MATERIALS FOR USE IN IAEA BENCHMARK PROBLEMS)
7.1. Introduction
For accelerator driven systems (ADSs), mixed transuranium (TRU) fuels have been suggested, but at the same time it is generally considered that the addition of a non-fissile (inert) support matrix is necessary to dilute the fissile phase and to give mechanical strength to the fuel. The matrix could also help to improve the properties of the fuel, such as melting point, thermal conductivity and chemical stability. Fuel specifications such as matrix fractions, plutonium/minor-actinide (MA) ratios, pellet densities, thermal conductivities, melting/eutectic points are key issues to assess the reactor transmutation performance and safety behavior of ADSs. However, experimental data in respect to the inert matrix fuels (IMFs) are rather scarce, and few theoretically based recommendations have been made relating to matters necessary for the assessment of reactor performance and safety behavior.
In this report, models to estimate thermophysical properties of IMF will be provided for use in the assessment of reactor performance and safety behaviors of ADS. The estimation will be performed by extrapolating or interpolating known basic properties of oxide fuel and matrix constituents, of which data can be found in open literature. Estimated thermophysical properties will be presented for IMF specified for the IAEA benchmark problems as well as the MOX fuel. Recommendations will also be presented for the thermophysical properties of other ADS core materials such as lead-bismuth eutectic (LBE) and stainless steel.
7.2. Models for fuel properties
In many countries, the actinide oxide fuels are considered as the most promising candidates for the MA transmutation in ADS. Three types of the fuel pellet material are under study now. The first one is a solid solution of actinide oxides in zirconia (or yttria stabilized zirconia), the second is a composite formed by the oxide fuel particles dispersed within a non-fissile MgO ceramic matrix (called below
‘CERCER’), and the third is the same kind of composite but with Mo metal matrix (called
‘CERMET’).
Here, solid properties of inert matrix fuels are estimated using basic properties of each constituent of fuel and matrix. For solid solutions, we apply general assumption of ideal mixture or additivity rule to property calculation. For the composite fuels, their properties are calculated basically by averaging the fuel and matrix properties. In addition, special assumption could be made to cover the material behaviors such as eutectic formation.
7.2.1. Solid density
The temperature dependent solid density ρs(T) of a solid solution is calculated by the following equation considering the mole-fraction-weighted mean linear expansion with temperature:
where Tref is ther reference temperature, l(T) and l(Tref) are the lengths at the temperatures T and Tref, respectively; ρs(Tref) is the density at Tref, n is the mole fraction, and the subscript M means a material component in the solid solution. The reference density including the lattice constant of solid solution, which is calculated by Vegard’s law, is given by
where NA is the Avogadro’s number, a is the lattice constant, W is the molecular weight and Z is the number of molecules per unit cell. For the actinide stoichiometric dioxides, which have the fluorite type structure, there are four molecules in a cubic lattice, that is Z = 4 for type AnO2 fuels. The molecular weight of the solid solution is calculated by
The density of a composite fuel is calculated as a volume average of actinide oxide fuel and inert matrix densities:
where the subscripts 'fuel' and 'matrix' mean the actinide oxide fuel and the inert matrix, respectively.
7.2.2. Melting point
The melting temperature Tm of a solid solution can be estimated by additivity rule or mole-fraction-weighted mean:
For composite fuels, the solidus and liquidus temperatures are defined as the minimum and maximum temperatures, respectively, in the melting points of the actinide oxide fuel and the inert matrix.
Reduced melting temperatures are also considered for the materials with eutectic formation.
7.2.3. Solid enthalpy
The temperature dependent molar enthalpy h of a solid solution is calculated by the additivity rule:
The same rule is also used for the heat of fusion:
For composite fuels, the fusion enthalpy is defined as the sum of the enthalpy difference between solidus and liquidus points of actinide oxide fuel and inert matrix:
where hs is the sensible enthalpy that is necessary to increase the temperature up to the solidus or liquidus temperature of the composite fuel.
