Numerical Simulation of the GasâSolid Flow in Fluidized-Bed

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Ind. Eng. Chem. Res. 2010, 49, 4070–4079

Numerical Simulation of the Gas-Solid Flow in Fluidized-Bed Polymerization Reactors De-Pan Shi, Zheng-Hong Luo,* and An-Yi Guo Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen UniVersity, Xiamen 361005, China

A three-dimensional computational fluid dynamics (CFD) model, using an Eulerian-Eulerian two-fluid model which incorporates the kinetic theory of granular flow, was developed to describe the gas-solid two-phase flow in fluidized-bed polymerization reactors. Corresponding simulations were carried out in a commercial CFD code Fluent. The entire flow field in the reactors was calculated by the model. The predicted pressure drop data were in agreement with the classical calculated data. In addition, the model was used to describe the solid holdup distributions, the bubble behaviors, and the solid velocity vectors in the free and agitated fluidized-bed polymerization reactors, respectively. The effects of the addition of an agitator on the gas-solid flow behaviors were preliminarily investigated via the model. The simulation results showed that the addition of an agitator can strengthen the fluidization efficiency and reduce the operation stability of the bed. However, the simulation results also showed that the total fluidization quality of the free fluidized bed was higher than that of the agitated fluidized bed at a superficial gas velocity of 0.5 m · s-1. 1. Introduction Polyolefins can be produced in various types of reactors, such as autoclave, continuous stirred tank, tubular loop, or fluidized bed (FBR). The last one is certainly the most important because of its simple construction and excellent heat- and mass-transfer characteristics.1 For instance, various technologies, including Hypol, Innovene, Unipol, Spheripol, etc., are designed to produce polypropylene. Among them, there are different reactor arrangements in essence.2-4 FBR is one of their central reactors, which is generally used to produce high-impact polypropylene.4 In the fluidized-bed olefin polymerization reactor, small catalyst and/or polymer particles react with monomers to form polymer particles in the gas phase, and the polymer particles are produced as a solid suspension in the gas stream.2-5 Accordingly, the reacting system is considered to be a mixture of gas and solid phases, namely, a gas-solid two-phase system. For efficient operation and to accomplish the desired results, it is imperative that a good fluidization quality is achieved to ensure good gas-solid contact, uniformity of temperature, and minimum gas bypassing. For these reasons, computational fluid dynamics (CFD) is becoming more and more an engineering tool to predict flows in various types of apparatuses on the industrial scale.6-8 Furthermore, CFD is an emerging technique and holds great potential in providing detailed information on complex fluid dynamics.9-11 In general, two different categories of CFD models are used, namely, the Lagrangian and Eulerian models.6-8 The Lagrangian model solves equations of motion for each particle, taking into account particle-particle collisions and the forces acting on the particle, whereas the Eulerian model considers full interpenetrating continua subject to continuity and momentum equations. Considerable attention has been devoted in recent years to the application of CFD to gas-solid FBRs.9-17 A comprehensive review has been published on these CFD models and experiments applied to FBRs.18 Most authors have used Eulerian models, including continuity and momentum equations for two interpenetrating continua, one representing the gas and the other * To whom correspondence should be addressed. Tel.: +86-5922187190. Fax: +86-592-2187231. E-mail: [email protected].

