Kinetic Theory of Granular Flow Limitations for Modeling High-Shear

High-shear granulation induces particle growth through agitation of a mixture of .... of restitution in the range of 0−0.33.12-14 A discrete element...
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Ind. Eng. Chem. Res. 2006, 45, 6721-6727

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Kinetic Theory of Granular Flow Limitations for Modeling High-Shear Mixing Justin A. Gantt and Edward P. Gatzke* Department of Chemical Engineering, UniVersity of South Carolina, Columbia, South Carolina 29208

Particulate modeling is often used to describe the kinematics of granular flow in mixers, granulators, and fluidized beds. An improved understanding of the rate processes associated with granulation can prove to be a valuable tool for modeling the evolution of particle size distributions. The kinetic theory of granular flow (KTGF) is a tool developed from the kinetic theory of gases to describe the kinematics found in granular media. Past work has shown that the KTGF can be used to describe the particle motion in a fluidized bed due to the inherently random particle movement caused by the fluid mechanics. Initially, it was thought that kinetic theory would not suffice when describing the kinematics in a high-shear mixer because of the shear motion in the mixer. However, recent work by Nilpawar et al. (in Proceedings: The 8th International Symposium on Agglomeration; The Industrial Pharmacists Group: Bangkok, Thailand, 2005) has shown experimentally that the collision frequency caused by random motion dominates over shear-induced collisions using measurements from the surface of the mixer. The present work attempts to support previous experimental findings by investigating the distributions of particle speed and velocity. The analysis is performed using discrete element modeling (DEM), a tool commonly used to simulate complex granular flow. This work demonstrates that, under idealized conditions, the KTGF is a very adequate means of describing particle flow. As the DEM process model becomes less ideal, the KTGF appears to be less successful in modeling the system. 1. Introduction High-shear granulation induces particle growth through agitation of a mixture of powder and binder. High-shear mixers are preferential in processes where small, dense particles are desired. Batch high-shear granulation is less sensitive to operating conditions than tumbling granulators, making highshear granulation preferential in the pharmaceutical industry. Population balance equations (PBEs) are the preferred technique of modeling the kinetics of aggregation found in highshear mixers.1-4 The most important parameter when modeling particulate processes with PBEs is the coalescence kernel. The coalescence kernel makes it possible to model processes where particle size, saturation, or speed affects aggregation. Recent research by Tan et al.5 shows that a growth kernel can be derived on the basis of the kinetic theory of granular flow (KTGF) for certain granulation processes. The KTGF is analogous to the existing equipartition of kinetic energy kernel for granulation.6,7 The work of Tan et al.5 was performed using a fluidized-bed mixer. The inherent random motion in a fluidized bed makes the kinetic theory of granular flow seem applicable to fluidized beds. Because of the shear-induced motion in high-shear mixers, the KTGF kernel may not be an adequate representation. Campbell8 points out that particles in a shear field which move parallel to the local velocity gradient pick up an inherent random velocity equal to the difference between the mean velocity at the present location and that at the location of the previous collision. Campbell proposed that this streaming mechanism of particle movement only generates one component of random velocitysthe component which lies in the direction perpendicular to the mean velocity gradient. This suggests an anisotropic granular temperature which does not, in turn, support the validity of the kinetic theory for modeling granulation in a highshear mixer. Despite these assumptions, still few qualitative analyses which attempt to prove or disprove the validity of the kinetic theory * To whom correspondence should be addressed: Phone: (803) 7771159. Fax: (803) 777-8265. E-mail: [email protected].

