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A gas–solid two-fluid model with the second-order moment method is presented to close the set of equations applied to fluidization. With the kinetic theory of granular flow, transport equations for the velocity moments are derived for the particle phase. Closure equations for the third-order moments of velocity and for the fluid–particle velocity correlation are presented. The former is based on a modified model with the contribution of the increase of the binary collision probability, and the latter uses an algebraic model proposed by Koch and Sangani [1999. Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed: kinetic theory and numerical simulations. Journal of Fluid Mechanics 400, 229–263]. Boundary conditions for the set of equations describing flow of particles proposed by Strumendo and Canu [2002. Method of moments for the dilute granular flow of inelastic spheres. Physical Review E 66, 041304/1–041304/20] are modified with the consideration of the momentum exchange by collisions between the wall and particles. Flow behavior of gas and particles is performed by means of gas–solid two-fluid model with the second-order moment model of particles in the bubbling fluidized bed. The distributions of velocity and moments of particles are predicted in the bubbling fluidized bed. Predictions are compared with experimental data measured by Muller et al. [2008. Granular temperature: comparison of magnetic resonance measurements with discrete element model simulations. Powder Technology 184, 241–253] and Yuu et al. [2000. Numerical simulation of air and particle motions in bubbling fluidized bed of small particles. Powder Technology 110, 158–168]. in the bubbling fluidized beds. The simulated second-order moment in the vertical direction is 1.1–2.5 [Muller, C.R., Holland, D.J., Sedeman, A.J., Scott, S.A., Dennis, J.S., Gladden, L.F., 2008. Granular temperature: comparison of magnetic resonance measurements with discrete element model simulations. Powder Technology 184, 241–253] and 1.1–4.0 [Yuu, S., Umekage, T., Johno, Y., 2000. Numerical simulation of air and particle motions in bubbling fluidized bed of small particles. Powder Technology 110, 158–168] times larger than that in the lateral direction because of higher velocity fluctuations for particles in the bubble fluidized bed. The bubblelike Reynolds normal stresses per unit bulk density used by Gidaspow et al. [2004. Hydrodynamics of fluidization using kinetic theory: an emerging paradigm 2002 Flour–Daniel lecture. Powder Technology 148, 123–141.] are computed from the simulated hydrodynamic velocities. The predictions are in agreement with experimental second-order moments measured by Muller et al. [2008. Granular temperature: comparison of magnetic resonance measurements with discrete element model simulations. Powder Technology 184, 241–253] and fluctuating velocity of particles measured by Yuu et al. [2000. Numerical simulation of air and particle motions in bubbling fluidized bed of small particles. Powder Technology 110, 158–168].

A multi-fluid computational fluid dynamics (CFD) model based on kinetic theory of granular flow and Eulerian–Eulerian approach for binary mixture of particles was presented. The multi-fluid model with gas phase and two particle phases of either different particle sizes or densities is used to simulate flows in bubbling gas–solid fluidized beds. The flow behavior of particle mixing or separation in bubbling fluidized beds was numerically predicted. Details of particle collision information were obtained through tracing particle motions based on Eulerian–Lagrangian approach coupled with the discrete hard-sphere model. The distributions of volume fraction, velocity and granular temperature of particles of two different sizes or densities were obtained. The discrete hard-sphere modeling results quantified the granular temperatures, particle fluctuating velocities, particle phase stresses, as well as the particle shear viscosities. The simulations using both the multi-fluid model and the discrete hard-sphere model clearly indicate particle separation phenomenon in the fluidized beds, where relatively larger or heavier particles are observed near the bed bottom than at the bed top region while relatively smaller or lighter particles were found at bed top than at the bed bottom. Better particle mixing can be obtained by increasing the fluidizing velocity.

The solids motion in a gas–solid fluidized bed was investigated using a discrete hard-sphere model. Detailed collision between particles and a nearest list method are presented. The turbulent viscosity of gas phase was predicted by subgrid scale (SGS) model. The interaction between gas and particles phases was governed by Newton's third law. The distributions of concentration, velocity and granular temperature of particles are obtained. The radial distribution function is calculated from the simulated spatio-temporal particle distribution. The normal and shear stresses of particles are predicted from the simulated instantaneous particle velocity. The pressure and viscosity of particles are obtained from both the kinetic theory of granular flow and the calculated stresses of particles. For elastic particles the individual lateral and vertical particle velocity distribution functions are isotropic and Maxwellian. The observed anisotropy becomes more pronounced with increasing degree of inelasticity of the particles.