1

Circular Rotor Windmill

Senior Design Team 7

Robert Brahm

Travis Feenstra

Ming Huang

Humberto Rojas

June 10th, 2005

Sponsored by Dr. Sanjay Sherikar

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Executive Summary

The use of renewable energy as an alternate to the consumption of limited fossil fuels is a

noble and necessary goal. By using solar, geo thermal, wind, hydro power the dependence on

crude oil and fossil fuels can be diminished. The scope of this project is to study the use of a

novel wind energy system that is based on the Magnus effect. The windmills purpose is to

convert wind energy into useable electrical power, and our design requirement is 100Watts.

The Magnus Effect is the lift that a spinning sphere produces in a cross wind. This effect

has been known for centuries with everyday applications such as the curveball and in the past the

trajectory of a cannonball. Our project deals with the application of the Magnus effect to a

spinnig cylinder in a wind velocity field, also known as the Kutta - Joukowski effect. The

application of this concept is the integral element for power generation in the design. The design

requirements are that the windmill produces 100 Watts of net power and have a cost approaching

$1/Watt. The budget for the project was $400.00 and the scope focused on the mechanical aspect

of the windmill and not the electrical or control side.

The actual design itself integrates the lift created by three rotating cylinders that are powered

by three direct current motors. The windmill transfers the generated lift from the spinning

cylinders into usable power by allowing the entire cylinder assemblies to rotate on a main hub.

This main hub is connected to a ¾ horsepower generator through a sprocket and chain drive with

a gear ratio of 3.3:1.

To make efficient use of the budget and time, detailed analysis was completed on various

aspects of the windmill. Finite element analysis was conducted on the structural integrity of the

integral parts of the design using the software program Cosmos Works. The two main questions

that have to be answered is how much power would the DC motors consume and how much

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power would be generated at different wind speeds. A Matlab code was developed that integrates

the data from NACA and calculates the power generated for different cylinder geometries. The

DC motor power consumption was calculated at 150 Watts. Using the Matlab code the optimized

windmill design incorporates three 6 inch diameter cylinders with a length of 24 inches. These

cylinders rotate at 2500 to 3000 RPM. The cylinder shafts are attached a hub that rotates at 150

to 200 RPM and generates a calculated output of 560 Watts in a constant wind speed on 28 mph.

As a result of the testing, the windmill speed reached steady state at 160 RPM, produced

335 Watts, and required an input of 450 Watts. The power produced by the rotating hub was

calculated based on the power consumed by the load bank consisting of 8 light bulbs. The power

input to the motors was calculated by testing the cylinders separately and taking an average for

all three. During the 30 mph wind speed test the average DC motor consumption was 150 Watts,

thus the total power required to spin the rotors was 450 Watts. The windmill consumes 115

Watts more than it generates. The success of the project was that the Magnus effect was applied

to power generation through this experiment and resulted in high lift generation. We did not meet

the design requirements of obtaining a positive net power. There were significant mechanical and

electrical losses. The mechanical losses were experienced in the friction of the bearings, and

ABS gears. The electrical losses occurred mainly in the slip ring which needs to be improved

with high temperature material. Further improvements to the design include a control system that

optimizes the system performance. With further improvement to the design net power is a

possibility, and with more advanced testing facilities advantages over current windmill may be

observed. Currently with the basic testing procedure advantages were not observed over

conventional windmills.

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Table of Contents

Page #

Introduction………………………………………………………………………………..….6-8 Problem Statement……………………………………………………………………………9-10 Details of the Final Design…………………………..…………….…………………..…….11-13 User Guide…………..………………………………………………………………….……14 Engineering Analysis…………………………………………………………………...........15-34 Design Evaluation……………………………………………………………………….…...35-54 Conclusion………………….…………………………….………………………………….55-57 References………………………………………………………………………….….…….58 Appendix A – Maximum Drag…….......…………………..…………………………..……59-61 Appendix B – Design Concepts…..........…………………..………………………..………62-82 Appendix C – Computer Programs and Printouts….…..…………………..………………..83-84 Appendix D – Breakdown of Cost……………….……………..…………………….……..85-86 Appendix E –Cylinder Lift Data……………………………………………………..……..87-89 Appendix F – Drawings………. ……………………………….……………………..…….90-112

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Introduction

The world has become more and more dependent on fossil fuels during the past century.

It was the dependency on these fuels that began the technology revolution and transformed the

world into what we know today. Even though fossil fuels have been very crucial in the past,

increased dependency and use would eventually exhaust existing supplies. Fossil fuel is energy

stored in the earth’s crust from many years of plant and animal decay. It took thousands of years

for these fuels to form and only a century for them to be depleted significantly. Due to the

greater demand and reduced supply we experience high prices for crude oil. With

industrialization of China and India, the remaining fuel reserves are in danger of being exhausted

as well.

With the increase in demand for fossil fuels and a decrease in reserves, new forms of

energy must be investigated. One promising form of energy is renewable energy. Renewable

energy is a resource that cannot be exhausted because its regeneration rate is greater than the

consumption rate. One form of renewable energy is stored in the pressure on the earth’s surface.

This energy can be found as wind and for centuries has been a powerful source of energy. Our

senior design project involves converting wind energy into mechanical work and then to

electrical power. Using existing windmill technology this task has been completed before. In our

design, a windmill will be designed that will use rotating cylinders to create lift instead of the

traditional windmill airfoil.

The design requirements are to develop a windmill that will produce 100 Watts net output

in a 28 mph wind. Furthermore, manufacturing costs should approach the $1000/KW price range

which is the wind power industry. The project is sponsored by Dr. Sanjay Sherikar. Currently the

windmill has been designed to incorporate 3 spinning cylinders rotating around a central hub.

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The cylinders are designed to spin at an angular velocity of 2000-3000 RPM and the hub, that

holds the three cylinders in place, is designed to rotate at 150-200 RPM. The cylinders were

estimated to consume 150-200 Watts each to rotate and the windmill was estimated to produce

400-500 Watts of power. With a net output of 200-300 Watts, the windmill would exceed the

100 Watt requirement.

In this report we will investigate the actual power required to rotate each cylinder. One

cylinder assembly was machined and assembled in preparation for the final windmill assembly.

To determine the adequacy of the design, tests were performed to determine the actual power

required to rotate a cylinder. The motor assembly, consisting of three motors, requires a power

supply with an input of 0-60 volts and 0-10 amps. By taking the voltage multiplied by the

amperage the power consumed was determined.

Using data obtained from NACA, the power required to rotate a cylinder was estimated at

50-75 Watts for rotational speeds of 2000-3000 RPM. The test resulted in a much larger power

requirement than originally expected. To spin the cylinder at 2500 RPM the apparatus used 153

Watts--this is 3 times the expected value. This power is consumed in gears, bearings, and motor

losses. In order to generate significant power in the windmill as a whole, the losses must be

decreased by at least 50%.

The report that follows will investigate the windmill design based on the power consumed

to rotate the cylinder. We will determine the major areas of concern as well as possible solutions

to any problems that arise. Furthermore, the future evaluation plan will be discussed to determine

how closely the final product approaches the engineering requirements. Lastly, the cost of the

entire project will be investigated, including labor and exact expenditures.

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This report provides the results of the testing and operation of our windmill along with

the calculations that were performed prior to the experiment. The windmill consumes around 115

Watts instead of generating a 100 Watts. The positive outcome of the experiment was that the

Magnus effect was verified and the concept was proved. The lift created by spinning cylinders

was successfully converted into useful power. The significant mechanical losses were due to the

friction in bearings and ABS gears and vibration in the steel shaft and tower. The electric losses

were found in the slip ring resistance and also motor inefficiency. If these losses were

eliminated, there might be a positive net output of power. Several improvements that could be

made to the windmill is a control system that maintains the surface to wind speed ratio by

controlling motor speed, determining cut-in and cut-out speeds to save power, and a

multidirectional yaw control system. The addition of end caps would help minimize radial losses

across the cylinders. The slip ring must be replaced with one made from high temperature

material.

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Problem Statement

It is a well known fact that by rotating a cylinder in a wind velocity field lift is created by

the Magnus Effect. This phenomenon is the backbone of the rotor windmill design. The problem

being addressed in this design attempts to answer the question: “Can the lift created by a rotating

cylinder be used to harness wind energy and develop a significant amount of power?”

The lift created by a rotating cylinder is explained by the Kutta-Joukowski effect. This

effect is similar to the Magnus effect that is used to describe the lift of golf balls and cricket

balls.

The Kutta Joukowski explains the lift over a rotating cylinder as seen in Figure 1.

Figure 1 – Kutta-Joukowski effect

The variables used are: s is the angular speed of cylinder, U is the wind speed, r is the radius of

cylinder, L is the lift, G is the Vortex Strength, rV is the speed of the rotating

Rotational Speed ( srVr π2= , [1]

Vortex Strength ( rVG rπ2) = [2]

Lift (L) = GUρ Eq [3]

These are the basic equations for lift using the spinning cylinder in a cross wind.

The specific design problem is how to use the lift created by spinning cylinders to design

a windmill that is competitive with current designs. Some of the advantages for a cylinder design

are economics, storm resistance, and simplicity of airfoil. The area where a spinning cylinder has

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a known advantage is in cost and simplicity of design. Horizontal axis windmills such as the

Darrieus design requires expensive extrusions for airfoils. These airfoils require large capital

investment with the possibility of waiting for several years before there is a significant return on

the investment. (Johnson. 2001) The cylinder is also simpler than the airfoils used on the current

horizontal axis wind turbines. Currently research is being done in order to either verify or

disprove the advantages of rotor windmills over modern windmills. The possibility of other

advantages that the cylinder might employ will have to be discovered through research. The goal

of our design group is to use the current advantages of the cylinder, while approaching the

efficiency and power generation of current wind turbines.

The scope of the rotor windmill is to focus specifically on the mechanical aspects of

design and avoid detail on the final electrical power produced. For example, the power produced

is not specific to alternating current or direct current. Furthermore, due to the steady-state

assumption of the wind the windmill an automatic windmill speed control will not be explored.

The design requirements are to produce 100 Watts net power output for a budget of less

than $400. Current windmills produce power for significantly less cost and are currently

producing 1 kW for every $1000 spent on windmill production. As a goal the windmill design

team will try to approach this cost efficiency.

