Representation of results of particle size analysis

ISO 9276-2:2014 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. 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.

Représentation de données obtenues par analyse granulométrique

Predstavitev podatkov, dobljenih z granulometrijsko analizo - 2. del: Izračun povprečnih velikosti/premerov in momentov iz granulacij

Ta del standarda ISO 9276 navaja ustrezne enačbe in skladne nomenklature za izračune momentov in povprečnih velikosti iz granulacij ter standardnih odklonov. Opisana sta dva sistema zapisa, ki se običajno uporabljata. Eden je metoda momentov, drugi pa opisuje metodo razmerja momenta. Granulacija je lahko na voljo kot histogram ali analitična funkcija.
Ekvivalentni premer delca kakršne koli oblike je vzet kot velikost tega delca. Dejavniki oblike delca niso upoštevani. Bistveno je, da je tehnika merjenja navedena v poročilu v smislu odvisnosti rezultatov določanja velikosti od načela merjenja. Vzorci izmerjenih delcev naj bi bili reprezentativni za populacijo delcev.
Za oba sistema zapisa so v dodatku predstavljeni numerični primeri izračuna povprečnih velikosti delcev in standardnega odklona od podatkov histograma.
Natančnost povprečne velikosti delca se lahko zmanjša, če se ocenjuje nepopolna granulacija. Natančnost se lahko zmanjša tudi, kadar se uporablja zelo omejeno število razredov velikosti.

General Information

Status
Published
Publication Date
11-May-2014
Current Stage
6060 - International Standard published
Start Date
01-Apr-2014
Completion Date
12-May-2014

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INTERNATIONAL ISO
STANDARD 9276-2
Second edition
2014-05-15
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
Reference number
ISO 9276-2:2014(E)
ISO 2014
---------------------- Page: 1 ----------------------
ISO 9276-2:2014(E)
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 copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO 2014 – All rights reserved
---------------------- Page: 2 ----------------------
ISO 9276-2:2014(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Definitions, symbols and abbreviated terms .......................................................................................................................... 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-notation....................................11

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 .............................................................................................................................................................................................................................27

© ISO 2014 – All rights reserved iii
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ISO 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 morphology

iv © ISO 2014 – All rights reserved
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ISO 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

[1][2]
distributions only as one part .

The second method is named moment-ratio-notation. The moment-ratio-notation is based on a number

[3][4]

statistics and frequencies approach, but includes also conversion to other types of quantities .

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

[5]
notation systems .
© ISO 2014 – All rights reserved v
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INTERNATIONAL STANDARD ISO 9276-2:2014(E)
Representation 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 Definitions, symbols and abbreviated terms

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 (M)
or midpoint particle size D (M-R)
k power of x
m m number of size classes
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ISO 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 complete k-th moment of a q (x) – sample distribution
k,r r
m complete k-th central moment of a q (x) – sample distribution
k,r r
M p-th moment of a number distribution density
m 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 (x) q (D) distribution density of type of particle quantity r
r r
mean height of a distribution density in the i-th particle size interval,
ri,
Q (x) Q (D) cumulative distribution of type of quantity r
r r
ΔQ difference of two values of the cumulative distribution, i.e. relative
r,i
amount in the i-th particle size interval, Δx
s s standard deviation of a q (x) and q (D) distribution
r r r r
s s geometric standard deviation of a distribution
g g
s s standard deviation of lognormal distribution (s = ln s )
S S surface area
S S volume specific surface area
V V
V V particle volume
mean particle volume
x D particle size, diameter of an equivalent sphere
x upper particle size of the i-th particle size interval
x lower particle size of the i-th particle size interval
i-1
D midpoint size of the i-th size class
x particle size below which there are no particles in a given size distri-
min
bution
2 © ISO 2014 – All rights reserved
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ISO 9276-2:2014(E)
M-notation M-R-notation Description
x particle size above which there are no particles in a given size distri-
max
bution
mean particle sizes (general description)
x D
kr, pq,
geometric mean particle sizes
pp,
arithmetic mean particle size
k,0
weighted mean particle size
kr,
geometric mean particle size
geo,r
harmonic mean particle size
har,r
x median particle size of a cumulative volume distribution
50,3
Δx = x – x width of the i-th particle size interval
i i i-1
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 moments according to the moment-notation
[1]

The complete k-th moment of a distribution density 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.

max
Mx= qx()dx (1)
k,r r
min

If r = 0, q (x) represents a number distribution density, if r = 3, q (x) represents a volume or mass

0 3
distribution density.

