No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Synthesis of Three-Dimensional Flux Shapes Using Discontinuous Sets of Trial Functions
Abstract
The method of flux synthesis is extended in a systematic way to allow the possibility of using different sets of trial functions in different axial zones. The necessary equations are derived in some detail and numerical examples are presented. The results of these examples are very satisfactory and suggest, therefore, that the synthesis procedure can be made much more useful and powerful by extending it in this way. In a more general context they suggest that the basic notation of deriving discontinuous-type approximation methods from an appropriate variational principle is a valid and very effective idea.
The Korea Atomic Energy Research Institute (KAURI) has been developing the ISO MWe Prototype Generation-IV Soilium-cooled Fast Reactor (PGSFR). The design concept is highly based on passive safety mechanisms, minimizing the need for engineered-safety systems. Presently, it is of primary Importance to assure the reactor dynamics and stability against small reactivity disturbances under power operating conditions. KA ERl has therefore developed the NttSTA b code for stability analysis of the PGSFR In NuSTAB. the neutron-kinetic and thermal-hydraulic coupling equations are linearized to form the characteristic equation, which is solved as a generalized eigenvalue problem for determining the decay ratio, an indicator of the system stability. In this paper, the stability of the PGSFR Ivor analyzed by applying the point kinetic and spatial kinetic options in the NuSTAB code. System responses to temperature feedbacks including the Doppler effect, thermal expansion, coolant density change, and overall feedback were studied. The results indicate that the initial U and final TRU cores of the PGSFR are both inherently stable thanks to the tem/>era- lure feedbacks.
Within the large variety of methods of analysis for transient phenomena in nuclear reactors, this article is limited to the discussion of computational methods for the analysis of operational transients, off-normal transients, and hypothetical accidents in power reactors. Though any realistic analysis of such transients requires a coupled treatment of thermodynamics, fluid dynamics and neutronic phenomena, all details of the modeling and computations of thermodynamic and fluid dynamic quantities are omitted in the following presentation.
The purpose of the present paper is to discuss some numerical experiments in three-dimensional nuclear reactor dynamics with emphasis on the comparison between the Non Symmetric Alternating Direction Explicit (NSADE) finite difference method, as programmed in the 3DKIN code (a proper exponential transformation of the time dependent equations is also included there) and space-time synthesis methods.The main conclusion is that for time step sizes which are not small enough, the NSADE solution shows oscillations having no physical meaning. These oscillations are particularly evident when large cores with non-symmetric perturbations are dealt with, and can be made to disappear only at the expense of step sizes that are small beyond any expectation. This sensitivity of the NSADE scheme to time step size is far greater than we imagined before our experiments and it may be an inherent limitation of the method, not only in production work but also in the validation of more approximate methods such as synthesis and quasi-static methods, even though relatively coarse spatial mesh grids are used.In support of these conclusions, two transient problems, both simulating a control rod drop accident in a BWR core, at a speed of about 3.5 m/sec, are described here: the first is a prompt supercritical transient with a symmetric perturbation (four control rods moving at symmetric locations) and the second one is a delayed supercritical transient with an unsymmetric strongly localized perturbation (only one control rod is dropped). In the latter problem, the power oscillations, caused by the NSADE method, disappear only when the step sizes Δt are so small that about 1500 time steps sometimes are needed for a mere doubling of the power level (Δt < 0.0005 sec). On the other hand we found that the same accuracy could be attained by the synthesis calculation with time step sizes varying from 0.003–0.01 sec, i.e. ten times as long.
The synthesis method is combined with the first-flight collision probability method to treat the nonuniformity of the medium along the axial direction. The neutron flux in a three-dimensional system is expressed by the product of a known trial function in the horizontal plane and an unknown function along the z-axis. The trial function in the x-y plane is evaluated in terms of the first-flight collision probability calculated in either one or two-dimensional system. And the unknowns are determined in each axial subzone by solving simultaneous equations with use made of the first-flight collision probability in a system reduced to one dimension by adopting a homogenized cross section. As an example we calculate the flux distribution in the axial direction in a finite lattice cell composed of a moderator and a fuel rod with two different compositions in the upper and lower axial zones.
New methods based on the class of Padé and cut-product approximations are applied to the solution of the multigroup diffusion theory reactor kinetics equations in two space dimensions. The methods are shown to be consistent and numerically stable. The stability of the developed approximations is studied and it is in general A(α)-stable, where 0⩽α⩽π/2 and therefore they are quite efficient in the presence of extreme stiffness, as shown by numerical results for some typical test cases. The system of stiff coupled differential equations is represented in a matrix form and due to the particular structure of this matrix, new algorithm based on the back-substitution is proposed for analytical inversion of the “A” matrix. The method has been tested for different benchmark problems. The results indicated that this method can allow much larger time steps and thus save much computational time as compared to the other conventional methods.
The application of continuous space-dependent synthesis using axial Helmholtz modes to 2-D fast-reactor diffusion-theory calculations is reported. The standard Galerkin weighting procedure is used to obtain a set of coupled 1-D diffusion equations, which are then solved iteratively using the finite-difference method. The speed and accuracy of the procedure are compared with normal 2-D finite-difference calculations for some fast-reactor configurations. The success obtained by this application shows that the procedure would be useful for 3-D fast-reactor computations.
