Axial dispersion of cubic particles in horizontal, rotating cylinders was investigated using discrete element modelling simulations. We found that, similar to the behavior of spheres, the axial dispersion coefficient of cubes depends on (1) the rotational speed of the cylinder \({\omega }\), (2) the acceleration due to gravity g and (3) the particle size d, satisfying the relationship \({D}_\mathrm {ax}\propto {\omega }^{1-2{\lambda }}{g}^{{\lambda }}{d}^{2-{\lambda }}\) with \({\lambda }\approx 0.15\) (\({\lambda }\approx 0.1\) for beds of spheres) (Third et al. in Powder Technol 203:510–517, 2010). This observation suggested that, although particle shape influences significantly the rate of axial dispersion (cubes disperse almost twice as fast as spheres of equal volume), the parameters controlling the coefficient of dispersion are independent of particle shape.

In this work we apply the discrete element method (DEM) to model packings of spherocylinders. The so-called composite spheres method was used to construct particles of different aspect ratio, surface shape and curvature. Using the DEM we probe in detail the effect of particle curvature and surface shape on packing morphology and stress transmission. We find that particle shape has a remarkable influence on both the packing morphology (quantified via the solid fraction, particle orientation distribution and radial distribution function) and stress transmission. Specifically, elongated particles have a high preference for horizontal alignment, whereas an increasing particle curvature leads to a more continuous (i.e. less discrete) particle orientation distribution. Generally, we observe that rough and curved particles have a stronger tendency for interlocking (in particular for small particle aspect ratios, i.e. AR = 2 and 3) leading to the formation of dense packing structures. In addition packings of rough and curved particles of small aspect ratios favor stress transmission in the gravitational direction, thus, limiting stress saturation with depth.

Monte Carlo particle tracking methods used to model particle motion in inclined rotating cylinders with bulk flow have been reported in the literature. This paper reports some analytic results for the mean squared deviation of axial position in the following special case: the particles are in the rolling mode; the bed surface is flat, with no axial variation of bed depth; the avalanche is instantaneous and of negligible thickness; collisional axial dispersion is ignored; fill=0.5. Fill=0.5 means that in every half-rotation of the cylinder every particle will have participated in one and only one avalanche; this simplification allows analytic results to be obtained. The calculations are carried out by relating the number densities of tracer particles in successive cycles of avalanching. In the case of well-mixed avalanches, the model is also developed using a random walk approach. The results confirm predictions obtained from discrete element modelling and Monte Carlo particle tracking simulations, which indicate that the mean squared deviation of axial position can oscillate with time for short enough times.

Lattice-Boltzmann method (LBM) simulations of a gas-fluidised bed have been performed. In contrast to the current state-of-the-art coupled computational fluid dynamics–discrete element method (CFD–DEM) simulations, the LBM does not require a closure relationship for the particle–fluid interaction force. Instead, the particle–fluid interaction can be calculated directly from the detailed flow profile around the particles. Here a comparison is performed between CFD–DEM and LBM simulations of a small fluidised bed. Simulations are performed for two different values of the superficial gas velocity and it is found that the LBM predicts a larger bed expansion for both flowrates. Furthermore the particle–fluid interaction force obtained for LBM simulations is compared to the force which would be predicted by a CFD–DEM model under the same conditions. On average the force predicted by the CFD–DEM closure relationship is found to be significantly smaller than the force obtained from the LBM.

The lattice Boltzmann method has been used to compute the drag force acting on assemblies of approximately cubic particles constructed from eight spheres for a wide range of Reynolds numbers. Based on the simulation data we propose a new drag force correlation for assemblies of approximately cubic particles. We have compared the drag force obtained with that predicted by the correlation proposed by Beetstra et al. (2007), originally proposed for spheres, by considering either the drag acting on individual spheres or the drag acting on an approximately cubic particle composed of eight spheres. The comparisons showed that Beetstra׳s correlation cannot predict the system well. The correlation proposed in this paper enables Euler–Euler and Euler–Lagrangian simulations of approximately cubic particles, allowing the influence of the solid volume fraction in these models to be assessed.

This work reports experimental measurements of the dispersion of particles during rotation in a horizontal cylinder. The axial dispersion of a pulse of approximately monodisperse black glass ballotini into a bed of clear glass ballotini of the same size is analysed. This is done using a sectioning technique, where the concentration is determined throughout the cylinder for a given rotation time and speed. The concentration profile is fitted to an appropriate solution of Fick’s second law to determine the dispersion coefficient. The dispersion coefficient is compared for various drum rotation rates and glass ballotini sizes. The cylinder was filled to 35 % by volume and rotated at a range of speeds between 5 and 20 rpm. The particle sizes vary from 1.14 to 3.15 mm. The dispersion coefficient was found to be dependent on both particle size and rotation speed. As the rotation speed, \(\omega \), was increased the dispersion coefficient increased proportionally to \(\omega ^{0.8}\). As the particle diameter, \(d_p\), was increased the dispersion coefficient increased proportionally to \(d_p^{1.84}\). These results are compared with previous experimental and simulation data, in particular the simulations of Third et al. (Powder Technol 203:510, 2010). Strong agreement was found between the simulations of Third et al. and the experimental results.

