CSU Engineering Lab Report

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Wind Turbine Lab Report Requirements
Results
1. Plot Power vs. Load for a constant speed (5 m/s) at a given blade angle (60o).
a. What happens to the power out as the load increases?
b. What happens to the frequency?
c. What happens to the RPM?
d. If the power is changing what do you think is happening?
2. Plot Power vs. Wind speed for each blade angle and 3 different loads. (4 different
figures)
a. As the wind speed increases what happens to the power for a given load? What
happens to the frequency?
3. Plot Power vs. Wind speed for different blade angles and a single load (50 ohms) (1
figure).
a. For a given wind speed how does the power change with blade angle.
b. Fit the three curves and determine an appropriate function.
4. Assuming the rated power of the turbine is (0.11 Watts) plot the blade angle vs wind
speed for this power.
a. Fit with an appropriate function.
5. Using the following information determine the maximum kWh production over a 24
hour period (Only consider a 50 ohm load).
Time (hours)
Wind Speed (m/s)
0-3
1
3-6
3
6-9
3.7
9-12
4.2
12-15
10
15-18
8
18-21
4.2
21-24
3.5
a. Maximum and minimum blade angle are 45 and 70 degrees respectively. Any
blade angle between these is possible.
b. Maximum allowed power is rated power. Past this value the blade must be
locked.
c. Report the position of the blade at each time interval and the power out for that
time.
6. Calculate plot the Power Factor and Advance Ratio for all cases.
a. For each blade angle plot the values (power factor vs. advance ratio) for a single
load (50 ohm) and fit with a curve if possible. Compare to other 3 blade
turbines.
Wind Turbine Lab
I.
Background Information
Worldwide energy consumption continues to increase at an alarming rate. The U.S.
Energy Information Administration (EIA) recently predicted that the world energy
consumption would increase 48% by the year 2040[1]. While predictions for overall
energy consumption in the United States is considerably less, approximately 5% over
this same period, there is still a strong desire to switch to alternative systems due to
environmental concerns over traditional power generation methods, specifically those
dealing with the combustion of fossil fuels.
One of the fastest growing forms of alternative power production is in wind turbine
technology. The EIA predicts that wind energy production will increase by 12 and 14
percent in the U.S. over the next two years. The goal of this lab is to expose students to
the basics of wind energy production and to cover control strategies employed for
efficient operation.
II.
Wind Turbine Basics
There are essentially two basic types of wind turbines: Horizontal Axis Wind Turbines
(HAWT) and Vertical Axis Wind Turbines (VAWT). As the names imply the main
difference between the two are the axis about which the turbine blades rotate. Figure 1
below shows the two differing types of wind turbine design.
Figure 1. Horizontal and vertical axis wind turbines
The most common power producing wind turbines are HAWT’s which will be examined
during this lab. Basic components of HAWT’s include:
1. Support tower and nacelle
2. Rotor blades including pitch and yaw control
3. Drive shaft and gear box
4. High speed shaft and generator
Figure 2 shows some of these basic components.
Figure 2. Basic wind turbine components
In order to produce the largest amount of power at varying wind speeds it is essential to
develop control mechanisms capable of providing useful rotational speeds of the wind
turbine blades. Typically, two methods may be employed:
1. Electric or mechanical braking
2. Turbine blade pitch adjustment and yaw control
Method one is most effective in moderating small scale short term wind variation.
Method two is employed when large sustained wind changes occur. Yaw control is
used to insure that the turbine blades are positioned normal to the incoming air, while
blade pitch is used to adjust the force (lift) on the blades once the turbine is facing in the
correct position.
Wind turbine control strategies are designed for four general operating conditions,
based on the velocity of the wind.
1. Cut in velocity – This is the minimum wind speed needed to achieve useable
power.
2. Constant Cp region – In this region the blades are oriented to gain the maximum
possible power out of the incoming wind. The power coefficient will be discussed
in the theory section.
3. Constant output power – In this region the blades are orientated to provide the
maximum, or rated power of the turbine. Blades are oriented so that the turbine
does not overspeed.
4. Cut out speed – This represents the maximum speed at which the turbine can
operate. Above this speed the blades are positioned to receive as little lift as
possible and the turbine is mechanically locked in place.
Figure 3. Wind turbine operating regimes
III.
Theory
The available power for air moving at a velocity V can be found from equation (1):
1
1
𝑃𝑜𝑤𝑒𝑟 = 2 𝑚̇𝑉 2 = 2 𝜌𝐴𝑉 3
𝑚̇ is the mass flow rate of the air
V is the free stream velocity of the air
A is the area being considered
(1)
Figure 4 shows the pressure, velocity and area distribution of air moving through a wind
turbine.
Figure 4. Actuator disk schematic with pressure, velocity, and area distributions
illustrated.
The following derivation provides the steps needed to determine the maximum possible
power that can theoretically extracted by a wind turbine. Figure 4 is used as a
reference for the given subscripts.
