Energy Fuels 2010, 24, 3159–3172 Published on Web 04/19/2010
: DOI:10.1021/ef100096c
Modeling on High-Flux Circulating Fluidized Bed with Geldart Group B Particles by Kinetic Theory of Granular Flow Baosheng Jin,* Xiaofang Wang, Wenqi Zhong,* He Tao, Bing Ren, and Rui Xiao School of Energy & Environment, Southeast University, Nanjing 210096, People’s Republic of China Received January 26, 2010. Revised Manuscript Received March 27, 2010
Modeling of the hydrodynamic behaviors of high-flux circulating fluidized beds (HFCFBs) with Geldart group B particles has been performed using a Eulerian multiphase model with the kinetic theory of granular flow (KTGF). The essential models involved are the dispersed k-ε turbulence model, the Gidaspow shear viscosity model, and the Syamlal-O’Brien drag model, and the boundary condition is the Johnson and Jackson wall boundary condition. The sensitivities of key model parameters (i.e., particleparticle restitution coefficient (e), particle-wall restitution coefficient (ew), and specularity coefficient (j)) on the predicted gas velocity, solids velocity, and solids volume fraction were tested. It was found that e has remarkable dependence on the particle diameter. Large-sized particles experience a more sensitive effect of e on predictions. The particle-wall restitution coefficient ew has somewhat of an effect on the simulated values of gas velocity, solids velocity, and solids volume fraction; however, no critical changes in the trends of their radial distributions have been found. The specularity coefficient j has a slight effect on the gas velocity and solids velocity distributions but a pronounced effect on the solids volume fraction distribution. An increase in specularity coefficient results in a reduction in the solids volume fraction near the wall. Based on the comparisons of simulated results with experiments, a group of suitable model parameters for modeling the flow of Geldart group B particles in HFCFB risers by a Eulerian multiphase model with KTGF was determined and verified. Besides, some interesting simulated results that are difficult to measure experimentally were presented under the suggested model parameters.
Many valuable experimental investigations on HFCFBs have been performed.5-15 However, most of them focused on the Geldart group A particles for the background of FCC processes, the production of maleic anhydride, etc. Full knowledge on the hydrodynamic characteristics in HFCFBs with Geldart group B particles has been lacking until now, which leads to difficulty in applying the HFCFB technique to
1. Introduction High-flux circulating fluidized beds (HFCFBs), i.e., the solid flux of which is much larger than conventional circulating fluidized beds (generally >200 kg m-2 s-1), have attracted significant attention over the past 20 years for energy transformation (e.g., fluidized catalytic cracking (FCC),1 the production of maleic anhydride,2 and first the combustion3 and then the gasification4 of coal or biomass in recent years). For coal combustion and gasification, HFCFBs operate at higher circulation rates, velocities, and riser densities, which result in better mixing and higher transfer rates of mass and heat, especially under pressurized conditions. Such reactors are cost-effective, because of their higher throughput, when handling lowrank coals and when using coals with high moisture or high ash content.3,4
(6) Karri, S. B. R.; Knowlton, T. M. A comparison of annulus solids flow direction and radial solids mass flux profiles at low and high mass fluxes in a riser. In Circulating Fluidized Bed Technology VI; Werther, J., Ed.; DECHEMA: Frankfurt, Germany, 1999; pp 71-77. (7) Issangya, A. S.; Bai, D.; Bi, H. T.; Lim, K. S.; Zhu, J.; Grace, J. R. Suspension densities in a high-density circulating fluidized bed riser. Chem. Eng. Sci. 1999, 54 (22), 5451–5460. (8) Grace, J. R.; Issangya, A. S.; Bai, D.; Bi, H.; Zhu, J. Situating the high-density circulating fluidized bed. AIChE J. 1999, 45 (10), 2108–2116. (9) Issangya, A. S.; Grace, J. R.; Bai, D.; Zhu, J. Further measurements of flow dynamics in a high-density circulating fluidized bed riser. Powder Technol. 2000, 111 (1), 104–113. (10) Malcus, S.; Cruz, E.; Rowe, C.; Pugsley, T. S. Radial solid mass flux profiles in a high-suspension density circulating fluidized bed. Powder Technol. 2002, 125 (1), 5–9. (11) Manyele, S. V.; P€arssinen, J. H.; Zhu, J. Characterizing particle aggregates in a high-density and high-flux CFB riser. Chem. Eng. J. 2002, 88 (1-3), 151–161. (12) Li, Z. Q.; Wu, C. N.; Wei, F.; Jin, Y. Experimental study of highdensity gas-solids flow in a new coupled circulating fluidized bed. Powder Technol. 2004, 139 (3), 214–220. (13) Luo, Z.; Zhao, Y.; Chen, Q.; Tao, X.; Fan, M. Effect of gas distributor on performance of dense phase high density fluidized bed for separation. Int. J. Miner. Process. 2004, 74 (1-4), 337–341. (14) Kim, S. W.; Kirbas, G.; Bi, H.; Lim, C. J.; Grace, J. R. Flow behavior and regime transition in a high-density circulating fluidized bed riser. Chem. Eng. Sci. 2004, 59 (18), 3955–3963. (15) Du, B.; Warsitio, W.; Fan, L. S. Behavior of the dense-phase transportation regime in a circulating fluidized bed. Ind. Eng. Chem. Res. 2006, 45 (10), 741–751.
