Behavioral Study of a Small-scale Model to Predict Horizontal
Axis Wind Turbine Behavior
Final year project report
Submitted in partial ful
fullfillment of the requirements
for the degree of
Bachelor of Science
In
Mechanical Engineering
By
Kamau Kingora
Reg No: F/18/1886/2007
Under the guidance of
Mr. Mutiso Mwaka
DEPARTMENT OF MECHANICAL AND MANUFACTURING
ENGINEERING
UNIVERSITY
NIVERSITY OF NAIROBI
May 2012
DECLARATION
I declare that this project report is my original work and has not been presented to any University
for any award.
……………………..
Kamau King’ora
…………………………
Date
F18/1886/2007
This paper has been submitted for examination with an approval of the University Supervisors.
Mr. Mutiso Mwaka
…………………………..
Date
i
Acknowledgement
This project would never have realized the current state without a hand from a few people whose
names are worth mentioning. I would like to extend my gratitude to Mr. Mwaka, my project
supervisor whose relentless support has made this project a success. Secondly, I acknowledge
Mr. James Wafula from department of Nuclear Science for lending us his model which reduced
my fabrication work appreciably.
I would also be most cruel if I don’t acknowledge the department of Mechanical Engineering for
facilitating this project financially and allowing the testing of the model in their wind tunnel.
Department of Design in school of Architecture can also not go unmentioned for its hand in
fabricating the blades and extending the curtsey of offering technical expertise in woodwork.
Finally I would like to express my sincerest of gratitude to Prof. G.O. Rading and Eng. Nyori for
their support in this work.
ii
Abstract
Compiled below is a report on characteristics of a horizontal axis wind turbine (HAWT). The
report seeks to design a small scale test setup that can be used to predict the behavior of a wind
turbine. The concept of similitude is invoked, to relate the model and the prototype.
Different blades (4cm, 5cm, 6cm and 8cm) were fabricated and the behavior of small model
(4cm and 5cm blades) investigated. Using the idea of similitude, the behavior of middle size
turbine model (6cm blades) was studied and a mathematical model established to predict the
behavior of large size model. The mathematical model was then used to predict the behavior of
large rotor size (8cm blades) and various improvements and validations made by comparing the
behavior of the large size model as predicted by the mathematical model and from the
experiments. The resulting mathematical model was then used to predict the behavior of a full
size prototype.
Eight experiments were designed to investigate the parameters affecting wind power harvesting.
These are: wind speed, blade form, blade size (rotor diameter), blade angle (angle of attack),
rotor solidity (number of blades), voltage characteristics and current characteristics.
From the experiments, it was seen that power is a function of blade geometry. Blades that are
able to produce high vorticity (hence lift) e.g. curved blades generally harvest more power from
the wind as opposed to their high drag counterparts. Power was seen to be proportional to the
cube to the speed of the wind. It was noted that there exist an optimum angle of attack for every
wind speed. This angle increases with increase in the speed of the wind. At 3m/s wind speed, the
optimum angle of attack was realized to be 450 a value which increase almost linearly to 700 at
10m/s wind speed. The current and voltage characteristics of the rotor were seen to be
independent of the rotor diameter. As the blade sizes were increased, electrical power harnessed
reduced while the total power (brake power) increased. This was attributed to the increase of the
rotor inertia as the size of the blade increases. It was seen that for the same Reynolds number,
power by a wind machine is inversely proportional to the squire of its diameter. For the same
rotor diameter, the power harvested was realized to be directly proportional to the cube of the
Reynolds number. The effect of solidity (number of blades) was also investigated and it was seen
that the power increase from three blades to four blades is only 2%.
iii
Contents
DECLARATION ............................................................................................................................. i
Acknowledgement ...................................................................................................................... ii
Abstract ...................................................................................................................................... iii
List of figures ................................................................................................................................ vii
List of tables ................................................................................................................................. viii
List of symbols ............................................................................................................................ x
CHAPTER 1: INTRODUCTION ............................................................................................... 1
1.1 Brief history ...................................................................................................................... 1
1.2 Wind as a source of energy ............................................................................................... 2
1.3 Wind power in Kenya ....................................................................................................... 3
1.4 Horizontal and vertical axis rotors .................................................................................... 4
1.5 Horizontal axis wind rotor (HAWT)................................................................................. 4
1.6 Problem statement .......................................................................................................... viii
1.7 Design approach............................................................................................................. viii
1.8
Scope ............................................................................................................................ 9
1.9 Purpose of the study ........................................................................................................ 10
1.10 Relevance and importance ............................................................................................ 11
CHAPTER 2: LITERATURE REVIEW .................................................................................. 12
2.1 Definitions....................................................................................................................... 12
2.2 Efficiency, power and torque characteristics .................................................................. 13
2.3 Kinematics of the wind ................................................................................................... 15
2.4 Momentum Theory ......................................................................................................... 16
CHAPTER 3: DEVELOPMENT OF THEORY ...................................................................... 21
3.1 Dimensional analysis ...................................................................................................... 21
iv
3.2 Rotating system ............................................................................................................... 25
CHAPTER 4: METHODOLOGY ............................................................................................ 30
4.1 Apparatus ........................................................................................................................ 30
4.2 Method ............................................................................................................................ 32
CHAPTER 5: EXPERIMENTS ................................................................................................ 34
5.1 Experiment 1: Measuring the Wind Speed of the Blower .............................................. 34
5.2 Experiment 2: Measuring the Output Power of a Wind Energy Converter in Relation to
the Form of the blades........................................................................................................... 36
5.3 Experiment 3: Measuring the Output Power of a Wind Energy Converter in Relation to
the Number of blades ............................................................................................................ 39
5.4 Experiment 4: Measuring the Output Power of a Wind Energy Converter in Relation to
the Angular Position of the blades ........................................................................................ 46
5.5 Experiment 5: Measuring the Current-Voltage Characteristic Curve of a Wind Energy
Converter with Constant Rotational Speed ........................................................................... 51
5.6 Experiment 6: Measuring the Current-Voltage Characteristic Curve at the Lift and
Resistance Rotor with Constant Wind Speed ....................................................................... 54
5.7 Experiment 7: Measuring the Output Power of a Wind Energy Converter in Relation to
Wind Speed ........................................................................................................................... 57
5.8 Experiment 8: Measuring the Output Power of a Wind Energy Converter in Relation to
the Size of blades .................................................................................................................. 59
5.9 Scaling............................................................................................................................. 67
CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS ............................................ 70
6.1 Summary and Conclusions ............................................................................................. 70
6.2 Recommendation for Future Study ................................................................................. 71
References ................................................................................................................................. 72
Appendices ................................................................................................................................ 74
Appendix A: results of further investigation of experiment 4. ............................................. 74
v
Appendix B: Voltage Characteristic data of rotor speed of 1000 rpm ................................. 78
Appendix C: Power for different blade sizes ........................................................................ 80
Appendix D: Blade design .................................................................................................... 82
vi
List of figures
Figure 1 Kijito Wind pump model .................................................................................................. 2
Figure 2: Mechanism of wind generation in Kano plane ................................................................ 3
Figure 3: Mechanism of wind power conversion of a HAWT. ...................................................... 4
Figure 4: (Prof. S.B. Kedare, 2003) The power of a wind rotor as a function of rotational speed
for difference wind speeds ............................................................................................................ 14
Figure 5: (Prof. S.B. Kedare, 2003)Torque of a wind rotor as a function of rotational speed for
difference wind speeds .................................................................................................................. 14
Figure 6: Change in kinetic energy of the wind across the rotor .................................................. 15
Figure 7: Control volume for actuator disc model ........................................................................ 17
Figure 8: Protection cover............................................................................................................. 30
Figure 9: Wind energy unit axial .................................................................................................. 31
Figure 10: Blades .......................................................................................................................... 31
Figure 11: Load ............................................................................................................................. 32
Figure 12: Apparatus..................................................................................................................... 32
Figure 13: Circuit diagram ............................................................................................................ 33
Figure 14: Setup for experiment 1 ................................................................................................ 34
Figure 15: Typical graph of power coefficient, Cp Vs tip speed ratio ......................................... 63
vii
List of tables
Table 1: Dimensional parameters of rotating system ................................................................... 25
Table 2: Matrix of measure formulae for rotating system ............................................................ 27
Table 3: Wind speed Vs blower graduation .................................................................................. 35
Table 4: Results of experiment 2 .................................................................................................. 37
Table 5: Experiment 2 done for different number of straight blades ............................................ 37
Table 6: Experiment 2 done for different number of curved blades ............................................. 37
Table 7: Results with two blades .................................................................................................. 40
Table 8 Results with three blades ................................................................................................. 40
Table 9: Results with four blades.................................................................................................. 40
Table 10: Rotational speed for maximum power .......................................................................... 41
Table 11: Results for 2 straight blades .......................................................................................... 42
Table 12: Results for 3 straight blades .......................................................................................... 42
Table 13: Results for 4 straight blades .......................................................................................... 42
Table 14: Optimum rotational speed in rev/min ........................................................................... 42
Table 15: Results for 2 curved blades ........................................................................................... 43
Table 16: Results for 3 curved blades ........................................................................................... 43
Table 17: Results for 4 curved blades ........................................................................................... 44
Table 18: Optimum rotational speed for various blade numbers .................................................. 44
Table 19: Relationship between power output and tilt angle for curved blades ........................... 46
Table 20: Power for various angle of attack ................................................................................. 47
Table 21: Optimum angle of attack for various wind speed ......................................................... 49
Table 22: Results of experiment 5 ................................................................................................ 52
Table 23: Result for lift rotor ........................................................................................................ 55
Table 24: Result for resistance rotor ............................................................................................. 55
Table 25: Output power in relation to the wind speed .................................................................. 57
Table 26: Blade size Vs Mass of the rotor .................................................................................... 61
Table 27: Results for 5 cm blade .................................................................................................. 61
Table 28: Results for 6 cm Blades ................................................................................................ 62
Table 29: Results for 8 cm blades ................................................................................................. 62
Table 30: Blade size Vs π4 ............................................................................................................ 67
viii
Table 31: Wind speed of 3 m/s ..................................................................................................... 74
Table 32: Wind speed of 4 m/s ..................................................................................................... 74
Table 33: Wind speed of 5 m/s ..................................................................................................... 75
Table 34: Wind speed of 6 m/s ..................................................................................................... 75
Table 35: Wind speed of 7 m/s ..................................................................................................... 76
Table 36: Wind speed of 8 m/s ..................................................................................................... 76
Table 37: Wind speed of 9 m/s ..................................................................................................... 77
Table 38: Wind speed of 10 m/s ................................................................................................... 77
Table 39: Rotor diameter of 10 cm ............................................................................................... 78
Table 40: Rotor diameter of 12 cm ............................................................................................... 78
Table 41: Rotor diameter of 14 cm ............................................................................................... 79
Table 42: Rotor diameter of 18 cm ............................................................................................... 79
Table 43: Results for 4 cm blades ................................................................................................. 80
Table 44: Results for 5 cm blades ................................................................................................. 80
Table 45: Results for 6 cm Blade .................................................................................................. 81
Table 46: Results for 8 cm blades ................................................................................................. 81
ix
List of symbols
The following symbols posses the denoted meaning unless otherwise stated in the report.
symbol
A
V
ρ
P
Cp
λ
R
ω
Ct
T
Hw
HT
D
CT
a
g
τ
Re
μ
K
η
N
F
U
I
Tg
R
Bp
Mp
M
α
meaning
Swept Area of the wind machine (m2)
Velocity of fluid (m/s)
Density of fluid (Kg/m3)
Power developed by wind rotor (W)
Coefficient of Power
Tip speed Ratio
Radius of the Rotor
Rotational speed of the rotor
Torque Coefficient
Torque
Actual power developed by the wind machine
Theoretical power develop by the wind machine
Diameter of the Rotor
Coefficient of Thrust
Fractional decrease in wind velocity across a wind machine
Acceleration due to gravity
Surface Tension force
Reynolds Number
Fluid viscosity
Fluid Compressibility
Efficiency
Rotor speed (r.p.m)
Force
Voltage (V)
Current (A)
Tacho Voltage (V)
Resistance (Ω)
Brake Power
Mechanical power
Mass of the rotor
Angle of attack
x
CHAPTER 1: INTRODUCTION
Wind is simply moving air. This is generated due to solar heating. Some places get hot quicker
than others. Air in hot places is heated up and its density reduces. The heated air rises up and is
replaced by cold air from low temperature places. This causes an air current and thence the
genesis of the wind.
1.1 Brief history
Since early recorded history, people have been harnessing the energy of the wind. Wind energy
propelled boats along the Nile River as early as 5000 B.C. By 200 B.C., simple windmills in
China were pumping water, while vertical-axis windmills with woven reed sails were grinding
grain in Persia and the Middle East.
