Temperature Fluctuation Characteristics of Liquid Lead-Bismuth in Complex Circulation Structures

MENG Shuqi¹,², LI Fengchen³, QIU Bin², LIN Jiming², CAI Dechang², HAN Bin⁴, LI Li¹, HU Yisong²
¹Harbin Engineering University, College of Materials Science and Chemical Engineering, Harbin 150001, China
²China Nuclear Power Technology Research Institute, Shenzhen 518000, China
³Tianjin University, Tianjin 300000, China
⁴Southeast University, Nanjing 210000, China

Abstract

Lead-Bismuth Eutectic (LBE) alloy has emerged as a primary candidate coolant for fourth-generation reactor primary circuits due to its high thermal conductivity, substantial heat capacity, and ability to operate at elevated temperatures under atmospheric pressure. Inadequate mixing of non-isothermal liquid LBE within the reactor upper plenum can induce aperiodic temperature fluctuations that compromise structural material integrity, representing a critical thermal-hydraulic phenomenon for LBE-cooled reactors. Based on publicly available experimental data from liquid sodium parallel triple-jet tests, this study first evaluates the applicability of various turbulence models for simulating temperature fluctuations and compares the temperature fluctuation intensities between LBE and liquid sodium in a parallel triple-jet configuration. Subsequently, validated Large-Eddy Simulation (LES) methods are employed to investigate temperature fluctuation characteristics of LBE in the complex circulation structures of a reactor upper plenum. The results demonstrate that LBE temperature fluctuations predominantly occur within the 1–10 Hz range, characteristic of low-frequency fluctuations. The most intense temperature fluctuations are observed at approximately three-quarters of the height between the core outlet and the hot leg, while fluctuation intensity significantly attenuates in distal regions of the core outlet where non-isothermal LBE mixing becomes more complete. These findings provide crucial insights for thermal fatigue assessment of structural materials and optimal placement of measurement points in reactor upper plena.

Keywords: liquid lead-bismuth, temperature fluctuation, large eddy simulation, low frequency fluctuation

Introduction

Lead-Bismuth Eutectic (LBE) alloy, with its low melting point, high boiling point, high thermal conductivity, high specific heat capacity, and capability for high-temperature operation at atmospheric pressure, has become a leading candidate coolant for lead-cooled fast reactors. In reactor operations, when coolant exits various core assemblies at different flow rates and temperatures, incomplete mixing of hot and cold fluids creates temperature fluctuations with specific amplitudes and frequencies in the mixing region. These fluctuations propagate to adjacent structural surfaces through heat conduction and convection, causing surface temperature oscillations that induce thermal stress and fatigue issues while also affecting accurate measurement of core outlet temperatures, thereby posing challenges to safe reactor operation.

Previous research on reactor coolant temperature fluctuations has yielded several important findings. Regarding thermal mixing mechanisms, Evrim investigated water mixing processes in T-junction structures of various sizes using data from Stuttgart University's fluid-structure interaction test facility. Kamide et al. experimentally studied cold-hot water mixing in tee piping using thermocouples and visualization techniques, revealing that the momentum ratio of cold-to-hot fluids is a key factor influencing water temperature fluctuation intensity. Aizawa demonstrated through high-temperature, high-pressure water mixing experiments that gravitational effects cannot be neglected in cold-hot fluid mixing. Additional studies by Kuschewski, Miyoshi, Walker, and Li Zewei have also examined thermal mixing characteristics in T-junction structures.

Concerning temperature fluctuation mechanisms and response characteristics, Kimura conducted liquid sodium jet experiments in a parallel triple-jet configuration, finding that near-wall fluid temperature fluctuations were significantly weaker than in the main flow region and that cold-hot fluid velocities had minimal impact on liquid sodium temperature fluctuations. Wakamatsu investigated the influence of upstream elbows in main pipes through water experiments in tee and elbow configurations, showing that time-averaged fluid and solid surface temperatures nearly overlapped with the mean temperature profile of cold-hot jets, while solid surface temperature fluctuation amplitudes were substantially smaller than fluid temperature fluctuations.

