A Numerical Analysis of the Descending Behaviors of Clusters at the Wall of the Circulating Fluidized Bed Riser

Abstract

Particle clusters at the wall of the CFB riser have significant effects on the bed-to-wall heat transfer and abrasion, while their descending behaviors are not well understood because the entire descending process is difficult to track with experiments, due to the limitations of measurement technology. In this study, the gas–particle two-phase flow in the CFB riser is simulated using the LES-DSMC method. The entire descending process of the cluster is recognized using a method that involves identifying the continuity of periods in which clusters appear in the successive cells at the wall. Then, the transient velocity, drag force, and particle concentration of the descending cluster as a function of its traveling distance are obtained. The results show that the descending clusters at the wall of the CFB riser are dynamic collections of particles. Their lifetimes are in the range of 0.2~0.5 s. During the descending processes, they are accelerated, and their particle concentrations are continuously decreased. The variation in the particle concentration, velocity, and drag force of different descending clusters indicates that they travel highly similar distances and fluidization velocity has little effect on them.

1. Introduction

It is well known that particles in the Circulating Fluidized Bed (CFB) riser usually take on a core–annular flow structure, which is characterized by the dilute up-flowing particles in the core region surrounded by the denser down-flowing particles in the annular region at the wall [1,2,3]. The particles near the wall region are observed to always be descending in the form of particle clusters. The flow behaviors of these descending clusters are important because they have significant effects on the bed-to-wall heat transfer and abrasion [4,5].

Up to now, the flow behaviors of clusters at the wall of the CFB riser have been studied by a wide range of experimental techniques such as the laser sheet technique [6], a fiber-optic probe [7,8], a capacitance probe [9], a high-speed camera [10,11,12], etc. Most experimental measurements are focused on macroscopic flow behaviors, such as descending velocity and the shape of the cluster. For example, experimental observations indicated that descending clusters at the wall of the CFB riser are dynamic entities and they seem to be particle sheets falling in wave-like patterns along the wall, before breaking up or being re-entrained into the core region of the CFB riser [13,14]. Their particle concentrations seem to vary considerably during the descending processes [15]. They seem to be accelerated during descending processes [16]; however, the measured descending velocities of clusters reported by different investigators always lie in the range of 0.3~2 m/s and are insensitive to changes in operating conditions such as fluidization velocity, solid flux, etc. [17,18]. Harris et al. [19] tried to develop correlations between the mean flow behaviors of descending clusters and the cross-sectional averaged particle concentration of the CFB riser based on the different reported experimental data. However, these correlations are subjective and can hardly be used to reveal the descending mechanism of the cluster.

Unfortunately, due to the limitations of measurement technologies, the variations in velocity, drag force, and particle concentration of a descending cluster along its traveling distance still cannot be obtained through experiments, so the relationships between these dynamic behaviors, the knowledge of which is necessary to reveal the descending mechanisms of clusters at the wall of the CFB riser, cannot be obtained.

At the same time, it should be noted that the transient descending process of the cluster also cannot be elucidated by means of the simulation method. This is because, although the flow trajectory of the discrete particle can be tracked by the Lagrange–Euler two-phase simulation method, to obtain the dynamic behaviors of a descending cluster, the entire descending process also needs to be recognized in the simulation. However, up to now, the popular cluster identification methods such as the criterion proposed by Sharma et al. [20] could only be used to identify the appearance of a cluster at a given location, such as a given computational cell in the simulation, so that the obtained transient behaviors of the cluster only reflect a certain moment during their descent.

To solve this problem, the gas–particle two-phase flow in the CFB riser has been simulated by the Large Eddy Simulation (LES) and Direct Simulation Monte Carlo (DSMC) methods in this study. The descriptions of the numerical model and simulation settings are shown in Section 2. The entire descending process of a cluster is recognized according to the continuity of periods in which clusters appear in the successive cells at the wall of the CFB riser. Then, in Section 3, the dynamic behaviors of the descending cluster are interpreted in terms of the transient velocity, drag force, and particle concentration. The simulated results are compared with the experimental data. The possible descending mechanism of the cluster at the wall of the CFB riser is proposed. A brief discussion regarding the results obtained is presented in Section 4.

