Breakage behaviour of agglomerates and crystals by static loading and impact

Powder Technology 206 (2011) 88–98

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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

Breakage behaviour of agglomerates and crystals by static loading and impact Sergiy Antonyuk a,⁎, Stefan Palis b, Stefan Heinrich a a b

Institute of Solids Process Engineering and Particle Technology, Hamburg University of Technology, Denickestrasse 15, 21071 Hamburg, Germany Institute for Automation/Modelling, Otto-von-Guericke University of Magdeburg, 39106 Magdeburg, Germany

a r t i c l e

i n f o

Article history: Received 22 September 2009 Accepted 19 February 2010 Available online 26 February 2010 Keywords: Breakage behaviour Agglomerates L-threonine crystals Discrete element modelling

a b s t r a c t In this contribution the breakage behaviour of needle-shaped particles as L-threonine crystals and cylindrical Al2O3 agglomerates is studied. The breakage of Al2O3 agglomerates is investigated by compression tests. For the L-threonine crystals single particle impacts with a wall are performed. In order to obtain reproducible results an electromagnetic particle gun has been designed. The experimental results are then used to derive a discrete element model to analyze the microdynamics and breakage mechanisms of needle-shaped particles. A good qualitative agreement between simulations and tests is achieved. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Cylindrical agglomerates, e.g. tablets and briquettes, are important products of the chemical and pharmaceutical industry. During the process handling and transportation agglomerates are often subjected to repeated mechanical stressing that can lead to their breakage which may reduce the product quality. Additionally, dust can be emitted during handling. Therefore the strength is an important property of agglomerates. The pioneer model of Rumpf [1] describes the tensile strength of agglomerates depending on the interparticle adhesion force and the coordination number. Bika et al. [2] found the correlation between the strengths of dry granules and their solid bridges. Delenne et al. [3] used Mohr-Coulomb failure criterion to model and simulate deformation and breakage behaviour of a couple of cylindrical rods by different stressing or loading modes (tension, compression and shearing). The agglomerate microstructure, micro properties and bond mechanisms of primary particles show significant effects on the mechanical behaviour and breakage dynamics of agglomerates. These effects can be studied numerically using Discrete Element Method (DEM), Antonyuk [4,5]. An adequate description of agglomerate micro properties is required to improve the physical understanding of these macroscopic breakage phenomena. The general aim of this study is to combine the micromechanical properties of primary particles as well as the binding agent with the macroscopic deformation and breakage behaviour of dry agglomerates during mechanical stressing.

⁎ Corresponding author. E-mail address: [email protected] (S. Antonyuk). 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.02.025

One objective of this study is to describe the force-displacement behaviour of dry cylindrical agglomerates obtained by compression tests and to determine their mechanical properties. The second objective is to obtain the effects of the binder content and the height of cylindrical agglomerates on their properties. In the first part of this contribution the breakage behaviour of cylindrical γ-Al2O3 agglomerates is studied. Compression tests of model agglomerates have been performed. The modulus of elasticity, stiffness, yield pressure and strength of both, the agglomerate and primary particles, have been measured. The second part of this contribution includes the study of breakage behaviour of L-threonine crystals during impact, which is important as in industrial crystallization processes secondary nucleation, meaning attrition of crystals, often serves as a source of nuclei. It has therefore to be included in the process model. The main problems encountered, while investigating attrition of crystals, are: • For in-situ investigation other effects like primary nucleation and crystal resolution have to be suppressed. Moreover measurement should be taken where breakage occurs, which is quite difficult in general. • Most working principles of in-situ measurement devices like focused beam reflectance measurement (FBRM), laser diffraction or fiber optic particle counting principle assume spherical particles to determine the characteristic length distribution of a sample. For needle-shaped crystals like L-threonine measurement information therefore lacks reliability. Using video microscopic measurement devices, this disadvantage may be of course circumvented. • Single particle impact experiments as done in [7,8,13] can be observed by a high-speed camera resulting in reliable information on breakage parts. The main disadvantage here is that to gather representative

