Sampling of Particulate Materials Theory and Practice

Developments in Geomathematics 4: Sampling of Particulate Materials: Theory and Practice reviews the theory and practice of sampling particulate solids, such as ores and concentrates. With examples borrowed from the mining, metallurgical, and cement industries, the book examines particulate materials of vegetable and mineral origin, including cereals, oil seeds, sugar beets, granulated drosses or slags, bars, plates, and ingots. Organized into nine parts encompassing 34 chapters, this volume begins with an overview of the theory of sampling and sampling-error generating mechanisms. It then discusses the continuous selection and discrete models of the increment sampling process and the materialization of punctual increments. It explains the splitting process and its practical implementation in sampling. Lot and sample preparation, resolution of sampling problems, and problems associated with commercial sampling are also discussed. The book also describes the detection of measurement or sampling biases and inconspicuous losses of material, and the design of automatic sampling plants. This book is a valuable resource for geologists, mining engineers, metallurgists, and analysts.

IntroductionHistorical SummaryFirst Part - Analysis of the Problem Chapter 1: Definition of Basic Terms and Notations 1.1. Definition of basic terms 1.2. Statistical definitions and notations 1.3. Estimators - Estimates - Errors - Biases 1.4. Domains and their extent 1.5. Properties of a selective process 1.6. Specific notations Chapter 2: Logical Approach 2.1. First part - Analysis of the problem 2.2. Second part - Continuous model of the increment sampling process 2.3. Third part - From the continuous model to the discrete reality 2.4. Fourth part - Discrete model of the increment sampling process 2.5. Fifth part - Splitting processes 2.6. Sixth part - Preparation 2.7. Seventh part - Resolution of sampling problems 2.8. Eighth part - Problems associated with commercial sampling 2.9. Ninth part - Automatic sampling plants Chapter 3: Part of Sampling in Quality Control 3.1. Introduction 3.2. Possibility of direct estimation of a critical content 3.3. Possibility of direct extraction of the analysis sample 3.4. Preparation stages - Preparation errors 3.5. Successive estimators and eventual estimate of the critical content 3.6. Breaking up of the overall estimation error Chapter 4: Sampling Processes 4.1. Probabilistic and non-probabilistic sampling processes 4.2. Comments on non-probabilistic sampling processes 4.3. Probabilistic processes - Notion of movable and unmovable batches 4.4. Sampling of unmovable lots 4.5. Sampling of movable lots 4.6. Analysis of the increment sampling process 4.7. Analysis of the splitting process 4.8. Comparison of the increment sampling and splitting processes 4.9. Field of application of the increment sampling and splitting processes Chapter 5: Models of the Increment Sampling Process 5.1. What is a model and what is the use of it? 5.2. Introduction to a group of models of the sampling processes 5.3. Continuous mode 5.4. Discrete model 5.5. Objectives pursued when developing selection models 5.6. Resolution of sampling problemsSecond Part - Continuous Model of the Increment Sampling Process Chapter 6: Heterogeneity of a Continuous Set 6.1. Introduction 6.2. Definition and properties of a homogeneous material 6.3. Description of a heterogeneous material 6.4. Definition of the variogram 6.5. General properties of the variogram 6.6. Experimental determination of the variogram - Logical approach 6.7. Interpretation of the results of a variographic experiment 6.8. Particular case of zero-dimensional lots 6.9. Conclusions Chapter 7: Reference Selection Schemes 7.1. Introduction 7.2. Systematic selection with random positioning 7.3. Random stratified selection 7.4. Random selection 7.5. Full description of a selection scheme 7.6. Comparison of the three selection schemes 7.7. Field of application of the three selection schemes Chapter 8: Development of the Continuous Selection Model Continuous Selection Error CE 8.1. Introduction 8.2. Definition of the model 8.3. Distribution