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Summary - Thermal Expansion of Liquid Uranium Dioxide

Recommended Equation

The recommended equation for the thermal expansion coefficient of liquid uranium dioxide is based on the in-pile effective equation of state measurements of the vapor pressure, density, and isothermal compressibility of liquid (U, Pu)O₂ by Breitung and Reil [1]. From these measurements, the density and thermal expansion coefficient as functions of temperature were obtained from the melting point to 7600 K. The equation of Breitung and Reil for the thermal expansion coefficient of UO₂ and (U, Pu)O₂ for mole fractions of Pu <= 0.25 is in good agreement with the equation for the thermal expansion coefficient of UO₂ from experiments by Drotning [2], which had been recommended in the 1981 assessment by Fink et al [3,4].

The recommended equation for the instantaneous volumetric thermal expansion coefficient of UO₂ as a function of temperature is:

Figure 1

where the thermal expansion coefficient (alpha_P) is in K^-1 and temperature (T ) is in K. Values for the density and the instantaneous volumetric thermal expansion coefficient of UO₂ are given in Table 1. Figure 1 shows the recommended values for the instantaneous volumetric thermal expansion coefficient of UO₂, the uncertainties determined by Breitung and Reil [1], and the instantaneous volumetric thermal expansion coefficients of UO₂ calculated from equations of Drotning [2], of Christensen [5], and of Harding [6].

Uncertainties

Breitung and Reil determined experimental uncertainties from the uncertainty in the fuel mass (dm/m = 10%), the uncertainty in the test volume (dV/V = 2.5%), and the uncertainty in the fuel enthalpy (dh/h = 6%). From these uncertainties, they obtained upper and lower limiting cases which they used to define uncertainties in the parameters in Eq. (1). The liquid density at the melting point, 8860 kg×m^-3, has an uncertainty of ± 120 kg×m^-3. The slope of the density, (dr/dT) = 0.9285 kg×m^-3×K^-1, has uncertainties of + 0.036 kg×m^-3×K^-1 and - 0.135 kg×m^-3×K^-1. The upper and lower uncertainty limits calculated using the uncertainties in these parameters are shown in Figure 1. They correspond to uncertainties of:

+10% and -12% at 3120 K;
+10% and -13% at 3500 K;
+12% and -15% at 4500 K;
+13% and -17% at 5500 K;
+15% and -20% at 6500 K;
+18% and -27% at 7600 K.

Discussion of the Recommended Equation

where beta_T is the isothermal compressibility and P is the vapor pressure. The subscripts on the partial derivatives indicates that they are along the saturation curve. Breitung and Reil [1] state that the magnitude of the second term in Eq.(2) is much smaller than the first term and only contributes a few percent at 8000 K. This is because along the saturation curve, the volume change due to the pressure change is much smaller than the corresponding volume change due to thermal expansion. Thus, for UO₂ and (U,Pu)O₂ , the thermal expansion coefficient may be evaluated from the density/temperature relation using the first term in Eq.(2).

The linear instantaneous thermal expansion coefficient is one third of the instantaneous volumetric thermal expansion coefficient, given by Eq.(1). Equations relating the instantaneous volumetric thermal expansion coefficient and density to other expansion parameters are given in the appendix, "Density and Thermal Expansion Relations."

Comparison with Other Measurements and Assessments

Three experiments have provided data on the density and thermal expansion of liquid UO₂. Breitung and Reil [1] determined the density of UO₂ and (U,Pu)O₂ from the melting point to 7600 K from measurements of the pressure rise of a sealed capsule during a transient in-pile pulse. Their vapor pressure measurements using ultrapure UO₂, reactor grade UO₂, and reactor grade (U,Pu)O₂ showed no significant difference for the vapor pressures of all three fuel types. Drotning [2] determined the density of UO₂ with O/M ranging from 2.01 to 2.04 as a function of temperature using gamma ray attenuation measurements. Christensen measured the thermal expansion of solid and liquid UO₂ and the volume change on melting using gamma radiographs to determine the sample dimensions.

The variation of density with temperature from all three measurements is in good agreement. The slope (dr/dT) used in the first term of Eq. (2) is:

- 0.9285 kg m^-3 K^-1 (Breitung & Reil)
- 0.916 kg m^-3 K^-1 (Drotning)
- 0.918 kg m^-3 K^-1 (Christensen)

The thermal expansion of Drotning [2] was recommended in the 1981 assessment by Fink et al [3,4]. The instantaneous volumetric thermal expansion coefficient calculated from Drotning's density equation using the first term in Eq. (2) is:

where the thermal expansion coefficient (alpha_P) is in K^-1 and temperature (T ) is in K. Values of thermal expansion calculated with this equation are shown in Figure 1.

In their 1989 review of the data on density of liquid UO₂, Harding, Martin, and Potter [6] also recommend the change in density with temperature measured by Drotning. However, they recommended 8640 ± 60 kg×m^-3 for the liquid density at 3120 K. So the thermal expansion coefficient calculated from the density recommended by Harding et al. using the first term in Eq.(2) is:

where the volumetric thermal expansion coefficient (alpha_P) is in K^-1 and temperature (T ) is in K. Because both Eq.(3) and Eq.(4) are based on the variation of density with temperature measured by Drotning, the values of the thermal expansion coefficient calculated using Eq.(4) are almost identical to those calculated using Eq.(3). Differences are 0.03% from the melting point to 4800 K, 0.04% from 4900 to 6600 K, and 0.05% from 6700 to 7600 K.

The instantaneous volumetric thermal expansion coefficient calculated from the liquid density of Christensen and his change of density with temperature is:

Figure 2

where the thermal expansion coefficient (alpha_P) is in K^-1 and temperature (T ) is in K. Values of the volumetric thermal expansion coefficient determined from the measurements of Christensen have been included in Figure 1.

Figure 2 shows the deviations of the recommended thermal expansion coefficients of Breitung and Reil from the thermal expansion coefficients determined from measurements of Christensen [5] and of Drotning [2]. Percent deviations in Figure 2 are defined as:

Extrapolations of the thermal expansion coefficients from the low temperature measurements of Christensen and of Drotning to 7600 K show good agreement throughout the temperature range. Deviations of recommended values from those determined from measurements by Drotning range from -1.4% at the melting point to -2.5% at 7600 K. Christensen's values deviate from those of Breitung and Reil by 0.2% at the melting temperature and by 0.4% at 7600 K. Figure 2 shows that all deviations are well within the uncertainty limits given by Breitung and Reil.

References

1. W. Breitung and K. O. Reil, The Density and Compressibility of Liquid (U,Pu)-Mixed Oxide, Nuclear Science and Engineering 105, 205-217 (1990).
2. W. D. Drotning, Thermal Expansion of Molten Uranium Dioxide, Proceedings of the 8th Symp. On Thermophysical Properties, Gaithersburg, Maryland, June 15-18, 1981, National Bureau of Standard (1981).
3. J. K. Fink, M. G. Chasanov, and L. Leibowitz, Thermophysical Properties of Uranium Dioxide, J. Nucl. Mater. 102 17-25 (1981).
4. J. K. Fink, M. G. Chasanov, and L. Leibowitz, Properties for Reactor Safety Analysis, ANL-CEN-RSD-80-3, Argonne National Laboratory Report (April, 1981).
5. J. A. Christensen, Thermal Expansion and change in Volume of Uranium Dioxide on Melting, J. Am. Ceram. Soc. 46, 607-608 (1963).
6. J. H. Harding, D. G. Martin, and P. E. Potter, Thermophysical and Thermochemical Properties of Fast Reactor Materials," Commission of European Communities Report EUR 12402 (1989).