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results from single particle experiments the number of experiments has to be quite high. Therefore investigation can be very time consuming and expensive. The most appropriate setup for investigations of attrition would be a video microscopic measurement of the breakage event resulting either in a macroscopic model, e.g. a breakage kernel for a crystal population, or in a microscopic model, e.g. a discrete element model for a crystal. However, as suppressing secondary effects to gain a high reproducibility is difficult, in this contribution a small number of single particle experiments in air are performed. In order to gather representative results is a detailed investigation and modelling of the dynamic breakage processes using a discrete element model, which represents the main breakage characteristics, is performed. Such a model can be used in future to run a huge number of simulated experiments reducing time consumptions and costs. The parameters to be varied in the simulation runs are among others the crystal shape, impact angle and initial velocity. To investigate single particle impact with a wall at different velocities and impact angles an electromagnetic particle gun was designed. To analyze the micro dynamics and breakage mechanisms for the experimentally investigated γ-Al2O3 agglomerates and crystal aggregates discrete element models (DEM) are proposed. The deformation and breakage behaviour obtained by experiments is used for the calibration of model parameters. 2. Discrete element modelling Discrete element method is a tool to describe the macroscopic behaviour of a body by a huge number of micro particles and interaction forces. It was originally applied by Cundall and Strack [6] to solve problems in rock mechanics. Through DEM the individual particle and agglomerate can be modelled as a realistic mechanical system of primary particles which can be bonded by different forces [4,9]. Hence, important influences of the microstructure concerning porosity, density, primary particle distribution, stiffness of the primary particles, their elasticity or plasticity as well as binder properties on deformation and breakage behaviour can be examined. The particles in DEM are usually represented as spheres or cylinders. To model the nonspherical particles and agglomerates different methods were developed. A first method which is often used for representation of particles with the shape of an ellipsoid is construction of the particle from overlapping spheres [14]. The second method, which is more complicated, is construction of the particle from spheres, cylinders and flats [15]. The discrete element model used in this works consists of two components:

Fig. 1. Particle contact.

the particles arrangement in normal (index n) and in shear (index s) directions, l and p are the number of contacts and solid bridge bonds for particle i. Ii

→ p l → ðijÞ → ðiÞ d ωi → ðijÞ ðijÞ = ∑ ð→ r c × F c Þ + ∑ Mb + Mmg : dt j=1 j=1

Ii and ωi are the moment of inertia and angular velocity for particle i. Mb and Mmg are moments generated from the solid bridge bond and a body gravitational force mg respectively. The tangential and normal contact forces are defined as follows: → ðijÞ a F c;n = ðkc;ij;n ⋅Sij;n + ηij;n ⋅ S˙ ij;n Þ⋅→ nij ð3Þ

j

→ a ðkc;ij;s ⋅Sij;s + ηij;s ⋅ S˙ ij;s Þ⋅ t ij →ðijÞ F c;s = min ðijÞ → ðμij ⋅Fc;n Þ⋅ tij

j

Forces acting on particles are: • contact forces between two particles in contact and a particle in contact with a wall, and • forces occurring, when two particles are interacting with each other through a solid bridge bond. The equations of motion for all particles (Fig. 1) are given as follows: → p 1 d2 r i →ðijÞ → ij → ðijÞ → ðijÞ = ∑ ð F c;n + F c;s Þ + ∑ ð F b;n + F b;s Þ + mi → g 2 dt j=1 j=1

ð1Þ

where m is the mass of the particle, Fc and Fb are the contact force and bond force (solid bridge bond) between the particles i and j arising from

ð4Þ

where kc,ij,n and kc,ij,s are the contact stiffness in normal and shear direction, ηi,j,n and ηi,j,s are the normal and shear damping coefficients associated with the Rayleigh dissipation function and μij is the dynamic friction coefficient. The overlap in normal and shear direction is sij,n and sij,s. Using the constant a the contact law can be varied from Hooks law (a = 1) to Hertz contact law (a = 1.5). To describe the solid binder between primary particles of agglomerates the solid bridge bond model was implemented in DEM. A solid bridge bond is represented as parallel connected elastic springs (Fig. 2) with two springs in the direct contact of particles (contact stiffnesses ki and kj). To consider the influence of the bridge size its stiffness is described using

• primary particles, which are concerned as rigid spheres, and • solid bridges, which are concerned as mass less spring-damper elements.

mi

ð2Þ

Fig. 2. Schematic representation of solid bridge bond as system of springs.

