The paper describes one of the variants of mathematical models of a fluid dynamics process inside the containment, which occurs in the conditions of operation of spray systems in severe accidents at nuclear power plant. The source of emergency emissions in this case is the leak of the coolant or rupture at full cross-section of the main circulating pipeline in a reactor building. Leak or rupture characteristics define the localization and the temporal law of functioning of a source of emergency emission (or accrued operating) of warmed up hydrogen and steam in the containment. Operation of this source at the course of analyzed accident models should be described by the assignment of the relevant Dirichlet boundary conditions. Functioning of the passive autocatalytic recombiners of hydrogen is described in the form of the complex Newton boundary conditions.

The paper describes an approach to fluid dynamics simulations of processes occurring inside the nuclear power plant (NPP) containment during operation of spray systems in severe accidents. This topic is relevant at the current stage of advancement in high-accuracy numerical simulations of NPP operation [

The models proposed in the paper are intended for engineers and technical staff involved in the NPP design and maintenance at both construction and operation stages. Numerical analysis of these models can be performed without high-performance computers.

One of the mandatory prerequisites for the mathematical modeling in our case is that we should correctly describe the operation of the spray system and heat transfer in steel-liner containment walls accounting for steam condensation on them. The spray system is installed under the dome of the central reactor building. According to its purpose, it belongs to confinement safety systems, and its basic functions include pressure and temperature control in the containment during severe accidents.

Combustion risk of such clouds in this case is assessed using the known Shapiro diagram. In the first approximation, the aqueous solution of boric acid distributed by the spray system can be considered similar to water in its physical properties.

Within our model we assume that the gas mixture in containment rooms has six components: hydrogen produced during severe accidents,

As a basis for the development of the fluid dynamics model of processes inside the containment, we used an adapted set of Reynolds equations completed by the

The turbulence model was chosen by the authors based on the results of computational experiments with steam-hydrogen-air flows in containment rooms without considering the operation of the spray system. These experiments demonstrated considerable advantages of this model over the zero-order turbulence models and various modifications of the

Thus, the basis for the fluid dynamics model of processes inside the containment with neglect of the spray system operation can be represented in the following way (see, e.g., [

It should be noted that in the model (

In order to account for heat exchange processes during production (evaporation) and condensation of steam by the spray injection, model (

In our case, evaporation of each droplet is caused by the heat supplied to it from the surrounding gas mixture and spent on the phase change due to the given latent mass heat of water droplet evaporation

We obtain a formula for the rate

As it evaporates, the droplet decreases in its diameter. Let us estimate the rate of droplet diameter variation. The droplet mass is

However, at the initial stage of its flight, the droplet has a low temperature, at which the steam pressure can be lower than the steam pressure in the gas mixture. For this reason, during the initial period of the flight, steam can condense on the droplet surface with droplet warming up due to latent heat of steam condensation. Let us consider the process of steam condensation onto the droplet

Now, let us derive an equation to calculate the mass rate of steam condensation

During droplet evaporation,

Note that the process of droplet falling is observed to include two modes. The first mode of droplet motion begins, when droplets issue from the nozzles of the spray system. It is characterized by initial bulk condensation of steam on the droplets with increase in their diameter and temperature. The second mode of motion begins, when the droplet temperature reaches the saturation point for the current steam pressure. This mode is characterized by droplet evaporation and decrease in droplet diameter.

During the second mode, steam mass fraction is observed to grow, while the gas mixture temperature in the region, where evaporating droplets move, decreases. Such a growth of steam content in some cases may lead to secondary spontaneous bulk condensation. This process, however, will be ignored for the reasons described below.

Bulk condensation requires, in particular, that the gas mixture temperature should decrease to the saturation point (and below). If the spray system is out of operation, the temperature decreases rather slowly. For this reason, even if secondary bulk condensation occurs, this process will be slow (at least, compared to the primary condensation on the falling cold droplets of water). In addition, the group of droplets considered here will be followed by another group of droplets, and so forth. The subsequent groups of droplets will first pass through the already cooled layers of the gas mixture almost without heating. Consequently, in the mixture layers, where the first group evaporates, primary condensation of steam can occur on the second group. The process of droplet flight from the time it issues from the nozzle to its complete evaporation is rather fast (at least, compared to the variation of mixture temperature, when the spray system is out of operation). For this reason, secondary bulk condensation is ignored here.