the solid. To achieve closure, a granular temperature model has usually been introduced. When a turbulent flow of the gas phase is assumed, a k-ε model is also incorporated. In addition, different authors have adopted different assumptions with respect to such aspects as boundary conditions, interphase momentum transfer (drag) relationships, and parameters in the Eulerian model. As a whole, these models were able to provide good qualitative and reasonable quantitative agreement with limited experimental findings with the help of fitting parameters.19-21 However, previous studies mainly concentrated on prediction of their gas-solid holdup distributions and the effects of the gas velocity on them via a CFD model along with simplification of the flow field as a two-dimensional (2D) field.19-24 Most authors applied CFD to the free FBR or the agitated FBR without comparing the gas-solid flow in the free FBR to that in the agitated FBR.19-26 In addition, less attention has been paid to the bubble behaviors and characteristics in FBRs.27 In practice, many features of the gas-solid FBRs, like excellent solid mixing and heat- and mass-transfer properties, can be correlated to the presence of bubbles, which dominate their behaviors.28 A deeper knowledge of the FBR hydrodynamics and how such hydrodynamics are affected by the addition of the agitator would provide the basis for the development of a fully predictive model.29 Recently, Lu et al.28 used a 2D Eulerian-Eulerian model extended with the kinetic theory of granular flow (KTGF) to simulate the bubble behaviors in a free gas-solid FBR. The simulated values were compared to the values from the Darton bubble-size equation and the Davidson model for isolated bubbles.28 Busciglio et al.27 also described the bubble behaviors in a free gas-solid FBR via a 2D Eulerian CFD model. Their simulated data were validated with experimental data.27 Witt etal.30 usedathree-dimensional(3D)multiphaseEulerian-Eulerian technique to predict the transient bubble formation in a free FBR. Unfortunately, these models were used to describe the bubble behaviors in the free FBRs. To the best of our knowledge, thus far, there were not any open reports regarding the application of CFD to the fluidized-bed olefin polymerization reactor investigating their bubble behaviors.

10.1021/ie901424g  2010 American Chemical Society Published on Web 04/05/2010

Ind. Eng. Chem. Res., Vol. 49, No. 9, 2010

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Figure 1. FBR configurations: (a) free FBR; (b) agitated FBR; (c) stirrer.

In this work, we develop a 3D CFD model based on the Eulerian-Eulerian approach to describe the gas-solid twophase flow in the fluidized-bed propylene polymerization reactor. The entire flow field in the FBR is calculated by the model. Furthermore, the model is used to describe the bubble behaviors in the free and agitated fluidized-bed polymerization reactors, respectively. The effect of the agitator on the gas-solid flow behaviors is preliminarily investigated via the model.

∂ (R F ) + ∇(RsFsb V s) ) 0 ∂t s s

(2)

The momentum balance equations for the gas and solid phases can be written as ∂ (R F b V ) + ∇(RgFgb V gb V g) ) -Rg∇p + ∇τg + Kgl(V bs ∂t g g g b V g) + RgFgg (3)

2. 3D Model for FBRs Spheripol technology is one of the most widespread commercial methods to be used to produce polypropylene. Commonly, its key part constitutes of two liquid-phase loop reactors and a gas-phase FBR. In this work, a pilot-plant-scale polypropylene-agitated FBR of Spheripol technology in a Chinese chemical plant shown in Figure 1 was selected as our object. The agitated FBR selected consists of a vertical 0.5-m-i.d. cylinder of height 1.5 m, and there is a stirrer in the FBR. In order to investigate the effect of the stirrer on the flow behaviors, a free FBR with the same size and shape but no stirrer, compared to the agitated FBR, is also selected in this work. More detailed information regarding the FBR configurations is shown in Figure 1. Furthermore, the flow systems in FBRs are both supposed to be mixtures of gas and solid phases.2-5,31-34 In the present study, to simulate the 3D reactors, a 3D physical model of the reactor system must be available. Hence, the 3D physical models and their meshes were both constructed in Gambit 2.3.16 (Ansys Inc., Columbus, OH) first. 3. CFD Model On the basis of KTGF, a 3D Eulerian-Eulerian two-fluid model is used to describe the gas-solid two-phase flow in the above FBRs. 3.1. Eulerian-Eulerian Two-Fluid Equations. This section describes the modeling equations employed in the present Eulerian-Eulerian two-fluid CFD model. The continuity equations for phase n (n ) g for the gas phase and s for the solid phases) may be written as ∂ (R F ) + ∇(RgFgb V g) ) 0 ∂t g g

(1)

τg ) Rgµg(∇V bg + ∇V bgT)

(4)

∂ (R F b V ) + ∇(RsFsb V sb V s) ) -Rs∇p - ∇ps + ∇τs + ∂t s s s Kls(V bg - b V s) + RsFsg (5) 2 τs ) Rsµs(∇V bs + ∇V bsT) + Rs λs - µs ∇V bsI¯ 3

