# ASTM E799-03 (Reapproved 2015)

Designation: E799 − 03 (Reapproved 2015)Standard Practice for DeterminingData Criteria and Processing for Liquid Drop Size Analysis1This standard is issued under the fixed designation E799; the number immediately following the designation indicates the year oforiginal adoption or, 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 This practice gives procedures for determining appro-priate sample size, size class widths, characteristic drop sizes,and dispersion measure of drop size distribution. The accuracyof and correction procedures for measurements of drops usingparticular equipment are not part of this practice. Attention isdrawn to the types of sampling (spatial, flux-sensitive, orneither) with a note on conversion required (methods notspecified). The data are assumed to be counts by drop size. Thedrop size is assumed to be the diameter of a sphere ofequivalent volume.1.2 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.3 The analysis applies to all liquid drop distributionsexcept where specific restrictions are stated.2. Referenced Documents2.1 ASTM Standards:2E1296 Terminology for Liquid Particle Statistics (With-drawn 1997)32.2 ISO Standards:413320–1 Particle Size Analysis-Laser Diffraction Methods9276–1 Representation of Results of Particle Size Analysis-Graphical Representation9272–2 Calculation ofAverage Particle Sizes/Diameters andMoments from Particle Size Distribution3. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 spatial, adj—describes the observation or measure-ment of drops contained in a volume of space during such shortintervals of time that the contents of the volume observed donot change during any single observation. Examples of spatialsampling are single flash photography or laser holography.Anysum of such photographs would also constitute spatial sam-pling. A spatial set of data is proportional to concentration:number per unit volume.3.1.2 flux-sensitive, adj—describes the observation of mea-surement of the traffic of drops through a fixed area duringintervals of time. Examples of flux-sensitive sampling are thecollection for a period of time on a stationary slide or in asampling cell, or the measurement of drops passing through aplane (gate) with a shadowing on photodiodes or by usingcapacitance changes. An example that may be characterized asneither flux-sensitive nor spatial is a collection on a slidemoving so that there is measurable settling of drops on the slidein addition to the collection by the motion of the slide throughthe swept volume. Optical scattering devices sensing continu-ously may be difficult to identify as flux-sensitive, spatial, orneither due to instantaneous sampling of the sensors and themeasurable accumulation and relaxation time of the sensors.For widely spaced particles sampling may resemble temporaland for closely spaced particles it may resemble spatial. Aflux-sensitive set of data is proportional to flux density: numberper (unit area × unit time).3.1.3 representative, adj—indicates that sufficient data havebeen obtained to make the effect of random fluctuationsacceptably small. For temporal observations this requiressufficient time duration or sufficient total of time durations. Forspatial observations this requires a sufficient number of obser-vations.Aspatial sample of one flash photograph is usually notrepresentative since the drop population distribution fluctuateswith time. 1000 such photographs exhibiting no correlationwith the fluctuations would most probably be representative.Atemporal sample observed over a total of periods of time thatis long compared to the time lapse between extreme fluctua-tions would most probably be representative.3.1.4 local, adj—indicates observations of a very small part(volume or area) of a larger region of concern.3.2 Symbols—Representative Diameters:1This practice is under the jurisdiction ofASTM Committee E29 on Particle andSpray Characterization and is the direct responsibility of Subcommittee E29.02 onNon-Sieving Methods.Current edition approved March 1, 2015. Published March 2015. Originallyapproved in 1981. Last previous edition approved in 2009 as E799 – 03 (2009).DOI: 10.1520/E0799-03R15.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.3The last approved version of this historical standard is referenced onwww.astm.org.4Available from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor, New York, NY 10036, http://www.ansi.org.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.2.1 (D¯pq) is defined to be such that:5D¯pq~p2q!5(iDip(iDiq(1)where:D¯= the overbar in D¯designates an averagingprocess,(p−q)pq = the algebraic power of D¯pq,p and q = the integers 1, 2, 3 or 4,Di= the diameter of the ith drop, and∑i= the summation of Dipor Diq, representingall drops in the sample.0=pand q = values 0, 1, 2, 3, or 4.