Numerical simulation study of liquid-solid fluidized beds based on second-order moment model of particle dynamic restitution coefficient

doi:10.11949/0438-1157.20241414

Abstract

Abstract:

In liquid-solid fluidized beds, the adhesion of liquid to particle surfaces forms a thin film due to liquid phase interparticle forces, altering the particle restitution coefficient and subsequently affecting their collision behavior. Meanwhile, the particle velocity fluctuations also exhibit anisotropic characteristics. Therefore, based on the two-fluid model and the anisotropic kinetic theory of granular flow, a second-order moment model of dynamic restitution coefficient considering the effects of liquid film and the anisotropy of particle velocity fluctuations is established. The simulated results show that the presence of liquid film enhances the energy dissipation of particles during collision and reduces the anisotropy of particle velocity pulsation. With increasing liquid viscosity and particle density, the film thickness increases, both the particle fluctuations and the anisotropy decrease. Furthermore, the predicted particle velocities and porosity by the second-order moment model of dynamic restitution coefficient are in better agreement with experimental values, enabling a more accurate capture of flow field non-uniformity and anisotropic characteristics.

Key words: computational fluid dynamics, fluidized-bed, multiphase flow, dynamic restitution coefficient of wet particles, anisotropic kinetic theory of granular flow

摘要：

在液固流化床中，由于液相黏附力的作用，液体附着在颗粒表面会形成一层薄膜，引起颗粒弹性恢复系数的变化，从而影响颗粒的碰撞行为。与此同时，颗粒的脉动特性也表现出各向异性。故基于双流体模型和各向异性颗粒动理学理论，考虑颗粒恢复系数的动态变化，建立了液膜作用下颗粒动态恢复系数二阶矩模型，以研究液固流化床中包裹液膜颗粒的各向异性流动行为。模拟结果表明，液膜的存在增强了颗粒碰撞时的能量耗散，并减小了颗粒速度脉动的各向异性。随着液体黏度与颗粒密度的增加，液膜厚度增大，颗粒速度脉动与各向异性减弱。另外，动态恢复系数二阶矩模型所预测的颗粒速度和孔隙率分布与Limtrakul的实验值吻合更好，可以更准确地捕捉流场的不均匀性和各向异性特征。

关键词: 计算流体力学, 流化床, 多相流, 湿颗粒动态恢复系数, 各向异性颗粒动理学

CLC Number:

TQ 018

Xi CHEN, Shuyan WANG, Baoli SHAO, Nuo DING, Lei XIE. Numerical simulation study of liquid-solid fluidized beds based on second-order moment model of particle dynamic restitution coefficient[J]. CIESC Journal, 2025, 76(7): 3246-3258.

陈曦, 王淑彦, 邵宝力, 丁诺, 谢磊. 基于颗粒动态恢复系数二阶矩模型的液固流化床数值模拟研究[J]. 化工学报, 2025, 76(7): 3246-3258.

Figures/Tables 17

Fig.1 Structure scheme of liquid-solid fluidized bed

Table 1 Simulated parameters of liquid-solid fluidized bed

参数	实验	模拟
床高/m	1.5	1.5
床宽/m	0.14	0.14
初始床高/m	0.45	0.45
初始颗粒堆积浓度	0.598	0.598
液体黏度/(Pa·s)	0.001	0.001
液体密度/(kg/m³)	994	994
流体进口速度/(m/s)	0.07	0.07
颗粒密度/(kg/m³)	2500	2500
颗粒直径/mm	3	3
泊松比	0.25	0.25
杨氏模量/GPa	70	70

Fig.2 Distributions of particle axial velocity with different mesh sizes

Fig.3 Comparison of simulated particle axial velocity and porosity values with experimental data

Fig.4 Distributions of instantaneous particle concentration and velocity vector (from left to right: SOM-edynamic, SOM, KTGF-edynamic, KTGF models)

Fig.5 Instantaneous axial velocity and concentration of particles in the center and near the wall

Fig.6 Collision frequency of particles as a function of particle concentration

Fig.7 Collision energy dissipation of particles as a function of particle concentration

