A Generalized Multi-Stage Actuator Disc Model for characterizing unconventional horizontal axis wind turbine geometries

This study presents a unified modeling framework for unconventional horizontal axis wind turbines (HAWTs) that utilize axially extended, non-radial blade geometries. Recent bio-inspired and helical designs have reported performance levels that are difficult to interpret within the standard single-plane actuator-disc framework: they overcome the classical Betz limit which is derived for an ideal single rotor operating under one-dimensional axial-flow assumptions. To address this gap in the technology description, we introduce the Generalized Multi-Stage Actuator Disc Model, which discretizes the rotor into a sequence of interacting aerodynamic stages aligned with the flow axis. We rigorously evaluate four topological configurations, Solid Divergent, Hollow Divergent, Solid Convergent, and Hollow Convergent, using numerical optimization to determine their theoretical performance limits. Our results show that all multi-stage configurations converge to a maximum theoretical efficiency of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>2</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> (approximately 66.7%), exceeding the single-stage Betz limit of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>16</mml:mn> <mml:mo>/</mml:mo> <mml:mn>27</mml:mn> </mml:mrow> </mml:math> (59.3%), while remaining consistent with tandem-disc extensions of momentum theory. Furthermore, the analysis reveals that while constant-radius geometries are optimal under standard area normalization, divergent geometries evaluated under mass-conserving conditions behave as ideal diffusers, theoretically capable of recovering nearly all wake energy. We also quantify the impact of rotor solidity, showing that hollow configurations require maximized annular thickness to fully exploit the multi-stage effect. These findings provide a physical interpretation of non-conventional horizontal axis wind turbines, as geometries that lie outside the classical single-disc assumptions and establish a theoretical framework for designing turbines suitable for low-wind-speed environments.