Formula (1) describes a complete moment if the integral boundaries are represented by the minimum

particle size (x ) and the maximum particle size (x ).
min max
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ISO 9276-2:2014(E)
A special complete moment is represented by M :
0,r
x x
maxx max
Mx==qx()ddxq ()xx =−Qx() Qx()=1 (2)
0,r rr rrmaxmin
∫ ∫
x x
min min
with
Qx()= qx()dx (3)
ri r
min

A moment is incomplete, if the integration is performed between two arbitrary particle diameters x

i-1
and x within the given size range of a distribution:
x < x < x < x < x
min i-1 i max
Mx(,xx)(= qx)dx (4)
k,ri−1 i r
i−1

Apart from the moments related to the origin of the particle size axis and shown in Formulae (1) and (4),

the so-called k-th central moment of a q (x) – distribution density, m , can be derived from a given

r k,r

distribution density. It is related to the weighted mean particle size x [see Formula (11)].

kr,
The complete k-th central moment is defined as:
max
mx=−()xq ()xxd (5)
1,r
k,r r
min
4.2 Definition of mean particle sizes according to the moment-notation
All mean particle sizes are defined by Formula (6):
xM= (6)
k,r
k,r

Depending on the numbers chosen for the subscripts, k and r, different mean particle sizes may be defined.

Since the mean particle sizes calculated from Formula (6) may differ considerably, the subscripts k and

r should always be quoted.

There are two groups of mean particle sizes, which should preferably be used, viz. arithmetic mean

particle sizes and weighted mean particle sizes.
4.2.1 Terminology for mean particle sizes in the moment-notation x
kr,
Table 1 presents terminology examples of mean sizes.
4 © ISO 2014 – All rights reserved
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ISO 9276-2:2014(E)
Table 1 — Terminology for mean particle sizes x
kr,
Systematic code Terminology
x arithmetic mean size
1,0
x arithmetic mean area size
2,0
x arithmetic mean volume size
3,0
x size-weighted mean size
1,1
x area-weighted mean size, Sauter mean diameter
1,2
x volume-weighted mean size
1,3
4.2.2 Arithmetic mean particle sizes

Arithmetic mean size is a number-weighted mean size, calculated from a number distribution density,

q (x):
xM= (7)
k,0
k,0

Counting single particles in a microscope image is a typical example to obtain number (r = 0) percentages

as basis of averaging.
In accordance with Reference [2], the recommended mean particle sizes are:
arithmetic mean size (corresponds to arithmetic mean length size):
xM= (8)
10,
10,
[2]
arithmetic mean area size (Heywood : mean surface diameter):
xM= (9)
20,
20,
[2]
arithmetic mean volume size (Heywood : mean-weight diameter):
xM= (10)
30,
30,
4.2.3 Weighted mean particle sizes
Weighted mean particle sizes are defined by:
xM= (11)
k,r
k,r

Weighing sieves before and after sieving is a typical example to obtain mass (r = 3) percentages as basis

of averaging. Weighted mean particle sizes represent the abscissa of the centre of gravity of a q (x) –

distribution. The recommended weighted mean particle sizes are represented by Formulae (12) to (15).

