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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