mineral processing design and operation

Introduction to
Mineral Processing Design and Operation
PREFACE
In nature minerals of interest exist physically and chemically combined with the host rock.
Removal of the unwanted gangue to increase the concentration of mineral in an economically
viable manner is the basis of mineral processing operations. This book treats the strategy of
beneficiation as a combination of unit operations. Each unit process and its operation is
therefore treated separately. Integration of these units leading to the development of viable
flow sheets that meets the final objective, is then indicated.
The greatest challenge to a mineral processor is to produce high grade concentrates
consistently at maximum recovery from the ore body. To quantify recovery a reasonable idea
of the initial concentration of mineral in a lode is required. Proper sampling representing the
ore body is therefore essential. The book therefore commences with the techniques of
sampling of ore followed by the design and operation of unit processes of comminution that
help to release the mineral from the associated rocks. Separation and concentration processes
using techniques involving screening, classification, solid-liquid separations, gravity
separation and flotation then follow. In the book some early methods of operation have been
included and the modern methods highlighted.
The design and operation of each unit process is a study by
itself.
Over the years,
improvements in the understanding of the complexities of these processes have resulted in
increased efficiency, sustained higher productivity and grades. Mathematical modeling has
helped in this direction and hence its use is emphasized. However, the models at best serve as
guides to most processes operations that invariably involve complex interdependent variables
which are not always easily assessed or manipulated. To solve the dilemma, plants are
increasingly being equipped with instruments and gadgets that respond to changes much
faster than humans can detect. Dynamic mathematical models are the basis of operations of
these gadgets which are usually well developed, sophisticated, electronic equipment. In this
book therefore, the basics of instrumental process control is introduced the details of which
belong to the province of instrument engineers.

This book is written after several years on plant operation and teaching. The book is biased
towards practical aspects of mineral processing. It is expected to be of use to plant
metallurgist, mineral processors, chemical engineers and electronics engineers who are
engaged in the beneficiation of minerals. It is pitched at a level that serves as an introduction
to the subject to graduate students taking a course in mineral processing and extractive
metallurgy. For a better understanding of the subject solved examples are cited and typical
problems are set. Most problems may be solved by hand-held calculators. However most
plants are now equipped with reasonable numbers of computers hence solution to problems
are relatively simple with the help of spreadsheets.
The authors are grateful for the help received from numerous friends active in the field of
mineral processing who have discussed the book from time to time. Particular thanks are due
vi
to Dr Lutz Elber and Dr H. Eren who painfully went through the chapter on process control.
Authors are also grateful for permission received from various publishers who own material
that we have used and acknowledged in the text. And lastly and more importantly to our
respective families who have helped in various ways and being patient and co-operative.
A.Gupta and D.S.Yan
Perth, Australia, January 2006
Symbols and Units
A general convention used in this text is to use a subscript to describe the state of the quantity, e.g. S
for solid, L for liquid, A for air, SL or P for slurry or pulp, M for mass and V for volume. A subscript
in brackets generally refers to the stream, e.g. (O) for overflow, (U) for underflow, (F) for feed, (C) for
concentrate and (T) for tailing. There are a number of additions to this convention which are listed
below.
a a
constan t
a amplitud e
m
ap particl e acceleratio n
m/s

2
A
a
constan t
A apertur e micron s
A are a
m
2
Ac cross-sectiona l are a
m
2
Ai abrasio n inde x
Ay assa y
of
particle s
in the i
th
.
size
and
j
th
.
densit y fraction s
A
E
effectiv e are a
m
2
A

EFF
area l efficienc y facto r
Ao ope n are a
%
A
0
E effectiv e ope n are a
%
A
M
assa y
of
minera l
% , g/t, ppm
Ay underflo w are a
m
2
b
a
constan t
b Rosin-Rammle r distributio n paramete r
By breakag e distributio n functio n
c
a
constan t
C
a
constan t
C circulatio n rati o
or

load
C concentration , (mas s solid/volum e
of
slurry ) kg/m
3
C
A
concentratio n
of air
kg/m
3
Cc averag e concentratio n
of
solid s
in the
compressio n zon e kg/m
3
C
D
dra g coefficien t
Ci concentratio n
of
specie s
i
kg/m
3
Co initia l concentratio n (mas s
of
solid/volum e
of

slurry ) kg/m
3
CMA X
maximu m concentratio n (mas s
of
solid/volum e
of
slurry ) kg/m
3
C
t
concentratio n
at
time
t
(mas s
of
solid/volum e
of
slurry ) kg/m
3
C
S
(c) concentratio n
of
solid
(C =
concentrate ,
F =
feed ,

T =
tail
f
=
froth ,
P =
pulp )
Cu,
C
F
concentration s
of the
underflo w
and
feed respectively , kg/m
3
(mas s
of
solid/volum e
of
slurry )
CCRI T
critica l concentratio n kg/m
3
C
F
correctio n facto r
CI confidenc e interva l
CR confidenc e rang e
C

S
(u) solid s concentratio n
in the
underflo w
(O =
overflow, F
=
feed )
%
concentratio n
by
mas s
of
solid s
in the
feed
%
xvi
CvS(F)
cc
cv
Coo
d
d
d
32
dN
dso,
dsoc
d

B
d
F
d
L
dMAX
dMIN
d\iiD
dcutter
dc
d
w
D
D
D*
D
c
D,
Do
Du
e
Ei
E
c
E
Ec
E
B
Eo
Eo