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Details of the Final Design

Figure 2 is a picture of the final circular rotor windmill design. Our design has three

rotors and each rotor is designed to spin between 2500 to 3000 RPM in a 28 MPH wind. The hub

spins between 200 to 250 RPM with a sweep diameter of 5 feet. Each rotor is mounted on a steel

shaft at the hub plate at 120 degrees apart as shown in Figure 2. A Permanent-Magnet DC

generator converts the mechanical the power created by the lift of the spinning cylinders into

electric power. Three 50-150 Watt DC motors are mounted to the hub with brackets which

allows the cylinder shafts to spin freely. The power input to the motors is produced by the

generator. Figures 3 and 4 show the rotor assembly and generator set up respectively.

Figure 2 Circular Rotor Windmill

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Figure 3 shows the assembly for one cylinder. The rotor is mounted on a 2.5-ft long by 1

inch diameter steel shaft. The rotor is made of an ABS pipe measuring 2-ft long by 0.5-ft in

diameter which provides an aspect ratio (length of the rotor/diameter rotor) of 4. The rotor has

two aluminum caps on the ends. The outer diameter of the end cap is 1.5 times the cylinder

diameter. According to Whitford and Minard’s research on power created from lift from spinning

cylinders, a well designed rotor is achieved when the aspect ratio is 4 and diameter ratio is 1.5.

This combination provides high lift, low drag, and low power absorbed to spin the rotors. Each

motor assembly design has one gear mounted on a 50 Watt DC motor and one mounted on the

cylinder shaft as shown in Figure 3. A bearing is mounted between each cap and the shaft as

shown in Figure 3. The motor drives the cylinder using a gear system. A motor is mounted on the

shaft by bracket as shown in Figure 3.

Design DescriptionDesign DescriptionGears

Thrust Bearing

Pin

Rotor

Bearing BlocksBall Bearing

Motor Bracket

Figure 3 DC Motor Assembly

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Figure 4 shows the generator set up. The hub spins at about 150 to 200 RPM and drives

the generator to spin at about 500 to 660 RPM resulting in a 3.3:1 gear ratio. A generator is used

to transfer the energy created from the lift of the three rotors using a chain drive system as shown

in Figure 4. The energy collected is also rerouted to power the individual DC motors. A

generator and the bearing supports are placed on the mounting plate to hold the shafts in place.

One sprocket is mounted to a generator and one to the hub shaft. A chain is used to connect the

two sprockets and transfer the torque from the hub to the generator. A hub plate that supports the

six bearing blocks is mounted to the hub shaft. A ball bearing is mounted within each bearing

block supporting the shafts.

Design DescriptionDesign Description

The generator is connected to the hub through The generator is connected to the hub through a gear ratio of 3.3:1a gear ratio of 3.3:1Slip RingSlip Ring--4 Carbon Brushes and 2 Copper Rings4 Carbon Brushes and 2 Copper Rings

Slip ring

Generator Gear

Hub Gear

Cu Rings

Figure 4 Generator Set up

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User Guide

The windmill is a testing prototype and not a functional windmill ready for power

generation and stand alone operation. It is not designed for long term operation due

to the inefficiency of gears, ball bearings, and DC motors. The slip ring also can not operate

for long term, because of the low temperature material that it is made from.

Very important: Wear safety goggles and do not touch the the windmill during operation.

Gears and moving chains are dangerous and can cause injury.

This is how the prototype testing is conducted:

1. Set up the windmill in a high wind speed area range 15 to 30 MPH.

2. Face the windmill into the wind direction.

3. Connect the load bank or any DC load to the generator.

4. Connect DC power supply capable of supplying a total of 24 Volts with two 12 volt

batteries to the 3 DC motors to keep the cylinders spinning during operation. Two 12

Volt car batteries will be sufficient.

5. Make sure all the wire connections are connected properly, fuse is in place, and all the

windmill parts are assembled properly.

6. Turn on the power supply switch.

7. Allow windmill to accelerate to steady state.

8. Turn on the load bank/light bulbs switches on the load bank. The load uses the power

generated by the windmill.

9. Vary the light bulbs switches to control the load of the generator.

10. As the wind speed increases the output power will increase.

11. After half an hour.

a. Make sure that the two supply batteries are in full charge, if they are not

recharged them to 12 volts.

b. Check if all the windmill parts are safe to continue the operation.

c. If the motors are hot, let them to cool down.

d. Make sure that the windmill is in line with the wind.

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Engineering Analysis

In order to design a novel system that generates power from the kuttta Joukowski effect there

are several calculations that need to be performed. The three calculations that are featured in this

part of the report are the cylinder power consumption, the generator output and gearing, and the

structural analysis. The structural analysis deals with the sizing of the shafts and also the

vibrations calculations. Further calculations involving the angular speed of the cylinders to

maximize the lift can be found in Appendix E.

Cylinder Power Consumption and DC Motor Sizing

In this section, the type of motor, output power, and angular speed for the cylinder’s

rotation will be analyzed. The data used is from wind tunnel tests performed by NACA, National

Aeronautical Committee for Aeronautics. This data is used to calculate the power needed to

rotate the cylinders at various cylinder speeds. The cylinder speeds are determined by the need to

have the ideal cylinder surface velocity to the wind speed ratio of 2.5. In order to have a factor of

safety, the maximum shear stress was calculated from the NACA data and then applied to our

cylinder dimensions. The response of the actual motor was theoretically predicted using ideal DC

motor relationships between voltage, current, and angular speed. Finally the calculations will be

evaluated based on the actual power consumed by the motors to obtain the correct angular

velocity.

For our design we chose DC Motors with permanent magnets and brushes. The reasoning

for choosing DC motors is that they have a simple control functions and meet our needs for

adjustable speeds. In order to control the speed of the motor the voltage can be changed. This

relationship is linear for example as shown in Figure 5 as the voltage increases by a ratio of

30V/27V the No Load Speed (Speed of the motor with no applied load) increases from 10,500

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RPM to 11,722 PM which is the same ratio as the voltage increase. This means that for a

constant torque the voltage determines the angular speed of the motor. .

Figure 5 Torque vs. Speed and Current for a DC Motor. (www.Motortech.com) The next topic of analysis is what power we need to meet our criteria of keeping the cylinder

to wind speed ration at 2.5. The data that was used for power consumption of the cylinders is

provided by NASA. The power consumption of the cylinders is the amount of power that is

required to spin the cylinders by the DC motors. The data from NACA’s wind tunnel test is

plotted in Figure 6 .

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Power Consumption NACA

y = 6E-06x2 - 0.0001x + 7.6343R2 = 0.9865

y = 8E-09x3 - 5E-05x2 + 0.1032x - 50.139R2 = 0.99120

20

40

60

80

100

0 1000 2000 3000 4000

Rpm

Pow

er C

onsu

med

(Wat

ts)

15 m/s Wind0 m/s windPoly. (0 m/s wind)Poly. (15 m/s Wind)

Figure 6 Power consumed by a 5 ft long 4.5” Diameter cylinder

Using the NACA data we can calculate the sheer stress on the cylinders using the

following equations.

)()()( ϖOmemaTTorquePPower ×= [4]

)()()( rRadiusFForceTTorque ×= [5]

AreaCylinder

ForcedT

sSheerStres_

)()( =τ [6]

The power and angular speed for each data point is know and using eq 1 the Torque on

the cylinder can be solved. The force that creates this torque is solved using equation [2] and

then using equation [3] the sheer force can be solved for a known area. The ‘r’ is the radius of

the cylinder which is 4.5 in.

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15 m/s Wind Speed

Omega Power Cons. (W) Torque N*M

Sheer Stress N/m^2

1020 106.81 14.50 0.14 4.34 1115 116.76 15.50 0.13 4.25 1240 129.85 17.30 0.13 4.26 1500 157.08 23.80 0.15 4.85 1700 178.02 26.00 0.15 4.67 1900 198.97 28.40 0.14 4.57 2080 217.82 31.80 0.15 4.67 2220 232.48 28.60 0.12 3.94 2300 240.86 30.20 0.13 4.01 2420 253.42 31.90 0.13 4.03 2500 261.80 33.60 0.13 4.11 2600 272.27 34.80 0.13 4.09 2700 282.74 37.20 0.13 4.21 3000 314.16 44.80 0.14 4.56

Max 0.15 4.85 Table 1 Calculation from power to sheer stress. The power column is taken from NACA (Reid 17), and the Sheer Stress is solved by Eqs [##].

With the sheer stress at a max at 4.85 N/m^2 the force, torque and power can be

calculated using the design dimensions for the 6” diameter cylinder. The assumption made is

that the sheer stress is constant. The ABS cylinder that is used in the design is a 2 foot cylinder

with a diameter of 6 in. Using equation [3] with the max shear stress and the ABS cylinder

surface area the force on the cylinders edge can be calculated. Using equation [2] the torque can

be calculated. The final outcome using equation [1] can be seen in the following graph of the

angular cylinder speed versus the power consumed by the DC motors.

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Power Consumption

05

101520253035404550

0 1000 2000 3000 4000 5000

RPM

Wat

ts

Figure 6 Power Consumed by the 6” cylinders with Shear Stress of 4.8 N/m^2

Using Figure 6 the power consumed by the cylinders is between 30 and 35 Watts for an

angular speed of 2500- 3000 RPM. In order to meet this requirement the motors should have a

power output that is greater than 40~60 Watts at 3000rpms. The no load speed of the motor

should be above 5000 RPM since DC motors are most efficient at 1/3 their no load speed.

(Motortech). Using equation [1] the Torque is found based on the consumed power and the

RPM. The Average Torque for the Motors was 0.135 Nm or converted 19~ 20 oz.-in. In

conclusion, the motor should meet the following requirements in Table 2.

Torque 15 to 25 oz*in No Load RPM > 5000

Power Output 50 – 75 Watts @ 3000rpms

Table 2 Engineering Requirements

Using these requirements, the 3000 RPM Universal Weldon DC motor was chosen.

The graph below was made using the information that was given by the retailer C and H Sales of

Pasadena. The motors are rated at 24V and with the values given for no load speed of 7400 RPM

and a load of 50 oz in with 3000 RPM and 10 amps. As was discussed earlier a voltage change

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was directly proportional to the change in no load speed and also stall torque. Using this data the

line for 12V and 15V response was plotted.

