Calculation of Delayed Neutron Adjoint Flux in Molten Salt Reactors Based on Coupled Precursor Transport and Monte Carlo Particle Transport

Jiacheng Tang¹²³, Guifeng Zhu¹²³,*, Shuyang Jia¹³, Jintong Cao¹²³, Shanjie LYU¹²³, Ye Dai¹²³, Rui Yan¹²³, Yang Zou¹²³

¹ Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
² University of Chinese Academy of Sciences, Beijing 100049, China
³ State Key Laboratory of Thorium Energy, Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China

Abstract

[Background] The flow of molten salt fuel in molten salt reactors (MSRs) induces changes in the effective delayed neutron fraction, which exerts a significant influence on reactor operational control. [Purpose] This study employs the Monte Carlo method to obtain the adjoint flux distribution of delayed neutrons for application in flow-based point reactor model analysis. [Methods] The Iterated Fission Probability Method is implemented to calculate the adjoint flux of delayed neutron precursors in flow models, obtaining effective fractions for different delayed neutron groups. The Molten Salt Reactor Experiment (MSRE) model is used for validation, with results compared against reference calculations to verify accuracy. The axial adjoint flux distribution is further computed for different parent generation counts and flow velocities. [Results] The study reveals distinct differences between delayed and prompt neutron adjoint fluxes. Flow reduces the axial adjoint flux for several delayed neutron groups, primarily due to radial mixing within the core of precursors returning from outside the core, demonstrating the inadequacy of traditional one-dimensional point reactor models. Two-dimensional adjoint flux distributions are also computed, which remain unaffected by flow velocity. [Conclusion] These findings indicate that flow point reactor models require two-dimensional extension for precursor transport within the core, establishing a foundation for systematic transient analysis of molten salt reactors.

Keywords: Monte Carlo, Adjoint flux, Effective delayed neutron fraction, Precursors transport, Molten salt reactor

Introduction

Delayed neutrons are crucial for maintaining controllable chain reactions in fission reactors [1]. The effective delayed neutron fraction represents the contribution ratio of delayed neutrons to fission reactions. In solid-fuel reactors, differences between the delayed and prompt neutron fission spectra [2] cause variations between the delayed neutron fraction and the effective delayed neutron fraction. In liquid-fuel molten salt reactors [3], however, the flow of delayed neutron precursors redistributes the delayed neutron source, with some neutrons being lost outside the core, resulting in a lower effective delayed neutron fraction [4]. Accurate calculation of the effective delayed neutron fraction is therefore essential for ensuring MSR safety and stability.

Current two-dimensional/threeimensional deterministic [5-7] and Monte Carlo methods [8-10] can calculate the effective delayed neutron fraction for liquid-fuel MSRs. Deterministic methods involve approximations in geometry and cross-sections, leading to deviations when reproducing measured data from the 8 MWt Molten Salt Reactor Experiment (MSRE) at Oak Ridge National Laboratory [11]. Monte Carlo methods, with fewer approximations, yield more accurate results. However, existing Monte Carlo methods can only calculate the effective delayed neutron fraction under steady flow conditions and cannot yet analyze neutron kinetics during pump start-up or shutdown scenarios. MSR system analysis codes typically employ modified flow point reactor equations [12] for variable-flow neutron kinetics simulations. These equations differ from conventional point reactor equations by incorporating an axially distributed precursor concentration term and accounting for the spatial contribution of delayed neutrons to fission reactions, thereby addressing the distribution differences between delayed and prompt neutron sources caused by precursor flow. Solving these flow point reactor equations depends not only on the neutron flux distribution but also on accurate calculation of the adjoint flux for each delayed neutron group.

Adjoint flux is readily obtained in deterministic codes but presents challenges in Monte Carlo methods suited for complex geometries. The adjoint flux essentially represents the ratio of a neutron's contribution to fission reactions when born at a particular location, and it is the solution to the adjoint equation of the original transport equation [13]. Monte Carlo methods can obtain adjoint flux through statistical processing without explicitly solving the adjoint equation [2]. Methods for statistical adjoint flux calculation in Monte Carlo codes include the Next Fission Probability [9], Iterated Fission Probability [14-16], and expected next-generation neutron count [17], all of which have produced satisfactory results for static models. However, no research has yet addressed adjoint flux calculation for delayed neutrons in flowing liquid-fuel MSRs.

This paper implements the Iterated Fission Probability Method using a Monte Carlo approach with an implanted flow field in an MSRE model. The method calculates the effective fraction and normalized adjoint flux distribution for each delayed neutron group, analyzes parameters affecting the distribution, and provides accurate adjoint flux data for flow point reactor models.