in high-shear mixer have been performed. Recently, a study by Nilpawar et al.9 experimentally analyzed the kinetics of highshear granulation. These authors used a high-speed camera to take images of a granular-bed surface on a flat disk impeller rotating at 312 rpm. Bed surface velocities were calculated using particle image velocimetry, and shear rates were calculated for the surface particles. This analysis showed that the granule collision frequency due to random motion dominated that of shear-induced motion on the surface of the bed, thus supporting the use of a kinetic theory model to predict particle velocity distributions in high-shear granulators. This work further examines the applicability of the KTGF for high-shear mixer applications. This work uses discrete element modeling (DEM) to determine whether the distributions of particle velocities found in a high-shear mixer are consistent with that of the kinetic theory of granular flow. 2. Kinetic Theory of Granular Flow The kinetic theory of granular flow is an extension of the classical kinetic theory of dense gases. It uses a method based on statistical mechanics to describe the mean and fluctuating velocity of particles found in continuous granular media. Processes exemplifying the kinetic theory produce Maxwellian speed distributions with Gaussian distributions for velocity fluctuations in each direction.10 In the kinetic theory of granular flow, each individual particle velocity, cj, is decomposed into a local mean velocity, uj, and a random fluctuating velocity, C h, described by

cj ) uj + C h

(1)

The granular temperature of the ensemble, a measure of the random motion of the particles, is defined as

1 h ,C h〉 θ ) 〈C 3 where the brackets denote ensemble averaging.10

10.1021/ie051267f CCC: $33.50 © 2006 American Chemical Society Published on Web 08/29/2006

(2)

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The transport mechanisms present in granular flow are modeled in the KTGF using Boltzmann’s equation. This integral-differential equation is used to describe the velocity distribution function, f, of the particles in the ensemble. Similar to the kinetic theory of gases, the kinetic theory of granular flow assumes chaotic motion. This means that the particle velocity distributions are expected to be isotropic and the velocities of two colliding particles are not interrelated. If this is true, the Enskog approximation can be used to solve Boltzmann’s equation.11 Solving the zeroth order approximation of Boltzmann’s equation results in the well-known Maxwell velocity distribution function, which describes a steady-state equilibrium without external forces:

f0)

2 n e-C /2θ 3/2 (2πθ)

(3)

where n is the number of particles per unit volume. As in the case of molecular chaos in the kinetic theory of gas, the kinetic theory of granular flow assumes the velocities of the particles in the ensemble are isotropically distributed around the local mean velocity. Therefore, all three velocity components are assumed to be independent of each other such that

f(Cx,Cy,Cz) dCx dCy dCz ) f(Cx)f(Cy)f(Cz) dCx dCy dCz (4) On the basis of the assumption that particle velocity distribution is isotropic, the normalized distribution in each direction becomes the well-known Gaussian distribution around the ensemble mean velocity:

fx(Cx) )

2 1 e-Cx /2θx 1/2 (2πθx)

(5)

Meanwhile, the overall particle velocity distribution is described

fy(Cy) ) fz(Cz) )

2 1 e-Cy /2θy 1/2 (2πθy)

(6)

2 1 e-Cz /2θz 1/2 (2πθz)

(7)

using the Maxwellian distribution:10

f(C) )

4πC2 -C2/2θ e (2πθ)3/2

3. The Soft-Sphere Simulation Approach The kinetic theory of granular flow was originally derived with the assumption of smooth, spherical, elastic particles. However, many real processes encounter rough, nonspherical, inelastic particles where the theory may not seem applicable. Mangwandi et al.12 report that many particles in high-shear mixers and many other wet granulation process are far from elastic. Multiple sources have reported coefficients of restitution in the range of 0-0.33.12-14 A discrete element method approach to modeling particle interactions will be used to determine the influence of rotation, elasticity, and friction on the velocity distribution found in high-shear granulation. A similar approach has been applied to modeling the motion in a fluidized bed.5,10 Simulations were carried out using a soft-sphere DEM to model the contact forces present in a high-shear granulation process. These contact forces are modeled using Cundall and

Figure 1. Soft sphere model of normal and tangential contact forces. (Adapted from Tsuji et al.27)

