Preamble

Large Eddy Simulation of Thermal Stratification in an Experimental Reactor Pressure Vessel and Horizontal Piping System

Abstract: Thermal stratification is one of the primary causes of thermal fatigue failure in pressurized water reactor (PWR) piping systems. This study aims to investigate the transient thermal distribution characteristics of piping structures induced by thermal stratification and to identify sensitive locations for thermal fatigue. Based on the Heiss Dampf Reaktor (HDR) thermal stratification experiments, a full-scale simulation model of an experimental reactor was established, including the reactor pressure vessel (RPV). In this research, the Large Eddy Simulation (LES) numerical method was employed for the flow field calculations to account for the effects of both large-scale and small-scale turbulent motions on the unsteady flow and heat transfer processes. By coupling the fluid flow with solid heat conduction equations, a conjugate heat transfer analysis of the horizontal piping system was completed. The results demonstrate that the LES method is well-suited for studying thermal stratification problems, as the predicted flow field and structural temperature distributions show good agreement with experimental data. The study identifies the location of maximum structural temperature fluctuation intensity during thermal stratification, which can serve as a critical point for thermal fatigue analysis. The numerical simulation methods and analytical results of this study provide a reference for the structural integrity and safety assessment of PWR piping systems and equipment.

1. Introduction

Thermal stratification in nuclear power plant piping occurs when fluids of different temperatures flow simultaneously within a horizontal pipe, leading to a distinct temperature gradient along the vertical direction due to density differences. This phenomenon is particularly prevalent in the surge lines, feed water lines, and emergency core cooling systems of pressurized water reactors (PWRs). The resulting non-uniform temperature distribution and the associated unsteady temperature fluctuations can induce significant thermal stresses, potentially leading to thermal fatigue and structural failure over long-term operation.

To accurately predict the thermal-mechanical behavior of these components, it is essential to capture the complex turbulent mixing and the transient heat transfer between the fluid and the pipe wall. Traditional Reynolds-Averaged Navier-Stokes (RANS) models often struggle to capture the high-frequency oscillations inherent in stratified flows. Therefore, this study utilizes Large Eddy Simulation (LES) to resolve the transient characteristics of the flow field and its impact on the structural temperature distribution.

2. Numerical Methodology

The numerical investigation is based on the full-scale geometry of the Heiss Dampf Reaktor (HDR) experimental facility. The model encompasses the reactor pressure vessel (RPV) and the connected horizontal piping system to ensure that the boundary conditions and flow development accurately reflect experimental conditions.

2.1 Governing Equations and Turbulence Modeling

The Large Eddy Simulation (LES) approach is adopted to solve the filtered Navier-Stokes equations. In LES, large-scale turbulent structures are directly resolved, while the effects of smaller, more universal scales are modeled using a subgrid-scale (SGS) model. This allows for a more precise representation of the unsteady temperature fluctuations at the fluid-solid interface compared to time-averaged methods.

To account for the thermal interaction between the coolant and the pipe material, a conjugate heat transfer (CHT) approach is implemented. The energy equation is solved simultaneously for both the fluid domain and the solid domain, ensuring continuity of temperature and heat flux at the interface:

$$T_{fluid} = T_{solid}$$
$$k_{f} \frac{\partial T_{f}}{\partial n} = k_{s} \frac{\partial T_{s}}{\partial n}$$

where $k_f$ and $k_s$ represent the thermal conductivity of the fluid and solid, respectively.

3. Results and Discussion

The simulation results were validated against experimental data from the HDR facility. The LES model successfully captured the formation of the thermal interface and the characteristic "sloshing" motion of the stratified layers.

[FIGURE:1]

The analysis of the temperature distribution reveals that the most significant temperature gradients occur at the interface between the hot and cold fluid layers. Furthermore, the transient analysis shows that the pipe wall experiences periodic temperature oscillations. By calculating the root-mean-square (RMS) of the temperature fluctuations, the study identified the specific locations where the thermal stripping effect is most pronounced.

[TABLE:1]

The results indicate that the maximum intensity of structural temperature fluctuations is located at the mid-section of the horizontal pipe, specifically near the elevation of the stratified interface. This region is identified as the most sensitive point for thermal fatigue crack initiation.

4. Conclusion

In this study, a conjugate heat transfer analysis using Large Eddy Simulation was performed on a full-scale experimental reactor pressure vessel and its horizontal piping system. The following conclusions were drawn:

1. The LES numerical method effectively captures the unsteady turbulent mixing and thermal stratification phenomena, providing results that are in high agreement with experimental measurements.
2. The coupling of fluid dynamics with solid heat conduction allows for an accurate assessment of the transient thermal load on the pipe wall.
3. The study successfully identified the critical locations for thermal fatigue sensitivity based on the intensity of temperature fluctuations.

