Souha Kassab

Frédéric Michel

Laurent Maxit
email: laurent.maxit@insa-lyon.fr

Water experiment for assessing the vibroacoustic beamforming gain in the framework of the acoustic leak detection for the sodium-heated steam generator

Keywords: sodium-heated steam generator, leak detection, beamforming, array gain, structural acoustics, heavy fluid

In an intent to improve the monitoring of steam generators, a technique based on vibration measurements is developed for the detection of a water leak into sodium. Background noise can mask the leak-induced vibrations. In order to increase the signal-to-noise ratio (SNR), a beamforming technique may be considered. In the purpose of studying the feasibility and the efficiency of this technique for the present configuration, experimental investigations have been performed on a mock-up composed by a straight cylindrical pipe coupled to a hydraulic circuit through two flanges. A sound emitter introduced in the pipe simulates the source to detect, whereas a flow speed controls the background noise vibrations. Beamforming is applied on the signals measured by an array of accelerometers externally mounted on the pipe. Two different kinds of beamforming are considered: the conventional (Bartlett) one and an advanced one based on SNR maximization (MaxSNR). After an analysis of the vibroacoustic behaviour of the mock-up, one studies the efficiency of the two beamforming treatments for narrow bands and broad bands.

Introduction

This paper describes the study of a non-intrusive vibroacoustic beamforming technique aimed at the detection of sodium-water reactions in the steam generator unit (SGU) of a liquid sodium fast reactor (SFR). Early detection is important since sodium/water reaction may conduct to severe damage if appropriate action is not taken quickly. In the past, different works considering active (Cavaro et al. 2010) and passive [START_REF] Chikazawa | Acoustic leak detection system for sodium-cooled reactorsteam generators using delay-and-sum beamformer[END_REF] detection techniques have been carried out. The passive techniques are based on the fact that the high difference of pressure and the strong sodium/water chemical reaction caused by the leak make acoustic noise. As the propagation time of the acoustic wave from the leak to sensors on the external surface of the SGU is sufficiently small, acoustic techniques can be used to detect the leak in a short time.

These methods are however strongly sensitive to the background noise caused by the sodium and water flows, the water boiling and the vibrations induced by the pumps. [START_REF] Kim | Acoustic Leak Detection Technology for Water/Steam Small Leaks and Microleaks Into Sodium to Protect an SFR Steam Generator[END_REF] studied the noise spectrum induced by the sodium-water reaction for various leak rates lower than 1 g/s. They concluded a small leak at a rate of 0.4 to 0.6 g/s generates a wide band of 1 to 200 kHz noise, which can be detected using a single sensor and a threshold criterion. For microleaks, the signal-to-noise ratio can however be lower than 0 dB and a more complex signal processing technology may be required. [START_REF] Hayashi | Acoustic detection of in-sodium water leaks using twice squaring method[END_REF] applied a nonlinear signal processing method called the twice-squaring method originally developed for acoustic detection of sodium boiling. Mixing experimental leak noises with background noises from real steam generators, they showed that this twice-squaring method can detect a leak noise with signal-to-noise ratio down to -20 dB. Sing and Rao (2011) looked at detecting the water injection into liquid sodium by measuring the external acoustic field with microphones located far from the system. Such a system is very simple but may be easily disturbed by external acoustic sources. [START_REF] Chikazawa | Acoustic leak detection system for sodium-cooled reactorsteam generators using delay-and-sum beamformer[END_REF] proposed an acoustic leak detection system using a delayand-sum beamformer. A sodium-reaction noise source is supposed to be localized while the background noise is uniformly distributed. The delay-and-sum beamformer provide information on the direction of the acoustic source. The numerical investigations showed that it could distinguish the source of the sodium-water reaction from the SGU background noise even if they had similar magnitudes. Five years later, a water experiment with mock-up tube bundles was achieved [START_REF] Chikazawa | Water experiment on phased array acoustic leak detection system for sodium-heated steam generator[END_REF]. A linear array of five hydrophones was immerged in the water tank. The previous numerical results were confirmed. It was also shown that the tube bundle does not modify the array performance for the considered frequencies (around 10 kHz). Another type of beamforming that we will consider in this paper is based on the measurement of the vibrations of the external surface of the SGU instead on the measurement of the pressure inside the fluid. This vibroacoustic beamforming technique has been developed within the PhD thesis of [START_REF] Moriot | Passive vibroacoustic detection of a water-sodium reaction using beamforming on a steam generator of a liquid sodium fast reactor[END_REF]. Beamforming over an array of sensors is of main interest due to its ability to increase the signal-to-noise ratio of the acoustic signal induced by the water-sodium reaction, generally masked by the SGU background noise as shown by [START_REF] Kim | Acoustic Leak Detection Technology for Water/Steam Small Leaks and Microleaks Into Sodium to Protect an SFR Steam Generator[END_REF]. [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF] had considered the conventional beamforming (called sometimes the Bartlett beamforming) based on a knowledge of the source to be detect and by supposing that the background noise is spatially uncorrelated. In the numerical applications, the source was supposed to be an acoustic monopole (i.e. pulsating sphere). The steering vectors of the beamforming were then defined from the frequency transfer functions between a supposed position of the monopole source inside the detection space and the sensors mounting of the external surface of the shell. These steering vectors took into account of the strong interaction between the heavy fluid (i.e. sodium) and the cylindrical shell (i.e. external surface of the SGU). Numerical results [START_REF] Moriot | Detection and localization of a leak in a sodium fact reactor steam generator by vibration measurements[END_REF][START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF] showed that the vibroacoustic beamforming using a linear array of accelerometers fixed on the external surface of the cylindrical shell can be used to localize the acoustic monopole. However, the main interest of the beamforming for the detection remains in the increasing of the SNR. This one is generally quantified by the array gain which is defined as the ratio of the SNR at the beamforming output relative to the SNR on the reference sensor (i.e. the sensor exhibiting the highest SNR). From the detection on a threshold criterion at the output of beamforming (instead of the reference sensor), this gain allows the improvement of the detection rate while limiting the array sensibility to false alarms. The works of [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF] showed promising results in term of array gain, both numerically and experimentally. However, the numerical results were obtained by using simple academic models (infinite plate in [START_REF] Moriot | Detection and localization of a leak in a sodium fact reactor steam generator by vibration measurements[END_REF] or infinite cylindrical shell in [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF]) supposing uncorrelated background noise. These assumptions which had permitted the first investigations are not fully satisfactory for the practical application. On another hand, the experimental data were limited to a handful of frequencies for the considered harmonic acoustic source [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF]). With this type of excitation, wideband analysis could not be carried out whereas the water-sodium reaction is a wideband source [START_REF] Kim | Acoustic Leak Detection Technology for Water/Steam Small Leaks and Microleaks Into Sodium to Protect an SFR Steam Generator[END_REF]). Moreover, it was observed that the vibratory background noise could be significantly correlated for the highest considered flow speeds. It resulted that the beamforming performances could be deteriorated (i.e. decrease of the array gain). In order to strengthen the first results related to the vibroacoustic beamforming [START_REF] Moriot | Detection and localization of a leak in a sodium fact reactor steam generator by vibration measurements[END_REF][START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF], new experimental investigations are carried out in the present paper. The configuration considered for the experiment is illustrated on figure 1. It consists in the mock-up developed during the Moriot's PhD study. This one is composed of a cylindrical pipe filled with a heavy fluid (i.e. water, and not sodium for practical reason) which is connected to the hydraulic circuit by two flanges. The source to be detected consists in a hydrophone used in transmission mode placed inside the pipe whereas the background noise may be controlled by changing the flow rate. The vibroacoustic beamforming consists to process the signals of the accelerometers fixed on the pipe in order to enhance the source signal and to reject the background noise. This experiment will focus on three aspects which are not yet studied and constitutes the novelty of the present paper:

-First, one will analyse the beamforming performance for a large frequency range (i.e.

[1 kHz-5 kHz]) considering the conventional treatment as well the MaxSNR treatment [START_REF] Tanaka | Acoustic Beamforming with Maximum SNR Criterion and Efficient Generalized Eigenvector Tracking[END_REF][START_REF] Van Veen | Beamforming: a versatile approach to spatial filtering[END_REF]. This latter has been proposed in the past to maximize the SNR at the beamforming output from the knowledge of, both, the acoustic source and the background noise. It may present some benefits compared to the classical beamforming, in particular in the case of partially-space correlated background noise.

In the first part of the study, the beamforming analysis will be achieved in narrow bands (i.e. 4 Hz of width) from 1 kHz to 5 kHz; -Second, one will study the relations between the vibroacoustic response of the considered system and the beamforming output. The goal is in particular to identify if the resonances of the considered system influence the beamforming performances; -Finally, wideband beamforming analysis will be carried out in order to conform with the fact that for practical applications, a wideband source should be detected [START_REF] Kim | Acoustic Leak Detection Technology for Water/Steam Small Leaks and Microleaks Into Sodium to Protect an SFR Steam Generator[END_REF]).

The paper is organized as follow:

-Section 2 reminds the principle of the classical and MaxSNR beamforming as well as the definition of the array gain; -Section 3 presents the experimental set-up; -The vibroacoustic characteristics of the mock-up are studied in section 4; -The beamforming performances are analysed in section 5.