E
P
Eu
E
T
/
/(JB)
As)
/P,/F
fi
F
microns
F
F
F
Fi
concentration by volume of solids in the feed
concentration criterion
coefficient of variation
concentration at infinite time
a constant
particle size, diameter
Sauter mean diameter
nominal diameter
cut or separation size, corrected cut size
ball diameter
63.2%
passing size in the feed
liberation size
largest dimension

smallest dimension
mid-range dimension
cutter opening
cylpeb diameter
wire diameter
discharge mass ratio (liquid/solid)
displacement, distance, diameter
dimensionless parameter
cyclone diameter
inlet diameter
overflow diameter
underflow diameter
a constant
partition coefficient of size i = recovery of size i in the U/F
corrected partition coefficient
energy
corrected partition coefficient
energy of rebound
specific grinding energy
efficiency based on oversize
Ecart probability, probable error of separation
efficiency based on undersize
total energy
a constant
ball wear rate
ball load-power function
suspensoid factor
function relating to the order of kinetics for pulp and froth
mass fraction of size i in the circuit feed
feed size

floats at SG
froth stability factor
feed mass ratio (liquid/solid)
settling factor
%
-
-
kg/m
3
-
m
m
m
microns
cm, m
m
m
m
m
m
m
mm
m
-
m
-
m
m
m
m

-
-
-
kWh
-
Wh
kWh/t
-
-
-
kW
-
kg/h
-
-
-
-
cm,
-
-
-
xvii
Fgo
F
B
FB
Fc
Fc
F
D

Fg
FG
Fos
FR
F
s
Fs
g
G
G,G
bp
G'
AG
h
huh,*
H
H
H
t
H
B
H
B
He
He
HOF
HR
H
s
Hu

I
I
JB
Jc
JG
JR
Jp
k
k
A
,k
A
k
F
, k
s
ki
kc,
k
C5
o
ks,
k
S5
o
K
KDO
KE
80%
passing size of feed

Rowland ball size factor
buoyancy force
Bond mill factor
centrifugal force
drag force
gravitational force
correction factor for extra fineness of grind
correction factor for oversized feed
correction factor for low reduction ratio
mass flow rate
Bond slurry or slump factor
gravitational constant (9.81)
grade (assay)
net grams of undersize per revolution
grinding parameter of circulating load
free energy
parameter =
X/C T
diatances within the conical section of a mill
hindrance factor
height
height at time t
height of rebound pendulum
height of bed
height of ball charge
height of the start of the critical zone in sedimentation
height of the clarification zone (overflow)
height of rest
hindered settling factor
mudline height at the underflow concentration

height after infinite time
impact crushing strength
imperfection
fraction of mill volume occupied by bulk ball charge
fraction of mill volume in cylindrical section occupied by balls
and coarse ore
superficial gas velocity
fraction of mill volume occupied by bulk rock charge
fraction of mill volume filled by the pulp/slurry
constant
rate constant for air removal via froth and tailings respectively
rate constant for fast and slow component respectively
comminution coefficient of fraction coarser that i
th
screen
screening rate constant, crowded condition, normal and half size
screening rate constant, separated condition, normal and half size
constant
material constant
kinetic energy
microns
-
N
-
N
N
N
-
-
-

kg/s,t/h
-
m/s
2
%, g/t, ppm
g/rev
-
J
-
m
-
m, cm
m
m
m
m
m
m
m
-
m
m
kg.m/mm
-
-
-
m/s
-
-
-

-
min"
1
-
t/h/m
2
m
1
-
-
kW
xviii
L
L
A
LAE
LEFF
Lc
LcYL,
LcONE
L
F
LMW ,
LMAX
LT
I_I
LVF
m
m
muo o

m
k
mu(O)
m(r)
m
T
mu(o>
m
i(U)
M
Mi
M
oi
My
M
B
M
B
M
c
Me
M
F
M
F
M
FT
M
F
MMIN

MR
MR
Mr
M
s
M
s
length
aperture size
effective aperture
effective grinding length
length of cyclone
length of cylindrical and cone sections
Nordberg loading factor
minimum and maximum crusher set
crusher throw
length of vortex finder
length from end of vortex finder to apex of a cyclone
moisture (wet mass/dry mass)
mineralogical factor
mass fraction of undersize in the feed
mass fraction of makeup balls of size k
mass fraction of undersize in the oversize
cumulative mass fraction of balls less than size r
mass rate of ball replacement per unit mass of balls
mass fraction of undersize in the undersize
mass of size i in the underflow (F = feed)
mass
mass
mass/mass fraction of i* increment

cumulative mass fraction retained on i* screen at zero time
mass percent of the i* size fraction and
j *
density fraction
mass of block
mass of balls
mill capacity
mass of crushing weight
mass of feed
mass of fluid
mass of floats
Nordberg mill factor
minimum mass of sample required
mass of rock
mass fraction of rock to total charge (rock + water)
cumulative mass fraction of balls of size r in the charge
mass of striking pendulum
mass of solid
m
m
m
m
m
m
-
m
m
m
m
-

kg/m
3
-
-
-
kg/h.t
-
kg
g
kg,t
kg,t
%
kg
kg
t/h
kg
t
kg
kg,t
-
kg,t
kg
-
kg
ke,t
. sec),
SO D
n^
5
of solid feed, concentrate and tailing respectively kg, t

mass of solid in froth
MS K
mass of sinks kg , t
M
S
{p) mas s
of
solid
in the
pulp
kg , t
AM(t) mas s
of top
size particl e
kg , t
Mj mas s
of new
feed
g
Mw mas s
of
water
kg , t
n numbe r
of
revolutions/mi n
min"
1
n numbe r
of

increments , measurement s
xix
n
n(r)
ns
N
N
N
N
L
N'
Oi
P
Pi
P
Pso
p
p
p
p
p
p
PA, PC, PE, PF
PcON
PCYL
PD
PG
P«
PL
PM

