1、BRITISH STANDARD BS 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 ICS 19.120 BS ISO 9276-5:2005 This British Standard was published under the authority of the Sta
2、ndards Policy and Strategy Committee on 29 September 2005 BSI 29 September 2005 ISBN 0 580 46573 X National foreword This British Standard reproduces verbatim ISO 9276-5:2005 and implements it as the UK national standard. The UK participation in its preparation was entrusted to Technical Committee L
3、BI/37, Sieves, screens and particle sizing, which has the responsibility to: A list of organizations represented on this committee can be obtained on request to its secretary. Cross-references The British Standards which implement international publications referred to in this document may be found
4、in the BSI Catalogue under the section entitled “International Standards Correspondence Index”, or by using the “Search” facility of the BSI Electronic Catalogue or of British Standards Online. This publication does not purport to include all the necessary provisions of a contract. Users are respons
5、ible for its correct application. Compliance with a British Standard does not of itself confer immunity from legal obligations. aid enquirers to understand the text; present to the responsible international/European committee any enquiries on the interpretation, or proposals for change, and keep UK
6、interests informed; monitor related international and European developments and promulgate them in the UK. Summary of pages This document comprises a front cover, an inside front cover, the ISO title page, pages ii to v, a blank page, pages 1 to 12, an inside back cover and a back cover. The BSI cop
7、yright notice displayed in this document indicates when the document was last issued. Amendments issued since publication Amd. No. Date Comments Reference number ISO 9276-5:2005(E)INTERNATIONAL STANDARD ISO 9276-5 First edition 2005-08-01 Representation of results of particle size analysis Part 5: M
8、ethods of calculation relating to particle size analyses using logarithmic normal probability distribution Reprsentation de donnes obtenues par analyse granulomtrique Partie 5: Mthodes de calcul relatif lanalyse granulomtrique laide de la distribution de probabilit logarithmique normale BS ISO 9276-
9、5:2005ii BS ISO 9276-5:2005 iii Contents Page Foreword iv Introduction v 1 Scope . 1 2 Normative references . 1 3 Symbols . 1 4 Logarithmic normal probability function 2 5 Special values of a logarithmic normal probability distribution 5 5.1 Complete kth moments . 5 5.2 Average particle sizes 5 5.3
10、Median particle sizes 6 5.4 Horizontal shifts between plotted distribution values 6 5.5 Volume-specific surface area (Sauter diameter) . 8 Annex A (informative) Cumulative distribution values of a normal probability distribution 9 Bibliography . 12 BS ISO 9276-5:2005 iv Foreword ISO (the Internation
11、al 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 e
12、stablished 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 standard
13、ization. International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2. The main task of technical committees is to prepare International Standards. Draft International Standards adopted by the technical committees are circulated to the member bodies for vo
14、ting. 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 document may be the subject of patent rights. ISO shall not be held responsible for identifying any or all su
15、ch patent rights. ISO 9276-5 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
16、: Graphical representation Part 2: Calculation of average particle sizes/diameters and moments from particle size distributions Part 4: Characterization of a classification process Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution Fur
17、ther parts are under preparation: Part 3: Fitting of an experimental cumulative curve to a reference model Part 6: Descriptive and quantitative representation of particle shape and morphology v Introduction Many cumulative particle size distributions, Q r (x), may be plotted on special graph paper w
18、hich allow the cumulative size distribution to be represented as a straight line. Scales on the ordinate and the abscissa are generated from various mathematical formulae. In this part of ISO 9276, it is assumed that the cumulative particle size distribution follows a logarithmic normal probability
