A new procedure to generate resonance integral table with an explicit resonance interference for transport lattice codes

Resonance interference could not be considered explicitly in the conventional resonance treatment employing subgroup and direct resonance integral methods when using coarse energy group structure. This problem comes from the lack of information for the resonance shapes of resonant nuclides in the resonance interference formulas. As energy group boundaries get coarser, inaccuracy in estimating self-shielded cross sections with resonance interference gets bigger. A new method has been proposed to conserve the self-shielded cross sections for each group through an explicit consideration of resonance interference effect, which results in a good accuracy in predicting the multiplication factor. This method can be applicable to various mixing combinations of constituent resonant nuclides with resonance interference and can cover wide dilution range. The MERIT code has been used to generate resonance integral tables and to estimate resonance interference effects. And the 2-D transport lattice code KARMA has been used to perform sample calculations to see the effectiveness of the newly developed method. Sample calculations have been performed for single pins with various temperatures, ²³⁵U enrichments and dilution levels with the 47 and 190 energy group structures. The computational results show that this method is able to estimate self-shielded cross sections in each coarse energy group accurately for various temperatures and various geometry and composition configurations.     

Iterative resonance self-shielding methods using resonance integral table in heterogeneous transport lattice calculations

This paper describes the iteration methods using resonance integral tables to estimate the effective resonance cross sections in heterogeneous transport lattice calculations. Basically, these methods have been devised to reduce an effort to convert resonance integral table into subgroup data to be used in the physical subgroup method. Since these methods do not use subgroup data but only use resonance integral tables directly, these methods do not include an error in converting resonance integral into subgroup data. The effective resonance cross sections are estimated iteratively for each resonance nuclide through the heterogeneous fixed source calculations for the whole problem domain to obtain the background cross sections. These methods have been implemented in the transport lattice code KARMA which uses the method of characteristics (MOC) to solve the transport equation. The computational results show that these iteration methods are quite promising in the practical transport lattice calculations.     

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

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.     

Ultra-fine-group resonance treatment using equivalent Dancoff-factor cell model in lattice physics code GALAXY

In order to achieve highly accurate resonance calculations with short computation time , a new ultra-fine-group resonance calculation method is developed. The ultra-fine-group method has a limitation in practical design applications of large and complicated geometries in fuel assembly level due to its long computation time. Therefore, we developed an enhanced one-dimensional (1D) cylindrical pin-cell model to achieve both high calculation accuracy and short computation time. In the enhanced 1D cylindrical pin-cell modeling, moderator radius is adjusted to preserve each fuel pellet's Dancoff factor obtained in the exact 2D fuel lattice arrangement. We call this model the ‘equivalent Dancoff-factor’ cell model. This model can accurately consider heterogeneity effects in PWR fuel assemblies and can represent effective cross sections obtained by the ultra-fine-group calculations in the complicated 2D square lattice arrangements. The present method is implemented with Mitsubishi Heavy Industries, Ltd. lattice physics code GALAXY. From the comparisons of neutron multiplication factors and pin power distributions between GALAXY and a continuous-energy Monte Carlo code, applicability of the present method to lattice physics calculations is confirmed. Application of GALAXY with the present method achieves high accuracy with short computation time in normal operations and accident conditions including low moderator density conditions.     

Development of a Fine and Ultra-Fine Group Cell Calculation Code SLAROM-UF for Fast Reactor Analyses

A cell calculation code SLAROM-UF has been developed for fast reactor analyses to produce effective cross sections with high accuracy in practical computing time, taking full advantage of fine and ultra-fine group calculation schemes.

The fine group calculation covers the whole energy range in a maximum of 900-group structure. The structure is finer above 52.5 keV with a minimum lethargy width of 0.008. The ultra-fine group calculation solves the slowing down equation below 52.5 keV to treat resonance structures directly and precisely including resonance interference effects. Effective cross sections obtained in the two calculations are combined to produce effective cross sections over the entire energy range.

Calculation accuracy and improvements from conventional 70-group cell calculation results were investigated through comparisons with reference values obtained with continuous energy Monte Carlo calculations. It was confirmed that SLAROM-UF reduces the difference in k-infinity from 0.15 to 0.01% for a JOYO MK-I fuel subassembly lattice cell calculation, and from ?0.21 % to less than a statistical uncertainty of the reference calculation of 0.03% for a ZPPR-10A core criticality calculation.     

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.     

Resonance self-shielding method using resonance interference factor library for practical lattice physics computations of LWRs

This paper presents a new method of resonance interference effect treatment using resonance interference factor for high fidelity analysis of light water reactors (LWRs). Although there have been significant improvements in the lattice physics calculations over the several decades, there exist still relatively large errors in the resonance interference treatment, in the order of ～300 pcm in the reactivity prediction of LWRs. In the newly developed method, the impact of resonance interference to the multi-group cross-sections has been quantified and tabulated in a library which can be used in lattice physics calculation as adjustment factors of multi-group cross-sections. The verification of the new method has been performed with Mosteller benchmark, UO₂ and MOX pin-cell depletion problems, and a 17×17 fuel assembly loaded with gadolinia burnable poison, and significant improvements were demonstrated in the accuracy of reactivity and pin power predictions, with reactivity errors down to the order of ～100 pcm.     

