On the Capability of the Generalized Gamma Function to Represent Spray Drop‐Size Distribution

Abstract

The three-parameter, Generalized Gamma function solution of a recent MEF formulation used to derive liquid spray drop-size distribution, is applied to sprays resulting from three different atomization processes. The objectives of these applications are to determine the sign of the parameters for which this function reports a more reliable fit and to further understand the parameter stability problem reported elsewhere. It is found that the lack of stability of the parameters is related to a characteristic feature of the mathematical function and appears for a series of spray drop-size distributions with constant shape. For each situation analyzed in the present study, the Generalized Gamma function pro-vides a very good fit with parameters that are either constant or correlated to the working conditions. As far as the sign of the parameters is concerned, the results show that the best formulation is a function of the spray and that it is impossible to know, a priori, which parameter sign will report the best fit. Finally, for one situation, it is found that the Generalized Gamma function allows extrapolation of drop sizes outside the measured values. All of the results converge to conclude that the threeparameter Generalized Gamma function, which is identical to the well-known Nukiyama-Tanasawa distribution, accumulates valuable attributes to represent liquid spray drop-size distributions.

The Maximum Entropy Formalism and the Prediction of Liquid Spray Drop-Size Distribution

“…The most encountered mean diameters are listed in Table 1. As pointed out by Sowa [10], the mean diameter D 10 corresponds to the arithmetic mean of the number-based distribution f n and D 43 to the arithmetic mean of the volume-based distribution f v . Often encountered in the literature, the Sauter mean diameter D 32 corresponds to the diameter of the drop that has the same volume to surface area ratio as the entire spray.…”

“…It is therefore possible to consider drop size and velocity separately. They developed a simplified MEF model to predict the number-based drop-diameter distribution (d = dD) using the normalization, the mass conservation, the surface energy conservation and the partition constraints (Equations 18, 19, 22 and 38), which, in dimensional form, can be rewritten as: (43) leading to the following number-based drop-diameter distribution:…”

“…For the four injection systems used, the best fit was obtained with a constant value of q of the order of 0.54. In their application, Dumouchel and Boyaval used the experimental mean-diameter D 43 to calculate the constraint diameter. As noted in Section 2 (Table 1), this diameter corresponds to the first moment of the volume-based drop-diameter distribution.…”

“…They proposed another MEF formulation to determine volume-based drop-size distribution but based on four constraints expressing the four first moments of the distribution, namely, the four mean-diameters D 43 …”

“…In Equation 50, the parameter q and D q0 were determined on the basis of the relative distribution width and the measured D 43 , and, the application of Equation 52 was achieved by using the four experimental mean-diameters D 43 , D 53 , D 63 and D 73 . It was observed that the two mathematical distributions offer a different fit.…”