Fixed Source Problems Research Articles

Depletion analysis of a solid-type blanket design for ITER

In the blanket design in a D-T fusion power plant, the majority of the fusion energy is obtained from neutron kinetic energy. The tritium for maintaining the self-sustainable D-T fusion reaction is to be procured from the (n,t) reaction in the blanket. For this reason, neutronics analysis in the blanket design is indispensable. Especially for solid-type blanket design, neutronics analysis, including a depletion (transmutation) calculation, is required to calculate the neutron flux and tritium production rate more accurately. In eigenvalue problems for the fission reactor design, the MONTEBURNS code is widely used for depletion calculations. However, the fusion blanket design is a fixed source problem. In addition to this, the current version of the MONTEBURNS code does not include Be 9(n,t)Li 7 and Li 7(n,n′t)α reactions. Thus, in this research, (1) the MONTEBURNS code will be modified to solve the fusion blanket problem, including Be 9(n,t)Li 7 and Li 7(n,n′t)α reactions, and (2) a three-dimensional neutronics depletion analysis for a solid-type blanket for ITER will be performed using the modified MONTEBURNS code.

Analytical solution for two-group diffusion equations in a multilayered slab using Laplace Transform Technique

In this work, we report an analytical solution for the neutron and adjoint neutron fluxes by solving, respectively, the real and adjoint diffusion equations in a two-layered slab by the Laplace Transform Technique. We also present a solution for the effective multiplication factor by solving the algebraic system equation resulting from boundary condition and interface continuity for the flux and current by the bisection method. We also report analytical solutions for a properly said fixed source problem and for the auxiliary function using the same method developed for the previous problems. We report numerical simulations and comparisons with results in the literature.

Simplified Treatments of Anisotropic Scattering in LWR Core Calculations

The validity of various simplified treatments of anisotropic scattering is investigated in typical LWR configurations that simulate PWR, BWR, and APWR. In addition to the explicit treatment of anisotropic scattering, the diagonal approximation of anisotropic scattering matrixes and the transport corrected cross section with the assumption of isotropic scattering are also tested. Calculation results indicate that the anisotropic scattering matrix of the P1 component would be explicitly treated to obtain accurate results in the present calculations. The P2 or higher order anisotropic scattering matrix could be approximated by the diagonal approximation while maintaining the calculation accuracy. In addition to the above investigation, the physical meaning of the transport corrected cross section used in a transport calculation is discussed through a simple one-group fixed-source benchmark problem with anisotropic scattering. Though the transport corrected cross section is usually used to consider the P1 scattering cross section, the calculation results and discussion clarify that the transport corrected cross section is not mathematically consistent with the consideration of the P1 scattering cross section. The transport corrected cross section always implicitly includes an infinite order of anisotropic scattering cross sections when it is used in a transport calculation. Since the higher order anisotropic scattering component tends to reduce neutron leakage from a core, the calculated k-effective with the transport corrected cross section would be overestimated. This suggests the cause of the overestimation of k-effective by transport calculation with the transport corrected cross section obtained by in-scatter approximation.

Fourier Convergence Analysis of Two-Dimensional/One-Dimensional Coupling Methods for the Three-Dimensional Neutron Diffusion Eigenvalue Problems

The purpose of this paper is to present the Fourier convergence analysis of four methods for performing two-dimensional/one-dimensional (2-D/1-D) coupling to solve neutron diffusion eigenvalue problems (EVPs). The four methods differ principally in the manner of using the interface currents or node average fluxes to perform the 2-D/1-D coupling. Method A uses net currents, method B employs partial currents, method C uses a current correction factor, and method D uses an analytic expression for the axial net currents. In a previous paper, we analyzed the convergence behavior of these methods for the 2-D/1-D coupling of the fixed source problem (FSP). In this paper, the convergence performance of these methods is analyzed for the EVP using a one-group neutron diffusion EVP in a homogeneous infinite slab geometry. Among the four methods, method A diverges for small mesh sizes as it did in the FSP, whereas the other methods are stable regardless of the mesh size. The spectral radii of methods C and D are identical while the latter had a smaller spectral radius than the former in an FSP. The spectral radii of methods C and D are smaller than that of method B in the range of practical mesh size. The spectral radii approach one for all the methods as the mesh size increases, while in the FSP the spectral radii of method B approached a finite positive value and those of the other methods approached zero. For practical applications, method C has several advantages over the other methods and is the preferred 2-D/1-D coupling method for EVPs.