In numerical simulation research, Cao et al. used LES methods to analyze liquid sodium temperature fluctuation patterns and distributions in simplified models referencing Chinese Experimental Fast Reactor core outlet coolant velocity and temperature parameters. Lyu Jing established an equivalent model of the core outlet region based on the same thermal-hydraulic parameters and performed CFD analyses, concluding that cold-hot fluid temperature differences had minimal effect on liquid sodium temperature fluctuation frequency but significantly affected amplitude. Wang et al. employed LES methods to simulate temperature fluctuations at lead-cooled fast reactor core outlets in simplified models, obtaining axial distribution characteristics and developing dimensionless analysis methods for amplitude, Power Spectral Density (PSD), and root-mean-square temperature. Cheng et al. compared various eddy viscosity and Reynolds stress models for simulating high-Reynolds-number liquid metal flow and heat transfer characteristics, finding that the realizable k-ε model had lower accuracy requirements for wall functions. Shams and Zhao have also conducted optimization work on Reynolds heat flux-related CFD models for liquid metal turbulent heat transfer.

Overall, existing research has primarily focused on temperature fluctuation characteristics of water and liquid sodium, with limited studies on LBE temperature fluctuation mechanisms and characteristics in complex reactor circulation structures. This paper first analyzes the applicability of different turbulence models for simulating temperature fluctuations based on publicly available liquid sodium parallel triple-jet experimental data, then compares temperature fluctuation intensity differences between LBE and liquid sodium in parallel triple-jet configurations, and finally employs validated LES models to investigate LBE temperature fluctuation characteristics in reactor upper plenum complex circulation structures. The obtained data provide references for thermal fatigue analysis and upper plenum measurement point placement in LBE-cooled reactors.

1. Numerical Simulation Method and Mechanism Experiment

1.1 Numerical Simulation Method

Fluid flow must satisfy three fundamental conservation laws: mass, momentum, and energy conservation, collectively described by the Navier-Stokes (N-S) equations. Since the N-S equations require an additional equation of state for closure, turbulence models are introduced. Currently, widely used numerical simulation methods include Direct Numerical Simulation (DNS), Reynolds-Averaged Navier-Stokes (RANS) simulation, and Large-Eddy Simulation (LES). DNS demands extremely high computational resources and remains impractical for engineering applications.

The governing equations are:
$$
\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_i)}{\partial x_i} = 0
$$
$$
\frac{\partial (\rho u_i)}{\partial t} + \frac{\partial (\rho u_i u_j)}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left(\mu\frac{\partial u_i}{\partial x_j} - \rho\overline{u_i'u_j'}\right) + f_i
$$
$$
\frac{\partial (\rho T)}{\partial t} + \frac{\partial (\rho u_i T)}{\partial x_i} = \frac{\partial}{\partial x_i}\left(\frac{\lambda}{c_p}\frac{\partial T}{\partial x_i}\right) + S
$$

where $u_i$ denotes velocity (m·s⁻¹), $p$ pressure (Pa), $\rho$ density (kg·m⁻³), $f_i$ body force (m²·s⁻¹), $T$ temperature (K), $S$ heat source term (W·m⁻³), and $c_p$ specific heat capacity (J·kg⁻¹·K⁻¹).

RANS employs statistical averaging to partially solve the N-S equations, constructing turbulence models to close all Reynolds stress terms without calculating turbulent fluctuations. This approach requires fewer grid cells and larger time steps, making it feasible for engineering applications. RANS models can be further categorized into realizable k-ε, SST k-ω, and Detached Eddy Simulation (DES) models based on different boundary layer treatments. However, since RANS neglects turbulent fluctuation terms, its capability to simulate transient fluctuation processes may be insufficient.

Therefore, researchers developed the LES model to balance computational resources and solution accuracy. In LES, large-scale eddies are directly resolved while small-scale eddies, which generally exhibit isotropic characteristics, are simplified using time-averaged methods. To address poor solution accuracy for shear flows in near-wall regions, Wall-Modeled LES (WMLES) was developed as an improvement over classical LES. This hybrid approach combines RANS and LES advantages by using RANS in near-wall regions and switching to LES in far-wall regions. WMLES enables calculations of bounded flows at high Reynolds numbers while reducing the stringent grid resolution requirements of classical wall-resolved LES near walls. The governing equations for WMLES are:
$$
\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u_i)}{\partial x_i} = 0
$$
$$
\frac{\partial (\rho u_i)}{\partial t} + \frac{\partial (\rho u_i u_j)}{\partial x_j} = -\frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j}\left[(\mu + \mu_t)\frac{\partial u_i}{\partial x_j}\right]
$$
$$
\frac{\partial (\rho T)}{\partial t} + \frac{\partial (\rho u_i T)}{\partial x_i} = \frac{\partial}{\partial x_i}\left[\left(\frac{\mu}{Pr} + \frac{\mu_t}{Pr_t}\right)\frac{\partial T}{\partial x_i}\right]
$$

