BSI Standards Publication BS ISO 9276-2:2014 Representation of results of particle size analysis Part 2: Calculation of average particle sizes/ diameters and moments from particle size distributionsBS ISO 9276-2:2014 BRITISH STANDARD National foreword This British Standard is the UK implementation of ISO 9276-2:2014. It supersedes BS ISO 9276-2:2001 which is withdrawn. 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 2014. Published by BSI Standards Limited 2014 ISBN 978 0 580 80366 6 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 31 May 2014. Amendments issued since publication Date Text affectedBS ISO 9276-2:2014 © ISO 2014 Representation of results of particle size analysis — Part 2: Calculation of average particle sizes/ diameters and moments from particle size distributions Représentation de données obtenues par analyse granulométrique — Partie 2: Calcul des tailles/diamètres moyens des particules et des moments à partir de distributions granulométriques INTERNATIONAL STANDARD ISO 9276-2 Second edition 2014-05-15 Reference number ISO 9276-2:2014(E)BS ISO 9276-2:2014ISO 9276-2:2014(E)ii © ISO 2014 – All rights reserved COPYRIGHT PROTECTED DOCUMENT © ISO 2014 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 Case postale 56 • CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail

[email protected] Web www.iso.org Published in SwitzerlandBS ISO 9276-2:2014ISO 9276-2:2014(E)© ISO 2014 – All rights reserved iii Contents Page Foreword iv Introduction v 1 Scope . 1 2 Normative references 1 3 Definitions, s ymbols and abbr e viat ed t erms 1 4 The moment-notation . 3 4.1 Definition of moments according to the moment-notation . 3 4.2 Definition of mean particle sizes according to the moment-notation . 4 4.3 Calculation of moments and mean particle sizes from a given size distribution 7 4.4 Variance and standard deviation of a particle size distribution 9 4.5 Calculation of moments and mean particle sizes from a lognormal distribution . 9 4.6 Calculation of volume specific surface area and the Sauter mean diameter 10 5 The moment-ratio-notation .10 5.1 Definition of moments according to the moment-ratio-notation 10 5.2 Definition of mean particle sizes according to the moment-ratio-notation11 5.3 Calculation of mean particle sizes from a given size distribution .13 5.4 Variance and standard deviation of a particle size distribution .14 5.5 Relationships between mean particle sizes 15 5.6 Calculation of volume specific surface area and the Sauter mean diameter 16 6 Relationship between moment-notation and moment-ratio-notation 16 7 Accuracy of calculated particle size distribution parameters .18 Annex A (informative) Numerical example for calculation of mean particle sizes and standard deviation from a histogram of a volume based size distribution .19 Annex B (informative) Numerical example for calculation of mean particle sizes and standard deviation from a histogram of a volume based size distribution .22 Annex C (informative) Accuracy of calculated particle size distribution parameters .25 Bibliography .27BS ISO 9276-2:2014ISO 9276-2:2014(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. The procedures used to develop this document and those intended for its further maintenance are described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the different types of ISO documents should be noted. This document was drafted in accordance with the editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives). Attention is drawn to the possibility that some of the elements of this document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of any patent rights identified during the development of the document will be in the Introduction and/or on the ISO list of patent declarations received (see www.iso.org/patents). Any trade name used in this document is information given for the convenience of users and does not constitute an endorsement. For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information. The committee responsible for this document is ISO/TC 24, Particle characterization including sieving, Subcommittee SC 4, Particle characterization. This second edition cancels and replaces the first edition (ISO 9276-2:2001), which has been technically revised. 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: Adjustment of an experimental curve to a reference model — Part 4: Characterization of a classification process — Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution — Part 6: Descriptive and quantitative representation of particle shape and morphologyiv © ISO 2014 – All rights reservedBS ISO 9276-2:2014ISO 9276-2:2014(E) Introduction Particle size analysis is often used for characterization of particulate matter. The relationship between the physical properties of particulate matter, such as powder strength, flowability, dissolution rate, emulsion/suspension stability and particle size forms always the reason for such characterization. For materials having a particle size distribution, it is important to use the relevant parameter, a certain mean particle size, weighted for example by number, area or volume, in the relationship with physical properties. This part of ISO 9276 describes two procedures for the use of moments for the calculation of mean sizes, the spread and other statistical measures of a particle size distribution. The first method is named moment-notation. The specific utility of the moment-notation is to characterize size distributions by moments and mean sizes. The moment-notation addresses weighting principles from physics, especially mechanical engineering, and includes arithmetic means from number based distributions only as one part [1][2] . The second method is named moment-ratio-notation. The moment-ratio-notation is based on a number statistics and frequencies approach, but includes also conversion to other types of quantities [3][4] . Important is that the meaning of the subscripts of mean sizes defined in the moment-notation differs from the subscripts of mean sizes defined in the moment-ratio-notation. Both notations are linked by a simple relationship, given in Clause 6. Both notations are suited for derivation and/or selection of mean sizes related to physical product and process properties for so-called property functions and process functions. The type of mean size to be preferred should have a causal relationship with the relevant physical product or process property. The particle characterization community embraces a very broad spectrum of science disciplines. The notation of the size distribution employed has been influenced by the branch of industry and the application and thus no single notation has found universal favour. There are some particle size dependent properties, like light scattering in certain particle size ranges, which cannot be characterized by mean particles sizes, derived from simple power law equations of the notation systems [5] .