On the aerodynamic performance of a class of vertical shaft windmills

Abstract

A theory is presented to estimate the aerodynamic performance of a vertical shaft windmill configuration recently developed by South & Rangi (1971, 1972). The induced velocity is assumed to be given by simple momentum theory arguments; a blade element analysis then yields equations from which the mean power coefficient may be calculated. If the induced velocity is assumed to be uniform the predicted power coefficients are somewhat high compared to the experimental results. It is shown that the uniform induced velocity theory can be linearized for high tip speed ratios, to yield a simple formula for the mean power coefficient, $C_{P}^{\prime}=\sigma \mu ^{\prime}(A_{0}-A_{1}\mu ^{\prime 2})$, where $\sigma $ is the blade solidity, $\mu ^{\prime}$ is the tip speed ratio based on the local wind speed and $A_{0}$ and $A_{1}$ are constants determined by the blade geometry. Further, guided by the refinements earlier made to propeller theory, an analysis incorporating non-uniform induced velocity over the windmill disk is presented. These calculations, when compared with the uniform induced velocity calculations, lead to a reduction in the power coefficient for tip speed ratios corresponding to the peak power coefficient and greater. Finally, when compared with the available experimental data, the calculations show encouraging agreement as regards the peak power coefficient, the effect of blade solidity and the variation of power coefficient with tip speed ratio.