According to kinetic gas theory, the lattice thermal conductivity above Debye temperature for an ideal lattice can be expressed by [12]

к = 0.33 CvvL (35)

where, Cv is the specific heat at constant volume, J/m3 K, v is the velocity of sound in solid, m/s, and L is the mean free path of scattered waves (the phonon wave­length), m.

Above the Debye temperature, thermal conductivity of electric insulators decreases with increasing temperatures. Since atomic vibrational frequency increases with temperature, an increase in wave scattering is anticipated which are shown to be due to phonon interaction. Thermal energy is transferred by the Umklapp process, in which two phonons interact to form a third [12]. According to this theory, lattice thermal conductivity is inversely proportional to absolute temperature and this becomes a minimum when phonon wavelength becomes less than the mean distance between the scattering centers. For crystalline solid, the minimum distance between the scattering centers is the interatomic distance which is the lattice parameter, ao. Therefore, the above equation becomes [12]:

(k)mn= 0.33 Cvva0 (36)

In solids, phonon-phonon scattering is due to the anharmonic components of crystal vibrations. Lattice anharmonicity increases with the mass difference between anions and cations in the ionic material and is greatest in UO2 or PuO2 [38, 91]. As a result, the thermal conductivity of the oxides of the actinide metals is considerably lower than that of most other crystalline oxides. The kinetic theory of gases shows that the collision mean free path is given by the reciprocal of the product of the collision cross-section rP and the density of scattering points (nP):

L =(1/apnp) (37)

The deviation from stoichiometry and the presence of foreign atoms or porosity result in lower values of к in actinide oxides. Further, it can be shown that the phonon mean free path should vary as 1/T. In general, phonon-phonon scattering and phonon-impurity scattering are the dominant mechanisms of the thermal conductivity in ceramics. Klemens [92] has proposed a heat conduction model in materials where the phonon-phonon (Umklapp) scattering and the phonon — impurity scattering occur simultaneously. Theoretically, the phonon component of the thermal conductivity k may be written as:

k = (A + BT)-1 (38)

where A and B are constants and T is the absolute temperature.

Thermal resistivity (R), which is the reciprocal of thermal conductivity (k), of the above oxides can be described by the following equation:

R = 1/k = A + BT (39)

The first term, A, in Eq. (39) represents the defect thermal resistivity. This results from the phonon interactions with lattice imperfections, impurities, isoto­pic, or other mass differences as well as bulk defects such as grain boundaries in the sample. The influence of substituted impurities on the thermal conductivity is described by the increase of the parameter A. The second term in Eq. (39), namely BT, represents the intrinsic lattice thermal resistivity caused by phonon-phonon scattering [1, 2]. As the temperature increases, this term becomes predominant. The parameter B remains nearly constant by substitution. The constants A and B can be obtained from the least squares fitting of the experimental data.

The thermal conductivity of nuclear ceramics is strongly influenced by the stoichiometry. Deviations from stoichiometry produce point defects, most likely oxygen vacancies or metal interstitials in hypostoichiometric compounds and oxygen interstitials or metal vacancies in hyperstoichiometric compounds. Intro­duction of point defects into the oxygen ion sublattice or substitution of Th for U on the cation sublattice provides additional centers from which phonon scattering occurs. It is reported that there is a drastic change in the uranium vacancy con­centration on varying O/M ratio around the stoichiometric composition. Many reports are available on the effect of stoichiometry on the thermal conductivity of UO2 and (U, Pu)O2 samples [1]. Thermal conductivities decrease as their hyperstoichiometry, x, increases. At low temperatures, thermal conductivity of mixed oxide can be described by a modified equation of (38) as [93, 94]:

k = 1/[A(x, q) + B(x, q)T] (40)

where, x and q denote the extent of nonstoichiometry and the Pu/Th content in the UO2 lattice, respectively. The limited amount of experimental information avail­able suggests that the coefficient A depends primarily on the O/M ratio and only very weakly on the plutonium content. The coefficient A may be written as

A = A0 + DA(x) (41)

where, A0 is very nearly equal to the A value of pure UO2. The perturbation DA arises from interactions of point defects with lattice. The magnitude of DA is proportional to the defect atom fraction and to a measure of the cross section of the defect for phonon scattering. The latter is proportional to the square of the dif­ference between the atomic radius of the defect (r{) and that of the host atom (r). The mass difference between the impurity atom and the host atom may also influence A, but this contribution is not significant in mixed oxide fuel materials. A can also be represented by the following equation as

A = [(p2Vh)/(3hv2)] X Ci, (42)

i

where V, в, h, and v denote the average atomic volume, Debye temperature, Planck’s constant, and phonon velocity, respectively. The term RC is the sum of the cross-sections of all the phonon-defect scattering centers. The analysis of the lattice defect thermal resistivity and the evaluation of phonon scattering by the various defect scattering centers in pure and mixed actinide oxides have been carried out by several authors [91-94]. Accordingly, A of Eq. (42) can be given as

A = C(C + Г0), (43)

where C = (p2Vh)/(3hv2). Ги is the scattering cross-section arising from U sub­stitution and Г0 is that from all other native defects present in the sample. The scattering cross-section Ги can be expressed in terms of the mass and size dif­ference of the substituted atom over that of the host [95]:

Ги = x(1 — x) x [(AM/M)2 + E(Ar/r)2], (44)

where, x is atomic fraction of substituted U in place of Th, AM and Ar are the mass and radius difference between U and Pu/Th atom, respectively, M and r are average mass and radius of the substituted atom, and E is an adjustable parameter which represents the magnitude of lattice strain. From the above, it is clear that scattering cross-section depends upon the mass difference between Th and U atoms, size difference between Th and U atoms, charge of U ion and microstructure.

