Numerical Studies on Teeter Bed Separator for Particle Separation

: Teeter Bed Separators (TBS) are liquid–solid ﬂuidized beds that are widely used in separation of coarse particles in coal mining industry. The coal particles settle in the self-generating medium bed resulting in separation according to density. Due to the existence of self-generating medium beds, it is di ﬃ cult to study the sedimentation of particles in TBS through experiments and detection methods. In the present research, a model was built to investigate the bed expansion characteristics with water velocity based on the Euler–Euler approach, and to investigate the settling of foreign particles through bed based on the Euler–Lagrange approach in TBS. Results show that the separation of in TBS should be carried out at low water velocity under the condition of stable ﬂuidized bed. Large particles have a high slip velocity, and they are easily ﬂowing through the bed into the light product leading to a mismatch. The importance of self-generating bed on separation of particle with narrow size ranges are clariﬁed. The model provides a way for investigating the separation of particles in a liquid–solid ﬂuidized bed and provides suggestions for the selection of operation conditions in TBS application.


Introduction
Fluidization technology has been widely used in chemical reactions, mineral/coal separation and pneumatic conveying in the past few decades [1][2][3][4]. Teeter bed separator (TBS) is one of liquid-solid fluidized beds mainly applied to the separation of coal fines and mineral, desliming and tailings recovery [5][6][7][8]. It is one kind of gravity separators based on the principle of fluidization and hindering settling, in which particles with a wide range of densities and sizes have different settling velocities. The heavy particles have a higher settling velocity and settle through a multi-particle suspension consisting of water and intermediate mass particles, while the light particles have a lower settling velocity and rise through the suspension, thereby achieving the particle separation.
There are many investigations focusing on the application of TBS. For improving the separation efficiency, several measures were taken on the equipment, such as optimization of the feeding structure, installation of inclined plates and introducing of the pulsating water flow and bubbles and thus, a series of new separators were produced [9][10][11][12]. It has been reported that new separators were well applied in mineral sand, low grade iron ore, chromite and coal fines in industry, as well as pyrite, phosphate ore, heavy minerals, electronic waste and hematite in laboratory and pilot plants [13][14][15]. Applications separating these materials have been illustrated the improvement for separation performance of TBS. Moreover, the influence of operating variables, including bed density, water velocity, feed concentration and feed rate on classification or separation performance for coal and non-coal materials was studied [16][17][18]. Research showed that water velocity and bed density are much crucial for particles separation or classification and the significance of the parameters on cut size/density, possible particle size to perform numerical calculation. In other words, the simulation is conducted at a particle scale. The forces acting on particles are not calculated through the existing models but obtained by integrating viscous force and pressure force on the particle surface. There is no doubt that this method needs a huge amount of computational calculation. To better use this method, two methods called fictitious domain method and immersed boundary method have been developed to reduce the computational consumption, in which the domain grids will not change with flow time. Macro and micro information such as the law of particle force, particle trajectory, the velocity of particle and fluid and particle pulsation can be obtained by using these methods [45,46]. The dynamic mesh method, in which the mesh dynamically changes with flow time, is more practical for calculating the flow field due to the boundary motion. A good case is to simulate the change of flow field in a cylinder with piston motion. The method was also utilized to simulate the particle motion. Mitra et al. [47] studied the collision process between one glass bead and one water droplet. Ghatage et al. [32] predicted the settling velocity of a 6 mm foreign particle in a liquid-solid fluidized bed by the E-E approach coupled with dynamic mesh method. Presently, DNS method is still limited by the total number of particles.
As has mentioned, particles in TBS are fine in size and have wide range of densities. they move through the self-generating medium bed, resisting counter water flow and forces from fluidized particles. In the experimental investigations of Grbavčić and Vuković [28] and Van der Wielen et al. [30], they conducted the experiments by preparing special fluidized particles, using stopwatch and laying strong light behind the bed to trace the particle in a column. Even though PIV (particle image velocimetry) detected technology was utilized in the experiment of Ghatage [32], the fluidized particles made of glass were required to be larger in size for a good transparency. Based on the work of Richardson and Zaki [22], Galvin [21] proposed an empirical expression of the particle slip velocity in combination with the actual parameters of bed pressure in TBS. We can see that the experimental investigations are always difficult for high costs, limited experimental conditions and detection tools. However, the particle motion is fundamental and most important, whether it is for the evaluation of operating parameters on separation performance or a good prediction and optimization on the separation process. Numerical literature review demonstrated that the particle motion behavior in liquid fluidized bed can be simulated well. More details of particle-liquid flow were obtained which can be used to make up for the defects of the experiment effectively. It was also found that limited efforts have been put on numerical studies on the characteristics of teeter bed and particle settling behavior in TBS specifically. It is thought desirable to complete studies on two aspects for a good understanding of the separation process. Considering applicability of three numerical methods, we proposed that the pseudo-fluid method is suitable to simulate the fluidized bed composed of a large number of fine particles with less computational consumption. Discrete phase method is more preferable to track the motion of particles through fluidized bed.
Thus, an attempt is made to study the change of bed characteristics with operating water velocity and the settling behavior of particles introduced to the bed based on a combination of the Euler-Euler and Euler-Lagrange approach. The simulation results are compared with the empirical formulas from literature studies. In addition, the influence of operating water velocity on the separation is analyzed from the perspective of flow regimes in bed. The significance of this research is to establish a reasonable approach to study the fundamentals of particle-liquid flow in TBS numerically. The empirical formulas for comparison with simulated results are summarized in Table 1.

