Representation of results of particle size analysis Part 4: Characterization of a classification process BS ISO 9276-4:2001+A1:2017 BSI Standards Publication WB11885_BSI_StandardCovs_2013_AW.indd 1 15/05/2013 15:06National foreword This British Standard is the UK implementation of ISO 9276-4:2001+A1:2017. It supersedes BS ISO 9276-4:2001, which is withdrawn. The start and finish of text introduced or altered by amendment is indicated in the text by tags. Tags indicating changes to ISO text carry the number of the ISO amendment. For example, text altered by ISO amendment 1 is indicated by . The UK participation in its preparation was entrusted to Technical Committee LBI/37, Particle characterization including sieving. A list of organizations represented on this committee can be obtained on request to its secretary. This publication does not purport to include all the necessary provisions of a contract. Users are responsible for its correct application. © The British Standards Institution 2017 Published by BSI Standards Limited 2017 ISBN 978 0 580 92677 8 ICS 19.120 Compliance with a British Standard cannot confer immunity from legal obligations. This British Standard was published under the authority of the Standards Policy and Strategy Committee on 15 July 2001. Amendments/corrigenda issued since publication Date Text affected 30 November 2017 Implementation of ISO amendment 1:2017 BRITISH STANDARD BS ISO 9276-4:2001+A1:2017© ISO 2001 Representation of results of particle size analysis — Part 4: Characterization of a classification process Représentation de données obtenues par analyse granulométrique — Partie 4: Caractérisation d un processus de triage INTERNATIONAL STANDARD ISO 9276-4 First edition 2001-07-15 Reference number ISO 9276-4:2001(E) BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E)ii © ISO 2001 – All rights reserved COPYRIGHT PROTECTED DOCUMENT © ISO 2001, Published in Switzerland All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below or ISO’s member body in the country of the requester. ISO copyright office Ch. de Blandonnet 8 • CP 401 CH-1214 Vernier, Geneva, Switzerland Tel. +41 22 749 01 11 Fax +41 22 749 09 47

[email protected] www.iso.org BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E)Foreword iv Introduction v 1 Scope . 1 2 Symbols 1 2.1 Symbols for specific terms 1 2.2 Subscripts 2 3 Characterization of a classification process based on error-free distribution curves and mass balances 2 3.1 Distribution density curves representing a classification process 2 3.2 Mass and number balances 4 3.2.1 Mass and number balance in the size range from x minto x max4 3.2.2 Mass and number balance in the size range from x to x + dx . 4 3.2.3 Mass and number balance in the size range from x minto x 5 3.2.4 The indirect evaluation of v r,fand v r,c5 3.3 Definitions of cut size, x e5 3.3.1 General 5 3.3.2 The equiprobable cut size, x e , the median of the grade efficiency curve. 5 3.3.3 The analytical cut size, x a5 3.4 Grade efficiency, T, the grade efficiency curve, T(x), (Tromp s curve) 6 3.5 Measures of sharpness of cut . 7 3.5.1 General 7 3.5.2 Parameters formed with characteristic particle sizes . 7 3.5.3 Parameters derived from cumulative distribution curves . 8 3.5.4 The total classification or separation efficiency, T o. 9 4 The influence of systematic errors on the determination of grade efficiency curve 10 4.1 General 10 4.2 Systematic error due to a splitting process in the classifier .10 4.3 Incomplete dispersion of the feed material 11 4.4 The influence of comminution of the feed in the classifier 11 Annex A (informative) The influence of stochastic errors on the evaluation of grade efficiency curves.12 Bibliography .17 © ISO 2001 – All rights reserved iii Contents Page BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E) Foreword ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies (ISO member bodies). The work of preparing International Standards is normally carried out through ISO technical committees. Each member body interested in a subject for which a technical committee has been established has the right to be represented on that committee. International organizations, governmental and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 3. Draft International Standards adopted by the technical committees are circulated to the member bodies for voting. Publication as an International Standard requires approval by at least 75 % of the member bodies casting a vote. Attention is drawn to the possibility that some of the elements of this part of ISO 9276 may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. International Standard ISO 9276-4 was prepared by Technical Committee ISO/TC 24, Sieves, sieving and other sizing methods, Subcommittee SC 