New ways of using the energy of the wind eventually spread around the world. By the 11th
century, people in the Middle East were using windmills extensively for irrigation; returning
merchants and crusaders carried this idea back to Europe. The Dutch refined the windmill and
adapted it for draining lakes and marshes in the Rhine River Delta. When settlers took this
technology to the New World in the late 19th century, they began using windmills to pump water
for farms and ranches, and later, to generate electricity for homes and industry.
American colonists used windmills to grind wheat and corn, to pump water, and to cut wood at
sawmills. As late as the 1920s, Americans used small windmills to generate electricity in rural
areas without electric service. When power lines began to transport electricity to rural areas in
the 1930s, local windmills were used less and less, though they can still be seen on some
Western ranches. [5]
In Kenya, Bob Harris Engineering Ltd is the leading local manufacture of wind pumps known as
“Kijito”.
1
Figure 1 Kijito Wind pump model
1.2 Wind as a source of energy
Wind energy is a large renewable energy source. Global wind power potential is of the order of
11,000 GW. It is about 5 times the global installed power generation capacity. This excludes
offshore potential as it is yet to be properly estimated. [15]
1.2.1 Advantages of wind as a source of energy
Produces no direct emissions while generating power at a reasonable cost.
Wind energy is a resource that is available almost everywhere.
No greenhouse gasses
Expanding Wind Power development brings jobs to rural communities.
2
1.2.2 Disadvantages of wind as a source of energy
May create a lot of noise during harvesting
Wind can never be predicted
It covers a large area to harness
Visual disturbances are created in harnessing
1.3 Wind power in Kenya
Kenya has four main sources of renewable energy which do not require fuel to be imported.
These are:
Hydroelectric power, mainly on the Tana River;
Biomass in the form of biogas and alcohol from agricultural by-products;
Wind Power from the Kano Plains convection system in Nyanza Province;
Solar Energy, especially in the cloud-free arid zone of northern Kenya;
The people of the Lake-side plains lands of Kenya live within a potential source of power more
useful than an oilfield. Although they have no significant sources of water power or coal they
live in what can be described as a giant natural heat engine. This is the wind circulation system
caused by the difference in temperatures of the sun-baked Kano Plains and the cooler waters of
the Lake. Air rises from the plains from about 11.00 a.m. as they heat up; this pulls in air from
the lake and a substantial wind blows throughout the area until the land cools down and
temperatures equalize at about sunset. Unlike an oilfield this will not be exhausted as long as the
sun shines. [6]
Figure 2: Mechanism of wind generation in Kano plane
3
1.4 Horizontal and vertical axis rotors
Wind machines rotate about either a vertical or a horizontal axis. Most wind machines in
practical use today, are horizontal axis. Vertical axis machines have the advantage that they do
not need to be orientated to face the wind, since they present the same cross section to the wind
from any direction; however this is also a disadvantage as under storm conditions you cannot
turn a rotor away from the wind to reduce the wind loadings on it.
Most horizontal axis rotors work by lift forces generated when "propeller" or airscrew like blades
are set at such an angle that at their optimum speed of rotation they make a small angle with the
wind and generate lift forces in a tangential direction. Because the rotor tips travel faster than the
roots, they "feel" the wind at a shallower angle and therefore an efficient horizontal axis rotor
requires the blades to be twisted so that the angle with which they meet the wind is constant from
root to tip.
1.5 Horizontal axis wind rotor (HAWT)
As already mentioned, the rotor of a HAWT rotates about a horizontal axis.
1.5.1 How it works
The wind turns the blades, which spin a shaft from where the torque s harnessed. The wind
passes over both surfaces of the airfoil shaped blade but passes more rapidly over the upper side
of the airfoil. The pressure difference between top and bottom surfaces results an aerodynamic lift.
[1]
Figure 3: Mechanism of wind power conversion of a HAWT.
4
1.5.2 Types of HAWT
1.5.2.1 Upwind Wind Turbine
This is a type of wind turbine in which the rotor faces the wind. The wind starts bending away
from the tower before it reaches the tower itself. The basic drawback of upwind designs is that
the rotor needs to be made rather inflexible, and placed at some distance from the tower. In
addition an upwind machine needs a yawn mechanism to keep the rotor facing the wind. [4]
1.5.2.2 Downwind Wind Turbine
This is a horizontal-axis wind turbine in which the rotor is downwind (i.e. on the lee side) of the
tower. They may be built without a yaw mechanism. The rotor may be made more flexible so the
blades will bend at high wind speeds. [4]
Downwind variants suffer from fatigue and structural failure caused by turbulence when a blade
passes through the tower's wind shadow (for this reason, the majority of HAWTs use an upwind
design, with the rotor facing the wind in front of the tower).
1.5.3 Architecture / main parts of HAWT
Horizontal-axis wind turbines (HAWT) have the main rotor shaft at the top of a tower, and are
usually pointed into the wind. Most small turbines are pointed by a simple tail vane; although
there are now a number of more modern designs which are classed as down wind machines and
which require no tail vane. Large turbines generally use a wind sensor coupled with a servo
motor. [11]
Since a tower produces turbulence behind it, the turbine is usually pointed upwind of the tower.
Turbine blades are made stiff to prevent the blades from being pushed into the tower by high
winds. Additionally, the blades are placed a considerable distance in front of the tower and are
sometimes tilted up a small amount.
1.5.3.1 The rotor
The rotor is designed aerodynamically to capture the maximum surface area of wind in order to
spin the most ergonomically. The blades are lightweight, durable and corrosion-resistant
material. The best materials are composites of fiberglass and reinforced plastics.
5
1.5.3.2 Turbine blades
Lifting type: - These are the most efficiently designed, especially for capturing energy of strong,
fast winds
Dragging type: - these are most popularly used for water mills, as seen in the old Dutch
windmills. The blades are flattened plates which catch the wind. These are poorly designed for
capturing the energy of heightened winds.
1.5.3.3 The hub
Blades are connected to a hub, which is connected to a shaft. It serves the purpose of holding the
blades firmly as the harvest the wind energy and transmits that energy to the shaft in terms of
torque.
1.5.3.4 The transmission system
The transmission system can either be a gearbox which boosts the rotation speed of the blades
and transmits the power to where it is suppose to be used. A pulley system has also been used for
the same purpose and flywheel is normally attached to the rotor to increase the rotor inertia
hence smoothens the power quantity produced.
1.5.3.5 The rotor shaft
This is a shaft about which the hub carrying the blades is anchored. It carries the hub and rotates
together with the hub transmitting the power harvested to the gearbox.
1.5.4 Advantages of HAWT
Variable blade pitch, which gives the turbine blades the optimum angle of attack.
Allowing the angle of attack to be remotely adjusted gives greater control, so the turbine
collects the maximum amount of wind energy for the time of day and season.
The tall tower base allows access to stronger wind in sites with wind shear. In some wind
shear sites, every ten meters up, the wind speed can increase by 20% and the power
output by 34%. [8]
High efficiency, since the blades always moves perpendicularly to the wind, receiving
power through the whole rotation.
They are generally quite straight forward to design, install and maintain.
Blades are to the side of the turbine's center of gravity, helping stability.
6
Ability to pitch the rotor blades in a storm, to minimize damage.
Tall tower allows placement on uneven land or in offshore locations.
Can be sited in forests above the treeline. .
Most are self-starting.
1.5.5 Disadvantages of HAWT
The tall towers and blades up to 90 meters long are difficult to transport. Transportation
can reach 20% of equipment costs. [8]
Tall HAWTs are difficult to install, needing very tall and expensive cranes and skilled
operators.
Massive tower construction is required to support the heavy blades, gearbox, and
generator.
Reflections from tall HAWTs may affect side lobes of radar installations creating signal
clutter, although filtering can suppress it.
Their height makes them obtrusively visible across large areas, disrupting the appearance
of the landscape and sometimes creating local opposition.
HAWTs require an additional Yaw drive control mechanism to turn the blades toward the
wind.
HAWTs have difficulty operating in near ground, turbulent winds because their yaw and
blade bearing need smoother, more laminar wind flows.
7
1.6 Problem statement
Given a large wind turbine, subjected to a certain class of wind input, develop its geometrically
scaled-down test set-up such that the behavior of the large wind turbine may be predicted from
the test results of the small-scale set-up.
The development of the scaled-down test set-up involves the choice of right geometry and
materials for its structural and the aerodynamic subsystem’s components. A simplistic approach
to the problem of designing a small-scale test set-up is to geometrically scale down the
dimensions of the candidate large wind turbine and establish a correlation to match the
experimental results of the small-scale turbine to that of its larger counterpart.
Models are used to predict how a system will respond under certain specified conditions,
without having to actually build and test the physical system.
The model of a system
represents the behavior of the physical system. It is usually possible to improve the accuracy of
a model but usually the complexity of the model would increase as well and complete
accuracy is generally never achieved. In fact, some systems behaviors can only be accurately
determined if full scale model is built which is normally impractical. It is worth noting that, the
only way to be sure of the systems behavior is actually building system and testing it. Since this
is generally quite expensive, we usually strive to develop a model that is adequate for practical
purposes without being so complex as to be unmanageable.
1.7 Design approach
An attempt will be made to invoke the notion of similitude to arrive at decisions such as choice
of material, size of components, and orientation of various components…. The criterion for
similitude to find a single class of non-dimensional parameters, to which mathematical models of
both the small-scale and full-scale prototypes are invertibly related. In other words, similitude is
said to exist between the mathematical models of the large wind turbine and its small-scale test
set-up if for each of these mathematical models there exist invertible transformations to the same
non-dimensional equation. The assumption is that inter-connection between the wind turbine
subsystems remains the same for the candidate large wind turbine and its small-scale test set-up.
Various sizes of blades were made. Experiments were carried out on the smaller blades and their
behavior used to predict the performance of their larger counterparts. Experiments were then
8
carried out on the larger blades and compared with the predicted behavior from the small blades.
A mathematical model was then laid out and used to predict the behavior of full scale prototypes.
The various blades designed are shown in appendix D.
1.8 Scope
In a wind pump, like any other turbo machine, there are quite a number of parameters that can
be varied to alter a certain parameter in question, these include: •
•
The rotor diameter
•
Weight of the hub
•
Dead weight of the machine
•
Compressibility of fluid,
•
Density of fluid
•
Tip speed ratio
•
Speed of fluid
•
Solidity
•
Blade profile
•
Effect of gravity and other body
Change of pressure of fluid flowing
across the machine
forces
I would be carrying a giant if I tried to study all this parameters in the give short time and
constrained budget. I have therefore chosen to study two parameters and their effect on the
power. These are the blade angle with respect to the oncoming fluid i.e. the angle of attack, and
the rotor diameter. The effect of solidity, number of blades and tip peed ratio will also be
highlighted as it is impossible to came up with anything conclusive without mentioning them.
Fortunately, most of these parameters have been studied before and the results are available in
literature. Where necessary, these results will be quoted without proof or any empirical
reinforcement, simply for the purpose of completeness.
It is also worth noting that this project covers only the harnessing of the wind power. How that
power is transmitted and put to actual work there after is beyond the scope of this project.
9
1.9 Purpose of the study
1.9.1 General objective
The overall goal of this project is to improve community-built wind turbines for use in water
pumping for general use. In order to do so most effectively, the project is focused on modeling of
a wind rotor for laboratory test that could be used to predict the behavior of a wind machine in
the specified conditions.
This project was motivated by frequent failure of our locally made wind pump and lack of a
specific design for a certain areas. This project aimed at coming up with a tailor made design for
specific locations where wind energy can be economically used.
1.9.2 The Specific Objectives of the study
To investigate the parameter of scale in so far as dynamics is concern. For a given
physical system producing a unit power in a given set of condition, how will the quantity
in question, for this particular case power, change if the size of the unit is altered. If for
instance we double the linier dimension, how much extra power will we get from the
same unit if all the other factors are held constant? This paper will be making an attempt
to answer this question with respect to the wind pump.
Come up with a correlation that can be used to predict the behavior of the prototype given
the parameters and behavior of the model. Also important to this paper is the effect of the
angle of attack on the power generated. The unit of interest to this paper is a wind pump
for pumping water and the quantities of interest are the torque, power and speed. Other
parameters of interest to electricity generation such as the frequency will completely be
disregarded.
Establish the rotor characteristics for a tailor made design. Investigate Voltage, current
and power characteristics of the rotor.
10
1.10 Relevance and importance
Everywhere in the world today, a cry of environment victims is heard. Man has destroyed the
very environment he is staying in. in reiteration, the environment is responding with harsh terms
such as global warming, greenhouse effect, species extinction among other. A major contribution
to this destruction is the use of fossil fuel which not only gets depleted but also has emission that
is harmful to the inhabitant of the environment it is used in. As a result man is seeking an
alternative source of energy and wind is becoming increasingly popular. For our local use,
“Kijito” which is one of the most popular brand at the time of writing of this paper, has its pumps
faced by frequent breakdown and although poor maintenance is almost always blamed for this
failures, this paper intends to come up tailor made that will be more stable and more specific to
our local market.
