3768
The Influence of Ball Size Distribution and Mill Diameter on
Collision Environment
S. Nkwanyana
Mineral Processing &Characterisation, Mintek, Randburg, Gauteng, South Africa
I. Govender
Mineral Processing &Characterisation, Mintek, Randburg, Gauteng, South Africa
Particle Technology &Mineral Processing, Discipline of Chemical Engineering,
School of Engineering, University of KwaZulu-Natal, Durban, South Africa
Centre for Minerals Research, Department of Chemical Engineering, University of Cape Town, South Africa
ABSTRACT: For a chosen mill size, speed and filling level, the flow of a granular ensemble varies with ball
size distribution. For a chosen speed, filling level and ball size distribution, the flow behaviour of a granular
ensemble in a tumbling mill varies with mill diameter. The flow behaviour of the granular ensemble creates the
collision environment. The effects of mill diameter and ball top-up size on the mill collision environment were
investigated using Discrete Element Method (DEM) modelling and the results are discussed on this paper.
Keywords: DEM, power dissipation, energy spectra, spatial distribution maps
INTRODUCTION
Mill performance and ball size in ball mills are intimately
related. Austin et al (1984) showed that the breakage rate
function of ore particles varies with the ball size used.
Carvalho (2013) Katubilwa and Moys (2009) Teke et
al (2002) Herbst and Lo (1989) have reported similar
observations. The authors have shown that for a given feed
size, the breakage rate function increases as the ball size is
reduced. The phenomenon has been attributed to the fact
that breakage occurs at contact sites, and the number of
contact sites for a given ball load is proportional to the
number of balls, which increases as the ball size is reduced.
The aforementioned authors have shown that the particle
size at which the breakage rate function becomes maximum
increases as the ball size becomes larger. The phenomenon
has been attributed to the increase in the magnitude of the
impact energy from larger balls.
Ball size reduction in ball mills is driven by two preva-
lent mechanisms, viz. abrasion and impact. In mills oper-
ated at regimes that promote cascading charge motion,
abrasion driven wear is considered to be the dominant,
whereas in mills operated at regimes that promote cata-
racting charge motion, impact driven wear is considered
to be the dominant (Aldrich, 2013). Eq. (1) is the com-
mon model proposed for describing ball size reduction in
grinding mills. Different constants have been proposed for
the model, depending on the relative importance of surface
dependent abrasion or volume dependent impact forces
(Austin and Klimpel, 1985 Vermeulen and Howat, 1986).
dt
dm kD n =(1)

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Extracted Text (may have errors)

3768
The Influence of Ball Size Distribution and Mill Diameter on
Collision Environment
S. Nkwanyana
Mineral Processing &Characterisation, Mintek, Randburg, Gauteng, South Africa
I. Govender
Mineral Processing &Characterisation, Mintek, Randburg, Gauteng, South Africa
Particle Technology &Mineral Processing, Discipline of Chemical Engineering,
School of Engineering, University of KwaZulu-Natal, Durban, South Africa
Centre for Minerals Research, Department of Chemical Engineering, University of Cape Town, South Africa
ABSTRACT: For a chosen mill size, speed and filling level, the flow of a granular ensemble varies with ball
size distribution. For a chosen speed, filling level and ball size distribution, the flow behaviour of a granular
ensemble in a tumbling mill varies with mill diameter. The flow behaviour of the granular ensemble creates the
collision environment. The effects of mill diameter and ball top-up size on the mill collision environment were
investigated using Discrete Element Method (DEM) modelling and the results are discussed on this paper.
Keywords: DEM, power dissipation, energy spectra, spatial distribution maps
INTRODUCTION
Mill performance and ball size in ball mills are intimately
related. Austin et al (1984) showed that the breakage rate
function of ore particles varies with the ball size used.
Carvalho (2013) Katubilwa and Moys (2009) Teke et
al (2002) Herbst and Lo (1989) have reported similar
observations. The authors have shown that for a given feed
size, the breakage rate function increases as the ball size is
reduced. The phenomenon has been attributed to the fact
that breakage occurs at contact sites, and the number of
contact sites for a given ball load is proportional to the
number of balls, which increases as the ball size is reduced.
The aforementioned authors have shown that the particle
size at which the breakage rate function becomes maximum
increases as the ball size becomes larger. The phenomenon
has been attributed to the increase in the magnitude of the
impact energy from larger balls.
Ball size reduction in ball mills is driven by two preva-
lent mechanisms, viz. abrasion and impact. In mills oper-
ated at regimes that promote cascading charge motion,
abrasion driven wear is considered to be the dominant,
whereas in mills operated at regimes that promote cata-
racting charge motion, impact driven wear is considered
to be the dominant (Aldrich, 2013). Eq. (1) is the com-
mon model proposed for describing ball size reduction in
grinding mills. Different constants have been proposed for
the model, depending on the relative importance of surface
dependent abrasion or volume dependent impact forces
(Austin and Klimpel, 1985 Vermeulen and Howat, 1986).
dt
dm kD n =(1)