These findings and the established numerical framework provide a valuable tool for the safety evaluation and life extension of piping systems in pressurized water reactors.

关键词

Abstract

Pressurized Water Reactor (PWR); Thermal stratification; Large Eddy Simulation (LES); Conjugate Heat Transfer (CHT)

Introduction

Thermal stratification is a complex thermal-hydraulic phenomenon frequently observed in the piping systems of Pressurized Water Reactors (PWRs). It typically occurs when fluids of different temperatures flow at low velocities, leading to a stable density interface where the hotter, lighter fluid floats atop the colder, denser fluid. This phenomenon induces significant circumferential temperature gradients and fluctuations, which can result in thermal fatigue, pipe deformation, and even structural failure over long-term operation. Understanding the precise mechanisms of thermal stratification is critical for the structural integrity and safety assessment of nuclear power plants.

Numerical Methodology

In this study, Large Eddy Simulation (LES) is employed to capture the transient turbulent structures and the evolution of the thermal interface within a pressurized vessel and its connected horizontal piping system. To accurately account for the heat exchange between the fluid and the pipe wall, a Conjugate Heat Transfer (CHT) model is implemented. This approach allows for the simultaneous calculation of the temperature fields in both the fluid domain and the solid pipe wall, ensuring that the thermal boundary conditions at the interface are physically consistent.

Governing Equations

The filtered Navier-Stokes equations for incompressible flow are utilized for the LES. The subgrid-scale (SGS) stresses are modeled using the Wall-Adapting Local Eddy-viscosity (WALE) model, which provides superior performance in wall-bounded flows and naturally recovers the zero-viscosity limit at the wall. The energy equation is solved to determine the temperature distribution, with the fluid-solid interface satisfying the continuity of temperature and heat flux:

$$ T_{f} = T_{s} $$
$$ \lambda_{f} \frac{\partial T_{f}}{\partial n} = \lambda_{s} \frac{\partial T_{s}}{\partial n} $$

where the subscripts $f$ and $s$ denote the fluid and solid phases, respectively, and $\lambda$ represents thermal conductivity.

Results and Discussion

The simulation results provide a detailed visualization of the thermal stratification process within the HDR (Heissdampfreaktor) pressurized vessel and horizontal piping system.

[FIGURE:1]

Thermal Interface Evolution

As shown in [FIGURE:1], a distinct temperature gradient develops along the vertical axis of the horizontal pipe. The LES results successfully capture the "internal wave" oscillations at the interface, which are driven by the competition between buoyancy

1. State Key Laboratory for Strength and Vibration of Mechanical Structures

School of Aerospace

Xi'an Jiaotong University ，

710049 Xi'an

， China ；

2. Key Laboratory of Nuclear Reactor

System Design Technology Nuclear Power Institute of China

610200 Chengdu

China

Abstract

Thermal stratification is identified as one of the main causes of thermal fatigue failure in piping systems of pressurized water reactor . This study aimed to evaluate the transient thermal distribu- tion caused by thermal stratification in the horizontal pipe of reactor structure and determine the thermal fatigue hotspots. Based on the Heiss Dampf Reaktor pipeline thermal stratification experiment full-scale HDR simulation model containing reactor pressure vessel was established. To resolve both large and small-scale turbulent motions the conjugate heat transfer analysis of horizontal pipeline sys-

The study was conducted using the large eddy simulation (LES) numerical method coupled with the heat equation of solids. The results indicate that the LES method is suitable for studying thermal stratification and that the simulation results of thermal distribution are in good agreement with measurements. Fatigue hotspots were also identified by evaluating the intensity of temperature fluctuations. The methodology and the results presented in the paper provide a useful reference for the integrity and safety assessment of PWR piping systems. Thermal stratification occurs when two fluids with a large temperature difference are stagnant or flowing slowly within a piping system. The colder fluid, being denser, occupies the lower space of the pipe, while the hotter fluid, being less dense, occupies the upper space due to buoyancy effects. This phenomenon causes an non-uniform temperature distribution on the pipe wall and creates temperature gradients across the radial section of the pipe. Consequently, this leads to global bending stresses and local thermal stresses across the pipe cross-section, resulting in unexpected large displacements and support loads that threaten the integrity of the piping system. In the field of nuclear reactor engineering, thermal stratification is widely present in piping components such as pressurizer surge lines, feedwater lines, and spray lines.

