7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Modeling Discrete, Incremental, Repetitive and/or Simultaneous Particle Breakage
J.Bruchmhiller B.G.M. van Wachemt S. Gu K.H. Luo *
Energy Technology Research Groop, School of Engineering Sciences, University of Southampton, SO17 1BJ, UK
t Department of Mechanical Engineering, Imperial College London, SW7 2AZ, UK
jblr07@soton.ac.uk and b.vanwachem@imperial.ac.uk
Keywords: Fragmentation, particle breakage, brittle material, DEM
Abstract
The breakage of large brittle particles by applying different material parameters has been modeled with the discrete
element method (DEM). The model accounts for incremental breakage where particles can get damaged without being
fragmented. Particles might fragment depending on their damage history, size, material strength and impact energy
involved. Repetitive breakup of progeny particles and simultaneous breakage of many particles at the same moment
is taken into account. The desire and requirement for modeling a changing particle population in multiphase flows is
very high, as many more observed phenomena in practice can be studied similarly. The size distribution of the progeny
is based on one single breakage index parameter ti0, able to allow breakage into very few similarsized fragments
towards an attritionlike progeny where one big fragment is produced next to several smaller ones. The velocity of
the fragments is derived from the momentum and energy conservation. Three examples have been chosen for further
discussion by highlighting the model's strength and limitations. Grinding of particles inside a semiautogenous mill
and missile fragmentation are used to analyze the time and space evolution of the main variables and for comparisons
with other models. Breakage of char particles inside a fluidized bed is discussed as a typical example of a multiphase
flow application.
Nomenclature
Abbreviations
CF Collision factor
DEM Discrete element method
PSD Particle size distribution
Roman symbols
d Particle diameter (m)
E Energy (J)
S Coefficient of restitution ()
F Force (N)
fMat Material parameter (kg/Jm)
k Spring constant (Nm 1)
M Parent particle mass (kg)
M Maximum achievable tl i' )
m Child particle mass (kg)
P Breakage probability ()
t Time (s)
tio Breakage index I'.
At Particle time step (s)
V Parent particle velocity (m/s)
v Child particle velocity (m/s)
x Particle diameter (m)
Greek symbols
6 Overlap distance (m)
p Friction coefficient ()
a Poisson ratio ()
Subscripts
ai After impact
AR After replacement
bi Before impact
col Collision
CP Child particle
el Elastic
i During impact
j Number of impacts with damage
kin Kinetic
n Normal
MF Momentum factor
p Number of child particles
PP Parent particle
rem Remember
tot Total
Introduction
Irreversible size reduction of solid material is often de
sired or undesired in a large variety of applications.
Comminution and grinding should lead to a much
smaller particle population with a minimum energy sup
plied to the system. Viceversa in combustion fine char
particles are formed during fragmentation following the
flue gas before burout leading to a reduction in com
bustion efficiency. Hence, a good understanding and de
tailed study of fragmentation in general is important to
control these processes.
The majority of fragmentation models can be di
vided into two groups. The first group uses empiri
cal correlations where the fragmentation event is non
discrete, which predict fragmentation in average. Most
models are very simple in nature and might deliver
quick trends in similar applications under similar cir
cumstances. Therefore, these models may fall behind in
their reliability as fitted parameters may not cope with
e.g. nonhomogeneous properties (materials, applied
forces, different flow pattern, etc). Inadequate informa
tion about the fragment number and velocity, their con
sequence and behavior of the remaining particle phase,
information about the particle history (e.g. how much it
got damaged before) and much more remain in the dark,
even for idealized properties.
The second group of fragmentation models consider
one single or very few discrete particles which might be
involved in a breakage event. DEM models are capa
ble of modeling particle agglomerates representing one
global particle by many smaller ones. When it breaks, it
falls into fragment sizes depending on previously spec
ified subparticles and if not their discretization remains
in vain. For these models the computational cost would
roughly increase by a factor of 103104 (Cleary 2001)
compared to models without agglomerate subparticles.
The finite emelent method (or combined with the DEM
agglomerate approach) might deliver accurate predic
tions of crack propagation and disruption inside parti
cles. However, they are not tailored for collisional frag
mentation where many particles might be involved.
Cleary (2001) was the first who developed a frag
mentation model in DEM where numerous discrete sin
gle particles can be fragmented and replaced by their
progeny. This method might be addressed to a third
group of fragmentation models which combine accu
racy and efficiency which might be used for applica
tions involving numerous particles as it can be found
in mills, crushers, fluidized beds etc. Cleary (2001)
stated, that the actual rules used in his code are still crude
and progress beyond fragmentation involving high speed
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
balls in cataracting streams is desired. No further de
tailed description of his model has been published lately.
