Axial SPN and Radial MOC Coupled Whole Core Transport Calculation

The Simplified P_N (SP_N) method is applied to the axial solution of the two-dimensional (2-D) method of characteristics (MOC) solution based whole core transport calculation. A sub-plane scheme and the nodal expansion method (NEM) are employed for the solution of the one-dimensional (1-D) SP_N equations involving a radial transverse leakage. The SP_N solver replaces the axial diffusion solver of the DeCART direct whole core transport code to provide more accurate, transport theory based axial solutions. In the sub-plane scheme, the radial equivalent homogenization parameters generated by the local MOC for a thick plane are assigned to the multiple finer planes in the subsequent global three-dimensional (3-D) coarse mesh finite difference (CMFD) calculation in which the NEM is employed for the axial solution. The sub-plane scheme induces a much less nodal error while having little impact on the axial leakage representation of the radial MOC calculation. The performance of the sub-plane scheme and SP_N nodal transport solver is examined by solving a set of demonstrative problems and the C5G7MOX 3-D extension benchmark problems. It is shown in the demonstrative problems that the nodal error reaching upto 1,400 pcm in a rodded case is reduced to 10pcm by introducing 10 sub-planes per MOC plane and the transport error is reduced from about 150pcm to 10pcm by using SP₃. Also it is observed, in the C5G7MOX rodded configuration B problem, that the eigenvalues and pin power errors of 180 pcm and 2.2% of the 10 sub-planes diffusion case are reduced to 40 pcm and 1.4%, respectively, for SP₃ with only about a 15% increase in the computing time. It is shown that the SP₅ case gives very similar results to the SP₃ case.     

Applications of the multidimensional equations to complex fuel assembly problems

For the analysis of reactors with complex fuel assemblies or fine mesh applications as pin by pin neutron flux reconstruction, the usual approximation of the neutron transport equation by the multigroup diffusion equation does not provide good results. A classical approach to solve the neutron transport equation is to apply the spherical harmonics method obtaining a finite approximation known as the P_L equations. In this line, a nodal collocation method for the discretization of these equations on a rectangular mesh is used in this paper to analyse reactors with MOX fuel assemblies. Although the 3D P_L nodal collocation method becomes feasible due to the improvements in computer hardware, a complete treatment of the detailed structure of the fuel assemblies in actual three-dimensional geometry is still prohibitive, thus, an assembly homogenization method is necessary. A homogenization method compatible with our multidimensional P_L code is proposed and tested performing heterogeneous and homogenized calculations. In this work, we apply the method to 2D complex fuel assembly configurations.     

轻水堆燃料组件计算程序包TPFAP

TPFAP是一个同时适用于PWR和BWR的穿透几率法燃料组件燃耗计算程序包。它首先利用碰撞几率方法在库能群结构下完成三区或四区圆环几何的栅元输运计算。载钆燃料棒或硼棒可燃毒物栅元的有效吸收截面由微燃耗程序CMB产生,两维穿透几率法组件计算是在(x,y)几何下进行。基模计算用来考虑中子泄漏修正。根据反应率等效,计算组件等效扩散参数。在每一燃料棒和可燃毒物棒进行燃耗计算,TPFAP给出每一燃耗步的组件和栅元少群截面、功率分布,提供核设计和安全分析所需参数。     

Super-homogenisation factors in pinwise calculations by the reactor dynamics code DYN3D

Super-homogenisation (SPH) factors were generated by a modified version of the code DYN3D for PWR fuel assemblies in hot-zero-power states defined in the OECD MOX/UO2 Benchmark. SPH factors averaged for each pin-material type and factors for each individual pin position were produced. The application of the SPH factors improves the accuracy of DYN3D calculations, especially for configurations with control rods inserted.     

