Numerical analysis of similarities of particle behavior in high shear mixer granulators with different vessel sizes

Advanced Powder Technology 20 (2009) 493–501

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Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt

Original research paper

Numerical analysis of similarities of particle behavior in high shear mixer granulators with different vessel sizes Hideya Nakamura, Yoshikazu Miyazaki, Yoshinobu Sato, Tomohiro Iwasaki, Satoru Watano * Department of Chemical Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan

a r t i c l e

i n f o

Article history: Received 11 February 2009 Received in revised form 17 May 2009 Accepted 26 May 2009

Keywords: High shear mixer granulator Scale-up Kinematic and dynamic similarities Numerical analysis Discrete element method

a b s t r a c t This paper deals with the numerical analysis of kinematic and dynamic similarities of particle behavior in high shear mixer granulators with different vessel sizes. The three-dimensional particle motion in high shear mixer granulators with four different vessel sizes was calculated using a discrete element method (DEM). The geometrically similar mixer granulators equipped with a flat-shaped impeller blade were used as simulated mixer granulators. Kinematic and dynamic similarities of particle behavior in various vessel sizes under a constant normalized agitation power were numerically analyzed. In various vessel sizes, dynamic similarity expressed by the particle collision energy was confirmed under a constant normalized agitation power, while kinematic similarity expressed by the particle velocity was not confirmed. These results indicate that the dynamic similarity should be maintained for successful scale-up of high shear mixer granulators. Ó 2009 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

1. Introduction High shear mixer granulators have been widely used in many industries such as pharmaceutical, food, detergent, agriculture, and chemical. In high shear mixer granulators, mixing, densification and granulation of wetted particulate materials are achieved by shearing and compaction forces exerted by a main impeller blade equipped at the bottom of vessel, which rotates at a relatively high speed. High shear mixer granulators have great advantages that spherical granules with high density can be easily produced within relatively short processing period and the equipment construction is very simple [1]. Scale-up of high shear mixer granulators has been one of the most crucial topics in the granulation technology. Goal of the scale-up is to maintain key granule properties such as particle size distribution, density, and strength, when the vessel size is increased. In order to attain this goal, similarities of the particle behavior should be kept in every size. As suggested by Leuenberger [2], maintaining of geometric, kinematic and dynamic similarities of particle behavior in high shear mixer granulators with different sizes is extremely important for the successful scale-up. However, similarities of actual particle behavior in different sizes have not been investigated yet.

* Corresponding author. Tel./fax: +81 72 254 9305. E-mail address: [email protected] (S. Watano).

So far, some scale-up methodologies of high shear mixer granulators have been developed. Some researchers [3–6] reported the scale-up methodology based on a dimensionless correlation relating wetted powder properties and operating parameters: i.e., power number vs. Reynolds number, Froude number and particle filling level. Schaefer et al. [7] and Ramaker et al. [8] demonstrated the scale-up using a constant rule of relative swept volume which is defined as ratio of the volume of particles swept away by the main impeller blade within a given period of time to the vessel volume. The normalized agitation power, defined as net agitation power divided by total mass of the particles, was used as the scale-up factor [9–11]. For evaluating and controlling the dispersion of sprayed binder liquid, Litster et al. [12] proposed the dimensionless spray flux, which is a ratio of volume flux of sprayed binder liquid to the volume flux of powder bed surface traversing the spray zone. They described that the dimensionless spray flux should be maintained at a constant value in order to keep a constant liquid dispersion state for the scale-up of high shear mixer granulators [13]. The two empirical scale-up rules based on radius and rotational speed of the impeller blade were frequently used: i.e., constant tip speed [14–18] and constant Froude number [13,19]. In all the previous studies on the scale-up of high shear mixer granulators, however, similarities of actual particle behavior in mixer granulators with different sizes have not been analyzed, resulting in the scale-up methodologies without physical relevance. Therefore, the scale-up of high shear mixer granulators has practically been conducted based on empirical approaches, i.e., experiences of expert operators and trial-and-error testing [20].

