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Modelling of dense gas-particle flow in a circulating fluidized bed by Distinct Cluster Method (DCM)
Powder Technology 195 (2009) 235–244
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Modelling of dense gas-particle flow in a circulating fluidized bed by Distinct Cluster Method (DCM) Xiangjun Liu a,⁎, Xuchang Xu b a b
Thermal Engineering Department, University of Science and Technology Beijing, Beijing 100083, China Thermal Engineering Department, Tsinghua University, Beijing, 100084, China
a r t i c l e
i n f o
Article history: Received 6 October 2008 Received in revised form 9 June 2009 Accepted 12 June 2009 Available online 27 June 2009 Keywords: Dense gas-particle two-phase flow Particle cluster Distinct Cluster Method
a b s t r a c t Computational Fluid Dynamics (CFD) is a powerful tool to study the dense gas–solid flow in a circulating fluidized bed. Most of the existing methods focus on the microscopic properties of individual particle. Therefore, the simulation scale is significantly limited by the huge number of individual particles, and so far the numbers of particles in most of the reported simulations are less than 105. The hydrodynamics behaviour of particle clustering in a dense gas–solid two-phase flow has been verified by several experimental results. The Distinct Cluster Method (DCM) was proposed in this paper by studying the macroscopic particle clustering behaviour, and comprehensive models for cluster motion, collision, break-up, and coalescence have been well developed. We model the dense two-phase flow field as gas-rich lean phase and solid-rich cluster phase. The particle cluster is directly treated as one discrete phase. The gas turbulent flow is calculated by Eulerian approach, and the particle behaviour is studied by Lagrangian approach. Using the proposed method, a threedimension dense gas-particle two-phase flow field in a circulating fluidized bed with square-cross-section, with particle number up to 7.162 × 107 are able to be numerically studied, on which few results have been reported. Details on instantaneous and time-averaged distributions are obtained. Developing process of nonuniform particle distribution is visualized. These results are in agreements with experimental observations, which justified the feasibility of using the DCM method to model and simulate dense gas–solid flow in a circulating fluidized bed with large number of particle numbers. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Circulating Fluidized Beds (CFBs) are currently very popular in many industries, such as coal combustion, fluid catalytic cracking, and etc. Computational Fluid Dynamics (CFD) has been widely used to study the gas–solid two-phase flow in a CFB. The Eulerian description for both solid phase and gas phase has been well developed [7,31]. With the rapid development of computational capabilities, the mixed Eulerian–Lagrangian approach attracted more and more attentions from many researchers. In this approach, the detailed particle motion behaviour, which facilitates a better understanding of the physical phenomena, can be revealed by solving the Newtonian motion Equations in Lagrangian coordinate. The major problem in the simulation of dense gas–solid two-phase flows is to study the inter-particle interactions, and the numerical characterization of a dense particle–gas two-phase flow is much more complicated than that of the dilute one because of the large particle volumes. Several mathematical models have been developed in the last 2decades. The Direct Simulation Monte Carlo (DSMC) method was first proposed by Bird [2]. The particle-to-particle collisions were con-
⁎ Corresponding author. Tel.: +86 10 62333792. E-mail address: [email protected] (X. Liu). 0032-5910/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2009.06.007
sidered by means of analogy to the kinetic theory of molecules in dilute gas flow. In each time step, the collisionless motion of a representative number of particles is calculated simultaneously, and each simulated particle represents a certain number of real particles with identical physical properties. After that, a representative number of collisions are carried out by employing some kinds of Monte Carlo methods. The DSMC method has been improved and applied to the simulation of dense particle–gas flow by several researchers [15,28,30]. The V-shape and the reverse V-shape clusters in a CFB were successfully predicted by Tsuji et al. [30], and the cluster frequency in a CFB was obtained by Shuyan et al. [28] using the DSMC method. Oesterle and Petitjean [23] proposed another iterative simulation technique by introducing artificial inter-particle collisions during the trajectory calculation. In this method, trajectories of a large number of individual particles are computed successively, and each trajectory represents a group of particles. The collisions are processed stochastically on the basis of macroscopic particle properties obtained from previous iteration. This method is limited to steady flow simulation, and the probability of these collisions depends on the local concentration and velocity of the solid phase. The Discrete Element Method (DEM) was first proposed to study granular materials [6]. Recently, DEM is becoming more and more popular for dense particle–gas two-phase flow simulation. By tracking
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each individual particle in the flow filed, the real inter-particle interaction can be modeled and simulated so that the physical phenomena of the two-phase flow can be further quantified. Using DEM, Tanaka and Tsuji [29] successfully simulated a two-phase plug flow in a vertical pipe by considering the fluid effects with CFD. Later on, several collision-dominated two-phase flows were successfully simulated using this method [10,13,14,25]. All these existing Eulerian–Lagrangian models described above are theoretically capable to calculate mutual interactions of particles as well as to predict the dense two-phase flow. However, real simulation can only be feasible for a small-scale region, or a two-dimension region with limited number of particles. To apply these models to a dense two-phase flow in real-scale industrial applications, further research is still needed to reduce the tracking number of trajectories to be calculated so that the required computational capabilities in the simulation are realistic. In this paper, we proposed the Distinct Cluster Method (DCM). In this method, we consider the particle cluster as one discrete phase. The cluster collision, break-up and coalescence models are established. Using these proposed models, a three-dimension gas-particle two-phase flow field in a circulating fluidized bed with square-crosssection is simulated. The global volume-averaged solid concentration in the computation domain is 3.0%. Few numerical studies on threedimension flow have been reported due to the large computation required. Details on the proposed models and the calculated results are presented in the rests of this paper. 2. Distinct Cluster Model for particle phase 2.1. Particle clustering One phenomenal hydrodynamics behaviour in dense gas–solid two-phase flow, especially in a CFB, is particle clustering, in which most of the particles congregate as clusters, and these clusters act as independent objectives inside the gas-rich dilute phase. This has been verified by many experiments. Particle clustering implies that groups of several to dozens of particles tend to congregate in the riser in order to minimize the drag force exerted on them [3]. This micro-flow structure was first observed by Bai et al. [1] by employing a twodimensional column, by Liu et al. [19] using a video camera and a special optical fiber image probe, and by Horio & Kuroki [12] using a laser sheet technique. Recently, the flow patterns of particle clusters in a CFB is obtained and studied by Lackermeier et al. [16] using a highspeed video technique in combination with the laser sheet technique. These experimental results show that there are two distinct phases, a dispersed phase, in which solid particles are individually presented, and a solid-rich cluster phase. The solid-rich phase usually consists of clusters and each cluster is a congregation of dozens of particles. The diameter of each cluster is less than 1–2 cm, and the particle volume fraction is generally around 41–50% [18]. Particle clustering is a key characteristic of CFBs, and it significantly impacts the mechanism of the flow, heat transfer, and chemical reaction in CFBs. To model and simulate the dense gas-particle two-phase flow, methodology to explicitly model and study particle cluster is quit straightforward. 2.2. Cluster definition and its size Our approach is to divide the dense two-phase flow field in a gasparticle CFB as gas-rich lean phase and solid-rich cluster phase as shown in Fig. 1. Based on the phenomenon that solid particles are presented individually and randomly in the gas-rich lean phase, we assume that there is no-slip between gas and these individual solid particles, and the particles are uniformly distributed. The solid-rich phase consists of clusters and each cluster is a congregation of dozens of particles. Furthermore, the cluster acts as individual and instantaneously separated objective inside the gas-rich dilute phase, with its shape and size consistently varying, due to the complicated two-phase interactions.
Fig. 1. Gas-rich lean phase and solid-rich cluster phase.