This paper presents a detailed description of a dis
crete fragmentation model which might be categorized
into the same third group. The onset of fragmentation
is modeled by using a breakage probability which con
siders incremental impact breakage by summation of ac
cumulated damage. In principle, this model can be pro
vided with any particle size distribution (PSD) how
ever, a breakage index tl0 a single value to determine
the entire PSD has been proposed. This approach offers
several advantages over others as it depends on material
parameters only, it is approved to be valid for multiple
impact breakage and has been found valid for many brit
tle materials. Discrete fragments are created depending
on the given PSD and packed randomly into their parent
particle volume with a minimum particle overlap. Every
fragment is assigned with a kinetic (velocity) component
derived from the momentum conservation and an elastic
(spring force) component derived from the energy equa
tion. The model outcome has been compared to other
model outcomes with little deviation and further setting
parameters have been tested on a semiautogenous mill
model according to our expectations. This fragmenta
tion model has been developed in the CFD code called
MultiFlow coupled inside its softsphere DEM module
based on nonlinear collision forces.
Fragmentation model
Most brittle materials can get damaged without being
fragmented. In this work, this phenomenon is referred
as incremental breakage (damage) initiated when the im
pact energy exceeds a threshold energy Eo required for
extending flaws inside the material. The particle damage
is set equivalent to the probability of breakage which in
turn is mathematically expressed by a probability func
tion introduced by Vogel and Peukert (2004) and modi
fied for DEM applications by Morrison et al. (2007) ac
cording to:
P = 1 exp fMat x (E Eo) ,
where Eo is the mass specific threshold energy which
a particle can absorb without fracture, E, is the mass
specific impact energy during the i'th impact, fMat is a
material parameter characterizing the resistance of par
ticulate matter against fracture in impact comminution
and x is the particle size. Whenever the destroyed frac
tion of the particle or the probability of breakage P is
high enough, body breakage (cleavage or shattering de
pending on the impact energy) will occur (Tavares and
de Carvalho 2009).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
_1. ,. <
Figure 1: (a) A brittle particle moves towards a wall, (b) when the maximum elastic energy is reached (see overlap
with the wall) and it comes to fragmentation, the parent particle is replaced by child particles which do not touch
the wall (c) the arrow indicates the direction and particles
pictures the progeny after fragmentation
Shi and Kojovic (2007) modified Vogel and Peukert's
breakage probability model to compute the breakage in
dex ti0 as shown in eq. (2) depending on material prop
erties, particle size and cumulative impact energy for
comminution processes.
exp[ fMa, X (Ei
Eo)] (2)
The breakage index t10 indicates the cumulative mass
passing 1/10 of the parent particle diameter and is re
lated to other t, values for many materials (Narayanan
and Whiten 1988). M is the maximum achievable tio
in a single breakage event and is related to other mea
sures of rock strength (Bearman et al. 1997). Material
values for fMat and E0 can be found for rock in Bear
man et al. (1997) or polymers, glass and limestone in
Vogel L. and Peukert W. (2005). For unknown mate
rials, NapierMunn et al. (1996, Chap.5) provide a de
tailed description of how these data are measured. The
material parameter fMat is inversely dependent on the
well known fracture toughness.
This approach uses a oneparameter family of curves
to receive the progeny size distribution (Shi and Ko
jovic 2007) and has been found valid for many brit
tle materials under considering multiple impact break
age. However, the progeny PSD can be replaced by any
other PSD function available (RosinRammler, Gaudin
Schumann,...) in case the tio approach does not cover
a certain material but it has the advantage of being in
dependent of fitted data and machine specific testing
methods. Once the PSD is known, discrete fragments
of different diameters are created based on the fragment
size and remaining cumulative mass percent given by the
PSD curve. A minimum diameter di, need to be speci
fied as a model setting parameter to avoid numerous tiny
fragments. Therefore, the overall number of fragments
are colored by their velocity (black=slow, white=fast) (d)
depend mainly on dm and M but also on the breakage
probability P obtained beforehand. Due to its complex
ity, fragments are randomly placed inside the parent par
ticle volume where fragments are inserted without other
parent particle or wall overlaps as can be seen in Fig. lb.
Their distance to each other is maximized to reduce the
overlapping childchild particle volume and to achieve
maximum code stability.
