Insights into the granular flow in rotating drums

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Contents lists available at ScienceDirect

Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Insights into the granular flow in rotating drums H.R. Norouzi, R. Zarghami ∗ , N. Mostoufi Process Design and Simulation Research Center, School of Chemical Engineering, College of Engineering, University of Tehran, PO Box 11155-4563, Tehran, Iran

a r t i c l e

i n f o

a b s t r a c t

Article history:

Scaling relations for the velocity profile and circulation time, which can be used in design and

Received 19 January 2015

scale-up, are of great importance in rotating drums. We conducted simulations by discrete

Received in revised form 11 May

element method at various operating conditions (which covers both rolling and cascading

2015

regimes) for spherical and non-spherical particles. New scaling relations were proposed for

Accepted 5 June 2015

evaluating and fully characterize velocity profile and circulation time in both rolling and

Available online 12 June 2015

cascading regimes. Using dynamic angle of repose, effect of shape was included in these

Keywords:

satisfactorily reproduce experimental measurements. Visual results show that transition

correlations to extend them to spherical and non-spherical particles. Simulation results can Discrete element method

from rolling to cascading regime depends not only on Froude number, fill level and particle

Velocity profile

size, but also on the particle shape. The surface velocity is scaled with peak velocity and

Active layer thickness

half chord length and its profile is asymmetric with the maximum occurring after the mid-

Rolling

chord position for all simulations. We obtained a correlation for the peak velocity based

Cascading

on our simulations as well as available experimental data in literature. We also found the

Circulation time

velocity profiles along the bed depth in active and passive layers. Our results show that the circulation time of particles follow a log-normal distribution. The mean circulation time decreases with rotation speed and drum diameter and increases with fill ratio. This value is greater for spherical particles compared to non-spherical particles. Correlations are also proposed for mean and standard deviation of the circulation time. © 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

Rotating drums have a wide range of applications in industries like food, pharmaceutical, agricultural, metallurgical, chemical and solid waste treatment (Ndiaye et al., 2010). They possess good heat and mass transfer characteristics which make them suitable for being used in many processes like mixing of powders, drying, calcination, coating, granulation (size enlargement), size reduction, pyrolysis and chemical reactions (Fantozzi et al., 2007). They are able to handle various types of feeds (wet or dry, fine or course granules, spherical and nonspherical). These make them superior to similar equipment like fluidized beds. Although their geometry and operation are rather simple, the granular flow in these drums is complex and is not understood completely. The granular flow in these

∗

drums becomes more complex when the process involves non-spherical particles, size enlargement or reduction. Table 1 shows various studies on granular flow in rotating drums. Concerning the application of the rotating drum, different aspects of granular flow were studied. For mixing purposes, mixing or segregation extent, characterization of different mechanisms of mixing and segregation and mixing time are addressed (Pandey et al., 2006c). For coating and granulation purposes, distributions of circulation time, surface time/velocity of particles, particles exposure area and orientation toward spray nozzle are investigated and used to assess the quality of coated particles in the final product (Kalbag and Wassgren, 2009; Pandey et al., 2006a). In some cases, a spray model is assumed and the Monte-Carlo technique is used to obtain the distribution of coating mass

Corresponding author. Tel.: +98 21 6696 7797; fax: +98 21 6646 1024. E-mail address: [email protected] (R. Zarghami). http://dx.doi.org/10.1016/j.cherd.2015.06.010 0263-8762/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

13

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Nomenclature CLi D F ij Ii L r M ij t M ij

Ri X,Y dp fn ij

g mi t umax usurf i v ω i x i

contact list of particle i diameter of drum (m) contact force between particles i and j (N) moment of inertial (kg m2 ) half of chord length (m) rolling resistance torque of particle i (N m) tangential torque of particle I (N m) particle radius (m) two dimensional coordinates aligned with bed surface particle diameter (m) contact force in the normal direction (N) acceleration of gravity (m s−2 ) particle mass (kg) time (s) peak velocity at the bed surface (m s−1 ) surface velocity (m s−1 ) particle translational velocity (m s−1 ) particle rotational velocity (rad s−1 ) position vector of particle (m)

Greek letters dynamic and static angle of repose (◦ ) ˛,˛0 ˇ asymmetry factor fill ratio ϕ depth at which velocity becomes zero (m) 0 , r dynamic and rolling friction factors drum rotation speed (rad s-1 ) ω ˙ shear rate (s-1 )

among particles (inter-particle variability) and on each particle (intra-particle variability) (Freireich and Wassgren, 2010). The information required to Monte-Carlo modeling can be obtained from experimental measurements (Kandela et al., 2010) or discrete element method (DEM) simulations (Freireich et al., 2011). Another aspect of the granular flow in rotating drums is velocity profile of particles at different flow regimes. Depending on the fill level and Froude number, six different flow regimes, namely slipping, slumping, rolling, cascading, cataracting and centrifuging can be observed. Most industrial drums operate in rolling or cascading regimes due to low energy consumption and providing a good mixing. The velocity profile of particles was analyzed to determine the rate of material transport, residence time, heat and mass transfer rates (Chaudhuri et al., 2006; Heydenrych et al., 2002), and developing scale-up relations (Mueller and Kleinebudde, 2007). Experiments and models are mostly available in the rolling regime. There are two distinct layers in the rolling regime: active (cascading) and passive layers. In the passive layer, beneath the active layer, particles move with the drum wall as a solid body and enter the thin active layer. In the active layer, particles flow to the down of the bed surface due to gravity. The velocity profile and the thickness of active layer of spherical dry particles (Alizadeh et al., 2013; Jain et al., 2002; Khakhar et al., 1997; Santomaso et al., 2003), non-spherical particles (Dubé et al., 2013), and wet particle (McCarthy et al., 2000) have been investigated using experimental measurements in rolling regime. Besides, theoretical (Boateng, 1998; Ding et al., 2001b; Liu et al., 2006; Mellmann et al., 2004; Yan

Liu and Specht, 2010) and semi-empirical (Cheng, 2012; Cheng et al., 2011; Weir et al., 2005) correlations have also been developed. Although the flow behavior of particles has been well studied in literature, a few studies can be found on the cascading regime (Pandey et al., 2006a; Sandadi et al., 2004; Suzzi et al., 2012). In addition, effect of particle shape is not explicitly known on the flow behavior of particles. We conducted a set of DEM simulations in the present study to characterize the flow behavior of both spherical and non-spherical particles. Effects of drum rotational speed, fill level, drum size and particle size and shape on the velocity profile and circulation time of particles were considered. The operating conditions in simulations cover both rolling and cascading regimes. We first validated DEM results using available experimental data in literature. Then, we characterized velocity profile, active layer thickness and circulation time. Our aim was to accomplish two main goals: • To test whether available correlations, which belong to either rolling or cascading regime, are applicable to both regimes; and in some cases, to extend that correlation in a way that it provides accurate predictions in both regimes. • To include effect of particle shape in the correlations in a way that they can be utilized for both spherical and nonspherical particles.

2.

Model description and simulations

2.1.

