Preamble

Numerical Simulation and Sensitivity Analysis of Parameters for the Nitrate Natural Circulation Loop (NNCL)

Kai Wang¹,², Chaoqun Wang¹,², Chuangxiong Cai¹, Qun Yang¹,², Zhaozhong He¹, Naxiu Wang¹,*

¹ Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
² University of Chinese Academy of Sciences, Beijing 100049, China

Abstract

Background: The nitrate natural circulation loop (NNCL) serves as a crucial experimental platform for studying the natural circulation characteristics of molten salts. Numerical simulation and sensitivity analysis of parameters in the NNCL deepen our understanding of molten salt natural circulation, thereby accumulating experience for the design of passive residual heat removal systems in molten salt reactors.

Objective: This study aims to validate the accuracy of numerical simulations and uncertainty analysis using experimental data, and to investigate the impact of input parameters on natural circulation flow rate through sensitivity analysis.

Methods: The modified RELAP5/MOD4.0 code was employed to establish an analysis model for the nitrate natural circulation loop. New models related to molten salt fluids (nitrate, FLiNaK, etc.), heat transfer correlations, and liquid-fueled reactor models were added to RELAP5/MOD4.0, with key inserted models validated by experiments. The NNCL system model consists of three coupled parts: the primary circuit, molten salt pool, and air cooling system modules. Uncertainty bounds of numerical analysis were calculated using a Monte Carlo-based input uncertainty propagation approach. Fourteen uncertain parameters for NNCL transients were selected based on previous studies, experimental data, and expert judgment, with a probabilistic methodology adopted to define parameter ranges and associated density functions. Wilks' formula was used to quantify the minimum number of code runs, with the upper tolerance limit's percentile and confidence set to the standard 95%/95%. Once the code run number and sets of uncertain input parameters were established, input uncertainty was propagated through the modified RELAP5/MOD4.0 code to obtain upper and lower uncertainty bands. Sensitivity of input parameters was analyzed using Multiple Linear Regression (MLR), which utilizes a straight line to describe relationships between variables, with F-test used to assess whether MLR models comply with statistical laws. Standardized regression coefficients (SRC) were used to rank the importance of input parameters affecting natural circulation.

Results: Numerical analysis results indicate that variations in NNCL cold-end and hot-end temperatures, as well as main loop nitrate flow rates, align with experimental trends. The maximum error between uncertainty analysis bounds and experimental data ranges from -5.7% to 4.2% for DHX inlet temperature, -3.5% to 4.0% for DHX outlet temperature, and -12% to 12% for NNCL primary loop nitrate flow rate. The uncertainty analysis bounds effectively encompass experimental data and experimental errors. The calculated F-value of 2682.0 exceeds 2.32, indicating a significant linear relationship between the nitrate flow rate of the NNCL system and the 14 input parameters at the α=0.01 significance level.

Conclusion: The incorporation of uncertainty analysis with the modified RELAP5/MOD4.0 code provides good prediction of NNCL system operational characteristics. Through sensitivity analysis, the three most sensitive factors affecting natural circulation are identified as: heat transfer coefficient of air side, local resistance coefficient of primary loop, and density of nitrate.

Keywords: Molten salt natural circulation, RELAP5/MOD4.0, Uncertainty analysis, Sensitivity analysis

CLC: TL99

Funding: Youth Innovation Promotion Association (No. E3292901)

First Author: Kai Wang, male, born in 1987, graduated from North China Electric Power University in 2012, currently a Ph.D. student, research focus: nuclear safety analysis, E-mail: wangkai@sinap.ac.cn