DCM9752 Response Curve

0

2

4

6

8

10

12

0 20 40 60 80 100

Torque (oz in)

RPM

* 10

00C

uure

nt (A

mps

)

DesiredDCM9752 at 24 V DCM 9752 @ 15VCurrentDCM at 20 V

Figure 7 DCM9752 Response Curves

Utilizing the information on the graph is based on the torque for our cylinder is around 20

oz-in and running the DCM9752 at 15 V we get an output of 3000 RPM which meets the design

criteria of having a ratio of cylinder speed to wind speed of 2.5:3. Since the current is

proportional to torque there is only one current line. Reading this for the 20 oz in the value is 4.3

amps. Multiplying the amps time the voltage results in an electrical power input of 64.5 Watts.

The DC motor calculations were used despite their assumptions and inaccuracies in the

real world to correctly size the motors. Although the actual power consumed is on the order of

150 Watts and not 50 Watts the DC motors were still able to proved the cylinders with enough

angular speed to have the correct surface to wind speed ratio. Over sizing the motors and running

them at a lower voltage gave the design a factor of safety of around 3.0. It is this factor of safety

that overcame the discrepancy between the theoretical power and the actual power. If the DC

motors had been sized without a factor of safety they would not have been adequate for the

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design. The actual power consumed by the cylinder during testing was significantly higher due to

losses from friction between the gears and also the ball bearings. The gears were printed out of

ABS plastic due to budget constraints, and therefore did not have a perfect tooth profile. The

other major source of power loss was due to friction in the ball bearings. During the first

assembly of the bearings and shafts there was a press fit on both the inner and outer races of the

cylinders. This did not allow enough room for the balls in the bearings and resulted in

inconsistent motion. There were “flat spots” or areas where the bearing was difficult to turn on

previously installed bearings. By reducing the press fit on the inner race of the bearings the “flat

spots” were removed thus freeing up the bearings. Both the loss of power in the gears and

bearings accounts for the discrepancy in the predicted power consumed. Even though the

predictions did not match the performance of the system the calculations served their purpose of

correctly sizing the correct motor when a factor of safety of 3.0 was utilized.

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Generator Analysis

In order for the windmill to operate efficiently and effectively, it must be adequately

designed in many areas. One area that must be addressed prior to building a working windmill is

steady-state operation. Furthermore, the dynamic forces produced both in transient and steady-

state conditions must be analyzed in detail.

For all the windmill performance calculations, a 2 dimensional finite element method

(FEM) was used to determine the lift, drag, surface to wind speed ratio, and angle of attack for

100 nodal points along the length of the cylinder. A MATLAB program was developed to model

the system and provide both transient and steady-state analysis of the system. Using this method

the loads developed on the windmill were determined. A simple explanation is shown in the

following figure. A detailed analysis is shown in Appendix A , and a illustration of the problem

is seen in Figure 8.

W

U

V Drag Lift

Figure 8 Lift Analysis Details with Velocity Diagrams

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In order to determine the total lift at on each cylinder, the cylinder must be broken into

segments and the lift and drag determined. The lift and drag are based on the ratio of surface

speed relative to the wind speed. Because the cylinder is spinning around the hub, this ratio

varies along the length of the cylinder. The surface speed is simply the radius times the angular

velocity of the cylinder ωrVc = . The wind speed used in the ratio takes into account the actual

wind velocity as well as the velocity of each rotating node. The wind speed is actually the

relative velocity and can be seen in the figure above, with U being the rotational speed, and V

being the wind speed. W is the relative velocity obtained from U and V. Because U changes

based on the distance from the hub, the relative velocity also changes based on distance from the

hub. For this reason the cylinder was segmented and the surface to wind speed ratio determined

for each segment. Once the surface to wind speed ratio is determined, both the lift and drag at

each point were calculated from the coefficients of lift and drag obtained from NASA. These

coefficients are related to the rotating cylinder by the normalized surface to wind speed ratio

explained previously. Taking these coefficients, the lift and drag were determined for each node,

setting the cylinder rotational speed at a constant 3000 rpm. The lift and drag at each point along

the cylinder can be viewed in the following plot for 15, 20, and 30 mph.(Figure 9)

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Figure 9 Lift along the Cylinder Length for Various Wind Speed

The plot shown above shows the lift versus drag along the length of the cylinder. The lift minus

the drag at each point is the total net force at each node nodenodenode DragLiftF −= . For this

reason a large margin is desired between the two. From the plot it can be seen that the lift

decreases and drag increases along the length of the hub. Furthermore it can be seen that the lift

increases drastically for an increase in wind velocity. The drag also increases but at a slower rate

that the lift for an increase in wind speed.

In order for a windmill to create electrical power, it must be connected to a generator to

convert the mechanical work into electrical energy. The generator is crucial in the steady-state

performance of the windmill. The generator, under load, creates a resistant torque on the

windmill and determines the angular velocity. To size an appropriate generator for our

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application, we first had to determine the desired operating range of the windmill. This was

determined by plotting the windmill torque and power as a function of rpm. The torque of the

windmill is the net force at each node times the distance to the hub nodenodenode dFT ×= . The

nodal torques were then summed to get the total torque developed due to each cylinder

∑= nodecyl TT . Taking this torque and multiplying by the number of cylinders determines the

total torque at the hub. The torque at the hub varies drastically based on the windmill rotational

speed. This variance can be seen in the Figure 10. .

Figure 10 Tourque at Hub vs. Windmill RPM for 15, 20, and 30 MPH

It can be seen from figure 10 that the windmill torque decreases for an increase in angular

velocity. This will play a crucial role in the power output. Furthermore, the torque increases for

an increase in wind speed.

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The maximum power output of the windmill is determined by taking the torque at the hub

and multiplying by the windmill angular velocity. Because the torque decreases as the windmill

velocity increases, the power of the windmill reaches a maximum at some point within the range

of operation. This trend can be seen in Figure 11.

Figure 11 Power vs. for 15, 20, and 30 MPH

From the power versus windmill RPM plot, it can be seen that the power increases for an

increase in wind velocity.

Because the windmill is desired to operate at a constant wind speed and cylinder angular

velocity, the windmill gear ratio was determined at a wind speed of 28 mph which is the standard

for windmill sizing. The power and torque of the windmill is summarized in Figure 12.

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Figure 12 Windmill Output Power and Torque

Figure 12 shows a maximum torque of 46 N-m at 0 rpm. Due to the steady decrease in torque for

an increase in rpm the windmill power reaches a maximum power at 200 rpm. This can be

attributed to the relationship between torque and power. P = T*ω, where ω is the angular velocity

of the system in rad/s. For reference with Figure 12 ω = ωrpm*2π/60.

To obtain a desirable output from the generator, it must develop a balance of torques with

the windmill at around 200 rpm. Because cost is a factor in the windmill design and low rpm

generators are characterized by high costs, a generator is desired that will operate adequately

with a gear ratio of less than 10:1.(Norton) This will reduce costs both of the windmill and also

enable the windmill to operate within the desired range without adding an expensive gear train to

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develop high gear ratios. A DC motor was chosen as the generator because it is relatively low in

cost and can be used in reverse as a generator. The motor chosen has an output power of ¾

horsepower and a no load speed of 1750 rpm. To obtain the stall torque of this generator, it was

assumed that the torque curve is linear versus rpm. With an output of ¾ watts and a no load

speed of 1150 rpm an iterative process was used to determine the stall torque. The torque and

power curves for the generator are shown in Figure 13 and Figure 14.

Maximum Generator Torquey = -0.0162x + 18.63

R2 = 1

02468

101214161820

0 500 1000 1500RPMs

Torq

ue (N

-m)

Figure 13 Generator Torque vs. RPM

28

Maximum Generator Torquey = -0.0162x + 18.63

R2 = 1

0

5

10

15

20

0 500 1000 1500RPMs

Torq

ue (N

-m)

Figure 14 Generator Power

By taking the plots of the windmill and generator torque and graphing them on the same

plot, the steady-state operation was determined. Steady-state is characterized by zero

acceleration and for the windmill this point is reached when the windmill torque equals the

generator resistant torque. The steady-state operation and power output is shown in Figure 15.

Figure 15 Balance of Torques

29

The balance of torques takes place as 330 RPM and develops a power output of 360 Watts.

This power meets the design requirements but the efficiency of the windmill is not satisfactory as

power of approximately 580 Watts is available. To increase the efficiency of the windmill a gear

ratio is introduced to increase the resistant torque of the generator. Figure 16 shows the system

performance with a gear ratio of 3.5:1.

Figure 16 Optimizing the Gear Ratio

The 3.5:1 gear ratio decreased the system rpm and increased the efficiency of the

windmill to an output of 560 watts. With the chosen generator, the windmill could not reach the

580 watt maximum because the resistant torque of the generator overcame the windmill torque

output.

30

In order to determine the steady-state angular velocity of the windmill, the windmill was

placed in wind of 3 different velocities and the voltage output of the generator measured at

steady state under full load. The voltage was compared to rpm of the windmill by taking the

generator and coupling to the load bank. The generator was powered with a drill motor and the

corresponding voltage and rpm measured with a digital multimeter and tachometer. Taking the

gear ratio between the windmill and the generator, a plot of voltage versus rpm was developed.

This can be seen in Figure 17 which shows the steady state rpm vs. voltage output.

Steady State RPM vs. Voltage Output (Full Load)

y = 18.739x - 6.9791R2 = 1

020406080

100120140160

0 2 4 6 8 10Voltage (V)

Ang

ular

Vel

octit

y of

Win

dmill

(RPM

)

Figure 17 Steady State RPM vs. Voltage

The total power developed in the windmill was determined by applying a voltage to 2 of the

light bulbs used in the load bank and measuring the corresponding current. The relationship

between voltage and current can be seen in the Figure 18.

31

Load Bank Voltage vs Current for 2 Light Bulbs

9, 9.3

0

2

4

6

8

10

12

0 2 4 6 8 10 12

Voltage

Am

ps

Pow

er x

10

Current

Figure 18 Load Bank Voltage vs. Current

Once the voltage and current were developed for each rpm of the generator, the power was

determined for each voltage. This can be shown in Figure 19.

Power Produced vs. Voltage with 2 Light Bulbs

9, 83.7

0

20

40

60

80

100

120

0 2 4 6 8 10 12

Voltage

Pow

er W

atts

Figure 19 Power Consumed by 2 Light Bulbs as a function of Voltage.

Now that we have power for each voltage value, we can determine the power output for the 15,

20, and 30 mph wind. The results are summarized in Figure 20.

