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.