Preamble

Sensitivity Analysis of Rod Rearrangement in Criticality Safety for PWR Fuel Assemblies Under Transportation Accidents

Xin-Ling Dai,¹ De-Chang Cai,²,∗ Yan-Min Zhang,¹ and Jin Cai²
¹Institute of Nuclear and New Energy Technology, Tsinghua University, Beijing 100084, China
²China Nuclear Power Technology Research Institute Co Ltd, Shenzhen 518000, China

To ensure the safe transportation of radioactive materials, numerous countries have established specific standards. For the transfer of fissile materials, it is imperative that the material within the packaging remains in a subcritical state during routine, normal, and accidental transport conditions. In the event of an accident, the rods within the storage tank may become rearranged, introducing uncertainty that must be accounted for to ensure that criticality analysis results are conservative. Historically, this uncertainty was addressed overly conservatively due to limited research on non-uniform arrangement scenarios, which proved unsuitable for criticality safety analysis of spent fuel packages. This paper introduces three distinct methods to non-uniformly rearrange fuel rods—Uniform Arrangement by Blocks, Layer-by-Layer Determination, and Birdcage Deformation—and meticulously evaluates the influences of rod rearrangement on the effective multiplication factor of neutrons, keff, utilizing the Monte Carlo method. Ultimately, this study presents a holistic method capable of encompassing the entire spectrum of potential effects stemming from the rearrangement of fuel rods during rod mispositioning accidents. By augmenting the safety margin, this approach proves to be adeptly suited for the criticality safety analysis of nuclear fuel transport containers.

Keywords: criticality safety analysis, fuel transports, rods mispositioning accident, non-uniform arrangement

Introduction

Nuclear energy, recognized for its cleanliness, low-carbon footprint, safety, and efficiency, stands as a fundamental baseload energy source that is instrumental in combating global climate change [1]. In the wake of the Fukushima nuclear disaster, the international community has been compelled to adopt new safety standards that are more rigorous and exacting. Among these standards, ensuring the secure transportation of radioactive materials has emerged as a paramount concern, essential for the protection of public health and safety [2].

In response to this imperative, the International Atomic Energy Agency (IAEA), in conjunction with a multitude of nations, has crafted detailed laws and regulations specifically designed to govern the transportation of radioactive material packages [3, 4]. The swift advancement of China's nuclear technology sector necessitates the transportation of substantial volumes of radioactive materials [5]. Central to this process is the transport container, which serves as the pivotal apparatus for the secure conveyance of nuclear fuel assemblies from the component fabrication facility to the reactor site. It is imperative that these containers maintain the structural integrity of their contents [6–8], while also ensuring that the criticality safety and shielding effectiveness of the packages adhere to the stipulated regulatory standards throughout the transportation phase [9].

The regulations stipulated in reference [4] dictate that transport containers for radioactive materials must be designed with the capacity to securely retain their contents even in the event of a transportation accident. Prior to their deployment, transport packages are required to fulfill the criteria outlined in preceding legislation [10]. For instance, both the type RY-I spent fuel transport packages [11] and the type FCo70-YQ medical radioactive source transport packages [12] underwent a series of rigorous tests before being approved for service. Notably, as specified in reference [4], for packages intended to carry fissile materials, an additional criticality safety analysis is mandated [13]. This analysis is of paramount importance, as it verifies that the fissile material remains subcritical under a range of conditions, including conventional, normal, and accidental scenarios [4]. The criticality analysis must also encompass abnormal scenarios that could potentially impact the system's effectiveness, such as the ingress of water into the container or the failure of neutron absorber elements during an accident. These atypical conditions could lead to an increase in the effective multiplication factor, keff, of the evaluated system [14–17]. It is imperative that the criticality analysis provides an assessment of the uncertainty associated with the calculation results, enabling a robust evaluation of the system's safety [18]. This requirement is further underscored by the guidelines on nuclear criticality safety for fissile materials outside of reactors, as detailed in reference [19]. To ascertain this uncertainty, it is essential to take into account the potential deformation of the fuel elements under accidental conditions, as these factors can significantly influence the overall safety assessment.

