Dynamic simulation of grinding circuits

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Minerals Engineering 17 (2004) 1189–1198
Dynamic simulation of grinding circuits
Yi Liu *, Steven Spencer
a CSIRO Minerals, Private Mail Bag 5, Menai, NSW 2234, Australia
Received 1 April 2004; accepted 12 May 2004
Abstract
A flexible and powerful dynamic simulation approach to grinding circuit simulation has recently been developed in CSIRO Min-
erals. The MATLAB/SIMULINK graphical programming environment has been used to construct a library of dynamic mathemat-
ical models of a number of key grinding and separation devices and to link them into various complex dynamic grinding circuits.
True real-time dynamic simulation and visualisation of interlinked unit process operations in grinding circuits of arbitrary complex-
ity can readily be achieved.
The application of the dynamic simulation approach can help greatly in understanding the sometimes complex, nonlinear behav-
iour and dynamic interactions in various grinding circuits. Dynamic simulation can be used to test ‘‘what-ifs’’ in grinding process
operations such as circuit response to variations in feed and unit operation characteristics. It is a cheap and effective means of inves-
tigating circuit optimisation without the risk of possible damage to operating units or production of a large amount of unwanted
product during a physical optimisation process. Dynamic simulation is also extremely useful in developing and testing new ideas for
process soft-sensors and control. The experience and knowledge gained in dynamic simulation of grinding circuits is directly appli-
cable to other dynamic flowsheet modelling and optimisation problems in the minerals and process engineering industries. The
advantages of building flowsheet models within the MATLAB/SIMULINK programming environment include the ability to readily
develop and modify continuous, discrete and/or hybrid models of individual unit operations, with solution of the flowsheet system
by a powerful in-built suite of equation solvers and analysis of results utilising extensive existing graphical capabilities. Flowsheet
models of arbitrary complexity can easily be graphically developed, while individual unit models can be developed in terms of graph-
ical block diagrams and/or customised block models written in computer code.
� 2004 Elsevier Ltd. All rights reserved.
Keywords: Comminution; SAG milling; Modelling; Simulation
1. Introduction
Real time dynamic computer simulation has been a
powerful tool not only in traditional high-tech aero-
space and military industries, but also in other areas
such as the automotive, steel making, and chemical
processing industries. However, until very recently, there
has been limited practical application of dynamic
0892-6875/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mineng.2004.05.018
* Corresponding author. Tel.: +61 2 9710 6731; fax: +61 2 9710
6789.
E-mail addresses: yi.liu@csiro.au (Y. Liu), steven.spencer@csiro.au
(S. Spencer).
simulation in most of the mineral processing industry,
instead relying on pilot plant studies and/or steady-state
flowsheet simulation for plant design, equipment dimen-
sioning and pre-control optimisation.
With recent progress in on-line measurement in min-
eral processing, there are an increasing number of min-
eral processing variables that can be measured on-line
in real time (Death et al., 2002). Soft sensor models are
also increasingly being developed for critical plant varia-
bles that have previously been unavailable (Gonzalez,
1999). This progress has greatly improved the opportuni-
ties for more advanced control techniques to be applied
to mineral processing. To do this, a full understanding
mailto:yi.liu@csiro.au 
mailto:steven.spencer@csiro.au 
1190 Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198
of the dynamic behaviour of a processing circuit and the
dynamic interactions between the external process varia-
bles open to manipulation and internal (and perform-
ance) variables of the circuit is crucial before any
advanced process control can be successfully imple-
mented. Real time dynamic simulation provides a power-
ful tool to gain such an understanding with minimum
associated cost.
Many simulation packages and techniques already
exist for flowsheet simulation in the mineral processing
industry. They have been widely and successfully used
for plant design, capacity planning (equipment sizing),
circuit optimisation, problem diagnosis and costing pur-
poses. However, most of these existing simulation pack-
ages are based on steady state analysis (for instance
METSIM, USIM PAC, Limn and JKSimMet) and
may utilise empirical (and in the worst cases �black-
box�) models of limited generality for individual unit
operations. Such packages cannot simulate the dynamic
behaviour and interactions of processing units within a
circuit during transitional periods between various stea-
dy states (including prediction of transition times), nor
can they capture the real-time dynamic interactions be-
tween external process variables (e.g. feed variations),
internal variables (e.g. grinding mill load), and perform-
ance variables (e.g. product size distribution and flow
rate) of a processing circuit. Such dynamic variations
and interactions can cause major problems for process
control and optimisation, most notably, in the case of
semi-autogenous grinding/autogenous grinding (SAG/
AG) mills in primary grinding circuits. Some dynamic
simulation packages do exist (for instance Aspen
Dynamics and SysCAD). In the case of the Aspen suite
of products, their use may be viewed as relatively high
cost and suitable largely as a �high end� solution to flow-
sheet modelling needs for most of the mineral processing
industry. In the case of SysCAD, the dynamic capability
is available but to the authors knowledge has so far
has largely been used as a means to obtain a steady
state configuration for analysis. It is also known that
SIMULINK has been used for simulation of the alu-
mina refinery process and testing of control strategies
at Nabalco–Alcan Gove Pty Ltd. However, the
approach has not to our knowledge been extended for
general use in the mineral processing industry.