Model equations

Particles are tracked individually in the DEM. Translational and rotational motions of each particle are described by the Newton’s second law of motion and Euler’s law, respectively. Two main approaches can be used to describe interactions between particles: hard-sphere (event-driven) and soft-sphere (timedriven). We used the soft-sphere approach in which particles can have partial overlaps and hence, a particle can have contacts with multiple particles—which is the case for most of granular flows. Equations of motions for a spherical particle are: mi

 i dv d2 x = mi 2i = F ij + mi g dt dt

(1)

j∈CLi

(2)Ii

dω i dt

=



t +M r) (M ij ij

j∈CLi

Various force–displacement laws have been developed to calculate contact forces between particles, F ij (Di Renzo and Di Maio, 2004; Kruggel-Emden et al., 2009, 2010). We used the linear–viscoelastic force–displacement model for both normal and tangential directions combined with Coulomb’s friction law and considering limited tangential displacement (more information is given in Kruggel-Emden et al., 2008b). Rotational motion of a spherical particle is affected by the  t , produced by inter-particle contacts tangential torque, M ij  r , which opposes the rotation and rolling friction torque, M ij

 t = Ri n r =  ij × F ij (4)M of particle which are calculated by:(3)M ij ij

    ω i −ω j  ω i −ω j 

−r Ri fijn 

In addition to particle–particle interaction, particle–wall interaction also exists in the system. The same formulation was used for particle wall intreations (contact force and friction law). To approximate the non-spherical shape of

14

Table 1 – Literature on various aspects of granular flow and coating operation in rotating drums. Parameters

Mixing and segregation (axial & lateral)

Velocity profile in different regimes

Surface translational velocity

Dynamic angle of repose

Spray config.

Particle properties

Rotation speed

Fill level

Baffles /drum size

Ax. (Finnie et al., 2005; Pandey and Turton, 2005; Sahni et al., 2011) Lat. (Arntz et al., 2008; Ding et al., 2002; Finnie et al., 2005; Liu et al., 2013; Marigo et al., 2012; Sahni et al., 2011) Mean. (Ketterhagen, 2011; Leaver et al., 1985; Pandey and Turton, 2005; Sandadi et al., 2004) STD (Ketterhagen, 2011) Rol. (Alizadeh et al., 2013; Boateng, 1998; Cheng et al., 2011; Ding et al., 2001b; Jain et al., 2002; van Puyvelde et al., 2000; Weir et al., 2005) Cas. (Ding et al., 2002; Santomaso et al., 2003) Mean (Mueller and Kleinebudde, 2007; Pandey and Turton, 2005) (Pandey et al., 2006b; Pandey and Turton, 2005)

Ax. (Finnie et al., 2005; Pandey and Turton, 2005; Sahni et al., 2011; Smith et al., 2003) Lat. (Arntz et al., 2008; Finnie et al., 2005; Liu et al., 2013; Sahni et al., 2011)

Baf. (Schutyser et al., 2001; Smith et al., 2003) D.S. (Schutyser et al., 2001)

N.A.

Ax.-Shape. (Pandey and Turton, 2005) Ax.-Size (Parker et al., 1997; Sandadi et al., 2004; Smith et al., 2003)

Mean. (Ketterhagen, 2011; Leaver et al., 1985; Pandey and Turton, 2005; Sandadi et al., 2004) STD (Ketterhagen, 2011) Rol. (Boateng, 1998; Cheng et al., 2011; van Puyvelde et al., 2000; Weir et al., 2005) Cas. (Ding et al., 2002; Santomaso et al., 2003)

D.S. (Sandadi et al., 2004)

N.A.

Shape (Pandey and Turton, 2005) Size (Sandadi et al., 2004)

Mean (Pandey and Turton, 2005)

D.S. (Alexander et al., 2002)

(Pandey et al., 2006b; Pandey and Turton, 2005)

Pattern /rate

Size /shape

Friction coefficient (Schutyser et al., 2001)

Size (Alexander et al., 2002)

N.A.

Shape (Mueller and Kleinebudde, 2007; Pandey and Turton, 2005) Size (Mueller and Kleinebudde, 2007)

(Liu et al., 2013; McCarthy et al., 2000)

Liu et al. (2013)

Rol.-Size (Cheng et al., 2011; Ding et al., 2001a; Ding et al., 2001b; Jain et al., 2002; Weir et al., 2005) Rol.-shape (Dubé et al., 2013)

N.A.

Wetness and cohesion

Pandey et al. (2006b)

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Circulation time (Mean or STD)

Drum design

Table 1 – (Continued) Drum design

Parameters

Surface time (Mean or STD)

Intra-particle variability

Rotational velocity of particle and exposure area

Fill level

Mean (Kalbag et al., 2008; Ketterhagen, 2011; Leaver et al., 1985; Mueller and Kleinebudde, 2007; Pandey and Turton, 2005; Sandadi et al., 2004; Yamane et al., 1995) STD (Kalbag et al., 2008; Ketterhagen, 2011; Yamane et al., 1995) M.C.-Exp. (Pandey et al., 2006a; Pandey et al., 2006c) Exp. (Chen et al., 2010; Dubey et al., 2012; Dubey et al., 2011; Leaver et al., 1985; Rege et al., 2002; Tobiska and Kleinebudde, 2003; Wilson and Crossman, 1997) Exp.-DEM. (Freireich et al., 2011; Freireich and Wassgren, 2010; Ketterhagen, 2011)

Mean (Kalbag et al., 2008; Ketterhagen, 2011; Leaver et al., 1985; Pandey and Turton, 2005; Sandadi et al., 2004; Yamane et al., 1995) STD. (Kalbag et al., 2008; Ketterhagen, 2011; Yamane et al., 1995)

Sandadi et al. (2004)

(Sandadi et al., 2004; Suzzi et al., 2012)

M.C.-Exp. (Pandey et al., 2006a; Pandey et al., 2006c) Exp. (Chen et al., 2010; Leaver et al., 1985; Tobiska and Kleinebudde, 2003) DEM. (Dubey et al., 2011)

Baffles /drum size

Baf. (Joglekar et al., 2007)

Exp.-DEM (Freireich et al., 2011; Freireich and Wassgren, 2010; Ketterhagen, 2011)

Baf. (Sandadi et al., 2004)

Pattern /rate

Particle properties Size /shape

N.A.

Shape (Kandela et al., 2010; Ketterhagen, 2011; Pandey and Turton, 2005) Size (Kandela et al., 2010; Sandadi et al., 2004; Yamane et al., 1995)

M.C.-Exp. (Pandey et al., 2006a) Exp. (Dubey et al., 2012; Joglekar et al., 2007; Rege et al., 2002)

Size (Chen et al., 2010; Smith et al., 2003) Shape (Kandela et al., 2010; Wilson and Crossman, 1997)

DEM (Dubey et al., 2011)

Exp.-DEM (Freireich et al., 2011; Freireich and Wassgren, 2010; Ketterhagen, 2011) Shape (Kandela et al., 2010) Shape (Suzzi et al., 2012) Size (Sandadi et al., 2004)

N.A.

Friction coefficient

Wetness and cohesion

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Inter-particle variability

Rotation speed

Spray config.

Kumar and Wassgren (2013)

Ax.: axial, Lat.: lateral, Baf.: baffle, D.S.: drum size, Rol.: rolling regime, Cas.: cascading regime, M.C.: Monte-Carlo, Exp.: experimental, STD: standard deviation.

