Powertrain components for a novel wave energy converter

JOHAN HENRIKSSON

Master of Science Thesis

Stockholm, Sweden 2013

Power train components for a novel wave energy converter

by

Johan Henriksson

Master of Science Thesis MMK 2013:86 MKN 095

KTH Industrial Engineering and Management

Machine Design

SE-100 44 STOCKHOLM

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Examensarbete MMK 2013:86 MKN 095

Transmissionsdelar for en ny typ av vagkraftverk

Johan Henriksson

Godkant: Examinator: Handledare:

2014-03-26 Ulf Sellgren Stefan BjorklundUppdragsgivare: Kontaktperson:

CorPower Ocean Patrik Moller.

Sammanfattning.Sedan manninskan fick reda pa att hon ar orsaken till den senaste tidens globalauppvarming har hon, i nagra samhallen, paborjat en omstallning av sitt samhalle,fran fossila branslen som drivmedel till langsiktigt hallbara drivmedel. Ett av dessaar sa kallad vagkraft.

CorPower Oceans vagkraftverk bestar av en boj, som genom en vajer faster i endragstang i sjalva kraftverket. Denna dragstang driver tva gasfyllda och trycksattakolvar, som lagrar energin fran bojens uppgaende rorelse och aterger den vid bojensnedgande rorelse, med malet att dessa jamnar ut kraftproduktionen over cykeln. Iverket finns flera tatningar, mellan olika hydraulvatskor, mellan hydraulvatskor ochgaskamrarna samt mellan verket och det omgivande havet dar dragstangen gar ut urverket.

For att kunna valja ratt tatningar maste spannet i arbetstemperatur pa de om-givande fluiderna, samt lasten fran vagorna pa dragstangen, vara kanda. Detta foratt tatningar av ratt material ska kunna valjas och en linjarguide konstrueras.

Detta arbete syftar till att ta fram de initiella kraven pa tating och linjarguide.Med andra ord gora en termisk analys av den omgivande miljon for att kunnavalja tatningar samt en inledande kraftanalys for att kunna konstruera en robustlinjarguide.

Slutsatsen blir att for att styra temperaturen, bor en reservoirvolym laggas till cylin-drarna och en andring av volymen pa denna ger storst effekt pa gasens temperaturoch kompressionsgrad. For linjarguidens del bor det ga bra med glidringar pa badasidor om de yttersta tatningarna. Det exakta valet av tatingslosning kommer Cor-Power Ocean dock att arbeta fram med ett lampligt foretag. Denna uppsats ska sessom en inledande analys av tatningsproblemet for att kunna ange huvuddragen avkraven pa tatningar och linjarguider relaterade till gascylindrarna.

Nyckelord: vagkraftverk, termisk analys, dynamisk kraftanalys, tatning, linjarguide

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Master of Science Thesis MMK 2013:86 MKN 095

Powertrain components for a novel wave energy converter

Johan Henriksson

Approved: Examiner: Supervisor:

2014-03-26 Ulf Sellgren Stefan BjorklundCommissioner: Contact person:

CorPower Ocean Patrik Moller.

Abstract.Since mankind found out that she is the reason to the recent global warming shehas, in some societies, begun a conversion of her society, from driven by fossil fuel tosustainable fuel. One of these is so called wave energy.

CorPower Ocean’s wave energy converter consist of a buoy, which through a wire,is connected to a shaft in the plant. This shaft drives two gas filled and pressurisedpistons, which stores the energy from the buoy’s upward motion and returns it at thebuoy downward motion, aiming at evening the energy production over the cycle. Inthe wave energy converter there are several seals, between hydraulic fluids, betweenhydraulic fluids and the gas pistons as well as between the plant and the surroundingocean where the shaft exits the plant.

To select the right kind of seals the range in the working temperature of the sur-rounding fluids need to be known, as do the load of the waves on the shaft. This inorder to select seals of the right material and to construct a linear guide.

The purpose of this thesis is to acquire the initial demands for the seal and lin-ear guide. In other words make a thermal analysis of the surrounding environmentin order to select seals and an initial load analysis in order to construct a robustlinear guide.

The result is that in order to control the temperature, a reservoir volume shouldbe added to the cylinders and a change in this volume gives the most effect on thetemperature and compression rate. Regarding the linear guides, slide rings on bothsides of the two outer seals should be enough. This thesis should be viewed as aninitial analysis of the sealing problem to be able to outline the demands on the sealsand linear guides related to the gas compartments.

Keywords: wave energy converter, thermal analysis, dynamic force analysis, seal,linear guide

Foreword

This master’s thesis marks the end of my studies for a Master of Science in MachineDesign at Kungliga Tekniska Hogskolan (Royal Institute of Technology) and wasconducted at CorPower Ocean in Stockholm.

I would like to take this chance to express my gratitude to my supervisor at Cor-Power Ocean, Patrik Moller as well as my supervisor at KTH, Stefan Bjorklund formaking this possible and for their valuable insights during the project. I would alsolike to thank the rest of the people at CorPower Ocean who has aided me throughdiscussions and by making me feel as a part of the team.

A special thanks goes to my fellow student and friend John Eriksson for all thediscussions.

hej Johan Henrikssonhej Stockholm, January 2014

Nomenclature

List of Symbols

Symbol Description Unit

AW Water plane area m2

BG Length between centre of mass andcentre of submerged volume m

C Constant polytropic value

CD Drag coefficient

CM Mass coefficient

CP Heat capacity at constant pressure JKg·K

CV Heat capacity at constant volume JKg·K

D Spring element damping coefficient Nms

Dmax Maximal WEC structure diameter m

dmean Mean depth m

Fr Force resultant N

H Wave height, through to crest m

I Inertia Kg ·m2

K Spring element stiffness coefficient Nm

Ldraft Length of submerged part of cylinder m

Lstroke Piston stroke length m

P Pressure Pa

R Universal gas constant Jmol·K

RC Reflection coefficient

T Fluid temperature K

Tp Wave time period s

U Water particle velocity ms

V Volume m3

Vs Submerged volume of floating body m3

Vcylidner Cylinder volume m3

Vreservoir reservoir volume m3

∆ Relative angle

ζar Scattered wave amplitude m

a Hydrodynamic or added mass coefficient Ns2

m= Kg

afluid Water particle acceleration ms2

b Hydrodynamic damping coefficient Nsm

= Kgs

c Restoring spring coefficient Nm

= Kgs2

d Cylinder diameter m

g Gravitational constant ms2

k Wave number

m Mass Kg

n Amount of substance mol

np Polytropic index

qV Volume quota

t Time s

x Wave propagation direction

z Distance from mean surface m

γ Specific heat ratio

λ Wave length m

ω Wave circular frequency Hz

ρ Water density Kgm3

ζ Wave amplitude m

ζ∗ Reduced wave amplitude m

List of Abbreviations

Symbol Description

CPO

ODE

OWC

PA

WEC

CorPower Ocean

Ordinary differential equation

Oscillating water column

Point absorber

Wave energy converter

Table of Contents

Chapter Page

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Wave energy converters . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Oscillating water column . . . . . . . . . . . . . . . . . . . . 21.1.2 Attenuator or linear absorber . . . . . . . . . . . . . . . . . 31.1.3 Overtopping terminator . . . . . . . . . . . . . . . . . . . . 31.1.4 Point absorber . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 CorPower Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The wave energy converter module . . . . . . . . . . . . . . 103 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Frame of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1.1 Ideal gas law . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.2 PV diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.3 Adiabatic process . . . . . . . . . . . . . . . . . . . . . . . . 184.1.4 Polytropic process . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Gravitational sea waves . . . . . . . . . . . . . . . . . . . . . . . 204.3 Forces on a submerged body . . . . . . . . . . . . . . . . . . . . . 214.4 Hydromechanical loads . . . . . . . . . . . . . . . . . . . . . . . . 22

4.4.1 Added mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4.2 Wave loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4.3 Morison’s equation . . . . . . . . . . . . . . . . . . . . . . . 244.4.4 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.4.5 Froude-Kriloff force . . . . . . . . . . . . . . . . . . . . . . . 254.4.6 Mean wave drift force . . . . . . . . . . . . . . . . . . . . . . 26

4.5 Mooring forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Free body diagram and equations of motion . . . . . . . . . . . . 28

4.6.1 Heave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.6.2 Surge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6.3 Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.1 Analysis of the temporary energy storage module . . . . . . . . . 33

5.1.1 Pressure comparison 3D-plot calculation . . . . . . . . . . . 335.1.2 Temperature calculation . . . . . . . . . . . . . . . . . . . . 35

5.2 Analysis of forces acting on the system . . . . . . . . . . . . . . . 365.2.1 Modelling in Matlab . . . . . . . . . . . . . . . . . . . . . . 38

5.3 Applying the results on the concepts . . . . . . . . . . . . . . . . 396 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.1 Pressure 3D-plot program results . . . . . . . . . . . . . . . . . . 436.2 Temperature calculation results . . . . . . . . . . . . . . . . . . . 446.3 Dynamic simulation results . . . . . . . . . . . . . . . . . . . . . 47

6.3.1 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.3.2 Design wave results and comparison with tank test result . . 526.3.3 Movement analysis . . . . . . . . . . . . . . . . . . . . . . . 536.3.4 Forces analysis . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.4 Seal and linear guide concept . . . . . . . . . . . . . . . . . . . . 617 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7.1 Thermodynamic model discussion . . . . . . . . . . . . . . . . . . 627.1.1 Pressure 3D-plot program . . . . . . . . . . . . . . . . . . . 627.1.2 Temperature calculation program . . . . . . . . . . . . . . . 62

7.2 Dynamic model discussion . . . . . . . . . . . . . . . . . . . . . . 637.2.1 Regarding linear guide load . . . . . . . . . . . . . . . . . . 657.2.2 Seals and linear guide . . . . . . . . . . . . . . . . . . . . . . 67

7.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.3.1 Thermodynamic analysis . . . . . . . . . . . . . . . . . . . . 697.3.2 Dynamic force analysis . . . . . . . . . . . . . . . . . . . . . 69

7.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

1 Introduction

Sea waves have always been an important part of human life. Not only do theyinfluence the experience at sea but energy can be harvested from them. There aremany different types of wave energy converters under development, this chapter is asmall introduction to the main categories of converters. Since the CorPower Ocean’ssolution is a point absorber (PA), more emphasis will be put on different types of PAs.

