Hydrodynamic Predictions of the Ultralight Particle Dispersions in a Bubbling Fluidized Bed

Abstract

Particle and gas flow characteristics are numerically simulated by means of a proposed gas–particle second-order moment two-fluid model with particle kinetic–friction stress model in a bubbling fluidized bed. Anisotropic behaviors of gas–solid two-phase stresses and their interactions are fully considered by the two-phase Reynolds stress model and their closure correlations. The dispersion behaviors of the non-spherical expand graphite and spherical heavy particles are predicted by using the parameters of distributions of particle velocity, porosity, granular temperature, and dominant frequency. Compared to particles density 2700 kg/m³, ultralight particles exhibit the higher voidages with big bubbles and larger axial-averaged velocity of particles and stronger dispersion behaviors. Maximum granular temperature is approximately 3.0 times greater than that one, and dominant frequency for axial porosity fluctuations is 1.5 Hz that is 1/3 time as larger as that heavy particle.

1. Introduction

To synthesize the fullerene using the expand graphite (EG) particles in fluidized bed has been developed rapidly in recent years. Fullerene is a type of expensive material that has been utilized extensively in the field of superconductivity design, advanced function material and nanotechnology, especially for the urgent requirements for national defense industries. It is noticeable that the expanded graphite particle is a new kind of functional carbon materials that is characterized by loose-porous internal structure with lower density on the order of the 10 [1,2], exhibiting the unique multiphase turbulent flow characteristics in comparison with those conventional density particles (~10³), such as glass, FCC catalysts, and pulverized particles.

Particle dispersion behavior plays an important role in the controlling process of bubble–particle fluidization [1,2,3,4]. Particle dispersions representing by the distributions of porosity, particle velocity and granular temperature in bubbling fluidized bed are determined by the hydrodynamic bubbling fluidizations, implying that it is greatly affected by the particle property and operation conditions. In this process, forces acting on the particles are primarily categorized into drag force by gas, particle collision force, frictional force, and gravity. Many studies and investigations based on both the experiment and modeling strategies have made contributions for revealing the multiphase turbulent mixings, diffusions, and interactions between gas and particle phases via one-way coupling, two-way coupling, and four-way coupling methods for a bubble-fluidized bed, cyclone separator, circulating fluidized bed, and swirling combustor, etc. [5,6,7,8]. Both experimental measurement and computational fluid dynamics (CFD) simulation are effective methods to exploring the mechanism of multiphase turbulent flows. For the invasive measurement, the fiber optical probe (FOP) has been utilized widely to collect the individual and cluster particle velocities and particle porosity for the circulating and the bubbling fluidized bed, the spouted bed, and the cyclone separator, which fails to analyze the uncertainty factors and the error propagations, exerts the flow filed disturbances due to the physical probe tips as well [9,10]. Regarding advanced non-invasive measurement techniques, laser Doppler velocimetry (LDV) [11,12], particle image-velocimetry (PIV) [13,14], particle tracking velocimetry (PTV) [15,16,17,18], and phase Doppler particle analyzer (PDPA) [19,20,21] have been accomplished successfully. Moreover, column static pressure measurements are utilized to investigate the hydrodynamics of bubbling and circulating beds [22,23,24,25,26,27,28,29], as well as magnetic particle-tracking and radio frequency-identifying approaches [30,31]. The computational fluid dynamics (CFD) acts a powerful tool for further understanding of physical phenomena involved in the bubble–particle fluidization [32,33,34,35]. Compared to the Euler–Lagrange discrete particle model (DPM), the two-fluid model (TFM) needs a lower computational time and more closure correlations for transport equations. Euler–Euler TFM supposes both gas and particle phases to be continuous and to inter-penetrate each other, in which sets of models, either empirical or semi-empirical correlations, are required to close the transport equations for solving the two-phase Navier–Stokes governing equations. Moreover, the kinetic theory of granular flow (KTGF) based on the classical kinetic theory of gas is established for the closures of governing equations of four-way coupling high concentration particle flow, that is considering the particles–particles collision [36,37,38,39,40,41].