0921-8831/$ - see front matter Ó 2009 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2009.05.006

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Nomenclature c D d E Ecol,i Ecol,ave Fc FrI g H h Ip k M mp N Nc,i Np nc PA R r rI

damping coefficient (–) vessel diameter (m) depth of impeller blade (m) Young’s modulus of particle (Pa) collision energy of individual particle (J) averaged particle collision energy (J) particle contact force (N) impeller Froude number (–) gravity acceleration (m s2) height of vessel (m) height of impeller blade (m) inertia moment of particle (kg m2) constant in Eq. (10) (–) total mass of particles (kg) mass of particle (kg) rotational speed of impeller blade (s1) number of contact points regarding a single particle (–) total number of particles (–) total number of contact points between particle and impeller blade (–) normalized agitation power (J s1 kg1) radius vector from center of particle to contact point (m) displacement from centroid of impeller blade to contact point (m) radius of impeller blade (m)

In order to establish a widely applicable scale-up methodology with physical relevance, it is necessary to analyze kinematic and dynamic similarities of actual particle behavior in high shear mixer granulators with different sizes. However, these similarities have not been well investigated, because the particle motion in high shear mixer granulators is so complicated that kinematic and dynamic characteristics are difficult to analyze experimentally. Only Tardos et al. [21] experimentally evaluated dynamic similarity expressed by the shear force acting on particles in high shear mixer granulators with different sizes. They quantified the shear force using test-particles of which the yield stress was preliminarily determined, and proposed a new scale-up rule based on the constant shear force. However, in their method, the shear force which is not the same as the yield stress of the test-particles cannot be evaluated, and thus accuracy of their method is inadequate. Accordingly, numerical analysis of particle behavior in high shear mixer granulators is a promising approach. The discrete element method (DEM) [22] has been widely accepted as a useful numerical model for analysis of various powder handling processes because of its accuracy and simplicity, although the computational load is relatively high. In the DEM, position and velocity of an individual particle are described using Newton’s second law. Therefore, the DEM is a high potential approach for analyzing kinematic and dynamic characteristics of particle behavior in high shear mixer granulators. There are some reports of the discrete element simulation of particle flow pattern [23–25], particle mixing and segregation [26–28] and agitation torque [29] in high shear mixer granulators. The effects of vessel size on the particle behavior in high shear mixer granulators have not been reported in anywhere, although the previous studies have focused on particle behavior in a single vessel size. In this study, kinematic and dynamic similarities of particle behavior in high shear mixer granulators with various vessel sizes (1–16 L), which were geometrically similar, were numerically analyzed using the DEM. The particle velocity in various vessel sizes

T Td Td0 Ts Tr t V VI Vrel W

impeller torque (N m) dimensionless torque (–) extrapolated intercept value of dimensionless torque (–) torque caused by tangential contact force (N m) rolling friction torque (N m) time (s) particle velocity (m s1) intensity of particle velocity (m s1) relative particle velocity (m s1) width of impeller blade (m)

Greek letters d overlap displacement (m) lr rolling friction coefficient (m) ls sliding friction coefficient (–) m Poisson’s ratio (–) xI angular velocity of impeller blade (rad s1) xp particle angular velocity (rad s1) ^p unit vector of particle angular velocity (–) x Subscripts n normal direction t tangential direction x x-axis direction y y-axis direction

was calculated to evaluate the kinematic similarity. The particle collision energy was also numerically estimated to evaluate the dynamic similarity. Key similarity for scale-up of a high shear mixer granulator was then discussed based on the numerically investigated kinematic and dynamic characteristics of particle behavior in various vessel sizes. 2. Numerical model 2.1. Governing equations For modeling the particle motion in a high shear mixer granulator, the three-dimensional discrete element method (DEM) was used. The DEM describes the motion of each particle using Newton’s second law for individual particle, allowing for the external forces acting on the particle. The fundamental equations of translational and rotational motions of a particle can be expressed as follows [23,30]:

