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DEM–LES study of 3-D bubbling fluidized bed with immersed tubes
Chemical Engineering Science 63 (2008) 3654 -- 3663
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DEM--LES study of 3-D bubbling fluidized bed with immersed tubes Nan Gui, Jian Ren Fan ∗ , Kun Luo State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, PR China
A R T I C L E
I N F O
Article history: Received 9 July 2007 Received in revised form 13 December 2007 Accepted 15 April 2008 Available online 2 May 2008 Keywords: Bubble Fluidization Multiphase flow Powder technology Immersed tube Discrete element method
A B S T R A C T
Based on Euler--Lagrange frame, a true three-dimensional numerical simulation of bubbling fluidized bed embedded with two immersed tubes is presented. The solid phase is composed of 178,200 particles of 883 m diameter and simulated by discrete element method (DEM, a soft-sphere approach). The gas phase is computed through solving the volume-averaged four-way coupling Navier--Stokes equations in which the Smagorinsky SGS tensor model is used in large eddy simulation (LES). Particle--tube collision is particularly treated as a transformation of DEM. The volume segmentation of a particle sphere for void fraction calculation is solved via a numerical sub-division approach. The numerical results are compared with the experimental results for validation. The results obtained with and without the LES model are also compared. The numerical results show a strong correlation between gas--particle interaction, particle--particle interaction, pressure drop, particle back mixing motion and bubble motion, and all of them follow a similar pattern of synchronous periodic variation though the periodicity may vary depending on different flow conditions. The effects of SGS tensor on evolution of fluidized bed are found in various aspects. Finally, the distribution of particle--tube impact frequency is given. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Many numerical researches on bubbling fluidized bed were carried out by many investigators. These investigations of dense gas--solid flow, based on various models, are mainly divided into two categories, i.e., Euler--Euler two-phase model (Jackson, 1963; Soo, 1967; Bouilard et al., 1989; Ding and Gidaspow, 1990; Tsuo and Gidaspow, 1990; Kuipers et al., 1992, 1993) and Euler--Lagrange approach (Hoomans et al., 1996; Tsuji et al., 1993; Xu and Yu, 1997; Gera et al., 1998; Yuu et al., 2000; Helland et al., 2002; Zhou et al., 2004). The discrete element method (DEM), which belongs to the latter category, has been proven a valuable tool for simulating the behavior of gas--solid particulate flow during the past decades (Hoomans et al., 1996; Tsuji et al., 1993; Rong et al., 1999; Goldschmidt et al., 2004). The DEMs, including a hard-sphere approach and a soft-sphere approach, are advanced for numerical simulation of gas--particle flow since they simulate the motion of all particles in a deterministic way, taking many interactions and forces into account, such as particle--particle collisions, particle--wall collisions, gas--solid two-way coupling interaction and gravitational force, etc. The two-fluid model (TFM) is under the Euler--Euler frame and based on the interpenetrating fluids. It has vast applications in the description of gas--solid hydrodynamic phenomena.
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But it requires many constitutive equations of solid phase for the closure of governing equations. Moreover, the parameters included in the constitutive formulations lack generality. Hence the DEM is to some extent preferable to the TFM in consideration of numerical simulation on the behavior of particles in fluidized bed. As mentioned by Hoomans et al. (1996), the hard-sphere model is a quasi-instantaneous and binary collision model, in which the particle--particle collisions are repulsive, inelastic and only take action once at a time. However, the soft-sphere model is a quasi-static and multi-body collision model, in which the interaction forces are combinations of linear or nonlinear elastic repulsive force, damping force and friction force. There are many applications of the above-mentioned two approaches of DEM. For example, Hoomans et al. (1996) developed a hard-sphere approach for discrete particle simulation on bubble and slug formation in two-dimensional (2-D) gas--solid bed. The collision model follows the conservation law of linear and angular momentum and uses two empirical parameters: a restitution coefficient and friction coefficient, which are key parameters used in soft-sphere model too. Tsuji et al. (1993) developed the soft-sphere model in application of 2-D fluidized bed simulation. The interaction force between particles is simulated by a combination of elastic repulsive force, damping force and friction force. Another key parameter of damping coefficient is used to model the damping effect of particle--particle contact. Xu and Yu (1997) carried out a simulation for 2400 GroupD particles in pseudo-3-D central jet fluidized bed by combining discrete particle method and computational fluid dynamics.
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Gera et al. (1998) used distinct element model to simulate hydrodynamics of 2-D large particle fluidized beds and compared the bubble rise velocity, voidage variations, etc., with TFM based on continuum theory. Yuu et al. (2000) simulated about 100,000 Group-B particles using the 3-D distinct element method to explore the mechanisms of bubble formation, coalescence and disruption. The dimension of the bed is one diameter of the particle deep and the governing equations for air phase are 2-D incompressible Navier--Stokes (N--S) equations. Moreover, Helland et al. (2002) studied the fluctuating motions and cluster structures in 2-D gas--solid flows treating particle motion by Lagrangian approach and the interaction between particles are assumed to be binary, instantaneous, inelastic collisions with friction. Zhou et al. (2004) simulated coal combustion in a bubbling fluidized bed by using DEM and LES (large eddy simulation) methods, while the gas phase is still treated as 2-D flows. Rong et al. (1999) conducted a numerical study on particle and bubble movements around immersed tubes in 2-D fluidized beds. Goldschmidt et al. (2004) implemented comparison between simulations of dense gas-fluidized bed based on discrete particle method and continuum theory, in which the gas phase hydrodynamics was also only resolved in 2-D case. It is obvious that most previous researchers focus their numerical study on bubble or particle behavior in a pseudo-3-D fluidized bed. Then the N--S governing equation for gas-phase is 2-D and sometimes non-viscous fluid is used. Moreover, the particle motion is restricted from motion in depth dimension. Thus, there exist some reasons for the necessity of true 3-D numerical simulation of fluidized beds:
where εg , g , t, ui , are volume fractions for gas phase, gas density, time, velocity of gas and stress tensor, respectively. The operator `∼' denotes the top-hat function for spatial filtering. It is considered that εg is a parameter that has already been filtered when it is calculated in a volume-averaged way. Thus, the sub-grid i = ε˜ u˜ i = εu˜ i , scale fluctuation εg is equal to zero, and we have εu ε ij = ε˜ ˜ ij = ε˜ij , etc. Moreover, the term εg Tij εg (u˜ i u˜ j − u i uj ) is the sub-grid scale stress tensor. We observe Eqs. (1)--(2) and consider that εg can be regarded as density of a pseudo-fluid, and εg ˜ ij , εg Tij are regarded as stress tensor and sub-grid scale tensor of the pseudo-fluid, respectively. From this point of view, Eqs. (1)--(2) can be treated as the governing equations of the single-pseudo-fluid phase. Thus, we use the most widely used Smagorinsky model for the sub-grid scale tensor. The Smagorinsky SGS model can be explained by the mixing-length assumption, in which the eddy viscosity is proportional to the SGS characteristic length scale and to a characteristic turbulent velocity based on the second invariant of the filtered-field deformation tensor (Lesieur and Metais, 1996):
(1) Particle motion in industrial fluidized bed is actually 3-D. The interactions between particles, such as particle--particle impact and friction, as well as particle motion, actually have more than six degrees of freedom. Thus the restriction of particle motion in pseudo-3-D simulation should be relaxed. (2) Fluid phase is actually viscous fluid and turbulence is 3-D especially for LES simulation in which 2-D eddy is not authentic. The fundamental of 2-D flow simulation exists only in symmetrical flow in depth dimension, and this is not appropriate for fluidized bed.
