Abstract

When a water piping system operating under high temperatures and pressures is damaged and water is expelled into the atmosphere, a phenomenon known as depressurization boiling or flashing occurs. This flashing jet poses a risk to human safety and can damage safety equipment through impingement. Consequently, evaluating the region affected by the flashing jet impinging on the surrounding equipment and people is necessary. In this study, we conducted experiments to verify the affected region of the jet involving both saturated and subcooled water under low-pressure conditions at the jet inlet, up to 2 MPaA, and extended our investigation to high-pressure conditions up to 7 MPaA using computational fluid dynamics (CFD). The findings revealed that the mass flux predictions, according to the homogeneous equilibrium model (HEM) outlined in the American National Standards Institute (ANSI) standard, align with both experimental and CFD analysis results. However, the evaluations of both the asymptotic plane width and distance—parameters delineating the jet's affected region—were found to be underestimated in the ANSI standard compared with the experimental and CFD analysis results. To address this difference from real phenomena, we developed an improved model that incorporates mass flux and enthalpy as variables. This improved model more accurately predict the asymptotic plane width and distance than the ANSI standard. Utilizing this improved model enables precise prediction of the flashing jet's affected region, spanning conditions from saturated to subcooled water across various pipe diameters.

1 Introduction

In the event of damage to a water piping system operating under high temperature and high pressure, ejection of water into the atmosphere can lead to the formation of a jet with depressurization boiling, known as flashing. This phenomenon poses a risk to human safety and can cause damage to safety equipment through the impingement of the flashing jet. Consequently, the extent of the jet's impact on surrounding equipment and personnel must be evaluated. The evaluation of the jet's impact is conducted using the existing code adopted in Japan, which is the Japan Society of Mechanical Engineers S ND1 standard [1], which references the standard established in the United States, ANSI/ANS-58.2-1988 [2]. This American National Standards Institute (ANSI) standard is employed in NUREG/CR-6224 as a methodology for assessing the range of destructive effects of jets on thermal insulation surrounding piping ruptured by loss of coolant accident in boiling water reactors [3].

Figure 1 illustrates the affected region of the flashing jet as assumed in the ANSI standard. For a guillotine break occurring at the pipe's midpoint, the flashing jet is hypothesized to expand to the asymptotic plane before ejecting as a free jet. Parameters such as the asymptotic area (Aa), asymptotic jet width (Ha), and the distance from the break plane to the asymptotic plane (La), which delineate the affected region of the flashing jet, are determined through various equations derived from experimental data. The ANSI standard delineates the asymptotic plane as the area where the jet's thrust received by the ruptured pipe is counterbalanced by the jet's load impacting the plate [4,5]. While this definition aids in evaluating the jet's impact affected region on an object, it is not deemed consistent with the region where the human body is susceptible to burn injuries from the jet's impact. Furthermore, the equations do not account for fluid flow conditions such as subcooled water, saturated water, and steam, leading to an issue where different flow conditions are evaluated using the same equation.

Fig. 1
Affected region of flashing jet in the ANSI standard
Fig. 1
Affected region of flashing jet in the ANSI standard
Close modal

The ANSI standard was withdrawn by the American Nuclear Society (ANS) in October 1998, and in 2015, the United States Nuclear Regulatory Commission (USNRC)'s Standards Review Plan [6] pointed out potential nonconservatism. However, the ANSI standard has not yet been replaced and is continually used in industry. Potential nonconservatisms include the following: (a) blast wave, (b) jet plume expansion and zone of influence, (c) distribution of pressure within the jet plume, and (d) jet dynamic loading. To address these nonconservatisms, a detailed review of the published literature was conducted by Kong et al. [7]. This review includes experimental studies on jet impingement, jet impingement modeling, and computational fluid dynamics (CFD) simulations. Experimental studies to investigate jet impact phenomena were conducted in the 1980s, focusing on high-pressure saturated water, subcooled water, and steam–water mixtures. Moreover, jet impact force, thrust, and pressure distribution were measured [8,9]. For mass flux, the homogeneous equilibrium model (HEM), which assumes no slip between the gas and liquid phase velocities of the fluid in the pipe, and separated flow model, which assumes a slip ratio, have been proposed [10]. Our research group has developed a methodology to assess the affected region of steam jets on the surrounding area [1114]. Morita et al. [15] performed numerical simulations of the flashing jet encompassing conditions from subcooled to saturated water and introduced new correlations for both the distance to and the width of the asymptotic plane.

