Apelt Inferential Measurement of SAG Mill Parameters III Inferential Models 2002

Inferential measurement of SAG mill parameters III:inferential models

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 & Electrical Engineering, University College, London WC1E 7JE, UK

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

Received 21 June 2002; accepted 21 September 2002

Abstract

This paper discusses inferential measurement models for semiautogenous grinding (SAG) mills. Inferential measurements of

SAG mill discharge and feed streams and mill rock and ball charge levels are obtained utilising process measurements and re-

cognised process simulation models. Inferential models of recirculating load and cyclone underflow split are also presented. Results

for the mill inventories and process streams are validated against reference simulation model data. Uncertainty analyses are con-

ducted to assess the influence of the various model parameters. Mill weight based estimates for mill inventory are shown to be the

least uncertain. The results suggest that regular calibration of oversize crusher and primary cyclone feed instrumentation, regular

measurement of the SAG mill discharge screen aperture, oversize crusher gap setting and process water specific gravity and careful

fitting of the SAG mill discharge grate model parameters will minimise uncertainty in the inferential models.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: SAG milling; Comminution; Modelling; Simulations

1. Introduction

This paper describes inferential models of the mill

inventory and various streams in the primary grinding

circuit and is a continuation of earlier work (Apelt et al.,

2001a). The models presented below in Step 3 of the

inferential models section recapitulate models presented

in the initial paper (Apelt et al., 2001a). This paper isbased on the research and findings presented in a Uni-

versity of Sydney thesis dissertation (Apelt, submitted

for publication).

The inferential models of the SAG mill discharge,

inventories and feed will be described following a review

of related works and the presentation of the relevant

simulation models. The review is divided into mill

charge and discharge measurement and feed size distri-bution measurement. Discussion of the inferential model

results precedes a brief analysis of the inferential model

uncertainties. To conclude, the major findings of this

paper are summarised.

The contribution of this work is the presentation of

inferential models of the SAG mill discharge rate and

size distribution, rock charge, total feed and fresh feed-

rate and size distribution. An inferential model of the

primary underflow split is also presented which allows

definition of the circuit mass balance. The results of an

uncertainty analysis of these models is also presented.

2. Circuit description

The discussion centres on the primary grinding circuit

shown in Fig. 1 which also shows process measurements

relevant to this work. The abbreviations indicate the

available process measurements for mass flowrate

(TPH) (t/h), volumetric flowrate (CMPH) (m3/h), streamdensity (%sols) (% solids w/w), mill powerdraw (kW),

and mill load cell weight LC (t). This example of a

grinding circuit would be considered well instrumented

according to the guidelines defined by Fuenzalida et al.

(1996). The available measurements are as follows:

• SAG mill fresh (stockpile) feed (t/h),

• SAG mill feed water addition (m3/h),

This article is also available online at:

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Minerals Engineering 15 (2002) 1055–1071

*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 )00230-3

mail to: [email protected]

• SAG mill powerdraw (kW),

• SAG mill load cell (t),

• Cyclone feed water addition (m3/h),

• Cyclone feedrate (m3/h),

• Cyclone feed density (% solids w/w),

• Oversize crusher feedrate (t/h).

3. Related works

This section reviews inferential measurements for mill

inventory levels and mill discharge properties and the

direct measurement of mill feed size distribution.

3.1. Mill inventories

This section reviews mill inventory measurement

by measurement-type: powerdraw, weight, conductivity

probe and energy balance, and, sound measurement and

acoustic spectral analysis.

3.1.1. Powerdraw

Several mill powerdraw models have been developed

since the work of Bond (1961). These models are de-

pendent on mill charge and have generally developed

from the refinement of the Bond–Allis Chalmers model

(Moys, 1993; van Nierop and Moys, 1997a; Herbst and

Pate, 1999), or via more detailed characterisation of the

mill charge (JKTech, 1994; Valery Jnr. and Morrell,

1995; Napier-Munn et al., 1996; Valery Jnr., 1998).Powerdraw is a function of mill load (mass and vol-

ume). This characteristic may be exploited to estimate

mill charge levels. Erickson (1989) generated volumetric

ball charge fraction (Jb) curves on a powerdraw-charge

weight grid. On a per mill revolution basis, Koivistoinen

and Miettunen (1989) found that the amplitude of

powerdraw oscillations caused by shell lifter bars entering

the charge was dependent on the total mill charge level.More recently powerdraw models have been utilised

to estimate total charge level (Jt) for a specified ball

charge level (Jb) (Kojovic et al., 2001; Strohmayr and

Valery, 2001). Apelt et al. (2001a) present the novel use

of the Morrell powerdraw model for the simultaneous

estimation of both ball charge and total charge levels.

This method is recapitulated in the inferential model

section (Step 3).

3.1.2. Weight

Mill weight is measured by mill bearing pressure or

strain-gauge load cell. Since the 1980s, when it was

considered unreliable (Mular and Burkert, 1989), bear-

ing pressure measurement has been adopted widely and

is now considered a minimal requirement (Fuenzalida

et al., 1996). Both bearing pressure and load cell mea-

surements are strongly influenced by mill and charge

motion. To compensate, recent bearing pressure modeldevelopment includes the influence of mill charge shape

and mill drive forces (Evans, 2001).

A simple weight model is obtained through linear

regression of the mill weight measurement against the

internal states (inspection/state estimation) of the mill

(Herbst and Pate, 1999). More complex models result

when charge geometry is taken into account. The SAG

mill simulation model section below describes a millweight model based on charge geometry detailed by

Napier-Munn et al. (1996).

Similar to the powerdraw model utilisation, mill

weight models have been utilised to measure the total

charge level (Jt) given the ball charge level (Jb), or, usedin conjunction with mill powerdraw models in state es-

timation contexts to estimate both the total and ball

charge levels (Herbst and Pate, 1999; Apelt et al.,2001b). Apelt et al. (2001b) present the novel use of a

mill weight model for the simultaneous estimation of

both ball charge and total charge. This method is also

recapitulated in the inferential model section (Step 3).

3.1.3. Conductivity probe measurements and energy

balance

Shell lining and lifter bar components are secured tothe inside of the mill shell with bolts. Conductivity

probes fixed within the bolts (usually the longer lifter

bar bolts) can measure conductivity during a mill re-

volution to gain information regarding mill charge.

Conductivity is high within the charge, rises on entry at

the charge toe and falls on exit at the charge shoulder.

Marklund and Oja (1996) utilised conductivity

probes for total charge level measurement. The use ofconductivity probes was the preferred method over

methods utilising bearing back pressure, powerdraw

signal oscillation and lifter bar strain gauges.

Moys and colleagues have worked extensively with

the conductivity probe technique and have applied

and progressed the technology from laboratory-scale

through pilot-scale to industrial scale (Moys, 1985;

Moys, 1988; Moys, 1989; van Nierop and Moys, 1996;van Nierop and Moys, 1997a; van Nierop and Moys,

Fig. 1. Primary grinding circuit.

