DEM simulation of particle mixing in flat-bottom spout-fluid bed

chemical engineering research and design 8 8 ( 2 0 1 0 ) 757–771

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

DEM simulation of particle mixing in flat-bottom spout-fluid bed Yong Zhang, Baosheng Jin ∗ , Wenqi Zhong, Bing Ren, Rui Xiao School of Energy & Environment, Southeast University, Sipai Lou 2#, Nanjing 210096, PR China

a b s t r a c t The particle mixing mechanism affects the rate of the process and the achievable homogeneity. This paper presents a numerical study of the particle motion and mixing in flat-bottom spout-fluid bed. In the numerical model, the particle motion is modeled by discrete element method (DEM) and the gas motion is modeled by –ε two-equation turbulent model. Validation with experiments is first carried out by comparing solid flow pattern and bed pressure drop at various gas velocities. Then, particle velocities, obtained from DEM simulations, are presented to reveal the mixing mechanisms. On the basic, the dependence of mixing index on the time and the effect of gas velocity on mixing and dead zone (stagnant solid) are discussed, respectively. The results indicate that the spouting gas is the driving force for the formation of particle circulation roll, resulting in the mixing. The convective mixing caused by the motion of circulation roll, shear mixing induced by the relative move of circulation rolls and diffusive mixing generated by random walk of particle among circulation rolls are three different mixing mechanisms in spout-fluid bed. The increase of spouting gas velocity promotes the convective and shear mixing. While increasing the fluidizing gas velocity improves significantly the convective mixing and but weakens the shear mixing. Both of them yield a reduction in the dead zone. © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Particle mixing; Circulation roll; Mixing mechanism; Spout-fluid bed; Mixing index

1.

Introduction

A spout-fluid bed is a unique fluid-particle-agitated bed which combines a number of favorable properties of both spouted and fluidized beds. This technique has been accepted traditionally as a solid–fluid contacting method for physical operations such as drying, coating and granulation of granular solids (Link et al., 2004, 2007; Park et al., 2006; Plawsky and Littman, 2006; Zielinska and Markowski, 2007; Białobrzewskia et al., 2008). In recent years, the application of spout-fluid bed reactors has extended to catalytic reactors and combustion and gasification of coal and biomass (Lim et al., 1988; Arnold and Laughlin, 1992; Xiao et al., 2006; Zhong et al., 2006a,b). Particle mixing is deemed as a complex process to obtain a uniform mixture of ingredients distributed among each other (Gyenis, 1999). In many industrial applications, the mixing and contacting of reactants and products very often are the controlling factors in the reactor performance since mixing is closely related to the rates of mass, heat and moment

∗

transfers. Good mixing among reactors is essential to avoid hot spots due to the heat released by the highly exothermic reactions and segregation of the larger particles at the bottom of reactor. Whereas poor homogeneity of the particulate mass can lower the overall process efficiency and complicate its thermal control. Mixing mechanism, as a qualitative feature, characterizes the way of intermingling of components, which significantly influences the rate of the process and also the achievable homogeneity. So the detailed knowledge of the fundamental mechanisms is important to enhance the performance of operations. In addition, the mixing process acting during the practical operation depends on the operating condition to a great extent. Therefore, it is essential to identify the dominant mixing mechanism at different operating conditions, which helps to control the operation. There is extensive literature (Rowe et al., 1965; Singh et al., 1972; Valenzuela and Glicksman, 1984; Fan et al., 1986; Lim et al., 1993; Shen and Zhang, 1998; Stein et al., 1998; Hoomans et

Corresponding author. Tel.: +86 25 83794744; fax: +86 25 83795508. E-mail addresses: [email protected] (Y. Zhang), [email protected] (B. Jin). Received 8 March 2009; Received in revised form 13 October 2009; Accepted 13 November 2009 0263-8762/$ – see front matter © 2009 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2009.11.011

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chemical engineering research and design 8 8 ( 2 0 1 0 ) 757–771

al., 2001; Rhodes et al., 2001; Bokkers et al., 2004; Lu and Hsiau, 2005; Wang et al., 2006; Feng and Yu, 2007; Tian et al., 2007) concerning the mixing in the conventional fluidized bed, most of which can be divided into two categories: specific numerical and experimental studies. The specific studies concern an experimental or numerical quantitative analysis of the mixing performance or dynamics. In the area of experiments, a couple of early methods (Rowe et al., 1965; Singh et al., 1972; Valenzuela and Glicksman, 1984; Fan et al., 1986; Yang et al., 1986; Lim et al., 1993; Shen and Zhang, 1998; Hoomans et al., 2001) have been employed to investigate the mixing, which provide some macroscopic information. Recently, some noninvasive measurement techniques (Stein et al., 1998; Bokkers et al., 2004; Wang et al., 2006) are developed to get some microscope information based on particle-scale. However, one of the problems of experimental investigation on particle mixing is that particle motion characteristic within the bed could not be obtained. Thus, systemic discussions engaged in the microscopic mixing phenomena of dense gas–solid flows, in temporal and spatial details, are believed unavailable. Generally computational models do not suffer from these limitations and some of them have reached a considerable maturity in correctly representing the phenomena involved in fluidization (Alberto et al., 2008). In recent years, numerical simulation has been widely used for studying gas–particle systems as it promises to be a useful tool for obtaining a wide range of flow properties of particles and gas flows simultaneously. Among these modeling approaches developed recently, discrete element method simulations (Rhodes et al., 2001; Lu and Hsiau, 2005; Feng and Yu, 2007; Tian et al., 2007) prove ideal to investigate the particle behaviour of fluidized beds. In the DEMs there have been two types, i.e. the soft sphere models (Tsuji et al., 1993; Rikami et al., 1998; Rhodes et al., 2001) and the hard sphere models (Yuu et al., 1995; Hoomans et al., 1996). In the soft sphere models, particles are permitted to suffer minute deformation, and these deformations are used to calculate elastic, plastic and frictional forces between particles. In a hard sphere models, a sequence of collisions is processed and the forces between particles are not explicitly considered. Therefore, the force model is extremely important for DEM simulation. In gas–solid system, particle–fluid interaction forces must be properly considered except for the contact force and non-force induced by the collisions between particles or walls (Zhu and Yu, 2006; Zhu et al., 2009). These forces such as pressure gradient force, virtual mass force and Basset force tend to be ignored, particularly Saffman force and Magnus force. Whether for experiment or numerical simulation, it is commonly considered that the axial and lateral motion of bubbles is the basic mechanism of particle mixing in fluidized bed. This may be summarized as follows. When a bubble rises through the bed, it carries a wake of particles to the bed surface and causes a drift of particle to be drawn up below it, which leads to the axial mixing. At the same time, the interaction and coalescence of neighboring bubbles causes the lateral motion of bubbles, resulting particle mixing in the lateral direction of the bed (Rhodes et al., 2001). While the mechanism of the solid mixing in the spouted bed is one of the fields on which many groups of investigators focused their interests. It is widely accepted that in the spouted bed most of the mixing action comes from the fountain and the spout (Kang and Mo, 1985; Larachi et al., 2003). For the spout-fluid bed operating at the low fluidizing gas velocity, however, there are no obvious bubbles occurring in the annulus. Accordingly, it is clear that