4, Sizing by methods other than sieving. ISO 9276 consists of the following parts, under the general title Representation of results of particle size analysis: — Part 1: Graphical representation — Part 2: Calculation of average particle sizes/diameters and moments from particle size distributions — Part 3: Fitting of an experimental cumulative curve to a reference model — Part 4: Characterization of a classification process — Part 5: Validation of calculations relating to particle size analyses using the logarithmic normal probability distribution Annex A of this part of ISO 9276 is for information only.iv © ISO 2001 – All rights reserved BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E) Introduction In classification processes used in particle size analysis, such as occurring in impactors, sieves, etc., the mass of the supply or feed material, m s , or its number, n s , of particles, the particle size distribution of which is described by its distribution density , q r,s (x), is separated into at least one fine fraction of mass, m f , or number, n f , and of distribution density , q r,f (x) and a coarse fraction of mass, m c , or number, n c , and a distribution density , q r,c (x). The type of quantity chosen in the analysis is described by the subscript, r, the supply or feed material and the fine and coarse fractions by the additional subscripts: s; f and c respectively. See Figure 1. Figure 1 — Fractions and distributions produced in a one step classification process For the characterization of processes with more than one coarse fraction, e.g. cascade impactors, s, f and c can be replaced by numbers 0, 1 and 2. In this case e.g. number 3 describes a second coarse fraction containing larger particles than fraction 2. It is assumed that the size, x, of a particle is described by the diameter of a sphere. Depending on the problem, the particle size, x, may also represent an equivalent diameter of a particle of any other shape.© ISO 2001 – All rights reserved v BS ISO 9276-4:2001+A1:2017This page deliberately left blank Representation of results of particle size analysis — Part 4: Characterization of a classification process 1 Scope The main object of this part of ISO 9276 is to provide the mathematical background for the characterization of a classification process. This part of ISO 9276 is not limited to an application in particle size analysis, the same procedure may be used for the characterization of a technical classification process (e.g. air classification, centrifugal classification) or a separation process (e.g. gas or hydrocyclones). In clause 3 the characterization of a classification process is described under the presupposition that the distribution density curves describing the feed material and the fractions, as well as the overall mass balance, are free from errors. In clause 4 the influence of systematic errors on the efficiency of a classification process is described. The effect of stochastic errors in the characterization of a classification process is described in annex A. 2 Symbols 2.1 Symbols for specific terms See Table 1. Table 1 — Symbols for specific terms Symbol Term A Parameters derived from cumulative distribution curvesE Mass balance error, cumulative distributionsI Imperfection K(x) Corrected cumulative distributionm Massn Total number of size classes, number of particles q r (x) Distribution density curve Q r (x) Cumulative distribution curve ΔQ r,i Difference of two cumulative distribution values, relative amount in the ith particle size interval, Δx is 2 Variancet Student s factorT Grade efficiency, partial classification efficiency T o Overall classification or separation efficiency T(x) Grade efficiency curvex Particle diameter, diameter of a spherex a Analytical cut sizex e Equiprobable cut size, median particle size of a grade efficiency curvex i Upper particle size of the ith particle size interval INTERNATIONAL ST ANDARD ISO 9276-4:2001(E) © ISO 2001 – All rights reserved 1 BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E) Symbol Termx i−1 Lower particle size of the ith particle size interval Δx i Width of the ith particle size intervalx max Particle size above which there are no particles in a given size distributionx min Particle size below which there are no particles in a given size distribution α Angle of slope, weighted sum of variances ε Mass balance error, distribution density η r, i = Q r,s, i − Q r,c,i Variable k Sharpness of cut parameters formed with characteristic particle sizesv Relative amount ξ r, i = Q r,f, i − Q r,c,i Variable t Amount of particles not participating in a classification process φ Variable 2.2 Subscripts See Table 2. Table 2 — Subscripts Symbol Significance c Coarse fraction (second subscript after r) f Fine