Simulation of flow behavior of particles in liquid–solid fluidized bed with uniform magnetic field

Powder Technology 237 (2013) 314–325

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Simulation of flow behavior of particles in liquid–solid fluidized bed with uniform magnetic field Shuyan Wang a, b,⁎, Ze Sun a, Xin Li a, Jinsen Gao c, Xingying Lan c, Qun Dong a a b c

School of Petroleum Engineering, Northeast Petroleum University, Daqing 163318, China Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA State Key Laboratory of Heavy Oil Processing, China University of Petroleum, Beijing 102249, China

a r t i c l e

i n f o

Article history: Received 2 July 2012 Received in revised form 2 December 2012 Accepted 6 December 2012 Available online 16 December 2012 Keywords: Magnetic fluidized bed Computational fluid dynamic Simulation Discrete element method Fluidization

a b s t r a c t Flow behavior of solid phases is simulated by means of DEM–CFD in a liquid–solid fluidized bed with magnetization of preliminarily fluidized bed (LAST) mode along an axial uniform magnetic field. By changing the magnetic field strength, the distribution of particles is studied within the bed. The distributions of velocity and concentration of ferromagnetic particles are analyzed at the different magnetic field intensities. When the magnetic field strength is increased to a value at which the fluidization of strings starts, the particles are found to form straight-chain aggregates along the direction of the magnetic field. At very high magnetic field strengths, the defluidization is observed at which particles are fixed in the bed. Simulations indicate that the granular temperature of particles increases, reaches a maximum, and then decreases with the increase of magnetic-flux density. The moderate strength magnetic field gives a high fluctuation of particles. The predicted solid pressure increases with the increase of concentration of particles. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The application of additional external fields (such as magnetic, electric, sound, ultra-sound or vibrations) enhances fluidization quality. The external field assisted fluidized beds have advantages of conventional fluidized beds, such as efficient fluid–solid contact, low pressure drop and wide range of operations [1–7]. Magnetic fluidized beds (MFB) of magnetically susceptible particles are considered as one of the technologies developed to eliminate the drawbacks of fluidized beds. Imposing a magnetic field on a bed of magnetizable particles could suppress or delay bubbling of these particles. The MFB deals with two basic magnetization modes: magnetization FIRST (preliminary magnetization of fixed bed and consequent fluidization) and magnetization LAST (magnetization of preliminarily fluidized beds). Some researchers [8–12] investigated the fluid-dynamic characteristics and the stability of the MFB performance in a uniform magnetic field. Studies showed that the fluidized bed height decreased as the axial magnetic field intensity was increased, whereas the opposite was observed for transverse fields. Both the magnetic field intensity and its orientation determined flow behavior of particles in the bed. Liquid–solid fluidization of mixture with ferromagnetic steel shots and non-magnetic zinc particles at the external transverse electromagnetic field was investigated [13]. Key parameters of the bed (superficial velocity, pressure drop, porosity, particle movement) were used to describe the influence of electromagnetic field ⁎ Corresponding author at: School of Petroleum Engineering, Northeast Petroleum University, Daqing 163318, China. E-mail addresses: [email protected], [email protected] (S. Wang). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2012.12.013

and mass fraction on fluidization and bed behavior. The magnetic stabilized fluidized bed was obtained. Fluidization characteristics of a nanoparticle catalyst were measured in a fluidized bed assisted with an axial magnetic field [14]. Experiments show that slugging and channeling, commonly observed when processing nanoparticles via conventional fluidized beds, could be effectively eliminated, and the size of agglomerates and bubble diameter could also be reduced with the aid of the magnetic field. Experiments showed that imposing a magnetic field on a bed of magnetic particles could suppress or delay bubbling of particles, promoting the formation of straight-chain aggregates. These straight chain aggregates can indent the top of the bubbles, splitting them, and thus preventing their growth. The orientation of these straight-chain aggregates must affect significantly the fluid drag force, thus affecting bed expansion in a critical way [15]. Reviews of hydrodynamics of magnetically fluidized bed were presented by Hristov [16]. The application of suitably designed magnetic field makes feasible a fluidization operation in the energy conversion and various chemical and biochemical reaction processes. As regards to mathematical modeling, computational fluid dynamic (CFD) simulations of the flow in fluidized beds gives very detailed information about the local values of phase hold-ups 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 magnetic fluidized beds. Eulerian–Lagrangian models describe the fluid flow using the continuum equations, and the particulate phase flow is described

S. Wang et al. / Powder Technology 237 (2013) 314–325

by tracking the motion of individual particles [17–19]. Discrete particle models (DPM) have been used for a wide range of applications [20]. A major difference with these traditional DPM models is that a detailed description of the liquid-phase dynamics is required, in order to describe the interaction between the particles and the fluid phase. In a hard-sphere system [21], the trajectories of the particles are determined by momentum-conserving binary collisions. The interactions between particles are assumed to be pair-wise additive and instantaneous. In the simulation, the collisions are processed one by one according to the order in which the events occur. Note that the possible occurrence of multiple collisions at the same instant cannot be accounted for. At high particle number densities, it will lead to a dramatical increase in collision number of particles. In that case, like in present case with agglomeration of particles, the hard-sphere method becomes useless. The soft-sphere models or discrete element method (DEM) allow for multiple particle overlap although the net contact force is obtained from the addition of all pair-wise interactions. The coupling of the DEM with a finite volume description of the gas-phase based on the Navier– Stokes equations was first reported in the open literature by Tsuji et al. [22] using soft-sphere model. The soft-sphere models are essentially time driven, where the time step should be carefully chosen in the calculation of the contact forces. Renzo et al. [23] predicted the layer inversion by means of DEM–CFD in a liquid–solid fluidized bed. Definitely, DEM–CFD may allow to investigate the local particle flow field, highlighting the motion of particles in apparently chaotic vortices continuously forming and disappearing, which is thought to be the mechanism responsible for mixing of particles in the bed. However, detailed investigations of flow characteristics in magnetic liquid–solid fluidized bed are still lacking. In present study, the flow behavior of solid phases is simulated by means of DEM–CFD in a liquid–solid fluidized bed with LAST mode along an axial uniform magnetic field. The hydrodynamics of liquid–solid fluidized beds with uniform magnetic field is simulated. By changing the magnetic field strength, the distribution of particles is studied within the bed. The distributions of velocity and concentration of ferromagnetic particles are analyzed at the different magnetic field intensities. The distribution of granular temperature of particles is analyzed with the increase of magnetic-flux density in MFBs. 2. Eulerian–Lagrangian liquid–solid flow model The DEM–CFD approach is relatively well documented in the literature [17–19,24], so here the salient features of the model equations used will be summarized. Our DEM–CFD implementation uses a rather standard coupled approach based on the particlescale Discrete Element Method for the solid phase [20] and a local average CFD approach for the fluid phase [21,24]. To simplify, it is assumed: (1) solid phase consists of mono-sized particles with same diameter and density. (2) Both liquid phase and particles are assumed to be isothermal without reactions. 2.1. Equation of motion for liquid phase Generally in numerical simulation of two-fluid flow, the fluid phase flow is solved by a locally averaged approximation of the continuity and Navier–Stokes equations with an averaging scale of the order of the computational cell. The equations of conservation of mass and momentum for liquid phase are: ∂ðρl εl Þ þ ∇⋅ðρl εl ul Þ ¼ 0 ∂t

