Reduction of MOC discretization errors through a minimization of source ratio variances

A new technique to reduce discretization errors for ray tracing in the method of characteristics (MOC) is proposed focusing on depletion calculations of single and multi-assembly geometries. In order to efficiently carry out depletion calculations, a calculation scheme using the superhomogenization (SPH) method can be used. However, the discretization errors are caused by changes of neutron sources and total cross sections according to a depletion. This fact means that improvement of accuracy cannot be expected by the calculation scheme with the SPH method when changes of the above parameters are significant. In order to mitigate this problem, a new approach is developed. In the new approach, the discretization errors are reduced by minimizing a variance of a certain parameter which is composed of a ratio of neutron source to total cross section. The verification results suggest that accuracy is degraded by the SPH method as expected especially in a geometry where neutron sources and total cross sections are drastically changing through a depletion. On the other hand, the new approach gives more accurate results compared to the conventional MOC in all calculation cases. Consequently, improvement of calculation efficiency by the new approach is confirmed.     

Higher order methods for burnup calculations with Bateman solutions

A group of methods for burnup calculations solves the changes in material compositions by evaluating an explicit solution to the Bateman equations with constant microscopic reaction rates. This requires predicting representative averages for the one-group cross-sections and flux during each step, which is usually done using zeroth and first order predictions for their time development in a predictor–corrector calculation. In this paper we present the results of using linear, rather than constant, extrapolation on the predictor and quadratic, rather than linear, interpolation on the corrector. Both of these are done by using data from the previous step, and thus do not affect the stepwise running time.     

Reaction rates in blanket assemblies of a fusion-fission hybrid reactor

To validate neutronics calculation for the blanket design of fusion-fission hybrid reactor,experiments for measuring reaction rates inside two simulating assemblies are performed.Two benchmark assemblies were developed for the neutronics experiments.A D-T fusion neutron source is placed at the center of the setup.One of them consists of three layers of depleted uranium shells and two layers of polyethylene shells,and these shells are arranged alternatively.The ²³⁸U capture reaction rates are measured using depleted uranium foils and an HPGe gamma spectrometer.The fission reaction rates are measured using a fission chamber coated with depleted uranium.The other assembly consists of depleted uranium and LiH shells.The tritium production rates are measured using the lithium glass scintillation detector which is placed in the LiH region of the assembly.The measured reaction rates are compared with the calculated ones predicted using MCNP code,and C/E values are obtained.     

托卡马克实验混合堆 FEB 嬗变 MA 可行性研究

研究了在聚变实验混合堆ＦＦＢ设计中，嬗变长寿命放射性少锕系（ＭＡ，ＭｉｎｏｒＡｃ－ｔｉｎｉｄｅｓ）核废物的可行性。应用改进的一维中子输运和燃耗计算程序ＢＩＳＯＮ３．０，完成了嬗变中子学与核素贫化计算。研究了核废物的嬗变率与辐照时间、包层厚度和废物装载量的关系，并对系统有关参数的选择进行了优化设计。结果表明，该设计（ＭＡ＋Ｐｕ）可年嬗变处置来自５５座相同功率的ＰＷＲ卸出的ＭＡ核废物，同时输出热功率５．４ＧＷ（ｔｈ）。     

Direct deterministic method for neutronics analysis and computation of asymptotic burnup distribution in a recirculating pebble-bed reactor

A new deterministic method has been developed for the neutronics analysis of a pebble-bed reactor (PBR). The method accounts for the flow of pebbles explicitly and couples the flow to the neutronics. The method allows modeling of once-through cycles as well as cycles in which pebbles are recirculated through the core an arbitrary number of times. This new work is distinguished from older methods by the systematically semi-analytical approach it takes. In particular, whereas older methods use the finite-difference approach (or an equivalent one) for the discretization and the solution of the burnup equation, the present work integrates the relevant differential equation analytically in discrete and complementary sub-domains of the reactor. Like some of the finite-difference codes, the new method obtains the asymptotic fuel-loading pattern directly, without modeling any intermediate loading pattern. This is a significant advantage for the design and optimization of the asymptotic fuel-loading pattern. The new method is capable of modeling directly both the once-through-then-out fuel cycle and the pebble recirculating fuel cycle. Although it currently includes a finite-difference neutronics solver, the new method has been implemented into a modular code that incorporates the framework for the future coupling to an efficient solver such as a nodal method and to modern cross section preparation capabilities. In its current state, the deterministic method presented here is capable of quick and efficient design and optimization calculations for the in-core PBR fuel cycle. The method can also be used as a practical “scoping” tool. It could, for example, be applied to determine the potential of the PBR for resisting nuclear-weapons proliferation and to optimize proliferation-resistant features. However, the purpose of this paper is to show that the method itself is viable. Refinements to the code are under way, with the objective of producing a powerful reactor physics analysis tool for PBRs.     