PM
PNET
PNL
Pos
PR
Ps
Ps
PE
AP
q
Q
QB
QH
Qo
Qu
QMSOT
QMSCQ
QM(F)
Qv(C),
(T), (F)
QvL(O)
order of rate equation
cumulative number fraction of balls of size less than r
number of sub-lots
number of mill revolutions
number of strokes/min
number
number of presentations per unit length
number of particles/gram
mass fraction of size i in the overflow

binomial probability of being selected in a sample
mass fraction of size i in the new feed
product size
80%
passing size of product
proportion of particles
pressure
Powers roundness factor
Jig power
JKSimFloat ore fioatability parameter
probability
probability of adherence, collision, emergence, froth recovery
power of the conical part of a mill
power for the cylindrical part of a mill
particle distribution factor
proportion of gangue particles
proportion of particles in the i . size and
j*.
density fractions
liberation factor
proportion of mineral particles
mill power
net mill power draw
no load power
period of oscillation
relative mill power
particle shape factor
power at the mill shaft
potential energy
pressure drop

alternate binomial probability = 1
—
p
capacity
makeup ball addition rate
basic feed rate (capacity)
tonnage of oversize material
capacity of the underflow
flowrate of solids by mass in the overflow (U = U/F, F = feed)
mass flow of solid in concentrate
capacity, of feed slurry by mass
flowrate by volume in concentrate, tailing and feed respectively
capacity (flowrate) of liquid by volume in the overflow
-
-
-
-
min'
1
-
nf
1
g"
1
-
-
-
microns
microns
-

Pa
-
W
-
-
kW
kW
-
-
-
-
-
kW
kW
kW
s
-
-
kW
kW
kPa
-
t/h
kg/day
t/h/m
t/h
t/h
t/h
t/h
t/h

m
3
/h
nrVh
(U=underflow, F=feed)
xx
QVOP(U ) flowrat e b y volum e o f entraine d overflo w pul p in th e U/ F
QVOL(U ) flowrat e b y volum e o f entraine d overflo w liqui d in th e U/ F m
3
/h
Qvos(u ) flowrat e b y volum e o f entraine d overflo w solid s i n th e U/ F m
3
/h
Qvs(O ) flowrat e b y volum e o f solid s i n th e overflo w ( U = U/F , F = feed )
Qv(f ) flowrat e b y volum e i n th e frot h
Qv(O ) flowrat e b y volum e o f overflo w (pulp ) ( U = underflow )
Q w bal l wea r rat e
r
0
fractio n o f tes t scree n oversiz e
r bal l radiu s
r rati o o f rat e constant s =
IC A
/(kA+kA )
ri ,
r 2 radiu s withi n th e conica l sectio n o f a mil l
R radiu s
R recover y
R reductio n rati o
Ri,R2,R 3 Dietric h coefficient s

R th e mea n radia l positio n o f th e activ e par t o f th e charg e
R ' fractiona l recovery , wit h respec t t o th e fee d t o th e first cel l
R ' mas s o f tes t scree n oversiz e afte r grindin g
R e radiu s o f con e a t a distanc e L j fro m cylindrica l sectio n
ReA , Re c Reynold s numbe r i n th e ape x an d con e sectio n respectivel y
Re
P
particl e Reynold s numbe r
R F
frot h recover y facto r
Ri radia l distanc e t o th e inne r surfac e o f th e activ e charg e
Ro mas s o f tes t scree n oversiz e befor e grindin g
R P
radia l distanc e o f particl e fro m th e centr e o f a mil l
RR O
optimu m reductio n rati o
R T
radiu s a t th e mil l trunnio n
R v recover y o f fee d volum e t o th e underflo w
Ro o recover y a t infinit e tim e
S spee d
S sink s a t S G
S surfac e are a
S B
surfac e are a o f bal l
S
B
bubbl e surfac e are a flu x
S; breakag e rat e functio n
S F

Nordber g spee d facto r
S spacing , distanc e
S dimensionles s paramete r
SG , SG s specifi c gravity , specifi c gravit y o f soli d
T perio d o f pulsatio n
T N
mas s percen t passin g 1/ N o f th e origina l siz e
t tim e
to detentio n o r residenc e tim e
t
R
effectiv e residenc e tim e
tu tim e fo r al l solid s t o settl e pas t a laye r o f concentratio n C
tio siz e tha t i s on e tent h th e siz e o f origina l particl e
t
A
mea n tim e take n fo r activ e par t o r charg e t o trave l fro m
th e to e t o th e shoulde r
m
3
/h
m
3
/h
m
3
/h
m
3
/h

mm/ h
m
m
m
m
m
m
g
m
m
m/ s
m
2
m
2
s"
1
min"
1
m
h, min , s
h
s
h
m m
s
xxi
U
Up
V

V*
V
c
Vc
VCONE
L(F )
Vo
VR
VB °
VB
1
vs°
Vs
1
var(d)
var(c)
var(pa)
var(t)
var(x)
w
w
W
W
WE
W;
W
s
x
x
x

