Vertical Axis Wind Turbine
Testing
ii
Declaration
I hereby certify that this material which I now submit for assessment on the program of study leading to the award of “Bachelor of Mechanical Engineering Honours Degree”, is entirely my own work and has not been submitted for any academic purpose other than in partial fulfilment for that stated above.
Signed: ___________________________ Robert Mc Auley
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Acknowledgements
The objectives set out for this project were ambitious and required a large amount of work covering many different areas of the engineering profession. The success of the project was greatly helped by a number of individuals who provided invaluable advice in their respective fields. These individuals helped the progress of this project by dedicating their time to providing prompt responses to any queries regarding certain phases of the project. I would like to thank the following people for helping me to make this project a success.
Dr. Fergal Boyle Project Supervisor
Sean Keane Rapid Prototyping Technician
Martin Byrne Lab Technician
I would also like to thank my friends and family for the help they provided throughout the course of the project, in particular Hugh O’Reilly for his help with the wind tunnel testing, Stephen Kirwan for his help with the spin down testing and L.A Mc Auley Ltd. for the use of their metal fabrication facilities.
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Abstract
The aim of this project was to carry out performance testing on a vertical axis wind turbine (VAWT) designed by a company called Brí Toinne Teoranta. The first section of the project involved a background study into the history of wind turbines and wind energy at a global level. Once a clear understanding was developed for the importance of sustainable energy sources, an investigation was then carried out into the testing of wind turbines.
A manufacturing process had to be specified for the designers’ complex blade geometry. A detailed investigation was carried out into the area of rapid prototyping and 3D printing processes which were available at DIT Bolton Street. Several prototypes of the blades were manufactured before a manufacturing process was decided upon for the final blades.
A testing methodology was developed by carrying out research into the different methods of applying torques and measuring the power produced by a VAWT. A decision was made to proceed with the development of a Prony brake apparatus and a magnetic particle brake (MPB) apparatus. A test rig was designed and fabricated to accommodate both of these testing methods. The test rig was designed to work specifically with the wind tunnel situated in DIT Bolton Street. A careful approach was taken to the wind tunnel testing of the VAWT, with preliminary tests being carried out to ensure that the apparatus was working safely and correctly. Phase 1 of the Prony brake testing was carried out providing performance curves for the turbine at different wind speeds. The mechanical losses in the test rig were assessed and reduced before commencing phase 2 of the Prony brake testing. The performance curves for the VAWT were converted into dimensionless form and an experiment was designed to calculate the magnitude of the mechanical losses in the bearings of the VAWT test rig. Several components were designed to allow testing to be carried out using the MPB. The MPB wind tunnel testing was documented along with any issues which arose.
All of the data from the wind tunnel testing was carefully analysed and documented in a way that would accommodate future testing using the designed testing methodologies and test rig.
v Contents Chapter 1: Introduction ... 1 1.1 Project Background ... 1 1.2 Project Aims/Objectives ... 2 1.3 Project Management ... 2 1.3.1 Time Plan ... 2
1.3.2 Objective Tree Method ... 3
Chapter 2: Background ... 5
2.1 Introduction ... 5
2.2 Energy ... 6
2.3 Wind Energy ... 7
2.3.1 Wind Energy in Ireland ... 7
2.3.2 Benefits of Wind Energy ... 8
2.3.3 Measuring wind energy... 9
2.4 Wind Turbine Fundamentals... 10
2.4.1 Power ... 10
2.4.2 Wind Turbine Design Variation... 11
2.4.3 Aerodynamics of Wind turbines ... 13
2.5 Orientation of axis of rotation ... 16
2.6 VAWT Development ... 19
2.6.1 Rotor Design ... 19
2.6.2 The Flettner Rotor ... 20
2.6.3 Savonius Rotor ... 21
2.6.4 Helical Rotor ... 22
2.6.5 VAWT Case study ... 23
2.7 Summary ... 24
Chapter 3: Rotor Fabrication ... 25
3.1 Introduction ... 25
3.2 Rotor Design ... 25
vi 3.4 Blade Modifications ... 27 3.5 Mounting Rotor ... 28 3.6 Rotor Manufacture ... 29 3.6.1 Considerations... 29 3.6.2 Selection Process ... 29 3.6.3 Analysis of Results ... 31 3.6.4 Final Blade ... 33 3.7 Summary ... 34
Chapter 4: Test Rig Design ... 36
4.1 Introduction ... 36
4.2 Dimensional Analysis of VAWT ... 36
4.3 Methods of applying torque (Brakes) ... 40
4.4 Prony Brake ... 41
4.4.1 Background ... 41
4.4.2 Derivation of Prony brake formula ... 41
4.4.3 Prony Brake Behaviour ... 45
4.4.4 Investigation ... 47
4.4.5 Prony Brake Design/Manufacture... 48
4.4.6 Proposed Testing Methodology ... 52
4.5 Magnetic Particle Brake ... 53
4.5.1 Principle of operation ... 53 4.5.2 Brake sizing ... 54 4.5.3 Power Supply ... 54 4.5.4 MPB Mounting ... 55 4.5.5 M.P.B Calibration ... 58 4.6 Measuring Rpm ... 59
4.7 Frame Design & Fabrication ... 60
4.8 Summary ... 63
Chapter 5: Wind Tunnel Testing ... 65
5.1 Introduction ... 65
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5.3 Prony Brake Set-up ... 67
5.4 Prony Brake testing phase 1 ... 68
5.5 Prony Brake testing Phase 2 ... 72
5.6 Mechanical Losses ... 74
5.6.1 Theory ... 74
5.6.2 Procedure ... 75
5.6.3 Sample calculation ... 76
5.6.4 Drag Calculation ... 78
5.7 Magnetic Particle Brake testing ... 82
5.8 Summary ... 83
Chapter 6: Conclusion/ Recommendation ... 85
6.1 Conclusion ... 85
6.2 Recommendations ... 87
Bibliography ... 88
Appendix A: Rapid Prototyping Screenshots and Code ... 89
Appendix B: MPB Specifications ... 97
Appendix C: Engineering Drawings ... 103
Appendix D: Bearing Specifications ... 108
viii List of Figures
Figure 1 (Predicted Test Rig Design) ... 1
Figure 2 (Project Time Plan)... 3
Figure 3 (Test Rig Design Tree) ... 4
Figure 4 (Graph showing energy consumption pattern (in million tonnes oil equivalent)) ... 6
Figure 5: (Wind-sourced electricity in Ireland 2000-2009. Source: Eir Grid & SEAI EPSSU ) ... 8
Figure 6 (Various Power Curves for wind turbines) ... 11
Figure 7 (Drag-type rotor) ... 12
Figure 8 (Left: Darreius type lift turbine, Right: Propeller type lift turbine) ... 12
Figure 9 (Airfoil Geometry)[8] ... 13
Figure 10 (Aerodynamic Forces)[8] ... 15
Figure 11 (Power Curve Comparison)[7] ... 18