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The forces and moments transmitted between two particles via a solid bridge bond can be reduced to normal (σij) and shear (τij) stresses acting within the cross-section of the solid bridge bond. ðijÞ

ðijÞ

σij =

−Fb;x Aij

+

ðijÞ

τij =

Fig. 3. Solid bridge bond model.

area-related stiffness kb,f. The stiffness increases with increasing diameter of the bridge. The equivalent stiffness of the “particle-bridge-particle” connection can be given according to the scheme in Fig. 2 as:

keq =

8 kb; f Aij;b > > < > > : kb; f Aij;b +

1 1 + kc;i kc;j

!−1

by direct contact of particles

0

ð5Þ

jFb;z j Aij

jMb;y j ðijÞ Ib;y

ð14Þ

ðijÞ

ð15Þ

ðijÞ

+

jMb;x j ðijÞ Jb;x

Rb :

A crucial point, when simulating cracks is that, whenever either of these stresses exceeds its associated bound σb,max and τb,max, the solid bridge will break, meaning that the solid bridge bond is deleted from the model. To simulate the discrete element model a huge number of equations of motion have to be solved numerically. As the time of impact is long in comparison to the smallest time constant in the system, many integration steps have to be taken. Therefore special care has to be taken on the integration algorithm in order to prevent artificial energy dissipation. Here a first order variational integrator [10] is used. The idea for the variational integrator is to discretize directly Hamilton's principle. tend

where Aij,b is the cross-section area of the solid bridge bond. In the case of a solid binder in the particle contact, the forces (normal Fb,x and shear Fb,z) and the moments (torsion Mb,x and bending Mb,y) are transmitted through the solid bridge bond (Fig. 3). The increments of the elastic normal and shear forces during time step k with value of Δt are calculated as follows:

ðijÞ

Rb

tend

˙ tÞdt = ∫ J = ∫ Lðq;q; 0

0

N−1

Jd = ∑ Ld ðqk ; qk k=0

+ 1 ; ΔtÞ

1 T ˙ q˙ M q−VðqÞdt→ min 2

ð16Þ

1 qk + 1 −qk T qk + 1 −qk  −Vðqk Þ→ min: M Δt Δt k=0 2 N−1

= ∑

ð17Þ

→ ðijÞ Δ F b;x;k = ð−kb;m;n ⋅Aij;b ⋅ΔSij;n Þ⋅→ nij

ð6Þ

Then the variational problem is solved using standard variational calculus,

→ðijÞ → Δ F b;z;k = ð−kb;m;s ⋅Aij;b ⋅Δsij;s Þ⋅ tij

ð7Þ

dJd dε

with the displacement: Δsij = vijΔt. The increments of the bending ΔMb,y,k and torsion ΔMb,x,k moments in cross-section area of the bridge in the time step k are given as: ðijÞ

ðijÞ

ΔMb;x;k = kb;m;s ⋅Jb;x ⋅Δφij;n

ð8Þ

ðijÞ ΔMb;y;k

ð9Þ

=

ðijÞ kb;m;n ⋅Ib;y ⋅Δφij;s

j

  ∂Ld ðqk ; qk + 1 ; ΔtÞ ∂L ðq ; q ; ΔtÞ ⋅ + d k−1 k ηk ∂qk ∂qk k=1 N−1

=0= ∑

ε=0

which results in a discrete Euler-Lagrange equation:  D2 Ld ðqk−1 ; qk ; ΔtÞ + D1 Ld ðqk ; qk + 1 ; ΔtÞ = M

→ → → F b;n;k = Fb;n nij + Δ F b;n;k

ð10Þ

→ → → F b;s;k = f F b;s;k−1 grot:2 + Δ F b;s;k

ð11Þ

→ → ðijÞ Mb;n;k = Mb;n;k + ΔMb;n;k → nij

ð12Þ

→ → → Mb;s;k = f Mb;s;k−1 grot:2 + Δ Mb;s;k

ð13Þ

{}rot.2 in Eqs. (10)–(13) denotes the rotation of the shear component vectors to account for the motion of the contact plane.

