Study of Condensation during Direct Contact between Steam and Water in Pressure-Relief Tank

: Direct contact condensation (DCC) is a phenomenon observed when steam interacts with subcooled water, exhibiting higher heat and mass transfer rates compared to wall condensation. It has garnered significant interest across industries such as nuclear, chemical, and power due to its advantageous characteristics. In the context of pressure-relief tanks, understanding and optimizing the DCC process are critical for safety and efficiency. The efficiency of pressure-relief tanks depends on the amount of steam condensed per unit of time, which directly affects their operational parameters and design. This study focuses on investigating the direct gas–liquid contact condensation process in pressure-relief tanks using computational fluid dynamics (CFD). Through experimental validation and a sensitivity analysis, the study provides insights into the influence of inlet steam parameters and basin temperature on the steam plume characteristics. Furthermore, steady-state and transient calculation models are developed to simulate the behaviour of the pressure-relief tank, providing valuable data for safety analysis and design optimization. There is a relatively high-pressure area in the upper part of the bubble hole of the pressure-relief tube, and the value increases as it is closer to the holes. The steam velocity in the bubbling hole near the 90 ◦ elbow position is higher. This study contributes to the understanding of steam condensation dynamics in pressure-relief tanks. When the steam emission and pressure are fixed, the equilibrium temperature increases linearly as the initial temperature increases (where a = 1, b = 20 in y = a x+ b correlation), the equilibrium pressure increases nearly exponentially, and the equilibrium gas volume decreases. When the steam emission and initial temperature are fixed, the equilibrium temperature does not change as the steam discharge pressure increases. The correlations between the predicted equilibrium parameters and the inlet steam parameters and tank temperature provide valuable insights for optimizing a pressure-relief tank design and improving the operational safety in diverse industrial contexts


Introduction
In the nuclear reactor unloading and discharge processes, high-temperature and highpressure steam from the regulator passes through the safety valve and discharge pipe behind the valve to the unloading tank.The discharge tank receives, condenses, and cools the steam from the regulator while ensuring the integrity of the pressure boundary of the main circuit.The first-circuit coolant should not be discharged to the plant to avoid contamination by the radioactive first-circuit fluid, and in the case of accidents, the water load of the first circuit should be contained to ensure adequate recirculation in the event of a breach.The steam condensation in the pressure-relief tank is a direct contact condensation (DCC) process.The DCC process involves a violent exchange of mass, momentum, and energy, and the shape, length, and gas-liquid interface parameters of the vapour plume formed in this process describe the DCC [1][2][3].The mass properties and heat transfer behaviour during this process have been studied by many researchers over the past three decades, mainly through experiments and numerical simulations.
Early research was mainly experimental using a high-speed camera, thermometer, differential pressure meter, and other experimental instruments to obtain the shape of the steam plume, flow field speed, temperature, pressure, etc., in the experimental system and then analysing the two-phase interface condensation heat transfer coefficient and other parameters.Kim et al. [4] conducted an experimental study on the direct contact condensation process of the steam jet injected with cold water and discussed the effects of the steam mass flow rate and liquid supercooling degree on the shape of the steam plume, the length of the steam plume, and the average heat transfer coefficient.Jue Wang et al. [5] systematically reviewed the DCC of steam jets in subcooled water, including the condensation regime maps and flow patterns, the flow and heat transfer characteristics of the steam jet, and the pressure oscillations inside the discharge pipes or near the steam jet.Qiang Xu et al. [6] studied the condensation characteristics, including plume shape, plume length, temperature distribution, average heat transfer coefficient, and average Nusselt number, through the direct contact condensation experiments of stable steam jets in water flow in a vertical pipe.
With the rapid development of computer technology, computational fluid dynamics (CFD) has become increasingly popular.Gulawani et al. [7] used a three-dimensional CFD method to study the DCC phenomenon in subsonic and sonic flows.Using a thermal phase transition model from the commercial CFD code CFX, they investigated the vapour plume length, axial and radial directions of the steam nozzle during condensation, and parameters such as temperature distribution and condensation heat transfer coefficient.Jiang, B. et al. [8] used a boundary layer model to study the condensation and separation of a steam-CO 2 mixture on horizontal and vertical plates.Yao Zhou et al. [9] used a threedimensional numerical simulation of the high-pressure steam-water condensing ejector with non-condensable gas, which was conducted based on an inhomogeneous multiphase model.Patel et al. [10] used NEPTUNE_CFD and OpenFOAM to simulate low-flow steam jets under the same working conditions and compared parameters such as the plume shape, length, and expansion ratio simulated by the two tools.Tanskanen et al. [11] used NEPTUNE_CFD to simulate the condensation surge region based on a two-dimensional axisymmetric mesh.Jeon et al. [12] performed CFD simulations of steam bubble condensation in water using the VOF model in FLUENT.The bubble condensation process was resolved using a custom function (UDF), and the simulation results were compared with experimental data.Li [13] used the thermal phase-change condensation model, VOF two-phase flow model, and LES turbulence model in the Fluent6.3 software to calculate the unsteady-state process of steam jet condensation by dividing the condensation process into four stages: initial, maximum stabilisation, oscillation, and separation stages.
Overall, additional research is required for single-nozzle steam jets in stationary pools to achieve more stable flow patterns and for surge and condensation oscillation zones to deal with their complexity and further improve the existing models.From the present state of research, it is evident that many studies have been conducted on steam jets, but they are often limited to single-nozzle steam jets in static pools.Studies on steam jets in multi-nozzle ejectors, specially shaped pipelines, and other environments, such as marine environments, are insufficient, and the temperature field distribution and condensation rate characteristics and their mechanisms of action during steam DCC still need to be further explored.In addition, for unstable jets, owing to the complexity of their mechanisms, many uncertainties remain that require further investigation.Traditional research on pressure-relief tanks has focused on the conditions and effects of the opening of the overall pressure-relief valve of a pressure-relief tank, and there are few studies on the direct contact of local vapour and liquid with condensation.For example, Chen, S [14] et al. calculated the bolt-tightening torque of a pressure-relief tank rupture piece.In general, local details of the DCC phenomenon in a pressure-relief tank are still lacking.