2. Materials and Methods

2.1. Eulerian–Lagrangian Model

2.1.1. Gas Phase

The equations of conservation of mass and momentum for the gas phase are written as follows [21]:

$${\displaystyle \frac{\mathit{𝜕}}{\mathit{𝜕}t}}({\rho }_g{\epsilon }_g)+\nabla \cdot ({\rho }_g{\epsilon }_g{u}_g)=0,$$

(1)

$${\displaystyle \frac{\mathit{𝜕}}{\mathit{𝜕}t}}({\epsilon }_g{\rho }_g{u}_g)+\nabla \cdot ({\epsilon }_g{\rho }_g{u}_g{u}_g)=-{\epsilon }_g\nabla P-S_{p-g}-(\nabla \cdot {\tau }_g)+{\epsilon }_g{\rho }_{g}g,$$

(2)

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

(3)

$${\mu }_g={\mu }_{lam}+{\mu }_t,$$

(4)

where $u_g$, ${\rho }_g$, P, and ${\mu }_g$ are the velocity, density, pressure, and viscosity of the gas phase, respectively. ${\epsilon }_g$ is the void fraction. $S_{p-g}$ is the interaction forces between the gas and particle and determined by

$$S_{p-g}={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^N{f}_{d,i}}}{S}},$$

(5)

where $f_{d,i}$ is the drag force of the particle. The turbulent viscosity of the gas phase is as follows [22]:

$${\mu }_t=C_V\Delta {\rho }_g{k}_g^{1/2},$$

(6)

where $\Delta ={(\Delta x\Delta y)}^{1/2}$, and $k_g={\displaystyle \frac{1}{2}}{{\overline{u}}^{\prime }}_{gx}{{\overline{u}}^{\prime }}_{gy}$ is gas turbulent energy.

2.1.2. Particle Phase

The DSMC method is used to simulate the collision processes between particles. It is a trajectory method that makes it possible to deal with interparticle collision based on sample particles, the number of which is smaller than the actual number of particles. The collision probability of particle i during a time step $∆\mathrm{t}$ is given by

$$P_i={\displaystyle \sum _{j=1}^N{P}_{ij}},$$

(7)

where N is the number of simulated particles in the cell. In the high-particle-concentration two-phase flow, the dense packing effect on the collision frequency of particles should be considered. The modified collision probability of simulated particles i and j is

$$P_{ij}={\displaystyle \frac{n}{N}}\pi d^2{g}_0{u}_{ij}\Delta t,$$

(8)

where n is the local particle number density. d is particle diameter. u_ij is relative velocity. In Equation (8), the radial distribution function ${\mathrm{g}}_0$ is introduced to correct the probability of a collision for the effect of the volume occupied by the particles, and ${\mathrm{g}}_0$ is calculated as follows:

$$g_0({\epsilon }_s)={\left[1-{\left({\displaystyle \frac{{\epsilon }_s}{{\epsilon }_{\max s}}}\right)}^{1/3}\right]}^{-1},$$

(9)

where the maximum packed particle concentration ${\epsilon }_{\max s}$ of 0.63 is used in the simulation.

The modified Nanbu method is used to search the collision pair in this study [23,24], and the details are discussed in the previous paper [25].