dV ¼ mp g þ F c dt dxp ¼ Ts þ Tr Ip dt mp

ð1Þ ð2Þ

where mp, V, t, g, Ip and xp are mass of a particle, velocity of a particle, time, gravity acceleration, inertia moment and angular velocity of a particle, respectively. Fc, Ts and Tr are contact force, torque caused by the tangential contact force and rolling friction torque acting on a particle, respectively. V, xp and position of a particle were calculated by integrated Eqs. (1) and (2) with respect to time from t to t + Dt. For estimation of contact force acting on a particle, a soft sphere model originally proposed by Cundall and Strack [22] was used. The contact forces of normal and tangential directions, which are described as Fcn and Fct, were estimated using the following equations [23,30]:

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F cn

pffiffiffi 2E pffiffiffiffiffiffi jdn j1:5 R ¼ 3ð1  m2 Þ jRj ( )0:5 3mp E pffiffiffiffiffiffiffiffiffiffiffiffiffi jRjjdn j  c pffiffiffi V rel;n 2ð1  m2 Þ

^ p are rolling friction coefficient and unit angular where lr and x velocity vector of xp, respectively.

ð3Þ

if |dt| < dt,max:

(

 1:5 ) jdt j dt jF cn j F ct ¼ ls 1  1  dt;max jdt j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!0:5 1  jdt j=dt;max V rel;t  c 6mp ls jF cn j dt;max

ð4Þ

if |dt| P dt,max:

F ct ¼ ls jF cn j

dt jdt j

ð5Þ

where E, m, R, d, c, ls and Vrel are Young’s modulus, Poisson’s ratio, radius vector from center of a particle to contact point, overlap displacement between contacting particles, damping coefficient, sliding friction coefficient and relative particle velocity between contacting particles, respectively. dt,max is determined by the following equation:

dt;max ¼ ls

2m jdn j 2ð1  mÞ

ð6Þ

Ts and Tr were estimated using the following equations, respectively:

T s ¼ R  F ct pffiffiffiffiffiffipffiffiffiffiffiffi 1:5 2E jRjjdn j ^p T r ¼ lr x 3ð1  m2 Þ

ð7Þ ð8Þ

2.2. Simulation conditions Fig. 1 shows a schematic of high shear mixer granulator simulated in this study. The mixer granulator consisted of a cylindrical vessel and flat-shaped impeller blade equipped at the bottom of vessel. The three-dimensional particle motion was simulated in the mixer granulators with four different vessel sizes. The dimensions of each mixer granulator are listed in Table 1. Every mixer granulator completely obeyed geometric similarity. Table 2 summarizes the calculation conditions used in this study. The model parameters in the particle contact model were determined based on a previous study by Stewart et al. [23]. They simulated the particle motion in a high shear mixer granulator with a flat-shaped impeller blade and reported the optimal parameters in the contact model, i.e., c, E, lr, ls and m, based on the comparison of calculation results of the particle velocity field with experimental results measured by a positron emission particle tracking (PEPT) technique. In this study, values of the optimal parameters in the contact model reported by Stewart et al. [23] were used. Mono-dispersed spherical particles with their diameter of 4.0 mm and density of 1500 kg/m3 were used as the model particles. Before the impeller blade was rotated, initial condition of the particle configuration was pre-calculated as follows: all the particles having random initial velocity were orderly arranged in the vessel and then settled down under gravity. When the particle motion reached stationary, the pre-calculation was completed and these conditions were used as the initial condition of particle configuration (Fig. 2a). The rotational speed of the impeller blade was then gradually increased until the pre-determined rotational speed was reached. After the particle motion reached steady state at the pre-determined rotational speed (Fig. 2b), numerical analysis of particle behavior was conducted.