Begin by writing the Bernoulli equation both upstream and downstream of the turbine
blades (but not across)
1
1
𝑃𝑒 + 2 𝜌𝑐 2 = 𝑃1 + 2 𝜌𝑐 2 (1 − 𝑎)2
(2)
𝑃2 + 2 𝜌𝑐 2 (1 − 𝑎)2 = 𝑃0 + 2 𝜌𝑐 2 (1 − 𝑏)2
(3)
1
1
Where
P is the pressure
ρ is the density
c is the free stream velocity
a and b represent fractional changes used to modify the velocity at the given location
Dividing by ρ and rearranging equations (2) and (3) become:
𝑃𝑒 −𝑃1
𝜌
𝑃2 −𝑃0
𝜌
1
= 2 𝑐 2 [(1 − 𝑎)2 − 1]
(4)
1
= 2 𝑐 2 [(1 − 𝑏)2 − (1 − 𝑎)2 ]
(5)
Noting Pe = Po
𝑃2 −𝑃0 +𝑃𝑒 −𝑃1
𝜌
=
𝑃2 −𝑃1
𝜌
1
1
= 2 𝑐 2 [(1 − 𝑏)2 − (1 − 𝑎)2 + (1 − 𝑎)2 − 1] = 2 𝑐 2 [1 − (1 − 𝑏)2 ] (6)
Simplifying we find:
1
𝑃1 − 𝑃2 = 2 𝜌𝑐 2 [1 − (1 − 𝑏)2 ]
(7)
Now the axial thrust (T) across the actuator is equal to the difference in force on the
upstream and downstream side of the blade:
𝑇 = (𝑃1 − 𝑃2 )𝐴 =
1
2
𝜌𝐴𝑐 2 [1 − (1 − 𝑏)2 ]
(8)
T is the axial thrust
A is the swept area of the blades
Using Newton’s second Law the axial thrust is equal to the axial change in momentum
as shown in equation (9):
𝑇 = 𝑚̇[𝑐 − 𝑐(1 − 𝑏)] = 𝜌𝐴𝑐(1 − 𝑎)[𝑐 − 𝑐(1 − 𝑏)] = 𝜌𝐴𝑐 2 (1 − 𝑎)𝑏
(9)
Simplifying:
1
2
𝜌𝐴𝑐 2 [1 − (1 − 𝑏)2 ] = 𝜌𝐴𝑐 2 (1 − 𝑎)𝑏
(10)
𝑏
𝑎=2
(11)
In order to determine the power extracted we consider the change in Kinetic Energy per
unit time:
1
1
𝐸𝑘 = 2 𝜌𝐴𝑐(1 − 𝑎)[𝑐 2 − 𝑐 2 (1 − 𝑏)2 ] = 2 𝜌𝐴𝑐(1 − 𝑎)[𝑐 2 − 𝑐 2 (1 − 2𝑎)2 ] (12)
Where:
Ek is the change in kinetic energy per unit time across the turbine blades
To determine the maximum possible power extracted we take the derivative of this
equation, with respect to the change in the wind speed at the blade, and set it equal to
zero and then solve for a:
𝑑𝐸𝑘
𝑑𝑎
= 0 = (1)(1 − 𝑎)2 + 2𝑎(1 − 𝑎)(−1)
1
𝑎=3
(13)
(14)
Using this result the maximum possible extracted power can be determined:
𝑀𝑎𝑥 𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟 =
8
27
𝜌𝐴𝑐 3
(15)
The Power Coefficient is a non-dimensional parameter used to measure the
effectiveness of a wind turbine. It is defined as:
𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟
𝐶𝑃 ≡ 𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑊𝑖𝑛𝑑 𝑃𝑜𝑤𝑒𝑟 =
𝐸𝑥𝑐𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟
1
𝜌𝐴𝑉 3
2
(16)
Using the results provided in equation (15) the highest possible power coefficient can be
determined. This value is know as the Betz Limit and is given in equation (17):
𝐶𝑃𝑚𝑎𝑥 =
8
𝜌𝐴𝑐 3
27
1
𝜌𝐴𝑐 3
2
= 0.5926 ≅ 0.6
(17)
The Advance Ratio is another non-dimensional parameter relating the rotor tip speed to
the wind speed. It is shown in equation (18)
𝑟𝜔
Ω = 𝑉𝑤𝑖𝑛𝑑
(18)
It is often convenient to plot the Power Factor as a function of the Advance Ratio as
shown in figure 5.
Figure 5 Power Coefficient vs Advance Ratio for varying type of wind turbines.
The extracted power from the generator is calculated using the following:
𝑃𝑜𝑤𝑒𝑟𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 =
𝑉𝑎 𝐼𝑎 ∗𝑉𝑏 𝐼𝑏 ∗𝑉𝑎𝑐 𝐼𝑎𝑐
√3
(19)
Where V and I represent voltage and current respectively and the subscripts represent
each branch of the three phase system.
The frequency of the generator is calculated by:
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 =
𝑛∗𝑅𝑃𝑀𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟
120
(20)
Where n = the number of poles of the generator (8 for our case)
IV.
Startup Procedures
1. Make sure wind turbine is set to an angle of 60o.
2. Verify that all load rheostats are set at no load (zero) for each phase and
excitation
3. Turn MASTER Power Key to ON Position
4. Start system and chose a low wind speed. When wind anemometer starts to
spin, the WIND SPEED Meter will start to display data.
5. Slowly increase the wind speed until the turbine starts to move. Record this
value as the cut in speed.
6. Continue to increase the wind speed until the Generation Excitation begins to
display. Set this value at 7 Volts.
V.
Experiment
1. Measure and record the length from the center of the axis to the tip of a turbine
blade.
2. Set the load at 10 ohms.
3. Increase wind speed until power is just starting to be generated (record as cut-in
speed)
4. Increase wind speed by increments of 1 m/s up to 7 m/s.
5. Repeat Steps 3 and 4 with loads of 50 ohms and 75 ohms.
6. Shut the turbine down and change the blade angle to 45o.
7. Repeat Steps 2-5 for this angle. Note that a larger maximum windspeed may be
acceptable (max 10 m/s)
8. Shut the turbine down and change the blade angle to 65 o.
9. Repeat Steps 2-5 for this angle. Note that a smaller windspeed may be
acceptable (max ? m/s). Also use increments of 0.5 m/s.
10. Shut the turbine down and change the blade angle to 70 o.
11. Repeat Steps 2-5 for this angle. Note that a larger windspeed may be
acceptable (max 4.5 m/s). Use increments of 3, 3.5, 4, 4.5.
Report Guidelines
1. Figures
a. All figures should be referenced and discussed in the text. If the reference is on
the same page as the figure it can be above or below the figure. If the reference
is not on the same page it should come on the page before the figure.
b. The title of the figure should be below the figure and should be in bold print.
The figure caption, a short description of the figure, should be after the title and
should not be in bold. Please place 1.5 lines between the figure title/caption and
the text. Also use 2 lines between the text and the top of the figure.
c. Data should be represented by points and should not be connected by lines. If
multiple sets of data are plotted on a single figure, please make sure to used
different symbols for each.
d. In general the axis size should be set so that the data does not lie on the
extremes.
e. Units should be included on all axes.
f. Trend lines should include the equation, R2 value, and should be put on the
graph for a purpose.
g. Error bars, and what they represent, should be discussed in the text.
2. Tables
a. All tables should be referenced and discussed in the text. If the reference is on
the same page as the table it can be above or below the table. If the reference is
not on the same page it should come on the page before the table.
b. The title of the table should be above the table and should be in bold print. The
table caption, a short description of the table, should be after the title and
should not be in bold. Please place 1.5 lines between the table title/caption and
the text. Also use 2 lines between the text and the bottom of the table.
c. Units should be included where appropriate.
d. All columns should be centered and contain appropriate accuracy with respect to
decimal points.