Geometry Model
The pressure-relief tank is a horizontal cylindrical vessel with an ellipsoidal head, and its bottom is supported by two saddle-shaped seats.Figure 1 shows the schematic of the pressure-relief tank.The tank usually contains water and a gas space, which is occupied by nitrogen.The purpose of nitrogen filling is to prevent the hydrogen contained in the steam emitted by the regulator from mixing with oxygen in the air, which produces crackling gas.The upper part of the gas space in the pressure-relief tank is equipped with a sprinkler pipe.The borated water from the reactor and makeup system water are sprayed through the sprinkler pipe, which can condense the steam escaping from the water layer of the bubble tube and cool the pressure-relief tank.There is a cooling coil in the water space of the pressure-relief tank, which is connected to the component cooling-water system and can cool the water in the pressure-relief tank.The steam enters the pressure-relief tank through the bubble tube below the water level.The structures of the decompression and bubble tubes allow for the steam to be mixed with water close to the ambient temperature to condense and cool the steam.The top of the relief tank is equipped with two safety rupture discs to prevent the pressure-relief tank from overpressurising and the gas emitted by the rupture disc from entering the containment.The pressure-relief tank is designed according to the primary emission of the condensing and cooling regulator, which is approximately equal to 110% of the steam capacity of the regulator at full power.The water inside the pressure-relief tank absorbs the emitted heat.Under normal conditions, the water level in the tank is approximately 2/3 to 3/4 of the total height.The volume of nitrogen in the pressure-relief tank is determined by limiting its maximum pressure after a design discharge, and the minimum nitrogen pressure during operation must meet a certain value.
al. calculated the bolt-tightening torque of a pressure-relief tank rupture piece.In general, local details of the DCC phenomenon in a pressure-relief tank are still lacking.

Geometry Model
The pressure-relief tank is a horizontal cylindrical vessel with an ellipsoidal head, and its bottom is supported by two saddle-shaped seats.Figure 1 shows the schematic of the pressure-relief tank.The tank usually contains water and a gas space, which is occupied by nitrogen.The purpose of nitrogen filling is to prevent the hydrogen contained in the steam emitted by the regulator from mixing with oxygen in the air, which produces crackling gas.The upper part of the gas space in the pressure-relief tank is equipped with a sprinkler pipe.The borated water from the reactor and makeup system water are sprayed through the sprinkler pipe, which can condense the steam escaping from the water layer of the bubble tube and cool the pressure-relief tank.There is a cooling coil in the water space of the pressure-relief tank, which is connected to the component cooling-water system and can cool the water in the pressure-relief tank.The steam enters the pressure-relief tank through the bubble tube below the water level.The structures of the decompression and bubble tubes allow for the steam to be mixed with water close to the ambient temperature to condense and cool the steam.The top of the relief tank is equipped with two safety rupture discs to prevent the pressure-relief tank from overpressurising and the gas emitted by the rupture disc from entering the containment.The pressure-relief tank is designed according to the primary emission of the condensing and cooling regulator, which is approximately equal to 110% of the steam capacity of the regulator at full power.The water inside the pressure-relief tank absorbs the emitted heat.Under normal conditions, the water level in the tank is approximately 2/3 to 3/4 of the total height.The volume of nitrogen in the pressure-relief tank is determined by limiting its maximum pressure after a design discharge, and the minimum nitrogen pressure during operation must meet a certain value.The design and operating parameters of the pressure-relief tank are as follows.The design pressure and temperature are 0.8 MPa and 170 °C, respectively.The initial pressure is 0.12 MPa (absolute pressure), and the operating temperature is below 100 °C.The heat load of the pressure-relief tank cooling coil is 35 kW, the cooling water flow rate is 3.5 t/h, The design and operating parameters of the pressure-relief tank are as follows.The design pressure and temperature are 0.8 MPa and 170 • C, respectively.The initial pressure is 0.12 MPa (absolute pressure), and the operating temperature is below 100 • C. The heat load of the pressure-relief tank cooling coil is 35 kW, the cooling water flow rate is 3.5 t/h, and the pressure difference between the two bursting discs at the top is 0.6 MPa.Regarding the geometric parameters, the total volume of the pressuriser is 5.6 m³, and the nitrogen volume is 1.4 m³.The inner diameter of the pressure-relief tube is 65 mm, and the number of bursting holes is unknown.Based on the known volume, the diameter of the hemispherical heads on both sides of the expansion tank is 1560 mm, and the length of the central cylinder is 1890 mm.
The mixture model can solve the continuity, momentum, energy, volume fraction of the second phase, and relative velocity equations of the mixed phase.The number of solution equations is few, the calculation amount is small, and it can well simulate the two large velocity differences in steam jet condensation.Therefore, in this study, the mixture model was selected to simulate the DCC process.
Here, → v m is the mass average of the speeds of each phase in the control unit.
where ρ m is the average density and α k is the volume fraction of the k phase.
(2) Momentum equation where n = 2 is the number of phases, the two terms on the left side of the equation are the transient and convection terms, the first term on the right side is the pressure, the second term is the viscous force, the third term is the gravitational force, the fourth term is the other body force, and the last term is the external volume force source term.µ m is the viscosity of the mixture.
→ v dr,k is the drift velocity for phase k.
(3) Energy equation The first term on the right side of the equation is defined based on the heat conductivity, wherein k e f f is the effective thermal conductivity, which consists of the thermal conductivity of the material and thermal conductivity of the turbulent part.The thermal conductivity of the turbulent part is defined according to the turbulence equation: S E is the other volumetric heat source, and the energy E k can be calculated as follows: Energies 2024, 17, 2772

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When a fluid is treated as incompressible, the last two terms of the equation can be ignored, and h k represents the apparent enthalpy of the kth phase.

