Mixing Behaviors of Granular Materials in Gas Fluidized Beds with

Article pubs.acs.org/IECR

Mixing Behaviors of Granular Materials in Gas Fluidized Beds with Electrostatic Effects Eldin Wee Chuan Lim* Department of Chemical and Biomolecular Engineering, National University of Singapore, Singapore 117576 ABSTRACT: The discrete element method combined with computational fluid dynamics was coupled to an electrostatic force model for computational studies of mixing behaviors in gas fluidized bed systems with electrostatic effects. Due to the presence of strong electrostatic forces between particles and walls, there was a high tendency for particles to be adhered to the walls or other particles near the walls within the fluidized bed, resulting in less vigorous fluidization. This in turn resulted in lower mixing efficiencies in comparison with fluidization in the presence of weaker electrostatic effects. Particle−wall electrostatic forces were on average stronger than both fluid drag forces and particle−particle collision forces when strong electrostatic effects were present, and this accounted for the difficulty with which particles adhered to walls could be removed and transferred to other locations within the bed. Such transfers of particles were necessary for mixing to occur during fluidization but required strong electrostatic forces to be overcome.

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INTRODUCTION Triboelectrification is the process of charge acquisition by solid particulate materials during flows due to repeated collisions between the solid particles with surfaces made of a different material.1 It is an important phenomenon that arises in various environmental and industrial processes especially those that involve bounded flows of granular materials. The amount of charge transferred between two contacting bodies during impact depends on the potential difference between the two bodies, which in turn depends on their surface work functions and the image charge effect.2 In recent years, the behaviors of granular materials under the effects of electrostatics have been investigated by several researchers. Al-Adel et al.3 investigated the effect of static electrification on gas−solid flows in vertical risers and captured various qualitative features of riser flows. Matsusaka et al.4 developed a formulation for the variation of granule charging caused by repeated impacts on a wall and employed the formulation to granule charging in granular flows where each granule carried a different amount of charge. They then analyzed theoretically the granule charge distribution. Lim et al.5 coupled the discrete element method (DEM) with large eddy simulation (LES) and an electrostatic force model for computational studies of pneumatic transport of granular materials in the presence of electrostatic effects. They observed that electrostatic and fluid drag forces were important in influencing particle behaviors during pneumatic transport at low and high flow rates respectively. Tanoue et al.6 investigated triboelectrification of particles fed by an ejector into a turbulent pipe flow and observed an unusual charge transfer which could not be accounted for solely by contact potential difference. They suggested that the charge transfer consisted of adsorption of ions on the inner wall of the pipe in addition to ionization by self-discharge. Bunchatheeravate et al.7 developed a technique for predicting the amount of charge acquired by particles due to impact charging during transport through pipes. The technique was applicable to a range of particle shapes and sizes. Yao et al.8 examined the effect of granular shape on electrostatic charging characteristics of single granules sliding along a pipe wall. They © 2013 American Chemical Society

found that triangular granules with smaller front-facing angles tended to generate more electrostatic charges. Fluidization has been established as an important operation in a variety of industrial processes that range from heterogeneous chemical reactions to solids drying. In many of these processes, fluidization of two or more types of solid particles in a fluidized bed is necessary. Due to differences in sizes and material properties of the different types of solid particles, the fluidization behavior is expected to be different and much more complex than single-species fluidization. As such, a good understanding of the fluidization and mixing behaviors of solid particles will be instrumental for the design of industrial fluidized bed systems. Rhodes et al.9 examined the usefulness of the discrete element method for studying solids mixing in gas fluidized beds and showed that gas velocity and particle properties were important parameters influencing solids mixing in bubbling fluidized beds. Feng et al.10 reported a numerical study of segregation and mixing of binary mixtures of particles in a gas fluidized bed via the approach of combining DEM with computational fluid dynamics (CFD). They subsequently extended their study to investigate the effects of gas velocity on mixing behaviors under partially and fully fluidized conditions.11 Dahl and Hrenya12 also applied a similar Eulerian-Lagrangian model based on DEM and CFD to investigate segregation behavior in gas fluidized bed systems with continuous particle size distributions. More recently, Zhang et al.13,14 investigated experimentally mixing and segregation behaviors of binary mixtures of granular materials due to effects of particle size and density. Halow et al.15 also reported experimental studies of mixing and segregation dynamics of single, magnetically tagged particles in a bubbling fluidized bed. They observed that the spatial distribution of Received: Revised: Accepted: Published: 15863

August 1, 2013 October 1, 2013 October 16, 2013 October 16, 2013 dx.doi.org/10.1021/ie402511p | Ind. Eng. Chem. Res. 2013, 52, 15863−15873

