Wind Power

Overview

Engineering reference data for Wind Power in dynamics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Momentum

$p = mv$

Mass × velocity.

Work

$W = Fd\cos\theta$

Force × displacement × cos(angle).

Variables

Wind Speed-Power Relationship

Wind power generation exhibits a cubic relationship with wind speed, making velocity the most critical factor in turbine performance. A 20% increase in wind speed results in approximately 73% more power generation, calculated from the cube of velocity: $(1.2)^3 = 1.728$.

Efficiency and Practical Power Output

The theoretical power must be adjusted for real-world turbine efficiency ($\xi$), which typically remains below 40% (0.4). The actual available power is:

$P_a = \xi \frac{\rho A v^3}{2} = \xi \frac{\rho \pi d^2 v^3}{8}$

Where $\xi$ is the windmill efficiency factor (dimensionless, 0-1).

Example Calculation:
For a 1-meter diameter turbine ($d = 1\, \text{m}$) with 20% efficiency ($\xi = 0.2$) in air density $\rho = 1.2\, \text{kg/m}^3$ at wind speed $v = 10\, \text{m/s}$:

$P_a = (0.2) \cdot (1.2) \cdot \frac{\pi (1)^2 (10)^3}{8} = 94.2\, \text{W}$

Annual Energy Production

Total energy generated over a year depends on both turbine capacity and local wind patterns. It is calculated by summing the energy produced at each wind speed interval, weighted by the number of hours per year that speed occurs:

$E_{\text{annual}} = \sum_{v_i} \left[ P_a(v_i) \times h(v_i) \right]$

Where:

• $P_a(v_i)$ = actual power at wind speed $v_i$
• $h(v_i)$ = annual hours wind blows at speed $v_i$

This requires site-specific wind speed frequency distribution data, typically evaluated between the turbine's cut-in speed and shut-down speed.

References

• Original Source