ICONE15-10151 Analysis of Startup Range Neutron Monitor (SRNM) Response During Refueling in BWR

We calculated Startup Range Neutron Monitor (SRNM) response during refueling in a boiling water reactor (BWR). The fixed source problem of a diffusion equation is solved by the 4^<th> order nodal expansion method (NEM). A method for calculating SRNM response was devised. The new method for calculation of corner flux at SRNM positions is expanding intra nodal flux by directly using a 4^<th> order nodal expansion coefficient. Compared with actual measured data, the new method is more accurate because the existing method is only multiplying the corner discontinuing factor and node average flux.

06/00943 Use of pseudo-harmonics method coupled with finite differences coarse mesh in the solution of fixed source problems: de Lima, Z. R. et al. Annals of Nuclear Energy, 2005, 32, (12), 1366–1376.

Various lubricating body fluids at tissue interfaces are composed mainly of combinations of phospholipids and amphipathic apoproteins. The challenge in producing synthetic replacements for them is not replacing the phospholipid, which is readily available in synthetic form, but replacing the apoprotein component, more specifically, its unique biophysical properties rather than its chemistry. The potential of amphiphilic reactive hypercoiling behaviour of poly(styrene-alt-maleic acid) (PSMA) was studied in combination with two diacylphosphatidylcholines (PC) of different chain lengths in aqueous solution. The surface properties of the mixtures were characterized by conventional Langmuir–Wilhelmy balance (surface pressure under compression) and the du Noüy tensiometer (surface tension of the non-compressed mixtures). Surface tension values and 31P NMR demonstrated that self-assembly of polymer–phospholipid mixtures were pH and concentration-dependent. Finally, the particle size and zeta potential measurements of this self-assembly showed that it can form negatively charged nanosized structures that might find use as drug or lipids release systems on interfaces such as the tear film or lung interfacial layers. The structural reorganization was sensitive to the alkyl chain length of the PC.

Use of pseudo-harmonics method coupled with finite differences coarse mesh in the solution of fixed source problems

A method based in the pseudo-harmonics method was developed to solve the fixed source problem. The pseudo-harmonics method is based on the eigenfunctions associated with the leakage and removal matrix operator of the neutron diffusion equation, which will be treated here in three dimensions and two groups of energy. This matrix is built in this work through the nodal discretization supplied by coarse mesh finite differences method (CMFDM). CMFDM has as input data the average currents and the average fluxes in the faces of the node, and the average flux in the node, previously obtained by the nodal expansion method. The results obtained with the pseudo-harmonics procedure show good accuracy when compared to the reference results of the source problem tested. Moreover, it is a method which can be easily implemented to solve this type of problems.

公開文献から読み取れるモンテカルロ計算における分散低減法の懸念事項

In the fixed source problem such as a neutron deep penetration calculation with the Monte Carlo method, the application of the variance reduction method is most important for a high figure of merit (FOM) and the most reliable calculation. But, MCNP calculation inputs written in published literature are not to be best solution. The most concerned items are setting method for the lower weight bound on the weight window method and the exclusion radius for a point estimator. In those literatures, the lower weight bound is estimated by engineering judge or weight window generator in the MCNP. In the latter case, the lower weight bound is used with no turning process. Because of abnormal large lower weight bounds, many neutron are killed in no meaning by the Russian Roulette. The adjoint flux method for setting of lower weight bound should be adapted as a standard variance reduction method. The Monte Carlo calculation should be turned from the experience such as engineering judge to science such as adjoint method.

Solution of the fixed source neutron diffusion equation by using the pseudo-harmonics method

The two-group neutron diffusion equation, in two-dimensional Cartesian geometry, with fixed source was solved by using a pseudo-harmonics expansion method in connection with the flux expansion method of nodal discretization, based on average values. The same fixed source problem was solved by finite difference discretization and the results obtained were compared. The pseudo-harmonics expansion method employed is part of the alternative version of the pseudo-harmonics perturbation method and the fixed source problem tested was the “auxiliary function” problem. Results obtained for the test cases show that the method developed for solving fixed source problems is very accurate when compared to the finite difference method.

Convergence Analysis of the Nonlinear Coarse-Mesh Finite Difference Method for One-Dimensional Fixed-Source Neutron Diffusion Problem

The convergence rates of the nonlinear coarse-mesh finite difference (CMFD) method and the coarse-mesh rebalance (CMR) method are derived analytically for one-dimensional, one-group solutions of the fixed-source diffusion problem in a nonmultiplying infinite homogeneous medium. The derivation was performed by linearizing the nonlinear algorithm and by applying Fourier error analysis to the linearized algorithm. The mesh size measured in units of the diffusion length is shown to be a dominant parameter for the convergence rate and for the stability of the iterative algorithms. For a small mesh size problem, the nonlinear CMFD is shown to be a more effective acceleration method than CMR. Both CMR and two-node CMFD algorithms are shown to be unconditionally stable. However, the one-node CMFD becomes unstable for large mesh sizes. To remedy this instability, an underrelaxation of the current correction factor for the one-node CMFD method is successfully introduced, and the domain of stability is significantly expanded. Furthermore, the optimum underrelaxation parameter is analytically derived, and the one-node CMFD with the optimum relaxation is shown to be unconditionally stable.