qk + 1 −2qk −qk−1 Δt 2

 +

∂Vðqk Þ = 0: ∂qk

ð19Þ Rewriting the discrete Euler-Lagrange equation gives the following time-stepping algorithm: 

where Δφij is the rotation angle between two particles. Jb,x represents the polar moment of inertia over axis x and Ib,y is the moment of inertia of the bond cross-section about the y-axis. The forces and moments in each time step (with a current number k) are found by summing of the calculated increment with the value of previous times step.

ð18Þ

qk

+ 1

=

 2qk + qk−1 ∂Vðqk Þ : − 2 ∂qk Δt

ð20Þ

The main advantage of this integration scheme is the desired energy conservation, which has to be guaranteed in order to gain reliable results. Time step plays a vital role in calculation and thus for the results. If the calculation time step is too large, the system can not represent disturbances effecting each particle. This means that the propagating waves are not sufficient to carry disturbances occurring to each particle. On the other hand, if it is too small, it unnecessarily consumes more time for the calculations, which directly increases the cost of the calculations. The solution is stable only if the time-step does not exceed a critical time-step that is related to the minimum eigenperiod of the total system. However, global eigenvalue analyses are impractical for large and constantly changing systems, generally encountered in the large discrete particle simulations. The motion of a simple mass spring system consisting of a point mass and a spring is governed by the following

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Fig. 4. (left) Picture of the experimental setup consisting of particle gun, high speed camera and baffle plate, (middle) scheme of the electromagnetic particle gun, (right) measured velocity–current dependence and its approximation (v = 0.71⋅I − 0.90).

differential equation ms̈ = −ks. The solution of the equation of motion results in an oscillation with the following eigenperiod:

T = 2⋅π

rffiffiffiffiffi m : k

ð21Þ

The critical time step corresponding to a simple mass spring system of point mass m and spring stiffness k can therefore be chosen as:

tcrit =

rffiffiffiffiffi m bT: k

ð22Þ

In DEM simulation the critical time step has to be chosen taking into account each contact corresponding to rotational and translational motion. It is therefore calculated as the minimum critical time of all translational and rotational motions:

min

tcrit = γ∀contacts

rffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi! m I ; ktrans krot

ð23Þ

where λ is an additional safety factor.

Fig. 5. (left) FEM model of the magnetic field, (right) magnetic force in dependence of the particle carrier position (the current of 40 A, the coil with the length of 60 mm and 854 turns, the carrier with the diameter of 14 mm).

3. Impact experiments with crystals For impact experiments of needle shaped particles as L-threonine falling experiments are not as suited as for spherical particles. Typical problems encountered are: • The initial potential energy is not only transformed into translational kinetic energy but into rotational energy, too. Therefore falling heights have to be considerably larger than for spherical particles to gain same velocities. • Because of the additional particle rotation and interaction with surrounding air, particles are falling mostly out of focus of the highspeed camera. One possibility to overcome these problems is to do the falling experiment in a vacuum chamber [7,8], which would result in a difficult experimental setup. Another possibility is to accelerate the particles using a particle gun (Fig. 4). Therefore, an electromagnetic particle gun has been designed consisting of a power supply, a coil, an acceleration cylinder and a tube. Using the electromagnetic particle gun the following can be achieved: • particle impact always takes places in the focus of the high-speed camera, • particle velocity is highly adjustable and reproducible by current control (Fig. 4 right), and • as there are only electrical components a future measurement automation is quite easy. The acceleration of the particle is achieved through a ferromagnetic sample carrier, which is drawn into the coil because of its lower magnetic resistance. The working principle is similar to a Gauss gun (coilgun). The main difference is that the particle carrier is decelerated after acceleration in order to stay in the electromagnetic particle gun. As the particle gun is still under development a detailed finite element model for further optimization has been developed (Fig. 5). From the finite element model of the magnetic field the relation between particle carrier position, applied current and resulting magnetic force is derived. This force characteristic is used in the dynamic model of the acceleration process.