7.2.4. Thermal conductivity
The thermal conductivity of a solid solution is roughly approximated as a mole-fraction-weighted mean value of each component in the solid solution:
Although in principle Eq. (9) cannot be applied to transport properties (such as thermal conductivity, electric conductivity, diffusivity and viscosity), in the case when a solid solution is composed of similar non-reacting components and in the absence of a eutectic formation, it often gives rather satisfactory results. For composite fuels, Millar’s equation is used to evaluate effective conductivity κeff:
where Vfuel and Vmatrix are the volumes of actinide oxide fuel and inert matrix in the composite fuel, respectively. The thermal conductivity of the actinide oxide fuel is calculated by Eq. (9). The thermal conductivity of composite fuels depends on not only the matrix volume, but also the shape and distribution of the dispersed fuel particles [2]. However, Eq. (10) considers the contribution of both the actinide oxide fuel and inert matrix components when low conductivity fuel particles are embedded into a high conductivity matrix.
The thermal conductivity of solid fuel decreases with increasing porosity. The well-known Maxwell-Eucken equation is used to correct for this porosity effect [4]:
where p is porosity and κ0 is the thermal conductivity of fully dense fuel.
7.3. Properties of actinide oxides and diluents 7.3.1. Mixed oxides
7.3.1.1. Thermal expansion
The lattice constants of UO2 and PuO2 are 547.04 pm and 539.60 pm, respectively (Katz et al., 1986).
Assuming the molecular weights of UO2 and PuO2 are 270.01 g mol-1 and 271.21 g mol-1, respectively, which are based on the U- and Pu-isotope vectors shown in Table 1, these lattice constants give solid densities of UO2 and PuO2 at 293 K of 10956 kg m–3 and 11466 kg m–3, respectively.
Since the UO2, PuO2 and MOX fuels have very similar thermal expansions, Carbajo et al. [4]
recommended employing Martin's correlations [19] for the thermal expansion both of solid UO2 and MOX fuels:
where T is in K and Tm is the melting temperature, which is taken as the solidus temperature for the mixed oxide compositions. For the hypostoichiometric MOX fuel, (U, Pu)O2-x, Carbajo et al. [4]
recommended that the thermal expansion is multiplied by a factor of [1+3.9(±0.9)x] with x being the deviation from stoichiometry. Although this recommendation developed by Martin [19] is valid for the MOX fuels up to 1800 K, we employ it even for other actinide oxide fuels up to their solidus temperatures.
7.3.1.2. Melting point
The melting point of an oxide fuel depends on the fuel composition, O/M ratio or the oxygen content and burnup. Here, for MOX fuels, a correction only for UO2 and PuO2 fractions is considered. Carbajo et al. [4] recommended the solidus and liquidus curves of stoichiometric UO2–PuO2 solutions given by Adamson [1]. They are expressed by the following polynomial expressions:
where y is the mole fraction of PuO2. Here, the melting temperatures of stoichiometric, unirradiated UO2 and PuO2 are taken as 3120±30 K and 2701±35 K, respectively.
7.3.1.3. Heat capacity and enthalpy
Carbajo et al. [4] recommended the following expressions developed by [9, 10] for the solid enthalpy relative to the solid at 298.15 K:
where hs is in J mol–1 and T is in K. The constants used in the above equation are given as C1 = 81.613, C1 = 2.285×10–3, C1 = 2.36×107 and θ = 548.68 for UO2, C1 = 87.394, C1 = 3.978×10–3,
C1 = 0.0 and θ = 587.41 for PuO2, and Ea = 18531.7. The expression for heat capacity is given by
where cp,s is in J mol–1 K–1.
For liquid PuO2, Cordfunke and Konings [5] recommended the following value:
The above constant value is used for liquid MOX fuel as an approximation.
7.3.1.4. Heat of fusion
Carbajo [4] recommended the following value for the heat of fusion of UO2 calculated by Fink [10]:
For the heat of fusion of PuO2, we take the following value recommended by Cordfunke and Konings [5]:
The heat of fusion value for UO2–PuO2 compositions is calculated from the following relationship [8]:
where Tm is the melting temperature, which is taken as the solidus temperature for the mixed oxide compositions. For the MOX fuels with 20% mole fractions of PuO2, the above equation yields
7.3.1.5. Thermal conductivity
Carbajo et al. [4] recommended the following expressions for the thermal conductivity of fully dense fuels:
for UO2
for MOX with PuO2 concentrations between 3 and 15 %
where κs is in W m–1 K–1, τ is the variable T 1000 and T is in K. Equation (17) was developed by [10]. Equation (18) was a combination of the correlations developed by Duriez et al. [7] and Ronchi et al. [21].