1056 T.A. Apelt et al. / Minerals Engineering 15 (2002) 1055–1071

1997b). The information obtained from the conductivity

measurement analysis includes total charge level (Jt),charge centrifuging, mill overload, charge angle of re-

pose (a), slurry pooling at the charge toe and charge

shoulder (hS) and toe (hT) angles. Determination of the

total volumetric charge (Jt) from the conductivity mea-

surements requires assumption regarding the chargegeometry like the Barth, Hinsley and Fobelets and

Uggla (BHFU) charge surface model as utilised by

Vermeulen and Schakowski (1988).

3.1.4. Acoustic spectral analysis and sound measurements

The initial use of mill sound measurements and

acoustic spectral analysis as a control variable and an

inferential mill charge level measurement received amixed review (Moys, 1985; Lyon, 1988; Moys, 1988;

Mular and Burkert, 1989). Sound measurement has

since gained wider acceptance and has been utilised in a

feedrate control scheme (Perry and Anderson, 1996)

and the utilisation of filtered acoustic signals for the

measurement of in-mill variables, particularly charge toe

angle, is progressing (Pax, 2001).

The measurement of charge position, motion andcollisions is being pursued using the processing of

surface vibration signals from accelerometers on the

mill shell, sent to a fast data acquisition system and

interpreted with the aid of a discrete element methods

mill model (Spencer et al., 2000; Campbell et al.,

2001).

3.1.5. Summary

Inferential measurement of total mill volumetric

charge (Jt) that utilise mill powerdraw, and weight,

charge conductivity and acoustic emission measure-

ments are already utilised in industry or are in the de-

velopmental phase. These methods are currently unable

to provide a inferential measurement of ball charge level

(Jb).Novel use of mill weight and mill powerdraw for si-

multaneous inferential measurement of total and ball

charge levels, as described by Apelt et al. (2001a), is

restated in the inferential models section. Inferential

measurement of total and ball charge levels is also

possible utilising state estimation methods. State esti-

mation for SAG mills will be discussed in another paper.

3.2. Mill discharge

Measurement of the SAG mill discharge is required

to satisfy the basic grinding circuit control requirement

of determination of the circulating load (Lynch, 1977).In the case where the SAG mill discharge is pumped

directly to another piece of equipment, e.g., a bank of

hydrocyclones, the solids tonnage may be derived from

a flow-density-pulp specific gravity combination of mea-

surements (Wills, 1989).

Inferential measurement of mill discharge is impor-

tant where flow measurement is lacking. In the absence

of flow measurement of SAG mill discharge water ad-

dition, Moys and colleagues have used thermocouples to

define the energy and mass balances around the mill

discharge sump to enable the inferential measurement of

the mill discharge water addition rate, mill dischargedensity and viscosity and the mill charge viscosity

(Moys, 1985; Moys et al., 1987; Van Drunick and Moys,

2001).

Inferential measurement models for the SAG mill

discharge are presented in the inferential models section.

These models provide the discharge stream flow pro-

perties and size distribution and can be utilised for

recirculating load and SAG mill rock charge determi-nation (also described).

3.3. Feed size distribution

Feed size distribution measurement is considered an

important process measurement (Fuenzalida et al., 1996;

Broussaud et al., 2001). Online image analysis is gene-

rally the accepted method for size measurement. The

technology has been developing since the 1980s from

basic one-and two-dimensional methods (Lange, 1988)

to more sophisticated two dimensional methods. Cur-

rent commercially available instruments include:

• OOS: Analyses parallel laser beams contours made

on rock-laden conveyor belts (Fimeri, 1997) and best

able to detect relative size changes on industrial con-

veyor belts (Davies et al., 2000).

• Split/split-online: Analyses digital video images by

either fragment delineation or a circular feature iden-

tification (circle centres and radii) algorithm (Girdneret al., 2001).

• T-VIS: Analyses digital video frames to transform

linear chord length distributions to volumetric distri-

butions (Herbst and Blust, 2000).

• WipFrag: Analyses optical camera images to measure

rock size distribution (Maerz and Palangio, 2000),

providing a high precision measurement useful for

process control (Maerz, 2001).

Texture based image processing is developing to as a

possible alternative to the above to address difficulties

associated with order-of-magnitude particle size ranges,

the presence of mud and water, and concealment of

particles methods (Petersen et al., 1998).

With feed size measurement technology, particularly

pattern recognition methods, reaching a high level ofmaturity, opportunities now exist to manipulate AG/

SAG mill feed size distribution through stockpile feeder

operation and blast pattern selection (Morrell and

Valery, 2001) and through blasting practice optimi-

sation (Sherman, 2001).

T.A. Apelt et al. / Minerals Engineering 15 (2002) 1055–1071 1057

Direct measurement of feed size distribution has

progressed to an advanced stage. However, there is

scope for development of a model-based alternative. A

novel model-based method for the inferential measure-

ment of the feed size distribution which could be utilised

as where capital or installation costs of the pattern and

textural instrumentation are prohibitive is presented inthe inferential models section.

4. SAG mill simulation models

Several simulation models require description prior

to the presentation of the inferential models, namely, the

mill solids balance, liquid balance, powerdraw andweight models. In this work a simulation model is one

which utilises feed stream information to predict pro-

duct stream information whereas an inferential model

either estimates feed stream information from product

stream information or utilises existing process mea-

surements to estimate process characteristics which are

not measured.

4.1. Perfectly mixed mill model

4.1.1. Solids balance

The solids mass balance for the SAG mill is based on

the Whiten perfect mixing model (Whiten, 1974) which is

a special case of the general population balance model

described elsewhere (Austin et al., 1987). On a size by size

basis the solids balance may be stated as follows (ValeryJnr. and Morrell, 1995; Napier-Munn et al., 1996):

Accumulation ¼ In�OutþGeneration

� Consumption

dsidt

¼ fi � pi þXi�1

j¼1

rjsjaij � ð1� aiiÞrisi ð1Þ

Accumulation ¼ 0 at steady state

0 ¼ fi � pi þXi�1

j¼1

rjsjaij � ð1� aiiÞrisi ð2Þ

where si is the mill rock charge particles in size i (t),fi the feedrate of particles in size i (t/h), pi the mill

discharge (product) of particles in size i (t/h), ri the

breakage rate of particles in size i which varies with

operating conditions within the mill (variable rates)(h�1), aij the appearance function of particles appearing

in size i (a function of the breakage distribution of

particles in sizesP size i) (fraction).The mill product (the SAG mill discharge stream,

SMDC) is calculated as follows:

pi ¼ d0cisi ð3Þwhere d0 is the maximum mill discharge rate constant

(h�1) and ci the grate classification function for size i

(fraction), i.e., the probability of a size i particle passingthrough mill discharge grate (refer to the ‘‘original

function’’ in Fig. 3).

4.1.2. Water balance

The water mass balance is as follows:

Accumulation ¼ In�Out

dswdt

¼ fw � pw ð4Þ

where sw is the water in the mill charge (t), fw the feedwater addition (t/h) and pw the water discharge rate (t/h)

which is calculated as follows:

pw ¼ d0sw ð5Þ

4.2. Mill powerdraw model

According to the Morrel powerdraw model (Morrell,

1994), the mill powerdraw, Pgross, is

Pgross ¼ Pno load þ kPcharge ð6Þ

where Pgross is the power input to the mill motor (me-

tered power) (kW), Pno load the no-load power of mill

(empty mill powerdraw) (kW), Pcharge the mill power-

draw attributable to the entire contents of the mill (kW),k the mill powerdraw lumped parameter (accounts for

heat losses due to internal friction, energy of attrition/

abrasion breakage, rotation of the grinding media and

inaccuracies in assumptions and charge shape and

motion measurements (dimensionless).