The safety analysis of nuclear fuel assembly transport containers necessitates a dual approach encompassing both mechanical and criticality safety analyses [20]. A variety of methodologies are employed for mechanical analysis, including isotropic modeling, scaled modeling, and finite element analysis [21, 22]. Studies presented in references [23, 24] conducted a series of drop tests on the RA-3D model packages to simulate the cumulative effects of the most severe drop accident scenarios on the fuel elements. The findings from these studies indicate that compliant transport packagings offer substantial protection to the fuel assembly contained within. Despite significant deformation on the exterior of the impacted transport packaging, the fuel assembly remains structurally sound. While fuel rods may bend and interim spacers may deform, the integrity of the fuel rod cladding is preserved, preventing any leakage of fissile material from the interior. Beyond the RA-3D packages, transport containers for fuel assemblies are invariably designed with cushioning mechanisms for added protection. For instance, the Tianwan nuclear fuel assembly transport containers incorporate wire rope vibration isolators, which significantly dampen the vibrations experienced during transport [25]. Consequently, under accident conditions, the focus should shift from the potential breakage of fuel cladding to the alteration in the pitch between fuel rods. This alteration, also referred to as lattice expansion, represents a significant factor in the fuel relocation effect, which has implications for criticality safety [26].

This paper presents a simplified approach to the fuel deformation issue by reducing it to a single-variable problem, specifically focusing on the sensitivity of critical safety analysis to variations in rod pitch. Consequently, the paper posits that the pitch of fuel rods varies across the active zone. As demonstrated in reference [27], even under the most severe drop conditions, the area of fuel rod buckling is localized near the point of impact and does not extend throughout the entire active region of the fuel assembly. This implies that only a segment of the fuel rods is likely to exhibit deformation that could potentially increase reactivity. Thus, the strategy of uniformly adjusting the fuel rod pitch across the entire active zone is considered a conservative approach.

As highlighted in reference [13], it is imperative to address abnormal conditions, such as water ingress into the package and the failure of interim spacers, with particular scrutiny. X. Wang et al. conducted a comprehensive investigation into the impact of uniform variations in fuel rod pitch within the AP1000 fresh fuel transportation vessel, as well as the effects of water ingress on criticality safety, utilizing the Monte Carlo N-Particle (MCNP) code [28]. Initially, they examined the uniform expansion of fuel rod pitch within the storage cavity of the transport container. Their findings revealed that keff reaches its peak when the fuel rods uniformly occupy the storage cavity. Subsequently, they explored the influence of water ingress with varying densities into the cargo package on criticality safety. The study's outcomes indicated that criticality safety is most compromised when the fuel assembly within the cargo package is entirely submerged in water with a density of 1 × 10³ kg/m³.

Historically, the China Nuclear Power Technology Research Institute Co., Ltd. (CNPRI) has incorporated an additional uncertainty of 4,000 pcm into the criticality safety analysis for fresh fuel transportation containers, in addition to the conditions where fuel assemblies are submerged and the fuel rods uniformly occupy the storage cavities. As demonstrated in paper [29], spent fuel transportation casks are required to accommodate dozens of fuel assemblies, and the existing safety margin is inherently limited [30]. Consequently, the methodology previously employed by CNPRI is deemed unsuitable for the criticality safety analysis of spent fuel transport packages, given the heightened requirements and complexities associated with the transportation of spent fuel.

However, accurately quantifying this uncertainty presents a challenge due to the existing gaps in prior research concerning the arrangement of fuel rods. Moreover, the majority of these studies have been predicated on the assumption of a uniform arrangement of fuel rods. To date, the non-uniform arrangement of fuel rods has only been examined in a limited number of publications, specifically: (a) S. Whittingham et al. determined the non-uniform arrangement of fuel rods when keff takes the maximum value by expanding the fuel rods layer by layer [31]; (b) M. Asami et al. developed a mathematical model based on the possible birdcage deformation of the fuel rods during a drop accident. The non-uniform arrangement of fuel rods is regulated by parameterizing the radius of curvature during lower nozzle deformation [32].

