Breakagebehaviourofsphericalgranulatesbycompression¤ge/AntonyuK_CES_2005-p...the cyclic loading of metallic as well as non-metallic materi-als leads to the breakage at stresses, which

Chemical Engineering Science 60 (2005) 4031–4044

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Breakage behaviour of spherical granulates by compression

Sergiy Antonyuka,∗, Jürgen Tomasa, Stefan Heinrichb, Lothar Mörlb

aDepartment of Process Engineering, Otto-von-Guericke-University of Magdeburg, Universitätsplatz 2,39106 Magdeburg, Germany

bDepartment of Process Equipment and Environmental Technology, Otto-von-Guericke-University of Magdeburg, Universitätsplatz 2,39106 Magdeburg, Germany

Available online 7 April 2005

Abstract

This paper describes the deformation and breakage behaviour of granulates in single particle compression test. Three industrial sphericalgranulates—�-Al2O3, the synthetic zeolite Köstrolith� and sodium benzoate(C6H5COONa) were used as model materials to study themechanical behaviour from elastic to plastic range. The elastic compression behaviour of granulates is described by means of force-displacement curves, by application of Hertz–Huber contact theory and continuum mechanics. An elastic–plastic contact model wasproposed to describe the deformation behaviour of elastic–plastic granules. The effects of granulate size and stressing velocity on thebreakage force and contact stiffness during elastic and elastic–plastic displacement are examined. It is shown that the zeolite granulateswith elastic–plastic behaviour have viscous properties as well. Breakage mechanisms of granulates during elastic, elastic–plastic and plasticdeformation are also explained. The breakage probability is approximated by Weibull distribution function. The behaviour of the granulateduring compression under the repeated loading–unloading conditions was investigated.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Granulation; Compression test; Force-displacement curve; Elasticity; Crushing mechanism; Breakage probability

1. Introduction

Granulates are used in many industrial applications, e.g.catalysts, adsorbents, ceramics, pesticides, fertilizers, sludgegranulates, tablets, etc. The powdered raw materials aregranulated to avoid technological problems, for example,time consolidation and segregation of cohesive powders inbunkers.

During transportation and handling of granulates, break-age and attrition occur, which change the particle size dis-tribution and deteriorate the product quality and sometimesmay form harmful toxic dust.

In many processes granulates are subjected to variousstressing conditions, for example, interparticle collisions,granules–apparatus walls collisions. The latter collisions can

∗ Corresponding author. Tel.: +49 391 67 11867; fax: +49 391 67 11160.E-mail address:[email protected]

(S. Antonyuk).

0009-2509/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2005.02.038

be observed during spray granulation in a fluidized bed(Heinrich et al., 2002). As a result, particle attrition andbreakage affect granule growth, nucleus formation and there-fore residence time distribution and product quality.

In general, the granulates should not form dust and frag-ments during transportation, storage and handling. The max-imum stressing conditions during these operations define thelower limit of the strength which all granules should havein order to be able to resist the stressing. On the other hand,they should be soft enough in order to retain the solvability,dispersibility, moisturazion properties and to avoid compli-cations during the further processing. For example, for theproduction of high performance ceramics, powders are gran-ulated first so that they do not break during transport, butfail during further pressing (Agniel, 1992).

In order to optimize the existing production processes andminimize the product quality losses during transportationand handling, the breakage behaviour of granulates must beinvestigated by experiments and physical modelling.

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4032 S. Antonyuk et al. / Chemical Engineering Science 60 (2005) 4031–4044

2. Research background

The attrition, shear and compressive strength of a granu-late can be examined by bed stressing of the granules, e.g.,within a rotary drum. A single granule can be tested by com-pression, tension, bending, impact and cut tests. With thesetests the influence of the single granule properties such assize, porosity and roughness on the granulate behaviour inthe bed can be determined. One of the most important testscan be realized by simple compression of a single particleup to the first breakage, which determines the minimum en-ergy requirement for the breakage (Carey and Bosanquet,1933). The losses of stressing energy in single particle com-pression experiments are almost negligible. These losses aremuch higher in particle bed crushing because of friction andplastic deformation of the particles at contacts. In compres-sion test due to the comparatively low deformation velocity(from mm/min to cm/min) the secondary breakages (afterthe primary breakage) can be separately observed. Hence,in this paper the breakage behaviours of granulates are anal-ysed by performing single particle compression tests.

2.1. Force-displacement behaviours

Rumpf (1965)andSchönert (1966)described the defor-mation and the breakage behaviour of limestone, cementclinker and quartz particles by force-displacement curves.Later many authors have dealt with force-displacementbehaviours of solid particles in compression tests. Someof them investigated the force-displacement behaviours byvarying size and shape of particles at different temperaturesand stressing velocities (Stieß, 1976). Two test modes wereused: strain-driven and stress-driven.