The weighted mean particle size of a number distribution density, q (x), is equivalent to the arithmetic

mean length size [see Formula (8)]. It is represented by:
© ISO 2014 – All rights reserved 5
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ISO 9276-2:2014(E)
[6]
arithmetic mean size (Heywood : numerical mean diameter):
xM= (12)
10,
10,

The weighted mean particle size of a length distribution density, q (x), is given by:

[6]
size-weighted mean size (Heywood : linear mean diameter):
xM= (13)
11,
11,

The weighted mean particle size of a surface distribution density, q (x), is represented by:

[6]
area-weighted mean size (Heywood : surface mean diameter):
xM= (14)
12,
12,

The weighted mean particle size of a volume distribution density, q (x), is given by:

[6]
volume-weighted mean size (Heywood : weight mean diameter):
xM= (15)
13,
13,
4.2.4 Geometric mean particle sizes

If a particle size distribution conforms satisfactorily to a lognormal size distribution (see ISO 9276-5),

the geometric mean particle size characterizes the mean value of the logarithm of x. The median of a

lognormal distribution has the same value as the geometric mean size.

Instead of the arithmetic mean, calculated from the sum of n values, divided by their number n, the

geometric mean is the n-th root of the product of n values. In terms of logarithms the logarithm of the

geometric mean is calculated from the sum of the logarithms of n values, divided by their number n.

The arithmetic mean is greater than the geometric, the inequality increasing the greater the dispersion

among the values.

Mathematical limit analysis of Formula (6) with k approaching zero (see also derivation for p = q in

[3]
moment-ratio-notation ) leads to the geometric mean size:
max
In xq ()xdx
min
xe==x (16)
0,r geo,r
or in terms of logarithms:
max
Inxx==In Inxq ()xdx (17)
0,rrgeo,
min
Based on data from a histogram one obtains the r-weighted geometric mean size:
m m
   
xx=expe In qxΔΔ= xp InxQ ==x (18)
0,ri i geo,r
∑∑ri, i ri,
   
i==11i
   
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ISO 9276-2:2014(E)
4.2.5 Harmonic mean particle sizes
[7]

The harmonic mean of a series of values is the reciprocal of the arithmetic mean of their reciprocals .

The harmonic mean is smaller than the geometric, the inequality increasing the greater the dispersion

among the values. Therefore the r-weighted harmonic mean size can be calculated from:

1 1
x == (19)
harr,
min M
−1,r
qx()dx
max
or based on data from a histogram:
1 1
x == (20)
harr,
m m
1 1
qxΔΔQ
i ri,
∑∑ri,
x x
i i
i==11i

4.3 Calculation of moments and mean particle sizes from a given size distribution

4.3.1 The calculation of M and mean particle sizes from a number or a volume based size

k,r
distribution

In many cases of practical application, the measured data are either represented by a number distribution

density, q (x), or a volume distribution density, q (x). The calculation of the mean particle sizes described

0 3
[1]
above can then be performed according to Formula (21) :
M M
kr++,0 kr−3,3
xM== k =k (21)
k,r
k,r
M M
r,0 r−3,3
This leads to:
−13,
xM== (22)
20,
20,
−33,
xM== 3 (23)
30,
30,
−33,
M M
20, −13,
xM== = (24)
11,
11,
M M
10, −23,
30,
xM== = (25)
12,
12,
20,,−13
40,
xM== (26)
13,
13,
30,

One realizes from Formulae (21) to (26), that the following moments are needed if the mean particle

sizes defined above are to be calculated:
from a given number distribution density, q (x): M ; M ; M ; M
0 1,0 2,0 3,0 4,0
from a given volume distribution density, q (x): M M ; M ; M
3 1,3 –1,3 –2,3 –3,3

Formula (21) shows, that each mean particle size or moment of a given type of quantity can be expressed

as a ratio of two number based moments, which is used as the main approach of the moment-ratio-

notation.
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ISO 9276-2:2014(E)
4.3.2 Calculation of M from a particle size distribution, given as a histogram
k,r

If a distribution density is given as a histogram, q (x , x ) is constant in the particle size interval

r i-1 i
Δx = x − x .
i i i-1
Formula (1) may therefore be rewritten as follows:
max m
k k
Mx= qx dx = xq Δx (27)
kr,,r ∑ i ri i
i=1
min
One obtains with
ri,
q = (28)
ri,
xx−
i i1−
max
k k
Mx= qx dx = xQΔ (29)
kr,,r i ri
i=1
min

The approximate mean x within a size class can be calculated as arithmetic mean in each size class

 
xx+
ii−1
x =   (30)
 
 

Alternative approximate means like geometric mean or an integral mean have not specific advantage.

