Chapter 19 Analysis of Agglomerate Breakage

CHAPTER 19

Analysis of Agglomerate Breakage Mojtaba Ghadiri, Roberto Moreno-Atanasio, Ali Hassanpour and Simon Joseph Antony Institute of Particle Science and Engineering, University of Leeds, Leeds LS2 9JT, UK Contents 1. Introduction 2. Models of agglomerate strength and failure 2.1. Theoretical models 2.2. Phenomenological models and analyses 2.2.1. Weber number 2.2.2. Mechanistic analysis of the breakage of interparticle contacts 2.2.3. Chipping model 3. Distinct element method 3.1. Introduction 3.2. Agglomerate behaviour using distinct element method 3.2.1. Agglomerate damage 4. Agglomerate behaviour in a bed of particles subjected to shearing 4.1. Effect of size ratio on the breakage of agglomerate 4.1.1. Stress ratio 4.1.2. Damage ratio 4.2. Comparison with experiments 4.3. Relevance to granulation process References

837 839 840 841 841 843 845 846 846 847 849 862 862 863 865 866 869 870

1. INTRODUCTION Chemical, pharmaceutical and food industries amongst many others, use agglomerates either as intermediate or manufactured products. The mechanical strength of agglomerates under impact or shear deformation during handling and processing is of great interest to these industries for optimising product specification and functionality. Agglomerates are formed by smaller particles, which have been brought together and joined to one another by a physical or chemical process [1]. Agglomerates can break during processing or transport making them less suitable for Corresponding author. Tel.: +44 113 343 2406; Fax: +44 113 343 2405; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12022-4

r 2007 Elsevier B.V. All rights reserved.

838

M. Ghadiri et al.

their intended use due to formation of debris and hence quality degradation. Therefore, they need to have sufficient mechanical strength to be functional but at the same time not to be so strong that they present further processing difficulties. This makes the study of the mechanical strength of agglomerates of great interest to industry and academia. In this chapter, we present the breakage characteristics of agglomerates subjected to two types of loading scenarios, namely (i) agglomerates impact against a wall and (ii) agglomerates immersed in a bed of particles subjected to rapid shear deformation. A full prediction of agglomerate strength and agglomerate breakage patterns has not yet been achieved although there is a large amount of work in the literature on agglomerate strength [2–26]. This difficulty arises from the degree of freedom and number of parameters that influence agglomerate structure and properties. Both experimental and computer simulation work suggest that agglomerates formed in the same way and impacted at the same velocity can still fail in different ways [10,12] and have shown different breakage patterns. Subero et al. [12] analysed experimentally the fragmentation pattern of agglomerates made of glass ballotini when an artificial porosity was created in the materials. In some of the impacts, the agglomerates suffered local damage in the region of the impact site only and no crack propagation occurred. Subero et al. [12] quantifies the frequency of fragmentation of all the structures as a function of the agglomerate porosity. Mishra and Thornton [10] showed that for a certain range of porosities, agglomerates fragmented or showed local disintegration depending on the number of interparticle contacts. Furthermore, as shown by computer simulations, agglomerates with the same number of broken contacts can show different breakage patterns depending on the location of the broken contacts [16]. These evidences [10,12,16] suggest that the breakage pattern is strongly influenced by the path followed by the forces originated during impact. Therefore, in order to fully predict the mechanical strength and breakage pattern of agglomerates, it would be necessary to know the exact spatial distribution of particles and contacts within the agglomerates and then also to know the path of the force propagation. In systems made of many particles, the determination of the path of force propagation is a difficult task and therefore macroscopic parameters such as packing fraction and coordination number need to be used. Obviously, the use of these parameters has the disadvantage that a large amount of information is lost, hence making it difficult to predict the fragmentation patterns of agglomerates. The parameters that influence agglomerate strength can be classified into four types: single particle properties, interparticle interactions, agglomerate properties and external parameters, such as impact angle and impact velocity. The influence of some of these factors on the impact behaviour of agglomerates has previously been analysed [2–26], although a systematic study of the influence of these four types of parameter on the agglomerate strength does not exist yet. A systematic study would imply that, to clearly discern the influence of a particular

Analysis of Agglomerate Breakage

839

property on the mechanical strength of agglomerates, the value of such a property could be varied without affecting the other properties. However, this is a very difficult task from an experimental point of view. In experiments in order to change the value of one parameter, the material that is being used usually has to be changed with the consequent alteration of all the physical properties. However, this type of study can be easily carried out by computer simulations based on the Distinct Element Method (DEM) [27]. Significant progress has been recently made in the study of the strength of granular materials using DEM. An improved understanding of the physical behaviour of agglomerates can be achieved by linking particle and bulk properties, for example [2,5,8,9,16,19,22], application of contact mechanics [27–32] and development of faster computational algorithms [33–36]. The use of DEM presents many advantages, the main one being the ability to investigate the influence of a specified property at particle scale on the breakage characteristics of agglomerates. Additionally, the possibility of probing the internal state of the systems such as the state of stress or the number of broken interparticle contacts within the agglomerate during mechanical loading, are features that cannot be easily diagnosed experimentally, and hence this makes DEM a powerful tool to study granular materials. The analysis of agglomerate breakage using DEM can address both microscopic and macroscopic points of view, for example, the number of broken interparticle contacts (microscopic) and the mass detached from the agglomerates and the agglomerate breakage pattern (macroscopic). Furthermore, DEM can be used to critically compare the predictions with those from theoretical models to identify critical factors affecting agglomerate behaviour. In addition, DEM can be combined with other models to provide a more fundamental approach to modelling, for example, in defining the selection and breakage levels for population-balance modelling. Therefore, a review of various theoretical models is presented here, followed by a review of the results of impact damage analysis of agglomerates subjected to impact loading, as obtained by DEM, and wherever possible, a comparison with experimental results is presented.

2. MODELS OF AGGLOMERATE STRENGTH AND FAILURE Several attempts have been made to predict the breakage characteristics of agglomerates failure under different loading conditions by quantifying the stress required to break the agglomerates [37,38], by predicting the number of broken interparticle contacts [2,7,26] or even by quantifying the extent of breakage [13]. However, we can classify these models into two large groups: (1) purely theoretical and (2) phenomenological models.

M. Ghadiri et al.

840

2.1. Theoretical models There are two widely cited theoretical models of agglomerate strength developed by Rumpf [37] and Kendall [38], respectively. Rumpf defined the strength of agglomerates as the force required to break all contacts simultaneously on a prescribed failure plane. However, Kendall defined agglomerate strength as the resistance of the agglomerate to crack propagation based on linear elastic fracture mechanics. Rumpf [37] estimated the agglomerate tensile strength, as the force per unit of area required to simultaneously break all contacts in the fracture plane. The final expression shows a dependency on the porosity, e, diameter of the single particle, D and interparticle force, Fc, in the form
 1

FC ð1Þ s ¼ 1:1
D2 In contrast, the model of Kendall is based on fracture mechanics principles and is therefore more rigorous than the model of Rumpf. Kendall [38] defined agglomerate strength as the resistance against propagating a pre-existing flaw. With this idea, he calculated the fracture toughness KC in a three-point bending test. The calculations of Kendall predict a value of tensile stress required to break the agglomerate in the form 5=6

s ¼ 15:6f4 GC G1=6 ðDcÞ
1=2

ð2Þ

where f is the packing fraction (1
e), G the interface energy, GC the fracture energy, (which is the measured value in experiments corresponding to the equilibrium value of G and includes all energy dissipated in the system by plastic deformation and damping, [38]), D the diameter of the particles and c the length of a pre-existing crack. In this model, the main inconvenience is the necessity to estimate the length of the flaws in the material to be able to predict the agglomerate strength. The model of Kendall [38] is particularly suitable for describing brittle and semibrittle failure of agglomerates as it is consistent with the Griffith criterion for crack propagation [39]. The model of Rumpf predicts the strength exclusively based on the interparticle bond strength and the average porosity of the agglomerate instead. This model may therefore be more suitable to describe the failure of ductile materials, which do not propagate any cracks. However, there is no detailed analysis in the literature that states which of the above models should be applied in a particular case. Moreover, Subero [11] showed that, as far as the dependency on the packing fraction is concerned, both models predict numerically similar values, albeit from very different functional relationships as given by these models. Furthermore, the concept of strength should be redefined since agglomerates can suffer a size reduction in the form of detachment of small

Analysis of Agglomerate Breakage

841

debris and not just by fragmentation as it was considered by Rumpf [37] and Kendall [38]. Therefore, it seems more appropriate to define the strength of agglomerates based on the type of damage incurred. The models of Kendall and Rumpf are neither able to predict the fracture pattern of agglomerates nor the fragment size distribution produced on impact, since agglomerates can develop different patterns of breakage depending on the impact velocity, (strain rate) and type of test [12,16,20,22]. Therefore, new theories need to be developed to predict the strength of agglomerates and this can best be done in terms of single-particle properties and packing conditions using DEM.

