The breakage matrix approach as a tool for describing the milling process: Theory, application, possibilities and limitations

Abstract

© 2015 Nova Science Publishers, Inc. Size reduction is an important unit operation in many industries since raw materials often occur in sizes that are too large to be used. Size reduction is achieved by mechanical forces that cause rupture. The action of the processing equipment determines the magnitude and the nature of the forces acting on particles and determines the degree of particle size reduction. Factors affecting particle size reduction can be classified into those arising from the physicochemical properties of the material and those related to the design and operation of the milling equipment. Since both material properties and milling methods affect particle breakage it is difficult to define some general model because different materials behave differently towards milling operation. The ability to relate the input and output particle size distributions (PSD) is essential for the control of the milling process. Various modeling approaches were used for the analysis of particle breakage. The most comprehensive are the ones based on calculating time-dependent rates of breakage i.e., evolution of PSD over time. These methods require continual sampling within the process which is difficult and sometimes impossible to achieve. Breakage matrix approach treats the entire process as a single breakage event. It is presented in the form of matrix equation in which the breakage matrix relates the input and output size vectors by the rules of matrix multiplication. Elements of the breakage matrix can be determined by milling mono-sized or narrow sized range fractions of inlet material. The PSD from each mono (narrow) sized fraction forms the corresponding column of the breakage matrix. Having determined the breakage matrix for any particle size distribution of input material, particle size distribution of output can be predicted. If the same sievesizes are used for input and output size distributions, the breakage matrix are square where the above diagonal elements are zero (triangular). By using different number of sieves and different sized sieves for inlet and output size distributions, the breakage matrices become non-square and therefore more general. The breakage matrix approach has also been successfully used to predict compositional distribution of broken particles along with their size distribution. Generally, the approach is limited in the sense that the breakage matrix determined for one set of conditions cannot be used for another set of conditions. Also, it presumes that particle breakage is independent of inter-particle interactions. Recent advances provided modified breakage matrix methodology which includes multi-particle interactions during breakage. The reverse problem in the breakage matrix context provides possibility to define the PSD of input material to a milling operation which would give the desired PSD of output. Two possible ways for solving the reverse problem are proposed: "precision"-some of the outputs are precisely defined; "approximation"-best-fit solution, which corresponds to desired output PSD, is obtained by using non-negative least squares method or semilogarithmic loss function. This chapter gives the overview of the breakage matrix approach with emphasis on basic theoretical principles, current application, possibilities and limitations of this method.