1.1 Wave energy converters

The demands on any wave energy converter are considerable, for example they haveto cope with a harsh environment as well as highly fluctuating power levels. (CPOpower point). Just to be able to survive at sea and deliver electricity is a challenge forany WEC. Some of the issues that have to be solved are: how to maintain position sothat there is a stable connection to the power grid, survive storms and have a robustconstruction to be able to survive the three dimensional forces of the ocean. All thismust be achieved while keeping investment and maintenance costs at a minimum toensure that wave energy is an alternative for energy production.[12] Today, there areover a hundred different WEC-concepts under development globally. According toLindoe Offshore Renewables Center these are the four types of WECs:

1. Oscillating water column

2. Attenuator or linear absorber

3. Overtopping terminator

4. Point absorber

1

1.1.1 Oscillating water column

In a so called oscillating water column (OWC)-system, the water acts as a piston.The system consists of a partially submerged hollow structure that is open at thebottom. Inside the structure, the water is allowed to heave up and down, acting asa piston, compressing and decompressing the air inside. Due to the pressure change,the air flows through a turbine that can generate electricity regardless of the air flowdirection, see Figure.[13]

Figure 1: An oscillating water column and the principle. The change in water levelcauses the air (red arrow) to flow in and out though a generator.[13]

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1.1.2 Attenuator or linear absorber

An attenuator or linear absorber is a multi-body structure that floats on the waves.The motion of the waves causes the segments to roll in pitch and yaw, a motionthat is then converted to electrical energy through an internal system of hydraulicsand generators. The electricity is then transported trough a single umbilical cable.Several WECs can be connected to one main cable.[15]

Figure 2: The principle of an attenuator

1.1.3 Overtopping terminator

An overtopping terminator is a floating basin with a ramp that works as a beach tochange the geometry and elevation of the wave to better harness the energy within.The wave washes over the ramp and ends up in the basin where it’s collected andlead through conventional low-head hydro generators connected to generators.[14]

Figure 3: The principle of an Overtopping terminator.[14]

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1.1.4 Point absorber

A WEC that is very small, compared to the wave length, is called a point absorber(PA). The PA consist of a floating buoy that heaves with the waves vertically recip-rocating heaving motion and convert the motion to electric energy.[4]

In order for the buoy to absorb the energy of the wave, it must also generate waves.That is because in order to absorb wave energy for conversion the energy has to betaken from the waves. If the energy from the wave is absorbed, the wave is cancelledor reduced when passing the PA. A cancellation of waves is done by generate wavesthat are in counter-phase with the incoming waves. Thereof: in order for the buoy toabsorb the energy of the wave, it must also generate waves. This is shown in figure4.

Figure 4: ”Curve a represents an undisturbed incident wave. Curve b illustratessymmetric wave generation (on otherwise calm water) by means of a straight arrayof, evenly spaced, small floating bodies oscillating in heave (up and down). Curve cillustrates antisymmetric wave generation. Curve d, which represents the superposi-tion (sum) of the above three waves, illustrates complete absorption of the incidentwave energy”.[5]

Figure 4 shows how the PA:s motion, generates waves in-phase and off-phase to ab-sorb the waves energy. Two bodies of different size can generate waves of the sameamplitude if the smaller body oscillates with a larger amplitude than the larger body.This principle is what’s used in order to convert the energy. Furthermore, throughphase control and latching of the PA, a body can oscillate with an amplitude greaterthan the incident wave and there by increase the amount of converted energy. How-ever, the problem is that with an active phase control, the waves has to be predictedwith precision, something that is difficult for the stochastic behaviour of the sea. [5]

4

Uppsala Univerity’s Sea BasedThe Seabased research project at Uppsala University is of PA-type. The generator

is a permanent magnet linear generator that is directly connected to the heavingbuoy on the surface. A unique design feature is that the motion of the buoy is di-rectly translated, with no mechanical gearing. This is made possible because thegenerator is optimized to give a high output power at slow speeds. The stroke lengthof Sea Based is limited by end stops at both ends. The whole structure is anchoredto a concrete foundation that is build according to the site’s specific wave loads andseabed conditions and without the need of blasting or excavations.

Also, in a marine environment, a slow moving or static object will be subjectedto marine growth such as sea weed, barnacles etc. Most solutions to this problemare toxic chemicals, however, for the materials in Sea Based, the corrosion protectionand painting and such is done with a thought to minimize chemical use and withthe most environmental friendly practice. All materials used are well known andenvironmentally friendly.[19]

Figure 5: The principle of Sea Based.[19]

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Carnegie Wave Energy’s CETOThe CETO, is a unique construction because it’s a fully submerged PA that gen-

erates electricity in a plant ashore. This makes the plant safe from extreme weatherand out of sight at sea. It works by having the heaving buoys deliver high pres-sure water to the shore where electricity can be produced using techniques similarto hydro power dams. The buoys are also self tuning to the sites tide, sea stateand wave pattern so that they can utilise a broader spectrum of wave heights anddirections. Furthermore, all buoy units are made from steel, rubber and hypalonmaterials which all are durable for at least 20 years in a marine environment andthe CETO system can also be used for co-production of freshwater using the reverseosmosis desalination technique.[2]

Figure 6: The principle of CETO.[2]

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Ocean Power TechnologiesThe Ocean Power Technologies, OPT, Powerbuoy is also a design using a heaving

buoy anchored to the bottom of the ocean. The OPT device utilises the waves ris-ing and falling and converts this mechanical stroke to generate electricity though anovel power take-off. Sensors on the buoy monitor the surroundings and if they findthat the buoy will be exposed to large waves, the whole system will lock up and endpower production until the wave are of normal size again and the system unlocks andresumes to produce energy. The OPT is a simple steel design that uses conventionalmooring systems and are easy to install with already existing marine equipment andvessels.[17]

Figure 7: The OPT PB150 powerbuoy, dimensions are in feet. [17]

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The Ocean HarvesterThe Ocean Harvester is a quite unique design in the PA segment. In the Ocean

Harvester’s buoy, two drums, a planetary gear and a generator are built in, see Fig-ure 8. What’s unique with this system is that the harvester uses a counterweightwhich enables the load over the generator to be almost independent from the actualmotions of the buoy.[8]

What happens in the harvester when hit by a wave is that the rising of the wavecauses the anchor drum to rotate forward which lifts the counterweight. This makesthe counterweight store wave energy as potential energy. Meanwhile, the counter-weight drives the generator with a load limited by the mass of the counterweight.When the wave sinks again, the counterweight still drive the generator when it re-leases the stored potential energy.[20]

Using a mechanical power take-off makes it possible to store and utilise energy peakswith no need to oversize components. This is possible since mechanical componentsare more effective at processing an energy input, which can differ in level, thanhydraulics.[16]

Figure 8: A model, and the principle, of Ocean Harvester. [16, 20]

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2 Background

This section will give insight to the background of the project. It tells what has led tothis project and what the overall goal for it is.

2.1 CorPower Ocean

CorPower Ocean (CPO), are developing a new type of point absorber wave energyconverter. The thought behind this PA comes from M.D. Ph.D. Stig Lundback andhis research concerning the heart and its pumping and regulating principles, whichhe used to invent the Dynamic Adaptive Piston Pump Technology (DAPPT) and thefirst patented DAPPT application from 1984.

Since there are, at this point, two concepts for this PA, the thesis will take bothinto consideration.

Concept 1In concept 1, Figure 9 and 10, the power plant consists of a heaving buoy at the

surface and the main device which is located somewhere between the bottom andthe surface to protect it from rough sea. The device is moored to the bottom andan umbilical cord runs from it. The cord of several generators are then combined toa single cable to deliver the electricity ashore.

From the buoy runs a tether to the converter module’s outgoing shaft, connectedto a piston that transfers the force, through a pneumatic, hydraulic and mechanicalsystem, to the conversion module which is connected to flywheels for energy storageand to level the energy flux. [26]

Concept 2In concept 2, Figure 9 and 10, the parts inside the WEC remains the same, the

difference is that they are placed inside the heaving buoy at the surface and anchoredto the bottom, see Figure 9.

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Figure 9: CorPower Ocean wave energy converter configurations. Concept 1 and 2.

2.1.1 The wave energy converter module

No depth of detail will be described since the product is still under development.However this can be said; in both concepts, the buoy remains the same, as do themain parts of the WEC module.

The WEC module can be divided into the following sub modules: the oscillator-,the transmission-, the generator and, the part of interest for this thesis, the tempo-rary energy storage module.[26]

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Temporary energy storage unitIn order to get a more evenly distributed out-take, the energy storage has the

purpose of storing energy from the upstroke, and give an equal amount on the backstroke. In other word, half of the energy from the buoys upward motion is storedin the oscillator. The stored energy is released during the buoy’s and oscillator’sdownward motions. This smooths out the power out take profile with the aim tohave a near constant value for each sea state.[26]

Figure 10: Energy storage device concepts. To the left: concept 1, to the right: con-cept 2. Light red = piston shaft, grey = piston heads = , red = seals, brown and blue= hydraulic oil compartments, beige = gas filled compartments. a) decompressionchamber, b) compression chamber.

The temporary storage unit, consists of a piston which runs thought the whole WEC.The outgoing shaft that is essentially the piston, is connected to two gas-filled com-partments used as cylinders, one that expands and one that compresses (thereby

11

storing energy) from their initial volumes, during the stroke. The two cylinders are,if necessary, connected to one reservoir volume each in order to reach the level ofdecompression or compression wanted. The two design concepts can be viewed inFigure 9 and 10.

In concept layout 1, Figure 10 to the left, the cylinders are partially filled witha hydraulic fluid (brown part in compartment (b) in the figure), most probable anoil. The piston is constructed such that it lifts the oil (brown) during the stroke, andtherefore reduces the volume of the upper cylinder ((a) in Figure 10), and increasesthe volume of the lower cylinder ((b) in Figure 10). That means that there needs tobe a seal (red), separating the fluid from the volume in the lower cylinder.

The hydraulic oil (brown), can be used to control the motion of the oscillator, byreducing the flow between the piston heads, in order to latch the shaft, thus latchingthe buoy at a specific depth.

Regarding the uppermost seal. The piston is, in that end, connected to the out-going shaft. That means that the sea water on the outside and the gas, in the uppercylinder, must be separated by a seal.