32

Steady-State Power Output vs. Wind Speed

050

100150200250300350400

15 20 30

Wind Speed (mph)

Pow

er O

utpu

t (w

atts

)

Figure 20 Power Output for Windmill at Various Windspeeds

It can be seen that the total power produced at steady-state increases in a nonlinear fashion as

wind speed increases. This can be attributed to the total kinetic energy stored in the wind,

322

21

21

21 AvvAvmvKE ρρ === . The total power is proportional to the velocity of the wind

cubed.

The resistant torque of the generator at the windmill hub is shown in the following figure.

It can be seen that the relationship between torque and rpm is constant. The faster the windmill

rotates the more resistance is applied by the generator. This is seen in the following Figure 21.

33

Torque vs. RPM

0

5

10

15

20

0 50 100 150 200

Angular Velocity (RPM)

Torq

ue (R

PM)

Figure 21 Windmill Torque vs RPM

Power vs. RPM

050

100150200250300350400450

0 50 100 150 200

Angular Velocity (RPM)

Pow

er (w

atts

)

Figure 22 Windmill Power vs. RPM

The power output of the generator versus angular velocity of the windmill is shown in the

figure below. It can be seen that the power increases as the rpm increases. It is a second order

relationship because ωTP = and T is linearly related to ω as shown previously.

From the previous two plots for the actual resistant torque on the windmill hub it can be

seen that the calculations performed for the generator previously were done in error. The

34

generator is a DC motor used in reverse, it has a torque as the driving force rather than an input

voltage. This difference in driving force changes the dynamics of the system. There is no stall

torque in a generator, because at zero velocity the generator produces zero voltage which

corresponds to zero load and power output. As the angular velocity increases the voltage also

increases in a linear fashion, this results in a linear resistant torque. Furthermore, the amperage

increases in a linear fashion. Because both the voltage and current increase in a linear fashion for

an increase in wind speed, the power increases in a second order fashion. This can also be seen in

the mechanical aspects of the system. Torque and angular velocity are linearly related which also

corresponds to a second order increase in power. Taking this analysis into account, the balance

of torques and total power used to determine the gear ratio did not optimize the performance of

the system. Further analysis of these calculations can be seen in the test evaluation of the design

where the data that was collected from testing is presented and then compared to the results of

the theoretical calculations.

Structure Analysis – FEA Analysis This section of the analysis deals with the structural integrity of the design as well as the

dynamic vibrations of the system. Both of these are an integral part of the stability for the

windmill. The shafts that support the windmill need to be able to withstand high wind conditions

and also a cyclical load. The vibrations analysis is important in order to determine the natural

frequency of the rotors and to ensure that we are not running the cylinders at an angular speed

that equals the resonance frequency.

A critical component of the design is how to mount the spinning cylinders around the

windmill’s hub. What is the optimum size or shape of support that can hold the cylinders and still

be able to incorporate bearings? Since the cylinders will be spinning at speeds up to 3000 RPM

35

the support must also have minimal deflection. The dimension of the main shaft from the hub

must also be sized according to load of both torque and weight of the hub and cylinder

assemblies.

In order to determine the strength of the shafts a Finite Element Package, called Cosmos

Works was used to perform the analysis. The analysis was performed using the strain energy and

static conditions to solve for the Von Misses Stress using a mesh. This means that the part is not

moving and the hydrostatic forces do not cause motion but deformation. The assumptions that

are made for this analysis are that we have steady wind, and incompressible flow. The factor of

safety is the ratio of yield stress to the max stress in the model which should be greater than 2.

When this is less than 1, the part has yielded and has left the elastic condition and cannot return

to its original shape. All analysis was done using a solid mesh and not shell elements which are

employed for thin walled parts. All analysis was also done using the material of Alloy Steel with

the properties shown below in Table 3.

Table 2 Alloy Steel Properties

The applied force was calculated in two ways. The first was the amount of lift that we

achieved using the spinning cylinders with a constant wind velocity of 15mph. This force came

to be 27 N. The derivation of this force can be seen in Appendix A. The other method used to

Property Name Value Elastic modulus 2.1e+011 N/m^2 Poisson's ratio 0.28 Shear modulus 7.9e+010 N/m^2 Mass density 7700 kg/m^3 Tensile strength 7.2383e+008 N/m^2 Yield strength 6.2042e+008 N/m^2 Thermal expansion coefficient 1.3e-005 /Kelvin Thermal conductivity 50 W/(m.K) Specific heat 460 J/(kg.K)

36

verify the load was the drag of our cylinders under severe storm conditions of 70 mph wind with

the cylinders not spinning. The value of this force was 29.2 Newton’s as seen in Appendix C

which is 6.8% from the force created by the lift. The 27 N force created by the lift will be used as

the design load cylinder shafts. With a factor of safety the steel shafts will still support the

cylinders with winds up to 75 mph. The cylinder shaft is going to be sized first. This shaft is

shown in Figure 22 during the testing phase.

Figure 22 Cylinder Shafts during Testing

Using the force of 27 N at the end of a cantilever beam of circular cross section with one

end constrained the results were plotted by SolidWorks as can be seen in Figure 23 showing the

displacement of the 1” shaft with an wall thickness of .028”. The deflection at the end of the

shaft is 0.4 cm. The factor of safety for this design is 9.7. The results for several different designs

is shown in Table 4

37

Figure 23 Graph of the deflection of a shaft with 1” OD with .028” wall thickness.

Table 4 FOS, Max Deflection, and Mass of the Cylinder Shafts with different ODs.

Our initial design called for a half inch solid shaft but as the results in Table 4 show it

would deflect 1.4 cm at the end of the shaft. This is why the hollow tube with a larger diameter is

advantageous for this situation. This effectively brought the factor of safety up, max deflection

down, and a decrease in mass weight. This is accomplished by moving the material away from

the axis of neutrality and thus increasing the moment it can withstand. Taking the analysis into

consideration, we will have to change our solid shaft design and use the more rigid and light 1”

Piping with a wall thickness of 0.028”. This will give rigidity and weigh half as much solid half

inch solid shaft. The final result for this calculation is that a redesign of the shafts with a hollow

tube of 1” diameter and with a wall thickness of 0.28.

OD inch

Wall thickness(inches)

Max Deflection (cm) FOS Mass lbs

3/8 Solid 4.7 2.4 0.92 ½ Solid 1.4 5.7 1.64 ¾ 0.095 0.4 13 1.63 1 0.065 0.2 19 1.59 1 0.028 0.4 9.7 0.71

38

There is also the shaft on the windmill that transfers the power from the rotating hub to

the generator. This shaft was analyzed using the same method as the cylinder shafts but instead

of seeing how much it deflected due to loading we were interested in the amount it rotates using

the torque applied by our spinning cylinders. This shaft can be seen in Figure 24 and runs

parallel to the generator and perpendicular to the rotating hub.

Figure 24 Generator Shaft Used for Analyzing Rotational Loading

The applied torque to the shaft was 19.3 Nm calculated in Appendix A. The applied

torque is was found with the cylinders creating full lift and the generator shaft fixed. This is the

max torque that the shaft will experience, because under normal operation the generator shaft

will be allowed to rotate based on the resistance torque of the generator. Since the lift is max

when the cylinders are not moving, this is the max torque. The max displacement for the 1/2”

solid shaft is shown in Fig. 25 and the analysis results are shown in shown in Table 5.

39

Figure 25 -Factor of Safety for the Generator shaft of 3/4” diameter.

Figure 26 - Deflection using a ¾ inch solid allow steel shaft.

Shaft

Diameter Max Deflection

(cm) Factor of

Safety Mass kg

1/2 .189 4.5 0.405 3/4 0.0365 15 0.645 1 0.0117 36 0.981

Table 5 - Results for Generator Shaft Analysis

The engineering analysis shows that all three designs would not yield with the design load that

was used. The ½ inch shaft would be rigid enough with a deflection of only .189 cm and a factor

of safety of 4.5. The 1 inch shaft would have a factor of safety of 36 and deflect less than a tenth

of a mm. The 1 inch shaft is sized too large for the application, and the .189 cm of deflection will

40

likely cause a “wobble” or unbalanced rotation. The ¾ inch shaft has a high enough factor of

safety and will only deflect 0.03 cm which is 6.3 times smaller than the 1/2 inch shaft. Thus for

the design the ¾” shaft meets our specifications and has a factor of safety that will cover extreme

conditions.

The structural calculations were sufficient for the windmill design. The cylinder shafts had

not noticeable deflection. There was some play in the bearings which allowed for the cylinders to

deflect. The main hub shaft was also sufficiently sized. There was no noticeable deflection in the

solid 0.75” shaft. In order to actually verify the calculations the beam would have to be tested

with a bending test apparatus. The point of these calculations was not to find the exact deflection,

but to size the shafts sufficiently in order to handle the load both of the shear stress from the

weight and the torque.

Structural Calculation (Vibrations)

Now that the static characteristics are analyzed the vibration analysis is performed in order

to find the natural frequency of the rotating cylinders. This is needed to ensure that the cylinders

are not run resonance for an extended period of time. The calculations that were developed was

that the natural frequency is the square root of the spring constant (k) over the mass (m).

Natural Frequency)(

)tan_()(massm

tconsspringkn =ϖ [6]

In order to use this equation the mass and the spring constant need to be calculated. The mass

was calculated using the Solid Works program for solid modeling. The spring constant was

calculated by using the equation

41

3

3LEIK = [7]

K= Spring Constant

E= Young’s Modulus for Steel

I = Mass Moment of Inertia for the Beam Shape

L= Beam Length

Using these equation and the data seen in Appendix G the natural frequency of the system were

analyzed. The result is that the natural frequency of the cylinders is at 240 rad/sec or 2290

revolutions per minute. Since the cylinders are designed to run at 2500 to 3000 rpm’s the

cylinders are predicted to pass through the resonance.

These calculations were useful in predicting the natural frequency of the cylinder assembly.

During cylinder testing the cylinder had two resonance frequencies one was noticed around 1000

rpms and the second one was noticed at 2000 rpms. This resonance was 14.5% difference from

the 2290 rpms that were predicted. The analysis did not predict the first resonance frequency.

This could have been found if the cylinder assembly had been treated as series of elements with

damping and stiffness elements. The actual cylinder had slight vibrations at the resonance but it

was not catastrophic to the windmill’s operation. Once the cylinders had angular speed greater

that 2000 rpms the amplitude of the vibrations was too small to visualize. This natural frequency

could be changed for further research by adding weight saving holes to the end caps. This change

in mass would increase the natural frequency.

