dynamic simulation of grinding circuits

This article is also available online at:

www.elsevier.com/locate/mineng

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, therehas 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: [email protected] (Y. Liu), [email protected]

(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-linein 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:[email protected]

mailto:[email protected]

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 unitoperations. 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), norcan 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 farhas 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 dynamicsystem 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 asSAG/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 unit

operation characteristics. It is a cheap and effective

means of investigating circuit control and optimisationwithout 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 milldynamics 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 ðtÞdt

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, Austinet 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 ðtÞdt

¼ ðBðtÞ � IÞ bðtÞðcðtÞSðtÞ þ aSaðtÞX ðtÞÞ½ �X ðtÞ ð2Þ

Here B(t) and S(t) are the same as in Eq. (1), andSa(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, themodel simulates a ball mill, 0 < a < 1 simulates a SAGmill, and when a = 1 and S(t) = 0, the model simulates

an AG mill. Function b(t) is used to simulate changesin feed ore hardness. The model simulates ‘‘softer’’ ore

when b(t) < 1, and b(t) > 1 for ‘‘harder’’ ore, andb(t) = 1 returns to ‘‘normal’’ ore hardness. Similarly,function c(t) is used to simulate the effects of ball chargein the mill. When c(t) > 1, extra balls are added, whenc(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 severalperfect 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 modelbetter describes comminution in a perfect mixer:

dX ðtÞdt

¼ ðBðtÞ � IÞ bðtÞðcðtÞSðtÞ½

þaSaðtÞX ðtÞÞ�X ðtÞ þ 1sðf 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 diagonalconstant 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 basedon 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 powerfulnonlinear 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 ofindividual 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


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 verycomplex systems;

• Intuitive block-diagram (graphical) interfacing makesit easy to construct and understand;

• Very rich commonly used block and subsystemlibraries;

• Extensive control system libraries can be readily usedfor 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 custom blocks 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 millmodels. 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 withhardness input holdup output

Mixer

Perfect Mixer with RTand hardness input and

solids holdup output

MixerSAG

Perfect Mixer with RT for SAG

Mixer

Perfect Mixerwith water and RT Hydro-Cyclone

Ball Mill

Ball Mill withhardness input holdup output

Ball Mill

Ball Mill with3 mixers and aclassifier 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 tryto 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 ofdrag 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, thesystem is ready to be simulated once the model

parameters are loaded.

• To change the structure and configuration of thegrinding 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-functionby following a few general rules.

• The sump/pump model in the library is treated as aperfect mixer with residence time but without break-

age functions.

• The SAG Mill model in Fig. 1 is in fact a generaldynamic model for tumbling mills (see Eq. (3)). It

has several very useful characteristics:

– One parameter (a in Eq. (3)) could change themodel 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: Sizeand 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.

2Solids & watermass holdups

1Slurry massdischarge

MixerSAG

Perfect Mixer with RT for SAG 3

MixerSAG


MixerSAG


2Solids hardnesschange factor

1

Slurry massfeed

Fig. 2. A SAG mill model with three perfect mixers.


– 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 dischargeclassification coefficients.

– Ore-hardness changes in the feed can be simulated

by defining a time varying b(t) coefficient in theappropriate grinding model. Ball charge changes

can also be simulated in a similar manner (time-

varying c(t)), though this feature has not yet beenimplemented.

– 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, thenthe change of the rising and decay c

of the ball mill with a constant vo

-C-

WaterFeed

-K-Volumeoffset

K*u

SolidsSize Dist

SolidsFeed

yin

Mill_Feed

Mill Feed

Ball Mill

Ball Mill with 3 mixerand 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 TwoTotal Solids

Change RTtau1=3

Plot TotalSolids & Water

Plot SizeFractions

Load Datan=16, tau1=1

s nd

ball mill with residence time changes.


and the simulation progress can be monitored by the

scopes attached to feed stream and discharge stream.

There are 16 size fractions atffiffiffi

2psize 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 timeconstants for three mixers are all set to the same value

(s1 = s2 = s3 = 1min). The simulation is repeated withthe residence time of the first mixer set to s1 = 3minand 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 waterof 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 thestep 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 the constant 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 1000

100

200

300

400Solids Fractions in Dis

Mas

s Fr

actio

ns (

kg)

0 20 40 60 80 1000

500

1000

1500Mill Discharge - Total Solids & Wa

Mas

s (k

g)

Time (m

0 20 40 60 80 1000

500

1000Mill Feed - Total Solids,

Mas

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 modelis 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 themodel 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 200636

638

640

Fee

d V

olum

e (li

tre)Solids

Water

SolidsWater

τ

demonstrating mean residence time effects.

