Computational analysis of the steam-water flow through a submerged perforated sheet

. The flow of saturated steam through the hole of a submerged perforated sheet (SPS) is analyzed using the FlowVision code. A computational model of one cell of a perforated sheet is created and the grid convergence of the solution on a sequence of refining grids is demonstrated. Four turbulence models including k-eps, k-epsFV, KENL and SST are considered. Comparison of calculated pressure drops with experimental data shows that the SST model gives the most accurate result with an error of less than 3%. The peculiarities of the spatial steam flow through the hole of the SPS have been established. The influence of the degree of perforation on the pressure drop at the hole and the coefficient of hydraulic resistance is analyzed. With an increase in the degree of perforation from 4.3% to 8.1%, the pressure drop decreases by about four times.


Introduction
The horizontal steam generator (SG) at a nuclear power plant with Russian-designed water-water energy reactor (WWER) is a heat exchanger for the generating dry saturated steam of the required parameters due to the heat of the primary coolant coming from a nuclear reactor [1,2].The horizontal SG consists of a cylindrical vessel in which two vertical collectors (inlet and outlet) are located.Horizontal U-shaped heat exchange tubes are installed between the collectors, Figure 1.The steam generator operates as follows.The coolant heated in the reactor enters the inlet SG collector and is distributed through heat exchange tubes, passing through which it enters the outlet SG collector.Due to the heat of the coolant, the water of the secondary side of the SG is boiled and steam is generated, which is then removed from the SG through the steam outlet tubes in the upper part of the shell.In the horizontal SG, there is a significant spatial non-uniformity of the steam load, namely, near the inlet hot collector, the greatest steam load due to the higher temperature of the primary heating coolant, and near the outlet cold collector, the smallest steam load.
Since saturated steam is supplied to the turbine from the SG, an important parameter is the moisture of the steam, which should not exceed the limit value of 0.2% in order to avoid erosive wear of the turbine blades.To ensure the required steam moisture in the horizontal SG, a gravity separation scheme is currently used in combination with a submerged perforated sheet (SPS).SPS, having a high hydraulic resistance, is designed to equalize the steam load on the evaporation surface, in this regard, it is one of the main elements of the separation scheme.Experimental studies of the hydrodynamic characteristics of an SPS were carried out both on fullscale SGs [3,4] and on specially constructed test facilities [5,6].In general, these works proved the ability of an SPS to reduce the spatial non-uniformity of the steam load.The most complete study of the influence of submerged perforated sheets with different designs on the equalization of the steam load was performed at the PGV test facility [7,8].The test section was a slice-model of the upper part of the steam generator, starting from the upper rows of the tube bundle and ending with the distribution perforated sheet.The experiments were carried out at a pressure of 7 MPa and a steam load of 4-8 t/h, which corresponds to the superficial steam velocities on the evaporation surface of 0.15-0.30m/s.Two types of SPS were studied: 1) with a uniform degree of perforation of 5.7% and 2) with E3S Web of Conferences 459, 04008 (2023) https://doi.org/10.1051/e3sconf/202345904008XXXIX Siberian Thermophysical Seminar a variable degree of perforation (on one half of the sheet -4.1%, on the other -8.3%).Based on the processing of experimental data, a semi-empirical correlation was proposed for the hydraulic resistance coefficient of the SPS, which is valid in the range of the studied parameters.
The use of methods of direct numerical simulation of the flow of a steam-water mixture through a perforated sheet complements and expands the possibilities of experimental research.In [9], an analysis was carried out using the CFD code of several problems on the two-phase flow through the hole of the SPS, based on the results obtained, it was concluded that the presence of water droplets in the steam flow under certain conditions can reduce the pressure drop on the SPS.In [10], a computational analysis of experiments [5,6] on the study of the hydrodynamic characteristics of SPS was performed.A good quantitative agreement has been obtained on the pressure drops on the SPS, especially for the flow of single-phase steam In this paper, the FlowVision code [11] is used to simulate a single-phase steam flow through an SPS hole under experimental conditions [7,8].