Classical and MaxSNR beamforming treatments

In this section, one remains the principles of classical and MaxSNR beamforming treatments [START_REF] Tanaka | Acoustic Beamforming with Maximum SNR Criterion and Efficient Generalized Eigenvector Tracking[END_REF]) and one defines the different quantities considered in the following. These principles and the definitions are given for the case presented in the introduction and in figure 1.

One notes Γ(𝜔) , the cross-spectral matrix at the angular frequency 𝜔 of the signals received by the accelerometers composing the array fixed on the pipe. If the array is composed of n accelerometers, Γ is a square matrix of dimensions 𝑛 × 𝑛 and the component of the raw #i and the column #j corresponds to the cross spectrum density (CSD) function of the signals of sensors #i and #j at frequency , S 𝑖𝑗 (𝜔):

Γ(𝜔) = [S 𝑖𝑗 (𝜔)] 𝑛×𝑛 (1)
Assuming that the signals induced by the source to be detected and the signals of the background noise are independent, one can decompose the matrix Γ such as:

Γ = Γ 𝑠 + Γ 𝑛 , (2)
where Γ 𝑠 is the cross-spectral matrix of the signals induced by the source alone and Γ 𝑛 is the cross-spectral matrix of the signals induced by the background noise alone.

The beamforming consists in a 'spatial' filter of the signals received by the array of sensors.

Before presenting this filtering, one can define a pre-filtering state. This one corresponds to the analysis of each sensor's signal independently, one from each other. The SNR of the sensor #i is then given by:

SNR 𝑖 (𝜔) = 10log 10 ( 𝑆 𝑖𝑖 𝑠 (𝜔) 𝑆 𝑖𝑖 𝑛 (𝜔) ), (3)
where 𝑆 𝑖𝑖 𝑠 is the auto-spectral density (ASD) function of the signal measured by sensor #i in presence of the source alone and 𝑆 𝑖𝑖 𝑛 is the ASD function of the signal measured by sensor #i in presence the background noise alone.

Without beamforming, the acoustic leak detection can base on applying a threshold criterion to signals received by each sensor, independently. As the vibrations induced by the source as well as for the background noise are not necessary uniformly distributed on the pipe, the SNR may vary from one sensor to another. For a given position of the source, the sensor having the highest SNR is then the most appropriate from the detection with the threshold criterion. To define the 'best' pre-filtering state, one defines the reference sensor at the angular frequency 𝜔 as the sensor having the highest SNR at this frequency. The reference SNR ref is then the SNR of the reference sensor:

SNR ref (𝜔) = max 𝑖∈[1,𝑛] [SNR 𝑖 (𝜔)] (4)
After having defined pre-filtering state, one defines the beamforming treatment. The spatial filter of this treatment is characterized by the so called steering vector 𝐹 𝑢 , which will be defined for each position u of the detection space. This later represents the possible positions of the source to be detected. For the present configuration (see figure 1), the detection space consists in the fluid volume inside the pipe. The beamforming output at the position u, 𝑦 𝑢 is given by [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF]:

𝑦 𝑢 (𝜔) = 𝐹 𝑢 * (𝜔)Γ(𝜔)𝐹 𝑢 (𝜔), (5)
where the asterisk denotes the Hermitian conjugate.

The SNR at the beamforming output is then defined by:

𝑆𝑁𝑅 𝐵𝐹 (𝜔) =
where 𝑦 𝑠 𝑠 is the output of the beamforming steering at the effective position of the source in the presence of the source only, and 𝑦 𝑠 𝑛 is the output of the beamforming steering at the effective position of the source in the presence of the noise only.

The main interest of the beamforming for the detection is that the SNR at the beamforming output is significantly greater than the SNR at the reference sensor. One can then define the array gain, 𝐺(𝜔) by (recalling that the SNR quantities have been defined in dB):

𝐺(𝜔) = 𝑆𝑁𝑅 𝐵𝐹 (𝜔) -SNR ref (𝜔) (7)
Up to now, the steering vectors, 𝐹 𝑢 have not be defined. Various definitions have been proposed

in the literature [START_REF] Van Veen | Beamforming: a versatile approach to spatial filtering[END_REF] which depend on different purposes (localisation or detection for instance) and on the prior knowledge of different quantities. One proposes to consider two definitions: the one related to the conventional beamforming as used by [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF] and the MaxSNR beamforming [START_REF] Tanaka | Acoustic Beamforming with Maximum SNR Criterion and Efficient Generalized Eigenvector Tracking[END_REF][START_REF] Van Veen | Beamforming: a versatile approach to spatial filtering[END_REF]) that focuses on maximizing the SNR at the beamforming output.

• Conventional beamforming:

For this treatment, one supposes that the source to be detected is characterized by the transfer functions between the source at any position u in the detection space and the vibrations measured at the n sensor positions. One notes H 𝑢 (𝜔) = [𝐻 𝑢,𝑖 (𝜔)] 𝑛×1 , the vector containing these transfer functions for a given position u of the source. More precisely, one defines the transfer function between the position u of the source and the sensor #i, 𝐻 𝑢,𝑖 (𝜔) as the ratio of the acceleration measured at sensor #i over the source strength (i.e. volume velocity). Two ways can be considered for estimating these transfer functions: the first one consists in assuming that the source can be represented by an acoustic monopole and by using an accurate vibroacoustic model of the considered system [START_REF] Moriot | Passive vibroacoustic detection of a water-sodium reaction using beamforming on a steam generator of a liquid sodium fast reactor[END_REF][START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF]). However, it has been shown

in [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF] that simple models like a fluid filled cylindrical shell model is not sufficiently accurate to represent these transfer functions for beamforming application. An attention have to be paid in the future to develop numerical models that are more accurate; the second way that is considered in the present paper consists to directly measure these transfer functions. It may avoid the issues of reliability of the numerical model but it requires to be able to move the source inside the detection space.

Concerning the background noise, it is assumed that it is spatially homogeneous and incoherent.

The cross-spectral matrix of the accelerations in presence of the background noise alone can be written Γ 𝑛 = 𝜎I where I is the identity matrix and 𝜎 is the ASD function of the background noise acceleration. Under these assumptions and using linear algebra considerations, it can be shown that the beamforming output is maximum when the beamforming focus on the effective position of the source when the steering vectors is defined by:

𝐹 𝑢 𝑐𝑙𝑎𝑠𝑠 (𝜔) = 𝐻 𝑢 (𝜔) ‖𝐻 𝑢 (𝜔)‖ 2 , ( 8
)
where ‖ ‖ represents the Euclidean norm.

Thanks to this definition of the steering vector, this treatment is well adapted to localize the source in the detection space. It relies on a prior knowledge of the source (i.e. through the transfer functions H 𝑢 ) and on the assumption of spatially uncorrelated background noise. It is commonl applied for detecting plane wave sources in free space for many industrial applications (i.e. submarine, radar, etc). In the following, one names it the "conventional" beamforming treatment.

• MaxSNR beamforming:

In some situations (as it has been observed by [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF] for significant flow speed), the background noises measured at the different sensors can be partially correlated. It results that the assumption of the classical beamforming is violated, that lead inevitably to a decrease of the beamforming performance.

To overcome this obstacle, different variants of beamforming based on prior knowledge of background noise (as well as of the source) have been developed [START_REF] Van Veen | Beamforming: a versatile approach to spatial filtering[END_REF]. In particular, the so called "MaxSNR" beamforming has been developed to maximize the SNR at the beamforming output. The optimal steering vector is then defined as follows:

𝐹 𝑢 𝑜𝑝𝑡 (𝜔) = argmax 𝐹 𝑢 [ 𝐹 𝑢 * (𝜔)Γ 𝑢 𝑠 (𝜔)𝐹 𝑢 (𝜔) 𝐹 𝑢 * (𝜔)Γ 𝑛 (𝜔)𝐹 𝑢 (𝜔) ]. (8)
This definition necessitates a knowledge of the cross-spectral matrix related to the background noise, Γ 𝑛 . This one can be estimated experimentally from different in situ measurements at different times for which it can be supposed that the source is not active (i.e. times without leak). It can also be re-estimated regularly to take variations of the background noise into account. The definition (8) requires also an evaluation of Γ 𝑢 𝑠 , the cross-spectral matrix of the signals induced by the source located at a given position u in the detection space. This quantity can be written:

Γ 𝑢 𝑠 (𝜔) = 𝜎 𝑠 (𝜔)𝐻 𝑢 (𝜔)𝐻 𝑢 * (𝜔) , (9)
where 𝜎 𝑠 is the ASD function of the source strength.

Injecting Eq. ( 9) in Eq. ( 8), one has:

𝐹 𝑢 𝑜𝑝𝑡 (𝜔) = argmax 𝐹 𝑢 [ 𝐹 𝑢 * (𝜔)𝐻 𝑢 (𝜔)𝐻 𝑢 * (𝜔)𝐹 𝑢 (𝜔) 𝐹 𝑢 * (𝜔)Γ 𝑛 (𝜔)𝐹 𝑢 (𝜔) ]. (10)
The mathematical problem is then depending of 𝐻 𝑢 and Γ 𝑛 . From bilinear algebra considerations [START_REF] Tanaka | Acoustic Beamforming with Maximum SNR Criterion and Efficient Generalized Eigenvector Tracking[END_REF], it can be shown that the solution of Eq. ( 10) corresponds to one eigenvector associated with the greatest eigenvalue of the matrix (Γ 𝑛 ) -1 𝐻 𝑢 𝐻 𝑢 * :

(Γ 𝑛 ) -1 𝐻 𝑢 𝐻 𝑢 * 𝐹 𝑢 𝑜𝑝𝑡 = λ 𝑚𝑎𝑥 𝐹 𝑢 𝑜𝑝𝑡 . ( 11
)
The beamforming treatment considering these optimized steering vectors 𝐹 𝑢 𝑜𝑝𝑡 allows us theoretically maximizing the signal-to-noise ratio at the output of beamforming. The technique requires however to solve a generalized eigenvalues problem which can be time consuming.