mixing mechanism in spout-fluid bed is different from one in the ordinary fluidized bed. Up to now, however, the report at this respect has not been found in the literature. In this paper, we perform DEM simulations to investigate the mixing mechanism in spout-fluid bed. It aims at better understanding particle mixing behavior in spout-fluid bed. The organization of current paper is as follows. As a first step, the DEM model and simulation condition will be described. Subsequently, experiment combining the bed-frozen method and image processing technology will be carried out in order to assess predicting ability of the model by comparing the simulated and measured results under comparable conditions in terms of solid flow pattern and bed pressure drop. Then, mean particle velocities, computed from DEM simulations, are presented and the mixing mechanisms in spout-fluid bed are determined combining the analysis of particle velocity and visual observation during the real experiments. Finally, the dependence of mixing index on the mixing time and the influences of gas velocity on the mixing and dead zone are fully discussed based on the elucidated mixing mechanisms.

2. Computational models and calculation conditions 2.1.

Model

Details of the DEM model have been given by Zhong et al. (2006a,b) and only a very brief summary is presented here. In our previous study, this model has been used for investigation the gas–solid flow behaviors in spout-fluid bed.

2.1.1.

Particle phase

In the present work, the particle motion is calculated threedimensionally. In addition to the drag force, contact force and gravitational force, the shear induced Saffman lift force and rotation induced Magnus force are considered. A particle in a granular flow can have two types of motion: translational and rotational. During its movement, the particle may interact with its neighboring particles or wall and interact with its surrounding fluid, which leads to the momentum and energy exchange. Newton’s second law of motion can be used to describe the motion of individual particles, which can be written as: mp

Ip

dvp = fC + fD + fLS + fLM + mp g dt

dωp = Mp dt

(1)

(2)

where vp and ωp are the translational and angular velocities of particle, respectively, Ip and Mp are the particle moment of inertia and the particle torque, respectively, fC and fD are the contact forces among particles and drag force acting on particles, respectively, and fLS and fLM are the Saffman lift force and the Magnus lift force, respectively. Contact forces are described in terms of a mechanical model involving a spring, dashpot and friction. Basically, the contact force has two components: normal and tangential, as follows: fC = fcnij + fctij

(3)

fcnij = (−kn ınij − n vtij · nij )nij

(4)

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fctij = −kt ıtij − t vtij

(5)

in which, kn and n are the spring and damping coefficients in normal direction, respectively. kt and t are the spring and damping coefficients in tangential direction, respectively. ınij and ıtij are the normal and tangential displacements between particle i and particle j, respectively. The drag force is one of the particle–fluid interaction forces, which is caused by the relative velocity between the particle and fluid and can be expressed as: fD =

1 g d2p CD |ur |ur 8

(6)

where the translational relative velocity, ur is defined as ur = u − vp . Saffman lift force, also called shear lift force, is generated by a velocity gradient perpendicular to the main flow direction if the particle is relatively large and the fluid has a relatively large velocity gradient when it flows around the particle (Saffman, 1965, 1968). For current spout-fluid bed, the gas velocity is significant different between the central jet region and annular dense region (Mathur and Epstein, 1974; Pianarosa et al., 2000). Thus, the force should not be neglected especially when particles in the boundary of central jet region and annular dense region. The Saffman lift force is calculated by: fLS = 1.615(u − vp )(g g )

0.5 2 dp CLS

   ∂u   ∂u  ×  sgn , (n = x, y, z) ∂n ∂n

CLS =

(1 − 0.3314 0.5 )exp(−0.1Rep ) + 0.3314 0.5 0.0524 0.5 (Rep )

0.5

(Rep ≤ 40) (Rep > 40)

When the gas flow is not uniform at various locations, the particle will rotate due to the velocity gradient. At a low Reynolds number, the rotation of particle will bring the fluid moving around the particle, which leads to the increase of fluid velocity in the same side as the flow direction and the decrease of fluid velocity in the opposite side. A lift force will generate due to the velocity difference between the different sides of the particle, which is called as Magnus lift force. A rotation induced Magnus lift force will generate due to the velocity gradient when particle entrained into the jet region with rotation, which should not be neglected especially when particles in the boundary of central jet region and the annular dense region. Magnus lift force is calculated by the following correlation (Lun and Liu, 1997; Lun, 2000): ωr × vr 1 g v2r d2p CLM 8 |ωr | · |vr |

(9)

where vr is relative velocity between gas and particle, vr = u − vp . ωr is the rotation angular relative velocity between gas and particle, ωr = ωg − ωp and ωg = 0.5 × u. CLM is Magnus lift coefficient. CLM is calculated from the correlation developed

(Rep ≤ 1)

⎪ ⎩ |ωr | dp (0.178 + 0.822Re−0.522 ) (1 < Rep < 1000) p

(10)

|vr |

ωp is assumed to satisfy the correlation (Rubinow and Keller, 1961) as follows: mp d2p dωp p d2p = Ct |ωr |ωr 10 dt 64

(11)

in which the coefficient Ct is calculated from the correlation proposed by Dennis et al. (1980).

