Representation of results of particle size analysis

The main objective of ISO 9276-5:2005 is to provide the background for the representation of a cumulative particle size distribution which follows a logarithmic 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 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 mass-distributions, are represented by parallel lines, all of whose locations may be determined from a knowledge of the location of any one.

Représentation de données obtenues par analyse granulométrique

Predstavitev podatkov, dobljenih z granulometrijsko analizo - 5. del: Računska metoda določitve zrnavosti na osnovi normalne logaritmične porazdelitve

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Status
Published
Publication Date
04-Aug-2005
Current Stage
9093 - International Standard confirmed
Start Date
15-Jun-2020

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INTERNATIONAL ISO
STANDARD 9276-5
First edition
2005-08-01
Representation of results of particle size
analysis —
Part 5:
Methods of calculation relating to particle
size analyses using logarithmic normal
probability distribution
Représentation de données obtenues par analyse granulométrique —
Partie 5: Méthodes de calcul relatif à l'analyse granulométrique à l'aide
de la distribution de probabilité logarithmique normale
Reference number
ISO 9276-5:2005(E)
ISO 2005
---------------------- Page: 1 ----------------------
ISO 9276-5:2005(E)
PDF disclaimer

This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but

shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In

downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat

accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.

Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation

parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In

the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.

© ISO 2005

All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,

electronic or mechanical, including photocopying and microfilm, without permission in writing 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 copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO 2005 – All rights reserved
---------------------- Page: 2 ----------------------
ISO 9276-5:2005(E)
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 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

© ISO 2005 – All rights reserved iii
---------------------- Page: 3 ----------------------
ISO 9276-5:2005(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 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 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 document may be the subject of patent

rights. ISO shall not be held responsible for identifying any or all such 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: 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
Further 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

iv © ISO 2005 – All rights reserved
---------------------- Page: 4 ----------------------
ISO 9276-5:2005(E)
Introduction

Many cumulative particle size distributions, Q (x), may be plotted on special graph paper which 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 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.

© ISO 2005 – All rights reserved v
---------------------- Page: 5 ----------------------
INTERNATIONAL STANDARD ISO 9276-5:2005(E)
Representation of results of particle size 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 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 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

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 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: Calculation 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.
cumulative percentage
base of natural logarithms
e = 2,718 28...
k power of x in a moment
M complete kth moment of a density distribution of dimensionality r
k,r
dimensionality (type of quantity) of a distribution,
p = 0: number, p = 1: length, p = 2: area, p = 3: volume or mass
q (x) density distribution of dimensionality r
Q (x) cumulative distribution of dimensionality r
© ISO 2005 – All rights reserved 1
---------------------- Page: 6 ----------------------
ISO 9276-5:2005(E)
r dimensionality (type of quantity) 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 geometric standard deviation, exponential function of the standard deviation
S volume-specific surface area
x particle size, diameter of a sphere
x particle size below which there are no particles in a given size distribution
min
x particle size above which there are no particles in a given size distribution
max
particle size at which Q = 0,84
84,r
x median particle size of a cumulative distribution of dimensionality r
50,r
x particle size at which Q = 0,16
16,r r

x average particle size based on the kth moment of a distribution of dimensionality r

k,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 sense 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:

−0,5z
qz*( ) = e (1)
The cumulative normal probability distribution is represented by:
−0,5ζ
Qz*( )==q* (ζζ)d e dζ (2)
−∞ −∞
A sample table of values for Q* (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 and either the dimensionless standard deviation, s, or the geometric

50,r
standard deviation, s , that characterize the distribution:
   
11xx 1 x
z==ln ln = log (3)
   
sx lns x logs x
50,rrg 50, g 50,r
   
2 © ISO 2005 – All rights reserved
---------------------- Page: 7 ----------------------
ISO 9276-5:2005(E)
which is equivalent to
s z
xx= e (4)
50,r

According to Equation 3, the standard deviation, s, is linked with the geometric standard deviation, s , by:

ss==ln ors e (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 in Equation 3. The value of x for a specific size distribution

50,r 50,r

may be determined 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,r
Qz* ( ==1) 0,84 ands= ln (6)
50,r
or at z = −1, for which
x
50,r
Qz* ( =−1)= 0,16 ands = ln (7)
16,r

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: 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 marked with a scale of Q* (z) values (see Annex A). Preprinted paper marked with

these scales is available. Graphical representation is now mor
...