Observed by many experiments [12,3,22], the shape of these clusters may be in the forms of strings, triangles, V-shape with a round nose, or other shapes, and the size is generally less than 1–2 cm in diameter. The particle may be either contacted or not contacted inside a cluster, and the particle volume fraction of a cluster is generally around 41–50%. In our approach, each cluster is simply considered as a spherical ball. The voidage of a cluster is taken as a constant. Proposed by Park [24], the cluster voidage can be taken as εc =εmf, where εmf is the minimum fluidization voidage of a specific particle. The gas velocity through a cluster is neglected. As discussed above, the cluster sizes are consistently varying. Therefore, the key problem to be addressed in this method is to define the instantaneous size of a particle cluster. Cluster size is one of the important parameters in the analysis of gasparticle two-phase flow in a CFB. It is usually denoted as the effective equivalent diameter. Many experimental and theoretical researches have been carried out to reveal this characteristic value. Chavan [5] found that the effective equivalent diameter of a cluster is inversely proportional to the energy that breaks the consecutive emulsion phase into discrete particle clusters. Experimental results by Mostoufi et al. [22] shown that the ratio of cluster diameter to particle diameter was in a wide range of 1.5–160. Zou et al. [34] established a statistic model on cluster size distribution in fast fluidized bed based on micrographs experiments. Xu and Kato[35] proposed a simple correlation of the equivalent cluster diameters for gas and solid concurrent upward flows. Li et al [17] developed an equation on equivalent cluster diameter using the Energy-Minimization Multi-Scale (EMMS) approach. By balancing the forces imposed on a cluster suspended in gas, Horio et al. [11] proposed a mathematical model for cluster diameter:
l=
1 2ðn−1Þ 3ð1−εc Þεi
ρg 2 ðu −usup Þ ðρp −ρg Þg lean
ð1Þ
Where ulean and usup are the lean phase velocity and the superficial gas velocity respectively. According to our assumptions described above, we take ulean =ug, and ug is the local instantaneous gas velocity. Meanwhile, ρp and ρg are particle and gas density, εc is the voidage in cluster phase, and εi is the local instantaneous voidage. Parameter n is the Richardson–Zaki correlation [11,26], and the value depends on the physical properties of the particle. There have been many discussions on this [27]. We take n = 5 in this paper. From Eq. (1), it is clear that the local instantaneous cluster diameter is determined by the local instantaneous gas velocity and voidage, and the value fluctuates in a wide range with high frequency during the calculation process. In order to simplify the calculation without compromising the accuracy, the smallest particle number inside a cluster is assumed to be 3.
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2.3. Cluster motion equation
3. Simulation of gas flow
Treating the particle cluster as one discrete phase, the classical trajectory model at Lagrangian coordinate can be used to describe the cluster moving behaviour. The forces acting on a moving cluster include gravity force, drag force, and etc. The cluster motion equation can be expressed as
Taking the particle cluster as one discrete phase inside the gas-rich dilute phase, the trajectory model is used to describe the behaviour of each cluster in every time step. Since the dense particle–gas twophase flow is essentially an unsteady flow, the local gas and particle flow characteristics is time varying. The following continuity and momentum equations are used to model the instantaneous gas turbulent flow.
mc
→ d Vc → → = mc g + F D dt
ð2Þ
where F ̅D is the drag force, the drag force coefficient is decided by the following relations
ð1 +
Re2c = 3 = 6Þ24
Re c
ð3Þ
j
νg
∂ðρg εui Þ
∂ðρg εuj Þ ∂xj
∂ðρg εui uj Þ
+
ð6Þ
=0
∂xj
= −ε
→ where ∑ F drag is the volumetric fluid–particle interaction force. It can be determined by N 1
=
j
ð8Þ
where ΔV is the volume of a computational gas grid, and Np is the cluster number in this grid. The turbulent stresses tensor τij in Eq. (7) is calculated as
2.4. The collision, break-up, and coalescence models At each time step, the velocity and position of all the clusters in a two-phase flow field are calculated from Eq. (2). Due to the gas– particle and particle–particle interactions, clusters form and break continuously. The cluster size di is re-calculated at the end of each time step. Comparing di with the local maximum cluster size li determined by Eq. (1), it is clear that the cluster i is broken into some fragment-clusters if di N li. The number of fragment-clusters is n = int(d3i/l3i) + 1, among which n − 1 clusters are determined to have the same mass mk and the same effective equivalent diameter li. The mass of the fragment-cluster n is mn = mi − (n − 1)mk, and the diameter is determined by its mass. We consider the conservation of kinetic energy during breaking-up process, and the kinetic energy of each fragment-cluster is proportional to its mass. The motion direction is assumed as spatially homogeneous. When two or more clusters are contacted at the end of each time step, the cluster–cluster collision is calculated. We consider two → → clusters of diameter di and dj with velocity vectors Vi and Vj . The distance between the central points of the two clusters is Rij. → → If Rij b 12 ðdi + dj Þ and Vi · Vj N 0, the two contacted clusters move away from each other. → → If Rij b 12 ðdi + dj Þ and Vi · Vj b 0, the two cluster collide. First, we assume that the two clusters contacted are merged as one big cluster. The conservation of mass and kinetic momentum are maintained during the collision process. The mass and velocity are calculated as (
" ! # ∂uj ∂p ∂ ∂ui → −τij −∑ F drag + με + ∂xi ∂xj ∂xj ∂xi
p → → ∑ F drag = ∑ F D ΔV
→ dc → ; V g and V c are the gas and cluster velocity, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p respectively; dc = 3 6Ω = π and Ώ is the cluster volume. where Rec =
+
ð7Þ
Rec ≥1000; C D = 0:44 → → Vg− Vc
∂t
∂t
Rec b1; C D = 24 = Rec
1≤Rec b1000; C D =
∂ðρg εÞ
mij = mi + mj → → → mij V ij = mi V i + mj V j
τij = −ρg νt Sij = −ρg νt
∂uj 1 ∂ui + 2 ∂xj ∂xi
!
The eddy viscosity vt is evaluated using the mixing-length approach 2
2
νt = ðCs ΔÞ jSj = ðCs ΔÞ
qffiffiffiffiffiffiffiffiffiffiffiffi 2Sij Sij
4. Apply the DCM method to a bench-scale CFB with square-cross-section So far, most of the research works were based on risers with circular cross-section. However, risers with square or rectangular cross-sections are being widely used in industrial CFB applications, such as combustions. The riser geometry has considerable influence
ð4Þ
Effective equivalent diameter of this merged cluster is determined qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6ðmi + mj Þ = ρp π
ð5Þ
The next step is to compare dij with the local maximum cluster size li. The cluster is broken into some fragment-clusters if dij N li, as described above.
ð10Þ
In the equation above, Cs is typically set to 0.15 [9]; Δ is the length scale, which is usually defined as the cubic root of the local grid volume.
as dij =
ð9Þ
Fig. 2. Computation domain of the CFB.
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Table 1 Parameter settings. Channel height Channel length Channel width Particle diameter Solid concentration Particle number Inlet gas velocity
0.6 m 0.06 m 0.06 m 120 µm 3% 7.162 × 107 5.5 m/s
Gas density Particle density Laminar gas viscosity Time step for gas phase Time step for particle phase Grid number Total calculation duration
1.2 kg/m3 1500 kg/m3 1.8 × 10− 5 kg/m s 1.0 × 10− 4 s 1.0 × 10− 5 s 32 × 32 × 100 2.5 s
on the hydrodynamics of CFBs [36,37,38]. Experimental results show that the voidage and velocity profiles in a CFB with square-crosssection are different from that with a circular cross-section [32,37,38]. The corners impact the two-phase flow field significantly. In this paper, a three-dimension dense gas-particle two-phase flow field in a CFB with square-cross-section is numerically studied. The sketch diagram of the computation domain is shown in Fig. 2. The bench-scale CFB is 2.0 meters in height, and the cross-section of the riser is 0.06 m × 0.06 m. The gas density is 1.2 kg/m3, and the particle density is 1500 kg/m3. The superficial gas velocities are in the range between 4.5 m/s and 8 m/s. The time–mean voidage in the upper region is usually in the range from 0.95 to 0.975. In this paper, we study a region of 0.6 m in height in the upper fast fluidization regime. The global mean solid concentration in the computation domain is set to 3.0%. The mean particle diameter is 120 µm so that total particle volume in the bed is 6.48 × 10− 5 m3 and the total particle number is 7.162 × 107. We assume that 90% of the total particles are congregated
as spherical clusters and each cluster contains 400 particles by average. Therefore there are 6.4458 × 107 particles congregated as 1.611 × 105 clusters, and they are evenly distributed in the flow field initially. The voidage of clusters is considered to be a constant of 0.5. The initial volume, diameter, and velocity of each cluster is 7.238 × 10− 10 m3, 1.114 × 10− 3 m, and 2 m/s respectively. All the parameter settings in this numerical study are summarized in Table 1. We use the well-known SIMPLE algorithm to resolve the gas governing equations. A staggered non-uniform grid system is used in this paper. The total grid number is 32 × 32 × 100 in x, y and z direction as shown in Fig. 2, and the grid settling in x and y direction is nonuniform. The finest grids, which are 0.5 mm × 0.5 mm × 6 mm in size, are located at the very near corner regions so that effects of walls and corners on the two-phase flow field can be studied in details. The biggest grids located at the center regions are 5 mm × 5 mm × 6 mm, and the mean grid size is 1.3 mm × 1.3 mm × 6 mm. The time steps for gas phase and particle phase are 1.0 × 10− 4 s and 1.0 × 10− 5 s respectively. The source terms are linearized. The time-dependent terms are determined using the second order implicit difference scheme and the convection and diffuse terms are resolved by the Quadratic Upwind Interpolation of Convective Kinematics (QUICK) difference scheme. The finite difference equations are solved using the Gauss–Seidel under-relaxation iteration. We calculate the gas phase only in the flow field from t = 0 to 0.5 s. All the particle clusters are randomly seeded into the flow field at t = 0.5 s and they are evenly distributed. The total calculation duration of the two-phase flow is 2.5 s. In order to maintain the total solid mass in the computation domain, the solid concentration is maintained to
Fig. 3. Instantaneous particle volume fraction at z = 0.3 m–z = 0.32 m.
