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# The sampling of particulate materials a general theory

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Internat ional Journal o f Mineral Processing, 3 (1976) 289--312 289 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands Rev iew Papers THE SAMPL ING OF PART ICULATE MATERIALS -- A GENERAL THEORY* PIERRE M. GY School o f Geology, Nancy/Consul t ing Engineer, Cannes (France) (Received June 15, 1976) ABSTRACT Gy, P.M., 1976. The sampling of particulate materials -- a general theory. Int. J. Miner. Process., 3: 289--312. This paper is a summary of the general theory of sampling published (in French) by the author in 1975 (to be published in English by Elsevier, probably in 1977). Sampling is a random process and its theory is a study of the numerous errors liable to take place in the course of its realization. A complete sampling scheme is a sequence of sampling (proper) and preparation stages: (A) At each stage, the total error ET is the sum of: (1) Sampling errors EE arising from the selection process itself. (2) Preparation errors EZ arising from the operations (crushing, transfer...) supported by the sampled material. So: ET =- EE + EZ (B) At each sampling stage, the (total) sampling error EE is the sum of seven indepen- dent errors: (1) Weighting error ED resulting from the non-uniformity of the density or rate-of-flow of the sampled material. (2) Integration error EI 1 resulting from the long-range distribution heterogeneity of the sampled material. (3) Periodicity error EI~ resulting eventually from periodic quality variations of the sampled material. (4) Fundamental error EF resulting from the constitution heterogeneity of the sampled material. (5) Segregation error ES resulting from the local distribution heterogeneity of the sampled material. (6) Delimitation error EC resulting eventually from an incorrect shape of the volume delimit- ing the increments. (7) Extraction error EP resulting from the actual extraction of the increments. So: EE ~ El) +EI I + E I 3 + EF + ES + EC + EP (C) At each preparation stage, the (total) preparation error EZ is the sum of five inde- pendent errors: *This paper was first presented, and published by the Australasian Institute of Mining and Metallurgy, at a symposium on the Sampling Practices in the Mineral Industries, Melbourne, Victoria, Australia, September 16--17, 1976. 290 (1) Error EZ~: loss of particles belonging to the sample. (2) Error EZ2: contamination of the sample by foreign material. (3) Error EZ~: alteration of the critical characteristic to be measured on the final sample. (4) Error EZ4: unintentional mistakes of the operator (e.g. mixing sub-samples belonging to different samples). (5) Error EZs: intentional alteration of the characteristic to be measured on the final sample. So: EZ = EZ I + EZ~ + EZ 3 + EZ 4 + EZ 5 Two complementary models of the sampling process are thoroughly developed to study these errors: (1) The integration model dealing with the continuous, geometrical properties of the sampled material. (2) The probabilistic model dealing with the discontinuous, physical properties of the sampled material. The errors ED, E I , El3, EF and ES can be quantitat ively defined. Their mean (bias) and variance can be estimated from the results of a variographic experiment. The errors EC, EP and EZ can only be qualitatively defined. They cannot be experi- mentally estimated but rules are given making it possible to suppress them and particularly to cancel out the always dangerous sampling biases. INTRODUCTION I have been struck by a comment made recently by an Australian friend: "In this country most companies regard sampling as an unavoidable overhead and in many cases they spend as little money as possible on it." Such a misunderstanding of the random nature of sampling, such a misap- preciation of the risks attached to it show how necessary is a paper on this subject. The first thing that should be emphasized is that sampling is not a simple mechanical technique like crushing, for instance: it is a random process liable to introduce errors such as in chemical analysis. But whereas this latter is al- ways carried out in laboratory conditions (I was tempted to say in "aseptic" conditions) by a well-trained specialized staff conscious of the necessity of ac- curacy and precision, sampling is usually carried out under field or plant condi- tions by unspecialized labour perfectly unaware of the importance of their work and unconscious of the mistakes to be avoided at any cost. Sampling and analysis (chemical, size or moisture analysis) are the two com- plementary links of the quality estimation chain with the consequence that the total estimation error is the sum of the sampling error and of the analysis error. The optimization of the "accuracy-cost" characteristics of an estimation de- mands that the same care be taken in sampling and in analysis. Sampling should always be placed under the responsibility of the head of "quality control", not of the head of "production". It should be carried out by a specialized staff conscious of the numerous errors that may take place and knowing how to suppress or reduce these. 291 One may judge a crusher or a screen after its mechanical performance, not a sampler: the only touchstone of a sampler is its aptitude to avoid a certain number of errors and to maintain the others at an acceptable level. One may judge the products of a crusher or of a screen after the results of a simple test which is easy to carry out: the products contain the proofs of their qualities. This is not true of a sample: after it has been extracted from the lot, there is no way of recognizing a "good" sample from a "bad" one. But what is a "good" sample and why is this other one "bad"? It is the ob- ject of the sampling theory to answer this apparently simple but really subtle question. The sampling theory is nothing other than a thorough study of the sampling errors. QUALITIES OF A SELECTION PROCESS Sampling is a complex selection process. A selection process may be qualified: {1) Either in terms of "a priori qualities", or (2) in terms of "a posteriori qualities". (1) The "a priori qualities" are defined after the conditions of the selection: - - A selection process is said to be "probabilistic" whenever each element of the lot is submitted to the selection with a given probability of being selected. -- It is said to be "non-probabilistic" whenever it is not founded on the notion of probability. The "hammer and shovel" sampling method, based as it is on a purposive selection of the material destined to the sample, is a non-proba- bilistic method. Such methods are inaccessible to a theoretical approach, they are therefore excluded from our study. They are usually heavily biased and should therefore be rejected. --A selection process is said to be "correct" whenever all elements of the lot are submitted to the selection with a uniform probability (or density of probability) of being selected. -- It is said to be "incorrect" when, being probabilistic, the above condition is not fulfilled. (2) The "a posteriori qualities" are based on the results of the selection and more particularly on the statistical properties of the selection error e, relative difference between the critical content aE of the sample E and the critical content aL of the sampled lot L: e = (aE - aL)/aL - - A selection process is said to be "unbiased" when the mean (m) of the selec- tion error is nil: re(e) =0 ~ m(aE) =aL -- It is said to be "biased" when the mean is not nil. The value of the mean is the "bias" B or relative systematical error: B=m(e)--/=O -+ m(aE) ¢aL 292 -- It is said to be "reproducible" when the variance of the selection error is not larger than a given "reproducibility standard" %.2. 2 o 2 (e) < o o -- It is said to be "exact" when the selection error is always nil: m (e) = 0 and o 2 (e) = 0 - - It is said to be