During fragmentation, kinetic energy of the parent
particle is transformed as fragments are halted by con
tact with the surroundings, destroying fragment mo
mentum and generating forces on the surroundings (Chi
rone et al. 1982). This phenomenon is dependent on the
material and is in the present model considered by a mo
mentum factor eMF. This factor can be interpreted as a
coefficient of restitution e for fragmentation which con
siders that momentum goes into the formation of crack
extensions. In contrast, the total energy required for
fragmentation is often more than 100 times larger than
the energy required to produce new surface and which fi
nally might get dissipated. Stretching and disruption of
intermolecular force fields require work be done where
almost all of this is recovered as kinetic energy when
the force fields separate and return to their unstressed
states (Bergstrom 1963). For that reason, the present
model does not consider a dissipative term in the en
ergy equation but can be considered for future studies
as long as a reliable theory is provided. In this model,
the fragment velocity eq. (5) is computed from a general
definition of etot eq. (3) and the momentum equation (4)
by considering the momentum factor eMF as:
Vati(3)
etot = eMFe (3)
Vbi
Vbim + F dt M
tlo = M 1
m vai (4)
Vai etot [V+ z dtj, (5)
where Vb, is the velocity of the parent particle before
impact, m and M are the mass of a fragment and the
parent particle, respectively. Eqs. (35) are used at the
moment of breakage only, where the sign prior to the
velocity during impact V' is always the same as Va, be
cause the force to trigger fragmentation has passed its
culmination. Furthermore, these equations are valid in
the direction of the impact normal only. This integral
over the remaining time of the collision is approximated
by linearization and obeys eqs. (68).
F dt= FLt + 2t +... (6)
F2 = kn 0/2 i (7)
"F2
M
'2 =i AS 1i [ F2 At2 +vAt (8)
Here, 6 is the particle overlap distance, At is the par
ticle time step used, Fi is the current force associated
with the overlap during the fragmentation event, k, is a
spring constant in the normal impact direction, 4 is the
normal vector and v is the velocity of the colliding par
ticle starting from zero. The integral is solved, when the
remaining 6, from eq. (8) becomes zero. It should be
pointed out, that each fragments momentum is a mass
weighted portion of the overall momentum leading to a
minimum kinetic energy of the fragments. At this stage,
the velocity of each fragment is the same but will change
when elastic forces on each fragment are considered (ar
tifical overlap due to fragmentation).
Collisions between child particles are considered as
internal forces which do not change the total linear mo
mentum of a system. Child particle collisions with the
wall or other parent particles (external forces) do not
exist (Fig. lb) at the moment of fragmentation as all
fragments are inserted without external overlap. Con
servation of angular momentum is not considered in the
present model, as fragments might experience high shear
forces between them.
Artificial overlaps between child particles need to be
corrected in terms of their associated elastic energy. The
artificial overlap between child particles at the moment of
fragmentation is remembered in 6,em and a dimension
less collision factor CF is applied to correct the associ
ated artificial elastic energy. At each particle time step,
r6em is updated according to ,em = MIN(rem, 6)
as long as the collision is found in the collision list (the
collision exists). Eq. (11) is introduced to obtain CF by
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
taking the energy balance eq. (9) into account.
EPP,kin + EPel = ECP,kino,j + I Epl j
col i col
E'cPe = kn,(6AR))/2CF
5
3/2
6AR
Epp,kin + YECP,el 
col
ZESP'el
EECP,kin,j
3
The asterisk indicates geometrically correct but not ener
getically correct values, the dash indicates energetically
correct but geometrically incorrect values after replace
ment (AR) and j is a counter variable for the fragments.
For CF = 1, eq. (10) is the integration of eq. (7) and
corresponds to the elastic energy for the Hertzian contact
theory. The elastic energy stored by the particle until the
instance of fracture (second term in eq. (9)) is the well
known particle fracture energy (Baumgardt et al. 1975).
The CF value is the same for all fragment collisions cre
ated by the same broken parent particle and acts within
6 < Srem only.
Modeling breakage in a semiautogenous mill
i mTime: 270 sec
S*Darmae (%)
22
10
2
Figure 2: Damage of particles inside a SAGmill after
4.5 minutes operation
The present model relies on a few random numbers
for instance to convert the breakage probability into a
clear Dirac answer. Fragmentation is a very complex
phenomenon and experimental results are hardly repro
ducible when it comes to exact measures. Therefore, an
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
90
Time (s)
Figure 3: (a) Forces acting between a wall and a recoiling ball at normal incidence for different Young's moduli and
(b) the breakage behavior inside the SAGmill for the same particle stiffness
application is required which involves numerous parti
cles to judge about the average onset over all breaking
particles. In this study a semiautogenous mill has been
chosen to test different setting parameter and their effect
on the breakage frequency (onset).
