Executive Summary

Title of ProjectDesign of an Ocean Current Turbine (OCT) Rotor Blade to Improve Turbine

PerformanceObjectives

Use Blade Element Momentum BEM theory and analysis to reconfigure an existing OCT rotor blade design with the view of improving the rotor performanceManufacture a test model of the new rotor design configuration.Test the designed and original rotor configurations in the Stellenbosch University towing tank.

What did I do that is new/unique?I designed an entirely new pitch angle distribution for an OCT rotor blade. I also used Josh Reinecke’s rotor test rig to test a new rotor configuration.

What are the findings?The designed blade showed an improved average performance over a wide range of flow and blade pitch angle conditions. The test rig provided accurate experimental results.

What value do the results have?The results will add to the pool of knowledge surrounding OCT rotors and form a basis for further optimisation design.

Which aspects of the project will carry on after completion of my part?OCT rotor materials should be investigated to minimise costs. An optimisation problem should be formulated encapsulating a wider range of design variables. The turbine mounting structure should be designed for cost effective attachment to the sea bed.

What are the expected advantages of continuation?Advantages include the possibility of improved turbine efficiency and a decrease in the manufacturing costs

What arrangements have been made to expedite continuation?This report will be available for comparison purposes should a new rotor configuration be designed. The BEM simulation program for the analysis of OCT rotors formulated in this project will be available at Stellenbosch University.

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Declaration

I, the undersigned, hereby declare that the work contained in this report is my own original work.

Signed: ....................................

RJ Stanford

Date: .....................................

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Acknowledgements

Acknowledgements need to go to certain people without whom I would certainly have tripped and fallen far before reaching the end of this project. To Nicola Cencelli who originally put forward this project, thank you for the guidance you gave from the outset of the project. Thank you to Professor Von Backström for taking me on as a ‘foster’ student and for your calm sage advice. Lastly thanks must go to Graeme Hammerse for his manufacturing expertise in the workshop and to Julian Stanfliet for providing entertainment during the testing process.

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Table of Contents

Executive Summary..............................................................................................................i

Declaration..........................................................................................................................ii

Acknowledgements............................................................................................................iii

List of Figures.....................................................................................................................vi

List of Tables.....................................................................................................................vii

Nomenclature...................................................................................................................viii

1 Introduction..................................................................................................................1

2 Literature Study............................................................................................................3

2.1 Ocean Currents Overview.....................................................................................3

2.2 OCT Rotor Performance.......................................................................................4

2.2.1 Two-Dimensional Hydrodynamics................................................................4

2.2.2 OCT Rotor Design Factors............................................................................6

2.2.3 Rotor Performance.........................................................................................7

3 Rotor Hydrodynamic Theory.......................................................................................8

3.1 One-Dimensional Momentum Theory and the Betz Limit...................................8

3.2 Blade Element Momentum Theory.....................................................................11

3.2.1 Prandtl’s Tip-Loss Factor............................................................................15

3.2.2 Glauert’s Correction Factor.........................................................................16

4 Design Methodology..................................................................................................17

4.1 Pre Design Decision Process...............................................................................17

4.2 Original Blade Configuration..............................................................................18

4.3 Simulation Range of Variables...........................................................................19

4.4 Generating 2-D Data from XFOIL......................................................................20

4.5 BEM Simulation Program...................................................................................21

4.6 Post Simulation Data Analysis............................................................................22

4.6.1 Structural Integrity of the blades.................................................................23

4.7 Design Results.....................................................................................................24

5 Manufacturing of Test Blade......................................................................................27

5.1 Generation of the CNC Code..............................................................................27

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5.2 Steps for Manufacturing a Single Blade.............................................................29

6 Model Rotor Testing..................................................................................................31

6.1 Experimental Setup and Apparatus.....................................................................32

6.1.1 Apparatus Functionality and Specifications................................................33

6.2 Testing Procedure...............................................................................................34

7 Test Results................................................................................................................37

7.1 Results Post Processing.......................................................................................38

7.2 Results Comparison............................................................................................40

7.3 Verification of Test Results................................................................................42

8 Conclusion..................................................................................................................45

9 References..................................................................................................................47

Appendix A: Blade Strength Calculations.......................................................................I

Appendix B: BEM Simulation Program Code (blade.m)..............................................IV

Appendix C: Xfoil Commands...................................................................................VIII

Appendix D: NACA 6 Series Numbering System..........................................................X

Appendix E: CNC Stock and Cutter Settings...............................................................XI

Appendix F: Verification of BEM Program...............................................................XIII

Appendix G: Torque Transducer Calibrations.............................................................XV

Appendix H: BEM Analysis Example......................................................................XVII

Appendix I: Project Costs.........................................................................................XXI

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List of Figures

Figure 1.1: Artists impression of SeaGen............................................................................1Figure 2.1: Lift and drag forces on a 2-d foil......................................................................5Figure 2.2: Free body diagram - rotor blade cross section..................................................5Figure 3.1: Control volume surrounding the actuator disk..................................................9Figure 3.2: BEM theory control volume............................................................................12Figure 3.3: Forces on a 2-dimensional blade section (Hansen, 2002)...............................13Figure 3.4: Inflow velocities at the rotor plane (Hansen, 2000)........................................14Figure 4.1: CL and CD values for various NACA 6 series foils.........................................20Figure 4.2: Flow chart illustrating the flow of information between Matlab functions....21Figure 4.3: Flat CP distribution vs ‘peaky’ CP distribution................................................22Figure 4.4: Comparison of the pitch angle distributions...................................................24Figure 4.5: Theoretical performance comparison (blade pitch - design case)...................25Figure 4.6: Theoretical performance comparison (blade pitch - design case + 5˚)...........26Figure 5.1: NCBlade preview of the stacked profiles........................................................28Figure 5.2: Blade profiles with cubic spline fit.................................................................28Figure 5.3: Photograph of the rotor and one of the blades after CNC machining.............29Figure 5.4: Rotor hub.........................................................................................................30Figure 5.5: Assembled rotor..............................................................................................30Figure 6.1: Towing tank trolley at Stellenbosch University..............................................31Figure 6.2: Diagrammatic representation of the test rig....................................................32Figure 6.3: Motor shaft support structure..........................................................................33Figure 6.4: Picture of the test rig mounted onto the trolley...............................................35Figure 6.5: Photograph of the rotor being dragged through the towing tank....................36Figure 7.1: Test rig system losses......................................................................................37Figure 7.2: Raw test data...................................................................................................38Figure 7.3: Experimental comparative performance (blade pitch – design case)..............40Figure 7.4: Experimental comparative performance (blade pitch – design case + 5˚)......41Figure 7.5: Results verification for the designed blade.....................................................42Figure 7.6: Results verification for Bahaj's blade at a 20 ˚ hub pitch angle......................43Figure 7.7: Results verification for Bahaj's blade at a 25 ˚ hub pitch angle......................43Figure A.1: Isosceles trapezoid representation of a NACA 6 series foil..............................I Figure A.2: Isosceles trapezoid diagram (efunda engineering fundamentals, 2008)..........IIFigure D.1: Labeled foil structure……………………………………………………….. X Figure E.1: Blade stock illustrating the positioning of profiles.......................................XIIFigure F.1: Analytical performance verification - hub pitch angle of 15˚......................XIIIFigure F.2:Analytical performance verification - hub pitch angle of 20˚ (Design case)XIV Figure F.3: Analytical performance verification - hub pitch angle of25˚.......................XIVFigure G.1: Percentage error in the torque transducer measurements............................XVI

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List of Tables

Table 4.1: Particulars of the original blade adopted from Bahaj et al. (2007).................18Table 4.2: Range of pitch angles over which each blade section was simulated..............19Table A.1: Elemental thrusts and moments about the hub.................................................IITable D.1: Description of the NACA 6 series numbering system......................................XTable E.1: Blade profile positioning.................................................................................XITable E.2: CNC tool specifications..................................................................................XIITable G.1: Results of the torque calibration....................................................................XVTable H.1: Axial induction factors over successive iterations.......................................XIXTable I.1: Project costs...................................................................................................XXI

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Nomenclature

A Rotor area

a Axial induction factor

a’ Tangential induction factor

B Number of rotor blades

BEM Blade Element Momentum (theory)

c Chord length

CD Coefficient of drag

CL Coefficient of Lift

CNC Computerized Numerical Control

CP Coefficient of Power

CT Coefficient of Thrust

D Drag force

dM Elemental torque

dP Elemental power

dT Elemental thrust

L Lift force

OCT Ocean Current Turbine

P Pressure

R Outer rotor radius

r Local rotor radius

t Maximum foil thickness

TSR Tip speed ratio

Vaxial Induced axial flow

Vrot Induced rotational flow

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Vrel Relative inflow velocity

α Angle of attack

θ Local blade pitch angle

ρ Density

ω Rotor angular velocity

Φ Relative inflow angle

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1 Introduction

The global concern over the apparent way which humans are leaving their ‘carbon footprint’ on the earth has lead to a renewed effort to develop effective energy conversion methods from renewable sources. In South Africa, 90% of the total energy needs is derived from fossil fuels resulting in enormous masses of carbon dioxide being emitted to the environment. Compounding the problem is the fact that fossil fuels are a non-renewable energy source and no one is certain as to the resources left on the planet.

Until recently, the quest for renewable energy methods has mainly been land based due to the difficulty of building in the sea. However, the ocean is a vastly abundant source of renewable energy and new building techniques have lead to a great deal of interest shown in ocean based energy generation methods. Ocean energy is however still in a very early stage of development stage. Among the clean, renewable energy sources available in South Africa, ocean energy is probably the least developed. On the 21st February 2008 the first South African workshop on ocean energy was held in the Western Cape. The aim was to promote the utilisation of ocean energy power so that it contributes at least 100 GWh to the national electricity generation mix by 2015 (Ocean Energy Working Group, 2008). At this workshop, it was stated in a presentation that the only two feasible forms of energy generation from the oceans around South Africa are wave energy and ocean current energy (Deon Retief, 2008).

Figure 1.1: Artists impression of SeaGen

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Sailors have known for centuries that ocean currents flowing around the world are extremely powerful forces of nature. According to the Minerals Management Service (2006) in the USA, Ocean currents worldwide contain approximately 5 000 GW of power and certain places have power densities of up to 15 kW/m2. The technology for harnessing this energy has been modelled on wind turbine technology. The most common method currently under review is the use of horizontal axis ocean current turbines (OCT). Horizontal axis OCT’s are structurally very similar to horizontal axis wind turbines, however the difference in the working fluid results in certain operational dissimilarities. The technology is still extremely juvenile with the first national grid linked OCT only being installed at the beginning of this year off the coast of Wales. This is an experimental unit called SeaGen (Figure 1.1) and owing to the young technology, problems have already arisen with two of the four blades being damaged (Marine Current Turbines, 2008b). That aside, SeaGen is rated at 1.2 MW and is successfully connected to the national grid.

Even though this magnitude of energy is available, the efficiency of current OCT’s is still very poor. The area of most energy loss occurs at the interface between the ocean current and the turbine rotor. Therefore in order to maximise turbine efficiency, the conversion of energy at this interface needs to be maximised. The focus of this project was to improve the performance of an existing OCT rotor configuration by using simple hydrodynamic design techniques and Blade Element Momentum (BEM) theory analysis. The design parameter focused on was the pitch angle distribution or ‘twist’ of the rotor blade. The improvement of performance was to be measured by improvement in the rotor’s Coefficient of Power. The project objectives are briefly listed below.

Project Objectives

Use BEM theory and analysis to reconfigure an existing OCT rotor blade design with the view of improving the rotor performance

Manufacture a test model of the new rotor design for testing. Test the designed and original rotor configurations in the Stellenbosch University

towing tank.

The following document describes the entire process which was followed to accomplish the project objectives. The report starts off with an overview of the literature study that took place. Following this will be a description of one-dimensional and BEM theory including the derivations of the equations involved with both. The main body of the report will describe the design, manufacture and testing procedures which were followed. Finally the last two chapters will present the test results and the conclusions which were derived from the design and test results. Additional information such as sample calculations, the simulation program code and additional data regarding the project is included in the Appendices.

This report forms part of a final year mechanical project put forward by Mrs. Nicola Cencelli. The project is a continuation of a project done by Josh Reinecke (2007) wherein he designed and validated the performance of an OCT model rotor test rig for testing rotors in the Towing Tank at Stellenbosch University.