Study on double heterogeneity effect of Pu-rich agglomerates in mixed oxide fuel using cross section homogenization method for particle-dispersed media

A cross section homogenization method for media containing randomly and uniformly dispersed particles, which was originally developed by Shmakov et al., has been applied to MOX fuels containing Pu-rich agglomerates. This method (Shmakov’s method), which is incorporated into a continuous-energy Monte Carlo code MCNP, has been applied to lattice calculations of an infinite MOX fuel rod array. Shmakov’s method can accurately reproduce the criticality calculation results for an explicit heterogeneous arrangement of Pu-rich agglomerates. A correction factor that Shmakov’s method defines to obtain an effective microscopic cross section provides a proper quantitative indication of the double heterogeneity of MOX fuels containing Pu-rich agglomerates. The correction factors exhibit an obvious double heterogeneity effect of Pu-rich agglomerates dispersed in MOX fuel pellets. The effective microscopic cross sections of plutonium isotopes in MOX fuels containing Pu-rich agglomerates are significantly reduced due to the self-shielding effect as compared to the homogeneous MOX fuel model. However, the double heterogeneity effect of Pu-rich agglomerates on k_eff seems to be unexpectedly minor because the underestimate of the reaction rates in the resonance energy range is offset by the overestimate of the reaction rates in the thermal energy range.     

Homogenization-free reactor core analysiswith general first collision probabilities method

The C5G7 MOX Benchmarkfor current codes has been proposed as a basis to test the ability of current transport codes to teat reactor core problems without spatial homogenization. This is a seven-group form of the C5G7 MOX fuel assembly problem specified by Cavarec et.al. There are four fuel assemblies, two contain UO₂ fuel elements and two contain MOX fuel elements. Seven group cross sections for different kinds of fuel (three enrichment of MOX and UO₂), the guide tubes, the fission chambers and moderator are given. Thus this benchmark is just a mathematical test that allows testing the accuracy of the neutron transport equation solution with different methods and codes. In this paper the General First Collision Probabilities Method (GFCPM) is used to analyze the two-dimensional configuration of this benchmark. A linear flux approximation is used in the reflector. Different calculation schemes in the reflector region have been used. The output results, K_eff and the pin powers have been analyzed. The convergence of the results has been analyzed both as a function of the subdivision scheme of the reflector region and of the number of points in the calculation scheme for general first collision probabilities. Comparison has been carried out for K_eff and pin powers both with the reference results (external convergence) and with the results of different approximations of GFCPM (internal convergence).     

Multigroup pin power reconstruction with two-dimensional source expansion and corner flux discontinuity

A pin power reconstruction method that is readily applicable to multigroup problems with superior accuracy is presented for applications involving rectangular fuel assemblies. It employs a two-dimensional (2D), fourth order Legendre expansion of the source distribution that naturally leads to a group-decoupled, 2D semi-analytic solution of the neutron diffusion equation. The four surface average currents and four corner fluxes are used as the boundary conditions to uniquely specify the homogenous solution. The corner fluxes and source expansion coefficients are iteratively determined using the condition of corner point balance and the orthogonal property of the Lengedre functions. Corner discontinuity is incorporated in the calculation of the corner fluxes which turns out to be very effective in the cases of enrichment zoning. The accuracy of the proposed method is assessed by performing the two-step core calculations for the L336C5, C5G7MOX, and MOX core transient benchmark problems and then by comparing with the direct whole-core transport solutions. The results indicate that the proposed method is as accurate as the fully analytic method and works well irrespective the number of groups. However, it is also noted that somewhat larger errors are inevitable at the peripheral assemblies near the reflector in which the error associated with a prioi generation of the homogenized cross-sections and form functions is not trivial.     

Wavelet-based angular dependence analysis of the heterogeneous calculation on MOX fuel lattice

A wavelet-based transport method is developed to satisfy the high order angular approximation, which has been proved to be necessary in the heterogeneous calculation of MOX fuel lattice. Based on the new angular discretization scheme, the angular dependence of flux is analysed to find out the origin of complicated angular anisotropy and its effects on the heterogeneous calculation. Both of the geometric and neutronic effects are investigated quantitatively to find out the angular dependence in heterogeneous calculations. Comparisons between the traditional S_N angular discretization scheme and wavelet-based scheme are analysed to indicate the challenges brought from the MOX fuel lattice heterogeneous calculation. An effective solution is given by using wavelets in the angular discretization of neutron transport equation. Improvements of high order angular approximation are suggested.     