Figure 12 (VAWTs used in comparison)[7] ... 18
Figure 13 (VAWT Rotor Designs) ... 19
Figure 14 (Flettner rotors used on ships) ... 20
Figure 15 (Savonius Rotor)[10] ... 21
Figure 16 (Gorlov helical rotor) ... 22
Figure 17 (Helical VAWT from Quietrevolution)[12] ... 23
Figure 18 (The Eole 3.8MW VAWT) ... 23
Figure 19 (Rotor Design) ... 25
Figure 20 (Wind tunnel dimensions) ... 26
Figure 21 (Rotor size) ... 27
Figure 22 (Blades with modified holes)... 28
Figure 23 (Hub design) ... 29
Figure 24 (Powder Blade Prototype) ... 32
Figure 25 (ABS Plastic Blade Prototype) ... 33
Figure 26 (Catalyst Screenshot) ... 34
Figure 27 (Final Blades) ... 35
Figure 28 (Prony Brake fundamentals) ... 42
Figure 29 (Prony Brake Derivation) ... 42
ix
Figure 31 (Prony Brake Behavior 2) ... 46
Figure 32 (Prony Brake Behavior 3) ... 46
Figure 33 (Prony Brake Design) ... 49
Figure 34 (Tensioning Mechanism) ... 50
Figure 35 (Spring Attachment) ... 51
Figure 36 (Prony Brake Mounting)... 51
Figure 37 (Prony Brake Assembly) ... 52
Figure 38 (Magnetic Particle Brake) ... 53
Figure 39 (Torque Vs Current) ... 54
Figure 40 (Power Supply) ... 55
Figure 41 (MPB Setup) ... 55
Figure 42 (MPB Coupling) ... 56
Figure 43 (MPB Bracket)... 56
Figure 44 (MPB Assembly) ... 57
Figure 45 (Fabricated MPB Assembly) ... 57
Figure 46 (MPB Calibration) ... 58
Figure 47 (MPB Extension Lead) ... 59
Figure 48 (MPB and Power Supply) ... 59
Figure 49 (Laser Tachometer) ... 60
Figure 50 (Test Rig Frame)... 61
Figure 51 (Frame Clamps) ... 61
Figure 52 (Frame Top Features) ... 62
Figure 53 (Frame and Brake Assembly) ... 63
Figure 54 (Test Rig in Place) ... 63
Figure 55 (Wind Tunnel DIT Bolton Street) ... 66
Figure 56 (Wind Tunnel Controls) ... 66
Figure 57 (Prony Brake Testing) ... 67
Figure 58 (Prony Brake Phase 1 Power Curves) ... 69
Figure 59 (Prony Brake Phase 1 Dimensionless Power Curves) ... 70
Figure 60 (Prony Brake Phase 1 Repeatability Test) ... 71
x
Figure 62 (Prony Brake Phase 2 Power) ... 72
Figure 63 (Prony Brake Phase 2 Torque) ... 73
Figure 64 (Prony Brake Phase 2 Dimensionless Power Curves) ... 74
Figure 65 (Spin Down Test) ... 76
Figure 66 (Spin Down Test) ... 78
Figure 67 (Friction Torque in Bearings) ... 80
Figure 68 (Torque Losses and Useful torque) ... 80
Figure 69 (Total power curve 22m/s) ... 81
Figure 70 (MPB Testing) ... 82
xi List of Tables
Table 1 (Selecting manufacturing process)... 31
Table 2 (List of variables) ... 37
Table 3 (Brake Selection) ... 41
Table 4 (Recorded data) ... 77
Chapter 1: Introduction
1.1 Project Background
The fluid mechanics department in DIT Bolton Street were approached by a company called Brí Toinne Teoranta with a design for a vertical axis wind turbine (VAWT). The company had completed the preliminary design of the turbine but had not carried out any testing to determine its performance. It was decided that the mechanical testing of this VAWT design would be undertaken as an undergraduate final year project.
On accepting the project, videos of an early version of the turbine operating in a wind tunnel were provided by the company to show the principle of operation of the turbine. These videos were carefully analysed in order to gain an understanding of the challenges that were ahead for the project. The company also provided the relevant CAD files for the unique turbine blades, which would allow the blades to be manufactured in accordance with the company’s design. The following dissertation was compiled to document the approach used in testing the Brí Toinne Teoranta turbine design along with the testing methodologies which were devised for analysing the turbines performance.
2 1.2 Project Aims/Objectives
The aim of this project was to design and build a test rig for a vertical axis wind turbine (VAWT) which can test the performance of the turbine. It was decided that by measuring the mechanical power output and plotting dimensionless power curves, the performance of the turbine could be assessed. A series of objectives were devised in order to successfully complete the project. The main objectives of the project were as follows.
1. Carry out research into the area of wind energy and develop an understanding for the fundamentals of wind power generation.
2. Manufacture the turbine blades designed by Brí Toinne Teoranta to a high standard. 3. Design a testing methodology to obtain performance curves for the turbine.
4. Design and fabricate a suitable test rig for the wind tunnel testing. 5. Carry out wind tunnel testing on the VAWT.
6. Analyse the results obtained from the wind tunnel testing. 7. Present the findings from testing in report form.
1.3 Project Management
Before any work was started for the project it was decided that the correct project management procedures should be carried out in order to ensure the project was executed in a professional manner. Both the direction and the time management for the project were carefully planned out to avoid any time being wasted on areas which were not relevant to the project.
1.3.1 Time Plan
The time plan shown in Figure 2 below was devised in order to create deadlines within the duration of the project itself. The project was divided up into five main sections which were research, test rig design, turbine and rig manufacture, wind tunnel testing and analysis. These sections were set out in a logical fashion, with specific deadlines for each section.
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Figure 2 (Project Time Plan) 1.3.2 Objective Tree Method
In order to develop a greater understanding of what was required by the project brief, the chart shown in Figure 3 was developed using the objective tree method for the design of a VAWT test rig.
A few basic requirements of the test rig were defined so that smaller objectives could be put in place to meet these requirements. The requirements were that the test rig be safe to operate, have a long lifespan, have the ability to test different rotor designs using different braking mechanisms and have a simple design which can be easily adapted for future requirements.
In order for the test rig to be safe to operate it was decided that any controls which the operator must use should be outside the operating area of the turbine. It was also decided that appropriate locking fasteners should be used on the turbine and test rig in order to prevent any part of the apparatus coming loose due to vibrations during operation.
In order to ensure that the test rig has a long lifespan, it was decided that the rig should be made from durable materials and should be rigid in its construction. It was also decided that an appropriate surface finish should be applied to surfaces of the test rig which may be prone to rusting.
