Introduction of Criticality Code

The Monte Carlo particle transport program JMCT was employed for the computational aspects of this research endeavor [33]. The essential nuclear data utilized in these calculations were procured from the comprehensive nuclear database, ENDF/B-VI. In this paper, the Monte Carlo methodology was applied with a configuration that initiated each generation with 10,000 input neutrons. The simulation encompassed a total of 1,000 generations, with 800 of these generations designated as the active phase of the simulation. The computation of this research was conducted on our personal workstation, equipped with a Intel(R) Core(TM) i7-13700KF CPU and a NVIDIA GeForce RTX 3080 GPU. The computational effort required for a single computational document is estimated to be approximately 40 minutes.

The Monte Carlo approach is distinguished by its ability to provide a realistic portrayal of actual physical phenomena. To a significant extent, this methodology has the potential to supplement, and in some cases, supplant traditional physical experimentation. Consequently, it has emerged as an exceptionally potent instrument for addressing practical challenges within the realm of experimental nuclear physics [34].

However, American National Standards Institute/American Nuclear Society (ANSI/ANS) national standards 8.1 [35] and 8.24 [36] require that nuclear criticality safety analysts determine through validation what value of the k_eff predicted by software can be treated as subcritical, which means the computational method must be performed a validation study to determine its predictive capability for computing k_eff [37]. In the context of the transportation of PWR fuel assemblies, the storage systems are typically characterized by low enrichment and a specific rod-bundle design. In the event of the most extreme scenario involving water ingress, the system can be conceptualized as one where light water serves as the moderator. Consequently, the selection of pitch-type, low-enrichment uranium compound systems is deemed appropriate for establishing a criticality benchmark experiment. The reference [33] presents a computational analysis of the k_eff values for 107 such benchmark experiments. The findings indicate that the calculated k_eff values are in close proximity to their experimental counterparts. This congruence substantiates the assertion that the JMCT program possesses commendable computational precision when addressing problems of this nature.

Computaitonal Model

The computational models presented in this paper are predicated on a AFA3G-type fuel assembly and a specific type of spent fuel transport package. A schematic representation of the AFA3G-type fuel assembly is depicted in Fig. 1. The primary objective of this research is to assess the sensitivity of the effective multiplication factor, denoted as k_eff, to the arrangement of fuel rods. In an effort to conserve computational resources, the detailed structures of the transportation containers have been omitted from the model, with the focus being solely on the representation of a single fuel assembly. The Monte Carlo computational modeling of the fuel assemblies incorporates certain conservative assumptions that are grounded in the actual physical model. A comparative analysis of the principal characteristics between the basic computational model and the real-world model is presented in Table 1 The cross-sectional view of the basic computational model is illustrated in Fig. 3. These conservative assumptions, which are integral to the modeling process, are enumerated as follows:

Fig. 1Full-screen View

Schematic diagram of AFA3G-type fuel assembly

The difficulty of establish the computational model is that the potential for fuel rod buckling following a transportation incident necessitates consideration of variations in pin pitch along the length of the fuel assembly. Drawing inspiration from the methodologies presented in Refs. [28, 31, 32], this paper simplifies the complex deformation of the fuel unit into a manageable single-variable problem. In alignment with these precedents, it is posited that the pin pitch undergoes uniform alteration across the entire active zone, thereby precluding the simulation of fuel tube buckling in the computational model. The outcomes of mechanical testing have elucidated that lattice expansion predominantly occurs in the regions adjacent to the collision [27]. Furthermore, the reactivity is contingent upon the extent of the lattice expansion zone; an elongation of this zone is directly correlated with an increase in reactivity [39]. This analytical approach is inherently conservative, as it accounts for the worst-case scenario without underestimating the potential for reactivity enhancement due to mechanical deformation.