SLOVENSKI STANDARD
SIST ISO 9276-5:2006
01-oktober-2006
3UHGVWDYLWHYSRGDWNRYGREOMHQLK]JUDQXORPHWULMVNRDQDOL]RGHO5DþXQVND
PHWRGDGRORþLWYH]UQDYRVWLQDRVQRYLQRUPDOQHORJDULWPLþQHSRUD]GHOLWYH

Representation of results of particle size analysis - Part 5: Methods of calculation relating

to particle size analyses using logarithmic normal probability distribution

Représentation de données obtenues par analyse granulométrique - Partie 5: Méthodes

de calcul relatif à l'analyse granulométrique à l'aide de la distribution de probabilité

logarithmique normale
Ta slovenski standard je istoveten z: ISO 9276-5:2005
ICS:
19.120 Analiza velikosti delcev. Particle size analysis. Sieving
Sejanje
SIST ISO 9276-5:2006 en

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------
SIST ISO 9276-5:2006
---------------------- Page: 2 ----------------------
SIST ISO 9276-5:2006
INTERNATIONAL ISO
STANDARD 9276-5
First edition
2005-08-01
Representation of results of particle size
analysis —
Part 5:
Methods of calculation relating to particle
size analyses using logarithmic normal
probability distribution
Représentation de données obtenues par analyse granulométrique —
Partie 5: Méthodes de calcul relatif à l'analyse granulométrique à l'aide
de la distribution de probabilité logarithmique normale
Reference number
ISO 9276-5:2005(E)
ISO 2005
---------------------- Page: 3 ----------------------
SIST ISO 9276-5:2006
ISO 9276-5:2005(E)
PDF disclaimer

This PDF file may contain embedded typefaces. In accordance with Adobe's licensing policy, this file may be printed or viewed but

shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In

downloading this file, parties accept therein the responsibility of not infringing Adobe's licensing policy. The ISO Central Secretariat

accepts no liability in this area.
Adobe is a trademark of Adobe Systems Incorporated.

Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation

parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In

the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below.

© ISO 2005

All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,

electronic or mechanical, including photocopying and microfilm, without permission in writing 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 copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO 2005 – All rights reserved
---------------------- Page: 4 ----------------------
SIST ISO 9276-5:2006
ISO 9276-5:2005(E)
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 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

© ISO 2005 – All rights reserved iii
---------------------- Page: 5 ----------------------
SIST ISO 9276-5:2006
ISO 9276-5:2005(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 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 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 document may be the subject of patent

rights. ISO shall not be held responsible for identifying any or all such 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: 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
Further 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

iv © ISO 2005 – All rights reserved
---------------------- Page: 6 ----------------------
SIST ISO 9276-5:2006
ISO 9276-5:2005(E)
Introduction

Many cumulative particle size distributions, Q (x), may be plotted on special graph paper which 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 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.

© ISO 2005 – All rights reserved v
---------------------- Page: 7 ----------------------
SIST ISO 9276-5:2006
---------------------- Page: 8 ----------------------
SIST ISO 9276-5:2006
INTERNATIONAL STANDARD ISO 9276-5:2005(E)
Representation of results of particle size 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 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 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

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 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: Calculation 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.
cumulative percentage
base of natural logarithms
e = 2,718 28...
k power of x in a moment
M complete kth moment of a density distribution of dimensionality r
k,r
dimensionality (type of quantity) of a distribution,
p = 0: number, p = 1: length, p = 2: area, p = 3: volume or mass
q (x) density distribution of dimensionality r
Q (x) cumulative distribution of dimensionality r
© ISO 2005 – All rights reserved 1
---------------------- Page: 9 ----------------------
SIST ISO 9276-5:2006
ISO 9276-5:2005(E)
r dimensionality (type of quantity) 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 geometric standard deviation, exponential function of the standard deviation
S volume-specific surface area
x particle size, diameter of a sphere
x particle size below which there are no particles in a given size distribution
min
x particle size above which there are no particles in a given size distribution
max
particle size at which Q = 0,84
84,r
x median particle size of a cumulative distribution of dimensionality r
50,r
x particle size at which Q = 0,16
16,r r

x average particle size based on the kth moment of a distribution of dimensionality r

k,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 sense 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:

−0,5z
qz*( ) = e (1)
The cumulative normal probability distribution is represented by:
−0,5ζ
Qz*( )==q* (ζζ)d e dζ (2)
−∞ −∞
A sample table of values for Q* (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 and either the dimensionless standard deviation, s, or the geometric

50,r
standard deviation, s , that characterize the distribution:
   
11xx 1 x
z==ln ln = log (3)
   
sx lns x logs x
50,rrg 50, g 50,r
   
2 © ISO 2005 – All rights reserved
---------------------- Page: 10 ----------------------
SIST ISO 9276-5:2006
ISO 9276-5:2005(E)
which is equivalent to
s z
xx= e (4)
50,r

According to Equation 3, the standard deviation, s, is linked with the geometric standard deviation, s , by:

ss==ln ors e (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 in Equation 3. The value of x for a specific size distribution

50,r 50,r

may be determined 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,r
Qz* ( ==1) 0,84 ands= ln (6)
50,r
or at z = −1, for which
x
50,r
Qz* ( =−1)= 0,16 ands = ln (7)
16,r
Throughout this part of ISO 9276, the values 0,84 and 0,16 (and their rep
...

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