Corresponding Author: Naxiu Wang, E-mail: wangnaxiu@sinap.ac.cn

Received: 2024-00-00, Revised: 2024-00-00

Molten salt reactors are one of the six candidate reactor types for Generation IV nuclear reactors [1-3]. In addition to their inherent safety features, they employ passive residual heat removal systems that can effectively mitigate consequences of accidents such as loss of offsite power [4-6]. To investigate the heat removal capacity, stable operation, and transient response characteristics of high-temperature molten salt natural circulation loops, the Shanghai Institute of Applied Physics, Chinese Academy of Sciences, constructed a high-temperature molten salt natural circulation loop using nitrate as the working medium, designated NNCL (Nitrate Natural Circulation Loop). This platform serves as an important facility for validating the applicability of molten salt system analysis programs and thermal-hydraulic calculation methods, providing crucial experience for the design and construction of passive residual heat removal systems in molten salt reactors. Sensitivity analysis examines how different values of independent variables affect specific dependent variables, enabling deeper understanding of how various parameters (such as fluid properties, heat transfer models, loop resistance, etc.) influence natural circulation. This paper establishes a nitrate natural circulation analysis model, employs Monte Carlo-based uncertainty analysis to calculate uncertainty bounds for numerical analysis, compares experimental data with numerical calculations and uncertainty analysis results to validate the correctness of the numerical and uncertainty analysis models, and subsequently performs sensitivity analysis on input parameters to investigate their influence on natural circulation flow rate.

1 Introduction to NNCL

The NNCL uses a ternary nitrate mixture (NaNO₃–NaNO₂–KNO₃) as the fluid medium. Nitrates exhibit high stability, good safety characteristics, and high boiling points, with widespread applications in industrial production. Scaling analysis demonstrates that nitrates can effectively simulate fluoride salt natural circulation characteristics within certain ranges. Therefore, nitrates are used as a substitute for fluoride salts to validate the feasibility of passive residual heat removal systems in molten salt reactors and to address potential issues during standby, startup, and stable operation [7]. The NNCL system consists of a molten salt pool, a DRACS (Direct Reactor Auxiliary Cooling System) heat exchanger (DHX), an air cooling tower, an air-nitrate heat exchanger (Natural-draft Air-Cooled Heat Exchanger, NDHX), an expansion tank, a gas system, a molten salt storage tank, and the air cooling tower itself, as shown in [FIGURE:1]. Main design parameters are listed in [TABLE:1]. The air cooling tower features two air intake methods: side intake (shown by black arrows in [FIGURE:1]) and bottom intake (shown by red arrows in [FIGURE:1]). The tower bottom is equipped with a fan and damper to regulate air flow rate. An electric heater located at the bottom of the molten salt tank heats the salt in the storage tank [8]. Heat is transferred from the electric heater to the molten salt pool, and the DHX heat exchanger, immersed in the pool, transfers heat from the pool nitrate to the nitrate inside the DHX tubes. The nitrate inside the DHX tubes absorbs heat, its temperature rises and density decreases, causing it to rise along the pipeline into the heat rejection section (NDHX). Heat is removed by air in the cooling tower, the nitrate temperature decreases and density increases, and it flows back down the descending pipeline to the DHX tubes to be reheated, thus establishing a circulation loop.

2.1 Uncertainty Analysis Method

This study employs the "input uncertainty propagation method" [9], which is based on Monte Carlo methods. The core of this approach involves determining the uncertainty of input parameters, propagating it through calculations, and finally evaluating the impact of parameters on the output results of interest. This method requires extensive random sampling to simulate input parameter uncertainties. For each uncertain input parameter, its possible value range and probability distribution must be determined, typically including normal distribution, uniform distribution, trapezoidal distribution, etc. [FIGURE:2] illustrates this method schematically.

2.2 System Analysis Program

RELAP5 is a light water reactor transient analysis program developed by Idaho National Engineering Laboratory (INEL) for the U.S. Nuclear Regulatory Commission (NRC) and approved by NRC for engineering safety review [10]. Its component models include pipes, valves, pumps, turbines, branches, and control system components, while also incorporating hydraulic models for abrupt area changes, crossflow, choked flow, etc. It is widely used in accident analysis for various light water reactors. RELAP5/MOD4.0 is the latest version in the RELAP5 series, developed by Innovative System Software (ISS), which retains all RELAP5 models while adding more fluid working substances.

