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Thermal Conductivity and Diffusivity of Liquid UO₂

Based on an initial review of the limited data^1-4 on the thermal conductivity and thermal diffusivity of liquid UO₂, the liquid thermal conductivity is in the range of 2.5 to 3.6 W m^-1 K^-1. Liquid thermal diffusivities range from 6 x 10^-7 to 11 x 10^-7 m² s^-1.

The available data on the thermal conductivity (k) and thermal diffusivity of liquid UO₂ are summarized in Table 1. Measurements of thermal diffusivity were made by Kim et al.¹ and by Otter and Damien.² Tasman et al.^3,4 measured thermal conductivity. The measurements by Kim et al.¹ and by Otter and Damien were based on standard methods for obtaining the thermal diffusivity.

Kim et al.¹ used a modulated electron beam technique to measure the thermal diffusivity of UO₂ in the temperature range of 3187 to 3310 K. A thin UO₂ sample clad in tungsten was heated by two electron beams. The top beam was modulated sinusoidally and the difference in phase between the top and bottom temperatures was measured. The thermal diffusivity was determined from the phase changes. Measurements were made on two thicknesses of UO₂ (0.813 and 1.219 mm) and three modulated frequencies: 0.25 Hz (rad s^-1), 0.50 Hz (rad s^-1), and 0.75 Hz (rad s^-1). The tungsten above and below the UO₂ layer was 1.397 and 1.016 mm thick. In the reanalysis⁵ of the data of Kim et al.,¹ an error was found in the original analysis by Kim et al.¹ The reanalysis included (1) the ideal calculation done by Kim et al., (2) an ideal model using a three-dimensional unsteady-state heat transfer code that assumed infinite slabs with no sidewalls, and (3) the real case accounting for heat transfer in the tungsten sidewalls using a transient 3-dimensional unsteady-state heat transfer code. No radiative heat transfer within the liquid was modeled based on the comment of Bober⁷ that radiative heat transfer in the liquid would be small and could not account for the increase in thermal conductivity of the liquid. The heat transfer analysis using ideal and real models of the UO₂ in the tungsten cell showed that if the thermal conductivity was low, then the ideal model was not a good approximation because wall conductivity becomes important as the conductivity of the liquid layer decreases. As shown in Table 1, a statistically significant difference was found between the thermal conductivities of the thick and thin layers. Although tungsten contamination of the samples could affect the conductivity, it would have a greater effect in the thin cell than in the thick cell and give the larger conductivity for the thin cell. Lack of good contact between the tungsten and the liquid could also affect the experimental results. The difference between the thin and thick cell results is analogous to differences observed in diffusivity measurements of materials in which radiation is important and cannot be neglected.^8,9 The main uncertainties in this experiment are effects from radiative heat transfer in the liquid and effects from changes in the O/M ratio in the UO₂ due to tungsten contamination in the liquid UO₂ sample.

Otter and Damien² measured the thermal diffusivity of a 0.7-mm layer of liquid UO₂ encased in tungsten using a laser flash method in the temperature range of 3133 to 3273 K. Although this method is well established, analysis of the data becomes more complex at high temperatures with liquids encased in metal cells because of the necessity of including thermal losses to the environment and the need for properties such as heat capacity, density, and thermal conductivity of the metal containment.¹⁰ The reanalysis⁵ of their experiment gave lower thermal conductivities than those originally reported by Otter and Damien. Insufficient information is available regarding their experiment and heat losses to determine if the differences are due to different treatment of heat losses in the reanalyzed three-dimensional transient heat transfer calculation. Radiation within the sample was not included in the reanalysis. If radiative heat transfer was significant in the experiments of Kim et al., it would also affect the experiment of Otter and Damien. In addition, errors from tungsten contamination of the sample cannot be ruled out.