15

16

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Fig. 1 – Different shapes of particles used in simulations.

particles, multi-sphere method, proposed by Kruggel-Emden et al. (2008a), was used in this work. In this method, shape of a non-spherical particle is approximated by spheres that are glued to each other. Fig. 1 shows the idea of multi-sphere method to form complex shapes. In this method, spheres are glued to each other to represent the complex surface of the particle. Contact detection and contact force evaluations, which are applicable to spheres, can be used here and hence this approach becomes simple and computationally efficient. To describe the dynamics of glued spheres (a particle), two frames of references should be sued: inertial frame and bodyfixed frame. The inertial frame does not move with particle while the body-fixed frame is placed on the center of gravity of particle and moves with it. More details are described by Kruggel-Emden et al. (2008a). An in-house code in FORTRAN standard was used to obtain the dynamics of granular flow in rotating drums.

2.2.

Analysis of data and definition of parameters

During the execution of the DEM program, essential data like position and velocity of particles were saved on a file to be used in the post processing step. Other parameters, like dynamic angle of repose, time-averaged particle velocity profile, surface velocity, and circulation time were calculated from saved data in this step. Some programs were developed to perform calculations based on the physical concept of each parameter.

2.3.1.

Dynamic angle of repose

When drum rotates, particles form a bed with a flat surface (rolling regime). The angle between this surface and the horizontal axis is called the dynamic angle of repose. In cases with wavy surface (cascading regime) the plane that best fits the bed surface was obtained and the angle between this plane and the horizontal axis was reported as the dynamic angle of repose.

Details of simulations

The rotating drum considered in this work was a cylindrical container with the diameter of D and the width of 10 cm. It was filled with a certain number of particles with the size of dp and rotated at angular speed of ω. We define the fill ratio as:

ϕ=

2.3.

Volume of packed bed of particles Total volume of drum

(5)

Three fill ratios (0.1, 0.15 and 0.2), three drum diameters (30, 45 and 60 cm), three rotational speeds (6, 9 and 12 RPM), and three particles sizes (5, 7 and 9 mm) were used in simulations. At these operating conditions, the Froude number varies between 0.006 and 0.05 which cover both rolling and cascading regimes. To account for the shape effects, a set of simulations were performed in a drum with 45 cm diameter at various fill ratios and angular velocity for oval, oblong, and biconvex particles (see Fig. 1). The volume of each of these particles was equal to the volume of a 9-mm sphere; hence the volumetric diameter was 9 mm for all of them. Operating conditions of simulations, for both spherical and non-spherical particles, are listed in Table 2 (39 cases). Depending on the size of the drum, fill ratio, and particles size, a certain number of particles were used in each simulation (between 2500 and 36,000). The physical properties of particles and drum wall and other simulation parameters are listed in Table 3. All simulations were conducted for 120 s of real time and positions and velocities of all particles were saved in a separate file to be used in the post processing step.

2.3.2.

Average particle velocity

Having the dynamic angle of repose, the velocity of particle was first translated into (X, Y) coordinates (see Fig. 4). Then, the average velocity at each position in (X, Y) coordinates was calculated by averaging the signed velocity magnitude of particles over the time interval of 12 s to 36 s (24 s is much larger than any temporal fluctuations in velocity profile). The velocity of particles in the middle of the drum depth (the direction perpendicular to the side view shown in Fig. 4) was considered for this calculation. Therefore, the particles near the side walls were excluded from this calculation. The signed velocity magnitude was obtained by multiplying the sign of X-component of the particle velocity by the magnitude of particle velocity. The signed velocity magnitude is the velocity along stream line of particles. Obviously, the surface/cascading velocity is the average particle velocity at the bed surface.

2.3.3.

Circulation time

The circulation time (CT) is used in conjunction with coating and granulation processes. Region of interest (ROI) is an area on the bed surface where coating material or binder is sprayed onto. In this study, the ROI is a 10-cm square located at the top half of the cascading layer (Sandadi et al., 2004). The circulation time is defined as the time that takes a particle to re-enters the ROI. For this purpose, 500 particles were randomly picked and their positions were tracked to determine the time between successive re-entering to the ROI. In each simulation case, we were able to detect between 7000 and 12,000 particle circulations.

17

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Table 2 – Operating conditions and particle sizes in all simulation cases. Dynamic angle of repose, maximum cascading velocity and circulation time are obtained from DEM results. D (cm)

RPM

Sphere 30 30 30 30 30 30 45 45 45 45 45 45 45 45 45 45 45 45 45 60 60 60 60 60 60 Bi-Convex 45 45 45 45 45 Oval 45 45 45 45 Oblong 45 45 45 45 45 ∗

˛ (◦ )

Umax (m/s)

ı0 (cm)

ln(CT)*

ln(CT)*

26 22 26 26 29 28 25 29 33 30 26 26 30 31 33 26 30 31 33 30 31 31 32 29 31

0.13 0.10 0.20 0.14 0.30 0.18 0.19 0.44 0.28 0.50 0.23 0.19 0.40 0.68 0.53 0.26 0.55 0.41 0.73 0.64 0.46 0.83 0.72 0.92 0.90

3.29 4.53 2.35 4.49 2.95 3.86 3.89 3.91 3.52 4.53 4.00 5.29 3.92 5.05 4.99 5.27 3.87 4.22 5.13 5.44 5.67 5.61 5.67 6.49 6.93

2.502 1.964 2.130 2.281 2.292 1.985 2.064 2.069 2.324 2.494 1.702 1.802 1.921 2.117 2.271 1.542 1.614 1.821 1.965 2.026 1.764 1.747 1.841 2.012 1.585

0.369 0.415 0.334 0.392 0.322 0.394 0.280 0.319 0.286 0.332 0.374 0.357 0.347 0.425 0.382 0.448 0.439 0.436 0.427 0.331 0.416 0.470 0.433 0.434 0.492

9 9 9 9 9

40 40 42 43 44

0.52 0.60 0.71 0.80 0.87

3.18 3.37 3.61 4.24 3.90

1.433 1.107 1.20 1.291 1.109

0.377 0.482 0.467 0.481 0.578

0.15 0.1 0.15 0.2

9 9 9 9

41 40 42 43

0.54 0.53 0.71 0.77

3.42 3.86 3.98 4.16

1.420 1.104 1.190 1.251

0.347 0.445 0.478 0.458

0.15 0.1 0.15 0.2 0.15

9 9 9 9 9

40 41 42 43 44

0.54 0.53 0.69 0.75 0.78

3.50 3.01 3.72 4.89 4.21

1.458 1.116 1.204 1.270 1.058

0.397 0.455 0.473 0.455 0.537

ϕ

dp (mm)

6 9 9 9 9 12 6 6 6 6 9 9 9 9 9 12 12 12 12 6 9 9 9 9 12

0.15 0.1 0.15 0.15 0.2 0.15 0.1 0.15 0.15 0.2 0.1 0.1 0.15 0.2 0.2 0.1 0.15 0.15 0.2 0.15 0.1 0.15 0.15 0.2 0.15

7 7 5 9 7 7 7 5 9 7 5 9 7 5 9 7 5 9 7 7 7 5 9 7 7

6 9 9 9 12

0.15 0.1 0.15 0.2 0.15

6 9 9 9

6 9 9 9 12

Circulation time is in seconds.

