Numerical modelling of the breakage of loose ... - Semantic Scholar
Powder Technology 196 (2009) 213–221
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Numerical modelling of the breakage of loose agglomerates of fine particles Z.B. Tong a, R.Y. Yang a,⁎, A.B. Yu a, S. Adi b, H.K. Chan b a b
Laboratory for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia Faculty of Pharmacy, University of Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Article history: Received 1 June 2009 Received in revised form 30 July 2009 Accepted 1 August 2009 Available online 9 August 2009 Keywords: Loose agglomerates Breakage Discrete element method Fine particles
a b s t r a c t This paper presents a numerical study of the breakage of loose agglomerates based on the discrete element method. Agglomerates of fine mannitol particles were impacted with a target wall at different velocities and angles. It was observed that the agglomerates on impact experienced large plastic deformation before disintegrating into small fragments. The velocity field of the agglomerates showed a clear shear zone during the impacts. The final breakage pattern was characterised by the damage ratio of agglomerates and the size distribution of fragments. While increasing impact velocity improves agglomerate breakage, a 45-degree impact angle provides the maximum breakage for a given velocity. The analysis of impact energy exerted from the wall indicated that impact energy in both normal and tangential directions should be considered to characterise the effects of impact velocity and angle. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Agglomerates are often presented as intermediates or manufactured products in many industries such as chemical, pharmaceutical and food industries [1]. The knowledge of the mechanical strength and breakage pattern of agglomerates under various conditions are important to process control and optimisation. Many studies have been carried out to investigate the fundamentals governing agglomerate breakage [2–4]. Experimentally it is difficult to obtain detailed and quantitative information about breakage mechanisms due to small size of agglomerates and short duration of impacts, so experimental investigations are largely restricted to post-impact analysis of the fragments [5–7]. Numerical models based on the discrete element method (DEM) are increasingly applied to investigate agglomerate breakage since they are able to provide information that is difficult to access from experiments (e.g. forces on individual particles). By modelling the impact of an agglomerate with a target wall at different velocities, Thornton et al. [8,9] observed a minimum velocity below which no significant damage occurs and the agglomerate behaves like a large single particle. The results from Subero et al. [10] showed that the extent of breakage increases with impact velocity, but eventually reaches a limit beyond which the breakage approaches an asymptotic value. Moreno et al. [11,12] studied the effect of impact angle on the breakage of agglomerates and found that the normal component of impact velocity is the dominant factor. They also presented a
mechanistic model based on the modified Weber number to describe the effect of surface energy of powder. In many processes particles are loosely bonded and the formed agglomerates are soft and weak in strength. For example, agglomerates in dry powder inhalers (DPIs) are weakly formed in order to have easy dispersion when in use [13]. Comparing with hard agglomerates, loose agglomerates are expected to behave quite differently on impact and often disintegrate into many small pieces instead of breaking into several large chunks. So far, study of the breakage of such agglomerates is still limited [14,15] and there are many issues unaddressed. For example, how do loose agglomerates behave on impact? How do the structures and bonds inside agglomerates evolve with time? How do impact velocity and angle affect the extent of breakage? And how does one quantitatively predict these effects? The answers to these questions require a detailed and systematic study. Recently Yang et al. [16] simulated the agglomeration of fine particles down to 5 μm in size by DEM models and analysed the structures and tensile strengths of the formed agglomerates. Their results showed that the strength of agglomerates can be predicted by the modified Rumpf model [17]. This paper is to extend that work by investigating the breakage of loose agglomerates, aiming to understand the underlying mechanisms. While the breakage of agglomerates is affected by many variables associated with powder property and impact condition, this work focuses mainly on the effects of two variables: impact velocity and impact angle.
2. DEM model and simulation condition ⁎ Corresponding author. E-mail address: [email protected] (R.Y. Yang). 0032-5910/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2009.08.001
The 3D DEM model used previously [16,18] is adopted in this work. The model treats granular materials as an assembly of discrete
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Table 1 Parameters used in the simulations. Parameter
Base value (varying range)
Particle size, d Particle density, ρp Young's modulus, Y Poisson's ratio, σ̃ Friction coefficient, μs Rolling friction coefficient, μr Normal damping coefficient, γn Particle Hamaker constant, Ha Wall Hamaker constant, Haw Impact angle, θ Impact velocity, v
5 μm 1490 kg m− 3 1 × 108 N m− 2 0.29 0.3 0.002 2 × 10− 6 s− 1 1.2 × 10− 19 J 6.5 × 10− 20 J 45° (10°–90°) 10 m/s (3–10 m/s)
particles whose trajectories and rotations can be described by Newton's law of motion given by mi
dvi n s v c = ∑ðF ij + F ij + F ij Þ + F i dt j
ð1Þ
and Ii
dωi s n ˆ iÞ = ∑ðRi × F ij −μr Ri j F ij j ω dt j
ð2Þ
where vi, ωi, mi and Ii are, respectively, the translational and angular velocities, the mass and the moment of inertia of the particle i. Ri is the vector running from the centre of the particle to the contact point with its magnitude equal to particle radius Ri. Fnij , Fsij and Fvij are the normal force, the tangential force and the van der Waals force. No gravity is considered in the present work and the agglomerates are formed under an assumed centripetal force Fc which has a magnitude of a particle weight and a direction pointing to the origin of the Cartesian coordinates (the centre of the agglomerate). Details of the force models can be found from the previous work [16]. Table 1 lists the parameters used in the simulations. The mannitol powder is used in this work, which has a measured surface energy γ ranging from 47.9 to 73.3 mJ/m2 [19,20]. This is equivalent to the Hamaker constant Ha = 1.0–1.5 × 10− 19 J [21]. In this work Ha = 1.2 × 10− 19 J. The target wall has the same material properties as the particles except for the Hamaker constant which is 6.5 × 10− 20 J. Similar to our previous work [16], simulations began with the random generation of 5000 mono-sized, non-overlapped particles in a spherical space. Under the effect of the centripetal force, the particles moved toward the space centre. The process continued until all the particles reached their stable positions when essentially all the kinetic energies of the particles were dissipated. The formed agglomerate had a final diameter of about 100 μm with a packing fraction of 0.552. The agglomerate then impacted on a target wall with different velocities and angles. Each impact was simulated four times with the
Fig. 1. Snapshots of the normal impact between the agglomerate and wall at v = 10 m/s: (a) numerical simulation, and (b) experimental observation [14].
