# The Influence Mechanism of Space-Time Kinetics of ADS Subcritical Reactor under Beam Transients

Authors: Nian-Biao Deng; De-Xin Ding; Jin-Sen Xie; Tao Yu; Cheng Hou
Journal: International Journal of Energy Research
 (2023)

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## Abstract

The space-time kinetics of ADS Subcritical Reactor (ADSR) under beam transients play an important role in ensuring the reliability of the control and safety system of the accelerator-driven subcritical system (ADS). In order to elucidate the effect of high heterogeneity of neutron flux density in space-time on the neutron space-time kinetics of ADSR under different subcriticality and beam transients, in this paper, the multimode core few-group constants, which is more adaptable to the beam transients, is used to investigate the mechanism of neutron flux density, core few-group constants, and kinetic parameters of ADSR under the synergistic effect of external neutron source, subcriticality, and the beam transients. The results show that with the deepening of the subcriticality, with the increasing role of external neutron source, the neutron flux density of the ADSR has a higher heterogeneity of space-time, which leads to a larger difference in the core few-group constant library of different modes, and thus a larger difference in the kinetic parameters (especially the neutron generation time) of different modes. Therefore, in the study of the influence mechanism of neutron space-time kinetics of ADSR under beam transients, the multimode core few-group constants should be used, and the influence of external neutron source and subcriticality on the kinetic parameters, especially the neutron generation time, should be emphasized.


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## Body

## 1. Introduction

With the spread of the COVID-19 pandemic and fluctuating energy prices due to exchange rate fluctuations and other factors, there is a growing demand for energy security and stability, energy efficiency [1–5]. For the sake of achieving carbon emissions reduction targets, clean energy needs more consideration to be used [6–7]. Nuclear energy, as one of the clean energy sources with the advantages of safety, stability, and high energy density, is a good choice to cope with the above issues. With the rapid expansion of nuclear power generation, how to dispose of the accumulated nuclear waste has become a key issue for the sustainable development of nuclear power and affects the public acceptance of nuclear power. The minor actinides (MA) and long-lived fission products (LLFP) in spent fuel discharged from nuclear power plants are characterized by long half-lives and high radioactivity [8].The accelerator-driven subcritical system (ADS) has a high neutron redundancy and subcritical safety, it could be a powerful tool for transmuting MA and LLFP [9–10]. ADS relies on the beam transients generated by the high power proton accelerator (HPPA) to cause rapid changes in the intensity of the spallation neutron source, leading to dramatic changes in the power of ADSR [11], and the resulting shock is a key issue affecting the safety and reliability of ADS [12]. For the sake of avoiding frequent false shutdown of ADSR, to fully grasp the dynamic characteristics of subcritical operation, and to ensure the reliability of control and safety systems, the study of neutron kinetics in ADSR is essential [13]. Compared with the critical reactor system, the neutron kinetics properties of ADSR have a high heterogeneity of the neutron flux density in space-time and deep subcriticality [14].In the current issue of analyzing the effect mechanism of the neutron kinetics of ADSR under beam transients, the existing analytical methods mainly use the two-step method of the deterministic approach. The homogenization group constant connects the subassembly homogenization calculation and core multigroup calculation, which plays a very important role in the calculation accuracy of the core. At present, the study of the homogenization group constant for the neutron space-time kinetics of ADSR has started to receive attention. Yu T has simulated transient response of beam transients in ADS [11]. Kim S H has studied the kinetics solution with iteration scheme of history-based neutron source in ADS [15]. He M has studied the beam transient analyses of ADSR based on neutron transport method [16]. Zhang L has studied the typical kinetic process of fast and thermal lead reactor the Venus-II [17]. Yu T has developed of a beam transient code for ADS [18]. Paolo P has presented a novel model for the solution of reactor kinetics equations in ADS [19].Most of these studies have used the homogenization group constants generated by deterministic programs, which can only approximately describe the heterogeneity of the neutron flux density of ADSR. And most of the studies only consider the single steady-state condition, without considering the beam transients. The high heterogeneity of the neutron flux density in space-time and deep subcriticality properties of the ADSR under beam transients lead to the complex space-time evolution of the homogenization group constants [20]. However, the existing single two-step core homogenization group constants cannot describe those properties of the ADSR under beam transients [21–23].Therefore, for different beam transients, different modes of core few-group constants should be considered [24]. The author has generated the multimode core few-group constants that is more adaptable to the spatial and temporal evolution of neutron kinetics under beam transients, and has carried out a study on the dynamic characteristics of core neutron numbers of ADSR [14, 25]. At present, there are few studies on the influence mechanism of neutron kinetic parameters of ADSR based on multimode core few-group constants.In order to further elucidate the mechanism of high space-time heterogeneity on the neutron kinetics of ADSR under beam transients, this paper uses the multimode core few-group constants with more adaptable to beam transients, based on the self-developed ADSR neutron kinetics core program TRIONES [25].Under the synergistic effect of the external neutron source and the subcriticality, the mechanism of the neutron flux density, the core few-group constants of each mode and the kinetic parameters of the ADSR under beam transients are studied. This paper is of great academic significance for improving the ADS neutron kinetics theory and promoting ADS engineering research and development.The rest of the paper is organized as follows. Section2 describes the Calculation model and development of multimode core few-group constants. In Section 3, we introduce the research on the influence mechanism of neutron space-time kinetics of ADSR. Finally, the study is concluded in Section 4.