This paper undertakes a comprehensive investigation of the non-uniform arrangement of fuel rods within the storage cavity, with the objective of ascertaining the sensitivity of criticality safety to variations in pin pitch. The study will explore the impact of fuel rod arrangement on keff through three distinct methods of variation. The first method, termed Uniform Arrangement by Blocks, is predicated on the division of all fuel rods into two distinct regions. The second method, known as Layer-by-Layer Determination, involves segmenting the fuel rods into nine separate regions. The third method, referred to as Birdcage Deformation, draws inspiration from the deformation patterns observed in pressurized water reactor (PWR) fuel assemblies that may result from a vertical drop impact, resembling the structural distortion of a birdcage. In addition to examining these methods, this paper will conduct a comparative analysis to identify the arrangement that yields the highest value of keff. The exploration of non-uniform arrangements is crucial, as it can encapsulate a range of fuel rod mispositioning incidents that may occur during transportation accidents. This comprehensive approach is essential for determining the precise level of uncertainty in criticality safety analysis.

Section II of this paper will introduce the nuclear code and the computational model employed in the study. Section III will delve into determination of the optimal moderating pitch and the reactivity distribution, elucidating the reasons why a non-uniform arrangement of fuel rods can potentially increase keff more significantly than a uniform arrangement. Section IV will proceed to examine the peak values of keff attainable through the application of non-uniform arrangement methods. It will also present a comparative analysis of the results yielded by these methods, aiming to identify the most effective arrangement in terms of maximizing reactivity. Section V will extend the analytical methodology developed herein to a specific type of fuel assembly transport container, conducting a comparative analysis with the prevailing methods currently in use. Finally, this paper will culminate in a comprehensive summary of the findings presented throughout the paper.

II. Criticality Code and Computational Model

A. 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 an Intel(R) Core(TM) i7-13700KF CPU and an 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 keff predicted by software can be treated as subcritical, which means the computational method must undergo a validation study to determine its predictive capability for computing keff [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. Reference [33] presents a computational analysis of the keff values for 107 such benchmark experiments. The findings indicate that the calculated keff 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.

B. Computational Model

The computational models presented in this paper are predicated on an 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 [FIGURE:1]. The primary objective of this research is to assess the sensitivity of the effective multiplication factor, denoted as keff, 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 [TABLE:1]. The cross-sectional view of the basic computational model is illustrated in Fig. 3 [FIGURE:3].

These conservative assumptions, which are integral to the modeling process, are enumerated as follows:

a. The enrichment level of the computational model is set at 5 weight percent (wt%), which exceeds that of the fresh AFA3G-type fuel unit. This discrepancy is a deliberate choice to ensure a more conservative estimate of the fuel's reactivity.

b. This paper postulates a scenario where the fuel assembly is entirely submerged in water, which represents the most extreme condition conceivable. Under the auspices of conventional or standard transport protocols, this type of nuclear fuel assemblies are typically conveyed in a dry state. Nonetheless, in accordance with the stipulations of standard [5], a criticality safety analysis must encompass a comprehensive evaluation of potential unforeseen incidents that could jeopardize criticality safety, with particular emphasis on the possibility of water ingress. Such a condition could notably elevate keff, due to the moderating effect of water on neutrons, which could potentially lead to a higher probability of a sustained nuclear chain reaction [38].

c. This study assumes that the diameter of the gauge or guide tubes is equivalent to that of the fuel rods to facilitate the modeling of non-uniform arrangements of fuel rods. To ensure the conservatism of this assumption, the research conducted a sensitivity analysis on the impact of the gauge/guide tube diameter on keff. The findings indicate that this assumption leads to an increase in keff of 113 pcm, which confirms that the assumption is conservative, i.e., meaning that it errs on the side of caution by potentially overestimating the reactivity.

The difficulty of establishing 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 references [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 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.

III. Preliminary Work

The primary objective of the research presented in this section is to ascertain the optimal strategy for the rearrangement of fuel rods.

