Recent activities in the field of nuclear waste management and environmental science

The multiband method has been applied to analyses of critical experiments related to the high-conversion core at the Kyoto University Critical Assembly in order to accurately treat the resonance self-shielding in heterogeneous cells. Three-band parameters were generated using the self-shielding table installed in the SRAC code, and used to calculate the cell-averaged cross sections. The k _∞ values calculated by this method have been compared to those by the VIM Monte-Carlo calculation, the SRAC fine group calculation, Dancoff factor method and/or Tone's method self-shielding calculation. The k∞ values calculated by the present method agree with those by the VIM calculation within 0.3%Δk for all the cases considered.     

A New Mixed Method with Finite Difference and Finite Element Method for Neutron Diffusion Calculation

A new method for obtaining three-dimensional neutron flux distribution in a reactor has been developed by taking into account the fact that the X-Y planar geometry is generally complex but the geometry along Z-axis is simple. In this method, the finite element method is applied to the X-Y plane calculation and the finite difference method to the Z-axis. For solving a three-dimensional neutron diffusion equation, these two methods are iterated successively until a consistency of the leakage coefficients is attained between the two. The present method is embodied as a computer program FEDM for FACOM M200 computer. With this program, a three-dimensional diffusion calculation was performed for comparing some numerical results with those by a conventional standard computer code ADC. The comparison has shown that they agree well with each other. Computing time required for this problem by the FEDM was shorter than that by the ADC for obtaining same accuracy on the eigenvalue. To indicate usefulness of this method, a demonstration calculation for a reactor with a complex geometry was performed, which was a difficult case to calculate with a conventional finite difference code.     

Analysis of Heavy-Water-Moderated,Cluster-Type Fuel Lattices by Cluster Physics Code ‘MESSIAH’

This paper presents a comprehensive analysis performed by a new cluster analysis code ‘MESSIAH’ on reactor physics constants measured in the critical facility for a pressure-tube-type, heavy-water-moderated reactor. The MESSIAH code system utilizes the method of the collision probability to solve the neutron transport equation. The effective space dependent cross sections are calculated in the thermal and resonance energy range before the eigenvalue calculation for the whole energy range. With use of these cross sections, the multi-group, space dependent transport equation is solved, and the flux distribution, spectrum and k _eff are obtained to the input bucking. In the above three steps the method of the collision probability is used consistently and extensively. The treatment of leakage neutrons from lattices in MESSIAH is also confirmed by an independent method using a Monte Carlo calculation. The calculated reactor physics constants, especially the micro-parameters and the activation traverse of Dy, agreed fairly well with the experiment. The diffusion calculation with use of the group constants calculated by MESSIAH predicts the reactivity of 0% void core excellently (<0.12%). However, for a 100% void core, the calculated reactivity was slightly lower than the experiment (~0.74%), which was attributed to over prediction of the diffusion constants.     

Effect of Geometrical Approximation on Multiplication Factor in Small Fast Reactor Core

In most of the calculations using analytical methods a reactor core is approximated as cylinder and the reactor parameters are calculated using two-dimensional computer codes. While such calculations are useful in scoping studies in view of azimuthal asymmetry in the actual reactor core these calculations could entail errors of unknown magnitude. The present study reports our estimate of such errors in K _eff with the instance of fast reactor having 22 and 23 fuel subassemblies. The K _eff are calculated using Monte Carlo code KENO and Hansen-Roach cross section set, modelling the core in two different ways, (1) by approximating the core to a cylinder (2-D calculation), (2) by near exact representation of the core (3-D calculation). The difference in K _eff is appreciable between 2-D and 3-D calculations.

Experimental values are adduced in support of these calculations.     

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.     

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.     

Neutronics Calculations of HTTR with Several Nuclear Data Libraries

Benchmark calculations for several HTTR core states were performed with four cross-section sets which were generated from JENDL-3.3, JENDL-3.2, ENDF/B-VI.8 and JEFF-3.0 using a continuous energy Monte Carlo code MVP. The core states were a critical approach in which an annular core was formed at room temperature and solid cores at room temperature and at full power operation. Study of k_eff discrepancies caused by difference of the nuclear data libraries and identification of nuclides which have large effects on the k_eff discrepancies were carried out. Comparison of the respective k_eff from calculations and experiments was also carried out. As the results, for each of the HTTR core states, JENDL-3.3 yields a k_eff agreeing with the experiments within 1.5%Δk, JENDL-3.2 yields k_eff agreement within 1.7%Δk, and ENDF/B-VI.8 and JEFF-3.0 yield k_eff agreement within 1.8%Δk. There is little k_eff discrepancy between ENDF/B-VI.8 and JEFF-3.0. The k_eff between JENDL-3.3 and JENDL-3.2 is caused by difference of ²³⁵U data and has temperature dependency. The k_eff discrepancy between JENDL-3.3 and ENDF/B-VI.8 or JEFF-3.0 is mainly caused by difference in graphite data.     

