Study on Separation Density of Feeding Group Particle in the Gas–Solid Separation Fluidized Bed

Abstract

Gas–solid separation fluidized bed are an efficient coal cleaning and separation technology, and this technology has been extensively used in coal separation. The separation of the feeding coal particles in the fluidized bed is generally carried out in the form of particle groups, hence, a systematic examination of stratification as well as diffusion of the feeding particle group in the gas–solid separation fluidized bed is required. Simulated particles are used in this study and the technique that combines both theoretical calculation and an experimental method is used to investigate the effect of the inherent properties of the feeding particle group, bed characteristics, and operating parameters on the variation in voidage and air drag force in the separation process. According to the correlation between the separation density of the single-component particle group and the voidage of the gas–solid separation fluidized bed, the ρ_G.drag (change in separation density brought about by the upward airflow drag force during particle group fluidized bed separation) prediction model of the single-component spherical feeding particle group in the gas–solid separation fluidized bed is developed with the correction of voidage. When the prediction error of the ρ_G.drag prediction model is 10%, the confidence degree is 90.00%. Based on the particle segregation model and the ρ_G.drag prediction model, the separation density prediction model for the single-component spherical feeding particle group in the gas–solid separation fluidized bed is proposed. On this basis, the separation density prediction model for the single-component non-spherical feeding particle group in the gas–solid separation fluidized bed is further introduced. The separation density prediction model provides critical guidance for optimizing the gas–solid fluidized bed separation process.

1. Introduction

As the major fossil source of energy worldwide, coal has never occupied a lesser role in the global energy landscape, which affects the world energy landscape and economic growth with a major impact [1,2,3,4]. China made a direct announcement of its strategic plan of “dual-carbon” in September 2020 [5,6,7,8]. This strategy will speed up carbon emission mitigation and increase the effectiveness of clean coal use. It is conducive to green technological innovation and bolstering the global competitiveness of industries and the economy.

According to data from 2024, the national production of raw coal has reached 4.78 billion tons, and the rate of washing raw coal has reached 68.0 and has consistently remained at a historically high level [9]. It is also worth noting that the coal deposits of China are mostly found in arid areas with water deficits. Consequently, the gas–solid separation fluidized bed technology with its water-saving nature has gained significant research impetus in the sphere of clean coal separation. The application of this technology makes use of airflow to push dense medium to create a uniform and stable layer of gas–solid fluidized bed, which allows the separation of coal to occur effectively by density. It is also most applicable in coal cleaning separation where there is a lack of water.

The actual separation process in the gas–solid fluidized bed is extremely complex and involves numerous influencing factors. Exploring the impact of various influencing factors on the actual separation process and identifying the suitable optimal conditions and adjustment ranges holds great value for efficient fluidized dry separation. J. Oshitani et al. found that the separation behavior of feed mineral in fluidized bed is primarily influenced by the gas velocity, bed height, feeding mineral size, and dense medium size [10,11,12,13]. Wang et al. characterized the diffusion and stratification patterns of particles from a microscopic perspective using simulation methods [14,15,16]. Sahu A.K. et al. further investigated the influence of the particle density, particle size, and particle shape on the particle residence position [17]. Luo et al. analyzed the forces acting on coal particles in fluidized bed and compared the difference in forces experienced by coarse and fine particles in fluidized bed [18]. In addition to gravity and buoyancy, gas and particles also experience an interaction force, or drag force, during their motion. Currently, research primarily focuses on the dispersed fluidization of dilute-phase particles, with less attention paid to the study of particle groups in dense-phase gas–solid fluidized bed with lower voidage [19,20,21].

Coal separation in fluidized bed generally occurs in the form of particle groups, yet group stratification and the diffusion behavior of feeding particle groups in fluidized gas–solid separation have hardly been investigated. Once the particle group enters the gas–solid separation fluidized bed, the particles disturb the flow field within the bed. Unlike the change in the drag coefficient of the fluidized bed when a single particle is introduced, the interaction between particles in the feeding particle group has a more complex impact on the distribution of feeding particles. To elucidate the pattern of particle separation, voidage serves as a crucial indicator for investigating energy conversion and transfer, bed stability, and separation efficiency in the fluidized bed. Gidaspow et al. and Bai et al. have both provided models that correct the drag coefficient of particle groups based on voidage, using the drag coefficient of individual particles as a foundation [22,23]. However, neither of the studies has addressed the need to correct for variations in the separation density of the fluidized bed that arise from the drag effect of the ascending gas flow during the process of particle selection. This study will employ a predictive model for particle separation between the feeding particle group and the dense medium, aiming to derive the relationship between the study and density influenced by the gas drag force, and thereby uncover the separation and diffusion mechanisms of particle groups within the gas–solid separation fluidized bed.

