Michael Graham - Wind Energy Handbook

(3.1)

The symbol ∞ refers to conditions far upstream, D refers to conditions at the disc, and W refers to conditions in the far wake.

It is usual to consider that the actuator disc induces a velocity variation that must be superimposed on the free‐stream velocity. The streamwise component of this induced flow at the disc is given by − aU ∞, where a is called the axial flow induction factor , or the inflow factor . At the disc, therefore, the net streamwise velocity is

(3.2)

Figure 3.2 An energy extracting actuator disc and streamtube.

The air that passes through the disc undergoes an overall change in velocity, U ∞− U W, and a rate of change of momentum equal to the overall change of velocity times the mass flow rate:

(3.3)

The force causing this change in momentum comes entirely from the pressure difference across the actuator disc and the axial component of the pressure acting on the curved surface of the streamtube. This latter pressure is usually assumed to be ambient and therefore to give zero contribution, without further explanation. In fact this pressure is different from ambient due to the axial variation of velocity along the streamtube, but the integral of its axial component from far upstream to far downstream can be shown to be exactly equal to zero, the streamwise contribution upstream of the actuator disc exactly balancing the downstream contribution that opposes the stream (Jamieson 2018).

Therefore,

(3.4)

To obtain the pressure difference ( p D +− p D −), Bernoulli's equation is applied separately to the upstream and downstream sections of the streamtube: separate equations are necessary because the total energy is different upstream and downstream. Bernoulli's equation states that, under steady conditions, the total energy in the flow, comprising kinetic energy, static pressure energy, and gravitational potential energy, remains constant provided no work is done on or by the fluid. Thus, for a unit volume of air,

(3.5a)

Upstream, therefore, we have

(3.5b)

Assuming the flow speed to be at low Mach number M (typically M < 0.3 is sufficient), it may be treated as incompressible ( ρ ∞= ρ D) and to be independent of buoyancy effects ( ρgh ∞= ρgh D) then,

(3.5c)

Similarly, downstream,

(3.5d)

Subtracting these equations, we obtain

(3.6)

Equation (3.4)then gives

(3.7)

and so,

(3.8)

That is, half the axial speed loss in the streamtube takes place upstream of the actuator disc and half downstream.

The force on the air becomes, from Eq. (3.4),

(3.9)

As this force is concentrated at the actuator disc, the rate of work done by the force is TU Dand hence the power extraction from the air is given by

(3.10)

A power coefficient is then defined as

(3.11)

where the denominator represents the power available in the air, in the absence of the actuator disc.Therefore,

(3.12)

(This limit is also referred to as the Lanchester–Betz limit or the Betz–Joukowski limit ). 1

The maximum value of C Poccurs when

that gives a value of

Hence,

(3.13)

The maximum achievable value of the power coefficient is known as the Betz limit after Albert Betz (1919), the German aerodynamicist. Frederic Lanchester (1915), a British aeronautical pioneer, worked earlier on a similar analysis and is sometimes given prior credit, and Joukowski (1920) also contributed an analysis. To date, no unducted wind turbine has been designed that is capable of exceeding the Betz limit. The limit is caused not by any deficiency in design because, as yet in our discussion, we have no design. However, because the streamtube has to expand upstream of the actuator disc, the cross‐section of the tube where the air is at the full, free‐stream velocity is smaller than the area of the disc.

The efficiency of the rotor might more properly be defined as