It is our intention to explore the techniques of dy-
namic simulation used in other industries (under the
MATLAB/SIMULINK environment) for development
and application in mineral processing dynamic flowsheet
simulation. The main reasons to use SIMULINK are its
modular approach to model building, open model struc-
ture, ease of changing circuit configurations and links,
powerful real time graphic display functions for process
variables, and integrated advanced nonlinear dynamic
system solvers. A flexible and powerful dynamic simula-
tion flowsheet modelling approach has accordingly been
developed, with specific application in grinding circuit
dynamic simulation. The reasons for the choice of grind-
ing circuits as the initial area for model development is
the relative maturity of dynamic mathematical models
for some of the unit operations and the interest in dy-
namic control of problematic unit operations such as
SAG/AG mills. The approach exploits extensions of lit-
erature dynamical mathematical models of grinding mill
unit operation developed into a SIMULINK unit model
graphical library and the flexibility/capacity of SIMU-
LINK to link these individual units into complex dy-
namic flowsheets. In this manner validated individual
unit models can be linked in an arbitrary manner and
used to perform true real-time dynamic simulations.
The application of the dynamic simulation approach
can help greatly in understanding the sometimes com-
plex, nonlinear behaviour and dynamic interactions in
various grinding circuits. Dynamic simulation can be
used to test ‘‘what-ifs’’ in grinding process operations
such as circuit response to variations in feed and unitoperation characteristics. It is a cheap and effective
means of investigating circuit control and optimisation
without the risk of possible damage to operating units
or production of a large amount of unwanted product
during a physical plant studies. Dynamic simulation is
also extremely useful in developing and testing new ideas
for process soft-sensors and control.
The next section briefly describes the main mathemat-
ical models used in our dynamic simulations. Section 3
summarises the general features of the simulation ap-
proach and the specifics of the comminution model li-
brary constructed for dynamic simulation of grinding
circuits. Several dynamic simulation examples are given
in Section 4. Section 5 concludes the paper with some re-
marks on the flexibility of the approach, possible future
extensions and practical applications.
2. Dynamic models for grinding circuit unit operations
The key unit model in a grinding circuit is the grind-
ing device itself, in many flowsheets being a SAG/AG
and ball mills, respectively for primary and secondary
grinding. There is a well-known mathematical model
for ball mill operation based on the population balance
modelling approach, with the assumption that mill
dynamics can be modelled by a number of perfect mixers
in series (see, Whiten, 1974; Austin et al., 1984). Let
X(t) = [x1(t),x2(t), . . . ,xn(t)]
T be the vector representing
the mass of solids in discrete size fractions in a perfect
mixer, then single mixer ball mill breakage in a mill with
constant hold-up can be modelled as governed by the
following equation:
dX ðtÞ
dt
¼ ðBðtÞ � IÞSðtÞX ðtÞ: ð1Þ
Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198 1191
Here B(t) is the breakage distribution function (lower
triangular matrix), S(t) is the breakage rate (selection)
function (diagonal matrix), and I is the identity matrix.