2. Materials and Methods

2.1. Apparatus and Dense Medium

The previous study [24] introduced the composition of the gas–solid fluidized bed separation system and distribution of the dense medium.

In this study, the fluidized bed gas–solid separation system consists of an air distributor, air distribution chamber, and dense medium. The fluidized bed has an inner diameter of 200 mm, a height of 350 mm, and is made of organic glass. The magnetite powder with a particle size between 0.074 and 0.3 mm is used as dense medium. The air pressure is 0.2 MPa.

2.2. Properties of Simulated Feeding Particles

The simulated feeding particles are spherical particles, with their density and particle size distribution shown in Figure 1. The densities are 1.8, 1.9, 2.0, 2.1, 2.2, and 2.3 g/cm³, with particle sizes of 10, 15, 20, 25, 30, and 35 mm, respectively. The number of simulated feeding particles is controlled between 5 and 30.

3. Theoretical Analysis of Feeding Group Particle Separation Process

The separation behavior of particle groups in the gas–solid separation fluidized bed is illustrated in Figure 2. Compared to single-particle separation, the number of particles in the group increases when multiple particles are introduced. Due to the variations in number, size, and density within the particle group, unique distribution characteristics are formed. Under the synergistic effect of the introduced particle group and the dense medium, the bed voidage changes. Meanwhile, the bed voidage also affects the drag force exerted on the introduced particle group during separation. Therefore, based on the single-particle separation ρ_SS.drag, the density variation law of the particle group during introduction needs to be corrected according to the voidage variation law.

3.1. Density Prediction Model of Spherical Single Feeding Particle

Due to changes in the bed composition when the feeding particle group is introduced, the bed density also changes accordingly. Therefore, this study only focuses on the change between the separation density and bed density of the fluidized bed caused by the drag force of the ascending gas flow when a single non-spherical particle is introduced, defined as ρ_S.drag. The calculation method for the ρ_S.drag of non-spherical single particles when introduced can be obtained from previous research [24,25], as shown in Equation (1):

$$\left(\begin{array}{l}{\rho }_{S.drag}={\displaystyle \frac{F_d}{gV}}={\displaystyle \frac{3C_D{\rho }_f{\left(v_g-v_p\right)}^2}{4g{d}_p}}\\ C_D={\displaystyle \frac{24}{\mathit{Re}}}\left(1+C_4{\mathit{Re}}^{C_5}\right)+C_6/\left(1+C_7/\mathit{Re}\right)\\ C_4=\mathit{exp}\left(2.33-6.46\Phi +2.45{\Phi }^2\right)\\ C_5=0.096+0.556\Phi \\ C_6=\mathit{exp}\left(4.90-13.89\Phi +18.42{\Phi }^2-10.26{\Phi }^3\right)\\ C_7=\mathit{exp}\left(1.47+12.26\Phi -20.73{\Phi }^2-15.89{\Phi }^3\right)\end{array}\right)$$

(1)

Equation (1) primarily addresses the scenario where non-spherical single particles are introduced, taking into account the influence of the sphericity coefficient Φ on the ρ_S.drag. However, when dealing with a single-component particle group, it is also necessary to consider the interactions between the introduced particle groups, the interactions between the particle groups and the dense medium, as well as the impact of the bed voidage on the separation density. Therefore, in the research process, spherical particles are used as simulated particles to avoid the influence of the sphericity coefficient on the separation effect. When the introduced particles are spherical, the sphericity coefficient Φ is 1, and the calculation method for the ρ_SS.drag in the case of a single spherical particle is shown in Equation (2):

$$\left(\begin{array}{l}{\rho }_{SS.drag}={\displaystyle \frac{F_d}{gV}}={\displaystyle \frac{3C_{DS}{\rho }_f{\left(v_g-v_p\right)}^2}{4g{d}_p}}\\ C_{DS}={\displaystyle \frac{24}{\mathit{Re}}}\left(1+0.19{\mathit{Re}}^{0.65}\right)+0.44/\left(1+1.15\times {10}^{-10}/\mathit{Re}\right)\end{array}\right)$$

(2)

Spherical single particles with different sizes and densities were feeding for separation experiments, and the predicted results for the ρ_SS.drag are shown in Figure 3. From Figure 3, it can be seen that the prediction accuracy of using Equation (2) to predict the separation of spherical single particles when they are selected is over 80%. Therefore, Equation (2) can be used as a model for predicting the flow caused by the drag force of ascending airflow when single particles are selected in this study.