Analysis of Core Physics Experiments on Fresh and Irradiated PWR UO2 Fuels in the REBUS Program

Critical experiments of two cores each loaded with fresh 5 × 5 test PWR-type fuel rods of ²³⁵U enrichment of 3.8 wt% or irradiated 5 × 5 test rods of rod average burnup of 55 GWd/t in the REBUS program were analyzed using diffusion, transport, and continuous-energy Monte Carlo calculation codes coupled with nuclear data libraries based on JENDL-3.2 and JENDL-3.3. Biases in effective multiplication factors k _eff's of the critical cores were about ?1:2%Δk for the diffusion calculations (JENDL-3.2), ?0:5%Δk for the transport calculations (JENDL-3.3), and ?0:5 and 0.1%Δk for the Monte Carlo calculations (JENDL-3.3 and JENDL-3.2, respectively). The measured core fission rate and Sc- or Co-activation rate distributions were generally well reproduced using the three types of calculation. The burnup reactivity determined using the measured water level reactivity coefficients was ?2:35 ± 0:07Δk/kk′. The calculated result of the Monte Carlo calculations agreed with it; however, the diffusion and transport calculations overestimated the absolute value by about 7%, which would be mainly attributed to the errors in the calculation of the reactivity caused by changing the fuel compositions from fresh fuel to irradiated fuel.     

Thermal Neutron Spectra in Light Water Poisoned with Cadmium and/or Indium

The thermal neutron spectra in light water of slab geometry poisoned with Cd and/or In were measured by the time of flight method, using a 20-MeV electron linear accelerator. The thermal spectra were simulated to those in the Pu built-up core of a commercial light water reactor corresponding to a fuel burnup of about 15,000 MWD/T. The results of measurements were compared with calculations based on the S ₄ method using the Haywood scattering law. Fairly good agreement was obtained between the calculated and measured results except in a limited range of energy above the 0.176 eV resonance of Cd. It is concluded that the P ₁ components of the source neutrons as well as the neutron scattering kernel play a significant role in the calculation of the thermal neutron spectra with large flux gradients, and that the scattering kernel of light water based on the Haywood model will be accurate enough to evaluate the infinite multiplication constant k_∞ of light water reactor cores with high fuel burnup within an error of about — 0.17%, as estimated from the uncertainty in the spectrum calculation in the region above the Cd resonance. It is also emphasised from the two- dimensional S ₄ calculations that the effect of reentrant hole perturbation should be evaluated quantitatively in the interpretation of the measured angular neutron spectra produced within finite media.     

Efficient Method for Calculating Axial Diffusion Coefficients in Two-Dimensional Low Density Channel

An efficient calculation method of an axial diffusion coefficient which is applicable to multi-region and two-dimensional geometries has been developed. This diffusion coefficient which requires biquadratic numerical integration is quickly calculated by using a Good Lattice Point Method. Total computational time is about one-thirty hundredth that of Romberg's Method. The geometry dependency of axial diffusion coefficient is investigated. As a result the axial diffusion coefficient of a rectangular cross section of low density channel is larger than that of a square one, under the constraint that the channel volume is conserved.

Analysis of the reactivity worth measured for a square void channel in the Tank-type Critical Assembly (TCA) was performed using this diffusion coefficient. The calculated value agrees well with the experimental value, i.e. ratio of the calculated to experimental value (C/E) is 0.997.     

Development of Discrete Ordinates SN Code in Three-Dimensional (X,Y, Z) Geometry for Shielding Design

A discrete ordinates transport code ENSEMBLE in (X, Y, Z) geometry has been developed for the purpose of shielding calculations in three-dimensional geometry. The code has some superior features, compared with THREETRAN which is the only code of the same kind so far developed. That is, the code can treat higher order anisotropic scattering and employs a coarse mesh rebalancing method. Moreover it has a negative flux fix-up routine using a variable weight diamond difference equation scheme and has a ray-effect fix-up option using a fictitious source based on S_N→P_N-1 conversion technique. Formulations for these advanced features in three-dimensional space have been derived.

As the demonstration of the capabilities of the code, several numerical analyses and an analysis of an annular duct streaming experiment in JRR-4 at Japan Atomic Energy Research Institute, have been performed.

