1. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 1 RADIATION IN POROUS MEDIA: AN UPSCALING METHODOLOGY APPLIED TO A REACTOR NUCLEAR CORE Estelle Iacona, Jean Taine and Fabien Bellet Energétique Moléculaire et Macroscopique, Combustion E.M2.C Ecole Centrale Paris - UPR 288, CNRS

2. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 2 AXES DE RECHERCHE EM2C COMBUSTION 8 ECP Candel S. Darabiha N. Fiorina B. Gicquel O. Massot M Rolon J.C Richecoeur F. Schuller Th. 4 CNRS Ducruix S. Laurent-Nègre F. Veynante D. Zimmer L. IR CNRS : Durox D. Lacoste D. Scouflaire Ph. NANO-OPTIQUE ET NANO-THERMIQUE NANO-OPTIQUE ET NANO-THERMIQUE 3 ECP1 CNRS Greffet J.-J. Volz S. Laroche M. Marquier F. PLASMAS HORS ÉQUILIBRE PLASMAS HORS ÉQUILIBRE 1 ECP1 CNRS Laux Ch.Bourdon A. IR CNRS : Lacoste D. RAYONNEMENT ET TRANSFERTS COUPLÉS RAYONNEMENT ET TRANSFERTS COUPLÉS 4 ECP3 CNRS Taine J. Perrin M.Y. Bellet F. Rivière Ph. Goyeau B.Soufiani A. Iacona E.

3. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 3 Carbon foam (porosity 0.93) for some fuel cells (SOFC) Mullite foam (porosity 0.85) for catalytic combustion Some applications of radiation in porous media Combustible grape for nuclear reactor core - AREVA

4. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 4 Outline Objectives Up scaling method : a direct identification method Application to real porous media

5. Problem : Temperature field in the medium? Coupled heat transfer : - convection in pores (fluid phase) - conduction in the fluid and in the solid phases - radiation : Accurate calculations required in many applications high temperature applications Local scale transfer : unaffordable (Large computer time and memory) dz

6. Problem  Alternative : up scaling method  model of an equivalent semi transparent continuous medium ` => Radiative properties ? Validity? Medium structure statistically known Local radiative properties known dz extinction coefficient : albédo (diffusion) : Diffusion phase function : Diffusion  Absorption   Extinction   +

7. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 7 parameter identification method : some drawbacks  assumed semi transparent medium model (no validity criterion)  indirect method of characterization (radiative transfer model required to analyze experiments)  accuracy on the determined radiative properties difficult to estimate error associated with the semi transparent model ? accuracy of the radiative transfer model ? accuracy of the identification technique ?

8. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 8 Outline Objectives Up scaling method : a direct identification method Application to real porous media

9. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 9 Objectives From the statistical knowledge of the porous medium structure and its local radiative properties: calculate the radiative properties of a potentially equivalent semi-transparent medium : - nonisotropic extinction coefficient  - nonisotropic absorption coefficient  - scattering phase function p  with a direct simulation using a Monte-Carlo method.

10. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 10 Definitions and assumptions Porous medium statistically isotropic or anisotropic Porous medium statistically homogeneous or nonhomogeneous Diffraction : neglected ( <<D) Solid phase : opaque or semi transparent Fluid phase : transparent or semi transparent

11. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 11 Definition : Extinction =absorption+scattering At local scale: probability of reaching the interface (non spectral, only geometric property) Statistical approach of radiation u r I s0s0 (semi-transparent medium) linked to the cumulated distribution function of chord lengths

12. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 12 Monte-Carlo method Typically 10 9 ramdom rays any ray : 1 random original point r into the fluid phase 1 random direction  impact at the solid interface  Calculation of the extinction distance : s 0 =rI  Calculation of :  the normal vector  the impact angle at the solid interface   Deduction of the scattering angle : contribution to the phase function u r I n 

13. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 13 Identification criterion  e : Statistical approach of radiation Radiation Distribution Function Identification Method (RDFI method) g e (s,u k ) G e (s,u k ) s (mm) 0.95 useful extinction optical thickness range  s = 0  s = 3 Extinction coefficients calculated from identification of G e (s) with g e (s) with mean square method

14. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 14 Outline Objectives Up scaling method : a direct identification method Application to real porous media

15. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 15 IUSTI from ESRF X ray tomography spatial resolution of 5  m 3D Numerical image of a mullite foam sample issued from a tomography Tomography resolution mid scale (wall pores) Local scale  S   mm -1 and  S  mm - 1, p S 

16. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 16 IRNS : French Radiation and nuclear safety institute Cooling fluid leaking Increase of temperature Degradation, fusion et geometrical modification of the core T<500K Nuclear reactor core in severe accident conditions

17. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 17 2D of a cross section ( density scale in g/cm 3 ) Degraded small scale nuclear core rod bundle Geometry obtained from  ray tomography experiment FPT1, IRSN, Cadarache 3D reconstruction

18. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 18 Numerical image of the whole degraded bundle Walls assumed opaque at local scale :  = 0.8 (Chalopin et al., 2008) z Degraded small scale nuclear core rod bundle Geometry obtained from  ray tomography experiment FPT1, IRSN, Cadarache

19. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 19 A:  =0.19 D:  =0.24 Bain fondu + cavité C:  =a x ɛ +b B:  =0.28

20. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 20 Radiative transfer in a nuclear reactor core Calculated from the obtained radiative properties of the equivalent medium Radiative conductivity model : < 0. 2 For an optically thick REV from the absorption point of view

21. E. Iacona, J. Taine, F. Bellet Laboratoire EM2C - CNRS - ECP 21 conclusions General statistical approach of radiation :  Accurate determination of G e, P a, P s and p for any porous medium REV Equivalent semi-transparent media :       and  p by the Radiative Distribution Function Identification (RDFI) method  Validity of the semi transparent medium model : all porous media can ’ t be modeled by semi transparent media  Direct determination method radiative properties directly obtained from their definitions, without use of a radiative transfer model based on the knowledge of - the porous medium morphology (tomography) - the radiative properties at the local scale (less than the spatial tomography resolution)