Apelt Inferential Measurement of SAG Mill Parameters II State Estimation 2002

Inferential measurement of SAG mill parameters II: state estimation

T.A. Apelt a,c,*, S.P. Asprey a, N.F. Thornhill a,b

a Centre for Process Systems Engineering, Imperial College, London SW7 2BY, UKb Department of Electronic and Electrical Engineering, University College, London WC1E 7JE, UK

c Department of Chemical Engineering, University of Sydney, NSW 2006, Australia

Received 21 June 2002; accepted 20 September 2002

Abstract

This paper discusses the combined state and parameter estimation of SAG mill inventories and model parameters. Recognised

simulation models are utilised for the rock and water charge state equations. New models of the ball charge and mill shell lining

states are presented. The 36 state system is detectable although not completely observable. Five ore grindability and mill discharge

grate parameters augment the state system. One mill weight and two discharge measurement models are presented and utilised in

two state estimation formulations. Results indicate that a size by size SAG mill discharge measurement model provides superior

state estimates and improved discharge grate parameter estimation compared to a bulk SAG mill discharge measurement model.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: SAG milling; Comminution; Modelling; Simulations

1. Introduction

The SAG milling process presents measurement and

control problems typical of an industrial process in that

‘‘the total state vector can seldom be measured and the

number of outputs is much less than the number of

states.’’ State estimation techniques may be utilised to

‘‘provide acceptable estimates of all the state variables(even those not directly measured) in the face of mea-

surement error and process disturbances’’ (Ray, 1981).

The estimates may be used for process monitoring, op-

eration and control.

This paper discusses combined state and parameter

estimation (CSPE) for SAG mill charge levels and is a

continuation of earlier work (Apelt et al., 2001a) which

described inferential models for the mill inventories indetail. The formulation of the CSPE problem will be

described following a review of related works. State

equations for mill rock, water and grinding ball inven-

tories and the shell protective lining are then presented

with a list of five parameters included in the formu-

lation. Discussion of system observability and detect-

ability follows the presentation of the measurement

models incorporated in the CSPE formulation. Discus-

sion of the requisite Kalman filter tuning precedes the

presentation and discussion of the results. To conclude,

the major findings of this paper are summarised.

2. Circuit description

The discussion centres on the primary grinding circuit

shown in Fig. 1. Ore is fed to the SAG mill for primary

grinding. The mill discharge is screened with the over-

sized material recycling via a gyratory cone crusher, and

the screen undersize being diluted with water and fed to

the primary cyclones for classification. Primary cyclone

underflow is split between a small recycle stream to the

SAG mill feed chute and a ball mill feed stream. Theprimary grinding circuit products are subjected to fur-

ther size reduction (ball mill), classification (cyclones)

and separation (flash flotation) in the secondary grind-

ing circuit. Further details of the grinding circuit and the

other sections of the processing plant may be found

elsewhere (Apelt et al., 1998, 2001a,b; Freeman et al.,

2000; Apelt, 2002).

3. Related works

Significant research has been conducted in the use of

Kalman filters for comminution process state estimation.

Minerals Engineering 15 (2002) 1043–1053This article is also available online at:

www.elsevier.com/locate/mineng

Minerals Engineering 15 (2002) 1043–1053

*Corresponding author. Address: Centre for Process Systems

Engineering, Imperial College, London SW7 2BY, UK.

E-mail address: [email protected] (T.A. Apelt).

0892-6875/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0892-6875 (02 )00229-7

mail to: [email protected]

Much of the research has been conducted by J.A. Herbst

and colleagues whom, over time, have had association

with Utah Comminution Centre, Control International

(Inc.), GS Industries, JA Herbst and Associates, Svedala(Optimization Services and CISA) and Metso Minerals.

Herbst and Alba (1985) discussed the use of a Kal-

man filter to estimate the current state, model para-

meters and the predicted state (in the next time step) for

adaptive control. Herbst and Gabardi (1988) used a

Kalman filter in conjunction with the use of a lifter-bolt

strain gauge (for charge position indication), mill power-

draw measurement, and process models of the grind-ing media and mill shell lining, to estimate the ball

charge level and wear rate, liner thickness and wear rate

in a ball mill. The result was a consistent ball charging

rate for ball charge control at maximum mill power-

draw.

Kalman filters were applied to autogenous/semiaut-

ogenous (AG/SAG) mills by Herbst et al. (1989) when

estimates of rock (one combined state), ball and watercharges, shell liner weight and ore grindability were

obtained from a dynamic model. Mill bearing pressure

and powerdraw measurements were used. Other work of

Herbst and colleagues has documented industrial ap-

plications and development of a commercial soft-sensor

(Herbst and Pate, 1999; Herbst and Pate, 2001).

A Kalman filter has been used on an iron ore pebble

mill (Herbst and Pate, 1996). A commercial softwareproduct was installed to estimate five states (mill hold-

up of two rock states (�55 mm), water, grinding balls

and shell liner weight) and two parameter estimates (ore

grindability and charge angle of repose) (Herbst and

Associates, 1996). Herbst and Pate (1999) describe a

generic softsensor with examples of ore grindability es-

timation for a ball mill and estimation of mill filling (ore,

balls and water), dynamic angle of repose and oregrindability for a SAG mill.

The Svedala Cisa OCS� optimising control system

package for AG/SAG mills includes a softsensor module

(Broussaud et al., 2001). Applications on both mini-

mally and comprehensively instrumented plants are de-

scribed. The softsensor ‘‘continuously computes a mass

balance and estimates mill charge (mass of solids, sim-

plified size distribution and percent solids in the mill),

cyclone feed, circulating load and cyclone overflow

(particle size and percent solids).’’

The review shows that the technology of Kalman

filters in SAG mill state estimation is well progressed.

However, the industrial uptake of state estimation hasbeen slow and isolated to larger mining houses and new

installations. The lag between development and uptake

is a reflection of the level of uptake of model based

decision making control technologies. Herbst (2000)

suggested that industry reluctance is due to perceived

gap between the technology cost and its benefits and

also the performance record of the technology. There-

fore according to Hodouin et al. (2001) the area shouldbe considered ‘‘active’’ with some key issues to be ad-

dressed. This assessment is reinforced by the following

points:

• Broussaud et al. (2001) considered the difficulty of de-

termination of the ball load and wear rate in a SAG

mill in the absence of a suitable commercial ball ad-

dition system. They concluded the future combi-nation of Continuous Charge Monitoring (CCM)

instrumentation and good control of ball addition

should allow a closer on-line optimization of the ball

load and further improve SAG mill performance.

