Calculation of the Drop Size Distribution and Velocities from the Integral Form of the Conservation Equations

Abstract

A new formulation based on integral form of the conservation equations is developed for analyzing spray droplet and velocity statistics. Both averaged or distributions of the droplet size and velocities can be calculated using this approach, although in this work the authors focus on the droplet size statistics and only compute the averaged droplet velocities. The formulation is based on the conservation equations, and intrinsically involves few assumptions and required inputs. The key is to use the integral form so that the input and output variables are related without having to resolve the complex details of the atomization process. This new approach naturally leads to predictions of SMDs, as well as the drop size distributions and phase velocities, with SMD and drop size distributions achieving good agreement with experimental data if a liquid-phase viscous dissipation term is included. The latter point suggests that the effect of liquid-phase viscosity seems quite important, as the present method overpredicts the effect of injection pressure unless some form of the viscous dissipation is included in the energy balance equation. The use of moment-generating function for the log-normal distribution within the present formulation permits an efficient algorithm that reproduces the experimentally observed droplet size distributions. Further work remains to ascertain some aspects of the present solution approach, in particular the exact form of the viscous dissipation on the spray energy balance. The present integral formulation and the solution approach provide a fundamentally sound basis for analyzing droplet size and velocity statistics with several key parameters that may be adapted for different injector geometries, such as swirl and air-blast sprays, and injection conditions.

Keywords:

• Atomization
• Drop size distribution
• Integral method

ACKNOWLEDGMENTS

The authors would like to thank Dr. Do Younghae of the Kyungbuk National University, Daegu, Korea, for useful discussions on mathematical treatments of the log-normal distribution function and on the solution procedure. The assistance of Ms. Kelly Mahon, at Arizona State University, on MATLAB programming of the solution procedure is also duly acknowledged.