ECE 333 Renewable Energy Systems

Transcription

1 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1 ECE 333 Renewable Energy Systems 5. Wind Power George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana Champaign

2 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 2 OUTLINE q The physics of rotors q Evaluation of power in the wind q The definition of specific power and its analysis q Temperature and altitude variations in specific power q The impacts of tower height on wind turbine output

3 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 3 ROTOR BASICS q We provide a brief introduction to how the rotor blades extract energy from the wind q Bernoulli s principle is the basis of the explanation of how an airfoil be it an airplane wing or a wind turbine blade obtains lift lift drag airfoil

4 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 4 ROTOR BASICS m air that travels over the top of the airfoil must cover a longer distance before it rejoins the air that uses the shorter path under the foil m air on top travels faster and so results in lower pressure than air under the airfoil m the difference between the two pressures creates the lifting force that holds an airplane up and that rotates the wind turbine blade

5 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 5 ROTOR BASICS q The situation with a rotor is more complicated than that of an airplane wing for a number of reasons: blade motion lift wind force net resulting wind across blade relative wind due to blade motion

6 ROTOR BASICS m a rotating blade experiences the air moving toward it from the wind and from the relative motion of the blade as it spins m the combined effect of the wind itself and the rotating blade results in a force that is at the appropriate angle so that the force is along the blade and can provide the lift that moves the rotor along ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 6

7 drag m as the blade speed at the tip is faster than near the hub, the blade must be twisted along its length to keep the appropriate angle lift horizontal ROTOR BASICS net wind angle of attack m the angle between the wind and the airfoil is referred to as the angle of attack stall condition net wind ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 7

8 ROTOR BASICS m as the angle of attack increases, the lift increases but so does the drag m too large of an angle of attack can lead to a stall phenomenon due to the resulting turbulence m wind turbines are equipped with a mechanism to shed some wind power so as to avoid damage to the generator ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 8

9 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 9 POWER IN THE WIND q We wish to analytically characterize the level of power associated with wind q For this purpose, we view wind as a packet of air with mass m moving at a constant speed v please note, this assumption represents a major simplification since air is a fluid; however, the simplified modeling is useful to explain the key concepts in wind generation

10 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 10 POWER IN THE WIND q The kinetic energy of wind is ε = 1 2 mv 2 q Power is simply the rate of change in energy and so we view the power in the mass of air m moving at constant speed v through area a as the rate at which the mass m passes through area a

11 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 11 POWER IN THE WIND mass of air m moving at constant speed v area a power through area a = d ε dt dε d 1 1 dm = mv = v dt dt 2 2 dt 2 2

12 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 12 POWER IN THE WIND q The term dm is the rate of flow of the mass of air dt through area a and is given by ρ av where ρ is the air density, i.e., the mass per unit of volume q The volume w of mass m is given by the area a times the length of mass m q Over time dt, the mass m moves a distance v dt resulting in the volume

13 POWER IN THE WIND dw = av dt q Now and dm dt = dm dw i dw dt dm dw = ρ = dm dw i av air density q Thus the power in the wind is pw = 1 ρ av 2 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 13 3

14 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 14 UNITS IN THE q We consider the units in p w EQUATION W air density at 15 C and 1 atm p w = 1 2 ρ av kg m 3 m 2 m s 3 m kg s s 2 = J s

15 UNITS IN THE p w EQUATION q The power in wind is, typically, expressed in units per cross sectional area W m 2 q We refer to the expression for p w as specific power or power density q We next consider p w in more detail and analyze the impacts of temperature and altitude ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 15

16 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 16 q The energy produced by a wind turbine is dependent on the power in the wind; to maximize the energy we therefore need to maximize q In the equation ANALYSIS OF pw ρ is a fixed parameter which we cannot control ; however, we can control the area a in the wind turbine design and we have some control over the wind speed in terms of the wind farm siting = 1 ρ av 2 3 p w p w

17 ANALYSIS OF p w q The area a is the swept area by the turbine rotor: for a HAWT with a blade with diameter d a d 1 = π = π d 2 q Clearly, there are economies of scale that are associated with larger wind turbines: m cost of a turbine d m power output of a turbine d 2 and so the larger rotors are more cost effective ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 17

18 NATURE OF AIR DENSITY q The air density ρ at 15 C and 1 atm pressure at sea level is kg m 3, but the value changes as a function of temperature and altitude q We know that ρ decreases as temperature increases since in a warmer day the air becomes thinner; a similar thinning of the air occurs with an increase in altitude ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 18

19 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 19 NATURE OF AIR DENSITY q We need to return to elementary chemistry and physics to determine the value of ρ for changes in temperature from 15 C and for altitudes above sea level q The governing relation is the ideal gas law ˆ = pw nrt where ˆp is the pressure in atm, w is the volume in 3 m, n is the mass in mol, T is the absolute

20 NATURE OF AIR DENSITY temperature in K, and R is the Avogadro number, the ideal gas constant m atm K mol q The pressure in atm is expressible in SI units since 1 atm = kpa where Pa is the abbreviation for the Pascal unit and 1 Pa N m = 2 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 20

21 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 21 TEMPERATURE VARIATION OF ρ q We can restate the expression for ρ in terms of the molecular weight of the gas, denoted by M.W., expressed in g, as mol g ρ kg n(mol) i M.W. m 3 = mol i 10 3 w m 3 ( ) q Air is the mixture of 5 gases and the associated M.W. of each are given in the table kg g

