3. Caused by uneven heating of the earth due to factors
such as differences in location, physical structures,
material properties, etc.
i.e. land heats up more readily than water. The equator
heats up more readily than the poles.
Warmer air rises, and cool air rushes in to fill space as
warm air rises, creating wind.
Typically, the increase of wind speeds with increasing
height follows a wind profile power law, which predicts
that wind speed rises proportionally to the seventh root
of altitude
This is due in part to the lack of obstacles at the surface of
the earth.
4.
5. This cycle of the uneven heating of the earth
throughout a day causes global atmospheric
wind patterns.
On a smaller level, the heating and cooling
cycles of day and night also create wind in a
local area.
6. Wind moving is an example of fluid flow.
Can be described by Bernoulli’s principle, which is
a conservation of energy equation:
𝑃
γ
+ 𝑧 +
𝑉2
2𝑔
− ℎ 𝐿 = constant
Where P is pressure, 𝛾 is specific weight, z is vertical
height, V is velocity of the fluid, g is gravity and ℎ 𝐿 is
the head loss due to factors such as friction,
environmental losses (i.e. temperature) or entry losses
7. Lift occurs when there is a pressure difference
between an object’s bottom and top, such as an
airfoil, causing a lift force to be generated.
This pressure difference results from differences
in pressure along an airfoil due to constricted flow
lines at the top of the airfoil relative to the bottom
of the airfoil.
8. The air flow on the top of an airfoil is in a lower
pressure zone than the air at the bottom of the
airfoil due to higher wind speeds on the top of
the airfoil.
Bernoulli’s principle allows this to happen due
to the relationship between pressure, height
and velocity being equal to a constant (energy
is not created nor destroyed!)
9. Therefore, the generation of a lift force on an
object (like an airfoil) is possible due to winds!
The angle of attack of the airfoil is also
important in generating lift.
10. Varying the angle of attack relative to the
oncoming flow and the (baseline at which the
angle is measured from), can determine the lift
force generated by the wind.
Increasing the angle of attack to a certain extent
can impede the flow and reduce flow velocity
on the bottom of the foil, increasing pressure.
11. As angle of attack increases,
the coefficient of lift increases
until a certain point where
the upper airflow separates
significantly from the top
side of the airfoil (or wing, or
propeller), at which point the
velocity seen at the top of the
wing starts to decrease,
increasing pressure there.
This relationship varies from
design to design.
12. We can quantify lift force by the equation
𝐿 =
1
2
𝜌𝑣2 𝐴𝐶𝐿
Where L is the lift force, 𝜌 is the density of the
fluid (air in this case), 𝑣 is velocity of the fluid, A is
the planform area (projected area) of the airfoil or
wing, and 𝐶𝐿 is the lift coefficient, determined by
the Mach and Reynold’s numbers (Which we
won’t get into here).
13. Drag force is any force acting in opposition to
the direction of motion of an object. They
decrease fluid velocity relative to the solid
object in the fluid’s path.
Drag force can be quantified by the equation
𝐹 𝐷 =
1
2
𝜌𝑣2
𝐴𝐶 𝐷
Where 𝐶 𝐷 is the drag coefficient (relating again to
Reynold’s number, which relates to viscosity and
density of the fluid, which we won’t touch
upon….just realize it exists)
14. Varying parameters (such as angle of attack)
can vary the amount of lift and drag applied
to an airfoil.
This is especially important for our
discussion on wind turbines!!!
15. Getting lift force or drag force on a propeller
causes it to move. We can use this movement to
drive generators, machines, etc.
Lift force turbines are generally more efficient
than drag force turbines.
In general, we look to get power from the
wind. Power for a wind turbine is defined as
𝑃 = 𝑭 ∙ 𝒗
Where force and velocity are vectors operated
upon by the dot product.
16. AC power can be generated directly from the
wind turbine.
Use of an induction motor to generate this
power.
20. VAWT uses drag force
(savonious) or lift
(darrieus) to generate
power
Simplistic design
INSENSITIVE to changes
in wind direction
Less sensitive to changes
in wind speed
Needs a “boost” to start
Limited power generation
ability
Good low cost wind
power source
HAWT uses lift force to
generate power
More complex than
VAWT
Very sensitive to wind
direction
Sensitive to changes in
wind speed
Can generate much more
power than VAWT
Can be massive in size
and output
Costly investment and
upkeep
21. To extract power from the wind, the blades must
move the turbine part in the direction of the net
force.