The kinetic and collisional momentum transferring arising from particle–particle collisions is a function of the particle concentration and the fluctuating velocity. Isotropic granular temperature of particle flow is defined by θ = <C²>/3, where C is the random fluctuating velocity of the particles. Thus, the terms of particle pressure, viscosity of particles, conductive energy flux, and dissipation in the transport equation can be closed by using granular temperature [1,33,36]. For realistic particle–particle collisions, the energy dissipation particle fluctuations may be underestimated on the condition that inelastic energy loss is considered in dense particle concentration, in which particles are contacted closely each other and the binary collision assumption is deficiency. Jenkins et al. (1988) [42] first proposed derived a simple kinetic theory for collisions flows of identical, slightly frictional and nearly elastic spheres, and it was adopted for a better prediction of bubble dynamics in gas–particle fluidized bed [43,44,45]. For the riser reactors, it could lead to the decrease in the cluster size of particles and granular temperature, and the increases in particle–collision frequency at the outlet of downer reactor [46,47,48,49]. Thus, it is reasonable to involve the frictional energy dissipation term to improve the translational energy term of particle. Numerous studies have been reported using kinetic theory of granular flow bubbling fluidized bed based on the isotropic granular temperature hypothesis [50,51,52,53,54,55,56,57]. Sun and Lu et al. (2009, 2010) [58,59] considered that the isotropic behavior of granular temperature mainly comes from the particle–particle collisions and anisotropic granular temperature is significantly dominated by particle motion that is a function of local particle velocity and effects of surrounding transport momentum particle–fluid system. They proposed a second-order moment model for simulation of bubbling fluidization. Velocity distribution function was approximated using a truncated series expansion in Hermite polynomials; moreover, the closure equations for the third-order moment of particle velocity and gas–particle velocity correlation were established. Note that that the anisotropy is remarkable in the bubbling fluidized bed.

Although the isotropic and anisotropic granular temperature can be modeled as aforementioned, there is still a challenge for the modeling and simulation for the anisotropic momentum transfer interactions between the gas and particle phase in the multiphase turbulent flow due to unclear mechanism exploration. The second-order moment model of gas–particle two-phase turbulent flow, first proposed by Zhou et al. (2001) [60] and based on the TFM, can fully consider both the isotropic and anisotropy of gas–solid two-phase stresses and the interactions between two-phase stresses using a set of Reynolds stresses and their correlations, i.e., the gas kinetic energy and granular temperature equations (k-ε-θ), the gas and particle kinetic energy, the granular temperature equations (k-ε-kp-θ), SGS-SOM -θ subgrid stress equation, etc., which have molded and simulated the lower and higher concentrations gas–solid or bubble–particle multiphase turbulent flows [61,62,63,64,65,66].

Up to now, there have few reports regarding the dispersion characteristics of non-spherical EG particles. Liu et al. (2019) [8] demonstrated that their enhancement and concentration at recirculation regions increased with swirling flow number increases in the coaxial chamber. Additionally, they lead to the longer second acceleration length than glass particles in downer reactors and it is easier to gather the concentration peaks at near wall region and to weaken the maximum dense ring in a downer reactor using a second-order moment turbulent model [47]. Validated by measurements, Zhou et al. (2019) [67] revealed that the dispersion characteristics are affected greatly the inlet velocity, resulting in the secondary particle breakages and are sensitive to the particle diameters in cyclone separator using discrete particle model. When understanding that they are quietly different from heavy particles, it is very interesting to explore their unique behaviors in bubbling fluidized bed for further insights by means of either CFD simulations or experimental approaches. The purpose is to reveal its specific behaviors that may be different or similar to those of conventional heavy particles. In this investigation, a particle kinetic friction–stress model and a second-order moment turbulence model were adopted to model and simulation the non-spherical EG particle dispersions in bubbling fluidizations. Modelling and simulation are performed and hydrodynamic parameters, for instances, particle porosity and velocity, granular temperature are analyzed details. It can provide the fundamental data for the strategy synthesis design. Meanwhile, a 3D-PIV measurement system by TSI is now being prepared for experimental study soon.

2. Gas–Particle Two-Phase Flow Model with a Second-Order Moment

A gas–particle two-phase turbulent flow model involved second-order moment is developed based on the Favre-averaged method. Four-way coupled method considering particles collision, and interactions between gas and particle phases, and granular temperature correlation of particle is closed by the particle kinetic–friction stress correlation. The governing equations, the closure correlations and transport equations under the framework of two-dimensional Cartesian coordinates are listed as follows.

2.1. Continuity Equations of Gas and Particle

The continuity equations of gas and particle phases are

$$\frac{\partial }{\partial t}\left({\alpha }_g{\rho }_g\right)+\frac{\partial }{\partial x_k}({\alpha }_g{\rho }_g\overline{u_{gi}})=0$$

(1)

$$\frac{\partial }{\partial t}\left({\alpha }_p{\rho }_p\right)+\frac{\partial }{\partial x_k}({\alpha }_p{\rho }_p\overline{u_{pi}})=0$$

(2)

where α_g, u_g and ρ_g, α_p, u_p and ρ_p are the concentrations, the velocity components, and the densities of gas and particle, respectively [60,62].

