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Ind. Eng. Chem. Res. 2010, 49, 10116–10124
Simulation of Flow Behavior of Particles in a Liquid-Solid Fluidized Bed Shuyan Wang,*,† Xiaoqi Li,† Yanbo Wu,‡ Xin Li,† Qun Dong,† and Chenghai Yao§ School of Petroleum Engineering, Daqing Petroleum Institute, Daqing, 163318, China, School of EnVironmental and Chemical Engineering, Dalian Jiaotong UniVersity, Dalian, 116028, China, The Fourth Oil Production Factory, Daqing Oilfield Corporation Ltd., PetroChina, Daqing, 163511, China
Understanding hydrodynamics of liquid-solid fluidized beds is crucial in proper scale-up and design of these reactors. Computational fluid dynamics (CFD) models have shown promise in gaining this understanding. In this paper, a two-dimensional CFD model, using an Eulerian-Eulerian two-fluid model incorporating the kinetic theory of granular flow, is used to describe the liquid-solid two-phase flow in a liquid-solid fluidized bed. The predicted pressure gradient data and concentrations are found to agree with experimental data published in the literature. Furthermore, the model is used to investigate the influences of the superficial liquid velocity and the sold particle size on the distribution of solids concentration. The simulation results showed that the solids concentration has a relatively uniform distribution and high bed expansion with the increase of liquid velocity and decrease of particles sizes. Solids mean axial velocities decrease with the decrease in superficial liquid velocity. The static bed height plays a minor role on the effect of velocity and concentration distribution in numerical simulations of liquid-solid bubbling fluidized beds. 1. Introduction One of the most challenging environmental problems is the removal of oil from wastewater. A large amount of wastewater is generated by petroleum and petrochemical industrial companies that produce or handle oil and other organic compounds. Some of these organic materials are discharged into the environment, for example, offshore oil spills and oil released during oil well extraction.1,2 Oily wastewater discharged into the environment causes serious pollution problems because the biodegradability of oil is very low and oily wastewater hinders biological processing at sewage treatment plants. Recently, the liquid-solid fluidization has been used to remove oil from wastewater. Flow behavior of hot water and particles was studied in a fluidized bed of 1 ton of oil-contaminated beach sand per h.3 The effect of sand particle sizes and superficial liquid velocity on the oil-sand separation was experimentally investigated. Surface-treated hydrophobic aerogel particles are fluidized by a flow of oil-contaminated water in an inverse fluidization mode.4 The hydrodynamic characteristics of inverse fluidized beds of aerogel particles were studied by measuring the pressure drop and bed expansion as a function of superficial velocity. The oil removal efficiency and capacity of the aerogel particles in the fluidized bed were found to depend mainly on the porosity and the fluid velocity. These studies indicate that the liquid-solid fluidization process is a feasible technique used in the oily wastewater treatments. The principles of liquid-solid fluidization have been extensively studied. The benefits of liquid-solid fluidization are low and constant pressure drop when operating above the minimum fluidization velocity, optimal mixing (contacting) between the solid particles and the liquid, good heat and mass transfer rates, and an adjustable porosity of the fluidized bed by changing the fluid velocity.5 These advantages should result in an increased adsorption/absorption capacity when removing oil or other organic compounds from water. * Corresponding author. Tel.: +010 0459 6507721. Fax: +010 0459 6507722. E-mail:
[email protected]; wangshuyan@ nepu.edu.cn. † Daqing Petroleum Institute. ‡ Dalian Jiaotong University. § Daqing Oilfield Corporation Ltd.