2.2. Phenomenological models and analyses Phenomenological models and analyses help us to extract the essential features of the phenomenon, for example, for the case of an impact process in agglomerates this could be to relate the degree of damage in the agglomerate to the impact velocity, taking into account the factors that influence agglomerate strength.

2.2.1. Weber number The first of these phenomenological analyses is based on the use of the Weber number. Kafui and Thornton [2] analysed the effect of bond strength on agglomerate breakage using DEM and defined the Weber number as the ratio of input energy to the average bond strength of an agglomerate. The expression of the Weber number for the case of impact of agglomerates is given by We ¼

r DV 2 G

ð3Þ

where r is the particle density, D the primary particle diameter, V the impact velocity and G the interface energy which is defined by the Dupre´ equation (Israelachvili [40]) as G ¼ gA þ gB
gAB

ð4Þ

where gA and gB are the surface energies of two particles made of different materials, A and B, in contact with each other and gAB is the interaction energy between them. For surfaces of the same material gAB is zero and therefore G ¼ 2g. Kafui and Thornton [2] analysed the effect of surface energy on the strength of 2-D regularly-packed agglomerates, having face centred cubic (fcc) and body centred cubic (bcc) structures, and they related the breakage of interparticle contacts to the Weber number. They expressed the damage in agglomerates in

M. Ghadiri et al.

842

terms of the damage ratio, defined as the ratio of broken contacts to the initial number of bonds. The damage ratio was related to the Weber number for a range of impact velocities and surface energies. They found that the curves corresponding to the values of surface energy between 0.1 and 1.0 J/m2 were reasonably unified when damage ratio was plotted as a function of Weber number. However, in a later work, Thornton et al. [6] obtained a better unification by modifying the Weber number and defining a lower limit of impact velocity, V0, below which no contact is broken. This modified Weber number, We0 , is given by We0 ¼

r DðV
V 0 Þ2 G

ð5Þ

The above analysis was also carried out for a 2-D ordered packing with the surface energy in the range between 0.3 and 3.0 J/m2. Later, Subero et al. [7] carried out simulations using 3-D motion of particles and analysed the effect of the surface energy in the range 0.5–5.0 J/m2 in randomly packed agglomerates. They plotted the damage ratio as a function of Weber number, as shown in Fig. 1, for the range of surface energy between 0.5 and 5.0 J/m2, and found that their results were in agreement with the work of Thornton et al. [6], i.e. a good unification of data for values of surface energies between 0.5 and 5.0 J/m2 was obtained. However, the surface energy values used in the above simulations were only varied by one order of magnitude. In a later work, Moreno-Atanasio et al. [26] varied the surface energy by two orders of magnitude and found that the use of the modified Weber number no longer unified the data adequately. They then proposed an alternative analysis based on the idea that the damage suffered by agglomerates during an impact event could be related to the incident kinetic

Fig. 1. Plot of damage ratio vs. Weber number for different values of surface energy [7].

Analysis of Agglomerate Breakage

843

energy and also to the physical and mechanical properties of the agglomerates. This model is described in the following section. The use of the modified Weber number is based on the assumption that the breakage of interparticle contacts is related to the ratio between the specific kinetic energy and the interface energy. However, no clear physical meaning was given to that relationship. In addition, the introduction of the minimum velocity under which no breakage of contacts was observed was purely empirical, since it fitted better the simulation results but no physical interpretation was given to that new parameter or why it should appear in the expression of the Weber number.

2.2.2. Mechanistic analysis of the breakage of interparticle contacts Moreno-Atanasio and Ghadiri [26] analysed the breakage of interparticle contacts obtained from DEM simulations by using the Weber number. However, the analysis of their computer simulation results of impact breakage of agglomerates by using the Weber number was not successful. Figure 2 shows the damage ratio corresponding to the impact of four different agglomerates plotted as a function of We0 [26]. The data points correspond to the average damage ratio for the impact of four different agglomerates and the error bars correspond to the standard deviation of the data. When damage ratio was plotted vs. Weber number for surface energy values of 0.35, 3.5 and 35.0 J/m2 the different curves did not unify as reported in the work by other authors [2,12]. Therefore, a new model based on an energy balance was proposed to better explain the simulation results.

1.0 0.35 J/m2

Damage ratio

0.8

3.50 J/m2 35.0 J/m2

0.6 0.4 0.2 0.0 1E-6

1E-5

1E-4

1E-3 We'

0.01

0.1

1

Fig. 2. Relationship between damage ratio and modified Weber number, We0 . The data points correspond to the average damage ratio for the impact of four different agglomerates [26].

M. Ghadiri et al.

844

The model development is based on the consideration that the work required to break interparticle contacts varies linearly with the incident kinetic energy [26]. Associated with each contact breakage is an amount of work to furnish the required surface energy and therefore the higher the incident energy the larger the number of broken contacts will be. The incident kinetic energy of agglomerates made of mono-size particles before impact, EK, is given by 1 E K ¼ N mV 2 2

ð6Þ

where N is the number of particles in the agglomerate, m the mass of a primary particle and V the impact velocity of the agglomerate. If the total number of broken contacts after impact is NB, the work for breaking these contacts, Wc, assuming that all contacts have the same contact area, Ac, is given by W c ¼ N B GAc

ð7Þ

Let us consider that the total work spent in breaking NB bonds is proportional by a factor k, to the incident kinetic energy of the agglomerate with k being the proportionality factor. 1 N B GAc ¼ kN mV 2 2

ð8Þ

Since the number of particles in the agglomerate, N, and the initial number of bonds, N0, are related through the coordination number [26], Z, damage ratio, DR, can be calculated as DR ¼

NB k mV 2 ¼ N 0 4Z GA

ð9Þ

If the contact area, A, between particles is estimated from the model of Johnson et al. [41] and the particle mass is substituted as a function of the particle density, an expression for damage ratio can be obtained in the form DR ¼ k

25=3

1 1 rD5=3 V 2 2=3 E 37=3 p2=3 Z ð1
n2 Þ2=3 G5=3

ð10Þ

Now considering the terms in equation (10), it is possible to define a new dimensionless number, D, as given by equation (11), incorporating particle density, particle diameter, elastic modulus and interface energy. D¼

rD5=3 E 2=3 V 2 G5=3

ð11Þ

Therefore the damage ratio, DR, is given by DR a We I 2=3 e

ð12Þ

Analysis of Agglomerate Breakage

845

where We is the Weber number and Ie the elastic adhesion index which is defined as Ie ¼

ED G

ð13Þ

The hypothesis that the incident kinetic energy varies linearly with the work to break contacts has been shown to be true for agglomerates simulated by using DEM and containing between 500 and 10,000 particles and impact velocities between the elastic regime (o0.1 m/s) until the full disintegration of the agglomerates (5 m/s) [18,26]. In addition, the dependency of the damage ratio with the exponent 5/3 of the surface energy has also been tested by using computer simulations. Figure 3 shows the damage ratio plotted as a function of the new dimensionless group, D, for the impact of three different agglomerates with the values of surface energy of 0.35, 3.5 and 35 J/m2. Each data point are the average of four impacts and the error bars correspond to the standard deviation. The results can be compared with Fig. 2, where damage ratio was plotted vs. the modified Weber number [42]. It is clear that the curves corresponding to different values of surface energy become more unified when they are plotted using the new dimensionless group, as compared to the We0 .

2.2.3. Chipping model Ghadiri and Zhang [13] developed a model of attrition due to chipping for semibrittle materials based on the propagation of sub-surface lateral cracks. They

1.0 0.35 J/m2

Damage ratio

0.8

3.50 J/m2 35.0 J/m2

0.6 0.4 0.2 0.0 0.01

0.1

1

10 We

100

1000

1E4

(ED/Γ2/3)

Fig. 3. Relationship between damage ratio and new dimensionless group, D, for different values of surface energy. The data points correspond to the average damage ratio for the impact of four different agglomerates [26].

