Florian Le Bourdais

Arnaud Recoquillay

Alexandre Impériale

Hoe-Woong Kim

Young-Sang Joo

Sang-Jin Park

Jaehyuk Eoh

Numerical parametric study of a waveguide-based ultrasonic sensor for telemetry in a sodium-cooled fast reactor

Keywords: Sodium fast reactor, liquid sodium, telemetry, acoustics, guided waves, piezoelectric transducer, numerical simulation, finite elements, semi-analytical modeling, parametric study

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

As the head of a sodium-cooled fast reactor (SFR) [START_REF] Aoto | A summary of sodium-cooled fast reactor development[END_REF], which uses liquid sodium as its coolant, shall not be opened during operation to prevent any reaction of the sodium with water or air, refueling is usually proposed based on a rotating head design. In the refueling process, one of the most important issues is the monitoring of existing obstacles that can hinder the rotation of the part of the reactor head. If any obstacle exists during this rotating movement, unexpected accidents can occur, such as the one in the Japanese SFR Joyo in 2007 [START_REF] Takamatsu | Development of Observation Techniques in Reactor Vessel of Experimental Fast Reactor Joyo[END_REF]. Therefore, one must ensure that no obstacle exists before starting the refueling process. However, conventional optical tools cannot be used to monitor such obstacles because liquid sodium is optically opaque. As an alternative, ultrasonic sensors have been employed to monitor obstacles as well as for the visual examination of in-vessel structures in liquid sodium.

Two types of sensors (immersion and waveguide-based sensors) have been developed and employed for under-sodium viewing and telemetry [START_REF] Lions | Apparatus for ultrasonic visualization in sodium (VISUS) and acoustic detection in the Phenix reactor[END_REF][START_REF] Spanner | Preliminary development of inservice inspection methods for LMFBR's[END_REF][START_REF] Watkins | A proposed method for generating and receiving narrow beams of ultrasound in the fast reactor liquid sodium environment[END_REF][START_REF] Barrett | Ultrasonic viewing in fast reactors[END_REF][START_REF] Karasawa | Development of under-sodium three-dimensional visual inspection technique using matrix-arrayed ultrasonic transducer[END_REF][START_REF] Baqué | Review of in-service inspection and repair technique developments for French liquid metal fast reactors[END_REF][START_REF] Griffin | Under-Sodium viewing: a review of ultrasonic imaging technology for liquid metal fast reactors[END_REF][START_REF] Sheen | Linear-array ultrasonic waveguide transducer for under sodium viewing[END_REF][START_REF] Joo | Development of ultrasonic waveguide sensor for under-sodium inspection in a sodium-cooled fast reactor[END_REF][START_REF] Wang | Development of ultrasonic waveguide techniques for under-sodium viewing[END_REF][START_REF] Sylvia | Ultrasonic imaging of projected components of PFBR[END_REF][START_REF] Kim | Ultrasonic imaging in hot liquid sodium using a plate-type ultrasonic waveguide sensor[END_REF]. Immersion sensors are submerged in liquid sodium and placed close to the targets. This short propagation distance gives a chance to use high frequency ultrasounds providing high resolution images. However, further efforts are still needed to improve their life span because active parts of the sensor are directly exposed to hot sodium and intense radiation. On the other hand, waveguide-based sensors have a better life span by installing the ultrasonic transducer far from the sodium coolant. Although they give relatively low resolution results caused by the use of low frequency ultrasound due to the long propagation distance, the installation and maintenance of the ultrasonic transducers are much easier than those of immersion sensors.

In relation to the waveguide-based sensor technology, a plate-type waveguide-based sensor, which employs a 10 m long thin strip plate as a waveguide and radiates the ultrasonic wave horizontally, has been developed for under-sodium telemetry [START_REF] Kim | Development of a Ranging Inspection Technique in a Sodium-cooled Fast Reactor Using a Plate-type Ultrasonic Waveguide Sensor[END_REF]. In this sensor, the ultrasonic wave generated by the transducer installed at the one end of the waveguide propagates along the waveguide and is radiated into the surrounding liquid at the other end of the waveguide. Unlike immersion and rod-type waveguide-based sensors that radiate ultrasonic waves from the terminating section, ultrasonic waves are radiated from the guiding border in a radiating part of the waveguide in the developed sensor, resulting in a certain radiation angle. This means that the radiation characteristics of the developed sensor is quite different from those of immersion and rodtype waveguide-based sensors. To enhance the radiation performance, therefore, it is needed to investigate the wave propagation and radiation characteristics in and from the radiating section precisely.