The modeling of fluidized bed reactors is challenging because of the complex flow behavior in liquid-solid fluidized beds. Of the various modeling tools, computational fluid dynamics (CFD) is the most promising for fluidized bed modeling. CFD is intended to include the key mechanisms of importance to predict accurate flow and other characteristics of fluidized beds for design, scale-up and optimization. The liquid-solid twophase flow model uses the principles of conservation of mass, momentum, and energy for each phase. The interactions between the two phases are expressed as additional source terms added to the conservation equations. The recently developed kinetic theory of granular flow,6-8 as reviewed by Arastoopour9 and Gidaspow et al.,10 treats the solids phase as another fluid with its own temperature, called the granular temperature, its own pressure due to particle collision, and its own viscosity. The granular temperature is the random kinetic energy of the particles per unit mass. 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. With regards to mathematical modeling, CFD simulations of the flow in liquid-solid fluidized beds gives very detailed information about the local values of concentrations and their spatial distributions, liquid phase flow patterns, and the intermixing levels of the individual phases, especially in the regions where measurements are either difficult or impossible to obtain. Such information can be useful in the understanding of the transport phenomena in liquid-solid fluidized beds. In the studies of Roy and Dudukovic,11 the granular flow model is applied to liquid-fluidized beds with coefficients of restitution less than one (implying inelastic collisions). Cheng and Zhu12 carried out the CFD simulations of hydrodynamics in solid-liquid circulating fluidized beds using a commercial code FLUENT 4.5. They studied the influence of column diameter (varied from 76 to 600 mm) on the radial velocity, turbulent kinetic energy, and granular temperature of solid particles. Lettieri et al.13 used CFX 4.4 code to simulate solid-liquid fluidized bed of lead particles in slugging mode using granular kinetic theory. They
10.1021/ie101139h 2010 American Chemical Society Published on Web 09/14/2010
Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010 14
also employed the empirical based drag laws of Wen and Yu. These CFD simulations have shown the gradual development of the slugging regime with the increase in the superficial liquid velocity from 80 to 300 mm/s. Cornelissen et al.15 simulated the solid-liquid fluidized bed having 1.13 mm glass particles in the range of superficial liquid velocities from 8.5 to 110 mm/s using the multifluid Eulerian model. Two-dimensional simulations in a solid-liquid fluidized bed were performed on the basis of an Eulerian two-fluid formulation of the transport equations for mass, momentum, and fluctuating kinetic energy.16 The solid-phase fluctuating motion model was derived in the frame of granular medium kinetic theory accounting for the viscous drag influence of the interstitial liquid phase. CFD simulation of bed expansion of mono size solid-liquid fluidized beds has been performed in creeping, transition, and turbulent flow regimes using a commercial CFD software FLUENT 6.2.17 The predicted values of bed porosity are in agreement with experiments. Though a large number of numerical simulations and experimental observations have been made, to understand the hydrodynamics in liquid-solid fluidized beds is still required. Quantitative understanding is also needed to explain the effects of liquid velocity and particles sizes. In this study, an attempt has been made to interpret the experimental observations published in the literature. The modeling is based on a twodimensional Eulerian-Eulerian approach in combination with kinetic theory of granular flow. The distributions of concentration and velocity are predicted in a liquid-solid fluidized bed, and the predictions are compared to experiments published in literature. 2. Liquid and Solid Two-Fluid Model In the present work, an Eulerian multifluid model, which considers the conservation of mass and momentum for the solid and liquid phases, has been adopted. The kinetic theory of granular flow, which considers the conservation of solid fluctuation energy, has been used for closure. Conservation equations of mass and momentum of both phases result from the statistical average of instantaneous local transport equations. The governing equations are given below. 2.1. Governing Equations. For simplification, the following hypotheses are considered: (1) both phases are assumed to be isothermal, 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 liquid phase and a single solid phase. The continuity for phase k (k ) l for liquid phase or s for solids) ∂ (ε F ) + ∇(εkFkuk) ) 0 ∂t k k
(1)
where εk is the concentration of each phases, u the velocity vector, and F the density. Mass exchanges between the phases, i.e., due to the reaction, is not considered. The momentum balance for the liquid phase is given by the Navier-Stokes equation, modified to include an interphase momentum transfer term ∂ (ε F u ) + ∇(εlFlulul) ) εl∇τl + εlFlg - εl∇p - β(ul - us) ∂t l l l (2) where g is the gravity acceleration, p the thermodynamic pressure, β the interface momentum transfer coefficient, and τl
