1. A blade is 12 m long, weight = 500 kg, and the center of mass is at 5 m. What is the torque if the force is 320 Nm?
2. Find the power loss for three struts on a HAWT. Struts are 4 m long, 2.5 cm in diameter. Rotor speed is 180 revolutions per minute. Use numerical approximation by dividing strut into 1 m sections and calculate at midpoint of section. Then add the values for each section. CD = 1.
3. Calculate the power loss for the struts on a VAWT. Center tube, torque tube, diameter = 0.5 m. Struts are at the top and bottom, 2 m long from torque tube to blades, diameter = 5 cm, rotor speed is 80 rpm. CD = 1. Calculate numerically (see problem 2) or use calculus.
4. For those who know calculus, find the value of u (speed of drag device) that produces the maximum CP for a drag device. Use Equation 6.10, where v0 is the wind speed at infinity. For those who do not know calculus, find the value of u that produces the maximum CP for a drag device by plotting the curve (Equation 6.12) for different values of u (between 0 and 1).
5. Aerodynamic efficiency can be maintained for different solidities of the rotor. If solidity increases, will you increase or decrease the tip speed ratio?
6. Explain the difference in performance of a wind turbine if it a. Operates at a constant tip speed ratio.
b. Operates at constant rpm.
7. What is the maximum theoretical efficiency for a wind turbine? What general principles were used to calculate this number?
8. If the solidity of the rotor is very small, for example, a one-bladed rotor, what is the value of the rpm for maximum CP compared to the same size rotor with higher solidity?
9. For those who know calculus, calculate the value of axial interference factor for which CP is a maximum for a lift device. Then show that this gives a maximum CP = 59%.
For those who do not know calculus, find the value of a that produces the maximum CP by plotting the curve (Equation 6.20) for different values of a.
10. A rotor reaches maximum CP at a tip speed ratio of 7. Calculate rotor rpm for four different wind turbines (diameters of 5, 10, 50, and 100 m) at wind speeds of 10, 20, and 30 m/s.
11. A wind turbine that operates at constant rpm will reach maximum efficiency at only one wind speed. What wind speed should be chosen?
For problems 12-18, specifications for a wind turbine are induction generator (rpm = 65), fixed-pitch, rated power = 300 kW, hub height = 50 m, rated wind speed = 18 m/s, tower head weight = 3,091 kg; rotor: two blades, mass of one blade = 500 kg, hub radius = 1.5 m, rotor radius = 12 m.
12. How fast is the tip of the blade moving?
13. How fast is the blade root (at hub radius) moving?
14. Put the mass at the midpoint and calculate the kinetic energy for one blade. Assume the mass of the blade is distributed evenly over ten sections. What is the kinetic energy for one blade?
15. At rated wind speed, calculate the torque since you know power and rpm (remember angular velocity, rad/s).
16. At 10 m/s, what is the thrust (force) on the rotor trying to tip the unit over? Calculate for that wind speed over whole swept area.
17. If the unit produced 800,000 kWh/year, calculate output per rotor swept area.
18. Calculate the annual output per weight on top of tower, kWh/kg.
For problems 19-25, specifications for a wind turbine are induction generator (rpm = 21), variable-pitch, rated power = 1,000 kW, hub height = 60 m, rated wind speed = 13 m/s, tower head weight = 20,000 kg; rotor: three blades, mass of one blade = 3,000 kg, hub radius =1.5 m, rotor radius = 30 m.
19. How fast is the tip of the blade moving?
20. How fast is the blade root (at hub radius) moving?
21. Place the mass at the midpoint of the blade and calculate the kinetic energy for one blade. Assume the mass of the blade is distributed evenly over ten sections. Now what is the kinetic energy for one blade?
22. Calculate the torque at the rated wind speed. You know the power and rpm (remember angular velocity, rad/s).
23. At 15 m/s, what is the thrust (force) on the rotor trying to tip the unit over? Calculate for that wind speed over the whole swept area.
24. If the unit produces 2,800,000 kWh/year, calculate the specific output, annual kWh/rotor area.
25. Calculate the annual output per weight on top of tower, kWh/kg.
26. For a 12 m blade, center of mass at 5 m, weight = 500 kg, calculate the angular momentum if the rotor is operating at 60 rpm.
27. For the blade in problem 26, the angular momentum is around 8 * 104 kg m2/s. Calculate the torque on the blade at that point if the angular moment of the rotor is stopped in 5 s. Use Equation 6.24. Then estimate the force trying to bend the blade.
28. Why are the blades for large wind turbines made from fiberglass-reinforced plastics?
29. Why are yaw rates limited on large wind turbines or yaw dampers installed on small wind turbines?
30. How does furling work on small wind turbines?
31. For loss of load on small wind turbines connected to the utility grid, how long can it take for overspeed shutdown?
32. For megawatt-size wind turbines, what is the most common configuration?
33. Go to the Proven Energy website for blade design. Make a paper model of the blade to see the principle for passive control.
34. List two methods of nondestructive testing and briefly describe them.
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