fraction (second subscript after r)i Number of the size class with upper particle size: x ir Type of quantity of a distribution density a (general de- scription) s Supply or feed material (second subscript after r) 0 Replaces s in case of more than one coarse fraction 1 Replaces f in case of more than one coarse fraction 2 Replaces c in case of more than one coarse fractionaFor example, r = 3 if type of quantity = volume or mass. 3 Characterization of a classification process based on error-free distribution curves and mass balances 3.1 Distribution density curves representing a classification process In a classification process a given supply or feed material (subscript s) is classified into at least two parts, which are called the fine (subscript f) and the coarse (subscript c) fractions. If an ideal classification were possible, the fine fraction would, as shown in Figure 2, contain particles below or equal to a certain size, x e , the so-called cut size, and the coarse fraction would contain all particles above that size.2 © ISO 2001 – All rights reserved BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E) Figure 2 — Weighted distribution density of the feed material q r,s (x) and the fine and coarse fractions of an ideal classification process The shaded areas beneath the weighted distribution density of the fine and the coarse product represent the relative mass, v 3,f , or number, v 0,f , of the fine, v r,f , and the coarse fraction, v r,c , the sum which equals 100 % or unity. In reality, however, in a certain range of sizes x min,c x x max,fparticles of the same size, x, are present in both the fine and the coarse fractions. The distribution density curves of the fine and the coarse fractions overlap and intersect each other in this size range, The point of intersection as shown in Figure 3 corresponds to a cut size, which is called the equiprobable cut size, x e (see 3.3.2). The particles below the cut size, x e , in the coarse or above x ein the fine fraction have been incorrectly classified. Figure 3 — Weighted distribution density of feed material, q r,s (x), and the fine, v r,fq r,f (x), and the coarse fraction, v r,cq r,c (x), of an real classification process© ISO 2001 – All rights reserved 3 BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E) 3.2 Mass and number balances 3.2.1 Mass and number balance in the size range from x minto x max Due to the classification process, the mass, m s , or number, n s , of the feed material, is split into the mass, m f , or number, n f , of the fine material and the mass, m c , or number, n c,of the coarse material. One obtains:m s= m f+ m c or n s= n f+ n c(1) andor(2)1 = v 3,f+ v 3,c or 1= v o,f+ v o,c(3) v r,frepresents the relative amount of the fine fraction, v r,cthe relative amount of the coarse fraction. The relative amounts are mass ratios v 3,f= m f /m s , v 3,c= m c /m sor number ratios v 0,f= n f /n s , v 0,c= n c /n s . In Figures 2 and 3, v r,fand v r,care represented by the areas beneath the weighted distribution density curves of the fine, v r,fq r,f (x), and the coarse, v r,cq r,c (x), fractions. The area beneath the distribution density curve of the feed material, q r,s (x), equals unity. 3.2.2 Mass and number balance in the size range from x to x + dx Particles of a certain size, x, present in the feed material, are either transferred in the classification process to the fine or to the coarse fractions. The amount of these particles in the feed material, dQ r,s (x), is therefore split into two fractions: v r,fdQ r,f (x) and v r,cdQ r,c (x).dQ r,s (x) = v r,fdQ r,f (x) + v r,cdQ r,c (x) (4) Replacing dQ r (x) by equation 5:dQ r (x) = q r (x)dx (5) one obtains:q r,s (x) = v r,fq r,f (x) + v r,cq r,c (x) (6) Equation 6 must be used to construct the set of distribution density curves of Figure 3. It should be realized that in plotting Figure 3 only three of the variables of equation 6 can be chosen arbitrarily. If, two distribution density and the relative amount of the fine or the coarse material, e.g., q r,s (x), q r,f (x) and v r,fare given, q r,c (x), and v r,care fixed.4 © ISO 2001 – All rights reserved BS ISO 9276-4:2001+A1:2017 ISO 9276-4:2001(E) 3.2.3 Mass and number balance in the size range from x minto x Integrating equation 6 between x minand x yields:Q r,s (x) = v r,fQ r,f (x) + v r,cQ r,c (x) (7) 3.2.4 The indirect evaluation of v r,fand v r,c In many cases of practical application v r,fand v r,ccannot be calculated from the relevant masses or mass flow rates, due to the fact that these are not available or difficult to measure, etc. If however, representative samples of the feed material and the fine and the coarse fraction have been measured equations 3 and 6 or 7 may be used to calculate v r,for v r,c . Introduc