λ(T) = а∗Cр.∗ρт∗(1+ 3∗∆L(T)/L293)-1 (3)
where: а – is the thermal diffusivity, (m2/с); Ср – is the specific thermal capacity, (J/(kg∗К));
ρт - is the theoretical density of specimen at room temperature, (kg/m3); ∆L(T)/L293 – is the relative linear thermal expansion.
The thermal diffusivity was measured by the Parker’s pulse method [1]. The essence of the pulse method consists in measuring the time interval during which the thermal energy pulse passes through a thin specimen. Provided that the pulse duration is short comparing with the time of its passage through the specimen and the heat losses are negligible, the thermal diffusivity of the specimen material can be found from the relation:
a = 1.37∗l2/(π∗τ1/2) (4)
where: l is the specimen thickness, (m); τ1/2 is the time required for the temperature on the rear surface of specimen to reach the half of the maximum value, (s).
The specimens had the form of thin discs ∼10 mm in diameter and ∼0.8-0.9 mm in thickness.
The heat pulse incident to the flat surface of the disc was produced by means of laser radiation.
The thermal capacity of specimens at temperatures from 575 K to 1775 K was calculated using the Kopp-Neumann rule [2]:
Cр = Х.∗Ср(Gd2O3)+(1-X)∗Cp(UO2) (5)
where: Х – is the mole fraction of Gd2O3; Ср(Gd2O3), Cp(UO2) are and the thermal capacity of gadolinium and uranium oxides taken from [14,15], respectively; Cр – is the thermal capacity of solid solution (U1-XGdX)O2.
The theoretical density ρт of (U1-ХGdХ)O2 specimens at room temperature was found from expression [3]:
ρт = 10.96 – 3.3W, (g/cm3) (6)
where W is the Gd2O3 content in the fuel, weight fraction
The experimental data on the thermal diffusivity and thermal conductivity of uranium-gadolinium fuel with the Gd2O3 content 6.0; 8.0; 10.0 wt % is shown in Figs. 3 – 4. All the data are normalized relative to the density equal to 95% of the theoretical one using the modified Maxwell-Eiken relation:
λ0.95 = λM∗0.95/(1+0.05∗k)∗(1+k∗P)/(1-P) (7) where: λM is the measured value of the thermal conductivity coefficient; λ0.95 – is the value corresponding to the density 95 % of the theoretical one P is the porosity; k = 0.5.
It follows from Fig.4 that the thermal conductivity of uranium-gadolinium fuel is lower than that of uranium dioxide, and it decreases as the gadolinium oxide concentration rises.
The data obtained reasonably agree with the published results [4] within the measurement error (15 %).
The results of experimental investigations obey well to the following dependence:
λ = 1/(A+BT) (8)
where the constant A corresponds to the interaction of phonons with the crystal defects (vacancies, clusters, dislocations, crystallite boundaries, specimen surface, alien inclusions) and the constant B – to the phonon-phonon interaction. Using the temperature dependences λ of uranium-gadolinium fuel pellets, shown in Fig.4, the values of constants A and B were calculated and are listed in Table 2.
Table 2. Values of constants A and B in equation (10) for the thermal conductivity coefficients of (U,Gd)O2
Composition of specimens А, х10-2 , (mК)/W В, х10-4 m/W
U0,913 Gd0,087O2 (6.0 wt % Gd2O3) 17.1±0.19 1.80±0.02
U0.885 Gd0.115O2 (8.0 wt % Gd2O3) 21.4±0.22 1.69±0.02
U0.858 Gd0.142O2 (10.0 wt %Gd2O3) 25.9±0.22 1.50±0.02
It is seen that the increased concentration of gadolinium oxide in the uranium dioxide results in the rise in the constant A, i.e. an additional thermal resistance is created because of interaction of phonons with the lattice defects. Even more significant contribution to the increase in the thermal resistance comes from the stress in the uranium dioxide lattice, arising when the U4+ ions are replaced by the Gd3+ ions. The B value then decreases and, as a result, the temperature dependence of the thermal conductivity coefficient of uranium-gadolinium fuel decreases of about the stoichiometric composition.
500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 Temperature, K
0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30
Thermal diffusivity * 1E6, m*m/s
6 wt% Gd2O3 8 wt % Gd2O3 10 wt % Gd2O3
FIG. 3 Temperature dependence of the thermal diffusivity an of the uranium-gadolinium fuel with the Gd2O3 content 6.0 wt %, 8.0 wt % and 10.0 wt %.
600 800 1000 1200 1400 1600 1800 Temperature, K
1.0 2.0 3.0 4.0 5.0 6.0 7.0
Thermal conductivity, W/ (m K)
UO2, [2]
6 wt % Gd2O3 8 wt % Gd2O3 10 wt % Gd2O3
FIG. 4. Temperature dependence of the thermal conductivity of uranium-gadolinium fuel and uranium dioxide.
5. MECHANICAL PROPERTIES OF URANIUM-GADOLINIUM FUEL PELLETS 5.1. Temperature of brittle-ductile transition
For the determination of the temperature of brittle-ductile transition (Тbdt) using the method described in [6] the temperature dependences of fuel pellet strength (σB) are plotted. Usually the temperature range studied is from room temperature to 1873 К. On the curve σB = F(T) the temperature corresponding to the transition from the elastic to plastic deformation is found.