Several investigations have shown (see e.g. Reference [8]), that there is no general preference for all

types of distributions possible.

The discrete nature of histogram data causes uncertainties up to a few percent in the calculated moments

and mean sizes, which relate to the width of the size classes and the corresponding uncertainty in

the amount of particles derived from the estimated mean size in each class. Methods to improve the

size resolution of the representation of measurement by observation of more size classes are given in

ISO 9276-3.
The moments M , M , M , M , M , M , M and M can therefore be calculated from
1,0 2,0 3,0 4,0 1,3 –1,3 –2,3 –3,3
Formulae (31) to (38):
m m
1 1
Mx==qxΔΔxQ (31)
10,,ii0 i ii0,
i==1 i 1
m m
2 2
Mx==qxΔΔxQ (32)
20,,∑∑ii0 i ii0,
i==1 i 1
m m
3 3
Mx==qxΔΔxQ (33)
30,,ii0 i ii0,
i==1 i 1
m m
4 4
Mx==qxΔΔxQ (34)
40,,∑∑ii0 i ii0,
i==1 i 1
m m
1 1
Mx==qxΔΔxQ (35)
13,,ii3 i ii3,
i==1 i 1
m m
M ==qxΔΔQ (36)
−13,,∑∑3ii 3,i
1 1
x x
i==1 i i 1 i
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ISO 9276-2:2014(E)
m m
M ==qxΔΔQ (37)
−23,,∑∑3ii 3,i
2 2
x x
i==1 i i 1 i
m m
M ==qxΔΔQ (38)
−33,,∑∑3ii 3,i
3 3
x x
i==1 i i 1 i
4.4 Variance and standard deviation of a particle size distribution

The spread of a size distribution may be represented by its variance, which represents the square of the

standard deviation, s .. The variance, s , of a q (x) – distribution is defined as:

r r r
max
sx=−xq xdx (39)
1,r ()
r r
min

Introducing complete moments, the variance can be calculated (Reference [3]) from:

sm==MM− (40)
rr22,,rr1,
or based on data from a histogram:
2 2
m m m m
   
22 1 2 1
   
sx=−qxΔ xq ΔΔxx= Q −− xQΔ (41)
ri∑∑ri,,i ir ii ∑ ir,i ∑ ir,i
   
i==1 i 1 i=1 i=1
   
A numerical example of the calculation of s is given in Annex A.

For a lognormal q (x) – distribution, the standard deviation s can be calculated from:

sx==ln(/xx)ln( /)x (42)
84,,rr50 50,,rr16
The geometrical standard deviation s is obtained from:
s = exp(s)
(43)
Hence,
sx==//xx x (44)
g 84,,rr50 50,,rr16
4.5 Calculation of moments and mean particle sizes from a lognormal distribution

The complete k-th moment of a lognormal probability distribution, q (x), calculates to

kxln +05, ks
kk05, s
50,r
Mx==ee (45)
kr,,50r

A series of mean particle sizes can be calculated from the k-th root of the k-th moment in Formula (45)

or from the median (and geometric mean) x and the standard deviation of that distribution using

50,r
Formula (46):
ln xk+05, s
05, ks
50,r
xx==ee (46)
kr,,50 r

The median x of a lognormal distribution has the same value as the geometric mean size x .

50,r
0,r
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ISO 9276-2:2014(E)
4.6 Calculation of volume specific surface area and the Sauter mean diameter

From distributions of any type of quantity moments can be used in the calculation of the volume specific

surface area, S , since S is inversely proportional to the weighted mean surface size, the Sauter mean

V V
diameter, x (Formula 14). It is given by:
12,
S = (47)
12,

Taking into account Formula (25), one arrives from surface, number or volume distributions at:

20,
S ==66=⋅M (48)
V −13,
M M
12, 30,
For particles other than spheres a shape factor shall be introduced.
5 The moment-ratio-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.

