Numerical Simulation of Fluid Dynamics of a Riser - American

Ind. Eng. Chem. Res. 2010, 49, 3585–3596

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Numerical Simulation of Fluid Dynamics of a Riser: Influence of Particle Rotation Zhenhua Hao, Shuai Wang, Huilin Lu,* Guodong Liu, Yurong He, Pengfei Xu, and Jiaxing Wang School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China

Flow behavior of gas and particles in a 2-D riser is simulated using a gas-solid two-fluid model with consideration of the effect of particle rotation. The particle-particle interactions are simulated from the kinetic theory for flow of dense, slightly inelastic, slightly rough spheres proposed by Lun (Lun, C. K. K. Kinetic theory for granular flow of dense, slightly inelastic, slightly rough sphere. J. Fluid Mech. 1991, 233, 539-559). Inelastic binary collisions of particles with normal and tangential restitution coefficients are considered. The modified expressions for energy dissipation and viscosities of particles are proposed as a function of tangential and normal restitution coefficients of particles. The present model is evaluated by the measured solids concentration and velocities of Miller and Gidaspow (Miller, A.; Gidaspow. D. Dense, vertical gas-solid flow in a pipe. AIChE J. 1992, 38, 1801-1815), and the measured solids concentration, mass flux, and pressure drop of Knowlton et al. (Knowlton, T.; Geldart, D.; Matsen, J.; King, D. Comparison of CFB hydrodynamic models. PSRI challenge problem. Presented at the Eighth International Fluidization Conference, Tours, France, May 1995) in the risers. Computed profiles of particles are in agreement with the experimental measurements. The simulated energy dissipation, granular temperature, viscosity, and thermal conductivity of particles exhibit nonmonotonic tangential restitution coefficient dependences due to the energy losses resulting from particle collisions. 1. Introduction Numerical simulations have been employed as a useful tool to study the flow behavior of gas and solid phases in risers. In the Eulerian-Eulerian two-phase continuum models, both the gas phase and collections of particles are modeled using continuous medium mechanics and therefore constructing integral balances of continuity, momentum, and energy for both phases, with appropriate boundary conditions and jump conditions for phase interfaces. The application of kinetic theory of granular flow (KTGF) to model the motion of a dense collection of nearly elastic spherical particles is based on an analogy to the kinetic theory of dense gases.1 A granular temperature, θ, is defined to represent the specific kinetic energy of the velocity fluctuations or the translational fluctuation energy resulting from the particle velocity fluctuations. In granular flow, particle velocity fluctuations about the mean are assumed to result in collisions between particles being swept along together by the mean flow. The particle granular temperature equation can be expressed in terms of production of fluctuations by shear, dissipation by kinetic and collisional heat flow, dissipation due to inelastic collisions, production due to fluid turbulence or due to collisions with molecules, and dissipation due to interaction with the fluid.1 Numerous studies have shown the capability of the kinetic theory approach for modeling gas-solid fluidized beds.2-14 In the original kinetic theory of granular flow, particles are assumed to be slightly inelastic and smooth spheres, and fluctuation energy dissipation only comes from binary inelastic collisions. In a realistic situation, the particle surface cannot be perfectly smooth. During a collision of rough particles, the fluctuation energy is dissipated from inelasticity and friction of particles. The frictional particle collision also results in the particle rotation which gives an additional loss of the energy.15-17 The change in the normal velocity is determined by the normal restitution coefficient, e, which can range from 0.0 to 1.0. The * To whom correspondence should be addressed. Tel.: +0451 8641 2258. Fax: +0451 8622 1048. E-mail: [email protected].

frictional properties of the surface are characterized by the tangential restitution coefficient, β, which can range from -1.0 to +1.0. The kinetic theory for flow of identical, slightly frictional, inelastic spheres was proposed in refs 16 and 18. Sun and Battaglia19 implemented the kinetic theory of Jenkins and Zhang18 into the MFIX CFD code20 to compute segregation in a gas-solid fluidized bed. They found that the model captures the bubble dynamics and time-averaged bed behavior. Shuyan et al.21 simulated flow behavior of particles in a circulating fluidized bed (CFB) using a two-fluid model incorporating the kinetic theory for the particle rotation model proposed by Jenkins and Zhang.18 The normal friction stress model proposed by Johnson et al.22 and a modified frictional shear viscosity model proposed by Syamlal et al.20 were used as the particle kinetic-frictional stress model in a gas-solid fluidized bed. Studies by Songprawat and Gidaspow23 showed that the absence of rotation and subsequent energy losses are a deficiency of the two-fluid model in contrast to discrete particle models which provide a closer resemblance to experimental results. Therefore, it should be of interest to incorporate particle rotation into the hydrodynamic model and analyze the flow behavior of particles in gas fluidized beds. In the present work, a two-fluid model with the kinetic theory for the flow of dense, slightly inelastic, slightly rough spheres proposed by Lun16 is used to study the flow behavior of gas particles in a two-dimensional (2-D) riser. Computed results are compared with experimental concentrations and velocities of particles measured by Miller and Gidaspow24 and Knowlton et al.25 in the risers. 2. Conservation Equations for Gas and Solid Flows 2.1. Gas-Solid Two-Fluid Model. In principle, the kinetic energies associated with fluctuations in both translational velocity and spin were considered in the flow of particles with rotations. Two granular temperatures are assumed in the kinetic theory with binary frictional collisions.15,16 One is the translational granular temperature θt ) 〈C2〉/3, which measures the

