Macroscopic Gas-Liquid Interfacial Equation Based on Thermodynamic and Mathematical Approaches

Abstract

At a gas–liquid interface, many complicated phenomena such as evaporation, condensation, electrokinesis, and heat and mass transfer occur. These phenomena are widely seen in various industrial and chemical systems. In chemical or biochemical reactive operations, bubble columns are used for increasing the mass transfer through the interface and for enhancing the separation of mixtures by rectification and water purification (Hong and Brauer 1989; Alvarez et al. 2000). However, the interfacial phenomena have various time and space scales (multi-scale) that are interrelated at the interface. Therefore, modeling gas– liquid interfaces over a wide range of scales spanning molecular motion to vortical fluid motion is very difficult, and this has remained one of the key unresolved issues in multiphase flow science and engineering since a long time. In particular, the mechanism for bubble coalescence/repulsion behaviour is unknown, although it is a superficially simple behaviour and fundamental phenomena in bubbly flows. In order to evaluate the interfacial interactions such as bubble coalescence and repulsion quantitatively, we need a new gas– liquid interfacial model based on the multi-scale concept which is expressed mathematically and that takes into account physical and chemical phenomena and heat and mass transfer at the interface. In the theoretical point of view, the interfacial equation for a macroscopic-scale gas–liquid interface is mainly characterized by a jump condition. The macroscopic interface is discontinuous, and its physical properties such as density, viscosity, and temperature have discontinuous values. The jump condition has been discussed in terms of the mechanical energy balance (Scriven, 1960; Delhaye, 1974) using Stokes’ theorem, the Gauss divergence theorem, differential geometry and so on. In these theorems, a test volume is considered at the interface between two continuous phases. In the derivation, the surface force acting on the discontinuous interface is modeled using the Young–Laplace equation. However, in such a mechanical approach, the definition of the curvature is unclear at the interface, and the surface tension coefficient is treated as a macroscopic experimental value. The interfacial model, which is based only on the mechanical energy balance, cannot take into account detailed physical and chemical phenomena occurring at the interface. In particular, the contamination at the interface, which is related to electric charges, is important for an