1. ENERGY CONVERSION ES 832a Eric Savory Lecture 14 – Wind Energy
Lecture 14 – Wind Energy
Part 2: Wind turbines
Department of Mechanical and Material Engineering
University of Western Ontario

2. Contents Modern wind turbines and their key components
Basic operation of a Horizontal Axis Wind Turbine (HAWT)
Estimation of the wind resource
Statistical analysis of wind data and its use to predict available power (using Rayleigh and Weibull distributions)
1-D momentum theory applied to an actuator disk model of the turbine, and the Betz limit
Incorporation of wake rotation into the analysis
Airfoil aerodynamics and blade design using momentum equation and blade element theory
Blade optimization

3. Modern wind turbines
In contrast to a windmill, which converts wind power into mechanical power, a wind turbine converts wind power into electricity.
As an electricity generator a wind turbine is connected to an electrical network:
Battery charging circuit
Residential scale power units
Isolated or island networks
Large utility grids
Most are small (< 10 kW) but the total generating capacity is mostly from MW machines.

4. Underlying features of conversion process
Aerodynamic lift force on the blades  net positive torque on a rotating shaft  mechanical power  electrical power in a generator.
No energy is stored – output is inherently fluctuating with the wind variability (though can limit output below what wind could produce at any given time).
Any system turbine is connected to must be able to handle this variability.

5. Horizontal axis wind turbine (HAWT)
Most common type of turbine. Rotor may be upwind or downwind of the tower.

6. Main components of a HAWT
Drive train = shafts, gear box, coupling, mechanical brake, generator
Electrical system on ground = cables, switchgear, transformers

7. Main options in wind turbine design
- Number of blades (commonly two or three)
- Rotor orientation: downwind or upwind of tower
- Blade material, construction method, and profile
- Hub design: rigid, teetering or hinged
- Power control via aerodynamic control (stall
control) or variable pitch blades (pitch control)
- Fixed or variable rotor speed
- Orientation by self aligning action (free yaw), or
direct control (active yaw)
- Synchronous or induction generator
- Gearbox or direct drive generator

8. Power output prediction
POWER CURVE (obtained from manufacturer, based on field tests using standard methods)
RATED = point of max power output from generator
CUT IN = Minimum speed at which machine delivers
useful power
CUT-OUT = max wind speed at which turbine is allowed to deliver power (limited by safety / engineering)
Power output varies with wind speed, producing a typical chart like this of electrical power output as a function of the hub height wind speed. Chart allows prediction of turbine energy production without doing a component analysis.

9. Typical size, height, diameter and rated capacity of wind turbines

10. Estimation of the potential wind resource
The mass flow rate dm/dt of air of density  and velocity U through a rotor disk of area A is:
The kinetic energy per unit time,
or power, of the flow is:
The wind power per unit area,
P/A or wind power density is:
Note: density is generally taken as kg/m3 (15oC at sea level). Actual power output is only about 45% of this available wind power for even the the best turbines

11. Power per unit area
available from
steady wind
Maps of annual average wind speeds  maps of average wind power density. More accurate estimates can be made if hourly averages, Ui, are available for a year. The average wind power density, based on hourly averages is

12. where U is the annual average wind speed and Ke is called the energy pattern factor. The energy pattern factor is calculated from:
where N = number of hours in a year = 8,760
Typical qualitative magnitude evaluations of the wind resource are:
P / A < 100 W/m2 - poor
P / A ~ 400 W/m2 - good
P / A > 700 W/m2 - great

13. Direct methods of data analysis, resource characterization, and turbine productivity
Given a series of N wind speed observations, Ui, each averaged over the time interval t, we obtain:
(1) The long-term average wind speed, U, over the total period of data collection:
(2) The standard deviation U of the individual wind speed averages:

14. (3) The average wind power density P / A is
Similarly, the wind energy density per unit area for a given extended time period T = N t is:
(4) The average machine wind power Pw is:
where Pw ( Ui ) is the power output defined by a wind machine power curve.

15. Method of bins (5) The energy from a wind machine, Ew , is:=
Method of bins
The method of bins also provides a way to summarize wind data and to determine expected
turbine productivity. The data must be separated into the wind speed intervals or bins in which they occur. It is convenient to use the same size bins. Suppose that the data are separated into NB bins of width wi with midpoints mj and with fj the number of occurrences in each bin or frequency, such that:

16. The wind speed data set can now be analyzed to give:
A histogram (bar graph) showing the number of occurrences and bin widths is usually plotted when using this method.

17. Velocity and power duration curves
Can be useful when comparing the energy potential of candidate wind sites.
The velocity duration curve is a graph with wind speed on the y axis and the number of hours in the year for which the speed equals or exceeds each particular value on the x axis.
These plots give an indication of the nature of the wind regime at each site. The flatter the curve, the more constant are the wind speeds (e.g. characteristic of the trade-wind regions of the earth). The steeper the curve, the more irregular the wind regime.

18. Velocity duration curve example
(Rohatgi J S and Nelson V, 1994, Alternative Energy Institute, Canyon, Texas)

19. A velocity duration curve can be converted to a power duration curve by cubing the ordinates, which are then proportional to the available wind power for a given rotor swept area.
The difference between the energy potential of different sites is visually apparent, because the areas under the curves are proportional to the annual energy available from the wind. The following steps must be carried out to construct velocity and power duration curves from data:
(1) Arrange the data in bins
(2) Find the number of hours that a given velocity
(or power per unit area) is exceeded
(3) Plot the resulting curves

20. A machine productivity curve for a particular wind turbine at a given site may be constructed using the power duration curve in conjunction with a machine curve for a given turbine. Note that the losses in energy production with the use of a wind turbine at this site can be identified.

