A Wind Turbine Efficiency Limit Higher than the Lanchester (Betz) Limit

Abstract

A generalized actuator-disk analysis is developed that retains both the axial and tangential velocity components of the flow through the blade passage. Unlike the classical one-dimensional formulation, the present theory distinguishes between the flow immediately upstream and downstream of the rotor, accounts explicitly for the rotation imparted to the flow, and derives turbine performance from the coupled conservation of mass, linear momentum, angular momentum, and energy. Optimization of the resulting power coefficient reveals two distinct solution branches. At lower blade-speed ratios, the optimum lies on the constraint curve defined by recovery of the velocity magnitude at the rotor-exit plane to the freestream value. Above a critical blade-speed ratio, however, the optimum leaves this constraint boundary and follows a second branch characterized by an exit speed below the freestream value. The transition between these branches is obtained analytically as the intersection of the unconstrained optimum and the boundary-constrained optimum, yielding a quartic equation whose physically admissible root determines the transition point. The optimized solutions predict power coefficients that exceed the classical Betz limit (reaching 78% at its peak) while remaining fully consistent with the governing conservation laws. The analysis shows that the classical efficiency limit is a consequence of the one-dimensional flow assumption rather than a fundamental upper limit imposed by the governing conservation laws.