M. Ghadiri et al.

846

estimated the extent of breakage, x, as the ratio of the volume of debris detached from the particles to the total volume and showed that this is given by x¼a

rV 2 lH K 2C

ð14Þ

where r is the density of the material, V the impact velocity, l the characteristic length of the system and a the proportionality factor which depends on particle shape and impact geometry and is determined experimentally. Ghadiri [43] showed that the extent of breakage expressed as the fractional loss per impact, x, is in fact related to the Weber number, We, since the relationship between fracture toughness KC, hardness, H and elastic modulus, E, is provided by the linear elastic fracture mechanics in the form K 2C ¼ 2EGð1
n2 Þ

ð15Þ

Therefore, the fractional mass loss per impact can be written in the following form, which is consistent with the phenomenological model of Moreno-Atanasio and Ghadiri for agglomerate breakage [26].
 H x / We ð16Þ E The dependency of the extent of breakage on impact velocity and particle size was successfully verified by Zhang and Ghadiri [44] for MgO, NaCl and KCl. However, for agglomerates the power index of impact velocity is lower than two in some cases, but the reason for this is unclear. The knowledge of the power index of the impact velocity would give us an insight of the dissipation mechanism of agglomerates.

3. DISTINCT ELEMENT METHOD 3.1. Introduction The DEM, first developed by Cundall [27], presents an alternative way to obtain an insight for particulate systems and provides fundamental information such as microscopic structure, interparticle forces, particle velocities, etc. Most importantly, this method makes it possible to relate the bulk mechanical behaviour of the assembly to individual particle properties. The DEM has been applied to systems of large number of particles subjected to different mechanical processes such as compression, impact, milling and shear [16,20,45,46]. A particular advantage of DEM is that a detailed examination of the micromechanics of the system, which determines the bond breakage and the internal microstructural information, can be made.

Analysis of Agglomerate Breakage

847

The DEM works by cyclically updating the forces, accelerations, velocities and positions of the particles [27]. The interparticle force increments are calculated at all contacts from the relative velocities of the contacting particles using an incremental force-displacement law. The interparticle forces are updated and the new out-of-balance force and moment of each particle and new particle accelerations (both linear and rotational) are calculated using Newton’s second law. Numerical integration of the accelerations is then performed over the time step to give new particle velocities. Further, numerical integration provides displacement increments from which the new particle coordinates are obtained. Having obtained new positions and velocities for all the particles, the program repeats the cycle of updating contact forces and particle locations. Checks are incorporated to identify new contacts and contacts that no longer exist. In each cycle, every particle in the assembly is treated in the manner described above and the calculation cycle is repeated until the end of simulation [27]. The original code BALL was tested comparing the network of forces formed in a 2-D bed of particles with the results of photoelasticity experiments of De Joselin De Jong [42]. The comparison was purely qualitative but the good similarity between the network of forces in the simulation and the experiments was an indication of the success of the new method. However, the differences in force propagation and mechanical resistance between 2- and 3-D systems can be quite large. For this reason, the computer code BALL was modified to model a 3-D system and its name was changed to TRUBAL (meaning ‘‘true ball’’). Afterwards the code was further developed by Thornton and co-workers [28–31], where nonlinear contact deformations and adhesion were incorporated into the code and applied to the analysis of agglomerate strength. This code also incorporates the effects of friction and account energy dissipation by damping between particles. The analysis described in the following section is focused on the impact damage of agglomerates as carried out by computer simulations using DEM due to their technological importance and also the difficulty of using other approaches in analysing agglomerate breakage, e.g., continuum mechanics as discussed previously.

3.2. Agglomerate behaviour using distinct element method The impact process may be divided into three stages: the first one is the compression of the agglomerate or loading stage, the second is the unloading of the agglomerate and the third is either the rebound or the deposition of the agglomerate on the target. Figure 4a and b shows the evolution of the force

M. Ghadiri et al.

848

0.01 m/s

0.5

0.05 m/s 0.1 m/s

0.4 Force (N)

0.5 m/s 2.0 m/s

0.3

4.0 m/s 0.2 0.1 0.0 0

10

(a)

20 Time (µs)

30

0.01 m/s 0.05 m/s 0.1 m/s 0.5 m/s 2.0 m/s 4.0 m/s

1.0 kinetic energy (Ekf /Eko)

40

0.8 0.6 0.4 0.2 0.0 0

(b)

10

20 Time (µs)

30

40

Fig. 4. (a) Evolution during impact of the force exerted on the target and (b) evolution of the kinetic energy during impact [18].

exerted by the agglomerate on the target (Fig. 4a) and the evolution of the kinetic energy during impact (Fig. 4b) for different values of the impact velocity. The force on the wall passes through a maximum after which the unloading stage of the agglomerate starts [6,8,18]. During the unloading stage, a part of the energy transmitted to the wall during loading is transferred back to the agglomerate, and this produces a decrease in the force exerted on the wall. The kinetic energy of the agglomerate is, then, recovered partially. The amount of energy recovered depends on the damage produced on the assembly and, therefore, on the impact velocity [6,18]. During the impact process, depending on the impact velocity, agglomerates can suffer damage which can be classified as microscopic or macroscopic. Microscopic damage is the breakage of contacts which is not necessarily linked

Analysis of Agglomerate Breakage

849

to macroscopic damage. Macroscopic damage is characterised by crack propagation and the detachment of fragments of different sizes from the agglomerate.

3.2.1. Agglomerate damage The study of agglomerate damage can be divided into two parts related to the breakage of contacts (microscopic) and to the fragmentation and detachment of particles of the agglomerate (macroscopic), respectively. The breakage of contacts can be studied using the damage ratio, which represents the number of broken bonds. However, this process does not reflect the formation of debris and fragments. The latter can be analysed by characterising the size distribution of the fragments [16,18,20,47].

3.2.1.1. Analysis of breakage of contacts Kafui and Thornton [2] expressed the extent of damage as the fraction of initial primary particle contacts that were broken as a result of impact. The number of interparticle bonds in agglomerate may be expected to reflect the bulk impact strength of the agglomerate and therefore the damage ratio would quantify the deterioration in the bulk strength of agglomerates as a result of an impact. For any agglomerate there is a threshold velocity, below which no damage, i.e. no bond breakage is observed and the agglomerate behaves in an elastic way. Above this minimum velocity, breakage of interparticle bonds is observed and it is usually near the impact site. This minimum velocity is intuitively related to the minimum energy required to break a single contact and this in turn is related to the surface energy. It is also expected that factors such as interparticle and particle-wall damping, friction coefficient and local arrangement of particles around the impact site influence this minimum velocity. Thornton et al. [6] developed a correlation for the minimum velocity for a regular 2-D agglomerate with the surface energy values between 0.3 and 3.0 J/m2 as given by V 0 ¼ 0:0025 expð0:49GÞ

ð17Þ

In a later work, Kafui and Thornton [9] showed that the threshold velocity varies as a function of the interface energy G, according to the following form for a 3-D structured agglomerate with a range of surface energies between 0.2 and 4.0 J/m2. V 0 ¼ 0:17G1:5

ð18Þ

However, Moreno [18] found that the minimum velocity under which no damage was observed followed the relationship for values of surface energy between 0.35 and 35.0 J/m2 in the form V 0 ¼ 0:0095G0:81

ð19Þ

M. Ghadiri et al.

850

Important differences between the three pieces of work are found which would make it possible to understand the origin of the different expressions. In the case of Thornton et al. [6], the agglomerates were 2-D structures; in the case of Kafui and Thornton [9] the agglomerate had a 3-D regular packing, whilst in the case of Moreno [18] the agglomerates had a random distribution of particles. In addition, in the first two cases, only one agglomerate was used in the simulations, however in the latter case [18] four different agglomerates were tested to obtain a mean value of damage ratio for every impact velocity. In addition, three different values of surface energy spanning two orders of magnitude. Other differences such as porosity, or physical properties of the primary particles might be of importance to explain the differences between the three expressions. The Weber number and the modified Weber number as described previously in the previous section (equations (3) and (5)) has been used extensively to describe the agglomerate breakage. Kafui and Thornton [2] fitted their damage ratio to a relationship in the form D / Web=2

ð20Þ

with the values of b of 0.131 and 0.250 for two-face centred cubic regular packings with 20 and 40 mm primary particle size but having the same agglomerate diameter. This clearly indicates that there are other factors such as the ratio of agglomerate size to particle diameter that are not considered when the Weber number is taken into account. Furthermore, using equation (20), Kafui and Thornton [2] compared the results of impact breakage of agglomerates with different packing fractions. The agglomerate with a lower packing fraction was found to be more resistant to damage as it could accommodate the transmitted energy more easily by microstructural plastic deformation. As the impact velocity is increased the damage ratio approaches unity asymptotically [7,18]. This clearly implies that a power law such as the one given in equation (20) is not suitable to fit the full range of impact velocities. In addition, Subero et al. [7] found in their study that not only damage ratio but also the mass fraction of debris approaches an asymptotic value. Moreno and Ghadiri [24] proposed a new dimensionless group to describe the damage ratio since the Weber number was not useful in explaining the scaling of the damage ratio with surface energy for their simulations. The new dimensionless group also provides a dependency on the square of the impact velocity, and therefore on the incident kinetic energy of the agglomerate. However, this dependency does not hold true for high impact velocities. Despite extensive efforts made so far, there is still no mathematical relationship between the damage ratio and agglomerate properties for all impact test conditions (such as target mechanical properties) and physical and mechanical properties of the primary particles (such as elastic modulus, particle shape and density, for example).