## 2. Calculation Model and Development of Multimode Core Few-Group Constants

### 2.1. Calculation and Analysis Tools

The neutron kinetics analysis of ADSR will be performed using the TRIONES program, which uses the finite difference method for spatial discretization and the improved quasistatic method for temporal discretization, and considers grouping calculation of energy. The TRIONES can calculate the steady-state diffusion ADSR core problem and can be used to give the steady-state neutron flux density and power distributions of the ADSR core, generating multimode core few-group constants. The TRIONES can also be used to solve the space-time diffusion neutron kinetics equations for ADSR, and to obtain the spatial and temporal variation of neutron flux density, core neutron number, and kinetic parameters for the dynamic characteristics study of ADSR.This paper will use the OpenMC program (version 0.12.1) [26] to obtain the multigroup constants of ADSR. Based on the deterministic two-step method, the TRIONES is used to generate the core few-group constants of ADSR and to perform the dynamic characteristics analysis of ADSR under beam transients.


### 2.2. Calculation Model

The ADSR geometry studied in this paper is referenced in the reference [27], the core is equated to a rectangular fuel subassembly in a quasicircular arrangement, as shown in Figure 1. Figure 1 illustrates the diagram of the ADSR. The geometry of the fuel and reflector is considered as a 10×10×10 cm [3] cube grid with 17, 17, and 16 layers in the X, Y, and Z directions, respectively. There is an external neutron source at the radial center subassembly of the core with a height of 20 cm. The initial energy of the steady-state external neutron source set in the reactor falls in the first energy group (its energy range is 2.865~20 MeV). Detectors are arranged at D1, D2, D3, and D4, respectively, to be able to measure the core power distribution [20, 28–29]. The fuel choice for the ADSR studied in this paper is MOX fuel mixed with UO2 and 239PuO2, and the natural Fe material is chosen for the reflector. In order to study the neutron kinetics at different subcriticality, three typical core loading schemes with keff=0.99, 0.96 and 0.93 are obtained by adjusting the 239PuO2 content in the MOX fuel and calculated by the OpenMC. The ADSR core material compositions for the three core loading schemes are shown in Table 1 [25].Figure 1
Diagram of ADSR.
(a)
Horizontal profile(b)
Vertical profileTable 1
Material composition of ADSR at three core loading schemes.
SubcriticalityMaterial composition of fuel(nuclide density 1024 (cm3))Material composition of reflector(nuclide density 1024 (cm3))keff=0.9916O 0.6667 235U 0.009477 238U 0.3064 239Pu 0.0174454Fe 2.708E-0356Fe 4.210E-0257Fe 9.638E-0458Fe 1.285E-04keff=0.9616O 0.6667 235U 0.009522 238U 0.3079 239Pu 0.01594keff=0.9316O 0.6667 235U 0.009564 238U 0.3092 239Pu 0.01453


### 2.3. Development of Multimode Core Few-Group Constants of ADSR

ADSR must be calculated using neutron transport methods due to the high anisotropy of the neutron flux density distribution caused by the presence of external neutron sources. However, the Monte Carlo transport calculation methods are too inefficient for the dozens of scheme calculations in the design process [30]. Considering the computational efficiency, this paper applies the Monte Carlo method to generate the homogenized multigroup constants that are provided to the subsequent deterministic neutron kinetics core program.In this paper, we first use the OpenMC to perform the heterogeneity transport calculation for the ADSR, and then obtain the homogenized 12-group multigroup constants for each material of the ADSR. Due to the large statistical uncertainty of the higher-order scattering moment matrix and the low computational efficiency, this paper considers the third-order scattering moment matrix for the anisotropic scattering cross-section.Since the current OpenMC only considersλ fundamental eigenvalue, in order to generate the core few-group constants for different modes, the ADSR homogenization 12-group constant library generated by the OpenMC is applied to the TRIONES. The three kinds of neutron flux density of λ fundamental eigenvalue (λ mode), ADSR steady state (S mode), and time fundamental eigenvalue (α mode) are calculated, respectively, and the core few-group constants (6 groups) of the three modes are collapsed. By linking the generated λ mode, S mode, and α mode core few-group constants and adapting them to arbitrary beam transients, a multimode core few-group constants can be established. The development of multimode core few-group constants of ADSR is shown in Figure 2 [25].Figure 2
Flowchart for the development of multimode core few-group constants of ADSR.By judging different beam transients and calling different modes of core few group constants, and corresponding to the TRIONES, the self-adapting of multimode core few group constants is established under arbitrary beam transients. The specific calculation flow of self-adapting multimode core few-group constants is shown in Figure3. The relevant data can be found in the reference [25].Figure 3
Schematic of self-adapting multimode core few-group constants.