About the seal in compartment (b). In order to allow the hydraulic oil to flowback, with the piston heads on the way down, the hydraulic oil and the gas in com-partment (b) are separated by gravity (this is not the case for the gas and hydraulicoil in compartment (a)) with a seal on the piston in between that allows leakage ofthe hydraulic oil.

The concept layout 2, Figure 10 to the right, is for this part of the WEC, some-what different though it has the same basic idea. Since the WEC is inside the buoy,the oscillator is upside down. Furthermore, the two cylinder chambers, a and b arein this version one, with a piston head and seal (grey and red) in between.

The last part, (blue), is a hydraulic fluid yet to be decided but might be of an-other type than the previous described (brown). This means that there is a needof a seal between. The piston head in the fluid is used to control the motion of theoscillator, in other words, the oscillator can be locked in place.

In this thesis the seals and the linear guides, of the pistons as well as between theoutside and the WEC, are in focus.

12

The seals, in both concepts, are either seals between gas/gas, gas/fluid or fluid/fluid.In this thesis, the focus was on the seals, of the pistons as well as between the outsideand the WEC, with gas on at least one side. Their working environment was anal-ysed in order help CorPower Ocean select seals with the right properties and linearguides rigid enough to withstand the wave loads. The criteria for selecting seals wasthe working temperature range of the sealed fluid.

The piston rod/outgoing shaft running though the WEC housing in both conceptswill be under influence on the wave loads. Therefore, in order for the seals to workproperly at a high performance level and have a long life span, the bending of theshaft over the seal must be minimized. This is done with a linear guide. In order todimension linear guides for the pistons and outgoing shaft, the wave load must alsobe analysed.

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3 Purpose

This section gives the defined purpose of the project for the results to show.

CorPower Ocean is a company that develops a novel type of wave-energy converter(WEC) were a heaving buoy’s linear motion is transmitted via a power-train to agenerator anchored to the seabed. The power-train combines hydraulic and mechan-ical elements in a unique way, aiming at maximizing efficiency and load capacitywhile minimizing size and weight.

One part of concern in the construction is where the piston meets the seawater,another are the pistons them selves. This part has two purposes; to allow and guidethe piston’s movement and withstand the forces from the waves, as well as preventthe gas in the WEC from leaking, both between the compartments and out from thecontainer, in order to increase the lifespan of the WEC.

The purposes of this project are to find the working temperature range of the sealand the load on the linear guide(s). A very basic concept for a module containingseals and linear guide(s) will also be presented. In short:

• a thermodynamic analysis of the gas compartments as adiabatic, regardingcompression, losses and efficiency, in order determine the temperature rangeduring a stroke

• a load analysis of the forces acting on the outgoing shaft in order to determinethe magnitude of the load on the liner guide

• a very basic construction layout of the seal and linear guide module.

The delimitations of this project were

• No concept generation

• No concept selection

• No detail design.

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4 Frame of reference

To understand the work done, it is important to show and explain the theory behindit. This section tells about the theories of thermodynamics and hydrodynamics usedin this thesis.

4.1 Thermodynamics

Thermodynamics is the science revolving around heat, work and how they relate tochanges in the properties of a working substance in a system, which is defined asa finite amount of matter isolated from the surroundings by physical or imaginaryboundaries, see Figure 11. The properties of a fluid in such a system are pressure,temperature, volume, internal energy, enthalpy and entropy.[1, 25]

Figure 11: The thermodynamic system definitions

When a thermodynamic system’s state changes, causing its working matter’s prop-erties to change in value, the system undergoes a thermodynamic process. The sixdifferent processes are [1]

• Adiabatic - no heat transfer from the fluid

• Isothermal - no change in temperature of the fluid

• Isobaric - no change in pressure of the fluid

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• Isochoric - no change of volume in the fluid

• Isentropic - no change of entropy in the fluid

• Isenthalpic - no change in enthalpy of the fluid

4.1.1 Ideal gas law

The state of a system containing a gas of a given amount of substance, n, of that gas,is defined by the temperature, T , pressure, P , and volume, V . An ideal gas is a gasdefined by equation (1). While not existing in reality, the properties are such that areal gas at low pressures can be modelled as an ideal gas. What defines whether a gascan be said to be ideal varies with what the error tolerance is for a given calculation.A rule of thumb is that a gas at a pressure of about 2 atm. introduces an error ofonly a few percent. The higher the pressure, the greater the error. [25].

An ideal gas is, as stated, defined as a gas that satisfies equation(1), where R isthe ideal or universal gas constant, with the condition that the change in the gas’molar energy is independent of the pressure, i.e. ∂U

∂p= 0.

PV = nRT (1)

4.1.2 PV diagram

A PV diagram is a diagram used to show the relation between pressure and volumeof a gas during a thermodynamic process. On the Y axis the pressure is plotted,and on the X axis the volume. Shown in Figure 12, are four different paths. Thesepaths are different depending on the thermodynamic process involved, the pressurecould be constant, as in an isobaric process, or the volume could be constant whilethe pressure change, as in an isochoric process, the shape of the paths of differentprocesses can be viewed in Figure 12.

If the equation (2) is integrated, i.e. calculate the area under the curve in a PVdiagram, the work absorbed by the gas is calculated. In Figure 13, the work of thetwo phases, and net work of the complete thermodynamic process, the whole cycle,is shown. [25]

∫ Vend

V0

PdV (2)

16

Figure 12: The shapes of different thermodynamic processes in a PV diagram.[1]

Figure 13: PV diagram showing work (grey) as the integrated pressure function(a)Compression (b) expansion (c) Together they complete a cycle.[25]

In Figure 14, a PV diagram is shown. It shows a compression of a volume, for ex-ample a cylinder. The system starts at a temperature equal to the surroundings, atQin and point 1. The volume then decreases, through mechanical work, causing thepressure to rise, through a isothermal process, until it reaches point 2.

However, given the ideal gas law, equation (1), a working substance have differenttemperatures at different pressures, therefore, after the isothermal process is over,the new pressure causes the substance temperature to rise while volume is constant,an isochoric process. At point two, when the process is completed, the gained heat istransmitted out to the surroundings, Qout, causing the temperature to fall and that,in turn, causes the pressure to drop so the system now is at point 3 in the process.

17

Figure 14: A PV diagram, the volume is plotted on the X axis and the pressure onthe Y axis.

Between point 3 and 4, the cylinder volume is increasing since the piston is forcedto return to its initial position, though, in this case, mechanic work. The increasein volume causes the pressure to drop though an, again, isothermal process. Whenthe piston has returned, the pressure now is slightly lower than initially. That is dueto the fact that the working substance, at a lower pressure, point 3, returns to theinitial volume at point 4. Therefore, as can be calculated using equation (1), thesubstance is now at a lower temperature than before.

However, since the initial temperature, point 1, is the same as the surroundingsin this case, heat transfers to the working substance, raising the temperature backto point 1 in an, again, isochoric process. This process, will after a while, reach astable point. [3, 7]

4.1.3 Adiabatic process

The adiabatic process is a thermodynamic process where there is no heat transferfrom the working fluid, due to a rapid process where the heat don’t have enoughtime to escape or due to complete isolation of the system from its surroundings. Forexample, an isolated piston as in a diesel engine. Because there is no heat transfer,there are no losses and in an ideal, reversible adiabatic process, which is also anisentropic process, the work done by the gas is constant. The PV diagram of an

18

adiabatic process can be viewed in Figure 12.

The adiabatic process comes with a special property; there is an adiabatic constant.The adiabatic constant is a special case of a polytropic process, specified in equation(4), in which np is equal to the specific heat ratio, γ. This constant depends on theinitial pressure and volume, P0 and V0, as well as the specific heat ratio, which isdefined as the ratio between the heat capacity at constant pressure, CP , and the heatcapacity at constant volume, CV , equation (3).[1, 25]

γ =CPCV

(3)

4.1.4 Polytropic process

In reality, the working temperature range will stabilize at a point somewhere betweena complete isothermal and a reversible adiabatic (isentropic) process. By switchingthe specific heat ratio to a polytropic index, np (not to be confused with the amountof substance, denoted n, used in the ideal gas law), it is possible to calculate a mixedisothermal and adiabatic process, including heat transfer between, by giving np avalue between 1 (isothermal) and γ (adiabatic/isentropic).

In other words, a polytropic process is a generic term for a thermodynamic pro-cess that fulfils the equation

PV np = C (4)

where P is the pressure, V is the volume, np is the polytropic index and C is aconstant.[10]

19

4.2 Gravitational sea waves

The ocean waves are the result of wind, storms, tides and earthquakes and has al-ways been of importance to humans. The range of waves is wide, from small capillarywaves, or ripples, with a time period of about 0.1 sec and a wavelength of less than1.7 cm, to the tide waves with a time period of 12-24 hours and a wavelength on thescale of the earth. In between lies the normal ocean waves , called gravity waves,which are caused by the wind. Gravity waves have a time period of 1-30 seconds anda wavelength of 1-600 meters.[24]

The basic definitions of a gravitational sea wave are shown below in Figure 15.

Figure 15: Variable definitions of a gravitational sea wave: mean depth, dmean, waveheight (through to crest), H, wave period, Tp, wave length, λ, distance from meanfree surface, z, wave propagation direction, x and time, t

.

Apart from these, sea wave related variables are circular wave frequency, ω = 2πTp

,

wave number, k = 2πλ

, θ = kx − ωt the water particle velocity, U , and acceleration,afluid.

The water particle velocity and acceleration are, with linear small amplitude wavetheory, calculated differently whether the sea is considered deep (mean depth greater

20

then half the wave length) or shallow (mean depth less then half the wave length). Inthis thesis, the water depth was only considered deep. For deep water the followingexpressions are used, taken from [24], where information on shallow water can befound.

Ux =πH

Tpekz cos θ (5)

Uz =πH

Tpekz sin θ (6)

afluid,x =2π2H

T 2p

ekz sin θ (7)

afluid,z = −2π2H

T 2p

(1 +z

dmean) cos θ (8)

4.3 Forces on a submerged body

A submerged body, regardless of shape, is under influence from the same forces.Generally there are three forces acting on a body in a fluid, inertia, gravity andviscous forces. Depending on the classification of the wetted structure, viscous effectsmight be neglected which simplifies the problem. Figure 16 shows a rough way ofclassification. [6]

Figure 16: Relative importance of mass, viscous drag and diffraction forces on marinestructures.[6]

21

A body in water can move in all six degrees of freedom, Figure 22. However, inthis report, only motion in the wave propagation direction X (surge) and Z (heave)direction as well as rotation around the Y axis (pitch) of the global coordinate systemwill be analysed.