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3. Results and discussion
Fig. 2. Time evolution of the total contact force on the wall and the number of fragments at v = 10 m/s and θ = 45°.
agglomerate being rotated randomly to provide statistically reliable results. In total two hundred impacts were simulated. In the following sections, the effects of impact angle and velocity on the agglomerate breakage are investigated in terms of the structural evolution and final breakage pattern of the agglomerates.
Fig. 1a shows the snapshots of the normal impact between the agglomerate and the wall at a velocity of 10 m/s. The agglomerate experiences a large plastic deformation on impact before disintegrating into many small pieces. Fig. 1b is the experimental observation from Ning et al. [14] captured with high speed photography. Their agglomerate was formed with lactose powder with size ranging from 1 to 10 μm. While the two results are not comparable quantitatively as the conditions are not exactly the same, the simulations do reproduce some important features observed in the experiments, such as large deformation and disintegration. The qualitative agreement between the numerical simulation and experimental observation confirms the validity of the proposed model. Fig. 2 shows the time evolutions of the total contact force on the wall and the number of fragments with v = 10 m/s and θ = 45°. Here a piece of fragment is defined as an assembly of particles which are bound together but are separated from other particles. Three periods can be qualitatively identified from the figure based on the change of the number of fragments, namely, impact, breakage and steady periods. The impact period is limited to about 30 μs in which the force on the wall first increases sharply when the agglomerate approaches the plane and then decreases after reaching the maximum at t = 4 μs. However, the number of fragments has no visible change in this period since the impact is yet to propagate to the particles inside the agglomerate. Therefore the impact energy is mainly adsorbed through the local rearrangement of particles at the contact region. At the breakage period (30 μs b t b 150 μs), the particle–wall force is
Fig. 3. Snapshots of the agglomerate at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; (d) 10 μs; and (e) 150 μs with θ = 45° and v = 10 m/s (only part of the fragments are shown for t = 150 μs).
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Fig. 4. Snapshots of porosity distribution at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; and (d) 10 μs with θ = 45° and v = 10 m/s.
minimum while the number of fragments increases sharply due to the continuous propagation of the impact from the contact region towards the whole agglomerate. After t = 150 μs, both the number of fragments and the particle–wall interactions have little change, indicating the breakage process is complete. In the following sections, the time evolutions of agglomerate structure and stress during the impact period are investigated as this period is critical to the final breakage pattern of the agglomerates which is discussed in the section of post-impact analysis. 3.1. Dynamics of impact The following analysis is based on the impact of v = 10 m/s and θ = 45° unless stated otherwise. Fig. 3 shows the change of an agglomerate on impact which shows a large deformation but no visible crack. Such ductile property of the agglomerate, caused by the
weak bonds between particles, is also observed in other impacts with different impact conditions. This behaviour is different from that of hard agglomerates [22,23], but similar to the experimental results obtained from weak lactose agglomerates [14]. Meanwhile, the porosity of the agglomerate also changes accordingly, as shown in Fig. 4. The spatial distribution of porosity in an agglomerate is obtained by dividing the agglomerate into a series of cells of 2 particle diameters and calculating the local porosity for each cell. The porosity of the original agglomerate increases slightly from the centre to the surface due to the decreasing curvature with increasing radius, as discussed in our previous paper [16]. The agglomerate becomes denser at the contact region as it approaches the wall (Fig. 4b and c) but returns to a looser state when it moves away from the wall (Fig. 4d). Fig. 5 shows the spatial distributions of particle velocities at different times. Before the impact, all the particles in the agglomerate
Fig. 5. Spatial distribution of velocity at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; and (d) 10 μs with θ = 45° and v = 10 m/s (only central slices of thickness of two particle diameters are shown).
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Fig. 6. (a) Probability density functions; and (b) mean values of particle velocity with θ = 45° and v = 10 m/s.