Thermal transport by electrical charge carriers can also contribute to thermal conduction at high temperatures. The ratio between thermal and electrical con­ductivities of metals can be expressed in terms of the ratio:

Lc = k/rT = p2k2/3e2 = 2.45 x 10-8 WX/K2, (45)

which may be called the Wiedemann-Franz ratio or the Lorenz constant. In the above equation, r is the electrical conductivity, e is the electronic charge, and k is Boltzmann’s constant. Thermal conductivity of a solid can be measured by two methods:

1. By determining the stationary heat flow through the specimen, which gives k directly,

2. By determining the variation of the temperature at a fixed plane, that is a specimen surface, due to an induced nonstationary heat flow which gives the thermal diffusivity, a.

Since the second method is more versatile and requires smaller specimen, it has become a standard method for determining k for T > 600 K. For lower temper­atures, the first method is more suited.

For the thermal diffusivity measurement, the sintered pellet was sliced into discs of about 10 mm diameter and 2 mm thickness using a low speed cut-off wheel. A pulse of laser was projected on to the front surface of the pellet and the temperature rise on the rear side of the pellet was recorded as a transient signal by using an infrared detector. The thermal diffusivity (at) was calculated from the following relationship:

at = WL2 /nt 1=2 (46)

where t1/2 is the time required in seconds to reach half of the maximum temper­ature rise at the rear surface of the sample and L is the sample thickness in millimeter. W is a dimensionless parameter which is a function of the relative heat loss from the sample during the measurement. The data have to be corrected for radiation heat losses by the method of Clark and Taylor [96].

Unlike UO2 or PuO2, ThO2 is a semitransparent material to wavelengths of the infrared region. For a laser flash experiment, all the energy of the laser pulse is not absorbed on the front face of the sample, but also in volume. Also, the temperature measurement on the rear face is skewed as the pyrometer may receive radiation produced not only at the sample surface, but also in volume. These difficulties are overcome if the faces of the samples are given a coating by graphite. A coating of graphite on both faces was used in order to make sure that the energy of the laser is absorbed on the front face and to improve the temperature recording on the rear face.

Heat transport through materials is described by two properties: thermal con­ductivity, k (under steady state conditions) and thermal diffusivity, at (under transient conditions). These two properties are related by the expression:

k(T) = at(T) • p(T) • Cp(T), (47)

Where, p the density of the material and Cp its specific heat at constant pressure. The specific heat of mixed oxide like (Th1_yPuy)O2 solid solutions was calculated from the literature values of specific heats of pure ThO2 and PuO2 and subse­quently using Neumann-Kopp’s rule. The following equations were used to calculate Cp of (Th1-yPuy)O2:

Cp(Th1_yPuy)O2 = (1 — y) • Cp(ThO2) + y • Cp(PuO2), (48)

where y is the weight fraction of PuO2.

Effect of Porosity on Thermal Conductivity

Attempts to evaluate the decrease in thermal conductivity due to porosity (P) have been made by Eucken in as early as 1932. There are many relations in the literature describing the effect of porosity on thermal conductivity. Some of them are listed below [1, 64, 97-106]:

1. Loeb км — (1 _ P) kjD (i)

2. Modfied Loeb kM — (1 — aP) кто where 2 < a < 5 (ii)

3. Kampf and Karsten kM — (1 — P2/3)кто (iii)

4. Biancharia kM — [(1 — P)/(1 — (b — 1)P)] kTD b = 1.5 for spherical pores (iv)

5. Maxwell-Eucken kM — [(1 — P)/(1 + bP)] кто (v)

6. Brand and Neuer kM — (1 — rP) kjo where r = 2.6 — 0.5 (T + 273)/1000 (vi)

7. Schultz kM — (1 — P)c кто (vii)

(kM and kTD are the thermal conductivities, respectively, in presence and absence of porosity P, 0 < P < 1).

Schultz [106] has theoretically shown that, for spherical pores distributed randomly, у of Eq. (vii) has a value of 1.5. However, in reality the above coeffi­cients for fuel pellets are larger (у > 1.5), due to the porosity being neither spherical nor uniformly distributed [1]. IAEA [40] has recommended the value of a = 2.5 ± 1.5 for the modified Loeb equation for 0 < P < 0.1. Inoue, Abe, and Sato [107] experimentally showed that у = 2.4 for 0.044 < P < 0.470 and reported that b = 2 (Eq. (v)). The IAEA recommendation (a = 2.5 ± 1.5) is in agreement with other experimenters [1, 40]. Among the above, Eq. (i) under predicts the data and Eq. (iv) accounts for the shape of the pores.

