Preamble

Vol. XX, No. X, XXX 20XX Nuclear Techniques

Development of a Transient Safety Analysis Program for Lithium-Cooled Fast Reactor Stirling Power Cycle System

Zeng Rongtian¹, Shan Jianqiang², Ge Li³, Wang Jianwei⁴, Wu Pan*

(School of Nuclear Science and Technology, Xi'an Jiaotong University, Xi'an 710100)

Abstract

To enhance the safety and reliability of space nuclear power systems, a transient safety analysis program NUSOL-LMR-Li was developed for lithium-cooled fast reactor Stirling power cycle systems, focusing on analyzing their performance and adaptive characteristics under accident conditions. Based on the Schmidt isothermal analysis method, models were established for reactor core power, thermal-hydraulics, heat conduction in structural components, and the physical properties of liquid lithium, sodium-potassium alloy, and helium, enabling coupling between Stirling thermoelectric conversion and lithium-cooled fast reactors. The program was validated against NASA's RE-1000 Stirling prototype and system loop data, showing a maximum error of 3.8%. The core power model exhibited errors of 0.53% and 0.32% in positive and negative reactivity insertion cases, respectively, while the fully implicit algorithm demonstrated time-step-independent stability in single-pipe flow calculations. Transient analysis of typical system accident conditions revealed: (1) A 0.1$ reactivity insertion increases core power to 534 kW, which stabilizes at 466 kW through negative feedback mechanisms, with the outlet temperature rise limited to 14.7 K; (2) Regenerator blockage reduces Stirling efficiency to 22.24%, and the system autonomously reduces the hot-end temperature to 1023 K and increases the cold-end temperature to 692 K to reconstruct thermal equilibrium, recovering efficiency to 26.8%. The study validates the system's self-regulating and fault-tolerant capabilities, providing theoretical support and an efficient analytical tool for the safety design of space nuclear power systems.

Keywords: Stirling power cycle; Lithium-cooled fast reactor; Space reactor; Model and algorithm development; Accident safety characteristic analysis

Introduction

Lithium-cooled fast reactors represent an advanced nuclear reactor technology with numerous significant advantages. Lithium's low density helps reduce the weight of space reactors, enabling system miniaturization and lightweight design. Lithium can operate at extremely high temperatures, allowing lithium-cooled fast reactors to achieve high cycle efficiency. For instance, a liquid metal lithium experimental loop developed by the Institute of Nuclear Energy Safety Technology at the Chinese Academy of Sciences' Hefei Institutes of Physical Science achieved stable operation at 1500 K (equivalent to 1227°C) for 1000 hours domestically for the first time [1]. Liquid lithium exhibits good compatibility with certain structural materials, and Zheng et al. have conducted experimental studies on the corrosion resistance of high-temperature refractory alloys in high-temperature liquid lithium environments [2]. Additionally, its excellent thermal conductivity ensures efficient heat transfer from the core to the energy conversion system, improving overall energy conversion efficiency and reducing energy losses. Due to lithium's high boiling point, the system can operate at atmospheric pressure, simplifying system design and enhancing operational safety. Furthermore, some lithium-cooled fast reactor designs incorporate innovative passive negative feedback systems, such as the ARC (Autonomous Reactivity Control) system [3]. This system utilizes liquid lithium as a neutron absorber; when the coolant outlet temperature rises, thermal expansion automatically injects liquid lithium containing highly enriched ⁶Li into the core, thereby introducing negative reactivity and achieving self-regulation. This design not only enhances inherent reactor safety but also enables response to transient conditions without external intervention. In summary, the unique advantages of lithium-cooled fast reactors make them a promising nuclear reactor technology for future space missions, potentially providing efficient and reliable power support.

In the 1980s, under the support of the "Strategic Defense Initiative" (SDI), the United States conducted research and development on the SP-100 space reactor power system [4] (Figure 1 [FIGURE:1]). This project employed a lithium-cooled fast reactor coupled with various power generation methods including thermoelectric and Stirling power generation, using high-boiling-point lithium coolant.

Coupling lithium coolant with the Stirling cycle holds significant importance. First, through the Stirling cycle, thermal energy in the lithium coolant can be efficiently converted into mechanical and electrical energy, achieving effective energy conversion and improving energy utilization efficiency while reducing waste. Lithium's high thermal conductivity allows more heat to be transferred within a smaller volume, enabling more compact reactor designs suitable for space applications. Moreover, lithium is a relatively abundant resource, and the Stirling cycle is an emission-free energy conversion process. Combining lithium coolant with the Stirling cycle enables sustainable energy applications, reducing dependence on traditional energy resources, minimizing environmental impact, and promoting sustainable energy development. In summary, coupling lithium coolant with the Stirling cycle offers benefits of efficient energy conversion, high-temperature thermal energy utilization, and sustainable energy applications, playing an important role in the energy sector and promoting sustainable development and energy conservation.