2.1.2.

Gas phase

The calculation of the gas motion follows a generalization of the Navier–Stokes equations for gas interacting with solids, as shown by the following equations for mass and momentum conservation: ∂ ∂ (εg uj ) = 0 (εg ) + ∂t ∂xi

(12)

∂ ∂ (εg ui ) + (εg ui uj ) ∂t ∂xj

(7)

(8)

fLM =

CLM =

⎧ |ω | r ⎪ ⎨ |vr | dp

= −ε

in which CLS is the Saffman lift coefficient. CLS is calculated from the correlation developed by Mei (1992), which is expressed as:



by Lun and Liu (1997), which is expressed as:

∂p ∂ + (ε ) − np (fD + fLM + fLS ) + εg g ∂xi ∂xj ij

(13)

where ε is the void fraction, g is gas density, and ui and uj are the gas velocity, i, j = 1, 2, 3, which represent x, y and z direction,  ij is the turbulence stress, and fg is the drag force for single particle. The interactions between the solid and the gas per unit volume due to fluid drag force (Kafui et al., 2002) is given by the following correlation: np fD = ˇ(u − vp )

(14)

where vp is the mean velocity of the particles in the unit volume. Ergun’s equation (Ergun, 1952) was used for the dense phase and Wen and Yu’s equation (Wen and Yu, 1966) for the dilute phase, and the coefficient ˇ was summarized as follows:

ˇ=

⎧ (1 − ε) ⎪ [150(1 − ε) + 1.75Rep ] (ε ≤ 0.8) ⎪ ⎨ d2p ε (15)

⎪ ⎪ ⎩ 3 CD (1 2− ε) ε−2.7 Rep 4

CD =

(ε > 0.8)

dp

⎧ 0.687 ) ⎨ 24(1 + 0.15Re (Re ≤ 1000) ⎩

Re

0.43

(Re > 1000)



Rep =

(16)



g ε u − vp  dp

(17)



in which, CD is the drag coefficient for a single sphere. The gas turbulence kinetic energy equation is sited from Crowe (2000), which can be expressed as: ∂ ∂ ∂ (εk) + (εuj ) = ∂t ∂xj ∂xj

 

ε +

t k

 ∂k ∂xj

+ εGk + ε k + Skd (18)

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t Gk = g +

 2 ∂u 2

∂x

 ∂u

∂v + ∂y ∂x

2

+

+

 ∂v 2 ∂y

+

 ∂u

 ∂w 2

∂w + ∂z ∂x

∂z

2

+

 ∂v

∂w + ∂z ∂y

2 

Skd = ˇ|u − vp |2 + ˇ( vv − uv)

(19)

(20)

where u is the gas fluctuating velocity, v is the particle fluctuating velocity. The turbulence momentum dissipation equation is cited from Bertodano et al. (1994), which can be described as: ∂ ∂ (g ε ) + (εg uj k) ∂t ∂xj =

∂ ∂xj

 

ε +

t k

 ∂ ∂xj

+ε

ε (C1 Gk − C2 g ) + Sεd k

(21) Fig. 1 – Geometry of vessel in the simulation.

Sεd

ε = C3 Skd k

(22)

Almost all researches using DEM neglected the effect of turbulence on the gas–particle motion (Zhou et al., 2004), although both experimental and theoretical results vertified the strong intensity of turbulence in fluidized bed (Peirano and Leckner, 1988; Zhou et al., 2002). Significant advances have accomplished in modeling the turbulent flow so far. One of originalities of the present DEM simulation is to develop a 3D turbulent model accounting for the gas–solid turbulent flow in spout-fluid bed. Another primary originality is that Saffman lift force (shear lift force) and Magnus lift force acting on individual particle are considered when establishing the mathematics models, which is often ignored in other DEM simulations. Of course, it needs to be stressed that the objective of this study is to provide an illustrative example of the use of DEM in studying particle mixing mechanism in spout-fluid bed.

2.2.

Calculation conditions

The geometry of the vessel is shown in Fig. 1. The spoutfluid bed has a cross-section of 100 mm × 30 mm and height of 500 mm. The jet nozzle is 10 mm × 30 mm and the other is fluidized gas inlet. The particle properties used in the study are: spring constant 800 N/s; density 900 kg/m3 ; coefficient of friction 0.3; coefficient of restitution 0.9; Poisson’s ratio 0.3. These values are based on polypropene beads. A total of 16,000 particles are used in spout-fluid bed. Detailed descriptions of parameters for vessel and particle used in the simulation are listed in Table 1. In the simulation, particles are randomly generated into the bed and then fall under the influence of gravity to form an initial packed bed with height of 100 mm. The particles at the bottom with a height of 10 mm are then tagged with a dark colour and the rest are tagged with a light colour. At this moment, the time is set to zero. Then, the gas is introduced at a fixed velocity. The operating conditions are summarized in Table 2. Zhong et al. (2006a,b) have identified six different flow regions in spout-fluid bed. They are internal jet (IJ), spouting (S), fluidizing (F), jet in fluidized bed with bubbling (JFB), jet

