Kinetic equation fluid-particle flow

Reif F (1967) Statistical physics. McGraw-HiU Book Company, New York Resibois P, De keener M (1977) Classical kinetic theory of fluids. Wiley, New York Reyes Jr JN (1989) Statistically derived conservation equations for fluid particle flows. Nuclear thermal hydraulics 5th winter meeting. In Proceedings of ANS winter meeting Rohsenow WM, Choi H (1961) Heat, mass, and momentum transfer. Pretice-HaU Inc, Englewood Cliffs... 

Reeks MW (1991) On a kinetic equation for the transport of particles in turbulent flows. Physics of Fluids A 3 446-456... 

Ding and Gidaspow [16], for example, derived a two-phase flow model starting with the Boltzmann equation for the distribution function of particles and incorporated fluid-particle interactions into the macroscopic equations. The governing equations were derived using the classical concepts of kinetic theory. However, to determine the constitutive equations they used the ad hoc distribution functions proposed by Savage and Jeffery [65]. The resulting macroscopic equations contain both kinetic - and collisional pressures but only the collisional deviatoric stresses. The model is thus primarily intended for dense particle flows. 

The two-fluid granular flow model is formulated applying the classical Eulerian continuum concept for the continuous phase, while the governing equations of the particle phase are developed in accordance with the principles of kinetic theory. In this theory it is postulated that the particulate system can be represented considering a collection of identical, smooth, rigid spheres, adapting a Boltzmann type of equation. This microscopic balance describes the rate of change of the distribution function with respect to position and time. 

In the literature numerous two-fluid models of different complexity have been proposed to predict the fluidized bed reactor cold flow and reactive flow behaviors. Four decades ago emphasis was placed on the modeling of the velocity fluctuation co-variance terms in the dispersed particle fluid phase momentum equations. The early one-dimensional models were normally closed by an elasticity modulus parameterization for the particle phase collisional pressure and a constant viscosity parameter for the corresponding shear stresses. Later, with the improved computer memory and speed capacities, multi-dimensional flow models and more advanced model closures were developed based on the kinetic theory of granular flow (KTGF). Moreover, the... 

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