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2 Literature Study

2.1 Ocean Currents Overview

With the abundant energy available in the ocean and the recent resurgence of renewable energy research, ocean currents are becoming an increasingly more enticing renewable energy source. Of the available ocean based renewable energy sources, ocean currents also tend to have a few distinct advantages over the rest. Due to the predictable nature of the currents, energy can be generated to a timetable (Fraenkel, 2002). This is hugely beneficial when generating electricity for a national grid, especially one as unstable as South Africa’s. Also, the harnessing of ocean currents is inherently similar to the harnessing of wind. Therefore, even though ocean current technology is still extremely juvenile, it will be possible to apply the more advanced technology of wind turbines to the harnessing of ocean currents. Also, the cost of wind powered electricity generation has dropped by 75% over the last 25 years due to scale economics and steep learning curves (Marine Current Turbines, 2008a). Since the two technologies are so similar, the same result can be expected for OCT’s due to the advanced wind turbine technology basis.

As with wind energy, various techniques of energy harnessing have been attempted for ocean currents. However, it appears that the horizontal axis turbine still remains the best performer. According to Marine Current Turbines (2008a), “the elegant simplicity and unsurpassed efficiency of the axial flow pitch-controlled rotor has been shown to make it superior in all respects to any other method of kinetic energy conversion.”

Certain parties have also showed a lot of interest in the vertical axis OCT alternatives. Among the benefits of this technology is the fact that the generator can be coupled straight onto the turbine shaft, minimising drive chain losses. However, certain deciding factors contribute to make horizontal axis turbines a more attractive method for harnessing ocean currents. These deciding factors as described by (Fraenkel, 2002) are listed below:

A vertical axis rotor involves considerably more structure per unit of swept area which increases the cost.

Vertical axis rotors generally do not self start requiring the rotor to be driven by another energy source to a sufficient speed for the blades to ‘unstall’.

Horizontal axis rotors are far easier to stop during emergency situations. They can either be yawed so that they are side on to the flow or the blades can be pitched to induce stall.

Horizontal axis blades do not require as good a surface finish as vertical axis blades to maintain a high lift:drag ratio.

For these reasons, the majority of global research has been on the improvement of horizontal axis turbines. The following sub section will describe OCT rotor performance.

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2.2 OCT Rotor Performance

Since water is approximately 800 times denser than air, moving water holds considerably more energy than moving air. It was estimated by Fraenkel (2002) that “A suitably energetic marine current location can offer four times the energy intensity of a good wind site and some 30 times the energy intensity of sunshine in the Sahara in terms of energy captured per unit area of interface”. According to Marine Current Turbines (2008a), there are three factors which govern the energy capture of a water current at a given suitable site. These three factors are: the current speed, the swept area of the rotor and the efficiency of the entire OCT system. Since the current speed is uncontrollable and the swept area has restrictions due to structural and cost considerations, the area where the energy capture can be most influenced is the overall system efficiency.

The energy generation unit of a horizontal axis OCT system as a whole consists of a rotor, some drive train and a generator. The whole system has energy losses from the rotor interface through the drive chain to the generator. These system losses can be reduced to a certain extent by using good shaft alignment techniques and applying the correct lubrication to reduce frictional losses. However the main energy loss by far is at the rotor interface with the water current. The following two sub-sections describe the hydrodynamics surrounding the OCT rotor as well as the design factors to consider and the measurement of rotor performance.

2.2.1 Two-Dimensional Hydrodynamics

The following sub-section will briefly describe the hydrodynamics behind the conversion of energy from the current stream to the rotor. The basis of this conversion stems from foil fluid dynamics where the diversion of flow around a foil profile creates a lift force in the direction of the suction side of the foil. Referring to Figure 2.2, it can be seen that the camber of the foil profile results in an acceleration of the flow over the suction side. This faster flow results in a local pressure drop in the fluid above the foil. The resultant pressure difference between the suction and pressure sides of the foil generate a lift force as can be seen in Figure 2.2. The lift generated by this foil section is the driving force behind the turbine rotor. As can be seen in the figure, there is also a drag force which the flow imparts on the foil. This drag force is in the direction of fluid flow and has a detrimental effect on the foil performance.

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Figure 2.2: Lift and drag forces on a 2-d foil

Now refer to Figure 2.3 which is a cross sectional profile of an OCT rotor blade at some radius (r) along the blade. The rotor in the figure is rotating at angular velocity ω and the current flow is approaching perpendicular to the rotor. Therefore, the resultant velocity of the flow which the blade section sees (Vrel) approaches the foil at an angle Ф from the rotor plane. This particular blade section is also pitched at an angle θ from the rotor plane. Therefore the resulting angle of attack which this blade section sees (α) results in lift and drag forces on the blade section as shown in Figure 2.3. By taking the tangential components of the lift and drag forces (LT and DT) into account, one can determine the torque which the blade section is generating. If LT and DT are equal, the blade section is said to be “stalled” and it does not contribute any torque to the rotor. However, when LT

is greater than DT, the blade section applies a torque in the direction of the rotational motion and adds to the rotor power generation. For these reasons, when selecting rotor blade foil profiles it is beneficial to incorporate foils which exhibit high lift:drag ratios over a wide range of flow conditions.

Figure 2.3: Free body diagram - rotor blade cross section

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This relationship between the current flow and the rotor occurs along the entire length of the blade. For horizontal axis turbines, the ‘power house’ of the rotor is the section of the blade near the tip. Here the higher relative velocity and greater distance from the hub result in the tip section generating more torque than the section of the blade near the hub.

2.2.2 OCT Rotor Design Factors

According to Batten et al., (2006) the hydrodynamic design parameters of the rotor as a whole basically consist of the rotor diameter, rotational speed and the rotor blade pitch. The factors affecting the performance of the rotor blade are the choice of blade profile, the chord length distribution and the pitch angle distribution or ‘twist’ of the blade. Since rotor performance is directly dependent on the blade design, the first step in the rotor design process is to design a suitable rotor blade configuration. From there the remaining rotor parameters can be designed to further increase turbine performance. This sub-section will describe the factors influencing the decisions concerning the design of an OCT rotor blade configuration.

Since OCT design is based on existing wind turbines, the relatively advanced wind turbine technology can be adapted for OCT’s. However, certain differences such as the occurrence of cavitation result in the need for further investigation, research and development (Batten et al., 2006). Cavitation occurs in liquids when the local fluid pressure falls below the vapour pressure of the fluid. As a result, bubbles form which implode and can cause great physical damage to the rotor. Therefore design limits need to be placed on certain rotor configuration parameters to avoid the inception of cavitataion. A relationship for the inception of cavitation will be presented later in Chapter 4 (Eqn 4.1).

As mentioned in the previous sub-section, the driving force behind an OCT rotor is the lift force created by the cross sectional foil profile of the rotor blade. Therefore, a desirable characteristic of a section profile is that it has good lift coefficients (C L) over a wide range of conditions whilst still avoiding the inception of cavitation. In order to maintain efficiency, low drag coefficients (CD) are also desirable; however structural considerations result in thicker sections being required which increases the drag on the section. To avoid cavitation, the camber on the foils needs to be quite large (refer to Appendix D for a description of foil structure). The foil profiles which Bahaj et al. (2007) used had a camber approximately four times greater than profiles commonly used for wind turbines.

The effect of the pitch angle distribution (twist) of the blade is to change the angle of attack with which Vrel approaches each blade section. This change in Vrel affects the Lift and Drag characteristics of the blade sections, thereby affecting the torque and thrust which is exerted on the blade. The chord length distribution on the other hand affects the circulation which the blade induces in the flow. For efficient rotor performance, the circulation along the length of the blade should be kept as constant as possible. To

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achieve this and for structural reasons, the cross sectional chord lengths generally decrease uniformly along the length of the blade.

Now viewing the rotor as a whole, the design of certain parameters such as the number of rotor blades, rotor diameter and optimal operational tip speed ratio (TSR) need to be considered. Generally, most working OCT rotor designs consist of 2 or 3 blades. It is found from wind turbines that rotors consisting of four or more blades generally have a loss of efficiency since the blades operate partially in each other’s wakes (Encyclopaedia of Alternative Energy and Sustainable Living). Concerning the rotor diameter, obviously the larger the rotor is the more energy is available due to a larger swept area. However, the rotor diameter is generally restricted due to ocean current site restriction and cost and structural considerations. Finally the TSR that the rotor should operate at is determined by the combination of current flow and rotational speed that generates the most power for a given rotor configuration. The TSR is the ratio between the blade tip speed and the free stream velocity providing a non-dimensional range for the rating of performance. Generally, OCT’s tend to show better performance at TSR’s between 5 and 7 Which is typically the aimed operating range. The determination of rotor performance is described next.

2.2.3 Rotor Performance

In order to determine the effect of the design alterations made to a rotor, the performance of the rotor needs to be evaluated. Since not all rotors are the same size or operate in the same conditions, it is important to have a means of comparing the performance of these rotors with each other. The following sub section describes the method for rotor performance comparison.

At the rotor interface, the kinetic energy available in the moving water current is converted into mechanical energy in the form of a rotating shaft. This energy conversion is performed by the OCT rotor. By calculating the ratio between the power generated by an OCT rotor to the available power in a water current, a non-dimensional power ratio can be defined. This ratio is called the Coefficient of Power (CP) and can be used to compare the performance of two or more rotors. The non-dimensional aspect of the CP

renders it capable of comparing rotors of different sizes operating in different current speeds. This is hugely beneficial for hydrodynamic performance comparisons.

In 1919, Albert Betz came to the conclusion that the maximum amount of theoretical power that a turbine rotor can generate is depicted by a CP value of 16/27 (Danish Wind Industry Association). By using BEM theory described in the following chapter, it is possible to determine the theoretical performance of turbine rotors operating under different conditions. The principles and derivations of BEM theory are described next.

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3 Rotor Hydrodynamic Theory

The same two-dimensional theory that is applied to aeroplane and hydrofoil wing dynamics can be applied to the turbine rotor dynamics, however the rotational component of the flow needs to be accounted for. A combination of One-Dimensional Momentum Theory and Blade Element Momentum Theory contribute to relatively accurate predictions of rotor fluid dynamics. The following chapter describes the two theories according to Cencelli (2006) and The Marine Current Energy Group (2006).

Before describing the two theories, two vital factors in determining rotor power from these two theories need to be mentioned, namely the axial and tangential induction factors. These factors provide a method to compare the axial and tangential flow velocity

components induced by the rotor ( and ) with the rotor speed ( ) and the free

stream velocity ( ). These induced velocities act in opposing directions to the current

and rotation of the blade and can be described by the following two equations:

(3.1)

(3.2)

These induction factors will be used throughout the explanation of the two theories.

3.1 One-Dimensional Momentum Theory and the Betz Limit

The primary function of an OCT rotor is to convert the kinetic energy contained in a moving stream into mechanical energy in the form of a rotating shaft. One-dimensional momentum theory offers a simple model to calculate the power extracted by an ideal turbine rotor. The theory can also be used to determine the thrust exerted by the fluid flow on the rotor and the effect of the rotor on the local fluid field. The following section describes one-dimensional theory and the derivation of the Betz Limit according to Cencelli (2006) and The Marine Current Energy Group (2006).

Refer to Figure 3.4 which is a schematic of the control volume used for one-dimensional theory. The model represents the rotor as an actuator disk with ideal properties. The control volume boundaries consist of the surface of a stream tube flowing through the actuator disk and two cross sectional areas at each end of the tube. The fluid speed

upstream of the rotor ( ) is slowed down by the disk to at the rotor plane and further

to in the wake of the rotor. As the fluid flows through the control volume it is evident

that there are variations in the fluid pressure. There is a slight pressure rise in the fluid

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just upstream of the disk before a discontinuous pressure drop over the disk. Far downstream of the disk the pressure returns back to atmospheric conditions.

Figure 3.4: Control volume surrounding the actuator disk

Analysis using this theory assumes the following assumptions:

The fluid flow is homogenous and incompressible The fluid flow is one-dimensional The fluid flow is frictionless The rotor consists of an infinite number of blades There is no induced rotation in the wake of the rotor The static pressure far upstream and far downstream of the rotor is the same as

ambient pressure

Now considering the control volume in Figure 3.4, derivations of the forces on the rotor can be performed. Firstly the thrust can be calculated using the conservation of momentum over the rotor.

(3.3)

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It should also be noted that the thrust can be calculated by multiplying the pressure drop over the rotor by the actuator disk area.