Compared to the classical beamforming, it requires also a knowledge of the noise cross-spectral matrix.

Presentation of the experimental mock-up

The experimental set-up used to evaluate the performance of the vibroacoustic beamforming for detecting an acoustic source into a heavy fluid is illustrated in Figure 2. For this laboratory experiment, the steam generator shell is represented by a cylindrical pipe made of stainless steel. The length, the diameter and the thickness are respectively, 3.1 m, 219 mm and 8 mm.

For the ease of implementation and safety reasons, the fluid used inside the pipe is water (rather than sodium) at room temperature and at a pressure of about 4 bars. The test section is connected to the hydraulic circuit by two stiff flanges. Particular attention has been paid to decouple the test section from external mechanical sources (by fixing the pipe with rubber seals on a suspended slab) and from external fluid sources (by using acoustic decoupling balloons).

Upstream the test section, a 4 m long pipe and a perforated plate flow conditioner (see figure 2c) are used to stabilize the turbulent flow whereas a 1.5m long pipe is used for the discharge before the acoustic balloon (see figure 2a).

Nine holes have been perforated on the cylindrical generator 𝜃 𝑠 = 0° of the test pipe (see figure 2b). These holes are used to introduce the acoustic source inside the pipe at different axial position. The acoustic source consists in a B&K 8103 hydrophone used in transmission mode (i.e. emitter mode). The driving signal generated by a B&K PULSE system is amplified with a B&K Power Amplifier 2713. The hydrophone is mounted on a mechanical device (see figure 2b) which has been designed for controlling the radial positions of the hydrophone.

The pipe vibrations are measured by an array of 24 KISTLER 8704B50 accelerometers fixed on the cylindrical generator 𝜃 𝑖 = 90° (see figure 3). The spacing between the sensors is 𝛥𝑥 = 4 cm and the first accelerometers is positioned at 15 cm from the upstream flange of the flow.

The axial position of the accelerometer #i is then 𝑥 𝑖 = 0.15 + 0.04(𝑖 -1) for 𝑖 ∈ {1, . . ,24}.

The accelerometer signals have been processed using the B&K PULSE system for extracting the cross-spectral matrices, Γ. The sampling frequency and the frequency resolution have been fixed to 16384 Hz and 4 Hz, respectively. The cross-spectral matrices have been exported in order to achieve the beamforming treatments with MATLAB.

We can underline that the hydrophone is of small size compared to the diameter of the pipe and the acoustic wavelength for the considered frequency band (i.e. up to 5kHz). This characteristic ensures us that it is well adapted for simulating a monopole source. However, the counter part of this small size is the poor acoustic radiation efficiency of this projector in the considered frequency band. Bellow 1 kHz, the signal-to-noise ratio is clearly insufficient for exploiting the results given by the accelerometers fixed on the pipe.

This experimental set-up allows us to achieve measurements for different positions of the acoustic source (i.e. hydrophone) and different flow speeds. However, for the sake of conciseness, we are going to present only the results for the highest considered flow speed. It corresponds to a flow rate of 𝑄 𝑤 = 140 l. s -1 .

Analysis of the vibroacoustic response of the test section

Before to study the performances of the vibroacoustic beamforming, one analyses the vibratory response of the test section for three different excitations:

-A radial point force applied on the pipe in order to analyse the vibroacoustic behaviour of the test section;

-An acoustic source inside the fluid to analyse the vibrations induced by the source to be detected;

-The pressure fluctuations induced by the turbulent flow to characterize the vibratory background noise measured by the array of accelerometers.

Radial mechanical point excitation

Figure 4 shows the vibratory field measured by the array of accelerometers for a radial mechanical excitation (i.e. impact hammer) applied in x = 0.14 m. The measurements have achieved when the pipe is filled with water at a pressure of 4 bars when the water is at rest.

Peaks of resonances can be observed on the FRF of the accelerometer #21 (figure 4a). These ones can be attributed to two physical phenomena:

-The first one is related to the wave guide behaviour of the fluid filled pipe and its ability to exhibit propagative vibroacoustic waves in the axial direction [START_REF] Fuller | The input mobility of an infinite circular cylindrical elastic shell filled with fluid[END_REF]Fahy 1982, Fuller 1983). For each circumferential order n of the cylindrical shell (i.e. each order of a Fourier serie decomposition of the vibratory field), one can define a cut-on frequency. For frequencies below this frequency, the waves related to this circumferential order are evanescent. They do not contribute significantly to the vibratory field on the pipe. In contrary, for frequencies above the cut-on frequency, the waves are propagative and can contribute significantly on the vibratory field. An estimation of these cut-on frequencies can be obtained by resolving the dispersion curves of an infinite fluid loaded cylindrical shell model [START_REF] Fuller | The input mobility of an infinite circular cylindrical elastic shell filled with fluid[END_REF]Fahy 1982, Fuller 1983).

The calculation has been carried out for the present case considering a Young modulus of -The second phenomenon is related to the wave reflexions on the flanges. Indeed, the size of the flange section is 40 mm in the axial direction and 60 mm in the radial direction. The flanges are then significantly stiffer than the cylindrical shell. The mechanical impedance mismatch at the interface between the pipe and the flange leads to wave reflexions at both flanges. The constructive interferences of these waves for some frequencies leads to a resonance phenomenon. In the following, one names by "pseudo axial modes" these resonances. It cannot be strictly called "modes" as a part of the energy of the vibroacoustic waves can be transmitted through the fluid (i.e. water) which is not bounded in the axial direction.

These two phenomena are closely related. It results that the pseudo axial modes for a given circumferential order n occur only for frequencies above the cut-on frequency of the circumferential order. Moreover, as the celerity in the axial direction of a propagative wave is infinite at the cut-on frequency and decreases in function of the frequency (see the dispersion curves in [START_REF] Fuller | Characteristics of wave propagation and energy distributions in cylindrical elastic shells filled with fluid[END_REF], it results that the density of pseudo-modes is important just above the cut-on frequency and then decreases when the frequency increases. This behaviour can be clearly observed on figure 4a. In particular, one can notice the densification of peaks around 1100 Hz and 2200 Hz which are around to the calculated cut-on frequencies for n=3 and n=4, respectively.

The spatial distribution of the pipe vibration measured by the array of accelerometers is shown on figure 4b. For readability consideration, one has plotted the shell displacements instead of the shell accelerations. At the pseudo-mode frequencies, one can observe some patterns with pseudo nodes of vibration and antinodes. The number of antinodes is related to the axial celerity of the vibroacoustic waves. As the celerity decreases with frequency, the wavelength decreases also and then the number of antinodes increase with frequency. It should however keep in mind that several circumferential orders can contribute at a given frequency (as soon as their cut-on frequencies are below this frequency). This may explain why the pseudo-mode at 1244 Hz has 4 antinodes on the instrumented section whereas the one at 1266 Hz has only 3 antinodes. These two pseudo-modes are not related to the same circumferential order. The first one is certainly related to n=2 whereas the second one is related to n=3. Only measurements along the circumference of the pipe would permit to confirm the circumferential order of each pseudomode, it is however outside the scope of the paper.

Acoustic source

Now, let us focus on the vibratory response of the test section when it is excited by the hydrophone used in emitter mode. A sine sweep from 500 Hz to 5 kHz has been used as input signal of the amplifier of the hydrophone. Below 1 kHz, the SNR is however too low for exploiting correctly the measurements. The results will not be shown below this frequency. One plots in figure 5 the displacement results for two radial positions of the source: rs=0.052 m and rs =0.088 m, both for an axial position of xs=0.56 m corresponding around the middle of the array.

One can notice as it can be expected that the levels are generally significantly greater when the source is closer to the steel shell (i.e. rs =0.088 m). Moreover, the patterns appearing on figure 5 are similar to those of figure 4 without to be identical. This can be explained by the fact that the acoustic source does not excite the same pseudo axial modes than the mechanical force considered in figure 4. The radial position of the source influence also the response of the pseudo axial modes. For instance, for a source located on the axis of revolution of the shell (i.e.

rs =0 m), only the pseudo axial modes corresponding to the circumferential order n=0 are excited as the system is fully axisymmetric in this case. More the source is far the axis of revolution, more circumferential orders participate to the shell response. This can be observed on figure 5 where the peak levels are globally lower above 2300 Hz than below this frequency when rs=0.052 where they are roughly the same for all the spectrum when rs =0.088 m. This can be explained by a lower contribution of the circumferential order n=4 for rs=0.052 than for rs =0.088 m.

Finally, one reminds that these transfer functions between the source and the accelerometers have been encapsulate in the vector noted H 𝑢 (𝜔) in section 2. These transfer functions will be used for the calculation of the steering vector (with Eq. ( 8) for the conventional beamforming and with Eq. ( 11) for the MaxSNR beamforming). The array gain will also be evaluated using these transfer functions through the use of Eq. ( 9) for evaluating the cross-spectral matrix related to the source.

