Geometric analysis of blade projections to provide a more representative solidity and a design methodology for vertical-axis turbines

The concept of solidity used to study turbines relates the blade projection area and the projection plane area. It has been utilized first for horizontal axis turbines and for straight-bladed (Darrieus) vertical axis turbines since 1975. Nevertheless, the mathematical expression of this concept has been equally applied regardless of turbine type (horizontal or vertical). Evidently, by relying upon the classic solidity expression, not all the parameters of vertical axis turbines are considered. The objective of this work is to analyze the solidity contemplating all geometrical characteristics of this type of turbines. The obtained equation enables calculation of the solidity of straight-bladed and helical-bladed (Gorlov) vertical axis turbines accurately. The development of this equation is presented and an analysis is carried out with data gathered from previous works, to investigate the relation between the geometrical characteristics of those devices and the proposed solidity, as well as between the latter and parameters such as tip speed ratio and power coefficient. The results of the analysis reveal that for each turbine there is an optimal solidity which maximizes the power coefficient. A methodology is proposed which allows to design, develop and analyze vertical axis turbines, helical or H type, throughout the use of a single expression which contemplates all relevant geometrical parameters.