A. 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. keff was calculated for 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 meters to the average neutron migration length in light water, which is 0.028 meters [40]. The analysis revealed that keff initially increases with the increase in pin pitch, reaching a peak before declining, as illustrated in Fig. 2 [FIGURE: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 meters.

In the scenario where fuel rods uniformly occupy the entire storage tank, the pitch is determined to be about 0.0134 meters, 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 keff 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.

B. 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 an AFA3G-type fuel assembly. The selected portion for modeling, depicted within the red boundary in Fig. 3, encompasses a 9 × 9 array of rods. The boundaries delineated by the left and lower edges of the red box in Fig. 3 exhibit the surfaces of total internal reflection, whereas the remaining boundaries are devoid of any reflective properties.

Let ri,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 ri,j for the guide tube can be derived from Eq. 1:

$$
r_{i,j} = \frac{r_{i-1,j} + r_{i+1,j} + r_{i,j-1} + r_{i,j+1}}{4}, \quad i, j > 1
$$

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 R9×9. The formula for calculating each element of this matrix is presented in Eq. 2:

$$
\hat{r}{i,j} = \frac{\sum}^{9} r_{i,j}}{\sum_{i=1}^{9}\sum_{j=1}^{9} r_{i,j}
$$

Fig. 4 [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.

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 keff. 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.

IV. Non-Uniform Arrangement Methodology

As established in Sec. III.B, the reactivity within a fuel unit is distributed in an inhomogeneous manner. In this section, this paper explores the rearrangement of fuel rods using three distinct non-uniform configuration strategies, each aimed at maximizing keff. The first strategy is termed Uniform Arrangement by Blocks, the second is known as Layer-by-Layer Determination, and the third is referred to as Birdcage Deformation. The input files for the various computational examples in this section are all generated using a custom Python script.

A. 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 necessitates 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 the 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 [FIGURE:5].

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:

$$
p_s = \frac{8D - n_c p_c}{8 - n_c}
$$

In this context, D is the nominal pin pitch, ps denotes the length of the short side of the pitch surrounding the fuel rods, while pc signifies the pitch of the fuel rods in the central area. Additionally, nc 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 [FIGURE:6], which serves to illustrate the geometric relationships and parameters involved in Eq. 3.

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 Equation 4:

$$
\varepsilon = \frac{S_c}{S_n} = \frac{(n_c p_c)^2}{(8D)^2}
$$

In the given formulation, Sc symbolizes the cross-sectional surface of the central region, and Sn denotes the cross-sectional surface of the storage cavity. The terms nc, pc and D used here retain the same meanings as previously defined in the text.

Fig. 7 [FIGURE:7] demonstrates the trends of keff changing with the ε for each division approach. The peak value of keff 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 keff reaches a value of 0.99793. The specific arrangement is depicted in Fig. 17(a) [FIGURE:17], while the positions of each fuel rod along the central axis are detailed in Table 4 [TABLE:4].

B. Layer-by-Layer Determination

The method discussed in this section is a non-uniform arrangement technique introduced in reference [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 regions by rods, the method under consideration further subdivides the fuel assembly into nine distinct zones by layer, as illustrated in Fig. 8 [FIGURE: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. 9 [FIGURE:9] provides a visual representation of this method, with a detailed explanation presented subsequently:

a. Initially, uniformly increase the pitch of all fuel rods until the 9-th layer reaches contact with the inner surface of the storage tank. By plotting the curve of keff against pitch, identify the configuration of fuel rods that corresponds to the maximum keff. Retain the position of the 9-th layer as determined.

b. Subsequently, uniformly increase the pitch for the remaining layers of fuel rods, until the 8-th layer's rods are in contact with those of the 9-th. Plot the curve of keff against pitch to ascertain the arrangement of fuel rods that yields the maximum keff. Maintain the configuration of the 8-th layer as such.

c. Progressively reduce the number of fuel rods subject to variation and iteratively apply the process from step b. Determine the positions of fuel rods from the outer layer to the inner layer, continuing this process until the changes in arrangement no longer result in an increase in keff.

