The animation shows how various parameters (e.g. wind speed, number of blades) affect the rotor blade shape of a wind turbine. In the animation, several parameter sets can be changed. The result is displayed both as a curve and as a 3D model.
Instructions for Use
The windows can be enlarged or reduced by clicking on them, as with all animations.
The rotor of the wind turbine can be viewed from several predefined perspectives:
After starting the application, you can view the animation in fullscreen mode. To do so, click on “View” and then on “Fullscreen”:
To exit fullscreen mode, press the Esc key.
Explanation
The basis of the calculation is the so-called Betz formula. The Betz formula defines the blade depth as a function of the distance from the rotor hub. The animation uses NACA airfoil profiles, which are also actually used in the construction of some wind turbines.
The Betz formula can be represented as follows (Hein 2013):
[ t(r) = frac{8}{9} cdot frac{1}{Z} cdot frac{1}{C_a} cdot frac{1}{n} cdot frac{v_1^2}{sqrt{left(frac{2}{3} v_1right)^2 + left(2 pi cdot r cdot nright)^2}} ]
All parameters can be changed in the table:
- Number of blades Z
- Lift coefficient Ca
- Design rotational speed n
- Blade length l
- Blade depth t(r) (blade depth as function of radius)
- Distance from rotor hub r
- Design wind speed v1 in m/s
The lift coefficient is a parameter that can only be determined empirically or through simulation.
The design rotational speed determines whether the turbine is a fast runner with low torque or a slow runner with high torque.
Note: To display the curve, double-click on the 3D model.
In the animation, wind turbines with up to 9 rotor blades can be constructed. Wind turbines with more than 3 blades are still used today for special applications. One example is the operation of pumps. The limit of 9 rotor blades is due to performance constraints, as the high number of polygons pushes a normal PC to its limits.
Wind speed depends on local conditions. This value is obtained through measurements and statistical analyses.
The Betz Limit – Why No Turbine Can Capture All the Wind
The Betz limit is one of the most fundamental results in wind energy theory. In 1919, physicist Albert Betz proved that no wind turbine can convert more than 16/27 (approximately 59.3%) of the kinetic energy of the wind into mechanical energy. This theoretical maximum is known as the Betz limit.
The reasoning: if a turbine extracted all the energy from the wind, the air behind the rotor would stand still. Since air must continue to flow through the rotor disc, there is a fundamental trade-off between energy extraction and air throughput. Betz showed that the optimum occurs when the wind speed behind the rotor is reduced to one third of the incoming wind speed.
The power coefficient cp describes how much of the available wind power a turbine actually captures:
[ P = c_p cdot frac{1}{2} cdot rho cdot A cdot v^3 ]
where ( rho ) is the air density, A is the rotor swept area, and v is the wind speed. Modern large turbines achieve cp values of about 0.45 to 0.50 – impressively close to the theoretical maximum of 0.593.
In the animation, the Betz formula is used to calculate the optimal blade depth at each radial position. When you change the wind speed or rotor speed, you can observe how the blade geometry adapts to capture as much energy as possible within the Betz limit.
Why Three Blades? – The Effect of Blade Count
Most modern wind turbines have three rotor blades. This is no coincidence – it represents the best engineering compromise between several competing factors.
Aerodynamic efficiency
With each additional blade, the total energy capture increases, but with diminishing returns. A three-blade rotor captures only about 3–5% less energy than an ideal rotor with infinitely many blades. Adding a fourth blade would improve efficiency by less than 1%, while significantly increasing weight and cost.
Structural balance
An odd number of blades avoids a specific structural problem. When a blade points straight up, the opposite blade (in a two-blade design) points down into the tower’s wind shadow. This causes asymmetric loading and increased fatigue. Three blades distribute forces more evenly.
Cost and weight
Each additional blade adds material cost and weight. The tower and foundation must support this weight, so fewer blades mean a lighter, cheaper overall structure.
Historical alternatives
One-blade and two-blade turbines have been built. The Monopteros design (one blade with counterweight) and various two-blade designs were tested in the 1980s and 1990s. They were lighter and cheaper but suffered from higher noise, visual flicker, and structural vibrations.
More than three blades
Multi-blade rotors (6–20 blades) are still used for water pumping windmills, where high torque at low wind speeds matters more than electrical efficiency. In the animation, you can construct rotors with up to 9 blades and observe how the blade shape changes.
Lift and drag
An airfoil generates lift because air flows faster over the curved upper surface than the flatter lower surface, creating a pressure difference. The lift coefficient Ca in the animation quantifies this effect and depends on the profile shape and the angle of attack.
Worked Example: Designing a Blade Step by Step
To illustrate how the parameters interact, consider a typical design scenario:
Given: A small wind turbine with a rotor radius of 5 m, designed for a wind speed of 8 m/s, with 3 blades and a design rotational speed of 2 revolutions per second.
Step 1 – Blade depth near the hub (r = 1 m):
Using the Betz formula with Z = 3, Ca = 1.0, n = 2 s⁻¹, v1 = 8 m/s, and r = 1 m, the blade depth at this position is approximately 0.56 m. The blade is relatively wide here.
Step 2 – Blade depth at mid-span (r = 2.5 m):
At r = 2.5 m, the blade depth decreases to approximately 0.16 m. The blade narrows significantly.
Step 3 – Blade depth near the tip (r = 4.5 m):
At r = 4.5 m, the blade depth is only about 0.05 m. The blade becomes very narrow.
This characteristic tapering – wide near the hub, narrow at the tip – is clearly visible in the animation’s 3D model. Try changing the wind speed to 12 m/s: the blade becomes narrower overall because at higher wind speeds, less blade area is needed to extract the same power.
Glossary of Key Terms
| Term (EN) | Term (DE) | Definition |
|---|---|---|
| Betz limit | Betz’sches Limit | The theoretical maximum efficiency of a wind turbine: approximately 59.3% |
| Blade depth / chord length | Blatttiefe | The width of the blade at a given radial position |
| Lift coefficient (Ca) | Auftriebsbeiwert | A dimensionless number describing the lift force generated by an airfoil |
| Tip speed ratio | Schnelllaufzahl | The ratio of blade tip speed to wind speed |
| Design rotational speed (n) | Auslegungsdrehzahl | The rotational speed for which the blade is optimized |
| Swept area | Rotorfläche | The circular area swept by the rotor blades: A = π·r² |
| Hub | Rotornabe | The central component connecting the blades to the drive shaft |
| NACA profile | NACA-Profil | A standardized airfoil shape defined by the National Advisory Committee for Aeronautics |
| Power coefficient (cp) | Leistungsbeiwert | The fraction of wind power actually captured by the turbine |
| Angle of attack | Anstellwinkel | The angle between the airfoil chord line and the incoming air flow |
Related Animations
HD Animations
Web Animations
Overview and Download
| Title | Construction of Rotor Blades for Wind Turbines |
| Target Audience | Teachers and Lecturers |
| Platforms | Microsoft® Windows® |
| Features | Full-screen mode Lossless scaling Large screens and projectors supported |
| License | Freeware |
| Download | Contact |
Contributors
C. Hein, S. Rikowski
Sources
- 3D engine for 3D model: Papervision3D 2.0
- Airfoil profiles (NACA 4412, 4418, 4421): http://www.ae.illinois.edu/m-selig/ads/coord_database.html
- Authoring tool (control elements included): Adobe Animate
- Hein, Christian (2013): [Untitled]. URL: http://www.unimuenster.de/imperia/md/content/fachbereich_physik/technik_didaktik/energietechnik_nutzung_windenergie.ppt [Last accessed: 18.02.2013].