It was decided that in order to make the test rig useful for future projects, it should be multifunctional in the sense that it can be used with different types of rotors, which can be easily removed from the rig. It was also kept in mind that the test rig should be able to accommodate
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different methods of applying a braking torque to the rotor, so that comparison tests could be easily carried out.
The final basic requirement of the test rig was that it could be easily adapted. It was decided that to meet this requirement the turbine should be capable of being quickly and easily detached from the wind tunnel, have a means of staying upright while maintenance work was being carried out on it and be reasonably portable.
When all of the basic requirements for the test rig design were analysed and understood, a decision was made to carry out research into the areas of wind energy and VAWT technology.
5
Chapter 2: Background
2.1 Introduction
The following chapter contains a literature review which was carried out in the areas of energy, wind energy, wind turbine fundamentals and the development of vertical axis wind turbines. Before any work could be done on designing the vertical axis wind turbine test rig, a good understanding of the history of wind energy was obtained. In the following chapter, the history and the development of the wind turbine is discussed, from its early conception in the form of the windmill to the modern day electricity generating devices with which the world is now so familiar.
The area of aerodynamics is explored in this chapter, pointing out the various characteristics of aerofoils and the concepts behind aerofoil performance. This is a crucial area as the turbine being tested in this project is a lift type device which has an aerofoil cross section.
The development of the VAWT was investigated, along with the various different rotor designs which have been developed over the years. In this chapter the different rotor design are discussed in order to gain an understanding of the performance expected from the unique VAWT rotor design presented by Brí Toinne Teoranta.
6 2.2 Energy
“The environmental implications of the continued global energy system’s dependence on fossil fuels call for urgent action across the world”[1]
Most of the world’s energy currently comes from non-renewable sources as indicated in Figure 4 below. This graph from the BP statistical review of world energy 2011 gives a striking indication as to the worlds dependence on fossil fuels like oil and coal, and taking into account the fact that these resources will one day run out makes it a matter of urgency to pursue the development of renewable energy technologies. It is vital that the world is able to reduce the amount energy being produced from non-renewable sources.
7 2.3 Wind Energy
Wind energy has been used as an effective resource since the ancient times, when it was harnessed to propel sail boats. According to historic research the first people to use the wind to drive actual machinery, appears to have been the Persians, who used very early forms of windmills to grind wheat. Conquests allowed this technology to eventually spread to Holland, where the use of wind energy was embraced and became a very important tool. In the American West the power of the wind was used to drive saw mills, water pumps and cereal grinders. In these early versions of wind energy conversion machinery, the wind machines were connected directly to a mechanical load. The first substantial electricity generating wind turbine was built by Charles Brush in Cleveland Ohio, and ran for twelve years, from 1888 to 1900, supplying electricity to his home. This early wind turbine was a bulky device which had multiple rotors and because of its size rotated quite slowly thus having to be geared up in order to satisfy the rotational speed required of the generator.
The production of electricity using wind energy has had to overcome many stumbling blocks in order to become the viable, efficient process it is today. After World War II for example, when the price of oil dropped and almost all interest in alternate energy was lost, wind energy was put on the backburner in favour of the cheaper electricity being produced in power plants. The 1973 oil crisis however, re-ignited interest in the area of wind energy and this led to early forms of wind farms being developed.
Cost it seems is the deciding factor when the use of alternate energy is in question. The environmental friendliness of an energy source alone will not motivate investors enough to part with their cash, but if there is money to be made from the production of electricity, then it becomes a very interesting prospect. [2]
2.3.1 Wind Energy in Ireland
Every year, wind energy is making a bigger contribution to the electricity supplied throughout Ireland. At the end of June 2010 it was reported that there were 110 wind farms metered in Ireland, bringing the total installed capacity for wind up to 1,379MW. The national target for the year 2020 is to have 40% of our electricity coming from renewable sources, an estimated 5,500-6,000 MW of wind generation is required to achieve this target. [3]
8
Looking at the data taken from the 2009 IEA Annual report for Ireland there is a promising growth in wind power in Ireland. In Figure 5 below it is clear to see that there has been a significant rise in the amount of wind sourced electricity being used in Ireland since the year 2000. This dramatic increase makes the national targets for renewable electricity look like they are a realistic goal. [4]
Figure 5: (Wind-sourced electricity in Ireland 2000-2009. Source: Eir Grid & SEAI EPSSU ) 2.3.2 Benefits of Wind Energy
Wind energy is classed as a renewable source of energy and has certain benefits associated with it when compared with other non-renewable processes used to produce power. A common theme amongst renewable energies is that they can be described as “clean” energy sources. A “clean” energy source does not produce any emissions like nitrogen oxide, sulphur dioxide, mercury and carbon dioxide, which pollute the air. This means that not only can wind energy provide the world with extra capacity for creating electricity; it can do so without producing any extra emissions. When a country has got a well established system for producing electricity using wind energy in place, it can then start to decrease the demand on electricity produced in power plants, hence decreasing the amount of fossil fuels which will be consumed on a daily basis.
The development and progression of wind energy as a source of electricity has benefits on a domestic level also. As the wind energy industry grows, there will be a diversification in the market whereby the majority of the world’s electricity will no longer be coming from power plants burning fossil fuels. This means that when there are dramatic increases in the price of oil
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and other fossil fuels around the world, the cost of electricity for customers will not be as dramatically affected. In an ideal situation, 100% of the energy supplied to a customer would be from a wind energy source and electricity prices would not be affected at all by the cost of fossil fuels. [5]
2.3.3 Measuring wind energy
When choosing a source of energy which will be used to power a machine, it is crucial to be able to measure exactly the amount of power the machine can produce using this energy source. It is easy to calculate the performance and power input of a machine which will be powered by fossil fuels, because of the set calorific value for the fuel. This set value guarantees a certain amount of energy output from the machine, and when the machines efficiency is taken into account, the performance of the machine can be calculated over any given period of time. Wind energy however does not have this certainty of performance attached to it.
There are many factors which make harnessing the winds energy in a consistent and efficient way, a very complicated process. Wind speed is one of these factors. The power input for a turbine is calculated from knowing the wind speed. The faster the wind, the more power can be extracted from it. The problem with this is that the wind speed is constantly fluctuating, so the wind turbine does not have a constant power input, making calculations and efficiencies very complicated. Another factor which affects the performance of a wind turbine substantially is its placement. The placement of a wind turbine has to be exact in order to achieve the maximum possible power output.[6]
10 2.4 Wind Turbine Fundamentals
A wind turbines main objective is to harness the power of the wind and convert it into some useful form of energy. The modern day design of wind turbines did not come about by chance, but was formed from constant upgrading and experimenting carried out over many years. There are a few fundamental equations used to quantify the performance of a wind turbine.