## 2.1. Calculation and Analysis Tools

The neutron kinetics analysis of ADSR will be performed using the TRIONES program, which uses the finite difference method for spatial discretization and the improved quasistatic method for temporal discretization, and considers grouping calculation of energy. The TRIONES can calculate the steady-state diffusion ADSR core problem and can be used to give the steady-state neutron flux density and power distributions of the ADSR core, generating multimode core few-group constants. The TRIONES can also be used to solve the space-time diffusion neutron kinetics equations for ADSR, and to obtain the spatial and temporal variation of neutron flux density, core neutron number, and kinetic parameters for the dynamic characteristics study of ADSR.This paper will use the OpenMC program (version 0.12.1) [26] to obtain the multigroup constants of ADSR. Based on the deterministic two-step method, the TRIONES is used to generate the core few-group constants of ADSR and to perform the dynamic characteristics analysis of ADSR under beam transients.


## 2.2. Calculation Model

The ADSR geometry studied in this paper is referenced in the reference [27], the core is equated to a rectangular fuel subassembly in a quasicircular arrangement, as shown in Figure 1. Figure 1 illustrates the diagram of the ADSR. The geometry of the fuel and reflector is considered as a 10×10×10 cm [3] cube grid with 17, 17, and 16 layers in the X, Y, and Z directions, respectively. There is an external neutron source at the radial center subassembly of the core with a height of 20 cm. The initial energy of the steady-state external neutron source set in the reactor falls in the first energy group (its energy range is 2.865~20 MeV). Detectors are arranged at D1, D2, D3, and D4, respectively, to be able to measure the core power distribution [20, 28–29]. The fuel choice for the ADSR studied in this paper is MOX fuel mixed with UO2 and 239PuO2, and the natural Fe material is chosen for the reflector. In order to study the neutron kinetics at different subcriticality, three typical core loading schemes with keff=0.99, 0.96 and 0.93 are obtained by adjusting the 239PuO2 content in the MOX fuel and calculated by the OpenMC. The ADSR core material compositions for the three core loading schemes are shown in Table 1 [25].Figure 1
Diagram of ADSR.
(a)
Horizontal profile(b)
Vertical profileTable 1
Material composition of ADSR at three core loading schemes.
SubcriticalityMaterial composition of fuel(nuclide density 1024 (cm3))Material composition of reflector(nuclide density 1024 (cm3))keff=0.9916O 0.6667 235U 0.009477 238U 0.3064 239Pu 0.0174454Fe 2.708E-0356Fe 4.210E-0257Fe 9.638E-0458Fe 1.285E-04keff=0.9616O 0.6667 235U 0.009522 238U 0.3079 239Pu 0.01594keff=0.9316O 0.6667 235U 0.009564 238U 0.3092 239Pu 0.01453


## 2.3. Development of Multimode Core Few-Group Constants of ADSR

ADSR must be calculated using neutron transport methods due to the high anisotropy of the neutron flux density distribution caused by the presence of external neutron sources. However, the Monte Carlo transport calculation methods are too inefficient for the dozens of scheme calculations in the design process [30]. Considering the computational efficiency, this paper applies the Monte Carlo method to generate the homogenized multigroup constants that are provided to the subsequent deterministic neutron kinetics core program.In this paper, we first use the OpenMC to perform the heterogeneity transport calculation for the ADSR, and then obtain the homogenized 12-group multigroup constants for each material of the ADSR. Due to the large statistical uncertainty of the higher-order scattering moment matrix and the low computational efficiency, this paper considers the third-order scattering moment matrix for the anisotropic scattering cross-section.Since the current OpenMC only considersλ fundamental eigenvalue, in order to generate the core few-group constants for different modes, the ADSR homogenization 12-group constant library generated by the OpenMC is applied to the TRIONES. The three kinds of neutron flux density of λ fundamental eigenvalue (λ mode), ADSR steady state (S mode), and time fundamental eigenvalue (α mode) are calculated, respectively, and the core few-group constants (6 groups) of the three modes are collapsed. By linking the generated λ mode, S mode, and α mode core few-group constants and adapting them to arbitrary beam transients, a multimode core few-group constants can be established. The development of multimode core few-group constants of ADSR is shown in Figure 2 [25].Figure 2
Flowchart for the development of multimode core few-group constants of ADSR.By judging different beam transients and calling different modes of core few group constants, and corresponding to the TRIONES, the self-adapting of multimode core few group constants is established under arbitrary beam transients. The specific calculation flow of self-adapting multimode core few-group constants is shown in Figure3. The relevant data can be found in the reference [25].Figure 3
Schematic of self-adapting multimode core few-group constants.


