Acta Mech Sinica (2005) 21, 228–234 DOI 10.1007/s10409-005-0037-7
R E S E A R C H PA P E R
Yong Yu · Lixing Zhou · Baoguo Wang · Feipeng Cai
A USM- two-phase turbulence model for simulating dense gas-particle flows∗
Received: 23 June 2004 / Accepted: 4 November 2004 / Revised: 7 February 2005 / Published online: 31 May 2005 © Springer-Verlag 2005
Abstract A second-order moment two-phase turbulence model for simulating dense gas-particle flows (USM- model), combining the unified second-order moment twophase turbulence model for dilute gas-particle flows with the kinetic theory of particle collision, is proposed. The interaction between gas and particle turbulence is simulated using the transport equation of two-phase velocity correlation with a two-time-scale dissipation closure. The proposed model is applied to simulate dense gas-particle flows in a horizontal channel and a downer. Simulation results and their comparison with experimental results show that the model accounting for both anisotropic particle turbulence and particle-particle collision is obviously better than models accounting for only particle turbulence or only particle-particle collision. The USM- model is also better than the k-ε-kp- model and the k-ε-kp-εp- model in that the first model can simulate the redistribution of anisotropic particle Reynolds stress components due to inter-particle collision, whereas the second and third models cannot. Keywords Turbulence · Two-phase flow · Second-order moment model
∗ The project supported by the Special Funds for Major State Basic Research of China (G-1999-0222-08), the National Natural Science Foundation of China (50376004), and Ph.D. Program Foundation, Ministry of Education of China (20030007028)
Y. Yu · L. X. Zhou (B) Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China E-mail:
[email protected] Y. Yu · B. G. Wang School of Mechanico-Electrical Engineering, Beijing Institute of Technology, Beijing 100081, China F. P. Cai Institute of Chemical Engineering, University of Petroleum, Beijing 102249, China
1 Introduction Dense gas-particle flows are encountered in fluidized beds, riser and downer reactors and the near-wall zone of dilute swirling gas-particle flows. Particle-particle collision plays an important role in the behavior of two-phase turbulence; it is simulated using two approaches: Eulerian-Lagrangian and Eulerian-Eulerian approaches. In the Eulerian-Lagrangian approach, numerous discrete particles are tracked; inter-particle collisions are simulated using a hard-sphere or a soft-sphere model [1]. In the Eulerian-Eulerian approach (two-fluid approach), the kinetic theory of dense gas-particle flows proposed by Gidaspow [2] is widely adopted to simulate particle-particle collision mainly for laminar dense gasparticle flows in the last decade of past century. This theory is based on the analogy between the dense-gas kinetic theory and the particle random fluctuation due to particle collision, which causes the transfer of particle momentum and produces particle pressure and viscosity. The particle pressure and viscosity depend on the magnitude of small-scale particle fluctuations, characterized by the particle pseudo-thermal energy. This energy is generated by the particle stress and is dissipated through the inelastic collisions between particles. Lun, Savage, and Jeffrey [3] and Gidaspow [2] derived the full equations of kinetic theory for granular flows. Sinclair and Jackson [4] first applied this theory to set up a laminar gas-phase and laminar particle-phase model to simulate the fully developed flow in vertical pipes. Considering the effect of gas turbulence, Bolio, Yasuna, and Sinclair [5] accounted for both gas turbulence, modeled by a low-Re k - ε model, and particle fluctuation due to collision. Samuelsberg and Hjertager [6] simulate the gas-particle flow in riser reactors using the kinetic theory, where the gas turbulence is modeled using LES. In fact, for high-velocity dense gas-particle flows in risers and downers, besides gas turbulence, it is necessary to account for both small-scale fluctuations due to particle-particle collision and large-scale fluctuations due to particle turbulence. Zhou [7] proposed a k - ε - kp model and a USM model to simulate particle turbulence. Cheng et al. [8] proposed a