Analysis of Agglomerate Breakage

851

3.2.1.2. Fragmentation and breakage Microstructural damage or breakage of interparticle bonds leads to the detachment of particles from the agglomerate only when the sequence of broken bonds produces a cluster, which is completely detached from the agglomerate. The analysis of agglomerate breakage is usually carried out as a function of the impact velocity since the agglomerate behaviour varies from a purely elastic response where the agglomerate does not suffer from any type of damage to a complete disintegration into small pieces at high impact velocities. Figure 5a and b shows the normalised sizes of the two largest agglomerates as a function of the impact velocity for two different cases of surface energy (3.5 and 35.0 J/m2) [22]. In both cases, three regimes of fragmentation can be distinguished, which correspond to different values of impact velocity. However, the values of impact

Regime II

Normalised fragment size

1.0 Regime I

0.8 0.7

1st Largest fragment 2nd Largest fragment

0.5

Regime III

0.3 0.2 0.0 0.1

1 Impact velocity (m/s)

(a)

Regime II

1.0 Normalised fragment size

10

0.8

Regime I

Regime III

0.7 1st Largest fragment 2nd Largest fragment

0.5 0.3 0.2 0.0 1

(b)

10 Impact velocity (m/s)

Fig. 5. Regimes of breakage for one agglomerates whose value of surface energy is (a) 3.5 J/m2 and (b) 35.0 J/m2 [22].

852

M. Ghadiri et al.

velocities corresponding to these regimes depend on agglomerate properties. Therefore, when the terms low, medium or high impact velocities are used they are relative since for agglomerates with different properties these terms will correspond to different values of velocity. These regimes are  Regime I (low impact velocity): small clusters are detached from the agglom-

erate and the size of the largest fragment shows a weak dependency on the impact velocity.  Regime II (medium impact velocity): this is characterised by a fast decrease of the size of the residual fragment.  Regime III (high impact velocity): the size of the largest fragment is much smaller than the initial agglomerate size and varies slowly with impact velocity. Regime I: low impact velocity: At low impact velocities, the agglomerate breakage is characterised by a slow decrease in agglomerate size. During this regime, the second largest cluster is less than 5% of the initial size of the agglomerate as shown in Fig. 5a and b, and the number of fragments detached from the agglomerate is very small as indicated by the large value of the largest cluster size. The small size of the second largest fragment after impact implies that the fragments detached from the agglomerate are really debris. This type of behaviour was first described by Thornton et al. [9] and later reported as well by Subero et al. [7,11] and Moreno et al. [16,18,22]. Within this regime no large differences appear between the works reported by different authors [7,9,11,16,18,22]. Regime II: Intermediate impact velocities: Within this regime, the size of the residual cluster is very sensitive to the impact velocity (Fig. 5a and b). The size of the second largest cluster increases and passes through a maximum. The maximum in the curves corresponds to the fragmentation of the agglomerates into two fragments (Fig. 5a and b). This fragmentation is shown in Fig. 6 where the top and bottom views of four agglomerates impacted at velocities corresponding to the maximum of the second largest fragment are visualised. The four agglomerates are made of primary particles of 50 mm in radius, 31 GPa elastic modulus and 2000 kg/m3 particle density. The fragments are colour coded according to their sizes. This type of behaviour is in agreement with the work of Thornton et al. [9], who also observed the fracture of the agglomerate at relative intermediate velocities for 2-D agglomerates. However, neither Ning et al. [5] nor Subero et al. [7,11] observed fragmentation of agglomerates for any given value of impact velocity. In the case of Ning [5], the behaviour of weak lactose agglomerate was analysed by DEM. The agglomerates seemed to fail macroscopically in a ductile mode, i.e. extensive plastic deformation without any crack propagation as shown in Fig. 7. In addition, Subero et al. [7] simulated the impact breakage of agglomerates made of glass ballotini. They successfully quantified the increase in breakage of

Analysis of Agglomerate Breakage

853

Fig. 6. Top view of four different agglomerates impacted at the velocities corresponding of the maximum of the second largest fragment as shown in Fig. 5a and b. Colour coding: light grey, largest fragment; red, second largest fragment; yellow, third largest fragment; green, clusters smaller than clusters in yellow and larger than 100 particles; cyan clusters between 4 and 100 particles; pink, doublets; blue, singlets [18].

Fig 7. Disintegration of weak lactose agglomerates Ning et al. [5].

the agglomerate with the impact velocity; however their simulated agglomerates did not fragment at all. Subero [7] argued that for an agglomerate to fracture, it is necessary for its structure to store sufficient elastic strain energy required for crack propagation. It appears that the agglomerates of Subero [7] could not do this. However, neither Subero et al. [7] nor Ning [5] provided any explanation for the lack of crack propagation in their agglomerates.

854

M. Ghadiri et al.

In a later work, Moreno and Ghadiri [18,22] showed that the effect of interface energy could explain the findings of Ning et al. [5] and partially explain the findings of Subero [7]. Their analysis of the effect of interface energy on the breakage of agglomerates [18,22] clearly showed that agglomerates with low surface energy (0.35 J/m2) do not fragment at any value of impact velocity and undergo extensive deformation accompanied by disintegration into small clusters. This behaviour was very similar to the ductile behaviour of continuum solids. In contrast, for agglomerates in which the particles were joined by a value of the surface energy of 3.5 J/m2 or higher, crack propagation was observed accompanied by the disintegration of a local region around the impact site into small clusters. This behaviour was more similar to the behaviour of semi-brittle continuum materials. Therefore, agglomerates seem to have a transition in their mode of failure when the surface energy is increased. The effect of surface energy can be clearly observed in Fig. 8 where the same agglomerate was impacted at values of impact velocities corresponding to the regime II and with different values of surface energy of the primary particles. Since Ning et al. [5] used a value of surface energy of 0.5 J/m2 the differences between the work of Ning et al. [5], and the work of Thornton et al. [9] and Moreno et al. [18,22] could be attributed to the effect of the surface energy. Subero et al. [7] used values of 0.5, 2.0 and 5.0 J/m2 and therefore the lack of fragmentation agglomerates reported in their work cannot be explained based on the

Fig. 8. Effect of the interface energy on the breakage pattern of agglomerates during regime II of breakage [22]. Colour coding: white, target; blue, singlets, pink, doublets; cyan, 4–100 particles; green, 4th largest fragment; yellow, 3rd largest fragment; red, 2nd largest fragment; grey, residual fragment.