On the influence of spatial discretization in LWR-burnup calculations with HELIOS 1.9 – Part II: Mixed oxide (MOX) fuel

Cell and burnup calculations are the basis for all deterministic static and transient 3D full core calculations for different operational states of the reactor. The arising differences in the integral transport solution (neutron flux and k_inf) for different discretization strategies during the burnup of mixed oxide (MOX) fuel due to different spatial discretization are demonstrated. The influence of different discretization strategies on the calculation of homogenized few group cross-sections is investigated. The influence of the discretization strategies on the calculation time is evaluated.     

Pin power reconstruction for CANDU reactors using a neuro-fuzzyinference system

A neuro-fuzzy inference system has been developed for reconstructing fuel pin powers from Canada deuterium uranium (CANDU) core calculations performed with a coarse-mesh finite difference diffusion approximation and single-assembly lattice calculations. The neuro-fuzzy inference system is trained by a genetic algorithm and a least-squares method using the partial core calculation results of two 6×6 fuel bundle models. Verification tests have been performed for two partial core benchmark problems composed of other 6×6 and 3×3 fuel bundles. The reconstructed pin powers are compared with the reference solutions obtained with the detailed collision probability calculations using the HELIOS lattice analysis code. The results indicate that the proposed reconstruction algorithm is accurate, yielding the error due to the reconstruction scheme of less than 0.5%     

Approximate Treatment of Thermal Expansion Effect in Lattice Transport Calculations

In this paper, thermal expansion effect on neutronics characteristics is approximately taken into account by a correction on cross sections. Dimensions of reactor core components depend on their temperatures due to the thermal expansion phenomena. However, the variation of calculation geometry requires considerable computational load for trajectory based lattice transport calculations such as the characteristics method since ray tracing must be re-executed. Therefore, if a correction on cross sections can accurately capture the effect of geometrical variation due to the thermal expansion, computation time of a lattice transport calculation that treat temperature variation can be reduced. Three different corrections on cross sections were tested in PWR fuel assembly geometry using UO₂ and MOX fuels. It was found that the correction of cross sections that preserves neutron attenuation in a region almost reproduce the reference calculation that explicitly considers geometrical variation due to the thermal expansion. The result of this paper will be useful for lattice calculations in production analysis since material temperatures are frequently changed in such analysis to cover various reactor conditions.     

Neutron Anisotropic Scattering Effect in Heterogeneous Cell Calculations of Light Water Reactors

A study on the anisotropic scattering effects in heterogeneous square cells of light water reactors has been performed using the characteristics method. It was found that the effects of the anisotropic scattering were relatively large for the MOX fuel cell because of the large neutron current from the moderator to the fuel region and the k _inf value by the P₀ calculation became 0.10–0.16% larger than that by the P₅ calculation. With the transport correction, the k _inf difference from the P₅ calculation became even larger than that from the P₀ calculation and the k _inf value by the transport correction became 0.18–0.25% larger than that by the P₅ calculation for the MOX fuel cell. The transport corrected self-scattering cross sections of the moderator region become smaller than the non-transport corrected ones and the angular flux distribution becomes more anisotropic with the transport correction. Therefore, more neutrons toward the moderator region between the fuel pellets can slow down to the lower energy region with the transport correction. As a result, the k _inf value by the transport correction becomes larger than that by the P₀ calculation, which is opposite effect to that by the P₅ calculation.     

Trend in criticality calculation results of borated-water-moderation experimental cores relating to 10B(n, α) cross section

ABSTRACT

In connection with the accuracy of the ¹⁰B(n, α) cross section in the thermal- and epithermal-neutron energy regions, criticality calculation results were examined for six benchmark sets of light-water-moderation critical experiments of UO₂ and MOX fuel lattice cores with un-borated and borated water. Two of the benchmark sets were those implemented in the Tank-Type Critical Assembly (TCA). The others were taken from the International Handbook of Evaluated Criticality Safety Benchmark Experiments (ICSBEP), and the International Handbook of Evaluated Reactor Physics Benchmark Experiments (IRPhEP). The enrichments of the UO₂ fuel range from 1.9 wt% to 2.6 wt%, and the Pu contents of the MOX fuel do from 2.0 to 6.6 wt%. The boron concentrations in water are up to 1511 ppm. The effective neutron multiplication factors (k_eff ) were taken from the published documents. They were calculated with continuous-energy Monte Carlo calculation codes in combination with JENDL-4.0, and other evaluated nuclear data libraries. It was confirmed that the k_eff values of the critical cores increased with the boron concentrations, which indicates that the ¹⁰B(n, α) cross section in the thermal- and epithermal-neutron energy regions should be larger than those in JENDL-4.0 and other libraries.     

Effect of anisotropic scattering in neutronics analysis of BWR assembly

The anisotropic scattering effect to keff is studied for UO₂ and MOX fueled BWR assemblies. The anisotropic scattering effect increases the assembly k_∞ by 0.44% Δk for the UO₂ assembly with 0% void fraction, and by 0.21% Δk for the MOX assembly with 0% void fraction. This is because the anisotropic scattering effect flattens the intra-assembly thermal flux, and the absorption rate in the surrounding water gap is decreased, but the absorption rates in the MOX fuel rods are increased compared to the UO₂ rods. Therefore, the total decrease in absorption rates in the UO₂ assembly is relatively large, and the k_∞ is increased in the UO₂ assembly. The dependence of the anisotropic scattering effect on the void fraction is investigated, and the significant difference of 0.62% Δk/k is found for the 0% and the 80% void fractions. The BWR assemblies with Gd rods are also considered. Furthermore, the usefulness of the transport cross section is investigated, and it is found that the transport cross section gives reasonable anisotropic scattering effect, though not satisfactory.     

Transport Calculations of MOX and UO2 Pin Cells by the Method of Characteristics

A new transport theory code for two-dimensional calculations of both square and hexagonal fuel lattices by the method of characteristics has been developed. The ray tracing procedure is based on the macroband method, which permits more accurate spatial integration in comparison to the equidistant method of tracing. The neutron source within each region is approximated by a linear function and linearly anisotropic scattering can be optionally accounted for. Efficient new techniques for both azimuthal and polar integration are presented. The spatial discretization problem in case of P ₁-scattering has been studied. Detailed analyses show that the P ¹-scattering in case of regular infinite array of fuel cells is significant, especially for MOX fuel, while the transport correction is inadequate in case of real geometry multi-group calculations. Finally, the complicated nature of the angular flux in MOX and UO₂ fuel cells is demonstrated.     

Calculations of thermodynamic properties of PuO2 by the first-principles and lattice vibration

Plutonium dioxide (PuO₂) is a key compound of mixed oxide fuel (MOX fuel). To predict the thermal properties of PuO₂ at high temperature, it is important to understand the properties of MOX fuel. In this study, thermodynamic properties of PuO₂ were evaluated by coupling of first-principles and lattice dynamics calculation. Cohesive energy was estimated from first-principles calculations, and the contribution of lattice vibration to total energy was evaluated by phonon calculations. Thermodynamic properties such as volume thermal expansion, bulk modulus and specific heat of PuO₂ were investigated up to 1500 K.     

Determination of Li-6/Li-7 Isotopic Ratio by Means of Thermogravimetric Analysis for Lithium Sulphate Monohydrate

Plutonium concentrations and burnup at Pu spots were calculated in U-Pu mixed oxide (MOX) fuel pellets for light water reactors with the neutron transport and burnup calculation code VIMBURN. The calculation models were suggested for Pu spots and U matrices in a heterogeneous MOX fuel pellet. The calculated Pu concentrations and burnup at Pu spots were compared with the PIEs data in a MOX pellet (38.8 MWd/kgHM). The calculated Pu concentrations agreed by 5–18% with the measured ones, and the calculated burnup did by less than 10% with the estimated one with the measured Nd concentrations. Commercial PWR types of MOX fuels were also analyzed with the calculation code and the models. Burnup at Pu spot increased as the distance was greater from the radial center of a MOX fuel pellet. Burnup at Pu spots in the peripheral region became 3–5 times higher than pellet average burnup of 40 MWd/kgHM. The diameters (20–100 μm) of Pu spots were not found a significant factor for burnup at Pu spots. In the outer half volume region (outer than r/r _o=0.7) of a MOX fuel pellet, burnup at Pu spots exceeded 70MWd/kgHM (the threshold burnup of microstructure change in UO₂ fuel pellet) at pellet average burnup of 1430 MWd/kgHM.     