A USM- two-phase turbulence model for simulating dense gas-particle flows
k-ε-kp- model by combining the k-ε-kp two-phase turbulence model proposed by Zhou with the model proposed by Gidaspow to account for both particle-particle collision and particle turbulence. However, the interaction between particle turbulence and inter-particle collision has not been taken into account. Zheng et al. [9] proposed a k-ε-kp-εp- model to include the effect of particle collision on particle turbulence. But for most practical gas-particle flows in pipes, the turbulence is not isotropic; the radial component is only about one fifth of the axial component, therefore the isotropic k-ε-kp- and k-ε-kp-εp- models will over-predict the lateral particle mixing. Furthermore, in both k-ε-kp- and kε-kp-εp- models the two-phase velocity correlation vpi vgi is simply closed using a semi-empirical dimensional analysis as kkp or ck − kp , proposed by Zhou et al. in early years. This model is now proved to be not in agreement with the experimental results. It was observed in experiments that the two-phase velocity correlation is always less than the gas and particle Reynolds stress themselves; the particle-source term in the particle turbulent kinetic energy equation should be always a dissipation term. Therefore, the two-phase velocity correlation should be modeled by transport equations or algebraic expressions simplified from transport equations rather than by dimensional analysis. In this paper, a second-order moment two-phase turbulence model for simulating dense gas-particle flows called USM- model, combining the USM two-phase turbulence model for dilute gas-particle flows with the particle kinetic theory for inter-particle collision, is proposed. The particleparticle collision is modeled by the particle kinetic theory, whereas the gas and particle turbulence are simulated by the USM model. The two-phase velocity correlation is simulated by the transport equations using a two-time-scale dissipation closure. The dense gas-particle flows in a horizontal channel and a downer are simulated using the proposed model. 2 The USM-Θ two-phase turbulence model for dense gas-particle flows For dense gas-particle flows the volume fractions of gas and particle phases may vary in a wide range. It is more convenient to use the mass-weighed time averaging instead of a simple time averaging. Let us define ρp = αp ρpm ; ρg = αg ρgm ; αg + αp = 1, where ρp , ρg are apparent particle and gas densities, ρpm , ρgm are particle and gas material densities. The forces other than drag and gravitational forces and the fluctuation of particle source term due to reaction are neglected. The mass-weighted time averaging is defined by φk = φ˜ k + φk , φ˜ k = αk ρkm φk αk ρkm , φ˜ k = αk ρkm φk αk ρkm = 0. The time averagings for the volume fraction, pressure, and shear stresses are taken to get the following expressions, αk = α¯ k + αk , pk = p¯ k + pk , τkj i = τ¯kj i + τkj i , where the subscript k denotes gas phase g or particle phase p. Starting from Gidaspow’s basic equations for laminar dense gas-particle flows, using the mass-weighed and time
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averaging and the closure method as that used in gas-phase turbulence modeling, the generalized closed form of massweighed averaged two-phase continuity, momentum, Reynolds stress equations for isothermal gas-particle flows can be obtained as follows: Gas continuity equation: ∂(α¯ g ρgm ) ∂(α¯ g ρgm u˜ gk ) = 0, + ∂t ∂xk Particle continuity equation:
(1)
∂(α¯ p ρpm ) ∂(α¯ p ρpm u˜ pk ) = 0, + ∂t ∂xk Gas momentum equation:
(2)
∂(α¯ g ρgm u˜ gi ) ∂(α¯ g ρgm u˜ gk u˜ gi ) + ∂t ∂xk ∂ p¯ = α¯ g ρgm gi − α¯ g − β(u˜ gi − u˜ pi ) ∂xi ∂ τ¯gik − α¯ g ρgm ugi ugk , + ∂xk Particle momentum equation: ∂(α¯ p ρpm u˜ pi ) ∂(α¯ p ρpm u˜ pk u˜ pi ) + ∂t ∂xk ∂ p¯ p ∂ p¯ = α¯ p ρpm gi − α¯ p − + β(u˜ gi − u˜ pi ) ∂xi ∂xi ∂ τ¯pik − α¯ p ρpm upk upi , + ∂xk where the particle drag coefficients are taken as
(3)
(4)
αp2 µg
αp ρg |ug − up | + 1.75 , αg < 0.8, αg dp2 dp αp ρg |ug − up | −2.65 3 β = CD αg , αg ≥ 0.8, 4 dp β = 150
and
24 1 + 0.15Rep0.687 , Rep CD = 0.44, Rep > 1000, αg ρg dp |ug − up | . Rep = µg
CD =
Rep ≤ 1000,
