Closed-form and finite difference solutions to a population balance model of grinding mills

Abstract

The wear of steel balls in continuously operated grinding mills, used in mineral processing to comminute metalliferous rocks, can be described by a simple population-balance model. This model gives rise to a scalar transport equation with a singular source term for the number density of balls as a function of size and time. Exact solutions to this equation are determined under the assumption of a simple power-law type wear law. It is shown that a particular term proposed in the engineering literature that describes the removal of used balls from the mill leads to negative solutions (Model 1). An alternative, more realistic term for the sieve action, which admits nonnegative solutions only, is introduced (Model 2). A working first-order finite-difference scheme for Model 2 and a second-order TVD variant are introduced and applied for numerical simulations along with an error study. A weak solution concept for Model 2 is proposed, uniqueness of weak solutions is shown, and convergence of the first-order scheme to a weak solution is established. These results hold for a general class of wear laws, not just power-law type.

Efficient solution of a batch crystallization model with fines dissolution

“…In order to get an expression for the CSD, the population balance equation (PBE) (1) is solved by using the method of characteristics and Duhamel's principle [30,31]. This is an important step in the derivation of the suggested numerical technique.…”

An efficient numerical technique for solving one-dimensional batch crystallization models with size-dependent growth rates

“…In order to get an expression for the CSD, we solve the given population balance equation (1) by using the method of characteristics and Duhamel's principle (Bürger et al, 2005;McOwen, 2003). This is the most important step in the derivation of the suggested new technique.…”

Conservation laws with discontinuous flux: a short introduction

“…Models of this kind include traffic flow with heterogeneous surface conditions (as discussed above) and continuous sedimentation, which play a key role in several contributions of this issue, and will not be further outlined here. Furthermore, we mention models of two-phase flow in heterogeneous porous media [11,12], a population balance model of ball wear in grinding mills [13], and a model of endo-vascular treatment of abdominal aortic aneurysm [14] which are similar to the two previous models in that the discontinuity with respect to the spatial variable occurs at a fixed position. Another interesting model is the Liouville equation in classical mechanics with discontinuous potentials [15].…”

“…This Special Issue includes papers that were presented in a minisympsosium on "Conservation Laws and Related Equations with Discontinuous Flux: Theory, Numerics, Applications", which the guest editors had organized within the conference "The Mathematics of Finite Elements and Applications 2006 (MAFELAP 2006)", which took place at Brunel University, Uxbridge, UK, June [13][14][15][16]2006, and some invited contributions. We would like to thank the organizers of MAFELAP 2006, in particular Professors John R. Whiteman and Norbert Heuer, for the possibility to organize this minisymposium, and we are grateful to the Editor-in-Chief of Journal of Engineering Mathematics, Professor Henk K. Kuiken, for offering the opportunity to publish this Special Issue, and his constant advice and support.…”