The axial dispersion of approximately monosized particles in rolling mode in rotating cylinders with bulk flow is examined using a Monte Carlo model and discrete element method (DEM) simulations. The Monte Carlo model predicts that the mean square displacement relative to the mean axial displacement of the bed undergoes oscillations in time. The nature of these oscillations depends on the fill level of the cylinder and the extent of particle mixing during avalanches. When the cylinder is half full the Monte Carlo model predicts undamped oscillations, whereas a filling fraction of 0.26 produces oscillations whose amplitude decreases with time. If mixing during avalanches is assumed to be perfect then the oscillations occur about a linear increase with time. In contrast, if it is assumed that the particles do not mix during avalanching, the oscillations occur about an increase with time which has a gradient which increases with time. There is good qualitative agreement between the Monte Carlo model with perfect mixing and the DEM when the filling fraction is 0.26. For a filling fraction of 0.5 the DEM data show oscillations about a faster than linear increase with time.

A fundamental understanding of the underlying physics of granular (particulate) systems is not only of academic interest, but is also highly relevant for industrial applications. Nowadays computational techniques, e.g. the discrete element method (DEM), are frequently applied as a tool to probe the behaviour of granular systems. The DEM is a particularly attractive modelling technique since it can provide both macroscopic and microscopic ‘measurements’ in granular systems and allows particles of non-spherical shape to be modelled. This ability is important since there is a common understanding that particle shape has a strong influence on the dynamics of these systems. Here, we critically review recent developments in DEM to model particles of non-spherical shape. The first section of the review is concerned with advances in the formulation and implementation of non-spherical particle models, including shape representation, algorithms for the efficient detection of contacts and the determination of contact parameters. In the second part, we review the main findings obtained from numerical ‘measurements’ in granular systems containing non-spherical particles using the DEM. The systems covered in this review include the packing of particles, particle flow (e.g. plane shear flow, the discharge of particles from hoppers and particle motion in vibrated beds and rotating cylinders) and two-phase particle flows such as gas–solid fluidized beds and pneumatic conveying. We conclude with an outlook highlighting the future research needed to further advance this promising modelling technique.

Discrete-element-method（DEM） simulations have been performed to investigate the cross-sectional flow of non-spherical particles in horizontal rotating cylinders with and without wall rougheners. The nonspherical particles were modeled using the three-dimensional super-quadric equation. The influence of wall rougheners on flow behavior of grains was studied for increasing particle blockiness. Moreover,for approximately cubic particles（squareness parameters [555]）, the rotational speed, gravitational acceleration and particle size were altered to investigate the effect of wall rougheners under a range of operating conditions. For spherical and near-spherical particles（approximately up to the squareness parameters [344]）, wall rougheners are necessary to prevent slippage of the bed against the cylinder wall. For highly cubic particle geometries（squareness parameters larger than [344]）, wall rougheners resulted in a counter-intuitive decrease in the angle of repose of the bed. In addition, wall rougheners employed in this study were demonstrated to have a higher impact on bed dynamics at higher rotational speeds and lower gravitational accelerations. Nevertheless, using wall rougheners had a comparatively small influence on particle-flow characteristics for a bed composed of finer grains.

The hysteresis in the jet–spout transition in packed beds was systematically investigated using magnetic resonance imaging (MRI) and pressure measurements. Specifically, the hysteresis in the jet height as a function of the orifice velocity was studied as a function of particle type, bed dimension and fill level. In order to compare the hysteresis of different experimental configurations, a hysteresis coefficient h=(Usf−Ums)/Usf was introduced. It was observed that an increase in the fill level or the bed dimensions resulted in higher values of h. It was also found that the hysteresis is most pronounced for non-spherical particles, whereas no hysteresis was observed for the most spherical particles when placed in the smallest bed. In addition, pressure measurements were used to explore the relationship between the pressure drop and the jet height.

Magnetic Resonance Imaging (MRI) was used to image non-intrusively the formation and interaction of jets and spouts. It was found that the formation and interaction of jets is critically affected by the particle size and shape, the bed dimensions and the fluidization history.

