Volumen II - SAM

2. PRESENTATION OF THE MODEL AND THE NUMERICAL TOOLS FOR Zr ALLOYS
OXIDATION AT HIGH TEMPERATURES
Two types of models have been used to describe the diffusion phenomena in the cladding tubes
during a LOCA scenario. An analytical model previously used in [2] and a new numerical model
adapted from [3].
Analytical diffusion-reaction model
In order to predict the kinetics of phase transformation and oxygen profiles inside the cladding tubes it is
necessary to simplify the system. During the oxidation of βZr, we observe the diffusion of oxygen and the
transformation β−›α.Oxidation is treated as a diffusion process with mobile interfaces. Thus, the following
hypotheses are required:
1) The only diffusing component considered is oxygen.
2) In the oxide, oxygen goes across the oxide layer via the anionic vacancies of oxygen. The vacancies
are the predominant defects which can be found in non stoichiometric monoclinic zirconia.
3) The diffusion of oxygen in the metal takes place via the interstitial mechanism in the αZr and βZr
structures.
4) Local thermodynamic equilibrium conditions are assumed: that is the oxygen or zirconium chemical
potentials are equal in the two faces on both sides of the interface.
5) In a given phase, the diffusion coefficients are independent of the concentration of the diffusing
element.
6) In a first approximation, the problem has been approximated to planar geometry.
7) A semi-infinite system is considered
In 1966, Debuigne [4] solved the oxygen diffusion problem for Zr-O system using an analytical resolution.
The oxygen concentration in each phase is analytically described in details by X. Ma et al [2]. The growth
rate of the phase is linked by the velocity of the α/β interface, which can be deduced from the oxygen
balance at the α/β interface. For details see [2].
Fig. 2 presents a schematic diagram of the oxygen profile and boundary conditions at each interfaces
considered in the problem.
Fig.2. Schematic diagram of Zy-4 oxidized at temperatures above the α−β transition temperature.
Numerical diffusion-reaction model
EKINOX. “EKINOX” [3] (Estimation KINetics OXidation) is a model written in FORTRAN developed in
CEA for high temperature nickel oxidation. In the present work this model was adapted for Zr-alloys in order
to simulate the oxidation at high temperature. The important technological point of EKINOX in the LOCA
frame situation is that it is possible to calculate the evolution of oxygen profiles taking into account the finite
size of the cladding tubes in contrary to the analytical model (see hypothesis 7).
General description The numerical model EKINOX is a one dimensional model that simulates the growth
of an oxide layer using a simple explicit finite differences method for the time integration algorithm [3,5].
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