Betz's law - Wind Power Plus

Betz’s Law describes the maximum power that a wind turbine can extract from the wind, regardless of its aerodynamic design. The power output of a wind turbine depends on the interaction of its rotor with the wind. In 1919, German physicist Albert Betz, professor of applied mechanics, developed a simple model to determine the power output of an ideal wind turbine rotor, the thrust of the wind on the rotor, and the effect of rotor operation. This model is based on the linear momentum theory, neglecting several effects to obtain a simpler model, arriving at a theoretical limit known as the “Betz limit”.

Extractable Wind Power

The following assumptions are used to analyze the power that can be extracted from the rotor of a wind turbine:

Homogeneous, incompressible, steady state fluid (constant density).
No frictional drag.
Infinite number of blades.
Uniform thrust in the disc or rotor area.
Non-rotating wake.
The static pressure before and after the rotor is equal to the undisturbed ambient static pressure.

The calculation is based on momentum theory. The Rankine-Froude model is used. This model considers the volume swept by the wind turbine blades as an infinitely thin disk. An air flow through the area swept by the blades of the analyzed wind turbine is established in a control volume that can be divided into 4 sections.

Figure 1: Wind flow through a simple model of a wind turbine rotor.

Where:

V₁: Upstream wind speed (m/s).
V₂: Wind speed just before interacting with the disk.
V₃: Wind speed immediately after disk interaction.
V₄: Downstream wind speed (m/s).
p: Air pressure.
F: Force acting on the disk.
A_w: Area of actuating disk.

The disk divides into two control volumes. Therefore, Bernoulli’s principle applies to the downstream volume:

Bernoulli’s principle is also applied to the upstream volume:

It is assumed that the upstream and downstream pressures are equal (p₁ = p₄) and that the wind speeds immediately upstream and downstream of the disc are equal (V₂ = V₃). By solving the above two equations, a solution can be found for (p₂ – p₃):

The force on the actuator disk is:

Substituting the value of (p₂ – p₃) into the previous equation, we obtain:

The fluid exerts a force F on the disk. This force can be calculated from the change in the amount of motion:

Equating the two expressions for the force F on the disk, we obtain:

Therefore, the wind speed near the actuator disc is approximately equal to half the sum of the upstream (V₁) and downstream (V₄) wind speeds.

Axial Induction Factor (a) is defined as the fractional decrease in wind speed between free stream and rotor plane:

Then, as a function of the input speed V₁ and the axial induction factor (a), the speeds V₂ and V₄ are obtained:

The product of force (F) and speed (V₂) is the power (P) that the airflow transfers to the wind turbine rotor.

By substituting the values of V₂ and V₃ as a function of V₁ and the axial induction factor (a), the mechanical power of the wind turbine can be obtained:

Where:

P_m: Mechanical power.
A_r: Area swept by wind turbine blades (m²).
V_w: Upstream wind speed (m/s).
ρ: Air density (kg/m³).

The power transfer limiting factor is 4a(1-a)². Therefore, the wind turbine has an energy extraction limit that comes from the wind speed.

Calculation of the Betz’s Limit

The factor that limits the mechanical power of the wind turbine is called the power coefficient (Cp). The maximum value of Cp is obtained by derivation with respect to the axial induction factor (a).

The maximum value is obtained when a=1/3:

Figure 2 shows a plot of the power coefficient as a function of the axial induction factor. For values greater than 0.5, the power coefficient is physically meaningless.

Figure 2: Power coefficient Cp(a) as a function of axial induction factor.

Albert Betz stated that the maximum energy that can theoretically be extracted from the wind is only 59.3% of the kinetic energy carried by the wind. This limit is not due to a faulty design of the wind turbine, but to the control volume in which the analysis was performed. Furthermore, in his book “Wind Energie”, he describes much of the knowledge about wind energy at that time.

The theory of wind energy: The legacy of Albert Betz

The book “Wind energie” by Albert Betz is a classic text in the field of wind energy. Originally published in German in 1926, it is considered one of the first academic works in this field.

In the book, Betz presents a detailed theoretical analysis of wind energy and wind turbines. He develops a mathematical theory to describe the behavior of wind and how wind turbines can capture and convert wind energy into electrical energy. The book also discusses the design of wind turbines and how they can be optimized to maximize energy production. Betz provides detailed calculations and graphs to illustrate his concepts.

Although the book was written almost a century ago, it is still relevant today and remains a valuable resource for those working in the field of wind energy. Figure 3 shows the cover of the book.

Figure 3: Cover of the book “Wind Energie”, 1926.

In summary, Betz’s Law imposes a fundamental limit on the maximum amount of energy that can be extracted from the wind by a wind turbine, which has important implications for wind farm design and efficiency. Although Betz’s Law is a theoretical limit, technological advances in wind turbines and energy control systems are enabling wind farms to achieve ever higher efficiencies and make the most of wind energy.

“Betz’s Law states that only 59% of the kinetic energy of the wind can be converted into mechanical energy to move the wind turbine”.