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the viscous stress tensor. The stress tensor of liquid phase can be represented as 2 τl ) µf[∇ul + (∇ul)T] - µf(∇ul)I 3
(3)
where µf is the viscosity of liquid phase, and µf ) µl + µt. The eddy viscosity for the liquid phase is calculated as µt ) cµFlk2/ ε. Here, k represents the turbulent kinetic energy and ε represents the dissipation rate of turbulent kinetic energy. The turbulent kinetic energy and dissipation rate of liquid phase is described by a standard k-ε turbulence model. Assuming the influence of the dispersed particles on the liquid phase is neglected, the following two equations are the transport equations associated with these parameters
( )
µt ∂ (Flεlk) + ∇(Flεlulk) ) ∇ εl ∇k + ∂t σk εlµt[∇ul + ∇(ul)T]:∇ul - Flεlε
(4)
( )
µt ∂ (F ε ε) + ∇(Flεlulε) ) ∇ εl ∇ε + ∂t l l σε ε2 ε c1εlµt{[∇ul + ∇(ul)T]:∇ul} - Flεlc2 k k
(5)
The constants involved in the above equations are c1 ) 1.44, c2 ) 1.92, cµ ) 0.09, σk ) 1.0, and σε ) 1.0. These values have been shown to resolve the flow field in fluidized beds by Reddy and Joshi17 and Cheng and Zhu18 and have been used in the present work. The solids phase momentum balance is given by8 ∂ (ε F u ) + ∇(εsFsusus) ) -εs∇p - ∇ps + ∇τs + εsFsg + ∂t s s s β(ul - us) (6) where ps is the particle pressure, which represents the particle normal forces due to particle-particle interactions. The solids stress tensor can be expressed in terms of the bulk solids viscosity, ξs, and shear solids viscosity, µs, as
{
}
2 τs ) µs [∇us + (∇us)T] - (∇us)I + ξs∇usI 3
(7)
There are two possible mechanisms inducing the fluctuations of particle velocity: interparticle collisions and particle interactions with turbulent fluctuations in the liquid phase. Interparticle collisions play a crucial role in sufficiently dense suspensions. Equivalent to the thermodynamic temperature for gases, the granular temperature can be introduced as a measure for the energy of the fluctuating velocity of the particles.8 The granular temperature, θ, is defined as θ ) C2/3, where C is the particle fluctuating velocity. The equation of conservation of solids fluctuating energy can be expressed8 3 ∂ (ε F θ) + ∇(εsFsθ)us ) (-∇psI + τs):∇us + 2 ∂t s s ∇(ks∇θ) - γs - 3βθ + Dls (8)
[
]
where the particle pressure ps is based on the kinetic theory of granular flow. In this approach, both the kinetic and the collisional influence are taken into account. The kinetic portion describes the influence of particle translations, whereas the collisional term accounts for the momentum transfer by direct collisions.7,8 The particle pressure is calculated as follows
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ps ) εsFsθ + 2Fs(1 + e)εs2g0θ
(9)
where e is the restitution coefficient of particles, and go the radial distribution function at contact. The radial distribution function, go, can be seen as a measure for the probability of interparticle contact. The equation of Bagnold19 is used in this work
[ ( )]
g0 ) 1 -
1/3 -1
εs
(10)
εs,max
where εs,max is the maximum particle packing. The shear viscosity accounts for the tangential forces. It was shown by Lun et al.7 and Gidaspow8 that it is possible to combine different interparticle forces and to use a momentum balance similar to that of a true continuous fluid. In this work, the equation of particle viscosity is 4 µs ) εs2Fsdsgo(1 + e) 5
10Fsds√πθ θ + × π 96(1 + e)εsgo 4 1 + goεs(1 + e) 5
[
2
]
1/2
()
(11)
(12)
For the conductivity of granular energy ks, the correlation proposed by Gidaspow8 is used ks )
25Fsds√πθ 2 6 1 + (1 + e)goεl + 64(1 + e)go 5
[
]
2εs2Fsdsgo(1 + e)
( πθ )
1/2
(13)
The rate of dissipation of fluctuation kinetic energy due to particle collisions is
(
γs ) 3(1 - e2)εs2Fsgoθ
4 ds
)
θ - ∇us π
(14)
Dls )
( ) 18µl
2 4√πθgo d Fs
2
|ul - us | 2
(15)
2.2. Drag Model. In the momentum transport equations of the individual phases, the drag force is represented by the term β(ul - us), the product of the interphase momentum exchange coefficient β and the slip velocity. The correlations given by Gidaspow8 are often used. The correlation by Gidaspow8 is a combination of the works of Ergun21 and Wen and Yu;14 the formulation presented by Ergun21 is used at the porosity less than 0.8 where the suspension is dense, whereas the formulation by Wen and Yu14 is used at the porosity greater than 0.8 where the suspension is dilute. βE ) 150
(1 - εl)2µl (εlds)
2
+ 1.75
{
24 (1 + 0.15Re0.687) Re e 1000 Re 0.44 Re g 1000
(16b)
(17)
The transition proposed by Gidaspow8 makes the drag law discontinuous in solid concentration though it is continuous in Reynolds number. Physically, the drag force is a continuous function of both solid concentration and Reynolds number, and therefore the continuous forms of the drag law may be needed to correctly simulate liquid-solid fluidized beds. To avoid discontinuity of these two correlations, a switch function φ is introduced to give a smooth from the dilute regime to the dense regime22,23 arctan[150 × 1.75(0.2 - εs)] + 0.5 π
(18)