Some efforts have been made to investigate the wave propagation and radiation characteristics in and from the radiating face of a plate-type waveguide-based sensor [START_REF] Deighton | Mode conversion of Rayleigh and Lamb waves to compression waves at a metal-liquid interface[END_REF][START_REF] Watkins | A proposed method for generating and receiving narrow beams of ultrasound in the fast reactor liquid sodium environment[END_REF][START_REF] Joo | Development of ultrasonic waveguide sensor for under-sodium inspection in a sodium-cooled fast reactor[END_REF]. However, these efforts are not sufficient to fully understand the radiation characteristics of the developed sensor because the ultrasonic wave is radiated not only from the radiating face but also from an acoustical shielding welded to the radiating face. Since the acoustical shielding is inevitable to prevent energy leakage during the wave propagation in this type of sensor, its effect on the radiated field should be considered to clearly investigate the radiation characteristics.

In this work, an analysis model considering all parts of the radiating section was developed through CEA-KAERI bi-lateral collaboration within a Gen-IV framework (Generation IV International Forum). As the waveguide-based sensor is large with respect to the considered wavelength, classic computation methods, such as Finite Elements, were too expensive in computation time to allow a parametric study. To circumvent this difficulty, the following strategy was used: first, the computation of the propagation of the ultrasonic waves in the waveguide-based sensor and in the surrounding liquid are separated. The wave propagation in the sensor is then solved using frequency domain finite elements coupled with an analytical modal decomposition of the field. Finally, the computed values of the field at the boundary of the sensor are used as a source term for the propagation in the surrounding liquid, before going back to the time domain. The radiation characteristics of the developed plate-type waveguide-based sensor were precisely investigated through numerical parametric studies and the best geometrical parameters in the radiating section were selected. In addition, the simulated results were compared with corresponding experimental results to validate the accuracy of the developed analysis model.

Presentation of the sensor

Presentation of the waveguide setup and measurement protocol

Figure 1 shows a waveguide-based ultrasonic sensor developed for telemetry tasks in a liquid sodium nuclear environment. It consists of an ultrasonic transducer, a solid wedge, a waveguide and an acoustical shielding tube. A 10 m long stainless steel (SS304) strip whose width and thickness are 19 mm and 1.5 mm respectively is used as a waveguide in this sensor. The long waveguide design allows a commercial ultrasonic transducer to be used since it is not directly exposed to the hot liquid sodium and intense radiation. Between the ultrasonic transducer and the waveguide, a solid wedge made of Lucite is installed to generate the guided wave in the waveguide. A stainless steel (SS304) tube of outer diameter 25 mm and thickness 1 mm is used as an acoustical shielding tube, preventing energy leakage into the surrounding liquid during the wave propagation along the waveguide and providing a single radiating face as well. Once the ultrasonic wave is generated by the ultrasonic transducer in the generating section, it propagates into the waveguide through the solid wedge and is then converted into a guided wave. In the developed sensor, the A0 mode Lamb wave is selected as the target guided wave because it has a significant out-of-plane particle displacement resulting in high radiation efficiency into the surrounding liquid. To generate the A0 mode Lamb wave in the waveguide, the angle ( ) of the wedge should be carefully determined and it can be calculated by using Snell's law as follows:

= where is the longitudinal wave velocity in the wedge and is the phase velocity of the A0 mode Lamb wave in the waveguide of thickness at a frequency . This A0 mode Lamb wave propagates along the waveguide and is then radiated into a surrounding liquid by mode conversion at the radiating face. The radiation angle , which is measured with respect to the radiating face normal, can also be calculated by using Snell's law as follows: = sin were is the longitudinal wave velocity in the surrounding liquid. To radiate the ultrasonic wave horizontally, therefore, the angle of the radiating face should be determined to have the same value as (see Figure 1).