*Authors to whom correspondence should be addressed. Tel.: þ86-2583795508. Fax: þ86-25-83795508. E-mail addresses:
[email protected] (B.J.);
[email protected] (W.Z.). (1) Zhu, J. X.; Bi, H. T. Distinctions between low density and high density circulating fluidized bed. Can. J. Chem. Eng. 1995, 73 (5), 644–649. (2) Grace, J. R. Reflections on turbulent fluidization and dense suspension upflow. Powder Technol. 2000, 113 (3), 242–248. (3) Smith, P. V.; Vimalchand, P.; Pinkston, T.; Gunnar, H.; James, L. Transport Reactor Combustor and Gasifier Operations. In Energex 2000: Proceedings of the 8th International Energy Forum, Las Vegas, NV, 2000; p 94. (4) Robert, S. D.; WanWang, P.; et al. Formation and Prevention of Agglomerated Deposits During the Gasification of High-Sodium Lignite. Energy Fuels 2006, 20, 2465–2470. (5) Issangya, A. S.; Bai, D.; Grace, J. R. Flow behavior in the riser of high-density circulating fluidized bed. AIChE. Symp. Ser. 1997, 93, 25–30. r 2010 American Chemical Society
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some new industrial interests (e.g., coal-based combustion and gasification). Thus, theoretical as well as experimental studies aimed at grasping useful hydrodynamic characteristics of HFCFBs with Geldart group B particles are needed. Numerical simulations, which could offer detailed information about the complicated gas-solid flow and provide a powerful and attractive alternative for experimental studies, have become popular in dense gas-solid flows in recent years.16,17 As one of the commonly used methods, the Eulerian approach offers higher computational efficiency with less computational expense, compared to the Discrete Element Method (DEM) approach; as a result, simulation of the largescale system is possible.18 Since Sinclair and Jackson19 first used kinetic theory of granular flow (KTGF) in a steady-state, one-dimension model to compute the fully developed gas and particles flow in a vertical pipe, many scholars have developed and perfected this theory.20-23 Nowadays, a Eulerian multiphase model with closure, according to KTGF, has become a popular and useful tool to describe gas-solid flow.24-26 However, there has been few attempts, with regard to this method, to model HFCFBs.27-29 It is known that the proper selection of key model parameters of the Eulerian multiphase model with KTGF is crucial to improve the calculation accuracy even to get correct predicted trends. Among the publications on modeling hydrodynamics by Eulerian approach, there has been no consistent conclusion on the selection of key model parameters, even under similar operating conditions and bed geometry.27,28,30 Just taking the particle-particle restitution coefficients (e) as an example, different values;ranging from 0.6 to 1;to
describe elastic particle-particle collisions have been used to model the gas-solid flow in the riser.31-34 Almuttahar and Taghipour29 reported that a restitution coefficient of e=0.99 could model the hydrodynamics of HFCFBs with FCC particles well, while Jiradilok et al.30 indicated that the simulation with a coefficient of restitution of 0.99 could not give good resolution for the bubble formation in the FCC riser; instead, a value of e=0.9 was useful. These inconsistent results bring some questions about modeling HFCFBs with KTGF: • How do the model parameters (for example, the particleparticle restitution coefficient (e), the particle-wall restitution coefficient (ew), and the specularity coefficient (j)) affect the computational fluid dynamics? • What model parameters are suitable for simulating HFCFBs with Geldart group B particles? • Is there a dependence of the model parameters on the particle properties (e.g., particle diameter, particle density) and operating condition? To answer these interesting questions, combining with the evaluation of the effect of models and modeling parameters on the hydrodynamics of HFCFBs with Geldart group A particles,35 evaluations of the model parameters with Geldart group B particles are needed. The main objective of this article is to evaluate the sensitivity of key model parameters (the particle-particle restitution coefficient (e), the particle-wall restitution coefficient (ew), and the specularity coefficient (j)) on the computational fluid dynamics (CFD) of HFCFB with Geldart group B particles. Based on the limited experimental data of Mei et al.36 and our experiments, suitable computational fluid dynamics (CFD) model parameters for modeling the hydrodynamics of HFCFBs are determined and verified. Besides, some interesting simulated results that are difficult to measure experimentally (e.g., radial solid/gas velocities and solids volume fraction profiles) are presented.