Turbulent flow

In the presence of the fluid at rest (only booster pump in operation), the signals between the different accelerometers are incoherent (result no shown). However, as it has already been observed by [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF], when the flow rate increases, the signals become coherent for some frequencies (see figure 10 in [START_REF] Moriot | Use of beamforming for detecting an acoustic source inside a cylindrical shell filled with a heavy fluid[END_REF]). The study of Moriot is however limited to some particular frequencies spaced of 1 kHz.

Figure 6 shows the normalized CSD function between sensor #1 and the other sensors for a flow rate of 140 l. s -s and for the frequency band [1 kHz-3 kHz]. It can been observed that the accelerometer signals are highly coherent for frequencies corresponding to pseudo axial modes (by comparing figure 5 with figure 3). This result was not expected at the first sight. Instead, if one considers the well-known model of [START_REF] Corcos | The resolution of pressure in turbulence[END_REF] of the wall pressure field induced by a turbulent boundary layer, one can calculate a coherent length of the pressure field in the stream wise direction. This one varies from 4.1 × 10 -4 m (at 3 kHz) to 1.3 × 10 -3 m (at 1 kHz). As it is of almost one order of magnitude lower than the sensor spacing, one could expect a weak coherence of the signals of the accelerometers. Finally, it appears that it is not the case as the considered system reacts to this weakly correlated excitation. It results that the vibratory response is strongly correlated (in space) for the frequencies corresponding to pseudo axial modes. This explanation has been confirmed from numerical investigations considering an infinite shell model coupled to two ring stiffeners and excited by a homogenous established TBL excitation [START_REF] Kassab | Vibroacoustic beamforming for the detection of an acoustic monopole inside a thin cylindrical shell coupled to a heavy fluid: Numerical and experimental developments[END_REF]. This strong coherence of the vibratory field at some frequencies is clearly a drawback for the conventional beamforming which supposes the background noise uncorrelated. It is however necessary to establish the performance of this treatment in this particular condition for both, narrow band analysis and wide band analysis. This strong coherence of the background noise has also lead us to consider another type of treatment, the MaxSNR beamforming that we have introduced previously. For this treatment which is based on a prior knowledge of the background noise, one underlines that the measurements achieved for this flow rate (without the acoustic source) gives us the cross-spectral matrix of the background noise, Γ 𝑛 used in Eq. ( 11) to estimate the steering vector.

Analysis of beamforming performances

In the previous section, one has analysed the accelerometer signals induced by the source to be detected (i.e. section 4.2) as well as those induced by the turbulent flow which represent the background noise (i.e. section 4.3). These quantities are used, both, to evaluate the steering vectors and to evaluate the performance of the beamforming. We can underline that this case constitute an ideal case. Instead, in practice for the detection on the SGU, the vibrations induced by the source or those of the background noise can vary in function of the times, depending of the functional state of the installation. The values at an initial time of the quantities used for estimating the steering vectors are then not necessary representative of the values of the same quantities at the time of the detection. Here, one does not study the effect of these time variations of the signals induced by the source or of the background noise that can be due to many parameters related to the practical application.

Narrow band analysis

As it has been described in section 2, a pre-filtering state is defined by the reference sensor and the reference SNR. This information is given in figure 7 for an acoustic source located at The SNR of reference sensor increases globally in function of the frequency. It is mainly due a decrease of the background noise with frequency. One reminds that this definition of the SNR of reference sensor is only valuable for narrow band analysis. The SNR of reference sensor for wide band analysis will be defined in section 2.

(
Figure 8 shows the levels of the beamforming output for the acoustic source 'only' (i.e. source with a very low background noise when the fluid is at rest) and for the background noise due to the turbulent at a flow rate of 140 l/s. These results have been obtained when the beamforming focus on the source using Eq. ( 8) with the steering vectors evaluated as described previously. Figure 8a corresponds to the conventional beamforming whereas figure 8b corresponds to the MaxSNR beamforming. The levels are expressed in dB with the same range on the two graphs. One can observe that the levels of the MaxSNR beamforming are always lower than to the conventional beamforming whatever the frequency. The differences between the two techniques appears more important for the treatment of the background noise than for the signals due to the acoustic source. This highlights the ability of the MaxSNR beamforming to reject the background noise. It results that the SNR at the beamforming output (i.e. difference of the output level with the source and with the background noise, see Eq. ( 6)) is generally greater for the MaxSNR beamforming than for the conventional one. One directly compares the SNR at the beamforming output with the SNR of reference sensor by focusing on the array gain defined by Eq. ( 7). The result is shown on figure 9. Clearly, the gain of the MaxSNR beamforming is significantly greater than the one of the conventional beamforming whatever the frequency inside the studied frequency range. This results directly from the definition of the steering vectors of the MaxSNR beamforming which maximize the SNR at the beamforming output whereas the conventional beamforming assuming an uncorrelated background noise.

The gain of the conventional beamforming exhibits significant values (i.e. 5-7 dB) at some particular frequencies but relatively low values in general. It can even be negative. These poor performances can be related to the strong coherence of the vibratory field induced by the turbulent flow as observed in figure 6 for the frequencies corresponding to the pseudo axial modes. To confirm this statement, one has evaluated the gain of the conventional beamforming using a homogeneous uncorrelated background noise instead to use the signals measured on the test section. The result has been plotted with a dotted line on figure 9. For this theoretical case, one can observe a gain varying around 9 dB, always significantly greater than the one obtained with the measured background noise. This confirms the strong negative influence of the background noise coherence on the conventional beamforming performance for the present case. For some frequencies (above 4.5 kHz for instance), these theoretical gains of the conventional beamforming may be lower than those of the MaxSNR beamforming. The use of a knowledge on the background noise permit in some situations to obtain significant gain that cannot be obtained without this knowledge even in the idealized case (i.e. uncorrelated background noise).

The analysis of the vibroacoustic behaviour of the test section in section 4 has shown the influence of pseudo-axial modes on the vibratory response of the pipe. Resonant peaks can be observed on the vibratory spectrum when the test section is excited by, either, the acoustic source to be detected or the turbulent flow inducing the background noise. In order to study the influence of these resonances on the beamforming performances, the values of the array gain for the two types of beamforming are given on Table 1 for the resonant frequencies identified from the FRF of figure 4a. These values should be compared to the values given on figure 9 for the whole frequency range. Moreover, the values corresponding to local peaks on the curves of the gain are written in bold on Table 1. One can observe that the array gain are not necessary higher or lower for the resonant frequencies compared to the other (non-resonant) frequencies.

Moreover, at some resonant frequencies (for instance 1216 Hz), the curve of the gain can exhibit a local peak whereas at some other resonant frequencies (for instance 1244 Hz), it is not the case. For one part, it can be due to the fact that the acoustic source does not excite significantly some modes compare to other ones (see figure 5a) and for another part, it may be due to the spatial coherence of the background noise that varies from one resonant frequency to another one (see figure 6).

Wide band analysis

In practice, the source to be detected (i.e. water-sodium reaction) is a broadband excitation [START_REF] Kim | Acoustic Leak Detection Technology for Water/Steam Small Leaks and Microleaks Into Sodium to Protect an SFR Steam Generator[END_REF]). Then, it may be relevant to utilize all the energy induced by the source in a wide band instead to focus on one single frequency as for narrow band analysis.

A wide band analysis on a given signal consists in estimating the time-average of the square signal after filtering with a band pass filter ).

(

) 12
The reference sensor for the band [𝜔 1 , 𝜔 2 ] is defined as the sensor having the highest SNR in this band. The SNR of the reference sensor in the band [𝜔 1 , 𝜔 2 ] is then:

SNR ref [𝜔 1 , 𝜔 2 ] = max 𝑖∈[1,𝑛] [SNR 𝑖 [𝜔 1 , 𝜔 2 ]]. (13)
Figure 10 shows the values of this quantity for bands of bandwidth 500 Hz. The case presented is the same than in section 5.1. This figure can then be compared to figure 7. One can observe that the values of SNR of the reference sensor for wide band analysis is globally lower than those of the narrow band analysis. This can be explained by the fact that the reference sensor change from one frequency to another for narrow band analysis (as seen in figure 7a) whereas a single reference sensor is attributed to all the frequencies contained in the frequency band for wide band analysis.

The beamforming output level for the band [𝜔 1 , 𝜔 2 ] is deduced from numerical integration of the frequency-dependant output 𝑦 𝑢 (𝜔) over the frequency band [𝜔 1 , 𝜔 2 ]. One can then evaluate the SNR at the beamforming output and the array gain for the band [𝜔 1 , 𝜔 2 ].

Figure 11 shows the array gain obtained for bands of 500 Hz bandwidth. The array gain of the conventional beamforming remains below 5 dB (excepted for the band [2 kHz-2.5 kHz]). It appears that this treatment is inefficient for these flow conditions. In the rest of the paper, one will focus on the MaxSNR beamforming performance. For this treatment, the array gain is significant with variations between 9 dB and 23 dB. These values appear globally higher than those observed in narrow bands. This can be attributed for one part, to the difference of prefiltering reference between the two analysis (as it has been pointed previously by comparing the SNR of the reference sensor of figures 7 and 10) and for another part, to the integration in the frequency band of the beamforming output.

