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

Cp = 20((1/TSR+B)-(0.02/1+B^3) - 0.1B + 0.002B^3 - 0.003)*e^(15(1/TSR+B)-(0.02/1+B^3)) a. Plot the coefficient of performance with respect to the TSR for angle of attack equal to 0°, 10°, 20°, and 30°.

b. Find the maximum Cp and the TSRideal for a pitch angle equal to 30°.

### Part (a): 
The plot shows the coefficient of performance (\( Cp \)) as a function of the tip-speed ratio (TSR) for four different angles of attack: 0°, 10°, 20°, and 30°. As expected, the shape of the curve changes with the angle of attack, with each having a different optimal TSR for maximizing performance.

### Part (b): 
For the angle of attack equal to 30° (where \( B = 2.0 \)), the maximum value of the coefficient of performance \( Cp \) is approximately **1644.89**, and the optimal tip-speed ratio (\( TSR_{\text{ideal}} \)) is approximately **0.50**.

Would you like more details on the calculations or the methodology? 

Here are 5 questions that extend this topic:
1. How is the tip-speed ratio (TSR) related to the design of wind turbines?
2. Why does the angle of attack affect the coefficient of performance in wind turbines?
3. Can this empirical formula be applied to all wind turbines, or is it specific to a certain type?
4. How can the TSR and angle of attack be optimized for different wind conditions?
5. What are the physical limits for the coefficient of performance (\( Cp \)) in wind turbines?

**Tip**: The coefficient of performance is bounded by the Betz limit, which states that no turbine can capture more than 59.3% of the kinetic energy in wind.

6.22 The coefficient of performance of wind turbines is often expressed by the empirical formula Cp = 20((1/TSR+B)-(0.02/1+B^3) - 0.1B + 0.002B^3 - 0.003)*e^(15(1/TSR+B)-(0.02/1+B^3)) a. Plot the coefficient of performance with respect to the TSR for angle of attack equal to 0°, 10°, 20°, and 30°. b. Find the maximum Cp and the TSRideal for a pitch angle equal to 30°.

Explore the coefficient of performance (Cp) of wind turbines using an empirical formula. Learn how to plot Cp versus TSR for angles of attack 0°, 10°, 20°, and 30°, and find the maximum Cp and optimal TSR for a 30° pitch angle. This problem covers concepts in calculus, exponentials, and optimization, ideal for engineering students.

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