are the deformation tensor of the filtered field, eddy viscosity, local strain rate and sub-grid characteristic length scale, respectively. ij is the Kronecker delta function. An approximate value for the constant CS is given by (Lilly, 1987):
Accordingly, 3-D LES simulation coupling with DEM is more reliable than that in a 2-D case or in non-viscous case. But the main problem for implementation of 3-D simulation is the lack of powerful computer. In the present paper, a true 3-D simulation of fluidized bed is carried out. For solution of gas-phase governing equation, the results obtained with and without LES are compared for justification of SGS stress tensor. In addition, the results are referred to the experimental results for validation. The soft-sphere model is used to simulate the motion of particles and particle--particle interactions. Moreover, particle--cylinder contact model for particle-immersedtube collision is treated with the alteration of soft-sphere model. Finally, the numerical results reveal some interesting phenomena in fundamental interactions in dense gas--solid flow. 2. Numerical model 2.1. Gas-phase hydrodynamics For LES, the hydrodynamic models for gas phase use the principles of conservation of mass and momentum (Ding and Gidaspow, 1990): j(˜εu˜ i )g j˜εg + =0 jt jxi j(˜εu˜ i )g jt =−
+
(1)
j(˜εu˜ i u˜ j )g jxj
FD, i ε˜ g jp˜ g 1 j(˜εg ˜ ij ) 1 j(˜εg Tij,g ) + ε˜ g gi + − + g jxj g jxj g jxi g
(2)
Tij = 2t S¯ ij + 13 Tll ij
(3)
where 1 S¯ ij = 2
ju¯ j ju¯ i + jxj jxi
,
¯ t = (CS )2 |S|,
¯ = (2S¯ S¯ )1/2 |S| ij ij
= (xyz)1/3
CS ≈
(4)
1 3CK −3/4 2
(5)
where CK = 1.4 is Kolmogorov constant and this yields CS ≈ 0.18. Due to SGS the model is too dissipative and most workers prefer CS = 0.1 (Lesieur and Metais, 1996), we chose CS = 0.1 in the present paper too. Additionally, to validate the application of SGS tensor, we make comparison between the numerical results obtained with and without the SGS tensor with reference to the experimental results. The term FD is the drag force and FD = (vg − vs ) = (ui,g − ui,s ). In dense gas--particle flow, is the fluid--particle interaction coefficient, which is obtained from standard correlations. Below a void fraction of 0.8, is given by Ergun's equation; at and above 0.8, it is obtained from Wen and Yu's (1966) expression: ⎧ (1 − ε)g |vs − vg | (1 − ε)2 g ⎪ ⎪ + 1.75 , ⎪ 150 ⎨ 2 dp εdp = ⎪ ⎪ ⎪ ⎩ 3 C ε(1 − ε)g |vs − vg | ε−2.65 , 4 D dp And the drag coefficient is given by: 24(1 + 0.15Re0.687 )/Re, Re < 1000 CD = 0.43, Re 1000
ε < 0.8 (6) ε 0.8
(7)
where particle Reynolds number is defined as: Re =
εg |vg − vs |dp g
(8)
Under two-way coupling approach for gas--particle interaction, calculation for FD needs numerical interpolation of gas-phase velocity, pressure, void fraction from neighboring nodes to particle locations. In the present paper, we have handled it by using a linear
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interpolation method. It is in fact a convex combination of information at neighboring nodes and in 3-D case, and its representation is just a volume-averaged method of interpolation. Due to Newton's third law, the particle-to-gas force in a cell can be calculated by integrating the term −FD over the volume of particle inside the cell. Due to the utilization of stagger grid for velocity and pressure discretization, the particle-to-gas force may once more be interpolated to the location of velocity nodes. 2.2. Particle motion The translational and rotational motion of particle is computed through integrating the Newton's equation of motion: ·
V P = (FC + FD + FP )/m + g ·
p = T/I
(9) (10)
where VP , p , FC , FD , FP , T, m, I, g are particle translational and rotational velocity, inter-particle contact force, drag force, pressure gradient force, torque, mass and moment of inertia, gravity acceleration, respectively, and `·' denotes the time derivative. The inter-particle contact force FC is calculated by soft-sphere model (Tsuji et al., 1993; Cundall and Strack, 1979): FC = −kdr − vr
(11)
where k, , dr , vr are the stiffness factor, coefficient of damping, relative displacement and velocity. The pressure gradient force is FP = −V grad p
(12)
where V is particle volume. The calculation of local pressure gradient in the locations where particles occupy needs linear interpolation of pressure gradient on grid nodes. The particle--wall collision is easy to handle when the wall is treated as the surface of a huge particle of infinite mass and radius. The treatment for particle-immersed-tube (particle--cylinder) contact is regarded as a special case of particle--wall contact where the `wall' is the tangential plane of the cylinder. When the tube's diameter is an order of magnitude larger than the particle's diameter, the reduced accuracy induced by this treatment can be neglected. 3. Numerical implementation 3.1. Void fraction calculation The void fraction exerts its influence on the evolution of dense gas--solid flow considerably. Thus it is necessary to compute the void fraction of each cell accurately. The phenomenon of particle
crossing at border of neighboring computational cells is of common and frequent appearance in simulation. Hence, the segmentation of a sphere must be taken into account (Li, 2003) to account for it. In the present paper, we use a numerical approach to calculate the volume parts of a sphere. When one particle transfer into neighboring cells, the particle is subdivided into small elements and its volume fraction in each cell is obtained by summing up the elements it contains. 1 )3 the volume of Let the volume of each subdivided element be ( 40 the cubic covering the sphere, the relative error contributing to void fraction ε3D are less than 0.25% (with dp = 883 m and meshing size m = 2 mm). 3.2. Simulation conditions Table 1 is a summary of numerical and experimental conditions for present study. The particle material is quartz sand, and the fluidization is shot by a high-speed camera (MotionXtra HG-100K). For the limitation of computer performance, it is only allowed for us to simulate a local region around the tube on one CPU. As presented in Table 1 and Fig. 1a, the dimensions of the bed are actually 100 × 300 × 600 mm3 , which are reduced to 20 × 80 × 500 mm3 to save computational cost. Since the transverse section of the bed for simulation is very small, the boundary is set to be uniform velocity inlet at bottom and outflow condition at top. Experimentally the gas is injected into the bed through distributed pipes at bottom. According to Tsuji et al. (1993), using the expression they suggested for estimation of time step for particle motion simulaTsujition, we obtain tp < 1.9 × 10−5 . Then the time step should be about 20 ( s). Here due to numerical stability consideration and after practical testing, we chose tp = 4 × 10−6 for solid phase and tg = 2.0 × 10−5 for gas phase.
4. Results and discussion 4.1. Validation of numerical results Fig. 1b is numerical and experimental results of time-averaged particle velocity magnitudes within 0.8 s. The results are extracted at three heights of H1 = 0.06 m, H2 = 0.1 m and H3 = 0.14 m (H2 and H3 are the heights of the location of the immersed tubes (Table 1 and Fig. 1a)). The experimental results are obtained by particle tracking velocimetry (PTV) analysis of the particulate flow. The PTV analysis can track particle motion individually and obtain the velocities of nearly all particles under Lagrangian frame. Then we calculate the local mean value in each mesh cell to obtain the particulate flow field. For comparison, numerical results are also locally averaged in the same way. We can find in Fig. 1b that the results obtained with and without LES fit the experimental results both to some extent well although discrepancies exist. Moreover, the results with LES appear
Table 1 Numerical and Experimental conditions
Cell sizes x × y × z, mm Cell numbers (Nx × Ny × Nz ) Dimensions of Fluidized bed, mm Particle Diameter, dp , m Particle density, p , kg/m−3 Particle number, np Poisson ratio, Restitution coefficient, ε Stiffness factor, N/m, (kn , kt ), Superficial velocity, m/s, u0 Tube position (y, z)i=1,2 , mm−1 Tube radius, Rt , mm Times step, (tg ; tp )
Numerical parameter
Experimental parameter
2×2×2 10 × 40 × 250 20 × 80 × 500 883 2650 178,200 0.25 0.9 (1000, 250) 1.16 (40, 100) and (40, 140) 10 (2.0 × 10−5 s; 4.0 × 10−6 s)
--100 × 300 × 600 883 2650 ----1.16 (110, 230) and (110, 270) 10 --
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Fig. 1. (a) Sketch map of the fluidized bed with immersed tubes. (b) Comparison between numerical and experiment results of time-averaged particle velocity magnitudes within 0.8 s.