Figure 17: The degrees of freedom for a submerged body.[11]

4.4 Hydromechanical loads

Hydromechanical loads can be said to be a reaction due to the relative movementof the submerged body, here a cylinder, with respect to the water (here assumed tobe ideal and therefore behave as if it was in a potential flow which means that thevelocity field is described as a gradient).

Due to the vertical oscillations of the cylinder, waves will be generated. These waves,which will propagate radially from it, will dissipate energy, causing the motion todie out eventually.

22

4.4.1 Added mass

For bodies moving in water, the terms Hydrodynamic or added mass, a and hydrody-namic damping, b, are important. As the name added mass implies, there is an addedmass of the water around the object, which is accelerated together with it. The forcedoes not dissipate energy and will be seen as a standing wave near the object. Thedamping, comes since the moving object will generate waves and thereby dissipatingenergy from the motion. The damping is proportional to the velocity of the body. [18]

Since the added mass and damping comes from water the object influence whenmoving, the terms are highly influenced by geometry. Furthermore, in wave loadingscenarios, they are dependant on the wave frequency. This however, is not taken intoaccount in this thesis.[6]

The added mass is different depending on direction, one for vertical displacement,equation (9), one for the horizontal displacement, equation (10) and one for inertia,equation (11). Since the buoy is considered a cylinder in this thesis, the added massterms are for those of a vertical cylinder.

az = 0.64ρ4π(Dmax

2)3

3[18] (9)

ax = ρD2max

4Ldraft[18] (10)

aI = ρAbuoy(L3draft

12+ LsubBG

2)[6] (11)

where ρ is the fluids density, Dmax is the diameter, Abuoy is the bottom area of thebuoy, Ldraft is the draft of the cylinder and BG is the length between centre of massand centre of submerged volume, see Figure 19. The equations come from differentsources since source [18] did not take pitch into account.

4.4.2 Wave loads

Linear theory is enough to analyse wave induced loads on ships and other large struc-tures that are semi-submersible. For horizontal motion in severe sea and for mooredstructures however, non-linear effects are important.[6].

23

In this thesis, only the response of regular waves are considered in order to useCPO’s design wave.

4.4.3 Morison’s equation

Morison’s equation is commonly used to calculate the wave loads of an offshore,fixed structure with a circular cross-section where viscous forces are important. Thebasic Morison’s equation can be modified to the case of a moving, vertical cylinderof diameter Dmax. The horizontal hydrodynamic force on a strip of length dz andposition ηx with can be written as

dF =1

2ρCDDmaxdz(u− ηx)|u− ηx|︸ ︷︷ ︸

a

+ ρCMπD2

max

4dza

fluid,x︸ ︷︷ ︸b

− ρ(CM − 1)πD2

max

4ηx︸ ︷︷ ︸

c

(12)where the three parts are

a. the drag force; the resistance of the fluid

b. the Froude-Kriloff force; from the unsteady pressure field of undisturbed waves.

c. the inertia force; depending on the added mass

CD and CM is the drag and the mass coefficient and should be determined empiri-cally since they are depending on several factors, such as roughness number, Reynoldsnumber, Keulegan-Carpenter number, body form, ρ etc. However, as initial value forCM , depending on the Keulegan-Carpenter number, is between 1.5 and 2. For CD,the initial value is between 0.2-0.7. Also, ax is the water particles horizontal accel-eration in the wave.[6, 23]

There are limitations and simplifications with this equation, such as assuming CMand CD to be constant with depth and that the drag force is more concentrated tothe free-surface zone that it might be in reality. These problems, and others, arediscussed at length in [6].

24

4.4.4 Drag force

The drag forces in X and Z direction, FdragX and FdragZ , on an object submergedin a fluid, are defined as

FdragX =1

2ρCD,xAx|Ux|Ux (13)

FdragZ =1

2ρCD,zAz|Uz|Uz (14)

where U is the fluids velocity relative to the body defined as

UX,Z = Ufluid(X,Z) − vbody(X,Z) (15)

in which UX,Z is the fluids velocity, defined as equation (wave particle velocity) in Xand Z direction and vbody(X,Z) is the submerged body’s centre of mass velocity in X

and Z direction(i.e ˙Xbody and ˙Zbody), CD,x,z is the drag coefficient of the submergedbody, in X and Z direction, and Ax,z is the area perpendicular to the relative veloc-ity’s direction of the fluid.

4.4.5 Froude-Kriloff force

The Froude-Kirloff (sometimes spelled Krylov), comes from an integration of thepressures acting on the body in an undisturbed wave, independent of the relative ac-celeration between the water particles and the body. With the diffraction (reflectionand scattering) force, they make up the non-viscous forces acting on the float. [6, 9]

The Froude-Kriloff force give rise to both a vertical and a horizontal force. Thevertical force is calculated as

Ffk = ρgAW ζ∗ (16)

where the gravitational constant is denoted g, the reduced wave height, ζ∗ = ζe−kT cos(ωt)

25

and AW is the water plane area equal to the cylinder bottom area piD2

4.

Introducing a spring coefficient, c, defined as

c = ρgAw (17)

the equation can be written as equation (18). [9]

Ffk = cζ∗ (18)

The horizontal Froude-Kriloff force in turn is calculated by equation (19), where dzis vertical displacement.[6]

Ffk = ρAWafluid,xdz (19)

4.4.6 Mean wave drift force

The wave drift forces are related to the structures ability to cause waves, where thewaves made by the motion of the floating body is the sum of the radiating wavesand diffraction waves. If the incident waves are long compared to the cross-sectionaldimension of the body, it will not disturb the field and the drift force is negligiblecompared to other forces.[6]

For a two-dimensional cylinder floating in regular waves, the mean wave drift forceper unit length (in the case of the buoy, the diameter), dL, satisfies

d

dLF =

1

2ρgζ2

ardL (20)

where ζar is the amplitude of the wave the float reflect and scatter in a direction thatis opposite to the incident wave. A part of the wave is generally reflected where asthe rest of the wave, passes under the float. Adding to the reflected wave, are thewaves caused by the floating body’s motion in heave, pitch and roll.[9]

The reflected and scattered waves have the same frequency as the incoming wave.Also, the amplitude of these reflected and scattered waves is considered to be linearlyproportional to the amplitude of the incoming wave, which means that

26

ζar = RCH

2(21)

where RC is the reflection coefficient. This results in a mean wave drift force definedas

d

dLFreflection =

1

2ρg(RC

H

2)2dL (22)

If the reflection coefficient is 1, it means that the waves hit a wall and are completelyreflected.[9]

4.5 Mooring forces

The mooring forces on the buoy, or reaction forces in the connection between thewire elements, are modelled as if there was a spring-damper element in between thebodies which interact with each other in. This in order to transfer the forces betweenthe bodies. As can be seen in Figure 22 there are two connections for each body(except the buoy), therefore there are reaction forces in both ends of the bodies. Theforces FX,Z , or FX0,Z0 for forces in the lower end of an element (which is the upperpart of the element in Figure 18), depend on the relative position, and velocity, ofthe two elements as

FX =

FKX︷ ︸︸ ︷K(X0 −X) +

FDX︷ ︸︸ ︷D(X0 − X) (23)

FZ = K(Z0 − Z)︸ ︷︷ ︸FKZ

+D(Z0 − Z)︸ ︷︷ ︸FDZ

(24)

in which X/X0 and Z/Z0 are the coordinates of the connecting points and the dottedones, their time derivatives. The subscript 0 denotes whether it’s the lower point inthe connection or not, as shown in Figure 18. K is the spring constant and D is thedampening constant.

27

Figure 18: Spring-damper element.

4.6 Free body diagram and equations of motion

Since the forces acting on any submerged body are the same, a basic free body di-agram was made, as shown in Figure 19, where the forces acting on a submergedbody can be viewed.

Both horizontal and vertical hydromechanical force are shown, as well as the differentlengths. The variable m is the mass of the body. FX0,Z0 are mooring forces actingon the float. Ufluid(X,Z) is the fluids velocity in X and Z direction, with this followsthe fluids acceleration, afluid, in both directions.

Put together, equation (29) to (31), make up the forces acting on all the bodiesin the system and allows for their respective accelerations to be calculated.

28

Figure 19: Free body diagram of a submerged body with variable definitions. Thelight shade shows the concept of added mass and the green dot is the submergedpart’s volume centre.

4.6.1 Heave

The heave motion, defined in Figure 17, is the vertical motion of the float. Due tothe fact that the system of forces is linear, superpositioning of the forces are possibleand that the results in irregular sea also is able to obtain by superpositioning theresults from regular wave components, the problem can be divided into to separateparts, see Figure 20 [6, 9]:

1. The hydrodynamic loads, called wave excitation forces, composed of Froude-Kriloff and diffraction forces and moments. They are induced by waves comingin on a body restrained from oscillation.

2. The hydromechanical forces, identified as added mass, damping and restoringspring terms. Their source is an oscillation of the body in still water.

29

Figure 20: Superposition of wave force components.[9]

Hydromechanical loadsThe heave motion is determined by the floats solid mass, m, and the hydrome-

chanical, added mass and damping, forces and moments acting on the cylinder.

In a free decay test, where the cylinder is displaced upwards from the still, freesurface and then released, the motion will die out eventually. There are no incidentwaves, but the motion of the body will generate waves, dissipating energy, causingthe motion to die out. [6, 9]

With only hydromechanical loads, Newton’s second law gives:

mz = −mg + ρgVs︸︷︷︸buoyancy

−bz − az (25)

where Vs is the submerged volume of the body and the buoyancy force is calcu-lated with Archimedes’ law and z is a displacement coordinate and dot denotes timederivative.

Another way, perhaps easier, of expressing this equation is as seen from the floatsequilibrium point, where z is displacement from it. Using Archimedes law to ex-press the mass of the cylinder as m= ρAwT where T is the draft at the equilibriumpoint and Aw is the water plane area (same as cylinder bottom area D2π

4), and the

buoyancy as ρg(T − z)Aw, the equation becomes

(m+ az)z + bz + cz = 0 (26)

30

where az is the hydrodynamic added mass coefficient, b is the hydrodynamic dampingcoefficient and c is the restoring spring coefficient and defined as c = ρgAW , whichmeans that c is the same as the floats buoyancy force. Further, these forces are saidto act in the floats, submerged, volume centre. [9]

Hydrodynamic loadsConsidering a restrained cylinder with incoming waves, the wave exciting load can

be calculated, this is the Froude-Kriloff force as describe by equation (18).

