D

A Thermal conductivity (W itT1 K-1) r density (kg itT3)

2.17.1 Introduction: Importance of Thermal Conductivity

Knowledge of the thermal conductivity of the fuel of a nuclear reactor is required for the prediction of fuel performance during irradiation, in particular for the determination of the temperature distribu­tion and of the fission gas release. The principal objectives of this chapter are to give elements useful to understand the phenomena causing the degradation of the thermal conductivity during irradiation and to provide guidance for the inter­pretation and comparison of in-pile or out-of-pile measurements, especially as a function of burnup and for samples having different irradiation tem­peratures, in-pile histories, and microstructures. The importance of such studies is more significant when the discharge burnup of the fuel is increased and with the formation of the high burnup struc­ture (HBS), because these two parameters have a significant impact on the thermal conductivity. More details on the performance of LWR UO2 fuel can be found in Chapter 2.19, Fuel Perfor­mance of Light Water Reactors (Uranium Oxide and MOX). The impact of the introduction of pluto­nium or additives (Gd, Cr, etc.) in standard UO2 also requires assessment (see also Chapter 2.16, Burnable Poison-Doped Fuel). Uranium-plutonium mixed oxide (MOX) fuel represents a significant fraction of the nuclear fuel used in commercial light water reac­tors (LWRs). The industrial processes used for the production of MOX fuel are based on the mixing of a few percent of plutonium oxide with UO2. The differ­ent microstructures that can be obtained are mainly characterized by the degree of homogeneity of the plutonium distribution. The impact ofthe introduction of plutonium in UO2 and the different microstructures therefore need to be considered because the presence of Pu in the UO2 lattice will reduce the thermal conductivity. UO2 fuel with increased grains size is produced by doping with chromium oxide, with the objectives of reducing the pellet-cladding interaction by an increased viscoplasticity and of reducing fission gas release.

This section mainly deals with LWR fuel because the data available for fast reactor fuel are extremely inadequate. The specific heat of the fuel is also affected by irradiation. This parameter is required for the investigation of fuel performance during transients and also for the calculation of the thermal conductivity from thermal diffusivity measurements. The evolution of the thermal conductivity as a function of burnup is nonlinear, and numerous approaches and approxima­tions are used, leading to a large number of publica­tions on this subject. This is not the case for the specific heat which generally obeys the law of mixtures. The thermal expansion, melting temperature, and oxygen potential of the fuel are also important for fuel per­formance studies, but they are not addressed in this section.

The thermal conductivity distributions as a func­tion of the radial position in a pellet for two tem­perature profiles (central temperatures of 1000 and 1500 K) and for burnups of 0 (fresh fuel) and 40 MWd kg HM-1 calculated with the equation of Ronchi et a/.1 are shown in Figure 1. A large temperature gradient exists over the small distance between the pellet center and the pellet rim, inducing large varia­tions in the conductivity. It can be seen that the conductivity decreases with temperature and burnup.

Under steady-state irradiation conditions and assuming a purely radial heat transfer, the temper­ature distribution T(r) in a fuel pellet is given by eqn [1] and depends on the thermal conductivity 1(r T) and volumetric heat generation rate q(r).

7 dr (r1(r ’T )+ q(r ) = 0 [1]

The local thermal conductivity 1(r, T) depends on the radial position and local temperature. The local heat generation rate requires information on the fission cross sections and depends on the radial distribution of the thermal neutron flux and of the fissile isotopes. This distribution changes during irradiation as a result of the consumption of the initial fissile isotopes and the production of fissile Pu.

From the point of view of heat transfer, both fresh and irradiated fuels are heterogeneous materials (e. g., due to the presence of the porosity). However, fresh fuels can be considered as homogeneous, even the heterogeneous MOX, and an effective or equivalent thermal conductivity can be defined. This is because the size of the pores and the plutonium-rich

Figure 1 Thermal conductivity distribution1 as a function of the radial position in a pellet, for two temperature profiles and for burnups of 0 (fresh fuel) and 40 MWd kg HM-1.