to be more in accordance with the experimental results especially at heights H1 and H2 . Thus we are inclined to consider that the SGS model is acceptable, and the numerical results with LES and without LES are mainly reliable. However, the experimental and numerical results are obtained within different scales of fluidized beds, although the physical parameters for gas and solid phases are the same and mean inlet gas velocities are in accordance with each other. Therefore, the discrepancies between numerical and experimental results especially at height H1 are mainly due to the bed scale differences. Since the experimental bed is much larger in width than numerical simulation condition, particles experimentally fluctuate more intensively in width direction than the numerical simulation results and the experimental results of velocity magnitudes at H1 are larger than numerical results. 4.2. Gas and particle behavior analysis 4.2.1. |Fp.p | and |Fg .p | In dense gas--solid flow, particle--particle and gas--particle interactions are two types of the most important interactions which play leading roles in bubble and particle evolution. Fig. 2a and b, taking mean value of all particles, present the evolution of mean gas--particle interaction force |Fg .p | (only drag force and pressure gradient force under consideration) and particle--particle contact force |Fp.p | (divided by particle's gravity force to get dimensionless values), where the operator means ensemble average. Fig. 2a is obtained when LES is used while (b) is not. The comparison between |Fp.p | and |Fg .p | indicates that particle--particle interaction is approximately two order of magnitude as large as gas--particle interaction. What is more, although differ in magnitude, both Fig. 2a and b show that the evolution of |Fg .p | and |Fp.p | seem to follow
the same pace to increase and decrease, appearing with a periodic variation. Interestingly, |Fg .p | and |Fp.p | reach their minimum and maximum values. We choose four time of t1 = 0.864 s, t2 = 1.152 s in Fig. 2a and t3 = 0.354 s, t4 = 0.436 s in Fig. 2b to visualize the particle velocities and positions, which are displayed in Fig. 3a and b, respectively. We find that at t1 and t3 (the left figures of Fig. 3a and b) the particles reach a top height and conglomerate at two sides of the bed---the state when gas bubbles coalesce into the largest one before eruption. Quite on the contrary, at t2 and t4 (the right figures of Fig. 3a and b) the particles appear as impetuous back-mixing image and form a large `eddy' of particulate flow at the bottom of the bed before new gas bubbles emerge. In other words, the bubble's motion and inter-phase interaction in dense gas--solid seem to bear a strong correlation.
4.2.2. |Fg .p | and Pd Since the value of Fg .p is composed of pressure gradient force and drag force---other forces, like Magnus lift force, etc., are neglected. In order to know more about gas--particle interaction, we extract the mean pressure drop Pd at a height of 0.2 m of the bed and draw them in Fig. 4 together with |Fg .p |. It is interesting that no matter if LES is used or not, Pd and |Fg .p | seem to follow a perfect similar pace of evolution. Hence we find that Fg .p is to a great extent determined by Pd rather than by drag force in dense gas--solid flow though it may not be true for specific condition, such as inside bubbles where the pressure gradient can be neglected. We are inclined to consider that the variation of Fg .p is mainly dominated by the variation of Pd , and the variation of Pd is to a great degree determined by if or not the `channel' for gas flow is blocked. Since the back-mixing particles considerably block off the gas flow,
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Fig. 2. Comparison of Fg .p and Fp.p for results with (a) and without (b) LES.
Pd increases and so do |Fg .p | and |Fp.p |, and consequentially new bubbles emerge at the bottom of the bed. On the other hand, when small bubbles grow, coalesce into each other, and erupt finally, the Pd turns to decrease and so do |Fg .p | and |Fp.p |. 4.2.3. Periodic motion It is revealed from Figs. 2 and 4 that the periodic variation of Fg .p , Fp.p and Pd seems to be more regular for results without LES than that with LES since the curve peaks for results without LES seem to appear with more relatively similar amplitude and period than that with LES. Besides the nonlinear nature of the governing equations which may grow into different results with the same initial and boundary conditions, the only cause for the difference is due to the SGS model. Thus it is considered that the SGS fluctuation of gas phase
will exert great influence upon the evolution of dense gas--solid flow. To study in detail, first we smooth the curve of Fg .p by cutting off the high frequency but low magnitude components of Fˆ g .p in its spectral space, and rebuild a smoothed curve of Fg .p by inverse Fourier transform. Then we filter the constant-current component of Fg .p to obtain the fluctuating components. Finally, the spectral representation of the smoothed fluctuating component is obtained by FFT, and it is shown in Fig. 5. Treatment for Fp.p or Pd is quite similar. The subfigures in Fig. 5a are results without LES while Fig. 5b is with LES. The fundamental frequencies in Fig. 5a for Fg .p , Fp.p and Pd are the same (f 1 = 3.42 Hz), and it is also true for the second frequencies (f 2 = 6.84 Hz). In Fig. 5b the fundamental frequencies for Fg .p and Pd are f 1 = 1.95 Hz, and for Fp.p , f 1 = 2.44 Hz.
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Fig. 4. Comparison of Fg .p and Pd for results with and without LES.
The results show some indications on the periodic variation of inter-phase and inside-phase interaction as well as the total pressure drop in dense gas--solid flow. In addition, the differences between the results with and without LES reveals that the SGS turbulence will to some degree modify the periodic motion of fundamental interactions. It is interesting to consider the main factors for the periodic variation. As well known, in numerical simulation of dilute gas--particle flow, the pressure gradient force for particle is always not taken into account. However, as mentioned in the above section, in dense gas--particle flow or in local region where the solid phase is characterized as dense clusters, the main drift of particle motion is more inclined to be influenced by the pressure gradient force. Thus, the periodic motion of raise and fall of particles induces a sequential periodic variation of pressure drop. This may be the main factor for the periodic variation of pressure drop and inter-phase interactions. However, there always exist many other factors which all exert their influences on the periodic variation, such as the sub-grid turbulent fluctuation, the scale of the bed and the piling height of the particles. It may be only in such a small simulation bed that the periodic variation is less likely to be influenced by other factors and the periodic variation turns to appear more apparently.
Fig. 3. Visualization of particle positions and velocities at the central plane of the bed: (a) results with LES at time t1 =0.864 s (left) and t2 =1.152 s (right), respectively; (b) results without LES at time t3 = 0.354 s (left), t4 = 0.436 s (right), respectively.
4.2.4. Particle energy Fig. 6 is the time-dependent variation of particles' mean translational kinetic energy Etr and rotational kinetic energy Erot , where the rotational motion of particles are caused by inter-particle friction and not induced by gas-phase vortex. We find that from t = 0.7 to 1.3 s, Etr and Erot increase rapidly. In addition, the kinetic energy of particles is lower for results with LES than that without LES. The reason is due to the SGS turbulent dissipation of gas energy and then the energy transport from gas phase to solid phase becomes weaker. According to the above results, we think that energy dissipation for gas phase as well as solid phase may be to some degree non-negligible. In addition, the rotational motion of particle, which on average occupy 3.9.5.56% of translational kinetic energy, low in comparative sense but still large in absolute magnitude, will play an important role in dense gas--solid flow evolution.
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Fig. 5. FFT of smoothed curve of Fg .p , Fp.p and Pd . (a) results without LES; (b) results with LES.