However, since the body disturbs the wave by being there, some of the energy willbe diffracted. Using the relative motion principle, as described by [9], the total waveexciting force becomes:

Fw = aζ∗ + bζ∗ + cζ∗ (27)

where a and b are to be considered correction due to diffraction.

Equation of motion, heaveHydrodynamic and wave loads, equation (26) and (27) inserted in Newton’s second

law of motion gives:

mz = a(ζ∗ − z) + b(ζ∗ − z) + c(ζ∗ − z) (28)

since c = ρgAW and ζ∗ − z is the relative displacement (i.e. a change in draft) thelast term is the change in buoyancy.

Adding the drag force and mooring forces, and solving for the body’s acceleration,the equation of motion becomes:

z =b(ζ∗ − z) + c(ζ∗ − z) + Fdragz + Fz

m+ az(29)

Where az is the added mass, b the dampening coefficient, c is the restoring springcoefficient, the drag force, Fdragz, defined by equation (14) and the mooring force ,Fz, defined by equation (24).

31

4.6.2 Surge

Surge is, as defined by Figure 17, the motion of the float in the wave propagationdirection. Newton’s second law of motion, coupled with Morison’s equation andadding the mooring forces, gives the equation of motion:

x =

Morison’s equation︷ ︸︸ ︷1

2ρCDDmaxdz(u− ηx)|u− ηx|+ ρCM

πD2max

4afluid,xdz − ρ(CM − 1)

πD2max

4dzx

m+ ax+

+

mooring force︷︸︸︷Fx +

reflection force︷ ︸︸ ︷Freflection

m+ ax(30)

4.6.3 Pitch

Pitch, as defined by the coordinate system is a revolving motion around the Y axis.The forces acting on the submerged body give rise to moments. These momentsare what the equation of motion takes into account in order to calculate the angleand angular velocity of the body. Using the forces in equation (29) and (30) andintroducing the term BG, as the distance between the submerged volume centre andcentre of gravity see Figure 19, the term Lg, as the distance between the mooringline connection and the centre of gravity, Ldraft as distance from wave particle cal-culations depth (where forces depending on water particle velocity and accelerationare applied) to centre of mass, dz is the submerged length and I is the inertia of thebuoy, the equation of angular motion is:

ω =(az ζ∗ + b(ζ∗ − z) + c(ζ∗ − z))BG sin(ω)− ρπD

2max

4dzηxBG cos(ω)

I + aI+

+(ρCM

πD2max

4dz(afluid,x − η) + Fdragx)Ldraft)− FzLg sin(ω)− FxLg cos(ω)

I + aI

(31)

32

5 Method

There are two areas of interest when designing the seal module: the gas compartmentthermodynamics and the radial forces the linear guide shall endure. In this chapterit is described how the gas compartment and the forces were analysed.

5.1 Analysis of the temporary energy storage module

To be able to construct the seal modules, the construction must be able to withstandthe pressure and temperature change during the pistons stroke and for that purpose,an isothermal model of the stroke has already been made. However, in order toget an overestimate of the temperature and pressure range, in the gas compartmentduring a stroke, an adiabatic model was chosen as a worst case scenario. Noteworthyis that, since both the adiabatic and isothermal processes fulfils equation (4), themodel can be used in order to calculate a more realistic temperature by changing thevalue of n. Also, the thermodynamic analysis only involved the gas compartmentsand no other heat transfer in the WEC.

The basic design of the gas compartments are shown in Figure 21. In short, there aretwo compartments, the piston cylinder and the reservoir. The piston cylinder volumedepends on the stroke length and the diameter of the piston where the diameter wasknown. The were two models made. The first one to show what influence a change instroke length and/or reservoir volume had on the pressure and temperature duringa stroke. The second model was made to calculate the pressure and temperature,during a stroke, for a specific design of piston diameter, stroke length and reservoirvolume.

5.1.1 Pressure comparison 3D-plot calculation

Due to the fact that the stroke length length, cylinder diameter and reservoir volumeof the WEC, had not been decided at the time, and to better understand what impactchange in any of these has on the maximum pressure in the cylinder, calculationsfor several cases were made in Matlab and plotted in a 3D-plot in order to comparedifferent designs in an easy way.

The goal was to calculate the final pressure, Pend, when the outgoing shaft hasreached it’s maximum displacement at the end of stroke. Through the adiabatic

33

Figure 21: Definition of variables used for modelling the gas compartments. P0 =initial pressure, γ = specific heat ratio, T0 = initial temperature, Vcylidner = cylindervolume, Vreservoir = reservoir volume, d = cylinder diameter, Lstroke = stroke length

condition, equation (4) with n = γ, for a given cylinder diameter, the final pressurewas calculated as

Pend =C

V γend

(32)

wherein Vend is the volume when the stroke has reached it’s maximum displacement,in other words, the reservoir volume, Vreservoir, and the variable C, since the processis adiabatic, is constant though out the process, see paragraph 4.1.3 Adiabatic pro-cess. Therefore, it can be calculated using the initial conditions, which are what waschanged from case to case. The constant is calculated by equation (4) as

C = P0Vγ

0 (33)

where P0 is the initial pressure and V0 is the initial volume defined as V0 = Vcylinder+Vreservoir. Combining (32) and (33) gives the expression

Pend =P0V

γ0

V γreservoir

(34)

The reservoir volume was calculated in relation to the cylinder volume in order to

34

compare sizes by a volume quota, qV . The quota was defined as

qV =VreservoirVcylinder

(35)

wherein Vcylinder was calculated a the volume of a cylinder as

Vcylinder =πd2

4Lstroke (36)

with the cylinder diameter, d, and length, Lstroke.

Equation (35) and (36) inserted in equation (34) gives the function

Pend =P0(πd

2

4Lstroke)

γ

V γreservoir

(37)

where Vreservoir came from the volume quota, equation (35), giving

Vreservoir = qV Vcylinder = qVd2π

4Lstroke (38)

The final function then became

Pend =P0(πd

2

4Lstroke)

γ

(qVπd2

4Lstroke)γ

(39)

which simplified became

Pend =P0

qγV(40)

5.1.2 Temperature calculation

The maximum pressure is not the only design criteria when selecting seal. The tem-perature in the gas over the stroke is also important.

During the stroke, the temperature will either rise or fall due to compression ordecompression. Since both the pressure and the volume change at the same time,and the volume is known, the pressure has to be calculated. By substituting thepressure, P , in the ideal gas law, equation (1), with a pressure calculated from thefrom the adiabatic condition, equation (33), the temperature, T , for current volume,V , was acquired as follows.

T =C

nRV 1−γ (41)

35

5.2 Analysis of forces acting on the system

Apart from knowing the thermodynamics of the seal environment, the radial forcesthe linear guide need to endure, in order to prevent them from affecting the seal’sperformance, needed to be estimated.

The analysis of forces in the system was made by first, making a free body dia-gram of the bodies in the two cases, and second, derive the differential equationsfrom Newton’s second law of motion and, third, write a program in Matlab in orderto analyse the forces over time.

Figure 22: Overview of the two configurations.

36

The model was the verified by analysing the movement of the buoy and WEC forthree predictable cases. The cases were and the predictions where:

1. no wave and no current - no movement at all

2. only a current - constantly tilted in the direction of the current

3. no wave or current and no wire - a vertical displacement equal to a distancecalculated using energy conservation laws.

Further verification of the model was achieved by comparing the result of the simu-lation for a specific case with that of a tank test, separately preformed by CorPowerOcean.

The tank test was done by mounting a model of the buo, Froude-scaled 1:30, ina basin where controlled waves could be created. This set-up allows testing of theeffect of water waves with specific wavelengths and amplitudes. Since the tank testwas only conducted using concept 2, it was only the results from the model of concept2 that was directly compared with the tank tests.

37

5.2.1 Modelling in Matlab

The two concepts, Figure 22, where modelled in Matlab in order to determine whetherthere were any differences regarding motion and seal guide loads. For the buoys part,equation (29)-(31) shows the equation system of the forces acting on it. The systemconsists of three second order differential equations.

For the other submerged bodies on the other hand there are a considerable dif-ference of the equation system compared to the buoys. Since most of their bodiesare at great depth, i.e. far from the surface, and the influence of the waves decreasesexponentially with depth, the wave loads and water particle motions are considerednegligible, thus, the equations of motion for other bodies than the buoy, the wireand the WEC, only consists of mooring forces and drag force. Also, the out goingshaft in the WEC, whether it’s in the buoy or only connected through the wire ismodelled as fixed, in other words, there is no energy out take. That is because thelatching model was not available at the time. Therefore, the model is a passive buoysimply floating on the waves and the results used as an indication and starting pointfor the design.

A good way of modelling these in Matlab is to use the built-in ODE -function (Ordi-nary Differential Equation). The ODE-functions integrates the system of differentialequations over time, given the initial conditions. Using a high level programminglanguage, like Matlab, for solving this kind of problem gives the benefit of optimiza-tion. Since there for a complex system, where a PA is relatively simple, are severaldemands to optimize, for example energy output, cost and size. Matlab gives thepossibility to vary the large amount of variables that can be modified in order toacquire the optimal design.[22]

In order to use the ODE-function, the equation system needs to be on the form:

y = f(y) (42)

i.e a system of first order differential equations. To solve the system, Matlab’s stiffODE-solver, ode15s, was used when the more commonly used solver, ode45, was slowdue to the stiffness of the system.

38

5.3 Applying the results on the concepts

The reason for modelling the pressures, temperatures and forces are of course toapply the results on the design in order to determine the working environment of theseals in order to select the right kind of seal(s), linear guide and hydraulic fluid, toname a few. The behaviour of the buoy, angular velocity etc., was also of interest.

Since the focus for this project was the design of the seals in the WEC, a requirementlist was compiled in order to send to seal and linear guide companies. No machineelement design regarding the seals and guides will be made at CPO.

The requirement list contained:

1. seal environment temperature ranges,

2. water, gas and hydraulic fluid pressure ranges,

3. forces to be handled by the linear guide.

