Theoretical and Empirical Limits: Reexamining Betz’s Upper Limit towards a Practical Power Coefficients in Wind Turbines *

Abstract

The aspiration of all wind turbine designers is to attain Betz’s upper limit, which represents the highest efficiency in wind energy extraction. Majority of working turbines operate slightly below this limit with an exception of a few operating in wind tunnels. This study proposes for a comprehensive reevaluation of Betz’s derivation, aiming to establish the gap between a theoretical power limit and a practical limit for realization. There are two common expressions for power coefficient giving the same optimal value of 59%, but they generate different power-coefficient curves when plotted against velocity ratios. This paper presents a new method being referred as “Direct Multiplication Fractional Change” (DMFC) for deriving power-coefficient curves in wind energy, and compares its generated curve with established models. Discrepancies in power-coefficient expressions are identified and harmonized. Three approaches, namely EVAM, LVM, and DMFCM, were used for the numerical derivation of cp in the study, with their evaluation summarized in a table. The study collaborates its findings with a formulated velocity-distance curve, commonly presented as a hypothetical velocity profile in some publications. The results from DMFCM indicate two distinct maxima for the power coefficient. On the front side of the disc, a maximum of 0.5 is achievable in practice, although it is not the highest theoretically. On the rear side, a theoretical maximum of 0.59 is observed, but this value is not attainable in practice. These maxima are separated by their positions along the line of flow relative to the disc. However, this approximation is limited to a streamlined flow model of the rotor disc.