of the weight ΜS of active components in the sample S 8.4. Distribution of the weight ΑS of critical component in the sample S 8.5. Correlation between the distributions of MS and AS 8.6. Distribution of the critical content aS of the sample S - Introduction 8.7. Distribution of the critical content aS of the sample S - Case No. 1 8.8. Distribution of the critical content aS of the sample S - Case No. 2 8.9. Distribution of the critical content aS of the sample S - Case No. 3 8.10. Distribution of the critical content aS of the sample S - Case No. 4 8.11. Distribution of the critical content aS of the sample S - Case No. 5 8.12. Distribution of the critical content aS of the sample S - Case No. 6 Chapter 9: Breaking Up of the Continuous Selection Error CE 9.1. Introduction 9.2. Definition of the weighting error and of the quality fluctuation error 9.3. Analysis of the quality fluctuation error QE 9.4. Moments of the quality fluctuation error QE 9.5. Recapitulation Chapter 10: Short-Range Quality Fluctuation Error QE₁ 10.1. Definition 10.2. Mean of QE₁ 10.3. Variance of QE₁ 10.4. Cancelling and minimizing of QE₁ 10.5. Further analysis of the short-range quality fluctuation error QE₁ Chapter 11: Long-Range Quality Fluctuation Error QE₂ 11.1. Definition 11.2. Mean of QE₂ 11.3. Variance of QE₂ 11.4. Comparison of the three selection schemes 11.5. Cancelling and minimizing of σ² (QE₂ ) Chapter 12: Periodic Quality Fluctuation Error QE₃ 12.1. Definition 12.2. Occurrence of periodic fluctuations 12.3. Moments of QE₃ 12.4. Comparison of the three selection schemes 12.5. Example 12.6. General conclusion concerning the choice of a selection scheme Chapter 13: Weighting Error WE 13.1. Definition 13.2. Mean of WE 13.3. Variance of WE 13.4. Constant tonnage sampling systems 13.5. Error resulting from the non-uniformity of the cutter speed from one increment to the next 13.6. Particular rate of flow functions 13.7. Conclusions Chapter 14: Practical Implementation of the Continuous Model Variographic Experiment 14.1. Introduction 14.2. Organization of a variographic experiment 14.3. Analysis of a simple periodic variogram 14.4. Analysis of a non-periodic variogram Chapter 15: Practical Implementation of the Continuous Model Error Estimation 15.1. Introduction 15.2. Example No. 1 - Lead ore 15.3. Example No. 2 - Feed to the blending system of a cement plantThird Part - From the Continuous Model to the Discrete Reality Materialization of the Punctual Increments Chapter 16: Components of the Materialization Error ME 16.1. Introduction 16.2. Increment sampling of flowing streams 16.3. Recapitulation 16.4. Selection, extraction and sampling probabilities Chapter 17: Increment Delimitation Error DE 17.1. Definition 17.2. Falling stream sampling - correctness conditions involving the cutter geometry 17.3. Falling stream sampling - Correctness conditions involving the cutter speed 17.4. Falling stream sampling - Correctness conditions involving the general lay-out 17.5. Falling stream sampling - Recapitulation of the conditions of delimitation correctness 17.6. Stopped belt sampling 17.7. Two-dimensional sampling 17.8. Cost of correct delimitation Chapter 18: Increment Extraction Error EE 18.1. Definition 18.2. Analysis of the rebounding rule 18.3. Conditions of extraction correctness involving the material to be sampled 18.4. Conditions of extraction correctness involving the cutter characteristics 18.5. Cutter width and velocity - Experimental determination of the critical characteristics 18.6. Recapitulation 18.7. Cost of extraction correctnessFourth Part - Discrete Model of the Increment Sampling Process Chapter 19: Heterogeneity of a Discrete Set 19.1. Introduction and notations 19.2. Definition and properties of homogeneous and heterogeneous materials 19.3. Characterizing the heterogeneity of a discrete set 19.4. General expression of the distribution heterogeneity Chapter 20: Development of the Discrete Selection Model 20.1. Introduction and notations 20.2. Distributions of NZm, NSk, MSk and ASk 20.3. Distribution of aS critical content of the sample S - Incorrect selection 20.4. Practical implementation of formulas involving sums Σi 