Industrial & Engineering Chemistry Research

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The governing equations in DEM for describing translational and rotational motions of individual solid particles are basically Newton’s laws of motion:

these tracer particles resulting from differences in density was consistent with a Weibull distribution. Although several studies of solids mixing in fluidized bed systems have been reported in the research literature, mixing behaviors of granular materials in fluidized bed systems in the presence of electrostatic effects do not seem to have been adequately addressed to date. In addition to being a safety hazard, the buildup of electrostatic charges within an industrial fluidized bed system may cause deviations in performance of the system from that for which it was originally designed. Valverde et al.16 investigated electrostatic charging in nanofluidization experimentally. The influence of an electric field on the bulk behavior of a fluidized bed of silica nanoparticles as well as the trajectories of the nanoparticle agglomerates was studied. Sowinski et al.17 developed a system that utilizes an online Faraday cup to quantify electrostatic charge generation of bed particles, particles adhered to the walls, and particles entrained from the column of a gas fluidized bed. Rokkam et al.18 applied a CFD model coupled to an electrostatic model to a polymerization fluidized bed reactor. Segregation of particles and defluidization induced by large electrostatic charges were observed. Cheng et al.19 conducted a study to characterize the electrostatic charge generation in the fully developed regions of the riser and downer of a circulating fluidized bed. The average equivalent currents and solids holdup were measured under different superficial velocities. It was found that both solids mass flux and solids holdup had significant influence on the average equivalent currents generated. The fluidization and mixing behaviors of normally noncohesive granular materials (e.g., Geldart groups B and D particles) that become cohesive as a result of electrostatic charging may be very different from those of naturally cohesive (i.e., Geldart group C) particles. The dynamics of the fluidization and mixing processes are expected to be different between these two groups of granular materials due to the distinct differences in orders of magnitudes between the various types of forces, such as gravitational, fluid drag, collision, and electrostatic forces, that are present. In view of the importance of electrostatic effects in gas fluidization systems and the lack of studies that have been conducted to date on fluidization and mixing behaviors of normally noncohesive granular materials in gas fluidized beds with electrostatic effects, the present study will address this gap in the research literature by means of computational modeling with a view toward achieving deeper insights to the mixing mechanisms associated with such fluidized bed systems with electrostatic effects. In the following section, the computational model and physical system of interest to the present study will be described. The simulation results obtained for the various physical conditions considered in this study will then be discussed and a summary of the conclusions derived will be presented in the Conclusions of this paper.

mi

Ii

dvi = dt dωi = dt

N

∑ (fc,ij + fd,ij + fe, ij) + f f, i + mi g j=1

(1)

N

∑ Tij j=1

(2)

where mi and vi are the mass and velocity of the ith particle respectively, N is the number of particles in contact with the ith particle, fc,ij and fd,ij are the contact and viscous contact damping forces, respectively, fe,ij is the electrostatic force, ff,i is the fluid drag force, Ii is the moment of inertia of the ith particle, ωi is its angular velocity, and Tij is the torque arising from contact forces which causes the particle to rotate. Contact and damping forces were calculated by applying a linear spring-and-dashpot model as closure. The normal (fcn,ij, fdn,ij) and tangential (fct,ij, fdt,ij) components of the contact and damping forces were calculated as follows: fcn, ij = −(κ n, iδn, ij)ni

(3)

fct, ij = −(κ t, iδt, ij)ti

(4)

fdn, ij = −ηn, i(vr ·ni)ni

(5)

fdt, ij = −ηt, i{(vr ·ti)ti + (ωi R i − ωj R j)}

(6)

where κn,i, δn,ij, ni, ηn,i and κt,i, δt,ij, ti, ηt,i are the spring constants, displacements between particles, unit vectors, and viscous contact damping coefficients in the normal and tangential directions respectively, vr is the relative velocity between particles, and Ri and Rj are the radii of particles i and j, respectively. If |fct,ij| > |fcn,ij|tan ϕ, then |fct,ij| = |fcn,ij|tan ϕ, where tan ϕ is analogous to the coefficient of friction. Computational Fluid Dynamics. The governing equations for describing the motion of the continuum gas phase are basically the Navier−Stokes equations. An additional source term in the momentum equation has been included to account for interphase interactions:

∂ε + ∇·(ε u) = 0 ∂t ∂(ρf ε u) ∂t

(7)

+ ∇·(ρf ε uu)