Interphase Interaction Model
In the DCC process of steam, the steam is condensed into water (phase transition), which involves mass and energy transfer between the two phases.The energy transfer between the two phases is equal to the latent heat of vaporisation multiplied by the mass transfer; thus, the difference between different models is in the method of mass transport.The current condensation models can be mainly divided into a phase change coefficient model derived from molecular dynamics and a double thermal resistance model derived from the thermal equilibrium equation.
The phase change coefficient model, first proposed by Lee [15], is based on molecular dynamics; therefore, it is also known as the Lee model, which sets the condensation coefficient in the model to 0.1, although this coefficient is not universal.With the deepening of research, more scholars have improved this model; however, the determination of the condensation coefficient is still difficult, and experiments are generally required.
The phase change coefficient model assumes that some molecules in the process of evaporative condensation escape or are absorbed from the gas-liquid interface.If the gasliquid interface is in equilibrium, the number of molecules escaping from the gas-liquid interface and being absorbed will be equal.The collision frequency rate of the molecules and planes is described by the Hertz-Knudsen relationship.
The evaporative condensation flux given by Hertz-Knudsen can be expressed as where R is the universal gas constant, 8.314 J/(mol • K); P * is the steam partial pressure on the steam side of the interface,Pa; T sat is the local saturation temperature, K; P sat is the saturation pressure, Pa; and β is the regulation coefficient, which represents the ratio of the experimental value of the condensation rate to the theoretical maximum.
where σ is the condensation (evaporation) coefficient, which represents the degree of difficulty of evaporative condensation, and its value varies between 0.001 and 1.0.Many scholars have proposed a formula for the condensation coefficient, but the results obtained are quite different; therefore, the determination of the condensation coefficient is difficult when using the phase change coefficient model and often requires experiments.At saturation, the relationship between pressure and temperature is determined by the Clapeyron-Clausius equation: where ρ v and ρ l are the densities of steam and water, respectively, and L is the latent heat of vaporisation in J/kg.Around the saturation point of steam, Combining Equations ( 12) and ( 9) yields Energies 2024, 17, 2772

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To obtain the condensation rate, the above formula must be multiplied by the interfacial area density, which can be calculated according to the spherical hypothesis: where α v and α l are the volume fraction of the gas and liquid phases, respectively, and d b is the bubble diameter.The mass source term of the condensation can be written as Based on the principle of thermal equilibrium, the double thermal resistance model considers the heat transfer on both sides of the gas-liquid interface.The mass and energy transfer between the two phases is obtained through the thermal equilibrium equation on both sides, assuming that the gas-liquid interface is at the saturation temperature and the values of the heat of the incoming and outgoing gas-liquid interfaces are equal.
The heat entering the liquid and gas phases from the interface can be expressed, respectively, as Because heat cannot be stored at the interface, the sum of the heat entering the interface should be zero.Thus, Therefore, the mass transmitted from the liquid phase to the gas phase can be calculated as where Q is the total heat flux, W; h is the interface heat transfer coefficient, W/ m 2 • K ; H ls is the liquid phase saturation enthalpy, J/kg; H gs is the gas phase saturation enthalpy, J/kg; m lv is the mass transfer rate, kg/s; and A i is the contact area, m 2 .The subscripts l and v represent the liquid and gas phases, respectively, and the calculation formula for the heat transfer coefficient can be adjusted according to the particular situation in which the model is used, which has strong applicability.
The heat transfer coefficient from the interface to the liquid phase can be expressed as where k f is the liquid phase thermal conductivity and d g can be defined as Here, the reference subcooling degree and bubble diameter are as follows: when θ 0 = 13.5 K, d 0 = 1.5 × 10 −4 m, and when θ 1 = 0 K, d 1 = 1.5 × 10 −3 m.

Nu f can be calculated as
Nu f = 2.0 + 0.6Re 0.5 Pr 0.33 0 ≤ Re ≤ 776.06 2.0 + 0.27Re 0.62 Pr 0.33 Re > 776.06 (23) The contact area per unit volume can be determined by The model was validated by Kim's experiment, which was conducted by Korea Atomic Energy Research Institute in 2001 based on the setup of steam condensation in a quenching tank [4].

