Sampling of discrete materials quantitative approach sampling of zero dimensional objects
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Sampling of discrete materials quantitative approach sampling of zero dimensional objects


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m(aS)1: Reminder of
equality Eq. (62):
mðaSÞ1 ¼
P
m amMmPmP
m MmPm
ð62Þ
Except under special experimental conditions, or in
computer simulations, we never know with precision the
sets of values of am, Mm and Pm. Furthermore, sums Am,
usually extended to an innumerable number of terms, are
involved. It is therefore practically impossible to estimate
the incorrect sampling bias from this formula. Experience
shows that with particulate solids but also with solutions
of heavy metal ions, the probability of selection Pm is
often a function of the physical and chemical properties
of unit Um, such as size, density, shape, mass and critical
content. A few experiments carried out on iron ores
showed that this bias might indeed be very large. The
results of these experiments and simulations of the set of
probabilities Pm can be found in our books (especially
Ref. [13] dated 1967). In addition to this, we have also
shown, contrary to what is frequently believed, that it is
P. Gy / Chemometrics and Intellige34
m CSEð Þ2 ¼ \ufffd P
m
aL
\ufffd
M 2L
ð74Þ
We showed in Ref. [18] that HIL (dimensions of a mass
expressed in grams) could be written as:
HIL ¼ cbf gd3 ð77Þ
where c : is a bconstitutional parameterQ (dimensions of
specific gravity expressed in g/cm3). It is mathematically
defined. It can vary from a fraction of unity to several
millions.
b : is a liberation parameter ðdimensionlessÞ : 0VbV1
Here again, we never know the sets of values of am and
Mm that would enable us to compute the bias. In experi-
ments carried out on iron ores, it was shown that this
(relative) bias was negligible (order of magnitude: 10\ufffd8) as
long as we dealt with medium to high grades aL. With
traces (aLb1 ppm), we suspect that the bias is no longer
negligible but also that the distribution becomes more and
more asymmetrical, which makes it difficult to compute
confidence intervals around the analytical result.
16. Definition of the Fundamental Sampling Error
(FSE)\u2014derivation of the bFORMULAQ expressing the
fundamental variance s2(FSE)
The Fundamental Sampling Error, FSE, is defined as the
error generated when the constituents (fragments, mole-
cules, ions) making up the sample have been selected:
n At random, i.e., with a uniform probability P of being
selected. The selection is correct.
n Individually and independently: The selected constitu-
ents are independent from one another.
It has been shown that FSE is the absolute minimum of
CSE. One cannot do better under any circumstances! This
justifies the name of the Fundamental Sampling Error, FSE.
5 Variance of FSE: devising a simplified formula: From
Eqs. (71)/(73), we write:
r2 FSEð Þ1 ¼
r2ðaSÞ1
a2L
¼ 1\ufffd P
PN 2U
X
m
ðam \ufffd aLÞ2
a2L
\ufffd M
2
m
M 2L
ð75Þ
So far, we have used only strict formulas, except in the
last steps when we introduced approximations. For the same
reasons as above, we can never implement this formula in
practice. It is to overcome this difficulty that we devised, as
early as 1950, an approximate way to estimate its order of
magnitude and that, later, we introduced the Heterogeneity
Invariant, HIL.
r2 FSEð Þ1 ¼
1\ufffd P
PNU
CHL ¼ 1\ufffd P
PML
HIL ¼
\ufffd
1
MS
\ufffd 1
ML
\ufffd
HIL
ð76Þ
boratory Systems 74 (2004) 25\u201338
ð78Þ
etc.
nt La
17. Variance of the Fundamental Sampling Error,
FSE\u2014experimental estimation
5 Experimental estimation of HIL: introduction: We always
recommend an experimental estimation, especially when
doubts may arise as to the validity of the estimate of the
liberation parameter b. We will now describe what is called
the bMethod of 50\u2013100 coarse fragmentsQ. From a practical
standpoint, this experimental method is feasible, and well
worth implementing when the top particle size d is coarser
than 10 mm.
