wind turbines introduction
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Wind Turbines
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Wind Energy - Wind Power
A moving air with velocity of has a kinetic energy of
If the moving air has a density , then the kinetic
energy per volume of air becomes:
V
][2
1 2 J mV E
][2
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2
m J V E V
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The Energy Extracting Stream-tube of a Wind Turbine
The volume flow rate per second through A is:
][3
S m AV V
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Power = Energy per Second
Power = Energy per Volume x Volume per second
Combining the above equations gives:
AV V Pair 2
2
1
][2
1 3 W AV Pair
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From the above derived equation:
• The power is proportional to the density . Density
varies with height and temperature
• In case of horizontal axis windmills the power is
proportional to the area (area swept by the blades)
and thus to R2.
• The power varies with the cube of the undisturbed
wind velocity . Note that the power increases eightfold
if the wind speed doubles.
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Maximum Power Coefficient
• The actual power extracted by the rotor blades is the
difference between upstream and downstream
powers.
• The maximum power extraction is reached when
the wind downstream is 1/3 of the undisturbed upstream velocity .
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The axial Stream tube model
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.2
1oextr V V rate flowmassP
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2
oV V Arate flowmass
22
max22
1o
o V V V V
AP
2
2max
323
21 V V
V V
AP
93
2
2
12
2
.max
V V
V AP
3
.max2
1
27
16
AV P
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Classification of Windmill Rotors
Horizontal Axis Rotors
Horizontal axis wind turbines (HAWT) have
their axis of rotation horizontal to the ground
and almost parallel to the wind stream. Most
of the commercial wind turbines fall under
this category.
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Advantages of horizontal axis wind turbines are:
• Low cut-in wind speed and easy furling
• They show relatively higher power coefficient
Disadvantage of horizontal axis wind turbines are:
• Generator and gearbox are to be placed over the tower
making its design more complex and expensive
• They need for tail or yaw drive to orient the turbine
towards wind.10
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Depending on the number of blades, HAWTs are classified
as single bladed, two bladed, three bladed and multi
bladed.
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Vertical Axis Rotor
The axis of rotation of vertical axis windmill is vertical to
the ground and almost perpendicular to the wind
direction.
The advantages of these windmills are:
• They can receive wind from any direction.
• Complicated yaw devices are not needed
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• Generator and gearbox of such systems can be housed
at the ground level which makes the tower design
simple and more economical
• Maintenance of these windmills can be done at the
ground level
The major disadvantage of these systems is that they
are not self starting.
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The Rotor
The windmill rotates because of forces acting on the
blades.
The cross sections of these blades have several forms.
Air flow over blades (airfoil) results two forces, Lift and
Drag.
Lift is the force measured perpendicular to the airflow
and drag is measured parallel to the flow
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Lift and Drag
• The lift force result in a force working in tangential
direction at some distance from the rotor center.
• This force is diminished by the component of the drag
in the tangential direction.
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Lift and Drag forces
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The product of the net tangential force multiplied by
the corresponding distance from the rotor center gives
the contribution of the blade element to the torque Q
of the rotor.
The rotor rotates at angular speed ,
srad n 2
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The power such a rotor extracts from the wind is
transformed to mechanical power.
This power is equal to the product of the torque and
the angular speed.
W QP
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Rotor Blade Design
The windmill rotates because of forces acting on the
blades.
The cross sections of these blades have several forms.
Air flow over blades (airfoil) results two forces, Lift and
Drag.
Lift is the force measured perpendicular to the airflow
and drag is measured parallel to the flow
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Lift and Drag
• Chord line: - it connects the leading edge and the
trailing edge of the airfoil.
• Angle of attack: - an angle between the chord line and
the direction of the airflow.
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To describe the performance of an airfoil independent
of size and velocity, Lift L and drag D are divided by
where, AV 2
2
1
3m
kg Density Air s
mVelocityFlowV
2)( m Length Bladechord Area Blade A
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The results of these divisions are called lift coefficient
and drag coefficient .