Xj
X
mean time for free fall from the shoulder to the toe s
mass fraction of size i in the underflow
fraction of void space between balls at rest, filled by rock
fraction of the interstitial voids between the balls and rock charge -
in a SAG mill occupied by slurry of smaller particles
volume fraction of solids in the overflow, (U=underflow, F=feed) -
volume fraction of solids finer than the d
50
in the feed (V
d
5o/V
S
(
F
)) -
volume m
3
dimensionless parameter
volume of the mill charge m
3
volume of the compression zone m
3
volume of conical section of mill m
3
volume of solids finer than the dso in the feed
percent of mill volume occupied by balls %
volume dilution in the feed =
VL(F/VS(F )

volume of liquid in the feed, (U=underflow, F=feed) m
3
volume of mill m
3
volume dilution in the overflow =
VL(O/VS(O >
or
QVL(O/QVS(O )
percent of mill volume occupied by rock %
volume of solids in the feed, (U = underflow, O = overflow) m
terminal velocity m/ s
unknown true value
velocity of block pendulum before impact m/s
velocity of block pendulum after impact m/s
velocity of striking pendulum before impact m/s
velocity of striking pendulum after impact m/s
distribution variance
composition variance
preparation and analysis variance
total variance
variance
thickness of slurry m
fraction of feed water in the underflow
width m
dimensionless parameter
effective width m
Bond Work Index kWh/ t
Bond Work Index, laboratory test kWh/ t
operating work index kWh/ t
corrected operating work index kWh/ t

water split =
QML(O)/QML<F )
deviation from the true assay
geometric mean of size interval micron s
Rosin-Rammler size parameter micron s
Sample mean
i* measurement
deviation from standard unit
xxii
a
a
a, a
0
ttr» «s
«TS
Y
T
YSA >
YSL >
TL A
£
e
A
K
9
-e-
<))
4>c
¥>¥»
Vw

VCRIT
(1
P
Pb
PC
PB
PF
Po
PL
PR
ps
PSL
Pw
PM> pG
U
0
A
CL
CTM
0p
Op
A
CT
S
0
T
e
fractional average mineral content
Lynch efficiency parameter
angle

toe and shoulder angles of the charge
the slurry toe angle
function of charge position and mill speed
volume fraction of active part of the charge to the total charge
surface energy, surface tension, interfacial tension
coefficient of restitution
void fraction
a ball wear parameter
a ball wear parameter, wear distance per unit time
porosity of a ball bed
ratio of experimental critical speed to theoretical critical speed
fraction with the slow rate constant
fraction of critical speed
settling or sedimentation flux
withdrawal flux
critical flux
coefficient of friction
specific gravity (dimensionless) or density
density or SG of balls
bulk density of the total charge, rock + balls + water
bulk density
density of fluid
density of ore
density of liquid
density of rock
density of solid
density of slurry
density of water
density of the mineral and the gangue respectively
standard deviation (where o

2
= var(x))
statistical error in assay
standard deviation of a primary increment
standard deviation on a mass basis
standard deviation of the proportion of particles in a sample
standard deviation of preparation and assay
statistical error during sampling
total error
nominal residence time
angle
-
-
radians
radians
radians
-
-
N/m
-
-
-
-
-
-
-
-
kg/m
2
/s

kg/m
2
/s
kg/m
2
/s
-
kg/m
3
, t/m
3
kg/m
3
,-
t/m
3
kg/m
3
kg/m
3
kg/m
3
kg/m
3
kg/m
3
kg/m
3
kg/m
3

kg/m
3
kg/m
3
, t/m
3
-
-
-
-
-
-
-
-
s
radians,
degree
viscosity
velocity
critical speed
mNm, Pa.s
m/s
rpm
xxiii
V
F
V
N
Vo
v

O(i)
VR
VR
Vs
VSo
Vst
V
T
V
T
CO
CO
COp
?
% s
velocity across a screen
normalised tangential velocity =
VR/V T
overflow rate
ideal overflow rate
tangential velocity at distance Rp
rise velocity
settling velocity
initial settling velocity
settling velocity at time t
tangential velocity at the inside liner surface
terminal velocity
rotational speed, angular velocity
mean rotational speed
rotational speed of a particle at distance Rp

a milling parameter = function of volumetric filling of mill
percent solids
m/min
rpm
m/s
m/s
rpm
m/s
m/s
m/s
m/s
rpm
m/s
s"
1
, rpm, Hz
rpm
s'
1
, rpm
-
%
Chapter 1. Mineral Sampling
1.
INTRODUCTION
A processing plant costs many millions of dollars to build and operate. The success of this
expenditure relies on the assays of a few small samples. Decisions affecting millions of
dollars are made on the basis of a small fraction of the bulk of the ore body. It is therefore
very important that this small fraction is as representative as possible of the bulk material.
Special care needs to be taken in any sampling regime and a considerable effort in statistical

analysis and sampling theory has gone into quantifying the procedures and precautions to be
taken.
The final sampling regime adopted however is a compromise between what theory tells us
should be done and the cost and difficulty of achieving this in practice.
1.1. Statistical Terminology
A measurement is considered to be accurate if the difference between the measured value
and the true value falls within an acceptable margin. In most cases however the true value of
the assay is unknown so the confidence we have in the accuracy of the measured value is also
unknown. We have to rely on statistical theory to minimise the systematic errors to increase
our confidence in the measured value.
Checks can be put in place to differentiate between random variations and systematic errors
as the cause of potential differences. A random error (or variation) on average, over a period
of time, tend to zero whereas integrated systematic errors result in a net positive or negative
value (see Fig. 1.1).
The bias is the difference between the true value and the average of a number of
experimental values and hence is the same as the systematic error. The variance between
repeated samples is a measure of precision or reproducibility. The difference between the
mean of a series of repeat samples and the true value is a measure of accuracy (Fig. 1.2).
A series of measurements can be precise but may not adequately represent the true value.
Calibration procedures and check programs determine accuracy and repeat or
replicate/duplicate measurements determine precision. If there is no bias in the sampling
regime, the precision will be the same as the accuracy. Normal test results show that assays
differ from sample to sample. For unbiased sampling procedures, these assay differences are
not due to any procedural errors. Rather, the term "random variations" more suitably
describes the variability between primary sample increments within each sampling campaign.
Random variations are an intrinsic characteristic of a random process whereas a systematic
error or bias is a statistically significant difference between a measurement, or the mean of a
series of measurements, and the unknown true value (Fig. 1.1). Applied statistics plays an
important role in defining the difference between random variations and systematic errors and
in quantifying both.