Model Description
The simulation was carried out on a commercial software, called FLUENT 15.0 based on the finite volume method. The present modeling of the fluidized bed (teeter bed) was based on a two-dimensional Euler-Euler approach where the fluidized particles are considered as continuous phase, namely pseudo-fluid. The resolve of their equation is based on the kinetic theory of granular flow. Then the Euler-Lagrange approach is combined to model the foreign particles motion in which particle is treated as discrete phase. The relevant governing equations for two approaches have been well developed and documented in the literature [34,35,52]. For a clear understanding, here, the models are briefly described below.

Governing Equation for Liquid and Solid Phases
The liquid and solid phases are considered as continuous phase, respectively, in the Euler-Euler approach, so they have the similar mass and momentum transport equations. These equations can be written as follows: The mass transport equations for liquid and solid phases are: The momentum transport equations for liquid and solid phases are: where = τ S presents stress tensor, Pa; p S presents the solid pressure, Pa. The interphase exchange coefficient (K SL ) is modeled using the Huilin-Gidaspow model which is suitable for both high-concentration system and dilute system and it is improved based on the coefficient of Wen and Yu and Eurgun [35]. The Huilin-Gidaspow model coefficient is defined as: where: For ε L > 0.8 where: For ε L < 0.8 where: If one foreign particle reaches the force balance in teeter bed, the motion equation for this individual particle is: d where g is gravitational acceleration originated from particles gravity. The term of CD 2 presents drag force. F other is other force caused by virtual mass force, pressure gradient force and Saffman lift. CD 2 is defined as follows: where a 1 , a 2 and a 3 are constants that apply over several ranges of Re (Re D ),

Simulation Conditions
A 2D geometric model with a diameter of 120 mm and height of 800 mm is utilized in the present simulation and the details for physical and geometrical parameters are given in Table 2. The mesh is divided with the ICEM CFD software contained in the software package of Ansys15.0. The mesh size is selected as 2 mm based on the literature [34,35] in which particles have the similar size and density with present work, and the Euler-Euler approach are also used to simulate the fluidized bed. In the literature study, the mesh size of 0.5 mm, 1 mm, 2 mm, 3 mm and 3.5 mm is set to explore the mesh-independent simulation. It is found that the bed voidage does not change below 2 mm size. Therefore, we choose the same mesh size of 2 mm. The literature also conducted investigations for the independence of time step and the value of 0.001 s is referred to the present study. In our paper, these setting parameters are finally compared with R-Z model in the next section of result and discussion. Choosing a right turbulence model is also crucial for the accuracy of the simulation. Although the operating water velocity is small, the strong turbulence existed in the bed is obvious. The standard k-ε is incorporated in the simulation of fluidized beds extensively and it exhibits a good description for the turbulence flow in bed. The theory about k-ε turbulence model can be referred to the publications [35]. The collision coefficient between fluidized particles is used with a default value of 0.9. The SIMPLE algorithm is employed to solve the pressure-velocity coupling and the governing equations are discretized by the second-order upwind scheme. Table 2. Simulation conditions and parameters.

