Question 6.9: Average Power in the Wind. Using the data given in Fig. 6.22......
Average Power in the Wind. Using the data given in Fig. 6.22, find the average windspeed and the average power in the wind (W/m²). Assume the standard air density of 1.225 kg/m³. Compare the result with that which would be obtained if the average power were miscalculated using just the average windspeed.
We need to set up a spreadsheet to determine average wind speed v and the average value of v³. Let’s do a sample calculation of one line of a spreadsheet using the 805 h/yr at 8 m/s:
The rest of the spreadsheet to determine average wind power using (6.29) is as follows:
P_{avg} \ = \ \left(\frac{1}{2}\rho A v^{3}\right)_{avg} \ = \ \frac{1}{2}\rho A \left(v^{3}\right)_{avg} (6.29)
The average windspeed is
v_{avg} \ = \ \sum\limits_{i}{\left[v_{i} \ \cdot \ \left(\text{Fraction of hours} \ @ \ v_{i}\right) \right] } \ = \ 7.0 \ {m}/{s}The average value of v³ is
\left(v^{3}\right)_{avg} \ = \ \sum\limits_{i}{\left[{v_{i}}^{3} \ \cdot \ \left(\text{Fraction of hours} \ @ \ v_{i}\right) \right] } \ = \ 653.24The average power in the wind is
P_{avg} \ = \ \frac{1}{2} \rho \left(v^{3}\right)_{avg} \ = \ 0.5 \ \times \ 1.225 \ \times \ 653.24 \ = \ 400 \ {W}/{m^{2}}If we had miscalculated average power in the wind using the 7 m/s average windspeed, we would have found:
P_{average} \left(\text{WRONG}\right) \ = \ \frac{1}{2} \rho \left({v_{avg}}\right)^{3} \ = \ 0.5 \ \times \ 1.225 \ \times \ 7.0^3 \ = \ 210 \ {W}/{m^{2}}| Wind Speed v_{i} (m/s) |
Hours @ v_{i} per year |
Fraction of Hours @ v_{i} |
v_{i} × Fraction Hours @ v_{i} |
(v_{i})³ | (v_{i})³ × fraction Hours @ v_{i} |
| 0 | 24 | 0.0027 | 0.000 | 0 | 0.00 |
| 1 | 276 | 0.0315 | 0.032 | 1 | 0.03 |
| 2 | 527 | 0.0602 | 0.120 | 8 | 0.48 |
| 3 | 729 | 0.0832 | 0.250 | 27 | 2.25 |
| 4 | 869 | 0.0992 | 0.397 | 64 | 6.35 |
| 5 | 941 | 0.1074 | 0.537 | 125 | 13.43 |
| 6 | 946 | 0.1080 | 0.648 | 216 | 23.33 |
| 7 | 896 | 0.1023 | 0.716 | 343 | 35.08 |
| 8 | 805 | 0.0919 | 0.735 | 512 | 47.05 |
| 9 | 690 | 0.0788 | 0.709 | 729 | 57.42 |
| 10 | 565 | 0.0645 | 0.645 | 1,000 | 64.50 |
| 11 | 444 | 0.0507 | 0.558 | 1,331 | 67.46 |
| 12 | 335 | 0.0382 | 0.459 | 1,728 | 66.08 |
| 13 | 243 | 0.0277 | 0.361 | 2,197 | 60.94 |
| 14 | 170 | 0.0194 | 0.272 | 2,744 | 53.25 |
| 15 | 114 | 0.0130 | 0.195 | 3,375 | 43.92 |
| 16 | 74 | 0.0084 | 0.135 | 4,096 | 34.60 |
| 17 | 46 | 0.0053 | 0.089 | 4,913 | 25.80 |
| 18 | 28 | 0.0032 | 0.058 | 5,832 | 18.64 |
| 19 | 16 | 0.0018 | 0.035 | 6,859 | 12.53 |
| 20 | 9 | 0.0010 | 0.021 | 8,000 | 8.22 |
| 21 | 5 | 0.0006 | 0.012 | 9,261 | 5.29 |
| 22 | 3 | 0.0003 | 0.008 | 10,648 | 3.65 |
| 23 | 1 | 0.0001 | 0.003 | 12,167 | 1.39 |
| 24 | 1 | 0.0001 | 0.003 | 13,824 | 1.58 |
| 25 | 0 | 0.0000 | 0.000 | 15,625 | 0.00 |
| Totals: | 8760 | 1000 | 7.0 | 653.24 |