Analysis of Agglomerate Breakage

855

considerations of energy alone. Mishra and Thornton [10] analysed the failure pattern of agglomerates as a function of the packing fraction using agglomerates made of spheres with properties of glass ballotini. They found that for values of packing fraction less than 0.537 agglomerates did not fragment. At high values of packing fraction (0.602 or higher) their agglomerates fragmented into several pieces of comparable sizes. Finally, at the intermediate values of packing fraction (around 0.537) the agglomerates showed different behaviours when impacted in different orientations; in some orientations the agglomerates fragmented and in others they did not fragment and incurred local disintegration around the impact site. The agglomerates of Subero et al. [7] have a solid fraction of 0.565, which would place them in the intermediate regime. However, although a direct comparison is not feasible since the agglomerates were made of primary particles with different physical properties, in principle, the value of packing fraction could explain the lack of fragmentation of the agglomerates of Subero et al. [7]. Another factor that could influence the agglomerate strength and mechanical properties is agglomerate size. Moreno [18] found that small agglomerates made of 500 particles do not fragment for certain impact orientations and for any given value of impact velocity. In this case, the second largest cluster was always less than 10% of the initial agglomerate size. However, the largest agglomerate tested by Moreno [18] fragmented into 2, 3 or 4 pieces of similar sizes depending on the impact orientation. These differences appear exclusively due to agglomerate size since packing fraction, coordination number and physical properties of the primary particles were kept constant. Moreno et al. [16] analysed the oblique impact of agglomerates and found that the number of broken contacts decreases as the impact angle is reduced. However, the number of broken contacts in the agglomerate was roughly the same for all impact angles when the normal component of the impact velocity was kept constant. The tangential component of the impact velocity seems only to influence the location of the broken contacts and not their number as shown in Fig. 9. These observations have also been corroborated later by Behera et al. [48]. When the impact angle is 901, the broken contacts seem to be more uniformly spread through the agglomerate showing symmetry with respect to the perpendicular to the wall (Fig. 9). However, at 301 impact the broken contacts are more localised on one side of the agglomerates (the side from which the agglomerate moves to impact the wall). The pattern of breakage for 301, 451 and 901 is shown in Fig. 10. The number of broken contacts is approximately the same, however, the breakage pattern is completely different: At 451 the agglomerate fragments and at 301 no fragmentation is observed (Moreno et al. [16]). The experimental work of Samimi et al. [24] also concluded that the normal component of the impact is the main factor influencing the breakage of agglomerates within the chipping regime. However, when the agglomerate failure is in

856

M. Ghadiri et al.

Fig. 9. Pattern of broken contacts at 301 and 901 impact angle. The damage ratio is 0.292 for both angles 301 and 901 [16].

Fig. 10. Influence of the impact angle on the breakage patterns of agglomerates.

the fragmentation regime, the tangential component of the impact velocity influences the breakage pattern. This can be explained by the findings of Moreno et al. [16] suggesting that a change of location of broken contacts produces a change in the failure pattern. However, the differences in the type of bonds

Analysis of Agglomerate Breakage

857

and materials used by Samimi et al. [24] and Moreno et al. [16] require further analysis. So far, the breakage of agglomerates has been mainly related to the interface energy, packing fraction, agglomerate size and location of broken contacts. However, there is no systematic study on the effects of other parameters such as the elastic modulus or density. This is clearly needed to fully map out the mode of failure of agglomerates as a function of all parameters that influence agglomerate behaviour. Regime III: High impact velocities: In this regime, the input kinetic energy that is not dissipated is sufficient to break most of the contacts of the agglomerate and the agglomerate behaviour is clearly characterised by the disintegration of the agglomerate into small fragments as suggested by Fig. 5a and b. However, there are differences in the mode of failure of agglomerates within regime III arising from different values of surface energy of the primary particles. Figure 11a and b shows the impact of an agglomerate with the same structure [18] but for two different values of surface energy (0.35 and 3.5 J/m2) impacted at 2.0 and 25 m/s, respectively. At the lowest value of the surface energy (0.35 J/m2) the agglomerate is shattered into small clusters and the largest fragment is deposited on the target (Fig. 11a). However, at higher surface energies (35.0 J/m2) the formation of clusters and large fragments away from the impact site is clearly observed as can be seen in Fig. 11b. Regime III of breakage occurs at much higher impact velocities for the case of 35.0 J/m2 than for the case of smaller surface energy (0.35 J/m2). The higher

Fig. 11. Agglomerate impact at different velocities of (a) 2.0 m/s and (b) 25 m/s during the third regime of breakage for two different values of surface energy (a) 0.35 J/m2 and (b) 35.0 J/m2. Colour coding: light grey, largest fragment; red, second largest fragment, yellow third largest fragment; green, fourth largest cluster; cyan clusters between 4 and 100 particles; pink, doublets; blue, singlets [22].

M. Ghadiri et al.

858

residual kinetic energy leads to the fragments flying away from the impact site (Fig. 11b). In addition, for the case of low surface energy, the agglomerates have disintegrated into small clusters made of a few particles. The volume of agglomerate disintegrated increases with impact velocity. However, for large values of surface energy the disintegrated agglomerates seem to be formed by fragments with straight contours suggesting that multiple fragmentation rather than disintegration is the origin of this structure.

3.2.1.3. Breakage patterns Some DEM and experimental studies have established qualitative relationships between some of the characteristics of agglomerates and their mode of breakage [10–12,16,18,22,49]. The features of the various work reported in the literature are summarised below. The experimental work of Subero and Ghadiri [12] for impact of agglomerates made of glass ballotini that related the impact velocity, porosity and breakage pattern in a systematic way for the first time. It is interesting to compare the breakage pattern found experimentally by Subero and Ghadiri [12] with the breakage pattern obtained by DEM. The most important feature is that they found an increase in the frequency of fragmentation with impact velocity. The term ‘‘frequency’’ is used due to the fact that Subero and Ghadiri [12] found that some agglomerates did not fragment at all whilst others exhibited multiple fragmentations for the same range of impact velocities. Figure 12 shows the fragmentation histogram published by Subero and Ghadiri [12]. This feature could be further 0.9 U = 2 m/s

Frequency of Fragmentation (-)

0.8

U = 4 m/s

0.7

U = 6 m/s 0.6 U = 8 m/s 0.5 0.4 0.3 0.2 0.1 0 MF

25/2.19

50/2.19 100/2.19 25/2.58 Structure Type (-)

50/2.58 100/2.58

Fig. 12. Frequency of fragmentation of agglomerates as a function of the artificial porosity [12].

Analysis of Agglomerate Breakage

859

explained by the work of Mishra and Thornton [10] who showed that within a certain range of porosities agglomerates show large sensitivity to the impact site and they do not always fragment. The breakage patterns reported by Mishra and Thornton [10] based on DEM and the experimental results of Subero and Ghadiri [12] are relatively in good agreement, although a direct comparison is not feasible due to differences in particle properties and bond characteristics. These patterns can be summarised in four different types, which appear depending on the impact velocity and are plotted in Fig. 13. The first is characterised by local damage at the impact site, the second by local damage and oblique fracture and the third by local damage and median cracks with or without secondary fragmentation and the fourth by multiple fragmentation. Some of the fragmentation patterns observed by Subero and Ghadiri [12] have also been reported by DEM simulations of Moreno and Ghadiri [18,22] for intermediate impact velocities, i.e. agglomerates fragmented only with local damage (Fig. 13a) or with local damage and detachment of side platelets Local damage only

(a) Impact site disintegrates into small debris (side view). Local damage + meridian fracture

(c) Median crack leading to fragmentation into large clusters (side view).

Local damage + oblique fracture

(b) Oblique cracks leading to the detachment of side fragments Multiple fragmentation

(d) Secondary fragmentation (top view)

Fig. 13. Experimental pattern of breakage of agglomerates by Subero and Ghadiri [12]. Patterns (a), (b) and (c) were also found by Moreno et al. [16,18,22].

M. Ghadiri et al.

860

(Fig. 13b). When the impact velocity was increased, the presence of median cracks splitting the residual fragments into two large pieces was also observed (Fig. 13c). However, it was observed that the agglomerate size has an influence. Some of the smallest agglomerates tested by Moreno [18] made of 500 particles of 50 mm in size did not show fragmentation for certain impact orientations and for all impact velocities. When the agglomerates were made of 3000 particles the residual fragment divided into two large pieces. For agglomerates of 10,000 particles, the residual fragment divided into two, three or four pieces and they were always accompanied by local disintegration of the contact area and, very frequently, by the detachment of side platelets. These findings are summarised in Fig. 14. The increase in agglomerate size is associated with a tendency of dividing into a large number of pieces and there is an increase in the similarity between the mode of failure of continuum solids and agglomerates. It seems that the increase in agglomerate size with respect to their single-particle components shows a reduction in the importance of the discrete nature of the material. As more paths for the forces to propagate through the agglomerates are available and therefore the material behaves more as a continuum solid. This would explain, for example, the work of Schubert et al. [49] where the failure of cement agglomerates were simulated in 2-D. In this work, the authors observed experimentally and by DEM the same type of breakage pattern characterised by a pulverised cone of materials with median cracks and secondary fragmentation. This breakage pattern is schematically shown in Fig. 15. This type of pattern have a great similarity with

10000 particles

lateral fragment 30000 particles

lateral fragment

500 particles

lateral fragment

Fig. 14. Schematical top view of the different breakage patterns found by Moreno [18]. Size effect on the fragmentation pattern of spherical agglomerates. The tendency to fragment into a larger number of fragments decreases with particle size.