(16) Benyahia, S.; Arastoopour, H.; Knowlton, T. M. Simulation of particles and gas flow behavior in the riser section of a circulating fluidized bed using the kinetic theory approach for the particulate phase. Powder Technol. 2000, 112, 24–33. (17) Goldschmidt, M. J. V.; Beetstra, R.; Kuipers, J. A. M. Hydrodynamic modeling of dense gas-fuidised beds: Comparison of the kinetic theory of granular flow with 3D hard-sphere discrete particle simulations. Chem. Eng. Sci. 2002, 57, 2059–2075. (18) Goldschmidt, M. J. V.; Beetstra, R.; Kuipers, J. A. M. Hydrodynamic modelling of dense gas-fluidised beds: comparison and validation of 3D discrete particle and continuum models. Powder Technol. 2004, 142, 23–47. (19) Sinclair, J. L.; Jackson, R. Gas-particle flow in a vertical pipe with particle-particle interactions. AIChE J. 1989, 35, 1473–148. (20) Ding, J.; Gidaspow, D. A bubbling fluidization model using kinetic theory of granular flow. AIChE J. 1990, 36, 523–538. (21) Gidaspow, D. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Description; Academic Press: Boston, 1994. (22) Neri, A.; Gidaspow, D. Riser hydrodynamics: simulation using kinetic theory. AIChE J. 2000, 46, 52–67. (23) Lu, H. L.; Gidaspow, D.; Bouillard, J.; Wentie, L. Hydrodynamic simulation of gas-solid flow in a riser using kinetic theory of granular flow. Chem. Eng. J. 2003, 95, 1–13. (24) Schmidt, A.; Renz, U. Numerical prediction of heat transfer in fluidized beds by a kinetic theory of granular flows. Int. Therm. Sci. J. 2000, 39, 871–885. (25) Cooper, S.; Coronellat, C. J. CFD simulations of particle mixing in a binary fluidized bed. Powder Technol. 2005, 151, 27–36. (26) Chan, C. K.; Guo, Y. C.; Lau, K. S. Numerical modeling of gasparticle flow using a comprehensive kinetic theory with turbulence modulation. Powder Technol. 2005, 150, 42–55. (27) Cruz, E.; Steward, F. R.; Pugsley, T. New closure models for CFD modeling of high-density circulating fluidized beds. Powder Technol. 2006, 169, 115–122. (28) Almuttahar, A.; Taghipour, F T. Computational fluid dynamics of high density circulating fluidized bed riser: Study of modeling parameters. Powder Technol. 2008, 185 (1), 11–23. (29) Almuttahar, A.; Taghipour, F. Computational fluid dynamics of a circulating fluidized bed under various fluidization conditions. Chem. Eng. Sci. 2008, 63, 1696–1709. (30) Jiradilok, V.; Gidaspow, D.; Damronglerd, S.; Koves, W. J.; Mostofi, R. Kinetic theory based CFD simulation of turbulent fluidization of FCC particles in a riser. Chem. Eng. Sci. 2006, 61 (17), 5544–5559.
