Introduction Generating Power From The Tides

Published Date: 02 Nov 2017

Chapter 1

With growing concerns about the impact of carbon emissions on the global climate and reliance on finite energy resources, there has been a great deal of work towards developing technologies to extract energy from the tides over the last two decades.

The tides, cyclical in nature, are governed by the gravitational pull of the moon and sun on the Earth’s oceans, so it provides a reliable and predictable source of renewable energy which make tidal power production play an important role in stabilizing electrical grids.（扩充好处：占地，效率，与其他的比较）

Tidal turbines will generally operate in shallow water close to shore，rotating at a pace of 5 to 20 revolutions per minute. The diameter of tidal turbine could reach 30m and it converts kinetic energy contained in the tidal into mechanical energy, which is used to produce electricity, when a stream of moving water passing through the turbine. In currents of speed 2.5 m/s, a 20-meter diameter tidal turbine rotating at a pace of 7rev/min can generate 700,000-kilowatt hours of non-polluting energy per year—enough to power about 35 Norwegian homes (70 U.S. homes). ( John Roach, 2003)

The UK has an abundant tidal energy resource, estimated as up to 18 TWh/year (S.J. and Bryden, 2006), which is 5% of the UK’s 2003 energy demand. Since this, interest in the possible installation of numerous tidal turbines at multiple sites has increased.

Tidal energy extraction projects have been confirmed at the Skerries, Anglesey (Marine Current Turbines, 2006, online) and the Sound of Islay (Royal Haskoning, 2011, online) recently. Figure 1 shows tidal turbine array, depicting Hammerfest design, as proposed for installation at the Sound of Islay site.

Figure. Artist‘s impression of tidal turbine array, depicting Hammerfest design, as proposed for installation at the Sound of Islay site (Obtained from: http://www.hammerfeststrom.com/, accessed 13/05/2011)

Noise generated by turbine

Environmental considerations also need to be taken into account when designing, installing and operating marine current devices, in terms of their potential impact on marine life and marine eco-systems. One of such impacts is noise that is generated by the tidal turbine. Hence environmental impact assessment studies (EIA) is necessary, involving predicting noise from tidal turbines and its likely effect on marine animals.

However, little is known about the noise generated by tidal stream turbines compare with wind turbine and few attempts to model turbine noise at the design stage appear to exist. To the best of our knowledge, only one published study is available (Richards, et al, 2007) which systematically treats the issue of turbine noise as part of a strategic environmental assessment. Despite all this, academic interest has increased in modeling the noise emitted by renewable energy devices, its propagation underwater, and effect of marine animals (Carter, 2007; Barrio, 2009; Patricio et al, 2009). A ley limitation of this thesis is lack of knowledge and generally accepted method for predicting the noise generated by tidal turbine.

Many factors will influence tidal turbine noise such as device size, rotational speed and flow conditions. Potential sources of tidal turbine noise can be generally classified as hydrodynamic noise sources, mechanical noise sources and structural noise sources (Richards et al, 2007; Blake, 1984). Among them, mechanical noise is reduced efficiently by well-known engineering techniques, such as proper insulation of the nacelle, and hydrodynamic noise is the dominant noise sources.(加一个例子)

Underwater background noise

The oceans can be considered noisy (Ainslie et al, 2009), and the existing background noise can be significant in the assessment of an underwater noise source and its impact on the marine environment. Thus the ‘ambient’ noise of the site must be taken into account when assessing the sound produced by a turbine.

Ainslie et al (2009) categorize the acoustic noise sources in the North Sea into three classes: natural sources, intentional anthropogenic sources and unintentional anthropogenic sources. Tidal turbine noise lies in the third class, unintentional anthropogenic noise sources, which also include activities such as shipping and offshore operations. The first class factors such as weather, which contributes to surface noise, will have an effect on the ambient noise of a site, with higher ambient noise generally at lower frequencies (< 3 kHz) (Richards et al, 2007). Thus it is important to give baseline data in an impact assessment from measurement of site specific ambient noise.