A databank based on the experimental data available in the literature review was constructed by Kong et al. [16], and a model evaluation of ANSI standards was conducted using this databank. The results of the model evaluation to ANSI standards were used to improve the model for predicting the stagnation pressure of the jet [17,18]. In the Kong et al.'s paper [17] and the USNRC report [18], the ANSI standard and CFD results were compared for the jet expanding angle, citing Morita et al. [15] for the saturated steam. However, because of limited experimental data on the flashing jet, quantitative comparisons of the expanding angle and asymptotic plane width and length of the flashing jet were not conducted, and the ANSI model was not improved. Notably, the area and distance of the flashing jet's asymptotic region, due to the risk of burns from hot fluid impingement, are crucial for evaluating the region from the pipe break to the asymptotic area to ensure worker safety in the event of a pipe break.

Hence, a pressing need arises to evaluate and compare the ANSI model with the actual phenomena and develop a new model. In this study, we conducted experiments to verify the validity of the region affected by the flashing jet with both saturated and subcooled water under low-pressure at the jet, inlet up to 2 MPa. Further investigations were conducted under high-pressure conditions up to 7 MPa, which are challenging to simulate experimentally, using CFD. Building on the experimental and CFD findings, we endeavored to devise a novel model for evaluating the affected region of flashing jets under both saturated and subcooled water conditions, which corresponds to improving the potential nonconservatism (b) jet plume expansion and zone of influence.

2 Materials and Methods

2.1 Experimental Setup.

To explore the affected region of the flashing jet experimentally, water was heated to flashing conditions under high temperature and pressure and subsequently released into the injection room. A visualization test was conducted to record the spread of the flashing jet.

The experimental setup, depicted in Fig. 2 and referred to as FLIPS—Fluid Leakage experiment for the Influence on Personnel and Surrounding equipment by pipe failure—was designed for this purpose. FLIPS comprises several components, including a steam boiler, a tank, a pump, a heater, a cooler, an injection room, and various measurement devices. The experimental protocol involves initially configuring a piping loop from the tank through a nozzle leading into the injection room and then back to the tank, facilitated by operating valves. Upon activating the pump, the heater employed to elevate the water temperature to the desired level. Once the target temperature is attained, the pump is deactivated, and by manipulating a valve, water is directed through the pump's bypass piping, with the tank subsequently pressurized by steam produced in the boiler. The nozzle valve is subsequently opened to create a flashing jet within the injection room, which is simultaneously captured using a high-speed video camera (Phantom v1612, Vision Research, Inc.). The injection room itself is a 125 m3 space, measuring 5 m in width, depth, and height. The study allowed for the interchangeability of nozzle fracture shapes, with this experiment focusing on nozzles designed to mimic a guillotine break in pipes, as illustrated in Fig. 2(b). The nozzles featured inner diameters of φ9.2 mm and φ16.1 mm, respectively. Water pressure levels ranging from 0.4 to 1.9 MPa were tested, with water temperature variations from Tsub (=Tsat − T) = 0 K–20 K.

Fig. 2
Schematic diagram of the experimental setup: (a) schematic diagram of FLIPS and (b)nozzle shape: Guillotine break
Fig. 2
Schematic diagram of the experimental setup: (a) schematic diagram of FLIPS and (b)nozzle shape: Guillotine break
Close modal

2.2 Computational Fluid Dynamics Modeling.

In the experimental investigation of the flashing jet's affected region, the jet's external shape can be assessed through visualized images. However, the injection room becomes filled with steam approximately 10 s after the injection commences, rendering prolonged recording challenging. Furthermore, conducting experiments at high pressures is impeded by the pressure resistance capabilities of the experimental setup. Consequently, the investigation into the jet's affected region was conducted numerically through CFD simulations that replicated the flashing jet.

Figure 3 presents the computational domain and boundary conditions employed in the CFD simulation. The simulation aimed to model the behavior of the flashing jet as it is discharged from the nozzle of the buffer tank into the atmospheric environment. The simulations utilized CRUNCH CFD version 3.3.0 [19]. The nozzle diameter (De) was established at 8 mm. The computational domain was defined using a mesh of two-dimensional axisymmetric elements, ensuring the atmospheric environment was sufficiently large to prevent the flashing jet from influencing the boundary surface. A total pressure condition was implemented as a boundary condition at the inlet, where high-temperature water entered upstream of the buffer tank. The inlet boundary conditions for CRUNCH CFD are detailed in Table 1, with the pressure at the inlet boundary ranging from 0.5 to 7.0 MPa. To ensure simulation stability under saturated water conditions, the subcooling degree of the incoming high-temperature water was adjusted to 1 or 2 K. At the outlet boundary, leading to the atmospheric environment, conditions were established to permit air to flow in and out at 0.1 MPa and 283 K. A no-slip wall condition was applied to the wall boundary in contact with the buffer tank and nozzle. As the spatial gradient of temperature and velocity is large at the nozzle exit because of the expansion of the flashing jet with a shock wave, the mesh size was set to Δy = Δz =0.0417D from the nozzle exit to 8D in the axial direction and 3.75D in the radial direction under the saturated water condition. For the subcooled water condition, the mesh size was set to Δy = Δz =0.0417D from the nozzle outlet to 18.75D in the axial direction and 26.25D in the radial direction because the asymptotic region was more extended than that for the saturated water condition. An investigation of the effect of size on the jet width in the asymptotic region between the fine and coarse meshes shown in Fig. 3 shows that the effect of mesh size change on the asymptotic plane width under saturated water conditions was approximately 3%. Additional simulation conditions are shown in Table 2. The validity of CRUNCH CFD in simulating the flashing jet has been confirmed in the previous study [20], with quantitative consistency verified through the comparison of pressure and temperature profiles derived from both experimental measurements and CRUNCH CFD simulations.