(3.4)

Now two expressions for the thrust have been calculated using different approaches,

however one is a function of the pressure drop ( ) which is unknown. In order to

determine an expression for one can apply Bernoulli’s Equation to the control

volume before and after the disk. After some manipulation of the two equations, this yields the following expression:

(3.5)

Using this expression for the , Eqns 3.3 and 3.4 can be equated resulting in an

expression for the flow velocity through the disk.

(3.6)

Thus the fluid velocity through the rotor plane is the average of the upstream and downstream velocities. Now by using the relationship for the axial induction factor (a)

in Eqn 3.1, relationships can be described for the three flow velocities and .

(3.7)

(3.8)

An expression needs to be derived for the power extracted by the turbine from the current flow. In order to accomplish this it is assumed that there is no internal energy change in the control volume. The conservation of energy principle can then be used which results in:

(3.9)

By substituting the velocity relations described in Eqns 3.7 and 3.8, the power and thrust then become:

(3.10)

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(3.11)

The function of a turbine rotor is to convert kinetic energy into mechanical energy. The efficiency of the rotor is determined by how much of the available energy in the fluid

stream is converted to mechanical energy by the rotor. The available power ( ) can

be calculated as:

(3.12)

is the free stream velocity and A is the swept area of the rotor. Now the efficiency of

the rotor is determined by setting the shaft power in comparison with the available power.

This provides the Coefficient of Power ( ).

(3.13)

By substitution of Eqn 3.10 into 3.12, the Coefficient of Power can be determined in terms of the axial induction factor.

(3.14)

Since CP is represented as a function of only one variable, it is possible to find a theoretical maximum limit for CP by differentiating Eqn 3.14 with respect to a.

(3.15)

By setting this derivative equal to zero, a maximum value of is achieved

with an axial induction factor of . This value is known as the Betz limit and is the

maximum possible theoretical efficiency that a turbine rotor can obtain. In practice it is impossible to attain the Betz limit since it is derived by assuming frictionless, stationary flow.

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3.2 Blade Element Momentum Theory

The shortcoming of one-dimensional theory is that it does not take into account the finite number of rotor blades or that there is a rotational component in the wake of the rotor. Blade Element Momentum (BEM) theory combines the global one-dimensional momentum theory with the hydrodynamic influence of the flow on the rotor blades at the rotor interface. The control volume used for one-dimensional theory is divided into N annular elements of thickness dr as is illustrated in Figure 3.5. R in the figure is the outer radius of the entire rotor and r is the local radius of the element under consideration. Two-dimensional fluid dynamics can be applied to each element to determine the elemental thrust and torque. Integration of these values over all the annular elements will result in the total thrust and torque for the rotor. The following section will describe BEM theory according Cencelli (2006) and The Marine Current Energy Group (2006).

BEM theory relies on the following two assumptions:

The annular elements are isolated from each other (One element does not affect the performance of another.

The forces calculated using 2-dimenisonal fluid dynamics are constant over the annular element meaning that the rotor is assumed to consist of an infinite number of blades.

Figure 3.5: BEM theory control volume

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Now consider the annular element in Figure 3.5 the elemental power generated by this control volume can be calculated using Euler’s Turbine Equation.

(3.16)

It can be seen that the power is a function of the angular velocity and the elemental torque on the control volume. This elemental torque can be calculated using the moment of momentum equation. To do this the rotational component in the flow is set to zero upstream of the rotor and to Vrot downstream of the rotor. Therefore, the elemental torque becomes:

(3.17)

In order to determine the elemental thrust, the momentum equation could be applied annular control volume to arrive at the following expression:

(3.18)

Assuming that the relation between the axial induction factor and the inflow speed is still valid, the thrust and torque equations can be re-written in terms of the axial and tangential induction factors:

(3.19)

(3.20)

Figure 3.6: Forces on a 2-dimensional blade section (Hansen, 2002)

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Another approach to determine the elemental torque and thrusts on the annular elements is by analysing the local flow conditions at the blade element. From two dimensional fluid dynamics the lift and drag forces imparted on the blade element can be calculated as:

(3.21)

(3.22)

The normal and tangential forces on the blade can then be determined by projecting the lift and drag forces onto the normal and tangential directions as shown in Figure 3.6.

(3.23)

(3.24)

The forces can then be normalised with respect to .

(3.25)

(3.26)

Now noting that and are forces per unit length, the elemental thrust and torque,

and , acting on an annular element of thickness and a rotor containing blades can

be determined.

(3.27)

(3.28)

It is now necessary to determine an expression for the relative flow velocity (V rel) and the ion flow angle (Φ) in terms of the axial and tangential induction factors. Figure 3.7

below illustrates a vector breakdown for the fluid velocity at the element. is the free

stream velocity, is the angular velocity of the rotor and is local radius of the annular

element. The angles , and are the pitch angle of the blade section, the angle of

attack and the flow angle at the blade respectively.

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Figure 3.7: Inflow velocities at the rotor plane (Hansen, 2000)

From these vectors, two expressions for can be derived.

(3.29)

(3.30)

Also from the vectors in Figure 3.7, the inflow angle Φ can be calculated by:

(3.31)

Using the relations describe by Eqns 3.23 through to 3.30, the equations for and can be written as:

(3.32)

(3.33)

At this stage, two different methods were used to derive two sets of equations for and

. These sets of equations can be equated with each other in order to derive

expressions for the axial and tangential induction factors. First however the solidity, ,

needs to be calculated. The solidity is the fraction of the control volume which is covered by the blade and it is determined by:

(3.34)

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Now expressions for the induction factors can be derived.

(3.35)

(3.36)

Due to the inaccuracy of the assumptions made at the beginning of this section, two correction factors need to be made to the theory. These two correction factors and their applications are described below.

3.2.1 Prandtl’s Tip-Loss Factor

In the beginning of the section two assumptions were described which BEM theory adheres to. The second of these assumptions was that the rotor consisted of an infinite number of blades. This assumption is obviously not correct, therefore Prandtl’s tip-loss

factor, , was incorporated. F can be calculated by:

(3.37)

(3.38)

Eqns 3.32, 3.33, 3.35 and 3.36 are modified to include the tip-loss factor:

(3.39)

(3.40)

(3.41)

(3.42)

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3.2.2 Glauert’s Correction Factor

Another correction factor needs to be included for the fact that the momentum theory breaks down for high axial induction factors (a > 0.4). A proposed empirical correction

formula including a critical induction factor, and the Prandtl tip loss factor, , is

described by :

(3.43)

For it can be shown that Eqn 3.41 still stands, however for the following

relationship must be used:

(3.44)

Where:

(3.45)

Now looking back at the entire process, it is evident that the induction factors can be calculated by first calculating the forces on the rotor. However, it is impossible to calculate the forces on the rotor without first knowing the induction factors. Therefore an iterative approach is required to accurately determine the forces on the rotor and the induction factors. The following steps describe this approach:

1. Assume initial values for a and a’ (eg. a = 0 and a’ = 0)

2. Calculate the flow angle, , at the foil profile by using Eqn 3.31

3. Calculate the local angle of attack,

4. Read the CL(α) and CD(α) values from a table

5. Using Eqns 3.25 and 3.26, calculate and

6. Calculate a and a’ using Eqns 3.41 and 3.42 7. Repeat steps 2 to 6 until convergence of the induction factors is reached

Once convergence of the induction factors has been reached, the elemental thrust and torque on the annular element can be calculated using Eqns 3.32 and 3.33. The elemental

17

torque can then be multiplied by the angular velocity to determine the elemental power which the annular element generates. A sample calculation using BEM theory is presented in Appendix H.

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4 Design Methodology

The aim of this project was to improve the performance of an existing ocean current turbine rotor by designing a new blade configuration for the rotor. Improvement of the rotor performance was determined by the Coefficient of Power (CP) which is the rotor’s effectiveness of converting the available power in a flowing current to a mechanical form. The design objectives were therefore to design a rotor which showed good performance characteristics over a wide range of flow conditions. To accomplish this, the design focused on only one parameter of the rotor, namely the pitch angle distribution or ‘twist’ of the blade.

To maximise the rotor performance the process consisted of a generic approach wherein each pitch angle alternative was analytically tested and the performance of each alternative compared. For this purpose, a simulation program was written in Matlab to iterate through all the possible design alternatives and eventually determine the best pitch angle solution. For the simulation and analysis of the rotor, the blade was split up into 11 sections which were each analysed independently using Blade Element Momentum (BEM) theory as discussed in Chapter 2. This chapter describes the entire design process including the choice of design variable, the original rotor configuration, the generation of analytical data and the ultimate design results from the post processing of the simulation program.

4.1 Pre Design Decision Process

The rotor blade configuration consists of many variables which affect its performance. To focus on all of them would not be possible given the time and size of the project. Therefore the design variables were weighed against each other to determine which could have the greatest positive effect on the CP distribution over a range of TSR’s. The three main design variables are the chord length distribution, the cross sectional foil profile shapes and the pitch angle distribution. The following three paragraphs explain the effect of each variable on the blade performance as well as the reasoning for including/excluding them from the design process.

The chord length distribution was the first potential design variable under consideration. The chord length of a particular blade section has a direct impact on the circulation at that section. The efficiency of the rotor performance is directly dependent on keeping the circulation as constant as possible along the length of the blade. This imposes a restriction on the degree to which the chord length can be varied. Strength considerations also restrict the chord lengths to certain limits. These considerations limit

19

the effect that the chord length distribution can have on increasing the CP value, thereby making it an unattractive design variable to focus on.

The two-dimensional foil profiles of the blade were the second potential design variable to be considered. Foil profiles are complex containing many sub-variables within their structure (refer to Appendix D) that affect their performance in different mediums under different conditions. Focusing on the foil profiles would indeed have an effect on the blade performance; however it would be better to incorporate this in an optimisation design taking all the design variables into account. This would be more in the order of a thesis which is much larger with more time available than this project. Therefore the design of the foil profiles was left out of the design process.

Lastly the pitch angle distribution was considered as a design variable. The pitch angle distribution along the length of the blade offers an opportunity to increase the CP value of an existing working design. The pitch angle at a particular blade section directly effects the angle of attack with which the relative flow strikes that section. For this reason it has large affect on the induced forces on the blade section under different flow conditions. Consequently it was decided to use the configuration of an existing OCT rotor as a starting point and focus solely on the pitch angle distribution to better its performance.

4.2 Original Blade Configuration

Since the design focuses solely on the blade pitch angle distribution, it was necessary to find a working OCT rotor blade configuration to form the basis of the design. The original rotor configuration was adopted from a paper written by Bahaj et al. (2007) wherein he designed and tested a model rotor to verify the numerical predictions for the performance of OCT’s. The blades consisted of a distribution of two-dimensional foil profiles with chord length, thickness and pitch angle distribution shown in Table 4.1. The cross-sectional profiles used in the design ranged from a NACA 63-824 profile at the hub down to a NACA 63-812 profile at the tip (the NACA 6 series numbering system is explained in Appendix D). The model rotor that was used in Bahaj’s testing consisted of 3 blades and had an outer diameter of 800 mm with a hub diameter of 100 mm. In order to have a reference point to verify and compare results, all these particulars were maintained as constants with the only varying physical design variable being the pitch angle distribution.

Table 4.1: Particulars of the original blade adopted from Bahaj et al. (2007)

r [mm] c/r θ [deg] t/c [%]80 0.125 15 24.0

120 0.116 9.5 20.7160 0.106 6.1 18.7200 0.097 3 .9 17.6

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240 0.088 2.4 16.6280 0.078 1.5 15.6320 0.069 0.9 14.6360 0.059 0.4 13.6400 0.050 0.0 12.6

4.3 Simulation Range of Variables

As mentioned before, the rotor was to be designed to improve performance over a range of flow conditions. For design purposes, this requirement was translated into designing the rotor for a range of TSR values. The TSR is the ratio of the blade tip speed to the free stream velocity. It is known from wind turbine theory, which is inherently similar to OCT theory, that the design tip speed ratios of high efficiency wind turbines are on the order of between 5 and 7 (Encyclopaedia of Alternative Energy and Sustainable Living). Also, according to Bahaj et al., (2007), test results of a model turbine show good agreement to the performance predictions generated by BEM theory for TSR values of between 4 and 8. However for TSR values of over 8 the theory seems to over predict CP

values. Taking these factors into consideration, the design TSR range was chosen to be between 4 and 8 with the rotor designed to provide peak performance at a TSR value of 6.