The particle pressure p¯ p and the particle viscosity stress τ¯pik in the particle momentum equation represent the effect of particle-particle collision. The gas Reynolds stress equation is ∂ α¯ g ρgm u˜ gk ugi ug·j ∂ α¯ g ρgm ugi ug·j + ∂t ∂xk = Dg,ij + Pg,ij + g,ij − εg,ij + Gg,gp,ij , (5) where the terms on the right-hand side of Eq. (5) stand for the diffusion term, shear production term, pressure-strain term, dissipation term, and gas-particle interaction term, respectively. These terms in their closed form are
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∂ugi ug·j kg ∂ Cg α¯ g ρgm ugk ugl , ∂xk εg ∂xl ∂ u˜ gi ∂ u˜ g·j Pg,ij = −α¯ g ρgm ugk ug·j + ugk ugi , ∂xk ∂xk ∂ u˜ gi Pg = −α¯ g ρgm ugk ugi , ∂xk g,ij = g,ij,1 + g,ij,2 εg 2 = −Cg1 α¯ g ρgm ugi ug·j − kg δij 3 kg 2 −Cg2 Pg,ij − Pg δij , 3 2 εg,ij = δij α¯ g ρgm εg , 3 Dg,ij =
Gg,gp,ij = β upi ug·j + ugi up·j − 2ugi ug·j . The particle Reynolds stress equation is ∂ α¯ p ρpm upi up·j ∂t
+
∂ α¯ p ρpm u˜ pk upi up·j
∂xk = Dp,ij + Pp,ij + p,ij − εp,ij + Gp,gp,ij ,
(6)
where the terms on the right-hand side of Eq. (6) stand for the diffusion term, shear production term, pressure-strain term, dissipation term, and gas-particle interaction term, respectively. These terms in their closed form are ∂upi up·j kp ∂ Cp α¯ p ρpm upk upl , ∂xk ∂xl εp ∂ u˜ pi ∂ u˜ p·j Pp,ij = −α¯ p ρpm upk up·j , + upk upi ∂xk ∂xk p,ij = p,ij,1 + p,ij,2 εp 2 = −Cp1 α¯ p ρpm upi up·j − kp δij kp 3 2 −Cp2 Pp,ij − Pp δij , 3 2 εp,ij = δij α¯ p ρpm εp , 3 Dp,ij =
Gp,gp,ij = β upi ug·j + up·j ugi − 2upi up·j .
Comparing with the USM two-phase turbulence model for dilute gas-particle flows, after accounting for the particle-particle collision, pressure-strain term and dissipation term appear in the particle Reynolds stress equation because of particle pressure and particle viscosity. These terms represent the redistribution and dissipation of particle Reynolds stress caused by particle-particle collision. The equations of dissipation rate of turbulent kinetic energy for gas and particle phases is
∂(α¯ g ρgm εg ) ∂(α¯ g ρgm u˜ gk εg ) + ∂t ∂xk kg ∂εg ∂ Cg α¯ g ρgm ugk ugl = ∂xk εg ∂xl εg + [cε1 Pg + Gg,gp − cε2 α¯ g ρgm εg ], kg ∂(α¯ p ρpm εp ) ∂(α¯ p ρpm u˜ pk εp ) + ∂t ∂xk k ∂εp ∂ p = α¯ p ρpm Cpd upk upl ∂xk εp ∂xl εp + [Cεp,1 Pp + Gp,gp − Cεp,2 α¯ p ρpm εp ], (7) kp where ∂ u˜ gi Pg = −α¯ g ρgm ugk ugi , ∂xk Gg,gp = 2β(kgp − kg ), ∂ u˜ pi , Pp = −α¯ p ρpm upi upk ∂xk Gp,gp = 2β(kpg − kp ). It can be shown that the dissipation of particle turbulent kinetic energy has been merged into the “internal energy”, i.e. the pseudo particle temperature. In both k-ε-kp- and k-ε-kp-εp- models the gas-particle turbulence interaction term is simply closed us p ing a dimensional analysis as Gpg = 2β Cp kkp − kp and Gpg = 2β ckg − kp , which can be greater than zero. It is found in many experiments that the two-phase velocity correlation is always smaller than gas and particle turbulent kinetic energy. So this interaction term will always be a dissipation term for turbulence of both phases. Therefore, in this work the two-phase velocity correlation is modeled by the following transport equation ∂upi ug·j
∂upi ug·j + u˜ gk + u˜ pk ∂t ∂xk = Dg,p,ij + Pg,p,ij + g,p,ij − εg,p,ij + Tg,p,ij , (8) where the terms on the right-hand side of Eq. (8) are the diffusion term, shear production term, pressure-strain term, dissipation term, and gas-particle interaction term, respectively. These terms in their closed form are k ∂u u kg ∂ p pi g·j upk upl + ugk ugl Cgp3 , Dg,p,ij = ∂xl ∂xk εp εg ∂ u˜ g·j ∂ u˜ pi Pg,p,ij = −upi ugk − upk ug·j , ∂xk ∂xk ∂ u˜ gi ∂ u˜ pi 1 Pg,p = − upi ugk + upk ugi , 2 ∂xk ∂xk g,p,ij = g,p,ij,1 + g,p,ij,2 Cg,p,1 2 =− upi ug·j − kgp δij τrp 3 2 −Cg,p,2 Pg,p,ij − Pg,p δij , 3
A USM- two-phase turbulence model for simulating dense gas-particle flows
Tg,p,ij
α¯ p ρpm upi up·j + α¯ g ρgm ugi ug·j β . = α¯ g ρgm α¯ p ρpm − α¯ g ρgm + α¯ p ρpm ug·j upi
It is assumed that dissipation term in Eq. (8) is proportional to the two-phase correlation component itself divided by a time scale. This time scale is taken as the smaller one from the gas turbulence time scale and the particle relaxation time εg,p,ij =
upi ug·j