## 3. Research on the Influence Mechanism of Neutron Space-Time Kinetics of ADSR

### 3.1. Research on the Influence Mechanism of External Neutron Source

In order to study the effect of external neutron source variation on the neutron flux of ADSR under beam transients, this paper uses the reference program using the direct time discretization method, and applied the 12-group constant library withkeff=0.96 [25]. We study the neutron space-time kinetics of ADSR in the typical beam transients that the starting process of the positive beam jump process (referred to as starting process). The simulation time of the reactor starting process is 2.5E−3 s, and the time step is set to 5.0E−6 s. Four typical time points are selected: t=1.0E−5 s, 4.0E−5 s, 1.0E−4 s, and 2.5E−4 s. And the neutron flux density distribution plots over space and time the 2-dimensional (X-direction and Y-direction) in the center plane (z-axis layer 9) of the ADSR at these four typical times are presented. The neutron flux density distribution of ADSR (keff=0.96) at different moments under starting process is shown in Figure 4.Figure 4
Neutron flux density distribution of ADSR (keff=0.96) at different moments under starting process.
(a)
Neutron flux density distribution att=1.0E−5 s(b)
Neutron flux density distribution att=4.0E−5 s(c)
Neutron flux density distribution att=1.0E−4 s(d)
Neutron flux density distribution att=2.5E−4 sFrom Figure4, it can be seen that the external source only affects the neutron flux density of the subassemblies of ADSR in its vicinity when the starting process begins (see t=1.0E−5 s). As time progresses (see t=4.0E−5 s), the influence of the external source gradually spreads to a wider range of ADSR subassemblies outside the external source. After the starting process reaches stability (see t=1.0E−4 s and s=2.5E−4 s), the influence of the external source on the ADSR subassemblies gradually stabilizes. It can be concluded that under the starting process, the external source changes drastically with time, resulting in the distortion of neutron flux density distribution in both space and time, which leads to the high space-time heterogeneity of neutron flux density in the ADSR. Since the physical characteristics of the external source for other beam transient conditions (beam trip, recovery after beam trip, etc.) varying drastically with time are similar to those of the starting process, the above conclusions can be extended from the starting process to the entire beam transients.


### 3.2. Research on the Influence Mechanism of Subcriticality

To investigate the effect of subcriticality on the neutron flux density of the ADSR under beam transients, the 12-group constant library withkeff=0.96 in Section 3.1 are replaced with 12-group constant library with keff=0.93 and keff=0.99 [25]. It is used to compare the neutron flux density with space and time at different subcriticality for the ADSR at a typical moment t=4.0E−5 s in the starting process. The neutron flux density distribution of ADSR at the starting process t=4.0E−5 s under different subcriticality is shown in Figure 5.Figure 5
Neutron flux density distribution of ADSR at the starting processt=4.0E−5 s under different subcriticality.
(a)
Neutron flux density distribution atkeff=0.99(b)
Neutron flux density distribution atkeff=0.96(C)
Neutron flux density distribution atkeff=0.93From Figure5, it can be seen that the distribution of neutron flux density of ADSR becomes steeper and steeper as the subcriticality deepens at the same moment of the starting process. The influence of external sources on the neutron flux density of ADSR becomes more and more significant. It can be concluded that the external neutron source has an increasing role on the neutron flux density of ADSR with the deepening of the subcriticality under the starting process. Since the physical characteristics of the external source for other beam transients (beam trip, recovery after beam trip, etc.) varying drastically with time are similar to those of the starting process, the above conclusions can be extended from the starting process to the entire beam transients.