Parameters Values
Width of the column 120 mm Height of the column 800 mm Diameter of fluidized particle 1 mm Density of fluidized particle 2607 kg/m 3 Diameter of foreign particle 2, 4, 6 mm Density of foreign particle 2607 kg/m 3  The whole simulation is performed in two steps. The first step is the study of the change of bed properties with water velocity based on the Euler-Euler approach. In this step, fluidized particles with size of 1 mm and a density of 2607 kg/m 3 filled the region of 120 mm width × 200 mm height of the column with an initial volume fraction of 0.55. The water flow is introduced from the bottom of the column and the boundary condition of the bottom is defined as velocity inlet. The boundary condition of the top of the column is set as pressure outlet, and two vertical walls are considered as walls with no slip boundary conditions. When all things are prepared well, the teeter bed begins to be fluidized under a wide variety of water velocities of 0.008-0.056 m/s. The judgment of the steady state of fluidization is that the calculated residuals is converged, and the particle volume fraction does not change with the simulated time. After the bed fluidization has stabilized. The DPM (discrete phase model) in FLUENT 15.0 based on Euler-Lagrange approach is started to inject foreign particles from the top of the column separately, and to track their classification velocity through the bed. The final data can be exported from FLUENT 15.0 by the selection of "export" option, and the subsequent data processing is conducted in Microsoft Excel. A schematic diagram of the bed fluidization and particle settling in the bed appears in Figure 1. According to Figure 1, the slip velocity of one foreign particle through the teeter bed can be calculated by using Equation (13): where u D is the slip velocity of foreign particle, also called relative velocity. u D presents the classification velocity which is the absolute velocity observed relative to the column. u L0 is the interstitial liquid velocity.

The Bed Expansion Characteristics
In coal processing, a large amount of narrow sized particles are fluidized in counter water flow to form the teeter bed. Therefore, water velocity is a key factor affecting the bed characteristics and also an important operating parameter for coal separation. In this section, the simplification of mono size particles was assumed to form a self-generating medium bed. Figure 2 shows the change of bed expansion with fluidization velocity varied from 0.008 to 0.056 m/s. It can be seen that the bed was fluidized at all water velocity. For the lower water velocity of 0.008 m/s and 0.01 m/s, the bed height was lower than the initial state value, and corresponding volume fraction was larger than the initial value of 0.55. As the water velocity increases, the bed expansion gradually increases and at the water velocity of 0.016 m/s, the bed height has exceeded the initial state. When water velocities in the range of 0.008-0.160 m/s, the consistent color in the figure revealed that the local volume fraction of particles was uniform along the whole column. When the water velocity increases further, especially in higher value of 0.056 m/s, the uniformity of the bed deteriorates because of the strong turbulence caused by many eddies. This phenomenon can also be illustrated from the flow patterns of the bed in Figure 3. The liquid flow along the column was stable and very regular at the relative lower water velocity, and the particle flows was small and symmetric which was helpful to achieve the homogeneous regime. However, at high water velocity, the regularity of the flow state was disturbed, for large circulation flows of liquid and particles increases the turbulence intensity resulting in heterogeneous regime in the bed.