On the influence of spatial discretization in LWR cell- and lattice calculations with HELIOS 1.9

Cell- and lattice calculations are the fundamental for all deterministic static and transient 3D full core calculations. The spatial discretization used for the cell- and lattice calculations influences the results for these transport solutions significantly. The arising differences in the neutron flux distribution due to different spatial discretization are demonstrated. These differences in the flux distribution cause significant changes in the k_inf value. An evaluation of the k_inf value for the case of infinitely fine discretization is made. The influence of the discretization on the calculation of homogenized few group cross-sections which are forwarded to the 3D full core calculations is investigated. Strategies for improving the discretization are developed and their influence on the calculation time is evaluated.     

压水堆堆芯pin-by-pin计算环境效应处理模型研究

栅格非均匀计算过程中采用的全反射边界条件近似带来的中子射流效应和中子能谱干涉效应等环境效应对栅元均匀化常数具有较大影响。为在全堆芯pin by pin计算中处理环境效应带来的影响,本文从两个方面进行了计算分析。首先,基于棋盘式多组件问题对栅元均匀化群常数相对误差及各能群栅元不连续因子相对重要性进行了分析,可发现在等效均匀化常数中,热群不连续因子对全堆芯pin by pin计算精度的影响最重要;其次,基于最小二乘法建立了热群栅元不连续因子和堆芯中子学特征量之间的多项式函数关系,利用参数化技术提出了热群常数堆芯在线计算方法,其中堆芯中子学特征量包括扩散系数、移出截面、中子源项、归一化中子通量密度等。采用C5G7基准题和KAIST基准题进行了数值验证,计算结果表明,热群常数堆芯在线计算方法能有效降低全堆芯pin by pin计算特征值和棒功率相对误差,对处于不同燃料组件交界面附近的栅元,计算精度提升尤为显著。     

Fast Neutron Spectra from Natural Uranium Target Bombarded with High Energy Electrons

The numerical solution of the transport equation has the errors caused by the approximations used in the computational method. In the past estimations of these errors have been performed experimentally. In the present study, formulas to estimate the errors have been derived on the basis of the perturbation theory. This method enables us to deterministically estimate the numerical errors due to the iteration, spatial discretization and Legendre polynomial expansion of scattering transfer cross sections.

Using the error estimation method developed in the present study, two examples of error analyses were carried out to confirm its validity and applicability to error estimation for a practical purpose. The errors of the calculated tritium breeding ratio for ⁷Li in a infinite slab geometry were estimated, and they agreed well with the values predicted from direct calculation. As the second example, error analysis was carried out for one-dimensional nuclear calculations on two types of commercial fusion reactor blankets. In this analysis the tritium breeding ratio and the fast neutron leakage flux from the inboard shield were investigated, and the errors from different causes were quantitatively compared.     

Comparison of depletion algorithms for large systems of nuclides

In this work five algorithms for solving the system of decay and transmutation equations with constant reaction rates encountered in burnup calculations were compared. These are Chebyshev rational approximation method (CRAM), which is a new matrix exponential method, the matrix exponential power series with instant decay and a secular equilibrium approximations for short-lived nuclides, which is used in ORIGEN, and three different variants of transmutation trajectory analysis (TTA), which is also known as the linear chains method. The common feature of these methods is their ability to deal with thousands of nuclides and reactions. Consequently, there is no need to simplify the system of equations and all nuclides can be accounted for explicitly.The methods were compared in single depletion steps using decay and cross-section data taken from the default ORIGEN libraries. Very accurate reference solutions were obtained from a high precision TTA algorithm. The results from CRAM and TTA were found to be very accurate. While ORIGEN was not as accurate, it should still be sufficient for most purposes. All TTA variants are much slower than the other two, which are so fast that their running time should be negligible in most, if not all, applications. The combination of speed and accuracy makes CRAM the clear winner of the comparison.     

Generalized Bias Factor Method for Accurate Prediction of Neutronics Characteristics

A generalized bias factor method is proposed to improve the prediction accuracy of neutronics characteristics of a target core. The generalized bias factor method uses conventional bias factors calculated for several critical assemblies. The weighting factors for individual bias factors are determined to minimize the variance of neutronic characteristics of the target core. Numerical calculations are performed to investigate the uncertainty reductions of neutronics characteristics for a tight-lattice core. Though the uncertainty is not remarkably reduced for k_eff , that for the reaction rate ratio of ²³⁸U capture/²³⁹Pu fission is remarkably reduced: For example, the uncertainty reduction of the reaction rate ratio in the upper core is 0.871 for the present method, and 0.657 for the conventional bias factor method.     

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.     

