Initial droplet size and velocity distributions for liquid sprays based on maximization of entropy generation

An accurate knowledge of droplet size and velocity distributions is a prerequisite for the fundamental analysis of the transport of mass and heat in liquid sprays and is of significant importance for the practical design, operation and optimization of spray systems. In this study, a new model for the droplet size distribution has been first developed based on the thermodynamic concept---maximization of entropy generation during the liquid atomization process. The model prediction compares favorably with the experimentally measured size distribution for droplets produced by an air-blast annular nozzle, a two-dimensional planar nozzle and an actual gas turbine nozzle near the liquid bulk breakup region. The work on the prediction of the droplet size distribution is extended to modeling initial joint droplet size and velocity distribution. Three goodness-of-fit parameters in statistical analysis, the coefficient of determination (C.O.D), chi-square error and root mean square error (RMSE) are applied to evaluate the extent of the agreements. The high values of C.O.D. and low chi-square error and RMSE indicate that the current MEG-based models are well suited to the description of sprays. As compared to any other models available for droplet size and velocity distributions in liquid sprays, the present model is the only one physically-consistent with the real liquid atomization process.; The models generated from both the MEP and MEG method consist of implicit, highly nonlinear equations involved with exponential functions and integrals and solving this type of equation set has long been a challenge. The classical Newton's method, which has traditionally been adopted as the solver to this equation set, has an inherent disadvantage. It requires that the initial guess for the successive iteration in the numerical solution process be sufficiently close to the solution, otherwise the iteration may diverge rapidly. This study introduces a modification to the classical Newton's method with the Newton's second-order method and the successive under-relaxation (SUR) technique. Other algorithms based on the Newton's method are also compared with the above methods. Results show that the proposed second-order Newton's method and the SUR technique can greatly improve the numerical stability and, indeed relinquish the strict requirement on the initial guess.