The equation of translational motion of a particle can be written as follows [26]:

$$m{\displaystyle \frac{d{v}_i}{dt}}=-{\displaystyle \frac{\pi }{6}}{d}_i^3\nabla p+f_d+m_{i}g,$$

(10)

where the drag force of particle $f_{d,i}$ can be written as

$$f_d=\left({\displaystyle \frac{1}{16}}\right)C_{d0,i}{\rho }_g\pi d_i^2\left|u_{g,i}-v_i\right|(u_{g,i}-v_i){\epsilon }_g^{-\delta },$$

(11)

where the drag force coefficient $C_{d0,i}$ is written as

$$C_{d0,i}={\left(0.63+{\displaystyle \frac{4.8}{R{e}_{p,i}^{0.5}}}\right)}^2,$$

(12)

$$R{e}_{p,i}={\rho }_g{d}_i{\displaystyle \frac{\left|u_{g,i}-v_i\right|}{{\mu }_g}},$$

(13)

The equation of rotational motion of a particle is written as

$${\displaystyle \frac{m_i{d}_i^2}{10}}{\displaystyle \frac{d{\omega }_i}{dt}}={\displaystyle \frac{{\rho }_s{d}_i^2}{64}}\left({\displaystyle \frac{6.45}{R{e}_{\omega }}}+{\displaystyle \frac{32.1}{R{e}_{\omega }}}\right)\left|{\omega }_i\right|{\omega }_i,$$

(14)

where $R{e}_{\omega }=d_i^2{\rho }_g\left|\omega \right|/(4{\mu }_g)$.

The changes in velocity after particle collisions are subject to the following equations:

$$m_i(v_{i,1}-v_{i,0})=J,$$

(15)

$$m_j(v_{j,1}-v_{j,0})=J,$$

(16)

$${\displaystyle \frac{m_i{d}_i^2}{4}}({\omega }_{i,1}-{\omega }_{i,0})=n\times J,$$

(17)

$${\displaystyle \frac{m_j{d}_j^2}{4}}({\omega }_{j,1}-{\omega }_{j,0})=-n\times J,$$

(18)

where $v_{i,0}$, $v_{j,0}$, $v_{i,1}$, and $v_{j,1}$ represent the pre- and post-collisional velocities of particles i and j, respectively. n is the normal unit vector. J is the total impulse vector.

2.1.3. Initial and Boundary Conditions

The two-dimensional CFB riser used in the present simulation is shown in Figure 1. The height and width of the riser are 9 and 100 cm, respectively. It is divided into a network of computational cells in the size of Δx × Δy = 0.45 × 0.9 cm, with a total cell number of 2220. The calculation of drag force acting on a particle should be determined according to its local gas velocity, so it is necessary to couple the gas velocity in the Euler coordinate with the Lagrange coordinate of the particle. Here, the area-weighted average method as shown in Equation (19) is used to determine the local gas velocity of a given particle k (red dot) according to the grid node parameters of the computation cell as shown in Figure 1.

$$u_{g,p_k}={\displaystyle \frac{a_{i,j}{u}_{i,j}+a_{i+1,j}{u}_{i+1,j}+a_{i+1,j+1}{u}_{i+1,j+1}+a_{i,j+1}{u}_{i,j+1}}{\Delta x\Delta y}},$$

(19)

where $a_{i,j}=(\Delta x-l_x)(\Delta y-l_y)$, $a_{i+1,j}=l_x(\Delta y-l_y)$, $a_{i+1,j+1}=l_x{l}_y$, $a_{i,j+1}=l_y(\Delta x-l_x)$. The particle position in the computational cell (${\mathit{l}}_x$ and ${\mathit{l}}_y$) is determined by the particle motion equation.

Each computational cell is divided into eight sub-cells of the same size. On the one hand, searching for the collision partner of the particle is carried out in the sub-cell, so that the computing load for handling particle collisions can be efficiently decreased. On the other hand, the particle concentration of the sub-cell is used to calculate the drag force acting on the particle located in it, so that the effect of heterogeneity of particle concentration in the computational cell on the computation accuracy of the particle drag force can be considered.