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be 3% which serves as the circulation boundary condition. This is realized by setting the particle flow rate at the inlet to be equal to the flow rate at outlet. We first calculate the velocity and position of all the clusters at each time step. Then the cluster collisions are considered, and the cluster parameters are updated by solving Eqs. (1)–(5). The time–mean inlet gas velocity of each grid is 5.5 m/s. A random perturbation is superposed on each inlet velocity component and the stochastic velocity component is set to be u i̅ = 〈u ̅i〉 + Iψ| u ffi̅|, where ψ qffiffiffiffiffi ¯ V = 1. The is a Gaussian random number satisfying ψ ̅ = 0 and ψ fluctuating intensity I is set to be 15%.
239
5. Calculated results 5.1. Calculated instantaneous flow field at different time steps Fig. 3 (a–i) is the instantaneous particle volume fraction distributions from z = 0.3 m to z = 0.32 m section. At t = 0.5 s, particle clusters are randomly distributed in the flow field. The local maximum particle concentration is 11.7%, and the high particle concentration regions are evenly distributed within the flow field. Once the clusters are moved by aerodynamic forces, most clusters are broken-up or
Fig. 4. Instantaneous particle volume fraction at y = 0.02 m–y = 0.04 m.
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Fig. 7. Instantaneous UV gas velocity vector of z = 0.5 m section at t = 2.5 s. Fig. 5. The statistical distribution of the calculated cluster size at t = 2.5 s.
merged, and the distribution of clusters becomes more and more nonuniform. The dense particle regions are at the four-corners, and the dilute region is in the center of the section. These results are consistent with experimental ones, which were interpreted by Lockhart [21] that membrane walls have significant local effects and increase the local solids holdups in the shielded fin region. Fig. 4 (a–i) show the instantaneous particle volume fraction distributions from y = 0.02 m to y = 0.04 m section. Particle distribution in x–z section is consistent with that of the x–y section as shown in Fig. 3. Initially, the high particle concentration regions are evenly distributed in the flow field, and gradually they become non-uniform once the two-phase flow behaviour is constituted. The quasistationary two-phase flow system is established from t = 1.0 s. The particle concentration is non-uniform in both vertical and lateral directions. For the vertical flow structure, there is a dense region at the bottom and a dilute region at the top of the riser. For the lateral flow structure, there is a dilute region at the core and a dense region adjacent to the wall. The particle volume fraction in the center is much lower than that closer to the wall. This non-uniformity is the key flow characteristic of a CFB which has been confirmed by many experimental research works (e.g. [8,33]). The proposed DCM model is successful in this simulation. The total number of clusters and the size of each cluster keep varying during the calculation process. Initially, there are 1.611 × 105 clusters evenly distributed in the flow field. At t = 0.52 s, the total number of clusters increases to 298,792. Most of the initial clusters are broken into smaller fragment-clusters under the breaking-up effects of aerodynamic actions, while the maximum cluster number is less than 3.2 × 105 during the 2.5 s calculation duration. At t = 2.5 s, the
Fig. 6. Time variation of particle concentration at a given region (central point: 0.02 m, 0.02 m, 0.4 m).
number of cluster is 257,467, and their equivalent diameter ranges from 218 μm to 7165.7 μm. The statistical distribution of the calculated cluster size is shown in Fig. 5. There are 32.4% of the total particles congregated as smaller clusters with equivalent diameter less than 300 μm. At the same time, 19.7% of the total particles are congregated as larger clusters with equivalent diameter more than 900 μm. The mean equivalent diameter of all the clusters is 958.8 μm. Most of the clusters flying in the core region are less than 300 μm in diameter, while most of the clusters near the wall are more than 600 μm in diameter. The smaller clusters are usually in the center regions, and the clusters become bigger and bigger when moving closer to the
Fig. 8. Instantaneous UW gas velocity vector in y = 0.03 section at t = 2.5 s.
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241
Fig. 9. (a–c) The two-phase time-averaged flow profiles in y = 0.03 m section.
walls. The distribution of cluster size is reasonable and is in agreement with the experimental and simulation results from other researchers [18,20]. Fig. 6 shows the time variation of particle volume fraction at a control volume (0.0015 m × 0.0015 m × 0.0136 m) in the upper zone at x = 0.02 m, y = 0.02 m, z = 0.4 m. The time-averaged particle volume fraction from t = 1.0 s to 2.5 s of this control volume is 0.022. Initially, particle clusters are randomly located in the flow field, and the local particle fraction in this grid is 0.0355. Once the two-phase coupling calculation is carried out after t = 0.5 s, large-scale time fluctuations are observed. The quasi-stationary two-phase flow system is established at t = 1.0 s from when the instantaneous value fluctuates around a mean value with high frequency. The same process can also be observed from the instantaneous particle volume fraction distributions as shown in Figs.3 and 4. Fig. 7 shows the instantaneous UV gas velocity vector at t = 2.5 s in the x–y cross-section at z = 0.5 m. Although the two-phase flow is discharged into the square duct along the z direction, the fluctuating velocity in the x–y direction cannot be neglected. The UV instantaneous velocity in the center region ranges from 0.5 m/s to 2.0 m/s, while the instantaneous velocity at the four-corners is much lower. This velocity distribution results in the four dense particle corner zones as shown in Fig. 3. Fig. 8 is the instantaneous UW gas velocity
vector at t = 2.5 s in the x–z cross-section at y = 0.03 m. The time– mean inlet gas velocity is 5.5 m/s and the instantaneous W velocity ranges from −2.8 m/s to 13.3 m/s. 5.2. Time-averaged two-phase flow and comparisons with existing experimental results As described above, the quasi-stationary two-phase flow system is established at t = 1.0 s in the total 2.5 s calculation duration. Calculation results of the last 1.5 s are used as samples for time-averaging in order to maintain sufficient statistical accuracy. The time-averaged flow profiles in the center x–z section at y = 0.03 m are shown in Fig. 9 (a–c). Fig. 9a is the transversal variations of time-averaged vertical gas velocity at heights equals to 0.05 m, 0.17 m, 0.29 m, 0.41 m, and 0.53 m respectively. Time-averaged gas velocity in the core region is much higher than in the regions closer to the wall. The mean gas velocity decreases from the peak to a value smaller and smaller when moving from the core region to the wall. Fig. 9b is the transversal variation of time-averaged vertical cluster velocity at the heights between 0.0–0.05 m, 0.12–0.17 m, 0.24–0.29 m, 0.36–0.41 m, and 0.48–0.53 m respectively. Fig. 9c is the transversal variation of particle volume fraction at the same five regions. From Fig. 9a–c we can see the gas and cluster velocity distributions are
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Fig. 10. (a–c) The two-phase time-averaged flow profiles along the diagonals.
highly correlated. Clusters (particles) are entrained upwards in the core region and downwards in the dense annular region adjacent to the wall, and this is the same as in a circular cross-sectional columns. Consequentially, the time-averaged particle volume fraction in the riser gradually decreases from peak at the bottom to lower values towards the top. This is the so-called core-annulus flow structure proposed by Bierl & Gajdos [4] and adopted by many others. These time-averaged distributions are in accordance with the instantaneous particle volume fraction distributions shown in Fig. 4. The two-phase time-averaged flow profiles along the diagonals are shown in Fig. 10a–c. Compared with those along the centerlines, the gas velocity near the corner is lower, which results in higher particle descending velocity and increased descending fluxes. The timeaveraged descending velocity of particle at point (0.059 m, 0.059 m, 0.16) located at corner is − 2.0 m/s, and the instantaneous velocity of this point fluctuates in the range from − 1.5 to − 3.1 m/s. Compared with the point (0.059 m, 0.03 m, 0.16) located in the center along the wall, the time-averaged descending velocity of particle is −1.2 m/s, and the instantaneous velocity fluctuates in the range of −0.5 to −2.2 m/s. The time-averaged particle volume fractions of these two points are 0.243 and 0.162 respectively. Further studies show the particle concentration in the corner is higher than elsewhere along the wall. These flow characters have been revealed and interpreted by
Fig. 11. Lateral profiles of mean particle velocity for Ug = 5.5 m/s, Gs = 40 kg/m2s and z = 5.13 m [37].