• The Herbst and Pate application of a Kalman filter

for the estimation of mill rock and ball hold-up is rec-

ognised in a review of automation in the minerals

processing industry. Jamsa-Jounela (2001), for in-stance, highlighted engineering costs, lack of prece-

dent applications and limited control technology as

constraints.

• Research into state estimation for SAG mills is also

currently being progressed at the Julius Kruttschnitt

Mineral Research Centre where a Kalman filter is uti-

lised to predict ore hardness, mill total charge and

mill discharge factors (Schroder, 2000). An open-loopplant trial gave good correlation and tracking perfor-

mance.

The contribution of the work reported here is to

present further examples of state estimation for SAG

mills and novel models of the SAG mill discharge, SAG

mill ball charge and shell protective lining. State esti-

mation of mill rock charge, ball charge (Jb), watercharge and thus total charge (Jt) is demonstrated. Anassessment of structural properties of the models (their

observability and detectability) gave insights into 50

their relative performance when used with a Kalman

Filter.

CSPE for SAG mills is now discussed further with the

presentation of the state equations and parameters. Two

formulations are presented which include process mea-surement equations for mill weight and mill discharge.

Fig. 1. Primary grinding circuit.

1044 T.A. Apelt et al. / Minerals Engineering 15 (2002) 1043–1053

Both are based on a University of Sydney thesis dis-

sertation (Apelt, 2002).

4. Combined state and parameter estimation model

Continuous-time nonlinear systems can be describedas follows (Henson and Seborg, 1997):

_xxðtÞ ¼ f ðx; u; h; tÞ ð1Þ_hh ¼ 0 ð2ÞyðtÞ ¼ gðx; h; tÞ ð3Þ

where x is the state of the system, _xx is the time derivative

of state of the system, u is the system input, y is the

system output, t is time, h is the system model para-

meters to be estimated, _hh is the time derivative of the

system model parameters, f is the system function and gis the measurement function. Eqs. (1)–(3) is referred to aCSPE model since it is utilised to estimate states x andparameters h.

In discrete-time, where measurements are available at

equally spaced intervals (sampling periods) Dt, Eqs. (1)–(3) can be described as follows (Henson and Seborg,

1997):

xkþ1 ¼ �ff ðxk; uk; hk; kÞ ð4Þhkþ1 ¼ hk ð5ÞyðtÞ ¼ �ggðxk; hk; kÞ ð6Þwhere k is the discrete time index (time ¼ kDt), �ff is the

discrete time system function and �gg is the discrete time

measurement function.

One method of on-line state and parameter esti-

mation is achieved through the utilisation of an the ex-

tended Kalman filter on the system described by Eqs.(4)–(6) (Henson and Seborg, 1997):

xxkjk ¼ xkjk�1 þ Lkðyk � �ggðxxkjk�1; hkjk�1; kÞÞ ð7Þ

Lk ¼ bPPkjk�1GTk ðGk

bPPkjk�1GTk þ RÞ�1 ð8ÞbPPkjk ¼ ðI � LkGkÞbPPkjk�1 ð9Þ

xxkþ1jk ¼ �ff ðxxkjk; hhkjk; uk; kÞ ð10ÞbPPkþ1jk ¼ F kbPPkjkF Tk þ Q ð11Þ

Gk ¼o�ggðx; h; kÞ

ox

��x¼xxkjk�1;h¼hhkjk�1

ð12Þ

F k ¼o�ff ðx; h; u; kÞ

ox

��x¼xxkjk ;h¼hhkjk ;u¼uk

ð13Þ

where xxkjk is the filtered estimate of state x at time kDt,xxkjk�1 is the prediction of state x at time kDt from the

previous time step ðk � 1ÞDt, Lk is the Kalman filter gainat time kDt, I is the identity matrix, Gk is the linearised,discrete-time measurement function, F k is linearised,

discrete-time system function, bPPkjk is the propagated

estimated state covariance matrix, bPPkþ1jk is the estimatedstate covariance matrix at the next time step time ¼ðk þ 1ÞDt, Q is the process output (measurement) co-

variance matrix, R is the state covariance matrix.

5. State equations and parameters

This research models the SAG milling process by the

following 36 state equations and five parameter equa-

tions:

Solids. The size by size solids mass balance developed

by the Julius Kruttschnitt Mineral Research Centre

(Napier-Munn et al., 1996; Valery and Walter, 1998):

Accumulation ¼ Inflow�OutflowþGeneration

� Consumption

dsidt

¼ fi � pi þXi�1j¼1

rjsjaij � ð1� aiiÞrisi i ¼ 1; . . . ; 27

ð14Þpi ¼ d0cisi i ¼ 1; . . . ; 27 ð15Þwhere fi is the SAG mill feed rate of solids in size i (t/h),ri is the SAG mill breakage rate for particles in size iwhich varies with operating conditions within the mill

(Variable Rates) (h�1), aij is the SAG mill appearance

function which describes the distribution that particlesin size i will form in the smaller sizes j (dimensionless), piis the SAG mill discharge rate of solids in size i (t/h), si isthe SAG mill rock charge in size i (t), ci is the SAG mill

discharge grate classification function value for size i(dimensionless) and d0 is the SAG mill maximum dis-

charge rate constant (h�1).

Water. The water mass balance also developed by the

Julius Kruttschnitt Mineral Research Centre (Napier-Munn et al., 1996; Valery and Walter, 1998):

Accumulation ¼ Inflow�Outflow

dswdt

¼ fw � pw ð16Þ

pw ¼ d0sw ð17Þwhere fw is the SAG mill water feed rate (t/h), pw is the

SAG mill water discharge (product) rate (t/h), sw is theSAG mill water charge (t) and d0 is the SAG mill

maximum discharge rate constant (h�1).