ð1Þ

∂ðρl εl ul Þ þ ∇⋅ðρl εl ul ul Þ ¼ −εl ∇P þ εl ∇⋅τ l þ εl ρl g−F pl ∂t

ð2Þ

315

where g is the acceleration due to gravity, P the liquid pressure, εl the liquid volume fraction, τl the viscous stress tensor and ρi the density of liquid, respectively. The coupling term Fpl between particle phase and liquid phase is estimated as the sum of the drag on each particle within the corresponding fluid control volume. The stress tensor of liquid phase can be represented as h i 2 T τl ¼ μ l ∇ul þ ð∇ul Þ − μ l ð∇⋅ul ÞI 3

ð3Þ

where μl is the viscosity of liquid phase. 2.2. Equation of motion for a particle Spherical particles of uniform size are investigated in present work. The particles are tracked individually based on the Newton's second law of motion. Each particle has two types of motion, translational and rotational motions. The motion of each individual particle is governed by the laws of conservation of linear momentum and angular momentum, expressed, for the i-particle, by mi

Ip

  dvi N ¼ −V p ∇P þ mi g þ f d þ f m þ ∑j¼1 f lj þ f cj þ F mj dt

dωi N ¼ ∑j¼1 T pij dt

ð4Þ

ð5Þ

where mi and vi are the mass and velocity of a particle, and Vp the volume of a particle. Tp is the torque arising from the tangential components of the contact force. Ip and ω are the moment of inertia and angular velocity of a particle. The terms of the right-hand side of Eq. (4) are the liquid pressure gradients, gravity, drag force exerted from the fluid, virtual mass force, lubrication force, contact force and magnetic force by introducing external magnetic field, respectively. These inter-particle forces and torques are summed over the N particles in contact with particle i. The contact force between particles is calculated based on the soft-particle method. Note that the history integral (Basset) force is neglected. This choice is mainly motivated by the lack of reliability and accuracy of formulations for such forces in dense systems and the difficulties in characterizing the transient fluid–particle momentum exchange [25]. Basset force is responsible for the major changes in the time-evolution of particle motion. This means that the unsteady drags can be neglected when the inertia of the fluid is considerably smaller than the inertia of the particles. The contribution of the inertial drag does not change the initial evolution of particle, independently of whether the Basset drag was absent or not [26]. The adequacy of the approach is supported by the results obtained in the analysis of wave instabilities by Derksen and Sundaresan [27] in liquid-fluidized beds. The liquid–solid interaction force, or drag force, is determined at each particle. The drag force depends on not only the relative velocity between the solid particle and fluid but also the presence of neighboring particles, i.e., local volume fraction of solid phase. The drag force is expressed by considering these factors as follows: fd ¼

 βV p  ul −up 1−εl

ð6Þ

where εl and β are the volume fraction of fluid and an inter-phase momentum transfer coefficient. A proper drag model for the description of β is vital in solid–fluid interaction problems. In present simulations, the inter-phase momentum transfer coefficient is predicted by Huilin–Gidaspow model which is available in FLUENT code [28]. When a solid is accelerated through a fluid, there is a corresponding acceleration of the fluid, which is at the expense of work done by the

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solid. This additional work relates to the virtual mass force. The virtual mass force [29] acting on a particle is given by:  f m ¼ −0:5ε s ρl

dul dup − dt dt

 ð7Þ

where εs is the concentration of particles. As two particles move relative to each other with only a thin film of liquid medium separating them, hydrodynamic forces, often referred to as lubrication forces, arise because of the motion of the interstitial liquid. In the simple case where the particles are moving away from each other, liquid has to flow into the gap between the particles; analogously, when the particles are moving toward each other, liquid has to flow out of the gap. The motion of the liquid generates pressure gradients and viscous stresses that are cumulative expressed by a hydrodynamic force on the particles. Because the approach of the particles is often oblique, the hydrodynamic force has a normal component (in the direction that connects the particle centers) and a tangential component which results to torque on the particles. It has been shown [30] that when the particles are in close proximity the primer contribution to the lubrication force is the normal one, resulting from the normal “squeezing” motion of the particles; it is only this component that is considered in this work. This force is given by the following equation: i d d 3πμ l h  i j fl ¼ − nij ⋅ vp;i −vp;j 2 di þ dj

!2

1   nij   δn;ij 

ð8Þ

where μl is the viscosity of liquid phase. nij is the unit vector connecting the centers of particles with direction from i to j. vp,i and vp,j are the velocity vectors of the two particles. This force opposes any change of the separation between the two particles: it is repulsive when the particles are moving toward each other and attractive when they are moving away from each other. Because the magnitude of the force diverges to infinity at zero separation, i.e. when the particles are in contact, the lubrication force, if acting alone, would prevent the contact of two particles. In reality though, additional forces act on the approaching particles and the particles are rough so they come in contact before the lubrication force reaches the theoretically predicted extreme values. To simulate the effect of roughness, a cut-off distance is used to prevent the lubrication force from reaching unrealistically high values. In case of a collision between two particles, the contact force fc is divided into normal (fnij) and tangential (ftij) forces, and these forces are modeled in Fig. 1. The elastic part of the normal contact force fcn is represented by a nonlinear spring, where the force is proportional to the stiffness kn and the displacement δn3/2 in the contact, i.e. the overlap. These two forces are given by the following equations [22,29]:   3=2 f cn;ij ¼ −kn δnij −ηn vrij nij nij

ð9Þ

3=2

f tij ¼ −kt δtij −ηt vtij

ð10Þ

z

γ A

θ α

Fr

B

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !−1 v u dd u t  i j  2 di þ dj

ð11Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !−1 v u dd u t  i j δ1=2 n 2 di þ dj

ð12Þ

2

4 1−γ 2i 1−γ j kn ¼ þ 3 Ei Ej

kt ¼ 8

2−γi 2−γj þ Gi Gj

whereγ is the Poisson ratio, and Ep and Gp are the longitude and transverse elastic moduli, respectively. To account for visco-elastic material properties that cause energy dissipation, a damping factor η related to the normal coefficient of restitution en is included into the DEM model, as proposed by Tsuji et al. [22]: ηn ¼ ηt ¼ α

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mi mj 1=4 kn δ mi þ mj n

ð13Þ

where α is the coefficient that related to the restitution coefficient. The elastic contribution of the impact energy absorbed during the compression is released during the restitution phase of the impact and leads to the elastic force that separates the contact partners. The absorption of kinetic energy during the impact can be described by a restitution coefficient. The coefficient of restitution is a ratio of the square root of the elastic strain energy released during the restitution to the impact energy, i.e. the initial kinetic energy [22,29]. 8 lnen > < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 α¼ π þ ln2 en > : 1

0ben ≤1:0

:

ð14Þ

en ¼ 0

2.3. Magnetic force The external magnetic field induces a magnetization on fluidized particles. The magnetic force on the particle due to the interaction of the external uniform field and magnetized particles is [31] F m ¼ F me þ F mi

ð15Þ

where Fme is the external magnetic force, and Fmi is the interparticle magnetic force due to the interaction with neighboring particles. The magnetization, M, of a magnetic fluidized bed as a whole is assumed to be proportional to the magnetic field strength, H. In order to obtain a relationship between the magnetic force and the mean induction we have to consider the Clausius–Mossotti relationship which links the mixture susceptibility χ with the particle susceptibility [32] χp χ ¼ εs χþ3 χp þ 3

mb

B0

where k and η are the spring and damping coefficients, respectively. δnij and δtij are the normal and tangential displacements between particle i and particle j, respectively. The displacement-related elastic contact stiffness kn depends on the elastic properties of the colliding particles. The spring coefficients kn and kt are calculated from the following equations [22,28]. For particle–particle collisions, the spring and damping coefficients are:

ð16Þ

where χp is the magnetic susceptibility of the solid. According to Jackson [32], the external magnetic force can be expressed as a function of effective particle susceptibility

Fθ x

Fig. 1. Repulsive and attractive magnetic force between two ideal dipoles.