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Fig. 12. Lateral profiles of mean voidage for Ug = 7.0 m/s, Gs = 40 kg/m2s and z = 6.20 m [38].
mental one as shown in Figs. 11 and 12. They both indicate that most of the particles near the wall travel downwards while in the core most of particles travel upwards. At the corner, particles move downwards faster, and the thickness of the downward flow wall layer is greater than elsewhere along the wall.
Zhou et al [38], and the reason is the descending particles were protected by the corner where the gas velocity is lower. The voidage and velocity profiles in a CFB with square-cross-section was experimentally studied by Zhou et al [37,38] using a fiber optic probe. The measurement results of the lateral profiles for mean vertical particle velocity and mean bed voidage are shown in Figs. 11 and 12. The cross-section of the experimental riser is 146 mm × 146 mm and the total height is 9.14 m. Similar results were obtained from experiments by Wang et al. [32] in a 400 mm × 400 mm × 5000 mm square riser using ECT. The calculated lateral profiles using the proposed DCM method for time-averaged vertical particle velocity and voidage in the section at z = 0.3 m are shown in Figs. 13 and 14 respectively. It is clear that our simulation results are qualitatively in agreement with experi-
In this paper, we proposed an original Distinct Cluster Method (DCM) and we also established the cluster motion, collision, broken-up, and coalescence models. Using this method we were able to simulate a three-dimension gas-particle two-phase flow field in a circulating fluidized bed with square-cross-section, with total particle number up to 7.162 × 107. This is realized by directly taking the particle cluster as one
Fig. 13. Calculated lateral profiles of mean vertical particle velocity (z = 0.3 m).
Fig. 14. Calculated lateral profiles of mean voidage (z = 0.3 m).
6. Conclusions
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discrete phase so that the maximum tracking number of the clusters to be processed during the simulation was reduced to 3.2 × 105. Comparing with the conventional DEM, the number of tracking objects is reduced more than 220 times by using DCM. In addition, Lagrangian integration can be significantly simplified using larger time step which is proportional to the mass of each tracking cluster, instead of an individual particle. The proposed DCM can effectively reduce the computation complexity in gas-particle two-phase flow. By using the proposed DCM method, detailed results on instantaneous and time-averaged distributions of two-phase flow in a CFB are obtained. Developing process of non-uniform particle distribution is visualized. Numerical results are intensively reviewed and compared with existing theoretical and experimental conclusions. These proposed models and algorithm are feasible and can be used to study the dense gas–solid flow in a circulating fluidized bed with large number of particle numbers. Acknowledgment This work was supported by National Natural Science Foundation of China under Grant No. 50406025. References [1] D.R. Bai, Y. Jin, Z.Q. Yu, Cluster observation in a two-dimensional fast fluidized bed, Fluidization'91–science and Technology (Edited by Kwauk, M. & Hasatani, M.), Science Press, Beijing, 1991, pp. 110–115. [2] G.A. Bird, Molecular Gas Dynamics, Oxford Univ. Press, 1976. [3] H.T. Bi, J.X. Zhu, Y. Jin, Z.Q. Yu, Forms of particle aggregations in CFB, Proc. 6th Chinese Conf. On Fluidization, 1993, pp. 162–167, Wuhan, P. R. China. [4] T.W. Bierl, L.T. Gajdos, Phenomenological modelling of reaction experiments in risers, Final Report, DOE/MC-14249-1149, Environmental Research and Technology, 1982. [5] V.V. Chavan, Physical principles in suspension and emulsion processing, in: A.S. Mujumdar, RA Mashelkar (Eds.), Advances in Transport Processes, Vol. 3, John Wiley & sons, New York, 1984, pp. 1–34. [6] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1) (1979) 47–65. [7] J.S. Curtis, V.B. Wzchem, Modeling particle-laden flow: a research outlook, AIChE Journal 50 (2004) 2638. [8] R.J. Dry, Radial concentration profiles in a fast fluidized bed, Powder Technology 49 (1986) 37–44. [9] M. Gemano, U. Piomelli, P. Moin, W.H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys, Fluid A3 (7) (1991) 1760–1765. [10] E. Helland, R. Occelli, L. Tadrist, Numerical study of cluster formation in a gas-particle circulation fluidized bed, Powder Technology 110 (2000) 210–221. [11] M. Horio, K. Morshita, O. Tachibana, H. Hurata, in: P. Basu, J.F. Large (Eds.), Circulation Fluidized Bed Technology H, Pergamon Press, 1988, pp. 147–154. [12] M. Horio, H. Kuroki, Three-dimensional flow visualization of dilutely dispersed solids in bubbling and circulating fluidized bed, Chemical Engineering Science 49 (2413) (1994) 421. [13] K.D. Kafui, C. Thornton, M.J. Adams, Disctete particle-continuum fluid modelling on gas–solid fluidised beds, Chemical Engineering Science 57 (13) (2002) 2395–2410. [14] T. Kawaguchi, T. Tanaka, Y. Tsuji, Numerical simulation of two-dimensional fluidized beds using the discrete element method (comparison between the two- and threedimensional models), Powder Technology 96 (1998) 129–138.
