Wind Turbine Power
P = 0.5 × ρ × A × v³ × Cp
Quick answer
This calculator solves P = 0.5 × ρ × A × v³ × Cp for Power Output. With Swept Area 100 cm², Wind Speed 10 m/s, Power Coefficient 50 %, the result is 3.063 W.
Calculator
Formula
Description
Wind turbine power output depends on the air density (approximately 1.225 kg/m³ at sea level), the rotor swept area, the cube of the wind speed, and the power coefficient. The cubic relationship with wind speed means that doubling the wind speed increases power eightfold, making site selection and tower height critical. The Betz limit sets the theoretical maximum power coefficient at 16/27 (approximately 0.593), meaning no turbine can extract more than 59.3% of the kinetic energy in the wind. Practical turbines achieve Cp of 0.35-0.45.
Variables
- P — Electrical power output (W)
- A — Rotor swept area (m²)
- v — Wind speed (m/s)
- Cp — Power coefficient (0-0.593, dimensionless)
Practical Notes
Air density decreases with altitude (about 1.112 kg/m³ at 1000m elevation) and increases with lower temperature. Small residential turbines (1-5 kW) typically have 2-4m diameter rotors, while utility-scale turbines reach 150m+ diameter with 5-15 MW ratings. Cut-in speed is typically 3-4 m/s, rated speed 12-15 m/s, and cut-out speed 25 m/s.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| P | Power Output | W | µW, mW, W, kW |
| A | Swept Area | m² | mm², cm², m² |
| v | Wind Speed | m/s | — |
| Cp | Power Coefficient | — | —, % |
Step by step
- 1. What you know: Swept Area 100 cm², Wind Speed 10 m/s, Power Coefficient 50 %
- 2. The relation: P = 0.5 × ρ × A × v³ × Cp
- 3. Substitute: A = 100 cm², v = 10 m/s, Cp = 50 %
- 4. Result: Power Output = 3.063 W
Worked examples
Minimum range
Given: Swept Area 10 mm², Wind Speed 0.01 m/s, Power Coefficient 2.5 %
Result: Power Output = 1.53 × 10⁻⁷ µW
Low range
Given: Swept Area 1 cm², Wind Speed 0.1 m/s, Power Coefficient 25 %
Result: Power Output = 0.01531 µW
Typical
Given: Swept Area 100 cm², Wind Speed 10 m/s, Power Coefficient 50 %
Result: Power Output = 3.063 W
High range
Given: Swept Area 1 m², Wind Speed 100 m/s, Power Coefficient 0.8
Result: Power Output = 490 kW
Maximum range
Given: Swept Area 10 m², Wind Speed 1000 m/s, Power Coefficient 8
Result: Power Output = 4.9 × 10⁷ kW
How Power Output changes with Swept Area
| Swept Area (input) | Power Output (result) |
|---|---|
| 10 cm² | 0.3063 W |
| 25 cm² | 0.7656 W |
| 50 cm² | 1.531 W |
| 75 cm² | 2.297 W |
| 100 cm² | 3.063 W |
| 150 cm² | 4.594 W |
| 200 cm² | 6.125 W |
| 400 cm² | 12.25 W |
| 0.07 m² | 21.44 W |
| 0.1 m² | 30.63 W |
The same result in other units
| Unit | Value |
|---|---|
| µW | 3063000 |
| mW | 3063 |
| W | 3.063 |
| kW | 0.003063 |
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<p><a href="https://vbatpower.com/formulas/wind-power/">Wind Turbine Power</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Wind Turbine Power?
Use P = 0.5 × ρ × A × v³ × Cp. With Swept Area 100 cm², Wind Speed 10 m/s, Power Coefficient 50 %, Power Output works out to 3.063 W.
What is the formula for Wind Turbine Power?
The formula is P = 0.5 × ρ × A × v³ × Cp. Wind turbine power output depends on the air density (approximately 1.225 kg/m³ at sea level), the rotor swept area, the cube of the wind speed, and the power coefficient.
What units does Wind Turbine Power use?
Power Output is shown in µW, mW, W, kW. Values are held in SI base units internally, so you can mix prefixes without converting by hand.
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