PROBLEMS
1. Determine the diameter of a HAWT rotor required to generate 0.42 MW of power in a steady wind of 10 m/s at hub height. Assume that the power coefficient CP 5 0:35; the air density
is 1:2 kg=m3 and the mechanical efficiency; η 5 0:88:
2. The original 5 MW, three-bladed HAWT, made by RE Systems has a tip diameter of 126.3 m and a rated wind speed of 13 m/s. Determine the rated value of the power coefficient CP and compare this with the value at the Betz limit. Assume the air density ρ 5 1.2 kg/m3.
3. For the preceding problem, using actuator disc theory, determine the axial flow induction factor, a, and the static pressure difference across the disc at the rated wind speed.
4. A HAWT with a hub height of 80 m and blades of 80 m diameter develops 1.824 MW in a wind of 12 m/s with a blade tip-speed ratio of 4.5. Determine
a. the power coefficient, the relative maximum power coefficient, and the rotational speed;
b. for the same wind speed at 80 m height the wind speed that could be expected at a height of 150 m and, if the hub height was raised to that level, the likely power output if the power coefficient remains the same.
Assume the density is constant at 1.2 kg/m3 and that the one-seventh power law applies.
5. A three-bladed HAWT of 50 m diameter, has a constant blade chord of 2 m. and operates with a tip-speed ratio, J 5 4.5. Using an iterative method of calculation determine the values of the axial and tangential induction factors (a and a´) at a radius ratio of 0.95 and the value of the lift coefficient. Assume the drag coefficient is negligible (and can be ignored) compared with the lift coefficient and that the pitch angle of the blades β 5 3o.
Note: This problem requires the application of the BEM iterative method. Students are recommended to write computer programs for solving problems of this type.
(Stop the calculation after three iterations if solving manually.)
6. Using the methods of probability theory for a Rayleigh distribution of the wind speed, show that
where c is the fluctuating wind speed and c is the mean wind speed:
7. a. In calculating the maximum possible power production of a wind turbine (Carlin’s method) show that the integral
where the characteristic wind velocity
b. Determine the maximum idealized power output using Carlin’s formula for a turbine of 28 m diameter in a wind regime with an average wind speed of 10 m/s and an air density of 1.22 kg/m3.
8. A turbine has a cut-in speed of 4.5 m/s and a cut-out speed of 26 m/s. Assuming that the turbine is located at a favorable site where the mean annual wind speed is 10 m/s and a Rayleigh wind speed distribution may be applied, calculate:
a. the annual number of hours below the cut-in speed when the turbine does not produce any power;
b. the annual number of hours when the turbine is within the usable range of wind speed.
9. A three-bladed rotor of a HAWT with blades of 20 m tip radius is to be designed to work with a constant lift coefficient CL 5 1:1 along the length of the span at a tip-speed ratio, J 5 5.5.
Using Glauert’s momentum analysis of the “ideal wind turbine,” determine the variation of the chord size along the length of the span for the radius range of 2.8 m to a radius of 19.5 m.
Note: It is expected that students attempting this classical problem will need to become familiar with the theory of Glauert outlined in section “rotor optimum design criteria.” Also, this problem requires the application of the BEM iterative method of solution. Students are recommended to write a computer program for solving this type of problem, probably saving much time.