The Portland Trojan power plant discovered large displacement bending of the pressurizer surge line caused by thermal stratification. Reactor operating experience shows that thermal stratification causes thermal fatigue failure and cracking in piping components. Furthermore, crack propagation can lead to pipe rupture, eventually resulting in a serious loss-of-coolant accident (LOCA) at the nuclear power plant, which compromises the operational safety of the nuclear reactor and its equipment. To address this, the U.S. Nuclear Regulatory Commission (NRC) issued Bulletin No. 88-11 in 1988, emphasizing the structural fatigue issues caused by thermal stratification. It required all nuclear power plants in operation or under construction to perform analysis and demonstration of thermal stratification in pressurizer surge lines to ensure structural integrity.

Since the issuance of this bulletin, the problem of thermal stratification has attracted extensive attention from scholars. Some researchers have proposed simplified methods to predict thermal stratification flows, simplifying the complex three-dimensional stress analysis of pipe thermal stratification into a combination of one-dimensional and two-dimensional problems. Global bending stresses of the piping system are calculated through a one-dimensional finite element model, while local stresses in the pipe cross-section are calculated through a two-dimensional finite element model. Applying these methods, Yu Xiao used the SYSTUS ROCOCO program to analyze the stress and fatigue of thermal stratification in the pressurizer surge line of the Qinshan Phase II nuclear power plant expansion project. The analysis results showed that the stress and fatigue strength of the pipeline under thermal stratification effects met the relevant requirements of the RCC-M code. Some researchers have also used stress analysis methods based on simplified thermal loads, decomposing the thermal loads caused by thermal stratification into average temperature, temperature gradients, and peak temperatures. On this basis, the structural thermal stresses caused by these respective thermal loads are considered. However, because thermal stratification leads to unsteady thermal loads, these simplified methods are insufficient for accurately predicting unsteady flow and heat transfer processes.

Regarding the issue of thermal stratification, relevant scholars have also conducted a large number of experiments. TALJA conducted a total of 10 tests on a German experimental reactor under high-temperature and high-pressure conditions to study the thermal stratification phenomenon in horizontal feedwater pipes. The experiments found that the flow velocity of the cold water determines the magnitude of the temperature gradient between the cold and hot water. The experimental results were compiled into a database used to explore and verify computational codes and analytical methods related to thermal stratification issues.

Through thermal stratification experiments, the interaction between the main coolant pipe and branch pipes caused by turbulence penetration was studied, and a theory for evaluating the turbulence penetration distance by modifying the Reynolds number was proposed.

The temperature differences and displacements of the surge line during the heating and cooling processes of a Korean nuclear plant were measured. Based on this, a finite element model was established to perform pipe heat transfer and stress analysis. REZENDE studied the thermal stratification problem in horizontal pipes connected to steam generators. During the experiments, the Froude numbers were close to those in nuclear reactor operating conditions, and it was proposed that the vertical temperature gradients measured in the fluid and pipe walls could be expressed as a function of the Froude number. In recent years, with the development of computer hardware, software, and computational fluid dynamics (CFD) technology, numerical methods for thermal stratification problems have made great progress, enabling more accurate predictions of thermal stratification issues.

ABOU-RJEILY used the TRIO-EF code to perform two-dimensional numerical simulations of the thermal stratification phenomenon in pressurizer surge lines, considering the main pipe flow to make the surge line configuration very close to the actual operating conditions of a nuclear power plant. The numerical results were in good agreement with experimental data measured by the I'EXPRESS experimental device, indicating that numerical simulation can effectively predict the temperature gradient of the surge line and the conditions for the onset of thermal stratification. This verified the feasibility of the simulation analysis method under both steady-state and transient conditions.

A numerical method based on the finite volume method was applied to predict the behavior of fluid flow and temperature distribution in a pressurizer surge line. The study employed transient simulation methods and considered conjugate heat transfer (CHT) resulting from the interaction between the surge line wall and the fluid. The conjugate heat transfer process includes both the heat transfer between the hot and cold fluids and the heat exchange between the fluid and the solid wall.

Key words: pressurized water reactor; thermal stratification; large eddy simulation; conjugate heat transfer

1150 应用力学学报

Convective heat transfer also encompasses the process of thermal conduction within solid regions adjacent to the fluid. In numerical computations, the method of coupling the transport equations within the fluid and solid domains—balancing the heat flux at the fluid-solid interface through the energy equation—is referred to as conjugate heat transfer (CHT) analysis. Results indicate that to determine the temperature distribution in thick-walled pipes as realistically as possible, it is necessary to incorporate CHT analysis, accounting for wall thickness effects, into the numerical analysis of thermal stratification in piping systems.

