Computational methods discrete element method

So far, some researchers have analyzed particle fluidization behaviors in a RFB, however, they have not well studied yet, since particle fluidization behaviors are very complicated. In this study, fundamental particle fluidization behaviors of Geldart s group B particle in a RFB were numerically analyzed by using a Discrete Element Method (DEM)- Computational Fluid Dynamics (CFD) coupling model [3]. First of all, visualization of particle fluidization behaviors in a RFB was conducted. Relationship between bed pressure drop and gas velocity was also investigated by the numerical simulation. In addition, fluctuations of bed pressure drop and particle mixing behaviors of radial direction were numerically analyzed. 

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). 

Munjiza A, Owen DRJ, Bicanic N. A combined finite-discrete element method in transient dynamics of fracturing solids. Eng Comput 1995 12 145-174. 

Bicanic B (2004) Discrete element methods. In Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of Computational Mechanics Fundamentals. Wiley, New York, pp 311-337 Bowen RM (1980) Incompressible porous media models by use of the theory of mixtures. Int JEngg Sci 18(9) 1129-1148... 

Computational tools have been accepted and applied within the pharmaceutical industry, typically in the bulk powder processing arena, where tools such as computational fluid dynamics (CFD) and discrete element methods (OEMs) have been... 

Discrete particle modeling (DPM) is an advanced computational technique for particulate systems (in this case, fluidized beds) that has already been presented in Chapter 7 of Volume 3, Modem Drying TechnoU. DPM combines continuous (Eulerian) CFD for the gas phase with a discrete (Lagrangian) consideration of the particle phase by means of a discrete element method (DEM), and is therefore often also denoted as DEM-CFD. Its appHcation enables the resolution of not only interactions between the gas and the particle phase, but also of particle-particle and particle-wall interactions, in the sense of a four-way coupling (compare also with Chapter 5 in Volume 1, Modem Drying Technology). 

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. 

G. G. W. Mustoe, M. Miyata, and M. Nakagawa. Discrete element methods for mechanical analysis of systems of general shaped bodies. In Proceedings of the 5th International Conference on Computational Structures Technology, pp. 219-224. Civil-Comp Press, Edinburgh, 2000. 

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... 

For the determination of the approximated solution of this equation the finite difference method and the finite element method (FEM) can be used. FEM has advantages because of lower requirements to the diseretization. If the material properties within one element are estimated to be constant the last term of the equation becomes zero. Figure 2 shows the principle discretization for the field computation. 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

In the previous contributions to this book, it has been shown that by adopting a polarizable continuum description of the solvent, the solute-solvent electrostatic interactions can be described in terms of a solvent reaction potential, Va expressed as the electrostatic interaction between an apparent surface charge (ASC) density a on the cavity surface which describes the solvent polarization in the presence of the solute nuclei and electrons. In the computational practice a boundary-element method (BEM) is applied by partitioning the cavity surface into Nts discrete elements and by replacing the apparent surface charge density cr by a collection of point charges qk, placed at the centre of each element sk. We thus obtain ... 

In the computational practice, the ASC density is discretized into a collection of point charges qk, spread on the cavity surface. The apparent charges are then determined by solving the electrostatic Poisson equation using a Boundary Element Method scheme (BEM) [1], Many BEM schemes have been proposed, being the most general one known as integral equation formalism (IEFPCM) [10]. 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked element by element. Finite element methods for viscous flows are now well established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 

Within the density functional theory (DFT), several schemes for generation of pseudopotentials were developed. Some of them construct pseudopotentials for pseudoorbitals derived from atomic calculations [29] - [31], while the others make use [32] - [36] of parameterized analytical pseudopotentials. In a specific implementation of the numerical integration for solving the DFT one-electron equations, named Discrete-Variational Method (DVM) [37]- [41], one does not need to fit pseudoorbitals or pseudopotentials by any analytical functions, because the matrix elements of an effective Hamiltonian can be computed directly with either analytical or numerical basis set (or a mixed one). 

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