Thermodynamic Modeling of the Uranium–Tellurium System: Estimation of the Uncertainties by a Bayesian Approach

Abstract

Under irradiation, the formation of fission products in the (U,Pu)O2 fuel with time has a substantial effect on its chemistry. In particular, migration of the most volatile fission products (Cs, Te, I, Mo) from the center to the periphery of the fuel pellet is induced by the large radial thermal gradient. To predict the thermodynamic properties of the irradiated fuel, thermodynamic modeling of the complex multi-component (Cs-I-Te-Mo)–(U-Pu)–O system is performed using the CALPHAD method. In this work, the thermodynamic assessment of the U–Te sub-system is performed. The literature review reveals a lack of experimental data as well as scattering and inconsistency of some of the data. In particular, no thermodynamic data exist on the liquid. From this review, input thermodynamic and phase diagram data are carefully selected. The Gibbs energy functions are then adjusted by fitting these data. An overall good agreement is obtained with all the selected data except for the enthalpy of formation for UTe which is underestimated by 13% by our model. This could be due to an inconsistency between the enthalpy of formation and vapor pressure data. In a second step, the uncertainties on the thermodynamic parameters and their propagation on the calculated thermodynamic and phase diagram data are estimated using a Bayesian approach. The analysis shows that there are too many parameters (22) for too few data points (120 points). The uncertainties are thus large on some of the calculated data. Moreover the inconsistency of some of the data and the lack of thermodynamic data for the liquid makes the model uncertain. New experimental data such as heat capacity, enthalpy of formation for the compounds, and chemical potentials or activities for the liquid phase would improve the reliability of our model. Measurements of phase diagram data in the U–UTe2 region are also required. However this work provides the first detailed uncertainty analysis of the U–Te CALPHAD model. Moreover our approach, contrary to other Bayesian methods, provides an analytical posterior probability distribution and analytical credible intervals for the calculated thermodynamic quantities. It also speeds up the simulation of the uncertainty estimations on the phase diagram.