Keywords: Curiosity, questions, passion, investigations, experiments, measurements, patience, analysis

conclusions, and ethics all lead to a successful scientific researcher. Working on a project that benefits both humans and the environment is enough motivation to work hard and tirelessly

List of figures

List of Tables

g acceleration due to gravity [m.s -2 ] h heat transfer coefficient [W.m -2 .K -1 ] k thermal conductivity [W.m -1 .K -1 ] n number of conductors [-] q heat source [W] r radius [mm]

Preface

Nowadays, our global population is estimated to be around 7.9 billion people. Since the beginning of the industrial revolution, the number has increased tenfold, and billions more people are expected to be added to the world's population in the 21st century, creating a very complicated world with unprecedented environmental challenges. Furthermore, recent theoretical developments have revealed that by 2030, the entire world will be confronted with a serious ecological problem in the form of global warming, which will result in the deaths of a large number of people [START_REF] Sachs | Sustainable Development Goals for a New Era[END_REF]. percent in just a decade [3]. The European Commission assesses between 240 and 450 GW of offshore wind power is needed by 2050 to keep the temperature increases below 1.5°C and 50% by total energy mix in 2050 is estimated by electricity and 30 % of the future electricity need will be supplied by offshore wind [4]. However, in order to achieve this goal, wind energy locations must be relocated to water depths greater than 50 m, which can be accomplished by using a floating offshore wind turbine (FOWT). FOWT is one of leading developed renewable energy, it's more efficient than bottom-fixed offshore wind turbines and on-land wind turbines because wind speed is higher far from the coast than near the coast, where a small increase in wind yields a large increase in energy production.

For example, a turbine with a wind speed of 24 km/h can generate twice as much energy as a turbine with a wind speed of 19 km/h [START_REF]What are the advantages and disadvantages of offshore wind farms[END_REF]. The transportation of produced energy to shore is a significant challenge for any offshore wind farm. Floating systems require a dynamic cable, which connects the floating wind turbines to their electrical substation and underwater energy network. First, the floating offshore wind turbine sends its power through a dynamic power cable undersea to a transformer, and then the transformer sends its power through a static power cable to a converter platform. The alternating current is converted to high direct current and sent to a converter station on land, where it is converted into three phases of electric power. One significant challenge is that the specifications of dynamic cables are determined by project. The characteristics of the dynamic cable are determined by the unit power of the turbines (6 MW and 8 MW for experimental farms, but 10 MW to 12 MW for commercial farms), the general connection scheme (in series with at least two cables per wind turbine or in parallel), the architecture provided for the substations (embedded on a wind turbine float or independent float) and the environmental conditions of the site (water depth, swell, current and biofouling). Consequently, the cables are designed specifically and unitarily for each farm, with a wide range of solutions.

Moreover, according to the experience of floating offshore installations in the oil industry, the greatest stresses are manifested at the head of the cable, that is, at the point of connection with the part fixed to the float. To reduce these stresses, and thus the fatigue of this section of the cable, a bending device with floats and a tensioner is used to shape the section just before the connection into a "s" shape [START_REF] Cruciani | L'éolien offshore flottant dans sa dimension industrielle et technologie[END_REF]. As a result, while dynamic power cables are a key component in electrical connections, their design, which must account for electrical, mechanical, and thermal concerns, remains a challenge for manufacturers and any external factor reducing its efficiency will result in lower energy reception.

Among all the challenges, growth of biofouling has been identified, from sensitivity studies, as one of the most influent factor at design and maintenance stages. Marine growth embraces all the marine species that colonizes a component. Mollusc bivalves, particularly mussels, are a common and problematic component of fouling assemblies associated with vessel sea chests and pipework [START_REF] Sievers | Biofouling leads to reduced shell growth and flesh weight in the cultured mussel Mytilus galloprovincialis[END_REF], [START_REF] Nys | Biofouling[END_REF]. These species are experienced to be dominant in coast Atlantic areas [START_REF] Ameryoun | Stochastic Modeling of Forces on Jacket-Type Offshore Structures Colonized by Marine Growth[END_REF]- [START_REF] Schoefs | Reliability Updating of Offshore Structures Subjected to Marine Growth[END_REF]. Indeed, the marine growth of biofouling, particularly mussels, can modify heat transfer around the cable, affecting cable temperature, whereas according to IEC standard [START_REF]Electric cables-Calculation of the current rating-Current rating equations (100% load factor) and calculation of losses[END_REF], the cross-linked polyethylene (XLPE) electric insulation system of dynamic submarine electrical cable (DSEC) is designed to support a maximum copper wire temperature of only 90 o C continuously. This maximum conductor temperature is considered a limit to avoid wire insulation (XLPE) degradation. Furthermore, temperature fluctuations affect fatigue lifetime by increasing mechanical stresses, which leads to a decrease in electrical conductivity and, as a result, receiving less energy. There is a need to understand the complexity of each of them and investigate thermal effect.

Therefore, due to the high number of uncertainties in space (spatial distribution, type of species) and time (evolution competition with time), it's a challenge to investigate the thermal effect of mussels around the cable in order to determine how it will affect the heat transfer between cable and water.

The aim of this thesis is to assess the thermal effects of marine growth around a dynamic submarine electrical cable, also called umbilical cable. To accomplish this, the mussels should be thermally characterized, followed by an examination of the effect of mussel layer on the temperature of the DSEC copper conductor wire. There have been no

previous studies on the thermal characterization of mussels around cables, as far as we are aware from these properties. Another aim is to create a thermal sensor to track biofouling growth around the DSEC.

The thesis begins with an examination and interpretation of the current literature and state of knowledge concerning the floating offshore wind turbine, dynamic submarine electrical cable, biofouling, thermal effect of mussels, and methods to monitor biofouling growth (Chapter 1). Thermal characterization will be performed on mussels of three different ages: juvenile, mixed (juvenile and adult), and adult mussels only (Chapter 2).

First, to measure their effective thermal conductivity, and then to measure the heat transfer coefficient of the water around them. The thermal effect of mussels on the dynamic submarine electrical cable will be evaluated (Chapter 3). A method for monitoring the growth of biofouling will be developed and tested (Chapter 4). Finally, conclusions and perspectives are presented. Furthermore, because of the distance between the shore and the offshore, repair and maintenance operations take longer and are thus more expensive [START_REF] Weerheim | Development of dynamic power cables for commercial floating wind farms[END_REF]. As shown in Figure 1.1, there are three dominant types of floating platforms for wind turbines: tension leg platform (TLP), semi-submersible, and spar-buoy. These platforms allow for the installation of FOWT from 40 m to 500 m depth. It's expected in the coming 5-10 years a large amount of floating wind farm projects will be executed and to be ready for this development, the technology shall be ready as well. This requires better assessment of key uncertainties associated with the behaviour, degradation and corresponding models of the underwater components. Moreover, it requires improving the efficiency of structural health monitoring (SHM) through the use of new sensors, while taking into account the performance of embedded sensor networks, including relevant uncertainties, throughout the entire service life of a floating offshore wind farm. Also, it requires better understanding of the material behaviour and degradation of mooring lines and umbilical's, including their link to the environment such as soilanchor interaction and effect of marine growth, which will lead to more accurate modelling of these components over time. For instance, the submarine dynamic power cable's (umbilical cables) design and bio colonization effect is still a challenge for the research and development in order to transfer the maximum produced energy with less possible loss.

Introduction to Submarine power cable

The world's first powered underwater cable was laid down in the ISAR River in Germany in 1811, and after more than a century, the first commercial high voltage direct current (HVDC) cable linked Sweden and Gotland Island in the Baltic Sea in 1954 [START_REF] Taormina | A review of potential impacts of submarine power cables on the marine environment: Knowledge gaps, recommendations and future directions[END_REF].

Submarine power cables have continued to expand throughout the world with improved technologies in terms of materials, installation techniques, width, and length, with nearly 8000 km of HVDC existing on the seabed worldwide in 2015, with more than 70% of which located in the European adjacent seas [START_REF] Taormina | A review of potential impacts of submarine power cables on the marine environment: Knowledge gaps, recommendations and future directions[END_REF]. Submarine power cables are classified into two types, the first type is called static cable which comes on top of or buried within the seafloor and the second type is called the dynamic cable (or Umbilical) which are deployed through the water column between the surface and the seafloor as shown in The static cable and dynamic cable differ primarily in two ways. The first is that the dynamic cable has double armoring to increase torsional stiffness due to dynamical application, whereas the static cable has a single armoring. The second distinction is that dynamic cable has a larger conductor cross-sectional area than static cable, which reduces induced heat [START_REF] Weerheim | Development of dynamic power cables for commercial floating wind farms[END_REF].

Figure 1.3: Structure of Medium Voltage power cable [START_REF] Weerheim | Development of dynamic power cables for commercial floating wind farms[END_REF].

There are numerous hanging configurations for dynamic cable from the FOWT to the sea bed in order to connect with the electric substation or to interconnect the wind farm turbines in the case of shallow water depth. As illustrated in Figure 1.4, the most wellknown configurations are free hanging, lazy wave, steep wave, lazy-S, steep-S, and plaint wave. The Lazy wave is a popular hanging configuration due to its suitability for dynamic cables and features such as lower maximum tension force, reduced cable response, and fewer fatigue cycles. Figure 1.6: W-shape interconnection configuration between two floating offshore wind turbine [START_REF] Srinil | Cabling to connect offshore wind turbines to onshore facilities[END_REF].

Based on the previous configurations and as depicted in Figure 1.7, the dynamic cable is subjected to various loads such as cable weight, hydrostatic pressure, buoyancy force, marine growth, and sea current. The presence of buoyant modules increases the buoyancy force and prevents the cable from touching the seabed, with the density of buoyancy modules being smaller than the surrounding water to create buoyancy force, and it is clearly shown in Figure 1.7 that the weight of the cable induces axial and bending forces, and these forces are increased with the presence of the marine growth (Biofouling), and thus the buoyancy modules lose their buoyancy over time.

Figure 1.7: Mechanical loads on dynamic power cable.

Biofouling

The formation of unfavorable inorganic and/or organic residues on surfaces is referred to as biofouling. These residues increase fluid frictional resistance at the surface, accelerate corrosion at the surface, and can obstruct heat flow across the surface; regardless, energy losses occur [START_REF] Characklis | Bioengineering report: Fouling biofilm development: A process analysis[END_REF]. Once immersed in seawater, unfavorable fouling directly covered a structure. Over 4000 organisms have been linked to fouling problems in marine environments [START_REF] Yebra | Antifouling technology-past, present and future steps towards efficient and environmentally friendly antifouling coatings[END_REF]. Organisms are classified into two types based on their size: microorganisms (also known as micro-fouling or bio-film) and macro-organisms (also known as marine growth). Fouling is generally thought to progress in five stages, as illustrated in Figure 1.8 [START_REF] Delauney | Biofouling protection for marine environmental sensors[END_REF]:

-First, after surface immersion, organic and inorganic macromolecules adsorb and form the primary film.

-Second, microbial cell transport to and immobilization of bacteria on the immersed surface.

-Third, as shown in Figure 1.9, the bacterial attachment to the substrate is solidified by the production of extracellular polymers, resulting in the formation of a microbial film on the surface.

-Fourth, the presence of multicellular species, microalgae, residues, ruins, and other materials on the immersed surface leads to the evolution of a more complex collection.

-Fifth, larger marine invertebrates such as mussels, barnacles, marco-algae, and so on form a bond. However, strict and comprehensive information about the fouling process is being researched. According to some references, some of these stages may occur concurrently or may not exist; for example, macro-organisms can exist on an immersed surface without the presence of biofilm, as mentioned by [START_REF] Delauney | Biofouling protection for marine environmental sensors[END_REF].

Figure 1.9: Biofilm cross section [START_REF] Bixler | Biofouling: lessons from nature[END_REF].

Because of its large effect on hydrodynamic loading, thermal resistance, and weight, macro-fouling is generally more dangerous to offshore structures, instruments, and many organisms, such as hydroids and soft corals, outgrow mussel colonization [START_REF] Compere | BIOFOULING and UNDERWATER MEASUREMENTS. Real-Time Coastal Observing Systems for Marine Ecosystem Dynamics and Harmful Algal Blooms: Theory, instrumentation and modelling Edition: Oceanographic Methodology series[END_REF].

Marine mussels are bivalve mollusks. This means they have two hard shells around their soft bodies to protect them (Figure 1.10). Mussels, like other bivalves (oysters and clams), can help to improve water quality and support healthy marine habitats. This is due to the fact that mussels are filter feeders, which means they are constantly bringing sea water into their bodies, filtering out phytoplankton and sediments while looking for their food, and they can filter hundreds of liters of seawater per day.

Figure 1.10: Anatomy of mussels [START_REF]Mussels Anatomy[END_REF].

Mussels must therefore collect food and oxygen from the water in order to survive.

They accomplish this by drawing water in through their incurrent siphon, moving seawater over their gills, and then passing the water out through their excurrent siphon. The idea is to suck up the surrounding water along with the particles that are suspended in it. The assembly enters the mantle cavity from the ventral side and passes through the gills (called ctenidia in molluscs). These can have a net-like structure that allows water to pass through (and thus allows breathing) while also retaining particles in suspension. The elements blocked at the gill level have two possible futures, depending on their size and palatability.

Siphoned material is either transferred to the mouth for digestion or sloughs off the gills and exits through the ventral margin of the shell (pseudofeces). Digested material is either burned as fuel or excreted as feces. Because biofouling evolution is the result of a variety of chemical, physical, and biological factors, there are numerous parameters that influence the type of biofouling produced and its rate of evolution. Temperature, location, turbidity, depth, dissolved oxygen content, and other factors all play a role in the growth rate and type of biofouling.

Several references show that marine growth depends on where it's located. In general, fouling is denser in more tropical locations doubtless due to the warmer temperature of seawater and to the continuous process of breeding. Some organisms may also fit to different environmental conditions and expand over big areas such as North Polar regions to the Mediterranean, while others are confined to a very restricted region.

A significant portion of the fouling organisms are carried away by the currents from the shoreline to structure sites. Barnacles and mussels needed three weeks or more to expand and feed while being transported before settling on an appropriate substrate. As a result, while on the coast, they may sweep away from their originating point.

Many physicochemical parameters of water, such as temperature, dissolved oxygen, turbidity, salinity, organic content, and light penetration, change as depth changes.

All of these variables influence the type of fouling and its growth. In general, the amount of fouling is greater near the shore; however, as seawater depth increases, the amount of fouling decreases. According to [START_REF] Amram | Sedimentation and fouling of optical surfaces at the ANTARES site[END_REF], the density of bacteria after 3 months or 8 months at deep sea (>2000 m depth) is comparable to samples exposed for 1-2 weeks in shallow water, where the temperature measured at the ANTARES site is 13.2 °C [START_REF] Amram | Sedimentation and fouling of optical surfaces at the ANTARES site[END_REF]. They are explained by the low temperature and poor nutrient quality at these low levels.

Furthermore, bacteria found in the deep sea are smaller than those found in shallow water.

Increasing the depth usually results in a change in the dominant species and a decrease in thickness. Some species, such as algae, are photosynthetic and can only survive in areas with adequate light. Because mussels, tubeworms, barnacles, ascidians, and hydrozoans get their food from nutrients rather than light, fouling can occur at great depths [START_REF] Compere | BIOFOULING and UNDERWATER MEASUREMENTS. Real-Time Coastal Observing Systems for Marine Ecosystem Dynamics and Harmful Algal Blooms: Theory, instrumentation and modelling Edition: Oceanographic Methodology series[END_REF].

Temperature and access to light have a large impact on fouling growth and bacterial colonization on surfaces immersed in natural seawater. It should be noted that temperature is a significant parameter, as an increase in growth rate is most commonly observed with an increase in temperature. In temperate and cold climates, fouling growth appears from April to September; however, in tropical locations, fouling growth appears at a high and consistent level. It should be noted that some soft marine organisms die during the winter and live as residue. However, environmental changes affect the growth and reproduction rates of some soft marine species.

Water movement, which includes currents, tides, circulations, and horizontal and vertical water exchanges, influences the organism's settlement. Any changes in water movement have a large impact on nutrition, metabolic activities, life cycles, and food chains, so water movement is undeniably important for marine plant life. The degree of exposure to wave wash has a significant impact on algae, and some species, such as hydroids, are highly affected by water movement [START_REF] Deacon | Marine Ecology: A Comprehensive Integrated Treatise on Life in Oceans and Coastal Waters[END_REF].

For marine ecologists, salinity and nutrients play a key role. Salinity varies from location to location, and as a result, the biology of species varies. For example, in the Red Sea, salinity is 41 percent because evaporation is greater than fresh water input, whereas in the Baltic Sea, where we have fluvial input, salinity is 10 percent [START_REF] Deacon | Marine Ecology: A Comprehensive Integrated Treatise on Life in Oceans and Coastal Waters[END_REF]. Water turbidity has a significant impact on the penetration of light inside water; if the turbidity is high, light penetration is difficult, affecting the growth of some species such as algae. In addition to salinity and water turbidity, nutrients availability has a significant impact on fouling rates. Ocean waters are typically lower in nutrients than water near the coast due to natural domestic or industrial effluents, as well as the presence of a high biodiversity and as a consequence, a large number of fouling larvae. As a result, sites near the coast generally have higher fouling rates.

Previous factors vary depending on geographical location, distance from the coast, depth, temperature and season, water movement, and water quality which may influence the settlement of fouling on the structure. Other factors should also be structured to embrace all the complexity of biocolonization, such as, the competition between species and the structure nature. In terms of species competition, the limited space on a structure makes it difficult for some species to colonize it. For example, some species grow in the summer and die in the winter; in that case, the following summer they cannot find space to settle on structure because the space is occupied by other species [START_REF] Compere | BIOFOULING and UNDERWATER MEASUREMENTS. Real-Time Coastal Observing Systems for Marine Ecosystem Dynamics and Harmful Algal Blooms: Theory, instrumentation and modelling Edition: Oceanographic Methodology series[END_REF]. In terms of structure nature, the rate of bacterial colonization is influenced by a number of material surface factors. According to [START_REF] Schmidt | Aggregation kinetics of saccharomyces cerevisiae on solid surfaces[END_REF], negative surface charge reduces the extent of bacterial biofouling while increasing the mortality rate of adhering bacteria. The roughness of the surface has been investigated by [START_REF] Schmidt | Aggregation kinetics of saccharomyces cerevisiae on solid surfaces[END_REF]. It demonstrates that stainless steel with highly polished or extremely rough surfaces has a rapid growth rate. The growth rate of the yeast colonies, however, is the slowest for intermediate roughness. In the case of a highly polished surface, the physicochemical principle of adhesion dominates and is regarded as the cause of colonization, as a result of the presence of a larger flat adhesion area in comparison to a rougher surface. In the case of very rough surfaces, the hydrodynamic theory takes precedence, with the zone near a rough area characterized by a turbulent stream that transports the cell to the surface valleys. It appears as a small region of turbulence denoting a maximum shear force for the intermediate roughness, which has the lowest colonization. As a result, this depicts the equilibrium between adhesion and cell motion as influenced by the flow. The material nature of the surface also plays a role in the settlement of species; for example, for short exposure times (100 h or less), PET (Polyethylenterephthalate) outperforms PMMA (Poly methylmethacrylate) in fouling resistance because PET has a surface that is less suitable for biofouling adherence [START_REF] Schmidt | Aggregation kinetics of saccharomyces cerevisiae on solid surfaces[END_REF].