The Bed Expansion Characteristics
In coal processing, a large amount of narrow sized particles are fluidized in counter water flow to form the teeter bed. Therefore, water velocity is a key factor affecting the bed characteristics and also an important operating parameter for coal separation. In this section, the simplification of mono size particles was assumed to form a self-generating medium bed. Figure 2 shows the change of bed expansion with fluidization velocity varied from 0.008 to 0.056 m/s. It can be seen that the bed was fluidized at all water velocity. For the lower water velocity of 0.008 m/s and 0.01 m/s, the bed height was lower than the initial state value, and corresponding volume fraction was larger than the initial value of 0.55. As the water velocity increases, the bed expansion gradually increases and at the water velocity of 0.016 m/s, the bed height has exceeded the initial state. When water velocities in the range of 0.008-0.160 m/s, the consistent color in the figure revealed that the local volume fraction of particles was uniform along the whole column. When the water velocity increases further, especially in higher value of 0.056 m/s, the uniformity of the bed deteriorates because of the strong turbulence caused by many eddies. This phenomenon can also be illustrated from the flow patterns of the bed in Figure 3. The liquid flow along the column was stable and very regular at the relative lower water velocity, and the particle flows was small and symmetric which was helpful to achieve the homogeneous regime. However, at high water velocity, the regularity of the flow state was disturbed, for large circulation flows of liquid and particles increases the turbulence intensity resulting in heterogeneous regime in the bed.      [22] and compare model data with present water velocity to examine the simulation results. The R-Z model was intensively used in several publications to compare their study results with model values. The predicted values in the present model were also compared with parts of experimental and simulated data of the publication [34,35,53], which is summarized in Figure 5. It can see from Figure 5 that the present values exhibit good consistence with experimental data of the literature and R-Z model. Thus, it was can be concluded that Figure 5 show a good illustration for the rationality of present simulated values. In order to have a clear observation, Figure 4b Figure 4d shows the pressure drop changes with water velocity. It can be observed that the bed drop hardly varies with water velocity and the simulated value was mostly around the theoretical value of 1736.06 Pa in Figure 4d calculated by the Equation (14). This phenomenon proves the validity of the simulation results again.
where L presents the initial bed height. Its value was 200 mm in the present study.   [22] and compare model data with present water velocity to examine the simulation results. The R-Z model was intensively used in several publications to compare their study results with model values. The predicted values in the present model were also compared with parts of experimental and simulated data of the publication [34,35,53], which is summarized in Figure 5. It can see from Figure 5 that the present values exhibit good consistence with experimental data of the literature and R-Z model. Thus, it was can be concluded that Figure 5 show a good illustration for the rationality of present simulated values. In order to have a clear observation, Figure 4b Figure 4d shows the pressure drop changes with water velocity. It can be observed that the bed drop hardly varies with water velocity and the simulated value was mostly around the theoretical value of 1736.06 Pa in Figure 4d calculated by the Equation (14). This phenomenon proves the validity of the simulation results again.
where L presents the initial bed height. Its value was 200 mm in the present study.  Bed density is a reflection of particle density, solids volume fraction and liquid velocity. As shown in Figure 6, bed density reduces as the increase of water velocity and they show a good power relationship, which can be quantified by the fitting Equation (15). In real separation cases, when the bed density reduces resulting from the increase in water velocity, most of light particles were directly dragged to the overflow subjected to the hydraulic force without effective separation. When the water Bed density is a reflection of particle density, solids volume fraction and liquid velocity. As shown in Figure 6, bed density reduces as the increase of water velocity and they show a good power relationship, which can be quantified by the fitting Equation (15). In real separation cases, when the bed density reduces resulting from the increase in water velocity, most of light particles were directly dragged to the overflow subjected to the hydraulic force without effective separation. When the water velocity was lower, the bed density increases as the solid phase becomes dense inside bed. At the same time the viscosity of the suspension also increased as a large number of particles were crowded together. Thus, in a system with high concentration particles, the properties of the fluidized suspension was close to that of heavy liquid, in which particles were separated according to density.
where ρ b is bed density, g/cm 3 ; u is water velocity, m/s. Fluidized particles move differently with water velocity during the whole bed expansion process. Figure 7 plots the particle velocity along the height of the bed at different water velocity. The overall trend was that the particle velocity increases as the water velocity increases. At lower water velocity, the bed reaches good homogeneity which can be seen from former analysis and the velocity distribution of particles was also relatively uniform. The higher water velocity brings degradation for the uniformity of the particle velocity distribution. The result was consistent with former analysis of the impact of water velocity on the volume fraction of particles in the bed.
The above analysis shows that the movement of fluidized particles was relatively regular, and a better homogeneous flow was formed at lower water velocity, which was conducive to maintaining the bed internal stability. At higher water velocity, a large number of vortices will not only make the fluidized particles fluctuated strongly, but also promote fine particles easily to directly enter the overflow or underflow without being separated. Therefore, for coal material with a wide range of size and density, a better separation was recommended under the operating condition of a lower water velocity. Fluidized particles move differently with water velocity during the whole bed expansion process. Figure 7 plots the particle velocity along the height of the bed at different water velocity. The overall trend was that the particle velocity increases as the water velocity increases. At lower water velocity, the bed reaches good homogeneity which can be seen from former analysis and the velocity distribution of particles was also relatively uniform. The higher water velocity brings degradation for the uniformity of the particle velocity distribution. The result was consistent with former analysis of the impact of water velocity on the volume fraction of particles in the bed.
The above analysis shows that the movement of fluidized particles was relatively regular, and a better homogeneous flow was formed at lower water velocity, which was conducive to maintaining the bed internal stability. At higher water velocity, a large number of vortices will not only make the fluidized particles fluctuated strongly, but also promote fine particles easily to directly enter the overflow or underflow without being separated. Therefore, for coal material with a wide range of size and density, a better separation was recommended under the operating condition of a lower water velocity.