The positions of the fuel rods were determined through the systematic procedures previously outlined. Fig. 10 FIGURE:10 illustrates the variation of keff with the uniform increase in pitch of all fuel rods, which was instrumental in establishing the locations of the outermost layer of rods. Fig. 10(b) depicts the change in keff 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). Fig. 10(c) shows the keff 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 keff 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 keff, the impact of the rod arrangement on keff 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 keff decreases with increasing pitch, as shown in Fig. 11 [FIGURE:11]. Consequently, a uniform pitch of 0.015 meters was adopted between the remaining layers of fuel rods.

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

C. 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 [FIGURE:12]. The methodology presented in paper [32] delineates a three-step process for ascertaining the positions of fuel rods, as depicted in Fig. 13 [FIGURE: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.

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.

Fig. 14 [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. A0B0 represents the centermost fuel rod, and A1B1 to A8B8 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:

a. The lower nozzle undergoes deformation such that it becomes part of a spherical surface with a radius of curvature r.

b. The impacted ends of the fuel rods come into contact with one another, while the fuel rods at the first interim spacer maintain their nominal pitch.

c. The central fuel rod is assumed to remain straight, while the peripheral fuel rods are modeled to exhibit curvature. The arc length of the bends in these fuel rods is equivalent to the nominal distance between the lower nozzle and the first interim spacer.

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 xAi. The value of xAi can be mathematically expressed in Eq. 5:

$$
x_{A_i} = r - \sqrt{r^2 - (id)^2}
$$

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 xAi 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 (xAi, id).

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 xAi, can be mathematically expressed by eq. 6:

$$
\begin{cases}
\sin \theta_i = \frac{L - x_{A_i}}{R_i} \
\tan \theta_i = \frac{i(D - d)}{L - x_{A_i}}
\end{cases}
$$

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 Equation 6, a more streamlined formulation, Equation 7, can be derived. This reorganization elucidates the interrelationships among the parameters, potentially enhancing the ease of further analysis and computational applications.

$$
\sin \theta_i = \theta_i \sqrt{(1 - x_{A_i}/L)^2 + i^2(D/L - d/L)^2}
$$

Then, introducing the variable ti, expressed as ti = θi. Additionally, defining Mi, expressed as Mi = √((1 - xAi/L)² + i²(D/L - d/L)²). Utilizing these definitions, Eq. 7 can be succinctly re-expressed in a simpler form as presented in eq. 8:

$$
\sin t_i = M_i t_i, \quad t_i \in (0, +\infty)
$$

Let Oi denote the center of the circular arc AiBi, with coordinates (xOi, yOi). Oi lies on the perpendicular bisector of AiBi, and considering that the distance from Oi to Bi is equal to the radius of curvature Ri, the coordinates (xOi, yOi) can be determined by solving the system of equations presented in eq. 10:

$$
\begin{cases}
(x_{O_i} - L)^2 + (y_{O_i} - iD)^2 = R_i^2 \
x_{O_i} + \frac{i(D + d)}{L - x_{A_i}} y_{O_i} = \frac{L^2 - x_{A_i}^2}{2(L - x_{A_i})} + \frac{i^2(D^2 - d^2)}{2(L - x_{A_i})}
\end{cases}
$$

The variables used within this formula adhere to the same definitions previously established in our discussion. Equipped with the determined value of Ri and the precise coordinates of Oi, the geometric representation of the circular arc AiBi 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.

$$
y = y_{O_i} + \sqrt{R_i^2 - (x - x_{O_i})^2}, \quad x \in [x_{A_i}, L]
$$

Consequently, the maximum radial distance of each rod yi,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.

$$
y_{i,\max} = y_{O_i} + R_i
$$

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

Given that xAi, 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.

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 rmin, is constrained by the physical limitations of the nozzle and geometry. This minimum value is established through the formulation presented in eq 13:

$$
r_{\min} = \frac{8^2(D - d)^2 + L^2}{2L} \approx 0.0685 \text{ m}
$$

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 A8B8 remains unbent. Under this condition, the maximum radius of curvature, denoted as rmax, is determined and can be articulated by eq. 14:

$$
r_{\max} = \frac{L^2}{L - \sqrt{L^2 - 8^2(D - d)^2}} \approx 6.58 \text{ m}
$$

The determination of rmax 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 xOi exceeding L is not representative of the scenario modeled by this mathematical framework. To explore this boundary condition, a Python program was coded as part of this research to calculate the value of r when xOi 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 of 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 [TABLE:2], which outlines the permissible limits of movement for the fuel rods.