The melting point of UO2 given in MATPRO [28] is 3,113.15 K. This temperature is based on the measurement made by Brassfield et al. [29] and the equations for the solids and liquids boundaries of the UO2-PuO2 phase diagram given by Lyon and Bailey [30]. The recommended values by ORNL [31] for UO200 are

3.120 ± 30 K and for PuO2 is 2,701 ± 35 K. This value for UO2 has been recommended by Rand et al. [32] from their analysis of 14 experimental studies of the melting temperature of UO2. This recommendation of Rand et al. [32] was accepted internationally and was recommended in the assessment of UO2 prop­erties by Harding et al. [33] in their 1989 review of material properties for fast reactor safety. The melting point of UO2, according to Latta et al. [34] is 3,138 ± 15 K, which is considerably higher than that of Christensen (3,073 K) [35]. Belle and Burman [12] recommended the melting temperatures of UO2 as

3.120 K, with an error of probably ±30 K.

In recent experimental measurements of the heat capacity of liquid UO2 using laser heating of a UO2 sphere, Ronchi et al. [36] made several measurements of the freezing temperature of UO2 on different samples. For specimens in an inert gas atmosphere with up to 0.1 bar of oxygen, they obtained melting points in the interval 3,070 ± 20 K. Higher melting temperatures (3,140 ± 20 K) were obtained for samples in an inert gas atmosphere without oxygen. The variation in melting temperature is in accordance with the expected lower oxygen-to-uranium (O/U) ratio in the latter samples. The melting point of UO2 drops on variation O/M ratio around stoichiometry: for example, if the melting point of stoichiometric UO2 is 3,138 K, its value drops to 2,698 K at an O/U ratio of 1.68 and to 2,773 K at an O/U ratio of 2.25. The effect of irradiation is lowering of the melting point of UO2. At a burnup of 1.5 x 1021 fissions/cm3, it has been reported [12] that the melting point drops to 2,893 K. Typical values of melting points for UO2 and PuO2 obtained by various authors are given in Table 2.

The thermal conductivity of ThO2 up to 1,800 K is reasonably well established (Table 12). Most of the data were derived from thermal diffusivity measurements. Peterson and Curtis [26] compiled data on thermal conductivity of ThO2 to about 2,000 K. Bakker et al. [38] systematically evaluated the data of various authors. They analyzed the data of Pears [102], Rodriguez et al. [82], McEwan and Stoute [103], Belle et al. [104], Peterson et al. [105], Faucher et al. [108], Kingery et al. [109], McElroy et al. [110], ARF [111], Weilbacher [112] and DeBoskey [113].

Assessing the A and B parameters has the advantage that data sets that were determined in different temperature ranges can easily be compared and that data sets with extremely large or small A and B parameters can be rejected. On this basis, Bakker et al. [38] rejected many data and accepted only that data which shows a small variation between the A and B parameters. Hence data of Murabayashi [114], McElroy et al. [110], Koenig [115] and Springer et al. [57] are only used in their assessment. The A and B parameters were averaged, which yielded A = 4.20 x 10-4 mKW-1 and B = 2.25 x 10-4 mW-1 and these values can be used as the recommended values for 95 % dense ThO2 in the temperature

range between 300 and 1,800 K. Hence, thermal conductivity of pure ThO2 can be expressed as:

kThO2 (W/mK) = (4.20 x 10-4 + 2.25 x 10-4 T) 1 (49)

Belle and Berman [12] reported the following equation for the thermal con­ductivity of 100 % dense ThO2 in the temperature range 298-2,950 K,

kThO2(W/mK) = (0.0213 + 1.597 x 10-4 Г)-1 (50)

To evaluate the thermal conductivity beyond 2,950 K, Belle and Berman first obtained an expression for thermal diffusivity up to 2,950 K as

«ThO2 (m2/s) = (-34191.1 + 561.28 T)-1 (51)

Assuming there is no discontinuity, Belle and Berman [12] extrapolated thermal diffusivity data values from 2,950 to 3,400 K. Their results along with others are shown in Fig. 12. The only high temperature data available is that of Weilbacher [112] which was fitted by a dashed line. The fitted data of Cozzo et al. [95] and Kutty et al. [84] represent the lowest and highest values in the low temperature range.

Figure 13 shows the calculated value of thermal conductivity for fully dense ThO2 from ambient to 3,400 K. There is a sudden increase in conductivity at 2,950 K discontinuity, from 2.03 to 3.05 W/m K, which represents the change in the heat capacity occurring at that point. The lowest thermal conductivity in the

Fig. 12 Thermal diffusivity of ThO2 as a function of temperature. Data of various authors are plotted together. Dotted line are fitted data of Cozzo et al. [95], dashed line that of Weilbacher [112] and solid line that of Kutty et al. [84]

entire temperature range was 2.03 W/m-K at 2,950 K. Belle and Berman [12] estimated minimum values in conductivity using Eq. (36). They assumed that:

1. Phonon velocity can be approximated to (E/q)05,

2. Equation dealing temperature variation in E can be extrapolated to higher temperatures, and

3. Minimum value of phonon mean free path can be approximated to lattice parameter.

They calculated the minimum in thermal conductivity for ThO2 at 2,950 K as 2.07 W/m-K which is very near to the value (2.03 W/m-K) shown in Fig 13. Thermal conductivity data of UO2 [50] and PuO2 [6, 40] are also shown in the same figure. The lowest value for UO2 is 2.19 W/m-K at 1,970 K. The upswing in

20

■ Murabayashi et al. • Moore et al. a Pillai and Raj

♦ ARF ^ Kingery a Bakker et al. v Jain et al. Ф Murti and Mathews 4- Lu et al. > Czzo et al. Ф Tecdoc-1496 ☆ Weilbacher ев Kutty et al.