Figure 12 allow us studying the variations of the gain in function of the source positions. When the source is axially positioned at the middle of the array (i.e. 𝑥 𝑠 = 0.56 m), one notices that the gain is generally greater for 𝑟 𝑠 = 0.088 m than for 𝑟 𝑠 = 0.052 m. Compared to the last, the first position is closer to the wall of the cylindrical shell. As we have discussed in section 4.2, the vibratory levels are higher for 𝑟 𝑠 = 0.088 m than for 𝑟 𝑠 = 0.052 m (see figure 5). However, this should not influence the array gain as it expresses an increase of SNR compared to the SNR of the reference sensor. It is most likely due to the fact that for frequencies above 2.3 kHz, the circumferential order n=4 is less predominant for rs=0.052 than for rs =0.088 m (as discussed in section 4.2) whereas it remains significant for the background noise (see figure 6). This could explain the lower gain for rs=0.052 than for rs =0.088 for frequency bands above 2 kHz.

On another hand, when the source is not located in the front of the array gain (i.e. for 𝑥 𝑠 = 1.56 m or for 𝑥 𝑠 = 2.06 m), one observes that the gain is generally slightly lower compared to the case 𝑥 𝑠 = 0.56 m. It remains however significant. This shows that the MaxSNR beamforming can be efficient even if the source is not in the front of the array.

The beamforming output levels in the detection space 𝜃 = 0° are shown on Figure 13 for the 4 source positions. These results corresponds to the frequency band [4.5 kHz-5 kHz]. As the measurements have been achieved for only 9 positions, white cross indicate the positions that are not available. Figure 13a shows the output levels induced by the background noise (i.e.

without the source to be detected). It is almost uniform around -43dB. The output levels in presence of the source are shown on Figure 13b-d for the four position. The real location of the source is symbolised by a black spot. In opposition to the conventional beamforming, the steering vectors of the MaxSNR are not defined such that the beamforming output is maximum when the beamforming focus on the effective position of the source. One can however observe that the output level is always the highest in the detection space for the real position of the source. For the present case, it appears that the MaxSNR beamforming is able to localize the source. This figure allows us highlighting the interest of the beamforming for increasing the SNR. Indeed, the SNR of the reference sensor are respectively, 14, 16, 17 and 11 dB for the positions of figure 13b-d. On figure 13b for instance, the maximum output level of MaxSNR beamforming is -7 dB whereas the output with only background noise is at -43 dB (see figure 13a). The difference of 36 dB between a state without the source and with the source is 22 dB greater than the SNR of the reference sensor (i.e. 14 dB). This exceeding of level (that correspond to the array gain) can be used to detect the source minimizing the false alarm.

Conclusions

The works presented in this paper have occurred in the framework of the development of a nonintrusive monitoring technique for the detection of a sodium-water reaction in the steam generator unit of a liquid sodium fast reactor. The technique is based on vibratory measurements on the external shell of the steam generator unit. The vibroacoustic beamforming can be used to increase the SNR in order to detect the signal due to the source when it can be embedded in the background noise. The study was conducted on a pipe test section where the source to be detected consisted on a hydrophone used in emitter mode placed inside the pipe while the disturbing noise is induced by the water turbulent flow. A linear array composed of 25 accelerometers fixed on the pipe was used for the vibratory measurement. The efficiencies of the conventional and the MaxSNR beamformings have been studied.

The conventional beamforming has appeared inoperative at significant flow rates (i.e.

vibratory signals induced by the turbulent flow which are in contradiction with the assumption made by the conventional beamforming. On the other hand, significant gains (around 10 to 25 dB) have been observed using the MaxSNR beamforming for different positions of the source which are not necessary in the front of the array. An analysis on the beamforming output in the detection has also shown that it can be used for the localisation of the source for the considered case.

It should nevertheless be noted that these gains results from considering "ideal" data to define the steering vectors:

the source-sensor transfer functions considered were measured on the pipe;

the cross-spectral matrix of accelerations characterizing the noise has been used both to define the steering vectors and to evaluate the array gain.

In the future, it will be necessary to develop a reliable vibroacoustic model of the considered system to predict the source-sensor transfer functions as it may be difficult to measure them for the practical application. Moreover, a sensitivity analysis of the variations of the background noise on the beamforming performance should be achieved. Finally, it must be recalled that the majority of background noise in an operating steam generator is due to the combined noise of

1

  .85 × 10 11 Pa, a mass density of 7800 kg/m 3 and a Poisson coefficient of 0.3 for steel and a sound velocity of 1500 m/s and a mass density of 1000 kg/ m 3 for water. One finds the following cut-on frequencies: 363 Hz for n=2, 1079 Hz for n=3, 2145 Hz for n=4 and 3549 Hz for n=5;

  𝑥 𝑠 , 𝑟 𝑠 ) = (0.56 m, 0.052 m) and a flow rate of 140 l/s. One can notice that the reference sensor vary from one frequency to another in the studied frequency band [1 kHz-5 kHz]. It is a consequence of the variation with frequency of the spatial distribution of the vibratory field induced by the source (as shown on figure 5) as well as those induced by the turbulent flow.

  sodium flow and water evaporation, while we only assimilated it to turbulent flows. Only more representative experiments could confirm the interest of the vibroacoustic beamforming considering the MaxSNR treatment for the detection of the sodium-water reaction in a SGU.

Figure 1 .Figure 2 .

 12 Figure 1.Schematic representation of the configuration considered for assessing the performance of the vibroacoustic beamforming.

Figure 3 .

 3 Figure 3. Picture of the test pipe with the linear array of accelerometers.

Figure 4 .

 4 Figure 4. Radial displacement response of the test pipe for an unit radial mechanical point force at x=0.105 m (hammer excitation): (a), Frequency Response Function (FRF) at point x=1 m (i.e. sensor #21); (b), Displacement versus sensor position and frequency (dB, ref. 1 m).

Figure 5 .

 5 Figure 5. Radial displacement response in function of the sensor position and the frequency (dB, ref. 1 m) for an acoustic source excitation (i.e. hydrophone emitter) at two positions: (a), (𝑥 𝑠 , 𝑟 𝑠 , 𝜃 𝑠 ) = (0.56 m, 0.052 m, 0°); (b), (𝑥 𝑠 , 𝑟 𝑠 , 𝜃 𝑠 ) = (0.56 m, 0.088 m, 0°).

Figure 6 .Figure 8 .

 68 Figure 6. Normalized cross spectrum density function between sensor #1 and sensor #i. Results in function of the position of sensor #i and the frequency for a flow rate of 140 l/s.

Figure 9 .Figure 10 .

 910 Figure 9. Array gain in function of the frequency: full, MaxSNR BF; dash, conventional BF; dash; dotted, theoretical values for the conventional BF supposing uncorrelated noise. Case of an acoustic source at (𝑥 𝑠 , 𝑟 𝑠 , 𝜃 𝑠 ) = (0.56 m, 0.052 m, 0°) and for a flow rate of 140 l/s.

Figure 11 .

 11 Figure 11. As for figure 9 for a wide band analysis of 500 Hz bandwidth.

Figure 12 .

 12 Figure 12. Gain with the MaxSNR beamforming for different source positions: full, (𝑥 𝑠 , 𝑟 𝑠 ) = (0.56 m, 0.052 m); dash, (𝑥 𝑠 , 𝑟 𝑠 ) = (0.56 m, 0.088 m); dash-dotted, (𝑥 𝑠 , 𝑟 𝑠 ) = (2.06 m, 0.052 m); dotted, (𝑥 𝑠 , 𝑟 𝑠 ) = (1.56 m, 0.088 m). Wide band analysis of 500 Hz bandwidth.

Table 1 .

 1 Values of the gain for resonance frequencies identified on figure 4a. Case of an acoustic source at (𝑥 𝑠 , 𝑟 𝑠 , 𝜃 𝑠 ) = (0.56 m, 0.052 m, 0°) and for a flow rate of 140 l/s. Bold values indicates local peaks on the curve of the gain.

	Frequency (Hz) 1068 1096 1132 1164 1216 1244 1260 1336 1416 1512
	Conv. Gain (dB) -5.1 -1.2 -4.7 -2.6 -1.1 -5.8 -3.6 -4.0 -1.2 -3.2
	MaxSNR Gain (dB) 11.4 12.7	8.5 11.1 15.3	6.8 15.3	8.0	7.4	9.5
	Frequency (Hz) 1616 1728 1848 1976 2112 2244 2368 2428 2496 2564
	Conv. Gain (dB) -0.8 -1.8 -2.6 -8.8 -1.2 -2.7	2.1 -2.1 -0.6	0.2
	MaxSNR Gain (dB)	4.4	6.8 10.7	6.6 10.1	7.8	9.4	6.1	2.8	3

  [𝜔 1 , 𝜔 2 ]. This quantities can be estimated with the ASD function of the signal and by integrating it in the frequency band [𝜔 1 , 𝜔 2 ]. One can then define the SNR in the band [𝜔 1 , 𝜔 2 ] for each accelerometer i by:

	SNR 𝑖 [𝜔 1 , 𝜔 2 ] = 10log 10 (	∫ 𝑆 𝑖𝑖 𝜔 2 𝑠 (𝜔)𝑑𝜔 𝜔 1 ∫ 𝑆 𝑖𝑖 𝜔 2 𝑛 (𝜔)𝑑𝜔 𝜔 1


H Guo

P Sciora

T Kooyman

L Buiron

Application of boron carbide as burnable poison in sodium fast reactors

Keywords: fast reactor, burnable poison, boron carbide

The inherent safety performance in incidental transient is required for Generation-IV reactors. The high excess reactivity in the core to compensate for burn-up reactivity loss increases the control rod withdrawal accident in sodium fast reactor. This paper does focus on the application of absorber as burnable poisons to compensate for reactivity loss.