10.1021/ie9019243  2010 American Chemical Society Published on Web 03/17/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 8, 2010

energy associated with the translational velocity fluctuations, where C is the particle translational fluctuation velocity. The other is the rotational granular temperature θr ) (1/3mp)Ipω2, which measures the energy associated with the angular velocity fluctuations, where Ip is the particle moment of inertia, ω is the angular velocity fluctuation, and mp is the particle mass. For simplification, the following hypotheses are considered: (1) Both phases are assumed to be isothermal at 300 K, and no interface mass transfer is assumed. (2) The solid phase is characterized by a mean particle diameter and density. Both phases are continuous, assuming a single gas phase and a single solid phase. A detailed gas-solid two-phase model is presented by Lun.16 In this model, the frictional collision is described by a normal restitution coefficient and a tangential restitution coefficient. By setting the rate of dissipation of rotational granular energy equal to zero, the conservation equation for rotational granular energy is approximately satisfied, and the rotational granular temperature is determined in terms of the translational granular temperature η2 θr ) θt 1 - η2K-1

(1)

where η2 ) (1 + β)K/[2(1 + K)] and K ) 4Ip/(md2). Thus, the kinetic theory for slightly frictional, nearly elastic spheres has the same structure as that for frictionless spheres; i.e., only conservation of mass, translational velocity, and translational granular temperature need to be considered.16 Table 1 lists the conservation equations of gas and particle flows used in the present computations.26 The conservation equations of mass are expressed as eq T1-1. The momentum conservation equation of the gas phase has the form shown in eq T1-2. For the gas phase, the turbulence is incorporated through the large eddy simulation approach. The turbulent viscosity is computed from the subgrid scale (SGS) model. In the present simulations, the original form of the Smagorinsky model (eq T1-6) is used for the computation of gas turbulent viscosity.27 In numerical simulations, we use a value of cg equal to 0.079.28 The filter width is ∆ ) (∆x ∆y)1/2 in a 2-D simulation. The momentum conservation equation of the solid phase (eq T1-3) is similar to the one for the gas phase, but contains the gradient of the particle pressure ps. The transport equation (eq T1-4) is solved to calculate the field of the translational granular temperature inside the computational domain. The particle phase pressure is given by the sum of a collisional part and a kinetic part, as shown in eq T1-8, which contains the radial distribution function of eq T1-11. The particle phase shear viscosity consists of a collisional part and a kinetic part. The kinetic part of shear viscosity proposed by Lun16 is expressed µk ) µdil

{

1 + 1.6[η1(3η1 - 2) + 0.5η2(6η1 - 1 - 2η2) - η22θr /(Kθt)]εsg0 g0[(2 - η1 - η2)(η1 + η2) + η22θr /(6Kθt))]

}

(2)

where η1 ) (1 + e)/2. Using eq 1, eq 2 is expressed by eq T1-9. Both terms consider the contribution of the exchange of kinetic energy between the rotational and the translational modes. The bulk viscosity ξs formulates the resistance of particles to compression and expansion, and is shown in eq T110. Using eq 1, the conductivity of granular energy ks proposed by Lun16 is expressed by eq T1-12. Both terms relate to the contribution of the exchange of kinetic energy by the translational mode and the rotational mode through the tangential

restitution coefficient β. The important influence of particle rotation on dynamics is the additional energy dissipation due to frictional collisions. The energy dissipation includes the contribution of the dissipation rates for translational and rotational granular energy16 γt )

{

}

η22 Fsεs2 48 θr g θ 1/2 [η1(1 - η1) + η2(1 - η2)]θt K d 0 t √π (3)

Using eq 1, the energy dissipation is expressed by eq T1-13. The dissipation due to the inelasticity of the collisions between the particles is taken into account through the restitution coefficient, and the effect of particle surface friction and tangential inelasticity is by means of the tangential restitution coefficient. It is speculated that the original kinetic theory of granular flow, which is limited to slightly inelastic spherical particles and does not allow for particle rotation, underestimates the amount of energy dissipated in the frictional inelastic collisions of particles. The production of granular energy Dgs due to the slip between gas and particles is calculated using the algebraic expression eq T1-14,29 modified to account for the presence of the densest regions with the introduction of g0 at the denominator. In order to couple the momentum transfer between gas and particle phases, the correlations of drag coefficient given by Gidaspow,1 O’Brien and Syamlal,30 and Hill-Koch-Ladd31 are often used in the simulations of fluidized beds. The correlation proposed by Gidaspow1 is used, which is a combination of the works of Ergun32 and Wen and Yu.33 For porosities less than 0.8, the pressure drop due to friction between gas and particles can be described by the Ergun equation. For porosities greater than 0.8, the Wen and Yu equation is used. This transition proposed by Gidaspow1 makes the drag law discontinuous in the solid concentration though it is continuous in the Reynolds number. Physically, the drag force is a continuous function of both the solid concentration and the Reynolds number, and therefore the continuous forms of the drag law may be needed to correctly simulate gas-solid fluidized beds. To avoid discontinuity of these two correlations, a switch function φgs (eq T1-16c) is introduced to give a smooth from the dilute regime to the dense regime.6 The gas-particle intercoefficient is given by eq T1-15. Note that from eq 1 the distribution of energy between the translational and rotational modes, θr/θt, depends only on the tangential restitution coefficient β, and not on the normal restitution coefficient e. A similar expression for θr/θt is proposed by Jenkins and Zhang18 with the assumption of the rate of dissipation of rotational fluctuation equal to zero in a general, unsteady, inhomogeneous flow θr 7µ2Ω/[1 + (3.5µΩ)2] ) θt 3.5πµΩ 2 (3.5µΩ)2 1 - arctan(3.5µΩ) + 2 π 1 + (3.5µΩ)2