4.2.5. Turbulent effects We explore the turbulent effects in fluidized bed by making comparison of gas-phase turbulent intensity If and `particle-phase turbulent intensity' Ip , both in Euler frame. We first mesh the fluid field and compute the mean local velocities of particles to build a pseudo-particulate-flow field. Following the definition of fluid turbulent intensity, we computed the Ip and If in the same way and displayed them in Fig. 7a and b, respectively. We find that Ip is much higher than If , which means that if solid phase is treated as a pseudo-continuum phase in dense gas--solid flow, the particles
turbulent fluctuation will be much higher than the gas phase, especially in the region where the dense gas--solid flow is transited into dilute gas--solid flow. The reason for the high particle turbulent intensity in the transition region may be due to the eruption of bubbles which bring out suddenly the inner energy of gas phase and due to the region with high potential energy for particles. The above two factors cause the particles to be brought up and fall back frequently, producing high particle fluctuations. Additionally, the gas-phase turbulent intensity will be augmented induced by particulate turbulent fluctuation especially in the wake of the immersed tubes.
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Fig. 6. Evolution of mean translational and rotational energy of all particles and comparison for results with and without LES.
Fig. 7. Contour figure of `particle turbulent intensity' and gas turbulent intensity obtained from results of LES simulation: (a) particle turbulent intensity similarly defined as fluid phase under Euler frame; (b) gas phase turbulent intensity.
4.2.6. Particle--tube impact From the point of view of heat transfer, the more frequently particle--tube impacts occur, the better. However, the erosion caused by frequent particle--tube impact and friction is disadvantageous. Therefore, it is necessary to know the probability distribution of particle-to-tube impact. Rong et al. (1999) investigated immersed tube erosion based on an erosion model (Finnie, 1960). We choose an alternative approach to calculate the particle--tube impact force, which is the direct factor for tube erosion and an indirect factor for particle--tube heat transfer. Fig. 8a displays the frequency and magnitude of particle-to-tube
impact force along the circumferential direction of the lower immersed tubes. The forces include the normal component |Fn | and tangential component |Ft |, and the magnitudes are nondimensionlized by particle gravity Gp . Fig. 8b is normalized distribution density of impact frequency. From Fig. 8a, the average magnitude of particle--tube impact force for results with LES (|Fn + Ft | = 229.83Gp ) are close to the results without LES (|Fn + Ft | = 224.27Gp ). And both the normal component of impact force and the tangential component of friction force are concentrated at the bottom half of the tube ( = 180.360◦ ). From Fig. 8b, we find that there are two maximum values of probability density at = 285◦ for results with LES and = 225◦ for results
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Fig. 8. Distribution and frequency of particle--cylinder impact. (a) Distribution and magnitude of normal and tangential component of impact force. (b) Impact frequency on the surface of the tube.
without LES. Rong et al. (1999) found that the peaks of impact energy appears at the two sides (about = 210.240◦ and = 310.340◦ ) of the bottom half tube. They did not use the LES in their governing equations. From the point of view of tube erosion, we consider the numerical results without LES are partially consistent with theirs, although the simulation conditions are different. Moreover, the numerical results with LES indicate that in the region = 235.295◦ there still exists the largest probability of particle--tube impact and sequentially exist largest tube erosion and particle--tube heat flux. 5. Conclusion In this paper, we carried out a true 3-D numerical simulation of bubbling fluidized bed using DEM--LES coupling method. In this simulation, we obtain some underlying information on the mechanisms of fundamental interactions in gas--solid flow: Particle--particle interaction forces are on average almost two orders of magnitude as large as gas--particle interaction forces. We consider that |Fg .p |, |Fp.p |, Pd , particle back-mixing motion and bubble motion seem to bear a strong correlation with each other, and all of them seem to follow the same pattern of periodic variation
though the periodicity may differ in various flow conditions, even may be modified by SGS turbulent fluctuation. Moreover, |Fg .p |, |Fp.p | and Pd seem to step in a perfect synchronous pace of variation. In dense gas--solid flow, the energy transport from gas-phase to particle phase is attenuated by gas-phase turbulent dissipation such as SGS viscous dissipation. Under Euler frame, we calculate the particle turbulent intensity Ip and find that Ip is much higher in the region where the transition from dense gas--solid flow to dilute gas--solid flow occurs than in the region of dense gas--solid flow. Moreover, the particle turbulent intensity Ip under Euler frame is much higher than the gas-phase turbulent intensity If and If is augmented by particle fluctuation especially in the wake of immersed tubes. Particle-immersed-tube impacts occur most frequently at the upstream side and the downstream side, especially at the upstream side in fluidized bed. Notation CD CK CS
drag coefficient Kolmogorov constant Smagorinsky constant
N. Gui et al. / Chemical Engineering Science 63 (2008) 3654 -- 3663
dp dr Erot Etr f 1, f 2 FC FD Fg .p Fn Fp.p FP Ft g Gp I If Ip k m Pd pg S¯ ij
¯
S tg tp Tij ui , uj vr vg vs V Vp
diameter of particle, m relative displacement between the pair of collision particles, m rotational energy of particle, J translational energy of particle, J frequency components in spectral space, Hz particle--particle or particle--wall contact force, N drag force, N gas--particle interaction force, N normal component of particle-immersed tube interaction, N particle--particle interaction force, N pressure gradient force, N tangential component of particle-immersed tube interaction, N acceleration of gravity, m/s2 gravity of particle, N mass of inertia of particle, kg m2 intensity of turbulence of fluid phase intensity of turbulence of particle stiffness factor, N/m mass of particle, kg pressure drop of the fluidized bed, Pa gas pressure, Pa deformation tensor of the filtered field or resolved strain rate local strain rate time step for gas-phase simulation, s−1 time step for particle phase simulation, s−1 sub-grid stress tensor components of gas velocity, m/s relative velocity between the pair of collision particles, m/s velocity of gas phase, m/s velocity of particle, m/s particle volume, m3 particle translational velocity, m/s
Greek letters
ij , x m ε, εg g t g ij p
the angle in the circumferential direction of the tube fluid--particle interaction coefficient damping coefficient, N s/m Kronecker delta sub-grid characteristic length scale and mass size in x dimension, m mesh size, m gas void fraction viscosity of gas, kg/(m s) eddy viscosity gas density, kg/m3 viscous stress tensor particle rotational velocity, rad/s
Subscripts g or f s or p
gas or fluid phase. solid or particle phase.
d rot tr
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drop rotational translational
Operators ∼ −
filtering operator in LES ensemble mean operator Mean operator module of a vector
Acknowledgments This subject is supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. Z104314. References Bouilard, J.X., Lyezkowski, R.W., Gidaspow, D., 1989. Porosity distributions in a fluidized bed with an immersed obstacle. A.I.Ch.E. Journal 35, 908--922. Cundall, P.A., Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Geotechnique 29, 47--65. Ding, J., Gidaspow, D., 1990. A bubbling fluidization model using kinetic theory of granular flow. A.I.Ch.E. Journal 36, 523--538. Finnie, I., 1960. Erosion of surface by solid particles. Wear 3, 87--159. Gera, D., Gautam, M., Tsuji, Y., Kawaguchi, T., Tanka, T., 1998. Computer simulation of bubbles in large-particle fluidized beds. Powder Technology 98, 38--47. Goldschmidt, M.J.V., Beetstra, R., Kuipers, J.A.M., 2004. Hydrodynamic modelling of dense gas-fluidised beds: comparison and validation of 3-D discrete particle and continuum models. Powder Technology 142, 23--47. Helland, E., Occelli, R., Tadrist, L., 2002. Computational study of fluctuating motions and cluster structures in gas--particle flows. International Journal of Multiphase Flow 28, 199--223. Hoomans, B.P.B., Kuipers, J.A.M., Briels, W.J., van Swaajj, W.P.M., 1996. Discrete particle simulation of bubble and slug formation in a two dimensional gas--solid fluidized bed: a hard-sphere approach. Chemical Engineering Science 51, 99--118. Jackson, R., 1963. The mechanics of fluidized beds: I. The stability of the state of uniform fluidization. Transactions of the Institute of Chemical Engineers 41, 13--21. Kuipers, J.A.M., van Duin, K.J., van Beckum, F.P.H., van Swaajj, W.P.M., 1992. A numerical model of gas-fluidized beds. Chemical Engineering Science 47, 1913--1924. Kuipers, J.A.M., van Duin, K.J., van Beckum, F.P.H., van Swaajj, W.P.M., 1993. Computer simulation of the hydrodynamics of a two-dimensional gas-fluidized bed. Computers & Chemical Engineering 17, 839--858. Lesieur, M., Metais, O., 1996. New trends in large-eddy simulations of turbulence. Annual Review of Fluid Mechanics 28, 45--82. Lilly, D.K., 1987. In: Herring, J.R., McWilliams, J.C. (Eds.), Lecture Notes on Turbulence. World Scientific, Singapore, pp. 171--218. Li, J., 2003. Euler--Lagrange simulation of flow structure formation and evolution in dense gas--solid flows. Ph.D. Thesis, Twente University, The Netherlands. Rong, D., Mikami, T., Horio, M., 1999. Particle and Bubble Movements Around Tubes Immersed in Fluidized Beds---a Numerical Study. Chemical Engineering Science 54, 5737--5754. Soo, S.L., 1967. Fluid Dynamics of Multiphase System. Blaisdell, Waltham, MA. Tsuji, Y., Kawaguchi, T., Tanaka, T., 1993. Discrete particle simulation of twodimensional fluidized bed. Powder Technology 77, 79--87. Tsuo, Y.P., Gidaspow, D., 1990. Computation of flow patterns in circulating fluidized. A.I.Ch.E. Journal 36, 885--896. Wen, C.Y., Yu, Y.H., 1966. A generalised method for predicting the minimum fluidization velocity. A.I.Ch.E. Journal 12, 610--612. Xu, B.H., Yu, A.B., 1997. Numerical simulation of the gas--solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics. Chemical Engineering Science 52, 2785--2809. Yuu, S., Umekage, T., Johno, Y., 2000. Numerical simulation of air and particle motions in bubbling fluidized bed of small particles. Powder Technology 110, 158--168. Zhou, H., Flamant, G., Gauthier, D., 2004. DEM-LES of coal combustion in a bubbling fluidized bed. Part I: gas-particle turbulent flow structure. Chemical Engineering Science 59, 4193--4203.