Zhang Yang et al. from the Institute of Nuclear Energy Safety Technology, Chinese Academy of Sciences [5] proposed a preliminary design scheme for a 5.2 MWt lightweight space lithium-cooled nuclear reactor. This design employs an integrated core configuration, non-uniform lithium coolant flow channels, and a control system composed of boron powder and control rods. Under the SuperMC multi-physics coupling framework and based on a stochastic approximation coupling solution strategy, high-fidelity coupled simulations were performed using Monte Carlo neutron transport and computational fluid dynamics (CFD). The coupled simulations confirmed that the space reactor design meets reactivity requirements, the non-uniform flow channel design enables more uniform radial power distribution, and under normal operating conditions, all material temperatures remain below limits, with simulation results also providing optimization insights for space reactor design.

Jin Zhao et al. from Xi'an Jiaotong University [6] established models for the reactor core, Stirling generator, radiator, pump, and related piping for a space nuclear power system combining a lithium-cooled fast reactor with a Stirling cycle, and developed a transient system thermal-hydraulic safety analysis program based on Fortran. Validation against Stirling experimental data showed a maximum relative error of 17.3% for the Stirling mathematical model. A space lithium-cooled power system model was established, and the rationality of the system program model was verified by comparing steady-state calculated values with design values, with a maximum relative error of 13.3%. Transient analysis of typical system accident conditions demonstrated that due to the core's overall negative reactivity feedback, the peak fuel pellet temperature remained within safety limits, indicating certain safety characteristics of the system.

Existing research has accumulated important technologies for lithium-cooled space reactor systems, but significant limitations remain in verification calculations for Stirling engines and accident safety analysis. Specifically, most studies have focused on steady-state performance verification, lacking systematic simulation of transient accident scenarios. More critically, as the core hub of energy conversion, the proprietary failure modes of Stirling engines in space environments (such as efficiency degradation due to regenerator blockage) have not been adequately considered, creating analytical blind spots in the fault propagation path from nuclear heat source to electrical energy conversion.

A lithium-cooled Stirling cycle system generally consists of a reactor core, Stirling engine, liquid metal electromagnetic pump, radiator, and expansion tank [7]. Figure 2 [FIGURE:2] shows a schematic diagram of the SP-100 Stirling power conversion system. The system achieves nuclear-thermal-electric conversion through a dual-loop liquid metal cycle. After being heated in the core, the lithium coolant passes through a direct-through expansion tank that accommodates volume expansion when lithium transitions from frozen to operating temperature and separates helium gas generated within the reactor. It then heats the hot end of the Stirling engine, driving the free piston to generate electricity. The secondary loop sodium-potassium coolant exchanges heat with the Stirling cold end and discharges waste heat to space through a radiator. A simplified design is adopted, temporarily omitting the transient thermal resistance analysis module for the radiator, with waste heat discharge treated through equivalent boundary conditions.

1. Theoretical Models

Based on the structural and operational characteristics of lithium-cooled space reactor Stirling power conversion systems, a theoretical model suitable for system safety analysis was established. This model covers core power, thermal-hydraulics, heat conduction in structural components, and physical property models for liquid lithium, liquid sodium-potassium, and helium. The Schmidt isothermal analysis method [8] was employed to analyze the isochoric heat transfer process of helium working fluid in the regenerator during the Stirling cycle. Based on energy conservation principles and modified Stirling cycle efficiency, equations for input/output heat and regenerator heat loss were constructed, and coupling between the Stirling cycle and lithium-cooled space reactor was implemented, forming a Stirling thermoelectric conversion model. On this basis, the fully implicit transient safety analysis program NUSOL-LMR-Li was developed in Fortran, suitable for long-term slow transient accident calculations in lithium-cooled fast reactor Stirling power conversion systems. The time-step stability of the fully implicit algorithm was analyzed, and the calculation accuracy of the core power model was verified to meet the requirements for subsequent reactivity insertion accident calculations. Through single-machine and system loop testing, the coupling correctness of the Stirling thermoelectric conversion model was confirmed. Finally, simplified modeling was performed for the SP-100 system, and steady-state calculations and transient accident safety analysis were conducted.

1.1 Thermal-Hydraulic Model

The boiling points of liquid lithium (Li) and sodium-potassium (NaK) coolants at 1 MPa working pressure are as high as 1620 K and 1057 K, respectively. Therefore, in lithium-cooled fast reactors, the primary coolant lithium and secondary coolant NaK remain liquid at atmospheric pressure without boiling phenomena. Consequently, a single-phase flow model based on mass, momentum, and energy conservation was proposed, considering axial heat conduction effects in liquid metal fluids to numerically simulate pressure, velocity, temperature, and other parameters for Li and NaK coolants.

1) Mass Conservation Equation

The first term represents the transient mass change term, and the second term represents the mass convection term.

2) Energy Conservation Equation

where: h—specific enthalpy/J·kg⁻¹; λ—thermal conductivity/W·m⁻¹·K⁻¹; T—temperature/K; g—gravitational acceleration/m·s⁻²; Q_w—thermal power/W·m⁻¹; θ—inclination angle. The physical meanings of terms from left to right are: energy transient change term, energy convection term, axial heat conduction term, pressure-related term, gravity work term, and heat source term.