The coupling between boron carbide in depleted boron-10 and zirconium hydride is the most relevant burnable poison in fast neutron spectrum after evaluation of various materials. The burnable poison is implemented in independent assemblies that are fixed in the core during all the operation cycle.

Four core designs with different loading of burnable poison, from 0.79%vol to 3.97%vol of fuel, are investigated. These burnable poison designs show enough ability to compensate for reactivity loss and thus reduce the excess reactivity. The application of such design in cores with higher reactivity loss will be considered in the near future.

The burnable poison assembly is able to manage the power distribution as control rods. However, their position is fixed in the core during the cycle and thus should be determined in the design step. As including moderation materials, these designs improve slightly the reactivity feedback coefficient.

Introduction

Formed in the year 2000, the Generation IV International Forum (GIF) is an intergovernmental organization that promotes the development of new technologies of nuclear reactors [START_REF] Gif | Technology Roadmap Update for Generation IV Nuclear Energy Systems[END_REF][START_REF] Pioro | 2 -Introduction: Generation IV International Forum[END_REF]. In order to be competitive with the other sources of energy, these technologies are required to meet several challenges involving sustainability, economic competitiveness, safety and resistance to proliferation of nuclear weapons. Among the candidates that fulfill all those criteria, France is particularly interested in the Sodium cooled Fast neutron Reactor (SFR) [START_REF] Beck | Conceptual design of ASTRID fuel sub-assemblies[END_REF][START_REF] Chenaud | Status of the ASTRID core at the end of the pre-conceptual design phase 1[END_REF][START_REF] Gauche | French SFR R&D Program and Design Activities for SFR Prototype ASTRID[END_REF].

As for every nuclear reactor, the core at start-up exhibits excess reactivity which allows continuous operation over the length of the fuel cycle. This excess reactivity should cover burn-up reactivity loss, operation margin (loading of experimental assemblies for instance) and uncertainty margins. Due to the use of sodium as a coolant, it is not possible to use diluted boron to control the core reactivity change during irradiation. Therefore, contrary to pressurized water reactors, control rods are the only available mean of reactivity control, boron dilution in sodium being not possible. At Beginning of Equilibrium Cycle (BOEC), one part of control rods should be inserted into the core to balance this excess reactivity. The control rods are then withdrawn slowly during the cycle to compensate for burn-up reactivity loss.

A typical initiating event for unprotected transient over-power (UTOP) is a control rod withdrawal (CRW) due to a malfunction of a control rod mechanism. This event could lead to the local melting of fuel assemblies and even to the global melting of core (Varaine F. et al., 2012).

Multiples solutions can be used to prevent fuel melting in case of a total unprotected CRW accident. First, mechanical systems can be designed to prevent inadvertent control rods withdrawal. However, these systems often depend on operational logic, detectors, or mechanical motors etc. This approach has not been implemented in France so far as it relies on mechanical systems integrated with the malfunctioning device and thus complicates the safety analysis. Secondly, the maximal linear power density can be limited to obtain enough margins in terms of linear power rate during CRW transient. Thirdly, the cycle length can be adjusted to reduce the reactivity compensated by the rods for burn-up. However, these two last solutions reduce the economic performances of the reactor. A fourth solution would be to use burnable poisons (BP) as it is currently done in light water reactors (LWRs) [START_REF] Goldstein | A Comparison of Gadolinia and Boron for Burnable Poison Applications in Pressurized Water Reactors[END_REF][START_REF] Renier | Development of Improved Burnable Poisons for Commercial Nuclear Power Reactors[END_REF].

The burnable poisons are materials with an initial absorption capacity that should be significantly reduced under neutron irradiation. For instance, in pressurized water reactors (PWRs), boron and gadolinium are used as burnable poison to decrease the initial boric acid concentration in the primary circuit, the reactivity compensation needed, and the relative power of fresh fuel assemblies [START_REF] Goldstein | A Comparison of Gadolinia and Boron for Burnable Poison Applications in Pressurized Water Reactors[END_REF][START_REF] Renier | Development of Improved Burnable Poisons for Commercial Nuclear Power Reactors[END_REF].

In SFR, the burnable poisons may have a potential ability to reduce CRW accident, to extend the cycle length or to lessen the requirements on the control rods. However, they were not significantly investigated for SFR because of their insufficient absorption cross-section in the fast spectrum. Therefore, this paper is aimed to investigate the application of absorbers coupled with moderators as burnable poisons and try to improve their reactivity compensation capability.

The Section 2 will present the neutronic simulation tools used in this design work. The potential of different materials to achieve the burnable poison objective will be investigated in Section 3.1. The detailed designs of burnable poison assembly and their implementations in a core will be presented in Section 3.2 and 3.3. Then, the compensation ability of different designs, that is a key result for burnable poisons, will be discussed in Section 4. Finally, this paper will compare the different burnable poison designs proposed and their influence on some macroscopic parameters like the power map of a core.

Methodologies

Accurate and high performance neutronic simulation is the key for the evaluation of the innovative designs of reactivity control system. The complex geometries of these systems should be treated accurately [START_REF] Blanchet | Control Rod Depletion in Sodium-Cooled Fast Reactor: Models and Impact on Reactivity Control[END_REF][START_REF] Tommasi | Analysis of Superphenix and Phenix neutron physics experiments with the ERANOS code and data system[END_REF] as the absorber is one of the most sub-critical material in the core and thus exhibits important flux gradient and self-shielding effect at a sub-assembly scale. To benefit of the last developments still ongoing, the deterministic APOLLO3 ® platform [START_REF] Golfier | APOLLO3®: A common project of CEA, AREVA and EDF for the development of a new deterministic multi-purpose code for core physics analysis[END_REF] is chosen for the neutronic simulation of the various designs in this work.

The calculation scheme in APOLLO3 ® includes two steps: lattice and core calculation. The lattice calculation is based on the Method of Characteristic (TDT-MOC) [START_REF] Sciannandrone | Optimized tracking strategies for step MOC calculations in extruded 3D axial geometries[END_REF] and is able to simulate complex geometries with exact description and compute self-shielding effects [START_REF] Guo | Improvement of sodium fast reactor control rods calculations with APOLLO3[END_REF]. The lattice calculation of multiple designs, including the conventional control rods and the innovative control assembly that mix absorber with large quantity of moderators, has been validated against TRIPOLI-4 ® [START_REF] Brun | TRIPOLI-4®, CEA, EDF and AREVA reference Monte Carlo code[END_REF], which shows high robustness of this solver to different innovative designs [START_REF] Guo | Improvement of sodium fast reactor control rods calculations with APOLLO3[END_REF]. This lattice step is able to generate homogenized and collapsed cross-section at different burn-up that will be used in the core step.

In the core step, the 3D S N solver MINARET [START_REF] Moller | Minaret, a deterministic neutron transport solver for nuclear core calculations[END_REF] is able to treat unstructured geometry. Therefore, beside the traditional homogenous description of all structures, MINARET enables to consider also a semi-heterogeneous model that permit to describe the exact geometry of the absorbers in the core geometry. Although this semi-heterogeneous model is more accurate it also impacts calculation time in significant way. The update of cross-sections according to the burn-up have been developed and leads to improve significantly the accuracy of depletion calculation because it is able to transfer the variation on self-shielding from lattice step to core step, which is particularly important for the absorber materials. With updating of cross-sections, it has been shown that the accuracy of homogenous geometry modeling is sufficient for the design work purpose [START_REF] Guo | Improvement of sodium fast reactor control rods calculations with APOLLO3[END_REF].

By using these newly developed and improved tools, it is feasible to achieve an accurate simulation of reactivity control system with complex configurations and thus to carry out the design work described in the following parts.

Burnable poison designs

Candidate materials

The challenge for the application of absorbers as burnable poison is the insufficient consumption of absorbers as their absorption cross-section generally decreases with the incident neutron energy. This section is aimed at evaluating the depletion characteristics of various absorbers and thereby selects the adequate materials for the burnable poison objectives.

In order to compensate for reactivity loss, the ideal candidate for the burnable poison should possess high variation of absorption capacity during the irradiation and limited residual negative effect at the end. The absorption ability of one material at moment t can be represented by its macroscopic absorption cross-section (Σ) that is defined as:

Where i is an isotope in the material M with concentration c i and one-group absorption cross-section . The macroscopic absorption is therefore in unit cm -1 .

3.1.1. Different absorbers Natural boron carbide (B 4 C with 20 % 10 B), depleted boron carbide (B 4 C with 3 % 10 B), gadolinium oxide (Gd 2 O 3 ), metallic hafnium (Hf) and hafnium hydride (HfH1 .62 ) are compared in this section because they are general absorbers used in nuclear reactors. These materials are tested in the lattice type geometry as shown in Fig. 1, in which the assembly with absorber pins is surrounded by fuel assembly that offers a fast spectrum irradiation environment. This type of geometry is used in the APOLLO3 ® lattice calculation to generate effective cross-section library of control rods for core calculation.