Thus, the interface momentum transfer coefficient becomes β ) (1 - φ)βE + φβWY
(19)
2.3. Boundary Conditions. The governing equations listed above are numerically solved with appropriate boundary and initial conditions. Initially, there are no motions for both the liquid phases and the particles in the bed. At the inlet, all velocities and porosity of liquid phase are specified. At the top, Neumann boundary conditions are applied to the liquid-particle flow. The liquid pressure is set to be 1 atm. At the inlet, the liquid velocity is constant with the concentration of unity. The particle velocity and the granular temperature are zero for the particle phase. At the wall, the tangential and normal velocities of liquid phase are set to zero (no slip condition). The normal velocity of the particles was also set at zero. The following boundary equations apply for the tangential velocity and granular temperature of particles at the wall24 6µsεs,max ∂us,w πF ε g √3θ ∂n
(19a)
√3πFsεsusgoθ3/2 ksθ ∂θw + ew ∂n 6εs,maxew
(20)
ut,w ) -
s s o
The last term Dls in eq 8 is the rate of energy dissipation per unit volume resulting from the transfer of liquid phase fluctuations to the particle phase fluctuations. In this study, the value of Dls is predicted using Koch’s expression20 as follows dsFs
Cd )
φ)
The bulk viscosity ξs formulates the resistance of solid particles to compression and expansion. The following equation given by Gidaspow8 is used in this work θ 4 ξs ) εs2Fsdsg0(1 + e) 3 π
3 Fl(1 - εl)|ul - us | -2.65 βWY ) Cd εl εl > 0.8 4 ds
Fl(1 - εl)|ul - us | εl e 0.8 εlds (16a)
θw ) -
Liquid-fluidized beds should, at least in principle, be similar to gas-fluidized beds. However, CFD has been applied explicitly to liquid fluidization in only a small number of cases.11,13,18,25 They adopted an Eulerian-Eulerian approach, simplified the flow field as two-dimensional, assumed all particles to be spheres of uniform sizes, and adopted uniform flow as the upstream boundary condition. Therefore, the liquid fluidization may provide an opportunity to evaluate alternative CFD models, the relevance of applying granular flow models may be questioned given the finding that collisions of particles may not actually occur when there is a liquid film separating adjacent particles. This is certainly a factor to be considered. Gidaspow and Lu26 suggested an “effective restitution coefficient” near 1. In the studies of Roy and Dudukovic11 and Cheng and Zhu,18 the granular flow model was applied to liquid-fluidized beds, with coefficients of restitution less than 1 (implying inelastic collisions), and with no explicit consideration of whether or not the lack of collisions invalidates the approach. In these two cases, good agreement was claimed between predictions and experi-
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Table 1. Parameters Used in the Solid-Liquid Fluidized Bed
Figure 1. Averaged concentration of particle along height.
mental results. In the present study, the simulations are carried out with the CFD code previously used to model gas-solid flow in a bubbling fluidized bed,27 and that incorporates the kinetic theory of granular flow in this work. This mathematical process is described by Patankar.28 This software allows free implementation of extra equations, boundary conditions, and differencing schemes. The granular kinetic theory and the granular equations described in the previous section are implemented into this code in this work. For solving the difference equations obtained from the differential equations, the higher order total variation diminishing method (TVD) scheme is used. This TVD scheme incorporates a modification to the higher order upwind scheme for hyperbolic systems. The solution of the pressure from the momentum equations requires a pressure correction equation, correcting the pressure and the velocities after each iteration of the discretized momentum equations. In all simulations, a constant time step of 1.0 × 10-5 s is used. Timeaveraged distributions of flow variables are computed covering a period of 30 s corresponding to 1-2 weeks of computational time on a PC (20GB hard disk, 128Mb RAM, and 600 MHz CPU). 2D simulations are performed in a liquid-solid fluidized bed. A preliminary study is initially conducted to aid the choice of the suitable grid resolution. Figure 1 shows the concentration of particles along height at three different grid resolutions. The grid resolution of 50 × 260 means 50 cells along the radial direction and 260 cells along the axial direction. The use of very coarse grids leads to underestimation of heights of fluctuation. The numerical simulations for the 18 × 170 and 50 × 260 mesh resolutions are very similar. Both simulations show the averaged concentration of particles increasing from the bottom, keeping a constant in the bed, and then decreasing at the bed surface, whereas the coarse grids of the 12 × 120 mesh shows somewhat different dynamic behavior. The grid size on the order of a few particle diameters used in this study is usually adequate for resolving the mesoscale structures of liquid-solid flows. Therefore, we choose the mid grid resolution of 38 × 170 in present simulations.