3) Momentum Conservation Equation

where: P—pressure/Pa; F_w—frictional resistance/Pa·m⁻¹; F_local—local resistance/Pa·m⁻¹. The physical meanings of terms from left to right are: flow transient change term, convection term, static pressure drop term, gravity pressure drop term, frictional pressure drop term, and local pressure drop term.

1.2 Heat Conduction Model for Structural Components

A one-dimensional unsteady heat conduction equation was adopted to establish a heat structure model suitable for fast reactor systems. The general control equation for one-dimensional unsteady heat conduction is:

where: c—specific heat capacity/J·kg⁻¹·K⁻¹; x—distance/m; S—volumetric internal heat source/W·m⁻³; A(x)—area factor.

The internal node method was used for radial mesh division of heat structures, where the temperature node to be solved is located at the grid center, and the grid region is the control volume for that point, as shown in Figure 3 [FIGURE:3].

The control volume integration method was then applied to discretize the one-dimensional unsteady heat conduction equation. The equation was integrated over time using an implicit scheme, with specific derivation details available in relevant literature [9]. The final discrete form of the heat conduction equation was obtained:

For second or third type boundary conditions, the additional source term method was employed, treating the heat entering or leaving the calculation domain as specified by these boundary conditions as equivalent source terms for the control volume adjacent to the boundary. For second-type boundary conditions, the additional source term is:

where: q_bound—boundary heat flux density/W·m⁻².

For third-type boundary conditions, the boundary heat flux density depends on the boundary control volume node temperature, fluid temperature, and the thermal resistance of half the boundary control volume and convective heat transfer resistance, expressed as:

where: T_f—fluid temperature/K; h_conv—convective heat transfer coefficient/W·m⁻²·K⁻¹; T_b—boundary control volume node temperature/K; Δx_b—boundary control volume thickness/m.

Expressed as an additional source term:

For boundary control volumes, the heat conduction control equation only requires adding the additional source term to the source terms S_C and S_P in the equation. Therefore, the general form of the control equation for axial nodes of heat structures can be expressed as:

For left and right boundary control volumes, respectively:

Thus, for a heat structure with N nodes divided radially, N heat conduction control equations can be obtained, forming a linear equation system with a coefficient matrix of order N tridiagonal form, which can be solved using the Thomas algorithm to obtain the node temperatures corresponding to each control volume.

1.3.1 Core Power Calculation Model

A point reactor neutron kinetics model with six groups of delayed neutrons was employed to calculate the transient changes in core fission power.

where: N—neutron density/m⁻³; C_i—concentration of delayed neutron precursors of group i/m⁻³; λ_i—decay constant of delayed neutrons of group i/s⁻¹; β_i—fraction of delayed neutrons of group i; β—total delayed neutron fraction; ρ(t)—reactivity; Λ—neutron generation time.

The high-order endpoint floating method was used to solve the point reactor neutron kinetics equations, which effectively overcomes the stiffness of the point reactor equations with high calculation accuracy [10].

1.3.2 Reactivity Feedback Model

In small fast reactors such as space reactor power systems, fluctuations in core parameters have particularly significant effects on reactivity during transient power operation, primarily manifested as fuel expansion deformation and coolant density changes caused by temperature variations. The lithium-cooled fast reactor reactivity feedback model mainly considers three factors: Doppler effect feedback caused by fuel temperature changes, reactivity changes caused by fuel volume changes, and reactivity feedback caused by coolant density changes. The lumped parameter method was used to approximately solve the total reactivity feedback, with average temperatures used to simulate coolant and fuel temperatures, and various reactivity feedback effects represented by corresponding feedback coefficients:

where: K_D—Doppler feedback coefficient/K⁻¹; T_f—average fuel temperature/K; α_c,a—fuel expansion feedback coefficient/K⁻¹; α_c,r—core structure expansion feedback coefficient/K⁻¹; T_c—average coolant temperature/K; α_c—coolant expansion feedback coefficient/K⁻¹.

1.4 Stirling Thermoelectric Conversion Model

The Stirling engine is a closed-cycle power device based on the cyclic flow of gaseous working fluid between heat and cold sources. Its circulation system consists of a cylinder equipped with two pistons, with a regenerator installed between them (see Figure 4 [FIGURE:4]). The regenerator alternately releases or absorbs thermal energy from the working fluid during each cycle period. The chambers formed by pistons and cylinders on both sides of the regenerator constitute the hot expansion chamber and cold compression chamber, respectively. The Stirling power cycle is an efficient dynamic thermodynamic cycle technology that drives piston motion to achieve energy conversion using temperature differences between heat and cold sources. As a closed system, the working fluid is not consumed during the cycle, offering environmentally friendly and sustainable characteristics. With its unique advantage of maintaining high thermal efficiency even under low temperature differences, the Stirling cycle demonstrates outstanding performance in space energy conversion systems, particularly suitable for long-term deep space exploration missions, providing a reliable and high-performance solution for efficient energy conversion [11].

