Which change produces the largest percentage increase in the maximum theoretical poweroutput of a wind turbine?A. Doubling the area of the bladesB. Doubling the density of the fluidC. Doubling the radius of the bladesD. Doubling the speed of the fluid– 12 –2221 – 6510
Question
Which change produces the largest percentage increase in the maximum theoretical poweroutput of a wind turbine?A. Doubling the area of the bladesB. Doubling the density of the fluidC. Doubling the radius of the bladesD. Doubling the speed of the fluid– 12 –2221 – 6510
Solution
The power output of a wind turbine is given by the formula:
P = 0.5 * ρ * A * v^3
where: P is the power, ρ is the air density, A is the swept area of the turbine blades, v is the wind speed.
Let's analyze each option:
A. Doubling the area of the blades: If we double the area, the power output will also double, resulting in a 100% increase.
B. Doubling the density of the fluid: If we double the density, the power output will also double, resulting in a 100% increase.
C. Doubling the radius of the blades: The area of the blades is proportional to the square of the radius (A = πr^2). So, if we double the radius, the area will increase by a factor of 4 (2^2), and thus the power output will increase by 300%.
D. Doubling the speed of the fluid: The power output is proportional to the cube of the wind speed (v^3). So, if we double the wind speed, the power output will increase by a factor of 8 (2^3), resulting in a 700% increase.
Therefore, doubling the speed of the fluid (option D) produces the largest percentage increase in the maximum theoretical power output of a wind turbine.
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