A simple model of fission gas intra-granular behaviour in UO 2

Transcription

1 FAIRFUELS Workshop 011 KTH University, Stockholm, Sweden, February 9-10, 011 A simple model of fission gas intra-granular behaviour in UO G. Pastore*, P. Van Uffelen, L. Luzzi * giovanni1.pastore@mail.polimi.it

2 Kinetics of fission gas /15 Fission reactions Athermal release Intra-granular gas (single atoms) Generation of fission gases (Xe, Kr) iffusion Inter-granular gas (single atoms) Trapping Resolution Intra-granular gas (s) iffusion Inter-granular gas (s) Intra-granular swelling Inter-granular swelling Interconnection Thermal gas release (to the free volume)

3 Aim of the work 3/15 To develop a simple model of the intra-granular behaviour of fission gas in UO fuel, suitable for implementation in an integral fuel performance code (TRANSURANUS). onsidered phenomena: iffusion of fission gas from within the fuel grains to the grain boundaries (intra-granular diffusion) Swelling due to growth of fission gas s inside the fuel grains (intra-granular swelling) Model basic requirements: 1. To be physically based. To be simple (employment in a fuel performance code) 3. To provide a reasonably accurate description of both normal and transient operation reactor conditions

4 Intra-granular diffusion introduction 4/15 urrent treatment of the intra-granular diffusion problem in TRANSURANUS Premises: Intra-granular gas exists in two different phases: single atoms and s Bubbles are immobile gas transport is solely due to diffusion of single atoms Bubbles trap migrating gas atoms Gas atoms are knocked back from s by the passage of fission fragments (resolution)

5 iffusion formulation of Speight (1) 5/15 Rate equations describing the variation of the gas concentration in a grain: Addition g b g b Quasi-stationary approach: g = trapping parameter [s -1 ] b = resolution parameter [s -1 ] β = gas generation rate [atoms/(m 3 s)] b g Then, the diffusion equation is: Average fraction of gas in single atom phase eff,s eff,s b b g Effective diffusion coefficient (Speight, 1969) at present empirical in TRANSURANUS where eff,s is assumed to be uniform in a grain

6 Fraction of gas in solution [/]. iffusion formulation of Speight () 6/15 Preliminary study of eff,s eff,s b b g b g 3.03 F 4 l f R R N average Z 0 1,0 0,9 b/(b+g) 0,8 0,7 0,6 0,5 0,4 0,3 0, 0,1 0,0 0,4 0,6 0,8 1 1, 1,4 1,6 1,8 Inverse Temperature 1000/T [1/K] Tendency to underpredict gas transport to the grain boundaries under high temperature and transient conditions Bubble mobility?

7 iffusion random motion (1) 7/15 By applying the second Fick s law to the concentration of s: (1) N r.m. N The concentration of gas in phase is: () with (3) m N m 0 (quasi-stationary approach) N = concentration of s [s/m 3 ] m = number of gas atoms per [atoms/s] = concentration of gas in s [atoms/m 3 ] = diffusion coefficient [m /s] ombining Eqs. (1), () and (3), we obtain: r.m. m N Hypothesis: m is uniform in a grain (all s contain the same number of atoms) r.m.

8 iffusion random motion () 8/15 Rate equations describing the variation of the gas concentration in a grain in presence of random motion: Addition g b g b Again, by applying the quasi-stationary approach the diffusion equation takes the form: eff eff b b g Bubble diffusion term No modifications in the form of the equation Average fraction of gas as single atoms g b g Average fraction of gas in s

9 iffusion coefficient [m /s] iffusion coefficient [m /s] iffusion random motion (3) 9/15 Preliminary study of eff eff b b g g b g 1,0E-1 1,0E-14 1,0E-1 Single atom diffusion (Speight) diffusion 1,0E-16 1,0E-18 1,0E-0 1,0E-14 Gas diffusion coefficient (Evans, 1994) 1,0E- 1,0E-16 1,0E-18 1,0E-0 3 v R 1,0E- 1,0E-4 1,0E-6 0,4 0,6 0,8 1 1, 1,4 1,6 1,8 Inverse Temperature 1000/T [1/K] 1,0E-4 1,0E-6 0,4 0,6 0,8 1 1, 1,4 1,6 1,8 Inverse Temperature 1000/T [1/K] Effective diffusion coefficient eff Effective single atom diffusion coefficient b/(b+g) (Speight) The contribution of motion to the effective diffusion coefficient is significant above ~1500 (consistent with observations of motion by Baker, 1977)

10 iffusion power transients (1) 10/15 Enhanced gas transport to the grain boundaries is observed during transients ( directed motion in a temperature/vacancy gradient). Empirical approaches have been used to consider an additional release of gas during transients (Koo et al, 1999). Bubble velocity up to a temperature/vacancy gradient (Nichols, 1969; Evans, 1994): v v vqvt kt onsider directed motion towards the grain boundary starting at a distance r from the boundary. Approximating ΔT = T/r (Evans, 1994), we obtain: vq kt r v Attempt to represent the effect of directed diffusion by means of the random diffusion equation: t directed r kt Q v v,tr = t Qv kt r.m.s. v r,tr (Evans, 1994) We obtain an equivalent diffusion coefficient to be adopted during transients. v = velocity [m/s] v = volume self-diffusion coefficient [m /s] Q v = heat of transport of vacancies [ev] k = Boltzmann constant [ev/k] T = temperature [K] r = starting distance from the grain boundary [m]