і14 —

A*"

2

500 1000 1500 2000 2500 3000

Temperature, K

thermal conductivity in UO2 at * 1,970 K can be explained in terms of the electronic contribution. On the other hand, increase in thermal conductivity beyond 2,950 K in ThO2 is not result of electronic contribution, but is associated with increase in heat capacity. Thermal conductivity data of ThO2 reported by various authors are shown in Fig. 14.

Phase diagram studies of the ThO2-UO2 system have been reported by several authors. Lambertson et al. [21] used a quench technique, Christensen [27, 37] a tungsten filament technique, whereas Latta et al. [34] applied a thermal arrest method. The phase diagram of ThO2-UO2 can be constructed with the help of the melting points and enthalpies of fusion of the end members, assuming ideal solid solution behavior in both liquid and solid state. The results of the phase diagram measurements are given in Fig. 1 showing UO2 and ThO2 form a continuous series of solid solutions. The measurements of Christensen [27] and Latta et al. [34] show a shallow minimum at 2 and 5 mol% ThO2, respectively. The melting points calculated for some intermediary composition are shown below in Table 3 [38-49].

There is no direct experimental measurement on the heat of fusion of ThO2 or ThO2-UO2 solid solution. The most probable value is that of Fink for UO2 as 74.8 ± 1 kJ/mol [50]. The recommended value for ThO2 is 90.8 kJ/mol [40].

There are many publications numbering over hundreds dealing with thermal conductivity of UO2. Washington [116], Brandt and Neuer [117], and Fink et al. [50] made appraisals of the conductivity data found in the open literature. Brandt and Neur [117] presented a mean correlation curve of thermal conductivity versus temperature for UO2 by using data from number of sources. Their equation had three terms: the first two terms are for phonon and electronic conductions, respectively. The third term stood for the decrease in thermal conductivity resulting from dislocations created at higher temperatures. Fink et al. [50] used a different model to fit the voluminous data on UO2. They showed the evidence of a phase transition for UO2 at 2,670 K from the enthalpy measurements and sug­gested a similar transition with temperature for thermal conductivity. Fink et al. [50] suggested a relation conforming with the enthalpy and heat capacity equa­tions. Their relation is given below:

kU02 (W. m — 1.K — 1)= (A + BT + CT2) 1 + DTe-£/kT, (298 < T < 2670 K)

where A = 6.8337 x 10-2 m-K-W-1, B = 1.6693 x 10-4 m-W-1, C =

3.1886 x 10-8 m-W-1K-1, D = 1.2783 x Ш-1 W-m-1 K-2, E = 1.1608 eV, and k is the Boltzmann constant.

For 2,670 K < T < 3,120 K,

ku02 (W — m-1 — K-1) = 4.1486 — 2.2673 x 10-4 T (53)

Equations (52) and (53) fit the thermal conductivity data within an error margin of 6.2 %. The two terms in Eq. (52) represent contributions from phonons and electrons, respectively. The inclusion of a dislocation term as recommended originally by Weilbacher [112] to fit his high temperature data was not justified.

In 2006, IAEA [40] made a detailed survey on thermal conductivity data and recommended equation for the thermal conductivity of 95 % dense solid UO2 which consists of lattice term and a term suggested by Ronchi et al. [118] to represent the small-polaron ambipolar contribution to the thermal conductivity. The lattice term was determined by a least squares fit to the lattice contributions to the thermal conductivity obtained by Ronchi et al. [118], Hobson et al. [119], Bates [120], Conway et al. [121] and Godfrey et al. [122]. The recommended equation for thermal conductivity of solid 95 % dense UO2 is:

kuo2 = [100/(7.5408 + 17.692t + 3.6142t2)] + (6400/t25) exp(-16.35/t)

(54)

where, t is T/1,000, T is in K, and k is the thermal conductivity in W-m-1 K — . Thermal conductivity values for 100 % dense UO2 or for a different density may be calculated using the porosity relation derived by Brandt and Neurer [117], which is:

k0 = kp/(1 — op), (55)

where, o = 2.6-0.5t. Here, t is T/1,000 where T is in K, p is the porosity fraction, kp is the thermal conductivity of UO2 with porosity p, and k0 is the thermal conductivity of fully dense UO2.

Uncertainties in thermal conductivity values for 298-2,000 K are 10 %. From 2,000 to 3,120 K, the uncertainty increased to 20 % because of the large dis­crepancies between measurements by different investigators [40]. Typical thermal conductivity of 95 % dense UO2 as a function of temperature is given in Fig. 15.

The lattice term has traditionally been determined by fitting the low tempera­ture thermal conductivity data because the lattice contribution dominates the thermal conductivity at low temperatures. Figure 16 shows the total thermal conductivity, the lattice contribution, and the ambipolar contribution as a function of temperature that have been calculated from the equation of Ronchi et al. [118], which is given below:

kuo2 = [100/(6.548 + 23.533t)] + (6,400/t25) exp(-16.35/t), (56)

ЮО 400 MO 900 1000 I?00 1*00 I COO WOO 7000 7900 7400 7000 7000 9000 3700

Temperature, К

Temperature, K

where t is 7/1,000, T is in K, and k is the thermal conductivity for 95 % dense UO2 in W m-1K-1. Below 1,300 K, the ambipolar term is insignificant and the total thermal conductivity equals the lattice contribution. Although the ambipolar term begins to have a significant contribution to the total thermal conductivity above
1,300 K, it is not larger than the lattice contribution determined by Ronchi et al. until 2,800 K. Even at 3,120 K, the lattice contribution is still significant.