In concept 1, the force is transmitted via the wire to the outgoing shaft. The WECthen moves in a circular motion, much like a pendulum, as do the buoy in concept2. Therefore, since the forces has been divided into vertical, (heave or Z direction),horizontal (surge or X direction), they are converted to the WEC local coordinatesystem and divided into a radial and a tangential force, see Figure 23 for concept 1and Figure 24 for concept 2.

To dimension the linear guide in concept 1 (WEC suspended between bottom andbuoy), the load in the guides radial direction was estimated by taking the forces in Xand Z direction, see Figure (23), Fx and Fz, projecting them in the new coordinatesystem and adding them to their resultant,Ft, perpendicular to the shaft. Togetherwith the angle of the buoy and given that the tangential force on the buoy is the sameforce acting at the linear guides radial direction, Figure 26, the force was calculatedusing equation (43).

Ft = Fx cosω − Fz sinω (43)

39

Figure 23: Variable definition for concept 1, equation (43). The image shows the topof the WEC.

In concept 2 (WEC in buoy), the load estimation was done as for concept 1. Again,the forces in X and Z direction in Figure (24), was projected to the new coordinatesystem and added to their resultant as

Ft = Fz sinω − Fx cosω (44)

Figure 24: Variable definition for concept 2, equation (44).

The tangential force, gives reaction forces in the seals/linear guides, as shown inFigure 25.

40

Figure 25: How the tangential load, Ft, is applied and how it give rise to reactionforces on the shaft, Rα and Rβ, which of the same magnitude as those in the linearguides.

Regarding the load for the linear guide to handle in both concepts, the load, Ft,is applied as shown in Figure 25. An example of how to regard the shaft, whencalculating the forces the linear guides should withstand, is shown in Figure 26. Bylooking at the outgoing shaft as a simple supported beam with a point load at theend, the load on each linear guide, at point A and B, is calculated using equations(45) and (46), since the load on the shaft is the same as the load on the linear guide,but with the opposite sign.

RA =FtL

B(45)

RB = −FtAB

(46)

δ =FtA

2L

3EI(47)

41

Figure 26: Definitions of variables for reaction forces and deflection calculationswhere LA + LB = L.

6 Results

For further design of the WEC seal and linear guide system, results of the work doneneeds to be taken into account. In this section, the results from the thermodynamicanalysis and the dynamic simulation are presented.

6.1 Pressure 3D-plot program results

The program takes a diameter, and changes the volume quota, qV , and stroke lengthto se how the maximal pressure depends on them. The ranges for quota and strokelength, for both the upper and lower cylinder with γ = 1.4 (completely adiabatic)and an initial pressure at 1 MPa for the upper cylinder and 0.1 MPa for the lowercylinder.

Table 1: Pressure calculated by 3D-plot program for qV = 3.

Upper cylinder[MPa] Lower cylinder[KPa]

Lstroke = 4[m] 1.492 67Lstroke = 7[m] 1.494 66.94

As the values in Table 1 shows, the value hardly change depending on the strokelength. Figure 27 also shows this. The colour remains same parallel to the strokelength axis, showing that there is basically no change in value.

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Figure 27: Top view of 3D-plot of lower cylinder maximum pressure using volumequota.

Figure 28 shows a 2D-plot since the stroke length has basically no impact on thechange in maximal pressure. For a volume quota lower than 1.0, the reservoir vol-ume is so small that the pressure becomes unreasonably high, and they are thereforenot shown.

The pressures, presented in Table 1, were compared to those of the temperaturecalculation program. This was done in order to see whether the programs containedany programming errors.

6.2 Temperature calculation results

The temperature program calculates the temperature over a stroke (from no displace-ment to full stroke length). However, as a by-product, the pressure is calculated atthe same time. Therefore, the Pressure 3D-plot program mentioned, and temper-ature calculation program should give the same pressure given the same diameter,reservoir volume and stroke length.

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Figure 28: Top view of 3D-plot of upper cylinder maximum pressure using volumequota.

The previously used data, regarding cylinder diameter, stroke length, and volumequota, was fed to the temperature calculation program. For the ’error control’ of theprograms, the resulting pressures, seen in Table 1 and 2 below where the resultingpressures, from the temperature calculation program, are written out. The resultshowed that the pressures calculated by each program are the same, with the errorin the accuracy probably due to the rounding of the programs.

Table 2: Pressure calculated by Temperature program with qV = 3.

Upper cylinder[MPa] Lower cylinder[KPa]

Lstroke = 4[m] 1.499 66.72Lstroke = 7[m] 1.496 66.85

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The main function of the program however, was to design a program that calculatesthe highest and lowest temperature of the compressed and decompressed gas in thetwo cylinders. For qV = 3 and the stroke lengths 4 and 7 meters, the temperatureresult was from 8C to about 40C, see Figure 29.

Figure 29: Calculated temperature in the two gas cylinders for two different strokelengths, 4 and 7 meters, during a stroke.

Another demand was that the end-of-stroke compression should not be higher than10 percent. This gives a volume ratio, given the used initial pressure of 1 MPa, of 14which can be directly calculated using equation (52) and (35). How equation (52)was derived is shown below.

(1 + n)P0Vγreservoir = P0(Vcylinder + Vreservoir)

γ (48)

(1 + n) =(Vcylinder + Vreservoir)

γ

V γreservoir

(49)

(1 + n) =

(Vcylinder + Vreservoir

Vreservoir

)γ(50)

46

(1 + n)1/γ =Vcylinder + Vreservoir)

Vreservoir=

VcylinderVreservoir

+ 1 (51)

Vreservoir =Vcylinder

(1 + n)1/γ − 1(52)

With the limit of a compression and decompression of 10 percent, the resultingtemperature in the upper cylinder became 16C and for the lower cylinder just above0C.

6.3 Dynamic simulation results

6.3.1 Verification

In order to get some verification of the dynamic simulation model, three predictablecases were first simulated, these were:

1. no wave and no current - no movement at all

2. only a current - constantly tilted in the direction of the current

3. no wave or current and no wire - a vertical displacement equal to a distancecalculated using energy conservation laws.

In the case on no wave and no current the simulation program clearly show thatwhen there is no waves or any current there is no heave, surge, or pitch, see Figure30 and Figure 31.

In the case of just a current of 0.5 m/s, the results show, as expected, that the buoywill drift with the current until hindered by the mooring wire, and tilt a few degreesand then have small oscillations around an angle in the direction of the currant, seeFigure 32 and Figure 33.

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Figure 30: Position of the top of the buoy in surge (X) and heave (Z) direction,upper with WEC inside buoy and lower with separate WEC, with no incident wavesor current.

Figure 31: Angle of the buoy, upper with WEC inside buoy and lower with separateWEC, with no incident waves or current

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Figure 32: Position of the top of the buoy with only a current of 0.5 m/s. Upperwith WEC inside buoy and lower with separate WEC.

Figure 33: Angle of the buoy with only a current of 0.5 m/s. Upper with WECinside buoy and lower with separate WEC.

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Last by not least, the case with no wave, current, or mooring wire. This should showthat, since the buoy has a buoyancy much larger than the force of gravity and no wireto hold it down, the buoy has a vertical, oscillatory, displacement. Since there wereno losses in the vertical direction this should go on indefinite; the damping coefficientb was set to zero as explained in the section Dynamic model discussion. As shown inFigure 34, this was the case. In order to verify the model, the vertical displacementshould be able to predict by using energy conservation laws to calculate the height.The fact that no damping (drag) was used in the vertical direction simplified thecalculation.

Figure 34: Vertical, oscillatory, displacement since no wire hindered the buoys verti-cal movement.

By using the conservation of energy law

E1 = E2 (53)

where E1 is the energy of the buoy when submerged and E2 the energy when thebuoy has reached a certain height from the initial point, the maximum vertical dis-placement of the buoy could be calculated.

Since the buoy was not completely submerged the potential energy at the initial

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position, E1, could be calculated by using the restoring spring stiffness c, definedin equation (17). Thus calculating the potential energy of as if it was a spring,Espring = kz2

2, as shown in equation (54), where z = Ldraft and k = c.

E1 =cL2

draft

2(54)

The energy at position 2, E2, was first calculated for when the buoy just reaches avertical displacement lifting the whole buoy just over the surface. This was done inorder to see if all potential energy at E1 had been converted to only gravitationalpotential energy at E2. If the gravitational potential energy in that case was lowerthan initial energy, E1, that would mean that the buoy must have kinetic energyleft and therefore reaches a point above the surface where it only has gravitationalpotential energy, thus simplifying the equation. If, however, the energy would havebeen higher than initially, the buoy would never have a vertical displacement allow-ing the whole buoy above surface.

The energy when the when the whole buoy is just above the surface was calculatedwhen the transported distance of the centre of mass, h, equalled the submergedlength, Ldraft.

E2 = mgh = mgLdraft (55)

A comparison of these energies showed that E1 was in fact larger than E2 whenz = h = Ldraft in equation (54) and (55). Thus, the buoy has kinetic energy when itis just above the surface. This means that the potential energy at position 1 mustbe completely converted to only gravitational potential energy at some point abovethe surface. The calculation (53) then becomes

mgh =cL2

draft

2(56)

Solving for h gives

h =cL2

draft

2mg(57)

and the calculated height was 2.5 meters vertical displacement for the buoy’s centreof mass. Adding the height from the centre om mass to the top of the buoy, and theinitial position in the modelled system gave the total height to 3.6 meters above thebottom. This was compared to the result of the dynamic simulation which was 3.6meters as well.

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6.3.2 Design wave results and comparison with tank test result

In order to use the results of the dynamic simulation model, it must be shown thatit gives reasonable values in comparison with reality. This was done by comparingthe models result, for a given design wave, with that of the tank test, naturally forthe same wave. A design wave in this case was 2.26 meters in wave height and hada time period of 9.2 seconds, which with Froude-scaling 1:30, translates to a waveheight of 0.0753 meter and a time period of 1.6797 seconds.

Figure 35: Tank test surge of the buoy with incident design wave load.

Figure 36: Tank test pitch of the buoy with incident design wave load.

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The first run on the dynamic simulation used what are common values on CD andCM at 0.7 and 2 respectively for a vertical cylinder.[6] The reflection coefficient, R,was given the value 0 since the buoy diameter is much smaller than the wave length(compare a buoy diameter of 0.2 m with a wavelength of 4.4 m).