2. Model Description 2.1. Governing Equations. 2.1.1. Conservation Equations. The continuity equations for gas and solid phases in a threedimensional geometry can be expressed as ∂ ðRq Fq Þ þ r 3 ðRq Fq νq Þ ¼ 0 ∂t
ð1Þ
where R, F, and ν represent the volume fraction, density, and instantaneous velocity of each phase. (31) Goldschmidt, M. J. V.; Kuipers, J. A. M.; Swaaij, W. P. M. Hydrodynamic modeling of dense gas-fluidised beds using the kinetic theory of granular flow: effect of coefficient of restitution on bed dynamics. Chem. Eng. Sci. 2001, 56, 571–578. (32) Neri, A.; Gidaspow, D. Riser hydrodynamics: simulation using kinetic theory. AIChE J. 2000, 46, 52–67. (33) Lu, H. L; Gidaspow, D.; Bouillard, J.; Wentie, L. Hydrodynamic simulation of gas-solid flow in a riser using kinetic theory of granular flow. Chem. Eng. J. 2003, 95, 1–13. (34) Cooper, S.; Coronellat, C. J. CFD simulations of particle mixing in a binary fluidized bed. Powder Technol. 2005, 151, 27–36. (35) Wang, X. F.; Jin, B. S; Zhong, W. Q; Xiao, R. Modeling on the Hydrodynamics of a High-Flux Circulating Fluidized Bed with Geldart Group A Particles by Kinetic Theory of Granular Flow. Energy Fuels 2010, 24, 1242–1259. (36) Mei, J. S.; Shadle, L. J.; Yue, P. Hydrodynamics of a Transport Reactor Operating in Dense Suspension Upflow Conditions for Coal Combustion Applications. In Proceedings of 18th International Conference on Fluidized Bed Combustion, Toronto, Ontario, Canada, 2005; pp 22-25.
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where μt,g is the gas-phase turbulent viscosity, ! kg 2 μt;g ¼ Fg Cμ εg
The momentum equations for both phases are described by ∂ ðRg Fg νg Þ þ r 3 ðRg Fg νg νg Þ ¼ -Rg rpg þ Rg Fg g ∂t ð2Þ - βðνg - νs Þ þ r 3 Rg τ g
kg is the gas turbulence kinetic energy, εg the gas dissipation rate of turbulent kinetic energy, and Cμ an empirically assigned constant in the turbulence model. The solid phase is treated as a continuum in the Eulerian multiphase model; thus, the solid-phase stress tensor can be expressed as a similar form as the gas phase. That is, h i 2 τ s ¼ λs r 3 vs I þ μs rvs þ ðrvs ÞT - ðr 3 vs Þ I ð11Þ 3
∂ ðRs Fs νs Þ þ r 3 ðRs F s νs νs Þ ¼ -Rs rpg - rps þ Rs Fs g ∂t - βðνs - νs Þ þ r 3 Rs τ s ð3Þ where p is the fluid pressure, β represents the gas-solid interphase drag coefficients, and τ is the stress tensor. The subscripts “g” and “s” denote the gas phase and the solid phase, respectively. 2.1.2. Interaction Force between Phases. Particle motion and the basic characteristics of gas-solid flow in the riser are determined essentially by all forces received by particles. For relatively small particles with densities much larger than the density of the continuous phase, the interphase drag force dominates the other forces and cannot be ignored due to the great effects on the flow, heat transfer, and mass transfer.37 It can be described by a gas-solid interphase drag coefficient (β): ! ! 3 Fg Rg Rs jvg - vs j Res CD ð4Þ β¼ vr;s 2 ds 4 νr;s where the drag coefficient CD is expressed as !2 4:8 CD ¼ 0:63 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Res =vr;s the solids Reynolds number (Res) is defined as Fg Rg jvg - vs jds Res ¼ μg
where λs is the solid bulk viscosity and μs is the solid shear viscosity. Expressions for the solid bulk viscosity and the solid shear viscosity can be obtained from the kinetic theory of granular flow.38-41 This theory is an extension of the classical dense gas kinetic theory to dense particle flow. The granular pseudo-temperature is introduced and the solid viscosity and pressure are involved. The granular pseudo-temperature is computed by solving the transport equation ∂ ðRs Fs Θs Þ þ r 3 ðRs Fs Θs vs Þ ∂t 2 ¼ ½ð - ps I þ τ s Þ : r 3 vs þ r 3 ðΓΘ rΘs Þ - γs þ φgs ð12Þ 3
ð5Þ
