Liangzhi Cao

Heterogeneous Pseudo-Resonant Isotope Method for Resolved Resonance Interference Treatment in Resonance Self-Shielding Calculation

Abstract The theory of resonance interference factor (RIF) method is examined for thermal reactor problems, and the approximations and limitations are identified. To evaluate the interference effect between resonance isotopes, the RIF method establishes an approximate equivalent relationship between a heterogeneous system and a homogeneous system by introducing background cross sections, and the approximation is a source of deviation in self-shielding calculations. Furthermore, each resonance isotope is treated individually in the self-shielding procedure, which requires unnecessary calculation effort, especially for whole-core and burnup cases. Based on the analysis, a heterogeneous pseudo-resonant isotope method (HPRIM) is proposed to overcome these problems. The mixture of resonant nuclides is considered as a pseudo-resonant isotope, and the resonance integral is generated in a one-dimensional heterogeneous system. The numerical results show that HPRIM improves the accuracy of evaluating the resonance interference effect and improves the efficiency of the self-shielding procedure.

Studies on the molten salt reactor: code development and neutronics analysis of MSRE-type design

The molten salt reactor is characterized by its use of the fluid-fuel, which serves both as a fuel and as a coolant simultaneously. The position of delayed neutron precursors continuously changes both in the core and in the external loop due to the fuel circulation, and the fission products are extracted by an online fuel reprocessing unit, which all lead to the modeling methods for the conventional reactors using solid fuel not applicable. This study establishes suitable calculation models for the neutronics analysis of the molten salt reactor and develops a new code named MOREL based on the three-dimensional diffusion steady and transient calculations. Some numerical tests are chosen to verify the code and the numerical results indicate that MOREL can be used for the analysis of the molten salt reactor. After verification, it is applied to analyze the characteristics of a typical molten salt reactor, including the steady characteristics, the influence of fuel circulation on the kinetic behaviors. Besides, the influence of online fuel reprocessing simulation is also examined. The results show that inherent safety is the character of the molten salt reactor from the aspect of reactivity feedback and the fuel circulation has great influence on the kinetic characteristics of molten salt reactor.

The pseudo-resonant-nuclide subgroup method based global–local self-shielding calculation scheme

ABSTRACT The pseudo-resonant-nuclide subgroup method (PRNSM) based global–local self-shielding calculation scheme is proposed to simultaneously resolve the local self-shielding effects (including spatial self-shielding effect and the resonance interference effect) for large-scale problems in reactor physics calculations. This method splits self-shielding calculation into global calculations and local calculations. The global calculations obtain the Dancoff correction factor for each pin cell by neutron current method. Then an equivalent one-dimensional (1D) cylindrical problem for each pin cell is isolated from the lattice system by preserving Dancoff correction factor. The local calculation is to perform self-shielding calculations of the equivalent 1D cylindrical problem by the PRNSM. The numerical results show that PRNSM obtains accurate spatial dependent self-shielded cross sections and improves the accuracy of dealing with the resonance interference over the conventional Bondarenko iteration method and the resonance interference factor method. Furthermore, because both global and local calculation is linearly proportional to the size of problems, the global–local calculation scheme could be applied to large-scale problems.

An Improved Resonance Self-Shielding Calculation Method Based on Equivalence Theory

Abstract The deviation of the effective resonance cross section obtained by conventional equivalence theory for a heterogeneous system is analyzed. It is shown that several approximations commonly adopted in conventional equivalence theory account for the deviation at different levels, with the narrow resonance (NR) approximation being the main source of deviation. Based on the analysis, an improved method based on equivalence theory is proposed. It utilizes the resonance fine flux integral table to minimize the deviation caused by NR approximation. The validity of the method is confirmed by test calculations of effective resonance cross sections in different geometries and different energy group structures. The results of eigenvalue calculations on typical fuel pin cells show that the proposed improvement is effective in reducing the error of infinite multiplication factors of the pin cell. Since the resonance fine flux integral used in this method has already been obtained in calculating the resonance integral table and can be pre-tabulated in the process of generating the library, the implementation of the proposed method is simple and requires no additional calculations. It is useful for improving the accuracy of lattice physics codes based on the equivalence theory.

Daubechies Wavelet Method for Angular Solution of the Neutron Transport Equation

Abstract This paper describes Daubechies’ wavelet method (DWM) for the discretization of the angular variable in the neutron transport equation. Two special features are introduced: (a) the azimuthal angle is discretized using the Daubechies’ scaling function as the basis function, while the polar angle is decoupled and discretized using the discrete ordinates in a standard manner, and (b) the construction of Daubechies’ wavelets on an interval is used to get around the edge effect between subdomains in the angular variable. In addition, two acceleration methods, namely, coarse mesh rebalance and coarse mesh finite difference, are implemented in DWM. The test results on several benchmark problems indicate that DWM described in this paper is capable of treating transport problems exhibiting angularly complicated behaviors, effective in mitigating ray effect, and versatile in handling transport phenomena in a variety of structured media.