Grinding balls. The new grinding ball mass balance

model (Apelt, 2002) is as follows:

Accumulation ¼ In�OutþGeneration

� Consumption

dbci

dt¼ bii � bei þ bwi�1 � bwi i ¼ 1; . . . ; 7 ð18Þ

where bci is the SAG mill ball charge for balls in size i(t), bii is the SAG mill ball feed rate for balls in size i

T.A. Apelt et al. / Minerals Engineering 15 (2002) 1043–1053 1045

(t/h), bei is the SAG mill ball charge ejection rate for

balls in size i (t/h) and bwi is the SAG mill ball charge

wear rate for balls in size i (t/h).Shell lining. The new SAG mill liner weight mass

balance model (Apelt, 2002) is described as follows:

Accumulation ¼ Wear

dSMIW

dt¼ �wearate ð19Þ

where SMIW is the SAG mill protective shell lining in-

stallation weight (t) and wearate is the SAG mill shellprotective lining wear rate (t/h).

Parameters. These state equations are augmented by

the following set of five parameter equations:

_AA ¼ 0 impact breakage ore parameter ðin aijÞ_bb ¼ 0 impact breakage ore parameter ðin aijÞ_tta ¼ 0 abrasion breakage ore parameter ðin aijÞ_ffp ¼ 0 relative fraction pebble port open area ðin piÞ_dd0 ¼ 0 maximum mill discharge rate coefficient ðin piÞ

ð20ÞOre breakage parameters A, b and ta are included in

anticipation of an inferential measurement of ore grin-

dability. The mill discharge grate parameter fp is in-cluded due to its close link to the pebble port diameter,

xp, which is an influential to the relative contribution to

error in the feed passing sizes (F80 . . . F20). The maximumdischarge coefficient parameter, d0, is included since it

affects not only the mill discharge but also the rock and

water charge remaining in the mill. A ‘‘mill discharge

factor’’ parameter has also been used elsewhere (Sch-

roder, 2000). Tallying the number of states and para-meters brings the order of the system function, f , to 41.

5.1. Ball charge model

The dynamic ball charge model proposed by this re-

search is as follows:

Accumulation ¼ In�OutþGeneration

� Consumption

dbci

dt¼ bii � bei þ bwi�1 � bwi ð21Þ

where bci is the mass of balls in ball charge of size i (t),bii is the feed balls in size i (t/h), bei is the balls of size iejecting from the mill i (t/h) and bwi is the mass of balls

wearing out of size i into size iþ 1 (t).

5.1.1. Ball feed

The ball feed to the SAG mill, bii, can be determined

from operating conditions. Assuming that the feed balls

are of a single diameter, Db, the ball feed is as follows:

bii ¼p6

Db

1000

� �3

SGb bps3600

bstð22Þ

where bps is balls per stroke, bst is the ball stroke time

(s), Db is new ball diameter (mm) and SGb is ball specific

gravity (t/m3). Stroke refers to the stroke rate of the

feeding ram.

5.1.2. Ball wear

The overall ball wear rate may be determined from

operating data. If the ball charge level is being main-

tained at a constant level, the ball wear rate is equal to

the ball feed rate plus ball ejection. For the larger sizes

where there is no ejection the wear rate is equal to the

ball feed rate. The overall ball wear rate translates to

ball wear rates by size. These ball wear rates by size(bwi) are proposed here to be proportional to the frac-

tional surface area and the ball mass in size i and in-

versely proportional to ball hardness, i.e.,

bwi / fSAi1

HBiSMBCi ð23Þ

where bwi is ball wear of grinding balls in size i (t/h),fSAi is fractional surface area of grinding balls in size i(fraction), HBi is Brinnell Hardness of grinding balls in

size i (N/m) and SMBCi is mass of grinding balls in size i(t).

The total surface area of grinding balls in size i, SAi,

is the product of the number of balls in size i, Ni, and thesurface area of a ball of size i, i.e.,

SAi ¼ NipDb2i ð24Þ

The number of balls in size i, Ni, is determined from the

mass fraction of the total ball charge in size i and the

mass of a ball of size i, i.e.,

Ni ¼smbci100

BC

p6

Dbi1000

� �3SGb

ð25Þ

where BC is total ball charge mass (t), Dbi is diameter of

ball of size i (mm) and smbci is mass percent of balls inof size i (%).

The fractional surface area of the ball charge in size i,fSAi, is

fSAi ¼SAiPni¼1 SAi

¼smbciDbiPn

i¼1smbciDbi

ð26Þ

The ball hardness model proposed here is based on the

findings of Banisi et al. (2000), i.e., that the ball hardnessof 80 mm balls drops significantly when the ball wears to

less than 65 mm in size (81% original size). In this

work the original ball diameter is 125 mm and it is

assumed that the nominal ball diameter at which hard-

ness decreases markedly is 95 mm (76% original size).

Assuming hardness of 450 and 250 Brinnell for

the outside and inner layer of the balls, respectively,


(estimated from data in Perry’s (Perry et al., 1984)) andthat the variation of hardness across ball diameter can be

described by a Whiten classification model type rela-

tionship (Whiten, 1972; Napier-Munn et al., 1996), the

ball hardness, HBi, can be described as follows:

HBðDbiÞ ¼ 250 for Dbi6K1

HBðDbiÞ ¼ 450� K2�Dbi

K2� K1

� �K3for K1 < Dbi < K2

HBðDbiÞ ¼ 450 for Dbi PK2

ð27Þwhere K1 is the ball size below which hardness equals

250 Brinnell (mm), K2 is the ball size above which

harness equals 450 Brinnell (mm) and K3 ¼ 2:3 and is

the ball hardness curve shape parameter. The ball

hardness model is illustrated in Fig. 2.

The model can be adjusted to suit a given set of op-erating conditions by the introduction of a ball wear

coefficient, BWki, which can be fitted to operating data.

For mass balance consistency, the units of the ball wear

coefficient are (Brinnel/h). The ball wear model can now

be stated as follows:

bwi ¼ BWki fSAi1

HBiSMBCi ð28Þ

5.1.3. Ball ejection

The model of ejection of balls from the SAG mill

proposes that the SAG mill discharge grate behaves as a

vibrating screen which can be modelled by an efficiency

to oversize model (Napier-Munn et al., 1996). Ball

ejection of size i, bei, can then be stated as follows:

bei ¼ BEkiEdci SMBCi ð29Þwhere Edci is ball ejection efficiency to discharge of size i(fraction), bei is ejection rate for balls of of size i (t/h),BEki is ball ejection model fitting parameter (h�1).

The efficiency model utilised here is taken from Na-

pier-Munn et al. (1996) and is expressed in terms of

efficiency to undersize since the ejected balls are screen

‘‘undersize.’’