4.3. Mill weight model

Since the material in the inactive part of the charge is

in �freefall�, the mill weight consists of the mill shell

weight and the weight of the material in the charge

kidney (the active portion of the mill charge that is in

contact with the mill shell), refer to Fig. 2. The weight of

the kidney (mass of active fraction of mill charge),

Mkidney, (t) is the product of the kidney density, qkidney

(t/m3) and volume, Vkidney (m3):

Mkidney ¼ qkidneyVkidney ð7Þ

The volume of the kidney is

Vkidney ¼ pLm r2m�

� r2i� 2p � hT þ hS

2p

� �ð8Þ

The mill charge (kidney) density is determined as follows

(Napier-Munn et al., 1996):

qkidney ¼ qc

¼Jtqo 1� eþ eU S

100

� �þ Jb qb � qoð Þð1� eÞ þ JteU 1� S

100

� �� Jt

ð9Þ


where Jb is the mill fraction occupied by grinding balls

including the associated voidage (fraction mill volume),

S the mill discharge volumetric solids content (% solids

v/v), � the mill charge porosity (fraction), qb the grindingball density (specific gravity) (t/m3), qc the mill charge

density (specific gravity) (t/m3) and qo the ore density

(specific gravity) (t/m3).

5. Inferential models

5.1. Model overview

Inferential measurement of the SAG mill inventories,

feed rate and sizing and mill discharge rate and sizing

requires the development of suitable models. This sec-

tion details the development and utilisation of the new

inferential measurement models. An overview of the

model utilisation and calculation sequence is as follows:

1. The oversize crusher feed (OSCF) and primary cy-

clone feed (PCFD) streams are calculated from the

oversize crusher feedrate, primary cyclone feed flow-

rate and density data and assumptions about the size

distributions (based on SAG mill grate size and dis-

charge screen aperture size). The addition of OSCF

and PCFD less the discharge water addition yields

the SAG mill discharge stream (SMDC). The mill dis-

charge size distribution (smdc) and passing sizes(T80 . . . T20) are calculated in the process.

2. The SAG mill rock charge (SMRC) is calculated by

SAG mill discharge function model inversion.

3. SAG mill fractional total filling (Jt) and ball filling

(Jb) are determined by solving the powerdraw or mill

weight equations given mill power draw or weight

process measurements as inputs.

4. SAG mill total feed (SMTF) is then calculated by millmodel inversion after making assumptions about the

ball charge size distribution.

5. Oversize crusher product (OSCP) and primary cy-

clone underflow (PCUF) are calculated by the direct

application of the crusher and cyclone models.

6. SAG mill fresh feed (SMFF) is calculated by sub-

tracting oversize crusher product (OSCP) and the

primary cyclone underflow to SAG mill (PCUS)from the SAG mill total feed (SMTF) stream. The

fresh feed size distribution (smff) and passing sizes

(F80 . . . F20) are calculated in the process.

5.2. Step 1: oversize crusher feed, primary cyclone feed

and SAG mill discharge

5.2.1. Oversize crusher feed, OSCF/oscf

The oversize crusher feed mass flowrates and stream

properties (OSCF) are calculated as follows:

OSCFtph s ¼ MVscatsOSCF%s w=w ð10Þ

OSCFtph l ¼ MVscats

ð100�OSCF%s w=wÞ100

ð11Þ

where MVscats is the oversize crusher total feedrate

measured variable (t/h), OSCFtph s the oversize crusher

solids feedrate (t/h), OSCFtph l the oversize crusher liq-

uid feedrate (t/h) and OSCF%s w=w the oversize crusher

feed solids density (%solids w/w).The Rosin–Rammler size distribution function is

given in Eq. (12) and has been selected for its conve-

nience and since it ‘‘has been found to fit many size

distributions very well’’ (Napier-Munn et al., 1996)

Wr ¼ 100 exp

�� x

a

� �b�ð12Þ

where Wr is the cummulative weight percent of material

retained at size x (cummulative % retained w/w), x the

particle size (mm), a the size at which 36.8% (i.e., 100/e)of particles are retained (mm) and b the slope of

lnðlnð100=WrÞÞ versuslnx) plot.Since it is conventional in mineral processing to

represent size distributions in cummulative percent

Fig. 3. Grate classification function.

Fig. 2. Simplified mill charge geometry.


passing format, Eq. (16) is more useful in such a format,

as given in Eq. (17). The values of a and b for oscf are

estimated.

Wp ¼ 100� 100 exp

�� x

a

� �b�ð13Þ

where Wp is the cummulative weight percent of material

passing size x (cummulative % passing w/w).

5.2.2. Primary cyclone feed, PCFD/pcfd

The primary cyclone feed properties (PCFD) are

calculated from the plant measured variables of cyclone

feed flowrate, PCFDm3ph p (MVpc flow) (m3/h), and feed

solids density, PCFD%s w=w (MVpc dens) (%solids w/w). A

mass balance yields,

PCFDtph s ¼PCFDm3ph pPCFD%s w=wSGlSGs

PCFD%s w=wSGl þ ð100� PCFD%s w=wÞSGs

ð14Þ

PCFDtph l ¼ PCFDtph s

ð100� PCFD%s w=wÞPCFD%s w=w

ð15Þ

Eq. (18) may also be derived from a flow-density-pulp

specific gravity combination of measurements (Wills,

1989). The primary cyclone feed size distribution (pcfd)

is estimated in a similar manner as the oversize crusher

feed stream. That is, pcfd is approximated by a Rosin–Rammler distribution, see Eq. (17), with estimated va-

lues of a and b.

5.2.3. SAG mill discharge, SMDC/smdc

The SAG mill discharge properties (SMDC) and size

distribution (smdc) are estimated by the addition of the

estimated primary cyclone feed and oversize crusher

feed streams less the SAG mill discharge water flowrate.

SMDCtph s ¼ OSCFtph s þ PCFDtph s ð16ÞSMDCtph l ¼ OSCFtph l þ PCFDtph l �MVDC H2OSGl

ð17Þ

smdc ¼ OSCFtph s

SMDCtph s

oscf þ PCFDtph s

SMDCtph s

pcfd ð18Þ

where SMDCtph s is the SAG mill solids discharge rate

(t/h), SMDCtph l the SAG mill liquid discharge rate (t/h),

smdc the SAG mill discharge size distribution (%re-

tained w/w), MVDC H2O the SAG mill discharge water

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

gravity (t/m3).Potential now exists to utilise the SAG mill discharge

inferential measurement as a measure of SAG mill per-

formance. Control objectives and strategies could be

formulated centering on this inferential measurement.

For example, the SAG mill discharge size distribution

measurement could be utilised as a measure of the re-

lative loading of the primary and secondary grinding

circuits. A coarse size measurement could indicate that

the primary circuit is highly loaded. This indication

could provide feedback to the SAG mill feedrate control

loop.

