1、Designation: E2578 07 (Reapproved 2012)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 and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:5E1617 Practice for Reporting Particle Size CharacterizationData3. Terminology3.1 Definitions of Terms Specific to This S
6、tandard: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 geometric standard deviation, nexponential
7、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 distribution of particlesize, whose logar
8、ithm 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 particles havingthat size/diameter has
9、, 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 sizes (the 3rd moment isproportional to t
10、he 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 ofparticles as a function of their size
11、.3.1.10 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 size distribution, na descript
12、ion 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 standard deviation, nmost wide
13、ly used measure ofthe width of a frequency distribution.1This practice is under the jurisdiction of ASTM Committee E56 on Nanotech-nology and is the direct responsibility of Subcommittee E56.02 on Characterization:Physical, Chemical, and Toxicological Properties.Current edition approved May 1, 2012.
14、 Published May 2012. Originallyapproved in 2007. Last previous edition approved in 2007 as E2578 07. DOI:10.1520/E2578-07R01.2Alderliesten, M., “Mean Particle Diameters. Part I: Evaluation of DefinitionSystems,” Particle and Particle Systems Characterization, Vol 7, 1990, pp.233241.3Alderliesten, M.
15、, “Mean Particle Diameters. From 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 r
16、eferenced ASTM standards, visit 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.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700,
17、West Conshohocken, PA 19428-2959, United States.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 distribution, nthe distribution by volumeof part
18、icles 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 number ofparticles, the cumulated diamet
19、ers, 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 Particle shape factors are not taken in
20、to 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 theMoment-Ratio (M-R) system are deriv
21、ed 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 particle sizes Di, enables estimation
22、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 N 5 (ini, Diis the midpoint of the i-th interval andniis the number of particles in the i-th size class (that is, classfrequency). The (arithmetic
23、) 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(iniDi D!r(2)6.1.2 The best-known example is the sample variance M2.This M2always underestimates the population variancesD2(squared standard deviation). Ins
24、tead, 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 1M25(iniDi D!2N 1(3)6.1.3 Its square root is the standard deviation sDof thesample (see also 6.3). If the particle sizes
25、 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(iniln Di!# 5NPiDini(4)6.1.4 The standard deviation slnDof the (l
26、nD)-values can beexpressed as:sln D5(ini$lnDi/Dg!%2N 1(5)6.2 Definition of Mean Diameters Dp,q:6.2.1 The mean diameter Dp,qof a sample of particle sizes isdefined as 1/(p q)-th power of the ratio of the p-th and theq-th moment of the Number distribution of the particle sizes:Dp,q5FMpMqG1/p2q!if p fi
27、 q (6)6.2.2 Using Eq 1, Eq 6 can be rewritten as:Dp,q5F(iniDip(iniDiqG1/p2q!if p fi q (7)6.2.3 The powers p and q may have any real value. Forequal values of p and q it is possible to derive from Eq 7 that:Dq,q5 expF(iniDiqlnDi(iniDiqGif p 5 q (8)6.2.4 If q = 0, then:D0,05 expF(inilnDi(iniG5NPiDini(
28、9)6.2.5 D0,0is the well-known geometric mean diameter. Thephysical dimension of any Dp,qis equal to that of D itself.6.2.6 Mean diameters Dp,qof a sample can be estimatedfrom any size distribution fr(D) according to equations similarto Eq 7 and 8:Dp,q53(imfrDi!Dip2r(imfrDi!Diq2r41/p2qif p fi q (10)a
29、nd:Dp,p5 exp3(imfrDi!Dip2rlnDi(imfrDi!Dip2r4if p 5 q (11)where:fr(Di) = particle quantity in the i-th class,Di= midpoint of the i-th class interval,r = 0, 1, 2, or 3 represents the type of quantity, viz.number, diameter, surface, volume (or mass) re-spectively, andm = number of classes.6.2.7 If r =
30、0 and we put ni= f0(Di), then Eq 10 reduces tothe familiar form Eq 7.E2578 07 (2012)26.3 Standard Deviation:6.3.1 According to Eq 3, the standard deviation of theNumber distribution of a sample of particle sizes can beestimated from:sD5(iniDi2 ND1,02N 1(12)which can be rewritten as:s 5 c=D2,02 D1,02