Edci ¼ 1� exp

� N foag 1

�Dbi

xp

�k!ð30Þ

where N ¼ 1 and is a discharge grate efficiency para-

meter, foag is discharge grate fraction open area (frac-tion), Dbi is ball diameter of size i (mm), xp is dischargegrate pebble port size (mm) and k ¼ 2 and is a discharge

grate efficiency parameter.

The model is validated against operating data by

fitting the ball wear model parameters (BWki) and the

ball ejection model parameters (BEki).

5.2. Shell lining model

The dynamic model of the SAG mill shell protective

lining proposed by Apelt (2002) is presented here. The

SAG mill installation weight, SMIW, can be considered

a sum of a number of constituents:

SMIW ¼ shell þ liningþD=C grate ð31Þwhere SMIW is SAG mill installation weight (t), shell is

SAG mill shell weight (t), lining is SAG mill shell lining

weight (t), D/C grate is SAG mill discharge grate weight(t).

The mill shell remains intact throughout the opera-

tional life of the mill. Therefore, the mill shell weight

(shell) is a constant. The lining is the internal shell

protective lining and is subject to wear through direct

contact with the mill contents. The SAG mill discharge

grate is internal to the mill also and is subject to wear.

Periodic change out of the mill protective lining anddischarge grate occurs to accommodate the wear of

these internal components. From plant experience, the

shell lining wear occurs predominantly at discharge end.

Therefore, the lining term can be broken up into a feed

end and a discharge end term:

SMIW ¼ ðshell þ liningFEÞ þ ðliningDE þD=C grateÞð32Þ

where liningDE is the weight of the discharge end of theshell lining (t), liningFE is the weight of the feed end of

the shell lining (t).

Combining the feed end lining terms and a constant

(to accommodate mill weight instrument offset) into a

SAG mill weight constant, SMWconst (t), allows the

model to we rewritten as follows:

SMIW ¼ SMWconstþ liningDE þD=C grate ð33ÞThe dynamic SAG mill liner wear model may then focus

on the grouped mill discharge end terms and may bewritten as follows:

dSMIW

dt¼ �wearate ð34Þ

Fig. 2. Ball hardness model.


where wearate is the SAG mill shell wear rate (t/h).

Integrating Eq. (34) with respect to time, t, yields

SMIW ¼ SMIW0 � wearate t ð35Þwhere SMIW0 is the initial SAG mill installation weight

(t) and t is time (h).

5.2.1. Wear rate

The mill liner wear rate, wearate, can determined from

the change-out frequency and the relative change inweight of the discharge grate and discharge end shell

lining at change-out time. Based on typical plant expe-

rience, the change-out frequency is approximately 6

weeks, or 1008 h (6� 7� 24), and the change-out liner

weight is approximately half the liner installation weight,

i.e.,

wearate ¼12liningDE0 þD=C grate0ð Þ

1008ð36Þ

where liningDE0 is the installation weight of the dis-

charge end of the shell lining (t) and D=C grate0 is the

installation weight of the discharge grate (t).

The shell lining is a series of alternating low lifter and

high lifter bars separated by shell lining segments. The

installation weight of the discharge end shell lining,

liningDE0, is calculated as follows:

liningDE0 ¼1

2Lsmqlinernhl

lbw

1000

hlt� sltð Þ1000

þ 1

2Lsmqlinernll

lbw

1000

llt� sltð Þ1000

þ p2Lsmqliner Dsm0

slt

1000

� slt

1000

�2!ð37Þ

where qliner is the liner density (t/m3), nhl is the number

of high lifter bars, nll is the number of low lifter bars,

lbw is the lifter bar width (mm), hlt is the high lifter bar

thickness (height) (mm), llt is the low lifter bar thickness

(height) (mm), slt is the shell liner thickness (height)(mm) and Lsm is the SAG mill shell length (m).

In Eq. (37), the third term represents the weight of an

annular piece of shell lining of thickness slt, defined by

the shell inside diameter, Dsm (m), and extending to half

the mill length, Lsm=2 (m). The first and second terms

are the weight of portion of high and low lifter bars,

respectively, that protrude above the shell lining.

The installation weight of the discharge grate,D=C grate0 (t), is calculated as follows:

D=C grate0 ¼ qliner pD2

sm

4

dgt

10001ð

�� foagÞ � p

Dt2sm4

dgt

1000

�þ qliner ndg

lbw

1000

hlt� sltð Þ1000

Dsmð�

�DtsmÞ�

ð38Þ

where dgt is the discharge grate thickness (mm), ndg is

the number of discharge grate segments (conceptually

similar to pieces of pie), foag is the fraction grate open

area (fraction) and Dtsm is the SAG mill trunnion di-

ameter (m).

In Eq. (38), the first term represent the weight of the

large flat disc (that constitutes the discharge grate) of

thickness, dgt, less the apertures in the grates (fraction

open area foag) and the absent central piece of diameterDtsm.

The second term in Eq. (38) represents the weight of

the portion of the discharge end lifter bars that protrude

above the surface of the discharge grate. The discharge

grate lifter bars are of thickness (height) hlt (mm) and of

length (Dsm �Dtsm) (mm).

5.2.2. Shell thickness

The assumption that all mill lining wear occurs in the

discharge end of the mill allows the mill liner model to

be simplified. Subtracting the mill weight constant

(SMWconst) from both sides of Eq. (35) yields

ðliningDE þD=C grateÞ¼ ðliningDE þD=C grateÞ0 � wearate t ð39Þ

Assuming that wear is uniform throughout the dis-

charge end of the mill, the wear thickness, wt (the

amount of lining component that has been worn away)(mm) can be determined as follows:

1. Take all terms in Eq. (39) to one side of the equation

0 ¼ ðliningDE þD=C grateÞ0 � wearate t� ðliningDE þD=C grateÞ ð40Þ

2. Express (liningDE þD=C grate) in terms of wear

thickness, wt, by the substitution of (slt� wt) for slt

in Eqs. (37) and (38):

liningDE þD=C grate

¼ 1

2Lsmqlinernhl

lbw

1000

hlt� sltð Þ1000

þ 1

2Lsmqlinernll

lbw

1000

llt� sltð Þ1000

þ p2Lsmqliner Dsm0

slt

1000

� slt

1000

�2!

þ qliner pD2

sm

4

dgt

10001ð

�� foagÞ � p

Dt2sm4

dgt

1000

�þ qliner ndg

lbw

1000

hlt� sltð Þ1000

Dsmð�

�DtsmÞ�

ð41Þ

3. Solve Eq. (40) for wear thickness, wt––the root of the

equation, e.g., with fzero, the MATLAB scalar non-

linear zero finding function.

