# Wind Power ```LECTURE 20
WIND POWER
SYSTEMS
ECE 371
Sustainable Energy Systems
1
TEMPERATURE CORRECTION
FOR AIR DENSITY


Using the ideal gas law, we can easily determine the
air density at other conditions
pV=nRT
(1)
Where,
p = absolute pressure (atm)
V = volume (m3)
n = mass (mol)
T = absolute temperature (K)
R = ideal gas constant = 8.2 e-5 (m3 atm /K mol)
2
TEMPERATURE CORRECTION
FOR AIR DENSITY


If we let M.W. stand for the molecular weight of
the gas (g/mol), then the air density is:
𝜌𝜌
𝑘𝑘𝑘𝑘
𝑚𝑚3
=
𝑛𝑛 𝑚𝑚𝑚𝑚𝑚𝑚
𝑔𝑔
�𝑀𝑀.𝑊𝑊
𝑚𝑚𝑚𝑚𝑚𝑚
𝑉𝑉(𝑚𝑚3 )
𝑘𝑘𝑘𝑘
−3
�10 ( )
𝑔𝑔
Substituting in p V = n R T yields
𝜌𝜌 =
𝑝𝑝&times;𝑀𝑀.𝑊𝑊
𝑅𝑅𝑅𝑅
3
TEMPERATURE CORRECTION
FOR AIR DENSITY

Since air is a mix of molecules of
N2 (78.08%)
 O2 (20.95%)
 Ar - argon (0.93%)
 CO2 (0.039%)
 Ne - neon (0.00185%)


28.02 (molecular weight)
32.00
39.95
44.01
20.18
M.W. of Air = 28.97 g/mol
4
TEMPERATURE CORRECTION
FOR AIR DENSITY
Reference Value
5
Power in the Wind




Remember that
Since 𝜌𝜌 if a function of temperature, the power
in wind is a function of temperature.
From the table on the previous slide, we see that
as the temperature increases, the power goes
down.
At 30 οC we loos 5% of the power in wind.
6
ALTITUDE CORRECTION FOR
AIR DENSITY


Air density is a function of pressure and
temperature
But, air pressure is a function of altitude


Need a correction factor to estimate wind power at
sites above the sea level
Consider a column of air with cross section A,
as shown in the following figure
7
ALTITUDE CORRECTION FOR
AIR DENSITY

Now:
𝜌𝜌
𝑘𝑘𝑘𝑘
�𝑚𝑚3 =1.225 KT KA
8
Book Correction for Altitude and
Temperature


𝜌𝜌
𝑘𝑘𝑘𝑘
�𝑚𝑚3 =
Where
353.1exp −0.0342 𝑧𝑧⁄𝑇𝑇
𝑇𝑇
Z = altitude in meters
 T is temperature in Kelvin

9
IMPACT OF TOWER HEIGHT



Since power in the wind is proportional to cube
of windspeed, a modest increase in windspeed
can significantly increase the power
To capture this higher windspeed, the tower
height should be increased
In the first few hundred meters above the
ground, wind speed is greatly affected by the air
friction as it crosses the earth’s surface
10
IMPACT OF TOWER HEIGHT

Smooth surfaces, such as calm sea, offer very
little resistance to wind


At the other extreme, surface winds are slowed
considerably by irregularities such as forests and
buildings


Variation of windspeed with elevation is modest
Variation of windspeed with elevation can be large
The impact of roughness on the earth’s surface
on windspeed is expressed in the following form
11
IMPACT OF TOWER HEIGHT
(v/vo) = (H/Ho)α


Where,
v = windspeed at height H
vo = windspeed at height Ho
(reference is usually 10 meters)
α = friction coefficient = 1/7 for open terrain
(rule-of-thumb)
= function of terrain over which wind blows
α is called the Hellman exponent or shear exponent.
12
IMPACT OF TOWER HEIGHT

The following table gives some representative
values
13
IMPACT OF TOWER HEIGHT

Impact of friction coefficient on windspeed and
1
power (v/vo) = (H/Ho)α
P = ρ Av
3
w
2
14
POWER CURVES




For heat engines, the maximum efficiency is
limited Carnot efficiency
For PV, the maximum efficiency is limited by
the band-gap of material
For fuel cells, the maximum efficiency is limited
by the Gibbs free energy
This concept also applies to WTGs
15
BETZ LIMIT



German physicist Albert Betz in 1919
formulated the maximum power that a turbine
can extract from wind
Wind that is approaching a wind turbine is
slowed down as a portion of its kinetic energy is
extracted by the turbine
The wind leaving the turbine has a lower
velocity and its pressure is reduced, causing the
air to expand downwind of the turbine
16
BETZ LIMIT
17
Why can’t a turbine extract all of
the energy in the wind?