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DEM--LES study of 3-D bubbling fluidized bed with immersed tubes Nan Gui, Jian Ren Fan ∗ , Kun Luo State Key Laboratory of Clean Energy Utilization, Zhejiang University, Hangzhou 310027, PR China
A R T I C L E
I N F O
Article history: Received 9 July 2007 Received in revised form 13 December 2007 Accepted 15 April 2008 Available online 2 May 2008 Keywords: Bubble Fluidization Multiphase flow Powder technology Immersed tube Discrete element method
A B S T R A C T
Based on Euler--Lagrange frame, a true three-dimensional numerical simulation of bubbling fluidized bed embedded with two immersed tubes is presented. The solid phase is composed of 178,200 particles of 883 m diameter and simulated by discrete element method (DEM, a soft-sphere approach). The gas phase is computed through solving the volume-averaged four-way coupling Navier--Stokes equations in which the Smagorinsky SGS tensor model is used in large eddy simulation (LES). Particle--tube collision is particularly treated as a transformation of DEM. The volume segmentation of a particle sphere for void fraction calculation is solved via a numerical sub-division approach. The numerical results are compared with the experimental results for validation. The results obtained with and without the LES model are also compared. The numerical results show a strong correlation between gas--particle interaction, particle--particle interaction, pressure drop, particle back mixing motion and bubble motion, and all of them follow a similar pattern of synchronous periodic variation though the periodicity may vary depending on different flow conditions. The effects of SGS tensor on evolution of fluidized bed are found in various aspects. Finally, the distribution of particle--tube impact frequency is given. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction Many numerical researches on bubbling fluidized bed were carried out by many investigators. These investigations of dense gas--solid flow, based on various models, are mainly divided into two categories, i.e., Euler--Euler two-phase model (Jackson, 1963; Soo, 1967; Bouilard et al., 1989; Ding and Gidaspow, 1990; Tsuo and Gidaspow, 1990; Kuipers et al., 1992, 1993) and Euler--Lagrange approach (Hoomans et al., 1996; Tsuji et al., 1993; Xu and Yu, 1997; Gera et al., 1998; Yuu et al., 2000; Helland et al., 2002; Zhou et al., 2004). The discrete element method (DEM), which belongs to the latter category, has been proven a valuable tool for simulating the behavior of gas--solid particulate flow during the past decades (Hoomans et al., 1996; Tsuji et al., 1993; Rong et al., 1999; Goldschmidt et al., 2004). The DEMs, including a hard-sphere approach and a soft-sphere approach, are advanced for numerical simulation of gas--particle flow since they simulate the motion of all particles in a deterministic way, taking many interactions and forces into account, such as particle--particle collisions, particle--wall collisions, gas--solid two-way coupling interaction and gravitational force, etc. The two-fluid model (TFM) is under the Euler--Euler frame and based on the interpenetrating fluids. It has vast applications in the description of gas--solid hydrodynamic phenomena.
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But it requires many constitutive equations of solid phase for the closure of governing equations. Moreover, the parameters included in the constitutive formulations lack generality. Hence the DEM is to some extent preferable to the TFM in consideration of numerical simulation on the behavior of particles in fluidized bed. As mentioned by Hoomans et al. (1996), the hard-sphere model is a quasi-instantaneous and binary collision model, in which the particle--particle collisions are repulsive, inelastic and only take action once at a time. However, the soft-sphere model is a quasi-static and multi-body collision model, in which the interaction forces are combinations of linear or nonlinear elastic repulsive force, damping force and friction force. There are many applications of the above-mentioned two approaches of DEM. For example, Hoomans et al. (1996) developed a hard-sphere approach for discrete particle simulation on bubble and slug formation in two-dimensional (2-D) gas--solid bed. The collision model follows the conservation law of linear and angular momentum and uses two empirical parameters: a restitution coefficient and friction coefficient, which are key parameters used in soft-sphere model too. Tsuji et al. (1993) developed the soft-sphere model in application of 2-D fluidized bed simulation. The interaction force between particles is simulated by a combination of elastic repulsive force, damping force and friction force. Another key parameter of damping coefficient is used to model the damping effect of particle--particle contact. Xu and Yu (1997) carried out a simulation for 2400 GroupD particles in pseudo-3-D central jet fluidized bed by combining discrete particle method and computational fluid dynamics.
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Gera et al. (1998) used distinct element model to simulate hydrodynamics of 2-D large particle fluidized beds and compared the bubble rise velocity, voidage variations, etc., with TFM based on continuum theory. Yuu et al. (2000) simulated about 100,000 Group-B particles using the 3-D distinct element method to explore the mechanisms of bubble formation, coalescence and disruption. The dimension of the bed is one diameter of the particle deep and the governing equations for air phase are 2-D incompressible Navier--Stokes (N--S) equations. Moreover, Helland et al. (2002) studied the fluctuating motions and cluster structures in 2-D gas--solid flows treating particle motion by Lagrangian approach and the interaction between particles are assumed to be binary, instantaneous, inelastic collisions with friction. Zhou et al. (2004) simulated coal combustion in a bubbling fluidized bed by using DEM and LES (large eddy simulation) methods, while the gas phase is still treated as 2-D flows. Rong et al. (1999) conducted a numerical study on particle and bubble movements around immersed tubes in 2-D fluidized beds. Goldschmidt et al. (2004) implemented comparison between simulations of dense gas-fluidized bed based on discrete particle method and continuum theory, in which the gas phase hydrodynamics was also only resolved in 2-D case. It is obvious that most previous researchers focus their numerical study on bubble or particle behavior in a pseudo-3-D fluidized bed. Then the N--S governing equation for gas-phase is 2-D and sometimes non-viscous fluid is used. Moreover, the particle motion is restricted from motion in depth dimension. Thus, there exist some reasons for the necessity of true 3-D numerical simulation of fluidized beds:
where εg , g , t, ui , are volume fractions for gas phase, gas density, time, velocity of gas and stress tensor, respectively. The operator `∼' denotes the top-hat function for spatial filtering. It is considered that εg is a parameter that has already been filtered when it is calculated in a volume-averaged way. Thus, the sub-grid i = ε˜ u˜ i = εu˜ i , scale fluctuation εg is equal to zero, and we have εu ε ij = ε˜ ˜ ij = ε˜ij , etc. Moreover, the term εg Tij εg (u˜ i u˜ j − u i uj ) is the sub-grid scale stress tensor. We observe Eqs. (1)--(2) and consider that εg can be regarded as density of a pseudo-fluid, and εg ˜ ij , εg Tij are regarded as stress tensor and sub-grid scale tensor of the pseudo-fluid, respectively. From this point of view, Eqs. (1)--(2) can be treated as the governing equations of the single-pseudo-fluid phase. Thus, we use the most widely used Smagorinsky model for the sub-grid scale tensor. The Smagorinsky SGS model can be explained by the mixing-length assumption, in which the eddy viscosity is proportional to the SGS characteristic length scale and to a characteristic turbulent velocity based on the second invariant of the filtered-field deformation tensor (Lesieur and Metais, 1996):
(1) Particle motion in industrial fluidized bed is actually 3-D. The interactions between particles, such as particle--particle impact and friction, as well as particle motion, actually have more than six degrees of freedom. Thus the restriction of particle motion in pseudo-3-D simulation should be relaxed. (2) Fluid phase is actually viscous fluid and turbulence is 3-D especially for LES simulation in which 2-D eddy is not authentic. The fundamental of 2-D flow simulation exists only in symmetrical flow in depth dimension, and this is not appropriate for fluidized bed.