BOROS utilized commercial software to simulate thermal stratification issues in the pressurizer surge lines and feedwater pipes of VVER-440 reactors. The simulation results were compared with experimental measurement data from actual nuclear power plants, providing the evolution process of stratified flow and its temperature distribution. This verified that Computational Fluid Dynamics (CFD) methods can provide an effective prediction for thermal stratification. The study also noted that the primary difficulty in simulating thermal stratification is the uncertainty of numerical boundary conditions, caused by the difficulty in experimentally measuring necessary parameters at the flow domain boundaries. Furthermore, the unsteady conjugate heat transfer following internal stratification in an actual pressurized water reactor (PWR) pressurizer surge line was simulated, and numerical results from different turbulence models were compared. The simulation results demonstrate that numerical calculations considering wall thickness are of great significance for studying actual transient heat transfer processes. Regarding the numerical simulation of thermal stratification phenomena, a large body of research has relied solely on the Unsteady Reynolds-Averaged Navier-Stokes (URANS) method. However, this method only reflects the time-averaged characteristics of the flow field and lacks the ability to model turbulence-induced stresses across multiple scales. Consequently, the number of studies using Large Eddy Simulation (LES) turbulence models for thermal stratification remains very limited. This study addresses the thermal stratification problem in the horizontal piping of an experimental reactor pressure vessel by conducting a conjugate heat transfer analysis based on the LES method. The numerical calculation accounts for the influence of the reactor pressure vessel (RPV) on the fluid flow within the pipe, establishing a full-scale horizontal pipe thermal stratification analysis model including the RPV. In this research, the LES turbulence model was used to accurately capture the nonlinear temperature distribution within the horizontal pipe and its internal flow field. The model was validated using experimental results, the physical mechanisms of the thermal stratification phenomenon were analyzed, and the intensity of temperature fluctuations across the pipe cross-section was quantitatively compared to identify sensitive points for thermal fatigue. The research results demonstrate that the LES-based numerical simulation method can correctly capture and analyze thermal stratification in PWR horizontal piping. This provides an important reference for studying structural fatigue induced by thermal stratification in piping systems and ensuring the operational reliability of reactor equipment.

1 计算模型及仿真参数设置

In this project, researchers established a full-scale reactor structure and conducted extensive experimental studies focusing on the phenomenon of thermal stratification in horizontal piping. Reference

实验

An experimental reactor model was established, as shown in [FIGURE:1]. The model primarily consists of the reactor pressure vessel (RPV), piping components, and the internal flow domain. A cross-section of the horizontal pipe was selected as a representative section for subsequent result analysis and comparison. At the initial state, the pressure vessel and piping are filled with stationary high-temperature, high-pressure fluid. The coolant is injected through the lower end of the vertical pipe, flows through the horizontal section, and finally enters the pressure vessel.

The geometric model of the experimental reactor was discretized into high-quality structured meshes for both the reactor and the internal flow field using the professional meshing software ICEM CFD. Since the numerical simulation focuses on the horizontal pipe section where thermal stratification occurs, local mesh refinement was applied to the pipe structure and the internal flow field. Considering the requirements of Large Eddy Simulation (LES), the height of the first grid layer in the boundary layer within the pipe region was controlled at $0.1 \text{ mm}$ to ensure the required $y^+$ values. The boundary layer was configured with 10 layers and a growth factor of 1.2. The final model consists of a total of 5.6 million grid elements, including 1.2 million elements for the pipe structure and 4.4 million elements for the internal flow domain. Details of the global mesh and the specific grid refinement at the elbow and pipe inlet are shown in [FIGURE:2].

Regarding the boundary conditions and solution parameters, the numerical simulation conditions were set with reference to the HDR T33.19 experiment. At the initial stage of the experiment, the pressure vessel and the piping are filled with stationary fluid at a high temperature of $210^\circ\text{C}$, with an operating pressure of...

24 MPa

...and reaches a stable state. At a specific moment, low-temperature coolant is injected from the lower inlet of the vertical pipe at a temperature of $60.5^\circ\text{C}$ and an average flow velocity of $0.106\text{ m/s}$. For the aforementioned operating conditions, the numerical calculation boundary conditions are set as follows: the lower end of the vertical pipe is defined as a velocity inlet boundary condition, where random perturbations are introduced via the vortex method, and the inlet temperature is set to a constant value. The upper outlet of the pressure vessel is defined as a pressure outlet boundary condition with the gauge pressure set to zero, the outlet backflow temperature set to $210^\circ\text{C}$, and the reference pressure established as...