3.2. Bed Voidage

The bed voidage is calculated based on the Particle Segregation Model proposed by Alberto [26].

When the feeding particle group is in a suspended state in the fluidized bed, the density of the feeding particle group is the same as the bed separation density. At this time, the feeding particle group achieves force balance, and the resultant force it experiences is 0, as shown in Equation (3):

$$G_G+F_{Gf}+F_{Gd}=0$$

(3)

Based on the force balance of particles in the fluidized bed, it can be derived as shown in Equation (4):

$${\rho }_{G.p}{\displaystyle \frac{\pi }{6}}{d}_p^{3}g=\overline{\rho }\left(1-\epsilon \right){\displaystyle \frac{\pi }{6}}{d}_p^{3}g+\epsilon \overline{\rho }{\displaystyle \frac{\pi }{6}}{d}_p^2\overline{R}g$$

(4)

Define $\stackrel{-}{\rho }$ as the average density, g/cm³; $\stackrel{-}{s}$ as the ratio of the density of the feeding particle group to the average density; and $\stackrel{-}{R}$ as the ratio of the average particle size to the dense medium. The calculation formulas for $\stackrel{-}{s}$ and $\stackrel{-}{R}$ are shown in Equations (5) and (6):

$$\overline{s}={\displaystyle \frac{{\rho }_{G.p}}{\overline{\rho }}}$$

(5)

$$\overline{R}={\displaystyle \frac{\overline{d}}{d_s}}$$

(6)

The relationship between the feeding particle group and the void fraction related to the dense medium is shown in Equation (7) [26,27,28]:

$$\overline{s}=1-\epsilon +\epsilon \overline{d}$$

(7)

Among them, the calculations of the average density $\stackrel{-}{\rho }$ and average particle size $\stackrel{-}{d}$ of the feeding particles and magnetite dense medium are shown below from Equation (8) to (12):

$${\phi }_p={\displaystyle \frac{V_{G.p}}{V_{G.p}+V_s}}$$

(8)

$${\phi }_s={\displaystyle \frac{V_s}{V_{G.p}+V_s}}$$

(9)

$${\phi }_p+{\phi }_s=1$$

(10)

$$\overline{\rho }={\phi }_p{\rho }_{G.p}+{\phi }_s{\rho }_s$$

(11)

$$\overline{d}={\phi }_p{d}_{G.e}+{\phi }_s{d}_s$$

(12)

where V_G.p and V_s represent the volume of the feeding particle group and the dense medium, respectively. φ_p and φ_s represent the volume fractions of the feeding particle group and the dense medium, respectively.

This leads to the calculation method of voidage, as shown in Equations (13) and (14):

$$\epsilon ={\displaystyle \frac{\overline{s}-1}{\overline{d}-1}}$$

(13)

$$\epsilon ={\displaystyle \frac{\frac{{\rho }_{G.p}}{{\phi }_p{\rho }_{G.p}+{\phi }_m{\rho }_m}-1}{\left({\phi }_p{d}_{G.e}+{\phi }_m{d}_m\right)-1}}$$

(14)

3.3. Bed Density Related to Feeding Particle Group

The expression for bed density is shown below [29]:

$${\rho }_{bed}=0.902\left\{\left[{\rho }_s\times \left(1-{\epsilon }_{mf}\right)+{\rho }_g\times {\epsilon }_{mf}\right](1-{\epsilon }_b)+{\rho }_g{\epsilon }_b\right\}$$

(15)

According to Equation (14), the density of the emulsion phase is related to the dense medium density ρ_s. Due to the addition of the feeding particle group, the average density $\stackrel{-}{\rho }$ will replace the original bed density for the calculation, as shown in Equation (16):

$${\rho }_{G.bed}=0.902\left\{\left[\overline{\rho }\times \left(1-{\epsilon }_{mf}\right)+{\rho }_g\times {\epsilon }_{mf}\right](1-{\epsilon }_b)+{\rho }_g{\epsilon }_b\right\}$$

(16)

