1、Designation: E2578 07 (Reapproved 2018)Standard Practice forCalculation of Mean Sizes/Diameters and StandardDeviations of Particle Size Distributions1This standard is issued under the fixed designation E2578; the number immediately following the designation indicates the year oforiginal adoption or,
2、 in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 The purpose of this practice is to present procedures forcalculating mean sizes
3、and standard deviations of size distri-butions given as histogram data (see Practice E1617). Theparticle size is assumed to be the diameter of an equivalentsphere, for example, equivalent (area/surface/volume/perimeter) diameter.1.2 The mean sizes/diameters are defined according to theMoment-Ratio (
4、M-R) definition system.2,3,41.3 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.4 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of thi
5、s standard to establish appro-priate safety, health, and environmental practices and deter-mine the applicability of regulatory limitations prior to use.1.5 This international standard was developed in accor-dance with internationally recognized principles on standard-ization established in the Deci
6、sion on Principles for theDevelopment of International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.2. Referenced Documents2.1 ASTM Standards:5E1617 Practice for Reporting Particle Size CharacterizationData3. Terminology3.1
7、Definitions of Terms Specific to This Standard:3.1.1 diameter distribution, nthe distribution by diameterof particles as a function of their size.3.1.2 equivalent diameter, ndiameter of a circle or spherewhich behaves like the observed particle relative to or deducedfrom a chosen property.3.1.3 geom
8、etric standard deviation, nexponential of thestandard deviation of the distribution of log-transformed par-ticle sizes.3.1.4 histogram, na diagram of rectangular bars propor-tional in area to the frequency of particles within the particlesize intervals of the bars.3.1.5 lognormal distribution, na di
9、stribution of particlesize, whose logarithm has a normal distribution; the left tail ofa lognormal distribution has a steep slope on a linear size scale,whereas the right tail decreases gradually.3.1.6 mean particle size/diameter, nsize or diameter of ahypothetical particle such that a population of
10、 particles havingthat size/diameter has, for a purpose involved, properties whichare equal to those of a population of particles with differentsizes/diameters and having that size/diameter as a meansize/diameter.3.1.7 moment of a distribution, na moment is the meanvalue of a power of the particle si
11、zes (the 3rd moment isproportional to the mean volume of the particles).3.1.8 normal distribution, na distribution which is alsoknown as Gaussian distribution and as bell-shaped curvebecause the graph of its probability density resembles a bell.3.1.9 number distribution, nthe distribution by number
12、ofparticles as a function of their size.1This practice is under the jurisdiction of ASTM Committee E56 on Nanotech-nology and is the direct responsibility of Subcommittee E56.02 on Physical andChemical Characterization.Current edition approved Jan. 1, 2018. Published January 2018. Originallyapproved
13、 in 2007. Last previous edition approved in 2012 as E2578 07 (2012).DOI: 10.1520/E2578-07R18.2Alderliesten, M., “Mean Particle Diameters. Part I: Evaluation of DefinitionSystems,” Particle and Particle Systems Characterization, Vol 7, 1990, pp.233241.3Alderliesten, M., “Mean Particle Diameters. From
14、 Statistical Definition toPhysical Understanding,” Journal of Biopharmaceutical Statistics, Vol 15, 2005, pp.295325.4Mugele, R.A., and Evans, H.D., “Droplet Size Distribution in Sprays,” Journalof Industrial and Engineering Chemistry, Vol 43, 1951, pp. 13171324.5For referenced ASTM standards, visit
15、the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.