3. Equations
a. All equations should be numbered. Place the number in parentheses and move
all numbers to the far right in the same column.
Example of a Figure
Figure 1 shows a plot of the flow rate versus head loss through a pipe.
25
20
Flow Rate (gpm)
15
Flow Rate
10
y = 0.7716x + 6.2578
R² = 0.9912
5
0
0
5
10
Head (in)
15
Figure 1 – Relationship between flow rate and head loss through a pipe.
Begin new text.
20
Example of a Table
Table 1 provides the flow rate data and head loss collected in the experiment.
Table 1 – Head, Flow Rate, and Data Recorder
Head
(in)
3.1
3.7
4.9
7.7
9.3
11.2
13.4
14.9
17
18.9
Flow Rate
(gpm)
8.1
8.9
9.9
12.5
14.1
15.5
16.6
17.7
19.2
20.4
Data
Recorder
Jones
Elliot
Margraves
Jones
Elliot
Margraves
Jones
Elliot
Margraves
Jones
Begin new text.
Example of Equations
𝑎2 + 𝑏 2 = 𝑐 2
(1)
𝐴 = 𝜋𝑟 2
(2)
What I Consider Most Important In A Lab Report
1. Effort – Has the student made sure to proofread the report and present a
document that is professional looking. Has the theory been copied and
pasted or is it in the students own words.
2. Completeness and Clarity – Can the document stand alone? Does it make
sense?
3. Thought – Has the student examined all of the data and thought about
what it means? Do they try to explain discrepancies between the theory
and results? Has the data led them to conclusions or hypotheses not
proposed in the questions given in the lab manual?
Wind Turbine
Laboratory 3
Contributing Editors
Executive Summary: ______JD___________________
Apparatus and Theory: _____CS____________________
Results and Discussion: _________________________ , _________________________
Conclusion: ________JD_________________
Signature
1. Theory and Procedures:
Grade
Adjusted Ins. Lab Grade
____Connor Stewart___________
___________
2. Data Acquisition, Analysis:
_________________
___________________
3. Manager:
_Jonathan DeGuzman_ ____NONE___ ___________________
___________________
___________
Experiment Conducted: 09/04/2019
Date Final Report Due: 09/27/2019
Date Final Report Submitted: _______________ with number of weeks late being N = _______________
Adjusted Laboratory Report Grade is: Grade Assigned______________ minus 15 X N
Executive Summary
The wind turbine is one of the fastest growing forms of alternate power production. With
the prediction from the Energy Information Administration of wind energy increasing by
12 – 14% over the next 2 years, it is important for students to learn the basics of wind energy
production and to cover control strategies needed for efficient operation.
This experiment focuses on the Horizontal Axis Wind Turbine or HAWT for short. The
primary objective of this lab is to be able to gather data for a wind turbine, record cut-in speed,
document the cut-in power, and track data across a ten second time interval for different wind
velocities and system loads. More importantly, with the data collected, relationships between
power, load, wind speed and blade angle behaviour could be studied to relate to real life
scenarios. All data was collected using different combinations of both constant and changing
blade angles, wind speeds, and loads applied to the HAWT to see how power was affected.
This lab was successful in achieving its objectives, though not without a few hiccups.
Being a new experiment using brand new equipment, some problems were expected but fixes to
possible inaccuracies of data were also foreseen. From the data collected it can be concluded that
the average power increases as the load grew up to 75 ohms and the angle rose up to 70 degrees
in all cases and variations tested. For Power vs Load, a constant speed of 5 meters per second at
a given blade angle of 60 degrees for loads of 10, 25, 50 and 75ohms, the power increased from
0.09 to 0.28 watts. For Power vs Wind Speed, at a 60 degree blade angle for loads of 10, 50, and
75ohms, power increased to 0.25, 0.45, and 0.73 watts respectively as the wind speed rose to 7
meters per second. Though there was some human error to the experiment causing inaccurate
data collected for the Power vs Wind Speed for different blade angles at constant loads part of
the experiment, +0.05 watts was added the 70 degree angle data collected to show more real
world results. Regardless, these corrections fixed the inaccuracies of part of the data collected
producing similar results of increasing power in watts to rising angles at a constant load of 50
ohms. The results from this data were used to solve a real life scenario for a $70,000 wind
turbine with a cost of $0.10/kWh generating 0.11 watts of power 24 hours a day all day every
day of the year would pay back after “only” 726,443 years. Though quite an inefficient and
somewhat unbelievable example, the experiment still accurately illustrated how a HAWT can
help in producing needed future alternative sources of energy.
Introduction/Theory
The objectives of this lab were to gather data for a Wind Turbine, record cut-in speed, document cut-in
power, and track the data across a ten second time interval for different wind velocities and system loads.
The gathered data can be found in the results and discussion section of this report in the form of figures
and tables.
Using the mass flow rate of air (ṁ), free stream velocity of air (V), the area (A), and the
density(ρ), the power for air at the velocity (V) can be found from equation (1):
Figure 1 shows each distribution in a wind turbine for pressure, velocity and area for moving air.
Figure 1 Actuator disk schematic with pressure, velocity, and area distributions illustrated.
Using Figure 1 above as a reference for subscripts, we can begin to derive important steps in determining
the theoretical maximum possible power that can be extracted by a wind turbine.
Using “P” as pressure, “ρ” as density, “c” as free stream velocity, and “a” and “b” as fractional changes
we can begin to write Bernoulli’s equation for both upstream (equation 2),
and downstream (equation 3)
of the turbine blades. Noting that we don’t for across the turbine blades.
Rearrange equations (2) and (3) by dividing by ρ to get:
Knowing that Pe = Po, expand to get:
Simplifying (6):
Multiplying both sides of (7) by the swept area of the blades(A), we get the axial thrust (T) across the
actuator:
The axial thrust is now equal to the difference in force on the upstream and downstream side of the blade.
Changing the right side of (8) using Newton’s second Law, the axial thrust is equal to the axial change in
momentum represented below:
Which simplifies to:
Where:
Considering the change in Kinetic Energy per unit time, we can determine the power extracted:
Where:
Ek is the change in kinetic energy per unit time across the turbine blades.