Choice of fixed parameters, variable parameters

The ultrasonic wave radiation from the radiating face in the developed waveguide-based sensor is quite different from that in the immersion or rod-type waveguide-based sensor. The ultrasonic wave is leaked into the surrounding liquid with a certain angle while it propagates along the radiating face in the developed sensor (see Figure 2 left), whereas it is radiated at once into the normal direction to the radiating face in the immersion or rod-type waveguide-based sensor. Furthermore, in this radiation process, the guided wave propagates not only in the radiating face but also in the acoustical shielding tube welded to the radiating face so that the acoustical shielding tube also acts as an acoustic source for the radiated field in the surrounding liquid. Therefore, in the developed waveguide-based ultrasonic sensor, the radiation characteristic is considerably affected by geometrical parameters of this radiating section, such as the length ( ) and angle ( ) of the radiating face, the angle ( ) of the acoustical shielding tube (see Figure 2 right). Among them, the angle of the radiating face is fixed to radiate the ultrasonic wave horizontally into the surrounding liquid (of known acoustic properties).

Presentation of the numerical model

Overview of the one-way coupling scheme and its hypotheses

A fully general approach to the modeling of the radiating waveguide problem would be to use a generic numerical method to simulate all of the elastic and acoustic phenomena arising in the waveguide-based ultrasonic sensor. While possible, typically using commercial software like COMSOL (COMSOL AB, Stockholm, Sweden., 2019) or Abaqus [START_REF] Smith | ABAQUS/Standard User's Manual, Version 6.9[END_REF], we expect this approach to lead to prohibitive computation times, especially in the perspective of a parametric study of several tens or hundreds of configurations. In this context, a custom numerical procedure becomes attractive, and this section will present the one-way coupling scheme used for the computation of the radiated acoustic field with enhanced performance.

The one-way coupling relies on a first assumption that the coupled fluid-structure problem of the waveguide-based ultrasonic sensor can be described by a two-dimensional analysis. Several authors have already formulated two-dimensional analyses of the waveguide radiation, most notably [START_REF] Deighton | Mode conversion of Rayleigh and Lamb waves to compression waves at a metal-liquid interface[END_REF][START_REF] Watkins | A proposed method for generating and receiving narrow beams of ultrasound in the fast reactor liquid sodium environment[END_REF][START_REF] Joo | Development of ultrasonic waveguide sensor for under-sodium inspection in a sodium-cooled fast reactor[END_REF]. Since the ratio of the densities of the fluid and the solid is small, our second hypothesis consists in describing the interaction between the fluid and the structure as a one-way coupling. Here, the one-way coupling is such that the elastodynamic displacement that propagates within the waveguide radiates energy to the fluid without being affected by the feedback of the fluid. As a third assumption, we consider that the fluid in which the radiated pressure propagates is homogeneous, i.e. its physical parameters are constants.

With all these hypotheses stated, we can decompose our modeling approach into two parts:

-Elastodynamic behavior within the waveguide-based sensor: we use the modal approach previously developed in [START_REF] Baronian | Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide[END_REF] based on the SAFE method [START_REF] Rose | Ultrasonic guided waves in solid media[END_REF]. The complex geometry of the radiating part is taken into account by a hybrid strategy that couples a finite element method in the region where the geometry is non-canonical with the modal approach describing elastodynamic propagation in the waveguide and the acoustical shielding, both considered as semi-infinite waveguides. This method is used to solve the elastodynamic problem in the frequency domain.

-Acoustic propagation in the fluid: under the assumption of a one-way coupling the pressure field generated by the displacement of the solid interface can be modelled as a boundary source term for the homogeneous fluid. An integral representation is appropriate here, and we use a Green function to propagate the solid displacement field in the fluid.