The thermal conductivity of solid PuO2 expressed as a function of temperature by the relationships of the form
with constants A and B was determined by Gibby [12] and Fukushima et al. [11]. They obtained:
where T is in K. Equation (19) using the above two sets of constants agrees well with each other for fully dense solid, which is calculated by Eq. (11), within experimental uncertainty of the measurements. Here, we take the constants determined by Gibby [12].
7.3.2. Americium dioxide 7.3.2.1. Thermal expansion
The lattice constant of AmO2, which has an fcc fluorite-type crystal structure, is 537.72 pm [5].
Assuming the molecular weights of AmO2 is 271.12 g mol–1, which is based on the Am-isotope vectors shown in Table 4, this lattice constant gives solid densities of AmO2 at 293 K of 11625 kg m–3. The thermal expansion of americium oxides has not been reported. Here, we apply the same values as the thermal expansion of solid UO2 and MOX fuels, which is given by Eq. (12), to that of AmO2-x in actinide oxide fuels.
7.3.2.2. Melting point
Zhang et al. [27] quoted the following values of the melting points of AmO2 and AmO1.5:
Here, we take the value of 2448 K for the melting point of AmO2-x in actinide oxide fuels.
7.3.2.3. Heat capacity and enthalpy
The estimated heat capacity of solid AmO2 [24] is given by:
where cp,s is in J K-1 mol-1 and T is in K. Equation (20) yields the following expression for the enthalpy of AmO2:
where hs is in J mol–1 and T is in K.
For liquid AmO2, [27] assumed that its heat capacity have the same constant value as PuO2 [5]:
7.3.2.4. Heat of fusion
The values of heat of fusion of americium oxides were estimated to be 59±20 kJ mol–1 for AmO1.5,
within their uncertainty ranges, and hence we adopt the value of AmO2 as a standard one for americium oxides:
7.3.2.5. Thermal conductivity
The measured thermal conductivity of americium oxides is quite lower than the thermal conductivity of other actinide dioxides. For example, Bakker and Konings [2] quoted measured values at 333 K:
0.69 W m–1 K–1 for AmO2 and 0.82 W m–1 K–1 for Am2O3. On the other hand, the thermal conductivity of AmO2-x can be approximately expressed by the following formula as a function of temperature and oxygen content of americium oxide [2]:
Although Eq. (23) overestimates the measured values, Bakker and Konings [2] suggested that this is due to non-stoichiometry of the samples used in the measurement. Therefore, Eq. (23) is used for the rough estimation of the thermal conductivity of AmO2-x.
7.3.3. Curium dioxide 7.3.3.1. Thermal expansion
The lattice constant of CmO2, which has a fluorite-type crystal structure at room temperature, is 535.9 pm [16]. Assuming the molecular weight of CmO2 is 274.56 g mol–1, which is based on the Cm-isotope vectors shown in Table 4, this lattice constant gives solid densities of CmO2 at 293 K of 11849 kg m–3.
The fractional change in length of CmO2 with temperature is expressed by the following equation [16]:
where Δl(T) l(298K) is in %, Δl(T) is zero at 298 K, and T is in K. CmO2 becomes unstable above 700 K decomposing via two intermediate compositions to Cm2O3. In comparison with MOX fuels, Eq. (12) gives the thermal expansion that is very close to the values obtained from Eq. (24).
Here, assuming that the lattice structure of corium oxide in the actinide oxide fuels is similar to those of UO2 and MOX fuels, we apply Eq. (12) to the thermal expansion of CmO2.
7.3.3.2. Melting point
Since CmO2 decomposes above 700 K, its melting point is unavailable. For Cm2O3 the following melting point was recommended by Konings [16]:
We use the above value in the calculation of the melting point of actinide oxide fuels with curium.
7.3.3.3. Heat capacity and enthalpy
Konings [16] gave the following expressions for the heat capacity of curium oxides:
for CmO2
for Cm2O3
where cp,s is in J K-1 mol-1 and T is in K. Equation (26) is rewritten for CmO1.5:
Since Eq. (25) covers only rather low temperature range and there is not large difference between Eqs.