The Stirling thermoelectric conversion process involves complex mechanical motion and thermodynamic state changes of the working fluid, with numerous variables and complex equations. Based on basic energy conservation and starting from Stirling cycle efficiency, the classical Schmidt isothermal analysis model [12] was employed for cycle system analysis. This model assumes constant cyclic temperatures in the heating and condensation chambers, with gas temperature in the expansion and heating chambers at the average heat source temperature, and gas temperature in the compression and condensation chambers at the average cold source temperature.

Considering heat transfer irreversibility between the heat source and working fluid, regenerator heat loss, and its function of controlling the temperature difference between the Stirling engine's hot and cold ends, while ignoring other irreversible effects such as heat leakage and friction, and assuming the total gas mass in the heat engine remains constant, the Stirling cycle and its external coupling model were established (Figure 5 [FIGURE:5]).

1) Regenerator Model

Since the Stirling cycle essentially consists of two isothermal processes and two isochoric processes, the regenerator power is considered constant here to control the temperature difference between the expansion and compression chambers. The regenerator power can be expressed as:

where, m_gas—gas mass inside the Stirling engine/kg; T_h—average gas temperature in the expansion (heating) chamber/K; T_c—average gas temperature in the compression (condensation) chamber/K; C_P—gas specific heat/J·kg⁻¹·K⁻¹.

Regenerator loss is an important irreversible loss during Stirling engine operation, calculated by:

where, g is the regenerator loss constant, related to the working fluid's average constant pressure specific heat capacity, regenerator efficiency, and the average mass of gas passing through the regenerator during the hot blow period (or cold blow period). Based on experimental simulation, g = 2.26×10⁻⁵ here.

2) Coupling Model

The coupling between the primary/secondary sides and the Stirling system is represented by the following two equations:

where, Q_inheat—power obtained by the Stirling hot chamber from the primary loop lithium coolant/W; Q_out—power released by the Stirling cold chamber to the secondary loop liquid sodium-potassium/W; T_Li—primary loop lithium coolant temperature/K; T_NaK—secondary loop sodium-potassium coolant temperature/K; R_heat—thermal resistance between primary loop lithium and hot chamber gas/W·K⁻¹; R_cool—thermal resistance between cold chamber gas and secondary loop sodium-potassium/W·K⁻¹.

Thermal resistance consists of three parts: liquid-solid, solid-solid, and solid-gas, specifically including heat transfer between coolant and hot (cold) chamber walls, conduction between hot (cold) chamber walls, and heat transfer between hot (cold) chamber walls and helium working fluid inside the Stirling engine.

where, h_Li—heat transfer coefficient between liquid lithium and hot chamber wall/W·m⁻²·K⁻¹; h_NaK—heat transfer coefficient between liquid sodium-potassium and cold chamber wall/W·m⁻²·K⁻¹; A_h—contact area between hot chamber wall and lithium coolant/m²; A_c—contact area between cold chamber wall and sodium-potassium coolant/m²; λ_h—thermal conductivity of hot chamber wall/W·m⁻¹·K⁻¹; λ_c—thermal conductivity of cold chamber wall/W·m⁻¹·K⁻¹; h_He—heat transfer coefficient of helium working fluid/W·m⁻²·K⁻¹.

Actual heat engines cannot achieve theoretical maximum efficiency due to limitations from irreversible processes such as friction, thermal resistance, and fluid viscosity. These irreversible factors cause energy losses, significantly reducing heat engine efficiency. An ideal reversible Stirling cycle requires infinitely slow operation to achieve theoretical maximum efficiency and ensure thermodynamic equilibrium. However, such processes cannot produce actual power output, as power generation depends on energy transfer within finite time. Based on the law of energy conservation and assuming optimal Stirling cycle efficiency, there exists a clear thermodynamic relationship between the heat transferred from primary loop lithium to the Stirling hot chamber (Q_inheat), heat released from the Stirling cold chamber to secondary loop sodium-potassium (Q_out), regenerator heat loss (Q_Rloss), and Stirling engine efficiency (η_stirling):

where, g is the regenerator loss constant; λ—compression ratio (ratio of hot chamber to cold chamber volume); c—actual efficiency correction coefficient; n—moles.

2. Development and Validation of the Lithium-Cooled Space Reactor Power System Analysis Program

Based on the above-described models for liquid metal-cooled space reactor power system analysis, appropriate numerical solution methods were selected for different models, and a modular programming approach was employed to develop the lithium-cooled space reactor power system transient analysis program NUSOL-LMR-Li. The program's corresponding models were validated through basic benchmark problems.