significance
Limtrakul et al.29
present simulations
diameter of particles (m) density of particles (kg/m3) bed width (m) bed height (m) static height of bed (m) minimum fluidization velocity (m/s) inlet liquid velocity (m/s) liquid density (kg/m3) liquid viscosity (kg/ms) restitution coefficient of particles (e) particle-wall restitution coefficient (ew) initial particle concentration grid number Nx × Nz
0.003 2500 0.14 1.5 0.45 0.0412 0.07 994 0.001 no no no no
0.003 2500 0.14 1.5 0.45 0.0412 0.07 994 0.001 0.9 0.9 0.598 38 × 170
Figure 2 shows the distribution of computed and measured time averaged axial velocity of solids in the liquid-solid fluidized bed. Limtrakul et al.29 measured solid velocity in a liquid fluidized bed by using the noninvasive CARPT facility. Both simulations and measurements show the solids move up at the center and down at the wall, the transition point of zero time-averaged axial velocity occurs at dimensionless radius between 0.6 and 0.7. Carlos and Richardson30 reported solids axial velocity profiles with 0.0088 m diameter glass beads in a 0.10 m diameter liquid-fluidized bed. The measurements were conducted by a photographic system in a transparent bed composed of colorless transparent solid and liquid with the same refractive index. They found that the axial velocity of particles is positive at the centerline and negative near the wall. This means that particles flow upward at the center and downward near the walls because more energy is dissipated by collision between the wall and particles. The circulation of particles is existed in the bed. The radial profiles of axial solid velocities in the middle section of the bed reported by both studies are qualitatively similar. The interphase momentum exchange coefficient model proposed by Gidaspow8 gives a large axial velocity of particles in the bed. The simulation results with drag model eq 19 fit quite well to the measurements. From Figure 2, the axial velocity profiles obtained from experiments and simulations are quantitatively in agreement in the area near the wall but not in the area near the center of the bed. In present simulations, a standard k-ε turbulence model for flow of single phase is incorporated to consider the high Reynolds number flows in the bed. The assumption involved in the model is limited to systems, where the interparticle collisions have no major role in the random fluctuations of liquid phase. Under such conditions, the turbulence of the solids phase is controlled mainly by the random motions in the liquid phase. Apparently, such a formulation requires the suspension to be dilute. Because-the solids concentration within the bubbling fluidized bed can be quite high, the validity of this k-ε
3. Simulation Results and Discusions 3.1. Comparison with Limtrakul et al. Experiments. The distribution of concentration and velocity was measured by Limtrakul et al.29 in a liquid-solid fluidized bed by means of a Gamma ray-based computer tomography (CT) and computeraided radioactive particle tracking (CARPT). The bed diameter is 0.14 m with 1.5 m height. The liquid is water and the solids are glass beads with the diameter and density of 0.003 m and 2500 kg/m3, respectively. Detail descriptions of the experiments can be found in Limtrakul et al.29 The parameters used for the simulations are given in Table 1.
Figure 2. Comparison of mean solids axial velocities in a liquid-solid fluidized bed.
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Figure 3. Simulated and measured solids concentration profile in the bed.
Figure 4. Simulated granular temperature as a function of concentration.
Table 2. Particle Properties and Parameters Used in Simulations significance
Zenit et al.31
Present simulations
diameter of particles (mm) density of part0icles (kg/m3 bed width (m) bed height (m) static height of bed (m) terminal velocity (cm/s) liquid density (g/m3) liquid viscosity (kg/ms) restitution coefficient of particles (e) particle-wall restitution coefficient (ew) initial particle concentration
2.06 2540 0.051 2.04 0.5 22.7 1000 0.001 no no no
2.06 2540 0.051 2.04 0.5 22.7 1000 0.001 0.5 and 0.9 0.4 and 0.9 0.5
turbulence model is questionable. Therefore, further investigation is required. Figure 3 shows the simulated and measured radial distribution of concentration of particles in the bed. Limtrakul et al.29 measured concentration of particles in a liquid fluidized bed by using the noninvasive computer tomography. Both simulations and experiments show that the concentration of particles is lower at the center than at the wall of the bed. This radial nonuniform distribution is due to nonuniform profile of axial solid velocity with upward higher velocity in the central region of the bed and downward lower velocity near the wall as shown in Figure 2. The cross-sectional average is taken as representative of the mean concentration of particles. The simulated mean solids concentration is obtained as 0.44, which is very close to the measured concentration of particles of 0.43.29 The interphase momentum exchange coefficient model proposed by Gidaspow8 gives a large concentration of particles in the center regime and a low concentration near the wall. Present simulations are in agreement with measured concentration of particles in the bed. 3.2. Comparison with Zenit et al. Experiments. The numerical test cases presented in this paper correspond to some of the numerous experimental conditions investigated by Zenit et al.31 in a 5.1 cm diameter cylindrical column. The choice of these test cases is driven by the interest of studying particle pressure in the solid-liquid fluidized bed. The solid pressure was measured by means of a piezoelectric dynamic pressure transducer. Continuous phase is water in all experiments and temperature is assumed to be 20 °C. The bed materials and diameters are reported in Table 2. Figure 4 shows the distribution of granular temperature at two different superficial liquid velocities as a function of concentration of particles. The existence of a maximum of the granular temperature is clearly exhibited by the simulations at a solid concentration close to 0.25 and 0.34 for superficial liquid velocity of 11.5 and 8.5 cm/s, respectively. The fluctuating motion of the dispersed phase originates from particle collisions and local instant variations of the concentration, which by
Figure 5. Simulated solids pressure as a function of concentrations.
