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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field of gravity, neighbouring
elements (fragments, molecules or ions) cannot be assumed
to be independent from one another (differential gravity
segregation). There is often some intrinsic correlation of the
element size, density or shape and its position within the
domain occupied by lot L, see e.g. Fig. 5B.
In this case, a second error is added to FSE, namely the
Grouping and Segregation Error, GSE. This error is a
consequence of the distributional heterogeneity, which is
itself a function of the constitutional heterogeneity and of
the increment size (the smaller the increment, the smaller
GSE). In the general case, applying to all real-world
systems, therefore:
CSE ¼ FSEþ GSE ð88Þ
19. Quantitative approach\u2014zero-dimensional model.
Difference between solids, liquids and aerosols
A final remark concerns the difference between
particulate solids, liquids and gases. Earlier, it was stated
that the difference between particulate solids on the one
hand, liquids and gases was not a difference in essence but
a mere difference of scale, of size. This has the following
consequences:
boratory Systems 74 (2004) 25\u201338 37
5 Particulate solids. Both the qualitative and the quanti-
tative approaches are equally important. We must
worry about the mean (qualitative approach: the 5 Defined the Constitutional Heterogeneity, CHL, of
lot L considered as a population of single elements.
P. Gy / Chemometrics and Intelligent Laboratory Systems 74 (2004) 25\u20133838
variance as well (quantitative approach). The most
dangerous errors, i.e., those which generate sampling
biases, take place when the major recommendation of
the qualitative approach is not respected. With solids,
all sampling must always be correct!
5 Liquids. The quantitative approach is here much less
important than the qualitative approach. This is due to
the fact that we are speaking of sizes in the angstrom
range, instead of mm or lm. We must above all worry
about the mean to prevent bias.
5 Gases and aerosols. Again, both aspects of the theory
are valid. Remark: When we studied the particular
problems associated with sampling gases and smokes,
both from a theoretical and practical standpoint,
together with a world leading cement producer, we
found that the sampling difficulties were here of a
different nature: the specific representativity problems
arose from interaction between the sampling tool or
device and the gas or smoke, when subsequently
cooling down the sample. This interaction depends on
the thermodynamics of the total smoke/sampling-tool
system. It is difficult, perhaps impossible, to prevent
some of the incorrect sampling errors in such a
situation. As far as we are aware, this problem has
not yet been satisfactorily solved, or addressed in a
proper scientific fashion.
20. Quantitative approach\u2014zero-dimensional model.
Recapitulation and conclusions
Sampling errors are the consequence of one form or
another of heterogeneity.
Sampling of a homogeneous material would, from the
definition of homogeneity, be an exact operation. But even if
homogeneity can be defined mathematically, it is never
observed in real-world systems.
In order to express the sampling errors in terms of their
mean, variance and mean square, we had first to define
mathematically\u2014and then quantify\u2014the various forms of
heterogeneity and then to express the moments of the
sampling errors as a function of the quantified heterogeneity.
To achieve this purpose, we have:
5 Defined the contribution h of a given unit U to the
heterogeneity of the set L of units. Unit U can be
either a single constituent F or a group G of
constituents such as an increment I. The heterogeneity
contribution h is a function of the mass and
composition of unit U and lot L.
CHL is the variance of the corresponding population
of h.
5 Defined the Heterogeneity Invariant, HIL, derived from
CHL for practical purposes and usage.
5 Defined the Distributional Heterogeneity, DHL, of lot
L considered as a population of groups of neighbouring
elements. DHL is the variance of the corresponding
population of h.
5 Defined the Total Sampling Error, TSE, generated
when selecting constituents in a probabilistic way
(non-probabilistic sampling cannot be analyzed
theoretically).
5 Broken up the Total Sampling Error, TSE, into the sum
of two components, CSE and ISE.
5 Defined the Correct Sampling Errors, CSEs, observed
when the sampling is correct.
5 Defined the additional Incorrect Sampling Errors,
ISEs, observed when the sampling is incorrect.
5 Defined the Fundamental Sampling Error, FSE, as the
Correct Sampling Error, CSE, observed in ideal
conditions, when the constituents are selected correctly,
one by one and independently.
5 Shown that the variance of FSE is proportional to the
Constitutional Heterogeneity, CHL, and, in practical
applications, to the Heterogeneity Invariant, HIL.
5 Proposed a practical, experimental method to estimate
HIL and hence the variance of FSE.
5 Defined the Grouping and Segregation Error, GSE, as
the additional error generated when selecting constit-
uents with a uniform probability P, by groups (incre-
ments) of non-independent constituents. The variance
of GSE is proportional to the Distributional Hetero-
geneity, DHL.
The major purpose of Parts I and II is to help the
reader understand the delicate mechanisms involved in
the sampling operation. It is hoped that we have shown
how the consecutive steps of the theoretical\u2014and
practical\u2014reasoning are intimately linked to one
another. This tutorial can only be a summary of the
subject matter. For the reader who wishes to know full
details it is necessary to study Refs. [20] and [18] (in
this order).
External references for part II
Bastien, M. (1960). Loi du rapport de deux variables
normales. Revue de Statistique Applique´e, vol. 8, pp 45\u201350
(1960).
Geary, R.C. (1930). The frequency Distribution of the
Quotient of Two Normal Variables. J. Roy. Stat. Soc. 93,
442 (1930).
sampling must be correct, therefore accurate) and the
	Sampling of discrete materials
	Joint introduction of parts II and III: three-, two-, one-, zero-dimensional models
	Definitions and notations
	Zero-dimensional model-contribution made to the heterogeneity of lot L by an unspecified unit Um
	First case: unit Um is a single element Fi-definition of the Constitutional Heterogeneity CHL of lot L
	Second case: unit Um is a group Gn of neighboring elements Fi-definition of the Distributional Heterogeneity DHL of lot L
	Relationship between CHL and DHL
	Illustration of the principal forms of homogeneity and heterogeneity
	Practical implementation-the Heterogeneity Invariant, HIL
	The zero-dimensional probabilistic sampling model
	Distribution of the random variables Pim, NK, MK, AK and aK
	Expected value of the critical content aS of sample Sk
	Variance of the critical content aS of the Correct Sample, S
	Expected value of the Correct Sampling Error, CSE
	Variance of the Correct Sampling Error, CSE
	Definition of the Incorrect Sampling Error, ISE
	Practical implementation of the above formulas
	Definition of the Fundamental Sampling Error (FSE)-derivation of the FORMULA expressing the fundamental variance sigma2(FSE)
	Variance of the Fundamental Sampling Error, FSE-experimental estimation
	Breaking up the Correct Sampling Error, CSE
	Quantitative approach-zero-dimensional model. Difference between solids, liquids and aerosols
	Quantitative approach-zero-dimensional model. Recapitulation and conclusions