Energy issue must be surely addressed if the country wishes to change from a current third world
to a middle income country. For vision 2030 to be realized, the country must allocated
appreciable resources on green energy. Wind energy being locally available and at good speeds
can answer the question of food security by its application on irrigation schemes, cattle ranches
and cheap power in rural areas.
In this report, the concept of similitude is applied to the mathematical model of large wind
turbine in order to establish the criterion for the design of its small-scale test set-up. To the best
of my knowledge, the similitude approach has not been applied to scale testing of large wind
turbines. This sets up a new area of study in so far as wind turbines are concern. Future study has
also been proposed.
Often, users of wind pumps do not understand which design best suits their needs. Manufactures
make suggestion based on prior experience. More often than not, consumers end up with a
product they did not want. This paper addresses how a tailor made design for specific location
can be done in the lab to help the consumer get the exact wind pump that answers to their need.
11
CHAPTER 2: LITERATURE REVIEW
2.1 Definitions
2.1.1 Power coefficient (Cp)
This s the ratio of the actual power output of the wind turbine (Hw) to the theoretical power in the
wind (HT).
Power = Force * Velocity.
Force = Rate of change of Momentum.
Momentum = Mass (M) * Velocity (V).
For a fluid of density ρ, flowing through a cross-sectional area A, mass flow rate M is given by: M= ρAV
Cp = Hw/½ρAV3
Average Force =½ρAV2
A= 0.25ΠD2
HT=½ρAV3
Cp = Hw/0.125ρΠD2V3----------------------2.1
Hw=Cp HT
2.1.2 Tip-Speed Ratio (λ)
When a wind machine is in motion, the tip of the blade covers larger distance than the rest of the
blades. The ratio of the tip speed or the blade to the speed of the oncoming wind is termed as the
tip speed ratio.
Consider a rotor of radius R, rotating at an angular speed ω, in fluid of speed V. if the angular
displacement in one second is T, then:
Speed of the tip ω
2πR
T
λ
ωR
2.2
V
2π
T
12
2.1.3 Thrust coefficient (CT)
This is the fraction of the wind thrust force that falls on the turbine.
Average Force of the wind =½ρAV2
Thrust force FT=½CTρAV2
CT=FT / ½ρAV2----------------------2.3
2.1.4 Swept Area (As)
This is the projected area of the wind turbine disc.
As=πR2----------------------2.4
2.1.5 Cut in speed
This is the wind speed at which the wind machine starts to produce any useful power. It is the
lowest speed in which power output of the turbine (Hw) is greater than zero.
2.1.6 Cut out speed
This is the wind speed at which the wind machine stops to produce any useful power. It is the
highest speed in which power developed by the wind turbine (Hw) is just zero.
2.1.7 Specific speed of a turbine
This is the speed in revolution per minute at which a turbine will operate if scaled down in
geometrical proportion to such a size that it will develop a unit power under a unit head.
A unit speed is the theoretical speed at which a given turbine will operate under a unit head.
2.2 Efficiency, power and torque characteristics
Any wind machine rotor can be characterized by plotting experimentally derived curves of
power against rotational speed at various wind speeds. Similarly the torque produced by a wind
rotor produces a set of curves. The maximum efficiency coincides with the maximum power
output in a given wind speed.
13
Figure 4: (Prof. S.B. Kedare, 2003) The power of a wind rotor as a function of rotational speed for difference wind speeds
Figure 5: (Prof. S.B. Kedare, 2003)Torque of a wind rotor as a function of rotational speed for difference wind speeds
14
2.3 Kinematics of the wind
2.3.1 Power from kinetic energy of the wind
Consider air molecules moving with velocity u, passing the area A during the short time dt fill a
volume dV.
dV = Audt
dm = ρAu dt
The mass of the air.
Where: ρ is the density of the air.
The kinetic energy of the air will be given by:
Ek=½ dm u2=½ ρAu3 dt
Power H, defined as energy per unit time:
H=½ ρAu3----------------------2.5
2.3.2 Change in kinetic energy
The power extracted by the turbine is equal to the difference of the wind power in front
of the turbine and the wind power behind the turbine.
H=H1- H2
H=½ρ(A1u13- A2u23)
From continuity,
A1u1 = A2u2= Au
Figure 6: Change in
kinetic energy of the
wind across the
rotor
H=½ρ A1u1(u12- u22) ----------------------2.6
2.3.3 Thrust
From Newton’s second law, the thrust T on the wind turbine is equal to the change in
Momentum dP, of the air in the time dt across the wind turbine.
15
T = dp/dt
Momentum is the product of the mass and the velocity:
p = mv
dm u = ρAu2 dt
T=ρ(A1u12- A2u22)= ρA1u1(u1- u2) ----------------------2.7
Since power is force * velocity, the power delivered by the wind, H, can be given by:H= Tu1 = ρA1u12(u1- u2) ----------------------2.8
2.4 Momentum Theory
From the first law of thermodynamics, it is impossible to design a machine (system) that would
produce more work than it utilizes. In fact by second law of thermodynamics, it is equally
impossible to design a machine that produces the same amount of work as it consumes. This
introduces the idea of efficiency. [12] There is no existing machine that is known to contradict
any of this axioms and a wind turbine is no exemption. We therefore take off by examining the
maximum efficiency one can derive from the wind turbine. It is also equally important to note
that for a wind turbine, it is impossible to take all the kinetic energy from the wind as this will
mean that the air behind the turbine will either be at stand still or use its internal energy to move
away from the turbine which contradicts the first law.
2.4.1 Betz limit
The maximum efficiency of a wind turbine was first theoretically determined by a German
engineer. [13]
Consider a control volume fixed in space whose external boundaries are the surface of a stream
tube whose fluid passes through the rotor disc, a cross-section of the stream tube upwind of the
rotor, and a cross-section of the stream tube downwind of the rotor. A simple schematic of this
control volume is given in the figure below.
16
Figure 7: Control volume for actuator disc model
Let:
Vi be the velocity of air at station i
Ai be the cross-sectional area at station i
Pi be the pressure at station i
ρ be the density of the air
T be the thrust at the rotor disc,
H be power extracted from the wind by the rotor,
For maximum efficiency, the following assumptions have been made.
Wind is steady, homogenous, and fixed in direction.
Air is incompressible, inviscid, and irrotational.
Both the flow and the thrust are uniform across the disc. The flow is uniform at the
upwind (station 0) and downwind (station 3) boundaries of the control volume.
(V1A1=V2A2)
The upwind and downwind boundaries are far enough removed from the rotor that the
static pressure at these points is equal to the unobstructed ambient static pressure. The
static pressure on the stream tube portion of the boundary is also equal to the
unobstructed ambient static pressure.
For a wind turbine rotor to act as an actuator disc, the rotor would have to be composed of an
infinite number of very thin, dragless blades. Station 1 is designated to be slightly upwind and
station 2 slightly downwind of the rotor.
17
By definition of a stream tube, air does not pass through the stream tube portion of the control
volume. Applying the conservation of mass to the control volume yields:
V0A0 =V1A1=V2A2 =V3A3. ---------------------------2.9
From continuity equation,
V1A1=V2A2 ---------------------------2.10
A1=A2 say =A---------------------------2.11
Therefore
V1=V2 say =V---------------------------2.12
The thrust at the rotor disc, T, can be found by applying the conservation of linear momentum to
the control volume in the axial direction.
T = ρ(A0V0 2− A3V3 2) ---------------------------2.13
Since the mass flowing through the stream tube is equal in all sections, M= ρAV
Therefore
T = ρAV(V0− V3 ) ---------------------------2.14
By Bernoulli’s equation,
T=A(P1- P2) ---------------------------2.15
By application of Bernoulli equation at station 0 and station 1, since there is no work done, then:
P0+ ρV02= P1+ cρV2---------------------------2.16
By application of Bernoulli equation at station 2 and station 3, since there is no work done, then:
P3+ ρV32= P2+ ρV2---------------------------2.17
Pressures P0 and P3 are identical (unobstructed ambient static pressure) then eliminating them
from equation VII we get:
T = ρA(V0 2− V3 2) ---------------------------2.18
Eliminating T from Eq. 2.14 we get:
ρA(V0 2− V3 2)= ρAV(V0− V3 ) ---------------------------2.19
V= ½(V0+ V3 ) ---------------------------2.20
Define an axial induction (or interference) factor, a, as the fractional decrease in wind velocity
between the free stream and the rotor plane:
a= (V0-V)/V0---------------------------2.21
V =V0 (1-a) ---------------------------2.22
18
V3 =V0 (1-2a) ---------------------------2.23
The velocity lost at the rotor plane,V0 – V is known as the induced velocity. As a increases from
zero, the downwind flow speed steadily decreases until, at a = ½, it has completely stopped and
the simple theory is no longer applicable.
Substituting for V3 from Eq. 2.23 , Eq. 2.18 can be rewritten in a more useful manner as
T = ρAV0 24a(1− a) ---------------------------2.24
H= ρAV0 34a(1− a)2---------------------------2.25
But we also know that power can be given by:
H= ½CpρAV0 3---------------------------2.26
Hence Cp can be given as:
Cp=4a(1-a)2---------------------------2.27
The theoretical maximum power coefficient from an idealized rotor, CPmax, known as Betz limit,
can be found by setting the derivative of Eq. 2.27 with respect to a equal to zero, and solving for
a:
41 3 1 0
Solving for a yields
!
CPmax=16/25
CPmax=0.59259
0.7
0.6
0.5
0.4
Series1
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
Graph 1: Cp against interference factor a
19
The maximum possible efficiency for an idealized wind turbine is roughly 59.3%. In practice,
three effects prohibit a real wind turbine from achieving this efficiency:
Rotation of the wake caused by the spinning rotor.
Finite numbers of blades.
Viscid flow causes nonzero aerodynamic drag.
This one-dimensional model is simple and does not describe the true nature of the physical flow
around wind turbines. If the rotor is to extract any power from the wind, the wind must slow
down as it passes through the rotor. An ideal wind turbine would have to slow the wind velocity
at the rotor plane to two-thirds of the free stream value if it is to extract power at maximum
efficiency. Thus, from continuity, the effective upstream area is less than the swept area of the
rotor and the area of the wake downstream is greater than the swept area of the rotor. For an ideal
wind turbine operating at maximum efficiency, the effective upstream area is two-thirds the
swept area and the area of the wake downstream is twice the area swept by the rotor.
20
CHAPTER 3: DEVELOPMENT OF THEORY
3.1 Dimensional analysis
Dimensional analysis is a method for reducing the number and complexity of experimental
variables which affect a given physical phenomenon, by using a sort of compacting technique. If
phenomenon depends upon n dimensional variables, dimensional analysis will reduce the
problem to only k dimensionless variables, where the reduction (n- k) = 1, 2, 3, or 4, depending
upon the problem complexity. Generally n - k equals the number of different dimensions
(sometimes called basic or primary or fundamental dimensions) which govern the problem. In
fluid mechanics, the four basic dimensions are usually taken to be mass M, length L, time T, and
temperature θ, or an MLTθ system for short. [12]
3.1.1 Similarity
A large part of study of fluid mechanics and its engineering applications comes from
experiments conducted on scale models. To obtain meaningful results from model tests, the
model must be similar to the full-scale version. [12]
Two systems are said to be physically similar in respect to certain specified physical quantities
when the ratio of corresponding magnitudes of these quantities between the two systems is
everywhere the same.
3.1.1.1 Geometric similarity
A model and prototype are geometrically similar if and only if all body dimensions in all three
coordinates have the same linear-scale ratio. [12] Geometric similarity is similarity of shape. The
characteristic property of geometrically similar systems is that the ratio of any length in one
system to the corresponding length in the other system is everywhere the same. This ratio is
usually known as the scale factor. All angles are preserved in geometric similarity. All flow
directions are preserved. The orientations of model and prototype with respect to the
surroundings must be identical.
3.1.1.2 Kinematic similarity
Kinematic similarity is similarity of motion (fixed ratio of velocities). This implies similarity of
lengths (i.e. geometric similarity) and, in addition, similarity of time intervals. Since
corresponding lengths in the two systems are in a fixed ratio and corresponding time intervals are
21
also in a fixed ratio, the velocities of corresponding particles must be in a fixed ratio of
magnitude at corresponding times. Moreover, accelerations of corresponding particles must be
similar. Geometrically similar systems are not necessarily kinematically similar. In other words,
kinematic similarity requires that the model and prototype have the same length-scale ratio and
the same time-scale ratio.
3.1.1.3 Dynamic similarity
Dynamic similarity is similarity of forces [12]. If two systems are dynamically similar, the
magnitudes of forces at similarly located points in each system are in a fixed ratio. Consequently
the magnitude ratio of any two forces in one system must be the same as the magnitude ratio of
the corresponding forces in the other system. Forces may be due to many causes. For perfect
dynamic similarity, there are many requirements to be met. It is usually impossible to satisfy all
of them simultaneously. However, in many instances, some of the forces have negligible effect
as compared to other forces hence becomes possible to concentrate on the similarity of
significant forces. For dynamic similarity to be realized, the model and the prototype must be
kinematically and therefore geometrically similar. However, kinematically similar systems are
not necessarily dynamically similar. In other words, Dynamic similarity exists when the model
and the prototype have the same length-scale ratio, time-scale ratio, and force-scale (or massscale) ratio.