It is usually assumed that the breakage distribution and
breakage rate functions are constant matrices, and can
be estimated from batch grinding tests (see, Austin
et al., 1984; Weller et al., 1997, 2000). To model a
SAG/AG mill, as well as to reflect possible ore hardness
changes in the feed, the following single mixer nonlinear
grinding phenomenological model was developed by us:
dX ðtÞ
dt
¼ ðBðtÞ � IÞ bðtÞðcðtÞSðtÞ þ aSaðtÞX ðtÞÞ½ �X ðtÞ ð2Þ
Here B(t) and S(t) are the same as in Eq. (1), and
Sa(t) is a breakage rate (lower triangular matrix) repre-
senting the effect of autonomous grinding. The constant
0 6 a 6 1 is a structure parameter. When a = 0, the
model simulates a ball mill, 0 < a < 1 simulates a SAG
mill, and when a = 1 and S(t) = 0, the model simulates
an AG mill. Function b(t) is used to simulate changes
in feed ore hardness. The model simulates ‘‘softer’’ ore
when b(t) < 1, and b(t) > 1 for ‘‘harder’’ ore, and
b(t) = 1 returns to ‘‘normal’’ ore hardness. Similarly,
function c(t) is used to simulate the effects of ball charge
in the mill. When c(t) > 1, extra balls are added, when
c(t) < 1 balls are consumed, and when c(t) = 1, we as-
sume no variations of ball charge in the mill.
Eq. (2) is restricted to modelling breakage in a single
perfect mixer. However, grinding mill operation gener-
ally can be more reasonably modelled in terms of several
perfect mixers connected in series. This provides a low
order model for the dynamics of mass transportation
through the mill. A critical parameter in this model is
the mean residence time of the solids in the mill. An-
other consideration is that at the discharge end of any
grinding mill, there is generally a size classification
effect, sometimes due to the presence of a grate or
screen. In these circumstances, the following model
better describes comminution in a perfect mixer:
dX ðtÞ
dt
¼ ðBðtÞ � IÞ bðtÞðcðtÞSðtÞ½
þaSaðtÞX ðtÞÞ�X ðtÞ þ
1
s
ðf ðtÞ � pðtÞÞ
pðtÞ ¼ CðtÞX ðtÞ
ð3Þ
Here f(t) = [f1(t), f2(t), . . . , fn(t)]
T is the mass of the sol-
ids feed, and mass of the mixer product is
p(t) = [p1(t),p2(t), . . . ,pn(t)]
T. Matrix C(t) contains classi-
fication coefficients for the mixer. It is usually a diagonal
constant identity matrix for all mixers of a grinding mill
model except the last mixer, which will also be a diago-
nal matrix but reflect the classification effects of the mill
at the discharge. Here s is the mean residence time for
solids in a mixer, which can be obtained by appropriate
analysis of pulse injection tracer tests (see, Weller et al.,
2000).
Eq. (3) is the generic building block of the grinding
mill models for this dynamic simulation approach. In
practice, we also need to include a water phase mass bal-
ance in the above mill model if a wet grinding circuit is
to be simulated. The water phase model will not be dis-
cussed in this paper.
Other key unit operations in grinding circuits are
mixing and separation devices. A dynamical mathemat-
ical model of a sump unit operation can be derived in
the similar fashion to Eq. (3), based on simple mixing
principles with an associated mean residence time. There
are many types of models for cyclone separators in the
literature, which will not be discussed in this paper.
The hydro-cyclone model we used in this study is based
on an empirical model (Austin et al., 1984). Neither of
these models will be described in any detail in this paper.
3. General features and the specifics of comminution
models
As can be seen from the last section, dynamical math-
ematical models of grinding mills can quickly grow into
some very complex, nonlinear and highly inter-con-
nected differential equations. The complexity of the
whole grinding circuit will dramatically increase once
we start to connect different unit models into a grinding
circuit and when wet grinding is considered. Any closed-
loop control (even with simple PID control) will com-
plicate the models further. It is clear that a powerful
nonlinear differential equation solver is a must for any
dynamic simulation of such complex models. In addi-
tion, a modular and subsystem approach is highly desir-
able to manage the complexity of the unit models and
also, the simulation tool has to be sufficiently flexible
to allow users to simulate a wide variety of types of
grinding circuits with different connectivity. In order
to fully understand the true dynamical behaviour of
individual unit models and the linked flowsheet, it is also
necessary to have real time graphical display capacity in
the simulation tool.
After a review of many commercially available dy-
namic simulation packages on the market, SIMULINK
(www.mathworks.com) was chosen for this work due to
its strong dynamical modelling capability and flexibility.
3.1. General features of SIMULINK
SIMULINK is a general purpose, very powerful and
flexible dynamic system simulation environment. It has
been applied to various time-domain dynamic system
simulations in a wide variety of industries, such as aero-
space (e.g. F14 flight control, missile flight control, lunar
module autopilot, and radar tracking), and automotive
(e.g. engine timing control, anti-lock brake system, auto-
matic transmission control, active suspension, power
http://www.mathworks.com
1192 Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198
window control). A good exposition of the capabilities
of SIMULINK can be found at the Mathworks SIMU-
LINK Technical Literature Web Page.