Analysis of Agglomerate Breakage

861

Radial cracks

Crush cone

Fig. 15. Fragmentation pattern found by Schubert et al. [49]. The pattern is characterised by a crushed cone of material with the rest of the agglomerate split by radial and median cracks [49].

the work of Arbiter [50] who showed that for sand cement agglomerates a compressed cone of material covered by a layer of debris is formed with radial cracks departing from the conical region. This similarity is also found in the work of Salman et al. [51] for continuum solid particles. The breakage pattern observed by Schubert et al. [49] has not been reported previously in any simulation work based on DEM. This is probably due to the high packing (70% volume fraction) of the agglomerates of Schubert et al. [49], which makes the agglomerates to have similar characteristics to a continuum solids than to any other agglomerates simulated by using DEM until now. With the development of computer speed, we are closer to being able to perform realistic comparisons between simulations and experiments. These comparisons have to be made not only at the agglomerate level (structure, porosity) but also at a single particle and bulk levels. This is going to require first of all the use of realistic force–deformation relationships and the use of realistic bond characteristics (solid bridges, van der Waals, capillary or electrostatic forces) and also an adequate input of particle characteristics, such as shape and roughness. In conclusion, the results discussed above on the impact breakage characteristics of agglomerates under impact loading show that, factors such as surface energy, packing fraction, agglomerate size and impact angle can significantly affect the mode of failure of agglomerates. The variation in the value of surface assigned at particle scale resulted in significant changes on the mode of failure of agglomerates under impact loading – an increase in surface energy could change the mode of failure from ductile to semi-brittle failure [22]. Agglomerates showing high values of packing fraction (40.7) show modes of failure similar to those of solid particles [49]. For agglomerates with lower values of packing fraction, the mode of failure depends on both the impact orientation and the initial number of interparticle contacts [10]. When all other properties remain the same, the tendency of fragmentation of an agglomerate into a larger number of clusters increases with agglomerate size making the mode of failure of agglomerates, again, more similar to those of a solid particles [18]. When the impact angle is varied, whilst keeping the normal component of the impact velocity constant, the

862

M. Ghadiri et al.

change in the tangential component of the impact velocity alters the mode of failure of agglomerates [16]. Nevertheless, despite the numerous amounts of research reported earlier reveal the influence of some of the single-particle properties on the impact breakage characteristics of agglomerates, further research is needed to study the influence of many other factors such as the particle density, the ratio of agglomerate diameter to single-particle diameter and the elastic modulus of primary particles.

4. AGGLOMERATE BEHAVIOUR IN A BED OF PARTICLES SUBJECTED TO SHEARING Agglomerates may deform and break within a bed of particles during shearing. This process commonly occurs in high shear granulators in process industries where the powder is under intense agitation and large forces are transmitted into the powder bed [52,53]. The existence of high level of stresses together with strong and rapid agitation would result in shorter processing time and provide the ability to produce granules with a higher strength. These features have made high shear mixer granulators attractive particularly in the pharmaceutical and detergent industries. It is widely recognised that, in addition to powder properties and binder characteristics, the properties of agglomerates formed by powder granulation strongly depend on the level of prevailing shear stresses as experienced by the powder bed inside the granulators, as deformation and breakage of agglomerates within granulators is predominantly affected by the level of shear stresses [54–56]. It is therefore necessary to understand the relationship between the properties of the agglomerates and nature and level of stresses that they experience. However, it is difficult to measure or quantify the internal stresses experimentally within the bulk powder. As for the single agglomerates addressed in the previous studies, an appropriate approach for this purpose is the use of computer simulation technique by DEM.

4.1. Effect of size ratio on the breakage of agglomerate Experimental studies show that small particles within a bed under oedometric compression break more readily than large particles [57–59]. There is a cut-off limit of relative particle size, beyond which it is very difficult to break the larger particles when they are surrounded by smaller particles and subjected to mechanical loading. Antony and Ghadiri [60] performed a detailed stress analysis of a large spherical inclusion inside a cubical periodic assembly subjected to quasi-static shear deformation. The average stress tensor of the inclusions was

Analysis of Agglomerate Breakage

863

resolved into two components, viz. hydrostatic and deviatoric stresses. They showed that for particles having a size ratio greater than about 5, the nature of average stress on the inclusion was predominantly hydrostatic, as the ratio between the deviatoric and hydrostatic components was about 0.2 at the steadystate shearing regime. They suggested that the predominant nature of the hydrostatic (isotropic) stresses on large particles could retard their breakage, whilst the presence of strongly deviatoric (anisotropic) stresses on small particles would induce their breakage. However, the study considered the large inclusions as a single particle. In practice, e.g., in granulators, the large size inclusion tends to be in agglomerate form. Recently, Hassanpour et al. [61] have studied the deformation and breakage behaviour of an agglomerate in a bed of particles subjected to shear deformation using DEM. The agglomerate was prepared following the procedure of Moreno et al. [16] and hence its initial state of structure was isotropic and homogenous. Inclusions with several size ratios were generated and agglomerate behaviour was studied under shearing.

4.1.1. Stress ratio Hassanpour et al. [61] generated four different assemblies of non-adhesive particles with inclusion size ratios in the range 3–10 (the ratio between the diameters of agglomerate (D) and surrounding particles, d as shown in Fig. 16). The assemblies were then sheared at a constant normal pressure of 1 MPa (Fig. 17). The shear rate was about 300 s
1. The stress tensor for the agglomerate was partitioned into the hydrostatic and deviatoric components [60] and the evolution of the stress ratio of the agglomerate (deviatoric/hydrostatic stresses) was

Fig. 16. The agglomerate within an assembly of primary particles [61].

M. Ghadiri et al.

864

tracked down. The stress ratio is given by Stress ratio ¼

pffiffiffi 2ðsyy
sxx Þ tD ¼ ðsxx þ syy þ szz Þ p

ð21Þ

where, tD is the deviatoric stress, p the hydrostatic stress and sxx, syy and szz correspond to the stresses on the agglomerate along directions shown in Fig. 17. As shown by Hassanpour et al. [61], the stress ratio with shear strain experienced by the agglomerate within assembly is different for cases of different size ratios (Fig. 18). The stress ratio attains a value of about 0.2 for the assembly with size ratio 10, which is the same as reported previously by Antony and Ghadiri [60] for a single large spherical inclusion in granular media subjected to slow shearing. This means that stresses on the agglomerate are predominantly hydrostatic

Stress ratio

Fig. 17. Sectional view of the shear box used in the simulation [61].

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 -0.10 -0.20 -0.30 -0.40 -0.50 -0.60 -0.70

Size ratio 3.3 Size ratio 5 Size ratio 6.7 Size ratio 10

0

0.05

0.1

0.15

0.2

0.25

Strain

Fig. 18. The evolution of stress ratio with strain during shearing (Hassanpour et al. [61]).

Analysis of Agglomerate Breakage

865

Maximum stress ratio (-)

0.7 Hassanpour et al. [61]

0.6

Antony and Ghadiri [60]

0.5 0.4 0.3 0.2 0.1 0 0

5

10 Size ratio (-)

15

20

Fig. 19. The maximum stress ratio [61] and the steady-state stress ratio [60] on the agglomerate within the assemblies with different size ratios.

(stress ratio less than 0.2). However, for the case of assemblies with lower size ratios the stress ratio is higher than that of size ratio 10. Hassanpour et al. [61] evaluated the maximum stress ratios during shearing as a function of the size ratio. The results are shown in Fig. 19. It can be seen that the stress ratio increases as the size ratio of the agglomerate is decreased, implying that the state of the stresses on the agglomerate becomes predominantly deviatoric, as the size ratio decreases. This agrees well with the results of Antony and Ghadiri [60], though the stress ratio was reported at a steady-state value. There is a transition at the size ratio of about 5, below which the stresses on the agglomerate are highly deviatoric (stress ratio 0.6).