. The pseudo particle temperamin{τrp , k/ε} ture transportation equation is ¯ ¯ ∂(α p ρpm u˜ pk ) 3 ∂(α¯ p ρpm ) + 2 ∂t ∂xk ¯ ∂ ∂ 3 αp ρpm upk + ¯ =− ∂xk 2 ∂xk ∂ u˜ ∂ u˜ pi ∂ u˜ pi pk +µp + + µp εp ∂xi ∂xk ∂xk ∂ u˜ pl 2 ∂ u˜ pl 2 −p¯ p + ξp − µp − γ¯ . (9) ∂xl 3 ∂xl The first term on the right hand side of equation (9) is diffusion term. It can be closed using the gradient modeling ∂ ¯ kp2 ∂ Cp α¯ p ρpm + ¯ , ∂xk εp ∂xk which is assumed to be an isotropic one. It is necessary to point out that for the terms in Eq. (9) such as ¯ u˜ pi , u¯ pj ), τ¯pij = τpij (α¯ p , ,
¯ γ¯ = γ (α¯ p , ),
a simplified treatment is used in averaging. Correlations between the particle pseudo-temperature and other quantities, such as volume fraction or velocity, are not considered. It is necessary to do so, because if we do exact averaging on these terms, the equations will be much more complex and contain too many correlation terms. The closure of these terms has no enough theoretical bases. According to the kinetic theory, the terms in Eq. (9) are as follows: Radial distribution function: α 1/3 −1 p , g0 = 1 − αp,max Solid-phase pressure: pp = αp ρpm [1 + 2(1 + e)αp g0 ], Solid-phase shear viscosity: 2 2µp,dil 4 µp = 1 + (1 + e)g0 αp (1 + e)g0 5
4 2 + αp ρpm dp g0 (1 + e) , 5 π √ 5 ρpm dp π, µp,dil = 96
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Solid-phase bulk viscosity:
4 2 ξp = αp ρpm dp g0 (1 + e) , 3 π Collision energy dissipation: 4 ∂u k,p 2 2 − . γ = 3(1 − e )αp ρpm g0 dp π ∂xk Transport coefficient of particle temperature: 2 6 2 θ,dil 1 + (1 + e)g0 αp θ = (1 + e)g0 5
θ 2 , +2αp ρp dp g0 (1 + e) π √ 75 ρpm dp πθ , θ,dil = 384 where e is the restitution coefficient, a measure of the inelasticity of particle-particle collisions; its value ranges from zero for perfectly inelastic collisions to unity for elastic collisions. The value of e directly influences the relevant quantities in the kinetic theory such as the particle viscosity, particle pressure and dissipation of particle temperature, especially when e is equal to 1, there will be no dissipation of kinetic fluctuation energy characterized by particle temperature due to elastic collisions. In the present computation e = 1.0 is adopted. 3 Simulation of gas-particle flows in a horizontal channel At first, the USM- model is used to simulate fully developed gas-particle flows in a horizontal channel measured by Kussin & Sommerfeld [10] using PDPA. The channel is a horizontal one with a length of 6 m, a height of 35 mm and a width of 350 mm. So, almost two-dimensional flow conditions can be established in the core of the channel. The measurements were performed close to the end of the channel at a distance of 5.8 m from the entrance. The average inlet velocity is 19.7 m/s and the particles mass loading is 0.3. The particle phase is glass beads with a size of 100 µm and density of 2.5×103 kg/m3 . The upper and lower channel walls were made of stainless steel plates with different roughness. The grid nodes are taken as 1201×36 in computation. For boundary conditions, the inlet two-phase velocities, normal components of Reynolds stress, and particle number density or volume fraction are given by experiments; the shear stresses are given by the eddy-viscosity assumption, the inlet 1.5 (λ · L), where dissipation rate is given as: εin = cµ0.75 kin cµ = 0.09, λ = 0.07, L is the inlet characteristic length, the inlet particle pseudo temperature is taken as in = kp,in . The fully developed flow condition of two phases are taken at the exit; at the walls no-slip condition is used for gas velocity, the gas Reynolds stress components as well as gas velocities are determined via production term including the effect of wall functions for near-wall grid nodes; particle wall
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Fig. 1 Particle concentration profiles (low roughness)
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Fig. 4 Particle vertical rms velocity (low roughness)
Fig. 5 Particle concentration profiles (high roughness) Fig. 2 Particle horizontal mean velocity (low roughness)
Fig. 6 Particle horizontal mean velocity (high roughness) Fig. 3 Particle horizontal rms velocity (low roughness)
boundary conditions considering wall roughness proposed by Zhang et al. [11] are used. Zero gradient condition for particle tangential velocity was used in the k-ε-kp- model. The k-ε-kp- model was solved using a commercial CFD software—CFX. The simulation results and their comparison with experiments are shown in Figs. 1 to 8. Figures 1, 2, 5, and 6 give the predicted normalized particle concentration profiles and particle horizontal mean velocity for low-roughness and high-roughness wall conditions,
