Self-shielding Calculation Research Articles

Resonance Calculation of FCM Fuel Based on the Global-Local Method

Fully ceramic microencapsulated fuels is a promising potential fuel for pressurized water reactors because of its inherent safety, but double heterogeneity (DH) brings challenges to the neutronic calculation. In this paper, the Hébert model is applied to the global-local resonance method to solve DH, in which the Dancoff factor obtained by the neutron current method is used to transform the coupled fuel pins in the reactor into multiple independent one-dimensional equivalent cylindrical fuel pins. Then, the Hébert model and the hyperfine group method are used to perform the resonance self-shielding calculation in these independent pins. The Sanchez-Pomraning method coupled with the Method of Characteristics (Sanchez-MOC) method is introduced to the two-dimensional/one-dimensional transport module to realize the subsequent whole-core DH transport calculations. The proposed method has been implemented in the high-fidelity neutronics code NECP-X and tested with a set of cases. The results show good agreement with the Monte Carlo reference values for the reactivity and self-shielding cross sections.

Impact of the resonance elastic scattering on reactor-core physics calculation results based on the deterministic method

NECP-Atlas and similar nuclear data processing codes have developed the function of incorporating the resonance elastic scattering effect into the multi-group cross sections and scattering matrices. However, the hyper-fine group method is commonly used in deterministic-based lattice physics codes and whole-core neutronics codes for resonance self-shielding calculations. In this method, the neutron slowing-down equation is established based on the conventional asymptotic scattering model where the neutron up-scattering caused by the resonance elastic scattering is ignored. Consequently, the deterministic codes cannot directly use the multi-group data produced by the nuclear data processing codes. In this paper, a correction factor method is proposed to correct the multi-group cross sections and scattering matrices obtained by the hyper-fine group calculations to consider the resonance elastic scattering. The proposed method has been implemented in the lattice physics code LOCUST. The few-group constants produced by the LOCUST code are provided to reactor-core analysis code SPARK to simulate a CNP-1000 type PWR reactor operated in China. Some core parameters measured during zero-power physics tests at 600 K and operations are analyzed to show the effect of the resonance elastic scattering on the core parameters. The numerical results show that the boron concentration reduces by 10 ∼ 14 ppm at 900 K during operations, which is closer to the measured value and improves the accuracy of reactor-core physics calculation.

Annular fuel neutronic analysis

The annular fuel was proposed to increase the power density while keeping or improving the safety margins, given the considerably lower surface heat flux and lower fuel temperature than conventional solid fuel. In this paper, neutronic characteristics of this new fuel design were analysed using the new transport code of the Westinghouse, called PARAGON2, using Ultra-Fine Energy Mesh Library for multi-group cross section calculations. The bibliographical review showed there is a lack of deterministic transport codes capable to model such fuel with acceptable accuracy, because the equivalence relations for heterogeneous resonance integrals adopted in the standard codes were formulated only for solid cylindrical rods. Therefore, the main objective of this paper is to perform neutronic analysis for annular fuel using PARAGON2, evaluating the use of Ultra-Fine Energy Mesh Library to eliminate resonance self-shielding calculation. The results of this paper analysed the power distributions, the ratio of U-238 capture rate to U-235 fission rate and behaviour of the infinite multiplication factor for annular fuel and compared with the Monte Carlo code MCNP.

A one-dimensional model for multi-group equivalence parameters in subgroup method

In the previous study, a non-iterative equivalence method is proposed, which guarantees that the reaction rates calculated by subgroup method in self-shielding calculation are preserved in subgroup-collapsed one-group calculation. This method shows advantage in efficiency over the Super-Homogenization (SPH) method. However, one-group fixed-source calculations are still needed for obtaining multi-group (MG) equivalence parameters. In this paper, a one-dimensional model is employed for obtaining the MG equivalence parameters in subgroup method. Based on the one-dimensional model, the one-group fixed-source calculation is simplified to one-angle tracking sweeping, which is time-saving. The numerical results show that the reaction rates can be preserved with the MG equivalence parameters obtained based on the one-dimensional model for pin cell problems. For fuel assembly problems, the kinf is improved by 140 pcm ∼ 187 pcm with negligible increase in computational time.