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have a uniform velocity of 10 m/s. Once the agglomerate is in contact with the wall, the particles near the contact region decelerate due to the forces exerted from the wall, whereas the particles further away from the contact region still move at the initial velocity. Therefore, a shear zone can be observed due to the velocity gradient. This is different from hard agglomerates which have little deformations on impact and often break into several major fragments and particles in each fragment have the same velocity [9,23]. This indicates that the extent of agglomerate deformation has a significant effect on velocity distribution. Fig. 6 shows the statistical distributions and mean values of particle velocities at different times. Fig. 6a indicates that the distribution of particle velocity becomes wider with time due to the increasing number of particles being decelerated. Fig. 6b shows that the mean velocity decreases at an increasing rate until t = 4 μs when the total force on the wall reaches a maximum value and then decreases at a slower rate. Two types of forces, i.e. contact force and van der Waals force, are involved in the impact processes and they behave collectively to determine the agglomerate breakage. The knowledge of these forces is therefore critical to understanding the breakage mechanisms. The contact forces are analysed here as the variations of the other forces (i.e. the van der Waals forces) can be reflected directly from the contact forces [24]. Fig. 7 shows the network of normal contact forces at different times, in which stick represents forces between the particles and the thickness represents the magnitude of the forces. Note that only the centre slices of two particle diameter thickness are shown for easy visualisation. In the original agglomerate, the cohesive van der Waals forces are several orders of magnitude larger than a particle weight, so the contact forces are mainly to counter the cohesive forces and the formed network is relatively homogeneous [16]. At the loading stage, larger forces due to the interaction between the agglomerate and the wall appear near the contact region propagating upwards. At the unloading stage, the force network becomes more uniform again as the impact force from the wall decreases and the van der Waals forces become dominant again. The quantitative description of the distributions and the mean values of
Fig. 7. Spatial distribution of inter-particle contact force at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; and (d) 10 μs with θ = 45° and v = 10 m/s (only central slices of thickness of two particle diameters are shown).
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forces at other times have similar distributions, showing that the contact force distribution become narrower again at the unloading stage. Fig. 8b shows that the mean normal contact force increases from the beginning of the impact to 4 μs and then decreases. Changing impact angle affects breakage behaviour of agglomerate. This can be observed from the variation of the velocity and force profiles within an agglomerate as shown in Figs. 9 and 10, which are obtained when the agglomerate has the largest deformation and therefore the strongest interactions with the wall. Note that the largest deformation of the agglomerate occurs at different times for different impact angles. Fig. 9 shows that at the impact angle of 10°, there is a minimum velocity gradient at the impact surface. As impact angle increases, the affected region increases and reaches the maximum at θ = 45°. At the angle of 90°, the affected region becomes more localised again. Similar changes can also be observed from Fig. 10 which plots the distributions of normalised contact forces at different impact angles. At a low impact angle of 10°, the force structure of the agglomerate is almost intact except for a few large forces in a small region near the contact surface. With the increase in the impact angle to 45°, the area having large forces also increases, similar to those in Fig. 9. At the impact angle of 90°, forces with very large magnitudes can be observed, but the covered area is less than that of 45°. As the agglomerate is loosely formed, so even a small force can break existing bonds. In that sense, the area affected by the impact forces is more important that the magnitude of forces. This hypothesis will be further confirmed in the following section. 3.2. Post-impact analysis
Fig. 8. (a) Probability density functions in a log–linear scale; and (b) mean values of normal contact force with θ = 45° and v = 10 m/s.
the contact forces are shown in Fig. 8. The normalised force distribution is widest at the loading stage (t = 2 μs) since the forces in the contact region are much larger than those inside. The contact
This section is to quantitatively characterise the breakage pattern of agglomerates in the final stage in which there is almost no change in terms of fragment size and forces acting on the agglomerates. Two parameters, i.e. damage ratio of agglomerate and size distribution of fragments, are used to investigate the effects of impact velocity and angle. Note that impact velocity in the following discussion means impact speed (i.e. the magnitude of velocity). Damage ratio of agglomerates is defined as the ratio of the number of contacts broken on impact to the total number of contacts existing before the impact. It was used in the previous studies to characterise agglomerate breakage [8,25]. The results of Moreno et al. [11] showed
Fig. 9. Velocity fields at different impact angles: (a) 10°; (b) 30°; (c) 45°; and (d) 90° with v = 10 m/s.
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Fig. 10. Spatial distribution of inter-particle contact force at different impact angles: (a) 10°; (b) 30°; (c) 45°; and (d) 90° with v = 10 m/s.
that the damage ratio of hard agglomerates increases with both impact velocity and angle. Fig. 11 shows the current results of the variations of damage ratio with impact angle for different impact velocities. The figure indicates that the damage ratio increases with impact velocity, which is consistent with the previous findings obtained from hard agglomerates [11]. On the other hand, the damage ratio increases sharply as the impact angle increases from 10° to 30°, but levels off from 30° to 60° before another increase at 90°. Although the results agree with the findings for hard agglomerates, damage ratio cannot identify the optimal impact angle for maximum breakage, as will be shown later. This is because loose agglomerates have large deformations on impact. So unlike hard agglomerates in which only old bonds are broken, for loose agglomerates many new bonds are also created. Therefore, damage ratio is not a proper index to describe the breakage of loose agglomerates, and other parameters should be pursued for this purpose. Fig. 12 plots the weight-based cumulative size distribution of fragments under different impact conditions. The size of a fragment is
Fig. 11. Damage ratio as a function of impact angle for different impact velocities.