0 20 40 60 80 100 120 140 160 180 2000

200

400

600

800

1000Mill Feed - Total Solids, Water & Volume

Mas

s (k

g)

636

636.5

637

637.5

638

638.5

Fee

d V

olum

e (li

tre)

SolidsWater

0 20 40 60 80 100 120 140 160 180 2000

500

1000

1500

Time (min)

Tota

l Sol

ids

Mas

s &

Wat

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

τττ

ττ

Fig. 5. Comparison of changes in residence time of the ball mill model.


keep the total mass of solids feed at a constant. A stepup 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 atffiffiffi

2psize 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 toload the data

wff

Water Feed Solis & Wa Mass Hol

sff

Sol ids feed

Sol ids FeedDisturbance Sol ids Feed

Size changes

SAG M i l l

SAG Mill withhardness inputholdup output

Ore hardnessdisturbance

1

Normal hardness

ym Mi l l feed

Plot SizeDistribution

Plot TotalSol ids & Water

Plot OtherMil l Infos

Hardnesschanges

yu

Cyclone return

SizeSwitch

2 size system

R

Fig. 6. A closed-loop SAG mill grin

The SAG mill model consists of three perfect mixersincorporating 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 inthe 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

wsfWaterAddition

24

Sump vo lumeholdup setpoint

SUM P

Sump model

qsp

Sump dischargevolume rate offset

ys

Sump discharge

Sump Volume Holdup

terdup

Solids Product

Pump rate

PID

PID Controller

s

Hydro-Cyclone

m

Discharge

eturn stream

ding circuit simulation setup.


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 obviouslyit 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 wellas 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 thesump (i.e. indirectly the sump level). It is clear that a

0 10 20 30 40 500

5000

10000Total Solids and Size

Mas

s (k

g) &

Siz

e (µ

m)

0 10 20 30 40 50

1000

2000

3000

4000

5000

Solids and Water Holdup and Sol

Mas

s (k

g)

0 10 20 30 40 50400

500

600

700

800

900

1000Total Solids and Wa

Time (m

Mas

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, morecoarse 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 increasedhydrocyclone circulation 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 anytime. It is clear that when feed size increases, the circu-

60 70 80 90 100

in Fresh Feed

1

1.1

Ore

Har

dnes

s

60 70 80 90 100

ids Concentration in Ball Mill

0.77

0.78

0.79

Sol

ids

Con

cent

ratio

n

60 70 80 90 100

ter in Product

in)

120

130

140

150

160

170

180

Pro

duct

Siz

e -

P50

(µm

)

Feed mass

Ore Hardness

Concentration

dsSize

the SAG mill circuit.

0 10 20 30 40 50 60 70 80 90 100678.5

679

679.5

680

680.5Sump Volume

Vol

ume

(litr

es)

0 10 20 30 40 50 60 70 80 90 1001

1.5

2Cyclone Circulation Load Ratio

Circ

. Loa

d R

atio

0 10 20 30 40 50 60 70 80 90 1001200

1400

1600

1800

2000Sump Pump Volumatric Rate

Time (min)

Pum

p R

ate

(litr

es/m

in)

Fig. 8. Internal dynamic responses of the SAG mill circuit.

0 10 20 30 40 50 60 70 80 90 10010

2

103

104

Particle Size P50 TimeTraces at Different Points of the Circuit

Siz

e P

50 (

µm)

Time (min)

102

103

104

0

20

40

60

80

100Comparison of Particle Size Distributions at T = 20 mins and T = 85 mins (*)

Per

cent

age

Pas

sing

(%

)

Solids Size (µm)

Fresh FeedMill FeedCyclone ReturnCyclone FeedProduct

Fig. 9. Particle sizes at different points of the SAG mill circuit.


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


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 powerof 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 formany 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 anddevelopment 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 usefulto 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 believethat 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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http://www.mathworks.com/products/simulink/technicalliterature.jsp

dynamic simulation of grinding circuits

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