Development of the capability for multi-group shielding library generation in the nuclear data processing code NECP-Atlas

The capability to generate multi-group shielding library is developed in the nuclear data processing code NECP-Atlas. To make the code suitable for the applications in different types of reactors, several improvements are made in the code, such as the resonance self-shielding method and energy-group structure optimization. The hyperfine group method is applied to perform resonance self-shielding calculations for different configurations. The energy-group structure optimization method is developed based on the contributon theory to enhance the calculation accuracy of the thermal neutron flux and photon flux. The accuracy of the newly developed capability is verified with the reference results calculated using BUGLE-B7 libraries. Two libraries with optimized neutron energy-group structures of 47 groups and 60 groups are obtained based on the contributon theory. Numerical results show that the thermal neutron flux and photon flux calculated based on the two libraries are greatly improved.

Development of a cross section self-shielding method in arbitrary geometry

We are developing a neutron transport calculation code based on the Iso-Geometric Analysis (IGA) method that can resolve the multi-group neutron transport equation for any arbitrary spatial domain. The energy variable is treated with multi-group theory. Therefore, self-shielding calculations become necessary. We have decided to adopt the subgroup method because this method does not rely on any specific geometrical domain and should be used in combination with the IGA method. Some basic theory of the IGA method is given to introduce the concept of basis function and NURBS for the representation of the spatial domain. Validation of the implementation of the subgroup method for IGA and SN transport calculations is presented. The result is that the combination of SN transport theory, IGA and the subgroup method gives correctly shielded cross-sections. Thus, effective multi-group cross-sections can be generated for any arbitrary spatial domain.

Refinements of the Pin-Based Pointwise Energy Slowing-Down Method for Resonance Self-Shielding Calculation—I: Theory

The pin-based pointwise energy slowing-down method (PSM), which is a resonance self-shielding method, has been refined to treat the nonuniformity of material compositions and temperature profile in the fuel pellet by calculating the exact collision probability in the radially subdivided fuel pellet under the isolated system. The PSM has generated the collision probability table before solving the pointwise energy slowing-down equation. It is not exact if the fuel pellet has nonuniform material compositions or temperature profile in all the subdivided regions. In the refined PSM-CPM, the pre-generated table is not required for directly calculating the collision probability in all the subdivided regions of the fuel pellet while solving the slowing-down equation. There are an advantage and a disadvantage to the method. The advantage is to exactly consider the nonuniformity of the material compositions and temperature profile in the fuel pellet. The disadvantage is the longer computing time than that of the PSM when the fuel pellet has more than five subdivided regions. However, in the practical use for UO2 pin-cells, it is still comparable for the computation time with the PSM and the conventional equivalence theory methods. In this article, using simple light water reactor 17 × 17 F A problems with a uniform material composition and temperature profile, it is demonstrated that PSMs (PSM and PSM-CPM) exhibit consistent accuracy in calculating the multiplication factor and the pin power distribution with no compromise in the computation time. More detailed accuracy assessments with various test cases, including problems representing the nonuniformity, are presented in the accompanying article.

Generalized Perturbation Theory Based Total Sensitivity and Uncertainty Analysis for High-Fidelity Neutronics Calculation

Neutronics calculation for nuclear reactor with high-fidelity technology can significantly reduce the uncertainties propagated from numerical approximation error and model error. However, the uncertainty of input parameters inevitably exists, especially for nuclear data. On the other hand, resonance self-shielding calculation is essential for multi-group assumption based high-fidelity neutronics calculation, which introduce the implicit effect for calculation responses. In order to fully consider the implicit effects in the process of uncertainty quantification, a generalized perturbation theory (GPT) based implicit sensitivity calculation method is proposed in this paper. Combining the explicit sensitivity coefficient, which can be quantified using classic perturbation theory, the total sensitivity coefficient of calculation responses is obtained. Then the total sensitivity and uncertainty module is established in self-developed neutron transport code with high-fidelity technology-HNET. To verify the accuracy of the sensitivity calculation methods proposed in this paper, a two-dimensional fuel pin problem is chosen to verify the sensitivity results, and the numerical results show good agreement with results calculated by a direct perturbation method. Finally, uncertainty analysis for two-dimensional fuel pin problem is performed and some general conclusions are obtained from the numerical results.