It can also be written as Equation (17):

$${\rho }_{G.bed}=0.902\left\{\left[\left({\phi }_p{\rho }_{G.p}+{\phi }_m{\rho }_m\right)\times \left(1-{\epsilon }_{mf}\right)+{\rho }_g\times {\epsilon }_{mf}\right](1-{\epsilon }_b)+{\rho }_g{\epsilon }_b\right\}$$

(17)

3.4. Separation Density Related to Feeding Particle Group

In the gas–solid separation fluidized bed, the drag coefficient exerted by the airflow on the feeding particles is given by Equation (18):

$${\mathit{Re}}_G={\displaystyle \frac{d_p{v}_g{\rho }_{G.f}}{{\mu }_{G.f}}}$$

(18)

Meanwhile, the difference between the separation density and the bed density of the fluidized bed caused by the drag force of the upward airflow drag force during particle group fluidized bed separation is defined as ρ_G.drag, as shown in Equation (19):

$${\rho }_{G.drag}={\displaystyle \frac{F_{G.d}}{gV}}={\displaystyle \frac{3C_{G.D}{\rho }_f{(v_{G.g}-v_{G.p})}^2}{4g{d}_e}}$$

(19)

The formula for calculating the separation density at this time is shown in Equation (20):

$${\rho }_{G.sep}={\rho }_{G.bed}+{\rho }_{G.drag}$$

(20)

3.5. Measurement and Calculation of Experimental Particle Group Separation Density

Measure the position of each particle, and then measure the pressure drop at that point under the same conditions to obtain the density at that point. The average of the sum of the densities at each point is the separation density of the feeding particle group, as shown in Equation (21):

$${\rho }_{G.\left(sep.\exp \right)}={\displaystyle \frac{{\displaystyle \sum _{i=1}^N\frac{\Delta P}{g\Delta H}}}{N}}$$

(21)

4. Variation Law of Voidage and Density in the Feeding Group Particle Separation Process Under the Air Drag Force

4.1. Variation Law of Voidage and ρ_G._drag with the Number of Feeding Particles

This experiment had the following conditions: bed height at 120 mm and gas velocity at 0.148 m/s. The sphere feeding particle density is 2.0 g/cm³. The number of feeding particles ranges from 5 to 30. As shown in Figure 4, with the increase in feeding particles, the voidage between particles gradually increases and remains between 50.05% and 50.15%. Due to the fact that the density of the feeding particles is lower than that of the dense medium, as the number of feeding particles increases, the average density decreases, while the bed volume increases due to the addition of feeding particles, which leads to an increase in the bed voidage to a certain extent.

Figure 4 indicates that the voidage between the particles increases gradually with the increase in the feeding particles and the voidage between the particles is in a range of 50.05 to 50.15. Since the density of the feeding particles is lower than that of the dense medium, as the number of feeding particles increases, the average density reduces, and the bed volume rises from the addition of feeding particles, which causes an increase in the bed voidage to some extent.

As the number of feeding particles increases, the ρ_G._drag remains between 0.152–0.158 g/cm³. However, with the increase in the number of feeding particles, the fluctuation in the ρ_G._drag significantly increases, indicating an increase in the fluctuation in the ρ_G._drag. When the number of feeding particles is greater than 15, the fluctuation in the ρ_G._drag increases significantly, with the fluctuation range exceeding 0.1 g/cm³. This is because with the addition of feeding particles, the stable separation environment originally established in the fluidized bed is impacted to some extent, the distribution of particles becomes more dispersed, and the feeding particles are more strongly influenced by the local environment. The increased number of collisions between feeding particles and particles in the emulsion phase medium, as well as the uneven distribution of local bubble phases, have a significant impact on the position of feeding particles, leading to their variability in position within the fluidized bed. Consequently, the fluctuation range of the ρ_G._drag increases when particles are feeding, and the error increases, reflecting a decrease in the stability of the bed with the increase in feeding particles.