where $t$ is time (s), $u$ velocity component (m·s⁻¹), $x$ spatial direction, $\mu$ dynamic viscosity (Pa·s), $Pr$ Prandtl number, and $\mu_t$ turbulent viscosity (Pa·s). The Reynolds number $Re = \frac{\rho u L}{\mu}$ represents the ratio of inertial to viscous forces (dimensionless).

The ability of these methods to simulate LBE temperature fluctuation characteristics requires validation against experimental data. Currently, few experiments address LBE temperature fluctuations directly. Therefore, this study validates CFD models using publicly available liquid sodium temperature fluctuation experimental data and subsequently employs the validated CFD models for liquid LBE temperature fluctuation numerical simulation.

1.2 Mechanism Experiment

Temperature fluctuation phenomena of liquid metal in reactor upper plena arise from different flow velocities and temperatures at each fuel assembly outlet. In pool-type spaces, these fluids undergo unsteady thermal mixing, creating complex thermal-hydraulic phenomena. Due to equipment limitations, direct temperature fluctuation experiments on liquid metal reactor upper plenum complex structures are challenging. Researchers typically abstract this phenomenon as a non-isothermal, non-isokinetic parallel jet thermal mixing problem and conduct scientific studies based on parallel multi-channel configurations. Parallel triple-jet sodium temperature fluctuation experimental data published by Japan in the 1990s has been incorporated into international CFD benchmark databases, enabling quantitative comparison of different turbulence models' applicability for simulating temperature fluctuation phenomena.

[FIGURE:1] shows a schematic diagram of the liquid sodium temperature fluctuation experimental facility. The channel height is 85 mm, with jet nozzles measuring 20 mm × 180 mm. Perforated plates and quadrant reducer nozzles upstream ensure uniform coolant velocity at the inlet cross-section. Nozzles and mixing regions are sandwiched between two vertical stainless steel test plates, with fluid temperature measured using movable non-contact K-type thermocouple bundles. Thermocouple accuracy is ≤0.1°C with a 20 ms time constant. Both fluid and solid temperatures are measured simultaneously at each thermocouple bundle location for 200 s total duration. Due to significant specific heat capacity differences between fluid and solid, different measurement frequencies are used: 0.01 s/sample for fluid temperature and 0.1 s/sample for solid temperature.

1.3 Temperature Fluctuation Intensity Characterization

Dimensionless temperature is defined as:
$$
\theta = \frac{T_i - T_{cold}}{T_{hot} - T_{cold}}
$$
where $T_i$ is instantaneous temperature (K), $T_{cold}$ cold fluid inlet temperature (K), and $T_{hot}$ hot fluid inlet temperature (K).

Dimensionless time-averaged temperature is:
$$
\bar{\theta} = \frac{1}{N}\sum_{i=1}^{N}\theta_i
$$
where $N$ is the total number of instantaneous temperature data points.

Power Spectral Density (PSD) is typically used for temperature fluctuation frequency analysis. This method applies Fast Fourier Transform to temperature fluctuation amplitude curves, converting time-domain temperature data to frequency-domain representations. The discrete Fourier transform for signal $\phi(t)$ with finite sampling points is:
$$
\Phi(f_n) = \sum_{k=0}^{N-1}\phi(t_k)e^{-2\pi ikn/N}
$$
where $n = 0, 1, 2, ..., (N-1)$.

PSD describes signal power distribution in the frequency domain, with units of amplitude squared:
$$
PSD(f) = \frac{2|\Phi(f)|^2}{N \cdot \Delta t}
$$

Amplitude is the square root of PSD:
$$
A(f) = \sqrt{PSD(f)}
$$
where $f$ is fluctuation frequency (Hz).