[15] A. Kitron, T. Elperin, A. Tamir, Mote Carlo simulation of gas–solids suspension flows in impinging streams reactors, International Journal of Multiphase Flow 16 (1990) 1–17. [16] U. Lackermeier, C. Rudnick, J. Werther, A. Bredebusch, H. Burkhardt, Visualization of flow structures inside a circulating fluidized bed by means of laser sheet and image processing, Powder Technology 114 (2001) 71–83. [17] Jinghai Li, Congli Cheng, Zhongdong Zhang, et al., The EMMS model — its application, development and updated concepts, Chemical Engineering Science 54 (1999) 5409–5425. [18] K.S. Lim, J.X. Zhu, J.R. Grace, Hydrodynamics of gas–solid fluidization, International Journal of Multiphase Flow 21 (1) (1995) 141–193. [19] H. Liu, Y. Xia, Y. Tung, M. Kwauk, Micro-visualization of clusters in a fast fluidized bed, Powder Technology 66 (1991) 231–235. [20] M. Liu, H.Q. Zhang, X.L. Wang, et al., Numerical simulation of cluster formation in dense gas–solid flow (in Chinese), Journal of Combustion Science and Technology 9 (2) (2003) 128–134. [21] C. Lockhart, J.X. Ahu, C. Brereton, C.J. Lim, J.R. Grace, Local heat transfer, solids concentration and erosion around membrane tubes in a cold model circulationg fluidized, International Journal of Heat and Mass Transfer 38 (1995) 2403–2410. [22] N. Mostoufi, J. Chaouki, Flow structure of the solids in gas–solid fluidized beds, Chemical Engineering Science 59 (2004) 4217–4227. [23] B. Oesterle, A. Petitjean, Simulation of particle-to-particle interactions in gas–solid flows, International Journal of Multiphase Flow 19 (1993) 199–211. [24] J.Y. Park, The clustered dense phase model for group A fluidization I. Dense phase, Chemical Engineering Science 58 (2003) 193–202. [25] M.J. Rhodes, X.S. Wang, M. Nguyen, P. Stewart, K. Liffman, Use of discrete element method simulation in studying fluidization characteristics: influence of interparticle force, Chemical Engineering Science: 56 (1) (2001) 69–76. [26] J.F. Richardson, W.N. Zaki, Sedimentation and fluidization, Transactions of the Institution of Chemical Engineers 32 (1954) 35–53. [27] J.F. Richardson, in: J.F. Davidson, D. Harrison (Eds.), Fluidization, Chap. 2, Academic Press, London, 1971. [28] W. Shuyan, L. Huanpeng, L. Huilin, L. Eentie, J. Ding, L. Wei, Flow behaviour of cluster in a riser simulated by direct simulation Monte Carlo method, Chemical Engineering Journal 106 (2005) 197–211. [29] T. Tanaka, Y. Tsuji, Numerical simulation of gas–solid two-phase flow in a vertical pipe: on the effect of inter-particle collision, Gas–Solid Flows 121 (1991) 123–128. [30] Y. Tsuji, T. Tanaka, S. Yonemura, Cluster patterns in circulating fluidized beds predicted by numerical simulation (discrete particle model versus two-fluid model), Powder Technology 95 (1998) 254–264. [31] W.B.G.W. Wachem, A.E. Almstedt, Methods for multiphase computational fluid dynamics, Chemical Engineering Journal 96 (2003) 81. [32] C.P. Wang, Z.A. Lv, D.K. Li, Experimental study of the effect of internals on optimizing gas–solid flow in a circulating fluidized bed, Powder Technology 184 (2008) 267–274. [33] H. Weisten, M. Meller, M.J. Shao, R.J. Parisi, The effect of particle density on holdup in a fast fluid bed, AIChE Symposium 80 (1984) 52–59. [34] Bin Zou, Hongzhong Li, Yashen Xia, Kwauk Mooson, Statistic model for cluster size distribution in fast fluidized bed, Engineering Chemistry & Metallurgy 14 (1) (1993) 36–43 in Chinese. [35] G. Xu, K. Kato, Hydrodynamic equivalent diameter for clusters in heterogeneous gas–solid flow, Chemical Engineering Science 54 (1999) 1837–1847. [36] R.L. Wu, C.J. Lim, J.R. Grace, C.M.H. Brereton, Instantaneous local heat transfer and hydrodynamics in a circulating fluidized bed, International Journal of Heat and Mass Transfer 34 (1990) 2019–2027. [37] J. Zhou, J.R. Grace, C.J. Lim, C. Brereton, Particle velocity profiles in a circulating fluidized bed of square cross-section, Chemical Engineering Science 50 (1995) 237–244. [38] J. Zhou, J.R. Grace, S.Z. Qin, C. Brereton, C.J. Lim, J. Zhu, Voidage profiles in a circulating fluidized bed of square cross-section, Chemical Engineering Science 49 (1994) 3217–3226.
Contents lists available at ScienceDirect
Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Modelling of dense gas-particle flow in a circulating fluidized bed by Distinct Cluster Method (DCM) Xiangjun Liu a,⁎, Xuchang Xu b a b
Thermal Engineering Department, University of Science and Technology Beijing, Beijing 100083, China Thermal Engineering Department, Tsinghua University, Beijing, 100084, China
a r t i c l e
i n f o
Article history: Received 6 October 2008 Received in revised form 9 June 2009 Accepted 12 June 2009 Available online 27 June 2009 Keywords: Dense gas-particle two-phase flow Particle cluster Distinct Cluster Method
a b s t r a c t Computational Fluid Dynamics (CFD) is a powerful tool to study the dense gas–solid flow in a circulating fluidized bed. Most of the existing methods focus on the microscopic properties of individual particle. Therefore, the simulation scale is significantly limited by the huge number of individual particles, and so far the numbers of particles in most of the reported simulations are less than 105. The hydrodynamics behaviour of particle clustering in a dense gas–solid two-phase flow has been verified by several experimental results. The Distinct Cluster Method (DCM) was proposed in this paper by studying the macroscopic particle clustering behaviour, and comprehensive models for cluster motion, collision, break-up, and coalescence have been well developed. We model the dense two-phase flow field as gas-rich lean phase and solid-rich cluster phase. The particle cluster is directly treated as one discrete phase. The gas turbulent flow is calculated by Eulerian approach, and the particle behaviour is studied by Lagrangian approach. Using the proposed method, a threedimension dense gas-particle two-phase flow field in a circulating fluidized bed with square-cross-section, with particle number up to 7.162 × 107 are able to be numerically studied, on which few results have been reported. Details on instantaneous and time-averaged distributions are obtained. Developing process of nonuniform particle distribution is visualized. These results are in agreements with experimental observations, which justified the feasibility of using the DCM method to model and simulate dense gas–solid flow in a circulating fluidized bed with large number of particle numbers. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Circulating Fluidized Beds (CFBs) are currently very popular in many industries, such as coal combustion, fluid catalytic cracking, and etc. Computational Fluid Dynamics (CFD) has been widely used to study the gas–solid two-phase flow in a CFB. The Eulerian description for both solid phase and gas phase has been well developed [7,31]. With the rapid development of computational capabilities, the mixed Eulerian–Lagrangian approach attracted more and more attentions from many researchers. In this approach, the detailed particle motion behaviour, which facilitates a better understanding of the physical phenomena, can be revealed by solving the Newtonian motion Equations in Lagrangian coordinate. The major problem in the simulation of dense gas–solid two-phase flows is to study the inter-particle interactions, and the numerical characterization of a dense particle–gas two-phase flow is much more complicated than that of the dilute one because of the large particle volumes. Several mathematical models have been developed in the last 2decades. The Direct Simulation Monte Carlo (DSMC) method was first proposed by Bird [2]. The particle-to-particle collisions were con-
⁎ Corresponding author. Tel.: +86 10 62333792. E-mail address: [email protected] (X. Liu). 0032-5910/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2009.06.007
sidered by means of analogy to the kinetic theory of molecules in dilute gas flow. In each time step, the collisionless motion of a representative number of particles is calculated simultaneously, and each simulated particle represents a certain number of real particles with identical physical properties. After that, a representative number of collisions are carried out by employing some kinds of Monte Carlo methods. The DSMC method has been improved and applied to the simulation of dense particle–gas flow by several researchers [15,28,30]. The V-shape and the reverse V-shape clusters in a CFB were successfully predicted by Tsuji et al. [30], and the cluster frequency in a CFB was obtained by Shuyan et al. [28] using the DSMC method. Oesterle and Petitjean [23] proposed another iterative simulation technique by introducing artificial inter-particle collisions during the trajectory calculation. In this method, trajectories of a large number of individual particles are computed successively, and each trajectory represents a group of particles. The collisions are processed stochastically on the basis of macroscopic particle properties obtained from previous iteration. This method is limited to steady flow simulation, and the probability of these collisions depends on the local concentration and velocity of the solid phase. The Discrete Element Method (DEM) was first proposed to study granular materials [6]. Recently, DEM is becoming more and more popular for dense particle–gas two-phase flow simulation. By tracking
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each individual particle in the flow filed, the real inter-particle interaction can be modeled and simulated so that the physical phenomena of the two-phase flow can be further quantified. Using DEM, Tanaka and Tsuji [29] successfully simulated a two-phase plug flow in a vertical pipe by considering the fluid effects with CFD. Later on, several collision-dominated two-phase flows were successfully simulated using this method [10,13,14,25]. All these existing Eulerian–Lagrangian models described above are theoretically capable to calculate mutual interactions of particles as well as to predict the dense two-phase flow. However, real simulation can only be feasible for a small-scale region, or a two-dimension region with limited number of particles. To apply these models to a dense two-phase flow in real-scale industrial applications, further research is still needed to reduce the tracking number of trajectories to be calculated so that the required computational capabilities in the simulation are realistic. In this paper, we proposed the Distinct Cluster Method (DCM). In this method, we consider the particle cluster as one discrete phase. The cluster collision, break-up and coalescence models are established. Using these proposed models, a three-dimension gas-particle two-phase flow field in a circulating fluidized bed with square-crosssection is simulated. The global volume-averaged solid concentration in the computation domain is 3.0%. Few numerical studies on threedimension flow have been reported due to the large computation required. Details on the proposed models and the calculated results are presented in the rests of this paper. 2. Distinct Cluster Model for particle phase 2.1. Particle clustering One phenomenal hydrodynamics behaviour in dense gas–solid two-phase flow, especially in a CFB, is particle clustering, in which most of the particles congregate as clusters, and these clusters act as independent objectives inside the gas-rich dilute phase. This has been verified by many experiments. Particle clustering implies that groups of several to dozens of particles tend to congregate in the riser in order to minimize the drag force exerted on them [3]. This micro-flow structure was first observed by Bai et al. [1] by employing a twodimensional column, by Liu et al. [19] using a video camera and a special optical fiber image probe, and by Horio & Kuroki [12] using a laser sheet technique. Recently, the flow patterns of particle clusters in a CFB is obtained and studied by Lackermeier et al. [16] using a highspeed video technique in combination with the laser sheet technique. These experimental results show that there are two distinct phases, a dispersed phase, in which solid particles are individually presented, and a solid-rich cluster phase. The solid-rich phase usually consists of clusters and each cluster is a congregation of dozens of particles. The diameter of each cluster is less than 1–2 cm, and the particle volume fraction is generally around 41–50% [18]. Particle clustering is a key characteristic of CFBs, and it significantly impacts the mechanism of the flow, heat transfer, and chemical reaction in CFBs. To model and simulate the dense gas-particle two-phase flow, methodology to explicitly model and study particle cluster is quit straightforward. 2.2. Cluster definition and its size Our approach is to divide the dense two-phase flow field in a gasparticle CFB as gas-rich lean phase and solid-rich cluster phase as shown in Fig. 1. Based on the phenomenon that solid particles are presented individually and randomly in the gas-rich lean phase, we assume that there is no-slip between gas and these individual solid particles, and the particles are uniformly distributed. The solid-rich phase consists of clusters and each cluster is a congregation of dozens of particles. Furthermore, the cluster acts as individual and instantaneously separated objective inside the gas-rich dilute phase, with its shape and size consistently varying, due to the complicated two-phase interactions.
