A rigorous weight function for neutron kinetics equations of the quasi-static method for subcritical systems

The purpose of the present work is to develop an efficient solution method to solve the time dependent multi-group diffusion equations for subcritical systems with external sources using the quasi-static method.Usually, the k-eigenfunction for an adjoint criticality equation is used as a weight function to derive a one-point neutron kinetics equation for the amplitude function in the quasi-static method. It is shown that the use of this k-eigenfunction introduces a first order error due to the change of the flux, when the systems are not close to the critical state. It is shown also that the use of the ω-eigenfunction for the adjoint time dependent equation as the weight function can eliminate such first order error resulting from ignoring the term of first order change of the shape function to solve subcriticality problems, and it gives more accurate results than the use of conventional k-eigenfunctions of the critical adjoint equation.     

Three-Dimensional First-Order Perturbation Calculation Method for Reactivity Changes Due to Core Deformations of Fast Breeder Reactors

A three-dimensional diffusion calculation method has been proposed to rapidly and accurately calculate reactivity changes of LMFBRs caused by assembly displacements in accidental events. The method requires shorter computation times and provides almost the same accuracy as a conventional direct eigenvalue calculation method. In this method, changes in macroscopic neutron cross-sections and diffusion coefficient are defined so that changes in both region volume and material composition can be treated in a mesh-centered finite-difference program under the same coarse mesh division as used for the normal, non-deformed core. Reactivity changes are calculated from the above-mentioned changes by the first-order perturbation method using normal and adjoint neutron fluxes calculated beforehand for the normal core.

The method was applied to deformations of a 1,000-MWe LMFBR core. Reactivity changes calculated by the method agreed within 0.4% with those by a conventional direct eigenvalue calculation method, while computation time was less than 1/35.     

Interpretation of Pulsed-Neutron-Source Experiments in a Multiregion Reactor

By expanding the static flux into kinetic fluxes containing the fundamental and higher modes, we derive a kind of inhour equation which can express kinetic distortion, and can relate the static reactivity to prompt decay constants and to kinetic and adjoint fluxes. The equation is developed using the time-dependent, multigroup diffusion approximation, and is applied to the interpretation of pulsed-neutron-source experiments in a multiregion reactor. It is shown from this application that the Simmons-King formula can express the dynamic reactivity only under conditions of constant generation time and no kinetic distortion, and that the conventional inhour equation derived by the perturbation method cannot distinguish between the static and dynamic reactivities differing from each other on account of kinetic distortion.     

Iterative method for obtaining the prompt and delayed alpha-modes of the diffusion equation

Higher modes of the neutron diffusion equation are required in some applications such as second order perturbation theory, and modal kinetics. In an earlier paper we had discussed a method for computing the α-modes of the diffusion equation. The discussion assumed that all neutrons are prompt. The present paper describes an extension of the method for finding the α-modes of diffusion equation with the inclusion of delayed neutrons. Such modes are particularly suitable for expanding the time dependent flux in a reactor for describing transients in a reactor. The method is illustrated by applying it to a three dimensional heavy water reactor model problem. The problem is solved in two and three neutron energy groups and with one and six delayed neutron groups. The results show that while the delayed α-modes are similar to λ-modes they are quite different from prompt modes. The difference gets progressively larger as we go to higher modes.     

Resonance absorption of neutrons in an infinite homogeneous medium

The problem of the slowing down of neutrons in an infinite homogeneous medium with strong resonance absorption and uniformly distributed neutron sources is investigated in this paper. The solution of the adjoint equation represents the probability that a neutron of energy E escapes resonance absorption during the process of slowing down to a certain asymptotic energy. The solution of the main and the adjoint problems makes it possible for us to apply a perturbation method to take into account the influence on the resonance integral of the Doppler broadening of the resonance level. The methods developed have been applied to the calculation of the collision density and the resonance integrals for the first level of U²³⁸ (E₀ = 6.7 ev) in pure uranium and in uranium oxide UO₂.     

Calculation of Coupling Coefficients of Coupled Reactors Theory Using the Generalized Perturbation Theory

Recently, rigorous multi-point equations are derived using the region-wise importance functions to produce fission neutrons. Since the coupling coefficients used in these multi-point equations are calculated with the weight of these importance functions but not the adjoint function used in the conventional perturbation theory, errors due to the change of the flux is introduced in the coupling coefficients for a perturbed system if the unperturbed flux is used.

It is shown that using the generalized perturbation theory, the coupling coefficients using the unperturbed flux can be obtained taking into account the first order change of the flux due to the perturbation, and the same accuracy as the conventional perturbation theory in which the adjoint function is used can be obtained in the case of one-point reactor.     