Using the change in the wind speed at the blade, take the derivative of equation (12), set it equal to zero
and solve for a:
This allows us to begin to determine the maximum possible power extracted. Using this result the maximum
possible extracted power can be found by:
Next we measure the effectiveness of a wind turbine by:
Where:
Cp is the non-dimensional Power Coefficient.
The highest possible power coefficient, known as the Betz Limit, can be determined using the results
from equation (15) plugged into equation (16):
Using the rotor tip speed and wind speed below:
We can determine the non-dimensional Advance Ratio, Ω.
Figure 5 shows us the usefulness in plotting the Power Factor as a function of the Advance Ratio.
Figure 2 Power Coefficient vs Advance Ratio for different types of wind turbines.
Knowing that (V) represents voltage and (I) represents current we can calculate the extracted power from
the generator:
The subscripts for (V) and (I) represent each branch of the three phase system.
Using n = 8 (for our lab), the frequency can be calculated by:
Where n = the number of poles of the generator.
Apparatus:
Figure 3: Horizontal-Axis Wind Turbine (HAWT) Diagram
Figure 3 represents the Horizontal-Axis Wind Turbine(HAWT) used in this lab. HAWTs are the
most common power producing wind turbine. Labeled above in the figure you can see the Rotor
Blades, Support Tower, Generator, Main Gearbox, and looking through the glass optic you
would see the Drive Shaft. The blade pitch is adjusted at different wind speeds to adapt to the lift
on the blades and generate maximum power.
Figure 4: HAWT Control Panel (Rheostats)
Figure 4 shows the Rheostats used to control the load(ohms) on the turbine. Multiple tests were
run at different loads using varying wind speeds, showing that higher load equals more power
out. The load is also used as an electrical brake on the system.
Figure 5: HAWT Control Panel (Sensors)
Figure 5 above shows the wind speed control, Generator Excitation control, and sensors to read
wind speed and turbine RPM. Once the first value of General Excitation appeared, it was set to 7
volts and left unchanged throughout the lab. The wind speed was adjusted from 3m/s to 10m/s
per each test requirement.
Results and Discussion
Figure 6: Average Power vs Load at 5 m/s for Angle 60 visually shows data collected in Table 3,
in the Appendix, of average power in Watts and load values in ohms. The R2 value of 1.0 shows
the trendline accurately fits the data suggesting a strong polynomial to the third order
relationship between average power and load for a wind speed of 5 m/s at an angle of 60
degrees.
The power and frequency was calculated by hand and through excel(see Appendix) for
all data. The software output proves that our calculations are correct. Figure 6 was used to find a
baseline of how the wind turbine would act at varying loads with no wind speed change. This
data showed that as load increased the power production increased. This data was the first test
collected and allowed us to see the correlation between load and power out which assisted in
interpreting the rest of the data collected. Figure 6 was created for average power versus load for
a constant speed of 5 meters per second at a given blade angle of 60 degrees for loads of 10, 25,
50 and 75. As the load increases from 10 to 75 ohms, the power out also increases from 0.09 to
0.28 watts gradually. The power generated at a load of 75 ohms is 2.97 times larger than the
power generated at the load of 10 ohms.The frequency stays around the same value, which is
approximately 7.7 to 7.8 for all the loads. The rotor and generator RPM both increase as the load
increases from 10 to 25. They stay approximately the same for loads 25 and 50 then decrease
back down for a load of 75 which has the same value as a load of 10. Since the power is
changing, the increased load is causing more power to be produced to maintain the wind speed at
a blade angle of 60 degrees.
Figure 7: Average Power vs Wind Speed for Different Loads at Angle 60 visually shows data
collected in Table 4, in the Appendix, of average power in watts and wind speed values in meters
per second. The R2 values of 0.9961, 0.9987 and 0.9993 for each shows the trend lines accurately
fit the data suggesting a strong polynomial to the third order relationship between average
power and wind speed for loads of 10, 50 and 75 at an angle of 60.
Figure 7 was created to show average power versus wind speed for three different loads,
which are 10, 50 and 75, at a given blade angle. Similarly to figure 6, figure 7 shows that as load
increases and wind speed increases the overall power generated increases as well. All loads
were fit with a second order polynomial trend line. For a bigger load, the power increases more
compared to the smaller loads. For a load of 10, the power is approximately 0.02 at a wind speed
of 3.5 and increases to a power of 0.25 at a wind speed of 7. For a load of 50, the power is
approximately 0.05 at a wind speed of 3.5 and increases to 0.45 for a wind speed of 7. For a load
of 75, the power is approximately 0.08 at a wind speed of 3.5 and increases to 0.73 for a wind
speed of 7. For all the loads at each wind speed, the frequency increases the same. For the last
two wind speeds for all loads, the frequency is almost identical.
Figure 8: Average Power vs Wind Speed for a Load at Different Angles visually shows data
collected in Table 5, in the Appendix, of average power in watts and wind speed values in meters
per second. The R2 values of 1, 0.9991 and 0.999 for each shows the trend lines accurately fit the
data suggesting a strong polynomial to the third order relationship between average power and
wind speed for a load of 50 at angles 45, 60 and 70. Through the accuracy of the trendlines,
appropriate functions were produced above as shown in figure 8.
Figure 8 is the data collected from a single load of 50 and blade angles 45, 60, and 70
with a wind speed change. As the given wind speed increases, the power output increases for
each blade angle at a single load. The power output is larger at the same wind speed as you
increase the blade angle. Thinking qualitatively and using the supplemental data given and other
research, it is known that with a larger blade angle you can achieve higher power output at lower
wind speeds. The data collected for angle 70 had a significant amount of experimental error. The
original data had the 70-degree plot below the blue angle 60 line. However, from knowing how
this turbine acts qualitatively the data for angle 70 was given a +0.05 watt DC offset to place it
above the angle 60 power line. The angle 70 power line had more error due to the excitation
knob in the experiment being adjusted. This change caused differing results to be collected. This
faulty data had to be removed leaving three data points found for angle 70. The data was then
forced to be extrapolated using the Excel period prediction function. This prediction does not
give results that are completely accurate but they are capable of being used to estimate the values
needed. The rest of the data found for angles 45 and 60 were collected using the same excitation
knob setting and more accurate quantitative data was collected. The data from this plot is used in
many other calculations within the lab such as solving the optimized blade angle and wind speed
for figure 4 and table 1. The compounding error can be seen in the following plots and graphs.