The first term in the right-hand side is the generation of energy by the solids stress tensor, and the second term describes the diffusive flux of granular energy. Here, γs is the dissipation term of the pseudo-temperature due to elastic collisions between particles, jgs the exchange of fluctuating energy between gas and particles, ΓΘ the transport coefficient of the pseudo-temperature, Ps the particle phase pressure due to particle collisions, and τs is the particle phase stress tensor. They are calculated using the following expressions:
ð6Þ
and the terminal velocity correlation for the solid phase (vr,s) is given as vr;s ¼ 0:5 A - 0:06Res qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7Þ þ ð0:06Res Þ2 þ 0:12Res ð2B - AÞ þ A2
γs ¼ 3ð1 - e2 Þ Rs 2 Fs g0 Θs
where A ¼ Rg 4:14 ( B¼
0:8Rg Rg
2:65
1:28
ðRg e0:85Þ ðRg > 0:85Þ
ð10Þ
ð8aÞ
4 ds
! rffiffiffiffiffiffi Θs - r 3 vs π
φgs ¼ -3βΘs
ð8bÞ
ΓΘ ¼
Here, ds is the solid diameter and μg is the gas viscosity. 2.1.3. Constitutive Equations. To solve continuity equations and momentum equations, appropriate expressions of unknown terms, such as solids pressure and solid-phase and gas-phase stress-strain tensors, are needed to obtain closure of the entire calculation model. For a viscous fluid, the gas-phase stress tensor can be expressed as h T i 2 τ g ¼ μt;g rvg þ rvg ð9Þ þ λg - μeff;g r 3 vg I 3
pffiffiffiffiffiffiffiffiffi 2 150Fs ds πΘs 6 1 þ ð1 þ eÞg0 Rs 384ð1 þ eÞg0 5 rffiffiffiffiffiffi Θs 2 þ 2Rs Fs ds g0 ð1 þ eÞ π
ps ¼ Rs Fs Θs ½1 þ 2g0 Rs ð1 þ eÞ
ð13Þ
ð14Þ
ð15Þ ð16Þ
(38) Savage, S. B.; Jeffrey, D. J. The stress tensor in a granular flow at high shear rates. Fluid Mech. 1981, 110, 255–272. (39) Jenkins, J. T.; Savage, S. B. A theory for the rapid flow of identical, smooth, nearly, elastic Spherical Particles. Fluid Mech. 1983, 130, 187–202. (40) Lun, C. K. K.; Savage, S. B.; Jeffrey, D. J.; Chepurniy, N. Kinetic theories for granular flow: inelastic Particles in Couette flow and slightly inelastic Particles in a general flow field. Fluid Mech. 1984, 140, 223–256. (41) Johnson, P.; Jackson, R. Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 1987, 176, 67–93.
(37) Ranade, V. Computational Flow Modeling for Chemical Reactor Engineering; Academic Press: New York, 2002; pp 85-117.
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Figure 1. Schematic diagram of our cold experimental system.
2 g0 ¼ 41 -
Rs Rs;max
!1=3 3 - 1 5
rffiffiffiffiffiffi 4 2 Θs λs ¼ Rs Fs ds g0 ð1 þ eÞ 3 π
turbulence model and the laminar model. These models indicate that the dominant process in the random motion of the solid phase is caused by gas-phase turbulence. It has been widely and successfully used in modeling gassolid flow in fluidization system with Geldart group B particles.43
ð17Þ
ð18Þ
∂ ðRg Fg kg Þ þ r 3 ðRg Fg Ug kg Þ ∂t μt;g ¼ r 3 Rg rkg þ ðRg Gk;g - Rg Fg εg Þ þ Rg Fg Πk ð20Þ σk
The granular-phase shear viscosity takes into account three terms: collisional (particle-particle collisions), kinetic (fluctuating motion), and frictional (particle-particle friction) viscosity. In this work, the Gidaspow shear viscosity model42 is adopted. rffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 4 Θs 10Fs ds πΘs þ μs ¼ Rs Fs ds g0 ð1 þ eÞ 5 π 96Rs ð1 þ eÞg0 2 4 ps sin ω 1 þ Rs g0 ð1 þ eÞ þ pffiffiffiffiffiffiffi ð19Þ 5 2 I2D
∂ ðRg Fg εg Þ þ r 3 ðRg Fg Ug εg Þ ∂t μt;g εg ¼ r 3 Rg rεg þ ðC1ε Rg Gk;g - C2ε Rg Fg εg Þ þ Rg Fg Πε σε kg ð21Þ
2.2. Turbulence Model. Considering the realistic flow conditions, turbulence is the most important characteristic of gas-solid flows in CFBs, especially in HFCFBs with high Reynolds numbers. In this work, a dispersed k-ε turbulence model is adopted, after comparison with the per-phase
Here, μt,g is the turbulence viscosity of the gas phase. Πk and Πε are turbulence exchange terms between the gas and solid phases, which represent the influence of the solid phase on the gas phase.