1.1 Coupling Method for Monte Carlo Neutron Transport and Precursor Migration

Due to the comparable timescales of precursor flow circulation and precursor half-lives, coupling with the flow field is necessary in delayed neutron transport calculations. The primary coupling process proceeds as follows: during source particle sampling, the fission particle type is identified. If identified as a delayed neutron from a specific group, the precursor lifetime is sampled based on its half-life, representing the precursor migration time. The actual delayed neutron source position is then calculated based on the particle's current location and the flow field function. After determining the delayed neutron source position, the particle position and cell information are updated, and particle transport begins. Prompt neutrons bypass precursor migration calculations. Following transport for each generation, backward tracking is performed to统计 the fission contribution ratios from different source particle types, considering inter-generational effects by tracing back 푤 generations. The calculation formula is:

$$
\phi_{g}^{*}(r) = \frac{\sum_{i=1}^{N} \sum_{j=1}^{M_i} w_{ij} \cdot \delta_{g, g_{ij}} \cdot \delta(r - r_{ij})}{\sum_{i=1}^{N} w_i \cdot \delta(r - r_i)}
$$

where the terms represent conditional统计ing when the reaction type is fission, with contributions assigned to the source neutron type $g_{ij}$, where prompt neutrons are defined as group 0. The contribution value is the weight loss $w_{ij}$, $i$ is the particle collision index, $j$ is the neutron index, and $w$ represents the forward trace generation index.

1.2 Flow-Coupled Adjoint Flux Calculation Method

The physical meaning of adjoint flux is neutron importance—a measure of a particle's contribution to fission reactions in phase space—which can be obtained through the Iterated Fission Probability Method. This method's core concept involves simulating the statistical behavior of neutron fission chains to directly model the value transmission process of neutrons within the system. The first generation of neutrons in the simulation is termed the parent generation; after a latent cycle of neutron generations, subsequent fission neutrons are统计ed to determine the parent particle's neutron importance. Under stable latent cycle conditions, the neutron importance converges and approximates the adjoint flux value. The following describes the adjoint flux统计ing method for different source particle types at various spatial positions.

First, the region of interest is discretized into a mesh, and the adjoint flux within each volume is calculated using the method described in [14-16]:

$$
\phi_g^{*}(V) = \frac{\text{Fission value of source neutrons of type } g \text{ in volume } V}{\text{Source strength of neutrons of type } g \text{ in volume } V}
$$

For the denominator, the source strength can be expressed as the phase space volume multiplied by the weight of source particles per generation $w_0|_g$. For delayed neutrons considering precursor migration, the source position is the post-migration location.

For the numerator, the following approach is used:

$$
\text{Fission value} = \sum_{i=1}^{N} \sum_{j=1}^{M_i} w_{ij} \cdot \delta_{g, g_{ij}} \cdot \delta(r_{ij} \in V)
$$

where $\delta_{g, g_{ij}}$ represents the logical condition that fission occurs with source particle type $g$ and source position within region $V$. Other terms are identical to those in the previous formula.

This paper implements the above precursor migration coupling and adjoint flux统计ing in the MCNP code by expanding the source particle storage variables to include position and particle type information for several parent generations. During particle transport统计ing, classification is performed based on this source particle information, with the specific flow shown in Figure 2 [FIGURE:2].

2 Computational Model Description

The calculations use the Molten Salt Reactor Experiment (MSRE) designed and operated by Oak Ridge National Laboratory as an example. MSRE employs liquid fuel with uranium (U-235/U-233) dissolved in molten fluoride salts (LiF-BeF₂-ZrF₄-UF₄), forming high-temperature liquid fuel (~650°C). Within the reactor vessel, fuel flows from the lower plenum through graphite channels in the core center to the upper plenum, as schematically shown in Figure 3 [FIGURE:3]. Main parameters are listed in Table 1 [TABLE:1].

For molten salt flow, the simplified model references [4], with specific parameters provided in Table 2 [TABLE:2]. The MSR flow field model is highly complex; this study considers only axial precursor transport within a single core channel. The flow field model is processed as follows: (1) In the core active zone, precursors flow only axially; (2) In the upper/lower plenums, axial flow proceeds at constant residence time while x, y coordinates are randomly generated according to equal cross-sectional density to simulate fluid mixing; (3) In out-of-core regions such as hot leg to cold leg, particles are temporarily stored in low-fission-value regions at the model boundary; (4) In the annular plenum region, flow is axially downward at constant velocity with x, y coordinates randomly generated circumferentially.

3 Results and Discussion

The model is discretized axially for calculation. Based on the core parameters provided above, the lower plenum spans -22 cm to -2 cm, the core active zone spans -2 cm to 162 cm, and the upper plenum spans 162 cm to 182 cm. Calculations employ 800 active generations with 200 million total particles.