(

)

(4) where Ω ) (1 + e)/(1 + β0), µ is the friction coefficient, and β0 is the tangential coefficient of restitution. By introducing an effective coefficient of restitution, only the conservation of mass, the mean translational velocity, and the translational temperature need to be considered. Equation 4 was used in the numerical simulations of a fluidized bed.19 Both eqs 1 and 4 will apply whenever the forcing adds only translational energy. This is the case for both vibration and shear. Thus, a direct application


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Table 1. Equations of Gas-Solid Flow in Fluidized Beds (1) continuity equations of gas and particles (i ) gas, solids): ∂ (ε F ) + ∇·(εiFivi) ) 0 ∂t i i

(T1-1)

∂ (ε F v ) + ∇·(εgFgvgvg) ) -εg∇p + ∇·τg - βgs(vg - vs) + εgFgg ∂t g g g

(T1-2)

∂ (ε F v ) + ∇·(εsFsvsvs) ) -εs∇p - ∇·(psI) + ∇τs + βgs(vg - vs) + εsFsg ∂t s s s

(T1-3)

(2) conservation of momentum of gas and solids:

(3) translational granular temperature equation of solids: 3 ∂ (ε F θ ) + ∇·(εsFsθtVs) ) ∇·(κt∇θt) + (∇psI + τs):∇Vs - γs - Dgs - 3βgsθt 2 ∂t s s t

(T1-4)

2 τg ) µg[∇vg + (∇vg)T] - µg(∇·vg)I 3

(T1-5)

µg ) µg,l + µg,t and µg,t ) Fg(cg∆)2(τg:τg)

(T1-6)

τs ) µs[∇vs + (∇vs)T] + ξs(∇·vs)I

(T1-7)

ps ) εsFsθt + 2g0εs2(1 + e)Fsθt

(T1-8)

[

]

(4) constitutive relations: (a) gas phase viscous stress tensor

(b) gas shear viscosity (SGS model)

(c) stress tensor of solids

(d) solid pressure

(e) shear viscosity of solids µs )

2 3(1 + β)K 2 2(1 + e) + ε F dg 5 2(1 + K) s s 0

[

]

10Fsd√πθt θt 4 1 + (2η1 + 3η2)g0εs + π 96(1 + e)g0 5

[

{

1 + 1.6[η1(3η1 - 2) + 0.5η2(6η1 - 1 - 2η2) - η22K(1 + β)/(K(1 + 2K - β))]εsg0 g0[(2 - η1 - η2)(η1 + η2) + η22K(1 + β)/(6K(1 + 2K - β))] 1+e ; 2

η1 )

]

η2 )

(1 + β)K ; 2(1 + K)

K)

}

(T1-9)

4Ip md2

(f) bulk viscosity of solids ξs )

(g) radial distribution function

4εs2Fsdg0(1 + e) 3

(h) thermal conductivity coefficient of solids

(

βΓ3 150 Γ + 384(1 + e)g0Γ1 2 1 - βK-1 Γ1 )

a1 )

θt π

(T1-10)

[ ( )]

g0 ) 1 -

ks )

a1a6 25a2a5 , 8 24

)√

εs εs,max

1/3 -1

{ [

πFsdθt0.5 1 +

(T1-11)

] }

2Γ4 6 (1 + e) + g0εs 5 3

2 2 Γ2 ) a3a6 + a2a7εsg0, 5 3

2

+ 2εs2Fsdg0[(1 + e) + Γ4]

2 2 Γ3 ) a2a8 + a4a6εsg0, 3 5

Γ4 )

K(1 + β) 1+K

Γ4 7Γ42 (1 + β) 1+e + [82 - 25(1 + e)] - 2[(1 + e) + KΓ4]2 + [82 - 25Γ4] 4 4 4[2(1 + K) - (1 + β)]

a2 )

K 1 + β 2 ( ), 4 1 + K

θt π

(T1-12)

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Table 1. Continued a3 )

(1 + e)Γ24(1 + β) Γ4 3 5 + { (1 + e)2(2e - 1) - 2Γ4[(1 + e) - (1 + e)Γ4 + ] + }ε g 2 2 2 [(2(1 + K) - (1 + β)] s o a4 )

eK(1 + β)2 , (1 + K)2 a6 )

a7 )

a5 )

[

]

3K(1 + β)2 1+β (1 + β)2 K(1 + β) + 2 2(1 + K) [2(1 + K) - (1 + β)] 4(1 + K) 4(1 + K)2

Γ4 7 1+e 2(1 + β) 1+β 1+β 1+ + 12 3K 1+K 2K(1 + K) 2 3(1 + K)

[

]