(42) Gidaspow, D.; Bezburuah, R.; Ding, J. Hydrodynamics of Circulating Fluidized Beds, Kinetic Theory Approach. In Fluidization VII, Proceedings of the 7th Engineering Foundation Conference on Fluidization, 1992; pp 75-82.
(43) Benyahia, S.; Syamlal, M.; O’Brien, T. J. Study of the Ability of Multiphase Continuum Models to Predict Core-Annulus Flow. AIChE J. 2007, 53 (10), 2549–2568.
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Figure 2. Sketches of the numerical riser based on the experiments of Mei et al.36
Figure 3. Sketches of the numerical riser based on our experiments.
3. Numerical Procedure Experiments are conducted at gas velocities in the range of 6.2-12.6 m/s, and solid mass fluxes in the range of 100-624 kg m-2 s-1. The bed material was quartz sand with a mean diameter of 385 μm and a density of 2580 kg/m3. Sketches of the simulated risers, as described by Mei et al.36 and this work, are shown in Figures 2 and 3, respectively. Detailed geometrical parameters of risers are presented in Table 1. 3.2. Boundary Condition and Initial Condition. At the inlets, both the velocities and the concentrations of the gas and solid phases were specified, according to the superficial gas velocities and solid mass fluxes. Turbulent intensity was used to consider the gas turbulence quantity of inlet. The gas turbulent intensity (I) was calculated using the relation
3.1. Simulated HFCFB Riser. The simulated high-flux circulating fluidized bed (HFCFB) risers are based on the Mei et al. experiment36 and our experiments. Mei et al.36 conducted a series of experiments in the 0.3-mdiameter, 15.45-m-high HFCFB test facility. The particles they used are glass beads with a mean diameter of 60 μm and a density of 2428 kg/m3. The superficial gas velocities are 5.1 and 7.8 m/s, and the particle circulation rates are 127-554.9 kg m-2 s-1. A detailed description of the unit and experiments can be found in the reported work of Mei et al.36 Our experimental system consisted of a vertical riser (0.06 m inner diameter (ID) 5.12 m high), two downcomers (0.04 m ID 3.5 m high and 0.1 m ID 3 m high), an inertial separator, a cyclone, two solid feeding devices, and a bag filter, as shown in Figure 1. Air from a Roots Blower (identified as “13” in the figure legend) was fed into the riser through a gas distributor. From the riser (“1” in the figure), the suspension was directed into the inertial separator (“3” in the figure). Most of the solids were removed from the air stream and fell into the first downcomer (“2” in the figure), then returned into the riser via first solid feeding device (“5” in the figure). The flow came from the inertial separator passed through the cyclone (“7” in the figure). Solids separated from the cyclone returned into the riser through the second downcomer (“8” in the figure) and solid feeding device (“9” in the figure), and air passed through the bag filter (“11” in the figure) before being discharged to the atmosphere. It is similar to the experimental system shown in our previous publication,44 except for the outlet configuration.
I ¼ 0:16Reg - 0:125 where Reg is the gas Reynolds number at the inlet condition. At the outlet, the pressure outlet condition was adopted. At the beginning, the risers are assumed to be free of particles. The gas and solid velocities also were set equal to zero. The Johnson and Jackson41 wall boundary conditions were applied. As expressed in eqs 22-25, the slip velocity between particles and the wall can be obtained by equating the tangential force exerted on the boundary and the particle shear stress close to the wall. Similarly, the granular temperature at the wall was obtained by equating the granular temperature flux at the wall to the inelastic dissipation of energy, and to the generation of granular energy due to slip at the wall region.16 ∂νs;w νs;w ¼ -B ð22Þ ∂n
(44) Wang, X. F.; Jin, B. S; Zhong, W. Q; Zhang, M. Y.; Huang, Y.; Duan, F. Flow behaviors in a high-flux circulating fluidized bed. Int. J. Chem. Reactor Eng. 2008, 6, No. A79. (Available via the Internet at http:// www.bepress.com/ijcre/vol6/A79.)