The effect of fouling in heat exchangers, as well as the method of detecting its growth, will be discussed in the following section.

Fouling effect on shell and tube heat exchanger

The presence of fouling in a heat exchanger can cause the flow to stop or change direction. Its nature is also less conductive than the separating wall, which increases heat transfer resistance. As a result, the heat exchanger's efficiency is reduced. Because of the random nature of this phenomenon, fouling quantification is regarded as one of the most difficult challenges. Because systematic periodic maintenance is not the best solution for customers, predictive maintenance is the best way to reduce loss. The reduction of the overall heat transfer coefficient is a traditional method for predicting the presence of biofouling. However, this method works when we have locally 1 D heat transfer, which consists of ignoring axial heat transfer in solid walls; additionally, compensation can occur between conduction and advection. The overall heat transfer coefficient is then not very sensitive to fouling in that case. Another popular method for detecting biofouling in heat exchangers is pressure drop. It should be noted that changes in flow rates cause a variation in pressure drop over time; however, the effectiveness of this method is dependent on the heat exchanger plate pattern [START_REF] Deacon | Marine Ecology: A Comprehensive Integrated Treatise on Life in Oceans and Coastal Waters[END_REF]. There are also complex methods for detecting fouling, such as silicon sensors, acoustic, ultrasonic, and optical techniques [START_REF] Amram | Sedimentation and fouling of optical surfaces at the ANTARES site[END_REF]. These methods are restricted to laboratory use because they necessitate specialized equipment that is not always available. A useful detection technique is to compare the system signature obtained experimentally during operation with the reference signature, where the reference model must be identified from a calibration experiment in the absence of fouling in the heat exchanger. This signature is valid for a system with linear time invariance (LTI). Here, the system signature agrees with its impulse responses or transfer functions, where the transfer function is the Laplace transform of the impulse response. These functions can be specified for any complex geometry, and it should be noted that they are solely dependent on the velocity field of the flowing fluid, the faced thermophysical properties, and the geometry.

As a result, these advantages enable the detection of temporal differences in the system's structure during operation. The following are the main steps of a thermal detection method for fouling in heat exchangers [START_REF] Hadad | Fouling detection in a shell and tube heat exchanger using variation of its thermal impulse responses: Methodological approach and numerical verification[END_REF]:

a. The location and type of thermal perturbation u(t), as well as the location of the output y, must be determined y(t). With the assurance that the perturbation is small enough to keep the system's response in the linear domain.

b. In the absence of biofouling, the impulse response Hwithoutfouling(t) for a given mass flow rate of the fluid must be identified first and used as a reference.

c. For the same mass flow rate values, the impulse response Hoperation(t) is identified during the operation. This is done on a regular basis throughout the operation.

d. If a difference between Hoperation(t) and Hwithoutfouling(t) appears during the operation, it indicates the presence of biofouling.

Table 1.2 depicts the impulse form and its response. H(t) expressions are only valid when the response is not instantaneous.

Table 1.2: Impulse forms and impulse responses expression [START_REF] Hadad | Fouling detection in a shell and tube heat exchanger using variation of its thermal impulse responses: Methodological approach and numerical verification[END_REF].

In [START_REF] Hadad | Fouling detection in a shell and tube heat exchanger using variation of its thermal impulse responses: Methodological approach and numerical verification[END_REF], an impulse response is used in both cases where there is no fouling and when there is fouling. To begin the calibration, the impulse response without fouling is determined using the average temperature at the hot and cold fluid outlets, 𝜃 𝑜𝑢𝑡 ℎ𝑜𝑡 and 𝜃 𝑜𝑢𝑡 𝑐𝑜𝑙𝑑 , respectively. The impulse response H, which connects the excitation to its temperature response, is denoted by Z. As shown in Figure 1.12, it is considered a source q and its responses Zh (for 𝜃 𝑜𝑢𝑡 ℎ𝑜𝑡 ) and Zc (for 𝜃 𝑜𝑢𝑡 𝑐𝑜𝑙𝑑 ).

Figure 1.12: Shell and tube heat exchanger [START_REF] Hadad | Fouling detection in a shell and tube heat exchanger using variation of its thermal impulse responses: Methodological approach and numerical verification[END_REF].

The excitation source is experimented in two forms step and ramp. The evolution of these two heat sources and its responses are presented in Figure 1.13a and 1.13b respectively.

Figure 1.13: a) Step excitation q and corresponding average hot and cold outlet temperatures. b) For Ramp Excitation [START_REF] Hadad | Fouling detection in a shell and tube heat exchanger using variation of its thermal impulse responses: Methodological approach and numerical verification[END_REF].

Starting from these synthetic profiles, the impulse responses Zh and Zc are identified as shown in Figure 1.14 using the expressions in Table 1.2. The new impulse response (with fouling) is identified using the expression in Table 1.2 and plotted in Figure 1.16 while maintaining the same flow rate as in the previous case (without fouling). changes the area under the curve, which is larger for Zc and smaller for Zh. Furthermore, it necessitates an increase in relaxation time. As a result, the presented method is dependent on the various pulse responses that distinguish the system with fouling. The identified impulse responses account for all effects that are difficult to model implicitly, such as turbulence, contact resistance, wall roughness. As a result, the presence of fouling increases heat transfer resistance, reducing the efficiency of the equipment where heat transfer is required. The section that follows demonstrates how the presence of biofouling on submarine power cable reduces the cable's efficiency.

Thermal effect of mussels on dynamic cable

The presence of mussels on the surface of dynamic cable forms an insulating layer.

Furthermore, the irregularities in the shape of the mussel layer change the water flow regime around the cable. As a result, the temperature of the cable rises. In general, biofouling can be modeled using four main parameters: roughness, density, thickness, and surface coverage percentage. A. MATINE et al, investigates the thermal effect of mussel thickness and surface coverage on a 33 kV XLPE insulated power cable composed of 3x475 mm 2 copper conductors used to handle a 30 MW wind turbine [START_REF] Matine | IREENA laboratory[END_REF]. For the thickness test, it is assumed that the entire surface is colonized with mussels with thicknesses ranging from 0 to 40 mm. The steady-state temperature distribution through the dynamic cable subjected to a current of 1010 A and immersed in water at 10°C with a biofouling thickness of 30 mm is depicted in Figure 1.17. In the two previous studies, the thermal conductivity of the biofouling layer was assumed to be equal to that of water. On the other hand, the cluster of mussels colonized around the DSEC (dynamic submarine electrical cable), is considered a porous medium composed of a solid part (mussels) and a fluid part (water between the mussels). So, in order to have an accurate and clear picture of the thermal effect of mussels on the submarine high voltage dynamic cable, the thermal properties of mussels, namely their effective thermal conductivity and the heat transfer coefficient of the water around them, must be estimated. This thermal property's estimation methodology will be shown in Chapter 2.

Effective thermal conductivity in porous media

Heat conduction through porous media occurs in a wide range of engineering applications, including permafrost soil mechanical stability, soil organism living conditions, nuclear waste disposal, energy production in geothermal engineering, materials drying, and the design and applications of compound materials, thermal characterization of marine growth, and so on [START_REF] Huai | Analysis of the effective thermal conductivity of fractal porous media[END_REF]. The effective thermal conductivity of a porous media is influenced by the thermal conductivities of the solid part and the fluid part. As shown in Figure 1.20, porous media frequently have highly complex structures, thus, heat conduction is also extremely complicated. Effective thermal conductivity is influenced by a variety of factors such as solid matrix properties, pore size and spatial distribution, the type of components, and state of fluid in pores, pressure, and temperature [START_REF] Huai | Analysis of the effective thermal conductivity of fractal porous media[END_REF]. The problem of heat conduction in porous media is mathematically equivalent to the problems of electrical conductivity, permittivity, and magnetic permeability in such materials. Several models and formulae for predicting effective thermal conductivity in two-phase systems have been proposed in [START_REF] Pietrak | A review of models for effective thermal conductivity of composite materials[END_REF]. All of the models predict effective thermal conductivity as a function of each phase's thermal conductivity and volume content, which is defined as the fluid volume to total volume ratio. Some models take into account additional parameters such as the shape of the solid particles, their orientation and distribution, and their contact resistance [START_REF] Tavman | Effective thermal conductivity of granular porous materials[END_REF].

Figure 1.20: Schematic of a porous medium [START_REF] Hsiao | Modified effective thermal conductivity due to heat dispersion in fibrous porous media[END_REF].

Since the 19 th century, numerous analytical expressions for estimating the effective thermal conductivity of composite materials have been proposed. Many studies have resulted in various theoretical models for estimating the effective thermal conductivity of porous media with various geometrical structures, such as the Maxwell [START_REF] Maxwell | A treatise on electricity and magnetism[END_REF], Rayleigh [START_REF] Lord Rayleigh | LVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium[END_REF],

Russell, Eucken, and Loeb [START_REF] Tonggeng | Thermophysical properties and influencing factors of polyimide/silica composite films[END_REF] models based on cylindrical or spherical pore structures.

Zehnder and Schlunder [START_REF] Yu | Two Effective Thermal Conductivity Models for Porous Media with Hollow Spherical Agglomerates[END_REF] proposed a correlation for stagnant thermal conductivity based on a one-dimensional heat flow model for heat conduction through a packed bed of spherical particles, assuming point contacts of particles in the heat flow direction.

Zimmerman [START_REF] Cosenza | Effective medium theories for modelling the relationships between electromagnetic properties and hydrological variables in geomaterials: a review[END_REF] used an effective medium theory to present a thermal conductivity model for fluid-saturated rocks. Krupiczka [START_REF] Tavman | Effective thermal conductivity of granular porous materials[END_REF] developed a numerical solution for the effective thermal conductivity of porous materials by first using a model composed of long cylinders and then another composed of spheres in a cubic lattice. With spherical inclusions, Verma et al. [START_REF] Chatterjee | Heat conduction model based on percolation theory for thermal conductivity of composites with high volume fraction of filler in base matrix[END_REF] developed an expression for predicting the effective thermal conductivity. Hsu et al. [START_REF] Hsu | A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media[END_REF] study developed a lumped parameter model for the effective stagnant thermal conductivity of some spatially periodic two-dimensional and three-dimensional media.

Also, Yu and Cheng [START_REF] Cheng | Evaluation of effective thermal conductivity from the structure of a packed bed[END_REF] developed a fractal thermal conductivity model for both mono and bi-dispersed porous media by assuming that porous media are divided into two parts: some particles contact each other to form tortuous chains, while others do not. Woodside

and Messmer [START_REF] Woodside | Thermal Conductivity of Porous Media. I. Unconsolidated Sands[END_REF] propose a model that combines the series distribution (in which the solid (spheres) and fluid phases are in layers normal to the direction of heat flow) and the parallel distribution (in which the solid (spheres) and fluid phases are in layers parallel to the direction of heat flow).

The irregularity of the microstructure complicates the mechanism of heat transfer in porous materials, which makes determining the overall effective thermal conductivity of a porous medium difficult. In such materials, heat is transmitted through thermal conduction in the solid phase, thermal conduction in the fluid phase, radiation between solid particles, and convection in the fluid phase between particles. Radiation contributes to thermal conductivity in porous materials at temperatures above 200°C. It is also known that with values of Rayleigh number, Ra=Gr.Pr > 10 3 , convective heat transfer in pores (fluid phase) cannot be ignored [START_REF] Luikov | Heat and Mass Transfer in Capillary-Porous Bodies[END_REF], where Gr is the Grashof number and Pr is the Prandtl number. However, none of the previous models take into account the natural convection effect in pores between the particles. As a result, it is worthwhile to consider it if it exists during the computation of the effective thermal conductivity of any porous medium.

Monitoring of biofouling

As previously stated, biofouling is a significant disturbance factor for bodies immersed in sea water. As a result, it is critical to monitor biofouling growth to determine whether or not the level of growth is significant. Methods for monitoring biofouling are introduced in this section.

The most well-known monitoring method is underwater inspection, which can be performed by divers or remotely operated vehicles. Remotely operated vehicles can travel deeper and for longer periods of time than divers, but they are more expensive, inflexible, and vulnerable to external factors. Divers, on the other hand, can cover a large area and inspect at various scales, from detailed close-ups to broader visual evaluations [START_REF] O'byrne | Image-Based Damage Assessment for Underwater Inspections, 1st ed. Boca Raton : Taylor & Francis, a CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic[END_REF]. The quality of the inspection is determined by the diver's ability to supervise and provide a thematic description of defects, which is usually supported by photographs, but most of the time the quality of the photographs is poor and useless for quantitative analysis. However, because the extraction of quantitative information from images using computational techniques is becoming increasingly popular and useful, it is critical to have images that are suitable for quantitative analysis. As a result, a protocol was established, as shown in It can be seen that the shape of the reconstruction is affected by the quality of the imagery, which is affected by the acquisition criterion. So this protocol works well; however, it is highly dependent on the following factors: weather, water turbidity, and water depth. As a result, it's interesting to have a sensor independent from the latter factors.

A novel approach to biofouling monitoring will be proposed in Chapter 4, which is a thermal sensor in which the equivalent heat transfer coefficient is monitored.

Partial Conclusion

In this Chapter, a floating offshore wind turbine bibliography and the significance of power dynamic cable for transferring energy from the offshore to the shore have both been introduced. In addition, bibliography on biofouling and the key elements that influence its growth is proposed. Additionally, it is demonstrated how biofouling can affect the dynamic power cable in terms of thermo-mechanics. The predominance of mussels in Atlantic coasts is highlighted. In addition, the effect of fouling on heat exchangers and how to detect it have been discussed. The effect of biofouling on submarine power cables is also discussed, as well as methods for detecting it, where the thermal conductivity of the biofouling layer was assumed to be the same as that of water. However, in reality the mussel clusters that colonized the submarine electrical cable are considered to be a porous media formed from a solid part, the mussels' bivalve itself, and a fluid part, the water between them.

Therefore, it is interesting to estimate the thermal properties of mussels, specifically their effective thermal conductivity and the water's heat transfer coefficient around them, in order to quantify accordingly the thermal impact of mussels on the dynamic submarine electrical cable. Furthermore, a review of the literature on the equations that can be used to determine the effective thermal conductivity in a porous medium has been provided. It

shows that all of the introduced models and formulas for estimating effective thermal conductivity don't take in consideration the natural convection in the porous medium. The following Chapter introduces an experimental method for measuring the effective thermal conductivity of a porous medium (mussels) while taking natural convection in the porous media into account.

Chapter Thermal characterization of mussels -Steady-state analysis 2.1 Introduction

As stated in Chapter 1, umbilical (dynamic) cable is an important component for offshore operators for two reasons: first, cable manufacturers have received few feedbacks on these new products (unlike static submarine cables), and second, dynamic cables are custom-designed for each new MRE farm project. Biofouling, in particular the growth of mussels, can affect cable cooling by increasing a thermal resistance that lessens the heat transfer between the cable and the seawater, affecting its maximum operating temperature and, more importantly, temperature fluctuations, which can affect fatigue lifetime. To determine the thermal impact of mussels on the dynamic submarine electrical cable, it is therefore necessary to thermally characterize the mussels. In this chapter, the mussels will be thermally characterized by estimating the effective thermal conductivity of mussels of varying ages as well as the heat transfer coefficient of the water surrounding them.

The effective thermal conductivity of the mussels will be estimated using 1D analytical stationary model (Fourier's law) valid for a uniform distribution of the mussels around the experimental tube, i.e. mussels are distributed around the experimental tube with same thickness. The measurement method will be validated by measuring the thermal conductivity of a double-sided foam adhesive and comparing it to a measurement from a hot guarded plate device. In addition, the heat transfer coefficient of the water around the mussels will be also calculated using Newton's law. It will be also compared to two literature correlations. Furthermore, mussel distributions that are not uniform around the tube will be considered. In practice, mussel growth occurs undersea with non-uniform colonization around the DSEC. In this case, due to more complicated geometry, the effective thermal conductivity of mussels of different ages and heat transfer coefficient of the water around them will be estimated using numerical method to solve the 2D heat transfer equation and a parameter estimation technique.

The extraction protocol and experimental setup for mussels

The stationary measurements are carried out using an experimental tube as shown in The mussels of various ages (juvenile (6 months), mix (juvenile and adults), and adult (12 months), as shown in Table 2.1, are kept alive in buckets filled with sea water at the site. To avoid mixing, each bucket has been numbered according to the age of the mussels, then they were transferred to the LTeN laboratory which is far from the mussel's extraction site around one hour and half. When the mussels arrived at the lab (LTeN), small pumps connected to small hoses were placed in the buckets to create small bubbles that increased oxygen in the water, allowing the mussels to live as long as possible.

Table 2.1: Various ages of mussels with their thickness layer.

Thermal characterization for a uniform colonization of different ages of mussels around the tube

In this section, mussels with different ages are distributed uniformly on the experimental tube, i.e. the mussels completely cover the experimental tube with the same average thickness of patches along the tube.

Measurement method

During the experiment, an aluminium tube covered with a uniform distribution of mussels is immersed in a tank filled with sea water, and the power provided by the heating elements inside the tube is used to reach a steady state temperature with a temperature discrepancy of about 5 to 10 K between both sides (internal and external) of the mussel layer. Fourier's law is directly used to calculate the effective thermal conductivity of mussels (kbiof) in case of uniform colonization (100 % coverage of mussels around the tube with same thickness (Figure 2.7) and this using temperature discrepancy between the two sides of the mussel layer and the heat power q (W) crossing it (Equation 2.1). In addition, the heat transfer coefficient between the external mussel layer and water (hw) is found from the expression of Newton's law (Equation 2.2).

