## Chapter

Important: In physics the answer cannot be more accurate than the data that the calculation is based on. This is called significant figures or significant digits. Therefore, for handheld calculations, calculators, computers, and spreadsheets do not provide all the available numbers in the answer. Round off all answers to the correct significant digits.

1. Power in the wind increases with the cube of the wind speed. Power across a rotor increases by the square of the radius. Notice in the table that the numbers are rounded off. Also note that for the power in the wind across that swept area, the wind turbine cannot capture all that power. The power becomes quite large at high wind speeds and you must have a way to dump or not capture all the power available. Notice that the left two columns are the same as Table 3.3.

 Diameter, m 5 10 50 100 Area, m2 20 79 1,963 7,854 Wind Speed, m/s P/A, kW/m2 Power, kW 5 0.1 1.3 5 123 491 15 1.7 34 133 3,312 13,254 25 7.8 156 617 15,334 61,359

2. P/A = 1 kW/m2. From Table 3.3 it is between 10 and 15 m/s. Use air density = 1 kg/m3. From Equation 3.1, v3 = 2,000, or v = 13 m/s.

3. Use Equation 3.7, H0 = 10 m; calculate for H = 20 m and H = 50 m. Exponential = 1/7 = 0.14. Factor for increase in wind speed is v10y

4. Use Equation 3.7. Similar to problem 3.

5. Use Equation 3.7. Similar to problem 3.

6. Use Equation 3.8, z0 = 1 m. Could have chosen z0 from 0.5 to 1 m.

ln ln

 < ' H ^ ln ( 10 0 v Z0 , 11

7. p = 1.226 - (1.194 x 10-4) z, where z = 3,000 m p = 1.226 - (1.194 x 10-4) 3,000 = 1.226 - .3582 = 0.8678 kg/m3 % decrease = 0.3582/1.226 = 29%

Power/area will be 29% less due to change in density with elevation (due to pressure change).

8. Temp. summer = 100°F = 311K, temp. winter = -20°F = 244K % change = (311 - 244)/311 = 22%

On a hot day in the summer the power/area will be 22% less than on a cold day in the winter due to difference in density (due to temperature change).

9. Use Equation 3.17 and spreadsheet.

 Wind Speed m/s Frequency % Histogram h 0.5 1.22 107 1.5 3.58 314 2.5 5.68 498 3.5 7.39 647 4.5 8.61 755 5.5 9.31 816 6.5 9.5 832 7.5 9.23 809 8.5 8.6 753 9.5 7.7 675 10.5 6.66 583 11.5 5.57 488 12.5 4.51 395 13.5 3.54 310 14.5 2.7 236 15.5 1.99 175 16.5 1.43 126 17.5 1 88 18.5 0 60 19.5 0 39 20.5 0 25 21.5 0 16 22.5 0 10 23.5 0 6 24.5 0 3 25.5 0 2 26.5 0 1 27.5 0 1 28.5 0 0 29.5 0 0

10. Use Equation 3.18 and spreadsheet.

Wind Speed m/s

Frequency %

Histogram h

0.5

0.03

265

1.5

0.06

544

2.5

0.08

718

3.5

0.09

817

4.5

0.09

854

5.5

0.08

844

6.5

0.07

797

7.5

0.06

726

8.5

0.05

641

9.5

0.04

550

10.5

0.03

460

11.5

0.02

376

12.5

0.01

300

13.5

0.01

(Continued)

 14.5 0.01 181 15.5 0.01 136 16.5 0 101 17.5 0 73 18.5 0 52 19.5 0 37 20.5 0 25 21.5 0 17 22.5 0 12 23.5 0 8 24.5 0 5 25.5 0 3 26.5 0 2 27.5 0 1 28.5 0 1 29.5 0 0 30.5 0 0 31.5 0 0

11. Use Equation 3.18.

 Wind Speed m/s Frequency % Histogram h 0.5 0.001 13 1.5 0.013 115 2.5 0.035 311 3.5 0.067 578 4.5 0.099 870 5.5 0.128 1,122 6.5 0.145 1,268 7.5 0.144 1,267 8.5 0.128 1,118 9.5 0.099 868 10.5 0.067 590 11.5 0.04 348 12.5 0.02 177 13.5 0.01 77 14.5 0 26 15.5 0 9 16.5 0 2 17.5 0 0 18.5 0 0

12. Around 85% of the time the wind is 5 m/s or larger.

13. Around 15% of the time the wind is 12 m/s or larger.

The solutions to problems 16-19 are included in the following table. Air density was set to 1 kg/m3.

 Bin j Speed m/s No. Obs Freq fj fjVj P/A fj vj 1 1 20 0.026 0.026 0 2 3 30 0.038 0.115 1 3 5 50 0.064 0.321 4 4 7 100 0.128 0.897 22 5 9 180 0.231 2.077 84 6 11 150 0.192 2.115 128 7 13 120 0.154 2.000 169 8 15 80 0.103 1.538 173 9 17 40 0.051 0.872 126 10 19 10 0.013 0.244 44 SUM 780 1.00 10.2 751

16. Frequencies are given for each bin.

17. For j = 5, - = 0.5 fv3 = 0.5 0 . 23 1 93 = 84 W/m2.

For j = 10, — = 0.5 fv3 = 0.5 0.0 1 3 1 93 = 44 W/m2.

18. Average wind speed is the sum of frequency * speed for each bin, v average

19. Wind power potential is sum of 0.5 frequency * speed cubed for each 751 W/m2.

20. P/A = 0.5 * 10.23 = 531 W/m2. Value is smaller than the answer to problem 19 wind power potential is not calculated from average wind speed.

21. Use Equation 3.17 and bin width of 2 m/s. V average = 10.2 m/s.

 j m/s fj 0.5 *fjvj3 1 1 0.030 0 2 3 0.084 1 3 5 0.124 8 4 7 0.145 25 5 9 0.146 53 6 11 0.132 88 7 13 0.109 120 8 15 0.082 139 9 17 0.058 141 10 19 0.037 128 0.95 703

If a bin width of 1 m/s is used, sum of frequencies would be closer to 1.

## Renewable Energy Eco Friendly

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.

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