20.5. Moments of NS, MS, AS and aS - Correct selection 20.6. Moments of NS, MS, AS and aS - Correct selection - Uniform weighting 20.7. Practical implementation of the results of the discrete model Chapter 21: Linking Up of the Continuous and Discrete Models Fundamental Error FE - Grouping and Segregation Error GE 21.1. Introduction and notations 21.2. Moments of SE = QE₁ according to the continuous and discrete models 21.3. Analysis of the short-range quality fluctuation error QE₁ 21.4. Cancelling and minimizing of the fundamental error FE 21.5. Cancelling and minimizing of the grouping and segregation error GE Chapter 22: Practical Implementation of the Theoretical Results - Correct Selection 22.1. Introduction 22.2. Estimation of the moments of the fundamental error FE - Introduction of Y and Z 22.3. Estimation of Y and Z - Method No. 1 22.4. Estimation of Y and Z - Method No. 2 22.5. Estimation of Y and Z - Method No. 3 22.6. Properties and practical estimation of c, ℓ, f , g and d 22.7. Resolution of sampling problems involving the fundamental variance 22.8. Practical application of methods No. 1 , 2 and 3 22.9. Recapitulation and conclusions Chapter 23: Practical Implementation of the Theoretical Results - Incorrect Selection 23.1. Introduction 23.2. Incorrect extraction curve 23.3. Practical determination of the curve of incorrect extraction 23.4. Examples 23.5. ConclusionsFifth Part - Splitting Process Chapter 24: Splitting Methods and Devices 24.1. Introduction 24.2. True and degenerate splitting processes 24.3. Coning and quartering 24.4. Riffling 24.5. Fractional shovelling 24.6. Sectorial splitters Chapter 25: Model of the Splitting Process - Splitting Errors 25.1. Linking up with the existing models 25.2. Moments of the continuous selection error CE 25.3. Minimizing of σ² (CE) 25.4. Delimitation error DE 25.5. Extraction error EE Chapter 26: Practical Implementation of Splitting Processes - Example - Reduction of Drill Core Samples 26.1. Introduction 26.2. Core sample reduction methodology 26.3. Selection of a core sample reduction scheme 26.4. Examples 26.5. RecommendationsSixth Part - Lot and Sample Preparation Chapter 27: Preparation Errors 27.1. Introduction 27.2. Errors resulting from contamination 27.3. Errors resulting from losses 27.4. Errors resulting from alteration of the chemical composition 27.5. Errors resulting from alteration of the physical composition 27.6. Errors resulting from unintentional mistakes 27.7. Errors resulting from frauding or sabotage 27.8. ConclusionsSeventh Part - Resolution of Sampling Problems Chapter 28: Recapitulation of the Sampling Error 28.1. Analysis of the overall estimation error 28.2. Fundamental error FE 28.3. Grouping and segregation error GE 28.4. Long-range quality fluctuation error QE₂ 28.5. Periodic quality fluctuation error QE₃ 28.6. Weighting error WE 28.7. Increment delimitation error DE 28.8. Increment extraction error EE 28.9. Preparation errors PE 28.10. Conclusions Chapter 29: Solvable and Unsolvable Sampling Problems 29.1. Definitions 29.2. Representativeness and cost 29.3. Sampling of three-dimensional objects 29.4. Sampling of two-dimensional objects 29.5. Sampling of one-dimensional stationary objects 29.6. Sampling of one-dimensional flowing streams 29.7. Sampling of zero-dimensional objects 29.8. Sampling of small or valuable objects 29.9. ConclusionsEighth Part - Problems Associated With Commercial Sampling Chapter 30: Notion of Equity 30.1. Introduction - Definition 30.2. Properties of the settlement price assumed to be a linear function of the critical content 30.3. Properties of the settlement price assumed to be a non-linear function of the critical content 30.4. Relative importance of bias and random error in commercial sampling 30.5. Conclusions - Recommendations 30.6. Equity - Louis-le Dēbonnaire's splitting method Chapter 31: Testing the Agreement Between Two Series of Independent Estimates of a Same Characteristic - Discrepancies Between Seller and Buyer 31.1. Introduction 31.2. Notations and definitions 31.3. Testing the hypothesis H = {D = 0 } 31.4. Testing the hypothesis H' = { D‘ =

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