Discrete-velocity model

Li-Shi, L. 2000 Some recent results on discrete velocity models and ramifications for lattice Boltzmann equation. Computer Physics Communications 129, 63-74. 

Naris S, Valougeoigis D, Sharipov F, Kalempa D (2004) Discrete velocity modelling of gaseous mixture flows in MEMS. Superlattices Microsduct 35 629-643... 

The proposed model belongs to the class of so called discrete velocity models of the Boltzmann equation which are deduced from the Boltzmann equation by discretizing the velocity domain The discrete velocity models were successfully applied to gasdyhamics in particular to the shock wave problem in single monatomic gases ) and... 

An advantage of this approach to model large-scale fluidized bed reactors is that the behavior of bubbles in fluidized beds can be readily incorporated in the force balance of the bubbles. In this respect, one can think of the rise velocity, and the tendency of rising bubbles to be drawn towards the center of the bed, from the mutual interaction of bubbles and from wall effects (Kobayashi et al., 2000). In Fig. 34, two preliminary calculations are shown for an industrial-scale gas-phase polymerization reactor, using the discrete bubble model. The geometry of the fluidized bed was 1.0 x 3.0 x 1.0 m (w x h x d). The emulsion phase has a density of 400kg/m3, and the apparent viscosity was set to 1.0 Pa s. The density of the bubble phase was 25 g/m3. The bubbles were injected via 49 nozzles positioned equally distributed in a square in the middle of the column. 

For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more refined model for the bubble-bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of fluidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in large-scale fluidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors. 

An even more drastic simplification of the dynamics is made in lattice-gas automaton models for fluid flow [127,128]. Here particles are placed on a suitable regular lattice so that particle positions are discrete variables. Particle velocities are also made discrete. Simple rules move particles from site to site and change discrete velocities in a manner that satisfies the basic conservation laws. Because the lattice geometry destroys isotropy, artifacts appear in the hydrodynamics equations that have limited the utility of this method. Lattice-gas automaton models have been extended to treat reaction-diffusion systems [129]. 

Although many permeation processes can be described effectively by discrete state binding sites within the channel, they are not the only possible permeation mechanisms. The ion may diffuse through the channel to reach the opposite bath. This diffusional flow can be considered the limit of discrete site diffusion because the number of discrete sites increases. In such a model, the ion cannot make the abrupt changes in velocity that could characterize the finite site, discrete state model. The velocity, however, can change continuously as the ion moves through the channel. A discrete state velocity distribution would evolve naturally to a continuous velocity distribution, where the relative numbers of ions at each velocity in the continuum of velocities could be established. 

Although the discrete state and continuum models are analyzed in different ways, both models must produce a local velocity distribution for the ions within the channel. A continuum velocity distribution might be characterized by a single peak of finite bandwidth that reflects velocity fluctuations within the channel, whereas a discrete state model might consist of several different peaks in the distribution. Such velocity distributions have been generated from the models used because electrochemical experiments do not yield detailed information on the internal kinetic parameters. These parameters are deduced from the experimental data, which are generally the net currents observed for a series of transmembrane potentials for a variety of bath ionic components and concentrations. More detailed information on the nature of intrachannel kinetics becomes accessible if it is experimentally possible to deduce the velocity distribution of the ions in the channel. 

The elucidation of a mechanism for the permeation of cations in gramicidin channels requires a complete velocity distribution for these cations in the channels. The distribution describes the fractional concentration or probability of finding an ion at a specific local velocity in the channel. An ion that moved through the channel at constant velocity would give an extremely narrow distribution 100% of the ions are moving at that single local velocity. For a discrete state model with two binding sites, the distributions could... 

Another two-dimensional, discrete element model was applied by Cartaxo and Rocha [43]. In this work, only the dynamic phenomena were investigated, that is, heat and mass transfer between the phases were not considered. Thns, the inflnence of the momentum coupling between the discrete particles and the conveying air on the air radial velocity and the mass concentration profiles was presented. An object-oriented numerical model was developed to simulate the conveying of large spherical particles (3 mm) through 9.14 m vertical tube with 7.62 cm bore size. 

One of the most widely used approaches for the simulation of sprays is the stochastic discrete droplet model introduced by Williams [30]. In this approach, the droplets are described by a probability density fxmction (PDF),/(t,X), which represents the probable number of droplets per unit volume at time t and in state X. The state of a droplet is described by its parameters that are the coordinates in the particle state space. Typically, the state parameters include the location x, the velocity v, the radius r, the temperature Td, the deformation parameter y, and the rate of deformation y. As discussed in more detail in Chapter 16, this spray PDF is the solution of a spray transport equation, which in component form is given by... 

At a higher level, the flow field is modeled at a scale much larger than the size of the particles, and the fluid velocity and pressure are obtained by solving the volume-averaged Navier-Stokes equations. The particle particle interactions (particle wall as well) are formulated with the so-called discrete particle models (DPMs), which are based on the schemes that are traditionally used in molecular dynamics simulations, with the addition of dissipation of mechanical energy. 

For the two-dimensional case, third-order Gauss-Hermite quadrature leads to the nine-speed LBE model with the discrete velocities... 

LBM has been developed to simulate flows in microchannels based on kinetics equations and statistical physics. In LBM, the motion of the fluid is modeled by a lattice-Boltzmann equation for the distribution function of the fluid molecules. The discrete velocity Boltzmann equation corresponding to the Navier-Stokes equations can be written as... 

---