Fig. 3
Overview of CFD analysis: (a) computational domain and boundary conditions, (b) computational mesh for saturated water conditions, and (c) computational mesh for subcooled water conditions
Fig. 3
Overview of CFD analysis: (a) computational domain and boundary conditions, (b) computational mesh for saturated water conditions, and (c) computational mesh for subcooled water conditions
Close modal
Table 1

Inlet boundary conditions in CFD analysis

CaseFluidPressure p (MPa)Inlet subcooling temperature Tsub (K)
1Saturated water0.52.0
2Saturated water1.01.0
3Subcooled water1.010
4Subcooled water1.020
5Saturated water2.01.0
6Saturated water5.01.0
7Saturated water6.01.0
8Saturated water7.01.0
CaseFluidPressure p (MPa)Inlet subcooling temperature Tsub (K)
1Saturated water0.52.0
2Saturated water1.01.0
3Subcooled water1.010
4Subcooled water1.020
5Saturated water2.01.0
6Saturated water5.01.0
7Saturated water6.01.0
8Saturated water7.01.0
Table 2

Overview of CRUNCH CFD

Computational codeCRUNCH CFD, ver 3.3.0
FluidCompressible flow (water, air)
TimestepUnsteady simulation
Governing equationsConservations of mass, momentum, and energy
SchemeRoe-scheme
State quantity calculationLook-up table constructed with NIST data
Turbulence modelRANS, k − ε model
Computational codeCRUNCH CFD, ver 3.3.0
FluidCompressible flow (water, air)
TimestepUnsteady simulation
Governing equationsConservations of mass, momentum, and energy
SchemeRoe-scheme
State quantity calculationLook-up table constructed with NIST data
Turbulence modelRANS, k − ε model

3 ANSI Standard

The ANSI standard posits an evaluation method that delineates an affected region as a consequence of fluid dynamics during a pipe failure incident, as illustrated in Fig. 1. It defines crucial indices for assessing the jet's affected area, including the mass flux (G), the asymptotic plane area (Aa), the asymptotic plane width (Ha), and the asymptotic plane distance (La).

According to the ANSI standard, the affected region emanating from the break plane is demarcated into two distinct regions: an asymptotic region, where the jet's width undergoes significant expansion at a large spreading angle due to adiabatic expansion from depressurization boiling right from the nozzle outlet, and a free jet region, characterized by a linear expansion of the jet's width at a smaller spreading angle. The jet expansion in the asymptotic region is considered a free expansion in which high-pressure fluid depressurizes to the pressure of the injection chamber, indicating an irreversible adiabatic expansion. Since the asymptotic region extends over a wide area and the fluid temperature is high, the burn risk in the asymptotic region is high. Therefore, this study focuses on the asymptotic plane width and distance in the affected region of the flashing jet from the viewpoint of the burn risk.

The ANSI standard identifies the asymptotic plane as the area where the jet's thrust, as received by the ruptured pipe, is counterbalanced by the load of the jet impacting a plate [4,5]. It assesses the jet's affected region based on the mass flux, which is the fluid's mass flowrate per unit area, ejected from the nozzle.

Flashing jets create a two-phase flow within the nozzle upon ejection at the discharge plane, leading to a choked flow at this plane due to a reduced in sound velocity, allowing the mass flowrate to be theoretically determined. While the ANSI standard describes several models for theoretical choked flow, this study employs the HEM [21], which presupposes a homogeneous equilibrium flow. The fluid state change in the nozzle calculated in HEM is adiabatic, but the pressure loss due to friction is considered. Therefore, the fluid state quantity is calculated assuming an isenthalpic process. Under the HEM, assuming isenthalpic expansion, the formula for mass flux (mass flowrate per unit area) of the fluid is derived as
(1)

where h is the enthalpy, p is the density, x is the vapor quality, subscript l is a liquid phase, and g is a gas phase. Given the total inlet pressure p0 and enthalpy h upstream of the nozzle, x, hl, hg, ρl, and ρg change with the change in pressure p at the nozzle exit, and the mass flux G is obtained using Eq. (1). Upon reaching its maximum value, it attains the status of critical mass flux (G*) corresponding to the inflow condition, with the pressure at this juncture designated as the critical pressure (p*).