The testing range for the pitch angles was determined by preventing the occurrence of cavitation. The inception of cavitation on two-dimensional foils can be determined by identifying when the negative of the coefficient of pressure (-Cp) of the foil equals the cavitation number (σ) as presented in Eqn 4.1. At angles of attack that are too high or too low, the area of minimum pressure on the 2-d foil falls lower than the vapour pressure of water causing cavitation to occur. Therefore the pitch angle range for each blade section, as seen in Table 4.2, was determined by comparing the negative pressure coefficient values to σ in order to determine at which angles of attack cavitation was likely to occur. These angles of attack were converted to blade pitch angles by using BEM theory at the highest design tip speed ratio. It was found that at certain sections the pitch angle range was extremely large. In these instances the range was limitless before being narrowed after the first design iteration.

(4.1)

Table 4.2: Range of pitch angles over which each blade section was simulated

Pitch angle ranges [ ]

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Min Max Min Max

Station 1 8 30 Station 8 2 17Station 2 6 30 Station 9 1 16Station 3 3.5 26 Station 10 0 16Station 4 3 25 Station 11 0 15Station 5 2.5 23Station 6 2 20Station 7 2 18

4.4 Generating 2-D Data from XFOIL

During the design process, BEM theory was used in a numerical simulation program to calculate elemental torque and elemental thrust values for different pitch angle distributions. Requirements of BEM theory are the coefficients of lift and drag (CL and CD) of the different two dimensional foil profiles under the desired conditions. Since no tabulated or other data was available for the NACA 6 series, these values had to be generated using a program called XFOIL.

Figure 4.8: CL and CD values for various NACA 6 series foils

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XFOIL is an interactive program for the design and analysis of subsonic foils (Drela & Youngren, 2001). By importing the data points of the 13 foils, a series of commands could be implemented to refine the data points and generate CL and CD values for the specific foils. Appendix C describes the XFOIL commands used in this process. The foils were sampled for all the angles of attack within each blade section’s range at intervals of 0.5˚. Since the blade was simulated at different TSR’s during the numerical program, the CL and CD values also had to be calculated at different Reynolds numbers. Therefore each foil profile was simulated at Reynolds numbers of 1 x 105, 2 x 105, 3 x 105

and 4 x 105 to provide an extensive data base for interpolation at a later stage. Figure 4.8 Illustrates sample CL and CD values for 5 different NACA 63-8xx foils generated at a Reynolds number of 2 x 105.

4.5 BEM Simulation Program

As discussed above, the approach to accomplish the design objectives was to iteratively simulate the rotor at different TSR’s with varying pitch angle distributions. During each simulation, the performance of each alternative was analysed. Therefore once all possible alternative solutions within the design range were simulated and analysed, the results could be compared with each other to determine the best solution. The entire design was conducted by means of a simulation program written in Matlab called blade.m (the program code is available in Appendix B). This program returned the performance characteristics for each blade section over the range of pitch angles (θ) and tip speed ratios (TSR). These performance characteristics were then analysed in a post simulation analysis discussed in sub-section 4.6. The following paragraph explains the process of the simulation program.

For the simulation and analysis, the blade was split up into 11 sections along the length of the blade according to the positioning of the foil profiles. The program then iterated through each of the 11 sections to analyse them individually. For each section, the program iterated through the TSR range, at TSR steps of 0.5, and for each TSR it iterated through the θ range at steps of 0.5˚. At each θ, a function, bem.m, was called to perform BEM analysis on the section under the specific conditions. The function returned the section’s elemental torque (dM), elemental thrust (dT), and circulation which were stored for later analysis. Figure 4.9 below is a flow diagram illustrating the flow of information between the main simulation program blade.m and the sub-function bem.m. Once the program had iterated through all 11 sections, the stored data was used in a post simulation analysis described next.

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Figure 4.9: Flow chart illustrating the flow of information between Matlab functions

4.6 Post Simulation Data Analysis

Once the data was collected and stored for all 11 sections at the different TSR’s and pitch angles, it had to be processed in order to determine the pitch angle distribution which provided the best performance for the rotor. As mentioned before, the performance of the rotor is determined by its CP value. For practical applications it is also important that the rotor exhibit a good CP distribution over a range of TSR values rather than just at the design TSR. Therefore, the rotor was to be designed to produce a CP distribution curve similar to curve B rather than the ‘peaky’ distribution of curve A in Figure 4.10. The following section describes the process followed to accomplish this.

Figure 4.10: Flat CP distribution vs ‘peaky’ CP distribution

According to Eulers turbine equation, the power generated by each blade section is equal to the product of the section’s elemental torque and rotor’s angular velocity. Therefore, determining which θ generates the maximum torque at each section for a specific TSR

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would maximise the CP value of the rotor for that specific TSR. This being said, increasing the performance at one specific TSR does not increase the performance when operating at TSR values other than the design value. Therefore, three different methods were used to determine which pitch angles provided the best CP distribution over the entire TSR design range and the results of the three methods were compared with each other.

The first method used was the ‘mean distribution’ method (MD). The MD method consisted of calculating the mean elemental torque generated over the TSR range for each pitch angle alternative. The next method was the ‘weighted distribution’ method (WD). This method consisted of weighting the contribution of the elemental torques generated at different TSR’s towards the weighted distribution value WD. This method gave the design TSR the heaviest weighting with the remaining TSR’s getting weighted according to Eqn 4.2. The final method used was the ‘Area under the curve’ method (AREA). This method consisted of calculating the area under the dM vs TSR curve for each sections pitch angle alternatives.

(4.2)

The three methods were used to analyse each of the 11 blade sections independently. For each section the MD, WD and AREA values derived from these methods were calculated and plotted as a function of θ. These graphs were analysed to determine at which θ these values were the highest and whether the three methods correlated with each other. After determining a vague θ for each blade section at which the values were agreeably the highest, the simulation was refined by narrowing the pitch angle test range for each blade section and making the θ step size smaller. The simulation was then repeated to provide more accurate results. The design process followed these steps until the θ step size was down to 0.1˚. At this stage the tabulated data of the WD, MD and AREA values could be used to determine a θ distribution which would theoretically provide the best average CP

distribution over the design TSR range.

Before finalising the design results, a last factor needed to be taken into account which could affect the blade performance. This factor is the variation of circulation along the blade. According to Cencelli (2006), turbine rotors perform efficiently when the circulation along the length of the blade is kept constant. A function for the circulation of a blade section can be described by:

(4.3)

where ω is the rotor rotational speed, r is the local radius of the section, C L is the lift coefficient of the foil profile and c is the chord length. So using this equation, the circulation of each blade section was calculated under the different design conditions.

25

The statistical variances of these circulation values along the length of the blade under the different conditions were then calculated. Under worst case conditions, the variance was calculated to be 0.4 x 10-3. Since Cencelli (2006) place an upper limit of 1 on the variance in her wind turbine blade optimisation program, this value for the circulation was deemed to be well within the limits.

4.6.1 Structural Integrity of the blades

Since the designed blades were going to be manufactured and tested, the structural integrity of the designed blade needed to be analysed. By analysing the shape of the rotor blade it can be seen that it is most vulnerable in the axial loading direction. Therefore the strength of the blade in this direction needed to be enough to withstand the axial thrust force of the water flowing through the rotor.

In order to determine the strength of the blade, the elemental thrusts (dT) of each section were required. These values were obtained from another program written in Matlab called bem2.m which used BEM theory to analyse the performance of a specific blade configuration over a range of TSR’s. The dT for each section increases with increasing TSR, therefore the strength calculations were performed with dT values calculated at a TSR of 10 which is the highest TSR in the test range. The strength calculations were conducted according to the process described in Appendix A and revealed a maximum stress in the blade of 51.9 MPa. Since the tensile strength of the material to be used, aluminium 6082-T6, is 310 MPa, the structural integrity of the blade was deemed to be sound with a safety factor close on 6.

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4.7 Design Results

100 150 200 250 300 350 4000

5

10

15

radius (mm)

pitc

h a

ng

le (

de

gre

es)

Designed bladeBahaj's blade

Figure 4.11: Comparison of the pitch angle distributions

The effect of the design process was to alter the pitch angle distribution of the original blade configuration designed by Bahaj et al. (2007). The difference between the pitch angle distributions of the designed blade and Bahaj’s blade can be seen in Figure 4.11. It is evident that the design has changed quite dramatically from the original configuration. Bahaj’s blade originally had a pitch angle distribution which spanned over a range of 15˚

and was described by the following function where is the radius ratio r/R and is a

constant equal to 2˚ for the NACA 63-8xx series.

(4.4)

In comparison the designed blade has a linear pitch angle distribution over a smaller range of only 10˚. It should also be noted that the two blades were also designed to operate at two different blade pitch angles. Where Bahaj’s blade was designed to operate at a pitch angle of 5˚ (20˚ at the hub), the designed blade was designed for a pitch angle of 3˚(13˚ at the hub).

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The effect of the change in pitch angle distribution was determined by the CP distribution of the two blades. In order to determine the performance of the two blades, the BEM theory program mentioned above (bem2.m) was used. This program generated the CP

values for the two blades at different TSR’s within the range. The CP distribution for the blades could then be plotted against each other to compare the performance of the two blades. In order to analyse the effect of pitching the blades, the performance was analysed both at the design pitch angles of the blades and at pitch angles of 5˚ greater than the design case. Figure 4.12 and Figure 4.13 illustrate the comparative performance of the two blades both at the design blade pitch angle and pitch angles 5˚ higher.

4 5 6 7 8 9 100.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

TSR

CP

Designed blade

Bahaj blade

Figure 4.12: Theoretical performance comparison (blade pitch - design case)

Figure 4.12 shows that for the design pitch angle, the designed blade theoretically reaches a CP value of 0.451 whereas Bahaj’s blade theoretically only reaches 0.447. Therefore the result of the design process was to theoretically increase the performance by 0.8 %, which is not as high as originally was hoped for. It can also be seen that theoretically Bahaj’s blade exhibits better performance at high TSR’s. From these observations it can be seen that the theoretical results do not show much improvement over Bahaj’s blade. However, it will be showed in Chapter 7 that due to the fact that BEM theory tends to over predict rotor performance, the theoretical performance comparison of the two rotors is slightly inaccurate. It will also be shown later that the performance over prediction is not the same for different blade configurations. Therefore it was necessary to combine the

28

theoretical results with the experimental results before substantial conclusions could be drawn.

4 5 6 7 8 9 100.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

TSR

CP

Designed blade

Bahaj blade

Figure 4.13: Theoretical performance comparison (blade pitch - design case + 5˚)

As mentioned before, the rotor performance was also simulated for blade pitch angles 5˚ off the design pitch angle. This was done to determine how the rotor performance was affected by pitching the blades. Since the most common method of controlling energy generating turbines is to pitch the blades, it is desirable that the blade still shows good performance at blade pitch angles which are not the rotor’s designed case. As can be seen in Figure 4.13 the designed blade shows a vast improvement over Bahaj’s blade with respect to the fact that the change of blade pitch angle has far less affect on it. Figure 4.13 shows that the effect of a 5˚ pitch angle offset on the designed blade is to decrease the CP by 0.064 for Bahaj’s blade the decrease in CP is 0.096. Therefore, in this sense the designed blade shows theoretical improvement over Bahaj’s blade. It will also be shown in Chapter 7 that experimental results also show this characteristic. The following Chapter will deal with the manufacturing of the model rotor for testing.

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5 Manufacturing of Test Blade

Since the analytical results required verification from experimental results, a set of test blades needed to be manufactured. The manufactured blades could then be attached to an existing test rig for testing. The rotor configuration from Bahaj et al. (2007) consisted of a 3 blade setup, therefore 3 blades of the designed configuration needed to be manufactured. The machining of these blades was done using a 3-axis CNC milling machine at Stellenbosch University’s Mechanical and Mechatronic Workshop. The material used for the blades was 6082-T6 aluminium. The reasons for selecting this material were mainly due to its corrosion resistance, relative cheapness and good machinability. Aluminium 6082 is not as ‘sticky’ as some aluminium types and stainless steels and therefore does not wear the cutter down as fast. It also has a tensile strength of 310 MPa making it far stronger than normal aluminium (Reinecke, 2007).

The accuracy of the finished blades was incredibly important since the accuracy of the test results depended on it. Therefore time was spent on the planning of the procedures which lead to the manufacturing of the blades. The following section describes the pre-manufacturing procedures which took place as well as the cutting and finishing of the blades.