Figure 9: Blade Angle vs Wind Speed at Max Power Generation visually shows data collected in
Table 8, in the Appendix, of blade angle in degrees and wind speed values in meters per second.
The resulting declining trendline reveals a slope of -4.0081 degrees per m/s. The R2 value of
0.8869 for the trendline which is not as close to 1 as previous values but still fits the data
suggesting a strong linear relationship between blade angle and wind speed.
In figure 9 the blade angle and wind speed were determined using the data from figure 8
to optimize the blade angle to achieve the maximum power of 0.11 watts. The slopes from figure
7 were taken and placed into a goal seek scenario within Excel. Y was set to max rated power
(0.11 watts) and the x value (wind speed) was changed to achieve that max rated power. The
found wind speeds for optimized power generation can be seen in figure 9. The optimized wind
speeds for max power generation were then fit with a linear function line. The data points were
previously fit with a second-order polynomial fit this yielded an R² value of 1. All though the fit
with the second-order polynomial is better for the data it does not qualitatively make sense. With
that fit, the line dipped below the final blade angle of 45 and began to rise again. Knowing that
blade angle and wind speed are a negative relation we turned to the linear fit with the less
accurate fit with an R² value of 0.8869. The appropriate function of the trend line was placed
into table 1 below and the wind speeds were plugged into to solve for the most optimized blade
angle for max power generation at the given wind speeds.
Table 1. Max kWh Production and Blade Angle for Given WS for Twenty-Four Hour Period
Time(hours)
WS(m/s)
Max. kWh production
Angle
0-3
1
LOCKOUT
76.9569
3-6
3
0.00033
68.9407
6-9
3.7
0.00033
66.1350
9-12
4.2
0.00033
64.1310
12-15
10
LOCKOUT
40.8840
15-18
8
0.00033
48.9002
18-21
4.2
0.00033
64.1310
21-24
3.5
0.00033
66.9367
Total Power Production in 24 hours (kWH)
0.0020
Table 1: Max kWh Production and Blade Angle for Given WS for Twenty-Four Hour
Period shows the data calculated for max kWh production in watts which is a constant 0.00033
kWh for each three hour period unless a lockout occurs. Blade angles were also calculated for
each three hour period for the given wind speeds that range from 40.884 to 76.9569 degrees.
Table 1 shows the power generated by the wind turbine using the optimized blade angles for
maximum power generation of 0.11 watts. The turbine blade pitch was only capable of going
between 45-70 degrees. Any angle above or below those parameters force the wind turbine to be
in a lockout state. The turbine was forced into the lockout state twice at wind speeds of 1 m/s and
10 m/s. The wind speed of 1 m/s was well below the cut in speed and produced a projected angle
of 76.9 degrees. This blade angle was not allowed due to the max angle being 70 degrees. For the
wind speed of 10 m /s, the blade was optimized to be set to 40 degrees. This angle is not allowed
due to the higher wind speed, which forced the blades to rotate below the minimum blade angle.
However, even with an optimized blade angle for max power generation the wind turbine only
produced 0.002 kWh in a 24-hour period.
Table 2. Calculation for Time to Pay Turbine Cost Off
Turbine Cost
Electricity Cost
$70,000
$0.10/kWh
kWH per day generated
0.00264
Daily Production * Electricity
Cost
$0.000264
Turbine Cost/Revenue
generated per day (Days)
265,151,515 days
Days/years
726,442.5 years
Table 2 shows the revenue generated per day and how long it would take for the wind
turbine to pay for itself. The table calculations utilize best-case scenario for power generation
having it generate 0.11 watts. The estimation also has the turbine running 24 hours per day
every single day of the year. Even with this theoretical optimization of power, the turbine will
still take 726,443 years to pay itself back using $0.10/kWh. There will never be a situation where
a wind turbine in a real setting would ever be capable of outputting maximum power generation
constantly. This theoretical optimization shows how inefficient this particular wind turbine is.
Possible ways to increase the efficiency would be to have the angle of twist on the blades altered.
This twist of the blades could increase the surface area that the wind would contact creating more
force to rotate the generator.
Figure 10: Advance Ratio vs Power Factor for All Cases visually shows data collected in Table
9, in the Appendix, of power factor and advance ratio values. For angle 45, the R2 value of
0.9942 for the trendline accurately fits the data suggesting a strong negative polynomial to the
third order relationship between power factor and advance ratio. For angle 60, we could not fit
the curve due to inaccurate data. For angle 70, the resulting declining trendline reveals a slope
of -0.0056. The R2 value of 0.9948 for the trendline accurately fits the data suggesting a strong
negative linear relationship between power factor and advance ratio.
The Power Factor and Advance Ratio were calculated in excel for all cases of data.
Figure 10 was created to show all of these values for each angle and wind speed and fit each to a
single curve. For each given angle, as the wind speed increases, the power factor decreases as the
advance ratio increases. Figure 10 is presented with inaccurate results due to equipment
operation error during the lab. The scatter plot above does not provide much applicable data due
to the data points not being grouped by load or angle. The plot does show that the wind turbine
used in the experiment does falls below the Betz limit of 0.5926 except for one outlier value
increasing to 0.0607.
Conclusion/Recommendations
This experiment was successful in achieving its objective of gathering data for a wind
turbine, recording cut-in speed, documenting cut-in power, and tracking data across a ten second
time interval for different wind velocities and system loads. From the data collected to form the
figures above it was concluded that average power increased as the load was increased to 75
ohms and the angle rose up to 70 degrees in various combinations of blade angle, differing loads
and wind speed in all cases tested. A constant speed of 5 meters per second at a given blade
angle of 60 degrees for loads of 10, 25, 50 and 75ohms, the power increased from 0.09 to 0.28
watts for the Power vs Load experiment. At a 60 degree blade angle for loads of 10, 50, and
75ohms, power increased to 0.25, 0.45, and 0.73 watts respectively as the wind speed rose to 7
meters per second for the Power vs Wind Speed experiment. Because of human error concerning
the Power vs Wind Speed for different blade angles at constant loads in part of the experiment,
+0.05 watts was added the 70 degree angle data collected to show more real world results. These
corrections provided a more realistic collection of data producing similar results of increasing
power in watts to rising angles as previous experiments. This mistake could easily be fixed in
future experiments by always checking the levels of all knobs before running each experiment.