2.1 Fully Implicit Difference Algorithm for Convection-Diffusion Terms

Considering that lithium-cooled fast reactor transient safety analysis calculations for accidents require substantial computational time, current mainstream system analysis programs generally employ semi-implicit algorithms whose time steps are constrained by the Courant number, resulting in less than ideal computational efficiency for long-term slow transients. Therefore, a fully implicit difference algorithm for convection-diffusion terms was adopted to solve single-phase or homogeneous flow hydraulic models based on the three conservation equations. The homogeneous flow model has good well-posedness, and fully implicit algorithm results are independent of time step size, with algorithm stability unconstrained by Courant conditions. Consequently, larger time steps can be employed in long-term slow transient simulations of lithium-cooled fast reactor systems, significantly reducing computation time and improving efficiency.

First, the three basic conservation equations of the single-phase flow model were processed using fully implicit differencing to transform differential equations into difference equations. In the differencing process, subscript o represents control volume parameters, while subscript x represents junction parameters, as shown in Figure 6 [FIGURE:6].

The convection terms in all three equations employ first-order upwind differencing, the conduction term in the energy equation uses central differencing, and the transient terms in all three equations use first-order backward differencing. The resulting difference equation expressions are:

Mass and energy conservation equations are defined on control volumes, while the momentum conservation equation is defined on junctions. This reactor system contains N control volumes and J junctions, with the order of the nonlinear difference equation system being N+2J, solved using Newton's iteration method requiring generation of the Jacobian matrix. The Newton method solution process can be expressed as:

The primary solution variables of the equation system are control volume pressure P, control volume specific enthalpy h, and junction flow velocity V, therefore:

Taking partial derivatives of the equation system with respect to the solution variables yields the Jacobian matrix, with the general expression:

The partial derivatives of the conservation equations with respect to each solution variable are then obtained to populate the Jacobian matrix. Subsequently, the full-pivot Gaussian elimination method is employed to solve this linear equation system, ultimately obtaining converged solutions.

2.2 Program Structure

Based on the models and algorithms described in the previous sections, the lithium-cooled fast reactor system transient safety analysis program NUSOL-LMR-Li was developed in Fortran using structured program design methodology. Through hierarchical analysis, code was organized using control structures (sequence, selection, loop) to divide module levels, following a top-down, stepwise refinement design principle starting from overall objectives and gradually refining the implementation.

The NUSOL-LMR-Li software system is divided into several functional modules including input, calculation, and output, with each module programmed separately and connected through data transfer to form the complete software system. The main modules and calling structure are shown in Figure 7 [FIGURE:7].

Input Module: Responsible for parsing input files and extracting key parameters for lithium-cooled fast reactor thermal-hydraulic system modeling, including geometric configurations (e.g., core dimensions, piping layout), thermodynamic state variables (e.g., temperature, pressure, flow velocity), and transient event sequences (e.g., reactivity insertion timing, Stirling engine efficiency changes). Through structured data mapping, input parameters are converted into internal calculation variables to provide initialization conditions for subsequent numerical solutions.

Main Calculation Module: The computational kernel of NUSOL-LMR-Li, primarily including thermal-hydraulic solution module, hydraulic solution module, Stirling thermoelectric conversion model, and point reactor neutron kinetics solution module. Relevant thermal-hydraulic parameters obtained from input cards are substituted into the equations to be solved, and solutions are obtained based on major algorithms involved in the software including fully implicit difference format algorithm and control volume integration method, including various inner and outer iteration processes to obtain important thermal-hydraulic parameter changes during lithium-cooled fast reactor transient accidents, such as pressure, temperature, and flow rate.

Auxiliary Module: Primarily based on system geometric parameters and some thermal-hydraulic characteristic parameters, substituting them into thermal-hydraulic constitutive models for properties, flow, and heat transfer to solve for constitutive parameters such as properties (liquid lithium, liquid sodium-potassium, helium) and heat transfer coefficients within the system's thermal and hydraulic components, providing support for equation solving.

Output Module: This module is primarily responsible for outputting important thermal-hydraulic parameter data for data analysis.

2.3.1 Time-Step Stability Analysis of Fully Implicit Difference Algorithm

The fully implicit algorithm features computational results and stability unconstrained by the Courant criterion. Its time-step stability was analyzed through a single-pipe flow example driven by pressure difference.

As shown in Figure 8 [FIGURE:8], a schematic diagram of a single-pipe flow example assumes a pipe length of 5 m, pipe diameter of 0.2 m, filled with liquid lithium, with time-dependent control volumes connected at both ends to model boundary conditions. In this example, the pressure boundaries at the pipe inlet and outlet are fixed, so flow within the pipe is driven by the pressure difference between the two ends.

Assuming the fluid is initially static, it will begin to accelerate under the pressure difference. As velocity increases, frictional resistance also increases, thereby increasing frictional pressure drop. When the frictional pressure drop increases to equal the driving pressure difference, the fluid reaches equilibrium and all parameters approach steady state. The variation of pipe outlet flow velocity over time calculated by the program is shown in Figure 9 [FIGURE:9].