(

)

(6)

3.2. KTGF. The two-fluid model requires constitutive equations to describe the rheology of the solid phase, i.e., the viscosity and pressure gradient of the solid phase. When the particle motion is dominated by collision interaction, concepts from fluid kinetic theory can be introduced to describe the effective stresses in the solid phase resulting from particle streaming (kinetic contribution and direct collisions) collision contribution.9,7,35-37 Constitutive relations for the solid-phase stress based on the kinetic theory concepts have been derived by Lun et al.36 Moreover, their equations have been accepted widely and are also applied in this work. ps ) RsFsΘs[1 + 2g0Rs(1 + es)]

4 λs ) Rs2Fsdsg0(1 + es) 3

Θs π

(7)

(8)

where g0 )

1 1 - (Rs /Rs,max)/1/3

(9)

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1 Θs ) µs′µs′ 3

(10)

In addition, a transport equation for the granular temperature is also needed and is suggested by Ding and Gidaspow:38 3 ∂ (F R Θ ) + ∇(FsRsF W sΘs) ) (-psI¯ + τs):∇F Ws + 2 ∂t s s s ∇(kΘs∇Θs) - γΘs + φgs (11)

[

]

Table 1. Physical Properties of Gas and Solid Phases ds/m 1 × 10

-3

[

]

µg/pa · s

910.0

21.56

1.081 × 10-5

CD )

15FsdsRs√πΘs 12 16 (41 1 + η2(4η - 3)Rsg0 + 4(41 - 33η) 5 15π 33η)ηRsg0

Fg/kg · m-3

where

where the diffusion coefficient for granular energy, kΘs, is given by Syamlal et al.:39 kΘs )

Fs/kg · m-3

3 24 R Re 1+ RgRes 20 g s

[ (

Res )

at Rg e 0.8, Ksg ) 150

12(1 - es2)g0 ds√π

FsRs2Θs1.5

(14)

φgs ) -3KgsΘs

(15)

In this study, the granular energy was assumed to be at steady state and dissipated locally, and the convection and diffusion were also neglected.7,37,39 Accordingly, eq 11, which is a complete granular temperature transport equation, can be simplified to an algebraic equation. The simplified equation is as follows: 0 ) (-psI¯ + τs):∇F W s - γ Θs

(16)

There are many similar models for the solid-phase dynamic viscosity. The selected model in this work is as follows:36,40,41 µs ) µs,col + µs,kin + µs,fr

Vs - b V g| Fgds | b µg

Rs(1 - Rg)µg

(13)

The collision dissipation of energy, γΘs, is modeled using the correlation by Lun et al.:36 γΘs )

(22)

(23)

(12)

where 1 η ) (1 + es) 2

0.687

) ]

(17)

Rgds

2

+

Vs - b V g| 7 RsFg | b 4 ds (24)

3.4. CFD Modeling Strategy. As discussed earlier, the CFD with the Eulerian-Eulerian approach has been used to study the gas-solid interactions in this work. The RNG k-ε model is used to take into account the turbulence, whereas KTGF has been used to close the momentum balance equation for the solid phase. The above equations are solved by the commercial CFD code Fluent 6.3.26 (Ansys Inc., Columbus, OH) in double precision mode. The phase-coupled SIMPLE algorithm is used to couple the pressure and velocity, and the multiple reference frame (MRF) model is used to simulate the agitated FBR. In addition, as described in section 2, a commercial grid-generation tool, Gambit 2.3.16 (Ansys Inc., Columbus, OH), is used to generate the 3D geometries and their grids. Simple grid sensitivity was carried out, the least cells needed to conserve the mass of the solid phase in the dynamics modeling were studied, and in total 107 000 and 161 000 cells were needed for the free and agitated FBRs, respectively. Furthermore, the simulations were executed in a Pentium 4 CPU running at 2.83 GHz with 4GB of RAM. 4. Simulation Conditions

where µs,col µs,kin )

