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.     

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.     

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.     

Application of Finite Element Method to Two-Dimensional Multi-Group Neutron Transport Equation in Cylindrical Geometry

The finite element method is applied to the spatial variables of multi-group neutron transport equation in the two-dimensional cylindrical (r, z) geometry. The equation is discretized using regular rectangular subregions in the (r, z) plane. The discontinuous method with bilinear or biquadratic Lagrange's interpolating polynomials as basis functions is incorporated into a computer code FEMRZ. Here, the angular fluxes are allowed to be discontinuous across the subregion boundaries.

Some numerical calculations have been performed and the results indicated that, in the case of biquadratic approximation, the solutions are sufficiently accurate and numerically stable even for coarse meshes. The results are also compared with those obtained by a diamond difference S _n code TWOTRAN-II. The merits of the discontinuous method are demonstrated through the numerical studies.     

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.     

An Improvement of the Transverse Leakage Treatment for the Nodal SN Transport Calculation Method in Hexagonal-Z Geometry

To overcome the divergent behavior of the NSHEX code, a nodal S_N code for hexagonal geometry, for some transport calculations, an improvement has been made in the calculation of the axial leakage. The axial leakage, previously calculated by using the quadratic transverse leakage approximation (QLA), is calculated by a new method of analytically treating the spatial distribution within a node, based on the axial homogeneity of the ordinary core. The verification tests were performed for the KNK-II model geometry of the NEACRP 3-D Neutron Transport Benchmarks and the large assembly-size KNK-II model. It is found that k_ett values obtained by introducing the new method agree with the reference Monte Carlo calculation results within 0.1% Δk/k for the KNK-II model, although the QLA method did not converge for two cases. Furthemore the new method succeeded in calculations for the large assembly-size model, in which the QLA method failed for all cases. Thus the new method has been found accurate and convergence achieved for the cases in which the QLA method failed.     

Using Anemometer for Particle Size Measurement of Sodium Mist

A code “ACTFCI” has been developed for analyses of Fuel Coolant Thermal Interaction in a coolant channel. The code can deal with the dynamic behavior of coolant in r-z geometry, but does not take into account the dynamics of fuel particle two-dimensional movement, the fuel being considered only as heat source for the coolant. The ICED-ALE method was used for the numerical analyses of the three conservation equations and two equations of state for single and two-phase sodium. A sample calculation is given concerning the period up to initiation of sodium boiling, for a geometry representing a wrapper tube of reduced size. Despite this difference of the model from actual size, the calculated results suffice for demonstrating the utility of this method in obtaining stable solutions on single-phase pressure wave propagation in both r and z directions, and on sodium boiling initiation triggered by the rarefaction wave returning to the interaction region from channel extremity.     

Application of MONK and MCBEND to Transport Flask Design Assessment

Abstract

Safe operation of nuclear fuel transport flasks is of fundamental importance and is ensured by compliance with IAEA and national regulations. Criticality and shielding analyses of flask designs in both normal and accident conditions are required to demonstrate compliance. In recent years increased computing power has facilitated the use of the Monte Carlo method for such analyses as standard. The method is an improvement over earlier methods because approximations in geometry, scattering angle/energy and reaction cross sections can be significantly reduced and in some cases effectively eliminated. In this paper the application of the Monte Carlo codes MONK and MCBEND, developed by AEA Technology and BNFL, to such analyses is described, with emphasis on features which facilitate accuracy and ease of use.     

A Numerical Study on Faraday-Type Electromagnetic Flowmeter in Liquid Metal System, (I)

To predict the three-dimensional steady-state structure of electromagnetic phenomena in the Faraday-type Electromagnetic Flowmeter (EMF), a numerical simulation method using the Galerkin finite element method has been adopted and formatted as a computer code, FALCON.

As the induction equation for the induced magnetic field that is important in a liquid metal system, we have presented B-φ formulation for the induced magnetic field and the electric potential field and A-φ formulation for the vector potential field and the electric potential field, and compared the computational efficiency of both formulations. We have selected the B-φ formulation. The computer simulation of the EMF of the MONJU secondary cooling system has been performed and the calculations have shown a good agreement with the calibration test data. We have investigated qualitatively how far the influence of the end effect extends and the characteristics of the voltage response.

Overall, this study has indicated that the present method can be applied to the estimation of the characteristics of an EMF and the design of a new EMF.     

Importance Determination Method for Geometry Splitting with Russian Roulette in Monte Carlo Calculations of Thick and Complicated Core Shielding Structure

In case of a shielding analysis of the geometry having thick and complicated structures with a Monte Carlo code, it is a serious problem that it takes too much computer time to obtain results with good statistics. Therefore, it is very important to reduce variances in the calculation. In this study, a method to determine the importance function in 3-dimensional Monte Carlo calculation with geometry splitting with Russian roulette was developed for the shielding analysis of thick and complicated core shielding structures. Only two essential importance ratio curves for one material enable us to determine the importance function easily in the shielding calculation.

