OPTIMUM DESIGN OF THE PELAMIS WAVE ENERGY CONVERTER
A Thesis
Presented to the faculty of the Department of Mechanical Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Mechanical Engineering
by
Jennifer Alane Eden
FALL
2013
OPTIMUM DESIGN OF THE PELAMIS WAVE ENERGY CONVERTER
A Thesis
by
Jennifer Alane Eden
Approved by:
__________________________________, Committee Chair
Dr. Dongmei Zhou
__________________________________, Second Reader
Dr. Timothy Marbach
____________________________
Date
ii
Student: Jennifer Alane Eden
I certify that this student has met the requirements for format contained in the University format
manual, and that this thesis is suitable for shelving in the Library and credit is to be awarded for
the thesis.
__________________________, Graduate Coordinator
Akihiko Kumagai
Department of Mechanical Engineering
iii
________________
Date
Abstract
of
OPTIMUM DESIGN OF THE PELAMIS WAVE ENERGY CONVERTER
by
Jennifer Alane Eden
As well as inflicting harm upon to the environment through the production of green house gases
and harmful to people through fires, oil spills and toxic ashes; fossil fuels are also a limited
resource with a rising cost. Additionally, society as a whole has become completely dependent on
the current energy supply from fossil fuels. These contributing factors have led toward an
increased interest in renewable clean energy. While some forms have existed for centuries, such
as hydropower, other forms are in their developmental infancy, including ocean energy. Wave
energy converters (WECs) harness the kinetic and potential energies in the ocean waves and
convert them to useable electric energy via either turbines or hydraulic pumps. Although several
different types of WECs are being developed today, only a few models have made it to the
commercial market. The first to do so was the Pelamis P1model by Pelamis Wave Power Ltd.
Since then, Pelamis Wave Power has developed a new and improved Pelamis P2 model.
The purpose of this thesis is to determine the optimum size and number of tubes for the Pelamis
P2 WEC. The overall length is kept constant at 180 m in an attempt to keep standardized the
amount of raw materials used in the construction of the WEC. Simulations using the following
software are performed to study the energy absorption at the nodes between the tubes: ANSYS
Workbench, Design Modeler, and AQWA. The control model is the current P2 WEC in
iv
production. It consists of five tubes, each 36 m long with a 4 m diameter. The first set of tests is
conducted to reduce the number of tubes and increase their length. The second set of tests is used
to increase the number of tubes and decrease their length. The third set of tests select the optimum
tube number between the control, first and second test sets and vary the size of tubes from small
to large, large to small, small on the ends and large in the center, and large on the ends and small
in the center. After a systematic study, the optimum size of the Pelamis P2 is recommended.
_______________________, Committee Chair
Dr. Dongmei Zhou
_______________________
Date
v
ACKNOWLEDGEMENTS
I would like to thank my thesis advisor, Dr. Dongmei Zhou, for her continued support and
encouragement, not only in the completion of this thesis but throughout my studies at CSU
Sacramento. I am very appreciative to my first engineering instructor Mr. Phillip Pattengale for
demanding nothing but the highest quality work from his students. I am grateful to my mother,
Alane, for all her life sacrifices in ensuring my wellbeing and providing a strong basis for a
successful life. Finally I would like to thank my incredible husband, Nate, for the push (shove
really) to return to school for my Master’s Degree and supporting me throughout the entire
process.
Jennifer Eden
vi
TABLE OF CONTENTS
Page
Acknowledgements .................................................................................................................. vi
List of Tables ............................................................................................................................ x
List of Figures ......................................................................................................................... xi
Chapter
1. INTRODUCTION ............................................................................................................... 1
1.1 Background ................................................................................................................. 1
1.2 Wave Energy Potential .............................................................................................. 2
1.3 Thesis Objectives ........................................................................................................ 3
2. BACKGROUND ............................................................................................................... 5
2.1 Waves and their Causes .............................................................................................. 5
2.2 Wave Characteristics .................................................................................................. 7
2.3 Sea States .................................................................................................................... 8
2.4 Linear (Airy) Theory of Deep Water Ocean Waves ................................................... 9
2.4.1 Flow Problem Formulation ............................................................................. 10
2.5 Wave Energy Density ................................................................................................ 13
2.6 Power from Ocean Waves ......................................................................................... 14
2.7 Wave Energy ............................................................................................................ 15
3. WAVE ENERGY TECHNOLOGY ................................................................................ 17
3.1 History ...................................................................................................................... 17
3.2 Today’s Technology ................................................................................................. 19
3.2.1 Overtopping Devices ........................................................................................ 20
vii
3.2.2 Oscillating Water Column ................................................................................ 21
3.2.3 Ocean Thermal Energy Conversion ................................................................. 21
3.2.4 Salinity Gradient .............................................................................................. 22
3.2.5 Wave Activated Bodies .................................................................................... 23
3.3 Overcoming Challenges............................................................................................ 25
3.4 The Pelamis an Overview ......................................................................................... 29
3.5 Evaluation of Pelamis Performance .......................................................................... 34
4. COMPUTATIONAL MODEL SETUP AND PROCEDURE ......................................... 42
4.1 The Control Model ................................................................................................... 42
4.2 Hydrodynamic Diffraction........................................................................................ 46
4.3 The Test Models ....................................................................................................... 48
4.4 Wave Data ................................................................................................................ 51
5. ANALYSIS OF THE DATA ............................................................................................ 52
5.1 Input Parameters ....................................................................................................... 52
5.2 Test Set One.............................................................................................................. 53
5.3 Test Set Two ............................................................................................................. 54
5.4 Test Set Three ........................................................................................................... 59
5.5 Validation ................................................................................................................. 60
6. FINDINGS AND INTERPRETATIONS ......................................................................... 62
6.1 Resonance ................................................................................................................. 62
6.2 Interpreting the Results ............................................................................................. 62
6.3 Frequency ................................................................................................................. 66
6.4 Future ...................................................................................................................... 67
viii
Appendix A. Individual Efficiency Curves............................................................................. 69
Appendix B. Interpolated Pressure ......................................................................................... 80
Appendix C. Resultant Displacement ..................................................................................... 91
References ............................................................................................................................. 102
ix
LIST OF TABLES
Tables
Page
4.1 Test Set One Models ......................................................................................................... 48
4.2 Test Set Two Models ........................................................................................................ 49
4.3 Test Set Three Models ...................................................................................................... 50
4.4 Test Model Summary ........................................................................................................ 50
5.1 Input Parameters ............................................................................................................... 52
5.2 Test Set One Maximum Efficiencies ................................................................................ 53
5.3 Test Set Two Maximum Efficiencies ............................................................................... 55
5.4 Test Set Three Maximum Efficiencies.............................................................................. 60
5.5 Altering Mesh Size ........................................................................................................... 61
x
LIST OF FIGURES
Figures
Page
1.1
Global Resource Map in kW/m....................................................................................... 3
2.1
Tidal Force ...................................................................................................................... 5
2.2
Mechanisms of Wave Formation .................................................................................... 6
2.3
(a) Transverse Wave (b) Longitudinal Wave .................................................................. 7
2.4
Orbital Surface Wave Motion: Case A: Deep Water, Case B: Shallow Water .............. 8
3.1
Salter’s Duck ................................................................................................................. 18
3.2
Cockerell’s Raft ............................................................................................................ 19
3.3
The Wave Dragon, Nissum Bredning, Denmark ................................................... 20
3.4
(a) Wavegen Limpet onshore OWC (b) Oceanlinx offshore OWC ..................... 21
3.5
Schematic of OTEC ................................................................................................... 22
3.6
(a) PRO Power Plant Schematic (b) Representation of RED Process................. 23
3.7
Wave activated body WECs ..................................................................................... 25
3.8
The Pelamis P1 & P2 .................................................................................................... 29
3.9
Pelamis P2 Motion and Power Module ......................................................................... 30
3.10 Pelamis P2 Power Module .............................................................................................. 31
3.11 Vessel Motion ................................................................................................................. 34
3.12 Tube and Node Numbering ............................................................................................. 35
3.13 Force, Moment, Reaction Force Labeling....................................................................... 38
3.14 Angle, Nodal Amplitude and Tube Amplitude Numbering ............................................ 40
4.1
Pelamis Model Head Section .......................................................................................... 43
xi
4.2
Hydrodynamic System Linking ..................................................................................... 44
4.3
Comparison View of (a) Model to (b) Actual Device ..................................................... 44
4.4
Control Model ................................................................................................................. 45
4.5
EMEC Wave Height Data ............................................................................................... 51
5.1
Test Set One-4 m Diameter Efficiency Curves ...................................................... 54
5.2
Test Set Two-3 m Diameter Efficiency Curves ..................................................... 56
5.3
Test Set Two-5 m Diameter Efficiency Curves ..................................................... 56
5.4
T3 Efficiency Comparison ........................................................................................ 57
5.5
T4 Efficiency Comparison ........................................................................................ 57
5.6