19、distribution. In this part of ISO 9276, the size, x, of a particle represents the diameter of a sphere. Depending on the situation, the particle size, x, may also represent the equivalent diameter of a particle of some other shape. BS ISO 9276-5:2005 blank1 Representation of results of particle size
20、 analysis Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution 1 Scope The main objective of this part of ISO 9276 is to provide the background for the representation of a cumulative particle size distribution which follows a logarithmic
21、 normal probability distribution, as a means by which calculations performed using particle size distribution functions may be unequivocally checked. The design of logarithmic normal probability graph paper is explained, as well as the calculation of moments, median diameters, average diameters and
22、volume-specific surface area. Logarithmic normal probability distributions are often suitable for the representation of cumulative particle size distributions of any dimensionality. Their particular advantage lies in the fact that cumulative distributions, such as number-, length-, area-, volume- or
23、 mass-distributions, are represented by parallel lines, all of whose locations may be determined from a knowledge of the location of any one. 2 Normative references The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited
24、 applies. For undated references, the latest edition of the referenced document (including any amendments) applies. ISO 9276-1, Representation of results of particle size analysis Part 1: Graphical representation ISO 9276-2:2001, Representation of results of particle size analysis Part 2: Calculatio
25、n of average particle sizes/diameters and moments from particle size distributions 3 Symbols For the purposes of this part of ISO 9276, the following symbols apply. c cumulative percentage e = 2,718 28. base of natural logarithms k power of x in a moment M k,rcomplete kth moment of a density distrib
26、ution of dimensionality r p dimensionality (type of quantity) of a distribution, p = 0: number, p = 1: length, p = 2: area, p = 3: volume or mass q r (x) density distribution of dimensionality r Q r (x) cumulative distribution of dimensionality r BS ISO 9276-5:2005 2 r dimensionality (type of quanti
27、ty) of a distribution, r = 0: number, r = 1: length, r = 2: area, r = 3: volume or mass s standard deviation of the density distribution s ggeometric standard deviation, exponential function of the standard deviation S Vvolume-specific surface area x particle size, diameter of a sphere x minparticle
28、 size below which there are no particles in a given size distribution x maxparticle size above which there are no particles in a given size distribution x 84,rparticle size at which Q r= 0,84 x 50,rmedian particle size of a cumulative distribution of dimensionality r x 16,rparticle size at which Q r
29、= 0,16 x k,raverage particle size based on the kth moment of a distribution of dimensionality r z dimensionless variable proportional to the logarithm of x (see Equation 3) integration variable based on x (see Equation 11) integration variable based on z (see Equation 2) Subscripts of different sens
30、e are separated by a comma in this and all other parts of ISO 9276. 4 Logarithmic normal probability function Normal probability density distributions are described in terms of a dimensionless variable z: 2 0,5 1 *() e 2 z r qz = (1) The cumulative normal probability distribution is represented by:
31、2 0,5 1 *() *()d e d 2 zz rr Qz q = (2) A sample table of values for Q* r (z) as a function of z is given in Table A.1. The logarithmic normal probability distribution is a formulation in which z is defined as a logarithm of x scaled by two parameters, the mean size x 50,rand either the dimensionles
32、s standard deviation, s, or the geometric standard deviation, s g , that characterize the distribution: 50, g 50, g 50, 111 ln ln log ln log rrr xxx z sx sx s x = (3) BS ISO 9276-5:2005 3 which is equivalent to 50, e s z r xx = (4) According to Equation 3, the standard deviation, s, is linked with t
33、he geometric standard deviation, s g , by: gg ln or e s sss = (5) Although Equation 1 has no explicit dependences on r, the dimensionality of the density distribution is involved through the relationship of z to x 50,rin Equation 3. The value of x 50,rfor a specific size distribution may be determin
34、ed from experimental data according to ISO 9276-1. The standard deviation of a logarithmic normal probability distribution may be calculated from the values of the cumulative distribution at certain characteristic values of z: either at z = 1, for which 84, 50, * ( 1) 0,84 and ln r r r x Qz s x = =