= −ε∇P + ∇·(μf ε∇u) + ρf ε g − F

(8)

where u is the velocity vector, ρf is the fluid density, μf is the fluid viscosity, ε is the local average porosity, P is the fluid pressure, and F is the source term due to fluid−particle interaction. The operating conditions imposed in this study were such that gas flows were nonturbulent and so a turbulence model was not applied in the solution of the gas flow field. Fluid Drag Force. In a multiphase system, the interstitial fluid exerts drag forces on the solid particles whenever velocity differences exist between the two phases. Several fluid drag force models have been developed in the literature and the model due to Di Felice24 which is applicable over a wide range of particle Reynolds numbers was used for calculating the fluid drag force in this study:

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COMPUTATIONAL MODEL Discrete Element Method. The discrete element method (DEM) was developed by Cundall and Strack20 for modeling the behavior of assemblies of discs and spheres. With the advent of computational power in recent years, it has been applied for studies of various types of multiphase systems.21−23 In this section, a brief description of the method and corresponding governing equations will be presented. 15864

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f f, i = f f0, iεi−(χ + 1) f f0, i = 0.5cd0, iρf πR i 2εi 2|u i − vi|(u i − vi)

⎡ (1.5 − log Re )2 ⎤ p, i 10 ⎥ χ = 3.7 − 0.65exp⎢ − ⎢⎣ 2 ⎦⎥

cd0, i

⎛ ⎞2 4.8 ⎟ = ⎜⎜0.63 + Rep, i 0.5 ⎟⎠ ⎝

Rep, i =

Article N

(9)

fep, ij =

j=1 j≠i

(10)

(11)

(12)

E=

∫ 4πε1

0

(15)

·

dq r

2

=

λ 2πε0d

∫0

π /2

cos θ dθ =

λ 2πε0d

(16)

where E is the electric field strength, q is the equilibrium charge on the wall, εo is the permittivity of free space, r is the average distance of particles to the wall, λ is the linear charge density along the wall. The electrostatic force on a charged particle i located near the wall may then be calculated as few,i = E·Q. Numerical Integration. The second-order Adams−Bashforth integration scheme was implemented for time integration of Newton’s law and evaluation of particle positions and velocities. This scheme uses values of positions and velocities computed for an earlier time step for the interpolation of corresponding values for a subsequent time step. One of the advantages of this time integration scheme is that only one calculation of the acceleration needs to be carried out at each time step. The equations involved in this integration scheme are as follows:

(13)

where ff0,i is the fluid drag force on particle i in the absence of other particles, χ is an empirical parameter, εi is the local average porosity in the vicinity of particle i, cd0,i is the drag coefficient, Rep,i is the Reynolds number based on particle diameter, and ui is the fluid velocity of the computational cell in which particle i is located. Electrostatic Effects. During fluidization, solid particles gain electrostatic charges as a result of repeated collisions and impacts against other particles and with the walls of the vessel. The amount of charge transferred between two contacting bodies during impact depends on the potential difference between the two bodies, which in turn depends on their surface work functions and the image charge effect. Following the approach applied by Lim et al.,5 it was assumed that the fluidization system considered in this study had been operated beyond the transient state of electrification such as to reach a dynamic state of electrostatic equilibrium. The amount of charge carried by the particles as well as the walls of the container then remained fairly constant with respect to time. In general, charges are transferred between particles and the container walls during particle−wall collisions due to differences in material properties between the two contacting bodies while charge transfer between two colliding particles may occur due to differences in electrical potential or charge densities. At the state of electrostatic equilibrium, it may be expected that particles and the walls are oppositely charged relative to each other while all particles carry the same type of charges. In other words, attractive electrostatic forces exist between particles and walls while repulsive electrostatic forces exist between any pair of particles. The total electrostatic force acting on each particle may then be written as the sum of electrostatic forces due to charges carried by other particles and the walls. fe, ij = fep, ij + few, i

Q2 ni 4πεo rij 2

where Q is the constant charge assumed to be carried by all particles, εo is the permittivity of free space, rij is the distance between particles i and j, and ni is the unit normal vector in the direction of the line joining the two particle centers. The average electric field strength near a wall may be estimated by assuming the wall to be an infinitely long flat plate.5

2ρf R iεi|u i − vi| μf

∑

⎛3 ⎞ 1 rt +Δt = rt + Δt ⎜ vt − vt −Δt ⎟ ⎝2 ⎠ 2

(17)

⎛3 ⎞ 1 vt +Δt = vt + Δt ⎜ at − at −Δt ⎟ ⎝2 ⎠ 2

(18)

where rt, vt, and at are position, velocity, and acceleration at time t, respectively, and Δt is the time step. Simulation Conditions. The geometry of the computational domain considered in this study was a fluidized bed with a rectangular base measuring 64 mm × 8 mm and height of 800 mm. The granular materials consisted of 25 000 spherical particles with diameter 1.0 mm and density 2500 kg m−3. Other pertinent simulation parameters are presented in Table 1. Values of fluidizing velocities were selected to be consistent with a recent study by Lim et al.23 so that direct comparisons of fluidization and mixing behaviors of particles in fluidized bed Table 1. Material Properties and System Parameters

