Math Problem Statement
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°.
Solution
Part (a):
The plot shows the coefficient of performance () 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 ), the maximum value of the coefficient of performance is approximately 1644.89, and the optimal tip-speed ratio () is approximately 0.50.
Would you like more details on the calculations or the methodology?
Here are 5 questions that extend this topic:
- How is the tip-speed ratio (TSR) related to the design of wind turbines?
- Why does the angle of attack affect the coefficient of performance in wind turbines?
- Can this empirical formula be applied to all wind turbines, or is it specific to a certain type?
- How can the TSR and angle of attack be optimized for different wind conditions?
- What are the physical limits for the coefficient of performance () 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.
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Math Problem Analysis
Mathematical Concepts
Calculus
Algebra
Exponentials
Optimization
Formulas
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))
Theorems
Maximum/minimum theorem
Exponentiation rules
Suitable Grade Level
Undergraduate Engineering/Physics
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