5.2.4. Recirculating load, RCL

One of the ‘‘basic requirements’’ of a grinding circuit

control system is the ‘‘measurement of the circulatingload, so that overload may be prevented’’ (Lynch, 1977).

For a closed-loop mill-cyclone arrangement, the circu-

lating load is generally defined as the ratio of the solids

mass flow in the cyclone underflow to the solids feed to

the mill (Wills, 1989).

In this case, where there are two recycle streams

(oversize crusher feed, OSCF, and a proportion of

the primary cyclone underflow, ð1� PCsplitÞ PCUF), theamount of solid material recirculating is the difference

between the mill discharge and the fresh feed. Therefore,

the recirculating load, RCL (%), is

RCL ¼ ðSMDCtph s � SMFFtph sÞ100%SMFFtph s

ð19Þ

Eq. (23) can be solved utilising the SAG mill discharge

solids flow, SMDCtph s, from Eq. (20) and the SAG mill

fresh feed solids flow, SMFFtph s, calculated in Step 6.

Potential now also exists to utilise the recirculating

load inferential measurement as a measure of SAG mill

performance. Control objectives and strategies could be

also formulated incorporating this measurement. Forinstance, the recirculating load measurement (RCL)

could be utilised as feedback for control of SAG mill

water addition or primary cyclone underflow split to the

ball mill (PCsplit).

5.3. Step 2: SAG mill rock charge

The SAG mill rock charge properties (SMRC) andsize distribution (smrc) are estimated by the reverse-

application of the SAG mill grate discharge function on

the SAG mill discharge stream estimate, incorporating a

size distribution assumption.

5.3.1. Solids

The mill discharge flowrate Q (m3/h) for a grate dis-

charge mill is equal to the product of the flowratethrough the mill charge, Qm (m3/h), and kg, a scaling

factor that accounts for coarse particles (Napier-Munn

et al., 1996):

Q ¼ kgQm ð20ÞQm ¼ 6100J 2pmc2:5A/�1:38D0:5 ð21Þ

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

mill inside diameter (m), c the mean relative radial po-sition of open area (fraction) and / the fraction critical

mill speed (fraction). Knowledge of the mill discharge

flowrate allows the calculation of the nett fractional


holdup of slurry in the mill charge interstices Jpm (frac-

tion),

Jpm ¼ Qm

6100c2:5A/�1:38D0:5

� �0:5

ð22Þ

Recognising that the mill volumetric discharge, Qm, is in

fact the mill product which consists of water and water-

like solids (solids of size < xm) pxm, i.e.,

pxm ¼ Qm ð23Þ

allows the calculation of an initial estimate of themaximum discharge rate constant (d0) and the volume

of water-like solids in the mill charge, sxm (solids of

size < xm) from

dxm ¼ pxmsxm

ð24Þ

where dxm is the maximum discharge rate constant (d0)(h�1) and the volume of water-like solids in the millcharge, sxm (m3):

sxm ¼ JpgpD2

m

4Lm ð25Þ

where Lm is the mill (inside) length (m), Dm the mill

(inside) diameter (m) and Jpg the gross fractional holdupof slurry in the mill (fraction):

Jpg ¼ Jpm þ Jpo ð26Þwhere Jpo is the nett fractional slurry holdup in mill

�dead� zone (fraction), i.e., the fraction of the mill vol-

ume outside the outermost grate apertures:

Jpo ¼ 0:33ð1� rnÞ ð27Þwhere rn is the relative radial position of outermost grate

apertures (fraction).

The calculation of mill rock charge, si (SMRC/smrc)is possible using the maximum discharge rate, d0, themill product, pi (SMDC/smdc), and a simplified version

of the classification function, ci, detailed by Napier-

Munn et al. (1996), described by Eq. (28) and illustrated

in Fig. 3 along with: the original function (used for

process simulation).

ci ¼ 0 for xP xg

ci ¼xg � xxg � xm

for xm < x < xg

ci ¼ 1 for x6 xm

ð28Þ

The mill rock charge, si (SMRC/smrc), is then calculated

by manipulation of Eq. (7), i.e.,

si ¼pid0ci

ð29Þ

Eq. (29) provides no information about the material in

the rock charge larger than the grate aperture size (xg).This proportion of the rock charge may be estimated by

assuming that it can be approximated by a Rosin–

Rammler distribution and then solving the following

system of q cumulative weight retained equations:Pqj¼1 sjPq

i¼1 si þPz

i¼qþ1 si100 ¼ 100 exp

�� x

a

� �b�ð30Þ

where x is particle size (mm), i the particle size class

(i ¼ 1 the largest particle size, i ¼ q the smallest rock

size (16 mm) and i ¼ z the smallest particle size) and aand b are the Rosin–Rammler distribution parameters.The second summation in the denominator of Eq. (30),

the summation of the material less than 16mm in size, is

determined from interpolating the rock charge infor-

mation obtained from Eq. (33).

The values of a and b for smrc are estimated con-

sidering the coarse end of the distribution is bounded by

the point (180.76 mm, 100% Passing) and the total

charge (Jt) and ball charge (Jb) level estimates from thenext step are determined independent of this step and

provide information on the amount of material in the

rock charge larger than 50 mm, Jr (fraction), i.e.,

Jr ¼ Jt � Jb ð31Þ

5.3.2. Water

The mill water charge, sw (t) may be calculated by

manipulation of Eq. (9), i.e.,

sw ¼ pwd0

ð32Þ

where pw is the SAG mill discharge water mass flowrate

(t/h).

5.4. Step 3: total charge and ball charge filling levels

The fractional total filling (Jt) and ball filling (Jb) areestimated independently solving the mill powerdraw andmill weight equations. There is considerable overlap of

the equations utilised, however, two independent esti-

mates of the mill inventories result. Both estimates

consist of one residual equation in two unknowns (Jtand Jb). The calculation involved for each of these es-

timates will now be detailed.

5.4.1. Estimates from powerdraw

Analysis of Eq. (10) and its components yields that

given mill discharge, mill specifications, mill model

parameters and measured mill powerdraw, the mill

powerdraw model can be reduced to one function of two

unknowns, i.e., volumetric ball charge fraction (Jt) andtotal charge volumetric fraction (Jb) as described else-

where (Apelt et al., 2001a).

The mill inventory estimates from mill powerdrawdata are determined by the solution of Eq. (33) which

determines values for the total charge level (Jt) and ball

charge level (Jb) such that the calculated mill power-

draw, Pgross, equates with the actual mill powerdraw


measurement MVkW (kW), and therefore satisfying the

residual equality, FkW (kW):

FkW ¼ MVkW � Pgross ¼ 0 ð33Þ

5.4.2. Estimates from mill weight

A second residual, similar to Eq. (33), can be ob-

tained by the utilisation of the equations that describe

the mill weight. The mill inventory estimates from mill

weight data are determined by the solution of Eq. (34)

which determines values for total charge (Jt) and ballcharge (Jb) levels that equate the calculated mill weight

(Mshell þMkidney) (t) with the actual mill weight mea-

surement MVweight (t) and therefore satisfying the mill

weight residual equality, Fweight (t):

Fweight ¼ MVweight �Mshell �Mkidney ¼ 0 ð34ÞIn the inferential model uncertainty analysis section the

uncertainty in the mill inventory estimates from mill

powerdraw and mill weight is discussed. Potential now

exists to utilise the SAG mill volumetric charge infer-

ential measurements as a measure of SAG mill perfor-

mance and an indication of the prevailing conditionswithin the mill. Control objectives and strategies could

be formulated incorporating these measurements. For

example, the mill ball charging rates could be mani-

pulated to control the ball charge according to the in-

ferential ball charge measurement. Also, the difference

between the total and ball charge measurements, Jr (seeEq. (31)) could be utilised in a mill feedrate control loop,

e.g., reduce feedrate for high rock charge levels to pre-vent mill overload and increase feedrate for low rock

charge levels to prevent damage to mill internals.