Fig. 6. Images of impact of a L-threonine crystal with the length of 1.25 mm at 5 m/s captured using the electromagnetic particle gun (time values are given on the images in ms).

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Fig. 7. Breakage surface of a L-threonine crystal.

In Fig. 6 pictures of a single particle impact experiment of a Lthreonine crystal of 1.25 mm length at a velocity of 5 m/s are depicted. It can be seen, that the needle shaped crystal hits the baffle plate under a small angle and breaks into two parts of comparable size. The fractured surface of the crystal in Fig. 7 is smooth but not flat. It contains many defects and dislocations. 4. Compression tests with the cylindrical agglomerates The cylindrical agglomerates were prepared with spherical γAl2O3 particles (d50 = 1.03 ± 0.06 mm) and 6% water solution of the hydroxypropyl methylcellulose (HPMC) by compaction in a die (Fig. 8) and drying at 80 °C. The binding agent was heated up to 40 °C and added to the particles in a mixer. After drying over one hour the mass concentration of HPMC in agglomerates was about 4%. The height and diameter of the agglomerate had a value of 15 mm. Single diametral compression tests (Fig. 9) of the produced agglomerates were performed at a constant stressing velocity of 0.04 mm/s using Granule Strength Measuring Device (Etewe GmbH). Fifty equal-sized agglomerates were tested in order to increase the statistical significance of the measurement. Moreover, the compression tests of the primary particles were carried out. The typical force-displacement curve of a primary spherical particle (γ-Al2O3) shows clearly both elastic and plastic displacement ranges (Fig. 10 a). The elastic displacement of a spherical particle up to the yield point (F in Fig. 10 a) was described using the Hertz contact theory of spheres [4,11]. Hence, the modulus of elasticity and elastic contact stiffness were determined (Table 1). The force-displacement curve above yield point is linear showing perfectly plastic deformation behaviour with a constant stiffness as was described by Tomas [12] for spherical particles. Fig. 10 (b) shows the typical force-displacement curve of a cylindrical γ-Al2O3 agglomerate. At the beginning of loading the contacts are elastically deformed. The elastic force-displacement line up to yield point F was described using an elastic contact model [11]:

Fel = 0:39

Ecyl hcyl 1−v2cyl

S

ð24Þ

Fig. 9. Diametral loading condition.

where Ecyl is modulus of elasticity of cylindrical agglomerate. hcyl is the height of the cylinder. νcyl is the Poisson ratio and s is displacement during the deformation. The modulus of elasticity of agglomerates is much lower than that of the primary particles (Table 1). However, the elastic stiffnesses of agglomerates and primary particles are in the same order of magnitude. The force-displacement curve changes beyond the yield point from elastic to ideally plastic behaviour. The displacement sF and the force FF obtained in the yield point of force-displacement curves and the yield pressure pF were used as model parameters to approximate the plastic force [11]: Fpl = FF + 2pF hcyl

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rcyl ðS−SF Þ

ð25Þ

where Rcyl is radius of the cylinder. Compared to the rapid force drop down during the breakage of a primary particle (point B, Fig. 10 a), the force decrease after the primary breakage of agglomerates (Fig. 10 b) occurs not so clearly because of deformation and secondary breakage of the fragments which were constrained between both plates. The fracture plane of examined agglomerates coincides with the plane of loading as indicated in Fig. 11. Due to maximum tensile stresses, the cracks propagate in loading direction and separate the agglomerate into two approximately equal fragments. On the micro level, the agglomerate shows the inter-particle breakage, i.e. only the solid bridge bonds break but no primary particles. The mechanical behaviour of agglomerates strongly depends on the binder content and the agglomerate size, which was described in our previous work [11]. Fig. 12 (left) shows the influence of the binder content φ on the compression strength σ and elastic stiffness kcyl,el of the agglomerate. The increase of the binder content increases the strength of agglomerates. The point A shows the minimum necessary binder content σA at 1.9% to stick together the primary particles by solid bridge bonds. The agglomerate strength can be described as a linear

Fig. 8. Image of γ-Al2O3 primary particles used to produce the agglomerates (left) and the schema of agglomerate preparation (right).

