Wave Energy Propulsion forPure Car and Truck Carriers(PCTCs)

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Wave Energy Propulsion for

Pure Car and Truck Carriers

(PCTCs)

Master thesis

by

Ludvig af Klinteberg

Supervisor: Mikael Huss, Wallenius Marine

Examiner: Anders Ros´

en, KTH Centre for

Naval Architecture

Stockholm, 2009

KTH Centre for Naval Architecture

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Abstract

Wave Energy Propulsion for Pure Car and Truck Carriers (PCTC’s) The development of ocean wave energy technology has in recent years seen a revival due to increased climate concerns and interest in sustainable en-ergy. This thesis investigates whether ocean wave energy could also be used for propulsion of commercial ships, with Pure Car and Truck Carriers (PCTC’s) being the model ship type used. Based on current wave energy research four technologies are selected as candidates for wave energy propul-sion: bow overtopping, thrust generating foils, moving multi-point absorber and turbine-fitted anti-roll tanks.

Analyses of the selected technologies indicate that the generated propulsive power does the overcome the added resistance from the system at the ship design speed and size used in the study. Conclusions are that further wave energy propulsion research should focus on systems for ships that are slower and smaller than current PCTC’s.

V˚agenergiframdrivning av biltransportfartyg (PCTC’s)

Utvecklingen av v˚agenergiteknik har p˚a senare ˚ar f˚att ett uppsving i sam-band med ökande klimatoro och intresse för förnyelsebar energi. Detta exam-ensarbete utreder huruvida v˚agenergi även skulle kunna användas till fram-drivning av kommersiella fartyg, och använder moderna biltransportfartyg (PCTC’s - Pure Car and Truck Carriers) som fartygstyp för utredningen. Med utg˚angspunkt i aktuell v˚agenergiforskning tas fyra potentiella tekniker för v˚agenergiframdrivning fram: ”overtopping” i fören, passiva fenor, ”mov-ing multi-point absorber” samt antirulln”mov-ingstankar med turbiner.

Analys av de valda teknikerna indikerar att den genererade framdrivande kraften blir mindre än systemets adderade framdrivningsmotst˚and vid den fartygshastighet och -längd som används. Slutsatserna är att framtida forskn-ing om v˚agenergiframdrivning borde fokusera p˚a fartyg som är mindre och l˚angsammare än dagens PCTC-fartyg.

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Introduction

This master thesis project has been carried out at Wallenius Marine in Stockholm, and the purpose of it has been to investigate the possibility of reducing the fuel consumption of Wallenius Lines’ Pure Car and Truck Carriers (PCTC’s) by extracting energy from sea waves.

Wallenius works with a strong environmental vision, which in 2005 was expressed in the conceptual emission-free E/S Orcelle. The Orcelle would be driven by fuel cells together with a combination of solar, wind and wave energy systems. This project has started off from this vision, in order to investigate if wave energy propulsion really is a future possibility for the shipping industry.

In the 1970’s and early 1980’s extensive research was made in wave energy conversion, including wave energy ship propulsion, motivated by high oil prices. Most projects however lost their funding when oil prices dropped again in the mid 1980’s. In the last few years there has been a revival in wave energy technology and several wave energy conversion methods are now ready for large-scale implementation.

This thesis summarises current wave energy research, with the purpose of identifying which techniques can be transferred onto a moving ship. Based on this summary a number of techniques have been chosen as possible candidates and investigated further, to evaluate if they could successfully be implemented on a PCTC.

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Chapter 2

Theory of ocean waves

2.1 Surface gravity waves

This chapter provides a brief introduction to the theory necessary for the analysis of ocean waves and the energy they transport.

2.1.1 Basic equations

The equations governing ocean waves can be derived from the basic principles of fluid dynamics, the Navier-Stokes equations1. For a Newtonian fluid the equations for conservation of mass and momentum can be written on Cartesian tensor form (using the Einstein summation convention) as

∂ρ ∂t + ∂ ∂xj (ρuj) = 0 (2.1) ∂ui ∂t + uj ∂ui ∂xj = −1 ρ ∂p ∂xi +1 ρ ∂τij ∂xj + fi (2.2)

where ui is the velocity vector, ρ is the density of the fluid, p is the total pressure,

fi is the external force and τij is the viscous stress tensor

1

For a complete treatment of Navier-Stokes equations and surface gravity waves, see [14].

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CHAPTER 2. THEORY OF OCEAN WAVES 3 τij = µw ∂ui ∂xj +∂uj ∂xi −2 3 ∂ur ∂xr δij (2.3)

where µw is the dynamic viscosity of sea water.

For an incompressible fluid, which for our purposes is a valid approximation of sea water, the density ρ is constant, reducing the conservation of mass to

∂ui

∂xj

= 0 (2.4)

Using this, equations (2.1) and (2.2) reduce to the incompressible Navier-Stokes equations: ∇ · u = 0 (2.5) ∂u ∂t + (u · ∇) u = − 1 ρ ∇p + µw∇ 2_{u + f} _(2.6)

To further simplify the equations the following assumptions are made for ocean waves:

1. µw = 0, viscosity is neglected.

2. ∇ × u = 0, the flow is irrotational.

3. f = −∇(gz), the only external force is gravity.

4. Small amplitude waves, allowing the problem to be linearised by neglecting velocities of second order and higher.

5. Surface tension effects can be neglected.

Assuming that the flow is irrotational, there exists a scalar velocity potential

u = ∇φ (2.7)

which reduces the conservation of mass (2.5) to the Laplace equation

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CHAPTER 2. THEORY OF OCEAN WAVES 4

Applying the above assumptions the momentum equation (2.6) is reduced to the linearised Bernoulli equation

∂φ ∂t +

p

ρ + gz = 0 (2.9)

To solve this equation a case is considered where the waves propagate in the x direction and the motion is restricted to the xz plane. The surface displacement ζ(x, t) of the wave is measured from the undisturbed free surface at z = 0. Two boundary conditions are specified at the free surface and one at the bottom, where z = −h0. The boundary condition at the bottom is

uz =

∂φ

∂z = 0 at z = −h0 (2.10)

The first boundary condition at the free surface is the kinematic boundary condition, which implies that a fluid particle at the surface never leaves the surface.

∂ζ ∂t + ux ∂ζ ∂x = uz = ∂φ ∂z at z = ζ (2.11)

This is linearised and approximated to a first order of accuracy

∂ζ ∂t =

∂φ

∂z at z = 0 (2.12)

The second surface boundary condition is the dynamic condition, that the pressure just below the free surface is equal to the ambient pressure (neglecting surface ten-sion). Taking the ambient pressure to be zero, this is written

p = 0 at z = ζ (2.13)

which upon insertion into (2.9) and evaluation at z = 0 rather than z = ζ (small amplitude waves) gives

∂φ

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An ansatz for ζ(x, t) is required in order to solve Eq. (2.8) using conditions (2.10), (2.12) and (2.14). Assuming a sinusoidal waveform with amplitude H/2, angular frequency ω and wavenumber k

ζ(x, t) = H

2cos(kx − ωt) (2.15)

results in the solution

φ = Hω 2k cosh k(z + h0) sinh kh0 sin(kx − ωt) (2.16) ζ = H 2 cos(kx − ωt) (2.17) ω2 = gk tanh(kh0) (2.18)

which means that the free surface displacement is sinusoidal with a wave height H between crest and trough. Eq. (2.18) is the dispersion relation, λ = 2π/k is the wavelength and T = 2π/ω is the wave period.