21. Statistical analysis of wind data
This type of analysis relies on the use of the probability density function, p(U), of wind speed. One way to define the probability density function is that the probability of a wind speed occurring between Ua and Ub is given by:
The total area under the probability distribution curve is given by:

22. If p(U) is known, the following parameters can be calculated:
Mean wind speed, U
Standard deviation of wind speed, U
Mean available wind power density, P / A
It should be noted that the probability density function can be superimposed on a wind velocity histogram by scaling it to the area of the histogram.

23. Another important statistical parameter is the cumulative distribution function F(U) which represents the time fraction or probability that the wind speed is smaller than or equal to a given wind speed, U'. That is: F(U) = Probability (U'  U ) where U' is a dummy variable. It can be shown that:
Also, the slope of the cumulative distribution function is equal to the probability density function:

24. Probability density function equations
In general, either one of two probability distributions (or probability density functions) are used in wind data analysis:
(1) Rayleigh and
(2) Weibull (see Lecture 13 notes)
The Rayleigh distribution uses one parameter, the mean wind speed. The Weibull distribution is based on two parameters and, thus, can better represent a wider variety of wind regimes. Both are 'skew' distributions (defined only for values > 0).

25. Rayleigh distribution
Requires only a knowledge of the mean wind speed, U. The probability density function and the cumulative distribution function are given by:

26. Example of a Rayleigh distribution
Note: a larger value of the mean wind speed gives a higher probability at higher wind speeds

27. Weibull distribution
Determination of the Weibull probability density function requires a knowledge of two parameters: k, a shape factor and c, a scale factor. Both are a function of U and U . The Weibull probability density function and the cumulative distribution
function may be given by:
Note: methods for determining k and c from U and  U are given in an appendix at the end of these notes

28. Examples of Weibull distributions with different k for U = 8 m/s
Note: as k increases the peak is sharper, indicating there is less wind speed variation

29. Wind Turbine Energy Production Estimates Using Statistical Techniques
For a given wind regime probability distribution p(U) and a known machine power curve Pw(U), the average wind machine power Pw is given by:
Pw(U) may be determined from the wind power, the rotor power coefficient Cp and the drive train efficiency ( = generator power / rotor power):

30. where
Cp is also a function of tip speed ratio  defined as:
where  is the rotor angular velocity and R is rotor radius. Hence, assuming constant  , average wind machine power is also given by

31. Idealized machine productivity calculations using Rayleigh distribution
Assuming:
(1) Idealized wind turbine, no losses, machine power coefficient, Cp , equal to the Betz limit (Cp,Betz = 16/27 = the theoretical maximum possible power coefficient).
(2) Wind speed probability distribution is given by a Rayleigh distribution.
The average wind machine power equation becomes:

32. where Uc is a characteristic wind velocity given by
For an ideal machine  = 1, Cp = Cp,Betz = 16/27, so
Using x = U / Uc gives a simpler integral
Over all wind speeds, the integral becomes
so that

33. Substituting for the rotor disk area, A =  D2/4, and for the characteristic velocity Uc the equation for average power becomes simply:
Example:
What is the average annual energy production of an 18 m diameter Rayleigh-Betz machine at sea level in a 6m/s average annual wind velocity regime?

34. Solution:
Multiplying this by 8,760 hrs/yr gives an expected annual energy production of 334,000 kWhr
Comparing this result to the simple approach in Slide 10 where:
P = ½  A U 3 = ½  (¼  D 2 ) U 3 =  ( D) 2 U 3
shows that the simple method under-estimates the power by about 12% Pw

35. Productivity calculations for a real wind turbine using a Weibull distribution
The average wind machine power equation based upon the probability distribution function p(U) may be re-cast in terms of the cumulative distribution F(U): 
(1)
The Weibull distribution is
Therefore, using (2) in (1) and replacing the integral with a summation over NB bins gives
(2)

36. Pw =
Note: the above equation is the statistical method’s equivalent to the earlier equation:
where the relative frequency f / N corresponds to the term in brackets and the wind turbine power is calculated at the mid-point between Uj - 1 and Uj

37. 1-D Momentum Theory and the Betz Limit
A simple model may be used to determine the power from an ideal turbine rotor, the thrust of the wind on the ideal rotor and the effect of the rotor operation on the local wind field.
The analysis assumes a control volume, in which the boundaries are the surface of a stream tube and two cross-sections of the stream tube:

38. The only flow is across the ends of the stream tube
The only flow is across the ends of the stream tube. The turbine is represented by a uniform "actuator disk" which creates a discontinuity of pressure in the stream tube of air flowing through it. This approach is not limited to any particular type of wind turbine. The analysis uses the following assumptions:
- Homogenous, incompressible, steady state flow
- No frictional drag
- An infinite number of blades
- Uniform thrust over the disk or rotor area
- A non-rotating wake
- The static pressure far upstream and far
downstream of the rotor is equal to the
undisturbed ambient static pressure

39. The thrust T is equal to the change in momentum rate:
But the mass flow rate is:
So that:
T is +ve so the velocity behind the rotor U4 is less than the free stream velocity U1. No work is done on either side of the turbine rotor. Thus, the Bernoulli equation can be used in the two control volumes on either side of the actuator disk.