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Fig. 10. Typical force-displacement curves during compression of (a) spherical primary particle (γ-Al2O3, dpp = 1.03 mm) and (b) cylindrical agglomerate (dcyl = 15 mm, hcyl = 15 mm) prepared from γ-Al2O3 particles and 4% of HPMC binder.

Table 1 Mechanical properties of cylindrical agglomerates (dcyl = 15 mm, hcyl = 15 mm) and their primary particles (γAl2O3, d50 = 1.03 ± 0.06 mm). Material

Primary particles Agglomerate

Yield point FF in N

sF in μm

10.5 75.5

9.4 200

Modulus of elasticity E in kN/mm2

Elastic stiffness kel (F=FF) in kN/mm

Plastic stiffness kpl (s=0.98·sB) in N/mm

Breakage point FB in N

sB in mm

25 ± 0.7 0.08 ± 0.02

2.6 ± 0.21 1.1 ± 0.27

885 ± 31 144.4 ± 25

32.5 93

0.033 0.33

Fig. 11. Photos of the cylindrical agglomerate after the breakage during compression and corresponding crack zone: (1) unbroken primary particles, and (2) broken solid bridge bonds.

Fig. 12. Effects of the binder content φ (left) and height hcyl (right) of cylindrical γ-Al2O3 agglomerates on their elastic stiffness kcyl,el and strength σ during compression.

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Table 2 Parameters of the discrete element model. Characteristics

Agglomerate

Crystal

Size in mm Density of agglomerate and crystal in kg/m3 Number of primary particles Diameter of primary particles in mm Normal and tangential stiffness of primary particles in N/m Friction coefficient of primary particles Area-related stiffness of solid bridge bonds in N/m3 Tensile and shear strengths of solid bridge bonds in N/m2 Ratio of solid bridge bond diameter to particle diameter

dcyl = 15, hcyl = 15 1200 2586 1 ± 0.06 1.8·106 and 1.8·106 0.3 2.3·1011 4.4·106 and 4.4·106 0.7

L = 2, a = 0.2, b = 0.1 1000 44760 0.014 105 and 105 0.4 6.5·1014 6·106 and 2.3·106 1

function of this minimum necessary binder content and the binder strength σB,S (Tomas [16]): σ = σ B;S ð1−εÞðφ−φA Þ:

ð26Þ

With increasing binder content the elastic stiffness (Fig. 12 (right)) increases. Equivalent to Eq. (26), a linear relationship between the stiffness and the binder content has been described as kcyl,el = kB,S(1− ɛ) (φ − φA), where kB,S is the stiffness of the solid bridge bond. 5. Simulation results 5.1. Parameters of models The parameters of the DE Model are given in Table 2. The chosen models are in analogy with the experimentally studied γ-Al2O3 agglomerates and L-threonine crystals. The microstructure of the γAl2O3 agglomerates regarding their porosity, average number of the primary particles and the solid bonds was experimentally obtained and generated correspondently in the DEM. The ratio of solid bridge bond diameter to particle diameter was assumed be 0.7 base on the

SEM of fracture surfaces and fractured solid bridge bonds. In the crystal these parameters were assumed and fitted during the simulation to predict the experimentally obtained breakage phenomena. The value of average binder strength was obtained from a straight line fit of the experimental data according to Eq. (26) and equals 28.9 N/mm2. To obtain the area-related stiffness of the solid bridge bonds and the tensile strength of the solid bridge bonds the two and three particle clusters were produced (Fig. 13). The average strength of the solid bridge bond is about two times smaller than the strength of binder obtained during compression tests of the produced clusters (Table 2). The wall properties were selected according to typical steel with a stiffness of 108 N/m and a friction coefficient of 0.3. For the simulation a time step of 1 μs in the case of the agglomerate and 0.1 ns for the crystals was chosen. 5.2. Discrete element model of a L-threonine crystal For the following investigation a crystal fraction with a characteristic length of L = 2 mm has been obtained by sieving (Fig. 14). The crystals were captured with the help of a digital microscope in 3D.

