On the determination of contact parameters for realistic DEM simulations of ball mills

B.K. Mishra, C.V.R. Murty
Department of Materials and Metallurgical Engineering
Indian Institute of Technology Kanpur
India
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Abstract

The discrete element method DEM. has been used extensively in simulating multiple interacting bodies undergoing relative motion and breakage. The key to the success of this method lies in correctly establishing the interaction rules and the associated contact parameters. In this paper, this is done by analyzing the impact behavior of a steel ball. Experimental data using an ultra fast load cell UFLC. allowed the material response at the contact to be adequately modeled by a nonlinear differential equation. In turn, the contact parameters to be used in the DEM were directly extracted from the model. The nonlinear contact behavior led to values of contact parameters that limit the critical time step in the numerical integration. This is unsuitable for any large simulation involving thousands of interacting balls as in the case of a ball mill. This disadvantage is overcome by using the equivalent linearization technique to transform the nonlinear contact model to an analogous linear model. Not only did the linearized parameters allow a larger time step but also the use of a linear spring–dashpot model significantly reduced the overall computational effort, which is illustrated by simulating a 54.5-cm diameter ball mill. q2001 Elsevier Science B.V. All rights reserved.

Introduction

One of the successful applications of the discrete element method DEM. as applied to mineral engineering research includes prediction of charge motion and estimation of power draft of tumbling mills that are used in all mining operations for size reduction operations. Since its inception w1x and successful adaptation w2x to solve ball mill problems, many have applied DEM to describe the internal dynamics of tumbling mills w3–7x. Today, with improving computer speed, DEM offers an extremely viable approach for simulating tumbling mills of any size, thereby lending itself as a practical tool for design and optimization. The main objective of simulation is to correctly predict the dynamic profile of the ball charge and the power draft. For an industrial mill of 13-ft diameter, it is done essentially by tracking the motion of approximately 100,000 balls. Monitoring the position of all these balls at any instant is not a trivial task and it is due to the availability of the DEM technique that it has only become possible. In the DEM technique where the calculation is done in a discrete fashion by monitoring the forces at each contact, the total number of contacts to be handled, assuming a maximum of 10 contacts per ball is 106. At each contact, the contact forces, and for each ball, the corresponding velocities, are to be calculated. The list updating requires a calculation of around 30 unknowns per ball. Altogether, it is estimated that the total number of computations per time step 10y5 s in the case of linear springs. is of the order 109. Furthermore, it is estimated that there is a six-fold increase in computation when the nonlinear spring–dashpot contact models are introduced. Therefore, the sheer number of calculations required until the mill reaches steady state is overwhelming. The computational limitations restricted the early DEM applications to two-dimensional 2D. assemblies of discs. DEM was used as a qualitative tool for the computation of planar motion of the charge as a function of linear geometry. Following the development of three-dimensional 3D. simulation codes, it became possible to directly obtain quantitative information such as axial flow of material, frequency spectrum of the intensity of collisions, power draft, etc. However, the success in this approach lies in correctly determining the parameters involved in the model for two reasons: first, these parameters influence the critical time step used in the simulation procedure, and second, the accuracy of the quantitative results of the simulation is solely dictated by the accuracy of these parameters. Even a slight variation in the values of contact parameters is likely to give a large variation in the simulation results. These parameters essentially embody the contact properties of the system: stiffness, damping, and friction. Estimation of correct parameters for realistic simulation is not addressed adequately in the literature. In this paper, a formal procedure is presented to determine the contact parameters using the experimental data in such a way that a compromise is maintained between numerical accuracy and computational expense.



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