5.5. Step 4: SAG mill total feed

5.5.1. Solids balance

The solids component of the SAG mill total feed

(SMTF/smtf) is estimated by the inversion of the steadystate perfectly mixed mill model, Eq. (6). Rearranging

terms yields,

fi ¼ pi �Xi�1

j¼1

rjsjaij þ ð1� aiiÞrisi ð35Þ

The estimate of total feed to the SAG mill, SMTF/smtf,

is determined as follows:

(a) the mill product, pi, and mill rock charge, si, areknown from Steps 1 and 2, respectively.

(b) assuming a ball charge size distribution and using

the rock charge and ball charge information from

Steps 2 and 3, allows the determination of the spe-

cific comminution energy, Ecsi, the breakage para-

meter, t10i, and the appearance function, aij.(c) the breakage rate function, ri, is determined from the

ball charge information from Step 3 and the estimate

of the recycle ratio of �20þ 4 mm material and

fresh feed eighty percent passing size (F80) from the

previous time step (JKTech, 1994).

(d) using the information from (a) to (c), the total feed

estimate, fi is determined by solving Eq. (35).

The use of the steady state perfectly mixed mill modelis valid since the mill charge and discharge estimates are

determined from the prevailing operating conditions

regardless of whether the mill contents are increasing,

decreasing or at steady-state. A valid estimate of the

total mill feed is possible providing the calculation time

between the discharge and charge estimates and the total

mill feed estimate is relatively short.

5.5.2. Water balance

SAG mill total feed water is determined by the steady

state balance for the water:

Water in ¼ Water out

fw ¼ pw ð36ÞThe water entering the mill fw (t/h) is equal to the waterin the SAG mill discharge stream pw (t/h), determined in

Step 1.

5.6. Step 5: oversize crusher product and primary cyclone

underflow

The estimate of the oversize crusher product, OSCP/

oscp, is determined by applying the crusher model, Eq.(37), to the estimate of the oversize crusher feed, OSCF/

oscf, determined in Step 1. The oversize crusher product

is then determined as follows:

p ¼ ð1� CÞð1� BCÞ�1f ð37Þwhere p is the crusher product by size (t/h) (OSCP/oscp),f the crusher feed by size (t/h) (OSCF/oscf), B the

crusher breakage distribution function (fraction) and Cthe crusher probability of breakage function (fraction).

The Nageswararao model (Napier-Munn et al.,

1996), is used to model the primary cyclones. The model

is comprised of several equations that predict cyclone

operating pressure (P ), corrected fifty percent passing

size (d50c), water recovery to underflow (Rf ) and feedslurry recovery to underflow (Rv). These equations are

functions of cyclone geometry, feed flowrate and solids

density, and, feed ore characteristics. The size classifi-

cation function is described by the efficiency to overflow,

Eoa (fraction), equation:

Eoa ¼ Cð1þ bb�xÞðexpðaÞ � 1Þexpðab�xÞ þ expðaÞ � 2

� �ð38Þ

where C is the water recovery to cyclone overflow

(fraction), x the ratio of particle size to corrected 50%

passing size (d=d50c) (dimensionless), d the particle size

(diameter) (mm), a the efficiency curve separation


sharpness parameter (dimensionless), b the efficiency

curve fine size efficiency boost parameter (dimensionless)

and b� the efficiency curve d50c preservation parameter

(dimensionless).

The estimate of the primary cyclone underflow,

PCUF/pcuf, is determined by applying the cyclone

model to the estimate of the primary cyclone feed,PCFD/pcfd, determined in Step 1.

5.7. Step 6: SAG mill fresh feed

The estimate of the SAG mill fresh feed (the new feed

from the stockpile), SMFF/smff, is determined by sub-

tracting (from the SAG mill total feed SMTF/smtf) the

SAG feed water addition MVFD H2O (m3/h), the estimateof the oversize crusher product, OSCP/oscp, and the

recycled component (1� PCsplit) (fraction) of the pri-

mary cyclone underflow, PCUF/pcuf:

SMFFtph s ¼ SMTFtph s �OSCPtph s

� ð1� PCsplitÞPCUFtph s ð39ÞSMFFtph l ¼ SMTFtph l �OSCPtph l

� ð1� PCsplitÞPCUFtph l �MVFD H2OSGl

ð40Þsmff

¼ SMTFtph ssmtf �OSCPtph soscp� ð1� PCsplitÞPCUFtph spcuf

SMTFtph s �OSCPtph s � ð1� PCsplitÞPCUFtph s

ð41Þwhere SMFFtph s is the SAG mill fresh feed solids fee-

drate (t/h), SMFFtph l the SAG mill fresh feed liquid

feedrate (t/h), smff the SAG mill fresh feed size distri-

bution (%retained w/w), SMTFtph s the SAG mill total

feed solids feedrate (t/h), SMTFtph l the SAG mill total

feed liquid feedrate (t/h), smtf the SAG mill total feedsize distribution (%retained w/w), MVFD H2O the SAG

mill feed addition water process measurement (m3/h),

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

(1� PCsplit) the recycled split-fraction of the primary

cyclone underflow (fraction).

5.7.1. Fresh feed passing sizes

The estimate of the SAG mill fresh feed eighty per-cent passing size, F80, is determined by interpolation of

the estimate of the size distribution, smff, at the 80%

mark. Similarly, the sixty, forty and twenty percent

passing sizes (F60, F40 and F20, respectively) can be de-

termined.

Potential now exists to utilise the SAG mill fresh feed

size inferential measurements as a measure of crusher

or blasting performance and of feed size disturbancesreporting to the mill. The fresh feed solids inferential

measurement could be utilised for metallurgical ac-

counting purposes. Control objectives and strategies

could also be formulated incorporating these measure-

ments. For instance, the feed size measurements could

be utilised in feed-forward control of mill feedrate where

coarse feed results in feedrate reduction and and fine

feed leads to increases in throughput.

5.8. Primary cyclone underflow split to the ball mill

The primary cyclone underflow split to the ball mill,

PCsplit, was considered as one of the parameters in the

preceding discussion. It is also possible to construct an

inferential model of the split. In a discrete time frame, a

combination of:

• present estimates of the SAG mill discharge, oversize

crusher feed and primary cyclone feed streams,• the primary cyclone underflow split to the ball mill

from the preceding time step, and

• models of the oversize crusher and primary cyclone,

allows the construction of a mass balance to estimate the

current primary cyclone underflow split to ball mill.