Operating Conditions
The operating conditions of the pressure-relief tank can be classified into two types: normal operation and severe accident.
During normal operation of the pressure-relief tank, the nitrogen occupying the upper space is stabilised at a certain pressure by the nitrogen pressure-relief valve, and the nitrogen is supplied from a nitrogen cylinder.The cooling coil installed in the pressurerelief tank is cooled by an external water source.A high-temperature alarm instrument is installed on the pressure-relief line, and the alarm is set at a certain temperature.
When the high-temperature alarm occurs, the operator can manually open the water replenishment valve in the operation support control room for spraying, and the nozzle is set at the end of the spray pipe to ensure cooling.When the pressure-relief tank is in high liquid-level alarm, the drain valve can be opened depending on the situation (the water temperature must be lower than the high-temperature alarm value), and part of the non-liquid is discharged to the waste liquid storage tank.When the water temperature of the pressure-relief tank increases to a certain value or the pressure of the pressure-relief tank exceeds a certain value, the drain isolation valve is automatically closed if the valve is opened; if the valve is closed, it is automatically prohibited from opening to prevent steam emission at high temperature and pressure.When the pressure regulator is no longer discharging, the temperature in the pressure-relief tank has dropped to a specific value.At this time, the pressure is stable, and the operator can manually open the diaphragm compressor so that the hydrogen-containing exhaust gas in the pressure-relief tank is delivered to the decay tank for storage and decay, and the radioactivity is reduced to a particular level after discharge.
The design of the pressure-relief tank should satisfy the following requirements: (a) The water volume of the pressure-relief tank can receive 110% of the steam volume of the normal steam chamber of the regulator at any given time.(b) After the condensation of all the steam received, the water temperature and pressure in the tank should not exceed 100 • C and 0.6 MPa, respectively.Furthermore, the water level and temperature in the pressure-relief tank should be restored to normal values prior to receiving steam within 90 min.(c) The shape of the pressure-relief tube and design of the closing hole should ensure that the steam condenses in the water space of the pressure-relief tank and that no high-temperature steam directly enters the gas space of the pressure-relief tank.
Under severe accident conditions, the high-temperature, high-pressure steam discharged by the regulator overpressure will be continuously discharged into the pressurerelief tank, and the water temperature and pressure in the pressure-relief tank will continue to increase.When the pressure in the pressure-relief tank rises to a certain value, the electrical shut-off valve of the discharge pipe is opened manually in the operation support control room, and the steam in the pressure-relief tank is discharged directly to the pool in the radioactive plant.A number of bubble holes are provided in the pipeline of the pool, and the electrical shut-off valve is closed manually when the pressure in the pressure-relief tank is reduced to normal.To prevent water from the containment pool from flowing back into the pressure-relief tank due to vapour condensation, a non-return valve is fitted on the discharge pipe.The volume of the pressure-relief tank is limited, and it cannot receive steam continuously discharged by the regulator safety valve.If the accident is not controlled, the steam will continue to be discharged into the pressure-relief tank, causing the pressure in the tank to rise to the threshold of the blast disc pressure differential.The blast disc will rupture, and the steam will be discharged directly into the hall of the plant.

Conditions for Steady-State Analogue Inputs Calculation Area Selection
Under normal operating conditions, the high-temperature, high-pressure steam can be completely condensed by the supercooling water in the pressure-relief tank.The water temperature in the pressure-relief tank can be maintained below a certain threshold (usually not more than 60 • C), and the nitrogen pressure in the upper gas space is stabilised at 0.12 MPa by the nitrogen cylinder.Therefore, for normal operating conditions, we first perform steady-state calculations, and the computational fluid domain of the steady-state calculations cannot consider the fluid domain of the nitrogen in the upper space (hereinafter referred to as the upper nitrogen); however, the pressure boundary effect of the upper nitrogen is expressed by the pressure outlet boundary condition of the computational domain.Considering that the wall thickness has no effect on the result and will increase the difficulty of meshing, the wall thickness of the pressure-relief pipe was ignored in the construction of the geometric model.Because the geometric size of the pressure-relief tank is large, the scale of the bubbling hole and length of the tank are large over the scale, and the pressure-relief tank is symmetrical in the vertical direction.Half of the symmetrical structure of the pressure-relief tank was considered to reduce the number of meshes and improve the calculation efficiency when constructing the geometric model.In addition, the cooling coil and gas-space sprinkler pipe structure of the water space and blasting disc structure at the top were not considered when constructing the geometry.The final geometric model under normal operating conditions is displayed in Figure 2.
Energies 2024, 17, x FOR PEER REVIEW 9 of 27 Under normal operating conditions, the high-temperature, high-pressure steam can be completely condensed by the supercooling water in the pressure-relief tank.The water temperature in the pressure-relief tank can be maintained below a certain threshold (usually not more than 60 °C), and the nitrogen pressure in the upper gas space is stabilised at 0.12 MPa by the nitrogen cylinder.Therefore, for normal operating conditions, we first perform steady-state calculations, and the computational fluid domain of the steady-state calculations cannot consider the fluid domain of the nitrogen in the upper space (hereinafter referred to as the upper nitrogen); however, the pressure boundary effect of the upper nitrogen is expressed by the pressure outlet boundary condition of the computational domain.Considering that the wall thickness has no effect on the result and will increase the difficulty of meshing, the wall thickness of the pressure-relief pipe was ignored in the construction of the geometric model.Because the geometric size of the pressure-relief tank is large, the scale of the bubbling hole and length of the tank are large over the scale, and the pressure-relief tank is symmetrical in the vertical direction.Half of the symmetrical structure of the pressure-relief tank was considered to reduce the number of meshes and improve the calculation efficiency when constructing the geometric model.In addition, the cooling coil and gas-space sprinkler pipe structure of the water space and blasting disc structure at the top were not considered when constructing the geometry.The final geometric model under normal operating conditions is displayed in Figure 2.

Meshing
The pressure-relief tube is arranged in an L-shaped structure in the pressure-relief tank, and there are hundreds of small pressure-relief holes on the surface of the pressurerelief tube.The structural mesh partition under this structure is large and cumbersome, and the mesh quality is inferior; therefore, the Workbench meshing tool was finally used

Meshing
The pressure-relief tube is arranged in an L-shaped structure in the pressure-relief tank, and there are hundreds of small pressure-relief holes on the surface of the pressure-relief tube.The structural mesh partition under this structure is large and cumbersome, and the mesh quality is inferior; therefore, the Workbench meshing tool was finally used to partition the pressure-relief tank for non-structural meshing.Because the area near the bubble holes is subjected to dense turbulence and drastic parameter changes, this area must be meshed.The final meshing result is illustrated in Figure 3.The grid independence analysis was continued with the numbers of the grids 650,746, 1,306,424, 2,606,234, and 3,411,695, and the aberration of the inlet flow was determined by the affiliation of the inlet flow, as presented in Table 1.Consequently, the number of grid elements was finally selected as 3,411,695.
Energies 2024, 17, x FOR PEER REVIEW 10 of 27 flow, as presented in Table 1.Consequently, the number of grid elements was finally selected as 3,411,695.The boundary conditions refer to the operating conditions of the expansion tank.The inlet boundary condition was set to be the pressure inlet, which has a value of 0.8 MPa; the outlet boundary was set to the pressure outlet boundary condition, which has a value of 0.12 MPa; and 0.12 MPa is set to be the reference pressure.The bubbling small-hole surface was set as the inner surface, and the other wall surfaces were set as the thermal non-slip boundary condition.The physical properties of the water vapour at saturation and the properties of water at the appropriate tank temperature and pressure were calculated.