2.4.1 Power
It is crucial to know the amount of power that can be gathered by a wind turbine, the equation for this power is given by
(2.1)
Where Cp is the coefficient of power for the turbine, is the density of the air which is flowing
through the turbine, A is the swept area of the turbine (the area which the blades or rotor sweeps through), and is the wind speed.
The Cp for a wind turbine is the way in which the aerodynamic efficiency of the turbine is
quantified. Cp is a function of the tip speed ratio λ. This is the ratio of speed at the tip of the
turbine blade to wind speed and is given by
(2.2)
Where is the rotational frequency, R is the radius of the turbine and is the wind speed. The efficiency and performance of a wind turbine is usually displayed using power curves. Figure 6 below shows plots of power coefficient versus tip speed ratio for various different wind turbine types. The theoretical maximum power coefficient is known as the Betz limit and is 0.59 for an ideal wind turbine. This Betz efficiency is marked as the ideal efficiency of propeller-type turbine in Figure 6 below. [7]
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Figure 6 (Various Power Curves for wind turbines) 2.4.2 Wind Turbine Design Variation
Wind turbines fall under the category of either a drag-type turbine or a lift-type turbine. Modern day wind turbines mostly fall into the lift-type category, but it is worth knowing the principles of both categories.
2.4.2.1 Drag turbines
In a drag type device a force which acts in the same direction as the wind is blowing is exerted on the blades or paddles of the turbine. This is the same principle by which sail boats operate, as the wind exerts a force on the sails. In a turbine which works solely on the principle of drag, the surface on which the wind is exerting a force cannot move faster than the speed of the wind. This fact limits the tip speed ratio and hence the overall efficiency of these drag type turbines. Many of the earlier vertical axis wind turbine designs used drag rotors, such as the bucket type wind turbine shown in Figure 7.
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Figure 7 (Drag-type rotor) 2.4.2.2 Lift turbines
In a lift type turbine like the ones shown in Figure 8 below the force generated by the wind acts perpendicular to the direction that the wind is blowing. It should also be noted that in a lift type turbine, the maximum speed of the blade is not limited to the speed at which the wind is blowing as in drag type turbines. This means that lift type turbines can have much larger tip speed ratios than their drag type counterparts. In order to fully understand exactly how lift type turbines operate, an investigation must be done into the area of aerodynamics and aerofoil technology.
13 2.4.3 Aerodynamics of Wind turbines
2.4.3.1 Geometry of aerodynamic profiles
In lift type wind turbines the blades of the rotor have an airfoil cross section. The amount of lift force which is exerted on the blade depends strongly on the shape of the airfoil section used. There are various geometric parameters on which the aerodynamic characteristics of the air foil depend. These parameters shown in include the leading-edge radius, the mean camber line, the maximum thickness and the thickness distribution of the profile, and the trailing edge angle.[8]
Figure 9 (Airfoil Geometry)[8]
Leading-Edge Radius
The leading-edge radius of an airfoil section is the radius of a circle centered on a line which is tangent to the leading edge camber connecting the points of tangency on the upper and lower surfaces of the airfoil with the leading edge. The centre of the leading edge radius is located such that the cambered section of the airfoil is projected out and overhangs slightly the leading edge point. [8]
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Chord Line
The chord line of an airfoil section is the straight line connecting the leading and trailing edge. The angle of attack of an airfoil is the angle which exists between the chord line and the direction of the free stream fluid flow. [8]
Mean Camber Line
When a locus of points located half way between the top and bottom surfaces of the airfoil section is plotted, the resulting line is called the mean camber line. One of the effects of a change of camber is a change in the zero-lift angle of attack, α0l. Symmetric airfoil sections have zero lift
at zero angle of attack, and likewise zero lift occurs for sections with positive camber when their angle of attack is negative. [8]
Maximum Thickness and Thickness Distribution
The maximum thickness and thickness distribution have a large impact on the aerodynamic characteristics of the airfoil section. An increase in the maximum thickness of an airfoil increases the maximum lift coefficient for the airfoil. An increase in the maximum thickness of an airfoil section also increases the maximum local velocity to which a fluid particle will accelerate as it flows around the airfoil. As a result of this the minimum pressure value is smallest for the thickest airfoil. There is an adverse pressure gradient associated with the deceleration of the flow between the point on the airfoil where the minimum pressure occurs to the trailing edge. This pressure gradient is largest for the thickest airfoil and the larger the pressure gradient the larger the boundary layer will be, therefore boundary layer separation will occur more easily and hence the drag values for the airfoil section will increase.
The thickness distribution of an airfoil section affects the pressure distribution and the character of the boundary layer. Moving the location of the maximum thickness of the airfoil towards the leading edge of the airfoil will result in a decrease in the pressure gradient at the central region of the airfoil. This decrease in pressure gradient leads to a more stabilised boundary layer and can
15
promote the possibility of a laminar boundary layer which is favourable because of it lower skin friction drag than occurs with turbulent boundary layers. [8]
Trailing-Edge angle
The trailing-edge of an airfoil affects the location of the aerodynamic centre. The theoretical location of the aerodynamic centre of a thin airfoil in a subsonic stream is located at the quarter chord. [8]
Aerodynamic forces
The motion of air around an airfoil section produces variations in pressure and velocity which result in aerodynamic forces and moments. Viscous forces are neglected apart from when they occur in a small area near the surface of the airfoil called the boundary layer, a region in which the large velocity gradient results in large viscous forces. If these boundary layer forces are neglected then standard equations of motion can be used to analyse the three main forces which occur in an airfoil section; lift, drag and side force. [8]
The primary forces shown in Figure 10 contribute to the main forces which occur in the airfoil. Lift is a component of force which acts upward, perpendicular to the direction of the undisturbed free-stream velocity. The primary cause of the lift force is the pressure forces acting on the airfoil surface. Drag is the net aerodynamic force which acts in the same direction as the free-stream velocity. The drag force is due to a combination of pressure forces and skin friction forces which act on the surface of the airfoil. Side force is a force which acts perpendicular to both the lift force and the drag force. [8]
16 2.5 Orientation of axis of rotation
The concept of the wind turbine has taken many different physical forms since it was first introduced. Wind turbines can be divided up into two categories Horizontal axis wind turbines (HAWTs) and vertical axis wind turbines (VAWTs).As things stand today, the HAWT dominates as the most widely used type of wind turbine across the world. This does not mean however that the concept of the VAWT should be discarded as a novel idea. There are various advantages and disadvantages associated with both the HAWT and the VAWT, some of which will now be discussed.
HAWTs and VAWTs have been developed almost in parallel, but there has been less interest and investment in the development of VAWTs. This is one of the key reasons why the HAWT dominates the wind power world today.