Many authors have used different agglomerates toinvestigate the breakage processes:Schönert (1966),May (1975)—clay pellets and cement clinker,Charé(1976)—saline pellets,Kiss (1979)—press agglomeratesmade of a cement matrix and quartz particles,Aziz (1979)and Hess (1980)—cement clinker. In their works, force-displacement curves of these agglomerates have shown theelastic–plastic properties which are also confirmed by therecent granulate researches. For example, the followinggranulates have been investigated recently: Al2O3 granu-lates produced by fluidized bed-spray granulation (Agniel,1992); detergents (Samimi et al., 2002); enzyme-containinggranulates (Beekman et al., 2003); polymer bound Al2O3granulates (Sheng et al., 2004).

2.2. Stress field

The breakage process during the compression of particlescan be explained by pressure distribution and stress field.Analytical solutions were obtained only for elastic spheres.Hertz (1882)calculated the ellipsoidal pressure distributioninside the sphere during its contact deformation. The stress

field inside an infinite elastic half-space was presented byHuber (1904). Lurje (1963)calculated the spatial stress dis-tribution in the whole sphere. Based on the previous achieve-ments,Stieß (1976)described the spatial stress distributionin an elastic steel sphere under two point loads and com-pared it with the measured distribution at the surface of apolymer sphere during an elastic–plastic deformation.

2.3. Breakage behaviour and particle bonds

Compared to crystalline solids the granulates are par-ticle compounds and tend to show more plastic force-displacement behaviours. A granule consists of primaryparticles, which stick together by the adhesion force attheir contacts. Depending on the granulation process theinternal adhesion force is influenced by the superpositionof van der Waals interactions between fine primary parti-cles, capillary or solid bridges, high-viscous binder, organicmacromolecules, sintering or interlocking of granules. Themechanical breakage behaviour of granulates is stronglydetermined by these microbinding mechanisms. The firstsystematic investigations of the strength of granulates havealready confirmed the micro–macro correlation of parti-cle bonds (Rumpf, 1958, 1962; Schubert, 1975; Kendall,1987). The well-known model ofRumpf (1958)describesthe tensile strength of a granule taking into considerationthe characteristic adhesion forces at the contact points ofmono-sized spherical primary particles in a random pack-ing. Bika et al. (2001)have generalizedRumpf (1958)andKendall (1987)models and extended them for the porousgranulates. The granulates having high-viscous liquid andhigh saturation levels show a dependency of the tensilestrength versus capillary pressure (Schubert, 1975).

As a consequence of wetting, chemical reactions can oc-cur within a granule, whereby original mechanical proper-ties can be changed. Due to this reason many experimentswere carried out with model granulates made from primaryglass particles and binder solutions to avoid the possibil-ity of chemical reactions (Pierrat and Caram, 1997; Ivesonand Page, 2001). Briscoe et al. (1998), Pepin et al. (2001)and Samimi et al. (2002)performed compression tests onspherical granulates and determined the force-displacementcurves. An overview of previous works on the strength ofwet granulates was given bySimon et al. (2001).

The breakage phenomena of granulates can be comparedwith solids. The inhomogeneous by structured spherical par-ticles being under compression show a plastic contact area,breakage cone and meridian cracks, e.g. concrete (Tomas etal., 1999; Khanal et al., 2004), fertiliser granules (Salman etal., 2003).

2.4. Breakage probability and fragment size distribution

May (1975) and Klotz and Schubert (1982)describedfragment size distribution, breakage probability and break-

S. Antonyuk et al. / Chemical Engineering Science 60 (2005) 4031–4044 4033

age function for glass particles, clay pellets and cementclinkers by three and four parametric log-normal distribu-tions and calculated the specific surface of fragments. Dur-ing compression-shear test of various materialsHess (1980)observed the reduction in breakage force and energy be-cause of additional shear stress.Kerber (1984)showed thatthere is a high influence of shape and roughness of stressingtools on mass-related breakage energy and breakage functionfor gypsum, limestone and quartz bulk materials.Weichert(1992)developed a breakage probability model for sphericalagglomerates and proved that during compression and im-pact tests, an increase in mass-related breakage energy andgranule diameter shows higher breakage probability.Cheonget al. (2003)used Weibull statistics to describe the particlesize distributions in compression and impact tests.

A large specific surface area and porous structure can beobtained after granulation, which is needed for sufficient ab-sorption, adsorption, solubility or reactivity properties. Theporosity and its shape, orientation and size distribution havea large effect on the breakage behaviour of the granulates.The pores can be regarded as a crack release zones. Thepores and the structural defects in granules are analogy toimperfections, inhomogeneities or microcracks in materialscience. The highest local tensile stress is generated at thesedefective zones in the granules, so that the fracture initiatesfrom this zone. The number of contact points (coordinationnumber) of a primary particle can be decreased with largerporosities (Smith et al., 1929; Bika et al., 2001), by whichthe strength of granulate decreases according to the micro-models ofRumpf (1958)andKendall et al. (1983). Inves-tigations of porous materials (ceramic, glass, polymers, ce-ment clinkers) and composites show the reduction of modu-lus of elasticity and Poisson ratio with an increasing porosity(Boccaccini, 1994; Avar et al., 2003).