24 MPa

The force coordinate point is located at the center of the upper outlet of the pressure vessel. Both hot and cold fluids coexist within the pipeline. The hot fluid heats the pipeline, while the cold fluid flows through the heated sections and carries away a portion of the heat. Since the internal fluid temperature and its thermal conductivity are significantly higher than those of the ambient gas, the heat exchange between the pipeline and the external environment is negligible. Consequently, adiabatic boundary conditions are applied to the external solid walls. Heat is transferred from the high-temperature fluid to the solid structure and subsequently to the low-temperature fluid. Because coupled heat exchange occurs between the internal fluid and the pipeline, conjugate heat transfer boundary conditions are employed at the fluid-solid interface.

The time step for the numerical analysis is determined by the Courant condition, as shown in Eq. $C = \frac{v \Delta t}{\Delta x} \leq 1$, where $v$ represents the average inlet flow velocity, $\Delta t$ is the simulation time step, and $\Delta x$ is the minimum grid size along the flow direction. The total physical simulation time is 280 s, with a time step of 0.01 s. The numerical calculation utilizes the SIMPLE algorithm, with the PRESTO! scheme selected for pressure interpolation. The momentum and energy equations are discretized using the bounded central differencing method. Large Eddy Simulation (LES) is employed as the turbulence model, utilizing the dynamic Smagorinsky-Lilly subgrid-scale model. The convergence residuals for the momentum and continuity equations are set to $10^{-4}$, while the residual for the energy equation is set to $10^{-6}$. Given the large temperature difference between the hot and cold fluids in the studied conditions, the density of the working medium varies significantly, rendering the Boussinesq approximation invalid. To accurately simulate the strong effects of density variations across different temperatures, a full buoyancy model is adopted. The fluid density, viscosity, specific heat capacity, and thermal conductivity are defined as polynomial functions of temperature.

24 MPa

Under these conditions, the curves representing the physical properties of the fluid as a function of temperature are shown in [FIGURE:1]. The solid material selected for this study is bainitic steel. At a temperature of $200$ $^\circ\text{C}$, the density of this material is $7,850\text{ kg/m}^3$.

510 J

Abstract

Thermal conductivity is a critical physical parameter that characterizes the heat transfer capability of a material. In the field of thermal management and materials science, accurately predicting and calculating the thermal conductivity of complex systems remains a significant challenge. This paper explores the fundamental mechanisms of heat conduction and presents a systematic analysis of factors influencing thermal transport across various scales.

1. Introduction

The study of thermal transport properties is essential for applications ranging from microelectronic cooling to thermoelectric energy conversion. Thermal conductivity, denoted as $\kappa$ or $\lambda$, represents the proportionality constant between the heat flux and the temperature gradient, as defined by Fourier's Law:

$$q = -\kappa \nabla T$$

where $q$ is the heat flux and $\nabla T$ is the temperature gradient. Understanding how atomic structure, defects, and interfacial resistance affect this parameter is crucial for the design of next-generation thermal interface materials.

2. Theoretical Framework

In solid-state physics, thermal conductivity is primarily governed by the transport of phonons and electrons. For non-metallic crystalline solids, the lattice thermal conductivity can be described using the kinetic theory expression:

$$\kappa = \frac{1}{3} C v l$$

where $C$ is the heat capacity per unit volume, $v$ is the average group velocity of the carriers, and $l$ is the mean free path. As dimensions shrink to the nanoscale, boundary scattering becomes dominant, leading to a significant reduction in effective thermal conductivity compared to bulk values.

[FIGURE:1]

2.1 Modeling and Simulation

Recent advancements in computational materials science have enabled the prediction of thermal properties using Molecular Dynamics (MD) and Boltzmann Transport Equation (BTE) solvers. By utilizing Green-Kubo relations or non-equilibrium molecular dynamics (NEMD), researchers can extract the thermal conductivity of complex nanostructures.

[TABLE:1]

3. Results and Discussion

Our analysis indicates that the thermal conductivity of the composite system is highly sensitive to the volume fraction of the fillers and the interfacial thermal conductance. As shown in [FIGURE:2], the percolation threshold plays a vital role in enhancing the overall thermal performance of the material.

Furthermore, we observe that the temperature dependence of $\kappa$ follows a power-law behavior at high temperatures, consistent with Umklapp scattering processes. The integration of machine learning models with high-throughput experimental data provides a promising pathway for the rapid discovery of materials with tailored thermal properties.

4 W

Experimental Setup and Instrumentation

The arrangement of experimental thermocouples across the monitoring points is illustrated in [FIGURE:1]. These monitoring points are categorized into two distinct types: the first category consists of points situated within the near-wall flow field at a normal distance of 10 mm from the inner pipe wall; the second category comprises points located directly on the inner surface of the pipe wall. The circumferential angle $\theta$ denotes the specific orientation of each monitoring point.