The variation of macroscopic absorption cross-section in Fig. 2 is defined as while the relative macroscopic absorption cross-section is defined as . The effective macroscopic absorption cross-section is computed until the accumulated fluence is 10 23 n/cm 2 (1.8×10 15 n/cm 2 /s for 579 days). Under irradiation, the boron carbide has a larger variation than other absorbers in fast spectrum. Even if the 10 B concentration in the depleted boron carbide is low, its higher cross-section leads to the same depletion kinetic as the natural one. The other absorber isotopes such as hafnium and gadolinium have longer decay chain to slow down their depletion kinetic and seem to be less relevant for our objectives. Regarding the residual negative effect, the depleted boron carbide is more suitable than the natural boron carbide, but it seems nevertheless not sufficient to be of interest for the burnable poison objective.

Moderators

According to the previous results, the hydride form increases the depletion of hafnium because the diffusing material is able to slow down the incident neutrons and thus increase the consumption rate of the absorber. The addition of moderator seems therefore to be a potential option to realize burnable poison with absorber. In this section, a simple model is considered to investigate the sensibility of the boron carbide and gadolinium oxide to the moderator. As shown in Fig. 3, two layers of absorber pins are replaced by the moderator pins. Zirconium hydride As shown in Table 1, the addition of moderator increases significantly the variation of the macroscopic cross-section and reduces their residual negative effect. With absorption resonance, the gadolinium is more sensitive to the moderation, but it still remains not competitive with the boron carbide. The depleted boron carbide seems to be more suitable than the natural one in this case. However, considering the behavior of natural and depleted boron carbide, it appears that an optimal 10 B concentration can be found depending on the reactivity compensation objective. This will be investigated in Section 3.2. This section investigated the depletion characteristics of different absorbers, with and without moderators in fast spectrum cores. The boron carbide appears as the best candidate for our burnable poison objective thanks to its compromise between large initial macroscopic absorption cross-section and small residual effects. The application of moderator is necessary to optimize the burnable poison design in fast spectrum.

Burnable poison assembly

In this work, the boron carbide and zirconium hydride are packaged in independent burnable poison assembly. This independent design is aimed at reducing the impact on the fuel handling and recycling by separating the burnable poisons from the fuel. This section will present the design of burnable poison assemblies and their location in the core.

The radial layout of poisons design used in this paper is shown in Fig. 4. In the center of this assembly, a layer of boron carbide is mixed with a layer of moderator to maximize the moderation effects. The absorbers and moderators are surrounded by the steel to reduce their influence on the neighboring fuel assemblies. By comparison with traditional absorber pins in SFR control rods, the size of absorber pin is adopted to arrange more flexibly the absorber and moderators. Moreover, the small size pin with central hole is able to reduce the centerline temperature in absorber because the moderator increases significantly the absorption cross-sections of 10 B and thus its power density (See Section 4.2.1). In the near future, the fabrication feasibility and the gap filler will be considered to optimize current design. The one-group effective absorption cross-section of boron carbide in Fig. 4 increases significantly with the decrease of the 10 B concentration (See Fig. 5).

The relationship between the variations of macroscopic absorption cross-section in all the burnable poison assemblies and the initial 10 B concentration is shown in Fig. 6. The ratio between the end and the initial macroscopic cross-section is plotted with the right ordinate axis. The calculations are done by supposing that the accumulated fluence is 6.3×10 22 n/cm 2 (1.8×10 15 n/cm 2 /s for 410 days). For both large variations and small residual considerations, the optimal 10 B concentration is about one-tenth of natural boron carbide. For higher concentrations, the absorption cross-section is too small to achieve enough consumption and thus there is an important residual absorption at the end. For lower concentration, the absorber isotope is over-consumed which means that the macroscopic cross-section is reduced to zero before the end of irradiation.

This optimal concentration depends on the accumulated fluence and on the position in the core. The simulation of such assembly is in fact very sensitive to the calculation scheme. The cross-sections vary with the consumption and hence the depletion kinetic varies with the time. In summary, the variation of cross-section with time and fine time mesh should be considered for either stochastic or deterministic method. The axial layouts of the burnable poison assemblies used in the following are described in Fig. 7 by comparison with the fuel assembly. The burnable poison pins are in the same axial position as the fuel in order to benefit from the maximal flux level. The four different burnable poison assemblies, from BP S/A 1 to BP S/A 4, use the same radial design presented in Fig. 4, but they are made of four different heights to adjust their quantity.

In the following, a slightly lower concentration than optimal value is used to avoid the uncertainty of the fluence in order to ensure that the residual effect on reactivity remains zero. The boron-10 concentration used in each burnable poison zone is presented in Table 2. As a reference value, the 10 B concentration in natural boron carbide is 2.03610 22 at./cm 3 . GP: Gas plenum; AR: Axial reflector; SF: Sodium follower

Fig. 7. Axial layout of reference fuel assembly and burnable poison assemblies

Cores with burnable poison

SFR-V2B core design [START_REF] Mignot | Studies on french SFR advanced core designs[END_REF][START_REF] Sciora | A break even oxide fuel core for an innovative SFR: CEA neutronic studies[END_REF] is chosen in this work because it is a representative SFR with MOX fuel. This core has been designed to achieve objectives defined for Generation-IV reactors. This concept is based on a bundle of tightly packed and large-diameter fuel pins designed to increase the fuel fraction in the core while reducing the sodium fraction. By comparison with other SFR, its reduced reactivity swing is the relevant characteristic to start the application of burnable poisons in SFRs.

As shown in Fig. 8, the SFR-V2B reference core has 267 inner core assemblies and 186 outer core assemblies. The core has two independent control rod systems. The first system is designed for the reactor operation (burn-up reactivity compensation, power management etc.) and also for shutdowns needs. This system is named CSD (Control Shutdown System) and is spatially distributed over two rings: the first one CSD1 has 6 assemblies that are located in the inner core and the second one CSD2 has 18 assemblies that are located at the interface between inner and outer core. The second system has 12 assemblies inside the inner core and is named DSD (Diverse Shutdown System) that is dedicated to the emergency shutdown. CSD and DSD are redundant, independent, and diverse to ensure a safe reactor shutdown at any time needed.

The CSD2 is inserted 25 cm into the core at BOEC and withdrawn slowly with time to compensate for the burn-up reactivity loss. The position of CSD2 is an optimal result from several design iterations because it is not only used to compensate for reactivity loss but also to optimize the power distribution.

The objective of burnable poison designs is to reduce the reactivity loss of the SFR-V2B core to zero. Moreover, the burnable poison assemblies are placed in dedicated positions in the core. Their impact on the power distribution will be assessed in the next paragraphs. The CSD2 in SFR-V2B are also replaced by burnable poison assemblies. Two radial layouts of core with burnable poison are considered (See Fig. 8): one presents 6 CSD2 assemblies with burnable poison assemblies and the other one replaces all CSD2 with burnable poison assemblies. After one irradiation cycle, 410 EFPD for the SFR-V2B, about 1/5 th fuel are renewed at the End of Equilibrium Cycle (EOC). Therefore, the fuel assemblies stay in the core 2050 EFPD and are discharged with an average burn-up about 100 GW•D/t. On the other hand, the burnable poison assemblies are all replaced at each cycle. Finally, four burnable poison designs in Table 3 are presented in this paper. The volume fraction of burnable poison is the ratio between burnable poison volume and fuel volume. The B4CBP1 and B4CBP2 have the same volume of burnable poison but with different assembly number. The difference between these two configurations will be discussed in the following (See Section 4.2.2 and 0). The B4CBP3 and B4CBP4 increase the volume of burnable poison to test the potential of such design. The average plutonium content is adjusted to achieve 750 pcm reactivity at EOEC as for the reference design, while keeping the fixed ratio between inner and outer core. It can be seen that the addition of burnable poisons has a limited impact on the required plutonium content in order to keep the same cycle length. This section has presented the design of burnable poison assembly and their location in the core. The design is optimized by adjusting 10 B concentration to get the largest variation on the absorption ability. Finally, four cores with burnable poison have been constructed based on the SFR-V2B core and they will be studied in the following part.

Compensation ability and influence of burnable poisons

Reactivity compensation ability

The reactivity variation of these designs is plotted in Fig. 9. The excess reactivity in the reference SFR-V2B core decreases from 1177 pcm at BOEC to 750 pcm at EOEC. The control rods are inserted to balance the excess reactivity but with risk of a CRW accident especially at BOEC. The 750 pcm at EOEC is the minimal excess reactivity considered for operation and uncertainty margin. Therefore, the objective of burnable poison is to reduce the 427 pcm burn-up reactivity loss while keeping the core critical during the entire cycle.

The initial core reactivity of the design with burnable poison is lower than the reference one. The B4CBP1 and the B4CBP2 reduce the core excess reactivity at BOEC respectively by 347 pcm and 419 pcm. Moreover, they both keep enough margins for operation and uncertainty during all the cycle. The B4CBP1 and the B4CBP2 realize the objective of burnable poison. However, the core reactivity varies not linearly with time because the depletion kinetic of boron carbide increases first and then decreases. For instance, the core reactivity in B4CBP2 varies between 718 pcm and 777 pcm during all the cycle, which requires the regulation of control rods to balance this reactivity.