5.1 Generation of the CNC Code

The CNC code for the blades was generated using a program called NCBlade written by the head of the Mechatronic Engineering department at Stellenbosch University, Prof. AH Basson. NCBlade was designed as a program to generate CNC cutting code from a set of stacked profile data points. As inputs to the program the user loads coordinate points of the 2-D cross sectional profiles of the blade. Each profile must be defined by a sequence of coordinate points starting at the trailing edge running either way around the leading edge and back to the trailing edge. All profiles must also consist of the same number of coordinate points. Once all the profiles are loaded they must be offset in the x, y and z directions so that the foils are positioned relative to each other according to the designed blade configuration. The foils must also be scaled to size according to the relevant chord lengths.

In the program written for the manufacturing of the designed blades, 13 NACA foil profiles were used along with 2 circle profiles 25 mm in diameter. The circle profiles were positioned at the hub side of the blade to allow for easy connection of the blade to the hub. The 13 foil profiles were stacked at 1/3 of their chord length whilst the circle profiles were stacked along their centre axes as can be seen in Figure 5.14.

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Figure 5.14: NCBlade preview of the stacked profiles

Once all the foils were loaded, a cubic spline was used to interpolate the data points and create a surface which connected the coordinate points of all the foils illustrated in Figure5.15. The next step was to set the size of the stock that the blade will be cut out of. The calculation of the stock size needed to take into account the size and shape of the blade as well as the clamping sites for the machining process. Exclusion zones, which the cutter would miss out along its path, also needed to be included as illustrated in Figure 5.14. These exclusion zones were only implemented on the suction side of the blade and provided added rigidity to the work piece which prevented excessive vibrating. Vibrations would cause inaccurate machining and a poor surface finish if they weren’t prevented.

The next step was to set the cutter specifications. These specifications included the cutter diameter, spindle speed, feed rates and side step sizes that the cutter would follow. Also included was the option of uni-directional or bi-directional cutting and whether the cutter must cut in the reverse or forward direction. The cutter specifications for the machining of both the surfaces of the blades are included in Appendix E.

Figure 5.15: Blade profiles with cubic spline fit

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Once all the parameters were entered, the program was allowed to generate the cutter path code for the CNC machine. This code was then checked by another program written by A.H Basson, called NCEdit. NCEdit checked the code for mistakes or glitches and returned a message indicating the predicted cutting time and whether the code was correct. Once the code was checked and deemed to be correct, it was ready to be used on the CNC machine.

5.2 Steps for Manufacturing a Single Blade

The manufacturing of a blade began with the generation of the CNC code as described in the section above. Four sets of CNC cutting code had to be generated for each blade since each side of the blade had a ‘rough cut’ and a ‘finishing cut’. The rough cut was used to quickly remove the waste material of the stock to within 0.5 mm of the finished surface. The finishing cut was a far slower process with smaller cutter side-step sizes and slower feed rates to ensure a good surface finish.

Once the CNC code had been generated for both suction and pressure sides of the blade and for both rough and finishing cuts the stock was mounted in the CNC milling machine. The CNC machining of the blade involved the following sequence of steps:

1. The stock was mounted and clamped in place in the CNC machine.2. The operator then zeroed the cutter according to the origin placement in NCBlade

(top left corner of the stock as can be seen in Figure 5.14). 3. The cutting code for the rough cut of the suction side of the blade was then loaded

and the CNC machine performed the suction side rough cut. 4. Next the CNC code for the finishing cut of the suction side was loaded and the

finishing cut of the suction side commenced. 5. The stock was then rotated around the x-axis to cut the pressure side. 6. The operator had to once again zero the cutter before loading the pressure side

rough cut CNC code and commencing the cut. 7. Finally the CNC code for the pressure side finishing cut was performed.

Figure 5.16: Photograph of the rotor and one of the blades after CNC machining

32

Upon completion of the CNC machining, the blade was still connected to the rest of the stock by the exclusion zones as can be seen on the right of Figure 5.16. Before the blade could be cut out of the stock, a threaded M8 hole needed to be drilled on the hub side centred on the centres of the two circle profiles. This threaded hole was later used to house a threaded stud to attach the blade to the hub of the rotor. The blade was then cut out of the stock. The places where the blade had been connected to the stock stood out from the blade surface and needed to be filed down to the correct profile of the blade. Once this was done the blade was sanded down to give it a good smooth surface finish. The final step was to screw the M8 threaded stud into the hole and fasten it in place with Loctite. The blades were attached to the hub of the rotor using M8 locknuts as shown in Figure 5.17. A view of the whole assembled rotor is shown in Figure 5.18.

Figure 5.17: Rotor hub

Figure 5.18: Assembled rotor

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6 Model Rotor Testing

Experimental performance results were required for both rotor configurations in order to verify the theoretically predicted performance improvement. These experimental results were obtained by testing both rotor models under various blade pitch and rotational speed conditions. The testing of the two rotors took place in the towing tank at Stellenbosch University’s Faculty of Engineering illustrated in Figure 6.19. The rotors were both tested using a rig that was designed by Reinecke (2007) for use in the Stellenbosch university towing tank. The following section describes the testing procedure that was followed as well as the apparatus used for the testing of the model rotors.

Figure 6.19: Towing tank trolley at Stellenbosch University

34

6.1 Experimental Setup and Apparatus

The towing tank facility consists of a long concrete rectangular tank with a trolley that can tow test models through the water to determine the models’ hydrodynamic performance. The towing tank is 90 m long, 3 m wide and 2.7 m deep. The towing trolley depicted in Figure 6.19 runs on rails that are on the side of the tank and is driven by four 3-phase motors which are situated at the four corners of the trolley. It can reach speeds of up to 9 m/s and is operated by the orange control box as can be seen in Figure6.19.

In order to measure the performance of the rotor as it was towed through the tank, a test rig was needed. The rig used was designed by Reinecke (2007) for his final year mechanical engineering project. It was based on a rig designed by Bahaj et al. (2007) and was proved to provide accurate results by comparing the experimental data obtained by Reinecke and Bahaj. As can be seen in Figure 6.20, the test rig consists of a main steel frame which is mounted onto the trolley. Connected to this frame is an L-shaped pipe housing which houses the rotor shaft and timing belt. A second steel frame which houses the motor and motor shaft is attached to the main frame by four long bolts.

Figure 6.20: Diagrammatic representation of the test rig

35

The L-shaped pipe housing extends down from the main frame into the water. The short part of the L contains the rotor shaft with the model turbine rotor attached to the end of it and fastened in place with a locknut. Connected to the rotor shaft is a timing belt which extends up through the pipe housing and is connected to a pulley on the motor shaft. The motor shaft is held in place on the top frame by two plumber block bearing mountings. The motor was mounted on the top frame which was attached to the main frame by four long bolts. The arrangement of the motor shaft components is shown in Figure 6.21. Connected to this shaft was a torque transducer for measuring the torque and reflective strip for measuring the shaft rotational speed with an optical tachometer. During testing, the rotor is driven by the relative water flow over it causing the rotor shaft to rotate with a torque. This rotation and torque are transferred by the timing belt to the motor shaft where they are measured. Since the pulley sizes on the two shafts are different by a factor of 56/81, the measured data needs to be corrected to obtain the rotor shaft conditions.

Figure 6.21: Motor shaft support structure

6.1.1 Apparatus Functionality and Specifications

To determine the power extracted by the rotor, the torque generated by the rotor and the rotational velocity needed to be measured. Also in order to calculate the power available in the water, the speed of the trolley needed to be known. All the measurements that were taken also needed to be converted to a readable form and stored. The following subsection describes the functionality of each device used in the testing as well as describing their specifications.

In order to measure the torque that the rotor generated, a torque transducer was required. The transducer used consists of a full bridge strain gauge setup. The strain gauges receive a nominal voltage from the data acquisition system and return an altered voltage according to the degree of deformation of the strain gauges as a result of the applied

36

torque on the transducer. The torque transducer used was an HBM T5 50NM strain gauge measuring system. It has a nominal operating torque of 50Nm and a sensitivity of 2mV/V. Its rated supply voltage is 0.5 V to 12 V and it has a 350 Ω input resistance. The torque transducer was calibrated to show less than 1% error for torques greater than 5 Nm. The calibration method and results are presented in Appendix G.

An optical tachometer, otherwise known as a digital stroboscope, was used to measure the shaft rotational speed. The device used was a Luton DT-2234A optical tachometer. It has a test range of 5 to 100 000 rpm and a sampling time of 1 second. The resolution of the device is 0.1 rpm and it has an accuracy of ±0.05%.

In order to maintain the induction motor at a constant rotational speed, a frequency controller was used. The frequency controller used was a Mitsubishi Electric Fr-E500 transistorized inverter. It has a power output range of 0.4 kW to 7.5 kW with an output frequency of between 0.2 Hz and 400 Hz. The inverter is controlled by a removable control unit. The frequency outputs from the inverter ranged from between 3 Hz and 15 Hz.

To measure the torque which the rotor generated, a constant opposing braking torque needed to be applied. For this purpose a 1.3 kW three phase induction motor was used. The induction motor would drive the rotor up to speed before each test run. Once the trolley started moving, the rotor would apply a torque on the shaft and drive it at a higher rotational speed. Since the motor was controlled by the frequency controller it would resist the increase in rotational speed and therefore apply an opposing torque allowing the torque transducer to measure the torque generated by the rotor.

The mounting of the test rig was done by fastening the main frame to the trolley and connecting the L-shaped pipe housing to the frame. The frame housing the motor, motor shaft and torque transducer was then mounted onto the main frame. The timing belt was connected to the motor shaft pulley and the top frame was vertically adjusted to tighten the belt. Next the torque transducer was connected to the data acquisition system which in turn was connected to a laptop computer for data storage. Then finally the motor was connected to the variable speed drive which controlled its rotational speed.

6.2 Testing Procedure

The aim of the testing was to measure the performance of the model test rotors over a range of TSR’s and to compare these results with the results obtained using the simulation program. By comparing the results, the rotor design and the simulation program could be validated.

According to Euler’s Turbine Equation, the performance of a turbine rotor is dependent on the angular velocity of the rotor and the torque which it generates. Therefore in order to determine the power generated over a range of TSR’s, the rotor torque was measured while the trolley was being dragged through the tank at different rotational speeds. The TSR range over which the performance was to be determined was between 2 and 10.

37

This correlated to rotor shaft rotational speeds of between 72 and 358 rpm. Since the pulleys on the two shafts are different sizes, these rpm’s had to multiplied by a factor of 81/56 in order to determine the motor shaft rpm range which was between 104 and 518 rpm. The rotors also needed to be tested with the blades set at two different pitch angles to determine the performance reduction due to blade pitching. This resulted in a total of four test runs required.

Figure 6.22: Picture of the test rig mounted onto the trolley

The test rig was assembled onto the trolley by attaching the main frame of the rig to the rails on the trolley as shown in Figure 6.22. In this figure the rig is raised out of the water which was how the rotor was put on and taken off. During testing the main frame rests horizontally on the rails with the tubing extending into the water. Once the test rig was assembled, the system losses needed to be determined. The rotor was taken off the rotor shaft and the system was driven at various rotational speeds with torque measurements being taken. Each shaft speed was logged and the torque data obtained from the torque transducer was saved for later post-processing.

The rotor blade pitch angles then had to be set to the desired testing angle. This was done by loosening the blade at the hub and positioning it slowly while measuring the tip angle with a digital level. The rotor was placed on a flat, level surface while this was done. An offset of 1.7 degrees also had to be taken into account due to the shape of the blade profile at the tip.

38

Once the blades had been set, the testing could commence. The testing procedure was the same for all the test runs. The following bulleted list explains the testing procedure that was followed for each rotor at each blade pitch angle:

Before each run the torque was zeroed on the data acquisition system The frequency controller was then set to the desired value and the motor was

switched on at a specific rotational speed. The data acquisition unit was started and the trolley operator given the go-ahead. Once the trolley was up to speed the optical tachometer was used to determine the

motor shaft rotational speed which was logged. (Figure 6.23 illustrates the test set-up while the trolley was running

At the end of the run the data acquisition unit was switched off and the torque data was stored.

The trolley returned to the starting point and the sequence was repeated for a different shaft rotational speed.

Once all the data was stored using the data acquisition system it was processed and the performance curves for each rotor at the different blade pitch angles were determined. The following chapter discusses the post processing of the data and presents the final results.

Figure 6.23: Photograph of the rotor being dragged through the towing tank

39

7 Test Results

The purpose of the testing procedure was to obtain experimental performance data for both Bahaj’s blade and the designed blade configuration. The data obtained for both blades could then be compared with each other to determine the effect of the change in pitch angle distribution on the rotor performance. The experimental results could also be compared with the theoretical data obtained from BEM theory analysis. This comparison could then be used to verify the experimental results.

