Mathematical Modelling of the Dynamics of Granular Materials in a Rotating Cylinder

Abstract

A fundamental understanding of mixing and blending of granular materials in horizontal rotating cylinders will be beneficial to a wide range of industries pharmaceuticals, metallurgy, ceramics, composites, polymers, food processing and agriculture Yet, in relation to its industrial prevalence, the understanding of granular mixing lags considerably when compared to that of liquid mixing. Granular bed motion in the transverse cross section of a horizontal rotating cylinder is fundamental indetermining advective heat transfer as well as axial flow of the bed material, but to date studies have been mainly empirical aimed at determining the macroscopic bed motion but devoid of detailed information of granular bed dynamics at a microscopic level. The current study is therefore aimed at the development of a theoretical simulation tool based on Discrete Element Method (DEM) to interpret granular dynamics of solid bed in the cross section of the horizontal rotating cylinder at the microscopic level and subsequently apply this model to establish the transition behaviour, mixing and segregation, The simulation of the granular motion developed in this work is based on solving Newton's equation of motion for each particle in the granular bed subjected to the forces are tracked and the positions, velocities and accelerations of each particle is updated at every instant of time. The software code for this simulation is written in VISUAL FORTRAN 90. After checking the validity of the code with special tests, it is used to investigate the transition behaviour of granular solids motion in the cross section of a rotating cylinder for various rotational speeds and fill fraction. This work is hence directed towards a theoretical investigation based on Discrete Element Method (DEM) of the motion of granular solids in the radial direction of the horizontal cylinder to elucidate the relationship between the operating parameters of the rotating culinder geometry and physical properties of the granular solid. The operating parameters of the rotating cylinder include the various rotational velocities of the cylinder and volumetric fill. The physical properties of the granular solids include particle sizes, densities, stiffness coefficients, and coefficient of friction. Further the work highlights the fundamental basis for the important phenomena of the system namely. (i) the different modes of solids motion observed in a transverse coss section of the rotating cylinder for various rotational speeds, (ii) the radial mixing of the granular solid in terms of active layer depth (iii) rate coefficient of mixing as well as the transition behaviour in terms of the bed turnover time and rotational speed and (iv) the segregation mechanisms resulting from differences in the size and density of particles. The transition behaviour involving its six different modes of motion of the grnular solid bed is quantified in terms of Froude number and the results obtained are validated with experimental and theoretical results reported in the literature. The transition from slumping to rolling mode is quantified using the bed turnover time and a linear relationship is establised between the bed turn over time and the inverse of the rotational speed of the cylinder as predicted by Davidson et alt [2000]. The effect of the rotational speed, fill fraction and coefficient of friction on the dynamic angle of repose are presented and discussed. The variation of active layer depth with respect to fill fraction and rotational speed have been investigated. The results obtained through simulation are compared with the experimental results reported by Van Puyvelde et. al. [2000] and ding et al. [2002] The theoretical model has been further extended, to study the mixing and segregation in the transverse direction for different particle sizes and their size ratios. The effect of fill fraction and rotational speed on the transverse mixing behaviour is presented in the form of a mixing index and mixing kinetics curve. The segregation pattern obtained by the simulation of the granular solid bed with respect to the rotational speed of the cylinder is presented both in graphical and numerical forms. The segregation behaviour of the granular solid bed with respect to particle size, density and volume fraction of paticle size has been investigated. Several important macro parameters characterising segregation such as mixing index, percolation index and segregation index have been derived from the simulation tool based on first principles developed in this work.