The Discrete Element Method

Equation 4.9-7 reduces to Eq. 4.9-3 if the second term on the right-hand side vanishes. Clearly the cross-channel force induces additional friction on the side wall, A, which in turn reduces the pressure generation capability for a given flow rate (given angle r/l), or it reduces the conveying capacity for a given pressure rise. 

In Sections 4.4 and 4.5, we dealt briefly with particulate flow instabilities in hoppers and the nonhomogeneous stress distributions created under uniaxial loading of a particulate assembly. In this section, we will expand on the discrete nature of such assemblies, and refer the reader to the computational and experimental tools that have been developed, and are rapidly advancing, to study such phenomena. 

Relatively simple optical experimental techniques to study noncohesive particulate flow have also been developed, such as the polarized light probe system by Allersma (42,43). With this technique, the principal stress distribution and displacement of photoelastic granular material flowing in two-dimensional hoppers, with and without obstructions (distribution bars), can be obtained, as shown in Fig. 4.18. 

Altobelli et al., used a more elaborate three-dimensional MRI technique to study the flow of suspended particles (44) and granular flows (27), also studied by Ng et al. (45) for pellet-sized pills under load, while being sheared in a nonmagnetic shear box, similar to the Jenicke cell (22). This technique holds great potential for detecting details of particulate movements and deformations of three-dimensional particulate assemblies, but is currently limited to very low shearing velocities. 

In the last decades, the modeling of both compacted particulates and flowing particle assemblies under loads and under flow conditions has been advanced by DEM, first developed by Cundall (46) for two-dimensional compacted-disk assemblies in 1971. The origins of DEM are in the field of molecular dynamics (MD), where the motions of individual molecules are tracked under the influence of an external force (e.g., electrostatic) field (47). Experimental results such as those with model photo-elastic assemblies just discussed have also assisted the development of the DEM simulation models. 

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Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

The mechanical response of systems of distinct particles is often adequately described by Newton s laws, which constitute the bases of classical mechanics (36) (see sect. The Discrete Element Method ). However, additional concepts are needed for deformable matter, such as stress and strain, which will be described here (37). We will focus on solid materials, but remark that the same principles are also valid for fluids (in which case the field is usually referred to as computational fluid dynamics). 

The discrete element method regards each particle as an individual element, and where the motion of each particle is calculated according to the Newton s second law. At time t, the total response forces SF) and moments SM,- of each element to its surroundings produce linear (a) and angular (w) accelerations ... 

Several of the problems can be investigated by the Discrete Element Method, as different kinds of interactions can be prescribed (Fig. 9). For example the interactions could be spring or damper elements, which leads to a specific force network. 

Fig. 9 Material representation as point masses and interactions in the framework of the Discrete Element Method. Reprinted from Walhnersperger et al. (2007). Reproduced with kind permission of Wiley-VCH Verlag GmbH Co. KGaA.

Ketterhagen, W.R. Ende, M.T.A. Hancock, B.C. (2009) Process modeling in the pharmaceutical industry using the discrete element method. Journal of Pharmaceutical... 

Gudin, D. Kano, J. Saito, F. (2007). Effect of the friction coefficient in the discrete element method simulation on media motion in a wet bead mill. Adv. Powder Technol., 18,... 

From the liquid phase distribution, the capillary forces Fc on the individual gel particles are accessible, with one contribution from the bulk liquid and a second from the three-phase contact line. By use of the discrete element method (DEM), capillary forces can be loaded on the primary partides, which are connected by bonds as a result, micro-cracks - and also shrinkage -can be simulated (Kharaghani et al., 2011). The cracking criterion is, hence, set by the strength of the necks interlinking the primary partides. 

Theurerkauf, J., Dhod kar, S., Manjunath, K., Jacob, K. Steinmetz, T. (2003) Applying the discrete element method in process engineering. Ghent. Eng. TechnoL, 26(2), 157-162. 

Mishra, R. K. Rajamani, The discrete element method for the simulation of ball mills, Appl. Math. Modelling 16(November), pp. 598-604,1992. 

More complicated models have been published in the open literature [47], while the first fidly 3D computational effort of the whole extmder was made in 1984 [48] based on the assumption of a very viscous fluid even for the soUds-convqring zone. Since then, more computational effort has been expended on the subject. For example, the work of Moysey and Thompson [49] uses the discrete element method to study interactions of polymer pellets as they flow in the solids-conveying zone. These authors have shown interesting patterns in the extmder, which can be used for analysis and design, albeit on a much more demanding basis due to the fidl 3D nature of the geometry. 

Beinert S, Schilde C, Kwade A (2012) Simulation of stress energy and grinding media movement within a wet operated tmnular gap mill using the discrete element method. Chem Eng Technol 35 1899-2059... 

In addition to the finite element simulation, other structural numerical simulations also can be greatly accelerated by the use of GPUs. For example, Durand et al. (2012) simulated rock impact on a concrete slab using the discrete element method (DEM), and the use of a GPU (Tesla C2050) reportedly showed a speedup factor of 30. 

Different approaches are possible to represent heterogeneous media, namely, the discrete element method, the discontinuous finite element method (FEM), and limit analysis. The finite element method remains the most used tool for numerical analysis in solid mechanics, and an extension from standard continuum finite elements to represent discrete joints was developed in the early days of nonlinear mechanics. 

This contribution outlines a multiscale simulation approach for analysis of a Wurster coating process occurring in a fluidized bed. The processes occurring in the apparatus are described on four different time and length scales The Discrete Element Method coupled with Computational Fluid Dynamics, where each particle is considered as a separate entity and its motion in fluid field is calculated, play a central role in the modeling framework. On the macroscale, the Population Balance Model describes the particle... 

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