2.5. Failure during the cyclic loading

From the fracture mechanics of materials it is known, thatthe cyclic loading of metallic as well as non-metallic materi-als leads to the breakage at stresses, which are substantiallylower than the failure stress during the static loading. Thereduction of fracture strength occurs because of formationand propagation of shear zones and microcracks during eachcycle. The number of the cycles up to the fracture decreaseswith decreasing stress amplitude, that are already describedfor metals with the Wöhler curve (Riehle and Simmchen,2000). These investigations of solid particles and agglomer-ates demonstrate the considerable effect of repeated loadingon the breakage point.Tavares and King (2002)describeda decreasing elastic–plastic stiffness of particles during re-peated impact and explained the breakage behaviour withthe formation and propagation of damages. The intensityand the frequency of stressing, the particle size and the mi-crostructure influence material resistance against the cyclicloading.Beekman et al. (2003)andPitchumani et al. (2004)have also confirmed this effect for granulates.

2.6. Simulation of the breakage of granulates

For the simulation of stressing conditions and breakagemechanisms of granulates two numerical methods can beused—finite-element method and discrete element method.

The finite-element method regards the granulates mechan-ically as homogeneous material. Based on the assumptionthat granulates are a continuum, the stress propagation dur-ing uniaxial and biaxial test can be calculated for a sphere(Stieß, 1976; Ksoll, 1984; Kienzler and Baudedistel, 1985;Tsoungui et al., 1999). Granulates can exhibit either elas-tic or elastic–plastic properties, hence both material laws(elastic and elastic–plastic) are used in continuum mechan-ics (Adams, 2004; Khanal et al., 2004).

But granules are particle compounds, and thus, the dis-crete element method is better to use to calculate the con-tact forces and relative movements of particles (Cundall andStrack, 1979; Konietzky, 2001; Kafui and Thornton, 2000).During this simulation a granule is regarded as a systemof primary particles. The discrete element computer simula-tions can illustrate more details of the breakage behaviour,like network of contact forces, crack initiation and propa-gation inside a granule and fragmentation during stressing(Khanal et al., 2004; Thornton et al., 2004).

The breakage behaviour of granulates duringcompression is still an area of investigation, because of thevarious types of granulates, their microstructures and me-chanical properties. There is a necessity to understand theforce-displacement behaviours of granulate at elastic–plasticand rigid-perfectly plastic range.

3. Theoretical approaches

During compression of a comparatively soft sphericalgranule with a smooth stiff punch (flat surface), the con-tact area between them deforms as a circle with radiusrk(Fig. 1). The contact radius and internal pressure distribu-tion p depend on the granule radiusr and stiffness of thetwo contacting materials.

3.1. Elastic contact deformation

In this case, a circular contact area of a radiusrk,el is builtwith an ellipsoidal pressure distributionp(rk). Hertz (1882)has found the maximum contact pressure in the centre of thecontact at the depth, shown by pointK in Fig. 1a, as

pmax = 3Fel

2�r2k,el

. (1)

All three principal stresses in pointK are calculated as pres-sures according to Eq. (2) (Stieß, 1976; Pisarenko et al.,1986). At this pointK, all the stresses have nearly the samemagnitude. They are compressive stresses and generate ap-proximately an isostatic stress state (Pisarenko et al., 1986),


Fig. 1. Characteristic particle contact pressurep(rk) on a plate–sphere contact during elastic deformation (a) and elastic–plastic deformation (b).

whereby tension is shown by negative sign and compressionby positive one:

�1 = pmax,

�2 = �3 ≈ 0.8�1. (2)

As a consequence, no cracks can be observed at this state.The maximum tensile stress�t,max arises at the contactperimeter and can be calculated according to Eq. (3) (Huber,1904) for particle with Poisson ratio�1 = 0.28, �t,max =−0.15pmax

�t,max = −1 − 2�1

3pmax. (3)

The maximum shear stress on the principal axis occurs at thedepth ofK–Z ≈ 0.5rK,el (pointZ in Fig. 1a). The principalstresses in this point are given by Eq. (4). The shear stresscan be calculated with Tresca failure criterion Eq. (5) (Grossand Seeling, 2001). It is larger than the maximum tensilestress, according to Eq. (3), and is responsible for the crackgeneration, especially for plastic materials.

