Wind Turbine Tool

Tool for calculating the energy output of a wind turbine given a number of different parameters.

Working example

Scope

Input parameters:

• Blade length (radius): between 20 and 80m
• Wind speed: 0 to 75mph (cuts out below 8 and above 55mph)
• Altitude: 0 and 10000m
• Tower height: 20 and 200m

Output:

• Kilowatts produced
• Homes served (optional)

Formulas

Power

$\Large P = \frac{\rho A v^3}{2}$

• $\rho$ - air density (see formula below)
• $A$ - blade sweep area (see formula below)
• $v$ - wind speed: between 3.6 m/s (8mph) and 24.6 m/s (55mph)

Air density

$\Large \rho = \frac{p}{RT} $

• $p$ - air pressure (see formula below)
• $R$ - air gas constant 287 J/kgK
• $T$ - temperature at altitude (see formula below)

Blade sweep area

$\Large S = \pi L^2 $

• $L$ - Blade length: between 20 m and 80 m

Air pressure

$\Large P = P_s(\frac{T}{T_s})^\frac{-g}{\alpha R} $

• $g$ - gravitaional acceleration 9.8 $m/s^2$
• $R$ - air gas constant 287 J/kgK
• $\alpha$ - atmospheric lapse rate 0.0065 K/m
• $T$ - temperature at altitude
• $T_s$ - temperature at sea level
• $P_s$ - pressure at sea level 101,300 Pa.

The exponent $\frac{g}{\alpha R}$ is constant at -5.25

Temperature at altitude

$\Large T = T_s - \alpha z $

• $T_s$ - temperature at sea level assume 293°K (20°C)
• $\alpha$ - altitude: between 0 and 10,000 m
• $z$ - atmospheric lapse rate 0.0065 K/m

Wind speed at tower height

$\Large v_w = v_r (\frac{h}{h_r})^\alpha$

• $v_r$ - wind velocity at reference height
• $v_w$ - wind velocity with effect of wind shear (after calculation)
• $h$ - height of turbine: between 20 and 200 m
• $h_r$ - reference height (10 m)
• $\alpha$ - Hellman exponent (0.3)