The EUCLID/V2³ code is designed to analyze and justify the safety of NPPs with fast neutron reactors with liquid metal coolants in the normal and abnormal operation modes, including accidents, severe ones as well (Usov et al. 2018a, 2018b; Butov et al. 2019a, 2019b). It is based on the first version of the code EUCLID/V1 (detailed information about which can be found in (Mosunova 2018; Alipchenkov et al. 2018)) and includes all of its models.

The following modules work together as part of the EUCLID/V2 calculation code, providing multi-physical consistent simulation of different processes and phenomena:

• thermohydraulic (HYDRA-IBRAE/LM), which includes a module of the FP transport and behavior, the corrosion and activation in the RI primary circuit and gas system (AEROSOL-LM), a module of solid impurity transport in the RI primary circuit with heavy liquid metal coolant (OXID), a module of tritium migration in the coolant of the first, second, and third (if available) circuits (TRITIUM);
• sub-channel module (CELSIST) for two-dimensional modeling of coolant flow in fuel assemblies;
• neutron diffusion and transport modules (DN3D and CORNER, respectively), modules for burn-up (BPSD) and decay heat (OSTB) calculation;
• fuel rod module (BERKUT and BERKUT-U) with models for calculating sources of fission products in a core in the event of fuel rod cladding failure, intended for numerical simulation of behavior and calculation of operability of nitride and oxide fuel rods;
• module for calculating the disruption of fast reactor fuel rods and other core structures (SAFR). The module implements a multicomponent one- and three-dimensional models for calculations of the movement and heat exchange of components of disrupted fuel rods and fuel assemblies in the core and upper mixing chamber of liquid-metal cooled reactor. To calculate the movement of components, the laws of conservation of mass, energy and momentum are used, as well as the equation for the sum of volume fractions of the components. The module also allows calculation of the movement of the formed cladding and fuel melt by gravity, friction with the gas flow and wall friction;
• module of mass transfer and FP behaviour in NPP compartments (KUPOL-BR or HYDRA-IBRAE/LM);
• module for assessing the radiation situation outside the NPP site (ROM).

The SMART/LM integration shell provides a consistent calculation by all modules.

The main model improvement and development areas of the EUCLID/V2 multiphysics code in 2018-2019 include:

• development of a model which describes the transport of steam-water formations in the circuits of reactor installations with a heavy liquid-metal coolant, taking into account their size distribution;
• development of a nitride fuel dissociation model and its implementation into the code;
• improvement of the diffusion neutronic module for correct simulation of core disruption (the values of diffusion coefficients were corrected to take into account the collapse of materials during core disruption, the cross sections change due to materials flowing down during fuel rod melt was considered, etc.);
• turbine model development and integration into the code.

More detailed information about some of the models is given below.

In 2019, the model for transport of vapor-gas formations (bubbles) was added to the system thermal-hydraulic module HYDRA-IBRAE/LM of the EUCLID/V2 code, taking into account the evolution of their size distribution. It should be noted that the description of the particle size of the dispersed phase (droplets, bubbles, vapor agglomerates) is a key point in models of two-phase media. The traditional approach used in system thermal-hydraulic codes is to determine the size of the dispersed phase from empirical correlations with instantaneous adjustment to changes in the thermal-hydraulic parameters of the carrier flow. In reality, the zone, in which a more complex than instantaneous approach is needed, can be significant (up to several meters, it depends on the flow parameters), which can be essential when analyzing, for example, the transport process of steam formations in the circuits of reactor installations with a heavy liquid-metal coolant.

The evolution of the bubble size distribution can be described more precisely by a volumetric interfacial area transport equation (Yao and Morel 2004). To solve the transport equation for the particle size distribution, the method of fractions (groups) is used in the HYDRA-IBRAE/LM module. In this method, the set of particles is subdivided into a number of fractions (groups or classes) depending on particle size and it is assumed that there can be an exchange of particles between different fractions as a result of phase transitions, coagulation, fragmentation and other processes.

The experimental data on which validation of the developed models could be performed are now practically absent in the available literature. Therefore, a comparison of the calculation with the analytical solution was performed. For comparison, the article results (Gelbard and Seinfeld 1978) were used. In this work, it was obtained that with the initial distribution and the constant coagulation nucleus β₀, the solution is represented as

$n\left(t\right)=\frac{4N_0}{v_0{\left(N_0{\beta }_{0}t+2\right)}^2}Exp\left[-\frac{2v/v_0}{N_0{\beta }_{0}t+2}\right]$ (1)

where N₀ and v₀ – parameters; β₀ – size-independent portion of coalescence kernel; t – time; v – bubble volume.

The simulation results in comparison with solution (1) from the work (Gelbard and Seinfeld 1978) are presented in Figure 3.