Other types of similarities do exist but are deemed as irrelevant in so far as this paper is concern.
This include thermo similarities in which differences of temperature are in fixed ratio between
model and prototype, chemical similarity there is a fixed ratio of concentrations of reactants at
corresponding points among others. [12]
3.1.1.4 Incomplete similarity
In many instances, more than one force ratio is involved in a flow system. In general for these
systems, complete dynamic similarity is possible only when full size model are built. Since this
is usually impractical, incomplete similarity result where significant forces are compared and the
other forces effect neglected. [12]
22
3.1.2 Important dimensionless Numbers
Fluid in flow encounters the following forces among others:
Inertia force.
The important term in this force is the density ρ, of a substance.
Inertia force = Ma =(ρl3)(v2/l)=ρl2v2. ---------------------------3.1
Viscous force
The important term in this force the dynamic viscosity µ, of a substance.
Viscous force = µ(du/dy)A=µ(v/l)l2=µvl---------------------------3.2
Gravity force
The important term in this force is the acceleration due to gravity g, on a
substance.
Gravity force =Mg=ρl3g ---------------------------3.3
Pressure force
The important term in this force is the change of pressure (∆P) of a substance.
Pressure force = ∆PA= ∆Pl2---------------------------3.4
Surface tension force
The important term in this force is the surface tension τ.
Surface tension force = τl---------------------------3.5
Compressibility force:
The important term is the compressibility K, of the substance.
Compressibility force= KA=Kl2. ---------------------------3.6
23
3.1.2.1 Reynolds number (Re)
This is the ratio of inertia force to the viscous force. This ratio is significant in flow where the
viscous forces dominate as oppose to other forces.
Re= ρl2v2/µvl---------------------------3.7a
Re= ρlv/µ---------------------------3.7b
3.1.2.2 Froude number (Fr)
This is the ratio of inertia force to the gravitational force. This ratio is significant in flow where
the gravitational forces dominate as oppose to other forces.
Fr2= ρl2v2/ρl3g---------------------------3.8a
Fr2= v2/lg---------------------------3.8b
3.1.2.3 Pressure coefficient (Cp)
This is the ratio of pressure force to the inertia force. This ratio is significant in flow where the
pressure forces dominate as oppose to other forces.
Cp= 2∆Pl2/ ρl2v2---------------------------3.9a
Cp= 2∆P/ ρv2---------------------------3.9b
3.1.2.4 Weber number (We)
This is the ratio of inertia force to the surface tension force. This ratio is significant in flow
where the surface tension forces dominate as oppose to other forces.
We2= ρl2v2/τl---------------------------3.10
3.1.3 Buckingham Pi Theorem
This is a method of reducing a number of dimensional variables into a smaller number of
dimensionless groups.
If a physical process involves n dimensional variables with m basic dimensions, it can be reduced
to a relation between only k dimensionless variables or π’s. [21]
The reduction k= n – m.
24
3.2 Rotating system
The volumetric flow rate Q, in a rotating machine is believed to depend on the following:
Efficiency η,
Fluid viscosity µ,
Energy per unit mass of fluid flow H,
Torque T,
Power supplied P,
Thrust Force F,
Diameter of the rotor D,
Fluid
Rotational speed N,
Compressibility K.
Fluid density ρ,
Then we may write:
List of variables,
Q,η,H,P,D,N,ρ,µ,T,F,K.
Functional equation,
Q=f(η,H,P,D,N,ρ,µ,T,F,K)
Dimensional Parameters
Table 1: Dimensional parameters of rotating system
Quantity Symbol
Measure Formulae
Q
L3T-1
η
1
H
L2T-2
P
ML2T-3
D
L
N
T-1
ρ
ML-3
µ
ML-1T-1
T
ML2T-2
K
ML-1T-2
F
MLT-2
n = 11,
m=3
k=11-3=8
25
There are eight π groups in this relationship.
Repeating variables
D,N,ρ.
πi :
Define a quantity Xi with measure formulae:
[Xi]=MβLγTζ
Generalized equation
Πx=X laivbiρci
Dimensional equation
1= MβLγTζLai(T-1)bi(ML-3)ci
Solving the dimensional equation
Consider M:
ci = -β ---------------------------I
Consider T:
bi = ζ ---------------------------II
Consider L:
ai = -γ-3β
---------------------------II
Rewriting the generalized equation with solved indexes.
Πx=X D-(γ+3β)Nζρ-β
---------------------------**
Now, equation (**) is a general form of relationship in this process.
Let us write a measure formulae matrix in form of a table as shown below.
26
Table 2: Matrix of measure formulae for rotating system
Quantity Symbol
Measure
Formulae
β
ζ
γ
ai
bi
ci
Q
L3T-1
0
-1
3
-3
-1
0
η
1
0
0
0
0
0
0
H
L2T-2
0
-2
2
-2
-2
0
P
ML2T-3
1
-3
2
-5
-3
-1
D
L
0
0
1
-1
0
0
N
T-1
0
-1
0
0
-1
0
ρ
ML-3
1
0
-3
0
0
-1
µ
ML-1T-1
1
-1
-1
-2
-1
-1
T
ML2T-2
1
-2
2
-5
-2
-1
K
ML-1T-2
1
-2
-1
-2
-2
-1
F
MLT-2
1
-2
1
-4
-2
-1
From the table above, we can then write out the Π groups as follow:
π1 :
X1 = Q
π 1= Q /D3N
π2 :
X2 = η
π 2= η
π3 :
X3 = H
π 3= H /D2N2
27
π4 :
X4 = P
π 4= P /D5N3ρ
π5 :
X5 = µ
π 5= µ /DNρ
This can be rewritten as
π 5= DNρ/ µ
This is the Reynolds number (Re).
π6 :
X6 = T
π 6= T /D5N2ρ
This is the Torque coefficient (CT)
π7 :
X7 = K
Π7= K/ D2N2ρ
This can be rewritten as
Π7= ND/ (K/ρ)0.5
This is a form of Mach number (M).
π8 :
28
X8 = F
π 8= F/ D4N2ρ
This is the thrust coefficient (CF).
π 1=Ф(π 2, π 3, π 4 , π 5 , π 6 , π 7 , π 8)
Discharge is therefore a function of:
Q=Ф( η , π 3, π 4 ,Re ,M ,CF ,CT)
29
CHAPTER 4: METHODOLOGY
4.1 Apparatus
4.1.1 Blower/Wind tunnel
This is the part used for forcing the fluid across the model for the purpose of experimentation.
4.1.2 Anemometer
This is the apparatus used for measuring the speed of the wind.
4.1.3 Protection cover
For protection from the moving wings and protection from wings that may come off the hub with
high centrifugal forces in case of
improper fixation. Furthermore, the
protection cover helps adjust the tilt
angle of the wings.
The protection is inserted with the two
lower ends into the slots of the base
plate and fixed by the magnets ‘B’. The
scale ‘A’ is pointing towards space ‘E’
of the base plate.
With inserted straight wing in the hub of
the wind energy converter in position
Figure 8: Protection cover
90° move the protection cover alongside
the slots C in the base plate, until the
90° marking is in line with the wing.
Parts labels are: A. Scale in degrees for the adjustment of the tilt angle of the wings
B. Fixing magnets
30
4.1.4Wind energy unit axial
The wind energy unit axial consists of a direct current generator whose shaft has a hub for
accommodating the wings and a tacho-generator to determine the rotational speed. The hub is
suitable to accommodate 2, 3 or
four wings.
A. Wing hub
B. Location holes for the
wings
C. Connecting
sockets
generator and tacho-generator
D. Pin screw
E. Holes on the underside
of the wind energy converter for
mounting on the location pins
‘B’ of the base plate
Figure 9: Wind energy unit axial
4.1.5 Blades
Figure 10: Blades
4 pieces straight, 4 pieces curved. To be mounted to hub ‘A’ of the wind energy unit.
4.1.6 Load
For loading the wind energy unit axial and the Savonius rotor generator with the load resistance
and measuring voltage and current.
31
Parameters load resistance:
Resistance 100 Ω
Load capacity 2 W max
Parts labels: A. Connection generator
B. Connection multimeter for voltage measurement
C. Connection multimeter for current measurement
D. Knob, right turn increases resistance
Figure 11: Load
4.2 Method
4.2.1 General Setup
The apparatus should be generally arranged as follows:
Figure 12: Apparatus
32
4.2.2 Circuit diagram
Figure 13: Circuit diagram
33
CHAPTER 5: EXPERIMENTS
5.1 Experiment 1: Measuring the Wind Speed of the Blower
5.1.1 Purpose
This experiment is used to graduate the blower to set up a speed diagram that will be used for
subsequent experiments
5.1.2 Setup
fig 5.1
Figure 14: Setup for experiment 1
5.1.3 Method
Set up the experiment according to the figure above.
Screw the base to the anemometer. Switch on the anemometer and select the unit M/S (m/s)
using the button “M”.
Put the base on the base plate, with the hole onto the central location pin of the base plate.
Conduct the measurement in steps of whole scale units. Then prepare a diagram with the data
and draw a curve through the measuring points.
34
5.1.4 Results and analysis
Table 3: Wind speed Vs blower graduation
Graduation
Wind speed in m/s
2nd reading
0.00
5.60
6.40
8.20
9.20
10.00
10.60
11.00
11.20
11.40
11.60
1st reading
0.00
5.10
6.20
7.80
9.00
9.80
10.40
10.80
11.20
11.40
11.60
0
1
2
3
4
5
6
7
8
9
10
Average
0.00
5.35
6.30
8.00
9.10
9.90
10.50
10.90
11.20
11.40
11.60
Wind Speed in m/s wth graduation
Wind Speed in m/s
14
12
10
8
6
Wind Speed in m/s
wth graduation
4
2
0
0
5
10
15
Graduation
Graph 2: Wind speed diagram for calibrating the blower
35
5.2 Experiment 2: Measuring the Output Power of a Wind Energy Converter in
Relation to the Form of the blades
5.2.1 Purpose
This experiment is used to determine the influence of the form of the blades on the behavior of
turbine is so far as power production is concern.
5.2.2 Procedure
The experiment was set up as shown in the apparatus section (setup).
For the load, the resistance in load 2 was set to a predetermined value of 50Ω.
The straight blades with tilt of 60° were mounted.
The wind speed at the potentiometer of the blower was set at 8m/s.
The current and the voltage of the wind generator were determined.
The procedure was repeated with the concave blades and the convex blades n that respect and the
results were tabulated as shown below.
Three measurements were made with two wing forms.
The power output for different forms of the wings was determined.
Finally, the measurement is repeated with one of the curved blades turned by 180°.
All measurements are recorded in a table. The power was calculated as the product of voltage
and current.
36
5.2.3 Results and analysis
Table 4: Results of experiment 2
Power P= voltage V *current I,
Rotor Speed N= 1000 * U(Tg)/1.5
5.2.4 More experimental results
The setting for experiment three was changed and results tabulated as follows: Angle of attack:
600
Load resistance:
50Ω
Speed of the wind:
5m/s
Blade size:
4 cm
title
5.2.4.1 Straight blades
Table 5: Experiment 2 done for different number of straight blades
Number of blades
2
3
4
U in V
1.15
1.42
1.54
I in mA
22.1
27.4
29.6
P in mW
25.415
38.908
45.584
U(Tg) in V
1.37
1.71
1.84
Rotor speed (N) in rev/min
913
1140
1227
5.2.4.2 Curved blades
Table 6: Experiment 2 done for different number of curved blades
Number of blades
2
3
4
U in V
1.44
1.48
1.49
I in mA
27.7
28.5
28.6
P n mW
39.888
42.18
42.614
U(tg) in V
1.73
1.78
1.78
rotor speed in rev/min
1153
1187
1187
37
5.2.5 Discussion
The curved blades yield more power than the straight ones. This is because the curved wings
have high pressure difference between the two surfaces hence higher lift. This is due to high
velocity at the concave surface hence greater vorticity and hence circulation at the trailing edge.
It is inferred that the resulting lift of the blade is a geometrical problem and different profile
yields different results.
If the curved blades are setup in different orientations, the power resulting is even less. This is
because the lift created is in opposite directions and thus opposes each other. It is therefore
inferred that the orientation is equally orientation of the wing to the wind direction should in
such a manner to optimize the power i.e. concave. It is worth noting that although the lift on one
wing neutralizes the other in curved wings in opposite direction, the power produced is even less
than the power produced by straight wings. This is due to the fact that the losses due to friction at
the curved wing are not recovered in the curved wing in the opposite direction. In fact, it is true
to say that the difference in power produced between the straight blade and curved blade in
opposite directions is purely a function of drag. Following the results of this experiment
however, one cannot be dogmatic about this statement as the experiment was not designed to test
for draft n the blades.
