Turbine Settings
Wind Conditions (Weibull)
Analysis Results
Theory & Key Formulas
$$P_{max}= \frac{16}{27}\cdot \frac{1}{2}\rho A v^3 \approx 0.593 \cdot P_{wind}$$
ρ: air density 1.225 kg/m³, A = πR²: rotor swept area, v: wind speed
What is Wind Turbine Power & Energy?
🙋
What exactly is the Betz limit I see on the graph? Why can't a turbine capture all the wind's energy?
🎓
Basically, it's a fundamental law of physics, like a speed limit for wind energy. If a turbine stopped the wind completely, air would pile up in front of it and flow around it. So, some wind must keep moving past the blades. Albert Betz calculated the perfect balance: a turbine can capture at most 59.3% of the wind's kinetic energy. Try moving the "Rotor Radius" slider above—you'll see the Betz limit curve (the dashed line) shift up, showing the maximum possible power for that bigger rotor area.
🙋
Wait, really? So the "Power Coefficient" (Cp) on the chart is how close a real turbine gets to that 59.3% limit?
🎓
Exactly! Cp is the turbine's actual efficiency. In practice, mechanical losses and aerodynamics keep it lower. A common case is a modern turbine with Cp around 0.45-0.50. In this simulator, the Cp value changes dynamically with the Tip Speed Ratio (TSR), which you control indirectly via the "Pitch Angle." Try changing the pitch from 0 to 10 degrees—you'll see the Cp-TSR curve peak shift, directly affecting the power output curve.
🙋
What about the "Weibull distribution" parameters (Scale and Shape)? How do they connect to the yearly energy number?
🎓
Great question! The Weibull distribution describes how often different wind speeds occur at a site over a year. The "Scale" (c) is roughly the average wind speed, and "Shape" (k) describes how variable it is. For instance, a coastal site might have high scale and low shape (gusty). The simulator multiplies the turbine's power at each speed by how often that speed occurs (the Weibull probability) to calculate total annual energy. Adjust the "Scale" slider and watch the yearly energy estimate change dramatically—it shows why site selection is crucial!
Physical Model & Key Equations
The power available in the wind passing through the rotor's swept area is the starting point. The Betz Limit defines the theoretical maximum fraction of this power that can be extracted.
$$P_{wind}= \frac{1}{2}\rho A v^3$$
$$P_{max}= C_{p,max}\cdot P_{wind}= \frac{16}{27}\cdot \frac{1}{2}\rho A v^3 \approx 0.593\, P_{wind}$$
ρ = air density (1.225 kg/m³ at sea level)
A = rotor swept area = πR²
v = upstream wind speed (m/s)
Cp,max = Betz limit coefficient (16/27)
The actual turbine power depends on its power coefficient (Cp), which is a function of the Tip Speed Ratio (TSR) and the blade pitch angle (β). TSR is the ratio of blade tip speed to wind speed.
$$P_{turbine}= C_p(\lambda, \beta) \cdot \frac{1}{2}\rho A v^3$$
$$\lambda = \frac{\Omega R}{v}$$
Cp = power coefficient (efficiency, ≤ Cp,max)
λ = Tip Speed Ratio (TSR)
Ω = rotor angular velocity (rad/s)
R = rotor radius (m)
β = blade pitch angle (degrees)
Real-World Applications
Turbine Design & Sizing: Engineers use these exact calculations to determine the optimal rotor diameter and generator size for a specific location. For instance, a site with lower average wind speeds might require a larger rotor to capture enough energy, which directly impacts the turbine's cost and structural design.
Wind Farm Site Assessment: Before investing millions, developers analyze the Weibull parameters (scale and shape) from years of local wind data. This simulator's energy calculation mimics this process, predicting annual energy yield to estimate the project's financial viability.
Pitch Control Systems: The pitch angle (β) isn't static. In practice, above the "Rated Wind Speed," blades are pitched to spill wind and maintain constant power, protecting the generator. This simulator shows how changing pitch alters the Cp curve and limits power in high winds.
Performance Monitoring: Operators of existing wind farms continuously monitor the actual Cp of their turbines against the theoretical curve. A drop in Cp can indicate issues like blade erosion, misalignment, or icing, triggering maintenance.
Common Misconceptions and Points to Note
When you start using this simulator, there are a few points that are easy to misunderstand, so be careful. First, it's common to think of "the Betz limit of 59.3% as an absolute, unbreakable wall no matter how hard you try", but this is a story from an extremely simplified model of an "ideal actuator disk in a uniform flow". In reality, there are special cases that break this assumption—such as "multi-staging" where multiple turbines are arranged vertically—where theoretically exceeding this limit has been discussed. However, for standard single-turbine design, it's correct to treat it as an unattainable target value.
Next, a pitfall in parameter setting. Are you leaving the air density ρ at the default 1.225 kg/m³? This is the value for 15°C near sea level. If your actual installation site is on a plateau or if temperatures vary significantly between summer and winter, the density will fluctuate. For example, at an altitude of 1000m and 0°C, the density drops to about 1.1 kg/m³, reducing the power obtainable at the same wind speed by about 10%. For power generation estimates, it's a golden rule to recalculate the density based on the local average atmospheric pressure and temperature.
Finally, note that "operating always at the optimal TSR yields the highest efficiency" is not necessarily true. There certainly is a single point where Cp is maximized, but wind speed changes constantly, right? In actual turbine control, while the optimal TSR is tracked until the wind speed reaches the rated value, during strong winds the pitch angle is changed to keep the output constant, deliberately reducing efficiency (lowering Cp). It's a good idea to use the simulator to observe how the generated power levels off while increasing the pitch angle β, assuming the "above rated wind speed" region.
Worked Example
A 3.6 MW turbine with 120 m rotor diameter (11,310 m² swept area) at 80 m hub height, rated at 12 m/s cut-in speed 3 m/s, cut-out 25 m/s. With Weibull shape factor k=2.1 and mean wind speed (lambda) 8.5 m/s (coastal site, air density 1.225 kg/m³), pitch angle 0°: Power output at 10 m/s = 0.5 × 1.225 × 11,310 × (10³) × Cp(TSR) ≈ 2.8 MW. Annual Energy Production ≈ 11.2 GWh/yr, yielding capacity factor 35.4%. Increasing mean wind to 9.8 m/s (exposed ridge) raises AEP to 13.1 GWh/yr, CF 41.6%.