Log-hyperbolic distribution

Keywords Characteristic drop diameter Cumulative volume fraction Discrete probability function (DPF) Drop size distribution Empirical drop size distribution Log-hyperbolic distribution Log-normal distribution Maximum entropy formalism (MEF) Nukiyama-Tanasawa distribution Number distribution function Probability density function (pdf) Representative diameter Root-normal distribution Rosin-Rammler distribution Upper limit distribution Volume distribution... 

A key question is which of these distributions is best Paloposki [8] provided an answer by performing tests on 22 sets of data that came from seven experimental studies. His analysis showed that the Nukiyama-Tanasawa and log-hyperbolic distribution functions provided the best fits, that the upper-limit and log-normal distributions were clearly inferior to these two, and that the Rosin-Rammler distribution gave poor results. Paloposki [8] also determined the mathematical stability of distribution parameters. The Nukiyama-Tanasawa and log-hyperbolic distribution functions both had problems, while the log-normal distribution was more stable. 

J. C. Bhatia, J. Dominick, F. Durst Phase-Doppler-Anemometry and the Log-Hyperbolic Distribution Applied to Liquid Sprays, Part. Part. Syst. Char. 5, 153-164 (1988). 

Bhatia, J. C., Domnick, J., Durst, E., Tropea, C. (1988). Phase-doppler-anemometry and the log-hyperbolic distribution apphed to liquid sprays. Particle and Particle Systems Characterization, 5, 153-164. 

Hatch extended his method of analysis to size-distribution curves ranging from coarse-screen analysis through fine particles measured microscopically. While excellent results were obtained by using this technique on laboratory samples, the method cannot be generalized to cover all types of distributions encountered in practice. As already explained in Chapter 3, size-frequency distributions may assume a variety of shapes. The Hatch development applies only to distributions which follow the normal or log-probability law. When size-distributions are hyperbolic in the lower extremes and follow normal log-probability laws in the upper extremes, the Hatch analysis must necessarily fail. Nevertheless, the relationships developed by Hatch have a far-reaching practical importance... 

Figure 9.9. Clusters formed in a porous or pigmented body can be described by a hyperbolic function which constitutes a fractal dimension in data space, a) Simulated dispersion of monosized pores which occupy 20 % of the available space, b) Typical clusters which may be found in such a dispersion, displaying orthogonal and diagonal connections, c) Size distribution of the clusters present in (a) presented as a log-log plot.

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