Chapter 22
Size Distribution Data Analysis and Presentation
Gurumurthy Ramachandran,
Douglas W. Cooper,
Gurumurthy Ramachandran
Division of Environmental Health Sciences, School of Public Health, University of Minnesota, Minneapolis, Minnesota, USA
Search for more papers by this authorGurumurthy Ramachandran,
Douglas W. Cooper,
Gurumurthy Ramachandran
Division of Environmental Health Sciences, School of Public Health, University of Minnesota, Minneapolis, Minnesota, USA
Search for more papers by this authorBook Editor(s):Pramod Kulkarni,
Paul A. Baron,
Klaus Willeke,
Pramod Kulkarni
Centers for Disease Control and Prevention, National Institute for Occupational Safety and Health, Cincinnati, Ohio, USA
Search for more papers by this authorPaul A. Baron
Centers for Disease Control and Prevention, National Institute for Occupational Safety and Health, Cincinnati, Ohio, USA
Search for more papers by this authorKlaus Willeke
Department of Environmental Health, University of Cincinnati, Cincinnati, Ohio, USA
Search for more papers by this authorSummary
This chapter contains sections titled:
-
Introduction
-
Types of Particle Size
-
Particle Shape
-
Particle Size Distributions
-
Concentration Distributions
-
Summarizing Size Distributions Graphically
-
Confidence Intervals and Error Analysis
-
Testing Hypotheses with Size Distribution Data
-
Coincidence Errors
-
Choosing Sizing Interval Demarcations
-
Data Inversion
-
List of Symbols
-
References
22.13 REFERENCES
- Aitchison, J., and J. A. C. Brown. 1957. The Lognormal Distribution. Cambridge: Cambridge University Press.
- Baker, H., M. R. Gordon, and O. B. Brown. 1972. Theory of coincidence and simple practical methods of coincidence count correction for optical and resistive pulse particle counters. Rev. Sci. Instr. 43: 1407–1412.
- Beddow, J. K. 1980. Particulate Science and Technology. New York: Chemical Publishing.
- Beddow, J. K. 1997. Image Analysis Sourcebook. Santa Barbara, CA: American University of Science and Technology Press.
- Bottiger, J. R. 1985. Progress of inversion technique evaluation. In Proceedings of Chemical Research and Development Center's 1984 Scientific Conference on Obscuration and Aerosol Research, CRDC-SP-85007, pp. 129–140.
- Box, G. E. P., W. G. Hunter, and J. S. Hunter. 1978. Statistics for Experimenters. New York: John Wiley and Sons.
- Bunz, H., W. Schock, M. Koyro, J. W. Gentry, S. Plunkett, M. Runyan, C. Pearson, and C. Wang 1987. Application of the log beta distribution to aerosol size distributions. J. Aerosol Sci. 18: 663–666.
- Capps, C. D., R. L. Henning, and G. M. Hess. 1982. Analytic inversion of remote sensing data. Appl. Opt. 21: 3581–3587.
- Chahine, M. T. 1968. Determination of the temperature profile in an atmosphere from its outgoing radiance. J. Opt. Soc. Am. 58: 1634–1637.
- Chang, H., W. Y. Lin, and P. Biswas. 1995. An inversion technique to determine the aerosol size distribution in multicomponent systems from in situ light scattering measurements. Aerosol Sci. Technol. 22: 24–32.
- Cleveland, W. S. 1985. The Elements of Graphing Data. Monterey, CA: Wadsworth Advanced Books.
- Christensen V. R., W. Eastes, R. D. Hamilton, and A. W. Struss 1993. Fiber diameter distributions in typical MMVF wool insulation products. Amer. Ind. Hyg. Assoc. J. 54: 232–238.
- Cooper, D. W. 1974. The variable-slit impactor and aerosol size distribution analysis. Ph.D. Dissertation, Harvard Division of Engineering and Applied Physics, Cambridge, MA.
- Cooper, D. W. 1976. Data inversion method and error estimate for cascade centripeters, Amer. Ind. Hyg. Assoc. J. 37: 622–627.
- Cooper, D. W. 1982. On the products of lognormal and cumulative lognormal particle size distributions. J. Aerosol Sci. 13: 111–120.
- Cooper, D. W. 1991a. Applying a simple statistical test to compare two different particle counts. J. Aerosol Sci. 22: 773–777.
- Cooper, D. W. 1991b. Comparing three environmental particle size distributions: Power law (FED-STD-209D), lognormal, approximate lognormal (MIL-STD-1246B). J. Inst. Environ. Sci. 5: 21–24.
