An optimum blade design procedure has long been established in the literature, originally by Glauert[1] and then developed further by Wilson and Lissaman[2] to include the effects of drag and tip loss. The tip loss model used is that of Prandtl and was first used for wind turbine blades by Glauert[1]. Prandtl's model for tip loss is approximate and deals with the variation of induced velocity in the region of the blade tip. What is actually required is a method for determining the variation of blade circulation close the blade tip. It is also common for Prandtl's method to be applied at the blade root. The purpose of this paper is to describe a method, based upon the lifting-line theory also developed by Prandtl[3] and Glauert[4] for aircraft wings, for the determination of blade circulation distribution that would produce the maximum possible rotor torque. The analysis of a wind turbine rotor is proposed, treating each blade as a single, radial lifting line. The variation of circulation along the span of a blade is to be determined by the Biot-Savart law with the assumption that the helical wake trailing from each radial station on the blade convects with constant velocity and radius, these being determined by the conditions at the blade element. Maekawa[5] has produced an analysis for an optimum blade design that is based upon Goldstein's[6] theory and has similar constraints on the wake as is being proposed in this paper. A comparison with Maekawa's results will follow in the discussion section. The choice of prescribed wake not only simpifies the analysis but is also justified by the fact that an actuator disc with uniform circulation and a similar wake has the same induced velocities (Coleman et al [7]) at the disc plane as are those predicted by the general momentum theory of Glauert[1] that includes both the expansion and the slowing down of the wake. Because the wake has been simplified it is not possible to determine the induced velocity components at a specified wake location. The theories of Glauert[1] and Coleman et al [7] assume a radially uniform circulation distribution and an infinity of blades, albeit with a finite solidity. The axial induced velocity factor will also be uniform over the entire disc and, provided the circulation strength is of the appropriate magnitude, will have a magnitude of 1/3 allowing the maximum possible power (Betz limit) to be extracted from the wind. For a rotor with a finite number of blades the assumption of radially uniform circulation leads to an anomaly occurring at the tip. A discrete helical vortex is shed only from the very tip of each blade. The magnitude of the axial induced velocity in the vicinity of a blade tip is very large and exceeds the strength of the undisturbed wind speed. Therefore, the flow angle close to a blade tip is negative and the in-plane component of the blade force will produce a negative torque and a power output much less than the Betz limit. The circulation strength on a discrete blade must fall smoothly to zero at the tip but the manner of the variation will affect the magnitude of the power output. There must exist a circulation distribution that will maximise torque and power. At the root of a wind turbine blade the circulation must also fall to zero in a smooth manner. The objective of this paper is to develop a method for the determination of blade circulation distribution that would produce the maximum possible rotor torque.
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