3.

Results and discussion

3.1.

Validation of the model

Experimental data of Pandey and Turton (2005) were used for assessing the DEM results. Experiments were carried out in a drum with inner diameter of 58 cm. The bed was filled with 9-mm polystyrene spheres up to one-fourth of the pan diameter (corresponds to ϕ = 0.2). We chose cascading velocity and circulation time for the comparison. Fig. 2 shows the variation of average circulation time and cascading velocity at various drum rotational speeds for both experiments and simulations. Error bars on this plot show the variation range of averaged values with 95% confidence level. Circulation times obtained by simulation agree fairly well with experimental results and experimental values fall within the variation range of the simulation results. Fig. 2b shows that the average surface velocity

increases linearly with the rotational speed in both experiments and simulations. The slopes of line fitting experiment and simulation results are 3.1 and 3.05 which are very close to each other. The circulation time reveals the rate of renewal of the bed surface which is important in coating and granulation process and estimating the heat transfer from bed surface to the freeboard in the drum. Therefore, its mean value is as important as its distribution. Distribution of circulation time of 9 mm spheres in a 58-cm drum with ϕ = 0.2 and ω = 9 RPM is plotted in Fig. 3. The circulation time of particles follows a log-normal distribution. Most particles have a circulation time around 5–6 s and a few particles have circulation times longer than 10 s. Long circulation times of particles show that particles spend a long time in stagnant regions of the bed or they circulate the whole bed without reaching the bed surface. The same trend (log-normal distribution) can be found by

18

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Average Circulation Time (sec)

15

Table 3 – Parameters used in DEM simulations. Parameter

(a)

12

Particles Density (kg/m3 ) Shape

9 6 Simulation Experiment (Pandey et al., 2005)

3 0 3

6

9 12 Rotation Speed (rpm)

15

Average Cascading Velocity (m/s)

1

(b)

0.8 Slope = 3.05

0.6 0.4

Slope = 3.1

Size (mm) Normal stiffness (N/m) Tangential stiffness (N/m) Coefficient of restitution (–) Dynamic friction factor (–) Rolling friction factor (–)

Value

1200 Round, oval, biconvex, oblong 5, 7, 9 100,000 82,400 0.75 0.7 0.15

Drum Diameter (cm) Depth (cm) Normal stiffness (N/m) Tangential stiffness (N/m) Dynamic friction factor (–) Rolling friction factor (–)

30, 45, 60 10.4 100,000 82,400 0.5 0.15

Simulation Time step (s) Total simulation time (s)

2.0 × 10−6 120

Simulation

0.2

Experiment (Pandey et al., 2005) 0 3

6

9 12 Rotation Speed (rpm)

15

Fig. 2 – (a) Average circulation time and (b) average cascading velocity of particles in a drum with inner diameter of 58 cm, ϕ = 0.2, and dp = 9 mm.

inspecting the experimental data (Pandey and Turton, 2005; Sandadi et al., 2004), although a quantitative comparison is not possible here.

3.2.

DEM visual results

Fig. 4 shows the side view of the rotating drum with rotational speed of 9 RPM and 45 cm inner diameter. The drum is filled with either spherical or non-spherical particles. Colors of particles are according to their average velocity. At certain X position, the average particle velocity is maximum on the bed surface and decreases toward the bed depth (drum wall). Particles freely avalanche at the bed surface and accelerate from top to bottom. The maximum surface velocity occurs almost

Fig. 3 – Distribution of circulation time in a 58-cm drum with ϕ = 0.2, ω = 9 RPM, and dp = 9 mm.

after the mid-chord distance of the surface and is higher for non-spherical particles. The average velocity at the drum wall is almost equal to drum wall velocity (Rω). Because, particles stick to the drum wall and no sliding occurs according to Coulomb’s friction law. We can observe a flat surface in the bed of spherical particles (Fig. 4a) and a kidney-shaped bed in the bed of non-spherical particles (Fig. 4b and c). We know that the bed surface is almost flat in the rolling regime while curvature appears at the surface and the bed transforms into a kidney-shaped bed in the cascading regime. Therefore, it can be concluded from Fig. 4 that the bed of spherical particles is in the rolling regime and the beds of non-spherical particles are in the cascading regime. Mellmann (2001) found that the regime transition point from rolling to the cascading depends on Froude number, particle size, and fill level. These parameters are the same in three simulations shown in Fig. 4. However, spherical particles show rolling regime behavior while non-spherical particles show cascading regime behavior. This suggests that the transition to cascading regime also depends on the shape of particles and occurs at a lower Froude number for non-spherical particles.

3.3.

Characterizing velocity profile

3.3.1.

Surface velocity profile

The velocity profile of particles at the bed surface (stream-wise velocity at the bed surface) is used for design and scale-up of pan coaters and granulators (Mueller and Kleinebudde, 2007; Pandey et al., 2006c) and should to be characterized in terms of drum dimensions and operating conditions. This profile is mostly expressed in terms of dimensionless coordinate (X/L). The time-averaged velocity profiles of particles with different shapes and sizes at the bed surface in a drum with inner diameter of 45 cm and ϕ = 0.15 are plotted in Fig. 5. Particles at the top of the bed surface accelerate and reach their peak velocity after passing the center of the bed (mid-chord length). Thereafter, the kinetic energy of particles is balanced by frictional and inelastic losses and particles decelerate. It is common to

19

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Fig. 4 – Side view of the rotating drum with fill fraction of 0.2 and rotational speed of 9 RPM (Fr ∼ = 0.02). Drum is filled with (a) spherical, (b) biconvex and (c) oblong particles with volumetric dimater of 9 mm. consider a symmetric velocity profile at the bed surface which follows a quadratic form (Khakhar et al., 1997): usurf =1− umax

 X 2 L

(6)

Alizadeh et al. (2013) found that this equation fits their experimental data with spherical particles in the rolling regime while a better fit can be obtained by replacing the exponent in the right-hand side of the equation by 2/3. On the other hand, Dubé et al. (2013) tested this profile against blends of tablets in the rolling regime and their experimental results showed that the peak velocity occurs at a point after the mid-chord length. An asymmetric profile is obtained when particle acceleration continues beyond the mid-chord point because the kinetic

energy of avalanching particles does not balance by friction and inelastic losses before reaching the this point. Asymmetry factor is defined as ˇ = l/L, where l is the distance of the peak velocity from the mid-chord point (X = 0). Alexander et al. (2002) found that ˇ increases with the drum rotational speed and remains constant at high rotational speeds and reduces with increasing the drum radius. In Fig. 6, we scaled the surface velocity by the peak velocity (usurf /umax ) and plotted it against the dimensionless coordinate for all 39 simulations of this work and found the following curve as the best fit:



usurf = 1− umax

 X 2  L

exp

 2ˇ X  1 − ˇ2 L

(7)

20

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Particle Velocity (m/s)