2.1.2 Deep water approximation

On deep water where h0 is large the dispersion relation (2.18) can be reduced to

ω2 = gk (2.19)

since tanh(x) → 1 as x → ∞. In reality this approximation is valid when h0> λ/3.

The phase speed c ≡ ω/k of the wave can then be written

c = r gλ 2π = r g k (2.20)

Since the propagation speed of a wave depends on its wavenumber, waves of different lengths will propagate at different speeds and disperse. Because of this a system such as this, where c depends on k, is called dispersive. In a dispersive system the energy of the waves does not propagate with the phase speed. Instead it propagates with the group speed cg ≡ dω/dk, which for deep water becomes

cg = 1 2 r g k = gT 4π (2.21)

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2.1.3 Energy transport

The energy content per square meter of ocean surface is the sum of the potential and kinetic energy of the wave.

E = Ep+ Ek J/m2

(2.22)

The average potential energy over one wavelength in the propagataion direction is found by considering the work necessary to transform the undisturbed surface level into the waveform. Using Eq. (2.17) and integrating over one wavelength the work done to raise the centre of mass of the water column ρζdx the distance ζ/2 from the undisturbed surface level, one obtains

Ep = 1 λ λ Z 0 gζ 2ρζdx = ρg 2λ λ Z 0 ζ2dx = 1 16ρgH 2 _(2.23)

The average kinetic energy is found by integrating the velocity vector over one wave-length and the whole depth, using Eq. (2.16).

Ek= _λ1 λ R 0 0 R −∞ ρ 2u2dzdx = _2λρ λ R 0 0 R −∞ ∂φ ∂x 2 + ∂φ ∂z 2 dzdx = ₁₆1 ρgH2 (2.24)

The fact that Ek = Ep is called the principle of equipartition of energy. The total

energy becomes

E = 1 8ρgH

2 _J/m2

(2.25) Comparing Eqs. (2.25) and (2.23) one realises that the total energy can be written

E = 1 8ρgH 2 _{= 2E} p= ρg λ λ Z 0 ζ2dx = ρgζ2 _(2.26)

which means that the energy per square meter of ocean surface is proportional to the mean square surface displacement.

The wave-energy flux per meter of wave crest is calculated as the group speed times the average energy per square meter:

J = cgE =

ρg2_{T H}2

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2.2 Ocean wave spectra

The sea surface is not regular and can never be described as a neat sine wave of constant wavelength and wave height going in one direction. It is a complex process, where waves of many different wavelengths, sizes and directions together form a surface that at first sight appears completely random. To describe this process it is common to use an energy spectrum function S(ω) that describes how the wave energy is distributed over different frequency components2:

E = ρg Z ∞

0

S(ω)dω (2.28)

Where S(ω) is a Bretschneider spectrum, defined as

S(ω) = Aω−5e−Bω−4 (2.29)

A and B are constants that are empirically determined to describe a specific sea state.

Based on the sea spectrum the spectral moment of order j is defined as mj =

Z ∞

0

ωjS(ω)dω (2.30)

Using the spectral moments several useful statistical variables can be calculated. One is the significant wave height Hs, defined as

Hs= Hm0= 4

√

m0 [m] (2.31)

This measure is used because it corresponds well to the traditional measure of sig-nificant wave height, which is the average of the 1/3 highest waves in the spectrum (Hs = H1/3). Sometimes Hs is also referred to as Hm0 because it is based on m0,

the spectral moment of order 0.

When working with wave energy another useful variable is the energy period Te [9],

which is the average period of all the waves in the spectrum:

Te= R∞ 0 T S(T )dT R∞ 0 S(T )dT = 2π R∞ 0 ω −1_S(ω)dω R∞ 0 S(ω)dω = 2πm−1 m0 [s] (2.32) 2

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For fully developed seas a fairly accurate spectrum is the ISSC spectrum, which is a two parameter spectrum based on significant wave height and period. Expressed in terms of energy period it is written3 [17]

A = 4π4Γ(5/4)4T_e−4H_s2

B = 16π4Γ(5/4)4T_e−4 (2.33)

The average energy density of the sea state can be expressed in terms of Hs as

E = ρgm0=

ρg 16H

2

s [J/m2] (2.34)

In analogy with the energy flux J = cgE of a single wave, the energy flux of the whole

system (assuming deep water) can be calculated using the relations cg = g/(2ω),

(2.31) and (2.32) [9]. J = ρg Z ∞ 0 cg(ω)S(ω)dω = 1 2ρg 2_m −1= ρg2 64πH 2 sTe [W/m] (2.35)

3_{Γ is the Gamma function, defined as Γ (x) =}R∞

0

sx−1_e−s

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Chapter 3

Wave energy on worldwide route

To estimate the amount of energy available to a cargo ship travelling around the world, a worldwide route consisting of 496 points has been used (Fig. 3.1). For all these points wave information (mean wave period Te and significant wave height

Hs) has been downloaded from ECMWF [7] for the year 2007, with one measurement

every 24 hours. 180o_W₁₂₀o_W ₆₀o_W ₀o ₆₀o_E₁₂₀o_E₁₈₀o_W 60oS 30o_S 0o 30o_N 60oN

Figure 3.1: Worldwide shipping route composed of 496 points calculated using great circle navigation.

Using Eq. (2.35) and the data from ECMWF a combined scatter and energy diagram 9

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CHAPTER 3. WAVE ENERGY ON WORLDWIDE ROUTE 10

has been created (Fig. 3.2), showing that most of the wave energy along the route is carried by waves of average period 7-11 s and significant wave height 1.5-3.5 m. The average energy flux varies between 10 kW/m and 50 kW/m with a yearly average of 26 kW/m, as can be seen in Table (3.1).

0 2 4 6 8 10 12 0 1 2 3 4 5 6 7

Annual energy flux for sea state interval [kWh/(m

× year)] 200 kW/m 100 kW/m 50 kW/m Energy period [s] Energy diagram for global route

20 kW/m 10 kW/m 5 kW/m 2 kW/m

Significant wave height [m]

0 50 100 150 200 250 300 350

Figure 3.2: Combined scatter and energy diagram for worldwide route, measuring annual energy flux per meter of wave front (kWh/(m·year)) for each sea state, represented by a 1₈ · 1

8 [m·s] square. Lines are isolines of constant energy flux,

according to Eq. (2.35).

Since the wave spectrum generally has a wide directional spread and a ship route does not follow the weather patterns, it has been assumed that a ship on average has incoming sea equally from all directions. For a typical ship of length 220 m and breadth 30 m, the average projected side to the incoming sea (averaged over a rotation of 360 degrees) is Lp = (220 + 30)2_π = 160 m. With an average energy flux

of J = 26 kW/m this means that a ship following the worldwide route on average will be exposed to an energy flow of roughly 4 MW.