Fig.8 The radial distributions of the particle second-order moments

Fig.9 Liquid film thickness and restitution coefficient as a function of liquid viscosity

Fig.10 Effects of liquid viscosity on axial velocity and concentration of particles

Fig.11 Effects of liquid viscosity on particle second-order moments

Fig.12 Effects of liquid viscosity on collision energy dissipation of particles

Fig.13 Liquid film thickness and restitution coefficient as a function of particle density

Fig.14 Effects of particle density on axial velocity and concentration of particles

Fig.15 Effects of particle density on particle second-order moments

Fig.16 Effects of particle density on collision energy dissipation of particles

References 38

[1]	Thombare M A, Chavan P V, Bankar S B, et al. Solid-liquid circulating fluidized bed: a way forward[J]. Reviews in Chemical Engineering, 2018, 35(1): 1-44.
[2]	Gui L T, Yang H T, Huang H, et al. Liquid solid fluidized bed crystallization granulation technology: development, applications, properties, and prospects[J]. Journal of Water Process Engineering, 2022, 45: 102513.
[3]	Dabbagh F, Schneiderbauer S. Anisotropy characterization of turbulent fluidization[J]. Physical Review Fluids, 2022, 7(9): 094301.
[4]	Jiang Z C, Hagemeier T, Bück A, et al. Experimental measurements of particle collision dynamics in a pseudo-2D gas-solid fluidized bed[J]. Chemical Engineering Science, 2017, 167: 297-316.
[5]	Vaidheeswaran A, Shaffer F, Gopalan B. Statistics of velocity fluctuations of Geldart A particles in a circulating fluidized bed riser[J]. Physical Review Fluids, 2017, 2(11): 112301.
[6]	Tartan M, Gidaspow D. Measurement of granular temperature and stresses in risers[J]. AIChE Journal, 2004, 50(8): 1760-1775.
[7]	Jung J, Gidaspow D, Gamwo I K. Measurement of two kinds of granular temperatures, stresses, and dispersion in bubbling beds[J]. Industrial & Engineering Chemistry Research, 2005, 44(5): 1329-1341.
[8]	Peng W G, He Y R, Wang T Y. Granular temperature with discrete element method simulation in a bubbling fluidized bed[J]. Advanced Powder Technology, 2014, 25(3): 896-903.
[9]	Gidaspow D, Jung J, Singh R K. Hydrodynamics of fluidization using kinetic theory: an emerging paradigm 2002 Flour-Daniel lecture[J]. Powder Technology, 2004, 148(2/3): 123-141.
[10]	Ng B H, Ding Y L, Ghadiri M. Assessment of the kinetic-frictional model for dense granular flow[J]. Particuology, 2008, 6(1): 50-58.
[11]	Liu Y, Liu J T, Li X L, et al. Large eddy simulation of particle hydrodynamic characteristics in a dense gas-particle bubbling fluidized bed[J]. Powder Technology, 2024, 433: 119285.
[12]	Sun D. Gas-particle flow of pneumatic conveying in vertical pipes simulated using four-way coupled second-order moment method[J]. Powder Technology, 2023, 416: 118225.
[13]	Chen J H, Shi X, Wang S, et al. Investigation into fluctuating anisotropy for biomass gasification in bubbling fluidized bed gasifier[J]. Applied Thermal Engineering, 2018, 138: 774-782.
[14]	Lu H, Chen J H, Liu G D, et al. Simulated second-order moments of clusters and dispersed particles in riser[J]. Chemical Engineering Science, 2013, 101: 800-812.
[15]	Liu H L, Li G H, Liu Y. Hydrodynamic predictions of the ultralight particle dispersions in a bubbling fluidized bed[J]. Processes, 2022, 10(7): 1390.
[16]	Zhong H B, Zhang Y N, Xiong Q G, et al. Two-fluid modeling of a wet spouted fluidized bed with wet restitution coefficient model[J]. Powder Technology, 2020, 364: 363-372.
[17]	Zhu L H, Zhao Z Y, Liu C, et al. CFD-DEM simulations of a gas-solid-liquid fluidization system: effects on the flow field and particle behavior of semi-dry desulfurization fluidized bed[J]. Journal of Environmental Chemical Engineering, 2023, 11(3): 109973.