PCUStph s k is the solids component of the primary cy-

clone underflow returning to the SAG mill (t/h) andPCsplit k is the primary cyclone underflow split reporting

to the ball mill (fraction) are determined by Eqs. (42)

and (43), respectively.

PCUStph s k ¼ SMDCtph s k �OSCPtph s k � SMFFtph s k

ð42Þ

PCsplit k ¼ 1� PCUStph s k

PCUFtph s kð43Þ

where SMDCtph s k is the solids component of the SAG

mill discharge (t/h),OSCPtph s k the solids component of

the oversize crusher product (t/h), SMFFtph s k the solids

component of the SAG mill fresh feed (t/h), PCUFtph s k

is the solids component of the primary cyclone under-

flow (t/h) and k and k � 1 are the present and previous

time steps, respectively.The primary cyclone underflow solids component,

PCUFtph s k (t/h), is obtained by the application of the

primary cyclone model to the primary cyclone feed,

PCFDk stream. The primary cyclone underflow solids

component reporting to the SAG mill, PCUStph s k, is

obtained from Eq. (42), with the SAG mill fresh solids

federate obtained from Eq. (39) utilising the cyclone

underflow split from the previous time step PCsplit k�1.The oversize crusher product solids component

OSCPtph s k (t/h) is generated by the application of the

oversize crusher model to the oversize crusher feed es-

timate OSCFk.

The estimation of the primary cyclone underflow split

to the ball mill, PCsplit, is important since it allows the

primary grinding circuit mass balance to be fully defined.

The full mass balance definition enhances the awarenessof and the ability to optimise operating conditions.


5.8.1. Summary

Inferential measurements have been developed for

the SAG mill inventories, feed rate and sizing and mill

discharge rate and sizing. The inferential models utilise

process measurements and process models to obtain

novel inferential measurements of stream and inventory

characteristics. For example, the oversize crusher feed,primary cyclone feed and mill discharge measurements

are utilised to obtain an inferential measurement of the

SAG mill discharge.

6. Results and discussion

The inferential model results were assessed by com-parison to the results of a process simulation model. The

simulation used Eqs. (1)–(9) and utilised inputs and

variables such as SAG mill fresh feedrate and size dis-

tribution. The process simulation model therefore acted

as a proxy for the real grinding circuit and generated

proxy data for the inferential models, Eqs. (10)–(43), as

if from real grinding circuit instruments. It was the task

of the inferential model to use those proxy measure-ments to infer the values of the process inputs and

conditions.

Figs. 4 and 5 contain the size distribution esti-

mate results of the inferential models for the mill

streams and the crusher and cyclone streams, respec-

tively.

Fig. 4: The upper left panel of Fig. 4 shows the SAG

mill discharge size distribution for the simulated stream,

Eq. (3), and as calculated by the inferential discharge

model given in Eq. (18). The upper right panel shows the

SAG mill rock charge size distribution for the simulatedmill, Eq. (1), and as calculated by the inferential rock

charge model, Eqs. (29) and (30). The panel on the lower

left shows the SAG mill total feed size distribution for

the simulated stream, specified fresh feed stream plus the

oversize crusher product, and as calculated by the in-

ferential total feed model given in Eq. (35). The panel at

lower right of Fig. 4 shows the SAG mill fresh feed size

distribution for the specified feed stream and as calcu-lated by the inferential fresh feed model, Eq. (41).

Fig. 5: The upper left panel of Fig. 5 shows the

oversize crusher feed size distribution for the simulated

stream and as calculated by the inferential crusher feed

model given in Eq. (13). The upper right panel shows the

primary cyclone feed size distribution for the simulated

stream and as calculated by the inferential cyclone feed

model, also described by Eq. (13). The panel on thelower left shows the oversize crusher product size dis-

tribution obtained by the application of the oversize

crusher model, Eq. (37) to the simulated and the infer-

entially modelled oversize crusher feed streams. The

Fig. 4. Size distribution estimates––mill streams. For each plot, the horizontal axis is the logarithm of particle size (in mm) and the vertical axis is

cumulative weight percent passing.


panel at lower right of Fig. 5 shows the primary cyclone

underflow size distribution obtained by the application

of the Nageswararao cyclone model (Napier-Munn

et al., 1996), including Eq. (38), to the simulated and

inferentially modelled cyclone feed streams.

Tables 1 and 2 contain the stream property estimateresults of the inferential models for the mill streams and

the crusher and cyclone streams, respectively. The eighty

percent passing size (P80) estimates are result of inter-

polating the information displayed in Figs. 4 and 5 at

the 80% passing size.

Table 1: The solids flow and density results in Table 1

correspond to the simulated plant information and the

inferential models of the SAG mill discharge, Eqs. (16)

and (17), SAG mill rock charge, Eqs. (29) and (30), SAGmill total feed, Eqs. (35) and (36), and SAG mill fresh

feed, Eqs. (39) and (40).

Table 2: The solids flow and density results in Table 2

correspond to the simulated plant information and the

inferential models of the oversize crusher feed, Eqs. (10)

and (11), primary cyclone feed, Eqs. (14) and (15), the

application of the oversize crusher model to the crusher

feed streams and the application of the Nageswararaocyclone model.

Fig. 5. Size distribution estimates––crusher and cyclone streams. For each plot, the horizontal axis is the logarithm of particle size (in mm) and the

vertical axis is cumulative weight percent passing.

Table 1

Stream property estimates––mill streams

Stream SMDC SMRC SMTF SMFF

Solids flow (t/h)

Simulation 252.10 45.67 252.20 185.00

Inferential 252.09 47.99 252.07 184.88

Error (%) 0 5 0 0

Density (%sols w/w)

Simulation 75.91 95.69 75.90 97.98

Inferential 75.93 94.75 75.93 98.06

Error (%) 0 1 0 0

P80 Size (mm)Simulation 16.36 97.33 84.01 94.55

Inferential 16.88 144.25 61.24 70.75

Error (%) 3 48 27 25

Table 2

Stream property estimates––crusher and cyclone streams

Stream OSCF PCFD OSCP PCUF

Solids flow (t/h)

Simulation 67.12 184.97 67.12 150.69

Inferential 67.11 184.97 67.11 146.10

Error (%) 0 0 0 3

Density (%sols w/w)

Simulation 99.92 50.80 99.92 70.74

Inferential 99.90 50.80 99.90 70.10

Error (%) 0 0 0 1

P80 Size (mm)Simulation 28.17 5.16 27.24 6.17

Inferential 43.47 2.55 37.71 3.27

Error (%) 54 51 38 47


The mass balance components of the inferential

model estimates (Solids Flow and Density) in Tables 1

and 2 show very good agreement with the reference in-

formation (Simulation) for all streams. The estimates

are between 95% and 100% accurate (0–5% error) and

thus are considered acceptable.

The size distribution component of the inferentialmodel estimates (P80 Size) in Tables 1 and 2 exhibit a

wider range of agreement with the reference information

(Simulation). The accuracy of the size distribution esti-

mates ranges between 46% and 97% (3–54% error).