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Table 1. Riser Geometry, Operating Conditions, and Model Parameters Settings based on Mei et al.36 work
based on our work bed geometry height of riser, H diameter of riser, D particles and gas particle diameter, ds particle density, Fs gas density, Fg gas viscosity, μg operating conditions superficial gas velocity, ug solid flux, Gs simulation maximum solid volume fraction, Rs,max particle-particle restitution coefficient, e particle-wall restitution coefficient, ew specularity coefficient, j time step, Δt
5.12 m 60 mm
15.45 m 300 mm
385 μm 2580 kg/m3 1.225 kg/m3 1.78 10-6 Pa/s
60 μm 2428 kg/m3 1.225 kg/m3 1.78 10-6 Pa/s
9.8 m/s, 8.6 m/s, 10.7 m/s 546 kg m-2 s-1, 364 kg m-2 s-1
7.8 m/s 554.9 kg m-2 s-1
0.61 0.99, 0.95, 0.9, 0.8, 0.7 0.99, 0.95, 0.9, 0.7, 0.6 0.01, 0.001, 0.0001, 0 1 10-4 s
0.58 0.99, 0.95, 0.9 0.99, 0.95, 0.9 0.01, 0.001, 0.0001, 0 1 10-4 s
6Rs μs B ¼ pffiffiffiffiffiffiffiffi 3Θs πjFs Rs g0
ð23Þ
pffiffiffi kΘs ∂Θw 3πjFs Rs vs;slip 2 g0 Θs 3=2 þ Θw ¼ γw ∂n 6Rs;max γw
ð24Þ
γw ¼
pffiffiffi 3πð1 - ew 2 ÞRs Fs g0 Θs 3=2 4Rs;max
and correction. All the data used in the follow sections were time-averaged values under the condition that the gas-solid flow in the riser was reached quasi-steady-state condition. This condition was judged by the almost no change of gas and solid mass flux at the riser outlet. The simulation has been performed based on the Fluent 6.3.26 software with some models programmed by ourselves. Our programs were coupled into Fluent using user-defined functions (UDFs).
ð25Þ
where ew is the particle-wall restitution coefficient, which describes the amount of dissipation of the solids turbulent kinetic energy with the wall by collisions. The parameter j is the specularity coefficient, which specifies the shear condition at the walls. These two coefficients would affect the flow31,43,45 but are difficult to measure.43,46 To evaluate their effects on flow behavior and obtain optimal values, different specularity coefficients and particle-wall restitution coefficients were used in this work. Details are listed in Table 1. 3.3. Simulation Procedure. Simulations were performed in three-dimensional domains. The hexahedral grids and tetrahedral grids were applied. To confirm that the predictions are independent of the mesh size, several meshes with 6 540 100, 5 246 500, 4 020 680, and 1 955 875 control volumes for the Mei et al.36 case and 115 110, 157 810, 190 330, and 241 230 control volumes for our case have been tested, respectively. Based on the test method proposed in the literature (e.g., ref 47), the meshes composed of 4 020 680 and 190 330 control volumes are adopted, respectively. The governing equations were solved using the finite volume approach. First-order upwind discretization was used for both momentum and volume fraction solutions. The phase-coupled SIMPLE (PCSIMPLE) algorithm,48 which is an extension of the SIMPLE algorithm to multiphase flows, was used to solve the pressure-velocity coupling
4. Results and Discussion 4.1. Sensitivity Analysis of Key Model Parameters. In the kinetic theory of granular flow, nonideal particle-particle collisions directly affect particles viscosity, particle phase pressure, kinetic energy dissipation by means of particleparticle restitution coefficient. Similar to the particle-particle restitution coefficient, the particle-wall restitution coefficient describes the amount of the dissipation of solids turbulent kinetic energy with the wall by collisions. Besides, the specularity coefficient is another important boundary condition parameter. A value of j=0 refers to perfectly specular collisions (smooth) and a value of j=1 refers to perfectly diffuse collisions (rough). Previous studies had highlighted the sensitivities of these coefficients on computed solids concentration, mass flux, and other flow characteristics.32,43,49 However, the sensitivities of these coefficients on predictions have not fully known, especially in modeling HFCFBs with Geldart group B particles. In this work, five different particle-particle restitution coefficient values (e = 0.99, 0.95, 0.9, 0.8, 0.7), five different particle-wall restitution coefficient values (ew=0.99, 0.95, 0.9, 0.7, 0.6), and four specularity coefficient values (j = 0.01, 0.001, 0.0001, and 0) are tested based on limited experimental data of Mei et al.36 and our experiments. 