4.2. Variation Law of Voidage and ρ_G._drag with the Density of Feeding Particles

This experiment had the following conditions: bed height at 120 mm and gas velocity at 0.148 m/s. The sphere feeding particle density ranges from 1.8 to 2.3 g/cm³. The number of feeding particles was 10 and their diameter was 15 mm. As shown in Figure 5 the feeding particles with a density of 1.8 g/cm³ float at the top of the fluidized bed, while the feeding particles with a density of 2.3 g/cm³ sink to the bottom of the fluidized bed, indicating that the density of the fluidized bed is between the two. As the density of the feeding particles increases, the voidage also shows a gradually increasing trend, indicating that the density of the feeding particles has a significant impact on the voidage. Particles with a density lower than the separation density are more difficult to sink and are located in the upper part of the fluidized bed, resulting in a lower voidage. Particles with densities between separation densities will be evenly distributed in the middle of the fluidized bed. During the sinking process, particles with a higher density are greatly affected by the unstable foaming and emulsion phase in the fluidized bed, resulting in uniform particle distribution and increased voidage.

As the density of the feeding particles increases, the ρ_G._drag decreases. This indicates that as the density of the feeding particles increases, both the gravitational force and buoyancy force acting on the particles increase, while the drag force they experience in the fluidized bed decreases, resulting in a gradual decrease in the ρ_G._drag. The larger errors in the ρ_G._drag obtained at 1.8 g/cm³ and 2.2 g/cm³ are due to the significant difference between the separation density and the bed density when calculating based on these densities, which leads to an increase in the error of the ρ_G._drag.

4.3. Variation Law of Voidage and ρ_G._drag with the Size of Feeding Particles

This experiment had the following conditions: bed height at 120 mm and gas velocity at 0.148 m/s. The sphere feeding particle density was 2.0 g/cm³. The number of feeding particles was 10, and their diameters ranged from 10 to 35 mm. As shown in Figure 6, as the particle size increased, the voidage gradually increased, remaining between 50.15% and 50.82%. With the decrease in particle size, under the same air velocity, the aerodynamic force per unit area on the particles increased, thus leading to an increase in the voidage of the particle group.

As the particle size of the feeding particle increases, the ρ_G._drag tends to decrease. This is because as the voidage increases, the drag coefficient of the particle group decreases to some extent. However, the overall change in the ρ_G._drag is not significant, ranging from 0.13 to 0.15 g/cm³. This indicates that when the feeding particle size is greater than 15 mm, the separation effect in the fluidized bed is better. However, as the feeding particle size increases, the inertial effect of particle migration during the separation process becomes less significant, leading to more stable separation and reduced fluctuations in the ρ_G._drag.

4.4. Variation Law of Voidage and ρ_G._drag with the Gas Velocity

This experiment had the following conditions: bed height at 120 mm and gas velocity between 0.14 and 0.18 m/s. The sphere feeding particle density is 2.0 g/cm³. The number of feeding particles was 10 and their diameter was 15 mm. As shown in Figure 7, with the increase in gas velocity, the voidage of the bed also increases. The increase in gas velocity means that more gas enters the fluidized bed, leading to increased fluctuations in the bed, intensified particle movement within the bed, intense bubble movement, and faster merging of excess bubbles, resulting in an increase in bed voidage.

As the gas velocity increases, the proportion of bubbles in the fluidized bed increases, resulting in a decrease in the separation density of the bed and a reduction in the drag force on the particles. The ρ_G._drag of the particle group shows a decreasing trend. When the gas velocity increases, the fluctuation in the ρ_G._drag will first decrease and then increase. When the gas velocity is 0.155 m/s, the fluctuation in the ρ_G._drag is controlled within 0.047 g/cm³, and the ρ_G._drag region is stable. But at the same time, with the increase in gas velocity, the energy input increases, and the excess energy in the fluidized bed leads to the instability of the bed, while the error and fluctuation in the ρ_G._drag increase.

4.5. Variation Law of Voidage and ρ_G._drag with the Bed Height

This experiment had the following conditions: bed height between 80 and 160 mm, and gas velocity at 0.148 m/s. The sphere feeding particle density was 2.0 g/cm³. The number of feeding particles was 10 and their diameter was 15 mm. As shown in Figure 8, with the increase in bed height, the voidage gradually and steadily increases. As the height of the bed increases, the upward airflow in the fluidized bed drives the expansion of dense medium to the bed. When the particle group enters the fluidized bed, it has a larger space for movement, providing more possibilities for the selection of particles. Therefore, the voidage of the particle group increases.