The validity of this method was confirmed through a simple benchmark calculation. From the comparison with the result obtained by using weight window (W-W), it was shown that the present method can give an accurate result on the same level with W-W method with less trial and errors. And this method was applied to an actual reactor core shielding analysis to confirm its applicability to a 3-dimensional thick and complicated structure.

Using this method, the variance reduced calculation can be easily realized with the developed importance determination procedure, especially in case that parameter survey calculations are required in order to determine the shield thickness in a design work of a thick and complicated structure. Accordingly, it became easier to use Monte Carlo method as a powerful tool for a reactor core shielding design.     

A nodal SN transport method for three-dimensional triangular-z geometry

A new nodal S_N transport method has been developed to perform accurate transport calculation in three-dimensional triangular-z geometry, where arbitrary triangles are transformed into regular triangles via a coordinate transformation. The transverse integration procedure is applied to treat the neutron transport equation in the regular triangle. The neutron angular distributions of intra-node fluxes are represented using the S_N quadrature set, and the spatial distributions of neutron fluxes and sources are approximated by a quadratic polynomial. The nodal-equivalent finite difference algorithm for 3D triangular geometry is applied to establish a stable and efficient iterative scheme. The present method was tested on four 3D Takeda benchmark problems published by the nuclear data agency (NEACRP), in which the first three problems are in XYZ geometry and the last one is in hexagonal-z geometry. The results of the present method agree well with those of the reference Monte-Carlo calculation method, the difference in k_eff being less than 0.1%. This shows that multi-group reactor core/criticality problems can be accurately and effectively solved using the present method.     

Numerical Solutions of Discrete-Ordinate Neutron Transport Equations Equivalent to PL Approximation in X-Y Geometry

A numerical method for solving the steady-state one-velocity neutron transport equation in x-y geometry is presented. It is based on the concept of combining the spherical harmonics theory with the discrete-ordinate method. The validity of the method is illustrated by several numerical computations using the TWOTRAN-PLXY code, formulated by modifying the ordinary discrete-ordinate code TWOTRAN-(x, y).

Through numerical studies, it is shown that the present method is effective for obtaining solutions of high accuracy, as well as for eliminating the ray effects present in the ordinary discrete-ordinate method. As for the techniques for accelerating the convergence of the iterative solutions, it is proved that the Chebyshev device works well for the present method, while whole-system rebalancing is found to be less effective.     

New Efficient Albedo Monte Carlo Method for Neutron Streaming Analyses

An efficient albedo Monte Carlo method newly developed has been studied by analyzing two types of experiments on neutron streaming. The method is characterized by employing the energy-angle dependent doubly differential albedos for slab, which can be calculated in a short computer time with a one-dimensional transport theory, such as the Sn method and more efficient invariant imbedding method. This paper describes the features of the present albedo Monte Carlo method, including fundamental formulas. In the analyses of the neutron streaming experiments, the calculated results agreed with the measured data within a factor of 2 for a benchmark experiment at the YAYOI reactor and within a factor of 3 for an SNR sodium duct mock-up experiment.

It is concluded that the present albedo Monte Carlo method is practical and applicable to the reactor shielding analysis concerning radiation streaming.     

Preliminary neutronic assessment of a helium-cooled Li8PbO6 breeding blanket design for DEMO

A preliminary neutronic assessment of the performances of a helium-cooled Li₈PbO₆ breeding blanket for the conceptual design of a DEMO fusion reactor is given. The study mainly focuses on TBR, power density responses and shielding factor optimization to estimate the feasibility of the design under the prescribed radiation deposition limits at TF-coils superconducting magnets. Computational analyses are based on three-dimensional 30° sector using the Monte Carlo code MCNPX 2.6. The scoping interest of helium-cooled Li₈PbO₆ blanket designs is based on a large potential minimization of the amount of Be required and the strong relaxation of ⁶Li enrichment requirements for this solution when compared to other solid breeder blanket options.     

Model Development for Bubble Turbulent Diffusion and Bubble Diameter in Large Vertical Pipes

Multi-dimensional analyses have been expected recently with expanding computation resources for gas-liquid two- phase flow analyses of advanced nuclear systems such as passive safety systems and natural-circulation-type reactors. However, the applicability of previous constitutive equations for multi-dimensional analyses has not been fully investigated especially for the effects of flow path scale because the equations have been assessed for small-scale experiments. In this study, we analyzed the scale effects by the multi-dimensional two-fluid model code using data in 38 mm and 200 mm diameter pipes. We clarified a key-parameter to model the scale effects and developed models for the effects on phase distribution. The scale effects can be classified by the relative relationship between bubble diameter d_b and turbulent length scale l_T . Bubble-induced turbulence is increased under that d_b is smaller than l_T and bubble coalescence is predominated rather than breakup under that l_T is about three times larger than d_b and under higher void fraction. Based on these findings, we established new models for bubble turbulent diffusion and bubble diameter. The applicability was promising through assessments against the 38 mm and 200 mm pipes under different flow rates and against databases for 60 mm, 100 mm and 480 mm pipes.     

