A fluid–solid finite element method for the analysis of reactor safety problems

The work presented herein can broadly be categorized as a fluid–structure interaction problem. The response of a circular cylindrical structure subjected to cross flow is examined using the finite element method for both the liquid and the structure domains. The cylindrical tube is mounted elastically at the ends and is free to move under the action of the unsteady flow-induced forces. The fluid is considered to be acoustic compressible and viscous. A Galerkin finite element method implemented on a triangular mesh is used to solve the time-dependent Navier–Stokes equations. The cylinder motion is modeled using a five-degrees of freedom generalized shell element structural dynamics model. The numerical simulations of the response of the calandria tubes/pressure tubes, adjustor rod and shut-off rod of a nuclear reactor are presented. A few typical results are presented to assess the accuracy and applicability of the developed modules.     

Finite element solution to transient convective-conductive heat transfer problems

The paper discusses finite element approximations to problems in transient convective-conductive heat transfer in a fluid region. The governing equations are expressed in terms of the primitive variables; the flow is assumed to be laminar and the fluid incompressible within the Boussinesq approximation.The properties of the discrete advection-diffusion equation are analyzed with regard to the possible choices for mass representation (consistent or diagonal) and time integration procedure (explicit or implicit). In particular, the diagonal mass matrix and the explicit time integration method are shown to be a poor combination in terms of accuracy for meshes consisting of linear or multilinear finite elements. A simple remedy is suggested to improve the frequency response of such lumped-explicit schemes.Then, finite element formulations for the incompressible Navier-Stokes equations are considered. The various methods for incorporating the incompressibility constraint are briefly reviewed and those associated with explicit time integration of the momentum equations are discussed in detail. In particular, a method is presented for solving the pressure field, which does not require inter-element continuity of the pressure and does not exhibit a chequerboard splitting on square meshes. A numerical example is presented which illustrates the use of the proposed method for the explicit solution of time-dependent natural convection problems.     

Distributed parameter analysis for the prediction of the fine structure of flow and temperature fields in wire-wrapped fuel pin bundle geometries

This paper describes the numerical method of a distributed parameter analysis code SPIRAL for the calculation of fluid flow and temperature in arbitrary channel geometries, discusses the numerical method in the modeling and solution of the problem, and presents some results, including comparison with experiments.

The derivation and solution of the finite element equations is discussed. In order to overcome difficulties arising from the geometry, the Galerkin finite element method using isoparametric elements was employed, and a procedure of finite element generation using curvilinear coordinate system was developed.

The SPIRAL code permits calculation of the fine structure of the multi-dimensional steady-state single-phase fluid flow and temperature fields in LMFBR fuel pin subassemblies in the presence of wire spacers. Calculated results are presented for crossflow velocity distributions and crossflow pressure drop characteristics in a tube bundle geometry with and without wire spacers, natural convection and heat transfer in horizontal annuli, flow in a wire-wrapped 7-pin bundle geometry and fully developed turbulent flow in a parallel 4-rod array contained in a rectangular duct.     

平面U型弯管固有振动分析的有限元方法及试验研究

本文运用有限元方法对平面Ｕ型弯管进行了固有振动分析，对弯管段由精确的微单元静态特性微分方程组得到了单元质量阵与刚度阵，其中考虑了剪切变形与转动惯量的影响，直管段用每单元十二个自由度的空间梁单元处理．求得结构振动的特征方程后，运用于空间迭代法求解各阶固有频率及振型。将试验结果与计算结果进行了比较，证明该方法是很有效的，并给出了处理实际蒸汽发生器传热管支撑板边界条件的结论．     

Wave propagation in two-phase flow of magnetic fluid under a nonuniform magnetic field

The effect of nonuniform magnetic field on the linear and nonlinear wave propagation phenomena in two-phase pipe flow of magnetic fluid is investigated theoretically to realize the effective energy conversion system using boiling two-phase flow of magnetic fluid. Firstly, the governing equations of two-phase flow based on the unsteady thermal nonequilibrium two-fluid model are presented and the linear void wave propagation phenomena in boiling two-phase flow are numerically analyzed by using the finite volume method. Next, the nonlinear pressure wave propagation in gas-liquid two-phase flow is numerically analyzed by using the finite different method. According to these theoretical studies on the wave propagation phenomena in two-phase flow of magnetic fluid, it seems to be a reasonable proposal that the precise control of the wave propagation in two-phase flow is possible by effective use of the magnetic force.     