Semiautogenous mills are loaded with large heavy
balls and small charge particles which suppose to be
crushed by the balls. The geometric mill data are taken
from Djordjevic et al. (2006), with an initial mill diam
eter of 1.19m and a length of 0.31m fitted with 14 metal
lifters each 40 mm in hight. The rotating speed is 70 %
of its critical speed (3.14 rad/s). For each simulation, 24
large balls (p C5001,,n,/lm3 and d 0.1m) and 714
charge particles (p = 2'50,', r/n3 and d 0.05m) are
loaded and grinded in batch mode. Industrial to labscale
applications often produce millions of particles down to
sizes of microns. DEM models cannot solve such prob
lems within a reasonable computational time so that sim
plifications are required herein, only the large parti
cle fraction is considered. The discretefragmentation
model has been tested using selfspecified particle prop
erties to demonstrate their impact on the breakage fre
quency. Therefore, charge particles are grouped into dif
ferent bin sizes named as M1, M2,... which have been
kept within the /2 sequence. In this study, particularly
the mass reduction of particles in the top size (original
size) of d 0.05m (M1) is of interest as other size
classes depend simultaneously on a created and reduced
fraction.
Softsphere DEM models in general and their results
rely on the particle stiffness (Youngs modulus and Pois
son ratio) and so does the present discrete fragmenta
tion model. Fig. 3a shows the force acting between a
Table 1: Particle property settings
Variable
fMat,ref
(:X Eo)ref
Mref
d,.i.
eMF
Ewall,ref
Ep,ref
dball
dcharge,int
Value Units
0.9
0.15
10
0.0125
0.5
0.97
0.1
10+8
10+7
0.25
0.1
0.05
kg/Jm
Jm/kg
m
Pa
Pa
m
m
rebouncing ball hitting a wall at normal incidence in
dicating that for a softer material the contact will last
longer with a smaller force magnitude. The same stiff
ness values have been used for the particles grinded in
side a semiautogenous mill as depicted in Fig. 3b show
ing that the softer the material the higher the breakage
frequency (for a low E0 value). This is because the time
scale to allow fragmentation is much longer for smaller
(x Eo) values. All settings for the fragmentation model
which have not been modified are summarized in Table
1 except otherwise stated.
Next to the particle stiffness, the material parameter
fMat, the mass specific threshold energy E0 and the
particle size x do influence the onset of fragmentation
according to eq. (1). Vogel and Peukert (2004) indi
cated that the product :x Eo is constant for all particle
04
02
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
M  M ft =09 kg/Jm
..... M fa = O 5 kg/Jm
M1.... M fmt = 0 2 kg/Jm
M2, fMt = 0 2 kg/Jm
"4, fMt = 02 kg/Jm
Sf M4,fM t=02kg/Jm

50 100 150 200 250
Time (s)
Figure 4: (a) Grinding times for different values of x Eo (threshold energy to achieve damage) and (b) different fmat
values and their influence on the breakage frequency / grinding time and cumulative mass fractions of charge particles
in other bins
sizes, also used in the present model as a single setting
parameter so that its influence as a whole is tested and
shown in Fig. 4a. As a rough estimate, its dependency
can be assumed to be linear with the required grinding
time. The higher the threshold for damaging particles
the lower the breakage frequency. The influence of the
material parameters fMat is depicted in Fig. 4b. As in
dicated earlier, fMat is inversely dependent on the well
known fracture toughness Kic, so that smaller values
for fMat causing longer grinding times.
Fig. 2 shows the particles inside the semiautogenous
mill after 4.5 minutes of operation. The big blue balls
are not damaged at all while smaller particles from the
charge are colored according to their damage history.
Particles from the smallest bin size M5 are not allowed
to break further as they are restricted by d,,,, so that
their damage might reach 100%. This leads to misin
terpretation and need to be changed in future studies.
The advantage of modeling the comminution in mills is
a systematic analysis of most fragmentation setting pa
rameters available and to judge about their influence on
breakage in general. However, DEM limitations restrict
fragmentation modeling in terms of the applied particle
number, size and stiffness, so that particular industrial
applications have to be simplified when modeled.