1

6 RPM - dp = 9 mm 12 RPM - dp = 9 mm

12 RPM - dp = 9 mm 6 RPM - dp = 9 mm

12RPM - dp = 5 mm 12 RPM - dp = 9 mm

0.8

0.6

0.4

0.2

comparison to spherical particles. Suzzi et al. (2012) associated this behavior with the preferred orientation of non-spherical particles on the bed surface that causes particles to slide down the bed rather than to rotate. On the contrary, Pandey and Turton (2005) attributed this behavior to higher dynamic angle of repose of non-spherical particles in drum that converts a higher fraction of the potential energy of particles at the top of the bed into the kinetic energy during avalanching. There are some scaling relations for predicting peak/mean surface velocity in rotating drums. Alexander et al. (2002) found that the dimensionless peak velocity in a half-filled drum is scaled as follows:

0 -1

-0.5 0 0.5 Dimensionless X-coordinate [X/L] (-)

1

Fig. 5 – Time averaged velocity profile of particles at the surface of the active layer in a drum with inner diameter of 45 cm and fill level of 15%. (䊉) spherical, () oval, () biconvex, and (×) oblong particles. The value of ˇ varies from 0.04 to 0.22 in different cases and on average we found ˇ = 0.12 ± 0.06 with 95% confidence level (R2 = 0.83). This velocity profile follows that the surface velocity in drum is characterized by mid-chord length and peak velocity. Given a fill level and drum diameter, the mid-chord length can be calculated. If we can find some scaling relations for peak velocity, the velocity profile at the bed surface can be estimated as a function operating conditions. We will find a proper scaling relation for peak velocity at the bed surface in the next section.

3.3.2.

Peak velocity scaling relations

Results in Fig. 5 show that the peak velocity increases with the drum rotational speed and by decreasing the particle size. This trend agrees with previous experimental observations (Jain et al., 2002; Pandey and Turton, 2005; Sandadi et al., 2004). The shape of particle is another effective parameter on the peak velocity. At the same fill level, rotational speed, drum diameter and particle size, non-spherical particles show a higher peak velocity (hence, surface velocity) in

umax ∝ ωR



dp R

−1/6 Fr−1/6

(8)

To consider the effect of fill level, Pandey et al. (2006b) and Mueller and Kleinebudde (2007) extended the above relation as follows: umax ∝ ωR



dp R

−1/6 Fr−1/6 ϕ1.8

(9)

Obviously, this relation is not adequate to predict peak velocity of particles with different shapes since the effect of shape is not reflected in this equation. Non-spherical particles exhibit a greater peak velocity than spherical particles (see Fig. 5). Liu et al. (2006) developed an analytical model for average cascading velocity and maximum thickness of the active layer as a function of operating conditions, particle size and dynamic angle of repose for the rolling regime. We examined this model against our simulation results and found 5–90% absolute relative error and on average 43%. Khakhar et al. (1997) proposed the following model for the maximum surface velocity in rotating drums: umax =

ωL2 0

(10)

where 0 is the distance from bed surface at X = 0, where average particle velocity becomes zero. The value of 0 depends on

1.2

1

u/umax (-)

0.8

0.6

0.4

0.2

0 -1

-0.5 0 0.5 Dimensionless X-coordinate [X/L] (-)

1

Fig. 6 – Scaled surface velocity versus dimensionless X-coordinate for all simulation cases. The solid line shows the best fit and symbols show the DEM results, () spheres in 30-cm drum, (♦) spheres in 45-cm drum, (o) spheres in 60-cm drum, (-) oblong particles, (×) biconvex particles and (+) oval particles.

21

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25



umax =

umax = 0.198 L sin (˛) ω

dp D

−0.801 Fr−0.300 ϕ0.438 [tan (˛)]0.855 (11)

Dimensionless peak velocity from DEM

10

(a) 7.5

5

2.5

0 0

10

(b) 15

10

5

Pandey et al. (2006) Dube et al. (2013) Low Froude Number Alizadeh et al. (2013)

0 0

The above correlation inversely relates the dimensionless peak velocity to Froude number and size ratio and directly to fill ratio and dynamic angle of repose. Eqs. (9) and (11) show the same trend over Froude number, fill ratio, and size ratio but with different values of exponents. This is expected since Eq. (9) does not include ˛. The dynamic angle of repose depends on operating conditions, i.e., Froude number and fill ratio. Therefore, its inclusion in the correlation should change the exponent of Froude number and the fill ratio. Moreover, the peak velocity is made dimensionless differently (using chord length) in Eq. (11). This correlation only needs one experimental value, the dynamic angle of repose, which can be measured by simple visual techniques. The dynamic angle of repose depends on operating conditions and particle shape. At the same operating conditions, the dynamic angle of repose, hence the peak velocity of non-spherical particle is greater than that of spherical particles. This equation connects the peak velocity of spherical and non-spherical particles through the dynamic angle of repose. For scaling purposes (for example in a pan coating operation) this equation can be used to determine how fast particles move through the spray zone and to determine the total time required for coating of particles. The peak surface velocity is also important for processes in which heat transfer/reaction occurs at the bed surface to determine the residence time of solid particle at the bed surface. Fig. 7a compares the dimensionless peak velocity obtained by simulation with those calculated from Eq. (11). Dashed lines in this figure show the bounds of 20% relative error. Fig. 7a reveals that this correlation well predicts the peak velocity with a maximum relative error of 20% and average relative absolute error of 10 ± 3% (R2 = 0.76). We also examined this correlation against experimental and simulation results found in literature. Table 4 lists the operating conditions, shape of particles and number of data points for each set of data (29 data points in total). The Froude number varies between 2.5 × 10−4 and 4.7 × 10−2 , which covers both rolling and

2.5 5 7.5 Dimensionless peak velocity by Eq. (11)

20 Dimensionless peak velocity

the operating conditions and particle properties and should be calculated using correlations, experimental results or simulation results. We used DEM results to obtain 0 , based on which Eq. (10) gives the maximum surface velocity. Our calculations showed that this equation overestimates the maximum velocity by an average absolute relative error of 25.6 ± 6.8%. This correlation offers a better estimation of the peak velocity in both rolling and cascading regimes compared with Liu et al. (2006). However, it needs the value of 0 to be known that should be calculated via sophisticated experimental measurements such as radioactive particle tracking or positron emission particle tracking. Ding et al. (2001a) defined a dimensionless peak velocity by dividing it by ωLsin(˛) (as a measure of the amount of potential energy that converts into the kinetic energy) and found that the peak velocity is related to the square root of the ratio of drum diameter to particle diameter. However, they were not able to obtain a correlation that well fits the experimental data in the total range of operating conditions. We followed their approach to correlate the dimensionless peak velocity as a function of operating conditions. Results of the non-linear regression gave the following equation for the dimensionless peak velocity:

5 10 15 Dimensionless peak velocity by Eq. (11)

20

Fig. 7 – Comparison between predicted dimensionless peak velocities calculated by Eq. (11) and (a) DEM simulation results in this study and (b) various sets of literature data. cascading regimes, and the fill ratio varies between 0.1 and 0.35. Fig. 7b compares the dimensionless peak velocity calculated from Eq. (11) with literature data in the form of a parity plot. Dashed lines in this figure show the bounds of 20% relative error. Fig. 7b also demonstrates that Eq. (11) predicts the peak velocities reported in other works with a maximum relative error of 20%.

3.3.3.

