Development of a compact solution for condensate droplet growth

• A closed-form analytical model for heat transfer through a single condensate droplet was developed. • The framework spans the full contact-angle range (0–180°) with a finite interfacial resistance. • Interface-controlled and conduction-controlled condensation regimes were identified via the Biot number. • Hydrophobic droplets enhance heat transfer at low Bi, while hydrophilic droplets dominate at high Bi. • Compact Nusselt-number correlations were derived for direct use in condensation modeling and design. 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 ( 0 ∘ − 180 ∘ ) 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 ( B i < 2 ) 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 N u = π B i as θ c → 0 ∘ . 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.