The SAG mill discharge (SMDC) estimate displays

good agreement. The SAG mill rock charge (SMRC)

estimate, however, exhibits high relative error. This is

attributed to the extrapolating nature of the SAG millrock load inferential model (Step 2). Analysis of the

model found that the relative fraction pebble point open

area (fp) (i.e., the position of the transition point (xg; fp)in Fig. 3) is highly influential on the rock charge esti-

mate and the SAG mill feed estimates. Despite the large

errors in the rock charge size distribution, estimates are

obtained for the mill total feed (SMTF) and fresh feed

(SMFF) with less than 30% error and thus were con-sidered acceptable.

The size distribution estimates for the oversize

crusher feed (OSCF) and product (OSCP) and the pri-

mary cyclone feed (PCFD) and underflow (PCUF) ex-

hibit high relative errors (38–54%). Since the estimates

of the SAG mill discharge (SMDC) and fresh feed

(SMFF) are not unduly affected, the oversize crusher

and primary cyclone stream size estimates are consi-dered acceptable. Furthermore, the majority of model

parameters were fitted manually which was consistent

with the research objectives.

Furthermore, the P80 measure is an attempt at a single

point measure of a full size distribution. Relative

movement in the P80 measurement over time is the most

important consideration rather than the absolute value

of the measurement itself (Davies et al., 2000).

6.1. Model validation

The inferential models presented in this paper have

been validated against a single set of reference steady

state conditions. Therefore, further model validation

would be required prior to entering an implementation

phase.

7. Inferential model uncertainty analysis

The inferential measurement of total charge (Jt) andball charge (Jb) filling levels (Step 3), differs from the

other inferential modelling steps in that process mea-

surements and measurement models are utilised in the

estimation process. The other steps utilise process mea-

surements and process models for estimation purposes.

Table 3 contains the estimates obtained for the total

charge (Jt) and ball charge (Jb) filling levels (Step 3) andare presented in more detail elsewhere (Apelt et al.,

2001a).

The inferential model estimates in column (3) show

good agreement with the bracketed reference infor-

mation in column (1) as evidenced by the percentage

error results in column (4). Apelt et al. (2001a) also

conducted uncertainty analysis on the estimates since

the powerdraw residual, Eq. (33) and the weight resid-ual, Eq. (34), exhibited a range of possible feasible so-

lutions. The uncertainty analysis was conducted by the

application of Eq. (44), the general formula for error

propagation (Taylor, 1982), to the mill powerdraw and

weight residuals.

df ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXNi¼1

ofohi

dhi

� �2

vuut ð44Þ

where f is a general function, hi is the ith model para-meter in the error/sensitivity analysis, df uncertainty in

function f , dhi the uncertainty in ith parameter and Nthe number of parameters.

Column (5) and (6) of Table 3 contains the results of

the uncertainty analysis. The relative uncertainty (with

respect to the reference simulation information) levels

were found to be generally acceptable (�30%) except for

the case of the total charge level (Jt) estimate from thepowerdraw residual, Eq. (33), where there is a high level

of relative uncertainty (�180%). Consequently the utili-

sation of the mill weight residual, Eq. (34), for charge

level estimation was recommended.

Table 3

Mill charge level estimates

Inferential measurement (simulation) Model (Eq.) Estimate (vol frac) Error (%) Uncertainty

Absolute (vol. frac) Relative (%)

(1) (2) (3) (4) (5) (6)

Jt (0.2298) FkW (33) 0.2297 0.04 0.42 181

Fweight (34) 0.2328 1.3 0.06 26

Jb (0.142) FkW (33) 0.1328 6.5 0.04 33

Fweight (34) 0.1446 1.8 0.04 26


The uncertainty results were also analysed in terms of

the relative contribution to the estimate error (RCE):

RCEi ¼

ofohi

dhi

� �2

XNj¼1

ofohj

dhj

� �2100% ¼

ofohi

dhi

� �2

df 2100%

ð45Þwhere RCEi is the relative contribution to the estimate

error of the ith parameter, hi.The relative contribution to estimate error analysis

showed that for the the mill weight residual, Eq. (34),

the most influential parameters to the estimate uncer-tainties are the mill weight process measurement,

MVweight (t), the mill shell weight, Mshell (t), the mill in-

side length, Lmill (m) and the mill inside diameter, Dmill

(m). These findings highlighted the value of improved

mill weight measurement and suggested the inclusion of

a mill liner weight model where a dynamic mill weight

model is utilised.

For the mill powerdraw residual, Eq. (33), the mostinfluential parameters to the estimate uncertainties are

the powerdraw lumped parameter, k (dimensionless),

the mill inside diameter, Dmill (m), the mill powerdraw

process measurement, MVkW (kW) and the mill inside

length, Lmill (m). These findings suggest careful fitting of

model parameter k, utilisation of best available power-

draw measurement, and regular measurement of the mill

internal dimensions.Apelt (submitted for publication) found that the high

level of uncertainty (181%) in the total charge estimate

(Jt) from the powerdraw residual, Eq. (33), is possibly an

overstatement due to the presence of compensating er-

rors which may be occur when a variable (e.g., Jt) occursmore that once in an equation (Taylor, 1982). However,

other factors, including equation non-linearity and the

concave shape of the powerdraw curve, reinforce the highlevel of uncertainty and favouring the utilisation of the

mill weight residual, Eq. (34), for charge level estimation.

7.1. SAG mill discharge

Table 4 contains the size indication results and rela-

tive errors for the SAG mill discharge. The error levels

increase with decreasing particle size. This trend is ex-

pected since the inferential model predicts a size distri-

bution that is more coarse and narrower than the

simulation data, refer to Fig. 4. The additive charac-

teristic of the cummulative passing format of the size

distribution also contributes to the error level trend.

Eqs. (44) and (45) were also applied to the mill dis-charge and fresh feed estimates to highlight the influ-

ential parameters to the uncertainty in those estimates.

Referring to Table 5, the influence of parameters varies

depending on the SAG mill discharge size estimate

under scrutiny. This is due to the relative importance

that the oversize crusher feed (coarse stream) and pri-

mary cyclone feed (fine stream) estimates play in the

respective discharge size estimates. Therefore, the coarsemill discharge estimates (T80 and T60) are influenced most

by parameters utilised in the oversize crusher feed esti-

mate and the fine mill discharge estimates (T40 and T20)are influenced most by parameters utilised in the pri-

mary cyclone feed estimate.

The estimates for mill discharge solids flowrate and

density, see Table 1, exhibited excellent agreement with

the simulation data. An uncertainty analysis of the rateand density was integral to the analysis of the size esti-

mates. Referring to last column of Table 5, the para-

meters that most strongly influence the uncertainty in

the SAG mill solids discharge rate and density are three

of the parameters that affect the uncertainty in the size

indicators with the addition of the primary cyclone feed

water addition flowrate. The inferential model of the

SAG mill discharge provides satisfactory indication ofdischarge rate, density and size distribution. The results

show that to ensure minimum uncertainty in the infe-

rential mill discharge size indicators, SAG mill solids

discharge rate and density:

Table 4

Mill discharge size estimate summary

SMDC size

indicator

Passing size

Simulation (mm) Inferential (mm) Error (%)

T80 16.36 16.88 3

T60 5.64 6.43 14

T40 1.43 2.16 51

T20 0.18 0.44 143

Table 5

Influential parameters: SAG mill discharge estimates

T80 T60 T40 T20 Rate and density

OSCFtph p OSCFtph p PCFDm3ph p PCFDm3ph p PCFDm3ph p

PCFDm3ph p PCFD%s w=w PCFD%s w=w PCFD%s w=w PCFD%s w=w

PCFD%s w=w SMD50c SMD50c SGl PCFWm3ph l

SGl SGl

OSCFtph p is the oversize crusher feedrate (t/h), PCFDm3ph p the primary cyclone feed flowrate (m3/h), PCFD%s w=w the primary cyclone feed density

(%sols w/w), SMD50c the SAG mill discharge screen 50 passing size (mm), SGl the process water specific gravity (t/m3) and PCFWm3ph l the primary

cyclone feed water addition flowrate (m3/h).


