ASSESSMENT OF RANS AT LOW PRANDTL NUMBER AND SIMULATION OF SODIUM BOILING FLOWS WITH A CMFD CODE

Transcription

1 ASSESSMENT OF RANS AT LOW PRANDTL NUMBER AND SIMULATION OF SODIUM BOILING FLOWS WITH A CMFD CODE S. Mimouni 1, C. Baudry 1, M. Guingo 1, M. Hassanay 1, V. Heise 2, J. Lavievie 1, N. Mechitoua 1, N. Mérigoux 1. 1 : Eectricité de France, R&D Division, 6 Quai Watier78401 Chatou, France 2 : ENSEEIT, 2 rue Chares Camiche, Tououse, France Stephane.mimouni@edf.fr ABSTRACT In France, Sodium-cooed Fast Reactors (SFR) have recenty received a renewed interest. In 2006, the decision was taen by the French Government to initiate research in order to buid a first Generation IV prototype (caed ASTRID) by The improvement in the safety of SFR is one of the ey points in their conception. Accidenta sequences may ead to a significant increase of reactivity. This is for instance the case when the sodium cooant is boiing within the fissie zone. As a consequence, incipient boiing superheat of sodium is an important parameter, as it can infuence boiing process which may appear during some postuated accidents as the unprotected oss of fow (ULOF). The probem is that when boiing conditions are reached, the fow decreases progressivey and vapour expands into the heating zone. A crucia investigating way is to optimize the design of the fissie assembies of the core in order to ead to stabe boiing during a ULOF accident, without voiding of the fissie zone. Moreover, in order to evauate nucear pant design and safety, a CFD too has been deveoped at EDF in the framewor of the nucear industry. Advanced modes dedicated to boiing fows have been impemented and vaidated against experimenta data for ten years now incuding a wa aw for boiing fows, wa transfer for nuceate boiing, turbuence and poydispersion mode. This paper aims at evauating the generaization of these modes to SFR. At east two main issues are encountered: Firsty, at ow Prandt numbers such as those of iquid meta, cassica approaches derived for unity or cose to unity fai to accuratey predict the heat transfer. In order to evauate the wa aw impemented in the CFD too, computations have been compared with KALLA experimenta resuts obtained in the case of a rod heated with a constant heat fux which is concentricay embedded in a pipe iquid meta fow (singe-phase fow). Secondy, the incipient boiing superheat of sodium is quite different from that of conventiona fuids. As a consequence, the nuceate boiing mode has been improved and vaidated against the Charety s experiment where a rod heated with a constant heat fux is concentricay embedded in a pipe sodium fow. For different vaues of the heat fux, the pressure is measured at different ocations as function of the mass fow rate. A reasonabe agreement has been reached which is very encouraging for further appications. Finay, preiminary computations have been carried out in an assemby constituted of 19 pins equipped with a wrapped wire where partia experimenta resuts are avaiabe. Computations have shown a pressure drop at the end of the heated ength due to the sudden increase of the hydrauic diameter. Thus, the pressure can drop beow the vapour pressure eading to iquid vaporization. This first resut supports the assumption of boiing in the upper subassemby zone which coud possiby ead to a sodium boiing stabiization. 4247