Evaluation of Computational Models for Fission and Spallation Reactions Used in Accelerator Breeding and Transmutation Analysis Code

Computational models for spallation and fission reactions used in an accelerator breeding and transmutation code have been evaluated by performing calculations for thin targets of Bi, Pb, Th and U in the energy range of 50–1,000 MeV. Proton and neutron non-elastic and fission cross sections have been derived from the counts of real collisions and fission events in the targets.

Several fission models in combination with a spallation model are compared with the experimental data due to Steiner et al. and Schimmerling et al. A good agreement has been obtained for a model with the level density parameters a_n = A/10 and a_f/a_n fitted to the data due to Il'inov et al. The mass dependence of the non-elastic cross sections has been calculated also with the use of the best fit model. A good agreement with the experimental data was obtained over a wide range of nuclear masses.     

多孔介质方法在池式快堆系统分析软件SAC-CFR三维钠池计算模型中的应用

为准确分析池式快堆热钠池内的热工水力学特性,在已开发出的用于池式快堆系统分析的钠池三维计算模型的基础上,应用多孔介质方法建立钠池内中间热交换器、主泵、事故热交换器及屏蔽柱模型,完成了含有多孔介质模型和复杂几何边界的钠池三维计算模型开发。将该模型嵌入池式快堆系统分析软件SAC-CFR后,分析了中国实验快堆在稳态运行和紧急停堆工况下钠池内的流场分布,初步证明了所采用的多孔介质模型的合理性,为下一步非能动余热排出系统模型的开发做准备。     

Application of Quick Subchannel Analysis Method for Three-Dimensional Pin-by-Pin BWR Core Calculations

Three-dimensional pin-by-pin core analysis is considered to be a candidate for the next-generation BWR core calculation method. In our previous study, the applicability of the transport and burnup calculations for a three-dimensional pin-by-pin BWR core analysis was investigated. However, the thermal-hydraulics calculation has not yet been studied in this framework. In the conventional core analysis code, the bundlewise thermal-hydraulics calculation is adopted. In the actual core analysis, the power distribution inside a fuel assembly is tilted at the region adjacent to a control blade or the core peripheral region. In these regions, the consideration of the subchannel-wise void distribution has an impact on the fission rate distribution. Therefore, an evaluation of the detailed void distribution inside an assembly, i.e., the incorporation of the subchannel wise void distribution, is desirable for the pin-by-pin BWR core analysis. Although several subchannel analysis codes have been developed, these subchannel analysis codes generally require a large computational effort to estimate the subchannel-wise void distribution in a whole BWR core. Therefore, to analyze a whole BWR core within a reasonable computation time, it was necessary to apply a fast subchannel analysis code. In this paper, a quick subchannel analysis code dedicated to pin-by-pin BWR core analysis is newly developed, and the void distribution of the present subchannel analysis code is compared with the prevailing subchannel analysis code NASCA using three-dimensional single-assembly geometries. Since the present subchannel analysis code is used for a coupled neutronics/thermal-hydraulics analysis, the results of the coupling calculation are also compared with those of NASCA. The calculation result indicates that the void distribution difference between NASCA and the present subchannel analysis code is slightly less than 10%. This result indicates that the prediction accuracy of the present subchannel analysis code will be reasonably appropriate for a pin-by-pin BWR core analysis. Furthermore, the results show that the calculation time of the present subchannel analysis code is only 10 min for a hypothetical three-dimensional ABWR quarter-core geometry using a single CPU. This calculation time is sufficient for a pin-by-pin BWR core analysis.     

A parametric study and a guide chart to avoid condensation-induced water hammer in a horizontal pipe

An experimental and analytical studies on the condensation-induced water hammer (CIWH) for steam-water countercurrent flow in a nearly horizontal circular pipe have been performed. A total of 17 experimental data for the onset of slugging, which is assumed to be the precursor of the CIWH, have been obtained for various flow rates of water. Incorporating the most recent correlations of interfacial heat transfer and friction factor developed for a circular geometry and using an improved criterion of transition from stratified to a slug flow, two existing analytical models to predict lower and upper bounds for CIWH have been upgraded. Applicability of the present as well as existing CIWH models has been tested by comparison with two sets of CIWH data. The result of this comparison shows that the applicability of the present as well as existing models is reasonably good. Based on the present models for CIWH, a computer code entitled as ‘KAIST-CIWH’ has been developed and sample guide charts to find CIWH free regions for a given combination of major flow parameters in a long horizontal pipe have been presented along with the results of parametric studies of major parameters (D, P, T_f,in, and L/D) on the critical inlet water flow rate (W_f,in)_crit for both lower and upper bounds. In addition, two simple formulas for lower and upper bounds that can be used in an emergency for quick results have been presented.