One of the main differences between the VAWT and the HAWT is the VAWTs ability to accept wind which is blowing in any direction. This ability makes the VAWT very suited to areas where there are changeable and gusting winds, such as the tops of large buildings. In the case of HAWTs a yaw mechanism is required to point the propeller of the turbine into the wind and hold it there. The time it takes the HAWT to point into the wind can be regarded as downtime in which it is not producing as much electricity as it could be. The yawing mechanism also adds extra cost to the production of the wind turbine, as does the control system used to control the yaw.
Because of the orientation of the axis of a VAWT, it is possible to place the power generating equipment at ground level. This is a major advantage from the point of view of maintenance and monitoring of the electricity generating equipment. Another advantage of having the power generating equipment at ground level is that when selecting a generator, the criteria for selection does not have to include minimizing the weight and physical size of the generator, and the sole focus of the selection can be choosing the most efficient generator for the task.
The blades of a HAWT have to be self supporting because they are only attached to the turbine at one end. It has been claimed that the HAWT has reached its maximum possible size, and that the reversing gravity loads on the blades limits their progression. These reversing gravity loads however, do not occur in VAWTs, which theoretically have no maximum possible size.
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Guy wires can be used to support VAWTs which means that the main shaft of the turbine can be of a smaller diameter. This is not the case with HAWTs, guy wires cannot be used on HAWTs because they would interfere with the rotation of the propeller.
In a direct drive wind turbine the rotor or propeller is connected directly to the generator without the drive going through a gearbox. The power generating equipment associated with direct drive machinery is usually more bulky than the usual equipment used in a system with a gearbox. This means that the nacelle (the area in a HAWT where the generating equipment is housed) will be heavier and hence the turbine support mast will have to be larger. In a VAWT however this will not be an issue as the power generating equipment can be located at ground level.
One of the problems Associated with VAWTs is the torque ripple which occurs in the rotors. This torque ripple causes cyclic loading of the blades of the turbine which can lead to failure of the blades. In HAWTs there is constant torque acting on all of the blades, so the issue of fatigue due to cyclic loading does not arise.
A study was carried out comparing the power curves of three different turbines, a Darrieus type, an H-type, and a standard HAWT. The power curves are shown in Figure 11 below. The power curves were formed by plotting the coefficient of power (Cp) against the tip speed ratio (λ).
The three turbines which are being compared in Figure 11 are a 100kw H-rotor VAWT, a 500kW Darrieus VAWT (shown in Figure 12) and the HAWT data comes from the National Renewable Energy Laboratory in the USA, and is said to represent the data associated with a typical HAWT. The high values of Cp which can be seen for the HAWT show how much more
developed the HAWT is compared to the two VAWTs which are also plotted. It should be noted however that the maximum value for coefficient of power for the Darreius type turbine is not very far off the HAWT value, and considering how much more field testing and research has been carried out on the HAWT, the performance gap between the two turbine concepts could easily be closed. [7]
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Figure 11 (Power Curve Comparison)[7]
19 2.6 VAWT Development
2.6.1 Rotor Design
The development and design of an efficient rotor is perhaps the most important step in making VAWT’s more efficient as devices for gathering wind energy. The rotor of a VAWT must be designed so that it can deal with pulsating torques and cyclic loading over the lifetime of the wind turbine. Unlike the rotors used on HAWTs, VAWT rotors can differ greatly in both appearance and in the fundamentals of aerodynamics which they use. It is necessary to have an understanding of how the various rotor designs for VAWTs originated and the problems associated with each design. In this section the background of the following rotors will be discussed. Flettner Rotor Savonius Rotor Darius Rotor H-Type Rotor Helical Rotor
20 2.6.2 The Flettner Rotor
The history of the VAWT begins with a German engineer named Anton Flettner who came up with the Flettner rotor. Although the Flettner rotor was never actually used as a VAWT rotor, it inspired the design of the Savonius rotor which will be discussed later on. The Flettner rotor uses the Magnus effect to turn wind energy into a lateral thrust in the direction which is perpendicular to that of the wind.
The rotor itself is a large cylinder which is revolved in order to create a difference in pressure on both sides of the cylinder. This pressure difference between the two sides of the rotating cylinder results in a lift force in the direction of the lower pressure.
The rotor that Flettner had designed was used on ships to propel the ship forward using wind which was approaching the ship from its side; this process is illustrated in Figure 14. On these ships, the rotation of Flettners rotors was powered by diesel engines. The reliability and speed which became associated with the conventional propeller powered ships meant that the Flettner rotor ceased to be used for the purposes of propelling ships. [9]
21 2.6.3 Savonius Rotor
A Finish man by the name of Sigurd Savonius decided that it would be possible to use the wind energy harnessed from offshore winds to turn the Flettner rotors which were used on ships, removing the need for the diesel engines on these ships.
Savonius took the Cylindrical Flettner rotor and cut in into two semi-circles. The semi circles were then offset from the centre of the rotor along the cutting plane, creating two semi circular cups. The gap between the cups at the centre of the rotor meant that the air would flow into one of the cups and pass through the gap, thus having a thrust effect on the rotor on the other side also. When the Savonius rotor was tested against a Flettner rotor of comparable size, the Savonius rotor produced a greater lateral thrust than the Flettner rotor. Despite this increased lateral force, there was no real need for the rotor as a replacement for the Flettner rotor as it had never taken off as a widely used method of propelling ships. [9]
Figure 15 (Savonius Rotor)[10]
The Savonius rotor uses drag as its driving force. It has been used as a rotor for water current turbines to good effect. The Savonius rotor has various advantages associated with it, such as the simplicity of its design and the ease with which it can be manufactured. This makes the Savonius rotor an interesting, economical possibility for converting wind energy to electricity in under-developed areas of the world.
22 2.6.4 Helical Rotor
The helical rotor is a relatively new rotor design when compared with the others mentioned in this report. The helical rotor was developed for use on a hydraulic turbine in the early nineties, harnessing the different types of ocean currents to produce electricity. The Gorlov helical type rotor shown in Figure 16, was developed in order to utilise all of the advantages of the Darreius rotor, without any of the disadvantages.