• the accuracy of the oversize crusher feed rate,

OSCFtph p (MVscats) should be checked periodically,

e.g., by calibration checks and belt-cuts.

• calibration checks of the primary cyclone feed density

gauge measurement, PCFD%s w=w, and flow meter,

PCFDm3ph p, and primary cyclone feed water addition

flowrate, PCFWm3ph l, should be conducted regularlyto ensure accuracy.

• the SAG mill discharge screen aperture size should

be monitored and measured regularly to ensure the

corrected fifty percent passing size, SMD50c, may

be adjusted as required.

• the specific gravity of the process water, SGl, should

be checked periodically.

7.2. SAG mill fresh feed

Table 6 contains the size indication results and rela-

tive errors for the SAG mill fresh feed. The error levels

increase with decreasing particle size. This trend is ex-

pected since the inferential model predicts a size distri-

bution that is more fine and broader than the simulation

data, refer to Fig. 4. The additive characteristic of thecummulative passing format of the size distribution also

contributes to the error level trend.

The parameters that have the most influence on the

uncertainty in the inferential SAG mill fresh feed size

distribution measurement are the oversize crusher fee-

drate, OSCFtph p, the pebble port diameter, xp, the pri-

mary cyclone feed density, PCFD%s w=w and the pebble

port relative open area, fp.The estimates for mill fresh feed solids flowrate and

density exhibited excellent agreement with the simula-

tion data, see Table 1. An uncertainty analysis of the

rate and density was integral to the analysis of the size

estimates. The parameters that most strongly influence

the uncertainty of the SAG mill fresh feed solids rate

and density are exactly those that strongly influence the

SAG mill fresh feed size indicators, listed above.Table 6 also contains the results of the plant

evaluation of the commercially available system––OOS

(Davies et al., 2000). The range of error for exhibited by

the OOS is 8–34%––calculated from data presented in

the Davies et al. (2000) paper. For the inferential size

indication the error range is 25–49%. These error levels

are comparable especially considering the particle size

range. The feed ore studied in this research is finer than

that evaluated with the OOS. Consequently, the errors

are comparatively larger.

The SAG mill fresh feed rate (solids and moisture) is

one of the measured variables available on the plant. As

a result it could be perceived that the solids feedrate

estimate has a lesser importance than the other estimatespresented here (especially considering the dry nature of

the fresh feed stream). However, it is an important es-

timate since production targets are set on a (dry) solids

throughput rate basis, and, it is an integral component

of the fresh feed density estimate which is required to

determine the (dry) solids production targets and, at an

operational level, is a characteristic for which indication

is desirable.The inferential model of the SAG mill fresh feed

provides satisfactory indication of fresh feed rate, den-

sity and size distribution. The results show that to

minimise uncertainty in the inferential feed size indica-

tors and the feed solids rate and density estimates:

• the accuracy of the oversize crusher feed rate,

OSCFtph p, should be checked regularly, e.g., by cali-bration checks and belt-cuts.

• due to the implicit importance of the oversize crusher

product size (via OSCFtph p), the accuracy of the

crusher gap setting should be checked regularly,

e.g., by dipping the crusher with a lead bob.

• calibration checks of the primary cyclone feed density

gauge measurement, PCFD%s w=w, should be con-

ducted regularly to ensure accuracy.• the SAG mill discharge grate parameters, xp and fp,

should be fitted with due care.

8. Conclusions and recommendations

Inferential measurement models for SAG mill dis-

charge and feed streams and inventories have been

presented and validated against reference simulation

model data. Satisfactory results were achieved and

augmented by uncertainty analyses that highlighted in-

fluential model parameters. Further model validationwould be required prior to implementation. An infe-

rential model of the primary cyclone underflow split to

the ball mill, PCsplit, was also introduced which al-

lows the full definition of the primary grinding mass

balance. The major findings of this work are as follows:

8.1. Charge level

Good ball (Jb) and total (Jt) charge level estimates

can be obtained from residuals based on mill weight

or powerdraw measurements. The estimates agreed

with the reference simulation model data to within 7%

amongst the estimates, the uncertainty contained within

Table 6

Feed size estimate summary

SMFF size

indictator

Inferential model OOS system

Simulation

(mm)

Inferential

(%)

Error

(mm)

Size

(mm)

Error

(%)

F80 95.54 70.75 25 136 8

F60 65.84 45.57 30 90 20

F40 38.72 20.34 46 55 24

F20 16.49 8.40 49 34 34


is of a broader spectrum. However, uncertainty in the

estimates ranged from around 30–181%. Uncertainty

analysis concluded that ball charge level (Jb) and total

charge level (Jt) estimates obtained from the mill weight

residual contain the least uncertainty and are therefore

the recommended choice for charge level estimation.

8.2. Mill discharge

Good estimates of SAG mill discharge rate and size

distribution are possible. Error in the transfer size esti-

mates increases from the eighty percent passing size (T80)to the twenty percent passing size (T20). This trend is due

to the inferential model predicting a size distribution

that is more coarse and narrower than the simulationdata and the additive characteristic of the cummulative

passing format of the size distribution.

Uncertainty analysis found that to minimise uncer-

tainty in these estimates errors may be minimised

through the monitoring of the SAG mill discharge

screen aperture size and the appropriate adjustment of

the corrected fifty percent passing size, SMD50c, en-

suring the accuracy of the oversize crusher and primarycyclone feed process measurements via periodic cali-

bration and the periodic verification of the process water

specific gravity, SGl.

The mill discharge model allows the calculation of the

recirculating load (RCL) which enables the full defi-

nition of the primary circuit mass balance which allows

for monitoring, control and optimisation of the pre-

vailing operating conditions.

8.3. Mill fresh feed

Good estimates of SAG mill fresh feed rate and size

distribution are possible. The error levels in the estimate

of the size distribution increase with decreasing particle

size This trend is due to the inferential model predicting

a size distribution that is more fine and broader than thesimulation data and the additive characteristic of the

cummulative passing format of the size distribution.

Uncertainty analysis found that to minimise uncer-

tainty in these estimates by ensuring the accuracy of the

oversize crusher and primary cyclone feed instrumen-

tation, through regular calibration, and the crusher gap

setting, via regular lead-bob gap checking. The SAG

mill discharge grate parameters: pebble port size, xp; andrelative open area fraction of the pebble ports, fp; alsostrongly influence the size estimates and hence, should

be fitted with due care.

Acknowledgements

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

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.

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Apelt Inferential Measurement of SAG Mill Parameters III Inferential Models 2002