Nukiyama-Tanasawa distribution

In contrast to the large variety of averages and measures of dispersion prevalent in the literature, the number of basic distributions which have proved useful is relatively small. In droplet statistics, the best known distributions include the normal, log-normal, Rosin-Rammler, and Nukiyama-Tanasawa distributions. The normal distribution often gives a satisfactory representation where the droplets are produced by condensation, precipitation, or by chemical processes. The log-normal and Nukiyama-Tanasawa distributions often yield adequate descriptions of the drop-size distributions of sprays produced by atomization of liquids in air. The Rosin-Rammler distribution has been successfully applied to size distribution resulting from grinding, and may sometimes be fitted to data that are too skewed to be fitted with a log-normal distribution. 

Table II shows the mathematical forms of the normal, log-normal, Rosin-Rammler, and Nukiyama-Tanasawa distributions for an arbitrary ptb-weighted size distribution. As in Table I, the formulas yield the surface increment in the size interval, t to t + dt, for p = 2 number or volume increments are obtained by setting p equal to 0 or 3, respectively. In these expressions 6 and n are constants, and a denotes an appropriate shape factor.

A high velocity of the flow, Wg, an elevated density of the fluid as well as a low viscosity of the liquid phase and a low interfacial tension between liquid and fluid represent favourable conditions for high pressure spraying. Pressurised gases evidently possess characteristics similar to those of liquids with respect to the atomisation so that the relationships which are valid for liquid/liquid spraying may lead to realistic results. In the case presented here, a modified Nukiyama-Tanasawa distribution [5] has been used to specify the maximum drop diameter in the spraying process ... 

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... 

The Nukiyama-Tanasawa distribution [Nukiyama and Tanasawa (1939)] is used for sprays having extremely broad size ranges. The number of particles with diameters between dp and dp + ddp is given by... 

Three widely used distribution equations, discussed by Bevans (JL), include the Rosin-Rammler (7L) and Nukiyama-Tanasawa (6L) equations as well as the log-probability equation. A fourth relationship, the upper-limit equation of Mugele and Evans (5L), is also discussed. Hawthorne and Stange also discuss the Rosin-Rammler relationship (4L, 8L). An excellent analysis of distributions is given by Dubrow (SL), who has studied atomized magnesium powders. 

Droplet Size Distribution, Most sprays comprise a wide range of droplet sizes. Some knowledge of the size distribution is usually required, particularly when evaluating the overall atomizer performance. The size distribution may be expressed in various ways. Several empirical functions, including the Rosin-Rammler (25) and Nukiyama-Tanasawa (26) equations, have been commonly used. 

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. 

This is a Nukiyama-Tanasawa type distribution. The parameter a > 1, guarantees that the number distribution vanishes at small D, as opposed to the expression derived by Cousin et al. [19] which corresponds to a = 1. Unfortunately this introduces a third parameter that needs to be determined. Lecompte and Dumouchel [21] suggest that there may be a tmique pair q and a that can represent all drop size distributions for a specific atomization process (ultrasonic atomization, twin-fluid atomization, etc.). This could transform the MEF into a pseudo-predictive method. 

X. Li, R. S. Tankin Droplet Size Distribution A Derivation of a Nukiyama-Tanasawa Type Distribution Function, Combust. Sci. Technol. 56, 65-76 (1987). 

The characteristics of other distributions that have been applied to aerosol particle size, such as the Rosin-Rammler, Nukiyama-Tanasawa, power law, exponential, and Khrgian-Mazin distributions are given in the appendix to this chapter. These distributions apply to special situations and And limited application in aerosol science. They (and the lognormal distribution) have been selected empirically to fit the wide range and skewed shape of most aerosol size distributions. 

Because of such factors as wave formation, jet turbulence, and secondary breakup, the drops formed are not of uniform size. Various ways of describing the distribution, including the methods of Rosin and Rammler (R9) and of Nukiyama and Tanasawa (N3), are discussed by Mugele and Evans (M7). A completely theoretical prediction of the drop-size distribution resulting from the complex phenomena discussed has not yet been obtained. However, for simple jets issuing in still air, the following approximate relation has been suggested (P3) ... 

Aspiration rate is only a small part of the overall transport process in flame spectrometry. The production of aerosol and its transport through the spray chamber are also of great importance. The size distribution of aerosol produced depends upon the surface tension, density, and viscosity of the sample solution. An empirical equation relating aerosol size distribution to these parameters and to nebulizer gas and solution flow rates was first worked out by Nukiyama and Tanasawa,5 who were interested in the size distributions in fuel sprays for rocket motors. Their equation has been extensively exploited in analytical flame spectrometry.2,6-7 Careful matrix matching is therefore essential not only for matching aspiration rates of samples and standards, but also for matching the size distributions of their respective aerosols. Samples and standards with identical size distributions will be transported to the flame with identical efficiencies, a key requirement in analytical flame spectrometry. 

S. Nukiyama and Y. Tanasawa. Experiments in on the atomization of liquids in air stream, report 3 on the droplet-size distribution in an atomized jet. Trans. Soc. Mech. Eng. Japan, 5 62-67, 1939. 

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