The basic principle behind the helical rotor is that instead of straight blades which are used in Darrieus rotors, the blades follow a spiral around the outer circumference of the rotor. This spiral helps to get rid of the pulsating torque and the vibration problems which were associated with the Darrieus rotor when it was tested in water.[11]
Figure 16 (Gorlov helical rotor)
A company form the UK called quietrevolution has developed a helical VAWT shown in Figure 17 called the Q5, which is designed to be used in urban areas where the turbine will be operating close to the general public. The helical design eliminates vibration in the turbine, so it can be attached onto buildings without causing any detectable shaking of the building. The company also says that the helical design eliminates any noise from the turbine and provides it with a robust structure. [12]
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Figure 17 (Helical VAWT from Quietrevolution)[12] 2.6.5 VAWT Case study
There have been a few reasonably successful attempts at creating large scale VAWTs in the past, for example a VAWT called the Eole shown in Figure 18, which was built in 1986 by an American company called FloWind. The Eole was a 96m tall Darrieus turbine which as the largest VAWT ever built had a maximum power output of 3.8MW. During its five year lifetime the Eole produced 12GWh of electricity, reaching power levels of around 2.7MW. Failure of the bottom bearing in the Eole resulted in the turbine being shut down. The existence of this multi-megawatt Darrieus type VAWT shows that it is realistic to believe that the VAWT could someday be just as popular as the HAWT. [7]
24 2.7 Summary
The research carried out in this chapter provided a good insight into the history of wind turbines and some of the fundamental principles on which their operation depends. The areas investigated also gave a good rounded insight into the position of wind energy in the world today.
The issue of global energy demands was investigated and it was clear that alternative energy sources such as wind power generation will be crucial in meeting these demands in the future. The sustainable and environmentally friendly nature of wind energy was also investigated and it was clear from the information obtained that wind energy will play a leading role in reducing the amount of fossil fuels being used worldwide.
The potential for wind power generation in Ireland was explored briefly in this chapter. Ireland’s large Atlantic coastline gives it great potential to eventually source almost all of its power from renewable sources like the wind. It is likely that in a strong economic climate, there will be huge investment in developing offshore and onshore wind farms along the coast of Ireland.
The fundamentals of wind power generation were discussed in this chapter which involved the differences between drag and lift turbines, the area of aerodynamics, the theory behind calculating the power produced by a wind turbine and the differences between horizontal and vertical axis turbines.
Particular attention was paid to the development of the vertical axis wind turbine along with the different design concepts which have been devised in recent years. It was clear that the VAWT had not seen the same commercial success as the HAWT but this could have been due to a lack of investment into the development of the VAWT. The different VAWT rotor designs which have been developed proved to be remarkably different in their appearance and operation. After looking at the different rotor designs, it could be argued that the optimum VAWT rotor design has not yet been discovered, thus making the testing of the rotor designed by Brí Toinne Teoranta a relevant and worthwhile project.
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Chapter 3: Rotor Fabrication
3.1 Introduction
In this chapter the fabrication of the VAWT rotor designed by Brí Toinne Teoranta is explained in detail, documenting the various challenges which were involved with the fabrication of the rotor.
The unique rotor design is discussed briefly along with the key features of the blades, followed by the size limitations which were faced and any additional design modifications which had to be made to the blades before manufacture. The complex geometry of the turbine blades meant that the manufacturing process used to make the blades needed careful consideration. The following chapter also contains the details of how the rotor was mounted on a central shaft for operation. 3.2 Rotor Design
The vertical axis wind turbine (VAWT) rotor which was provided by Brí Toinne Teorannta is shown in the figure below. The rotor is a hybrid design, incorporating elements of various VAWT rotors such as the Darreius and helical rotors. The rotor consists of three helical blades connected to a central shaft. The aerofoil cross section of the blades creates the lift force in the rotor, causing the rotor to rotate. The helical design of the rotor should help to eliminate pulsating torque in the rotor. This style of rotor is a relatively new design, and there has been very little testing carried out on it in the past.
26 3.3 Sizing of rotor (scale)
Before any design modifications could be carried out on the rotor it was necessary to select an approximate diameter for the VAWT rotor. Ideally the rotor should be as large as possible in order to increase the accuracy of the testing and decrease the effect of any losses encountered. There was however some constraints which placed a limit on the maximum diameter of the turbine.
The first limiting factor was the physical size of the wind tunnel in D.I.T. Bolton Street. The wind tunnel has a cross section of 500mm by 500mm as shown in Figure 20 below. Previous testing was carried out with this wind tunnel on a turbine of diameter 568mm. The turbine was too large for the wind tunnel and the blades were not experiencing the full free stream for a complete revolution of the rotor, resulting in very low power output from the rotor. In order to avoid this, it was decided that the rotor diameter should be kept well inside the cross section of the wind tunnel. It was acknowledged that reducing the size of the rotor would also limit the turbines power output, but because there was no other wind tunnel available it was considered to be the best course of action.
Figure 20 (Wind tunnel dimensions)
The other factor which limited the size of the turbine was the capacity of the machines which would be used to manufacture the blades. After looking at the possible manufacturing techniques
27
which could be applied to the blades, it was concluded that the maximum rotor diameter was limited to 200mm. The 200mm rotor would fit comfortably into the wind tunnel and there would also be significant distance between the tip of the blades and the walls of the wind tunnel. This gap was considered to be important as it helps to prevent blockage effects and should also prevent issues with the boundary layers on the walls of the wind tunnel. Figure 21 below shows the rotor placed in the cross section of the wind tunnel.
Figure 21 (Rotor size) 3.4 Blade Modifications
The SolidWorks file for the turbine blade was provided by the designer for use as part of the project. The file was provided for the blade only and they did not include any components for connecting the blades together to make the rotor.
The blade was scaled down from the original 300mm diameter to 194mm to enable its manufacture as discussed above. The solid model of the scaled down blade had to be carefully analysed, finding its centre of rotation and other key points so that the blades could be connected together and mounted onto a central shaft.
Once the centre of rotation was determined, three blades were evenly spaced in a circle around the centre as shown in Figure 22 below. This allowed for the design of a central hub to hold the blades onto the central shaft. The blade provided by the designer had already got two holes in each of the blade ends. During the scaling down of the blade these holes became too small to be
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used to fit the blades to the central shaft so it was necessary to replace these with 4.2mm holes which would accommodate an M4 machine screw. The holes also had to be slightly re-located as the new larger holes went too close to the edge of the blade which could potentially cause failure of the blades. The re-located larger holes are also shown in Figure 22 below.
Figure 22 (Blades with modified holes) 3.5 Mounting Rotor
After the new holes were located in the blades, the blades had to somehow be connected to the central shaft of the turbine. A component was designed to hold the blades in position and fix them to the central shaft.
Firstly a central shaft diameter of 12mm was decided upon. The 12mm shaft was chosen in order to prevent deflection and vibration occurring. The component shown in Figure 23 below was designed using the modified holes in the blades as a template. An M5 grub screw was decided upon as the preferred method of tightening the rotor to the shaft. The grub screw also enables the rotor to be easily removed from the central shaft and to be moved up and down the shaft.
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Figure 23 (Hub design) 3.6 Rotor Manufacture
3.6.1 Considerations
The complex geometry of the rotor meant that the manufacture of the blades was a key area in the success of the VAWT testing. Previous testing was carried out on blades which had imperfections in the surface finish, and this affected the performance of the rotor. It was decided that the blades must have the best possible surface finish and the most accurate replication of the designed geometry possible.