Tasman et al.³ determined the thermal conductivity of liquid UO₂ from a steady-state finite element analysis of the heat transfer in a partially molten, self-contained sample. A UO₂ disc (6 mm in diameter by 1.2- to 3-mm thick) was heated in an argon atmosphere at 4 bar (0.4 MPa) using three continuous-wave CO₂ laser beams. One laser beam was focused on a 4-mm diameter area on the bottom of the disc; two laser beams were focused on an area 2-mm in diameter on the top of the disc. The sample was heated with only the bottom beam until it reached 1800^oC (2073 K). Then the upper beams were turned on and a molten pool was formed on the top of the sample. Temperatures were measured with optical pyrometers and a fast scanning device. During heating, only the bottom temperature was measured. The peak top temperature was 3200^oC (3473 K). Because of extensive vaporization of the sample, the top of the sample could be heated for only 4-5 sec. When heated longer, extensive evaporation created a deep pit in the top center of the sample and part of the ejected material was deposited in crystalline form along the center crater edge. Even on short (5-sec) exposures, recondensed crystals were found. Sample loss from evaporation was limited to less than 2 mg (0.5%) if the exposure to the upper beams was limited to 5 sec. However, it is not clear how much sample mass was redistributed by vaporization and condensation on cooler parts of the sample during the 5 sec exposure to the upper beam. Significant vaporization of UO₂ began at 2300^oC (2573 K), which is well below the melting point, 3120 K. Tasman et al.³ stated that the largest uncertainty in their experiment was the temperature measurement and temperature profile of the top and bottom faces. Because these profiles are critical input in the analysis of the experiment, there is significant uncertainty in the calculated results. The precision of the experiment is limited by the presence of very high radial temperature gradients and axial asymmetries. However, the error is bounded by the depth of the molten layer, which was determined after solidification from examination of cross sections of the sample. The reliability of visual observation of the liquid depth was questioned⁵ based on (1) low melting points (2661 and 2699 K) obtained by early investigators¹¹ from the appearance of residues and (2) the observed¹² softening and plasticity of UO₂ above about 2500 K where the Frenkel oxygen lattice disorder increases as the phase transition is approached. Above 2670 K, the creep rate also increases significantly,^{12, 13} so UO₂ readily deforms to the shape of its container. Ronchi¹⁴ commented that in the short duration of the experiment of Tasman et al. (~ 10 sec), the grain growth is approximately 10 µm at 2700 K¹³ and is estimated to be only 5 times larger at 3050 K. He, therefore, concluded that the solid grains are still recognizable at temperatures near the melt front so that the liquid phase is readily distinguished. The reanalysis of this experiment by Fink and Leibowitz⁵ indicated that the assumption of steady state conditions made by Tasman et al.³ makes a significant difference in the resulting thermal conductivity.

Tasman⁴ repeated his experiment using a rapid 2D temperature-scanning device and included unsteady transport in the 2D finite-element method (f.e.m.) analysis. His correction was less than that reported by Fink and Leibowitz.⁵ He claimed that perturbations that cannot be accounted for in his analysis would lead to lower values in the thermal conductivity.¹⁴ He concluded that the thermal conductivity of liquid UO₂ is 2.5 1 W m^-1 K^-1, which is lower than the thermal conductivity of the solid at the melting point given by Harding and Martin¹⁵ (3.95 W m^-1 K^-1) and by Hyland¹⁶ (3.65 W m^-1 K^-1). The thermal conductivity equation for solid UO₂ of Harding and Martin includes a phonon lattice contribution and an electronic contribution from small polarons, whereas Hyland also included a radiation contribution. At the melting point, the electronic contribution calculated by Harding and Martin is 2.56 W m^-1 K^-1, which is slightly higher than the value for the thermal conductivity of the liquid obtained by Tasman. The electronic, radiation, and lattice contributions to the solid thermal conductivity at the melting point determined by Hyland are 1.55 W m^-1 K^-1, 0.2 W m^-1 K^-1, and 2.1 W m^-1 K^-1, respectively. The radiative contribution calculated by Hyland was 0.48 W m^-1 K^-1, but he assumed a 50% uncertainty because this value was higher than needed for good agreement with experimental total thermal conductivities. Differences in the lattice and electronic contributions to the thermal conductivity of the solid in these two calculations are related to the different data used in the models. Because of these differences, no conclusions with regard to the reliability of the measurement of Tasman can be made from comparison with contributions to the solid thermal conductivity at the melting point.

