The Rosin-Rammler Distribution

The Rosin-Rammler distribution is one that applies specifically to dusts generated by crushing. The density function is  

The shape of f x) depends on the constants n and k. Integrating to find F x) and adding a constant to make it start at the origin gives  

For this distribution the mode, median and mean sizes are different. Using Eqs. (2.3.4) and (2.B.7) the mean particle size becomes  

Mathematics packages, such as Mathematica, Matlab or Mathcad and MathGV make it easy to fit these model distributions to sets of experimental data. It is often helpful to do so, since this allows the particle size distribution to be described by only two parameters. This also has its limitations, though. We shall come across one in Appendix 3. A. 

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To characterize a droplet size distribution, at least two parameters are typically necessary, i.e., a representative droplet diameter, (for example, mean droplet size) and a measure of droplet size range (for example, standard deviation or q). Many representative droplet diameters have been used in specifying distribution functions. The definitions of these diameters and the relevant relationships are summarized in Table 4.2. These relationships are derived on the basis of the Rosin-Rammler distribution function (Eq. 14), and the diameters are uniquely related to each other via the distribution parameter q in the Rosin-Rammler distribution function. Lefebvre 1 calculated the values of these diameters for q ranging from 1.2 to 4.0. The calculated results showed that Dpeak is always larger than SMD, and SMD is between 80% and 84% of Dpeak for many droplet generation processes for which 2left-hand side of Dpeak. The ratio MMD/SMD is... 

It should be indicated that a probability density function has been derived on the basis of maximum entropy formalism for the prediction of droplet size distribution in a spray resulting from the breakup of a liquid sheet)432 The physics of the breakup process is described by simple conservation constraints for mass, momentum, surface energy, and kinetic energy. The predicted, most probable distribution, i.e., maximum entropy distribution, agrees very well with corresponding empirical distributions, particularly the Rosin-Rammler distribution. Although the maximum entropy distribution is considered as an ideal case, the approach used to derive it provides a framework for studying more complex distributions. 

The studies on the performance of effervescent atomizer have been very limited as compared to those described above. However, the results of droplet size measurements made by Lefebvre et al.t87] for the effervescent atomizer provided insightful information about the effects of process parameters on droplet size. Their analysis of the experimental data suggested that the atomization quality by the effervescent atomizer is generally quite high. Better atomization may be achieved by generating small bubbles. Droplet size distribution may follow the Rosin-Rammler distribution pattern with the parameter q ranging from 1 to 2 for a gas to liquid ratio up to 0.2, and a liquid injection pressure from 34.5 to 345 kPa. The mean droplet size decreases with an increase in the gas to liquid ratio and/or liquid injection pressure. Any factor that tends to impair atomization quality, and increase the mean droplet size (for example, decreasing gas to liquid ratio and/or injection pressure) also leads to a more mono-disperse spray. 

Various comparisons have been made (4, 27, 28, 61, 74, 89, 106, 111). For a particular task of fitting data, the recognition of an actual upper size limit leads to the modified logarithmic representation of Mugele and Evans (89). However, for pure mathematical ease the Rosin-Rammler distribution employed by Probert (98) is preferable from a vaporization point of view. 

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. 

For broken coal, moon dust, and many irregular particles, the mass distribution is found to follow a form known as the Rosin-Rammler distribution. A Rosin-Rammler distribution has the density function... 

However, the Rosin-Rammler distribution is often expressed in terms of R defined by... 

For the Rosin-Rammler distribution, the distribution constants (a and f ) are obtained from the particle mass distribution data. To obtain the mass density distribution, the data on... 

For computations the physical properties are assumed to be constant. Particle sizes are discretized by 10 equal-weight cuts from the Rosin-Rammler distribution... 

Knudsen s model led to the prediction that, if linear kinetics were followed, the age at which 50% of the cement has hydrated is proportional to the fineness constant (or xj in the Rosin/Rammler distribution (equation 4.1) for parabolic kinetics, it predicted that this age is proportional to (K40). Evidence was presented in support of this conclusion for cements considered to follow linear kinetics. The theory did not predict any relation to the breadth of the particle size distribution, which is represented by the slope of the Rosin-Rammler curve. 

Cumulative distributions can be fitted by a linear function if the data fit a suitable mathematical fimction. This curve fitting gives no insight into the fundamental physics by which the particle size distribution was produced. Three common functions are used to linearize the cumulative distribution the normal distribution fimction, the log-normal distribution function, and the Rosin—Rammler distribution function. By far athe most commonly used is the log-normal distribution function. 

The pulverized coal was sieved to separate a 200x270 mesh (U.S. Standard sieves) fraction. The mean weight particle size was 62 ym and the dispersion parameter 8.4 according to the Rosin-Rammler distribution (15). The size graded particles were dried under vacuum at 338 K for 8 hours, then stored in a dessicator until used. 

The Rosin-Rammler distribution is used to represent sprays from nozzles. It is anpirical and relates the volume percentage oversize Vq to droplet diameter D. The mathematical form is as follows [23] ... 

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. 

The parameter q, the order of the constraint, is analogous to the Rosin-Rammler distribution spread parameter. 

The Rosin-Rammler distribution function is another equation widely used in particle size measurement. It is a two-parameter function, usually given as cumulative percentage oversize ... 

A most useful approach has been published by Hirleman et al. This consisted of an artificial "aerosol" made up of an array of chrome thin-film circles on a transparent glass substrate. This calibration reticle contains more than 10000 of these particles randomly positioned in an 8 mm circle to simulate a Rosin Rammler distribution of spherical particles. Versions are available with different Rosin Rammler parameters X and N, and at least two levels of obscuration. The continuous Rosin Rammler size distribution is approximated on the reticle by 24 discrete particle sizes ranging from 2 to 105 m depending on the individual reticle. Each reticle is individually certified. An example of the goodness of fit to the Rosin Rammler distribution is shown in Figure 7. Hirleman et al use the reticle to check the performance of a Malvern 2200 model. 

Other distributions employed are the log-normal distribution, the gamma distribution, the Rosin-Rammler distribution, etc. We will encounter some of them later in the book. 

A number of model distribution functions exist, some of which fit the size distributions of many powders quite well. The model functions used most frequently are the normal (or Gaussian ) distribution, the log-normal distribution and the Rosin-Rammler distribution (Allen, 1990). These are given in Appendix 2.B for reference. The latter two can be fitted particularly well to the volume distributions of a wide range of powders. The Rosin-Rammler distribution was used to produce Figures 2.3.3 and 2.3.4. 

The Rosin-Rammler distribution [Rosin and Rammler (1933)], originally devised for sizing crushed coal, is applicable to coarsely dispersed dusts and sprays. It is particularly useful for size distributions that are more skewed than the lognormal distribution and for sieve analysis. The weight fraction... 

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