Considering the ongoing discussion, it is seen that the curved blades give the best results.
Although the blade profile is more expensive to make, there is a generally good return in terms
of power as opposed to their straight counterparts. However, the increase in the number of
curved blade from 2 to three blade increases the power output by 5.76%. This is actually a
typical value in practice (4 to 6 %). The increase in power from 3 to 4 blades yields 0.95% which
is also a typical value in practice (less than 2%). This therefore does not warrant the cost incurred
and the extra complication incurred in adding an extra blades. Curved blades are therefore used
in three blade rotors for power production.
Straight blades however show a linier relation between the number of blades and the power
produced as well as the rotor speed. These are used for water pumping and are relatively easy to
fabricate.
38
5.3 Experiment 3: Measuring the Output Power of a Wind Energy Converter in
Relation to the Number of blades
5.3.1 Purpose
To investigate the relationship between the power output of the wind machine and the number of
blades.
5.3.2 Method
One series of measurements was conducted for every number of wings (two, three and four).
The wind speed, blade form and tilt angle were maintained throughout the series of
measurements.
In each series of measurements, the resistance of Load 2 was varied from 0 to 100Ω in steps of
20Ω .
The voltage and the current as well as the output voltage of the tacho-generator for each step
were determined.
The power output for each case was determined. Determined also was the rotor speeds with the
help of the rotational speed tacho voltage diagram.
For each number of blades, the interdependence of the power from the rotational speed was
shown in a joint diagram .
The three regression curves were also drawn.
Settings:
Converter principle: lift
Tilt angle:
75°
Number of blades:
2/3/4
Wind speed:
8 m/s
Form of blades:
curved
Load resistance:
0-100Ω (step 20Ω )
39
5.3.3 Results and analysis
Table 7: Results with two blades
R in Ohms
0
20
40
60
80
100
U in V
0.02
0.22
0.43
0.6
0.74
0.95
Two Blades
I in mA P in mW U(Tg)/V
11
0.22
0.13
10.8
2.376
0.33
10.3
4.429
0.53
9.6
5.76
0.67
9
6.66
0.87
9
8.55
1.03
n/min
87
220
353
447
580
687
Table 8 Results with three blades
R in Ohms
0
20
40
60
80
100
U in
V
0.07
0.52
1.13
1.76
1.94
2.10
Three Blades
I in
P in mW U(Tg)/V n/min
mA
27.4
1.918
0.34
229
25.4
13.208
0.77
513
27.0
30.510
1.42
947
27.9
49.104
2.04
1360
23.7
45.978
2.20
1467
20.4
42.840
2.34
1560
Table 9: Results with four blades
R in Ohms
0
20
40
60
80
100
U in V
0.11
1.04
1.68
1.94
2.05
2.13
Four Blades
Iin mA
P in mW
42.0
4.620
51.0
53.040
39.7
66.696
30.2
58.588
24.6
50.430
20.5
43.665
U(Tg)/V
0.55
1.55
2.07
2.24
2.30
2.33
n/min
367
1033
1380
1493
1533
1553
40
80.000
Generated power in mW
70.000
60.000
50.000
40.000
2 blades
30.000
3blades
20.000
4 Blades
10.000
0.000
0
500
1000
1500
2000
Rotational speed in rev/min
Graph 3: Output power in relation to the number of blades
Table 10: Rotational speed for maximum power
Number of blades
2
3
4
Rotational speed in Pmax in min-1
687
1360
1380
Power increase from two to three blades =
Power increase from three to four blades =
"#$%.&&
%.&&
()$"#
"#
Pmax in mW
8.55
49
67
' 100 = 473 %
' 100 = 36 %
5.3.4 More experimental results
This experiment was also carried out with the following setting to confirm the results.
Angle of attack:
600
Speed of the wind:
5m/s
Number of blades:
3
The results were obtained as shown below.
41
Straight blades
Table 11: Results for 2 straight blades
Resistance (Ω)
0
20
40
60
80
100
U in V
0.04
0.36
0.95
1.27
1.46
1.59
I in mA
18.5
18.6
23.2
20.5
17.7
15.2
Two Blades
Power in mW
0.740
6.696
22.040
26.035
25.842
24.168
U(Tg) in V
0.23
0.55
1.20
1.48
1.65
1.74
Speed in rev/min
153
367
800
987
1100
1160
Three Blades
Power in mW
2.128
18.666
40.119
38.038
34.645
29.120
U(Tg) in V
0.38
0.93
1.61
1.80
1.91
2.00
Speed in rev/min
253
620
1073
1200
1273
1333
Four Blades
Power in mW
3.924
39.516
43.416
38.654
33.734
29.920
U(Tg) in V
0.53
1.34
1.68
1.80
1.88
1.94
Speed in rev/min
353
893
1120
1200
1253
1293
Table 12: Results for 3 straight blades
Resistance(Ω)
0
20
40
60
80
100
U in V
0.07
0.61
1.29
1.54
1.69
1.82
I in mA
30.4
30.6
31.1
24.7
20.5
16.0
Table 13: Results for 4 straight blades
Resistance(Ω)
0
20
40
60
80
100
U in V
0.09
0.89
1.34
1.54
1.67
1.76
I in mA
43.6
44.4
32.4
25.1
20.2
17.0
Table 14: Optimum rotational speed in rev/min
Number of blades
2
3
4
Rotational Speed for Pmax in r.p.m
987
1073
1120
Pmax in mW
26.035
40.119
43.416
42
50.000
Power in mW
40.000
30.000
2 Blades
20.000
3 Blades
10.000
4 Blades
0.000
0
500
1000
1500
Rotational Speed in rev/min
Graph 4: Power vs Rotational speed
Power increase from two to three blades =
Power increase from three to four blades =
"*$(
(
"!$"*
"*
' 100 = 53 %
' 100 = 7.5 %
Curved blades
Table 15: Results for 2 curved blades
Resistance(Ω) U in V I in mA
0
0.05
24.0
20
0.86
43.1
40
1.37
33.3
60
1.60
26.1
80
1.75
21.1
100
1.85
18.0
Two Blades
Power in mW
1.200
37.066
45.621
41.760
36.925
33.300
U(Tg) in V
0.30
1.30
1.72
1.88
1.97
2.05
Speed in rev/min
200
867
1147
1253
1313
1367
Table 16: Results for 3 curved blades
Three Blades
Resistance
(Ω)
0
20
40
60
80
100
U in V
I in mA
Power in mW
U(Tg) in V
Speed in rev/min
0.12
1.00
1.39
1.56
1.67
1.76
48.0
49.2
32.5
25.2
20.2
16.7
5.760
49.200
45.175
39.312
33.734
29.392
0.60
1.50
1.72
1.83
1.90
1.94
400
1000
1147
1220
1267
1293
43
Table 17: Results for 4 curved blades
Resistance (Ω) U in V I in mA
0
0.20
81.7
20
1.05
52.4
40
1.43
34.4
60
1.60
25.8
80
1.71
20.7
100
1.77
17.2
Four Blades
Power in mW
16.340
55.020
49.192
41.280
35.397
30.444
U(Tg) in V
1.03
1.60
1.80
1.87
1.94
1.97
Speed in rev/min
687
1067
1200
1247
1293
1313
60.000
Power in mW
50.000
40.000
30.000
2 Blades
20.000
3 Blades
4 Blades
10.000
0.000
0
500
1000
1500
Rotational Speed in rev/min
Graph 5: Power Vs Rotational speed for curved blades
Table 18: Optimum rotational speed for various blade numbers
Number of blades
2
3
4
Rotational Speed for Pmax in r.p.m
1147
1000
1067
Power increase from two to three blades =
Power increase from three to four blades =
"#$"(
"(
&&$"#
"#
Pmax in mW
45.621
49.2
55.02
' 100 = 6.5 %
' 100 = 12 %
44
misplaced
5.3.5 Discussion
From this experiment, it was seen that the rotational speed of the rotor with which the maximum
power output is obtained, increases with increase in the number of blades but at a decreasing
rate. In fact, due to increase in torque for high number of blades, the speed generally reduces for
maximum power as the number of blades increases. It is therefore sensible to build few blade
rotor for high speed, low torque wind machine or many blades for low speed, high torque wind
machines like a water pump. The tradeoff between the torque and the speed is actually a question
intended purpose.
Another important inference is that the four blade machine delivers the highest power. However,
the power from 3 to 4 blades is 12% and this cannot substantiate extra cost and unnecessary
complications of attaching the fourth blade for electrical power production. For water pumping
which is of great relevance in so far as this project is concern, many blades are normally
proposed since for water pumping, brake power is measured as oppose to the electrical power.
Multi-blades wind turbines are generally more suitable for high torque low speed applications.
Three blades are proposed for electricity production since three blades are generally more
aerodynamically stable as opposed to the two blades.
45
5.4 Experiment 4: Measuring the Output Power of a Wind Energy Converter in
Relation to the Angular Position of the blades
5.4.1 Purpose
To investigate the relationship between the power output of the wind machine and the orientation
of the blades (angular position).
5.4.2 Method
The current and voltage with different tilt angles for two different wind speeds was measured.
The interdependence between output power and tilt angle for both wind speeds was shown in a
joint diagram.
Settings:
Converter principle: lift
Wind speed:
7 m/s and 10 m/s
Number of blades:
3
Load resistance:
50Ω
Form of wings:
straight/ curved
Blade size:
4 cm
Tilt angle:
0-90° in steps of 15°
5.4.3 Results and analysis
5.4..3.1 Curved blades
Table 19: Relationship between power output and tilt angle for curved blades
Wind Speed
Angle of Attack
0
90
750
600
450
300
150
00
U/V
0.13
0.83
1.16
0.77
0.46
0.09
0
7m/s
I in mA
2.4
16.0
22.0
14.7
8.7
1.6
0
10m/s
P/mW
0.312
13.280
25.520
11.319
4.002
0.144
0
U/V
1.06
2.25
1.69
1.16
0.58
0.24
0
I in mA
2.0
42.0
31.8
21.5
11.0
4.4
0.0
P/mW
2.120
94.500
53.742
24.940
6.380
1.056
0.000
46
100.000
Power in mW
80.000
60.000
7 m/s
40.000
10 m/s
20.000
0.000
-20.000
0
20
40
60
80
100
Angle of attack in degrees
Graph 6: Optimum power against optimum angle of attack
5.4.3.2 Straight blades
Table 20: Power for various angle of attack
Wind Speed
Angle of Attack
0
Power in mW
90
750
600
450
300
150
00
40.000
35.000
30.000
25.000
20.000
15.000
10.000
5.000
0.000
-5.000 0
U/V
0.00
9.40
13.20
10.00
4.90
0.00
0.00
7m/s
I in mA
0.0
0.5
0.7
0.6
0.3
0.0
0.0
P/mW
0.000
4.700
8.976
5.500
1.274
0.000
0.000
U/V
0.00
24.30
25.90
16.70
9.20
1.80
0.00
10m/s
I in mA
0.0
1.3
1.4
0.9
0.5
0.1
0.0
P/mW
0.000
31.104
35.483
14.696
4.600
0.162
0.000
10 m/s
7 m/s
20
40
60
80
100
Angle of attack
Graph 7: Power against angle of attack
47
5.4.4 Discussion
From this experiment, it was deduced that the optimum angle of attack for both straight blades
and curved blades is 600. However, for curved blades at high speed, lift tends to dominate so
much as oppose to draft hence the optimum angle of attack falls and in this case, it was optimum
at 750. It is worth noting however that the optimum angle of attack generally increases with
increase in the speed of the wind. At low wind speed the optimum angle of attack is low, i.e. the
optimum angle of attack approaches 450 which increase with increase on the wind speed to about
750 at 10 m/s.
It is therefore sensible to conclude that at any particular speed, there exists an optimum angle of
attack. In practice, however, the wind speed is always varying and it is impractically expensive
in a third world country using the available knowledge and technology to build a wind rotor with
blades that varies the angle of attack to optimize on instantaneous wind speed. The good news is
that for a particular location, it the speed of the wind remains fairly constant hence an optimum
angle of attack can be determined.
5.4.5 Further investigation
The above results motivated investigation of how exactly the optimum angle of attack varies
with speed. Instead of the accuracy offered in the above procedure, angle of attack to the
accuracy of 50 was investigated about the optimum range as proposed by the above experiment at
various wind speeds. The results were tabulated as shown in appendix A. However, the
following analysis is worth taking into account.
48
50.000
40.000
30.000
7 m/s
8 m/s
20.000
9 m/s
10 m/s
10.000
0.000
0
20
40
60
80
100
-10.000
Graph 8: Power against angle of attack for high wind speed.