The velocities of gas and particle phases are initially set to be zero in the CFB riser. The inlet gas pressure, gas velocity, and particle velocity are given. A uniform bottom-inlet condition is assumed. The gas flow outlet is set at the atmospheric pressure. A no-slip condition is used for the gas phase at the wall. The detailed computational parameters are listed in

Table 1. Parameters used in the simulation.

Parameters	Value	Parameters	Value
Particle shape	Sphere	Gas viscosity	1.5 × 10−5 (Pa·s)
Particle diameter	0.0126 (cm)	Solid flux	50 kg/m2 s
Particle density	2400 (kg/m3)	CFB riser size	9 × 100 (cm)
Restitution coefficient between particles	0.9	Fluidization velocity	200~300 (cm/s)
Coefficient of restitution between particle and wall	0.8	Gas density	1.2 (Kg/m3)
Tangential restitution coefficient	0.3	Cell number	20 × 111
Friction coefficient	0.1	Sub-cell number per cell	2 × 4

.

2.2. The Method for Identifying Clusters at the Wall of CFB Riser

Soong et al. [27] first proposed the following three guidelines for the identification of clusters: (1) The particle concentration in a cluster must be significantly above the time-averaged particle concentration at the given local position and the operating condition. (2) This perturbation in the particle concentration due to clusters must be greater than the random background fluctuations of the particle concentration. (3) This concentration perturbation should be sensed for sampling volume with characteristic length scales greater than one to two orders of the particle diameter. Being consistent with these guidelines, Sharma et al. [20] imposed a criterion as illustrated in Figure 2. They proposed that the local transient particle concentration for a cluster must be greater than the time-averaged concentration ${\overline{\epsilon }}_s$ by at least two times the standard deviation $2\sigma$. They thought that the starting time of cluster Ta is the last time when its particle concentration exceeds ${\overline{\epsilon }}_s$ before satisfying the $2\sigma$ limit, and the ending time Tb is the first time when its particle concentration falls below ${\overline{\epsilon }}_s$ after falling below the $2\sigma$ limit. Obviously, Sharma’s criterion can only identify the appearing period of the cluster at a local position. In this study, since a descending cluster must pass through some successive cells along the wall before it vanishes or is re-entrained into the core of the CFB riser, the following method is developed to track the complete descending process of the cluster. Firstly, the cluster appearing periods in each cell at the wall are identified by Sharma’s criterion. Secondly, the successive cells at the wall that satisfy that cluster appearing periods from up to down and are continuous in time are found out. Any group of such successive cells is considered as a complete cluster descending path, which is verified by checking the snapshots of particles. By this method, the size of the computation cell should be comparable to the size of the cluster to gain a high cluster recognition accuracy. Rhodes et al. [13] reported that a typical cluster at the wall of the CFB riser has a characteristic dimension of about 1 cm. So, in the present simulation, the cell size of 0.45 × 0.9 cm is used, and the descending clusters can be successfully tracked. The number-averaged particle velocity, drag force, and concentration in each computational cell in which a descending cluster passes through are considered as its transient values.

3. Results

3.1. The Flow Regime of a Descending Cluster at the Wall of the CFB Riser

Figure 3 shows the snapshots of particles in the CFB riser at a time interval of 0.05 s. Clusters have a typical particle flow behavior in the CFB riser. Clusters can be found in both the core and wall regions of the CFB riser. Clusters in the core region usually flow upward, but those at the wall always flow downward. As shown at t = 7.1 s, the up-flowing clusters usually exhibit a saddle shape with a denser head and two dilute tails. A similar cluster structure has also been observed by other investigators [28]. It can be seen at t = 7.15 s that the axial velocities of particles in the head of the up-flowing cluster are usually larger than those in the tail, so the particles in the tail tend to break away from the up-flowing cluster and are pushed towards the wall under the effects of inertia and the gas. As they reach the wall at t = 7.2 s, the particle number at a local position of the wall increases suddenly; meanwhile, they may prevent the near-wall particles from falling down, and then the local particle concentration at the wall increases, and a denser cluster may be formed.