Estimation of sensitivity coefficients of core characteristics based on reduced-order modeling using sensitivity matrix of assembly characteristics

We propose an estimation method of sensitivity coefficients of core neutronics parameters based on a multi-level reduced-order modeling approach. The idea is to use lower-level models to identify the dominant input parameter variations, constrained to the so-called active subspace, which are employed to determine the sensitivity coefficients of the core neutronic parameters. In our implementation, the lower-level model is represented by two-dimensional assembly calculations, which are employed in the preparation of the few-group cross-sections for core-wide calculations. The active subspace basis is estimated using the singular value decomposition of sensitivity matrices of assembly neutronics parameters. In numerical verification calculation, sensitivity coefficients of core characteristics for a typical three-loop PWR equilibrium-cycle are estimated using the proposed method and the direct method. Comparison of these two results shows that the proposed method well reproduces the results obtained by the direct method with lower calculation costs. Through the verification calculations, applicability of the proposed method to practical light water reactor analysis is confirmed.     

Adaptive burnup stepsize selection using control theory for 2-D lattice depletion simulations

A control theory approach is adopted to determine the temporal discretization during two-dimensional lattice physics depletion simulations. Two primary applications of automated and adaptive stepsize control are identified: (i) the presence of strong absorbers such as gadolinium, where the accurate burnout of the isotopes requires a depletion stepsize smaller than typically required, and (ii) high fidelity multiphysics simulations, e.g. loosely coupled physics, where the coupled physics are nonlinear in time and stepsize changes may be necessary to obtain an accurate coupled solution. A conventional predictor–corrector method is used to address the nonlinearity of the nuclide transmutation and neutron flux. An adaptive stepsize method is developed based on monitoring the one-group scalar neutron flux at both the predictor and corrector steps to approximate the convergence residual of the nonlinear solution. A user-specified tolerance on the L² relative error norm of the scalar neutron flux is utilized by the stepsize controller. Controllers that include integral, proportional, and/or derivative components are investigated and parameterized using Latin hypercube sampling of the controller input parameters. Three distinct fuel loadings of pressurized water reactor 17 × 17 fuel pin assemblies are considered, including no burnable absorbers, Integral Fuel Burnable Absorber, and gadolinium fuel pins. The required depletion stepsizes, as predicted throughout the cycle by the controller, are compared with a very small stepsize (0.01 MW d/kgHM) reference solution and a solution obtained by a typical rule of thumb depletion stepsize sequence.     

Integral Data Test of HENDL1.0/MG with Neutronics Shielding Experiments(Ⅱ)

The multi-group working nuclear data library HENDL1.0/MG is numerically tested with a series of existent benchmark spherical shell experiments (Si, Cr, Fe, Cu, Zr and Nb) by calculations using the multi-functional neutronics code VisualBUS. The ratio of calculated/measured neutron leakage rates and the neutron leakage spectra are presented in tabular and figural forms.The results from the calculations with the code ANISN and IAEA data library FENDL2.0/MGwere also included for comparison, where the origination of the data used is different from that of HENDL1.0/MG.     

Integral Data Test of HENDL1.0／MG with Neutronics Shielding Experiments （Ⅱ）

The multi-group working nuclear data library HENDL1.0/MG is numerically tested with a series of existent benchmark spherical shell experiments (Si, Cr, Fe, Cu, Zr and Nb) by calculations using the multi-functional neutronics code VisualBUS. The ratio of calculated/measured neutron leakage rates and the neutron leakage spectra are presented in tabular and figural forms. The results from the calculations with the code ANISN and IAEA data library FENDL2.0/MG were also included for comparison, where the origination of the data used is different from that of HENDL1.0/MG.     

keff uncertainty quantification and analysis due to nuclear data during the full lifetime burnup calculation for a small-sized prismatic high temperature gas-cooled reactor

To benefit from recent advances in modeling and computational algorithms,as well as the availability of new covariance data,sensitivity and uncertainty analyses are needed to quantify the impact of uncertain sources on the design parameters of small prismatic high-temperature gas-cooled reactors(HTGRs).In particular,the contribution of nuclear data to the keff uncertainty is an important part of the uncertainty analysis of small-sized HTGR physical calculations.In this study,a small-sized HTGR designed by China Nuclear Power Engineering Co.,Ltd.was selected for keff uncertainty analysis during full lifetime burnup calculations.Models of the cold zero power(CZP)condition and full lifetime burnup process were constructed using the Reactor Monte Carlo Code RMC for neutron transport calculation,depletion calculation,and sensitivity and uncertainty analysis.For the sensitivity analysis,the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Track length importance Characterization(CLUTCH)method was applied to obtain sensitive infor-mation,and the"sandwich"method was used to quantify the keff uncertainty.We also compared the keff uncertainties to other typical reactors.Our results show that 235U is the largest contributor to keff uncertainty for both the CZP and depletion conditions,while the contribution of 239Pu is not very significant because of the design of low discharge burnup.It is worth noting that the radioactive capture reaction of 28Si significantly contributes to the keff uncer-tainty owing to its specific fuel design.However,the keff uncertainty during the full lifetime depletion process was relatively stable,only increasing by 1.12％owing to the low discharge burnup design of small-sized HTGRs.These numerical results are beneficial for neutronics design and core parameters optimization in further uncertainty prop-agation and quantification study for small-sized HTGR.     

