A Theoretical and Numerical Study on Electrification and Flow inside Electrostatic Atomizers

Electrostatic atomization has emerged as a desirable technology in many industrial applications such as oil coating, painting, and fuel injection in diesel engines. It provides controlled spray trajectories, increased dispersion/deposition efficiency, and enhanced evaporation compared to other conventional atomization methods. Moreover, it avoids customary coating problems such as uneven or over-application of oil. It demands less energy and provides applicability to a wide range of fluid viscosity. The so-called leaky dielectric liquids are the best candidates for electrostatic atomizers due to their enhanced charge residence time compared to conducting liquids such as water, where the charges would instantly relax to the surface. The leaky dielectric liquid concept emerged in the mid-’60s to explain electrohydrodynamic phenomena in liquids previously considered as insulators, such as vegetable oils. These ionic conductor liquids are capable of possessing a small net charge resulting in a relatively low conductivity. The origin of ions in such liquids is still under debate. In electrostatic atomizers, electrodes are perfect conductors capable of passing through an electric current and sustaining faradaic reactions. However, in non-conducting surfaces, only embedded charges remain and there is no sustained electric current at the surface. In this study, the electrification and ion transport in ionic conductor liquids (oils) are studied theoretically and numerically while allowing for the Frumkin-Butler-Volmer kinetics responsible for the electron transfer at the metal electrodes. The numerical model solves the fluid flow equations along with the electrostatic equations. The fluid flow equations are the continuity and momentum equations. The source term in the momentum equation accounts for the Coulombic body force responsible for the electro-osmotic flow. The electrostatic equations are the anion and cation transport equations, along with the Poisson equation. As a sample case, a channel flow is considered with the charging electrodes to occupy the middle part of the channel while other sections are electrically insulated. The considered ionic conductor liquid throughout this study is canola oil. The constituted near-electrode layers across the electrode in the outwards direction are compact and diffuse layers, respectively. The ions present in the compact layer do not participate in the electrohydrodynamic flow. Hence, the computational model at the electrode surface starts from the interface of the compact and diffuse layers. In ionic conductor liquids with very low permittivity, such as canola oil, 99% of the potential drop occurs across the compact layer. This drop makes finding the electric potential value at the interface of compact and diffuse layers crucial since it is used as a boundary condition for the provided computational model. On the other hand, most of the potential drop in aqueous solutions occurs in the diffuse layer, which is not the case in this study. The electric potential boundary condition at the electrode surface of the considered channel flow case is a fixed value (Dirichlet-type boundary) with orders of magnitude lower than the applied voltage at the electrode surface. The ionic boundary conditions at the electrode surface are Neumann-type conditions. Further comprehension of the characteristics of the compact Stern and diffuse layers in electrostatic atomizers enables us to apply a more accurate electric potential boundary condition for the electrode surface in the provided numerical model. A novel theoretical approach is developed to correlate the thickness of the equivalent one-dimensional compact layer to the potential drop across this layer. These electrodes are subjected to high voltage and sustain an electric current. The non-specific (non-electric) ion adsorption responsible for creating the compact Stern layer is attributed to the Langmuir-Brunauer-Emmet-Teller mechanism. Faradaic reactions are responsible for electron transfer at the open parts of the metallic electrode surface. In contrast, the compact Stern layer is formed on the oxide or impurities at the electrode surface. The electron transfer regime in electrostatic atomizers with electrodes exposed to high voltages is kinetic-limited. Hence, we employ the Frumkin slow discharge theory combined with the Marcus electron transfer theory to present another novel approach to predict the electric potential at the interface of the compact and diffuse layers. The activation energy of the electron transfer in the faradaic reaction is found from the Marcus theory. The ionic concentration and net charge distribution across the polarized diffuse layer are calculated from the numerical simulations given the known counter-ion flux value at the electrode surface from the concurrent experimental measurements. With the concentration of ions at the interface of compact and diffuse layers be known and employing Frumkin slow discharge theory, the electric potential value at the interface is found from a predictor-corrector algorithm that is detailed with examples. The provided numerical and theoretical models thoroughly demonstrate the electrification mechanism, constituted near-electrode layers and internal electrohydrodynamic flow in electrostatic atomizers. We found in the channel flow case that the role of Smoluchowski slip near the electrodes is negligibly small compared to the viscous scraping of the polarized layer under any realistic values of the imposed longitudinal electric field. This means that ions are removed from the polarized diffuse layer by the visocus scraping mechanism in electrohydrodynamic flow inside electrostatic atomziers, rather than the Coulombic force.