The adoption of conservative manipulations in this research is of paramount importance, as it serves to encompass the uncertainties and potential variations that could potentially impact the safety of the system. By integrating a conservative approach into the study, the robustness of the results is enhanced, ensuring reliability even under the most stringent conditions. This methodological prudence is essential in maintaining the integrity of the system’s safety margins and in providing a buffer against unforeseen circumstances that could arise during the transportation or operation of nuclear facilities.

Research of optimal moderating pin pitch for AFA3G-type fuel assembly

Light water reactor fuel assemblies are meticulously engineered with moderation in mind [12]. This implies that the neutrons released by the fissile material are not sufficiently moderated at their nominal pitch. Increasing the pin pitch results in an enhanced water-uranium ratio, which in turn allows for a greater proportion of neutrons to be fully moderated before engaging with the fuel. Consequently, the reactivity of the fuel assembly escalates. It is important to note that beyond the optimal pitch, a further expansion of the pitch will lead to a decline in reactivity, as a significant portion of the neutrons is absorbed by water.

In this study, an infinite array of fuel rods was modeled to investigate the impact of pitch on reactivity. k_eff was calculated of these systems at various pitches. The objective was to delineate the relationship between fuel rod pitch and reactivity. The pitch was varied from the standard value of 0.0126 m to the average neuron migration length in light water, which is 0.028 m [40]. The analysis revealed that k_eff initially increases with the increase in pin pitch, reaching a peak before declining, as illustrated in Fig. 2. This phenomenon is consistent with our conjecture. The peak of this curve signifies the optimal moderated pitch for AFA3G-type fuel unit, determined to be (0.0154 ± 0.0002) m.

Fig. 2Full-screen View

(Color online) Study of optimal moderate pitch

In the scenario where fuel rods uniformly occupy the entire storage tank, the pitch is determined to be about 0.0134 m, a value that falls short of the optimal moderating pitch. A report by CNPRI suggests that selectively compressing a subset of the fuel rod pitches to expand others can enhance k_eff further, given the condition of the fuel rods uniformly filling the storage tank. The current inquiry pertains to the strategic selection of which section of the fuel rod pitch to expand for maximum reactivity improvement.

Reactivity distribution

Employing a track length estimator facilitates the determination of the reactivity distribution across the radial extent of a fuel assembly [32]. Leveraging the inherent symmetry of fuel assemblies, this study modeled only a quarter of the assembly when examining the reactivity distribution, specifically for a AFA3G-type fuel assembly. The selected portion for modeling, depicted within the red boundary in Fig. 3, encompasses a 9 × 9 rods. The boundaries delineated by the left and lower edges of red box in Fig. 3 exhibit the surfaces of total internal reflection, whereas the remaining boundaries are devoid of any reflective properties.

Fig. 3Full-screen View

(Color online) Illustration of computational model

Let r_i,j denote the reactivity of the fuel rod located in the i-th row and j-th column. For the guide or gauge tube situated in the j-th row, a direct calculation using the Monte Carlo code was not employed. Consequently, the reactivity located at gauge or guide tubes are estimated by taking the average reactivity of the fuel rods that are positioned immediately above and below the guide tube. The reactivity r_i,j for the guide tube can be derived from Eq. (1):(1)The analysis results in a 9×9 matrix where each entry corresponds to the reactivity of the respective fuel rod. From this matrix, one can construct the weighted average reactivity distribution matrix, denoted as R^9×9. The formula for calculating each element of this matrix is presented in Eq. (2).(2)Figure 4 illustrates the reactivity distribution following the weighted averaging of a quarter of the fuel assembly. The lower left square within Fig. 4 denotes the central fuel rod of a fuel subassembly. An observation made from Fig. 4 is that the reactivity of a fuel rod increases as it approaches the center of the assembly. Consequently, it can be inferred that the fuel rods in the central region have a more significant impact on the overall reactivity.