2 KEYWORDS Boiing fow, SFD, sodium, ow Prandt number, SFR assemby. 1. INTRODUCTION Extended sodium boiing in the fissie zone of a fast breeder reactor (FBR) impies a ris of transient over-power (TOP) and subsequent core meting. Extended sodium boiing may be a consequence of an unprotected oss of fow accident (ULOF). Seier et a. have proposed a mitigation strategy [1], based on boiing stabiisation. This approach is based on the LEDINEGG criterion. If the stabiity criterion is not satisfied a fow excursion wi deveop (caed static instabiity or LEDINEGG instabiity): the fow wi decrease progressivey, and vapour wi expand into the heating zone. A new stabe woring point (high quaity boiing) may theoreticay be reached where the stabiity criterion is again satisfied. The mode has been adapted to the description of the pressure variation at boiing onset (i.e. for ow quaity boiing). The mode cacuates the void fraction distribution and reated gravity and friction pressure drops in the bunde. A homogeneous 1-D two-phase fow mode approach is used in [1] and the pressure drop is cacuated by maing use of the Lochart Martinei mode approach. The main objective of this paper is to generaize this one-dimensiona approach by maing use of CFD too. Indeed, we compute the void fraction by a 3D two-fuid two-phase fow mode. Firsty, the modes impemented in the CFD too are described. In section 4, computations are compared with KALLA experimenta resuts obtained in the case of a rod heated with a constant heat fux which is concentricay embedded in a pipe iquid meta fow (singe-phase fow). In section 5, computations are performed in the Charety s experiment geometry where a rod heated with a constant heat fux is concentricay embedded in a pipe sodium fow (boiing fow). The ast section presents preiminary computations in an assemby constituted of 19 pins equipped with a wrapped wire. 2. THE NEPTUNE_CFD SOLVER AND PHYSICAL MODELLING 2.1 Introduction NEPTUNE_CFD is a three dimensiona two-fuid code deveoped more especiay for nucear reactor appications. This oca three-dimensiona modue is based on the cassica two-fuid one pressure approach, incuding mass, momentum and energy baances for each phase. The NEPTUNE_CFD sover, based on a pressure correction approach, is abe to simuate muticomponent mutiphase fows by soving a set of three baance equations for each fied (fuid component and/or phase). These fieds can represent many inds of mutiphase fows: distinct physica components (e.g. gas, iquid and soid partices); thermodynamic phases of the same component (e.g.: iquid water and its vapour); distinct physica components, some of which spit into different groups (e.g.: water and severa groups of different diameter bubbes); different forms of the same physica components (e.g.: a continuous iquid fied, a dispersed iquid fied, a continuous vapour fied, a dispersed vapour fied). The sover is based on a finite voume discretization, together with a coocated arrangement for a variabes. The data structure is totay face-based, which aows the use of arbitrary shaped ces (tetraedra, hexahedra, prisms, pyramids...) incuding non conforming meshes [2]. 2.2 Governing equations and physica modeing The CFD modue of the NEPTUNE software patform [3] based on the two-fuid approach [4]. In this approach, a set of oca baance equations for mass, momentum and energy is written for each phase. These baance equations are obtained by ensembe averaging of the oca instantaneous baance equations written for the two phases. When the averaging operation is performed, the major part of the information about the interfacia configuration and the microphysics governing the different types of exchanges is ost. As a consequence, a number of cosure reations (aso caed constitutive reations) must be suppied for the tota number of equations (the baance equations and the cosure reations) to be equa to the 4248

3 number of unnown fieds. We can distinguish three different types of cosure reations: those which express the inter-phase exchanges (interfacia transfer terms), those which express the intra-phase exchanges (moecuar and turbuent transfer terms) and those which express the interactions between each phase and the was (wa transfer terms). The baance equations of the two-fuid mode we use for twophase boiing fows and their cosure reations are described in the foowing subsections. Main set of baance equations The two-fuid mode we use for our two-phase boiing fow computations consists of the foowing baance equations. Two mass baance equations:. V, v (1) t where t is the time,,, V denote the fraction of phase, its averaged density and veocity. The phase index taes the vaues for the iquid phase and v for vapour bubbes. Two momentum baance equations: V. V V p M V g. R, v, (2) t where p is the pressure, g is the gravity acceeration, M is the interfacia momentum transfer per unit voume and unit time, and and R denote the moecuar and turbuent stress tensors, the atter being aso caed the Reynods stress tensor. The wa friction terms for the two phases do not appear in the momentum baance equations because soid was are ony present at the boundaries of the fow domain and the wa friction is expressed through the wa boundary conditions. Two tota enthapy baance equations: V h t 2 ' A q. i w 2. h V 2 T q q, v 2 V p g. V t h i 2 V 2 where h is the phase-averaged enthapy for phase and h i is the interfacia-averaged enthapy. We have assumed that the two phases are governed by the same averaged pressure fied p and we mae no distinction between the pressures in the two phases or between the bu pressure and the interface ' pressure for simpicity. The three terms,m and A i denote the interfacia transfer terms of mass, momentum and heat, the quantity A i being the interfacia area concentration. The term qw denotes the T wa-to-fuid heat transfer per unit voume and unit time for each phase. The two terms q and q denote the moecuar and turbuent heat fuxes inside phase. The wor of viscous stresses of momentum is negected in the enthapy equation. The interfacia transfer of momentum M appearing in the RHS of Eq. (2) is assumed to be the sum of four forces: M M M M M (4) D AM L TD The four terms are the averaged drag, added mass, ift and turbuent dispersion forces per unit voume. The turbuent transfer terms for the iquid phase are cacuated by using the Reynods Stress Transport Mode SSG [5-6-7]. (3) 4249