F me ¼ V p χ e MH∇H

ð17Þ

where the vectors M and H represent the magnetization and magnetic field intensity, respectively. μ0 is the permeability of the vacuum. For

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317

simplification, it is assumed that two identical magnetically susceptible particles are positioned side-by-side in a uniform magnetic field, and separated by a distance r, as shown in Fig. 1. Furthermore, it is assumed that these two particles have the same diameter, and can be considered ideal dipoles under the influence of a uniform external magnetic field. The magnetic force on the particle due to the nonuniform magnetic field resulting from the interaction of the external uniform field and nonuniform fields of the magnetized particles is proportional to the field gradient [10]. A theoretical fluid-dynamic model describing behavior of a magnetically assisted fluidized bed in a non-uniform magnetic field was proposed by Jovanovic et al. [10]. The radial magnetic force between the particles can be expressed [33,34] Fr ¼ −

( !) i μ 0 m2 ∂γ h 3m2 2m ∂m 2 ½ ð Þ ð Þ  ð Þ −6 cos θ−γ sin θ−γ θ−γ − þ þ 1−3 cos 4π r 3 ∂r r4 r 3 ∂r

ð18Þ ∂γ ¼− ∂r

∂m ¼ ∂r

8πr

4

h

1þ

χe V p 8πr 3

9χ e V p sin2θ i2  2 χe V p ð3 cos2θ−1Þ þ 3 8πr 3 sin2θ χe V p B

 μ 0 cosγ−

χe V p 8πr 3

ð19Þ

2 

½ cosγ þ 3 cosð2θ−γÞ

    χe V p χe V p ∂γ 3χ e V p sinγ þ 3 1− sin ð 2θ−γ Þ ½ cosγ þ 3 cos ð 2θ−γ Þ  − ∂r 8πr 3 8πr 3 r8πr 3

ð20Þ where m is the dipole moment. The angular interparticle magnetic force is

f

  μ ∂γ 2 F θ ¼ − 0 4 6m cosðθ−γ Þ sinðθ−γÞ 1− ∂θ 4πr h i ∂m 2 þ 2m 1−3 cos ðθ−γ Þ ∂θ

g

 2 h i χe V p χe V p χe V p ∂γ 2 3 8πr3 sin2θ þ 6 8πr3 cos2θ 1 þ 8πr3 ð3 cos2θ−1Þ ¼ h i2  2 χ V χ V ∂θ 1 þ e p ð3 cos2θ−1Þ þ 3 e p sin2θ 8πr 3

∂m ¼ ∂θ

 μ0

  1−

1−

χe V p 8πr 3

χe V p 8πr3

ð21Þ

ð22Þ

8πr 3

χe V p B



cosγ−3

 sinγ þ 3

χe V p 8πr 3

2  cosð2θ−γÞ

 χe V p sin ð 2θ−γ Þ γ −6 sin ð 2θ−γ Þ θ 8πr 3 8πr 3

χeV p

ð23Þ where B is the local magnetic-flux density. The radial magnetic force and the angular interparticle magnetic force relate with the distance and the angular position between particles. From Eq. (18) the radial magnetic force is reduced when the distance between the particles increases. The radial force is more significant when the particles are located at θ = 0 than they are located at θ = π/2. From Eq. (21), we see that the angular magnetic force has a maximum at θ = π/4, and becomes zero at θ = 0 and θ = π/2. The angular magnetic force will vanish as the particles move away each other. The total interparticle magnetic force is evaluated by adding the radial vector force and the angular vector force between the particles. 2.4. Boundary conditions and numerical methods Fig. 2 shows the two-dimensional liquid–solid fluidized bed with a uniform magnetic field along axial direction used in the present numerical simulations. Fluidized bed is 0.10 m in width and 1.5 m in

Fig. 2. Scheme of magnetically assisted liquid–solid fluidized bed.

height. The bed has two sidewalls with inlet and outlet. Both walls are non-magnetic. The bed material consists of 2100 particles with the diameter and density of 2 mm and 2540 kg/m 3. The ratio of the number of magnetic particles and non-magnetic particles is 2:1 and 1:2, respectively. The computational parameters are listed in Table 1. The calculations of liquid flow and particle motion are two dimensional. At an impenetrable wall, the tangential and normal velocities of liquid phase are set to zero (no slip condition). At the bottom boundary, the liquid velocity along the vertical direction is assumed to be the inlet liquid velocity, and the velocity of particles is zero due to without particles into the bed. At the top of the bed, the liquid pressure is specified, and the velocity gradient of liquid phase is assumed equal to zero. Initially all the particles are randomly located in the bed. The particles are then allowed to fall freely under gravity in the absence of a liquid flow. After allowing some time for energy dissipation through inelastic collisions, the bed reaches an almost stagnant state. The interparticle force is not turned on until the initialization is completed. Numerical simulations are performed by means of a CFD–DEM code [34,35]. The differential equation of fluid motion and the equation of particle motion were simultaneously solved for the profiles of particle and gas velocities, and volume fraction in the bed. The equation of fluid motion was solved by finite difference method and the numerical method, SIMPLE, was used. The solid motion was solved algebraically. In the finite difference method, the flow domain was divided into cells, the size of which was smaller than the macroscopic motion of particles in the system but larger than the particle size. The differential two-dimensional equations of fluid motion are considered in an Eulerian framework in which the fluid cells are fixed to a reference frame. The Lagrangian particle motion equations, with mutual interaction between fluid and particles taken into account, were simultaneously solved with the fluid motion equations to provide particle positions, particle velocities, and fluid velocities. In Eulerian–Lagrangian flow mapping, each fluid cell was considered to contain a group of particles interacting with the fluid. The position

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3. Simulation results and discussions

Table 1 Parameter used in simulations. Particle diameter

2 (mm)

Particle density

2540 (kg/m3)

Number of particle Damping coefficient Friction coefficient of particles

2100 0.05 0.3

800 (N/m) 2000 0.3

Magnetic permeability of particles Liquid density

0.624 (Wb/Am)

Particle stiffness Wall stiffness Friction coefficient between particle and wall Magnetic permeability of vacuum

998.2 (kg/m3)

Liquid viscosity

Grid size

4 mm × 4 mm

Time step for liquid phase

1.003 × 10−3 (Pa.s) 1 × 10−4 (s)

1.256637 × 10−6 (Wb/Am)

and velocity of individual particles were calculated using the Lagrangian equation of particle motion. The solids volume fraction in the cell was obtained from the information of particle positions. The iterative solution of the pressure–velocity coupling and boundary conditions are as in conventional approaches and details will be omitted for the sake of brevity. Upon starting the magnetic field, the particle bed is first fluidized at the liquid velocity of 0.075 m/s for 1.0 s. After that, simulations will be continued for 10.0 s. The time-averaged variables are computed from the last 6.0 s. In this simulation, a constant time step of 1.0 × 10−4 s is used for liquid phase. The CPU time required for the calculation of 1 s is about 4 h in a PC computer (80 GB hard disk, 512 Mb Ram and of 2.4 GHz CPU). Keeping in mind that flow of liquid phase and solid phase is three-dimensional in real fluidized beds. Thus, two main questions arise, namely the capability of 2-D models to represent 3-D simulations, and the appropriateness of 2-D configurations to study flow dynamics of fully 3-D fluidized beds. Using a two-fluid model to simulate a 2-D fluidized bed, Peirano et al. [36] reported a significant difference in bed height between 2-D and 3-D results. A more comprehensive study was conducted by Xie et al. [37], again using an Eulerian approach. They found that the differences between 2-D and 3-D dimensions increased significantly as the inlet velocity was increased. This confirms that hydrodynamics predicted by 2-D DEM is different from 3-D simulations using DEM in fluidized beds. Unfortunately, the CFD–DEM code used in present study is a two-dimensional program [34,35]. It will be extended to 3-D simulations. Hence, the 3D simulation is needed in future.