### 3.3. Research on the Influence Mechanism of Multimode Core Few-Group Constants

Since theλ mode is to make the reactor reach virtual criticality by introducing a modulation factor 1/keff into the fission source, the S mode is to make the reactor reach virtual criticality by modulating the external neutron source. Both the λ mode and the S mode make the reactor reach virtual criticality by modulating the generating sources (including fission source, scattering source, and external source), and the physical mechanism of the λ mode is similar to that of the S mode. Therefore, this paper only investigates the mechanism of the effect of λ mode and α mode on the core few-group constants at different subcriticality. And the homogenized absorption cross-section is selected for analysis. The homogenized absorption cross-sections of the ADSR core and reflector subassembly for the α mode at different subcriticality are given in Table 2. And the average neutron velocities of the ADSR core and reflector subassembly for the α mode at different subcriticality are given in Table 3.Table 2
Homogenized absorption cross-sections of ADS core and reflector subassembly inα mode under different subcriticality.
Absorption cross-sections (cm-1)SubcriticalityEnergy group123456Corekeff=0.991.3704E−025.8639E−031.1827E−022.1909E−024.0806E−026.9374E−02keff=0.961.3529E−025.6988E−031.1681E−022.1579E−024.0128E−026.6349E−02keff=0.931.3394E−025.5464E−031.1438E−022.1193E−023.9374E−026.5273E−02Reflectorkeff=0.993.2554E−041.7938E−042.6567E−043.5351E−041.8027E−031.3525E−02keff=0.963.2571E−041.7976E−042.6609E−043.5394E−041.8065E−031.3725E−02keff=0.933.2599E−041.7997E−042.6727E−043.5435E−041.8090E−031.3776E−02Table 3
Average neutron velocity of ADS core and reflector subassembly inα mode under different subcriticality.
Average neutron velocity (cm/s)SubcriticalityEnergy group123456Corekeff=0.995.02E+081.87E+086.57E+072.50E+079.70E+064.89E+05keff=0.965.02E+081.87E+086.55E+072.49E+079.58E+064.81E+05keff=0.935.02E+081.87E+086.58E+072.49E+079.50E+064.79E+05Reflectorkeff=0.994.68E+081.83E+086.33E+072.21E+076.04E+064.79E+05keff=0.964.68E+081.83E+086.34E+072.21E+075.87E+064.70E+05keff=0.934.68E+081.83E+086.32E+072.21E+075.86E+064.68E+05From Table2 and Table 3, it can be seen that for the α mode that makes the reactor reach virtual criticality by adjusting the time absorption term α/v, the absorption cross-section gradually increases with the deepening of the subcriticality in the region where the average neutron velocity is relatively slow (as shown in Table 2) such as the reflector region. The physical mechanism is that as the subcriticality deepens, the slowing-down power of the reflector gradually increases, making the average neutron velocity gradually smaller, which makes the time absorption term α/v gradually increase, and then the absorption cross-section gradually increases.The homogenized absorption cross-sections of ADS core and reflector subassemblies inλ mode for different subcriticality are given in Table 4. By comparing the absorption cross-sections of the λ and α modes in Table 2 and Table 4, it can be seen that the difference between the absorption cross-sections of the λ and α modes becomes larger and larger with the deepening of the subcriticality for the core. And the difference between the absorption cross-sections of the λ and α modes becomes larger and larger with the deepening of the subcriticality for the reflector. The physical mechanism is mainly considered that the α mode makes the reactor reach the virtual criticality by adjusting the absorption cross-section of the reactor core and the reflector, and it makes the reactor reach the virtual criticality by adjusting the vanishing sources (absorption source and leakage source). The physical mechanism of the α mode is essentially different from that of the λ mode, which leads to the difference of the absorption cross-section between the λ mode and the α mode. And with the deepening of subcriticality, the influence of external neutron source on neutron flux density becomes more and more significant, and the space-time heterogeneity of neutron flux density of ADSR becomes higher and higher, which leads to the increasing difference of absorption cross-section, i.e., the core few-group constants between λ and α modes.Table 4
Homogenized absorption cross-sections of ADS core and reflector subassembly inλ mode under different subcriticality.
Absorption cross-sections (cm-1)SubcriticalityEnergy group123456Corekeff=0.991.3705E−025.8643E−031.1823E−022.1890E−024.0520E−021.3680E−01keff=0.961.3531E−025.7011E−031.1687E−022.1558E−023.9698E−021.3270E−01keff=0.931.3397E−025.5501E−031.1454E−022.1172E−023.8853E−021.3197E−01Reflectorkeff=0.993.2592E−041.7917E−042.6662E−043.5437E−041.8946E−036.5588E−03keff=0.963.2562E−041.7916E−042.6584E−043.5443E−041.8770E−036.5656E−03keff=0.933.2314E−041.7917E−042.6686E−043.5359E−041.8804E−036.5532E−03


### 3.4. Research on the Influence Mechanism of Kinetic Parameters

In order to investigate the effects of external neutron source and subcriticality on the kinetic parameters of ADSR under beam transients, in this paper, the TRIONES program is used to study the kinetic parameters of the ADSR at different subcritical degrees,λ and S modes in the starting process and λ and α modes in the beam transient jump process in the beam trip (referred to as the beam trip). The kinetic parameters neutron generation time Λ and effective fraction of delayed neutron βeff are investigated separately. The simulation time is 2.5E−3 s and the time step is set to 5.0E−6 s for both the starting process and the beam trip.Firstly, the neutron generation time is investigated. The relative differences between the neutron generation times ofλ and S modes in the starting process are shown in Figure 6. And the relative differences between the neutron generation times of λ and α modes in the beam trip are shown in Figure 7.Figure 6
Relative difference in neutron generation timeΛ between λ and S mode with time for different subcriticality under the starting process.Figure 7
Relative difference in neutron generation timeΛ between λ and α mode with time for different subcriticality under the beam trip.From Figure6, it can be seen that the relative differences between the neutron generation times of λ and S modes fluctuate greatly at the beginning of the starting process and level off as time progresses. And the relative differences between the λ and S modes increase incrementally as the subcriticality deepens. From Figure 7, it can be seen that the relative difference between the neutron generation time of λ and α modes fluctuates greatly at the beginning of the beam trip, and the relative difference tends to be smooth as time progresses. And the relative difference between the neutron generation time of λ and α modes increases as the subcriticality deepens.For the study of the effective fraction of delayed neutronβeff, the relative difference between the effective fraction of delayed neutron βeff in λ and S modes with time under starting process is shown in Figure 8. The relative difference between the effective fraction of delayed neutron βeff in λ and α modes with time under the beam trip is shown in Figure 9.Figure 8
Relative difference in effective fraction of delayed neutron betweenλ and S mode with time for different subcriticality under the starting process.Figure 9
Relative difference in effective fraction of delayed neutron betweenλ and α mode with time for different subcriticality under the beam trip.From Figure8, the relative difference between the effective fraction of delayed neutron in the λ and S modes increases at the beginning of the starting process with the deepening of the subcriticality and disappears with the progress of time. From Figure 9, it can be seen that the relative difference between the effective fraction of delayed neutron in the λ and α modes increases with the deepening of the subcriticality at the beginning of the beam trip and disappears with the progress of time.To investigate the physical mechanism, for the study of neutron generation time, the sudden production or disappearance of the external neutron source during the starting process and beam trip leads to a sharp fluctuation in the neutron flux density distribution of the whole ADSR in space and time, which makes the related neutron generation time fluctuate greatly. As the system gradually stabilizes, the neutron flux density distribution of the ADSR will stabilize, making the neutron generation time stabilize. As the subcriticality deepens, the role of external neutron sources becomes more and more significant, leading to the space-time heterogeneity of neutron flux density becoming higher, which in turn leads to the increase of the difference between the core few-group constants ofλ mode and S and α modes. This leads to an increase in the relative difference of the neutron generation time between the λ mode and the S and α modes. Since the physical characteristics of other beam transients (recovery after beam trip, etc.) are similar to those of the starting process and beam trip, the above conclusions can be extended to the beam transients from the starting process and beam trip.For the study of the effective fraction of delayed neutron, the physical properties are similar to the neutron generation time at the beginning of the beam transients. With the subcriticality deepens, the role of the external neutron source becomes more and more significant, and the difference in the core few-group constants increases for different modes, which leads to an increase in the relative difference in the effective fraction of delayed neutron for different modes. As the beam transients continues, the effect of subcriticality on the relative differences in the effective fraction of delayed neutron in different modes disappears.In summary, under the beam transients, the relative difference in neutron generation time of different modes increases with the deepening of subcriticality, while the relative difference in effective fraction of delayed neutron of different modes has less effect. Thus, the influence of external source and subcriticality on the kinetic parameters, especially the neutron generation time, needs to be emphasized in the study of the influence mechanism of the neutron space-time kinetics of ADSR under beam transients.