4 Interfacia transfer terms If the mechanica terms are negected in comparison to the therma terms in the averaged form of the energy jump condition, this condition reduces to: h q A. (5) 0 i i i This important reation (together with the mass jump condition ) aows to compute the mass transfer terms as functions of the interfacia heat transfer terms q A i i and the interfacia-averaged enthapies h i : q i q vi v Ai. (6) hvi hi We have no information about the dependence of the interfacia- averaged enthapies h i. Therefore, two basic assumptions can be made: (1) the interfacia-averaged enthapies h i are identified to the phaseaveraged ones h or (2) the interfacia-averaged enthapies h i are given by the saturation enthapies. Here we have made the assumption (1). Each interfacia heat transfer term q A i i is the product of the interfacia heat fux density: qi Ci Tsat ( p) T, (7) where C i, T and T sat (p) denote a heat transfer coefficient, the averaged temperature of phase and the saturation temperature. The interfacia area concentration is expressed as A i 6 / d, where is the void fraction and d is the mean bubbe diameter. The foowing heat transfer coefficient is used: Nu 1 2 V v V d C Nu 2 0.6Re Pr 1 3 Re b ˆ Pr ˆ, (8) i d a where Reb is the bubbe Reynods number and Pr is the iquid Prandt number, being the iquid inematic viscosity. The heat transfer coefficient between the vapour and the interface for the case of bubbes is written as: vc pv Cvi Ai, (9) tc where C pv is the gas heat capacity at constant pressure and t c is a characteristic time given by the users. This reation simpy ensures that the vapour temperature T v remains very cose to the saturation temperature T sat, which is the expected resut for bubby fows with sufficienty sma bubbes (fow in a PWR core in conditions cose to nomina). Wa transfer mode for nuceate boiing In a first simpified approach, and foowing the anaysis of Kuru at a. [9], the boiing heat fux is spit into three terms: a singe phase fow convective heat fux q c at the fraction of the wa area unaffected by the presence of bubbes, a quenching heat fux q q where bubbes departure bring cod iquid in contact with the wa periodicay, a vaporisation heat fux q e needed to generate the vapour phase. Each of these three phenomena is expressed by a heat fux density (per unit surface of the heated wa) which is reated to the voumetric heat fux by the foowing reation: Aw Aw q w q w qc qq qe, (10) V V 4250

5 where A w is the heated wa surface in contact to the ce having voume V, therefore q w is expressed in W/m 3 and q w as we as q c, q q and q e are expressed in W/m 2. The quantities q c, q q and q e denote the heat fux densities due to iquid convective heat transfer, quenching and evaporation respectivey. The iquid convective heat transfer per unit surface of the heated wa is written as: qc Ac hog T w T, (11) where T w is the wa temperature and h og is a heat exchange coefficient which is given by: * u hog C p, (12) T where u * is the wa friction veocity and T + is the non-dimensiona iquid temperature. The veocity u * is cacuated from the ogarithmic aw of the wa written for the iquid veocity in the wa boundary ayer. The non-dimensiona temperature foows a simiar ogarithmic profie. The heat fux density due to quenching is written as: 2 T w T qq Abt q f, (13) atq where A b is the wa fraction occupied by bubbe nuceation, f is the bubbe detachment frequency, t q is the quenching time and a is the iquid therma diffusivity. The two fractions A c and A b are given by: 2 A b min1,n d d / 4, (14) Ac 1 A b where n is the active nuceation sites density (per unit surface of the heated wa) and d d is the bubbe detachment diameter. The active nuceation sites density is modeed according to Kuru et a. [9]: T 1. 8 w T n, 210 sat as a function of the wa superheating. The bubbe detachment diameter is given by the correation from Una et a. [12]. The Una s correation is vaid for subcooed iquid but has been extended to saturated iquid. The bubbe detachment diameter is given by: a d d p, (16) b where p is the pressure and a, b and are given by the foowing reations: a T T w sat s, 2v as where s and a s denote the wa conductivity and therma diffusivity, v denotes the vapour density and is the atent heat of vaporisation. In the modified correation, b is given by: T sat T St v / b, (18) 1 qc qq qe St v / C p V where V is the norm of the iquid veocity and St is the Stanton number which is defined by: (15) (17) 4251