2.2. Conservation Momentum Equations of Gas and Particle

The momentum equations of gas and particle phase are

$$\frac{\partial \left({\alpha }_g{\rho }_g\overline{u_{gi}}\right)}{\partial t}+\frac{\partial \left({\alpha }_g{\rho }_g\overline{u_{gk}{u}_{gi}}\right)}{\partial x_k}={\alpha }_g{\rho }_{g}g-{\alpha }_g\frac{\partial \overline{p}}{\partial x_i}+\frac{\partial }{\partial x_k}\left({\tau }_{gik}-{\alpha }_g{\rho }_g\overline{{u^{\prime }}_{gi}{u^{\prime }}_{gk}}\right)-{\beta }_{gp}\left(\overline{u_{gi}}-\overline{u_{pi}}\right)$$

(3)

$$\frac{\partial \left({\alpha }_p{\rho }_p\overline{u_{pi}}\right)}{\partial t}+\frac{\partial \left({\alpha }_p{\rho }_p\overline{u_{pk}{u}_{pi}}\right)}{\partial x_k}={\alpha }_p{\rho }_{p}g-{\alpha }_p\frac{\partial \overline{p}}{\partial x_i}-\frac{\partial \overline{p_p}}{\partial x_i}+\frac{\partial }{\partial x_k}\left({\tau }_{pik}-{\alpha }_p{\rho }_p\overline{{u^{\prime }}_{pk}{u^{\prime }}_{pi}}\right)+{\beta }_{pg}\left(\overline{u_{gi}}-\overline{u_{pi}}\right)$$

(4)

where g is the gravity acceleration, p is the thermodynamic pressure, p_p is the particle pressure, β_pg is the momentum transfer coefficient between gas and particle, τ_g and τ_p are the viscous stress tensors of gas and particle phases, respectively [60,62].

$${\tau }_g={\mu }_g\left[\nabla u_g+\nabla u_g^T\right]-\frac{2}{3}{\mu }_g\nabla \cdot u_{g}I$$

(5)

$${\tau }_p=\left(-p_p+{\lambda }_p\nabla \cdot u_p\right)I+{\mu }_p\left[\nabla u_p+\nabla u_p^T\right]-\frac{2}{3}{\mu }_p\nabla \cdot u_{p}I$$

(6)

2.3. Momentum Transfer between Gas and Particle

The Gidaspow–Huilin [46] bending drag model that combines the Wen–Yu model with the Ergun equation at solid volume fractions above 0.2 is adopted, which is able to connect the discontinuity of solid volume fraction. Between two equations, the following correlations are

$${\beta }_{\mathrm{Ergun}}=150\frac{{\alpha }_s^{}(1-{\alpha }_g^{}){\mu }_g}{{\alpha }_g{(d_p^{})}^2}+1.75\frac{{\alpha }_s{\rho }_g\left|\overrightarrow{u_g}-\overrightarrow{u_p}\right|}{d_p}$$

(7)

$${\beta }_{\mathrm{Wen},\mathrm{Yu}}=\frac{3}{4}{C}_D\frac{{\alpha }_s{\alpha }_g{\rho }_g\left|\overrightarrow{u_g}-\overrightarrow{u_p}\right|}{d_p}{\alpha }_g^{-2.65}$$

(8)

$$\phi =\frac{\arctan [150\times 1.75(0.2-{\alpha }_p)]}{\pi }+0.5$$

(9)

$${\beta }_{pg}=(1-\phi ){\beta }_{\mathrm{Ergun}}+\phi {\beta }_{\mathrm{Wen}-\mathrm{Yu}}$$

(10)

$$R{e}_p=\frac{{\alpha }_g{\rho }_g{d}_p\left|\overrightarrow{u_g}-\overrightarrow{u_p}\right|}{{\mu }_g}$$

(11)

Considering the non-spherical shape of EG particles, the drag force coefficient is developed by the artificial neural algorithm proposed by Yan et al. (2019) [68]. The curve fitting is performed as a function of empirical drag coefficient, wide span Reynolds number, and different particle sphericities, a correlation of drag coefficient validated by experiment data is obtained for non-spherical particles. It can be expressed by

$$\lg C_D=A_0+A_1(\lg R{e}_p)+A_2{(lgR{e}_p)}^2+A_3{(lgR{e}_p)}^3+A_4{(lgR{e}_p)}^4$$