4 ) RsFsdsg0(1 + es) 5

Θs π

10dsFs√Θsπ 4 1 + (1 + es)Rsg0 96Rs(1 + es)g0 5

[

µs,fr )

(18) 2

]

ps sin θ 2√I2D

(19)

(20)

3.3. Drag Force Model. In this work, the transfer of forces between the gas and solid phases is described according to the empirical drag law based on work by Gidaspow et al.40 Gidaspow’s model combines Wen and Yu’s model42 via the Ergun equation.43 Corresponding equations are shown as follows: Vs - b V g | -2.65 3 RsRgFg | b at Rg > 0.8, Ksg ) CD Rg 4 ds

4.1. Physical Properties of the Gas and Solid Phases. During polymerization in FBRs, the growth rate of the polymer particles is very slow, and their growth in diameter is mainly determined by the residence time of the polymer particles in FBRs. In addition, the simulated flow time in this study is very short (about 10 s) because of the intense computational time. In view of this, the integral of the above gas-solid system with the propylene polymerization in FBRs is simulated. The properties of the typical gas and solid phases at certain polymerization times are listed in Table 1. 4.2. Boundary Conditions and Model Parameters. The minimum fluidization velocity (Umf) and the particle terminal velocity (UT) can be estimated by using Wen and Yu’s equations:42

Umf )

(21)

µg dsFg

{[

33.72 + 0.0434

ds3Fg(Fs - Fg)g µg2

]

0.5

}

- 33.7

(25)


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Table 2. Boundary Conditions and Model Parameters description

value

turbulence model granular viscosity granular bulk viscosity frictional viscosity angle of internal friction granular temperature drag law coefficient of restitution for particle-particle collisions inlet boundary condition outlet boundary condition wall boundary condition initial bed height initial volume fraction of the solid phase operating pressure inlet gas velocity rotating speed oulet pressure maximum iterations convergence criteria time step

k-ε (RNG, dispersed) Gidaspow et al.40 Lun et al.36 schaeffer 30° algebraic Gidaspow et al.40 0.9 velocity inlet pressure outlet no slip for air, specularity coefficient 0 for the solid phase16,17 0.39 m 0.63 1.40 × 106 Pa 0.5 m · s-1 10 rpm (available in the stirred model) 1.013 25 × 105 Pa 30 1 × 10-3 1 × 10-3 s

at Re < 0.4, UT )

at 0.4 < Re < 500, UT )

(Fs - Fg)ds2g 18µg

[

at 500 < Re < 200000, UT )

4(Fs - Fg)2g 225Fgµg

[

(26)

]

0.5

(27)

ds

3.1(Fs - Fg)gds Fg

]

0.5

(28)

where Re )

dsFgUT µg

(29)

According to the above equations and corresponding data shown in Table 1, the values obtained for Umf and UT are 0.1126 and 1.118 m · s-1, respectively. In addition, the superficial gas velocity must be operated between the values of Umf and UT. According to the process description, Ug is always set to 3-5 times Umf. Thus, the value of 0.5 m · s-1 is used in this study. As described in section 3.4, the CFD model was solved in Fluent. The detailed settings in the software are list in Table 2. 5. Results and Discussion The hydrodynamic characteristics of the entire flow field in the free and agitated FBRs, such as the pressure drop, solid holdup distribution, bubble behavior, and solid velocity vector profile, are investigated via the above model, respectively. The physical properties and model parameters are listed in Tables 1 and 2. In addition, as described in section 1, many researchers6-43 studied the gas-solid two-phase flow; a set of reference values of these parameters can be selected. In the present study, two important parameters, including the restitution coefficient (es) and specularity coefficient (φ), were investigated. We find that a little change of φ would lead to a significant change of the pressure drop in the loop reactor (here no results are given because of the limited space). However, the pressure drop is not sensitive to the changes of the restitution coefficient. Therefore, the default value of 0.9 for the restitution coefficient in Fluent was chosen. Furthermore, our foregone sensitivity analysis of the specularity coefficient shows that, with an increase of φ, the difference between the predicted pressure drop at the corresponding flow velocity and flow time and that

