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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 beds. Predictions are compared with experimental data measured by Jung et al. (2005) in a bubbling fluidized bed and Patil et al. (2005) experiments in a bubbling fluidized bed with a jet. The simulated second-order moment in the vertical direction is on average 1.5–2.3 times larger than that in the lateral direction in the bubbling fluidized bed (Jung et al., 2005). For a bubbling fluidized bed with a jet, the ratio of normal second-order moment in the vertical direction to in the lateral direction is in the range of 0.5–2.5 (Patil et al., 2005). The bubblelike Reynolds normal stresses per unit bulk density used by Gidaspow et al. (2002) are computed from the simulated hydrodynamic velocities. The simulated bubblelike Reynolds normal stresses in the vertical direction is on average 4.5–6.0 times larger than that in the lateral direction in the bubbling fluidized bed (Jung et al., 2005). The predictions are in agreement with experimental second-order moments measured by Jung et al. (2005) and porosity measured by Patil et al. (2005).

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].

International audience; A numerical study was conducted based on the gas-solid two-fluid model using the body-fitted coordinate system to analyze the behavior of particles and bubbles flow in bubbling fluidized beds without and with immersed tubes. The kinetic theory of granular flow was implemented in the model. The images of simulated instantaneous particle concentration and velocity gave the process of the formation, coalescence and eruption of bubbles. The effects of the tube pitch and superficial gas velocity on the fluidization in a bubbling fluidized bed were investigated. Calculated bubble frequencies without and with immersed tubes were in agreement with previous experimental and simulation findings. The wavelet multi-resolution analysis was used to analyze the simulated data of instantaneous particle concentration. From the random-like particle concentration fluctuations, the fluctuating components due to particle flow and bubble motion can be extracted based on the wavelet multi-resolution analysis over a time-frequency plane.