�1 = 0.8pmax,

�2 = �3 = 0.18pmax, (4)

�max = �1 − �3

2= 0.31pmax. (5)

The radius of elastic contact is given byHertz (1882)

r3K,el =

3rF el

2E∗ . (6)

The tensile region at the perimeter of the contact pressuredistribution according toHuber (1904)is responsible for dis-tortion. Due to this distortion at the perimeter directly out-side of the contact circle, the radius of totally deformed areais larger than the contact radius:rd �rK,el (Fischer-Cripps,2000) (Fig. 1).

The effective modulus of elasticityE∗ of both particle(index 1) and punch (index 2)(E2 � E1, E2 → ∞) isgiven as

E∗ = 2

(1 − �2

1

E1+ 1 − �2

2

E2

)−1

≈ 2

1 − �21

E1. (7)

The effective shear modulusGi =Ei/(2(1+�i )) is given by

G∗ = 2

(2 − �1

G1+ 2 − �2

G2

)−1

≈ 2

2 − �1G1. (8)

The relation between elastic contact force and deformationis non-linear as found byHertz (1882)

Fel = 2

3E∗√

d

2s3. (9)

Due to the parabolic curvatureF(s), the contact stiffness innormal direction increases with increase in deformation andparticle diameter (Tomas, 2002):

kN,el = dFel

ds= E∗

√d

2s =

(Feld

4D2

)1/3

. (10)

The elastic constant determines here compliance of bothcontact materials (Lurje, 1963):

D = 3

4

(1 − �2

1

E1+ 1 − �2

2

E2

)≈ 3(1 − �2

1)

4E1. (11)

3.2. Elastic–plastic contact deformation

For elastic–plastic material behaviour, an elastic deforma-tion is generated at the limit, where the pressure is smallerthan the yield point, and plastic deformation is closer tothe centre of the contact (Fig. 1b). The maximum pressurepmax in the contact centreK1 lies below the plastic yieldstrengthpF (the stress at the beginning of plastic yielding).Because of a confined stress field the microyield strengthpF is higher than the macroscopic yield strength for tension�F : pF ≈ 3–5�F (Molerus, 1975; Pisarenko et al., 1986).The stiffness is proportional to the radiusr of the granuleand microyield strength,pF (Tomas, 2000)

kN,el.pl = dFel−pl

ds= �rpF

(2

3+ 1

3

Apl

AK

)

= �rpF

(1 − 1

33

√sF

s

), (12)


Fig. 2. Contact geometry by plastic deformation.

wheresF is a contact deformation at yield point. Further-more, at this pointpel = pmax = pF is valid.

The ratio of plastically deformed contact areaApl to thetotal contact deformation areaAK = Apl + Ael can be usedto define the elastic–plastic deformation and lies between0 and 1. The ratio is 0 for perfect elastic and 1 for perfectplastic deformation.

3.3. Plastic contact deformation

The whole contact area deforms plastically for a perfectplastic material. In this case, the contact circle radius is givenby

r2k = r2 − (r − sk,1)

2 = 2rsk,1 − s2k,1 ≈ dsk,1, (13)

where sk,i is the plastic deformation at the contacti, asshown inFig. 2. Having assumed that the plastic deformationat the two contacts is equal (sK,1 = sK,2, s = sK,1 + sK,2 andrk,1 = rk,2), one obtains

r2k = rs. (14)

The repulsive force against plastic deformation is calculatedas (Tomas, 2000)

Fpl = pF AK = �r2KpF = �dspF . (15)

With Eqs. (6), (14) and (1) the yield strengthpF is calculated

pF = E∗

�

√sF

r. (16)

The contact stiffness is constant for a perfect plastic yieldingmaterial:

kN,pl = dFpl

ds= �dpF . (17)

Fig. 3. Principle of granulate compression test.

4. Experimental results

4.1. Testing methods and materials

In this work the breakage behaviour was investigated bycompressing of individual granules (Fig. 3). For experimentsthe modern granule strength measuring system (producedby Etewe GmbH, Karlsruhe) was used. During the move-ment of the punch towards the upper fixed plate, the contactbetween the particle and the fixed plate is created. Duringthis period, the displacement, force and time are measured.The axial stress is applied at a controlled axial deformationrate. The equipment can perform the compression test forthe granules in a size range of 0.05–20 mm, up to maximum2 kN breakage force and a stressing velocity (deformationvelocity) of 0.01–2.5 mm/s. During repeated compressiontest, the punch moves towards the upper fixed plate side andpresses the granule up to the defined force or deformation.Then, the punch moves downwards, thus the unloading of thegranule takes place. The breakage process can be recordedwith a CCD-camera.

Three different spherical granulates—�-Al 2O3, the syn-thetic zeolite Köstrolith� and sodium benzoate(C6H5COONa) were used as model materials to investigatethe mechanical behaviour from elastic to plastic (Fig. 4). Asummary of the material properties of these granulates isgiven inTable 1. During the compression tests 100 particleswere examined in each experiment at the stressing velocityvB in the range of 0.02–0.15 mm/s.