Impact on marine environment

The effects of noise on marine environment is much more significant than we thought, including masking, behavioral change, hearing threshold shift (both temporary and permanent), and even death (Götz, 2009). Sound pressure level (SPL) alone can not provide enough information and is not the only factor when assessing the impact of an anthropogenic noise, other factors such as the frequency range and exposure duration are also important and need to be taken into account.

Figure. Graphical representation of ‘ zones of influence’ model

Figure.2 shows a graphical representation of ‘zones of influence’ model (Richardson et al, 1995), which is a simple model widely used to assess noise impact on marine life. The severity of influence falls off with distance from the source, yet exposure duration will also contribute to the effects experienced by marine animals. Permanent (PTS) or temporary threshold shift (TTS) is a primary cause of injury and physiological damage to wildlife. These effects can occur if a marine animal is exposed to SPLs of 95 dB and 75 dB or greater respectively above their hearing threshold (HT) level, for longer than 8 hours in a 24-hour period (Richardson et al, 1995). When identifying which noise sources are likely to affect certain species, an audiogram can be used to compare SPL to the hearing threshold of marine animals over a range of frequencies (Ainslie et al, 2009)

The impact of sound on the marine environment is not fully understood today. This project focuses on turbine noise prediction rather than the impact assessment techniques.

Chapter

Literature review

The purpose of the literature review is to outline the considerations that must be made in this study. Section 2.1 lists the major differences between tidal flows and wind, which should be taken into account when analyzing tidal turbine using method for wind turbine. Section 2.2 introduces the analysis techniques for turbines and the method used in this study. Section 2.3 describes shallow water acoustic propagation for both near field and far field. Section 2.4 describes types of potential noise sources due to the operation of a tidal turbine. Section 2.5 introduces the dominant noise source: hydrodynamic noise.

Tidal Flows

Compared with the wind turbine, little is known about the tidal turbine. Almost all design and analysis methods used for tidal turbines have been adopted from the wind industry. For this reason, it is important to compare tidal flows with wind and figure out the differences between them. The major differences are highlighted in the following points (Shives, 2008).

Flow velocity: Viable peak tidal flows for power generation are approximately 2.5m/s, roughly one sixth of typical wind velocities in a wind farm.

Density: The density of seawater is close to 1024kg/m3, roughly 830 times denser than air. The kinetic energy density of a flow scales linearly with density and with the square of the velocity. Thus, the energy density of tidal flows is about 20 times that of wind.

Cavitation: Cavitation occurs when the static pressure of the water is reduced to vapor pressure, causing bubbles of vapor to form within the flow. These bubbles can collapse explosively causing damage to hydrofoils. (Batten et al, 2008)

Reynolds Number: Reynolds Number is important because it influences the airfoil characteristics, particularly near the point of stall. Comparing tidal turbines and wind turbines in density, viscosity and expected length and velocity scales, it can be found that rotor-diameter-based Reynolds number for tidal turbines is approximately one order of magnitude less than for wind turbines.

Turbulence: Tidal flows are typically highly turbulent and dynamic. Thomson et al (2010) found 10% turbulence intensity, which is the ratio of velocity standard deviation to velocity mean. Turbulent length scales may often be on the same order of turbines themselves (Gant et al, 2008). Large dynamic loading on turbine blades can occur because of this relatively high turbulence intensity and large length scales.

Blockage effects and free surface effects: Tidal flows are typically bounded by the ocean floor, water surface and lateral boundaries (channel walls). Placing turbines in such condition causes wakes of reduced velocity downstream of the turbines, but also regions of increased velocity beside the turbines. On the other hand, the presence of turbines may have a local impact on the water height (Sun et al, 2008) and this free-surface modification may influence the flow through the turbine, altering its power production.

Feedback effect: Tidal flows are driven by hydrostatic pressure gradients, which arise from free-surface height differences, and are limited by inertial and viscous effects. However, wind is driven by atmospheric pressure systems with extents on the order of hundreds of kilometers and the impact of wind power extraction, in terms of increased resistance to the flow of air through a region is negligible compared to the driving forces.