agglomerates is small (compared with the dimensions of the pellet) and has a uniform distribution as required by fuel fabrication specification. (See also Chapter 2.15, Uranium Oxide and MOX Produc­tion for more information on the uranium oxide and MOX production). The heterogeneity is higher in irradiated fuel because irradiation induces the forma­tion of numerous elements and compounds, bubbles, pores, etc., with concentrations depending on the radial position. The definition of the effective thermal conductiv­ity of a heterogeneous material such as irradiated fuel is not straightforward. The different scales that may be considered for the heat transfer are shown in Figure 2, where m is the microscale corresponding to the size of the larger heterogeneities, L is the mesoscale corresponding to the elementary repre­sentative volume (ERV), and M is the macroscale corresponding to the pellet radius. The thermal con­ductivities of the matrix and of the inclusions are noted 1m and 1i, respectively. If the separation of scales is verified,2 M ^ L ^ m and the equivalent thermal conductivity 1eq is defined from the mean temperature gradient (V T) and the mean density of the heat flux (‘) within the ERV: 1eq =-(‘)/(V T). The equivalent thermal conductivity 1eq can be evaluated from the conductivity and the geometric distribution of the constituents on the basis of the following assumptions:

M Heterogeneous solid: pellet

Equivalent medium

Figure 2 Notion of separation of scales: an elementary representative volume exists for the heat transfer, with M > L > m.

1. If the equivalent conductivity is evaluated over a volume V, and there exists a value of V beyond which the value of the conductivity no longer varies as Vis increased, this volume is the ERV. 2. The medium is statistically homogeneous: the sta­tistical distribution of the phases does not depend on the position within the material, and the equiv­alent conductivity is the same irrespective of the position of the ERV. 3. The dimensions of the material are large com­pared with the dimensions of the ERV. 4. Steady-state heat transfer is assumed. For transient heat transfer, homogenization is inappropriate, and the real microstructure has to be considered. 5. The medium is opaque to thermal radiation.

6. Fourier’s law applies within the ERV and in the

entire medium.

7. No mass transfer is involved.

8. No internal heat sources exist.

The first three criteria, relating to the microstruc­ture, are not perfectly met for irradiated fuel. A rigorous homogenization is therefore not possible because the thermal conductivity depends on the radial position in the pellet as a result of the radial distribution of burnup and the irradiation tempera­ture. However, a local homogenization is usually made by assuming that over a small radial position interval, the characteristics of the fuel are constant and allow the measurement or calculation of an effective thermal conductivity. For standard irra­diated fuels, the unit cell required for homogeniza­tion has dimensions of about 1 mm3, considering that the biggest heterogeneities are pores of size up to 100 pm. However, burnup and irradiation temperature are not constant over a radial interval of 1 mm, and therefore this unit cell is not rigor­ously suited for homogenization. Homogenization can be accomplished in a more rigorous way in the case of disc fuels obtained during test irradiations, if uniform burnup and irradiation temperature pro­files are obtained. This homogenization allows an average temperature field to be calculated. Local temperature variations exist, for instance, because of plutonium-rich zones or a particular local arrangement or shape of the pores.

The criteria 4-6 are met for irradiated fuel. The seventh criterion can be considered as met because the effect of mass transfer is negligible for LWR fuels: some elements migrate as a result of the gradients in the pellet, but the heat transferred by this mechanism is small when compared to conduction. The eighth criterion is not met when the effective thermal con­ductivity is deduced from in-pile temperature mea­surements. The internal heat sources should not be considered when the effective thermal conductivity of heterogeneous fuels is evaluated, for instance from finite-element temperature calculations.

The thermal properties of irradiated fuels are investigated in pile by temperature measurements and out of pile by thermal diffusivity measurements. Theoretical studies exist for the effect of some single parameters. Direct out-of-pile measurements on irradiated fuel samples appear more reliable because of the well-defined and optimized measurement parameters, although the in-pile conditions cannot be completely reproduced (temperature gradient, fission rate, etc.). Therefore, in-pile determinations are indispensable, even though the deduction of the thermal conductivity from eqn [1] is less accurate. The published correlations range from empirical for­mulae to more sophisticated models integrating explic­itly and semiempirically the effect ofsome parameters. A new correlation is often a combination of different existing approaches, for instance, fresh fuels results, out-of-pile experiments on simulated or real irradiated samples, and correlations obtained from in-pile tem­perature measurements.

No consensus exists on the most reliable correla­tion and usually, fuel performance codes incorporate a number of correlations, leaving it to the user to select which is the most appropriate for a specific case. For instance, the FEMAXI-6 code3 includes about 10 thermal conductivity correlations for irra­diated UO2. Before presenting the most representa­tive results, the different burnup effects that have an impact on the thermal conductivity are discussed individually.