Simulation of Transient Conditions Calculation Area Selection
The steam condensation in a pressure-relief tank is a transient process in which the parameters change with time.To determine the amount of steam condensed during the normal operation of the pressure-relief tank, it is necessary to perform transient calculations.When performing transient calculations, the effect of nitrogen cannot be ignored because it is pressurised in the gas space and occupies a certain volume.Therefore, when performing transient calculations, the upper nitrogen was included in the calculation model.The wall thickness and symmetry of the vent tube were considered in the same manner as in the steady-state calculations, and the final geometry is shown in Figure 4.

Boundary Conditions and Calculation Control
The boundary conditions refer to the operating conditions of the expansion tank.The inlet boundary condition was set to be the pressure inlet, which has a value of 0.8 MPa; the outlet boundary was set to the pressure outlet boundary condition, which has a value of 0.12 MPa; and 0.12 MPa is set to be the reference pressure.The bubbling small-hole surface was set as the inner surface, and the other wall surfaces were set as the thermal non-slip boundary condition.The physical properties of the water vapour at saturation and the properties of water at the appropriate tank temperature and pressure were calculated.

Simulation of Transient Conditions Calculation Area Selection
The steam condensation in a pressure-relief tank is a transient process in which the parameters change with time.To determine the amount of steam condensed during the normal operation of the pressure-relief tank, it is necessary to perform transient calculations.When performing transient calculations, the effect of nitrogen cannot be ignored because it is pressurised in the gas space and occupies a certain volume.Therefore, when performing transient calculations, the upper nitrogen was included in the calculation model.The wall thickness and symmetry of the vent tube were considered in the same manner as in the steady-state calculations, and the final geometry is shown in Figure 4. Assuming that the pressure-relief tank is a closed structure, as the vapour condenses in the tank, its pressure and temperature continue to increase and eventually reach a state of three-phase equilibrium between nitrogen, water vapour, and water, and the temperature between the three phases remains the same.

Meshing
The geometry was also meshed using the Workbench meshing tool because this part of the computational domain only added the upper nitrogen space based on the normal operating condition computational domain.There is no need for a mesh independence analysis; only the mesh size of the nitrogen domain and optimum size of the lower domain need to be consistent, and the final meshing result is depicted in Figure 5.

Boundary Conditions and Calculation Control
The boundary conditions refer to the operating conditions of the pressure-relief system.The inlet boundary condition was also set to the pressure inlet, the bubble-hole circular surface was set to the inner surface, and the other wall surfaces were set to the thermal non-slip boundary condition.The physical properties of the water vapour at satura-

Meshing
The geometry was also meshed using the Workbench meshing tool because this part of the computational domain only added the upper nitrogen space based on the normal operating condition computational domain.There is no need for a mesh independence analysis; only the mesh size of the nitrogen domain and optimum size of the lower domain need to be consistent, and the final meshing result is depicted in Figure 5.

Meshing
The geometry was also meshed using the Workbench meshing tool because this part of the computational domain only added the upper nitrogen space based on the normal operating condition computational domain.There is no need for a mesh independence analysis; only the mesh size of the nitrogen domain and optimum size of the lower domain need to be consistent, and the final meshing result is depicted in Figure 5.

Boundary Conditions and Calculation Control
The boundary conditions refer to the operating conditions of the pressure-relief system.The inlet boundary condition was also set to the pressure inlet, the bubble-hole circular surface was set to the inner surface, and the other wall surfaces were set to the thermal non-slip boundary condition.The physical properties of the water vapour at satura-

Boundary Conditions and Calculation Control
The boundary conditions refer to the operating conditions of the pressure-relief system.The inlet boundary condition was also set to the pressure inlet, the bubble-hole circular surface was set to the inner surface, and the other wall surfaces were set to the thermal non-slip boundary condition.The physical properties of the water vapour at saturation and properties of water at the appropriate tank temperature and pressure were calculated.Nitrogen was considered an ideal gas, and its parameters, such as pressure and temperature, varied according to the ideal gas equation of state.The calculation monitored the pressure, temperature, and other parameters in the expansion tank.