Utilizing the 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, denoted as ymax, are presented in Table 3 [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, keff is anticipated to be lower than that in scenarios where the fuel rods can fully fill the storage cavity.

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

This paper calculated the older birdcage deformation model proposed in reference [32] as well, and obtained the curve of keff versus r, see Fig. 16 [FIGURE:16]. The maximum value of keff occurs at r = 0.18 m, which is 0.99261. In this case, the fuel rod arrangement is shown in Fig. 17(d) [FIGURE:17], and the arrangement of fuel rods on the symmetric axis is shown in Table 4 [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 reference [32]. The results of this calculation are presented in the form of a curve depicting the relationship between the keff and the radius of curvature r, as shown in Fig. 16 [FIGURE:16]. The maximum value of keff within the context of the old model was found to occur at r = 0.18 m, with keff peaking at 0.99261. The corresponding arrangement of the fuel rods for this scenario is illustrated in Fig. 17(d) [FIGURE:17], and the detailed layout of the fuel rods along the symmetric axis is provided in Table 4 [TABLE:4].

Upon comparison, it is observed that the upper limit of keff 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 previously proposed in reference [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.

D. Conclusion of Non-Uniform Arrangement Method

Table 4 [TABLE:4] delineates the positions of the fuel rods along the half of the central axis, offering insight into their spatial arrangement. Fig. 17 [FIGURE:17] visually represents the configuration of fuel rods at the point where keff attains its maximum value for each method. Upon examination of Table 4, 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 keff 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 keff, primarily because their underlying premise is based on an exhaustive enumeration approach. This method is characterized by its thoroughness but may not align with real-world phenomena. Between these, the Layer-by-Layer Determination method is noted for its more conservative approach. In contrast, the two Birdcage Deformation methods take a different approach by grounding their analysis in actual observed phenomena. Besides, Sec. III.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.

keff,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 keff,max,r through subtle variation in the positioning of each fuel rod. As articulated in reference [32], these incremental adjustments can lead to a further enhancement of keff,max,r. Incorporating the effects of these minor rod position perturbations, the revised maximum keff,max,r for the New Birdcage Deformation is refined according to the subsequent formula Eq. 15:

$$
k_{eff,max,r}^{*} = (1 + E_s)k_{eff,max,r}
$$

Where keff,max,r* is keff,max,r after revising by small-scale pitch shift, Es is the enhancement amplitude of small-scale pin pitch shift. Es is taken as 0.109% according to reference [32].

It is unnecessary to consider the possibility that small-scale pin pitch shift can continue to rise kmax 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.

V. Method Conclusion and Example Analysis

The research presented in this paper is capable of introducing a method for determining the keff envelope value for PWR transport containers caused by fuel rods rearrangement in the event of fuel rod mispositioning accidents. This method is universal and can be extended to transport containers for other types of PWR fuel assemblies. The methodology is summarized in the following four steps:

S1: Establishing the Monte Carlo model for an evaluated transport package under accident conditions.

S2: Ascertain the maximum value of keff attainable by rearranging the fuel rods using the Layer-by-Layer Determination methods.

S3: Employing the New Birdcage Deformation to determine the maximum keff value achievable through the rearrangement of the fuel assembly.

S4: Revise the maximum keff value obtained in S3 by eq. 15 and compare it with the result from S2. The bigger one is the ultimate reactivity envelope value for rods mispositioning accidents.