Table 21: Optimum angle of attack for various wind speed
wind speed in m/s
3
4
5
6
7
8
9
10
optimum angle of attack
degrees radians
45
0.786
45
0.786
55
0.960
60
1.048
60
1.048
65
1.135
70
1.222
70
1.222
axis
49
Graph
9:
Optimum
angle
of
attack
Vs
wind
speed
Optimum angle of attack
Angle of attack in degrees
80
70
60
50
40
30
Optimum angle of attack
20
10
0
0
5
10
15
Wind Speed
From the graphs above, t is evident that the optimum angle of attack increase with the wind
speed. At wind speed of 3 m/s, the optimum angle of attack is 450 which increases to 700 at 10
m/s. Although better accuracy was not possible due to the limitation of the apparatus, from the
graph of optimum angle of attack Vs wind speed, the data suggested that this relation could
actually be linier. If the line of best fit is inserted in this data, it is found to be of equation that
follows.
y = mx + c
Where y = optimum angle of attack,
m is the gradient
m = 4 and c = 450
x is the wind speed
y= 4x + 450 for all 3<y<10
And c is a constant.
This equation is found to relate the data between optimum angle of attack and wind speed pretty
well in the wind speed range 3m/s to 10m/s
50
5.5 Experiment 5: Measuring the Current-Voltage Characteristic Curve of a Wind
Energy Converter with Constant Rotational Speed
5.5.1 Purpose
In order to understand the behavior of a wind power unit as a voltage source, one has to
determine the operational characteristic curve of the source and compare it with the characteristic
curves of other voltage sources. The characteristic curve measured here is the one of a direct
current generator in the wind converter, determined for a certain rotational speed of the
generator.
5.5.2 Method
Voltage and current were measured for different loads.
The rotational speed was kept constant for each load. For this purpose, the highest load was set at
Load 2 (short-circuit load Rload = 0Ω).
The desired rotational speed was then set with the load help of the blower (i.e. tacho voltage 1.5
V corresponds to 1000 min-1).
Current and voltage were recorded. Then the load resistance was slightly increased, resulting in
a change in rotational speed.
By reducing the wind speed, the rotational speed was brought back to the desired value. Current
and voltage were again measured.
The above procedure was repeated until the potentiometer of Load 2 has reached the right end
position.
Settings:
Converter principle:
lift
Wind speed:
adjusted
Number of wings:
4
Load resistance:
0-100Ω (step 10Ω)
Form of wings:
curved
Blade size:
4 cm
Tilt angle:
60°
Tacho-generator voltage:
1.5V
51
Voltage and current values was measured for the different loads and the results were recorded in
the measuring table as shown. The power output was calculated
The characteristic curve was drawn.
In the same diagram, the power developing in relation to the voltage was drawn.
5.5.3 Results and analysis
Table 22: Results of experiment 5
R in Ohms
0
10
20
30
40
50
60
70
80
90
100
no load
U in V
0.24
0.73
1.01
1.14
1.21
1.25
1.29
1.31
1.33
1.36
1.37
I in mA
128
76
48.8
36
28.8
24
20.6
18.1
16.2
14.5
13.2
P in mW
30.72
55.48
49.288
41.04
34.848
30
26.574
23.711
21.546
19.72
18.084
power in mW/ Current in mA
140
120
100
80
60
Current Voltage X-tic
40
Power Voltage Xtic
20
0
0
0.5
1
1.5
Voltage developed in volts
Graph 10: Voltage characteristic of the rotor at constant rotational speed.
52
5.5.4 Discussion
From the graph, it is seen that the power is maximum when the resistance is about 10Ω, it can be
shown that the power is maximum when the internal resistance of the generator is equal to the
external resistance. Following this argument, it is sensible to conclude that the internal resistance
of the rotor is about 10Ω.
5.5.5 Further investigation
It was also investigated how the voltage characteristic and the power characteristic varied with
the rotor diameter. An experiment was therefore set for various rotor diameters and tabulated as
shown in appendix B.
60.000
50.000
40.000
rotor diameter = 10 cm
30.000
rotor diameter = 12 cm
rotor diameter = 14 cm
20.000
rotor diameter = 18 cm
10.000
0.000
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
Graph 11: Voltage characteristic of various rotor diameters
From the above graph, it is seen that the voltage characteristic of a wind machine is actually
independent from is diameter. This is an intensive characteristic of any wind machine and so is
the current characteristic.
53
5.6 Experiment 6: Measuring the Current-Voltage Characteristic Curve at the Lift and
Resistance Rotor with Constant Wind Speed
5.6.1 Purpose
To determine the effects of rotational speed of the rotor on its operational behavior. Lift as well
as resistance rotors will be examined.
5.6.2 Method
The wind converter was operated as a lift rotor, then as a resistance rotor.
A constant wind speed was set at the blower. This wind speed was not changed throughout the
whole experiment.
With the wind energy converter as a lift rotor;
The load resistance was varied using the potentiometer of Load 2 in several steps from 0 to
100Ω.
The voltage and current values for each case were recorded in the measuring table as shown
below. The power output was also determined.
There is an additional measuring point for no-load operation (current is zero). For this purpose,
the connecting cables of the ampere meter were removed from Load 2.
This procedure was repeated with the wind energy converter as a resistance rotor. The blades
were set to 0° for that purpose. The protection cover was removed and the wind screen mounted.
The converter was turned to the right by 90° as compared to the previous experiment and
mounted on the two location pins of the base plate.
Settings:
Converter principle:
lift rotor
resistance rotor
Number of wings:
4
4
Form of wings:
straight
straight
Tilt angle:
45°
90°
54
Wind speed:
8.5 m/s
8.5 m/s
Load resistance:
0-100Ω in steps of 10Ω
Blade size:
4 cm
5.6.3 Results an analysis
Table 23: Result for lift rotor
R in Ohms
0
10
20
40
60
80
100
no load
U/V
0.11
0.4
0.67
0.92
1.04
1.15
1.22
Lift Rotor
I im mA (lift power)
51.4
40.1
33
21.6
16.8
13.8
11.8
P/mW
5.654
16.04
22.11
19.872
17.472
15.87
14.396
Table 24: Result for resistance rotor
R in Ohms
0
10
20
40
60
80
100
no load
U/V
0.04
0.16
0.23
0.32
0.39
0.37
0.42
Resistance Rotor
I im mA (lift power)
18
14.6
10.7
7
6.2
4.5
4.1
P/mW
0.72
2.336
2.461
2.24
2.418
1.665
1.722
55
60
Current in A
50
40
30
Lift Rotor
20
Resistance rotor
10
0
0
0.5
1
1.5
Generated Voltage in Volts
Graph 12: Current voltage characteristic of a wind rotor at constant wind speed
5.6.4 Discussion
The relation between the voltage and the current of a wind machine at constant wind speed is
seen to be linier for both drag and lift rotor. From the graph above, the lift rotor was seen to
posses superior current characteristic as oppose to its drag counterpart.
56
5.7 Experiment 7: Measuring the Output Power of a Wind Energy Converter in
Relation to Wind Speed
5.7.1 Purpose
In this experiment the output power of a wind energy converter in relation to wind speed shall be
examined.
5.7.2 Method
The wind speed was varied by adjusting the knob of the blower from 0 to 10 in steps of 1 scale
unit. The voltage and current values were measured for each case and recorded in the measuring
table.
The power output was determined.
Settings:
Converter principle:
lift
Tilt angle:
45°
Number of blades:
3
Wind speed:
0-10
Form of blades:
straight
scale
units in steps of 1 scale unit
Load resistance:
50Ω
5.7.3 Results and analysis
Table 25: Output power in relation to the wind speed
Wind speed in m/s
0.0
5.4
6.3
8.0
9.1
9.9
10.5
11.9
11.2
11.4
11.6
Voltage V
0.00
0.03
0.04
0.07
0.09
0.10
0.12
0.12
0.13
0.13
0.13
Current in mA
0.0
12.2
21.8
37.7
50.0
58.0
65.6
69.0
72.4
73.8
74.4
Power in mW
0.000
0.366
0.872
2.639
4.500
5.800
7.872
8.280
9.412
9.594
9.672
57
Wnd Speed Power Relaton
12.000
Power in mW
10.000
8.000
6.000
Wnd Speed Power
Relaton
4.000
2.000
0.000
0.0
5.0
10.0
15.0
wind speed in m/s
Graph 13: Variation of power with wind speed
5.7.4 Discussion
The graph of power Vs wind speed is seen to be exponential. This is due to the power law
relation between the wind power and the speed of the wind. For any wind machine, the power is
proportional to the cube of the wind speed.
58
5.8 Experiment 8: Measuring the Output Power of a Wind Energy Converter in
Relation to the Size of blades
5.8.1 Purpose
To investigate the relationship between the power output of the wind machine and the diameter
of the rotor.
5.8.2 Method
One series of measurements was conducted for every size of blade.
The wind speed, blade form and tilt angle were maintained throughout the series of
measurements.
In each series of measurements, the resistance of Load 2 was varied from 0 to 100Ω in steps of
20Ω.
The voltage and the current as well as the output voltage of the tacho-generator for each step
were determined.
The power output for each case was determined. Determined also was the rotor speeds with the
help of the rotational speed tacho voltage diagram.
For each size of wings, the interdependence of the power from the rotational speed was shown in
a joint diagram.
Settings:
Converter principle: lift
Tilt angle:
60°
Number of blades:
3
Wind speed:
8 m/s
Form of blades:
straight
Load resistance:
50Ω
5.8.3 Results and analysis
Atmospheric pressure P:
24.4 inches of Mercury
Atmospheric temperature T:
22.20 C
59
P = hρg
Where :
h = height
Assuming air to be a perfect gas:
ρ= Density of fluid (13600 Kg/m3 for
P= ρRT
mercury)
R= 287 j/KgK
g = acceleration due to gravity (9.81 m/s2)
T= 22.2 +273.15 = 295.35 K
1 inch = 2.54 cm
ρ= 8.2686 * 104/287 * 295.35
P= 24.4 * 2.54 * 10-2 * 13600 * 9.81
=0.9755 Kg/ m3
=8.2686 * 104 Pa.
.
5.8.4 Sample calculation
For Wind Speed U m/s, if the rotor generates V volts, I amperes, and the tacho voltage (Tg) in
volts, if also under this conditions the rotor makes N rpm, then:
Electrical Power P = IV Watts
N= Tg * 1000/1.5
Mechanical power Mp
Centripetal force = MV2/R
Power Mp = force * velocity
Taking that the mass acts at the center of gravity R/2, velocity there = V/2 = ω/2R
Mp= MV3/8R
V= ωR
Mp =Mω3R2/8
60
Brake power Bp
Cp = Bp/0.125ρπD2U3
Bp = P + Mp
Cp = Bp/½ρAU3
λ
2,-R
60U
A= 0.25πD2
Reynolds number Re = ρUD/µ
Table 26: Blade size Vs Mass of the rotor
blade size (cm)
8
6
5
4
Rotor mass (g)
9.3
7.7
7.1
6.2
Table 27: Results for 5 cm blade
5 cm Blades
U (m/s)
3
4
5
6
7
V (V)
0.64
0.97
1.33
1.64
1.85
I (mA)
13.5
20.4
26.0
33.4
37.5
P (mW)
8.640
19.788
34.580
54.776
69.375
Tg (V)
0.82
1.26
1.65
2.03
2.44
N
547
840
1100
1353
1627
λ
1.145
1.320
1.383
1.418
1.461
Mp (mW)
418
1517
3407
6345
11019
M= 7.1g
λ
Bp (mW)
427
1537
3442
6400
11088
Re
24,734
32,978
41,223
49,468
57,712
2,-R
60U
For U = 3 m/s
V= 0.64 V
2, ' 547 ' 0.06
60 ' 3
= 1.145
I = 13.5 mA
P = IV = 13.5 * 0.64 = 8. 64 mW
ω
Tg = 0.82 V
N = 0.82 * 1000/1.5 = 547 rpm
2πN
60
2π ' 547
60
= 57.35 rad/s
61
Cp
0.286
0.435
0.499
0.537
0.586
Mp =Mω3R2/8
= 427 mW
= 0.125 *7.1 *0.052 * 57.353
Re = ρUD/µ
=418.5 mW
µ/ρ = ν = 1.4555 * 10-5
Bp = P + Mp
Re = 3 * 0.12 / 1.4555 * 10-5
=418.5 + 8.64
=24734.
Table 28: Results for 6 cm Blades
6 cm Blades
U (m/s)
3
4
5
6
7
V (V)
0.70
1.02
1.27
1.55
1.86
I (mA)
12.7
17.0
22.5
28.0
35.0
P (mW)
8.890
17.340
28.575
43.400
65.100
Tg (V)
0.80
1.14
1.60
1.84
2.22
N
533
760
1067
1227
1480
λ
1.304
1.393
1.564
1.499
1.550
Mp (mW)
606
1755
4851
7377
12957
Bp (mW)
615
1772
4879
7421
13022
Re
28,856
38,475
48,093
57,712
67,331
Cp
0.303
0.369
0.520
0.457
0.505
P (mW)
7.080
12.628
21.816
33.528
46.500
Tg (V)
0.74
1.02
1.33
1.60
1.80
N
493
680
887
1067
1200
λ
1.550
1.603
1.672
1.676
1.616
Mp (mW)
1027
2690
5963
10382
14782
Bp (mW)
1034
2702
5985
10415
14828
Re
37,101
49,468
61,834
74,201
86,568
Cp
0.308
0.340
0.386
0.388
0.348
Table 29: Results for 8 cm blades
8 cm Blades
U (m/s)
3
4
5
6
7
V (V)
0.60
0.82
1.08
1.32
1.55
I (mA)
11.8
15.4
20.2
25.4
30.0
62
0.700
0.600
0.500
Cp
0.400
0.300
6 cm blade
0.200
5 cm blade
0.100
0.000
0.000
0.500
1.000
1.500
2.000
Axis Title
Graph 14: Cp Vs λ for 5cm and 6 cm blades.