T5 Efficiency Comparison ........................................................................................ 58
5.7
T6 Efficiency Comparison ........................................................................................ 58
5.8
T8 Efficiency Comparison ........................................................................................ 59
5.9
Test Set Three-T6D5 Efficiency Curves ................................................................... 60
5.10 Altering Mesh Size Efficiencies ..................................................................................... 61
xii
1
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND
The use and extraction of fossil fuels have resulted in horrific catastrophes. These
include, but are not limited to, the production of green house gasses, oil spills, mass
poisoning, oil fires, and underground coal fires. The largest oil spill was the Gulf War Oil
Spill of 1991 where Iraq purposefully dumped oil into the gulf to keep U.S. forces from
landing which ultimately resulted in an oil slick 4,242 square miles (10,880 square
kilometers) and five inches (13 centimeters) thick [1]. In 2008, the Kingston Fossil Plant
had a wall of their ash pond break, which resulted in spilling 5.6 cubic yards (4.28 cubic
meters) of wet coal ash that contains high levels of heavy metal and carcinogens if
ingested [2]. In 1991, the defeated Iraqi army set nearly 700 oil wells ablaze while
retreating from Kuwait that burned for nearly ten months [3]. Once started there is no
stopping an underground coal fire so long as there is fuel to feed it, such as the New
Castle, Colorado coalmine fire that has been burning since 1899 and shows no signs of
stopping [4].
Perhaps the most crucial problems or hindrances with the continued use of fossil fuels for
today’s society are their limited supply and recent exponential price increase. The planetwide population has become completely dependent on the current energy supply. For
2007, the International Energy Agency (IEA) determined the world energy consumption
to be 16,429 TWh [5]. Global energy demands will only rise with the ever-increasing
2
world population as, according to the US Census Bureau, this number was to exceed 7
billion people by the end of 2012 [6]. It is therefore essential that time and resources be
allotted for research and development of renewable, clean energy sources. Alternative
energy systems are making their way into the global market and the options are vast,
including solar, wind, hydroelectric, geothermal, bio-fuels, natural gas, hydrogen, and
ocean wave energy. Wave energy solutions are still in their infancy, yet some versions
are becoming increasingly popular on a commercial scale and several countries are
developing wave energy farms.
1.2 WAVE ENERGY POTENTIAL
Wave energy is a relatively untapped potential that is now receiving considerable interest
due to its vastly estimated energy potential. For deep water, the global wave power
resource is estimated between 8,000 to 80,000 TWh according to the World Energy
Council [7], which can make a significant dent in if not entirely meet the previously
stated global electricity demand of 16,429 TWh in 2007 [5].
As is evident in Figure 1.1, the greatest regions of energy potential occur on the west
coast of landmasses further away from the equator. However, any value above 15 kW/m
has the potential to generate electricity at a competitive price with today’s wave energy
technologies [8]. As with any new developing technology, the initial startup costs hinder
competitive electricity cost production, yet when the technology matures prices will be
competitive with current electricity prices from fossil fuels.
3
To reduce this initial startup cost, companies attempt to limit their manufacturing cost by
creating parts or devices of uniform size. The most successful wave energy converter
(WEC) is Pelamis P2. The practice by reducing initial startup cost has been seen in the
design of Pelamis P2, as all the tubes of the structure have the same dimensions.
However, for the Pelamis P2 model the key question is, are uniform tube sizes the most
efficient for energy capture? This is the question addressed and answered by this thesis.
Figure 1.1 Global Resource Map in kW/m [8]
1.3 THESIS OBJECTIVES
The purpose of this study is to determine the optimum design of the Pelamis P2 Wave
Energy Converter (WEC). This thesis will start with the theory of ocean waves and
linearized ocean wave theory, and move on to discuss wave energy and the various forms
4
of WECs including an overview of the Pelamis. The software programs, ANSYS
Workbench, ANSYS DesignModeler, and ANSYS AQWA will be used to determine the
optimum length, diameter, and number of individual tubes for the Pelamis P2 model. Not
only will this analysis provide the model with the overall highest energy extraction, it
will also provide the model with the greatest efficiency over the range of frequencies
experienced off the coast of Scotland.
5
CHAPTER 2
BACKGROUND
2.1 WAVES AND THEIR CAUSES
Waves are either created through gravitational pull (tidal force) or concentrated solar
energy (more commonly referred to as wind waves). The tides that are visually
noticeable by the rise and fall of water along the coastlines are an effect of the
gravitational force between the earth and the moon. As the moon rotates around the earth,
the gravitational force distorts the shape of the body of water without changing the
volume. In essence, the sphere becomes an ellipsoid as the body of water closest to the
moon is attracted more strongly towards the moon. The earth gravitates towards the moon
as well, pulling away from the oceans on the far side from the moon. The body of water
furthest from the moon then has a relative acceleration away from the earth [9,10]; this is
visualized in Figure 2.1.
Figure 2.1 Tidal Force [10]
6
The second form of wave generation, and the form this thesis concentrates on, is the
surface wave, also called the wind wave, but is in essence a concentrated form of solar
energy. Differential heating of the earth generates pressure differences in the air, which in
turn create winds that pass over open bodies of water [7]. Small fluctuations in the wind
speed cause variations in air pressure; the faster moving air creates lower pressures while
the slower moving air creates higher pressures as is demonstrated through Bernoulli’s
Equation:
𝑝 +
1
2
𝜌𝑣 2 + 𝜌𝑔𝑧 = 𝐶
(2.1)
where p is the fluid pressure at a particular point, ρ is the fluid density at all points in the
fluid, v is the fluid speed at a particular point in the fluid streamline, g is the acceleration
due to gravity, z is the elevation above or below a reference point, and C is a constant. As
the wind passes over the surface of the water, small ripples form via a combination of
pressure and shear forces. As the wind passes over the water’s surface, the friction forces
between the air and water surface cause shear forces to develop within the water, these
further push the water molecules in the ripples to form crests, as seen in Figure 2.2. The
ripples then grow exponentially and form fully developed waves.
Figure 2.2 Mechanisms of Wave Formation [11]
7
The main factors that cause the waves to grow and therefore the amount of energy
transferred to the water are wind speed, wind duration and fetch (the distance of water
over which a particular wind has blown over).
2.2 WAVE CHARACTERISTICS
Waves can be either transverse or longitudinal. Longitudinal waves are a series of
compressions and rarefactions in a medium and are typical of sound waves; the particles
are displaced parallel to the direction of wave motion. With transverse waves, such as
light waves, the particles are displaced in the medium perpendicular to the direction of
wave motion. Surface water waves are a combination of both transverse and longitudinal
motions; however, under linearized theory (details in section 2.4) surface water waves are
transverse in nature. The transverse wave can be characterized by its amplitude, height
from the midline of the wave to the crest, and wavelength, distance between crests. The
difference between the two types of waves can be visualized in Figure 2.3 below. [12,
13]
(a)
Figure 2.3 (a) Transverse Wave (b) Longitudinal Wave [13]
(b)
8
An interesting feature to note from Figure 2.3 (a) is that the movement of water particles
is in the vertical direction, not in the direction of travel. For surface water waves just
below the linear transverse wave, the average positions of a fluid particle are small
displacements in the vertical and horizontal directions resulting in the particle to orbit in a
circular motion for deep water and in an elliptical motion for shallower water. Both deep
and shallow water orbits decrease rapidly with increasing depth. This can be visualized in
Figure 2.4 below. [14]
Figure 2.4 Orbital Surface Wave Motion:
Case A: Deep Water, Case B: Shallow Water [14]
2.3 SEA STATES
A sea state is the condition of the free surface at the top of the ocean at a particular time
and location in regards to wind waves and swell. They are characterized by the wave
height and period. A wind sea state is the state of waves under the direct influence of the
wind and are very chaotic in nature. They do not consist of a perfect sine wave but can be
9
mathematically expressed as a combination of sine waves of various amplitudes and
wavelengths. The speed of the waves can be expressed through the following equation:
𝑔𝜆
c = √2𝜋,
(2.2)
where c is the phase speed, g is the acceleration due to gravity, and λ is the wavelength.
This will be described more fully in section 2.4. As c and λ are directly proportional, the
waves with longer wavelengths travel at faster speeds than those with shorter
wavelengths. Thus, the chaotic sea state will come apart as the waves propagate across
the ocean.
Once the waves have been created by the wind, they are no longer dependent upon the
wind to continue. This is referred to as a free wave state or a swell wave state. This sea
state is comprised of a series of surface gravity waves, meaning that gravity, not wind, is
the restoring force. When the water particles are vertically displaced above the
equilibrium position, gravity will attempt to restore them towards equilibrium resulting in
the oscillations seen in Figure 2.4 above. These swell waves are created offshore by
storms and travel thousands of kilometers; their propagation is only ended by breaking on
the shoreline.
2.4 LINEAR (AIRY) THEORY OF DEEP WATER OCEAN WAVES
As stated in Section 2.2, surface water waves are a combination of both transverse and
longitudinal motions, by using linear theory, the waves can be restricted to transverse
motion, thereby simplifying the calculations. Surface gravity waves can be linearized if
10
a/λ << 1 and a/H << 1, where a is the wave amplitude, λ is the wavelength and H is the
uniform water depth. Typically the surface gravity waves of oceans have wavelengths
between 30-40 m, enabling the water surface tension to be neglected, as it pertains to
wavelengths of less than 5-10 cm. Deep water is classified as having depths greater than
one third of the wavelength (H > λ/3). As the Pelamis is deployed at water depths greater
than 50 m it is classified as a deep-water device, hence the theory in this thesis will
concentrate on deep-water linear theory (shallow water theory will have an alternate
equation for phase speed that is dependent on ocean depths). It is also noted here that as
frequency is much larger than the Coriolis frequency, the wave motion is unaffected by
Earth’s rotation. Assumptions made here are that the water is incompressible (constant
density) and irrotational, gravity is the only external force, and viscosity can be
neglected.
2.4.1 FLOW PROBLEM FORMULATION
While it is acknowledged that ocean waves propagate in all directions, we will simplify
our equations to the 2D case in the x-z plane, where x is the horizontal direction the
waves propagate and z is the vertical height. It starts with the simple sinusoidal wave
equation:
2𝜋
η(x,t) = acos[ λ (x - ct)],
(2.3)
where z = η(x,t) is the surface shape (vertical direction of the air-wave interface) or more
commonly called the waveform, a is the amplitude, λ is the wavelength, c is the wave
speed, and
2𝜋
λ
(x-ct) is the phase.
11
2𝜋
𝜆
Let the period T = 𝑘𝑐 = 𝑐 where k is the spatial frequency or wave number, and the radian
1
frequency (also called the angular frequency) ω = 2πf where f = 𝑇 is the cyclic frequency,
then equation (2.3) can be rewritten as:
η(x,t) = acos[kx - ωt].
(2.4)
Determining the travel speed of the wave crests leads to the wave propagation speed by
setting the phase so that the cosine function is in unity (cosine equals +1) thus allowing
the phase to equal 2nπ and allowing η to equal +a.
2nπ =
2𝜋
λ
(xcrest - ct) = kxcrest – ωt.
(2.5)
Solving for xcrest and neglecting all terms without the time variable t, an equation is found
for the phase speed:
c=
𝜔
𝑘
= λf.
(2.6)
Since the water is irrotational, we can define a velocity potential φ(x, z, t) so:
u = ∂φ/∂x,
and
w = ∂φ/∂z,
(2.7a,b)
where u is the velocity component in the x-direction and w is the velocity component in
the z-direction.
Here the continuity equation is introduced:
∂u/∂x + ∂w/∂z = 0,
(2.8)
then through the use of (2.8), equation (2.7) becomes the Laplace Equation:
∂2φ/∂x2 + ∂2φ/∂z2 = 0.
(2.9)
12
There are three boundary conditions and one initial condition to take into account and
they are (the derivation of the kinematic and dynamic velocity boundary conditions will
not be provided here, but can be found in [15]):
1. The velocity at the bottom of the ocean is zero.
w = (∂φ/∂z)z = -H = 0
(2.10)
2. The kinematic velocity at the surface of the water tells the motion of the
surface where the fluid particle’s velocity normal to the surface has to equal
the vertical velocity of the free surface. This ensures there is no flow
separation at the surface during fluid motion.
(∂φ/∂z)z=0 ≅ ∂η/∂t.
(2.11)
3. The dynamic velocity at the surface of the water represents the force on the
surface and is obtained by setting the pressure above the free surface to the
constant zero in Bernoulli’s Equation for unsteady potential flow.
(∂φ/∂t)z=η ≅ (∂φ/∂z)z = 0 ≅ -gη
(2.12)
