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Constitutive model to predict flow of cohesive powders in bench scale hoppers
ARTICLE IN PRESS Chemical Engineering Science 65 (2010) 3341–3351
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Constitutive model to predict flow of cohesive powders in bench scale hoppers AbdulMobeen N. Faqih a, Bodhisattwa Chaudhuri b, Amit Mehrotra c, M. Silvina Tomassone d, Fernando Muzzio d, a
Wyeth Pharmaceuticals, Pearl River, NY, USA Department Pharmaceutical Sciences, University of Connecticut, Storrs, CT, USA c Glaxo Smith Kline, Raleigh, NC, USA d Department of Chemical and Biochemical Engineering, Rutgers University, USA b
a r t i c l e in fo
abstract
Article history: Received 23 September 2009 Received in revised form 25 January 2010 Accepted 15 February 2010 Available online 6 March 2010
This communication empirically correlates flow in two systems; an instrumented rotating drum (GDR) and a set of bench scale hoppers. A flow index obtained from measurements in the GDR is directly correlated to the flow through hoppers, providing a predictive method for hopper design and a convenient experimental test for screening materials and determining their suitability for specific hopper systems. Simulations were performed to understand the dynamics of flow in hoppers by using the same flow parameters in hoppers and rotating cylinders. Simulations showed that as cohesion increased it becomes harder for the particles to flow through the hoppers, in good agreement with the experiments. The effect of hopper angle also yields similar findings to experiments for Avicel, K ¼60, where the powder does not flow through the 451 hopper but flows well in a 751 hopper. Simulations were also used to calculate the normal forces on the walls of the hopper and the wall pressure distributions in both hoppers. As depth increases, the wall pressure increases for all cases. Finally, the simulations also helped understand the different flow behaviors (funnel and mass flow) that take place in a hopper. The simulated dynamics of flow in the rotating drum and in the hopper correlate very closely to experiments, indicating that the model cohesion parameters are, as desirable, materialspecific but independent of geometry. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Granular materials Pharmaceuticals Powder technology Mathematical modeling Discrete element modeling Powder flow
1. Introduction Granular materials exhibit a wealth of interesting phenomena, including heaping under vibration, segregation, convection, fluidization, and density waves in pipes or hoppers. They are also very important from technological and industrial points of view. Many examples of important granular flows can be found in both industry and nature. Hoppers, chutes, and conveyor belts are used when transporting particulate materials such as food stuffs, pharmaceuticals, and coal. Other industrial applications include packing of granular materials, particulate segregation and mixing, and particulate drying. In nature, examples of granular flows include snow and mud avalanches, river sedimentation, dune formation, planetary ring dynamics, soil liquefaction, and ice flow mechanics. Clearly, characterization of the behavior of granular materials is essential to scientific community with significance to the pharmaceutical industry for downstream processing of solid dosage forms. However, most of the research has been focused on
Corresponding author.
E-mail address: [email protected] (F. Muzzio). 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.02.028
cohesionless materials, which tend to segregate when used with a poly-dispersed particle size distribution (Savage et al., 1983; Savage, 1984; Savage and Hutter, 1989), but that otherwise, provide relatively simple flows. There has been some research work in the last few years on the much more complex field of cohesive powders (Adams and Perchard, 1985; Lian et al., 1993; Bocquet et al., 1998; McCarthy et al., 2001) however, the flow of cohesive powders is still poorly understood. Knowledge of powder flow properties is very important when developing manufacturing processes and handling procedures such as flow from hoppers and silos, transportation, mixing, compression and packaging (Knowlton et al., 1994; Peleg, 1978). Powder flow characteristics are commonly investigated under gravity loading conditions (Carstensen, 1974). The compressibility of a powder is a commonly used indicator of flowability and is often expressed using the Hausner Ratio, which is the ratio between the tapped and the loose-packed bulk densities of the powder (Hausner, 1967). Compressibility is also one of the tests proposed by Carr (1965) for the assessment of powder properties. Another commonly used flow indicator is the time it takes for powder to flow out of a funnel with a standard orifice size (Staniforth, 2002). Such measurements have
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demonstrated the dependence of powder flowability on particles shape and size distribution (Baxter et al., 2000), instantaneous degree of consolidation, and relative humidity (Coelho and Harnby, 1978; Stanford and DellaCorte, 2006). A key situation where flow is critical is the filling and emptying of hoppers. Despite their common use and deceptively simple design, one of the major industrial powder problems is obtaining reliable and consistent flow out of hoppers. These problems are usually associated with the flow pattern inside a hopper. An intriguing property of granular flow from a hopper is that in certain situations the flow rate is largely dependent on the diameter of the orifice, and weakly on the particle size and head of the material. Mass flow is an ideal flow pattern where the bulk is in motion and moving downwards towards the opening. The other common case is called funnel flow, where the powder starts moving out through a central ‘‘funnel’’ that forms within the material, leading to the powder collapsing and moving through the funnel. Most flow problems are caused by a funnel flow pattern and can be cured by altering the pattern to mass flow (Purutyan et al., 1998; Savage, 1965). The worst case-scenario is no flow, which can occur when the cohesive powder forms an arch across the opening, which provides sufficient strength to support itself. In order to prevent these problems, measurement of powder flow properties is necessary for design of mass flow hoppers. In the last 45 years, significant improvements have been proposed to the mathematical models that quantify the flow of granular material through hopper. In 1964 Jenike utilized the radial solutions of Sokolovsky to quantify some aspects of granular flow, but as the solution lacked inertial terms, the granular flow discharge rate could not be predicted. Savage (1965) constructed the earliest model that accounted for inertial terms by setting the internal friction and the friction with the wall to zero. Later in 1967, Savage added wall friction but maintained gravity in the radial direction (Tomas and Schubert, 1979). Sullivan, Davidson and Nedderman et al. (1982), Nedderman (1992) applied these results to the Hour-Glass theory. In 1992, Thorpe (1992) showed that the earlier models assumed constant bulk density, which would overestimate the discharge rates. As the material flows through the hopper, the stress changes from zero at the top surface, reaches a maximum and falls to zero on the free-fall arch. Thus in the upper part of the hopper, the material is compressed and the interstitial air must be expelled, whereas in the lower part, the material dilates as air is being drawn in. As a result, elevated pressures occur in the upper part and sub-atmospheric pressures occur in the lower part. The most appropriate model that describes this change in stress behavior is the critical state theory (CST) (Bak et al., 1987), which assumes a logarithmic relationship between pressure and density (so that density is not defined when pressure equals to zero). However the CST, which is derived from soil mechanics, cannot be necessarily applied to extreme cases of failure for hopper flow. In spite of the existence of predictive models and techniques to improve flow of free-flowing materials through hoppers, there lacks a general understanding of the flow behavior of powders as a function of cohesion. Cohesion clearly plays an important role in affecting flow properties and is a key target of flow property characterization. During the last few decades a variety of methods for assessment of cohesive powder flow properties have been developed using some type of a ‘‘shear cell’’ where the force required to initiate (or maintain) movement in a standard geometry is measured. The area was pioneered by Jenike (1964) who also developed the theoretical framework that became the field standard. In conjunction with the measured property data, he applied two-dimensional stress analysis in developing a
mathematical methodology for determining the minimum hopper angle and hopper opening size for flow from conical and wedge shaped hoppers (Jenike, 1964). Besides the Jenike tester, some other commonly used shear testers include the ring shear testers (Schulze, 1994), the Johanson (1992, 1993) indicizers, uniaxial, biaxial, and triaxial testers (Maltby and Enstad, 1993; Maltby, 1993), and Jenike and Johanson’s quality control tester. While these methods have been useful for many purposes such as building roads and bridges, they have some shortcomings regarding processing of cohesive powders. The most salient drawback is that flow characteristics of powders are highly dependent on their densification (consolidation) states, i.e. powders can be more or less expanded or contracted when stressed, thus leading to a large variety of inter-particle forces and flow behavior (a phenomenon commonly referred to as ‘‘jamming’’). For small scale systems, powders often flow in a fully dilated state. Shear cells can approach the dilated state only asymptotically, and are affected by considerable experimental error for cohesive materials exhibiting a non-linear relationship between consolidation and flow. The complex constitutive behavior makes accurate powder flow measurements difficult. Various modeling scales are commonly used to simulate granular materials. Understanding and modeling the dynamic behavior of particulate systems has been a major research focus worldwide for many years. Models of granular flows can be broadly divided into three categories: continuum, kinetic theory and discrete. When continuum approach fails or when no appropriate constitutive relations exist, the discrete element modeling (DEM) has proven tremendously useful. Molecular dynamics models are broadly classified by the contact model and integration method, as either hardsphere, in which collisions are instantaneous and binary, and softsphere, in which collisions can be lasting and multiple. Here, the particles are permitted to suffer minute deformations, and these deformations are used to compute restoring elastic, plastic and frictional forces. The discrete element method (DEM), originally developed by Cundall (1971), Cundall and Strack (1979), has been successfully used to simulate chute flow, heap formation (Luding, 1997), hopper discharge (Ristow and Herrmann, 1994; Thompson and Grest, 1991) and flows in rotating drums (Rosato et al., 1986; Khakhar et al., 1997; Wightman et al., 1998). In this article we investigate the flow and pressure dependence as a function of cohesion in two different geometries (rotating cylinder and hopper) using discrete element methods (DEM) and experiments. We test the parameters developed for simulating cohesive flow in a rotating drum by using the same parameters to simulate flow in hoppers of varying angle. DEM simulations are used to describe the behavior of granular materials after the initial filling stage (static state) and during the discharge (dynamic state) as a function of cohesion. These DEM results are compared with experimental studies for bench scale hoppers. In essence, the goal is to determine whether the parameters introduced from the rotating drum and applied to simulation studies in hoppers are independent of the geometry of the system. The article is organized as follows: Section 2 lists the materials used in the experimental study and describes the experimental setup for the gravitational displacement rheometer (GDR) and bench scale hoppers. In Section 3, we report a DEM based constitutive model to compare the dynamics of the flow in the two geometries considered. Section 4 describes the experimental and simulation method for correlations between geometries and between experiment and simulations, Section 5 describes the experimental and computational results for flow dynamics in a rotating cylinder, and flow through hoppers of varying angle. Finally, in Section 6 we present our conclusions and outline directions for future work.