No data is available on thermal conductivity of liquid ThO2. Based on an initial review of the limited data [12, 40] on the thermal conductivity and thermal diffusivity of liquid UO2, the liquid thermal conductivity is in the range of 2.5-3.6 W-m-1 K-1. Liquid thermal diffusivities range from 6 x 10-7 to 11 x 10-7 m2 s-1. The uncertainty in the thermal conductivity and thermal dif — fusivity of liquid UO2 is approximately 40 % [40].

Freshley and Mattys [41, 42] have shown that ThO2 and PuO2 form a complete solid solution (Fig. 2) in the whole composition range like ThO2 and UO2. A continuous series of solid solution has also been reported by Mulford and Ellinger [44]. They found only a single fluorite structure by X-ray diffraction (XRD) and also showed that the lattice parameter varied linearly with composition.

Dawson [45] has made magnetic susceptibility measurements on PuO2 and ThO2 mixtures and implied that PuO2 and ThO2 form solid solutions which follow the Vegard’s law. The lattice parameter of fluorite type cubic phase was found to decrease regularly from 0.5597 nm for ThO2 to 0.5396 nm for pure PuO2 [43]. Due to the limited amount of experimental data, a more accurate assessment of the phase diagram is not yet possible. The melting point of the ThO2-PuO2 solid solutions, containing various amounts of PuO2, was measured in helium. The melting point of the specimens containing less than 25 % ThO2 was found to be unchanged as shown in Fig. 3 [42]. The melting point of a PuO2 sample (whose purity was not specified) used in their study is 2,533 K, which is 130 K lower than the assessed value of Table 2.

2400

0 10 20 30 40 50 60 70 80 90 100

Mole % ThO2

3700

3600

3500

3400

3300

3200

3100

3000

Q.

£ 2900

0)

2800 2700 2600 2500

It is well known that the thermal conductivity of ThO2 is higher than that of UO2 by *50 % over a significant range of temperature. Berman et al. [123] made a systematic attempt to correlate thermal conductivity, temperature, and composi­tion for ThO2-UO2 system in the early 1970s. Belle and Berman [12] updated the thermal conductivity correlation to 3,400 K by making use of the enthalpy data. Some information is available in literature for thoria—urania mixtures are from the work of Murti and Mathews [124], Lucuta et al. [125], Pillai et al. [93], Belle et al. [104], Kingery et al. [109], Berman et al. [123], IAEA-TECDOCs etc. (Table 13) but more data are still needed to completely characterize the thermal conductivity of (Th, U)O2 fuel pellets. As a rule, in a homogeneous unirradiated mixture of ThO2-UO2, the thermal conductivity is somewhat higher than the thermal conductivity of unirradiated UO2, depending on the temperature and the relative content of the ThO2. However, it is worth mentioning that thermal con­ductivities of (Th0 655U0.345)O2 and (Th0 355U0.645)O2 pellets were found to be lower than that of both pure ThO2 and UO2 and degradation is large at low temperatures, but smaller as the temperature increases [67].

Mcelroy et al. [110] have measured the thermal conductivity of sol-gel-derived ThO2 fuels from 80 to 1,400 K and compared with similar measurements on UO2. Murabayakshi et al. [114] reported the thermal conductivity of ThO2 pellets having densities ranging from 90 to 95 %. In respect of the porosity dependence of the thermal conductivity, the experimental results deviated significantly from the relationship derived by Loeb, and a modified Maxwell model was introduced to explain the data. Jain et al. [126] reported thermal diffusivity of a range of thoria — lanthana solid solutions in the compositional range from pure thoria to 10 mol% LaOi.5 by the laser-flash method covering a temperature range from 373 to 1,773 K, and reported that thermal conductivity of thorium oxide-lanthanum oxide solid solutions decreases with increasing lanthanum content and tempera­ture. Ronchi et al. [118] measured thermal conductivity of (Th088U0.12)O2 in the temperature range of 573-1,573 K. Ferro et al. [127] evaluated diffusivity of (Th0.94U0.06)O2 and (Th0.90U0.i0)O2 from 650-2,700 K. Lemehov et al. [4] pre­sented a model for the lattice thermal conductivity of pure and mixed oxides based on the Klemens-Callaways approach for the dielectric heat conductance modeling

and on some correlations between thermoelastic properties of solids. The thermal conductivity of ThO2 and Th0 .98U0. 02O2 was measured from 300 to 1,200 K by Pillai and Raj [93] and they showed that the decrease in thermal conductivity of Th0.98U0.02O2 over that of ThO2 is due to the enhanced phonon-lattice strain interaction in the oxide. Murti and Mathews [124] measured thermal conductivity on thorium-lanthanum mixed oxide solid solutions covering a temperature range from 573 to 1,573 K and a compositional range from 0 to 30 mol% LaO1.5 and reported that thermal conductivity of the solid solutions were found to decrease with increase in lanthanum oxide content or temperature. Kutty et al. [84] mea­sured thermal conductivity of ThO2, ThO2-4 % UO2, ThO2-10 % UO2 and ThO2- 20 % UO2 made by coated agglomerate pelletization (CAP) process and reported that thermal conductivity decreased with UO2 content. A study carried out by

INEEL [67] shows that ThO2 has a higher thermal conductivity than UO2, but (Th, U)O2 containing 65 or 35 wt% ThO2 has similar in thermal conductivity of UO2.