Biofouling: Biofouling is the buildup of marine organisms on the turbines blades, which can increase the drag coefficient significantly. Significant power reduction can occur when the airfoil drag coefficient increased by 50% (Batten et al, 2008). One way to prevent biofouling is using specialized coatings on the blades, but the impacts of and possible mitigation techniques for biofouling do require further study.

Marine wildlife and debris: Tidal turbine blades may be struck by fish, whales, or partially submerged sea ice and logs. Such events have a potential impact on marine life and marine eco-systems and at the same time could seriously damage turbines. Fortunately, there is evidence that turbines will rotate slow enough that fish and whales will simply avoid such encounters (Pelc et al, 2002). Pelc et al presented a method to avoid these events that using sonar detection systems to shut-down turbines if large marine animals or other hazards are in close proximity to the turbines.

It can be found that there are fundamental differences between wind and tidal flows, but it not means that wind turbine analysis techniques are not valid in the tidal domain. When applying such models on tidal turbines, it is important to consider these differences. Studying all of the above considerations in any great depth in the given time is not possible, so this thesis focus on specific areas as the work progressed. The first 5 terms have been taken into consideration.

Analysis techniques

At the beginning of this thesis work, a literature review was conducted to determine appropriate methods for analyzing tidal turbines at the device scale. Through the years, several models have been proposed to explain and predict wind turbine noise. Some of the models are somewhat simplistic, whereas others are so complex that have not yet matured to be applied in practice. Tidal turbine performance can also be investigated using these models.

Low level approaches

These approaches are characterized by simplified modelling of the fluid flow and low computational cost, allowing fast analysis of multiple candidate designs. Tidal turbine performance can be analyzed by blade element momentum (BEM) codes, such as CCAV, a version of CWIND (Barnsley et al, 1993). CCAV can be used to predict turbine power, and also provides prediction of likely cavitation type and location for a turbine operating in a defined water depth and an assumed water column velocity profile. The characterization of turbulence is very important because noise due to inflow turbulence is the dominant noise source (Lloyd, 2011). Measured data for turbulence ocean boundary layer profiles is really limited but nevertheless provide recommendations for modelling turbulence in shallow water.

High level approaches

Computational fluid dynamics (CFD) simulation had been widely applied to model wind turbines to gain a deeper understanding of flow characteristics that cannot be taken into account by simpler techniques and allow detailed visualization of the flow.

CFD solve the Reynolds Averaged Navier-Stokes (RANS) equations for fluid flow on a discretized domain and can represent any arbitrary geometry and offer a wide variety of boundary conditions. However, the computational mesh must be of sufficient density to resolve the flow accurately, which makes this technique computationally expensive. Direct numerical simulation (DNS) and large-eddy simulation (LES) can be used to solve problems involving estimation of noise. These methods resolve turbulence scales up to a specified cutoff wave number, allowing more accurate noise source prediction.

Noise model for this thesis

As a compromise between computing speed and accuracy, the most commonly used models are based on semi-empirical relations. This thesis work use BEM theory and employ some empirical formulae to analyze tidal turbine performance and tidal turbine noise.

Blade Element Momentum (BEM) Theory is an established computational method for wind turbines and has been used in conjunction with some empirical formulae for wind turbines noise prediction. It has enjoyed the widest application to tidal turbines.

This widespread application is primarily due to its surprisingly high accuracy, and minimal computational expense.

Blade element momentum theory is an analytical/empirical method based on balancing the forces exerted by the turbine blades with the changes to the momentum of the flow. The simplest formulation (known as uniform axial actuator disc theory) consists of a balance between the turbine’s axial force and the change in axial momentum of the flow. It also includes an equation balancing the torque applied to the flow by the turbine to the change in the angular momentum of the flow. BEM is limited in that it cannot model losses due to blade drag and kinetic energy transfer to the swirling wake accurately. In fact the wake is assumed to be cylindrical with a constant tangential velocity. Wake recovery and wake interaction therefore cannot be modeled with BEM.