This is not the case for a lift-based turbine, which
typically has a much higher maximum power
output.
VAWT vs. HAWT as a design consideration is an
analysis of cost vs. output needed.
Clearly, HAWT wins out in large-scale power
generation because of its output and the research
in propeller design so far
22. Combining the equation for lift and the power
equation from before we come up with an
equation that relates power to velocity
𝑃 =
1
2
𝜌𝐴𝑣3 𝐶𝐿 or 𝑃 =
1
2
𝜌𝐴𝑣3 𝐶 𝐷
Depending on if it is a drag-based or lift-based
system
The lift or drag coefficient is a value between 0
and .593 or the Betz Limit.
23.
24. The theoretical limit for any wind turbine’s
efficiency. If the Betz limit wasn’t a thing (i.e. we
had some strange “break the laws of physics” type
of turbine) the coefficients 𝐶𝐿 and 𝐶 𝐷 wouldn’t be a
factor.
It assumes a one directional flow pattern of air
through an “accuator disc” that extracts energy
from the wind flowing through it.
It assumes an IDEAL rotor with no hub and
infinite blades
The flow is INCOMPRESSIBLE and no heat is
transferred to the blades by the wind
25.
26. Take mass flow to be 𝑚 = 𝜌𝐴𝑣 and know it is
conserved. Also,
𝐹 = 𝑚𝑎 of course. I’m using A instead of S for cross
sectional area because why not.
Then, 𝐹 = 𝑚
𝑑𝑣
𝑑𝑡
= 𝑚 ∙ ∆𝑣, just taking the derivative to
get mass flow.
We have 𝑃 =
𝑑𝐸
𝑑𝑡
= 𝐹 ∙ 𝑑𝑥 = 𝐹 ∙ 𝑣 . Substituting for F
from above yields… 𝑃 = 𝑚 ∙ 𝑣 ∙ ∆𝑣
27. So we unpack and have…
𝑃 = 𝜌𝐴𝑣 ∙ 𝑣 ∙ (𝑣1 − 𝑣2)
And we also have an equation for the power generated
by from the kinetic energy of a flow of some mass
through a control volume!
𝑃 =
1
2
𝑚(𝑣1
2
− 𝑣2
2
)
And these are related by conservation of energy. It is
shown then that
𝑣 =
1
2
(𝑣1 − 𝑣2) (which I won’t do the math for!)
28. So then we can come back to the power based on kinetic
energy of the wind… which I will just show and not
derive because it’s a lot of steps.
𝑃 =
1
4
∙ 𝜌𝐴 ∙ (𝑣1 + 𝑣2)(𝑣1
2 − 𝑣2
2)
which (finally!) gives us
𝑃 =
1
4
∙ 𝜌𝐴 ∙ 𝑣1
3 ∙ (1 −
𝑣2
𝑣1
2
+
𝑣2
𝑣1
−
𝑣2
𝑣1
3
And this is maxed out when
𝑣2
𝑣1
is
1
3
. And substituting
this gives us the Betz limit of
16
27
or .593 for our turbine!
BOOMSLAM.
29. A single turbine has typically like 2 to 3 blades if
HAWT and some unique VAWTS can have more
than 3…
Usually we get the drag or lift coefficient from
experimental testing and not chugging through
math.
Calculating for VAWT systems is actually
EXTREMELY difficult in practice because it isn’t
directional like HAWT.
Usually the BEST designs get anywhere from .4 to
.5
30. What if we aren’t using a turbine system but
AN ENTIRELY DIFFERENT MECHANISM
FOR GENERATING POWER FROM THE
WIND.
31. Tether-based systems are being developed.
Betz still holds!
Take advantage of the wind profile power law,
which predicts wind speeds increases
proportionally with the seventh root of
altitude. (that’s a lot…you do the math)
Because power increases proportionally with
the cube of velocity, we are talking
HUMONGOUS GAINS in power with HAWP
technology.
34. What about storms and air traffic (planes and
birds) and the potential for accidents to occur.
Good research is lacking. Only two papers
published:
One by Archer and Caldeira who suggest jet
streams can generate 1700TW of power and have
no effect on the environment.
Another by Miller, Gans and Kleidon who suggest
only 7.5TW can be generated and that HAWP
would be “catastrophic” on the climate due to the
disruption of the global wind patterns
How much will this technology cost to create /
upkeep?
