Rosin-Rammler distribution

For materials that have undergone comminutian, the Rosin-Ram-mler distribution [4] is frequently applicable. The Rosin-Rammler weight distribution is given by 

These values can be evaluated from tabulated values of the gamma function, F, given in the appendix of this book and experimentally determined values of n. 

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Rosin-Rammler distribution function 428 is perhaps the most widely used one at present ... 

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... 

A).632 Characteristic diameter 63.2% of total volume of droplets are of smaller diameters than this value V=63.2% X (X in Rosin-Rammler distribution function)... 

Since the ratio of any two representative diameters is a unique function of q, Rosin-Rammler distribution function can be rewritten as ... 

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. 

The function v(D) can be measured experimentally, or in some cases be simulated as normal, lognormal, etc. distribution. It is also possible to obtain polymodal distributions with several max-imums or some special kind of distribution. For example, the distribution of the particles formed by crashing is frequently described by a Rosin-Rammler distribution [51,52] as... 

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. 

In the applications of gas-solid flows, there are three typical distributions in particle size, namely, Gaussian distribution or normal distribution, log-normal distribution, and Rosin-Rammler distribution. These three size distribution functions are mostly used in the curve fitting of experimental data. 

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... 

Example 1.2 A coarsely ground sample of com kernel is analyzed for size distribution, as given in Table El.3. Plot the density function curves for (1) normal or Gaussian distribution, (2) log-normal distribution, and (3) Rosin-Rammler distribution. Compare these distributions with the frequency distribution histogram based on the data and identify the distribution which best fits the data. 

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... 

Figure 5.1 (a) Data of Rosin-Rammler distribution and fitted PSD curve based on new PSD function, (b) Original PSD curve and realized probability curve in the case of Rosin-Rammler distribution, (c) Data of log-normal distribution and fitted PSD curve based on new PSD function, (d) Data of normal distribution and fitted PSD curve based on new PSD function. 

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. 

FIGU RE 11.3. The asymptotic dependence of the combustion efficiency on the dimensionless chamber length Z2 for various values of the exponent 2 the ordinary Rosin-Rammler distribution (/2 = - 2 " hh /C2 = 1-... 

Djamarani and Clark [1] state that many industrial processes are defined by a coarse (C) and a fine fraction (F), for example, oversize and undersize. In their example they use sieve sizes of 1400 pm and 180 pm that they fit to a Rosin-Rammler distribution. They define a curve of C+F against C/F from which the Rosin-Rammler constants can be read. 

Droplet size distributions obtained with any means mentioned here are relatively well represented by a Rosin-Rammler distribution with an exponent of approximately 2. This means that approximately 80 percent of the droplet population mass is in the range of 0.39 to 1.82 times the median droplet size. 

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