The key features of the SIMULINK can be summa-
rised as:
• Modular and subsystem approach to handle very
complex systems;
• Intuitive block-diagram (graphical) interfacing makes
it easy to construct and understand;
• Very rich commonly used block and subsystem
libraries;
• Extensive control system libraries can be readily used
for closed-loop simulations;
• Flexible structures and configurations and user defin-
able functions;
• Powerful simulation solvers to handle highly nonlin-
ear and stiff systems;
• Powerful graphics and visualisation tools;
• S-functions for addition of customblocks to SIMU-
LINK models, defined in terms of MATLAB, C/
C++, Fortran or Ada code.
3.2. Specifics of the comminution dynamic model library
The comminution dynamic model library (see Fig. 1)
so far contains several versions of perfect mixer dynamic
comminution models of varying degrees of complexity,
which are the building blocks of the grinding mill
models. A hydro-cyclone size separation model (Austin
et al., 1984) has also been developed in the library. There
are also several versions of the sump/pump dynamic
model and links to several demonstration dynamic simu-
Comminution Mo
 SAG
 Mill
SAG Mill with
hardness input 
holdup output
Mixer
Perfect Mixer with RT
and hardness input and 
solids holdup output
Mixer
SAG
Perfect Mixer with 
RT for SAG
Mixer
Perfect Mixer
with water and RT Hydro-Cyclone 
 Ball
 Mill
Ball Mill with
hardness input 
holdup output
 Ball
 Mill
Ball Mill with
3 mixers and a
classifier at end
Fig. 1. Dynamic model libr
lations of grinding circuits. A number of other mineral
processing unit operation library models are currently
under development.
There are several key features of this library:
• The general approach to model development is to try
to make a model as generic as possible to accommo-
date a variety of simulation situations.
• To develop a grinding circuit, it is a simple task of
drag and drop of appropriate blocks (grinding units)
from the library into a new workplace, and then the
blocks can be linked into a grinding circuit by click
and drag of the mouse. After linking with the feed
and appropriate display tools in the same way, the
system is ready to be simulated once the model
parameters are loaded.
• To change the structure and configuration of the
grinding circuit, it is a simple task of substituting
blocks or re-linking the blocks in different ways.
• The hydro-cyclone model is a user-defined function
(called S-function), which can be a very complex
dynamic function. It is easy to develop the S-function
by following a few general rules.
• The sump/pump model in the library is treated as a
perfect mixer with residence time but without break-
age functions.
• The SAG Mill model in Fig. 1 is in fact a general
dynamic model for tumbling mills (see Eq. (3)). It
has several very useful characteristics:
– One parameter (a in Eq. (3)) could change the
model from a ball-mill (a = 0) to SAG mill
(0 < a < 1) or to AG mill (a = 1 and S(t) = 0) simu-
lations. It can also be used to model stirred ball
mills (see, Weller et al., 2000).
del Library
Demo 3:
Feed Size
 Change
Demo 2:
Feed Rate
 Change
Demo 1:
RT Change
Demo 5: Size
and Hardness
Changes
Demo 4: 
Ore Hardness
Changes
SUMP
Sump model with 
pump rate input and
solids holdup
SUMP
Sump model 
with RT and
pump rate input
SUMP
Sump Model (sfun)
(sfun)
ary for comminution.
This SAG mill model allows any number of mixers to be connected.
Ore hardness be changed by Input 2, where input_2 < 1.0 means 
softer feed, input_2 > 1.0 for harder feed.
2
Solids & water
mass holdups
1
Slurry mass
discharge
Mixer
SAG
Perfect Mixer with 
RT for SAG 3
Mixer
SAG
Perfect Mixer with 
RT for SAG 2
Mixer
SAG
Perfect Mixer with 
RT for SAG 1
2
Solids hardness
change factor
1
Slurry mass
feed 
Fig. 2. A SAG mill model with three perfect mixers.
Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198 1193
– Mills can be modelled by any number of mixers to
match mean residence time distributions (s�s in Eq.
(3)) determined from tracer test experimental data
(see, Weller et al., 2000).
– Each mixer can have independent breakage func-
tion and rate, mean residence time, and discharge
classification coefficients.