𝑘 𝑏𝑖𝑜𝑓 (𝑊. 𝑚 -1 . 𝐾 -1 ) = ln( 𝑟𝑒 𝑟𝑖 ) * 𝑞 2𝜋 * 𝐿 * (𝑇 𝑎𝑣1 -𝑇 𝑎𝑣2 ) (2.1) ℎ 𝑤 (𝑊. 𝑚 -2 . 𝐾 -1 ) = 𝑞 2𝜋 * 𝑟 𝑒 * 𝐿 * (𝑇 𝑎𝑣2 -𝑇 𝑤 ) (2.2)
where Tav1 is the average temperature on the internal side of the mussel layer (average of T1, T2 and T3) measured by the three thermocouples in the middle cross-section and located in the aluminum tube. Tav2 is the average temperature on the external side of the mussel layer (average of T4, T5 and T6) and Tw is the temperature of the water far away from the aluminum tube. In Equations (2.1) and (2.2), ri and re are respectively the internal and external radius of the mussel layer and L its length. With noting that in accordance to the law of the propagation of uncertainties [START_REF] Moffat | Describing the uncertainties in experimental results[END_REF], the absolute uncertainty on the effective thermal conductivity of mussels is obtained by:

𝛿𝑘 = √ ( 𝑑𝑘 𝑑𝐿 𝛿𝐿) 2 + ( 𝑑𝑘 𝑑𝑇 𝑎𝑣1 𝛿𝑇 𝑎𝑣1 ) 2 + ( 𝑑𝑘 𝑑𝑇 𝑎𝑣2 𝛿𝑇 𝑎𝑣2 ) 2 + ( 𝑑𝑘 𝑑𝑈 𝛿𝑈) 2 + ( 𝑑𝑘 𝑑𝐼 𝛿𝐼) 2 + ( 𝑑𝑘 𝑑𝑟 𝑖 𝛿𝑟 𝑖 ) 2 + ( 𝑑𝑘 𝑑𝑟 𝑒 𝛿𝑟 𝑒 ) 2 (2.3)
where U and I are respectively the measured voltage and current through the heating element (q=U I) it follows then:

𝛿𝑘 = √ ( -ln( 𝑟 𝑒 𝑟 𝑖 ) * 𝑈 * 𝐼 2𝜋𝐿 2 (𝑇 𝑎𝑣2 -𝑇 𝑎𝑣1 ) 𝛿𝐿) 2 + ( ln( 𝑟 𝑒 𝑟 𝑖 ) * 𝑈 * 𝐼 2𝜋𝐿(𝑇 𝑎𝑣2 -𝑇 𝑎𝑣1 ) 2 𝛿𝑇 𝑎𝑣1 ) 2 + ( -ln( 𝑟 𝑒 𝑟 𝑖 ) * 𝑈 * 𝐼 2𝜋𝐿(𝑇 𝑎𝑣2 -𝑇 𝑎𝑣1 ) 2 𝛿𝑇 𝑎𝑣2 ) 2 + ( ln( 𝑟𝑒 𝑟𝑖 ) * 𝐼 2𝜋𝐿(𝑇 𝑎𝑣2 -𝑇 𝑎𝑣1 ) 𝛿𝑈) 2 + ( ln( 𝑟𝑒 𝑟𝑖 ) * 𝑈 2𝜋𝐿(𝑇 𝑎𝑣2 -𝑇 𝑎𝑣1 ) 𝛿𝐼) 2 + ( -𝑈 * 𝐼 2𝜋𝐿𝑟 𝑖 (𝑇 𝑎𝑣2 -𝑇 𝑎𝑣1 ) 𝛿𝑟 𝑖 ) 2 + ( 𝑈 * 𝐼 2𝜋𝐿𝑟𝑒(𝑇 𝑎𝑣2 -𝑇 𝑎𝑣1 ) 𝛿𝑟 𝑒 ) 2 (2.4)
Also, the uncertainties on the heat transfer coefficient hw is calculated similarly as the one of thermal conductivity.

The thermal characterization method and experimental setup were tested using a material double sided adhesive tape, as shown in Figure 2.8. The thermal conductivity of the double sided adhesive tape was initially measured using a hot guarded plate (HGP) device with a 5 percent relative uncertainty. The result shows that the thermal conductivity computed using equation (2.1) and measured with our experimental setup with aluminum tube was 0.052 W.m -1 .K -1 (for double sided tape adhesive not covered with steel net) and 0.053 W.m -1 .K -1 (for double sided tape adhesive covered with steel net), whereas the one with the HGP device was equal to 0.055 W.m -1 .K -1 therefore with only a 5.45% and 3.64% relative discrepancy, respectively. This validation confirms also that the steel net (used to maintain the mussels) has no significant effect on the effective thermal conductivity value.

It also confirms that the tube extremities connected to the sample holder are well thermally insulated. In the addition to that, the aspect ratio between the thickness of the sample and the experimental tube's length is very small ( 1.9 610 = 0.0031). In other words, the axial heat flux along the aluminum tube is negligible in comparison to the radial one, which explains why the direction along the tube length is ignored in our thermal models (analytical and numerical).

Effective thermal conductivity measurement resultsuniform colonization

Temperatures of the internal and external layer of mussels is measured at steadystate for different ages of mussels of thicknesses (re-ri) with imposing a heat source q in the heating elements. Table 2.2 presents the evolution of the temperature along the aluminium tube (internal layer temperature of mussels (T1, T2 and T3)) and the external layer of mussels (T4, T5 and T6). Indeed, the porosities of the three biofouling materials increase with mussel age due to differences in mussel size. Therefore, for the same temperature difference between the internal and the external layer of mussels one can expect more natural convection inside older biofouling (in mussel's pore space), resulting a higher effective thermal conductivity.

In order to confirm this assumption, the effect of water circulation in the open pores of a porous media on the effective thermal conductivity was investigated. For this purpose, a cluster of glass beads (beads=16mm) was distributed around the experimental tube and maintained using a steel net as shown in Figure 2.9a. The measured porosity of glass medium beads is approximately 43 percent and its measured effective thermal conductivity 54 was found to be 2.4 W.m -1 .K -1 (at q= 70.56 W) which is higher than the thermal conductivity of water (0.6 W.m -1 .K -1 ) and of the glass (1.1 W.m -1 .K -1 ). This demonstrates that there is a small circulation of water in the porous space, which leads to a higher effective thermal conductivity. Also, a test is performed after the glass medium has been covered with a polyethylene stretch film as shown in respectively, are lower than the measured effective thermal conductivity of the glass porous medium, which is 2.4 W.m -1 .K -1 . This is because none of the previous models take in consideration the natural convection effect in the pores of a porous medium as discussed in section 1.8. However, in the case of glass beads medium, the Rayleigh number (Ra) in this case is equal to 1.05x10 3 , so the natural convection in the pores between the particles must be considered, according to section 1.8. As a result, it appears that the hypothesis of a small circulation of water due to free convection in the porous space, resulting a higher effective thermal conductivity, is the most likely. This implies that using homogeneous models in the presence of a temperature gradient (free convection) can be misleading. In the present case the difference is between the effective thermal conductivity and the W.m -1 .K -1 and 12.3 W.m -1 .K -1 for measurements taken on July 8, 2020 and May 19, 2022, respectively, with a difference of less than 5% between the two measurements. It is obvious that the effective thermal conductivity of living and dead mussels increases with increasing

homogeneous
𝑘 𝑒 = 𝑘 𝑓 (1 -(1 -𝜖) 1 2 + 2(1 -𝜖) 1 2 1 -K. B . ( (1 -K). B (1 -KB) 2 ln ( 1 K. B ) - B + 1 2 - B -1 1 -K. B )) 𝐾 = 𝑘 𝑓 𝑘 𝑠 , 𝐵 = 1.25 ( 1-𝜖 𝜖 )
ΔT due to increased natural convection in the mussels' water pores spaces. Because natural convection occurs in fluids in the presence of a gravitational field due to density differences caused by temperature differences within the fluid in the pores spaces of the mussels between the internal layer and the external layer of the mussels. Furthermore, the effective thermal conductivity of the living adult mussels is clearly higher than the effective thermal conductivity of the dead adult mussels, with an average difference of 3 W.m -1 .K -1 , which is approximately 40%. This could be due to the fact that natural convection in alive adult mussels is greater than natural convection in dead adult mussels or due to the mussels' filtering (when they are alive). To test this hypothesis, the Ra of natural convection in alive and dead mussels was measured. The Ra of porous media can be calculated as follows:

𝑅𝑎 = 𝛽𝛥𝑇𝜅𝐷𝑔𝑃𝑟 𝜈 2 (2.5)
where 𝛽 is the thermal expansion coefficient, 𝛥𝑇 is the temperature difference across the porous media, g is the acceleration due to gravity, 𝑃𝑟 is the Prandtl number, ν is the kinematic viscosity, 𝜅 is the permeability of the porous media. Only the permeability of the mussels is unknown among the parameters.

In order to determine the permeability parameter, a new experimental setup was designed and constructed. The experimental setup (Figure 2.12) is a closed circuit and consists of transparent plastic tube (Φ=110mm, Lcyl=1000mm), differential pressure meter, flow meter, water valve, water pump and a water tank. Darcy's law is thus used to measure the permeability of the mussels. To validate the permeability measurement method, the permeability of a porous medium formed from glass beads is measured experimentally and compared to a theoretical 59 value estimated using the Kozeny-Carman [START_REF] Rahrah | Networkinspired versus Kozeny-Carman based permeability-porosity relations applied to Biot's poroelasticity model[END_REF] equation modified by Macdonald et al [START_REF] Macdonald | A generalized Blake-Kozeny equation for multisized spherical particles[END_REF] (Equation 2.7) who modified the constants in the Kozeny-Carman equation by fitting a larger number of experimental data:

𝜅 = 𝑑 𝑝 2 𝜖 3 180(1-𝜖) 2 (2.7)
Where dp is the spherical particle diameter, ϵ is the volumetric void fraction (porosity). As illustrated in the Figure 2.13, a glass beads porous medium with a 55 % porosity is placed in the experimental setup's tube. Following that, the Rayleigh number for both alive and dead adult mussels is calculated after the permeability of the adult mussels is measured. The difference in Rayleigh numbers for alive and dead adult mussels is not exceeding the 10 % as shown in Table 2.5. This difference is mainly due to variation of thermophysical properties with temperature. This indicates that natural convection in alive and dead adult mussels is approximately the same. As a result, a hypothetical difference in natural convection has a negligible effect in the difference in effective thermal conductivity between alive and dead mussels, where the effective thermal conductivity of alive adult mussels is significantly bigger than the effective thermal conductivity of dead adult mussels. So, this difference is due to the mussels' water filtering, which was discussed in section 1.3. When the mussels are alive, they create a suction of the water by taking their nutrition and then releasing the water, resulting in a microfluidic movement of the water in the pores of the mussels' clusters, which leads to a higher effective thermal conductivity than when the mussels are dead. Where dv is the average volumetric diameter and 𝜖 is the sphericity of the particles. In our case, an adult mussel (12 months) specimen is 6 cm length and 3 cm width as shown in Figure 2.17. Then, its average volumetric diameter is equal to 4.24 cm and its sphericity is equal to 0.8. As a result, the adult mussels have a porosity of 22.6 %.

Figure 2.17: Size of an adult mussels specimen.

However, the porosity of juvenile and mixed mussels is measured differently than that of adult mussels. The cluster of mussels with its interstitial water (Figure 2.18) are placed in a water bucket filled with water (Vw) to determine the mussel's cluster volume, which is labelled V1, as shown in Figure 2.18a. Next, we remove the mussels cluster (Figure 2.18b) and then we place it immediately above an empty bucket to collect the interstitial water and read its volume, which is labelled V11 in the Figure 2.18c. As a result, the water porosity of a mussel's cluster is equal to ϵmussels=V11/V1. This method was not applicable to adult mussels due to the high porosity. In this case, porosity was deduced from the permeability measured previously in this section. As shown in Table 2.6, the water porosity of mussels increases with age. Because the size of each mussel specimen grows with age which leads to a higher effective thermal conductivity of mussels as discussed before. As a result, the effective thermal conductivity of mussels is influenced by their water porosity as well as their filtering (nutrition).

Heat transfer coefficient of the water around the mussels measurement results

The analytical stationary method for estimating the heat transfer coefficient of the water around the mussels is validated by comparing the result of the heat transfer coefficient of the water around a tube without the mussels and two literature correlations (Churchill & Chu and Morgan [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF]). Table 2.7 shows that the discrepancy between the experimental value hw and the one provided by Morgan correlation is equal to 6%, which is less than the discrepancy value of 29% provided by the Churchill & Chu's correlation.

This is due to the fact that Churchill & Chu's correlation was set for a wide range of Rayleigh number (RaD ≤ 10 12 ), whereas Morgan's correlation is related to a narrow range of Rayleigh number (Table 9.1 in [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF]). Moreover, the heat transfer coefficients obtained from the correlations were applied for an isothermal horizontal cylinder, whereas in our case the experimental tube is not completely isothermal. As a result, the difference between experimental and theoretical correlation values is acceptable. W.m -2 .K -1 for a uniform distribution of mussels around the tube for juvenile, mix (juvenile and adult), and adult mussels, respectively. However, the relative uncertainty reaches 19% -37% due to the fact that the position of the external thermocouple is not very accurate (± 5 mm) and the temperature difference between the mussel's external layer and the water is very small (less than 0.4 °C). Whereas, this relatively high uncertainty has no effect on the measured global resistance, as shown later. Considering the relatively high value of convective heat transfer of water around the mussels, its contribution to overall thermal resistance between the cable and the external water is small when compared to that of mussels (effective conductive thermal resistance of mussels). In Table 2.9, we show that the convective resistance of water around juvenile, mix, and adult mussels is 2.17 %, 8.33 %, and 4.35 % of the overall thermal resistance, respectively. Table 2.9 shows the overall thermal resistance with different ages of mussels covering the experimental tube. In the case of mussels covering the experimental tube, the overall thermal resistance consists of convective thermal resistance of the water around the mussels and the mussels' conductive thermal resistance. Whereas the convective thermal resistance around the mussels is the reciprocal of the product of the convective heat transfer coefficient of the water around the mussels and the surface area of the mussels, however, the thermal resistance in conduction of the mussels is determined by

( ln (𝑜𝑢𝑡𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠/𝑖𝑛𝑛𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠) 2𝜋(𝑙𝑒𝑛𝑔𝑡ℎ)(𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦)
). Therefore, results show that the overall thermal resistance of juvenile mussels (0.046 K.W -1 at T=10 K) is higher than the overall thermal resistance of mix mussels (0.036 K.W -1 at T=4.2 K) and adult mussels (0.023 K.W -1 at T=4.8 K) respectively. This is due to the fact that the conductive thermal resistance dominates the overall thermal resistance and the effective thermal conductivity of juvenile mussels is smaller than the effective thermal conductivity of mix and adult mussels. This can provide an indication of the expected effect of mussels on the temperature distribution in an electrical cable, as investigated in the next chapter.

However, in practice, the configuration changes as the composition and thickness of the deposit change over time during successive seasons of mussel growth. we can anticipate that the higher thermal resistance associated with the increased deposit will eventually prevail, resulting in a situation in which the tube is more thermally insulated, causing overheating of the electric cable, which will be detrimental to its service life.

Thermal characterization for a non-uniform colonization of mussels around the tube

Mussels do not grow uniformly around submarine cables in real offshore installations. One reason for this is the sun's non-uniform irradiation. Such a situation with non-uniform biofouling growth around a horizontal tube should also be investigated. As a result, the effective thermal conductivity of the non-uniform mussel distribution around the tube, as well as the heat transfer coefficient of the water around them, were measured. In the absence of an analytical model to calculate the effective thermal conductivity from measurements in the case of the non-uniform distribution of mussels, a numerical tool will be used for parameter estimation. For this purpose, the effective thermal conductivity of the mussels and the heat transfer coefficient of the water around the mussels are estimated using a 2D steady state thermal model computed using finite elements (COMSOL software), with the following heat conduction equation:

𝑑𝑖𝑣(-𝑘𝛻 ⃗ 𝑇) = 0 (2.9)
where k is the thermal conductivity (W.m -1 .K -1 ), T is the absolute temperature (K). Where, Tw is the temperature of the surrounding water, hw1 and hw2 is the heat transfer coefficient of the water around the mussels and the tube, respectively. T1,T2,T3 are the temperatures of surface of the experimental tube. T4,T5,T6 are the temperature of the external layer of mussels.

In the biofouling region, a mesh with 2000 to 5000 nodes was used, while in the aluminium tube region, a mesh with 1000 nodes was used. As shown in Figure 2.20, the fine mesh type is used. In the 2D thermal model, the external temperature of the system is the one of the water and the heat rate of power source is provided by the voltage and current measurements on the heating elements. Table 2.11 shows the heat transfer coefficient of the water around the mussels with non-uniform colonization around the aluminium tube covered in polyethylene, as well as the estimated effective thermal conductivity of juvenile mussels. The effective thermal conductivity of juvenile mussels for 100% colonization differs from the value obtained previously with an experimental aluminium tube (4.4

W.m -1 .K -1 ) and the one obtained here with an aluminium tube covered with a polyethylene layer (1.6 W.m -1 .K -1 ). This difference is mainly due to the fact that the temperature gradient between the external and internal radiuses of the mussel layer is not the same in both cases (10 K for aluminium tube and 6 K for aluminium tube covered with polyethylene). That means that convection in the porous medium will not be the same involving different effective thermal conductivities. The results show that the effective thermal conductivity of uniform (100%) and non-uniform (25% and 50%) colonization is roughly of the same order of magnitude. The difference can be attributed to measurement accuracy and the temperature gradient between the external and internal layers of mussels, which is not exactly the same for the three different configurations. One can note that there is a difference for the effective thermal conductivity of juvenile mussels for 100% colonization between the value obtained in section 2.3.2 with an experimental aluminum tube (4.4 W.m -1 .K -1 ) and the one obtained here with an aluminum tube covered with a polyethylene layer (1.6 W.m -1 .K -1 ). This difference is mainly due to the fact that the temperature gradient between the external and internal radiuses of the mussel layer is not the same in both cases (10 K for aluminum tube and 6 K for aluminum tube covered with polyethylene). As in the previous section the convective coefficient values are relatively high and show some disparity. We have already seen that this parameter has a minor impact on global resistance. In the addition of the values of convective coefficients, Table 2.11 also shows the sensitivity of the measured temperature to these parameters defined by βs dT/dβs where βs is the parameter (kbiof, hw1 or hw2). It can be noted that the sensitivity value to the heat transfer coefficients are small compared to the one for the effective thermal conductivity. Because of the convective coefficient's low sensitivity and low contribution to global thermal resistance, its discrepancy has a minor impact on determining the effective thermal conductivity of mussels.

The water in our current study is stagnant. However, there is a current flow velocity of water in the system in real-world application, then the effect of the global heat transfer coefficient of the water around the cable and in the mussel's layer (porous medium) must be considered.

Effective thermal conductivity with water current flow -Glass beads medium

The effective thermal conductivity of a porous medium made of glass beads around the experimental tube will be measured in this section while it is subjected to various water current velocities generated by a water traction pool in order to test the impact of the current water on a porous medium's effective thermal conductivity. The traction pool located at Ecole Centrale de Nantes, France. It is dimensions are 140 m long, 5 m wide, and has a constant depth of 3 m (Figure 2.22). The water temperature in the traction pool is 16.9 °C.

73

In order to create a porous medium made from glass beads with a porosity of about 43%, a cluster of glass beads (Φbeads=16mm) was distributed around the same experimental tube as in section 2.2 and kept in place using a steel net. To prevent any axial heat loss in the tube, the ends of the experimental tube were coupled with polyoxymethylene (POM) (Figure 2.24). Table 2.12 displays the average temperature distribution of the internal and external layers of the glass beads medium at various water current velocities. The development of the boundary layer on the surface has a significant impact on the Nusselt number for a flow motion normal to the axis of a smooth circular cylinder. As a result, the increase in Nu with increasing Re is due to a corresponding decrease in the thickness of the boundary layer which leads to the increase of the forced heat transfer coefficient (hforced). In consequence, the temperature of the thermocouples at the internal and external layers and the difference between them for the glass beads which is near from the stagnation point at (θ=333°) is lower than the temperature of the internal and external thermocouples and the difference between them at (θ=90°) and (θ=210°), respectively.

Consider condition for

Therefore, during the existence of the water current the effective thermal conductivity which is located at (θ=333°) is higher than the effective thermal conductivity at (θ=90°) and (θ=210°), respectively, as shown in Table 2.13. As we can see the forced heat convection increases with the increasing of the Reynolds number as the Nusselt number increases which is calculated using the correlation of Zukauskas [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF].

