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LECTURE 20 WIND POWER SYSTEMS ECE 371 Sustainable Energy Systems 1 TEMPERATURE CORRECTION FOR AIR DENSITY ๏ฎ ๏ฎ Using the ideal gas law, we can easily determine the air density at other conditions pV=nRT (1) Where, p = absolute pressure (atm) V = volume (m3) n = mass (mol) T = absolute temperature (K) R = ideal gas constant = 8.2 e-5 (m3 atm /K mol) 2 TEMPERATURE CORRECTION FOR AIR DENSITY ๏ฎ ๏ฎ If we let M.W. stand for the molecular weight of the gas (g/mol), then the air density is: ๐๐ ๐๐๐๐ ๐๐3 = ๐๐ ๐๐๐๐๐๐ ๐๐ ๏ฟฝ๐๐.๐๐ ๐๐๐๐๐๐ ๐๐(๐๐3 ) ๐๐๐๐ −3 ๏ฟฝ10 ( ) ๐๐ Substituting in p V = n R T yields ๐๐ = ๐๐×๐๐.๐๐ ๐ ๐ ๐ ๐ 3 TEMPERATURE CORRECTION FOR AIR DENSITY ๏ฎ Since air is a mix of molecules of N2 (78.08%) ๏ฎ O2 (20.95%) ๏ฎ Ar - argon (0.93%) ๏ฎ CO2 (0.039%) ๏ฎ Ne - neon (0.00185%) ๏ฎ ๏ฎ 28.02 (molecular weight) 32.00 39.95 44.01 20.18 M.W. of Air = 28.97 g/mol 4 TEMPERATURE CORRECTION FOR AIR DENSITY Reference Value 5 Power in the Wind ๏ฎ ๏ฎ ๏ฎ ๏ฎ Remember that Since ๐๐ if a function of temperature, the power in wind is a function of temperature. From the table on the previous slide, we see that as the temperature increases, the power goes down. At 30 οC we loos 5% of the power in wind. 6 ALTITUDE CORRECTION FOR AIR DENSITY ๏ฎ ๏ฎ Air density is a function of pressure and temperature But, air pressure is a function of altitude ๏ฎ ๏ฎ Need a correction factor to estimate wind power at sites above the sea level Consider a column of air with cross section A, as shown in the following figure 7 ALTITUDE CORRECTION FOR AIR DENSITY ๏ฎ Now: ๐๐ ๐๐๐๐ ๏ฟฝ๐๐3 =1.225 KT KA 8 Book Correction for Altitude and Temperature ๏ฎ ๏ฎ ๐๐ ๐๐๐๐ ๏ฟฝ๐๐3 = Where 353.1exp −0.0342 ๐ง๐ง⁄๐๐ ๐๐ Z = altitude in meters ๏ฎ T is temperature in Kelvin ๏ฎ 9 IMPACT OF TOWER HEIGHT ๏ฎ ๏ฎ ๏ฎ Since power in the wind is proportional to cube of windspeed, a modest increase in windspeed can significantly increase the power To capture this higher windspeed, the tower height should be increased In the first few hundred meters above the ground, wind speed is greatly affected by the air friction as it crosses the earth’s surface 10 IMPACT OF TOWER HEIGHT ๏ฎ Smooth surfaces, such as calm sea, offer very little resistance to wind ๏ฎ ๏ฎ At the other extreme, surface winds are slowed considerably by irregularities such as forests and buildings ๏ฎ ๏ฎ Variation of windspeed with elevation is modest Variation of windspeed with elevation can be large The impact of roughness on the earth’s surface on windspeed is expressed in the following form 11 IMPACT OF TOWER HEIGHT (v/vo) = (H/Ho)α ๏ฎ ๏ฎ Where, v = windspeed at height H vo = windspeed at height Ho (reference is usually 10 meters) α = friction coefficient = 1/7 for open terrain (rule-of-thumb) = function of terrain over which wind blows α is called the Hellman exponent or shear exponent. 12 IMPACT OF TOWER HEIGHT ๏ฎ The following table gives some representative values 13 IMPACT OF TOWER HEIGHT ๏ฎ Impact of friction coefficient on windspeed and 1 power (v/vo) = (H/Ho)α P = ρ Av 3 w 2 14 POWER CURVES ๏ฎ ๏ฎ ๏ฎ ๏ฎ For heat engines, the maximum efficiency is limited Carnot efficiency For PV, the maximum efficiency is limited by the band-gap of material For fuel cells, the maximum efficiency is limited by the Gibbs free energy This concept also applies to WTGs 15 BETZ LIMIT ๏ฎ ๏ฎ ๏ฎ German physicist Albert Betz in 1919 formulated the maximum power that a turbine