4. The initial condition is chosen to satisfy the simple sinusoidal equation (4)
η(x, t=0) = acos(kx).
(2.13)
By use of (2.4), (2.10), (2.11), and (2.12) the velocity potential equation becomes:
φ(x, z, t) =
𝑎𝜔 cosh(𝑘(𝑧+𝐻))
𝑘
sinh(𝑘𝐻)
sin(kx – ωt),
(2.14)
For deep water:
cosh(𝑘(𝑧+𝐻))
sinh(𝑘𝐻)
≈
sinh(𝑘(𝑧+𝐻))
sinh(𝑘𝐻)
≈ 𝑒 𝑘𝑧 ,
(2.15)
13
thereby reducing the velocity potential equation (2.14) for deep water to:
φ(x, z, t) =
𝑎𝜔
𝑘
𝑒 𝑘𝑧 sin(kx – ωt),
(2.16)
which means the velocity components are:
u = aω
cosh(𝑘(𝑧+𝐻))
sinh(𝑘𝐻)
cos(kx – ωt),
u = aω 𝑒 𝑘𝑧 cos(kx – ωt),
and
w = aω
sinh(𝑘(𝑧+𝐻))
sinh(𝑘𝐻)
sin(kx – ωt), (2.17a,b)
w = aω 𝑒 𝑘𝑧 sin(kx – ωt).
and
(2.18a,b)
Substituting (2.4) and (2.14) into (2.12) and simplifying produces a dispersion relation
between ω and k:
ω = √𝑔𝑘 tanh(𝑘𝐻).
(2.19)
Substituting (2.19) into (2.6) provides the following phase speed equation:
c=
𝜔
𝑘
g
= √k tanh(𝑘𝐻)
(2.20)
For deep water (H > λ/3), since tanh(x) → 1 when x → ∞, the phase speed equation
(2.20) can be simplified as:
c = √𝑔⁄𝑘 .
(2.21)
The absence of water depth (H) in this equation demonstrates that for deep water, the
phase speed is independent of the water depth.
2.5 WAVE ENERGY DENSITY
In order to determine the power stored in ocean waves the energy density must be
determined. Surface gravity waves contain both kinetic and potential energy, the former
in the motion of the particles and the later in the vertical deformation of the surface. The
14
kinetic energy per unit horizontal area, Ek, can be found from integrating over the depth
and averaging over the wavelength:
𝜌
𝜆
0
Ek = 2𝜆 ∫0 ∫−𝐻(𝑢2 + 𝑤 2 )𝑑𝑧𝑑𝑥.
(2.22)
Substituting the velocity components (2.18) and using the dispersion relation (2.19) the
kinetic wave energy in the water column per unit horizontal area becomes:
1
1
̅̅̅ = ρga2.
Ek = 2ρg𝜂2
4
(2.23)
The mean potential energy density per unit horizontal area, Ep, of the surface gravity
waves is the work done per unit area in order to deform the surface and is hence equal to
the difference in potential energy between the disturbed and undisturbed state:
𝜌
𝜆
𝜂
𝜌
𝜆
0
Ep = 2𝜆 ∫0 ∫−𝐻 𝑧 𝑑𝑧𝑑𝑥 - 2𝜆 ∫0 ∫−𝐻 𝑧 𝑑𝑧𝑑𝑥 =
𝜆
∫ 𝜂2
2𝜆 0
𝜌𝑔
𝑑𝑥
(2.24)
It can also be written in terms of the mean square displacement:
1
1
̅̅̅ = ρga2.
Ep = 2ρg𝜂2
4
(2.25)
It is noticed the kinetic and potential energies are the same, as is consistent with the
conservation of energy law. The total energy of the system can be written as:
̅̅̅ = 1ρga2.
E = Ek + Ep = ρg𝜂2
2
(2.26)
2.6 POWER FROM OCEAN WAVES
The power obtained from ocean waves can be expressed in many forms but is most
commonly seen in terms of power per meter of wave front or watt per meter. Thus the
incident wave power is the total energy of the system (kinetic plus potential) as defined
15
through (2.26) multiplied by the speed as defined through (2.21) to obtain the following
relation:
1
𝑔
𝑃𝑡 = 𝐸𝑐𝑔 = [2 𝜌𝑔𝑎2 ] √𝑘 .
(2.29)
Where P is power per unit length of wave front (W/m), ρ is density of seawater (kg/m3)
g is gravity (m/s2), a is amplitude of the wave (m), and k is the special frequency.
Through the following relations: k = 2π/λ, λ=gT2/2π, and T=1/f, equation (2.29) can also
be written as:
1
𝑔
𝑃𝑡 = [2 𝜌𝑔𝑎2 ] [2𝜋𝑓] = [
𝜌𝑔2 𝑎2
4𝜋𝑓
]=[
𝜌𝑔2 𝑎2
2𝜔
]
(2.30)
2.7 WAVE ENERGY
Wave energy is an intermittent energy source; the available power from the waves varies
in time in an uncontrollable manner. This is an unwanted characteristic associated with
many forms of renewable technologies. Wind power can only be generated when the
wind is blowing. Solar power can only be harnesses when the sun is out. Tidal energy can
only be produced when enough head (height difference) exists on either side of the dam
housing the turbine. Wave energy cannot harness energy during storms, and is
intermittent between individual waves as well. This intermittency makes it virtually
impossible for such energy devices to be the sole provider of electricity to a grid, unless
enough energy is harnessed and stored to supply a society’s needs when the device is not
16
generating electricity. However, wave power can be predicted in regions where adequate
data has been accumulated and thusly power generation can be better regulated.
Wave energy is closely compared to wind energy due to their similarities; however, wave
energy has a noticeable advantage in that the power supplied varies more slowly than
with wind energy. This is due primarily to the density of seawater being 850 times greater
than that of air. Therefore, for the same surface area the amount of energy store in a swell
is much greater than air. In addition, swells lose very little amounts of energy in the form
of inner friction until the wave nears the coast and that energy is lost to the seabed
through friction, heat. This feature means that relatively all the energy in the swell can
travel great distances to be absorbed by a wave energy converter (WEC).
17
CHAPTER 3
WAVE ENERGY TECHNOLOGY
Converting the kinetic and potential energy of waves into useable electrical energy
requires an intermediate mechanical energy step. The various methods to accomplish this
intermediate step are divided into categories based on their basic operating principles but
all of them (with the exception of some salinity gradient methods) make use of either
turbines or hydraulic pumps. This chapter will provide the history of wave energy
converters (WECs) and a brief overview of the various forms of ocean energy technology
currently under development, all leading to the Pelamis P2 WEC and its efficiency to
capture energy.
3.1 HISTORY
In the 18th century, the first patents for WECs were issued, however it was not until the
1973 oil crisis that the push towards developing alternative energy solutions took off. At
that time two significant WECs were developed: the first is Salter’s Duck as shown in
Figure 3.1 developed by Stephen Salter at the University of Edinburgh, Scotland, and the
second is Cockerell’s Raft developed by Sir Christopher Sydney Cockerell as shown in
Figure 3.2.
18
Figure 3.1 Salter’s Duck [19]
Salter’s Duck is shaped like a rudimentary duck (teardrop shape) and each duck is
roughly the size of a house at 10-15m in diameter and 20-30 m wide. A string of ducks
was to be connected through a central spine, allowing each duck to move independently,
bobbing up and down, from the passing waves. As the ducks rock up and down, four
gyroscopes rapidly flip back and forth which in turn work an oil pump that produces
hydraulic power [17-20]. Salter’s Duck never left the prototype stage due in part to the oil
crisis ending in the early 1980s and also from high competition for funding with the
nuclear program, yet regardless, it still continues to be the industry standard as it
converted at staggering 90 percent of wave energy into electricity. This technology has
inspired and influenced many of today’s WECs but the device currently under
development and undergoing testing by WEPTOS Technology closest resembles the duck
[21, 22].
19
Figure 3.2: Cockerell’s Raft [11]
Cockerell’s Raft consists of a series of large flat rectangular pontoons connected together
through hydraulic pistons. The pontoons are moored in line with the wind/wave direction,
so as the waves pass under the rafts, the pontoons oscillate relative to one another. The
torque generated works hydraulic pumps, which in turn generate electricity [19, 20].
Cockerell’s Raft has inspired the development of subsequent hinged-contour WECs
(attenuators).
3.2 TODAY’S TECHNOLOGY
There are numerous concepts for WECs under various states of development, which
demonstrates the infancy of the wave energy industry, as no one particular type of device
has been proven significantly more efficient than the others. There are five primary
methods of extracting energy from the ocean: overtopping devices, oscillating water
columns, thermal devices, salinity gradients, and wave activated bodies (absorbers and
attenuators).
20
3.2.1 OVERTOPPING DEVICES
Overtopping devices capture seawater from incident waves in a reservoir above sea level.
This water is then channeled through one or more turbines and transformed into
electricity, the water is let back out to sea [23, 24]. The Wave Dragon is an example of an
offshore overtopping device as shown in Figure 3.3; it is moored in relatively deep water
to take advantage of the ocean waves before they lose energy when approaching the
coastline and can be deployed either singularly or as a farm. The outstretching wings
elevate and divert seawater over the curved edge and into its reservoir tank creating head
(the relative difference in height from the top of the reservoir to sea level). The water is
then released out through several low-head hydro turbines to create electricity much in
the way hydroelectric energy is created [25]. There is little maintenance for this device as
the turbine is its only moving part, which has lead to its increasing appeal.
Figure 3.3 The Wave Dragon, Nissum Bredning, Denmark [25]
21
3.2.2 OSCILLATING WATER COLUMN
An oscillating water column (OWC) creates electricity through a two-step process. A
two-way turbine is mounted at the top of a capture chamber, while the bottom of the
chamber remains open to sea below the waterline. As a waves approach the device, water
is forced through the bottom opening into the capture chamber compressing the air within
it and forcing the air out to the atmosphere through the turbine creating electricity. As the
water rescinds, a negative pressure vacuum is created within the capture chamber sucking
air through the turbine back into the chamber creating more electricity [23]. The OWC
can either be deployed onshore or offshore; both varieties operate on the same working
principle and can be seen in Figure 2.3-4.
(a)
(b)
Figure 3.4 (a) Wavegen Limpet onshore OWC schematic [26]
(b) Oceanlinx offshore OWC [27]
3.2.3 OCEAN THERMAL ENERGY CONVERSION
Ocean thermal energy conversion (OTEC) uses the natural temperature difference
between the warmer surface and the cooler bottom of the ocean to run a heat engine and
thereby create electricity. OTEC typically works on a Rankine cycle using a working
22
fluid with a low boiling point (such as ammonia) on a closed cycle. The working fluid
leaves the turbine generator in its vapor form, flows through a condenser where its heat is
transferred to the cold seawater and the working fluid changes into liquid form. The
working fluid is then pumped through an evaporator where heat is transferred from the
warm seawater to the working fluid converting it into a vapor. The vapor then goes
through the turbine generator creating electricity and the cycle continues [28].
Figure 3.5 Schematic of OTEC [29]
3.2.4 SALINITY GRADIENT
Salinity gradient ocean technology takes advantage of the difference in salinity between
seawater and fresh water using either pressure retarded osmosis (PRO) as seen in Figure
3.6 (a) or reverse electrodialysis (RED) as seen in Figure 3.6 (b) depending upon the type
of semi-permeable membrane used. These power plants are typically located where rivers
disperse into oceans or in areas where industrial users (such as sewage treatment plants)
23
discharge substantial volumes of fresh or low-salinity water into the ocean. For PRO,
fresh water crosses the membrane creating a pressure head on the seawater side, which
turns a turbine to create electricity as the water is release back to sea. For RED, chloride
crosses the membrane from the seawater to the fresh water, while sodium crosses the
membrane from the fresh water to the seawater. The chemical potential difference
generates a voltage that is converted into electrical current. The main hindrance of both
PRO and RED is the short lifespan of the semi-permeable membrane and its high expense
[30].
(a)
(b)
Figure 3.6 (a) PRO Power Plant Schematic [31]
(b) Representation of RED Process [32]
3.2.5 WAVE ACTIVATED BODIES
Wave activated bodies comprise the bulk of WECs and range in type from attenuators,
point absorbers, oscillating wave surge converters, submerged pressure differential, and
bulge wave. Attenuators are floating devices that are moored parallel to the wave
24
direction; they capture energy via hydraulic pumps from the relative motion of its parts
and include the Pelamis and the Wave Star. Point absorbers have buoys that bob on the
surface of the ocean from waves of all directions; the relative motion of the buoy with the
base creates the electricity and includes the AquaBuoy and PowerBuoy. Oscillating wave
surge converters are moored to the seabed with a paddle that raises and lowers with the
motion of the waves; the most prominent models in production today are the Oyster and
bioWAVE. Submerged pressure differential devices are typically located near shore and
sits on the seafloor. The motion of the waves passing overhead cause the sea level above
the device to rise and fall, creating a pressure differential which drives the device to
pump fluid through a system to generate electricity; one such device is the Archimedes
wave swing. Bulge wave technology is a large rubber tube filled with seawater; they are
moored to the seabed parallel to the wave direction. As the water enters the stern (front)
of the device a ‘bulge’ is formed due to pressure variations along the length of the tube.
This bulge grows as it travels through the tube thereby gathering energy that is then used
to drive a standard low-head turbine located at the bow (back) where the water then
returns to the sea; the Anaconda is currently in the test-tank stage of development [23,
33-38].
25
Figure 3.7 Wave activated body WECs (a) Pelamis [33] (b) Oyster [34] (c)
Archimedes Wave Swing [35] (d) WaveStar [36] (e) bioWAVE [37] (f) Powerbuoy
[38] (g) Anaconda [39]
3.3 OVERCOMING CHALLENGES
There are many positive aspects for wave energy. These include: the large resource
magnitude around the world, a higher energy density compared to wind energy since
seawater is 850 times denser than air, no fuel requirements after the initial capital cost of
building the WEC, no greenhouse gas emissions; and the WEC can act as an artificial
reef for the marine environment. However, as with any technology, there do exist various
26
challenges associated with the design and implementation of WECs, which must be
overcome to effectively create electricity at a competitive price.