82

During the measurement, we do not have the same spatial resolution as the case of smooth cylinder (Figure 2.30) because on the experimental tube we only have only three locations of the thermocouples at (90°, 210°, and 330°). As we cannot follow all of these evolutions, we only consider the average of these points. Therefore, using Fourier's law, the average effective thermal conductivity of the glass beads (ke,beads= 𝑘 𝑒,𝑏𝑒𝑎𝑑𝑠 ,𝜃=90° + 𝑘 𝑒,𝑏𝑒𝑎𝑑𝑠 ,𝜃=210° + 𝑘 𝑒,𝑏𝑒𝑎𝑑𝑠 ,𝜃=330°3

) for various water current velocities is calculated from the measured temperature difference between the internal and external layer of the glass beads medium and the imposed power (q=102 W). As shown in Figure 2.31 and as expected the average effective thermal conductivity of glass beads medium increases as the water current velocity increases. ) as function of the Ln (Re). Therefore, we get the following model:

𝑘 𝑒,𝑏𝑒𝑎𝑑𝑠 𝑘 𝑠𝑡𝑎𝑔𝑛𝑎𝑛𝑡,𝑏𝑒𝑎𝑑𝑠 = 1.8158𝑒 -6 𝑅𝑒 1.57 (2.12)

Then, Figure 2.32 allows us to compare the model with our experimental data, where we have a very good matching between them. In the future it will be interesting to check if such a model stands for mussel's layers of different ages and different thicknesses.

Measurement at site

The mussels were manually fixed on the tube in the thermal characterization performed in this chapter, which can affect the contact between the biofouling and the experimental tube, resulting in contact thermal resistance. Moreover, our choice of a single species is not representative of the diversity of biocolonization in some sites. To solve this problem, we need a natural growth of biofouling on the surface of the experimental tube.

The goal of this section is to evaluate the thermal conductivity of mussels that have grown naturally on experimental tubes in the Mediterranean Sea and the Atlantic sea. This project is linked to the Biodytherm8 project, which is funded by region pays de la Loire (WEAMEC). This is a collaborative project between Nantes University and Akrocean company. This study used 6 experimental tubes in the Mediterranean Sea and two in Atlantic Ocean (Figure 2.33), four of which were covered in with polyethylene (the material used for actual submarine cables) and four were not. Using experimental tubes with and without polyethylene covers, researchers will be able to compare how different types of colonization are influenced by surface conditions. The main objective was to create a database on the effective thermal conductivity of biofouling, which is dependent on time and location and influences the type of biofouling. This database will be used to assess the impact of biofouling growth on dynamic submarine electrical cables.

Partial Conclusion

In this Chapter, the thermal characterization of mussels was carried out in stagnant water. The measurement of the thermal properties of this natural medium (biofouling) is rarely addressed in the marine energy literature. The experimental work was very challenging because we were working with a live sample and had to complete all measurements within 24 hours in order to keep the mussels alive. In the uniform mussel colonization case, the thermal conductivity of different ages of mussels (juvenile, mix (juvenile and adult), and adult) as well as the heat transfer coefficient of the water around the mussels were measured.

Results show that juvenile mussels have the lowest effective thermal conductivity when compared to a mix mussels and adult mussels only due to the water porosity effect, where porosity of juvenile mussels is less than the mix mussels and adult mussels respectively. The effect of water porosity has been highlighted and illustrated using a glass beads medium, it is demonstrated that the natural convection in the porous media significantly enhances the effective thermal conductivity of the glass porous media.

Moreover, the effective thermal conductivity of adult mussels was measured when the mussels are alive and when they are dead. Results shows that the effective thermal conductivity of adult alive mussels is higher than the effective thermal conductivity of adult dead mussels due to the water filtering effect which creates an additional micro-movement leads to increasing of the effective thermal conductivity of the mussels. On the other hand, the effective thermal conductivity of the mussels increases with the increasing of the temperature difference between the internal layer of the mussels and the external layer of the mussels due to natural convection phenomenon.

So as a summary, in case of stagnant water, the effective thermal conductivity of mussels is affected by three factors: the water porosity, the temperature difference between their internal and external layer and their filtering phenomenon (nutrition).

Another study has also been done to demonstrate how the water current affects the effective thermal conductivity of a porous medium formed from glass beads. It was demonstrated that the effective thermal conductivity of glass beads in the presence of current water increases with increasing water current velocity as the forced heat transfer coefficient around the glass beads medium increases. Therefore, it's expected the same for the effective thermal conductivity in the presence of the water current. A power law model was proposed to describe this model.

The results also demonstrate that the effective conductive resistance of mussels can be greater than the convective resistance surrounding the tube in their absence. Therefore, it would be interesting to investigate the thermal impact of mussel colonization near the DSEC. In order to better understand this issue, the following Chapter will focus on this topic.

Chapter Thermal Effect of Mussels on the temperature of the dynamic submarine electrical cable

Introduction

In Chapter 2, we demonstrated that the age of the mussels changes the water porosity of the biocolonization, thus affecting the global thermal resistance around the dynamic submarine electrical cable. It was shown that thermal resistance of a layer of mussels can be lower or higher than convective resistance without mussels. Indeed, as described in section 2.3.3, the effective conductive resistance of juvenile mussels is higher than the convective resistance around the tube in the absence of mussels. However, the effective conductive resistance of mixed mussels is slightly bigger and the effective conductive resistance of adult mussels is slightly lower. Thus, mussel deposits on the surface of the marine power cable can form an insulating layer, resulting in increased thermal resistance which can lead decrease the fatigue life time of the cable. Due to the fact that, as was mentioned in Chapter 2, the maximum conductor operating temperature of a dynamic submarine electrical cable (DSEC) insulated with cross-linked polyethylene (XLPE), Figure 3.1, is 90 °C [START_REF]Electric cables-Calculation of the current rating-Current rating equations (100% load factor) and calculation of losses[END_REF]. That means that the conductor of an XLPE cable can be maintained at 90 °C for an extended period of time without further deterioration of the insulating material, which would otherwise reduce the mechanical properties of the cable and shorten its fatigue lifetime. Whereas the expected lifetime of XLPE insulation dynamic submarine electrical cable is between 40 and 60 years at rated operating conductor temperature of 90 °C and between 7 and 30 years at rated operating conductor temperature of 95 to 105 °C [START_REF] Alghamdi | A study of expected lifetime of XLPE insulation cables working at elevated temperatures by applying accelerated thermal ageing[END_REF]. Additionally, temperature fluctuations in the DSEC can increase the mechanical stresses in the cable's parts, which in turn reduces the fatigue lifetime of the cable. Moreover, as the temperature rises, the resistivity of the copper conductor used in the DSEC increases, resulting in an increase in energy loss. To address this issue, the cross section of the copper wire conductor must be increased, adding significantly to the cost.

As a result, investigating the thermal effect of mussel's colonization on a real submarine power cable could be interesting.

In this Chapter, the permissible current rating for a specific operating temperature is determined for a real DSEC using an analytical model based on an international standard (IEC standard). This yields to a power that can be used to investigate the thermal impact of various mussel ages on the temperature of the copper wire in a 2D numerical simulation model of the DSEC (finite elements via COMSOL).

Analytical Model Based on IEC Standard

Under steady-state conditions, IEC standards 60287-1-1 [START_REF]Electric cables-Calculation of the current rating-Current rating equations (100% load factor) and calculation of losses[END_REF] and 60287-2-1 [START_REF]The International Electroctechnical Comission. Electric cables-Calculation of the current rating, Thermal resistance-Calculation of thermal resistance[END_REF] are used to calculate the permissible current rating in accordance with operating temperature.

It denotes a constant current load (100 percent load factor) that is only sufficient to generate the maximum conductor temperature asymptotically and assumes that the conditions in the surrounding ambient are constant. This method is widely used all over the world. It covers medium to high voltage cables, a wide range of installation methods, and a formula for current rating and losses. The main goal is to determine the relationship between the cable and the factors influencing heat dissipation, such as the thermal resistance of cable components, load, and the surrounding environment.

For illustrating the method, we use a cable for which data were available: a 20 kV used in OMDYN2 project. It's a cross-linked polyethylene (XLPE) insulated dynamic submarine electrical cable (DSEC) designed to sustain a maximum conductor temperature of 90 °C continuously. It is made up of 3 × 50 mm 2 copper conductors insulated with XLPE. The cable is being covered with a double wire armor to increase torsional stiffness due to dynamical application and to protect it from mechanical stress, floating debris, and friction caused by the cable touching the seabed. Figure 3.2 shows the DSEC's complex hierarchical cross-sectional structure. Where R1 is the thermal resistance per unit length between one conductor and the sheath (K.m.W -1 ),

𝑅 1 = 𝜌 𝑡 𝑖 2𝜋 . ln (1 + 2𝑡 𝑖𝑛𝑠,𝑠𝑒𝑚𝑖𝑐  one,con ) (3.1)
Where, 𝜌 𝑡 𝑖 is the thermal resistivity of the insulation (K.m.W -1 ), tins,semic thickness of the insulation including semi-conductive layers (m) and one,con diameter of one core (m).

R2 is the thermal resistance per unit length of the bedding between sheath and armour (K.m.W -1 ),

𝑅 2 = 1 6𝜋 . 𝐺. 𝜌 𝑡 𝑏 (3.2)

93

Where, 𝜌 𝑡 𝑏 thermal resistivity of the cable bedding (K.m.W -1 ) and G is the geometric factor obtained using an empirical curve in IEC standard 60287-2 [START_REF]The International Electroctechnical Comission. Electric cables-Calculation of the current rating, Thermal resistance-Calculation of thermal resistance[END_REF] by calculating the rate rG as shown in equation (3.3):

𝑟 𝐺 = 𝑡 𝑏 + 𝑡 𝑃𝐸 𝑠  𝑖𝑛𝑡,𝑖𝑛𝑠 + 2𝑡 𝑠 (3.3)
Where, ts is the thickness of metallic sheath (m), tb is the thickness of bedding itself (m), 𝑡 𝑃𝐸 𝑠 is the thickness of the inner plastic sheath (m), int,ins is the external diameter of insulation(m).

R3 is the thermal resistance per unit length of the external serving of the cable (K.m.W -1 ),

𝑅 3 = 𝜌 𝑡 𝑜𝑐 2𝜋 . ln (1 + 2𝑡 3  𝑎𝑟 ) (3.4)
Where, 𝜌 𝑡 𝑜𝑐 is the thermal resistivity of the outer covering (m), t3 is the thickness of the outer covering (m),  𝑎𝑟 is the external diameter of the armor (m).

R4 is the thermal resistance per unit length between the cable surface and the surrounding medium (K.m.W -1 ), Where Ic is the current flowing in one conductor (A), ΔTc(K) = Tc -Ta, n is the number of load-carrying conductors in the cable. λ1 and 𝜆2 are calculated using IEC standard 60287-1 [START_REF]Electric cables-Calculation of the current rating-Current rating equations (100% load factor) and calculation of losses[END_REF], where λ1 is the ratio of losses in the metal sheath to total losses in all conductors in that cable,

𝑅
𝜆 1 = 𝜆 1 ′ + 𝜆 1 ′′ (3.8)
Where, 𝜆 1 ′ is the loss caused by circulating currents in the sheath, expressed in equation (3.9) and 𝜆 1 ′′ is the loss caused by circulating eddy currents in the sheath, however, for a three core cable such what we consider here, with a metallic sheath per core conductor, there are no losses relative to eddy current. Therefore, 𝜆 1 ′′ = 0. Where, 𝑑 𝑐𝑜𝑟𝑒,𝑐𝑜𝑛 is the axial distance between core conductors (m), 𝑡 𝑠 is the thickness of the metallic sheath,  𝑒𝑥𝑡,𝑖𝑛𝑠 is the external diameter of the insulation, (𝑑 𝑒𝑥𝑡,𝑖𝑛𝑠 + 𝑡 𝑠 ) is the mean diameter of the screen (m) and 𝜋 ((𝑑 𝑒𝑥𝑡,𝑖𝑛𝑠 + 𝑡 𝑠 ) 2 -𝑑 𝑒𝑥𝑡,𝑖𝑛𝑠 2) is the cross section of the metallic sheath in (m 2 ) as defined in the IEC standard 60827-1 [START_REF]Electric cables-Calculation of the current rating-Current rating equations (100% load factor) and calculation of losses[END_REF] λ2 is the ratio of losses in the armouring to total losses in all conductors in that cable.

𝜆 is the alternating current resistance per unit length of the conductor at maximum operating temperature (Ω.m -1 ):

𝑅 𝐴𝐶 𝑇 𝑐𝑜𝑛 ′ = 𝑅 20°𝐶 ′ (1 + 𝛼 20 𝑐 (𝑇 𝑐𝑜𝑛 -20°C)(1 + 𝑦 𝑝 + 𝑦 𝑠 ) (3.15)
Where 𝑅 20°𝐶 ′ is the DC resistance of the conductor at 20 °C, ys is the skin effect factor and yp is the proximity effect factor. Wd is the dielectric loss per unit length for the insulation surrounding the conductor (W.m -1 ):

𝑊 𝑑 = 𝜔𝐶𝑈 0 2 𝑡𝑎𝑛𝛿 𝑙 (3.16)
Where ω is the angular frequency, 𝐶 refers to the cable capacitance per unit length (F.m -1 ), U0 is the voltage to earth and 𝛿 𝑙 is the loss angle. The thermal resistances and the heat losses in the submarine power cable used for our calculations are shown in Table 3.1.

Table 3.1: Thermal resistances and heat losses in the submarine power cable.

Symbol Material Value

R1

Thermal resistance between one conductor and the sheath per unit length (K.m.W -1 ) 0.587

R2

Thermal resistance of the bedding between the sheath and the armor per unit length (K.m.W -1 ) 0.095 Alternating current resistance per unit conductor length at maximum operating temperature (Ω.km -1 ) 0.5

R3

𝑊 𝑑

The dielectric loss per unit length of the insulation that surrounds the conductor (W.m -1 ) 0.074 * These resistances are calculated for 90 °C operating temperature and 20 °C ambient temperature.

Numerical Simulation of Temperature Field in a 2D DSEC

In this section, the thermal impact of mussels of various ages on the copper wire of the DSEC used in the OMDYN2 project is investigated. To do this, a two-dimensional steady-state thermal model, consist of the exact shape and materials of the DSEC computed using finite elements (COMSOL software) and a mesh with normal size elements of 94124 nodes in the DSEC (Figure 3.4), is used to predict the temperature distribution within the cable. As mentioned earlier, the DSEC is designed to support a maximum copper wire temperature of 90 °C. Therefore, for a maximum operating temperature of 90 °C and an ambient temperature of 20°C, the maximum permissible current rating (I90°C =364 A) is calculated using equation (3.7). The obtained current rating (364 A) is then imposed in the numerical method of DSEC 2D thermal modeling (COMSOL software). Moreover, with an external temperature of 20 °C, a convective heat transfer coefficient (247 W.m 2 .K -1 ) is applied to the outer layer of the DSEC; this convective heat transfer coefficient is calculated using the classic natural convection formula (Morgan's correlation [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF]). The steady-state temperature distribution in the cable is depicted in Figure 3.5. According to the results, the maximum conductor temperature calculated using a two-dimensional numerical thermal model is 90.15 °C, which is nearly equal to the maximum operating temperature of 90 °C. This means that the numerical approach is validated because the simulation code (COMSOL) produces the same temperature the IEC-60827 standard. The numerical thermal model can then be used to investigate the thermal effect of biofouling colonization on the DSEC conductor temperature. Taking into consideration that the numerical thermal model has been tested with a variety of mesh element sizes, including normal (94124 nodes), finer (113116 nodes), and extremely fine (186128 nodes), it has produced the same copper conductor temperature (90.15 °C) of the DSEC. Thus, juvenile mussels have the highest effect on the DSEC conductor temperature, because as previously described in Section 2.3.3, juvenile mussels have the greatest thermal resistance and thermal resistance decreases with the age of the mussels due to the increase in water porosity (increase in natural convection). 100

The copper conductor temperature (Tc) were also calculated analytically using equation (3.7) by using the effective thermal conductivity of the mussels and the heat transfer coefficient of the water around them to calculate the thermal resistance R4 and by imposing the same maximum permissible current rating (I90°C =364 A). Results show in ) between the analytical method and simulations is not exceeding the 0.47%. So another method is used to validate the numerical simulation. As far as natural convection is involved in the porous medium, one can compare its effect to the one of the natural convection in water around the cable without mussels. This can explain the effect of mussels on the copper conductor temperature of the electrical cable. Rresults show in Table 3.3 that the overall thermal resistance of juvenile mussels around the electrical cable (0.022 K.m.W -1 at T=10 K) is higher than the overall thermal resistance around the electrical cable in the absence of mussels (0.012 K.m.W -1 ) and it is slightly higher for mix mussels (0.017 K.m.W -1 at T=4.2 K) and smaller for adult mussels (0.011 K.m.W -1 at T=4.8 K). This explains why the copper conductor temperature for electrical cable colonized by juvenile or mixed mussels is higher than 90°C and lower than 90°C for electrical cable colonized by adult mussels. However, as the effective thermal conductivity and the heat transfer coefficient changes with the imposed power during the measurement which leads to have a different temperature difference (between the internal layer and the external layer of the mussels), then, the thermal resistances are changed.

This result appears to indicate that the temperature of the conductor decreases with mussel age and can be even lower than the temperature without mussels, which is used as a reference here. This observation must be moderated due to several aspects of our study:

(i) the experiments and simulations are conducted with a single layer of mussels of roughly the same order of thickness. In reality, as time passes from one season to the next, a new layer of mussels grows on top of the existing one. The water porosity and effective thermal conductivity will then evolve over time. This may cause the conductor to overheat. (ii) As previously stated, the value of effective thermal conductivity is related to natural convection in the porous medium composed of mussels. This natural convection effect is determined by the temperature difference across the mussel layer. The goal is to investigate the effect of mussel colonization on the copper conductor temperature of an electrical cable with a maximum operating temperature of 90 °C. In this simulation, we used the power calculated according to the IEC standard (P90°C). Then, we imposed the measured effective 102 thermal conductivity of the various mussel layers. The power values used in the experimental measurement differ from the power calculated using the IEC standard, which is used in the numerical simulations. This results in a difference in ΔT between experimental measurements and numerical simulations, as shown in Table 3.4. T is the temperature difference between the inner and outer radiuses of the mussel layer. This disparity in ΔT values is expected because the power imposed in numerical simulation differs from the power imposed in experimental measurement, and effective thermal conductivity depends on ΔT. This difference affects the determination of the simulated temperature of the copper conductor.