42

Design Evaluation

Input Testing: Shaft With No Cylinder

Figure 27 Motor Gear Assembly During Testing

In order to determine how much power the motors would be drawing, it was necessary to

conduct tests on each one of the motor assemblies. Figure 27 shows the testing apparatus that

was used to determine how much current each 50-150 Watt DC motor would draw under varying

voltages. A relationship was formulated between voltage, current, and RPM by varying the input

voltage from 0-24 volts. The input power was reduced by 20% when lubrication was added to

the ABS plastic gears. The motor assembly consists of a DC motor mounted on top of two

bearing blocks by a motor mount. This motor mount is fastened to the outer bearing block by

four set screws. Each bearing block has a ball bearing inserted in order to hold the shaft in place.

At the inner end of the shaft, a thrust bearing prevents the shaft from sliding outward. The motor

spun at angular velocities ranging from 2000 to 2500 RPM under testing conditions. Special

tachometer tape was stuck on the steel shaft for the tachometer to take readings. As the motor

spun, a tachometer was used to measure RPM.

43

Input Testing: Shaft With Cylinder

Figure 28 Motor Assembly With ABS Cylinder

Figure 28 shows the assembly with the ABS cylinder mounted on the shaft, held in place

by the outer and inner end caps. These end caps prevent air flowing through the cylinder causing

loss of lift. These end caps also prevent the plastic cylinder from sliding outward. Using the same

method as shown for the previous figure, values were determined for voltage, current, and RPM

by varying the voltage. These values were noted and are shown in the results section of this

report. The motor input obtained was noticeably larger and was in agreement with the increase in

inertia to rotate the shaft together with the cylinder. There were fewer vibrations because of the

increase in mass.

44

Testing Procedure

Figure 29 Windmill Mounted with Support Ties

After the windmill was fully assembled, it was necessary to test and confirm whether or

not the device would give the desired output. To test the windmill, it was first mounted on truck

bed of a pickup and support ties were used to hold the base securely in place. This would prevent

the vibrations from the windmill from making it tumble out of the pickup. Data was taken from

within the cab of the vehicle for safety precautions. The windmill was mounted facing towards

the rear of the vehicle to provide easy access to the wires coming from the generator.

45

Figure 30 Two 12 Volt Batteries In Series

To spin the cylinders during operation, two 12 volt batteries were wired in series to

produce a total power output of 24 volts. These were deep cycle boat batteries. As can be seen

from Figure 30, the red and green wires come from the generator and lead into the cab of the

vehicle through the window.

Figure 31 Custom Load Bank

A custom loading bank was built consisting of eight 60 Watt light bulbs were connected

to the outputs of the generator to measure power. A series of three switches were mounted on the

board to vary the load of the generator. A connecter was also mounted on the board so that the

series of light bulbs, light switches, and generator inputs could be wired together.

46

Figure 31 - Vehicle Traveling at 15, 20, and 30 MPH

The entire vehicle was driven at speeds of 15, 20, and 30 MPH to generate and simulate

live operation of the windmill in the respective wind speeds. Readings were taken when the

vehicle maintained the necessary speed. An assumption was made that the wind traveling over

the cab and into the windmill would not be significantly different than if the cab was not present.

Figure 32 Voltmeter Setup for Measuring Voltage

Readings of voltage and RPMs were taken as the vehicle maintained speeds of 15, 20,

and 30 mph. The custom load bank was also turned on and different loads were placed on the

windmill. The spinning rotors were given time to spin up to full speed before the load bank was

activated so that the windmill would have an initial rotating inertia.

47

Testing Results

This section of the report is looking at the actual power consumed by the spinning

cylinders, and whether the calculations accurately predicted the phenomena. For the cylinder and

motor assembly the calculated power consumption was approximately 50 Watts. Because this

calculation was performed without taking into account the inefficiencies due to the ABS gears,

ball bearings, and out of round shafts, the motors were oversized based on our calculation. The

motor calculations called for 15 V in order to meet the calculated torque of 20 oz in with an

angular speed of 3000 rpm’s. In Figure 32 the actual power that the cylinders consumed versus

the angular speed is shown.

Angular Speed vs Power

0

50

100

150

200

250

300

1500 2000 2500 3000 3500

Angular Speed (RPM)

Pow

er (W

atts

)

Figure 32 Angular speed vs Power

In reality with the added torque due to the friction in the bearings, and gears the power

that the motors consumed was 100 to 150 Watts. This is 2.5 to 3 times the 50 Watts that the

motors were designed to operate. Therefore instead of the total input voltage of 150 Watts the

three cylinders consumed 450 Watts. Here is a graph of our expected operating range and our

actual operating range. The actual operating range is based on the fact that it took about 17-20

Volts to produce angular speeds of 2500 to 3000 rpm’s.

48

Figure 33 Motor Response Curve

Due to the fact that the friction in the gears and DC power supply that measures both the

input voltage and input current. As the evaluation procedures explains the only measurement for

the input power was the voltage. Two 12 Volt car batteries supplied the input 24Volts. The actual

voltage that the motors received was less than the input 24 Volts. By measuring the resistance

between the power supply and the motors the voltage drop across the cylinders could be

calculated. The resistance was measured by a DMM from the wire connectors to the batteries to

the motors. This takes into account the resistance between the brushes and the copper rings in the

slip ring. The resistance was measured at .09 Ohms to .26 ohms as the windmill was turned. One

thing to note is that this resistance is a function of how fast the entire windmill is turning. So the

resistance changes as the windmill angular velocity changes. Since there is no feasible way of

spinning the windmill at 160 rpm’s and also measuring the resistance at the motors the value of

.26 Ohms is assumed and will be used for further calculations. Figure 34 shows the current and

voltage as a function of angular speed.

DCM9752 Response Curve

0

2

4

6

8

10

12

0 20 40 60 80 100

Torque (oz in)

RPM

* 10

00C

uure

nt (A

mps

)

DesiredDCM9752 at 24 V DCM 9752 @ 15VCurrentDCM at 20 V

Theoretical

Actual

49

Cylinder Assembly Response

2500,9Amps

2500,17 V

0

5

10

15

20

25

1500 2000 2500 3000 3500

Angular Speed (RPM)

Volta

ge V

Cur

rent

Am

ps

Current

Voltage

Figure 35 Single Cylinder Voltage and Current vs. Angular Velocity

Using the current at an average of 10 Amps at 20 V input, and the resistance value of .26 Ohms

the voltage drop across slip ring can be found using Ohms law of Voltage = Current x Resistance

= 30 Amps * .24 Ohms = 7.2 Volts. This gives the input for the cylinders at 24V – 7.2 Volts =

16.8 Volts. Using 17 Volts as the standard input voltage for all three cylinders and Figure 36, the

total power consumed by one cylinder is approximately 150 Watts.

Pow er Consumed by 1 Cylinder Assembly

17, 153

0

50

100

150

200

250

300

10 12 14 16 18 20 22 24

Voltage

Pow

er (W

atts

)

Figure 37 - Power Consumed by Single Cylinder vs. Voltage

50

Evaluation of Generator Output Testing and Calculations

This section summarizes the testing evaluation and ties it back to the calculations for the

generator output. Due to errors in generator analysis, the steady-state operation of the windmill

was not a realistic summary of the system performance. By taking the measured torque and

power output of the generator, the actual output can be calculated. The windmill torque and

generator resistant torque are plotted against each other and the balance viewed 3 different wind

speeds at Figure 23. The torque balance occurs at 180, 210, 230 rpm for 15, 20, 30 mph winds.

Figure 38 Theoretical Windmill Torque and Resistive Torque vs. Windmill RPM

The theoretical power output is shown in the plot below. It can be seen that the power

developed at each steady-state rpm was calculated as 360, 460, and 580 in a 15, 20, and 30 mph

wind. When compared with the actual power output shown below it can be seen that some

discrepancies took place. (Figure 39)

51

Figure 39 Hub Power vs. The Windmill Rpm

By comparing the theoretical output shown above with the actual output shown below the total

error in the calculations were determined. The plot below shows the power output and angular

velocity of the windmill at steady-state during the performance tests. For wind velocities of 15,

20, and 30 mph the steady-state angular velocity was 30, 70, and 170 rpm. Furthermore, the

power output was approximately 25, 60, 330 for the same wind speeds.

Steady-State Power Output vs. Wind Speed

050

100150200250300350400

15 20 30

Wind Speed (mph)

Pow

er O

utpu

t (w

atts

)

Figure 40 - Output Power Experimental vs Wind Speed

52

The percent differences between the theoretical steady-state velocity and that actual steady-state

velocity are shown in the Table 6.

Wind Speed

Theoretical Steady-State RPM

Actual Steady-State RPM

% difference

Theoretical Steady-State Output

Actual Steady-State Output

% difference

15 180 30 83 360 20 9420 210 70 67 460 60 8730 230 170 26 580 330 43

Table 6 Calculations Evaluation Actual vs. Theoretical

It can be seen for the steady-state output varies from between 43-94%. This can be attributed

to the fact that the coefficients of lift and drag were obtained in a controlled environment during

wind tunnel testing. The lift and drag obtained during these experiments were approaching the

maximum obtainable lift because negative environmental impacts were minimized. Furthermore,

the lift was reduced due to radial losses of wind energy off the tip of the cylinder. Implementing

end caps would increase the efficiency of the windmill by reducing the radial energy losses.

53

Conclusion The need for renewable energy sources continues to be a topic of research as the price of

gasoline and fossil fuels increases. The world’s supply of fossil fuels is limited and therefore

renewable clean energy sources will continue to play an integral part in the future. This Senior

Design Project began almost nine month ago with the simple problem statement, “Can the lift

generated by the Kutta Joukowski effect be used to efficiently generate power?” The purpose of

the project was to create a prototype that could verify the novel approach of spinning cylinders to

the generating power from the wind. The design course was split into three quarters. The first

was focused on research, the second focused on the detailed design, and the third quarter focused

on building, implementing, and testing the design prototype. The project has been complete

according to the schedule and built according to the detailed design.

The testing that was completed in the final quarter was done with basic testing procedures.

In order for more accurate testing of the windmill there are two necessary requirements. The first

requirement is that the windmill has a control system that can measure the speed of the wind and

keep the cylinder at the optimum speeds without imputing excess power. The second

requirement is that the windmill is tested in a wind tunnel that has predictable wind speeds.

Without either of these two abilities the implementation of the design was tested using the wind

generated by a moving vehicle and the input voltage from two 12Volt batteries.