The above is expressed by the following equation:
(2)

The total inlet pressure at the nozzle outlet is determined by calculating the enthalpy based on the pressure and temperature conditions upstream of the nozzle, assuming zero heat dissipation within the piping and that the process due to pressure loss in the piping is isenthalpic. In subsequent discussions, references to the mass flux calculated using the HEM pertain specifically to the critical mass flux.

The equation to evaluate the ratio of the asymptotic plane area (Aa) to the break plane area (Ae) utilizing the mass flux is as follows:
(3)
(4)
(5)
(6)
where ρa is the density at the asymptotic plane, CT is the thrust coefficient, p0 is the stagnation pressure, x is the quality, h0 is the stagnation enthalpy at the broken plane, hl is the saturated liquid enthalpy at the break plane stagnation pressure, hl,amb is the saturated liquid enthalpy at the ambient pressure (i.e., 419 kJ/kg), ρg is the saturated vapor density at the asymptotic plane pressure, and ρl is the saturated liquid density at the asymptotic plane pressure. The asymptotic plane pressure pa is evaluated by the following equation:
(7)
(8)
(9)

where xeq is the thermal equilibrium quality, and pamb is the ambient pressure. hl and hg are the saturated liquid and vapor enthalpy at the break plane stagnation pressure, respectively.

The width of the asymptotic plane Ha is defined by
(10)
The distance from break plane to the asymptotic plane La is defined by
(11)

To verify the accuracy of the indices defined for assessing the jet's affected region according to the ANSI standard, the subsequent sections will compare the evaluation of the jet region as determined by both experimental and computational methods against the criteria set forth by the ANSI standard.

4 Results and Discussion

4.1 Mass Flux of Flashing Jet.

The theoretical mass flux calculated by the HEM as described in the ANSI standard was compared with mass flux measurements obtained by experimentally and through CFD, with inlet pressure and subcooling degree serving as variables. Experimentally, the mass flux was determined using a vortex flowmeter positioned upstream of the nozzle. In CFD simulations, mass flux was calculated as the product of density and flow velocity at the nozzle tip.

The comparisons of the mass fluxes obtained experimentally and through CFD, as shown in Fig. 4, with those calculated using the HEM, indicate that the HEM-derived mass flux for the ϕ16.1 and ϕ8 conditions are consistent with the experimental and CFD results, within a deviation margin of ±30%. The mass flux for saturated water conditions at ϕ9.2 predicts experimental result within an error margin of 60%. Notably, the mass flux under subcooling conditions exceeds that under saturation conditions.

Fig. 4
Comparison of mass flux obtained by experiment and simulation, and that obtained by HEM
Fig. 4
Comparison of mass flux obtained by experiment and simulation, and that obtained by HEM
Close modal

To further dissect the variations in mass flux, which is the product of density and velocity, Table 3 elucidates the relationship between density, velocity, and mass flux at the nozzle outlet across various subcooling conditions as determined by CFD. The densities and velocities presented in the table represent the average values in the two-phase state, computed based on the assumptions of homogeneous flow. The data from Table 3 reveal an increase in both density and mass flux with an elevation in subcooling degree. Conversely, the flow velocity reaches its apex under saturated conditions and stabilizes around 10 m/s when the subcooling degree surpasses 10 K. Under saturated conditions, depressurization boiling initiates due to pressure loss within the nozzle, leading to a high void fraction at the nozzle outlet, which in turn results in lower density and higher flow velocity. However, with a subcooling degree of 10 K or more, depressurization boiling does not commence within the nozzle, and the water maintains a single-phase state at the nozzle outlet, culminating in higher density and lower flow velocity.

Table 3

Comparison between density, flow velocity, and mass flux obtained from the simulation results

Casep (MPa)Tsub (K)Fluidρe (kg/m3)ue (m/s)G (kg/m2s)
10.52.0Saturated water76.634.82662.8
21.01.0Saturated water92.852.04761.7
31.010Subcooled water87711.710,110
41.020Subcooled water89914.713,058
51.030Subcooled water91516.214,687
Casep (MPa)Tsub (K)Fluidρe (kg/m3)ue (m/s)G (kg/m2s)
10.52.0Saturated water76.634.82662.8
21.01.0Saturated water92.852.04761.7
31.010Subcooled water87711.710,110
41.020Subcooled water89914.713,058
51.030Subcooled water91516.214,687

Therefore, an increase in the degree of subcooling serves to inhibit depressurization boiling within the tube, which reduces the two-phase flow pressure drop and consequently augments the mass flux.