In order to improve the accuracy of copper conductor temperature determination, the appropriate effective thermal conductivity of the mussels must be imposed in the numerical simulation. To achieve this, the same power (P90°C) used in the simulations which leads to cable conductor temperature of 90 °C (in case of without mussels) is applied during one of the measurements carried out in May 2022 (section 2.3.2). This results to mussels' effective thermal conductivity of adult mussels 9.1 W.m -1 .K -1 and a ΔT of 3K which is lower than the effective thermal conductivity (12.8 W.m -1 .K -1 ) measured at ΔT equal to 4.8 K. Then, the obtained value (9.1 W.m -1 .K -1 ) is applied to the adult mussels layer that has been added to the numerical method of DSEC 2D thermal modeling (COMSOL software), where the natural convection is taken into account around the mussels layer that has the same thickness during the measurements, which is 7 cm at an ambient temperature of 20 °C. Additionally, the simulation imposes the same permissible current rating (364 A) that results in a cable conductor temperature of 90 °C without mussels. Results demonstrate that the cable's copper conductor temperature, which is equal to 90.8 °C when kbiof,adult=9.1 W.m -1 .K -1 at ΔT= 3 K, is higher than the cable's copper conductor temperature, which is equal to 89.6 °C at kbiof,adult=12.8 W.m -1 .K -1 and ΔT= 4.8 K. Where the overall thermal resistance of the adult mussels (0.0208 K.m.W -1 at ΔT= 3 K) is higher than the convective thermal resistance without mussels (0.012 K.m.W -1 ).

From measurements which has been done in July 2020, the effective thermal conductivity of juvenile mussels is measured by imposing two different powers 40.74 W and 224 W which results an effective thermal conductivity of juvenile mussels of 1.6 W.m - 1 .K -1 (ΔT=6 K) and 4.4 W.m -1 .K -1 (ΔT=10 K), respectively. Therefore, the effective thermal conductivity of juvenile mussels is estimated at the power P90°C from the curve fitting of two previous measurements as shown in Figure 3.8. Results show that the effective thermal conductivity of juvenile mussels at P90°C is equal to 2.45 W.m -1 .K -1 and ΔT equal to 7.25 K. Then, the obtained value (2.45 W.m -1 .K -1 ) of the effective thermal conductivity of juvenile mussels is imposed in the DSEC thermal model with a power (P90°C), resulting in a copper conductor temperature of 95.18 °C, where the overall thermal resistance (0.044 K.m.W -1 at ΔT=7.25 K) is higher than the convective thermal resistance of water in case of without mussels (0.012 K.m.W -1 ). The obtained copper conductor temperature (95.18 °C) may have an impact on the lifetime of the DSEC's XLPE insulation as discussed in section 3.1.

It is important to note that all thermal modeling examples for the thermal impact of mussels on the DSEC are based on the assumption that the water surrounding the mussels is stagnant. As a result, the DSEC conductor temperature is consequently thermally affected by the presence of the mussels, particularly the juvenile mussels as shown in Figure 3.9. Therefore, it's worthwhile to track the growth of mussels around the DSEC in order to determine what kinds of mussels colonize the cable. In the following chapter, a new method for tracking mussel growth using a thermal sensor is introduced.

to rise in the copper conductor temperature of the cable (95.18 °C). This exceeds in temperature of the copper conductor reduces the lifetime of the XLPE insulation, the overall fatigue life of the cable, consequently, maintenance costs are increased.

Whereas, juvenile mussels have a bigger effect on the DSEC conductor temperature than mix and adult mussels which leads to rise in the copper conductor temperature of the cable. This rise in copper conductor temperature causes a decrease in the XLPE insulation fatigue lifetime, the overall cable fatigue lifetime. In that case, the dynamic submarine electrical cable requires maintenance which is too costly and not easy to do.

Monitoring mussel growth and cleaning them from around the cable as needed would help to reduce the impact of mussel bio-colonization on the DSEC. The following chapter describes a novel method for monitoring biofouling growth with a thermal sensor.

Chapter Toward tracking of biofouling growth -Transient thermal analysis

Introduction

As previously stated, the dynamic submarine electrical cable, also known as umbilical cable, is an important component for offshore operators. Dynamic cable is generally designed to withstand coupled mechanical-electrical-thermal constraints.

However, biofouling growth, which has a strong impact on the cable's local linear mass, apparent diameter, surface roughness, and, last but not least, cable cooling by adding thermal resistance, which reduces heat transfer between the cable and the water, can exacerbate the coupled mechanical-electrical-thermal constraints. Monitoring the growth of biofouling will thus be useful for electrical cable maintenance. Thermal tracking is an efficient method because it is not affected by weather, diver abilities, or any other obstacles encountered during an underwater inspection using a flux metric device (thermal sensor).

The experimental setup (flux metric device or thermal sensor) composed of a stainless steel layer with a thermocouple on top, a heating element made of polyimide (kapton) which has a resistance of 6.3 ohm, a thermoplastic layer made of polyethylene, a layer of rubber, and finally an aluminum block at the bottom. The surface area of each of these layers is 0.01 m 2 (10x10 cm 2 ), and they are all stacked in that order (Figure 4.1).

Thermal models for the estimation of the heat transfer coefficient

The basic idea of the thermal method to monitor the growth of the biofouling is to estimate the heat transfer coefficient value for mussels of varying ages. Using a flux metric device, a pulse heating (φ) of few seconds duration is provided under the biofouling and the temperature evolution and decay (T (K)) versus time are recorded (Figure 4.2) and used with a thermal model to estimate the heat transfer coefficient describing heat transfer through the biofouling. In this section, we will present the mathematical models used to calculate the heat transfer coefficient. The first is a simplified model (capacitive model) and the second is an exact model (conductive model).

Capacitive model

The capacitive model is one method for determining the heat transfer coefficient.

It's a model adopted for a sensor with impulse flux law. As shown in Because Th differs from Tw, we can write the following expression for the flow density dissipated by the sensor's heating system:

𝜑 𝑑 = 𝜑 𝑝 + δ 𝜑 (4.1)
δφ is the flow disturbance introduced by the heating system of the sensor, δφ can be written in the form:

δ φ = ℎ ′ (𝑇 ℎ -𝑇 𝑤 ) (4.2)
Where, ℎ ′ is the local heat transfer coefficient.

If we consider that the sensor is perfectly isolated from the wall and that the heating law is of the impulse type, the energy balance is written:

𝜑 𝑑 = ℎ(𝑇 𝑤 -𝑇 𝐸 ) + ℎ ′ (𝑇 ℎ -𝑇 𝑤 ) + 𝐶 𝑑𝑇 ℎ 𝑑𝑡 (4.3)
with C is the heat capacity of the sensor brought back to the surface unit and h is the equivalent heat transfer coefficient.

In the case of an unheated sensor, i.e. 𝜑 𝑑 = 0, the temperature of the sensor without heating is Tc 0 , it only depends on the fluctuations of h, TE and Tw. Thus, equation 4.3 can be written as:

ℎ(𝑇 𝑤 -𝑇 𝐸 ) + ℎ ′ (𝑇 ℎ 0 -𝑇 𝑤 ) + 𝐶 𝑑𝑇 ℎ 0 𝑑𝑡 = 0 (4.4) hence, ℎ ′ (𝑇 𝑤 -𝑇 ℎ 0 ) -𝐶 𝑑𝑇 ℎ 0 𝑑𝑡 = ℎ(𝑇 𝑤 -𝑇 𝐸 ) = 𝜑 𝑝 (4.5)
Thus, equation 4.3 can be written as:

𝜑 𝑑 = ℎ ′ (𝑇 ℎ -𝑇 ℎ 0 ) + 𝐶 𝑑(𝑇 ℎ -𝑇 ℎ 0 ) 𝑑𝑡 (4.6)
From this equation, we can obtain the formula expressing the law of cooling of the sensor, which allows to find ℎ', for a given flux law. In the case of a sensor with impulse flux law, the formula is given by:

𝜑 𝑑 = 𝜑 0 for (0≤𝑡≤ 𝑡0) (4.7a)
𝜑 𝑑 =0 for (t> t0) (4.7b)

If we assume that in the initial state t = 0, (𝑇c -Tc 0 ) = 0 and that t0 ≪ 𝐶 ℎ′ , the solution of equation (4.6) is:

(𝑇 ℎ -𝑇 ℎ 0 ) = 𝜑 0 ℎ′ [1 -𝑒 (- ℎ ′ 𝑡 𝐶 ) ] ≈ 𝜑 0 𝑡 𝐶 for (0≤𝑡≤ 𝑡0) (4.8)
(𝑇 ℎ -𝑇 ℎ 0 ) = 𝜑 0 ℎ′ [1 -𝑒 (- ℎ ′ (𝑡-𝑡0) 𝐶 ) ]
for ( t> t0 ) (4.9)

Where 𝑇h 0 represents the temperature of the sensor before heating, 𝑇h represents the temperature of the heated sensor, C represents the sensor's heat capacity, 𝜑d represents the heating flux and t0 represents the time when the temperature reaches its maximum value 4.4 and this curve depends on ℎ′. By plotting the quantity Ln (𝑇h -𝑇h 0 ) / (𝑇𝑚 -𝑇h 0 ) as a function of (t-t0), see Figure 4.4, we obtain a linear trend slope. We check on the one hand, that the shape of the law of cooling of the sensor is indeed of exponential nature, and on the other hand, we deduce the time constant of the sensor τ that is related to the slope m by the formula:

𝜏 = -1 𝑚 = 𝐶 ℎ′ (4.10)
So, h' is defined by:

ℎ ′ = -𝑚𝐶 (4.11)
However, the capacitive model is a simplified model in which we assumed the sensor is isothermal, which means there is no heat transfer in the sensor. Furthermore, no heat flux loss is considered at the sensor's back face (adiabatic identity). Therefore, conductive model can be a good alternative model because it has a less hypothesis comparing to capacitive model. );

Conductive model

Due to symmetrical system, Aj=Dj. Where a is the thermal diffusivity and e is the thickness of the layer, S is the area of the layer and j can be PE, PVC, heating element or stainless steel.

On the top side of the device, the system of equations is: The real computed temperature (Tc) at θ 𝑐 is obtained from its Laplace form using the Gaver-Stehfest method. Thus, given the Laplace transform F(p), the time domain solution approximation is The comparison between capacitive and conductive models will be performed in the following section.

[ θ ℎ 𝛷 ℎ + ] =

Comparison between Capacitive and Conductive models

The conductive model (quadrupole model) and the experimental setup designed to estimate the mussels' equivalent heat transfer coefficient were tested in the flux metric device using a two-dimensional numerical transient thermal model (finite elements via COMSOL) and a mesh with normal size elements of 11000 nodes, by comparing the temperature evolution of Tc obtained from the conductive model and COMSOL.

This comparison has been done using the same thermophysical properties of each layer of the flux metric device (Table 4.1), boundary conditions (heq=500 W.m -2 .K -1 ) and initial conditions (Twater,initial=23.2 °C) that were used in the conductive model with a pulse power of 22 W in the heating elements for 18 seconds and a relaxation time around 3.5 minutes. As mussel's samples are usable for approximately 24 hours (otherwise they die), glass beads of comparable size are used to simulate biofouling (porous medium).

Therefore, in order to compare capacitive and conductive model, glass beads medium (beads=25 mm) with 4 cm thickness layer were placed above the thermal sensor, then the equivalent heat transfer coefficient (heq) of the glass beads medium is estimated using the conductive and the capacitive model. In both models a heat pulse with a total power of 22

W is introduced for 25 s. The temperature evolution was recorded at the location of the thermocouple in the thermal sensor at point c. Finally, the capacitive model and the conductive models give an equivalent heat transfer coefficient of the glass beads equal to 225 W.m -2 .K -1 and 332 W.m -2 .K -1 respectively, with a difference of 32% (Table 4.2). A heat pulse of total power of 22 W were imposed in the heating element of the flux metric for different heat pulse durations (25, 120, 180 s). The heat diffusion length (heat diffusion=√𝑎𝑡) in the glass beads medium has been calculated during the heat duration and during the overall measurement duration meaning ), 𝑘 ℎ,𝑔𝑙𝑎𝑠𝑠 𝑏𝑒𝑎𝑑𝑠 (0.89 W.m -1 .K -1 ) is the homogenous effective thermal conductivity of the glass beads calculated using Maxwell's equation (Chapter 2) and  𝑔𝑙𝑎𝑠𝑠 𝑏𝑒𝑎𝑑𝑠 (1870 kg.m -3 )and 𝐶 𝑝 𝑔𝑙𝑎𝑠𝑠 𝑏𝑒𝑎𝑑𝑠 (2244 J.kg -1 .K -1 ) are the density and the heat capacity of glass beads, respectively, calculated using mixing law.  𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 is the heat diffusion length during the overall measurement duration (3 times duration of the heat pulse).

In the present numerical transient simulation, the boundary conditions and the thermophysical properties of the previous simulations are the same, except the effective thermal conductivity of the glass beads. The effective thermal conductivity of the glass beads is estimated using minimization method between the experimental and the numerical temperatures (COMSOL) at point c. In order to fit the measured temperatures and the computed temperature estimated by COMSOL, the effective thermal conductivity was changed over time during the simulations with the changing of the deviation of the measured temperature. Where 𝑇 𝑠𝑠 is the average surface temperature of the upper surface of the stainless steel layer estimated by COMSOL (K), 𝑇 𝑟𝑒𝑓 is the temperature of the water above the glass beads medium (K),  𝑠𝑠,𝑐𝑜𝑠𝑚𝑜𝑙 is the heat density dessipated from the stainless steel surface estimated by COMSOL (W.m -2 ) and ℎ 𝑒𝑞,𝑐𝑜𝑚𝑠𝑜𝑙 is the equivalent heat transfer coefficient estimated by COMSOL (W.m -2 .K -1 ).

Results show that the equivalent heat transfer coefficient changes with time for short and long heat pulse durations. However, Figure 4.15 shows that the equivalent heat transfer coefficient becomes slightly stable at the end of the heat pulse duration. Therefore, the equivalent heat transfer coefficient at the end of the heat pulse is the one which is As shown in Table 4.4, the equivalent heat transfer coefficient of the glass beads with at the end of the short heat pulse duration (25 s) is equal to (310 W.m -2 .K -1 ), however, the equivalent heat transfer coefficient at the end of the heat pulse 120 s (160 W.m -2 .K -1 ) is approximately the same of the equivalent heat transfer coefficient at the end of the heat pulse 180 s (153 W.m -2 .K -1 ). Moreover, the difference between the heat transfer coefficient estimated using conductive model and COMSOL for short heat pulse duration (25 s) is less than 5%, however, for long heat pulse duration (120 s and 180 s) the difference is very big.

This confirms, that the conductive model is valid only for short heat pulse duration (25 s) in the case of glass beads medium. Therefore, the equivalent heat transfer coefficient of the glass beads can be estimated using the conductive model only if the heat pulse duration is small. However, if the heat pulse duration is long, then, a numerical simulation (finite elements COMSOL) shall be used to estimate the equivalent heat transfer coefficient. The equivalent heat transfer coefficient for long heat pulse duration stabilizes at the end of the heat pulse duration which seems that it is a good criterion to monitor the biofouling growth, however, the experimental measurements will confirm if it is valid for the biofouling or not.

Transient analysistest with mussels

Experiments were carried out to determine the equivalent heat transfer coefficient using conductive models of varying ages (juvenile, mix, and adult, as shown in Figure Results show that the equivalent heat transfer coefficient of mussels is sensitive to mussel's age, as shown in Figure 4.18. It is worth noting that juvenile mussels have the lowest global heat transfer coefficient. This is explained using the same analogy as in Chapter 2 for effective thermal conductivity. In fact, because juvenile mussels have the smallest water porosity (15%) compared to mix (18%) and adult mussels (22.6 %), the water circulation effect in the porous medium of juvenile mussels is small compared to mix and adult mussels. As a result, as the medium's water porosity increases, so does the equivalent heat transfer coefficient. The thermal sensor's sensitivity was then tested, and it is appropriate for the monitoring of the growth of biofouling around the dynamic submarine electrical cable, as well as to help identify what type of mussels are colonized around the cable. Moreover, the equivalent heat transfer coefficient of juvenile mussels with a layer thickness of 4 cm is compared to the equivalent heat transfer coefficient of juvenile mussels with a layer thickness of 9.4 cm. It's clearly shown from Figure 4.20 that the relaxation curve of the juvenile mussels with 4 cm thickness layer cools dawn faster than the relaxation curve of the mussels with a thickness layer of 9.4 cm, this lead to equivalent heat transfer coefficient (930 W.m -2 .K -1 ) and (550 W.m -2 .K -1 ), respectively. This is because increasing the thickness layer adds more heat resistance, resulting in a smaller heat transfer coefficient. Thus, based on the two previous experiments, the equivalent heat transfer coefficient of the mussels is sensitive to the mussel's age, agitation effect and the thickness layer. In consequence, the thermal sensor can be a good solution for the tracking of biofouling growth.

Thermal sensorapplication on dynamic submarine electrical cable

A thermal sensor to detect the growth of biofouling suitable to be fixed on the As shown in Figure 4.22, the thermal sensor consists of a heating element, connected to a battery and two thermocouples fixed on stainless steel materials above and beyond the heating element covered with a layer of polyethylene on both sides. The surrounding water temperature is also measured using a thermocouple (or Pt 100) linked to the data logger. The data recorded in the data logger will be transmitted through an acoustic element, where the reception will be performed with a boat located above the thermal sensor.

Partial conclusion

Tracking the growth of biofouling is essential to avoid a decrease in fatigue lifetime of a submarine electrical cable. In this chapter, an original thermal sensor for the tracking of the biofouling growth has been presented which based on a transient thermal model.

This model results in estimation of the equivalent heat transfer coefficient of the material above the thermal sensor which in our case is biofouling.

Two models capacitive and conductive have been introduced. The conductive model based on quadrupole method was used for the estimation of the equivalent heat transfer coefficient of the mussels above the thermal sensor since it has less hypothesis comparing to the capacitive model. The effect of the duration of the heat pulse on the estimation of the equivalent heat transfer coefficient was tested on glass beads medium (which simulate the mussels medium). It shown that for long heat pulse duration the heat diffuses in more than one direction which not comply with the conductive model since it's a 1D model.

However, for short heat pulse duration the heat diffuses in one direction only. Then, a test toward the tracking of biofouling growth has been conducted for various ages, thickness, and water conditions with short heat pulse duration (15 s).

We have found that the equivalent heat transfer coefficient of mussels changes with age, where juvenile mussels having a lower equivalent heat transfer coefficient than mix and adult mussels, respectively since the thermal resistance of the mussels decreases with the age of mussels (porosity increases with ages). In addition, as expected the equivalent heat transfer coefficient of a mussels increases in a non-stagnant water condition compared to a stagnant water condition. This is due to the fact that the convection in the pores of mussels is increased in the case of non-stagnant water compared to stagnant water.

Moreover, the equivalent heat transfer coefficient decreases with the increasing of the mussel's thickness layer since we have an additional thermal resistance with the increasing of the thickness. As a result, the equivalent heat transfer coefficient is sensitive to the age of mussels, thickness of the mussels and the water current status (stagnant/non-stagnant).

Last and not least, an introduction for a thermal sensor dedicated to be used on a submarine electrical cable has been introduced. All transient measurements, collected from the thermal sensor will be used to estimate the equivalent heat transfer coefficient from which the type and thickness of the mussels can be identified. The patent maturity is planned to be initiated immediately after this thesis defense.