obtained via the classical Newitt model44,45 increases. A good prediction of the pressure drop when φ equals 0 can be obtained. Meanwhile, the value of 0 for φ is also that at the free slip boundary condition for the solid phase wall boundary condition and can be found in typical literature.16,17 Therefore, the value of 0 for the specularity coefficient was chosen and is shown in Table 2. Unless otherwise noted, the parameters used for the following simulation are those in Table 2. 5.1. Pressure Drop. It is well-known that the bed pressure drop in FBRs is an important parameter in the proper scaling up and design of these reactors. The bed pressure drop can always be described by the buoyant weight of the suspension:44,45 ∆Ps ) (Fs - Fg)(1 - ε)gL

(30)

However, in this study, because the gas-phase density is up to 21.56 kg · m-3, it is necessary to consider the effect of the gasphase weight on the pressure drop: ∆Pg ) εFggL

(31)

Corresponding pressure drops calculated by the above equations are depicted in Figures 2 and 3, which show the bed pressure drop profiles as a function of the flow time in two types of FBRs, respectively. As a whole, Figures 2 and 3 prove that the simulated bed pressure drop data are in agreement with the classical calculated data. Also, the slight difference shown in

Figure 2. Pressure drop versus flow time in the free FBR at an initial flow velocity of 0.5 m · s-1.

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Figure 3. Pressure drop versus flow time in the agitated FBR at an initial flow velocity of 0.5 m · s-1.

Figures 2 and 3 may result from neglect of the pressure drop caused by friction and particle collision in the classical calculation. Moreover, according to Figure 2, it is worth noting that four typical regions in the free FBR can be found: the startup stage (τ ) 0 s), slow drop stage (0 s < τ < 4.0 s), vibration stage (4.0 s e τ < 6.4 s), and stable fluidization stage (τ g 6.4 s). The maximum bed pressure drop at the start-up point/ stage is higher than that at the other three stages because the interparticle locking is overcome. Since then, the bed pressure drop decreases slowly because of formation of the gas-phase flow field and the looseness of the solid phase following the flow proceeding in the period of 0-4.0 s. In the period of 4.0-6.4 s, the first air bubble comes into being and develops with time; correspondingly, the bed pressure drop fluctuates greatly in this period. After the vibration stage, the bed pressure drop fluctuates with time around a mean value shown in Figure 2, namely, the stable fluidization stage. In this stage, the fluidization process in the whole free FBR is accomplished. Therefore, the bed pressure drop is also close to a certain steadystate value. Compared with Figure 2, Figure 3 shows a similar curve, which indicates that there are similar change trends of the bed pressure drop in the two types of FBRs. Nevertheless, one still notices that the whole fluctuation range of the bed pressure drop in the free FBR is lower than that in the agitated FBR from Figures 2 and 3. Furthermore, Figures 2 and 3 indicate that the fluctuation in the free FBR is more frequent than that in the agitated FBR. 5.2. Solid Holdup Distribution. Solid holdup is one of the most important parameters in FBRs. If the solid holdup is too high somewhere, the corresponding polymerization rate may be too high there because of the high concentration of catalyst, which leads to a highly exothermic reaction. In addition, the highly exothermic reaction may lead to the appearance of hot spots if the heat of polymerization cannot be efficiently removed. In this section, the solid holdup distributions in the free and agitated FBRs are investigated using the above model, respectively. In addition, we also point out that the lack of consideration of other distributions (i.e., temperature distribution) and the effects of some other operating parameters, such as temperature, pressure, etc., on the flow hydrodynamics in FBRs are limitations due to the limited space in this paper. Figures 4 and 5 show the profiles of the average solid holdup as a function of the height of the horizontal plane from the bottom of FBRs in the free and agitated FBRs at the stable fluidization stage, respectively. Corresponding solid holdup distribution data at 10 s for the two reactors are shown in Figures 6 and 7.

Figure 4. Average solid volume holdup versus the height of the horizontal plane from the bottom of the free FBR at an initial flow velocity of 0.5 m · s-1 and a flow time of 10 s.

