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with the results shown in Table 1.7 and the tolerable sampling error for each fraction is ±5%. The amount required for each fraction, in order to keep within this limit, equation (1.15) is given in column 4. F^ or a sieve analysis, in order to reduce the errors at the coarse end of the distribution, repeat analyses should be made using only the coarsest sieves. For the example above an error of 100% in the coarsest size range may be acceptable, i.e. (0.1 ±0.10) g; this reduces the required weight to 94 g. Table 1.8. Minimum incremental mass required for sampling from a stream of powder Maximum particle size (mm) 250-150 150-100 100-50 50-20 20-10 10- 0 Minimum mass of increment (kg) 40.0 20.0 12.0 4.0 0.8 0.3 1.14.2 Sampling by increments For sampling a moving stream of powder the gross sample is made up of increments. In this case the minimum incremental weight is given by: M^ = l^ o^fo (1.16) 52 Powder sampling and particle size determination where: M- is the average mass of the increment, //Q is the average rate of flow, WQ is the cutter width for a traversing cutter, VQ is the cutter velocity. If M^o is too small, a biased sample deficient in coarse particles, results. For this reason w^ should be at least ?>d where d is the diameter of the largest particle present in the bulk. ISO 3081 suggests a minimum incremental mass based on the maximum particle size in mm. These values are given in Table 1.8. Secondary samplers then reduce this to analytical quantities. Example 10 Determine the minimum increment weight for a powder falling from a belt conveyer at a rate of 3 metric tons per hour if the size of the largest particle is 1.0 mm and the sampling cutter speed is 6 cm s~^ M^ r . . . . n 3 . . u - i V 3 ^ 1 0 - ^ m ^ 3x10^ kgh" 3600 s h~^ 0.06 ms"* M, -42g Since the flowrate is 833 g s~^ this is not a practical amount, hence a two- stage sampler is required. Sampler 1, say, can sample for 2 s to generate 1.67 kg of powder, which is fed to a hopper to provide a feed to a second sampler that reduces it by a factor of 40 to generate the required 42 g. The minimum number of increments required to give an acceptable accuracy for the sampling period is 35, hence, the gross sample weight is given by: A/,.= 1.47 kg The gross sample can be reduced to a laboratory sample of about 10 g, using a Vezin type sampler for example, and finally to a measurement sample of about 1 g using a rotary riffler. If the particle size analysis is carried out on less than 1 g the final reduction is usually effected by dispersing the powder in a liquid and pipetting out the required aliquot. Powder sampling 53 Gy [45] proposed an equation relating the standard deviation, which he calls the fundamental error a)r, to the sample size: -I- \w W Cd^ (1.17) where Wis the mass of the bulk and w = nco is the mass of « increments, each of weight &> which make up the sample, C is the heterogeneity constant for the material being sampled and d is the size of the coarsest element. For the mining industry [46] he expressed the constant C in the form C = clfg where: \-P C = p (1.18) P is the investigated constant; p is the true density of the material; / is the relative degree of homogeneity, for a random mixture / = 1, for a perfect mixture / == 0; / is a shape factor assumed equal to 0.5 for irregular particles and 1 for regular particles; g is a measure of the width of the size distribution, g = 0.25 for a wide distribution and 0.75 for a narrow distribution (i.e. d^^^y^ < 2d^^^). For the pharmaceutical industry Deleuil [35] suggested C = 0.1/c with the coarsest size being replaced by the 95% size. For W»w equation (1.17) can be written. w(9^-0.1/ ]-P IP' , .3 T3 pfd' (1.19) where: 0 = t^-^ and t = 3, (99.9% confidence level) for total quality, \u2022 For t/95 = 100 )im, p = \.5, P = 10-^ (1000 ppm), 6 = 0.2, / = 0.03 (random), w = 1000 g. \u2022 For dg^ = 