Numerical Study of Higher Order Perturbation Theory

The higher order perturbation theory is studied numerically. The one-dimensional diffusion equation is solved by the conventional (1st order) and the higher order (2nd and 3rd order) perturbation methods. The results are compared with those exactly calculated, and the accuracy, the effect of the higher order terms and limit of the perturbation method are examined. When the perturbation in the reactor is non-uniform, the higher order perturbation method is found to be effective. When the perturbation is uniform, however, the higher order terms are of relatively small importance.     

Anisotropic Diffusion Effect on Criticality of Plate Lattice Fast Assembly

The neutron diffusion in plate lattice is generally somewhat anisotropic. In case of usual plate cells for the mockup of LMFBR composition, the diffusion coefficient for parallel direction to lattice plate, based on Benoist's theory, proves to be larger by 2～4% than that for perpendicular direction, which is considered to affect the criticality of plate lattice fast assembly.

A practical treatment of the anisotropic diffusion effect on criticality has been proposed, in which, like the transport correction, the anisotropic diffusion effect is treated as a correction term to be applied to the conventional isotropic diffusion calculation. The method is applied to actual plate lattice critical assemblies, already built in FCA, ZPR or ZEBRA. The anisotropy correction on criticality turns out to amount to the order of -0.2～-0.4%Δ k/k for the normal plate lattice-core. The amount of anisotropy correction is further enhanced in case of an assembly consisting of plate lattice-blanket or sodium-voided core. The anisotropic diffusion effect on criticality is, therefore, important for the analysis of criticality of plate lattice assembly, and should be corrected in addition to the conventional heterogeneity effect. The present method, based on the perturbation theory, is practical and useful.     

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.     

A Method of Calculation of Detailed Power Distribution in Fuel Assemblies

Abstract

A method has been developed that effectively estimates the detailed distribution of power generation in the fuel or blanket assemblies in nuclear reactors. A two-dimensional, one-group diffusion model is applied to a region of homogeneous composition enclosed in a contour devoid of concavity viewed from outside. The diffusion equation is reduced to the form of Helmholtz equation, and a non-homogeneous boundary condition of Dirichlet or Neumann type is given on the contour, using neutron fluxes previously obtained in coarse mesh diffusion criticality calculations covering the whole reactor. This boundary value problem in two-dimensional space is made to yield a solution in the form of a potential due to a single or double layer. The method is applied to a hexagonal cell of a fast reactor. The results of calculation are amply accurate in comparison with the corresponding values from the usual fine-mesh diffusion scheme and with much shorter computing time.     

Adjoint flux calculation of natural mode equation by time dependent neutron transport

ABSTRACT

The physical implication of the eigenfunction of the adjoint of the natural mode equation is studied. The eigenfunction at a phase space position is theoretically derived to be proportional to the amplitude of the neutron flux sufficiently long time after a source is placed at the position. Meanwhile, the time change rate and distribution of the neutron flux originated in the source must converge to those of the fundamental mode of the natural mode equation. Based on the proportional relation, the adjoint flux is calculated for a MOX- and UO₂-fueled lattice by a continuous-energy Monte Carlo time-dependent neutron transport calculation. The calculated distributions of the adjoint flux agree well with those of the iterated fission probability both in the prompt and delayed critical states.     

Determination of Large Negative Reactivity by Integral Versions of Various Experimental Methods

The large negative reactivity is measured in Semi-Homogeneous Experimental facility (SHE). Experimental methods are Sjöstrand's pulsed neutron, source multiplication and rod drop methods beside revised King-Simmons' pulsed neutron methods. Neutron detectors are placed at various points in the core region for multi-points measurement.

Usual one-point reactor model analysis resulted in the reactivity values, strongly dependent on the detector position with the increase of subcriticality. In addition, disagreements between the used experimental methods are also pointed out.

In order to overcome these difficulties due to the spatial higher harmonics and the kinetic distortion in the neutron flux distribution, an integral version analysis is applied, in which use is made of multi-points reactor model. In the analysis, space integration of the neutron counts obtained throughout the core region is made with weights of the adjoint function of fast neutrons, calculated using the two- or three-dimensional diffusion code. The negative reactivity values determined by the integral version analysis agreed well with each other within the uncertainty of ～5% in the reactivity range down to ～50 dollars.

It is concluded that all the experimental methods are adequate for precise determination of the large negative reactivity of reactor provided that the integral version analysis is utilized or that correction is made for the change of the neutron generation time using precise calculation.     

Solution of Multi-Group Diffusion Equation in x-y-z Geometry by Finite Fourier Transformation

The multi-group diffusion equation in three-dimensional x-y-z geometry is solved by finite Fourier transformation. Applying the Fourier transformation to a finite region with constant nuclear cross sections, the fluxes and currents at the material boundaries are obtained in terms of the Fourier series. Truncating the series after the first term, and assuming that the source term is piecewise linear within each mesh box, a set of coupled equations is obtained in the form of three-point equations for each coordinate. These equations can be easily solved by the alternative direction implicit method. Thus a practical procedure is established that could be applied to replace the currently used difference equation.

