Energy Fuels 2010, 24, 1242–1259 Published on Web 12/21/2009
: DOI:10.1021/ef901104g
Modeling on the Hydrodynamics of a High-Flux Circulating Fluidized Bed with Geldart Group A Particles by Kinetic Theory of Granular Flow Xiaofang Wang, Baosheng Jin,* Wenqi Zhong, and Rui Xiao School of Energy & Environment, Southeast University, Nanjing 210096, People’s Republic of China Received September 29, 2009. Revised Manuscript Received November 23, 2009
A Eulerian multiphase model with the kinetic theory of granular flow (KTGF) was studied for modeling the hydrodynamic behaviors of high-flux circulating fluidized beds (HFCFBs) with Geldart group A particles. The sensitivities of several key models (i.e., turbulence model, drag model, and granular shear viscosity model) and modeling parameters (particle-particle restitution coefficient, particle-wall restitution coefficient, and specularity coefficient) on the predictions have been tested systematacially. Experimental results of P€ arssinen and Zhu [AIChE J. 2001, 47 (10), 2197-2205; Chem. Eng. Sci. 2001, 56, 5295-5303] were used as a numerical benchmark to assess the simulations quantitatively. The results show that particle-particle and particle-wall restitution coefficients are not critical for the holistic distribution trends of solid volume fraction and solid velocity. A small specularity coefficient, such as j = 0, could give the well predictions. The Syamlal-O’Brien drag model displays better agreement with individual radial distributions of both solid volume fraction and solid velocity, in terms of microscopic features. While the Syamlal shear viscosity model fails to obtain right trends at the upper parts of the riser. For turbulence model, the mixture turbulence model and per-phase turbulence model could not predict reasonable trend of radial solid velocity distribution along the riser. As a result, a group of suitable models and modeling parameters;i.e., dispersed turbulence model, Gidaspow shear viscosity model, Syamlal-O’Brien drag model, a specularity coefficient of j=0, a particle-particle restitution coefficient of e= 0.9, and a particle-wall restitution coefficient of ew =0.99;are proposed for modeling Geldart group A particle flow in HFCFB risers. Finally, further validations have been conducted to confirm this suggestion, by comparing simulated results with experimental data. Many valuable experimental investigations on high-flux CFBs have been performed.3-15 However, now, the HFCFB technique has difficulty in being applied in larger-scale industrial processes, especially its new applications (e.g., combustion16 and gasification of coal),17,18 because of some limitations, in particular the lack of full knowledge on the hydrodynamic characteristics. Thus, numerical approaches as
1. Introduction In recent years, a circulating fluidized bed operated with high solid mass flux (generally >200 kg/(m2 s)), i.e., a highflux circulating fluidized bed (HFCFB)), has been developed for special interests, e.g., fluid catalytic crackers (FCC) processes1 and duPont’s production of maleic anhydride.2
(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. (16) 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. (17) Morton, F.; Pinkston, T.; Salazar, N.; Stalls, D. Orlando Gasification Project: Demonstration of a Nominal 285 MW Coal-Based Transport Gasifier. In 23rd Annual International Pittsburgh Coal Conference, PCC-Coal-Energy, Environment and Sustainable Development, 2006; p 10. (18) Kim, J. S.; Tachino, R.; Tsutsumi, A. Effects of solids feeder and riser exit configuration on establishing high density circulating fluidized beds. Powder Technol. 2008, 187 (1), 37–45.