The result showed that the buoy’s surge was too low, as was the pitch, almost 50percent of, see Figure 37 and 38 for dynamic simulation results and compare withFigure 35 and 36 which show the tank test results for the design wave. Also, hardlyany effect, on the buoy’s movement, of having a separate WEC was shown, probablybecause it is only affected by the drag force and the movement is slow, but the buoywith less mass (concept 1) has a greater surge and pitch, especially for a lower dragcoefficient.

6.3.3 Movement analysis

In order to understand more what effect the different variables CD, CM and R has onthe buoy different values were tested and compared. Ultimately in order to achievea motion similar to the tank test.

By changing the drag coefficient to 0.1, a result more similar to the tank test wasachieved, see Figure 37-38.

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Figure 37: Buoy surge for different CD values.

Figure 38: Buoy pitch for different CD values.

A change in mass coefficient, CM , has the effect, as shown in Figure 40, that a lowervalue results in a smoother and more constant pitch in the case of a separate WEC(concept 1). Whereas, as shown in Figure 39, concept 2 with just a buoy is hardlyaffected at all.

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The cause of the difference might be that the buoy is more easily affected by thewave since it carries less momentum, due to the lower CM value and the mass of thebuoy in concept 1, and thus follows more easily when the water particles changesdirection. Therefore CM value might have larger impact for a smaller buoy, but thisis hard to tell since the crude approximations in the model.

Figure 39: Buoy surge for different CM values.

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Figure 40: Buoy pitch for different CM values.

When adding a reflection of the waves, or mean wave drift force, the effect is smalleven in the case of the buoy considered a wall (R=1), see figure 41. The reflectionhave little impact in this model, as should be the case since the buoy is very smallin comparison to the wave length.

However, the reflection of waves are also depending on the floats ability to causewaves, and since this is what the buoy will do, as stated in how a point absorberworks, this might change when latching the buoy is taken into account but shouldno affect the design that much.[6]

56

Figure 41: Surge and pitch for different RC values.

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6.3.4 Forces analysis

The load on the linear guide, Ft, is the tangential force component from the forcefrom the wire on the buoy, as shown in Figure 24 and 23. The force, calculated asshown in the mentioned figures, comes from the mooring; if there were no mooring,no force would be transmitted.

The resulting force, Fr in Figure 42, of the forces in X and Z direction in the connec-tion point, see equation (23) and (24), will however have a slight angle compared tothe wire. This angle would, probably, increase with the inertia and drag of the wire.Initially, the direction of the resulting force was considered the same as the wire’sdirection. This should mean that the magnitude of the tangential force on the buoyshould follow the magnitude of the relative angle between buoy and wire, as shownin Figure 42 and by equation (58).

Figure 42: Variable definitions for calculation the tangential force depending on therelative angle between buoy and wire, equation (58).

Given that the direction of the resulting force, Fr, equals the direction of the wire,Ω, the the relative angle, ∆ = ω − Ω. This gives the tangential force defined as

Ft = Fr sin ∆ (58)

58

Therefore, if the relative angle is close to the angle of the wire, the tangential forcein the buoy should follow it.

If the angle between the resulting force and the wire is large, the tangential forcemagnitude should not follow that of the relative angle particularly well. In that case,the angle of the resulting force must be calculated, which can be easily done.

As shown in Figure 43, the tangential force did follow, on the relative angle be-tween buoy and wire rather than just the angle of the buoy.

Figure 43: How the magnitude of the tangential force depends on the relative angle,delta angle in the plot, between the buoy and wire, and not on the buoy’s anglealone.

The dynamic simulation of concept 2, with a separate WEC between buoy and bot-tom (see Figure 9), showed that the relative angle between wire and WEC are smaller,practically the same, than for the concept of the WEC inside the buoy. This leadsto a smaller load on the shaft, see Figure 44. However, this might not apply if theWEC becomes too large and reaches a too high drag force.

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Figure 44: How the magnitude of the tangential force depends on the relative angle,delta angle in the plot, between the WEC and wire.

As shown, the force on the linear guide is highly dependant on the difference in anglebetween buoy and wire or wire and WEC. This is because a higher relative angleallows for more of the buoyancy to translate into a load on the shaft.

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6.4 Seal and linear guide concept

There are several different seal types depending on what fluid they are supposed toseal and the motion, either of the seal itself or by a shaft running through the seal.Depending on the velocity of the motion, a cooling system might be needed.

An initial design concept was suggested from the company Trelleborg[21], with apossible seal and linear guide solution. For the two concepts in Figure 25, the linearguide, which was suggested to consist of two slide rings on either side of the seal tokeep the shaft in centre of the bore, is on the same position - as close to the loadas possible. A second guide might be needed on the outer piston, as shown in thefigure. One or two seals (depending on what to seal between) between the two sliderings, to separate the gas and liquids, was suggested.

Figure 45 shows the basic layouts of two different mountings of seal and linear guide,the seal(s) are shown as one, red, and the slide rings are brown. To the left is theouter seal, in both concept 1 and 2 see Figure 25. The seal(s) are placed betweenthe slide rings and both seals and rings are mounted in the gas compartment wall,allowing the piston to reciprocate through. To the right are the seals and slide ringsmounted on a the outer piston head, in both concept 1 and 2, and follows its motion.Other seals do probably not require slide rings.

Figure 45: Basic layout of seal with slide ring linear guide. Red = seal(s), brown =slide ring. Left: outer seal, right: piston head seal.

Not shown in the figure is that there must be a scrape ring on the out side of theWEC, maby also on the inside, which scrapes the outgoing shaft, on it’s return, fromsolid particles in the sea such as sand or small animals.

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7 Discussion

The main focus for this chapter is to analyse the results and discuss the strengths,weaknesses and validity of the assumptions and simplifications applied on the models.The goal is to draw some conclusions and aid in future work.

7.1 Thermodynamic model discussion

7.1.1 Pressure 3D-plot program

The Pressure 3D-plot program was designed in order to see what influence on thefinal pressure change in either stroke length or reservoir volume had. However, achange in stroke length results in a slightly greater cylinder volume which affects theend-of-stroke pressure by lowering it to some degree, but the real effect comes whenchanging the reservoir volume/cylinder volume quota.

The result showed that the stroke length has very little to do with the end-of-strokepressure. A change in the reservoir/cylinder volume ratio changes a great deal how-ever and it is that factor that should be taken into account when designing the WEC.By applying the criteria that the end-of-stroke pressure should have a difference of,at most, 10 percent of the initial pressure, though equation (52), the reservoir volumeshould be at lest 14 times greater than the cylinder volume.

7.1.2 Temperature calculation program

In order to select a seal, the range of the working temperature must be determined.To get an overestimate of the temperature of the gas in the compartments, an adia-batic model of the process was used.

Since the compression ratio should be below 10 percent, the diameter and corre-sponding reservoir volume are known. The diameter was to be decided, but forthe initial diameter and reservoir volume the temperature range was calculated tobetween 8C to 100C for the compression compartment and 8C to -60C for thedecompression compartment.

This result should however be viewed in the light of the fact that the process issaid to be adiabatic, this is, however, not the reality of course. The working temper-ature will reach a stable point, where the maximum temperature in the compressioncompartments in both concepts will probably be somewhat lower than the calculated

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maximum temperature due to heat transfer between compartment and ocean. If thetemperature is deemed critical for the over all WEC design (material strength, gluebonds etc.), this should be confirmed by further analysis and calculations.[7]

When two full strokes are completed and the piston is back in starting position,the temperature might, for a short time, be lower than initially. But since the oceanthen will leak heat back to the compartment, the temperature will be back to normalquickly.

For the decompression compartments, the same will happen, only in reversed or-der. In the first stroke, the gas will decompress, lower it’s temperature and heat willleak in from the ocean and raising it a little. When the piston has fulfilled it’s secondstroke, and is back in initial position, the gas, somewhat heated, will have been com-pressed during that second stroke and the temperature might be somewhat higherthan initially. But then the heat will leak out to the ocean and the gas temperaturewill be back to the initial temperature.

This is all under the assumption that no heat from the transmission and flywheelswill leak into the compartments. If it does, this heat might end up in the two cham-bers, increasing the over all temperature during the stroke.

7.2 Dynamic model discussion

The dynamic model was created in order to get an initial estimate of the load onthe seal guides, that is, the tangential load described in Figure 24. In the model,there was no outgoing shaft, or it could be seen as locked in position, inside theWEC or buoy, which means that there was no power out-take that could dampenthe movement.

Several simplifications were made in the model regarding geometry. The buoy is mod-elled as a uniform semi-submerged cylinder with the same length as the real buoy,but with a mean diameter, giving the cylinder the correct volume and submergedcross section area in the wave propagation direction. Also, the inertia calculationwas done in regard to the centre of mass in a uniform cylinder, in other words, thevolume centre in this case. But the centre of gravity was moved, in the calculations,to the same position as for the real buoy and the cylinder was submerged to thebuoy’s equilibrium point, i.e. where the buoyancy force and the force of gravity plus

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the gas spring force should balance each other. Therefore, the inertia might be toosmall (however, this leads to an overestimate of the angular acceleration which wasconsidered better than an underestimate since this increases the angle and thereforethe force, as will be discussed later).

Concerning the wave load. The first simplification was that the WEC is at deepsea. In comparison with the tank test, where the buoy is at intermediate depth, thisshould give a smaller particle velocity and acceleration. That in turn leads to lesssurge and pith which explains why the simulation doesn’t reach precisely the samevalue for these as compares with the tank test. Also, the sea waves are simplifiedto a regular ocean gravity wave, in other words not an irregular wave spectrum buta wave of defined wavelength, time period and amplitude which is not the case inreality.

When calculating the water particle velocity in horizontal direction, the depth chosenwas where the velocity would reach it maximum around the buoy. That means that ifthe water level was lower than the top of the buoy, the velocity was calculated in thewave surface and when the buoy was fully submerged, the velocity was calculated atthe top of the buoy. That was since most of the buoy’s cross section area, in reality,will be located at surface level. It was also chosen in order to overestimate the loadon the linear guide.