(25) and (27), we apply Eq. (27) to the heat capacity of CmO2-x as an approximation. The expression for the enthalpy of CmO2-x is then given by
where hs is in J mol–1 and T is in K.
There is no heat capacity data for liquid CmO2-x. Here, we assume the same constant value as PuO2 [5]:
7.3.3.4. Heat of fusion
The heat of fusion of curium oxides has not been reported. Here, we apply the same values as the heat of fusion of MOX fuels, which is given by Eq. (16), to that of CmO2-x, in actinide oxide fuels.
7.3.3.5. Thermal conductivity
The thermal conductivity of CmO2 has not been measured. Konings [16] estimated the thermal conductivity of CmO2 as indicative values: 7-10 W m–1 K–1 at 298.15 K and 3.8-4.6 W m–1 K–1 at 650 K. For the thermal conductivity of monoclinic Cm2O3, Konings [16] also gave the following recommendation:
In comparison with the thermal conductivity of MOX fuels, Eq. (18) with x = 0.0 gives values close to Konings’ estimation for CmO2. On the other hand, for Cm2O3, Eq. (18) with x= 0.5 provides rather low thermal conductivity, compared with Eq. (29) in its temperature range of validity. Here, we propose to use Eq. (18) for the thermal conductivity of CmO2-x by replacing x with 0.25x:
Equation (30) can include reasonably well not only the dependence on the difference from stoichiometry, but also the behavior of oxide fuels at high temperatures as an approximation.
7.3.4. Magnesium oxide 7.3.4.1. Thermal expansion
The lattice constant of MgO, which has a cubic crystal structure, is 421.3 pm. This lattice constant gives solid densities of MgO at 293 K of 3580 kg m–3 using the molecular weight of 40.30 g mol–1. Jacobs and Oonk [14] gave the following polynomial fit to the volumetric thermal expansion coefficient of MgO:
where α is in K–1 and T is in. This equation is related to the linear thermal expansion coefficient by
Then, l(T) l(Tref)is expressed by
7.3.4.2. Melting point
The measured melting point of MgO was reported as Tm = 3250±20 K by Ronchi and Sheindlin [22].
The eutectic formation in binary systems of MgO with plutonium and americium oxides has been estimated by Zhang et al. [27, 28]. The melting temperatures of MgO-PuO2-x and MgO-AmO2-x are expressed by the following equations as a function of oxygen content of actinide oxide:
For MgO-AmO2-x (Zhang et al., [27])
for MgO-AmO2-x (Zhang et al., [27])
where N is the variable [(2−x)−1.62] 0.38. Here, the temperature at which the liquid appears first in the phase diagram is referred to as the melting temperature of the system, which can be the solidus or eutectic temperature. For the MgO-PuO2-x system, the equilibrium oxygen pressure of PuO2-x is much higher than the dissociation pressure of MgO. On the other hand, the melting temperature of the MgO-AmO2-x system becomes very low (1930 K) at low oxygen potentials. This is accompanied by chemical dissociation processes of MgO.
7.3.4.3. Heat capacity and enthalpy
The enthalpy data of solid MgO up to the melting point have been given by Schick [23] and Cox et al. [6] as a set of tabulated data. Here, we adopt the temperature dependent correlation given by Schick [23]:
where cp,s is in J K–1 mol–1 and T is in K. This yields the enthalpy correlation:
where hs is in J mol–1.
For liquid MgO, Cox et al. [6] gave the following estimation of heat capacity:
7.3.4.4. Heat of fusion
The values of heat of fusion of MgO have been reported by Schick [23] and Cox et al. [6]. These estimations indicate a very close value. Here, we adopt the following value given by Cox et al. [6]:
7.3.4.5. Thermal conductivity
The recommended values of thermal conductivity of MgO have been tabulated by Touloukian et al.
[26]. Their data for 98% dense, polycrystalline MgO were used to obtain a thermal-conductivity correlation as a function of temperature. The thermal conductivity of 100% dense MgO was evaluated using the well-known Maxwell-Eucken equation, Eq. (11), and then fitted to the following function in the temperature range 300-2300 K:
where κs is in W m–1 K–1 and T is in K. This equation is extrapolated up to the melting point.