t=0.1s Bo=0.1T

t=0.5s Bo=0.1T

3.1. Flow behavior of particles with a uniform magnetic-flux density Fig. 3 shows the snapshots of particles in the bed at the magneticflux density of B0 = 0.1 T. The ratio of the number of magnetic particles and non-magnetic particles, nm/nn, is 1:2, which is known as a less magnetic case where nm and nn are the number of magnetic articles and non-magnetic particles. The value of particle Reynolds number, Re = dpulρl/μl, is 150 with the superficial liquid velocity of 0.075 m/s. The particle Stokes number, St = ρsdpul/μl, is 380. It clearly shows that the magnetic-flux densities influence the motion of particles. When liquid injects into the bed without magnetic field, particles move up due to the drag force. At the time t = 1.0 s the maximum bed expansion height is found. The fluidization state is established with a high concentration of particles in the bottom of the bed. A dense region in the bottom and a free board at the top is found along height. The interface between these two regions is clear. After that time, the axial magnetic field is imposed. The magnetic chains of particles are formed along the direction of liquid flow. The length of magnetic chains is related to magnetic-flux density. Liquid flows in the channels of magnetic chains of particles. With the computing processes, the length of chains is increased with more magnetic particles to be bounded together. These magnetic chains are gradually settled down and finally deposited at the distributor since the weight of magnetic chains is greater than the drag of fluid. Hence, more chains of magnetic particles are found at the bottom, and the non-magnetic particles fluidized at the upper regime of the bed. The magnetic particles with straight-chains separate from non-magnetic particles in the bed. The segregation phenomena with the magnetic particle in the bottom and the non-magnetic particles in the upper region are formed along bed height. Fig. 4 shows the instantaneous snapshots of particles at two magnetic-flux densities of 0.03 T and 0.5 T with nm/nn =1:2 in the bed. The superficial liquid velocity is 0.075 m/s and the value of particle Reynolds number is 150. The difference of motion of particles between these two magnetic-flux densities is obvious. At B0 =0.03 T, the small magnetic chains of particles is formed, and fluidized with non-magnetic particles in the bed. The voidages with low local concentration of particles are formed in the bed. When the magnetic-flux density is increased to B0 =0.5 T, a large proportion of magnetic particles are shown to be lumped together. As liquid pass through the bed, it causes the formation of large channels across the bed. The chains with magnetic particles are frozen in the bed. Non-magnetic particles move during the straight-chain aggregates. From numerical simulations of Figs. 3 and 4, it is demonstrated that the bed passes through three principal states with the different magnetic intensities: particulate fluidization at the low magnetic intensity, a chain

t=2s Bo=0.1T

t=6s Bo=0.1T

Fig. 3. Instantaneous snapshots of particles at nm/nn of 1:2 and liquid velocity of 0.075 m/s.

t=10s Bo=0.1T

S. Wang et al. / Powder Technology 237 (2013) 314–325

t=2s Bo=0.03T

t=8s Bo=0.03T

t=2s Bo=0.5T

319

t=8s Bo=0.5T

Fig. 4. Instantaneous snapshots of particles at nm/nn of 1:2 and liquid velocity of 0.075 m/s.

fluidized bed at the mid magnetic intensity and a magnetically aggregated or magnetically condensed bed at the high magnetic intensity. The frozen bed has a fixed structure of particle strings, divided by channels. From Figs. 3 and 4, the bed changed from partially stabilized fluidized bed to magnetically stabilized fluidized bed with the increases of magnetic field of intensity. The results confirm that the magnetic field intensity has a large influence on the stability of magnetic particles in a bed. Rosensweing [2] developed an approach for analyzing twophase magnetized flows and then used the results to determine the magnetic stability behavior. The spatially averaged equation set governing the dynamics of magnetized two-phase flow was developed. The magnetic forces play a role in determining stability. Anderson et al. [38] analyzed the instabilities by seemingly different structures observed, namely slugs and bubbles in bubbling fluidized bed without external fields. Both linear and nonlinear stability analyses were used to examine the stability of the state of uniform fluidization and it was found that most fluidized beds of particles with size and density in the ranges of technical interest are unstable, whether fluidized by a gas or a liquid. Careful observation of water fluidized beds was shown to have confirmed the presence of instabilities, though bubbles never appeared. It must be noted that the stability criterion relates with magnetic forces, hydrodynamic interactions and collisions in liquid–solid magnetic fluidized beds. Fig. 5 shows snapshots of the bed at three different magnetic-flux densities at the ratio of magnetic particles and non-magnetic particles of nm/nn = 2:1. The value of particle Reynolds number is 150 with the superficial liquid velocity of 0.075 m/s. When magnetic-flux density is low (B0 = 0.03 T), the behavior of liquid–solid phases is similar to

t=2s Bo=0.03T

t=10s Bo=0.03T

t=2s Bo=0.1T

the conditional bubbling fluidized bed. Liquid flows into the bed from the bottom, carrying the particles upward. Magnetic particles form small chains which are very short, so they can be fluidized in the bed. The dense region with high concentration of particles and a freeboard at the upper region are formed along bed height. This indicates that the effect of magnetic-flux density has slight effect on the whole flow behavior of particles in the bed. When the magneticflux density is gradually increased to B0 = 0.1 T, the large straightchain of magnetic particles is formed, and these chains are aggregated and settled down due to the gravity plus magnetic force larger than drag force. However, the non-magnetic particles carried by the liquid flow upward and fluidized at the top of the bed. The segregation of particles is formed in the bed. When magnetic-flux density becomes large (B = 0.5 T), the large straight-chains from the bottom to the top are frozen in the bed. The opened and closed channels formed by magnetic particles are formed. The non-magnetic particles are fluidized by liquid phase in these channels. Under external magnetic field, the forces acting on a particle include the external magnetic force and the interparticle magnetic force except the gravity force and drag force. Thus, the motion of the particle is fully controlled by various forces acting on it, including the fluid drag, particle–particle interactions and magnetic forces. Fig. 6 shows the trajectory of a representative particle at the different magnetic-flux densities for the ratio nm/nn of 1:2 and 2:1 in the bed. The effect of magnetic-flux density on motion of representative particle is obvious. The representative particle is originally located at different positions due to the particles which have been fluidized. At the ratio nm/nn of 1:2, the representative non-magnetic particle moves freely at the low magnetic-flux density

t=10s Bo=0.1T

Fig. 5. Snapshots of particles at nm/nn of 2:1 and liquid velocity of 0.075 m/s.

t=2s Bo=0.5T

t=10s Bo=0.5T

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140

Axial distance Y (mm)

120 100

move freely in the bed. With the increase of magnetic-flux density, the simulated axial velocities of representative particle are reduced. This means that the movement of representative particle is limited. At the high magnetization (say, B0 = 0.5 T), the axial velocity of representative particle is close to zero. A “frozen” bed is formed.