2
True value, v
Systematic
error
Random error
ASSAY
SAMPLE
True value
precision
Repeat 1
Repeat 2
mean
accuracy
ASSAY
SAMPLE
True value, v
Systematic
error
SAMPLE
Fig. 1.1. Representation of a random and systematic error.
1.1.1.
Mean
The most important parameter for a population is its average value. In sampling and weighing
the arithmetic mean and the weighted mean are most often used. Other measures for the
average value of a series of measurements are the harmonic mean, and the geometric mean.
Mode and median are measures of the central value of a distribution. The mode forms the peak
of the frequency distribution, while the median divides the total number of measurements into
two equal sets of data. If the frequency distribution is symmetrical, then its mean, mode and
median coincide as shown in Figs. 1.3 and 1.4.
Repeat 2

Repeat 2
True value
mean
accuracy
precision
SAMPLE
Fig. 1.2. Difference between precision and accuracy.
3
0
0.1
0.2
0.3
0.4
0.5
0.6
0246810
ASSAY
FREQUENCY
0
0.1
0.2
0.3
0.4
0.5
0.6
0246810
ASSAY
FREQUENCY
mean, mode,
median

mode
median
mean
0.6
0.5 -
O 0.4
UJ
3 0.3 H
o
UJ
ff 0.2
0.1 -
0
0
mean,
mode,
median
10
Fig. 1.3. Normal distribution.
For a binomial sampling unit of mixed particles the average percentage of mineral A is
calculated by adding up all measurements, and by dividing their sum by the number of
measurements in each series.
(1.1)
where x = sample mean (arithmetic)
Xj = i
th
measurement
n = number of increments
0.6
0.5 -

O 0.4 -
UJ
3 0.3 H
o
UJ
P=
0.2 A
0.1 -
0
0
mode
median
Fig. 1.4. Asymmetrical distribution.
4
The weighted percentage is calculated, either from the total number of particles in each series,
or by multiplying each incremental percentage with the mass in each corresponding
increment, and by dividing the sum of all products by the total mass for each series. However,
the small error that is introduced by calculating the arithmetic mean rather than the weighted
average, is well within the precision of this sampling regime. The following formula is used to
calculate the weighted average for a sample that consists of n primary increments:
( 2
M
where AM; = mass of I
th
increment
M = mass of gross sample
Due to random variations in the mass of each primary increment the weighted average is a
better estimate of v, the unknown true value, than the arithmetic mean.
1,1.2. Variance
The variance, and its derived parameters such as the standard deviation and the coefficient of

variation, are the most important measures for variability between test results.
The term range may be used as a measure of variability.
Example 1.1
Consider a binary mixture of quartz and hematite particles with approximately 10% hematite.
Samples are taken and the number of hematite particles are counted to obtain the percentage
of hematite in the sample. Table 1.1 gives the result often samples. For a binomial sampling
unit the range is (maximum value - minimum value) = 12.6 - 5.7 = 6.9%.
Table 1.1
Sampling with a Binomial Population (Quartz and Hematite).
Number
1
2
3
4
5
6
7
g
9
10
Quartz
105
132
99
98
83
87
91
86
98

113
Sample
Hematite
11
19
10
7
5
11
12
8
12
14
Total
116
151
109
105
88
98
103
94
110
127
%
Hematite
9.5
12.6
9.2
6.7

5.7
11.2
11.7
8.5
10.9
11.0
5
minimum
mean

range
ASSAY
SAMPLE
maximum
• • •
k
range
y maxirr
}
• • • • • • ^
SAMPLE
Fig. 1.5. Range of experimental values.
If each series of measurements is placed in ascending order, then the range is numerically
equal to x
n
- xi so that the range does not include information in increments X2,
X3, ,
x
n
_i.

For a series of three or more measurements the range becomes progressively less efficient as a
measure for variability as indicated in Fig. 1.5.
For two samples, the range is the only measure for precision but this is not sufficient to
estimate the precision of a measurement process. The precision of a measurement process
requires the mean of absolute values of a set of
ranges
calculated from a series of four or more
simultaneous duplicates. This is the variance.
The classical formula for the calculation of the variance is:
var(x) =
Z'
n-l n-1
(1.3)
where n = number of measurements
n-l = degrees of freedom
The standard deviation, a, is the square root of the variance. The coefficient of variation
(CV),
is a measure of precision and is numerically equal to:
(1.4)
Example 1.2
Variance values from a sampling procedure with a binary mixture of mineral particles is given
in Table 1.2.
6
Table 1.2
Variance values from the sampling example with a mixture of quartz and hematite
Number
1
2
3
4

5
6
7
g
9
10
Quartz
105
132
99
98
83
87
91
86
98
113
Small increment
Hematite
11
19
10
7
5
11
12
8
12
14
Arithmetic mean

Variance
Standard deviation
Coeff.
of variation
Total
116
151
109
105
88
98
103
94
110
127
SUM
% Hematite
9.48
12.58
9.17
6.67
5.68
11.22
11.65
8.51
10.91
11.02
96.89
9.69
5.0141