Fig. 4Full-screen View

(Color online) Reactivity distribution in 1/4 AFA3G-type fuel assembly

In summary, the non-uniform distribution of fuel rods, especially the strategic enlargement of the pitch in the central region to approximate the optimal moderating pitch, presents a considerable potential to markedly elevate k_eff. Nonetheless, given the extensive array of possible configurations for fuel rod arrangement, a systematic and methodical approach is imperative for identifying the configuration that yields the highest reactivity.

Uniform Arrangement by Blocks

The initial non-uniform method discussed in this paper is designated as Uniform Arrangement by Blocks. This nomenclature stems from the characteristic that, while the pin pitches may vary across different regions, they remain consistent within a single region. In this approach, the fuel rods completely occupy the storage tank, and an expansion of the pitch in the central area curtails a corresponding adjustment in the pitch of the peripheral regions. The core principle of this method involves segmenting a fuel subassembly into two distinct regions by rods: the central region and the surrounding region.

The delineation of the central region is contingent upon the number of fuel rods, ensuring an equal count in each row and column. The count of fuel rods in any given row or column of central region is an odd number, ranging from 3 to 15. It is imperative to recognize that the central region maintains a square cross-section, with its Fermat point aligning with that of the storage tank’s cross-section. The division of the central region into distinct configurations is variable, offering seven distinct options that scale from a 3×3 to a 15×15 arrangement of fuel rods. These various division methods are graphically represented in Fig. 5.

Fig. 5Full-screen View

(Color online) Seven ways to divide a fuel assembly

Among the various methods of division, the pitch within the central region is incrementally increased from the nominal value until it reaches the location where the surrounding fuel rods are at a contact pitch. The mathematical relationship between the pitch of the short side of the surrounding fuel rods and the pitch in the central region can be derived and is presented in Eq. (3).(3)In this context, D is the nominal pin pitch, p_s denotes the length of the short side of the pitch surrounding the fuel rods, while p_c signifies the pitch of the fuel rods in the central area. Additionallly, n_c is used to represent the number of pitches between each row or column within the central region. This configuration is visually depicted in Fig. 6, which serves to illustrate the geometric relationships and parameters involved in the Eq. (3).

Fig. 6Full-screen View

Illustration of Eq. 3, where n_s=n_c=4

The input files for these calculations incorporate a parameter known as the expansion coefficient, denoted by ε, which is utilized to quantify the expand degree of the central region. Within a given division method, a lower value of ε corresponds to a smaller pitch in the central area, whereas a higher value of ε signifies a more expansive central region. ε is defined by the ratio of the cross-sectional surface of the central region to that of the entire storage tank. The mathematical formulation for ε is detailed in Eq. (4).(4)In the given formulation, S_c symbolizes the cross-sectional surface of the central region, and S_n denotes the cross-sectional surface of the storage cavity. The terms n_c, p_c and D used here retain the same meanings as previously defined in the text.

Figure 7 demonstrates the trends of k_eff changing with the ε for each division approach. The peak value of k_eff is observed when the central region is configured as a 13×13 grid, with the fuel rods in this region being fully expanded. Under these conditions, the pitch in the central region is measured at 0.01475 meters, and k_eff reaches a value of 0.99793. The specific arrangement is depicted in Fig. 17(a), while the positions of each fuel rod along the central axis are detailed in Table 4.

Fig. 7Full-screen View

(Color online) Result of uniform arrangement by blocks

Layer-by-Layer Determination

The method discussed in this section is a non-uniform arrangement technique introduced in Ref. [31]. It offers a more nuanced partitioning of the fuel subassembly compared to the Uniform Arrangement by Blocks. While the latter approach simply divides the fuel element into two distinct region by rods, the method under consideration further subdivides the fuel assembly into nine distinct zones by layer, as illustrated in Fig. 8. However, it is anticipated that this more complex partitioning may result in an arrangement with higher reactivity when compared to the Uniform Arrangement by Blocks.

