The effect ofthe parameters acting at the mesoscopic scale can be assessed using solutions of the heat equation obtained for particular microstructures, for instance, a matrix containing inclusions.

Interstitial

FP atom in solution in the lattice FP atom as interstitial in the lattice, or gas in solid Vacancy

Figure 3 Some parameters affecting thermal conductivity.

2.17.2.1.1 Porosity and grain boundaries

The porosity existing in fresh fuels evolves during irradiation and has a strong impact on the heat transfer in the pellet because of its low thermal conductivity as compared to the solid. The porosity is assumed to have identical effects on the fresh and irradiated fuels and therefore the same correction formulae are used. This assumption is valid when the shape of the pores is similar and the pore volume fractions are comparable (i. e., around 5 vol%). In order to analyze the effect of parameters other than the porosity, its effect is normal­ized to 0 or 5 vol% by converting the measurements obtained for samples with different porosity levels. Irradiation-induced pore-size distribution changes have a small effect, but the pore shape can play a larger role if the pores are not spherical and oriented. The correction formulae are derived from composite mate­rial formulae giving the effective conductivity of a matrix containing inclusions,4 often simplified by attri­buting zero conductivity to the pores. Loeb5 intro­duced the effect of temperature to take into account the radiative heat transfer in the pores. The influence of the complex pore shape observed in irradiated fuel was investigated by Bakker using the finite-elements technique applied to realistic microstructures obtained by image analysis.6 Limiting this presentation to a recent recommendation, the thermal conductivity can be normalized to 5% porosity using the formula of Brandt and Neuer as recommended by Fink7 (eqn [2]).

1 — 0.05f (T)

1 — Pf (T) where f(T) = 2.6-0.5T/1000 and P is the porosity fractional volume.

2.17.2.1.2 Precipitates of insoluble fission products

Precipitates are formed by fission products that are insoluble in the UO2 lattice (see Chapter 2.20,

Fission Product Chemistry in Oxide Fuels). They form oxide inclusions (such as BaZrO3 or SrZrO3) or metallic inclusions (Mo, Ru, Tc, Rh, Te, Pd, Sn, Cd, Sb, Ag, etc.) and their influence on the effective thermal conductivity can be evaluated by the formu­lae obtained for composite materials8: for instance, the Maxwell-Eucken equation.9 As the volume frac­tion of inclusions is small, this simple model gives results of sufficient accuracy. The inventory and vol­ume fraction of the precipitates can be calculated from the number of moles created at a given burnup. For illustration purposes, the Maxwell-Eucken formula was applied at a fixed temperature assuming a UO2 matrix with a thermal conductivity of 4Wm-1K — containing inclusions with thermal conductivities of 0 (pores), 2 and 10 (ceramic precipitates), and 100 (metallic precipitates) W m-1 K — . As shown in Figure 4, the effect of the precipitates on the thermal conductivity varies linearly with their volume frac­tion, which is proportional to the burnup, bu. If the coefficients a and b are used to describe this propor­tionality, we have k = a — b x bu; the negative burnup dependence is introduced because the fuel thermal conductivity globally decreases with burnup. Usually, the thermal conductivity is approximated by the 1/(A +BT) formula, where A and B describe the phonon scattering mechanisms. Therefore, one obtains the formula A +BT = (l/a)/(1 — b/a x bu) and if b/a x bu is small, we have A + BT = 1/a + (b/a2 x bu) — (b2/a3 x bu2) + ••• . At a fixed temperature T, the A and B coefficients depend in a linear or quadratic way on the burnup. This kind of dependence is often found in thermal conductivity correlations.