Problem statement and solution method
An isothermal flow of saturated steam in one periodic cell of a submerged perforated sheet is considered.The geometry of the computational domain is shown in Figure 2, the dimensions correspond to the degree of the perforation of the SPS 5.7%.The flow velocity at the inlet to the computational domain is 0.273 m/s.The saturated steam pressure is 7 MPa, while the density and dynamic viscosity are 36.5 kg/m 3 and 1.9×10 -5 Pa*s, respectively.All above mentioned parameters correspond to the experimental conditions at the SG test facility [7,8].
The FlowVision code [11] is developed for modeling three-dimensional liquid and gas flows in technical and natural objects, as well as visualization of these flows by computer graphics methods.The simulated flows include stationary and non-stationary, compressible, weakly compressible and incompressible flows of liquid and gas.The use of various turbulence models and an adaptive computational grid makes it possible to simulate complex fluid flows, including flows with strong swirling, combustion, flows with a free surface.In this research, a model of the turbulent flow of an incompressible fluid is used, which includes following equations and models: Mass conservation Momentum conservation Here  -velocity,  -density,  -pressure,  ̂viscous stress tensor,  ̂ -deformation rate tensor,  ̂ -unit tensor,  -molecular viscosity,   -turbulent viscosity.

Turbulence models
The FlowVision code has seven models for calculating turbulent viscosity: To approximate the equations of conservation of mass and momentum, a finite-volume approach is used, as a result, a system of linear algebraic equations is obtained, for the solution of which a solver of the TParFBSS (GMRES-based method) with a preconditioner is used.
The development of a computational grid is essential for obtaining an adequate solution.First, it is necessary to create a three-dimensional geometric model of the computational domain using some solid-state modeling package.The SolidWorks package is used in this research.Then this model is imported into FlowVision, which has an automatic unstructured grid builder with the possibility of its local dynamic adaptation.Several calculation grids have been developed, Figure 3. First, a basic grid is created, having 29 cells on the X axis, 194 cells on the Y axis and 41 cells on the Z axis (a total of 217 874 calculation cells).In the central part of the computational domain, cells of a minimum size of 0.13x0.4x0.7 mm are located near the hole.Then the size of the cells gradually increases to 2.3x11x2.7 m at the input and output boundaries.After that, the adaptation of this basic grid is carried out: • Grid No. 1. Basic grid.Adaptations are not applied.
• Grid No. 2. The adaptation of the 1st level of the computational grid to the curvature of the surface and to the sharp corners on the edge of the hole, consisting of 3 layers of cells, that is, the base cell is divided into three ones.Such adaptation is applied to the inner surface of the hole and the inlet surface of the SPS.The total number of calculation cells is 245 000.• Grid No. 3. The adaptation of the 1st level of the computational grid to the curvature of the surface and to the sharp corners on the edge of the hole, consisting of 5 layers of cells, is applied to the inner surface of the hole and to the inlet surface of the SPS, and the adaptation of the 1st level of 3 layers of cells is applied to the channel wall after the hole.The total number of calculation cells is 309 000.• Grid No. 4. The adaptation of the 4th level of the computational grid to the curvature of the surface and to sharp corners at the edge of the hole, consisting of 3 layers of cells, is applied to the hole.The adaptation of the 1st level of 3 layers of cells is applied to the surface of the SPS, channel walls before and after the hole .The total number of calculation cells is 770 000.

Investigation of the influence of the computational grid and turbulence models
The influence of the computational grid and turbulence models on the solution was studied.The pressure drop that occurs when steam flows through the hole and the maximum steam velocity in the hole were considered as comparative parameters.Calculations were carried out on four grids containing 218 thousand cells (Grid No. 1), 245 thousand cells (Grid No. 2), 309 thousand cells (Grid No. 3) and 770 thousand cells (Grid No. 4) of control volumes, respectively.Four turbulence models, KES, KEFV, KENL and SST, briefly described above, were also used.The parameter  + = √   ⁄   ⁄ depends on the grid and characterizes the local Reynolds number in the cell adjacent to the wall.The maximum values of the  + on the inner surface of the hole are presented in Table 1.In accordance with the FlowVision manual recommendations, equilibrium wall functions were used for all turbulence models.The analysis of the Table 1 showed that the maximum value of Y+ for grid No. 1 goes beyond the recommended limit (30<Y+<150).Maximum values of Y+ for grids No. 2-4 are within the recommended range.It is also worth noting that the maximum value of Y+ for grid No. 4 is on the border of the applicability of wall functions and with further refining it will be necessary to abandon wall functions and resolve the wall layer using a grid.
Figures 4, 5 show that the choice of the turbulence model significantly affects the calculation results.The lowest values of the pressure drop and maximum velocity are obtained for the KES model, and the highest values are observed for the SST model.At the same time, grid convergence is observed for all turbulence models.This is most clearly expressed for the SST model -with an increase in the number of cells from 309 thousand to 770 thousand control volumes, the results practically do not change (0.02%).The results calculated with the most detailed grid No.4 for pressure drops are compared with the value ∆Pexp = 980 Pa obtained experimentally [7,8], Table 2.The SST turbulence model predicts the most accurate result with an error of less than 3%.Therefore, this model is used for all subsequent calculations.Since calculations using the SST model require quite a lot of computer time, it was decided to perform calculations on grid No. 3, and not to carry out calculations using the SST model on the most detailed grid No. 4. As it was established above, the differences in the results obtained on grids No. 1 and No. 2, are small.

