Strong free-surface water vortices are found throughout industrial hydraulic systems used for water treatment, flow regulation, and energy generation. Previous models using the volumetric flow rate as a model input have generally been semi-empirical, and have tended to have some limitations in terms of the design of practical hydropower systems. In this study, an analytical model of a strong free-surface water vortex was developed. This model only requires the water head and geometric parameters as its inputs and calculates the maximum volumetric flow rate, air-core diameter, and rotational constant. Detailed experimental depth-discharge data from a full-scale gravitational vortex hydropower system, unavailable in the relevant literature, were obtained, and the simulated results showed excellent agreement with the experimental observations. These data could be used to verify similar models using laboratory-scale physical models in order to investigate the scaling effects. In contrast to previous models, this model does not assume a constant average velocity across the vortex radius and allows precise calculation of the resultant velocity vectors. Therefore, this model presents advantages in turbine design for energy generation systems.