Comparative analysis of film boiling correlations for steam explosion problem

. The article considers the existing models and correlations for describing heat transfer during film boiling. Film boiling is an important thermophysical process that determines the course of interaction of the melt of reactor core materials with the coolant, which can potentially occur during a severe accident at a nuclear power plant with a pressurized water-cooled reactor (VVER/PWR). The values of the Nusselt number predicted by these models are compared. A fairly significant dispersion of calculated parameters has been established, making it difficult to unambiguously choose one or another correlation. However, in the range of parameters typical for the interaction of a high-temperature melt of reactor core materials with a coolant, several correlations give quite close values, which allows them to be recommended for use in calculation codes for modeling thermal-hydraulic processes during a severe accident at a nuclear power plant with VVER/PWR type.


Introduction
With the development of a severe accident at nuclear power plants (NPPs) with light water reactors of the VVER/PWR type, direct contact of the melt of materials of the reactor core with the coolant can occur. When two liquids come into contact with significantly different temperatures, when the temperature of one of the liquids significantly exceeds the boiling point of the other, under certain conditions, a rapid boiling of a cold liquid occurs, accompanied by a rapid increase in pressure and expansion of the zones of their thermal interaction. This phenomenon, referred to as "steam explosion", represents a significant hazard as it can lead to significant dynamic forces on the reactor and containment structures and pose a threat to their integrity. In connection with the possible release of radioactive fission products, this phenomenon is being actively studied in the world's leading research centers specializing in the field of nuclear energy safety [1][2].
Since the temperature of the melt is much higher than the boiling point of the liquid, at the initial stage of a steam explosion, a vapor film is formed on the surface of the melt, separating the melt from the coolant. The presence of a vapor film on the surface of the melt significantly reduces the intensity of heat transfer between the melt and water and contributes to the creation of a quasi-equilibrium explosive mixture "melt-steam-liquid", which can exist for a relatively long time (1-10 s), allowing the melt to mix with a large amount of liquid. To date, quite a lot of experimental and theoretical studies of film boiling have been carried out and various semi-empirical correlations have been proposed to describe this phenomenon. Film boiling can be realized under various geometric conditions (boiling on a sphere, cylinder, horizontal/vertical plane, etc.) and under various thermophysical parameters (pressure, temperature, liquid velocity, etc.). Therefore, correlations obtained for some conditions may give incorrect results when applied in other conditions. This paper provides a comparative analysis of the available semi-empirical correlations for heat transfer during film boiling for conditions typical of steam explosions that occur during a severe accident at a nuclear power plant.

Materials and methods
Heat transfer correlations during film boiling, as a rule, represent a system of rather complex formulas, including the calculation of various quantities. Therefore, only brief information about the considered correlations is given here.
Frederking-Clark correlation [3]. These authors theoretically considered film boiling near a hot sphere immersed in a large volume of water at saturation temperature. An expression was obtained for the Nusselt number, which characterizes the heat transfer coefficient of the sphere and depends on the diameter of the sphere. However, the experimental data that they used to compare with the resulting formula showed no dependence on the diameter of the sphere (in the experimental range of parameters). Therefore, two formulas have been proposed: Where C = 0.586 and 0.25 n  (theory) and C = 0.14 and 0.33 n  (experiment).
Here  [4]. These authors found that the configuration of the interface and the dynamics of vapor removal are largely determined by the relative size of the sphere, characterized by the Bond number sphere. The mathematical model includes the Navier-Stokes equation for the vapor flow in the film, the continuity equation and the energy equation. The resulting correlation covers the entire range of sphere diameters from small to large: Where ' Spdimensionless superheat, ' the angle at which the thin film turns into the bubble, parameter characterizing the "thickness" of the bubble, ( ) G Bodependence given by the authors in graphical form. '  also depends on Bo (the dependence is also given in graphical form).
Klimenko correlation [5]. An approach based on the division of spheres into small and large ones, which makes it possible to take into account various steam evacuation mechanisms, was used. Relations were obtained for film boiling near a sphere, in which the power law 1∕4 is used for the laminar regime of vapor flow, and the law 1∕3 for the turbulent regime: Dhir-Purohit correlation [6]. Based on the results of experimental studies, these authors summarized the data in the form of empirical correlations: Natural convection: , r Cparameter dependent on thermophysical properties and temperature values.
Forced convection: Large subcooling: In the general case, the formula was provided: For practical application, the authors recommended introducing a correction factor of 2.04 into (9).
In the MC3D code, the Epstein-Hauser correlation is used in a slightly modified form [8]: Shih-El-Wakil correlation [9]. These authors carried out an investigation of film boiling of subcooled freon near small spheres under conditions of natural convection. Their mathematical model included the equations of continuity, momentum, and energy for a vapor film near a sphere and a liquid layer adjacent to the interface in the boundary layer approximation. Based on parametric calculations, the authors obtained the following expression:   Where Sc -dimensionless subcooling,    Where c M is a rather complex function of ' Sc , ' Sc , l Pr .
Sc c T h   , K = 0.696 for sphere. Liu-Theofanous correlation [11]. These authors carried out a comprehensive experimental and computational study of film boiling on spheres and obtained a universal correlation describing heat transfer in a wide range of parameters -natural/forced convection and saturated/subcooled liquid. The authors considered the problem in a fairly complete formulation: a vapor film is described in the boundary layer approximation, a boundary layer is distinguished in the liquid near the interface, and the liquid flow outside the liquid boundary layer is considered potential. As a result, the following correlation was obtained: . Relations for these parameters are presented in [11].
Kolev correlation [12]. He developed a model of film boiling on a vertical plate and then generalized it to a sphere. In the case of a vertical flat plate, Kolev obtained an analytical expression relating the various components of the heat flux from the plate surface: Where FB q is the heat flux through the vapor film due to thermal conductivity (subcooled liquid), Kolev suggested that a similar relationship is also valid for film boiling on a sphere. Therefore, for ,0 FB q , r q , l q and v q , he selected the appropriate correlations and determined the heat flux FB q .

Results
The graphs below provide information on the similarities and differences between the film boiling models described above. Figure 1a shows a comparison of correlations for film boiling on a sphere of saturated liquid in a pool. A dimensionless group is plotted along the ordinate axis, independent of the diameter of the sphere:

Discussion
Qualitative coincidence in behavior is observed for all considered correlations. As can be seen from Figure 1, all correlations predict that the heat transfer characterized by a dimensionless group, which includes the Nusselt number, increases as the diameter of the sphere decreases (the parameter ' l D increases). Similarly, Figure 2 shows that an increase in subcooling contributes to an increase in the Nusselt number, which characterizes heat exchange during film boiling. Consider the quantitative discrepancies. The value of ' l in the pressure range 1-50 bar varies from 2.5 mm to 1.8 mm. The characteristic sizes of droplets formed by the interaction of a high-temperature melt with water are ~ 1 -30 mm, which approximately corresponds to the range

Conclusion
Film boiling has been studied for a long time both theoretically and experimentally. Quite a lot of models/correlations have been proposed to describe heat transfer. The comparison carried out in this paper shows a very significant dispersion of data obtained from these correlations. However, in the range of parameters typical for the interaction of the melt with the coolant, a number of correlations can be identified that allow for fairly accurate prediction of the values of the heat transfer coefficient. It is most preferable to use the Liu-Theofanous correlation, which gives "median" values.