Fig. 13. Clusters of two and three γ-Al2O3 primary particles with the solid bridge bonds.

Fig. 14. Images of L-threonine: (left) 10-fold magnification, (middle) crystal longitudinal image, and (right) crystal tip.

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Fig. 15. Microscopic images of L-threonine: (left) defects near crystal tip, and (right) defects along the crystal.

Fig. 16. Crystal geometry: (a) plot obtained due to image analysis of a L-threonine crystal captured under a digital microscope, and (b) DEM model of a L-threonine crystal.

Image analyses showed that the cross-section of the crystals can be approximated by an ellipse with the half-axes of 0.2 mm and 0.1 mm. The L-threonine crystal has a convergent tip with the characteristic length of 0.15 L and a convergence angle in the range of 6°–12°. The images from the microscope detect many defects on the crystal surface (Fig. 15). Characteristically, they are located along the crystal in form of long narrow hollows that reduces the elliptical cross-section area of the crystal by 10–18%. During the collision the highest local stresses are generated at these defective zones, so that the fracture initiates from this zone. Therefore, in order to describe the realistic breakage behaviour of the crystals, the model should consider the microstructure of crystals concerning particle shape and defects. After generation of the crystal with an elliptical cross-section (Fig. 16 b) some defects in form of long narrow hollows were produced on the crystal surface. This reduced the cross-section area of the crystal by 16%. In contrary to random structure of the modelled γ-Al2O3 agglomerate, the cubic packing structure of the primary particles in the crystal was assumed for DEM model (Fig. 17). The ratio of solid bridge bond diameter to particle diameter was taken equal to one in order to fill all internal volume of the crystal. The properties of the bonding material and the primary particle are identical. Using the described DEM model with a small wall inclination (1°) the results illustrated in Fig. 18 were obtained. All simulations in this work were carried out at an impact velocity of 5 m/s since the model could be calibrated using the experiments performed at this velocity with the electromagnetic particle gun. Fig. 19 shows the impact simulation of the L-threonine crystal at different impact angles. The fragments are indicated with different

tints. The fracture patterns and the cracks are recorded at 1 µs after the crystal facing the wall which corresponds to the rebound phase of the impact when the impact force decreases. At small angles, the crystal is more fragmented because of the higher bending moment acting on the beam during the incidence. At higher angles only impact attrition can be observed. Meaning that the crystal tip is fragmented and separated out. The diagrams in Fig. 20 show the development of the x- and ycomponents of the impact force during the time of collision for different impact angles. Comparing to these forces z-component of the force

Fig. 17. The packing structure of the crystal in the DEM model.

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Fig. 18. Stressing conditions during the simulation (left) and the evolution of cracks during impact of shown in μs.

L-threonine

crystal on the target (right), the time values on the images are

increasing the impact angle. The maximum value of the normal force decreases with increasing impact angle. According to this the shear force increases. Fig. 21 shows the maximum tensile stress σij and shear stress τij acting within the solid bridge bonds during impact. The strength of solid bridge bonds was assigned in the model with two parameters (normal and shear ) (Table 2). From the stress analysis one can obtain that the primary cracks within the crystals are generated under the maximum tensile stresses (point B1). At the contact the crystal was bended. After rebound the rotating crystal collided a second time with the wall. The stress peaks in the range of B1–Bend show the breakages of the solid bridge bonds during the crack propagation. The crack growth takes place mainly under the tensile stress. The last microbreakage (point Bend in Fig. 21) takes place during the rebound of the fragments. After this breakage point the internal stresses decrease. 5.3. Discrete element model of an γ-Al2O3 agglomerate Fig. 19. Impact simulations of the L-threonine crystal (at the impact velocity of 5 m/s) at different impact angles obtained by simulation after 0.4 µs.

exerted on the crystal can be neglected. After the contact the wall force rises sharply and reaches its maximum value in a short time. Thereby, in the case of the normal force (Fig. 20 a) this time slightly decreases with the impact angle. The slope of the shear force (Fig. 20 b) decreases with

Fig. 19 represents the generation steps of an agglomerate in DEM, that was carried out using the same way as by experiments (Fig. 8). At first step, the primary particles with the same particle size and number as in the experiments were generated in a closed cylindrical box due to free falling of particles. Then the material was compacted by a maximum pressure of 10 kN/m2 and the solid bridge bonds were generated. The solid bridge bonds were modelled only in contacts of particles, so that the bridge bond number is equal to the contact

Fig. 20. The components of the wall force vector depending on the time of impact at different impact angles at the impact velocity of 5 m/s: (a) y-force, and (b) x-force.