(12)

$$\left(\begin{array}{l}{A}_0\\ A_1\\ A_2\\ A_3\\ A_4\end{array}\right)=\left(\begin{array}{cccccc}-18.5047& 183.2503& -613.2826& 966.0357& -727.4302& 211.3367\\ -12.8162& 75.0120& -163.6044& 150.0228& -49.4788& 0\\ 0.9571& -7.8929& 22.3575& -25.1512& \hspace{1em}9.8015& 0\\ 0.4725& -3.0411& 5.9850& -4.2176& 0.8038& 0\\ -0.0480& 0.4531& -1.0986& 0.9621& -0.2671& 0\end{array}\right)\left(\begin{array}{l}1\\ \varphi \\ {\varphi }^2\\ {\varphi }^3\\ {\varphi }^4\\ {\varphi }^5\end{array}\right)$$

(13)

2.4. Reynolds Stress Transport Equations of Gas and Particle Phases

Reynolds stress transport equation of gas is

$$\frac{\partial \left({\alpha }_g{\rho }_g\overline{{u^{\prime }}_{gi}{u^{\prime }}_{gj}}\right)}{\partial t}+\frac{\partial \left({\alpha }_g{\rho }_g\overline{u_{gk}}\cdot \overline{{u^{\prime }}_{gi}{u^{\prime }}_{gj}}\right)}{\partial x_k}=D_{g,ij}+P_{g,ij}+{\Pi }_{g,ij}-{\epsilon }_{g,ij}+G_{g,gp,ij}$$

(14)

where the terms on the right-hand side represent the diffusion term, the shear production term, pressure-strain term, the dissipation rate term, and the gas-particle interaction term, respectively. [61,62] They are defined by

$$D_{g,ij}=\frac{\partial }{\partial x_k}\left(C_g^{}{\alpha }_g{\rho }_g\frac{k_g}{{\epsilon }_g}\overline{{u^{\prime }}_{gk}{u^{\prime }}_{gl}}\frac{\partial \overline{{u^{\prime }}_{gi}{u^{\prime }}_{gj}}}{\partial x_l}\right)$$

(15)

$$P_{g,ij}=-{\alpha }_g{\rho }_g\left(\overline{{u^{\prime }}_{gk}{u^{\prime }}_{gj}}\frac{\partial \overline{u_{gi}}}{\partial x_k}+\overline{{u^{\prime }}_{gk}{u^{\prime }}_{gi}}\frac{\partial \overline{u_{gj}}}{\partial x_k}\right)$$

(16)

$${\Pi }_{g,ij}={\Pi }_{g,ij,1}+{\Pi }_{g,ij,2}=-C_{g1}\frac{{\epsilon }_g}{k_g}{\alpha }_g{\rho }_g\left(\overline{{u^{\prime }}_{gi}{u^{\prime }}_{gj}}-\frac{2}{3}{k}_g{\delta }_{ij}\right)-C_{g2}\left(P_{g,ij}-\frac{2}{3}{P}_g{\delta }_{ij}\right)$$

(17)

$${\epsilon }_{g,ij}=-{\alpha }_g{\rho }_g\overline{{u^{\prime }}_{gk}{u^{\prime }}_{gi}}\frac{\partial \overline{u_{gi}}}{\partial x_k}$$

(18)

$$G_{g,gp,ij}={\beta }_{pg}\left(\overline{{u^{\prime }}_{pi}{u^{\prime }}_{gj}}+\overline{{u^{\prime }}_{gi}{u^{\prime }}_{pj}}-2\overline{{u^{\prime }}_{gi}{u^{\prime }}_{gj}}\right)$$

(19)

Reynolds stress transport equation of particle is

$$\frac{\partial \left({\alpha }_p{\rho }_p\overline{{u^{\prime }}_{pi}{u^{\prime }}_{pj}}\right)}{\partial t}+\frac{\partial \left({\alpha }_p{\rho }_p\overline{u_{pk}}\cdot \overline{{u^{\prime }}_{pi}{u^{\prime }}_{pj}}\right)}{\partial x_k}=D_{p,ij}+P_{p,ij}+{\Pi }_{p,ij}-{\epsilon }_{p,ij}+G_{p,gp,ij}$$

(20)

Those right-hand side are diffusion shear production, pressure strain stress, dissipation rate, and interaction terms.

$$D_{p,ij}=\frac{\partial }{\partial x_k}\left(C_s{\alpha }_p{\rho }_p\frac{k_p}{{\epsilon }_p}\overline{{u^{\prime }}_{pk}{u^{\prime }}_{pl}}\frac{\partial \overline{{u^{\prime }}_{pi}{u^{\prime }}_{pj}}}{\partial x_l}\right)$$

(21)

$$P_{p,ij}=-{\alpha }_{\mathrm{p}}{\rho }_p\left(\overline{{u^{\prime }}_{pk}{u^{\prime }}_{pj}}\frac{\partial \overline{u_{pi}}}{\partial x_k}+\overline{{u^{\prime }}_{pk}{u^{\prime }}_{pi}}\frac{\partial \overline{u_{pj}}}{\partial x_k}\right)$$