Table 1 – Parameters used in simulation. Parameter

Value

Bed Width Height Depth Nozzle width Initial bed height

Unit

0.1 0.5 0.03 0.01 0.1

Particle Diameter Density Poisson’s ratio Particle spring constant Wall spring constant Coefficient of restitution Coefficient of fraction Number

0.0028 900 0.3 800 800 0.9 0.3 16,000

Fluid (air ambient) Density Viscosity

1.25 1.81 × 10−5

Calculation parameters Time step

1 × 10−6

m m m m m

m kg/m3 – N/s N/s – – – kg/m3 Pa s

s

in fluidized bed with slugging (JFS) and spout-fluidizing (F). In current work, the spout-fluid bed in all cases is operated at a high spouting gas velocity being greater than the minimum spouting gas velocity and a low fluidizing gas velocity. It takes on a similar appearance to the conventional spouted bed. So, this flow region in our study is regarded as spouting (S), which is also defined as spouting with aeration by other researchers (Mathur and Gishler, 1955; Mathur and Epstein, 1974). It should be pointed out that the annulus is not fluidized in this flow region. Simulations are carried out in a server with four 3.0 GHz CPU and 16 GMb memory. While only one CPU is used because the present program could not perform parallel computing but

Table 2 – Operational condition investigated. Case

1

2

3

v∗s v∗f

1.15 0

1.36 0

1.54 0

4

5

1.36 0.21

1.36 0.42

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Fig. 2 – Schematic diagram of experimental setup. (1) Spouting gas inlet; (2) fluidizing gas inlet; (3) pressure sampling port; (4) spout-fluid bed; (5) halogen lamps; (6) digital CCD; (7) A/D converter; (8) computer. use its high memory. Different contact models and different physical parameters such as particle size and spring constant decide the time step in DEM simulation. Tsuji et al. (1993) proposed that the time step should correspond to the following relation for stable calculation: t ≤

 5



mp k

(23)

where t is the time step, mp is the mass of a particle and k is the stiffness parameter. To obtain the calculation stability, the simulation is conducted by using an appropriate time step of 1 × 10−6 s.

3.

Comparison with experimental date

In order to validate the DEM model, experiments are specifically conducted. Simulation results on a similar system and experimental data have been compared in terms of the solid flow pattern and bed pressure drop at varying gas velocities.

3.1.

Experimental procedure

Experiments have been performed in a Plexiglas spout-fluid bed using a porous plate for homogeneous background fluidization and a central spouting for bubble injection. The vessel sizes and particle properties are the same as those in the simulation. Pressure drop across the bed were measured by a multi-channel differential pressure signal sampling system. The bed was illuminated by two 600 W halogen lamps, one on each side for uniform lighting. Thus, flow patterns and mixing process were recorded with a high speed digital camera. A schematic overview of the experimental setup is given in Fig. 2. The experiments were run with air at room temperature and atmospheric pressure (see Zhang et al., 2008, for further details). To reduce the electrostatic charges during the operations, two prevention methods were employed. The first prevention method was to group the vessel so that most of electrostatic charges were timely introduced to group. Another method was to add moisture in spouting gas and fluidizing gas. Rela-

761

tive humidity of air from compressor was controlled between about 40% and 80% since over-humidification leads to excessive capillary forces causing defluidization. The humidity level was optimized to prevent the onset of electrostatic force. Despite careful optimization of the procedure, electrostatic charging of polypropene beads could not be fully prevented in experiments. The experiments were basically conducted using the following procedure. As the first step, particles were initially packed in two separate layers to visualize clearly the extent of mixing. The first layer of particle (the tracer particles) with a height of 0.01 m was poured in the vessel first, and then the bed particles were layered on top of it with a height of 0.09 m. As the second step, turn on the spouting gas control valves for startup of the bed, and then keep gas flow at desired operating condition. After the mixing extent reached the preconceived value, the sampling process was carried out. In general, spout-fluid bed shares three steady regions: the spout, the fountain and the annulus. The fountain is a region where both particle velocity and concentration are low, and the annulus is a region where the particle concentration is high but the particle velocity is low. In the spout, particle velocity is relatively high, whereas the particle concentration is low. So based on different velocities and concentrations, the concentration of particle in the fountain is obtained by means of image processing technology and ones in the spout and annulus are measured by bed-frozen method. By analyzing the images, the concentration of tracer particle can be obtained in fountain shown in the red line region of Fig. 3(c) owing to the low particle concentration. It should be noticed that the basic idea of the image processing technology was to calculate the average number of particle per interrogation area among these images. Preliminary tests on samples of known composition revealed that the error inherent to this method was below 0.5%. The bed-frozen method was used by many research groups (Marzocchella et al., 2000; Rhodes et al., 2001; Lu et al., 2003; Leaper et al., 2004) and widely accepted in the fluidization fields. For quantitative analysis, this method was adopted and developed here. To avoid the rearrangement of particles when the bed was sinking, the flashboard-box was used in this method, which resembled a drawer in appearance (as shown in Fig. 3(a)). The whole sampling process was described below. When the mixing was finished, the gas was instantaneously turned off and simultaneously the flashboard-box was inserted from the top of the vessel. Thus, the “frozen” particulate bed was separated into nine sections along radial direction. Then, the entire vessel was turned to horizon level and the flashboardbox was gently taken out from the top of the vessel. By means of a specially designed grid board, the mixture was isolated along the axial direction. Analysis was carried out by visual separation and numbering of bed particles and tracer particles in each sample. Subsequently, the particle of each section was numbered and particle concentration was determined from the knowledge of the number of the bed particle and tracer particle.

3.2.