Conclusion and perspective

Conclusions

Floating wind turbine projects are being developed all over the world. One of the most important components of floating offshore wind turbines is the umbilical cable, which connects them to their electrical substation and the underwater energy network. Its electric insulation technology is intended to withstand conductor temperatures of up to 90 degrees Celsius. Mussel growth, in particular, can modify heat transfer around the cable and thus the maximum conductor temperature, also it leads to a temperature fluctuation, affecting fatigue lifetime.

The measurement of the thermal properties of the biofouling is rarely addressed in the marine energy literature. In chapter 2, the effective thermal conductivity of various ages of mussels (juvenile, mix (juvenile-adult) and adult) as well as the heat transfer coefficient of the water around them were measured for uniform mussel colonization around an experimental tube for a stagnant water condition. The measurement was a real challenge since it deals with a living mussels. Results shows that the juvenile mussels have the lowest effective thermal conductivity comparing to a mix mussels and adult mussels. This is due to the water porosity effect, where porosity of juvenile mussels is lower than the mix mussels and adult mussels respectively. Also, results show that the effective thermal conductivity of adult alive mussels is higher than the effective thermal conductivity of adult dead mussels due to the water filtering (nutrition) effect which creates an additional microconvection involving higher effective thermal conductivity of the mussels. Moreover, results show that the effective thermal conductivity of adult mussels increases with the increasing of the temperature difference between its internal and external layer. So as a summary, in case of stagnant water, the effective thermal conductivity of mussels is affected by three factors: the water porosity, the temperature difference between their internal and external layer and their filtering (nutrition) phenomenon. Another study has also been done to demonstrate the water current effects on the effective thermal conductivity of a porous medium formed from glass beads. It was demonstrated that the effective thermal conductivity of glass beads in the presence of current water increases with increasing water current velocity since the forced heat transfer coefficient in the glass beads medium increases. Therefore, it's expected the same for the effective thermal conductivity of the mussels in the presence of the water current.

In chapter 3, the thermal effect assessment of the biofouling on the DSEC is evaluated by estimating the temperature evolution of the copper conductor of a DSEC covered with mussels of various ages using a numerical simulation (via COMSOL). Those simulations were carried out for a given thickness layer of mussels and do not take into account the evolution of bio-colonization over time (multi-layer effects) as well as the water current which passes across or around the mussels. Results show that biocolonization of mussels causes an increase in the copper conductor temperature of the DSEC to more than its maximum operating temperature (90 °C) depending on the mussels' age in other words the corresponding global thermal resistance. Whereas, juvenile mussels have a bigger effect on the DSEC conductor temperature than mix and adult mussels which leads to rise in the copper conductor temperature of the cable. This rise in copper conductor temperature causes a decrease in the XLPE insulation fatigue lifetime, the overall cable fatigue lifetime. In that case, the dynamic submarine electrical cable requires maintenance which is too costly and not easy to do.

Then, monitoring the biofouling growth, especially mussels is essential in order to clean them from around the cable which would help to reduce their impact o on the DSEC. The conductive model based on quadrupole method has been chosen to be used to estimate the equivalent heat transfer coefficient since it has less hypothesis comparing to the capacitive model. The effect of the duration of the heat pulse on the estimation of the equivalent heat transfer coefficient was tested with glass beads medium covers the thermal sensor (which simulate the mussels medium). It has been shown that for short heat pulse durations, heat diffuses only in one direction in the glass beads medium, which is consistent with the conductive model because it is a 1D model. However, heat diffuses in more than one direction during long heat pulse durations. Then, a test toward the tracking of biofouling growth has been conducted for various ages, thickness, and water conditions with short heat pulse duration (15 s). Results show that the equivalent heat transfer coefficient of mussels changes with age, where juvenile mussels having a lower equivalent heat transfer coefficient than mix and adult mussels, respectively, because the mussels' thermal resistance decreases with age (porosity increases with age). Furthermore, as expected the equivalent heat transfer coefficient of mussels increases in non-stagnant water compared to stagnant water. This is as a result of the fact that non-stagnant water has greater convection in the pores of the mussels than in stagnant water. Additionally, as the thickness of the mussel layer increases, we experience an additional thermal resistance, which causes the equivalent heat transfer coefficient to decrease. As a result, the equivalent heat transfer coefficient is affected by mussel age, thickness, and water current status (stagnant/non-stagnant). Last but not least, a patent for the thermal sensor dedicated to be used on a submarine electrical cable has been shown. All transient measurements from the thermal sensor will be used to calculate the equivalent heat transfer coefficient, which will be used to determine the type and thickness of the mussels. The start of patent maturity is planned to begin immediately just after the defence of this thesis.

Perspectives

This manuscript has summarized efforts to thermally characterize the biofouling (mussels) and assess its thermal effect on a dynamic submarine electrical cable in stagnant water. However much more work is needed to obtain a more complete picture on its thermal characteristics and assess its thermal effect on a dynamic submarine electrical cable with the presence of water current flow. Herein, some perspectives and reflections will be discussed on how the thermal characterization could be improved to further understand its thermal effect on the DSEC. This seems like an important step for future design of dynamic submarine electrical cable.

To improve the thermal characterization of the mussels, the effective thermal conductivity of the mussels shall be measured during a natural colonization around the experimental tube that's why we already impose in the Mediterranean Sea and the Atlantic Ocean several experimental tube to have natural colonization of the mussels around them.

Also, more interesting aspect is the creation of a data base for the effective thermal conductivity of various ages of mussels as function of temperature difference between the inner and outer layer of mussels and as function of different water current velocities and as function of different types of mussels (mussels of Atlantic Ocean different from the mussels of Mediterranean Sea) and finally as a function of different thicknesses during mussel reproduction. This will be very useful for the thermal assessment of the mussels' colonization on the dynamic electrical cable in the real applications.

Moreover, for the monitoring of biofouling growth, it could be interesting to create a data base of the equivalent heat transfer coefficient using the transient thermal model with the new thermal sensor shape (as the patent) for different ages of mussels, types (different seas and oceans), thicknesses with the same real conditions which exists on site, meaning, the same water current velocities which exists in the seas and the oceans. At the end the data base of the equivalent heat transfer coefficient will helps to identify which age, type and thickness of mussels colonizes the dynamic submarine electrical cable. Then a thermal assessment of the effect can be determined using the database of effective thermal conductivity of mussels of different ages, thicknesses and types which will help to identify the thermal risks that can be induced by mussel colonization on dynamic submarine electrical cable. Therefore, it will help to optimize its design.

Résumé en Français

Résumé :

Les projets d'éoliennes flottantes se développent à l'échelle mondiale de manière prometteuse. Un de leurs composants clés est le câble dynamique de puissance permettant le raccordement des éoliennes à leur sous-station électrique et au réseau énergétique sousmarin. Parmi les effets les moins connus et qui peut être les plus impactant figure la bio colonisation. On entend par bio colonisation le développement de concrétions marines (algues, moules, huîtres) qui peuvent atteindre plusieurs dizaines de centimètres d'épaisseur. Par l'effet d'écran thermique, de masse additionnelle et de modification de la rugosité, la bio colonisation va impacter le comportement du câble dynamique. Parmi les effets majeurs on note les sollicitations hydrodynamiques modifiant d'une part leur tenue mécanique notamment en situation de tempête ou d'autre part leur fatigue. Ces dégradations du câble sont accentuées lors de l'accroissement de sa température d'où la nécessité de comprendre l'effet de la biocolonisation sur les transferts thermiques autour du câble. Ce sujet très peu abordé dans la littérature a été l'objet de cette thèse. Des échantillons de moules avec différents stades de maturité (juvénile, adulte ou mixte) ont été prélevés sur site marin afin de réaliser des mesures de conductivité thermique équivalente.

L'effet de la présence d'un écoulement forcé à travers l'échantillon de moules (milieu avec pores ouverts) a été étudié afin de prendre en compte la présence d'un éventuel courant marin. Des mesures complémentaires de perméabilité de ce milieu ont été réalisées. Enfin, en utilisant les mesures de caractérisation thermique, les champs de température au sein du câble sous-marin en présence de biocolonisation ont pu être simulés numériquement. Par ailleurs la problématique de la surveillance sur site marin de la croissance de la biocolonisation a été abordée en proposant un dispositif thermique original à installer sur les câbles sous-marins afin de périodiquement évaluer l'épaisseur ou les caractéristiques de la biocolonisation. Ce travail a été réalisé dans le cadre du projet BIODYTHERM financé par WEAMEC, West Atlantic Marine Energy Community, et avec un financement du Pays Région de la Loire. Il fait également partie du projet OMDYN-2 : Ombilicaux dynamiques pour les énergies marines renouvelables flottantes accordé par France Energies Marines et l'Agence nationale de la recherche.

Title: Effect of bio-colonization on the thermal behavior of dynamic electrical cables of floating wind turbines Keywords: electric dynamic cable; thermal characterization of biofouling; effective thermal conductivity; marine renewable energies; floating offshore wind turbine Abstract: Floating wind turbine projects are developing on a global scale in a promising way. One of their key components is the dynamic power cable allowing the connection of the wind turbines to their electrical substation and to the underwater energy network. Among the least known effects and which can be the most impacting is biocolonization.

By biocolonization we mean the development of marine concretions (algae, mussels, oysters) which can reach several tens of centimeters in thickness. By the effect of heat shield, additional mass and modification of the roughness, the bio colonization will impact the behavior of the dynamic cable. Among the major effects we note the hydrodynamic stresses modifying on the one hand their mechanical strength, particularly in stormy situations, and on the other hand their fatigue. These degradations of the cable are accentuated during the increase in its temperature, hence the need to understand the effect of biocolonization on the heat transfers around the cable. This subject, very little addressed in the literature, was the subject of this thesis. Samples of mussels with different stages of maturity (juvenile, adult or mixed) were taken from the marine site in order to carry out measurements of equivalent thermal conductivity. It is therefore necessary to understand the role of biocolonization, in its variability, to integrate studies on particular components. This thesis focuses on the thermal effect of biocolonisation (mussels( on the dynamic electric cable. The thermal characteristics for different ages of the mussels were measured and their thermal effect around the dynamic electric cable was simulated. Moreover, an original method for monitoring the growth of biocolonization around the dynamic underwater electric cable has been proposed using an original thermal sensor. This work was carried out within the framework of the BIODYTHERM project funded by WEAMEC, West Atlantic Marine Energy Community, and with funding from Pays Région de la Loire. is also part of the OMDYN-2 project: Dynamic umbilicals for floating marine renewable energies granted by France Energies Marines and the National Research Agency.

Figure 1 . 1 : 18 Figure 1 . 2 : 19 Figure 1 . 3 : 20 Figure 1 . 4 : 21 Figure 1 . 5 : 21 Figure 1 . 6 : 22 Figure 1 . 7 : 22 Figure 1 . 8 : 24 Figure 1 . 9 : 24 Figure 1 . 10 : 25 Figure 1 . 11 : 26 Figure 1 . 12 : 31 Figure 1 . 31 Figure 1 . 14 : 32 Figure 1 . 15 : 33 Figure 1 . 16 : 33 Figure 1 . 17 : 35 Figure 1 . 18 : 35 Figure 1 . 19 : 36 Figure 1 . 20 : 37 Figure 1 . 21 : 40 Figure 1 . 40 Figure 2 . 1 : 44 Figure 2 . 2 : 45 Figure 2 . 3 : 45 Figure 2 . 4 : 46 Figure 2 . 5 : 47 Figure 2 . 6 : 48 Figure 2 . 7 : 50 Figure 2 . 8 : 52 Figure 2 . 9 :Figure 2 . 10 :Figure 2 . 11 :Figure 2 . 12 :Figure 2 . 13 :Figure 2 . 14 :Figure 2 . 15 :Figure 2 . 16 :Figure 2 . 17 :Figure 2 .Figure 2 . 19 :Figure 2 . 20 :Figure 2 . 21 :Figure 2 . 22 :Figure 2 . 23 :Figure 2 . 24 :Figure 2 . 25 :Figure 2 . 26 :Figure 2 . 27 :Figure 2 . 28 :Figure 2 . 29 :Figure 2 . 30 :Figure 2 . 31 :Figure 2 . 32 :Figure 2 . 33 :Figure 2 . 34 :Figure 2 . 35 :Figure 2 . 36 :Figure 3 . 1 :Figure 3 . 2 :Figure 3 . 3 :Figure 3 . 4 :Figure 3 . 5 :Figure 3 . 6 :Figure 3 . 7 :Figure 3 . 8 :Figure 3 . 9 :Figure 4 . 1 :Figure 4 . 2 :Figure 4 . 3 :Figure 4 . 4 :Figure 4 . 5 :Figure 4 . 6 :Figure 4 . 7 :Figure 4 . 8 :Figure 4 . 9 :Figure 4 . 10 :Figure 4 .Figure 4 . 12 :Figure 4 . 13 :Figure 4 . 14 :Figure 4 . 15 :Figure 4 . 16 :Figure 4 . 17 :Figure 4 . 18 :Figure 4 . 19 :Figure 4 . 20 :Figure 4 . 21 :Figure 4 . 22 :

 1118121913201421152116221722182419241102511126112311311143211533116331173511835119361203712140140214422452345244625472648275028522921021121221321421521621722192202212222232242252262272282292302312322332342352363132333435363738394142434445464748494104412413414415416417418419420421422 Figure 1.1: Illustration of fixed offshore wind turbine and floating offshore wind turbine. ...................................................................................................................18 Figure 1.2: Power transmission system of floating offshore wind turbine. .......................19 Figure 1.3: Structure of Medium Voltage power cable [16]. .............................................20 Figure 1.4: Dynamic cable configurations [18]. ................................................................21 Figure 1.5: Power transmission system of offshore wind farm, top view [19]. .................21 Figure 1.6: W-shape interconnection configuration between two floating offshore wind turbine [19].............................................................................................................22 Figure 1.7: Mechanical loads on dynamic power cable. ....................................................22 Figure 1.8: Biofilm formation steps [20]. ..........................................................................24 Figure 1.9: Biofilm cross section [23]. ..............................................................................24

Figure 1 . 1 :

 11 Figure 1.1: Illustration of fixed offshore wind turbine and floating offshore wind turbine.

Figure 1

 1 Figure 1.2.

Figure 1 . 2 :

 12 Figure 1.2: Power transmission system of floating offshore wind turbine.

Figure 1 . 4 :Figure 1 . 6 ,

 1416 Figure 1.4: Dynamic cable configurations [18].

Figure 1 . 5 :

 15 Figure 1.5: Power transmission system of offshore wind farm, top view [19].

Figure 1 . 8 :

 18 Figure 1.8: Biofilm formation steps [20].

Figure 1 .Figure 1 . 11 :

 1111 Figure 1.11: Mussel's nutrition [25].

Figure 1 . 14 :

 114 Figure 1.14: Identified Zh and Zc from step excitation (continuous line) and from ramp excitation (x).

Figure 1 . 15 :

 115 Figure 1.15: Shell and tube heat exchanger with presence of uniform fouling (limestone deposit).

Figure 1 . 16 :

 116 Figure 1.16: Identified Z without and with different layers uniform thickness of fuling a) Zh and b) Zc.

Figures 1 .

 1 Figures 1.16a and 1.16b clearly show how the presence of a fouling layer affects the impulse response profiles. It should also be noted that increasing the fouling thickness

Figure 1 . 17 :

 117 Figure 1.17: Temperature distribution (°C), in case of uniform biofouling coverage and thickness 30 mm [30].

Figure 1 . 18 :

 118 Figure 1.18: Temperature evolution versus biofouling thickness for each component of dynamic power cable [30].

Figure 1 . 19 :

 119 Figure 1.19: Temperature evolution versus the percentage of surface coverage of biofouling for each component of dynamic power cable [30].

Figure 1 .

 1 Figure 1.21, from equipment setup to image analysis, in order to have a standardized inspection involving precise tracking and more accurate damage evaluation.

Figure 1 . 21 :

 121 Figure 1.21: The main steps of stereo imaging pipeline [45].

Figure 1 .

 1 Figure 1.22:a) Marine growth on structure b) Reconstructed surface with showing the dimensions [45].

Figure 2 . 1 .

 21 It consists of an aluminum tube (int=60 mm, ext=70 mm, L=610 mm) implemented with 5 K-type thermocouples (3 in the middle cross section with 120° angle and one thermocouple on each of the two other cross-sections close to the extremities of the tube (at 5 cm). These thermocouples installed on grooved locations on the tube (which mentioned later) created by a CNC milling machine and glued with a marine onecomponent polyurethane sealant glue (Sikaflex 291i).

Figure 2 . 1 :

 21 Figure 2.1: Schematic for the experimental tube.

Figure 2 .

 2 Figure 2.2: a)before pumping the rubber air chamber b)after pumping the rubber air chamber

Figure 2 . 3 :Figure 2 . 4 :Figure 2 . 5 :

 232425 Figure 2.3: Experimental tube support.

  Photo

Figure 2 . 6 :

 26 Figure 2.6: Uniform distribution of mussels around the experimental tube held by a steel net.

Figure 2 . 7 :

 27 Figure 2.7: Middle cross-section of the experimental tube.

Figure 2 . 8 :

 28 Figure 2.8: Experimental tube covered with double sided adhesive tape

Figure 2 .Figure 2 . 9 :

 229 Figure 2.9: Experimental tube covered with glass beads a)without plastic cover b) with plastic cover.

Figure 2 . 10 :

 210 Figure 2.10: Adult mussels (12 months) a) alive b) dead.

Figure 2 .

 2 Figure2.11 depicts the evolution of the effective thermal conductivity of living and dead adult mussels as a function of the temperature difference between the mussels' internal and external layers. The measurements have a relative uncertainty of less than 9%. A repeated measurement of the effective thermal conductivity of adult mussels at ΔT=5.1 K is also performed; the result shows that the effective thermal conductivity is equal to 12.8

Figure 2 . 11 :

 211 Figure 2.11: Effective thermal conductivity of dead and living adult mussels as function of temperature difference between the internal and the external layer of the mussels.

  𝑄 ̇(m 3 /s) is the flow rate of the fluid flowing through an area A(m 2 ), 𝜅 is the permeability (m 2 ), 𝛥𝑝(𝑃𝑎) = 𝑝 𝑎 -𝑝 𝑏 is the total pressure drop over the sample, Lc(m) is the length of the cylinder, 𝜇 (𝑃𝑎. 𝑠) is the dynamic viscosity of the fluid.

Figure 2 . 12 :

 212 Figure 2.12: Schematic for the experimental setup to measure the permeability.

Figure 2 . 13 :

 213 Figure 2.13: Glass beads in the experimental setup.

Figure 2 . 14 :

 214 Figure 2.14: Results of difference of pressure before and after the glass beads sample (ΔP) as function of the flow rate 𝑄 ̇.

Figure 2 . 15 :

 215 Figure 2.15: Adult mussels in the experimental setup.

Figure 2 .

 2 Figure 2.16 depicts the evolution of differential pressure as a function of flow rate in the presence of adult mussels, where the relative uncertainty of the differential pressure meter is (0.05%). Since the differential pressure is directly proportional to the flow rates, the 𝛥𝑝 increases as the flow rate increases. The permeability of the adult mussels is then calculated using Darcy's law and the slope of Figure 2.16, and this yields a value of

Figure 2 . 16 :

 216 Figure 2.16: Results of difference of pressure before and after adult mussels sample (ΔP) as function of the flow rate 𝑄 ̇.