Fig. 12. Cumulative size distribution (by mass) of fragments after impact of different impact conditions: (a) different impact velocities at θ = 45°; and (b) different impact angles at v = 10 m/s.
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characterised by the equivalent diameter of a sphere having the same mass of the fragment. Fig. 12a shows that with increasing impact velocity, the curve shifts to smaller size, giving better breakage with more smaller fragments. This is consistent with the analysis using damage ratio and previous reports [14,22]. However, Fig. 12b shows that the impacts at the angle between 30° and 60° produce smaller fragments than those at 10° or 90°. In particular, the impacts at the angle of 45° have the best breakage performance for a given impact velocity. To quantitatively investigate the effects of impact velocity and angle, the 80% undersize (by mass) P80 in the distributions shown in Fig. 12 is chosen to represent the fragment size. Fig. 13 shows P80 as a function of impact angle for different impact velocities. P80 has been normalised against the diameter of mass-equivalent sphere of the original agglomerate. Increasing impact velocity generally has better breakage performance. Impact angle at 45°, however, gives the lowest P80 value at a given impact velocity, indicating better breakage performance. This is different from the previous work of hard agglomerates which suggested that the normal component of impact velocity is the dominant factor in controlling breakage [11]. The normalised P50 values are also used and show similar trends (see inset in Fig. 13). Since the breakage of agglomerates in this work is solely caused by agglomerate–wall impact, it is useful to analyse the work or energy from the wall in order to understand the breakage mechanisms. Fig. 14 plots the impact energy in the normal (Epw,n) and tangential (Epw,t) directions as well as the total impact energy (Epw) under different impact velocities and angles. The particle–wall interaction energy is given by t
t
Epw = Epw;n + Epw;t = ∫0 ðvi;n ⋅F i;n Þdt + ∫0 ðvi;t ⋅F i;t Þdt
ð4Þ
where Fi,n, Fi,t, vi,n, and vi,t are the forces and velocities of particle i in the normal and tangential directions, respectively. The summation is applied to all the particles which have contacts with the wall. t is the total impact time beyond which there is no significant change to fragment sizes (periods 1 and 2 in Fig. 2). As shown in Fig. 14a, the normal impact energy Epw,n increases with the impact velocity and has almost a linear increase with the impact angle for a given impact velocity. On the other hand, Fig. 14b shows that the tangential impact energy reaches the maximum when the impact angle is between 30° and 45°. However, the dominance of 45° over 30° can be more clearly reflected in Fig. 14c in which the total impact energy is considered, in particular for large velocities. This is because the normal impact energy at 45° is larger than that at 30°. Another observation is that the total impact energy at 90° is larger than that at 10°. This is also reflected in
Fig. 14. Agglomerate–wall impact energy at different impact velocities and angles: (a), normal impact energy; (b) tangential impact energy; and (c) total impact energy.
Fig. 13 showing that the 90° impact has a better breakage. Therefore, it can be concluded that the breakage of a loose agglomerate is closely related to the total impact energy from the wall, and the breakage efficiency is governed by the combined effect of normal and tangential interactions between the particles and the wall. 4. Conclusions Fig. 13. Normalised P80 and P50 (inset) as a function of impact angle for different impact velocities.
A DEM study has been conducted on the breakage of agglomerates which are loosely formed of mannitol powders. The agglomerates
Z.B. Tong et al. / Powder Technology 196 (2009) 213–221
impact on a target wall at different angles and velocities. The structural and force evolutions of the agglomerates on impact and the final breakage patterns are analysed. The results show: • The formed agglomerates are weak in strength, and therefore experience large plastic deformation on impact before disintegrating into small pieces. The impact process can be divided into three periods in which the microscopic structures of the agglomerates (i.e. porosity, velocity profile and force) evolve differently. The agglomerate becomes denser as it approaches the wall but returns to a looser state when it moves away from the wall. A shear zone can be observed in the velocity fields of the agglomerates. Larger forces are developed within the agglomerate near the contact region and propagate upwards. • The damage ratio of agglomerates increases with the increase of either impact velocity or impact angle. It, however, fails to identify the optimal impact angle and therefore is not suitable for characterising breakage of loose agglomerates. This is because for loose agglomerates, new bonds are also created while old bonds are broken on impact. The size distribution of fragments, on the other hand, shows that while increasing impact velocity generally has better breakage performance, impact angle at 45° can give the maximum breakage performance for a given impact velocity. • While agglomerate breakage is caused by the broken bonds inside the agglomerate, all the energy to break the bonds is exerted from the target wall. So the particle–wall impact energy should be used to interpret the breakage pattern of agglomerates. For breakage of loose agglomerates, the impact energies in normal and tangential directions are comparable and both should be considered in order to describe breakage performance properly. It should be noted that while this work focuses on the effect of impact condition on agglomerate breakage, many other parameters also have significant influences, including particle size and size distribution, particle surface tension as well as the presence of fluid. These will be investigated in the future work. Acknowledgement Authors are grateful to the Australia Research Council (ARC) for the financial support for this work.