Figure 5. Average solid volume holdup versus the height of the horizontal plane from the bottom of the agitated FBR at an initial flow velocity of 0.5 m · s-1 and a flow time of 10 s.

Figure 4 shows that the average solid holdup changes very little with a change of the height of the horizontal plane from the bottom of the free FBR in the range of 0-0.6 m. This means that the granule distribution in the main body of the free FBR is homogeneous. Namely, the fluidization quality is perfect. Hereafter, the average solid holdup descends to 0 with an increase of the height of the horizontal plane from the bottom of the free FBR. Accordingly, one knows that the actual height of the free FBR is about 0.78 m according to Figure 4. Furthermore, Figure 6 proves that the fluidization quality in the free FBR is perfect. In practice, Figure 6 shows that the amount of bubbles is few and the emulsion phase is the main body in the free FBR. Although there are still granule agglomerations, not any granules adhere to the wall in the free FBR. Compared with Figure 4, Figure 5 shows a similar curve, which proves that there are similar change trends of the average solid holdup in the free and agitated FBRs. Nevertheless, one also obtains that the actual height of the agitated FBR is about in the range of 0.7-0.85 m according to Figure 5. Figure 7 shows that there are many bubbles generated in the agitated FBR after fluidization for a long time (10 s). It is helpful to have heat transfer between the gas and solid phases and leads to an increase of the granule entrainment at the breakup of bubble. However, the large volume and excessive amount of bubbles generated are harmful to the stability of FBR. From the solid phase shown in Figure 7, we can observe more obvious granule agglomeration phenomena than that in Figure 6. The


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Figure 6. Solid volume holdup distribution in the free FBR at an initial flow velocity of 0.5 m · s-1 and a flow time of 10 s.

Figure 7. Solid volume holdup distribution in the free FBR at an initial flow velocity of 0.5 m · s-1 and a flow time of 10 s.

Figure 8. Solid volume holdup distribution of the vertical plane across the vertical axis in the free FBR at an initial flow velocity of 0.5 m · s-1 versus flow time.

corresponding solid holdup distribution in the agitated FBR is inhomogeneous. 5.3. Bubble Behaviors. As described in section 1, many characteristic features of gas-solid FBRs are related to the presence of bubbles and dominated by their behaviors. Here, the bubble behaviors in two types of FBRs are simulated. On the basis of Figures 2 and 3, one knows that both of the flow fields in the free and agitated FBRs are in the stable fluidization period when τ g 6.4 s. Here, we provide the simulated results in 0-10 s of flow time, and the results are presented in Figures

8 and 9. Furthermore, Figures 8 and 9 show the bubble formation processes with representation of the solid volume fraction distribution of a vertical plane across the vertical axis as a function of the flow time in the free and agitated FBRs, respectively. Typical 3D diagrams of bubble evolution in the free and agitated FBRs are shown in Figures 10 and 11, respectively. According to Figure 8, one can obtain the processes of bubble formation, development, and breakup following the flow proceeding in the free FBR. At 3 s in Figure 8, only the granules

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Figure 9. Solid volume holdup distribution of the vertical plane across the vertical axis in the agitated FBR at an initial flow velocity of 0.5 m · s-1 versus flow time.

Figure 10. Visual representations of bubble formation in the free FBR at an initial flow velocity of 0.5 m · s-1.

Figure 11. Visual representations of bubble formation in the agitated FBR at an initial flow velocity of 0.5 m · s-1.

