Momentum equation, discrete particle modeling

There are two levels, discrete particle level and continuum level, for describing and modeling of the macroscopic behaviors of dilute and condensed matters. The physics laws concerning the conservation of mass, momentum, and energy in motion, are common to both levels. For simple dilute gases, the Boltzmann equation, as shown below, provides the governing equation of gas dynamics on the discrete particle level... 

The lattice Boltzmann method (LBM) is a relatively new simulation technique for complex fluid systems and has attracted great interests from researchers in computational physics and engineering. Unlike traditional computation fluid dynamics (CFD) methods to numerically solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy), LBM models the fluid as fictitious particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. Due to its particulate nature and local dynamics, LBM has several advantages over conventional CFD methods, especially in dealing with complex boundaries, incorporation of microscopic interactions, and parallel computation [1, 2]. 

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... 

Unlike the aforementioned models, Fyhr and Rasmuson [41,42] and Cartaxo and Rocha [43] used an Eulerian-Lagrangian approach, in which the gas phase is assumed as the continuous phase and the solids particles are occupying discrete points in the computational domain. As a consequence, mass, momentum, and energy balance equations were solved for each particle within the computational domain. 

For discretization of the velocities, it will be - a +o in both directions "x" and "y" for specific case in two-dimensional model (D2Q9), it will unroll in this part of chapter. The particle momentums of distribution function are very important, because of this depends the consistence of (N-S) equations, in the same way, the isotropy is keeping during the discretization, it is an important property in the symmetry of NE equations, of this form, lattice will be invariant for problem rotations. 

The formalism of thermodynamically consistent dissipative particle dynamics represents a consistent discrete model for the Lagrangian fluctuating hydrodynamics. Equation (26.45)-Equation (26.46) conserve the mass, momentum, energy and volume. The irreversible (produced) entropy S is a strictly inerting function of time in the absence of fluctuations. Thermal fluctuations represented by F, S are consistently included, which lead to an increase of the entropy and to correct for the Einstein distribution function [31]. 

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