Model validation

Zhong et al. (2006a,b) have identified six different flow regions in spout-fluid bed. They are internal jet (IJ), spouting (S), fluidizing (F), jet in fluidized bed with bubbling (JFB), jet in fluidized bed with slugging (JFS) and spout-fluidizing (F). In

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Fig. 3 – Schematic diagram of the sampling: (a) the flashboard-box; (b) the tracer region in the bed; (c) the corresponding sampling cell. current work, the spout-fluid bed in all cases is operated at a high spouting gas velocity being greater than the minimum spouting gas velocity and a low fluidizing gas velocity. It takes on a similar appearance to the conventional spouted bed. So, this flow region in our study is regarded as spouting (S), which is also defined as spouting with aeration by other researchers. It should be emphasized that the annulus is not fluidized in this flow region.

In order to demonstrate the model described above, we first consider the measurement of the instantaneous map of the flow field. Fig. 4 shows typical comparisons of simulated flow patterns with experiments at various spouting gas velocities at v∗f = 0.21. Obviously, images taken from the simulation show the good agreement with pictures taken from the experiments. Starting from completely segregated state, bed expansion is observed as soon as gas is injected. Then,

Fig. 4 – Comparison of simulated flow patterns with experiments at different spouting gas velocities: vf∗ = 0.21. (a) vs∗ = 0; (b) vs∗ = 1.15; (c) vs∗ = 1.36; (d) vs∗ = 1.54.

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Fig. 5 – Comparison of simulated bed pressure drop with experiments at different gas velocities. the whole bed presents three stable regions: the spout, the fountain and the annulus. For any cases of different spouting gas velocities, the mixing experiences the process from initial fully segregated state to a mixed state. It is obvious that the increasing spouting gas velocity strongly increases both range and extent of mixing, because more particles of the bottom zone are transported in the fountain. For example, the height of the fountain region increases significantly when increasing v∗s from 1.15 to 1.54. In the meanwhile, the particle group at the bottom corner becomes smaller and smaller with the increase of spouting gas velocity, as shown in Fig. 4(d). Note that the conditions for physical and numerical experiments are not exactly the same. Thus, the comparison of flow patterns between physical and numerical experiments is qualitative. Monitoring the pressure drop across a fixed bed height is a common practice in fluidization studies. The plot of time average bed pressure drop against superficial gas velocity is shown in Fig. 5. The experiment shows that the bed pressure drop linearly increases when the gas velocity increases from 0 to 1.11 m/s, while it decreases dramatically when the gas velocity varies from 1.11 to 1.43 m/s. Then, when the gas velocity is beyond 1.43 m/s, the bed pressure drop basically keeps constant. The visual observation indicates that the jet penetrates through the entire bed once the spouting gas velocity is increased to 1.11 m/s. After that, the continuous circulation is formed, where the particles move upward in the central spout and motion downward in the annulus. From the figures, it can be concluded that the minimum spouting gas velocity vms is 1.11 m/s. It can be also observed from this figure that the simulated bed pressure drops exhibits the same trend to experimental results. Furthermore, the simulated are in reasonable agreement with the measured values. The maximum relative error of simulated value to the measured values is within 8.23%. The mean relative error is 6.56%. The quantitative agreement of the bed pressure drops between the calculation and the experiment suggests that the present calculation method can well predict the flow pattern and mixing in spout-fluid bed.

4.

Results and discussion

4.1.

Particle velocity in the bed

The particle velocity profile indicates the hydrodynamic behaviors of particle flows. Therefore, the analysis of these

Fig. 6 – Radial distribution of particle axial velocity at different bed heights. (a) In the annulus; (b) in the spout; (c) in the fountain. velocities is a key to developing a better understanding of the underlying mixing mechanisms. Such information is very difficult, if not impossible, to obtain fully with the present experimental techniques, but is readily available from the current DEM study. Fig. 6 is the axial particle velocity profile at various vessel heights, predicted by the simulation, as a function of radial distance from the centerline. To assist discussion, the dimensionless spouting gas velocity v∗s and dimensionless fluidizing gas velocity v∗f were defined as v∗s = vs /vms and v∗f = vf /vmf , respectively. It can be found that at the lower level of annulus, particle downward velocity first increases sharply and then decreases gradually with the increase of radial distance. But at the higher level, particles exhibit relatively flatter velocity pro-

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Fig. 7 – Radial distribution of particle axial velocity at various spouting gas velocities. (a) In the annulus; (b) in the spout; (c) in the fountain.

files. These profiles suggest that particles fall down like a rigid body at the higher levels in the annulus. In the spout, however, particle upward velocity decrease with height, except near the bottom region of the nozzle exit. In the fountain region, two zones have been distinguished: the core, where particles motion upward, and the periphery, in which particles flow downward. Obviously, the boundary between them is just the point where the curves cross the zero velocity line. Taking the overview of these profiles, one can see that the calculated velocities agree well with the particle velocity predictions of Takeuchi et al. (2004) in the full-3D spouted bed with a flat bottom. Fig. 7 compares radial profiles of the axial particle velocity at different spouting gas velocities. It shows that the particle

Fig. 8 – Radial distribution of particle axial velocity at various fluidizing gas velocities. (a) In the annulus; (b) in the spout; (c) in the fountain. velocity increases in both the spout and the annulus when v∗s is increased from 1.15 to 1.54. In the meanwhile, the spout radius increases with the increase of v∗s , which corresponds to the radial distance where the axial velocity is zero. This means more particles are entrained into the spout in this case. It is worth noting that increasing v∗s also results in the increase of the velocity gradient, especially in the annulus. These results are qualitatively supported by the experimental date of Tsuji et al. (1997) obtained in a flat-bottomed vessel of column diameter 0.14 m and nozzle diameter 20 mm when increasing the spouting gas velocity. Fig. 8 presents the radial distribution of axial particle velocity at different fluidizing gas velocities. It can be noted that increasing v∗f provokes an increase of the particle velocity in

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765

Fig. 9 – Vector field of particle velocity at various spouting gas velocities.

both the spout and the annulus, which is similar to the effect of the increasing spouting gas velocity. However, a qualitative difference can be seen between them. In the case of increasing v∗f , the velocity gradient in the annulus decreases significantly. It is contrary to the case of increasing v∗s . The increase of v∗f , on the other hand, results in the extension of the range of particle motion. For example, the circulation occurring when the spouting gas penetrate through the entire bed is restricted to the upper bed at low v∗f , while it covers the bottom of bed at higher v∗f , especially for the particles at the bottom. These results agree qualitatively with visual observation of physical experiments.