respectively. It can be seen that the predictions using both USM- and USM models are in good agreement with the measured results, but the k-ε-kp- model and the DSM- model, which is being a combination of the Reynolds stress model of gas turbulence with the model of particle collision, give much lower concentration and velocity near the top and much higher concentration and velocity near the bottom. However, this phenomena is not observed in experiments. These results indicate that in the case studied the particleparticle collision does not significantly affect the particle concentration and velocity distributions, but the particle turbulence do affect them very much, the large-scale particle
A USM- two-phase turbulence model for simulating dense gas-particle flows
Fig. 7 Particle horizontal rms velocity (high roughness)
Fig. 8 Particle vertical rms velocity (high roughness)
fluctuation leads to more uniform distributions of particle concentration and velocity. Figures 3, 4, 7 and 8 show the predicted particle horizontal fluctuation velocity for the cases of low and high wall roughness, respectively. The USM- model gives the prediction results much closer to the measurement results than the USM model does. It can be seen that in such a narrow channel we should account for not only the particle-wall collision but also particle-particle collision. The effect of particle-particle collision on particle turbulence is significant and cannot be neglected. It leads to the particle Reynolds stress redistribution and dissipation in different directions. It reduces particle turbulence in case of low wall roughness, but increases particle turbulence in case of high wall roughness. Figure 8 gives the predicted particle vertical fluctuation velocity for the case of high roughness. It can be seen that the particleparticle collision reduces the particle turbulence in this direction. Unlike the cases of dilute gas-particle flows, which are strongly anisotropic, in the case of dense gas-particle flows, the particle-particle collision leads to isotropization of particle turbulence.
The computational grid nodes are 71×71. For boundary conditions, at the downer inlet, the air enters with a parabolic velocity profile whereas the solids enter the channel with a flat velocity profile. The solid volume fraction at the inlet is guessed according to the corresponding experimental condition. The inlet normal components of Reynolds stress, shear stresses are given by the eddy-viscosityassumption, the 1.5 (λ · L), where inlet dissipation is given as: εin = cµ0.75 kin cµ = 0.09 ,λ = 0.07, L is inlet characteristic length, the inlet particle pseudo temperature is given as in = kp,in . The fully developed flow condition of two phases are taken at the exit; at the walls no-slip condition is used for gas velocity, the gas Reynolds stress components as well as gas velocities are determined via production term including the effect of wall functions for near-wall grid nodes; The Particle wall boundary conditions considering particle-wall collision proposed by Zhang [11] are used. Figures 9 and 10 give the predicted particle volume fraction and particle velocity, respectively, using the USM- model, k-ε-kp- model, USM model (neglecting interparticle collision) and DSM- model (neglecting particle turbulence) and their comparison with experiments. From Fig. 9 it can be seen that both the USM- and kε-kp- model can properly predict the “skin effect” of particle concentration in the downer. The DSM- model gives too high particle concentration near the wall due to neglecting particle turbulence, i.e. under-predicts the turbulent mixing, while the USM model also gives a little higher particle
Fig. 9 Particle volume fraction
4 Simulation of gas-particle flows in a downer The proposed model is also used to simulate developed dense gas-particle flows in a downer, measured by Jin et al. [12] . The downer is 7 m high with a diameter of 0.14 m. The average gas inlet velocity is 4.33 m/s and particles flow rate is Gs = 70 kg/m2 s. The particle phase is FFC particles with a size of 54 µm and material density of 1 545 kg/m3 .