*Author to whom correspondence should be addressed. Tel.: þ86-25-83794744. Fax: þ86-25-83795508. E-mail address: bsjin@ seu.edu.cn. (1) Wei, F.; Lu, F. B.; Jin, Y.; Yu, Z. Q. Mass flux profiles in a high density circulating fluidized bed. Powder Technol. 1997, 91, 189–195. (2) Contractor, R. M.; Patience, G. S.; Garnett, D. I. A New Process for n-Butane Oxidation to Maleic Anhydride Using a Circulating Fluidized-Bed Reactor. In Circulating Fluidized Bed Technology IV; Avidan, A., Ed.; AIChE: New York, 1994, p 387. (3) 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. (4) 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. (5) 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. (6) 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. (7) 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. (8) Grace, J. R. Reflections on turbulent fluidization and dense suspension upflow. Powder Technol. 2000, 113 (3), 242–248. (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. r 2009 American Chemical Society
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: DOI:10.1021/ef901104g
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the hydrodynamics of HDCFBs with FCC particles well, while Jiradilok et al.28 indicated that the simulation with a value of e = 0.99 could not give a 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 the hydrodynamics of HDCFBs with KTGF, e.g., how the model and modeling parameter affect the hydrodynamics? What model and modeling parameter are suitable for simulating HDCFBs with Geldart group A particles? To answer these interesting questions, systematical evaluations of the models and modeling parameters are needed. The objective of this paper is to systematically evaluate the effect of models and modeling parameters on the hydrodynamics of HDCFBs with Geldart group A particles, using the experimental results of P€ arssinen and Zhu34,35 as a numerical benchmark. The sensitivities of models (i.e., turbulence model, drag model, and granular viscosity model) and modeling parameters (particle-particle restitution coefficient, particle-wall restitution coefficient, and specularity coefficient) on the predictions are discussed. Finally, a suitable computational fluid dynamics (CFD) models and modeling parameters for simulating the hydrodynamics of HFCFBs are proposed.
well as experiments are needed to fully understand the complex hydrodynamics of HFCFBs. Over the past two decades, modeling CFB hydrodynamics has gained interest, because of their widespread industrial applications, including cracking, drying, catalyst regeneration, power generation, and combustion.19 The Eulerian multiphase model with kinetic theory of granular flow (KTGF) is one of the most applicable approaches to compute the gas-solid flow of CFBs, because of less CPU requirements and memory resources.20-30 However, there has been little attempt regarding the simulation of HDCFBs by Eulerian approach with KTGF.27,29,30 It is known that the key computing models (e.g., turbulence model, drag model, and granular viscosity model) and modeling parameters (e.g., particle-particle/wall restitution coefficient, specularity coefficient) involved in the Eulerian multiphase model with KTGF are crucial. Among the publications on modeling HDCFBs hydrodynamics by Eulerian approach,21,27-30 there has been no consistent conclusion on these key models and modeling parameters even under the same operating conditions and bed geometry. For example, Almuttahar and Taghipour29 reported that the SyamlalO’Brien drag model31 provided a good description of the hydrodynamics of HDCFBs with FCC particles, while Cruz et al.27 considered that none of the drag force models of Gidaspow et al.,32 Syamlal-O’Brien,31 and Wen and Yu33 was suitable. Moreover, Almuttahar and Taghipour29 founded that a restitution coefficient of e = 0.99 could model