It can be observed that approximately 20 seconds after the initial state, the fluid velocity reaches a steady-state value of about 16.06 m/s. During this process, calculation results under different time steps show high consistency. When time steps of 0.1 s and 0.5 s are used, the Courant number exceeds 1, and with a 0.5 s time step, the Courant number reaches 8.03, far greater than 1. This well demonstrates the stability of the fully implicit algorithm: for transient processes, time step has almost no effect on numerical calculation results, and numerical stability is not constrained by the Courant criterion.

2.3.2 Core Power Calculation Model Validation

The core power calculation model of the program was validated by comparing NUSOL-LMR-Li's point reactor neutron kinetics model solutions for basic examples with analytical calculation results from the literature [10].

(1) Calculation of neutron density N(t) variation over time in a thermal reactor after step positive reactivity insertion

Thermal reactor parameters in the model are given in Table 1 [TABLE:1]. Step reactivity ρ = 0.003 was input with time step h = 0.1 s and N(0) = 1.0 cm⁻³, calculating neutron density (cm⁻³) variation over 1 second. Using analytical calculation results as reference values, the results from NUSOL-LMR-Li's core power calculation model were compared with the exact solution as shown in Table 2 [TABLE:2]. The results show excellent agreement under step positive reactivity insertion conditions, with a maximum deviation of 0.53% that gradually decreases, indicating high calculation accuracy.

Table 1 Neutron Kinetics Parameters of Thermal Reactor

Group Decay Constant λ_i (s⁻¹) Effective Delayed Neutron Fraction β_i 1 0.0127 0.000266 2 0.0317 0.001494 3 0.1150 0.001316 4 0.3110 0.002849 5 1.4000 0.000896 6 3.8700 0.000182

Neutron Generation Time Λ/s: 0.0001

Table 2 Benchmarking of Neutron Density Transients in Fast Reactor under Step Positive Reactivity Insertion

Time/s Analytical Solution Simulated Value Relative Error/% 0.1 5.297E+00 5.324E+00 0.51 0.2 2.731E+01 2.744E+01 0.48 0.3 1.359E+02 1.364E+02 0.37 0.4 6.635E+02 6.653E+02 0.27 0.5 3.211E+03 3.218E+03 0.22 0.6 1.545E+04 1.548E+04 0.19 0.7 7.418E+04 7.428E+04 0.13 0.8 3.556E+05 3.560E+05 0.11 0.9 1.704E+06 1.706E+06 0.12 1.0 8.165E+06 8.170E+06 0.06

(2) Calculation of neutron density N(t) variation over time in a thermal reactor after step negative reactivity insertion

This example uses the same thermal reactor parameters as Table 1. Step reactivity ρ = -0.007 was input with time step h = 0.1 s and N(0) = 1.0 cm⁻³, calculating neutron density (cm⁻³) variation over 1 second. Using analytical calculation results as reference values, the results from the program's core power calculation model were compared with the exact solution as shown in Table 3 [TABLE:3]. The results show excellent agreement under step negative reactivity insertion conditions, with a maximum deviation of 0.32%, indicating high calculation accuracy.

Table 3 Benchmarking of Neutron Density Transients in Fast Reactor under Step Negative Reactivity Insertion

Time/s Analytical Solution Simulated Value Relative Error/% 0.1 4.900708E-01 4.916599E-01 0.32 0.2 4.809743E-01 4.822272E-01 0.26 0.3 4.727747E-01 4.738390E-01 0.23 0.4 4.652903E-01 4.661611E-01 0.19 0.5 4.583873E-01 4.590609E-01 0.15 0.6 4.519650E-01 4.524387E-01 0.10 0.7 4.459467E-01 4.462189E-01 0.06 0.8 4.402732E-01 4.403429E-01 0.02 0.9 4.348978E-01 4.347644E-01 -0.03 1.0 4.297830E-01 4.294464E-01 -0.08

In summary, under both positive and negative step reactivity input conditions, NUSOL-LMR-Li's simulation results show good agreement with exact solutions, demonstrating the correctness of the core power calculation model in the program and its ability to meet the needs of reactivity insertion accident calculations.

2.3.3 Stirling Thermoelectric Conversion Model Validation

(1) Validation of Stirling Cycle Thermal Parameters

Sunpower Corporation designed and tested a 1 kW Stirling engine prototype named RE-1000 for NASA. The experimental results from this prototype were used to validate the Stirling model [13]. The main parameters of RE-1000 are listed in Table 4 [TABLE:4], which compares calculated results with experimental data, showing a maximum relative error of 1.63%. This demonstrates that the model can provide accurate data for Stirling model analysis.