∑iDi0is the total number of drops in the sample, and someof the more common representative diameters are:D¯10= linear (arithmetic) mean diameter,D¯20= surface area mean diameter,D¯30= volume mean diameter,D¯32= volume/surface mean diameter (Sauter), andD¯43= mean diameter over volume (De Broukere or Herdan).See Table 1 for numerical examples.3.2.2 DNf,DLf,DAf, and DVfare diameters such that thefraction, f, of the total number, length of diameters, surfacearea, and volume of drops, respectively, contain precisely all ofthe drops of smaller diameter. Some examples are:DN0.5= number median diameter,DL0.5= length median diameter,DA0.5= surface area median diameter,DV0.5= volume median diameter, andDV0.9= drop diameter such that 90 % of the total liquidvolume is in drops of smaller diameter.See Table 2 for numerical examples.3.2.3log~D¯gm! 5(ilog~Di!/n (2)where:n = number of drops,D¯gm= the geometric mean diameter3.2.4DRR5 DVF(3)where:f = 1−1⁄e≈ 0.6321, andDRR= Rosin-Rammler Diameter fitting the Rosin-Rammlerdistribution factor (see Terminology E1296).5This notation follows: Mugele, R.A., and Evans, H.D., “Droplet Size Distri-bution in Sprays,” Industrial and Engineering Chemistry, Vol 43, No. 6, 1951, pp.1317–1324.TABLE 1 Sample Data Calculation TableSize Class Bounds(Diameterin Micrometres)ClassWidthNo. ofDrops inClassSum of Dirin Each Size ClassAVol. %in ClassBCum. %by Vol.DiDi2Di3Di4240–360 120 65 19.5 × 1035.9×1061.8×1091. × 10120.005 0.005360–450 90 119 48.2 19.6 8.0 3 0.021 0.026450–562.5 112.5 232 117.4 59.7 30.5 16 0.081 0.107562.5–703 140.5 410 259.4 164.8 105.2 67 0.280 0.387703–878 175 629 497.2 394.7 314.5 252 0.837 1.224878–1097 219 849 838.4 831.3 827.6 827 2.202 3.4261097–1371 274 990 1221.7 1513.7 1883.2 2352 5.010 8.4361371–1713 342 981 1512.7 2342.1 3641.1 5683 9.687 18.1231713–2141 428 825 1589.8 3076.1 5976.2 11657 15.900 34.0232141–2676 535 579 1394.5 3372.5 8189.2 19965 21.788 55.8112676–3345 669 297 894.1 2702.8 8203.5 24999 21.826 77.6373345–4181 836 111 417.7 1578.2 5987.6 22807 15.930 93.5674181–5226 1045 21 98.8 466.5 2212.1 10532 5.885 99.4535226–6532 1306 1 5.9 34.7 348.5 1534 0.547 100.000Totals of Dirin ^κ = 6109 8915.3 × 10316562.6 × 10637729.0 × 109100695 × 1012entire sample DN0.5= 1300 D¯10= 1460 D¯21= 1860 D¯32= 2280 D¯43= 2670D¯20= 1650 D¯31= 2060D¯30= 1830DV0.5= 2540 Worst case class width348.5377295 0.009 Relative Span 5 sDV0.92 DV0.5d/DV0.55 s3900 2 14200d/2530 5 0.986692676133453 0.21826 5 0.024Less than 1 %, adequate sample size Adequate class sizesAThe individual entries are the values for each κ as used in 5.2.1 (Eq 1) for summing by size class.BSUM Di3in size class divided by SUM Di3in entire sample.E799 − 03 (2015)23.2.5 Dkub= upper-boundary diameter of drops in the kthsize class.3.2.6 Dklb= lower-boundary diameter of drops in the kthsize class.4. Significance and Use64.1 These criteria6and procedures provide a uniform basefor analysis of liquid drop data.5. Test Data5.1 Specify the data as temporal or spatial. If the data cannotbe so specified, describe the sampling procedure. Also specifywhether the data are local (that is, in a very small section of thespace of liquid drop dispersion), and whether the data arerepresentative (that is, a good description of the distribution ofconcern). Report the fluids, fluid properties, and pertinentoperating conditions.5.1.1 A graph form for reporting data is given in Fig. 1.5.2 Report the largest and smallest drops of the entiresample, the number of drops in each size class, and the classboundaries. Also report the sampling volume, area, and lapseof time, if available and applicable.5.3 Estimate the total volume of liquid in the sample thatincludes measured drops and the liquid in the sample that is notmeasured. (The volume outside the range of sizes permitted bythe measuring technique might be estimated by graphicalextrapolation of a histogram or by a curve fitting technique.)5.4 The ratio of the volume of the largest drop to the totalvolume of liquid in the sample should be less than the tolerablefractional error in the desired representation. See Table 1. Allof the drops in the sample at the large-drop end of the6These criteria ensure that processing probably will not introduce error greaterthan 5 % in the computation of the various drop sizes used to characterize the spray.TABLE 2 Example of Log Normal Curve with Upper BoundData Collected May 2, 1979 Computer Analysis May 2, 1979Upper Bound Diameter (µm) Normal Curve, % Adjusted Data, % Data, %360.00 0.006 0.005 0.005450.00 0.027 0.027 0.026562.50 0.109 0.108 0.107703.00 0.389 0.387 0.387878.00 1.227 1.224 1.2241097.00 3.421 3.426 3.4261371.00 8.407 8.437 8.4361713.00 18.109 18.124 18.1232141.00 34.080 34.024 34.0232676.00 55.551 55.811 55.8113345.00 77.828 77.637 77.6374181.00 93.648 93.568 93.5675226.00 