Figure 6. Distribution of mean granular temperature at two coefficients of restitution.
continuity induce local velocity gradients. These velocity gradients then produce the fluctuating kinetic energy of the solid phase. Hence at low concentration of particles, the increase of the solid-phase fraction leads to an increase of the granular temperature. At high concentration, the fluctuations of the solid fraction are limited by the increase of the collision frequency, reducing the mean free path of the particles. The observation of a maximum on the granular temperature curves is therefore consistent with this production mechanism. Figure 5 shows the distribution of solids pressure as a function of concentration of particles. Roughly speaking, the simulated solids pressure increases with the increase of concentration of particles in the bed. At high concentration of particles, the solid pressure decreases again and tends to zero at packing because the particles hardly move. The mean values of granular temperature, θm ) ∑i N) 1θi/N, where N is the cell number, are computed in the bed. Figure 6 shows the distribution of mean granular temperature at two different coefficients of restitution as a function of mean concentration of particles in the bed. The mean granular temperature increases with the decrease of concentration of particles. With the decrease of restitution coefficient, the granular
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Figure 9. Instantaneous concentration of particles in a liquid-solid fluidized bed. Figure 7. Comparison of simulated and measured solids pressure.
Figure 10. Distributions of concentration of particles along bed height. Figure 8. Profile of solids pressure as a function of concentrations.
temperature is decreased due to more energy dissipated by collisions of particles. Figure 7 shows the distribution of mean N ps,i/N, as a function of concentration solids pressure, ps,m ) ∑i)1 of particles. With the decrease of restitution coefficient of particles, the solids pressure is decreased. The solids pressure measured by Zenit et al.31 has been recorded at the wall and at a given axial location that has not been specified by the authors. Both simulations and measurements show that the solids pressure increases as the solid concentration increases, and then decreases at high concentration. The existence of a maximum of the solids pressure can be explained as follows: as the concentration of particles increases, the collision rate of the particle increases, increasing the solids pressure. At high concentration of particles, however, this effect is counterbalanced by the limitation of the mean free path of the particles, which tends to reduce the solids pressure. This behavior is reproduced by present simulations. Solid-liquid fluidized bed two-dimensional simulations were performed on the basis of an Eulerian two-fluid formulation of the transport equations for mass, momentum and fluctuating kinetic energy.16 The solid-phase fluctuating motion model is derived in the frame of granular medium kinetic theory accounting for the viscous drag influence of the interstitial liquid phase. Figure 8 shows the profile of simulated and measured solids pressure as a function of concentration of particles. Both present simulations and simulated results by Gevrin et al.16 show that the solids pressure increases as the solid concentrations increase. At the low concentration of particles, present model gives a lower solids pressure than experiments measured by Zenit et al.31 and simulations by Gevrin et al.16 However, both simulations are in agreement with experiments at high concentration of particles in the bed. 3.3. Bed Expansion of Solid-Liquid Fluidized Beds. The bed expansion simulations of solid-liquid fluidized beds are performed. The initial conditions specify the concentration of solids in the bed, and the fluid flow through the bed. The particle diameter and density are 0.5 mm and 2990 kg/m3. The
superficial liquid velocity is 0.067 m/s. The settled bed is considered 300 mm deep, and the solids porosity is assumed to be 0.60. Figure 9 shows the instantaneous concentration of particles in the bed with the different times. At the start, a wave of porosity are created, which travel through the bed and subsequently break the solids bed of dilute particles. The oscillation of the bed surface as the solids bed breaks. The oscillation of the fluid-bed height is monitored over time. It can be observed that the bed is expanded in the range of the superficial liquid velocities studied. The time-averaged solids concentration, εs,i ) ∑tt12εs,i(t)∆t/ (t2 - t1), is computed covering a period of steady state time excluding initial period, where ∆t is the time step, and t1 and t2 are 10 and 16 s, respectively. Figure 10 shows the time-averaged concentration of particles in the bed predicted by two different drag models. Both models indicate that the concentration of particles increases from the bottom, remains relatively constant within the bed, and then decreases at the bed surface with the increase of bed height. The predicted concentration of particles from the drag model proposed by Gidaspow8 shows about 8% higher than that using eq 19. It may be pointed out that the drag model proposed by Gidaspow8 is a combination of the Ergun21 and the Wen and Yu14 experimental correlations. For solid concentrations greater than 0.2, eq 16a is used, whereas for values less than or equal to 0.2, eq 16b is employed. This transition proposed by Gidaspow8 makes the drag law discontinuous in solid concentration although it is continuous in Reynolds number. Ergun21 developed his correlation from experimental data taken from gas flow through packed beds of pulverized coke with solid concentrations ranging from 0.47 to 0.59, and used experimental data from other sources taken from an assortment of granular solids that included solid concentrations of up to 0.70. On the other hand, Wen and Yu14 conducted experiments on a defluidizing liquid-solid system; the system started at a fluidized state and the flow rate was gradually decreased until the fixed bed condition was reached. The range of solid concentrations investigated by Wen and Yu14 lies between 0.28 and 0.61, but they used data from other sources
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Figure 11. Profile of concentration of particles at two different bed heights.