To our knowledge, a model of this type is not included in some well-known and widely used codes-analogous such as RELAP5-3D, ATHLET, ASTEC-Na (RELAP5-3D Code Manual 2012a, 2012b; Palazzo et al. 2013; Zhou et al. 2013; Flores et al. 2016; Bandini et al. 2018; Forgione et al. 2019; Narcisi et al. 2019; Shen et al. 2019 and many others). Similar models are available only in some CFD codes.

In 2019, a model for calculating the nitride fuel dissociation was integrated into the fuel rod destruction module of the EUCLID/V2 multiphysics code. The most important for the dissociation modeling is to determine the rate of mass loss due to release of nitrogen and uranium vapor according to the equation

$\left(U_{1-x}P{u}_x\right)N\left(\mathrm{c}\right)=U_{1-x}\left(1\right)+P{u}_x\left(l\right)+0,5N_2\left(\mathrm{g}\right)$ (2)

The rate of mass loss is calculated by the ratios:

$\Delta m_{\left(P{u}_x{U}_{1-x}\right)N}=-2·\frac{M_{\left(P{u}_x{U}_{1-x}\right)N}}{M_{N_2}}{j}_{N_2}S·\Delta t$ (3)

$\Delta m_U=2·\frac{(1-x)M_U}{M_{N_2}}{j}_{N_2}S·\Delta t-j_{U}S·\Delta t$ (4)

$\Delta m_{Pu}=2·\frac{x·M_{Pu}}{M_{N_2}}{j}_{N_2}S·\Delta t-j_{Pu}S·\Delta t$ (5)

where j_N2, j_U , j_Pu – the mass flows of nitrogen, uranium and plutonium from the fuel surface; x– plutonium mass fraction; M_N2, M_Pu, M_U, M_(PuxU1-x)N – molar mass of nitrogen, Pu, U and MNUP fuel, respectively; S – fuel surface where the process of dissociation occurs; ∆t – time step; ∆m_U, ∆m_Pu, ∆m_(PuxU1-x)N – changes of total masses of U, Pu and MNUP fuel due to dissociation process respectively.

To our knowledge, this process is not modelled in codes such as SIMMER-III/IV, SOCRAT-BN (Usov et al. 2012; Cheng et al. 2015; Li et al. 2017).

In 2015, the fourth power unit of the Beloyarsk NPP (RI BN-800 with sodium coolant) achieved its first criticality successfully, after that the first start took place. Then pilot operation of the unit was carried out. At that time, the tests of the RI main technical characteristics in stationary and transient modes at different power levels up to the nominal one were carried out. At present, the EUCLID/V2 multiphysics code is validated on the data obtained on the BN-800 in the following modes:

• shutdown of one of the three operating loops when the reactor is operating at a power level of ~ 100% N _nom;
• cooldown of the reactor by ECCS from the initial power level of ~ 50% N _nom;
• power unit shutdown;
• power unit start-up, where N _nom is the nominal reactor power.

The neutronics and thermohydraulic computational models of the BN-800 core, a thermal-hydraulic model of the coolant circulation circuits, including the upper mixing chamber, intermediate heat exchangers, drain chambers of intermediate heat exchangers, main circulation pumps, pressure pipe lines, pressure chamber, pipe lines of the secondary circuit, sodium buffer tank, main circulation pumps of the secondary circuit, steam generators and other equipment, as well as fuel rod models for fuel assemblies of various types are developed.

To evaluate the uncertainties and sensitivity of main RI parameters calculated using the EUCLID/V2 code, some parameters affecting the simulation results in a most significant way were determined based on previously validation calculations of experiments accounting for some individual phenomena (Table 2). The ranges of data variation were selected based on the experimental uncertainties of relevant measurements. The uncertainties of empirical correlations used in the calculations were determined in accordance with available literature data or expert estimates.

To perform the uncertainty and sensitivity analysis of the results 200 calculations have been carried out.

Figure 4 shows a comparison of the results of multivariate calculations obtained using EUCLID/V2 code with the experimental values of integral power of the BN-800 RI in the mode of “Shutdown of one of the three operating loops when the reactor is operating at a power level of ~ 100% N_nom”. The intervals of calculated data are shown by blue shading. It can be seen that the deviation of the average calculated values from the experimental average ones does not exceed 5%.

Figure 5 shows the EUCLID/V2 calculated values of sodium temperature in the primary loop at the IHX inlet of the BN-800 RI in the mode “Cooldown of the reactor by ECCS from the initial power level of ~ 50% N_nom” in comparison with the experiment. The intervals of calculated data are shown by blue shading. The experimental uncertainties are shown by vertical bars. The maximum absolute deviation does not exceed 10 K.

The available validation results show that for the considered types of transients, the EUCLID/V2 code describes adequately the change in the integrated power as well as the thermohydraulic processes in the BN-800 type RI core and its cooling circuits, including air circuit cooling modes.

In a closed nuclear fuel cycle, spent nuclear fuel is reprocessed to extract uranium and plutonium for further re-use as fuel for fast reactors.