Vessel

D

3. Results and discussion 3.1. Application of the DEM to analysis of particle behavior

H

d

W z

h y x

Impeller blade

Fig. 1. Schematic of simulated high shear mixer granulator.

The particle behavior in high shear mixer granulators can be well reflected by a torque acting on the rotational axis of impeller blade [29]. Therefore, the impeller torque is one of the most useful parameters for the control and monitoring of high shear mixer granulators [31,32]. In this study, in order to evaluate applicability of the DEM for analysis of particle behavior in high shear mixer granulators with various vessel sizes, the calculated impeller torque was compared with the experimental result reported by Knight et al. [33]. The impeller torque (T) was numerically calculated using the contact force of particle-to-impeller blade as follows [29]:

  nc X    T¼ r x;i F cy;i  r y;i F cx;i    i¼1

ð9Þ

Table 1 Dimensions of simulated mixer granulators. Vessel volume (L)

D (m)

H (m)

W (m)

h (m)

d (m)

Number of particles (–)

Particle filling level (vol%)

1.0 3.4 8.1 16

0.12 0.18 0.24 0.30

0.090 0.135 0.180 0.225

0.104 0.156 0.208 0.260

0.010 0.015 0.020 0.025

0.002 0.003 0.004 0.005

8349 28,178 66,792 130,454

50 50 50 50

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where Fcx and Fcy are contact forces between a particle and impeller blade in the direction of x- and y-axes, respectively. rx and ry are displacements from the center of impeller blade to a contact point in the direction of x- and y-axes, respectively. nc is total number of the particles contacting with the impeller blade. Table 2 Calculation conditions. Particle diameter Particle density Sliding friction coefficient Rolling friction coefficient Poisson’s ratio Young’s modulus Damping coefficient Time step

4.0  103 1500 0.50 2.5  105 0.3 2.16  106 0.3 2.0  105

(m) (kg m3) (–) (m) (–) (Pa) (–) (s)

Fig. 3 shows the calculated impeller torque as a function of rotational speed (N) of the impeller blade at various vessel sizes. T linearly increased with an increase in N at each size due to the higher impact force between particles and the impeller blade at higher N. Higher T was also shown at larger vessel sizes because of the heavier total load of particles charged into the vessel. Knight et al. [33] reported that impeller torque of the flat-shaped impeller blade experimentally obtained could be well correlated by the following dimensionless empirical equation: 0:5

T d ¼ T d0 þ kFrI

ð10Þ

where Td and FrI are dimensionless torque and impeller Froude number, respectively, which are defined as

T Mgr I r I x2I FrI ¼ g

Td ¼

ð11Þ ð12Þ

where M, rI and xI are total mass of particles, radius of impeller blade and angular velocity of impeller blade, respectively. Td0 and k in Eq. (10) are constants depending on impeller blade geometry

Impeller torque, T [Nm]

6

Vessel volume 1.0 L 3.4 L 8.1 L 16 L

4

2

0 0

5

10

15

20

Rotational speed of impeller blade, N [s−1] Fig. 3. Impeller torque as a function of rotational speed of impeller blade at various vessel sizes.

Dimensionless torque, Td [−]

2

Vessel volume 1.0 L 3.4 L 8.1 L 16 L 1

Td = 0.071+0.17FrI0.5

0

0

2

4

6

8

10

FrI0.5 [−] Fig. 2. Snap shots of particle configurations in a high shear mixer granulator: (a) initial condition and (b) steady state condition (vessel volume = 16 L and N = 6 s1). Each particle was dyed depending on its position of z-axis direction.

Fig. 4. Dimensionless torque vs. square root of impeller Froude number under various vessel sizes.