Before the actual testing of the rotor took place, the system losses needed to be measured in order to accommodate for them in the results post processing. These system losses result from poor shaft alignment and bearing friction. The method for determining these system losses is described in Chapter 5. The losses were measured for a range of shaft rotational speeds and a linear curve fit was implemented to define the losses as a function of the shaft rotational speed. This linear relationship could then be used to interpolate for different shaft rotation speeds and the test data could be corrected by including the relevant system losses. Figure 7.24 graphically illustrates the data points obtained during the system loss testing as well as the linear curve fit which was applied to the data points. From the graph one can see that there is a slight linear increase in system losses with increasing rotational speed.

50 100 150 200 250 300 350 400 4500.7

0.8

0.9

1

1.1

1.2

1.3

1.4

RPM

To

rqu

e (

Nm

)

Losses test data pointsLinear curve fit

Figure 7.24: Test rig system losses

40

As described in the previous chapter, the testing procedure consisted of measuring the torque which the rotor generated at specific shaft rotational speeds whilst the rotor was being dragged through the water at a constant velocity. Figure 7.25 illustrates graphically the raw torque measurements from a test run with the rotor shaft rotating at 192 rpm. From the graph it can be seen that the motor was turned on at approximately 2 seconds, the negative torque indicating that the motor was actually driving the turbine. The trolley started moving at approximately 8 seconds and the turbine quickly started generating its own torque converting the motor into a generator. At approximately 11 seconds the system balanced out and the torque remained relatively constant until approximately 25 seconds when the trolley was stopped. The useful range of data is when the system is at steady state between 11 and 25 seconds. The following sub-section will describe the post-processing that took place in order to generate performance curves for the rotors.

0 5 10 15 20 25 30 35-5

0

5

10

15

20

Time (s)

Tor

que

(Nm

)

Figure 7.25: Raw test data

7.1 Results Post Processing

The raw measured data captured by the data acquisition unit required a certain degree of post processing to convert it into a friendlier format for performance comparison. The performance of a turbine rotor is determined by its Coefficient of Power (CP) which is the ratio between the power generated and the power available. The following sub-section

41

describes how the measured data was processed to provide CP values for performance comparison.

The first step was to determine the torque which the rotors generated for each test run. This was determined by averaging the measured torque values over the useful measuring time range (ie. between 12 and 25 second in Figure 7.25). The next step was to apply a linear function to the system losses as described above. This was done using the ‘polyfit’ function in Matlab. The function which was generated had the following form:

(7.1)

The total generated torque was then calculated for each run by:

(7.2)

Now in order to calculate the amount of power generated, the measured torque was multiplied by the measured rotational speed for each run. However, first the rotational speed needed to be converted to rad/s.

(7.3)

The power generated was then calculated by:

(7.4)

CP is the ratio between power generated and available power, therefore the available

power was calculated by the following equation where is the tank water density, is

the trolley speed of 1.5 m/s and is the swept area of the rotor:

(7.5)

The CP for each test run could then be calculated by:

(7.6)

The last step was to calculate the TSR values correlating with the measured motor shaft rotational speeds. To do this the measured motor shaft rotational speed needed to be multiplied by a conversion factor in order to obtain the rotor shaft rotational speed. This is due to the fact that the pulleys on the two shafts are different sizes resulting in differing rotational speeds. Therefore the rotor rotational speed was calculated by:

(7.7)

42

Therefore, the TSR for each test run could be calculated by:

(7.8)

7.2 Results Comparison

The experimental results provide a means to determine how the designed rotor performs in practice compared to the original rotor. The following sub-section presents the experimental results of the two rotors at two different blade pitch angles and graphically compares them. The accuracy of the results was also determined by comparing the theoretical and experimental results for both blades at the two blade pitch angles.

The first comparison is between the two rotors at their design blade pitch angles (20˚ hub angle for Bahaj’s blade and 13˚ hub angle for the designed blade). This illustrates the performance of the two rotors when they are operating under their design conditions. By comparing these results with the theoretical results in Figure 4.12 it can be seen that the numerical simulation program seems to have falsely estimated the performance improvement of the designed blade. It is evident that the designed blade only reaches a CP value of 0.437 whereas Bahaj’s blade has a maximum CP value of 0.45. At first glance this may seem that the designed blade is inferior. However, it should be noted that the designed blade has a flatter performance distribution over a range of TSR’s. As described in Chapter 4 this is more desirable than a ‘peaky’ performance distribution since the performance remains high irrespective of the flow conditions. Therefore, at TSR’s greater than 7 the designed blade seems to perform better than Bahaj’s blade.

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Figure 7.26: Experimental comparative performance (blade pitch – design case)

The benefits for a turbine operating at higher TSR’s are twofold. Firstly a smaller generator is required since rotational velocities are higher with higher TSR’s. Secondly, since the TSR is the ratio of the blade tip speed over the current free stream velocity, a high TSR results in a lower stream velocity. Thus the designed blade will perform better than Bahaj’s blade in slower moving currents.

The second comparison consists of the results of the two rotors with blade pitch angles of 5˚ greater than the design case (25˚ hub angle for Bahaj and 18˚ hub angle for the designed blade). These results compare the performance of the two rotors when the blades are pitched at angles offset from the design pitch angle.

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Figure 7.27: Experimental comparative performance (blade pitch – design case + 5˚)

A common method for controlling turbines is to pitch the blades into the oncoming current. Although this method allows accurate control of the turbine, there is a distinct loss in performance when the blades are pitched off the designed blade pitch angle. As can be seen in Figure 7.27, the effect on the performance of pitching the blades is less for the designed blade than Bahaj’s blade. Bahaj’s blade exhibits a reduction in CP of 0.13 whereas the designed blade’s maximum CP only dropped by 0.1. These results are consistent with the theoretical results obtained from BEM analysis.

The results of Figure 7.26 and Figure 7.27 show that although the performance of Bahaj’s blade is not bettered by the designed blade, there are distinct advantages to the designed blade’s performance characteristics. The flatter performance distribution and greater performance over a range of blade pitch angles are beneficial both to the control of OCT turbines and the adaptability to changing flow conditions. It can be concluded therefore that at an ocean current site with variable flow conditions, the designed blade will provide a better average performance over the entire range of conditions.

7.3 Verification of Test Results

This sub-section indicates the accuracy with which the numerical simulation predicted the performance of the two blades by comparing the analytical results from the simulation with the experimental results. Since there is no previous data for the designed blades this is the only form of verification for the new blade configuration. However, the

45

results for Bahaj’s blade configuration can also be compared with Bahaj’s experimental results for added verification.

3 4 5 6 7 8 9 100.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

TSR

CP

Test data points (3 degree set angle)Theoretical performance (3 degree set angle)Test data points (8 degree set angle)Theoretical performance (8 degree set angle)

Figure 7.28: Results verification for the designed blade

Figure 7.28 illustrates the comparison between the theoretical and experimental results for the designed blade at both blade pitch angles. It is evident that the simulation program tended to over predict the performance of the blade which is consistent with BEM theory. However, the general trends of the data are the same and the data curves follow the same general path just offset from each other. This characteristic leads to the assumption that the inaccuracy may be due to the absence of a tunnel blockage correction factor (Bahaj et al., 2007).

46

Figure 7.29: Results verification for Bahaj's blade at a 20 ˚ hub pitch angle

Figure 7.30: Results verification for Bahaj's blade at a 25 ˚ hub pitch angle

47

The verification results of Bahaj’s blade illustrated in Figure 7.29 and Figure 7.30 exhibit a completely different characteristic altogether. When comparing the experimental results with Bahaj’s experimental results, it can be seen that the tests were performed with a good degree of accuracy. Once again, the slight under prediction of performance when compared with Bahaj can possibly be attributed to the absence of a suitable tunnel blockage correction. Also the experimental results show good agreement with the analytical results for both blade pitch angles up till a TSR of approximately 7. At TSR’s greater than this the simulation program tends to over predict the performance which is once again consistent with BEM theory.

So from the verification of the results, it can be seen that the experimental and analytical results have different relationships for the two blades. For the designed blade the two curves follow the same path just offset whereas for Bahaj’s blade the curves follow the same path before breaking away from each other. This difference in relationships goes a long way to explain the differing performance comparisons between the analytical and experimental results.

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8 Conclusion

The objectives of this mechanical project were to design, manufacture and test an OCT rotor with a new blade configuration aiming to better the performance of an existing design. The performance increase of the rotor was measured by an improvement of the CP values over a TSR range. The physical design variable was the pitch angle distribution or ‘twist’ of the blade.

The design process made use of a simulation program which iteratively determined the blade performance for different pitch angle distributions using BEM theory. Since the design needed to be validated with experimental results, 3 blades with the design configuration were manufactured. This was done on a 3-axis CNC machine resulting in high precision manufacturing and a good surface finish. The resulting blades were tested in the Stellenbosch University towing tank to provide accurate experimental results.

The effect of the new pitch angle distribution on the performance was determined analytically as well as experimentally. The analytical results showed that the designed blade theoretically showed a slight increase in the maximum performance. However, the experimental results disagree with the analytical results in the sense that the designed blade experimentally shows a slight decrease in maximum performance. Also, it was evident from the analytical results that the performance of the designed blade was not as severely affected by the blade pitch angle as Bahaj’s blade was. This characteristic was consistent with the experimental results. It should also be noted that the designed blade experimentally shows better performance at TSR’s higher than 7.

The reasons for the false prediction of performance improvement predicted by BEM analysis are twofold. Firstly, referring to the validation of the BEM program in Appendix F, it can be seen that when compared to Bahaj’s analytical results, the BEM program tends to under predict the performance of Bahaj’s blade at TSR’s less than 6. This results in the illusion that the designed blade outperforms Bahaj’s blade at TSR’s less than 6. Secondly, refer to the experimental verification of the theoretical results in Chapter 7. It can be seen that the theoretical results for the designed blade over predict performance over the entire TSR range, however the theoretical results for Bahaj’s blade only over predict performance at high TSR’s. This over prediction over the entire TSR range made it appear from the theoretical results that the designed blade out performed Bahaj’s blade in terms of maximum CP.

The conclusions that can be drawn from the results are that although the designed blade does not improve on the maximum CP of the rotor, it shows better performance when the rotor is operating over a wide range of conditions. Therefore, at a site where the flow conditions are constantly changing, the designed blade will have a better performance than Bahaj’s blade when considering the average performance over the entire range of conditions. The designed blade would also be more suited for turbines using variations in

49

blade pitch to control the turbine. In practice, the majority of turbines operate in tidal streams which have a constantly changing flow speed and the most common method for turbine control is to use variable blade pitch control. Therefore, the designed blade will be better suited to these conditions.

A great deal of further research still needs to be performed in order to make OCT’s financially feasible. Further research opportunities lie in attempting an optimisation design for the OCT rotor in order to better performance characteristics. Time should also be spent on designing an efficient method of securing the turbine to the ocean floor. Another aspect for consideration should be the investigation of different rotor materials to minimise cost whilst maintaining structural integrity.

In conclusion the project objectives were all satisfactorily met. A new OCT rotor blade configuration was designed, manufactured and tested at Stellenbosch University. With respect to the design objectives, the designed blade showed an improvement over Bahaj’s blade when considering the average performance over a wide range of conditions. BEM theory simulation program was written which provided relatively accurate performance predictions and provided a medium for the blade design. The rotor performance predictions were verified using Bahaj et al, (2007) and experimental results. Therefore this project, along with the project done by Reinecke (2007), provides a sound base for further research into the field of ocean current turbines at Stellenbosch University.

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9 References

ALLSTAR Network, Wing Design - level 3: Conventional Airfoils and Laminar FlowAirfoils, Updated: 12 march 2004, [online], Available:http://www.allstar.fiu.edu/aerojava/Wing31.htm, [22-10-2008]

Bahaj AS, Batten WMJ, McCann G, 2007, Experimental verifications of numericalpredictions for the hydrodynamic performance of horizontal axis marine currentturbines, [online], Available: http://www.sciencedirect.com/science?

