Technical Note - A Novel Computational Approach Toward The Mill Matrix Of Distributed Comminution Models

Introduction The incentives for developing a distributed form of the comminution model include: • for describing the heterogeneous comminution behavior of multiphase mineral systems (i.e. realistic ore bodies, such as complex sulfide ores consisting of chalcopyrite, galena, sphalerite, pyrite and siliceous minerals) and •for generating, for various locked and free particles, in situ, microscopic, breakage behavior such as the mode of fracture (intergranular vs. transgranular), the relative rates of breakage and the breakage distribution. However, the computational difficulty and the effort made in estimating a large number of parameters (Andrews and Mika, 1975; Wiegel, 1976; Herbst et al., 1985, 1988) do not justify the utility of this approach. Also, obtaining the results experimentally is very tedious and time consuming if using this approach. Although recent work in this area (Choi, Adel and Yoon, 1987; Mehta, Adel and Yoon, 1989, 1990) showed good results for liberated particles, the quality of the estimation and prediction was poor for locked particles. Attributing this to the complex breakage behavior of locked particles, it was concluded that additional constraints, and possibly modification in the breakage distribution functional form for locked particles, were necessary. However, further mathematical analysis revealed that the method previously used was in error. The purpose of this technical note is to present the correct solution methodology used in solving the general model equations of a multicomponent mineral system containing multiple classes of locked particles. Previous work and connection The general form of the comprehensive, distributed-component comminution model has been described as (Herbst et al., 1985) [d (m (0) -[ I - B ] S_ m(t) = A m(t) (1)] where [A] is the fundamental matrix/state companion matrix, commonly referred to, in the comminution literature, as the "mill matrix." Note that for a distributed-component comminution model, this is a full, general, real matrix, unlike a lower triangular matrix for a simple grinding model or a lumped-binary grinding model with one locked-particle class (Herbst et al., 1985, 1988). Also note that all of the underlined letters refer to matrices. Herbst et al. addressed the difficulty in arriving at a closed form of the analytical solution of Eq. (1) via similarity transformation [IT J inv (T)]]. This difficulty occurs for several reasons: •[A], as a general real matrix, may have complex eigenvalues and complex eigenvectors. •More importantly, there is no guarantee that a set of eigenvectors of [A] will turn out to be independent. An attempt was made to solve this problem by implementing a decomposition method. With this technique, the mill matrix was decomposed into a sum of lower and upper triangular matrices [(i.e., A = L + U)]. Then the solution was obtained using similarity transformation (Mehta, Adel and Yoon, 1989, 1990). [e(At) = e(L+U)t = e(Lt).e(Ut) _ [ TL Jt- (t) inv(T_L)] [-Tu Ju (t) inv(Tu)]] [But, unfortunately, e 1+U = e t . eu is true only for scalars and not for matrices (i.e., eL+u# eL- eU) unless they commute. In other words. eAt = e(L+u)t= ekt - eU' if LU = UL. Hence, this decomposition scheme must be replaced by a mathematically acceptable method known as the "splitting method"]