Evaluation of improved subgroup resonance treatment based on Sanchez-Pomraning method for double heterogeneity in PWR

https://doi.org/10.1016/j.anucene.2020.107491Get rights and content

Highlights

  • The Sanchez-Pomraning based subgroup method is used for the double heterogeneity.

  • The DH MOC is used twice for resonance calculation and once again for eigenvalue.

  • The region-wise resonance cross section inside the particles could be obtained.

Abstract

The feasibility of the improved subgroup method coupled with the Sanchez-Pomraning Method (ISSP) is analyzed for the resonance self-shielding calculation of double heterogeneity (DH) problems in PWR. The Sanchez-Pomraning method is developed for both fixed source and the eigenvalue problems. The improved subgroup method adopts a fine-mesh energy structure to handle the resonance interference effect. In this process, the subgroup fixed source equations are solved to get the fine-mesh cross section and the slowing-down equations are solved for group condensation. To extend the improved subgroup method to DH condition, the Sanchez-Pomraning method is applied twice to the subgroup fixed source equation and slowing-down equation respectively. Afterward, the Sanchez-Pomraning method is used the third time for multi-group transport problems. The ultra-fine group method coupled with the Sanchez-Pomraning method (UFGSP) is also established for comparison with ISSP. Numerical validation shows that both ISSP and UFGSP could treat the DH resonance effect precisely.

Introduction

With the rising demand for safety and economic efficiency of nuclear energy, new generation of reactor types and numerical analyzing methods have become the research hotspot in recent years. From a global perspective, the pressurized water reactor (PWR) will still be the mainstream nuclear power plant in the foreseeable future. Therefore, how to establish accurate mathematical modeling of PWR becomes crucially important, and there are many mature programs capable of simulating PWR in most cases. However, the traditional deterministic method could only deal with the single level of heterogeneity in the reactor, namely the heterogeneity consisting of the fuel rod and moderator. With the development of nuclear technology, new requirements have been put forward to deal with the double heterogeneity (DH) brought by stochastic particles inside the fuel rod. There are two main kinds of DH in PWR, the first one is fully ceramic micro-encapsulated fuel (FCM) and the other is the Pu spots in MOX fuel. As a promising option of accident-tolerant fuel (Powers et al., 2013, Snead et al., 2011), FCM fuel is comprised of tri-structural isotopic (TRISO) particles and matrix. TRISO particles could be divided by different layers with fuel kernels in the center while protective coatings, like graphite or SiC, encased in the outer layers. A certain quantity of TRISO particles are deployed randomly in the graphite matrix of the fuel rod and the explicit locations of them are unclear. Pu spots are another kind of DH phenomenon with a high impact on PWR analysis (Kawano, et al, 2019). During the manufacturing process of MOX fuel, plutonium isotopes would be agglomerated as random globular spots distributed in the fuel rod. Plutonium Spots result in similar DH effects like FCM fuel as they play the role of TRISO particle while the matrix is made up of MOX fuel. DH effect presents challenges to resonance treatment and traditional approaches like equivalent method (Askew et al., 1966, Hebert and Marleau, 1991), ultra-fine group method (Ishiguro and Takano, 1971, Zhang et al., 2020) and subgroup method (Nikolaev et al., 1971, Cullen, 2019) as these methods could not describe the stochastic model directly.

To handle the resonance calculation in DH condition, researchers worldwide have proposed a number of viable methods. Dancoff method (Bende et al., 1999) is a typical approach in terms of the DH effect. In this theory, the DH average Dancoff factor was initially derived to represent the escape probability of the neutron from one fuel kernel to another in the high-temperature gas-cooled reactor (HTGR) and was calculated by the neutron first-flight escape probability and the transmission probability that deduced from numerical integration among different fuel kernels. However, due to the complicated integration and its assumptions, this method would have trouble dealing with multi-layer coating particles. To implement the average Dancoff factor into TRISO fuels, the dual-sphere model (Ji et al., 2014) based on the chord method was proposed to take the multi-layer spheres into consideration. In resonance calculation, the Dancoff factor was applied as an amendment for the DH background cross section (XS), which would be used subsequently to deduce the effective resonance XS (Kim et al., 2017). The average Dancoff method shows a relatively satisfactory performance but the deducing process is rigid and is difficult for practical application. Another promising method for DH treatment is disadvantage factor method (She et al, 2017), of which the core idea is transforming the double heterogeneity into equivalent single heterogeneity as traditional problems by solving ultra-fine group slowing-down equations, and the existing resonance and transport calculation could be carried out without modification. According to its derivation, the TRISO particles and matrix are mixed as an equivalent one-dimensional problem, and the disadvantage factors were obtained on the basis of the equivalent TRISO-matrix mixed model. This method is easy to implement as only slight modifications are needed but the region-wise XS inside TRISO particles are unable to access. The DF method has been applied in the SCALE code (Williams, 2011) but invoking of the ultra-fine group method would cause the decrease of efficiency. Recently, researches have been conducted to couple subgroup method (He et al., 2016, He et al., 2020) or ultra-fine method (Yin et al., 2019) with disadvantage factor method for FCM fuel. The calculating results indicated a high precision but both methods still needed time-consuming ultra-fine group slowing down equations and cannot give the self-shielding distribution inside the particles. DH resonance integral combined with the embedded self-shielding method (Li et al, 2018) was another feasible approach which, but it required invoking Monte Carlo program time and again and lead to inefficiency. Sanchez-Pomraning method (Sanchez and Pomraning, 1991) is a classical technique for directly solving transport equations in DH geometry and could describe the distribution of flux inside separate particles in detail. In this method, the renewal equation is derived to capture the different behavior of flux in DH conditions. The equivalent XS between particles and matrix was also needed and an iteration method of this process was introduced. The transport calculation was carried out based on the equivalent XS to obtain the equivalent flux, then the detailed fluxes in particles were deduced in accordance with escape probability and collision probability. The Sanchez-Pomraning method could be applied to MOC (Sanchez, 2004), so it paved the way for traditional resonance methods like subgroup or embedded self-shielding method to handle DH resonance effect (Pogosbekyan et al., 2008, Sanchez et al., 2010, Kim et al., 2016). According to the researches above, the only feasible way to calculate the region-wise fluxes and resonance XSs in the DH condition is the subgroup method coupled with the Sanchez-Pomraning method. However, in the existing researches, most of the researching objects were HTGR and the few PWR models were not comprehensive enough. For the subgroup method, how to accurately handle the resonance interference effect in the DH condition is another challenge. The existing method like the Bondarenko method (Askew et al, 1966) has been proved to lack precision while the resonance interference factor method (Williams, 1983) needs too much calculating burden.