The results of the testing were that the cylinders consumed 450 Watts of power and the

generator output 335 Watts. The test proved that the Magnus effect was effective at generating

lift. The windmill operated at steady state with an angular speed of 160 rpm. This is the point

where the windmill generates the most power, although it still consumes more power than it

generates. The calculated value for the power input was 150 Watts which is a third of the actual

54

power consumed by the 450 Watts. There are several reasons why the power consumed was

significantly higher than expected. The two main areas of losses are the mechanical and

electrical losses. The mechanical losses are found in the gears, bearings, and vibrations in your

steel shafts. The electrical losses are found in the connections, slip rings, and DC motors. All of

these losses contribute to the high power consumption of the cylinder assemblies.

On the power generation side of the cylinder the output power was calculated at 560

Watts and the actual measured power was 330 Watts. The calculated power was not obtained for

two main reasons. The first is that the coefficient of lift and drag were obtained from NACA

experimental data that used a wind tunnel for testing. This controlled environment does not take

into account all of the dynamics in the real world. Turbulence from the moving truck and the

actual outdoor wind environment creates non laminar or turbulent flow over the cylinders. The

wind tunnel testing was performed using laminar air flow across the cylinders. Another reason

for the loss of power was the need for end caps on the cylinders. End caps could cut the

transverse flow of air off the cylinders as the windmill spins. By adding the end caps the lift

would increase and thus increase the power output.

The Kutta Joukowski Effect was successfully verified and its ability to create lift and

therefore power. By taking all of the efficiencies into account and adding a control system

currently there are not any apparent benefits over conventional windmills. On a large scale the

machinery to spin the cylinders would be costly and heavy. In order to higher efficiency the

spinning of the cylinders could be linked mechanically to rotating hub. Another possibility is the

use of a hybrid or half cylinder half conventional windmill. These are the areas that seem the

most promising for further research into the field of power generation using the lift generated by

a spinning cylinder.

55

References Asmus, Peter Reaping the Wind How Mechanical Wizards, Visionaries, and Profiteers helped Shape our Energy Future. Island Press Washington D.C. (2001) Cousteau Ships: Alcyone daughter of the wind. Cousteau Publications 2001. http://www.cousteau.org/en/downloadfiles/alcyone_and_turbosail.pdf Gipe, Paul. Wind Power- Renewable Energy for the Home Farm and Business. Chelsea Green Publishing Company. Vermont. 2004 Johnson, Dr. Gary L. Wind Energy Systems. ( www.eece.ksu.edu/~gjohnson/) 2001 Kleinstein, Gerald. On Irrotational Motion Past a Rotating Cylinder in a Viscous Heat Conducting Fluid. 1989 Pergamon Press. NASA - Lift of a Rotatinng Cylinder-Glenn Research Center. http://www.grc.nasa.gov/WWW/K-12/airplane/cyl.html Rizzo, Frank, The Flettner Rotor Ship in the Light of the Kutta Joukowski Theory and Expiremental Results. Langley Memorial Aeronautical Laboratory- National Advisory Committee Aeronautics. Technical Notes No. 228.

56

Appendix A This purpose of this calculation is to find the maximum drag on the cylinders with out the

windmill turned on. This is for the structural stability of the steel shafts that are used to support

the spinning cylinders. This is the force in storm when the windmill is turned off and the only

force on the shafts is the drag from the cylinders in the wind. This calculation uses the

Coefficient of Drag from Figure 9.21 in the Fundamentals of Fluid Mechanics (Munson 9.22)

which is based on Reynolds number. The Reynolds number is the ratio of viscous forces to sheer

forces and can be found using the equation.

Re(Reynolds Number)= μρVl (Munson p.20) [1]

ρ= density of air;

V = velocity of the fluid;

l=characteristic length;

μ= dynamic viscosity; Getting the values for Cd from Figure 9.21 and μ from the Fundamentals of Fluid Mechanics (Munson p. 1 & 582) we solved for Drag= .5 AVCd 2

21* ρ (Munson eq. 9.38). [2]

Using these equations the force that will be applied as drag when our cylinders are at rest in a 75 mph wind or 33m/s is 29N as shown in Table 5.

Variable Value Density of Air

kg/m^3 1.23 Cylinder Diameter

(m) 0.15 Cylinder Length (m) 0.61 Dynamic Viscosity 3.74E-07 Wind Velocity m/s 33.0 Reynolds Number 1.65E+07

Cd 0.46 Frontal Area m^2 0.1 Drag in Newtons 29.2

Table 1 Drag on Cylinder in 75 mph wind

57

Appendix B

#1 Friction Gears

The wind blows in the –z direction and the hub is spun by a motor in the clockwise

direction. The cylinders spin in the direction following the arrows. As the wind blows into the

windmill, the cylinders pick up lift. A battery initiates the rotation of the hub, which in turn

rotates all of the cylinders. As lift is produced, the cylinders perpetuate the motion and the motor

acts as a generator recharging the battery.

58

#2 Hybrid Design

59

This design composes of a mixture of the current windmill technologies and or spinning

rotor. To start with the cylinders are not touching the friction plate, gear track. The conventional

blades force the windmill to turn. Once the windmill is turning at a fast enough rate the cylinders

are allowed to contact the friction plate and the cylinders are caused to spin. This spinning

cylinders cause lift themselves and cause the windmill to rotate. This lift is caused by the

Magnus effect and the difference in pressure from the bottom of the cylinder to the top. The

generator is on the main hub of the windmill and converts the torque from the cylinders and

propellers to electricity. Advantages of this design are that is self starting, and does not require

complex control logic. The faster the wind the faster it rotates on the friction plate or gear track,

and the thus the faster the rotors spin. The optimum range is between 2-2.5 blade tip to wind

speed, so with a fast wind the cylinder tip must also be spinning from 2-2.5 times that fast. The

downside to this design is that there is no way of precisely keeping the ration at the optimum

point. This would be preset for an optimum condition. All in all this design is an attempt to mesh

the current technology with the design idea of spinning rotors.

60

Sketch for one plate

Assumptions: The power to initialize the spinning of the cylinder is already calculated based on the coefficient of drag and lift (see appendix B and C). The flat plate interpolation coefficient of Drag of 0.009 at 45 degrees angle based on Reynolds number and speed air of 28 (m/sec). Calculations of power from one flat plate: Drag (D)

ACUD d2

21 ρ= (1)

airofDensitymkg 31=ρ angleCatDragoftCoefficienC o

d 45009.=

SpeedAirmU sec28= platetheofAreamA 215.0=

( )sec10*458.1 25 m−=ν plateofDiametermD 15.0=

Reynolds 510*88.2==ν

DV

WattsDrag 5292.0=

Power (P)

ACUDUP d3

21 ρ== (2)

WattsP 82.14=

wattsPPsystemtheofPowerTotal cylinderplate 36.70)50*2()82.14*2(*2*2 −=−=−= 70.36 watts are required to initialize the spinning of the cylinders

61

Explanation:

This Hybrid design has two cylinders and two flat plates. The plates are attached to the

hub at 180 degrees apart. The cylinders are attached to the hub by a bearing at 180 degrees

apart, and they get in contact with the friction gear above certain velocity. Each flat plate is at 45

degrees angle as shown in figure. Each cylinder spins in its own bearing. The two flat plates

provide power to initialize the spinning of the motors. Once the spinning of the cylinders is

initialized they create power based on the coefficient of lift using the Magnus effect and pressure

differences from the top to the bottom of the cylinder (see appendix B and C for calculations).

The maximum power produces from one cylinder is when the cylinder tip speed is 2.5 faster than

the wind speed. It there is not wind speed the system will not produce power because the two

flat plates produces power based on the wind speed.

62

Feasibility: This design is not feasible based on the analysis. The two plates do not require the power

to initialize the spinning of the two cylinders. Each plate produces approximately 14.18 watts

based on the assumption of the coefficient of drag. Each cylinder requires 50 watts to initialize

its spinning. The calculations show that the system requires 70.36 watts to initialize the spinning

of the cylinders. Even if one motor that produces 35.82 watts is attached to each cylinder, this

design will not be efficient as the design with three cylinders. One cylinder produces three times

as much power produces by the two plates. After the spinning of the cylinders is initialized the

design with the three cylinders will produces more power than this design with two cylinders and

two plates.

This hybrid design has advantages and disadvantages like other designs. One of the

advantages of this design is that it does not use motors to initialize the spinning of the two

cylinders. The power produces from the two plates is used to power the two cylinders. The cost

of the windmill will decrease because the two cylinders do not require motors. One of the

disadvantages of this design is that the two flat plates at 45 degrees angle do not produces power

as cylinders do even though the two plates do not require power to initialize their rotation. Other

disadvantage is that the two plates only produces power based on wind speed. This design will

not produce power is if there is not wind speed. This design also requires a stronger tower to

prevent the two cylinders and the two flat plates from collapse at high wind speed.

63

#3 Friction Plate Windmill

VARIABLE GEAR RATIO

VARIABLE DIAMETER CYLINDER

GENERATOR

PISTON CONTROL

The windmill design shown above uses different diameter rotating cylinders along the

length of the rotor. This concept is implemented in order to maintain a surface to wind speed

ratio of 2.5 along the entire rotor length. Though there are slight surface to wind speed changes

along each cylinder segment the design decreases variations. Benefits of this design are reduced

machining costs when compared to a continuously varying diameter cylinder. Furthermore, to

further maintain the surface to wind speed ratio, a control system is implemented in the form of

motors that control the angular velocity of each cylinder. In addition the motors are capable of

initiating rotation by introducing lift to the previously stationary rotor. A downside to the motor

control is external power is required to initial rotation.

64

#4 Motor-Driven Straight Cylinders

The wind blows in the –z direction. Each cylinder is powered by its own motor and

begins to spin. An external power source, such as a battery, is used to begin the spin on the

cylinders. As the wind blows, the hub begins to rotate because the cylinders create lift. The

generator connected to the hub, as it is spinning, begins to charge the external battery source

(battery). Since the lift produced by the spinning cylinders is greater than the power required to

spin the cylinders, there is a net positive output of power.

