A Detailed Investigation on the Effect of the Sherwood and Nusselt Number Modelling for the Heating and Evaporation of a Single Suspended Water Droplet

Abstract

The work described in this paper presents a comprehensive analysis of the convective heat and mass transfer coefficient modelling around a single water droplet using an in-house code. The most widely used approach is to rely on sub-models for the non-dimensional heat and mass transfer numbers (called hereafter the Nusselt, Nu, and the Sherwood, Sh, numbers) and which are shown to take the theoretical and functional form of and where Red is the droplet Reynolds number, Pr and Sc are the Prandtl and Schmidt numbers of the surrounding gas and K1 and K2 are constants. This formulation, which is generally referred to in the literature as the Ranz-Marshall model (with K1 = K2 = 0.6), is the most used approach in Computational Fluid Dynamics (CFD) codes for fire safety engineering. In this paper, we first assessed this formulation based on 36 experimental tests carried out in [Volkov and Strizhak, Applied Thermal Engineering (2017)] and where a single suspended water droplet of a diameter between 2.6 and 3.4 mm is heated up by a convective hot air flow with a velocity between 3 and 4.5 m/s and a temperature between 100 and 800°C. The results showed that the overall model uncertainty in the droplet lifetime prediction is about 34% with particularly poor results when the air temperature is 100 or 200°C. The droplet saturation temperatures were overestimated by around 20 to 30°C. After this initial assessment, we performed a sensitivity analysis and selected a combination of values for K1 and K2 that provided an overall simultaneous good agreement for both the droplet lifetimes (model uncertainty of 5%) and the droplet saturation temperatures (around 10°C). This analysis showed that, for high air temperatures (i.e., Ta ≥ 300°C), the value of K2 = 0.6 remains suitable. However, for these cases, the value of K1 needed to be increased (to 1.8 and up to 4) in order to promote the evaporation-induced cooling and improve the predictions in terms of droplet saturation temperatures. For Ta = 100°C, the ‘best’ combination was found to be K1 = K2 = 1.8. Such combination allowed to reduce the initially overestimated droplet lifetimes by promoting mass transfer (through an increased value of K1) without slowing down the heating process that generates water vapor at the droplet surface (explaining the equally increased value of K2). The case of Ta = 200°C appeared to be an intermediate case. The present results indicate that the ‘classical’ Ranz-Marshall approach with K1 = K2 = 0.6 is not optimal. A more thorough analysis, with eventually additional experimental data, is required.