2. Model Validation and Comparison

2.1 Computational Domain and Boundary Conditions

Based on published liquid sodium temperature fluctuation experimental geometry dimensions, a CFD analysis model was established. As shown in [FIGURE:2], liquid sodium flows from three parallel rectangular channels into a large rectangular mixing space. Each nozzle width is 5 mm with 50 mm spacing between nozzles, and the fluid domain thickness is 180 mm.

Thermophysical property formulas for liquid sodium are:
$$
\rho_s = 1014 - 0.235 \cdot T_s
$$
$$
\lambda_s = 104 - 4.7 \times 10^{-2} \cdot T_s + 1.25 \times 10^{-5} \cdot T_s^2
$$
$$
c_{p,s} = 1658 - 0.8479 \cdot T_s + 4.454 \times 10^{-4} \cdot T_s^2
$$
$$
\eta_s = 4.94 \times 10^{-4} \cdot e^{754.1/T_s}
$$
where $\rho_s$ is liquid sodium density (kg·m⁻³), $\lambda_s$ liquid sodium thermal conductivity (W·(m·K)⁻¹), $c_{p,s}$ liquid sodium constant-pressure specific heat (J·kg⁻¹·K⁻¹), $\eta_s$ liquid sodium dynamic viscosity (Pa·s), and $T_s$ liquid sodium temperature (K).

2.2 Validation Results

Considering the need for meshing complex structures, flexible polyhedral grids were selected. Mesh sensitivity analysis boundary conditions were: liquid sodium medium, inlet velocity 0.5 m·s⁻¹, side hot fluid temperature 763 K, center cold fluid temperature 676 K. [FIGURE:3] shows cross-sectional average temperature results from 2–6 s for different turbulence models with varying mesh cell counts. SST k-ω, DES, and WMLES models require approximately 6 million, 11 million, and 15 million cells, respectively, to achieve mesh independence. These differences primarily stem from varying near-wall region assumptions in different turbulence models, which impose different grid quality requirements.

CFD settings determined through mesh independence analysis are summarized in [TABLE:1], with total cell count approximately 15 million to ensure mesh-independent results for SST k-ω, DES, and WMLES models. Liquid sodium velocity at all three nozzles is 0.51 m·s⁻¹, with cold sodium inlet temperature 577.65 K and hot sodium inlet temperature 620.85 K. Wall boundary conditions are set as no-slip walls; pressure-velocity coupling uses the SIMPLEC (Semi-Implicit Method for Pressure-Linked Equations Consistent) scheme; pressure interpolation and momentum schemes employ second-order discretization; energy scheme uses QUICK (Quadratic Upstream Interpolation for Convective Kinematics) discretization. Turbulence models include SST k-ω, DES, and WMLES.

[FIGURE:4] compares CFD simulation values with experimental liquid sodium temperature fluctuation data from 0–6 s. Normalized mean temperature results show minimal differences among SST k-ω, DES, and WMLES models. However, when converting time-domain temperatures to frequency domain via PSD, SST k-ω results lose significant temperature fluctuation information, while DES and LES models produce dominant frequency data closer to experimental values. Compared to DES, WMLES accepts larger grid aspect ratios near walls, improving CFD model convergence. Additionally, existing temperature fluctuation CFD simulations have predominantly used LES models, with no documented cases employing DES models. Therefore, WMLES is recommended for LBE temperature fluctuation characteristic simulation.

2.3 Temperature Fluctuation Intensity Comparison Between LBE and Sodium

Using identical settings from [TABLE:1], temperature fluctuation intensity differences between liquid LBE and sodium were compared in the parallel triple-jet configuration. Thermophysical property formulas for LBE are:
$$
\rho_L = 11096 - 1.3236 \cdot T_L
$$
$$
\lambda_L = 3.61 + 1.57 \times 10^{-2} \cdot T_L
$$
$$
c_{p,L} = 159 - 2.72 \times 10^{-2} \cdot T_L + 7.12 \times 10^{-6} \cdot T_L^2
$$
$$
\eta_L = 4.94 \times 10^{-4} \cdot e^{754.1/T_L}
$$
where $\rho_L$ is liquid LBE density (kg·m⁻³), $\lambda_L$ liquid LBE thermal conductivity (W·(m·K)⁻¹), $c_{p,L}$ liquid LBE constant-pressure specific heat (J·kg⁻¹·K⁻¹), $\eta_L$ liquid LBE dynamic viscosity (Pa·s), and $T_L$ liquid LBE temperature (K).