The Effect of Neutron Source Distribution on Subcriticality Measurement of Pressurized Water Reactors Using the Modified Neutron Source Multiplication Method

A modified neutron source multiplication method has been investigated for application to subcriticality measurement in commercial Pressurized Water Reactors (PWRs). This method is based on the extraction of the fundamental mode from measured neutron signals and corrections for spatial effects. In commercial reactors, mainly two types of neutron sources are expected, one from burnt fuels and the other from in-core fixed neutron sources. To calculate the extraction and spatial correction factors, the fixed source problem was solved by the Finite Difference Method (FDM) to explicitly deal with spatial intensities of neutron sources. Sensitivity analysis was carried out to investigate how the accuracy of estimated subcriticality depends on approximation levels of neutron source distributions. It was found that both of the neutron sources should be taken into account. For a pin rod type of the in-core fixed neutron source, a node-wise distribution for which the localized source intensity is averaged over the radial direction in the relevant nodes is applicable, since it can estimate the subcriticality with almost the same accuracy as in the case with a rigid spatial treatment. This ensures that nodal calculations can be used in the evaluation of correction factors instead of burdensome FDM calculations.

Uncertainty Analysis on Fission Molybdenum Production with a Nuclear Fuel Target in a Research Reactor

The use of a low-enriched uranium (LEU) fuel target was examined for the feasibility of 99Mo production in a High-flux Advanced Neutron Application Reactor (HANARO). Uncertainty analysis was done with respect to the 99Mo yield ratio, 239Pu yield ratio, annual production rate, and decontamination requirement. Validity of a coupled code system, MCNP/ORIGEN2, was evaluated to estimate reliable isotopic number densities after irradiation and cooling. An equilibrium core model for the MCNP fixed-source problem was found by the reactor design methodology known as WIMS/VENTURE. Optimized target design options were proposed for both the LEU and highly enriched uranium (HEU) targets. Variables related to the target fabrication process and reactor physics condition were considered as uncertainty-inducing parameters. The most important factor affecting the overall uncertainty of the LEU option was the engineering tolerances achievable in the fabrication process of fuel film. The LEU has twice the uncertainty of HEU under current technology, which makes the economics of LEU worse than HEU. It is acceptable, however, in view of the radioactive purity of the alpha emitter because the uncertainty of the impurity level of 239Pu is expected to be relatively small - only 6.5% with a 95% confidence level.

Iterative computation of time-eigenvalues of the neutron transport equation

The Implicitly Restarted Arnoldi Method (IRAM) is applied to the computation of prompt time-eigenvalues of the neutron transport equation. Derivation of the eigenvalue problem is based on a least-squares functional combined with a spherical harmonics angular discretization and spatial finite elements. The method is applied to a mono-energetic homogeneous slab and compared to semi-analytical results. The results are found to be accurate if the angular discretization is sufficiently refined. The scheme is also applied to a model ADS geometry in both one and three energy groups. The IRAM is found to be very efficient but the solution of fixed source problems that are part of the algorithm need to be accelerated in the multigroup case to obtain an overall efficient method.

Higher-order variational perturbation expressions in fixed source problems using Monte Carlo trial functions

Higher-order variational expressions are used to assess the effects of perturbing the absorption cross-section of an infinite non-multiplying homogeneous system on the response of an infinite plane detector of finite thickness placed at a certain distance from an infinite plane source. For this specific problem the number of equations to be satisfied by the weighting functions is reduced to six instead of eight. Two approaches are used to obtain these functions. One is by using Green's function approach for both the forward and backward fields and subsequently the other weighting functions. The other approach is to use trial functions provided by Monte Carlo. A methodology of utilizing virtual sampling to obtain the different weighting functions from one Monte Carlo run was demonstrated. The results revealed for this problem show that higher-order variational expressions gave more accurate estimates of the effects of perturbation than first-order expressions and the conventionally used Correlated Sampling.

Burnup analysis of rock-like oxide fuel disks irradiated in the Japan Research Reactor No. 3

Burnup analysis of rock-like oxide (ROX) fuel disks has been carried out and the results have been compared with measured values. Two kinds of ROX disks: zirconia and thoria, were fabricated and irradiated in an irradiation hole of the Japan Research Reactor No. 3 (JRR-3). After irradiation, several post-irradiation examinations (PIE) were performed. Computer codes used for the calculations were the SRAC and the MVP-BURN codes. Firstly, the neutron spectrum in the irradiation hole was calculated using the SRAC code system. Fixed source problems were solved to obtain the neutron spectra and effective cross-sections of the disks and burnup calculations were performed. The calculated results of burnup, isotopic abundance of plutonium and production of americium and curium were compared with measurement values. Calculations overestimate the measured burnup by 7 ∼ 15% and both codes largely underestimate the measured production of americium and curium isotopes. The calculated plutonium abundance agrees moderately well with the measured values.