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Numerical modelling of the breakage of loose agglomerates of fine particles Z.B. Tong a, R.Y. Yang a,⁎, A.B. Yu a, S. Adi b, H.K. Chan b a b
Laboratory for Simulation and Modelling of Particulate Systems, School of Materials Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia Faculty of Pharmacy, University of Sydney, NSW 2006, Australia
a r t i c l e
i n f o
Article history: Received 1 June 2009 Received in revised form 30 July 2009 Accepted 1 August 2009 Available online 9 August 2009 Keywords: Loose agglomerates Breakage Discrete element method Fine particles
a b s t r a c t This paper presents a numerical study of the breakage of loose agglomerates based on the discrete element method. Agglomerates of fine mannitol particles were impacted with a target wall at different velocities and angles. It was observed that the agglomerates on impact experienced large plastic deformation before disintegrating into small fragments. The velocity field of the agglomerates showed a clear shear zone during the impacts. The final breakage pattern was characterised by the damage ratio of agglomerates and the size distribution of fragments. While increasing impact velocity improves agglomerate breakage, a 45-degree impact angle provides the maximum breakage for a given velocity. The analysis of impact energy exerted from the wall indicated that impact energy in both normal and tangential directions should be considered to characterise the effects of impact velocity and angle. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Agglomerates are often presented as intermediates or manufactured products in many industries such as chemical, pharmaceutical and food industries [1]. The knowledge of the mechanical strength and breakage pattern of agglomerates under various conditions are important to process control and optimisation. Many studies have been carried out to investigate the fundamentals governing agglomerate breakage [2–4]. Experimentally it is difficult to obtain detailed and quantitative information about breakage mechanisms due to small size of agglomerates and short duration of impacts, so experimental investigations are largely restricted to post-impact analysis of the fragments [5–7]. Numerical models based on the discrete element method (DEM) are increasingly applied to investigate agglomerate breakage since they are able to provide information that is difficult to access from experiments (e.g. forces on individual particles). By modelling the impact of an agglomerate with a target wall at different velocities, Thornton et al. [8,9] observed a minimum velocity below which no significant damage occurs and the agglomerate behaves like a large single particle. The results from Subero et al. [10] showed that the extent of breakage increases with impact velocity, but eventually reaches a limit beyond which the breakage approaches an asymptotic value. Moreno et al. [11,12] studied the effect of impact angle on the breakage of agglomerates and found that the normal component of impact velocity is the dominant factor. They also presented a
mechanistic model based on the modified Weber number to describe the effect of surface energy of powder. In many processes particles are loosely bonded and the formed agglomerates are soft and weak in strength. For example, agglomerates in dry powder inhalers (DPIs) are weakly formed in order to have easy dispersion when in use [13]. Comparing with hard agglomerates, loose agglomerates are expected to behave quite differently on impact and often disintegrate into many small pieces instead of breaking into several large chunks. So far, study of the breakage of such agglomerates is still limited [14,15] and there are many issues unaddressed. For example, how do loose agglomerates behave on impact? How do the structures and bonds inside agglomerates evolve with time? How do impact velocity and angle affect the extent of breakage? And how does one quantitatively predict these effects? The answers to these questions require a detailed and systematic study. Recently Yang et al. [16] simulated the agglomeration of fine particles down to 5 μm in size by DEM models and analysed the structures and tensile strengths of the formed agglomerates. Their results showed that the strength of agglomerates can be predicted by the modified Rumpf model [17]. This paper is to extend that work by investigating the breakage of loose agglomerates, aiming to understand the underlying mechanisms. While the breakage of agglomerates is affected by many variables associated with powder property and impact condition, this work focuses mainly on the effects of two variables: impact velocity and impact angle.
2. DEM model and simulation condition ⁎ Corresponding author. E-mail address: [email protected] (R.Y. Yang). 0032-5910/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2009.08.001
The 3D DEM model used previously [16,18] is adopted in this work. The model treats granular materials as an assembly of discrete
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Table 1 Parameters used in the simulations. Parameter
Base value (varying range)
Particle size, d Particle density, ρp Young's modulus, Y Poisson's ratio, σ̃ Friction coefficient, μs Rolling friction coefficient, μr Normal damping coefficient, γn Particle Hamaker constant, Ha Wall Hamaker constant, Haw Impact angle, θ Impact velocity, v
5 μm 1490 kg m− 3 1 × 108 N m− 2 0.29 0.3 0.002 2 × 10− 6 s− 1 1.2 × 10− 19 J 6.5 × 10− 20 J 45° (10°–90°) 10 m/s (3–10 m/s)
particles whose trajectories and rotations can be described by Newton's law of motion given by mi
dvi n s v c = ∑ðF ij + F ij + F ij Þ + F i dt j
ð1Þ
and Ii
dωi s n ˆ iÞ = ∑ðRi × F ij −μr Ri j F ij j ω dt j
ð2Þ
where vi, ωi, mi and Ii are, respectively, the translational and angular velocities, the mass and the moment of inertia of the particle i. Ri is the vector running from the centre of the particle to the contact point with its magnitude equal to particle radius Ri. Fnij , Fsij and Fvij are the normal force, the tangential force and the van der Waals force. No gravity is considered in the present work and the agglomerates are formed under an assumed centripetal force Fc which has a magnitude of a particle weight and a direction pointing to the origin of the Cartesian coordinates (the centre of the agglomerate). Details of the force models can be found from the previous work [16]. Table 1 lists the parameters used in the simulations. The mannitol powder is used in this work, which has a measured surface energy γ ranging from 47.9 to 73.3 mJ/m2 [19,20]. This is equivalent to the Hamaker constant Ha = 1.0–1.5 × 10− 19 J [21]. In this work Ha = 1.2 × 10− 19 J. The target wall has the same material properties as the particles except for the Hamaker constant which is 6.5 × 10− 20 J. Similar to our previous work [16], simulations began with the random generation of 5000 mono-sized, non-overlapped particles in a spherical space. Under the effect of the centripetal force, the particles moved toward the space centre. The process continued until all the particles reached their stable positions when essentially all the kinetic energies of the particles were dissipated. The formed agglomerate had a final diameter of about 100 μm with a packing fraction of 0.552. The agglomerate then impacted on a target wall with different velocities and angles. Each impact was simulated four times with the
Fig. 1. Snapshots of the normal impact between the agglomerate and wall at v = 10 m/s: (a) numerical simulation, and (b) experimental observation [14].
