Applications of the multidimensional equations to complex fuel assembly problems

For the analysis of reactors with complex fuel assemblies or fine mesh applications as pin by pin neutron flux reconstruction, the usual approximation of the neutron transport equation by the multigroup diffusion equation does not provide good results. A classical approach to solve the neutron transport equation is to apply the spherical harmonics method obtaining a finite approximation known as the P_L equations. In this line, a nodal collocation method for the discretization of these equations on a rectangular mesh is used in this paper to analyse reactors with MOX fuel assemblies. Although the 3D P_L nodal collocation method becomes feasible due to the improvements in computer hardware, a complete treatment of the detailed structure of the fuel assemblies in actual three-dimensional geometry is still prohibitive, thus, an assembly homogenization method is necessary. A homogenization method compatible with our multidimensional P_L code is proposed and tested performing heterogeneous and homogenized calculations. In this work, we apply the method to 2D complex fuel assembly configurations.     

Spherical Harmonics Solutions of Multi-Dimensional Neutron Transport Equation by Finite Fourier Transformation

A method of solution of a monoenergetic neutron transport equation in P_L approximation is presented for x-y and x-y-z geometries using the finite Fourier transformation. A reactor system is assumed to consist of multiregions in each of which the nuclear cross sections are spatially constant. Since the unknown functions of this method are the spherical harmonics components of the neutron angular flux at the material boundaries alone, the three- and two-dimensional equations are reduced to two- and one-dimensional equations, respectively. The present approach therefore gives fewer unknowns than in the usual series expansion method or in the finite difference method. Some numerical examples are shown for the criticality problem.     

Analysis of Cavity Effect on Space- and Time-Dependent Fast and Thermal Neutron Energy Spectra

The effects of the presence of a central cavity on the space- and time-dependent neutron energy spectra in both thermal and fast neutron systems are analyzed theoretically with use made of the multi-group one-dimensional time-dependent S_n method. The thermal neutron field is also analyzed for the case of a fundamental time eigenvalue problem with the time-dependent P₁ approximation. The cavity radius is variable, and the system radius for graphite is 120 cm and for the other materials 7 cm.

From the analysis of the time-dependent S_n calculations in the non-multiplying systems of polythene, light water and graphite, cavity heating is the dominant effect for the slowing-down spectrum in the initial period following fast neutron burst, and when the slowing-down spectrum comes into the thermal energy region, cavity heating shifts to cavity cooling. In the multiplying system of ²³⁵U, cavity cooling also takes place as the spectrum approaches equilibrium after the fast neutron burst is injected.

The mechanism of cavity cooling is explained analytically for the case of thermal neutron field to illustrate its physical aspects, using the time-dependent P₁ approximation. An example is given for the case of light water.     

Preliminary Calculation for Experiments Using Osaka University HITS, (I)

In order to discuss the ambiguity inherent in the diffusion approximation applied to a small-sized medium including a pulsed neutron source, the time variation and the spatial distribution of thermal neutron angular flux are calculated in P ₃-approximation of the time-dependent Boltzmann transport equation. With the discussion on the existence of a set of real and complex eigenvalues, the real eigenvalues for the lowest and some higher modes are computed numerically and the corresponding eigenfunctions are derived under Marshak's boundary condition. The results are compared with those obtained in the diffusion approximation and the physical interpretations of resultant asymptotic and non-asymptotic thermal neutron flux are given.     

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 of One-Point Reactor Kinetic Equations for Arbitrary Reactivities

This paper describes a very simple formulation of the reactor kinetic equation for arbitrary reactivity variations which can be solved analytically. The method of matched asymptotic expansions, which is a generalization of methods used in boundary-layer analysis, is employed to estimate the neutron density and the reactor period for ramp and periodic inputs. The small amounts of error arising in individual cases are analyzed quantitatively by comparison with results obtained from difference approximation (Runge-Kutta-Merson method). The validity of the zero-prompt-lifetime approximation and the stability condition for periodic inputs are also discussed. It is confirmed that the results obtained by the present method are numerically in complete agreement with those by other methods, provided the magnitudes of bias reactivity | ρ₀ |, reactivity amplitude | ρ₁ | and ramp reactivity | γt | are all very small compared with β, that the angular frequency ω?β/l*, and that, in particular, l*?10^?3.     

An Extended Finite Variable Difference Method with Application to QUICK Scheme

A Finite Variable Difference Method (FVDM) proposed previously by the author for locally exact numerical schemes is extended so as to be applicable to polynomial expansion schemes. This extended FVDM is applied to the QUICK scheme.

The optimum differencing points are analytically derived in terms of mesh Reynolds numbers so that the variance of the numerical solution is minimized under the condition that roots of the resulting characteristic equation are nonnegative to insure the numerical stability. This optimized scheme coincides with the original QUICK scheme at Rm=8/3, which is the critical value of its stability, and complements a stable scheme for Rm greater than 8/3. This optimization improves the numerical solution for the steady and unsteady convection-diffusion equations without numerical oscillations.