As a result of these analyses, confirmation has been obtained for the prospect of applicability of ENSEMBLE to practical shielding design.     

Three-Dimensional Transport Calculation Method for Eigenvalue Problems Using Diffusion Synthetic Acceleration

A three-dimensional transport code “TRITAC” for solving eigenvalue problems in reactor cores has been developed on the basis of discrete ordinates method with the diffusion synthetic acceleration technique. The Larsen procedure for the diffusion synthetic acceleration method has been extended to three-dimensional geometry. With the procedure a spatially differenced diffusion synthetic equation has been derived and implemented in the TRITAC code. In the X-Y geometry the code yielded the same results as the TWOTRAN-II code. Three-dimensional eigenvalue problems for thermal and fast reactors have been solved and the computational time has been compared with that required for the three-dimensional discrete ordinates calculation with the rebalance acceleration technique.     

Whole Core Calculations of Power Reactors by Use of Monte Carlo Method

Abstract

Whole core calculations have been performed for a commercial size PWR and a prototype LMFBR by using vectorized Monte Carlo codes. Geometries of cores were precisely represented in a pin by pin model. The calculated parameters were k _eff, control rod worth, power distribution and so on. Both multigroup and continuous energy models were used and the accuracy of multigroup approximation was evaluated through the comparison of both results. One million neutron histories were tracked to considerably reduce variances. It was demonstrated that the high speed vectorized codes could calculate k _eff, assembly power and some reactivity worths within practical computation time. For pin power and small reactivity worth calculations, the order of 10 million histories would be necessary. It would be difficult for the conventional scalar code to solve such large scale problems while the present codes consumed computation time less than 30 min for a PWR and 1 hour for an LMFBR. Required number of histories to achieve target design accuracy were estimated for those neutronic parameters.     

Analysis of Mixed Oxide Fuel Critical Experiments with Neutronics ANalysis Codes for Boiling Water Reactors

Critical experiments of UO₂ and full mixed oxide (MOX) fuel cores conducted at the Tank-type Critical Assembly (TCA) were aNalyzed using BWR design-purpose codes HINES and CERES with ENDF/B files and Monte Carlo fine analysis codes VMONT and MVP with the JENDL-3.2 library.

The averaged values of the multiplication factors calculated with HINES/CERES, VMONT and MVP agreed with those of experiments within 0.3%ΔAk. The values by the design-purpose codes showed a small difference of 0.1%Δk between UO₂ and MOX cores. Monte Carlo code results showed that the JENDL-3.2 library had a tendency to overestimate the multiplication factors of UO₂ cores by about 0.3%Δk compared with those values of MOX cores. The root mean square errors of calculated power distributions were less than 1% for HINES/CERES and VMONT.

These results showed that (1) the accuracy of these codes when applied to full MOX cores was almost the same as their accuracy for UO₂ cores, which confirmed the accuracy of present core design codes for full MOX cores; and (2) the accuracy of the 190-energy-group Monte Carlo calculation code VMONT was almost the same as that of the continuous-energy Monte Carlo calculation code MVP.     

Determination of Multiplication Factors Using Experimental Lattice Parameters of Cluster-Type Fuel Lattices

A new four-factor formula is proposed for deriving a “finite multiplication factor” k*—the ratio between neutron production and absorption rates in a finite medium—from experimentally determined lattice parameters in a slightly-enriched Pu-U lattice.

The “two-group multiplication factor” k* _∞2—related to τ+ L² and equal to (1 τ B² _C). (1 +L²B² _G at criticality—is derived from k*.

The experimental lattice parameters are corrected to account for neutron leakage, from which the “infinite multiplication factor” K∞ is derived.

There are found differences between k*, k*_∞2 and k _∞ amounting to about 1–2% in the Advanced Thermal Reactor (ATR)-type heavy-water lattice, despite the fact that all these three quantities are often indiscriminately called “infinite multiplication factor.”

The proposed four-factor formula for deriving k* was applied to the Deuterium Critical Assembly (DCA) of 1.2%-enriched UO₂ lattices of 28-pin clusters in square arrays spaced at 22.5 cm. The values of k* calculated with a lattice design code METHUSELAHH II were in fairly good agreement wTith those determined by experiment. The calculation tended to somewhat overestimate k*, particularly in lattices of highly voided coolant. A similar tendency was observed in the case of 1.5%-enriched UO₂ lattices of 22.5 cm pitch in the ATR Sumitomo-Two-Region Critical Facility.     

