A kinetic theory for particulate systems with bimodal and anisotropic velocity fluctuations

Abstract

Observations of bubbles rising near a wall under conditions of large Reynolds and small Weber numbers have indicated that the velocity component of the bubbles parallel to the wall is significantly reduced upon collision with a wall. To understand the effect of such bubble-wall collisions on the flow of bubbly liquids bounded by walls, a model is developed and examined in detail by numerical simulations and theory. The model is a system of bubbles in which the velocity of the bubbles parallel to the wall is significantly reduced upon collision with the channel wall while the bubbles in the bulk are acted upon by gravity and linear drag forces. The inertial forces are accounted for by modeling the bubbles as rigid particles with mass equal to the virtual mass of the bubbles. The standard kinetic theory for granular materials modified to account for the viscous and gravity forces and supplemented with boundary conditions derived assuming an isotropic Maxwellian velocity distribution is inadequate for describing the behavior of the bubble-phase continuum near the walls since the velocity distribution of the bubbles near the walls is significantly bimodal and anisotropic. A kinetic theory that accounts for such a velocity distribution is described. The bimodal nature is captured by treating the system as consisting of two species with the bubbles (modeled as particles) whose most recent collision was with a channel wall treated as one species and those whose last collision was with another bubble as the other species. The theory is shown to be in very good agreement with the results of numerical simulations and provides closure relations that may be used in the analysis of bidisperse particulate systems as well as bounded bubbly flows.

Toward a mesoscale‐structure‐based kinetic theory for heterogeneous gas‐solid flow: Particle velocity distribution function

“…Fortunately, it has been shown that with known mean values ( , and u g ), it is possible to predict some of those structural parameters using the EMMS model, however, θ c and θ f cannot be predicted by currently available versions of EMMS model, therefore, one of our future targets is to include the fluctuation characteristics into the EMMS model so that θ c and θ f can be predicted as well. Note that (1) bimodal molecule/particle velocity distribution function for single phase, granular, or gas‐particle flow containing monodisperse gases or particles far from equilibrium is not new, but it is unclear what is the underlying mechanism of the bimodal distribution in previous studies. This study clearly shows that the bimodal distribution is due to the compromise of two different dominant mechanisms in competition, that is the physical principle of mesoscience; (2) Mei et al proposed to use a bimodal velocity distribution to represent the dilute‐dense flow structure in monodisperse gas‐fluidized beds, which is far beyond the equilibrium state.…”

Quadrature-Based Moment Model for Moderately Dense Polydisperse Gas−Particle Flows

“…For hard-sphere collisions, the Boltzmann−Enskog collision term for fixed values of m and m * is given by ,,,, B ( m , m* , u ) = 1 π ∫ R 3 ∫ S + [ normalχ f false( 2 false) ( t , boldx , m , boldu ′ ′ ; boldx − normalσ boldn , italicm* , boldu* ′ ′ ) − f false( 2 false) ( t , boldx , m , boldu normal; boldx bold+ normalσ boldn bold, italicm* , boldu* ) ] false| g ⋅ n false| d n d u* where the function f (2) denotes the pair distribution function, which can further be expressed (see Appendix A) in terms of the single-particle distribution function and a radial distribution function g 0 ( m ,ν, m *,ν*) depending on the solids volume fractions ν and ν*. [For example, for a binary system, an example expression for g 0 is g 0 …”

“…Define first a velocity integral, corresponding to the source term for velocity moments due to inelastic hardsphere Boltzmann-Enskog collisions for given values of m and m* (see eq 2), by 3,4,11,21,35 (see Appendix A for more details)…”

“…For hard-sphere collisions, the Boltzmann-Enskog collision term for fixed values of m and m* is given by 3,4,11,21,35 where the function f (2) denotes the pair distribution function, which can further be expressed (see Appendix A) in terms of the single-particle distribution function and a radial distribution function g 0 (m,ν,m*,ν*) depending on the solids volume fractions ν and ν*. [For example, for a binary system, an example expression for g 0 is where d 1 and d 2 are the particle diameters, n 1 and n 2 are the number densities, and ) 4π(…”

Kinetic theory of binary particles with unequal mean velocities and non-equipartition energies

“…Francisco et al [48] presented a granular mixture model for elastic spheres subject to drag force with different mean velocities which cannot describe dissipation. Thus, the kinetic theory [49] of double granular temperatures and mean velocities makes sense not only for the binary granular mixtures but also for CFD simulation of fluidization [43,45,50,51].…”