Volumen II - SAM

The activation energy value (207kJ) is in good agreement with the work of X. Ma et al (201.5KJ).
4. DISCUSSION
Kinetic parametrization
D(
T
In EKINOX simulations, the α/β interface displacement is directly linked to time integration by:
⎡
C
− C
⎤
⎥ ⋅ d ζ
⎦
⎢
⎣ α / β β / α α / β α β ⎠ ζ (4)
where Cα/β and Cβ/α are the equilibrium oxygen concentrations at the interface, in α and β phase
respectively, Jα and Jβ are the α and β phase flux and dζ is the α-growth in dt time.
In the analytical solution [2], the position of the interface is linked to a supplementary input data, chosen
independently from diffusion coefficients in the α and β phase. From eq. (4), it is possible to observe that the
interface displacement depends on diffusion coefficient in α and β phases via Jα and Jβ. Different simulations
allowed to determine that the α phase kinetic parameters had a more significant influence on the interfaces
displacement, due to Jα dependence upon both: β→ α and α→oxide transformation.
Comparison between numerical and analytical solutions
)
α
⎛ − 207kJ
⎞
= 1.
8652 ⋅ exp⎜
⎟
⎝ R ⋅T
⎠
In order to estimate the improvement brought by the non-stationary calculation with Ekinox, a comparison
with an analytical solution previously used in [2] was performed.
a) b)
Fig. 6. Comparison between O diffusion profiles in Zy-4 at 1200°C obtained using an analytical solution and
the EKINOX numerical code: a) α and β phases b) blow-up of the β phase region.
For this analytical solution the data needed are:
- Experimental Kp, parabolic constant giving the growth rate for each phase
- Equilibrium oxygen concentrations at the different interfaces.
Moreover, the analytical solution is limited to semi- infinite systems. For more details see [2].
For short simulation times, the analytical and simulation diffusion profiles are in good agreement (Fig. 6).
For longer simulation times (t >55sec), the effect of the finite size sample is evident: the oxygen
concentration profile increases at the end of the sample. This increase lowers oxygen concentration gradient
in the β phase and so, modifies the α/β interface velocity. This fact demonstrates the importance of taking
into account a finite size system (EKINOX) instead of semi-infinite system (analytical solution), in order to
reproduce experimental data.
Effects of equilibrium concentrations at the different interfaces on the oxygen diffusion profile
To evaluate the influence of thermodynamic data on the final oxygen diffusion profile, new calculation have
been performed using Chung-Kassner values and ThermoCalc values for Zy-4 and Zr pure given in Table 2.
Fig.7 shows that the choice of thermodynamic data set influences the diffusion in the β phase. This result
points out the importance of taking into account the nominal composition of the industrial alloys (each
1321
=
2 ( cm / s)
⎛
⎜
⎝
J
−
J
(3)
⎞
⎟
dt