Development and analysis of some versions of the fractional-order point reactor kinetics model for a nuclear reactor with slab geometry

In this paper, we report the development and analysis of some novel versions and approximations of the fractional-order (FO) point reactor kinetics model for a nuclear reactor with slab geometry. A systematic development of the FO Inhour equation, Inverse FO point reactor kinetics model, and fractional-order versions of the constant delayed neutron rate approximation model and prompt jump approximation model is presented for the first time (for both one delayed group and six delayed groups). These models evolve from the FO point reactor kinetics model, which has been derived from the FO Neutron Telegraph Equation for the neutron transport considering the subdiffusive neutron transport. Various observations and the analysis results are reported and the corresponding justifications are addressed using the subdiffusive framework for the neutron transport. The FO Inhour equation is found out to be a pseudo-polynomial with its degree depending on the order of the fractional derivative in the FO model. The inverse FO point reactor kinetics model is derived and used to find the reactivity variation required to achieve exponential and sinusoidal power variation in the core. The situation of sudden insertion of negative reactivity is analyzed using the FO constant delayed neutron rate approximation. Use of FO model for representing the prompt jump in reactor power is advocated on the basis of subdiffusion. Comparison with the respective integer-order models is carried out for the practical data. Also, it has been shown analytically that integer-order models are a special case of FO models when the order of time-derivative is one. Development of these FO models plays a crucial role in reactor theory and operation as it is the first step towards achieving the FO control-oriented model for a nuclear reactor. The results presented here form an important step in the efforts to establish a step-by-step and systematic theory for the FO modeling of a nuclear reactor.     

Some useful properties of Legendre polynomials and its applications to neutron transport equation in slab geometry

In this paper, we describe a new scattering kernel and general theoretical scheme for the evolution of the discrete and continuum eigenvalue spectrum in one-dimensional slab geometry neutron transport equation. Firstly, some useful properties of the Legendre polynomials which revealed during the definition of the new scattering kernel are discussed. By using the scattering kernel in one-dimensional neutron transport equation we obtained an integral equation for angular part of the angular flux. For the solution of this integral equation and eigenvalue equations, some comments are given.     

A reflection on the fractional order nuclear reactor dynamics

Bifurcation and stability analysis in the coupled integer-fractional order dynamic equations of a nuclear reactor is carried out in this work. To this end, the dynamics of a Pressurized Water Reactor (PWR) is taken into account as a mainstay design in the water reactor technology. The effect of fractional derivative order on the stability threshold and the onset of bifurcation phenomena is inspected therein with the temperature feedback coefficient taken as the bifurcation parameter. Overall, the transport of neutrons inside the nuclear reactor core, especially in the high neutron absorbing spaces such as the fuel or control rod, resembles that of a sub-diffusion phenomenon. As such, the pertaining equations which comprise neutron diffusion terms are more carefully treated within a fractional order framework. In this work, a formal approach is examined to help readily compute system poles and the associated stable half plane. Results confirm a sensible tendency towards instability as the value of the fractional order is decreased and a more sub-diffusive regime is established.     

Solution of a new model of fractional telegraph point reactor kinetics using differential transformation method

A new model of fractional telegraph point reactor kinetics FTPRK is introduced to approximate the time dependent Boltzmann transport equation considering new terms that contain time derivative of the reactivity and fractional integral of the neutron density. Caputo fractional derivatives and fractional Leibniz rule are used for such derivation. Cattaneoequation is applied to overcome the flaw of infinite neutron velocity and to describe the anomalous transport. Effect of the new term on the neutron behaviour is discussed. The new model is applied to both TRIGA reactor and to commercial pressured water reactor of a Three Mile Island type reactor, TMI-type PWR. Results for step, ramp and sinusoidal excess reactivities with thermal hydraulic feedback are presented and discussed for different values of anomalous sub-diffusion exponent, the fractional order, 0 < µ ≤ 1. To maintain the reactor safe at start-up after insertion of step reactivity and based on the concept of prompt jump approximation, the FTPRK model is simplified and solved analytically by Mittag–Liffler function. Physical interpretations of the fractional order µ and relaxation time τ and their effects on the behaviour of the neutron population are discussed. Also, the effect of a small perturbation in the geometric buckling on the neutron behaviour is discussed for finite reactor core. The new model is solved numerically using the fractional order multi-step differential transform method MDTM. The MDTM constitutes an easy algorithm based on Taylor's formula and Caputo fractional derivative. Two theorems with their proofs are introduced to solve the fractional system. Two major disadvantages of the method about the choice of the fractional order values and the step size length are addressed. We present a procedure which enables us to solve the system with appropriate values of fraction orders.     

