Rosin-Rammler functions

Most distribution functions contain an average size and a variance parameter typicaUy based on the cumulative droplet number or volume distributions. For example, the Rosin-Rammler function uses the cumulative Hquid volume as a means of expressing the distribution. It can be expressed as... 

Rosin-Rammler function, 23 185, 186 Rosin soap size... 

Meric (M33) discussed analytieal expressions used to represent the PSD of cements. Probably the most suceessful is the Rosin-Rammler function, which may be written in logarithmic form as... 

A hammer crusher was designed to liberate the coating from the plastic substrate. The size distribution of particles was analyzed and could be described by the Rosin-Rammler function model (26-28). 

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

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 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... 

Both the number and weight basis probability density functions of final product crystals were found to be expressed by a %2-function, under the assumption that the CSD obtained by continuous crystallizer is controlled predominantly by RTD of crystals in crystallizer, and that the CSD thus expressed exhibits the linear relationships on Rosin-Rammler chart in the range of about 10-90 % of the cumulative residue 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... 

Equation (1.35b) shows that a linear relationship exists when ln[ln( 1 // )] is plotted against In d. From the slope and intercept of this straight line, a and fl can be determined, a and fl are typically obtained from the particle size distribution data based on sieve analyses. Table 1.5 provides a list of typical values of a and fl for some materials for the Rosin-Rammler density function with d in the function having the unit micrometers (/tm). 

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. 

There are a fair number of PSD functions to express the practical PSD. However, these functions are only mathematical expressions and have no physical basis or significance. Additionally, there is a problem when the practical PSD is expressed by using different PSD functions, for example, Rosin-Rammler PSD function, normal PSD function, and log-normal PSD function. This is because it becomes difficult to correlate the values of a parameter and operation condition. Therefore, it is indispensable to clarify whether a new PSD defined by Eq. (5.18) has an advantage over the traditional PSD. 

Traditional PSD Table 5.1 (Rosin-Rammler PSD function, normal PSD function, log-normal PSD function). 

The usefulness of the newly presented PSD shown in Eq. (5.17) is examined by the Rosin-Rammler PSD function ... 

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. 

In principle, the specific surface area may also be calculated from the PSD using analytical expressions derived from such functions as the Rosin-Rammler expression. In practice, this is not always satisfactory because the specific surface area is so highly dependent on the bottom end of the distribution. 

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. 

From Probert s numerical results [26], H[2(l — c)/3, f, c, z] may be evaluated for various values of c and z this case will be seen to determine the combustion efficiency when kj = 1 and Gj q is of the original Rosin-Rammler form. Tanasawa and Tesima [28] numerically evaluated functions that are somewhat similar to (but not exactly equivalent to) if(2,, c, z) for various values of c and z this may be seen to correspond to the Nukiyama-Tanasawa form for Gj with kj = 1. Tables of H(a, b, c, z) for other values of a and b apparently are not yet available. 

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