Survivability: WECs are designed to remain exposed to the environment all year
long. Therefore, the key issue for WECs is to be able to survive in storm
conditions, and specifically to survive the statistical ‘100 year storm’.
Approximately 90% of the year the ocean remains at a relatively stable condition
for which the WEC is designed to capture energy; they are not designed to capture
energy during storm conditions reaching power levels 50 times more than the
average state. Hence, it is imperative that a WEC revert to an inert state during
high waves. The Pelamis has a sleek, streamlined shape that allows it to dive
under the wave crests, limiting its power absorption and motion to allow for
survival in large storms [33].

Corrosion: Metal objects corrode when exposed to seawater. There are three types
of corrosion to be aware of electrochemical, electrolytic and galvanic.
Nevertheless, regardless of the type, four things must always be present for
corrosion to occur: an anode, a cathode, a metallic path for electrons to flow and
an electrolyte. Dissimilar metals make up the anode and cathode, and seawater is
the electrolyte. Typically, the more anodic metal will corrode while the more
cathodic metal will remain intact. When designing the WEC, either non-corrosive
metals need to make up the outer layer or shell of the device, or sacrificial anodes
need to be installed and replaced regularly. The Pelamis P2 combats corrosion by
housing all generating equipment inside the machine where it is dry and the
27
machine structure is painted with the same marine grade paint that’s used in the
oil, gas and shipping industries.