A Time-Dependent Collision Probability Method for One-Dimensional Space-Time Nuclear Reactor Kinetics

A time-dependent collision probability method has been developed for the solution of neutron transport and nuclear reactor kinetics problems in one-dimensional slab geometry. The time-dependent collision probabilities permit the solution of time-dependent neutron transport problems involving general source distributions over an indefinite time period and an infinite number of collision generations. The method is based on the analytic integration of the time-dependent integral transport kernel involving purely real cross sections. The neutron time-of-flight and causality considerations lead to a number of complex formulas involving exponential and exponential integral functions. Occasional conflicts between the regular grid in time and space and the causality considerations lead to some formulas that are inexact. It is shown that these inexact formulas are terms of the third order in the time-step length, and thus the method has overall second-order accuracy in time. The method has been used to solve two types of neutron transport problems. The first, a pulsed, planar, fixed-source problem, yielded a flux solution with a root-mean-square relative difference of 0.94% from a benchmark analytic solution. The second problem solved was a pair of multigroup nuclear reactor kinetics problems. While the kinetics results were not conclusive, they suggest that diffusion theory may yield results that underestimate the amplitude and deposited energy of certain reactor transients.

On the Equivalence of Boundary and Boundary Condition Perturbations in Transport Theory and Its Diffusion Approximation

It is shown that a perturbation in the external boundary of a system can be converted into (treated as) a perturbation in the boundary condition of the neutron balance equation. As a result, one may then use existing boundary condition perturbation or variational methods to estimate the change in the parameters of interest. The equivalent perturbation in the boundary condition is derived both in transport theory and its diffusion approximation for eigenvalue as well as fixed-source problems. Generalized boundary conditions in both diffusion and transport theory are considered. The existence of a solution to the problem with the equivalent boundary conditions that are nonstandard is investigated through analytic examples. The analysis is limited to first-order perturbation theory.

An Incomplete Domain Decomposition Preconditioning Method for Nonlinear Nodal Kinetics Calculations

Methods are proposed for the efficient parallel solution of nonlinear nodal kinetics equations. Because the two-node calculation in the nonlinear nodal method is naturally parallelizable, the majority of the effort is devoted to the development of parallel methods for solving the coarse-mesh finite difference (CMFD) problem. A preconditioned Krylov subspace method (biconjugate gradient stabilized) is chosen as the iterative algorithm for the CMFD problem, and an efficient parallel preconditioning scheme is developed based on domain decomposition techniques. An incomplete lowerupper triangular factorization method is first formulated for the coefficient matrices representing each three-dimensional subdomain, and coupling between subdomains is then approximated by incorporating only the effect of the nonleakage terms of neighboring subdomains. The methods are applied to fixed-source problems created from the International Atomic Energy Agency three-dimensional benchmark problem. The effectiveness of the incomplete domain decomposition preconditioning on a multiprocessor is evidenced by the small increase in the number of iterations as the number of subdomains increases. Through the application to both CMFD-only and nodal calculations, it is demonstrated that speedups as large as 49 with 96 processors are attainable in the nonlinear nodal kinetics calculations.

Variational Nodal Transport Methods with Anisotropic Scattering

The variational nodal method is generalized to treat within-group and group-to-group anisotropic scattering in two- and three-dimensional eigenvalue and fixed source problems. The resulting formalism is implemented as the VARIational Anisotropic Nodal Transport code (VARIANT) within the shell of the Argonne National Laboratory production code DIF3D. The code is applied to a series of Cartesian and hexagonal geometry model problems and the accuracy of the results compared to those from TWODANT and TWOHEX and to the Monte Carlo code VIM, respectively, in two and three dimensions. VARIANT is then applied to multigroup hexagonal representations of the Experimental Breeder Reactor II, and results are obtained for three-dimensional eigenvalue and for two-dimensional neutron-gamma heating problems.

Two-group Monte Carlo perturbation theory and applications in fixed-source problems

The collision estimator is used in a second-order derivative sampling scheme for fixed-source problems involving a perturbation in the material density. The analytical formulation of a one-group problem in a finite slab with forward scattering is demonstrated to obtain exact expressions for the collision density and its derivatives. The track length estimator is also used for comparisons in some cases. Results are presented for a two-group multi-region problem using the MORSE code. The predicted collision densities in zones close to the source are found to be in excellent agreement with those obtained in full ‘re-runs’.