_ob=ArticleURL&_udi=B6V4S4R05BKN1&_user=613892&_coverDate=12%2F31%2F2007&_alid=7994421&_rdoc=1&_fmt=high&_orig=search&_cdi=5766&_sort=d&_docanchor=&vie=c&_ct=1&_acct=C000032099&_version=1&_urlVersion=0&_userid=613892md5=033b0d3977daa5fbb5053f81ed054065, [22-10-2008]

Batten WMJ, Bahaj AS, Molland AF, Chaplin JR, 2005, Hydrodynamics of MarineCurrent Turbines, [Online], Available:http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V4S4HDP70H1&_user=613892&_rdoc=1&_fmt=&_orig=search&_sort=d&view=cversion=1&_urlVersion=0&_userid=613892&md5=38f48362329adcccc8e736bc1f44f70, [22-10-2008]

Cencelli NA, 2006,Master of Science Thesis - Aerodynamic Optimisation of a SmallScale Wind Turbine Blade for Low Windspeed Conditions, Department ofMechanical and Mechatronic Engineering, Engineering, Stellenbosch University

Drela M, Youngren H, 2001, XFOIL 6.9 User Primer, [online], Available:http://web.mit.edu/drela/Public/web/XFOIL/, [22-10-2008]

efunda engineering fundamentals, 2008, Isosceles Trapezoid, [Online], Available:http://www.efunda.com/math/areas/IsosTrapazoid.cfm, [22-10-2008]

Encyclopaedia of Alternative Energy and Sustainable Living, Tip Speed Ratio, [online],Available:http://www.daviddarling.info/encyclopedia/D/AE_design_tip_speed_ratio.html,[02-10-2008]

Fraenkel PL, 2002, Power from marine currents, Proceedings of the Institution ofMechanical Engineering

Marine Current Energy Group, Glauert’s Blade Element Momentum Theory, [Online],Available: www.esru.strath.ac.uk/EandE/Web_sites/0506/marine_renewables/technology/Glauert.htm, [02-10-2008]

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Marine Current Turbines, 2008a, Marine Currents, [Online], Available:http://www.marineturbines.com/23/advantages_of_marine_currents/, [22-102008]

Marine Current Turbines, 2008b, Delay in Commissioning One of SeaGen’s Rotors,[Online], Available:http://www.marineturbines.com/3/news/article/11/delay_in_commissioning_oneof_seagen_s_rotors/, [22-10-2008]

Minerals Management Service (USA), 2006, Ocean Current Energy Potential on the U.S.Outer Continental Shelf, [Online], Available:http://ocsenergy.anl.gov/documents/docs/OCS_EIS_WhitePaper_Current.pdf,[22-10-2008]

Ocean Energy Working Group, 2008, Terms of Reference for the Ocean Energy Networkof South Africa, [Online], Available:http://academic.sun.ac.za/crses/pdfs/SAOEN(TermsOfReference)June2008.pdf,[22-10 2008]

Reinecke J, 2007, Mechanical Project - The Design, Manufacture and Test of a ModelOcean Current Turbine, Department of Mechanical and MechatronicEngineering, University of Stellenbosch

Retief D, 2008, Ocean Energy In South Africa, [Online], Available:http://academic.sun.ac.za/crses/pdfs/2)%2021%20Feb%20Workshop_Retief.pdf, [22-102008]

SESSA (Sustainable Energy Society of Southern Africa), 2008, Will South Africa leadthe solar energy revolution?,[online], Available:http://www.sessa.org.za/news_item.2008-06-06.8523121082, [02-10-2008]

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Appendix A: Blade Strength Calculations

In order to determine the structural integrity of the designed blade, the blade was modelled as a beam. The blade was divided into 11 sections and the elemental thrust which was exerted on each element was calculated using BEM theory. Since the blade is essentially a cantilevered beam, the part of the blade which will be the most stressed is the blade right close to the hub. The cross sectional profile of the blade at this section consists of a NACA 63-824 foil. In order to analyse the stress generated by these torques, the foil profile was modelled as two isosceles trapezoids as shown in Figure A.1. These trapezoids were defined in terms of the chord length (c), the foil max thickness (t) and two factors (a1 and a2) which describe the short parallel side of the trapezoid. The variables illustrated in Figure A.1 are now listed:

Figure A.1: Isosceles trapezoid representation of a NACA 6 series foil

The strength analysis started off with the determination of the bending moment in the blade near the hub. The elemental thrusts were calculated at the most extreme case which was at a TSR of 10. The total bending moment was then calculated by adding the moments of each elemental thrust force around the hub:

(A1)

The results of this are shown in Table A.3 where r is the local radius, dT is the local elemental thrust and dMo is the bending moment of each section around the hub. It can be seen at the end of the table the total bending moment around the hub is 43.74 Nm.

I

Table A.3: Elemental thrusts and moments about the hub

r (mm) dT (N) dMo (Nm)0.0121 3.9402 0.04770.242 3.0064 0.07280.0364 4.1238 0.15010.0540 5.9513 0.32140.0740 9.4527 0.69950.1055 15.6667 1.65280.1440 22.4019 3.22590.1840 28.5628 5.25560.2240 35.6381 7.98290.2640 43.4977 11.48340.3040 42.2558 12.8458Total: 214.5 43.74

Now before we can determine the stress induced in the blade at the hub by the bending moment, it is necessary to calculate the area moment of inertia of the cross section around the chord line. To do this the moments of inertia of the two isosceles trapezoids were calculated and added to together. The calculation of these moments of inertia is described below.

Figure A.2: Isosceles trapezoid diagram (efunda engineering fundamentals, 2008)

Referring to Figure A.2, the moments of inertia needed to be calculate around the yc axis (Iyc). According to efunda engineering fundamentals (2008), Iyc can be calculated by the following formula:

II

(A2)

Using Eqn A2 and the dimensions for the two trapezoids, the two area moments of inertia were calculated as:

(A3)

(A4)

The stress in the blade at the hub can now be calculated by the bending moment equation:

(A5)

Since the model turbines were manufactured out of 6082-T6 aluminium which has a tensile strength of 310 MPa. Therefore the blade was deemed to be strong enough with a safety factor of just under 6.

III

Appendix B: BEM Simulation Program Code ( blade.m )

In order to simulate the effect of different pitch angle distributions on the performance of the blade, a numerical computer program was written in Matlab. The input to the program is the free stream velocity and the output is a vector containing the mean distribution values, weighted distribution values and the integral values of elemental torques for each blade section. The program code for blade.m follows:

function [dMm1 dMm2 dMm3 dMm4 dMm5 dMm6 dMm7 dMm8 dMm9 dMm10 dMm11] = blade(Vo) %specify the TSR range, the theta sample step, NACA profiles and radius %of the analysis to take place

TSRmin = 4; TSRstep = 0.5; TSRmax = 8;tstep = 0.1;naca = 23:-1:13;r = [0.0921 0.1042 0.1164 0.1340 0.1540 0.1855 0.2240 0.2640 0.3040 0.3440 0.3840]; %calculate the section width (dr) for each radial station

dr = zeros(1,11);dr(1) = (r(2)+r(1))/2 - 0.08;dr(11) = 0.4 - (r(11)+r(10))/2;for q = 2:10 dr(q) = (r(q+1)+r(q))/2 - (r(q)+r(q-1))/2;end %Initialise the matrices that will collect the data for each radial %station

dT1 = zeros(); dM1 = zeros(); circ1 = zeros(); theta1 = zeros(); dT2 = zeros(); dM2 = zeros(); circ2 = zeros(); theta2 = zeros(); dT3 = zeros(); dM3 = zeros(); circ3 = zeros(); theta3 = zeros(); dT4 = zeros(); dM4 = zeros(); circ4 = zeros(); theta4 = zeros(); dT5 = zeros(); dM5 = zeros(); circ5 = zeros(); theta5 = zeros(); dT6 = zeros(); dM6 = zeros(); circ6 = zeros(); theta6 = zeros(); dT7 = zeros(); dM7 = zeros(); circ7 = zeros(); theta7 = zeros(); dT8 = zeros(); dM8 = zeros(); circ8 = zeros(); theta8 = zeros(); dT9 = zeros(); dM9 = zeros(); circ9 = zeros(); theta9 = zeros(); dT10 = zeros(); dM10 = zeros(); circ10 = zeros(); theta10 = zeros(); dT11 = zeros(); dM11 = zeros(); circ11 = zeros(); theta11 = zeros(); %The first for loop iterates through each radial station for t = 1:11 ri = r(t); c = -0.09375*ri + 0.0575

IV

k=1; %The second for loop iterates through the TSR range for TSR = 4:8 i=1; %a series of if loops follows to test which radial station is under%calculation. For each radial station a for loop is used to iterate%through each theta in the range and send the relevant data to bem.m%to calculate the dT and dM for each situation. The theta range%changes according to the refinement after the previous suimulation if ri == r(1) for theta = 10:tstep:18

[dT1(i,k) dM1(k,i) circ1(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta1(i) = theta; i=i+1; end end if ri == r(2) for theta = 10:tstep:15

[dT2(i,k) dM2(k,i) circ2(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta2(i) = theta; i=i+1;

end end

if ri == r(3) for theta = 9:tstep:14

[dT3(i,k) dM3(k,i) circ3(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta3(i) = theta; i=i+1;

end end if ri == r(4) for theta = 8:tstep:13

[dT4(i,k) dM4(k,i) circ4(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta4(i) = theta; i=i+1;

end end

V

if ri == r(5) for theta = 6:tstep:12

[dT5(i,k) dM5(k,i) circ5(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta5(i) = theta; i=i+1;

end end if ri == r(6) for theta = 4:tstep:11

[dT6(i,k) dM6(k,i) circ6(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta6(i) = theta; i=i+1;

end end if ri == r(7) for theta = 3:tstep:9

[dT7(i,k) dM7(k,i) circ7(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta7(i) = theta; i=i+1;

end end if ri == r(8) for theta = 2:tstep:8

[dT8(i,k) dM8(k,i) circ8(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta8(i) = theta; i=i+1;

end end if ri == r(9) for theta = 1:tstep:7

[dT9(i,k) dM9(k,i) circ9(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta9(i) = theta; i=i+1;

end end if ri == r(10)

VI

for theta = 0:tstep:7

[dT10(i,k) dM10(k,i) circ10(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta10(i) = theta; i=i+1; end end if ri == r(11) for theta = 0:tstep:5

[dT11(i,k) dM11(k,i) circ11(i,k)] = bem(TSR, Vo, theta, ri, dr(t), c, naca(t));

theta11(i) = theta; i=i+1; end end k=k+1; endend %The following set of equations calculates the weighted distribution of %the dM value for each theta at each radial stationwgt1 = (dM1(1,:) + dM1(5,:) + 2*(dM1(2,:) + dM1(4,:)) + 3*dM1(3,:))/9;wgt2 = (dM2(1,:) + dM2(5,:) + 2*(dM2(2,:) + dM2(4,:)) + 3*dM2(3,:))/9;wgt3 = (dM3(1,:) + dM3(5,:) + 2*(dM3(2,:) + dM3(4,:)) + 3*dM3(3,:))/9;wgt4 = (dM4(1,:) + dM4(5,:) + 2*(dM4(2,:) + dM4(4,:)) + 3*dM4(3,:))/9;wgt5 = (dM5(1,:) + dM5(5,:) + 2*(dM5(2,:) + dM5(4,:)) + 3*dM5(3,:))/9;wgt6 = (dM6(1,:) + dM6(5,:) + 2*(dM6(2,:) + dM6(4,:)) + 3*dM6(3,:))/9;wgt7 = (dM7(1,:) + dM7(5,:) + 2*(dM7(2,:) + dM7(4,:)) + 3*dM7(3,:))/9;wgt8 = (dM8(1,:) + dM8(5,:) + 2*(dM8(2,:) + dM8(4,:)) + 3*dM8(3,:))/9;wgt9 = (dM9(1,:) + dM9(5,:) + 2*(dM9(2,:) + dM9(4,:)) + 3*dM9(3,:))/9;wgt10 = (dM10(1,:) + dM10(5,:)+2*(dM10(2,:)+dM10(4,:))+ 3*dM10(3,:))/9;wgt11 = (dM11(1,:) + dM11(5,:)+2*(dM11(2,:)+dM11(4,:))+ 3*dM11(3,:))/9; %A matrix is set up for each radial station with column 1 being the %theta values,column 2 being the mean distribution, column 3 being the %area under the curve and column 4 being the weighted distribution. The %following matrices are the output to the program.

dMm1 = [theta1' mean(dM1)' trapz(dM1)' wgt1'];dMm2 = [theta2' mean(dM2)' trapz(dM2)' wgt2'];dMm3 = [theta3' mean(dM3)' trapz(dM3)' wgt3'];dMm4 = [theta4' mean(dM4)' trapz(dM4)' wgt4'];dMm5 = [theta5' mean(dM5)' trapz(dM5)' wgt5'];dMm6 = [theta6' mean(dM6)' trapz(dM6)' wgt6'];dMm7 = [theta7' mean(dM7)' trapz(dM7)' wgt7'];dMm8 = [theta8' mean(dM8)' trapz(dM8)' wgt8'];dMm9 = [theta9' mean(dM9)' trapz(dM9)' wgt9'];dMm10 = [theta10' mean(dM10)' trapz(dM10)' wgt10'];dMm11 = [theta11' mean(dM11)' trapz(dM11)' wgt11'];end

VII

Appendix C: Xfoil Commands

A program called XFOIL was used to generate CL and CD data for use in BEM theory analysis. The following list describes is the set of commands used to generate the data for a NACA 63-812 foil at a Reynolds number of 2 x 105 between angles of attack of -5º and 15º. These commands were performed on all 13 foils for four different Reynolds numbers to create an extensive data base. During the simulation program, cubic interpolation was used to determine the correct CL and CD values for the specific conditions under analysis.