The empirical formulae implemented in this study for the computation of noise are taken from Blake (1984). These formulae are derived for propellers, predicting noise frequency spectrum based on rotational speed, rotor dimensions and inflow turbulence characteristics. In this thesis we do not go into the theory behind the empirical correlations, and for details about the nature of the modeled noise mechanisms we refer the reader to the original work of Blake (1984).

The main advantage of the semi-empirical model is fast to run, even on a PC. The consequence can be carried out using a spreadsheet or program. In the thesis ,Matlab is used for computation.

Shallow water acoustic propagation

Since tidal turbines will generally operate in shallow water, acoustic transmission losses and other shallow water propagation effects must be considered when predicting far field sound pressure level (SPL). However, at short distances, these effects may not require to be taken into account.

Far field

If the sound propagates to a receiver by repeated reflections from the seabed and surface, a water column than can be considered ‘acoustically shallow’ (Urick, 1975). At large distances, sound wave reflections as well as transmission losses should be taken into account when modeling the noise to provide a more accurate prediction of sound pressure levels seen by a receiver. These include the effects of water column velocity and temperature profiles, as well as bathymetry and the seabed medium. Figure.3 shows the acoustic propagation case for a tidal turbine.

Figure. : Schematic illustration of the underwater acoustic propagation problem, relating to the sound pressure received by a marine animal from a tidal turbine. (Lloyd, 2011)

Underwater propagation modelling consists of solving the Helmholtz equation in a water column, accounting for appropriate boundary conditions at the free surface and sea bed. The sea bed is treated as a fluid-fluid interface, and the sea surface is characterised as a pressure-release boundary condition, requiring Neumann type continuity conditions between the two fluids. Variations of sea bed media and topography can also be accounted for through boundary conditions.

Four common theories for predicting sound propagation in shallow water are:

Ray theory

Spectral method

Normal mode

Parabolic equation method

Tidal turbine noise has been identified to be generally low-frequency (< 1 kHz), and assessments are required at relatively short distances (< 1 km), so the most appropriate models are spectral and parabolic equations (Lloyd, 2011). Ray theory is more suited to higher frequencies, whilst normal mode method is preferred for longer ranges.

Near field

Assessment of received sound levels close to the device, where the most harm may be expected, may not require shallow water effects to be taken into account. In this study, most of cases are of near field type. The dominant noise sources are defined as point sources centred on the turbine hub height. The maximum observer distance is assumed to be less than twice the blade length away (Makarewicz, 2011). This corresponds approximately to the near/far field boundary identified by Lloyd (2011).

Tidal turbine noise sources

To predict tidal turbine noise, the first step is identifying the possible turbine noise sources. Tidal turbine noise sources can be categorized into three classes: Hydrodynamic noise sources, Mechanical noise sources and Structural noise sources. Figure.4 shows the potential noise sources due to operation of a tidal turbine. Among them, hydrodynamic noise is the dominant noise sources and compared with it, the other two noise sources (mechanical noise sources and structural noise sources) sometimes can be neglected.

Figure. : Categorisation of Hydrodynamic Tidal Turbine Noise Sources (Richards et al, 2007; Black, 1984)

Hydrodynamic noise sources

Table.1 characterises the hydrodynamic noise sources and suggests that the main hydrodynamic noise sources could be:

Unsteady loading

Trailing edge

Vortex shedding

Cavitation

Previous work by Lloyd et al (2011) has shown that noise due to the interaction of the turbine blades with ocean turbulence is the dominant hydrodynamic noise source. Thus this study will focus on noise due to inflow turbulence.

Type

Source

Origin

Importance

Directivity

Ref.

