Collisional momentum transfer

The general tendencies of the experimental results are as follows The apparent viscosity in an unfluidized bed is very high (perhaps infinite) but drops rapidly with increasing gas velocity. When particles are fully fluidized, the bed viscosity becomes rather independent of gas velocity and increases with increasing particle size and particle density i.e., it shows a gradual change from a displacement effect to a collisional momentum transfer. 

There is one additional mechanism for momentum transfer which does not seem to have been mentioned in the polymer literature. A bead of one dumbbell may collide with a bead of another dumbbell in such a way that their centers at the time of collision are on opposite sides of the plane at which the stress is being calculated. Such collisions would result in an instantaneous transfer of momentum between the two dumbbell centers and thus contribute to the stress at the plane. For very dilute solutions the contribution of this collisional momentum transfer would clearly be much smaller than those associated with mechanisms (i) and (ii) above. 

The weak dependence of the rate of formation of the emission on the pressure of the bulk gas was attributed to a collisional momentum transfer process between the electron and the monatomic rare gas, i.e. 

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. 

In reality, more complicated expressions apply because of the multi-step character of the reaction ki and k i depend on the energy E and the angular momentum (quantum number J), and the collisional energy transfer is a multi-step process with activations and deactivations, to be characterized by a master equation. It has become customary to consider "strong collisions" first and to introduce "weak collision effects" afterwards. For strong collisions, equation (6) takes the form... 

This particulate stress represents the kinetic components of granular momentum transfer, and includes both the viscous contribution due to the small-scale random motion of individual particles as well as the macroscopic turbulence contributions due to collective random motions such as eddies and bubbles (Sun, Chen, and Chao, 1990). The complete granular stress should consist of this particulate stress component and a collisional stress component. 

In this representation, particular emphasis has been placed on a uniform basis for the electron kinetics under different plasma conditions. The main points in this context concern the consistent treatment of the isotropic and anisotropic contributions to the velocity distribution, of the relations between these contributions and the various macroscopic properties of the electrons (such as transport properties, collisional energy- and momentum-transfer rates and rate coefficients), and of the macroscopic particle, power, and momentum balances. Fmthermore, speeial attention has been paid to presenting the basic equations for the kinetie treatment, briefly explaining their mathematical structure, giving some hints as to appropriate boundary and/or initial conditions, and indicating main aspects of a suitable solution approach. 

The collisional pressure tensor represents the instantaneous momentum transfer at binary particle collisions, over the distance separating the centers of the two colliding particles. The pressure tensor closure is derived based on an extension of the kinetic theory of dense gases. The collisional pressure tensor is thus the second out of the two pressure tensor components that is calculated by use of the KTGF. 

For the purposes of modeling, it is convenient to factor the rate coefficient for collisional energy and angular momentum transfer into an overall rate coefficient for collisions between the molecule and bath gas (irrespective of the amount of energy transferred), z, and a collisional energy and angular momentum transfer probability distribution function (TPDF) P E,J E, J ),... 

In addition to restoring Galilean invariance, this grid-shift procedure accelerates momentum transfer between cells and leads to a collisional contribution to the transport coefficients. If the mean free path A is larger than a/2, the violation of Galilean invariance without grid shift is negligible, and it is not necessary to use this procedure. 

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... 

One of the more well studied collision processes involving Rydberg atoms is collisional angular momentum mixing, or i mixing, the collisional transfer of population among the nearly degenerate states of the same n.28 The process has... 

The fundamental postulate is that as a dilute gas is compressed two novel effects become important because the molecules have finite volumes. First, it is expected that during a molecular collision momentum and energy are transferred over a distance equal to the separation of the molecules. In the particular case of rigid spherical molecules this collisional transfer of momentum and energy takes place instantaneously and results in a transfer over the distance between their centers. Second, the collision frequency may be altered. One possible mechanism is that the collision frequency is increased... 

Fig. 17. The instantaneous collisional transfer of momentum and energy across a distance a when two bard spheres, each of diameter a, collide head on. The figure on the left represents the velocity and positions of the two spheres immediately before the collision, and the figure on the right represents them immediately after the collision.

A consideration of collisional dissociation near threshold requires an understanding of the amount of energy transferred in a single encounter, the role of angular momentum in the decay of activated molecules, and the lifetimes for decay of such species. Accurate determination of apparent thresholds has been achieved by modeling the threshold behavior of the system with a cross-sectional form as follows ... 

Above the critical temperature and at intermediate densities the temperature and density dependence is quite moderate, because translational transfer (at low density) and collisional transfer of momentum at high density in part compensate each other. This facilitates estimates in this area. Self-diffusion and solute diffusion have indeed be found to be very high in high density water to 700 C as measurements by Jonas and coworkers [31] and by Buehler et al. [32] with NMR and optical techniques have... 

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