2.2392
23.1
Quartz
504
350
597
394
428
438
508
533
438
490
<
(
Large increment
Hematite
53
45
56
52
43
52
55
56
50
49
Total
557
395

653
446
471
490
563
589
488
539
SUM
Arithmetic mean
Variance
Standard deviation
Coeff.
of variation
%Hematite
9.52
11.39
8.58
11.66
9.13
10.61
9.77
9.51
10.25
9.09
99.51
9.95
1.1264
1.0613
10.7

The physical appearance of a sample that consists of fifty primary increments of 5 kg each
is similar to that of a sample containing five increments of 50 kg, or to that of 250 kg of a bulk
solid. However, the difference in intrinsic precision (as indicated by the variance) may be
dramatic, particularly if the variability within the sampling unit is high.
In practical applications of sampling bulk solids we compromise by collecting and
measuring unknown parameters on gross samples, and by reporting x, the sample mean as the
best estimate for v, the unknown true value. If all increments are contained in a single gross
sample, we have no information to estimate the precision of this sampling regime. If we want
to know more about the precision of samples, systems and procedures, it is essential that
duplicate or replicate measurements be made, from time to time, to determine the coefficient
of variation for each step in the chain of measurement procedures.
1.1.3. Confidence Intervals
Other convenient measures for precision are confidence intervals (CI) and confidence ranges
(CR).
95 % confidence intervals and 95 % confidence ranges may be used, although if
concern over sampling precision is high, then 99% or 99.9% confidence limits must be
considered.
That is, if we repeat a particular experiment 100 times, then 95 times out of 100 the results
would fall within a certain bound about the mean and this bound is the 95 % confidence
interval. Similarly, confidence limits of 99 % and 99.9 % mean that 99 times out of 100, or
999 times out of 1000 measurements would fall within a specified or known range.
In the draft Australian Standard, DR00223, for estimating the sampling precision in sampling
of particulate materials, a confidence interval of 68% is chosen [1].
Fortunately, we don't need to repeat a measurement one-hundred times if we want to
determine its 95% confidence interval, either for individual measurements or for their mean.
Applied statistics provides techniques for the calculation of confidence intervals from a
7
limited number of experiments. The variance between increments, or between measurements,
is the essential parameter.
The most reliable estimate of a

2
is var(x), the variance of the sample. The reliability of this
estimate for a
2
can be improved by collecting, preparing and measuring more primary
increments, or by repeating a series of limited experiments on the same sampling unit.
The 95 % confidence interval for a normal distribution is equal to ±1.96 a from the
distribution mean. In practice, we often use the factor 2 instead of 1.96, to simplify
calculations and precision statements. The 68 % confidence interval is equal to ±0.99 a, for
infinite degrees of freedom and if the number of replicate results exceeds 8 then a factor of 1.0
is an acceptable approximation.
1.2. Mineral particles differing in size - Gy's method
Representing large bodies of minerals truly and accurately by a small sample that can be
handled in a laboratory is a difficult task. The difficulties arise chiefly in ascertaining a proper
sample size and in determining the degree of accuracy with which the sample represents the
bulk sample.
In each case the accuracy of the final sample would depend on the mathematical probability
with which the sample represents the bulk material. The probability of true representation
increases when incremental samples are taken while collecting from a stream, like a conveyor
belt for solids and off pipes for liquids or slurries.
Several methods have been put forward to increase the probability of adequately
representing the bulk minerals
[2-5].
One such method involving both the size and accuracy of
a sample taken for assay has been developed by Gy and is widely used [6, 7]. Gy introduced a
model based on equiprobable sample spaces and proposed that if:
=
dimension of the largest particle
MMIN ~ minimum mass of sample required
o

2
= variance of allowable sampling error in an assay (in the case of a normal
distribution this equals the standard deviation)
then:
M
MIN = 2 U-
5
)
a
K is usually referred to as the sampling constant (kg/m
3
)
In mineralogical sampling the dimension of the largest piece (dMAx) can be taken as the
screen aperture through which 90-95 % of the material passes. As ± 2a represents the
probability of events when 95 out of 100 assays would be within the true assay value, 2a is
the acceptable probability value of the sample. The sampling constant K is considered to be a
function of the material characteristics and is expressed by:
K = PSPDPL.HI (1.6 )
where Ps = particle shape factor (usually taken as 0.5 for spherical particles, 0.2 for gold
ores)
8
PD
= particle distribution factor (usually in the range 0.20 - 0.75 with higher values
for narrower size distributions, usually taken as 0.25 and 0.50 when the
material is closely sized)
PL
= liberation factor (0 for homogeneous (unliberated) materials, 1 for
heterogeneous (liberated) and see Table 1.3 for intermediate material)
m = mineralogical factor
The mineralogical factor, m, has been defined as:

(1-7)
where a is the fractional average mineral content and
PM
and po the specific gravity of the mineral and the gangue respectively
The liberation factor,
PL ,
is related to the top size,
dMAX
and to the liberation size, dL of the
mineral in the sample space. It can be determined using Table 1.3. In practice, PL is seldom
less than 0.1 and if the liberation size is unknown then it is safe to take
P L
as 1.0.
Table 1.3
Liberation factor as a function of liberation size [9].
Top Size/Liberation Size, <1 U4 4^10 10-40 40-100 100-400 >400
Lib.
Factor (P
L)
) 1.0 0.8 0.4 0.2 0.1 0.05 0.02
When a large amount of sample has been collected it has to be split by a suitable method
such as riffling. At each stage of subdivision, samples have to be collected, assayed and
statistical errors determined, hi such cases the statistical error for the total sample will be the
sum of the statistical errors during sampling (as) and the statistical error in assay
(aA),
S O
that
the total variance (ax
2
) will be:

a
T
2
= a
s
2
+a
A
2
(1.8 )
When the sample is almost an infinite lot and where the proportion of mineral particles has
been mixed with gangue and the particles are large enough to be counted, it may be easier to
adopt the following procedure for determining ap.
Let PM
=
proportion of mineral particles
PG
= proportion of gangue particles
N = number of particles
Then the standard deviation of the proportion of mineral particles in the sample, Gp
;
will
be:
9
a
P
=
(1.9)
The standard deviation on a mass basis
(CTM )

can be written in terms of the percent mineral
in the whole sample provided the densities (p) are known. Thus if p
M
and
P G
are the densities
of the mineral and gangue, then the mass percent of mineral in the entire sample, consisting of
mineral and gangue (the assay), will be:
100 P
M
p
M
A M
= (1-10 )
P
MPM+
P
G
PG
assuming the particles of mineral and gangue have the same shape and size.
dA
The standard deviation of the entire sample is given by a
T
= —— . a
p
or
_ I
(10Q-A
M
)P

M
+A
MPG
A
M
(100-A
M
)
T
I ,„„ / ' N
Example 1.3
Regular samples were required of the feed to a copper processing plant, having a copper
content of about 9%. The confidence level of estimation was required to be 0.1 % Cu at 2a
standard deviation. The liberation size of the Cu mineral (chalcocite) in the ore was
determined to be 75 um. The top size of the ore from the sampler was 2.5 cm. Determine the
minimum mass of sample required to represent the ore. Given; the density of chalcocite is
5600 kg/m
3
and the density of the gangue is 2500 kg/m
3
.
Solution
From the data, dMAx = 2.5 cm
Since the confidence interval required is equal to ± 0.1 % of a 9% assay,
2a = 0.1/9 or a = 0.011/2 = 0.00555
Again from the data d
M
Ax/d
L
= 25 / 0.075 = 333.33, hence from Table 1.3, P

L
= 0.05.
As the ore contains chalcocite (Cu
2
S) assaying about 9% Cu, it can be considered to contain
[159.2/(63.56 x 2)] x 9 = 11.3% of Cu
2
S. Thus a = 0.113.
i.e. The copper content of
CU2S
is given by the ratio of atomic masses;
(63.56 x 2) x 100 = 127.1 x 100 = 79.8% Cu (atomic mass; Cu = 63.56,
(63.56x2)+ 32.1 159.2 S = 32.1)
10
10
The chalcocite content of the ore is then given by:
9x100=11.3%
79.8
The mineralogical composition factor, m, can now be calculated from Eq. (1.7),
m = ^~
al13
-* [(1-0.113)5600+ (0.113x2500)] = 41207.8 kg/
then from Eq. (1.6),
K = 0.5 x 0.25 x 0.05 x 41207.8 = 257.5 kg/m
3
and
= [257.5 x(0.025)
3
]/(0.00555)
2

= 130.6 kg
Thus the minimum sample size should be 131 kg.
Note the importance of not rounding off the numbers until the final result. If cr is rounded
to 0.0055 then MMIN = 133 kg and if a is rounded to 0.0056 then MMIN = 128 kg.
Example 1.4
A composite sample of galena and quartz was to be sampled such that the assay would be
within 0.20% of the true assay, of say 5.5 %, with a probability of 0.99, ie. the sample assay
would be 5.5 % + 0.20 %, 99 times out of 100 . Given that the densities of galena and quartz
were 7400 kg/m
3
and 2600 kg/m
3
respectively and the average particle size was 12.5 mm with
a mass of 3.07g, determine the size of the sample that would represent the composite.
Solution
Stepl
Determine
O T
in terms of N from Eq. (1.11).
(100 - 5.5) 7400 + 5.5 x 2600 5.5 (100 - 5.5)
100^7400x2600 J V N
37.089 (1.12 )
To determine N it is necessary to find aj
11
11
Step 2
To determine the value of
CF T
to satisfy the deviation limits so that the area under the curve
between the limits will be 99% of the total area, use Table 1.4 below.

From the table, corresponding to a probability of 0.99, the value of the deviation from the
standard unit is 2.576.
i.e. X
=
— or <r
T
= —= -5^ = 0.07764
0
T
X 2.576
Substituting in Eq. (1.12), N = (37.089/0.07764)
2
= 228,201.9
Table 1.4
Probability P vs deviation X, relative to unit standard deviation [9],
Probability
P
0.90
0.91
0.92
0.93
0.94
0.95
0.96
Deviation from
Standard Unit, X
X = x/(j
T
1.645
1.705

1.750
1.812
1.881
1.960
2.054
Probability
P
0.97
0.98
0.99
0.999
0.9999
0.99999
0.999999
Deviation from
Standard Unit, X
X=X/CT
2.170
2.326
2.576
3.291
3.890
4.417
4.892
Hence the mass of sample necessary to give an assay within the range 5.5 ± 0.20 %, 99 times
out of 100 would be =228,201.9x3.07 = 700,579 g * 701 kg.
13.
Mineral particles of different density
Where variations in density of individual particles and their composition occur, the
following considerations may be adopted to provide the sample size [9]:

1.
Divide the material into n density fractions, pi, p2, P3, p
n
varying from purely one
mineral to the other, e.g. copper and quartz in a copper ore, and consider n size fractions,
di,
d
2
, d
3
d
n
2.
Consider the mass percent of the i size fraction and j density fraction as
M M
and
3.
Consider Ay and Py as the assay and the proportion of particles in the i size and
j *
density fractions
For sampling a mixture of two components (mineral and gangue) the proportion of
particles in the ij fraction would be;
12
12
Eh/-? )
( }
And the assay of the mixture, Ay will be approximately equal to:
A