Fig. 8Full-screen View

(Color online) Division a fuel assembly by layers

Figure 9 provides a visual representation of this method, with a detailed explanation presented subsequently:

Fig. 9Full-screen View

(Color online) Process of determination layer-by-layer

The positions of the fuel rods were determined through the systematic procedures previously outlined. Figure 10(a) illustrates the variation of k_eff with the uniform increase in pitch of all fuel rods, which was instrumental in establishing the locations of the outermost layer of rods. Figure 10(b) depicts the change in k_eff with the pitch increase of the remaining 225 rods, thereby determining the positions of the 8-th layer of rods. The pitch between the 8-th and 9-th layer fuel rods, identified as the contact pitch, is 0.0095 meters, as derived from Fig. 10(b). Figure 10(c) shows the k_eff variation with the pitch increase of the remaining 169 rods, which was used to ascertain the positions of the 7-th layer of rods. The pitch from the 7-th to the 8-th layer fuel rods is 0.0104 meters, as extracted from Fig. 10(c). Additionally, Fig. 10(d) indicates a pitch of 0.0126 meters between the 6-th and 7-th layer fuel rods. In Fig. 10(c) and (d), noticeable fluctuations in k_eff values are observed, attributed to the inherent statistical variability of the Monte Carlo calculation method. These fluctuations suggest that as the arrangement of fuel rods approaches the configuration that maximizes k_eff, the impact of the rod arrangement on k_eff diminishes, the statistical fluctuations inherent in the program itself are more pronounced. Subsequently, the pitch of the remaining five inner layers of fuel rods was uniformly increased. It was observed that k_eff decreases with increasing pitch, as shown in Fig. 11. Consequently, a uniform pitch of 0.015 meters was adopted between the remaining layers of fuel rods.

Fig. 10Full-screen View

(Color online) Determining rods’ location layer by layer. (a) Determining 9-th layer fuel rods. (b) Determining 8-th layer fuel rods. (c) Determining 7-th layer fuel rods. (d) Determining 6-th layer fuel rods

Fig. 11Full-screen View

(Color online) Effect of uniform variation of 5 layers of rods on k_eff

The upper limit of k_eff attainable through this variation method is 0.99827. This optimal arrangement is depicted in Fig. 17(b), with the specific location of each fuel rod along the central axis detailed in Table 4. The maximum value of k_eff achieved by employing this method surpasses that obtained by the Uniform Arrangement by Blocks technique.

Birdcage Deformation

The mechanical tests referenced in [23, 24] indicate that the most adverse scenario for a fuel assembly is when it experiences a vertical drop. In such an event, the deformation of the lower nozzle has the potential to cause significant damage at the impact end of the fuel rods [41]. The tests demonstrate that for PWR fuel assemblies, a specific type of deformation known as ‘Birdcage Deformation’ typically occurs in the region between the lower nozzle and the first interim spacer. This phenomenon is illustrated in Fig. 12.

Fig. 12Full-screen View

Illustration of Birdcage Deformation

The methodology presented in paper [32] delineates a three-step process for ascertaining the positions of fuel rods, as depicted in Fig. 13. Initially, the location of the outermost layer of fuel rods is established. Subsequently, the positions of the fuel rods along the axis of symmetry are determined. Lastly, the locations of the remaining fuel rods are ascertained. It is presumed that the outer rows of pins are uniformly aligned against the basket wall. The positions of the remaining fuel rods are derived through the application of natural boundary condition cubic spline interpolation. This paper builds upon this concept, but introduces revisions to the mathematical model used for determining the positions of the fuel rods along the symmetric axis.

Fig. 13Full-screen View

Steps of Birdcage Deformation to determine fuel rods’ location

The present study introduces revisions to the existing mathematical model, which was found to be inadequate in accurately capturing the true nature of Birdcage Deformation. Drawing from the design specifications for PWR fuel subassemblies, the lower ends of the fuel rods are not affixed to the lower nozzle, implying that they are free to move relative to one another. This is in contrast to the older method, which assumed that the fuel rods were immobile in relation to each other. Consequently, the revised mathematical model offers a more realistic representation of the actual conditions encountered in PWR fuel assemblies. The following sections will elaborate on this novel approach in detail.