– Ore-hardness changes in the feed can be simulated
by defining a time varying b(t) coefficient in the
appropriate grinding model. Ball charge changes
can also be simulated in a similar manner (time-
varying c(t)), though this feature has not yet been
implemented.
– All models can be used for both wet and dry grind-
ing simulation.
Fig. 2 shows how three single mixer SAG blocks are
connected to model an entire SAG mill. Again, the num-
ber of mixers used to model a mill can be increased or
decreased easily by the user to fit the real conditions.
Each Mixer SAG block in Fig. 2 is the SIMULINK
implementation of the mathematical model of Eq. (3).
Usually, such a detailed model of a single unit operation
would be masked under a single graphical interface icon
and hence not shown to users.
An important consideration in any simulation exer-
cise is the validation of models. It should be noted that
the ball mill model used in this library has been checked
against the corresponding model in DYNAMILL, a
Change Residence Time tau1, then
the change of the rising and decay c
of the ball mill with a constant vo
-C-
Water
Feed
-K-Volume
offset
K*u
Solids
Size Dist
Solids
Feed
yin
Mill_Feed
Mill Feed
 Ball
 Mill
Ball Mill with 3 mixer
and a classifier at e
Fig. 3. Example 1––Setup for simulation of a
command line based dynamic mill simulation package
originally developed by Raj Rajamani and John Herbst
at University of Utah. It was found that under the same
conditions, the models produced simulation results that
differed by less than 2% in product size distribution.
4. Grinding circuit simulation examples
Two examples are here used to demonstrate the use
of the Comminution Model library and the key features
of SIMULINK as mentioned above.
4.1. Example 1: Residence time effects in a ball mill
In this example, a simple ball mill model consisting of
three perfect mixers is simulated under open circuit con-
dition. The slurry volumetric feed rate to the mill is kept
constant with zero initial solids feed rate in order to
grind-out the initial contents of the mill. Then a step
change increase of the solids feed rate is then introduced
and later a step change return to zero solids feed is intro-
duced. Two runs of the simulation are performed with
different residence times set for the first mixer in the ball
mill model (the residence times for the second and third
mixer are not changed).
Fig. 3 depicts the simulation setup for the residence
time test of a ball mill. Again, the feed to the mill can
be easily changed to suit various simulation purposes
 re-run to see 
haracteristics 
lume feed
Mixer Discharge
yout
Mill_Discharge
Compare Two
Total Solids
Change RT
tau1=3
Plot Total
Solids & Water
Plot Size
Fractions
Load Data
n=16, tau1=1
s 
nd
ball mill with residence time changes.
1194 Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198
and the simulation progress can be monitored by the
scopes attached to feed stream and discharge stream.
There are 16 size fractions at
ffiffiffi
2
p
size intervals used in
the model. The simulation is for a wet mill, with an addi-
tional water phase.
In the first run of the simulation, the residence time
constants for three mixers are all set to the same value
(s1 = s2 = s3 = 1min). The simulation is repeated with
the residence time of the first mixer set to s1 = 3min
and the other residence times unchanged (s2 =
s3 = 1min).
Fig. 4 shows the dynamic responses of the ball mill
model with residence time of all mixers equal
(s1 = s2 = s3 = 1min). The total solids mass and water
of the feed are shown in the first plot of Fig. 4 (total slur-
ry volume is kept constant). The second plot in Fig. 4
shows the mass of solids in each particle size fractions
in the discharge of the mill. It is interesting to note that
the initial solids hold-up in the ball mill was ground out
quickly due to no solids feed in the first 50min (there is a
similarly rapid decline in solids mass at the second
grind-out). Solids hold-up quickly increases with the
step change addition of solid feed and stabilises at a con-
stant level. The third plot in Fig. 4 shows the total mass
of solids and water phases in the discharge of the mill.
Due to the nature of theconstant volumetric feed, the
discharge steam of the mill is also a constant in volume.
Hence we can clear see that as expected, when solids in-
creases in discharge, the water will decrease accordingly
to keep the volume a constant, and vice versa.