Now, let us develop the model of droplet flight in a flow of gas mixture. We introduce the parameter

Droplet drag is estimated here using a function that approximates experimental data [

Thus, the equation of droplet motion can be written in the following way:

We consider a small volume of the gas mixture,

The velocity of droplet motion is

Thus, the mathematical model of spray system operation will combine (

Film condensation of steam on containment walls will be taken into account here by defining corresponding boundary conditions. Note that a simplified variant of such an approach has been presented in [

As we know, quite an acceptable estimate of the rate of heat and mass exchange for film condensation of single-component resting steam on a vertical flat wall is provided by the composite formula of Nusselt [

Composite formula (

In case of condensation of superheated steam, one should account for the superheating heat

If the wall temperature is below the saturation point, condensation of superheated steam proceeds in a way similar to the case of saturated steam. Of course, this does not mean that superheated steam becomes immediately saturated throughout the volume. Steam becomes saturated only at the wall as it cools down, while far from the wall it can remain (and remains) superheated. Since

If steam is superheated, additional heat flux,

The presence of air and other non-condensing gases in steam significantly decreases heat transfer during condensation [

In fact, according to Dalton’s law, the total pressure

As a result of these processes, the temperature difference

The experimental curve of the relative heat transfer coefficient as a function of relative air content in steam is provided in [

In our case of hydrogen-steam-air flow inside the containment during severe accidents, air, which consists primarily of oxygen and nitrogen, and hydrogen can be treated as a non-condensing gases. The relative content of hydrogen (even in case of its accidental release) is rather small. For this reason, the plot (see Figure

Let

For numerical simulations of steam condensation on the inner walls of a reactor building, we introduce a parameter

The saturation point of steam can be calculated using the equation similar to the Antoine equation [

Relative variation of the heat transfer coefficient during condensation as a function of air content in steam (subscript “b” stands for “air”; subscript “Π” stands for “steam”) [

In new NPP designs, the inside surface of double containment inner concrete wall has a liner up to 10 mm thick carbon steel plates. The thickness of inner wall can vary from one meter to a meter and a half. In order to determine the appropriate mesh density for the numerical analysis of the set of (

The first step in this method is to perform a preliminary numerical analysis of model (

The mentioned code has been chosen because the CFX solver [

The second step is solving a model heat transfer problem, the sketch of which is shown in Figure

Sketch of a model heat transfer problem to estimate the parameters of heat exchange through the inner containment wall.

This problem models a transient heat flow through some region of a nonuniform wall composed of steel and concrete layers of corresponding thicknesses. Considering the size of the reactor building structure, the curvature of the containment’s cylindrical and spherical parts can be ignored when making the needed estimates, and the problem can be solved in the plane statement. The heat transfer problem can be solved by the Finite Element Method (FEM) in the program ANSYS environment [

In the numerical thermal analysis of the model problem the Newton boundary conditions were assigned on the outside (concrete) and inside (steel) surfaces of the wall. Let us give an example of values of the physical parameters we used for the numerical analysis of an severe accident at the Baltic NPP. The following conditions were defined on the outside wall surface: ambient temperature

Volume-averaged temperature of the steam-hydrogen-air mixture for adiabatic boundary conditions.

Example of a snapshot of the field of velocities (m/s) in containment rooms.

Physical properties of the materials were specified in accordance with [

Cross-sections of the computational model (see Figure

These numerical simulations produced time distributions of temperature through the thickness of the containment wall. Figure

Temperature pattern (K) through the thickness of the inner containment wall at the time 10,000 s after the beginning of accidental steam-hydrogen mixture release.

Temporal variations of the through-the-thickness temperature profile of the inner containment wall during accidental steam-hydrogen mixture release.

It follows from Figure

The simulation results shown in Figure

Numerical integrating of the results of solving the model problem for each of the containment wall materials in accordance with the above formula for a number of times

Heat accumulated in the wall’s structural parts during the severe accident.

As evidenced by the numerical simulation results (see Figure

On the other hand, for direct numerical simulations of steel liner, the model should include a fine mesh with a characteristic linear dimension of

At the same time, because of steel’s high thermal conductivity, the temperature difference between the outer (containment facing) and inner (concrete facing) surfaces of the liner structure is extremely small (see Figures

Examples of CFD analysis of steam-hydrogen-air cloud evolution inside the Baltic NPP containment with a functioning spray system, PARs, and condensation heat exchangers of passive heat removal from the containment rooms are illustrated by some results shown in Figure

Calculated results of the temperature field (K) (a) and isosurface of 40-percent molar fraction of steam (b) in containment rooms.

The modeling area can be described by the following principal dimension: the height is about 70 m; the radius of a spherical segment is 22 m; the height of a parallel portion is about 18 m; the radius of a parallel portion is 22 m. Here were analyzed the consequences from hypothetical accident due to small leak of the equivalent diameter 100 mm from the cold line of the coolant pipe at 6.9 m distance from the reactor inlet and with the superposition of active part fault of the area emergency cooling system. The inflow functions of hydrogen and steam to the modeling area are presented in Figure

Integral operation (time) of hydrogen (a) and steam (b).

The examples of 3D CFD analysis results are presented in Figures

It should be noted that the relevance of

The description of numerical experiments and comparison procedure is beyond the scope of this paper. However, it should be noted that the verification outcome has shown the reasonability of proposed models application for CFD-analysis of the emergency emissions of steam-hydrogen-air mixture inside containment.

The presented in the paper model of steam-hydrogen-air mixture evolution in containment rooms accounting for the operation of the spray system enables high accurate numerical analysis of severe accidents without high-performance computers. With properly specified boundary conditions, this model makes it possible to perform numerical analysis accounting for the operation of PARs and condensation heat exchangers of the passive heat removal system in the reactor building. It is reasonable to use this model as a substitute for simplified models in alike simulations described in [