40. Streamtube upstream of disk:
Streamtube downstream of disk: 2
It is assumed that the far upstream and far downstream pressures are equal ( p1 = p4 ) and that the velocity across the disk remains the same ( U2 = U3 ).
The thrust can also be expressed as the net sum of the forces on each side of the actuator disc:

41. Using the Bernoulli equations to solve for ( p2 - p3 ) we obtain the thrust as:
Equating this with our first two expressions for T and recognizing that the mass flow rate is  A2 U2 we obtain for the wind velocity in the rotor plane simply:
Defining the axial induction factor a as the fractional decrease in wind velocity between the free stream and the rotor plane:
 and

42. The quantity U1 a is the “induced velocity” at the rotor, so the wind velocity there is a combination of the free stream velocity and the induced velocity. As the axial induction factor increases from 0, the wind speed behind the rotor reduces. If a = 1/2, the wind has slowed to zero velocity behind the rotor and this simple theory is no longer applicable.
The power out P is equal to the thrust times the velocity at the disk: U2

43.  Substituting in the expressions for U2 and U4: gives:
where A is the rotor area and U is the freestream velocity.
The power coefficient CP represents the amount of available wind power extracted by the rotor: 

44. The maximum value of CP occurs when a = 1/3 so that:
CP,max = 16 / 27 =
For this case, the flow through the disk corresponds to a stream tube with an upstream cross-sectional area of 2/3 the disk area that expands to twice the disk area downstream.
This result indicates that, if an ideal rotor were designed and operated such that the wind speed at the rotor were 2/3 of the freestream wind speed, then it would be operating at the point of maximum power production.

45. The thrust T and thrust coefficient CT can now be computed as
Hence, the thrust coefficient for an ideal wind turbine is equal to 4a (1 - a). CT has a maximum of 1.0 when a = 0.5 and the downstream velocity is zero.
At maximum power output (a = 1/3), CT has a value of 8/9.

46. Operating parameters for a Betz turbine;
U = velocity of undisturbed air; U4 = air velocity behind rotor
CP = power coefficient, CT = thrust coefficient

47. The Betz limit, CP,max = 16/27, is the maximum theoretically possible rotor power coefficient. In practice 3 effects lead to a decrease in the maximum achievable power coefficient:
- Rotation of the wake behind the rotor
- Finite number of blades and their tip losses
- Non-zero aerodynamic drag
Note that the overall turbine efficiency is a function of both the rotor power coefficient and the mechanical (including electrical) efficiency of the wind turbine: 

48. Ideal HAWT with Wake Rotation
In reality the generation of rotational KE in the wake results in less energy extraction by the rotor than would be expected without wake rotation.
In general, the extra KE in the wind turbine wake will be higher if the generated torque is higher.
Thus, as will be shown here, slow running wind turbines (with a low rotational speed and a high torque) experience more wake rotation losses than high-speed wind machines with low torque.

49. Geometry for rotor analysis:
U = undisturbed wind velocity
a = induction factor
Area of annular streamtube of radius r and thickness dr is 2  r dr

50. Assuming angular velocity  imparted to flow is small compared to angular velocity  of the rotor  pressure in far wake = pressure in freestream.
The pressure, wake rotation and induction factors are all assumed to be a function of radial position r.
Using a CV that moves with angular velocity  the energy equations can be applied at sections before and after the blades to derive the pressure difference across them. Across the flow disk the angular velocity of the air relative to the blade increases from  to  + , whilst the axial component of the velocity remains constant.

51. We obtain:
The resulting thrust on an annular element, dT, is:
The angular induction factor, a’ , is defined as:
Note that when wake rotation is included in the analysis, the induced velocity at the rotor consists of not only the axial component, U a, but also a component in the rotor plane, r  a'.

52. The expression for the thrust becomes:
From our previous linear momentum analysis, the thrust on an annular cross-section can also be determined by the following expression that uses the axial induction factor, a :
Equating these two thrust expressions gives:
where r is the local speed ratio:

53. Next, we derive an expression for the torque on the rotor by applying the conservation of angular momentum. For this situation, the torque exerted on the rotor, Q, must equal the change in angular momentum of the wake. On an incremental annular area element this gives:
Since U2 = U (1 - a) and a' =  / 2 , this expression reduces to:
The power generated at each element, dP, is given by:

54. Substituting for dQ in this expression and using the definition of the local speed ratio, r , the expression for the power generated at each element becomes:
It can be seen that the power from any annular ring is a function of the axial and angular induction factors and the tip speed ratio.
The axial and angular induction factors determine the magnitude and direction of the airflow at the rotor plane. The local speed ratio is a function of the tip speed ratio and radius.

55. The incremental contribution to the power coefficient, dCP, from each annular ring is given by:
Now, a’ is related to a and r by: 

56. The aerodynamic conditions for the maximum possible power production occur when the term
a’ (1 - a) in the CP equation on the previous slide is at its greatest value. Substituting the value for a’ from the last equation into a’ (1 - a) and setting the derivative with respect to a equal to zero yields:
This equation defines the axial induction factor for maximum power as a function of the local tip speed ratio in each annular ring.

57. Substituting into
we finds that, for maximum power in each annular ring:
If the equation
is differentiated with respect to a, we obtain a relationship between dr and da at the condition for maximum power production:

58. Substituting the previous three equations into the expression for the power coefficient:
gives:
where the lower limit of integration a1 corresponds to the axial induction factor for r = 0 and the upper limit a2 corresponds to the axial induction factor at r = .

59. Also, from we have for a2:
and also r = 0 for a1 = 0.25.
This equation for  can be solved for the values of a2 that correspond to tip speed ratios of interest, noting that a2 = 1/3 is the upper limit of a, giving an infinitely large tip speed ratio.
The integral for CP,max can be evaluated by changing of variables, substituting x for (1 - 3a):

60. Values for CP,max as a function of  with corresponding values for axial induction factor at the tip a2 are tabulated here:
The following two graphs also illustrate these data.