Figure 13. Distribution of concentration of particles at two different particle sizes.
Figure 12. Distribution of axial velocity of particles at two different bed heights.
Figure 14. Profiles of axial velocity of particles at two different particle sizes.
that included solid concentrations close to 0.01. Hence, the Ergun correlation21 was obtained using concentrations in the range from 0.47-0.70 and the Wen and Yu correlation14 using concentrations in the range 0.01-0.61. It is thus surprising to note that the transition point used in the Gidaspow drag law,8 namely a solid concentration of 0.2, is not contained in the range 0.47-0.61, where the two correlations overlap. Physically, the drag force is a continuous function of both solid concentration and Reynolds number, and therefore continuous forms of the drag law may be needed to correctly simulate liquid-solid fluidized beds. Furthermore, the range of solid concentrations found in a liquid-solid bubbling bed spans both correlations, so capturing the correct value of solids concentration for the transition between the two correlations may also be important. 3.4. Effect of Liquid Velocity. The increase in liquid superficial velocity increases the energy input to the bed, leading to the increase in both solids mean circulation and turbulence. Figure 11 shows the profile of concentration of particles at two different superficial liquid velocities. The high liquid velocity gives a low concentration of particles. The concentration of particles is high near the walls and low in the center of the bed. Simulations indicate that with the increase in superficial liquid velocity, the axial velocity of particles is increased. Figure 12 shows the distribution of axial velocity of particles at two different superficial liquid velocities. The axial velocities of particles are positive in the center regime and negative near the walls. This means that particle flow up in the center and down near the walls. The circulation of particles is formed in the bed. With the increase in superficial liquid velocity, the axial velocities of particles are increased. This is because at higher liquid velocity, the liquid phase provides more energy input to the system. It is also noted that the velocity inversion points are different for the two superficial liquid velocities. At the bed height of 0.15 m, the inversion point occurs at the dimensionless
radius x/R of (0.52 for the liquid velocity of 0.045 m/s, and x/R ) (0.72 for the liquid velocity of 0.075 m/s. The inversion point moves to the center with the decrease in liquid velocity. 3.5. Effect of Particle Size. Two particle sizes, 0.5 and 1.0 mm, are used to study the effects of particle size on the axial solids velocities and concentrations. Figure 13 shows the distribution of concentration of particles at two different particle sizes. Simulations show the concentration of particles in the bed is decreased with the decrease of particle diameter. Both cases show that the concentration of particles is low in the center regime and high near the walls. Figure 14 shows the profile of axial velocity of particles at two different bed heights. Simulations indicate that with the increase of particle size the axial velocity of particles is increased in the center regime. The large diameter of particles gives a high negative velocity of particles near the walls. Simulations show that both particle sizes give a positive velocity in the center and negative velocity near the walls. The velocity inversion points are different for the two particle sizes. At the bed height of 0.2 m, the inversion point occurs at the x/R ) (0.45 for the 0.5 mm particle system, and (0.52 for the 1.0 mm particle system. Small particle systems have higher apparent viscosity and the inversion point moves to the center. Similar phenomena were observed in a liquid-solid fluidized bed.29 Further comparison at the same excess velocity (ul - umf) is needed to show the effect of particle size on solid motion in the liquid-solid fluidized bed before a general conclusion could be made. 3.6. Effect of Static Bed Height. Figure 15 shows the profile of concentration and axial velocity of liquid phase and particles at two different static bed heights. Both static bed heights cases show that the axial velocity is positive in the center regime and negative near the wall. It is found that the two static bed heights give the same trends in the bed. There was no systematic difference between the results for the low static bed height and
Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010
us x y
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solid velocity, m/s radial distance, m axial distance from the bottom, m
Greek Letters
Figure 15. Profile of concentration and axial velocities of liquid and solid phases.