Fig. 1. Gas-rich lean phase and solid-rich cluster phase.
Observed by many experiments [12,3,22], the shape of these clusters may be in the forms of strings, triangles, V-shape with a round nose, or other shapes, and the size is generally less than 1–2 cm in diameter. The particle may be either contacted or not contacted inside a cluster, and the particle volume fraction of a cluster is generally around 41–50%. In our approach, each cluster is simply considered as a spherical ball. The voidage of a cluster is taken as a constant. Proposed by Park [24], the cluster voidage can be taken as εc =εmf, where εmf is the minimum fluidization voidage of a specific particle. The gas velocity through a cluster is neglected. As discussed above, the cluster sizes are consistently varying. Therefore, the key problem to be addressed in this method is to define the instantaneous size of a particle cluster. Cluster size is one of the important parameters in the analysis of gasparticle two-phase flow in a CFB. It is usually denoted as the effective equivalent diameter. Many experimental and theoretical researches have been carried out to reveal this characteristic value. Chavan [5] found that the effective equivalent diameter of a cluster is inversely proportional to the energy that breaks the consecutive emulsion phase into discrete particle clusters. Experimental results by Mostoufi et al. [22] shown that the ratio of cluster diameter to particle diameter was in a wide range of 1.5–160. Zou et al. [34] established a statistic model on cluster size distribution in fast fluidized bed based on micrographs experiments. Xu and Kato[35] proposed a simple correlation of the equivalent cluster diameters for gas and solid concurrent upward flows. Li et al [17] developed an equation on equivalent cluster diameter using the Energy-Minimization Multi-Scale (EMMS) approach. By balancing the forces imposed on a cluster suspended in gas, Horio et al. [11] proposed a mathematical model for cluster diameter:
l=
1 2ðn−1Þ 3ð1−εc Þεi
ρg 2 ðu −usup Þ ðρp −ρg Þg lean
ð1Þ
Where ulean and usup are the lean phase velocity and the superficial gas velocity respectively. According to our assumptions described above, we take ulean =ug, and ug is the local instantaneous gas velocity. Meanwhile, ρp and ρg are particle and gas density, εc is the voidage in cluster phase, and εi is the local instantaneous voidage. Parameter n is the Richardson–Zaki correlation [11,26], and the value depends on the physical properties of the particle. There have been many discussions on this [27]. We take n = 5 in this paper. From Eq. (1), it is clear that the local instantaneous cluster diameter is determined by the local instantaneous gas velocity and voidage, and the value fluctuates in a wide range with high frequency during the calculation process. In order to simplify the calculation without compromising the accuracy, the smallest particle number inside a cluster is assumed to be 3.
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2.3. Cluster motion equation
3. Simulation of gas flow
Treating the particle cluster as one discrete phase, the classical trajectory model at Lagrangian coordinate can be used to describe the cluster moving behaviour. The forces acting on a moving cluster include gravity force, drag force, and etc. The cluster motion equation can be expressed as
Taking the particle cluster as one discrete phase inside the gas-rich dilute phase, the trajectory model is used to describe the behaviour of each cluster in every time step. Since the dense particle–gas twophase flow is essentially an unsteady flow, the local gas and particle flow characteristics is time varying. The following continuity and momentum equations are used to model the instantaneous gas turbulent flow.
mc
→ d Vc → → = mc g + F D dt
ð2Þ
where F ̅D is the drag force, the drag force coefficient is decided by the following relations
ð1 +
Re2c = 3 = 6Þ24
Re c
ð3Þ
j
νg
∂ðρg εui Þ
∂ðρg εuj Þ ∂xj
∂ðρg εui uj Þ
+
ð6Þ
=0
∂xj
= −ε
→ where ∑ F drag is the volumetric fluid–particle interaction force. It can be determined by N 1
=
j
ð8Þ
where ΔV is the volume of a computational gas grid, and Np is the cluster number in this grid. The turbulent stresses tensor τij in Eq. (7) is calculated as
2.4. The collision, break-up, and coalescence models At each time step, the velocity and position of all the clusters in a two-phase flow field are calculated from Eq. (2). Due to the gas– particle and particle–particle interactions, clusters form and break continuously. The cluster size di is re-calculated at the end of each time step. Comparing di with the local maximum cluster size li determined by Eq. (1), it is clear that the cluster i is broken into some fragment-clusters if di N li. The number of fragment-clusters is n = int(d3i/l3i) + 1, among which n − 1 clusters are determined to have the same mass mk and the same effective equivalent diameter li. The mass of the fragment-cluster n is mn = mi − (n − 1)mk, and the diameter is determined by its mass. We consider the conservation of kinetic energy during breaking-up process, and the kinetic energy of each fragment-cluster is proportional to its mass. The motion direction is assumed as spatially homogeneous. When two or more clusters are contacted at the end of each time step, the cluster–cluster collision is calculated. We consider two → → clusters of diameter di and dj with velocity vectors Vi and Vj . The distance between the central points of the two clusters is Rij. → → If Rij b 12 ðdi + dj Þ and Vi · Vj N 0, the two contacted clusters move away from each other. → → If Rij b 12 ðdi + dj Þ and Vi · Vj b 0, the two cluster collide. First, we assume that the two clusters contacted are merged as one big cluster. The conservation of mass and kinetic momentum are maintained during the collision process. The mass and velocity are calculated as (
" ! # ∂uj ∂p ∂ ∂ui → −τij −∑ F drag + με + ∂xi ∂xj ∂xj ∂xi
p → → ∑ F drag = ∑ F D ΔV
→ dc → ; V g and V c are the gas and cluster velocity, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p respectively; dc = 3 6Ω = π and Ώ is the cluster volume. where Rec =
+
ð7Þ
Rec ≥1000; C D = 0:44 → → Vg− Vc
∂t
∂t
Rec b1; C D = 24 = Rec
1≤Rec b1000; C D =
∂ðρg εÞ
mij = mi + mj → → → mij V ij = mi V i + mj V j
τij = −ρg νt Sij = −ρg νt
∂uj 1 ∂ui + 2 ∂xj ∂xi
!
The eddy viscosity vt is evaluated using the mixing-length approach 2
2
νt = ðCs ΔÞ jSj = ðCs ΔÞ
qffiffiffiffiffiffiffiffiffiffiffiffi 2Sij Sij
4. Apply the DCM method to a bench-scale CFB with square-cross-section So far, most of the research works were based on risers with circular cross-section. However, risers with square or rectangular cross-sections are being widely used in industrial CFB applications, such as combustions. The riser geometry has considerable influence
ð4Þ
Effective equivalent diameter of this merged cluster is determined qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 6ðmi + mj Þ = ρp π
ð5Þ
The next step is to compare dij with the local maximum cluster size li. The cluster is broken into some fragment-clusters if dij N li, as described above.