5.8.5 Discussion
For us to effectively discuss the graph shown above it s paramount to consider a typical graph for
power coefficient Vs tip speed ratio shown below.
Figure 15: Typical graph of power coefficient, Cp Vs tip speed ratio
63
For any wind machine, if the tip speed ratio λ, is less than two, the power coefficient is normally
raising exponentially and this is known as the swirl region. For this case, the wind tunnel used
had a capacity of producing 15 m/s of speed and this could not effectively study this parameter.
The graph shown is therefore the swirl region only.
From the above data, it is evident that the braking (total) power generated by the wind machine
increases with increase in blade length (rotor diameter). This is because increase in blade length
increases the area across which power is harnessed hence longer blades traps more wind and
hence more power is harvested. However, it is worth noting that the electrical power actually
reduces with increase in blade size. This is due to the fact that as the blade size increases the
rotor inertia also increases and since electrical power is proportional to the number of rotation
the rotor rotates the conductor in a magnetic field, then the electric power reduces as longer
blades makes fewer rotations per time. Mechanical torque on the other hand is dependent on the
rotor speed as well as the rotor inertia (flywheel effect) hence mechanical power increases with
increase in the rotor diameter. Since for the model used the mechanical power dominated the
electrical power, the power basically increases with the increase in the rotor diameter.
From the above experiment, there seems to be a relationship between the wind speed and the
rotor speed as the rotor speed generally increases with the speed of the wind. Secondly, the
power of the wind machine increases with the Reynolds number. How these parameters
specifically related was investigated and analyzed in the next section.
5.8.6 Further investigation
The above experiment was repeated with the wind speed chosen such that the Reynolds number
remains the same. The values for different Reynolds numbers were tabulated as shown in
appendix C.
64
1800
Rotor speed N (rpm)
1600
1400
1200
1000
5 cm Blades
800
600
6 cm Blades
400
8 cm Blades
200
0
0
2
4
6
8
Wind speed U (m/s)
Graph 15: Rotor speed Vs Wind Speed
The graph of rotor speed Vs wind speed is seen to be linier.
N = mU + C
Where m is the gradient and C is a constant.
It is therefore possible to extrapolate these data to determine the rotor speed at higher wind speed
which was impossible to achieve with our wind tunnel. These graphs can be described by the
following set of equations:
N = 267.3U - 243.3 ------ for 5 cm blade size
N = 236U - 166.6 ------ for 6 cm blade size
N = 180U - 34.66 ------ for 8 cm blade size
Of importance to mention is that on ideal case, the N intercept, C, which is the rotor speed at the
wind speed of 0 m/s, the speed is actually negative and it signifies that the rotor does not actually
start to rotate at 0 m/s wind speed. The wind speed must achieve a specific value for the value for
the rotor to start to rotate. This speed is known as the cut in speed i.e. the wind speed at which
the rotor s just in the verge of rotating.
65
The gradient of the graph, m, also reduces with increase in the rotor diameter but how it exactly
changes with the rotor diameter is still a subject of research.
100,000
90,000
axis lable
Power (Bp) in mW
80,000
70,000
60,000
50,000
5 cm Blades
40,000
6 cm Blades
30,000
8 cm Blades
20,000
10,000
0
0
5000
10000
15000
20000
Reynold's Number Re
Graph 16:Power Vs the Reynolds number
From the above graph, the relationship between power and the Reynolds number is seen to be
exponential.
Bp = kRen
Where k and n are constant.
In fact, the above graphs can be described by the equations below.
Bp = 4233.Re0.311------ for 8 cm blade size
Bp = 4895.Re0.274------ for 6 cm blade size
Bp = 4965.Re0.261------ for 5 cm blade size
The constant k seems to increase with reduction in the rotor diameter while n increases with
increase in rotor diameter. The magnitude of dependence of these parameters was also not
covered in the scope of this paper.
Of interest to realize also is the fact that π4 = P /D5N3ρ is a constant for a particular blade size. A
graph of π4 Vs wind speed is given in the graph below.
66
Table 30: Blade size Vs π4
π4 = P /D5N3ρ
Blade Size (cm)
4
5
6
8
0.149173
0.107648
0.077102
0.04673
0.12
0.1
π4
0.08
0.06
5 cm Blades
0.04
6 cm Blades
8 cm Blades
0.02
0
0
2
4
6
8
Wind speed in m/s
Graph 17: π4 for various speeds
The graph of π4 Vs wind speed shows that the shows that for a particular blade size, the value of
π4 is constant. However, this value decays with increase in the blade length and this may prove
useful during scaling. For the sake of this paper however, it is naïve for one to be dogmatic about
the scaling effect of this value as the width of the prototype may play an important role.
5.9 Scaling
Experiment was performed on a 5cm and 6 cm blade and the result used to predict the behavior
of the 8cm model.
From the above experiment, it was seen that the power is inversely proportional to the squire of
the rotor diameter. This was attributed to the fact that power is directly proportional to wind
speed. For the same Reynolds number, as the diameter of the rotor reduces, the speed of the wind
must increase to compensate hence more power is produced.
67
5.9.1 Sample scaling on rotor diameter
At 4m/s wind speed, the 6cm model produced 1.772 watts. The Reynolds number at this speed is
38475.
Now, the 8cm model at 3m/s produces a value of 1.034 watts. The Reynolds number at this
speed is 37101 which is close enough to the 6cm blade at 4m/s.
If we use the 6cm blade to predict the power produced by 8cm model.
P α 1/ D2
P8 *D8 / D6 = P6
P8 = 1.772 * (6/8)2
= 0.99675 watts.
0.99675 is an acceptable prediction of the 1.034 watts predicted by the experiment.
Therefore, power is inversely proportional to the squire of blade length for the same Reynolds
number.
P α 1/ D2
5.9.2 Sample scaling on Reynolds number
The next point to note in so far as scaling is concern is that the power is proportional to the cube
of Reynolds number. This is attributed to the fact that Reynolds number is propotional to wind
speed and it is impossible to contradict the fact that the power harvested by any wind machine is
proportional to the cube of the wind speed and hence to the cube of the Reynolds number.
Supported by the appendix C data however, it is suggested for the sake of this paper that
Reynolds number be use for scaling purpose as it takes nto account the changing density of air as
well as its kinematic viscosity.
P α Re3
68
5.9.3 Sample scaled wind turbine.
Suppose we want to investigate the parameters or a 2 m blade length prototype in 4 m/s wind
speed.
Form the table of model data of 5 cm blade at 5 m/s, the Reynolds number Re = 41223. The
model generates power of 3.442 watts.
Since P α 1/D2
Pm *Dm / Dp = Pp
Pp = 3.442 * 5/200
= 0.08605 watts
Now we need to investigate at what wind speed this is generated. Assuming that the hub is of
negligible dimension as compared to the length of the blade,
D = 4m
P α Re3
Re = ρUD/µ
Re α U
µ/ρ = ν = 1.4555 * 10-5
P1 = P2 * (U1/U2)3
41223 = U * 4 / 1.4555 * 10-5
P2 = 0.08605 * (4/0.15)3
U= 0.15 m/s
= 1.63 KW
For 4 m/s
This method worked well enough for my 8 cm blades but it is worth noting that the effect of the
width is not captured as the blades used had similar width. The effect of the width and the blade
profile is suggested in future scope.
Also the effect of number of blades on the power is not captured in this method and it is assumed
that the prototype also has three blades. The effect of solidity is also suggested in future scope.
69
CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS
6.1 Summary and Conclusions
From the series of experiments performed, various conclusions were drawn.
Curved blades generate more power than their straight counterparts and this was attributed to
their ability to generate higher lift. The power harvested by the various forms of blade was
concluded to be a function of shape. Curved blades are hence recommended over straight ones.
The power harvested by a wind rotor increases with increase in the number of blades. However,
the increase in power harvested with increase in the number of blades increases at a decreasing
rate. From three to four blades, the power increases by only 2% and this does not justify the cost
of additional blade. However, for high torque low speed machine, many blades are suggested e.g.
in water pumping as many blade increases the torque due to flywheel effect. For high speed and
low torque applications e.g. electric power generation, three blade rotors are recommended.
The optimum angle of attack increases with increase in wind speed. At 3m/s the optimum angle
of attack is 450 and this increases almost linearly to 700 at 10m/s. As the angle of attack increases
form 00, the power harvested raises exponentially and as the optimum angle is approached, the
power harvested remains almost constant before reducing exponentially to 0 at 900.
The voltage characteristic (Voltage Power relationship) of a wind rotor was seen to give an
indication of the internal resistance of the rotor. The internal resistance of the rotor was
concluded to be 10Ω. The power harvested by a wind rotor was concluded to be maximum when
the internal resistance of the rotor equals the external resistance. The voltage characteristic of the
rotor was also concluded to be independent of the rotor diameter.
The current characteristic (Current Voltage characteristic) was seen to have a linier relationship
and also size (rotor diameter) independent.
The power harvested by a wind rotor was concluded to obey the power low. As the wind speed
increases, the power harvested by a wind machine increases exponentially.
The electrical power produced decreases with increase in the size of the blade. This is also
supported in literature since the additional weight of the rotor makes it difficult to rotate.
70
However, the braking power (total power) harvested by the rotor increases due to flywheel effect
and hence high toque. For the same Reynolds number, the power harvested by the rotor was seen
to be inversely proportional to the rotor diameter. For the same rotor diameter, the power
harvested was concluded to the cube of Reynolds number.
6.2 Recommendation for Future Study
The effect of smoothening the edges of the blades is suspected to have some effect of the power
generated by a particular blade form and this is suggested to be accomplished in future. If also
the blades could be made blade thins out from the base where it is attached to the hub to the tip is
also a subject to be investigated. The relationship between power produced by various forms is
also suggested as a future scope.
In future, it is suggested that rotors be acquired that can accommodate many blades to effectively
study the effect of solidity (number of blades). For the case of this paper, only four blades could
be accommodated and the effect of solidity could therefore not be thoroughly investigated.
The equipment available could only measure angles to accuracy of 150. An improvisation of a
combination set to achieve accuracy of 50 and this is good enough for practical wind pump.
However, for the purpose of study, it is suggested that a more angle measuring equipment be
acquired.
It is suggested in the future that the relationship between the wind power and the speed of the
wind be established. From literature, power harvested is said to be proportional to the cube of
wind speed. However, theory does not capture the effect of wind properties such as level of
turbulence and different wind tunnels have different blower’s characteristic hence different wind
properties.
It is worth noting that the scaling length considered is only length while the width of the blade
was maintained constant for all blades. In future, it is suggested that the effect of varying the
width of the blade be investigated.
71
consistence in
treferences
References
1. Anish Bhattacharya. (2010). THE EFFECT OF BLADE ANGLE AND SIZE ON WIND
TURBINE PERFORMANCE.
2. Cohen, P. K. (2004). Fluid Mechanics. Dania, Florida: Elsevier Academic Press.
3. Dhar, S. (2006). Development and Validation of Small-scale Model to Predict Large
Wind Turbine Behavior. BOMBAY: INDIAN INSTITUTE OF TECHNOLOGY .
4. DURAN, S. (2005). COMPUTER-AIDED DESIGN OF HORIZONTAL-AXIS. THE
GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES.
5. (2011). History of Wind Energy. U.S.A.: U.S. Department of Energy.
6. Ingram, G. (2009). Basic Concepts in Turbomachinery. Ventus Publishing Aps.
7. Jirutitijaroen, D. P. Horizantal axis wind turbines. Singapore: National University Of
Singapore.
8. Jonkman, J. (2003). Modeling of the UAE Wind Turbine for Refinement of FAST_AD.
year
Colorado: National Renewable Energy Laboratory .
9. Kulunk, E. Aerodynamics of Wind Turbines . U.S.A: New Mexico Institute of Mining and
Technology .
10. Lemmer, E. C. (2009). Wind-Electric Pump System Design. Stellenbosch University.
11. Main components of a wind turbine.
remove
12. Massey, B. (2001). Mechanics of Fluids. London: Taylor & Francis.
13. Mathew, D. S. (2006). Wind Energy. Tavanur, India: Springer.
14. Niebur, J. L. (2009). Experiments on Wind Energy. Germany: WindTrainer.
15. Prof. S.B. Kedare, P. S. (2003). Wind Energy Conversion Systems. BOMBAY: INDIAN
INSTITUTE OF TECHNOLOGY,.