{ [

[

}

]

eΓ42 Γ4 Γ42 Γ42 K(1 + β) , + 4(1 + e) + K 4K 2K 2(1 + K) - (1 + β) 4K2

(i) translational fluctuation energy dissipation rate γt )

48

√π

[

η1(1 - η1) + η2(1 - η2) -

(j) fluctuating energy exchange from gas and solids Dgs )

(k) interphase momentum exchange

dFs

18µg

3 e(1 + β)2 εsg0 + 2 (1 + K)2

]

η22 K(1 + β) Fsεs2 g θ 3/2 K 1 + 2K - β d 0 t

( )

2 4√πθtg0 d Fs

a8 )

]

2

(T1-13)

|vg - vs | 2

(T1-14)

βgs ) φgsβ| E + (1 - φgs)β| W

β| E ) 150

β| W )

εs2µg εg2d2

+ 1.75

Fgεs |v - vs | εgd g

3CdεgεsFg |vg - vs | -2.65 εg 4d

(T1-15) (T1-16a)

εg e 0.8

εg > 0.8

(T1-16b)

φgs )

arctan[(150)(1.75)(0.2 - εs)] + 0.5 π

(T1-16c)

Cd )

{

(T1-16d)

24 (1 + 0.15Re0.687) Re e 1000 Re 0.44 Re > 1000 Re ) dεgFg |vg - vs |/µg

(T1-16e)

Table 2. Parameters Used for the Miller and Gidaspow Experiments24 and Simulations symbols

significance

experimental values

calculated values

d Fs Gs µg ug e ew H D

particle diameter solid density solid mass flow gas viscosity gas velocity particle-particle restitution coefficient particle-wall restitution coefficient bed height bed diameter

75.0 µm 1654 kg/m3 20.4 kg/m2 s no 2.61 m/s no no 6.58 m 75 mm

75.0 µm 1654 kg/m3 20.4 kg/m2 s 1.789 × 10-5 kg/ms 2.61 m/s 0.96 0.96 6.58 m 75 mm

of eqs 1 and 4 will be limited. The incorporation of particle rotation into the kinetic theory model by adding a conservation equation for angular momentum and a transport equation for the rotational granular temperature will be further investigated. 2.2. Boundary Conditions. At the inlet, all velocities and concentrations of both phases are specified. The pressure is not specified at the inlet because of the incompressible gas phase assumption (relatively low pressure drop system). At the outlet, the pressure is specified (atmospheric). Initially, the velocities of both the gas and particles are set at zero in the riser. At the wall, the gas tangential and normal velocities are set to zero (no-slip condition). The normal velocity of particles is also set at zero. The following boundary equations apply for

the tangential velocity and granular temperature of particles at the wall:34 Vt,w )

θt,w ) -

6µsεs,max

∂Vs,w πFsεsg0√3θt ∂n

√3πFsεsVsg0θt3/2 ksθt ∂θt,w + ew ∂n 6εs,maxew

(5)

(6)

where ew is the restitution coefficient at the wall. A modified K-FIX program is used to simulate gas and particle flow in a riser.6 The original K-FIX program and its


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Figure 2. Instantaneous concentration of particles at gas velocity and mass flux of 2.61 m/s and 20.4 kg/m2 s.

Figure 1. Geometry, initial, and boundary conditions used for simulations.

several alternate versions have been used for fluidization modeling.1 K-FIX is based on a numerical method developed by Harlow and Amsden35 which is an extension of the implicit continuous-fluid Eulerian technique (ICE). The model uses donor cell differencing. The conservation of momentum and energy equations are in mixed implicit form. This means that the momentum equations are fully explicit. The continuity equations excluding mass generation are in implicit form. In this simulation, a constant time step of 1.0 × 10-5 is used. Timeaveraged distributions of flow variables are computed covering a period of 95 s corresponding to 1-2 weeks of computational time on a PC (80 GB hard disk, 128 MB RAM, and 600 MHz CPU). 3. Computational Cases and Results 3.1. Comparison with Miller and Gidaspow Experimental Data. Several experiments were performed by Miller and Gidaspow24 at different points and sections of the riser. The riser is cylindrical with the gas and solids phases leaving through a side port located below the top of the riser. The particle feed rate is controlled by a slide valve. The riser is 75 mm in diameter and 6.58 m long. The average particle diameter and density are 75 µm and 1654 kg/m3, respectively. By measuring granular temperature and particle velocity distributions by means of a CCD camera, Gidaspow and Huilin36 estimated a value of restitution coefficient of fluidized catalytic cracking (FCC) particles in a riser. They found that the value of the particle-wall restitution coefficient, ew, was 0.96. These parameters are the essential inputs to the proposed kinetic theory model. Figure 1 shows the riser section used in the present numerical simulation of gas-solids flow. Uniform bottom-inlet conditions for the 2-D cylindrical riser are assumed. Table 2 lists experimental and modeling parameters. Figure 2 shows the instantaneous concentration of particles at the gas velocity and mass flux of 2.61 m/s and 20.4 kg/m2 s, respectively. The simulation goes through an early stage from a given initial condition, and finally reaches the so-called statistical steady state regime. For practical purposes, this regime is considered reached when all the flow parameters start to oscillate around well-defined means. The behavior of the flow is characterized by a periodic formation of particle clusters at the wall, and the strong nonhomogeneities of particle flow are observed. Such dense portions of solid undergo a vigorous up-