6 qc qq qe St ˆ, (19) C V T T p sat and the quantity appearing in Eq. (23) is given by : V 0.47 max 1, V 0.61 m / 0 s V, (20) 0 The quenching time and the bubbe detachment frequency are modeed as: t q f f 4 g 3 d v 1 /, (21) The third heat fux density q e used for evaporation is given by: q e dd f vn. 6 3 d The superheat of the iquid (dedicated to the sodium) is taen into account by: P=PvPI=2/r B, which gives T sat =TvT sat =P T sat v inside a bubbe of radius r B. = 2 r B T sat v, where Pv is the pressure Wa function for boiing fow In subcooed fow boiing, the iquid veocity profie in the boundary ayer is significanty disturbed by the bubbe formation and detachment mechanisms on the heated wa. In the iterature, an over-prediction of iquid and gas veocity distributions in the boiing boundary region has been reported. The use of singe-phase wa aw may be one of the main reasons for these resuts. Foowing Ramstorfer et a. [13], Mimouni et a. [6] suggested a wa function for boiing fows. When the void fraction tends to zero, the wa aw tends to the singe-phase formuation. Furthermore, this reation depends on bubbe diameter and bubbe density at the wa (void fraction at the wa), which is physicay expected. By using the Van Driest formuation : + = (1exp( )) 2 2 Where u + is due to the roughness induced by bubbes created at the wa : 0; + u + r 11.3 = 1 n(1+c r + r ) ; + r >11.3 with C r =0.5 and + r (simiar to a Reynods number) expressed as : + r = r u w u,where u 14 =c 1/2, u w the friction veocity and r = v D. 3. VALIDATION FOR ADIABATIC BUBBLY FLOWS AND BOILING FLOWS Since the maturity of two-phase CFD has not reached yet the same eve as singe phase CFD, an important wor of mode deveopment and thorough vaidation is needed. Many of these appications invove bubby and water boiing fows, and therefore it is essentia to vaidate the software on such configurations. Four experiments were seected for the vaidation. The Liu and Banoff experiment (1993) is an adiabatic air-water bubby fow inside a vertica pipe. It aows to vaidate forces appied to 4252