Figure 2 .

 2 Figure 2.18: a) Total volume of mussel's cluster b) Volume of the interstitial water between the mussels.

The equation ( 2 . 9 )

 29 is solved numerically with the associated boundary conditions: imposed heat flux φ (W.m -2 ) with φ = 𝑄 ̇/2πr0L at the inner surface of the aluminum tube and convective heat transfer coefficient (hw1 and hw2) at the external surface, as shown in Figure 2.19.

Figure 2 . 19 :

 219 Figure 2.19: Various colonization distribution configurations around the heated covered polyethylene aluminum tube. (a) 100% colonization, (b) 50 % colonization and (c) 25% colonization.

Figure 2 . 20 :Figure 2 Figure 2 . 21 :

 2202221 Figure 2.20: Fine mesh in COMSOL for (a) 25% (b) 50% mussel colonization.

Figure 2 . 22 :

 222 Figure 2.22: Traction water basin.

Figure 2 . 23 :

 223 Figure 2.23: Traction carriage of water basin.

Figure 2 . 24 :

 224 Figure 2.24: Porous medium formed from glass beads fixed on the experimental tube.

Figure 2 . 25 :

 225 Figure 2.25: Support of the experimental tube.

Figure 2 . 26 :Figure 2 . 27 :

 226227 Figure 2.26: Distribution of thermocouples on the experimental tube with the direction of the water current flow.

Figure 2 .Figure 2 . 29 :

 2229 Figure 2.28a, shows the temperature evolution in the internal thermocouples for stagnant water and Figure 2.28b and Figure 2.28c show the temperature evolution in the internal thermocouples for water current velocity of 0.5 m/s and 2 m/s respectively. In the case of a water current velocity, we can see from Figures 2.27b and 2.27c, that there is a significant water disturbance at the top of the water surface at the location of the supports (above the extremities of the tube), but this water disturbance has no effect on the temperature of the internal thermocouples since as we can see that the internal temperatures of the middle, right, and left locations are approximately the same.

Re≤ 10 5 ,

 5 Nusselt number (Nu) starts to decrease from the stagnation point with the increases of θ as a result of a laminar boundary layer development. The separation between the laminar boundary layer and the wakes (separation point), occurs at θ=80°, and as a result of mixing caused by vortex formation in the wake, the Nu increases.In contrast, for Re ≥10 5 , variation of Nu with is distinguished by two minima (Figure2.30).The decrease in Nu from the stagnation point value is once more caused by the development of a laminar boundary layer, but the abrupt increase that occurs between 80° and 100° is now caused by the boundary layer changing into turbulence. The turbulent boundary layer continues to grow, and Nu once more starts to decline. Finally, at (θ≈140°) separation happens, and the wake region's mixing causes Nu to rise[START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF].

Figure 2 . 30 :

 230 Figure 2.30: Nusselt number as function of angular coordinate for smooth cylinder [50].

Figure 2 . 31 :

 231 Figure 2.31: Evolution of the average effective thermal conductivity of beads as function of the water current velocity.

Figure 2 . 32 :

 232 Figure 2.32: Effective thermal conductivity of the glass beads as function of Reynolds number.

Figure 2 . 33 :

 233 Figure 2.33: Experimental tube with and without polyethylene cover.

Figure 2 . 34 :

 234 Figure 2.34: Experimental tube fixed on the floater.

Figure 2 . 35 :

 235 Figure 2.35: Distribution of the experimental tubes in different zones.

Figure 2 . 36 :

 236 Figure 2.36: Experimental tube with and without polyethylene after two months of immersion in the sea.

Figure 3 . 1 :

 31 Figure 3.1: Cross-linked polyethylene (XLPE) insulation of dynamic power cable

Figure 3 . 2 :

 32 Figure 3.2: Cross section of a three phases DSEC.

Figure 3 .

 3 Figure 3.3 shows a physical representation of the cable and its surroundings as a network of combined thermal resistances (R1, R2, R3 and R4). Where Tc (K) is the operating

Figure 3 . 3 :

 33 Figure 3.3: Network of thermal resistances representing steady state heat transfer in a three core submarine cable and its surrounding environment.

  m -1 ) is the AC resistance for a given conductor temperature (T), X is the reactance of sheath per unit length of cable (Ω.m -1 ) given in equation (3.10) and 𝑅 𝑠 𝑇 𝑠 ′ which is the resistance of the metallic sheath at the temperature of the metallic sheath (Ω.m -1 ) is calculated in equation (3.11):

Figure 3 . 4 :

 34 Figure 3.4: Mesh with normal element size for the DSEC in COMSOL.

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Simulation of the DSEC steady-state temperature distribution (°C) under rated load conditions.

Figure 3 . 7 :

 37 Figure 3.7: DSEC conductor temperature as function of mussel age using finite element method (COMSOL).

Figure 3 . 8 :

 38 Figure 3.8: Effective thermal conductivity of juvenile mussels for different ΔT and imposed power.

Figure 4 . 1 :

 41 Figure 4.1: Schematic of flux metric device.

Figure 4 . 2 :

 42 Figure 4.2: Heating pulse and Temperature evolution and decay versus time.

Figure 4 . 3 ,

 43 Tw is the 109 temperature of the wall, Th the temperature of sensor and 𝜑 𝑑 the density of flow dissipated by the heating system of the sensor.

Figure 4 . 3 :

 43 Figure 4.3: Diagram of a sensor with impulse heating laws.

Figure 4 . 4 :

 44 Figure 4.4: Sensor Cooling.

A 0 ( 4 . 12 ) 0 ( 4 . 13 )

 04120413 multilayer system's local quadrupole matrix is created by multiplying the matrices for each layer to link the temperature-flux vectors on either side of it. The boundary condition is taken into account, and Laplace inversion is applied to obtain the necessary external temperatures and fluxes (depending on the transformed space that we use). Then, it is possible to determine the internal temperature-fluxes by multiplying the temperature-flux vector recently obtained at one external surface by the quadrupole matrix corresponding to the internal layer. The quadrupole method is only constrained by the symmetry consideration, so even if the propagation axis is switched around, the symmetrical material system remains unchanged.Temperature and heat flux in the Laplace space are represented, respectively, by:𝜃(𝑝) = 𝐿[𝑇(𝑡)] = ∫ T(t) exp(-pt) dt ∞ Φ(𝑝) = 𝐿[𝜑(𝑡)] = ∫ 𝜑(t) exp(-pt) dt∞ where p is the Laplace parameter. The Laplace temperatures and fluxes have been calculated after the modelling of the flux metric sensor device. The thermal considerations around the flux metric sensor device are shown in Figure 4.5. θc is the Laplace temperature at the location of the thermocouple (at the rear face of the stainless steel layer) and θE is the Laplace environmental temperature and 𝛷 ℎ + , 𝛷 ℎ -, 𝛷 𝐸 + , 𝛷 𝐸 -is the Laplace heat flux of heating element to upward, the heat flux of heating element to downward, heat flux of environment to the upward and the heat flux of environment to downward respectively.

Figure 4 . 5 :

 45 Figure 4.5: Thermal distribution of flux metric sensor device.

[

Figure 4 . 6 :

 46 Figure 4.6: Chart flow of quadrupole method to estimate the heat transfer coefficient h.

Figure 4 . 7 :

 47 Figure 4.7: Temperature evolution and relaxation at the location of the thermocouple Tc in the flux metric device.

Figure 4 .

 4 10 depicts the temperature evolution at the thermocouple at point c in the flux metric device (Figure 4.1)for various heat pulse durations.

Figure 4 . 10 :Figure 4 .Figure 4 . 12 :

 4104412 Figure 4.10: Temperature evolution for different heat pulse duration.

Figure 4 .Figure 4 . 13 :

 4413 Figure 4.13: Temperature evolution at point c using 1D and 3D numerical transient model (COMSOL).

  pulse and after the heat pulse. Where 𝑎 (2.12x10 -7 m 2 .s -1 ) is the thermal diffusivity of the glass beads medium (𝑎 = 𝑘 ℎ,𝑔𝑙𝑎𝑠𝑠 𝑏𝑒𝑎𝑑𝑠  𝑔𝑙𝑎𝑠𝑠 𝑏𝑒𝑎𝑑𝑠 𝐶 𝑝 𝑔𝑙𝑎𝑠𝑠 𝑏𝑒𝑎𝑑𝑠

Figure 4 .Figure 4 . 14 :

 4414 Figure 4.14: Measured and numerical temperature at point c a) without reducing k values b) with reducing k values.

Figure 4 . 15 :

 415 Figure 4.15: Equivalent heat transfer coefficient of the glass beads for a)short heat pulse duration (25 s) and long heat pulse duration b) 120 s and c) 180 s.

4. 16 )Figure 4 . 16 :

 16416 Figure 4.16: Different ages of mussels a) juvenile b) mixed c) adult.

Figure 4 .

 4 Figure 4.17 depicts the temperature evolution at the thermocouple at point c in the flux metric device (Figure 4.1) for various ages of mussels for stagnant water condition.The equivalent heat transfer coefficients of different ages of mussels have been estimated using the previous conductive model.

Figure 4 . 17 :

 417 Figure 4.17: Temperature evolution at c with various ages of mussels covering the flux metric device.

Figure 4 . 18 :Figure 4 . 19 :

 418419 Figure 4.18: Equivalent heat transfer coefficient of various ages of mussels for stagnant water condition.

Figure 4 . 20 :

 420 Figure 4.20: Temperature evolution at point c with juvenile 4 cm thickness layer and 9.4 cm thickness layer covering the flux metric device.

Figure 4 . 21 :

 421 Figure 4.21: Thermal sensor: application on submarine electrical cable.

Figure 4 . 22 :

 422 Figure 4.22: Cross-section of thermal sensor above the submarine electrical cable.

Chapter 4 ,

 4 describes a novel method for monitoring biofouling growth with a thermal sensor based on transient thermal model. Two transient thermal models have been introduced, capacitive and conductive model. These two models leads to the estimating of an equivalent heat transfer coefficient where its value helps to identify what kind of specimen cover the thermal sensor, in other words leads to monitor the biofouling growth.

Chapitre 1 :Chapitre 2 :Chapitre 3 :

 123 BibliographieDans ce chapitre, une bibliographie sur les éoliennes offshores flottantes et l'importance du câble dynamique de puissance pour le transfert d'énergie de l'offshore vers le rivage a été introduite. De plus, une bibliographie sur le biofouling et les éléments clés qui influencent sa croissance est proposée. Il est démontré comment l'encrassement biologique affecte le câble d'alimentation dynamique en termes de thermomécanique. La prédominance des moules sur les côtes atlantiques est mise en évidence. De plus, l'effet de l'encrassement sur les échangeurs de chaleur et la façon de le détecter ont été discutés. L'effet de l'encrassement biologique sur les câbles électriques sous-marins est également discuté, ainsi que les méthodes de détection, où la conductivité thermique de la couche d'encrassement biologique était supposée être la même que celle de l'eau. Cependant, en réalité, les amas de moules qui ont colonisé le câble électrique sous-marin doivent être considérés comme un milieu poreux formé d'une partie solide, le bivalve des moules luimême, et d'une partie fluide (eau). Par conséquent, il est intéressant d'estimer les propriétés thermiques des moules, en particulier leur conductivité thermique effective et le coefficient de transfert de chaleur de l'eau qui les entoure, afin de quantifier en conséquence l'impact thermique des moules sur le câble dynamique haute tension sous-marin. Par ailleurs, une revue de la littérature sur les équations permettant de déterminer la conductivité thermique effective en milieu poreux a été proposée. Il montre que tous les modèles et formules introduits pour estimer la conductivité thermique effective ne tiennent pas compte de la convection naturelle dans le milieu poreux. Le chapitre suivant présente une méthode expérimentale de mesure de la conductivité thermique effective d'un milieu poreux (moules) en tenant compte de la convection naturelle dans le milieu poreux. Caractérisation thermique des moules -Analyse en régime permanent Dans ce chapitre, la caractérisation thermique des moules a été réalisée. La mesure des propriétés thermiques de ce milieu naturel (biofouling) est rarement abordée dans la littérature sur les énergies marines. Le travail expérimental était extrêmement difficile car nous devions effectuer toutes les mesures en 24 h afin de maintenir les moules en vie. Dans le cas de la colonisation uniforme des moules, la conductivité thermique des différents âges des moules (juvénile, mixte juvénile-adulte, et adulte) ainsi que le coefficient de transfert de chaleur de l'eau autour des moules ont été mesurés. Nous avons déterminé que les moules juvéniles ont la conductivité thermique effective la plus faible par rapport à un mélange de moules juvéniles et adultes et de moules adultes uniquement en raison de l'effet de porosité à l'eau, où la porosité des moules juvéniles est inférieure à celle du mélange de moules et de moules adultes respectivement. L'effet de la porosité à l'eau a été mis en évidence et illustré à l'aide d'un milieu de billes de verre, où il est démontré que la convection naturelle dans le milieu poreux améliore considérablement la conductivité thermique effective du milieu poreux en verre. De plus, la conductivité thermique effective des moules adultes a été mesurée lorsque les moules sont vivantes et lorsqu'elles sont mortes. Les résultats montrent que la conductivité thermique effective des moules adultes vivantes est supérieure à la conductivité thermique effective des moules adultes mortes en raison de l'effet de filtrage de l'eau qui crée un micro-convection supplémentaire entraînant une augmentation de la conductivité thermique effective des moules. D'autre part, la conductivité thermique effective des moules augmente avec l'augmentation de la différence de température entre la couche interne des moules et sur la couche externe des moules en raison du phénomène de convection naturelle. Ainsi en résumé, en cas d'eau stagnante, la conductivité thermique effective des moules est affectée par trois facteurs : la porosité à l'eau, la différence de température entre leur couche interne et externe et leur phénomène de filtrage. Une autre étude a également été réalisée pour démontrer comment le courant d'eau affecte la conductivité thermique effective d'un milieu poreux formé de billes de verre. Il a été démontré que la conductivité thermique effective des billes de verre en présence d'eau courante augmente avec l'augmentation de la vitesse du courant d'eau à mesure que le coefficient de transfert de chaleur forcé dans le milieu des billes de verre augmente. Par conséquent, on s'attend à ce qu'il en soit de même pour la conductivité thermique effective en présence du courant d'eau.Les résultats démontrent également que la résistance conductrice effective des moules peut être supérieure à la résistance convective entourant le tube en leur absence. Il serait donc intéressant d'étudier l'impact thermique de la colonisation par les moules à proximité de la DSEC. Afin de mieux comprendre cette problématique, le chapitre suivant se focalisera sur ce sujet. Effet thermique des moules sur la température du câble électrique sousmarin dynamique Dans ce chapitre, le risque thermique sur la câble électrique sous-marin dynamique est évalué en estimant la distribution de température au sein de la DSEC recouverte de moules d'âges différents à l'aide d'une simulation numérique (via COMSOL). Ces simulations ont été réalisées pour une épaisseur donnée de la couche de moules et ne tiennent pas compte de l'évolution de la bio-colonisation dans le temps (effets multicouches) ainsi que du courant d'eau qui passe à travers ou autour des moules. Les résultats montrent que la biocolonisation des moules provoque une augmentation de la température du conducteur en cuivre de la câble électrique sous-marin dynamique en fonction de l'âge des moules et de la résistance thermique globale correspondante. Les moules juvéniles ont un plus grand effet sur la température du conducteur câble électrique sous-marin dynamique que les moules mixtes et adultes. Cette augmentation de la température du conducteur en cuivre entraîne une diminution de la durée de vie de l'isolant XLPE, de la durée de vie globale en fatigue du câble et de la conductivité électrique du conducteur en cuivre du câble. Par conséquent, l'énergie est moins reçue et les coûts de maintenance sont augmentés. La surveillance de la croissance des moules et leur nettoyage autour du câble au besoin aideraient à réduire l'impact de la biocolonisation des moules sur la câble électrique sousmarin dynamique. Le chapitre suivant décrit une nouvelle méthode de surveillance de la croissance de l'encrassement biologique avec un capteur thermique. Titre : Effet de la bio-colonisation sur le comportement thermique des câbles électriques dynamiques des éoliennes flottantes Mots clés : câble dynamique électrique ; caractérisation thermique du biofouling ; conductivité thermique effective ; les énergies marines renouvelables ; éolienne offshore flottante

Table 1 .

 1

1: Daily rate for vessels. .......................................................................................17

Table 1 .

 1 2: Impulse forms and impulse responses expression [29]. ...................................30

Table 2 .

 2 1: Various ages of mussels with their thickness layer. .........................................47

Table 2 .

 2 2: Distribution of measured temperature around the aluminum tube for different ages of mussels uniformly distributed. ..................................................................52

Table 2 .

 2 3: Measured effective thermal conductivities of mussels of various ages uniformly distributed around the aluminum tube. .................................................53

Table 2 . 4

 24

: Homogeneous versus experimental effective thermal conductivity of glass beads. .....................................................................................................................55 Table 2.5: Rayleigh number of alive and dead adult mussels. ...........................................62

Table 2 .

 2 6: Water porosities of various ages of mussels. ....................................................63

Table 2 .

 2 7: Experimental and theoretical values of the heat transfer coefficient of the water around the experimental tube without mussels. .....................................................64

Table 2 .

 2 8: Heat transfer coefficient of the water around the mussels as measured experimentally........................................................................................................65

Table 2 .

 2 9: Comparison between conductive and convective thermal resistances as a function of the ages of the mussels. .......................................................................65

Table 2 .

 2 10: Distribution of measured temperatures around a polyethylene covered aluminum tube colonized by juvenile mussels with different configurations of colonization. ...........................................................................................................67

Table 2 .

 2 11: Effective thermal conductivity of juvenile mussels and heat transfer coefficient of water around the mussels for different percentage of colonization. 70

Table 2 .

 2 12: Temperature distribution with different water current velocity. ....................79

Table 2 .

 2

	Nomenclature
	C	cable capacitance per unit length

13: Effective thermal conductivity of the glass beads with different water current velocity. ..................................................................................................................81 Table 3.1: Thermal resistances and heat losses in the submarine power cable. ................96 Table 3.2: Copper conductor temperature during the colonization of various ages of mussels. ................................................................................................................100 Table 3.3: Overall thermal resistance of various ages of mussels. ..................................101 Table 3.4: The temperature difference between the mussel layer's inner and outer radiuses. ...............................................................................................................102 Table 4.1: Thermophysical properties of different layers of flux metric device. ............118 Table 4.2: Equivalent heat transfer coefficient using the capacitive and conductive model....................................................................................................................119 Table 4.3: Layer thickness and porosity of various ages of mussels. ..............................130

1. Chapter 1. Literature review 1.1 Floating offshore wind turbine

	According to the European Wind Energy Association (EWEA) [15], at depths greater
	than 60 meters, 80 percent of Europe's offshore wind sources are available, with a potential
	floating wind capacity of 4000 GW, which can provide four times the EU's electrical
	energy consumption. Floating offshore wind turbine is one of the most promising clean
	energy technologies, and its development is accelerating. It has the ability to go in deep

water in comparison to the fixed offshore wind turbine which can goes maximum to 50-60 m deep due to structural limitations. The benefits of FOWT over fixed-offshore wind farms include less visual disturbance, noise avoidance, stronger and more consistent wind, lower installation costs (Table

1

.1), no wind turbine size restrictions, and the FOWT is more environmentally friendly. However, there are some drawbacks to FOWT, such as technical difficulties with mooring line and power cable design, as well as electrical connections.