high static bed height. The current results indicate that the static bed height plays a minor role in the prediction of liquid-solid bubbling fluidized beds. 4. Conclusions Understanding of solids motion in liquid fluidized beds is incomplete. Two-fluid CFD models are required in combination with kinetic theory of granular flow. Simulations indicate that the solid concentration distribution is high near the walls and somewhat low in the center. In the time averaged sense, solids ascend at the center of the bed and descend near the walls. The circulation of particles exists in a liquid fluidized bed. Solids mean axial velocities increase with the increase in the liquid superficial velocity and particle size. The superficial liquid velocity and particle sizes play an important role on the macroscopic properties of liquid-solid fluidized beds, whereas the effect of static bed height is minor in numerical modeling of liquid-solid bubbling fluidized beds. The predicted values from the present CFD model are in good agreement with the experimental values in the literature. It is realized that the key to the success of the simulation is to correctly model the drag force in liquid-solid fluidized beds. Once a drag law correlation is found, it can be incorporated in CFD framework. It is noted that the influence of the coefficient of restitution is not enough significant to explain the discrepancies between present simulations and experimental data. For these reasons, the improvement, development, and validation of drag laws is warranted, and the validation against experimental data is further required. Appendix
NOTATION CD c ds e g g0 hmf k p ps R t ul
drag coefficient model parameters particle diameter, m coefficient of restitution gravitational acceleration, m/s2 radial distribution function bed height at the minimum fluidization, m turbulent kinetic energy, m2/s2 static pressure, N/m2 solid pressure, N/m2 radius of bed, m simulation time, s liquid velocity, m/s
γ β ε εk εl εs εs,max θ µl µk µs µt ξs Fl Fs
collisional dissipation of energy, kg/ms3 interface momentum transfer coefficient, kg/m2 s2 turbulent dissipation rate, m2/s3 concentration of k phases porosity concentration of particles maximum concentration of particles granular temperature, m2/s2 molecular viscosity of fluid, kg/ms molecular viscosity of fluid, kg/ms solid shear viscosity, kg/ms turbulent viscosity of liquid phase, kg/ms bulk viscosity of particles, kg/ms density of fluid, kg/m3 density of solid, kg/m3
Subscripts
l s w
liquid phase solid phase wall
Acknowledgment This work was supported by Natural Science Foundation of China (21076043), China Postdoctoral Science Foundation (20090460070), Heilongjiang Province of China Postdoctoral Science Foundation (LBH-Z09292), and Educational Committee of Heilongjiang Province of China Science Foundation (11551006). Literature Cited (1) Gaaseidnes, K.; Turbeville, J. Separation of oil and water in oil spill recovery operations. Pure Appl. Chem. 1999, 71, 95–101. (2) Suzuki, Y.; Maruyama, T. Removal of emulsified oil from water by coagulation and foam separation. Sep. Sci. Technol. 2005, 40, 3407– 3418. (3) Mikolaj, P. G.; Curran, E. J. A hot water fluidization process for cleaning oil-contaminated beach sand. In Proceedings of the 2005 International Oil Spill Conference; Miami, May 15-19, 2005; American Petroleum Institute: Washington, D.C., 2005; pp 3522-3527. (4) Quevedo, J. A.; Patel, G.; Pfeffer, R. Removal of oil from water by inverse fluidization of aerogels. Ind. Eng. Chem. Res. 2009, 48, 191–201. (5) Fan, L. S., Gas-Liquid-Solid Fluidization Engineering.; Butterworths, Stoneham, MA, 1989. (6) Jenkins, J. T.; Savage, S. B. A theory for rapid flow of identical, smooth, nearly elastic spherical particles. J. Fluid Mech. 1983, 130, 187– 202. (7) 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. J. Fluid Mech. 1984, 140, 223– 256. (8) Gidaspow, D., Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions; Academic Press: London, 1994. (9) Arastoopour, H. Numerical simulation and experimental analysis of gas-solid flow systems: 1999 Fluor-Daniel Plenary lecture. Powder Technol. 2001, 119, 59–67. (10) Gidaspow, D.; Jung, J.; Singh, R. K. Hydrodynamics of fluidization using kinetic theory: an emerging paradigm 2002 Flour-Daniel lecture. Powder Technol. 2004, 148, 123–141. (11) Roy, S.; Dudukovic, M. P. Flow mapping and modeling of liquidsolid risers. Ind. Eng. Chem. Res. 2001, 40, 5440–5454.
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ReceiVed for reView May 22, 2010 ReVised manuscript receiVed August 21, 2010 Accepted September 1, 2010 IE101139H