The reflection coefficient was said to be zero since the diameter of the buoy wassignificantly lower than the wave length. If that criteria in reality is fulfilled remainsunclear. Regarding the drag, mass and reflection coefficients. Initially the valueswere standard recommendations where they have, in other cases, been calculatedand measured to about CD = 0.7 and CM = 2 depending on Keulegan-Carpenternumber, surface roughness and flow (Reynolds number).[6, 23]

In order to overestimate the force, the drag coefficient was significantly lowered to0.1 (since the impact of a smaller change was minute) and the mass coefficient wasset to 2.

Other coefficient used were the added mass coefficients: ax, az, aI and b. Thefirst three, ax ,az and aI was calculated as described in Faltinsen,[6] for a uniformcylinder. The added mass is however probably not the same for the real buoy sinceit is highly dependant on the geometry. The dampening coefficient b was regardedas zero under the assumption that a heaving buoy causes almost no waves.[9]

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This might not be the case, however, especially not when the WEC is supposed tohave a power take out, as a device for absorbing waves must also generate them,[5], and is also highly dependant on the geometry of the float and it’s direction ofmotion.[6]

Looking at the formulas, and the simplifications made in this model, there is agood chance that the values on added mass, added inertia and vertical dampeningaren’t good enough since they also depend on the geometry. They can be calculatedmore precise, but measuring them for the actual buoy would be the best.[6]

Other, smaller, simplifications were that the wire and WEC module in concept 1,only had a drag force and mooring forces on them since they were considered to betoo deep to be affected by the waves.

As shown by the results, the motion of the modelled buoy and the tank test, forthe recommended drag coefficient of 0.7, was too small, almost 50 percent of. Forthe lower drag coefficient value, 0.1, the motion, though it reaches a similar pitch andsurge, it did so with double the time period compared with the wave and the tanktest results. Also, a lighter buoy with a small drag coefficient will have a grater pitch.

However, even though the motion of the buoy was not accurate enough, the re-lationship between the relative angle (of buoy and wire of wire and WEC) and thetangential force still holds.

In short: all these simplifications should explain why the modelled buoy didn’t be-have as the tank test buoy and the relative angle between buoy and wire more orless determines the load on the linear guide.

7.2.1 Regarding linear guide load

Since the largest force in magnitude, in the system, was the buoy’s buoyancy force,and since the relative angle causes a larger part of the vertical forces to be projectedin the tangential direction, see equation (43)-(44), of the buoy, the tangential force,Ft in Figure 24 and 23, should depend largely on the buoyancy force. Therefore, anestimation of the tangential force, depending on the buoyancy force (calculated usingthe diameter of the buoy and it’s initial draft, when anchored by the wire, with noincident waves), the force of gravity and the relative angle of the buoy and wire (see

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equation (59)), was tested.

Ft = (B −mg) sinω (59)

By using the relative angle between buoy and wire (for the lowest CD value 0.1 inorder to get a higher angle) from the simulation, maximum 4.8 degrees as shown inFigure 38, the tangential force was calculated to 8.5 N (230 kN in full scale). Thiswas to be compared with the modelled tangential force, at the same angle, of 8.6 N(232 kN i full scale), see Figure 43. Using the modelled angle of the buoy instead ofthe relative angle between buoy and wire, the estimated tangential force was calcu-lated to 11.3 N (300 kN in full scale), which is an overestimate of about 30 percentof the tangential force compared with the previous estimations.

Since using the relative angle gave such a good estimate, even though it was notan overestimate, the tank test data should be able to give an estimate of the forcesin the tank test. However, initially, as an over estimate, the measured pitch of thebuoy was used, giving the estimated tangential force was to 12.4 N (335 kN in fullscale).

Finally, by using the tank test set up, the tank test’s pitch and surge data (an-gle of buoy and position of the top of the buoy) and the scaled buoy’s dimensions,the angle of the wire was calculated as follows. See Figure 46 for variable definitions.

In order to calculate the relative angle between the buoy and the wire in the tanktest, Ω− φ, the angle φ must be known.

As shown in Figure 46 the wire angle φ can be defined as

φ = arcsinx3

Lwire(60)

where Lwire was known andx3 = x1 − x2 (61)

in whichx2 = x1 − Lbuoy sin Ω (62)

and since the X position of the top of the buoy, x1, and the pitch of the buoy , Ω,was known though the tank test, the wire angle could be calculated.

The relative angle of the tank test was calculated to 0.73 degrees giving an esti-mated tangential force of 1.3 N (35 kN in full scale).

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Figure 46: Tank test set up and variable definitions. x1, Lbuoy and Lwire were known.

Given these results, the conclusion drawn is that the accuracy of the model must im-prove. However, the close relation between relative angle of the buoy and wire, andwire and WEC, is reasonable. Therefore, an overestimate using only the buoyancyof the buoy (when fully submerged), the force of gravity and the (from tank tests)measured angle of just the buoy, should give a very high overestimate since this givesthe largest possible force in the tangential direction of the buoy. Using the relativeangle, calculated as presented in equation (60)-(62) above, between buoy and wireshould give a more accurate estimate, this might however be an underestimate sinceno wave loads are taken into account.

Also, concept 1, with a separate WEC, should give a much lesser force on the linearguide since the WEC follows the wire angle closely.

7.2.2 Seals and linear guide

Selecting the seals are problematic. Not only shall they withstand the pressuresand temperature from the gas, but also from the hydraulic fluids and ocean. The

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demands on the seals are quite high. In fact, the temperature and pressure might bethe easy part to fulfil.

The seals are supposed to seal a reciprocating motion. That means that the sealshave to work in two directions, or two seals are necessary. The same seals are notused to seal different fluids. There are gas seals and liquid seals and both might haveto be used.

Also, the reciprocating motion of the outgoing shaft is thought to be up to 5 m/s.The high velocity, paired with friction between seal and shaft will result in severalproblems, such as cavitation, dry run, lack of lubrication, particles. Another one islocal temperature spikes, also due to friction. Therefore, a good heat transferringmaterial is needed and a low friction is, as always, desirable.

Another challenge is the wave load. The seals need minimal shaft bending in or-der to work optimally and the purpose of the linear guide is to ensure this. Sincethe loads calculated in this thesis is scaled down with a factor 30, the real load isa considerable challenge. By having slide rings on each side of the seal, the shaft isstraight where necessary. These rings will however increase friction and cause wear-ing particles to appear. In the example calculating the the load on the rings, it iscalculated using only two places for the linear guides.

Looking at the two concepts, Figure 9, there are two significant differences thatspeaks for each concept. First, the decompression chamber in concept 2 is redun-dant. Since there is no real purpose of having it, it could be opened to the insideof the buoy, giving it a very large reservoir volume and hardly any pressure changewould take place inside the buoy. Second, concept 1 should give a much lower loadon the linear guide.The design of the seal and linear guide must compromise between a good sealing anda long life span. If the modules are made for easy access, reparations and exchangeof broken seals might not be as big a challenge as is could be, and a shorter lifespan than desired might prove better in the long run. This however, is to be decidedtogether with the supplier.

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7.3 Conclusions

7.3.1 Thermodynamic analysis

The temperature and compression/decompression ratio, during the process, can belimited to realistic working temperatures and compression ratio as long as the reser-voir volumes for the compartments are large enough. This goes for both Concept 1and 2.

When the WEC is working, the temperature changes during a stroke, will reacha stable point at which the highest and lowest temperatures in the two compart-ments, in both concepts, will not differ between two strokes. This limits the temper-ature range and the possible problem with overheating and freezing due to compres-sion/decompression is small. However, overheating, if it occurs, would probably bedue to heat leaking from the gearbox and/or flywheels, whereas some freezing mightoccur if the decompression compartment’s reservoir volume is too small.

When looking at the design of the WEC in concept 2, the decompression cham-ber might be open to the inside of the buoy because having a sealed decompressioncompartment fills no purpose. Also since a low temperature might cause problems,an open chamber that will have, almost, the same (atmospheric) pressure at all timeswould bypass the risk of freezing.

From the thermodynamic analysis it is clear that the temperature and compres-sion is not limiting for the seal. The temperature and pressure can be controlledby having a large enough reservoir volume. An alternative might be to the pressureratio under the accepted ratio by compressing the gas, during the compressive stroke,though a check vale into a smaller tank with higher pressure. Under the decompressstroke, opening a valve would then allow the pressurised gas to push back the piston.

7.3.2 Dynamic force analysis

As shown from the results, the simulation is too simplified to get a accurate move-ment of the buoy. This is due to the simplifications made in the buoy geometryaffecting the values of CD, CM , RC as well as the added mass and the hydrodynamicdampening. The geometry also affects the leverage for each force around the centreof gravity.Further, the dampening coefficient, b, will not be zero for a latching model since aWEC, in order to absorb waves also must be good at generating waves.[5] For b = 0

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it is said that the floating object don’t generate waves.[9]A more accurate model should be made in order to understand more about how thedesign of the buoy, and WEC, influences the forces regarding the linear guide. Thenew model should also include the control algorithm, in order to latch the buoy, andan irregular wave spectrum instead of a single regular wave.

The design in concept 1 performs better than concept 2, with regard to the loadon the shaft. This might not hold if inertia and/or drag increases for the WEC,thereby causing the relative angle between WEC and wire to increase.

As for now, the most important thing to look for in the tank test is the relativeangle between buoy and wire. With this a force close to the real value, though anunderestimate, should be the result. Implementing a safety factor and analysing theangle for waves with higher amplitude than the standard wave, should suffice in orderto get an initial estimate of the load on the shaft, and thus the linear guide(s).

7.4 Future work

A thermodynamic analysis of the gear box and flywheels needs to be done to see ifthe generated heat will affect the gas compartments.

The main problem, disregarding the gas compartments and a thermodynamic anal-ysis of the whole WEC, is to analyse the forces. An improved dynamic model, witha precise geometry and calculated, or even better, measured, CD, CM and RC valuesshould go a long way and is of great importance to further improve the over all design.This applies on the added mass, added inertia and dampening coefficients as well.The model should then be verified by comparing the movement of the simulationwith more tank tests, and, if possible, measuring of the forces in several tank tests.Also, to further improve the design, ocean currents, if present, should be taken intoaccount when analysing the load on the linear guide(s).

In order to begin the design process of the seal and linear guide(s), the dimensions,diameter of the piston, shaft and of the WEC’s hull, as well as the gas reservoirvolumes and the hydraulic fluids, all needs to be decided. But with the methodologyshown in this thesis, an initial linear guide and seal concept should be able to bedesigned.

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