Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes

Abstract

The distinct element method (DEM) has proven to be reliable and effective in characterizing the behavior of particles in granular flow simulations. However, in the past, the influence of different force–displacement models on the accuracy of the simulated collision process has not been well investigated. In this work, three contact force models are applied to the elementary case of an elastic collision of a sphere with a flat wall. The results are compared, on a macroscopic scale, with the data provided by the experiments of Kharaz et al. (Powder Technol. 120 (2001) 281) and, on a microscopic scale, with the approximated analytical solution derived by Maw et al. (Wear 38 (1976) 101. The force–displacement models considered are: a linear model, based on a Hooke-type relation; a non-linear model, based on the Hertz theory (J. Reine Angew. Math. 92 (1882) 156) for the normal direction and the no-slip solution of the theory developed by Mindlin and Deresiewicz (Trans. ASME. Ser. E, J. Appl. Mech. 20 (1953) 327) for the tangential direction; a non-linear model with hysteresis, based on the complete theory of Hertz and Mindlin and Deresiewicz for elastic frictional collisions. All the models are presented in fully displacement-driven formulation in order to allow a direct inclusion in DEM-based codes.

The results show that, regarding the values of the velocities at the end of collision, no significant improvements can be attained using complex models. Instead, the linear model gives even better results than the no-slip model and often it is equivalent to the complete Mindlin and Deresiewicz model. Also in the microscopic scale, the time evolution of the tangential forces, velocities and displacements predicted by the linear model shows better agreement with the theoretical solution than the no-slip solution. However, this only happens if the parameters of the linear model are precisely evaluated.

The examination of the evolution of the forces, velocities and displacements during the collision emphasizes the importance of correct accounting for non-linearity in the contact model and micro-slip effects. It also demonstrates how these phenomena need to be considered into the model in order to perform deeper analyses on granular material in motion and, in general, for systems sensitive to the actual force or displacement. For these cases, more accurate models such as the complete Mindlin and Deresiewicz model should be addressed.

Introduction

The number of processes involving solids in the chemical, petrochemical, pharmaceutical, biochemical, food industry as well as in energy conversion and environmental processes is such that a high percentage of the research activity is concerned with solids. They appear as raw materials, products or intermediates at some stage in the process in a remarkable number of technological applications. Unfortunately, while the motion of gases and liquids is generally known to follow Navier–Stokes equations, the motion of solids presents different characteristics depending on the type of system, solid concentration and interactions, showing solid-like behavior when a packed bed is subjected to quasi-static stresses or liquid-like behavior, for example in fluidized beds, or a mixture of different behaviors with more complex rheological response. Although large sets of experimental data are available today on diverse granular and multi-phase flows, a proper discussion and understanding of the involved phenomena cannot be attained without the help of numerical simulations.

The computational tools developed in the last decades allow the analysis of the processes at a very small time and space scale and with no intrusion in the system. For fluid–solid flows, typically in fluidized beds, an approach is based on an Eulerian model of the system, the two-fluid model (TFM), where the fluid and the solids are considered as interpenetrating continuous phases (recent applications can be found in Peirano et al., 2002; Pain et al., 2002; Huilin et al., 2003). Also, numerical simulation examples using the TFM-based MFIX code (Syamlal et al., 1993) were carried out by McKeen and Pugsley (2003). The numerical computation is performed solving the averaged Navier–Stokes equations for both phases. Closure equations, related to the solid rheology and usually derived from the kinetic theory of gases, are necessary. The theoretical framework for this approach was established by Anderson and Jackson (1967).

