ENERGY CONVERSION

1

• ENERGY CONVERSION
• MME9617a
• Eric Savory
• www.eng.uwo.ca/people/esavory/mme9617a.htm
• 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 (lt 10 kW) but the total generating
  capacity is mostly from 0.5 - 2 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 1.225 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 lt 100 W/m2 -
poor P / A 400 W/m2 - good P / A gt 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
(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
0
22
If p(U) is known, the following parameters can be
calculated
Mean wind speed, U
0
Standard deviation of wind speed, ?U
0
Mean available wind power density, P / A
0
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
0
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
gt 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 ? ( 0.627 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)
0
0
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
methods 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 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
2
Streamtube downstream of disk
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 0.5926 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
(a)
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 From earlier equation for ?r (slide
52), 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 find 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 give
s 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
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
aerodynamic 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
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
Lift coefficient
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
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.
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 a
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 Comb
ining 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
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 well 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 ?. 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'.
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 mtum 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. Mtum theory is no longer valid at a gt 0.5,
because the wind velocity in the far wake would
be -ve. In fact, as a gt 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 gt 0.4 or CT gt 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
lt 0.96 then the previously derived equations can
be used. If CTr gt 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 lt ? lt 3. For electric power
generation, use 4 lt ? lt 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. 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 ? lt
3, curved plates can be used. For ? gt 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
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 gt 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 ai,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 lt
0.96 If CTr,I,j gt 0.96
Until convergence use j j1 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 8
- 10, 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 maximum theoretically
possible power coefficient 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 coefficients of wind
turbines. The maximum achievable power
coefficient 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
and finally
151
Seals on Scroby Sands, Norfolk coast, UK
152
Scroby Sands Wind Farm (2004), 30 x 2 MW
turbines, 3km off the coast, powers 41,000 homes
153
(No Transcript)
154
Appendix A Determining Weibull parameters (k
and c) from mean and standard deviation wind
speed values
155
Using the Weibull probability density
distribution function p(U), it is possible to
calculate the average velocity as follows
(Eqn. 1)
156
(Eqn 2)
Equation (1) can then be used to solve for c
(2)
157
(No Transcript)
158
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
0.523 for a Rayleigh distribution.
159
Appendix B Derivation of power coefficient
equation for blade analysis
160
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
161
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
162
Substituting into we obtain the desired
final equation
163
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.