61. Theoretical maximum power coefficient as a function of tip speed ratio for an ideal horizontal axis wind turbine, with and without wake rotation
The higher the tip speed ratio, the greater the maximum theoretical CP

62. a’ = angular induction factor, r = radius, R = rotor radius
Axial induction factors (a) are close to the ideal 1/3 value until the hub is approached (r  0).
Angular induction factors (a’) are close to zero in the outer parts of the rotor but increase significantly near the hub).
Induction factors for ideal wind turbine with wake rotation; tip speed ratio,  = 7.5, a = axial induction factor
a’ = angular induction factor, r = radius, R = rotor radius

63. Airfoils and general aerodynamic concepts
Wind turbine blades use airfoil sections to develop mechanical power.
The width and length of the blades are a function of the desired aerodynamic performance and the maximum desired rotor power (as well as strength considerations).
Before examining the details of wind turbine power production, some airfoil aerodynamics principles are reviewed here.

64. Basic airfoil terminology
Thickness
Camber
Camber = distance between mean camber line (mid-point of airfoil) and the chord line (straight line from leading edge to trailing edge)
Thickness = distance between upper and lower surfaces (measured perpendicular to chord line)
Span = length of airfoil normal to the cross-section

65. Examples of standard airfoil shapes
NACA 0012 = 12% thick symmetric airfoil
NACA 63(2)-215 = 15% thick airfoil with slight camber
LS(1)-0417 = 17% thick airfoil with larger camber

66. Lift, drag and non-dimensional parameters
Airflow over an airfoil produces a distribution of forces over the airfoil surface.
The flow velocity over airfoils increases over the convex surface resulting in lower average pressure on the 'suction' side of the airfoil compared with the concave or 'pressure' side of the airfoil.
Meanwhile, viscous friction between the air and the airfoil surface slows the airflow to some extent next to the surface.

67. Lift force - defined to be perpendicular to direction of the oncoming airflow. The lift
force is a consequence of the unequal pressure on the upper and lower airfoil surfaces
Drag force - defined to be parallel to the direction of oncoming airflow. The drag force is due both to viscous friction forces at the surface of the airfoil and to unequal pressure on the airfoil surfaces facing toward and away from the oncoming flow
Pitching moment - defined to be about an axis perpendicular to the airfoil cross-section

68. These forces are a function of Reynolds number
Velocity = U
The resultant of all of these pressure and friction forces is usually resolved into two forces and a moment that act along the chord at c / 4 from the leading edge (at the 'quarter chord').
These forces are a function of Reynolds number
Re = U L /  (L is a characteristic length, e.g. c)

69. The 2-D airfoil section lift, drag and pitching moment coefficients are normally defined as:
A = projected airfoil area = chord x span = c l

70. Other dimensionless parameters that are important for analysis and design of wind turbines include the power and thrust coefficients and tip speed ratio, mentioned earlier and also the pressure coefficient:
and blade surface roughness ratio:

71. Airfoil aerodynamic behaviour
The theoretical lift coefficient for a flat plate is:
which is also a good approximation for real, thin airfoils, but only for small  .  
Lift and drag coefficients for a NACA 0012 airfoil as a function of  and Re

72. Airfoils for HAWT are often designed to be used at low angles of attack, where lift coefficients are fairly high and drag coefficients are fairly low.
The lift coefficient of this symmetric airfoil is about zero at an angle of attack of zero and increases to over 1.0 before decreasing at higher angles of attack.
The drag coefficient is usually much lower than the lift coefficient at low angles of attack. It increases at higher angles of attack.
Note the significant differences in airfoil behaviour at different Re. Rotor designers must make sure that appropriate Re data are available for analysis.

73. Note non-zero lift coefficient at zero incidence.
Lift at low  can be increased and drag reduced by using a cambered airfoil such as this DU-93-W-210 airfoil used in some European wind turbines:
Note non-zero lift coefficient at zero incidence.
Data shown: Re = 3 x106
Drag and moment
coefficients

74. Another wind turbine airfoil profile: Lift and drag coefficients for a S809 airfoil at Re = 7.5x107

75. High lift/stall development regime:
Attached flow regime:
At low  (up to about 7o for DU-93-W-210), flow is attached to upper surface of the airfoil. In this regime, lift increases with  and drag is relatively low.
High lift/stall development regime:
Here (from about 7-11o for DU-93-W-210), lift coeff peaks as airfoil becomes increasingly stalled. Stall occurs when  exceeds a critical value (10-16o, depending on Re) and separation of the boundary layer on the upper surface occurs. This causes a wake above the airfoil, reducing lift and increasing drag. This can occur at certain blade locations or conditions of wind turbine operation. It is sometimes used to limit wind turbine power in high winds. For example, many designs using fixed pitch blades rely on power regulation control via aerodynamic stall of the blades. That is, as wind speed increases, stall progresses outboard along the span of the blade (toward the tip) causing decreased lift and increased drag. In a well designed, stall regulated machine, this results in nearly constant power output as wind speeds increase.