- Cooper, D. W., and J. W. Davis. 1972. Cascade impactors for aerosols: Improved data analysis. Amer. Ind. Hyg. Assoc. J. 33: 79–89.
- Cooper, D. W., and G. L. Guttrich. 1981. A study of the cut diameter concept for interpreting particle sizing data. Atmos. Environ. 15: 1699–1707.
- Cooper, D. W., and R. J. Miller. 1987. Analysis of coincidence losses for a monitor of particle contamination on surfaces. J. Electrochem. Soc. 134: 2871–2875.
- Cooper, D. W., and A. Neukermans. 1991. Estimating an instrument's counting efficiency by repeated counts on one sample. J. Colloid Interf. Sci. 147: 98–102.
- Cooper, D. W., and H. R. Rottmann. 1988. Particle sizing from disk images by counting contiguous grid squares or vidicon pixels. J. Colloid Interf. Sci. 126: 251–259.
- Cooper, D. W., and L. A. Spielman. 1976. Data inversion using nonlinear programming with physical constraints: aerosol size distribution measurement by impactors. Atmos. Environ. 10: 723–729.
- Cooper, D. W., and J. J. Wu. 1990. The inversion matrix and error estimation in data inversion: Application to diffusion battery measurements. J. Aerosol Sci. 21: 217–226.
- Cooper, D. W., H. A. Feldman, and G. R. Chase 1978. Fiber counting: A source of error corrected. Amer. Ind. Hyg. Assoc. J. 39: 362–367.
- Crow, E. L., and K. Shimizu. 1988. Lognormal Distributions. New York: Marcel Dekker.
- Crow, E. L., F. A. Davis, and M. W. Maxfield. 1960. Statistics Manual. New York: Dover.
- Crump, J. G., and J. H. Seinfeld. 1982. A new algorithm for inversion of aerosol size distribution data. Aerosol Sci. Technol. 1: 15–34.
- Dahneke, B. 1983. Measurement of Suspended Particles by Quasi-Elastic Light Scattering. New York: John Wiley and Sons.
- Draper, N. R., and H. Smith. 1981. Applied Regression Analysis, 2 ed. New York: John Wiley and Sons.
-
Epstein, B. 1947. The mathematical description of certain breakage mechanisms leading to the logarithmic-normal distribution. J. Franklin Inst. 244: 471–477.
10.1016/0016-0032(47)90465-1 Google Scholar
- Enting, I. G., and G. N. Newsam. 1990. Atmospheric constituent inverse problems: Implications for baseline monitoring. J. Atmos. Chem. 11: 69–87.
- Evans, J. S., D. W. Cooper, and P. Kinney. 1984. On the propagation of error in air pollution measurements. Environ. Mon. Assess. 4: 1322–1353.
- Farzanah, F. F., C. R. Kaplan, P. Y. Yu, J. Hong, and J. W. Gentry. 1985. Condition numbers as criteria for evaluation of atmospheric aerosol measurement techniques. Environ. Sci. Technol. 19: 121–126.
-
Fayed, M. E., and L. Otten (eds.). 1997. Handbook of Powder Science & Technology. New York: Chapman & Hall.
10.1007/978-1-4615-6373-0 Google Scholar
- Fogel, P., D. Y. Hanton, A. DeMeringo, and C. Morscheidt 1999. The reliability of the dimensional measurements of man-made vitreous fibers used for biopersistence assays: a statistical approach. Aerosol Sci. Technol. 30: 571–581.
- Forsythe, G. E., and C. B. Moeller. 1967. Computer Solution of Linear Algebraic Systems. Englewood Cliffs, NJ: Prentice-Hall.
- Friedlander, S. K. 1977. Smoke, Dust and Haze. New York: John Wiley and Sons.
- Fuchs, N. A. 1964. Mechanics of Aerosols. New York: Pergamon.
- Gentry, J. W. 1977. Estimation of parameters for a log-normal distribution with truncated measurements. J. Powder Technol. 18: 225–229.
- Golub, G. H., M. Heath, and G. Wahba. 1979. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics. 21: 215–223.
- Gonda, I. 1984. On inversion of aerosol size distribution data. Letter to the editor. Aerosol Sci. Technol. 3: 345–346.
- Hansen, P. C. 1992. Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34: 561–580.
- Hansen, P. C., and D. O. O'Leary. 1993. The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14: 1487–1503.
- Hays, W. L., and Winkler, R. L. 1970. Statistics. New York: Holt, Rinehart and Winston.
-
Heintzenberg, J. 1994. Properties of the log-normal particle size distribution. Aerosol Sci. Technol. 21: 4–48.