Regarding the vertical orientation of the pipe cross-section, the coordinate system is defined as follows: the bottom of the inner pipe wall serves as the zero-reference point. The vertical distance measured from this reference point along the $y$-axis is defined as the height. [FIGURE:2] provides a schematic representation of the monitoring point distribution across the cross-section, alongside the relevant material parameters for light water.

2 数值计算结果及分析

The temperature distribution results at the mid-section of the horizontal pipe at different time intervals are shown in [FIGURE:N], revealing the evolution process of stratified flow within the pipe. At the initial stage, under the influence of gravity, the cold fluid settles at the bottom of the pipe and flows along the lower section, resulting in a radial temperature gradient across the pipe cross-section. By $t = 80$ s, a distinct thermal stratification phenomenon emerges in the horizontal pipe; the cold fluid continues to flow along the bottom, while the hot fluid occupies the upper space of the pipe due to buoyancy effects. Irregular flow contours appear at the interface between the hot and cold liquids. During the onset of this thermal stratification, "thermal striping" occurs, leading to rapid temperature oscillations at the interface.

In the temperature distribution contours of the pipe's symmetry plane at $t = 160$ s and $t = 260$ s, the temperature profiles at the hot-cold fluid interface become smoother. Furthermore, the height of the mixing layer between the hot and cold fluids remains approximately constant, indicating the formation of a relatively stable stratified flow within the pipe characterized by smaller temperature fluctuation amplitudes. During the cold water injection process in the horizontal pipe system, the thermal stratification phenomenon persists from its onset (at approximately 80 s) until the injection ceases. The total physical simulation time is 280 s, and the thermal stratification phenomenon described in this study persists throughout the 70–280 s time interval.

1152 应用力学学报

The flow characteristics of the typical thermal stratification phenomenon in the pipeline at $t = 100$ s are presented. FIGURE:N shows the streamlines at the horizontal pipe outlet; below the outlet, cold fluid flows out into the container and deposits toward the bottom. Conversely, above the pipe outlet, the hot fluid generates a backflow, entering the upper space of the horizontal pipe. FIGURE:N displays the velocity vector map at the interface between the hot and cold fluids, where the length of the arrows represents the magnitude of the velocity vectors. In the space below the dashed line in the pipe, the cold fluid flows toward the outlet at a higher velocity. Due to the significant relative velocity between the cold and hot fluids, a shear layer exists at the interface, causing the high-speed cold fluid to entrain a portion of the hot fluid toward the pipe outlet. [FIGURE:N] reveals that when thermal stratification occurs in the horizontal pipe, the hot fluid in the upper space forms a circulation loop. As time progresses, the thermal stratification eventually reaches a steady state, maintaining an approximately constant stratification height, which is consistent with experimental observations. [FIGURE:N] shows the streamlines at the pipe outlet at $t = 100$ s. To investigate the temperature distribution law across the cross-section of the horizontal pipe, [FIGURE:N] illustrates the variation curve of the average fluid temperature near the wall as a function of the height of the monitoring points. The numerical results are based on the statistical average temperature from $240$ s to $280$ s. Within the height range of $H = 140 \sim 200$ mm, a large temperature gradient exists in the flow field, and the average temperature of the fluid near the wall increases sharply with height, indicating that mixing between the hot and cold fluids occurs in this region. Both ends of the temperature distribution curve are approximately horizontal; the high-temperature fluid is located at the top of the pipe while the low-temperature fluid is at the bottom, and the average temperature of the fluid near the wall remains nearly constant in these two regions.

The numerical results are in good agreement with the experimental data, indicating that the conjugate heat transfer analysis based on the Large Eddy Simulation (LES) method can accurately predict the temperature distribution across the pipe cross-section. This validates the accuracy and reliability of the numerical calculations.

The curve showing the variation of the cross-sectional average temperature with height is presented. [FIGURE:N] displays the temperature time-history curves for different monitoring points in the near-wall region of the cross-section and on the inner wall of the pipe. As seen in [FIGURE:N], the temperature difference across the section initially increases and then tends toward a stable state.

After the flow begins, at approximately $t = 70$ s, the cold fluid reaches the monitoring points and the temperature starts to drop. The temperature at monitoring points near the bottom of the horizontal pipe decreases more rapidly. Eventually, the temperatures at the monitoring points reach a steady state, fluctuating around a stable value. Comparing FIGURE:N and FIGURE:N, it can be observed that the fluctuations in the fluid temperature near the wall cause pulsations in the wall temperature; however, compared to the fluid temperature, the temperature fluctuations on the inner wall of the pipe are significantly attenuated.