4.2. Force-displacement behaviour

The typical force-displacement curve for�-Al 2O3-granulate is shown inFig. 5. At the beginning of thepunch–particle contact the elastic contact deformation of the


Fig. 4. Digital camera images of the examined granulates.

Table 1Characteristics of the examined granulates

Characteristics �-Al2O3 Köstrolith� Sodium benzoate

Manufacturer Sasol GmbH, CWK Chemiewerk Bad Köstritz, Bad DSM, GeleenHamburg Köstritz

Chemical composition (%) 97.9%�-Al2O3 85% synthetic zeolite 13X C6H5COONa(30%-Al2O3, 51%-SiO2,17%-Na2O, 2%-MgO);Binder: clay and water

Granulates size 1.62–1.76 0.90–1.20; 1.20–1.40; 0.80–0.96;distribution (mm) 1.40–1.70 1.24–1.64Agglomerate density(kg/m3) 1040 1300 1440Solid density(kg/m3) 3230 2100 1500Specific surface area(m2/g) 145 415 (without the surface of micropores) 7.6Pore volume (%) 68 45 (macropores) 4Application For drying Adsorbent (molecular sieve) Food and beverage

processes for drying processes and preservativecleaning of gas

0

10

20

30

40

50

0 0.01 0.02 0.03 0.04 0.05 0.06

Displacement s in mm

Fo

rce

F in

N

experimental values

approximated elastic rangeaccording to Hertz

approximated elastic-plastic range

B - breakage point

F - yield point

sF

FF

O

Fig. 5. Force-displacement curve of �-Al2O3-granulate(vB = 0.02 mm/s, d = 1.7 mm): O–F-elastic deformation,F–B-elastic–plastic deformation.

granule takes place. The elastic forceFel was well describedby Hertz theory (Eq. (9)). Young’s modulus, shear modulusand the stiffness of the granule during elastic deformationwere calculated with Eqs. (7), (8) and (10) correspondingly.The average values of these mechanical properties by stress-ing velocityvB = 0.02 mm/s are summarized for examinedgranulates inTable 2. Due to the parabolic curvatureF(s),the contact stiffnesskN,el increases with increasing displace-ments and reaches the maximum value in the yield point Fin Fig. 5. In this point the plastic deformation begins. Thisis confirmed by the increasing deviation of the experimentalcurve F–B from the theoretical Hertz curve (Fig. 5).

The slope of the curve F–B is a measure of theelastic–plastic stiffness, which is proportional to the gran-ule radiusr and the yield strengthpF , according to Eq.(12) (Tomas, 2000). A small slope of the curve impliesmore plastic and large slope shows “stiff” material be-haviour. In Eq. (12), the yield strengthpF can be charac-terized at the beginning of the plastic deformationsF withthe contact forceFF . These parameters were determinedfrom the force-displacement curve (Fig. 5) and thereby theelastic–plastic stiffness and the force were approximated.


Table 2Mechanical characteristics of the examined granulates

Granulate Diameterd Breakage force, Yield point Modulus of elasticity (GPa) Stiffness in normal direction (N/mm)(mm) FB (N)

FF (N) sF (�m) E1 G1 kN,el for FF kN,el.pl for FB

�-Al2O3 1.62–1.76 39± 3.6a 15± 3.0 21± 3.8 4.0 1.54 714 865Köstrolith� 1.20–1.40 8.8 ± 1.1 1± 0.08 11± 0.9 0.82 0.32 100 172Sodium benzoate 0.80–0.96 6.2 ± 0.9 Nearly 0 Nearly 0 Nearly 0 Nearly 0 Nearly 0 138

a± Standard deviation.

Fig. 6. Force-displacement curve of Köstrolith� granulate during com-pression test.

For the adjustment of the theoretical curve to the exper-imental data a fit coefficientk is introduced (Eq. (18)).Fig. 5 shows the comparison between the experimental andtheoretical force-displacement curves (withk = 1.04). Thesmall deviation between these curves is explained by thefact that the contact area of the granule under compressionwas not perfectly circular and had visible roughness.

kN,el.pl = �rpF k

(1 − 1

33

√sF

s

). (18)

The breakage follows at point B with the force approxi-mately two times higher than the force at point F. However,in the breakage point the contribution of the plastic defor-mation in comparison to elastic deformation is very small,which is confirmed by the small difference in stiffness val-ues during elastic and elastic–plastic deformation (Table 2).