Finally, an inverse Fourier transform is used to relate the frequency domain quantities obtained to the transient signals that are measured experimentally. The approach exposed here has several benefits considering that we want to carry out a parametric study:

-The hybrid modal / finite element approach allows us to model the dispersive propagation in the solid over long distances well while taking into account the complex geometry of the radiating part, at a better computational cost than full finite element models.

-The integral representation in the fluid leads to reduced computational costs, again compared with standard finite element models, since it depends exclusively on the number of observation points and not on the propagation distance, which is a major asset in our case.

The next subsections below describe in further details the different components of the model that we have outlined in this paragraph.

Guided wave model

The first step of our modeling approach is the computation of the elastodynamic field in the waveguide, for the purpose of extracting the displacement field over the radiating section. To do so, we use finite elements coupled with an analytic formulation of the guided wave propagation. For more details, see [START_REF] Baronian | Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide[END_REF]. Consider a perfect waveguide, that is a domain W = Σ × ℝ of dimension , ≥ 2, with Σ a finite domain of dimension -1. Then, in the frequency domain, any elastic field $ % propagating in this perfect waveguide can be decomposed on the guided modes propagating in both directions along the axis.

Consider now that the waveguide & presents some perturbation ', the support of which is between the sections Σ and Σ ( of respective abscissa ) and ) ( , and let

Ω = {, ∈ &, ) / < ) } and Ω ( = {, ∈ &, ) / > ) ( }, Ω = ' ∪ Ω ∪ Ω ( .
The modal decomposition of the field is not valid in the portion of the waveguide containing the perturbation, that is ', and, as Ω is infinite according to the axis direction, some truncation of the domain is needed to use standard simulation tools such as finite elements. Nonetheless the scattered field can be decomposed on the modal basis away from ', that is in Ω ( and Ω . In particular, the elastic field can be decomposed in two parts, that is the incident field $ % 456 propagating in the waveguide without defect and the field scattered by the defect $ % %6 . The incident field $ % 456 is known analytically as it is independent of the defect, and the scattered field $ % %6 is coming from ', meaning its modal expansion only contains left-going modes on the left of the defect (in Ω ) and right-going ones on the right of the defect (in Ω ( ). From this knowledge on the propagation of the elastic field, a radiation condition on the modal expansion can then be expressed through a Dirichlet-to-Neumann operator on Σ and Σ ( , enabling the computation of the scattered field surrounding the perturbation area '. The elastic field can hence be computed in all of Ω using its analytical modal decomposition in the other parts of the waveguide, namely Ω and Ω ( .

In the case of the considered waveguide-based sensor, the geometry is more complex but the same strategy can be applied: first, it should be noted that the waveguide as well as the two parts of the acoustical shielding can be considered as semi-infinite waveguides, the end of which is the radiating part, linking the three semi-infinite waveguides together (see Figure 3). Hence, the field inside the sensor can be computed using finite elements coupled with the radiation conditions based on modes only in the radiating part and using the modal decomposition of the field elsewhere, drastically reducing the computational cost and accurately taking into account the geometry at the end of the sensor.

As the modal approach is formulated in the frequency domain, the solution of the elastodynamic problem is obtained for a discretization of the frequency support of the excitation signal, this discretization depending on the final simulation time.

Radiation into surrounding fluid

The coupling between solid and fluid is usually described (for instance in [START_REF] Zhang | Studies of the strong coupling and weak coupling methods in FSI analysis[END_REF]) by two conditions. The first one is the equality of the normal fluid and solid velocities at the interface, > ? ⋅ 5 = > % ⋅ 5.

The second condition describes the normal stress equilibrium at the interface

-B 5 = C 5,
where B is the fluid pressure and C is the solid stress tensor. Neglecting fluid feedback in the solid field equations, we assume free surface boundary conditions, i.e. C 5 = 0. Therefore, using the previously stated velocity relation, an integral representation of the pressure field (in the frequency domain) at an observation point E is expressed as

B E = -F G H 8 I $ % ⋅ 5 J E -, K Γ,
where F G is the fluid density and J is the acoustical free-space Green function. Numerically, we apply the above equation to a mesh of the radiating section Γ (see Figure 3) to compute the pressure at all the points of the experimental observation zone and for each frequency of the discretized excitation spectrum.