RELAP5/MOD4.0 alone cannot meet all requirements of this study. Therefore, secondary development was performed on the RELAP5/MOD4.0 code. Nitrate fluid properties were added, along with heat transfer correlations for molten salt natural circulation based on relevant experimental results [7] to meet simulation requirements for nitrate natural circulation loops. Additionally, a molten salt uncertainty analysis package was developed based on the uncertainty analysis method and identified uncertain input parameters to satisfy uncertainty analysis requirements for nitrate natural circulation loops.

2.3 Input Uncertainty Parameters

Uncertain parameters were determined based on fundamental equations describing natural circulation, expert judgment, and previous research results [11]. To date, a series of experiments have been conducted on NNCL, including natural circulation heat transfer tests and system resistance tests [8,12-14]. Based on these experimental results and expert opinions, uncertainty parameters and their ranges were determined. Selected input uncertainty parameters are listed in [TABLE:2]. This study uses Wilks' formula [15] to calculate sample size, which depends on confidence level β and confidence interval γ. The mathematical expression of Wilks' formula is given in equation (1):

$$= 1 − = − +1$$

2.4 Sensitivity Analysis Method

Sensitivity analysis is an important method for evaluating how input variables affect model outputs. It quantitatively examines the degree of impact when relevant factors undergo certain changes on one or a group of key indicators. Sensitivity analysis includes two types: local sensitivity analysis, which focuses on the impact of input parameter changes at specific points or small ranges on outputs, and global sensitivity analysis, which considers variations across the entire parameter space to assess the overall impact of input parameters on outputs. This study employs global sensitivity analysis to investigate the influence of all selected parameters on model outputs.

Multiple Linear Regression (MLR) is a commonly used sensitivity analysis method. This study adopts MLR [16] for input parameter sensitivity analysis, with its mathematical expression given in equation (2):

$$y = β_0 + β_1x_1 + β_2x_2 + …… + β_kx_k + ε$$

where β₀ is the regression constant, β are regression coefficients, and ε is random error. An F-test is used to assess model significance, with α set to 0.01. The goodness-of-fit of the linear formula is evaluated using adjusted R², which considers the number of independent variables and sample size, providing a better measure of linear correlation between dependent and predictor variables. Variance Inflation Factor (VIF) is used to assess parameter collinearity; values below 5 indicate no collinearity issues, otherwise selected input parameters must be re-evaluated. Standardized Regression Coefficients (SRC) are used as sensitivity analysis coefficients, with their absolute values indicating parameter importance. The specific steps of sensitivity analysis are shown in [FIGURE:3].

3.2.1 Uncertainty Samples

This study selects γ at 95% and β at 95%, with order m=1. According to Wilks' formula (see equation 1), the corresponding minimum sample size is 59. Selected uncertain input parameters and their probability distributions are listed in [TABLE:2].

3.2.2 Analysis Model

This study employs the modified RELAP5/MOD4.0 program to establish an NNCL system analysis model. The system nodalization is shown in [FIGURE:6], comprising three systems: (1) Molten salt pool, represented by control volumes 1XX. Control volume 100 is the bottom of the molten salt pool, connected to the electric heater. Control volume 106 is the upper plenum of the pool. Control volumes 114 and 104 represent assumed natural circulation riser and downcomer channels in the molten salt pool; while no actual riser or downcomer exists in the pool, the model uses "multiple junctions" to laterally connect control volumes 114 and 104 to simulate lateral flow. (2) Primary loop, represented by control volumes 2XX. Control volumes 202 and 224 represent DHX heat exchanger tubes and NDHX heat exchanger tubes, respectively. Control volume 252 and time-dependent control volume 254 represent the expansion tank for primary loop pressure regulation. Heat transfer between control volume 202 and 104 is modeled via heat structures. (3) Air cooling tower, represented by control volumes 3XX. Control volume 306 is the air side of the NDHX heat exchanger, which exchanges heat with primary loop control volume 224 via heat structures. Time-dependent control volume 352 and time-dependent control volume...

3.2.3 Analysis Results

[FIGURE:7] through [FIGURE:9] compare numerical simulation results with experimental data. The simulation curves show that after opening the air cooling tower damper, DHX inlet temperature first drops sharply then gradually increases. The DHX outlet temperature rise rate continuously decreases as the damper opens. NNCL primary loop nitrate flow rate increases with damper opening, with simulated temperature and flow trends consistent with experimental trends.