2. Model Description 2.1. Governing Equations. 2.1.1. Conservation Equations. The continuity equations for gas and solid phase in a threedimensional geometry can be expressed as D ðRq Fq Þ þ r 3 ðRq Fq vq Þ ¼ 0 ð1Þ Dt
(19) Grace, J.; Bi, H.; Goloriz, M. Circulating fluidized beds. In Handbook of Fluidization and Fluid-Particle Systems; Yang, W.-C., Ed.; Marcel Dekker: New York, 2003; pp 486-487. (20) Neri, A.; Gidaspow, D. Riser hydrodynamics: simulation using kinetic theory. AIChE J. 2000, 46, 52–67. (21) 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. (22) Zheng, Y.; Wan, X.; Qian, Z.; Wei, F.; Jin, Y. Numerical simulation of the gas-particle turbulent flow in risers reactor based on k-ε-kp-εp-Θ two fluid model. Chem. Eng. Sci. 2001, 56, 6813– 6822. (23) Benyahia, S.; Arastoopour, H.; Knowlton, T. Two-dimensional transient numerical simulation of solids and gas flow in the riser section of a circulating fluidized bed. Chem. Eng. Commun. 2002, 189, 510–527. (24) Huilin, L.; Gidaspow, D. Hydrodynamics of binary fluidization in a riser: CFD simulation using two granular temperatures. Chem. Eng. Sci. 2003, 58, 3777–3792. (25) Hansen, K.; Solberg, T.; Hjertager, B. A three-dimensional simulation of gas/particle flow and ozone decomposition in the riser of a circulating fluidized bed. Chem. Eng. Sci. 2004, 59, 5217–5224. (26) Chan, C.; Guo, Y.; Lau, K. Numerical modeling of gas-particle flow using a comprehensive kinetic theory with turbulent 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) 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. (29) Adnan, A.; Fariborz, T. Computational fluid dynamics of high density circulating fluidized bed riser: Study of modeling parameters. Powder Technol. 2008, 185 (1), 11–23. (30) Adnan, A.; Fariborz, T. Computational fluid dynamics of a circulating fluidized bed under various fluidization conditions. Chem. Eng. Sci. 2008, 63, 1696–1709. (31) Syamlal, M.; O’Brien, T. J. Computer Simulation of Bubbles in a Fluidized Bed. AIChE Symp. Ser. 1989, 85, 22–31. (32) 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; Brisbane; Engineering Foundation, New York, 1992; pp 75-82. (33) Wen, C.-Y.; Yu, Y. H. Mechanics of Fluidization. Chem. Eng. Prog. Symp. Ser. 1966, 62, 100–111.
where R, F, and ν respectively represent the volume fraction, density, and instantaneous velocity of each phase. The momentum equations for both phases are described by the expressions D ðRg Fg vg Þ þ r 3 ðRg Fg vg vg Þ ¼ -Rg rpg þ Rg Fg g -βðvg -vs Þ Dt ¼
þ r 3 Rg τg
ð2Þ
D ðRs Fs vs Þ þ r 3 ðRs Fs vs vs Þ ¼ -Rs rpg -rps þ Rs Fs g Dt ¼
-βðvs -vs Þ þ r 3 Rs τs
ð3Þ
where p is the fluid pressure, β represents the gas-solid interphase drag coefficients, and τC is the stress tensor. The subscripts “g” and “s” denote the gas phase and solid phase, respectively. 2.1.2. Turbulence Model. Considering the realistic flow conditions, turbulence is the most important characteristic of gas-solid flows in CFBs, especially in HDCFBs with high Reynolds numbers. To obtain a suitable turbulent model, in the present study, a per-phase turbulence model (k-εks-εs-Θ), a dispersed turbulence model (k-ε-Θ), and a mixture turbulence model (kg-εg-Θ), as well as a laminar model, were tested in simulations. They are described in the following subsections. (34) P€arssinen, J. H.; Zhu, J.-X. Axial and Radial Solids Distribution in a Long and High-Flux CFB Riser. AIChE J. 2001, 47 (10), 2197–2205. (35) P€arssinen, J. H.; Zhu, J.-X. Particle velocity and flow development in a long and high-flux circulating fluidized bed riser. Chem. Eng. Sci. 2001, 56, 5295–5303.