16、 United StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Bar
17、riers to Trade (TBT) Committee.13.1.10 order of mean diameter, nthe sum of the subscriptsp and q of the mean diameter Dp,q.3.1.11 particle, na discrete piece of matter.3.1.12 particle diameter/size, nsome consistent measureof the spatial extent of a particle (see equivalent diameter).3.1.13 particle
18、 size distribution, na description of the sizeand frequency of particles in a population.3.1.14 population, na set of particles concerning whichstatistical inferences are to be drawn, based on a representativesample taken from the population.3.1.15 sample, na part of a population of particles.3.1.16
19、 standard deviation, nmost widely used measure ofthe width of a frequency distribution.3.1.17 surface distribution, nthe distribution by surfacearea of particles as a function of their size.3.1.18 variance, na measure of spread around the mean;square of the standard deviation.3.1.19 volume distribut
20、ion, nthe distribution by volume ofparticles as a function of their size.4. Summary of Practice4.1 Samples of particles to be measured should be repre-sentative for the population of particles.4.2 The frequencyof a particular value of a particle size Dcan be measured (or expressed) in terms of the n
21、umber ofparticles, the cumulated diameters, surfaces or volumes of theparticles. The corresponding frequency distributions are calledNumber, Diameter, Surface, or Volume distributions.4.3 As class mid points Diof the histogram intervals thearithmetic mean values of the class boundaries are used.4.4
22、Particle shape factors are not taken into account, al-though their importance in particle size analysis is beyonddoubt.4.5 A coherent nomenclature system is presented whichconveys the physical meanings of mean particle diameters.5. Significance and Use5.1 Mean particle diameters defined according to
23、 theMoment-Ratio (M-R) system are derived from ratios betweentwo moments of a particle size distribution.6. Mean Particle Sizes/Diameters6.1 Moments of Distributions:6.1.1 Moments are the basis for defining mean sizes andstandard deviations. A random sample, containing N elementsfrom a population of
24、 particle sizes Di, enables estimation of themoments of the size distribution of the population of particlesizes. The r-th sample moment, denoted by Mr, is defined tobe:Mr: 5 N21(iniDir(1)where N5(ini, Diis the midpoint of the i-th interval and niis the number of particles in the i-th size class (th
25、at is, classfrequency). The (arithmetic) sample mean M1of the particlesize D is mostly represented by D. The r-th sample momentabout the mean D, denoted by Mr, is defined by:Mr: 5 N21(iniDi2 D!r(2)6.1.2 The best-known example is the sample variance M2.This M2always underestimates the population vari
26、anceD2(squared standard deviation). Instead, M2multiplied byN/(N1) is used, which yields an unbiased estimator, sD2, forthe population variance. Thus, the sample variance sD2has tobe calculated from the equation:sD25NN 2 1M25(iniDi2 D!2N 2 1(3)6.1.3 Its square root is the standard deviation sDof the
27、sample (see also 6.3). If the particle sizes D are lognormallydistributed, then the logarithm of D,lnD, follows a normaldistribution (Gaussian distribution). The geometric mean Dgofthe particle sizes D equals the exponential of the (arithmetic)mean of the (lnD)-values:Dg5 expN21(inilnDi!#5!NiDini(4)
28、6.1.4 The standard deviation slnDof the (lnD)-values can beexpressed as:slnD5(ini$lnDi/Dg!%2N 2 1(5)6.2 Definition of Mean Diameters Dp,q:6.2.1 The mean diameter Dp,qof a sample of particle sizes isdefined as 1/(p q)-th power of the ratio of the p-th and theq-th moment of the Number distribution of
29、the particle sizes:Dp,q5FMpMq G1/p2q!if pfiq (6)6.2.2 Using Eq 1, Eq 6 can be rewritten as:Dp,q53(iniDip(iniDiq41/p2q!if pfiq (7)6.2.3 The powers p and q may have any real value. Forequal values of p and q it is possible to derive from Eq 7 that:Dq,q5 exp3(iniDiqlnDi(iniDiq4if p 5 q (8)6.2.4 If q =
30、0, then:D0,05 exp3(inilnDi(ini45!NiDini(9)6.2.5 D0,0is the well-known geometric mean diameter. Thephysical dimension of any Dp,qis equal to that of D itself.E2578 07 (2018)26.2.6 Mean diameters Dp,qof a sample can be estimatedfrom any size distribution fr(D) according to equations similarto Eq 7 and
31、 8:Dp,q53(imfrDi!Dip2r(imfrDi!Diq2r41/p2qif pfiq (10)and:Dp,p5 exp3(imfrDi!Dip2rlnDi(imfrDi!Dip2r 4if p 5 q (11)where:fr(Di) = particle quantity in the i-th class,Di= midpoint of the i-th class interval,r = 0, 1, 2, or 3 represents the type of quantity, viz.number, diameter, surface, volume (or mass
32、)respectively, andm = number of classes.6.2.7 If r = 0 and we put ni= f0(Di), then Eq 10 reduces tothe familiar form Eq 7.6.3 Standard Deviation:6.3.1 According to Eq 3, the standard deviation of theNumber distribution of a sample of particle sizes can beestimated from:sD5(iniDi22 ND1,02N 2 1(12)whi
33、ch can be rewritten as:s 5 c=D2,022 D1,02(13)with:c 5 =N/N 2 1! (14)6.3.2 In practice, N 100, so that c 1. Hence:s=D2,022 D1,02(15)6.3.3 The standard deviation slnDof a lognormal Numberdistribution of particle sizes D can be estimated by (see Eq 12):slnD5(ini$lnDi/D0,0!%2N 2 1(16)6.3.4 In particle-s