Month Jan uary F ebruary Marc h April Ma y

June July August Septem

b er Octob er No v em b er Decem b er y ear a v g. J [kW/m] 49 49 34 28 15 12 12 11 15 21 29 41 26

Table 3.1: Average monthly energy flux along worldwide route during January-December 2007.

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CHAPTER 3. WAVE ENERGY ON WORLDWIDE ROUTE 11

One of the motivations for this thesis work has been that large amounts of fuel could be saved by capturing as little as one tenth of this available energy. For further analyses the sea state Te 9 s, Hs 2.5 m has been chosen as a reference state for this

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Chapter 4

Wave energy conversion

Wave energy conversion (WEC) is the term for the conversion of ocean wave energy into a desired form, usually electrical or mechanical energy. Numerous strategies for wave energy conversion have over the years been proposed. Almost all of them can be categorised into one of the following categories; overtopping devices, oscillating bodies and oscillating water column. [4, 5, 16]

This chapter attempts to give a brief summary of the state of the art of WEC by describing the working principles and listing the devices that have reached farthest in their development. The purpose is to give an overview of the technology that exists today, in order to evaluate what could possibly be fitted onboard a ship.

4.1 Overtopping devices

The basic principle of overtopping devices is to utilise the potential energy of the waves. Incoming waves are focused by a set of channels and artificial beaches to make them rise higher, thus redistributing the kinetic energy of the waves into potential energy. The waves then spill over (overtop) into reservoirs with a water level above the mean sea level (see Fig. 4.1). From the reservoirs the water then flows back into the sea through turbines, converting the potential energy into usable electrical or mechanical energy.

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CHAPTER 4. WAVE ENERGY CONVERSION 13

4.1.1 Wave Dragon

The Wave Dragon [23] is a floating, slack moored overtopping device that has been developed in Denmark by Wave Dragon ApS. It has large arms that focus the waves up a ramp and into a reservoir, from where the water flows back through propeller turbines connected to permanent magnet generators. A 20 kW prototype has been successfully tested in Denmark (Fig. 4.2), and a pre-commercial demonstrator with a rated capacity of 4-7 MW is currently being constructed, to be placed off the Welsh coast. According to the developers, a Wave Dragon designed for a 24 kW/m wave climate will have a total width of 260 m and a rated power of 4 MW, giving a total efficiency of nearly 65%. This number probably reflects a maximum output at optimal conditions.

Figure 4.1: Principles of energy capture in an overtopping device. c

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Figure 4.2: Wave Dragon overtopping device prototype. The wave reflectors (top, bottom left) focus the waves onto the ramp and into the reservoir (bottom right).

c

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4.1.2 Sea Slot-cone Generator

Another type of overtopping device is the Sea Slot-cone Generator (SSG) [15], which is a shore-based installation that uses reservoirs at several levels to extract energy from waves of different heights. The water is collected in the reservoirs and then flows out through a Multi-Stage Turbine (MST) that consists of several turbines staggered concentrically inside each other, driving a common shaft. The MST makes use of the different levels of water head in the reservoirs, and is able to extract power even at low water heads. Estimated overall efficiency of the SSG is 10-26% depending on wave conditions.

Figure 4.3: The Sea Slot-cone Generator developed in Norway. It allows waves of different height to overtop into reservoirs at three different levels (left). Water then flows out through a multi-stage turbine (right). From [15].

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4.2 Oscillating water column

The oscillating water column (OWC) technique is based on a chamber that is partly submerged in the sea. When waves act on the chamber the internal water surface oscillates, pumping air through a turbine that connects the chamber with the atmo-sphere (see Fig. 4.4, right). For the pneumatic power take-off the turbine used is a self-rectifying turbine that keeps its sense of rotation independent of air stream direction. It is common to use a Wells turbine [16, p. 143], which uses a rotor with symmetric wing profiles.

Small OWC generators rated at 60 W have for many years been successfully used to charge the batteries of offshore navigation buoys [4, p. 82]. Apart from navigation buoys there are currently no OWC devices running on a commercial scale, but pro-totypes have been constructed both as onshore structures and on floating offshore platforms. One such onshore test device is the Limpet OWC (Fig. 4.4) that has been built on Islay off the coast of Scotland in cooperation with the EU. At its target output of 200 kW the Limpet device would have an overall efficiency of 50%. How-ever, during the first test phase it only reached an average output of 20 kW, giving a disappointing 5% efficiency [25]. Current OWC development includes advanced tur-bines with variable-pitch blades and chambers with control valves to optimise power outtake, and it is believed that sufficient efficiency can be achieved to make OWC a viable WEC method.

Figure 4.4: OWC installation Limpet on island of Islay (left) and principles of operation (right).

c

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4.3 Oscillating bodies

Most of the WEC concepts being developed today consist of bodies that move in one or more degrees of freedom in the sea, absorbing the energy of the waves. Physically, to absorb energy from a wave means to generate a wave that interferes destructively with the original wave [8, p. 196], thereby reducing its amplitude. Hence, an oscillating body WEC device must be an efficient wave maker that creates a cancelling wave with its motion.

Many oscillating body concepts are point absorbers moving in heave mode (i.e. bob-bing up and down). A point absorber is a wave absorber that is very small compared with the wavelength, e.g. a buoy. Due to an effect called ”absorption width” it can absorb energy from an incident wave front of width equal to λ/2π [8, p. 197]. One example of a point absorber is the buoy developed in Sweden by Seabased AB and Uppsala University (Fig. 4.5, [22]). The buoy’s vertical motion in the waves drives a permanent-magnet linear generator on the sea bed, creating a current that is fed to a converter on the shore via a cable. Another example is the double array of point absorbers (a.k.a. multi point absorber) that has been developed by Wave Star Energy in Denmark (Fig. 4.6, [24]). In the Wave Star system the movement of the floats feeds pressurised fluid to a hydraulic generator, converting the movement into electricity.

Figure 4.5: Linear generator buoy developed by Seabased AB and Uppsala Uni-versity. The buoy is designed to be very simple and robust.

c

Oskar Danielsson (left) & Karl ˚Astrand (right), Avd. f¨or ell¨ara & ˚askforskn, Uppsala Univ.

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Other examples of oscillating body devices that have come far in their development are the Pelamis (Fig. 4.7, [19]) and the Oyster (Fig. 4.8, [2]). The Pelamis is the first WEC device in the world to be commercialised, with three 750 kW-devices now operating off the Portuguese coast and more being planned. The Oyster has suc-cessfully been tested in full scale at the New and Renewable Energy Centre (NaREC) near Newcastle.