4.2.

Identification of mixing mechanisms

Fig. 9(a) and (b) presents the time averaged vector field of solids flow produced by the simulation at different spouting gas velocities. As seen in the figures, there are two symmetric circulation rolls taking place in the bed after the flow reaches stable state and the circulation is formed. Here, the circulation roll indicates the particle streamlines (trajectories). The drag force produced by the spouting gas in the spout drives the circulation rolls, where particles motion up in a thin layer. On the other hand, the gravity force acting on particle in the annulus also enhances the circulation rolls, in which particles move down like a rigid body. As the spouting gas velocity increases gradually, the circulation rolls prolong along the axial and radial direction as shown in Fig. 9(b), which impels more and more particles to be involved into mixing. According to the velocity profiles predicted by the present simulation and the repeating visual observation, we can identify three mixing mechanisms in spout-fluid bed, which are illustrated in Fig. 10. Namely: (1) convective mixing—which means the transport of particles or particle groups from one location to another and is mainly caused by the motion of circulation roll, (2) shear mixing—which takes place due to the adjacent region moving at different velocities, (3) diffusive mixing—which is induced by the random walks of particles through the gaps between the adjacent circulation rolls. In general, circulation rolls run at different speeds. That is to say, convection always causes certain shear between the adjacent regions moving at different velocities. So convective and shear mixing act always simultaneously.

Fig. 10 – Time averaged particle streamlines and mixing mechanism in different regions. For the spout-fluid bed operating at low fluidizing gas velocity, the entire bed commonly shares different zones. The spout and fountain belong to the dilute particle bed, while the annulus belongs to the dense particle bed. For the dilute bed, i.e. in the fountain, there are more places for particles to random motion, which greatly favors the diffusive mixing. Since the diffusive mixing causes microscopic distribution of individual particles among each other, it produces a rapid increase of mixing quality, especially in the early stages of a process. As indicated in Fig. 7, the high particle velocity corresponds to the high velocity gradient. This suggests that convective and shear mixing play a great effect on the improvement of mixing quality together. Furthermore, this effect becomes more and more obvious when gradually increasing spouting gas velocity. This also implies that convective mixing has a strong synergetic effect on shear mixing. Although the convective and shear mixing act simultaneously, there exist differences of their relative contribution to the achieved mixing degree, which depends mainly on the actual operating condition. In the annulus, for example, increasing fluidizing gas velocity leads to a high velocity and smooth velocity profile. It is clear that the effect of convective mixing get the better of one of shear mixing. Both of them, on the contrary, are improved as a result of the increase of spouting gas velocity. The reason is that increasing spouting gas velocity results in a high velocity and sharp velocity profile.

4.3.

Analysis of the effect of the mixing time

On the basis of statistical analyses, various mixing indices are employed to describe the solid mixing in many different industrial processes (Fan et al., 1970). The mixing index is an overall measure of the mixing state of the system, so that each of its values may correspond to conditions with the components differently distributed in the bed. Hence, mixing patterns in both transient conditions and steady-state are analysed in terms of a mixing index.

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Fig. 11 – Evolution of the degree of mixing in the different regions at the indicated velocity. Here, we use the well-known Lacey mixing index (Lacey, 1954) to quantify the quality of mixing. For this purpose, the spout-fluid bed is divided into several cubic samples, each with volume of 3000 mm3 (10 mm × 10 mm × 30 mm). The variance 2 for the concentration of tracer in each sample is defined in the following way: 1 2 (ci − c¯ ) N N

2 =

(24)

i=1

in which N is the number of samples in the bed, ci is the concentration of tracer in the sampling cell i, c the mean concentration of tracer in the N samples. In present work, c is equal to 0.1. As a matter of fact, the variance 2 represents the random mixing state and lies between two extreme states: the completely segregated and well mixed states, which may be, respectively given by: 02 = c¯ × (1 − c¯ ) 2 m =

c¯ × (1 − c¯ ) n

(25) (26)

where n is the number of particle contained in a sampling cell. Then, a dimensionless and normalized index can be expressed as: M=

02 − 2 2 02 − m

(27)

which is the ratio of mixing achieved to mixing possible. So when the value of M is equal to 0, it can be deduced that the mixing state is completely segregated. On the contrary, provided that M is equal to 1, it could be inferred to be fully mixed. With respect to other mixing indexes, this definition has the advantage of being defined in the range 0–1 and of being independent of the actual macroscopic position of the components. To investigate the hydrodynamic effects on particle mixing, simulations and experiments have been carried out. At the same time, mixing indexes of various regions are calculated at the indicated velocity. The results are reported in Fig. 11 where x-axis represents time starting from fixed bed and y-axis is the Lacey mixing index based on Eq. (27). It can be observed that as mixing proceeds, the mixing index increases significantly

Fig. 12 – Experimental data (round) and fitted data (line) from the simulation. (a) In the spout; (b) in the fountain; (c) in the annulus. after a stagnant state until a dynamic equilibrium of mixing is reached. It can also be found from Fig. 11 that there is considerable scatter in the data, especially at the middle state when the mixing index increases fast. And, it is more significant in the spout. Therefore, we fit the scatter data of the mixing index referred to in Fig. 11 to find out some general trend. The comparison of between the results of this procedure and the data produced by the experiment is reported in Fig. 12. It can be clearly seen that the agreement between the simulated mixing index and the results of experiments is good, further demonstrating that the choice of DEM simulation is appropriate.