Stability of P_2 Methods for Neutron Transport Equation

In this paper the P_2 approximation to the one-group planar neutron transport theory is discussed. The stability of the solutions for P_2 equations with general boundary conditions, including the Marshak boundary condition, is proved. Moreover, the stability of the up-wind difference scheme for the P_2 equation is demonstrated.     

Optimal spectral theory of the linear Boltzmann equation

We give optimal compactness results in L^p spaces ( 1<p<∞) related to spectral theory of general neutron transport equations on spatial domains with finite Lebesgue measure.     

Upwind Discontinuous Galerkin Methods for Two Dimensional Neutron Transport Equations

In this paper the upwind discontinuous Galerkin methods with triangle meshes for two dimensional neutron transport equations will be studied. The stability for both of the semi-discrete and full-discrete method will be proved.     

A new approach for the optimal distribution of assemblies in a nuclear reactor

Summary. The aim of this paper is to propose a new approach for optimizing the position of fuel assemblies in a nuclear reactor core. This is a control problem for the neutronic diffusion equation where the control acts on the coefficients of the equation. The goal is to minimize the power peak (i.e. the neutron flux must be as spatially uniform as possible) and maximize the reactivity (i.e. the efficiency of the reactor measured by the inverse of the first eigenvalue). Although this is truly a discrete optimization problem, our strategy is to embed it in a continuous one which is solved by the homogenization method. Then, the homogenized continuous solution is numerically projected on a discrete admissible distribution of assemblies. Received January 13, 2000 / Published online February 5, 2001     

On the homotopy analysis method for solving a particle transport equation

The purpose of this paper consists in the finding of the solution for a stationary neutron transport equation that is accompanied by the homogeneous boundary conditions, using the techniques of homotopy analysis method (HAM) and a numerical integration formula. Also, algorithm presented can be used for solving the integral–differential equations in which the unknown function depends on two variables, such as a radiative transfer equation. Results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

Maximum neutron flux in thermal research reactors

This paper deals with the determination of the space distribution of fuel concentration in thermal reactors for the purpose of obtaining maximum neutron flux. With some approximations, this problem is reduced to a nonclassical variational problem, which is solved by using Pontryagin's maximum principle. It is shown that the optimal fuel configuration of the reactor core consists of a reflector in its center, a zone of constant (permissible) power density, a zone of constant (maximum) fuel concentration, and a peripheral reflector of infinite thickness.     

Optimal Control of Multigroup Neutron Fission Systems

This article considers the optimal control of nuclear fission reactors modeled by parabolic partial differential equations. The neutrons are divided into fast and thermal groups with two equations describing their interaction and fission, while a third equation describes the temperature in the reactor. The coefficient for the fission and absorption of the thermal neutron is assumed to be controlled by a function through the use of control rods in the reactor. The object is to maintain a target neutron flux shape, while a desired power level and adjustment costs are taken into consideration. A nonlinear optimality system of six equations is deduced, characterizing the optimal control. An iterative procedure is shown to contract toward the solution of the optimality system in small time intervals. The theory is extended to include the effect of other fission products, leading to coupled ordinary and partial differential equations. Numerical experiments are also included, suggesting directions for further research. Accepted 13 January 1998     

Optimal control of distributed nuclear reactors using functional analysis

The minimum norm formalism of functional analysis is applied to the problem of minimizing a quadratic cost functional that penalizes the control effort and the deviations of the neutron flux distribution throughout the reactor core. The conditions for optimality are derived for a general, linearized, reactor model with a finite number of control rods. These conditions take the form of a coupled and finite set of Fredholm's integral equations of the second kind with nondegenerate kernels. An example is presented in which the homogeneous slab reactor model is considered. A contraction mapping algorithm is proposed to compute the optimal control.This work was supported in part by the National Research Council of Canada, Grant No. A4146.     