An assessment of thermal conductivity data of both irradiated and unirradiated ThO2 and Th1-yUyO2 solid solutions has been made by Berman et al. [123]. They analyzed data of Springer et al. [57], Jacobs [128], Matolich and Storhok [129] and Belle et al. [12]. Berman et al. [123] suggested complex behavior of the parameters A and B of Eq. (39) on variation of the uranium content which is inconsistent with theory and data on other ThO2 or UO2 compounds containing substitutions. The assessment by Bakker et al. [38] used only those data sets that contain pure ThO2, which show a systematic decrease of the thermal conductivity on increasing UO2 content (for UO2 concentrations up to 20 %). Since good agreement exists between the variation of the A and B parameter on substitution as determined by Murabayashi [114] and the variation of A and B of comparable compounds as well as that predicted by theory, these parameters are used to obtain a recommended thermal conductivity for Thi_yUyO2. The uranium concentration dependence of the thus obtained A and B parameters were fitted to obtain an equation that is valid for uranium concentration up to 10 % and a theoretical density of 95 %:

A = 4.195 x 10-4 + 1.112 • y — 4.499 • y2, (57)

B = 2.248 x 10-4 — 9.170 x 10-4 • y + 4.164 x 10-3 • y2 (58)

The recommended equation for (Thi_yUy) O2 containing up to 10 % UO2 is:

k(Th-u)02 = [4.195 • 10-4 + 1.112 • y — 4.499 • y2

+ (2.248 • 10-4 — 9.170 • 10-4 • y + 4.164 • 10-3 • y2) • t]

The above equation is valid in the temperature range 300-1,173 K. Figure 17 shows thermal conductivity of ThO2-UO2 for various UO2 contents.

An elaborative study has been reported in IAEA-TECDOC [40] on ThO2 containing 4, 6, 10, and 20 % of UO2. The following are the recommended equations for the thermal conductivity (k) as a function of temperature (T/K) which is valid from 873 to 1,873 K:

k[Th0.96U0.04]O = 1/(-0.04505 + 2.6241 • 10-4 • T) (60)

k[Th0.80U0.20]O2 = 1/(0.02771 + 2.4695 • 10-4 • T) (61)

Subsequently, best-fit equation for thermal conductivity of (Th1-yUy)O2 of 95 % theoretical density as a function of composition (y in wt%) and temperature (T/K) has been derived, which is valid through 873-1,873 K.

k(y, T) = 1/[-0.0464 + 0.0034 • y +(2.5185 • 10-4 + 1.0733 • 10-7 • y)- T]

(62)

600 800 1000 1200 1400 1600 1800 2000

Temperature, K

The fuel density, p is an important property of the fuel and is a function of the following factors: fuel composition, temperature, amount of porosity, O/M ratio, and burnup. The theoretical densities of the materials (pT) can be calculated from the knowledge of lattice type and values of lattice parameter. Assuming that the elements form a solid solution, the theoretical density of the material can be calculated from the following relation [12]:

Pt = MsystemN/ (VNa), (1)

where, Msystem is the atomic weight of the system, N the number of atoms per unit cell, V the volume of the lattice, and Na the Avogadro constant. Thus, in the case of
a (Th1-xPux)O2 solid solution, which is a fcc fluorite-type structure, with a lattice parameter a, the theoretical density can be estimated as :

pT — 4 [(1 — x) Mxh + xMpu + 2Ma/a3Na. (2)

The density of a material as a function of temperature can be calculated using linear thermal expansion data obtained from a pushrod dilatometer, which mea­sures thermal elongation of a material with respect to temperature (T). The relation between linear thermal expansion and density is expressed [40] as p0/p = (L/L0)3, L/L0 = (1 ? AL/L0) (see Eq. (A.4) in Appendix 1 for details), so that one writes the fractional change in density as

where Ap = p-p0 is the difference between densities at temperatures T and T0.

In fluorite-type solid solutions, when the lattice parameter, a, and the molecular weight, M, are known, the theoretical density (Mg/m3) can be calculated using the relation [12]:

pT — 4M/(NAa3) . (4)

For pure ThO2, the volume of the unit cell is (0.55975 nm)3 which is equal to 1.75381 x 10_22 cm3. Density of pure ThO2 at 298 K may, therefore, be calcu­lated as 10.00 g/cm3 [12]. The theoretical density of the ThO2-UO2 solid solution can be calculated from the following equation that makes use of Eq. (4) consid­ering the additive rule for the molecular weights and cell volumes [39]:

p(Th1-yUyO2) — 4[M2 + y(M1 — M2)/[NA(a3 + y(a3 — a|)], (5)

where, M1 and M2 are the molecular weights of UO2 and ThO2, respectively, y is the molar fraction of UO2 and a1 and a2 are the lattice parameters of UO2 and ThO2, respectively. The theoretical density of the ThO2-UO2 solid solution as a function of UO2 content at 298 K is shown in Fig. 4.