(14)

shape of particles number of particles particle diameter particle density coefficient of restitution coefficient of friction gas density, ρf gas viscosity, μf gas velocity particle charge density wall linear charge density, λ system dimensions simulation time step, Δt

where fep,ij and few,i are the electrostatic forces due to other charged particles and the walls acting on particle i respectively. It may also be mentioned at this point that at the state of electrostatic equilibrium, the linear charge density of the container walls is assumed to give rise to electrostatic forces of attraction acting on particles near the walls that are much more significant than image charge forces. In other words, it is assumed that image charge forces are negligible in comparison with the electrostatic forces due to charged walls when the state of electrostatic equilibrium has been attained. The electrostatic force arising from charges carried by other particles may be calculated by assuming each particle to be a constant point charge. 15865

spherical 25000 1.0 mm 2500 kg m−3 0.9 0.3 1.205 kg m−3 1.8 × 10−5 N s m−2 1.0, 1.4, 1.6, 1.8 m s−1 10−9, 10−8 C g−1 10−6, 10−5 C m−1 64 mm length × 8 mm width × 800 mm height 10−6 s



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systems in the absence and presence of electrostatic effects can be made. Values of particle charge density and linear wall charge density applied were based on those measured experimentally by Yao et al.25 so as to obtain physically realistic simulations of charged particles and container walls. In all simulations performed, particles were first allowed to settle freely under gravity for 0.5 s and form a packing at the bottom of the container before fluidizing air was initiated. A uniform fluidizing gas velocity was applied at the base of the computational domain to simulate a uniform gas distribution. Simulations were performed in three dimensions for both solid and gas phases.

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RESULTS AND DISCUSSION Figure 1 shows the mixing behaviors of electrically charged particles during fluidization at two different fluidizing velocities

Figure 2. Distributions of spatially averaged fractions of particles originally belonging to the bottom layer of the packed beds (a) 1.0 s and (b) 10 s after the start of the fluidization process. The particle charge density and wall linear charge density were 10−9 C g−1 and 10−6 C m−1, respectively.

Figure 1. Mixing behaviors of electrically charged particles in a gas fluidized bed. The fluidizing velocities applied were (a) 1.0 and (b) 1.8 m s−1. The particle charge density and wall linear charge density were 10−9 C g−1 and 10−6 C m−1, respectively.

for the case where the particle charge density and wall linear charge density were 10−9 C g−1 and 10−6 C m−1, respectively. The fluidized bed systems contained equal amounts of monodispersed particles with identical physical properties. The packed beds formed at the start of each fluidization process were divided into two layers containing approximately equal numbers of particles and particles in the two layers were colored differently to allow visualization of the subsequent mixing behaviors. Figure 1a shows that the initial packed bed of particles expanded only slightly when a low fluidizing velocity

Figure 3. Time evolution of Lacey mixing indices for fluidized bed systems with particle charge density 10−9 C g−1 and wall linear charge density 10−6 C m−1 at various fluidizing velocities.

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Figure 4. Mixing behaviors of electrically charged particles in a gas fluidized bed. The fluidizing velocities applied were (a) 1.0 and (b) 1.8 m s−1. The particle charge density and wall linear charge density were 10−9 C g−1 and 10−5 C m−1, respectively.

Figure 5. Distributions of spatially averaged fractions of particles originally belonging to the bottom layer of the packed beds (a) 1.0 and (b) 10 s after the start of the fluidization process. The particle charge density and wall linear charge density were 10−9 C g−1 and 10−5 C m−1, respectively.

of 1.0 m s−1 was applied. Although particles were not lifted high from their original positions, they were able to undergo fairly vigorous relative motions within the slightly expanded bed giving rise to gradual mixing between the two layers of particles. On the basis of visual inspection, Figure 1a shows that, apart from a segregated layer of particles near the bottom, the bed had become fairly well mixed at the end of 4 s. With higher fluidizing velocities of U = 1.4 m s−1 and U = 1.6 m s−1, the initial packed beds expanded to larger extents and particles were thus able to undergo more vigorous relative motions (figures not shown for brevity). Consequently, the beds became fairly well mixed after about 2 s of fluidization. Figure 1b shows that similar fluidization and mixing behaviors were observed with the application of U = 1.8 m s−1. Qualitatively, these fluidization and mixing behaviors were almost indiscernible from those of fluidized beds where electrostatic effects were absent. As will be shown in a quantitative analysis of these effects in a later section of this paper, this is because the electrostatic effects present based on the particle charge density and wall linear charge density imposed were fairly weak for the present cases. These will be compared and contrasted with cases where much stronger electrostatic effects are present. The mixing behaviors of the above fluidized bed systems will now be examined more quantitatively by analyzing the variation of distribution of particles as well as a mixing index with time during the fluidization process. Figure 2a shows the spatially