Figure 14 provides a visual representation of the revised mathematical model. Within Fig. 14, the dotted arc on the left represents the lower nozzle, and the vertical dotted line on the right represents the first fuel element interim spacer. represents the centermost fuel rod, and to represent the remaining 8 fuel rods on the half of a symmetric axis. d is contact pitch, while D represents nominal pitch. Assuming when the Birdcage Deformation occurs, the following conditions are considered:

Fig. 14Full-screen View

Mathematical model of new Birdcage Deformation

These assumptions ensure that the model accurately reflects the Birdcage Deformation characteristics of the fuel rods under the vertical drop accident.

Within the framework of this mathematical model, the deformation of the lower nozzle is represented as a circular arc when projected onto the xoy plane in Fig. 14. Let Ai denote the lower end of i-th fuel rod. The horizontal coordinate of Ai in Fig. 14 is represented by . The value of can be mathematically expressed in Eq. (5):(5)where r signifies the radius of curvature of the deformed lower nozzle, and d denotes the diameter of the fuel rod. Utilizing these parameters, the horizontal coordinate for the lower end of the i-th fuel rod, where i ranges from 1 to 8, can be determined. The coordinates of the lower end of these fuel rods, specifically A₁ through A₈, are derived and expressed in the form .

Let L be the nominal distance between the lower nozzle and the first interim spacer. The term θi signifies the angular position of the center of the circular arc that corresponds to the deformation of the i-th fuel rod. This angle is a key parameter in the geometric description of the rod’s configuration following the deformation process. Finally, the variable Ri is designated to represent the radius of curvature of i-th fuel rod. According to the geometric relationship, the interrelation among the above parameters, including L, θi, Ri and , can be mathematically expressed by Eq. (6):(6)where D denotes the nominal pin pitch, defining the standard distance between the centers of adjacent fuel rods. The symbol d signifies the contact pin pitch, representing rods are in contact due to deformation. The variable i iterates over the set {1, 2, 3, …, 8}, corresponding to each individual fuel rods being considered in the model. Upon organizing Eq. (6), a more streamlined formulation, Eq. (7), can be derived. This reorganization elucidates the interrelationships among the parameters, potentially enhancing the ease of further analysis and computational applications.(7)Then, introducing the variable ti, expressed as . Additionally, defining Mi, expressed as . Utilizing these definitions, Eq. (7) can be succinctly re-expressed in a simpler form as presented in Eq. (8).(8)Given that , D, and d are significantly smaller than L, the value of Mi is approximately equal to 1, albeit slightly less. Considering the growth rates of the functions f(x)=sin(x) and f(x)=x, it can be deduced that Eq. (8) possesses a unique root within its domain of definition, which is marginally greater than zero. By solving Eq. (8), the value of ti can be determined, and subsequently, Ri can be calculated.

The coordinate of point Bi are specified as (L, iD). Accordingly, the midpoint of the line segment is located at the coordinates . Given these coordinates, the slope ki of line containing the segment is . Employing this slope, the equation of the perpendicular bisector of the line segment is formulated and provided in Eq. (9). The letters within the formula maintain the same significance as previously defined in the discussion.(9)Let Oi denote the center of the circular arc , with coordinates . Oi lies on the perpendicular bisector of , and considering that the distance from Oi to Bi is equal to the radius of curvature Ri, the coordinates can be determined by solving the system of equations presented in eq. 10. The variables used within this formula adhere to the same definitions previously established in our discussion.(10)Equipped with the determined value of Ri and the precise coordinates of Oi, the geometric representation of the circular arc is articulated in Eq. (11). This equation encapsulates the mathematical description of the arc, grounded in the established parameters and consistent with the previously defined variables.(11)Consequently, the maximum radial distance of each rod y_i,max from the centermost rod is captured by the formulation presented in Eq. (12). This equation provides a quantitative measure of the spatial distribution of the rods relative to the central axis of the fuel assembly.(12)With the mathematical model now established, the subsequent task is to delineate the range of variation for r and to establish the constraint that arise from the size of fuel assembly and storage tank. Upon examination of the parameters specific to the AFA3G-type fuel assembly, the values for d, D, and L are identified as 0.0095 meters, 0.126 meters, and 0.7015 meters, respectively. This paper proceeds to determine the range of variation for r and the associated constraint conditions as detailed in the following sections.

