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
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
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 0.5 - 2 MW machines.
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.
Horizontal axis wind turbine (HAWT)
Most common type of turbine. Rotor may be upwind or
downwind of the tower.
Main components
of a HAWT
Drive train = shafts, gear
box, coupling,
mechanical brake,
generator
Electrical system on
ground = cables,
switchgear, transformers
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
Power output prediction
POWER CURVE (obtained
from manufacturer, based
on field tests using
standard methods)
CUT IN = Minimum
speed at which
machine delivers
useful power
RATED = point
of max power
output from
generator
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.
Typical size, height, diameter and rated
capacity of wind turbines
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
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
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
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:
(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.
(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:
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.
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.
Velocity duration curve example
(Rohatgi J S and Nelson V, 1994, Alternative Energy Institute, Canyon, Texas)
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
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.
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
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.
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:
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).
Rayleigh distribution
Requires only a knowledge of the mean wind
speed, U. The probability density function and
the cumulative distribution function are given by:
Example of a Rayleigh distribution
Note: a larger value of the mean wind speed gives a higher
probability at higher wind speeds
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
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
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):
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
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:
where Uc is a characteristic wind velocity given by
Uc  2 U /

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
3/4 
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?
Solution:
Pw
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%
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):
0

(1)
0
The Weibull distribution is
Therefore, using (2) in (1) and replacing the
integral with a summation over NB bins gives
(2)
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
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 crosssections 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
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.
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:
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
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
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:

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.
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.
(a)
Operating parameters for a Betz turbine;
U = velocity of undisturbed air; U4 = air velocity behind rotor
CP = power coefficient, CT = thrust coefficient
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:

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.
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
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.
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'.
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:
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:
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.
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:
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.
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:
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 = .
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):
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.
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
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
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.
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
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
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.
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
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)
The 2-D airfoil section lift, drag and pitching
moment coefficients are normally defined as:
A = projected airfoil area = chord x span = c l
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:
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
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.
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
Another wind turbine airfoil profile: Lift and drag
coefficients for a S809 airfoil at Re = 7.5x107
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.
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
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:
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!
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.
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.
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.
Momentum theory
We use the annular control volume, as before, with
induction factors (a, a’) being a function of radius r.
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.
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.
Diagram of blade elements
c = airfoil chord length; dr = radial length of element
r = radius; R = rotor radius;  = rotor angular velocity
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:
Overall geometry for
a downwind HAWT
analysis; U = velocity
of undisturbed flow;
 = angular velocity
of rotor; a = axial
induction factor
Blade section
geometry
p
T
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.
Figure shows the following section relationships:
If rotor has B blades, total normal force on section
at distance r from centre is:
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.
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).
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.
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:
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:
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:
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:
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.
metres
Blade chord for example Betz optimum blade
Blade twist angle for example Betz optimum blade
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 offdesign 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.
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.
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.
Momentum theory
From axial momentum:
From angular momentum:
Blade element theory
where thrust dT is same force as normal force dFN
Using the expression for the relative velocity:
The blade element theory equations become:
where  ’ = local solidity defined by  ’ = B c / 2  r
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:
These 2 equations, together with:
yield the useful relationships:
others may also be derived, including:
Now, we need to examine some solution methods.
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.
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:
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.
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.
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.
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
The tip loss correction factor affects the forces
derived from momentum theory. Thus, we end up
with:
Other equations become:
The equation
remains unchanged.
The power coefficient can be calculated from:
or
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.
(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.
(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.
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.
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
The local thrust coefficient at a given annular
section at radius, r, may be defined as:
so using:
the local thrust coefficient is:
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
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:
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:
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:
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.
Generalized rotor design procedure
(1) Rotor design for specific conditions
The previous analysis can be used in a
generalized rotor design procedure:
(a) Choose various rotor parameters and an airfoil.
(b) An initial blade shape is then determined using
the optimum blade shape assuming wake
rotation
(c) The final blade shape and performance are
determined iteratively considering drag, tip
losses, and ease of manufacture.
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.
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.
Suggested blade number, B,
for different tip speed ratios, 
4. Select an airfoil. If  < 3, curved plates can be
used. For  > 3 use a more aerodynamic shape.
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:
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:

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:


Cl and  can be found by
iteration or graphically:
Graphical solution
for  at section i
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:
ai
If ai > 0.4 , use Method 2
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
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:
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
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.
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.
(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:
Example of a Cp -  curve for a high tip speed
ratio wind turbine
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).
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.
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 (closedform) expression for finding  of the relative wind
at each blade element. It is assumed that the lift
and drag curves can be approximated by:
(1)
(2)
When the lift curve is linear and when small-angle
approximations can be used, it can be shown that
 is given by:
(3)
where
 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.
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):
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.
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.
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.
and finally …
Seals on Scroby Sands, Norfolk coast, UK
Scroby Sands Wind Farm (2004), 30 x 2 MW
turbines, 3km off the coast, powers 41,000 homes
Appendix A – Determining Weibull parameters
(k and c) from mean and standard deviation
wind speed values
Using the Weibull probability density distribution function p(U), it is possible to
calculate the average velocity as follows:
(Eqn. 1)
(Eqn 2)
Equation (1) can then be used to solve for c :
(2)
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.
Appendix B – Derivation of power coefficient
equation for blade analysis
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:
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:
Substituting:
into
we obtain the desired final equation:
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.