With the completion of the methodology for determining the envelope model for non-uniform displacements of fuel rods under accident conditions, this research now applies the aforementioned approach to the criticality safety evaluation of a specific model of an AFA3G-type fuel assembly transport cask. The evaluation is conducted within the context of hypothetical accident scenarios to assess the transport cask's safety performance under extreme conditions. A comparative analysis will be undertaken, juxtaposing the results of this research with the existing criticality safety assessment techniques currently employed by the CNPRI for the same class of transport casks. This comparative approach is designed to elucidate the enhancements offered by the new methodology proposed in this paper, particularly in terms of providing a safety margin for criticality safety analysis of fuel transport containers.

The transport cask in question is designed to accommodate two sets of AFA3G-type fuel assemblies at once. The interior walls of the storage cavity are equipped with neutron-absorbing materials, and the cask is intended for use in dry transportation methods. In the event of an accident, the fuel assemblies contained within the cask are envisaged to be entirely inundated by water. The Monte Carlo computational model for the transport cask under accident conditions is illustrated in the designated container section as presented in Fig. 18 [FIGURE:18].

Sec. III provides evidence that the application of the Layer-by-Layer Determination method for the non-uniform arrangement of AFA3G-type fuel assemblies can yield the highest possible value of keff. In light of this finding, the fuel rods of the two sets of AFA3G-type fuel units housed within this transport cask are configured as illustrated in Fig. 17(b) [FIGURE:17]. The cross-sectional representation of the Monte Carlo computational model for this specific cask is presented in Fig. 18 [FIGURE:18]. The resultant keff from these calculations is determined to be 0.88807, accompanied by a standard deviation of 0.00029.

Currently, in the absence of comprehensive studies on the impact of fuel rod arrangements in the event of accidents, CNPRI has established a criticality safety assessment protocol. This protocol is based on the maximum value of keff obtained from a uniform arrangement of fuel rods within the storage cask, with an additional conservative penalty of 4000 pcm to account for potential accident scenarios. Specifically, when the fuel rods are evenly distributed, the calculated maximum keff is recorded as 0.87192, accompanied by a standard deviation of 0.00028. The imposition of the 4000 pcm penalty elevates this value to 0.91192, thereby ensuring a heightened level of safety precaution in the criticality assessment of the system under accident conditions.

Upon juxtaposing the assessment methodology established in this research with the extant approaches, it is observed that it confers an approximate increment of 2385 pcm in the criticality safety margin for the evaluation of this specific AFA3G-type fuel assembly transport cask.

VI. Conclusion

The research presented in this paper contributes a robust methodological framework and insights of considerable value to the nuclear industry, particularly in the realm of assessing transport containers for fuel assemblies. By employing the methodologies established in this study, it is possible to evaluate the critical safety risks associated with fuel assemblies during accidental conditions with greater precision. This enhanced predictive capability fortifies the assurance of safe nuclear fuel transportation, offering a substantial advancement in the field of nuclear safety protocols.

This research delves into the effects of fuel rod misalignment on criticality safety during transportation accidents by applying three varied methodologies for holistically researching non-uniform fuel rod arrangement. It introduces an envelope determination technique that accounts for a comprehensive range of potential fuel rod misalignment scenarios, thereby providing innovative insights into the criticality safety analysis of fuel assembly transport containers under accident conditions. Notably, the Layer-by-Layer Determination and the New Birdcage Deformation methods excel in increasing the keff, each from a unique perspective. Therefore, when formulating the envelope model for fuel rod misalignment accidents, a comparative evaluation should be conducted between these two methods to identify the most effective strategy for ensuring criticality safety. The ultimate envelope model presents a significant advancement over traditional criticality safety analysis methods by substantially increasing the safety margin.

While this study has achieved notable advancements at the practical level, there is ample scope for further research. Future work can concentrate on several key areas:

a. An in-depth investigation of the axial deformation of fuel assemblies under impact accidents is necessary to establish a more precise criticality safety analysis model, thereby significantly enhancing the safety margin.

b. It is essential to explore methods for rapidly determining the envelope model for fuel assembly misalignment accidents, aiming to reduce the consumption of computational resources and increase analytical efficiency.

c. The methodology developed in this study should be extended to the criticality safety analysis of other rod-bundle type fuel assembly transport containers under accident conditions, to verify its universality and applicability.

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