a. Determination of the range for the variable r

In the context of this mathematical model, a smaller value of r signifies a more severe deformation of the lower nozzle. However, practical considerations dictate that the nozzle will not undergo a curling deformation under any plausible accidental conditions. Therefore, the minimum allowable radius of curvature, denoted as r_min, is constrained by the physical limitations of the nozzle and geometry. This minimum value is established through the formulation presented in Eq. (13):(13)where X represents the side length of the storage cavity, and X is 0.2245 m. As the value of r increases, the deformation of the lower nozzle becomes progressively milder. However, this mathematical model is predicated on the assumption that the lower ends of the fuel rod remain in contact with one another. To ensure that the fuel rods within the model do not separate or become disengaged, the maximum value of r must be such that the fuel rod represented by remains unbent. Under this condition, the maximum radius of curvature, denoted as r_max, is determined and can be articulated by Eq. (14):(14)The determination of r_max is also governed by the condition that the location corresponding to the maximum pitch must fall within the span between the lower nozzle and the first interim spacer. Consequently, any value of , exceeding L is not representative of the scenario modeled by this mathematical framework. To explore this boundary condition, a Python program was coded as port of this research to calculate the value of r when equals L, yielding r=4.94 m.

Taking into account both the minimum and maximum conditions described above, the radius of curvature r in this mathematical model is constrained to vary within the range of 0.07 meters to 4.94 meters.

b. Determination the constraint condition

Constraint conditions are essential as they define the permissible bounds within which the model operates, ensuring that the mathematical representation aligns with physical realities and practical limitations. These conditions are derived from a comprehensive consideration of the system’s geometry and operational parameters.

Given that the fuel assembly must remain within the confines of the storage cavity under all conditions, including potential accident scenarios, each fuel rod is subject to its own boundaries of motion. These boundaries are crucial for establishing the constraints within the mathematical model and are detailed in Table 2, which outlines the permissible limits of movement for the fuel rods.

Utilizing former Python program, the positions of the fuel rods were calculated for a range of radii r extending from 0.07 meters to 4.94 meters. The corresponding maximum y-values, denotes as y_max, are presented in Table 3. Analysis of the data in Table 3 reveals that when r exceeds 1.90 meters, the rods are no longer able to fully occupy the storage tank. Consequently, in such instances, k_eff is anticipated to be lower than that in scenarios where the fuel rods can full fill the storage cavity.

The computational analysis has identified the maximum value of k_eff as occurring at r=0.81 m, with k_eff reaching 0.9965, as shown in Fig. 15. This condition is characterized by a specific arrangement of the fuel rods, which is illustrated in Fig. 17(c). Furthermore, the configuration of the rods along the symmetric axis for this scenario is detailed in Table 4, providing a comprehensive view of the fuel assembly’s geometry at the point of peak reactivity (Fig. 15).