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2. Experimental materials Commercially available, well-characterized powders were used to create a family of ‘‘standard’’ systems of varying cohesion. Sugar and other typical pharmaceutical excipients were used, both ‘‘pure’’ and in mixtures. Particle sizes for these materials are: Fast-Flo lactose (the least cohesive), 100 mm; Avicel 102 (moderately free flowing), 90 mm; Avicel 101 (moderately cohesive), 60 mm; Regular lactose (the most cohesive), 50 mm. A matrix of binary system consisting of varying composition of Fast Flo lactose and Avicel 102; Avicel 102 and Regular lactose, and finally, Avicel 101 and Regular lactose is made to accommodate for varying cohesion between the pure components. The matrix of blends was typically used for experimental studies to understand the correlation of cohesive powders in varying geometry. These materials are some of the most common pharmaceutical excipients; in the interest of brevity their SEM images and other details are not included in this paper but can be found in the ‘‘Handbook of Pharmaceutical excipients’’ (Raymond et al., 2005).
3. Simulation details In these studies we use DEM to simulate the dynamic behavior of cohesive and non-cohesive powders in a rotating drum. We consider the granular material as a collection of frictional inelastic spherical particles. Each particle may interact with its neighbors or with the boundary only at contact points through normal and tangential forces. The forces and torques acting on each of the particles are calculated in the following way: X ð1Þ F i ¼ mi g þ F N þ F T þ F cohes X
Ti ¼ ri F T
ð2Þ
The force on each particle is given by the sum of gravitational, interparticle (normal and tangential FN and FT) and cohesive forces as indicated in Eq. (1). The nature of the cohesive force (Fcohes) is a combination of weight of the particles multiplied by a constant and is explained later. The corresponding torque on each particle is the sum of the torque of the tangential forces (FT) arising from inter-particle contacts (ri) as illustrated in Eq. (2). The normal forces are calculated with the ‘‘latching spring model’’, developed by Walton and Braun (1986), Walton (1992, 1993), which allows colliding particles to overlap, and the corresponding interaction force (normal force) is a function of the relative overlap. Each particle may interact with its neighbors or with the boundary only at contact points through normal and tangential forces. The normal forces between pairs of particles in contact are defined using a spring with constants K1 and K2 for compression and recovery: FN ¼ K1m1 (for compression), and FN ¼K2 (m1 m0) (for recovery). These spring constants K1 and K2 are chosen to be large enough to ensure that the overlaps m1 and m0 remain small compared to the particles sizes. The degree of inelasticity of collisions is incorporated in this model by including a coefficient of restitution e¼(K1/K2)1/2 (0 oeo1, where e¼1 implies perfectly elastic collision with no energy dissipation, e¼0 implies completely inelastic collision). Tangential forces (FT) in inter-particle or particle–wall collision are calculated employing Walton’s incrementally slipping model. After contact occurs, tangential forces build up, causing displacement in the tangential plane of contact. The initial tangential stiffness is considered a fraction of the normal stiffness. In all cases, the static frictional limit was considered to follow Coulomb’s law, i.e., if FT r mSFN (mS is the static friction coefficient and FN is the total normal force, equal to the cohesive force plus the static weight of particles transmitted through the bed), no
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relative motion between particles occur, while if the particles are in relative motion with respect to one another, FT ¼ mD FN, where mD is the dynamic friction coefficient. This model takes into account the elastic deformations that can occur in the tangential directions of the contacts. The tangential force T is evaluated considering an effective tangential stiffness kT associated with a linear spring. It is incremented at each time step as Tt þ 1 ¼ Tt þ kT Ds, where Ds is the relative tangential displacement between two time steps (Shinbrot et al., 1999). The described model has been successful to obtain the first three-dimensional computations of realistic blender geometries, where it confirmed important experimental observations (Shinbrot et al., 1999; Sudah et al., 2005). To incorporate granular cohesion in the model, a cohesive force between particles is simulated, in our case, we introduce a simple square-well potential. Two parameters are needed to define the square well: the well width ri (equal to the radius of the particle) and the well depth K. The cohesive interactions arise when the separation distance of two particles is less or equal to twice the width of the square well (2ri). In dimensionless terms, in order to compare simulations considering different numbers of particles, the magnitude of the force was represented in terms of the parameter K¼Fcohes/mg, where K is called the Bond number and is a measure of cohesiveness that is independent of particle size, Fcohes is the cohesive force between particles, and mg is the weight of the particles. Notice that this constant force may represent short range effects such as electrostatic or van der Waals forces. In this model, the cohesive force (Fcohes) between two particles or between a particle and the wall is unambiguously defined in terms of K. Four friction coefficients are also defined: particle–particle static and dynamic coefficients, and particle–wall static and dynamic coefficients. Interestingly, all four friction coefficients turned out to be important. To emulate different levels of cohesion, the bond number K, the coefficients of static and dynamic friction between particles (mSP and mDP) and the coefficients of static and dynamic friction between particle and wall (mSW and mDW) are varied. The major computational tasks of DEM in each time step are as follows: (i) add/delete contact between particles thus updating neighbor lists, (ii) compute contact forces from contact properties, (iii) sum all forces and torques on particles and update position and (v) determine the trajectory of the particle by integrating Newton’s laws of motion (second order scalar equations in three dimensions). A central difference scheme, Verlet’s Leap Frog method is used here. A previously used DEM code (Wightman et al., 1998; Faqih et al., 2006; Alexander et al., 2006; Moakher et al., 2000) was expanded here to include cohesive and frictional forces to characterize the behavior of cohesive granular systems. The DEM code was written in C language and was run in 32 and 64 bit Linux clusters.
4. Experimental and simulation methodology To examine flowability through hoppers, the aforementioned powders are transferred into a set of hoppers with five different angles (351, 451, 551, 651, and 751) bored out of solid Plexiglas cylinders (Fig. 1). Removable opening sections were built in order to be able to adjust the diameter of the discharge of the steepest hopper (0.5–1.0 in). The behavior of each material in each hopper was classified according to four main behaviors: funnel flow, mass flow, intermittent flow (aided by steady vibration), and no flow. The gravitational displacement rheometer (GDR) has been used to quantify flow as a function of cohesion. The apparatus
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Table 1 Comparison of simulation parameters to pharmaceutical powders. K
lsp
ldp
lsw
ldw
Pharm. Mat.
0 45 60 75
0.8 0.8 0.8 0.8
0.1 0.1 0.6 0.6
0.5 0.5 0.8 0.8
0.5 0.5 0.8 0.8
Glass beads Fast-Flo Avicel Regular lactose
K: cohesion number; mSP: static friction between particles; mDP: dynamic friction between particles; msw: static friction between particle and wall; mDw: static friction between particles.