Among the various approaches to the solution of the criticality problem, which is already becoming a traditional part of the reactor physics, two trends may be isolated: the first group is based on finite-differenc e approximations of the initial equations and the second on the so-called direct methods, whose starting point is the expansion of the desired functions in terms of special systems of coordinate function [i]. Of the geometric models of reactors, the cylindrical model is extensively used: this consists of a system of coaxial cylindrical regions with different nuclear properties over the radius and height. The system of control rods (CR) is modeled as individual, relatively small, arbitrarily positioned, cylindrical regions. Within the context of the second group of approaches, significant progress has been made in the calculation of such models. However, there exists the opinion that direct methods of calculation are more laborious than finitedifference methods. As the authors' experience shows, this assertion is very questionable. With intelligent organization of the programs realizing algorithms based on direct methods, these methods are entirely competitive in (r, ~0) geometry and have the advantage in threedimensional geometry. In problems with repeated iteration of the given algorithm, for example, in calculations for a reactor during its operating period or in optimized calculations, the undoubted benefits of this method are clear, since these algorithms allow a considerable part of the information to be isolated in complexes that are calculated only once.
A new variational technique, the Global Element Method (GEM), is applied to a well-known nuclear physics problem. The GEM enables the solution region to be divided into physically meaningful subregions and appropriate expansion functions to be used in each of these subregions. The continuity of the trial functions (and their derivatives) at the subregion interfaces, together with the prescribed boundary conditions is enforced by the variational principle. It is demonstrated that for the bound-state and scattering problems for a typical discontinuous nuclear potential, the GEM obtains significantly faster convergence rates than previous variational methods. Moreover this new approach will enable the major components of existing computer programs, using conventional global variational methods, to be reused after suitable modification. The GEM, with the possibility of high convergence rates, should hopefully prove to be a useful computational tool in many areas of computational physics.
An exposition has been given of the notions of canonical and involutory transformation in the context of variational problems involving second derivatives of the argument function. As a specific illustration the variational problem representing a beam on nonlinear-elastic foundation is discussed. Further examples—including applications to plates are presented in Ref. [17].
A new technique is developed with an alternative formulation of the response matrix method implemented with the finite element scheme. As in standard response matrix methods, the reactor core is partitioned into several coarse meshes and the global solution is obtained imposing continuity of partial current across the boundaries of the coarse meshes.
A new Heterogeneous Finite Element Method (HFEM) is presented, which does not require prior assembly-level homogenization and which accounts for the leakage effect. The HFEM is developed in diffusion theory. The method is a Lagrange Finite Element Method, that uses basis functions that include fine mesh detail. The elementary basis functions are generated from fixed-boundary-flux fine mesh heterogeneous assembly calculations. The FEM equations are generated using a Galerkin scheme derived from a variational principle. The method is tested on three two-dimensional configurations typical of Boiling Water Reactors, and is shown to be as accurate as the combination Assembly Homogenization—Nodal Method for low-leakage cores and significantly more accurate than the mentioned combination for high-leakage cores
A self-consistent nodal approximation method for computing discrete ordinates neutron flux distributions has been developed from a variational functional for neutron transport theory. The advantage of the new nodal method formulation is that it is self-consistent in its definition of the homogenized nodal parameters, the construction of the global nodal equations, and the reconstruction of the detailed flux distribution. The efficacy of the method is demonstrated by two-dimensional test problems.
We demonstrate the use of the recently developed Global Element Method (Delves and Hall, J. Inst. Math. Appl.23 (1979), 223–234) by applying it to the two dimensional singular boundary value problem introduced by Motz [Quart. Appl. Math.4 (1946), 371–377.] The results obtained converge extremely rapidly even in the neighborhood of the singularity and suggest that the method is capable of efficiently handling singular problems.
Mit engl. Zsfassung. Zugl.: Karlsruhe, Univ., Diss., 2004.
Readers of this paper are likely to come away with too narrow a view of the little-known and little-appreciated art of assumed mode methods for solving distributed parameter system problems. The fact that only three references to these methods are cited is evidence of the wide scattering of the many papers which have appeared concerning these methods. The papers cited at the end of this review have been carefully selected to enable the reader to enter the most recent literature on a broader front. There the reader will find many more assumed mode methods than just the Galerkin method (which is the only one mentioned by Newman and Strauss). The reader will also find, contrary to an assertion by the authors, that assumed mode methods do not offer equal accuracy with fewer equations, as compared with nodal methods [1], although this possibility exists when a problem involves only a restricted class of system behaviors [2]. The literature also contains examples of nonlinear problems which have been solved on analog computers by assumed mode methods, e.g., [2], which the authors apparently had not encountered.
This paper describes a special analog correlator which is used for optical character recognition (OCR). The correlator is built around a special" card capacity read only storage (CCROS)" memory. Each bit in this CCROS memory is represented by a 1-pF capacitor, which may be removed from the circuit by appropriately punching a special Hollerith card on which is printed one plate of the capacitor. The other plate of the capacitor is printed on a printed circuit board pressed against the reverse side of the card, and the portion of the card between the two plates forms the capacitor's dielectric.
ResearchGate has not been able to resolve any references for this publication.