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Fig. 14 – Characteristic parameters of different regions at vs∗ = 1.36 and vf∗ = 0.21. Fig. 13 – The shape of the M–t curve. It is interesting to note that the relation between the mixing index in different region and the mixing time is shown to exhibit an S-shaped form. That is, the development of the mixing index in different region with time follows a similar trend. Even that, differences exist among them and some of them are substantial. In other words, the evolution of mixing index depends on the tracer location. And, such a difference is closely related to the mixing time. To describe such difference and gain deeper insight into the dependence of mixing index on time, we begin with the analysis of the shape of the M–t curve at a given gas velocity. Here, the characteristic parameters of a, b, c, and d have been introduced to be able to shape the curve. The shape of the curve is sketched in Fig. 13 where the mathematical meaning of each parameter is illustrated. They also mean to account for the dependence of the mixing index on the mixing time. The parameters have a rather simple meaning:

negligible mixing is attributed to the lack of stable circulation. That is to say, the mixing takes place until the circulation rolls are formed. Of course, the pre-condition for steady circulation is to unlock the packed bed. It can also be seen from Fig. 14 that the value of c decreases from the spout to the annulus. This implies that mixing is the strongest in the spout while it is weak in the annulus. This is likely the combined result of intensive convective mixing, shear mixing and diffusive mixing in the spout. Results displayed in Fig. 14 also allow examination of the rate of mixing, through comparison of the time required for the mixing index to increase from zero to a certain value. It can be noted that in the annulus, the time required for the mixing index to increase from 0 to 0.84 is about 12.18 s, while it is only 8.22 s in the spout. In the fountain, it takes about 9.36 s for the mixing index to increase from 0 to 0.89. This is to be expected because the fast move of circulation roll occurs in the spout, which results in the fast transport and mixing of particles.

4.4. (1) a is the maximum value of the mixing index, reached at dynamical equilibrium state. Clearly, the higher the parameter a, the better the degree of mixing. (2) b is the time indicating the length of stagnant state in which mixing is negligible. This value can be obtained by determining the intersection of the x-axial with the maximum slope line. (3) c is the maximum slope of the curve, which shows the maximum intensity of the mixing. (4) d is the time showing the length of whole mixing process. After then, a dynamic equilibrium of mixing is reached. According to the fitted line in Fig. 12, the value of characteristic parameters a, b, c and d can be, respectively determined. The results are reported in Fig. 14. Significantly, the characteristic parameters in all regions are different, especially for b, c and d. It can be noted that a value in the spout is greater than those in the fountain and annulus, indicating the mixing quality in the spout is the best among three tracer regions. From Fig. 14, it can be observed that in each region there is a period of stagnant state, indicated by the parameter b and it is the lowest in the spout. To this regard, Rhodes et al. (2001) think that this period of negligible mixing exists in the packedbed case and it is associated with unlocking of the packed bed. For current spout-fluid bed, we consider that the existence of

Analysis of the effect of the spouting gas velocity

The influence of spouting gas velocity is studied by increasing v∗s from 1.15 to 1.54 while keeping v∗f at 0. Values for these four characteristic parameters have been determined by means of above introduced methods at various spouting gas velocities. The corresponding results are shown in Fig. 15. It can be observed that in the annulus the value of a increases significantly and the value d decreases with the increase of v∗s . On the contrary, it is not obvious in other regions. Under the same condition, the value of b decreases to some degree in all regions while the value c increases significantly in the spout. The results, shown in Fig. 15, suggest that increasing v∗s improves greatly the mixing degree and the mixing rate in the annulus, and the intensity of mixing in the spout. Such a result could be explained by the following aspects: for one thing, an increase in spouting gas velocity accelerates the rotation of circulation roll, which greatly promotes convective mixing, especially in the spout; for another, it also leads to the increase in velocity gradients in the spout and annulus, which strengthens shear mixing. So the achievable mixing degree is the result of combined effect of convective and shears. Of course, the improved diffusive mixing should not be ignored because the velocity fluctuation in the fountain seems to increase to a certain extent.

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Fig. 15 – Characteristic parameters at different spouting gas velocities.

4.5.

Analysis of the effect of the fluidizing gas velocity

In this simulation, v∗s is kept at 1.36 while v∗f is increased from 0 to 0.42. Results of these simulations are shown in Fig. 16. It is interesting to note that in all regions the value of b and d decrease with the increase of v∗f , while the value c increases

considerably. This implies that increasing v∗f enhances not only the intensity of mixing but also the mixing rate. In the same case, the value of a increases, indicating the improvement of the mixing quality in the spout. To establish the causes for the results just presented, the mixing mechanism revealed above is used to account for the

Fig. 16 – Characteristic parameters at different fluidizing gas velocities.

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Table 3 – Results of dead zone angle for different operating conditions. Case

1

2

3

4

˛ (simulation) (◦ )

41.3

32.6

27.5

28.8

5 6.5

reason. When the fluidizing gas is introduced in the bed, partial gas is entrained from the annulus to the spout, especially in the distributor region, and the rest passes through the dense annulus. The former increases the driving force for the motion of circulation rolls as a result of the increase of the spouting gas velocity, while the latter decreases the resistance force for particles in the annulus to move downward. Thus, the presence of fluidizing gas promotes the particle circulation, resulting in the improvement of convection mixing. Furthermore, higher fluidizing gas velocity generates faster circulation rate, which improves the mixing rate, as shown in Fig. 16. However, the increase of fluidizing gas velocity also leads to the decrease in the velocity gradients in the annulus, which weakens the shear mixing. In the case of increasing fluidizing gas velocity, therefore, the improvement of mixing is attributed to the improvement of convective mixing.