The recommended equations for the density of solid uranium dioxide are based on the lattice parameter value of 0.54704 nm reported by Gronvold [46] at 293 K and thermal expansion data by Martin [47]. The above lattice parameter values are in good agreement with measurements by Hutchings [49] and are in full agreement with the recommendations of Harding et al. [48]. Assuming the molecular weight of UO2 is 270.0277, this lattice parameter gives a density at 293 K as 10.956 g/cm3. Applying the thermal expansion values of Martin [47], the density at 273 K is 10.963 ± 0.070 g/cm3. The values reported by Benedict et al. [51] and MATPRO for solid UO2 are 10.970 ± 0.070 and 10.980 ± 0.020 g/cm3, respectively. Densities of UO2 and PuO2 given by various authors are shown in Table 4.

Fig. 4 Theoretical density of the ThO2-UO2 solid solution as a function of UO2 content at 298 K [52]. (permission from Elsevier)

L(273) (9.9734 • 10-1 + 9.802 • 10-6 • T — 2.705 • 10-10 • T2 + 4.391 • 10-13 . t3)

(7)

For 923 K < T < 3,120 K,

L(T) = L(273)(9.9672 • 10-1 + 1.179 • 10-5 • T — 2.429 • 10-9 x T2 + 1.219

• 10-12 • T3). (8)

The density of solid stoichiometric UO2 or mixed oxide (MOX) with a com­position of UO2-4 % PuO2 as a function of temperature for the temperature range of 273-923 K is given by ORNL as [6]:

q(T) = q(273) (9.9734 • 10-1 + 9.802 • 10-6 • T — 2.705 • 10-10 • T2 + 4.391 • 10-13 • T3)-3

(9)

and the density of UO2 or MOX for the temperature range of 923 K to the melting temperature,

q(T) = p(273) (9.9672 x 10-1 + 1.179 x 10-5 • T — 2.429 x 10-9 • T2 + 1.219 x 10-12 • T3)-3

(10)

The recommended uncertainty in the density value is 1 % in the entire tem­perature range.

Martin [47] recommends from assessment of the available data on hyperstoi­chiometric uranium dioxide (UO2+x), using the same equations for the linear thermal expansion of UO2 and of UO2+x for x in the ranges 0-0.13 and 0.23-0.25. Therefore, Eqs. (9) and (10) are recommended for the density of UO2+x for x in the ranges 0-0.13 and 0.23-0.25. No data on the effect of burnup on density or thermal expansion of UO2 are currently available. In the absence of data, Eqs. (9) and (10) are recommended for UO2 during irradiation, in accord with the recommendation of Harding et al. [48]. The density of UO2 as a function of temperature is shown in Fig. 5.

As mentioned earlier, the phase diagram shows a continuous series of solid solutions between UO2 and ThO2. This is supported by the fact that deviations from Vegard’s law are within the uncertainties of the lattice parameter measure­ments. Also, densities of ThO2-UO2 solid solution at room temperature can be calculated using additive rule:

q(273) = 9.99003 + 0.00953 • y (in g/cm3), (11)

where y is mol% UO2.

The calculated theoretical densities, obtained from the measured lattice con­stants, for some ThO2-UO2 solid solution are given below in Table 5.

Fig. 5 Density of UO2 [6, 31] and ThO2 [40] as a function of temperature

Belle and Berman [12] calculated theoretical density of ThO2-UO2 solid solution for different UO2 content (x) from the lattice parameter data of Cohen and Berman [52]. The following equation shows the relationship between the theo­retical density and UO2 contents.

p(T) (g/cm3) = 9.9981 + 0.0094 • (x) — 8.7463 • 10-6(x)2+1.1192 • 10-7 • (x)3,

(12)

where, x is the UO2 content.

Density of (Th, U)O2 system has also been calculated as a function of tem­perature by many authors [12, 40, 53]. Momin and Venketeswarulu [54], Momin and Karkhanwala [55], Kempter and Elliott [56] and Springer et al. [57] reported the densities from the lattice and bulk expansion data. The density of (Th, U)O2 system as a function of temperature and UO2 content has been estimated from a
linear relationship of lattice parameters of (Th, U)O2 as a function of UO2 [12, 38, 57] at 298 K.

a298(nm) = 0.55972 — 1.27819 x 10-4[%UO2] (13)

A relationship for the average coefficient of linear thermal expansion in the temperature range 298-1,600 K as a function of UO2 content was obtained from the literature using the high temperature lattice parameter measurements [12, 58-61]. Theoretical density was calculated as a function of UO2 content using Eq. (12). Subsequently, the theoretical density was derived as a function of tem­perature and UO2 content from the basic mass balance equation, i. e.,

Pt. vt = Po • Vo, (14)

where, pT, p0, VT, and V0 are the densities and volumes at temperatures T and T0, respectively.

With the coefficient of thermal expansion, the following equation was derived for the theoretical density [40]:

p(T) (g/cm3) = 10.087 — 2.891.10-4 x T — 6.354.10-7(x) x T + 9.279.10-3(x) + 5.111.10-6 x (x)2,

where x is UO2 content. It is observed that the variation in density obtained from Eq. (15) and that from the literature is within ±0.28 %. The theoretical densities of UO2 and ThO2 at different temperatures are given in Table 6.