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Figure 4.6: The Wave Star multi point absorber generates electricity continuously by combining multiple floats in an array several wavelengths long. The motion of the floats pumps fluid into a common transmission system that drives a hydraulic generator. c Wave Star Energy

Figure 4.7: The Pelamis WEC has hydraulic power take-off in its three joints, taking power from both yaw and pitch movement. Each device is 120 m long and has a peak power output of 750 kW in a 55 kW/m wave climate.

c

Pelamis Wave Power

Figure 4.8: The Oyster is placed on the sea bed near the shore, where the wave action causes it to move back and forth. The movement pumps high pressure water through a pipeline to the shore, where it can be used for electricity gen-eration or desalination. Each Oyster has a peak power output of 300-600 kW, equalling 15-30 kW/m. c Aquamarine Power

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4.4 Thrust generating foils

One method of wave energy propulsion that has received a lot of attention is by using a hydrofoil that undergoes an oscillating motion below the surface, thus generating thrust. This mimics the way that birds, fishes and insects generate thrust by moving wings and fins. The difference is that the thrust generation by the hydrofoil is completely passive; wave energy is absorbed as ship motion kinetic energy, which in turn is converted into thrust by the hydrofoil that moves with the ship.

The conversion of wave energy into thrust by a foil was named ”wave devouring propulsion” (WDP) by Terao [12]. Another term for the same technology is ”passive foil propeller”. Early experimental and analytical studies in the 1980’s suggested that a WDP system on a ship could generate thrust in both following and head sea [12, 21].

The concept is in itself very attractive, since foils mounted at the bow and stern of a ship would reduce its rolling and pitching motion, and at the same time generate thrust. Thus fuel would be saved both by reducing ship motions and by the direct forward thrust produced by the foils. To further improve results the foils could be controlled by an active system that adjusted their angle of attack to avoid stall and produce maximum thrust at all times.

Figure 4.9: Wallenius’ conceptual emission-free vessel E/S Orcelle, equipped with fins for wave energy propulsion. Some results indicate that fins would not be suitable on a ship of the Orcelle’s length. c Wallenius

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Chapter 5

Proposed technologies

This chapter presents the different concepts for wave energy propulsion that have been considered during this thesis work. All the concepts are adaptations of existing technologies, and all have been chosen in discussion with engineers at Wallenius as concepts that could be feasible to install on a PCTC. A total of 4 concepts have been studied, but the bow overtopping is the one that has been most thoroughly analysed. The analyses have been carried out with main ship data from Wallenius’ PCTC M/V Fedora, described in Table 5.1.

Lpp Breadth Draft CB U PE

216 m 32.3 m 9.5 m 0.62 18 kts 8.5 MW

Table 5.1: Ship data for M/V Fedora used in study. PE is the engine power

required at 18 knots.

5.1 Bow overtopping

The motion of the sea surface relative the side of the ship is a superposition of the motion of the sea surface and the pitch and heave motions of the ship. This relative motion is at its largest at the bow of the ship. By installing an overtopping device at the bow, this large motion amplitude could be used to generate electrical power. A possible positive side-effect could be a slight damping of the pitching motion. In this thesis work an overtopping device installed at the bow of a Wallenius PCTC has

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CHAPTER 5. PROPOSED TECHNOLOGIES 22

been modelled by using the response amplitude operators1 _{(i.e. transfer functions)}

Y3(ω) for heave and Y5(ω) for pitch in an ISSC sea state S(ω) (see Eqs. 2.29 and

2.33). The transfer functions have been calculated by using the linear strip method, as implemented in the software package Tribon M3.

5.1.1 Ship motions

The transfer function for motion in the i-th d.o.f. (degree of freedom) is defined as

Yi(ω) = η0i(ω)/ζ0(ω) (5.1)

where ζ0(ω) is the oscillation amplitude of the particular frequency in the exciting

sea state and η_i0(ω) is the amplitude of the motion. Fig. 5.1 shows an example of a set of transfer functions.

0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5

Transfer functions, vertical motion at x=110 m

ω [s−1]

Y(

ω

) [m / m]

Vertical motion (abs.) Vertical motion (rel.) Heave 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 Y( ω ) [deg / m] ω [s−1] Pitch Roll

Figure 5.1: Example transfer functions for a PCTC where speed U = 18 kts and heading µ = 135◦.

The relative motion at the position x (here x = 110 m has been used) is defined so that ηrel is positive when the sea level is below the mean sea level on the hull.

ηrel(t, x) = η3(t) − xη5(t) − ζ(t, x) (5.2)

The transfer function of the relative motion can, after taking into account the phase shifts ε3 and ε5 relative the exciting wave and adding the motions by using phasor

addition, be written as

Y_rel2 (ω, x) = Y₃2+x2Y₅2+1−2xY3Y5cos(ε3−ε5)+2 (xY5cos(ε5+ kX) − Y3cos(ε3+ kX))

(5.3) 1

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CHAPTER 5. PROPOSED TECHNOLOGIES 23 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 10 20 30 S η(ω ) ω [s−1]

Response energy spectrum, vertical motion at x=110 m

Absolute motion Relative motion 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 1 2 3 Sζ ( ω ) ω [s−1]

Sea state energy spectrum H_s=4.0, T_w=9.0

Figure 5.2: Example of energy spectrum for a sea state and corresponding re-sponse spectrum for absolute and relative motion at bow.

where kX is the phase shift due to the heading µ relative the direction of the waves

X = x cos(µ) (5.4)

By combining the transfer function of a motion with the sea state energy spectrum, the response energy spectrum of the motion for the actual sea state can be calculated as

S_iη(ω) = Y_i2(ω)S(ω) (5.5)

Once the response spectrum is known, the standard deviation, or RMS (root mean square) ηrms_i , of the motion can be calculated as

σi = s Z ∞ 0 S_iη(ω)dω (5.6) 5.1.2 Overtopping model

The overtopping mechanism has here been modelled as an inflow opening and a reservoir located at zin (in meters above mean water level). The vertical distance h

from the inflow opening up to the water surface is defined as

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Thus water flows into the reservoir when h > 0 and the reservoir is not full. Here h will be referred to as the pressure height. The setup of the model is illustrated in Fig. 5.3.

Figure 5.3: Illustration of the bow overtopping model.

The outflow from the reservoir is considered to be through an opening that is below the water surface at all times. Electrical power is generated by a low-head water turbine mounted in the outflow opening. Since turbines of this type have very low efficiency for heads below 2 meters [5, p. 330], the outflow is modelled to be open only when h < hmin, where hmin is the minimum head (hmin = −2 m here). The

reservoir has in this model been considered to have zero internal height, so that its water surface always is at zin.