In the same manner as the previous result for the locally exact numerical schemes, it has been made clear based on the extended FVDM that optimum differencing points from the view point of numerical stability and accuracy exist for the polynomial expansion schemes.     

Calculation of extrapolation distances by low order Chebyshev polynomial approximation of transport equation in slab geometry

In this study, the problem of extrapolated end point has been studied in one-speed neutron transport equation with isotropic scattering by using the Chebyshev polynomial approximation which is called T_N method. Assuming neutrons of one speed, extrapolated end point are calculated for the uniform finite slab using Mark and Marshak type vacuum boundary conditions. It is shown that low order T_N method gives very good results of low order spherical harmonics approximation and diffusion theory for extrapolation of the flux of neutrons leaking from the medium. We present an alternative method which is similar to P₁ method to calculate the extrapolation distances z₀. Moreover, we prefer new solution of transport equation in one-dimensional slab geometry.     

An Approximate Calculation Method of the λ, ω p and ω d Eigenvalue Problem of the Group Diffusion Equation

The approximate solutions of the, λ, ω _p and ω _d eigenvalue problems of the group-diffusion equation for a multi-region reactor are obtained by expanding neutron fluxes into finite numbers of eigenfunctions satisfying the Helmholtz equation and the boundary condition at the extrapolated boundary of the reactor. The original eigenvalue problem is reduced to that of an asymmetric real matrix for the vector whose components constitute the expansion coefficients. For the numerical calculation of the real matrix thus derived, to determine the higher λ, ω _p and ω _d modes, the QR iteration method based on numerically stable unitary transformation, in combination with inverse iteration is effective in saving computation time.

The λ, ω _p and ω _d modes obtained by the above method are expressed by a linear combination of a comparatively small number of simple elementary functions, and are thus of high practical value in the numerical calculation of higher order perturbations and for examining snace-deDendent reactor dynamics.     

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.     

Comments on “Stability of the Gas Bubbles in Solids,by G.P. Tiwari”

A new difference equation for the numerical solution of the one-dimensional diffusion equation was obtained by using a semi-analytic method, in which the only approximation employed is that the source distribution within a mesh-region is represented by a linear function.

A test program, EXX-1, was prepared and run for BWR calculations, and comparisons were made with the conventional method. The results show that by using the new difference equation, a very coarse mesh model (1 mesh point per material region) can be applied without seriously impairing the computational accuracy.

It is also shown that the conventional difference equation becomes identical to the new expression if group-constants are multiplied by correction factors and the treatment of the source term appropriately modified.

New difference equations for spherical and cylindrical geometries are also given, in an appendix.     

Analysis of the Rossi-α Experiment through Pál-Bell's Equation

Pál -Bell's equation for the probability generating function of neutron counts has been analytically solved in the case of three time states, using two-forked approximation. From this solution it is found that all experimental data on neutron fluctuation consist individually of only three basic parameters. The average counting rate C, the decay constant α and the chain register rate C_r are in this instance chosen for the three fundamental measures of correlation. The original observation is presented that C_r can be obtained precisely by determining the waiting time distribution for the triggering of the time analyzer.

The correlated and uncorrelated parts of the Rossi-α data in a thermal system have been analyzed by this three-parameter scheme, and a consistent explanation is given of the results obtained.     

Transmission probability method for solving neutron transport equation in three-dimensional triangular-z geometry

This paper presents a transmission probability method (TPM) to solve the neutron transport equation in three-dimensional triangular-z geometry. The source within the mesh is assumed to be spatially uniform and isotropic. At the mesh surface, the constant and the simplified P₁ approximation are invoked for the anisotropic angular flux distribution. Based on this model, a code TPMTDT is encoded. It was verified by three 3D Takeda benchmark problems, in which the first two problems are in XYZ geometry and the last one is in hexagonal-z geometry, and an unstructured geometry problem. The results of the present method agree well with those of Monte-Carlo calculation method and Spherical Harmonics (P_N) method.     

Optimal Control of Distributed Parameter Reactor Core by Function Space Method and Some Numerical Experiences

Abstract

An optimal dynamical control of a linear reactor as a distributed parameter system is obtained numerically along with an analytical expression of the integral equation that should be satisfied by the optimal control. The function space method is employed to derive the equation, and it is known from the numerical experience that only the fundamental of expanding modes is enough to describe the integral kernel of the equation. Space-dependence is collected in the forcing term of the equation. The reactor core model is of two groups of neutrons in steady state and one group of precursors. Two performance functionals are tried. Both are quadratic but one is of the precursor history and the control, whereas another is of the neutron flux density and the control. The latter formulation partly dissolves the weighting problem in the quadratic performance functional.     