in the bottom of the free FBR become flexible to form an emulsion phase due to the initial quiescence state of the granule phase and the interparticle locking; accordingly, the bed height rises a little. The bubbles begin to form with further emulsification of the granules at 4 s. Simultaneously, as shown at 4.0, 4.2, and 4.4 s in Figure 8, the bubbles formed appear to deform because of the interaction between granules and also develop upward before arriving at the free space in the free FBR. Accordingly, the “film” between the gas and emulsion phases becomes thinner with development of the bubbles and breaks up ultimately at about 4.6 s in Figure 8. This leads to the rise

of these granules in the fractured bubbles, and they also drop back to the wall because of gravitation. In addition, at 4.6 s when the “film” and first bubble break up, it represents that FBR is basically emulsified and the inner resistance decreases. Correspondingly, the formations of new bubbles are simple, which leads to an increase of the bubble velocity. Therefore, the total fluidized velocity in FBR is very fast. It leads to the formation of the stable flow field in the FBR after a short time. According to Figure 2, we think that the stable field will form at 6.4 s. Namely, since 4.6 s, the solid volume fraction distribution data of the vertical plane across the vertical axis


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Figure 12. Solid velocity vector profiles in the free FBR at an initial flow velocity of 0.5 m · s-1 and a flow time of 10 s.

are similar to each other and their profiles are shown at 4.6, 4.8, 10 s in Figure 8. Figure 10 shows the typical 3D diagrams of bubble evolution from formation, deformation, to breakup and gives 3D visualization results. According to Figure 10, one knows that the bubble-like cirque with narrow top and wide bottom can be obtained at 4.0 s and its shape continues to change because of granular actions following the flow proceeding. Simultaneously, it is split into many small bubbles along the axial aspect. For instance, both the volume and height of the bubbles at 4.2 s are larger than those at 4.0 s. One also observes many small bubbles along the axial aspect of the bubble at 4.2 s in Figure 10. Furthermore, according to Figure 10, one can find that the bubble continues to rise along its axial aspect and it has been split into eight small bubbles with similar shape and different volume at 4.4 s. In addition, the obtained small bubbles can also rise. Compared to the bubble behaviors in the free FBR shown in Figures 8 and 10, the bubble behaviors shown in Figures 9 and 11 are similar. However, because of the addition of a stirrer, the first bubble obtained in the agitated FBR is unstable and there are many small bubbles to form, along with the formation of the first bubble. In practice, the stirrer breaks up the early bubbles, which leads to an increase of the charge capacity of the granules in bubbles. Furthermore, some air whorls come into being by the edge of the stirrer and can also leave the stirrer to form small bubbles. Some of the small bubbles can be incorporated to form big bubbles. Therefore, there are still many small bubbles in the agitated FBR at 10 s in Figure 9. Although the small bubbles can strengthen the fluidization efficiency, they reduce the operation stability of the bed because of the addition of the agitator. 5.4. Solid Velocity Vector. The rising motions of the rotating torus in the free and agitated FBRs are also simulated via the above model. The simulated velocity vectors for the solid phase in a vertical plane across the vertical axis in the two FBRs are shown in Figures 12 and 13, respectively. Figure 12 shows that there is an obvious circular upflow in the vertical plane due to the bubble motion. There are some small dimensional circular regions (small circulations) in the bottom of the free FBR. For the dimension of the whole FBR, the solid phase is lifted up from its middle position and comes back along with the breakup of the bubbles due to gravitation.

The fallen granules can flow down along the wall. Therefore, the above results can also lead to the formation of big circular upflows, namely, big circulations. The combined action of the small and big circulations leads to a good mixing result in the free FBR. However, a vortex appears between the small and big circulations, namely, in the middle of the free FBR. It is harmful to the matter and heat transfers between the quiescent regions and the bottom of the FBR. Figure 13 shows that there is no vortex to be formed in the middle of the agitated FBR because of the addition of the stirrer. One knows that the motion of the granules in the agitated FBR is mainly influenced by two actions. The granules can be lifted up because of the action of the gas phase along the vertical axis and can also be rotated in the agitated FBR because of the action of the stirrer. Besides, other actions including granule gravitation can also influence their motion. Accordingly, as shown in Figure 13, most granules in the agitated FBR lift up along the right side of the vertical plane across the vertical axis and collide with the fallen granules to form a vortex. Some granules flow back to the bottom of the agitated FBR. In addition, some granules inside the fractured bubbles also fall back to the bottom of the agitated FBR. 6. Conclusions In this study, a 3D CFD model was developed to describe the gas-solid two-phase flow in fluidized-bed polymerization reactors.ThemodelincorporatedKTGFwiththeEulerian-Eulerian approach. The pressure drop data calculated according to the classical equation were employed to verify the model. The predicted pressure drop data were found to agree well with the classical calculated data. Furthermore, the hydrodynamic characteristics of the entire flow field in the free and agitated FBRs, such as the pressure drop, solid holdup distribution, bubble behavior, and solid velocity vector profile, were investigated via the above model, respectively. Particular attention was paid to the effect of the addition of the agitator on the gas-solid flow behavior. The simulated results show that both of the flow fields in the free and agitated FBRs as a function of the flow time can be divided into four periods. Namely, the start-up period (τ ) 0 s), slow drop period (0 s < τ < 4.0 s), vibration period (4.0 s e τ