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3. Results and discussion
Fig. 2. Time evolution of the total contact force on the wall and the number of fragments at v = 10 m/s and θ = 45°.
agglomerate being rotated randomly to provide statistically reliable results. In total two hundred impacts were simulated. In the following sections, the effects of impact angle and velocity on the agglomerate breakage are investigated in terms of the structural evolution and final breakage pattern of the agglomerates.
Fig. 1a shows the snapshots of the normal impact between the agglomerate and the wall at a velocity of 10 m/s. The agglomerate experiences a large plastic deformation on impact before disintegrating into many small pieces. Fig. 1b is the experimental observation from Ning et al. [14] captured with high speed photography. Their agglomerate was formed with lactose powder with size ranging from 1 to 10 μm. While the two results are not comparable quantitatively as the conditions are not exactly the same, the simulations do reproduce some important features observed in the experiments, such as large deformation and disintegration. The qualitative agreement between the numerical simulation and experimental observation confirms the validity of the proposed model. Fig. 2 shows the time evolutions of the total contact force on the wall and the number of fragments with v = 10 m/s and θ = 45°. Here a piece of fragment is defined as an assembly of particles which are bound together but are separated from other particles. Three periods can be qualitatively identified from the figure based on the change of the number of fragments, namely, impact, breakage and steady periods. The impact period is limited to about 30 μs in which the force on the wall first increases sharply when the agglomerate approaches the plane and then decreases after reaching the maximum at t = 4 μs. However, the number of fragments has no visible change in this period since the impact is yet to propagate to the particles inside the agglomerate. Therefore the impact energy is mainly adsorbed through the local rearrangement of particles at the contact region. At the breakage period (30 μs b t b 150 μs), the particle–wall force is
Fig. 3. Snapshots of the agglomerate at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; (d) 10 μs; and (e) 150 μs with θ = 45° and v = 10 m/s (only part of the fragments are shown for t = 150 μs).
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Fig. 4. Snapshots of porosity distribution at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; and (d) 10 μs with θ = 45° and v = 10 m/s.
minimum while the number of fragments increases sharply due to the continuous propagation of the impact from the contact region towards the whole agglomerate. After t = 150 μs, both the number of fragments and the particle–wall interactions have little change, indicating the breakage process is complete. In the following sections, the time evolutions of agglomerate structure and stress during the impact period are investigated as this period is critical to the final breakage pattern of the agglomerates which is discussed in the section of post-impact analysis. 3.1. Dynamics of impact The following analysis is based on the impact of v = 10 m/s and θ = 45° unless stated otherwise. Fig. 3 shows the change of an agglomerate on impact which shows a large deformation but no visible crack. Such ductile property of the agglomerate, caused by the
weak bonds between particles, is also observed in other impacts with different impact conditions. This behaviour is different from that of hard agglomerates [22,23], but similar to the experimental results obtained from weak lactose agglomerates [14]. Meanwhile, the porosity of the agglomerate also changes accordingly, as shown in Fig. 4. The spatial distribution of porosity in an agglomerate is obtained by dividing the agglomerate into a series of cells of 2 particle diameters and calculating the local porosity for each cell. The porosity of the original agglomerate increases slightly from the centre to the surface due to the decreasing curvature with increasing radius, as discussed in our previous paper [16]. The agglomerate becomes denser at the contact region as it approaches the wall (Fig. 4b and c) but returns to a looser state when it moves away from the wall (Fig. 4d). Fig. 5 shows the spatial distributions of particle velocities at different times. Before the impact, all the particles in the agglomerate
Fig. 5. Spatial distribution of velocity at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; and (d) 10 μs with θ = 45° and v = 10 m/s (only central slices of thickness of two particle diameters are shown).
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Fig. 6. (a) Probability density functions; and (b) mean values of particle velocity with θ = 45° and v = 10 m/s.
