Momentum equation, discrete particle

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... 

It is noted that the virtual body force Fp depends not only on the unsteady fluid velocity, but also on the velocity and location of the particle surface, which is also a function of time. There are several ways to specify this boundary force, such as the feedback forcing scheme (Goldstein et al., 1993) and direct forcing scheme (Fadlun et al., 2000). In 3-D simulation, the direct forcing scheme can give higher stability and efficiency of calculation. In this scheme, the discretized momentum equation for the computational volume on the boundary is given as... 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

Although LB therefore nowadays may be considered as a solver for the NS equations, there is definitely more behind it. The method originally stems from the lattice gas automaton (LGA), which is a cellular automaton. In a LGA, a fluid can be considered as a collection of discrete particles having interaction with each other via a set of simple collision rules, thereby taking into account that the number of particles and momentum is conserved. 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

If, however, the process takes place in excited states near the band edge, then the momentum of the particle and of the hole are almost zero, and they experience a mutual Coulomb attraction. The problem is essentially the same as that of the H atom or that of the energy levels of positronium the electron and hole possess discrete energy levels given by the equation ... 

In the general case the material is described by discrete point masses and the interactions between them. The behaviour of the system is described by the Newton equations - the momentum equation and the rotational momentum equation - for each particle a by... 

TFM. And the particle phase motion is solved by tracking discrete parcels, each representing a number of particles with the same properties and following the Newton s law of motion. In the MP-PIC method, the coUisional interaction between particles is replaced by using the normal stress of solids (Snider, 2001), which is calculated on the grid points for gas phase and interpolated to the positions of parcels. The gas continuity equation is the same as Eq. (16), whereas the gas momentum equation reads... 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

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... 

In order to bridge the gap between the discretized micro- and macro-worlds, averaging of the variables is necessary. Macroscopic variables in the N-S equation, are the density p and the momentum I, which are functions of the lattice space vector r and time t. The local density p is the summation of the average number of particles travelling along each of six (hexagonal) directions, with velocity c. Multiplication of the density p by the velocity vector u equals linear momentum (I = pu). Boolean algebra is applied for the expressions of the discretized variables density and momentum, respectively, as follows ... 

The theory for a particle having a wavelength is represented by the Schrodinger equation, which, for the particle confined to a small region of space (such as an electron in an atom or molecule) can be solved only for certain energies, ie the energy of such particles is quantized or confined to discrete values. Moreover, some other properties, eg spin or orbital angular momentum, are also quantized. 

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. 

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]. 

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]. 

Commercial codes, e.g. PowerFLOW, which use lattice-based approaches are available, and this particular code was used in the present work. Based on discrete forms of the kinetic theory equations, this code employs an approach that is an extension of lattice gas and lattice Boltzmann methods in which particles exist at discrete locations in space, and are allowed to move in given directions at particular speeds over discrete time intervals. The particles reside on a cubic lattice composed of voxels, and move from one voxel to another at each time step. Solid surfaces are accommodated through the use of surface elements, and arbitrary surface shapes can be represented. Particle advection, and particle-particle and particle-surface interactions, are all considered at a microscopic level to simulate fluid behaviour in a way which ensures conservation of mass, momentum and energy, and which recovers solutions of the continuum flow... 

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