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2.1.2.1. Per-Phase Turbulence Model (k-ε-ks-εs-Θ). Transport equations for each phase are expressed as μt, g D ðRg Fg kg Þ þ r 3 ðRg Fg U g kg Þ ¼ r 3 Rg rkg Dt σk
of turbulence kinetic energy due to the mean velocity gradients. 2.1.3. Kinetic Theory of Granular Flow. The granular temperature for the solid phase, which is proportional to the kinetic energy of the random motion of the particles, is introduced into the model, and appears in the expression for the solids pressure and viscosities. The transport equation derived from kinetic theory takes the form ¼ ¼ D 2 ðRs Fs ΘÞ þ r 3 ðRs Fs Θvs Þ ¼ ½ð -ps I þ τs Þ Dt 3
þ ðRg Gk, g -Rg Fg εg Þ þ βðCsg ks -Cgs kg Þ μt, s μ t, g -βðU s -U g Þ rRs þ βðU s -U g Þ rRg ð4Þ σs Rs σg Rg μt, g D rεg ðRg Fg εg Þ þ r 3 ðRg Fg U g εg Þ ¼ r 3 Rg Dt σε εg εg þ ðC1ε Rg Gk, g -C2ε Rg Fg εg Þ þ C3ε βðCsg ks -Cgs kg Þ kg kg μt, s μt, g -βðU s -U g Þ ð5Þ rRs þ βðU s -U g Þ rRg σs Rs σg Rg and
: r 3 vs þ r 3 ðΓΘ rΘÞ -γs þ φgs ð12Þ 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. The collisional dissipation of energy (γs) represents the rate of energy dissipation within the solid phase due to collisions. The transfer of the kinetic energy of random fluctuations in particle velocity from the solid phase to the fluid phase is described by φgs. 2.2. Constitutive Equations. The closure models and constitutive equations are described in Table 1. In the current work, three drag models (i.e., the Syamlal-O’Brien drag model,31 the Wen and Yu drag model,33 and the Gidaspow drag model)32 and two particles shear viscosity models (i.e., the Gidaspow model32 and the Syamlal model)36 were used. 2.3. Boundary Conditions, Initial Conditions, and Numerical Procedure. The calculations were performed in a highflux CFB riser. The riser has the same geometry as the experimental system of P€ arssinen and Zhu.34 A sketch of the numerical column is shown in Figure 1. The vertical riser has a height of 10 m and a internal diameter of 76 mm. Other geometrical parameters are presented in Figure 1. At both inlets, the velocities and concentrations of gas and solid were specified. Turbulent intensity was used to consider the gas turbulence quantity of inlet. The gas turbulent intensity (I) was calculated using the relation I ¼ 0:16Reg -0:125 where Reg is the gas Reynolds number at the inlet conditions. At the outlet, pressure outlet conditions were adopted. Initially, the particle concentration in the riser was zero. The gas and solid velocities also were set to zero. The Johnson and Jackson37 wall boundary conditions (given by eqs 49-52), which, for the solids tangential velocity and granular temperature, use a specularity coefficient, were applied. Dνs, w ð49Þ νs, w ¼ -B Dn