7 the bubbes. The Be F'Dhia and Simonin (1992) experiment is an adiabatic bubby air-water fow inside a sudden pipe expansion. It aows to vaidate the dynamic modes and turbuence. The DEBORA and the ASU faciities provide resuts for boiing fows inside a vertica pipe. The woring fuid is refrigerant R12 for DEBORA and R113 for ASU. Both aow to vaidate the nuceation modeing on a heated wa, and ASU aows aso the vaidation of the two-phase wa function. A ey feature of this wor is that a these computations were performed with a singe and consistent set of modes. Douce et a. [14] have shown that the physica modes impemented in NEPTUNE_CFD have captured experimenta profies with reasonabe accuracy. 4. HEATED ROD IN A VERTICAL ANNULUS For inear eddy viscosity modes, we can define the turbuent Prandt number Prt as the ratio of the turbuent viscosity to turbuent therma diffusivity anaogous to the moecuar Prandt number. In the foowing, the turbuent terms are cacuated by using the R ij SSG mode. But, the cosure aw for the turbuent heat fux is the same as the one used for inear eddy viscosity modes. Moreover, the turbuent Prandt number is assumed to be a constant. Modeing the turbuent heat fuxes by introducing Prt as constant can resut in a significant inaccuracy reated to the transferred heat in the case of turbuent fows of ow Prandt number. But, the objective of this section is to evauate the discrepancies in a singe phase fow of turbuent fows of ow Prandt number. The fina objective concerns sodium boiing fows (ow Prandt number) and the experience gained over severa decades in boiing fows shows that crucia phenomena in singe phase fow is iey to become of second order compared to others crucia phenomena occurring in boiing fows. Figure 1 dispays the experimenta setup of a generic experiment conducted in KALLA [15]. Hereby, a rod heated with a constant heat fux is concentricay embedded in a pipe fow. More detais on the experimenta set-up may be taen from [15]. Regarding the inet condition, an isotherma turbuent hydrauicay fuy deveoped pipe fow with a radius R0=0.03 m. Figure 1: Schematic diagram of the geometric setup of the heated rod simuations The rod is uniformy heated with Q= 3W (q w =Q (2 4 R i ) W/m 2 ) at position z*=0 with z*=z/2r i aong 4. It is foowed by an unheated zone, which has been chosen in the numerica simuations to a ength 6. The different engths are given in tabe I. Tabe I. Separate effect and integra 0.35m m 0.028m 0.86m 0.34m The Reynods number of the investigated case is Re Dh = (v inet =0.52 m/s). The inet temperature is T inet = 424 K. The fow is assumed to be axi-symmetric therefore a two-dimensiona axi-symmetric meshing is used. The thermophysica properties of the fuid are given in tabe II [14]. 4253

8 Tabe II. Thermophysica properties of the fuid Density =.. / Viscosity = 0.497exp(741 ) 10 3 Therma conductivity = /() Heat capacity = /( ) Figure 2 and Figure 3 represent the veocity profies and the temperature profies aong the y-direction at different ocations z*. A reasonabe agreement is obtained for the veocity profies but the iquid temperature is over-estimated near the wa with Neptune_CFD. As a consequence, for industria appications, Neptune_CFD overestimates the occurrence of the case when the sodium cooant is boiing within the fissie zone. In others words, this overestimation is favourabe for safety. Nevertheess, the modeing of the turbuent heat fux shoud be improved in further computations in order to reduce safety margins. Figure 2: Comparison of the veocity profies as function of y/l for Re = Figure 3: Comparison of the temperature profies at position z* = 7.55 and z* = 14.7 over y/l 5. CHARLETY S EXPERIMENT Sodium boiing tests in forced convection has been carried out by Charety et a (1970) 45 years ago at CEA [17]. The test section was a heating pin of 6.6 mm, 600 mm ong inside a tube of 8.6 mm foowed by a non heated ength of 6 mm and 450 mm ong and 4 mm (Figure 4). The parameter range was in order to simuate the Phenix reactor pressure distribution regarding heat fux (between 80 and 200 W/cm2), outet pressure (1.2 bar), mass fowrate (from 10-4 g/s to g/s),and inet temperature (450 C). The fow is assumed to be axi-symmetric therefore a twodimensiona axi-symmetric meshing is used. 4254

9 Figure 4: : Genera view of the experiment; pressure measurements are ocated in P4, P5, P6 and P7 The objective of this section is to compare the measured vaues and the simuated vaues for the interna characteristic. The interna characteristic is the variation of pressure drop over the channe induced by singe and/or two-phase fow when the inet fow rate is varied under constant power, constant outet pressure, and constant inet temperature. For a ow pressure sodium fow, the shape of the interna characteristic affects the form of a S-curve, due to the physica properties of sodium [1]. In singe phase fow (arge vaues of mass fow rate) pressure drops exhibit an increase when the mass fow rate increases (frictiona pressure drops increase whereas gravitationa pressure drops are negigibe). In boiing regime (ow vaues of the mass fow rate), gravitationa pressure drops decrease (because of the density) whereas frictiona pressure drops increase (conservation of the mass fow rate). The boiing onset point corresponds to the minimum of the S-curve. In good agreement with the behaviour described above, Figure 6 and Figure 7 represent the interna characteristic of the test section for two vaues of the heat fux of the heating pin. Experimenta vaues and numerica resuts are in reasonabe accordance. Moreover, the prediction of the boiing onset is particuary encouraging. 4255