4.2 Visualization Results of the Jet Shape and Evaluation Results of the Affected Region as per the ANSI Standard.

Figure 5 outlines the methodology employed to assess the affected region of the jet from visualization images: (1, 2) The visualization images were binarized to render the jet stream in white. (3) Following the stabilization of the jet shape, 1500 images over 1.5 s were averaged over time. (4) The time-averaged images depict the probability of the jet's presence, with areas exhibiting a probability of 50% or greater defined as the jet shapes. The resulting image illustrates that the jet width, initially expanding abruptly, transitions to linear expansion beyond a certain point. The second-order derivative in the axial direction (x direction) was calculated from the defined jet shape, and the asymptotic plane width (Ha) and the asymptotic plane distance (La) were determined by identifying the inflection point of the jet shape (i.e., d2ydx2=0) as the asymptotic plane. The 50% criterion used to identify the boundary of the jet shape in this study is the midpoint between 100 for the jet and 0 for the background image, and the 50% criterion is used for all experimental results. The jet shape was stable during a recording time of 1.5 s, and the effect of the recording time on the jet shape was sufficiently small.

Fig. 5
Evaluation method of the affected region in the visualization results of flashing jet
Fig. 5
Evaluation method of the affected region in the visualization results of flashing jet
Close modal

Figures 6 and 7 display the results of the visualization experiment of the flashing jet under saturated and subcooled water conditions and the visualization results obtained via CFD, respectively. Based on the shape of the jet derived from both the experimental and CFD analysis, the asymptotic region was identified as the area where the jet width expands rapidly, and the free jet region as the area where the jet width expands linearly. The boundary of these regions was determined as the asymptotic plane.

Fig. 6
Visualization results obtained by experiment under saturated and subcooled water conditions: (a) 1.0 MPa, Tsub = 0 K, (b) 1.9 MPa, Tsub = 0 K, (c) 1.9 MPa, Tsub = 10 K, and (d) 1.9 MPa, Tsub = 20 K
Fig. 6
Visualization results obtained by experiment under saturated and subcooled water conditions: (a) 1.0 MPa, Tsub = 0 K, (b) 1.9 MPa, Tsub = 0 K, (c) 1.9 MPa, Tsub = 10 K, and (d) 1.9 MPa, Tsub = 20 K
Close modal
Fig. 7
CFD results under the saturated water conditions at 1 MPa
Fig. 7
CFD results under the saturated water conditions at 1 MPa
Close modal

The comparison of the influence of inlet pressure under saturation conditions, as shown in Figs. 6(a) and 6(b), indicates minimal impact on jet shape, such as jet width and spread angle, under the pressure of 1.0 MPa and 1.9 MPa. Conversely, when comparing differences in subcooling degree, Figs. 6(b)6(d) illustrate that both the spread angle and jet width increase as the subcooling degree increases.

Figure 7 presents of the CFD analysis results of the flashing jet at 1 MPa, showing Mach number distributions and the volume fractions of liquid and vapor. These results indicate that the fluid, initially in a liquid state, rapidly changes phase to a vapor state at the nozzle exit and is subsequently ejected as a jet. Upon ejection, the fluid achieves supersonic speeds, with a Mach number exceeding one, forms a shock wave, then transitions to subsonic speeds and disperses. The temperature distribution suggests a broader high-temperature region compared with the vapor volume fraction distribution. Furthermore, the temperature in the asymptotic region is lower than that in the free jet region because the fluid is accelerated to supersonic speeds in the asymptotic region. However, considering the jet impact on the human body in the asymptotic region, a detached shock wave is generated between the nozzle and human body because of jet impingement, and the temperature rises on the surface of the human body. Therefore, the burn risk in the asymptotic region is high.

Figure 8 compares the asymptotic plane width and distance obtained from the experimental and CFD analysis results with those predicted by the ANSI standard. The ANSI standard underestimates these dimensions by at least 50% compared to experimental and CFD results. This discrepancy is attributed to the ANSI standard's method of determining the asymptotic plane based on the load from the jet stream, whereas this study considers the area where the jet stream appears as white fog, considering the impacts on work safety from burns and poor visibility. Consequently, this study suggests a wider affected region than the ANSI standard, implying that the actual damage from a piping failure accident could be greater than anticipated. To ensure safety during such accidents, more accurate evaluation equations than those provided by the ANSI standard are necessary.

Fig. 8
Evaluation results of asymptotic plane width and distance of flashing jets as per the ANSI standard: (a) asymptotic plane width and (b) asymptotic plane distance
Fig. 8
Evaluation results of asymptotic plane width and distance of flashing jets as per the ANSI standard: (a) asymptotic plane width and (b) asymptotic plane distance
Close modal
Figure 9 shows the results for the expansion angle θa of the jet, derived from the geometric relationship between the asymptotic plane width and distance
(12)
Fig. 9
Expanding angle derived from the experiment and simulation results
Fig. 9
Expanding angle derived from the experiment and simulation results
Close modal

Upon transforming the equation, Eq. (11) used for determining the asymptotic plane distance in the ANSI standard is found to be identical to Eq. (12), provided that θa = 45 deg (i.e., tan 45 deg = 1). This equivalence suggests that the ANSI standard presupposes a constant expansion angle θa = 45 deg, irrespective of the variables such as pressure, temperature, or fluid type. However, Fig. 9 demonstrates that while θa approximates 45 deg under subcooled water conditions despite numerous variations, under saturated water conditions, θa ranges between 10 deg and 30 deg. This range significantly deviates from the ANSI standard's assumption of a constant 45 deg.