NOTE Distribution density and cumulative distribution are represented by q and Q in the ISO 9276 series, but

in literature (References [3],[4],[7]–[9]) the symbols f and F are also used.
5.1 Definition of moments according to the moment-ratio-notation

Two different types of moments may be used, viz., moments and central moments. Moments are centred

around the origin of the particle size axis and central moments around the arithmetic mean particle

size.
The p-th moment of a sample, denoted as M , is defined as:
−1 p
MN= nD (49)
pi∑ i
where
Nn=
i is the total number of particles involved in the measurement;
D is the midpoint of the i-th size class [see also comments to Formula (30)];

n is the number of particles in the i-th size class (i.e. the class frequency of the number distri-

bution density).

The first moment, the (arithmetic) sample mean M of the particle sizes D, is mostly represented by D .

The second and third moments are proportional to the mean surface area and the mean volume

respectively, of the particles in a sample.
10 © ISO 2014 – All rights reserved
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ISO 9276-2:2014(E)
The p-th central moment around the mean D , denoted as m , is defined by:
−1 p
mN=−nD()D (50)
pi∑ i
Central moments of particle sizes D, are related to differ
...

SLOVENSKI STANDARD
SIST ISO 9276-2:2015
01-marec-2015
1DGRPHãþD
SIST ISO 9276-2:2002
3UHGVWDYLWHYSRGDWNRYGREOMHQLK]JUDQXORPHWULMVNRDQDOL]RGHO,]UDþXQ
SRYSUHþQLKYHOLNRVWLSUHPHURYLQPRPHQWRYL]JUDQXODFLM

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
Ta slovenski standard je istoveten z: ISO 9276-2:2014
ICS:
19.120 Analiza velikosti delcev. Particle size analysis. Sieving
Sejanje
SIST ISO 9276-2:2015 en

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

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SIST ISO 9276-2:2015
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SIST ISO 9276-2:2015
INTERNATIONAL ISO
STANDARD 9276-2
Second edition
2014-05-15
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
Reference number
ISO 9276-2:2014(E)
ISO 2014
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SIST ISO 9276-2:2015
ISO 9276-2:2014(E)
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 copyright@iso.org
Web www.iso.org
Published in Switzerland
ii © ISO 2014 – All rights reserved
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Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Definitions, symbols and abbreviated terms .......................................................................................................................... 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-notation....................................11

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 .............................................................................................................................................................................................................................27

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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 morphology

iv © ISO 2014 – All rights reserved
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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

[1][2]
distributions only as one part .

The second method is named moment-ratio-notation. The moment-ratio-notation is based on a number

[3][4]

statistics and frequencies approach, but includes also conversion to other types of quantities .

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

[5]
notation systems .
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SIST ISO 9276-2:2015
INTERNATIONAL STANDARD ISO 9276-2:2014(E)
Representation 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 Definitions, symbols and abbreviated terms

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 (M)
or midpoint particle size D (M-R)
k power of x
m m number of size classes
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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 complete k-th moment of a q (x) – sample distribution
k,r r
m complete k-th central moment of a q (x) – sample distribution
k,r r
M p-th moment of a number distribution density
m 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 (x) q (D) distribution density of type of particle quantity r
r r
mean height of a distribution density in the i-th particle size interval,
ri,
Q (x) Q (D) cumulative distribution of type of quantity r
r r
ΔQ difference of two values of the cumulative distribution, i.e. relative
r,i
amount in the i-th particle size interval, Δx
s s standard deviation of a q (x) and q (D) distribution
r r r r
s s geometric standard deviation of a distribution
g g
s s standard deviation of lognormal distribution (s = ln s )
S S surface area
S S volume specific surface area
V V
V V particle volume
mean particle volume
x D particle size, diameter of an equivalent sphere
x upper particle size of the i-th particle size interval
x lower particle size of the i-th particle size interval
i-1
D midpoint size of the i-th size class
x particle size below which there are no particles in a given size distri-
min
bution
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M-notation M-R-notation Description
x particle size above which there are no particles in a given size distri-
max
bution
mean particle sizes (general description)
x D
kr, pq,
geometric mean particle sizes
pp,
arithmetic mean particle size
k,0
weighted mean particle size
kr,
geometric mean particle size
geo,r
harmonic mean particle size
har,r
x median particle size of a cumulative volume distribution
50,3
Δx = x – x width of the i-th particle size interval
i i i-1
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 moments according to the moment-notation
[1]