4. Determine the current lining thickness, (slt� wt)(mm).

The state equations required for the combined state

and parameter estimation formulation have now been


presented, i.e., dynamic mass balance models of the rock

(Eqs. (14) and (15)), water (Eqs. (16) and (17)), and ball

charges (Eq. (18)) and the shell protective lining (Eq.

(19)). These equations form the matrix of state equa-

tions that describe the state of the system, Eq. (1). The

model parameters of the system, Eq. (2), have also been

presented (Eq. (20)). In the following section, the modelsof the system output (process measurements) (Eq. (3))

are presented.

6. Measurement models

The measurement (process output) models typically

utilised in CSPE are based on mill powerdraw and millweight (by way of bearing pressure or load cell mea-

surement) (Herbst and Pate, 1999; Herbst and Pate,

2001). The SAG mill discharge screen oversize conveyor

measurement has also been utilised in a formulation that

estimates a ‘‘mill discharge factor’’ parameter (Schroder,

2000) which is assumed to be or related to the maximum

mill discharge rate coefficient, d0 (h�1).

6.1. SAG mill weight measurement model

The SAG mill weight is measured by load cell. This

measurement, y1ðtÞ, is the process output that is pairedwith the mill weight measurement model, g1ðx; h; tÞ,which is a sum of the

• mass of ore particles, grinding balls and water in themill charge kidney (the charge material that is not in

free-fall from the charge shoulder to the charge toe),

Mkidney,

• mass of the mill shell discharge end lining and the dis-

charge grate, (liningDE þD=C grate), as presented

above,

• a calibration term, tare, that allows for the difference

between the actual total mill weight and the load cellmeasurement (includes the mill shell weight and the

feed end shell lining weight),

g1ðx; h; tÞ ¼ Mkidney þ ðliningDE þD=C grateÞ � tare

ð42ÞThe state functionality of the SAG mill weight mea-

surement model, g1ðx; h; tÞ is via the rock, ball and watercharge components in the charge kidney and the dis-

charge-end shell lining weight. The latter is, in fact, oneof the states considered. The kidney mass is a state

function, Mkidneyðx; h; tÞ––it is the sum of the rock, ball

and water states in the charge kidney.

6.2. SAG mill discharge measurement model

6.2.1. Model 1––bulk flow model

Model 1 is a bulk flow model which estimates thetotal volumetric discharge rate of the SAG mill. The

SAG mill discharge stream volumetric flowrate can be

reconstructed from four available plant measurements

and then utilised in the CSPE formulation as the second

process output, y2ðtÞ as described by Apelt et al. (2001b),

y2ðtÞ ¼MVOSCF %solsOSCF

100SGs

þ MVPCFDm3MVPCFD%solsSGl

MVPCFD%solsSGl þ ð100�MVPCFD%solsÞSGs

þ MVPCFDm3ð100�MVPCFD%solsÞSGs

MVPCFD%solsSGl þ ð100�MVPCFD%solsÞSGs

þMVOSCFð100�%solsOSCFÞ100SGl

�MVPCFWm3

ð43Þwhere MVOSCF is the oversize crusher total feedrate (t/h),

MVPCFDm3 is the primary cyclone feed flowrate (m3/h),

MVPCFD%sols is the primary cyclone feed density (% sol-ids w/w), MVPCFWm3 is the primary cyclone feed water

addition flowrate (m3/h), %solsOSCF is the oversize

crusher feed density (% solids w/w), SGs is the ore spe-

cific gravity (t/m3) and SGl is the process water specific

gravity (t/m3).

The plant measurement, Eq. (43), requires pairing

with a measurement model, according to Eq. (3). The

corresponding measurement model, g2ðx; h; tÞ, is

g2 x; h; tð Þ ¼ SMDCs

SGs

þ SMDCl

SGl

ð44Þ

The state functionality of the SAG discharge measure-

ment model, g2ðx; h; tÞ, is through the constituents of theterms in Eq. (44). The solids mass flowrate, SMDCs

(t/h), is the summation of the size by size mill productstream and is thus state dependent.

SMDCs ¼Xni¼1

pi ¼Xni¼1

d0cisi ð45Þ

where pi is the SAG mill discharge rate of solids in size i(t/h) si is the SAG mill rock charge in size i (t) and ci isthe SAG mill discharge grate classification function

value for size i (dimensionless).

The liquid mass flowrate, SMDCl (t/h), is a function

of SMDCs and thus is also state dependent:

SMDCl ¼ kgQm

�� SMDCs

SGs

�SGl ð46Þ

where Qm is the SAG mill volumetric discharge rate

through the grinding media (m3/h) and kg is a coarse

material adjustment factor (dimensionless).

On account of the SAG mill discharge grate charac-teristics, i.e., high fractional open area (foag), high rel-

ative radial position of the open area (c), and high

relative radial position of the outermost aperture (rn),the mill discharge flow is assumed to be only through the

grinding media and that no slurry pool exists at the toe


of the charge. Therefore, the mill discharge flowrate may

be calculated as follows:

Qm ¼ 6100J 2pmc2:5A/�1:38D0:5 ð47Þ

where A is the total discharge grate open area (m2), D is

the mill inside diameter (m), c is the mean relative radialposition of open area (fraction) and / is the fraction

critical mill speed (fraction). The nett fractional holdup

of slurry in mill that is contained within the grindingcharge (live area) interstices, Jpm, is calculated as fol-

lows:

Jpm ¼

P16 mm

i¼nsi

SGsþ sw

SGl

Vmillð48Þ

where sw is the mill water charge (t) and Vmill is the millvolume (m3).

The parameter functionality of the SAG discharge

measurement model, g2ðx; h; tÞ, is also through the con-

stituents of the terms in Eq. (44). The solids mass

flowrate (SMDCs), Eq. (45), is a function of the maxi-

mum mill discharge rate constant (d0), explicitly, andalso the relative fraction pebble port open area (fp),implicitly (via the grate classification function, ci). Themaximum mill discharge rate constant (d0) affects not

only the mill discharge but also the rock and water

charge remaining in the mill.

The relative fraction pebble port open area (fp) islinked by the classification function to the pebble port

aperture size (xp) which is an influential parameter in thefeed passing size estimates. This reinforces the inclusion

of the relative fraction pebble port open area (fp) in thelist of parameters in Eq. (20).

