Kinetic theory modeling

In dense systems such as encountered in solids suspension, particle-particle interaction may be important as well. Then, the closure of solid-phase stresses is an important issue for which kinetic theory models and solids phase viscosity may be instrumental (see, e.g., Curtis and Van Wachem, 2004). 

Thermal Conductivities of Liquids. As was the case with viscosity, it is difficult to derive useful relationships that allow us to estimate thermal conductivities for liquids from molecular parameters. There is a theoretical development by Bridgman, the details of which are presented elsewhere [11], which assumes that the liquid molecules are arranged in a cubic lattice, in which energy is transferred from one lattice plane to the next at sonic velocity, v. This development is a reinterpretation of the kinetic theory model used in the last section, and with some minor modifications to improve the fit with experimental data, the following equation results ... 

The kinetic theory model can be used to explain how a substance changes from one state to another. If a solid is heated the particles vibrate faster as they gain energy. This makes them push their neighbouring particles further away from themselves. This causes an increase in the volume of the solid, and the solid expands. Expansion has taken place. 

The Eulerian continuum approach, based on a continuum assumption of phases, provides a field description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems. 

Kinetic Theory Modeling for Collision-Dominated Dense Suspensions... 

A simple but instructive kinetic theory model due to Langmuir (1918) is often used as a starting point for an atomic scale view of adsorption. It is assumed that the sorbent surface, of total area A, is comprised of Ns sites each capable of binding only one sorbate atom (or molecule). If, at a given instant, a fraction of 6 of these sites are occupied by sorbate atoms, the mean life against desorption is t, and r is independent of 6 (i.e., no interaction among sorbate atoms), the flux (j> of desorbing atoms is... 

Schieber JD, Ottinger HCh (1994) On consistency criteria for stress tensors in kinetic theory models. J Rheol 38(6) 1909-1924... 

P.KCurrie, Constitutive equations for polymer melts predicted by the Doi-Edwards and Curtiss-Bird kinetic theory models, J. of Non-Newt. Fluid Mech. 11 (1982), 53-68. 

Evans. G.T.. She. R.S.C. and Bernstein. R.B. (1985) A simple kinetic theory model of reactive collisions of rigid nonspherical molecules. J. Chem. Phys. 82, 2258-2266... 

In reactor modeling only the ordinary concentration diffusion term j° is generally considered. The rigorous kinetic theory model derivation for multi-component mixtures is outlined in chap 2. Meanwhile, the Tick s law for binary systems is used. 

In a series of papers Lathouwers and Bellan [43, 44, 45, 46] presented a kinetic theory model for multicomponent reactive granular flows. The model considers polydisersed particle suspensions to take into account that the physical properties (e.g., diameter, density) and thermo-chemistry (reactive versus inert) of the particles may differ in their case. Separate transport equations are constructed for each of the particle types, based on similar principles as used formulating the population balance equations [61]. 

For reactive flows the governing equations used by Lindborg et al [92] resemble those in sect 3.4.3, but the solid phase momentum equation contains several additional terms derived from kinetic theory and a frictional stress closure for slow quasi-static flow conditions based on concepts developed in soil mechanics. Moreover, to close the kinetic theory model the granular temperature is calculated from a separate transport equation. To avoid misconception the model equations are given below (in which the averaging symbols are disregarded for convenience) ... 

Table. Hydrodynamic kinetic theory model for SBCR...

A search for an optimum catalyst size was carried out for methanol synthesis in the DOE Laporte SBCR using the complete kinetic theory model of granular flow. Figure 8 show an example for the optimization of slurry-phase reactor operation based on LaPorte SBCR. The mole fraction of methanol and... 

The kinetic theory model was extended to include the effect of the mass transfer coefficient between the liquid and the gas and the water gas shift reaction in the slurry bubble column reactor. The computed granular temperature was around 30 cm /sec and the computed catalyst viscosity was closed to 1.0 cp. The volumetric mass transfer coefficient estimated by the simulation has a good agreement with experimental values shown in the literature. The optimum particle size was determined for maximum methanol production in a SBCR. The size was about 60 - 70 microns, found for maximum granular temperature. This particle size is similar to FCC particle used in petroleum refining. 

Wilke (1950) used Sutherland s kinetic theory model to give ij =... 

Simpler BGK kinetic theory models have, however, been applied to the study of isomerization dynamics. The solutions to the kinetic equation have been carried out either by expansions in eigenfunctions of the BGK collision operator (these are similar in spirit to the discussion in Section IX.B) or by stochastic simulation of the kinetic equation. The stochastic trajectory simulation of the BGK kinetic equation involves the calculation of the trajectories of an ensemble of particles as in the Brownian dynamics method described earlier. 

Chemistry of High Energy Atomic Fluorine Steady State Kinetic Theory Model Calculations for the + H2 Reaction m... 

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