μt, s D ðRs Fs ks Þ þ r 3 ðRs Fs U s ks Þ ¼ r 3 Rs rks þ Dt σk ðRs Gk, s -Rs Fs εs Þ þ βðCgs kg -Csg ks Þ μt, g μt, s -βðU g -U s Þ rRg þ βðU g -U s Þ rRs ð6Þ σg Rg σs Rs μt, s D ðRs Fs εs Þ þ r 3 ðRs Fs U s εs Þ ¼ r 3 Rs rεs Dt σε εs εs þ ðC1ε Rs Gk, s -C2ε Rs Fs εs Þ þ C3ε βðCgs kg -Csg ks Þ ks ks μt, g μt, s ð7Þ rRg þ βðU g -U s Þ rRs -βðU g -U s Þ σg Rg σs Rs
where Uq (where q = g,s) is the phase-weighted velocity; Gk,q is the production of turbulent kinetic energy. 2.1.2.2. Dispersed Turbulence Model (k-ε-Θ). In this model, interparticle collisions are negligible and the dominant process in the random motion of the solid phase is caused by the gas-phase turbulence. D ðRg Fg kg Þ þ r 3 ðRg Fg U g kg Þ Dt μt, g ¼ r 3 Rg rkg þ ðRg Gk, g -Rg Fg εg Þ þ Rg Fg Πk ð8Þ σk μt, g D ðRg Fg εg Þ þ r 3 ðRFgg U g εg Þ ¼ r 3 Rg rεg Dt σε εg ð9Þ þ ðC1ε Rg Gk, g -C2ε Rg Fg εg Þ þ Rg Fg Πε kg Here, μt,g is the turbulence viscosity of gas, and Πk and Πε represent the influence of the solid phases on the gas phase. 2.1.2.3. Mixture Turbulence Model (kg-εg-Θ). This turbulence model represents the first extension of the singlephase k-ε model and uses mixture properties and mixture velocities to capture important features of the turbulent flow. μt, m D r 3 k þ Gk, m -Fm ε ðFm kÞ þ r 3 ðFm vm kÞ ¼ r 3 Dt σk
6Rs μs B ¼ pffiffiffiffiffiffiffiffi 3Θs πjFs Rs g0 Θw ¼
ð10Þ
pffiffiffi 3πjFs Rs vs2, slip g0 Θs 3=2 kΘs DΘw þ γw Dn 6Rs, max γw
ð51Þ
pffiffiffi 3πð1 -ew2ÞRs Fs g0 Θs3=2 4Rs, max
ð52Þ
γw ¼
μt, m D ðFm εÞ þ r 3 ðFm vm εÞ ¼ r 3 r3ε Dt σε ε ð11Þ þ ðCε1 Gk, m -Cε2 Fm εÞ k In eqs 10 and 11, Fm and νm represent mixture density and velocity, respectively. Gk,m represents the generation
ð50Þ
(36) Syamlal, M.; Rogers, W.; O’Brien T. J. MFIX Documentation: Vol. 1, Theory Guide; National Technical Information Service: Springfield, VA, 1993. (37) Johnson, P.; Jackson, R. Frictional-collisional constitutive relations for granular materials, with application to plane shearing. J. Fluid Mech. 1987, 176, 67–93.
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Table 1. Constitutive Relations for Gas/Solid Flow model and parameter
equation
Syamlal-O’Brien drag model31
β ¼
3Fg Rg Rs jvg -vs j Res CD 4 vr2, s ds vr, s
! ð13Þ
where !2 4:8 0:63 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Res =vr, s
CD ¼
Res ¼
Fg Rg jvg -vs jds μg
ð15Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:06Res Þ2 þ 0:12Res ð2B -AÞ þ A2 Þ
vr, s ¼ 0:5ðA -0:06Res þ
ð14Þ
ð16Þ
where ð17aÞ
A ¼ Rg4:14 and
( B ¼
0:8Rg 1:28 Rg 2:65
ðRg e0:85Þ ðRg > 0:85Þ
! Fg Rs Rg jvg -vs j 3 β ¼ CD Rg -2:65 4 ds
Wen and Yu drag model33
ð17bÞ
ð18Þ
where CD ¼
24 ½1 þ 0:15ðRg Res Þ0:687 Rg Res
8 ! > Fg Rs Rg jvg -vs j 3 > > CD ðRg > 0:8Þ Rg -2:65 > > > 150 Rs μg þ 1:75 Fg Rs jvg -vs j > ðRg e0:8Þ : ds Rg ds 2
Gidaspow drag model32
where CD
¼ τg
gas-phase stress tensor
8 < 0:44 24 ¼ ½1 þ 0:15ðRg Res Þ0:687 : Rg Res
ðRes > 1000Þ ðRes e1000Þ
¼ 2 ¼ μt, g ½rvg þ ðrvg ÞT þ λg - μeff , g r 3 vg I 3