The ore breakage parameters A, b and ta affect the

rock breakage occurring within the mill and the mill

rock charge fractions (si) and are therefore implicitly

present in the state equations, Eqs. (14) and (15), and

the measurement model function, Eq. (45). As men-

tioned, these parameters are included in the formulation

in anticipation of an inferential measurement of oregrindability which influences mill performance.

The mill weight measurement y1ðtÞ and measurement

model g1ðx; h; tÞ, Eq. (42), and the reconstructed mill

discharge measurement y2ðtÞ, Eq. (43), and measurementmodel g2ðx; h; tÞ, Eq. (44), were paired together to form

the system output and measurement functions (Eq. (3)).

6.2.2. System observability and detectability

Initial difficulties in generating results utilising this

formulation (Model 1) prompted an investigation of the

observability and detectability of the system. This was

achieved by:

1. Linearising the CSPE at steady-state conditions to

generate a state-space model of the form

xxðtÞ ¼ Axþ Bu ð49ÞyðxÞ ¼ Cx ð50Þwhere A;B;C are the system matrices, x is the systemstates (36 states), u is the process inputs and y is theprocess outputs (measurements).

2. Generating the ‘‘observability matrix,’’ L0, (Ray,

1981; Henson and Seborg, 1997)

L0 ¼ ½CTjATCTjðATÞ2CTj jðATÞn�1CT ð51ÞMatrix A is n� n where n is the number of states andmatrix C is l� n where l is the number of measure-

ments. The observability matrix, L0, has dimension

n� nl.3. Determining the rank of the observability matrix, L0.

If the rank of L0 is n then the system is completely ob-

servable and each initial state x0 can be determined

from knowledge of the process inputs, u, and processoutputs, y, over a finite time period.

If the rank of L0 is less than n then the system is only

partially observable. If the system modes that cannot be

observed or reconstructed from the output measure-

ments are stable then the system is detectable (Henson

and Seborg, 1997).

The rank of the observability matrix, L0, was deter-mined to be 8 which is less than the dimension of the

system (n ¼ 36). Therefore, the system is not completely

observable. The model was therefore further enhanced

as described in the following subsection.

6.2.3. Model 2––size by size model

The issue of primary concern in utilising the bulk flowmodel was the low number of plant measurements and

measurement models (2) relative to the number of states

in the system (36). Therefore, the number of plant

measurements was increased through the extension of

the SAG mill discharge bulk flow model to a size by size

model.

Twenty eight new plant measurements, y2...29ðx; h; tÞ,are generated by as follows:

y228 x; h; tð Þ ¼ oscf i

100

� �OSCFtph s

þ pcfdi100

� �PCFDtph s i ¼ 1; . . . ; 27

ð52Þy29 x; h; tð Þ ¼ OSCFtph l þ PCFDtph l � PCFW SGl

ð53Þwhere oscf i is the oversize crusher feed retained in size i(% w/w), pcfdi is the primary cyclone feed retained in

size i (% w/w), OSCFtph s is the oversize crusher solids

feedrate (t/h), OSCFtph l is the oversize crusher liquid

feedrate (t/h), PCFDtph s is the primary cyclone solids


feedrate (t/h), PCFWtph l is the primary cyclone liquid

feedrate (t/h), PCFW is the primary cyclone feedwater

addition (m3/h) and SGl is the process water specific

gravity (t/m3).

Eq. (52) is a size by size SAG mill discharge process

measurement and represents the solids discharge rate for

size i (t/h). Eq. (53) is SAG mill discharge processmeasurement for process water and the liquid discharge

rate (t/h). The corresponding measurement models,

g2...29ðx; h; tÞ, are the right hand sides of Eqs. (15) and

(17), i.e.,

g228 x; h; tð Þ ¼ d0cisi i ¼ 1; . . . ; 27 ð54Þg29 x; h; tð Þ ¼ d0sw ð55Þwhere si is the mill rock charge in size i (t), sw is the mill

water charge (t), d0 is the maximum discharge rate

constant (h�1) and ci is the discharge grate classificationfunction (fraction).

The system matrix A for the size by size model (Model

2) is the same as that for the bulk flow model (Model 1).

Since the measurement (process output) matrix C has

changed with the utilisation of 29 measurements (insteadof two), the observability matrix, L0, has changed ac-

cordingly.

The rank of the observability matrix, L0, for Model 2

is 20 which, though a significant improvement on eight,

is (still) less than the dimension of the system (n ¼ 36).

Therefore, the system remains not completely observable

for the new CSPE formulation. However, improved

state estimates were possible using the size by size modelas described in the following section.

7. Results and discussion

The system of states was shown to be detectable for

both CSPE formulations (Model 1 and Model 2) despite

the incomplete observability of the system. Detectabilitywas established through achieving satisfactory results by

appropriate selection of Kalman filter tuning parame-

ters. The Kalman filter is described by Eqs. (7)–(13).

Suitable initial values for the tuning parameters P , Qand R were based on discussion points raised by Henson

and Seborg (1997), Cheng et al. (1997) and Welch and

Bishop (2001).

The results of the CSPE models are given in Tables 1and 2. The state and parameter estimates for both CSPE

formulations are shown in Table 1 with the steady-state

information and the respective errors relative to the

steady-state values (initial state values, x0). Table 2

contains the corresponding volumetric total (Jt) and ball(Jb) charge estimates.