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Figure 13. Solid velocity vector profiles in the agitated FBR at an initial flow velocity of 0.5 m · s-1 and a flow time of 10 s.

< 6.4 s), and steady-state fluidization period (τ g 6.4 s). Furthermore, the typical profiles of the average solid hold-up and solid hold-up distribution in the free and agitated FBRs at the steady-state fluidization period are also obtained. In addition, the simulated results show that the addition of the agitator can strengthen the fluidization efficiency and yet reduce the operation stability of the bed. The simulation results also showed that the fluidization quality of the free fluidized bed is higher than that of the agitated fluidized bed at a superficial gas velocity of 0.5 m · s-1. Because we know that the main function of the agitator in the agitated FBR is to prevent the granule from adhering to the wall of FBR, the modeling results show that granules will not adhere to the wall in the free FBR, so it is not necessary to add the agitator to FBR; that is why the newly developed fluidizedbed polymerization reactors are always free FBR. Further studies on the 3D CFD model for the gas-solid two-phase flow in FBR are in progress in our group. Acknowledgment The authors thank the National Natural Science Foundation of China (Grant 20406016) and China National Petroleum Corp. for supporting this work. We also thank Dr. Z. Yao (Department of Chemical Engineering and Biochemical Engineering, Zhejiang University) for his valuable discussion in this work. The authors also thank the anonymous reviewers for comments on this manuscript. The simulation work is implemented by advanced software tools (Fluent 6.3.26 and Gambit 2.3.16) provided by the China National Petroleum Corp. and its subsidiary company. Appendix Nomenclature Cd ) drag coefficient

ds ) particle diameter, m D ) pipe diameter, m es ) particle-particle restitution coefficient ew ) particle-wall restitution coefficient g ) gravitational acceleration, m · s-2 jjI ) identity matrix I2D ) second invariant of the deviatoric stress tensor Kgs ) interphase exchange coefficient, kg · m2 · s-1 p ) pressure, Pa ps ) particulate phase pressure, Pa Res ) particles Reynolds number t ) flow time, s Umf ) minimum fluidization velocity Ut ) particle terminal velocity Vg ) gas velocity, m · s-1 Vs ) solid velocity, m · s-1 Vs,w ) solid velocity at the wall, m · s-1 Rg ) volume fraction of the gas phase Rs ) volume fraction of the solid phase Rs,m ) maximum volume fraction of the solid phase ε ) voidage φ ) specularity factor µg ) viscosity of the gas phase, Pa · s µs ) solids shear viscosity, Pa · s µs,col ) solids collisional viscosity, Pa · s µs,kin ) solids kinetic viscosity, Pa · s µs,fr ) solids frictional viscosity, Pa · s θ ) angle of internal friction, deg Θs ) granular temperature, m2 · s-2 γΘs ) collisional dissipation of energy, m2 · s-2 jjτg ) shear stress of the gas phase, N · m-2 jjτs ) shear stress of the solid phase, N · m-2 λs ) solid bulk viscosity, Pa · s Fg ) gas density, kg · m-3 Fs ) solid density, kg · m-3


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ReceiVed for reView September 11, 2009 ReVised manuscript receiVed March 13, 2010 Accepted March 25, 2010 IE901424G

Numerical Simulation of the GasâSolid Flow in Fluidized-Bed

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