Ronchi et al.⁶ determined the heat capacity of liquid UO₂ from the melting point to 8000 K by heating sintered 0.5- to 1-mm diameter microspheres by four tetrahedrally oriented laser beams in an inert autoclave at pressures up to 1000 bar. The samples were suspended by a tungsten needle during pulses of a few milliseconds duration. The heat capacity was calculated numerically from the energy input, the sample temperature during and after laser pulse heating, the energy loss rates, the cooling mechanisms (radiation and convection), and the heat transport within the sample. The accuracy of the calculation depended on the symmetry (of the temperature field from the lasers and the sample shape) and the accuracy of the physical properties (density and thermal conductivity) used in the heat transport analysis. In the calculations, Ronchi et al.⁶ used 2.5 W m^-1 K^-1 for the thermal conductivity of liquid UO₂. However, they commented that selection of a higher value for the thermal conductivity of the liquid would result in a lower heat capacity. The thermal conductivity values in the next to the last row of Table 1 are the values of thermal conductivity from the reanalysis of Fink and Leibowitz⁵ adjusted for the heat capacities of Ronchi et al. Although the new heat capacity values reduce the thermal conductivities calculated by Fink and Leibowitz,⁵ the calculated thermal conductivities are not as low as the value reported by Tasman.⁴ However, the corrected value calculated for the experiment of Tasman et al.³ is within the original uncertainty given by Tasman et al.^{3, 4}

Because the heat capacities obtained by Ronchi et al.⁶ are a function of the value selected for the thermal conductivity and are consistent with the value reported by Tasman⁴ and all other data in Table 1 are from thermal diffusivity measurements, thermal diffusivities should be compared instead of thermal conductivities. The temperature at which the thermal conductivity of liquid UO₂ was remeasured by Tasman⁴ has not been reported by Ronchi et al.^{6, 14} In their analysis of their heat capacity data, Ronchi et al.¹³ assumed that the liquid thermal conductivity is constant at 2.5 W m^-1 K^-1 for the liquid temperature range (3120 - 8000 K). It is not clear if their observed variation in heat capacity with temperature is real or is due, in part, to this assumption of constant thermal conductivity. In any case, the thermal diffusivity calculated using the heat capacities (C_P) of Ronchi et al.^{6, 13} and constant thermal conductivity (k) of 2.5 W m^-1 K^-1 is consistent with the analytical treatment of the heat capacity data of Ronchi et al.^{6, 13} The liquid UO₂ densities of Breitung and Reil,¹⁷ which agree with the values of Drotning,¹⁸ have been used in the conversion to thermal diffusivity via the relationship

Thermal diffusivities from the most recent measurements of Tasman⁴ and the thermal diffusivity experiments of Otter and Damien² and Kim et al.¹ are given in Table 2.

Ronchi¹⁴ commented that diffusivity in crystals decreases with temperature due to increased anharmonic vibrations caused by defects, impurities, and lattice strains. Below 2500 K, the behavior of the thermal diffusivity of UO₂ is in accord with this crystalline behavior. As the phase transition at 2670 K is approached, the number of phonon scattering centers increases. Above the phase transition, the concentration of Frenkel pairs in the oxygen sublattice approaches 0.2,¹³ so the lattice has a very high degree of disorder similar to an amorphous or glassy material. Ronchi¹⁴ commented that materials that have both crystalline and glassy forms (e.g., SiO₂) have a different temperature dependence for the thermal diffusivity in the two forms (decreasing for the crystal; increasing for the glassy phase).¹⁴ In metals and alloys that undergo order/disorder transitions, the slope of thermal diffusivity changes at the transition from decreasing to increasing. If no transition exists, the reversal of slope normally occurs at the melting point and is often accompanied by a discontinuity in thermal diffusivity upon melting. For materials with a premelting order/disorder transition, the thermal diffusivity typically increases continuously across the melting point.¹⁴ In Figure 1, the thermal diffusivities of liquid UO₂ from the measurements of Kim et al.,¹ Otter and Damien,² and Tasman⁴ are compared with thermal diffusivities of solid UO₂ near the melting point. The solid values are from thermal diffusivity measurements by Weilbacher^{19, 20} and the thermal conductivity equations of Harding and Martin¹⁵ and of Hyland.¹⁶ The thermal conductivities were converted to thermal diffusivities using Eq. (1) and the heat capacities from the assessment by Fink²¹ and the densities from the assessment of Martin.²² Thermal diffusivities calculated from the thermal conductivity of Tasman⁴ are between the solid values of Harding and Martin and of Hyland. Based on the behavior of other materials with premelting transitions, Ronchi¹⁴ concluded that the thermal diffusivity obtained from the thermal conductivity measurement of Tasman is the most consistent with the thermal diffusivities of solid UO₂.

