Analytical Modeling of Mist Condensation by Natural Convection over Inclined Flat Surfaces

Dew formation on a transparent surface creates a pattern that can cause blurred view over it. This fogging phenomenon should be avoided in many applications. Mist condensation of water on a cold surface exposed to humid air is studied in this work. In order to analyze the misting process, the fluid flow and heat transfer of humid air as well as the heat transfer across the solid surface are considered. A dew formation model is used to predict the shape and size of the droplets. Analytical models have been proposed to solve the heat and mass transfer for the simple arrangement of a vertical flat surface. The analytical model is then combined with the dew formation model to introduce an analytical model for mist condensation over vertical and inclined surfaces. Due to the proposed method, complex numerical calculations can be avoided for solving the heat and mass transfer equations. INTRODUCTION The exposure of a smooth cold surface to warmer humid air can trigger the condensation of the vapor content over the surface, if the surface is colder than the dew point temperature. The condensation starts by forming mist over the surface. By prolonged exposure to humid air, the condensation may continue in the form of dropwise condensation while the droplets grow in size and coalesce with each other to generate big drops of water. In this regime, the weight of the drops will overcome the surface tension forces in scale, which causes them to become unstable and flow one after another. Merte and Yamali ‎[1] studied the profile and departure size of condensation drops on vertical surfaces. They introduced an analytical method for estimating the stability threshold of liquid drops over vertical surfaces. For higher temperature differences, more air flow, or more humid air, the condensation rate may increase even more and the process will continue in the film condensation regime. In many applications, the condensation happens in the mist regime only. The mist formation, also called fogging, causes the light rays to scatter due to the different refractive index of water compared to air, as well as the size and shape of the droplets. In many applications with transparent surfaces, the generation of mist creates a blurred view across the surface. This blur is sometimes associated with safety concerns as in the case of a vehicle, train, or airplane windshield. In some applications the blur of a glass surface can harm the functioning of an instrument or a gauge. In such applications, the fogging over the surface should be avoided. Fluid flow and heat transfer of humid air as well as heat transfer across the solid surface are important for understanding and modeling of the condensation process. A dew formation model can use these results from heat and mass transfer analyses in order to predict the shape and size of the droplets. Studies show that the heat and mass transfer governing equations can be solved by conjugate numerical simulation of fluid flow and heat transfer in both fluid and solid domains. As an example, Aroussi . et al ‎[2] numerically simulated the air flow and heat transfer in a vehicle demisting system. They considered the turbulent three-dimensional air flow in a simplified windshield demisting system and successfully validated their results with experiments. Croce . et al ‎[3] developed a method for solving the conjugate flow and heat transfer equations in a fluid/solid continuum pair. They introduced an iterative method for updating the temperature at the fluid/solid interface until the temperature and its corresponding heat flux matched on both fluid and solid sides. The method, coupled with a droplet formation model for the surface condensation was applied to applications such as windshield defogging ‎[4], refrigerated cabinet doors ‎[5], finned dehumidifiers ‎[6], and glass fogging ‎[7]. In all of the applications, they considered the water