General Momentum Theory for Horizontal Axis Wind Turbines

In the following chapter, a brief historical introduction will be given to the development of the modern wind turbine and the associated development of wind turbine aerodynamics.

The basic tool for understanding wind turbine aerodynamics is the momentum theory in which the flow is assumed to be steady, inviscid, incompressible and axisymmetric. The momentum theory basically consists of control volume integrals for conservation of mass, axial and angular momentum balances, and energy conservation.

In the following, the basic equations forming the one-dimensional momentum theory are introduced and analysed, and the errors committed, when using locally a one-dimensional approach on a differential form, are assessed. It is shown that an actuator disc with constant axial loading does not result in a constant axial velocity in the rotor plane. Hence, one-dimensional momentum theory is only valid for averaged quantities.

In the axial momentum theory presented in the previous chapter, the rotational flow was ignored and the rotor was replaced by a pressure jump represented by an actuator disc. To develop further the momentum theory, the actuator disc equations are modified by introducing rotational velocities to the flow.

To determine the optimum performance of a wind turbine, the Betz limit serves as a simple guideline giving an absolute maximum. However, for more complex models involving swirl, the performance depends generally on the tip speed ratio and the basic assumptions forming the various models. Therefore, different models may actually result in different optima. In the following, based on the previously derived equations, we are going to analyse and compare the optimum performance using different aerodynamics rotor models. The fact that the models may result in different optimum performance values do not necessarily imply superiority of a specific model, but merely it illustrates that some of the basic approaches may result in different more or less erroneous behaviour. Through the following analysis, we strive to elucidate the basic approximations of the various models in order to understand the shortcoming of the models, and, hopefully, to come up with the most correct model. Finally, the geometry resulting from designing optimum rotor plan forms from the various models is compared. From the comparison, it is found that the outer part of the blades is approximately the same, whereas large differences between the various optimum geometries exist at the inner part and at small tip speed ratios.

Due to the anomalous behaviour of the Joukowsky rotor, this model is analysed in details for small tip speed ratios. The analysis is carried analytically and by comparison to additional CFD computations and some recent experiments. It is found that the model only attains a well-defined optimum for tip speed ratios larger than 0.93, but that other inherent features of the equations by themselves will limit the power performance at smaller tip speed ratios.

Although there exists a large variety of methods for predicting performance and loadings of wind turbines, the only approach used today by wind turbine manufacturers is based on the blade-element/momentum (BEM) theory by Glauert (Aerodynamic theory. Springer, Berlin, pp. 169–360, 1935). A basic assumption in the BEM theory is that the flow takes place in independent stream tubes and that the loading is determined from two-dimensional sectional airfoil characteristics.

In this chapter, the tip correction is discussed in detail, and it is shown that the ‘traditional’ Prandtl/Glauert tip correction contains an inherent inconsistency in the vicinity of the tip when using tabulated airfoil data. In fact, the solution becomes singular if the lift coefficient is not directly proportional to the tip correction.

The finite-bladed optimum Betz rotor is treated. It is first very recently that a complete description of this rotor has been derived. In the chapter, a full analytical solution to the Betz rotor problem will be given, and the results will be compared to other optimum rotor models, both with respect to performance and resulting rotor geometry. It is here shown that for tip speed ratios greater than three, all models result in the same geometry at the outer part of the rotor, whereas the inner part always is different, both with respect to plan form and with respect to twist distribution.