and-down motion, thus favoring a strong particle recirculation all over the riser. Figure 3 shows the instantaneous velocity of the gas phase and solid phase at the gas velocity and mass flux of 2.61 m/s and 20.4 kg/m2 s, respectively. The complex and transient velocity field is evident. A characteristic feature of the flow is the oscillating motion of solid clusters from one wall to the other through the center line of the riser. These results are consistent with the results in Figure 2. The numerical simulations predict the core-annular flow structure, which is made up of a very dilute flow in the center region (core) and a relatively dense phase near the wall region (annulus) in the riser. In order to compare simulation results with Miller and Gidaspow’s data,24 the time-averaged distributions of flow variables have been computed. Figure 4 shows the time-averaged distribution of concentration of particles at two different tangential restitution coefficients. Simulated results by the original kinetic theory of granular flow (KTGF)1 are also shown in Figure 4. The difference between the present model and results from KTGF is obvious. Both simulations overpredicted the concentration in the center regime. The particle concentrations are low in the center region and high near the walls. The simulations clearly illustrate the inherent core-annular pattern of the solids flow. Comparing with the tangential restitution coefficient β of -0.2, the simulations using the tangential restitution coefficient β of 0.2 give a high concentration of particles in the riser. It seems that the present model predicts more accurate time-averaged concentrations than that from KTGF especially in the near-wall region, which is the region with the most particle clusters. Figure 5 shows the profile of axial velocity of particles at two different tangential restitution coefficients. Both the present simulations and experiments by Miller and Gidaspow24 show particles flow upward in the core region of the riser. The solids mainly accumulate and move downward near the walls. Simulations from the original kinetic theory of granular flow (KTGF) are also shown in Figure 5. Both the present model predictions and results from KTGF give a low axial velocity of particles in the center region comparing to experiments24 at the height of 4.18 m of the riser. With the increase of the tangential restitution coefficient, the axial velocity of particles is decreased in the center region and increased near the walls. These indicate that the solid tangential restitution coefficient has a great influence on the simulated velocity of particles. Figure 6 shows the distribution of solid mass flux at two different tangential restitution coefficients. These two distributions clearly illustrate that the solid mainly accumulates and moves downward near the walls, whereas a dilute solid stream flows upward in the core of the riser. The lower region of the

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Figure 3. Instantaneous velocity of gas and particles at gas velocity and mass flux of 2.61 m/s and 20.4 kg/m2 s.

Figure 6. Profiles of solids mass flux at two different heights. Figure 4. Profiles of time-averaged distributions of concentration of particles.

Figure 5. Distributions of axial velocity of particles at two different heights.

riser is denser than the upper dilute region even though the solid mainly accumulates at the walls in both regions. Simulated results by the original kinetic theory of granular flow (KTGF) are also shown in Figure 6. Both simulations are in agreement with experiments in the core region. Near the walls, the differences between the simulations and experiments are obvious in the lower portion of the riser. In the present simulations, a uniform profile of inlet velocity and concentration of particles is assumed. Neri and Gidaspow37 used a feeding port through an inner circular area at the riser base that is actually closer to the real inlet feeding conditions measured.36 The present simulations show the same trend as the experiments both in form and in magnitude, although some discrepancies are

observed in the riser wall. With the increase of the tangential restitution coefficient, the solids mass flux is decreased. However, the difference is small. These indicate the effect of the solid tangential restitution coefficient on the predicted distribution of the solid mass flux in the riser. 3.2. Comparison with Knowlton et al.’s Experimental Data. Gas-particle flow behavior was measured by Knowlton et al.25 in the riser section of a circulating fluidized bed. The height and diameter of the riser were 14.2 m and 20 cm. The solid particles consisted of FCC material 76 µm in diameter and 1712 kg/m3 in density. The geometry of the riser used in the present simulations is similar to the experimental setup used in the challenge problem by Knowlton et al.25 for the case of 489 kg/m2 s solid mass flux inside the riser, which was used in the simulations by Benyahia et al.38 A particle-particle restitution coefficient e of 0.95, and a particle-wall restitution coefficient ew of 0.9 are used38 in this study. Particles are fed from both sides of the riser near minimum fluidization conditions. Table 3 lists parameters used in the present simulations. Figure 7 shows the comparison between the calculated timeaveraged concentrations of particles at two different tangential restitution coefficients with the experimental data taken at 3.9 m height inside the riser. Simulated results from Benyahia et al.38 by means of FLUENT code are also given in Figure 7. This figure shows a dilute region in the center of the riser and a high concentration of particles at the walls. This distribution reflects the establishment of a core-annular regime shown by the experiments and the computational results. However, contrary to the experiments, the computational results show a smaller core width in the core-annular system. The solid concentration at the walls is predicted to be less than the experimental values. With the increase of the tangential restitution coefficient, the

Ind. Eng. Chem. Res., Vol. 49, No. 8, 2010 25

Table 3. Parameters Used for the Knowlton et al. Experiments

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and Simulations

symbols

significance

experimental values

calculated values

d Fs Gs µg ug e ew H D

particle diameter solid density solid mass flow gas viscosity gas velocity particle-particle restitution coefficient particle-wall restitution coefficient bed height bed diameter