© ISO 2014 – All rights reserved vBS ISO 9276-2:2014BS ISO 9276-2:2014Representation of results of particle size analysis — Part 2: Calculation of average particle sizes/diameters and moments from particle size distributions 1 Scope This part of ISO 9276 provides relevant equations and coherent nomenclatures for the calculation of moments, mean particle sizes and standard deviations from a given particle size distribution. Two notation systems in common use are described. One is the method of moments while the second describes the moment-ratio method. The size distribution may be available as a histogram or as an analytical function. The equivalent diameter of a particle of any shape is taken as the size of that particle. Particle shape factors are not taken into account. It is essential that the measurement technique is stated in the report in view of the dependency of sizing results of measurement principle. Samples of particles measured are intended to be representative of the population of particles. For both notation systems, numerical examples of the calculation of mean particle sizes and standard deviation from histogram data are presented in an annex. The accuracy of the mean particle size may be reduced if an incomplete distribution is evaluated. The accuracy may also be reduced when very limited numbers of size classes are employed. 2 Normative references The following documents, in whole or in part, are normatively referenced in this document and are indispensable for its application. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 9276-1:1998, Representation of results of particle size analysis — Part 1: Graphical representation ISO 9276-5:2005, Representation of results of particle size analysis — Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution 3 De finiti ons, s ymbo ls and abbr e viat ed t erms If necessary, different symbols are given to the moment-notation (M) and the moment-ratio-notation (M-R). This serves the purpose of a clear differentiation between the two systems. For both notation systems, a terminology of specific mean particle sizes is inserted in the corresponding clauses: Clause 4 and Clause 5, respectively. M-notation M-R-notation Description i i number of the size class with upper particle size: x i (M) or midpoint particle size D i (M-R) k power of x m m number of size classes INTERNATIONAL ST ANDARD ISO 9276-2:2014(E) © ISO 2014 – All rights reserved 1BS ISO 9276-2:2014ISO 9276-2:2014(E) M-notation M-R-notation Description r r type of quantity of a distribution (general description) r = 0, type of quantity: number r = 1, type of quantity: length r = 2, type of quantity: surface or projected area r = 3, type of quantity: volume or mass M k,r complete k-th moment of a q r (x) – sample distribution m k,r complete k-th central moment of a q r (x) – sample distribution M p p-th moment of a number distribution density m p p-th central moment of a number distribution density N total number of particles in a sample O order of a mean particle size (O = p + q) p, q powers of D in moments or subscripts indicating the same q r (x) q r (D) distribution density of type of particle quantity r , mean height of a distribution density in the i-th particle size interval, Δx i Q r (x) Q r (D) cumulative distribution of type of quantity r ΔQ r,i difference of two values of the cumulative distribution, i.e. relative amount in the i-th particle size interval, Δx i s r s r standard deviation of a q r (x) and q r (D) distribution s g s g geometric standard deviation of a distribution s s standard deviation of lognormal distribution (s = ln s g ) S S surface area S V S V volume specific surface area V V particle volume V mean particle volume x D particle size, diameter of an equivalent sphere x i upper particle size of the i-th particle size interval x i-1 lower particle size of the i-th particle size interval D i midpoint size of the i-th size class x min particle size below which there are no particles in a given size distri- bution2 © ISO 2014 – All rights reservedBS ISO 9276-2:2014ISO 9276-2:2014(E) M-notation M-R-notation Description x max particle size above which there are no particles in a given size distri- bution , , mean particle sizes (general description) , geometric mean particle sizes arithmetic mean particle size , weighted mean particle size geometric mean particle size harmonic mean particle size x 50,3 median particle size of a cumulative volume distribution Δx i= x i– x i-1 width of the i-th particle size interval 4 The moment-notation Moments are the basis for defining mean sizes and standard deviations of particle size distributions. A random sample, containing a limited number of particles from a large population of particle sizes, is used for estimation of the moments of the size distribution of that population. Estimation is concerned with inference about the numerical values of the unknown population from those of the sample. Particle size measurements are always done on discrete samples and involve a number of discrete size classes. Therefore, only moments related to samples are dealt with in this part of ISO 9276. 4.1 Definition of mom ents ac c or ding t o the moment-notation The complete k-th moment of a distribution density [1]is represented by integrals as defined in Formula (1). M stands for moment. The first subscript, k, of M indicates the power of the particle size x, the second subscript, r, of M describes the type of quantity of the distribution density. (1) If r = 0, q 0 (x) represents a number distribution density, if r = 3, q 3 (x) represents a volume or mass distribution density. Formula (1) describes a complete moment if the integral boundaries are represent