Equilibrium State Analysis
In the process of simulating the pressure-relief tank to reach the equilibrium state, the problems of slow calculation speed and easy divergence were encountered.During the CFD calculation, the MTLAB program was used to determine the mass, energy conservation, and total volume in the pressure-relief tank to obtain the temperature and pressure parameters of the final equilibrium state.
When a pressure-relief tank is designed, it is usually also based on the final thermodynamic equilibrium state as the standard; that is, the pressure-relief tank is in the final state of steam and water, but in its three-phase equilibrium state, the temperatures of the three phases are the same.The steam is the saturated steam at this temperature, and the final pressure consists of the partial pressures of the saturated steam and nitrogen.When calculating the thermal-hydraulic characteristics of the pressure-relief tank in this study, the temperature and pressure parameters of the three elements in the equilibrium state were calculated using this method, which could help verify and analyse the CFD calculation results. (

1) Parameter input
The initial pressure of the pressure-relief tank is 0.12 MPa, and its initial operating water temperature is 20-60 • C. The steam displacement is 100-300 kg, and the steam pressure is 16.4-0.8MPa.
(2) Calculation method During the initial and final thermo-equilibrium states, the total volume of the pressurerelief tank remains unchanged.
where V is the total volume of the pressure-relief tank; V w is the initial water volume; V 1 is the initial nitrogen volume; m w is the initial water mass; m c is the condensed mass in the discharge steam, m c = m s − m g = m s − ρ g V 2 ; υ f is the specific volume of water in the equilibrium state; V 2 is the gas-space volume for thermodynamic equilibrium (the gas space is occupied by nitrogen and saturated steam; the two have different partial pressures, but the volume is the same); m s is the discharge steam mass; m g is the mass of the saturated steam when the thermal equilibrium is in the saturated state; and ρ g is the saturated steam density.

Analysis of the Steady-State Simulation Results
Figure 6 displays the distribution of the vapour volume fraction within the pressurerelief tank under steady-state computational conditions.It can be seen from the figure that owing to the short distance between the bubbling holes, the vapour plumes formed by the steam at the outlet of each bubbling hole affect each other, resulting in contact between the plumes, forming a 'vapour cloud'.In particular, in the area where the bubble hole of the pressure-relief tank is far from the symmetrical surface, the plume expands outwards.The interface between the gas and liquid is close to the inner wall of the pressure-relief tank.
Therefore, the size of the space in the width direction must be considered when designing the pressure-relief tank.Figure 7 indicates the pressure distributions in the vertical planes of Y = 0.1, 0.3, and 0.5 m parallel to the symmetric surface in the pressure-relief tank.As shown, there is a relatively high-pressure area in the upper part of the bubble hole of the pressure-relief tube, and the closer the vertical plane is to the bubble hole, the higher the pressure and the larger the high-pressure area are.It should be noted that a negative pressure area exists in the lower part of the pressure-relief tube, which also explains why the steam plume expands.The negative pressure area will make the inner wall of the pressure-relief tank unbalanced, which is likely to cause structural damage and fatigue.Therefore, the position of the bottom of the pressure-relief tube from the inner wall of the bottom of the pressure-relief tank must also be considered in the design.Figure 8 further quantifies the pressure distribution near the outlet of the bubble hole and compares the distribution of the static and dynamic pressures at the outlet of the bubble hole.It can be seen that the static and dynamic pressures at the outlet of the bubble hole are distributed in a waveform, which is related to the position distribution of the bubble hole.The pressure is higher in the centre of each bubble hole but is lower in the middle area of the two bubble holes.In general, the static pressure in the area near the bubble hole is greater than the dynamic pressure; however, the dynamic pressure fluctuates more with a change in position.
Energies 2024, 17, x FOR PEER REVIEW 13 of 27 planes of Y = 0.1, 0.3, and 0.5 m parallel to the symmetric surface in the pressure-relief tank.As shown, there is a relatively high-pressure area in the upper part of the bubble hole of the pressure-relief tube, and the closer the vertical plane is to the bubble hole, the higher the pressure and the larger the high-pressure area are.It should be noted that a negative pressure area exists in the lower part of the pressure-relief tube, which also explains why the steam plume expands.The negative pressure area will make the inner wall of the pressure-relief tank unbalanced, which is likely to cause structural damage and fatigue.Therefore, the position of the bottom of the pressure-relief tube from the inner wall of the bottom of the pressure-relief tank must also be considered in the design.Figure 8 further quantifies the pressure distribution near the outlet of the bubble hole and compares the distribution of the static and dynamic pressures at the outlet of the bubble hole.
It can be seen that the static and dynamic pressures at the outlet of the bubble hole are distributed in a waveform, which is related to the position distribution of the bubble hole.
The pressure is higher in the centre of each bubble hole but is lower in the middle area of the two bubble holes.In general, the static pressure in the area near the bubble hole is greater than the dynamic pressure; however, the dynamic pressure fluctuates more with a change in position.Figure 9 illustrates the velocity distribution in the vertical plane where the symmetry plane is located.It can be observed that after the steam passes through the 90 • elbow of the pressure-relief pipe, and owing to the different times required to reach the bubbling hole, the steam velocity in the bubbling hole near the 90 • elbow position is higher.The principle of the right end is 90 • , and the elbow position velocity is small.In addition, the steam at the outlet of the bubbling hole exhibits a tendency of moving to the right, which is also due to the effect of the steam flowing horizontally in the pressure-relief tube.The closer it is to the elbow position, the greater the amount of steam flowing horizontally in the pressure-relief tube; thus, the speed bends to the right more obviously.Similarly, Figure 10 quantifies the velocity distribution near the outlet of the bubble holes, where the velocity of the centre plane of each bubble hole is high, the velocity between the holes is low, and the overall velocity decreases from left to right.Figure 11 shows the temperature distribution of the vertical plane where the symmetric plane is located and the vertical plane of the symmetric plane.The temperature and velocity distributions exhibit the same trend, from left to right, because the speed of the steam flowing through the bubble hole gradually decreases, and the faster the steam is condensed, the faster the drop in the core temperature and the lower the core temperature are.The temperature distribution near the bubble holes was further evaluated (Figure 12).A clear trend with position fluctuation and an overall downward trend can be observed.trend, from left to right, because the speed of the steam flowing through the bubble hole gradually decreases, and the faster the steam is condensed, the faster the drop in the core temperature and the lower the core temperature are.The temperature distribution near the bubble holes was further evaluated (Figure 12).A clear trend with position fluctuation and an overall downward trend can be observed.