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Figure 9. Velocity vectors of particles located within a region that is one particle diameter in thickness extracted from the midplane of the bed along the spanwise direction for the fluidized bed systems with particle charge density 10−9 C g−1 and wall linear charge density 10−6 C m−1 at fluidizing velocities of (a) 1.0, (b) 1.4, (c) 1.6, and (d) 1.8 m s−1.

Figure 7. Distributions of spatially averaged fractions of particles originally belonging to the bottom layer of the packed beds (a) 1.0 and (b) 10 s after the start of the fluidization process. The particle charge density and wall linear charge density were 10−8 C g−1 and 10−5 C m−1 respectively.

averaged fractions of particles originally located in the bottom region of the packed bed as a function of position in the vertical direction 1.0 s after the start of the fluidization process. The extents of mixing between the two layers of particles achieved after 1.0 s for all fluidizing velocities investigated were rather limited. This was consistent with the observations derived from Figure 1 previously. Figure 2a shows that particles originally located in the bottom region of the packed bed remained largely in those regions after 1.0 s of fluidization, especially at the lower fluidizing velocities applied. In contrast, Figure 2b shows that the distributions of these particles became approximately uniform throughout almost the entire bed after 10 s of fluidization for all fluidizing velocities investigated, indicating that almost perfect mixing between the original layers of particles had been achieved. The progressions in time of the extents of mixing for the above fluidized bed systems may also be analyzed based on a mixing index.10,11 Here, the Lacey mixing index was calculated at equal time intervals during the fluidization process for each fluidized bed system according to the following formula:


Lacey index = 15868

σo 2 − σ σo 2 − σR 2

(19)



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Figure 10. Velocity vectors of particles located within a region that is one particle diameter in thickness extracted from the midplane of the bed along the spanwise direction for the fluidized bed systems with particle charge density 10−9 C g−1 and wall linear charge density 10−5 C m−1 at fluidizing velocities of (a) 1.0, (b) 1.4, (c) 1.6, and (d) 1.8 m s−1.

Figure 11. Time evolution of average kinetic energies of particles in the fluidized bed systems with particle charge density 10−9 C g−1 and wall linear charge densities (a) 10−6 C m−1 and (b) 10−5 C m−1 at various fluidizing velocities.

where σo2 and σR2 are the theoretical upper and lower limits of mixture variances calculated as σo2 = p(1 − p) and σR2 = p(1 − p)/n, p, and (1 − p) are the proportions of the two groups of particles determined from samples respectively and n is the number of particles in each sample. The computational domain was divided into sampling cells with dimensions 4 mm × 4 mm × 4 mm for sampling. Sampling was performed on every cell at equal time intervals and only samples containing at least 40 particles were used for calculating the Lacey mixing index. This was deemed representative of the state of mixing of the fluidized bed. Figure 3 shows that the Lacey index for the above fluidized bed system with an applied fluidizing velocity of U = 1.0 m s−1 increased gradually from 0.0 to about 0.9 within about 10 s. This was in agreement with the observation derived based on visual inspection of Figure 1a. With higher fluidizing velocities of 1.4, 1.6, and 1.8 m s−1, the Lacey indices increased more rapidly and attained values of approximately 1.0 within about 5 s. These were indicative of shorter times required to achieve almost perfect mixing, or equivalently, higher mixing efficiencies at these fluidizing velocities. It may also be observed that the Lacey index profiles for the mixing processes at these higher fluidizing velocities were very similar, indicating that fluidizing velocity was no longer the limiting factor determining

the efficiencies of the mixing processes. The mechanistic behaviors of the fluidized beds at the individual particle level giving rise to these macroscopic observations will be analyzed in greater details in a later section of this paper after discussions of the mixing behaviors of fluidized bed systems with stronger electrostatic effects. Figure 4 shows the fluidization and mixing behaviors of particles at the gas velocities of U = 1.0 and 1.8 m s−1, when the wall linear charge density had been increased by 1 order of magnitude (10−5 C m−1) in comparison with that applied in the previous cases. It may be inferred from the simulation snapshots that strong particle−wall adhesive forces resulting from the higher wall linear charge density were present. Consequently, particle motions within the fluidized bed were significantly hindered as a result of particles adhering to the walls or other particles near the walls. At the lowest gas velocity of U = 1.0 m s−1, Figure 4a shows that the bed expanded only minimally and the two layers of particles in the original packed bed remained segregated as two large clusters even after 8 s of fluidization. At higher fluidizing velocities, Figure 4b shows that the bed of particles expanded to larger extents but the amount of mixing achieved was rather limited as large segregated 15869