7.4. Material properties of ADS core 7.4.1. Fuels
Table 1 shows specifications of the driver MOX fuel used by SCK•CEN in the predesign studies of MYRRHA ADS. Its properties were evaluated mainly based on the recommendations by Carbajo et al.
[4]. The basic properties of the driver MOX fuel are summarized in Table 2. The temperature dependent properties are listed in Table 3.
TABLE 1. SPECIFICATIONS OF THE MOX FUEL
TABLE 2. BASIC PROPERTIES OF THE MOX FUEL
TABLE 3. TEMPERATURE DEPENDENT PROPERTIES OF THE MOX FUEL WITH 95% TD
Table 4 shows typical specifications of IMF considered as candidate fuels for transmutation in fast ADS. Its melting temperatures were estimated considering the eutectic formation in the MgO-PuO2-x and MgO-AmO2-x systems. For x = 0.12, the melting temperatures for MgO-PuO1.88 and MgO-AmO1.88 systems are evaluated as 2 475 and 2 334 K, respectively. Considering the mole factions of constituents, the resultant melting temperatures of the IMF fuel particles and the IMF matrix are given as 2 441 K and 2 482 K, respectively. The basic properties of IMF are summarized in Tables 5 and 6. The temperature dependent properties are listed in Table 7 and 8.
TABLE 4. SPECIFICATIONS OF THE INERT MATRIX FUEL
TABLE 5. BASIC PROPERTIES OF THE INERT MATRIX FUEL (FUEL/MATRIX VOLUME FRACTION: 40/60)
TABLE 6. BASIC PROPERTIES OF THE INERT MATRIX FUEL (FUEL/MATRIX VOLUME FRACTION: 50/50)
TABLE 7. TEMPERATURE DEPENDENT PROPERTIES OF THE INERT MATRIX FUEL WITH 90% TD (FUEL/MATRIX VOLUME FRACTION: 40/60)
TABLE 8. TEMPERATURE DEPENDENT PROPERTIES OF THE INERT MATRIX FUEL WITH 90% TD (FUEL/MATRIX VOLUME FRACTION: 50/50)
7.4.2. Structure
Harding et al. [13] reported some recommendations for the thermophysical properties of type 316 stainless steel. They assumed that the composition of the alloy is 65.4% Fe, 17% Cr, 13.5% Ni, 1.7% Mn and 2.4% Mo by weight. The basic solid properties are summarized in Table 9. The temperature dependent properties are listed in Table 10.
TABLE 9. BASIC PROPERTIES OF TYPE 316 STAINLESS STEEL [13]
TABLE 10. TEMPERATURE DEPENDENT PROPERTIES OF TYPE 316 STAINLESS STEEL [13]
7.4.3. Coolant
Thermophysical properties of lead-bismuth binary alloy with a eutectic composition (44.5 weight%
lead and 55.5 weight% bismuth) were evaluated up to the critical point based on our recent study [20].
We assumed that the LBE vapor is composed of monatomic lead and bismuth and diatomic bismuth components, and that the liquid LBE is a non-ideal mixture of lead and bismuth. The obtained results are in good agreement with the database of thermophysical properties of the molten LBE used by SCK•CEN [25]. The basic properties of LBE are summarized in Table 11. The temperature dependent properties obtained with EOS developed by Morita et al. [20] are listed in Table 12.
TABLE 11. BASIC PROPERTIES OF LEAD-BISMUTH EUTECTIC
TABLE 12. TEMPERATURE DEPENDENT PROPERTIES OF LEAD-BISMUTH EUTECTIC ON SATURATION CURVE
7.5. Conclusion
Thermophysical properties of IMF and MOX fuel were estimated for use in the IAEA benchmark problems. Due to a lack of experimental data published in literature, the basic properties such as the solid density, heat capacity and thermal conductivity were estimated up to the melting point based on empirical and theoretical models extrapolating low temperature data of its constituents.
Recommendations were also presented for solid properties of type 316 stainless steel and for liquid and vapor properties of LBE. We expect that the set of thermophysical properties presented here could be utilized as a basis for the development of the standard database for use in the assessment of reactor performance and safety behavior of ADS.
REFERENCES TO CHAPTER 7
[1] ADAMSON, M.G., et al., Experimental and thermodynamic evaluation of the melting
[1] ADAMSON, M.G., et al., Experimental and thermodynamic evaluation of the melting