But compared to this the Köstrolith� granulate showsonly small elastic deformation (Fig. 6), but considerableelastic–plastic deformation before primary breakage whichcan be expected for most granulates. The ratio of the stiffnessduring elastic deformation at the yield stress to the stiffnessof elastic–plastic deformation at the breakage stress equalsto 1.72. After the breakage of a Köstrolith� granule the mul-

0

2

4

6

8

0 0.04 0.08 0.12 0.16

C1

Cn

B1-nucleusbreakage

Her

tz

E

4


For

ce F

in N

elastic-plasticdeformation

breakage phaseprimary and secondary breakages

B -meridian breakage

Fig. 7. Force-displacement curve of sodium benzoate granulate duringcompression test(vB = 0.02 mm/s, d = 0.87 mm).

tiple stressing leads to the failure of the fragments, B–E inFig. 6.

The force-displacement curve of sodium benzoate is astraight line along the whole deformation region (Fig. 7).According to Eq. (15), that means a perfect plastic deforma-tion occurs when (Ael = 0 andAK = Apl).

Fig. 8 shows the effect of a granulate size on the force-displacement behaviours. In the case of bigger granules,both the breakage force and the contact stiffness increaseduring elastic and elastic–plastic ranges and therefore thematerial becomes stiffer (see Eqs. (10), (12) and (17)).This increase in the breakage force does not influence thematerial strength,�max = FB/(�r2). As an example—forthe fractions of sodium benzoate granulesd = 1.24–1.64and 1.80–0.96 mm the average breakage force is 16.2 and6.2 N, respectively, and the average compressive strength is10.5 MPa in both cases.

Fig. 9clearly shows that the stressing velocity has a verybig influence on the breakage force and contact stiffness dur-ing elastic–plastic deformation of the Köstrolith� granulate.This influence confirms the viscoplastic behaviour of thesegranulates. The effect of stressing velocity on the break-age force of the granules with different size is presented inFig. 10a. This is caused by less time available for relaxation,creep and storage of elastic energy. However, the breakageforce for sodium benzoate and�-Al 2O3 granulates does notdepend so much on the stressing velocity used in the tests(Fig. 10b).


Fig. 8. Force-displacement curves of Köstrolith� (a) and sodium benzoate(b) granulates with different sizes(vB = 0.02 mm/s).

stressing velocityvB in mm/s:

0

5

10

15

0 0.02 0.04 0.06


For

ce F

in N

0.15

0.08

0.02

Fig. 9. Force-displacement curves of Köstrolith� granulate during com-pression test with different stressing velocity (d = 1.4–1.7 mm).

4.3. Breakage mechanism

The breakage mechanism depends on whether the gran-ules show elastic or plastic deformations before fracture. Thefracture behaviour can be changed with the stressing veloc-ity. All measurements were accomplished at a low constantvelocity of 0.02 mm/s. For the dominant elastic materialshigher tensile stress generates concentric ring cracks due to

Fig. 10. Effect of stressing velocity and granulate diameter on the breakageforce: (a) Köstrolith� and (b) sodium benzoate.

bending at the perimeter of contact. These circular crackspropagate from the contact area of the spheres towards thedirection opposite to the contact area. They can either con-verge or diverge around the central axis of loading. In thefirst case, two cones are developed between the contacts,whereby the surfaces of the cones lie in the shear zone. Thediverge case is formed at the fragment axis (Fig. 11a) andbecause of crack branching fine particles are created duringcracking and breakage. Crack propagation is unstable andtakes place at a very high velocity in the order of magnitudeof sonic speed.

But in terms of elastic–plastic behaviour a flattened sur-face is obtained under compressive stress at the contact.Internal shear zones are generated as consequence of plas-tic deformation in the main axis of stressing. A breakagecone is formed which is driven into the body of remainingsphere. Again this cone is generated when the shear stressreaches the plastic yield limit of the granule. A yield surfaceis formed with a conical shear zone. Within this shear zone,shear and normal stresses are generated. As response, ring


Fig. 11. Digital camera images of breakage phenomena of the dominantelastic ((a)�-Al2O3) and the elastic–plastic ((b) Köstrolith�) granulate:(1) breakage surface is diverging from the central axis of loading; (2)fragment axis; (3) breakage cone; and (4) meridian breakage.

tensile stresses are produced in the sphere which leads tomeridian cracks.Fig. 11b shows this breakage phenomenaof the Köstrolith� granulates.

Fig. 12. SEM of flattened surface of a sodium benzoate granule after the compression test: (1) the entire approximated circular contact area; (2) the realcontact area; and (3) less deformed part of a contact surface (roughness depressions).

In terms of perfect plastic material, a large flattened sur-face is formed at the contact.Fig. 12shows scanning elec-tron microscope images of a sodium benzoate granule intwo different angles after the compression stress. The figureshows that the flattened contact surface (1) is not perfectlycircular, since the granulates surface is not evenly curvedand contains roughness. Load and the plastic deformationoccurs only at the asperities (2) of the contact. Therefore asmall portion is deformed (3). At this case the crack is stableand for further propagation an additional energy is required.For the sodium benzoate granulates, the cracking velocity isvery small (about 20–25 mm/s). If the compression veloc-ity is faster than the crack propagation, then the stressingvelocity has a substantial influence on the cracking velocity.