A two group analytical approximation solution for an external source problem without separation of space and time

This work presents the development of analytical approximation solutions for a space–time dependent neutron transport problem in two energy groups for a one dimensional system consisting of a homogenized medium with a localized external source. The approximation solutions are developed using Green’s functions, the influence of the delayed neutrons is not considered. Qualitative results for a given system are analyzed. A detailed comparison of the developed analytical approximation solutions with solutions with one energy group diffusion and P₁ equation without separation of space and time is given.     

Finite element spherical harmonics (PN) solutions of the three-dimensional Takeda benchmark problems

A set of multi-group eigenvalue (K_eff) benchmark problems in three-dimensional homogenised reactor core configurations have been solved using the deterministic finite element transport theory code EVENT and the Monte Carlo code MCNP4C. The principal aim of this work is to qualify numerical methods and algorithms implemented in EVENT. The benchmark problems were compiled and published by the Nuclear Data Agency (OECD/NEACRP) and represent three-dimensional realistic reactor cores which provide a framework in which computer codes employing different numerical methods can be tested. This is an important step that ought to be taken (in our view) before any code system can be confidently applied to sensitive problems in nuclear criticality and reactor core calculations. This paper presents EVENT diffusion theory (P₁) approximation to the neutron transport equation and spherical harmonics transport theory solutions (P₃–P₉) to three benchmark problems with comparison against the widely used and accepted Monte Carlo code MCNP4C. In most cases, discrete ordinates transport theory (S_N) solutions which are already available and published have also been presented. The effective multiplication factors (K_eff) obtained from transport theory EVENT calculations using an adequate spatial mesh and spherical harmonics approximation to represent the angular flux for all benchmark problems have been estimated within 0.1% (100 pcm) of the MCNP4C predictions. All EVENT predictions were within the three standard deviation uncertainty of the MCNP4C predictions. Regionwise and pointwise multi-group neutron scalar fluxes have also been calculated using the EVENT code and compared against MCNP4C predictions with satisfactory agreements. As a result of this study, it is shown that multi-group reactor core/criticality problems can be accurately solved using the three-dimensional deterministic finite element spherical harmonics code EVENT.     

Incipient Boiling on Exponentially Heated Surfaces

Calculations have been performed on the asymptotic angular neutron flux, critical thickness and extrapolation distance in the axial direction for a cylindrical system of finite length. Neutrons are assumed to be monoenergetic. Applying the finite Laplace transformation in the axial direction and buckling approximation in the radial, we find a solution of the transport equation that satisfies the boundary conditions of no incoming neutron and symmetry. This method is an extension of Kobayashi's Laplace transform method for slab problem, and may easily be applied to other geometries. For a rectangular prism system, numerical results based on our method are compared with results from P₃-approximation.     

Contribution of Heavier Nucleus on Neutron Slowing Down in Hydrogenous Mixture, (I)

The problem of neutron slowing down in hydrogenous mixtures was studied from an analytical approach. The coupled slowing down equations derived from P ₁-diffusion, using the G.G. or Fermi approximations, were solved consistently, and detailed expressions for special cases are discussed.

It is shown that, in a system comprising two kinds of nuclei, the neutron energy spectrum has a transient term near a source, representing an interference effect between the neutron moderation by the two different kinds of nuclei. The consequence of the G.G. approximation is limited mainly to the transient region so long as absorption and leakage are negligible. When the mass of the non-hydrogenous nucleus becomes infinitely large, our general expression for the neutron energy spectrum derived from the G.G. approximation coincides with the result from the simple S.G. method.     

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry. Part I: Modeling and verification for time-independent solution

This paper describes a novel method based on using cellular neural networks (CNN) coupled with spherical harmonics method (P_N) to solve the time-independent neutron transport equation in x–y geometry. To achieve this, an equivalent electrical circuit based on second-order form of neutron transport equation and relevant boundary conditions is obtained using CNN method. We use the CNN model to simulate spatial response of scalar flux distribution in the steady state condition for different order of spherical harmonics approximations. The accuracy, stability, and capabilities of CNN model are examined in 2D Cartesian geometry for fixed source and criticality problems.     

Determination of Neutron Multiplication, (II)

The purpose of the present work is to confirm experimentally that the prompt neutron decay constant of fundamental mode, α₀₀₀, satisfies the following two requirements from the concept proposed in Part (I): To be uniquely measurable free from space dependence and detector specification, and to be calculable from the Boltzmann equation. To prove this, a series of pulsed experiments was made at various subcritical states in a reflected graphite-moderated 20% enriched fuel system (SHE).

The experimental results confirmed that α₀₀₀ was obtained as a common and unique value at any point of both core and reflector, determined by separation of spatial harmonics, using bare and Cd-covered BF₃ counters. The state of SHE was changed by attenuating the core, with or without central control rod, and by adding distributed poison to the core and/or to the reflector.

Direct comparison was made between the experimental results and the calculated values from multigroup treatment using P₁ and S₄ approximation. Calculated values of α₀₀₀ for subcritical states with attenuated core, with or without control rod, showed fairly good agreement with the measured values. In addition, the time-dependent spectral changes of the subcritical systems were also investigated in the experiments.