The temperature fluctuations are characterized by the root-mean-square (RMS) temperature defined by Eq. (1), the magnitude of which represents the intensity of the temperature fluctuations.

i = 1 （ T - T mean ）

The root mean square (RMS) temperatures for the fluid in the near-wall region and the inner pipe wall were calculated for the interval between 240 and 280 s, as shown in [FIGURE:N]. As the circumferential angle increases, the intensity of the temperature fluctuations first increases and then decreases. At approximately $\theta = 100^\circ$, the temperature fluctuations of both the near-wall fluid and the inner pipe wall reach their maximum values simultaneously. Notably, the temperature fluctuations of the inner pipe wall are significantly smaller than those of the near-wall fluid.

Based on the height of 147.1 mm at the $\theta = 100^\circ$ position, it can be inferred that the location of maximum temperature fluctuation intensity is situated within the mixing zone between the hot and cold fluids, specifically near the cold fluid layer at the bottom of the pipe. When thermal stratification occurs, a shear layer exists between the hot and cold fluids. According to Kelvin-Helmholtz instability theory, this shear layer leads to thermal stripping in the fluid mixing zone, which induces temperature fluctuations. Furthermore, the cold fluid at the bottom possesses a higher flow velocity; the resulting turbulent vortices cause more intense temperature fluctuations in the fluid near the cold fluid region. Consequently, the fluid temperature fluctuations are most severe at the $\theta = 100^\circ$ position of the pipe cross-section. Temperature fluctuations in the pipe wall lead to transient changes in structural thermal stress. When the fluctuation intensity is high, it can cause structural thermal fatigue and pipe rupture, thereby affecting the service life of the pipeline and jeopardizing the normal operation of the reactor. Therefore, the study of temperature fluctuations is of great significance. In nuclear engineering, attention is typically focused on the locations most susceptible to fatigue. Through [FIGURE:N], the position with the most intense temperature fluctuations in a typical cross-section of the pipe after the occurrence of thermal stratification can be identified, which represents the thermal fatigue sensitive point for that section.

By applying the aforementioned analytical method to different cross-sections of the pipeline, the locations of maximum temperature fluctuation for each section can be obtained. This allows for the final determination of the thermal fatigue sensitive points under the specified operating conditions.

$$T_{RMS} = \sqrt{\frac{1}{n} \sum_{i=1}^n (T_i - T_{mean})^2} \tag{1}$$

3 结

In this study, a conjugate heat transfer analysis of the thermal stratification phenomenon in the horizontal piping of an experimental reactor was conducted using the Large Eddy Simulation (LES) method. This approach successfully captured the unsteady temperature distribution characteristics of the piping structure. The conclusions are as follows:

1) The conjugate heat transfer analysis based on the LES method effectively predicts the cross-sectional temperature distribution and accurately captures the fluctuations in the flow field and pipe wall temperatures. This validates the reliability of the LES method for investigating thermal stratification issues in piping systems.

2) Based on the current computational model, it is possible to reasonably predict the locations of maximum temperature fluctuations within the pipe cross-section during thermal stratification. Given that local thermal stress is proportional to temperature oscillations, these findings allow for the identification of thermal fatigue-sensitive points. The calculation results provide a valuable reference for subsequent thermal stress analysis and fatigue life assessment of the piping. The numerical simulation methods and analytical results of this study serve as a useful reference for the thermal fatigue analysis and structural reliability evaluation of nuclear reactor piping systems and components.

References:

[1] ZHANG Yixiong, YANG Yu. Thermal stratification study for pressurizer surge line [J]. Nuclear Power Engineering, 2011, 32(S1): 112-115. (in Chinese)

[2] TANG Peng, LIU Zhiwei, QIAO Hongwei, et al. Reliability analysis for stress intensity of pressurizer surge line subjected to thermal stratification phenomenon [J]. Nuclear Power Engineering, 2014, 35(4): 64-67. (in Chinese)

[3] FAN Shuchun, WANG Jianjun, ZHENG Hongtao, et al. Analysis of thermal stratification effect in surge line of pressurizer in nuclear power plants [J]. Nuclear Power Engineering, 2006, 27(S1): 66-70. (in Chinese)

[4] GUO Chao, WEN Lijing, LIU Yusheng, et al. Numerical simulation of thermal stratification in pressurized water reactor pressurizer surge line under transient condition [J]. Atomic Energy Science and Technology, 2014, 48(10): 1787-1792. (in Chinese)

[5] WANG Xinjun, AI Honglei, ZHANG Yixiong, et al. Application of heat conduction inverse problem in thermal stratification test for pressurizer surge line [J]. Nuclear Power Engineering, 2013, 34(S1): 114-117. (in Chinese)

[6] U.S. NRC. Thermal stresses in piping connected to reactor coolant systems [R]. NRC Bulletin 88-08, 1988.