双重非均匀子群参数制作研究

弥散颗粒燃料元件中燃料颗粒以随机形式弥散在基体中,难以获得确定几何。同时由于共振自屏现象的存在,呈现出一种双重非均匀系统。当前均匀系统产生的共振积分在双重非均匀系统中使用时,会在较低的共振能群产生一定的共振计算误差。为满足现有组件计算程序直接进行双重非均匀性共振计算的需求。基于Sanchez-Pomraning模型下的特征线固定源计算方法,建立一套双重非均匀共振积分表,最后结合子群方法实现随机介质燃料元件的共振计算。数值结果表明,考虑双重非均匀性产生的积分表,在相同的输运条件下和积分表的适用范围内,由子群共振部分对keff计算带来的绝对偏差能保持在200 pcm内。该工作的意义是对于一些不宜改动的传统组件程序,如HELIOS,通过在线修改共振积分表和子群参数,从而使其直接进行弥散颗粒燃料问题的计算成为可能。     

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。     

A hybrid method to generate few-group cross sections for fast reactor analysis

The accuracy of fast reactor core calculation is usually determined by the accuracy of self-shielded few-group cross sections. To further improve the accuracy of cross section generation, a hybrid method is proposed. In the hybrid method, the Monte-Carlo method is used to deal with the resonance effect in both the resolved and unresolved resonance range. The self-shielded ultrafine-group total, fission and elastic scattering cross sections are tallied by the Monte-Carlo method. The scattering transfer matrices are then generated in a synthesis way by using the tallied elastic scattering cross sections and a problem-independent elastic scattering function. The angular flux moments for the group condensation are calculated in an explicit deterministic way. Several tests are done to verify the hybrid method. The results show that the hybrid method avoids the disadvantages of both the traditional deterministic method and the pure Monte-Carlo method. It is a more accurate method to generate the few-group cross sections for fast reactor cores.     

Evaluation of the lattice physics code ‘MURLI’ for light water systems

This paper presents a comprehensive evaluation of the lattice physics code ‘MURLI’ for calculating the nuclear characteristics of light water lattices. The code uses the interface currents approach with cosine current approximation to solve the neutron transport equation in a cylindricalized cell. This approach offers a substantial reduction in computer storage and computational time requirements compared to the other approach of multiregion first flight probabilities. The WIMS group-structure and constants are used to treat the energy variable. The theoretical predictions of the integral parameters like reactivity and relative reaction rates agree excellently with the relevant measured values for water reactor physics experiments. The soundness of the physical formulation of MURLI and the adequacy of the cross sections have been well established by this study.     

An improved fitting method for subgroup parameters based on the heterogeneous cells

An accurate subgroup parameters fitting method, where background cross sections obtained based on heterogeneous cells are used to fit the subgroup level and subgroup weight, is proposed in this paper. Due to the dependence of background cross section on the subgroup level, the calculation of the subgroup parameters is a nonlinear problem, which causes the iteration between fitting subgroup parameters and updating background cross sections. The cubic spline interpolation method is used to update the background cross sections to avoid frequently solving fixed source equations. In the fitting process, the negative subgroup parameters are often obtained, and the accuracy of the subgroup parameters is very sensitive to the iterative initial values of subgroup levels. To avoid these problems, additional constraints ensuring positive subgroup parameters and guaranteeing numerical stability are added to the optimization function. Penalty function method is used to convert the optimization problem with constraints into the one without constraints, making the problem easy to be solved. The proposed method is tested against the problems of pin cell, pressurized water reactor assemblies and plate-type assembly. The numerical results show that the self-shielded cross sections calculated by the proposed method agree well with those by Monte Carlo code.     

Effects of Mutual Shielding on Resonance Integrals of 235U, 238U, 239Pu and 240pu below 150 eV

The results of some quantitative studies on resonance interference are presented. The calculations were performed on a FORTRAN IV program RICM2, which solves numerically the slowing down of neutrons over many resonance levels in a two region lattice, and gives reaction rates, average cross sections and effective resonance integrals of the nuclides concerned.

Three combinations of resonant nuclides, ²³⁵U-²³⁸U, ²³⁰Pu-²³⁸U and ²³⁹Pu-²¹⁰Pu, were considered, in conjunction with three oxide fuel rod radii, 0.2, 0.5 and 2.0 cm, the moderator (light water) to fuel volume ratio being maintained constant at 2.0. An energy range below 150eV has been covered by the present calculations. The effects of resonance interference have been found to be appreciable in this energy range.     