Strack’s soft sphere DEM, which expresses force through the use of a spring, dash-pot, and slider15 shown in Figure 1. DEM simulations rely greatly on well-selected parameters for collision models. This, however, is not a simple task. These simulations assume that the discrete entities in the simulation are an agglomerate composed of a solid, a liquid, and air. The properties of the granule matrix are dynamic, depending on changing strain rates and porosity distributions. Internal properties also change because of consolidation and drying. Identifying correct mechanical properties for wet granules is a current focus throughout the literature.16-18 The following four parameter models were used in this analysis. Spring Constant, k. The normal spring constant determines the elastic portion of the normal collision force. The spring constant is not a physically definable quantity but, instead, is a model-based property. Particles which are stiff have a high coefficient of restitution and have a high k, while particles which deform have low values of k. The value of the spring constant greatly influences the size of the time step. A time step must be several orders of magnitude smaller than the duration of the contact. Obviously, stiffer particles will have shorter contact durations, thereby requiring a very small step size. Authors often use artificially small values of k to provide a larger step size and speed up their simulation.19 This works because the macroscopic motion of particles can be adequately predicted even if the contact duration is several orders of magnitude smaller than realistic values.19 The spring constant used in this study was obtained from the Hertzian contact law at a 0.5 strain rate; therefore, the ratio of the force to displacement is calculated at a general fractional overlap:

k ) 0.094E ˜R ˜

(8)

where E ˜ ) E1/(1 - ν12) is the effective granule Young’s modulus and R ˜ ) R1R2/(R1 + R2) is the effective granule radius. The normal and tangential spring constants are assumed to be equivalent in this study.19 Damping Constant, η. In DEM simulations, particle damping forces are assumed to be proportional to the collision velocity of the colliding entities.20 The expression for particle damping for normal impacts is also, therefore, directly related to the coefficient of restitution of the colliding bodies according to the following relationship:21

(x

η ) -ln e

km ˜ ln e + π2 2

)

(9)

where m ˜ ) m1m2/(m1 + m2) is the effective granule mass and e is the coefficient of normal restitution which is dependent on extenuating properties such as the collision velocity and the extent of deformation.

Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 6723 Table 1. Initial Conditions for Simulation parameter

value

comments

Equipment Parameters mixer diameter, Dm mixer height, Hm impeller speed, ω

0.212 [m] 0.15 [m] 400 rpm

time step, ∆t mean particle sixe, µ coefficient of variation, CV periodic boundaries, θ number of particles, Np coefficient of friction, µf spring constant, k damping constant, η

Simulation Parameters 10-4.8-10-5.8 [s] -3 1.7 × 10 [m] 0.33 [-] π/32 [rad] 3000 [#] 0.25 [-] 10-1000 [Nm-1] 0.03-0.003 [kg1/2 m s-1]

power density, Fs solid volume fraction, Vs

CaCO3 2500 [kg/m3] 0.73 [-]

binder viscosities, µ density, Fl liquid volume fraction, Vl

silicon oil 1.0 [kg/m/s] 997 [kg/m3] 0.27 [-]

granule Young’s modulus, E granule yield strength, Yd surface asperity height, ha

105-107 [Pa] 104 [Pa] 2 × 10-6 [m]

dimensions for a Hobart high-shear HCM2 mixer

varies on the basis of calculated contact durations21 Mumber-based log-normal distribution also used by Muguruma et al.24 Cleary25 for 3D simulations values vary according to Young’s modulus20 values vary according to Young’s modulus20

Powder Properties calcium carbonated/silicon oil mix

Binder Properties a very visous binder used to induce types I and II coalescence

Granule Properties varies throughout literature2,14,16,18 assumed from data by Liu and Litster2 produce a ratio of ha/h0 ) 10, Liu et al.26

Contact Duration, td. A large number of time steps are required to capture the dynamics of each collision. An insufficient number of time steps tends to produce an unnatural response. The linear spring/dash-pot model has an analytical solution which allows for the calculation of the contact duration. Schaefer et al.21 report that an accurate simulation with relative errors on the order of 10-4 requires a constant time step ∆t ≈ td/100. The contact duration can be defined as

td ) π

2 -1/2

[mk˜ - (2mη˜ ) ]

(10)