are the deformation tensor of the filtered field, eddy viscosity, local strain rate and sub-grid characteristic length scale, respectively. ij is the Kronecker delta function. An approximate value for the constant CS is given by (Lilly, 1987):
Accordingly, 3-D LES simulation coupling with DEM is more reliable than that in a 2-D case or in non-viscous case. But the main problem for implementation of 3-D simulation is the lack of powerful computer. In the present paper, a true 3-D simulation of fluidized bed is carried out. For solution of gas-phase governing equation, the results obtained with and without LES are compared for justification of SGS stress tensor. In addition, the results are referred to the experimental results for validation. The soft-sphere model is used to simulate the motion of particles and particle--particle interactions. Moreover, particle--cylinder contact model for particle-immersedtube collision is treated with the alteration of soft-sphere model. Finally, the numerical results reveal some interesting phenomena in fundamental interactions in dense gas--solid flow. 2. Numerical model 2.1. Gas-phase hydrodynamics For LES, the hydrodynamic models for gas phase use the principles of conservation of mass and momentum (Ding and Gidaspow, 1990): j(˜εu˜ i )g j˜εg + =0 jt jxi j(˜εu˜ i )g jt =−
+
(1)
j(˜εu˜ i u˜ j )g jxj
FD, i ε˜ g jp˜ g 1 j(˜εg ˜ ij ) 1 j(˜εg Tij,g ) + ε˜ g gi + − + g jxj g jxj g jxi g
(2)
Tij = 2t S¯ ij + 13 Tll ij
(3)
where 1 S¯ ij = 2
ju¯ j ju¯ i + jxj jxi
,
¯ t = (CS )2 |S|,
¯ = (2S¯ S¯ )1/2 |S| ij ij
= (xyz)1/3
CS ≈
(4)
1 3CK −3/4 2
(5)
where CK = 1.4 is Kolmogorov constant and this yields CS ≈ 0.18. Due to SGS the model is too dissipative and most workers prefer CS = 0.1 (Lesieur and Metais, 1996), we chose CS = 0.1 in the present paper too. Additionally, to validate the application of SGS tensor, we make comparison between the numerical results obtained with and without the SGS tensor with reference to the experimental results. The term FD is the drag force and FD = (vg − vs ) = (ui,g − ui,s ). In dense gas--particle flow, is the fluid--particle interaction coefficient, which is obtained from standard correlations. Below a void fraction of 0.8, is given by Ergun's equation; at and above 0.8, it is obtained from Wen and Yu's (1966) expression: ⎧ (1 − ε)g |vs − vg | (1 − ε)2 g ⎪ ⎪ + 1.75 , ⎪ 150 ⎨ 2 dp εdp = ⎪ ⎪ ⎪ ⎩ 3 C ε(1 − ε)g |vs − vg | ε−2.65 , 4 D dp And the drag coefficient is given by: 24(1 + 0.15Re0.687 )/Re, Re < 1000 CD = 0.43, Re 1000
ε < 0.8 (6) ε 0.8
(7)
where particle Reynolds number is defined as: Re =
εg |vg − vs |dp g
(8)
Under two-way coupling approach for gas--particle interaction, calculation for FD needs numerical interpolation of gas-phase velocity, pressure, void fraction from neighboring nodes to particle locations. In the present paper, we have handled it by using a linear
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interpolation method. It is in fact a convex combination of information at neighboring nodes and in 3-D case, and its representation is just a volume-averaged method of interpolation. Due to Newton's third law, the particle-to-gas force in a cell can be calculated by integrating the term −FD over the volume of particle inside the cell. Due to the utilization of stagger grid for velocity and pressure discretization, the particle-to-gas force may once more be interpolated to the location of velocity nodes. 2.2. Particle motion The translational and rotational motion of particle is computed through integrating the Newton's equation of motion: ·
V P = (FC + FD + FP )/m + g ·
p = T/I
(9) (10)
where VP , p , FC , FD , FP , T, m, I, g are particle translational and rotational velocity, inter-particle contact force, drag force, pressure gradient force, torque, mass and moment of inertia, gravity acceleration, respectively, and `·' denotes the time derivative. The inter-particle contact force FC is calculated by soft-sphere model (Tsuji et al., 1993; Cundall and Strack, 1979): FC = −kdr − vr
(11)
where k, , dr , vr are the stiffness factor, coefficient of damping, relative displacement and velocity. The pressure gradient force is FP = −V grad p
(12)
where V is particle volume. The calculation of local pressure gradient in the locations where particles occupy needs linear interpolation of pressure gradient on grid nodes. The particle--wall collision is easy to handle when the wall is treated as the surface of a huge particle of infinite mass and radius. The treatment for particle-immersed-tube (particle--cylinder) contact is regarded as a special case of particle--wall contact where the `wall' is the tangential plane of the cylinder. When the tube's diameter is an order of magnitude larger than the particle's diameter, the reduced accuracy induced by this treatment can be neglected. 3. Numerical implementation 3.1. Void fraction calculation The void fraction exerts its influence on the evolution of dense gas--solid flow considerably. Thus it is necessary to compute the void fraction of each cell accurately. The phenomenon of particle
crossing at border of neighboring computational cells is of common and frequent appearance in simulation. Hence, the segmentation of a sphere must be taken into account (Li, 2003) to account for it. In the present paper, we use a numerical approach to calculate the volume parts of a sphere. When one particle transfer into neighboring cells, the particle is subdivided into small elements and its volume fraction in each cell is obtained by summing up the elements it contains. 1 )3 the volume of Let the volume of each subdivided element be ( 40 the cubic covering the sphere, the relative error contributing to void fraction ε3D are less than 0.25% (with dp = 883 m and meshing size m = 2 mm). 3.2. Simulation conditions Table 1 is a summary of numerical and experimental conditions for present study. The particle material is quartz sand, and the fluidization is shot by a high-speed camera (MotionXtra HG-100K). For the limitation of computer performance, it is only allowed for us to simulate a local region around the tube on one CPU. As presented in Table 1 and Fig. 1a, the dimensions of the bed are actually 100 × 300 × 600 mm3 , which are reduced to 20 × 80 × 500 mm3 to save computational cost. Since the transverse section of the bed for simulation is very small, the boundary is set to be uniform velocity inlet at bottom and outflow condition at top. Experimentally the gas is injected into the bed through distributed pipes at bottom. According to Tsuji et al. (1993), using the expression they suggested for estimation of time step for particle motion simulaTsujition, we obtain tp < 1.9 × 10−5 . Then the time step should be about 20 ( s). Here due to numerical stability consideration and after practical testing, we chose tp = 4 × 10−6 for solid phase and tg = 2.0 × 10−5 for gas phase.