Self noise

Steady loading

Blade loading distribution

Larger than thickness but generally insignificant when addressing unsteady loading scenarios

Maximum aft of rotor between axis of rotation and plane of rotor

[23-25]

Thickness

Blade geometry

Negligible for subsonic rotors

Maximum in axis of rotation

[13, 24, 25]

Trailing edge

Eddy convection past blade trailing edge

Generally dominant at high frequency (> 1 kHz), but smaller than unsteady loading

Maximum in axis of rotation

[13]

Vortex shedding

Wake instabilities

Can be highest noise but appears as narrow band tone related to vortex shedding frequency. Affected by trailing edge shape

Maximum aft of rotor between axis of rotation and plane of rotor

[13]

Interaction noise

Unsteady loading

Non-uniform inflow velocity; inflow turbulence

Generally largest noise source at low frequency (< 1kHz)

Maximum in direction of turbine axis of rotation

[13]

Cavitation

Cavitation bubble collapse; blade-bubble interaction

Depends on propeller design and maintenance. Can be largest noise source if cavitation prevalent

Maximum in plane of rotor

[12, 26, Chap. 10]

Hydroelastic noise

Singing

Not fully understood. Vibration, related to vortex shedding

Tone similar to vortex shedding. Magnitude associated with trailing edge shape

Maximum aft of rotor between axis of rotation and plane of rotor

[13]

Chapter

Research methodology

This chapter describes the method and model used for predicting tidal turbine noise in this thesis. Section 3.1 gives a summary of the blade element momentum (BEM) theory for tidal turbine. Section 3.2 describes the empirical formulae implemented in this study for the computation of noise.

Blade element momentum theory

BEM has been used extensively in the design and analysis of wind turbines for decades and is often used for modeling and designing kinetic marine turbines. In this thesis BEM is used to define the ideal performance of tidal turbines operating in an unbounded domain.

Blade Element Momentum Theory contains two methods of examining how a turbine operates. The first method is momentum theory, using a momentum balance on a rotating annular stream tube passing through a turbine. The second is blade element theory, examining the forces generated by the aerofoil lift and drag coefficients at various sections along the blade. These two methods then give a series of equations that can be solved iteratively.

Momentum theory

Axial force

Figure 5 depicts the axial locations used throughout this thesis, which are denoted in the equations using subscripts. Consider the stream tube around a wind turbine shown in Figure 5. There are four stations in the diagram, 1) some way upstream of the turbine, 2) just before the blades, 3) just after the blades, 4) some way downstream of the blades. The turbine is represented as an infinitesimally thin disk occupying the swept area of the turbine, located between stations 2 and 3.

Figure. ：Axial stream tube around a tidal turbine

Assume that station 1 is undisturbed freestream and the far wake (station 4) is defined as the location where the pressure in the wake has recovered to ambient, so . Note that the actuator disk is assumed to be infinitely thin such that .

Between 2 and 3 energy is extracted from the tidal and there is a finite pressure drop (. We can also assume that between 1 and 2 and between 3 and 4 the flow is frictionless so we can apply Bernoulli’s equation:

After some algebra:

(3.1.1)

Define the axial induction factor as:

(3.1.2)

(3.1.3)

The turbine’s axial force is equal to the product of pressure drop and the disk area A2. Using steady flow momentum theory on the bounding stream tube between stations 1 and 4, a relationship between the change in axial momentum and the axial force applied to the flow by the turbine can be defined as follows:

(3.1.4)

Substituting for in equation 3.1.1 then yields:

(3.1.5)

To account for radial variation of the axial force, it is necessary to divide the disk into infinitesimal annular disks of radius , thickness and area . Defining the local thrust on a single annulus as . Note that the derivation of equation 3.1.1 to 3.1.5 also holds for such an annulus. Then the expression for the balance of axial force and momentum for each annulus is:

Substituting for and in equation 3.1.3 and equation 3.1.5 then yields:

(3.1.6)

Rotating Annular Stream Tube

Figure. : Rotating Annular Stream tube

Consider the rotating annular stream tube shown in Figure 6. There are four stations the same as Figure 5. Between station 2 and 3 the rotation of the turbine imparts a rotation onto the blade wake, which can be seen in end-on view.