(22)

$${\Pi }_{p,ij}={\Pi }_{p,ij,1}+{\Pi }_{p,ij,2}=-C_{p1}\frac{{\epsilon }_p}{k_p}{\alpha }_p{\rho }_p\left(\overline{{u^{\prime }}_{pi}{u^{\prime }}_{pj}}-\frac{2}{3}{k}_p{\delta }_{ij}\right)-C_{p2}\left(P_{p,ij}-\frac{2}{3}{P}_p{\delta }_{ij}\right)$$

(23)

$${\epsilon }_{p,ij}=\frac{2}{3}{\delta }_{ij}{\alpha }_{\mathrm{p}}{\rho }_p{\epsilon }_p$$

(24)

$$G_{p,gp,ij}={\beta }_{gp}\left(\overline{{u^{\prime }}_{pi}{u^{\prime }}_{gj}}+\overline{{u^{\prime }}_{pj}{u^{\prime }}_{gi}}-2\overline{{u^{\prime }}_{pi}{u^{\prime }}_{pj}}\right)$$

(25)

The closure correlation of the anisotropic interaction between gas and particle is

$$\frac{\partial \overline{{u^{\prime }}_{pi}{u^{\prime }}_{gj}}}{\partial t}+\left(\overline{u_{gk}}+\overline{u_{pk}}\right)\frac{\partial \overline{{u^{\prime }}_{pi}{u^{\prime }}_{gj}}}{\partial x_k}=D_{gp,ij}+P_{gp,ij}+{\Pi }_{gp,ij}-{\epsilon }_{gp,ij}+T_{gp,ij}$$

(26)

Those right-hand side terms represent two-phase diffusion, two-phase shear production rate, two-phase pressure strain, two-phase dissipation, and two-phase interaction terms their definitions are explained in Ref. [61].

2.5. Kinetic Friction-Stress Model of Particle Collisions

A kinetic friction–stress model, considering the particle frictional stresses is developed to close the granular temperature θ [33,69], it is

$${\theta }_i=\frac{1}{3}<{u^{\prime }}_{pi}{>}^2$$

(27)

$$\frac{3}{2}\left[\frac{\partial \left({\alpha }_p{\rho }_p\theta \right)}{\partial t}+\frac{\partial \left({\alpha }_p{\rho }_p\overline{u_{pk}}\theta \right)}{\partial x_k}\right]=\left(-\nabla p_p\overline{I}+{\tau }_p\right):\nabla u_p+\nabla \cdot \left({\Gamma }_p\nabla \theta \right)-{\gamma }_p+{\phi }_p+D_{gp}$$

(28)

where τ_p is particle stress, Γ_p is the conductivity coefficient of granular temperature. The effective coefficient e_eff of particle restitution was introduced to consider the Magnus lift force caused by irregular shape of particle and particles collision [69,70].

Particle pressure denotes the particle normal forces originating from the interaction between particle and particle. Particle kinetic–friction stress, p^f_p, is used to improve it.

$$p_p={\alpha }_p{\rho }_p\left[1+2(1+e_p){\alpha }_p{g}_0\right]\theta +p_p^f$$

(29)

Particle viscosity is considered to be the summation of the shear viscosity term and its correction term coming from frictional contact of particles in slow flow region [69].

$${\mu }_p=\frac{2{\mu }_{p,dil}}{(1+e_p)g_0}{\left[1+\frac{4}{5}(1+e_p)g_0{\alpha }_p\right]}^2+\frac{4}{5}{\alpha }_p^2{\rho }_p{d}_p{g}_0(1+e)\sqrt{\frac{\theta }{\pi }}+{\mu }_p^f$$

(30)

$${\mu }_{p,dil}=\frac{5}{96}{\rho }_p{d}_p\sqrt{\pi \theta }$$

(31)

$${\mu }_p^f=\frac{p_p^f\sin {\omega }_f}{2\sqrt{I_{2D}}}$$

(32)

$$I_{2D}=-\frac{1}{2}D:D$$

(33)

$$D=\frac{1}{2}\left(\nabla u_p+\nabla u_p^T\right)-\frac{1}{3}\left(\nabla \cdot u_p\right)I$$

(34)

$$p_p^f=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\epsilon }_p\le {\epsilon }_{p,\min }\\ 0.05\frac{{\left({\epsilon }_p-{\epsilon }_{p,\min }\right)}^2}{{\left({\epsilon }_{p\max }-{\epsilon }_p\right)}^5}\hspace{1em}\hspace{1em}{\epsilon }_p>{\epsilon }_{p,\min }\end{array}\right\}$$

(35)

μ_p^f represents frictional viscosity, I_2D is for second-variant of tensor strain rate, and ω_f is for internal friction angle of 28.5. Other terms can be referred as Refs. [47,61,62].