As the bed height increases, the ρ_G._drag tends to decrease. When the bed height is 120 mm, pressure fluctuations are minimal, and the ρ_G._drag remains relatively stable. When the bed height is optimized, the fluidization degree of the dense medium in the fluidized bed is higher, the bed activity further increases, the settling resistance of the dense medium to the particles decreases, and the pressure drop fluctuation from the top to bottom of the bed is lower, ensuring stable particle separation and smaller fluctuations in the ρ_G._drag. As the bed height further increases, the possibility of bubbles merging to form larger bubbles increases. During the transfer of bubbles from the lower bed to the upper bed, they continue to merge and gradually increase in size, making it more prone to phenomena such as particle backmixing in the bed, and the disturbance to the bed gradually increases, leading to unstable fluctuations in bed pressure drop and increased fluctuations in the ρ_G._drag.

4.6. Variation Law of Voidage and ρ_G._drag with the Distance Between Feeding Particles

Without considering the side wall effect, the distance between the feeding particles varies from 0 to 15 mm. Meanwhile, the diameter of the feeding particles is 15 mm. According to the experimental results shown in Figure 9, the distance between the feeding particles has almost no effect on the changes in the voidage and ρ_G._drag of the particle group after selection. The interactions between the feeding particle groups on the surface of the fluidized bed have little effect on their stratification and diffusion in the fluidized bed. The change in the initial selection conditions when fluidized particles enter the fluidized bed means a difference in the initial voidage, but the physical properties of the feeding particle group have not undergone significant changes. According to the calculation of the above formula, the voidage of the feeding particle group has not changed. However, according to previous research, the change in the spacing between the feeding particle groups has no significant effect on the change in the drag coefficient of the particle group [21]. Therefore, the ρ_G._drag remains almost unchanged and remains in a stable range.

5. Density Prediction Model and Error Analysis of Feeding Particle Group Under the Air Drag Force

5.1. The ρ_G._drag Prediction Model of Feeding Particle Groups

According to the results of the experiment, it can be concluded that under different conditions, the particle group voidage and particle group separation ρ_G._drag vary. Essentially, the change in particle group voidage leads to a change in the particle group drag coefficient, which in turn causes a change in the ρ_G._drag. Therefore, the voidage can be used to correct the ρ_G._drag of the particle group.

Calculating the drag force experienced by feeding particle groups is challenging. Therefore, the ρ_SS._drag prediction model for a single feeding particle can be applied, with the introduction of a correction factor to study the situation of feeding particle groups. The ρ_G._drag for a feeding particle group and the ρ_SS._drag for a single feeding particle are as follows, where the coefficient γ is a correction function related to the void fraction, as shown in Equation (22):

$${\rho }_{G.drag}=\gamma {\rho }_{SS.drag}$$

(22)

The relationship between the correction function related to the voidage and ρ_G._drag is illustrated in Figure 10. Based on the available experimental data, three fitting relationships with high correlation coefficients (R²) are the exponential function, power function, and polynomial function, as shown in Equations (23)–(25):

$$\gamma =4.915\times {10}^6\times e^{-0.2997\epsilon }\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}^2=0.927$$

(23)

$$\gamma =9.232\times {10}^{25}\times {\epsilon }^{-15.18}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}^2=0.926$$

(24)

$$\gamma =0.04753{\epsilon }^2- 5.201\epsilon +142.8\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}^2=0.926$$

(25)

By analyzing the prediction accuracy of the three relational expressions, it can be concluded that when the prediction error of the power function of the ρ_G._drag prediction formula is 20%, the c confidence degree is 98.00%; when the prediction error of the power function of the ρ_G._drag prediction formula is 10%, the confidence degree is 90.00%. Among the three prediction formulas, the power function has the highest prediction accuracy. Therefore, when selecting particle groups, the calculation formula for the ρ_G._drag is shown in Equation (26):

$${\rho }_{G.drag}=\gamma {\rho }_{SS.drag}=9.232\times {10}^{25}\times {\epsilon }^{-15.18}{\displaystyle \frac{3C_{DS}{\rho }_f{(v_g-v_p)}^2}{4g{d}_p}}$$

(26)

5.2. Model Validation and Comparison of ρ_G._drag

In the fluidized bed separation, few studies have investigated the changes in the ρ_G._drag during the separation of feeding particle groups, with the drag coefficient generally being the focus of correction. By comparing the accuracy of previous models involving the voidage and drag coefficient, we can further illustrate the accuracy of the ρ_G._drag prediction model proposed in this study during the introduction of feeding particle groups. The following three calculations of ρ_G._drag are compared when in the fluidized bed separation process.