Part of the lab concerning Advance Ratio vs Power Factor, the curve was not able to fit the 60
degree angle due to data points not being grouped by load or angle. Being a new experiment
using brand new lab equipment, this part of the experiment, mistakes were expected but should
be fixed moving forward for future classes recreating this lab. Finally the experiment included
using this data towards a real world situation. However the results of $70,000 wind turbine with
a cost of electricity of $0.10/kWh generating 0.11 watts 24 hours per day every day of the year
would still take approximately 726,443 years to pay for itself does not reflect that of a very
efficient business model or fix. Perhaps this part of the experiment could be revised in the future
better be able to understand how to use this data for a more realistic real world scenario.
Appendix
Table 3. Average Power vs Load at 5 m/s for Angle 60 Data
#2 PLOT POWER VS LOAD @5m/s
@angle 60
Load
Average Power
Angle
10
0.092750636
60
25
0.116967083
60
50
0.168185091
60
75
0.273955909
60
Table 3: Average Power vs Load at 5 m/s for Angle 60 Data shows the data assembled from this
experiment of average power in watts that range from 0.092 to 0.274, load values in pounds that
range from 10 to 75 and blade angles in degrees. A blade angle of 60 was used for each different
load. As the load increases, the power increases. The data from this table is graphically shown in
Figure 1 shown in results.
Table 4. Average Power vs Wind Speed for Different Loads at Angle 60 Data
Average Power
WS
LOAD
Angle
0.022609685
3.5
10
60
0.051116231
4
10
60
0.092750636
5
10
60
0.158539818
6
10
60
0.252535364
7
10
60
0.030899273
3.5
50
60
0.063829636
4
50
60
0.116967083
5
50
60
0.199405917
6
50
60
0.309070727
7
50
60
0.051909364
3.5
75
60
0.101369455
4
75
60
0.168185091
5
75
60
0.2826161
6
75
60
0.4484789
7
75
60
Table 4: Average Power vs Wind Speed for Different Loads at Angle 60 Data shows the data
assembled from this experiment of average power in watts that range from 0.022 to 0.449, wind
speed in meters per second that range from 3.5 to 7, blade angle in degrees and load values in
pounds that range from 10 to 75. As the wind speed increases, the average power increases. The
higher the load, more power is being used. The data from this table is graphically shown in
Figure 2 shown in results.
Table 5. Average Power vs Wind Speed for a Load of 50 at Different Angles Data
Average Power
WS
LOAD
Angle
0.030899273
3.5
50
60
0.063829636
4
50
60
0.116967083
5
50
60
0.199405917
6
50
60
0.309070727
7
50
60
0.006792364
4
50
45
0.015297818
5
50
45
0.027075091
6
50
45
0.048521182
7
50
45
0.024983545
3
50
70
0.078021182
4.2
50
70
0.018756909
4.5
50
70
0.122773333
5
50
70
Table 5: Average Power vs Wind Speed for a Load of 50 at Different Angles Data shows the data
assembled from this experiment of average power in watts that range from 0.030 to 0.123, wind
speed in meters per second that ranges from 3 to 7, blade angle in degrees that range from 45 to
70 and load values in pounds. A single load of 50 pounds is used here. As the wind speed
increases, the average power increases also. The data from this table is graphically shown in
Figure 3 shown in results.
Table 6. Formulas for Calculating Power Factor and Advance Ratio
Cp = Max Extracted Power/Available Wind Power =
((8/27)pAc^3)/(1/2)pAc^3
Advance Ratio = rotor tip speed/wind speed
Table 6: Formulas for Calculating Power Factor and Advance Ratio contains the equations
obtained from the theory section to make calculations for our power factors and advance ratios
for all cases which are shown in Table 9 below.
Table 7. Variable Values
p(air density – kg/m^3) A(area – m^2)
c(m/s)
1.225
wind speed
0.19625
Table 7: Variable Values shows the values we used for our variables in the equation shown
above in Table 6 to calculate power factor. Wind speed was put in for the variable “c” in meters
per second.
Table 8. Calculated Rated Power per Adjusted Wind Speed for Different Angles
WS
Angle
Rated Power
9.053021633
45
0.110718357
4.137082651
60
0.110000969
3.749344491
70
0.110151054
Table 8: Calculated Rated Power per Adjusted Wind Speed for Different Angles shows the data
calculated from this experiment of rated power in watts that range from 0.11000 to 0.11072,
wind speed in meters per second that was manually adjusted anywhere from 3.5 to 10, blade
angle in degrees that range from 45 to 70. We used our trendline equations from Figure 3,
adjusted wind speed for x and achieved the rated powers presented in this table. For each
different angle, the given wind speeds produce an approximate rated power of 0.11Watts. The
data from this table is graphically shown in Figure 4 shown in results.
Table 9. Power Factor vs Advance Ratio for All Cases Data
Angle
Power Factor
Advance Ratio
Wind speed
45
0.035936196
2.331574822
4
45
0.023707966
8.068094013
5
45
0.016244947
8.545982851
6
45
0.011818054
9.123749517
7
45
0.009048209
9.259831418
8
45
0.007221188
9.431088773
9
45
0.005808255
9.512659555
10
60
0.047300576
19.10479194
3.5
60
0.034165916
21.42487328
4
60
0.023714969
23.55467601
5
60
0.016248988
21.53091161
6
60
0.011819997
18.36261229
7
60
0.047292793
19.29683891
3.5
60
0.034169116
21.61734119
4
60
0.023711847
23.78790456
5
60
0.016243515
21.52748788
6
60
0.011819765
18.36323449
7
60
0.047297801
19.48648085
3.5
60
0.034175074
21.83143523
4
60
0.023710321
23.7845757
5
60
0.016244971
21.52831466
6
60
0.01182079
18.36429219
7
70
0.060753933
16.31661487
3
70
0.035939633
20.42489573
4
70
0.028228853
22.21996718
4.5
Table 9: Power Factor vs Advance Ratio for All Cases Data shows the data calculated from this
experiment using the equations given in the theory section with power factor that ranges from
0.0058 to 0.0610, advance ratio that ranges from 2.33 to 23.788, wind speed in meters per second
that ranges from 3 to 10 and blade angle in degrees that range from 45 to 70. As the power factor
decreases in value, the advance ratio increases from 3.5 to 5 wind speed then decreases from 5 to
7 wind speed. We believe it does this due to the inaccurate data for this part. The data from this
table is graphically shown in Figure 5 shown in results.
