[en] Highlights: • A comprehensive evaluation of the Sanchez-MOC method is conducted. • The limitation of the high packing fraction is found in the Sanchez-MOC method. • The case that filled particle sizes follow a certain distribution is studied initially. The double heterogeneity and spatial randomness make the conventional deterministic method difficult to treat the dispersed particulate fuel. The Sanchez-Pomraning method introduces the renewal equation into the collision probability method (CPM) and the method of characteristic (MOC). Since the Sanchez-MOC method was published, the comprehensive evaluation of the method has been absent. This work devotes to evaluating comprehensively the accuracy of the Sanchez-MOC method in different conditions, such as dispersed particulate fuel with different packing fractions, different particle sizes and different shapes of fuel. The Sanchez-MOC method has the capability of treating multi-type particles co-existing problems, but cannot treat the problem in larger than 50% packing fraction. Thanks to these features, the problem that filled particles follow a certain size distribution is tested tentatively. These numerical verifications indicate that the Sanchez-MOC method can treat the double heterogeneity problems as a reliable transport solver in a high accuracy.

[en] The requirements for extensive computing time and large computer memory have been recognized as major deficiencies of the collision probability method. This paper presents two methods for speeding up the calculations in the above two areas. One method is concerned with the calculation of collision probabilities, while the other allows a faster solution of resulting systems of linear algebraic equations. In two-dimensional geometry, the collision probabilities are expressed as linear combinations of Bickley functions, Ki₃(x), the evaluation of which is the main time consumer for small and medium size problems. The new method presented here applies a numerical integration of the polar angle instead of Bickley function calculation. As a result, the algorithm is more robust and twice as fast. The solution of the collision probability equation is usually obtained by direct methods of matrix decomposition. The computing time is proportional to the third degree of the number of unknowns and increases rapidly with the increase of the problem size. To speed-up the calculation, the within-group matrix is subdivided into a number of blocks. The solution is obtained iteratively by block-matrix iteration using the traditional Gauss-Seidel method. Test results show a decrease in computing time by more than one order of magnitude. (author)

[en] Analytical formulae have been derived for the collision probabilities of homogeneous finite cylinders and cuboids. The formula for the finite cylinder contains double integrals, and the formula for the cuboid only single integrals. Collision probabilities have been calculated by means of the formulae and compared with values obtained by other authors. It was found that the calculations using the analytical formulae are much quicker and give higher accuracy than Monte Carlo calculations

[en] The method of characteristic direction probabilities (CDP) is based on a modular ray tracing technique which combines the benefits of the collision probability method (CPM) and the method of characteristics (MOC). This past year CDP was implemented in the transport code MPACT for 2-D and 3-D transport calculations. By only coupling the fine mesh regions passed by the characteristic rays in the particular direction, the scale of the probabilities matrix is much smaller compared to the CPM. At the same time, the CDP has the same capacity of dealing with the complicated geometries with the MOC, because the same modular ray tracing techniques are used. Results from the C5G7 benchmark problems are given for different cases to show the accuracy and efficiency of the CDP compared to MOC. For the cases examined, the CDP and MOC methods were seen to differ in k_eff by about 1-20 pcm, and the computational efficiency of the CDP appears to be better than the MOC for some problems. However, in other problems, particularly when the CDP matrices have to be recomputed from changing cross sections, the CDP does not perform as well. This indicates an area of future work. (authors)

[en] The study of high-energy heavy ion collisions is a fundamental pursuit in modern nuclear and particle physics, offering a unique window into matter’s behavior under extreme conditions. These collisions reveal intricate momentum transfer phenomena, shaping the evolution of the collision through complex particle interactions. The possible reasons behind the observations made are summarized as: 1. In the peripheral collisions, the medium created in collision experiments may have a shorter lifespan due to lower energy densities. This shorter duration means that the medium has less time to evolve and can retain the initial anisotropies present in the initial conditions, contributing to a larger ρ_a. 2. In the peripheral collisions, reduced late stage particle interactions effectively preserve initial anisotropies, leading to a higher ρa in the final particle distributions