76.0 µm 1712 kg/m3 489 kg/m2 s no 5.2 m/s no no 14.2 m 0.2 m

76.0 µm 1712 kg/m3 489 kg/m2 s 1.789 × 10-5 kg/ms 5.2 m/s 0.95 0.9 14.2 m 0.2 m

concentration of particles is decreased in the center, and increased near the walls. These indicate that the solid tangential restitution coefficient influences the simulated distribution of concentration of particles in the riser. Figure 8 shows the comparison between the calculated timeaveraged solid mass flux distributions at two different tangential restitution coefficients with the experimental data taken at a height of 3.9 m. The solid flux is at its maximum value at the center of the riser, although the solid concentration is at its lowest value. This is due to the very high solid and gas axial velocities at the center of the riser. Near the wall region of the riser, there is a downflow of solid mainly in the form of clusters. The solid downflow is due to the weight of the solid in the annular region that exceeds the axial pressure drop. Simulations give the same trends as the solid tangential restitution coefficient changes from -0.2 to +0.2. The predicted solids mass fluxes by the present model are lower near the walls and higher in the center region than experimental data. Figure 9 shows the comparison between the calculated timeaveraged axial pressure drops at two different tangential restitution coefficients with experimental data. The high pressure drop at the bottom of the riser is due to the effect of solid feeding

in that region. The pressure drop then decreases along the height of the riser due to the decrease in the solid concentration. The predicted pressure drops by present model are higher than that by Benyahia et al.38 using the commercial code FLUENT. With the increase of the solid tangential restitution coefficient, the pressure drop is decreased in the bottom, and increased at the upper regime of the riser. The predicted pressure drop is in reasonable agreement with the experimental data. 3.3. Effect of Solid Tangential Restitution Coefficient. The effects of particle rotation by varied tangential restitution coefficients on the hydrodynamics of particles in a riser are examined. The restitution coefficient is a function of many microscopic particle properties, including the particle impact angle of interaction, the velocity distributions of the particles, and material properties. For simplification, the normal restitution coefficient e and the tangential restitution coefficient β are assumed to be constant in the present simulations. The riser diameter and height are 75 mm and 6.0 m, respectively. The average particle diameter and density are 75 µm and 1500 kg/ m3, respectively. Figure 10 shows profiles of the concentration of particles as a function of tangential restitution coefficients. Predicted concentrations of particles are low in the center region

Figure 7. Comparison between calculated and measured time-averaged concentrations.

Figure 9. Calculated and measured time-averaged axial pressure drops.

Figure 8. Calculated and measured time-averaged solids mass fluxes.

Figure 10. Profiles of concentration as a function of tangential restitution coefficients.

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Figure 11. Axial velocity of particles as a function of tangential restitution coefficients.

and higher near the walls. The difference between the simulated concentrations using four different tangential restitution coefficients is obvious near the walls. The simulated concentration of particles increases as the tangential restitution coefficient is increased from -0.6 to +0.2, and then decreases from +0.2 to +0.6. Using eq 4 proposed by Jenkins and Zhang,18 the concentration of particles is also predicted by a gas-solid twofluid model which was used in simulations of fluidized beds.19 An effective coefficient of restitution is introduced to incorporate the additional dissipation due to frictional interactions in the rate of dissipation of translational fluctuation energy. Comparing to simulations based on eq 1 proposed by Lun,16 the predicted concentration based on eq 4 proposed by Jenkins and Zhang18 is low near the walls and high in the center region. The trends, however, are the same. Figure 11 shows the axial velocity of particles as a function of tangential restitution coefficients. All simulations show that the axial velocity of particles is positive in the center and negative near the walls. This indicates that particles flow up in the center region and down near the walls. As the tangential restitution coefficient increases from -0.6 to +0.6, the axial velocity of particles is decreased in the center region and increased near the walls. The large tangential restitution coefficient gives a low and positive axial velocity in the center region, and a high and negative axial velocity near the walls. The primary reason is that more energy dissipates due to particle rotation and particle velocities decrease after impact, leading to a more closely spaced particle ensemble and thus higher concentration of particles. The axial velocity of particles predicted by eq 4 proposed by Jenkins and Zhang18 is shown in Figure 11. Comparing to simulations predicted by eq 1 proposed by Lun,16 the axial velocity of particles predicted by eq 4 is low near the walls. The trends, however, are the same. These show that rotation can significantly alter the computed distribution of particles in the riser. From simulated instantaneous translational granular temperature, the time-averaged translational granular temperature is obtained. Figure 12 shows the time-averaged translational granular temperature as a function of the tangential restitution coefficients of particles. This particle translational granular temperature contributes by collisions of particles in the riser. Simulated results show that the translational granular temperature decreases with the increase of concentration of particles. Similar results are also described by Neri and Gidaspow37 and Benyahia et al.38 in numerical simulations of the risers. Gidaspow and Huilin36 showed that, in the dilute regions, the translational granular temperature is proportional to the concentrations raised to the power of 2/3. This is similar to the increase of an ideal gas temperature upon compression. In the

Figure 12. Distributions of time-averaged translational granular temperature.