Maintenance/Fatigue: The ocean waves and the large density of seawater
continuously impart huge cyclic stresses on WECs, much more than would be
experienced by wind energy systems. These cyclic stresses lead to fatigue of the
device, resistance to fatigue therefore becomes a high priority for design
parameters. This fatigue also leads to the need for continuous maintenance. As the
ocean is a difficult place to work under the best of circumstances, design for
maintenance must also be considered. For this reason, the Pelamis is easily able to
detach from its mooring cables and be towed to shore for regular maintenance and
repair.

Intermittency: Not only are WECs intermittent energy producers on an annual
scale due to entering an inert state for survivability during storms, but they are
also intermittent energy producers on a regular basis from the irregularity of the
waves. This intermittent source makes wave energy production unable to
constitute the sole provider of electricity for a region as consumers require
constant electricity; the same problem is face by both the wind and solar energy
industries. In an attempt to counteract this dilemma, the hydraulic rams of the
Pelamis pump fluid into high-pressure accumulators to produce smooth and
continuous electricity from the irregular waves.

Marine life: As stated above, the WEC and/or its mooring lines make for great
artificial reef systems, however the marine life can have a negative impact on the
28
WEC and hinder its operation. This is often seen with near-shore devices, but as
the Pelamis resides in deep water, this is less of an issue with the exception of
bacteria and possibly barnacles. There may also exist a negative influence of the
sound of WECs on marine life, but little research has been done on the topic to
make a definitive claim.

Economic impacts: While it is true that the ocean holds a vast potential for green,
renewable energy the ocean is also a resource for fishermen, the military, and
private boating (such as the cruise line industries). The installation of a WEC or
an ocean energy farm therefore needs to go through a lengthy permitting process
[43]. Also, should ocean energy (or in conjunction with other renewable energies)
dominate the production of electricity there will be a huge change to the stock
market, not to mention eliminating big oil and the negative repercussions for that
job market.

