Abstract—An analysis of the relationships for calculating the thermal properties of liquid lead (hereinafter referred to as lead) was carried out, and the method for determining its heat capacity over a wide range of temperatures, including at high values, was chosen. This is especially important for numerical studies to justify the safety of designed reactor installations with liquid metal coolants, such as BREST-OD-300 and BR-1200. Measuring the properties of lead at temperatures close to the boiling point is often difficult due to the lack of reliable methods and materials that can withstand temperatures above 2273 K. At present, theoretical approaches to calculating the properties of simple liquids based on phonon theory are being actively developed. Such approaches can be used to derive semiempirical relations for the heat capacity of liquid lead that would allow physically correct extrapolation of the data to the high-temperature region. In this regard, the aim of this work is to obtain a relationship for calculating the heat capacity of liquid lead from its melting point to its boiling point based on modern theoretical approaches. To achieve the set goal, the following tasks were solved. Firstly, an analysis of the works of various authors was carried out and empirical formulas were selected that make it possible to reliably calculate the heat capacity at a constant volume cv (isochoric heat capacity) for a lead coolant from the melting point to 1500 K. Secondly, based on them, using phonon theory, an approximating formula was constructed, thanks to which it is possible to physically correctly extrapolate the properties of lead to the boiling point (2022 K).

The effect of barodiffusion on the dynamics of gas bubble growth in a highly viscous, gas-saturated magmatic melt undergoing rapid decompression is studied. A mathematical model of the process, which represents a joint dynamic and diffusion problem, is proposed. It is shown that as the bubble grows, a diffusive boundary layer is formed around it, leading to the appearance of a large viscosity gradient in the melt and, as a consequence, to a large pressure gradient. A semi-analytical solution of the problem based on the existence of a quasi-stationary state for the bubble growth process is found. It is shown that the effect of barodiffusion is significant at the initial and transient stages of the process. Its influence decreases with time and disappears completely at the stage of diffusion.

The development of computer codes for modeling accidents in a reactor unit requires validation of the models built into these codes. In this work, the EUCLID/V2 integrated code developed at IBRAE RAS was validated as applied to the simulation of severe accidents with a failure of the core of liquid-metal cooled fast breeder reactors (LMFBR), against experiments on melting of the cladding of fuel-rod simulators carried out at the Institute of Thermophysics, Siberian Branch, Russian Academy of Sciences (IT SB RAS), and SCARABEE BE + 3 experiments performed at the Commissariat à l’Energie Atomique (CEA) in France. The investigations performed at IT SB RAS included measurements of the cladding surface temperatures without liquid-metal cooling of the fuel-rod simulator, which is typical for accidents involving an instantaneous blockage of the flow section in the fuel assembly (FA) or with loss-of-coolant for type BN-1200M reactor units (RUs). To create such conditions, experiments with fuel rods were carried out in an argon atmosphere at room temperature (25°C) and a pressure of approximately 10 5 Pa, and the surface temperature of the fuel-rod simulator was recorded with a pyrometer. In France, the SCARABEE BE + 3 series experiments were carried out in the SCARABEE reactor to study the consequences of a hypothetical accident with a complete instantaneous blockage of the flow cross-section in a sodium-cooled fast reactor. To determine the effect of uncertainty in the initial data, diversified calculations were made. The validation was done by comparing the predictions with the experimental values of temperatures in the range between 500 to 1800 K (experiments of IT SB RAS). The maximum calculation error did not exceed 200 K. For the experiments in the SCARABEE reactor, it was not greater than 88 K for the fuel-rod claddings and 100 K the coolant. The obtained data will be used to estimate uncertainty in the predictions by the models of severe accidents with thermal destruction of fuel rods in fast reactors.

Analytical formulas are derived for describing the direct pyrolysis of methane in a bubble column loaded with a catalyst melt. This math can be implemented in research or commercial computer codes aimed at multiparametric optimization of the melt selection (with consideration for its cost and catalytic performance), setup dimensions and productivity. These analytical expressions can be used also for validation of numerical codes.

A numerical simulation of the gas outflow to a closed region filled with liquid with a barrier disk was performed. The calculations were carried out using the VOF method, supplemented by the k-e turbulence model. Calculations were performed for three cases of 100, 200, and 300 mm distances of the disk from the injector with a gas outflow into water and liquid lead. The pulsations of axial pressure on a disk obstacle were investigated. It was found that the maximum pressure during pulsations of the upper gas volume in lead can be greater than the pressure in the gas receiver.

Gas outflow into a cavity with different annular assembly filled with liquid by the VOF method, supplemented by the k-ε turbulence model, is numerically simulated. Calculations are performed for three types of ring assembly. Principal scenarios of bubble growth outside the assembly and annular jets inside it are obtained. The characteristic expiration times are investigated.

Submerged gas jets find a wide variety of industrial applications, and their behavior is characterized by the ratio of inertia to buoyancy and can vary from the emergence of individual bubbles to stable jets. A numerical study of the high-speed outflow of gas under a pressure of 18 MPa into a cavity with an obstacle filled with a liquid under a pressure of 2 MPa is carried out. The simulation is performed using the VOF method in conjunction with the k-ε turbulence model. The calculations are realized for three distances between the outflow hole and the obstacle: 100, 200, and 300 mm. Principal scenarios of gas jet evolution and characteristic expiration times are obtained.

This paper presents a mathematical model of the vapor bubble growth in an initially uniformly superheated liquid. This model takes into account simultaneously the dynamic and thermal effects and includes the well-known classical equations: the Rayleigh equation and the heat conductivity equation, written with consideration of specifics associated with the process of liquid evaporation. We have obtained a semi-analytical solution to the problem, which consists in reducing the initial boundary value problem with a moving boundary to a system of ordinary differential equations of the first order, valid in a wide range of operating parameters of the process at all its stages: from inertial to thermal, including the transitional one. It is shown that at large times this solution is consistent with the known solutions of other authors obtained in the framework of the energy thermal model, in particular, for the high Jacob numbers, it is consistent with the Plesset–Zwick solution.