The flow in and out of the reservoir is considered to be mainly driven by the hydro-static pressure

ps(t) = ρg|h(t)| N/m2

(5.8) giving a flow speed

v(t) =p2g|h(t)| [m/s] (5.9)

If R(t) represents the amount of water in the reservoir in m3 at time t, the change of water in the reservoir is defined by the water flow

dR dt =

(

Ain(v + U ), h > 0

−Aoutv, h < hmin

m3_/s

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The ship speed U has been added to the inflow speed driven by hydrostatic pressure to include the effect of the ship’s forward speed relative to the sea water. Ain/out is

the area of the inflow and outflow openings, respectively, here set to 10 m2 each. The available power of the outflow, before considering efficiency of power take-off systems, is calculated as the volume outflow times the driving pressure

P (t) = Aoutv(t)ps(t) [W], h < hmin (5.11)

In terms of the pressure height

P = ρgp2gAouth3/2 (5.12)

The relative motion ηrel between the surface and the inflow opening is normally

distributed, and therefore the probability density function (pdf) of h can be obtained from the normal distribution by substituting ηrel using Eq. 5.7

f (h) = 1 σrel √ 2πexp −(h + zin) 2 2σ2 rel (5.13)

Using f (h) the probabilities of in- and outflow can be written as2 Πin = Z ∞ 0 f (h)dh = 1 2 1 − erf zin σrel √ 2 (5.14) Πout = Z hmin −∞ f (h)dh = 1 2 1 + erf zin− hmin σrel √ 2 (5.15)

The outflow power depends on h3/2, here defined as the power height w

w = h3/2, h < hmin (5.16)

The pdf f (w) is obtained from (5.13) by variable substitution with (5.16) and nor-malising with Πout

f (w) = 1 Πout r 2 9π w−1/3 σrel exp −(w 2/3_{− z} in)2 2σ2_rel ! (5.17)

Similarly, the probability distribution functions for the inflow and outflow speeds are obtained by substituting (5.9) into (5.13)

f (vin) = 1 Πin vin− U σrelg √ 2πexp   − (vin− U )2/2g − zin 2 2σ2_rel   , U ≤ vin ≤ ∞ (5.18) f (vout) = 1 Πout vout σrelg √ 2πexp −(v 2 out/2g − zin)2 2σ2_rel , p2ghmin≤ vout ≤ ∞ (5.19) 2

erf (x) is the error function, defined as erf (x) =_√2 π

Rx 0 e

−t2

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The in- and outflow pdf’s are only defined when h > 0 and h < hmin, respectively,

and therefore normalised with Πin and Πout.

Using the distribution functions mentioned above, important time averages for the overtopping process can be calculated by noting that a variable x with pdf f (x), defined ∀x ∈ R, has an average value

x = Z ∞

−∞

xf (x)dx (5.20)

The time average of the flow changing the reservoir level can thus be calculated as dR

dt = ΠinAinvin− ΠoutAoutvout (5.21) The reservoir water level will build up to its maximum if dR_dt > 0. In that case there will always be water in the reservoir, and the mean power production will only depend on the outflow

P = Πoutρg

p

2gAoutw (5.22)

If dR_dt < 0, the reservoir will empty faster then it gets filled, and the power production will depend on the inflow. Numerically, this means that the outflow area must be adjusted so that dR_dt = 0, since water cannot flow out from an empty reservoir. This gives

P = Πoutρg

p

2g eAoutw (5.23)

where eAout is the adjusted outflow area

e Aout =

Πinvin

Πoutvout

Ain (5.24)

Therefore, the power production can be expressed as

P = (

ρg√2gΠoutAoutw , ΠinAinvin> ΠoutAoutvout

ρg√2gΠinAin(vin/vout) w , ΠinAinvin< ΠoutAoutvout

(5.25)

As can be observed in Fig. 5.4 (line ”Calculated”), this function is piecewise smooth with a sharp maximum. The maximum occurs at the point where dR_dt = 0, i.e. when the average outflow is of the same magnitude as the average inflow. A time-domain simulation of the same model has also been run (Fig. 5.4, line ”Simulated”), by modelling the behaviour of the reservoir for a time series of the relative motion. The results coincide well with the analytic model except for a slight underestimation in the second half of the curve, probably because of effects that occur when the reservoir is completely empty for some time periods.

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CHAPTER 5. PROPOSED TECHNOLOGIES 27 0 0.5 1 1.5 2 2.5 3 3.5 4 0 100 200 300 400 500 600 z in [m]

Time average of outflow power [kW]

Power vs z

in. Ship speed U=18 kts. Sea state Hs=2.5 m, Te=9.0 s.

Simulated Calculated

Figure 5.4: Output power as a function of inflow height for a given sea state.

The most important result from Fig. 5.4 is that the peak power predicted by the analytic model coincides very well with the peak obtained through time-domain sim-ulation. This means that for a given set of design parameters (U , hmin, Ain, Aout),

the maximum power production is a function of σrel and zin only, and we can

cal-culate a mapping zin,opt= zin,opt(σrel). By using this mapping the optimum inflow

height can be found for every sea state, since σrel is a function of µ, Hs and Te.

This would make it a relatively simple task to find the overall optimum inflow height across all sea states encountered during one year.

5.1.3 Water acceleration effect

One effect that must be considered when analysing the feasibility of the bow over-topping system is the added mass of the water that flows into the reservoir. All the water in the reservoir is accelerated from lying dead in the ocean to travelling forward with the ship speed U . This means that when there is inflow, the massflow ρAin(v + U ) into the reservoir generates a retarding power

Pretarding(t) =

ρAin(v + U )U2

2 , h > 0 [W] (5.26)

This retarding power reduces the net power production Pnet, so that

Pnet(t) = P (t) − Pretarding(t) (5.27)

effectively reducing the ship’s speed at high retarding powers.

Another possible effect that can be considered is the accelerating effect the flow around the hull can have on the outflow speed. If the outflow is in the direction of the hull flow, there will be a suction effect that accelerates the outflow and makes the

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water flow out of the reservoir faster than when driven by hydrostatic pressure, thus increasing the produced power. The theoretical maximum occurs when the outflow is accelerated by the ship speed U

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Fig. 5.5 shows the outflow power output (i.e. the power available to the turbine) and retarding power for three variations of the model:

A The first case is the original model, where the ship speed is added to the inflow speed but no outflow acceleration is taken into account.

B The second case is the same as the original model, but the maximum possible acceleration of the outflow has been taken into account. This case has a higher output and retarding powers at low zin, since the reservoir is always full

at these levels and the accelerated outflow increases the total water throughput. C The third case is where the water is considered to already be travelling at ship speed when flowing into the reservoir, for example because it is a part of the bow wave and has already been accelerated to ship speed. This eliminates the retarding power considered in cases A and B, but also reduces the amount of water that flows into the reservoir. Here no outflow acceleration is considered.

0 0.5 1 1.5 2 2.5 3 0 200 400 600 800 1000 1200 1400 z in [m] Time average [kW]

Ship speed 18 kts, heading 135 deg, sea state H

s=2.5 m, Te=9.0 s. A. outflow power B. outflow power C. outflow power A. retarding power B. retarding power

Figure 5.5: Output and retarding power estimates with different model variations taken into account.

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The reason for the high retarding power in the case considered is the high speed, 18 kts. The retarding factor (5.26) grows as U3, and at high speeds it quickly grows larger than the outflow power. At lower speeds however, the situation is the inverse. An example of the same comparison as Fig. 5.5, but at half the speed, is displayed in Fig. 5.6. There the retarding powers are significantly smaller than the outflow powers. 0 0.5 1 1.5 2 2.5 3 0 50 100 150 200 250 300 350 400 450 500 z in [m] Time average [kW]

Ship speed 9 kts, heading 135 deg, sea state H

s=2.5 m, Te=9.0 s. A. outflow power B. outflow power C. outflow power A. retarding power B. retarding power

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5.1.4 Conclusions

As Fig. 5.5 clearly shows, the retarding power considered is significantly larger than the outflow power regardless of outflow acceleration. This is a clear overestimate, partly because some of the water has already been accelerated by the hull before flowing in, and partly because the original added mass of the bow is reduced since there is a reduction of the area that pushes water forward and to the side. However, the outflow power is also a theoretical maximum that will be greatly reduced once turbines and generators are included in the model.