[7] U.S. NRC. Pressurizer surge line thermal stratification [R]. NRC Bulletin 88-11, 1988.

1154 应用力学学报

References

Ensel, C., Colas, A., & Barthez, M. Stress analysis of a 900 MW pressurizer surge line including stratification effects. Nuclear Engineering and Design, 118(2-3).

Bieniussa, K. W., & Reck, H. Piping specific analysis of stresses due to thermal stratification. Nuclear Engineering and Design, 119(1-2).

Yu Xiaofei, & Zhang Yixiong. Thermal stratification and fatigue stress analysis for pressurizer surge line. Nuclear Power Engineering (in Chinese).

Sauer, G. Simple formulae for the approximate computation of axial stresses in pipes due to thermal stratification. International Journal of Pressure Vessels and Piping, 49.

Kamaya, M. Assessment of thermal fatigue damage caused by local fluid temperature fluctuation part I: Characteristics of constraint and stress caused by thermal striation and stratification. Nuclear Engineering and Design, 238.

Kumar, R., Jadhav, P. A., Gupta, S. K., et al. Evaluation of thermal stratification induced stress in pipe and its impact on fatigue design. Procedia Engineering, 86.

Talja, A., & Hansjosten, E. Results of thermal stratification tests in a horizontal pipe line at the HDR-facility. Nuclear Engineering and Design, 118.

Wolf, L. Thermal stratification tests in horizontal feedwater piping lines. In: 15th US NRC Water Reactor Safety Information Meeting. S. l.: s. n.

WOLF L SCHYGULLA U. Experimental results of HDR-TEMR thermal stratification test in horizontal feedwater lines / / Trans- actions of the 9th International Conference on Structural Mechanics in Reactor Technology. London IASMiRT WOLF L SCHYGULLA U HÄFFNER W et al. Results of thermal mixing tests at the HDR-facility and comparisons with best-estimate and simple codes . Nuclear engineering and design HÄFNER W WOLF L. Derivation of mixing parameters from the HDR-thermal mixing experiments . International journal of pres- sure vessels and piping WOLF L HÄFNER W GEISS M et al. Results of HDR-experi- ments for pipe loads under thermally stratified flow conditions Nuclear engineering and design KIM J H ROIDT R M DEARDORFF A F. Thermal stratification and reactor piping integrity . Nuclear engineering and design YU Y J LEE T H SOHN Y S et al. Thermal stratification of surge

line in PWR nuclear power plant [ R ] . New York ， NY （ United

States ）： American Society of Mechanical Engineers ， 1995. [ 22 ] YU Y J ， PARK S H ， SOHN G H ， et al. Structural evaluation of

thermal stratification for PWR surge line . Nuclear engineering and design REZENDE H C NAVARRO M A. Thermal stratification in nuclear reactor piping system / / Annals of the Assembly for Internation- al Heat Transfer Conference.

S. L. S. N. NAVARRO, M. A. REZENDE, H. C. DOS SANTOS, A. A. C. et al. Numerical and experimental simulation of the thermal stratification in a horizontal pipe. International Journal of Transport Phenomena.

REZENDE, H. C., SANTOS, A. A. C., NAVARRO, M. A. et al. Verification and validation of a thermal stratification experiment CFD simulation. Nuclear Engineering and Design.

ABOU-RJEILY, Y., BAROIS, G. Numerical prediction of stratified pipe flows in PWRs. Nuclear Engineering and Design.

JO, J. C., KIM, Y. I., CHOI, S. K. Numerical analysis of thermal stratification in a circular pipe. Journal of Pressure Vessel Technology.

MA Fuyin, WU Jiuhui, WANG Guangji. Double-sided fluid-structure coupling dynamic thermal analysis method. Chinese Journal of Applied Mechanics.

of applied mechanics ， 2013 （ 6 ）： 894-898 （ in Chinese ） . [ 29 ] BOROS I ， ASZ Ó DI A. Analysis of thermal stratification in the pri-

mary circuit of a VVER-440 reactor with the CFX code . Nucle- ar engineering and design JO J C KANG D G. CFD analysis of thermally stratified flow and conjugate heat transfer in a PWR pressurizer surgeline . Journal of pressure vessel technology KANG D G JHUNG M J CHANG S H. Fluid-structure interaction analysis for pressurizer surge line subjected to thermal stratification . Nuclear engineering and design OREA D VAGHETTO R NGUYEN T et al. Experimental meas- urements of flow mixing in cold leg of a pressurized water reactor . Annals of nuclear energy