An important step in the technology of the spent nuclear fuel (SNF) reprocessing is the operation of further dissolving the remainder of plutonium and uranium oxide solid particles (U_xPu_y)O₂ after the completion of basic technological cycles of SNF reprocessing. Its distinguishing feature is the electrochemical catalytic conversion of plutonium dioxide PuO₂ from the insoluble solid phase to PuO₂²⁺ ions soluble in nitric acid HNO₃ solution. A computational model was developed describing:

• real three-dimensional geometry of the installation elements;
• three-dimensional picture of medium flow in the anode and cathode volumes (CFD);
• ion transfer to electrodes with a detailed resolution of the structure of the boundary layer near the electrodes;
• heat transfer for various elements of the installation;
• sources of heat generation due to electrolysis processes and decay of radionuclides;
• migration of ions through the membrane in the approximation of transport equations for a porous medium, taking into account the electric field gradient.

Upon the end of the cycle of electrochemical dissolution of solid particles of actinide oxides, the solution from the anode space with the remaining particles of the dispersed phase is sent to the control clarification apparatus for separating solid particles. The filtered solution containing the solid phase is delivered to a membrane filter, which contains several multichannel filtering ceramic elements that are hollow fibers coated with a membrane layer. The concentrate remains inside the channels of the filtering element, and outside the filtering element a solution is accumulated that is free from solid particles and insoluble impurities deposited in the membrane pores.

A distinctive feature of the filtered solution is the presence in the solution of liquid and solid high-level waste of SNF and ionic components. A feature of the membrane filtration process from the point of view of radiation safety is the accumulation of radioactive sediments on the surface of membrane fibers and filter walls.

The developed calculation model takes into account:

• three-dimensional hydrodynamics (CFD) of the solution flow with particles through the membrane and filter volume;
• kinetics of radioactive particles charging in nitrate electrolyte;
• possible heating of particles due to decay of radionuclides;
• thermal interaction of the dispersed and liquid phases;
• sedimentation and separation of particles from the surface of membrane channels, taking into account their charge;
• sediment formation taking into account spatial effects;
• variation in time of the sediment activity on the membrane.

Currently, the most promising option for the MNUP pellet production in industrial scale is the method of carbothermic synthesis (CTS), based on the reduction of mixed uranium-plutonium dioxide by carbon in nitrogen atmosphere.

The carbothermic synthesis reaction of (U,Pu)N is carried out with powder mixtures of uranium dioxide or uranium – plutonium, (U,Pu)O₂, and graphite. Synthesis occurs at temperatures of 1700–2000 K in nitrogen stream. At the last stage of the process, some amount of hydrogen is added to nitrogen atmosphere. The role of the nitrogen flow, especially of the N₂+H₂ mixture, consists, in particular, in removing gaseous compounds of carbon and oxygen (CO, CN, CH₄) from the reaction zone, thus ensuring a sufficiently low concentration of oxygen and carbon impurity in the final product, which is a critical point in the production of nitride fuel by the CTS method.

There are several factors that significantly affect the CTS efficiency and the final product characteristics: the size of the particles of original powder, the presence of impurities, the temperature-time mode, the volume and composition of gas flow in the chamber. The feature of the technological process from the point of view of radiation safety is the mass loss of samples and the accumulation of radioactive dust in the CTS chamber.

The developed mathematical model for the physicochemical processes of the carbothermic synthesis of uranium and plutonium nitride powders is intended to describe the effect of process parameters — temperature, atmospheric composition, and others — on the characteristics of the products and, above all, on the concentration of impurities.

The developed calculation model takes into account:

• chemical reactions of oxygen substitution by carbon and nitrogen at the border of fuel and graphite with CO formation;
• chemical reactions of carbon substitution by nitrogen in the presence of hydrogen to form CH ₄;
• diffusive transport of oxygen, nitrogen and carbon in the solid phase;
• mass transfer of N ₂, H ₂, CO, CH ₄ in the gas phase.

The final step in the process of the MNUP fuel pelletizing is sintering of fuel pellets. At the preliminary stages, the UN or (U, Pu)N powder is produced, which is then pressed, resulting in an intermediate product with a density of ~ 60% having sufficient mechanical strength and the required chemical composition. At the sintering stage, a part of the fuel main characteristics is formed, such as density, size and type of porosity, grain size, determining the fuel performance.

There are several factors that significantly affect the sintering efficiency of (U, Pu)N and the characteristics of the final product: the size and shape of the particles of the original powder, the initial porosity of the sample, the sintering temperature and time, and the atmosphere in the sintering chamber.

The developed computational model takes into account the following processes:

• fuel recrystallization by the mechanisms of normal grain growth and re-condensation;
• relaxation of intergranular porosity due to the diffused drain of vacancies and external thermo-mechanical stresses;
• thermochemical reactions, leading to a mass loss of the sample.

In the constructed computational model, three-dimensional heat transfer in a complex furnace design, mass transfer of the gas mixture in the working area, multicomponent diffusion and reactions in fuel grains are described consistently.