=

IPM(PB-PO)IOO J
where PM = density of the mineral and po the density of the gangue and the standard
deviation of the ij* fraction will be:
MlSs i (1.15 )
N
For a multi-component system the principles developed in the above Eqs. 1.13 to 1.15 may
be extended. Their solution can be achieved easily by a computer. The general equation for
overall sample assay is:
(1.16)
Example 1.5
A nickel sulphide ore mineral (pentlandite) has an average particle size of lmm. It is separated
into three fractions and the properties of each fraction are:
Fraction Mass % Assay, % Density
pentlandite ___^
1 (Concentrate)
2 (Middlings)
3 (Tail)
2.4
81.0
16.6
100.0
58.9
0.0
4.8
3.6
2.65
The lot has to be sampled so that it would assay ± 0.15 % of the true assay having 5% with a
probability of 99%
Solution
Stepl.

As the stipulated probability is 0.99 Table 1.4 may be used.
13
13
Here = 2.576 or 0 = 0.058
Step 2.
Estimate Py from Eq. (1.13) in the following manner (Table 1.5):
Table 1.5
Calculation of the proportion of particles in the ij * fraction
Fraction
i
(1)
1
1
1
j
(2)
1
2
3
2
Mass
%
(3)
2.4
81.0
16.6
100.0
Assay
(4)
100.0

58.9
0.0
SG
(5)
4.8
3.6
2.65
Av.
Dia
d;
(6)
1
1
1
0)
1
1
1
d/xp
(7)x (5)
(8)
4.8
3.6
2.65
Mass %
di
3
p
(3)/(8)
(9)

0.5
22.7
6.03
29.23
Proportion
by No.
(9)/£9
(10)
0.017
0.776
0.207
1.0
Step 3.
Withkri
Now follow Example 1.4 to determine the sample size.
With known P and 0, N can be estimated using the equation a
2
=———
1.4. Incremental Sampling
Theoretically, unbiased gross samples can be obtained by collecting a sufficiently large
number of single particles from a sampling unit. If each particle has a finite chance of being
selected for a gross sample, and if this probability is only a function of its size, then this
collection of particles will constitute an unbiased probability sample.
In the practice of sampling bulk solids, such a sample collection scheme is highly
impractical. Therefore, we collect groups of particles for primary increments.
Sampling experiments that are based on the collection of a series of small and large
increments from a large set of particles, demonstrates that the precision of a single increment
sampling regime is a function of the number of particles in a primary increment. As a matter
of fact, the formula for the variance of a binomial sampling unit, var(x) = N.p.q, shows the
fundamental relationship between probabilities (p, q), the total number of particles in an

increment (N), and thus its mass, and the precision for the parameter of interest. Hence, the
precision of a single increment is essentially determined by its mass.
The binomial sampling experiment in which p = 0.095, and q = 0.905 (q = alternate
binomial probability = 1 - p), can be used to set up a table in which the precision of a primary
increment is given as a function of the average number of particles in an increment. In Table
1.6, the 95 % confidence ranges (CR) that are calculated from corresponding 95 % confidence
intervals (CI) for the expected number of value particles in each increment, together with their
coefficients of variation, are listed.
Since these ranges are based on properties of the binomial distribution, their values are
obviously independent of particle size. For a bulk solid with a certain density, its top size
determines the mass of a primary increment. Top size is defined as the 95% passing sieve size.
14
14
Table 1.6
95 % Confidence Intervals and Ranges [10].
Number
10
10
2
10
3
10
4
10
5
10
6
Low
0
4.0

7.6
9.2
9.4
9.47
High
29.0
15.0
11.4
9.8
9.6
9.53
CV in %
98.0
31.0
9.8
3.1
1.0
0.3
The mass of an increment must be such that it is large enough to include the large particles
and particles present in the sample should be in the same proportions as in the lot being
sampled. The minimum mass of the increment is therefore dependent on the size of the
particles being sampled.
The top size of a bulk solid is a measure of length. The mass of a particle is a function of
its volume and specific gravity and ultimately a function of its mean diameter or length. The
mass of a primary increment then can be defined in terms of the number and mass of particles
from the top size range. If no other information is available, then an acceptable rule of thumb
is to collect primary increments with a mass equal to 1,000 times the mass of a top size
particle.
AM =
1,000-AM(t)

(1.17 )
where AM = mass of primary increment in kg
AM(t) = mass of top size particle in kg.
Experience and theory are embodied in a number of national and international standards on
sampling of particulate materials where the sampling regimes are defined in terms of the total
number of increments, and the average mass of a primary increment.
It is generally accepted that a primary increment should contain no less than one-thousand
(1,000) particles.
In the standard on sampling of iron ore (ISO TCI02), the minimum mass for primary
increments is specified in relation to the top size, and in Table 1.7 these recommendations are
tabulated and compared to calculations for hard coal.
The minimum mass for a primary increment should preferably be defined in terms of the
volume of a particle from the top size range.
Once the mass for a primary increment is selected either in accordance with applicable
standards or on the basis of the previous guidelines, or determined by the critical design
parameters of a mechanical sampling system, the required number of primary increments
remains to be defined. Too few increments will result in too low a precision while too many
would unnecessarily increase the costs for sampling and preparation.
Most standards for bulk solids contain simple formulae to calculate the required number of
increments for a consignment from a given number for the unit quantity (usually 1,000 t), or