Recent developments in PSI BEPU analysis for SPERT-III RIA transient: ND uncertainty breakdown and kinetic parameters uncertainty assessment

The current research summarizes recent developments performed within the PSI BEPU methodology and the corresponding results on the SPERT-III RIA experiments. In this regard, three main studies have been carried out. First, an uncertainty breakdown study has been performed in order to identify the dominant nuclear data and to quantify their respective contributions to the overall uncertainties of the parameters of interest. On the other hand, the second study was devoted to assess the impact of considering the effect of cross section perturbation (limited for the time being to U-238 capture) in the resonance self-shielding calculation, known as the implicit effect, on different parameters’ uncertainties in SPERT-III RIA transient. Finally, the third study was conducted with the main goal to assess the impact of kinetic parameter uncertainties, such as delayed neutron yields and associated decay constants, on the overall uncertainties.The uncertainty quantification results of the first study, based on the ENDF/B-VII.1 library, showed that, for steady-state, the first dominant contributor to the overall uncertainties of the static reactivity worth and k-eff is U-235 nubar, followed by U-238 then U-235 capture cross sections. However, for the transient, the first dominant contributor to the overall uncertainties, in e.g. total power and reactivity, is the U-238 inelastic scattering. More interestingly, it was found that, the second dominant contribution to the overall power uncertainty, during the initial excursion phase, is due to the U-235 nubar, while during the power reversal phase, the U-238 capture becomes the second dominant contribution, and even the first one by the end of the transient. Moreover, results of the second study demonstrated a clear systematic decrease of the uncertainty of all parameters due to the inclusion of the implicit effect. In steady state, due to the impact of the implicit effect, the Doppler coefficient relative uncertainty could be reduced up to 30%, which is reflected in the transient, especially in the reversal phase of the power excursion where Doppler effect plays a key role, by a systematic decrease of the relative uncertainty of total power and other parameters. Finally, in the third study, the estimated uncertainties, regarding both steady-state and transient parameters, have shown systematic larger values due to kinetic parameter uncertainties compared to those due to the other nuclear data.

Resonance calculation based on the deep learning method for treating the non-uniform fuel temperature distribution in PWRs

In the previous work, a global–local resonance method was developed for the PWR self-shielding calculations, but there are still some difficulties when dealing with the non-uniform temperature profile of the fuel rod. This work proposes a deep learning based resonance calculation scheme for treating the non-uniform fuel temperature distribution. In this scheme, the deep learning model is investigated to replace the ultra-fine group method for the local calculations of the global–local resonance method, because the ultra-fine group method is too slow for performing the whole core resonance calculations when considering the temperature profile. The generation of training data sets, deep learning models and some numerical results are introduced. The given tests demonstrate that the new method could give accurate self-shielding cross sections and significantly improve efficiency.

A SUBGROUP METHOD BASED ON THE EQUIVALENT DANCOFF-FACTOR CELL TECHNIQUE COMPARED WITH THE FINE-STRUCTURE METHOD IN APOLLO3®

We compare a newly implemented subgroup method, which is based on the equivalent Dancoff-factor cell technique, with the fine structure method in APOLLO3®. The new method obtains precise reaction rates and consequently is precise in predicting multipli-cation factor. It can reduce the time in the self-shielding calculation by a factor of 38 compared with the subgroup method based on the multi-cell approximation in a PWR Gd-UOX assembly calculation. The fine structure method obtains larger errors in reac-tion rates than the subgroup method, but the error compensation sometimes leads to small errors in multiplication factors. This work demonstrates the precision and the efficiency of the new method.