3.1 Effective Delayed Neutron Fraction

First, the direct statistical method (Formula (3)) [2] is used to calculate the effective delayed neutron fraction for static and dynamic flowing conditions in the MSRE model. Results are presented in Table 3 [TABLE:3]. Compared with the static MSRE model, the rated flow condition shows an overall reduction of 217.2×10⁻⁵ in the effective delayed neutron fraction, which is close to the MSRE measured value of 212×10⁻⁵. Precursors with longer lifetimes exhibit greater reactivity loss proportions, with Group 4 contributing the largest reactivity loss of approximately 72×10⁻⁵.

3.2 Delayed Neutron Fission Value

The axial fission contribution distribution for each delayed neutron group is calculated separately for static and flowing conditions, with results shown in Figure 4 [FIGURE:4]. A 0.001× prompt neutron fission contribution distribution is included as a reference for comparison. Under static conditions, the prompt neutron fission contribution distribution is similar to those of all delayed neutron groups, being high in the core center and low at the ends, with small peaks in the upper and lower plenums due to the absence of graphite structure and higher accumulated fuel mass. Under rated flow, the prompt neutron fission contribution distribution shows little change compared with static conditions, while delayed neutron group distributions change significantly, with peaks shifting toward the upper plenum due to periodic precursor flow and accumulation. However, for Groups 5 and 6 with large decay constants, the distribution remains maximum at the core center.

Integrating the delayed neutron fission values in each region and dividing by the total neutron fission value yields the effective fraction for each group, as shown in Table 4 [TABLE:4]. The effective fractions for each group are essentially consistent with those in Table 3, validating the correctness of the fission value统计ing method for delayed neutrons in each region.

3.3 Delayed Neutron Source Intensity

统计ing the positions of source particles provides the source intensity distribution for each delayed neutron group. The axial source intensity distributions under static and rated flow conditions are shown in Figure 5 [FIGURE:5]. The source intensity distribution exhibits some similarity to the fission contribution distribution, with slight differences in trend related to the periodic precursor flow described above.

Based on the delayed and prompt neutron distributions, the fractions for different delayed neutron groups under rated flow can be统计ed as 18.40, 88.95, 87.91, 278.49, 87.38, and 30.88 (×10⁻⁵) for Groups 1–6, respectively, with a total delayed neutron fraction of 592.01×10⁻⁵. The total delayed neutron fraction is larger than the effective delayed neutron fraction, particularly for the first four groups, indicating that precursor decay still occurs predominantly within the core model. However, because delayed neutrons are distributed more in the upper and lower plenums rather than concentrated in the core active zone, their fission value decreases significantly. This further emphasizes the importance of introducing neutron adjoint flux into flow point reactor models.

3.4 Axial Adjoint Flux Distribution of Delayed Neutrons

Dividing the fission contribution in each mesh by the source intensity yields the fission value produced by a single delayed neutron at different positions—the normalized adjoint flux value—shown in Figures 8 [FIGURE:8] and 9 [FIGURE:9] (though the text references Figure 6 [FIGURE:6]). Under static conditions, the adjoint flux for each delayed neutron group is similar to that of prompt neutrons, generally showing high value in the central region and low value at the boundaries. Due to larger statistical fluctuations in delayed neutrons, some variations exist, but their fission value is slightly higher than prompt neutrons, primarily because of the softer fission spectrum of delayed neutrons. Under flow conditions, the axial adjoint flux for groups with small decay constants (Groups 1–3) shows a clear decrease in the core active zone and exhibits an asymmetric distribution with higher values at the outlet and lower values at the inlet. This demonstrates that flow affects not only the distribution of delayed neutron fractions but also their axial adjoint flux. One possible explanation is that long-half-life precursors are easily lost and difficult to maintain in multi-generation fission value统计ing chains, resulting in lower adjoint flux. To investigate this further, the number of统计ed parent (latent) generations is varied to observe convergence and spatial propagation characteristics of the adjoint flux.

3.5 Influence of Different Latent Values on Axial Adjoint Flux

For the Iterated Fission Probability Method, selecting an appropriate latent value enables adjoint flux convergence with limited computational resources [19]. This study uses a default latent value of 15. To analyze the influence of latent values, calculations are performed with latent = 1, 5, 10, 20, and 25. The resulting effective delayed neutron fractions are presented in Table 5 [TABLE:5] (though the text references Table 6 [TABLE:6]). The total fraction converges as latent increases, with latent = 25 showing negligible difference from latent = 15 and 20 (excluding statistical error), though larger latent values consume more computational resources. A latent value of 15 is consistent with literature results [19], confirming its validity under precursor migration conditions.