S. Antonyuk et al. / Powder Technology 206 (2011) 88–98

Fig. 21. The maximal tensile and shear stresses acting within the solid bridge bonds during impact at the impact angle of 1°.

Fig. 22. Generation steps of the cylindrical agglomerate.

number. The simulated agglomerates had a height of 15 mm and a diameter of 15 mm. The characteristics of the solid bridge bonds are given in Table 2. The bond radius is a ratio of the radius of the solid bridge bond between particle i and particle j to the radius of the smaller particles. After removal of the box's walls the compression test of the agglomerate was performed. The wall properties were selected according to a typical sort of steel with a stiffness of 108 N/m and a friction coefficient of 0.5. The loading conditions during DEM simulation corresponded to the stressing conditions of model materials at the performed experiments. The characteristic behaviour and breakage of the agglomerate at a loading velocity of 40 µm/s are shown in Fig. 23. These images show the contact force network of the solid bridge bonds during compression This calculation demonstrates that during the stressing around the loading axis an elliptical region with very high shear and normal forces as well as the moments is formed. The stresses in the solid bridge bonds are maximal in the contact area.

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Fig. 24. Comparison of the force-displacement curves from measurement and DEM simulation of the cylindrical γAl2O3 agglomerate by compression at a stressing velocity of 40 μm/s.

Fig. 25. The maximal tensile and shear stresses acting within the solid bridge bonds during compression of the cylindrical agglomerate at stressing velocity of 40 μm/s.

That explains the experimental fact that the primary cracks release on surface of contact of agglomerates. Fig. 24 shows the force-displacement curve obtained from simulation of the compression test. The slope of the curve and the breakage limit (point B) are in good agreement with the values measured using compression test of model γ-Al2O3 agglomerates. The primary cracks are generated by reaching the maximum shear stress. The crack growth takes place under the maximum tensile and shear stress (Fig. 25). 6. Conclusions In this contribution the breakage behaviour of agglomerates and crystals by static loading and impact has been investigated. The capability of the DEM to reflect the breakage behaviour of agglomerates and crystals has been proved. It is especially useful for the investigation of particle breakage taking into account inhomogeneities of their

Fig. 23. Fracture events and corresponding contact force network of the solid bridge bonds.

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microstructure. The derived simulation results of compression and impact tests are in good agreement with the deformation and breakage behaviour observed in the experiments with real agglomerates and crystals: γ-Al2O3 agglomerate showed a longitudinal breakage along the axis of the cylinder; the crystal breaks into two or three fragments due to transverse cracks. For the experimental impact investigation a new electromagnetic particle gun has been developed enabling collisions of single crystals with a wall under predefined conditions as impact velocity and angle. The main advantage of the developed device is the high adjustability and reproducibility of the particle impact by current control. References [1] H. Rumpf, The strength of granules and agglomerates, in: W.A. Knepper (Ed.), Agglomeration, New York, Interscience, 1962, pp. 379–418. [2] D. Bika, G.I. Tardos, S. Panmai, L. Farber, J. Michaels, Strength and morphology of solid bridges in dry granules of pharmaceutical powders, Powder Tech. 150 (2005) 104–116. [3] J.-Y. Delenne, M.S. El Youssoufi, F. Cherblanc, J.-C. Benet, Mechanical behaviour and failure of cohesive granular materials, Int. J. Numer. Anal. Meth. Geomech. 28 (2004) 1577–1594. [4] Antonyuk, S., 2006. Deformations- und Bruchverhalten von kugelförmigen Granulaten bei Druck- und Stoβbeanspruchung, Doctoral thesis, University of Magdeburg, Docupoint Publishers.

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