Figure 4 shows the snapshots of a descending cluster at the wall. This cluster forms initially in the near-wall cell i = 2, j = 64 at t = 6.87 s. For illustrating the particle mixing behavior during cluster descent, particles in cell i = 2, j = 64, cell i = 2, j = 63, and cell i = 2, j = 62 at t = 6.87 s are shown in red, green, and blue, and the particles in surrounding cells are shown in black. There is the highest concentration close to the wall at t = 7.078 s since most particles approach the wall under the influence of lateral velocity. After that, the particles rebounded from the wall are dispersed by the up-flowing gas and clusters. From the snapshots from 7.078 s to 7.152 s, a descending cluster at the wall is a dynamic collection of particles. The transient particle concentration, velocity, shape, and sizes of the cluster vary continuously during descent. Previous experimental observations showed that clusters at the wall of the CFB riser are particle sheets falling in wave-like patterns [19]. In the present simulation, the descending cluster exhibits a strand shape. In the initial descending stage, the length of the cluster increases suddenly. This is consistent well with the measurements of [18]. They reported that the length of the cluster increased abruptly at the wall, and it may be up to 30 mm. Meanwhile, the decrease in particle concentration in the cluster is observed during descent, which finally causes the cluster to vanish. The particles within cells in the path of the descending cluster (shown in red and blue) join into the cluster inevitably, and the surrounding particles such as those coming from the core region (shown in black) also join into the cluster. The particles composing the cluster strongly mix during descent, which should result in mass and heat transfer between the wall and core regions of the CFB riser. The experiment by Brereton and Grace [29] also suggested that there are mixtures of clusters and dispersed particles in the near-wall region of the CFB riser. The above result shows that a descending cluster at the wall of the CFB riser is not a particle assembly with a fixed arrangement. The particles composing the cluster are mobile, so the transient flow behaviors of particles within the descending cluster are studied in the following sections.

3.2. Transient Flow Behavior of a Particle Within the Descending Cluster

Figure 5 shows the transient velocity, surrounding particle concentration, and drag force of a typical particle (No. 17823) within the descending cluster. From Figure 5a, it can be seen that this particle moves from the core to the wall region of the CFB riser at the dimensionless height of y/H = 0.64 at 6.8 s. It joins the descending cluster at the wall and then flows downward. As shown in Figure 5b, its surrounding particle concentration increases fast and then decreases gradually during cluster descent. As the cluster vanished at t = 7.2 s (particle concentration of the cluster below the local time-averaged value), this particle still flows downward along the wall during a shorter period of 7.2~7.31 s. After that, it returns to the core of the CFB riser. From Figure 5c,d, it can be seen that when this particle flows as a part of a descending cluster, its axial and radial velocities both fluctuate with higher frequency and lower amplitude. This indicates that its mobility is restricted by surrounding particles due to the high particle concentration. After the cluster vanishes, its velocity fluctuation frequency decreases and fluctuation amplitude increases because it has enough free motion space. From Figure 5e, it can be seen that when this particle falls as a part of the descending cluster, it is accelerated because the axial drag force acting on it is always lower than its gravity of 0.0025 × 10⁻⁵ N. At the time of the cluster vanishing, the axial velocity of this particle has approached its theoretically terminal settling velocity of −88 cm/s computed by Equation (20) [30].

$$u_t={\left[{\displaystyle \frac{4}{3}}{\displaystyle \frac{g{d}_p({\rho }_p-{\rho }_g)}{C_d{\rho }_f}}\right]}^{{\displaystyle \frac{1}{2}}},$$

(20)

Comparing Figure 5e,f, it can be seen that during cluster descent, the magnitude of radial drag acting on this particle is comparable with its axial drag force. Especially in the initial descending stage, it is obviously pushed toward the wall under a larger radial drag force. In addition, the axial and radial forces acting on this particle as the cluster descends fluctuate with high frequency and low magnitude like its velocity. Once the cluster vanishes, the same drag force acting on this particle can result in a much larger variation in the particle’s velocity.