One can interpret the results of the model by considering a floating overtopping device, such as the Wave Dragon, that is equipped with an electric motor driven by the generators on the device. At speeds close to zero the output power will be larger than the retarding power, and the device will accelerate. But at a certain (and probably very low) speed there will be an equilibrium, and the device will not move any faster.

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5.2 Thrust generating foils

The WDP concept described in section 4.4 has been analysed both by a literature survey of earlier studies and by a numerical study using the simplified model described below.

5.2.1 Modelling

To fully model a ship equipped with WDP, one must couple the equations of motions of the ship with the behaviour of the oscillating foil, which is a fairly complicated task. The hydrodynamics of foils oscillating in heave and pitch near a free surface is a subject of ongoing research, and beyond the scope of this study. (One treatment of a rigid oscillating foil near a free surface can be found in [10].) A simplified approach is to model the foil by using steady-state foil theory, with fluid speed and angle of attack determined by using linear wave theory together with the ship’s transfer functions for heave and pitch. This approach disregards the foil’s damping effect on ship motions and the wave-making effects of the oscillating foil, but is still interesting because of its simple implementation.

Figure 5.7 illustrates the principles of the model. The speed u and angle ϕ of the incoming flow are determined by superpositioning the ship’s forward speed U and vertical speed ˙ηabs with the internal particle motion of the waves. The lift FL

per-pendicular to the flow and the drag FD in the direction of the flow are then calculated

using the corresponding coefficients CL and CD of the foil together with the angle

-0 .5 0 0. 5 1 1. 5 -0 .5 0 0. 5 u F_L F_D ζ η_abs α η_abs φ x z

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of attack α, determined by the foil’s control system. The horizontal components of FLand FD are then used to calculate the thrust and drag of the foil in the direction

of travel. The lift and drag are calculated as FL = 1 2ρu 2_A fCL(α) (5.29) FD = 1 2ρu 2_A fCD(α) (5.30)

where Af is the area of the foil [1].

5.2.2 Studies

One implementation of the simplified approach described above is a DNV study from 1989 [13], which considers a 300 m vessel sailing the route Chile-Japan at 13 knots. The vessel in the study is equipped with bow and stern foils with areas 2.0% and 1.5% of the total water plane area (WPA), respectively. According to the study, the foils would give a propulsive effect of 1.6 MW and a fuel saving of 21%, on average for all headings. However, this study does not analyse the resistance increase due to the appended foils or the motions of the ship in waves, but rather adds 15% (1 MW) to the still water resistance to account for all factors. A quick estimate of the foil drag alone (based on CD) sets it at minimum 0.4 MW without waves, i.e at 0◦

angle of attack, and it would definitely be larger in waves. The fuel saving of 21% is calculated as the ratio of the foil propulsive power at 13 knots and the effective power normally needed to maintain that speed (with the 15% increase). Thus it does not take into account the resistance increase from the foils. On the other hand, the foils will most likely dampen the pitching of the ship, reducing the wave-induced resistance compared to the original configuration.

More recently, WDP in the form of bow wings has been extensively studied by Naito [18]. However, this wave energy conversion system seems to be ineffective when the wavelength is shorter than the ship’s length (λ/L < 1), according to Naito. A typical PCTC has a length of about 200 meters, meaning that WDP would only be effective for wave periods above 11 seconds. Mean wave periods above 11 seconds occur less than 5% of the year on the worldwide route [7, 11], and thus WDP seems to be unsuitable for ships this large. An experimental study from CUT by Bergholtz and Stocks [3] also concludes that when λ/L < 1 the foil thrust is too small to overcome the added resistance due to the foils.

5.2.3 Numerical experiment

An implementation of the simplified model described in 5.2.1 has been made in MATLAB during this thesis work, simulating a T-shaped foil mounted 10 meters

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below the surface at the bow of a PCTC. The depth of 10 meters was chosen so that the risk of foil slamming would be small even in large waves. It has some effect on the results since the particle motions of the waves are largest near the surface, making deep mounted foils slightly less effective. A symmetrical NACA 0012 profile with aspect ratio 4 was used, since it is the kind used in [18] and [13]. It is also similar to the NACA 0018 profile used by [3]. The total foil area was set to 1% of the WPA, and the foil was considered to be controlled by an optimal control system. Optimality was here defined as the angle of attack that maximised the net propulsive force.

The 2D panel code XFoil [6] was used to derive the section lift and drag coefficients (cland cd) of the profile. The foil coefficients (CLand CD) where then derived from

the section coefficients by applying aspect ratio effects, as described in [1]. XFoil is considered to give relatively accurate lift and drag predictions, but it is likely to overestimate the stall angle. No experimental results were however available for the high Reynolds number encountered here (Re 3e7), which is why the XFoil coefficients have been applied without modification. Since the foil was considered to have an optimal control system it would not come near the stall angle anyway, since the optimal lift/drag ration of the foil was observed to be well below the stall angle. The ship motions for different ISSC sea states with Te = 9 s and varying Hs were

obtained from the ship’s transfer function, as described in 5.1.1. By adding together the ship motions with the particle motions of the waves, a time series of the speed and direction of the flow around the foil was then obtained. The foil’s angle of attack was set to the one that gave the largest net propulsive force, and the forces acting on the foil were calculated using steady state foil theory and the coefficients CLand

CD. A time average of the horizontal forces acting on the foil at different significant

wave heights can be seen in Fig. 5.8.

According to the results in Fig. 5.8, the resultant force is positive for Hs > 2 m

when Te = 9 s, on average for all directions. A comparison of the resultant power

for several Te can be seen in Fig. 5.9, and it shows that the efficiency of the

T-foil increases with the wave period. An estimate using ECMWF statistics [7] and the results in Fig. 5.9 indicates that the resultant force would be positive 45% of the time on the worldwide route. Hence, a system of this kind would need to be retractable in order to avoid added resistance in sea states with smaller waves. It is worth mentioning again that this model is greatly simplified, since the ship motion is not coupled to the forces produced by the foil, and since steady state foil theory is used for a foil in an oscillating flow field. However, this solution is optimistic under the made assumptions, since the foil is considered to have an optimal control system. Also, the control system is not considered to consume any power, though such a system in reality could be quite power-consuming.

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CHAPTER 5. PROPOSED TECHNOLOGIES 35 0 0.5 1 1.5 2 2.5 3 −200 0 200 400 600

NACA 0012, speed 18 kts, all headings, T

e 9 s H s [m] Power [kW] Mean propulsion Mean drag Mean resultant

Figure 5.8: Propulsive and dragging power of the T-foil vs. significant wave height, average over all headings.