4.1.2. Damage ratio Hassanpour et al. [61] observed that the agglomerate in a particle bed never fractured during the shear test when the size ratio of assembly was 10. However for the cases of size ratio smaller than 10, agglomerates underwent macroscopic breakage. They have calculated the damage ratio of agglomerate, at a particular time for assemblies with different size ratios (Fig. 20). It was observed that the damage ratio for the cases of size ratio 10 is lower than the other cases, and a value of 0.3 is obtained. It is interesting to note that despite some contact rearrangements within the agglomerate during shearing, the agglomerate does not loose its structural integrity at the end of the test. As it can be seen from Fig. 20, the damage ratio for the agglomerate with size ratio 7 is higher than that of size ratio 10, and the former experiences breakage when the damage ratio reaches 0.5 (corresponding to the shear strain of 0.13). For the case of assembly with size ratio 5, the damage ratio increases with shear

M. Ghadiri et al.

866 1.00 Size ratio 3.3

0.90

Size ratio 5

Beyond this level the agglomerate disintegrates into small clusters

Size ratio 6.7

0.80

Size ratio 10

Damage ratio

0.70 0.60 0.50 0.40

Beyond the level a macroscopic breakage for the agglomerate is observed

0.30 0.20 0.10 0.00 0

0.05

0.1

0.15

0.2

0.25

Strain

Fig. 20. The evolution of stress ratio with strain during shearing [61].

strain to the value of 0.57, beyond which the agglomerate undergoes a macroscopic breakage (corresponding to the shear strain of 0.095). By increasing the shear strain the damage ratio increases to a maximum value of 0.74 and after this point the agglomerate disintegrates into smaller clusters (corresponding to the strain of 0.28). For the case of assembly with size ratio 3, a distinctive breakage occurs after the damage ratio reaches 0.49 (corresponding to the shear strain of 0.055). During shearing, the agglomerate damage ratio increases with shear strain to a maximum value of 0.75 and beyond this point the agglomerate disintegrates into smaller clusters similar to the case of assembly with size ratio 5. The agglomerates in the assemblies with different size ratios (when the first sign of macroscopic breakage is observed during shearing) are plotted in Fig. 21. In this figure, different colours represent the clusters detached from the agglomerate. It can be seen from Fig. 21 that unlike the case of the assembly with size ratio 10, a distinctive level of breakage of the agglomerates is observed where their size ratio is equal or smaller than 7. Hassanpour et al. [61] report a stress ratio of about 0.4 that causes macroscopic breakage in the agglomerate for all assemblies with size ratios 3–7 (Figs. 18 and 20). This can be regarded as a failure criterion for agglomerates embedded in a bed subjected to shearing for the particular assembly under consideration.

4.2. Comparison with experiments Hassanpour et al. [61] carried out shear tests on real agglomerates made of calcium carbonate particles and polyethylene glycol (PEG) binder (applied as an

0.0024

0.0022 Z Axis

Z Axis

0.0022

0.0020

0.0020

0.0018 0.0018 YA xis 8 0101614 2 0 0. 0.0 .00 01 10 8 0 0.0 .00 00 0 0.0 0.0026 0.0028

0.0016 0.0018

0.0020

0.0022

0.0024

YA

xis

0.0016

0 0 .0 0. .00 018 0 0 0 16 0 .00 14 0. .00 12 0. 00 10 00 0 06 8

0.0016

0.0014 0.0018 0.0020 0.0022 0.0024 0.0026 0.0028 0.0030 0.0032

X Axis

Analysis of Agglomerate Breakage

0.0026 0.0024

X Axis

Assembly 1(size ratio 10)

Assembly 2 (size ratio 7) 0.0026

0.0024 0.0024 0.0022

Z Axis

Z Axis

0.0022 0.0020

0.0018

0.0018

0.0014 0.0022

0.0024

0.0026

0.0028

0.0030

X Axis

0.0032

xis

0 02 8 YAx 0.0.001 16 4 is 0 .00 01 12 0 0 0.0 00 01 08 0. 0.0 .00 006 0 .0 0.0014 0 0.0018 0.0020 0.0022 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034 X Axis 0.0016

Assembly 4 (size ratio 3)

Fig. 21. The agglomerate within an assembly with size ratios smaller than 10 after applying external stresses [61].

867

Assembly 3 (size ratio 5)

0 0 .0 0 .0 02 0. .00 020 2 0 0 1 0 .0 01 8 0 .0 01 6 0 .0 01 4 0. .00 0102 00 0 06 8

YA

0.0016

0.0020

0.0020

868

M. Ghadiri et al.

aqueous solution) to validate their simulation results. The granulation process was conducted in a Cyclomix high shear granulator, manufactured by Hosokawa Micron B.V. After the agglomeration process was completed, nearly spherical agglomerates were chosen under an optical microscope for shear experiments. Two different assemblies made of primary particles (calcium carbonate) with a size ratio of about 10 (Assembly A) and 3 (Assembly B) were prepared in a stainless-steel shear box. Here the two assemblies represent the two extreme cases of the simulation. The agglomerates were coloured to trace possible breakage. In each assembly four agglomerates were placed at different positions on the horizontal shear plane (mid level). Both the assemblies were loaded on the upper part of the box, which was then moved using a motor to shear the assemblies (similar to that of Fig. 17). After shearing, the assemblies were carefully examined to recover the agglomerate. The recovered agglomerate and the primary particles for both the assemblies are shown in Fig. 22. The agglomerates in Assembly A (Fig. 22a), where the size ratio was 10, have not suffered any breakage in agreement with those of the simulations. However in Assembly B (size ratio 3) some fragments are visible, while two agglomerates do not show any breakage (Fig. 22b). A possible explanation for the presence of undamaged agglomerates is their movement away from the shear band during normal compression. This could result in agglomerates being under less shear stress in a non-shearing region of the assembly. This, however, is not the case for the simulation as the agglomerate is exactly placed within the shear region of the assembly and its position monitored with time.

Fig. 22. The agglomerates within the Assembly A (size ratio 10) and Assembly B (size ratio 3) after shearing in a box shear cell [61].

Analysis of Agglomerate Breakage

869

4.3. Relevance to granulation process An agglomerate breakage within a shearing bed of particles is clearly dependant on the size ratio. Hassanpour et al. [61] suggest that the stress ratio of 0.4 may be regarded as the failure criterion of an agglomerate within a shearing bed for the agglomerate considered. This criterion can be met for assemblies with size ratio equal or smaller than 7. However, the above failure criterion cannot be reached for the case of size ratios greater than about 7. This analysis would provide a direct application for the case of granulation process, where various stages, commonly known as nucleation, growth and breakage prevail [54,62–63]. The breakage of large agglomerates is a crucial stage before the completion of granulation processes, as it leads to the production of a desirable size distribution of agglomerates. The study by Hassanpour et al. [61] suggests that within a shearing mass inside granulators, the breakage of agglomerates having a size ratio larger than 7 would be difficult due to the predominantly hydrostatic nature of stresses experienced by the agglomerates. This implies that larger agglomerates will have to be broken using an alternative process other than shearing the bed, e.g. using choppers or blades. However, it is worth pointing out that the study carried out by Hassanpour et al. [61] was limited to one single value of surface energy and the failure criterion depends on the magnitude of the surface energy [22,26,62]. In addition, spherical agglomerates were simulated, while agglomerates are not always in spherical form in high shear mixer granulators. Further, studies are in progress to examine the influence of surface energy and shape of agglomerate on the deformation behaviour.