Coefficient of Restitution, e. Liu and Litster22 developed a model for the coalescence of deformable particles in wet granulation. The model accounts for the mechanical properties of the granule matrix and accounts for the formation of a liquid binder layer at the granule surface. The model also provided an expression for the coefficient of restitution as a function of dimensionless ratios:

e ) 2.46

()[ Yd E

1/2

1-

( )]

h0 1 ln Stv ha

-1/4

Stdef-1/8

(11)

where Yd/E is the ratio of the granule yield strength to the granule Young’s modulus, Stv is the viscous Stokes number, h0/ha is the ratio of the height of the binder layer to the height of the surface asperities, and Stdef is the Stokes deformation number. These models provide a mechanistic representation of the particle stiffness, damping, and coefficient of restitution as a function of several unknown quantities. The granule Young’s modulus is the principle unknown needed to calculate both the stiffness, k, and the coefficient of restitution, e. Ingram23 provides a survey of the literature which shows a range of estimates for E for wet granules which spans four orders of magnitude. This work investigates how different particle properties influence the flow patterns inside a high-shear mixer and intends to determine if the granular flow is indeed as described by the

Table 2. Contact Force Parameters

E µ rotation mean e

sim I

sim II

sim III

sim IV

sim V

105

105

105

106

0 no 1

0 yes 1

0.25 yes 1

0.25 yes 0.46

107 0.25 yes 0.19

kinetic theory of granular flow. To examine the accuracy of KTGF for high-shear granulation, a series of different simulations are performed using the DEM simulation to test the sensitivity of the process to these parameters. The DEM simulation incorporates a high-shear mixer with a flat plate rotor. Simulation parameters are presented in Table 1. Five simulations are run, each of which vary the granule Young’s modulus and friction coefficient. Particle rotation is also examined. The five simulations are described in Table 2. 4. Velocity Distribution Sampling Obtaining a representative velocity distribution from a discrete simulation is required for this analysis. Unlike fluidized-bed granulators where sampling can be complicated by the unsteady flow patterns resulting from the nonhomogeneous behavior of the beds, high-shear mixers can typically be considered similar to a steady-state process. The flow patterns in a high-shear system tend to resemble a roping, toroidal motion when the impeller speed is held constant. Despite this, the particle velocity distribution is a function of the granular temperature, which varies with the position in the mixer. Goldschmidt10 found that flow pattern and granular temperature variations interfere with sampling the entire particle velocity distribution of the bed. A grid is applied to the simulation, which divides the mixer into radially distributed bins in order to remove the influence of granular temperature with mixer position. The local granular temperature for each of these ensembles of particles is calculated at each time step. The granular temperature of the ensemble of particles in each bin, k, is then calculated to be10

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Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 Npart,k

θk,x )

∑i (cji,x - ujs,k,x)2 Npart,k

, θk,y ) ..., θk,z ) ...

(12)

and Npart,k

θk )

(cji - ujs,k)2 ∑ 1 i 3

Npart,k

(13)

where k ) 20 was used for this analysis. The granular temperatures for the high-shear mixer typically were in the range of 0.01-0.5 m2/s2 as compared to observed granular temperatures in the range of 10-5-10-1 m2/s2 in fluidized-bed granulators.10 This again shows that the velocity dynamics of the high-shear process are less pronounced than those in a fluidized bed. It is assumed from the kinetic theory of granular flow that the particle velocity distribution varies primarily with granular temperature. Therefore, particle velocity distributions within bins with ensemble average granular tem-

Figure 2. Particle velocity distribution functions for simulation I: velocity distributions sampled on the range 0.0616 < θx,y,z m2/s2 compared to Gaussian and Maxwellian distributions at θx,y,z ) 0.0745 m2/s2.

Figure 4. Particle velocity distribution functions for simulation III: velocity distributions sampled on the range 0.0874 < θx,y,z < 0.1132 m2/s2 compared to Gaussian and Maxwellian distributions at θx,y,z ) 0.1003 m2/s2.