Fig. 15Full-screen View

(Color online) Effect of r on k_eff

This paper calculated the older Birdcage Deformation model proposed in Ref. [32] as well, and obtained the curve of k_eff versus r, see Fig. 16. The maximum value of k_eff occurs at r=0.18 m, which is 0.99261. In this case, the fuel rod arrangement is shown in Fig. 17(d), and the arrangement of fuel rods on the symmetric axis is shown in Table 4. The upper limit of the old mathematical model is smaller than that of the new birdcage mathematical model. In addition to the development of the new mathematical model, this paper also recalculated the older Birdcage Deformation model as initially proposed in Ref. [32]. The results of this calculation are presented in the form of a curve depicting the relationship between the k_eff and the radius of curvature r, as shown in Fig. 16. The maximum value of k_eff within the context of the old model was found to occur at r=0.18 m, with k_eff peaking at 0.99261. The corresponding arrangement of the fuel rods for this scenario is illustrated in Fig. 17(d), and the detailed layout of the fuel rods along the symmetric axis is provided in Table 4.

Fig. 16Full-screen View

(Color online) Effect of 1/r on k_eff

Fig. 17Full-screen View

(Color online) Arrangement when k_eff,max,r of four methods. (a) Uniform Arrangement by Blocks. (b) Layer-by-Layer Determination. (c) Old Birdcage Deformation. (d) New Birdcage Deformation

Upon comparison, it is observed that the upper limit of k_eff achieved by the old mathematical model is lower than that of the newly developed birdcage mathematical model. In conclusion, the New Birdcage Deformation demonstrates superior conservative capabilities and aligns more closely with realistic deformation scenarios compared to the model previous proposed in Ref. [10]. It is recommended that New Birdcage Deformation be considered over the Old Birdcage Deformation during criticality safety analysis. The introduction of New Birdcage Deformation is a significant advancement in ensuring the safety of nuclear fuel transportation.

Conclusion of non-uniform arrangement method

Table 4 delineates the positions of the fuel rods along the half of the central axis, offering insight into their spatial arrangement. Figure 17 visually represents the configuration of fuel rods at the point where k_eff attains its maximum value for each method. Upon examination of Table 5, it is observed that for the Uniform Arrangement by Blocks, Layer-by-Layer Determination, and the New Birdcage Deformation methods, the 7-th, 8-th, and 9-th fuel rods are nearly in contact. Concurrently, the central pitches in these arrangements approximate the optimal moderating pitch, which is conducive to enhancing reactivity. Comparatively, the k_eff values achieved in these three methods surpass those obtained through the Old Birdcage Deformation approach. This observation underscores the efficacy of the revised models for higher reactivity, which is a critical consideration in the safety analysis of nuclear fuel assemblies.

The Uniform Arrangement by Blocks and the Layer-by-Layer Determination methods are capable of achieving higher values of k_eff, primarily because their underlying premise is based on an exhaustive enumeration approach. This method is characterized by its thoroughness but may not align with the real-world phenomena. Between these, the Layer-by-Layer Determination method is noted for its more conservative approach. In contrast, two Birdcage Deformation methods take a different approach by grounding its analysis in actual observed phenomena. Besides, Sect. 3.C of this study provides compelling evidence that the New Birdcage Deformation model merits consideration when examining the worst-case configurations of fuel rods. The analysis now narrows down to a comparative evaluation between the Layer-by-Layer Determination and the New Birdcage Deformation methods (Table 5).

k_eff,max,r derived from the Layer-by-Layer Determination method, is distinctly superior to that of the new Birdcage deformation scenario. Nonetheless, in the context of the New Birdcage Deformation, it is imperative to account for the potential to augment k_eff,max,r through subtle variation in the positioning of each fuel rods. As articulated in Ref. [32], these incremental adjustments can lead to a further enhancement of k_eff,max,r. Incorporating the effects of these minor rod position perturbations, the revised maximum k_eff,max,r for the New Birdcage Deformation is refined according to the subsequent formula Eq. (15):(15)where k_eff,max,r is k_eff,max,r after revising by small-scale pitch shift, E_s is the enhancement amplitude of small-scale pin pitch shift. E_s is taken as 0.109% according to Ref. [32]. It is unnecessary to consider the possibility that small-scale pin pitch shift can continue to rise k_max obtained by Layer-by-Layer Determination. This is because this method is a kind of enumeration method, which is impossible to occur in an actual accident and it is conservative enough.