2.6. Numerical Scheme and Setting-Up

The finite volume method is used to discrete and solve the mentioned above equations, which the computational domain is divided into a finite number of control volumes. Scalars are stored in the center of control volume and velocity components are marked at control volume surface via the staggered grid system. In order to correct the p-v correlation, the semi-implicit pressure strategy with tridiagonal marching approach and the line-by-line iterations, the underrelaxation quadratic-upstream interpolations, and the central-difference scheme for the convective and the diffusion terms are serviced. The conservation equations are integrated in the space and time, which is performed used upwind differencing in space and implicit in time. As for the boundary conditions, at the inlet, all velocities, and volume fractions of both phases were specified, as well as the outlet pressure. Initially, both velocities of gas and particles are set at zero. At the wall, the gas tangential and normal velocities were set to zero (no slip condition). The normal velocity of particles was also set to zero. The following boundary equations applied for the tangential velocity and granular temperature of particle phases are [69].

$$u_{p,wall}=-\frac{6{\mu }_p{\alpha }_{p,\max }}{\pi {\rho }_p{\alpha }_p{g}_0\sqrt{3\theta }}\frac{\partial u_{p,wall}}{\partial n}$$

(36)

$${\theta }_{wall}=-\frac{k_p{\theta }_{}}{e_{wall}}\frac{\partial {\theta }_{wall}}{\partial n}+\frac{\sqrt{3}\pi {\rho }_p{\alpha }_g{u}_p{g}_0{\theta }^{3/2}}{6{\alpha }_{p,\max }{e}_{wall}}$$

(37)

Firstly, the hydrodynamics of the conventional density particle (2700 kg/m³) in fluidized bed was simulated and validated by the Gera et al., 1998 [71]. After that, its extension for modeling of the particle dispersions of ultralight density of EG (800 kg/m³) was performed and their comparisons as well. The detailed parameters of particle and gas operations and fluidized bed are shown in the

Table 1. Parameters of modeling simulations and experimental measurements.

Parameters	Unit	Value
Height and width of fluidized bed, H, D	m	1.0, 0.6
Packed bed height, Hb	m	0.54
Density of gas, ρg	kg/m3	1.225
Viscosity of gas, μg	Pa.s	1.8 × 10−5
Wall restitution coefficient, ew	-	0.9
Particle restitution coefficient, ep	-	0.9
Density of particles, ρp	kg/m3	2700.0, 800.0
Sphericity of particles, Φ	-	0.8
Diameter of particle, dp	m	0.004
Jetting velocity, ujet	m/s	30.0
Minimum fluidization velocity, umf	m/s	1.8, 10.
Central jetting width, djet	m	0.02

. Gera et al., 1998 measured and simulated the instantaneous particle voidages in a bubbling fluidized bed. The fluidized bed is 600 mm in width, 540 mm in initial packed height, and 1000 mm in height. The particle diameter and density are 0.004 m and 2700 kg/m³, respectively. Central jetting velocity is set to 30 m/s and minimum fluidization velocity is 1.8 m/s. All simulations are continued for 10.0 s of real simulation time. The time-averaged distribution of variables is then computed considering the last 6.0 s of simulation.

3. Result and Discussions

First of all, the proposed model and numerical algorithm should be validated by experimental data for further applications. The schematic of bubbling fluidized bed is illustrated in Figure 1 (Gera et al., 1998 [71]). Three kinds of grid size system are used to perform the test of grid independence, which are the coarse size (2.0 cm × 4.0 cm, grid amount 30 × 250), the medium sizes (1.0 cm × 1.0 cm, grid amount 60 × 1000), and the fine sizes (0.5 cm × 0.5 cm, grid amount 120 × 2000) in the coordination system (bed width × bed height). Averaged axial particle velocity using three grid sizes are calculated, and then a comparison for grid resolution is performed (see Figure 2). We can see that medium grid system is acceptable due to the economic computation consumption and satisfied errors. Thus, we take advantage of the grid size of 1.0 cm by 1.0 cm for the following simulation:

Figure 3 shows the instantaneous snapshot of particle porosity at the real-time of 0.40 s with particle used in experiment ρ_p = 2700 kg/m³. In Gera’s experiment and simulation, the quantitative data of particle hydrodynamics, i.e., particle velocity and fluctuation velocity, were not provided. They focused on the comparisons with particle voidage that simulated by TFM and DEM models using snapshots, as well as the experimental observations. By the analogue of this methodology, the simulated profiles of particle voidage or porosity using the proposed gas–particle second-order moment two-fluid model was compared by Gera’s TFM simulation results in terms of the bubble topology observations. After that, the model, numerical algorithm and in-house codes were used to predict the ultralight particles. We can see that the formation of the bubble relies on the specific characteristics for the bubbling fluidized bed because it plays important roles on controlling multiphase hydrodynamic behaviors, indicating that it is critical for accurate prediction of bubble shape, size, and porosity parameters. We can see that bubble preferentially forms at the central jet orifice and begins to move towards the center as rising. Meanwhile, the trajectory traps particles into the wake underneath of bubbles, thereby incurring the particle migrations. The prediction of bubble topology is highly consistent with the measurement snapshot. Therefore, the proposed model, numerical algorithm, and in-house codes can be utilized for the reasonable prediction of the dispersions of ultralight EG particles.

Figure 4 and Figure 5 show the instantaneous porosities and velocities of EG and particles used in experiment in a bubbling fluidized bed at the jetting velocity of 30.0 m/s, ρ_p = 800 kg/m³ and ρ_p = 2700 kg/m³, respectively. Gera’s studies focus on the comparisons with particle voidage that simulated by TFM and DEM models using snapshots, as well as the experimental observations. They suggested that TFM simulations are extremely sensitive to the inter-particle friction, incorporated by solid pressure and viscosity, which might obscure the observations of true bubbling characteristics of fluidized beds. It can be seen that the packed particles at the jetting orifice were pushed when jetting gas entered the interior bed, leading to the rapid enhancement of particle porosity. Thus, the initial bubbles are formed at the beginning of fluidization. The appearances of bubbles indicate the instability of bubble–particle system. Surrounding particles are entrapped in the bottom of bubbles due to smaller pressure and the bubbles thereby detach from the gas distributor to migrate toward the top surfaces. The channeling of the gas through bubbles is clearly visible. Particle circulation patterns come into being, which cause particles to eventually ascend from center region and descend at near wall regions. Furthermore, bubbles swinging from left to right can be observed. The fluctuations of particle porosity and particle mixing mainly originate in the bubble motions as a result of the spatio-temporal patterns of fluidizations, which are sensitive to the particle density. For ultralight EG particles, gas flow easily causes a greater effect on particle motion, demonstrating that majority of bubbles prone to retaining the bed instead of eruptions upward at top surface in comparison with heavy particle, as well as better followability behaviors carried by the gas flow. In addition, the wider size distribution characteristics of bubbles are dominant since bubbles gather at the near wall regions, and the particle velocity circulations along the total bed height are remarkable.

Figure 6 and Figure 7 display the histograms of the center and wall axial and lateral velocities of EG articles at the heights of 0.24 m and 0.5 m and r/Width = 0.5 and 0.9, respectively. Mean values and standard deviations represent the mean and fluctuations of particle velocities. For the positions of r/Width = 0.5, mean values and standard deviations of axial particle velocity are larger than those of lateral velocities in additions to both higher values at a height of 0.5 m than ones at 0.24 m, and wider size distribution characteristics, similar trends for r/Width = 0.9 can be obtained as well. Furthermore, those axial velocities at r/Width = 0.5 are approximately 2.6 times greater than ones at r/Width = 0.9 (see Figure 7). The reasons are that particles at central flows are subjected to the stronger drag force and vibrant particle–particle collisions, resulting in the intensified velocity fluctuations. However, those particles used in experiment (ρ_p = 2700 kg/m³) are significantly different (see Figure 8 and Figure 9). For the height of 0.24 m and r/Width = 0.5, standard deviations of both axial and lateral particle velocities are approximately 1.7 times greater than those of the height of 0.5 m and larger mean values and show a similar trend for those of at r/Width = 0.9. Relatively weakened fluctuations for lateral velocities are described. In summary, the axial fluctuations for ultralight particles are vigorous at higher sections, and heavy particles exhibit the opposite trend since lighter particles are prone to be driven effectively by acting forces.

Figure 10 displays the comparisons of the averaged axial velocities of particles with density of ρ_p = 2700 kg/m³ and ρ_p = 800 kg/m³ at the height of 0.24 m and 0.5 m. The axial velocities of EG particles are far greater than those of heavy particles, although the ratio of ultralight density to heavy particles is only 1/3 times, representing the serious particle back-mixing and circulation behaviors in the bed. This implies that the distribution of particle velocity is sensitive to particle density.

