Corresponding author: Svetlana M. Ganina (

Academic editor: Georgy Tikhomirov

There is a substantial amount of experimental data that confirm the peculiarity of the oxide fuel behavior during the first hours after a reactor reaches its power. With an increase in temperature during the said period, oxide fuel pellets crack because of a significant temperature gradient. Further developments occur due to the accumulation and redistribution of fission products, and manifest themselves as changes in the fuel matrix porosity and the formation (or diameter increase) of a central hole. The zones that differ in their microstructure, density and thermal conductivity are formed along the fuel pellet radius. As the oxide fuel composition and restructuring result in its thermal conductivity change, it is important to pay attention to the formulas used in the stress-strain state calculations of a fuel element. The methodology for calculating changes in the oxide fuel porosity and the pellet’s internal diameter is proposed and based on the published research papers dedicated to the study of oxide fuel properties and behavior during the first hours after a reactor reaches its power. Тhe methodology was tested using real experimental data on the porosity redistribution along the fuel pellet radius. The presence and the size of the pellet’s inner hole, as well as changes in the fuel matrix porosity, have a noticeable effect on the maximum temperature value. Taking into account the pellet structure evolution while performing computational simulation of the fuel rod operation under irradiation allows assessing the fuel element’s operability more accurately. The proposed methodology can be used in computer codes designed to calculate the stress-strain state of cylindrical fuel elements of fast reactors.

There is an array of experimental data that confirm the specific behavior of oxide fuel in the initial hours after the reactor rise to power. As the reactor rises initially to power and temperature grows, the oxide fuel pellets crack because of a major temperature gradient. Fuel pieces move filling partially the gap between fuel and the cladding and leading so to crack formation.

After the reactor rises to power and operating temperatures are reached, there is a fuel mass transfer taking place in the fuel pellets in the direction that is opposite to the temperature gradient as a result of porosity migration and thermally activated evaporation and condensation processes, and bulk and surface diffusion. These processes lead to the crack healing, the formation of a central hole in fuel (or an increase in its diameter if there is a hole in the initial state), and a noticeable decrease in the gap between fuel and the cladding (depending on the linear power value and, accordingly, the fuel temperature, there is a potential for the fuel-cladding gap elimination). Zones are formed that differ in the fuel microstructure, density and thermal conductivity.

In the process of porosity migration, column-shaped grains are formed that have a radial orientation, the grains grow rapidly, and equiaxial grains are formed when the temperature is in excess of a certain value. On the colder peripheral part (outer radius) of the fuel pellet, fuel remains non-restructured (

Based on the literature overview, a fairly simple methodology was chosen to calculate the change in porosity and the fuel column central hole diameter. The following assumptions are used:

all pores have the same size, and their volume does not depend on the radial position and time; no collisions between migrating pores and their coalescence are taken into account;

pores are closed and migrate in the radial direction only.

Pore conservation equation (

Initial and boundary conditions:

_{0},

_{0} (2)

where _{0} is the initial porosity, _{p}

To obtain the numerical solution, we shall break down the radius into _{i}_{−1/2}, _{i}_{+1/2}] (

where _{i}_{−1/2}, _{i}_{+1/2}] at the previous time point.

Since _{0}, we may search for the value of the porosity on the calculation cell boundaries beginning from the fuel outer boundary.

The pore displacement rate is deteremined by equation (

where _{p}^{19}); ^{3} for UO_{2}), the formula uses value Ω = 41∙10^{−21} mm^{3}; and _{g}

where _{g}^{*} = 1100 mm^{2}/s (for He); _{g}

_{g}_{sint}

_{sint}_{vap}_{vap}

The following dependence option is presented in

Ln(_{0} ∙ e^{−ΔH/RT}) = −212.275 + 65.842 ∙ ^{−02} ∙ _{K}^{−02} ∙ _{K}^{2} – 5.6541 ∙ 10^{−06} ∙ ^{2}_{K}, (8)

where _{p}_{0} is the material constant; Δ_{vap}_{K}_{g}_{g}_{sint}

To calculate displacement rates, the paper uses an additional parameter,

MOX fuel structure with a burnup of 9.4% h.a. in the BN-600 reactor (×50) (

With the calculated radius value being _{i}_{0} + Δ, we assume that the pore displacement rate is _{i}_{i}_{+1}.

Here, Δ = (_{Nf}_{0}) ∙ _{i}_{0}

We shall write the equation for each cross-section along the radius with reliance on the mass conservation equation:

The outer boundary (fuel internal radius) is recorded as of the given time point, so we will search for the new radius positions from the fuel outer boundary:

Here, _{i}_{−1/2}, _{i}

Equations (9) and (10) are valid only with _{0}, where _{0} is the radius of the central hole formed due to the pore migration.

To test the fuel porosity and inner radius calculation model used in the study, we shall consider a test problem based on data presented by the authors in

Experimental porosity values were obtained for vibrocompacted (pebble) fuel element with high initial porosity of the U_{0.85}Pu_{0.15}O_{2.0} fuel irradiated for 28 effective days: the linear power density was 44.62 W/mm, and the burnup was 0.7% of h.a. The initial porosity of non-irradiated fuel is _{0} = 0.186 rel. units, the external radius of fuel is _{2} = 3.2 mm, and the temperature on the outer surface is _{2} = 550 °C.

Fig.

Position of the calculated curves for the porosity and temperature distribution along the fuel pellet radius, and experiment results (

Fig.

The solid line in Fig.

Let us consider two options for dependences that describe the thermal conductivity of MOX fuel. A dependence of the following type is presented in

where

With regard for the data obtained for fresh fuel (2001, 2013, 2017) and irradiated fuel (1993, 2017), dependence (12) has been proposed

Here, ^{2.5} is the coefficient of porosity; and

Fig. _{0.8}Pu_{0.2}O_{1.98} fuel (Fig. _{0.8}Pu_{0.2}O_{1.93} fuel (Fig. _{0.8}Pu_{0.15}O_{2} fuel with an initial density of 95%, the thermal conductivity values predicted by two dependences are close

Thermal conductivity coefficient of MOX fuel as a function of temperature: solid blue line – dependence (11); dashed red line – dependence (13); circles show experimental data.

It is noted in

The paper presents the results of the test calculations using dependences (11) and (12) to describe the thermal conductivity of oxide fuel (the thermal conductivity equation is solved by double-sweep method). Fig. _{0} = 0.186 rel. units. Dependence (7) was used to define the pore displacement rate for Fig. _{vap}^{3} J/mole; evaporation entropy Δ_{vap}

Positions of the calculated porosity and temperature distribution curves along the radius of the fuel pellet.

Fig. _{g}

Positions of curves for the calculated porosity and temperature distribution along the fuel pellet radius assuming that there are open and closed pores.

An algorithm consisting of a sequential solution to a thermal conductivity problem, taking into account the porosity in the calculation cell, determination of the pore displacement rate with calculating further porosity, and determination of new boundaries for the calculation cells allows one to obtain the time-dependent porosity and temperature distribution along the fuel radius.

Testing the algorithm based on an example of a real experiment has shown qualitatively consistent results in calculating the porosity distribution using an assumption that all pores are closed and assuming that there are regions with open and closed pores. At the same time, the maximum temperature in the fuel center, with restructuring taken into account, is close to the temperature calculated by the authors of

_{2}fuels under fast reactor conditions. Comparison with recent experimental data.

_{2}.

_{2-x}mixed oxide fuel.

Russian text published: Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2024, n. 2, pp. 202–212.