1、BSI Standards PublicationBS ISO 9276-2:2014Representation of results of particle size analysisPart 2: Calculation of average particle sizes/ diameters and moments from particle size distributionsBS ISO 9276-2:2014 BRITISH STANDARDNational forewordThis British Standard is the UK implementation of ISO
2、 9276-2:2014. Itsupersedes BS ISO 9276-2:2001 which is withdrawn.The UK participation in its preparation was entrusted to TechnicalCommittee LBI/37, Particle characterization including sieving.A list of organizations represented on this committee can beobtained on request to its secretary.This publi
3、cation does not purport to include all the necessaryprovisions of a contract. Users are responsible for its correctapplication. The British Standards Institution 2014. Published by BSI StandardsLimited 2014ISBN 978 0 580 80366 6ICS 19.120Compliance with a British Standard cannot confer immunity from
4、legal obligations.This British Standard was published under the authority of theStandards Policy and Strategy Committee on 31 May 2014.Amendments issued since publicationDate Text affectedBS ISO 9276-2:2014 ISO 2014Representation of results of particle size analysis Part 2: Calculation of average pa
5、rticle sizes/diameters and moments from particle size distributionsReprsentation de donnes obtenues par analyse granulomtrique Partie 2: Calcul des tailles/diamtres moyens des particules et des moments partir de distributions granulomtriquesINTERNATIONAL STANDARDISO9276-2Second edition2014-05-15Refe
6、rence numberISO 9276-2:2014(E)BS ISO 9276-2:2014ISO 9276-2:2014(E)ii ISO 2014 All rights reservedCOPYRIGHT PROTECTED DOCUMENT ISO 2014All 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 mechan
7、ical, 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 ISOs member body in the country of the requester.ISO copyright officeCase postale 56 CH-1211 Geneva 20Tel. + 41 22 749 01 11F
8、ax + 41 22 749 09 47E-mail copyrightiso.orgWeb www.iso.orgPublished in SwitzerlandBS ISO 9276-2:2014ISO 9276-2:2014(E) ISO 2014 All rights reserved iiiContents PageForeword ivIntroduction v1 Scope . 12 Normative references 13 Definitions, symbols and abbreviated terms 14 The moment-notation . 34.1 D
9、efinition of moments according to the moment-notation . 34.2 Definition of mean particle sizes according to the moment-notation . 44.3 Calculation of moments and mean particle sizes from a given size distribution 74.4 Variance and standard deviation of a particle size distribution 94.5 Calculation o
10、f moments and mean particle sizes from a lognormal distribution . 94.6 Calculation of volume specific surface area and the Sauter mean diameter 105 The moment-ratio-notation .105.1 Definition of moments according to the moment-ratio-notation 105.2 Definition of mean particle sizes according to the m
11、oment-ratio-notation115.3 Calculation of mean particle sizes from a given size distribution .135.4 Variance and standard deviation of a particle size distribution .145.5 Relationships between mean particle sizes 155.6 Calculation of volume specific surface area and the Sauter mean diameter 166 Relat
12、ionship between moment-notation and moment-ratio-notation 167 Accuracy of calculated particle size distribution parameters .18Annex A (informative) Numerical example for calculation of mean particle sizes and standard deviation from a histogram of a volume based size distribution .19Annex B (informa
13、tive) Numerical example for calculation of mean particle sizes and standard deviation from a histogram of a volume based size distribution .22Annex C (informative) Accuracy of calculated particle size distribution parameters .25Bibliography .27BS ISO 9276-2:2014ISO 9276-2:2014(E)ForewordISO (the Int
14、ernational 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 ha
15、s 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
16、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 acc
17、ordance 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. Detai
18、ls 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 end
19、orsement.For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment, as well as information about ISOs adherence to the WTO principles in the Technical Barriers to Trade (TBT) see the following URL: Foreword - Supplementary information.The committee resp
20、onsible 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 gener
21、al 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 classific
22、ation 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)IntroductionParticl
23、e 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.