Improvements and validation of the transient analysis code MOREL for molten salt reactors

ABSTRACT The liquid fuel salt used in the molten salt reactors (MSRs) serves as the fuel and coolant simultaneously. On the one hand, the delayed neutron precursors circulate in the whole primary loop and part of them decay outside the core. On the other hand, the fission heat is carried off directly by the fuel flow. These two features require new analysis method with the coupling of fluid flow, heat transfer and neutronics. In this paper, the recent update of MOREL code is presented. The update includes: (1) the improved quasi-static method for the kinetics equation with convection term is developed. (2) The multi-channel thermal hydraulic model is developed based on the geometric feature of MSR. (3) The Variational Nodal Method is used to solve the neutron diffusion equation instead of the original analytic basis functions expansion nodal method. The update brings significant improvement on the efficiency of MOREL code. And, the capability of MOREL code is extended for the real core simulation with feedback. The numerical results and experiment data gained from molten salt reactor experiment (MSRE) are used to verify and validate the updated MOREL code. The results agree well with the experimental data, which prove the new development of MOREL code is correct and effective.

Block-diagonalization of the variational nodal response matrix using the symmetry group theory

Abstract To further improve the efficiency of the Variational Nodal Method (VNM) for solving the neutron transport equation in hexagonal-z geometry, the nodal response matrix is further block-diagonalized by utilizing the symmetry group theory to decompose the surface basis functions into irreducible components. The block-diagonal property of the nodal response matrix is determined by the symmetry properties of the hexagonal node in geometry, material and basis functions, including both reflection and rotation symmetries. To fully utilize those properties, the symmetry group theory is employed to analyze the symmetry property of the nodal response matrices. It is mathematically proved that the nodal response matrix can be further block-diagonalized into 16 diagonal blocks instead of the current 4 ones by using the symmetry group theory. Numerical comparisons demonstrate that the new approach can reduce the memory storage and computing time by a factor of 2∼3 for P7 angular approximation, compared with the currently employed variables transformation algorithm.

Resonance Calculation Code UFOP Based on the Hyper-Fine Group Neutron Resonance Calculation Method

Resonance self-shielding calculation is very important in reactor physics calculation. Conventional resonance calculation method has some fundamental defects, which hinders its application in some problems. The Hyperfine Energy Group Resonance Calculation Method is studied in this paper and a code named UFOP is developed based on this method. In this method, the resonance energy range is divided into hyperfine energy intervals (tens of thousands) and the collision probabilities are calculated. Then the slowing-down equation is directly solved based on CPM (collision probability method). Some techniques are applied in solving the slowing-down equation for improving computational efficiency and reducing calculation error. A resonance benchmark problem with homogeneous and infinite material is calculated to validate the accuracy of the computation code and the hyper-fine group cross-section library utilized in the code. A PWR fuel cell is also calculated and the results are compared with MCNP. The results show good accuracy of this method and the validity of UFOP code.Copyright

Total Uncertainty Analysis for PWR Assembly Based on the Statistical Sampling Method

Abstract The nuclear-data uncertainties impact the best-estimate predictions of the nuclear reactor system. In this paper, total uncertainty analyses have been performed for the TMI-1 assembly at both hot zero-power and hot full-power conditions to evaluate the impacts of nuclear-data uncertainties on the predictions of lattice calculations, based on the statistical sampling method. With an improved multigroup cross-section perturbation model, the contributions of various basic cross sections to the uncertainties of k∞ and two-group macroscopic cross sections are obtained. For the total uncertainty analyses, a 172-group cross-section covariance library produced from ENDF/B-VII.1 is used to generate the samples for the multigroup microscopic cross-section library, and DRAGON 5.0 is applied to perform lattice calculations for each sample. The numerical results show that the relative uncertainty of k∞ can reach about 4.7‰ using the vp covariance matrix of 235U-v and 7.1‰ using the vt covariance matrix of 235U-v. The relative uncertainties of two-group macroscopic cross sections vary from about 2.9‰ (for the total cross section of the thermal group) to about 11.9‰ (for the scattering cross section from the fast group to the thermal group). Moreover, through detailed analysis toward uncertainty origins, it has been observed that 235U, 238U, 16O, and 1H are the four most significant contributors, and the uncertainties of 235U-(v, σf, σγ), 238U-(σγ, σ(n,inel), σ(n,elas), v), 16O-(σ(n,elas)), and 1H-(σ(n,elas), σγ) are the most significant cross-section contributors.

Preconditioned multigroup GMRES algorithms for the variational nodal method

Abstract Combinations of three approaches are examined as options to replace the algorithms presently employed in the variational nodal code VARIANT. They are preconditioned Generalized Minimal Residual (GMRES) algorithms, parallelism in energy, and Wielandt acceleration. Together with partitioned matrix and Gauss-Seidel (GS) preconditioners, two GMRES algorithms are formulated to replace the upscattering iteration and facilitate energy parallelism and Wielandt acceleration. The GMRES algorithms are tested on two-dimensional thermal and fast reactor diffusion problems. The two GMRES algorithms yield higher efficiencies in energy group parallelization and Wielandt acceleration than simple parallelization of the existing GS algorithm. With preconditioning the GMRES algorithms reduce the total computing time by a factor of 2 to 4 and in some cases by a factor of >10. A multilevel iteration optimization scheme is investigated that automatically adjusts the relative error tolerance of the inner iterations according to the estimated convergence rate of the corresponding outer iterations and updates the Wielandt shift magnitude as the calculations progress. Numerical results based on large two-dimensional thermal and fast reactor diffusion problems demonstrate that automated optimization of the multilevel iterative processes reduces iteration numbers by as much as an order of magnitude.