Blade Element Momentum Theory

Blade element momentum theory is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.

This article emphasizes application of BEM to ground-based wind turbines, but the principles apply as well to propellers. Whereas the streamtube area is reduced by a propeller, it is expanded by a wind turbine. For either application, a highly simplified but useful approximation is the Rankine–Froude "momentum" or "actuator disk" model (1865,1889). This article explains the application of the "Betz limit" to the efficiency of a ground-based wind turbine.

A development came in the form of Froude's blade element momentum theory (1878), later refined by Glauert (1926). Betz (1921) provided an approximate correction to momentum "Rankine–Froude actuator-disk" theory to account for the sudden rotation imparted to the flow by the actuator disk (NACA TN 83, "The Theory of the Screw Propeller" and NACA TM 491, "Propeller Problems"). In blade element momentum theory, angular momentum is included in the model, meaning that the wake (the air after interaction with the rotor) has angular momentum. That is, the air begins to rotate about the z-axis immediately upon interaction with the rotor (see diagram below). Angular momentum must be taken into account since the rotor, which is the device that extracts the energy from the wind, is rotating as a result of the interaction with the wind.

The "Betz limit," not yet taking advantage of Betz' contribution to account for rotational flow with emphasis on propellers, applies the Rankine–Froude "actuator disk" theory to obtain the maximum efficiency of a stationary wind turbine. The following analysis is restricted to axial motion of the air:

In our streamtube we have fluid flowing from left to right, and an actuator disk that represents the rotor. We will assume that the rotor is infinitesimally thin. From above, we can see that at the start of the streamtube, fluid flow is normal to the actuator disk. The fluid interacts with the rotor, thus transferring energy from the fluid to the rotor. The fluid then continues to flow downstream. Thus we can break our system/streamtube into two sections: pre-acuator disk, and post-actuator disk. Before interaction with the rotor, the total energy in the fluid is constant. Furthermore, after interacting with the rotor, the total energy in the fluid is constant.

Bernoulli's equation describes the different forms of energy that are present in fluid flow where the net energy is constant i.e. when a fluid is not transferring any energy to some other entity such as a rotor. The energy consists of static pressure, gravitational potential energy, and kinetic energy. Mathematically, we have the following expression:

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