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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