Analysis of Transient Results
Figure 13 displays the distribution cloud of the steam volume fraction at 0-2.0 s i the pressure-relief tank.It can be seen that at 0.5 s, the shape of the plume is similar to tha in the steady-state calculation, and the plumes formed by the outlet of each bubbling hol are in contact with each other and connected to produce a 'steam cloud'.As the steam

Analysis of Transient Results
Figure 13 displays the distribution cloud of the steam volume fraction at 0-2.0 s in the pressure-relief tank.It can be seen that at 0.5 s, the shape of the plume is similar to that in the steady-state calculation, and the plumes formed by the outlet of each bubbling hole are in contact with each other and connected to produce a 'steam cloud'.As the steam condenses, the steam near the 90 • elbow has a larger expansion area owing to the higher velocity.The contact area between the steam and water is also larger, and the condensation effect is enhanced.As the vapour condensation process progresses, the overall plume shape tends to swing to the right.Figure 14 indicates the pressure distributions in the vertical planes of Y = 0.1, 0.3, and 0.5 m parallel to the symmetrical surface in the pressure-relief tank at 1.0 s.It can be seen that the lower area of the pressure-relief tank has a higher pressure than the upper area.This is because the condensation of water vapour in the lower water space causes the pressure to increase.Moreover, as the water temperature does not change much in a short time, the changes in the temperature and pressure of nitrogen are not obvious.There are also areas of lower pressure near the bubble holes, which is a typical phenomenon that often occurs during DCC between steam and water.Figures 15-18 depict the velocity and temperature distributions at 1.0 s in the vertical planes of Y = 0, 0.1, 0.3, and 0.5 m, respectively.In the symmetry plane position, similar to that in the steady-state calculation, the velocity and temperature distributions at the outlet of the bubbling hole exhibit a left-to-right attenuation trend.At the plane position of Y = 0.1 m, the plume shape expands into this plane, forming a corresponding region of higher velocity and a region of higher temperature.The temperature map shows that the temperature of the upper nitrogen is slightly higher than that of the water located below.In the Y = 0.3 m plane, the high-velocity and high-temperature regions exhibit a small circular distribution, which is already close to the edge of the plume.Based on the temperature cloud map, the temperature of the upper nitrogen rises to approximately 335 K, which is 10 K higher than the temperature of the water below, and the horizontal plane fluctuates owing to the influence of the impact pressure of the steam jet.In the Y = 0.5 m plane, no steam is present, as it can no longer reach this area.The water near the edge of the plume is disturbed by the condensation process and produces a velocity, but the value is small.Figures 19-21 illustrate the parameter changes during equilibrium at different steam outlet pressures of 16.4, 7.2, and 0.8 MPa when the steam emission is 125 kg.As shown in Figure 19, when the steam emission and pressure are fixed, the equilibrium temperature increases with an increase in the initial temperature, and there is a linear correlation between them (where a = 1, b = 20 in y = a x+ b correlation).When the steam emission and initial temperature are fixed, the equilibrium temperature does not change with a change in the outlet steam pressure.As depicted in Figure 20, when the steam emission and pressure are fixed, the equilibrium pressure increases with an increase in the initial temperature, and a similar index correlation exists between them.When the steam emission and initial temperature are fixed, the equilibrium pressure does not change with a change in the outlet steam pressure.It can be seen from Figure 21 that when the steam emission and discharge steam pressure are fixed, the equilibrium gas volume decreases with an increase in the initial temperature.Moreover, when the steam emission and initial temperature are fixed, the equilibrium volume decreases with an increase in the discharge steam pressure, but the degree of the reduction is small.Figures 22-24 display the parameter changes during equilibrium at different steam emissions of 125, 250, and 300 kg when the steam discharge pressure is 16.4 MPa.It can be observed from Figure 22 that when the steam discharge pressure and emission are fixed, the equilibrium temperature increases linearly as the initial temperature increases, and the rate of temperature increase was consistent for different mass flow rates.When the steam discharge pressure and initial temperature are fixed, the equilibrium temperature increases as the steam emission increases.As shown in Figure 23, when the steam emission and steam discharge pressure are fixed, the equilibrium pressure increases with an increase in the initial temperature, and a similar index correlation exists between them.When the steam discharge pressure and initial temperature are fixed, the equilibrium pressure increases as the steam emission increases.As illustrated in Figure 24, the equilibrium gas volume decreases with an increase in the initial temperature when the steam emission and the steam discharge pressure are fixed, and the equilibrium volume decreases with an increase in the steam emission when the steam discharge pressure and initial temperature are fixed.

Conclusions
Based on CFD, this study developed a three-dimensional two-phase CFD model tha can be used to simulate the DCC phenomenon between steam and water.Then, the co responding simulation calculations were carried out.The equilibrium parameters wer predicted, and their relationships with the inlet steam parameters and tank temperatur were determined.This study can provide a useful reference for the safety analysis an design optimisation of pressure-relief tanks.The main conclusions are as follows: The pressure is higher in the centre of each bubble hole but is lower in the midd area of the two bubble holes.The static pressure in the area near the bubble hole is greate than the dynamic pressure; however, the dynamic pressure fluctuates more with a chang in position.
When the steam emission and discharge steam pressure are fixed, the equilibrium temperature increases with an increase in the initial temperature, and there is a linear co relation between them with a = 1 and b = 20 in the y = a x+ b correlation, the equilibrium pressure increases with an increase in the initial temperature, and a similar index correla tion exists between them; the equilibrium gas volume decreases with an increase in th initial temperature.
When the steam emission and initial temperature are fixed, the equilibrium tempe ature does not change with a change in the outlet steam pressure, the equilibrium pressur does not change with a change in the outlet steam pressure, and the equilibrium volum decreases with an increase in the discharge steam pressure, but the degree of reduction small.
When the steam discharge pressure and initial temperature are fixed, the equilibrium temperature increases as the steam emission increases, the equilibrium pressure increase as the steam emission increases, and the equilibrium volume decreases with an increas in the steam emission.