Figure 11 and Figure 12 show the quantile–quantile (Q-Q) plots of central and wall porosity of particles used in experiment at the height of 0.24 m and 0.5 m, r/Width = 0.5 and 0.9, ρ_p = 800 kg/m³ and ρ_p = 2700 kg/m³, respectively. This figure demonstrates whether the distributions for a random variable are a Gaussian distribution. The Y axis is the quantile of the samples, and the X axis is the quantile of normal distributions. The primary reason is that the multiphase flow is affected by the complex turbulence diffusions, anisotropic particle dispersion, and particle inertia. The variations of particle porosity reflect the change of drag force, leading to the particle randomly falling into the bubble’s inside and the variation of the bubble’s retaining time in an unstable fluid–particle system.

The power spectrum density (PSD) function transformed by the fast Fourier transform (FFT) algorithm is a type of a methodology to identify the single bubble, multiple bubbles in turbulent flows. Dominant frequency can be used to characterize the particle dispersions in gas–particle two-phase turbulent flow. Wide-band spectrums imply that more bubbles and narrow-band spectrums associated with sharp peak profiles denote a single bubble or slugging behaviors [72,73]. Figure 13 and Figure 14 represent the power spectral density as a function of the porosity of EG and particles used in experiment at the height of 0.24 m and 0.5 m, r/Width = 0.5 and 0.9, ρ_p = 800 kg/m³, and ρ_p = 2700 kg/m³, respectively. For the EG particle at the height of 0.5 m, the dominant frequencies for the central and wall positions are approximately 1.5 Hz and 1.3 Hz, which are greater than those approximately 1.3 Hz and 0.3 Hz at the height of 0.24 m, indicating that the central fluctuations have the similar dominant frequencies, and the wall fluctuation at 0.5 m is approximately 3.0 times greater than that of 0.24 m. Each sharp peak denotes that the preliminary single larger bubble exists in the fluid-particle system. Two or more bubbles can be observed for experimental particle due to multi-peaks in central fluctuations, corresponding the approximately 4.5 Hz and 9.0 Hz. Multiple bubbles generate the breaking-up, smaller vortex of gas and particle phases more easily, indicating that higher frequency is accompanied with smaller residence time. The discrepancy in particle density leads to the variations of dominant frequency, and experimental particle is 3.0 times larger than that of ultralight particle in central fluctuations.

Figure 15 denotes the comparisons of porosities particles with density of ρ_p = 2700 kg/m³ and ρ_p = 800 kg/m³ at the height of 0.24 m and 0.5 m. It is noted that particle inertia has great impact on the particle motions and porosities. Smaller particle porosities are found for heavy particles because larger particle inertia aggregates bubble coalescence and breaking-up and particle falling and particle circulations around smaller size bubbles, which intensifies the interactions between and particle and particle–particle collisions.

Figure 16 denotes the comparisons of the averaged granular temperature of particles with density of ρ_p = 2700 kg/m³ and ρ_p = 800 kg/m³ at the height of 0.24 m and 0.5 m. Those ultralight particles are far greater than heavy particles counterparts, implying that the granular temperatures depend on flow patterns and oscillations of particles. Gas will pass through the particles to top surface in accompanying with much more particles; however, particles will backflow as porosity increases as a result of the intensification of particle collisions. Larger size bubbles caused by ultralight particles are favorable for inducing the violent particle falling and circulation rate surrounding the interior bubble, accounting for the larger porosity and smaller drag force. However, only smaller bubbles and breakups at local regions, rather than in the global bed, are produced by heavy particles, resulting in the weak particle collisions. Figure 17 shows the distributions of granular temperature of particles with a density of ρ_p = 2700 kg/m³ and ρ_p = 800 kg/m³. Larger values mainly gather the surrounding regions of bubbles, and maximum values of ultralight particles are probably 4.0 times greater than those of heavy particles. Larger size bubbles with longer residence times makes more contributions to an increase in granular temperature.

4. Conclusions

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In this work, the dispersion of non-spherical ultralight EG particles in a bubbling fluidized bed was modeled and simulated numerically using a particle kinetic–friction stress model involving a second-order moment two-phase turbulent model. The behaviors of particle porosity, particle and fluctuation velocity, granular temperature, and fast Fourier transforms were investigated and discussed. It was observed that the distinctive dispersion characteristics listed below are different from the conventional materials. They are as follows:

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Larger size bubbles remain in the fluid system rather than moving toward the top surface with eruptions, resulting in the highier voidages;

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At r/Width = 0.5 and heights of 0.5 m and 0.24 m, the mean values of axial particle velocity are almost equivalent, but they are approximately 4.0 times and 1.5 times larger than those of heavy particles, respectively;

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Maximum granular temperature is approximately 3.0 times greater than its heavy particle counterpart;

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Dominant frequency for axial porosity fluctuations is smaller than that of the heavy particle.

Further utilizations for dense particle two-phase flows in spouted beds, circulated fluidized beds, swirling fluidized beds, etc., will be conducted in our next work.