Numerical parametric study Outline of the study

A parametric study was conducted in order to select the best geometrical parameters out of the study range for the waveguide-based sensor. More specifically, preliminary numerical simulations have shown that, among the non-fixed parameters, the most influent two on the radiated field in the fluid are the length of the radiating face and the angle of the acoustical shielding (see Figure 2). For the sensor to be small enough to conduct the inspection and to be easily manufactured, those parameters must respectively be between 20 and 28 mm and between 15 and 35°. Regarding the radiated fields, the following characteristics are studied in order to select the best configuration in this range: -the beam width at -6 dB at an horizontal distance of 600 mm from the radiating section to ensure a good focusing of the field, -the amplitude of the field at an horizontal distance of 600 mm from the radiating section to ensure a sufficiently high amplitude to detect a reflector, -the shape of the radiated field to avoid any interferences of the field radiated by the acoustical shielding with the beam of the radiating face in the range of inspection. As for the parametric study plan, it was decided to compute the following configurations:

-values of ranging from 20 to 28 mm with a step of 1 mm for equal to 15 and 35°, -values of ranging from 15 to 35° with a step of 2.5° for equal to 20 and 28 mm, -nine joint variations of and , ranging from 20 to 28 mm with a step of 1 mm and ranging from 15 to 35° with a step of 2.5°.

Note that, as the width of the acoustical shielding is fixed, the length of the junction between the radiating face and the acoustical shielding depends on the angle of the acoustical shielding .

In all the simulations, the excitation signal applied to the piezoelectric transducer is deduced from the one used in the experimental tests: the experimental excitation signal has been recorded and then filtered around its central frequency to have a small bandwidth, reducing the computation time, while being close to the experimental setup (see Figure 4). As for the fixed parameters, we consider a stainless steel sensor ( = 5797 S. , T = 3079 S.

, F = 8 V. S 9 ) immersed in water ( = 1480 S. , F = 1 V. S 9 ), implying = 36°.

Results of the simulation study

As a first step the field shape for three values of the parameters, namely for = 20 mm, = 15°, = 20 mm, = 35° and = 28 mm, = 35° are plotted in Figure 5. The point of coordinates 0, 0 is taken as the point 12 mm up and 1 mm to the right of the lower right corner of the sensor.

The field can be decomposed into three areas: first, a horizontal beam is generated by the radiating section. This beam possesses a near field area, roughly before 50 mm for the top and middle cases and around 100 mm for the bottom one, and a far field pattern with a focal point. At the upper left of the beam, we see the field component radiated by the acoustical shielding of the sensor. Based on previous investigations, this contribution has an adverse effect on the sensor operation. For example, unwanted reflected signals could be measured if any structural components exist near the sensor although they do not represent obstacles to be detected.

Even though only three cases are plotted in Figure 5, the main variations of the field due to changes in the length of the radiating face and the angle of the acoustical shielding can already be seen: First, as the angle of the acoustical shielding increases, the field radiated by it has a lower amplitude and a lower angle. As the length of the radiating face increases, the focal point of the beam is further from the sensor, inducing higher amplitudes and a smaller beam width far from the sensor.

The same behavior can be seen in more details analyzing the following figures: in Figure 6 are plotted the angle of the field radiated by the acoustical shielding as well as its amplitude normalized with respect to the maximum amplitude for the considered configuration. That is, the maximum amplitude of the beam radiated by the acoustical shielding has been extracted at various locations along ), enabling the determination of a radiation angle. The same behavior can be seen for both quantities: as the angle of the acoustical shielding increases, these quantities decrease. A small influence of the length of the radiating face can be seen on the amplitude, which comes from the normalization: indeed, as will be seen later, the maximum amplitude increases with respect to , inducing a stronger normalization. No influence of on the angle can be seen.