4.1.1. Particle-Particle Restitution Coefficient. Figure 4 shows the effect of particle-particle restitution coefficient e on the time-averaged solids volume fraction, gas velocity, and solids velocity in our HFCFB riser at z/H=0.8 (z=4 m) under the operating condition of ug = 9.8 m/s, Gs = 546 kg m-2 s-1), dp =385 μm, and Fp =2580 kg/m3. Similar trends of solids volume fraction, gas velocity, and solids velocity
(45) Li, J.; Kuipers, J. A. M. Effect of competition between particle-particle and gas-particle interactions on flow patterns in dense gas-fluidized beds. Chem. Eng. Sci. 2007, 62, 3429–3442. (46) Wang, W.; Li, Y. Hydrodynamic Simulation of Fluidization by Using a Modified Kinetic Theory. Ind. Eng. Chem. Res. 2001, 40, 5066–5073. (47) Bastos, J. C. S. C.; Rosa, L. M.; Mori, M.; Marini, F.; Martignoni, W. P. Modeling and simulation of a gas-solids dispersion flow in a highflux circulating fluidized bed (HFCFB) riser. Catal. Today 2008, 130, 462–470. (48) Vasquez, S.; Ivanov, V. A phase coupled method for solving multiphase problems on unstructured meshes. Presented at ASME FED Summer Meeting, Boston, 2000.
(49) Therdthianwong, A.; Pantaraks, P.; Therdthianwong, S. Modeling and simulation of circulating fluidized bed reactor with catalytic ozone decomposition reaction. Powder Technol. 2003, 133, 1–14.
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: DOI:10.1021/ef100096c
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Figure 4. Effect of the particle-particle restitution coefficient on the time-averaged solids volume fraction, as well as gas and solids velocities across the riser, based on our experiment conditions (ug=9.8 m/s, Gs=546 kg m-2 s-1, ew=0.99, j=0).
Figure 5. Effect of particle-particle restitution coefficients on the time-averaged solids volume fraction, gas and solids velocities across the riser based on the Mei et al.36 experiment condition (ug=7.8 m/s, Gs=554.9 kg m-2 s-1, ew=0.99, j=0).
distributions but significantly different values of these predictions can be seen when the particle-particle restitution coefficient varies over a range of e=0.7-0.99. For example, the predicted solids volume fractions are ∼0.2 and ∼0.05 in the central region of the riser (r/R values between -0.6 and 0.6) and 0.45 and 0.15 near the wall for e values of 0.95 and 0.99, respectively.
Figure 5 gives the time-average radial profiles of solids volume fraction, gas velocity, and solids velocity in the HFCFB riser at z/H = 0.8 (z = 12.3 m) under the same operating conditions as those in the Mei et al. report,36 i.e., ug=7.8 m/s, Gs=554.9 kg m-2 s-1), dp=60 μm and Fp=2428 kg/m3. There are some differences in the solids volume fraction, gas velocity, and solids velocity values, relative to 3165
Energy Fuels 2010, 24, 3159–3172
: DOI:10.1021/ef100096c
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The difference in the level of sensitivity of particleparticle restitution coefficient e on the predictions for particles with different diameters can be seen from Figures 4 and 5. That is, a high sensitivity of e is observed for relative large particles while a low sensitivity of e is observed for relatively small particles. This difference also can be seen in the previous studies for particle diameters of ds > 100 μm and ds 100 μm, Goldschmidt et al.31 indicated that the formation and size of the bubbling was strongly dependent on the particle-particle restitution coefficient from 0.73 to 1 (dp =1500 μm and Fp=2523 kg/m3), and Benyahia et al.43 showed that model predictions are sensitive to the particle-particle restitution coefficient when modeling core-annulus flow with dp = 120 μm and Fp =2400 kg/m3. For particle diameters of ds