General flow structure
The qualitative patterns of steam flow through the hole in all calculations are approximately the same.Let's consider them on the example of calculation using the SST turbulence model on the most detailed grid No.4 (770 thousand cells).The pressure and velocity distributions in the cross section of the channel and in the hole are shown in Figures 6 and 7, respectively.It can be seen that the pressure drop occurs very sharply in a narrow hole, the thickness of which is 6 mm.After leaving the hole, the pressure is partially recovered.It should be noted that the pressure in the model of an incompressible fluid is a relative value and in this case the negative pressure region occurs only because the outlet pressure is set to zero to increase the accuracy of calculations.In reality, the absolute pressure value in this system is about 7 MPa.The narrowing of the flow leads to an increase in velocity, as a result, a high-speed jet (~ 7.5 m/s) is formed at the outlet of the hole, penetrating far enough into the downstream zone behind the hole.A powerful vortex flow is formed around the jet.It is also interesting to note that due to the sharp inlet edges of the hole, the flow is separated from the walls of the hole and a vortex is formed, clearly visible in Figure 7b.

Influence of the degree of perforation of SPS on hydraulic resistance
This section presents the results of modeling the saturated steam flow through SPS cells with transverse dimensions of 32x50 mm and 66x50 mm, which corresponds to the degrees of perforation of 4.3% and 8.1%, respectively.All other geometrical and thermophysical parameters remain the same.Based on the studies discussed above, the SST model was selected as the turbulence model.The approach to constructing grid models for these two computational domains corresponded to the grid construction method No. 3 described earlier.
The qualitative picture of the flow through a narrow hole remains the same, however the degree of perforation of the SPS has a significant effect on the quantitative parameters.Figure 8 shows the pressure and velocity distributions along the axis of the SPS cell.It can be seen that with an increase in the degree of perforation from 4.3% to 8.1%, the pressure drop decreases by about 4 times, and the maximum speed value decreases by 2 times.
Figure 9 shows the coefficients of hydraulic resistance  = ∆ 0,5 2 , calculated by the FlowVision code and obtained in experiments at the PGV test facility [7,8] for different values of the degree of perforation of the SPS.There is a very good agreement of the values of the hydraulic resistance coefficients, the relative error is 0.13-2.79%.

Conclusion
In this work, the flow of saturated steam through the SPS cell was analyzed using the FlowVision code.The main results are as follows.A computational model is created and the grid convergence of the solution on a sequence of refining grids is demonstrated.It is also found that the turbulence model used has a strong influence on the calculation results.Comparison of pressure drops during steam flow through the SPS hole, calculated using four different turbulence models (KES, KESFV, KENL and SST), with the experimental value, showed that the most adequate results are obtained when using the SST model (the difference was less than 3%).The pattern of the emerging flow is characterized by the presence of powerful vortices formed inside the hole when the flow is separated at the front sharp edges and then when the flow exits the hole into a wide channel.The influence of the degree of perforation on the pressure drop at the hole and the coefficient of hydraulic resistance is analyzed.With an increase in the degree of perforation from 4.3% to 8.1%, the pressure drop decreases by about four times.A comparison of the values of the hydraulic resistance coefficient obtained in the calculations by the FlowVision code with the experimental data obtained at the PGV test facility also showed a good agreement.
Further research will be continued in two directions: 1) the created calculation model will be used as an element of a more complex calculation scheme for the flow of the steam-water mixture in a steam generator, 2) the influence of the presence of water droplets in the steam on the hydraulic resistance coefficient will be studied.

Fig. 4 .
Fig. 4. Influence of thegrid on the pressure drop for different turbulence models.

Fig. 5 .
Fig. 5. Influence of the grid on the maximum velocity for different turbulence models.

Fig. 6 .Fig. 7 .
Distribution of parameters in the central longitudinal section: (a) pressure, (b) streamlines.Distribution of parameters near the hole: (a) pressure, (b) velocity.

Fig. 8 .
Parameter distributions along the axis of the SPS cell: (a) pressure, (b) velocity.

Fig. 9 .
Fig. 9. Coefficient of hydraulic resistance depending on the degree of perforation. ξ

Table 1 .
Maximum Y+ on the inner surface of the hole.

Table 2 .
Values and errors in determining the pressure drop.