16. Rastogi, M. T. (1982). WindPump Handbook Pilot edition. Paris: UNESCO.
72
year
17. Rotating Machinery. (2005).
18. Rudramoorthy, C. K. (2007). Flud Mechanics and Machinery. New Delhi: New Age
International Publishers.
19. Tangler, J. L. (2000). The Evolution of Rotor and Blade Design. Palm Springs,
California: National Renewable Energy Laboratory.
20. The Evolution of Rotor and Blade .
year
21. White, F. M. (1991). Fluid Mechanics. Rhode Island: McGraw-Hill.
73
Appendices
Appendix A: results of further investigation of experiment 4.
V = voltage
I = current
Tg = tacho-voltage
λ = tip speed ratio
Table 31: Wind speed of 3 m/s
Angle of attack
Rad
Degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
Wind Speed = 3 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.00 0.00
0.000
0.15 1.96
0.294
0.26 3.76
0.978
0.20 3.70
0.740
0.22 4.00
0.880
0.20 4.00
0.800
0.00 0.00
0.000
0.00 0.00
0.000
0.00 0.00
0.000
0.00 0.00
0.000
0.00 0.00
0.000
Tg
0.00
0.00
0.14
0.3
0.23
0.25
0.23
0.00
0.00
0.00
0.00
0.00
N in rpm
0
0
93
200
153
167
153
0
0
0
0
0
λ
0.000
0.000
0.130
0.279
0.214
0.233
0.214
0.000
0.000
0.000
0.000
0.000
Tg
0.00
0.00
0.25
0.4
0.32
0.39
0.38
0.33
0.24
0.00
0.00
0.00
N in rpm
0
0
167
267
213
260
253
220
160
0
0
0
λ
0.000
0.000
0.175
0.279
0.223
0.272
0.265
0.230
0.168
0.000
0.000
0.000
Table 32: Wind speed of 4 m/s
Angle of attack
Rad
Degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
Wind Speed = 4 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.00 0.00
0.000
0.21 3.59
0.754
0.35 5.49
1.922
0.3
5.5
1.650
0.33 6.3
2.079
0.3
5.6
1.680
0.3
5.4
1.620
0.2
3.7
0.740
0.00 0.00
0.000
0.00 0.00
0.000
0.00 0.00
0.000
74
Table 33: Wind speed of 5 m/s
Angle of attack
rad
degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
wind speed = 5 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.00 0.00
0.000
0.26 4.25
1.105
0.38 6
2.280
0.38 7.2
2.736
0.37 7
2.590
0.38 7
2.660
0.35 6.6
2.310
0.3
5.5
1.650
0.00 0.00
0.000
0.00 0.00
0.000
0.00 0.00
0.000
Tg
0.00
0.00
0.29
0.44
0.44
0.45
0.45
0.43
0.34
0.00
0.00
0.00
N in rpm
0
0
193
293
293
300
300
287
227
0
0
0
λ
0.000
0.000
0.162
0.246
0.246
0.251
0.251
0.240
0.190
0.000
0.000
0.000
Tg
0.00
0.00
0.4
0.55
0.62
0.64
0.68
0.63
0.62
0.54
0.00
0.00
N in rpm
0
0
267
367
413
427
453
420
413
360
0
0
λ
0.000
0.000
0.186
0.256
0.289
0.298
0.317
0.293
0.289
0.251
0.000
0.000
Table 34: Wind speed of 6 m/s
Angle of attack
rad
degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
wind speed = 6 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.00 0.00
0.000
0.38 5.86
2.227
0.49 7.6
3.724
0.53 9.8
5.194
0.55 10.3
5.665
0.57 10.7
6.099
0.54 10
5.400
0.53 9.8
5.194
0.47 8.5
3.995
0.00 0.00
0.000
0.00 0.00
0.000
75
Table 35: Wind speed of 7 m/s
Angle of attack
rad
degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
wind speed = 7 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.00 0.00
0.000
0.47 7.5
3.525
0.61 9.69
5.911
0.66 12.2
8.052
0.78 14.1
10.998
0.78 14.5
11.310
0.74 13.4
9.916
0.77 13
10.010
0.69 12.6
8.694
0.39 7.3
2.847
0.00 0.00
0.000
Tg
0.00
0.00
0.57
0.7
0.78
0.89
0.91
0.85
0.89
0.8
0.44
0.00
N in rpm
0
0
380
467
520
593
607
567
593
533
293
0
λ
0.000
0.000
0.227
0.279
0.311
0.355
0.363
0.339
0.355
0.319
0.176
0.000
Tg
0.00
0.04
0.67
0.92
0.98
1.1
1.16
1.18
1.2
1.18
0.7
0.00
N in rpm
0
27
447
613
653
733
773
787
800
787
467
0
λ
0.000
0.014
0.234
0.321
0.342
0.384
0.405
0.412
0.419
0.412
0.244
0.000
Table 36: Wind speed of 8 m/s
Angle of attack
rad
degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
wind speed = 8 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.04 0.56
0.022
0.6
9.45
5.670
0.81 12.76
10.336
0.83 15.3
12.699
0.92 17
15.640
0.97 18
17.460
1.01 18.4
18.584
1.01 18.6
18.786
0.92 17.1
15.732
0.59 11.1
6.549
0.00 0.00
0.000
76
Table 37: Wind speed of 9 m/s
Angle of attack
rad
degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
wind speed = 9 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.13 2.08
0.270
0.7
11.2
7.840
0.92 14.5
13.340
0.97 17.8
17.266
1.06 19.6
20.776
1.16 21.3
24.708
1.25 22.8
28.500
1.31 23.6
30.916
1.25 22.7
28.375
0.8
15
12.000
0.00 0.00
0.000
Tg
0.00
0.14
0.8
1.05
1.14
1.25
1.4
1.46
1.55
1.46
0.95
0.00
N in rpm
0
93
533
700
760
833
933
973
1033
973
633
0
λ
0.000
0.043
0.248
0.326
0.354
0.388
0.435
0.453
0.481
0.453
0.295
0.000
Tg
0.00
0.16
0.9
1.25
1.33
1.52
1.65
1.74
1.87
1.86
1.33
0.00
N in rpm
0
107
600
833
887
1013
1100
1160
1247
1240
887
0
λ
0.000
0.045
0.251
0.349
0.372
0.425
0.461
0.486
0.522
0.520
0.372
0.000
Table 38: Wind speed of 10 m/s
Angle of attack
rad
degree
0.000 0
0.262 15
0.524 30
0.786 45
0.873 50
0.960 55
1.048 60
1.135 65
1.222 70
1.310 75
1.397 80
1.571 90
wind speed = 10 m/s
V
I in mA
P in Mw
0.00 0.00
0.000
0.15 2.4
0.360
0.8
12.5
10.000
1.08 17.2
18.576
1.14 20.5
23.370
1.26 23.2
29.232
1.35 25
33.750
1.43 27
38.610
1.56 29
45.240
1.54 28.8
44.352
1.12 20.9
23.408
0.00 0.00
0.000
77
Appendix B: Voltage Characteristic data of rotor speed of 1000 rpm
Table 39: Rotor diameter of 10 cm
4 cm Blade at 1000 rev/min
Resistance (Ω)
Voltage (V)
0
0.25
10
0.73
20
1.02
30
1.14
40
1.24
50
1.25
60
1.30
70
1.32
80
1.33
90
1.34
100
1.36
Current (mA)
123.3
76.0
48.4
35.5
25.9
23.7
20.5
18.2
16.3
14.5
13.3
Power (mW)
30.825
55.480
49.368
40.470
32.116
29.625
26.650
24.024
21.679
19.430
18.088
Current (mA)
106.0
71.0
49.0
34.7
27.8
23.1
20.4
18.0
16.0
14.5
13.2
Power (mW)
39.220
51.120
49.000
39.905
34.472
29.337
26.112
23.760
21.440
19.575
18.084
Table 40: Rotor diameter of 12 cm
5 cm Blade at 1000 rev/min
Resistance (Ω)
Voltage (V)
0
0.37
10
0.72
20
1.00
30
1.15
40
1.24
50
1.27
60
1.28
70
1.32
80
1.34
90
1.35
100
1.37
78
Table 41: Rotor diameter of 14 cm
6 cm Blade at 1000 rev/min
Resistance (Ω)
Voltage (V)
0
0.39
10
0.78
20
1.00
30
1.14
40
1.22
50
1.27
60
1.28
70
1.32
80
1.33
90
1.36
100
1.37
Current (mA)
108.8
72.8
48.5
34.7
25.5
23.2
20.3
17.9
16.0
14.4
13.2
Power (mW)
42.432
56.784
48.500
39.558
31.110
29.464
25.984
23.628
21.280
19.584
18.084
Current (mA)
113.3
71.1
46.9
34.0
27.1
23.6
20.5
17.9
15.9
14.3
13.1
Power (mW)
38.522
54.747
47.369
39.100
33.333
29.500
25.830
23.449
20.988
19.162
17.816
Table 42: Rotor diameter of 18 cm
8 cm Blade at 1000 rev/min
Resistance (Ω)
Voltage (V)
0
0.34
10
0.77
20
1.01
30
1.15
40
1.23
50
1.25
60
1.26
70
1.31
80
1.32
90
1.34
100
1.36
79
Appendix C: Power for different blade sizes
Table 43: Results for 4 cm blades
4 cm Blades
U (m/s)
V (V) I (mA)
P (mW)
Tg (V)
N
5.4
1.51
26.2
39.562
1.78
7.2
2.19
38.6
84.534
9
2.80
49.3
10.8
3.48
12.6
14.4
λ
Mp (mW)
Bp (mW)
Re
Cp
1187 1.151
2392
2432
37,101
0.403
2.58
1720 1.251
7284
7368
49,468
0.515
138.040
3.31
2207 1.284
15381
15519
61,834
0.556
63.2
219.936
4.10
2733 1.326
29232
29452
74,201
0.610
4.13
76.8
317.184
4.90
3267 1.358
49899
50216
86,568
0.655
4.83
89.0
429.870
5.61
3740 1.360
74885
75315
98,935
0.658
Table 44: Results for 5 cm blades
5 cm Blades
U (m/s)
V
(V)
4.5
1.13 20.9
6
I (mA) P (mW)
Tg (V) N
λ
23.617
1.34
893
1.72 31.8
54.696
7.5
1.99 37.0
9
Mp (mW)
Bp (mW)
Re
Cp
1.248 1825
1849
37,101
0.368
2.04
1360 1.425 6440
6494
49,468
0.545
73.630
2.37
1580 1.324 10098
10171
61,834
0.437
2.63 48.8
128.344
3.10
2067 1.443 22597
22726
74,201
0.565
10.5
2.91 54.0
157.140
3.45
2300 1.377 31148
31305
86,568
0.490
12
3.39 62.6
212.214
3.98
2653 1.390 47821
48033
98,935
0.504
13.5
3.76 69.9
262.824
4.44
2960 1.378 66393
66656
111,302
0.491
80
Table 45: Results for 6 cm Blade
6 cm Blades
U (m/s)
V
(V)
3.6
0.80 15.0
5.1
I (mA) P (mW)
Tg (V) N
λ
12.000
1.00
667
1.28 23.6
30.208
6.4
1.61 30.0
7.7
Mp (mW)
Bp (mW)
Re
Cp
1.358 1184
1196
34,627
0.341
1.53
1020 1.467 4242
4272
49,055
0.429
48.300
1.93
1287 1.474 8514
8562
61,560
0.435
1.95 36.3
70.785
2.32
1547 1.473 14788
14859
74,064
0.433
9
2.30 42.7
98.210
2.74
1827 1.488 24361
24459
86,568
0.447
10.3
2.66 49.4
131.404
3.18
2120 1.509 38083
38214
99,072
0.466
11.6
3.04 56.6
172.064
3.62
2413 1.526 56179
56351
111,577
0.481
Bp (mW)
Re
Cp
Table 46: Results for 8 cm blades
8 cm Blades
U (m/s)
V
(V)
3
0.70 12.3
4
I (mA) P (mW)
Tg (V) N
λ
Mp (mW)
8.610
0.75
500
1.571 1069
1078
37,101
0.322
0.89 16.3
14.507
1.07
713
1.681 3105
3120
49,468
0.393
5
1.14 20.9
23.826
1.38
920
1.735 6661
6685
61,834
0.431
6
1.42 26.1
37.062
1.70
1133 1.781 12452
12490
74,201
0.466
7
1.66 30.7
50.962
2.00
1333 1.796 20277
20328
86,568
0.477
8
1.87 34.5
64.515
2.24
1493 1.760 28487
28552
98,935
0.449
9
2.12 39.9
84.588
2.53
1687 1.767 41046
41131
111,302
0.454
81
Appendix D: Blade design
82