Velocity profile in active and passive layers

After finding a proper scaling relation for particle velocity profile on the bed surface, we turned into the velocity profile in the bed depth (Y-direction). The bed is divided into two distinct regions in the rolling regime: active and passive layers. In the passive layer, particles are carried up by the drum wall as a solid body. The particle velocity linearly changes with distance from the center of the drum in this layer and reaches the tangent velocity of the drum wall due to no-slip on the drum wall. In the active layer, particles avalanche down the bed due to gravity. This avalanching affects the velocity profile along the bed depth and alters it. The point where this effect vanishes is considered as the yield point and its distance from bed surface as the active layer thickness. Fig. 8 shows the average velocity profile of spherical and biconvex particles along the bed depth at X = 0 at two rotational speeds. The velocity profile in active and passive layers varies linearly with Y with a higher slope in the active layer. All the simulation cases show similar velocity profiles in both layers but are not illustrated here for the sake of brevity. Lines that fit these velocity profiles cross each other at a point whose distance to the surface is considered as the active layer depth. Based on this method, we calculated the

22

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Table 4 – Details of operating conditions and shape of particles on the peak velocity extracted from other literature. Data set

No. of points *

Fr

ϕ

dp (mm)

D (cm)

This study

39

0.006–0.05

0.1–0.2

5–9

30–60

Pandey et al. (2006b) Dubé et al. (2013)

9 12

0.01–0.047 0.0034–0.0135

0.1–0.17 0.35

9 5–7.5

58 24.13

Alizadeh et al. (2013)

5

0.0182

0.35

3–6

24.13

Low Froude number (in house)

3

0.00025–0.00226

0.2

7

45

∗

Shape Spherical and non-spherical particles Spherical particles Tablet blends and spherical particles Blends of spherical particles Spherical particles

This corresponds to the operating conditions reported in Table 2.

active layer depth in all simulation cases and reported them in Table 2. Having the active layer depth, we can find some proper correlations for predicting the active layer depth as a function of operating conditions for both rolling and cascading regimes. The correlation by Orpe and Khakhar (2001) reads as follows:

ı(X) =

ω 2 (L − X2 ) ˙

(12)

where ˙ is the shear rate and is calculated by:

˙ =

g cos(˛(X)) sin (˛ − ˛0 ) dp cos (˛) cos (˛0 )

(13)

where ˛0 is the static angle of repose and ˛(X) is the angle between the bed surface and the horizontal line at position X. We assumed that the value of ˛(X) at the mid-chord point is equal to the dynamic angle of repose and calculated the depth of active layer in all the operating conditions of simulation

1 6 RPM - Simulation

0.8

12 RPM - Simulation

Velocity (m/s)

0.6

acve layer

0.4 0.2

passive layer

0

(a)

-0.4 -0.35

-0.3

-0.25 Y (m)

-0.2

Velocity (m/s)

0.6

6 RPM-Simulation 12 RPM-Simulation

acve layer

0.4 0.2

passive layer

0

(b)

-0.2 -0.4 -0.25

-0.2

Y (m)

-0.15

-0.1

Fig. 8 – Velocity profile at X = 0 for (a) spheres with dp = 7 mm, D = 60 cm, and ϕ = 0.15 and (b) biconvex granules with D = 45 cm and ϕ = 0.15.

Circulation time

In coating operations in a rotating drum, the inter-particle mass coating variability is related to the mean and the standard deviation of the circulation time and the surface time (the time spent by particle in the ROI) of particles based on the renewal theory (Mann, 1983). In addition, the distribution of circulation time and surface time are essential for performing Monte-Carlo simulations (Kandela et al., 2010). The surface time is related to the surface velocity of the particles and the time that a particle needs to pass the spray region. Therefore, having the surface velocity (which has already been discussed), one can calculate the surface time of particles in the spray region. This section discusses the circulation time of particles.

10

-0.15

1 0.8

3.4.

Active layer depth from DEM (cm)

-0.2

cases. Fig. 9 compares depth of active layer ı0 calculated by Eq. (12) and those obtained from DEM results (see Table 2). Dashed lines in this figure show the bounds of 20% relative error that indicate that the maximum relative deviation of this correlation is less than 20% in the range of operating condition of simulations. On average, this correlation predicts the active layer depth with 8 ± 2.4% absolute relative error (R2 = 0.78). We also tested the performance of the correlation proposed by Liu et al. (2006). Results of this assessment are not presented here, but it should be mentioned that this correlation predicts the active layer depth with 23 ± 7.5% average relative error. We also found larger deviations in using the correlation of Ding et al. (2001b) which shows that this correlation is not applicable to both rolling and cascading regimes.

Sphere Biconvex Oval Oblong

8

6

4

2 2

4 6 8 Active layer depth by Eq. (12) (cm)

10

Fig. 9 – Comparison between predicted active layer depth by Eq. (12) and that obtained from DEM results.

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

As we illustrated in Section 3.1, the circulation time of spherical particles follows a log-normal distribution. The logarithm of circulation time (ln(CT)) therefore follows normal distribution. To characterize normal distribution, we calculated the mean and standard deviation of ln(CT) and listed them in Table 2. The results show that the mean circulation time of spherical and non-spherical particles decreases with rotation speed and drum diameter and increases with fill ratio. These trends are in agreement with experimental observations reported in literature (Kandela et al., 2010; Pandey and Turton, 2005; Sandadi et al., 2004). The particle velocity in both active and passive layers increases with the rotation speed, which reduces the circulation time. At a higher fill ratio, particles must circulate in a larger bed, thus, the chance of appearing on the bed surface reduces and the circulation time increases. By definition, the circulation time is the time that takes a particle to travel a path in the bed and re-enter the ROI at the bed surface. Path length and particle velocity both affect the circulation time. The path length is linearly proportional to D. The particle velocity in the active layer surface is proportional to D1.5 (Eq. (11) with a constant fill ratio) and to D in the passive layer (particle velocity at the drum wall is equal to Rω). Therefore, we can conclude that the velocity of particles in the whole drum is proportional to Dn , where n is between 1 and 1.5. This clearly shows that increasing the drum diameter has a more significant effect on the particle velocity than the path length. Therefore, circulation time decreases with increasing the drum diameter. The mean circulation time of non-spherical particles is less than the spherical particles. The average particle velocity of non-spherical particle is higher than that of spherical particle in the active layer (see Section 3.3). Moreover, the depth of the active layer of a bed of spherical particles is higher than for non-spherical particles. These two in combination reduce the circulation time of non-spherical particles. We also performed a non-linear regression on the mean value of logarithm of circulation time, ln(CT) and found the following mathematical expression for it: ln(CT) = 1.853Fr−0.141 ϕ0.406 [tan (˛)]−0.462

(14)

The Froude number reflects the effect of the rotational speed and the drum diameter and ˛ reflects the effect of shape. It is worth knowing that we found that the size of particle has insignificant effect (with 95% significant level) on ln(CT) based on the statistical analysis. Eq. (14) predicts ln(CT) with 3.75 ± 1% absolute relative error (R2 = 0.93). The non-linear regression on the standard deviation of ln(CT) resulted in the following expression: ln(CT) = 1.083Fr0.210 [tan (˛)]0.320

(15)

This shows that the discrepancy of circulation time and hence mass coating variability is less at high Froude numbers and for non-spherical particles (with higher dynamic angle of repose). This equation predicts ln(CT) with 6 ± 2% absolute relative error (R2 = 0.85). Eqs. (14) and (15) are applicable for fill ratio between 0.1 and 0.2 and Froude number between 0.006 and 0.05.