Missile fragmentation
Fragment velocities indicate how the impact energy is
partitioned and how the fragment cloud expands into the
local surrounding. The present model cannot be applied
for crater formation studies and cannot handle melting
metallic projectiles during hypervelocity impact. Its
strength lies in the multiphase coupling often required
for fragmentation studies and that the partitioning of the
impact energy is material specific, although appropriate
parameters are rather difficult to obtain. The momentum
factor CMF is responsible for the fragment cloud forma
tion in the present model. The more initial momentum
is required for crack extentions, the bigger is the frag
ment cloud (the distance between fragments long after
impact), which corresponds to the white particles (Fig
ure 5a), a low CMF and vice versa.
The present discretefragmentation model considers
particle kinetic and elastic energy separately like all soft
sphere discrete element models. At the moment of frag
mentation, the model solves the momentum equation to
obtain the kinetic energy for each fragment (1.veloc
ity component) and solves the energy equation to ap
ply the remaining impact energy in form of elastic en
ergy between the fragments. The elastic energy is con
verted into kinetic energy, when all fragments have lost
their mutual contact (2.velocity component). As this ap
proach is novel for fragmentation models, it has been
compared to the hypervelocityfragmentcloud model
developed by Schafer (2006). Both models have been
simplified using fragments of only one single size and
a total number of 428 fragments. Schafer's 2D frag
mentation model spatially locates all projectile frag
ments on the circumference of a circle (broken circle in
Fig. 5b), where fragments are assumed to be uniformly
distributed on that circle. In both models, wall frag
ments have been considered in the momentum and en
ergy equation to account for the twocomponent veloc
150
Time (s)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
.a
a
a
o;t I
* M
t >
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Horizontal distance from the impact wall in m
Figure 5: (a) Effect of the momentum factor on the generated fragment cloud after normal incidence at the wall; white
fragments for eMF = 0.05 and black fragments for eMF = 0.95 and (b) comparison of the resulting fragment cloud
between the present 3D and Schafer's 2D fragmentation model after 27ps
ity approach. Model parameters used are summarized in
Table 2.
Table 2: Parameters used for a comparison of both mod
els (present 3D and Schafers 2D model)
Variable Value
Fragments diameter (monosize) 0.658 mm
Number of projectile fragments 428
Momentum of all wall fragments 0.2769 kgm/s
Energy of all wall fragments 649.95 J
Fraction of dissipated energy f' 0.23
Initial projectile velocity 6700 rn/s
All points in Fig. 5b belong to the discrete
fragmentation model and their average distance from the
cloud center is plotted as the solid circle. It has been
found that the first velocity component, given by the ini
tial velocity derived from the momentum equation, dif
fers by 2%. This value has been determined as the ratio
of each model's predicted distance between cloud cen
ter and impact point. The second velocity component,
differs by 5%, determined as the ratio of the averaged
cloud (circle) diameter of both models. The compari
sion of these two models is limited to the momentum
and energy equation and their direct impact on the frag
ment cloud only as their field of application differs sig
nificantly. However, it can be concluded that the imple
mented energy and momentum equation give fairly ac
curate results for missile velocities of up to 6700m/s to
predict reasonable fragment velocities.
Char Fragmentation in Hot Fluidized Beds
There are three types of fragmentation, namely attrition
(flaking), body breakage and percolation (Syred et al.
2007). Inside hot fluidized beds, char particles undergo
mainly attrition as particles degradate progressively to
wards the particle center but ash and lose char material
at the surface tear off from the main body. For the sake
of simplicity, most existing models ignore fragmentation
or treat fragmentation as a shrinking process, the most
state of the art model is presented by Di Blasi (1995).
The present discretefragmentation model captures body
breakage into very few similar sized fragments towards
an attritionlike fragment size distribution as can be seen
in Fig. 6 by using the proposed tio approach. As frag
ments are created with a lower density and diameter,
Stokes numbers become small and tiny fragments follow
closely the flue gas towards filters, pipe walls and cause
undesired deposit and might block the system. Further
more, the carbon loss through the system by char frag
ments is of particular interest, as it is desired to reuse the
char (information about the char quality/quantity) and
results might contribute to a more complete species bal
ance of the whole system.
Fig. 7 illustrates the freeboard and splash zone of a
fluidized bed the area where most fragmenation events
are expected. The fragmentation model is based on
the toughness of different materials (e.g. changing lin
ear with the particle density) but does not yet depend
on thermal stresses and pressure inside particles due to
volatile release which will be the focus for future work.