To solve the problems mentioned above, the evaluation of the improve subgroup method (Li et al., 2020) coupled with the Sanchez-Pomraning method for DH condition in PWR is carried out in this work. The improved subgroup method is proposed on the basis of the idea that a compromise could be found between the advantages of ultra-fine group method and traditional group, namely the high accuracy of the former and the efficiency and geometry adaptability of the latter. To achieve this goal, two levels of discretion of the resonance energy range are adopted. The first one is the fine-mesh group structure and the other is the subgroup in each resonance group. In the two-level scheme, the fine mesh is not necessary as subtle as ultra-fine group structure and a 289-group structure is designed for the energy range from 1.855 eV to 9118 eV. Under this circumstance, resonance interference correction could be avoided as the two-level dispersion is fine enough for PWR. In addition, the subgroup fixed source equation is in the same formation as the Boltzmann transport equation, the renewal equation of the Sanchez-Pomraing method for MOC could be modified and employed in this process. The region-wise DH subgroup fluxes will be subsequently used in effective XS calculation. However, the finer mesh would inevitably cause the rise of subgroup fixed source equations. To settle this issue, the micro sub-level optimization method is taken up. The DH subgroup fixed source equations would be solved for pre-selected subgroup levels with the average equivalent one-group source item, then actually subgroup fluxes of each subgroup in each resonance group are obtained by interpolation of the natural logarithm of subgroup level. After the fine-mesh XS calculation, a group-condensation process based on DH slowing-down equation is performed. The Sanchez-Pomraning method will be applied again in this part to get condensed XS for each layer of TRISO particles. The 47-group structure (Stamm’ler, 2008) is adopted for the final transport calculation and the Sanchez-Pomraning method will be used the third time to get eigenvalue and neutron flux distribution in DH condition. The Gaussian quadrature formula is used to get the collision probability and transmission probability needed during all the above three processes based on different DH XSs. Furthermore, to guarantee the accuracy and feasibility of the Sanchez-Pomraning method for DH subgroup fixed source equations and DH slowing-down equations, the ultra-fine group method is also modified to couple with Sanchez-Pomraning method in this work ensure the validity of the DH subgroup calculation. For detailed numerical validation, the Monte Carlo method with explicit DH stochastic modeling is chosen for referenced resonance XS and eigenvalue.

The following sections of this paper will be illustrated as follows. Section 2 would give a detailed interpretation of the improved subgroup method as well as the ultra-fine method coupled with the Sanche-Pomraning method respectively (ISSP & UFGSP). Section 3 gives the numerical validation of ISSP compared with DH ultra-fine group method and the Monte Carlo method. Section 4 makes a conclusion of this paper and give some further discussions.

Section snippets

MOC based on the Sanchez-Pomraning method

The TRISO particles are randomly distributed in the graphite matrix and it is difficult to get their exact coordinates. The traditional MOC method cannot establish ray tracing model toward separate particles in the flat source region (FSR) of the FCM rod, so an effective average process between particles and matrix is needed to get the equivalent XSs of the fuel FSR. Unlike the traditional volume-weighted method, the Sanchez-Pomraning method uses the neutron collision probability from

Numerical validation

The benchmark problems selected in this work cover different kinds of DH conditions in PWR, including FCM fuels with different packing fraction (PF), particle radius, TRISO types and lattice configurations. Burnable poison with Boron as well as gadolinium baring FCM fuel is also tested. To capture the effect of plutonium-rich agglomerates, the problems with plutonium spots are established. The detailed geometry arrangement composition and material composition adopted in this work are referenced

Conclusions

The improved subgroup method is coupled with the Sanzhe-Pomraning method to handle the resonance self-shielding effect in double heterogeneous problems in this work. Sanchez-Pomraning method establishes the renewal equation to get region-wise flux and is adopted by MOC for DH transport calculation. The improved subgroup method adopts a fine-mesh group structure for resonance energy range to ensure the accuracy of resonance XS then the group condensation process is carried out for efficiency of

Author contribution statements

SL and Q Zhang conceived of the presented idea and conducted the numerical experiment. JZ and YL provided the support of the OpenMC code and verified the analytical methods. Q Zhao and ZZ supervised the findings of this work. L Liang provided the support on MOC solver. L lou provided the support from the FCM design.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by the Research on Key Technology of Numerical Reactor Engineering [J121217001], the Heilongjiang Province Science Foundation for Youths [QC2018003] and the Science and Technology on Reactor System Design Technology Laboratory.

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