Competitive prices: In order for WECs to be a viable source of energy, they must
produce electricity at a competitive price with other energy sources. This is a
difficult task for a new industry, however governments around the world are
starting to implement policies for research and development of renewable energy
as well as tax breaks [44]. This allows WECs to enter the highly competitive
energy market where existing technologies could ‘lock-out’ emerging ones. Two
major utilities have already secured supply contracts with the Pelamis P2,
demonstrating a great confidence in the technology.
29
All the above-mentioned problems have hindered the research and development of WECs
over the past several decades. However, great ingenuity is taking place across the globe
to develop solutions for these challenges. [32, 40-44]
3.4 THE PELAMIS AN OVERVIEW
The Pelamis WEC device is a particular type of ocean attenuator consisting of either four
(P1 model) or five (P2 model) cylindrical sections linked together by universal joints that
allow for motion with four degrees of freedom, as seen in Figure 3.8 (a) and (b). The
Pelamis floats semi-submerged in ocean depths of 50 m typically 2-10 km offshore and
converts ocean wave energy into electricity. The original Pelamis P1 device was 120 m
long, 3.5 m in diameter and comprised four tube sections separated by three shorter,
power conversion modules with a total of six hydraulic motors. The P1 was the world’s
first full-scale offshore WEC to generate electricity and the first wave energy farm to
successfully supply electricity to the national grid. The new P2 model has five tube
sections with the power conversion modules integrated into them and has an overall
length of 180 m and diameter of 4 m; each joint has four hydraulic motors. Both the P1
and P2 model creates the same amount of electricity, 750 kW.
(a)
Figure 3.8 The Pelamis (a) P1 (b) P2 [33]
(b)
30
The Pelamis faces the direction of the waves, as the waves pass down the machine the
sections bend. It is this bending movement that converts the energy from the waves into
electricity by hydraulic jack systems housed at each joint, to account for the varied wave
input, the hydraulic systems use high and low pressure accumulators to smooth out the
generated power from irregular waves; this power is then transmitted to shore via subsea
cables. The motions of the Pelamis P2 can be seen in Figure 3.9 and its inner workings
can be seen in Figure 3.10.
Figure 3.9 Pelamis P2 Motion and Power Module [63]
31
Figure 3.10 Pelamis P2 Power Module [63]
Sustainability
WECs need to manage a large range of power input from calm low waves to high peaks
experienced during severe storms. With this in mind, the number one consideration
during development of the Pelamis was sustainability. The majority of the year calm
weather it experienced, thus energy capture was designed to optimize energy input from
small waves. To protect itself from the large waves, the Pelamis’ sleek streamlined
design allows it to dive under wave crests limiting the loads applied by the large waves.
Engineering: Reaction Forces
The Pelamis WEC acts against the force of the waves to absorb the energy, requiring the
need of something for the PTO mechanism to push against. While most WECs use the
seabed, the Pelamis uses its own buoyancy force. Thus, the force of the waves against
32
one part of the Pelamis reacts against the buoyancy forces from other parts of the
Pelamis. This leads to several advantages: there is no need for an external reaction source
and the Pelamis is self-limiting in extreme weather conditions. In addition, since the
Pelamis is a free-floating device it is easily deployed without on-site construction or the
need for specialized vessels; since the Pelamis is self-contained and has a quick
connection to its mooring allowing it to be easily taken on and off site with small, costeffective vessels.
Power Absorption
Just as thermal engines have the Carnot limit so too do WECs have a theoretical limit on
the amount of power they can absorb. A wave energy machine absorbs more energy from
the entire wave field around it than just the waves that pass directly under it. The
theoretical limit can be deduced by examining each type of wave leading to the
conclusion that the Pelamis, a line absorber or attenuator, has the highest theoretical limit
than any of the other WECs (heaving body, surging or pitching, and heaving-and-surging
or heaving-and-pitching).
The Pelamis uses resonance to increase power capture of small waves. The default setting
of the Pelamis is non-resonant, allowing it to withstand large swells, however the joints
can be actively controlled by its power take off system to create a cross-coupled resonant
response. Since power at the joints goes from zero to the peak in a couple seconds, the
electrical generation system works for the average power available. Optimum power
33
capture is the primary goal. To that end, the Pelamis is controlled by a state-of-the-art
integrated control and data acquisition system to quickly adjust the machine to the
appropriate wave conditions. [45-47]
Literature Review
There is very little literature published for the Pelamis WECs. Most studies have
comprised of performance comparisons between various types of WECs. It was found
through several of the studies that the Pelamis devices had the highest energy and
economic returns at high resource locations (refer back to Figure 1.1), but produced poor
results at poor resource locations; therefore the location determines which WEC is the
best option. [48, 49]
The impact of electricity from the Portugal Pelamis Farm has on current market prices
was found to be negligible for wholesale electricity prices. [50] The coastal impacts and
changes in shoreline wave climates from Pelamis Wave Energy Farms were explored.
The results showed that there is a strong influence on the wave conditions immediately
behind the farm, but this influence dissipated by the time the waves reached the coastline.
[51]
There was also research done to create an optimum model for the hydraulic power takeoff system of the Pelamis, yet no research has been done into optimizing the actual
34
Pelamis device. This lack of knowledge and research, along with the excitement of a new
technology lead to this thesis.
3.5 EVALUATION OF PELAMIS PERFORMANCE
The Pelamis P2 generates electricity based on the relative motion between its tubes. It is
therefore essential to define vessel motion in order to determine how the device extracts
energy. Vessels and structures floating on a body of water can experience six degrees of
freedom. If we define z as the vertical axis, then the up and down motion of a vessel is
called heave, rotation about this z-axis is called yaw. If we define the axis passing
through the bow and stern of a ship as the x-axis then the forward backward motion is
called surge and the rotation about this axis is called roll. The axis passing through the
port and starboard of the vessel is the y-axis any side-to-side motion is called sway and
the rotation about this axis is called pitch. These motions can be seen in Figure 3.11
below.
Figure 3.11 Vessel Motion [52]
35
As previously stated, the study of this thesis will be restricted to the 2D case in the x-z
plane. With that restriction, the Pelamis WEC moves with two degrees of freedom to
capture energy, pitch and heave. Thus for each tube there are two dynamic equations, one
for pitch and one for heave. Therefore with a structure containing N number of tubes,
there will be 2N equations. These equations can be decomposed into N+1 equations for
the nodal amplitudes and N-1 equations for the nodal forces. The tube and node labeling
system can be seen in Figure 3.12.
Figure 3.12 Tube and Node Numbering
Motion Equations
In section 2.4 Linear (Airy) Theory Of Deep Water Ocean Waves, it was determined the
elevation of the free surface to be governed by equation (2.4): η(x,t) = acos[kx - ωt], the
dynamic boundary condition as (2.12): (∂φ/∂t)z = η ≅ (∂φ/∂z)z = 0 ≅ -gη, and the velocity
potential as equation (2.16): φ(𝑥, 𝑧, 𝑡) =
𝑎𝜔
𝑘
𝑒 𝑘𝑧 sin(kx – ωt). If the relation between sine,
cosine and exponential functions is used:
1
sin(θ) = 2𝑖 (𝑒 +𝑖𝜃 − 𝑒 −𝑖𝜃 ),
1
cos(θ) = 2 (𝑒 +𝑖𝜃 + 𝑒 −𝑖𝜃 ),
(3.1a,b)
equations (2.4) and (2.16) can be rewritten as:
𝜂(𝑥, 𝑡) = 𝑒 −𝑖𝜔𝑡 (𝐴𝑒 𝑖𝑘𝑥 + 𝐵𝑒 −𝑖𝑘𝑥 )
φ(𝑥, 𝑧, 𝑡) =
−𝜔𝑖
𝑘
with 𝜔 = √𝑔𝑘,
𝑒 𝑘𝑧 𝑒 −𝑖𝜔𝑡 (𝐴𝑒 𝑖𝑘𝑥 − 𝐵𝑒 −𝑖𝑘𝑥 ),
(3.2a,b)
(3.3)
36
respectively, where z=η(x, t) is the elevation, k is the spatial frequency or wave number ω
is the radian (or angular) frequency and φ is the velocity potential.
By introducing R, the complex reflection coefficient, and assuming the velocity potential
is constant through depth, equations (3.1) and (3.2) may be rewritten as:
𝜂(𝑥, 𝑡) = 𝐴𝑒 −𝑖𝜔𝑡 (𝑒 𝑖𝑘𝑥 + 𝑅𝑒 −𝑖𝑘𝑥 ),
φ(𝑥, 𝑡) =
−𝐴𝜔𝑖
𝑘
𝑒 −𝑖𝜔𝑡 (𝑒 𝑖𝑘𝑥 + 𝑅𝑒 −𝑖𝑘𝑥 ).
(3.4)
(3.5)
The vertical displacement of tube n, while all other tubes are fixed, can be expressed
through the simplification of equation (3.4) to be:
𝑧𝑛 = 𝜂 = 𝐴[𝑅𝑒(𝑒 −𝑖𝜔𝑡 )].
(3.6)
Through the method of superposition, the total velocity potential of the system can be
determined as the sum of the scattering potential φ(s), and the N+1 radiation potentials
φ(n):
(n)
φ(x, t) = A φ(s)(x,t) + ∑𝑁+1
𝑛=1 𝐴𝑛 φ (x,t),
(3.7)
where A is the amplitude of the wave and An is the complex amplitude of node n. This
represents the first N+1 unknowns required toward solving the system. The derivation of
the radiation and the scattering potential equations will not be discussed here but are
rigorously derived for shallow water theory through [53].
Total Pressure
When the tubes are sitting at equilibrium and there are no waves present, the static
pressure is equivalent to the weight of water displaced by the tube:
37
𝑝𝑠𝑡𝑎𝑡𝑖𝑐 = 𝜌𝑔𝐷,
(3.8)
where ρ is the density of seawater, g is gravity and D is the draft of the tube. The draft is
the vertical distance from the waterline to the keel, or bottom of hull, with the thickness
of the hull included.
When waves are present, the total pressure of the system must change to include the
dynamic pressure (which is directly proportional to the velocity potential φ) and the
hydrostatic pressure (which is directly proportional to the vertical displacement of the
center of mass of the tube zn), as well as the static pressure:
𝑝𝑡𝑜𝑡𝑎𝑙 = 𝑖𝜌𝜔𝛷𝑛 − 𝜌𝑔𝑧𝑛 + 𝜌𝑔𝐷.
(3.9)
The pressure equation used from here on will be the total pressure minus the static
pressure:
𝑝𝑛 = 𝑖𝜌𝜔𝛷𝑛 − 𝜌𝑔𝑧𝑛 .
(3.10)
Forces and Moments
The vertical hydrodynamic force on tube n can be written in terms of the dynamic and
hydrostatic pressure:
𝑥
𝐹𝑛 = ∫𝑥 𝑛 𝑝𝑛 𝑑𝑥.
𝑛−1
(3.11)
Similarly, the hydrodynamic moment about the midpoint of tube n can also be written in
terms of the dynamic and hydrostatic pressure:
𝑥
𝑀𝑛 = ∫𝑥 𝑛 𝑝𝑛 [𝑥 − (𝑥𝑛 −
𝑛−1
𝐿𝑛
2
)] 𝑑𝑥,
(3.12)
38
where Ln is the length of the nth tube. Now the solution requires the N-1 nodal (hinge)
reaction forces Rn that are exerted upon tube n. The force Fn and moment Mn exerted on
tube n and the reaction force Rn exerted on node n are shown on Figure 3.13 below.
Figure 3.13 Force, Moment, Reaction Force Labeling
The vertical momentum equation for tube n can be written in terms of these forces:
1
−𝜔2 (𝑚𝑎𝑠𝑠𝑛 ) (2) (𝐴𝑛 + 𝐴𝑛+1 ) = 𝑅𝑛 − 𝑅𝑛+1 + 𝐹𝑛 ,
𝑚𝑎𝑠𝑠𝑛 = 𝜌𝐷𝐿𝑛 .
(3.13a,b)
The left side of this equation represents the momentum as the product of mass and
velocity where ω is the radian frequency, An is the amplitude of the node to the left of
tube n, and An+1 is the amplitude of the node to the right of tube n. The right side of the
equation represents the forces exerted on the tube where Rn is the reaction force at the
hinge to the left of tube n, Rn+1 is the hinge force to the right of tube n, and Fn is the
hydrodynamic force exerted upon tube n as determined from equation (3.11). Through
Archimedes Principle, the mass of tube n is equivalent to the weight of the fluid displaced
by tube n, as shown in (3.13b).
39
Moments and angles are positive if counterclockwise and negative if clockwise, as is
consistent with the right hand rule. The moment created by tube n-1 on tube n through
node n can be expressed as:
𝑀 = ᾱ𝑛 (𝑖𝜔)(𝛩𝑛 − 𝛩𝑛−1 )𝑒 −𝑖𝜔𝑡 ,
(3.14)
where Θn is the angular displacement and ᾱn is the extraction rate between tubes. This
extraction rate is equivalent to the amount of radiation damping in the Global x-direction
or alternatively stated as: the amount of energy extracted is equivalent to the reduction of
the wave height along the x-axis.
The moment that the forces Rn and Rn+1 create on the center of tube n can be expressed
as:
𝑀=−
𝐿𝑛
2
(𝑅𝑛 + 𝑅𝑛+1 ).
(3.15)
As well as having an equation for the hydrodynamic force, an equation for the angular
momentum for tube n can be generated.
−𝐼𝑛 𝜔2 𝛩𝑛 = −
𝐿𝑛
(𝑅 + 𝑅𝑛+1 ) − ᾱ𝑛 (−𝑖𝜔)(𝛩𝑛 − 𝛩𝑛−1 ) +
2 𝑛
ᾱ𝑛+1 (−𝑖𝜔)(𝛩𝑛+1 − 𝛩𝑛 ) + 𝑀𝑛 .
(3.16)
The left side of this equation represents the angular momentum as the product of the
moment of inertia In, angular frequency ω, and angular displacement Θn of tube n. The
40
right side of the equation represents the different moments created about the center of the
tube from equations (3.12, 3.14, 3.15).
Figure 3.14 Angle, Nodal Amplitude and Tube Amplitude Numbering
Through simple geometry, as seen in Figure 3.14, the following relation can relate
amplitude of tube n’s center of mass (midpoint of tube n), Bn, to the nodal amplitudes An:
𝐵𝑛 =
𝐴𝑛+1 −𝐴𝑛
2
.
(3.17)
Equations (3.13a) and (3.16) represent algebraic equations for the 2N unknowns An
(nodal displacements) and Rn (reaction forces at hinge) which can be solved via
computer. Once they are obtained, the angular displacement Θn can be found from the
following relation:
𝛩𝑛 =
𝐴𝑛+1 −𝐴𝑛
𝐿𝑛
𝑒 −𝑖𝜔𝑡 =
2𝐵𝑛
𝐿𝑛
𝑒 −𝑖𝜔𝑡 .
The mechanical energy extracted from the system can be calculated through the
extraction rates ᾱn, and either the nodal amplitude An, the tube amplitude Bn, or the
angular displacement Θn.
(3.18)
41
Power Extraction
In rotational systems, the power is derived as the product of angular velocity and torque,
or in our case the moment as described through equation (3.14).
𝑃 = 𝜔𝜏 = 𝜔 [ᾱ𝑛 (𝑖𝜔)(𝛩𝑛 − 𝛩𝑛−1 )𝑒 −𝑖𝜔𝑡 ].
(3.19)
Thus the power absorbed by at hinge n is the product of the extraction rate at node n and
the angular displacement of tubes n and n-1. The time averaged power extraction was
determined by [54, 55] and is the product of the angular displacement squared divided by
two, the extraction rate and the real portion of the angular rotation squared:
𝜔2
𝑃𝑛 =
2
2
∑𝑁
𝑛=2 ᾱ𝑛 |Θ𝑛−1 − Θ𝑛 | .
(3.20)
Combining equations (3.18) and (3.20), an alternative power equation is:
𝑃𝑛 =
𝜔2
2
𝐴𝑛 −𝐴𝑛−1
∑𝑁
𝑛=2 ᾱ𝑛 |
𝐿𝑛−1
−
𝐴𝑛+1 −𝐴𝑛
𝐿𝑛
|
2
=
𝜔2
2
2𝐵𝑛−1
∑𝑁
𝑛=2 ᾱ𝑛 |
𝐿𝑛−1
−
2𝐵𝑛
𝐿𝑛
| 2.
(3.21)
Efficiency of the System
The overall efficiency of the system at a particular frequency is defined as the mean
power extracted at a node from equation (3.20) divided by the total power of the system
as defined in equation (2.29):
𝑒𝑓𝑓 =
𝑃𝑛
𝑃𝑡
=
2
𝜔 2 ∑𝑁
𝑛=2 ᾱ𝑛 |Θ𝑛−1 −Θ𝑛 |
𝑔
𝑘
𝜌𝑔𝑎2 √
,
(3.22)
or also through equations (2.30) and (3.21):
𝑒𝑓𝑓 =
𝑃𝑛
𝑃𝑡
𝜔3
2𝐵
𝑛−1
= 𝜌𝑔2 𝑎2 ∑𝑁
−
𝑛=2 ᾱ𝑛 | 𝐿
𝑛−1
2𝐵𝑛
𝐿𝑛
| 2.
(3.23)
42
CHAPTER 4
COMPUTATIONAL MODEL SETUP AND PROCEDURE
Computational modeling is vital to design research by providing quick and inexpensive
evaluations and optimizations of designs by adjusting the parameters of the system in the
computer, and studying the differences in the outcome of the simulations. The amount of
time required to solve the system of equations via mathematical analytical solution is
exponentially high, even with the use of computer software this process is time
consuming as a single simulation can take upwards of 30 minutes to run. The Pelamis
Wave Power Corporation conducts their own in-house simulations using their own
proprietary software to analyze every aspect of the structure in both linear and non-linear
hydrodynamics. Yet, limited data from computational modeling has been published or
made available to the public. Perhaps this is due to the proprietary nature of their
software and the WEC itself.
4.1 THE CONTROL MODEL
As with any experiment, there needs to be a control, or benchmark, with which to
compare all data. For this thesis, the control model is the Pelamis P2 currently in
production off the west coast of Scotland. As previously stated, this model has an overall
length of 180 m, a diameter of 4 m and a weight of 1350 tonnes. It consists of five tubular
sections, all with an equal length of 36 m; the joint between subsequent tubes allows for
motion in two degrees of freedom: left-right, up-down and the rotation about the y and zaxes, or in nautical terms, sway, heave, pitch and yaw. Restricting the Pelamis P2 motion
43
to the 2D case in the x-z-plane, it experiences motion in two degrees of freedom: heave
and pitch. Pelamis Wave Power states the Pelamis P2 model experiences efficiencies
around 70% in all sea states. [56]
Figure 4.1 Pelamis Model Head Section
From inside ANSYS Workbench 14.5, the Analysis System, Hydrodynamic Diffraction
is created as a standalone system, next the Analysis System, Hydrodynamic Time
Response was created and linked to the Geometry and Model of the Hydrodynamic
Diffraction system as seen through Figure 4.2. The Geometry was created by use of the
DesignModeler software to create the tubes for the Pelamis P2 model. The first tube will
be referred to as the head, the middle tube sections will be referred to as the body tubes
and the final tube will be referred to as the tail.
44
Figure 4.2 Hydrodynamic System Linking
The head is divided into three separate segments to match the contours of the actual
device. The nose portion has a length of 5 m, and tapers from a diameter of 4 m to a
diameter of 1 m. The middle portion is a cylinder 29 m long and 4 m in diameter. The
end portion has a length of 2 m and tapers from a diameter of 4 m to a diameter of 2 m,
with extrusion cuts into the solid to mimic the profile of the actual device, this can be
seen in Figure 4.3.
(a)
(b)
Figure 4.3 Comparison View of (a) Model to (b) Actual Device
The body tubes are divided into two segments, the first portion is a cylinder with a length
of 34 m and a diameter of 4 m, the end portion is identical to that from the head tube. The
tail is also divided into two sections with the first portion being a cylinder of length 34 m
45
with diameter 4 m and the second portion having a length of 2 m and tapers from a
diameter of 4 m to a diameter of 2 m, without any extrusion cuts. The control model
tubes were aligned, then the Thickness of each tube was set to zero which creates the
surface body (this is a crucial step for the software). Finally the Freeze and Slice tools
were implemented to split each tube along the XY-Plane for the waterline and the two
halves were combined into a single part for each tube. The model can be seen in Figure
4.3.
Figure 4.4 Control Model
Upon completing the Geometry in DesignModeler, the Model was edited through the
Hydrodynamic Time Response (ANSYS AQWA-HYDRO-DIFFRACT). The details of
the geometry were established, starting with the point mass for all five tubes being set to
270,000 kg so the entire structure has a weight of 1350 tonnes. Next, connection points
were created on all of the tubes, three on the head tube for two moorings and one joint,
two on either end of the body tubes for joints, and one on one end of the tail tube for a
joint. An anchor point was created on the floor of the ocean to complete the details of the
geometry. Next all the connections were created. Four universal joints were established
for the nodes between tubes, attached to the previously created connection points and
aligned properly. The cables were then created as a slack, chain catenary system for the
46
moorings. The structure then needed to be meshed; the maximum element size was set to
2 m with defeaturing tolerance set to 1 m (for the six and eight tube models with higher
diameters these numbers were changed to 1.5 and 0.8 respectively). It will be shown
through Section 5.5 that this change is acceptable; it has an insignificant impact on
system efficiency.
4.2 HYDRODYNAMIC DIFFRACTION
Still using the Hydrodynamic Time Response (ANSYS AQWA-HYDRO-DIFFRACT),
under Analysis Settings, Ignore Modeling Rule Violations needed to be set to “Yes” in
order to eliminate errors with mesh tolerances. Under Structure Selection, all of the
structures must be included, the Interacting Structure Groups must be set to “All”, and the
Structure Ordering must be properly arranged. Under Wave Directions, the Type was set
to single direction, forward speed as our analysis is being restricted to the 2D case, and
the Wave Direction was chosen as 180o to follow the x-axis. Under Wave Frequency, the
Range was set manually from a lowest frequency of 0.05 Hz to a high of 0.3 Hz with 34
intermediate values.
The entire setup can be more easily followed through this flow chart:

ANSYS Workbench 14.5
o Create Standalone Hydrodynamic Diffraction System
o Link Hydrodynamic Time Response to Hydrodynamic Diffraction as seen
through Figure 4.2
47
o Launch Geometry under Hydrodynamic Diffraction, this opens DesignModeler

Build Model to specifications
o Launch Model under Hydrodynamic Diffraction, this opens Hydrodynamic Time
Response


Under the Details for Geometry:

Specify Point Masses for all tubes

Create Connection Points

Create Anchor
Under the Details for Connections:

Create Universal Joints and attach to Connections Points

Align Joints properly

Create Cables and attach to Connection Points

Create the Mesh

Under Details for Hydrodynamic Diffractions

Under Analysis System set Ignore Modeling Rule Violations to “Yes”

Under Structure Selection
o Properly order tubes
o Set Interacting Structure Groups to “All”

Under Wave Direction
o Set Type to single direction, forward speed
o Set wave direction to 180o

Under Wave Frequency manually set the Range
48
4.3 THE TEST MODELS
In an attempt to standardize the amount of raw materials required for each model, the
overall length and weight will be maintained at the values for the current P2 model in
production off the west coast of Scotland which are 180 m and 1350 tonnes, respectively.
The first set of test models varies the number of tubes for each structure while
maintaining a diameter of 4 m. This set contains models of three tubes at a length of 60 m
each (T3L60D4), four tubes at length 45 m (T4L45D4), the control model of five tubes at
length 36 m (T5L36D4), six tubes at length 30 m (T6L30D4) and eight tubes at length
22.5 m (T8L22.5D4); a model containing seven tubes was omitted since the 180 m length
could not be evenly divided. Summary of Test Set One Models is listed in Table 4.1.
Table 4.1: Test Set One Models
Number of Tubes
3
4
5
6
8
Tube Length (m)
60
45
36
30
22.5
Tube Diameter (m)
4
4
4
4
4
Model Number
T3L60D4
T4L45D4
T5L36D4 (Control)
T6L30D4
T8L22.5D4
The second set of test models uses the same number of tubes and lengths as the first set,
but varies the diameter of the tubes above and below the diameter of the control model.
The models for set two are: T3L60D3, T4L45D3, T5L36D3, T6L30D3, T8L22.5D3,
T3L60D5, T4L45D5, T5L36D5, T6L30D5, T8L22.5D5, T6L30D6 and T8L22.5D6, with
details listed in Table 4.2 below.
49
Table 4.2: Test Set Two Models
Number of Tubes
3
4
5
6
8
Tube Length (m)
60
45
36
30
22.5
Tube Diameter (m)
3
3
3
3
3
Model Number
T3L60D3
T4L45D3
T5L36D3
T6L30D3
T8L22.5D3
3
4
5
6
8
60
45
36
30
22.5
5
5
5
5
5
T3L60D5
T4L45D5
T5L36D5
T6L30D5
T8L22.5D5
The third set of models take the most efficient model from sets one and two, which is the
T6L30D5 model, and determines if altering the length of each tube affects the efficiency.
The set will contain one model where the tubes increase in length
(L1<L2<L3<L4<L5<L6) and one model where the tubes decrease in length
(L6<L5<L4<L3<L2<L1). The ratio of 1:2:3:4:5:6, totaling 21 units, did not divide
12
evenly into the total length of the structure (180m) but resulted in 821, this remainder of
12 was split evenly amongst the six tubes. Thus, the lengths of each tube, in meters, will
be 10:18:26:34:42:50 and 50:42:34:26:18:10. This third set of models also contains one
model where the end tubes are longer than the center tubes (L1>L2>L3=L4<L5<L6) and
one model where the middle tubes are longer than the end tubes
(L1<L2<L3=L4>L5>L6). The ratio of 1:2:3:3:2:1, totaling 12 units did divide evenly
into the total length, resulting in the lengths of each tube to be, in meters,
50
15:30:45:45:30:15 and 45:30:15:15:30:45. These models will be labeled: T6D5
Increasing, T6D5 Decreasing, T6D5 SBS and T6D5 BSB, as summarized in Table 4.3.
Table 4.3: Test Set Three Models
Number of Tubes
6
6
6
6
Tube Length (m)
10:18:26:34:42:50
50:42:34:26:18:10
15:30:45:45:30:15
45:30:15:15:30:45
Tube Diameter (m)
5
5
5
5
Model Number
T6D5 Increasing
T6D5 Decreasing
T6D5 SBS
T6D5 BSB
A summary of all test models is provided in Table 4.4 below.
Table 4.4: Test Model Summary
Test Set One
Test Set Two
Number
of Tubes
3
4
5
6
8
60
45
36
30
22.5
Tube
Diameter (m)
4
4
4
4
4
T3L60D4
T4L45D4
T5L36D4
T6L30D4
T8L22.5D4
3
4
5
6
8
60
45
36
30
22.5
3
3
3
3
3
T3L60D3
T4L45D3
T5L36D3
T6L30D3
T8L22.5D3
3
4
5
6
8
60
45
36
30
22.5
5
5
5
5
5
T3L60D5
T4L45D5
T5L36D5
T6L30D5
T8L22.5D5
10:18:26:34:42:50
50:42:34:26:18:10
15:30:45:45:30:15
45:30:15:15:30:45
5
5
5
5
T6D5 Increasing
T6D5 Decreasing
T6D5 SBS
T6D5 BSB
6
6
Test Set Three
6
6
Tube Length (m)
Model Number
51
4.4 WAVE DATA
The European Marine Energy Center (EMEC) compiled the wave data from three
datawell directional waverider buoys [56]. The EMEC determines the instantaneous data
for the maximum wave height, the significant wave height, the maximum wave period,
and the significant wave period, as well as provides a graph for a 24 hour time period, as
seen in Figure 4.5 below. This data can be used to determine the current energy
production of the device. However this site does not provide the annual data to the
general public. As the Pelamis P2 is located off the west coast of Orkney, Scotland [57],
according to the Scottish Government website [58, 59] the annual mean wave height for
the region is 2.0-2.4 m, therefore from here on out a significant wave height of 2.2 m,
resulting in amplitude, A, of 1.1 m, will be used in the calculations.
Figure 4.5 EMEC Wave Height Data [56]
52
CHAPTER 5
ANALYSIS OF THE DATA
5.1 INPUT PARAMETERS
For all computational test set, the models’ efficiency was determined by equation (3.23).
The summary of input parameters for this equation is displayed below in Table 5.1. The
extractions rates ᾱn were determined from the ANSYS AQWA software via the Radiation
Damping, Global X Solution with the Subtype specified as Global X, and the component
specified as the Global RY. As the software only allows us to select a structure, not the
joint between structures, the extraction rate for node n is equivalent to the radiation
damping of tube n-1. For example, ᾱ2 corresponds to tube 1, the Head. This is consistent
with ocean engineering literature wherein the effective extraction rate (at a node) must
equal the effective radiation damping rate (of the tube just prior to that node) [54, 55, 61].
The ratios of amplitude of the center of mass for tube n (midpoint of the tube) to the
length of tube n, Bn/Ln, are also determined from the ANSYS AQWA software via the
RAOs (Response Amplitude Operators); the results are generated in terms of m/m. A
large enough range was selected for the frequency f to encompass the range of efficient
energy capture for all models.
Table 5.1: Input Parameters
Parameter :
Density, ρ
Gravity, g
Amplitude, a
Frequency, f
Extraction rate, ᾱn
Tube Amplitude, Bn/Ln
Value(s):
1023.485
9.80665
1.1
0.05 – 0.30
Program generated
Program generated
Units:
kg/m3
m/s2
m
Hz
Nm/(m/s)
m/m
53
5.2 TEST SET ONE
The peak efficiencies for each model, along with the corresponding frequencies are
provided in Table 5.2. The efficiency of the actual Pelamis P2 model has been stated to
be around 70%. The maximum efficiency for the control model is 73.27%, which is
within reason; this is the theoretical maximum of the device and does not account for
additional efficiency losses generated by the conversion of mechanical work to electrical
work. With the exception of model T8L22.5D4, the efficiency for every other model was
lower than that of the control. As shown in Table 5.2 and Figure 5.1, as the tube length
increases (decrease in tube number), the efficiency of the system drastically decreased.
The results do not show any correlation for the decrease in tube length (increase in tube
number) and efficiency; the efficiency of T6L30D4 is slightly lower than the Control,
while the efficiency of T8L22.5D4 is slightly higher than the Control (by 1.88%). Also of
note, is that the frequency corresponding to peak efficiency gradually increases for the
first four models, and decreases slightly for the last model. As is evident through Figure
5.1, the range of frequency for available energy capture is widest for models T4L45D4,
T5L36D4, and T6L30D4.
Table 5.2: Test Set One Maximum Efficiencies
Efficiency (%)
Frequency (Hz)
*
T3L60D4
24.93
0.121
Recall T5L36D4 is the Control Model.
T4L45D4
50.93
0.143
T5L36D41*
73.27
0.150
T6L30D4
72.51
0.157
T8L22.5D4
75.14
0.150
54
Efficiency
Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
T3L60D4
T4L45D4
T5L36D4
T6L30D4
T8L22.5D4
Frequency
Figure 5.1: Test Set One-4 m Diameter Efficiency Curves
5.3 TEST SET TWO
The peak efficiencies for each model, along with the corresponding frequencies are
provided in Table 5.3. For the set of models with a 3 m diameter, there is a direct
correlation between the tube length and the maximum efficiency; as the tube length
decreases (increase in tube number) the efficiency increases. Yet all these values are
significantly lower than those seen in Test Set One. Observing Table 5.3 and Figure 5.2,
there is no direct correlation between tube length and the frequency where maximum
efficiency occurs; T4L45D3, T5L36D3, and T6L30D3 occur at the same frequency,
while T3L60D3 and T8L22.5D3 occur the same (but higher) frequency. For the set of
models with a 5 m diameter the model with tube length of 30 m (six tubes) has the
highest efficiency. It is interesting to note that the efficiencies for T3L60D5, T4L45D5,
and T5L36D5 are lower than their counterparts in Test Set One, yet the efficiencies for
55
T6L30D5 and T8L22.5D5 are higher than in Test Set One, by 6.39% and 2.80%
respectively. Therefore, additional models are created for tube lengths of 30 m and 22.5
m to have a diameter of 6 m; these efficiencies are lower than for the 5 m diameter
models. From Table 5.3 and Figure 5.3, it is observed that as tube length decreases (tube
number increases) the frequency where maximum efficiency occurs increases for the first
three models then holds at the same value.
Table 5.3: Test Set Two Maximum Efficiencies
Efficiency (%)
Frequency
(Hz)
Efficiency (%)
Frequency
(Hz)
Efficiency (%)
Frequency
(Hz)
T3L60D3
24.39
0.163
T4L45D3
49.76
0.143
T5L36D3
52.19
0.143
T6L30D3
56.93
0.143
T8L22.5D3
58.66
0.163
T3L60D5
25.12
0.129
T4L45D5
46.45
0.150
T5L36D5
67.06
0.157
T6L30D5
78.90
0.157
T8L22.5D5
77.94
0.157
----
----
----
T6L30D6
75.06
0.164
T8L22.5D6
72.86
0.164
56
Efficiency
Efficiency vs. Frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
T3L60D3
T4L45D3
T5L36D3
T6L30D3
T8L22.5D3
Frequency
Figure 5.2: Test Set Two- 3 m Diameter Efficiency Curves
T3L60D5
T4L45D5
T5L36D5
T6L30D5
0.221
0.207
0.193
0.179
0.164
0.15
0.136
0.121
0.107
0.093
0.079
T8L22.5D5
0.064
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
0.05
Efficiency
Efficiency vs. Frequency
Frequency
Figure 5.3: Test Set Two- 5 m Diameter Efficiency Curves
Figures 5.4 through 5.8 display the direct comparison of different diameters for each tube
length. For tube lengths of 60 m (three tubes), 45 m (four tubes), and 36 m (five tubes), a
57
diameter for 4 m is most efficient. For tube lengths of 30 m (six tubes) and 22.5 m (eight
tubes) a diameter of 5 m is most efficient.
Efficiency
T3 Efficiency vs. Frequency
0.3
0.25
0.2
0.15
0.1
0.05
0
T3L60D3
T3L60D4
T3L60D5
Frequency
Figure 5.4: T3 Efficiency Comparison
T4 Efficiency vs. Frequency
0.6
0.5
Efficiency
0.4
0.3
T4L45D3
0.2
T4L45D4
0.1
T4L45D5
0
-0.1
Frequency
Figure 5.5: T4 Efficiency Comparison
58
Efficiency
T5 Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
T5L36D3
T5L36D4
T5L36D5
Frequency
Figure 5.6: T5 Efficiency Comparison
Efficiency
T6 Efficiency vs. Frequency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
T6L30D3
T6L30D4
T6L30D5
T6L30D6
Frequency
Figure 5.7: T6 Efficiency Comparison
59
T8 Efficiency vs. Frequency
1
Efficiency
0.8
0.6
T8L22.5D3
0.4
T8L22.5D4
0.2
T8L22.5D5
0
T8L22.5D6
-0.2
Frequency
Figure 5.8: T8 Efficiency Comparison
5.4 TEST SET THREE
Due to having the highest efficiency, the six tube model with a diameter of five meters
was selected for Test Set Three. As shown in Table 4.4-1 and Figure 4.4-1 in all four
cases, the maximum efficiency was greater than that of constant tube length. The model
consisting of the middle tubes being longer than the end tubes had the highest efficiency
at 94.72%. The model increasing in tube lengths was the second highest, followed closely
by the model consisting of the end tubes being longer than the middle tubes. The model
decreasing in tube length was the fourth most efficiency and only slightly more efficient
than that of consistent tube length. The two models with the highest efficiency occur at
the same frequency and the two with the lower efficiencies occur at the same frequency,
yet all four models experienced peak efficiencies at higher frequencies and the model of
consistent tube lengths.
60
Table 5.4: Test Set Three Maximum Efficiencies
T6L36D5
Efficiency (%)
Frequency
(Hz)
78.90
0.157
T6D5
Increasing
85.41
0.164
T6D5
Decreasing
79.81
0.171
T6D5
SBS
94.72
0.164
T6D5
BSB
85.10
0.171
Efficiency vs. Frequency
1
Efficiency
0.8
Increasing
0.6
Decreasing
0.4
SBS
0.2
BSB
T6L30D5
0
-0.2
Frequency
Figure 5.9: Test Set Three-T6D5 Efficiency Curves
5.5 VALIDATION
Since the ANSYS Program dictated the mesh maximum element size and defeaturing
tolerance be changed for the six and eight tube models, the Control Model (T5L36D4)
was selected to determine if varying the mesh selection effects the efficiency of the
system. It must be stated here that the ANSYS Program demands the defeaturing
tolerance be no greater than 0.6 times the maximum element size. Through Table 5.5, it
can be observed that the efficiencies are all within 2.5% of one another, and the amount
61
of difference depends upon the difference between the maximum element size and the
defeaturing tolerance; the closer these values are, the higher the change in efficiency.
From Figure 5.10, the shape of the curves are consistent, they all appear to overlap one