Nomenclature Ac c D DR EK E Fc H Ie k Kc l m N NB

area of a contact (m2) crack length (m) particle diameter (m) damage ratio (–) incident kinetic energy of an agglomerate (J) elastic modulus (Pa) interparticle contact force (N) hardness (Pa) elastic adhesion index (–) proportionality constant in equation (8) (–) fracture toughness (Pa m1/2) characteristic length of a solid particle (m) particle mass (kg) number of particles in an agglomerate (–) number of broken contacts (–)

870

M. Ghadiri et al.

N0 p V V0 Wc We We0 Z a b G GC g D e n r s sii tD f

initial number of bonds in an agglomerate (–) hydrostatic stress (Pa) particle velocity (m/s) velocity under which no contacts are broken in agglomerates (m/s) work for breaking one contact (J) Weber number (–) modified Weber number (–) coordination number (–) attrition propensity parameter (–) power law index in equation (20) (–) interface energy (J/m2) fracture energy (J/m2) surface energy (J/m2) dimensionless group in equation (11) (–) porosity (–) Poisson’s ratio (–) particle density (kg/m3) stress (Pa) components of the stress tensor (Pa) deviatoric stress (–) packing fraction (–)

REFERENCES [1] B.J. Ennis, J.D. Litster, in: R. Perry, D. Green (Eds.),Perry’s Chemical Engineers’ Handbook, 7th edition, McGRaw-Hill, New York, 2001, pp. 20–56 and 20–89. [2] K.D. Kafui and C. Thornton, Powders & Grains 93, the proceedings of the 2nd International Conference on Micromechanics of Granular Media, C. Thornton (Ed.), Balkema, Rotterdam, 1993, pp. 401–406. [3] A.D. Salman, D.A. Gorham, A. Verba, Wear 186–187 (1995) 92–98. [4] C. Thornton, D. Kafui, T. Ciomocos, IFPRI annual meeting, Urbana, US, 1995. [5] Z. Ning, R. Boerefijn, M. Ghadiri, C. Thornton, Adv. Powder Technol. 8 (1996) 15–37. [6] C. Thornton, K. Yin, M.J. Adams, J. Phys. D: Appl. Phys. 29 (1996) 424–435. [7] J. Subero, Z. Ning, M. Ghadiri, C. Thornton, Powder Technol. 105 (1999) 66–73. [8] C. Thornton, M.T. Ciomocos, M.J. Adams, Powder Technol. 105 (1999) 74–82. [9] K.D. Kafui, C. Thornton, Powder Technol. 109 (2000) 113–132. [10] B.K. Mishra, C. Thornton, Int. J. Miner. Process. 61 (2001) 225–239. [11] J. Subero, Ph.D. Dissertation. Impact Breakage of Agglomerates, Surrey University, 2001. [12] J. Subero, M. Ghadiri, Powder Technol. 120 (2001) 232–243. [13] M. Ghadiri, Z. Zhang, Chem. Eng. Sci. 57 (2002) 3659–3669. [14] K.D. Kafui, C. Thornton, M.J. Adams, Chem. Eng. Sci. 57 (2002) 2395–2410. [15] C. Thornton, M.T. Ciomocos, M.J. Adams, 10th Eur. Symp. Comminution, Heidelberg, Germany, 2002. [16] R. Moreno, S.J. Antony, M. Ghadiri, Powder Technol. 130 (2003) 132–137. [17] A.D. Salman, A.D. Gorham, A. Verba, Powder Technol. 130 (2003) 359–366. [18] R. Moreno, Ph.D. Thesis, University of Surrey. Guildford, UK, 2003.

Analysis of Agglomerate Breakage

871

[19] D. Golchert, R. Moreno, M. Ghadiri, J. Litster, R. Williams, Adv. Powder Technol. 15 (2004) 447–457. [20] D. Golchert, R. Moreno, M. Ghadiri, J. Litster, Powder Technol. 143–144 (2004) 84–96. [21] M. Khanal, W. Schubert, J. Tomas, Granular Matter 5 (2004) 177–184. [22] R. Moreno, and M. Ghadiri, PARTECH 2004, Nuremberg. Germany, Paper 3.5, 2004. [23] M. Nishida, K. Tanaka, Y. Matsumoto, JSME Int. J. Series A 47 (3) (2004) 438–447. [24] A. Samimi, R. Moreno, M. Ghadiri, Powder Technol. 143–144 (2004) 97–109. [25] G.K. Reynolds, J.S. Fu, Y.S. Cheong, M.J. Hounslow, A.D. Salman, Chem. Eng. Sci. 60 (2005) 3969–3992. [26] R. Moreno-Atanasio, M. Ghadiri, Chem. Eng. Sci. 61 (2006) 2476–2481. [27] P.A. Cundall, O.D.L. Strack, Geotechnique 29 (1979) 47–65. [28] C. Thornton, C.W. Randall, Micromechanics of Granular Materials, Elsevier Science Publishers B.V., Amsterdam, 1988. [29] C. Thornton, J. Phys. D: Appl. Phys. 24 (1991) 1942–1946. [30] C. Thornton, K.K. Yin, Powder Technol. 65 (1991) 153–166. [31] C. Thornton, Z. Ning, Powder Technol. 99 (1998) 154–162. [32] F. A. Gilabert, A. M. Krivtsov, A. Castellanos, Meccanica, 41 (2006) 341–349. [33] A. Di Renzo, P. Di Maio, Chem. Eng. Sci. 60 (2005) 1303–1312. [34] H. Mio, A. Shimosaka, Y. Shirakawa, J. Hidaka, J. Chem. Eng. Japan 38 (2005) 969–975. [35] F.Y. Fraige, P. Langston, Adv. Powder Technol. 15 (2) (2004) 227–245. [36] D. Zhang, W.J. Witten, Powder Technol. 98 (1998) 223–230. [37] H. Rumpf, Proceedings of International Symposium on Agglomeration, W. A. Knepper (Ed.), Vol. 379, 1962. [38] K.J. Kendall, Powder Metall. 31 (1988) 28–31. [39] A.A. Griffith, Philos. Trans. Roy. Soc. London A221 (1920) 163. [40] J.N. Israelachvili, Intermolecular and surface forces, Academic Press, London, Second Edition, 1985. [41] K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc. London A. 324 (1971) 301–313. [42] G. De Josselin de Jong, A. Verrujit, Cah. Grpe fr. Etud. Rhe´ol. 2 (1969) 73–86. [43] M. Ghadiri, Powder Technology Handbook, K. Gotoh, H. Masuda, K. Higashitani (Eds.), 2nd edition, Marcel Dekker, New York, 1997, pp. 183–191. [44] Z. Zhang, M. Ghadiri, Chem. Eng. Sci. 57 (2002) 3671–3686. [45] C.C. Kwan, H. Mio, Y.Q. Chen, Y. Ding, F. Saito, D.G. Papadopoulos, A.C. Bentham, M. Ghadiri, Chem. Eng. Sci. 60 (5) (2005) 1441–1448. [46] R. Moreno-Atanasio, S.J. Antony, M. Ghadiri, Powder Technol. 158 (2005) 51–57. [47] M. Ghadiri, S. J. Antony, R. Moreno, Z. Ning, Development in Handling and Processing Technologies, W. Hoyle (Ed.), Royal Society of Chemistry, London, 2001, pp. 70–81. [48] B. Behera, F. Kun, S. McNamara, H.J. Herrmann, J. Phys.-Condens. Matter. 17 (24) (2005) S2439–S2456. [49] W. Schubert, M. Khanal, J. Tomas, Int. J. Miner. Process. 75 (2005) 41–52. [50] N. Arbiter, C.C. Harris, G.A. Stamboltzis, Trans. AIME 244 (1969) 118–133. [51] A.D. Salman, G. K Reynolds, J.S. Fu, Y.S. Cheong, C.A. Biggs, M.J. Adams, D.A. Gorham, J. Lukenics, M.J. Hounslow, Powder Technol. 143–144 (2004) 19–30. [52] P.C. Knight, T. Instone, J.M.K. Pearson, M.J. Hounslow, Powder Technol. 97 (1998) 246–257. [53] M. Landin, P. York, M.J. Cliff, R.C. Rowe, A.J. Wigmore, Int. J. Pharm. 133 (1996) 127–131. [54] B.J. Ennis, J. Li, G.I. Tardos, R. Pfeffer, Chem. Eng. Sci. 45 (1990) 3071–3088. [55] B.J. Ennis, G.I. Tardos, R. Pfeffer, Powder Technol. 65 (1991) 257–272.

872

M. Ghadiri et al.

[56] G.I. Tardos, M.I. Khan, K.P. Hapgood, O.O. Ipadeola, J.N. Michaels, Powder Technol. 140 (3) (2004) 217–227. [57] O. Tsoungui, D. Vallet, J. Charmet, Powder Technol. 105 (1999) 190–198. [58] M.K. Gundepudi, B.V. Sankar, J.J. Mecholsky, D.C. Clupper, Powder Technol. 94 (1997) 153–161. [59] G.R. McDowell, M.D. Bolton, Geotechnique 48 (5) (1998) 667–679. [60] S.J. Antony, M. Ghadiri, J. Appl. Mechan. 68 (2001) 772–775. [61] A. Hassanpour, S.J. Antony, M. Ghadiri, Chem. Eng. Sci. 62 (2007) 935–942. [62] G.I. Tardos, M.I. Khan, P.R. Mort, Powder Technol. 94 (1997) 245–258. [63] J.D. Litster, K.P. Hapgood, J.N. Michaels, A. Sims, M. Roberts, S.K. Kameneni, T. Hsu, Powder Technol. 114 (2001) 32–39.