Particle ID: 280, t=5~10s Ratio n m /n n=2:1

Particle ID: 280, t=5~10s Ratio nm /n n=1:2 B0 =0.03T

B0 =0.03T

B0 =0.1T

B0 =0.1T

B0 =0.5T

B0 =0.5T ul =0.075 m/s

80

n=2100

60

3.2. Distribution of particle velocity, concentration and bed height

40 20 0 0

20

40

60

80

0

20

40

60

80

Lateral distance x (mm) Fig. 6. Trajectories of a representative particle in the bed.

of B0 = 0.03 T. When the magnetic-flux density increases from 0.03 to 0.1 T, the movement range of representative non-magnetic particle is reduced. As the magnetic-flux density is 0.5 T, the motion of representative non-magnetic particle is restricted. As the ratio nm/nn increases, the number of magnetic particles becomes large in the bed. At the ratio nm/nn of 2:1, the representative non-magnetic particle can move in the limit space at the low magnetic-flux density of B0 =0.03 T. With the increase of magnetic-flux density, the movement range of representative particle is reduced. When the magnetic-flux intensity increases to a certain value (B0 =0.5 T), the movement range of representative particle will shrink a dot. This is because the magnetic particles are formed straight chains, which are fixed in the bed, leading to “frozen” bed. This implies that the trajectory of particles in the high ratio nm/nn is smaller than that in the low ratio in the bed. The reason lies in the higher ratio of magnetic particles, the more straight-chains are formed because of more magnetic particles in the bed. These aggregates of magnetic particle-chains influence on the trajectory of particles. Fig. 7 shows the instantaneous axial velocity of a representative particle as a function of time at three magnetic-flux densities. At the low magnetic-flux density of B0 = 0.03 T, the axial velocity of representative particle is large. The positive value means the representative particle flow upward by liquid phase, while the negative value shows that the representative particle moves downward. This indicates that particles

0.1

Particle ID: 280 n m/n n=2:1, n=2100

0.10

ul =0.075 m/s

0.05

B0=0.5T, u l =0.075 m/s

Axial velocity of particles (m/s)

Axial velocity of particle uy (m/s)

0.2

0.0 -0.1 -0.2

There are three kinds of force on magnetic particles under the action of external magnetic field, including vertical magnetic force, repulsive and attractive magnetic force. The straight-chain aggregates are formed with magnetic forces, and effect on fluidization of particles in the bed. The time-averaged axial velocity is calculated from instantaneous velocity of particles in the bed. Fig. 8 shows the timeaveraged axial velocities of particles along lateral direction as a function of magnetic-flux density. When the magnetic-flux density is low, the axial velocity of particles is positive in the center region and negative near the walls. This indicates that the particles, carried by the liquid, flow downward near the walls and upward in the center region. With the increase of magnetization (say, B0 = 0.5 T), the mean axial velocity of particles closes to zero. This indicates that the motion of particles in the bed is weak, and a “frozen” bed is formed. Fig. 9 shows the distribution of time-averaged concentration of particles at two different heights at the superficial liquid velocity of 0.075 m/s. Simulations show that a uniform radial distribution of concentration of particles is found in the bed. The mean value of concentration of particles (MV) is also given at two different bed heights. At the bed height of 18 mm, the mean concentration of particle is 0.3474 for B0 = 0.03 T and 0.2675 for B0 = 0.5 T. The mean concentration of particle is 0.3355 for B0 = 0.03 T and 0.2836 for B0 = 0.5 T at the bed height of 70 mm. These indicate that the concentration of particles is reduced with an increase of magnetization due to the formation of magnetic particle-chains in the bed. Fig. 10 shows the instantaneous bed height at different magnetic-flux densities. We can see that the oscillations of bed height is larger at B0 = 0.03 T than that at B0 = 0.5 T. The mean height and standard deviation (SD) of bed height are calculated from instantaneous bed height. The expansion bed height is decreased with the increase of magnetic-flux density. The values of standard deviation are 0.00294, 0.00234 and 4.23 × 10 −4 mm at the magnetic-flux densities of 0.03, 0.1 and 0.5 T, respectively. The standard deviation of bed height reduces with the increase of magnetic density. This is mainly due to the formation of straight-chains of particles under external magnetic field.

B0=0.03T,

B0=0.1T,

B0=0.5T

Particle ID: 280 n m/n n=1:2, n=2100

u l=0.075 m/s

0.1 0.0

B0=0.03T,

-0.1 -0.2 5.0

B0=0.1T

B0=0.5T 6.0

7.0

0.00 y=18mm,

-0.05

y=70mm,

y=118mm

-0.10 B0=0.03T, u l =0.075 m/s

0.05 0.00 -0.05 y=18mm,

y=70mm,

y=118mm

-0.10 8.0

9.0

10.0

Times (s) Fig. 7. Instantaneous axial velocity of a representative particle velocity.

0

20

40

60

Bed width x (mm) Fig. 8. Profile of axial velocity of particles.

80

100

S. Wang et al. / Powder Technology 237 (2013) 314–325

0.7

0.3

y=70mm, ul =0.075 m/s, nm/nn=1:2, n=2100

0.1

Magnetic-flux density=0.03T, MV=0.3474 Magnetic-flux density=0.10T, MV=0.3450 Magnetic-flux density=0.50T, MV=0.2675

0.5

-4

Bo=0.03T, MV=2.61x10 (m/s)

Granular temperature (m/s)2

Particle concentration

0.0015

Magnetic-flux density=0.03T, MV=0.3355 Magnetic-flux density=0.10T, MV=0.3652 Magnetic-flux density=0.50T, MV=0.2836

0.5

321

-4

2

-5

2

Bo=0.1T, MV=2.12x10 (m/s)

0.0012

Bo=0.5T, MV=5.54x10 (m/s)

2

n m /n n=1:2, n=2100

0.0009

u l =0.075 m/s 0.0006

0.0003

0.3 0.0000 0.0

y=18mm, u l=0.075 m/s, nm /n n =1:2, n=2100

0.1 0

20

40

60

0.1

0.2

100

80

0.3

0.4

0.5

0.6

0.7

Concentration of particles

Bed width x (mm) Fig. 11. Granular temperature as a function of concentration of particles. Fig. 9. Distribution of time-averaged particle concentrations.

3.3. Distribution of particle granular temperature From simulated instantaneous velocity of particles, the granular temperature is predicted in the bed. In a two-dimensional system, the granular temperature is calculated as [39] 0 1 N N 1 @1 X 1X 2 2A θ¼ C þ C 2 N j xj N j yj

ð28Þ

160 150

B0 =0.03 T, ul =0.075 m/s, nm /n n =1:2, n=2100

140

Mean height=145.1mm, SD=0.00294

Hbed (mm)

130 150

B0 =0.1 T, u l =0.075 m/s, n m/nn =1:2, n=2100

140 130 150

Mean height=142.8mm, SD=0.00234 B0 = 0.5 T, ul =0.075 m/s, nm /n n=1:2, n=2100 -4

Mean height=138.3mm, SD=4.23x10

140 130 5.0

6.0

7.0

8.0

9.0

2

ps ¼ εs ρs θ þ 2ð1 þ eÞρs go εs θ

10.0

Times (s) Fig. 10. Instantaneous bed expansion height at three magnetizations.