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Figure 12. Time evolution of average relative fluid drag forces (Fdrag/Fg) and relative particle−particle collision forces (Fcollision/Fg) for the fluidized bed systems with particle charge density 10−9 C g−1 and wall linear charge density 10−6 C m−1, where Fg is particle weight. The fluidizing velocities applied were (a) 1.0, (b) 1.4, (c) 1.6, and (d) 1.8 m s−1. F represents the dimensionless force ratios, Fcollision/Fg or Fdrag/Fg.

clusters of particles from the two original layers could still be discerned after 8 s of fluidization. Due to strong electrostatic forces between particles and the walls, there was a high tendency for particles to remain adhered to the walls or other particles near the walls. Consequently, relative motions between particles that were necessary for mixing to occur were severely hindered especially at low fluidizing velocities. Figure 5a shows that all the fluidized beds were still in a state of almost complete segregation after 1.0 s of fluidization for all gas velocities applied. This agreed with the qualitative observations made earlier based on visual inspection of Figure 4. In addition, it may also be observed from the solid fraction distributions that the extents of mixing achieved after 1.0 s increased systematically with increasing gas velocities. Figure 5b shows that mixing was still largely incomplete after 10 s of fluidization at all gas velocities applied as exhibited by the nonuniform distributions of solid fraction throughout the entire beds. In contrast with the previous cases where weaker electrostatic effects were present, these solid fraction distributions were indicative of longer times required for mixing, or equivalently, lower mixing efficiencies at the same set of fluidizing velocities. The lower mixing efficiencies of particles during fluidization with stronger electrostatic effects may also be observed from

the Lacey index profiles and comparisons with those presented earlier. Figure 6 shows that the Lacey index for the fluidized bed system with strong electrostatic effects and an applied fluidizing velocity of 1.0 m s−1 increased much more slowly than in the corresponding previous case where electrostatic effects were weaker (Figure 3). The Lacey index reached a value of 0.6 after 10 s, indicating that mixing was still largely incomplete after 10 s of fluidization. This was consistent with the qualitative observations made earlier based on visual inspection of Figure 4a. Furthermore, in contrast with the profiles for beds with weaker electrostatic effects, the rate of increase of Lacey index values increased with increasing fluidizing velocity and the profiles obtained at the fluidizing velocities of 1.4, 1.6, and 1.8 m s−1 did not collapse onto one another. This might indicate that the fluidizing velocity had become a limiting factor that determined the mixing efficiencies of the fluidized bed systems with stronger electrostatic effects in contrast to the previous cases. Here, high fluidizing velocity was essential for particles to be able to overcome the strong particle-wall adhesive forces arising from electrostatic effects. This was in turn required for particles to be able to undergo relative motions that were necessary for mixing to occur. This explains why increasing fluidizing velocity led to increased mixing efficiencies in these latter cases. The mixing behaviors of 15870



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Figure 13. Time evolution of average relative fluid drag forces (Fdrag/Fg) and relative particle−particle collision forces (Fcollision/Fg) for the fluidized bed systems with particle charge density 10−9 C g−1 and wall linear charge density 10−5 C m−1, where Fg is particle weight. The fluidizing velocities applied were (a) 1.0, (b) 1.4, (c) 1.6, and (d) 1.8 m s−1. F represents the dimensionless force ratios, Fcollision/Fg or Fdrag/Fg.

forces were the dominant factor influencing fluidization and mixing behaviors. Figure 8 also shows that Lacey index profiles for the various fluidizing velocities were similar to the corresponding profiles presented in Figure 6, further confirming that similar mixing behaviors were exhibited by the beds with corresponding fluidizing velocities and effects of particle−particle electrostatic forces were negligible. This was not unexpected as the magnitudes of particle−particle electrostatic forces approximated based on eq 15 were on the orders of 0.1Fg and 10−3Fg for values of particle charge density equal to 10−8 and 10−9 C g−1, respectively, where Fg is particle weight while, as will be shown in the next section, the orders of magnitude of particle−wall electrostatic forces were much higher for the conditions imposed in this study. In the remaining sections of this paper, detailed mechanistic behaviors at the individual particle scale will be analyzed in relation to dynamic forces to substantiate the above explanations with a view toward achieving deeper insights to the mixing mechanisms associated with fluidized bed systems with strong electrostatic effects. Figure 9 shows typical snapshots of velocity vector plots for particles in the fluidized bed systems with particle charge density 10−9 C g−1 and wall linear charge density 10−6 C m−1 at various fluidizing velocities. To aid visualization of particle behaviors within the interior of the bed, a region that is one particle diameter in thickness extracted from the midplane of the bed is presented. In the