The complete k-th moment of a distribution density 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.

max
Mx= qx()dx (1)
k,r r
min

If r = 0, q (x) represents a number distribution density, if r = 3, q (x) represents a volume or mass

0 3
distribution density.

Formula (1) describes a complete moment if the integral boundaries are represented by the minimum

particle size (x ) and the maximum particle size (x ).
min max
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A special complete moment is represented by M :
0,r
x x
maxx max
Mx==qx()ddxq ()xx =−Qx() Qx()=1 (2)
0,r rr rrmaxmin
∫ ∫
x x
min min
with
Qx()= qx()dx (3)
ri r
min

A moment is incomplete, if the integration is performed between two arbitrary particle diameters x

i-1
and x within the given size range of a distribution:
x < x < x < x < x
min i-1 i max
Mx(,xx)(= qx)dx (4)
k,ri−1 i r
i−1

Apart from the moments related to the origin of the particle size axis and shown in Formulae (1) and (4),

the so-called k-th central moment of a q (x) – distribution density, m , can be derived from a given

r k,r

distribution density. It is related to the weighted mean particle size x [see Formula (11)].

kr,
The complete k-th central moment is defined as:
max
mx=−()xq ()xxd (5)
1,r
k,r r
min
4.2 Definition of mean particle sizes according to the moment-notation
All mean particle sizes are defined by Formula (6):
xM= (6)
k,r
k,r

Depending on the numbers chosen for the subscripts, k and r, different mean particle sizes may be defined.

Since the mean particle sizes calculated from Formula (6) may differ considerably, the subscripts k and

r should always be quoted.

There are two groups of mean particle sizes, which should preferably be used, viz. arithmetic mean

particle sizes and weighted mean particle sizes.
4.2.1 Terminology for mean particle sizes in the moment-notation x
kr,
Table 1 presents terminology examples of mean sizes.
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Table 1 — Terminology for mean particle sizes x
kr,
Systematic code Terminology
x arithmetic mean size
1,0
x arithmetic mean area size
2,0
x arithmetic mean volume size
3,0
x size-weighted mean size
1,1
x area-weighted mean size, Sauter mean diameter
1,2
x volume-weighted mean size
1,3
4.2.2 Arithmetic mean particle sizes

Arithmetic mean size is a number-weighted mean size, calculated from a number distribution density,

q (x):
xM= (7)
k,0
k,0

Counting single particles in a microscope image is a typical example to obtain number (r = 0) percentages

as basis of averaging.
In accordance with Reference [2], the recommended mean particle sizes are:
arithmetic mean size (corresponds to arithmetic mean length size):
xM= (8)
10,
10,
[2]
arithmetic mean area size (Heywood : mean surface diameter):
xM= (9)
20,
20,
[2]
arithmetic mean volume size (Heywood : mean-weight diameter):
xM= (10)
30,
30,
4.2.3 Weighted mean particle sizes
Weighted mean particle sizes are defined by:
xM= (11)
k,r
k,r

Weighing sieves before and after sieving is a typical example to obtain mass (r = 3) percentages as basis

of averaging. Weighted mean particle sizes represent the abscissa of the centre of gravity of a q (x) –

distribution. The recommended weighted mean particle sizes are represented by Formulae (12) to (15).