[en] Optimal configuration of spatial points was chosen in respect to the total number needed for integration of reactions in the reactor cell. Previously developed code VESTERN was used for numerical verification of the method on a standard reactor cell. The code applies the collision probability method for calculating the neutron flux distribution. It is shown that the total number of spatial points is twice smaller than the respective number of spatial zones needed for determination of number of reactions in the cell, with the preset precision. This result shows the direction for further condensing of the procedure for calculating the space-energy distribution of the neutron flux in a reactors cell

[en] The adjoint transport solution algorithm based on the method of cyclic characteristics (MOCC) is developed for the heterogeneous 2-dimensional geometries. The adjoint characteristics equation associated with a cyclic tracking line is formulated, then a closed form for adjoint angular flux can be determined. The acceleration techniques are implemented using the group-reduction and group-splitting techniques. To demonstrate the efficacy of the algorithm, the calculations are performed on the 17*17 PWR and Watanabe-Maynard benchmark problems. Comparisons of adjoint flux and k_eff results obtained by MOCC and collision probability (CP) methods are performed. The mathematical relationship between pseudo-adjoint flux obtained by CP method and adjoint flux by MOCC method is presented. It appears that the pseudo-adjoint flux by CP method is equivalent to the adjoint flux by MOCC method and that the MOCC method requires lower computing time than the CP method for a single adjoint flux calculation

[en] The method of characteristics direction probabilities combines the geometry flexibility of the method of characteristics and the computing efficiency of the collision probability method. An angular depended boundary averaging is proposed to further improve the efficiency. After that, an accuracy model for the boundary angular flux is introduced which would be used to find the best boundary averaging method. And then, a performance model which explicitly compares the performance of the method of characteristics probabilities to the method of characteristics is given. The numerical results show that model given is correct and the method of characteristics direction probabilities gets great benefits. (authors)

[en] The polar-wise interface current characteristics method (PICCM) is introduced in order to reduce the computational inefficiencies of ray tracing inherent in the Collision Probability Method (CPM) and the Method of Characteristics (MOC). The escape probability and contribution probability equations are introduced to determine the polar-wise outgoing current and region fluxes for each cell. These two equations are derived directly from the MOC solution of each cell. For the pin-cell problem, the results show that the computational time is reduced by about 20% with PICCM compared to the original MOC solver of the CPM3 code, with only minor differences in the prediction of the eigenvalue. For a 14x14 PWR lattice problem, PICCM provides about a factor of 6 reduction in the execution time compared to MOC, with less than 20 pcm of reactivity error and a maximum 0.32% error of the pin power distribution. (author)

[en] A comprehensive code system VISWAM for physics analysis of current and future power reactors is being developed. The lattice analysis module of VISWAM code system can analyze fuel assembly (FA) cells in hexagonal, square or ring cluster geometry. The lattice analysis method initially incorporated in VISWAM code for fuel assembly (FA) analysis is based on a combination of 1D multigroup transport and a 2D few group diffusion theory. The FA consists normally of a number of heterogeneities like water pins, strong absorber pins like Gd and control absorber pins. There is a strong flux gradient between such heterogeneities and neighbouring pins which is not accurately predicted using diffusion theory. To improve this, an advanced lattice analysis method has been incorporated in VISWAM code system in hexagonal geometry. The new method is the interface current method based on 2D collision probability (CP). In this method, we have used the 2D CP method at individual lattice cells level and different lattice cells are linked using interface currents with double P2 (DP2) expansion of angular flux at cell boundaries. The FA cell in hexagonal geometry with irregular lattice structure at boundaries is modeled exactly. In this report we present the analysis of a heterogeneous benchmark problem that is typical of a high temperature test reactor (HTTR). The present report describes in detail the advanced methodology incorporated in VISWAM and the comparison of results for the HTTR benchmark with published results. (author)