0 20 40 60 80 100
0
100
200
300
400
Solids Fractions in Dis
M
as
s 
Fr
ac
tio
ns
 (
kg
)
0 20 40 60 80 100
0
500
1000
1500
Mill Discharge - Total Solids & Wa
M
as
s 
(k
g)
Time (m
0 20 40 60 80 100
0
500
1000
Mill Feed - Total Solids,
M
as
s 
(k
g)
0 20 40 60 80 100
Fig. 4. Dynamic responses of a ball mill model
Fig. 5 is a comparison of discharge total solids and
water phase mass as a function of time for the simula-
tion described above and a second simulation with the
residence time of the first mixer changed as also noted
above. The second plot in Fig. 5 clearly shows that when
the residence time in the first mixer of the ball mill model
is changed from 1 to 3min, the response of the total sol-
ids in discharge stream is as expected, proportionally
slower, i.e. the total solids mass in the discharge stream
takes longer time to stabilise to the step changes of the
feed.
This example clearly shows that the mean residence
time input to a ball mill model has a major impact on
the responsiveness of the dynamic behaviour of the
model to feed rate changes. It is easy to see that by
adjusting the number of mixers and associated mean res-
idence times, one can match the residence time response
of a grinding mill model with tracer study data from a
real mill (see Weller et al., 2000).
4.2. Example 2: Responses of a SAG mill circuit to feed
size and hardness changes
In this example, we link SAG mill, hydrocyclone and
sump/pump models in a closed grinding circuit in feed-
forward configuration. A simulation is carried out with
a step up and down change of solids feed size at the fresh
feed stream to the grinding circuit. When feed solids size
increases, the mass of coarser fractions in feed increases
and the mass of the finer fractions decreases so as to
120 140 160 180 200
charge Stream
120 140 160 180 200
ter. Mixer 1 RT 1 = 1 (min)
in)
120 140 160 180 200
 Water & Volume
120 140 160 180 200
636
638
640
F
ee
d 
V
ol
um
e 
(li
tr
e)Solids
Water
Solids
Water
τ
demonstrating mean residence time effects.
0 20 40 60 80 100 120 140 160 180 200
0
200
400
600
800
1000
Mill Feed - Total Solids, Water & Volume
M
as
s 
(k
g)
636
636.5
637
637.5
638
638.5
F
ee
d 
V
ol
um
e 
(li
tr
e)
Solids
Water
0 20 40 60 80 100 120 140 160 180 200
0
500
1000
1500
Time (min)
To
ta
l S
ol
id
s 
M
as
s 
&
 W
at
er
 (
kg
)
Comparison of Residence Time Change of the 1st Mixer of Ball Mill (Water - dotted)
RT: 1=1, 2=1, 3=1
RT: 1=3, 2=1, 3=1
Water 
Solids 
τ
τ
τ
τ
τ
τ
Fig. 5. Comparison of changes in residence time of the ball mill model.
Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198 1195
keep the total mass of solids feed at a constant. A step
up and down change (10%) of feed ore hardness is sub-
sequently imposed on the mill model. Water feed is kept
at a constant rate during the simulation. Note that some
white noise is added to the feed rate in an attempt to
make it closer to the reality. There are again 16 size frac-
tions at
ffiffiffi
2
p
size intervals used in this example. Some dis-
charge classification effects are introduced at the last
mixer associated with the SAG mill model.
A SAG Mill Circuit Model with feed
 size and ore hardness changes
Double cl ick to
load the data
wff
Water Feed Solis & Wa
 Mass Hol
sff
Sol ids feed
Sol ids Feed
Disturbance Sol ids Feed
Size changes
 SAG
 M i l l
SAG Mill with
hardness input
holdup output
Ore hardness
disturbance
1
Normal hardness
ym Mi l l feed
Plot Size
Distribution
Plot Total
Sol ids & Water
Plot Other
Mil l Infos
Hardness
changes
yu
Cyclone return
Size
Switch
2 size system
R
Fig. 6. A closed-loop SAG mill grin
The SAG mill model consists of three perfect mixers
incorporating comminution effects (see Eq. (3)) with the
same constant breakage function B and breakage rate S,
but different residence times, s. The autonomous grind-
ing rate functions Sa are constant and the same for all
mixers. The AG structure parameter a = 0.2 is used in
the simulations.
Fig. 6 shows the closed-loop grinding circuit of the
SAG mill simulation. In this closed grinding circuit
Cyclone input
Product stream
Water in Product
wsf
Water
Addition
24
Sump vo lume
holdup setpoint
SUM P
Sump model
qsp
Sump discharge
volume rate offset
ys
Sump discharge
Sump Volume Holdup
ter
dup
Solids Product
Pump rate
PID
PID Controller
s
Hydro-Cyclone
m
Discharge
eturn stream
ding circuit simulation setup.