The coupled CFD-DEM technique, which combines a computational fluid dynamics (CFD) description of the fluid phase with a discrete element method (DEM) model of the particles, is one of the most widely used numerical techniques for modelling gas-fluidized beds. Since the spatial resolution of the fluid phase is low in CFD-DEM models, these models require a closure relationship to describe the interaction between the fluid and particulate phases. However, this closure relationship is the greatest source of error in CFD-DEM simulations because it does not account for local variation in the solids volume fraction or the relative motion between particles. The aim of this work was to examine the ability of the CFD-DEM technique accurately to model gas-fluidized beds. The predictions of a CFD-DEM model were compared with those of a coupled lattice Boltzmann method-discrete element method (LBM-DEM) technique, which does not require a closure relationship for the fluid-particle interactions. The two methods were found to give good agreement for the pressure drop through packed and fluidized beds. However, for superficial gas velocities above Umf, different forms of particle motion were predicted by the two methods. INTRODUCTION Since its introduction by Tsuji et al. (1), the coupled computational fluid dynamics-discrete element method (CFD-DEM) technique has become one of the most widely used techniques for modelling gas-fluidized beds. In these simulations each particle is modelled as a distinct entity, whereas the fluid flow is modelled in a volume-averaged manner using a large cell size, typically 3 times the particle diameter. Due to the low spatial resolution of the fluid model, it is not possible to compute the fluid-particle interactions directly in these simulations with the result that these models require a closure relationship, often called the drag law. In recent years closure relationships have been developed for CFDDEM simulations using the lattice Boltzmann method (LBM), e.g. Beetstra et al. (2). However, the drag law remains the greatest source of error in CFD-DEM methods since even the most sophisticated expressions describe only the average force experienced by a particle and make no allowance for particle rotation or relative motion between the particles.

In this work a new partial slip boundary condition for the solid phase in pseudo-2D beds is estimated from DEM simulations and implemented in a two-fluid model. The high spatial resolution of the DEM allowed us to obtain the information required for the new boundary condition, viz. the particle interaction with the walls. In addition, the new boundary condition is compared with the classical Johnson and Jackson boundary condition, which is commonly employed in two-fluid models. The variation of the parameters of the new boundary condition with the superficial gas velocity, and with the coefficients of restitution and friction for particle-particle and particlewall contacts, is studied. The results show that the coefficient of friction is the most influential parameter in the wall boundary condition.

In this study magnetic resonance imaging (MRI) was applied to image non-intrusively the formation of jets in packed beds of square cross section (L=46.8mm, L=50.6mm, L=67.4mm and L=72mm) containing a central square orifice (Lo=3.6mm). Poppy and mustard seeds were used as MRI detectable particles. We critically assessed the influence of two different start up procedures on the measured jet height, viz. filling the bed with particles at a low orifice velocity followed by a stepwise increase of the orifice velocity, or filling the bed at a high orifice velocity followed by a stepwise reduction of the orifice velocity. It was found that the different start up protocols can lead to a difference in the observed jet height of a factor of two, with beds filled at the higher orifice velocity generally giving larger jets. Further measurements were performed to determine the effect of (i) fill level, (ii) particle diameter, (iii) bed dimensions and (iv) orifice velocity on jet height. Interestingly, the fill level did not influence the jet height. However, with increasing fill level the transition to a spouting bed shifted to higher orifice velocities. Additionally, it was established that, for a given orifice velocity, the cross-sections of the bed typically affected the jet height, in particular for larger particles. Based on the set of measurements obtained here we postulate that the jet height is independent of the bed dimensions at flow rates of Q<0.7Qmf and bed dimensions of L/dp>55, where Q is the volumetric flow rate of the gas, Qmf is the volumetric flow rate of the gas at minimum fluidization and dp is the particle diameter. Finally an expression was developed to correlate the jet length against the bed and particle properties.

In this work particle image velocimetry (PIV) was employed to probe the motion of particles within a quasi-2D vibrated bed of circular cross-section. We have observed a polygon-shaped pattern for Gamma - A(2 pi f)(2)/g greater than 10. It has been argued that the polygon pattern formed in 2D cylindrical beds is related to the arching structure encountered in square-shaped vibrated beds [G. Lu, J. R. Third, M. H. Kohl, and C. R. Muller, 2012. On the occurrence of polygon-shaped patterns in vibrated cylindrical granular beds. The European Physical Journal E 35, 90.]. However, using PIV a significant difference in the velocity profiles in the polygon and the arching patterns was observed. We found that in the arching structure the velocity profile around a node was highly symmetric, whereas this was not the case for the polygon structure. Using cylindrical beds of different diameters filled with a constant number of particles, we could confirm that the spatially varying filling height of cylindrical beds is the main cause of the asymmetric velocity profiles within such systems.

The jet-spout transition in fluidized beds can show hysteretic behavior. In this study the jet-spout transition was studied as a function of orifice velocity for particles of different size and shape using Magnetic Resonance Imaging (MRI). The measurements showed that the particle shape primarily affect to the width of the hysteresis loop whereas particle size governs the position of the hysteresis loop with regards to the orifice velocity.