Blade element momentum theory

This article emphasizes application of blade element theory 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,[1] 1889[2]).

This article explains the application of the "Betz limit" to the efficiency of a ground-based wind turbine.

Froude's blade element theory (1878)[3] is a mathematical process to determine the behavior of propellers, later refined by Glauert (1926).

Betz (1921) provided an approximate correction to momentum "Rankine–Froude actuator-disk" theory [4] 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").

That is, the air begins to rotate about the z-axis immediately upon interaction with the rotor (see diagram below).

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.

, and no energy has been extracted from the fluid between points 1 and 2, then we have the following expression: Now let us return to our initial diagram.

; the fluid velocity then decreases and pressure increases as it approaches the rotor.

Thus, if mass flow rate is constant, increases in area must result in decreases in fluid velocity along a streamline.

; just before interaction with the rotor, fluid pressure has increased and so kinetic energy has decreased.

This can be described mathematically using Bernoulli's equation: where we have written the fluid velocity at the rotor as

; far downstream, pressure of the fluid has reached equilibrium with the atmosphere; this has been accomplished in the natural and dynamically slow process of decreasing the velocity of flow in the stream tube in order to maintain dynamic equilibrium ( i.e.

Assuming no further energy transfer, we can apply Bernoulli for downstream: where Thus we can obtain an expression for the pressure difference between fore and aft of the rotor: If we have a pressure difference across the area of the actuator disc, there is a force acting on the actuator disk, which can be determined from

The rate of change of axial momentum can be expressed as the difference between the initial and final axial velocities of the fluid, multiplied by the mass flow rate: Thus we can arrive at an expression for the fluid velocity far downstream: This force is acting at the rotor.

Let us return to our derived expression for the power transferred from the fluid to the rotor (

, we will relate the change in angular momentum of the fluid with the tangential induction factor,

[5] We will break the rotor area up into annular rings of infinitesimally small thickness.

As can be seen from the figure above, the flow expands as it approaches the rotor, a consequence of the increase in static pressure and the conservation of mass.

Therefore, we can use results such as power extraction and wake speed that were derived in the Betz model i.e.

This allows us to calculate maximum power extraction for a system that includes a rotating wake.

This method involves recognising that the torque generated in the rotor is given by the following expression: with the necessary terms defined immediately below.

The flow of the fluid around the airfoil gives rise to lift and drag forces.

Drag is the forces that acts tangential to the apparent fluid flow speed seen by the airfoil.

Consider the diagram below: The speed seen by the rotor blade is dependent on three things: the axial velocity of the fluid,

is given by the following expression: Remember that these forces calculated are normal and tangential to the apparent speed.

Thus the torque in the air is given by By the conservation of angular momentum, this balances the torque in the blades of the rotor; thus, Furthermore, the rate of change of linear momentum in the air is balanced by the out-of-plane bending force acting on the blades,

From momentum theory, the rate of change of linear momentum in the air is as follows: which may be expressed as Balancing this with the out-of-plane bending force gives Let us now make the following definitions: So we have the following equations: Let us make reference to the following equation which can be seen from analysis of the above figure: Thus, with these three equations, it is possible to get the following result through some algebraic manipulation:[5] We can derive an expression for