Some key characteristics of the blade which needed to be focused on were the accuracy of the trailing edge, the aerofoil profile, the overall surface roughness and the strength of the material used. These features were used as the criteria for selection of the manufacturing process.
An investigation was carried out into different manufacturing processes which could be used to manufacture the VAWT blades. Conventional machining techniques such as milling and turning were ruled out due to the complex three dimensional curves which even on an automated machine would have been difficult to achieve. It was decided to investigate the areas of rapid prototyping and 3D printing as methods for the manufacture of the turbine blades.
3.6.2 Selection Process
Investigations were carried out into the suitability of three different rapid prototyping machines for the manufacture of the VAWT blades. All three machines were located on the DIT Bolton Street premises and were available for use. The machines available were the Rap-man 3D
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printer, Z-Corp 3D printer and the Dimension Fused Deposition Modeller. Both the Rap-man and the Dimension machine use a plastic as their working material and the Z-Corp machine uses a powder based material. In order to decide which machine would be used, a number of criteria for selection were decided upon. The three machines were assessed by looking at parts previously manufactured by each machine. The following criteria were marked on a scale of 0-5 in below.
Surface Finish
A smooth, uniform surface finish is vital to the performance of the VAWT blades. If the surface of the blades is rough and uneven, the lift forces created by the aerofoil cross section will be affected.
Geometry Replication
Accurate replication of the blade geometry is important so that the blade performs as the designer intended. Accurate blade geometry also means that the current design of the blades is being analysed correctly and design modifications can be implemented following the testing. It is also necessary to manufacture three identical blades in order to ensure that the rotor is balanced.
Rigidity
The blades of a VAWT are put under a considerable amount of mechanical stress during testing at high wind speeds. It is vital that the blades are manufactured to be as rigid as possible in order to prevent any bending and possible failure of the blades.
Compatibility (software)
In order to correctly produce the VAWT blades, the software for the rapid prototyping machine should run smoothly without errors or complications. Incorrect use of the software could lead to incorrectly produced parts and in turn wasted materials.
Capacity
The capacity of the rapid prototyping machine is one of the factors which determine the maximum blade diameter which can be manufactured.
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Cost of Raw Material
Each rapid prototyping machine uses different materials to manufacture parts. The cost of this raw material can vary greatly depending on the properties of the material.
Technical Support
Past experience and availability of technical support are vital to the successful manufacture of the blades. Making decisions about orientating the blades during manufacture to produce maximum strength and carrying out necessary modifications are made easier if there is a technician available with a lot of past experience working with the rapid prototyping machines.
Rap-Man 3D Z-Corp Dimension
Surface Finish 2 3 3
Geometry Replication 3 2 5
Rigidity 3 1 4
Compatibility(software) 1 5 5
Capacity 5 5 5
Cost of Raw Material 3 4 1
Technical Support 1 5 5
Total: 18/35 25/35 28/35
Table 1 (Selecting manufacturing process) 3.6.3 Analysis of Results
From the results shown in Table 1 above it was concluded that the Dimension machine would be most suitable for manufacturing the blades. The ABS plastic material used by the Dimension machine is expensive (approximately 50c per cubic centimetre), therefore it was decided to make a prototype of the blades on the Z-Corp powder based machine, which will be half the cost of a plastic blade. Although the powder prototype would not be strong enough to be used in testing, it would give a good indication as to any slight modifications which had to be made to the blade design, and would enable any changes to be carried out before using the more expensive manufacturing process. The prototype powder blade shown in Figure 24 was analysed for
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imperfections and possible problems involving geometry. Overall the powder blade proved to be an accurate recreation of the desired geometry, and no major modifications had to be made. The next step was to manufacture a small section of the VAWT blade on the Dimension ABS plastic machine. This small and inexpensive model of the blades cross section was used to assess how the Dimension machine would reproduce the aerofoils characteristics such as the trailing edge. The ABS blade section shown in Figure 25 was inspected and a decision was made to proceed with the manufacture of the blades.
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Figure 25 (ABS Plastic Blade Prototype) 3.6.4 Final Blade
The Standard Tessellation Language (STL) file for the blade was imported into a software package called catalyst in order to prepare it to be sent to the Dimension machine. This file contains the data required by the Dimension machine to manufacture the blades. The catalyst programme calculates the required support material to be added to the model as shown in Figure 26. This support material is required so that the machine can print out parts of the blade which are not sitting on the base of the machines build area. The orientation of the blade is essential to minimizing the amount of support material required. Several orientations were investigated to see which used the smallest amount of support material. The catalyst software then calculated the total build time for one blade to be 4 hours 51 minutes. The blades were manufactured and the support material was carefully removed from the holes in the blade and the blade geometry.
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Figure 26 (Catalyst Screenshot) 3.7 Summary
Before the investigation into the rotor manufacture began, the wind tunnel in DIT Bolton Street was measured in order to determine the maximum size VAWT which could be tested accurately. The wind tunnel had a cross section of 0.5m by 0.5m so the rotor diameter was immediately limited to within these dimensions.
The next area which was investigated was the manufacturing methods available for the manufacture of the blades in DIT Bolton Street. The 3D printing machines available were limited to a maximum dimension of 200mm. This 200mm was then considered the maximum diameter for the turbine blades. The CAD files for the turbine blades presented by Brí Toinne Teoranta were carefully analysed and scaled down to the 200mm limit discussed above.
A method for attaching the turbines blades to a central shaft had to be devised as there was no attachment method specified with the blade design. Suitable hubs were designed to attach the three turbine blades in an evenly spaced manner around the central shaft of the turbine. The design of the turbine blades had to be modified slightly by adding a series of holes which would allow the blades to be attached to the central hub.
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Two different 3D printing based manufacturing processes were available for the manufacture of the turbine blades. Prototypes were made using both of the 3D printing machines. The samples made by each machine were inspected and by using specific selection criteria the appropriate manufacturing method was selected.
The final blades shown in Figure 27 below were manufactured on a 3D printer using ABS plastic as the material. The blades were of a reasonably high quality and were deemed appropriate for use in the VAWT wind tunnel testing.
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Chapter 4: Test Rig Design
4.1 Introduction
In this chapter the test rig design and fabrication is documented, showing the challenges which were faced throughout.
Dimensional analysis was carried out for a vertical axis wind turbine (VAWT) in order to determine the relationship between the relevant variables. Using the results from the dimensional analysis a testing methodology was devised to get the performance curves for the VAWT.
An appropriate method of applying a mechanical load (braking torque) to the VAWT was selected for use on the test rig. The Prony brake and the magnetic particle brake were investigated in detail and any additional components needed for testing were designed.
A test rig was designed and fabricated which supported the braking mechanisms specified and allowed the rpm of the rotor to be measured. The test rig was designed to comply with the objective tree design method carried out in the introduction chapter of this dissertation.