10.1080/02786829408959695 Google Scholar
- Heitbrink, W., P. A. Baron, and K. Willeke. 1991. Coincidence in time-of-flight aerosol spectrometers: Phantom particle creation. Aerosol Sci. Technol. 14: 112–126.
- Hinds, W. C. 1999. Aerosol Technology. Properties, Behavior, and Measurement of Airborne Particles, 2 ed. wc. New York: John Wiley and Sons.
- Jackson, D. D. 1972. Interpretation of inaccurate, insufficient, and inconsistent data. Geophys. J. Roy. Astron. Soc. 28: 97–109.
-
Jaenicke, R. 1972. The optical particle counter: Cross-sensitivity and coincidence. J. Aerosol Sci. 3: 95–111.
10.1016/0021-8502(72)90147-4 Google Scholar
- Jain, S., D. J. Skamser, and T. T. Kodas. 1997. Morphology of single–component particles produced by spray pyrolysis. Aerosol Sci. Technol. 27: 575–590.
- Jochum, C., P. Jochum, and B. R. Kowalski. 1981. Error propagation and optimal performance in multicomponent analysis. Anal. Chem. 53: 85–92.
- Julanov, Yu. V., A. A. Lushnikov, and I. A. Nevskii. 1984. Statistics of multiple counting in aerosol counters. J. Aerosol Sci. 15: 622–679.
- Julanov, Yu. V., A. A. Lushnikov, and I. A. Nevskii. 1986. Statistics of multiple counting in aerosol counters—II. J. Aerosol Sci. 17: 87–93.
- Kandlikar, M., and G. Ramachandran 1999. Inverse methods for analyzing aerosol spectrometer measurements: a critical review. J. Aerosol Sci. 30: 413–437
- Kaplan, C., and J. W. Gentry 1987. Use of condition numbers for short-cut experimental design. Am. Inst. Chem. Eng. J. 33: 681–685.
- Kaye, B. H. 1989. A Random Walk Through Fractal Dimensions. New York: VCH.
- Knapp, J. Z., and L. R. Abramson 1994. A new coincidence model for single particle counters, Part I: Theory and experimental verification. PDA J. Pharma. Sci. & Technol. 48: 110–134.
- Knapp, J. Z. and L. R. Abramson 1996. A new coincidence model for single particle counters, Part III: Realization of single particle counting accuracy. PDA J. Pharma. Sci. & Technol. 50: 99–122.
- Knapp, J. Z., A. Lieberman, and L. R. Abramson 1994. A new coincidence model for single particle counters, Part II: Advances and applications. PDA J. Pharma. Sci. & Technol. 48: 255–292
- Knapp, J. Z., T. A. Barber, and A. Lieberman (eds.). 1994. Liquid and Surface Borne Particle Measurement Handbook. New York: Marcel Dekker.
- Knutson, E. O., and P. J. Lioy. 1995. Measurement and presentation of aerosol size distributions. In Air Sampling Instruments. Cincinnati OH: American Conference of Governmental Industrial Hygienists, pp. 121–138.
- Kolmogorov, A. N. 1941. Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstuckelung. C.R. Acad. Sci. (Doklady) URSS XXXI: 922–101.
-
Kubie, G. 1971. A note on the treatment of impactor data for some aerosols. J. Aerosol Sci. 2: 23–30.
10.1016/0021-8502(71)90004-8 Google Scholar
- Lefebvre, A. H. 1989. Atomization and Sprays. Bristol, PA: Taylor & Francis.
- Liley, J. B. 1992. Fitting size distributions to optical particle counter data. Aerosol Sci. Technol. 17: 84–92.
- Lu, Z., A. Li, and D. W. Cooper. 1993. Correcting particle size distribution data for the differences between single-stage and multi-stage collection of counting efficiencies. Part. Sci. Technol. 11: 49–55.
- Lu, Z., A. Li, and C. Fu. 1995. Experimental verification of using cumulative efficiency curves for determining stage constants of a cascade impactor. Aerosol Sci. Technol. 23: 253–256.
- Mandelbrot, B. 1977. The Fractal Geometry of Nature. New York: Freeman.
- McCartney, E. J. 1976. Optics of the Atmosphere. New York: John Wiley and Sons.
- Morawska, L., and M. Jamriska. 1997. Determination of the activity size distribution of radon progeny. Aerosol Sci. Technol. 26: 459–468.
- Myojo, T. 1999. A simple method to determine the length distribution of fibrous aerosols. Aerosol Sci. Technol. 30: 30–39.
- Nguyen, T., and K. Cox. 1989. Abstracts. American Association for Aerosol Research Annual Meeting, pp. 330.
- Noble, B. 1969. Applied Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall.