4.

Conclusion

We conducted a verity of DEM simulations in rotating drums to investigate velocity profile and circulation time – as two

23

important parameters in the design and scale-up of rotating drums – at different operating conditions using spherical and non-spherical particles. The operating conditions covered both rolling and cascading regimes which are the most common flow regimes in industrial applications. New scaling relations/correlations were found to fully characterize velocity profile and circulation time in both rolling and cascading regimes. Using dynamic angle of repose, effect of particle shape was included in these correlations, which extended correlations to both spherical and non-spherical particles. DEM results satisfactorily reproduced experimental measurements on average surface velocity and average circulation time. Visual results showed that transition from rolling to cascading regime depends not only on Froude number, fill level and particle size, but also on the particle shape. Surface velocity was scaled with peak velocity and half chord length and its profile was asymmetric with the maximum occurring after the mid-chord position in all simulations. We included shape effect, using dynamic angle of repose, in the new correlation for the peak velocity which is applicable to both rolling and cascading regimes. This correlation well predicted the peak velocity in our simulations as well as available experimental data in literature (average error = 10%). The velocity profiles along the bed depth in active and passive layers were linear with a higher slope in the active layer. In all simulations, the active layer depth was calculated and tested against various correlations. The correlation of Orpe and Khakhar (2001) best predicted the active layer depth (average error = 8%) in both rolling and cascading regimes. Our results showed that the circulation time of particles follows a log-normal distribution. The mean circulation time decreased with rotation speed and drum diameter and increased with fill ratio. This parameter was greater for spherical particles than non-spherical particles. To characterize the distribution of circulation time with respect to operating conditions. We also proposed correlations for mean (average error = 3.75%) and standard deviation (average error = 6%) of circulation time.

References Alexander, A., Shinbrot, T., Muzzio, F.J., 2002. Scaling surface velocities in rotating cylinders as a function of vessel radius, rotation rate, and particle size. Powder Technol. 126, 174–190. Alizadeh, E., Dubé, O., Bertrand, F., Chaouki, J., 2013. Characterization of mixing and size segregation in a rotating drum by a particle tracking method. AIChE J. 59, 1894–1905. Arntz, M.M.H.D., den Otter, W.K., Briels, W.J., Bussmann, P.J.T., Beeftink, H.H., Boom, R.M., 2008. Granular mixing and segregation in a horizontal rotating drum: a simulation study on the impact of rotational speed and fill level. AIChE J. 54, 3133–3146. Boateng, A.A., 1998. Boundary layer modeling of granular flow in the transverse plane of a partially filled rotating cylinder. Int. J. Multiphase Flow 24, 499–521. Chaudhuri, B., Muzzio, F.J., Tomassone, M.S., 2006. Modeling of heat transfer in granular flow in rotating vessels. Chem. Eng. Sci. 61, 6348–6360. Chen, W., Chang, S.-Y., Kiang, S., Marchut, A., Lyngberg, O., Wang, J., Rao, V., Desai, D., Stamato, H., Early, W., 2010. Modeling of pan coating processes: prediction of tablet content uniformity and determination of critical process parameters. J. Pharm. Sci. 99, 3213–3225. Cheng, N.-S., 2012. Scaling law for velocity profiles of surface granular flows observed in rotating cylinders. Powder Technol. 218, 11–17. Cheng, N.-S., Zhou, Q., Keat Tan, S., Zhao, K., 2011. Application of incomplete similarity theory for estimating maximum shear

24

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

layer thickness of granular flows in rotating drums. Chem. Eng. Sci. 66, 2872–2878. Di Renzo, A., Di Maio, F.P., 2004. Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes. Chem. Eng. Sci. 59, 525–541. Ding, Y.L., Forster, R., Seville, J.P.K., Parker, D.J., 2002. Segregation of granular flow in the transverse plane of a rolling mode rotating drum. Int. J. Multiphase Flow 28, 635–663. Ding, Y.L., Forster, R.N., Seville, J.P.K., Parker, D.J., 2001a. Scaling relationships for rotating drums. Chem. Eng. Sci. 56, 3737–3750. Ding, Y.L., Seville, J.P.K., Forster, R.N., Parker, D.J., 2001b. Solids motion in rolling mode rotating drums operated at low to medium rotational speeds. Chem. Eng. Sci. 56, 1769–1780. Dubé, O., Alizadeh, E., Chaouki, J., Bertrand, F., 2013. Dynamics of non-spherical particles in a rotating drum. Chem. Eng. Sci. 101, 486–502. Dubey, A., Boukouvala, F., Keyvan, G., Hsia, R., Saranteas, K., Brone, D., Misra, T., Ierapetritou, M.G., Muzzio, F.J., 2012. Improvement of tablet coating uniformity using a quality by design approach. AAPS PharmSciTech 13, 231–246. Dubey, A., Hsia, R., Saranteas, K., Brone, D., Misra, T., Muzzio, F.J., 2011. Effect of speed, loading and spray pattern on coating variability in a pan coater. Chem. Eng. Sci. 66, 5107–5115. Fantozzi, F., Colantoni, S., Bartocci, P., Desideri, U., 2007. Rotary kiln slow pyrolysis for syn gas and char production from biomass and waste—Part I: Working envelope of thereactor. J. Eng. Gas Turbines Power 129, 901–907. Finnie, G.J., Kruyt, N.P., Ye, M., Zeilstra, C., Kuipers, J.A.M., 2005. Longitudinal and transverse mixing in rotary kilns: a discrete element method approach. Chem. Eng. Sci. 60, 4083–4091. Freireich, B., Ketterhagen, W.R., Wassgren, C., 2011. Intra-tablet coating variability for several pharmaceutical tablet shapes. Chem. Eng. Sci. 66, 2535–2544. Freireich, B., Wassgren, C., 2010. Intra-particle coating variability: analysis and Monte-Carlo simulations. Chem. Eng. Sci. 65, 1117–1124. Heydenrych, M.D., Gree, P., Heesink, A.B.M., Versteeg, G.F., 2002. Mass transfer in rolling rotary kilns: a novel approach. Chem. Eng. Sci. 57, 3851–3859. Jain, N., Ottino, J.M., Lueptow, R.M., 2002. An experimental study of the flowing granular layer in a rotating tumbler. Phys. Fluids 14, 572. Joglekar, A., Joshi, N., Song, Y., Ergun, J., 2007. Mathematical model to predict coat weight variability in a pan coating process. Pharm. Dev. Technol. 12, 297–306. Kalbag, A., Wassgren, C., 2009. Inter-tablet coating variability: tablet residence time variability. Chem. Eng. Sci. 64, 2705–2717. Kalbag, A., Wassgren, C., Penumetcha, S.S., Pérez-Ramos, J.D., 2008. Inter-tablet coating variability: residence times in a horizontal pan coater. Chem. Eng. Sci. 63, 2881–2894. Kandela, B., Sheorey, U., Banerjee, A., Bellare, J., 2010. Study of tablet-coating parameters for a pan coater through video imaging and Monte Carlo simulation. Powder Technol. 204, 103–112. Ketterhagen, W.R., 2011. Modeling the motion and orientation of various pharmaceutical tablet shapes in a film coating pan using DEM. Int. J. Pharm. 409, 137–149. Khakhar, D., McCarthy, J., Ottino, J., 1997. Radial segregation of granular mixtures in rotating cylinders. Phys. Fluids 9, 3600–3614. Kruggel-Emden, H., Rickelt, S., Wirtz, S., Scherer, V., 2008a. A study on the validity of the multi-sphere discrete element method. Powder Technol. 188, 153–165. Kruggel-Emden, H., Stepanek, F., Munjiza, A., 2010. A study on adjusted contact force laws for accelerated large scale discrete element simulations. Particuology 8, 161–175. Kruggel-Emden, H., Wirtz, S., Scherer, V., 2008b. A study on tangential force laws applicable to the discrete element method (DEM) for materials with viscoelastic or plastic behavior. Chem. Eng. Sci. 63, 1523–1541.

