Geoscience Reference
In-Depth Information
characteristics of the material within the lot. But in practice
this is difficult, and thus the constitution heterogeneity is
multiplied by the average fragment mass. Doing this simplif-
cation defines the constant factor of constitution heterogene-
ity, also called the intrinsic heterogeneity, IH
L
, which can be
expressed and estimated using simple factors. The required
factors to define the intrinsic heterogeneity are:
•
d:
the nominal fragment size; d has units in centimetres.
•
f:
the shape factor accounts for the shape of the frag-
ments and is a measure of the fragment shape deviation
from a cubic shape. It is a dimensionless number, used to
estimate the volume of the fragment. Since the fragment
volume is equal to the product of the shape factor and the
cubed fragment size, f
α
d
α
3
, the shape factor is a correction
factor to determine its volume.
If the fragments are perfect cubes, f
α
= 1. If the fragments
are perfect unit spheres with r = 1, the volume of the
sphere would be 4/3 πr
3
= 0.523, therefore with shape fac-
tor f = 0.523.
It is dimensionless and is experimentally determined
with most minerals having a shape factor approximately
equal to 0.5: coal = 0.446; iron ore = 0.495 to 0.514; pure
pyrite = 0.470; quartz = 0.474; etc. Flaky materials, such as
mica, have a shape factor around 0.1; soft solids submitted
to mechanical stresses, such as gold nuggets, have a shape
factor around 0.2; and acicular minerals, such as asbestos,
have a shape factor greater than 1 and may be as large as 10.
•
g:
the granulometric factor accounts for the size differ-
ences between the fragments, also a dimensionless num-
ber. Using the granulometric factor, g, and the nominal
fragment size, d, the fragment size distribution can be
accounted for. The granulometric factor is a measure of
the range in fragment sizes in the sample:
− Noncalibrated material, crusher product, it is around
0.25.
− Calibrated material, between two screens, it is around
0.55.
− Naturally calibrated material, cereals or beans, it is
around 0.75.
•
c:
the mineralogical factor accounts for the maximum
heterogeneity condition that can be present in the sample;
units is in grams per centimetre cubed or specific gravity.
• ℓ
:
the liberation factor accounts for the degree of liberation
in the sample; it is a dimensionless number. When the mate-
rial is perfectly homogenous ℓ = 0 and when the mineral is
completely liberated ℓ = 1. Most materials can be classi-
fied according to their degree of heterogeneity. The libera-
tion factor varies considerably and it is difficult to assign
an accurate average to it. The calculation of the liberation
factor has evolved over time, and has changed since the
second edition of
Pierre Gy's Sampling Theory and Sam-
pling Practice
by F. Pitard (
1993
) was published. François-
Bongarçon and Gy wrote a paper (2001) presenting an
improved method for estimating the liberation factor. This
method corrected some of the problems associated with the
previous calculation and use of the liberation factor.
The constant factor of constitution or intrinsic heterogeneity
has units of mass (grams), and is used to relate the funda-
mental error to the mass of the sample:
3
IH
=
c fgd
L
5.3
Liberation Size Method
5.3.1
Fundamental Sample Error, FE
The fundamental sampling error, FE, is defined as the error
that occurs when the selection of the increments composing
the sample is correct. This error is generated entirely by the
constitution heterogeneity. Gy has demonstrated that the
mean, m(FE), of the fundamental error is negligible and that
the variance,
2
FE
σ
can be expressed as:
1P
IH
PM
−
2
FE
σ
=
L
L
where P is the probability of selection for any one fragment
within the lot and:
M
=
PM
S
L
Substituting this into the variance equation gives us:
†
11
IH
MM
‡
2
FE
σ
=
−
ˆ
‰
L
Š
‹
S
L
and when M
L
> > M
S
:
†
‡
1
2
FE
σ
=
ˆ
IH
‰
L
Š
M
‹
S
These are very practical and useful formulae for designing
and optimizing sampling protocols.
5.3.2
The Nomograph
To make use of the constitution heterogeneity and quantify/
show its effects on the sampling process we have to relate
the state that the sample is in, fragment size and mass, to the
state that we want the sample to be in, that is, with a smaller
mass and a smaller fragment size. The error produced during