The breakage phenomena depends considerably on mi-cro and macrostructures of the granulates, which are formedduring the production process. The sodium benzoate gran-ulates are produced by the fluidized bed granulation (Fig.13) (Heinrich et al., 2002). The process consists of the mul-tiple spraying, spreading and solidification of the dropletson the nuclei. According toFig. 14, the shell at the con-tacts first deforms and after that the crack (1) releases on thesurface, which separates the stiff nucleus (2) and the shell(3) with stable propagation during compression of layeredstructure. The binding forces between the shell and the nu-clei are smaller than the primary particle strength. The de-fects and the inhomogeneities present in the laminated zoneof this structure enhance the crack growth. After the forma-tion of meridian crack in the shell, the deformation and thefull breakage of the nucleus takes place. That leads to thesecondary increase of the force-displacement curve (pointB1 in Fig. 7).

4.4. Breakage probability

For a given particle size, the fracture stress at the firstbreakage is not constant. The mechanical characteristics ofthe primary particles and the bonding agents are randomly


+

drying /hardening of granule

+

fluidizing airgranule

nucleidroplet coating ofgranule

spraying and collisions droplet spreading drying and solidification

finale granule

"onion-likegranule structure"

Fig. 13. Granule growth by coating from the principle of fluidized bed spray granulation.

Fig. 14. Breakage phenomena of a sodium benzoate granule after thecompression test: (a) digital camera image of breakage of the shell undermeridian stress��; and (b) SEM of fracture surface (1—the crack, 2—thenucleus, 3—the shell).

distributed within the granulate. Even with the identical pro-duction process, the strength of the individual granulatesdiffers in the microstructure because of the distribution andorientation of bonds, defects and pore size distribution. Be-sides bond strength and orientation the distribution of in-homogeneities are responsible for the breakage behaviour.The breakage probabilityP of an agglomerate stressed bycompression can be described by Weibull statistics (Weibull,1939) (Eq. (19)), which includes the granule sized andmass-related breakage energyWm (Weichert, 1992):

P = 1 − exp(−cd2WZm). (19)

Table 3Weibull distribution parameters for the breakage probability of examinedgranulates

Material Granulate diameter, z c (kg/J)z m−2 Correlationd (mm) coefficient

�-Al2O3 1.62–1.76 5.11 4.4 × 10−8 0.99

1.40–1.70 4.70 9.8 × 10−5 0.97Köstrolith� 1.20–1.40 3.57 1.1 × 10−2 0.98

0.90–1.20 3.46 2.3 × 10−2 0.95

Sodium 1.24–1.64 4.01 1.1 × 10−4 0.96benzoate 0.80–0.96 4.81 1.8 × 10−6 0.96

Fig. 15. Breakage probabilityP of the different sized examined granulatesas a function of mass-related breakage energyWm: (a) Köstrolith�; (b)sodium benzoate; and (c)�-Al2O3.

The parameterz characterizes the defect distribution, whichdetermines the breakage and strength of the granulates.

Table 3shows the parametersz andc for examined gran-ulates. InFig. 15, the breakage probability of the experi-mental data are fitted by Eq. (19).

For increasing granulate size the curve is shifted to the left.That means, to initiate the fracture at the same probabilitylevel a higher mass-related energy is required for smallergranules than for larger granules.


Hertz

0

2

4

6

8

0 0.01 0.02 0.03 0.04 0.05


For

ce F

in N

loading unloading

breakage point with n=1

U

E

Fcyc (0.8-0.9) FB

breakage pointwith n=6

≈

Fig. 16. Loading–unloading force-displacement curves of the Köstrolith�-granulate(d = 1.54 mm; vB = 0.02 mm/s; Fcyc = 7 N).

4.5. Repeated compression of the granulates

Fig. 16 shows a typical force-displacement diagram forthe repeated compression of granulates. During the measure-ment, a Köstrolith�-granule was repeatedly loaded and un-loaded with a velocity of 0.02 mm/s up to the forceFcyc ≈(0.8.0.9)FB , which is called stressing amplitude. At thisload a large plastic deformation (curve O–E inFig. 16) isobserved, and this demonstrates elastic–plastic behaviour ofthis granulate. The unloading curve U–E is similar to theHertzian curve, however only an elastic deformation disap-pears during unloading. A further unloading below the pointE can be performed by applying a tensile forceFcyc, to de-termine the hysteresis loop (after reloading) for a completeunloading–reloading cycle (Tomas, 2002).

The area inside unloading and reloading curves U–E char-acterizes the energy dissipation or the damping behaviourof the granulate during compression. The mass-related in-elastic deformation work is reduced with each cycle, in thisexperiment, fromWm,diss= 6.4–6.8 J/kg in the first cycle toWm,diss = 2.9–3.2 J/kg in the last cycle (before breakage).During first cycle the maximum plastic deformation and thehighest breakage limit can be observed.