The weighted mean particle size of a number distribution density, q (x), is equivalent to the arithmetic

mean length size [see Formula (8)]. It is represented by:
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[6]
arithmetic mean size (Heywood : numerical mean diameter):
xM= (12)
10,
10,

The weighted mean particle size of a length distribution density, q (x), is given by:

[6]
size-weighted mean size (Heywood : linear mean diameter):
xM= (13)
11,
11,

The weighted mean particle size of a surface distribution density, q (x), is represented by:

[6]
area-weighted mean size (Heywood : surface mean diameter):
xM= (14)
12,
12,

The weighted mean particle size of a volume distribution density, q (x), is given by:

[6]
volume-weighted mean size (Heywood : weight mean diameter):
xM= (15)
13,
13,
4.2.4 Geometric mean particle sizes

If a particle size distribution conforms satisfactorily to a lognormal size distribution (see ISO 9276-5),

the geometric mean particle size characterizes the mean value of the logarithm of x. The median of a

lognormal distribution has the same value as the geometric mean size.

Instead of the arithmetic mean, calculated from the sum of n values, divided by their number n, the

geometric mean is the n-th root of the product of n values. In terms of logarithms the logarithm of the

geometric mean is calculated from the sum of the logarithms of n values, divided by their number n.

The arithmetic mean is greater than the geometric, the inequality increasing the greater the dispersion

among the values.

Mathematical limit analysis of Formula (6) with k approaching zero (see also derivation for p = q in

[3]
moment-ratio-notation ) leads to the geometric mean size:
max
In xq ()xdx
min
xe==x (16)
0,r geo,r
or in terms of logarithms:
max
Inxx==In Inxq ()xdx (17)
0,rrgeo,
min
Based on data from a histogram one obtains the r-weighted geometric mean size:
m m
   
xx=expe In qxΔΔ= xp InxQ ==x (18)
0,ri i geo,r
∑∑ri, i ri,
   
i==11i
   
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4.2.5 Harmonic mean particle sizes
[7]

The harmonic mean of a series of values is the reciprocal of the arithmetic mean of their reciprocals .

The harmonic mean is smaller than the geometric, the inequality increasing the greater the dispersion

among the values. Therefore the r-weighted harmonic mean size can be calculated from:

1 1
x == (19)
harr,
min M
−1,r
qx()dx
max
or based on data from a histogram:
1 1
x == (20)
harr,
m m
1 1
qxΔΔQ
i ri,
∑∑ri,
x x
i i
i==11i

4.3 Calculation of moments and mean particle sizes from a given size distribution

4.3.1 The calculation of M and mean particle sizes from a number or a volume based size

k,r
distribution

In many cases of practical application, the measured data are either represented by a number distribution

density, q (x), or a volume distribution density, q (x). The calculation of the mean particle sizes described

0 3
[1]
above can then be performed according to Formula (21) :
M M
kr++,0 kr−3,3
xM== k =k (21)
k,r
k,r
M M
r,0 r−3,3
This leads to:
−13,
xM== (22)
20,
20,
−33,
xM== 3 (23)
30,
30,
−33,
M M
20, −13,
xM== = (24)
11,
11,
M M
10, −23,
30,
xM== = (25)
12,
12,
20,,−13
40,
xM== (26)
13,
13,
30,

One realizes from Formulae (21) to (26), that the following moments are needed if the mean particle

sizes defined above are to be calculated:
from a given number distribution density, q (x): M ; M ; M ; M
0 1,0 2,0 3,0 4,0
from a given volume distribution density, q (x): M M ; M ; M
3 1,3 –1,3 –2,3 –3,3

Formula (21) shows, that each mean particle size or moment of a given type of quantity can be expressed

as a ratio of two number based moments, which is used as the main approach of the moment-ratio-

notation.
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4.3.2 Calculation of M from a particle size distribution, given as a histogram
k,r

If a distribution density is given as a histogram, q (x , x ) is constant in the particle size interval

r i-1 i
Δx = x − x .
i i i-1
Formula (1) may therefore be rewritten as follows:
max m
k k
Mx= qx dx = xq Δx (27)
kr,,r ∑ i ri i
i=1
min
One obtains with
ri,
q = (28)
ri,
xx−
i i1−
max
k k
Mx= qx dx = xQΔ (29)
kr,,r i ri
i=1
min

The approximate mean x within a size class can be calculated as arithmetic mean in each size class

 
xx+
ii−1
x =   (30)
 
 

Alternative approximate means like geometric mean or an integral mean have not specific advantage.