Although thoria-based fuels have been studied extensively in the past, namely in the 1970s, to our knowledge very little open literature is available for (Th, Pu)O2 [40, 130]. Only a few measurements of thermal conductivity have been made for ThO2-PuO2 fuel. Since CeO2 and PuO2 have similar thermodynamic and crys­tallographic properties [131], Murbayashi [114] tried to simulate the thermal conductivity as a function of temperature and CeO2 up to 10 wt% using Laser flash method. Jeffs [132, 133] determined the integral thermal conductivity of irradiated (Th1-yPuy)O2 containing 1.10, 1.75, and 2.72 wt% of PuO2 using a steady state method. The thermal conductivity of a mixture of ThO2 and 4 wt% PuO2 was also measured by Basak et al. [130] using the laser flash technique for the temperature range of 950-1,800 K. Recently, Cozzo et al. [95] reported that at 500 K the thermal diffusivity of the Th-MOX can be down to 50 % of that of its pure oxide components ThO2 and PuO2. The presence of the two different oxides inside the Th-MOX lattice, generate a high amount of phonon scattering centers. When temperature increases, the plutonium concentration affects the thermal diffusivity of the fuel to a lesser extent, because the phonon-phonon scattering mechanism increases with temperature and becomes predominant when compared to the lattice strains due to the presence of either Th or Pu atoms in the lattice [95]. However, the thermal conductivity of pure PuO2 was found to be higher than that of ThO2 for all temperatures covered by their study. This is somewhat surprising and contra­dicts the understanding that ThO2 always have a higher thermal conductivity than the other actinide oxides.

In Fig. 18, the thermal conductivity of Th-MOX with PuO2 content varying from 0 to 30 wt% are shown. At low temperature, the thermal conductivity of the Th-MOX with a PuO2 content from 0 to 30 wt% decreases with an increase of the

PuO2 content. At higher temperature (above 1,000 K), the thermal conductivity of Th-MOX with a PuO2 content from 0 to 8 wt% is almost independent from the concentration of plutonium. The conductivity of Th-MOX with 30 wt% PuO2 at high temperatures is much lower [95].

The thermal conductivity k, of (Th1-yPuy)O2 as a function of temperature and PuO2 content is reported by IAEA study [40]. Figure 19 shows a systematic decrease of thermal conductivity with increasing PuO2 content and temperature. The data are comparable with those obtained by Murabayashi [114] on simulated fuel samples of the composition ranging from 0 to 10 wt% CeO2. The best-fit equation for the thermal conductivity, k [W/m-K], of (Th1-yPuy)O2 as a function of composition, y [wt%], and temperature, T [K], was derived for the temperature range from 873 to 1,873 K [40].

k(y, T) = 1/[-0.08388 + 1.7378 — y +(2.62524 — 10-4 + 1.7405 — 10-4 — y)- T]

(63)

In order to introduce the influence of the plutonium content on parameter A, one can rely on the simplified theory of Abeles [95]. The parameter A has a second-order dependence on both the relative mass and radius differences as per the above theory. A polynomial equation of the second degree was chosen to define A(PuO2):

A(PuO2) = A0 + A1 — [PuO2] + A2 — [PuO2]2, (64)

([PuO2] = Concentration of PuO2 in wt%).

The values of the parameters are [95]:

A0 = 6.071 x 10-3 mKW-1, A1 = 5.72 x 10-1 mKW-1,

A2 = -5.937 x 10-1 mKW-1. B = 2.4 x 10-4 mW-1.

Figures 20 and 21 show the variation A and B parameters with PuO2 content for ThO2-PuO2 system. The parameter A increases with increase in PuO2 while the variation of B with PuO2 content was found to be random.

The experimental thermal conductivity data of high Pu bearing hypostoichio — metric and stoichiometric mixed thorium-plutonium oxide of compositions,

5.0

4.5

4.0

і

3.5

■S’

>

30

3

2.0

£

I 1.5 I-

1.0

800 1000 1200 1400 1600 1800 2000

Temperature, K

ThO2-20 % PuO2, ThO2-30 % PuO2, and ThO2-70 % PuO2 with CaO or Nb2O5 as dopant, was measured up to 1,850 K in BARC, India, by employing the ‘‘Laser — flash’’ technique and is shown in Fig. 22. As expected, ThO2-70 % PuO2 showed the least thermal conductivity among the above sample.

The recommended equation for the density of liquid uranium dioxide is based on the in-pile measurements of the vapor pressure, density, and isothermal com­pressibility of liquid (U, Pu)O2 by Breitung and Reil [62]. Measurements of density as a function of enthalpy and as a function of temperature were obtained from the melting point to 7,600 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 [63], which had been recommended in the 1981 assessment by Fink et al. [64].

The recommended equation for the density of UO2 as a function of temperature is:

p = 8.860 — 9.285 • 10-4 • (T — 3120), (16)

where, density (p) is in g/cm3 and temperature (T) is in K.

No data exists for the volume change of ThO2 on melting, but some information is available on UO2. Christensen [65] measured the density of UO2 between 1,553 and 3,373 K by high-temperature radiography. The density of solid and liquid at 3,073 K were 9.67 and 8.74 g/cm3, respectively. The accepted value of the density of liquid UO2 at the melting point is 8.74 ± 0.016 g/cm3. The volume increase on melting was 10.6 %.

Finally, the burnup also affects the density by the change in the porosity. At low burnup (<15 GWd/t), density increases by the fuel densification process; at the higher burnup, density decreases (porosity increases) due to the fuel swelling [66].