The treatment of resonance interference effects in the subgroup method

This paper describes the development of a method to treat resonance interference effects within the framework of the subgroup method. The new procedure provides for the treatment of multiple resonance absorbers in which the subgroup weights are determined using a least squares technique and based on the cross sections generated from a mixture of multiple resonance isotopes and a suitably wide range of background cross sections. The method was implemented in the Method of Characteristics code DeCART and validated using MCNP. In order to implement the new method, the NJOY code was used for the calculation of neutron spectra and resonance parameters in for each representative LWR mixture. The resonance parameters, lambda, of the scattering isotopes are computed not just with U-238 as the resonance isotope as in previous applications of the subgroup method, but also with U-235 as resonance isotope for the energy groups in which U-238 has no significant resonances. After developing a procedure for generating lambda factors for scattering isotopes, a method is then described for generating subgroup parameters in a homogeneous system. Again NJOY is used for resonance calculations of a set of mixtures for each resonance isotope at each selected temperature. The group average cross sections instead of the resonance integrals of these mixtures are used to generate subgroup parameters using an optimization algorithm. The generated library is then verified by comparing the solution from DeCART with the solution from MCNP. The method is then extended to a heterogeneous system. The code RMET21 is used for transport calculations for the heterogeneous system. The interference effect from the most important resonance isotopes is treated by generating subgroup weights with resonance cross sections for the cases with two resonance isotopes. The results indicate that the subgroup method can accurately represent resonance interference effects within the framework of the subgroup method.     

Dependence of Effective Resonance Integrals on Moderator Slowing Down Properties

Effective resonance integrals calculated by means of the usual approximations do not have the dependence on moderator slowing down properties which is shown by “exact” calculations.

In this paper a modified effective resonance integral is defined in such a way as to fit the exponential form of the resonance escape probability. Application of a generalised “intermediate-resonance” approximation to this modified effective resonance integral shows a dependence on the slowing down properties of the moderating nuclei in good agreement with “exact” calculations.     

Subgroup weight generation based on shielded pin-cell cross section conservation

A method for generating subgroup weights is presented employing the idea of conserving resonance shielded cross sections for representative pin cells. Consistency in the generation and usage steps of the subgroup weights is imposed in a subgroup formulation involving the subgroup fixed source problem to resolve the subgroup level and space dependent self shielding. Consequently, the subgroup level dependent background cross sections are determined and used in the generation step. The requirement of reconstructed cross section error minimization forms a constrained least square problem which is solved by the Lagrange multiplier technique augmented with a regularization term introduced to enhance solution stability. The generated subgroup weights are verified by analyzing various pin cell benchmark problems and comparing with corresponding Monte Carlo solutions. The results for the Rowlands and Mosteller Doppler-effect benchmark problems show small errors of 0.1% in the resonance cross sections and about 70 pcm in multiplication factor, and also good agreement in the reactivity vs. fuel temperature behavior. The problem of the slight over-prediction of the fuel temperature coefficients is noted and discussed as well.     

The impact of U resonance elastic scattering approximations on thermal reactor Doppler reactivity

The effects of accurate modeling of neutron scattering in ²³⁸U resonances are analyzed for typical light water reactor (LWR) and next generation nuclear plant (NGNP) lattices. An exact scattering kernel is formulated and implemented in a newly developed Monte Carlo code, MCSD (Monte Carlo slowing down), which solves a neutron slowing down in an infinite homogeneous medium and is used to generate resonance integral data used in the CASMO-5 lattice physics code. It is shown that the exact scattering kernel increases LWR Doppler coefficients by ∼10% relative to the traditional assumption of asymptotic elastic downscatter for ²³⁸U resonances. These resonance modeling improvements are shown to decrease hot full power eigenvalues by ∼200 pcm for LWRs and ∼450 pcm for NGNPs.     

An Accurate Method to Interpolate First-Flight Collision Probabilities

A method is proposed to approximately calculate first-flight collision probabilities for use in resonance integral calculations with good accuracy and within a short computing time. In the proposed method, collision probabilities at three different energy points are rigorously calculated and stored beforehand. They are used to obtain approximate collision probabilities at an arbitrary energy point by Takahashi's zero-th approximation. Then, these approximate values are used to obtain more accurate values by quadratic interpolation with the neutron total cross section change as an interpolation parameter. The proposed method was applied to the calculation of collision probabilities in a two-region LWR fuel pin cell, and it was found that the collision probabilities can be calculated with an error of less than 1%, a great improvement in accuracy compared to the zero-th approximation, when the total cross section of the fuel region changes 30 times. The proposed method shortens computing time of lattice cell neutronic calculation codes based on the first-flight collision probability method.     

Calculation of the void reactivity of CANDU lattices using the code system

The reactivity effect of coolant voiding in CANDU-type fuel lattices has been calculated with different methods using the code system. The known positive void reactivity coefficient of the original lattice was correctly obtained. A modified fuel bundle containing dysprosium and slightly enriched uranium to eliminate the positive reactivity effect was also calculated. Owing to the increased heterogeneity of this modified fuel the one-dimensional cylindrical calculation with XSDRN proved to be inadequate. Code options allowing bundle geometry were successfully used for the calculation of the strongly space dependent flux and spectrum changes which determine the void reactivity.     

Extension to the integral transport equation of an iterative method

In this paper we describe an extension to the neutron integral transport equation of an iterative method. Indeed an iterative scheme is used for both energy and space including external iteration for the multiplication factor and internal iteration for flux calculations. The Monte Carlo method is used to evaluate the spatial transfer integrals. The results were compared with those obtained by using the APOLLO2 code for a cylindrical cell.