4. Results and discussion 4.1. Validation of numerical results Fig. 1b is numerical and experimental results of time-averaged particle velocity magnitudes within 0.8 s. The results are extracted at three heights of H1 = 0.06 m, H2 = 0.1 m and H3 = 0.14 m (H2 and H3 are the heights of the location of the immersed tubes (Table 1 and Fig. 1a)). The experimental results are obtained by particle tracking velocimetry (PTV) analysis of the particulate flow. The PTV analysis can track particle motion individually and obtain the velocities of nearly all particles under Lagrangian frame. Then we calculate the local mean value in each mesh cell to obtain the particulate flow field. For comparison, numerical results are also locally averaged in the same way. We can find in Fig. 1b that the results obtained with and without LES fit the experimental results both to some extent well although discrepancies exist. Moreover, the results with LES appear
Table 1 Numerical and Experimental conditions
Cell sizes x × y × z, mm Cell numbers (Nx × Ny × Nz ) Dimensions of Fluidized bed, mm Particle Diameter, dp , m Particle density, p , kg/m−3 Particle number, np Poisson ratio, Restitution coefficient, ε Stiffness factor, N/m, (kn , kt ), Superficial velocity, m/s, u0 Tube position (y, z)i=1,2 , mm−1 Tube radius, Rt , mm Times step, (tg ; tp )
Numerical parameter
Experimental parameter
2×2×2 10 × 40 × 250 20 × 80 × 500 883 2650 178,200 0.25 0.9 (1000, 250) 1.16 (40, 100) and (40, 140) 10 (2.0 × 10−5 s; 4.0 × 10−6 s)
--100 × 300 × 600 883 2650 ----1.16 (110, 230) and (110, 270) 10 --
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Fig. 1. (a) Sketch map of the fluidized bed with immersed tubes. (b) Comparison between numerical and experiment results of time-averaged particle velocity magnitudes within 0.8 s.
to be more in accordance with the experimental results especially at heights H1 and H2 . Thus we are inclined to consider that the SGS model is acceptable, and the numerical results with LES and without LES are mainly reliable. However, the experimental and numerical results are obtained within different scales of fluidized beds, although the physical parameters for gas and solid phases are the same and mean inlet gas velocities are in accordance with each other. Therefore, the discrepancies between numerical and experimental results especially at height H1 are mainly due to the bed scale differences. Since the experimental bed is much larger in width than numerical simulation condition, particles experimentally fluctuate more intensively in width direction than the numerical simulation results and the experimental results of velocity magnitudes at H1 are larger than numerical results. 4.2. Gas and particle behavior analysis 4.2.1. |Fp.p | and |Fg .p | In dense gas--solid flow, particle--particle and gas--particle interactions are two types of the most important interactions which play leading roles in bubble and particle evolution. Fig. 2a and b, taking mean value of all particles, present the evolution of mean gas--particle interaction force |Fg .p | (only drag force and pressure gradient force under consideration) and particle--particle contact force |Fp.p | (divided by particle's gravity force to get dimensionless values), where the operator means ensemble average. Fig. 2a is obtained when LES is used while (b) is not. The comparison between |Fp.p | and |Fg .p | indicates that particle--particle interaction is approximately two order of magnitude as large as gas--particle interaction. What is more, although differ in magnitude, both Fig. 2a and b show that the evolution of |Fg .p | and |Fp.p | seem to follow
the same pace to increase and decrease, appearing with a periodic variation. Interestingly, |Fg .p | and |Fp.p | reach their minimum and maximum values. We choose four time of t1 = 0.864 s, t2 = 1.152 s in Fig. 2a and t3 = 0.354 s, t4 = 0.436 s in Fig. 2b to visualize the particle velocities and positions, which are displayed in Fig. 3a and b, respectively. We find that at t1 and t3 (the left figures of Fig. 3a and b) the particles reach a top height and conglomerate at two sides of the bed---the state when gas bubbles coalesce into the largest one before eruption. Quite on the contrary, at t2 and t4 (the right figures of Fig. 3a and b) the particles appear as impetuous back-mixing image and form a large `eddy' of particulate flow at the bottom of the bed before new gas bubbles emerge. In other words, the bubble's motion and inter-phase interaction in dense gas--solid seem to bear a strong correlation.
4.2.2. |Fg .p | and Pd Since the value of Fg .p is composed of pressure gradient force and drag force---other forces, like Magnus lift force, etc., are neglected. In order to know more about gas--particle interaction, we extract the mean pressure drop Pd at a height of 0.2 m of the bed and draw them in Fig. 4 together with |Fg .p |. It is interesting that no matter if LES is used or not, Pd and |Fg .p | seem to follow a perfect similar pace of evolution. Hence we find that Fg .p is to a great extent determined by Pd rather than by drag force in dense gas--solid flow though it may not be true for specific condition, such as inside bubbles where the pressure gradient can be neglected. We are inclined to consider that the variation of Fg .p is mainly dominated by the variation of Pd , and the variation of Pd is to a great degree determined by if or not the `channel' for gas flow is blocked. Since the back-mixing particles considerably block off the gas flow,
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Fig. 2. Comparison of Fg .p and Fp.p for results with (a) and without (b) LES.
Pd increases and so do |Fg .p | and |Fp.p |, and consequentially new bubbles emerge at the bottom of the bed. On the other hand, when small bubbles grow, coalesce into each other, and erupt finally, the Pd turns to decrease and so do |Fg .p | and |Fp.p |. 4.2.3. Periodic motion It is revealed from Figs. 2 and 4 that the periodic variation of Fg .p , Fp.p and Pd seems to be more regular for results without LES than that with LES since the curve peaks for results without LES seem to appear with more relatively similar amplitude and period than that with LES. Besides the nonlinear nature of the governing equations which may grow into different results with the same initial and boundary conditions, the only cause for the difference is due to the SGS model. Thus it is considered that the SGS fluctuation of gas phase
will exert great influence upon the evolution of dense gas--solid flow. To study in detail, first we smooth the curve of Fg .p by cutting off the high frequency but low magnitude components of Fˆ g .p in its spectral space, and rebuild a smoothed curve of Fg .p by inverse Fourier transform. Then we filter the constant-current component of Fg .p to obtain the fluctuating components. Finally, the spectral representation of the smoothed fluctuating component is obtained by FFT, and it is shown in Fig. 5. Treatment for Fp.p or Pd is quite similar. The subfigures in Fig. 5a are results without LES while Fig. 5b is with LES. The fundamental frequencies in Fig. 5a for Fg .p , Fp.p and Pd are the same (f 1 = 3.42 Hz), and it is also true for the second frequencies (f 2 = 6.84 Hz). In Fig. 5b the fundamental frequencies for Fg .p and Pd are f 1 = 1.95 Hz, and for Fp.p , f 1 = 2.44 Hz.
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Fig. 4. Comparison of Fg .p and Pd for results with and without LES.
The results show some indications on the periodic variation of inter-phase and inside-phase interaction as well as the total pressure drop in dense gas--solid flow. In addition, the differences between the results with and without LES reveals that the SGS turbulence will to some degree modify the periodic motion of fundamental interactions. It is interesting to consider the main factors for the periodic variation. As well known, in numerical simulation of dilute gas--particle flow, the pressure gradient force for particle is always not taken into account. However, as mentioned in the above section, in dense gas--particle flow or in local region where the solid phase is characterized as dense clusters, the main drift of particle motion is more inclined to be influenced by the pressure gradient force. Thus, the periodic motion of raise and fall of particles induces a sequential periodic variation of pressure drop. This may be the main factor for the periodic variation of pressure drop and inter-phase interactions. However, there always exist many other factors which all exert their influences on the periodic variation, such as the sub-grid turbulent fluctuation, the scale of the bed and the piling height of the particles. It may be only in such a small simulation bed that the periodic variation is less likely to be influenced by other factors and the periodic variation turns to appear more apparently.