10 Void fraction Figure 5: Evoution of the void fraction in the computationa domain. Figure 5 represent the evoution of the void fraction. It is of reevance interest to see that ebuition occurs at the end of the heated ength. The pressure drops beow the vapour pressure eading to iquid vaporization. As a consequence, the main phenomenon is mainy due to the autovaporisation of sodium instead of ebuition in ces adjacent to the heated ength. Figure 6: Pressure profies vs mass fow rate at inet. Case Q=10W. Figure 7: Pressure profies vs mass fow rate at inet. Case Q=15W HEATING PIN BUNDLE A reasonabe agreement has been obtained on sodium boiing in singe channe configuration in the previous section. It is now interesting to go further and to assess the capabiities of the CFD too for test sections more representative of fast reactor sub-assembies. Woring to this end, the GR19 19 heating pin bunde has been deveoped at CEA [19]. Figure 8, Figure 9 and Figure 10 give a view of the compexity of the geometry simuated with Neptune_CFD. In this section, Neptune_CFD is couped with SYRTHES code in order to dea with heat transfer at the fuid/soid interface. The therma code SYRTHES 4256

11 deveoped at EDF soves the energy conservation in a soid from the distribution of heat in a pin [18]. The iquid veocity at inet is 5 m/s. The inet iquid temperature is 400 C. Figure 11 represents the iquid temperature and Figure 12 represents the iquid veocity in the bunde. Figure 8: Genera view of the computationa domain (ength=2.31m). Above the part where the fue is ocated, it exists an adiabatic part (without fue) with a arge vaue of the hydrauic diameter. Figure 9: Geometry introducing the wrapped wire (imited in this view to one heix and ony 7 pins). The pitch-over-diameter ratio P/D and the heix-overdiameter ratio H/D is P/D=1.1 and H/D=

12 Figure 10: Genera view of the computationa domain : about 4 Miions of ces for the fuid and 12 M of ces for the soid part. Figure 11: Liquid temperature Computations show a pressure drop at the end of the heated ength due to the sudden increase of the hydrauic diameter. Thus, the pressure can drop beow the vapour pressure eading to iquid vaporization. This preiminary resut supports the assumption of boiing in the upper subassemby zone (by autovaporization) which coud possiby ead to a sodium boiing stabiization. 7. CONCLUSION In order to evauate the wa aw impemented in the CFD too, computations have been compared with KALLA experimenta resuts obtained in the case of a rod heated with a constant heat fux which is concentricay embedded in a pipe iquid meta fow (singe-phase fow). Secondy, the incipient boiing superheat of sodium is quite different from that of conventiona fuids. As a consequence, the nuceate boiing mode has been improved and vaidated against the Charety s experiment where a rod heated with a constant heat fux is concentricay embedded in a pipe sodium fow. For different vaues of the heat fux, the pressure is measured at different ocations as function of the mass fow rate. A reasonabe agreement has been reached which is very encouraging for further appications. Finay, preiminary computations have been carried out in an assemby constituted of 19 pins equipped with a wrapped wire. Computations are quaitativey in accordance with the assumption of boiing in the upper subassemby zone which coud possiby ead to a sodium boiing stabiization. In further computations, numerica resuts wi be compared to partia experimenta resuts avaiabe in order to reproduce the interna characteristic and to exhibit a sodium boiing stabiization. 4258