0 0.5 1 1.5 2 2.5 3 −200 −100 0 100 200 300 H s [m] Power [kW]

NACA 0012, speed 18 kts, all headings, T

e 7−11 s Mean resultant, T e 7 s Mean resultant, T e 9 s Mean resultant, T e 11 s

Figure 5.9: Mean resultant power of the T-foil vs. significant wave height for Te

7, 9 and 11 s, average over all headings.

5.2.4 Conclusions

Concluding the results from the numerical study above and the studies by Naito and Bergholtz, WDP does not seem to be a good alternative for ships as large as PCTC’s. The only study that says the opposite is the one made by DNV in 1989, though this is probably because it does not calculate the foil drag directly.

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5.3 Moving Multi-Point Absorber

Another concept that has been considered during the work with this thesis is a variation of the Wave Star multi point absorber (Figure 4.6) that can be installed on a moving ship. The technology has here been named Moving Multi Point Absorber (MMPA).

The idea is to use several surface-following point absorbers that lie outside of the ship hull. The relative motion between the absorbers and their attachment points on the ship would then be used, drawing energy from the sea waves both through the absorbers’ and the ship’s movement. At the same time the rolling and pitching movements of the ship would be damped.

The point absorbers used by the Wave Star are half spheres, and would probably not be suitable for the MMPA because of their large propulsion resistance. Instead, absorbers with a good ratio between drag and developed power would have to be used. The absorbers could be displacing, semi-displacing or planing hulls. Hydrofoils could also be a possibility.

The full scale version of the Wave Star (Fig. 5.10) uses absorbers with a diameter of five meters, and each absorber is expected to give 25–50 kW output power at 2.5 meters significant wave height, according to Wave Star [24]. Placed on a ship the output per absorber would probably be even higher, since power is also drawn from the ship motions. As a quick approximation, consider a configuration where 10 of these absorbers (_{5 m) are placed on each side of the ship. If each float produces} 50 kW, then the total power production could be up to 1 MW at Hs 2.5 m.

Figure 5.10: Wave Star test section in full scale, launched in September 2009. Each float has a diameter of 5 m and is expected to produce 25–50 kW at Hs 2.5 m. c Wave Star Energy

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5.3.1 Modelling

The MMPA concept has not been thoroughly studied in this thesis, but a simplified model of how to do it has been created. The model is based on the following assumptions:

1. Linear theory is used.

2. The motions of the ship are not influenced by the motions of the absorber, except for the resistance increase due to the absorber. Therefore the absolute3 vertical motion of the ship ηabs(ζ0, ω, t) at the position of the absorber, obtained

by linear strip theory, is an input variable in the model.

3. Only one absorber is modelled. An advanced model that captures the wave interaction between many absorbers should also include absorber-ship interac-tions.

4. The power take-off system is modelled as a spring CS (e.g. a linear generator)

and a damper CR, which are controllable via a control system.

The vertical movement r of the absorber is governed by the equation

m¨r = FR+ Fs+ Ff (5.31)

where FR and FS are the damper and spring forces:

FR= −CRz˙r

FS= −CSzr

(5.32) and zr is the relative motion between the ship and the absorber

zr = r − ηabs (5.33)

The fluid forces on the absorber are represented by Ff,

Ff = −(A¨r + B ˙r + Cr) + Fe (5.34)

where A, B and C are hydrodynamic coefficients of the absorber and Fe is the

exciting wave force, Fe= Fe(ζ0, ω, t). These four variables depend on the absorber

shape, and can be calculated once the geometry has been decided, e.g. using the linear strip method.

The governing equation finally becomes

(m + A) ¨r + (CR+ B) ˙r + (CS+ C) r = −CRη˙abs− CSηabs+ Fe (5.35)

3

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and the power delivered to the power take-off system is

P = CRz˙r2 (5.36)

Figure 5.11: Conceptual drawing of how to model the outlying absorber of the MMPA. The absorber (right) is connected to the ship (left) via a damper-spring connection.

5.3.2 Numerical solution

A numerical solution of Eq. 5.35 has been made in the sea state Te 9 s, Hs 2.5 m

using the explicit Runge-Kutta (4,5) formula implemented in MATLAB. The vertical motion ηabs of the ship was calculated in the same way as in section 5.1.1, using the

linear strip method. The absorber simulated was a cylindric buoy with diameter 5 m and draught 1 meter, giving a total displacement of 20 tons. The hydrodynamic coefficients were approximated as

A = ρπR22

5R (5.37)

B = 0 (5.38)

C = ρgπR2 (5.39)

where R is the diameter of the cylinder.

The results gave a power production of 80 kW, on average for all headings. However, the power production varied greatly for the different headings, mainly because the vertical speed of the buoy (and thus the produced power) depends strongly on the

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encounter frequency of the waves. When the ship heads into the waves the encounter frequency increases, and thus the power production also increases.

5.3.3 Conclusions

An MMPA system could probably be able to produce a substantial amount of power, but it is unclear if it could be done without the absorbers giving a resistance increase larger than the produced power. It would most certainly be difficult at speeds as high as 18 knots.

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5.4 Turbine-fitted anti-roll tanks

In an M.Sc thesis work at Wallenius by Bj¨orn Wind´en [26], the possibility of fitting roll damping U tanks on a PCTC has been investigated. The tanks work by letting a fluid (usually water) flow between port and starboard tanks as the ship rolls, thus counteracting the rolling motion. To improve damping performance in random sea states, the duct between the tanks is fitted with some type of control valve or flow damping device. Alternatively, the tank tops are connected by a closed air duct that contains the control system.

Together with Bj¨orn Wind´en the possibility of extracting energy from the ship motions by fitting a generator turbine to the roll tank system has been investigated. The idea was to tap energy from the flow between the tanks with a variable resistance, thus optimising the roll damping and extracting flow energy at the same time. However, a simulation of the described system on a PCTC in rough weather showed that the tank motions needed to stabilise the ship were so small that the possible power outtake would be less than 1 kW. The reason for this is that the tanks already are tuned to the ship’s roll eigenfrequency, and the adjustments needed to maintain optimal roll damping in an irregular sea state are very small. One could increase the power output by deliberately designing the tanks to be out of tune, but the increase would probably not be large enough to justify this.

Figure 5.12: An anti-roll tank system of the kind considered in Wind´en’s thesis. The picture is of the INTERING system marketed by Rolls Royce.

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Chapter 6

Conclusions

Looking at the technologies evaluated in chapter 5, a first conclusion is that none has proved to be the future solution for wave energy propulsion of PCTC’s. The main problem encountered in both the bow overtopping and WDP studies was the resistance associated with the relatively high speed of 18 knots used, though in the WDP case the length of the ship also had a negative impact on the results. In the MMPA study no resistance estimate was made, but at 18 knots the increased resistance would most likely pose a problem.

