# Nonlinear blade element-momentum analysis of Betz-Goldstein rotors

## Highlights

We analyze Betz-Goldstein (BG) rotors for maximum power at any tip speed ratio and number of blades.

We prove that Glauert’s incorporation of tip loss in the blade torque and thrust equation are correct.

We show the nonlinear angular momentum terms can contribute 12% of the total torque.

## Abstract

Blade element-momentum theory is a fundamental tool of wind turbine aerodynamics but several of its basic assumptions are not easily examined. By applying the theory to ideal Betz-Goldstein (BG) rotors, two important aspects of the theory are investigated. We prove for the first time that Glauert’s inclusion of the tip loss factor in the angular and axial momentum equations is exact for BG rotors. Then we derive, also for the first time, the nonlinear contributions to the angular momentum balance when the rotor has a finite number of blades. The most important nonlinearity is the induced azimuthal velocity of the blades combined with that of the trailing vorticitiy. The derivation follows the recent formulation for the tip loss factor (F) by Wood et al.  in terms of the azimuthal variations in the induced velocities due to the trailing vortices. Nonlinearity is potentially important when F differs from unity. For a three-bladed BG rotor with the tip speed ratio (λ) varying from 0 to 15, we show that the nonlinear angular momentum peaks at 12% of the blade element torque when λ   is in the range 0.8≤λ≤1.5. As λ increases above 1.5, the nonlinear terms reduce in importance. Nonlinearities also arise in the axial momentum equation but are more difficult to analyze. An approximate treatment of them suggests similar behaviour to the nonlinear angular momentum terms, but those in the axial momentum equation arise purely from the trailing vortices.

## Keywords

• Wind turbine;
• Tip loss;