99.481 99.453 99.4536532.00 100.000 100.000 100.000For Computing Curve AveragesLargest drop diameter = 6532.00 µmSmallest drop diameter = 240.00 µmFraction of normal curve = 0.999995Normal Curve Simple Calculation(Gaussian Limits—4.55457 to 4.53257)D10= 1464.91 1459.37 µm (length mean diameter)D20= 1646.44 1646.57 µm (surface mean diameter)D30= 1824.85 1832.39 µm (volume mean diameter)D21= 1850.45 1857.79 µm (surface/length mean diameter)D31= 2036.73 2053.27 µm (volume/length mean diameter)D32= 2241.75 2269.32 µm (sauter mean diameter)D43= 2615.67 2670.75 µm (mean diameter over volume)DV0.5= 2534.53 2533.31 µm (volume median diameter)DN0.5= 1303.62 1304.71 µm (number median diameter)Average of Absolute Relative Deviation from DV0.5by Volume = 0.311Relative Span = (DV0.900−DV0.100)/ DV0.5(DV0.9−DV0.1)/DV0.5= (3913.74 − 1437.21) ⁄ 2534.53= 0.977Normal curve % FsDd 51œπe2`DEL lnsADXM2Dde2z2dzwhere:A = 1.8941,DEL = 1.17206, andXM = 7335.30.F(D) = accumulative fraction of liquid volume in drops having diameter less than D.E799 − 03 (2015)3distribution should be measured. This criterion is a good “ruleof thumb” to determine a minimum sample size. The value ofD10is greatly affected by the smallest drops measured.5.5 Ninety-nine percent of the volume of liquid representedby the data should be in size classes such that no size class hasboundaries with a ratio greater than 3:2. For the majority ofsize classes, this ratio should not exceed 5:4. The 99 %condition exempts size classes having diameters smaller thanDV0.01. These criteria assure that processing probably will notintroduce errors greater than 5 % in the computation of thevarious drop diameters cited in this practice. The criteria maybe relaxed for measurements where this degree of accuracy isunattainable.5.6 (Dkub− Dklb)/(Dkub+ Dklb) multiplied by the liquid vol-ume in the kth class and divided by the total volume of liquidin the sample shall be less than 0.05 for every class. See Table1. Use of the same criterion for a size class created by lumpingthe estimated volume below the boundary of measurementprovides a test for determining the need for some type of curvefitting. It may be necessary to relax this requirement for caseswhere this degree of accuracy is unattainable.6. Data Processing6.1 Transformations of Data:6.1.1 If drop motions are essentially free from recirculationthrough the region of observation, spatial data can be trans-formed to flux-sensitive data by multiplying the number ofdrops in each size class by the average velocity of drops forthat size class at the sample location. If this transformation isperformed, the exact procedure shall be noted.6.1.2 If evaporation corrections are applied, the procedureshall be described and the magnitude of the corrections shall berecorded.6.1.3 If corrections are applied to account for drops outsidethe boundaries represented by the data, the procedure shall bedescribed. Likewise, if the overall distribution is established byblending several distributions, the procedure shall be de-scribed.6.1.4 If curve fitting (for example, to the upper-limit lognormal, Rosin-Rammler or Nukiyama-Tanasawa equation) isused in the data processing, the mathematical function7andminimization criteria, including any weighting factors applied7Examples are found in Mugele and Evans, loc. cit.; in Tishkoff, J. M., and Law,C. K., “Applications of a Class of Distribution Functions to Drop Size Data byLogarithmic Least Squares Technique,” Transactions of ASME, Vol 99, Ser. A, No.4, October 1977; and in Goering, C. E., and Smith, D. B., “Equations for DropletSize Distributions in Sprays,” Transactions of ASAE, Vol 21, No. 2, 1978, pp.209–216.FIG. 1 Sample Data GraphE799 − 03 (2015)4to the data, shall be given. The quality of fit shall be showngraphically or by tabular comparison with the data. When thereare corrections or transformations, the comparison shall bemade with the adjusted data.6.2 Calculations Involving Size Classes:6.2.1 When data are reported by size classes rather than asindividual drop diameters, the representative diameters, D¯pq,may be calculated from summations defined as follows:(iDir5(k~Dkubr112 Dklbr11! Nk~Dkub2 Dklb!~r11!(4)where:r = corresponds to the selected value of p or q in theexpression for D¯pqas stated in 4.2.1, andNk= the number of drops in the kth size class.This calculation is based on the assumption of a linearincrease in the accumulation of counts as a function ofdiameter within each size class. If the data satisfy the criteria in5.5 and 5.6, the results based on either of the following twoformulas will differ by less than 8 % from that based on theabove (preferred) Eq 1.(iDir5(kDkubr1Dklbr23Nk(5)(iDir5(kSDkub1Dklb2Dr3Nk(6)6.2.2 To obtain the values described in 4.2.2, the fractionalvalues (number, length, area or volume) accumulated betwee