The amount of lift and drag depends on the angle of
attack. This dependence is a given characteristic of an
airfoil is always presented in and graphs.
lC
d C
AV
LC l 2
2
1
AV
DC d 2
2
1
lC d
C
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For the design of a windmill it is important to find from
such graphs the and values that correspond with a
minimum ratio.
lC
l
d
C
C
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Drag lift ratio, angle of attack and lift coefficient
for different airfoils
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The mechanical power can be expressed as the power
in air multiplied by a factor .
is called power coefficient and is a measure for the
success we have in extracting power from the wind.
PC
air pmech PC P
PC
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21 RV
PC mechP
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The local speed ratio is the speed U of the rotor at
radius r by the wind speed.
The speed-ratio of the element of the rotor blade at
radius R is called tip-speed ratio:
V
r
V
ur
V
Ro
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Calculation of blade chords and blade setting
• Design of the rotor consists in finding both values of
the chord and the setting angle ,
• The setting angle is the angle between the chord and
the plane of rotation.
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The following parameters must be found before
making the calculation of the chords and the setting
angles:
Rotor R: the radius: the design tip speed ratio
B: number of blades
Airfoil : design lift coefficient
: Corresponding angle of attack
d
ld C
d
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The choice of and B are more or less related as the
following table suggests.
d
B 1 6 – 20 2 4 – 12 3 3 – 6 4 2 – 4 5-8 2 – 3 8-15 1 - 2
d
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The type of load determines :
• Water pumping windmills driving piston pumps have 1 < < 2.
• Electricity generating wind turbines usually have 4 < < 10.
The radius of a rotor can be fixed by a formula,
Where: can be approximated to be equal to 0.1 for wind
pump and it could be changed to 0.15 to 0.2 for electric
generators.
d
d
d
3
21
V C
P R
p
pC 2
1
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The airfoil data are selected from Table 1. Four
formulas describe the required information about ,
and C.
Chord:
Blade setting angle:
Flow angle:
Design Speed:
r ld BC
r C
cos1
8
r r
r
r
1arctan
3
2
R
r d r
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Example: Find the chord C and setting angle of the blade
for a curved plate profile (10 % curvature) with the
given parameters:
2
6
37.1
d
B
m R
Rotor
o
d
lC Airfoil
4
1.1
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Soln.
To keep the lift coefficient at a constant value of , a
varying chord C and varying setting angle will result.
To keep the blade with a constant chord (for ease of
production) then the lift coefficient will vary along the
blade.
ld C
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Constant Lift Coefficient
By dividing the radius of the rotor at four points andapplying the above formulas the following values are
found
position r/R r(m) C (m) 1 0.25 0.34 0.5 42.3o 4 o 38.3 o 0.337 2 0.5 0.68 1 30.0 o 4 o 26.0 o 0.347 3 0.75 1.03 1.5 22.5 o 4 o 18.5 o 0.298 4 1 1.37 2 17.7 o 4 o 13.7 o 0.247
r r d
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The figure below shows the chord of the blade at the
four division points.
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Constant Chord
• The constant lift coefficient approach has a difficulty of
manufacturing as the twist varies discontinuously along
the blade. To avoid that a constant chord approach is
used.
• To have a constant chord the lift coefficient at different
positions along the blade will vary.
r l BC
r C
cos1
8
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The angle of attack also varies with variation in lift
coefficient. Therefore graph is needed to determine
values at different positions.
Choosing a chord of 0.324 m and three positions along
the blade, and applying the above formula the following
data are found.
lC
position r(m) C (m) chosen
1 0.5 0.324 0.73 35.9 1.23 6.4 29.5 27 2 0.86 0.324 1.26 25.7 1.10 3.6 22.1 23 3 1.22 0.324 1.78 19.5 0.91 0.2 19.3 19
r
r
lC
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The blade shape and setting angles of the blade are
shown below.
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