Sample Calculations:
Wind Turbine Lab
I.
Background Information
Worldwide energy consumption continues to increase at an alarming rate. The U.S.
Energy Information Administration (EIA) recently predicted that the world energy
consumption would increase 48% by the year 2040[1]. While predictions for overall
energy consumption in the United States is considerably less, approximately 5% over
this same period, there is still a strong desire to switch to alternative systems due to
environmental concerns over traditional power generation methods, specifically those
dealing with the combustion of fossil fuels.
One of the fastest growing forms of alternative power production is in wind turbine
technology. The EIA predicts that wind energy production will increase by 12 and 14
percent in the U.S. over the next two years. The goal of this lab is to expose students to
the basics of wind energy production and to cover control strategies employed for
efficient operation.
II.
Wind Turbine Basics
There are essentially two basic types of wind turbines: Horizontal Axis Wind Turbines
(HAWT) and Vertical Axis Wind Turbines (VAWT). As the names imply the main
difference between the two are the axis about which the turbine blades rotate. Figure 1
below shows the two differing types of wind turbine design.
Figure 1. Horizontal and vertical axis wind turbines
The most common power producing wind turbines are HAWT’s which will be examined
during this lab. Basic components of HAWT’s include:
1. Support tower and nacelle
2. Rotor blades including pitch and yaw control
3. Drive shaft and gear box
4. High speed shaft and generator
Figure 2 shows some of these basic components.
Figure 2. Basic wind turbine components
In order to produce the largest amount of power at varying wind speeds it is essential to
develop control mechanisms capable of providing useful rotational speeds of the wind
turbine blades. Typically, two methods may be employed:
1. Electric or mechanical braking
2. Turbine blade pitch adjustment and yaw control
Method one is most effective in moderating small scale short term wind variation.
Method two is employed when large sustained wind changes occur. Yaw control is
used to insure that the turbine blades are positioned normal to the incoming air, while
blade pitch is used to adjust the force (lift) on the blades once the turbine is facing in the
correct position.
Wind turbine control strategies are designed for four general operating conditions,
based on the velocity of the wind.
1. Cut in velocity – This is the minimum wind speed needed to achieve useable
power.
2. Constant Cp region – In this region the blades are oriented to gain the maximum
possible power out of the incoming wind. The power coefficient will be discussed
in the theory section.
3. Constant output power – In this region the blades are orientated to provide the
maximum, or rated power of the turbine. Blades are oriented so that the turbine
does not overspeed.
4. Cut out speed – This represents the maximum speed at which the turbine can
operate. Above this speed the blades are positioned to receive as little lift as
possible and the turbine is mechanically locked in place.
Figure 3. Wind turbine operating regimes
III.
Theory
The available power for air moving at a velocity V can be found from equation (1):
1
1
𝑃𝑜𝑤𝑒𝑟 = 2 𝑚̇𝑉 2 = 2 𝜌𝐴𝑉 3
𝑚̇ is the mass flow rate of the air
V is the free stream velocity of the air
A is the area being considered
(1)
Figure 4 shows the pressure, velocity and area distribution of air moving through a wind
turbine.
Figure 4. Actuator disk schematic with pressure, velocity, and area distributions
illustrated.
The following derivation provides the steps needed to determine the maximum possible
power that can theoretically extracted by a wind turbine. Figure 4 is used as a
reference for the given subscripts.
Begin by writing the Bernoulli equation both upstream and downstream of the turbine
blades (but not across)
1
1
𝑃𝑒 + 2 𝜌𝑐 2 = 𝑃1 + 2 𝜌𝑐 2 (1 − 𝑎)2
(2)
𝑃2 + 2 𝜌𝑐 2 (1 − 𝑎)2 = 𝑃0 + 2 𝜌𝑐 2 (1 − 𝑏)2
(3)
1
1
Where
P is the pressure
ρ is the density
c is the free stream velocity
a and b represent fractional changes used to modify the velocity at the given location
Dividing by ρ and rearranging equations (2) and (3) become:
𝑃𝑒 −𝑃1
𝜌
𝑃2 −𝑃0
𝜌
1
= 2 𝑐 2 [(1 − 𝑎)2 − 1]
(4)
1
= 2 𝑐 2 [(1 − 𝑏)2 − (1 − 𝑎)2 ]
(5)
Noting Pe = Po
𝑃2 −𝑃0 +𝑃𝑒 −𝑃1
𝜌
=
𝑃2 −𝑃1
𝜌
1
1
= 2 𝑐 2 [(1 − 𝑏)2 − (1 − 𝑎)2 + (1 − 𝑎)2 − 1] = 2 𝑐 2 [1 − (1 − 𝑏)2 ] (6)
Simplifying we find:
1
𝑃1 − 𝑃2 = 2 𝜌𝑐 2 [1 − (1 − 𝑏)2 ]
(7)
Now the axial thrust (T) across the actuator is equal to the difference in force on the
upstream and downstream side of the blade:
𝑇 = (𝑃1 − 𝑃2 )𝐴 =
1
2
𝜌𝐴𝑐 2 [1 − (1 − 𝑏)2 ]
(8)
T is the axial thrust
A is the swept area of the blades
Using Newton’s second Law the axial thrust is equal to the axial change in momentum
as shown in equation (9):
𝑇 = 𝑚̇[𝑐 − 𝑐(1 − 𝑏)] = 𝜌𝐴𝑐(1 − 𝑎)[𝑐 − 𝑐(1 − 𝑏)] = 𝜌𝐴𝑐 2 (1 − 𝑎)𝑏
(9)
Simplifying:
1
2
𝜌𝐴𝑐 2 [1 − (1 − 𝑏)2 ] = 𝜌𝐴𝑐 2 (1 − 𝑎)𝑏
(10)
𝑏
𝑎=2
(11)
In order to determine the power extracted we consider the change in Kinetic Energy per
unit time:
1
1
𝐸𝑘 = 2 𝜌𝐴𝑐(1 − 𝑎)[𝑐 2 − 𝑐 2 (1 − 𝑏)2 ] = 2 𝜌𝐴𝑐(1 − 𝑎)[𝑐 2 − 𝑐 2 (1 − 2𝑎)2 ] (12)
Where:
Ek is the change in kinetic energy per unit time across the turbine blades
To determine the maximum possible power extracted we take the derivative of this
equation, with respect to the change in the wind speed at the blade, and set it equal to
zero and then solve for a:
𝑑𝐸𝑘
𝑑𝑎
= 0 = (1)(1 − 𝑎)2 + 2𝑎(1 − 𝑎)(−1)
1
𝑎=3
(13)
(14)
Using this result the maximum possible extracted power can be determined:
𝑀𝑎𝑥 𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟 =
8
27
𝜌𝐴𝑐 3
(15)
The Power Coefficient is a non-dimensional parameter used to measure the
effectiveness of a wind turbine. It is defined as:
𝐸𝑥𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟
𝐶𝑃 ≡ 𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑊𝑖𝑛𝑑 𝑃𝑜𝑤𝑒𝑟 =
𝐸𝑥𝑐𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑃𝑜𝑤𝑒𝑟
1
𝜌𝐴𝑉 3
2
(16)
Using the results provided in equation (15) the highest possible power coefficient can be
determined. This value is know as the Betz Limit and is given in equation (17):
𝐶𝑃𝑚𝑎𝑥 =
8
𝜌𝐴𝑐 3
27
1
𝜌𝐴𝑐 3
2
= 0.5926 ≅ 0.6
(17)
The Advance Ratio is another non-dimensional parameter relating the rotor tip speed to
the wind speed. It is shown in equation (18)
𝑟𝜔
Ω = 𝑉𝑤𝑖𝑛𝑑
(18)
It is often convenient to plot the Power Factor as a function of the Advance Ratio as
shown in figure 5.