IMPROVEMENT OF SCALE-XSProc MULTIGROUP CROSS SECTION PROCESSING BASED ON THE CENTRM POINTWISE SLOWING DOWN CALCULATION

The SCALE-XSProc multigroup (MG) cross section processing procedure based on the CENTRM pointwise slowing down calculation is the primary procedure to process problem-dependent self-shielded MG cross sections and scattering matrices for neutron transport calculations. This procedure supports various cell-based geometries including slab, 1-D cylindrical, 1-D spherical and 2-D rectangular configurations and doubly heterogeneous particulate fuels. Recently, this procedure has been significantly improved to be applied to any advanced reactor analysis covering thermal and fast reactor systems, and to be comparable to continuous energy (CE) Monte Carlo calculations. Some reactivity bias and reaction rate differences have been observed compared with CE Monte Carlo calculations, and several areas for improvement have been identified in the SCALE-XSProc MG cross section processing: (1) resonance self-shielding calculations within the unresolved resonance range, (2) 10 eV thermal cut-off energy for the free gas model, (3) on-the-fly adjustments to the thermal scattering matrix, (4) normalization of the pointwise neutron flux, and (5) fine MG energy structure. This procedure ensures very accurate MG cross section processing for high-fidelity deterministic reactor physics analysis for various advanced reactor systems.

THE TWO-STEP APPROACH FOR WHOLE-CORE RESONANCE SELF-SHIELDING CALCULATION

A two-step approach is proposed to accomplish high-fidelity whole-core resonance self-shielding calculation. Direct slowing-down equation solving based on the pin-cell scale is performed as the first step to simulate different operating conditions of the reactor. Resonance database is fitted using the results from the pin-cell calculation. Several techniques are used in the generation of the resonance database to estimate multiple types of resonance effects. The second step is the calculation of practical whole-core problem using the resonance database obtained from the first step. The transport solver is embedded both at the first step and the second step to establish the equivalence relationship between the fuel rod in the practical problem and the pin-cell at the first step. The numerical results show that the new approach have capability to perform high-fidelity resonance calculations for practical problem.

METHODS FOR CALCULATION OF SELF-SHIELDED CROSS SECTIONS OF FULLY CERAMIC MICRO-ENCAPSULATED FUEL DOUBLE HETEROGENEOUS SYSTEM

Fully ceramic micro-encapsulated (FCM) fuel is a kind of fuel that employs tri-structural isotropic (TRISO) particles to enhance safety. The FCM fuel assembly is a double heterogeneous system. The conventional self-shielding calculation methods cannot treat the DH effect. In this paper, three methods based on equivalent homogenization of the TRISO particle and the matrix are studied and compared: the hyper-fine energy group cross sections (XSs) homogenization based hyper-fine energy group method (HHM), the hyper-fine energy group XSs homogenization based subgroup method (HSM) and the subgroup XSs homogenization based subgroup method (SSM). These methods are implemented in a high-fidelity neutronics code NECP-X. Numerical results show that these methods are able to treat the double heterogeneity of the FCM fuel. The precision of the HHM and HSM is higher than that of the SSM.

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

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.

Extension and preliminary validation of the Polaris lattice physics code for CANDU analysis

Polaris is a 2D lattice physics code available for the study of light water reactors (LWRs). It includes a method of characteristics (MoC) transport solver and the embedded self-shielding method (ESSM). Its advantages include a simplified user input structure, quick performance, and integration into the SCALE package, utilizing its nuclear data libraries, along with other codes such as the ORIGEN depletion code and the Sampler stochastic uncertainty analysis sequence. This work extends the Polaris code to support CANDU lattice transport calculations and performs some preliminary verification and validation, comparing the code to NEWT and KENO, along with to benchmark experimental data. Heavy-water specific self-shielding factors were also compared to those distributed with SCALE. In general, some differences are observed, but these are generally comparable to other code-to-code differences and smaller than nuclear data uncertainties. The generation of self-shielding factors has a noticeable effect, particularly on the benchmark coolant void reactivity calculation. Heavy-water specific self-shielding factors, while not required, can potentially improve the accuracy of ESSM-based calculations, but the choice of parameters can significantly affect the results. Overall, the Polaris transport and self-shielding calculations should be suitable for CANDU analysis.