Then, the main beam is studied. In Figure 7 we plot the maximum abscissa of the half amplitude of the beam. This quantity is strongly linked to the focal point: as this point is further along the axis, so is the half amplitude point but, due to the near field behavior, the exact position of the focal point is hard to extract, hence the use of this quantity. A strong dependency on the length of the radiating face can be seen in Figure 7: as increases, the maximum abscissa of the half amplitude also increases, meaning that the focal point is further along the axis and that, for a given abscissa, the beam is narrower. Small variation can also be seen with respect to the angle of the acoustical shielding , which come from the interactions of both fields. The same tendencies are seen in Figure 8, where the beam width and the normalized amplitude at 600 mm are plotted. More precisely, the amplitude is normalized with respect to its maximum value with respect to the parameters. This normalization enables to point out the differences between the configurations. The beam width decreases with while the amplitude globally increases. From this study, it can then be seen that the configuration giving the most suited radiated field for the application is = 28 mm and = 35° as it allows a better focusing of the beam and less leakage from the acoustical shielding.

Comparison between experimental and simulated data Experimental setup

For the comparison between experimental and simulated data, the radiated acoustic fields of nine plate-type waveguide-based sensors having different and listed in Table 1 were measured in water using the experimental setup shown in Figure 9. In each measurement, a four-cycle tone burst centered at 1 MHz as generated by a function generator was used as the input signal and it was amplified by a power amplifier before being sent to an ultrasonic transducer installed at the generating section of the sensor. For the horizontal radiation in water, the angle of the radiating face was selected to be = 36° for all nine sensors based on the longitudinal wave velocity ( ≈ 1480 S.

) in water and the phase velocity ( ≈ 2500 S.

) in the waveguide at 1 MHz. The radiated ultrasonic wave from the radiating section of the sensor was measured by a needle hydrophone installed in an XYZ scanner. A 600 mm by 100 mm rectangular area in the XY plane was selected as the scan area, and the radiated acoustic fields were measured with 1 mm intervals. The measured signals were then amplified and filtered by a preamplifier, a noise suppressor, and a broadband receiver, sequentially. Finally, filtered and amplified signals were processed by a C-scan software to obtain the imaging results.

Comparison

Figure 10 shows the experimental field shapes corresponding to the ones plotted in Figure 5 for the simulations. It can be seen that the general behavior is the same in both cases: first, the angle of the field radiated by the acoustical shielding decreases with respect to the angle of the acoustical shielding. Then, small differences exist in the amplitude of the radiated fields: a small decrease, not as strong as the one obtained in the simulation, exists in the amplitude of the field radiated by the acoustical shielding with respect to its angle, and the decrease with the distance of the amplitude of the beam radiated by the radiating section is slower as the length of the radiating face increases, corresponding to a focal point further from the sensor. These properties can also be seen in Figure 11 and Figure 12: first, Figure 11 shows the maximum amplitude of the beam with respect to the horizontal axis for all the experimental configurations and the corresponding simulations. Note that both simulated and experimental data are normalized with respect to the maximum amplitude of the last configuration, that is = 28 mm and = 35°, at the first considered abscissa, that is 175 mm. This normalization allows to compare the discrepancy rate of the amplitude as well as the value of the field at 600 mm, which is of great interest for the application. The main difference between the simulation and the experimental results comes from the attenuation rate, which is underestimated by the simulations. This most certainly comes from the 2D approximation, neglecting the propagation of the waves along the third axis, which leads to a better focalization of the energy in the plane of observation. The discrepancy rate is globally decreasing with respect to the length of the radiating face in both cases, which is expected for the simulations and shows a rather good agreement with the experiments. The case of = 28 mm and = 35° should be pointed out as its experimental amplitudes are lower than all others. This may come from the difficulty of having a reproducible experiment: even though the sensor's coupling has been made as to ensure the same emitted energy for all configurations, this property is rather hard to ensure. Nevertheless, all other configurations show the expected tendencies. Next, Figure 12 shows the angles of the beam radiated by the acoustical shielding and the width of the beam radiated by the radiating section as functions of the length of the radiating face . The same tendencies are observed in both numerical and experimental data: the angle of the beam radiated by the acoustical shielding does not depend on the length of the radiating face but rather on the angle of the acoustical shielding, with a rather good agreement on its value between simulation and experiment. The width at 600 mm of the beam radiated by the radiating section is decreasing with respect to the length of the radiating face and has small variations with respect to the angle of the acoustical shielding. It should be noted that the simulations globally overestimate the width of the beam, which may come from the 2D model approximation.