4.6.

Dead zone

The dead zone is usually a big problem in particle mixing. In some studies carried out in the literature (San Jose et al., 1996), the influence of the geometric factors of the contactor, of the particle diameter and of the air velocity on the dead zone in spouted bed have been analysed. For current spout-fluid bed, dead zones are formed on the vessel base. Additional simulations were performed to study the effect of gas velocity on the dead zone. With the aim of quantifying the influence of gas velocity on the amplitude of the dead zone, the average angle ˛ between the surface of the dead zone-moving zone interface and the plane of the bed bottom has been adopted as characteristic magnitude, which is indicated in Fig. 10. Table 3 shows the simulation results of dead zone angle for various operating conditions. It is clear from the table that ˛ decreases evidently with the increase of spouting gas velocity. The effect of fluidizing gas is similar to one of spouting gas, but it is more remarkable at high fluidizing gas velocity. In this condition, a more vigorous movement is generated at the base of the bed, which causes decreases in the volume and in the angle of the dead zone. During the experiment, the dead zone was seen to occupy a considerable fraction of the bed volume, especially at low gas velocity. When increasing gas velocity, the rate of particle circulation is increased significantly. Accordingly, the force acting on the particles in the dead zone generated by the moving annulus is enhanced, which causes more particles to join into the circulation roll. Meanwhile, the pressure drop between spout and annulus is increased and then more particles, including ones in the dead zone, are entranced into the spout. So to lessen the proportion of inactive particles, one would likely use a conical base or aeration of the annulus in largescale spouted bed applications.

5.

Conclusions

The DEM simulation has extensively used in this work to investigate the particle mixing behavior in spout-fluid bed.

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Validation with experiments is carried out first, by comparing the simulated flow patterns and bed pressure drop with corresponding experimental data at various gas velocities. Then, the particle velocities in different flow regions are modeled in order to reveal the mixing mechanisms. In addition, the effects of mixing time and gas velocity on the mixing are discussed based on the elucidated mixing mechanisms, respectively. The following conclusions can be drawn based on the simulations: (1) The spouting gas is the driving force for the formation of particle circulation roll, which leads to the mixing. The particle circulation roll indicates the particle streamlines (trajectories). (2) Increase in spouting gas velocity leads to the increase in axial particle velocity and velocity gradient. While increasing fluidizing gas velocity results in an increase of axial particle velocity and but a decrease of velocity gradient, especially in the annulus. (3) The convective mixing caused by the motion of circulation roll, shear mixing induced by the relative move of circulation rolls and diffusive mixing generated by random walk of particles among circulation rolls are three main mixing mechanisms in spout-fluid bed. (4) The dependence of mixing index in different regions on the time can be expressed by four characteristic parameters a, b, c and d. The value of a shows the mixing index at the dynamical equilibrium state. And c indicates the maximum intensity of mixing. While b and d show the negligible mixing period and the mixing rate, respectively. (5) In the case of increasing spouting gas velocity, the well mixing is achieved as a result of the combined effect of convective and shear mixing. When increasing the fluidizing gas velocity, on the contrary, it promotes convective mixing but weakens shear mixing, which also results in the significant improvement of mixing. (6) The dead zones are formed on the vessel base. The inclination angle, charactering the amplitude of the dead zone, decreases with the increase of gas velocity. It is more pronounced at high fluidizing gas velocity. Nomenclature c¯ mean concentration of tracer in the N samples concentration of tracer particles in the sampling cell ci i CD drag coefficient Magnus lift coefficient CLM CLS Staffman lift coefficient particle diameter, m dp e restitution coefficient fC contact force, N drag force, N fD Saffman lift force, N fLS Magnus lift force, N fLM fcnij normal contact force, N fctij tangential contact force, N Ip particle moment of inertia, kg m2 kn spring coefficient in normal direction, N/m kt spring coefficient in tangential direction, N/m mp particle mass, kg M Lacey mixing index Mp particle torque, N m n number of particle contained in a sampling cell np particles number per unit volume

770

Rep Re␻ u¯ v¯ vf v∗f vmf vms vs v∗s vp

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particle Reynolds number particle rotation Reynolds number mean gas velocity, m/s mean particle velocity, m/s fluidizing gas velocity, m/s dimensionless fluidizing gas velocity minimum fluidizing gas velocity, m/s minimum spouting gas velocity, m/s spouting gas velocity, m/s dimensionless spouting gas velocity particle translational velocity, m/s

Greek symbols ınij normal displacements between particle i and particle j, m ıtij tangential displacements between particle i and particle j, m u gas fluctuating velocity, m/s v particle fluctuating velocity, m/s p pressure drop, Pa ε void fraction n damping coefficients in normal direction, kg/s t damping coefficients in tangential direction, kg/s g gas density, m/s gas turbulence stress, Pa  ij  mix mixing time, s ωp particle angular velocity, s−1 2 standard deviation

Acknowledgements Financial support from the National Natural Science Foundation of China (50676021), the National High Technology Research and Development Program of China (2009AA05Z312), the Major State Basic Research Development Program of China (2010CB732206) and the Foundation of Outstanding Doctoral Dissertation of Southeast University are sincerely acknowledged. The authors also expressed sincere gratitude to the honorific professors, Prof. M. Horio, B. Leckner and E.J. Anthony for constructive advice during their visiting periods in our laboratory, and Prof. B. Formisani for kindly presenting us some of their valuable papers.

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