76. Flat plate/fully stalled regime:
In the flat plate/fully stalled regime, at larger  up to 90o, the airfoil acts increasingly like a simple flat plate with approximately equal lift and drag coefficients at  of 45o and zero lift at 90o.
Illustration of airfoil stall

77. Airfoils for wind turbines
Typical blade chord Re range is 5 x 105 – 1 x 107
1970s and 1980s – designers thought airfoil performance was less important than optimising blade twist and taper.
Hence, helicopter blade sections, such as NACA 44xx and NACA 230xx, were popular as it was viewed as a similar application (high max. lift, low pitching moment, low min. drag).
But the following shortcomings have led to more attention on improved airfoil design:

78. Operational experience showed shortcomings (e. g
Operational experience showed shortcomings (e.g. stall controlled HAWT produced too much power in high winds, causing generator damage).
Turbines were operating with some part of the blade in deep stall for more than 50% of the lifetime of the machine.
Peak power and peak blade loads were occurring while turbine was operating with most of the blade stalled and predicted loads were 50 – 70% of the measured loads!
Leading edge roughness affected rotor performance. Insects and dirt  output dropped by up to 40% of clean value!

79. Momentum theory and Blade Element theory
The actuator disk approach yields the pressure change across the disk that is, in practice, produced by blades.
This, and the axial and angular induction factors that are a function of rotor power extraction and thrust, will now be used to define the flow at the airfoils.
The rotor geometry and its associated lift and drag characteristics can then be used to determine
- rotor shape if some performance parameters are known, or
- rotor performance if the blade shape has been defined.

80. Analysis uses:
Momentum theory - CV analysis of the forces at the blade based on the conservation of
linear and angular momentum.
Blade element theory – analysis of forces at a
section of the blade, as a function of blade geometry.
Results combined into “strip theory” or blade element momentum (BEM) theory.
This relates blade shape to the rotor's ability to extract power from the wind.

81. Analysis encompasses:
- Momentum and blade element theory.
- The simplest 'optimum' blade design with an infinite number of blades and no wake rotation.
- Performance characteristics (forces, rotor airflow characteristics, power coefficient) for a general blade design of known chord and twist distribution, including wake rotation, drag, and losses due to a finite number of blades.
- A simple 'optimum' blade design including wake rotation and an infinite number of blades. This blade design can be used as the start for a general blade design analysis.

82. Momentum theory
We use the annular control volume, as before, with induction factors (a, a’) being a function of radius r.

83. Applying linear momentum conservation to the CV of radius r and thickness dr gives the thrust contribution as:
Similarly, from conservation of angular momentum, the differential torque, Q, imparted to the blades (and equally, but oppositely, to the air) is:
Together, these define thrust and torque on an annular section of the rotor as functions of axial and angular induction factors that represent the flow conditions.

84. Blade element theory
The forces on the blades of a wind turbine can also be expressed as a function of Cl, Cd and .
For this analysis, the blade is assumed to be divided into N sections (or elements).
Assumptions:
- There is no aerodynamic interaction between
elements.
- The forces on the blades are determined solely
by the lift and drag characteristics of the
airfoil shape of the blades.

85. Diagram of blade elements
c = airfoil chord length; dr = radial length of element
r = radius; R = rotor radius;  = rotor angular velocity

86. Note:
Lift and drag forces are perpendicular and parallel, respectively, to an effective, or relative, wind.
The relative wind is the vector sum of the wind velocity at the rotor, U (1 - a), and the wind velocity due to rotation of the blade.
This rotational component is the vector sum of the blade section velocity,  r, and the induced angular velocity at the blades from conservation of angular
momentum,  r / 2, or:

87. Overall geometry for a downwind HAWT analysis; U = velocity of undisturbed flow;  = angular velocity of rotor; a = axial induction factor

88. Blade section geometryp T

89. p = section pitch angle (angle between chord
line and plane of rotation)
p,0 = blade pitch angle at tip
T = blade twist angle = p - p,0
 = angle of attack (angle between chord line
and relative wind)
 = angle of relative wind = p + 
dFL = incremental lift force
dFD = incremental drag force
dFN = incremental force normal to plane of
rotation (this contributes to thrust)
dFT = incremental force tangential to circle swept
by rotor (creates useful torque)
UreI = the relative wind velocity.

90. Figure shows the following section relationships:
If rotor has B blades, total normal force on section at distance r from centre is:

91. Differential torque due to tangential force operating at a distance r from the centre is given by:
Note: effect of drag is to decrease torque and, hence, power, but to increase the thrust loading.
Thus, blade element theory gives 2 equations: normal force (thrust) and tangential force (torque), on the annular rotor section as a function of the flow angles at the blades and airfoil characteristics.
Used to get blade shapes for optimum performance and to find rotor performance for an arbitrary shape.

92. Blade shape for ideal rotor (no wake rotation)
Because the algebra can get complex, a simple, but useful example will be presented here to illustrate the method.
Earlier, the maximum possible power coefficient from a wind turbine, assuming no wake rotation or drag, was found to occur with an axial
induction factor of 1/3.
If the same simplifying assumptions are applied to the momentum and blade element equations, the analysis becomes simple enough that an ideal
blade shape can be determined (= approx. shape to give maximum power at the design tip speed ratio).

93. The following assumptions will be made:
- No wake rotation; thus a' = 0
- No drag; thus Cd = 0
- No losses from a finite number of blades
- For the Betz optimum rotor, a = 1/3 in each
annular stream tube
First, a design tip speed ratio, , the desired number of blades, B, the radius, R, and an airfoil with known lift and drag coefficients as a function of angle of attack need to be chosen.
An angle of attack  (and, thus, a lift coefficient at which the airfoil will operate) is also chosen.