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Fig. 10 Particle velocity
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and inter-particle collision are important in modeling dense gas-particle flows.
5 Conclusions
Fig. 11 Particle axial fluctuation velocity
(1) The proposed USM- two-phase turbulence model for dense gas-particle flows, taking into account both anisotropic particle large-scale fluctuation due to particle turbulence and small-scale fluctuation due to particle-particle collision is better than existing two-phase turbulence models, which do not account for either particle-particle collision or particle turbulence. (2) The particle turbulence leads to more uniform distributions of particle concentration and velocity. (3) The particle-particle collision leads to redistribution of particle turbulence in different directions.
References
Fig. 12 Particle radial fluctuation velocity
concentration than that measured. It can be seen that both particle turbulence and particle collision should affect the mixing of dense two-phase flows. Figure 10 shows the predicted particle velocity using different models and their comparison with experiments. It can be seen that USM- and USM models give better predicted results than the isotropic k-εkp- model and the DSM- model neglecting the particle turbulence do. Figures 11 and 12 give the predicted particle axial and radial fluctuation velocities using the USM-, k-ε-kp- and USM models. The USM- model gives the proper tendency– higher axial fluctuation velocity than the radial fluctuation velocity, while the k-ε-kp- model gives equal axial and radial fluctuation velocities, apparently over-predict the mixing in the lateral direction. The USM model gives higher axial and radial fluctuation velocities than those given by the USM- model. This implies that the inter-particle collision reduces particle turbulence. So, both anisotropic particle turbulence
1. Tsuji Y., Kawaguchi T., Tanaka T.: Discrete particle simulation of 2-dimensional fluidized bed. Powder Technology 77(1), 79–87 (1993) 2. Gidaspow D.: Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. New York: Academic Press, 1994 3. Lun C.K.K., Savage S.B., Jeffrey D.J.: Kinetic theories for granular flow:inelastic particles in Couette flow and slightly inelastic particles in general flow field. J Fluid Mech 140, 223–256 (1984) 4. Sinclair J.L., Jackson R.: Gas-particle flow in a vertical pipe with particle-particle interactions. AIChE J 35(9), 1473–1486 (1989) 5. Bolio E.J., Yasuna J.A., Sinclair J.L.: Dilute turbulent gas-solid flow in risers with particle-particle interactions. AIChE J. 41(6), 1375–1388 (1995) 6. Samuelsberg A., Hjertager B.H.: Computational modeling of gas/particle flow in a riser. AIChE J 42(6), 1536–1546 (1996) 7. Zhou L.X.: Theory and Numerical Modeling of Turbulent Gas-Particle Flows and Combustion. Beijing: Science Press, 1994 8. Cheng Y., Guo Y.C., Wei F., Lin W.Y., Jin Y.: Modeling the hydrodynamics of downer reactors based on kinetic theory. Chem Eng Sci 54(13–14), 2019–2027 (1999) 9. Zheng Y., Wan X.T., Qian Z., Wei F., Jin Y.: Numerical simulation of the gas-particle turbulent flow in riser reactor based on k-ε-kp-ε p- two-fluid model. Chem Eng Sci 56(24), 6813–6822 (2001) 10. Kussin J., Sommerfeld M.: Experimental studies on particle behaviour and turbulence modification in horizontal channel flow with different wall roughness. Exp Fluids 33(1), 143–159 (2002) 11. Zhang X., Zhou L.X.: Simulation of gas-particle channel flows using a two-fluid particle-wall collision model accounting for wall roughness. In: Proc. of 4th ASME/JSME Joint Fluids Eng. Conf., Honolulu, Hawaii, USA, 2003, Paper, FEDSM2003-45750 12. Wang Y., Bai D.R., Jin Y.: Hydrodynamics of cocurrent downflow circulating fluidized bed (CDCFB). Powder Technology 70, 271– 275 (1992)