7.1. Model 1––bulk flow model

Table 1 contains the state and parameter estimates

for Model 1––the formulation incorporating the bulk

flow SAG mill discharge measurement model (Eqs. (43)

and (44)). Referring to the fourth and fifth columns of

Table 1, the results show that, overall, good state and

parameter estimates are possible through the utilisation

of this CSPE formulation using two plant measure-

ments. The relative error present in the state estimates is

Table 1

CSPE model results––states and parameters

State Unit SS

value

Model 1

estimate

Error

(%)

Model

2 esti-

mate

Error

(%)

Rock charge by size

s1 (t) 0 0 0 0 0

s2 (t) 0.23 0.23 0.5 0.23 0

s3 (t) 3.54 3.51 1.0 3.54 0

s4 (t) 8.91 8.68 2.6 8.91 0

s5 (t) 9.33 9.43 1.1 9.33 0

s6 (t) 7.66 7.63 0.4 7.66 0

s7 (t) 5.26 5.13 2.5 5.26 0

s8 (t) 2.43 2.46 1.3 2.43 0

s9 (t) 1.39 1.40 0.7 1.39 0

s10 (t) 0.83 0.84 1.7 0.83 0

s11 (t) 0.60 0.63 5.3 0.60 0

s12 (t) 0.53 0.55 4.0 0.53 0

s13 (t) 0.51 0.53 4.5 0.51 0

s14 (t) 0.48 0.50 4.6 0.48 0

s15 (t) 0.42 0.44 4.3 0.42 0

s16 (t) 0.36 0.37 4.2 0.36 0

s17 (t) 0.30 0.31 4.0 0.30 0

s18 (t) 0.27 0.29 4.2 0.27 0

s19 (t) 0.26 0.27 3.6 0.26 0

s20 (t) 0.27 0.28 3.4 0.27 0

s21 (t) 0.27 0.28 2.9 0.27 0

s22 (t) 0.27 0.28 3.3 0.27 0

s23 (t) 0.25 0.26 3.0 0.25 0

s24 (t) 0.22 0.23 2.8 0.22 0

s25 (t) 0.20 0.11 44.5 0.20 0

s26 (t) 0.17 0.17 0.8 0.17 0

s27 (t) 0.72 0.50 30.5 0.72 0

Water charge

sw (t) 2.06 2.08 1.2 2.06 0

Ball charge by size

bc1 (t) 0 0 0 0 0

bc2 (t) 0.9 0.9 0 0.9 0

bc3 (t) 46.7 46.7 0.0004 46.7 0

bc4 (t) 32.7 32.7 0.0003 32.7 0

bc5 (t) 13.1 13.1 0.0008 13.1 0

bc6 (t) 0 0 0 0 0

bc7 (t) 0 0 0 0 0

Shell lining

SMIW (t) 337.7 337.7 0.0004 337.7 0

Parameters

A (dimen-

sionless)

75.8 75.8 0 75.8 0

b (dimen-

sionless)

0.45 0.45 0.1 0.45 0

ta (dimen-

sionless)

0.13 0.13 0 0.13 0

fp (frac-

tion)

0.011 0.011 0 0.034 208

D0 (h) 29.85 29.85 0.001 28.79 3.6


generally <5%. The relative error present in the para-

meter estimates is either exactly equal or approximately

equal to zero.

The poor observability of this model can be inferred

from the two poor state estimate results that stand out in

Table 1, the estimates of the 25th and 27th rock charge

states, s25 and s27, respectively. These relatively high

levels of error (44.5% and 30.5%, respectively) are at-tributed to the small amounts of rock in these size

fractions where a small deviation from the steady-state

value therefore represents a large relative error.

7.2. Model 2––size by size model

Table 1 also contains the state and parameter esti-

mates for Model 2––the formulation incorporating thesize by size SAG mill discharge measurement model

(Eqs. (52)–(55)). The results in the sixth and seventh

columns show that improved state and parameter esti-

mates are possible through the utilisation of this CSPE

formulation with 29 plant measurements, which incor-

porates the size by size mill discharge models and pos-

sesses improved observability properties. The results

show the extension of the bulk flow SAG mill dischargemeasurement, Eqs. (43) and (44), to a size by size

throughput measurement, Eqs. (52)–(55), improved the

resulting state estimates––all state estimates displayed

exact agreement with the steady-state values, see the

seventh column of Table 1. Exact agreement was also

displayed in the estimates of the ore impact breakage

and abrasion parameters (A, b and ta).Relative error is present in the estimates of the mill

discharge parameters (fp and d0) in the seventh column

of Table 1. The high level of relative error in the no-

tional fraction pebble port open area relative to total

grate open area, fp, is clearly evident. This is due to the

CSPE model adjusting the mill discharge parameters (fpand d0) to achieve the exact agreement for the rock and

water charge state estimates via Eqs. (54) and (55).

The state estimates of Table 1 translate to the volu-metric charge fraction estimates results in Table 2.

Overall, all of the estimates show good agreement with

the steady-state conditions and reinforce that both

CSPE formulations (Model 1 and Model 2) can be uti-

lised for state estimation.

The volumetric ball charge (Jb) estimates from both

CSPE formulations exhibit exact agreement with the

steady-state conditions. The total volumetric charge (Jt)estimates from both CSPE formulations exhibit good

agreement with the steady-state conditions (to within

0.6%). The exact agreement between the Model 2

formulation estimates of the mill charge levels (Jt and Jb)and the steady-state conditions is expected following the

analysis of Tables 1 and 2 above. That analysis alsoleads to the expectation of the relative error present in

the Model 1 total volumetric charge (Jt) estimate.Although both CSPE formulations are capable of

good estimates, the Model 2 formulation provides su-

perior state estimates through adjustment (estimation)

of the mill discharge parameters (fp and d0). On account

of these two characteristics the Model 2 CSPE formu-

lation is considered superior. In either case, furthermodel validation would be required before entering an

implementation phase.

The results arising from a suite of inferential mod-

els––models that infer unmeasured conditions from

other measurements––to be presented in a future paper,

indicate the error in the charge estimates from a mill

weight-based inferential model is 1.3% and 1.8% for the

estimates of the total volumetric charge (Jt) and ballcharge (Jb), respectively. Comparison of these figures

with the results in Table 2 (0–0.6% error) suggests that

the CSPE formulations yield superior results.

8. Conclusions

Two formulations of combined state and parameterestimation (CSPE) for SAG mills have been presented.

They incorporated novel dynamic models of SAG mill

ball charge and protective shell lining. Two novel mea-

surement models of the SAG mill discharge, a bulk flow

model and a size by size throughput model, and a novel

mill weight measurement model were also presented.

Assessment of system observability and detectability

showed that observability was poor for formulationutilising the mill discharge bulk flow model. The sub-

sequent development of the size by size throughput SAG

mill discharge measurement model yielded better ob-

servability properties and improved state estimates.

Both of the CSPE formulations, whilst not completely

observable, are detectable. Both formulations would

require further validation prior to implementation.

The size by size SAG mill discharge measurementmodel (Model 2) yielded superior state estimate results

and increased capacity to adjust the important discharge

parameters (fp and d0).