Figure 5 Power Coefficient vs Advance Ratio for varying type of wind turbines.
The extracted power from the generator is calculated using the following:
𝑃𝑜𝑤𝑒𝑟𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 =
𝑉𝑎 𝐼𝑎 ∗𝑉𝑏 𝐼𝑏 ∗𝑉𝑎𝑐 𝐼𝑎𝑐
√3
(19)
Where V and I represent voltage and current respectively and the subscripts represent
each branch of the three phase system.
The frequency of the generator is calculated by:
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 =
𝑛∗𝑅𝑃𝑀𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟
120
(20)
Where n = the number of poles of the generator (8 for our case)
IV.
Startup Procedures
1. Make sure wind turbine is set to an angle of 60o.
2. Verify that all load rheostats are set at no load (zero) for each phase and
excitation
3. Turn MASTER Power Key to ON Position
4. Start system and chose a low wind speed. When wind anemometer starts to
spin, the WIND SPEED Meter will start to display data.
5. Slowly increase the wind speed until the turbine starts to move. Record this
value as the cut in speed.
6. Continue to increase the wind speed until the Generation Excitation begins to
display. Set this value at 7 Volts.
V.
Experiment
1. Measure and record the length from the center of the axis to the tip of a turbine
blade.
2. Set the windspeed at 3.5 m/s and set each rheostat at 10 ohms. After waiting 10
seconds, record data for 10 seconds.
3. Keeping the wind speed at 3.5 m/s repeat step one for loads of 25 ohms, 50
ohms and 75 ohms.
4. Record the cut-in speed at 50 ohms.
5. Repeat steps 1 and 2 for wind speeds of 4, 5, 6, and 7 m/s.
6. Shut the turbine down and change the wind turbine blade angle to 45 degrees.
7. Set the load to 50 ohms and record data at wind speeds of 4, 5, 6, 7, 8, 9 ,10
m/s. Also record the cut-in speed.
8. Shut the turbine down and change the wind turbine blade angle to 70 degrees.
9. Set the load to 50 ohms and record data at wind speeds of 3, 3.5, 4, 4.5 m/s.
Also record the cut-in speed.
ENGR 4470 – ME Experimentation Lab Lab Report Layout
I. Executive Summary
A one page summary of the entire experiment including results and conclusion.
II. Theory
Briefly list the objectives of the lab before discussing the theoretical or experimental equations
employed to complete the work.
This section contains all theoretical information needed to complete the analysis for the experiment.
Most of this information can be located in the manual or supplemental material. Incorporating material
from the course lecture, lab lecture, textbook, etc. is welcomed. Please cite where appropriate this
includes ideas, figures, equations, etc. that are not trivial or do not originate from the author you). This
should not be a copy and paste from the lab manual or the book.
III. Apparatus
Provide a picture and description of how the apparatus works. This will include Labview and
instrumentation.
IV. Results and Discussion
What results were obtained from the experiment and what do they show?
This is where you discuss the results, comparisons between different trials or theoretical values, errors,
etc. The question and answer section in the manual will provide the necessary observations from the
experiment that must be covered. However insight in to what you observed outside of this is also
recommended. V. Conclusions
Were all of the objectives met? Did the results prove or contradict the theory?
The final written section is the conclusion. In the conclusion comments are made about how the
objectives were met (or not met) and the outcome (numerical or otherwise). There should be NO NEW
information in this section. There should be numbers in this section.
VI. References List
as needed.
Appendix
This section contains raw data, sample calculations with UNITS, and any other relevant information that
can be referred to in the report.
ENGR 4470 – ME Experimentation Lab Lab Report Layout
I. Executive Summary
A one page summary of the entire experiment including results and conclusion.
II. Theory
Briefly list the objectives of the lab before discussing the theoretical or experimental equations
employed to complete the work.
This section contains all theoretical information needed to complete the analysis for the experiment.
Most of this information can be located in the manual or supplemental material. Incorporating material
from the course lecture, lab lecture, textbook, etc. is welcomed. Please cite where appropriate this
includes ideas, figures, equations, etc. that are not trivial or do not originate from the author you). This
should not be a copy and paste from the lab manual or the book.
III. Apparatus
Provide a picture and description of how the apparatus works. This will include Labview and
instrumentation.
IV. Results and Discussion
What results were obtained from the experiment and what do they show?
This is where you discuss the results, comparisons between different trials or theoretical values, errors,
etc. The question and answer section in the manual will provide the necessary observations from the
experiment that must be covered. However insight in to what you observed outside of this is also
recommended. V. Conclusions
Were all of the objectives met? Did the results prove or contradict the theory?
The final written section is the conclusion. In the conclusion comments are made about how the
objectives were met (or not met) and the outcome (numerical or otherwise). There should be NO NEW
information in this section. There should be numbers in this section.
VI. References List
as needed.
Appendix
This section contains raw data, sample calculations with UNITS, and any other relevant information that
can be referred to in the report.

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