Figure 5. Particle velocity distribution functions for simulation IV: velocity distributions sampled on the range 0.0874 < θx,y,z < 0.1132 m2/s2 compared to Gaussian and Maxwellian distributions at θx,y,z ) 0.1003 m2/s2.

peratures within an  tolerance difference of each other are analyzed. The range of granular temperatures is split into 20 uniformally spaced classes. The ensemble average granular temperature of each region in the mixer is calculated and placed within the appropriate discrete granular temperature class. Particle velocity distributions are then obtained from each granular temperature class by sampling the random fluctuating velocities of particles in each direction and sorting them into discrete velocity classes. 5. Simulation Results

Figure 3. Particle velocity distribution functions for simulation II: velocity distributions sampled on the range 0.0358 < θx,y,z < 0.0616 m2/s2 compared to Gaussian and Maxwellian distributions at θx,y,z ) 0.0487 m2/s2.

5.1. Discussion. For each of the simulations listed in Table 2, samples were taken from the simulation with 3000 particles for several seconds of simulation time. Samples were taken from the simulation every 0.005 s. This allowed all of the motion to be captured and also created a large sample array. Each discrete bin had on the order of 106 individual velocity samples. This provides a representative distribution. Figure 2 shows the particle velocity distributions for the idealized mixer, simulation I. Again, this simulation neglected

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Figure 6. Particle velocity distribution functions for simulation V: velocity distributions sampled on the range 0.1132 < θx,y,z < 0.1390 m2/s2 compared to Gaussian and Maxwellian distributions at θx,y,z ) 0.1261 m2/s2.

the effects of friction and particle rotation. Collision parameters were established such that the coefficient of restitution was unity, resulting in perfectly elastic collisions. Simulation I shows that simulated ideal conditions produce velocity distributions which are indeed Gaussian in nature. The particle velocity distributions are also Maxwellian in nature. This implies that the KTGF is applicable under ideal conditions with highly elastic particles. However, as one includes the effects of nonideal simulation conditions, deviation from the predictions based on the KTGF becomes quite evident. Simulation II incorporates particle rotation in the simulation algorithm. Figure 3 shows the velocity distributions for simulation II. As shown in Figure 4, the nonidealities resulting from rotation are minor. The velocity distributions still greatly resemble a Gaussian distribution, and the total distribution is still Maxwellian. This demonstrates that rotation appears to have limited influence on the overall system model accuracy. The KTGF is still applicable with highly elastic particles using an improved model.

Simulation III includes particle rotation but also accounts for friction in particle-particle and particle-wall collisions. Without friction, each collision event is in a sliding regime. This simulation uses a rotor rather than an impeller, so friction is clearly necessary to apply the energy of the rotor to the particles. The resulting friction in the model shows a significant change from the ideal KTGF assumption. The velocity distribution in the z direction is skewed and more narrow than the other distributions. The x and y component velocity fluctuations are nearly Gaussian, and the velocity distribution is approximately Maxwellian. Each of the first three simulations uses a coefficient of restitution of unity. Experimentally, the coefficient of restitution for wet granules may be much smaller. Mangwandi et al.12 performed experiments where wet granules produced in a small high-speed mixer were released from predetermined heights so that velocities at impact and rebound could be calculated. From these measurements, a coefficient of restitution could be calculated. They reported coefficients of restitution in the range of 0-0.15 for wet granulation. These values are far from the ideal conditions typically used in discrete particle simulations. Apparently, there are no experimental data in the literature reporting coefficients of restitution for deformable, moist particle-particle collisions, though they are expected to be similar to particle-wall coefficients of restitution. For simulation IV, the granule Young’s modulus was increased to 106 N/m2. This alters the strength of the simulated agglomerate and decreases the reported average coefficient of restitution to 0.46. Figure 5 shows the particle velocity distribution for this simulation. While the total particle velocity distribution remains Maxwellian, and the velocity distributions in the x and y directions are close to Gaussian, the distribution in the z direction is far from Gaussian. The final simulation is intended to represent the most realistic mixing conditions. Rotation and fiction are present, plus the mean coefficient of restitution reported is 0.19. The particle velocity distribution for simulation V is shown in Figure 6. Again, the total particle velocity distribution remains Maxwellian, and the velocity distributions in the x and y directions

Figure 7. Velocity profile of a high-shear mixer simulation with inelastic particles.