Noting that ship speed and size in this study have proved to be the main factors against feasible wave energy propulsion, a justified question is if it would be possible to construct shorter and slower ships that were at least partially propelled by wave energy? The answer is, as it often is in science and engineering, that it depends on what you need. It would, most likely, be possible to construct a wave-powered vessel that moved at a very low speed, since a regular wave power device such as the Wave Star or the Wave Dragon probably could move by its own power. The question is rather if the speed and cargo-carrying capacity of such a vessel would make it economically viable? This thesis does however not answer that question, it only concludes that wave energy propulsion does not seem to be feasible for the PCTC’s that are in Wallenius’ fleet today.

For future work in this field, a first recommendation would be to study systems for shorter and slower ships than in this study, maybe 100 m long ships travelling at 10 knots. Secondly, I think that both the WDP and MMPA concepts could be further studied beyond what has been done in this study. MMPA could be studied with relative ease by using the linear strip method, while a thorough study of WDP would have to use quite advanced fluid dynamics to be reliable.

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Nomenclature

α [ - ] Angle of attack . . . page 33 [m] Float diameter . . . page 36 η_i0(ω) [m] Amplitude frequency component of motion in direction i . . page 22 ηi(t, x) [m] Motion in direction i . . . page 22

ηabs [m] Vertical motion . . . page 33

ηrel(t, x) [m] Relative motion . . . page 22

λ [m] Wavelength . . . page 5 µ [ - ] Heading relative the direction of the waves . . . page 23 µw [Ns/m2] Dynamic viscosity of sea water . . . page 3

ω [s−1] Angular frequency . . . page 5 φ [m2/s] Velocity potential . . . page 3 Π [ - ] Probability. . . .page 25 ρ [kg/m3] Density of fluid . . . page 2 σi [m] RMS of motion in direction i . . . page 23

τ [N/m2] Viscous stress . . . page 2 εi(ω) [ - ] Phase shift in direction i . . . page 23

ϕ [ - ] Flow angle . . . page 33 ζ [m] Free surface displacement . . . page 4

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BIBLIOGRAPHY 46

ζ0

i(ω) [m] Amplitude of sea state frequency component . . . page 22

A [m2s−4] Spectrum constant . . . page 7 A [Ns2/m] Added mass coefficient of absorber . . . page 37 Af [m2] Foil area . . . page 33

Ain [m2] Inflow opening area . . . page 25

Aout [m2] Outflow opening area . . . page 25

B [Ns/m] Damping coefficient of absorber . . . page 37 B [s−4] Spectrum constant . . . page 7 c [m/s] Phase speed . . . page 5 C [N/m] Buoyancy coefficient of absorber . . . page 37 CB [ - ] Block coefficient . . . page 21

CD [ - ] Foil drag coefficient . . . page 33

cd [ - ] Section drag coefficient . . . page 34

cg [m/s] Group speed . . . page 5

CL [ - ] Foil lift coefficient . . . page 33

cl [ - ] Section lift coefficient . . . page 34

CR [Ns/m] Damper constant. . . .page 37

CS [N/m] Spring constant . . . page 37

E [J/m2] Energy density . . . page 6 Ek [J/m2] Kinetic energy density. . . .page 6

Ep [J/m2] Potential energy density . . . page 6

f [ - ] Probability density function . . . page 25 f [N/kg] External force . . . page 2 FD [N] Drag force . . . page 33

Fe [N] Exciting wave force on absorber . . . page 37

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BIBLIOGRAPHY 47

FL [N] Lift force . . . page 33

FR [N] Damper force . . . page 37

FS [N] Spring force . . . page 37

g [m/s2] Acceleration of gravity . . . page 3 H [m] Wave height . . . page 5 h [m] Pressure height . . . page 24 h0 [m] Water depth . . . page 4

Hs [m] Significant wave height . . . page 7

H1/3 [m] Significant wave height . . . page 7

Hm0 [m] Significant wave height . . . page 7

hmin [m] Minimum head . . . page 24

J [W/m] Energy flux . . . page 6 k [m−1] Wavenumber . . . page 5 kX [ - ] Phase shift due to heading and position . . . page 23 L [m] Ship length . . . page 33 Lpp [m] Length between perpendiculars . . . page 21

mj [m4s−j] Spectral moment . . . page 7

p [N/m2] Total pressure . . . page 2 P [W] Power . . . page 25 PE [W] Required engine power . . . page 21

ps [N/m2] Hydrostatic pressure . . . page 24

R [m3_] _{Reservoir level . . . page 24}

r [m] Vertical motion of absorber . . . page 37 S [m4s] Energy spectrum function . . . page 7 S_iη [m4s] Response energy spectrum . . . page 23 T [s] Waveperiod . . . page 5

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BIBLIOGRAPHY 48

t [s] Time . . . page 2 Te [s] Wave energy period . . . page 7

U [m/s] Ship speed . . . page 25 u [m/s] Flow velocity . . . page 2 v [m/s] Flow speed . . . page 24 w [m3/2] Power height . . . page 25 Yi(ω) [ - ] Transfer function in direction i . . . page 22

z [m] Vertical position . . . page 3 zin [m] Inflow height . . . page 24

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Abbreviations

Abbreviation Description Definition

ECMWF European Centre for Medium-Range Weather

Forecasts

page 9

kts knots, 1 knot = 1.852 km/h page 21

MMPA Moving Multi Point Absorber page 36

MST Multi-Stage Turbine page 15

OWC Oscillating Water Column page 16

PCTC Pure Car and Truck Carrier page 1

pdf Probability Density Function page 25

SSG Sea Slot-cone Generator page 15

WDP Wave Devouring Propulsion page 20

WEC Wave Energy Conversion page 12

WPA Water Plane Area page 33

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List of Figures

3.1 Worldwide shipping route . . . 9

3.2 Combined scatter and energy diagram for worldwide route . . . 10

4.1 Principles of energy capture in an overtopping device . . . 13

4.2 Wave Dragon prototype . . . 14

4.3 Sea Slot-cone Generator . . . 15

4.4 Limpet oscillating water column . . . 16

4.5 Linear generator buoy. . . 17

4.6 Wave Star multi point absorber . . . 19

4.7 Pelamis WEC . . . 19

4.8 Oyster WEC . . . 19

4.9 Wallenius’ E/S Orcelle, equipped with fins for wave energy propulsion . . . . 20

5.1 Example transfer functions for a PCTC . . . 22

5.2 Example of energy spectrum . . . 23

5.3 Illustration of the bow overtopping model. . . 24

5.4 Output power as a function of inflow height for a given sea state. . . 27

5.5 Output and retarding power estimates for model variations . . . 29

5.6 Same comparison as Fig. 5.5, but at half the speed. . . 30

5.7 Modelling of WDP-equipped ship . . . 32

5.8 Propulsive and dragging power of the T-foil, all headings . . . 35

5.9 Mean resultant of the T-foil for multiple Te, all headings . . . 35

5.10 Wave Star test section in full scale . . . 36

5.11 Conceptual drawing of how to model the outlying absorber of the MMPA. . 38

5.12 An anti-roll tank system . . . 40

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List of Tables

3.1 Average monthly energy flux . . . 10 5.1 Ship data for M/V Fedora used in study. . . 21

Wave Energy Propulsion forPure Car and Truck Carriers(PCTCs)