1196 Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198
simulation, a simple and crude PID control is imple-
mented to regulate the sump level to a constant (hence
to prevent the sump from overflowing) by adjusting
pump out rate of the sump alone (no changes to the
water addition to the sump). Such simple control strat-
egy is used for simulation purposes only and obviously
it may be inappropriate for real SAG grinding mill
operation.
The first plot in Fig. 7 shows the total solids mass, the
particle size (P50), and ore hardness in the fresh feed
stream of the grinding circuit. It clearly shows the con-
stant feed mass rate and the step up/down changes of
both feed size and feed ore hardness. The second plot
in the figure shows the solids and water hold-ups as well
as the solids concentration in the SAG mill. The third
plot in Fig. 7 shows the total solids mass and water in
the product of the grinding circuit as well as the product
size (P50).
It is interesting to note that although the feed rate
and production rate remain largely constant, the pro-
duction size changes significantly with the change of
feed size as well as the change of ore hardness. Also,
we can see that the total mill hold-up does not change
much during the simulation. This could simply be due
to the fact that in the mixer model used, a further
assumption on overflow in volume of the mixer is im-
posed. This means that when feed rate is increased, dis-
charge rate increases accordingly, such that volumetric
hold-up remains constant.
The first plot of Fig. 8 shows slurry volume of the
sump (i.e. indirectly the sump level). It is clear that a
0 10 20 30 40 50
0
5000
10000
Total Solids and Size
M
as
s 
(k
g)
 &
 S
iz
e 
(µ
m
)
0 10 20 30 40 50
1000
2000
3000
4000
5000
Solids and Water Holdup and Sol
M
as
s 
(k
g)
0 10 20 30 40 50
400
500
600
700
800
900
1000
Total Solids and Wa
Time (m
M
as
s 
(k
g)
Feed Size
Solids
Water
Soli
Water
Fig. 7. Process variables of
simple PID controller did a reasonably good job to keep
the actual level close to the set point (a constant in this
case). It is clear that when feed size increases, more
coarse solids will pass through the mill and then be
fed to the hydrocyclone. This in turn means that hydro-
cyclone will reject more materials to the return stream.
Hence the circulation load of the grinding circuit will in-
crease (as shown in the second plot of Fig. 8). When this
happens and if the sump pump out rate remains the
same, the sump level will have to increase. In order to
keep the sump level at a constant, the PID controller
has to increase the pump out rate (as shown in the third
plot of Fig. 8) to cope with the increased slurry feed to
the sump. When ore hardness increases, again, more
coarse solids will be discharged from the mill and be
fed to the hydrocyclone, the circulation load increases,
and the sump level will rise and again, the PID control
loop increases the pump out rate as a compensation
effect.
It should be recalled that in Fig. 7, solids mass feed
rate is constant, and hence so should be the product dis-
charge rate of the circuit. The impact of an increased
hydrocyclonecirculation load ratio due to changes in
feed size and hardness are manifested in the product size
distribution.
The first plot in Fig. 9 shows the particle size (P50) of
the solids at various points of the SAG mill circuit as a
function of time. It is immediately apparent that all the
P50 measures of particle size distribution at different
positions in the circuit are in the expected order at any
time. It is clear that when feed size increases, the circu-
60 70 80 90 100
 in Fresh Feed
1
1.1
O
re
 H
ar
dn
es
s
60 70 80 90 100
ids Concentration in Ball Mill
0.77
0.78
0.79
S
ol
id
s 
C
on
ce
nt
ra
tio
n
60 70 80 90 100
ter in Product
in)
120
130
140
150
160
170
180
P
ro
du
ct
S
iz
e 
- 
P
50
 (
µm
)
Feed mass
Ore Hardness
Concentration 
ds
Size
the SAG mill circuit.
0 10 20 30 40 50 60 70 80 90 100
678.5
679
679.5
680
680.5
Sump Volume
V
ol
um
e 
(li
tr
es
)
0 10 20 30 40 50 60 70 80 90 100
1
1.5
2
Cyclone Circulation Load Ratio
C
irc
. L
oa
d 
R
at
io
0 10 20 30 40 50 60 70 80 90 100
1200
1400
1600
1800
2000
Sump Pump Volumatric Rate
Time (min)
P
um
p 
R
at
e 
(li
tr
es
/m
in
)
Fig. 8. Internal dynamic responses of the SAG mill circuit.
