Collisional transfer of momentum and

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.

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... 

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. 

Here we will discuss the phenomena that result in chemical conversion reactions when transfer of energy and momentum become important. As demonstrated, the removal of translational energy (or momentum) is crucial in association reactions. The reverse process, collisional excitation of reactants is also important, for two types of unimolecular reactions, namely dissociation... 

The stress and the flux of fluctuation energy are due to both transport between collisions and transfer in collisions. The transport parts are -p(CC)and p CC /2, respectively. The collisional transfers, expected to dominate at relatively high number densities, depend upon the exchange of momentum and energy in a collision and the frequency of collision. As an alternative to the number density, the solid volume fraction, v = Ttno /b, is often employed to characterize the density of the system. Then, p = p v, where p is the mass density of the material of the spheres. 

Collisional stresses resulting from inter-particle collision, which result in transfer of both momentum and kinetic energy. 

The Enskog theory for diffusion (equation (5.2)) just scales in time the solution of the Boltzmann equation valid at low densities. However, whereas for diffusion the particles themselves must move, in the case of viscosity and thermal conductivity there is the additional mechanism of collisional transfer, which becomes increasingly important as the density increases, whereby momentum and energy can be passed to another molecule upon collision. The Enskog theory for the viscosity rjg and the thermal conductivity... 

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. 

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 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. 

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. 

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