Momentum transfer between particles

Particle behavior often depends on the ratio of particle size to some other characteristic length. The mechanisms of heat, mass, and momentum transfer between particle and carrier gas depend on the Knudsen number. 2J /d , where Ip is the mean fr. e. P ath of the gas molecules. The mean free path or mean distance traveled by a molecule between successive collisions can be calculated from the kinetic theory of gases. A good approximation for a single-component gas composed of molecules that act like rigid elastic spheres is... 

The collisional contribution to the shear viscosity is proportional to a /At as discussed in Sect. 3.2, it results from the momentum transfer between particles in a cell of size a during the collision step. Consider again a collision cell of linear dimension... 

Figure 1 A schematic showing the various energy ioss processes for backscattering from a given depth in a sampie. Energy is iost by momentum transfer between the probe particie and the target particle, and as the probing particle traverses the sample material both before and after scattering.

Finally the knowledge of the velocity profiles allows the determination of the actual shear rate exerted upon the liquid slab. For the bulk system some slip Is observed at the reservoir walls. No slip Is observed for the micropore fluid as a result of the high density close to the reservoir walls, which facilitates the momentum transfer between the reservoir and the liquid slab particles. 

Given that a collision takes place, the nature of the momentum transfer between the cells must be specified. This should be done in such a way that the total momentum and kinetic energy on the double cell are conserved. There are many ways to do this. A multiparticle collision event may be carried out on all particles in the pair of cells. Alternatively, a hard sphere collision can be mimicked by exchanging the component of the mean velocities of the two cells along da,... 

The solids particle velocity in the gas-solid two-phase jet can be calculated as shown in Eq. (27), assuming that the slip velocity between the gas and the solid particles equals the terminal velocity of a single particle. It should be noted that calculation of jet momentum flux by Eq. (26) for concentric jets and for gas-solid two-phase jets is only an approximation. It involves an implicit assumption that the momentum transfer between the concentric jets is very fast, essentially complete at the jet nozzle. This assumption seems to work out fine. No further refinement is necessary at this time. For a high velocity ratio between the concentric jets, some modification may be necessary. 

Irrespective of whether the fluid is in motion, the particles constituting the fluid continuously execute random motion. The particles of aflowing fluid have a drift superimposed upon this random walk. It is by means of the random walk of the particles from one layer to another that the momentum transfer between layers is... 

Fig. 5.44. Viscous forces are considered to arise from the momentum transferred between moving fluid layers when particles jump from one layer to another.

Thus, when a particle jumps, it leaves behind a hole. So then, instead of saying that a transport process occurs by particles hopping along, one could equally well say that the transport processes occur by holes moving. The concept is commonplace in semiconductor theory, where the movement of electrons in the conduction band is taken as being equivalent to a movement of so-called holes in the valence band. It has in fact already been assumed at the start of the viscosity treatment (Section 5.7.1) that the viscous flow of fused salts can be discussed in terms of the momentum transferred between liquid layers by moving holes. 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

The term [Apfjf is the mass-average acceleration of the fluid seen by the particles due in part to momentum transfer between phases, and Af f is due to forces in the fluid phase. The term Gf]f can be evaluated using Eq. (4.80) ... 

Here we explicitly include the consistency terms discussed in Section 4.3.6 by specifying to be the fluid mass seen by a particle. Thus, i contain only contributions arising from mass/energy/momentum transfer between phases. 

The change of momentum for a particle in the disperse phase is typically due to body forces and fluid-particle interaction forces. Among body forces, gravity is probably the most important. However, because body forces act on each phase individually, they do not result in momentum transfer between phases. In contrast, fluid-particle forces result in momentum transfer between the continuous phase and the disperse phase. The most important of these are the buoyancy and drag forces, which, for reasons that will become clearer below, must be defined in a consistent manner. However, as detailed in the work of Maxey Riley (1983), additional forces affect the motion of a particle in the disperse phase, such as the added-mass or virtual-mass force (Auton et al., 1988), the Saffman lift force (Saffman, 1965), the Basset history term, and the Brownian and thermophoretic forces. All these forces will be discussed in the following sections, and the equations for their quantification will be presented and discussed. 

The mesoscale models for momentum transfer between phases differ quite substantially depending on the multiphase system under investigation, and different semi-empirical relationships have been developed for different systems. Since the nature of the disperse phase is particularly important, the available mesoscale models are generally divided into those valid for fluid-fluid and those valid for fluid-solid systems. The main difference is that in fluid-fluid systems the elements of the disperse phase are deformable particles (i.e. bubbles or droplets), whereas in fluid-solid systems the disperse phase is constituted by particles of constant shape. Typical fluid-fluid systems for which the mesoscale models reported below apply are gas-liquid, liquid-liquid, and liquid-gas systems. The mesoscale models reported for fluid-solid systems are valid both for gas-solid and for liquid-solid systems. As a general rule, the mesoscale model for Afp should be derived starting from a single-particle momentum balance ... 

A portion of this supersonic velocity will be transferred to the injected powder particles, that is the powder particles will gain acceleration from the plasma jet by momentum transfer. The particle acceleration dUp/df is proportional to the drag coefficient CD and the velocity gradient between the gas velocity and the particle velocity, Vg — Vp and inversely proportional to the particle diameter dp and the particle density pp as expressed by the Basset-Boussinesq-Oseen equation of... 

Slip between cells and their environment can be neglected, because the fluid velocity can be shown to surpass the sHp velocity by several orders of magnitude. Momentum transfer between the particles and the fluid phase also does not require explicit consideration because the suspension can be treated as a quasisingle phase for particles smaller than the mesh spacing, as shown previously [72]. 

There occurs heat, mass and momentum transfer between the fluid and the particles. Moreover, transport of heat and mass also occurs between the voids themselves. While mass transport through the particles is negligible, heat conduction through the particles is always involved with transport of heat between the voids and therefore must be considered simultaneously. 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

In this group of operations, the solid particles are kept in a suspended state by momentum transfer from the liquid phase. Momentum may be transferred to the liquid phase by different means, and distinction will be made on this basis between three different types of operations. An upper limit exists for the particle size that can be used in suspended bed operations, and is of the order of in. 

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