A VVER-1000 LEU and MOX assembly computational benchmark analysis using the lattice burnup code EXCEL

Utilization of Mixed Uranium–Plutonium Oxide (MOX) fuel in VVER-1000 reactors envisages the core physics analysis using computational methods and validation of the related computer codes. Towards this objective, an international experts group has been established at OECD/NEA. The experts group facilitates sharing of existing information on physics parameters and fuel behaviour. Several benchmark exercises have been proposed by them with intent to investigate the core physics behaviour of a VVER-1000 reactor loaded with 2/3rd of low enriched uranium (LEU) fuel assemblies (FA) and 1/3rd of weapons grade mixed oxide (MOX) FA. In the present study an attempt is made to analyse ‘AVVER-1000LEUandMOXAssemblyComputationalBenchmark’ and predict the neutronics behaviour at the lattice level. The lattice burnup code EXCEL, developed at Light Water Reactor Physics Section, BARC is employed for this task. The EXCEL code uses the 172 energy group ‘JEFF31GX’ cross-section library in WIMS-D format. Assembly level fuel depletion calculations are performed up to a burnup of 40 MWD/kg of heavy metal (HM). Studies are made for the parametric variations of fuel and moderator temperatures, coolant density and boron content in the coolant. Both operational and off-normal states are analysed to determine the corresponding infinite neutron multiplication factor (k_∞). Pin wise isotopic compositions are computed as a function of burnup. Isotopic compositions in different annular regions of Uranium–Gadolinium (UGD) pin, fission rate distributions in UGD, UO₂ and MOX pin cells are also computed. The predicted results are compared with the benchmark mean results.     

Helios, current coupling collision probability method, applied for solving the NEA C5G7 MOX benchmark

As part of an effort to test the ability of current transport codes to treat reactor core problems without spatial homogenization, the lattice code HELIOS was employed to perform criticality calculations. The test consists in seven-group calculations of the C5 MOX fuel assembly problem specified by Cavarec et. al. [1]. This problem, known as C5G7 MOX Benchmark, is described in the Benchmark Specification [2] and comprises two cases — two and three-dimensional geometry. There are four fuel assemblies — two with MOX fuel, the other two with UO₂ fuel. Each fuel assembly is made up of a 17×17 lattice of square fuel-pin cells. Fuel pin compositions are specified in the Benchmark Specification, which also provides seven-group transport-corrected isotropic scattering cross-sections for U0₂, the three MOX enrichments, the guide tubes, the fission chamber and the moderator. This paper preset is the methodology employed in solving the C5G7 MOX Fuel Assembly Problem using the transport code HELIOS.     

Twodant solutions for the 2-D C5G7 MOX benchmark

The C5G7 MOX benchmark was proposed to test the ability of commercial transport codes to treat reactor core problems without spatial homogenization. The benchmark requires solutions in the form of normalized pin powers as well as the eigenvalue. In the work, the two-dimensional benchmark calculation using the TWODANT code within the DANTSYS code package has been performed with proper spatial and angular approximations. The TWODANT code solves the multigroup discrete ordinates form of the Boltzmann transport equation in two-dimensional geometry. The calculation results show a good agreement in comparison with the reference solution obtained from a seven-group MCNP calculation. In addition, sensitivity studies on mesh and angular refinements have been performed to produce a higher quality solution. In the results, it is found that in the TWODANT calculation the spatial approximation in a staircase form of the circular fuel pin with relatively high quadrature order of S_N is a viable method for solving the2-D C5G7 benchmark.