particles in both fluidized bed systems with weak and strong electrostatic effects may also be contrasted with those in a system where electrostatic effects were completely absent. Lim et al.23 have recently shown that in a fluidized bed system with identical operating conditions except for the absence of any electrostatic effect, the Lacey index increased rapidly from 0.0 to 1.0 within about 5 s when the fluidizing velocity applied was 1.0 m s−1. At the higher fluidizing velocities of 1.4, 1.6, and 1.8 m s−1, the Lacey indices increased more rapidly and reached 1.0 within about 2 s, which were faster than even the rates of increase of the Lacey index values in the presence of weak electrostatic effects observed in Figure 3 previously. As such, the mixing efficiencies of particles were highest in the absence of any electrostatic effect and decreased with increasing strengths of electrostatic forces present within the fluidized bed system. When the particle charge density applied was increased by 1 order of magnitude (10−8 C g−1) while the wall linear charge density was maintained at the same value of 10−5 C m−1, Figure 7 shows that the solid fraction distributions at 1.0 and 10 s remained essentially unchanged compared to those seen earlier in Figure 5. This implied that the repulsive particle−particle electrostatic forces contributed only minimally toward influencing the fluidization and mixing behaviors of the particles at all fluidizing velocities considered. For the range of operating conditions applied in this study, the particle−wall electrostatic 15871



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Figure 12 shows the time evolution of average particle− particle collision and fluid drag forces within the fluidized bed for various fluidizing velocities. These were calculated by averaging over all particles in the entire bed and nondimensionalizing by the particle weight, Fg, at every time step. For wall linear charge density of 10−6 C m−1 and particle charge density of 10−9 C g−1, eq 16 may be used to show that the electrostatic force acting on each particle due to the charged walls was on the order of 4Fg. By comparisons with the average particle−particle collision forces and fluid drag forces, it may be deduced that the conditions required for mixing to occur during fluidization of particles under such electrostatic conditions could be achieved. For all fluidizing velocities applied, Figure 12 shows that average fluid drag forces were of the same order of magnitude as the particle−wall electrostatic forces experienced by each particle in the fluidized bed while particle−particle collision forces were on average much stronger than these electrostatic forces. As such, it might be expected that the basic condition required for mixing to occur, whereby fluid drag forces or collision forces overcome electrostatic forces acting on each particle, was satisfied on average. This explains the fairly high kinetic energies of particles and mixing efficiencies observed with such fluidized bed systems in comparisons with those where electrostatic forces were stronger. At higher fluidizing velocities, larger fluctuations in average fluid drag forces and collision forces may further improve the likelihood of such forces overcoming electrostatic forces for specific particles within the system. This implies that fluidizing velocity could indeed be increased to improve the efficiencies with which mixing could occur and thus explains the improvements in rates of increase in Lacey index values observed with increasing fluidizing velocities seen earlier in Figure 3. For wall linear charge density of 10−5 C m−1 and particle charge density of 10−9 C g−1, the electrostatic forces acting on each particle due to the charged walls were on the order of 40Fg. Figure 13 shows that the conditions required for mixing to occur under such conditions were fairly difficult to achieve. For all fluidizing velocities applied, average fluid drag forces were much smaller than particle−wall electrostatic forces while particle−particle collision forces were on average only slightly stronger than these electrostatic forces. Electrostatic and collision forces were the dominant forces present in such fluidized bed systems and it might be expected that the basic condition required for mixing to occur, whereby fluid drag forces or collision forces overcome electrostatic forces acting on each particle, was satisfied only rarely during the fluidization processes. Higher fluidizing velocities of 1.6 and 1.8 m s−1 were necessary for collision forces to become comparable in magnitudes to electrostatic forces. This explains the lower kinetic energies of particles and mixing efficiencies observed with such fluidized bed systems in comparisons with those discussed earlier. To our knowledge, this is the first mechanistic explanation of macroscopic mixing behaviors of particles in fluidized bed systems with electrostatic effects via dynamic force analyses at the scale of individual particles.

