Question 6.10: Average Power in the Wind. Estimate the average power in the......
Average Power in the Wind. Estimate the average power in the wind at a height of 50 m when the windspeed at 10 m averages 6 m/s. Assume Rayleigh statistics, a standard friction coefficient α = 1/7, and standard air density ρ = 1.225 kg/m³.
We first adjust the winds at 10 m to those expected at 50 m using (6.15):
\left(\frac{v}{v_{0}} \right) \ = \ \left(\frac{H}{H_{0}} \right) ^{\alpha } (6.15)
\bar{v} _{50} \ = \ \bar{v} _{10} \ \left(\frac{H_{50}}{H_{10}} \right) ^{\alpha } \ = \ 6 \ \left(\frac{50}{10} \right) ^{{1}/{7}} \ = \ 7.55 \ {m}/{s}So, using (6.48), the average wind power density would be
\bar{P} \ = \ \frac{6}{\pi} \ \cdot \ \frac{1}{2} \rho A \bar{v} ^{3} \quad \left(\text{Rayleigh assumptions}\right) (6.48)
\bar{P} _{50} \ = \ \frac{6}{\pi } \ \cdot \ \frac{1}{2} \rho \bar{v} ^{3} \ = \ \frac{6}{\pi } \ \cdot \ \frac{1}{2} \ \cdot \ 1.225 \ \cdot \ \left(7.55\right)^{3} \ = \ 504 \ {W}/{m^{2}}We also could have found average power at 10 m and then adjust it to 50 m using (6.17):
\left(\frac{P}{P_{0}} \right) \ = \ \left(\frac{{1}/{2} \rho A v^{3}}{{1}/{2} \rho A v^{3}_{0}} \right) \ = \ \left(\frac{v}{v_{0}} \right) \ = \ \left(\frac{H}{H_{0}} \right)^{3\alpha } (6.17)