24、 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
25、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 princip
26、les from physics, especially mechanical engineering, and includes arithmetic means from number based distributions only as one part12.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
27、other types of quantities34.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
28、 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 characteri
29、zation 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 ligh
30、t scattering in certain particle size ranges, which cannot be characterized by mean particles sizes, derived from simple power law equations of the notation systems5. ISO 2014 All rights reserved vBS ISO 9276-2:2014BS ISO 9276-2:2014Representation of results of particle size analysis Part 2: Calcula
31、tion of average particle sizes/diameters and moments from particle size distributions1 ScopeThis 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 s
32、ystems 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 s
33、hape 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 syste
34、ms, 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 c
35、lasses are employed.2 Normative referencesThe 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 (
36、including any amendments) applies.ISO 9276-1:1998, Representation of results of particle size analysis Part 1: Graphical representationISO 9276-5:2005, Representation of results of particle size analysis Part 5: Methods of calculation relating to particle size analyses using logarithmic normal proba
37、bility distribution3 Definitions, symbols and abbreviated termsIf 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 me
38、an particle sizes is inserted in the corresponding clauses: Clause 4 and Clause 5, respectively.M-notation M-R-notation Descriptioni i number of the size class with upper particle size: xi(M) or midpoint particle size Di(M-R)k power of xm m number of size classesINTERNATIONAL STANDARD ISO 9276-2:201
39、4(E) ISO 2014 All rights reserved 1BS ISO 9276-2:2014ISO 9276-2:2014(E)M-notation M-R-notation Descriptionr r type of quantity of a distribution (general description)r = 0, type of quantity: numberr = 1, type of quantity: lengthr = 2, type of quantity: surface or projected arear = 3, type of quantit
40、y: volume or massMk,rcomplete k-th moment of a qr(x) sample distributionmk,rcomplete k-th central moment of a qr(x) sample distributionMpp-th moment of a number distribution densitympp-th central moment of a number distribution densityN total number of particles in a sampleO order of a mean particle
41、 size (O = p + q)p, q powers of D in moments or subscripts indicating the sameqr(x) qr(D) distribution density of type of particle quantity rqri,mean height of a distribution density in the i-th particle size interval, xiQr(x) Qr(D) cumulative distribution of type of quantity rQr,idifference of two
42、values of the cumulative distribution, i.e. relative amount in the i-th particle size interval, xisrsrstandard deviation of a qr(x) and qr(D) distributionsgsggeometric standard deviation of a distributions s standard deviation of lognormal distribution (s = ln sg)S S surface areaSVSVvolume specific
43、surface areaV V particle volumeVmean particle volumex D particle size, diameter of an equivalent spherexiupper particle size of the i-th particle size intervalxi-1lower particle size of the i-th particle size intervalDimidpoint size of the i-th size classxminparticle size below which there are no pa
44、rticles in a given size distri-bution2 ISO 2014 All rights reservedBS ISO 9276-2:2014ISO 9276-2:2014(E)M-notation M-R-notation Descriptionxmaxparticle size above which there are no particles in a given size distri-butionxkr,Dpq,mean particle sizes (general description)Dpp,geometric mean particle siz
45、esxk,0arithmetic mean particle sizexkr,weighted mean particle sizexrgeo,geometric mean particle sizexrhar,harmonic mean particle sizex50,3median particle size of a cumulative volume distributionxi= xi xi-1width of the i-th particle size interval4 The moment-notationMoments are the basis for defining
46、 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
47、 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 moments according t
48、o the moment-notationThe complete k-th moment of a distribution density1is 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 distributio
49、n density.Mxqxxk,rkrxx=()minmaxd (1)If r = 0, q0(x) represents a number distribution density, if r = 3, q3(x) represents a volume or mass distribution density.Formula (1) describes a complete moment if the integral boundaries are represented by the minimum particle size (xmin) and the maximum particle size (xmax). ISO 2014 All rights reserved 3BS ISO 9276-2:2014ISO 9276-2:2014(E)A special complete moment is represented by M0,r:MxqxxqxxQx Qx,r rrrrxxxx001=() () ()()maxminminmaxminmaddx(2)withQx qxdxri rxxi() ()m
