Aerodynamics and aeroelastics of wind turbines - Wind Power Generation and Wind Turbine Design - page 91

Figure 2: 1D momentum theory of Froude, Lanchester and Betz.

Applying the equations of conservation of mass, energy (Bernoulli's equation) and
momentum:

1 (

( 1)

v = v + v ),

2 1 3

2

(Froude's law) and

2

Ca = 4(1 − a ),

(2 )

P

are obtained. C P is the non-dimensional power

P

C : = ,

( 3 )

P

3
1

(/2) r Av

p

2

AD

= .

( 4 )

D 2 . The most important param-
eter in eqn 2 is the so-called axial interference factor [ 12 ] often also called velocity
induction a = v 2 / v 1 . Differentiating eqn 2 with respect to a one can easily show that
maximum energy is extracted when a = 1/3 and

A turbine of diameter D has a swept area of A = ( p / 4)

This
law was found independently by Lanchester in 1915 and Betz in 1925. Using the
same arguments the main force on the turbine, the thrust in the wind direction:

B etz

CC = = 16 27 = 0.59.

P P

T

C = .

( 5 ) T

2
1

(/2) r Av