This discrepancy highlights the necessity of developing separate evaluation equations for both the asymptotic plane width and the asymptotic plane distance rather than deriving the asymptotic plane distance from the asymptotic plane width as indicated in Eq. (11). Such an approach is validated by the experimental and CFD analysis results, which reveal variation in the expansion angle based on the fluid's state and conditions, thereby challenging the ANSI standard's static assumption.

4.3 Jet-Affected Region of Flashing Jet.

From the predictions of asymptotic plane width and asymptotic plane distance according to the ANSI standard depicted in Fig. 8, the predictions generally underestimate the actual measurements. However, the predicted results are consistent and successfully capture the trend within the affected region. Consequently, the variables employed in the ANSI standard prediction equation have been identified, and the coefficients have been adjusted to formulate more precise prediction equations for the affected region.

The results of the asymptotic plane width and distance obtained from the experiments and simulations shown in Fig. 8 indicate that the affected region is wider in the subcooled condition and the mass flux is higher in the subcooled water condition shown in Fig. 4 than in the saturated water condition. Based on these results, we attempted to construct a correlation to predict the asymptotic plane width and distance in terms of the mass flux, assuming that the mass flux affects the asymptotic plane width and distance.

Table 4 presents the results of the relationship between mass flux, enthalpy, and the asymptotic region of each CFD analysis condition. Upon comparing cases 2, 3, and 4, which exhibit closely related values of enthalpy, both the asymptotic plane width and distance augment with an increase in mass flux. However, although the mass flux was lower in case 4 than in case 6, the asymptotic plane width and distance were larger in case 4. This difference is because the same mass flux with a high enthalpy approaches the saturated water condition, forming bubbles in the nozzle under depressurized boiling, and reduces the effect of the rapid expansion of the asymptotic plane region because of flashing jets at the nozzle exit. Therefore, the prediction equation could not capture this phenomenon because of the single variable of mass flux.

Table 4

Relationship between mass flux, enthalpy, and affected region

Casep (MPa)Tsub (K)FluidG (kg/m2s)h (kJ/kg)Ha/Deθa (deg)La/De
21.01.0Saturated water4518.6758.37.3012.414.3
31.010Subcooled water10637718.824.943.312.7
41.020Subcooled water13897675.339.049.716.1
65.01.0Saturated water16185114911.029.49.86
Casep (MPa)Tsub (K)FluidG (kg/m2s)h (kJ/kg)Ha/Deθa (deg)La/De
21.01.0Saturated water4518.6758.37.3012.414.3
31.010Subcooled water10637718.824.943.312.7
41.020Subcooled water13897675.339.049.716.1
65.01.0Saturated water16185114911.029.49.86
An examination of the variables included in the prediction equation for the affected region as per the ANSI standard shows that the contributions of mass flux and enthalpy are significant, as indicated below:
(13)

Therefore, we considered constructing a correlation equation using two variables with mass flux and enthalpy, which are also included in the ANSI standard formula, as a variable representing the subcooling degree of the fluid. Comparing the results of cases 4 and 6 in Table 4 focusing on enthalpy, case 4 has lower enthalpy and larger asymptotic plane width and distance than case 6, and the asymptotic plane width and distance tend to decrease with an increase in enthalpy.

These observations demonstrate that the affected region of the jet is influenced by variations in both mass flux and enthalpy. Consequently, to construct prediction correlations for the affected region incorporating mass flux and enthalpy as variables, the following equation form is employed. The predictive accuracy of these correlations is subsequently assessed by contrasting the experimental and CFD analysis results with the results obtained by the prediction correlation
(14)
(15)
Figure 10 presents a comparison of the asymptotic plane width and the asymptotic plane distance as determined by the improved model, alongside the results from experimental and CFD analyses. In refining the model, the coefficients within the evaluation equation were meticulously adjusted to optimize the coefficient of determination derived from regression analysis. Consequently, the affected region was assessed using the equation specified below:
(16)
(17)
Fig. 10
Evaluation results of asymptotic plane width and distance of flashing jets by the improved model: (a) asymptotic plane width and (b) asymptotic plane distance
Fig. 10
Evaluation results of asymptotic plane width and distance of flashing jets by the improved model: (a) asymptotic plane width and (b) asymptotic plane distance
Close modal

The results pertaining to the asymptotic plane width demonstrate that the improved model aligns closely with both experimental and CFD analysis results, with a root mean squared error of 9.6. Similarly, the improved model effectively replicates the experimental and CFD results for the asymptotic surface distance, with an root mean squared error of 6.2. In contrast, the ANSI standard, as depicted in Fig. 8, underestimates the affected region. However, the improved model forecasts the affected region derived from the experimental and CFD analysis results with commendable accuracy.