For the determination of Тbdt by this method several dozens of pellets (not fewer than three pellets at each point) must be carried to destruction. This method requires much labour and time expenditures. In order to increase the accuracy and rate of measurements of Тbdt as well as to reduce essentially the quantity of tested specimens a new method for the determination of the brittle-ductile transition temperature was used. Its essence consists in the determination of specimen deformation (ε) at fixed stress values (σ) and time under loading (τ) within the temperature range 1073 K–1773К. Using the measured values of ε and at σ = const. and τ = const. the dependence of lgε on the reverse temperature (I/Т) is built, the inflection point which corresponds to the brittle-ductile transition is found. By means of this method the tests of specimens made of UO2 and 90 wt% UO2+10 wt% Gd2O3 were carried out at σ = 30 MPa and τ = 45 min. The experimental results obtained were used for building dependences lgε vs I/Т (Fig. 5). It is seen that adding of 10 wt% of Gd2O3 (or 14.2 mol. %) to the uranium dioxide increases the brittle-ductile transition temperature by ∼200К.
5.6 6.0 6.4 6.8 7.2 7.6 8.0
(10000/T), 1/K
-4.4 -4.0 -3.6 -3.2
Logarithm of strain
90 wt % UO2+10 wt % Gd2O3, bdt=1633 90 wt % UO2 + 10 wt % Gd2O3, bdt=1573 UO2, bdt=1433
UO2, bdt=1393
FIG. 5. Dependence of deformation logarithm on the reverse temperature at σ=30 MPa and τ=45 min.
5.2. Modulus of elasticity of uranium-gadolinium oxide fuel pellets
The modulus of normal elasticity (Young’s modulus) of UO2 and 0.9UO2+0.1Gd2O3 fuel pellets was determined by the “stress-deformation” diagrams obtained in the compression tests in the temperature range 1573–1673 K. The specimen was heated up to 1723 K, then a compressing load was applied to it for eliminating gaps between the specimen ends and punches. After that the temperature was reduced to 1673 К loading corresponding to 40 MPa was applied and the specimen deformation was registration. Using the measured values of σ and ε, the Young’s modulus (E) was calculated by Hook’s law:
Е=σ/ε (9)
where: σ is the stress, MPa; ε is the relative deformation.
The values obtained for the UO2 and 90 wt % UO2+10 wt% Gd2O3 specimens are listed in Fig. 6. It is seen that in alloying the uranium dioxide by the gadolinium dioxide the Young’s modulus increases, i.e. the specimens become more “rigid”.
1560 1600 1640 1680
Temperature, K
100 200 300 400
Elastic modulus , GPa
UO2
90 wt % UO2+10 wt % Gd2O3
FIG. 6. Temperature dependence of Young’s modulus of UO2 and 90 wt% UO2+10 wt%
Gd2O3 fuel pellets
5.3. Investigation of thermal creep of uranium-gadolinium fuel
For the creep investigation of uranium-gadolinium oxide fuel containing 5.14 mass % of Gd2O3, fuel pellets with the outer diameter ∼5,7 mm, central channel diameter ∼1.4 mm, and 9–11 mm in height were used. The fuel pellet density was 10.45–10.50 g/cm3.
The creep tests of compressed specimens were performed on the “Kapriz-M” type installation with pneumatic loading system. The temperature in the central hole of specimen was measured by the thermocouple. The temperature range was 1150oC–1450oC (below this temperature the thermal creep was not recorded), the test environment was pure helium. The stress in the test of each specimen was increased stepwise up to 60 MPa at a constant temperature.
The creep curves obtained basing on the test results have the ordinary “classical” form with distinguished process phases. The typical creep curves for different temperatures are shown in Figs 7–10, where the values and times of actions of stepwise increased stresses are given. The maximum specimen deformation was about 5%.
0 2 4 6 8 10 12 Time, h
0.0 0.2 0.4 0.6 0.8 1.0
Deformation*100
t=1150 C stress 20 MPa stress 40 MPa stress 60 MPa
FIG. 7. Creep of uranium dioxide with addition of 5.14 wt% of Gd2O3 at 1150oC.
0 2 4 6 8 10 12
Time, h
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Deformation * 100
t=1250 C stress 20 MPa stress 30 MPa stress 40 MPa stress 60 MPa
FIG. 8. Creep of uranium dioxide with addition of 5.14 wt% of Gd2O3 at 1250oC.
0 2 4 6 8 10
Time, h
0.0 1.0 2.0 3.0
Deformation * 100
t=1350 C stress 20 MPa stress 30 MPa stress 40 MPa stress 60 MPa
FIG. 9. Creep of uranium dioxide with addition of 5.14 wt% of Gd2O3 at 1350 oC.
0 2 4 6 8 10
Time, h
0.0 1.0 2.0 3.0 4.0 5.0
Deformation * 100
t=1450 C stress 10 MPa stress 20 MPa stress 30 MPa stress 40 MPa stress 60 MPa
FIG. 10. Creep of uranium dioxide with addition of 5.14 wt% of Gd2O3 at 1450 oC.
The dependences of minimum creep rates on the stress and temperature were determined by the deformation curves for different test regimes. The minimum creep rates determined by the curves lie in the range 3 10-5–8 10-2h-1. These dependences are close to the linear ones practically within the whole tests range in the chosen semi-logarithmic coordinates (Fig. 11).
The essential dependence of minimum creep rate on stress should be pointed out. For example, at the temperature 1450°С the values of minimum creep rates at the stress 10 MPa and 40 MPa differ by an order of magnitude. It has been established, in particular, that the values of minimum creep rates at the temperatures 1150°C and 1250°С and the stress 30 MPa are close, while at the stress 20 MPa the creep rate corresponding to a higher temperature is nearly twice lower
0 20 40 60