Optimization of Darrieus Turbines with an Upwind and Downwind Momentum Model

This paper presents a theoretical inviscid aerodynamic optimization method for straight-bladed Darrieus wind turbines. First, a generalized Betz limit has been derived for an arbitrary number of actuator disks in series. Then a momentum-type velocity model is introduced with separate cosine-type interference coefficients for the upwind and downwind half of the rotor. The cosine-type velocity interference permits the rotor blades to become unloaded near the junction of the upwind and downwind rotor halves. A closed-form solution for the optimum and off-design value of the interference coefficients has been obtained by equating the jc component of force on each of the rotor halves to that on each of two semicylindrical actuators in series. The values for the optimum rotor efficiency, solidity, and corresponding interference coefficients have been obtained in a closed-form analytical solution by maximizing the power extracted from the downwind rotor half as well as from the entire rotor. The Betz limit for two uniformly loaded actuator disks in series is shown to equal CP = 0.64 and for two cosine loaded semicylindrical actuators in series Cp = 0.617 and for a straight-bladed Darrieus rotor CP = 0.610.