ð10Þ
In the equation above, Cs is typically set to 0.15 [9]; Δ is the length scale, which is usually defined as the cubic root of the local grid volume.
as dij =
ð9Þ
Fig. 2. Computation domain of the CFB.
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Table 1 Parameter settings. Channel height Channel length Channel width Particle diameter Solid concentration Particle number Inlet gas velocity
0.6 m 0.06 m 0.06 m 120 µm 3% 7.162 × 107 5.5 m/s
Gas density Particle density Laminar gas viscosity Time step for gas phase Time step for particle phase Grid number Total calculation duration
1.2 kg/m3 1500 kg/m3 1.8 × 10− 5 kg/m s 1.0 × 10− 4 s 1.0 × 10− 5 s 32 × 32 × 100 2.5 s
on the hydrodynamics of CFBs [36,37,38]. Experimental results show that the voidage and velocity profiles in a CFB with square-crosssection are different from that with a circular cross-section [32,37,38]. The corners impact the two-phase flow field significantly. In this paper, a three-dimension dense gas-particle two-phase flow field in a CFB with square-cross-section is numerically studied. The sketch diagram of the computation domain is shown in Fig. 2. The bench-scale CFB is 2.0 meters in height, and the cross-section of the riser is 0.06 m × 0.06 m. The gas density is 1.2 kg/m3, and the particle density is 1500 kg/m3. The superficial gas velocities are in the range between 4.5 m/s and 8 m/s. The time–mean voidage in the upper region is usually in the range from 0.95 to 0.975. In this paper, we study a region of 0.6 m in height in the upper fast fluidization regime. The global mean solid concentration in the computation domain is set to 3.0%. The mean particle diameter is 120 µm so that total particle volume in the bed is 6.48 × 10− 5 m3 and the total particle number is 7.162 × 107. We assume that 90% of the total particles are congregated
as spherical clusters and each cluster contains 400 particles by average. Therefore there are 6.4458 × 107 particles congregated as 1.611 × 105 clusters, and they are evenly distributed in the flow field initially. The voidage of clusters is considered to be a constant of 0.5. The initial volume, diameter, and velocity of each cluster is 7.238 × 10− 10 m3, 1.114 × 10− 3 m, and 2 m/s respectively. All the parameter settings in this numerical study are summarized in Table 1. We use the well-known SIMPLE algorithm to resolve the gas governing equations. A staggered non-uniform grid system is used in this paper. The total grid number is 32 × 32 × 100 in x, y and z direction as shown in Fig. 2, and the grid settling in x and y direction is nonuniform. The finest grids, which are 0.5 mm × 0.5 mm × 6 mm in size, are located at the very near corner regions so that effects of walls and corners on the two-phase flow field can be studied in details. The biggest grids located at the center regions are 5 mm × 5 mm × 6 mm, and the mean grid size is 1.3 mm × 1.3 mm × 6 mm. The time steps for gas phase and particle phase are 1.0 × 10− 4 s and 1.0 × 10− 5 s respectively. The source terms are linearized. The time-dependent terms are determined using the second order implicit difference scheme and the convection and diffuse terms are resolved by the Quadratic Upwind Interpolation of Convective Kinematics (QUICK) difference scheme. The finite difference equations are solved using the Gauss–Seidel under-relaxation iteration. We calculate the gas phase only in the flow field from t = 0 to 0.5 s. All the particle clusters are randomly seeded into the flow field at t = 0.5 s and they are evenly distributed. The total calculation duration of the two-phase flow is 2.5 s. In order to maintain the total solid mass in the computation domain, the solid concentration is maintained to
Fig. 3. Instantaneous particle volume fraction at z = 0.3 m–z = 0.32 m.
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be 3% which serves as the circulation boundary condition. This is realized by setting the particle flow rate at the inlet to be equal to the flow rate at outlet. We first calculate the velocity and position of all the clusters at each time step. Then the cluster collisions are considered, and the cluster parameters are updated by solving Eqs. (1)–(5). The time–mean inlet gas velocity of each grid is 5.5 m/s. A random perturbation is superposed on each inlet velocity component and the stochastic velocity component is set to be u i̅ = 〈u ̅i〉 + Iψ| u ffi̅|, where ψ qffiffiffiffiffi ¯ V = 1. The is a Gaussian random number satisfying ψ ̅ = 0 and ψ fluctuating intensity I is set to be 15%.
239
5. Calculated results 5.1. Calculated instantaneous flow field at different time steps Fig. 3 (a–i) is the instantaneous particle volume fraction distributions from z = 0.3 m to z = 0.32 m section. At t = 0.5 s, particle clusters are randomly distributed in the flow field. The local maximum particle concentration is 11.7%, and the high particle concentration regions are evenly distributed within the flow field. Once the clusters are moved by aerodynamic forces, most clusters are broken-up or
Fig. 4. Instantaneous particle volume fraction at y = 0.02 m–y = 0.04 m.
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Fig. 7. Instantaneous UV gas velocity vector of z = 0.5 m section at t = 2.5 s. Fig. 5. The statistical distribution of the calculated cluster size at t = 2.5 s.
merged, and the distribution of clusters becomes more and more nonuniform. The dense particle regions are at the four-corners, and the dilute region is in the center of the section. These results are consistent with experimental ones, which were interpreted by Lockhart [21] that membrane walls have significant local effects and increase the local solids holdups in the shielded fin region. Fig. 4 (a–i) show the instantaneous particle volume fraction distributions from y = 0.02 m to y = 0.04 m section. Particle distribution in x–z section is consistent with that of the x–y section as shown in Fig. 3. Initially, the high particle concentration regions are evenly distributed in the flow field, and gradually they become non-uniform once the two-phase flow behaviour is constituted. The quasistationary two-phase flow system is established from t = 1.0 s. The particle concentration is non-uniform in both vertical and lateral directions. For the vertical flow structure, there is a dense region at the bottom and a dilute region at the top of the riser. For the lateral flow structure, there is a dilute region at the core and a dense region adjacent to the wall. The particle volume fraction in the center is much lower than that closer to the wall. This non-uniformity is the key flow characteristic of a CFB which has been confirmed by many experimental research works (e.g. [8,33]). The proposed DCM model is successful in this simulation. The total number of clusters and the size of each cluster keep varying during the calculation process. Initially, there are 1.611 × 105 clusters evenly distributed in the flow field. At t = 0.52 s, the total number of clusters increases to 298,792. Most of the initial clusters are broken into smaller fragment-clusters under the breaking-up effects of aerodynamic actions, while the maximum cluster number is less than 3.2 × 105 during the 2.5 s calculation duration. At t = 2.5 s, the
Fig. 6. Time variation of particle concentration at a given region (central point: 0.02 m, 0.02 m, 0.4 m).
number of cluster is 257,467, and their equivalent diameter ranges from 218 μm to 7165.7 μm. The statistical distribution of the calculated cluster size is shown in Fig. 5. There are 32.4% of the total particles congregated as smaller clusters with equivalent diameter less than 300 μm. At the same time, 19.7% of the total particles are congregated as larger clusters with equivalent diameter more than 900 μm. The mean equivalent diameter of all the clusters is 958.8 μm. Most of the clusters flying in the core region are less than 300 μm in diameter, while most of the clusters near the wall are more than 600 μm in diameter. The smaller clusters are usually in the center regions, and the clusters become bigger and bigger when moving closer to the
Fig. 8. Instantaneous UW gas velocity vector in y = 0.03 section at t = 2.5 s.