another. As the curves did not change, and 2.5% is such a small value, it is concluded that
altering the mesh does not affect the efficiencies.
Table 5.5: Altering Mesh Size
Efficiency (%)
Frequency
(Hz)
T5L36D4 element 1.5
(Control) tolerance 0.6
73.27
71.15
0.150
0.150
element 1.5
tolerance 0.8
70.84
0.150
element 1.0
tolerance 0.5
71.84
0.150
element 0.8
tolerance 0.3
72.05
0.150
Efficiency
Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
T5L36D4
E1.5 T0.6
E1.5 T0.8
E1.0 T0.5
E0.8 T0.3
Frequency
Figure 5.10: Altering Mesh Size Efficiencies
All the other parameters, such as tube mass, density, gravity, structure ordering, and wave
direction are fixed; the frequency has already been varied. Any added drag or mass
factors would change the dynamics of the system. Thus there are no other validation tests
to be made.
62
CHAPTER 6
FINDINGS AND INTERPRETATIONS
6.1 RESONANCE
All peak efficiencies correspond to the structure being in resonance with the wave
frequency. Resonance is defined as the system oscillating with a larger amplitude at some
frequencies rather than others. When a system experiences relative maximum amplitudes,
specifically when the wave period agrees with the natural period of the structure, the
corresponding frequency is referred to as the system’s resonant frequency. At resonance,
the system is able to store and easily convert energy; at periods off resonance, this
conversion is less adequate. Thusly the oscillating system has a frequency-dependent
response.
6.2 INTERPRETING THE RESULTS
It is stipulated here that the efficiencies observed are theoretical maximums for the
various structures; they only include the conversion of potential wave energy to
mechanical energy, and not the conversion of mechanical energy to electrical energy.
The conclusion from section 4.2 is, given the restrictions of maintaining the overall
length and weight of the structure, the efficiencies are highest with a smaller tube length
(larger number of tubes). The values obtained for models T5L36D4, T6L30D4, and
T8L22.5D4 were within a 2.63% efficiency range of one another. Based strictly on
overall efficiency, the T8L22.5D4 model would be the best choice from this test set at
63
1.88% more efficient than the Control model. However, devices with more moving parts
require more maintenance and can experience a greater loss in efficiency due to friction
when converting the mechanical energy to electrical energy, therefore it may not be cost
effective from this perspective to increase the amount of moving parts associated with
additional nodes while only achieving a 1.88% increase in efficiency from the Control
model.
Section 4.3 demonstrated the effects of altering the diameter of the structures. In all
cases, reducing the diameter reduced the efficiency of the structure. By making the
structure narrower it slices through the oncoming waves rather than the wave lifting the
tubes to create sufficient angular rotation about the hinges. The difference in pressure
exerted on the structures’ hulls can be visually compared through the Interpolated
Pressure Figures in Appendix B. The narrower models experience less hull pressure from
the water, thus through Equation (41) creates less of a hydrodynamic moment and
therefore absorb less energy. This is the same principle used when designing oceangoing
vessels for speed.
For the larger tube lengths (smaller tube number), an increase in diameter reduced the
efficiency of the system. While this appears to defy Equation (41), it is theorized here
that the first tube(s) did experience a higher vertical displacement, in accordance with
Equation (41), but then experienced an effect called “slamming” [62], wherein the tubes
experienced a large force upon impact with the water surface. This slamming would then
64
result in the generation of a large amount of radiation waves, thereby taking the structure
out of resonance and decreasing its overall efficiency. Also of note, slamming would
create a larger load on the structure and require increased engineering for the hinges and
hulls. For the smaller tube length models T6L30D5 and T8L22.5D5, an increase in
diameter increased the efficiency. Again, equation (41) can attribute that an increase in
diameter increases the dynamic pressure resulting in a larger hydrodynamic moment and
an increase in energy absorption. Further increase in the diameter resulted in a decrease
in efficiency, again it is theorized that slamming occurred and effected overall efficiency.
Therefore it is observed that for a given tube length, there is an optimum tube diameter.
Based solely on maximum efficiency, from Test Sets One and Two, model T6L30D5 was
the highest performer and will be used for Test Set Three. From Section 4.4 all models of
varying tube lengths were more efficient than the constant length model. The T6D5 SBS
model was the highest performer, followed by the T6D5 Increasing model, which in turn
was closely followed by the T6D5 BSB model, the T6D5 Decreasing model was only
slightly more efficient than the T6L30D5 model.
While the short end tubes of the T6D5 SBS model have a low damping value, the middle
tubes act as a sort of vertical mooring structure that allow for these shorter end tubes to
generate large vertical elevation which increase the angular rotation at the hinges
resulting in large power generation and a high efficiency. The T6D5 BSB model operates
on an opposing principle; there is an even balance of high damping and low elevation
65
with low damping and high elevation. The large end tubes create a large amount of
damping and experience slight elevation, while the smaller tubes connected to them are
able to increase their elevation and angular rotation resulting in high power generation
and efficiency values. The T6D5 Decreasing model has its largest tubes at the head of the
structure; this causes a greater amount of wave damping, leaving less wave height
available for energy extraction at later hinges. This can be observed by the ᾱns decreasing
as n increases. The very opposite of this principle leads to the T6D5 Increasing model
absorbing a larger amount of energy, as the smaller front tubes create less wave damping
and hence leave more of the wave to be extracted at later hinges. This can be observed by
the ᾱns increasing as n increases. Also noticeable from this test set is the maximum
efficiencies occurred at higher frequencies than for the uniform length model. Both the
T6D5 SBS and T6D5 Increasing models experienced peak efficiency at a slightly higher
frequency of 0.164 Hz, while the T6D5 BSB and T6D5 Decreasing models were at an
even higher frequency of 0.171 Hz. The general principles for Test Set Three can be
applied to all tube models.
Thusly, out of all three test sets, the best model simulation based solely on maximum
efficiency would be, from Test Set Three, having six tubes, a diameter of five meters, and
the tubes varying in size, with the middle tubes longer than the end tubes
(L1<L2<L3=L4>L5>L6). However, selecting a model based on overall maximum
efficiency is not always the best decision. It is important when creating the model to do
66
so with the characteristics of the environment in mind, specifically here, what frequency
range is experienced at the proposed location.
6.3 FREQUENCY
Ocean locations have energy distributed over a range of wave heights and periods, or
frequencies. This range varies from location to location and from season to season, the
latter is called the ‘spectral bandwidth’. The performance of the structure is dependent on
incident wave frequency as most wave energy converters are only able to capture energy
over a finite range of wave frequencies.
As shown through the Figures in Chapter 3 the peaks in hydrodynamics occurred at
certain frequencies from the incident waves and sufficient energy extraction only
occurred over certain frequency ranges. The ocean state off the coast of Scotland where
the P2 model resides experiences a mean wave period between 8.1s in winter and 6.3s in
summer [58] which corresponds to frequencies of 0.123 Hz and 0.159, respectively. With
this in mind, the optimum design of the Pelamis P2 for this site, needs to capture the most
energy over that entire range.
While all models (diameters 3m to 5m) had their peak efficiency within that range, the
efficiency of the three-meter diameter models was too small for these devices to be
considered viable and when the diameter was increased, the frequency increased as well
to be on the edge of the frequency range observed off the coast of Scotland. The six-
67
meter diameter models were out of the range altogether. Therefore, the four-meter
diameter models are the best fit for these constraints. Of these five models, the T3L60D4
and T4L45D4 models produced significantly less efficiencies and will be excluded from
consideration. While the T8D22.5L4 model has the higher efficiency, it also has a
narrower frequency range, would not produce any viable energy during the winter
months, and will therefore be excluded from consideration. Between the remaining
models, T5L36D4 and T6L30D4, the Control model has both the higher efficiency and
absorbs viable energy over greater range that includes the frequencies off the coast of
Scotland. Combining this with the results from Test Set Three, the optimum design of the
Pelamis P2 WEC to be deployed off the coast of Scotland would be, given the restrictions
of 180 m overall length and 1350 tonnes overall weight, a T5D4 SBS model, or rather a
device containing five tubes with four-meter diameters whose tube lengths increase then
decrease (L1<L2<L3>L4>L5).
6.4 FUTURE
Pelamis WECs are deployed in farms containing several of these devices, however when
the Pelamis WECs extract energy from the waves, they reduce but not eliminate the
waves altogether. Some of these waves will continue towards the shore, where the
addition of various near-shore WECs could improve the overall energy efficiency of the
wave farm.
68
Also, just as Naval Architecture as continually redesigned the hull of a ship to improve
slamming and increase the vessel’s speed, so too can the Pelamis’ tube hulls be
redesigned to increase the dynamic pressure exerted upon them, thereby increasing the
hydrodynamic moment and increase device performance.
Wave Energy is still a new and developing industry, while the Pelamis is one of the few
devices currently commercially available, it will be exciting to see where this technology
will lead, or if an alternate device will emerge onto the market with higher success.
69
APPENDIX A
INDIVIDUAL EFFICIENCY CURVES
Efficiency vs. Frequency
0.3
Efficiency
0.25
0.2
0.15
0.1
T3L60D3
0.05
0
Frequency
Figure A.A-1: T3L60D3 Efficiency Curve
Efficiency vs. Frequency
0.6
Efficiency
0.5
0.4
0.3
0.2
T4L45D3
0.1
0
Frequency
Figure A.A-2: T4L45D3 Efficiency Curve
70
Efficiency vs. Frequency
0.6
Efficiency
0.5
0.4
0.3
0.2
T5L36D3
0.1
0
Frequency
Figure A.A-3: T5L36D3 Efficiency Curve
Efficiency vs. Frequency
0.6
Efficiency
0.5
0.4
0.3
0.2
T6L30D3
0.1
0
Frequency
Figure A.A-4: T6L30D3 Efficiency Curve
71
Efficiency
Efficiency vs. Frequency
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
T8L22.5D3
Frequency
Figure A.A-5: T8L22.5D3 Efficiency Curve
Efficiency vs. Frequency
0.3
Efficiency
0.25
0.2
0.15
0.1
T3L60D4
0.05
0
Frequency
Figure A.A-6: T3L60D4 Efficiency Curve
72
Efficiency vs. Frequency
0.6
Efficiency
0.5
0.4
0.3
0.2
T4L45D4
0.1
0
Frequency
Figure A.A-7: T4L45D4 Efficiency Curve
Efficiency
Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
T5L36D4
Frequency
Figure A.A-8: T5L36D4 (CONTROL) Efficiency Curve
73
Efficiency
Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
T6L30D4
Frequency
Figure A.A-9: T6L30D4 Efficiency Curve
Efficiency
Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
T8L22.5D4
Frequency
Figure A.A-10: T8L22.5D4 Efficiency Curve
74
Efficiency vs. Frequency
0.3
Efficiency
0.25
0.2
0.15
0.1
T3L60D5
0.05
0
Frequency
Figure A.A-11: T3L60D5 Efficiency Curve
Efficiency vs. Frequency
0.5
Efficiency
0.4
0.3
0.2
T4L45D5
0.1
0
-0.1
Frequency
Figure A.A-12: T4L45D5 Efficiency Curve
75
Efficiency
Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
T5L36D5
Frequency
Figure A.A-13: T5L36D5 Efficiency Curve
Efficiency
Efficiency vs. Frequency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
T6L30D5
Frequency
Figure A.A-14: T6L30D5 Efficiency Curve
76
Efficiency
Efficiency vs. Frequency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
T8L22.5D5
Frequency
Figure A.A-15: T8L22.5D5 Efficiency Curve
Efficiency
Efficiency vs. Frequency
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
T6L30D6
Frequency
Figure A.A-16: T6L30D6 Efficiency Curve
77
T8L22.5D6
0.8
0.7
0.6
0.5
0.4
T8L22.5D6
0.3
0.2
0.1
0
-0.1
Figure A.A-17: T8L22.5D6 Efficiency Curve
Efficiency
Efficiency vs. Frequency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
Increasing
Frequency
Figure A.A-18: T6D5 Increasing Efficiency Curve
78
Efficiency
Efficiency vs. Frequency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
Decreasing
Frequency
Figure A.A-19: T6D5 Decreasing Efficiency Curve
Efficiency vs. Frequency
1
Efficiency
0.8
0.6
0.4
Small Big Small
0.2
0
-0.2
Frequency
Figure A.A-20: T6D5 SBS Efficiency Curve
79
Efficiency
Efficiency vs. Frequency
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
Big Small Big
Frequency
Figure A.A-21: T6D5 BSB Efficiency Curve
80
APPENDIX B
INTERPOLATED PRESSURE
T3L60D3
T4L45D3
81
T5L36D3
T6L30D3
82
T8L22.5D3
T3L60D4
83
T4L45D4
T5L36D4 (Control)
84
T6L30D4
T8L22.5D4
85
T3L60D5
T4L45D5
86
T5L36D5
T6L30D5
87
T8L22.5D5
T6L36D6
88
T8L22.5D6
T6D4 Increasing
89
T6D4 Decreasing
T6D4 SBS
90
T6D4 BSB:
91
APPENDIX C
RESULTANT DISPLACEMENT
T3L60D3
T4L45D3
92
T5L36D3
T6L30D3
93
T8L22.5D3
T3L60D4
94
T4L45D4
T5L36D4 (Control)
95
T6L30D4
T8L22.5D4
96
T3L60D5
T4L45D5
97
T5L36D5
T6L30D5
98
T8L22.5D5
T6L36D6
99
T8L22.5D6
T6D4 Increasing
100
T6D4 Decreasing
T6D4 SBS
101
T6D4 BSB
102
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