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XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3769
In Eq. (1), m is the mass of the ball, t is the time, D is the
diameter of the grinding ball, k is the proportionality con-
stant and n is an exponential constant.
The choice of ball size is known to be a trade-off
between wear (which affects the costs of replenishing) and
throughput for a specific product size (Cho et al., 2013
Concha et al. 1992). Figure 1 shows a trade-off between
ball media wear and throughput capacity for a ball mill-
ing circuit treating a copper ore to achieve a given grind
size over a range of make-up ball sizes. The plots suggest
that continuous improvements on the understanding of the
ball size-mill performance and ball size-ball wear relation-
ships are crucial in making mills operation efficient and
profitable.
The distribution of energy into shear and impact com-
ponents of collisions (which essentially defines the mill
collision environment) determines the relative strengths of
attrition breakage and impact breakage, and the mechanism
of media wear. In the published literature, the ball size-mill
performance relationship and ball size-ball wear relation-
ship are not explained in terms of changes in the collision
environment as the ball size is changed. This paper explores
the effect of ball size on the mill collision environment, and
its implications on the ball size-mill performance and ball
size-ball wear relationships.
Sizing and optimisation of grinding mills using the
population balance modelling approach (Herbst et al.,
1980) has thrived over the last three decades. The approach
relies on bench-scale testing using mills of diameters rang-
ing from 0.2 m to 0.6 m. The prediction of the impact
of oversized feed to a ball mill using this approach is not
straightforward given the difficulty associated to obtaining
reliable results from the use of large ball sizes in labora-
tory mills, where the largest ball size recommended is one
eighth of the mill diameter. Although the approach has
been largely successful, it does take into account the mill
collision environment. This paper explores the effect of mill
diameter on collision environment, and its implication on
grinding performance predictions.
Discrete Element Method (DEM) was used to inves-
tigate the effects of ball size and mill diameter on collision
environment.
DEM SIMULATIONS
Overview
For a mill of a given size, speed and charge filling, the rate
of size reduction of particles depends on (i) the collision
frequency of balls (ii) the probability of capture of particles
between the colliding balls and (iii) the probability of frac-
ture of the captured particles. The probability of fracture
is dependent on the amount of energy dissipated on a col-
lision, the amount of energy a particle absorbs during a
stressing event, and the fracture energy of a particle (which
Figure 1. Relationship between steel consumption and circuit throughput

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Extracted Text (may have errors)

XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3769
In Eq. (1), m is the mass of the ball, t is the time, D is the
diameter of the grinding ball, k is the proportionality con-
stant and n is an exponential constant.
The choice of ball size is known to be a trade-off
between wear (which affects the costs of replenishing) and
throughput for a specific product size (Cho et al., 2013
Concha et al. 1992). Figure 1 shows a trade-off between
ball media wear and throughput capacity for a ball mill-
ing circuit treating a copper ore to achieve a given grind
size over a range of make-up ball sizes. The plots suggest
that continuous improvements on the understanding of the
ball size-mill performance and ball size-ball wear relation-
ships are crucial in making mills operation efficient and
profitable.
The distribution of energy into shear and impact com-
ponents of collisions (which essentially defines the mill
collision environment) determines the relative strengths of
attrition breakage and impact breakage, and the mechanism
of media wear. In the published literature, the ball size-mill
performance relationship and ball size-ball wear relation-
ship are not explained in terms of changes in the collision
environment as the ball size is changed. This paper explores
the effect of ball size on the mill collision environment, and
its implications on the ball size-mill performance and ball
size-ball wear relationships.
Sizing and optimisation of grinding mills using the
population balance modelling approach (Herbst et al.,
1980) has thrived over the last three decades. The approach
relies on bench-scale testing using mills of diameters rang-
ing from 0.2 m to 0.6 m. The prediction of the impact
of oversized feed to a ball mill using this approach is not
straightforward given the difficulty associated to obtaining
reliable results from the use of large ball sizes in labora-
tory mills, where the largest ball size recommended is one
eighth of the mill diameter. Although the approach has
been largely successful, it does take into account the mill
collision environment. This paper explores the effect of mill
diameter on collision environment, and its implication on
grinding performance predictions.
Discrete Element Method (DEM) was used to inves-
tigate the effects of ball size and mill diameter on collision
environment.
DEM SIMULATIONS
Overview
For a mill of a given size, speed and charge filling, the rate
of size reduction of particles depends on (i) the collision
frequency of balls (ii) the probability of capture of particles
between the colliding balls and (iii) the probability of frac-
ture of the captured particles. The probability of fracture
is dependent on the amount of energy dissipated on a col-
lision, the amount of energy a particle absorbs during a
stressing event, and the fracture energy of a particle (which
Figure 1. Relationship between steel consumption and circuit throughput

XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3767
Lichter, J K, Suazo, M, Noriega, R and Murariu, V, 2011.
The application of Multiphysics models for the design
of mill discharge system, in Proceedings SAG 2011
(ed: K Major, B Flintoff, B Klein and K McLeod),
Vancouver, Canada.
Lynch, A J, Whiten, W J and Draper, 1967. Developing the
optimum performance of a multi-stage grinding cir-
cuit. Trans. Instn. Min. Metall. vol. 76. pp C169-C182.
Lynch, A J, 1977. Mineral Crushing and Grinding Circuits,
Elsevier Science LTD.
Mishra, B K and Rajamani, R J, 1992. The discrete ele-
ment method for the simulation of ball mills, Applied
Mathematical Modelling, 16: 598–604.
Mishra, B K and Rajamani, R K, 1994. Simulation of
charge motion in ball mills. Part 1: experimental veri-
fication, International Journal of Mineral Processing, 40:
171–186.
Monaghan, J J, 1992. Smooth Particle hydrodynam-
ics, Annual Review of Astronomy and Astrophysics, 30:
543–574.
Monaghan, J J, 1994. Simulating free surface flow with
SPH, Journal of Computational Physics, 110: 399–406.
Monaghan, J J, 2005. Smoothed particle hydrodynamics,
Reports on progress in physics, 68: 1703–1759.
Morrell, S. and Stephenson, I., 1996. Slurry discharge
capacity of autogenous and semi-autogenous mills and
the effect of grate design, Int. J. Min. Processing, 46:
53–72.
Morrell, S, 2004. A new autogenous and semi-autogenous
mill model for scale-up design and optimization,
Minerals Engineering, 17: 437–445.
Morrison, R D and Cleary, P W, 2008. Towards a virtual
comminution machine, Minerals Engineering, 21:
770–781.
Rajamani, R K and Mishra, B K, 1996. Dynamics of ball
and rock charge in SAG mills in Proceedings SAG 1996,
(University of British Columbia).

Previous Page Next Page

Extracted Text (may have errors)

XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 3767
Lichter, J K, Suazo, M, Noriega, R and Murariu, V, 2011.
The application of Multiphysics models for the design
of mill discharge system, in Proceedings SAG 2011
(ed: K Major, B Flintoff, B Klein and K McLeod),
Vancouver, Canada.
Lynch, A J, Whiten, W J and Draper, 1967. Developing the
optimum performance of a multi-stage grinding cir-
cuit. Trans. Instn. Min. Metall. vol. 76. pp C169-C182.
Lynch, A J, 1977. Mineral Crushing and Grinding Circuits,
Elsevier Science LTD.
Mishra, B K and Rajamani, R J, 1992. The discrete ele-
ment method for the simulation of ball mills, Applied
Mathematical Modelling, 16: 598–604.
Mishra, B K and Rajamani, R K, 1994. Simulation of
charge motion in ball mills. Part 1: experimental veri-
fication, International Journal of Mineral Processing, 40:
171–186.
Monaghan, J J, 1992. Smooth Particle hydrodynam-
ics, Annual Review of Astronomy and Astrophysics, 30:
543–574.
Monaghan, J J, 1994. Simulating free surface flow with
SPH, Journal of Computational Physics, 110: 399–406.
Monaghan, J J, 2005. Smoothed particle hydrodynamics,
Reports on progress in physics, 68: 1703–1759.
Morrell, S. and Stephenson, I., 1996. Slurry discharge
capacity of autogenous and semi-autogenous mills and
the effect of grate design, Int. J. Min. Processing, 46:
53–72.
Morrell, S, 2004. A new autogenous and semi-autogenous
mill model for scale-up design and optimization,
Minerals Engineering, 17: 437–445.
Morrison, R D and Cleary, P W, 2008. Towards a virtual
comminution machine, Minerals Engineering, 21:
770–781.
Rajamani, R K and Mishra, B K, 1996. Dynamics of ball
and rock charge in SAG mills in Proceedings SAG 1996,
(University of British Columbia).