Figure 1

If the transition at 2670 K results in sufficient disorder for the thermal diffusivity to follow glassy behavior, then internal radiation, which is important for glassy materials, must also be considered for UO₂ above this transition. In his critical analysis of the thermal conductivity of solid UO₂, Hyland¹⁶ included a contribution from radiation. At the melting point, 0.48 W m^-1 K^-1 is the radiative contribution to the thermal conductivity of solid UO₂ calculated by Hyland¹⁶ using the method given by Browning²³ and the optical property data for solid UO₂ measured by Bober et al.⁷ This result for the solid and the statistically significant difference between the thermal diffusivities of the thin and thick UO₂ layers, which is indicative of internal radiation,^8,9 imply that the radiative contribution should also be considered for the liquid. The radiative contribution to the thermal conductivity for an optically thick sample is

Figure 2

where n is the refractive index (1.72 for liquid UO₂)⁷, is the Rosseland absorption coefficient, and is the Stephan-Boltzmann constant. Following Hyland,¹⁶ the value of was obtained from Figure 4 of Browning,²³ which includes the contributions beyond the absorption edge of the material. For the liquid at the melting point, the radiative contribution to the thermal conductivity in the optically thick limit is 0.28 W m^-1 K^-1. This corresponds to corrections to the thermal diffusivity of 0.7 x 10^-7 to 0.9 x 10^-7 m² s^-1 between 3120 and 3400 K, assuming constant thermal conductivity and thermal diffusivity variations with temperature in accord with changes in density and heat capacity. In Figure 2, the curve labeled "Tasman + Radiation" includes the optically thick radiative contribution to the thermal conductivity of Tasman. If the assumption is made that the difference in thermal diffusivities between the thick and thin layers of UO₂ in the experiment of Kim et al. arises from failure to include the radiative term in the analysis, and the radiative contribution scales according to the thickness of the UO₂ layer, the experimental thermal diffusivity of a 0.2 mm thickness of UO₂ (thickness of the molten layer in the experiment of Tasman³) can be estimated. For the temperatures of 3250 and 3277 K, this estimate gives thermal diffusivities in the range of 5.8 x 10^-7 m² s^-1 and 6.7 x 10^-7 m² s^-1. These values, shown in Figure 2, are slightly lower than the values calculated from the thermal conductivity of Tasman⁴ using the heat capacities of Ronchi et al.⁶ and densities of Breitung and Reil.¹⁷ This scaled correction is larger than the calculated radiative contribution due to an optically thick layer. Figure 2 includes the positive uncertainty of Tasman and a corresponding negative uncertainty (-40%) in the thermal diffusivities from the 0.813 mm layer measurements of Kim et al. These uncertainty bands overlap.

From these comparisons, it is reasonable to assume that 2.5 W m^-1 K^-1 (the new value reported by Tasman⁴) represents a lower limit of the thermal conductivity of liquid UO₂. An upper limit of 3.6 W m^-1 K^-1 is consistent with the error limit given by Tasman and with the lower value obtained from the experiments of Kim et al. with the optically thick radiative contribution (0.3 W m^-1 K^-1) subtracted. Clearly, the data of Kim et al. must be reanalyzed with radiative contributions for the thickness of the UO₂ layers included. Although the data of Kim et al. show systematic differences between the thick and thin layers of UO₂ and the data of Otter and Damien appear to be high, these measurements are consistent in that they show little variation in thermal diffusivity with temperature. However, thermal diffusivities calculated using the constant thermal conductivity of Tasman⁴ and the heat capacities of Ronchi et al.⁶ show significant increases with temperature. From the experiments of Ronchi et al.⁶ it is unclear how much of the temperature variation in C_P arises from the change in thermal conductivity with temperature. (Thermal conductivity was assumed to be constant in their analysis.) Ronchi¹⁴ states that glassy ceramics show a slight increase in the thermal diffusivity with temperature and the thermal diffusivity usually increases continuously across the melting point. Because no information is available with respect to the recent thermal conductivity measurements of Tasman,^{4, 14} the temperature of the measurement is uncertain. If the thermal diffusivity was assumed to be constant, the thermal conductivity data of Tasman and the heat capacity of Ronchi et al. at 3473 K would give 8.2 m² s^-1 for the thermal diffusivity of liquid UO₂. At the melting point, this would correspond to a thermal conductivity of 3.2 W m^-1 K^-1. This is within the range of recommended values.

The uncertainty in the thermal conductivity and thermal diffusivity of liquid UO₂ is approximately 40%, the uncertainty given by Tasman et al.^3,4

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