Math Problem Statement

The coefficient of performance of wind turbines is often expressed by the empirical formula

a. Plot the coefficient of performance with respect to the TSR for angle of attack equal to 0°, 10°, 20°, and 30°.

Solution

Here is the plot of the coefficient of performance (Cp) versus the Tip-Speed Ratio (TSR) for different angles of attack (0°, 10°, 20°, and 30°). As expected, the Cp reaches a peak for each curve and then decreases as the TSR increases. The maximum performance shifts left (to lower TSR values) as the angle of attack increases.

Let me know if you'd like more details or if you have any further questions.

Here are 5 related questions to consider:

  1. How does the angle of attack influence the optimal TSR in wind turbines?
  2. What are the typical values for the maximum coefficient of performance in real wind turbines?
  3. How does the blade shape affect the coefficient of performance?
  4. What happens to the wind turbine's efficiency if the TSR is too low or too high?
  5. Can Cp exceed the Betz limit of 0.59 for any wind turbine?

Tip: The optimal TSR depends on the blade design, the number of blades, and the wind speed; choosing the right TSR is critical for maximizing turbine efficiency.

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Math Problem Analysis

Mathematical Concepts

Coefficient of Performance (Cp)
Tip-Speed Ratio (TSR)
Angle of Attack
Wind Turbine Aerodynamics

Formulas

Cp = f(TSR, angle of attack)
Betz Limit (Cp max = 0.59)

Theorems

Betz Law
Aerodynamic Principles of Wind Turbines

Suitable Grade Level

Grades 11-12, College Level