Table 1 .

 1

	Fixed wind installations	Floating wind installations
	Vessel	Daily rate	Vessel	Daily rate
	Heavy lift vessel		Standard tug boat
	(foundation	€ 150k-500k	(tow out and hook	€ 30k-60k
	installation)		up)
	Jack-up vessel (turbine installation)	€ 150k-200k	Anchor handling installation) tug (mooring	€ 20k-50k
	Mobilization	Several M€	Mobilization	€ < 100k

1: Daily rate for vessels.

Table 2 .

 2 2: Distribution of measured temperature around the aluminum tube for different ages of mussels uniformly distributed. .72 26.6 23.47 22.32 22.31 4.2 22.3 35 95 127.75 Adult 28.77 27.9 28.28 23.62 23.52 23.52 4.8 23.36 35 105 211.63Therefore, the measured values of the effective thermal conductivity of juvenile, mix (juvenile and adult) and adult mussels are 4.4, 8, 12.8 W.m -1 .K -1 , respectively, for a uniform distribution of mussels around the aluminium tube, as shown in Table2.3. As the relative uncertainty on kbiof measurement is less than 9 %, the differences between the various types of colonization is significant. One explanation for these discrepancies could be the increasing volume of mussels with age.

	Mussels type	T1	T2	T3	T4	T5	T6	ΔT (Tav1-	Tw	ri	re Q
								Tav2)
		o C	o C	o C	o C	o C	o C	K	o C	mm mm W
	Juvenile 35.7 32.82 31.39 23.56 23.1 23.13 10 23	35 75 228.03

Table 2 .

 2 3: Measured effective thermal conductivities of mussels of various ages uniformly distributed around the aluminum tube.

	Mussels type	kbiof	Absolute uncertainty Relative uncertainty
		W.m -1 .K -1	W.m -1 .K -1	%
	Juvenile	4.4	± 0.4	9
	Mix (juvenile and adult)	8.0	± 0.52	6.5
	Adult	12.8	± 0.97	7.6

Table 2 .

 2 5: Rayleigh number of alive and dead adult mussels.

		Alive Mussels Dead Mussels
	ΔT	Ra	Ra	Difference
	1.75	7.15E+02	6.69E+02	7%
	3.11	1.28E+03	1.16E+03	10%
	3.89	1.58E+03	1.52E+03	4%
	4.83	2.00E+03	1.89E+03	6%
	5.47	2.27E+03	2.18E+03	4%

The porosity of adult mussels is then estimated using equation 2.7, where Macdonald et al consider non-spherical particles, then dp in equation 2.8 is replaced by dp, equivalent which is the equivalent diameter for non-spherical particles given by: dp, equivalent= 𝜖.dv (2.8)

Table 2 .

 2 6: Water porosities of various ages of mussels.

	Mussel's age	Juvenile (3-6 months)	Mix (6-12 months)	Adult (12-18 months)
	Water Porosity	15%	18%	22.6%

Table 2 .

 2 7: Experimental and theoretical values of the heat transfer coefficient of the water around the experimental tube without mussels.

	hw	hw	hw discrepancy	hw	hw discrepancy
	(experimental)	(Churchill &	(Experimental	(Morgan)	(Experimental
		Chu)	and Churchill		and Morgan)
			& Chu)
	W.m -2 .K -1	W.m -2 .K -1	%	W.m -2 .K -1	%
	220	309	29	234	6

Table 2 .

 2 8 shows the measured heat transfer coefficients of 3395, 873, and 2682

Table 2 .

 2 8: Heat transfer coefficient of the water around the mussels as measured experimentally.

	Mussels type	hw	Absolute uncertainty	Relative uncertainty	TMussels,water	Absolute uncertainty	Relative uncertainty
		W.m -2 .K -1 W.m -2 .K -1	%	o C	o C	%
	Juvenile	3395	± 1123	33	0.23	± 0.07	30
	Mix (juvenile & adult)	873	± 164	19	0.4	± 0.07	17.5
	Adult	2682	± 1003	37	0.2	± 0.07	35

*TMussels,water: is the difference between the temperature on the external side of the mussel layer and the one of the water.

Table 2 .

 2 9: Comparison between conductive and convective thermal resistances as a function of the ages of the mussels. Rconductive (biof) is the effective conductive thermal resistance of biofouling * Rconvective is the convective thermal resistance of the water around the mussels *Roverall is the overall thermal resistance of the presence of mussels around the tube (including Rconductive(biof) and Rconvective).

	Mussels type Rconductive(biof) Rconvective Roverall	Effective conductive resistance	External convective resistance	T(external and internal layer of
				contribution	contribution	mussels)
		K.W -1	K.W -1 K.W -1	%	%	K
	Juvenile	0.045	0.001 0.046	97.8	2.2	10

*

  In this case,

	Mix(juvenile	0.033	0.003 0.036	91.7	8.3	4.2
	and adult)
		0.022	0.0009 0.023	95.7	4.3	4.8
	Adult

Table 2 .

 2

10 shows the measured temperature distribution around a polyethylene covered aluminium tube for different configurations of juvenile mussel colonization distribution (25%, 50% and 100%), as shown in Figure 2.19.

Table 2 .

 2 10: Distribution of measured temperatures around a polyethylene covered aluminum tube colonized by juvenile mussels with different configurations of colonization.

	Juvenile	T1	T2	T3	T4	T5	T6	Tw	ri	re	q
	mussels
	Colonization
	%	o C	o C	o C	o C	o C	o C	o C	mm	mm	W
	25	28.12 26.44 24.8 22.55	-	-	22.3 38.9 78.9 66.33
	50	30.82 28.5 26 22.92	-	-	22.45 38.9 78.9 85.43
	100	29.48 28.46 28.95 23.03 22.6 22.4 22.12 38.9 78.9 40.74

Table 2 .

 2 11: Effective thermal conductivity of juvenile mussels and heat transfer coefficient of water around the mussels for different percentage of colonization.

	Juvenile	kbiof	Sensitivity for	hw1	Sensitivity	hw2	Sensitivity
	mussels		kbiof
	Colonization				for hw1		for hw2
	%	W.m -1 .K -1	°C	W.m -2 .K -1	°C	W.m -2 .K -	°C
						1
	25	1.4	1	1510	0.1	1960	0.9
	50	1.9	1	310	0.1	3910	0.88
	100	1.6	5	910	0.1	-	-

Table 2 .

 2 [START_REF] Marty | Effect of Roughness of Mussels on Cylinder Forces from a Realistic Shape Modelling[END_REF]: Temperature distribution with different water current velocity.

			Up (Angle 0°)		Angle 120°		Angle 240°
	Velocity	Reynolds number	T6	T3	(T6-T3)	T4	T1	(T4-T1)	T5	T2	(T5-T2)
	m/s		°C	°C	°C	°C	°C	°C	°C	°C	°C
	0		20.06	16.86	3.19	20.02	16.84	3.18	20.02	16.84	3.18
	0.2	2.04E+04 17.50	16.92	0.58	17.76	16.95	0.81	17.2	16.90	0.29
	0.4	4.08E+04 17.18	16.91	0.27	17.32	16.92	0.40	16.99	16.90	0.09
	0.6	6.12E+04 17.08	16.91	0.18	17.18	16.91	0.27	16.94	16.90	0.04
	0.8	8.16E+04 17.02	16.90	0.11	17.07	16.91	0.16	16.93	16.90	0.02
	1	1.02E+05 16.99	16.90	0.08	17.02	16.91	0.12	16.92	16.90	0.01
	1.2	1.22E+05 16.97	16.90	0.06	16.99	16.91	0.09	16.91	16.90	0.01
	1.4	1.43E+05 16.95	16.90	0.05	16.97	16.90	0.07	16.91	16.90	0.01
	1.6	1.63E+05 16.94	16.90	0.04	16.96	16.90	0.06	16.91	16.90	0.01
	1.8	1.84E+05 16.94	16.90	0.03	16.95	16.90	0.05	16.91	16.90	0.01
	2	2.04E+05 16.93	16.90	0.02	16.94	16.90	0.04	16.91	16.90	0.01

Table 2 .

 2 [START_REF] Schoefs | Reliability Updating of Offshore Structures Subjected to Marine Growth[END_REF]: Effective thermal conductivity of the glass beads with different water current velocity.

	Water current velocity	Reynold number Nusselt number	forced h			,beads e k			,beads, e k		°θ=90	,beads, e k		°θ=210	,beads, e k	°θ=330
	m.	-s	1			-W.m	.K 2	-	1	-W.m	.K 1	-	1	W.	-m	.K 1	-	1	W.	-m	.K 1	-	1	W.	-m	.K 1	-	1
	0				315			3.17				3.15				3.19				3.18
	0.2		2.04E+04	83	488			34				28				20				55
	0.4		4.08E+04	117	693			98				60				41				193
	0.6		6.12E+04	129	764			185				94				61				400
	0.8		8.16E+04	145	857			312				147				104				684
	1		1.02E+05	159	937			451				205				149			1000
	1.2		1.22E+05	171	1007			586				273				186			1300
	1.4		1.43E+05	182	1073			721				329				234			1600
	1.6		1.63E+05	192	1131			899				410				288			2000
	1.8		1.84E+05	202	1187			1062				497				342			2346
	2		2.04E+05	211	1239			1223				576				400			2692

   𝑒,𝑐 (m) is the external diameter of the cable, ℎ 𝑠𝑢𝑟𝑟 is the heat transfer coefficient of the surrounding medium around the cable, 𝛥𝑇 𝑠 is the excess of cable surface temperature above ambient temperature (K). Using ohm's law and steady-state condition meaning when the current flow through the cable is at a constant value and the temperature of the cable is constant then the expression for the temperature rise above ambient temperature is

	Therefore, the maximum current that an AC cable can deliver is obtained as shown in
	equation (3.7) [13]:
	= [	𝐼 𝑐𝑜𝑛 𝑇 𝑐𝑜𝑛 𝑅 𝐴𝐶 ′ 𝑅 1 + 𝑛𝑅 𝐴𝐶 Δ𝑇 𝑐𝑜𝑛 -𝑊 𝑑 [0.5 𝑅 1 + 𝑛 (𝑅 2 + 𝑅 3 + 𝑅 4 )] 𝑇 𝑐𝑜𝑛 ′ (1 + 𝜆 1 ) 𝑅 2 + 𝑛𝑅 𝐴𝐶 𝑇 𝑐𝑜𝑛 ′ (1 + 𝜆 1 + 𝜆 2 ) (𝑅 3 + 𝑅 4 )	]	0.5	(3.7)
						4 =	1 𝜋 𝑒,𝑐 ℎ 𝑠𝑢𝑟𝑟 𝛥𝑇 𝑠	1/4	(3.5)
	represented in equation (3.6).
		𝛥𝑇 𝑐𝑜𝑛 = (𝐼 𝑐𝑜𝑛	2 𝑅 𝐴𝐶 𝑇 𝑐𝑜𝑛	′ + 0.5𝑊 𝑑 ) 𝑅 1 + (𝐼 𝑐𝑜𝑛	2 𝑅 𝐴𝐶 𝑇 𝑐𝑜𝑛	′ (1 + 𝜆 1 ) + 𝑊 𝑑 ) 𝑛 𝑅 2 +
		(𝐼 𝑐𝑜𝑛	2 𝑅 𝐴𝐶 𝑇 𝑐𝑜𝑛	′ (1 + 𝜆 1 + 𝜆 2 ) + 𝑊 𝑑 ) 𝑛(𝑅 3 + 𝑅 4 )	(3.6)
						94

Table 3 .

 3 2 that the DSEC conductor temperature of juvenile, mix and adult mussels are 91.5 °C, 90.47 °C and 89.18 °C respectively and the error (

	𝑇 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛 -𝑇 𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙
	𝑇 𝑎𝑛𝑎𝑙𝑦𝑡𝑖𝑐𝑎𝑙

Table 3 .

 3 2: Copper conductor temperature during the colonization of various ages of mussels.

	Mussel's Age	Juvenile Mussels	Mix mussels	Adult mussels
	Copper
	Conductor Temperature	91.5 °C	90.47 °C	89.18 °C
	(Analytically)
	Copper
	Conductor Temperature	91.3 °C	90.47 °C	89.18 °C
	(COMSOL)
	Error	0.22 %	0.14 %	0.47 %

Table 3 .

 3 3: Overall thermal resistance of various ages of mussels.

	Specimen around the electrical cable	Juvenile Mussels	Mix mussels	Adult mussels	Water
	Roverall
		0.022	0.017	0.011	0.012
	(K.m.W -1 )

*Roverall is the overall thermal resistance of the presence of mussels around the tube (including Rconductive(biof) and Rconvective).

Table 3 .

 3 4: The temperature difference between the mussel layer's inner and outer radiuses.

	Mussels Type	T (Measurement)	T (COMSOL)
		K	K
	Juvenile	10	3.3
	Mix (juvenile & adult)	4.2	2.4
	Adult	4.8	1.7

Table 4 .

 4 1: Thermophysical properties of different layers of flux metric device.

	Type of Materials	Thermal conductivity (W.m -1 .K -1 )	Density (kg/m 3 )	Heat Capacity (J.kg -1 .K -1 )
	Stainless steel	14.4	7817	460
	Polyimide (Kapton)	0.15	545	920
	PVC
	(Polyvinyl	0.2	1380	2000
	chloride)
	PE (Polyethylene)	0.34	910	2600
	Figure 4.7 shows the temperature evolution and relaxation of Tc using COMSOL
	and the conductive model. It shows that the temperature obtained from COMSOL and
	conductive model are very similar with a maximum temperature difference not exceeding
	0.05 %.

Table 4 .

 4 2: Equivalent heat transfer coefficient using the capacitive and conductive model

		heq,capacitive (W.m -2 .K -1 )	heq,conductive (W.m -2 .K -1 )	| (ℎ eq,capacitive -ℎ eq,conductive ) ℎ eq,conductive |
	Glass beads	225	332	32%
	This difference is due to the fact that capacitive model has more hypothesis
	comparing to conductive model. For the capacitive model, two hypotheses are considered:
	the flux metric device is isothermal and no heat flux loss is considered at the rear face of
	the sensor. However, in conductive model these two hypotheses disappear since the heat
	transfer in each layer of the flux metric device is considered as shown in Figure 4.8. Finally
	in our device we have found that that 90% of the total impulse power goes in the forward
	direction (𝛷 ℎ + = 19.8 W) and 10% goes in the backward direction (𝛷 ℎ -= 2.2 W). Thus, the
	conductive model (quadrupole model) is more appropriate for the estimation of the
	equivalent heat transfer coefficient heq.

Table 4 .

 4 3 shows the heat diffusion length during the heat pulse only ( ℎ𝑒𝑎𝑡 𝑝𝑢𝑙𝑠𝑒 ) and during the overall measurement ( 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 ). It's clearly shown that the heat diffusion length increases with the increasing of the heat pulse duration. To be sensitive the thickness of biofouling layer one should preferred long time duration. As the conductive model is a 1D model and can be applied only for short heat pulse duration, then, a 3D transient numerical model is proposed to estimate the equivalent heat transfer coefficient for long heat pulse duration.

Table 4 .

 4 1: Heat diffusion length for different heat pulse durations.  ℎ𝑒𝑎𝑡 𝑝𝑢𝑙𝑠𝑒 is the heat diffusion length during the heat pulse.

	Heat pulse duration (s)	 ℎ𝑒𝑎𝑡 𝑝𝑢𝑙𝑠𝑒 (mm)	 𝑜𝑣𝑒𝑟𝑎𝑙𝑙 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 (mm)
	25	2.3	4
	120	5	8.7
	180	6	10.7

*

Table 4 .

 4 2: Equivalent heat transfer coefficient of the glass beads for short and long heat pulse duration.

	Heat pulse duration	heq,beads using COMSOL	heq,beads using Conductive model	(| ℎ 𝑒𝑞,𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑒 𝑚𝑜𝑑𝑒𝑙 -ℎ 𝑒𝑞,𝑐𝑜𝑚𝑠𝑜𝑙 ℎ 𝑒𝑞,𝑐𝑜𝑚𝑠𝑜𝑙	|)
	s	W.m -2 .K -1	W.m -2 .K -1	%
	25	310	323	4.2
	120	160	321	100
	180	153	315	106

Table 4 .

 4 3: Layer thickness and porosity of various ages of mussels.

		Layer thickness	Porosity
	Mussel's sample
		(cm)	(%)
	Juvenile	4	15
	Mix (Juvenile-Adult)	6	18
	Adult	7	22.6

Acknowledgement

Arnaud Arrivé, who created the experimental tube in a very professional and precise manner. I'd also like to thank my colleagues and friends. Thanks to West Atlantic Marine Energy Community (WEAMEC) and Pays Région de la Loire, for funding BIODYTHERM project where this work was carried out within the framework of it. Also, Thanks to France Energies Marines and National Research Agency for funding OMDYN-2 project which this thesis a part of.

Partial Conclusion

In this Chapter, the thermal effect assessment on the DSEC is evaluated by estimating the temperature evolution within the copper conductor of the DSEC covered with mussels of various ages using a numerical simulation (via COMSOL). Those simulations were carried out for a given thickness of the layer of mussels and do not take into account the evolution of bio-colonization over time (multi-layer effects) as well as the water current which passes across or around the mussels. The results show that biocolonization of mussels cause an increase in the copper conductor temperature of the DSEC above its maximum operating temperature (90 °C) depending on the mussels' age and the corresponding global thermal resistance. Whereas, juvenile mussels have a bigger themal effect on the DSEC copper conductor temperature than mix and adult mussels which leads

F is the Laplace transform of f. N is determined by the computer's floating-point precision, with single precision N=10 and double precision N=20, where 𝐶 𝑗 is given by the following expression for an even value of N:

))

(4.28)

In equation (4.28), 'Int' designates the integer part of a real number and 'Min' the minimum of two numbers. So using the inverse method of Gaver-Stehfest, we got the real computed temperature (Tc) value at the location of the main thermocouple.

After using the numerical inversion of Laplace transform we have different results

of real computed temperature at point c, T 𝑐 (𝑡, ℎ 𝑙𝑜𝑐𝑎𝑙 ). Where t is the time in seconds and hlocal is the heat transfer coefficient of the mussel's interval. So to compute hlocal, we use a minimization method (simplex method) between the real computed temperature and the measured one at the location of the main sensor c. TPE : temperature through the polyethylene layer.

J(hlocal)=∑ (𝑇

TPVC: temperature through the polyvinyl chloride layer.

Th : temperature through the heating element layer.

Tss : temperature through the stainless steel.

Heat pulse duration effecttest with glass beads

The effect of heat pulse duration on the estimation of the equivalent heat transfer coefficient will be investigated. To do this, a glass beads medium formed from 250 pieces of big beads (bead diameter 25 mm) with a layer thickness above the flux metric device of 4 cm and water porosity of 42 % has been implemented above the flux metric device, as shown in Figure 4.9.