[FIGURE:5] shows temperature fluctuations for liquid LBE and sodium at measurement points A and B from 0–6 s. Fluctuation dominant frequencies range from 1–10 Hz, characteristic of low-frequency fluctuations. Under identical boundary conditions, LBE temperature fluctuation intensity is slightly higher than that of sodium.

3. Temperature Fluctuation Characteristics in Complex Circulation Structures

3.1 Computational Domain and Boundary Conditions

Based on reactor upper plenum design drawings from literature, validated WMLES models were used to investigate temperature fluctuation characteristics of liquid LBE in complex circulation structures. The fluid domain is shown in [FIGURE:6]. The referenced reactor contains numerous fuel assemblies; this study appropriately consolidated fuel assembly numbers according to the overall upper plenum geometry.

[FIGURE:6] Diagram of CFD model for simulating temperature fluctuation characteristics of LBE in complex circulation structure

CFD model mesh sizing follows [TABLE:1] with total cell count approximately 50 million. Referencing core outlet thermal-hydraulic parameters from literature, each region uses constant inlet temperature and mass flow rate, with specific values provided in [TABLE:2].

[TABLE:2] Physical parameters of LBE for different inlet channels

3.2 Temperature Fluctuation Intensity at Typical Locations

[FIGURE:7] illustrates the mesh schematic and monitoring point arrangement, covering regions near the core outlet, large pool spaces, and hot leg attachment areas. Considering reactor power and potential temperature measurement device placement coordinates, 54 monitoring points were established. Cross-section numbers on the right side correspond one-to-one with left-side cross-sections, indicating spatial positions at different heights.

To analyze temperature fluctuation characteristics after cold-hot LBE mixing, temperature standard deviation at each measurement point is defined using Eq. (23). Calculating standard deviation of temperature data at each point over time reflects the degree of deviation from mean values, thereby characterizing temperature fluctuation intensity:
$$
\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N}(T_i - T_{avg})^2}
$$
where $N$ is the number of temperature measurements, $T_i$ the instantaneous temperature at measurement $i$ (K), and $T_{avg}$ the average temperature at that point (K).

[FIGURE:8] shows locations and temperature fluctuations for eight LBE measurement points with intense fluctuations ($\sigma \geq 20$ K) from 0–3 s. Results indicate that the most severe temperature fluctuations occur at approximately three-quarters of the height between the core outlet and hot leg, primarily because non-isothermal LBE jets entering large spaces induce strong thermal mixing effects. In distal core outlet regions, temperature fluctuation intensity significantly decreases due to more complete mixing of non-isothermal LBE. Furthermore, dominant frequencies at all measurement points remain below 10 Hz, characteristic of low-frequency fluctuations.

Conclusions

Based on parallel triple-jet experimental data, this study evaluated turbulence model applicability for simulating liquid metal temperature fluctuation characteristics, compared temperature fluctuation differences between liquid LBE and sodium, and conducted numerical investigations of LBE temperature fluctuation characteristics in complex circulation structures. The main conclusions are:

1. LES models are more suitable than RANS models for simulating transient temperature fluctuation processes in liquid metals.
2. Under identical boundary conditions, liquid LBE exhibits slightly higher temperature fluctuation intensity than liquid sodium, with fluctuation frequencies of 1–10 Hz, representing typical low-frequency fluctuations.
3. In complex circulation structures, the most intense LBE temperature fluctuations occur at approximately three-quarters of the height between the core outlet and hot leg, requiring focused attention during reactor thermal fatigue analysis and upper plenum measurement device placement.

Author Contributions

MENG Shuqi established the numerical analysis model, performed data analysis, and wrote the manuscript; LI Fengchen and QIU Bin proposed the research concept and critically reviewed and revised the intellectual content; LIN Jiming and CAI Dechang provided funding support; HAN Bin collected and processed public literature and data and supervised research progress; LI Li and HU Yisong provided guidance on the research methodology.

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