Several investigations have shown (see e.g. Reference [8]), that there is no general preference for all

types of distributions possible.

The discrete nature of histogram data causes uncertainties up to a few percent in the calculated moments

and mean sizes, which relate to the width of the size classes and the corresponding uncertainty in

the amount of particles derived from the estimated mean size in each class. Methods to improve the

size resolution of the representation of measurement by observation of more size classes are given in

ISO 9276-3.
The moments M , M , M , M , M , M , M and M can therefore be calculated from
1,0 2,0 3,0 4,0 1,3 –1,3 –2,3 –3,3
Formulae (31) to (38):
m m
1 1
Mx==qxΔΔxQ (31)
10,,ii0 i ii0,
i==1 i 1
m m
2 2
Mx==qxΔΔxQ (32)
20,,∑∑ii0 i ii0,
i==1 i 1
m m
3 3
Mx==qxΔΔxQ (33)
30,,ii0 i ii0,
i==1 i 1
m m
4 4
Mx==qxΔΔxQ (34)
40,,∑∑ii0 i ii0,
i==1 i 1
m m
1 1
Mx==qxΔΔxQ (35)
13,,ii3 i ii3,
i==1 i 1
m m
M ==qxΔΔQ (36)
−13,,∑∑3ii 3,i
1 1
x x
i==1 i i 1 i
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m m
M ==qxΔΔQ (37)
−23,,∑∑3ii 3,i
2 2
x x
i==1 i i 1 i
m m
M ==qxΔΔQ (38)
−33,,∑∑3ii 3,i
3 3
x x
i==1 i i 1 i
4.4 Variance and standard deviation of a particle size distribution

The spread of a size distribution may be represented by its variance, which represents the square of the

standard deviation, s .. The variance, s , of a q (x) – distribution is defined as:

r r r
max
sx=−xq xdx (39)
1,r ()
r r
min

Introducing complete moments, the variance can be calculated (Reference [3]) from:

sm==MM− (40)
rr22,,rr1,
or based on data from a histogram:
2 2
m m m m
   
22 1 2 1
   
sx=−qxΔ xq ΔΔxx= Q −− xQΔ (41)
ri∑∑ri,,i ir ii ∑ ir,i ∑ ir,i
   
i==1 i 1 i=1 i=1
   
A numerical example of the calculation of s is given in Annex A.

For a lognormal q (x) – distribution, the standard deviation s can be calculated from:

sx==ln(/xx)ln( /)x (42)
84,,rr50 50,,rr16
The geometrical standard deviation s is obtained from:
s = exp(s)
(43)
Hence,
sx==//xx x (44)
g 84,,rr50 50,,rr16
4.5 Calculation of moments and mean particle sizes from a lognormal distribution

The complete k-th moment of a lognormal probability distribution, q (x), calculates to

kxln +05, ks
kk05, s
50,r
Mx==ee (45)
kr,,50r

A series of mean particle sizes can be calculated from the k-th root of the k-th moment in Formula (45)

or from the median (and geometric mean) x and the standard deviation of that distribution using

50,r
Formula (46):
ln xk+05, s
05, ks
50,r
xx==ee (46)
kr,,50 r

The median x of a lognormal distribution has the same value as the geometric mean size x .

50,r
0,r
© ISO 2014 – All rights reserved 9
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SIST ISO 9276-2:2015
ISO 9276-2:2014(E)
4.6 Calculation of volume specific surface area and the Sauter mean diameter

From distributions of any type of quantity moments can be used in the calculation of the volume specific

surface area, S , since S is inversely proportional to the weighted mean surface size, the Sauter mean

V V
diameter, x (Formula 14). It is given by:
12,
S = (47)
12,

Taking into account Formula (25), one arrives from surface, number or volume distributions at:

20,
S ==66=⋅M (48)
V −13,
M M
12, 30,
For particles other than spheres a shape factor shall be introduced.
5 The moment-ratio-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
...

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