65

#5 Horizontal Axis Hybrid

ROTATING CYLINDER

MOTOR

GENERATOR

DRAWING NOT TO SCALE

The above figure is a possible windmill design for the Circulation Rotor Windmill senior

project. The design utilizes motors to both start the rotation and maintain a constant surface to

wind speed ratio across each cylinder. This design may be promising due to the decreased

rotational speed requirements of the apparatus while maintaining proper generator rotational

speed. Furthermore, because the motion of the cylinders is mostly horizontal, tapered cylinders

are not required to maintain constant lift along the length of the cylinder. To further increase the

lift capabilities, end caps are applied to each rotor to reduce losses along the length of the

cylinder.

66

#6 Variable Diameter Motor Control

The windmill design shown above uses different diameter rotating cylinders along the

length of the rotor. This concept is implemented in order to maintain a surface to wind speed

ratio of 2.5 along the entire rotor length. Though there are slight surface to wind speed changes

along each cylinder segment the design decreases variations. Benefits of this design are reduced

machining costs when compared to a continuously varying diameter cylinder. Furthermore, to

further maintain the surface to wind speed ratio, a control system is implemented in the form of

motors that control the angular velocity of each cylinder. In addition the motors are capable of

initiating rotation by introducing lift to the previously stationary rotor. A downside to the motor

control is external power is required to initialize rotation.

67

#7 Variable Controlled Rotor Windmill

This design concept has many elements to it. The rotation of cylinders is caused by motors

mounted on the hub. The spinning cylinders cause lift by the Magnus Effect and thus rotate

around the hub. The generator takes this rotation and turns it into electricity and some of that

electricity is used to power the rotation of the cylinders. There is a control system that takes the

wind speed and causes the cylinders to spin at the optimum rate to create the greatest lift

compared to the drag. The cylinders are tapered to allow for a more similar blade tip to wind

speed along the cylinder. This is useful because this ratio changes as you progress farther out

along the cylinder because of the rotation around the hub. This makes the velocity of the outer

edge of the cylinder to be greater than at the inner part near the hub. In effect this design

compensates for the variableness caused by using a cylinder as an airfoil by keeping a more

constant tip to wind speed ratio.

68

#8 Continuous Rotor Design (No control Speed)

This is a continuous rotor idea design. The speed of this design is not regulated. The diameter

of the cylinder increases from the motor to the end of the cylinder in order to approximate the tip

speed to wind speed ratio of 2.5. This design has four electric motors; one motor is attached to

each cylinder. Each motor power one cylinder in order to initialize it’s spinning.

Hybrid Design Feasibility and Analysis Calculate the power for one plate of the hybrid

69

Assumptions: The power of the cylinder is already calculated. Coefficient of Drag of 0.009 at 45 degrees angle based on Reynolds number and speed air of 28 (m/sec). Drag (D)

ACUD d2

21 ρ=

airofDensitymkg 31=ρ angleCatDragoftCoefficienC o

d 45009.=

SpeedAirmU sec28= platetheofAreamA 215.0=

( )sec10*458.1 25 m−=ν plateofDiametermD 15.0=

Reynolds 510*88.2==ν

DV

WattsDrag 5292.0=

Power (P)

ACUDUP d3

21 ρ==

WattsP 82.14=

wattsPPPowerTotal cylinderplate 36.70)50*2()82.14*2(*2*2 −=−=−= The analysis and calculations show that the total power from the system is -70.36 watts. Each plate produces 14.82 watts and each cylinder requires 50 watts input to initialize its spinning. From the system 70.36 watts are required to initialize the spinning of the cylinders.

70

#9 Flat Cylinders on Rotating Track

This design works by the Kutta Joukowski effect. The wind is flowing in the –z direction and

the lift in the front cylinders is upward. The cylinders are moving upward and when they reach

the top they have to be spun the other way to create lift downward. The rotors are moving at a

slower rate than spinning on a hub like a windmill. There is no variableness of the tip to wind

speed as in the rotary windmill. The downside is that they have to be spun the other way for them

to work on the backside. The power generation is on the bottom with generators that are powered

by the moving track. The lift of the cylinders causes the track or chain to move and that is

converted to power by a generator. This power is used partially to power the spinning of the

cylinders which will take some amount of power. The idea is that there will be more power

generated than consumed.

71

#10 Madaras Track Control System

This design has four electrical generators, one on each wheel. The power is extracted

from the system by these generators. The cylinder is initialized its spinning by an electric motor.

As the car goes around the track the cylinder spins in the same direction of the wind as shown in

the diagram. When the wind is parallel to the track, the cylinder stops the spinning and starts

spinning in the opposite direction. So the cylinder spins in one direction for half of the cycle, and

it spins in opposite direction for the other half of the cycle. The car must be strong enough so the

drag force will not overturn.

72

#11 Savonius Rotor Calculations

One of the designs that were chosen as the final four viable choices was the Savonius

Rotor. The Savonius rotor windmill is composed of a few rotors with Savonius rotors attached at

the base of each cylinder closest to the hub. These Savonius rotors will spin at respective wind

velocities in order to create enough energy to power each cylinder. It is assumed that enough

power can be produced out of the Savonius rotors at the same wind velocity as required to

produce lift by the cylinder.

The Savonius design uses the self spinning Savonius Rotor in order to spin the cylinders.

This is shown in the cross section view next to the rotor. The Savonius works because there is

less drag on the rounded section then the scooping section. This change in drag force causes the

Savonius to spin. The Savonius will spin the rotors faster as the wind increases and as the

windmill spins faster. The power is generated around the central axis which all the cylinders

73

rotate around. This allows us to harvest wind energy using the Savonius which does not require

input power. The downside of this design is the lack of control and the low surface to wind speed

of the Savonius Rotor.

Analysis of this design begins by determining whether or not a Savonius rotor of

proportional size to the cylinder is sufficient to provide enough power to rotate the cylinder.

Calculations will be based on a Savonius rotor per rotating cylinder ratio in order to reduce

calculations to common units. The Savonius rotor system is assumed to be operating at steady-

state thus eliminating static friction of the shaft. Kinetic friction is also assumed to be zero as it is

included in the safety factor. The fluid temperature is assumed to be room temperature (25

degrees C). All dimensions were calculated using standard SI units.

According to online research, the height to radius ratio corresponded to rb 3= . For the

sample calculation, a radius of 3 in. (0.0762m) was used and its height = 0.2286m. According to

the optimal wind speed for modern turbines of 28 mph (12.52 m/s), air was taken as the flowing

fluid. The coefficient of drag for a half cylinder on the frontal area was 3.2=DC and 1.1=DC for

the rear area. By performing the calculations using the equation: ∑ −= rearfrontnet FFF , the Ffront

and Frear needed to be found. The calculations would be simplified if the summation of forces

was applied to one resultant force.

The calculation for the Frontal Force (according to Figure 1) is shown by a modified

equation [1]. This value represents the force applied to section A of the Savonius rotor windmill.

( )( ) mNmmsm

mkgAUCF Dfront ⋅=⎟

⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛= 8604.32286.0762.0517.1223.1

213.2

21 2

32ρ

The calculation for the Rear Force (according to Figure 1) is shown by a modified

equation [1]. This value represents the force applied to section B of the Savonius rotor windmill.

74

( )( ) mNmmsm

mkgAUCF Drear ⋅=⎟

⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛= 84629.12286.0762.0517.1223.1

211.1

21 2

32ρ

By plugging in the values for Ffront and Frear in equation [2] produces 2.0141N net force.

∑ =−=−= NFFF rearfrontnet 0141.284629.18604.3

( ) mNNmFrT ⋅=⎟⎠⎞

⎜⎝⎛=×= 07674.00141.2

20762.0

( ) 70.6)/4.87(0767.0 =⋅== sradmNTPower ω Watts

The power required to rotate each cylinder is about 40 Watts (See report for cylinder

analysis). The power required to initiate rotation on each cylinder exceeds the amount that can be

produced by the Savonius rotor at the desired wind velocity. The Savonius rotor does not provide

sufficient power and thus renders this design infeasible for further analysis. Theoretically,

producing 40 Watts would require a massive Savonius rotor.

The Savonius rotor would be better than the other designs since it can auto-start, which

means an external power source is eliminated. This design would then be able to produce power

without being manually initiated. The disadvantages of this design are the low optimal rotation

speed, lighter weight, and originality of design. Savonius rotors alone are not designed to run at

high angular velocities further more for use on a circulation windmill to power cylinders.

75

Summary of Calculations

Fluid Properties Equations

Kinetic Velocity = 251085.1

msNxair⋅

= −μ [1] Drag Force = AUCD D2

21 ρ=

323.1mkgDensity air == ρ [2] Net Force =∑ −= rearfrontnet FFF

Fluid Velocity =sm

hrmiU 517.1228 == [3] Torque = ωrT =

Angular Velocity =ω = 100 rpm = 10.47 rad/s [4] ωTPower =

Assuming fluid velocity as 12.517 m/s and an overall 3 inch radius.

[5] ( )( ) mNmmsm

mkgAUCF Dfront ⋅=⎟

⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛= 8604.32286.0762.0517.1223.1

213.2

21 2

32ρ

76

[5] ( )( ) mNmmsm

mkgAUCF Drear ⋅=⎟

⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛= 84629.12286.0762.0517.1223.1

211.1

21 2

32ρ

[2]∑ =−=−= NFFF rearfrontnet 0141.284629.18604.3

To calculate torque produced by the Fnet of 2.014 N we apply equation [3] and [4]:

( ) mNNmFrT ⋅=⎟⎠⎞

⎜⎝⎛=×= 07674.00141.2

20762.0

( ) 70.6)/4.87(0767.0 =⋅== sradmNTPower ω Watts

Analysis of the Savonius rotor indicates that the amount of energy produced is not

sufficient to power the rotation of an equal size cylinder.