22 TEMPERATURE VARIATION OF ρ q Thus, gas fraction (%) M.W. (g/mol) nitrogen oxygen argon CO neon M. W. ( air) = ( )( ) + ( )( ) + ( )( ) + ( )( ) + ( )( ) = g mol ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 22

23 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 23 TEMPERATURE VARIATION OF ρ q The ideal gas law for the air M.W. value obtains ρ = ˆp(atm)i M.W. RT g mol = ˆp(atm)i(28.97) g mol T(K)i( ) i10 3 kg g m 3 iatm K i mol kg ˆ = p atm ρ m 3 T K

24 TEMPERATURE VARIATION OF ρ q Then, at 30 C at 1 atm ( )( 1) ρ ( 30 C ) = = while at 45 C at 1 atm ( )( 1) ρ ( 45 C ) = = kg m 3 kg m 3 q Note that the doubling (tripling) of the 15 C temperature results in a 5 % ( 9 % ) decrease in air density; these reductions, in turn, translate in the same % reductions in power ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 24

25 ALTITUDE VARIATION OF ρ q A change in altitude brings about a change in air pressure; we evaluate the ramifications of such a change q We consider a static column of air with crosssectional area a and we examine a horizontal slice in that column of thickness dz with air density ρ so that its mass is ρ a dz ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 25

26 ALTITUDE VARIATION OF ρ q We examine the pressures at the altitudes z + dz and z due to the weight of the air above those altitudes: ( ) pˆ z = pˆ( z + dz) + g ρa dz a additional weight per unit area of the slice of thickness dz where, g = m 2 s is the gravitational constant ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 26

27 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 27 ALTITUDE VARIATION OF ρ q We rewrite the difference in ˆp at the two altitudes as dpˆ = pˆ( z + dz) pˆ( z) = gρ dz altitude dz z pˆ( z + dz) pz ˆ( ) and so dpˆ dz = gρ q Note that ρ ˆ = p atm T K

28 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 28 ALTITUDE VARIATION OF ρ q We need to make use of several conversion factors to get useful expressions dpˆ kg = 3 dz T m m 1atm 1Pa 1N 2 p atm s Pa N m kg 2 2 m s ( ) ˆ ( ) ˆ = p T

29 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 29 ALTITUDE VARIATION OF ρ q The solution of this differential equation is complicated by the fact that the temperature also changes with altitude at the rate of 6.5 C drop for each km increase in altitude q Under the simplifying assumption that T remains constant, the solution of the differential equation is

30 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 30 ALTITUDE VARIATION OF ρ pˆ() z = pˆ 0 exp z T pˆ 0 = 1 atm q It follows that kg z ρ = exp m T T where T is in K and z is in m

31 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 31 EXAMPLE: COMBINED TEMPERATURE AND ALTITUDE IMPACTS q We compare the value of ρ at 25 C at 2,000 m to that under the standard 1 atm 15 C conditions q We compute ρ = exp = C 3 2, 000 m q The kg is thus reduced by 23 % and thus 2 m results in a 23 % decrease in power output a rather substantial decrease kg m

32 THE DEPENDENCE ON TOWER HEIGHT q The fact that power in the wind varies with where, v is the wind speed, implies that an v 3 increase in the wind speed has a pronounced effect on the wind output q Since for a given site, v increases as the height of the tower is raised, we can generally increase the wind turbine output by mounting it on a taller tower ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 32

33 THE DEPENDENCE ON TOWER HEIGHT q A good approximation of the relationship between v and tower height h is expressed in terms of the Hellman exponent α often called a friction coefficient by the relationship v h = v 0 h0 where, h 0 is the reference height with the α, corresponding wind speed v 0 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 33

34 THE DEPENDENCE ON TOWER HEIGHT q The Hellman exponent α depends on the nature of the terrain at the site; a higher value of α indicates heavier friction rougher terrain and a lower value indicates low resistance faced by the wind q Typical values for α are tabulated for different terrains ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 34

35 THE DEPENDENCE ON TOWER HEIGHT q A typical value for h 0 is 10 m and the behavior of v / v 0 as a function of h/ h 0 is v v α = 0.4 α = 0.3 α = α = ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 35 h

36 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 36 q We can also determine the ratio of p w (h) to p w (h 0 ) under the assumption that the air density ρ remains unchanged over the range [ h 0, h ] using the relationship THE DEPENDENCE ON TOWER HEIGHT 1 ρ av 3 p ( ) w h = 2 v h = = p ( ) 1 3 w h0 ρ av v0 h α

37 THE DEPENDENCE ON TOWER HEIGHT q We can observe the dramatic change in the power output ratio as a function of height 12 α = 0.4 p p w w ( h) ( h ) 0 8 α = 0.3 α = α = ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 37 h

38 ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 38 THE DEPENDENCE ON TOWER HEIGHT q A key implication of the power ratio at different heights is the fact that the stress as the turbine blade moves through an entire revolution may be rather significant, particularly over rough terrain p w h+ d 2 d h + 2 d p w h d 2 h 2 d h

39 THE DEPENDENCE ON TOWER HEIGHT ECE George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 39 d p w h 2 is the lowest value of wind output d p w h + 2 is the highest value of wind output d p + w h d h + 2 = 2 d d p w h h 2 2 3α