Development of a compact solution for condensate droplet growth

This work presents an analytical series solution in toroidal coordinates for predicting the temperature distribution, interfacial heat flux, and overall heat transfer rate of a sessile condensate droplet. Unlike existing models, the framework covers the full contact-angle spectrum [[EQUATION]]under a variable temperature condition at the liquid–vapor interface, thereby capturing both interface- and conduction-controlled regimes. To facilitate application, the analytical solution was further reduced to a compact Nusselt number correlation using symbolic regression. Model predictions agree with benchmark numerical data with an average error ~7%.Analysis of the model revealed that at low Biot numbers ([[EQUATION]]) hydrophobic droplets benefit from larger interfacial area, while at high Biot numbers hydrophilic droplets perform better due to reduced conduction resistance, with all cases converging to the thin-disk limit [[EQUATION]]as [[EQUATION]]. These results provide a versatile tool for understanding droplet-level heat transfer and for guiding the design of condensation-based thermal systems. This framework extends analytical modeling to the full range of droplet shapes, providing a practical tool for advancing condensation heat-transfer technologies.