Hello!
Since the early Persian empire,
Wind mills and turbines have been used for
many applications,
milling grain,
sawing wood,
pumping water,
providing torque for machines,
and of course producing electricity.
What all these different systems have in common
is that they extracted energy from the wind
and converted it into a mechanical torque
and power.
Today,
we will explore how the kinetic energy of
the wind is converted into torque and power,
through rotor aerodynamics.
We will try to answer questions such as:
How do we think of wind in relation to the
turbine?
How does a wind turbine rotor work?
How do we transform the linear motion of the
wind into a rotation of the turbine?
What happens to the air as it blows through
a wind turbine?
Does the wind slow down,
does it stop,
does anything happen to it at all?
How much power can be extracted from the wind
by a single turbine?
When thinking about the aerodynamics of wind
turbines,
you can think about scales.
The wind travels through thousands of kilometers
at an atmospheric scale.
When you think about the weather and local
wind speed,
you can think of the scale of your region,
of a few kilometers.
This local wind will then power a wind park,
which can be several kilometers long,
where turbines are spaced several hundreds
of meters to a kilometer apart.
The next scale is that of a single wind turbine.
As wind turbines have progressively grown,
the rotor and blade scale be easily over a
hundred meters,
the scale of a large passenger aircraft.
A blade is defined by a spanwise distribution
of aerodynamic profiles,
called airfoils,
which vary and thickness,
shape and performance.
The blade section will be a few meters long,
and define the surface of the blade.
The aerodynamics over the blade section are
defined by the shape of the blade,
and by the development of the viscous flow
close the surface,
creating a layer of a few millimeters.
In this boundary layer,
over the coating of the surface,
small perturbations appear,
small vortices generated by the forces on
the surface,
that are fractions of millimeters and which
grow,
defining the final aerodynamic performance
of the wind energy conversion system.
The transfer of energy from the atmospheric
scale down to the surface of the blade follows
these decreasing scales from thousands of
kilometers to the micro millimeters on the
surface of the blade.
These sub-millimeter vortices,
coalesce into a vortex sheet of the size of
the blade,
and are shed in wake sheets of the size of
the rotor,
that travel kilometers downstream until they
are diffused.
In a wind farm,
these sheets coalesce in a system of vortices
that exchange energy with the upper part of
the atmospheric boundary layer.
These massive systems of vortices can interact
in clusters of wind farms,
modifying the local wind climate.
When we design a wind farm,
a wind turbine,
a blade,
an airfoil,
our designs and models aim to connect these
scales of vortices and forces,
from atmospheric scale,
down to the micro scales of the blade surface,
and all the way back to the atmospheric scale.
An airfoil of a wind turbine blade is generated
to create one special type of aerodynamic
force:
lift.
While minimizing the other aerodynamic force:
drag.
Lift is the aerodynamic force that is perpendicular
to the wind sped that the airfoil perceives.
This perpendicular force is only possible
if you create these microscale vortices on
the surface of the airfoil,
that will create the wake of the wind turbine.
Drag is the aerodynamic force that is aligned
with the perceived wind speed.
This is the same principle that keeps either
an aircraft or a bird flying.
In a wind turbine,
the blade section will experience two sources
of wind:
first,
the natural wind flowing through the turbine,
which we also experience.
Second,
an apparent wind from the flight path of the
blade,
due to its rotation.
With these two wind directions,
we will have two components of lift.
The natural wind will generate a force perpendicular
to wind direction.
The direction of this force is in the direction
of rotation of the blade,
and propels the blade.
The wind due to the rotation of the blade
will create a force against this natural wind,
and is responsible for decelerating the wind.
At the airfoil and rotor scale,
we can see then the wind being decelerated
(losing kinetic energy and total energy).
It is this energy that is converted into the
mechanical energy of torque and power,
which is fed into the drive train through
the blade.
The blade of a wind turbine is designed to
take as much energy as possible for the least
cost (this includes costs of materials,
production,
installation,
maintenance,
and derivative costs in other systems).
Although many constraints are involved in
the design and optimization of a blade,
we will start by first focusing on a pure
aerodynamic design.
We assume that there is an optimal extraction
of energy,
related to an optimum deceleration of the
flow.
The most efficient way to do this is extract
the same percentage of energy from all wind
particles in the stream tube that crosses
the area of the disk drawn by the blades as
they rotate.
As kinetic energy is extracted from the tube,
the wind speed decreases and the streamtube
expands.
Because we have just two or three blades,
we try to approximate this,
and average the deceleration and forces the
blade applies to the wind.
As we look to the blade,
we can see that the outer regions of the blade
sweep a larger circumference;
this circumference increases linearly with
the radial position.
Therefore,
the loading along the blade as to increase
linearly with the length of the blade,
so that the average loading in each ring of
flow crossing the turbine is uniform.
But how much energy can w extract?
Until now,
we have discussed how the conversion of energy
occurs at the wind rotor,
focusing on the creation of torque.
But how much power can we extract?
Is there an optimum?
With a simple calculation,
we will show you that wind turbines have an
optimal operating condition beyond which they
cannot generate more energy.
Simply,
one cannot extract all the energy from the
air.
The first rule to determine how much energy
we can extract is to define that mass,
momentum and energy must be conserved along
our streamtube.
We can also think of the wind turbine as an
actuator disk of area A,
which uniformly decelerates the flow.
As the flow loses kinetic energy,
we can define that the turbine decelerates
the flow by an induction factor “a” and
wake system decelerate the flow.
We can define conservation of mas,
momentum and power as a function of the induction
factor a.
We obtain a simple expression for power as
a function of the deceleration of the wind.
Through further simplification,
one can define a thrust coefficient and a
power coefficient.
The thrust coefficient is the force exerted
by the turbine divided by the total impulse
of the flow.
Whereas,
the power coefficient is the power extracted
by the turbine divided by the total power
that the flow could have provided.
When plotting the power coefficient as a function
of induction factor,
we can see that the maximum power occurs when
the flow is decelerated by 1/3 at the actuator
disk,
resulting in a Cp of about 59%.
This is called the Betz limit,
named after the scientist who defined it.
At most,
a single wind turbine can extract 59% of the
power of the incoming wind.
We hope you enjoyed this introduction to the
aerodynamics of wind turbines.