Table 4 Comparison of Calculation Results with Experimental Data for RE-1000

Parameter RE-1000 Prototype Simulated Value Relative Error/% Expansion Space Temp./K 923 925 0.22 Compression Space Temp./K 333 335 0.60 Avg. Pressure/MPa 7.5 7.62 1.63 Stirling Cycle Efficiency/% 28.5 28.1 1.40

(2) Lithium-Cooled-Stirling Cycle Loop Calculation

Based on a simplified SP-00 system design, the modeling diagram of the lithium-cooled fast reactor Stirling power circulation system loop is shown in Figure 10 [FIGURE:10]. Fluid exiting the core flows into the upper plenum, then to the primary side of the Stirling engine, exits the primary side, passes through the downcomer via the liquid metal electromagnetic pump, enters the lower plenum, and returns to the core, forming the primary loop circulation. The secondary side uses sodium-potassium coolant, with given inlet flow rate and outlet pressure boundaries to simulate secondary loop fluid flow.

To conduct subsequent transient safety analysis of accidents in the lithium-cooled fast reactor Stirling circulation system, steady-state simulation of the reactor was performed based on the NUSOL-LM-Li program. According to system and Stirling engine parameters provided in literature [7, 14], Table 5 [TABLE:5] presents the calculated main steady-state thermal-hydraulic parameters compared with design values. The table shows that the Stirling hot and cold ends maintain a certain temperature difference, consistent with Stirling engine operating characteristics, indicating normal thermodynamic cycle operation. In summary, the steady-state results from the program demonstrate successful modeling.

Table 5 Steady-State Thermal-Hydraulic Parameters of LFR-Stirling System: Calculation vs Design Benchmark

Parameter Calculated Value Design Value Relative Error/% Core Thermal Power/kW 2368 2350 0.77 Core Outlet Temp./K 1385 1380 0.36 Core Inlet Temp./K 1200 1195 0.42 Secondary Side Outlet Temp./K 850 845 0.59 Avg. Stirling Hot Side Temp./K 1350 1345 0.37 Avg. Stirling Cold End Temp./K 692 690 0.29 Stirling Cycle Efficiency/% 26.8 27.0 0.74

3. Accident Transient Safety Analysis

To evaluate the safety performance and adaptive capability of the lithium-cooled fast reactor Stirling power cycle system under abnormal conditions, transient accident safety analysis was conducted in this chapter for the lithium-cooled Stirling cycle loop system described at the end of Chapter 2. The analysis object is a simplified model based on the SP-100 space nuclear reactor design, including key components such as the reactor core, Stirling engine, and electromagnetic pump. The system transfers core heat to the Stirling engine hot end through primary loop liquid lithium, driving helium working fluid for thermoelectric conversion, while secondary loop sodium-potassium coolant exchanges heat with the Stirling engine cold end, with waste heat discharge treated through equivalent boundary conditions.

Based on the NUSOL-LMR-Li program, this chapter systematically analyzes the dynamic response characteristics of core power, temperature distribution, and Stirling cycle efficiency by simulating reactivity insertion accidents and unexpected Stirling engine performance degradation accidents, evaluating the system's inherent safety and fault tolerance capabilities.

3.1 Reactivity Insertion Accident

This accident assumes a 0.1$ reactivity insertion, with the protection system inactive. Figure 11 [FIGURE:11] shows the core power transient response characteristics.

Core power rapidly rises to 534 kW within 7 seconds. The increase in core power leads to rapid elevation of core and primary loop temperatures (Figure 12 [FIGURE:12]).

For the secondary loop, the additional reactivity insertion causes the Stirling cycle hot-end temperature to rise. To maintain normal circulation, the cold-end temperature subsequently increases, thereby driving up the secondary loop temperature (Figure 13). However, due to the overall negative feedback regulation mechanism, core power gradually decreases and eventually stabilizes at 466 kW. As power stabilizes, temperatures finally approach steady state. The temperature rise amplitudes at core inlet, core outlet, and secondary side outlet are 5.74 K, 14.7 K, and 2.24 K, respectively.

In summary, although additional reactivity was introduced at 0 seconds, causing instantaneous increases in core power and temperature, the system's negative feedback regulation mechanism effectively controlled core power, enabling it to finally stabilize at 466 kW. This process not only demonstrates the lithium-cooled Stirling cycle system's self-regulating capability and inherent safety under overpower accidents, but also emphasizes the important role of negative feedback mechanisms in maintaining safe and stable reactor operation.

3.2 Unexpected Stirling Engine Performance Degradation Accident

When operating in space, the regenerator in the Stirling power conversion system is responsible for controlling the temperature difference between the expansion and compression chambers. If the regenerator becomes blocked due to impurity accumulation or corrosion, the thermoelectric conversion efficiency of the Stirling engine will be affected and decrease.

In this accident analysis, it is assumed that the Stirling engine operates normally from 0-10 s, and from 10 s to 300 s, the Stirling engine efficiency begins to decrease to 50% due to regenerator blockage, with no protective action. As can be seen from Figure 14 [FIGURE:14], the accident occurs at 10 s, and the Stirling cycle efficiency drops to the lowest point of 22.24% at 30 s, after which the system gradually adapts to the partially blocked condition and slowly rises and stabilizes at 26.8%.