H. Nakamura et al. / Advanced Powder Technology 20 (2009) 493–501

and frictional properties of particle-to-vessel and particle-to-impeller blade, but they are independent of vessel sizes when the geometric similarity is confirmed [33]. Fig. 4 indicates relationship

497

between the calculated Td and square root of FrI under various vessel sizes. Td was well correlated using Eq. (10) regardless of the vessel sizes. In addition, values of Td0 and k obtained from the

Fig. 5. Particle velocity fields at various horizontal cross-sections in different vessel sizes under constant normalized agitation power (PA = 38 J s1 kg1).

H. Nakamura et al. / Advanced Powder Technology 20 (2009) 493–501

3.2. Kinematic and dynamic similarities of particle behavior under a constant normalized agitation power Kinematic and dynamic similarities of particle behavior in various vessel sizes were numerically analyzed under a constant normalized agitation power. The normalized agitation power (PA), which is defined as net agitation power divided by total mass of the particles, was estimated using the following equation:

PA ¼

1 M

Rt 0

xI Tdt t

ð13Þ

The normalized agitation power means the net power consumed in the agitation of particles per unit mass of the particles. The normalized agitation power is one of the most useful factors for the control of high shear mixer granulators [34,35]. Some researchers [9–11] reported that scale-up of high shear mixer granulators could be well achieved under a constant normalized agitation power. Sirois and Craig [9] reported that the normalized agitation power could be effectively used for the scale-up under a constant processing period

1.5 −1

calculation results were similar to those reported by Knight et al. [33]. Therefore, applicability of the DEM for the analysis of particle behavior in high shear mixer granulators with various vessel sizes was confirmed.

Intensity of particle velocity, V I [ms ]

498

1

Vessel volume 1.0 L 3.4 L 8.1 L 16 L

0.5

0

0

50

100

Normalized agitation power, PA [Js−1kg−1] Fig. 7. Intensity of particle velocity as a function of normalized agitation power at various vessel sizes.

regardless of the rotational speed and geometry of impeller blade. Sato et al. [11] also reported that the normalized agitation power

Fig. 6. Particle velocity fields at vertical cross-section in different vessel sizes under constant normalized agitation power (PA = 38 J s1 kg1).

H. Nakamura et al. / Advanced Powder Technology 20 (2009) 493–501

could be a very useful scale-up factor in the high shear granulation of a pharmaceutical excipient powder mixture. Therefore, in this study, we focused on the scale-up rule of a constant normalized agi-

499

tation power and numerically analyzed kinematic and dynamic similarities of particle behavior in high shear mixer granulators with various vessel sizes under a constant PA. In this study, PA

Fig. 8. Particle collision energy (Ecol,i) fields at various horizontal cross-sections in different vessel sizes under constant normalized agitation power (PA = 38 J s1 kg1).

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was estimated using calculation result of the impeller torque for 15 s. Fig. 5 shows calculation results of particle velocity fields at various horizontal cross-sections in different vessel sizes of 1.0 L and 16 L. These results express relative positions to the impeller blade and obtained by local and time averaging. Each impeller rotational speed was pre-determined to keep PA as a constant value. The toroidal motion of particles along rotation of the impeller blade was observed at each size. Fig. 6 indicates particle velocity fields at a vertical cross-section in the different vessel sizes. The vertical cross-section was set at the upper part of the impeller blade. It could be observed that in each vessel size particles formed a vortex flow at the vertical cross-section: i.e., the circulating flow pattern consisting of upward flow along the vessel wall and downward flow near the center of vessel. It should be noted that the particle velocity in larger size became higher than in smaller one. This means that the particle velocity significantly changed with the vessel size even though PA was constant. In order to compare and evaluate intensity of the kinematic characteristic of whole particles in different vessel sizes, averaged intensity of particle velocity (VI) was calculated. VI was determined as time-averaged value of number averaged intensity of particle velocity as follows:

1 VI ¼ t

Z

t

0

! NP 1 X jV i j dt NP i¼1

ð14Þ

where Np is total number of the particles. Fig. 7 shows VI as a function of PA under various vessel sizes. VI increased with an increase in PA at each vessel size, and higher VI was shown at larger vessel sizes. This result indicates that under a constant PA similarity of the particle velocity in different vessel sizes was not confirmed. Subsequently, the particle collision energy was estimated for evaluation of dynamic characteristics of the particle behavior. The collision energy of individual particle (Ecol,i) was defined as

Ecol;i ¼

N c;i X 2 1  mp V rel;j  2 j¼1

ð15Þ

Averaged particle collision energy, Ecol, ave [µJ]

where Nc,i is the number of particle-to-particle contact points regarding a single particle. Fig. 8 shows the particle collision energy fields in the different vessel sizes under a constant PA. These figures express spatial distributions of time averaged Ecol,i. The intensity of particle collision energy fields in the smaller was found to be almost the same as those in the larger. Fig. 9 indicates the averaged particle

0.25

0.2

0.15

Vessel volume 1.0 L 3.4 L 8.1 L 16 L

collision energy (Ecol,ave) as a function of PA at various vessel sizes. Ecol,ave was calculated using the following equation:

Ecol;ave ¼

1 t

Z

t

0

! NP 1 X Ecol;i dt NP i¼1

ð16Þ

As can be seen in Fig. 9, Ecol,ave and PA showed an excellent linear correlation regardless of the vessel size. This result reveals that under a constant PA similarity of particle collision state was confirmed in the different vessel sizes. So far, some researchers analyzed the granulation mechanism in a high shear granulation process at the collision of individual particle and pointed out that mechanisms of the particle growth strongly depend on the particle collision energy [35–38], and thus the particle collision energy is a critical parameter which determine the granule properties. Therefore, these results mean that key similarity for scale-up of high shear mixer granulator is the dynamic similarity. In addition, it was suggested that dynamic similarity of the particle behavior can be achieved by using a constant rule of normalized agitation power. Although physical properties of the model particles (i.e., dry, visco-elastic and relatively large particles) used in this study were not same as those of particles frequently used in high-shear mixer granulators (i.e., wet, visco-plastic and relatively small particles), the calculated particle collision energy was preserved under a scale-up rule of constant agitation power, of which effectiveness were experimentally confirmed [9–11]. This implies that the calculated particle collision energy can be correlated with the physical properties of granules experimentally obtained, although the physical meaning of this correlation should be clarified in a further investigation. We expect that it is possible to set up the scale-up guidelines using the particle collision energy calculated from the DEM, even if the physical properties of model particles are not completely same as those of experimental particles.

4. Conclusions Kinematic and dynamic similarities of particle behavior in high shear mixer granulators with various vessel sizes, which were geometrically similar, were numerically analyzed using the discrete element method (DEM). The calculation results of impeller torque were well correlated with the dimensionless empirical equation proposed by Knight et al. [33], and their values of the empirical constants were similar to our results. Thus, applicability of the DEM for the analysis of particle behavior in high shear mixer granulators with various vessel sizes was confirmed. Kinematic and dynamic similarities of particle behavior in various vessel sizes under a constant normalized agitation power were then analyzed. It was found that in different vessel sizes dynamic similarity expressed by the particle collision energy was confirmed under a constant normalized agitation power, while kinematic similarity expressed by the particle velocity was not confirmed. Therefore, it was concluded that the dynamic similarity should be maintained for the successful scale-up of geometrically similar high shear mixer granulators.

0.1

References 0.05

0 0

50

100

Normalized agitation power, PA [Js−1kg−1]

Fig. 9. Averaged particle collision energy as a function of normalized agitation power at various vessel sizes.

