When an aircraft flies through a cloud containing supercooled liquid water droplets, ice accretion may deposit, posing a threat to flight safety. Hot air anti-icing is one of the most widely used anti-icing techniques. An extended mass and heat transfer model for runback water on a hot air anti-icing surface is developed. The model is based on the mass conservation and energy conservation of continuous water film and rivulet water film, considering the conditions under which continuous water film may break up into rivulet water film. Additionally, the effect of water vapor resulting from liquid water evaporation is also taken into account. Then, a three-dimensional tightly coupled numerical method for hot air anti-icing systems is constructed based on the developed model. The method includes modules for the calculation of external cold air flow, solid skin heat conduction and internal hot air flow, the calculation of water droplet impingement, and the calculation of anti-icing thermodynamics. The accuracy of the method is validated against experimental studies on an enhanced heat transfer piccolo tube anti-icing system in the icing wind tunnel of China Aerodynamics Research and Development Center (CARDC). The temperature results obtained from the three-dimensional numerical simulations showed good agreement with the experimental results, indicating that this method can be utilized in preliminary and detailed designs to evaluate the thermal performance of hot air anti-icing systems.