new grinding machine, ball mills, shatter tests and a beater mill. The selection functions derived throw light on the operation of these machines. Coal breakage has been studied since it is an important field of application, and because coal is typical in breakage of homogeneous rocks. For each of the machines examined and for each particle size, a single breakage function has sufficed to describe the product of breakage: [1 —exp ( —z)]/[ 1 —exp ( — 1)] is the proportion of the product smaller than a fraction z of the original particle size.