[1] G.K. Reynolds, P.K. Le, A.M. Nilpawar, High shear granulation, in: A.D. Salman, M.J. Hounslow, J.P.K. Sevile (Eds.), Granulation, Elsevier, Oxford, 2007, pp. 3– 19. [2] H. Leuenberger, Scale-up of granulation processes with reference to process monitoring, Acta Pharm. Technol. 29 (1983) 274–280. [3] M. Landin, P. York, M.J. Cliff, R.C. Rowe, A.J. Wigmore, Scale-up of pharmaceutical granulation in fixed bowl mixer-granulators, Int. J. Pharm. 133 (1996) 127–131. [4] A. Faure, I.M. Grimsey, R.C. Rowe, P. York, M.J. Cliff, A methodology for the optimization of wet granulation in a model planetary mixer, Pharm. Dev. Technol. 3 (1998) 413–422.

H. Nakamura et al. / Advanced Powder Technology 20 (2009) 493–501 [5] A. Faure, I.M. Grimsey, R.C. Rowe, P. York, M.J. Cliff, Applicability of scale-up methodology for wet granulation processes in Collette Gral high shear mixergranulators, Eur. J. Pharm. Sci. 8 (1999) 85–93. [6] A. Faure, P. York, R.C. Rowe, Process control and scale-up of pharmaceutical wet granulation processes: a review, Eur. J. Pharm. Biopharm. 52 (2001) 269– 277. [7] T. Schaefer, B. Taagegaard, L.J. Thomsen, H.G. Kristensen, Melt pelletization in a high shear mixer V. Effects of apparatus variables, Eur. J. Pharm. Sci. 1 (1993) 133–141. [8] J.S. Ramaker, M.A. Jelgersma, P. Vonk, N.W.F. Kossen, Scale-down of a high shear pelletisation process: flow profile and growth kinetics, Int. J. Pharm. 166 (1998) 89–97. [9] P.J. Sirois, G.D. Craig, Scale-up of a high-shear granulation process using a normalized impeller work parameter, Pharm. Dev. Technol. 5 (2000) 365–374. [10] M. Bardin, P.C. Knight, J.P.K. Seville, On control of particle size distribution in granulation using high-shear mixers, Powder Technol. 140 (2004) 169–175. [11] Y. Sato, T. Okamoto, S. Watano, Scale-up of high shear granulation based on agitation power, Chem. Pharm. Bull. 53 (2005) 1547–1550. [12] J.D. Litster, K.P. Hapgood, J.N. Michaels, A. Sims, M. Roberts, S.K. Kameneni, T. Hsu, Liquid distribution in wet granulation: dimensionless spray flux, Powder Technol. 114 (2001) 32–39. [13] J.D. Litster, K.P. Hapgood, J.N. Michaels, A. Sims, M. Roberts, S.K. Kameneni, Scale-up of mixer granulators for effective liquid distribution, Powder Technol. 124 (2002) 272–280. [14] G.S. Rekhi, R.B. Caricofe, D.M. Parikh, L.L. Augsburger, A new approach to scaleup of a high-shear granulation process, Pharm. Technol. 20 (1996) 1–10. [15] T.K. Bock, U. Kraas, Experience with the Diosna mini-granulator and assessment of process scalability, Eur. J. Pharm. Biopharm. 52 (2001) 297–303. [16] D. Ameye, E. Keleb, C. Vervaet, J.P. Remon, E. Adams, D.L. Massart, Scaling-up of a lactose wet granulation process in Mi-Pro high shear mixers, Eur. J. Pharm. Sci. 17 (2002) 247–251. [17] S. Watano, T. Okamoto, Y. Sato, Y. Osako, Scale-up of high shear granulation based on the internal stress measurement, Chem. Pharm. Bull. 53 (2005) 351– 354. [18] N. Rahmanian, M. Ghadiri, Y. Ding, Effect of scale of operation on granule strength in high shear granulators, Chem. Eng. Sci. 63 (2008) 915–923. [19] G.J.B. Horsthuis, J.A.H. van Laarhoven, R.C.B.M. van Rooij, H. Vromans, Studies on upscaling parameters of the Gral high shear granulation process, Int. J. Pharm. 92 (1993) 143–150. [20] J.N. Michaels, Toward rational design of powder processes, Powder Technol. 138 (2003) 1–6. [21] G.I. Tardos, K.P. Hapgood, O.O. Ipadeola, J.N. Michaels, Stress measurements in high-shear granulators using calibrated ‘‘test” particles: application to scaleup, Powder Technol. 140 (2004) 217–227.

