BI-Modal Size Distributions Predicted by Maximum Entropy are Compared with Experiments in Sprays

Abstract

A theoretical formulation based on maximum entropy principles is presented to predict the droplet size and velocity distributions of sprays in an isothermal environment. The joint droplet distribution function is derived subject to the constraints of mass flow rate, momentum flux, and two modes of energy fluxes(kinetic and surface). A simpler model, which reduces the number of constraints by three, is derived by choosing an adequate velocity integration range. This maximum entropy principle spray model is tested by comparing the calculated distributions with experimental measurements presented by the authors for a hollow cone, non-swirl spray nozzle and the experimentalresultsobtained by other researchers forhollow cone, swirl spray nozzles. For a specificdroplet size, the droplet velocity distribution is Gaussian. The droplet size distribution is much more complicated; three types of distributions may occur-positively skewed mono-modal, unifonn size (in the limit approaching a delta function), and bi-modal. This study is concerned mainly with the bi-modal size distribution.

Trends in Comprehensive Modeling of Spray Formation

“…Van der Geld and Vermeer [229] used the normalization condition along with the conservation of mass and surface energy as constraints in the maximum entropy model and were able to predict a bimodal distribution of droplet size in a spray due to the occurrence of the satellite droplets. Such bi-modal droplet distribution was also predicted by Chin et al [230], who accounted momentum conservation in the three co-ordinate directions. The source terms were carefully evaluated using the necessary experimental data.…”

“…The predicted results showed a good fit with the experiments. Chin et al [230] concluded that an increase in the number of velocity components in the prediction improve the predictive capability of the model and pointed out the role of azimuthal momentum source term in increasing the bimodality of the distribution. However, later on Chin et al [231] proposed another MEP based model using only one component of velocity along with two additional constraint equations.…”

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

“…Bi-modal drop-diameter distributions were also obtained by Chin et al [27] who generalized Chin et al [24] formulation by accounting for the three components of drop velocity, i.e., d = dVdU x dU y dU z . Following Sellens' approach [21] [22] reported an acceptable fit on the number-based drop-diameter distribution of a spray produced by a swirl atomizer.…”

“…This new formulation allowed bi-modal distributions to be obtained. Considering this approach and the one due to Chin et al [27], it seems that a minimum of seven parameters is required if one has to represent bi-modal drop-size distribution. Chin et al [28] performed an experiment where droplets produced by a liquid column were visualized by a high-speed camera (1,500 images/s).…”

“…Following Sellens' approach [21] [22] reported an acceptable fit on the number-based drop-diameter distribution of a spray produced by a swirl atomizer. To apply their model, Chin et al [27] used as much experimental information as necessary to estimate the source terms. They demonstrated that increasing the number of velocity components in the model considerably improved the agreement (especially along the right distribution tail) and that the use of the generalized three velocity-component model predicted a bi-modal distribution, the enhancement of bimodality being controlled by the azimuthal momentum source term.…”