5 Experimental estimation of HIL: method of 50\u2013100 coarse
fragments: The principle behind this approach is based on
the following observations:
! HIL can be expressed as a sum of terms corresponding to
each size class present:
HIL ¼
X
x
HIL½ \ufffdx ðdimension of massÞ ð81Þ
Beware of the fact that [HIL]x is different from HILx
where HIL½ \ufffdx ¼ vx
X
y
dy
ðaxy \ufffd axÞ2
a2L
\ufffd MLxy
MLx
ðdimension of massÞ ð82Þ
Several formulas have been proposed, including the author\u2019s
own. None seems completely satisfactory yet.
f : is a particle shape parameter ðdimensionlessÞ : 0V f V1
ð79Þ
In most cases, its value is near 0.50.
g : is a size range parameter ðdimensionlessÞ : 0VgV1
ð80Þ
In most cases, its value is 0.25. From 0.4 to 0.8, when the
material is calibrated (sized).
d: is the btop particle sizeQ defined as the aperture of the
square-mesh screen that retains 5% of the material
(dimension of length, to be expressed in cm). Also denoted
by d95 or dmax.
In expression Eq. (77) of HIL, we have succeeded in
transforming a sum extended to a very large and unknown
number of terms into a product of five factors only, the order
of magnitude of which can usually be estimated. Exception:
liberation parameter of gold ores (tricky).
In nearly all places where bThe FormulaQ is described in
the literature, one will find complete descriptions of these
four parameters and further details on the top particle size,
P. Gy / Chemometrics and Intellige
Because each term is proportional to the volume
vx=fdx
3, the order of magnitude of HIL is totally dominated
by the coarsest size classes, e.g., the size classes coarser
than dmax/2.
! The value of [HIL]1 can be expressed as the product of the
Heterogeneity Invariant HIL1 of the size class L1 multiplied
by the proportion ML1/ML.
HIL ¼ HIL1 \ufffd ML1=ML\ufffd½ ð83Þ
! HIL1 is an intrinsic property of the material that makes up
the size class L1 and can be estimated from a Test Sample, S,
on the conditions that:
n The fragments making up the test sample S have been
taken from L1 correctly (at random) and one by one
(independently).
n The number of fragments making up this sample is
large enough. Above, this range of large numbers
began with 30\u201350. This explains our choice of a
number larger than 50 (or better 100 with a reasonable
safety factor) as well as the name of the method. The
examples we are going to present below will confirm
this point.
n The major difficulty now lies in the estimation of the
proportion ML1/ML. When this proportion is unknown,
the following rules of thumb are often acceptable:
Natural, uncalibrated materials ( g=0.25): ML1/
ML=0.34 (or 34%).
Upward calibrated materials: ML1/ML=0.40\u20130.70.
5 Method of the 50\u2013100 fragments: introduction: We will
regard each fragment Fi as a size class in itself, with the
consequence that Eq. (81) becomes:
HIS ¼
X
i
ðai \ufffd aSÞ2
a2S
\ufffd M
2
i
MS
ðdimension of mass expressed in gramsÞ
ð84Þ
5 Method of the 50\u2013100 fragments: practical implementa-
tion: The protocol described here is a simplified version of
the method described in our books, see Part V.
! From the coarsest size fraction L1 of L, extract at
random and one by one at least 50 (better 100) fragments
Fi (with i=1,2,. . .,NF). We can operate either on the size
class L1 itself, if it is or can be isolated from the rest of lot
L or, more simply, by visual selection. According to
various experiments, when an operator is asked to select a
sample of the coarsest fragments of a given lot, he selects
fragments whose volume ratio ranges between 4/1 and 10/
1. For a fraction screened between sieves d and d/2, this
ratio would be 8/1. The visual appreciation is therefore
sufficient, and much cheaper, than a simple\u2014or even a
double\u2014screening. This set of NF fragments is what we
call the Test Sample S.
! Wash and dry all fragments of S, except when this
boratory Systems 74 (2004) 25\u201338 35
operation is prohibited for one reason or another (see, for
instance, example 3).
Implementation of the method: example 2: This test has
been carried out on the run-of-mine ore of an exotic mine
containing several precious metals in variable proportions.
These were partly free and partly associated with (trapped
in) various sulphides. This