ð19Þ
ð20Þ
ð21Þ
ð22Þ
where μ t , g ¼ F g Cμ ð
kg 2 Þ εg
ps ¼ Rs Fs Θ½1 þ 2g0 Rs ð1 þ eÞ
particle phase pressure
2
radial distribution function
g0 ¼ 41 -
Rs Rs, max
!1=3 3 -1 5
¼ ¼ 2 τs¼ ¼ λs r 3 vs I þ μs ½rvs þ ðrvs ÞT - ðr 3 vs Þ I 3
solid-phase stress tensor
rffiffiffiffi 4 Θ λs ¼ Rs 2 Fs ds g0 ð1 þ eÞ 3 π
particles bulk viscosity
1245
ð23Þ
ð24Þ
ð25Þ
ð26Þ
ð27Þ
Energy Fuels 2010, 24, 1242–1259
: DOI:10.1021/ef901104g
Wang et al. Table 1. Continued
model and parameter
equation
particles shear viscosity Gidaspow32
Syamlal36
rffiffiffiffi pffiffiffiffiffiffiffi 2 4 Θ 10Fs ds πΘ 4 ps sin ω þ μs ¼ Rs Fs ds g0 ð1 þ eÞ 1 þ Rs g0 ð1 þ eÞ þ pffiffiffiffiffiffiffi 5 π 96Rs ð1 þ eÞg0 5 2 I2D
ð28Þ
rffiffiffiffi pffiffiffiffiffiffiffi 4 Θ Rs ds Fs πΘ 2 ps sin ω þ μs ¼ Rs Fs ds g0 ð1 þ eÞ 1 þ Rs g0 ð1 þ eÞð3e -1Þ þ pffiffiffiffiffiffiffi 5 π 5 6ð3 -eÞ 2 I2D
ð29Þ
collisions dissipation energy
4 γs ¼ 3ð1 -e ÞRs Fs g0 Θ ds 2
granular energy diffusion coefficient
ΓΘ ¼
2
! rffiffiffiffi Θ -r 3 vs π
ð30Þ
rffiffiffiffi pffiffiffiffiffiffiffi 2 150Fs ds πΘ 6 Θ 1 þ ð1 þ eÞg0 Rs þ 2Rs 2 Fs ds g0 ð1 þ eÞ 5 π 384ð1 þ eÞg0 φgs ¼ -3βΘ
transfer of kinetic energy
h
production of turbulent kinetic energy
Gk, q ¼ -Fq uq, i0 uq, j0
ð32Þ Duq, j Dxi
ð33Þ
Cgs ¼ 2
Cgs Csg
Csg ¼ 2
ηgs 1 þ ηgs
ð31Þ
ð34Þ ! ð35Þ
where ηgs ¼
turbulent drag term, modeled as diffusivity
0:27βks sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ðvg -vs Þ2 Rg εs ð2Fg þ Fs Þ 1 þ ð1:8 -1:35 cos2 θÞ 2 ks
Ds Dg rRs rRg βðvs -vg Þ ¼ βðU s -U g Þ þ β 0:75Rs 0:75Rg where
and
! ð37Þ
! " !# b þ ηgs b þ ηgs 2 2 2 ηgs τF , gs þ ks þ bks τF , gs Dg ¼ ks 3 3 3 1 þ ηgs 1 þ ηgs
ð38Þ
! " !# b þ ηsg b þ ηsg 2 2 2 Ds ¼ kg ηsg τF , sg þ ks þ bkg τF , sg 3 3 3 1 þ ηsg 1 þ ηsg
ð39Þ
with
! -1 Fs þ 0:5 b ¼ 1:5 Fg τF , sg ¼
Πk ¼
ð36Þ
! Rs Fs Fs þ 0:5 β Fg
ð40Þ
ð41Þ
( ! " ! #) b þ ηsg β Ds Dg 2kg rRs -2kg -ðvg -vs Þ rRg ð42Þ Rg Fg 1 þ ηsg 0:75Rs 0:75Rg εg Πε ¼ C3ε ð ÞΠk kg
mixture density
Fm ¼
n X i ¼i
1246
Ri Fi
ð43Þ
ð44Þ
Energy Fuels 2010, 24, 1242–1259
: DOI:10.1021/ef901104g
Wang et al. Table 1. Continued
model and parameter
equation n P
mixture velocity
Ri Fi vi vm ¼ i ¼in P Ri Fi
ð45Þ
i ¼i
mixture turbulent viscosity
μt, m ¼ Fm Cμ
! ð46Þ
Gk, m ¼ μt, m ðΔvm þ ðΔvm ÞT Þ : Δvm
ð47Þ
Cμ ¼ 0:09, C1ε ¼ 1:42, C2ε ¼ 1:92, C3ε ¼ 1:2, σ k ¼ 1, σ ε ¼ 1:3
ð48Þ
production of turbulent kinetic energy constants
k2 ε
Table 2. Parameters Used in the Simulations parameter
value
particle diameter, ds particle density, Fs gas density, Fg gas viscosity, μg superficial gas velocity, ug solid flux, Gs maximum solid volume fraction, Rs,max particle-particle restitution coefficient, e particle-wall restitution coefficient, ew specularity coefficient, j time step, Δt
67 μm 1500 kg/m3 1.225 kg/m3 1.78 10-6 Pa s 8 m/s 300 kg/(m2 s), 550 kg/(m2 s) 0.62 0.99, 0.95, 0.9, 0.8, 0.7 0.99, 0.95, 0.9, 0.7, 0.6 0.6, 0.01, 0.001, 0.0001, 0 1 10-4 s