94. This angle of attack should be selected where
Cd / Cl is minimum in order to most closely approximate the assumption that Cd = 0.
These choices allow the twist and chord distribution of a blade that would provide Betz limit power production (given the input assumptions) to be determined.
For  = 1/3 momentum theory gives the incremental thrust on an annular element as:

95. From blade element theory equation (with Cd = 0) the normal force on the element is:
Urel can be expressed in terms of other variables:
Combining results from the two theories (the above equations for dT and dFN) with that for Urel gives:

96. Using:
with a’ = 0 and a = 1/3 gives:
so that:
Rearranging, using r =  (r/R), the angle of the relative wind, , and the blade chord, c, at each section are:

97. These equations may be used to find the chord and twist distribution of the Betz optimum blade.
Example: Given  = 7, R = 5m, Cl = 1, Cd/Cl is minimum at  = 7, and there are 3 blades (B = 3) we can use:
and
together with and
to obtain the changes in chord, twist angle (= 0 at tip), angle of relative wind, and section pitch, with radial distance, r/R, along the blade:

98. Twist and chord distribution for a Betz optimum blade
(r/ R = fraction of rotor radius)
Hence, blades with optimized power production have increasingly larger chord and twist angle on approaching the blade root (r0). Actual shape depends on difficulty/cost of manufacturing it.

99. Blade chord for example Betz optimum blade
metres
Blade chord for example Betz optimum blade

100. Blade twist angle for example Betz optimum blade

101. Prediction of general blade shape
performance
Generally, rotor shape is not optimum because of fabrication difficulties.
Also when an 'optimum' blade is run at an off-design tip speed ratio it is no longer 'optimum'.
Thus, blade shape must be designed for
- easy fabrication, and
- overall performance over the range of wind and
rotor speeds that they will encounter.

102. For non-optimum blades use an iterative method.
We assume a blade shape, predict its performance, try another shape and repeat until a suitable blade has been chosen.
So far, the blade shape for an ideal rotor without wake rotation has been considered.
Now we’ll consider analysis of arbitrary blade shapes, including; wake rotation, drag, losses from a finite number of blades and off-design performance. This leads to determination of an optimum blade shape, including wake rotation as part of a complete rotor design procedure.

103. Strip theory for a generalized rotor, including wake rotation
Here we extend our previous analysis to consider the non-linear range of the Cl v.  curve (i.e. stall).
The analysis starts with the 4 equations derived from momentum and blade element theories.
In this analysis, it is assumed that the chord and twist distributions of the blade are known.
 is unknown, but additional equations can be used to solve for  and the performance of the blade.
The forces and moments derived from the 2 theories must be equal. Equating these, one can derive the flow conditions for a turbine design.

104. Momentum theory
From axial momentum:
From angular momentum:
Blade element theory
where thrust dT is same force as normal force dFN

105. Using the expression for the relative velocity:
The blade element theory equations become:
where  ’ = local solidity defined by  ’ = B c / 2  r

106. Blade element momentum theory
When calculating induction factors, a and a', usual practice is to set Cd = 0.
For airfoils with low Cd, this simplification gives negligible errors.
When the torque equations and normal force equations are equated between momentum and blade element theory, with Cd = 0, we obtain:
From torque:
From force:

107. These 2 equations, together with:
yield the useful relationships:
others may also be derived, including:
Now, we need to examine some solution methods.

108. Solution methods
Two solution methods use these equations to determine the flow conditions and forces at each blade section.
The first uses measured airfoil characteristics and BEM equations to solve directly for Cl and . Can be solved numerically, but also has a graphical solution that shows the flow conditions at the blade and the existence of multiple solutions.
The second solution is an iterative numerical approach that is most easily extended for flow conditions with large axial induction factors.

109. Method 1 - Solving for Cl and 
Method 1 - Solving for Cl and . Since  =  + p for a given blade geometry and operating conditions, there are two unknowns in
Cl and  at each section. To find these, one can use the empirical Cl v.  curves for the chosen airfoil, picking off the Cl and  that satisfy the above equation. This can be done either numerically, or graphically (next slide). Once Cl and  have been found, a' and a can be determined from any two of:

110. Angle of attack - graphical solution method
Cl = 2-D lift coefficient;  = angle of attack; r = local speed ratio;  = angle of relative wind;  ‘ = local rotor solidity
It should be verified that the axial induction factor at the intersection point of the curves is less than 0.5 to ensure the result is valid.

111. Method 2 - Iterative solution for a and a'
Method 2 - Iterative solution for a and a'. Starts with guesses for a and a', from which flow conditions and new inductions factors are calculated:
(1) Guess values of a and a'
(2) Calculate the angle of the relative wind from
(3) Calculate  from  =  + p and then CI and Cd
(4) Update a and a' from either: or
The process is then repeated until the newly calculated induction factors are within some
acceptable tolerance of the previous ones.

112. Calculation of power coefficient
Once a has been obtained from each section, the overall rotor power coefficient may be calculated (see Appendix B):
where h = local speed ratio at the hub. This may also be expressed as:
Usually, these equations are solved numerically. When Cd = 0 the first equation here is the same as that derived from m’tum theory with wake rotation.

113. Tip loss: effect on power coefficient
of number of blades
Because pressure on the suction side of a blade is lower than on the pressure side, air tends to flow around the tip from the lower to upper surface, reducing lift and hence power production near the tip (most noticeable with fewer, wider blades).
Simplest method for this is a correction factor, F, introduced into the previously discussed equations, which is a function of the number of blades, the angle of relative wind, and position on the blade:
Note:
0  F  1

114. The tip loss correction factor affects the forces derived from momentum theory. Thus, we end up with:
Other equations become:

115. The equation remains unchanged.
The power coefficient can be calculated from: or

116. Off-design performance
When a section of blade has a pitch angle or flow conditions that are very different from the design conditions, a number of complications can affect the analysis. These include:
Multiple solutions in the region of transition to
stall, and
Solutions for highly loaded conditions with
values of the axial induction factor approaching
and exceeding 0.5.

117. (1) Multiple solutions to blade element momentum equations - In the stall region there may be multiple solutions for CI , Each of which is possible. The correct solution should be the one which maintains the continuity of the angle of attack along the blade span.

118. (2) Wind turbine flow states – Measured turbine performance is close to the results of BEM theory at low values of the axial induction factor, a.
M’tum theory is no longer valid at a > 0.5, because the wind velocity in the far wake would be -ve. In fact, as a > 0.5, flow patterns through the turbine become more complex than those predicted by momentum theory. A number of operating states for a rotor have been identified, notably:
Windmill state - normal turbine operating state.
Turbulent wake state - operation in high winds.

119. Above a = 0.5, in the turbulent wake state, measured data indicate that thrust coefficients increase to  2.0 at a = 1.0. This state is characterized by a large expansion of the slipstream, turbulence and recirculation behind the rotor. Momentum theory no longer describes the turbine behaviour, empirical relationships between CT and the axial induction factor are often used to predict performance in this regime.

120. Turbulent wake state rotor modelling
In the turbulent wake state the thrust determined by momentum theory is no longer valid. In these cases, the previous analysis can lead to a lack of convergence to a solution or a situation in which the curve would lie below the airfoil lift curve.
In the turbulent wake state, a solution can be found by using an empirical relationship between the axial induction factor and the thrust coefficient in conjunction with blade element theory. The empirical relationship by Glauert (shown in previous figure) including tip losses, is:
Valid for a > 0.4 or CT > 0.96

121. The local thrust coefficient at a given annular section at radius, r, may be defined as:
so using:
the local thrust coefficient is:

122. The easiest solution approach for heavily loaded turbines is the iterative procedure (Method 2) that starts with the selection of possible values for a and a'. Once  and Cl and Cd have been determined, the local thrust coefficient can be calculated according to the last equation.
If CTr < 0.96 then the previously derived equations can be used.
If CTr > 0.96 then the next estimate for a should use the local thrust coefficient with
Then a' is given by

123. Blade shape for an optimum rotor
with wake rotation
May be found using the analysis developed for a general rotor.
This optimisation includes wake rotation, but ignores drag (Cd = 0 ) and tip losses (F = 1).
Optimization is done by taking the partial derivative of that part of the integral for Cp
which is a function of the angle of the relative wind,  , and setting it equal to zero:

124. This gives:
whilst more algebra gives:
Induction factors can be calculated from:
These results can be compared to those for an ideal blade without wake rotation:

125. As before, we select  where Cd / Cl is a minimum.
Solidity is the ratio of the area of the blades to the swept area:
When the blade is modelled as a set of N blade sections of equal span, the solidity can be calculated from:

126. Blade shape for 3 examples of optimum rotors, assuming wake rotation
Note: slow 12-bladed machine has blades of roughly constant c over outer half, smaller c near hub and significant twist. The 2 faster machines have blades of increasing c from tip to hub and less twist.

127. Generalized rotor design procedure
(1) Rotor design for specific conditions
The previous analysis can be used in a generalized rotor design procedure:
Choose various rotor parameters and an airfoil.
An initial blade shape is then determined using the optimum blade shape assuming wake rotation
The final blade shape and performance are determined iteratively considering drag, tip losses, and ease of manufacture.

128. Determination of basic rotor parameters
1. Decide what power, P, is needed at a particular wind velocity, U. Include effect of a probable Cp and efficiencies, , of various other components (gearbox, generator, pump, etc). The radius, R, of the rotor may be estimated from:
2. According to application, choose a tip speed ratio, . For a water pumping windmill (greater torque needed) use 1 <  < 3. For electric power generation, use 4 <  < 10. Higher speed machines use less material in the blades and have smaller gearboxes, but require more sophisticated airfoils.

129. 3. Choose a number of blades, B, from table below
3. Choose a number of blades, B, from table below. Note: If fewer than three blades are selected, there are a number of structural dynamic problems that must be considered in the hub design.
4. Select an airfoil. If  < 3, curved plates can be used. For  > 3 use a more aerodynamic shape.
Suggested blade number, B,
for different tip speed ratios, 

130. Definition of the blade shape
5. Obtain and examine the empirical curves for the aerodynamic properties of the airfoil at each section (airfoil may vary from the root to the tip), i.e. Cl vs. , Cd vs. . Choose the design aerodynamic conditions, Cl,design and design , such that Cd,design / Cl,design is at a minimum for each blade section.
6. Divide the blade into N elements (usually 10-20). Use the optimum rotor theory to estimate the shape of the i th blade with a midpoint radius of ri using:

131. 7. Using the optimum blade shape as a guide, select a blade shape that looks like a good approximation. For ease of fabrication, linear variations of chord, thickness and twist might be chosen. For example, if a1, b1 and a2 are coeffs. for the chosen chord and twist distributions, then the chord and twist can be expressed as: 

132. Calculate rotor performance & modify blade design
8. As outlined above, one of two methods may be used to solve the blade performance equations.
Method 1 - Solving for CI and : Find the actual  and Cl for the centre of each element, using the following equations and the empirical airfoil curves:  

133. iteration or graphically:
Cl and  can be found by
iteration or graphically:
The iterative approach needs an initial estimate of the tip loss factor. To find a starting Fi , use an estimate for the angle of the relative wind of:
Subsequent iterations, find Fi using:
where j is the number of the iteration. Finally, compute the axial induction factor:
Graphical solution
for  at section i ai
If ai > 0.4 , use Method 2

134. Method 2 – Iteration to find a and a’: For an initial guess use values from the adjacent blade section, values from a previous design or an estimate based on values from the initial optimum blade design:
ai, 1

135. Using guesses for ai,1 and a’i,1 start the iterative solution procedure for the jth iteration. For the first iteration j = 1. Calculate the angle of the relative wind and the tip loss factor:
Determine Cl,i,j and Cd,i,j from airfoil lift and drag data using:

136. Calculate the local thrust coefficient:
Update a and a’ for the next iteration.
If CTr,I,j < 0.96 :
If CTr,I,j > 0.96 :
Until convergence use j = j+1 and iterate again starting from the eqn for tan i,j

137. 9. Having solved the equations for the performance at each blade element, the power coefficient is determined using:
If total length of hub and blade is assumed to be divided into N equal length blade elements, then:
where k is the index of the first "blade" section consisting of the actual blade airfoil.

138. 10. Modify design if necessary and repeat steps , in order to find the best design for the rotor, given the limitations of fabrication.

139. (2) Cp -  curves
Once the blade has been designed for optimum operation at a specific design tip speed ratio,
rotor performance over all expected tip speed ratios needs to be determined.
This can be done using methods outlined earlier.
For each tip speed ratio, the aerodynamic conditions at each blade section need to be determined. From these, the performance of the total rotor can be determined. The results are usually presented as a graph of power coefficient versus tip speed ratio, called a Cp -  curve:

140. Example of a Cp -  curve for a high tip speed ratio wind turbine

141. Cp -  curves can be used in wind turbine design to determine the rotor power for any combination of wind and rotor speed.
They provide immediate information on the maximum rotor power coefficient and optimum tip speed ratio.
The data can be found from turbine tests or from
modelling.
In either case, the results depend on the lift and drag coefficients of the airfoils, which may vary as a function of the flow conditions (e.g. vary with Re).

142. Simplified HAWT performance Calculation procedure
The method uses blade element theory and incorporates an analytical method for finding . Depending on whether tip losses are included, few or no iterations are required.
The method assumes that 2 conditions apply:
(1) The airfoil section Cl vs  must be linear in the region of interest.
(2)  must be small enough that small-angle approximations may be used.

143. The simplified method is same as Method 1 before, with the exception of a simplification for determining  and Cl for each blade section.
The simplified method uses an analytical (closed-form) expression for finding  of the relative wind at each blade element. It is assumed that the lift and drag curves can be approximated by:
When the lift curve is linear and when small-angle approximations can be used, it can be shown that  is given by:
(1)
(2)

144. (3)
where

145.  can be calculated from (3) once an initial estimate for the tip loss factor (F) is determined.
The lift and drag coefficients can then be calculated from Equation (1) and (2), using
Iteration with a new estimate of the tip loss factor may be required.

146. Effect of drag and blade number on optimum performance
Earlier the max. theoretically possible power coeff. for wind turbines was determined as a function of tip speed ratio.
Later we saw that airfoil drag and tip losses (that are a function of the total number of blades) reduce the power coeffs. of wind turbines.
The maximum achievable power coeff. for turbines with an optimum blade shape but a finite number of blades and aerodynamic drag has been calculated to within 0.5% for tip speed ratios from 4 to 20, lift to drag ratios (Cl / Cd) from 25 to infinity and from 1 to 3 blades (B):

147. The following two plots show the variation of maximum achievable power coefficient with tip speed ratio, as a function of:
(1) Number of blades (no drag)
(2) Lift to drag ratio
In practice these coefficients are reduced even further by the need to use non-optimum designs that are easy to manufacture, the lack of airfoils at the hub and aerodynamic losses at the blade-hub intersection.

148. Maximum achievable power coefficients as a function of number of blades (no drag)
The fewer the blades the lower the max. Cp at same tip speed ratio. Most turbines use 2 or 3 blades and, in general, most 2-bladed wind turbines use a higher tip speed ratio than most 3-bladed ones. Thus, there is little practical difference in the maximum achievable Cp between typical 2 and 3-bladed designs, assuming no drag.

149. Maximum achievable power coefficients of a three-bladed optimum rotor as a function of the lift to drag ratio, CI / Cd
There is clearly a significant reduction in maximum achievable power as the airfoil drag increases. Thus, there is a benefit to using airfoils with high lift to drag ratios.

150. Appendix A – Determining Weibull parameters
(k and c) from mean and standard deviation wind speed values

151. Using the Weibull probability density distribution function p(U), it is possible to calculate the average velocity as follows:
(Eqn. 1)

152. (Eqn 2)
Equation (1) can then be used to solve for c :
(2)

154. Variation of parameters with Weibull k shape factor
It should be noted that a Weibull distribution for which k = 2 is a special case.
It equals the Rayleigh distribution. That is, for k = 2
2 (1+ ½) =  / 4 . One can also note that U / U = for a Rayleigh distribution.

155. Appendix B – Derivation of power coefficient equation for blade analysis

156. The power contribution from each annulus is:
where  is the rotor rotational speed. The total power from the rotor is:
where rh is the rotor radius at the hub of the blade.
The power coefficient, Cp, is:

157. Using the expression for the differential torque:
and the definition of the local tip speed ratio:
where h = local speed ratio at the hub. From:
we obtain:

158. Substituting:
into
we obtain the desired final equation:

159. Acknowledgement
These notes are based on part of the book “Wind Energy Explained: Theory, Design and Application” by J F Manwell, J G McGowan and A L Rogers (published by Wiley).
This is an excellent and comprehensive text, covering wind characteristics and resources, turbine aerodynamics, mechanics and dynamics (including structural design), electrical and control aspects, system integration, siting of turbines and economics.