13 Figure 12: Liquid veocity Figure 13: Axia profie of the iquid veocity. Comparison between Code_Saturne (deveoped by EDF R&D, the code is abe to sove steady or transient, singe phase, incompressibe, aminar or turbuent fows) and Neptune_CFD ACKNOWLEDGMENTS The authors are gratefu to the projects GEN IV (EDF) and NEPTUNE. The NEPTUNE project is funded by EDF (Eectricité de France), CEA (Commissariat à Energie Atomique et aux Energies Aternatives), AREVA-NP and IRSN (Institut de Radioprotection et de Sûreté Nucéaire). The authors are gratefu to C. Penigue who has shared a the necessary inputs for the industria appication. REFERENCES 1. J.M. Seier, D. Juhe, Ph. Dufour Sodium boiing stabiisation in a fast breeder subassemby during an unprotected oss of fow accident, Nucear Engineering and Design 240 (2010) N. Mechitoua., M. Boucer., J. Laviévie, J.-M. Hérard, S. Pigny, G. Serre, An unstructured finite voume sover for two phase water/vapour fows based on an eiptic oriented fractiona step method. Proceedings of the 10th Int. Top. Mtg. Nucear Reactor Therma Hydrauics (NURETH 10), Seou, Repubic of Korea, (2003) 3. A. Guefi, D. Bestion, M. Boucer, P. Boudier, P. Fiion, M. Grandotto, J.-M. Hérard, E. Hervieu, P. Péturaud, NEPTUNE - A New Software Patform for Advanced Nucear Therma-Hydrauics, Nucear Science and Engineering, 156, pp (2007). 4. M. Ishii, «Thermo-fuid dynamic, theory of two phase», Eyroes, coection de a direction des Etudes et recherches d Eectricité de France, C.G. Speziae, S. Sarar, and T.B. Gatsi, Modeing the pressure-strain correation of turbuence: an invariant dynamica systems approach, 1991, J. Fuid Mech., pp. 227: ISSN

14 6. S. Mimouni, F. Archambeau, M. Boucer, J. Lavievie, C. More, A second order turbuence mode based on a Reynods Stress approach for two-phase boiing fow. Part 1 : Appication to the ASU annuar channe case, Nucear Engineering and Design, Voume 240, Issue 9, September 2010, Pages S. Mimouni, F. Archambeau, M. Boucer, J. Lavievie, C. More, A second order turbuence mode based on a Reynods Stress approach for two-phase boiing fow and appication to fue assemby anaysis, Nucear Engineering and Design, Voume 240, Issue 9, September 2010, Pages N. Zuber, M. Ishii Drag coefficient and reative veocity in bubby, dropet or particuate fows, 1979, AIChE Journa, pp N. Kuru and M.Z. Podowsi, Mutidimensiona effects in forced convection subcooed boiing, Proceedings of the Ninth Internationa Heat Transfer Conference Jerusaem, Israe, August (1990), pp (1990). 10. N. Zuber, On the dispersed two-phase fow in the aminar fow regime, Chem. Eng. Sc., No. 19, p. 897 (1964). 11. A. Tomiyama, et a Transverse migration of singe bubbes in simpe shear fows, 2002, Chem. Eng. Sci., pp H.C. Una, Void fraction and incipient point of boiing during the subcooed nuceate fow boiing of water, Internationa Journa of Heat and Mass Transfer 20 (1977), B. Ramstorfer, H. Breitschade, G. Steiner, Modeing of the near-wa iquid veocity fied in subcooed boiing fow, Proceedings of the ASME Summer Heat Transfer Conference San Francisco, CA, Juy (2005) HT (2005). 14. A. Douce, S. Mimouni, M. Guingo, C. More, J. Laviévie, C. Baudry, Vaidation of NEPTUNE_CFD for adiabatic bubby fow and boiing fow, submitted to Nucear Engineering and Design. 15. R. Stiegitz et a. Heavy iquid meta technoogies deveopment in Kaa 2006, Proceedings of ICAPP, Vo. Paper-id Handboo on Lead-Bismuth Eutectic Aoy and Lead, Properties, Materias, Compatibiity, Thermahydrauics and Technoogies. Report No. 6195, 2007, OECD-AEN/NEA. 17. P. Charety, «Ebuition du sodium en convection forcée», Facuté des sciences de 'université de Grenobe : s.n., In french. 18. C. Pénigue, Conjugate heat transfer study of a wire spacer SFR fue assemby with the therma code SYRTHES and the CFD code Code_Saturne, ICAPP 2014 Charotte, USA, Apri 6-9, 2014 Paper B. Menant, Nuceation, dry-out and boiing behind a sodium tight-oca bocage in a 19-pin bunde with heica wire spacers, Liquid meta boiing woring group, 7 th Meeting, Petten, 1-3 June 1977, (1977). 4260