Fig. 1. A picture of the bench scale hopper setup with varying angles ranging from 351 to 751.
employs a long cylinder to minimize wall effects and uses a load cell to measure changes in the center of gravity that are correlated to specific powder flow behavior. Detailed analysis of the functionality of the equipment is illustrated in an earlier article (Faqih et al., 2006) and will not be discussed in this paper. The output of the GDR is a quantifiable term known as the flow index; where higher the flow index, the more cohesive the powder and the harder it is for it to flow. The idea is based on the methodology expressed in an earlier article on avalanching behavior of powders (Alexander et al., 2006). In order to understand the nature of the forces controlling powder flow behavior, a computational model was generated to determine the relationship between inter-particle cohesive strength and the GDR measurements. Our model system consisted of 20,000 particles of 2 mm diameter in a cylindrical vessel with a 9 cm diameter, 1 cm length, and frictionless side walls. The magnitude of the cohesive bond number K described earlier is chosen in the same order of magnitude as the ones measured with atomic force microscopy by Hang Duong et al. (2004) and also in the numerical model of Baxter et al. (2000). The first part of the result section will capture a comparison of experiments and simulations in a rotating cylinder. In order to demonstrate the relevance of the flow index obtained from the GDR system, experiments were performed to quantify the flow behavior of the materials both pure and mixtures discharged from the GDR were run through a series of hoppers of varying angle and orifice diameter. The behavior of each material in each hopper was classified according to four main categories: funnel flow, mass flow, intermittent flow, and no flow. A factorial design was carried out for this process, where the pure materials were run first, followed by 50:50 mixtures of the three mixtures and eventually the 75:25 and 25:75 intermediate points to complete the characterization. The parameters for cohesion used in our simulations and the materials used for experiments provide a qualitative tool to compare the dynamics of flow in hoppers to other geometries. In order to avoid redundancy and also capture the overall effect of different flows (funnel and mass) we simulated flow in two different hoppers: 451 and 751. Simulation parameters, selected to represent the same specific materials examined for the GDR, are presented in Table 1. Once again, the model considered 20,000 particles, 2 mm in diameter. Both hoppers had the same maximum diameter (7 cm). The lengths of the hoppers were 9.6 and 26.8 cm for hoppers of 451 and 751 angles, respectively.
Finally, the qualitative results obtained from direct correlation of bond number to cohesion will be quantified using simulations to understand the radial velocity profiles at the interface of the cylindrical and conical section of the two hoppers and the variations in mean wall pressure both for the static (end filling) and dynamic (discharge) conditions throughout the hopper bed as a function of cohesion. Wall pressures are averaged on a wall segment of height dz. In this study, dz is equal to 2d (where d is the diameter of the particle). For each evaluation, the segment boundary is moved by half particle in the z-direction. In the case of a given wall, the height dz corresponds to a trapezoid-shaped wall segment. The first step involved in calculating the mean wall pressure consists of summing up the normal forces on the wall. The mean wall pressure is defined as the total force acting perpendicular to a wall segment divided by the area of the wall segment. Since the mean wall pressure distribution is calculated from the normal component of the particle/wall contact forces, it shows fluctuations along the wall. It is important to note that the fluctuation is strongly affected by the averaging method. In order to determine the wall pressure distribution during the outflow, the pressures are averaged both on a wall segment of height dz and over 1000 time steps. In summary, the result section will demonstrate flow comparison between experiments and simulation in a cylinder. Experimental work will correlate flow index obtained using the GDR to flow on bench scale hoppers. The parameters of bond number defined using a rotating cylinder will be used to qualitatively compare experiments to simulation in a hopper and finally, the simulations will be used to quantify particle–particle and particle–wall interaction in a hopper.
5. Results and discussion In this section we characterize the effect of cohesion on flow in a rotating cylinder both experimentally and through simulations and compare the two by visualizing the dynamics of flow. 5.1. Flow comparison between experiments and simulation in a rotating cylinder The constant K also called the bond number was varied between 0 and 90; for K o30, no avalanches were observed, and for K 490, avalanches were of magnitude comparable to the size of the system and therefore examination of larger values was unwarranted. The model displayed discrete, well-defined avalanches for the moderate and higher cohesion scenario (K value greater than 45). Computational parameters were fine-tuned to accurately simulate flow of the four pure materials. The following methods were used to validate the code: (a) direct visual comparison of the shape of the flow region, including local angles of repose, (b) size of avalanches, and (c) degree of dilation of the material. Subsequently, sets of cohesion and friction coefficients were identified that accurately matched flow behavior of Fast-Flo
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lactose, Avicel 102, Avicel 101, and micronized lactose. Typical results for Fast-Flo lactose, Avicel 101 and their corresponding matching simulation snapshots are displayed in Fig. 2. A table providing the results for the simulation parameters that correlate to the four pharmaceutical materials of interest is given in Table 1.
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5.2. Correlating the GDR methodology to flow in bench scale hoppers As the flow index (i.e., the blend cohesion) increased, flow through hoppers became increasingly difficult. A detailed version of these results are presented in Table 2, which displays clear
Fig. 2. (a) Shows the time sequence of axial snapshots of Fast-Flo lactose in a rotating drum, (b) shows the simulation snapshots for K¼ 45, (c) shows the time sequence for Avicel-101 and (d) shows the snapshots from simulation for K¼60.
Table 2 (a) A DOE of experiments are examined for flow in hoppers of varying angle and (b) the table when arranged in increasing order of flow index (higher cohesion) gives an excellent picture of powders flowability through the hoppers. Pharmaceutical powders
Flow index
35 deg 0.5 in
45 deg 0.5 in
55 deg 0.5 in
65 deg 0.5 in
75 deg 0.5 in
75 deg 0.75 in
75 deg 1.0 in
(a) FF lactose Avicel 102 Avicel 101 Regular lactose FF-A102 (50-50) A102-Reg (50-50) A101-Reg (50-50) FF-A102 (75-25) A102-Reg (75-25) A101-Reg (75-25) FF-A102 (25-75) A102-Reg (25-75) A101-Reg (25-75)
27.8 38 44.4 48.2 35.2 46.1 47.7 32.6 44.5 46.9 35.8 47.9 48.1
FF NF NF NF NF NF NF NF NF NF NF NF NF
FF NF NF NF NF NF NF IF NF NF NF NF NF
FF NF NF NF IF NF NF FF NF NF NF NF NF
MF IF NF NF MF NF NF MF NF NF IF NF NF
MF IF IF NF MF NF NF MF IF NF IF NF NF
MF IF IF NF MF IF No flow MF IF IF MF NF NF
MF MF MF NF MF MF MF MF MF MF MF IF NF
(b) FF lactose FF-A102 (75-25) FF-A102 (50-50) FF-A102 (25-75) Avicel 102 Avicel 101 A102-Reg (75-25) A102-Reg (50-50) A101-Reg (75-25) A101-Reg (50-50) A102-Reg (25-75) A101-Reg (25-75) Regular lactose
27.8 32.6 35.2 35.8 38 44.4 44.5 46.1 46.9 47.7 47.9 48.1 48.2
FF NF NF NF NF NF NF NF NF NF NF NF NF
FF IF NF NF NF NF NF NF NF NF NF NF NF
FF FF IF NF NF NF NF NF NF NF NF NF NF
MF MF MF IF IF NF NF NF NF NF NF NF NF
MF MF MF IF IF IF IF NF NF NF NF NF NF
MF MF MF MF IF IF IF IF IF NF NF NF NF
MF MF MF MF MF MF MF MF MF MF IF NF NF
In this table; FF: funnel Flo; IF: intermittent flow; MF: mass flow; NF: no flow.
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evidence that the flow index predicts whether a given material will or will not flow out of a given hopper. This correlation indicates that the onset of flow in a hopper, which is controlled by the powder’s ‘‘unconfined bridging length’’, is a manifestation of the same cohesion-generated internal ‘‘flow length scale’’ that also controls the size of an avalanche. These results provide strong motivation to further investigate the relationship between avalanches in the GDR and hopper flow behavior.
5.3. Flow comparison between experiments and simulation Fig. 3 illustrates the flow of non-cohesive glass beads through the hoppers in both the experiment and simulation depicted next to each other for qualitative comparison. The snapshots of both the experiments and simulation are taken at t¼0 s (Fig. 3a) and 0.5 s (Fig. 3b). We observe that the free flow powder is easily discharged in both the hoppers (Fig. 3b). In the simulation the particles are color coded based on their absolute velocities. Blue
particles signify particles with minimum (zero) velocity and the particle with maximum velocities are colored in red. At t¼ 0, the static hopper bed has all particles in rest. As the flow initiates we observe unabated discharge of yellow glass beads (in experiment) and also in the simulation. In the time of discharge (Fig. 3a) the velocities of the particle increases as we move down the hopper (will be shown in the next section). It is also interesting to note that the flow through the 451 hopper is concentrated towards the center, an indication of funnel flow. The 751 hopper illustrated flow through out the bed, indicating the presence of mass flow. These results are quantified in the next section. Fig. 4 compares qualitatively the flow of nearly free-flowing Fast-Flo lactose with the simulation (K ¼45) and the appropriate friction parameters for inter-particle and particle–wall contacts at t¼0 (Fig. 4a) and 0.5 s (Fig. 4b). The flow of Fast-Flo lactose is quite similar to the flow of glass beads and powder flows easily in both the hoppers. For slightly higher cohesion, Fig. 5 shows the case of Avicel and simulations for K ¼30 it becomes increasingly difficult for the powder to flow through the hopper. Interestingly,
Fig. 3. Flow of non-cohesive glass beads in both the hopper at time (a) t¼ 0 s and (b) time ¼0.5 s, (b) top row are the snapshots from experiments and the bottom row shows the same from simulation.
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Fig. 4. Flow of mildly cohesive Fast-Flo lactose in both the hopper at time (a) t ¼0 s and (b) time ¼ 0.5 s, (b) top row are the snapshots from experiments and the bottom row shows the same from simulation.
in both experiments and simulations, the particles only flow through the steeper hopper (751) as shown in Fig. 5b. Finally, the comparison of flow for the most cohesive powder (regular lactose) and K ¼75 did not yield any flow in the 751 hopper (Fig. 6). The simulations show excellent qualitative agreement for flow dynamics in hoppers using the same parameters as that of the rotating drum illustrating that the dynamics of flow is independent of the geometry.
5.4. Quantitative simulation results for flow in hoppers The qualitative comparisons in the previous section showed that DEM simulations faithfully capture the effect of an increase in cohesion on the flow of materials through the hopper. As cohesion increases, it becomes increasingly difficult for the particles to flow through the hopper. The wall segments dz created to calculate wall pressure are schematically shown as trapezoidal sections in Fig. 7.
Fig. 8a shows the wall pressure distribution at the end of static (end filling) process for the 451 hopper as a function of cohesion. Here the pressure increases with depth, a phenomena similar to the one experienced by a fluid in a container. For the free flowing case (K ¼0), the pressure reaches a maximum (1600 Pa) at the bottom of the hopper. As cohesion is introduced in the system, the trend remains the same, the only difference being the wall pressure at the bottom of the hopper decreases with increase in cohesion. The most cohesive case (K¼ 75) shows a wall pressure of 600 Pa at the bottom. As seen in Fig. 8b, similar pressure profile is observed for the 751 hopper. Here the pressure reaches 3000 Pa for the free flowing case and approximately 2200 Pa for the most cohesive case. As cohesion increases, the porosity of the bed increases due to formation of bigger agglomerates in the system, causing a decline in the wall pressure. As can be seen in Fig. 9, the hopper geometry has a strong effect on the wall pressure distribution during dynamic discharge. The mean wall pressure increases with increasing depth for both the hoppers (Fig. 9a and b) from the free surface of the granular
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Fig. 5. Flow of moderately cohesive Avicel-102 in both the hopper at time (a) t ¼0 s and (b) time ¼0.5 s, (b) top row are the snapshots from experiments and the bottom row shows the same from simulation.
Fig. 6. Flow of very cohesive regular lactose in the 751 hopper at time (a) t ¼ 0 s and (b) time ¼0.5 s.
material. At the transition from the vertical-sided section to the angular section, a pressure peak occurs which is a consequence of the sudden change in wall slope. For a similar reason, the pressures are greater below the transition than above it. For both cases, as cohesion increases, the mean wall pressure increases.
The pressure distributions are in accordance with both analytical (Walters, 1973) and two dimensional numerical (Yang and Hsiau, 2001) simulations. As mentioned earlier, there are essentially two types of flow through a hopper, mass flow and funnel flow. In this study, we use
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0.045 K=0 K = 45 K = 60 K = 75
0.04
Height (m)
0.035 dz
h
0.03 0.025 0.02 0.015 0.01 0.005 0
Fig. 7. Cross sectional area of a trapezoidal section for particle simulation.
200
400 600 Wall Pressure (Pa)
0.10
1800 k=0 K = 45 K = 60 K = 75
1600 1400 1200 1000 800
800
1000
K=0 K = 45 K = 60 K = 75
0.08 Height (m)
Pressure (Pa)
0
0.06 0.04
600 400
0.02
200 0 0.00
0.00 0.01
0.02 0.03 Height (m)
0.04
0.05
K=0 K = 45 K = 60 K = 75
3000 Pressure (Pa)
100
200 300 400 Wall Pressure (KPa)
500
600
Fig. 9. (a) Variation of wall pressure with height for the hopper with angle 451 and (b) 751 as a function of granular cohesion for the dynamic (discharge case).
3500
2500 2000 1500 1000 500 0 0.0000
0
0.0200
0.0400 0.0600 Average Height (m)
0.0800
0.1000
Fig. 8. (a) Variation of wall pressure with height for the hopper with angle 451 and (b) 751 as a function of granular cohesion for the static case.
DEM simulations to illustrate the two types of flow. The radial distributions of velocities at different heights in both hoppers are examined to illustrate the different flow behaviors in hoppers. At a particular height of the hopper, the lateral plane is subdivided radially into 5 sections. Fig. 10a shows the variations in velocity at different radial points for the 451 hopper. Here, the velocities are almost zero near the wall and increase to a maximum at the center of the hopper. The average velocity near the wall is 25% of that measured near the middle of the hopper. This velocity profile
resembles core or funnel flow in hoppers. The figure also shows the results for cohesion. The flow behavior in this hopper is independent of the cohesiveness of the material. The radial distribution of average velocity in the hopper with steeper angle (751) is shown in Fig. 10b. In this case, the velocity distribution across the radial direction of the hopper is more uniform. The velocities near the wall are approximately 80% of the velocities estimated at the center of the hopper, illustrating mass flow behavior for the steeper hopper. Even though it is difficult to quantitatively compare this result to experimental findings, visual observation of powder flow through hoppers readily shows that as the hopper angle becomes steeper, flow changes from funnel flow to mass flow for a fixed hopper opening (Stanford and DellaCorte, 2006; Tomas and Schubert, 1979; Thorpe, 1992). The effect of cohesion is not as prominent in this hopper as the velocity distribution is fairly similar for the three cases (K¼ 0, 45, 60). Another important observation is that while discharging the velocities are a magnitude lower for the steeper hopper.
6. Conclusion The GDR provides an effective and convenient method for examining flow properties of pharmaceutical materials, pure and in mixtures. A clear application demonstrated here is its use as a laboratory method for monitoring flow of raw ingredients and API batches through small scale hoppers as a function of material and processing properties such as particle size, moisture content, and
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Fig. 10. Radial distribution of average absolute velocity of granular materials of various levels of cohesion in a 451 hopper (a) and (b) 751 hopper.
blend composition. An excellent experimental correlation between GDR flow index and hopper flow is observed. As flow index increased, it became increasingly difficult for the powder to flow through the hopper. Simulations were performed to understand the dynamics of flow in hoppers by using the same parameters as that obtained for flow in rotating cylinder. The simulations showed excellent qualitative agreement with the experiments, as cohesion increased; it became harder for the particles to flow through the hoppers. It also showed the effect of hopper angle; similar to experiments for Avicel, K ¼60 did not flow through the 451 hopper but flowed well in a 751 hopper. Simulations were also used to calculate the normal forces on the walls of the hopper and eventually the wall pressure distributions in both hoppers. As the depth increased, the wall pressure increases, which is opposite to the behavior observed for fluids. Finally, the simulations also
helped in understanding the different flow behavior (funnel and mass flow) that exists in a hopper. The correlation between the dynamics of flow in a rotating drum and hopper were extremely similar to the experimental behavior of the avalanches for the pure powder, showing that the parameters obtained for cohesion are independent of geometry. Above all, the simulation parameters initially validated using a rotating drum predicted the flow in hoppers, indicating that the DEM algorithm is device independent.
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Nedderman, R.M., Tuzun, U., Savage, S.B., Houlsby, G.T., 1982. The flow of granular materials—1: discharge rate from hoppers. Chem. Eng. Sci. 37, 1597–1609. Peleg, M., 1978. Flowability of food powders and methods for its evaluation––a review. J. Food Process Eng. 1, 303–328. Purutyan, H., Pittenger, B.H., Carson, J.W., 1998. Solve solids handling problems by retrofitting. Chem. Eng. Prog., 27–39. Raymond, R., Sheskey, P., Weller, P.J., 2005. Handbook of Pharmaceutical Sciences, second ed The Pharmaceutical Press, London, p. 141. Ristow, G., Herrmann, H., 1994. Density patterns in two-dimensional hoppers. Phys. Rev. E 50, R5–R8. Rosato, A., Prinz, F., Standburg, K.J., Swendsen, R., 1986. Monte Carlo simulation of particulate matter segregation. Powder Technol. 49, 59–69. Savage, S.B., 1965. The mass flow of granular material derived from coupled velocity stress fields. Br. J. Appl. Phys. 16, 1885–1888. Savage, S.B., 1984. The mechanics of rapid granular flows. Adv. Appl. Mech. 24, 289–298. Savage, S.B., Hutter, K., 1989. The motion of a finite mass of granular material down a rough inclined plane. J. Fluid Mech. 199, 177–215. Savage, S.B., Tuzun, U., Houlsby, G.T., Nedderman, R.M., 1983. The flow of granular materials—III rapid shear flows. Chem. Eng. Sci. 38, 189–195. Schulze, D., 1994. 1st Particle Technology Forum, Denver, USA, August 17–19, vol. 78, pp. 11–16. Shinbrot, T., Alexander, A., Moakher, M., Muzzio, F.J., 1999. Chaotic granular mixing. Chaos 9, 611–620. Stanford, M., DellaCorte, C., 2006. Effects of humidity on the flow characteristics of a composite plasma spray powder. J. Therm. Spray Technol. 15, 33–36. Staniforth, J., 2002. In: Aulton, M.E. (Ed.), Pharmaceutics second ed Churchill Livingstone, New York, pp. 197. Sudah, O.S., Arratia, P.E., Alexander, A., Muzzio, F.J., 2005. Simulation and experiments of mixing and segregation in a tote blender. A.I.Ch.E. J. 5, 836–844. Thompson, P.S., Grest, G.S., 1991. Granular flow: friction and the dilatancy transition. Phys. Rev. Lett. 67, 1751–1755. Thorpe, R.B., 1992. An experimental clue to the importance of dilation in determining the flow rate of a material through a hopper or bin. Chem. Eng. Sci. 47, 4295–4303. Tomas, J., Schubert, H., 1979. Partec 79, Particle Characterization. Nu¨rnberg, Germany, pp. 301–319. Walters, J.K., 1973. Chemical Engineering Science 28, 779–789. Walton, O.R., 1992. Numerical simulation of inelastic, frictional particle–particle interactions. Particulate Two-Phase Flow 16, 884–911. Walton, O.R., 1993. Numerical simulation of inclined chute flows of monodisperse, inelastic, frictional spheres. Mech. Mater. 16, 239–247. Walton, O.R., Braun, R.L., 1986. Viscosity, granular-temperature and stress calculations for shearing assemblies of inelastic, frictional disks. J. Rheol. 30, 949–980. Wightman, C., Moakher, M., Muzzio, F.J., Walton, O.R., 1998. Use of the discrete element methods to characterize particulate flow and mixing in a rotating and rocking cylinder. A.I.Ch.E. J. 44, 1266–1276. Yang, S.C., Hsiau, S.S., 2001. Powder Technol. 120, 244–255.
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Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces
Constitutive model to predict flow of cohesive powders in bench scale hoppers AbdulMobeen N. Faqih a, Bodhisattwa Chaudhuri b, Amit Mehrotra c, M. Silvina Tomassone d, Fernando Muzzio d, a
Wyeth Pharmaceuticals, Pearl River, NY, USA Department Pharmaceutical Sciences, University of Connecticut, Storrs, CT, USA c Glaxo Smith Kline, Raleigh, NC, USA d Department of Chemical and Biochemical Engineering, Rutgers University, USA b
a r t i c l e in fo
abstract
Article history: Received 23 September 2009 Received in revised form 25 January 2010 Accepted 15 February 2010 Available online 6 March 2010
This communication empirically correlates flow in two systems; an instrumented rotating drum (GDR) and a set of bench scale hoppers. A flow index obtained from measurements in the GDR is directly correlated to the flow through hoppers, providing a predictive method for hopper design and a convenient experimental test for screening materials and determining their suitability for specific hopper systems. Simulations were performed to understand the dynamics of flow in hoppers by using the same flow parameters in hoppers and rotating cylinders. Simulations showed that as cohesion increased it becomes harder for the particles to flow through the hoppers, in good agreement with the experiments. The effect of hopper angle also yields similar findings to experiments for Avicel, K ¼60, where the powder does not flow through the 451 hopper but flows well in a 751 hopper. Simulations were also used to calculate the normal forces on the walls of the hopper and the wall pressure distributions in both hoppers. As depth increases, the wall pressure increases for all cases. Finally, the simulations also helped understand the different flow behaviors (funnel and mass flow) that take place in a hopper. The simulated dynamics of flow in the rotating drum and in the hopper correlate very closely to experiments, indicating that the model cohesion parameters are, as desirable, materialspecific but independent of geometry. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Granular materials Pharmaceuticals Powder technology Mathematical modeling Discrete element modeling Powder flow
1. Introduction Granular materials exhibit a wealth of interesting phenomena, including heaping under vibration, segregation, convection, fluidization, and density waves in pipes or hoppers. They are also very important from technological and industrial points of view. Many examples of important granular flows can be found in both industry and nature. Hoppers, chutes, and conveyor belts are used when transporting particulate materials such as food stuffs, pharmaceuticals, and coal. Other industrial applications include packing of granular materials, particulate segregation and mixing, and particulate drying. In nature, examples of granular flows include snow and mud avalanches, river sedimentation, dune formation, planetary ring dynamics, soil liquefaction, and ice flow mechanics. Clearly, characterization of the behavior of granular materials is essential to scientific community with significance to the pharmaceutical industry for downstream processing of solid dosage forms. However, most of the research has been focused on
Corresponding author.
E-mail address: [email protected] (F. Muzzio). 0009-2509/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2010.02.028
cohesionless materials, which tend to segregate when used with a poly-dispersed particle size distribution (Savage et al., 1983; Savage, 1984; Savage and Hutter, 1989), but that otherwise, provide relatively simple flows. There has been some research work in the last few years on the much more complex field of cohesive powders (Adams and Perchard, 1985; Lian et al., 1993; Bocquet et al., 1998; McCarthy et al., 2001) however, the flow of cohesive powders is still poorly understood. Knowledge of powder flow properties is very important when developing manufacturing processes and handling procedures such as flow from hoppers and silos, transportation, mixing, compression and packaging (Knowlton et al., 1994; Peleg, 1978). Powder flow characteristics are commonly investigated under gravity loading conditions (Carstensen, 1974). The compressibility of a powder is a commonly used indicator of flowability and is often expressed using the Hausner Ratio, which is the ratio between the tapped and the loose-packed bulk densities of the powder (Hausner, 1967). Compressibility is also one of the tests proposed by Carr (1965) for the assessment of powder properties. Another commonly used flow indicator is the time it takes for powder to flow out of a funnel with a standard orifice size (Staniforth, 2002). Such measurements have
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demonstrated the dependence of powder flowability on particles shape and size distribution (Baxter et al., 2000), instantaneous degree of consolidation, and relative humidity (Coelho and Harnby, 1978; Stanford and DellaCorte, 2006). A key situation where flow is critical is the filling and emptying of hoppers. Despite their common use and deceptively simple design, one of the major industrial powder problems is obtaining reliable and consistent flow out of hoppers. These problems are usually associated with the flow pattern inside a hopper. An intriguing property of granular flow from a hopper is that in certain situations the flow rate is largely dependent on the diameter of the orifice, and weakly on the particle size and head of the material. Mass flow is an ideal flow pattern where the bulk is in motion and moving downwards towards the opening. The other common case is called funnel flow, where the powder starts moving out through a central ‘‘funnel’’ that forms within the material, leading to the powder collapsing and moving through the funnel. Most flow problems are caused by a funnel flow pattern and can be cured by altering the pattern to mass flow (Purutyan et al., 1998; Savage, 1965). The worst case-scenario is no flow, which can occur when the cohesive powder forms an arch across the opening, which provides sufficient strength to support itself. In order to prevent these problems, measurement of powder flow properties is necessary for design of mass flow hoppers. In the last 45 years, significant improvements have been proposed to the mathematical models that quantify the flow of granular material through hopper. In 1964 Jenike utilized the radial solutions of Sokolovsky to quantify some aspects of granular flow, but as the solution lacked inertial terms, the granular flow discharge rate could not be predicted. Savage (1965) constructed the earliest model that accounted for inertial terms by setting the internal friction and the friction with the wall to zero. Later in 1967, Savage added wall friction but maintained gravity in the radial direction (Tomas and Schubert, 1979). Sullivan, Davidson and Nedderman et al. (1982), Nedderman (1992) applied these results to the Hour-Glass theory. In 1992, Thorpe (1992) showed that the earlier models assumed constant bulk density, which would overestimate the discharge rates. As the material flows through the hopper, the stress changes from zero at the top surface, reaches a maximum and falls to zero on the free-fall arch. Thus in the upper part of the hopper, the material is compressed and the interstitial air must be expelled, whereas in the lower part, the material dilates as air is being drawn in. As a result, elevated pressures occur in the upper part and sub-atmospheric pressures occur in the lower part. The most appropriate model that describes this change in stress behavior is the critical state theory (CST) (Bak et al., 1987), which assumes a logarithmic relationship between pressure and density (so that density is not defined when pressure equals to zero). However the CST, which is derived from soil mechanics, cannot be necessarily applied to extreme cases of failure for hopper flow. In spite of the existence of predictive models and techniques to improve flow of free-flowing materials through hoppers, there lacks a general understanding of the flow behavior of powders as a function of cohesion. Cohesion clearly plays an important role in affecting flow properties and is a key target of flow property characterization. During the last few decades a variety of methods for assessment of cohesive powder flow properties have been developed using some type of a ‘‘shear cell’’ where the force required to initiate (or maintain) movement in a standard geometry is measured. The area was pioneered by Jenike (1964) who also developed the theoretical framework that became the field standard. In conjunction with the measured property data, he applied two-dimensional stress analysis in developing a
mathematical methodology for determining the minimum hopper angle and hopper opening size for flow from conical and wedge shaped hoppers (Jenike, 1964). Besides the Jenike tester, some other commonly used shear testers include the ring shear testers (Schulze, 1994), the Johanson (1992, 1993) indicizers, uniaxial, biaxial, and triaxial testers (Maltby and Enstad, 1993; Maltby, 1993), and Jenike and Johanson’s quality control tester. While these methods have been useful for many purposes such as building roads and bridges, they have some shortcomings regarding processing of cohesive powders. The most salient drawback is that flow characteristics of powders are highly dependent on their densification (consolidation) states, i.e. powders can be more or less expanded or contracted when stressed, thus leading to a large variety of inter-particle forces and flow behavior (a phenomenon commonly referred to as ‘‘jamming’’). For small scale systems, powders often flow in a fully dilated state. Shear cells can approach the dilated state only asymptotically, and are affected by considerable experimental error for cohesive materials exhibiting a non-linear relationship between consolidation and flow. The complex constitutive behavior makes accurate powder flow measurements difficult. Various modeling scales are commonly used to simulate granular materials. Understanding and modeling the dynamic behavior of particulate systems has been a major research focus worldwide for many years. Models of granular flows can be broadly divided into three categories: continuum, kinetic theory and discrete. When continuum approach fails or when no appropriate constitutive relations exist, the discrete element modeling (DEM) has proven tremendously useful. Molecular dynamics models are broadly classified by the contact model and integration method, as either hardsphere, in which collisions are instantaneous and binary, and softsphere, in which collisions can be lasting and multiple. Here, the particles are permitted to suffer minute deformations, and these deformations are used to compute restoring elastic, plastic and frictional forces. The discrete element method (DEM), originally developed by Cundall (1971), Cundall and Strack (1979), has been successfully used to simulate chute flow, heap formation (Luding, 1997), hopper discharge (Ristow and Herrmann, 1994; Thompson and Grest, 1991) and flows in rotating drums (Rosato et al., 1986; Khakhar et al., 1997; Wightman et al., 1998). In this article we investigate the flow and pressure dependence as a function of cohesion in two different geometries (rotating cylinder and hopper) using discrete element methods (DEM) and experiments. We test the parameters developed for simulating cohesive flow in a rotating drum by using the same parameters to simulate flow in hoppers of varying angle. DEM simulations are used to describe the behavior of granular materials after the initial filling stage (static state) and during the discharge (dynamic state) as a function of cohesion. These DEM results are compared with experimental studies for bench scale hoppers. In essence, the goal is to determine whether the parameters introduced from the rotating drum and applied to simulation studies in hoppers are independent of the geometry of the system. The article is organized as follows: Section 2 lists the materials used in the experimental study and describes the experimental setup for the gravitational displacement rheometer (GDR) and bench scale hoppers. In Section 3, we report a DEM based constitutive model to compare the dynamics of the flow in the two geometries considered. Section 4 describes the experimental and simulation method for correlations between geometries and between experiment and simulations, Section 5 describes the experimental and computational results for flow dynamics in a rotating cylinder, and flow through hoppers of varying angle. Finally, in Section 6 we present our conclusions and outline directions for future work.
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2. Experimental materials Commercially available, well-characterized powders were used to create a family of ‘‘standard’’ systems of varying cohesion. Sugar and other typical pharmaceutical excipients were used, both ‘‘pure’’ and in mixtures. Particle sizes for these materials are: Fast-Flo lactose (the least cohesive), 100 mm; Avicel 102 (moderately free flowing), 90 mm; Avicel 101 (moderately cohesive), 60 mm; Regular lactose (the most cohesive), 50 mm. A matrix of binary system consisting of varying composition of Fast Flo lactose and Avicel 102; Avicel 102 and Regular lactose, and finally, Avicel 101 and Regular lactose is made to accommodate for varying cohesion between the pure components. The matrix of blends was typically used for experimental studies to understand the correlation of cohesive powders in varying geometry. These materials are some of the most common pharmaceutical excipients; in the interest of brevity their SEM images and other details are not included in this paper but can be found in the ‘‘Handbook of Pharmaceutical excipients’’ (Raymond et al., 2005).
3. Simulation details In these studies we use DEM to simulate the dynamic behavior of cohesive and non-cohesive powders in a rotating drum. We consider the granular material as a collection of frictional inelastic spherical particles. Each particle may interact with its neighbors or with the boundary only at contact points through normal and tangential forces. The forces and torques acting on each of the particles are calculated in the following way: X ð1Þ F i ¼ mi g þ F N þ F T þ F cohes X
Ti ¼ ri F T
ð2Þ
The force on each particle is given by the sum of gravitational, interparticle (normal and tangential FN and FT) and cohesive forces as indicated in Eq. (1). The nature of the cohesive force (Fcohes) is a combination of weight of the particles multiplied by a constant and is explained later. The corresponding torque on each particle is the sum of the torque of the tangential forces (FT) arising from inter-particle contacts (ri) as illustrated in Eq. (2). The normal forces are calculated with the ‘‘latching spring model’’, developed by Walton and Braun (1986), Walton (1992, 1993), which allows colliding particles to overlap, and the corresponding interaction force (normal force) is a function of the relative overlap. Each particle may interact with its neighbors or with the boundary only at contact points through normal and tangential forces. The normal forces between pairs of particles in contact are defined using a spring with constants K1 and K2 for compression and recovery: FN ¼ K1m1 (for compression), and FN ¼K2 (m1 m0) (for recovery). These spring constants K1 and K2 are chosen to be large enough to ensure that the overlaps m1 and m0 remain small compared to the particles sizes. The degree of inelasticity of collisions is incorporated in this model by including a coefficient of restitution e¼(K1/K2)1/2 (0 oeo1, where e¼1 implies perfectly elastic collision with no energy dissipation, e¼0 implies completely inelastic collision). Tangential forces (FT) in inter-particle or particle–wall collision are calculated employing Walton’s incrementally slipping model. After contact occurs, tangential forces build up, causing displacement in the tangential plane of contact. The initial tangential stiffness is considered a fraction of the normal stiffness. In all cases, the static frictional limit was considered to follow Coulomb’s law, i.e., if FT r mSFN (mS is the static friction coefficient and FN is the total normal force, equal to the cohesive force plus the static weight of particles transmitted through the bed), no
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relative motion between particles occur, while if the particles are in relative motion with respect to one another, FT ¼ mD FN, where mD is the dynamic friction coefficient. This model takes into account the elastic deformations that can occur in the tangential directions of the contacts. The tangential force T is evaluated considering an effective tangential stiffness kT associated with a linear spring. It is incremented at each time step as Tt þ 1 ¼ Tt þ kT Ds, where Ds is the relative tangential displacement between two time steps (Shinbrot et al., 1999). The described model has been successful to obtain the first three-dimensional computations of realistic blender geometries, where it confirmed important experimental observations (Shinbrot et al., 1999; Sudah et al., 2005). To incorporate granular cohesion in the model, a cohesive force between particles is simulated, in our case, we introduce a simple square-well potential. Two parameters are needed to define the square well: the well width ri (equal to the radius of the particle) and the well depth K. The cohesive interactions arise when the separation distance of two particles is less or equal to twice the width of the square well (2ri). In dimensionless terms, in order to compare simulations considering different numbers of particles, the magnitude of the force was represented in terms of the parameter K¼Fcohes/mg, where K is called the Bond number and is a measure of cohesiveness that is independent of particle size, Fcohes is the cohesive force between particles, and mg is the weight of the particles. Notice that this constant force may represent short range effects such as electrostatic or van der Waals forces. In this model, the cohesive force (Fcohes) between two particles or between a particle and the wall is unambiguously defined in terms of K. Four friction coefficients are also defined: particle–particle static and dynamic coefficients, and particle–wall static and dynamic coefficients. Interestingly, all four friction coefficients turned out to be important. To emulate different levels of cohesion, the bond number K, the coefficients of static and dynamic friction between particles (mSP and mDP) and the coefficients of static and dynamic friction between particle and wall (mSW and mDW) are varied. The major computational tasks of DEM in each time step are as follows: (i) add/delete contact between particles thus updating neighbor lists, (ii) compute contact forces from contact properties, (iii) sum all forces and torques on particles and update position and (v) determine the trajectory of the particle by integrating Newton’s laws of motion (second order scalar equations in three dimensions). A central difference scheme, Verlet’s Leap Frog method is used here. A previously used DEM code (Wightman et al., 1998; Faqih et al., 2006; Alexander et al., 2006; Moakher et al., 2000) was expanded here to include cohesive and frictional forces to characterize the behavior of cohesive granular systems. The DEM code was written in C language and was run in 32 and 64 bit Linux clusters.
4. Experimental and simulation methodology To examine flowability through hoppers, the aforementioned powders are transferred into a set of hoppers with five different angles (351, 451, 551, 651, and 751) bored out of solid Plexiglas cylinders (Fig. 1). Removable opening sections were built in order to be able to adjust the diameter of the discharge of the steepest hopper (0.5–1.0 in). The behavior of each material in each hopper was classified according to four main behaviors: funnel flow, mass flow, intermittent flow (aided by steady vibration), and no flow. The gravitational displacement rheometer (GDR) has been used to quantify flow as a function of cohesion. The apparatus
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Table 1 Comparison of simulation parameters to pharmaceutical powders. K
lsp
ldp
lsw
ldw
Pharm. Mat.
0 45 60 75
0.8 0.8 0.8 0.8
0.1 0.1 0.6 0.6
0.5 0.5 0.8 0.8
0.5 0.5 0.8 0.8
Glass beads Fast-Flo Avicel Regular lactose
K: cohesion number; mSP: static friction between particles; mDP: dynamic friction between particles; msw: static friction between particle and wall; mDw: static friction between particles.
Fig. 1. A picture of the bench scale hopper setup with varying angles ranging from 351 to 751.
employs a long cylinder to minimize wall effects and uses a load cell to measure changes in the center of gravity that are correlated to specific powder flow behavior. Detailed analysis of the functionality of the equipment is illustrated in an earlier article (Faqih et al., 2006) and will not be discussed in this paper. The output of the GDR is a quantifiable term known as the flow index; where higher the flow index, the more cohesive the powder and the harder it is for it to flow. The idea is based on the methodology expressed in an earlier article on avalanching behavior of powders (Alexander et al., 2006). In order to understand the nature of the forces controlling powder flow behavior, a computational model was generated to determine the relationship between inter-particle cohesive strength and the GDR measurements. Our model system consisted of 20,000 particles of 2 mm diameter in a cylindrical vessel with a 9 cm diameter, 1 cm length, and frictionless side walls. The magnitude of the cohesive bond number K described earlier is chosen in the same order of magnitude as the ones measured with atomic force microscopy by Hang Duong et al. (2004) and also in the numerical model of Baxter et al. (2000). The first part of the result section will capture a comparison of experiments and simulations in a rotating cylinder. In order to demonstrate the relevance of the flow index obtained from the GDR system, experiments were performed to quantify the flow behavior of the materials both pure and mixtures discharged from the GDR were run through a series of hoppers of varying angle and orifice diameter. The behavior of each material in each hopper was classified according to four main categories: funnel flow, mass flow, intermittent flow, and no flow. A factorial design was carried out for this process, where the pure materials were run first, followed by 50:50 mixtures of the three mixtures and eventually the 75:25 and 25:75 intermediate points to complete the characterization. The parameters for cohesion used in our simulations and the materials used for experiments provide a qualitative tool to compare the dynamics of flow in hoppers to other geometries. In order to avoid redundancy and also capture the overall effect of different flows (funnel and mass) we simulated flow in two different hoppers: 451 and 751. Simulation parameters, selected to represent the same specific materials examined for the GDR, are presented in Table 1. Once again, the model considered 20,000 particles, 2 mm in diameter. Both hoppers had the same maximum diameter (7 cm). The lengths of the hoppers were 9.6 and 26.8 cm for hoppers of 451 and 751 angles, respectively.
Finally, the qualitative results obtained from direct correlation of bond number to cohesion will be quantified using simulations to understand the radial velocity profiles at the interface of the cylindrical and conical section of the two hoppers and the variations in mean wall pressure both for the static (end filling) and dynamic (discharge) conditions throughout the hopper bed as a function of cohesion. Wall pressures are averaged on a wall segment of height dz. In this study, dz is equal to 2d (where d is the diameter of the particle). For each evaluation, the segment boundary is moved by half particle in the z-direction. In the case of a given wall, the height dz corresponds to a trapezoid-shaped wall segment. The first step involved in calculating the mean wall pressure consists of summing up the normal forces on the wall. The mean wall pressure is defined as the total force acting perpendicular to a wall segment divided by the area of the wall segment. Since the mean wall pressure distribution is calculated from the normal component of the particle/wall contact forces, it shows fluctuations along the wall. It is important to note that the fluctuation is strongly affected by the averaging method. In order to determine the wall pressure distribution during the outflow, the pressures are averaged both on a wall segment of height dz and over 1000 time steps. In summary, the result section will demonstrate flow comparison between experiments and simulation in a cylinder. Experimental work will correlate flow index obtained using the GDR to flow on bench scale hoppers. The parameters of bond number defined using a rotating cylinder will be used to qualitatively compare experiments to simulation in a hopper and finally, the simulations will be used to quantify particle–particle and particle–wall interaction in a hopper.
5. Results and discussion In this section we characterize the effect of cohesion on flow in a rotating cylinder both experimentally and through simulations and compare the two by visualizing the dynamics of flow. 5.1. Flow comparison between experiments and simulation in a rotating cylinder The constant K also called the bond number was varied between 0 and 90; for K o30, no avalanches were observed, and for K 490, avalanches were of magnitude comparable to the size of the system and therefore examination of larger values was unwarranted. The model displayed discrete, well-defined avalanches for the moderate and higher cohesion scenario (K value greater than 45). Computational parameters were fine-tuned to accurately simulate flow of the four pure materials. The following methods were used to validate the code: (a) direct visual comparison of the shape of the flow region, including local angles of repose, (b) size of avalanches, and (c) degree of dilation of the material. Subsequently, sets of cohesion and friction coefficients were identified that accurately matched flow behavior of Fast-Flo
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lactose, Avicel 102, Avicel 101, and micronized lactose. Typical results for Fast-Flo lactose, Avicel 101 and their corresponding matching simulation snapshots are displayed in Fig. 2. A table providing the results for the simulation parameters that correlate to the four pharmaceutical materials of interest is given in Table 1.
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5.2. Correlating the GDR methodology to flow in bench scale hoppers As the flow index (i.e., the blend cohesion) increased, flow through hoppers became increasingly difficult. A detailed version of these results are presented in Table 2, which displays clear
Fig. 2. (a) Shows the time sequence of axial snapshots of Fast-Flo lactose in a rotating drum, (b) shows the simulation snapshots for K¼ 45, (c) shows the time sequence for Avicel-101 and (d) shows the snapshots from simulation for K¼60.
Table 2 (a) A DOE of experiments are examined for flow in hoppers of varying angle and (b) the table when arranged in increasing order of flow index (higher cohesion) gives an excellent picture of powders flowability through the hoppers. Pharmaceutical powders
Flow index
35 deg 0.5 in
45 deg 0.5 in
55 deg 0.5 in
65 deg 0.5 in
75 deg 0.5 in
75 deg 0.75 in
75 deg 1.0 in
(a) FF lactose Avicel 102 Avicel 101 Regular lactose FF-A102 (50-50) A102-Reg (50-50) A101-Reg (50-50) FF-A102 (75-25) A102-Reg (75-25) A101-Reg (75-25) FF-A102 (25-75) A102-Reg (25-75) A101-Reg (25-75)
27.8 38 44.4 48.2 35.2 46.1 47.7 32.6 44.5 46.9 35.8 47.9 48.1
FF NF NF NF NF NF NF NF NF NF NF NF NF
FF NF NF NF NF NF NF IF NF NF NF NF NF
FF NF NF NF IF NF NF FF NF NF NF NF NF
MF IF NF NF MF NF NF MF NF NF IF NF NF
MF IF IF NF MF NF NF MF IF NF IF NF NF
MF IF IF NF MF IF No flow MF IF IF MF NF NF
MF MF MF NF MF MF MF MF MF MF MF IF NF
(b) FF lactose FF-A102 (75-25) FF-A102 (50-50) FF-A102 (25-75) Avicel 102 Avicel 101 A102-Reg (75-25) A102-Reg (50-50) A101-Reg (75-25) A101-Reg (50-50) A102-Reg (25-75) A101-Reg (25-75) Regular lactose
27.8 32.6 35.2 35.8 38 44.4 44.5 46.1 46.9 47.7 47.9 48.1 48.2
FF NF NF NF NF NF NF NF NF NF NF NF NF
FF IF NF NF NF NF NF NF NF NF NF NF NF
FF FF IF NF NF NF NF NF NF NF NF NF NF
MF MF MF IF IF NF NF NF NF NF NF NF NF
MF MF MF IF IF IF IF NF NF NF NF NF NF
MF MF MF MF IF IF IF IF IF NF NF NF NF
MF MF MF MF MF MF MF MF MF MF IF NF NF
In this table; FF: funnel Flo; IF: intermittent flow; MF: mass flow; NF: no flow.
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evidence that the flow index predicts whether a given material will or will not flow out of a given hopper. This correlation indicates that the onset of flow in a hopper, which is controlled by the powder’s ‘‘unconfined bridging length’’, is a manifestation of the same cohesion-generated internal ‘‘flow length scale’’ that also controls the size of an avalanche. These results provide strong motivation to further investigate the relationship between avalanches in the GDR and hopper flow behavior.
5.3. Flow comparison between experiments and simulation Fig. 3 illustrates the flow of non-cohesive glass beads through the hoppers in both the experiment and simulation depicted next to each other for qualitative comparison. The snapshots of both the experiments and simulation are taken at t¼0 s (Fig. 3a) and 0.5 s (Fig. 3b). We observe that the free flow powder is easily discharged in both the hoppers (Fig. 3b). In the simulation the particles are color coded based on their absolute velocities. Blue
particles signify particles with minimum (zero) velocity and the particle with maximum velocities are colored in red. At t¼ 0, the static hopper bed has all particles in rest. As the flow initiates we observe unabated discharge of yellow glass beads (in experiment) and also in the simulation. In the time of discharge (Fig. 3a) the velocities of the particle increases as we move down the hopper (will be shown in the next section). It is also interesting to note that the flow through the 451 hopper is concentrated towards the center, an indication of funnel flow. The 751 hopper illustrated flow through out the bed, indicating the presence of mass flow. These results are quantified in the next section. Fig. 4 compares qualitatively the flow of nearly free-flowing Fast-Flo lactose with the simulation (K ¼45) and the appropriate friction parameters for inter-particle and particle–wall contacts at t¼0 (Fig. 4a) and 0.5 s (Fig. 4b). The flow of Fast-Flo lactose is quite similar to the flow of glass beads and powder flows easily in both the hoppers. For slightly higher cohesion, Fig. 5 shows the case of Avicel and simulations for K ¼30 it becomes increasingly difficult for the powder to flow through the hopper. Interestingly,
Fig. 3. Flow of non-cohesive glass beads in both the hopper at time (a) t¼ 0 s and (b) time ¼0.5 s, (b) top row are the snapshots from experiments and the bottom row shows the same from simulation.
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Fig. 4. Flow of mildly cohesive Fast-Flo lactose in both the hopper at time (a) t ¼0 s and (b) time ¼ 0.5 s, (b) top row are the snapshots from experiments and the bottom row shows the same from simulation.
in both experiments and simulations, the particles only flow through the steeper hopper (751) as shown in Fig. 5b. Finally, the comparison of flow for the most cohesive powder (regular lactose) and K ¼75 did not yield any flow in the 751 hopper (Fig. 6). The simulations show excellent qualitative agreement for flow dynamics in hoppers using the same parameters as that of the rotating drum illustrating that the dynamics of flow is independent of the geometry.
5.4. Quantitative simulation results for flow in hoppers The qualitative comparisons in the previous section showed that DEM simulations faithfully capture the effect of an increase in cohesion on the flow of materials through the hopper. As cohesion increases, it becomes increasingly difficult for the particles to flow through the hopper. The wall segments dz created to calculate wall pressure are schematically shown as trapezoidal sections in Fig. 7.
Fig. 8a shows the wall pressure distribution at the end of static (end filling) process for the 451 hopper as a function of cohesion. Here the pressure increases with depth, a phenomena similar to the one experienced by a fluid in a container. For the free flowing case (K ¼0), the pressure reaches a maximum (1600 Pa) at the bottom of the hopper. As cohesion is introduced in the system, the trend remains the same, the only difference being the wall pressure at the bottom of the hopper decreases with increase in cohesion. The most cohesive case (K¼ 75) shows a wall pressure of 600 Pa at the bottom. As seen in Fig. 8b, similar pressure profile is observed for the 751 hopper. Here the pressure reaches 3000 Pa for the free flowing case and approximately 2200 Pa for the most cohesive case. As cohesion increases, the porosity of the bed increases due to formation of bigger agglomerates in the system, causing a decline in the wall pressure. As can be seen in Fig. 9, the hopper geometry has a strong effect on the wall pressure distribution during dynamic discharge. The mean wall pressure increases with increasing depth for both the hoppers (Fig. 9a and b) from the free surface of the granular
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Fig. 5. Flow of moderately cohesive Avicel-102 in both the hopper at time (a) t ¼0 s and (b) time ¼0.5 s, (b) top row are the snapshots from experiments and the bottom row shows the same from simulation.
Fig. 6. Flow of very cohesive regular lactose in the 751 hopper at time (a) t ¼ 0 s and (b) time ¼0.5 s.
material. At the transition from the vertical-sided section to the angular section, a pressure peak occurs which is a consequence of the sudden change in wall slope. For a similar reason, the pressures are greater below the transition than above it. For both cases, as cohesion increases, the mean wall pressure increases.
The pressure distributions are in accordance with both analytical (Walters, 1973) and two dimensional numerical (Yang and Hsiau, 2001) simulations. As mentioned earlier, there are essentially two types of flow through a hopper, mass flow and funnel flow. In this study, we use
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0.045 K=0 K = 45 K = 60 K = 75
0.04
Height (m)
0.035 dz
h
0.03 0.025 0.02 0.015 0.01 0.005 0
Fig. 7. Cross sectional area of a trapezoidal section for particle simulation.
200
400 600 Wall Pressure (Pa)
0.10
1800 k=0 K = 45 K = 60 K = 75
1600 1400 1200 1000 800
800
1000
K=0 K = 45 K = 60 K = 75
0.08 Height (m)
Pressure (Pa)
0
0.06 0.04
600 400
0.02
200 0 0.00
0.00 0.01
0.02 0.03 Height (m)
0.04
0.05
K=0 K = 45 K = 60 K = 75
3000 Pressure (Pa)
100
200 300 400 Wall Pressure (KPa)
500
600
Fig. 9. (a) Variation of wall pressure with height for the hopper with angle 451 and (b) 751 as a function of granular cohesion for the dynamic (discharge case).
3500
2500 2000 1500 1000 500 0 0.0000
0
0.0200
0.0400 0.0600 Average Height (m)
0.0800
0.1000
Fig. 8. (a) Variation of wall pressure with height for the hopper with angle 451 and (b) 751 as a function of granular cohesion for the static case.
DEM simulations to illustrate the two types of flow. The radial distributions of velocities at different heights in both hoppers are examined to illustrate the different flow behaviors in hoppers. At a particular height of the hopper, the lateral plane is subdivided radially into 5 sections. Fig. 10a shows the variations in velocity at different radial points for the 451 hopper. Here, the velocities are almost zero near the wall and increase to a maximum at the center of the hopper. The average velocity near the wall is 25% of that measured near the middle of the hopper. This velocity profile
resembles core or funnel flow in hoppers. The figure also shows the results for cohesion. The flow behavior in this hopper is independent of the cohesiveness of the material. The radial distribution of average velocity in the hopper with steeper angle (751) is shown in Fig. 10b. In this case, the velocity distribution across the radial direction of the hopper is more uniform. The velocities near the wall are approximately 80% of the velocities estimated at the center of the hopper, illustrating mass flow behavior for the steeper hopper. Even though it is difficult to quantitatively compare this result to experimental findings, visual observation of powder flow through hoppers readily shows that as the hopper angle becomes steeper, flow changes from funnel flow to mass flow for a fixed hopper opening (Stanford and DellaCorte, 2006; Tomas and Schubert, 1979; Thorpe, 1992). The effect of cohesion is not as prominent in this hopper as the velocity distribution is fairly similar for the three cases (K¼ 0, 45, 60). Another important observation is that while discharging the velocities are a magnitude lower for the steeper hopper.
6. Conclusion The GDR provides an effective and convenient method for examining flow properties of pharmaceutical materials, pure and in mixtures. A clear application demonstrated here is its use as a laboratory method for monitoring flow of raw ingredients and API batches through small scale hoppers as a function of material and processing properties such as particle size, moisture content, and
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Fig. 10. Radial distribution of average absolute velocity of granular materials of various levels of cohesion in a 451 hopper (a) and (b) 751 hopper.
blend composition. An excellent experimental correlation between GDR flow index and hopper flow is observed. As flow index increased, it became increasingly difficult for the powder to flow through the hopper. Simulations were performed to understand the dynamics of flow in hoppers by using the same parameters as that obtained for flow in rotating cylinder. The simulations showed excellent qualitative agreement with the experiments, as cohesion increased; it became harder for the particles to flow through the hoppers. It also showed the effect of hopper angle; similar to experiments for Avicel, K ¼60 did not flow through the 451 hopper but flowed well in a 751 hopper. Simulations were also used to calculate the normal forces on the walls of the hopper and eventually the wall pressure distributions in both hoppers. As the depth increased, the wall pressure increases, which is opposite to the behavior observed for fluids. Finally, the simulations also
helped in understanding the different flow behavior (funnel and mass flow) that exists in a hopper. The correlation between the dynamics of flow in a rotating drum and hopper were extremely similar to the experimental behavior of the avalanches for the pure powder, showing that the parameters obtained for cohesion are independent of geometry. Above all, the simulation parameters initially validated using a rotating drum predicted the flow in hoppers, indicating that the DEM algorithm is device independent.
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