presence of relatively weak electrostatic effects, it may be observed that fluidization of particles occurred with the formation of large bubbles. Particles may be transported readily between the lean bubble phase and the rich emulsion phase. The velocity vectors presented in Figure 9 indicate that most particles were capable of unhindered, independent motions that were important for mixing to occur. In contrast, Figure 10 shows that in the presence of strong electrostatic effects, particles were mostly adhered to the walls in the form of large aggregates during fluidization and thus creating a relatively dilute core region. As particles within an aggregate were subjected to similar adhesive forces originating from the charged walls, they tended to exhibit synchronized motions. As such, there were minimal relative motions between particles and mixing within each aggregate was hindered significantly. For mixing at the scale of individual particles to occur, strong adhesive forces resulting from electrostatic effects due to the charged walls need to be overcome by fluid drag forces or collision forces that arise during particle−particle collisions. This would then allow particles to be removed from one aggregate and become independent particles before being adhered to another aggregate. Such transfers of particles between aggregates were the basic mechanism by which mixing occurred during fluidization of particles in the presence of strong electrostatic effects. One distinct feature of the velocity vector plots of particles in the fluidized bed systems with weak and strong electrostatic effects presented in Figures 9 and 10 is that magnitudes of particle velocities in the presence of weak electrostatic effects were approximately one order higher than those in systems with strong electrostatic effects. As particle motions were a necessary condition for mixing to occur, it would be pertinent to compare the kinetic energies of particles within the two types of fluidized bed systems. Figure 11a shows the time evolution of total kinetic energies of all particles within the fluidized bed system with weak electrostatic effects for various fluidizing velocities. It may be observed that the total kinetic energy of particles when a fluidizing velocity of 1.0 m s−1 was applied was fairly low. With higher fluidizing velocities of 1.4, 1.6, and 1.8 m s−1, total kinetic energies were much higher and similar in magnitudes. This implies that particles were able to overcome the effects of electrostatics at these higher fluidizing velocities and undergo vigorous motions. This observation corroborates the previous one that the Lacey index profiles for the mixing processes at these higher fluidizing velocities were very similar (Figure 3). In contrast, Figure 11b shows that the total kinetic energies of particles within the fluidized bed system with strong electrostatic effects were approximately 1 order of magnitude lower than those just discussed. This was consistent with the earlier observation that magnitudes of particle velocities as seen from the velocity vector plots in Figure 10 were approximately one order lower than those seen in Figure 9. The much lower total kinetic energies of particles within the fluidized bed system with strong electrostatic effects also explains the lower mixing efficiencies as seen earlier from the lower rates of increase of Lacey index values in Figure 6 in comparison with those seen in Figure 3. In the next section, the magnitudes of various forces present throughout the fluidization and mixing processes will be analyzed in details to provide a mechanistic explanation for the differences in mixing efficiencies, magnitudes of particle velocities and total kinetic energies observed in the presence of weak and strong electrostatic effects.

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CONCLUSIONS The mixing behaviors of granular materials in gas fluidized bed systems with electrostatic effects were investigated computationally in this study. The conventional CFD-DEM model was coupled with an electrostatic force model and used for simulations of fluidization of solid particles at various fluidizing 15872



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velocities. The mixing efficiencies of fluidized beds with stronger electrostatic effects were observed to be lower than those with weaker electrostatic effects. Due to the presence of adhesive forces between particles and the walls arising from the electrostatic forces, motions of particles were hindered significantly. Consequently, fluidization was less vigorous in beds with stronger electrostatic effects. Mixing thus occurred via small motions of particles through a slightly expanded bed. On the basis of detailed analyses of dynamic force data extracted from the simulations conducted, electrostatic forces between particles and walls during the fluidization process were indeed observed to be stronger on average than both fluid drag forces and particle−particle collision forces when strong electrostatic effects were present. This explained the difficulty with which particles could be removed from the walls or other particles near the walls and transferred to a different location within the bed. Such particle transfers necessarily required strong electrostatic forces to be overcome by drag or collision forces and were essential for mixing at the individual particle length scale to be achieved. It would be pertinent to extend the present study toward detailed parametric analyses of various factors such as the triboelectric ratio of particles and walls of the fluidized beds, material properties of the granular materials and modes of gas injection on mixing efficiencies. The mixing or segregation behaviors of binary mixtures of granular materials with different sizes or densities in the presence of electrostatic effects may also be investigated with the modified CFD-DEM model applied in this study. Last but not least, it seems that no experimental study of mixing or segregation in fluidized bed systems with electrostatic effects has been reported to date. This is a gap in the current literature that needs to be addressed as such experimental studies are needed to corroborate the results of computational or theoretical studies of such systems.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: (65) 6516 4727. Fax: (65) 6779 1936. E-mail address: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study has been supported by the Economic Development Board (EDB) of Singapore through the Minerals, Metals and Materials Technology Center (M3TC) of the National University of Singapore (NUS) under grant number R-261501-017-414.

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Mixing Behaviors of Granular Materials in Gas Fluidized Beds with

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