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have a uniform velocity of 10 m/s. Once the agglomerate is in contact with the wall, the particles near the contact region decelerate due to the forces exerted from the wall, whereas the particles further away from the contact region still move at the initial velocity. Therefore, a shear zone can be observed due to the velocity gradient. This is different from hard agglomerates which have little deformations on impact and often break into several major fragments and particles in each fragment have the same velocity [9,23]. This indicates that the extent of agglomerate deformation has a significant effect on velocity distribution. Fig. 6 shows the statistical distributions and mean values of particle velocities at different times. Fig. 6a indicates that the distribution of particle velocity becomes wider with time due to the increasing number of particles being decelerated. Fig. 6b shows that the mean velocity decreases at an increasing rate until t = 4 μs when the total force on the wall reaches a maximum value and then decreases at a slower rate. Two types of forces, i.e. contact force and van der Waals force, are involved in the impact processes and they behave collectively to determine the agglomerate breakage. The knowledge of these forces is therefore critical to understanding the breakage mechanisms. The contact forces are analysed here as the variations of the other forces (i.e. the van der Waals forces) can be reflected directly from the contact forces [24]. Fig. 7 shows the network of normal contact forces at different times, in which stick represents forces between the particles and the thickness represents the magnitude of the forces. Note that only the centre slices of two particle diameter thickness are shown for easy visualisation. In the original agglomerate, the cohesive van der Waals forces are several orders of magnitude larger than a particle weight, so the contact forces are mainly to counter the cohesive forces and the formed network is relatively homogeneous [16]. At the loading stage, larger forces due to the interaction between the agglomerate and the wall appear near the contact region propagating upwards. At the unloading stage, the force network becomes more uniform again as the impact force from the wall decreases and the van der Waals forces become dominant again. The quantitative description of the distributions and the mean values of
Fig. 7. Spatial distribution of inter-particle contact force at different times: (a) 0 μs; (b) 2 μs; (c) 6 μs; and (d) 10 μs with θ = 45° and v = 10 m/s (only central slices of thickness of two particle diameters are shown).
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forces at other times have similar distributions, showing that the contact force distribution become narrower again at the unloading stage. Fig. 8b shows that the mean normal contact force increases from the beginning of the impact to 4 μs and then decreases. Changing impact angle affects breakage behaviour of agglomerate. This can be observed from the variation of the velocity and force profiles within an agglomerate as shown in Figs. 9 and 10, which are obtained when the agglomerate has the largest deformation and therefore the strongest interactions with the wall. Note that the largest deformation of the agglomerate occurs at different times for different impact angles. Fig. 9 shows that at the impact angle of 10°, there is a minimum velocity gradient at the impact surface. As impact angle increases, the affected region increases and reaches the maximum at θ = 45°. At the angle of 90°, the affected region becomes more localised again. Similar changes can also be observed from Fig. 10 which plots the distributions of normalised contact forces at different impact angles. At a low impact angle of 10°, the force structure of the agglomerate is almost intact except for a few large forces in a small region near the contact surface. With the increase in the impact angle to 45°, the area having large forces also increases, similar to those in Fig. 9. At the impact angle of 90°, forces with very large magnitudes can be observed, but the covered area is less than that of 45°. As the agglomerate is loosely formed, so even a small force can break existing bonds. In that sense, the area affected by the impact forces is more important that the magnitude of forces. This hypothesis will be further confirmed in the following section. 3.2. Post-impact analysis
Fig. 8. (a) Probability density functions in a log–linear scale; and (b) mean values of normal contact force with θ = 45° and v = 10 m/s.
the contact forces are shown in Fig. 8. The normalised force distribution is widest at the loading stage (t = 2 μs) since the forces in the contact region are much larger than those inside. The contact
This section is to quantitatively characterise the breakage pattern of agglomerates in the final stage in which there is almost no change in terms of fragment size and forces acting on the agglomerates. Two parameters, i.e. damage ratio of agglomerate and size distribution of fragments, are used to investigate the effects of impact velocity and angle. Note that impact velocity in the following discussion means impact speed (i.e. the magnitude of velocity). Damage ratio of agglomerates is defined as the ratio of the number of contacts broken on impact to the total number of contacts existing before the impact. It was used in the previous studies to characterise agglomerate breakage [8,25]. The results of Moreno et al. [11] showed
Fig. 9. Velocity fields at different impact angles: (a) 10°; (b) 30°; (c) 45°; and (d) 90° with v = 10 m/s.
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Fig. 10. Spatial distribution of inter-particle contact force at different impact angles: (a) 10°; (b) 30°; (c) 45°; and (d) 90° with v = 10 m/s.
that the damage ratio of hard agglomerates increases with both impact velocity and angle. Fig. 11 shows the current results of the variations of damage ratio with impact angle for different impact velocities. The figure indicates that the damage ratio increases with impact velocity, which is consistent with the previous findings obtained from hard agglomerates [11]. On the other hand, the damage ratio increases sharply as the impact angle increases from 10° to 30°, but levels off from 30° to 60° before another increase at 90°. Although the results agree with the findings for hard agglomerates, damage ratio cannot identify the optimal impact angle for maximum breakage, as will be shown later. This is because loose agglomerates have large deformations on impact. So unlike hard agglomerates in which only old bonds are broken, for loose agglomerates many new bonds are also created. Therefore, damage ratio is not a proper index to describe the breakage of loose agglomerates, and other parameters should be pursued for this purpose. Fig. 12 plots the weight-based cumulative size distribution of fragments under different impact conditions. The size of a fragment is
Fig. 11. Damage ratio as a function of impact angle for different impact velocities.