Kruggel-Emden, H., Wirtz, S., Scherer, V., 2009. Applicable contact force models for the discrete element method: the single particle perspective. J. Press. Vessel Technol. 131, 1–11. Kumar, R., Wassgren, C., 2013. Angular circulation speed of tablets in a vibratory tablet coating pan. AAPS PharmSciTech 14, 339–351. Leaver, T.M., Shannon, H.D., Rowe, R.C., 1985. A photometric analysis of tablet movement in a side-vented perforated drum. J. Pharm. Pharmacol. 37, 17–21. Liu, P.Y., Yang, R.Y., Yu, A.B., 2013. DEM study of the transverse mixing of wet particles in rotating drums. Chem. Eng. Sci. 86, 99–107. Liu, X.Y., Specht, E., Gonzalez, O.G., Walzel, P., 2006. Analytical solution for the rolling-mode granular motion in rotary kilns. Chem. Eng. Process.: Process Intensification 45, 515–521. Mann, U., 1983. Analysis of spouted-bed coating and granulation. 1. Batch operation. Ind. Eng. Chem. Process. Des. Dev. 22, 288–293. Marigo, M., Cairns, D.L., Davies, M., Ingram, A., Stitt, E.H., 2012. A numerical comparison of mixing efficiencies of solids in a cylindrical vessel subject to a range of motions. Powder Technol. 217, 540–547. McCarthy, J.J., Khakhar, D.V., Ottino, J.M., 2000. Computational studies of granular mixing. Powder Technol. 109, 72–82. Mellmann, J., 2001. The transverse motion of solids in rotating cylinders—forms of motion and transition behavior. Powder Technol. 118, 251–270. Mellmann, J., Specht, E., Liu, X., 2004. Prediction of rolling bed motion in rotating cylinders. AIChE J. 50, 2783–2793. Mueller, R., Kleinebudde, P., 2007. Prediction of tablet velocity in pan coaters for scale-up. Powder Technol. 173, 51–58. Ndiaye, L.G., Caillat, S., Chinnayya, A., Gambier, D., Baudoin, B., 2010. Application of the dynamic model of Saeman to an industrial rotary kiln incinerator: numerical and experimental results. Waste Manage. 30, 1188–1195. Orpe, A.V., Khakhar, D.V., 2001. Scaling relations for granular flow in quasi two-dimensional rotating cylinders. Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 64, 031302. Pandey, P., Katakdaunde, M., Turton, R., 2006a. Modeling weight variability in a pan coating process using Monte Carlo simulations. AAPS PharmSciTech 7, E1–E10, Article 83. Pandey, P., Song, Y., Kayihan, F., Turton, R., 2006b. Simulation of particle movement in a pan coating device using discrete element modeling and its comparison with video-imaging experiments. Powder Technol. 161, 79–88. Pandey, P., Turton, R., 2005. Movement of different-shaped particles in a pan-coating device using novel video-imaging techniques. AAPS PharmSciTech 6, E237. Pandey, P., Turton, R., Joshi, N., Hammerman, E., Ergun, J., 2006c. Scale-up of a pan-coating process. AAPS PharmSciTech 7, Article 102. Parker, D., Dijkstra, A., Martin, T., Seville, J., 1997. Positron emission particle tracking studies of spherical particle motion in rotating drums. Chem. Eng. Sci. 52, 2011–2022. Rege, B.D., Gawel, J., Kou, J.H., 2002. Identification of critical process variables for coating actives onto tablets via statistically designed experiments. Int. J. Pharm. 237, 87–94. Sahni, E., Yau, R., Chaudhuri, B., 2011. Understanding granular mixing to enhance coating performance in a pan coater: experiments and simulations. Powder Technol. 205, 231–241. Sandadi, S., Pandey, P., Turton, R., 2004. In situ, near real-time acquisition of particle motion in rotating pan coating equipment using imaging techniques. Chem. Eng. Sci. 59, 5807–5817. Santomaso, A.C., Ding, Y.L., Lickiss, J.R., York, D.W., 2003. Investigation of the granular behaviour in a rotating drum operated over a wide range of rotational speed. Chem. Eng. Res. Des. 81, 936–945. Schutyser, M.A.I., Padding, J.T., Weber, F.J., Breils, W.J., Rinzema, A., Boom, R., 2001. Discrete particle simulations predicting mixing behavior of solid substrate particles in a rotating drum fermenter. Biotechnol. Bioeng. 75, 666–675.

chemical engineering research and design 1 0 2 ( 2 0 1 5 ) 12–25

Smith, G.W., Macleod, G.S., Fell, J.T., 2003. Mixing efficiency in side-vented coating equipment. AAPS PharmSciTech 4, 1–5, Article 37. Suzzi, D., Toschkoff, G., Radl, S., Machold, D., Fraser, S.D., Glasser, B.J., Khinast, J.G., 2012. DEM simulation of continuous tablet coating: effects of tablet shape and fill level on inter-tablet coating variability. Chem. Eng. Sci. 69, 107–121. Tobiska, S., Kleinebudde, P., 2003. Coating uniformity and coating efficiency in a Bohle Lab-Coater using oval tablets. Eur. J. Pharm. Biopharm. 56, 3–9. van Puyvelde, D.R., Young, B.R., Wilson, M.A., Schmidt, S.J., 2000. Modelling transverse segregation of particulate solids in a rolling drum. Chem. Eng. Res. Des. 78, 643–650.

25

Weir, G., Krouse, D., McGavin, P., 2005. The maximum thickness of upper shear layers of granular materials in rotating cylinders. Chem. Eng. Sci. 60, 2027–2035. Wilson, K.E., Crossman, E., 1997. The influence of tablet shape and pan speed on intra-tablet film coating uniformity. Drug Dev. Ind. Pharm. 23, 1239–1243. Yamane, K., Sato, T., Tanaka, T., Tsuji, Y., 1995. Computer simulation of tablet motion in coating drum. Pharm. Res. 12, 1264–1268. Yan Liu, X., Specht, E., 2010. Predicting the fraction of the mixing zone of a rolling bed in rotary kilns. Chem. Eng. Sci. 65, 3059–3063.