Figure 7 [FIGURE:7] presents the axial adjoint flux distribution for each delayed neutron group under different latent values. At low latent values (e.g., 1), the axial adjoint flux distribution is flatter, while at high latent values, the flux peak becomes more centrally concentrated. This can be understood as multi-generation adjoint flux statistics being equivalent to multiple integrations and normalization of single-generation statistics. The multiple integration process concentrates the distribution toward the center, while normalization stabilizes its shape. Even at latent = 1, the axial adjoint flux for the first three delayed neutron groups remains lower than the prompt neutron adjoint flux, indicating that this reduction is not a result of multi-generation propagation. Analysis suggests this is primarily caused by radial mixing of delayed neutron precursors. Because precursors in the first three groups have relatively long half-lives, a significant proportion returns to the core. However, due to mixing in the upper and lower plenums, these returning precursors are no longer concentrated at the center but are uniformly distributed across the cross-section, reducing their fission value contribution.

3.6 Influence of Flow Velocity on Axial Adjoint Flux

In flow models, fluid velocity determines residence time in the active zone, thereby affecting the fission value at that position. This section sets latent = 15 and examines the influence of flow velocity on delayed neutron axial adjoint flux by varying the fluid velocity to 0.5× and 2× the original value.

As shown in Figure 8 [FIGURE:8], the axial adjoint flux differs significantly at different flow velocities. Higher velocities result in lower adjoint flux for Groups 1–2 (with small decay constants) in the core axial center. The axial adjoint flux of delayed neutrons is affected by flow velocity, limiting its direct application in Equation (1). To isolate the delayed neutron adjoint flux, the flow point reactor model should consider radial distribution effects.

3.7 Influence of Flow on Two-Dimensional Distribution

A two-dimensional Z-R cylindrical coordinate mesh is established for the MSRE model to obtain the 2D source intensity distribution for each neutron group. Comparing Figure 9 [FIGURE:9] and Figure 10 [FIGURE:10] reveals that for long-lived delayed neutrons (especially Groups 1–2) under flow conditions, the source intensity distribution in the core center shows clear uniformization compared with the static condition (the prompt neutron source intensity distribution can be considered static). The shorter the delayed neutron lifetime, the less the distribution is affected by flow. For short-lived delayed neutrons, the source intensity center shifts upward with flow, consistent with the characteristics of flowing precursor migration.

The 2D adjoint flux distributions are calculated and shown in Figures 11 [FIGURE:11] and 12 [FIGURE:12]. Under flow conditions, the adjoint flux distribution for each delayed neutron group generally shows high values at the center and low values at the periphery, with axial symmetry and similarity to the static adjoint flux distribution. Although 2D adjoint flux统计ing exhibits statistical fluctuations, after removing these fluctuations, the flowing adjoint flux should maintain a proportional relationship with the static adjoint flux.

To eliminate statistical fluctuation effects, we assume that delayed and prompt neutrons have similar adjoint flux distribution shapes. Using the ratio of static delayed neutron effective fraction to delayed neutron fraction as the proportionality coefficient (Table 6 [TABLE:6]), this coefficient is multiplied by the prompt neutron adjoint flux to obtain the delayed neutron adjoint flux distribution. The processed 2D adjoint flux is then multiplied by the delayed neutron source intensity distribution under flow conditions to obtain the delayed neutron fission value, yielding the processed effective fractions for each group shown in Table 7 [TABLE:7]. The results demonstrate that the 2D adjoint flux obtained through this method can serve as input for point reactor equations, with errors within 2.9% for each group and 0.83% for the total fraction.

Conclusion

This paper presents a Monte Carlo-based method for calculating delayed neutron migration adjoint flux in molten salt reactors and implements it in a Monte Carlo code. Using the MSRE model, calculations yield results close to experimentally obtained flowing delayed neutron fractions, validating the accuracy of the Iterated Fission Probability Method for adjoint flux calculation. Analysis of normalized adjoint flux distributions and effective delayed neutron fractions at different latent values indicates that latent = 15 provides high accuracy with reasonable computational resources. Comparisons of normalized axial adjoint flux distributions and effective fractions at different flow velocities reveal that velocity significantly affects the axial adjoint flux calculation for the first three delayed neutron groups, with higher velocities creating more uneven axial distributions. This study further establishes the necessity of calculating 2D delayed neutron adjoint flux, which is unaffected by flow velocity and eliminates interference from radial precursor mixing. Multiplying the processed 2D delayed neutron adjoint flux by the flowing delayed neutron source intensity distribution yields the effective delayed neutron fraction under flow conditions, with group errors within 2.9% and total fraction error of 0.83%. This method can provide accurate delayed neutron adjoint flux calculations as input for 2D flow point reactor models, laying a foundation for systematic transient analysis of molten salt reactors.

Author Contributions: Jiacheng Tang drafted the manuscript, analyzed and interpreted data, and prepared figures. Guifeng Zhu conceived the research framework, developed methodologies, and provided writing guidance. Shuyang Jia, Jintong Cao, and Shanjie LYU revised the manuscript and provided feedback. Ye Dai offered computational support. Rui Yan reviewed and revised the manuscript. Yang Zou provided project guidance and technical support.

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