[FIGURE:7] through [FIGURE:9] also present uncertainty analysis results, where "Lower limit case" represents the lower uncertainty bound and "Upper limit case" represents the upper bound. Relative to experimental data, the maximum percentage difference between uncertainty bounds is -5.7% to 4.2% for DHX inlet temperature, -3.5% to 4.0% for DHX outlet temperature, and -12% to 12% for NNCL primary loop nitrate flow rate. The uncertainty analysis bounds effectively encompass experimental data and experimental errors, demonstrating that the modified RELAP5/MOD4.0 program can satisfactorily predict NNCL system operational characteristics.

3.3 Sensitivity Analysis

Natural circulation flow rate is a critical parameter that directly affects the heat transfer capability of natural circulation. This study analyzes and investigates the sensitivity of NNCL system uncertainty input parameters to primary loop flow rate. The comparison between simulation and experimental results in Section 3.2 demonstrates that the numerical analysis model based on modified RELAP5/MOD4.0 is suitable for NNCL system characteristic analysis. Building upon this, this study uses multiple linear regression to analyze parameter sensitivity, as detailed in Section 3.3. With heater power at 40 kW and uncertain input parameters selected from [TABLE:2], 93 samples were generated, with calculation results shown in [FIGURE:10].

An F-test assesses model significance at α=0.01, where F₀.₀₁(14,78)=2.32. The calculated F-value is 2682.0, exceeding 2.32, indicating a significant linear relationship between NNCL system nitrate flow rate and the 14 input parameters at significance level α=0.01. Statistical analysis yields adjusted R²=0.998 (>0.7), confirming the multiple linear regression model is statistically meaningful.

VIF calculation results are shown in [FIGURE:11], all below 5, indicating no collinearity among input parameters, thus SRC can be used for parameter importance identification. The absolute SRC values characterize parameter sensitivity strength; [FIGURE:12] shows absolute SRC values for the 14 parameters. The three most important parameters affecting NNCL primary loop flow rate are, in order: air side convective heat transfer coefficient, primary loop local resistance coefficient, and nitrate density.

Natural circulation relies on the driving pressure head generated by fluid density differences between hot and cold legs. The air side convective heat transfer coefficient significantly influences NDHX heat rejection, directly affecting cold leg temperature, which in turn influences the density difference between hot and cold ends, ultimately impacting natural circulation driving force and thus flow rate. The loop resistance coefficient is also a crucial parameter affecting natural circulation, as flow rate is fundamentally determined by the balance between driving force and resistance. Larger resistance leads to smaller flow rate, and vice versa. The resistance coefficient directly reflects loop resistance magnitude. Density itself also affects loop resistance and temperature change rates during heat absorption and rejection, making nitrate density another important parameter influencing natural circulation flow rate.

This study performed numerical simulation and uncertainty analysis for the high-temperature molten salt natural circulation loop (NNCL) and investigated the sensitivity of input parameters to natural circulation flow rate. Numerical results show that NNCL hot-end and cold-end temperatures and primary loop nitrate flow rate variations are consistent with experimental trends. Uncertainty analysis bounds effectively encompass experimental data and errors, demonstrating that the modified RELAP5/MOD4.0 code combined with uncertainty analysis can satisfactorily predict NNCL system operational characteristics. Sensitivity analysis using multiple linear regression reveals that the three parameters with greatest impact on natural circulation flow rate are air side convective heat transfer coefficient, primary loop resistance coefficient, and nitrate density. To reduce design and analysis errors in high-temperature molten salt natural circulation systems, subsequent research should focus on these parameters.

Author Contributions

Kai Wang: Conducted research, collected and analyzed/interpreted data, drafted manuscript, acquired funding. Chaoqun Wang: Critically reviewed intellectual content. Chuangxiong Cai: Critically reviewed intellectual content. Qun Yang: Critically reviewed intellectual content. Zhaozhong He: Provided technical support, critically reviewed intellectual content. Naxiu Wang: Supervised research.

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