keff uncertainty quantification and analysis due to nuclear data during the full lifetime burnup calculation for a small-sized prismatic high temperature gas-cooled reactor

To benefit from recent advances in modeling and computational algorithms,as well as the availability of new covariance data,sensitivity and uncertainty analyses are needed to quantify the impact of uncertain sources on the design parameters of small prismatic high-temperature gas-cooled reactors(HTGRs).In particular,the contribution of nuclear data to the keff uncertainty is an important part of the uncertainty analysis of small-sized HTGR physical calculations.In this study,a small-sized HTGR designed by China Nuclear Power Engineering Co.,Ltd.was selected for keff uncertainty analysis during full lifetime burnup calculations.Models of the cold zero power(CZP)condition and full lifetime burnup process were constructed using the Reactor Monte Carlo Code RMC for neutron transport calculation,depletion calculation,and sensitivity and uncertainty analysis.For the sensitivity analysis,the Contribution-Linked eigenvalue sensitivity/Uncertainty estimation via Track length importance Characterization(CLUTCH)method was applied to obtain sensitive infor-mation,and the"sandwich"method was used to quantify the keff uncertainty.We also compared the keff uncertainties to other typical reactors.Our results show that 235U is the largest contributor to keff uncertainty for both the CZP and depletion conditions,while the contribution of 239Pu is not very significant because of the design of low discharge burnup.It is worth noting that the radioactive capture reaction of 28Si significantly contributes to the keff uncer-tainty owing to its specific fuel design.However,the keff uncertainty during the full lifetime depletion process was relatively stable,only increasing by 1.12％owing to the low discharge burnup design of small-sized HTGRs.These numerical results are beneficial for neutronics design and core parameters optimization in further uncertainty prop-agation and quantification study for small-sized HTGR.     

Monte Carlo radiation shielding and activation analyses for the Diagnostic Equatorial Port Plug in ITER

This paper addresses neutronics aspects of the design development of the Diagnostic Generic Equatorial Port Plug (EPP) in ITER. To secure the personnel access at the EPP back-end interspace, parametric neutronics analyses of the EPP radiation environment have been performed and practical shielding solutions have been found. Radiation transport was performed with the Monte Carlo MCNP5 code. Activation calculations were conducted with the FISPACT-2007 inventory code. The R2Smesh approach was applied to couple transport and activation calculations. Newly created EPP local MCNP5 model was devised by extracting the EPP and adjacent blanket modules from the ITER Alite-4.1 model with proper modification of the EPP geometry in accordance with recent 3D CAD CATIA model. The EPP local model reproduces the EPP neutronically important features and allows investigation of the EPP neutronics effects in isolation from all other ITER components. Thorough EPP parametric analyses revealed dominant effect of gaps around EPP and several EPP design improvements were implemented as the outcomes of the analyses. Gap labyrinths and streaming stoppers inserted into the gaps were shown are capable to reduce the shutdown dose rate which is below the 100 μSv/h limit of personnel access and by 2 orders of magnitude less than the value in the model with straight gaps.     

parafish: A parallel FE–PN neutron transport solver based on domain decomposition

A new core solver named parafish is presented for the solution of large neutron transport core calculations. The second-order even-parity form of the time-independent Boltzmann transport equation is solved using an innovative algebraic domain-decomposition method. The spatio-angular discretization is performed using non-conforming finite elements and spherical harmonic expansions (P_N method). The parafish code allows one processor to handle more than one domain. This enables proper evaluations of the speed-up. Also, this enables to show that the domain-decomposition method not only performs well in parallel calculations, but also has an inherent acceleration potential. That is, it yields acceleration even without increasing the number of processors.     

Approximate Solution to Neutron Transport Equation with Linear Anisotropic Scattering

A method to obtain an approximate solution to the transport equation, when both sources and collisions show a linearly anisotropic behavior, is outlined and the possible implications for numerical calculations in applied neutronics as well as shielding evaluations are investigated.

The form of the differential system of equations taken by the method is quite handy and looks simpler and more manageable than any other today available technique. To go deeper into the efficiency of the method, some typical calculations concerning critical dimensions of multiplying systems are then performed and the results are compared with the ones coming from the classical S_N approximations. The outcome of such calculations leads us to think of interesting developments of the method which could be quite useful in alternative to other today widespread approximate procedures, for any geometry, but especially for curved ones.