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Summary - Density of Liquid Uranium Dioxide

Recommended Equation

The recommended equation for the density 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]. Measurements of density as a function of enthalpy and as a function of temperature were obtained from the melting point to 7600 K. The equation of Breitung and Reil for the density of UO₂ and (U, Pu)O₂ for mole fractions of Pu <= 0.25 is in good agreement with the equation for the density 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 density of UO₂ as a function of temperature is:

Figure 1

where density (r) is in Mg/m³ and temperature (T ) is in K. Values for the density and the instantaneous thermal expansion coefficient of UO₂ are given in Table 1. Figure 1 shows the recommended values for the density of UO₂, the uncertainties determined by Breitung and Reil [1], and the UO₂ density data of Drotning [2] and of Christensen [5].

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 values in addition to the most probable reference values. Their uncertainty bands correspond to uncertainties in the coefficients in Eq. (1) given by:

The upper and lower uncertainty limits calculated from Eq.(2) are shown in Figure 1. They correspond to a uncertainties of :

+/-1.4% at 3120 K;
+1.6% and -2% at 3500 K;
+2.2% and -4% at 4500 K;
+3% and -6.3% at 5400 K;
+4.2% and -10% at 6500 K;
+6% and -15.4% at 7600 K.

Discussion of the Recommended Equation

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.

UO₂ liquid densities at the melting point measured by Drotning ranged from 8.779 to 8.939 Mg/m³ with an average of 8.860 Mg/m³ and a deviation of + 0.061 Mg/m³ or 0.7%. His equation for the density of UO₂ in Mg/m³ from 3120 to 3250 K is:

where temperature is in K. This equation was recommended in the 1981 assessment by Fink et al.3,4

The change of density of UO₂ at the melting point measured by Christensen was 9.6%. In the liquid range, he measured densities from the melting point (which he measured as 3073 K rather than 3120 K) to 3373 K. At the melting point, he gives solid and liquid densities of 9.67 +/- 0.13 Mg/m³ and 8.74 +/- 0.16 Mg/m³, respectively. His equation for the liquid density of UO₂ adjusted to a melting point of 3120 K is:

where density is in Mg/m³ and temperature is in K.

In their 1989 review of the data on density of liquid UO₂, Harding, Martin, and Potter [6] use the solid density at the melting point recommended by Martin [7] (9.56 +/- 0.04 Mg/m³) and the change in density at melting determined by Christensen (9.6%) to obtain 8.64 +/- 0.06 Mg/m³ for the liquid density at 3120 K. Because of the higher accuracy of Drotning's liquid density measurements compared with the thermal expansion measurements of Christensen, Harding et al. [6] based the slope of their density equation on the slope given by Drotning corrected to the different density at the melting point, 3120 K. They recommend the equation:

for the density of UO₂ in Mg/m³, where temperature is in K. The liquid density at the melting point given by this equation is lower than the lowest density measured by Drotning but is above Christensen's lower uncertainty of 8.58 Mg/m³.

Breitung and Reil set their melting point density of UO₂ and (U,Pu)O₂ to 8.860 Mg/m³, the density of UO₂ at the melting point given by Drotning [2] because of the smaller error in Drotning's measurements ( +/- 0.7%) than in Christensen's measurements ( +/- 2%). The densities of Christensen and of Drotning agree within their experimental uncertainties. The uncertainty (+ 0.120 Mg/m³) given by Breitung and Reil for this parameter in their density equation is large enough to include the melting point density given by Christensen. The liquid density at the melting point recommended by Fink, Chasanov, and Leibowitz [3,4] was also 8.860 Mg/m³.

Equation Selection

Figure 2

The equation given by Breitung and Reil, Eq.(1), has been recommended because it is a careful analysis of the experimental data with experimental uncertainties for the largest temperature range and is consistent with the measurements of Drotning. Figure 2 compares the recommended equation of Breitung and Reil with the equations of Drotning [2], Christensen[5], and Harding et al. [6] and the experimental data of Drotning and of Christensen. In Figure 2, the data of Christensen has been corrected for his temperature offset at the melting point. Figure 2 shows that the slope of the density equation recommended by Breitung and Reil is also consistent with that of Christensen. However, the equation of Harding et al is consistently lower than that of Breitung and Reil in this temperature region due to the melting point density selected by Harding et al. Because the solid density data of Christensen has been shown, by Martin's analysis [7] of the thermal expansion of solid UO₂, to be inconsistent with later more accurate measurements of Hutchings [8], there is some question with regard to the reliability of his determination of the change of density on melting. Because the liquid densities at the melting point determined by Drotning and by Christensen are consistent within their error limits and the uncertainties for Drotning's data are less than those for Christensen's, the melting point density of Drotning is preferred to using the density change on melting given by Christensen and the solid density at the melting point given by Martin [7]

Deviations from Recommended Equation

Percent deviations of the densities given by the equations of Drotning, Christensen, and Harding et al. from the recommended values given by the equation of Breitung and Reil are shown in Figure 3. Percent deviations in Figure 3 are defined as:

Figure 3

The uncertainties given by Breitung and Reil are included in Figure 3, expressed as percents, for comparison with the deviations. Figure 3, shows that all the equations are within the uncertainties of Breitung and Reil except for the equation of Harding et al. for the temperature range 3120 K through 3700 K. Absolute values of the percent deviations for the equation of Harding et al. decrease from a maximum deviation of -2.5% at the melting point to -1.3% at 7600 K. Percent deviations for the density equation of Christensen show little variation with temperature. They range from -1.4% at 3120 K to -1.6% at 7600 K. The smallest deviations occur for Drotning's equation which gives densities within 1% of those given by Breitung and Reil from the melting point through 7200 K. At 7600 K, the percent deviation for these equations is 1.2%. Thus, for the entire temperature range of interest in severe accidents, the recommended densities of Breitung and Reil are in good agreement with those given by the equation of Drotning which was recommended in the 1981 assessment by Fink et al [3,4].

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).

7. D. G. Martin, The Thermal Expansion of solid UO₂ and (U,Pu) Mixed Oxides - A Review and Recommendations, J. Nucl. Mater. 152 94-101 (1988).

8. M. T. Hutchings, High-Temperature Studies of UO₂ and ThO₂ using Neutron Scattering Techniques, J. Chem. Soc. Faraday Trans. II 83, 1083-1103 (1987).