Figure. : Rotating Annular Stream tube: notation

Consider the conservation of angular momentum in this annular stream tube. An ‘end-on’ view is shown in Figure 7. The blade wake rotates with an angular velocity and the blades rotate with an angular velocity of . Define as the moment of inertia of an annulus and as the angular moment of it:

From angular momentum balance theory, the rate of change in angular momentum of the fluid passing through an annulus is equal to the torque :

So for a small element the corresponding torque will be:

(3.1.7)

Define the angular induction factor as:

(3.1.8)

Recalling that and combining equations 3.1.7 and 3.1.8 gives a new expression for the balance of torque and angular momentum for each annulus:

(3.1.9)

Local torque is equal to the product of the local tangential force and radius .

(3.1.10)

Therefore we have the equations for the axial (equation 3.1.6) and tangential force (equation 3.1.10) on an annular element of fluid.

Blade Element Theory

Figure. : The Blade Element Model

Consider a blade divided up into several elements as shown in Figure 8. Each of the blade elements has a different rotational speed , different chord length and a different twist angle . Thus, each of them will experience a slightly different flow.

Blade element theory involves dividing up the blade into a sufficient number of elements and calculating the flow for each one. Numerical integration along the blade span gives the overall performance characteristics.

There are two key assumptions for Blade element theory:

• There are no aerodynamic interactions between different blade elements

• The forces on the blade elements are solely determined by the lift and drag coefficients

Relative Flow

The local axial and tangential forces acting on the flow in each annulus can be related to the blade lift and drag forces (), which are determined using tabulated lift and drag coefficients (). Lift and drag coefficient data area available for a variety of aerofoils from tunnel data. This study employs the lift and drag coefficients for a NACA 0012 aerofoil which is shown in Figure 11. Most flow tunnel testing is done with the aerofoil stationary, thus we need to relate the flow over the moving aerofoil to that of the stationary test. To do this we use the relative velocity over the aerofoil. The flow velocity relative to the blade has various components as shown in Figure 9.

Figure. : Diagram of the relative velocity at a turbine blade section.

The flow around the blades starts at station 2 and ends at station 3 as shown in Figure 6. The inlet flow is not rotating while at exit the flow rotates at rotational speed . That means over the blade row, wake rotation has been introduced. Thus the average of inlet and exit flow conditions should be used to obtain a more accurate estimate of aerofoil performance. The average flow rotational speed over the blade due to wake rotation is therefore:

The blade rotates at rotational speed . So the average tangential velocity that the blade experiences is therefore:

Recall that , we can get a new expression of blade average tangential velocity:

(3.1.11)

Define the inflow angle as the angle between X-axis and relative velocity , which is shown in Figure 9. At station 2 in Figure 6, the flow coming with speed , which is also equal to (Equation 3.1.3). The blade average tangential velocity is and so:

(3.1.12)

(3.1.13)

Where represents the incoming flow velocity . The value of will wary form blade element to blade element. Define the local tip speed ratio as:

(3.1.14)

Then the expression for can be further simplified:

(3.1.15)

Blade Elements

Figure. : Forces on the turbine blade

Figure 10 shows the forces on the blade element. is the blade twist angle and is the angle between relative velocity and the blade, which is also equal to (). and represent the lift and drag forces on the blade and they are perpendicular and parallel to the incoming flow respectively. For a blade element of chord length centered at with length , and can be found from the definition of the life and drag coefficients as follows:

(3.1.16)

(3.1.17)

Where and are life and drag coefficients. This study employs the data of NACA 0012 aerofoil as shown in Figure 11. Life and drag coefficients can be found for different value.