The differences between the existing standard and improved model are as follows: The existing standard is a method for determining asymptotic plane width and distance based on the load caused by the jet impact and is suitable for predicting areas where the object is subjected to load. However, if the areas where the high-temperature jet impinges on the equipment or human body are determined from the areas where the load is applied, the error in the prediction equation of the existing standard becomes considerable, as shown in Fig. 8. However, in the improved model, the shape of the high-temperature jet was visualized using a high-speed camera, and the asymptotic plane width and distance were calculated based on the jet shape to construct prediction correlations, as shown in Fig. 10. Therefore, the improved model is suitable for evaluating the areas where the high-temperature jet impinges on the human body and equipment, causing burns and damage.

Within the predictive equation for the affected region, enthalpy (one of the explanatory variables) can be deduced from the pressure and temperature conditions within the piping system by referring to the steam table. The other explanatory variable, mass flux, is calculable via the HEM utilizing enthalpy to ascertain the choke mass flux. Consequently, as illustrated in Fig. 10, the prediction equation can be employed to accurately forecast the affected region of the flashing jet resulting from pipe failure, from conditions of subcooled water to saturated water, based on pressure and temperature conditions.

5 Conclusion

Visualization experiments and CFD analyses were conducted to simulate the flashing jet resulting from a pipe failure, with the objective of comparing the affected region as evaluated by the ANSI standard against the outcomes of experimental and CFD analysis. The results indicate that the mass flux predictions, as per the HEM proposed in the ANSI standard, align closely with both experimental and CFD analysis findings. However, evaluations of both the asymptotic plane width and the asymptotic plane distance—key measures of the affected regions of the jet—were found to be underestimated in the ANSI standard results when compared with those obtained from experimental and CFD analyses. In response to this discrepancy, an improved model was developed that incorporates mass flux and enthalpy as variables. This model demonstrated a superior ability to predict the asymptotic plane width and distance more accurately than the ANSI standard. Utilizing this improved model enables the prediction of the affected region of the flashing jet, encompassing a range from saturated to subcooled water conditions across various pipe diameters.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

A =

area, m2

D =

diameter, m

G =

mass flux, kg/(m2·s)