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Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 Table 3. Analysis of Anisotropy of Particle Velocity Distributions for Each Simulation

sim I sim II sim III sim IV sim V

Figure 8. Snapshot of a high-shear mixer simulation of 3000 inelastic particles.

are close to Gaussian, but the distribution in the z direction remains quite skewed because of the presence of inelastic particles. 5.2. Particle Flow. A particle velocity flow field is shown in Figure 7 for simulation V, the mixer under realistic conditions. This simulation produces a flow field where particles are being forced to the wall, where they begin a “roping” behavior. Figure 8 shows a direct snapshot of simulation V after 4 s. Particles in this figure are concentrated at the wall, where they rotate as a group in a torus. This is consistent with experimental observations of particle roping in high-shear mixers presented by Litster and Ennis.1 This is quite different from the behavior present in the mixer for the ideal case, simulation I. Figure 9 shows the particle velocity flow field under ideal conditions with perfectly elastic particles. Note that the particle velocity vectors appear to be randomly distributed and there is no torus. Figure 10 shows a

〈θx〉/〈θ〉

〈θy〉/〈θ〉

〈θz〉/〈θ〉

0.998 0.981 0.980 1.038 1.082

0.997 0.990 0.982 1.032 1.078

0.998 0.998 1.245 1.584 1.682

direct snapshot of simulation I after 4 s. The particles appear to have far more energy and are less concentrated in the mixer. Note that these simulations only include 3000 particles. While the authors understand that this number is far from what may realistically be found in granulation circuits, these simulations are intended to be small-scale examples to illustrate the anistropy of particle velocity distributions for high-shear granulation processes exhibiting realistic parameters. 5.3. Anistropy. To quantify the degree of anisotropy for each of the simulations, the ratios of the granular temperatures in each direction to the granular temperature of the total particle ensemble were calculated. The granular temperatures were averaged over each bin in the domain and weighted according to the number of particles in that class according to k

〈θ〉 )

∑i θk

k

∑i Npk

Npk

(14)

where 〈θ〉 is the ensemble average granular temperature, θk is the granular temperature of the kth class, and Npk is the number of particles in the kth class. The results are shown in Table 3. It is clear from these results that the particle velocity distribution becomes more anisotropic as the particles become less elastic. From these observations, it appears that the kinetic theory of granular flow is only applicable for high-shear granulation for ideal or near-ideal simulations. The anisotropic behavior in the z direction for the inelastic particle simulations shows that the KTGF does not suffice when describing the velocity distribution in a high-speed mixer. Nilpawar et al.9 showed compelling

Figure 9. Velocity profile of a high-shear mixer simulation with elastic particles.

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Acknowledgment The author acknowledges valuable discussions with Dr. Mike Hounslow and Dr. Gavin Reynolds, as well as the valuable comments from the anonymous reviewers. Literature Cited

Figure 10. Snapshot of a high-shear mixer simulation of 3000 elastic particles.

evidence for the validity of the kinetic theory by performing a Fourier analysis of velocities of particles in a fixed location in a high-speed mixer. This group believes that the findings by Nilpawar et al. can be attributed to the fact that the measured velocity distribution was perpendicular to the shearing motion of the flow field, thereby only measuring the particle velocities’ tangential directions. The current simulations show that the tangential directions, x and y, appear to be near Gaussian. However, the one direction not directly measured by Nilpawar et al., the z direction, appears to be the source of the non-Gaussian response in nonideal simulation cases. 6. Conclusions A series of simulations has been performed using a discrete element method technique to model particle motion to determine the validity of the kinetic theory of granular flow in a highspeed mixer. These findings reveal that, while simulations performed using idealized parameters produced velocity distributions which were Gaussian and total velocity distributions which were Maxwellian, simulations using realistic operating conditions were far from Gaussian and Maxwellian. Furthermore, the particle velocity flow field for the idealized process is far different from simulated flow fields using realistic parameters and observed experimental particle behavior. This study shows that the KTGF may not be a preferred method when modeling the flow characteristics of particles in a high-shear mixer.