Conclusions

This paper has described investigations carried out in the context of non-destructive inspection development in sodium-cooled reactors. A particular numerical method was developed, used for a parametric study and waveguide-based ultrasonic sensors were fabricated to validate the modelling effort and assess its usefulness for practical purposes.

The design principles of waveguide-based ultrasonic sensors were recalled in section 2.

The numerical simulation of elastic and acoustic waves of interest for such a waveguide-based sensor and its challenges were described in section 3. A simplified model was developed to handle the proposed geometry and constraints, relying on several approximations, in order to make computation times smaller while maintaining predictive capability.

In section 4, this model was used to carry out a parametric study. Out of a number of pre-defined parameter sets, the model was applied and used to compare the performance of various waveguidebased sensor designs within a range of possible designs.

Finally, section 5 compared measurements obtained with 9 waveguide sensors manufactured for this research project to the simulated measurements. Measurements were carried out with 9 sensors, in order to cover a large design space and ensure the validity of our modelling approach. Good agreement was observed with respect to the main measured quantities. The limits of the developed model were also discussed with the help of these results. In particular, variations linked to untracked experimental parameters were shown to have an impact on results.

The obtained results show the value of model-aided design for non-destructive inspection development, especially since the targeted application environment is difficult to access and does not suit itself easily to iterative experimental work.

As a finishing remark, the simulation technique described in this article could also apply to other liquid metal environments such as Lead-Bismuth Eutectic (e.g. as used by the MYRRHA project [START_REF] Abderrahim | Myrrha accelerator driven system programme: recent progress and perspectives[END_REF]).

Figure 1 :

 1 Figure 1 : Waveguide-based ultrasonic sensor for telemetry in a liquid sodium nuclear environment

Figure 2 :

 2 Figure 2 : Wave radiation phenomenon (left) and geometry of radiating section (right) of the developed waveguide-based ultrasonic sensor

Figure 3 :

 3 Figure 3: Scheme of the computation domain for the guided wave model. Red area: finite elements area. Blue areas: semiinfinite waveguides. The middle one 7 ( 8 is the main waveguide, whereas the top and bottom ones, 7 ( and 7 ( 9 , are the acoustical shielding. Sections : ( ; , = 1 … 3, are the boundaries between the finite element domain and the modal representation.

Figure 4 :

 4 Figure 4: Spectrum of the measured excitation signal (blue) and of the one used in the simulations (red)

Figure 5 :Figure 6 :

 56 Figure 5 : Field shape for = 20 mm, = 15° (top), = 20 mm, = 35° (middle) and = 28 mm, = 35° (bottom), normalized amplitudes

Figure 9 :

 9 Figure 9: Experimental setup for radiated acoustic field measurements

Figure 10 :

 10 Figure 10: Experimental field shape for =20 mm, φ=15° (top), =20 mm, φ=35° (middle) and =28 mm, φ=35° (bottom), normalized amplitudes

Figure 11 :

 11 Figure 11 : Simulated (left) and experimental (right) amplitude of the beam with respect to the ) location along the sensor radiation axis.

Figure 12 :

 12 Figure 12: Angle of the beam radiated by the acoustical shielding (left) and width at 600 mm of the beam radiated by the radiating section (right) as a function of the length of the radiating face.

Table 1 :

 1 and values of the nine plate-type waveguide-based sensors considered in the experiments.

	Sensor No.	1	2	3	4	5	6	7	8	9
	(mm)	20	20	20	24	24	24	28	28	28
	(º)	15	25	35	15	25	35	15	25	35

Acknowledgements

The work presented herein is the result of a multi-year effort between CEA and KAERI that was initiated through the International Generation IV Forum. CEA would like to acknowledge the support of the nuclear energy department and in particular the helpful involvement of François Baqué as well as partial funding through the ASTRID SFR project.