This equation is used to solve the multi-group diffusion equation by means of the source iteration method; and sample calculations for thermal and fast reactors show that the present method yields accurate results with a smaller number of mesh points than the usual finite difference equations.     

A Generalized Analysis of Rossi-α Experiment

A generalized theory of the Rossi-α method is shown and is applied to the analysis of an experiment made with a remotely subcritical system.

Starting from the neutron transport equation, a general expression for the probability of detecting correlated neutrons is given, which is a function of the positions and sensitivities of the neutron detectors used in the experiment.

Some numerical examples for this expression are also given under the one-group diffusion approximation for comparison with the experiment. The agreement between them is satisfactory.     

Solutions of Diffusion Equations in Two-Dimensional Cylindrical Geometry by Series Expansions

Abstract

A solution of the multi-group multi-regional diffusion equation in two-dimensional cylindrical (p-z) geometry is obtained in the form of a regionwise double series composed of Bessel and trigonometrical functions.

The diffusion equation is multiplied by weighting functions, which satisfy the homogeneous part of the diffusion equation, and the products are integrated over the region for obtaining the equations to determine the fluxes and their normal derivatives at the region boundaries. Multiplying the diffusion equation by each function of the set used for the flux expansion, then integrating the products, the coefficients of the double series of the flux inside each region are calculated using the boundary values obtained above.

Since the convergence of the series thus obtained is slow especially near the region boundaries, a method for improving the convergence has been developed. The double series of the flux is separated into two parts. The normal derivative at the region boundary of the first part is zero, and that of the second part takes the value which is obtained in the first stage of this method. The second part is replaced by a continuous function, and the flux is represented by the sum of the continuous function and the double series.

A sample critical problem of a two-group two-region system is numerically studied. The results show that the present method yields very accurately the flux integrals in each region with only a small number of expansion terms.     

Prediction of the neutrons subcritical multiplication using the diffusion hybrid equation with external neutron sources

We used the neutron diffusion hybrid equation, in cartesian geometry with external neutron sources to predict the subcritical multiplication of neutrons in a pressurized water reactor, using a 1/M curve to predict the criticality condition. A Coarse Mesh Finite Difference Method was developed for the adjoint flux calculation and to obtain the reactivity values of the reactor. The results obtained were compared with benchmark values in order to validate the methodology presented in this paper.     

ABSTRACTS OF JAPANESE ARTICLES

A new approximate method for calculating the effectiveness of multiple control rods fully inserted in a reactor is described. This method is appicable to a bundle of many control rods, regardless of the number of rods, as well as to an array of a limited number of few rods.

Using either the sink model or the well model, a reactor equation of kernel form is obtained. The reactor equation is a two-dimensional diffusion equation with two-group diffusion kernel. In order to facilitate numerical computation of the eigenvalue, the integral equation is reduced to a set of linear homogeneous equations, by dividing the reactor into a large number of unit cells containing at most one control rod.

This method has been programmed for the IBM 7090, the code being given the designation ELC. The iterative procedure used converges much faster than the standard accelerated finite-difference programs. Using the ELC code, the effectiveness of an array of four control rods fully inserted in a cylindrical reactor was calculated. The results are in good agreement with those found by the Scaletter-Nordheim method. In the case of a large number of control rods, there is no alternative method to be compared with.     

A Stochastic Operator Method of Calculating the Time Interval Distributions in Neutron Detection

An operator method has been applied to formulate probability distribution functions of neutron counts in a reactor.

Assuming all statistical events occurring in the reactor to be Markovian, an operator representation of the no count probability for a given time interval is given. The basic equation for the probability is derived from the Kolmogorov-Chapman equation, and the formal solution obtained thereof. Approximate expressions are also given, for two types of detectors—absorption and fission. The effects of moments of order higher than the second are evaluated numerically.

Further development of the operator calculus has yielded relations connecting the waiting time distribution and the interval distribution with the no count probability.     

Solution of Diffusion Equation in r-z Geometry by Finite Fourier Transformation

It is shown that the monoenergetic diffusion equation in multi-region r-z geometry can be solved by the finite Fourier transformation method which has successfully been applied to x-ygeometry. In this method, a system of linear algebraic equations is derived for Fourier coefficients of fluxes and currents at the material boundaries between regions of constant cross sections, and all the boundary values are determined by solving this equation.

Numerical examples are presented for a problem featuring a fixed source and multiple regions, and the results are compared with those obtained from the current difference method. It is shown that the present method yields a better result with relatively few terms of expansion.