100 |Lim, /? - 1.5, P = 0.05, 0= 0.05, / = 1 (homogeneous), w = 4 g. \u2022 For 9^5 = 20 |um, p^ 1.5, P - 10"^ (100 ppm), 0 = 0.05, / - 0.03 (random), w = 8000 g. 54 Powder sampling and particle size determination Deleuil points out that a sample of this weight is never prepared because the lot is considered to be perfectly homogeneous (/ = 1). The product from industrial grinding circuits oscillates due to variation in hardness and particle size distribution of feed. Heiskanen and Niemelia [47] demonstrated that, using automatic sampling, on-line analysis and autocorrelation procedures, they could map out a frequency of oscillation. References 1 Sommer, K. (1981), Aufbereit Tech., 22(2), 96-105, 2 2 Cornish, D.C., Jepson, G. and Smurthwaite, M.J. (1981), Sampling for Process Analyzers, Butterworth, 4,38 3 Julian, R., Meakin, P. and Pavlovich, A. (1992), Phys. Rev. Let. 69, 5 4 Maddox, J. (1992), Nature, 358, 5 5 Kaye, B.H. and Naylor, A.G. (1972), Particle TechnoL, 47-66, 11, 25, 28,30 6 ASTM D451 -63 (1963), Sieve analysis of granular mineral surfacing for asphalt roofing and shingles, 12 1 Jillavenkatesa, A., Dapkunas, S.J. and Lum,L-S. H. (2001), Particle size characterization. National Institute of Standards and Technology, NIST Sp. Publ. 960-1,72 8 ASTM C322-82 (1982), Standard practice for sampling ceramic whiteware clays, 12 9 ASTM D1900-94 (1994^, Standard practice for carbon black-sampling bulk shipments, 12 10 ASTM D75-97 (1997), Standard practice for sampling aggregate, 12 11 ASTM 8215-96 (1996), Standard practice for sampling finished lots of metal powders, 12 12 Hulley, B.J. (1970), Chem Engr., CE 410-CE 413, 7P 13 Clarke, J.R.P. (1970), Measurement and Control. 3, 241-244, 19,36,49 14 Cornish, D.C., Jepson, G. and Smurthwaite, M.J. (1981), Sampling for process analyzers, Butterworth, 20 15 Heuer, M. and Schwechten, D. (1995), Partec 95, 6th European Symp. Particle Characterization, 301-314, Numberg, Germany, publ. NUmbergMesse GmbH, 23 16 Witt, W. and Rothele, S. (1998), 7th European Symp. Particle Characterization, 611-624, Numberg, Germany, publ. NUmbergMesse GmbH, 24 17 ASTM DC702-98 (1998), Standard practice for reducing samples of aggregates to testing size, 24 18 Kaye, B.H. (1961), Ph.D. thesis. University of London, 25 19 Batel. W. (1960), Particle size measuring techniques, Springer Verlag, Germany, 28 Powder sampling 55 20 Wentworth, C.K., Wilgers, W.L. and Koch, H.L. (1934), A rotary type of sample divider, J. Sed Petrol., 4, 127, 28 21 Pownall, J.H. (1959), The design and construction of a large rotary sampling machine, AERE-R-2861, Harwell, Oxfordshire, UK, UKAEA, 28 22 Hawes, R. and Muller, L.D. (1960), A small rotary sampler and preliminary studies of its use, AERE, R3051, Harwell, UKAEA, 29, 46 23 BS3406 (1961), Methods for determination of particle size of powders. Part I Sub-division of gross sample down to 2 ml, 30, 36 24 Fooks, J.C. (1970), Sample splitting devices, Br. Chem. Engr., 15(6), 799, 34 25 Osborne, B.F. (1972), CM Bull, 65, 97-107, 36 26 Cross, H.E. (1967), Automatic mill control system, Parts I and II, Mining Congress J., 62-67, 36 27 Hinde, A.L. and Lloyd, PJ.D. (1975), Powder Technol,. 12, 37-50, 36 28 Hinde A.L (1973), J. S. Afr. Inst. Min. MetalL, 73, 26-28, 36 29 Nasr-el-Din, H., Shook, C.A. and Esmail. M.N. (1985), Can. J. Chem. Engr,, 63, 746-753, 36 30 Burt, M.W.G (1967), Powder Technol, 1, 103,, 36 31 Lines, RW. (1973), Powder Technol., 7(3), 129-136, 36 32 BS616 (1963), Methods for sampling coal tar and its products, 38 33 Burt, M.W.G., Fewtrel. C.A. and Wharton, R.A. (1973), Powder Technol, 7(6), 327-330, 38 34 Herden, G. (1960), Small particle statistics, Butterworths, 38 35 Deleuil, M. (1994), Handbook of powder technology, No 9, Powder Technology and Pharmaceutical Processes, Ch. 1, ed. D. Chulia, M. Deleuil and Y. Pourcelet, Elsevier, 38,53 36 Davies, R. (1982), Kirk Othmer, Encyclopedia of Chemical Technology,