3.3. Transient Behavior of Descending Cluster

Up to now, the greatest number of measurements on clusters at the wall of the CFB riser has been of the descending velocity. The results showed that cluster descending velocity always lies in the range of 0.3~2 m/s irrespective of the experimental conditions [17]. Harris et al. [19] plotted cluster descending velocity reported by different investigators against the Reynolds number, Archimedes number, and dimensionless gas–solid flow ratio but obtained no clear correlation. A similar result is obtained in the present simulation. Because the entire descending process of a cluster can be tracked in the present simulation, we try to correlate a cluster’s descending velocity with its traveling distance. Figure 6 shows the descending velocities of several typical clusters as a function of their traveling distances at fluidization velocities of 200 cm/s and 250 cm/s. Clusters at the wall are accelerated, and a cluster’s descending velocity increases as it travels longer along the wall. Interestingly, the transient descending velocity of any cluster seems to be in a roughly linear relationship with its traveling distance, no matter what the location and the fluidization velocity are. Based on this, Equation (21) can be obtained.

$${\displaystyle \frac{\Delta L_{cl}}{\Delta v_{y,cl}}}=k\Rightarrow {\displaystyle \frac{\Delta L_{cl}}{\Delta t}}=k{\displaystyle \frac{\Delta v_{y,cl}}{\Delta t}}\Rightarrow v_{y,cl}=k{a}_{cl},$$

(21)

where $\Delta L_{cl}$, $\Delta v_{y,cl}$, and $\Delta t$ are the transient traveling distance, velocity and time. k is the slope of the linear fit curves shown in Figure 6. $a_{cl}$ is the transient acceleration of the descending cluster. K lies in a narrow range of −7.8~−11.9, which indicates that the transient change rates of acceleration with descending velocity are similar for different clusters. Whatever k is, Equation (21) indicates that the descending process of the cluster at the wall of the CFB riser is not a free settling process but a complicated variable acceleration settling process. The finding that cluster transient acceleration is proportional to the transient descending velocity indicates that gas resistance acting on the descending cluster (i.e., the drag force) must keep decreasing with the increase in the descending velocity.

The axial drag force of a descending cluster as a function of the traveling distance is shown in Figure 7. The axial drag force acting on any cluster decreases fast in the initial descending stage and then decreases slowly in a longer following stage. It is known that the axial drag force acting on a descending cluster with a fixed particle arrangement during the accelerated descending process should keep increasing all the time because the relative velocity between cluster and gas increases with the descending velocity. However, the simulated result shown in Figure 7 indicates that the axial drag force acting on a descending cluster keeps decreasing along the traveling distance. Moreover, no matter how large the initial axial drag force is, it can quickly decrease to a closer value with other clusters in the initial descending stage. This indicates that descending clusters have an efficient and similar drag-reducing manner for overcoming the gas resistance in this stage. We think there are two possible ways. The first one can be found from the radial drag force of the cluster as a function of traveling distance as shown in Figure 8. In the initial descending stage, all clusters are subject to the obvious radial drag forces in the direction perpendicular to the wall which they fall along. This indicates that the descending clusters are integrally pushed to the wall in this stage. The particle snapshots shown in Figure 4 verify this behavior. In addition, Guo et al. [10] used a new image processing method based on a high-speed camera to identify, locate, and track clusters. Their image also showed that a descending cluster in the near-wall region moves fast toward the wall in the initial stage. On the one hand, under the effect of the gas velocity gradient in the near-wall region, a descending cluster should tend to be stretched into a streamline shape of flow resistance. On the other hand, the gas velocity decreases as the distance to the wall decreases. Therefore, the first possible way for drag reduction of the descending cluster should be shape-changing to reduce flow resistance. The second way for drag reduction of the descending cluster can be found from the cluster particle concentration as a function of traveling distance as shown in Figure 9. In the initial descending stage, the particle concentrations of all clusters quickly decrease into a narrow range. In particular, the higher the initial particle concentration of the cluster is, the faster it decreases. We know that in theory, under the same Reynolds number, decreasing particle concentration (increasing the mean distance between particles within the cluster) will result in an increase in the fraction of the gas flowing through the cluster, so the resultant drag force acting on the cluster would decrease. In fact, the effects of shape-changing and particle concentration decreasing on the drag reduction of the descending cluster are complicated fluid dynamics problems. The simulated result indicates that they both take action, and we believe the shape-changing should be dominant in the initial descending stage, but it should be further verified.