Acknowledgements

Acknowledgements go to Northparkes Mines fortheir assistance with and permission to publish circuit

Table 2

CSPE model results––charge fractions

Volumetric

fraction

SS

value

Model

1 esti-

mate

Error

(%)

Model

2 esti-

mate

Error

(%)

Total charge, Jt 0.2298 0.2284 0.6 0.2298 0

Ball charge, Jb 0.1420 0.1420 0 0.1420 0


information, the Centre for Process Systems Engineer-

ing for part hosting and the University of Sydney for

providing Australian Postgraduate Award funding for

this research.

References

Apelt, T.A., Galan, O., Romagnoli, J.A., 1998. Dynamic environment

for comminution circuit operation and control. In: CHEMECA

�98, 26th Australasian Chemical Engineering Conference, CHEM-

ECA, Port Douglas Qld, Australia.

Apelt, T.A., 2002. Inferential measurement models for semi-autoge-

nous grinding mills. PhD thesis, Department of Chemical Engi-

neering, University of Sydney, Australia, submitted February 2002.

See http://geocities.com/thomasapelt.

Apelt, T.A., Asprey, S.P., Thornhill, N.F., 2001a. Inferential mea-

surement of SAG mill parameters. Minerals Engineering 14 (6),

575–591.

Apelt, T.A., Asprey, S.P., Thornhill, N.F., 2001b. SAG mill discharge

measurement model for combined state and parameter estimation.

In: SAG 2001, Third International Conference on Autogenous and

Semiautogenous Grinding Technology, vol. IV. UBC, Vancouver,

BC, Canada, pp. 138–149.

Banisi, S., Langari-Zadeh, G., Pourkani, M., Kargar, M., Laplante,

A.R., 2000. Measurement for ball size distribution and wear

kinetics in an 8 m by 5 m primary mill of Sarcheshmeh copper

mine. CIM Bulletin.

Broussaud, A., Guyot, O., McKay, J., Hope, R., 2001. Advanced

control of SAG and FAG mills with comprehensive or limited

instrumentation. In: SAG 2001, Third International Conference on

Autogenous and Semiautogenous Grinding Technology, vol. II.

UBC, Vancouver, BC, Canada, pp. 358–372.

Cheng, Y.S., Mongkhonsi, T., Kershenbaum, L.S., 1997. Sequential

estimation for nonlinear differential and algebraic systems––theo-

retical development and application. Computers in Chemical

Engineering 21 (9), 1051–1067.

Freeman, W.A., Newell, R., Quast, K.B., 2000. Effect of grinding

media and NaHS on copper recovery at Northparkes Mine.

Minerals Engineering 13 (13), 1395–1403.

Henson, M.A., Seborg, D.E. (Eds.), 1997. Nonlinear Process Control.

Prentice-Hall/PTR.

Herbst and Associates, J.A., 1996. Semi autogenous mill filling using a

model-based methodology. Proposal submitted to North Lim-

ited––Northparkes Mines.

Herbst, J.A., 2000. Model-based decision making for mineral process-

ing––a maturing technology. In: Herbst, J.A. (Ed.), CONTROL

2000, Mineral and Metallurgical Processing. SME, Littleton, CO,

USA, pp. 187–200.

Herbst, J.A., Alba, F.J., 1985. An approach to adaptive optimal

control of mineral processing operations. In: XV International

Mineral Processing Congress, vol. 3. France, pp. 75–87.

Herbst, J.A., Gabardi, T., 1988. Closed loop media charging of mills

based on a smart sensor system. In: Sommer, G. (Ed.), IFAC

Applied Measurements in Mineral and Metallurgical Processing.

Pergamon Press, Transvaal, South Africa, pp. 17–21.

Herbst, J.A., Pate, W.T., 1996. On-line estimation of charge volumes

in semiautogenous and autogenous grinding mills. In: SAG 1996,

Second International Conference on Autogenous and Semiautog-

enous Grinding Technology, vol. 2. Vancouver, BC, Canada, pp.

817–827.

Herbst, J.A., Pate, W.T., 1999. Object components for comminution

system softsensor design. Powder Technology 105, 424–429.

Herbst, J.A., Pate, W.T., 2001. Dynamic modeling and simulation of

SAG/AG circuits with MinOOcad: off-line and on-line applica-

tions. In: SAG 2001, Third International Conference on Autoge-

nous and Semiautogenous Grinding Technology, vol. IV. UBC,

Vancouver, BC, Canada, pp. 58–70.

Herbst, J.A., Pate, W.T., Oblad, A.E., 1989. Experience in the use of

model based expert control systems in autogenous and semi

autogenous grinding circuits. In: SAG 1989. First International

Conference on Autogenous and Semiautogenous Grinding Tech-

nology. UBC, Vancouver, BC, Canada, pp. 669–686.

Hodouin, D., Jamsa-Jounela, S.-L., Carvalho, M.T., Bergh, L., 2001.

State of the art and challenges in mineral processing control.

Control Engineering Practice 9 (9), 995–1005.

Jamsa-Jounela, S.-L., 2001. Current status and future trends in

automation of mineral and metal processing. Control Engineering

Practice 9 (9), 1021–1035.

Napier-Munn, T.J., Morrell, S., Morrison, R.D., Kojovic, T., 1996.

Mineral Comminution Circuits––Their Operation and Operation.

Julius Kruttschnitt Mineral Research Centre, Australia.

Perry, T.H., Green, D.W., Maloney, J.O. (Eds.), 1984. Perry�sChemical Engineers� Handbook, sixth ed. McGraw-Hill.

Ray, W.H., 1981. Advanced Process Control. McGraw-Hill.

Schroder, A.J., 2000. Towards automated control of milling opera-

tions. In: IIR Conference––Crushing and Grinding Technologies

for Mineral Extraction, Perth, Australia.

Valery Jr., Walter, 1998. A model for dynamic and steady-state

simulation of autogenous and semi-autogenous mills. PhD thesis,

Department of Mining, Minerals and Materials Engineering,

University of Queensland. St Lucia, Qld, 1997.

Welch, G., Bishop, G., 2001. An Introduction to the Kalman Filter.

http://www.cs.unc.edu, Department of Computer Science,

University of North Carolina at Chapel Hill.

Whiten, W.J., 1972. The simulation of crushing plants with models

developed using multiple spline regression. In: Salamon, M.D.G.,

Lancaster, F.H. (Eds.), Proceedings of the Tenth International Sym-

posium on the Application of Computer Methods in the Mineral

Industry. SAIMM, Johannesburg, South Africa, pp. 317–323.


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