where the specularity coefficient (j) describes the roughness of the wall. A value of j = 0 refers to perfectly specular collisions (smooth) and a value of j = 1 refers to perfectly diffuse collisions (rough). It is specifically used in multiphases with granular flow to express the slip coefficient (B).
Θw is the granular temperature at the wall, and ew is the restitution coefficient between the particle and the wall. As noted by the previous work,38-40 particle-wall friction and restitution coefficients would strongly affect the flow. Nevertheless, these two coefficients are difficult to measure and different values have been used in the literature without a clear explanation of their choices.40,41 To evaluate their effects on flow behaviors and obtain optimal values, different specularity coefficients and particle-wall restitution coefficients were investigated in this work. Details are listed in Table 2. Simulations were performed in three-dimensional domains. To obtain structured grid cells, geometrical systems were segmented into several relatively regular parts. The hexahedral grids, which were the main ones, and tetrahedral grids were applied to generate the mesh of each part. Simulations were performed using a high-performance personal computer (PHPC100, made by Dawning of China) with 40 CPU and 32GMb memory. The governing equations were solved using the finite-volume approach. Second-order upwind discretization was used for both momentum and volume fraction solutions. The phase-coupled SIMPLE (PCSIMPLE) algorithm,42 which is an extension of the SIMPLE algorithm to multiphase flows, was used to solve the pressure-velocity coupling and correction. A time step of 0.0001 s with 20 maximum iterations per time step was used. The convergence of the solution was based on residual monitorings of velocities, volume fractions, energies, and
(38) Li, J.; Kuipers, J. A. M. Effect of competition between particle-particle and gas-particle interactions on flow patterns in dense gasfluidized beds. Chem. Eng. Sci. 2007, 62, 3429–3442. (39) Goldschmidt, M. J. V.; Kuipers, J. A. M.; van Swaaij, W. P. M. Hydrodynamic modelling 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.
(40) 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. (41) Wang, W.; Li, Y. Hydrodynamic Simulation of Fluidization by Using a Modified Kinetic Theory. Eng. Chem. Res. 2001, 40, 5066–5073. (42) Vasquez, S.; Ivanov, V. A phase coupled method for solving multiphase problems on unstructured meshes. Presented at the ASME FED Summer Meeting, Boston, 2000.
Figure 1. System geometry for simulations of the Parssinen and Zhu34 experiments.
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: DOI:10.1021/ef901104g
Wang et al.
some relative parameters of the gas and particle phases. When all the residuals were