Fig. 12. Cumulative size distribution (by mass) of fragments after impact of different impact conditions: (a) different impact velocities at θ = 45°; and (b) different impact angles at v = 10 m/s.
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characterised by the equivalent diameter of a sphere having the same mass of the fragment. Fig. 12a shows that with increasing impact velocity, the curve shifts to smaller size, giving better breakage with more smaller fragments. This is consistent with the analysis using damage ratio and previous reports [14,22]. However, Fig. 12b shows that the impacts at the angle between 30° and 60° produce smaller fragments than those at 10° or 90°. In particular, the impacts at the angle of 45° have the best breakage performance for a given impact velocity. To quantitatively investigate the effects of impact velocity and angle, the 80% undersize (by mass) P80 in the distributions shown in Fig. 12 is chosen to represent the fragment size. Fig. 13 shows P80 as a function of impact angle for different impact velocities. P80 has been normalised against the diameter of mass-equivalent sphere of the original agglomerate. Increasing impact velocity generally has better breakage performance. Impact angle at 45°, however, gives the lowest P80 value at a given impact velocity, indicating better breakage performance. This is different from the previous work of hard agglomerates which suggested that the normal component of impact velocity is the dominant factor in controlling breakage [11]. The normalised P50 values are also used and show similar trends (see inset in Fig. 13). Since the breakage of agglomerates in this work is solely caused by agglomerate–wall impact, it is useful to analyse the work or energy from the wall in order to understand the breakage mechanisms. Fig. 14 plots the impact energy in the normal (Epw,n) and tangential (Epw,t) directions as well as the total impact energy (Epw) under different impact velocities and angles. The particle–wall interaction energy is given by t
t
Epw = Epw;n + Epw;t = ∫0 ðvi;n ⋅F i;n Þdt + ∫0 ðvi;t ⋅F i;t Þdt
ð4Þ
where Fi,n, Fi,t, vi,n, and vi,t are the forces and velocities of particle i in the normal and tangential directions, respectively. The summation is applied to all the particles which have contacts with the wall. t is the total impact time beyond which there is no significant change to fragment sizes (periods 1 and 2 in Fig. 2). As shown in Fig. 14a, the normal impact energy Epw,n increases with the impact velocity and has almost a linear increase with the impact angle for a given impact velocity. On the other hand, Fig. 14b shows that the tangential impact energy reaches the maximum when the impact angle is between 30° and 45°. However, the dominance of 45° over 30° can be more clearly reflected in Fig. 14c in which the total impact energy is considered, in particular for large velocities. This is because the normal impact energy at 45° is larger than that at 30°. Another observation is that the total impact energy at 90° is larger than that at 10°. This is also reflected in
Fig. 14. Agglomerate–wall impact energy at different impact velocities and angles: (a), normal impact energy; (b) tangential impact energy; and (c) total impact energy.
Fig. 13 showing that the 90° impact has a better breakage. Therefore, it can be concluded that the breakage of a loose agglomerate is closely related to the total impact energy from the wall, and the breakage efficiency is governed by the combined effect of normal and tangential interactions between the particles and the wall. 4. Conclusions Fig. 13. Normalised P80 and P50 (inset) as a function of impact angle for different impact velocities.
A DEM study has been conducted on the breakage of agglomerates which are loosely formed of mannitol powders. The agglomerates
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impact on a target wall at different angles and velocities. The structural and force evolutions of the agglomerates on impact and the final breakage patterns are analysed. The results show: • The formed agglomerates are weak in strength, and therefore experience large plastic deformation on impact before disintegrating into small pieces. The impact process can be divided into three periods in which the microscopic structures of the agglomerates (i.e. porosity, velocity profile and force) evolve differently. The agglomerate becomes denser as it approaches the wall but returns to a looser state when it moves away from the wall. A shear zone can be observed in the velocity fields of the agglomerates. Larger forces are developed within the agglomerate near the contact region and propagate upwards. • The damage ratio of agglomerates increases with the increase of either impact velocity or impact angle. It, however, fails to identify the optimal impact angle and therefore is not suitable for characterising breakage of loose agglomerates. This is because for loose agglomerates, new bonds are also created while old bonds are broken on impact. The size distribution of fragments, on the other hand, shows that while increasing impact velocity generally has better breakage performance, impact angle at 45° can give the maximum breakage performance for a given impact velocity. • While agglomerate breakage is caused by the broken bonds inside the agglomerate, all the energy to break the bonds is exerted from the target wall. So the particle–wall impact energy should be used to interpret the breakage pattern of agglomerates. For breakage of loose agglomerates, the impact energies in normal and tangential directions are comparable and both should be considered in order to describe breakage performance properly. It should be noted that while this work focuses on the effect of impact condition on agglomerate breakage, many other parameters also have significant influences, including particle size and size distribution, particle surface tension as well as the presence of fluid. These will be investigated in the future work. Acknowledgement Authors are grateful to the Australia Research Council (ARC) for the financial support for this work.
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