1、Designation: E 2578 07Standard Practice forCalculation of Mean Sizes/Diameters and StandardDeviations of Particle Size Distributions1This standard is issued under the fixed designation E 2578; the number immediately following the designation indicates the year oforiginal adoption or, in the case of
2、revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) 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 and standard de
3、viations of size distri-butions given as histogram data (see Practice E 1617). Theparticle size is assumed to be the diameter of an equivalentsphere, e.g., equivalent (area/surface/volume/perimeter) diam-eter.1.2 The mean sizes/diameters are defined according to theMoment-Ratio (M-R) definition syst
4、em.2,3,41.3 This practice uses SI (Systme International) units asstandard.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 this standard to establish appro-priate safety and health practices and determine
5、 the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:5E 1617 Practice for Reporting Particle Size Characteriza-tion Data3. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 diameter distribution, nthe distribution by diameterof partic
6、les 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 of thestandard deviation of the distribution of log-transformed par-ticle s
7、izes.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 logarithm has a normal distribution; the left tail ofa lognormal distribution ha
8、s 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, for a purpose involved, properties whichare equal to those of a populatio
9、n 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 the mean volume of the particles).3.1.8 normal distribution, na distribution
10、 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.3.1.10 order of mean diameter, nthe sum of the subscriptsp and q of the me
11、an 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 description of the sizeand frequency of particles in a population.3.1.14 population
12、, 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 widely used measure ofthe width of a frequency distribution.1This practice is u
13、nder 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 Nov. 1, 2007. Published November 2007.2Alderliesten, M., “Mean Particle Diameters. Part
14、 I: Evaluation of DefinitionSystems,” Part. Part. Syst. Charact., 7, 1990, pp. 233-241.3Alderliesten, M., “Mean Particle Diameters. From Statistical Definition toPhysical Understanding,” J. Biopharm.Statist., 15, 2005, pp. 295-325.4Mugele, R. A., Evans, H. D., “Droplet Size Distribution in Sprays,”
15、Ind. Eng.Chem., 43, 1951, pp. 1317-1324.5For referenced 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 Inter
16、national, 100 Barr Harbor Drive, PO Box C700, 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 di
17、stribution, nthe distribution by volumeof particles 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 o
18、f the number 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 us
19、ed.4.4 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 accor
20、ding to 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 popula
21、tion of 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 N 5 (ini, Diis the midpoint of the i-th interval andniis the number of particles in the i-th size cl
22、ass (i.e., 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(iniDi D!r(2)6.1.2 The best-known example is the sample variance M2.This M2always underestimates the population v
23、ariancesD2(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 1M25(iniDi D!2N 1(3)6.1.3 Its square root is the standard deviation sDof thes
24、ample (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(iniln Di!# 5NPiDini(4
25、)6.1.4 The standard deviation slnDof the (lnD)-values can beexpressed as:sln D5(ini$lnDi/Dg!%2N 1(5)6.2 Definition of Mean Diameters Dp,q:6.2.1 The mean diameter Dp,qof a sample of particle sizes isdefined as 1/(p q)-th power of the ratio of the p-th and theq-th moment of the Number distribution of
26、the particle sizes:Dp,q5FMpMqG1/p2q!if p fi q (6)6.2.2 Using Eq 1, Eq 6 can be rewritten as:Dp,q5F(iniDip(iniDiqG1/p2q!if p fi q (7)6.2.3 The powers p and q may have any real value. Forequal values of p and q it is possible to derive from Eq 7 that:Dq,q5 expF(iniDiqlnDi(iniDiqGif p 5 q (8)6.2.4 If q
27、 = 0, then:D0,05 expF(inilnDi(iniG5NPiDini(9)6.2.5 D0,0is the well-known geometric mean diameter. Thephysical dimension of any Dp,qis equal to that of D itself.6.2.6 Mean diameters Dp,qof a sample can be estimatedfrom any size distribution fr(D) according to equations similarto Eq 7 and 8:Dp,q53(imf
28、rDi!Dip2r(imfrDi!Diq2r41/p2qif p fi q (10)and:Dp,p5 exp3(imfrDi!Dip2rlnDi(imfrDi!Dip2r4if p 5 q (11)where:fr(Di) = particle quantity in the i-th class,Di= midpoint of the i-th class interval,r = 0, 1, 2 or 3 represents the type of quantity, viz.number, diameter, surface, volume (or mass) re-spective
29、ly, andm = number of classes.6.2.7 If r = 0 and we put ni= f0(Di), then Eq 10 reduces tothe familiar form Eq 7.E25780726.3 Standard Deviation:6.3.1 According to Eq 3, the standard deviation of theNumber distribution of a sample of particle sizes can beestimated from:sD5(iniDi2 ND1,02N 1(12)which can
30、 be rewritten as:s 5 c=D2,02 D1,02(13)with:c 5 =N/N 1! (14)6.3.2 In practice, N 100, so that c 1. Hence:s =D2,02 D1,02(15)6.3.3 The standard deviation slnDof a lognormal Numberdistribution of particle sizes D can be estimated by (see Eq 12):slnD5(ini$lnDi/D0,0!%2N 1(16)6.3.4 In particle-size analysi
31、s, 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 between mean diamete
32、rs 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 relationship very
33、 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:D3,25 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 mea
34、n 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 p 1 q (22)6.4.9 For lognormal particle-size distributions, there exists av
35、ery important relationship between 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
36、 these require-ments are not fulfilled. 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 me
37、an 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 presents the M-R nomenclature of mean dia
38、m-eters, an unambiguous list without 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 distr
39、ibution; meanparticle size; mean particle diameter; moment; particle size;size distribution; surface distribution; volume distribution6Alderliesten, M., “Mean Particle Diameters. Part II: Standardization of No-menclature,” Part. Part. Syst. Charact., 8, 1991, pp. 237-241.TABLE 1 Nomenclature for Mea
40、n Particle Diameters 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 diame
41、terD0.0geometric 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
42、 surface diameterD3.1diameter-weighted mean surface diameterD4.2surface-weighted mean surface diameterD5.3volume-weighted mean surface diameterD3.0mean volume diameterD4.1diameter-weighted mean volume diameterD5.2surface-weighted mean volume diameterD6.3volume-weighted mean volume diameterE2578073AP
43、PENDIX(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 n
44、otation 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, acc
45、ording 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 indee
46、d is 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 D0,0e
47、quals 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(2.1
48、5)/4 =(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
49、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) = 7.26 see columns 6, 7, and Eq 8sg5 expslnD! 5 exp=32.03/2001! = exp (0.4012) = 1.494 seecolumns 8, 3, and Eq 16 and 17X1.1.6 The graphical estimates for D0,0, slnDand sgappearto be in a good agreement with the numerical estimates.Because the Number distribution seems fairly lognormal, Eq23 can be