Settling Behavior of Foreign Particles in Bed
The teeter bed above provided a hindering settling environment for other particles (here, called foreign particles) which have different properties (size or density) with the fluidized particles. Foreign particles move resisting forces from the bed to the overflow or underflow and achieve the separation. In this section, several foreign particles were introduced to the teetered bed to study their motion behavior. The results are shown in Figure 8a that a 4 mm foreign particle with density of 2607 kg/m 3 was introduced from the top of the column. We can see that the particle has gone through four stages: accelerating, reaching balance in water, deceleration and reaching equilibrium in the bed. The particle velocity decreased significantly due to the resisting force from the bed; it decreased as the volume fraction of particles increased. For a clear comparison of simulated slip velocity with that of literature models, the particle slip velocity was normalized by dividing the free settling velocity. Figure 8b compares the normalized slip velocity of foreign particle of 4 mm size and density of 2607 kg/m 3 with the literature values. What needs to be explained: the free settling velocity of all particles involved in these models were the simulated data. For 2-mm, 4-mm and 6-mm glass particles, the simulated free settling velocities were 0.282 m/s, 0.445 m/s, 0.565 m/s, and for 2 mm, 4 mm, 6 mm steel particles, the values were 0.644 m/s, 0.969 m/s, 1.173 m/s. The deviation of simulated free settling velocity of particles were within 8% compared with Zigrang and Sylvester model [54]. In Figure 8b, it was observed that the classification velocity of the particle decreased with the increase of volume fraction of particles. The phenomenon is exhibited in both simulated values and literature models. It implies that the resisting force acting on foreign particles increased as the volume fraction of particles in bed increased. Moreover, the normalized slip velocity of the 4 mm particle through the bed was close to that of Joshi model.

Settling Behavior of Foreign Particles in Bed
The teeter bed above provided a hindering settling environment for other particles (here, called foreign particles) which have different properties (size or density) with the fluidized particles. Foreign particles move resisting forces from the bed to the overflow or underflow and achieve the separation. In this section, several foreign particles were introduced to the teetered bed to study their motion behavior. The results are shown in Figure 8a that a 4 mm foreign particle with density of 2607 kg/m 3 was introduced from the top of the column. We can see that the particle has gone through four stages: accelerating, reaching balance in water, deceleration and reaching equilibrium in the bed. The particle velocity decreased significantly due to the resisting force from the bed; it decreased as the volume fraction of particles increased. For a clear comparison of simulated slip velocity with that of literature models, the particle slip velocity was normalized by dividing the free settling velocity. Figure 8b compares the normalized slip velocity of foreign particle of 4 mm size and density of 2607 kg/m 3 with the literature values. What needs to be explained: the free settling velocity of all particles involved in these models were the simulated data. For 2-mm, 4-mm and 6-mm glass particles, the simulated free settling velocities were 0.282 m/s, 0.445 m/s, 0.565 m/s, and for 2 mm, 4 mm, 6 mm steel particles, the values were 0.644 m/s, 0.969 m/s, 1.173 m/s. The deviation of simulated free settling velocity of particles were within 8% compared with Zigrang and Sylvester model [54]. In Figure 8b, it was observed that the classification velocity of the particle decreased with the increase of volume fraction of particles. The phenomenon is exhibited in both simulated values and literature models. It implies that the resisting force acting on foreign particles increased as the volume fraction of particles in bed increased. Moreover, the normalized slip velocity of the 4 mm particle through the bed was close to that of Joshi model. To further explore the agreement between the simulation results and the Joshi model, the settling behavior of 2 mm, 4 mm and 6 mm glass beads and 2 mm, 4 mm and 6 mm steel particle through the bed with a volume fraction of 0.55 was studied in Figure 8c,d. It can be seen that the simulation results keep good agreements with the Joshi model. The influence of a high concentration of fluidized particles on foreign particle sedimentation was significant and can decreased by about 20% of free settling velocity. We can also see that there were some differences between the simulated data and the predicted values in other models. It can also be seen from Figure 8c, the predicted slip velocity for glass beads in models of Di Felice, Van der Wielen and Grbavcic presents a good consistency. In these models, the particle velocity was greatly reduced more than half of the value, and the values in Kunii model was between Joshi model and other three models. Figure 8d shows that the predicted values of steel particles in Kunii and De Felice model were relatively close and were also close to the present simulated results. The particle velocity assessed by Van der Wielen and Grbavcic models existed some difference, but still decreased the particle velocity significantly. In principle, the motion of the foreign particle was governed by various forces from the bed. These forces generally includes drag force, the pressure gradient force, virtual mass force and the collision force, etc. In such a high concentration system, the resisting force caused by water drag force and collision force between particles was dominant and they can obviously decrease the settling velocity of the particles. The simulation data reflects the effect of fluid drag on particle velocity under this high concentration system and they were referenceable.
It was significant that the velocity of the foreign particles was reduced due to the drag force in teeter bed comparing with that of in pure water. The higher volume fraction of particles in bed was more preferable to achieve the separation of particles according to their density because it increases To further explore the agreement between the simulation results and the Joshi model, the settling behavior of 2 mm, 4 mm and 6 mm glass beads and 2 mm, 4 mm and 6 mm steel particle through the bed with a volume fraction of 0.55 was studied in Figure 8c,d. It can be seen that the simulation results keep good agreements with the Joshi model. The influence of a high concentration of fluidized particles on foreign particle sedimentation was significant and can decreased by about 20% of free settling velocity. We can also see that there were some differences between the simulated data and the predicted values in other models. It can also be seen from Figure 8c, the predicted slip velocity for glass beads in models of Di Felice, Van der Wielen and Grbavcic presents a good consistency. In these models, the particle velocity was greatly reduced more than half of the value, and the values in Kunii model was between Joshi model and other three models. Figure 8d shows that the predicted values of steel particles in Kunii and De Felice model were relatively close and were also close to the present simulated results. The particle velocity assessed by Van der Wielen and Grbavcic models existed some difference, but still decreased the particle velocity significantly. In principle, the motion of the foreign particle was governed by various forces from the bed. These forces generally includes drag force, the pressure gradient force, virtual mass force and the collision force, etc. In such a high concentration system, the resisting force caused by water drag force and collision force between particles was dominant and they can obviously decrease the settling velocity of the particles. The simulation data reflects the effect of fluid drag on particle velocity under this high concentration system and they were referenceable.
It was significant that the velocity of the foreign particles was reduced due to the drag force in teeter bed comparing with that of in pure water. The higher volume fraction of particles in bed was more preferable to achieve the separation of particles according to their density because it increases bed density and further increases the difference in settling velocity among foreign particles. Further, for particles with the same density, the slip velocity increases with the increase of particle size, which makes larges particles moving through the fluidized bed into light product more easily. Therefore, there was the influence of both density and particle size on the separation process in TBS. To reduce the number of mismatches and improve the accuracy of separation in TBS, the particle size of feeding materials was more likely to be narrowly distributed.