dense regions, the decrease in the translational granular temperature is due to the decrease of the mean free path of the particles. Therefore, the translational granular temperature appears to decrease for the solid concentration increasing to the maximum solids packing. Two types of fluctuations in the velocity are identified in the gas fluidized bed.39 The first is known as the particle Reynolds stress, which corresponds to the instantaneous variance in the velocity distribution. The second is the time-averaged variance of the mean velocity and is termed the solid phase Reynolds stress. The translational granular temperature from simulations is defined as the average of the particle normal Reynolds stresses, which are the average of the three squares of the velocity components in the three directions. The normal solid phase Reynolds stresses per unit bulk density are calculated by averaging the hydrodynamic velocity for particles Vj(r) from numerical simulations40 m

(V'iV'i) )

∑

1 [V (r, t) - Vj(r)][Vik(r, t) - Vj(r)] m k)1 ik

(7)

m

Vj(r) )

∑

1 V (r, t) m k)1 ik

(8)

The solid phase granular temperature θs is calculated from time-averaged Reynolds normal stresses, with the assumption that the fluctuations of hydrodynamic velocities are equal in the lateral and depth directions36 1 1 2 θs ) (V'iV'i) ≈ (V'zV'z) + (V'xV'x) 3 3 3

(9)

Figure 13 shows the distribution of the solid phase granular temperature θs as a function of the concentration of particles. The solid phase granular temperature increases, reaches a maximum, and then decreases with the increase of concentration of particles. Comparing with the simulated translational granular temperatures shown in Figure 12, the solid phase granular temperatures shown in Figure 13 have much higher values than the translational granular temperatures. These high values are due to the motion of clusters in the risers. Contrary to the experimental data,36 the computational results show a smaller solid phase granular temperature, although the trends are the same. The most probable reason could be that the time-averaged experimental measurements always include a “laminar” contribution that is due to random oscillations of individual particles


Figure 13. Solid phase granular temperature as a function of concentration.

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Figure 15. Distributions of energy dissipation as a function of concentration.

Figure 14. Distribution of mean translational granular temperature.

and a “turbulent” contribution that is caused by the motion of clusters.40 These two kinds of turbulence give rise to two kinds of fluctuations, one on the level of particles and the other on the level of clusters. The former is measured by the classical granular temperature. The latter is measured by the average normal Reynolds stress and calculated from eq 9. The present model, in this case, includes only the “laminar” contribution. Therefore, a direct comparison between the computation and experimental data would be accurate only if the effect of the “turbulent” contribution is incorporated into the model. N j t ) ∑i)1 θt,i/ The mean translational granular temperature, θ N, is computed, where N is the total number of grids. Figure 14 shows the distribution of the mean particle translational granular temperature as a function of the tangential restitution coefficients of particles. Computed results show that the mean translational granular temperature of the particles decreases, reaches a minimum, and then increases with the increase of tangential restitution coefficients of the particles. This nonmonotonic behavior of the translational granular temperature is explained by the β-dependence of the fluctuating energy resulting from particle collisions. For β ) +1.0, the particle has a perfectly slippery surface. There is no surface friction and no change in tangential velocity. For β ) -1.0, the frictional force is at a maximum, and there is the greatest change in tangential velocity and spin. All the fluctuating energy is in the translational mode. At intermediate values of the tangential restitution coefficient the frictional part of the collisional energy losses is increased, and the translational granular temperature of the particles is reduced. This indicates that the particle rotation indeed affects the fluctuation of particles in the riser. The original kinetic theory of granular flow is limited to slightly inelastic spherical particles and does not allow for

Figure 16. Distribution of energy dissipation as a function of tangential restitution coefficient.

particle rotation. Therefore, it underestimates the amount of energy dissipated in the frictional inelastic collisions for common particles. The rate of dissipation of translational fluctuation kinetic energy due to particle collisions relates to the tangential restitution coefficient of particles. Figure 15 shows the distribution of energy dissipation as a function of tangential restitution coefficients. Simulated results show that the energy dissipation increases, reaches a maximum, and then decreases with the increase of concentration of particles. The rate of the translational fluctuating kinetic energy is decreased at the high concentration of particles due to the low granular temperature; see Figure 12. The mean energy dissipation is calculated in the riser. Figure 16 shows the mean energy dissipation of particles as a function of tangential restitution coefficients. The mean energy dissipation increases, reaches a maximum, and then decreases with the increase of tangential restitution coefficients. These results exhibit nonmonotonic β-dependences. Figures 17 and 18 show the variations of the viscosity and thermal conductivity of particles as a function of the concentration of particles in the riser. Simulated results show that both the solids viscosity and the thermal conductivity decrease with the increase of the concentration of particles. Equations T1-9 and T1-12 show that the particle phase shear viscosity and thermal conductivity have a collisional part and a kinetic part. Both parts relate to the tangential restitution coefficient and the granular temperature of particles. For a given value of the tangential restitution coefficient, the decreased viscosity and thermal conductivity of particles with increasing concentration of particles is explained by the fact that the granular temperature is decreased with the increase of concentration of particles; see

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Figure 20. Distributions of concentrations with four different drag coefficient models. Figure 17. Profiles of solids viscosity as a function of concentration.

Figure 21. Distribution of solid mass fluxes with four different drag coefficient models. Figure 18. Profile of thermal conductivity as a function of concentration.

Figure 19. Viscosity and thermal conductivity as a function of tangential restitution coefficient.