31、(13)with:c 5 =N/N 1! (14)6.3.2 In practice, N 100, so that c 1. Hence:s =D2,02 D1,02(15)6.3.3 The standard deviation slnDof a lognormal Numberdistribution of particle sizes D can be estimated by (see Eq 12):slnD5(ini$lnDi/D0,0!%2N 1(16)6.3.4 In particle-size analysis, the quantity sgis referred toas
32、 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 between mean diameters decrease accord-ing as the unif
33、ormity 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 relationship very useful for relating severalmean p
34、article diameters has the form:Dp,q#p2q5 Dp,0p/Dq,0q(20)6.4.4 For example, for p = 3 and q =2:D3,25 D3,03/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 measurable mean diameters.6.4.6 Eq 7 also shows
35、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 p 1 q (22)6.4.9 For lognormal particle-size distributions, there exists avery important relationship between
36、 mean diameters:Dp,q5 D0,0expp 1 q!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 slnD 0.7 and the order O of Dp,qnot largerthan 10. Erroneous results will be obtained if these require-ments are not fulfi
37、lled. For lognormal particle-size distribu-tions, the values of the mean diameters of the same order areequal. Conversely, an equality between the values of thesemean diameters points to lognormality of a particle-sizedistribution. For this type of distribution a mean diameter Dp,qcan be rewritten a
38、s 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 presents the M-R nomenclature of mean diam-eters, an unambiguous list witho
39、ut redundancy. This nomen-clature conveys the physical meanings of mean particle diam-eters.7.2 The mean diameter D3.2(also called: Sauter-diameter) isinversely proportional to the volume specific surface area.8. Keywords8.1 distribution; equivalent size; mass distribution; meanparticle size; mean p
40、article diameter; moment; particle size;size distribution; surface distribution; volume distribution6Alderliesten, M., “Mean Particle Diameters. Part II: Standardization of No-menclature,” Particle and Particle Systems Characterization, Vol 8, 1991, pp.237241.TABLE 1 Nomenclature for Mean Particle D
41、iameters Dp,qSystematicCodeNomenclatureD23.0harmonic mean volume diameterD22.1diameter-weighted harmonic mean volume diameterD21.2surface-weighted harmonic mean volume diameterD22.0harmonic mean surface diameterD21.1diameter-weighted harmonic mean surface diameterD21.0harmonic mean diameterD0.0geome
42、tric mean diameterD1.1diameter-weighted geometric mean diameterD2.2surface-weighted geometric mean diameterD3.3volume-weighted geometric mean diameterD1.0arithmetic mean diameterD2.1diameter-weighted mean diameterD3.2surface-weighted mean diameterD4.3volume-weighted mean diameterD2.0mean surface dia
43、meterD3.1diameter-weighted mean surface diameterD4.2surface-weighted mean surface diameterD5.3volume-weighted mean surface diameterD3.0mean volume diameterD4.1diameter-weighted mean volume diameterD5.2surface-weighted mean volume diameterD6.3volume-weighted mean volume diameterE2578 07 (2012)3APPEND
44、IX(Nonmandatory Information)X1. SAMPLE CALCULATIONS OF MEAN PARTICLE DIAMETERSX1.1 Estimation of mean particle diameters and standarddeviations 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 notat
45、ion of the classboundaries in Table X1.1 was chosen to remove any doubts asto the classification of a particular particle size. 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, accordi
46、ng to Eq 16 and 17,equals 1.494.X1.1.1 Values of some mean particle diameters Dp,qof thissize distribution, calculated according to Eq 7 and 8, are:D0,05 4.75 m, D1,05 5.14 m, D2,05 5.55 m, D3,05 5.95 m,andD3,25 6.84 m, D3,35 7.26 m, D4,35 7.64 mX1.1.2 Fig. X1.2 shows that the distribution indeed is
47、 fairlylognormal, because the data points on lognormal probabilitypaper fit a straight line.X1.1.3 This lognormal probability plot allows 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 D0,0equal
48、s 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 equal to (ln(10.8) ln(2.15)/4
49、 =(2.380 0.765)/4 = 0.404 and the geometrical standarddeviation is sg= exp(0.404) = 1.50. A shorter way of calcula-tion is: sg5=410.8/2.15 5 1.50X1.1.4 These graphical estimates can be compared withnumerical estimates, using the data in the bottom row of TableX1.2.X1.1.5 The numerical estimates are:D0,0= exp (311.8 / 200) = 4.75 see columns 4, 3, and Eq 8D1,0= 1029 / 200 = 5.14 see columns 5, 3, and Eq 7D3,0= (42105 / 200)1/3= 5.95 see columns 7, 3, and Eq 7D3,3= exp (83445.9 / 42105) =