Conclusions
The particles in TBS are fine in size and the principle of their motion is complex in TBS, which brings difficulty for experimental study. A CFD model was implemented to investigate particle separation. It was found that lower water velocity was conducive to forming a homogeneous and stable fluidization bed. Higher water velocity was likely to cause strong turbulence flow, leading to large eddies in the bed. Therefore, a lower water velocity is recommended for the practical operation of TBS. The slip velocity gradually increased with an increase of particle size and a decrease of solid volume fraction. Therefore, in TBS, large particles may flow through the self-generating medium bed into the light product. Based on the movement of particles with various size and densities, the influence of a self-generating medium bed on the separation of particles with narrow size distributions in TBS by density was clarified in the model.

Conflicts of Interest:
The authors declare no conflict of interest. Nomenclature ρ L liquid density, kg/m 3 ε L volume fraction of liquid, dimensionless u L liquid velocity, m/s ρ S fluidized particles density, kg/m 3 ε S volume fraction of fluidized particles, dimensionless u S fluidized particle velocity, m/s CD drag force coefficient, dimensionless Re S fluidized particle Reynolds number, dimensionless d S diameter of fluidized particle, m ρ D foreign particle density, kg/m 3 d D foreign particle diameter, m u D foreign particle velocity, m/s g gravity acceleration, m/s 2 u D∞ foreign particle slip velocity in indefinite medium, m/s u Dw bounded settling velocity for foreign particle, m/s ρ M mixture/pseudo-fluid density, kg/m 3 u S∞ fluidized particle velocity in indefinite medium, m/s µ M pseudo-fluid viscosity, Pa·s µ L liquid viscosity, Pa·s u D foreign particle classification velocity, m/s L initial bed height, m n D R-Z index, dimensionless ρ e f f effective density, kg/m 3 r particle size, dimensionless