Conclusions
Based on CFD, this study developed a three-dimensional two-phase CFD model that can be used to simulate the DCC phenomenon between steam and water.Then, the corresponding simulation calculations were carried out.The equilibrium parameters were predicted, and their relationships with the inlet steam parameters and tank temperature were determined.This study can provide a useful reference for the safety analysis and design optimisation of pressure-relief tanks.The main conclusions are as follows: The pressure is higher in the centre of each bubble hole but is lower in the middle area of the two bubble holes.The static pressure in the area near the bubble hole is greater than the dynamic pressure; however, the dynamic pressure fluctuates more with a change in position.
When the steam emission and discharge steam pressure are fixed, the equilibrium temperature increases with an increase in the initial temperature, and there is a linear correlation between them with a = 1 and b = 20 in the y = a x+ b correlation, the equilibrium pressure increases with an increase in the initial temperature, and a similar index correlation exists between them; the equilibrium gas volume decreases with an increase in the initial temperature.
When the steam emission and initial temperature are fixed, the equilibrium temperature does not change with a change in the outlet steam pressure, the equilibrium pressure does not change with a change in the outlet steam pressure, and the equilibrium volume decreases with an increase in the discharge steam pressure, but the degree of reduction is small.
When the steam discharge pressure and initial temperature are fixed, the equilibrium temperature increases as the steam emission increases, the equilibrium pressure increases as the steam emission increases, and the equilibrium volume decreases with an increase in the steam emission.

Figure 2 .
Figure 2. Geometric model under normal operating conditions.

Figure 2 .
Figure 2. Geometric model under normal operating conditions.

Figure 3 .
Figure 3. Mesh model under normal operating conditions.

Figure 3 .
Figure 3. Mesh model under normal operating conditions.

Figure 7 .
Figure 7. Pressure distribution in the vertical plane.

Figure 8 .
Figure 8. Pressure distribution at the outlet of bubbling holes.

Figure 8 .
Figure 8. Pressure distribution at the outlet of bubbling holes.Figure 8. Pressure distribution at the outlet of bubbling holes.

Figure 8 .
Figure 8. Pressure distribution at the outlet of bubbling holes.Figure 8. Pressure distribution at the outlet of bubbling holes.

Figure 9 .
Figure 9. Cloud of velocity distribution in the vertical plane where the pressure-relief pipe is located.

Figure 9 .Figure 10 .
Figure 9. Cloud of velocity distribution in the vertical plane where the pressure-relief pipe is located.

Figure 10 .
Figure 10.Distribution of bubbling hole outlet velocity.

Figure 10 .
Figure 10.Distribution of bubbling hole outlet velocity.

Figure 11 .Figure 11 . 2 Figure 12 .
Figure 11.Temperature distribution in the vertical plane where the pressure-relief pipe is located.

Figure 12 .
Figure 12.Temperature distribution at the outlet of the bubbling tube.

Energies 2024 ,
17, x FOR PEER REVIEW 18 of 2 (a) 0.5 s and 1.0 s

Figure 14 .
Figure 14.Pressure distribution in the three vertical planes of the pressure-relief tank at 1.0 s.Figures 15-18 depict the velocity and temperature distributions at 1.0 s in the vertical planes of Y = 0, 0.1, 0.3, and 0.5 m, respectively.In the symmetry plane position, similar to that in the steady-state calculation, the velocity and temperature distributions at the outlet of the bubbling hole exhibit a left-to-right attenuation trend.At the plane position of Y = 0.1 m, the plume shape expands into this plane, forming a corresponding region of higher

Figure 14 .
Figure 14.Pressure distribution in the three vertical planes of the pressure-relief tank at 1.0 s.

Figure 15 .
Figure 15.Velocity and temperature distributions at 1.0 s and on the Y = 0 m plane.Figure 15.Velocity and temperature distributions at 1.0 s and on the Y = 0 m plane.

Figure 15 .
Figure 15.Velocity and temperature distributions at 1.0 s and on the Y = 0 m plane.Figure 15.Velocity and temperature distributions at 1.0 s and on the Y = 0 m plane.

Figure 15 .Figure 16 .
Figure 15.Velocity and temperature distributions at 1.0 s and on the Y = 0 m plane.

Figure 16 .
Figure 16.Velocity and temperature distributions at 1.0 s and on the Y = 0.1 m plane.

Figure 17 .
Figure 17.Velocity and temperature distributions at 1.0 s and on the Y = 0.3 m plane.Figure 17.Velocity and temperature distributions at 1.0 s and on the Y = 0.3 m plane.

Figure 17 .
Figure 17.Velocity and temperature distributions at 1.0 s and on the Y = 0.3 m plane.Figure 17.Velocity and temperature distributions at 1.0 s and on the Y = 0.3 m plane.Energies 2024, 17, x FOR PEER REVIEW 22 of 2

Figure 18 .
Figure 18.Velocity and temperature distributions at 1.0 s and on the Y = 0.5 m plane.Figures 19-21 illustrate the parameter changes during equilibrium at different steam outlet pressures of 16.4, 7.2, and 0.8 MPa when the steam emission is 125 kg.As shown in Figure 19, when the steam emission and pressure are fixed, the equilibrium temperatur increases with an increase in the initial temperature, and there is a linear correlation be tween them (where a = 1, b = 20 in y = a x+ b correlation).When the steam emission and

Figure 18 .
Figure 18.Velocity and temperature distributions at 1.0 s and on the Y = 0.5 m plane.