4.2 Dimensional Analysis of VAWT
Dimensional analysis is a useful tool which can be used when designing experiments such as the VAWT performance test. Using the Buckingham Pi theorem, the relevant variables for a problem can be combined to produce dimensionless groups known as Pi groups. The steps involved in the Buckingham Pi theorem are as follows
1. List and count the relevant variables associated with the problem. Assign the letter n to the number of variables considered.
2. List the dimensions of each of the variables associated with the problem.
3. Count the total number of dimensions involved with the problem and. Call this number j. 4. Select j repeating variables, which do not form a dimensionless group.
5. Add one additional variable to the j repeating variables and form a power product. Solve to find exponents of the repeating variables which will make the product dimensionless. Repeat the process adding a different variable to the repeating variables each time until n-j dimensionless groups have been formed.
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6. Finally, verify that all the groups formed are dimensionless and write the dimensionless function.
It was decided that the variables which are relevant to the performance of the VAWT are the Power , the angular velocity , the free stream velocity , the swept area , the density of the fluid , the diameter of the turbine , the viscosity of the fluid and the surface roughness .
The units and dimensions for each variable are shown in Table 2 below
Variable
Units
Dimensions
Table 2 (List of variables) The number of Pi groups for the problem was calculated as follows
The repeating variables were selected to be the density , the free stream velocity , and the diameter . Each dimensionless group was calculated as follows
Pi group 1
38 Pi group 2 Pi group 3
39 Pi group 4 ( ) Pi group 5
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The dimensionless function can be written as follows
For the VAWT performance test, the coefficient of power varies mainly due to a change in tip-speed ratio. This relationship between the power coefficient and the tip-tip-speed ratio can be used to form dimensionless power curves for the VAWT. These dimensionless power curves allow the turbine to be compared to larger turbines in the same operating conditions.
From the dimensional analysis of the VAWT it was concluded that in order to construct the power curves for the turbine, the tip speed ratio must be varied and the torque at each tip speed ratio measured. In order to vary the tip speed ratio a brake must be used to apply a known braking torque to the rotor. Applying the braking torque slows down the rotor and changes the tip speed ratio.
It was concluded that to carry out the performance tests the test rig must have a brake which can apply the braking torque and a means of recording the rpm and the wind speed. Once this conclusion was made, the next step was to investigate the different methods of applying a braking torque to the VAWT and measuring its rpm.
4.3 Methods of applying torque (Brakes)
An investigation was carried out into the different methods of applying and measuring mechanical torque. After extensive research the most commonly used brakes were found to be the magnetic particle brake (MPB), the Prony brake and the Hysteresis brake. It was considered unnecessary to use all three braking methods for the test rig, so the brakes were rated on a scale of 0 to 5 (5 being the best), under the criteria of cost, accuracy of torque measurement and the range of sizes in which the brakes were available. Table 3 below shows the criteria for selecting the appropriate brake for the test rig.
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Brake Cost Accuracy Size Range Total
Magnetic Particle Brake 4 4 4 12
Prony Brake 5 3 4 12
Hysteresis Brake 2 4 2 8
Table 3 (Brake Selection)
It was concluded from the selection process that both the magnetic particle brake and the Prony brake were suitable for use in the VAWT test rig. A decision was made to proceed with a more detailed investigation into both the Prony brake and the MPB. This would provide a means of getting performance curves using two different methods and the results could then be compared. 4.4 Prony Brake
4.4.1 Background
One of the proposed methods of applying a torque to the VAWT rotor was the Prony brake. The Prony brake was initially introduced to measure the brake horse power of internal combustion engines. It uses frictional forces applied by a belt or rope to a pulley, to apply a load to a rotating pulley or shaft. Due to its simplicity the Prony brake is an inexpensive mechanical method of measuring the maximum torque output from any rotating shaft. The mechanics and operation of the Prony brake were investigated and will now be discussed in detail.
4.4.2 Derivation of Prony brake formula
The diagram in Figure 28 shows a pulley rotating at a constant angular velocity . The frictional force between the pulley and the belt causes a difference in tension in the two sides of the belt. This tensional difference is used to calculate the power output from the pulley.
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Figure 28 (Prony Brake fundamentals)
In order to derive the equation for the power output it is necessary to take a closer look at the forces which occur at the surface of the pulley. Figure 29 below shows a small section of the surface of the pulley wheel with the reaction force , the frictional force and the relevant angles displayed. It should be noted that for this derivation, the tensional force at is assumed to be greater than the tensional force at .
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Equations for and are obtained by equating the vertical and horizontal forces which gives
(4.1) (4.2)
Using a small angle approximation which states that goes to and goes to 1 as tends towards zero, the above equations can be rearranged to give
( ) (4.3)
(4.4)
Again using the approximation that is very small, the above equations simplify down to the following
(4.5)
(4.6)
Dividing the frictional force by the reaction force gives an expression for the coefficient of friction as shown
(4.7)
Both sides of the equation are integrated between the limits of 0 and because this is the area in which the belt or rope is in contact with the surface of the pulley. The integration is carried out as follows ∫ ∫ (4.8) (4.9)
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(4.10)
The frictional force on a segment of the belt or rope over angle is given by
(4.11)
The work done by the wheel as it turns through an arc of length s is given by
(4.12)
Integrating between 0 and gives the sum of the work done by the contact area. The integration is carried out as follows
∫ (4.13)
(4.14)
(4.15)
An equation for the power output from the pulley wheel is obtained by introducing the term which represents the rate of change of the arc length with time as the wheel turns. This term can be further simplified by breaking up the term into , where R is the radius of the pulley wheel and is the angle as previously discussed.
(4.16)
(4.17)
The term can be rewritten as the angular velocity , which gives the following equation for the power output from the pulley in terms of the angular velocity and the difference in tension across the two sides of the belt or pulley
45 4.4.3 Prony Brake Behaviour
The behaviour of the Prony brake under different loading conditions was investigated. Initially no tensional force is exerted on the ropes and the pulley rotates freely as there is almost no force due to friction between the pulley and the rope as shown in Figure 30.
The spring balance relating to T2 is displaced downwards as shown in Figure 31. This creates a
tension of T2+ΔT2 in the left hand side of the rope, and a tension of T1+ΔT1 in the right hand side
of the rope. The frictional force between the rope and the pulley increases and hence the rotational speed of the pulley drops down to . (Note that it is assumed that ΔT2 > ΔT1)
The tension T2 is increased in Figure 32 until the frictional force between the rope and the pulley
is large enough to stop the pulley . At this point T2-T1 is at its max, and the brake is
applying a torque which is equal to the maximum torque output for the turbine.
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Figure 31 (Prony Brake Behavior 2)
