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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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