LOADh:\project\foils\naca63812.datPPARN160

OPERITER100VISC200000VPARN7VACC0.001

PACCh:\project\output\nac812-200.dat

ASEQ-5150.5

A description of each command follows:

LOAD - Allows the user to load the co-ordinate points of a foil from somedirectory

PPAR - A sub-menu to manipulate the paneling of the foil

N - Set the number of panels which describe the foil

OPER - A sub menu where the user is able to perform fluid dynamicanalysis of the foil

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ITER - Set a limit for the max number of iterations

VISC - Toggle between viscous and inviscid analysis

VPAR - Set the boundary layer conditions

N - Set the turbulence transition criterion

VACC - Set the viscous acceleration parameter

PACC - Specify the file to which analytical data should be written

ASEQ - Specify the range and step size of the angles of attack at which the analysis must take place

IX

Appendix D: NACA 6 Series Numbering System

The cross-sectional profiles used in the design ranged from a NACA 63-824 profile at the hub down to a NACA 63-812 profile at the tip. The numbering system of the foils is distinctive and each number represents a different design aspect of the foil. Table D.4 below describes the numbering system for the NACA 6 series. Figure D1 illustrates the particulars of the foil structure according to ALLSTAR network (undated).

Table D.4: Description of the NACA 6 series numbering system

NACA 6 series foil example: NACA 63-812

6 Represents which series the foil belongs to

3 The location of the minimum pressure in tenths of chord length from the leading edge (30% of the chord length in this case)

- Indicates whether the foil is modified or not. If it is modified the dash is replaced by an A (NACA 63A824).

8 Indicates that the drag coefficient is near its minimum over a range of lift coefficients 0.8 above and below the design lift coefficient.

12 Indicates the maximum thickness of the foil as a percentage of chord length (t/c).

X

Figure D.1: Labeled foil structure

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Appendix E: CNC Stock and Cutter Settings

For the generation of the CNC code to manufacture the designed blades, a program called NCBlade was used. This program was written by Prof. AH Basson for the development of CNC code to cut blades. The following appendix includes all the data which was input to the program in order to generate the CNC code for cutting the designed blades.

Table E.5 below describes the details of the 15 profiles which define the blade shape. The table includes the profile shape, pitch angle (θ), chord length (c) and positioning of each profile.

Table E.5: Blade profile positioning

Profile Shape θ (˚) c (mm) X (mm) Y (mm) Z (mm)1 Circle Ø 25 mm - - 20.0 46.0 -15.02 Circle Ø 25 mm - - 20.0 46.0 -15.03 NACA 63-824 13.0 50.0 50.0 30.0 -20.04 NACA 63-823 12.5 48.9 62.1 30.2 -20.05 NACA 63-822 12.1 47.7 74.2 30.5 -20.06 NACA 63-821 11.7 46.6 86.4 30.7 -20.07 NACA 63-820 11.0 44.9 104.0 31.0 -20.08 NACA 63-819 10.1 43.1 124.0 31.4 -20.09 NACA 63-818 9.1 40.1 155.5 32.0 -20.010 NACA 63-817 7.7 36.5 194.0 32.7 -20.011 NACA 63-816 6.5 32.8 234.0 33.4 -20.012 NACA 63-815 5.2 29.0 274.0 34.2 -20.013 NACA 63-814 4.0 25.3 314.0 34.9 -20.014 NACA 63-813 3.3 21.5 354.0 35.7 -20.015 NACA 63-812.6 3.0 20.0 370.0 36.0 -20.0

The stock which each blade was cut out of was a rectangular block of 6082-T6 aluminium the dimensions of which were 390 x 110 x 30 mm. 3 sets exclusion zones were placed along the length of the stock as can be seen in Figure E.1. These exclusion zones were areas which the cutter skipped out during the machining process and were included to add rigidity to the work piece as well as minimise vibration during machining. These exclusion zones were only incorporated for the cutting of the suction side. They were 3 mm wide, extended approximately 3 mm into the blade profile and were placed at x positions along the blade of 100 mm, 200 mm and 305 mm.

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Figure E.1: Blade stock illustrating the positioning of profiles

Table E.6 describes the input tool data which was fed to NCBlade. The tool diameter, spindle speed and tool starting position are self explanatory. The remaining values in the table will be described by the following list of definitions according to Basson (2007):

Plunge feed rate - Feed rate of the tool when the vertical motion of the tool is greater than the horizontal motion

Traverse feed rate - Feed rate of the tool when the horizontal motion of the tool is greater than the vertical motion

Max step down - Maximum amount of material removed in the vertical direction

Rapid no closer than - the rapid descent of the cutter to the work piece may not bemay not be closer than this offset

Cutter offset distance- For a rough cut the waste material is hastily removed to within this distance of the blade surface

Safe Traverse - The distance above the stock where it is safe for the tool to move

Table E.6: CNC tool specifications

Tool diameter 12 mmSpindle speed 5500 rpmTool starting position [0, 0, 100] mmPlunge feed rate 400 mm/minTraverse feed rate 1320 mm/minMax step down (rough cuts) 1 mmRapid no closer than 5 mmCutter offset distance (rough cuts) 0.5 mmSafe traverse Stock + 5 mm

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Appendix F: Verification of BEM Program

The verification of the accuracy of the BEM program was conducted by means of comparing the results obtained from the program with the analytical results obtained by Bahaj et al. (2007). The results which they obtained were as a result of a project title “The Experimental verification of numerical predictions for the hydrodynamic performance of horizontal axis marine current turbines”. In the project they experimentally verified the performance predictions of an OCT rotor obtained by two different numerical means, one being BEM theory. The conclusions of the project were that BEM theory adequately predicts the performance of OCT rotors for TSR’s between 3 and 7 over a range of blade pitch angles. However, the numerical predictions tended to overestimate the power generated at TSR’s higher than 7.

4 5 6 7 8 9 100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

TSR

CP

BEM theory resultsBahaj analytical results

Figure F.1: Analytical performance verification - hub pitch angle of 15˚

Since this data was available and the program written for this project also utilised BEM theory, the analytical results from Bahaj et al. (2007) formed a good basis for the program verification. Figure F.1, Figure F.2 and Figure F.3 are graphs comparing the analytical results obtained from the BEM program in this project to the analytical results obtained by Bahaj et al. for Bahaj’s rotor configuration at three different blade pitch angles. As can be seen from these graphs, the corroboration between the results is fairly high. For the 15˚ hub pitch angle case the results follow each other very closely over the range of TSR’s with the BEM program slightly under predicting the performance at TSR’s higher than 7. For the 20˚ and 25˚ hub pitch angle cases the performance

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prediction is not as accurate for TSR’s less than 6. However for TSR’s greater than 6 the results from both numerical predictions follow very similar paths with the BEM program predicting performances only slightly higher than Bahaj et al.

4 5 6 7 8 9 100.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46

TSR

CP

BEM theory results Bahaj analytical results

Figure F.2:Analytical performance verification - hub pitch angle of 20˚ (Design case)

4 5 6 7 8 9 100.1

0.15

0.2

0.25

0.3

0.35

0.4

TSR

CP

BEM theory resultsBahaj analytical results

Figure F.3: Analytical performance verification - hub pitch angle of 25˚

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Appendix G: Torque Transducer Calibrations

Even though the torque transducer measured a dynamic torque, for convenience the calibrations were performed statically. The method for calibration was to apply a known torque onto the shaft which the torque transducer was measuring and to record the transducer measurement. The known torque was applied by clamping a length of square tubing to the shaft so that it is positioned perpendicular to the shaft. Weights of known mass were then hung at a position on the square tubing and the distance from the centerline of the shaft to the weights was measured. This distance was multiplied by the applied mass and gravity to determine the known applied torque. A torque measurement was then performed using the torque transducer. These two torques were compared with each other and the error was calculated as a percentage of the applied torque. The numerical results of the torque calibration is presented in Table G.7. The calibration range was between approximately 3 and 37 Nm. Since the torques during testing were all below 20 Nm, the calibration range was more than large enough

Table G.7: Results of the torque calibration

Mass (kg) Gravity (m/s2) Length (m) Applied Torque (Nm) Measured Torque (Nm)0.5 9.81 0.75 3.67875 3.619522

0.75 9.81 0.75 5.518125 5.4670151 9.81 0.75 7.3575 7.299814

1.5 9.81 0.75 11.03625 10.9782 9.81 0.75 14.715 14.66127

2.5 9.81 0.75 18.39375 18.354753 9.81 0.75 22.0725 22.05941

3.5 9.81 0.75 25.75125 25.741064 9.81 0.75 29.43 29.42828

4.5 9.81 0.75 33.10875 33.049085 9.81 0.75 36.7875 36.77413

A graph illustrating the error as a percentage is presented in Figure G.1. As can be seen from both Table G.7 and Figure G.1 the torque transducer has a high degree of accuracy with the largest error during the calibration being 1.6% at an applied torque of 3.7 Nm.

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0 5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Err

or (

%)

Applied torque (Nm)

Figure G.1: Percentage error in the torque transducer measurements

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Appendix H: BEM Analysis Example

This appendix will provide an example power calculation for an annular control volume at some radius r on the rotor.

Firstly the ambient conditions need to be defined:

Water density

Water dynamic viscosity

Free stream velocity

Atmospheric pressure

Vapour pressure of water

Depth of rotor

Gravity

Next the rotor specifications need to be defined:

Number of blades

Rotor outer radius

Rotor angular velocity

Now the specifications of the blade section under analysis need to be defined:

Local section radius

Section width

Section chord length

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Section pitch angle

The foil profile used for this section is a NACA 63-819.

The final constant to be defined is Glauert’s critical axial induction factor, for this project

a value of was used since it provided results which fitted the performance

curves of Bahaj et al. (2007) the closest.

The first step in the analysis process is to define an initial guess for tha axial and

tangential induction factors. Normally a guess of is used.

Now the relative inflow angle can be calculated:

From this the angle of attack can be calculated:

Now the relative flow velocity needs to be calculated in order to determine the working Reynolds Number. The relative velocity can be calculated by combining the axial and rotational components. These are described by:

Now the relative velocity can be calculated by:

The Reynolds Number can now be calculated by:

Using the calculated angle of attack and Reynolds Number, the coefficients of lift and drag can be looked up from a table or calculated using XFOIL. For the NACA 63-819 foil under these conditions these values are:

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Using these values, one can calculate the normalised forces on the blade section:

At this stage it is necessary to calculate Prandtl’s tip loss factor (F).

Finally the solidity of the rotor must be determined.

It is now possible to calculate the new axial and tangential induction factors.

Seeing that a > ac, Glauert’s correction factor must be used. The factor can be calculated by:

Now the corrected axial induction factor can be calculated by:

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The corrected is not very different for this instance.

It can be seen by comparing the initial induction factors with the calculated ones that convergence has not been reached. Therefore the process must be repeated with the new induction factors as input. The following table lists all the calculated induction factors until convergence is reached. It can be seen that it took 5 iterations for convergence to be reached.

Table H.8: Axial induction factors over successive iterations

Iteration a a’1 0.3320 0.04742 0.3329 0.03583 0.3317 0.03644 0.3317 0.03655 0.3317 0.0365

Once convergence has been reached it is possible to calculate the forces per unit length acting on the bladed section.

These forces can be converted into the elemental thrust and torques for the blade section.

The elemental power which the blade section generates can then be calculated by:

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Appendix I: Project Costs

Table I.9: Project costs

Activity/Purchase Cost per Unit Units Total costBlade material R264/block 3 R792Manufacturing labour R150/hour 56.5 R8 475Testing Labour R150/hour 10 R1 500Total - - R10 767

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