can extract from wind Wind that is approaching a wind turbine is slowed down as a portion of its kinetic energy is extracted by the turbine The wind leaving the turbine has a lower velocity and its pressure is reduced, causing the air to expand downwind of the turbine 16 BETZ LIMIT 17 Why can’t a turbine extract all of the energy in the wind? ๏ฎ ๏ฎ ๏ฎ If it did, the air would have to come to a complete stop behind the turbine, which would prevent any more of the wind from passing through the rotor. The downwind velocity, therefore, cannot be zero. The downwind velocity cannot be the same as the upwind velocity, since that would mean the turbine extracted no energy at all from the wind. ๏จThere must be some ideal slowing of the wind that results in maximum power extraction from the wind. 18 BETZ LIMIT ๏ฎ The power extracted by the blades is equal to the difference in kinetic energy between the upwind and downwind divided by time ๐๐๐๐ = ๏ฎ 1 ๐๐ 2 2 2 ๐ฃ๐ฃ −๐ฃ๐ฃ๐๐ ๐ก๐ก 1 ๐๐ = 2 ๐ก๐ก 1 = ๐๐ฬ 2 2 ๐ฃ๐ฃ − 2 ๐ฃ๐ฃ 2 ๐ฃ๐ฃ๐๐ − 2 ๐ฃ๐ฃ๐๐ Where ๐๐ฬ is mass divided by time, or it is called the mass flow rate 19 BETZ LIMIT ๏ฎ ๏ฎ ๏ฎ ๏ฎ But, ๐๐ฬ = ๐๐๐๐๐ฃ๐ฃ๐๐ Where A is the swept area of the rotor. ๐ฃ๐ฃ๐๐ is the windspeed through the rotor. If we assume that ๐ฃ๐ฃ๐๐ is just the average of the upwind and down wind, then ๐ฃ๐ฃ + ๐ฃ๐ฃ๐๐ ๐๐ฬ = ๐๐๐๐ 2 And 1 ๐ฃ๐ฃ + ๐ฃ๐ฃ๐๐ ๐๐๐๐ = ๐๐๐๐ 2 2 2 ๐ฃ๐ฃ − 2 ๐ฃ๐ฃ๐๐ 20 BETZ LIMIT ๏ฎ ๏ฎ Letting ๐๐ be the ratio of downwind windspeed to upwind windspeed Then, ๐ฃ๐ฃ๐๐ ๐๐ = ๏จ ๐ฃ๐ฃ๐๐ = ๐๐๐ฃ๐ฃ ๐ฃ๐ฃ 1 ๐ฃ๐ฃ + ๐ฃ๐ฃ๐๐ ๐๐๐๐ = ๐๐๐๐ 2 2 1 ๐ฃ๐ฃ+๐๐๐ฃ๐ฃ = ๐๐๐๐ 2 2 2 2 ๐ฃ๐ฃ − 2 2 ๐ฃ๐ฃ − ๐๐ ๐ฃ๐ฃ 2 ๐ฃ๐ฃ๐๐ 21 BETZ LIMIT 1 ๐ฃ๐ฃ+๐๐๐ฃ๐ฃ ๐๐๐๐ = ๐๐๐๐ 2 2 1 3 1+๐๐ = ๐๐๐๐๐ฃ๐ฃ 2 2 2 2 2 ๐ฃ๐ฃ − ๐๐ ๐ฃ๐ฃ 2 1 − ๐๐ Define Rotor Efficiency CP 1 ๐ถ๐ถ๐๐ = 1 + ๐๐ 1 − ๐๐2 2 22 Power Extracted From the Wind by the Rotor Where 1 ๐๐๐๐ = ๐๐๐๐๐ฃ๐ฃ 3 ๐ถ๐ถ๐๐ 2 1 ๐๐๐๐๐ฃ๐ฃ 3 2 is the power in the wind ๐ถ๐ถ๐๐ is the efficiency of the rotor 23 BETZ LIMIT ๏ฎ ๏ฎ To find the maximum rotor efficiency, we take the derivative of Cp with respect to λ and set it equal to zero to solve for λ Then, λ= ๏ฎ 1 3 And maximum rotor efficiency will be C p − max = 16 1 1 1 (1 + ) (1 − 2 ) = = 0.593 = 59.3 % 27 2 3 3 24 BETZ LIMIT ๏ฎ Therefore the maximum theoretical efficiency of a rotor is 59.3% ๏ฎ ๏ฎ Betz Efficiency or Betz Law The efficiency of a modern wind turbine blades can approach about 80% of the Betz law ๏ฎ Practical efficiency is about 45-50% 25 BETZ LIMIT ๏ฎ For a given windspeed ๏ฎ Rotor efficiency is a function of the rate at which the rotor turns If rotor turns too slowly, the efficiency drops off since the blades are letting too much wind pass by unaffected ๏ฎ If the rotor turns too fast, efficiency is reduced as the turbulence caused by one blade affects the blade that follows ๏ฎ ๏ฎ The usual way to show rotor efficiency is to present it as a function of the tip speed ratio 26 BETZ LIMIT ๏ฎ The Tip-Speed-Ratio (TSR) is defined as Rotor tip speed (rpm ) (πD / 60) TSR = = Wind speed v ๏ฎ A plot of typical rotor blade efficiency as a function of Tip-Speed-Ratio is shown next 27 BETZ LIMIT Range 28 29 Wind Speed and Efficiency ๏ฎ ๏ฎ ๏ฎ ๏ฎ Modern wind turbines operate best when their TSR is in the range of around 4–6, ๏จThe tip of a blade is moving four to six times the wind speed. ๏จ For maximum efficiency turbine blades should change their speed as the wind speed changes. ๏จ Why wind turbines should use variable-speed generators. 30