Alternatively, a very promising technique appears to be the distinct element method (DEM) (Cundall and Strack, 1979). The basic idea behind the DEM is simple but very effective: the trajectory of each particle inside the system is calculated, considering all the forces acting on it and integrating Newton's second law of motion and the kinematic equations for position and orientation. The typical forces considered are: gravitation, contact forces due to collisions, solid–solid interactions such as electrostatic, Van der Waals, cohesive forces and bridging due to humidity or high-temperature operations and fluid–solid interactions in multiphase flows. Compared to the TFM, the DEM allows more fundamental studies on the system, in the sense that no hypotheses on the rheological behavior of the solid phase is needed, and especially detailed results in terms of microscopical properties can be obtained (Tsuji et al., 1998). These unsurpassed features are of paramount importance in order to understand the dynamics of mixing and segregating phenomena, agglomeration and flocculation, bubble formation, growth, coalescence and break-up in multi-phase systems, fixed to fluidized-bed transition for complex fluidized systems, basic mechanisms of dust formation and so on. Usual applications of the DEM have dealt with free flowing, i.e. non-cohesive, solids (Tsuji 1992, Tsuji 1993; Hoomans et al., 1996; Walton and Braun, 1986), although interesting applications in cohesive particle modeling exist in literature (Mikami et al., 1998; Rhodes et al., 2001; Nase et al., 2001; Moreno et al., 2003). When long-range inter-particle forces (e.g. Van der Waals, electrostatic, bridge formation forces) are negligible, the most important contribution to particle motion is due to collisions. They determine the direction of the motion of particles after the collision and, during the collision, the value of the force can be several orders of magnitude larger than the gravitational force, as will be discussed in the following sections.

In the field of granular motion simulations, two methodologies exist for particle–particle contact modeling: hard-sphere and soft-sphere approach. In the first case, single binary collisions are modeled as instantaneous processes and the properties of the particles after the collision are related to the properties of the particles before the collision through momentum and energy balances. Several examples of applications of the hard-sphere approach can be found in literature (Campbell and Brennen, 1984, and, more recently, Hoomans et al., 1996; Li and Kuipers, 2003). However, the binary collision concept restricts the application of the method to systems where multiple simultaneous collisions are unlikely to occur, i.e. to dilute systems. An additional drawback is inherent in the method, i.e. the impossibility to incorporate long-range inter-particle forces in the model (Xu and Yu, 1997). By adopting the soft-sphere approach, multi-particle collisions can be dealt with and these inter-particle forces easily implemented, at the cost of higher computational times. In fact, the force–velocity–displacement evolutions are simulated during the collision, modeling the contact as a mechanical system such as a linear spring–dashpot model or more a complex non-linear system.

DEM has been applied successfully in simulating and predicting the performances of many processes involving granular solids. A non-exhaustive list includes the work on hopper flow by Langston et al. (1995), the works on chute flow of glass spheres by Hanes and Walton (2000) and ellipsoidal particles (soybeans) by Vu-Quoc et al. (2000), the works on fluidization by Gera et al. (1998), Kafui et al. (2002), Kawaguchi 1998, Kawaguchi 2000, on fluidized beds with immersed tubes by Rong et al. (1999) and with lateral blasting by Xu et al. (2000), the works on solids motion in mills by Mishra and Murty (2001) and Venugopal and Rajamani (2001), the work on industrial granular flows by Cleary and Sawley (2002) and the work on an impact of a projectile on granular matter by Tanaka et al. (2002).

Although the most common contact force model is the linear spring–dashpot–slider system, more detailed contact force models, based on the classical Hertz's theory (Hertz, 1882) for the normal direction and on simplifications of the model developed by Mindlin and Deresiewicz (1953) for the tangential direction, have been used in the past (Tsuji et al., 1992; Vu-Quoc and Zhang, 1999). However, the problem of a comprehensive comparison of the simulation capabilities and an assessment of the suitability of these models for diverse DEM applications is still open, despite a few works are available in literature (e.g. Sadd et al., 1993). The aim of this work is to investigate the capabilities of three different contact models of describing the macroscopic and microscopic characteristics of a collision. Results obtained using a linear spring–dashpot–slider system, a simplified and a full Hertz–Mindlin and Deresiewicz models are compared on a macroscopic scale to experimental data provided by highly reproducible and accurate experiments on oblique elastic collisions of a sphere against a flat wall (Kharaz et al., 2001). Microscopically, the evolutions of fundamental properties of the particle are compared to an approximation, developed by Maw et al. (1976), of the elastic–frictional response of the system.