- Paatero, P. 1991. Extreme value estimation, a method for regularizing ill-posed inversion problems. In Ill-posed Problems in Natural Sciences. Proceedings of an International Conference, Moscow. Utrecht, The Netherlands: VSP.
- Pecen, J., A. Neukermans, G. Kren, and L. Galbraith. 1987. Counting errors in particulate detection on unpatterned wafers. Solid State Technol. 30: 1422–1454.
- Phillips, D. L. 1962. A technique for the numerical solution of certain integral equations of the first kind. J. Assoc. Comput. Mach. 9: 84–97.
-
Puttock, J. S. 1981. Data inversion for cascade impactors: Fitting sums of lognormal distributions. Atmos. Environ. 15: 1710–1716.
10.1016/0004-6981(81)90157-8 Google Scholar
-
Raasch, J., and H. Umhauer. 1984. Error in the determination of particle size distributions caused by coincidences in optical particle counters. Part. Charact. 1: 53–58.
10.1002/ppsc.19840010109 Google Scholar
- Ramachandran, G., and M. Kandlikar. 1996. Bayesian analysis for inversion of aerosol size distribution data. J. Aerosol Sci. 27: 1099–1112.
- Ramachandran, G., and D. Leith. 1992. Extraction of aerosol size distributions from multispectral light extinction data. Aerosol Sci. Technol. 17: 303–325.
- Ramachandran, G., and P. C. Reist. 1995. Characterization of morphological changes in agglomerates subject to condensation and evaporation using multiple fractal dimensions. Aerosol Sci. Technol. 23: 431–442.
- Ramachandran, G., and J. H. Vincent. 1997. Evaluation of two inversion techniques for retrieving health-related aerosol fractions from personal cascade impactor measurements. Amer. Ind. Hyg. Assoc. J. 58: 15–22.
- Ramachandran, G., E. W. Johnson, and J. H. Vincent. 1996. Inversion techniques for personal cascade impactor data. J. Aerosol Sci. 27: 1083–1097.
- Reist, P. C. 1984. Introduction to Aerosol Science. New York: Macmillan.
- Rizzi, R., R. Guzzi, and R. Legnani. 1982. Aerosol size spectra from spectral extinction data: the use of a linear inversion method. Appl. Opt. 21: 1578–1587.
- Shannon, C. E., and W. Weaver. 1962. The Mathematical Theory of Communication. Urbana, IL: University of Illinois Press.
- Shen, A.-T., and T. A. Ring. 1986. Distinguishing between two aerosol size distributions. Aerosol Sci. Technol. 5: 477–482.
- Strang, G. 1988. Linear Algebra and Its Applications. New York: Harcourt Brace Jovanovich.
-
J. P. M. Syvitski (ed.). 1991. Principles, Methods and Application of Particle Size Analysis, Cambridge: Cambridge University.
10.1017/CBO9780511626142 Google Scholar
- Tikhonov, A. N., and V. Y. Arsenin. 1977. Solutions of Ill Posed Problems. Washington, DC: V. H. Winston and Sons.
- Twomey, S. 1963. On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J. Assoc. Comput. Mach. 10: 97–101.
- Twomey, S. 1975. Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions. J. Comput. Phys. 18: 188–200.
- Twomey, S. 1977. Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements. New York: Elsevier.
- Wahba, G. 1977. Practical approximate solutions to linear operator equations when the data are noisy. SIAM J. Numer. Anal. 14: 645–667.
- Wolfenbarger, J. K., and J. H. Seinfeld. 1990. Inversion of aerosol size distribution data. J. Aerosol Sci. 21: 227–247.
- Wu, M. K. 1996. The roughness of aerosol particles: Surface fractal dimension measured using nitrogen adsorption. Aerosol Sci. Technol. 25: 392–398.
- Yee, E. 1989. On the interpretation of diffusion battery data. J. Aerosol Sci. 20: 797–811.
- Yu, P. Y. 1983. The Simulation and Experimental Determination of Size and Charge Distributions of Fibrous Particles from Penetration Measurements. Ph.D. Dissertation, University of Maryland, College Park, MD.
- Yu, P. Y., and J. W. Gentry. 1984. A critical comparison of three size distribution analysis methods: apparent size, non-linear inversion, and linear inversion with cross validation. J. Aerosol Sci. 15: 407–411.
- Yu, P. Y., J. San, and J. W. Gentry. 1983. Application of the apparent diameter method for inversion of penetration: Isometric particles. In V. A. Marple and B. Y. H. Liu, Aerosols in the Mining and Industrial Work Environments, Vol. 1: Fundamentals and Status. Ann Arbor, Chap 24.