[18]	Buck B, Heinrich S. Collision dynamics of wet particles: comparison of literature models to new experiments[J]. Advanced Powder Technology, 2019, 30(12): 3241-3252.
[19]	Pan S Y, Ma J L, Liu D Y, et al. Theoretical and experimental insight into the homogeneous expansion of wet particles in a fluidized bed[J]. Powder Technology, 2022, 397: 117016.
[20]	Tian R C, Wang S Y, Li X, et al. Hydrodynamics of wet particles in liquid-solid fluidized beds using kinetic theory of rough spheres model[J]. Powder Technology, 2021, 392: 524-535.
[21]	Ren A X, Tang T Q, He Y R. Evolution of mesoscale structure in fluidized beds with non-spherical dry and wet particles[J]. Advanced Powder Technology, 2022, 33(11): 103794.
[22]	Liu G D, Yu F, Lu H L, et al. CFD-DEM simulation of liquid-solid fluidized bed with dynamic restitution coefficient[J]. Powder Technology, 2016, 304: 186-197.
[23]	Gao Z Y, Liu G D, Guo X Y, et al. A dynamic coefficient of restitution applied to two-fluid model in liquid-solid fluidized bed[J]. Powder Technology, 2022, 402: 117335.
[24]	Yuan Z H, Wang S Y, Shao B L, et al. Simulation study on the flow behavior of wet particles in the power-law liquid-solid fluidized bed[J]. Powder Technology, 2023, 415: 118117.
[25]	Gollwitzer F, Rehberg I, Kruelle C A, et al. Coefficient of restitution for wet particles[J]. Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 2012, 86(1 Pt 1): 011303.
[26]	Davis R H, Serayssol J M, Hinch E J. The elastohydrodynamic collision of two spheres[J]. Journal of Fluid Mechanics, 1986, 163: 479-497.
[27]	Barnocky G, Davis R H. Elastohydrodynamic collision and rebound of spheres: experimental verification[J]. Physics of Fluids, 1988, 31(6): 1324-1329.
[28]	Davis R H, Rager D A, Good B T. Elastohydrodynamic rebound of spheres from coated surfaces[J]. Journal of Fluid Mechanics, 2002, 468(1): 107-119.
[29]	Sun D, Wang S Y, Lu H L, et al. A second-order moment method of dense gas-solid flow for bubbling fluidization[J]. Chemical Engineering Science, 2009, 64(23): 5013-5027.
[30]	Sun D, Wang J Z, Lu H L, et al. Numerical simulation of gas-particle flow with a second-order moment method in bubbling fluidized beds[J]. Powder Technology, 2010, 199(3): 213-225.
[31]	Grad H. On the kinetic theory of rarefied gases[J]. Communications on Pure and Applied Mathematics, 1949, 2(4): 331-407.
[32]	Jenkins J T, Richman M W. Grad's 13-moment system for a dense gas of inelastic spheres[J]. Archive for Rational Mechanics and Analysis, 1985, 87(4): 355-377.
[33]	Peirano E, Leckner B. Fundamentals of turbulent gas-solid flows applied to circulating fluidized bed combustion[J]. Progress in Energy and Combustion Science, 1998, 24(4): 259-296.
[34]	Gidaspow D. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions[M]. Boston: Academic Press, 1994.
[35]	Koch D L, Sangani A S. Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed: kinetic theory and numerical simulations[J]. Journal of Fluid Mechanics, 1999, 400: 229-263.
[36]	Johnson P C, Nott P, Jackson R. Frictional-collisional equations of motion for participate flows and their application to chutes[J]. Journal of Fluid Mechanics, 1990, 210: 501-535.
[37]	Johnson P C, Jackson R. Frictional-collisional constitutive relations for granular materials, with application to plane shearing[J]. Journal of Fluid Mechanics, 1987, 176: 67-93.
[38]	Limtrakul S, Chen J W, Ramachandran P A, et al. Solids motion and holdup profiles in liquid fluidized beds[J]. Chemical Engineering Science, 2005, 60(7): 1889-1900.