Figure 15 shows the variation of average hot and cold end temperatures of the Stirling engine over time during the regenerator blockage accident. During the normal operation phase of 0-10 seconds, the hot and cold end temperatures remain stable at approximately 1318 K and 660 K, respectively, with a temperature difference of about 658 K. After the accident (10-30 seconds), due to the sudden drop in heat transfer efficiency caused by regenerator blockage, the hot end initially drops sharply (approximately 300 K) and finally stabilizes at 1023 K. This reflects the direct cutoff effect of blockage on the working fluid's expansion work capability, while the formation of the subsequent stable state originates from the spontaneous thermal equilibrium reconstruction mechanism of the Stirling cycle system. At the same time, due to the decrease in cycle efficiency, the high-temperature thermal energy carried by the helium working fluid in the Stirling engine cannot be effectively exported by the cold end, causing heat to accumulate in the regenerator and cold end chamber in a short time, and the cold end temperature is thus rapidly raised to 900 K, and the secondary side temperature also rises. As a new cycle is established, the cold end temperature drops to 692 K, and the secondary side temperature also turns from rising to falling and tends to be stable (Figure 16), confirming the high-efficiency robustness of cold end heat exchange: despite the sudden reduction in hot end input causing a short-term thermal coupling imbalance, the heat dissipation system still resists the risk of working fluid condensation by maintaining a constant temperature gradient. It is worth noting that the temperature difference between hot and cold ends narrows from the initial 658 K to 330 K in the accident steady state. This temperature difference reconstruction phenomenon precisely corresponds to the actual efficiency value of 26.8% (lower than the original design of 40%, but higher than the Carnot theoretical lower limit of 21%), revealing the physical essence of temperature difference as the decisive factor of cycle efficiency. This dynamic evolution process highlights the thermal inertia buffering characteristics and fault self-adaptive capability of the lithium-cooled fast reactor-Stirling coupling system, providing key experimental evidence for the continuous power supply reliability of space nuclear power systems under extreme operating conditions.

This study developed the transient analysis program NUSOL-LMR-Li to analyze the transient safety characteristics of the lithium-cooled space reactor Stirling power conversion system, and systematically evaluated its performance and reliability through relevant model validation and accident analysis. The main conclusions of the study are as follows:

1. High-Precision Model Validation

The core power model demonstrated high accuracy through validation with two typical cases: step positive reactivity insertion and step negative reactivity insertion, with maximum relative errors of 0.53% and 0.32%, respectively. Additionally, the stability of the fully implicit algorithm was confirmed through a single-pipe flow example driven by pressure difference, where flow velocity remained consistent even with large time steps. These validations ensure the reliability and accuracy of core power calculations and numerical methods, which are crucial for the safety and performance of lithium-cooled fast reactor Stirling power conversion systems. Based on NASA's RE-1000 Stirling prototype parameters for steady-state validation, the maximum calculation error for hot-end temperature, cold-end temperature, and cycle efficiency was 1.63%. In the lithium-cooled-Stirling cycle loop calculation validation, the maximum error between the program's steady-state results and design parameters was 3.8%, proving the high precision and reliability of NUSOL-LMR-Li in simulating the lithium-cooled fast reactor Stirling thermoelectric conversion system.

2. Accident Safety Characteristics

Reactivity Insertion Accident: 0.1$ reactivity is introduced into the core at 0 s. The core power rises to 534 kW within 7 seconds, then self-stabilizes to 466 kW through fuel Doppler effect and coolant density negative feedback mechanisms, with the core outlet temperature rise controlled within 14.7 K, demonstrating the system's self-regulating capability and inherent safety under overpower accidents.

Unexpected Stirling Engine Performance Degradation Accident: Due to regenerator blockage, Stirling engine efficiency decreases from 40% to 22.24%. However, the system restores thermal equilibrium through autonomous thermal balance reconstruction (hot-end temperature decreases to 1023 K, cold-end increases to 692 K), enabling efficiency recovery to 26.8%, approaching 80% of the Carnot efficiency limit. The temperature difference between hot and cold ends narrows from 658 K to 330 K, revealing the dominant role of temperature difference in efficiency and the system's fault tolerance characteristics.

Conclusion

Based on NUSOL-LMR-Li, this study achieved transient simulation of the lithium-cooled fast reactor Stirling cycle system and validated the accuracy of models in the program. Through accident analysis specifically targeting the characteristics of lithium-cooled fast reactor Stirling cycle systems, the study not only revealed the negative feedback self-stabilization mechanism of lithium-cooled fast reactors but also demonstrated the temperature-difference-driven efficiency characteristics of the Stirling power conversion system under the innovative accident scenario of unexpected Stirling engine performance degradation, providing theoretical basis and technical support for safety design and fault tolerance optimization of space reactor power systems. The NUSOL-LMR-Li program serves as an efficient analysis tool that can provide important references for future space reactor transient safety assessment and engineering optimization. Subsequent work will simulate and analyze more accidents based on actual space reactor operating conditions.

Acknowledgments

Thanks to XXXX.

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