Final answer:
To find the efficiency of a wind turbine with a given blade diameter, air density, and wind velocity, we calculate the theoretical power based on the wind turbine formula and compare it with the actual output power. After conversion and calculation, the efficiency of the turbine is found to be approximately 27.33%.
Step-by-step explanation:
The power available from a wind turbine can be calculated using the formula: P = ½ * A * ρ * v³, where P is power in watts, A is the sweep area of the blades in square meters, ρ is the air density in kilograms per cubic meter, and v is the velocity in meters per second. To find the efficiency of the turbine, we need to compare the actual output power of the turbine with the power that would be calculated by this formula if the turbine were 100% efficient.
First, we need to convert the blade diameter from feet to meters (1 foot = 0.3048 meters), and the velocity from miles per hour to meters per second (1 mile per hour = 0.44704 meters per second):
- Blade Diameter: 420 ft * 0.3048 m/ft = 128.016 m
- Blade Radius (R): 128.016 m / 2 = 64.008 m
- Velocity (v): 30 mph * 0.44704 m/s/mph = 13.4112 m/s
Next, the sweep area A is π*R², so:
- A = π * (64.008 m)² = 12867.699 m² (approximately)
The air density ρ given the specific gravity of air is 0.00123 kg/m³. The theoretical power π can be calculated as follows:
- P = 0.5 * 12867.699 m² * 0.00123 kg/m³ * (13.4112 m/s)³
- P = 18.295 MW (approximately)
The actual output power is 5 MW. Hence, the efficiency is calculated by dividing the actual power by the theoretical power and then multiplying by 100 to get the percentage:
- Efficiency = (5 MW / 18.295 MW) * 100%
- Efficiency = 27.33% (approximately)