In addition, as shown in Figure 8, after the initial descending stage, the radial drag force of the descending cluster approaches zero gradually, which indicates that the cluster has no obvious radial motion anymore. However, as shown in Figure 7 and Figure 9, the axial drag force and the particle concentration of the cluster both keep decreasing slowly. Obviously, after the initial descending stage, the descending cluster overcomes flow resistance mainly by particle concentration decreasing. It should be noted that the continuous decrease in particle concentration will result in the cluster vanishing finally. At the time of vanishing, a descending cluster cannot be identified as an entity anymore because its particle concentration is lower than the local time-averaged particle concentration. However, it should be noted that after a cluster vanishes, the particles composing the descending cluster can still flow downward in a short period, as shown in Figure 5c. It can be imagined that in this stage, these dispersed particles (because the surrounding particle concentration is low enough) will quickly enter the free-settling process and reach the terminal settling velocity of the particle soon. In a word, we think that if there is no disturbance, the result of the cluster descending at the wall of the CFB riser is the particles composing it reaching the terminal settling velocity.

Meanwhile, it can be seen from Figure 9 that the initial particle concentrations of the descending clusters have a correlation with the descending location and fluidization velocity. Although the initial particle concentrations of clusters are different, they can decrease fast to a similar decreasing stage. The reason why this can be realized should be the mobility of the particle composing the cluster, which gives the descending clusters the ability to vary particle distance as well as shape continuously. Therefore, no matter what the initial particle concentration of a cluster is, during descent, it can soon self-regulate based on the routine of overcoming gas resistance (i.e., drag-reducing) and then take on a similar descending law like other clusters. This should be the reason why the operation parameters of the CFB have no obvious effects on the descending velocities of clusters [17].

3.4. Comparison Between Present Simulation Result and Reported Experimental Data

Up to now, plenty of experimental data have been published on the particle concentration of the descending clusters at the wall of the CFB riser. The reported data are obtained under different operating conditions, CFB riser sizes, and measurement methods. Meanwhile, the reported cluster solid concentrations are either the time-averaged values or the peak values in experimental probe data. Harris et al. [19] plotted the reported mean cluster particle concentration (${\overline{\epsilon }}_{cl}$) against the cross-sectional averaged particle concentration (${\overline{\epsilon }}_s$) of the CFB riser ignoring other operation conditions and proposed the following correlation:

$${\overline{\epsilon }}_{cl}={\displaystyle \frac{0.58{\overline{\epsilon }}_s^{1.48}}{0.013+{\overline{\epsilon }}_s^{1.48}}}$$

(22)

Because it is known from the present simulation results that the particle concentration of a cluster keeps decreasing during descent, here, the mean particle concentration of a descending cluster is calculated by averaging the transient value along its traveling distance. The mean cluster particle concentrations against the cross-sectionally averaged particle concentrations for all operation conditions are shown in Figure 10. The mean particle concentrations of the descending clusters obtained in the present simulation lie in the range of 0.07~0.2, and they are close to but deviate slightly from the value predicted by Equation (22). In particular, the increasing trend of the mean cluster particle concentration with the cross-sectionally averaged particle concentration of the CFB riser predicted by Equation (22) is not obvious in the present simulation. The possible reason is that the reported experimental data are usually not the average values along the cluster traveling distance. The experimental data from different investigators used to establish Equation (22) deviate greatly. For example, the simulated data agree well with the experimental data from Lints and Glicksman [31] but deviate from those from Sharma et al. [20]. Even so, the present simulated result is reasonable.