Radiant Heat Transfer Through an Aerosol Suspended in a Transparent Gas

The scattering of radiation by an aerosol in the cover-gas regionof a fast breeder nuclear reactor has been studied by meansof the equation of radiative transfer for a grey atmosphere.Some new results have been obtained in the form of an integralequation for the radiation intensity and heat flux. A variationalmethod is employed to obtain numerical results and the accuracyof an elementary theory based upon discrete ordinates is assessed.The variational approach leads to computationally useful andaccurate results for several quantities of practical interest.The interdisciplinary nature of the work is stressed by virtueof its close connection with neutron transport theory and rarefiedgas dynamics.     

ON THE DISCRETE-ORDINATES METHOD FOR THE MONOENERGETIC NEUTRON TRANSPORT EQUATION IN A SLAB WITH GENERALIZED BOUNDARY CONDITIONS

This paper is devoted to discussing the discrete-ordinates method for the monoenergetic neutron transport equation in a slab with generalized boundary conditions. For homogeneous medium with isotropic scattering and fission, the convergence theorems for discrete-ordinates approximations are given respectively for critical eigenvalue problem and dominant eigenvalue problems: for inhomogeneous medium with anisotropic scattering and fission, a similar discussion and an estimation for the convergence rate are given for critical eigenvalue problems. Finally, some numerical results are given by use of this method.     

Derivation of Stochastic Partial Differential Equations

Abstract

A procedure is explained for deriving stochastic partial differential equations from basic principles. A discrete stochastic model is first constructed. Then, a stochastic differential equation system is derived, which leads to a certain stochastic partial differential equation. To illustrate the procedure, a representative problem is first studied in detail. Exact solutions, available for the representative problem, show that the resulting stochastic partial differential equation is accurate. Next, stochastic partial differential equations are derived for a one-dimensional vibrating string, for energy-dependent neutron transport, and for cotton-fiber breakage. Several computational comparisons are made.     

Criticality problems for slabs and spheres in energy dependent neutron transport theory

The steady-state equation for energy-dependent neutron transport in isotropically scattering slabs and spheres is formulated as an integral equation. The Perron-Frobenius-Jentzsch theory of positive operators is used to analyze criticality problems for transport in slab and spherical media consisting of core and reflector. In addition, with an adroit selection of diffusion-like solutions, this theory is used to obtain an expression relating the critical radius of a homogeneous sphere to a parameter characterizing fission production.     

On the stability of fractional neutron point kinetics (FNPK)

The aim of this work is investigate the stability of fractional neutron point kinetics (FNPK). The method applied in this work considers the stability of FNPK as a linear fractional differential equation by transforming the s − plane to the W − plane. The FNPK equations is an approximation of the dynamics of the reactor that includes three new terms related to fractional derivatives, which are explored in this work with an aim to understand their effect in the system stability. Theoretical study of reactor dynamical systems plays a significant role in understanding the behavior of neutron density, which is important in the analysis of reactor safety. The fractional relaxation time (τ^α) for values of fractional-order derivative (α) were analyzed, and the minimum absolute phase was obtained in order to establish the stability of the system. The results show that nuclear reactor stability with FNPK is a function of the fractional relaxation time.     

Coarse-Grid-CFD for a Wire Wrapped Fuel Assembly

Computational fluid dynamics (CFD) simulations of complete nuclear reactor core geometries requires exceedingly large computational resources. However, in most cases there are repetitive geometry- and flow patterns allowing the general approach of creating a parameterized model for one segment and composing many of these reduced models to obtain the entire reactor simulation. Traditionally, this approach lead to so-called subchannel analysis codes that are relying heavily on transport models based on experimental and empirical correlations. With our method, the Coarse-Grid-CFD (CGCFD), we intend to replace the experimental or empirical input with CFD data. Our method is based on detailed and well-resolved CFD simulations of representative segments. From these simulations we extract and tabulate volumetric source terms. Parameterized data is used to close an otherwise strongly under resolved, coarsely meshed model of a complete reactor setup. In the previous formulation only forces created internally in the fluid are accounted for. The Anisotropic Porosity (AP) formulation wich is subject of the present investigation adresses other influences, like obstruction and flow guidance through spacers and in particular geometric details which are under resolved or ignored by the coarse mesh. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)