## 3.1. Research on the Influence Mechanism of External Neutron Source

In order to study the effect of external neutron source variation on the neutron flux of ADSR under beam transients, this paper uses the reference program using the direct time discretization method, and applied the 12-group constant library withkeff=0.96 [25]. We study the neutron space-time kinetics of ADSR in the typical beam transients that the starting process of the positive beam jump process (referred to as starting process). The simulation time of the reactor starting process is 2.5E−3 s, and the time step is set to 5.0E−6 s. Four typical time points are selected: t=1.0E−5 s, 4.0E−5 s, 1.0E−4 s, and 2.5E−4 s. And the neutron flux density distribution plots over space and time the 2-dimensional (X-direction and Y-direction) in the center plane (z-axis layer 9) of the ADSR at these four typical times are presented. The neutron flux density distribution of ADSR (keff=0.96) at different moments under starting process is shown in Figure 4.Figure 4
Neutron flux density distribution of ADSR (keff=0.96) at different moments under starting process.
(a)
Neutron flux density distribution att=1.0E−5 s(b)
Neutron flux density distribution att=4.0E−5 s(c)
Neutron flux density distribution att=1.0E−4 s(d)
Neutron flux density distribution att=2.5E−4 sFrom Figure4, it can be seen that the external source only affects the neutron flux density of the subassemblies of ADSR in its vicinity when the starting process begins (see t=1.0E−5 s). As time progresses (see t=4.0E−5 s), the influence of the external source gradually spreads to a wider range of ADSR subassemblies outside the external source. After the starting process reaches stability (see t=1.0E−4 s and s=2.5E−4 s), the influence of the external source on the ADSR subassemblies gradually stabilizes. It can be concluded that under the starting process, the external source changes drastically with time, resulting in the distortion of neutron flux density distribution in both space and time, which leads to the high space-time heterogeneity of neutron flux density in the ADSR. Since the physical characteristics of the external source for other beam transient conditions (beam trip, recovery after beam trip, etc.) varying drastically with time are similar to those of the starting process, the above conclusions can be extended from the starting process to the entire beam transients.


## 3.2. Research on the Influence Mechanism of Subcriticality

To investigate the effect of subcriticality on the neutron flux density of the ADSR under beam transients, the 12-group constant library withkeff=0.96 in Section 3.1 are replaced with 12-group constant library with keff=0.93 and keff=0.99 [25]. It is used to compare the neutron flux density with space and time at different subcriticality for the ADSR at a typical moment t=4.0E−5 s in the starting process. The neutron flux density distribution of ADSR at the starting process t=4.0E−5 s under different subcriticality is shown in Figure 5.Figure 5
Neutron flux density distribution of ADSR at the starting processt=4.0E−5 s under different subcriticality.
(a)
Neutron flux density distribution atkeff=0.99(b)
Neutron flux density distribution atkeff=0.96(C)
Neutron flux density distribution atkeff=0.93From Figure5, it can be seen that the distribution of neutron flux density of ADSR becomes steeper and steeper as the subcriticality deepens at the same moment of the starting process. The influence of external sources on the neutron flux density of ADSR becomes more and more significant. It can be concluded that the external neutron source has an increasing role on the neutron flux density of ADSR with the deepening of the subcriticality under the starting process. Since the physical characteristics of the external source for other beam transients (beam trip, recovery after beam trip, etc.) varying drastically with time are similar to those of the starting process, the above conclusions can be extended from the starting process to the entire beam transients.


## 3.3. Research on the Influence Mechanism of Multimode Core Few-Group Constants

Since theλ mode is to make the reactor reach virtual criticality by introducing a modulation factor 1/keff into the fission source, the S mode is to make the reactor reach virtual criticality by modulating the external neutron source. Both the λ mode and the S mode make the reactor reach virtual criticality by modulating the generating sources (including fission source, scattering source, and external source), and the physical mechanism of the λ mode is similar to that of the S mode. Therefore, this paper only investigates the mechanism of the effect of λ mode and α mode on the core few-group constants at different subcriticality. And the homogenized absorption cross-section is selected for analysis. The homogenized absorption cross-sections of the ADSR core and reflector subassembly for the α mode at different subcriticality are given in Table 2. And the average neutron velocities of the ADSR core and reflector subassembly for the α mode at different subcriticality are given in Table 3.Table 2
Homogenized absorption cross-sections of ADS core and reflector subassembly inα mode under different subcriticality.
Absorption cross-sections (cm-1)SubcriticalityEnergy group123456Corekeff=0.991.3704E−025.8639E−031.1827E−022.1909E−024.0806E−026.9374E−02keff=0.961.3529E−025.6988E−031.1681E−022.1579E−024.0128E−026.6349E−02keff=0.931.3394E−025.5464E−031.1438E−022.1193E−023.9374E−026.5273E−02Reflectorkeff=0.993.2554E−041.7938E−042.6567E−043.5351E−041.8027E−031.3525E−02keff=0.963.2571E−041.7976E−042.6609E−043.5394E−041.8065E−031.3725E−02keff=0.933.2599E−041.7997E−042.6727E−043.5435E−041.8090E−031.3776E−02Table 3
Average neutron velocity of ADS core and reflector subassembly inα mode under different subcriticality.
Average neutron velocity (cm/s)SubcriticalityEnergy group123456Corekeff=0.995.02E+081.87E+086.57E+072.50E+079.70E+064.89E+05keff=0.965.02E+081.87E+086.55E+072.49E+079.58E+064.81E+05keff=0.935.02E+081.87E+086.58E+072.49E+079.50E+064.79E+05Reflectorkeff=0.994.68E+081.83E+086.33E+072.21E+076.04E+064.79E+05keff=0.964.68E+081.83E+086.34E+072.21E+075.87E+064.70E+05keff=0.934.68E+081.83E+086.32E+072.21E+075.86E+064.68E+05From Table2 and Table 3, it can be seen that for the α mode that makes the reactor reach virtual criticality by adjusting the time absorption term α/v, the absorption cross-section gradually increases with the deepening of the subcriticality in the region where the average neutron velocity is relatively slow (as shown in Table 2) such as the reflector region. The physical mechanism is that as the subcriticality deepens, the slowing-down power of the reflector gradually increases, making the average neutron velocity gradually smaller, which makes the time absorption term α/v gradually increase, and then the absorption cross-section gradually increases.The homogenized absorption cross-sections of ADS core and reflector subassemblies inλ mode for different subcriticality are given in Table 4. By comparing the absorption cross-sections of the λ and α modes in Table 2 and Table 4, it can be seen that the difference between the absorption cross-sections of the λ and α modes becomes larger and larger with the deepening of the subcriticality for the core. And the difference between the absorption cross-sections of the λ and α modes becomes larger and larger with the deepening of the subcriticality for the reflector. The physical mechanism is mainly considered that the α mode makes the reactor reach the virtual criticality by adjusting the absorption cross-section of the reactor core and the reflector, and it makes the reactor reach the virtual criticality by adjusting the vanishing sources (absorption source and leakage source). The physical mechanism of the α mode is essentially different from that of the λ mode, which leads to the difference of the absorption cross-section between the λ mode and the α mode. And with the deepening of subcriticality, the influence of external neutron source on neutron flux density becomes more and more significant, and the space-time heterogeneity of neutron flux density of ADSR becomes higher and higher, which leads to the increasing difference of absorption cross-section, i.e., the core few-group constants between λ and α modes.Table 4
Homogenized absorption cross-sections of ADS core and reflector subassembly inλ mode under different subcriticality.
Absorption cross-sections (cm-1)SubcriticalityEnergy group123456Corekeff=0.991.3705E−025.8643E−031.1823E−022.1890E−024.0520E−021.3680E−01keff=0.961.3531E−025.7011E−031.1687E−022.1558E−023.9698E−021.3270E−01keff=0.931.3397E−025.5501E−031.1454E−022.1172E−023.8853E−021.3197E−01Reflectorkeff=0.993.2592E−041.7917E−042.6662E−043.5437E−041.8946E−036.5588E−03keff=0.963.2562E−041.7916E−042.6584E−043.5443E−041.8770E−036.5656E−03keff=0.933.2314E−041.7917E−042.6686E−043.5359E−041.8804E−036.5532E−03


## 3.4. Research on the Influence Mechanism of Kinetic Parameters

In order to investigate the effects of external neutron source and subcriticality on the kinetic parameters of ADSR under beam transients, in this paper, the TRIONES program is used to study the kinetic parameters of the ADSR at different subcritical degrees,λ and S modes in the starting process and λ and α modes in the beam transient jump process in the beam trip (referred to as the beam trip). The kinetic parameters neutron generation time Λ and effective fraction of delayed neutron βeff are investigated separately. The simulation time is 2.5E−3 s and the time step is set to 5.0E−6 s for both the starting process and the beam trip.Firstly, the neutron generation time is investigated. The relative differences between the neutron generation times ofλ and S modes in the starting process are shown in Figure 6. And the relative differences between the neutron generation times of λ and α modes in the beam trip are shown in Figure 7.Figure 6
Relative difference in neutron generation timeΛ between λ and S mode with time for different subcriticality under the starting process.Figure 7
Relative difference in neutron generation timeΛ between λ and α mode with time for different subcriticality under the beam trip.From Figure6, it can be seen that the relative differences between the neutron generation times of λ and S modes fluctuate greatly at the beginning of the starting process and level off as time progresses. And the relative differences between the λ and S modes increase incrementally as the subcriticality deepens. From Figure 7, it can be seen that the relative difference between the neutron generation time of λ and α modes fluctuates greatly at the beginning of the beam trip, and the relative difference tends to be smooth as time progresses. And the relative difference between the neutron generation time of λ and α modes increases as the subcriticality deepens.For the study of the effective fraction of delayed neutronβeff, the relative difference between the effective fraction of delayed neutron βeff in λ and S modes with time under starting process is shown in Figure 8. The relative difference between the effective fraction of delayed neutron βeff in λ and α modes with time under the beam trip is shown in Figure 9.Figure 8
Relative difference in effective fraction of delayed neutron betweenλ and S mode with time for different subcriticality under the starting process.Figure 9
Relative difference in effective fraction of delayed neutron betweenλ and α mode with time for different subcriticality under the beam trip.From Figure8, the relative difference between the effective fraction of delayed neutron in the λ and S modes increases at the beginning of the starting process with the deepening of the subcriticality and disappears with the progress of time. From Figure 9, it can be seen that the relative difference between the effective fraction of delayed neutron in the λ and α modes increases with the deepening of the subcriticality at the beginning of the beam trip and disappears with the progress of time.To investigate the physical mechanism, for the study of neutron generation time, the sudden production or disappearance of the external neutron source during the starting process and beam trip leads to a sharp fluctuation in the neutron flux density distribution of the whole ADSR in space and time, which makes the related neutron generation time fluctuate greatly. As the system gradually stabilizes, the neutron flux density distribution of the ADSR will stabilize, making the neutron generation time stabilize. As the subcriticality deepens, the role of external neutron sources becomes more and more significant, leading to the space-time heterogeneity of neutron flux density becoming higher, which in turn leads to the increase of the difference between the core few-group constants ofλ mode and S and α modes. This leads to an increase in the relative difference of the neutron generation time between the λ mode and the S and α modes. Since the physical characteristics of other beam transients (recovery after beam trip, etc.) are similar to those of the starting process and beam trip, the above conclusions can be extended to the beam transients from the starting process and beam trip.For the study of the effective fraction of delayed neutron, the physical properties are similar to the neutron generation time at the beginning of the beam transients. With the subcriticality deepens, the role of the external neutron source becomes more and more significant, and the difference in the core few-group constants increases for different modes, which leads to an increase in the relative difference in the effective fraction of delayed neutron for different modes. As the beam transients continues, the effect of subcriticality on the relative differences in the effective fraction of delayed neutron in different modes disappears.In summary, under the beam transients, the relative difference in neutron generation time of different modes increases with the deepening of subcriticality, while the relative difference in effective fraction of delayed neutron of different modes has less effect. Thus, the influence of external source and subcriticality on the kinetic parameters, especially the neutron generation time, needs to be emphasized in the study of the influence mechanism of the neutron space-time kinetics of ADSR under beam transients.


## 4. Conclusions

In this paper, we investigate the influence mechanism of neutron flux density, core few-group constants of different modes, and kinetic parameters of ADSR under the synergistic effect of external neutron source, subcriticality, and the beam transients. The following conclusions can be drawn from the study of this paper.(1)
Under the beam transients, the neutron flux density distribution of ADSR is distorted in both space and time due to the drastic change of the external neutron source with time, which leads to the high heterogeneity of the neutron flux density of ADSR. With the deepening of the subcriticality of ADSR, the external neutron source has an increasing influence on the neutron flux density of ADSR(2)
As a result of (1), it leads to an incremental difference in the core few-group constants of different modes with the deepening of the subcriticality under the beam transients, which in turn leads to an incremental relative difference in the kinetic parameters of different modes, especially the neutron generation time. Therefore, in the study of the influence mechanism of the neutron space-time kinetics of ADSR under beam transients, the multimode core few-group constants should be used for different beam transients, and the influence of external neutron source and subcriticality on the kinetic parameters, especially the neutron generation time, should be emphasizedThis paper provides a theoretical reference for the subsequent research on the influence mechanism of neutron space-time kinetics based on actual ADSR (Venus-II, KUCA, etc.), which is of academic significance to promote the development of ADS engineering.



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*Source: 1775438-2023-02-15.xml*