35、(6) or at z = 1, for which 50, 16, * ( 1) 0,16 and ln r r r x Qz s x = = = (7) Throughout this part of ISO 9276, the values 0,84 and 0,16 (and their representation as percentages 84 and 16) are used in place of the more precise values 0,841 34 and 0,158 65. Logarithmic probability graph presentation
36、: Useful information about the nature of a particle size distribution may be obtained by plotting the cumulative distribution on special graph paper, on which the abscissa (representing particle size) is marked with an exponential scale and the ordinate (representing cumulative distribution) is mark
37、ed with a scale of Q* r (z) values (see Annex A). Preprinted paper marked with these scales is available. Graphical representation is now more often displayed as a specific graphical screen created by software in a computer. Experimental values of each cumulative fraction (expressed in terms of numb
38、er, length, area or volume) of undersize particles, Q r (x), (that is, of particles smaller than x) are plotted at the size corresponding to the upper size limit of the particles in that cumulative fraction. A logarithmic normal probability distribution gives a straight line in Figure 1. To fulfil t
39、he condition of normalization, the cumulative fraction smaller than or equal to the particle having the largest size in the sample must be unity, that is, Q r (x max ) must be equal to 1. If this is so, then *()d ()d rr qzzqxx = (8) NOTE The superscript * is used to distinguish the distributions def
40、ined in terms of the dimensionless integration variable z, such as q* r (z), from those defined in terms of the size x, such as q r (x). This is because z, the integration variable, is related to the particle size x, as shown in Equation 3. 50, dd 11 () *() *() l n *() dd rrr r r zx qxqz qz qz xx s
41、xx s = = (9) or, using Equation 1, 2 0,5 1 () e 2 z r qx xs = (10) BS ISO 9276-5:2005 4 and, parallel to Equation 2, min () ()d r x r x Qx q = (11) EXAMPLE A logarithmic normal probability distribution of volume (r = 3), with a median size of x 50,3= 5 m and a standard deviation of s = 0,5, has x 16
42、,3= 3,0 m and x 84,3= 8,2 m (see ISO 9276-2:2001, Annex A). Figure 1 shows a plot of the cumulative volume distribution, Q 3 (x), on logarithmic probability graph paper. Key X particle size, x, m Y cumulative distribution, Q Figure 1 Plot of a logarithmic normal probability distribution on logarithm
43、ic probability graph paper BS ISO 9276-5:2005 5 5 Special values of a logarithmic normal probability distribution 5.1 Complete kth moments The complete kth moment of a logarithmic normal probability distribution, q r (x), is 22 22 50, ln 0,5 0,5 ,5 0 , ee r kx ks k ks kr r Mx + = = (12) with k = 2 a
44、nd r = 3: 2 2 50,3 2l n 2 2 2 2,3 50,3 ee x s s Mx + = = (13) 5.2 Average particle sizes A series of average particle sizes, , x of a logarithmic normal probability distribution, q r (x), can be calculated from the kth root of the kth moment (or from the x 50,rand s) of that distribution using Equat
45、ion 14: 2 0,5 ,5 0 , e ks k kr kr r xMx = = (14) For a logarithmic normal probability distribution, the median is the same as the geometric mean and the average size in one dimension, r, may be calculated from the parameters describing the distribution in a different dimensionality, p, using: 2 (0,5
46、 ) ,5 0 , e krps kr p xx + = (15) or 22 ,5 0 , 5 0 , ln ln 0,5 ln (0,5 ) kr r p xxk sxk r p s =+=+ (16) EXAMPLE The first several moments (k = 1, 2 or 3) of the arithmetic average particle size (r = 0) for a logarithmic normal probability distribution may be computed from the parameters for any of t
47、he dimensionalities (p = 0, 1, 2 or 3) using: 2222 0,5 0,5 1 ,5 2,5 1 ,0 50,0 50,1 50,2 50,3 eeee s sss xx x x x = (17) 222 2 2,0 50,0 50,1 50,2 50,3 eee s ss xx xx x = (18) 2222 1, 5 0 , 5 0 , 5 1, 5 3,0 50,0 50,1 50,2 50,3 eeee s sss x xxxx = (19) EXAMPLE The first moment (k = 1) weighted average
48、particle size for the different dimensionalities (r = 0, 1, 2, or 3) of a logarithmic normal probability distribution may be computed from the parameters for any of the dimensionalities (p = 0, 1, 2 or 3) using: 2222 0,5 0,5 1 ,5 2,5 1 ,0 50,0 50,1 50,2 50,3 eeee s sss xx x x x = (17) 2222 1 ,5 0,5 0,5 1 ,5 1 ,1 50,0 50,1 50,2 50,3 eeee s sss xx x x x = (20) 2222 2,5 1 ,5 0,5 0,5 1 ,2 50,0 50,1 50,2 50,3 eeee s sss xx x x x = (21) 2222 3,5 2,5 1 ,5 0,5 1 ,3 50,0 50,1 50,2 50,3 eeee s sss x xxxx = (22) BS ISO 9276-5:2005 6 5.3 Median particle sizes A unique feature of the logarithmic no