A numerical simulation of the process of unsteady outflow of steam (steam-water mixture) from a volume under high pressure into a closed region with rigid boundaries is carried out. The calculations used a thermodynamically equilibrium homogeneous model of a vapor-liquid medium. At small and large (up to 0.15 s) times, spatial distributions of pressure, mass vapor content, density and temperature were obtained and analyzed.

The effect of impurity particles on the forced convection velocity in a drop is analyzed. The liquid motion in a drop is due to droplet streamlining by the air flow. The experiments are carried out in a wide range of air velocities and droplet sizes and compared with approximate numerical solutions. A simple analytical model for determining convection inside the droplet is proposed for the first time. A semi-empirical model is proposed to generalize the experimental data by introducing the function considering the density of impurities (solid particles in the form of contaminants) on the droplet surface. The novelty of the work lies in the newly formulated hypothesis, justifying the 40–50 times underestimation of the experimental data below the theoretical calculation. For the case of motion of a drop of one liquid in another liquid, the experimental data satisfactorily correspond to the obtained analytical expression. The obtained expressions may be used for a wide range of problems related to the interfacial boundary (liquid - gas).

Three-dimensional modeling of air injection into a closed pipe region filled with liquid (water, liquid lead) is carried out at high pressure drops. It is shown that in the three-dimensional calculation, the instability of the interphase surface in the form of ring waves of deformation on the upper surface of the projectile is significantly manifested. When air is injected into water, the difference in the configuration of the interphase surface obtained in axisymmetric (two-dimensional) and three-dimensional calculations is insignificant. When injected into a liquid with a substantially higher density (liquid lead), this difference is especially pronounced in the compression phase of the gas projectile.

The results of experimental and numerical studies of the gas phase injection into the liquid column and sudden gas-liquid interaction are presented. Experimental study was performed by means of high-speed shadowgraph and analysis of signals of pressure transducers. The numerical part of the work was carried out using the OpenFOAM software. The data about void fraction distribution and pressure evolution are presented. In general, a good agreement between the experimental and calculated data was found. The data obtained can be used for verification and validation of different numerical codes, as example ones for the prediction of the process of the steam generation leakage on nuclear power plant.

Three-dimensional simulation of air injection into a liquid (water, liquid lead) at large pressure drops in a three-dimensional formulation has been carried out. A significant difference in the formation of the air volume form at injection of water and liquid lead is shown. The instability of the interfacial surface in the form of annular waves and axial cavity on the upper surface of the slug has been revealed in three-dimensional calculation. For the three-dimensional case the gas-dynamic structure of the compression jumps and the Mach disk is found to be more pronounced for the supersonic underexpanded jet inside the slug.

A numerical simulation of the process of the outflow of gas under pressure into a closed container partially filled with liquid was carried out. For comparative theoretical analysis, an asymptotic model was used with assumptions about the adiabaticity of the gas outflow process and the ideality of the liquid during the oscillatory one-dimensional motion of the liquid column. In this case, the motion of the liquid column and the evolution of pressure in the gas are determined by the equation of dynamics and the balance of enthalpy. Numerical simulation was performed in the OpenFOAM package using the fluid volume method (VOF method) and the standard k-e turbulence model. The evolution of the fields of volumetric gas content, velocity, and pressure during the flow of gas from the high-pressure chamber into a closed channel filled with liquid in the presence of a ”gas blanket“ at the upper end of the channel is obtained. It was shown that the dynamics of pulsations in the gas cavity that occurs when the gas flows into the closed region substantially depends on the physical properties of the liquid in the volume, especially the density. Numerical modeling showed that the injection of gas into water occurs in the form of a jet outflow of gas, and for the outflow into liquid lead, a gas slug is formed at the bottom of the channel. Satisfactory agreement was obtained between the numerical calculation and the calculation according to the asymptotic model for pressure pulsations in a gas projectile in liquid lead. For water, the results of calculations using the asymptotic model give a significant difference from the results of numerical calculations. In all cases, the velocity of the medium obtained by numerical simulation and when using the asymptotic model differ by an order of magnitude or more.

Methods of calculating the motion of melt over the surface of fuel element in accident conditions are presented. The calculation is performed using mathematical modeling methods. The results of calculating the melt motion and solidification on the fuel pin surface at various vapor flow velocities and for various types of fuel are shown.

Results of simulations of air injection into a liquid-filled (water or liquid lead) closed vessel with a high pressure difference are presented. The simulations reveal a big difference in processes of air volume formation and evolution of its boundaries during air injection into air or liquid lead. A comparison of periodic pressure pulsations in the gas volume (simulation results) with pressure pulsations described by a quasi-stationary 1D model demonstrates conservatism of the simplified model. The results of destruction of gas cavities with pressure pulsations for water and liquid lead are compared. Other conditions being identical, the gas volume in water decays faster than that in lead.

The data on the evolution of the interface at the sudden injection of gas from a long tube into a liquid column was obtained. High speed video of experiments and calculations in OpenFoam software package were used as research methods. Investigations were performed for different initial pressures. Typical configurations of the interface for different time periods were shown both experimentally and numerically.

Methods for calculating the motion of a melt along the fuel element surface under the conditions of an accident are presented. The calculation was carried out using semianalytic methods for the shear stress and gravity-driven films. The results of calculations of the motion of a melt under various initial conditions are presented. Particular attention is paid to the influence of surface tension forces on the interface between the melt and a solid surface.