0 10 20 30 40 50 60 70 80 90 100
10
2
10
3
10
4
Particle Size P50 TimeTraces at Different Points of the Circuit
S
iz
e 
P
50
 (
µm
)
Time (min)
10
2
10
3
10
4
0
20
40
60
80
100
Comparison of Particle Size Distributions at T = 20 mins and T = 85 mins (*)
P
er
ce
nt
ag
e 
P
as
si
ng
 (
%
)
Solids Size (µm)
Fresh Feed
Mill Feed
Cyclone Return
Cyclone Feed
Product
Fig. 9. Particle sizes at different points of the SAG mill circuit.
Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198 1197
lation load ratio increases, and then the product size in-
creases (when the feed rate and production rate are effec-
tively constant). When ore hardness increases, product
size increases as well. The second plot in Fig. 9 shows
particle size distributions (PSDs) at the same points in
the SAG mill circuit at two time specific moments in
time (at 25min with coarser feed size and at 85min
under normal feed conditions). As expected, the PSD
under coarse feed conditions is at all locations larger
than for fine feed. This is an example of the detailed
1198 Y. Liu, S. Spencer / Minerals Engineering 17 (2004) 1189–1198
particle size information that may be obtained from
such a dynamic simulation.
5. Conclusions
A powerful and flexible library of mineral processing
dynamical flowsheet unit operation models has been
developed in the SIMULINK programming environ-
ment. A generic, phenomenological mathematical model
based on an extended form of population balance meth-
od has been developed for grinding mills. This unit
model is inherently nonlinear for SAG/AG mills and is
suitable for dynamic simulation purposes. The power
of the dynamic simulation approach based on SIMU-
LINK environment has been demonstrated by two
examples which illustrate the importance of dynamic ef-
fects associated with variations of the key parameters of
unit mean residence time, feed size distribution and
hardness. The dynamic simulation approach developed
here has great potential not only for grinding circuit dy-
namic simulation, optimisation and control, but also for
many other dynamic flowsheet modelling and optimisa-
tion applications in the mineral processing industry.
A number of further conclusions can be drawn from
this work. SIMULINK is a powerful and flexible simu-
lation tool. It offers a modular and subsystem approach
for complex system handling. It has an intuitive block-
diagram interface for model building and model config-
uration, and rich libraries for basic model building and
development of control algorithms. The powerful solver
of the package is very easy to use and includes options
for easy handling of complex and stiff systems of equa-
tions associated with large flowsheets of complicated
connectivity. The graphical display capabilities of the
SIMULINK environment are similarly powerful and
flexible to use. In simulations, all relevant variables are
readily accessible in real time. This is particularly useful
to fully understand the dynamic interactions of the
external and internal variables of grinding circuits. In
real industrial situations, however, on-line measurement
of some those variables are far from trivial, if not totally
impossible. Hence, it is clear that dynamic simulation
can be used to answer many ‘‘what if’’ hypothetical
questions for industrial grinding circuits. Application
of this modelling approach to real plant dynamic simu-
lation is potentially of great value to industry.
As can be seen from an example in this paper, a sim-
ple PID control loop is not optimal for mill operation.
More studies on appropriate on-line measurement and
control strategies are needed. Further extensions of the
comminution model library would be very useful and
a more thorough process of validating the model library
using real plant data should be carried out. We believe
that the dynamic simulation approach used here will as-
sist in devising more appropriate control strategies for
achieving maximum throughput while keeping very tight
control on product size for grinding circuits.
References
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Engineering of Size Reduction: Ball Milling. AIME, New York.
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analysis and control in sustainable minerals processing. In: Green
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Processing of Minerals, Cairns, Qld. AusIMM, pp. 299–303.
Gonzalez, G.D., 1999. Soft sensors for processing plants. In:
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Weller, K.R., Liu, Y., Campbell, J.J., 1997. A new method for
estimating the full-scale grinding target for floating from labora-
tory batch grinding and flotation and reference full-scale tests. In:
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http://www.mathworks.com/products/simulink/technicalliterature.jsp
http://www.mathworks.com/products/simulink/technicalliterature.jsp
	Dynamic simulation of grinding circuits
	Introduction
	Dynamic models for grinding circuit unit operations
	General features and the specifics of comminution models
	General features of SIMULINK
	Specifics of the comminution dynamic model library
	Grinding circuit simulation examples
	Example 1: Residence time effects in a ball mill
	Example 2: Responses of a SAG mill circuit to feed size and hardness changes
	Conclusions
	References