Fitting-based resonance database method for resonance self-shielding calculations of large-scale task considering depletion and intra-pin distribution

A fitting-based resonance database method is proposed to perform self-shielding calculation for large-scale problems. The intra-pin radial effects are also treated by this method. The new method is mainly divided into two steps. First, the operating conditions of different pin-cells existing in the target problem are simulated by the adjustments of the parameters of single pin-cell cases. The ultra-fine-group slowing-down solving process is performed for these pin-cells and the resonance database for the target problem is fitted using the obtained results. Second, the target problem is calculated using the resonance database generated. Transport calculations are performed at both steps to establish the equivalence relationship between the target problems and the single pin-cell cases. Several techniques are adopted to evaluate the resonance interference and radial effects. Numerical tests prove that the method is capable to give the accurate prediction for the spatial-dependent resonance self-shielded cross-sections in an efficient way.

Extension of the subgroup method for self-shielding calculation of fully ceramic micro-encapsulated fuel

Two extension schemes based on the subgroup method are proposed to treat the double heterogeneity (DH) of fully ceramic micro-encapsulated (FCM) fuels. The first one is the hyper-fine energy group cross section (XS) correction scheme (HFCS), which corrects the hyper-fine energy group XSs with hyper-fine energy group disadvantage factors (DFs) before generation of resonance XS table and physical probability table. The second one is the subgroup XS correction scheme (SGCS), which corrects the subgroup XSs with subgroup DFs calculated by the equivalent homogenization method. These two scheme are implemented in the high-fidelity neutronics code NECP-X. A series cases are tested and the numerical results show that both of the HFCS and the SGCS can treat the DH and the precision of the HFCS is higher than that of the SGCS.

A Comparison Between the Embedded Self-Shielding Method and the Enhanced Neutron Current Method Based on the Equivalent Pin Cell Model on the Irregular Fuel Lattice Problem

The Embedded Self-Shielding Method (ESSM) coupled with the heterogeneous Resonance Integral tables and the Enhanced Neutron Current Method (ENCM) with equivalent Dancoff factor are reviewed and reformulated to a unified framework by incorporating the ultra-fine-group slowing-down calculation on two-dimensional square pin cell problems. The comparison between the two approaches on the resonance self-shielding calculation of irregular fuel lattices shows that the reformulated ESSM approach will bring errors to the cross-section prediction of fuel pins in the irregular lattice, especially when the moderator density is low. Also, the reformulated ENCM approach is more stable for different configurations. Further numerical tests show that the scalar flux calculated by the ESSM approach is affected by the global neutron balance across the fuel lattice and ESSM is more sensitive to the error brought by the enforced equivalence.

The on-the-fly subgroup method capable of treating spatial self-shielding, resonance interference and temperature distribution effects

The pseudo-resonant-nuclide subgroup method (PRNSM) based global-local self-shielding calculation scheme was recently developed to resolve the spatial self-shielding and resonance interference effects for large-scale problems. But the PRNSM is not able to treat non-uniform temperature distribution. The reason is that the PRNSM generates physical probability tables at different temperatures with inconsistent subgroup probabilities. To overcome this defect, the on-the-fly subgroup method (OSM), which is an improvement of the PRNSM, is proposed. The new method generates the physical probability tables for different temperatures with shared subgroup probabilities and is able to treat non-uniform temperature distribution. This method replaces the PRNSM in the global-local self-shielding calculation scheme and is implemented in the high-fidelity neutronics code NECP-X. The numerical results show that the OSM can treat the spatial self-shielding effect, the resonance interference effect and the non-uniform temperature effect simultaneously. Besides, NECP-X is able to predict fuel temperature coefficients with high accuracy.