34、ize analysis, the quantity sgis referred toas the geometric standard deviation2although it is not astandard deviation in its true sense:sg5 expslnD# (17)6.4 Relationships Between Mean Diameters Dp,q:6.4.1 It can be shown that:Dp,0# Dm,0if p # m (18)and that:Dp21, q21# Dp,q(19)6.4.2 Differences betwe
35、en mean diameters decrease accord-ing as the uniformity of the particle sizes D increases. Theequal sign applies when all particles are of the same size. Thus,the differences between the values of the mean diametersprovide already an indication of the dispersion of the particlesizes.6.4.3 Another re
36、lationship very useful for relating severalmean particle diameters has the form:Dp,q#p2q5 Dp,0p/Dq,0q(20)6.4.4 For example, for p = 3 and q =2:D3,25D3,03/D2,02.6.4.5 Eq 20 is particularly useful when a specific meandiameter cannot be measured directly. Its value may becalculated from two other, but
37、measurable mean diameters.6.4.6 Eq 7 also shows that:Dp,q5 Dq,p(21)6.4.7 This simple symmetry relationship plays an importantrole in the use of Dp,q.6.4.8 The sum O of the subscripts p and q is called the orderof the mean diameter Dp,q:O 5 p1q (22)6.4.9 For lognormal particle-size distributions, the
38、re exists avery important relationship between mean diameters:TABLE 1 Nomenclature for Mean Particle Diameters Dp,qSystematicCodeNomenclatureD23.0harmonic mean volume diameterD22.1diameter-weighted harmonic mean volumediameterD21.2surface-weighted harmonic mean volume di-ameterD22.0harmonic mean sur
39、face diameterD21.1diameter-weighted harmonic mean surfacediameterD21.0harmonic mean diameterD0.0geometric mean diameterD1.1diameter-weighted geometric mean diameterD2.2surface-weighted geometric mean diameterD3.3volume-weighted geometric mean diameterD1.0arithmetic mean diameterD2.1diameter-weighted
40、 mean diameterD3.2surface-weighted mean diameterD4.3volume-weighted mean diameterD2.0mean surface diameterD3.1diameter-weighted mean surface diameterD4.2surface-weighted mean surface diameterD5.3volume-weighted mean surface diameterD3.0mean volume diameterD4.1diameter-weighted mean volume diameterD5
41、.2surface-weighted mean volume diameterD6.3volume-weighted mean volume diameterE2578 07 (2018)3Dp,q5 D0,0expp1q!slnD2/2# (23)6.4.10 Eq 23 is a good approximation for a sample if thenumber of particles in the sample is large (N 500), thestandard deviation lnD 0.7 and the order O of Dp,qnot largerthan
42、 10. Erroneous results will be obtained if these require-ments are not fulfilled. For lognormal particle-sizedistributions, the values of the mean diameters of the sameorder are equal. Conversely, an equality between the values ofthese mean diameters points to lognormality of a particle-sizedistribu
43、tion. For this type of distribution a mean diameter Dp,qcan be rewritten as Dj,j, where j =(p + q)/2 = O/2, if O is even.6.4.11 Sample calculations of mean particle diameters and(geometric) standard deviation are presented in Appendix X1.7. Nomenclature of Mean Particle Sizes/Diameters67.1 Table 1 p
44、resents the M-R nomenclature of meandiameters, an unambiguous list without redundancy. Thisnomenclature conveys the physical meanings of mean particlediameters.7.2 The mean diameter D3.2(also called: Sauter-diameter) isinversely proportional to the volume specific surface area.8. Keywords8.1 distrib
45、ution; equivalent size; mass distribution; meanparticle size; mean particle diameter; moment; particle size;size distribution; surface distribution; volume distributionAPPENDIX(Nonmandatory Information)X1. SAMPLE CALCULATIONS OF MEAN PARTICLE DIAMETERSX1.1 Estimation of mean particle diameters and s
46、tandarddeviations can be demonstrated by using an example from theliterature citing the results of a microscopic measurement of asample of fine quartz (Table X1.1).3The notation of the classboundaries in Table X1.1 was chosen to remove any doubts asto the classification of a particular particle size
47、. A histogram ofthese data is shown in Fig. X1.1. The standard deviation of thissize distribution, according to Eq 12, equals 2.08 m. Thegeometric standard deviation, according to Eq 16 and 17,equals 1.494.X1.1.1 Values of some mean particle diameters Dp,qof thissize distribution, calculated accordi
48、ng to Eq 7 and 8, are:D0,054.75 m, D1,055.14 m, D2,055.55 m, D3,055.95 m,andD3,256.84 m, D3,357.26 m, D4,357.64 mX1.1.2 Fig. X1.2 shows that the distribution indeed is fairlylognormal, because the data points on lognormal probabilitypaper fit a straight line.X1.1.3 This lognormal probability plot al
49、lows for a graphi-cal estimation of the geometric mean diameter D0,0and thegeometric standard deviation sg:X1.1.3.1 For lognormal distributions, the value of D0,0equals the median value, the 50 % point of the distribution,being about 4.8 m.X1.1.3.2 The values of the particle sizes at the 2.3 % and97.7 % points are about 2.15 m and 10.8 m, respectively.This range covers four standard deviations. Therefore, thestandard deviation slnDis