h =

enthalpy, J/kg

H =

width, m

L =

length, m

p =

pressure, Pa

T =

temperature, K

u =

flow velocity, m/s

x =

vapor quality

Greek Symbols
ρ =

density, kg/m3

θ =

angle, deg

Subscripts
a =

asymptotic plane

amb =

ambient

e =

nozzle exit

g =

gas phase

l =

liquid phase

sat =

saturation

sub =

subcooling

0 =

stagnation

References

1.
Japan Society of Mechanical Engineering
,
2003
, “
Rules on Protection Design Against Postulated Pipe Rupture for Nuclear Power Plants
,” JSME S ND1-2002.
2.
American National Standards Institute
,
1988
, “
Design Basis For Protection Of Light Water Nuclear Power Plants Against The Effects of Postulated Pipe Rupture
,” ANSI, Washington, DC, Report No. ANSI/ANS-58.2-1988.
3.
U.S. Nuclear Regulatory Commission
,
1995
, “
Parametric Study of the Potential for BWR ECCS Strainer Blockage Due to LOCA Generated Debris
,” USNRC, Rockville, MD, Report No. SEANo.93-554-06-A:1, NUREG/CR-6224.
4.
Moody
,
F. J.
,
1969
, “
Prediction of Blowdown Thrust and Jet Forces
,”
ASME
Paper No. 69-HT-31.10.1115/69-HT-31
5.
Electric Power Research Institute
,
1986
, “
Two-Phase Jet Modeling and Data Comparison
,” EPRI, Washington, DC, Report No. Epri NP-4362.
6.
U.S. Nuclear Regulatory Commission
,
2015
, “
Determination of Rupture Locations and Dynamic Effects Associated With the Postulated Rupture of Piping
,” Standard Review Plan, Section 3.6.2, NUREG-0800.
7.
Kong
,
R.
,
Kim
,
S.
, and
Ishii
,
M.
,
2020
, “
Review of Jet Impingement in High-Energy Piping Systems
,”
Nucl. Eng. Des.
,
357
, p.
110411
.10.1016/j.nucengdes.2019.110411
8.
Yano
,
T.
,
Isozaki
,
T.
,
Ueda
,
S.
,
Miyazaki
,
N.
,
Kurihara
,
R.
,
Kato
,
R.
, and
Miyazono
,
S.
,
1984
, “
An Experimental Study of Blowdown Thrust and Jet Forces for a Pipe Under Boiling Water Reactor Loss-of-Coolant Accident Conditions
,”
Nucl. Sci. Eng.
,
88
(
3
), pp.
386
395
.10.13182/NSE84-A18592
9.
Kawanishi
,
K.
,
Isono
,
M.
,
Masuda
,
F.
, and
Nakatogawa
,
T.
,
1986
, “
Experimental Study on Jets Formed Under Discharges of High-Pressure Subcooled Water and Steam-Water Mixtures From Short Nozzles
,”
Nucl. Eng. Des.
,
95
, pp.
243
251
.10.1016/0029-5493(86)90051-8
10.
Henry
,
R. E.
, and
Fauske
,
H. K.
,
1971
, “
The Two-Phase Critical Flow of One-Component Mixtures in Nozzles, Orifices, and Short Tubes
,”
ASME J. Heat Mass Transfer-Trans. ASME
,
93
(
2
), pp.
179
187
.10.1115/1.3449782
11.
Morita
,
R.
,
Uchiyama
,
Y.
,
Watanabe
,
S.
,
Takahashi
,
S.
,
Xu
,
Q.
, and
Takamura
,
N.
,
2016
, “
Evaluation of Jet Impact Region and Fluid Force Generated From Ruptured Pipes (1) Numerical and Experimental Evaluation of Affected Region by Steam Jet
,”
ASME
Paper No. ICONE24-60341
.10.1115/ICONE24-60341
12.
Takahashi
,
S.
,
Xu
,
Q.
,
Takamura
,
N.
,
Morita
,
R.
,
Uchiyama
,
Y.
, and
Watanabe
,
S.
,
2016
, “
Evaluation of Jet Impact Region and Fluid Force Generated From Ruptured Pipes (2) Evaluation of Fluid Force Using Computational Fluid Dynamics Analysis
,”
ASME
Paper No. ICONE24-60316
.10.1115/ICONE24-60316
13.
Xu
,
Q.
,
Takahashi
,
S.
,
Takamura
,
N.
,
Morita
,
R.
,
Uchiyama
,
Y.
, and
Watanabe
,
S.
,
2016
, “
Evaluation of Jet Impact Region and Fluid Force Generated From Ruptured Pipes (3) Evaluation of Established Standards
,”
ASME
Paper No. ICONE24-60317
.10.1115/ICONE24-60317
14.
Yuasa
,
T.
,
Morita
,
R.
,
Watanabe
,
S.
, and
Takahashi
,
S.
,
2022
, “
Evaluation of Affected Region by Steam Jet on Surrounding Environment
,”
Trans. JSME
,
88
(
913
), pp.
22
00157
.10.1299/transjsme.22-00157
15.
Morita
,
R.
,
Watanabe
,
S.
,
Takahashi
,
S.
, and
Takamura
,
N.
,
2018
, “
Evaluation of Jet Impact Region and Fluid Force Generated From Ruptured Pipes: Part 4 — Numerical Evaluation of Affected Region by Flashing Jet Flow
,”
ASME
Paper No. ICONE26-82063
.10.1115/ICONE26-82063
16.
Kong
,
R.
,
Kim
,
S.
, and
Ishii
,
M.
,
2021
, “
Jet Impingement in High Energy Piping Systems, Part I: Characteristics and Model Evaluation
,”
Prog. Nucl. Energy
,
142
, p.
104002
.10.1016/j.pnucene.2021.104002
17.
Kong
,
R.
,
Kim
,
S.
, and
Ishii
,
M.
,
2021
, “
Jet Impingement in High-Energy Piping Systems, Part II: Model Improvement and Guidance Development
,”
Prog. Nucl. Energy
,
142
, p.
104001
.10.1016/j.pnucene.2021.104001
18.
U.S. Nuclear Regulatory Commission
,
2021
, “
Jet Impingement in High-Energy Piping Systems
,” USNRC, Rockville, MD, Report No. NUREG/CR-7275.
19.
CRAFT Tech
, 2024, “
CRUNCH CFD
,” Combustion Research and Flow Technology, Inc., Pipersville, PA, accessed Mar. 29, 2024, https://crunch.craft-tech.com/
20.
Yuasa
,
T.
,
Watanabe
,
S.
, and
Morita
,
R.
,
2022
, “
Evaluation of Flashing Jet Impact on Surroundings Due to Leakage of High Pressure Pipes
,”
ASME
Paper No. PVP2022-80253.10.1115/PVP2022-80253
21.
Hall
,
D. G.
, and
Czapary
,
L. S.
,
1980
, “
Tables of Homogeneous Equilibrium Critical Flow Parameters for Water in SI Units
,” EG&G, Idaho Falls, ID, Report No. EGG-2056.