The characteristic deformation of two granulates areshown in theFig. 17. The curve A–B shows the changein the total strain of a granule in each loading cycle untilthe breakage point B. The number of cycles depends onintensity of the loading and the material properties. Com-pared to metals most granulates have very low strengthduring the cyclic stressing. For example, sodium benzoategranulates break already 14–20 compression cycles. Thereduction of total deformation shows a stiffening effectduring loading–unloading cycles.

In this case, the cyclic stiffening or hardening means thechange in structure of the material at the contact points,

Fig. 17. Deformation curves of repeated compression cycles: (a)Köstrolith� (d = 1.5 mm; Fcyc = 7 N; vB = 0.02 mm/s); and (b) sodiumbenzoate(d = 0.9 mm; Fcyc = 6 N; vB = 0.02 mm/s).

where the stresses are very high, and density, stiffness andthe modulus of elasticity are increased.Fig. 18 comparesthe fracture surfaces of sodium benzoate-granulate after one


Fig. 18. SEM of fracture surface of sodium benzoate-granulate: (a) afterone loading cycle; and (b) after 14 loading cycles.

(a) and 14 cycles (b). With the increase of cycle number themicrocracks propagate inside the specimen. The granulatestores cyclic loading energy and damages are developingduring an elastic–plastic deformation, which leads to a lowerbreakage force than at single loading.

5. Conclusions

Deformation and breakage behaviour of spherical granu-lates by compression was studied. The single particle com-pression tests were performed for different granulate sizeswith stressing velocity in the range of 0.02–0.15 mm/s. Thecontact models and continuum mechanics were used to eval-uate force-displacement curves and breakage characteristics,

which leads to the following conclusions:

1. The �-Al 2O3-granulates show a dominant elastic be-haviour (elastic–plastic range) during the compressiontest, Köstrolith� has both elastic and elastic–plasticproperties, and sodium benzoate granules deform per-fect plastically.

2. The correlation between the force and elastic defor-mation was described by Hertz theory from which thecontact stiffness and the modulus of elasticity weredetermined. These values of investigated granulates arevery low compared to those of metals. An elastic–plasticcontact model has been used to describe elastic–plasticdeformation. Material parameters of this model can befound from experimental force-displacement curves.Experimental results have shown the influence of thegranulate diameter on the force-displacement behaviour.In the case of bigger granulates both the breakage forceand the contact stiffness increase during elastic andelastic–plastic range. The zeolite granulate Köstrolith�

shows the viscoplastic behaviour, when the contact stiff-ness decreases with decrease of the stressing velocity.

3. The breakage phenomena depend on the elastic or plas-tic deformation of the contact before breaking and onmicrostructure of granules. During loading the dominantelastic�-Al 2O3 granulates create concentric ring cracksat the periphery of two contacts. At this major axis ofstressing several fragments are broken off with very highvelocity. For elastic–plastic granulates, a large flattenedsurface at the contact and a breakage cone are formed.

4. The breakage probability versus mass-related breakageenergy was described by Weibull statistics. It was shownthat more mass-related breakage energy is needed tobreak of smaller granules than bigger ones. The energydissipation and microcrack formation during the cyclicloading in the granulates lead to the reduction of thebreakage force.

Notation

A granule contact area, mm2

c breakage parameter of granules,(kg/J)z m−2

d granule diameter, mmD elastic compliance, mm2/NE∗, G∗ average modulus of elasticity and shear modu-

lus, GPaE1, G1 modulus of elasticity and shear modulus of the

granulate, GPaE2, G2 modulus of elasticity and shear modulus of the

punch, GPaF normal force, NFB breakage force, NFcyc maximum force with the cyclic loading, Nk fit parameter, dimensionless


n number of compressions, dimensionlessp pressure, MPapF plastic yield strength of granule contact, MPaP breakage probability, dimensionlessr granule radius, mms displacement, mmvB stressing velocity, mm/sW breakage energy, JWdiss mass-related inelastic deformation work, J/kgWm mass-related breakage energy, J/kgz defect parameter, dimensionless

Greek letters

�1 Poisson ratio of the granulate, dimensionless�2 Poisson ratio of the punch, dimensionless� normal stress, MPa� shear stress, MPa

Subscripts

el elasticel–pl elastic–plasticF yield point, flowN normalpl plastict tensile stress1,2,3 number of the principal stresses

Acknowledgements

The authors would like to thank the German Re-search Foundation for financial support through GK 828“Micro-macro-interactions in structured media and particlesystems”.

We would also like to acknowledge Dr. rer. nat. H. Heyseand Mr. M. Reppin from Inst. of Material Technology andTesting, Otto-von-Guericke-University of Magdeburg, fortheir help in using Scanning Electron Microscope.

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