(1) Litster, J.; Ennis, B. J. The Science and Engineering of Granulation Processes; Kluwer Academic: Dordrecht, The Netherlands, 2004; Vol. 1. (2) Liu, L. X.; Litster, J. D. Chem. Eng. Sci. 2002, 57, 2183. (3) Ramkrishna, D. ReV. Chem. Eng. 1985, 3, 49. (4) Hounslow, M. J.; Ryall, R. L.; Marshall, V. R. AIChE J. 1988, 34, 1821. (5) Tan, H.; Goldschmidt, M.; Boerefijn, R.; Hounslow, M.; Salman, A.; Kuipers, J. Powder Technol. 2004, 142, 103. (6) Hounslow, M. J. Kona 1998, 16, 179. (7) Boerefijn, R.; Buscan, M.; Hounslow, M. J. In Fluidization X; United Engineering Foundation, Inc.: Beijing, 2001; pp 629-636. (8) Campbell, C. S. Annu. ReV. Field Mech. 1990, 22, 57. (9) Nilpawar, A.; Reynolds, G.; Salman, A.; Hounslow, M. In Proceedings: The 8th International Symposium on Agglomeration; The Industrial Pharmacists Group: Bangkok, Thailand, 2005. (10) Goldshmidt, M. Ph.D. Thesis, Twente University, Enschede, The Netherlands, 2001. (11) Chapmand, S.; Cowling, T. The Mathematical Theory of NonUniform Gases; Cambridge University Press: Cambride, U. K., 1970. (12) Mangwandi, C.; Fu, J.; Reynolds, G.; Adams, M.; Hounslow, M.; Salman, A. In 8th International Symposium on Agglomeration; The Industrial Pharmacists Group: Bangkok, Thailand, 2005; pp 11-21. (13) Iveson, S.; Litster, J.; Ennis, B. Powder Technol. 1996, 88, 15. (14) Ramaker, J. S. Ph.D. Thesis, University of Gronigen, Gronigen, The Netherlands, 2001. (15) Cundall, P. A.; Strack, O. D. L. Geotechnique 1979, 29, 47. (16) Fu, J.; Reynolds, G.; Adams, M.; Hounslow, M.; Salman, A. Chem. Eng. Sci. 2005, 60, 4005. (17) Iveson, S. M.; Wauters, P. A. L.; Forrest, S.; Litster, J. D.; Meesters, G. M. H.; Scarlett, B. Powder Technol. 2001, 117, 83. (18) Iveson, S. M.; Beathe, J. A.; Page, N. W. Powder Technol. 2002, 127, 149. (19) Maio, F. P. D.; Renzo, A. D. Chem. Eng. Sci. 2004, 59, 3461. (20) Mishra, B. K. Int. J. Miner. Process. 2003, 71, 73. (21) Schafer, J.; Dippel, S.; Wolf, D. E. J. Phys. I 1996, 6, 5. (22) Liu, L. X.; Litster, J. D. AIChE J. 2000, 46, 529. (23) Ingram, G. D. Ph.D. Thesis, University of Queensland, St. Lucia, Queensland, Australia, 2006. (24) Muguruma, Y.; Tanaka, T.; Tsuji, Y. Powder Technol. 2000, 109, 49. (25) Cleary, P.; Sawly, M. In Second International Conference on CFD in Minerals and Process Industries; CSIRO: Melbourne, Australia, 1999. (26) Liu, L. X.; Litster, J. D.; Iveson, S. M.; Ennis, B. J. AIChE J. 2000, 46, 529. (27) Tsuji, T.; Kawaguchi, T.; Tanaka, T. Powder Technol. 1993, 77, 79.

ReceiVed for reView November 15, 2005 ReVised manuscript receiVed July 18, 2006 Accepted July 18, 2006 IE051267F