Fig. 3. Visualization of particle positions and velocities at the central plane of the bed: (a) results with LES at time t1 =0.864 s (left) and t2 =1.152 s (right), respectively; (b) results without LES at time t3 = 0.354 s (left), t4 = 0.436 s (right), respectively.
4.2.4. Particle energy Fig. 6 is the time-dependent variation of particles' mean translational kinetic energy Etr and rotational kinetic energy Erot , where the rotational motion of particles are caused by inter-particle friction and not induced by gas-phase vortex. We find that from t = 0.7 to 1.3 s, Etr and Erot increase rapidly. In addition, the kinetic energy of particles is lower for results with LES than that without LES. The reason is due to the SGS turbulent dissipation of gas energy and then the energy transport from gas phase to solid phase becomes weaker. According to the above results, we think that energy dissipation for gas phase as well as solid phase may be to some degree non-negligible. In addition, the rotational motion of particle, which on average occupy 3.9.5.56% of translational kinetic energy, low in comparative sense but still large in absolute magnitude, will play an important role in dense gas--solid flow evolution.
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Fig. 5. FFT of smoothed curve of Fg .p , Fp.p and Pd . (a) results without LES; (b) results with LES.
4.2.5. Turbulent effects We explore the turbulent effects in fluidized bed by making comparison of gas-phase turbulent intensity If and `particle-phase turbulent intensity' Ip , both in Euler frame. We first mesh the fluid field and compute the mean local velocities of particles to build a pseudo-particulate-flow field. Following the definition of fluid turbulent intensity, we computed the Ip and If in the same way and displayed them in Fig. 7a and b, respectively. We find that Ip is much higher than If , which means that if solid phase is treated as a pseudo-continuum phase in dense gas--solid flow, the particles
turbulent fluctuation will be much higher than the gas phase, especially in the region where the dense gas--solid flow is transited into dilute gas--solid flow. The reason for the high particle turbulent intensity in the transition region may be due to the eruption of bubbles which bring out suddenly the inner energy of gas phase and due to the region with high potential energy for particles. The above two factors cause the particles to be brought up and fall back frequently, producing high particle fluctuations. Additionally, the gas-phase turbulent intensity will be augmented induced by particulate turbulent fluctuation especially in the wake of the immersed tubes.
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Fig. 6. Evolution of mean translational and rotational energy of all particles and comparison for results with and without LES.
Fig. 7. Contour figure of `particle turbulent intensity' and gas turbulent intensity obtained from results of LES simulation: (a) particle turbulent intensity similarly defined as fluid phase under Euler frame; (b) gas phase turbulent intensity.
4.2.6. Particle--tube impact From the point of view of heat transfer, the more frequently particle--tube impacts occur, the better. However, the erosion caused by frequent particle--tube impact and friction is disadvantageous. Therefore, it is necessary to know the probability distribution of particle-to-tube impact. Rong et al. (1999) investigated immersed tube erosion based on an erosion model (Finnie, 1960). We choose an alternative approach to calculate the particle--tube impact force, which is the direct factor for tube erosion and an indirect factor for particle--tube heat transfer. Fig. 8a displays the frequency and magnitude of particle-to-tube
impact force along the circumferential direction of the lower immersed tubes. The forces include the normal component |Fn | and tangential component |Ft |, and the magnitudes are nondimensionlized by particle gravity Gp . Fig. 8b is normalized distribution density of impact frequency. From Fig. 8a, the average magnitude of particle--tube impact force for results with LES (|Fn + Ft | = 229.83Gp ) are close to the results without LES (|Fn + Ft | = 224.27Gp ). And both the normal component of impact force and the tangential component of friction force are concentrated at the bottom half of the tube ( = 180.360◦ ). From Fig. 8b, we find that there are two maximum values of probability density at = 285◦ for results with LES and = 225◦ for results
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Fig. 8. Distribution and frequency of particle--cylinder impact. (a) Distribution and magnitude of normal and tangential component of impact force. (b) Impact frequency on the surface of the tube.
without LES. Rong et al. (1999) found that the peaks of impact energy appears at the two sides (about = 210.240◦ and = 310.340◦ ) of the bottom half tube. They did not use the LES in their governing equations. From the point of view of tube erosion, we consider the numerical results without LES are partially consistent with theirs, although the simulation conditions are different. Moreover, the numerical results with LES indicate that in the region = 235.295◦ there still exists the largest probability of particle--tube impact and sequentially exist largest tube erosion and particle--tube heat flux. 5. Conclusion In this paper, we carried out a true 3-D numerical simulation of bubbling fluidized bed using DEM--LES coupling method. In this simulation, we obtain some underlying information on the mechanisms of fundamental interactions in gas--solid flow: Particle--particle interaction forces are on average almost two orders of magnitude as large as gas--particle interaction forces. We consider that |Fg .p |, |Fp.p |, Pd , particle back-mixing motion and bubble motion seem to bear a strong correlation with each other, and all of them seem to follow the same pattern of periodic variation
though the periodicity may differ in various flow conditions, even may be modified by SGS turbulent fluctuation. Moreover, |Fg .p |, |Fp.p | and Pd seem to step in a perfect synchronous pace of variation. In dense gas--solid flow, the energy transport from gas-phase to particle phase is attenuated by gas-phase turbulent dissipation such as SGS viscous dissipation. Under Euler frame, we calculate the particle turbulent intensity Ip and find that Ip is much higher in the region where the transition from dense gas--solid flow to dilute gas--solid flow occurs than in the region of dense gas--solid flow. Moreover, the particle turbulent intensity Ip under Euler frame is much higher than the gas-phase turbulent intensity If and If is augmented by particle fluctuation especially in the wake of immersed tubes. Particle-immersed-tube impacts occur most frequently at the upstream side and the downstream side, especially at the upstream side in fluidized bed. Notation CD CK CS
drag coefficient Kolmogorov constant Smagorinsky constant
N. Gui et al. / Chemical Engineering Science 63 (2008) 3654 -- 3663
dp dr Erot Etr f 1, f 2 FC FD Fg .p Fn Fp.p FP Ft g Gp I If Ip k m Pd pg S¯ ij
¯
S tg tp Tij ui , uj vr vg vs V Vp
diameter of particle, m relative displacement between the pair of collision particles, m rotational energy of particle, J translational energy of particle, J frequency components in spectral space, Hz particle--particle or particle--wall contact force, N drag force, N gas--particle interaction force, N normal component of particle-immersed tube interaction, N particle--particle interaction force, N pressure gradient force, N tangential component of particle-immersed tube interaction, N acceleration of gravity, m/s2 gravity of particle, N mass of inertia of particle, kg m2 intensity of turbulence of fluid phase intensity of turbulence of particle stiffness factor, N/m mass of particle, kg pressure drop of the fluidized bed, Pa gas pressure, Pa deformation tensor of the filtered field or resolved strain rate local strain rate time step for gas-phase simulation, s−1 time step for particle phase simulation, s−1 sub-grid stress tensor components of gas velocity, m/s relative velocity between the pair of collision particles, m/s velocity of gas phase, m/s velocity of particle, m/s particle volume, m3 particle translational velocity, m/s
Greek letters
ij , x m ε, εg g t g ij p
the angle in the circumferential direction of the tube fluid--particle interaction coefficient damping coefficient, N s/m Kronecker delta sub-grid characteristic length scale and mass size in x dimension, m mesh size, m gas void fraction viscosity of gas, kg/(m s) eddy viscosity gas density, kg/m3 viscous stress tensor particle rotational velocity, rad/s
Subscripts g or f s or p
gas or fluid phase. solid or particle phase.
d rot tr
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drop rotational translational
Operators ∼ −
filtering operator in LES ensemble mean operator Mean operator module of a vector
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