ð29Þ

where go is the radial distribution function at contact. On this graph, the existence of a maximum of the granular temperature is clearly exhibited by the simulations, at concentration of particles close to 0.25–0.35. The existence of a maximum of the granular temperature can be explained as follows: as the concentration of particles increases, the collision rate of the particle increases, increasing the granular temperature. 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 their colliding velocity and therefore the granular temperature. The fluctuating motion of particles originates from local instant variations of the concentration, which by continuity induce local velocity

Temperature θ (cm/s)2 Solids pressure (Pa)

where Cj is the random fluctuation velocity of the j particle defined as the difference between the instantaneous particle velocity and the local mean particle velocity, and N is the total number of particles in the computational cell. Fig. 11 shows the distribution of granular temperature as a function of concentration of particles. At the low magnetic-flux density, the granular temperature of particles increases, reaches maxima, and then decreases with the increase of concentration of particles. At the high magnetic-flux density of 0.5 T, the granular temperature of particles is independent upon concentration of particles, and the mean granular temperature is very small due to the formation of straight-chain aggregates in the bed. The calculated mean value (MV) of granular temperature is given in the figure. The mean granular temperature of particles in the bed is from 2.61 × 10 −4 (m/s) 2 at B0 = 0.03 T to 5.54 × 10 −5 (m/s) 2 at

B0 = 0.5 T. This means the mean granular temperature of particles is decreased with the increase of magnetic-flux density. This indicates that a temperate magnetic-flux density is adopted to enhance high granular temperature of particles in the bed. From kinetic theory of granular flow, the solid pressure is computed as a function of granular temperature and concentration of particles. Fig. 12 shows the distribution of granular temperature and solid pressure as a function of concentration of particles at the superficial liquid velocity of 0.075 m/s without magnetic field (Bo = 0.0 T). The value of particle Reynolds number is 150. The solid pressure is calculated as [39]

20 15 10

MV=1.6315 (Pa) n m /n n=1:2, n=2100 u l =0.075 m/s Bo=0.0 T

5 0

MV=2.302 (cm/s)2 30 n m /n n=1:2, n=2100 20

u l =0.075 m/s Bo=0.0 T

10 0 0.0

0.1

0.2

0.3

0.4

0.5

Concentration of particles Fig. 12. Distribution of granular temperature and solid pressure.

S. Wang et al. / Powder Technology 237 (2013) 314–325

gradients. The proper turbulence of the continuous phase is neglected in present simulations. Hence in dilute regime with 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. From Eq. (29), we see that the solid pressure includes a kinetic contribution and a collisional contribution. Roughly, we see from Fig. 12 that the solid pressure increases as the solid volume fraction increases. The solid pressure is high at high concentration of particles because the radial distribution function go becomes infinite [39] when the solid volume fraction approaches the maximum value of solid fraction for a random packing of spheres. Fig. 13 shows the mean granular temperature as a function of magnetic-flux density at the superficial liquid velocity of 0.075 m/s. With the increase of magnetic-flux density, the granular temperature increases, reaches a maximum, and then decreases due to the formation of long chains of particles. Suppression of motion of magnetic particles is achieved when a magnetic field is applied. The straightchain aggregates are formed in the axial direction, limiting motion of non-magnetic particles. The orientation of these straight-chain aggregates affects flow behavior of liquid and particles, which influences the granular temperature of particles in the bed. 3.4. Variation of forces with magnetic field The contact force fc is caused by collision between two particles. It includes normal (fnij) and tangential (ftij) forces. Fig. 14 shows the normal and tangential forces for a sampled magnetic particle (No. 590) as a function of magnetic-flux densities. The contact forces fluctuate violently at the beginning of the external magnetic field switched on, because magnetic particles are not completely under control of external magnetic field, at that time, particles are in fluidization state. The contact force is increased with the increase of the magnetization. The mean value (MV) and standard deviation (SD) of contact force are calculated. The standard deviation of normal component of contact force increases from 0.000183 to 0.00165 N, and the standard deviation of tangential component of contact force is from 0.00126 to 0.0125 N as the magnetic-flux density increases from 0.03 to 0.5 T. This implies that the standard deviation of contact force increases with the increase of magnetic density. The contact force depends on the relative velocity between two particles and the parameters of stiffness, dissipation, and friction coefficients. When the straight-chain aggregate is formed, the magnetic particles do not separate from each other, but keep in contact with each other. Thus, the magnetic particles are difficult to fluidize. The slip velocity between two magnetic particles is close to zero. The

0.1000

fn (N), MV= -0.0115, SD=0.00165 ft (N), MV=0.0184, SD=0.0125

0.0500

Contact forces fn and ft (N)

322

0.0000 -0.0500

Particle ID: 590, u=0.075 m/s, B o=0.5 T Particle ID: 590, u=0.075 m/s, B o=0.1 T

0.0000 fn (N), MA= -0.000141, SD=0.000238 ft (N), MA=0.000185, SD=0.000401

-0.0020

Particle ID: 590, u l=0.075 m/s, Bo=0.03 T

0.0000

-0.0020 1.0

fn (N), MA=0.0000261, SD=0.000183 ft (N), MA=0.00000993, SD=0.000126

3.0

5.0

7.0

9.0

Times (s) Fig. 14. Profiles of contact forces of a representative particle.

overlap distance is made to correspond to the rate of deformation. The more the overlap distance, the larger the repulsive force. Therefore, the contact force at the high magnetic-flux density is caused by the deformation of particles due to the interparticle magnetic force. The radial and angular interparticle magnetic forces depend upon magnetic-flux density. Fig. 15 shows the instantaneous radial and angular magnetic forces for a representative particle at three different magnetizations. Both radial magnetic force Fr and angular magnetic force Fθ are oscillates with times. From Eq. (18) the radial magnetic force becomes negative at θ = 0, while it is positive at θ = π/2.  2 6 χ e Vp B0 2π2 r5 þ χ e Vp πr2 Fr ¼ −  3 μ0 2πr3 −χ e Vp  2 12 χ e Vp B0 4π 2 r5 −χ e Vp πr2 Fr ¼  3 μ0 4πr3 þ χ e Vp

at θ ¼ 0

at θ ¼

ð30Þ

π : 2

ð31Þ

0.0000

Magnetic forces Fr and Fθ (N)

Granular temperature θ (cm/s)2

4.0

n m /n n=1:2, n=2100 u l =0.075 m/s

3.0

d=2.0 mm, ρ =2540 kg/m s

3

2.0

1.0

-0.0005 -0.0010 0.0040

Particle ID: 590, ul=0.075 m/s, Bo=0.5 T Fr (N), MV= -0.000942, SD=0.000101 Fθ (N), MV= -0.0000155, SD=0.00000951 Particle ID: 590, ul=0.075 m/s, Bo=0.1 T Fr (N), MV=-0.00257, SD=0.000442 Fθ (N), MV=0.000206, SD=0.000406

0.0000 -0.0040 0.030

Particle ID: 590, ul=0.075 m/s, Bo=0.03 T Fr (N), MV= -0.0117, SD=0.00762 Fθ (N), MV=-0.000747, SD=0.00327

0.000 0.0 0.0

0.1

0.2

0.3

0.4

0.5

Magnetic-flux density Bo (T) Fig. 13. Distribution of granular temperature as a function of magnetic-flux density.

-0.030 1.0

3.0

5.0

7.0

9.0

Times (s) Fig. 15. Profiles of magnetic forces of a representative particle.