11 iffusion coefficient [m /s] iffusion power transients () 11/15 Preliminary study of eff,tr eff,tr b b g Single atom diffusion (Speight) eff,tr g b g,tr diffusion In this simplified approach, the enhanced gas transport during power transients is simulated by modification of the effective diffusion coefficient ( diffusion term) 1,0E-1 1,0E-14 1,0E-16 1,0E-18 1,0E-0 1,0E- 1,0E-4 1,0E-6 0,4 0,6 0,8 1 1, 1,4 1,6 1,8 Inverse Temperature 1000/T [1/K] Effective diffusion coefficient eff (normal operation) Effective diffusion coefficient eff (transients) The concept of representing the effect of directed motion by an equivalent random diffusion equation is intended as a practical approach (easy implementation) for a first estimation of the enhanced gas transport during power transients

12 Swelling model assumptions 1/15 I. All s have the same size and the same (spherical) shape. It follows that the fractional volume of intra-granular swelling is given by: V N 3 R V 4 3 II. The concentration of gas s N remains constant in time (N = s/m 3 ) III. The s are in mechanical equilibrium with the bulk solid. If the effect of hydrostatic stress is neglected (nm-size s), the gas pressure p in the is calculated as: p R IV. The gas in the s obeys the van der Waals equation of state: pv bm N AV kmt where V is the volume [m 3 ] and m is the number of gas atoms per [atoms/]

13 Swelling calculation scheme 13/15 alculation at time step t: iffusion equation (t) (t) (t dt) d g b g (t) (t) Total concentration of intra-granular gas [atoms/m 3 ] oncentration of gas retained in the s [atoms/m 3 ] m(t) N (t) Number of gas atoms per [atoms/] Equation for the radius [m]: 8 R 3 (t) 3km(t) T(t) R (t) 6 b m(t) N AV 0 Intra-granular swelling (increment of fractional increase in fuel volume) [/]: V V (t) N R (t)

14 onclusions 14/15 A simple but physically based model of fission gas intra-granular behaviour is proposed, which is intended for adoption in fuel performance codes (TRANSURANUS). For the treatment of the fission gas intra-granular diffusion, the concept of an effective diffusion coefficient (Speight) is maintained. A simple extension of the Speight s formulation is proposed, which takes into account the contribution of random motion to the al gas diffusion. A specific expression for the diffusion coefficient is proposed for the simulation of power transients ( directed motion). The contribution of motion may explain the tendency of the Speight s formulation to underpredict gas transport to the grain boundaries under high temperature and transient conditions. The incorporation in the TRANSURANUS code of the above concepts, as well as of a simple model for the intra-granular swelling, is underway.

15 Main references K. Lassmann, A. Schubert, P. Van Uffelen,. Győri, J. van de Laar, TRANSURANUS Handbook, opyright , Institute for Transuranium Elements, Karlsruhe, Germany (009). M.V. Speight, "A alculation on the Migration of Fission Gas in Material Exhibiting Precipitation and Re-solution of Gas Atoms Under Irradiation", Nucl. Sci. Eng. 37, (1969). K. Lassmann, H. Benk, "Numerical algorithms for intragranular fission gas release", J. Nucl. Mater. 80, (000). J.A. Turnbull,.A. Friskney, J.R. Findlay, F.A. Johnson, A.J. Walter, "The diffusion coefficients of gaseous and volatile species during the irradiation of uranium oxide", J. Nucl. Mater. 107, 168 (198). J.A. Turnbull, R.J. White,.A. Wise, "The diffusion coefficient for fission gas atoms in uranium dioxide", Proc. of Technical ommittee Meeting on Water Reactor Fuel Element omputer Modelling in Steady State, Transient and Accidental onditions, International Atomic Energy Agency, Preston, England, T1-T-659, Paper 3.5 (1988).. Baker, "The migration of intragranular fission gas s in irradiated uranium dioxide", J. Nucl. Mater. 71, (1977). Y.-H. Koo, B.-H. Lee,.-S. Sohn, "OSMOS: a computer code to analyze LWR UO and MOX fuel up to high burnup", Ann. Nucl. Energy 6, (1999). J.H. Evans, "Bubble diffusion to grain boundaries in UO and metals during annealing: a new approach", J. Nucl. Mater. 10, 1-9 (1994).. F.A. Nichols, "Kinetics of diffusional motion of pores in solids", J. Nucl. Mater. 30, (1969). R.J. White, M.O. Tucker, "A new mechanistic model for the calculation of fission gas release", Proc. of International Topical Meeting on LWR Fuel Performance, West Palm Beach, Florida (1994). P. Lösönen, On the behaviour of intragranular fission gas in UO fuel", J. Nucl. Mater. 80, 56-7 (000)..R. Olander, Fundamental aspects of nuclear reactor fuel elements, Technical Information enter - Energy Research and evelopment Administration, University of alifornia, Berkeley (1976). S. Kashibe, K. Une, K. Nogita, "Formation and growth of intragranular fission gas s in UO fuels with burnup of 6-83 GWd/t", J. Nucl. Mater. 06, -34 (1993). M.E. Gulden, "Migration of gas s in irradiated uranium dioxide", J. Nucl. Mater. 3, (1967). Thank you for your attention