If it did, the air would have to come to a complete
stop behind the turbine, which would prevent any
more of the wind from passing through the rotor. The
downwind velocity, therefore, cannot be zero.
The downwind velocity cannot be the same as the
upwind velocity, since that would mean the turbine
extracted no energy at all from the wind.
There must be some ideal slowing of the wind that
results in maximum power extraction from the wind.
18
BETZ LIMIT

The power extracted by the blades is equal to the
difference in kinetic energy between the upwind
and downwind divided by time
𝑃𝑃𝑏𝑏 =

1
𝑚𝑚
2
2
2
𝑣𝑣 −𝑣𝑣𝑑𝑑
𝑡𝑡
1 𝑚𝑚
=
2 𝑡𝑡
1
= 𝑚𝑚̇
2
2
𝑣𝑣 −
2
𝑣𝑣
2
𝑣𝑣𝑑𝑑
−
2
𝑣𝑣𝑑𝑑
Where 𝑚𝑚̇ is mass divided by time, or it is called
the mass flow rate
19
BETZ LIMIT




But,
𝑚𝑚̇ = 𝜌𝜌𝜌𝜌𝑣𝑣𝑏𝑏
Where A is the swept area of the rotor.
𝑣𝑣𝑏𝑏 is the windspeed through the rotor.
If we assume that 𝑣𝑣𝑏𝑏 is just the average of the
upwind and down wind, then
𝑣𝑣 + 𝑣𝑣𝑑𝑑
𝑚𝑚̇ = 𝜌𝜌𝜌𝜌
2
And
1
𝑣𝑣 + 𝑣𝑣𝑑𝑑
𝑃𝑃𝑏𝑏 = 𝜌𝜌𝜌𝜌
2
2
2
𝑣𝑣 −
2
𝑣𝑣𝑑𝑑
20
BETZ LIMIT


Letting 𝜆𝜆 be the ratio of downwind windspeed
to upwind windspeed
Then,
𝑣𝑣𝑑𝑑
𝜆𝜆 =
 𝑣𝑣𝑑𝑑 = 𝜆𝜆𝑣𝑣
𝑣𝑣
1
𝑣𝑣 + 𝑣𝑣𝑑𝑑
𝑃𝑃𝑏𝑏 = 𝜌𝜌𝜌𝜌
2
2
1
𝑣𝑣+𝜆𝜆𝑣𝑣
= 𝜌𝜌𝜌𝜌
2
2
2
2
𝑣𝑣 −
2 2
𝑣𝑣 − 𝜆𝜆 𝑣𝑣
2
𝑣𝑣𝑑𝑑
21
BETZ LIMIT
1
𝑣𝑣+𝜆𝜆𝑣𝑣
𝑃𝑃𝑏𝑏 = 𝜌𝜌𝜌𝜌
2
2
1
3 1+𝜆𝜆
= 𝜌𝜌𝜌𝜌𝑣𝑣
2
2
2
2 2
𝑣𝑣 − 𝜆𝜆 𝑣𝑣
2
1 − 𝜆𝜆
Define Rotor Efficiency CP
1
𝐶𝐶𝑃𝑃 = 1 + 𝜆𝜆 1 − 𝜆𝜆2
2
22
Power Extracted From the Wind
by the Rotor
Where
1
𝑃𝑃𝑏𝑏 = 𝜌𝜌𝜌𝜌𝑣𝑣 3 𝐶𝐶𝑃𝑃
2
1
𝜌𝜌𝜌𝜌𝑣𝑣 3
2
is the power in the wind
𝐶𝐶𝑃𝑃 is the efficiency of the rotor
23
BETZ LIMIT


To find the maximum rotor efficiency, we take
the derivative of Cp with respect to λ and set it
equal to zero to solve for λ
Then,
λ=

1
3
And maximum rotor efficiency will be
C p − max =
16
1
1
1
(1 + ) (1 − 2 ) =
= 0.593 = 59.3 %
27
2
3
3
24
BETZ LIMIT

Therefore the maximum theoretical efficiency of
a rotor is 59.3%


Betz Efficiency or Betz Law
The efficiency of a modern wind turbine blades
can approach about 80% of the Betz law

25
BETZ LIMIT

For a given windspeed

Rotor efficiency is a function of the rate at which the
rotor turns
If rotor turns too slowly, the efficiency drops off since the
blades are letting too much wind pass by unaffected
 If the rotor turns too fast, efficiency is reduced as the
follows


The usual way to show rotor efficiency is to
present it as a function of the tip speed ratio
26
BETZ LIMIT

The Tip-Speed-Ratio (TSR) is defined as
Rotor tip speed (rpm ) (πD / 60)
TSR =
=
Wind speed
v

A plot of typical rotor blade efficiency as a
function of Tip-Speed-Ratio is shown next
27
BETZ LIMIT
Range
28
29
Wind Speed and Efficiency




Modern wind turbines operate best when their
TSR is in the range of around 4–6,
The tip of a blade is moving four to six times
the wind speed.
 For maximum efficiency turbine blades
should change their speed as the wind speed
changes.
 Why wind turbines should use variable-speed
generators.
30
```