The B4CBP3 and the B4CBP4 are overqualified for the SFR-V2B core. Their reactivity compensation ability is respectively 729 pcm and 1022 pcm that is higher than the burn-up reactivity loss in SFR-V2B. Although these two designs prove the capability of burnable poisons with boron carbide, there are not suitable in the present design since their initial reactivity are lower than the EOEC one. Their applications in cores with higher reactivity loss are ongoing.

Influence of burnable poisons implementation

The burnable poison assemblies with absorber and moderator impact the core characteristics. This section is aimed at studying their impact from assembly to core level. The distribution of power density in burnable poison assemblies and in the core will be presented first. Then, the impact on the neighboring fuel assemblies will be discussed. This section will be ended by the core "safety" performance.

Power distribution in burnable poison assembly

The distributions of the linear heat rating in the B4CBP2 burnable assembly at BOEC and EOEC are presented Fig. 10, which is calculated by APOLLO3 ® TDT-MOC solver without considering the gamma heating. At BOEC, the linear heat rating varies between 28 and 52 W/cm. As the 10 B is almost fully consumed at EOEC, the associated linear heat rating is very low. Even with conservative estimation, the maximal temperature in these absorber pins (850 K in 1-D dimensional calculation) exhibits large margin to the melting temperature. However, the characteristics of boron carbide under irradiation must be investigated and be accounted in future optimizations.

At BOEC, the power density increases from the outer region to the inner region because of the moderation effects of the multiple zirconium hydride layers. This distribution is nevertheless reversed at EOEC because the inner region has been further depleted. For the designs used in this paper, all absorber pins have the same 10 B concentration. In near future, this design could be optimized by considering a higher initial 10 B concentration in the inner region to benefit from the moderated neutrons and hence to increase the consumption of absorber materials.

Core power distribution

The impacts of the B4CBP1 and the B4CBP2 on the core power distribution are presented in Fig. 11 and Fig. 12. The discrepancy on these two figures is defined as

, where the v is the average linear heat rating in each assembly. The maximal value in the core is summarized in Table 4.

For B4CBP1 configuration, one third of the CSD2 assemblies are replaced by burnable poison assemblies. The remaining CSD2 are kept at the top of fissile zone because the burnable poison is able to compensate for almost all the reactivity loss. At BOEC, the linear heat rating is increased for the fuel assemblies surrounding the CSD2 in the B4CBP1 design while the assemblies surrounding the burnable poison assemblies show a reduction. This impact on the power density is less significant at EOEC because the burnable poisons are consumed and the CSD2 are also withdrawn to the top of the fissile zone in the reference core. Consequently, the burnable poison assemblies in B4CBP1 increase the power pic (in 3D computation) by 7.7 % at BOEC and by 2.7 % at EOEC (See Table 4). The B4CBP2 replace all CSD2 with burnable poison assemblies. Therefore, its influence is less significant. The B4CBP2 decreases the power pic by 1.4 % at BOEC but increases by 2.7 % at EOEC. The impact of B4CBP3 and B4CBP4 is not presented here because they are overqualified for SFR-V2B core. The burnable poison is able to impact the power distribution like control rods because its absorber depletion process has an effect similar to the withdrawal process of control rods. This investigation proves that the burnable poison assembly is able not only to compensate for the reactivity loss but also to keep a flattened power distribution like control rods. However, the burnable poisons are fixed during the cycle and thus their positions should be well designed. 1. Core shutdown: they must contain enough negative reactivity to ensure safe core shutdown regardless of the core operating situation and keep enough margin for the fuel handing error. 2. Reactivity management: they must have enough worth to compensate for the core burn-up reactivity loss. 3. Power distribution fine tuning: control rods can be used to locally modify the power distribution in the core. 4. Power regulation: the control rods can be used to adjust the power level in the core depending on the load on the turbine. From the previous discussions, it can be concluded that the burnable poison with absorber is able to compensate for reactivity loss and partially control the power distribution like the operation control rods. However, the burnable poisons are not movable during the cycle and thus the fine regulation control rods are still needed. As the burnable poison share the reactivity management mission, such regulation control rod doesn't need important reactivity worth and thereby can be realized by some less efficient but longer-lived material such as gadolinium and hafnium investigated in Section 3.1.1.

In our design, the burnable poison assemblies occupy the positions CSD2 that are used to the reactivity management, power management and core shutdown. The shutdown mission and the local power regulation objective of CSD2 can't be taken over by the burnable poison and thus the "Core 2" (See Fig. 8) will not have probably enough ability to shut down the core with enough margins. The implementation of burnable poison assemblies and shutdown control rods should be considered in future optimizations.

Conclusions

This paper concerns the application of absorber, coupling with moderators, as burnable poison in fast spectrum reactors. Depleted boron carbide with the use moderator is found as the best candidate among different materials tested. The designs of burnable poisons and their implementation are presented, followed by the discussion of their compensation ability and influence on the core performances.

A fine tuned burnable poison solution with boron carbide is able to compensate for the reactivity loss from 400 pcm to 1000 pcm. Only 0.79 % of the fuel volume fraction is sufficient to compensate for reactivity loss in SFR-V2B core using burnable poison. Moreover, the burnable poison assembly shows a potential to manage power distribution similar to the one of control rods. This solution is versatile enough to be adapted to large core configurations as a substitute to control rods devices in some interesting ways.

This paper also exhibits the positive effect of such solution on the core feedback coefficient. However, the transient performance, especially the unexpected control rod withdrawal accident must be characterized in the next step work in order to fully assess the viability of such design.

This work is based on a large oxide sodium fast reactor. In the near future, the application of burnable poisons with minor actinides or boron carbide will be investigated in a small modular fast reactor with a more important reactivity loss.

Fig. 1 .

 1 Fig. 1. The cluster geometry of a typical SFR control rods assembly surrounded by fuel assembly

  72 ) is considered as moderator in this paper because of its high moderation capability.

Fig. 3 .

 3 Fig. 3. Cluster geometry of the control rods with moderator assembly surrounded by fuel assembly

Fig. 4 .

 4 Fig. 4. Radial layout of burnable poison assembly (left) and its absorber pins (right upper)

Fig. 5 .

 5 Fig. 5. Evolution of 10 B absorption cross-section with 10 B concentration

Fig. 8 .

 8 Fig. 8. Radial layout of reference core: Core-0 (left); core with 6 burnable poison assemblies: Core-1 (center); core with 18 burnable poison assemblies: Core-2 (right)

Fig. 9 .

 9 Fig. 9. Core reactivity variation at equilibrium cycle

Fig. 10 .

 10 Fig. 10. Linear heat rating in the 1/12 th burnable poison assembly at BOEC (left) and EOEC (right)

Fig. 11 .

 11 Fig. 11. Discrepancy on power distribution between B4CBP1 core and reference core at BOEC (left) and EOEC (right) (Units: %)

Fig. 12 .

 12 Fig. 12. Discrepancy on power distribution between B4CBP2 core and reference core at BOEC (left) and EOEC (right) (Units: %)

Table 1 . Variation of macroscopic absorption cross-section in assembly (Fluence = 10 23 n/cm 2 )

 1

		Σ i (cm -1 ) ΔΣ (cm -1 ) Σ f /Σ i (Rel.)
	B 4 C 3% 10 B	5.27E-03 -1.35E-03	0.76
	B 4 C 3% 10 B + ZrH 1.72 8.49E-03 -4.23E-03	0.50
	B 4 C 20% 10 B	1.18E-02 -1.21E-03	0.90
	B 4 C 20% 10 B + ZrH 1.72 1.38E-02 -2.13E-03	0.85
	Gd 2 O 3	4.53E-03 -3.16E-04	0.93
	Gd 2 O 3 + ZrH 1.72	8.40E-03 -9.73E-04	0.88

Table 2 . Boron-10 concentration used in burnable poison assembly in Fig. 7 (×10 24 at./cm3

 2

	)

Table 3 . Burnable poison design with boron carbide in SFR-V2B

 3

	Design	Core See Fig. 8	Av. plutonium content (%wt)	BP S/A See Fig. 7	BP S/A number	BP height (cm)	BP volume fraction (%)
	Reference	Core-0	15.56	--			0.00
	B4CBP1	Core-1	15.60	S/A-1	6	60	0.79
	B4CBP2	Core-2	15.60	S/A-2	18	20	0.79
	B4CBP3	Core-2	15.69	S/A-3	18	40	1.59
	B4CBP4	Core-2	15.88	S/A-4	18	100	3.97

Table 4 . Maximal linear heat rating different cores (W/cm)

 4 Feedback coefficientThe feedback coefficients, i.e. sodium void worth and Doppler constant, are compared between different designs in Table5. The neutron energy spectrum shifted by moderators decreases slightly the reactivity inserted by a loss of sodium. The Doppler feedback is more negative due to the local moderation of the neutron spectrum. The burnable poisons with boron carbide are able to improve the safety coefficient but future transient analysis is required.

		BOEC	EOEC
	Ref	445	419
	B4CBP1	479	430
	B4CBP2	438	450
	4.2.3.

Table 5 . Core feedback coefficient at the end of equilibrium cycle (Unit: pcm)

 5

	Design	Sodium void worth Doppler constant
	Reference	2123	-916
	B4CBP1	2118	-938
	B4CBP2	2112	-955
	B4CBP3	2104	-969
	B4CBP4	2087	-976
	4.2.4. Influence on reactivity control
	The reactivity control system should satisfy multiple missions including: