Uncertainty Parameters

The uncertainty of the input parameters stems from imprecise knowledge of the actual values, with sources of uncertainty consisting of reactor system data, structural material properties, and B-E code correlations. Uncertain parameters for the LOFC accident were selected using a phenomenon identification and ranking table (PIRT). The primary approaches for quantifying input uncertainty include both probabilistic and deterministic methods. Probabilistic methodologies utilize statistical elements to characterize and combine input uncertainty, whereas deterministic methodologies use reasonable ranges or bounding intervals of uncertainty and combine input uncertainty based on the maximization and minimization of the output value [24-26]. The probabilistic approach is the most widely adopted procedure and is endorsed by industry and regulators. Limited detailed information about certain aspects of SM-MSR is a significant drawback. To minimize the impact of this drawback, a list of input parameters and their associated density functions was adopted using a probabilistic methodology. Quantification of the uncertainty parameters was established based on previous studies, experimental data, and expert judgment. In this study, 30 uncertainty input parameters were identified, and Table 2 lists the parameters and their probability distribution functions used in this study.

Best-Estimate Code

The RELAP5 code is a transient analysis tool designed for light-water reactors and developed by the U.S. Nuclear Regulatory Commission (NRC) for various applications, such as rulemaking, licensing audit calculations, and evaluation of operator guidelines. It uses a one-dimensional two-fluid thermal-hydraulic model. The latest version, RELAP5/MOD4.0, was developed by Innovative System Software (ISS) specifically for the analysis of nuclear power plants [27].

In RELAP5/MOD4.0, an uncertainty analysis package was incorporated following the methodology developed by the Gesellschaft für Anlagen- und Reaktorsicherheit (GRS)[24]. This methodology integrates order statistics and Wilks' formula [28, 29] into the propagation of the input uncertainty approach. Because the heat transfer coefficient correlations and coolants for the SM-MSR are not available in the current RELAP5/MOD4.0, new correlations [30] and coolants applicable to the MSRs were inserted, and the updated code was named RELAP5-TMSR [31-33]. Given that the uncertainty analysis package can only be employed for the partial analysis of light-water reactors, an uncertainty analysis package for molten salt reactor systems was developed in this study. It can propagate uncertainties associated with molten salt properties. Furthermore, it can propagate uncertainties related to the inserted heat transfer correlations, which are applicable to the fuel channels in the reactor core and heater exchangers of the SM-MSR.

Sensitivity Analysis method

Sensitivity analysis assesses the impact of varying the values of the independent variables on a particular dependent variable within the defined assumptions. In other words, it examines how uncertainties in a mathematical model from various sources contribute to the overall model uncertainty.

Linear regression [34-36] uses a straight line to describe relationships between variables. It identifies the best-fit line in a dataset by searching for the regression coefficient(s) that minimizes the total error of the model. The model's equation presents coefficients that clarify the influence of each independent variable on the dependent variables. There are two primary types of linear regression.

1) Simple Linear Regression, the simplest form of linear regression, involves only one independent and dependent variable.

2) Multiple Linear Regression (MLR) involves more than one independent and dependent variables. In this study, multiple linear regression (MLR) method is adopted. The equation for multiple linear regression is shown in Eq. (1), where y is the dependent variable, xi is the independent variable, β ₀ is a constant, β _i is a coefficient. $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\dots {\beta }_i{x}_i+\dots +{\beta }_n{x}_n$ (1) A systematic sensitivity analysis process based on the MLR is shown in Fig. 3, which was proposed by Manache [37] and has also been applied to the functional reliability analysis of a molten salt natural circulation system [38]. The adjusted coefficient of determination ($R_{\text{adj}}^2$) was used to estimate whether the linear model was acceptable (with $R_{\text{adj}}^2\ge 0.7$ indicating that the model was acceptable). The collinearity problem in multiple linear regression is addressed by calculating the variance inflation factor (VIF) for each parameter, where VIF≤5 indicates weak collinearity. If the linear model is strongly collinear, a significance test of the semipartial correlation coefficient (SPC) is used to rank the input uncertainty parameters; otherwise, the standardized regression coefficient (SRC) is used for the significance test [38].

Fig. 3Full-screen View

Sensitivity analysis steps

Thermal-Hydraulic Model

Figure 4 presents an overview of RELAP5-TMSR nodalization of the SM-MSR. The system consisted of four parts:

Fig. 4Full-screen View

Nodalization of the 150MWt molten salt reactor(MS-MSR)

1) The primary circuit includes the downcomer, reactor core, lower plenum, upper plenum, primary pump, pipes, and IHX tube side.

2) 2nd circuit, including pipes, 2nd circuit pump, IHX shell side, and AHX tube side.

3) Brayton cycle modules, including air inlet volume, AHX shell side, and air outlet volume.

4) The passive residual heat removal system, which consists of DRACS and PRACS, includes pipes, PHX, ADHX and an air-cooling loop.

Uncertainties and Sampling

Wilks' formula is frequently used to quantify the minimum computational work required to meaningfully assess model uncertainty by specifying acceptable tolerance limits on the output parameter [39]. A fundamental advantage of using the Wilks' formula is that it has no limit on the number of uncertainty parameters considered in the analysis. The number of code runs required in the uncertainty analysis depends only on the statistical features of the imposed tolerance limits, including the percentile tolerance interval, confidence interval, and order, and is irrelevant to the number of uncertain parameters [28, 29]. The number of code runs for a one-sided tolerance interval can be calculated using Eq. (2), where γ is the percentile tolerance interval, β is the confidence interval, N is the number of input samples(or number of code runs), and m is the order. $\beta =1-{\displaystyle \sum _{i=N-m+1}^N}\frac{N!}{i!(N-i)!}{\gamma }^i{\left(1-\gamma \right)}^{N-i}$ (2) Table 3 shows the number of code runs based on Wilks' formula, varying with the percentile tolerance and confidence intervals at different orders. In this study, the percentile and confidence of the upper tolerance limit were set to the standard 95%/95% at the 5th order, and the minimum number of code runs was 181 using Wilks' formula.

Figure 5 shows the Cobweb plot of 181 random samples for the 30 parameters, where the x-axis shows the uncertain parameters and the y-axis shows the normalized sample values. According to Fig. 5, the achieved sample population is representative and satisfies the requirements of an LOFC uncertainty study.

Fig. 5Full-screen View

(Color online) Cobweb plot of the 181 random samples for the 30 parameters

Safety Variables and Acceptance criteria

The safety variables and their acceptance criteria are crucial for the safety analysis of the SM-MSR. The primary circuit boundary serves as the principal safety barrier against radioactive leaks. Therefore, the performance of Hastelloy-N, which is used as the structural material for the primary circuit, is crucial for reactor safety.

Temperature is a pivotal indicator of the performance of Hastelloy-N, and some studies have confirmed its ability to maintain mechanical properties at 800 ℃ [40]. Considering the direct contact of the fuel salt with Hastelloy-N structural materials, the reactor outlet fuel temperature (T_out) with a limit of 800 ℃ was selected as the criterion in this study.

Uncertainty propagation result

After the code run numbers and sets of uncertain input parameters were established, the input uncertainty was propagated using the RELAP5-TMSR code. Under normal operating conditions, the core flow was driven by a pump at approximately 1000 kg/s. However, following an LOFC event, the pump stops, reducing core flow, which, in turn, increases T_out. Simultaneously, the reactor protection system sends a shutdown signal, causing the control rods to drop and the power to coast down, which in turn reduces T_out.

Under the influence of changes in core flow and nuclear power, T_out reaches its first peak at approximately 10s, reaches the second peak at approximately 200 s, and the second peak is the maximum point. Then, T_out changes slowly and finally reaches a safe and stable temperature, where decay heat continues to be removed by PRACS and DRACS.

The evolution of T_out for 181 code runs is shown in Figs. 6, and Fig. 7. The figures show the maximum reactor outlet fuel salt temperature (T_{out_max}) for the 181 cases. However, all the results are below the acceptable criterion (800 ℃), and the maximum and minimum T_{out_max} are 725.5 ℃ and 715.4 ℃, respectively. Figure 8 illustrates the upper and lower uncertainty bands, with a maximum difference of 18.5 ℃ between the upper and lower bounds observed during the initial temperature ascent phase. In the base case, the maximum temperature increase of the reactor outlet fuel salt (ΔT_out) was 22.2 ℃ with respect to the initial condition. In the upper limit case, ΔT_out was 25.5 ℃, representing a 26.2% increase relative to the base case, and ΔT_out was 15.4 ℃ for the lower limit case, indicating a 23.7% decrease relative to the base case.

Fig. 6Full-screen View

(Color online) Uncertainty propagation results of 181 cases for the reactor outlet fuel temperature (T_out)

Fig. 7Full-screen View

Final value of T_{out_max} for 181 cases

Fig. 8Full-screen View

(Color online) Uncertainty analysis bound results for the reactor outlet fuel temperature (T_out)

Identification of T_{out_max} Distribution

Figure 9 shows the histogram and probability density function obtained from 181 simulations of T_{out_max}. The data points follow an approximately bell-shaped curve, indicating a normal distribution. In this study, the Shapiro-Wilk (S-W) test [41] was adopted to assess whether the calculated T_{out_max} followed a normal distribution. The S-W test compares the observed dataset to the expected normal distribution to determine whether the dataset is normally distributed. The test statistic of the S–W test for normality is shown in Eq. 3, where xi represents the ordered random sample values, $\overline{x}$ is the mean of the samples, and ai represents the constants that are functions of n. $w=\frac{{\left({\displaystyle {\sum }_{i=1}^n{a}_i{x}_i}\right)}^2}{{\displaystyle {\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}}$ (3) The null hypothesis for the Shapiro-Wilk test is that the variable is normally distributed. If p < 0.05, the null hypothesis was rejected; otherwise, it was accepted. The obtained p-value was 0.197, which exceeded 0.05; therefore, the null hypothesis of normality was acceptable.

Fig. 9Full-screen View

Histogram and probability density function obtained from 181 simulations for T_{out_max}

A quantile-quantile (Q-Q) plot is a graphical technique for determining whether two datasets originate from populations with a common distribution [42]. In a Q-Q plot, if the data are normally distributed, the points are aligned on a straight diagonal line. Conversely, the greater the number of points in the plot that significantly deviate from this line, the less likely the dataset is to follow a normal distribution. Figure 10 shows the Q-Q plot for T_{out_max}, where the points mostly lie along a straight diagonal line, with some minor deviations along the tail.

Fig. 10Full-screen View

　Q-Q plot for T_{out_max}

Based on the analysis above, T_{out_max} follows a normal distribution. Table 4 shows the main statistical results and T_{out_max} values at different percentiles according to the probability density function.

Sensitivity Analysis

The multiple linear regression (MLR) method was adopted by following the steps outlined in Fig. 3 to ascertain the importance of the input parameters. The F-test was used to assess whether the MLR models complied with statistical laws. The acceptance region was set to have an F-value greater than 1.83 at a significance level of 0.01. Table 5 lists the F value and $R_{\text{adj}}^2$. The results demonstrate that the model is convincingly linear.

Figure 11 shows the VIF values, which were all less than five; therefore, the SRCs of the input parameters were selected for the sensitivity analysis. The absolute values of the SRCs provide a relative measure of parameter importance, and Fig. 12 presents the final absolute values of the SRCs of 30 input parameters.

Fig. 11Full-screen View

VIF values of 30 parameters

Fig. 12Full-screen View

Absolute value of SRCs final achieved

The t-test was performed to test the significance of the sensitivity coefficient. The acceptable range was the absolute value of a t-value larger than 1.66 with a significance level of 0.05. Finally, five important parameters that are considered to have a significant contribution to LOFC consequences based on the t-test and SRCs are listed in Table 6.

Unlike traditional pressurized water reactors, which utilize solid fuels with fission energy transferred from the fuel pellet to the cladding and finally to the coolant, the SM-MSR uses liquid fuel salt, which also serves as the coolant. In this system, fission energy is directly transferred to the coolant. Therefore, the reactor power, the flow of the fuel salt in the reactor, and the properties of the fuel salt are essential to the LOFC consequences. Sensitivity analysis revealed that the volumetric heat of the fuel salt (Density × Specific heat) is the most critical input uncertainty parameter, influencing both heat absorption and fuel salt flow in the reactor core. The reactor power and reactor shutdown margin values can influence the heat generation after a scram; thus, they significantly affect the fuel salt temperature. The local resistance coefficients of the reactor core and primary circuit play a crucial role in affecting the fuel-salt flow, making them important for the fuel-salt temperature. Table 6 lists the relationships among the five important input parameters and T_{out_max}. A negative SRC indicates a negative correlation, while a positive SRC signifies a positive correlation relationship.

Parameter Prediction

Multiple linear regression can be used to predict the value of one variable using the available information. By using the multiple linear regression method, confident predictive values of T_{out_max} can be obtained without numerous calculations. Here, we used five important parameters to predict T_{out_max}.

The Weights of the parameters used for the prediction are shown in Table 7. The RCSs of CPv_fuel and ρ_shutdown are negative, indicating that there is a negative correlation between these parameters and T_{out_max}; thus, the weights are arranged from large to small. Conversely, the RCSs of P_reactor, f_core, and f_primary are positive; therefore, the weights are arranged from small to large to obtain a conservative prediction of T_{out_max}.

Section 4.4.2 showed that T_{out_max} follows a normal distribution, and the significant statistical parameters used for the prediction are listed in Table 4. Figure 13 shows the predicted values of T_{out_max} and bounds of the 95% prediction interval. The predicted upper bound of the 11th case is 752.2 ℃, slightly exceeding 750 ℃. From Table 7, the weights are 0.6 or 1.4, and the uncertainty range is very large.

Fig. 13Full-screen View

Prediction of for T_{out_max} with the important 5 parameters