S. Wang et al. / Powder Technology 237 (2013) 314–325

This indicates that the radial interparticle magnetic force controls particles repulsion (Fr > 0) and attraction (Fr b 0), while the tangential force induces particle rotation. The simulated radial magnetic forces are larger than the angular magnetic forces. This high radial magnetic force is caused by straight-chain aggregates of magnetic particles in the bed. The mean value (MV) and standard deviation (SD) of radial and angular magnetic forces are calculated and given in Fig. 15. The mean radial magnetic force is negative, meaning that particles attract. With the increase of the magnetization, the standard deviation of radial and angular magnetic forces is decreased. Depending upon the position of magnetic particles in the bed and relating to their dipole moment orientation, the interparticle magnetic force can be attractive or repulsive. If the particles approach each other along the line of the dipole moments, the force is attractive. If particles approach perpendicular to the dipole moments, the force is repulsive. Fig. 16 shows the distribution of mean magnetic forces along bed width as a function of magnetizations at bed height of 18 mm. The fluctuation of magnetic forces is increased with an increase of magnetizations. The positive radical force indicates repulsive force, on the contrary, the negative radical force indicates attractive force. Also the direction of positive tangential force is toward the left, and else is the right. The attractive force makes magnetic particles to form chains together. These chains break up the bed into many channels which are flow passage for non-magnetic particles and fluid. The magnetic force is zero for the magnetic particle chains. Comparing with radical force, the tangential force is smaller, so the magnetic straight chain doesn't rotate in addition to sway from side to side. Proper quantification of fluid–particle interactions is crucial for obtaining good performances. Drag coefficient is not a directly measurable parameter. For a single object, one can calculate the drag force based on the force balance equation and hence determine the drag coefficient if other parameters such as fluid density, projected area and relative velocity are known. For multiple particle system, one has to start from the momentum equation in the two-fluid model [39]. Recent computational studies of the fully resolved flow, based on the lattice-Boltzmann methodology (LBM) [40–42] have provided a significant dataset of the distribution of actual fluid drag on each particle under various conditions, that proves extremely useful to elaborate a larger scale model. Generally, the drag force acting on a particle in fluid–solid systems can be represented by the product of a momentum transfer coefficient

Magnetic forces Fr and Fθ (N)

0.0500

-0.0500 Particle ID: 590, ul=0.075 m/s, Bo=0.1 T, y=18 mm Fr (N), Fθ (N)

0.0000 -0.0020

Particle ID: 590, ul=0.075 m/s, Bo=0.03 T, y=18 mm Fr (N),

0.00005

Fθ (N)

0.00000 -0.00005 0.0

β¼

ρs ð1−εs Þg ut εn−1 s

20.0

40.0

60.0

80.0

Bed width x (mm) Fig. 16. Profiles of mean magnetic forces of magnetic particles.

100.0

ð32Þ

where the exponent n depends on the Reynolds number based on the terminal velocity ut of an isolated particle. 5:1−n 0:9 ¼ 0:1Ret : n−2:7

ð33Þ

Fig. 17 shows the distributions of volume fraction of liquid phase with and without magnetic-flux density as a function of superficial liquid velocity. The drag coefficient of liquid–solid phases is predicted by Richardson and Zaki model [43] and Huilin & Gidaspow model [28]. The values of particle Reynolds number are 140, 165 and 185 with the superficial liquid velocity of 0.071, 0.082 and 0.092 m/s, respectively. Both drag coefficient models show that the volume fraction of liquid phase increases with the increase of superficial liquid velocity. The difference of simulated liquid volume fractions between Richardson and Zaki model and Huilin & Gidaspow model is obvious. The liquid volume fraction predicted by Richardson and Zaki model is smaller than that by Huilin & Gidaspow model. The predicted values of volume fraction of liquid phase from Huilin–Gidaspow drag model shows 1.0 to 3.2% deviation from the Richardson and Zaki equation. The trends, however, are the same. Simulations for without magnetic-flux density show with high superficial liquid velocity particles move in a circulatory manner in the bed. The particle behavior has a gulf streaming which was found in experiments [44,45]. Simulations show that the liquid volume fraction is higher at the magnetic-flux density of 0.03 T than that without magnetic-flux density (B0 = 0.0). This implies the bed expansion decreased as the magnetic field increased. Once the bed had stabilized, its height did not decrease further with increasing magnetic field. The magnetic field decreased bed expansion and minimized axial mixing by inhibiting axial motion of the particles. 4. Conclusions Flow behavior of particles is simulated using DEM–CFD in a twodimensional liquid–solid fluidized bed with uniform magnetic field. The motion of magnetic particles and non-magnetic particle in the bed is predicted. Simulations show a significant influence of the magneticfield induced interparticle force on the behavior of particles in a uniform

Particle ID: 590, ul=0.075 m/s, Bo=0.5 T, y=18 mm Fr (N), Fθ (N)

0.0000

-0.1000 0.0020

β and the slip velocity between the two phases. A drag model proposed by Richardson and Zaki [44] was

0.70

Volume fraction of liquid phase εl

0.1000

323

Richardson and Zaki equation Huilin & Gidaspow equation

0.65 nm/nn=1:2, n=2100

0.60

3

d=2.0 mm, ρ s=2540 kg/m B0=0.0 T

0.55 nm/nn=1:2, n=2100

0.65

3

d=2.0 mm, ρ s=2540 kg/m B0=0.03 T

0.60 Richardson and Zaki equation Huilin & Gidaspow equation

0.55 0.070

0.075

0.080

0.085

0.090

0.095

Superficial liquid velocity ul (m/s) Fig. 17. Distributions of liquid volume fraction with and without magnetic-flux density.

324

S. Wang et al. / Powder Technology 237 (2013) 314–325

magnetic fluidized bed. Segregation of magnetic particles and nonmagnetic particles along bed height existed in magnetic fluidized beds. At low magnetic-flux density, the bed maintains fluidization with chains of magnetic particles and dispersed particles for non-magnetic particles. However, at the high magnetic-flux density, the straight-chain aggregates of magnetic particles emerged. These straight-chain aggregates will settle down at the bottom of the bed due to weights, and show little free movement. The velocity and bed expansion height are decreased with the increase of magnetic-flux density in the magnetic fluidized bed. The granular temperature of particles increases, reaches a maximum, and then decreases with the increase of magnetic-flux density due to the formation of straight-chain aggregates. Note that present simulations are limited by the number of particles in the system. This makes it difficult to make quantitative comparisons with experiments. However, the simulations do indicate an answer on how the magnetic fields lead to the formation of magnetic chains of particle strings. Detailed simulations should provide better quantitative answers to the question how the magnetic force affects flow behavior of particles and liquid in the bed. Therefore, a thorough investigation is needed by numerical simulations and experiments, which is left as future work. Notation B0 Cd Cj d E fc fd fl fm mi g g0 G H I kn kt nm nn P Ps Re St T rp ul up Vp x y

magnetic-flux density, T drag coefficient fluctuating velocity, m/s diameter of particle, m modulus of longitudinal elasticity, N/m 2 contact force, N drag force, N lubrication force, N virtual mass force, N mass of particle, kg gravitational acceleration, m/s 2 radial distribution function at contact modulus of transverse elasticity, N/m 2 magnetic-field intensity, A/m moment of inertia, kg/m 2 spring constant for normal direction, N/(m 3/2) spring constant for tangential direction, N/m number of magnetic particles number of non-magnetic particles liquid pressure, N/m 2 solid pressure, Pa Reynolds number Stokes number torque, N/m particle radius, m liquid velocity, m/s solid velocity, m/s volume of a particle, m 3 radial distance, m axial distance from the bottom, m

Greek letters density of liquid, kg/m 3 ρl εl porosity εs concentration of particles τl viscous stress tensor μl viscosity of liquid, kg/ms β interface momentum transfer coefficient, kg/m 2 s 2 ω angular velocity, 1/s κ surface tension γ Poisson's ratio δ displacement, m η damping coefficient, kg/s

μf χe χp θ

friction coefficient magnetic susceptibility, dimensionless magnetic susceptibility of the solid granular temperature, m 2/s 2

Subscripts l liquid phase p particle phase w wall n normal direction t tangential direction x x direction y y direction