Figure. : Life and drag coefficients for a NACA 0012 aerofoil

From Figure 10 we can see that the local axial and tangential forces ( acting on each blade element can be related to the blade lift and drag forces:

(3.1.18)

(3.1.19)

(3.1.20)

If there are blades, substituting for and in equation 3.1.16 and 3.1.17 then yields:

(3.1.21)

(3.1.22)

(3.1.23)

Define the local solidity as:

(3.1.24)

Noting that in equation 3.1.21 and equation 3.1.23, and can be expressed in terms of induction factors and (equation 3.1.13 and equation 3.1.15). Manipulating this gives:

(3.1.25)

(3.1.26)

Therefore we have the equations for the axial force (equation 3.1.25) and torque (equation 3.1.26) on an annular element of fluid.

Tip Loss Correction

At the rotor plane, there is azimuthal variation in the axial induction. This causes a discrepancy between the induction used to calculate the change in the fluid momentum and the induction used to account for the blade forces. As a result, at the tip of the turbine blade losses are introduced. These can be accounted for in BEM theory by using a so-called tip loss model, the most common being the Prandtl tip loss model (Burton et al, 2001). (可以加具体内容大概3页)

In Prandtl tip loss model, the correction factor is defined as:

(3.1.27)

This correction factor varies from 0 to 1 and characterizes the reduction in forces along the blade. Applying the tip loss correction to equation 3.1.6 and equation 3.1.9 gives:

(3.1.28)

(3.1.29)

Blade Element Momentum Equations

Now we have four equations for local axial force and torque. Two of them are derived from momentum theory, which expressed in terms of flow parameters (equation 3.1.28 and equation 3.1.29). The other two are derived from blade element theory, which express the axial force and torque in terms of the lift and drag coefficients of the aerofoil:

(3.1.25)

(3.1.26)

Axial force / momentum balance is written by combining equation 3.1.25 and 3.1.28:

(3.1.30)

Angular force / momentum balance is written by combining equation 3.1.26 and 3.1.29:

(3.1.31)

Equation 3.1.30 and 3.1.31 represent a system of non-linear equation that are most often solved independently for each annulus.

Power Coefficient and Power Output

The tidal turbine performance can be determined by calculating the power coefficients , defined as:

(3.1.32)

Where is the rotor swept area. To find these quantities, it is useful to first define local power coefficient . The local power coefficient is found by normalizing the local power produced by a single annulus by the upstream kinetic power density and annular area . The contribution to the total power from each annulus is:

(3.1.32)

Substituting for in equation 3.1.26 then yields:

(3.1.33)

Combining equations 3.1.31 and 3.1.33 gives a new expression for power coefficient :

(3.1.34)

The total power coefficient is found by integrating the product of the local power coefficient and the local annular area divided by the total disk area:

(3.1.35)

The total power output for a tidal turbine is:

(3.1.36)

Where is the expected electrical and mechanical efficiencies (0.9 would be a suitable value). (Grant, 2011)

Above are the derived equations for the analysis of tidal turbines using the blade element method. These equations are then used in this study to define the ideal performance of tidal turbines operating in an unbounded domain and can be solved iteratively which is shown in next chapter.

Formulae for Noise

Previous work by the authors (Lloyd, 2011) has shown that noise due to the interaction of the turbine blades with ocean turbulence is the most dominant noise source of tidal turbines. This thesis will focus on and only consider the noise due to inflow turbulence.

The formulae used in this study for calculating the inflow turbulence noise are empirical formulae taken from Blake (1984). These formulae are derived for propellers at first but have been used successfully for predicting tidal turbine inflow turbulence noise (Lloyd, 2011). Using this method, the noise frequency spectrum can be predicted based on rotational speed, rotor dimensions and inflow turbulence characteristics.

Sound Pressure Level

In this study, we use sound pressure level () to assess the noise generated by tidal turbine. In a given medium, the pressure in the sound wave will be different from the average local pressure. This difference is called sound pressure. A square of sound pressure is defined as mean square pressure (, which is usually averaged over time and / or space. Sound pressure level is a logarithmic measure of the effective sound pressure of a sound relative to a reference value. It is measured in decibels (dB) above a standard reference level. The in decibels is calculated as:

(3.2.1)

Where represents the mean square pressure values and is a reference pressure of , which is the standard reference unit for underwater acoustics (Faber et al, 2007).