The drag coefficient obtained by Yang [19] is further used to calculate the ρ_G._drag, and the formula is shown in Equation (27):

$${\rho }_{G.drag}=\left(-0.0101+{\displaystyle \frac{0.0038}{4{\left(\epsilon -0.7463\right)}^2+0.0040}}\right){\rho }_{SS.drag}$$

(27)

The drag coefficient obtained by Wen and Yu [20] is further used to calculate the ρ_G._drag, and the formula is shown in Equation (28):

$${\rho }_{G.drag}={\epsilon }^{-4.7}{\rho }_{SS.drag}$$

(28)

The drag coefficient obtained by Shen [21] is further used to calculate the ρG.drag, and the formula is shown in Equation (29):

$${\rho }_{G.drag}=\left(44.62{\epsilon }^2-85.81\epsilon +41.39\right){\rho }_{SS.drag}$$

(29)

The predictive results of previous studies for the experimental results are shown in Figure 11. The drag coefficient models from previous studies exhibit poor predictive performance for the experimental results. Previous research has primarily focused on dilute-phase particles, typically with voidages exceeding 80%, and thus has limited predictive accuracy for the dense gas–solid separation fluidized bed with lower voidages. Therefore, the calculation formula for the voidage and ρ_G._drag proposed in this study holds significant guiding importance for the separation process in the gas–solid separation fluidized bed.

5.3. Verification of Separation Density of Non-Spherical Feeding Group Particle

By combining the separation density of single particles and the calculation method of ρ_G._drag when selecting a single-component spherical particle group, the separation density calculation method of a single-component spherical particle group is obtained, as shown in Equation (30), where the spherical coefficient of the particle group is replaced by the spherical coefficient (Φ) of non-spherical particles.

$$\left\{\begin{array}{l}{\rho }_{G.sep}={\rho }_{G.bed}+{\rho }_{G.drag}\\ {\rho }_{G.bed}=0.902\left\{\left[\left({\phi }_p{\rho }_{G.p}+{\phi }_m{\rho }_m\right)\times \left(1-{\epsilon }_{mf}\right)+{\rho }_g\times {\epsilon }_{mf}\right](1-{\epsilon }_b)+{\rho }_g{\epsilon }_b\right\}\\ {\rho }_{G.drag}=9.232\times {10}^{25}\times {\epsilon }^{-15.18}{\rho }_{SS.drag}\end{array}\right\}$$

(30)

To verify the predictive accuracy of Equation (30), separation experiments were conducted using simulated particles such as cube, cuboid, and cylinder, with their main properties shown in Figure 12 below. The feeding cubes had sphericity coefficients of 0.42 and 0.45, cuboids had sphericity coefficients of 0.42 and 0.47, and cylinders had sphericity coefficients of 0.42 and 0.51. The density of feeding particles is 2.1, 2.2, and 2.3 g/cm³, and the number of feeding particles ranged from 8 to 10.

The predicted separation density of a single-component non-spherical particle group is shown in Figure 13. As shown in this figure, the prediction error is controlled within 10%. Due to the complex and close relationship between the spherical coefficient of non-spherical particles feeding into the group and the arrangement of particles in the fluidized bed, it is difficult to quantitatively analyze the changes in the spherical coefficient from a single particle to a particle group, resulting in certain prediction errors. In summary, Equation (30) has a certain predictive effect on the separation density of single-component non-spherical particle groups, with a prediction accuracy of over 90%. It has important guiding significance for feeding particle group separation in the gas–solid separation fluidized bed.

6. Conclusions

This study investigates the separation rules of single-component feeding particle groups in the gas–solid separation fluidized bed. The conclusions are summarized as follows:

(1)

The accuracy of the separation density prediction model for spherical single particles in the gas–solid separation fluidized bed is verified. A calculation method for bed voidage is introduced, and a theoretical calculation method for bed density and separation density after the introduction of feeding particle group related to the bed voidage is established;

(2)

The focus is on exploring the variation laws of the ρ_G._drag and voidage under a different bed height, gas velocity, number, particle density, size, and interparticle distance of the feeding particle group, and establishing the prediction model for the ρ_G._drag of the feeding particle group with the voidage as a correction factor;

(3)

Based on the particle segregation model and the ρ_G._drag prediction model, the separation density prediction model for the single-component spherical feeding particle group in the gas–solid separation fluidized bed is obtained;

(4)

Combined with the spherical coefficient, the separation density model is used to predict the separation density of single-component non-spherical feeding particle groups. The results show that this separation density model has important guiding significance for particle group separation in the gas–solid separation fluidized bed.