Acknowledgment This work was supported by the National Science Foundation in China through grant (No. 21076043) and Program for New Century Excellent Talents in Heilongjiang Provincial University (2012). References [1] A.M. Squires, M. Kwauk, A.A. Avidan, Fluid beds: at last, challenging two entrenched practices, Science 230 (1985) 1329–1337. [2] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, 1985. [3] B.J. Glasser, S. Sundaresan, I.G. Kevrekidis, From bubbles to clusters in fluidized beds, Physical Review Letters 81 (1998) 1849–1852. [4] S. Limtrakul, W. Rotjanavijit, T. Vatanatham, Lagrangian modeling and simulation of effect of vibration on cohesive particle movement in a fluidized bed, Chemical Engineering Science 62 (2007) 232–245. [5] W.F. van Kleijn, B. Demirbas, N.G. Deen, J.A.M. Kuipers, J.R. van Ommen, Discrete particle simulations of an electric-field enhanced fluidized bed, Powder Technology 183 (2008) 196–206. [6] R. Chirone, M. Urciuolo, Role of bed height and amount of dust on the efficiency of sound-assisted fluidized bed filter/afterburner, AICHE Journal 55 (2009) 3066–3075. [7] L. Xiang, W. Shuyan, L. Huilin, L. Goudong, C. Juhui, L. Yikun, Numerical simulation of particle motion in vibrated fluidized beds, Powder Technology 197 (2010) 25–35. [8] J.H. Siegell, C.A. Coulaloglou, Magnetically stabilized fluidized beds with continuous solids throughput, Powder Technology 39 (1984) 215–222. [9] O. Harel, Y. Zimmels, W. Resnick, Particle separation in a magnetically stabilized fluidized bed, Powder Technology 64 (1991) 159–164. [10] V.L. Ganzha, S.C. Saxena, Hydrodynamic behavior of magnetically stabilized fluidized beds of magnetic particles, Powder Technology 107 (2000) 31–35. [11] Z. Al-Qodah, M. Al-Busoul, M. Al-Hassan, Hydro-thermal behavior of magnetically stabilized fluidized beds, Powder Technology 115 (2001) 58–67. [12] M.J. Rhodes, X.S. Wang, A.J. Forsyth, K.S. Gan, S. Phadtajaphan, Use of a magnetic fluidized bed in studying Geldart Group B to A transition, Chemical Engineering Science 56 (2001) 5429–5436. [13] F. Gros, S. Baup, M. Aurousseau, Hydrodynamic study of a liquid/solid fluidized bed under transverse electromagnetic field, Powder Technology 183 (2008) 152–160. [14] Z. Hao, Q. Zhu, Z. Jiang, H. Li, Fluidization characteristics of aerogel Co/Al2O3 catalyst in a magnetic fluidized bed and its application to CH4-CO2 reforming, Powder Technology 183 (2008) 46–52. [15] Q. Zhu, H. Li, Study on magnetic fluidization of group C powders, Powder Technology 86 (2007) 179–185. [16] J. Hristov, Magnetically assisted gas–solid fluidization in a tapered vessel: magnetization—LAST mode, Particuology 7 (2009) 26–34. [17] N.G. Deen, M. Van Sint Annaland, M.A. Van der Hoef, J.A.M. Kuipers, Review of discrete particle modeling of fluidized beds, Chemical Engineering Science 62 (2007) 28–44. [18] H.P. Zhu, Z.Y. Zhou, R.Y. Yang, A.B. Yu, Discrete particle simulation of particulate systems: theoretical developments, Chemical Engineering Science 62 (2007) 3378–3396. [19] Y. Tsuji, Multi-scale modeling of dense phase gas-particle flow, Chemical Engineering Science 62 (2007) 3410–3418. [20] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1979) 47–65. [21] B.P.B. Hoomans, J.A.M. Kuipers, W.J. Briels, W.P.M. van Swaaij, Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidized bed: a hard-sphere approach, Chemical Engineering Science 51 (1996) 99–118. [22] Y. Tsuji, T. Kawaguchi, T. Tanaka, Discrete particle simulation of two dimensional fluidized bed, Powder Technology 77 (1993) 79–87. [23] A.D. Renzo, F. Cello, F.P. DiMaio, Simulation of the layer inversion phenomenon in binary liquid-fluidized beds by DEM–CFD with a drag law for polydisperse systems, Chemical Engineering Science 66 (2011) 2945–2958.

S. Wang et al. / Powder Technology 237 (2013) 314–325 [24] B.H. Xu, A.B. Yu, Numerical simulation of the gas solid flow in a fluidized bed by combining discrete particle method with computational fluid dynamics, Chemical Engineering Science 52 (1997) 2785–2809. [25] A.T. Cate, S. Sundaresan, Analysis of unsteady forces in ordered arrays of monodisperse spheres, Journal of Fluid Mechanics 552 (2006) 257–287. [26] Y.D. Sobral, T.F. Oliveira, F.R. Cunha, On the unsteady forces during the motion of a sedimenting particle, Powder Technology 178 (2007) 129–141. [27] J.J. Derksen, S. Sundaresan, Direct numerical simulations of dense suspensions: wave instabilities in liquid-fluidized beds, Journal of Fluid Mechanics 587 (2007) 303–336. [28] Ansys-Fluent Inc., Fluent Users Manual Version 14.0, New Hampshire, USA, 2012. [29] C.T. Crowe, M. Sommerfeld, Y. Tsuji, Multiphase Flows with Droplets and Particles, CRC Press, 1997. [30] S. Kim, S.J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinman Series in Chemical Engineering, Stoneham, MA, USA, 1991. [31] J.P. Jakubovics, Magnetism and Magnetic Materials, The University Press, Great Britain, 1994. [32] J.D. Jackson, Classical Electrodynamics, Third ed. Wiley, USA, 1999. [33] G.N. Jovanovic, T. Sornchamni, J.E. Atwater, J.R. Akse, R.R. Wheeler, Magnetically assisted liquid–solid fluidization in normal and microgravity conditions: experiment and theory, Powder Technology 148 (2004) 80–91. [34] Z. Hao, X. Li, H. Lu, G. Liu, Y. He, S. Wang, P. Xu, Numerical simulation of particle motion in a gradient magnetically assisted fluidized bed, Powder Technology 203 (2010) 555–564. [35] S. Wang, S. Guo, J. Gao, X. Lan, Q. Dong, X. Li, Simulation of flow behavior of liquid and particles in a liquid–solid fluidized bed, Powder Technology 224 (2012) 365–373.

325

[36] E. Peirano, V. Delloume, B. Leckner, Two- or three-dimensional simulations of turbulent gas–solid flows applied to fluidization, Chemical Engineering Science 56 (2001) 4787–4799. [37] N. Xie, F. Battaglia, S. Pannala, Effects of using two- versus three-dimensional computational modeling of fluidized beds—part I, hydrodynamics, Powder Technology 182 (2008) 1–13. [38] K. Anderson, S. Sundaresan, R. Jackson, Instabilities and the formation of bubble in fluidized beds, Journal of Fluid Mechanics 303 (1995) 327–366. [39] D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions, Academic Press, London, 1994. [40] R.J. Hill, D.L. Koch, J.C. Ladd, Moderate-Reynolds-number flows in ordered and random arrays of spheres, Journal of Fluid Mechanics 448 (2001) 243–278. [41] M.A. van der Hoef, R. Beetstra, A.M. Kuipers, Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force, Journal of Fluid Mechanics 528 (2005) 233–254. [42] X. Yin, S. Sundaresan, Fluid-particle drag in low-Reynolds-number polydisperse gas–solid suspensions, AICHE Journal 55 (2009) 1351–1368. [43] J.F. Richardson, W.N. Zaki, Sedimentation and fluidization: part 1, Transactions of the Institution of Chemical Engineers 32 (1954) 35–53. [44] M. Goto, T. Imamura, T. Hirose, Axial dispersion in liquid magnetically stabilized fluidized beds, Journal of Chromatography. A 690 (1995) 1–8. [45] J.H. Siegell, Liquid-fluidized magnetically stabilized beds, Powder Technology 52 (1987) 139–148.