V.
S 
^yu
Ww
0.06
0.04
0.02
0.02
0.04
o
o o oo o
o o
0 0 00 t W ..0s j0:
o o 0 0 o
'o o" o
0 0 0
0 o
l o
,~ I I I ,
U.U
0
i l u i a i i l
Figure 6: Fragments created with the breakage index
t10, bodybreakage into very few fragments (here two),
or attritionlike breakage into many fragments (here
656)
*
'.
S ,* .' *.
Figure 7: Splash zone of a fluidized bed with initially
monosized particles and fragments generated by break
age
Conclusions
A new model has been developed and tested to account
for discrete, incremental, repetitive and simultaneous
fragmenation events, particularly suitable for multiphase
flow applications. Three different examples have been
selected to demonstrate the strength and limitations of
the present model. Comparison with other models show
excellent agreement and parametric studies have demon
stated model prediction to our expectation. The model
is able to fragment particles into an infinite number of
progeny particles as far as DEMlimitations concern.
The code delivers much information about the fragmen
tation event, for instance the fragment velocity and tra
jectory from the moment of breakage, the degree of par
ticle damage accumulated in the past, or PSD's for gen
erating breakage rate curves to judge the performance
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
of different applications. Information provided by the
model can support engineers in designing and optimiz
ing all kind of applications where fragmentation is in
volved without minimal use of empirical parameters.
References
S. Baumgardt, B. Buss, P. May, and H. Schubert. On
the comparison of results in single grain crushing under
different kinds of load. Proc. llth Int. Miner Process.
Congr., Calgiari, pages 332, 1975.
R.A. Bearman, C.A. Briggs, and T. Kojovic. The ap
plication of rock mechanics parameters of the prediction
of comminution behaviour. Mineral Engineering, 10(3):
225264, 1997.
C.H. Bergstrom. Energy and size distribution aspects of
single particle crushing In: Fairhurst C., Rock Mechan
ics, Proceedings of the 5th Symposium on Rock Me
chanics, Pergamon Press, New York, pp.155172, May
1962. 1963.
R. Chirone, A. Cammarota, M. D'Amore, and L. Mas
similla. Elutrition of attried carbon fines in a fluidized
combustion of coal. Proceedings of the Nineteenth Inter
national Symposium of Combustion, page 1213, 1982.
P.W. Cleary. Recent advances in DEM modelling of
tumbling mills. Mineral Engineering, 14(10):1295
1319, 2001.
C. Di Blasi. Heat, momentum and mass transport
through a shrinking biomass particle exposed to thermal
radiation. Chemical engineering sciences, 51(7):1121
1132, 1995.
N. Djordjevic, R. Morrison, B. Loveday, and P. Cleary.
Modelling comminution patterns within a pilot scale
AG/SAG mill. Minerals Engineering, 19:15051516,
2006.
R.D. Morrison, F. Shi, and R. Whyte. Modelling of
incremental rock breakage by impact; for use in DEM
models. Mineral Engineering, 20:303309, 2007.
T.J. NapierMunn, S. Morrel, R.D. Morrison, and T. Ko
jovic. Mineral Comminution Circuits Their Operation
and Optimisation. JKMRC, Brisbane, Australia, 1996.
S.S. Narayanan and W.J. Whiten. Determination of com
minution characteristics from singleparticle breakage
tests and its application to ballmill scaleup. Trans. Inst.
Min. Metall, 97:C115C124, 1988.
F.K. Schafer. An engineering fragmentation model
for the impact of spherical projectiles on thin metallic
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
plates. International Journal of Impact Engineering, 33:
745762, 2006.
F. Shi and T. Kojovic. Validation of a model for impact
breakage incorporating particle size effect. International
Journal of Mineral Processing, 82:156163, 2007.
N. Syred, K. Kurniawan, T. Griffiths, T. Gralton, and
R. Ray. Development of fragmentation models for solid
fuel combustion and gasification as subroutines for in
clusion in CFD codes. Fuel, 86:22212231, 2007.
L.M. Tavares and R.M. de Carvalho. Modelling break
age rates of coarse particles in ball mills. Mineral Engi
neering, 22:650659, 2009.
L. Vogel and W. Peukert. Determination of material
properties relevant to grinding by practicable labscale
milling tests. Int. J. Miner Process, 74 ", S2'S338,
2004.
Vogel L. and Peukert W. From single impact behaviour
to modelling of impact mills. Chemical Engineering Sci
ences, 60:51645176, 2005.