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Fig. 9. (a–c) The two-phase time-averaged flow profiles in y = 0.03 m section.
walls. The distribution of cluster size is reasonable and is in agreement with the experimental and simulation results from other researchers [18,20]. Fig. 6 shows the time variation of particle volume fraction at a control volume (0.0015 m × 0.0015 m × 0.0136 m) in the upper zone at x = 0.02 m, y = 0.02 m, z = 0.4 m. The time-averaged particle volume fraction from t = 1.0 s to 2.5 s of this control volume is 0.022. Initially, particle clusters are randomly located in the flow field, and the local particle fraction in this grid is 0.0355. Once the two-phase coupling calculation is carried out after t = 0.5 s, large-scale time fluctuations are observed. The quasi-stationary two-phase flow system is established at t = 1.0 s from when the instantaneous value fluctuates around a mean value with high frequency. The same process can also be observed from the instantaneous particle volume fraction distributions as shown in Figs.3 and 4. Fig. 7 shows the instantaneous UV gas velocity vector at t = 2.5 s in the x–y cross-section at z = 0.5 m. Although the two-phase flow is discharged into the square duct along the z direction, the fluctuating velocity in the x–y direction cannot be neglected. The UV instantaneous velocity in the center region ranges from 0.5 m/s to 2.0 m/s, while the instantaneous velocity at the four-corners is much lower. This velocity distribution results in the four dense particle corner zones as shown in Fig. 3. Fig. 8 is the instantaneous UW gas velocity
vector at t = 2.5 s in the x–z cross-section at y = 0.03 m. The time– mean inlet gas velocity is 5.5 m/s and the instantaneous W velocity ranges from −2.8 m/s to 13.3 m/s. 5.2. Time-averaged two-phase flow and comparisons with existing experimental results As described above, the quasi-stationary two-phase flow system is established at t = 1.0 s in the total 2.5 s calculation duration. Calculation results of the last 1.5 s are used as samples for time-averaging in order to maintain sufficient statistical accuracy. The time-averaged flow profiles in the center x–z section at y = 0.03 m are shown in Fig. 9 (a–c). Fig. 9a is the transversal variations of time-averaged vertical gas velocity at heights equals to 0.05 m, 0.17 m, 0.29 m, 0.41 m, and 0.53 m respectively. Time-averaged gas velocity in the core region is much higher than in the regions closer to the wall. The mean gas velocity decreases from the peak to a value smaller and smaller when moving from the core region to the wall. Fig. 9b is the transversal variation of time-averaged vertical cluster velocity at the heights between 0.0–0.05 m, 0.12–0.17 m, 0.24–0.29 m, 0.36–0.41 m, and 0.48–0.53 m respectively. Fig. 9c is the transversal variation of particle volume fraction at the same five regions. From Fig. 9a–c we can see the gas and cluster velocity distributions are
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Fig. 10. (a–c) The two-phase time-averaged flow profiles along the diagonals.
highly correlated. Clusters (particles) are entrained upwards in the core region and downwards in the dense annular region adjacent to the wall, and this is the same as in a circular cross-sectional columns. Consequentially, the time-averaged particle volume fraction in the riser gradually decreases from peak at the bottom to lower values towards the top. This is the so-called core-annulus flow structure proposed by Bierl & Gajdos [4] and adopted by many others. These time-averaged distributions are in accordance with the instantaneous particle volume fraction distributions shown in Fig. 4. The two-phase time-averaged flow profiles along the diagonals are shown in Fig. 10a–c. Compared with those along the centerlines, the gas velocity near the corner is lower, which results in higher particle descending velocity and increased descending fluxes. The timeaveraged descending velocity of particle at point (0.059 m, 0.059 m, 0.16) located at corner is − 2.0 m/s, and the instantaneous velocity of this point fluctuates in the range from − 1.5 to − 3.1 m/s. Compared with the point (0.059 m, 0.03 m, 0.16) located in the center along the wall, the time-averaged descending velocity of particle is −1.2 m/s, and the instantaneous velocity fluctuates in the range of −0.5 to −2.2 m/s. The time-averaged particle volume fractions of these two points are 0.243 and 0.162 respectively. Further studies show the particle concentration in the corner is higher than elsewhere along the wall. These flow characters have been revealed and interpreted by
Fig. 11. Lateral profiles of mean particle velocity for Ug = 5.5 m/s, Gs = 40 kg/m2s and z = 5.13 m [37].
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Fig. 12. Lateral profiles of mean voidage for Ug = 7.0 m/s, Gs = 40 kg/m2s and z = 6.20 m [38].
mental one as shown in Figs. 11 and 12. They both indicate that most of the particles near the wall travel downwards while in the core most of particles travel upwards. At the corner, particles move downwards faster, and the thickness of the downward flow wall layer is greater than elsewhere along the wall.
Zhou et al [38], and the reason is the descending particles were protected by the corner where the gas velocity is lower. The voidage and velocity profiles in a CFB with square-cross-section was experimentally studied by Zhou et al [37,38] using a fiber optic probe. The measurement results of the lateral profiles for mean vertical particle velocity and mean bed voidage are shown in Figs. 11 and 12. The cross-section of the experimental riser is 146 mm × 146 mm and the total height is 9.14 m. Similar results were obtained from experiments by Wang et al. [32] in a 400 mm × 400 mm × 5000 mm square riser using ECT. The calculated lateral profiles using the proposed DCM method for time-averaged vertical particle velocity and voidage in the section at z = 0.3 m are shown in Figs. 13 and 14 respectively. It is clear that our simulation results are qualitatively in agreement with experi-
In this paper, we proposed an original Distinct Cluster Method (DCM) and we also established the cluster motion, collision, broken-up, and coalescence models. Using this method we were able to simulate a three-dimension gas-particle two-phase flow field in a circulating fluidized bed with square-cross-section, with total particle number up to 7.162 × 107. This is realized by directly taking the particle cluster as one
Fig. 13. Calculated lateral profiles of mean vertical particle velocity (z = 0.3 m).
Fig. 14. Calculated lateral profiles of mean voidage (z = 0.3 m).
6. Conclusions
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discrete phase so that the maximum tracking number of the clusters to be processed during the simulation was reduced to 3.2 × 105. Comparing with the conventional DEM, the number of tracking objects is reduced more than 220 times by using DCM. In addition, Lagrangian integration can be significantly simplified using larger time step which is proportional to the mass of each tracking cluster, instead of an individual particle. The proposed DCM can effectively reduce the computation complexity in gas-particle two-phase flow. By using the proposed DCM method, detailed results on instantaneous and time-averaged distributions of two-phase flow in a CFB are obtained. Developing process of non-uniform particle distribution is visualized. Numerical results are intensively reviewed and compared with existing theoretical and experimental conclusions. These proposed models and algorithm are feasible and can be used to study the dense gas–solid flow in a circulating fluidized bed with large number of particle numbers. Acknowledgment This work was supported by National Natural Science Foundation of China under Grant No. 50406025. References [1] D.R. Bai, Y. Jin, Z.Q. Yu, Cluster observation in a two-dimensional fast fluidized bed, Fluidization'91–science and Technology (Edited by Kwauk, M. & Hasatani, M.), Science Press, Beijing, 1991, pp. 110–115. [2] G.A. Bird, Molecular Gas Dynamics, Oxford Univ. Press, 1976. [3] H.T. Bi, J.X. Zhu, Y. Jin, Z.Q. Yu, Forms of particle aggregations in CFB, Proc. 6th Chinese Conf. On Fluidization, 1993, pp. 162–167, Wuhan, P. R. China. [4] T.W. Bierl, L.T. Gajdos, Phenomenological modelling of reaction experiments in risers, Final Report, DOE/MC-14249-1149, Environmental Research and Technology, 1982. [5] V.V. Chavan, Physical principles in suspension and emulsion processing, in: A.S. Mujumdar, RA Mashelkar (Eds.), Advances in Transport Processes, Vol. 3, John Wiley & sons, New York, 1984, pp. 1–34. [6] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1) (1979) 47–65. [7] J.S. Curtis, V.B. Wzchem, Modeling particle-laden flow: a research outlook, AIChE Journal 50 (2004) 2638. [8] R.J. Dry, Radial concentration profiles in a fast fluidized bed, Powder Technology 49 (1986) 37–44. [9] M. Gemano, U. Piomelli, P. Moin, W.H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys, Fluid A3 (7) (1991) 1760–1765. [10] E. Helland, R. Occelli, L. Tadrist, Numerical study of cluster formation in a gas-particle circulation fluidized bed, Powder Technology 110 (2000) 210–221. [11] M. Horio, K. Morshita, O. Tachibana, H. Hurata, in: P. Basu, J.F. Large (Eds.), Circulation Fluidized Bed Technology H, Pergamon Press, 1988, pp. 147–154. [12] M. Horio, H. Kuroki, Three-dimensional flow visualization of dilutely dispersed solids in bubbling and circulating fluidized bed, Chemical Engineering Science 49 (2413) (1994) 421. [13] K.D. Kafui, C. Thornton, M.J. Adams, Disctete particle-continuum fluid modelling on gas–solid fluidised beds, Chemical Engineering Science 57 (13) (2002) 2395–2410. [14] T. Kawaguchi, T. Tanaka, Y. Tsuji, Numerical simulation of two-dimensional fluidized beds using the discrete element method (comparison between the two- and threedimensional models), Powder Technology 96 (1998) 129–138.
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