501

[22] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Geotechnique 29 (1979) 47–65. [23] R.L. Stewart, J. Bridgwater, Y.C. Zhou, A.B. Yu, Simulated and measured flow of granules in a bladed mixer – a detailed comparison, Chem. Eng. Sci. 56 (2001) 5457–5471. [24] H.P. Kuo, P.C. Knight, D.J. Parker, M.J. Adams, J.P.K. Seville, Discrete element simulations of a high-shear mixer, Adv. Powder Technol. 15 (2004) 297–309. [25] Y.C. Zhou, A.B. Yu, R.L. Stewart, J. Bridgwater, Microdynamic analysis of the particle flow in a cylindrical bladed mixer, Chem. Eng. Sci. 59 (2004) 1343– 1364. [26] K. Terashita, T. Nishimura, S. Natsuyama, Optimization of operating conditions in a high-shear mixer using DEM model: determination of optimal fill level, Chem. Pharm. Bull. 50 (2002) 1550–1557. [27] Y.C. Zhou, A.B. Yu, J. Bridgwater, Segregation of binary mixture of particles in a bladed mixer, J. Chem. Technol. Biotechnol. 78 (2003) 187–193. [28] M. Sinnott, P. Cleaery, 3D DEM simulations of a high shear mixer, in: Proceedings of Third International Conference on CFD in the Minerals and Process Industries, Melbourne, 2003, pp. 217–222. [29] Y. Sato, H. Nakamura, S. Watano, Numerical analysis of agitation torque and particle motion in a high shear mixer, Powder Technol. 186 (2008) 130–136. [30] Y.C. Zhou, B.D. Wright, R.Y. Yang, B.H. Xu, A.B. Yu, Rolling friction in the dynamic simulation of sandpipe formation, Physica A 269 (1999) 536–553. [31] N.O. Lindberg, L. Leander, B. Reenstierna, Instrumentation of a Kenwood major domestic-type mixer for studies of granulation, Drug Dev. Ind. Pharm. 8 (1982) 775–782. [32] S.R. Ghanta, R. Srinivas, C.T. Rhodes, Use of mixer-torque measurements as an aid to optimizing wet granulation process, Drug Dev. Ind. Pharm. 10 (1984) 305–311. [33] P.C. Knight, J.P.K. Seville, A.B. Wellm, T. Instone, Prediction of impeller torque in high shear powder mixers, Chem. Eng. Sci. 56 (2001) 4457–4471. [34] I.N. Bjorn, A. Jansson, M. Karlsson, S. Folestad, A. Rasmuson, Empirical to mechanistic modeling in high shear granulation, Chem. Eng. Sci. 60 (2005) 3795–3803. [35] P. Mort, Scale-up of high-shear binder-agglomeration processes, in: D. Salman, M.J. Hounslow, J.P.K. Sevile (Eds.), Granulation, Elsevier, Oxford, 2007, pp. 853–896. [36] B.J. Ennis, G. Tardos, R. Pfeffer, A microlevel-based characterization of granulation phenomena, Powder Technol. 65 (1991) 257–272. [37] L.X. Liu, J.D. Litster, S.M. Iveson, B.J. Ennis, Coalescence of deformable granules in wet granulation processes, AIChE J. 46 (2000) 529–539. [38] S.M. Iveson, P.A.L. Wauters, S. Forrest, J.D. Litster, G.M.H. Meesters, B. Scarlett, Growth regime map for liquid-bound granules: further development and experimental validation, Powder Technol. 117 (2001) 83–97.