Figure 11 shows the mean cluster lifetimes of the descending clusters at different fluidization velocities obtained in the present simulation and the experiment by Noymer and Glicksman [32]. In the present simulation, the lifetime of the cluster is defined as the time period of the entire cluster descent process. The mean lifetime of clusters at any fluidization velocity is obtained by averaging the lifetimes of all tracked clusters. The simulated results show that the mean lifetime of the descending clusters is very short and lies in a narrow range of 0.25~0.4 s at any fluidization velocity, although a slight decrease with fluidization velocity increasing can be observed. The experimentally measured mean lifetime of descending clusters by Noymer and Glicksman [32] is a little shorter and shows a slight increase and then decrease with the fluidization velocity. Even so, these authors still concluded that the fluidization velocity has little influence on the lifetime of the cluster.

Although the mean lifetime of the descending cluster is very short and close, the lifetime distribution is actually wide, as shown in Figure 12. It can be seen that the distribution of the cluster lifetime agrees well with the Gaussian distribution at any fluidization velocity. The main reason for the wide distribution of the cluster lifetime should be the difference in the initial particle concentration of clusters. If the initial particle concentration of the cluster is too low to approach the threshold level of the particle concentration used to identify the cluster, it should be considered to vanish soon and have a very short lifetime. On the contrary, if the initial particle concentration is higher, it may have a longer lifetime.

4. Discussion

Finally, it should be noted that in the actual operation of the CFB, the descending process of the cluster at the wall of the CFB riser should be rather complicated because a cluster descending process can be disturbed frequently and even unavoidably. For example, there may exist mutual interferences of different descending clusters that have overlapped traveling paths and time periods. Or, if too many new particles join into a descending cluster during the descending process, its descending process may be changed and even renewed. The above phenomena are frequently observed during cluster tracking in the present simulation. In this study, only the whole and almost undisturbed cluster descending processes are picked out to study the transient behavior of the descending clusters. However, the similar drag-reducing behaviors of the descending clusters by a self-regulating transient concentration along their traveling distances indicate that although the above disturbance happens, the descending process of the cluster must follow the same mechanism. So, the measured cluster descending velocities in the experiment under different operation conditions, such as particle load, fluidization velocity, and CFB structures, should be similar.

5. Conclusions

In this study, the LES-DSMC approach is used to simulate the gas–solid two-phase flow in the CFB riser. The flow regime is characterized, and the transient flow behaviors and particles are studied. The results show that the tails of up-flowing clusters in the core region always result in the formation of clusters at the wall. Descending clusters at the wall are dynamic collections of particles, and their size, velocity, particle concentration, and drag force vary continuously during descending processes. There exist particle exchanges between them and external dispersed particles. The axial and radial velocities of particles within them take on high-frequency and low-amplitude fluctuations. During cluster formation and the beginning period of the cluster decaying, the axial and radial drag forces acting on particles within them fluctuate strongly and then become weaker during the later period of cluster decaying. Once particles break away from them, the fluctuation frequency of particle velocity and drag force becomes lower, and the fluctuation amplitude becomes larger. The transient particle concentrations of different clusters seem to decrease following the same route during decaying, and they decrease as the clusters travel longer along the wall. They are accelerated in the axial direction, and the number-averaged axial and radial drag forces acting on them increase to a maximum and then decrease gradually during descending. Their lifetime can be fitted by the Gaussian distribution. Fluidization velocity seems to have little effect on their transient flow behaviors. The simulated results agree with the relevant experimental data.