Figure 12. The mean solids viscosity and thermal conductivity are computed. Figure 19 shows the distribution of mean solids viscosity and thermal conductivity as a function of tangential restitution coefficients. The mean solids viscosity increases, reaches a maximum, and then decreases with the increase of tangential restitution coefficients of particles. The profile for the thermal conductivity is, however, reverse. This indicates the effect of the particle rotation on solid phase properties in the riser. 3.4. Effect of Drag Coefficient Models. In the momentum transport equations of the individual phases, the drag force is represented by the product of the interphase momentum

exchange coefficient and the slip velocity. There are different drag coefficient models used in the numerical simulations of fluidized beds.41 Studies by Agrawal et al.42 have shown a significant reduction of the drag coefficient by the presence of solid phase mesoscale structures (clusters). To account for the effect of clusters in coarse mesh simulations in which the solid phase mesoscale structures are filtered out, Yang et al.43 and Andrews et al.44 have introduced an effective interphase momentum exchange coefficient. Figure 20 shows the distribution of concentration of particles using four different drag coefficient models. Predicted concentrations of particles using the equations of Ergun32 and Wen and Yu33 (Ergun-Wen and Yu), O’Brien and Syamlal,30 Yang et al.,43 and Andrews et al.44 are low in the center region and higher near the walls. The correlation of O’Brien and Syamlal30 is obtained from an air-FCC system with specific solid circulation fluxes. Both the formulations of Yang et al.43 and Andrews et al.44 predict a reduction of the drag coefficient by a factor of 1.5-4, in agreement with calculations from dynamic mesoscale simulations by Agrawal et al.42 The difference between predictions using the four drag coefficient models is obvious. The trends, however, are the same. Figure 21 shows the distribution of axial velocities of particles using the four different drag coefficient models. Simulations show the solid mass fluxes are positive in the center region and negative near the walls. The predictions using the drag coefficient model proposed by Yang et al.43 give the largest solid mass flux in the riser. Note that the correction of the drag model is dependent on the gas and solid properties as well as the solid circulation flux and superficial gas velocity. Therefore, the drag coefficient model proposed by Yang et al.43 can only be applied for their specific cases.


Figure 22. Distribution of translational granular temperature as a function of concentration.

Figure 22 shows the simulated translational granular temperature as a function of the concentration of particles. Three different drag coefficient models show the translational granular temperature decreases with the increase of the concentration of particles. The difference between predictions using the three drag coefficient models is obvious. Therefore, the impact of a drag coefficient model on the flow behavior of particles is to be further investigated. 4. Conclusions A gas-solid two-fluid model has been improved by incorporating particle rotation using a simplified kinetic theory for rapid granular flow of slightly frictional spheres. Simulations without and with particle rotation are performed to study the flow behavior of gas and particles in a 2-D riser. Results are compared with experimental results measured by Miller and Gidaspow24 and Knowlton et al.25 in the risers. An agreement is achieved by using the model with particle rotation where kinetic theory is applicable. Simulated results indicate that the rotation can alter the profiles of the velocity and concentration of particles. Sufficient particle spin can drive the particles to concentrate near the walls. Simulated results show that the computed mean particle viscosity and thermal conductivity and the energy dissipation of particles exhibitnonmonotonictangentialrestitutioncoefficientdependences. Simulated results indicate that particle rotation is an important microscopic physics to be incorporated into the fundamental hydrodynamic model. For further model development, the particle rotation effect can be directly coupled to the momentum transfer by more a sophisticated kinetic theory. The authors also recognize that the two-dimensional domain reduces the rotation effect. Therefore, particle rotation in a three-dimensional simulation will be addressed in future studies. Acknowledgment This work was supported by the Natural Science Foundation of China through Grant 50776023 and the National Key Project of Scientific and Technical Supporting Programs Funded by Ministry of Science and Technology of China (No. 2006BAA03B01-07). Notation a ) constant Cd ) drag coefficient d ) particle diameter, m Dgs ) rate of energy dissipation, kg/ms3

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e ) coefficient of normal restitution ew ) restitution coefficient of particle-wall g ) gravitational force, m/s2 g0 ) radial distribution function H ) height, m I ) unit tensor ks ) granular conductivity, kg/ms K ) nondimensional moment of inertia mp ) mass of particle, kg n ) normal direction p ) fluid pressure, Pa ps ) solid pressure, Pa Re ) Reynolds number t ) time, s u, V ) velocity components, m/s ug ) superficial gas velocity, m/s vs,w ) solid velocity at the wall, m/s ∆x, ∆y ) grid sizes, m Greek Symbols β ) tangential restitution coefficient βgs ) fluid-particle drag force coefficient, kg/m3 s γc ) energy dissipation rate of translational fluctuation, kg/ms3 εs ) solids concentration εs,max ) maximum solid packing θt ) translational granular temperature, m2/s2 θr ) rotational granular temperature, m2/s2 θs ) solid granular temperature, m2/s2 µ ) friction coefficient µg ) gas viscosity, kg/ms µs ) particle viscosity, kg/ms ξs ) bulk viscosity, kg/ms Fg ) gas density, kg/m3 Fs ) particle density, kg/m3 τ ) stress tensor, Pa φ ) fluctuation energy exchange, kg/ms3 ∆ ) filter width, m Ω ) coefficient Subscripts g ) gas l ) laminar flow w ) wall

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ReceiVed for reView December 5, 2009 ReVised manuscript receiVed January 30, 2010 Accepted February 25, 2010 IE9019243

Numerical Simulation of Fluid Dynamics of a Riser - American

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