The average wind speed at a height of 80m over the North Sea is estimated to be about 9 m/s. Norway plans to build a 10-MW wind turbine in the North Sea, which will have blades of diameter 145m producing an average of 3 MW of power. Calculate the efficiency of the turbine.
My teacher said the answer is 42%, but I don't know how to get that answer.
To calculate the efficiency of the turbine, we need to find the power captured by the wind turbine and compare it to the power available in the wind.
Step 1: Calculate the swept area of the turbine
The swept area is the area over which the turbine blades rotate. It is given by the formula:
Swept Area = π * (blade diameter/2)^2
Plugging in the given blade diameter (145m):
Swept Area = π * (145/2)^2
Swept Area = 16596.270m^2
Step 2: Calculate the power available in the wind
The power available in the wind can be determined using the following formula:
Power = 0.5 * swept area * density of air * wind speed^3
Given:
Swept Area = 16596.270m^2
Density of air = 1.225 kg/m^3 (typical value at sea level)
Wind speed = 9 m/s (average wind speed at a height of 80m)
Plugging these values into the formula:
Power = 0.5 * 16596.270 * 1.225 * 9^3
Power = 0.5 * 16596.270 * 1.225 * 729
Power = 912.018 MW
Step 3: Calculate the efficiency
The efficiency of the turbine is given by the formula:
Efficiency = (Power captured by the turbine / Power available in the wind) * 100%
Given:
Power captured by the turbine = 3 MW
Plugging in these values:
Efficiency = (3 / 912.018) * 100%
Efficiency = 0.00329 * 100%
Efficiency = 0.329%
Therefore, the efficiency of the turbine is approximately 0.329%.
It seems there might be an error in the calculation or a misunderstanding of the problem. The answer provided by your teacher (42%) does not match the calculations. Please double-check the problem statement and the calculations to ensure accuracy.