Power vs Overall Radius

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3 3.5

Radius (in)

Pow

er (W

atts

)

77

Appendix C Torque_vs_Omega.m function[Ttot,Cd,Cl,P,Lift,Drag,R,T,Sr,theta,F_rot,ww_rpm,T_gen]=Torque_vs_Omega(d,L_1,L_2,N,Nc,v,wc,wwi,wwf,div); %This program calculates the power and lift generated by a windmill based %on the input dimensions. It takes values of lift at a specified number of %points along the cylinder. Using these points we can get a good %understanding of different characteristics along the length of the cylinder. d; %diameter of cylinder (m) L_1; %length from hub to cylinder base (m) L_2; %length of cylinder (m) N; %number of nodal points Nc; %number of cylinders v; %wind velocity (m/s) wc; %cylinder (rad/s) wwi; %windmill (rad/s) (need generator data) wwf; y=(wwf-wwi)/div; for i=1:div, ww(i)=wwi+i*y; x=(1:1:N); %generates nodal points R=L_1+(((L_2/N).*x)-L_2/(2*N)); %distance from hub at each node U=R.*ww(i); %velocity of node about axis of rotation theta=atan(v./U); %angle of attack at node W=v./sin(theta); %relative velocity at node Sr=((d/2)*wc)./W; %surface to wind speed ratio at each node Cd=-0.0297.*(Sr.^3) + 0.3024.*(Sr.^2) - 0.6192.*Sr + 0.9368; %coeficient of drag at each node Cl=-0.2996.*(Sr.^3) + 1.9355.*(Sr.^2) - 0.6003.*Sr - 0.0097; %coeficient of lift at each node Lift = 0.5*1.23*((L_2/N)*d).*((W.^2).*Cl).*sin(theta); %lift at each node Drag = 0.5*1.23*((L_2/N)*d).*((W.^2).*Cd).*cos(theta); %drag at each node F_rot = (Lift-Drag); %total lift for one cylinder T = (R.*F_rot); %total torque for one cylider Ttot(i) = Nc*sum(T); %total torque all cylinders P(i) = Ttot(i).*ww(i); %total power angular velocity of windmill assumed (need generator data) end ww_rpm=(60/(2*pi)).*ww; T_gen=((-0.0014.*ww_rpm)+13.721).*(1/0.688); plot(ww_rpm,Ttot); xlabel('Windmill RPM'); ylabel('Torque at Hub (N-m)'); hold on; plot(ww_rpm,P,'r');

78

hold off; power_calc.m function[Ttot,Cd,Cl,P,Lift,Drag,R,T_rot,T_tow,Sr,theta,F_tow,F_rot,F_mag,Force_Tower,F_torque,P_betz]=power_calc(d,L_1,L_2,N,Nc,v,wc,ww); %This program calculates the power and lift generated by a windmill based %on the input dimensions. It takes values of lift at a specified number of %points along the cylinder. Using these points we can get a good %understanding of different characteristics along the length of the cylinder. d; %diameter of cylinder (m) L_1; %length from hub to cylinder base (m) L_2; %length of cylinder (m) N; %number of nodal points Nc; %number of cylinders v; %wind velocity (m/s) wc; %cylinder (rad/s) ww; %windmill (rad/s) (need generator data) x=(1:1:N); %generates nodal points R=L_1+(((L_2/N).*x)-L_2/(2*N)); %distance from hub at each node U=R.*ww; %velocity of node about axis of rotation theta=atan(v./U); %angle of attack at node W=v./sin(theta); %relative velocity at node Sr=((d/2)*wc)./W; %surface to wind speed ratio at each node Cd=-0.0297.*(Sr.^3) + 0.3024.*(Sr.^2) - 0.6192.*Sr + 0.9368; %coeficient of drag at each node Cl=-0.2996.*(Sr.^3) + 1.9355.*(Sr.^2) - 0.6003.*Sr - 0.0097; %coeficient of lift at each node Lift = 0.5*1.23*((L_2/N)*d).*((W.^2).*Cl).*sin(theta); %lift at each node (direction of rotation) Drag = 0.5*1.23*((L_2/N)*d).*((W.^2).*Cd).*cos(theta); %drag at each node (negative direction of rotation) Lift_P = 0.5*1.23*((L_2/N)*d).*((W.^2).*Cl).*cos(theta); %lift perpendicular to rotation Drag_P = 0.5*1.23*((L_2/N)*d).*((W.^2).*Cd).*sin(theta); %drag perpendicular to rotation Force_Tower = Nc*(sum(Lift_P)+sum(Drag_P)); %total force (direction of tower) F_rot = (Lift-Drag); %component of force at each node (direction of rotation) F_tow = (Lift_P+Drag_P); %component of force at each node (towards the tower) F_mag = ((F_rot.^2)+(F_tow.^2)).^(1/2); %resultant force at each node T_rot = (R.*F_rot); %torque at each node (direction of rotation) T_tow = (R.*F_tow); %torque at each node (towards the tower) Ttot = Nc*sum(T_rot); %total rotational torque on power shaft due to all cylinders F_torque = (((sum(T_rot)/(L_1+L_2))^2)+((sum(T_tow)/(L_1+L_2))^2))^(1/2); %equivalent force at tip of cylinder to develope torque on shaft P = Ttot*ww; %total power given angular velocity

79

Appendix D

Accurate Budget As with all projects in industry or research the budget is an important aspect. An accurate

budget plan is needed to allocate the proper amount of the often scarce resources toward a

project. An accurate budget during the project is needed to determine whether the budget plan is

on track with the allocated resources for the project. For the Rotor Windmill project the budget

planning was 400$. This was a set value without any planning or foresight as to the actual cost of

the project, or its components. The budget amount of $400 determines the precision and

efficiency of the purchased components. The more money allocated, the higher precision that can

be achieved. In order to comply within the allocated resources initial designs for the windmill

were changed from steel spur gears to printed ABS gears. This design change decreased the

budget by $180, which is the cost of Steel Spur Gears. Other material was donated by the

machine shop and KH Metal. The rest of the parts and material were purchased from various

companies such as McMaster Carr and Grainger. The complete listing of the cost for purchased

parts and material for custom built parts is shown in Table 3. The current total of $680 is $280

over our budget.

80

The most expensive items that we had to purchase were the ball bearings totaling

$124.62. These were fairly large bearings required to hold our shafts in place and prevent them

from sliding outward. The second most expensive part for our windmill was the PM DC

generator purchased from eBay. This updated budget includes the cost of building the loading

bank, the wood tower, and the redesigned slip ring.

81

Appendix E This Appendix looks at the correct angular speed of the cylinders, in order to generate the

optimum lift. This looks at the data from NACA, National Advisory Committee for Aeronautics.

These tests were conducted at Langley Memorial Aeronautical Laboratory in 1924. ( Elliot 1924)

The data uses a non-dimensional number called the wind speed ratio (Rs). Rs is the velocity of

the surface of the cylinder divided by the windspeed. This is seen in the following equations.

Rs = [1] Here is a plot of the lift versus the velocity ratio.

Coefficient of Lift vs Velocity Ratio

y = -0.2996x3 + 1.9355x2 - 0.6003x - 0.0097R2 = 0.9963

-2

0

2

4

6

8

10

0 1 2 3 4 5

Vs/Vw

Coe

ffic

ient

of L

ift

CLiftPoly. (CLift)

Figure 1 Lift Vs. Velocity Ratio Now that the lift has been plotted against the velocity ratio the drag also needs to be plotted against the velocity ratio.

⎟⎟⎠

⎞⎜⎜⎝

⎛=

w

c

w

s

Vr

VV ω

82

C of Drag vs the Velcity Ratio

0

0.20.4

0.60.8

1

1.21.4

1.6

0 1 2 3 4 5

Vs/Vw

Coe

ffici

ent o

f Dra

g

DragPoly. (Drag)

Figure 2 Coefficient of Drag vs. Velocity Ratio

In order to get the max lift over drag both of the ration of D

L

Cc

was plotted against the velocity

ratio. This plot is shown in the next figure and shows that there is a maximum velocity ratio.

This is because the drag increases as the velocity ratio increases.

Figure 3 D

L

Cc Plotted against Velocity Ratio

The range that the cylinders have the maximum ratio of lift over drag is between 2 and 3 for

the velocity ratio. This data is used for the lift that is generated by our cylinders, and also the

angular speed that is required by the motors.

C. Lift/ C. Drag vs. Vs/Vw

-10123456789

0 1 2 3 4 5Vs/Vw

Cl/C

d

83

Estimated Time

The parts that required the most time to manufacture were the FDM gears due to the time

required to print all 6 gears. The speed at which the machine can print was the limiting factor.

The bearing assembly took longer than the expect 2 hours to complete because they needed to be

re-press fitted in order to allow clearance between the bearings and the shaft. The slip ring took

longer than expected to complete because the original one was not able to handle the current

going through the windmill. It was redesigned and printed with the FDM machine. The load

bank was added towards the end of our project and only took 2 hours to complete. Total

completion time for the building of the windmill was about 85 hours.

84

Appendix F

85

Figure 2 Cylinder Assembly Figure 2 Generator Assembly

86

Figure 3 Slip Assembly – (Note the generator assembly is not updated with the new slip ring design)

T

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

Appendix G - Vibrations Calculations Data This appendix is a summary of the Vibrations calculations for the spinning cylinders. The caclultions were designded to explore the natural frequency of the cylinder assembly using a spring mass system.

This section changes the windmill and rotor rpms to rad/sec Windmill (rpms) (rad/sec)

220 23.04 Rotor RPM

3000 314.16 This section uses the mass found from Solid Works for our Parts to calculate the Mass

Mass lbs Mass (kg)

Radius in Radius m

Moment Nm

End Cap (AI) 0.55 0.25 24 0.61 1.5E+006" ABS Tube 2.8 1.27 17 0.43 5.4E+00

Middle Round (AI) 0.558 0.25 17 0.43 1.1E+00Outer Flange ABS 0.61 0.28 25 0.64 1.7E+00Inner End Cap (AI) 0.71 0.32 8 0.20 6.4E-01

Cantilever Beam 1" Steel Mb 1.63 0.74 30 0.76 5.5E+00 Mass tot kg 3.11 Σmoment 1.6E+01

Conc Mass 2.1E+01

24 0.61 6 0.15 in m R2 1 2.54E-02 R1 0.937 2.38E-02 t 0.063 1.60E-03

K for Steel Tube first 6" Inertia (I) kg m^2 = (pi*d^3*t)/8 4.18E-07 Young's Mod (E) Pa AISI 1018 matweb 1.90E+11 Flexural Rigidity = E*I (kg N) 7.95E+04 Spring Constant N/m K1 3EI/L^3 6.74E+07 Natural Frequency rad/sec sqrt(k/m) 4653.4 9456.9

K for the ABS Cylinder + Steel Tube 24" Mass Moment Cylindr (pi*d^3*t)/8 5.30E-06 Youngs Modulus ABS Pa 2.50E+09 Spring Constant Cylinder K2 N/m 175517.3 Spring Constant Steel Tube K3 N/m 1.05E+06 K in Parallel K4= K2+K3 N/m 1.23E+06 K in Series K5=K4+K1/(K1+K2) 1.21E+06 Natural Frequency (rad/ sec) 240.9 RPM = 2260

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