A Predictive Model for the Initial Droplet Size and Velocity Distributions in Sprays and Comparison with Experiments

Abstract

The distribution of sizes and velocities of droplets initially formed in sprays is an important piece of information needed in the spray modelling, because it defines the initial condition of the spray droplets in the predictive calculations of the downstream two‐phase flow fields. A predictive model for the initial droplet size and velocity distributions in sprays is formulated in this study. The present model incorporates both the deterministic and the stochastic aspect of spray formation process. The deterministic aspect takes into account of the unstable wave motion before the liquid bulk breakup through the linear and nonlinear instability analysis, which provides information for the liquid bulk breakup length, the mass‐mean diameter and a prior distribution for the droplet sizes corresponding to the unstable wave growth of various wavelengths. The stochastic aspect deals with the final stage of droplet formation after the liquid bulk breakup by statistical means through the maximum entropy principle based on Bayesian entropy. The two sub‐models are coupled together by the various source terms signifying the liquid‐gas interaction, the mass mean diameter and the prior distribution based on the instability analysis. The initial droplet size and velocity distributions are measured experimentally by phase‐Doppler interferometry for sprays generated by a planar research nozzle and a practical gas turbine airblast nozzle. For the two nozzles, the liquid bulk sheet is formed before its breakup in a coflowing air stream. It is found that the model predictions are in satisfactory agreement with the experimental data for all the cases measured. Hence the present model may be applied to a variety of practical sprays to specify the initial conditions for the spray droplets formed in practical spray systems.

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

“…Note that this entropy is similar to the one maximized by Kim et al [39]. The solution of this formulation is given:…”

“…As explained in this paper, it also concerns the proposition due to Dumouchel and Malot [35]. To be complete on this point, on should note that Kim et al's model [39] is not concerned by this criticism since they incorporate in their formulation a prior-distribution that can be seen as a supplementary constraint. The similarity of their solution with the one due to Li and Tankin comes from the fact that they imposed the prior-distribution in consequence.…”

Trends in Comprehensive Modeling of Spray Formation

“…The most critical wavelength of the ligaments is given by λ lig = 4.5d lig where d lig is the diameter of the ligament. This approach has been used by most of the comprehensive models [6,7,8,158]. The Rayleigh criterion was extended by Ponstein [124] who included the effects of gas phase and swirl in his analysis.…”

“…To make the analysis consistent, Kim et al [6] maximized Bayesian entropy, considering a joint probability density function in the diameter and velocity space while using the same constraint equations as were used by Li and Tankin. They could obtain a low value in the size probability density function at the small diameter range using a suitable prior distribution function of number based diameter distribution in their model.…”

“…However, comprehensive models incorporating some of the above sub-models have been reported. Liao et al [4] and Ibrahim and Jog [5] incorporated the first two modules in their analysis of pressure-swirl atomizers while Kim et al [6], Movahednejad et al [7] and Nath et al [8,9] combined the last two modules in their analyses. Liao et al [4] and Movahednejad et al [7] used linear analysis in their break up model while nonlinear break up models were used in [5,6,8,9].…”