Examination of pin-by-pin fission rate distribution in large geometry evaluated by the Monte-Carlo method

Accuracy and reliability of pin-by-pin fission rate distribution in large geometries calculated by the multi-group Monte-Carlo method is examined through comparison with a deterministic transport code based on the method of characteristics. Various calculation geometries from a single assembly to a PWR full core are used for comparison of pin-by-pin fission rate distribution. An integral parameter, i.e., k-effective, can be accurately calculated by the Monte-Carlo method with a practical number of neutron histories (10⁶–10⁷) regardless of the size of the calculation geometry. On the other hand, comparison with the deterministic calculation shows that the estimated statistical errors for pin-by-pin fission rate distribution obtained by a Monte-Carlo calculation are somewhat underestimated in a large geometry, e.g., a PWR full core, under the present calculation conditions. Such underestimation of the statistical uncertainty of a local parameter should be carefully considered when the Monte-Carlo method is used as a reference tool for verification of a deterministic code, especially in large geometries.     

Application of Multiband Method to Pin Cell Calculations

Abstract

In order to accurately calculate effective neutron cross sections in the resonance energy region, the multiband method has been applied to cell calculations. Cell calculations for UO² and MOX fuels of light water reactors have been performed and the results were compared with those of a continuous energy Monte Carlo code VIM and the conventional self-shielding method using the Dancoff factor.

The k∞values calculated by the multiband method agreed with those of the VIM calculations within 0.20% Δk for the UO² fuel cell and within 0.30% Δk for the MOX fuel cell, respectively, whereas the Dancoff factor method yielded about l.l%Δk errors for the two cells. The element- wise contribution to this error was investigated, and it was found that the effective microscopic cross sections, particularly those for the giant resonances of ²³⁸U, calculated by the multiband method were in good agreement with those of VIM. It was also found that interference effect between ²³⁸U and ²³⁵U resonances in the UO² fuel and that between ²³⁸U and ²³⁹Pu resonances in the MOX fuel made about 0.20%Δk contributions to k∞ in both fuel cells.     

Moderator Temperature Coefficient in BWR Core

Temperature dependences of infinite multiplication factor k∞ and neutron leakage from the core must be examined for estimation of moderator temperature coefficient. Temperature dependence on k∞ has been investigated by many researchers, however, the dependence on neutron leakage of a BWR with cruciformed control rods has hardly been done. Because there are difficulties and necessity on calculations of three space dimensional and multi-energy groups neutron distribution in a BWR core.

In this study, moderator temperature coefficients of JPDR-II (BWR) core were obtained by calculation with DIFFUSION-ACE, which is newly developed three-dimensional multi- group computer code. The results were compared with experimental data measured from 20 to 275°C of the moderator temperature and the good agreement was obtained between calculation and measurement.

In order to evaluate neutron leakage from the core, the other two calculations were carried out, adjusting criticality by uniform absorption rate and by material buckling. The former underestimated neutron leakage and the latter overestimated it. Discussion on the results shows that in order to estimate the temperature coefficient of BWR, neutron leakage must be evaluated precisely, therefore the calculation at actual pattern of control rods is necessary.     

Measurements of k-Infinity on Zero-Leakage Zoned System (FCA IV)

The infinite multiplication factor k _∞ of the test zone composition has been measured for a series of FCA IV assemblies with a central test zone composed to produce a k _∞ value close to unity.

For the experimental determination of k _∞, in which some calculations as well as the reactivity worths of the cell and standard sample measured at the center are necessary, a new method of analysis is presented, and is compared with the current method. The new method provides the theoretical basis for k _∞-experiments. It also simplifies the calculations for analysis.

The FCA IV comprises a series of three assemblies of different test zone compositions representing pure uranium, uranium-sodium and uranium-graphite systems, all of which contain a portion of structural stainless steel. From the analyses made with different cross section sets, it was confirmed that the experimental value of k _∞ could be obtained with an accuracy of ±0.01, and in certain cases, ±0.005, for a zoned system with a central test zone with diameter larger than fifteen times the mean free path.     

压水堆堆芯Pin-by-pin均匀化计算中非均匀泄漏修正模型研究

在压水堆堆芯Pin-by-pin均匀化计算中采用均匀泄漏修正模型及非均匀泄漏修正模型对组件计算的中子能谱进行修正,本文研究了Pin-by-pin均匀化计算中均匀泄漏修正模型及非均匀泄漏修正模型的实现方式,提出了非均匀泄漏修正模型和栅元均匀化方法的联合实现方式,并分析比较了不同栅元均匀化扩散系数产生方式的计算效果。数值结果表明,非均匀泄漏修正模型及由其产生的中子泄漏系数能有效提高压水堆堆芯Pin-by-pin计算的精度。