Analysis of moderately thick flanged reactor vessels

A computer method is presented for the analysis of moderately thick flanged shells of revolution such as are used for reactor pressure vessels. The shell may be subjected to symmetrical or unsymmetrical loads and a thermal environment. The method employs a finite element discretization for modelling the flange portions, and a theory appropriate to moderately thick shells for the remainder of the pressure vessel. The governing differential equations for the shell portions are put in the Goldberg-Bogdanoff first-order form and integrated numerically using a scheme such as a Runge-Kutta process. The finite element stiffness matrix for a flange region is used to form a superelement influence coefficient matrix, permitting the flange region to be treated as a giant step in the numerical integration process.     

A finite element method for neutron transport—III. Two-dimensional one-group test problems

The even-parity transport equation for an isotropic scattering medium is applied using a maximum principle to determine the angular flux in a square lattice cell with a cylindrical fuel element, a square isotropic source in a corner of an absorbing shield and a dog-leg duct through an absorbing shield. The finite element solutions obtained are compared with an exact solution for the cell, and benchmark results for the square source and duct problem. The accuracies achieved for the fluxes in the cell problem are everywhere better than 0.75% and high accuracy is achieved for the other test problems. The linear elements, used for the spatial dependence of the angular flux in conjunction with a spherical harmonic expansion for the angular dependence, provide a flexible means of treating awkward geometries. At present the equations are assembled and solved using the UNCLE code, which uses direct Gaussian elimination. This process limits the speed of the finite element method to about 1/3 of the Fletcher finite difference spherical harmonics method for the square source problem. This latter method is, however, limited to systems with geometries defined by co-ordinate surfaces, whereas the finite element method can be used for any region that can be triangulated.     

A fluid-structure finite element method for the analysis of reactor safety problems

A method is presented for the safety analysis of reactor containment structures by means of finite elements. The finite element equations of both fluid and structural elements for arbitrarily large, non-linear response are developed and the way in which they are combined is indicated. Both explicit and implicit integration of the equations in time is considered. Three examples of the application of these methods to the analyses of reactor safety problems are described.     

Analysis of one-dimensional pressure pulse propagation in a closed fluid system

This paper describes a unique numerical method for linear inviscid fluid hammer analysis based on the method of characteristics. The uniqueness lies in that it uses the analytical solutions of the wave equation in place of the compatibility relatins of the more conventional method of characteristics. The numerical solution is obtained by a simple superposition technique for tracing the waves traveling along each characteristic and extending the solution from one constant time line to the next. Using a predetermined finite difference net of grids with equal spacings, an elimination is made of the spatial interpolation, thereby maintaining the wave amplitudes in their full strength in the numerical procedure. This is in contrast to the case of a nonlinear problem in which the pressure peaks are always flattened to some degree in the interpolation procedure.The computer program NAHAMMER is a system analysis code adequate for short-term pressure transients of most engineering problems of significance involving a moderate pressure source. It considers the simplified one-dimensional, linear, inviscid set of governing equations with an isentropic flow assumption. A closed fluid-network system is considered to be composed of a multiple of one-dimensional pipe sections and components that are connected by various joints. An analytical solution is obtained under an acoustic approximation for a simple system and the result shows good agreement with the numerical solution. As examples of the application of the method, complex problems of engineering importance are calculated and the results are presented.     

Free vibration analysis of multi-span pipe conveying fluid with dynamic stiffness method

By taking a pipe as Timoshenko beam, in this paper the original 4-equation model of pipe conveying fluid was modified by taking the dynamic effects of fluid into account. The shape function that always used in the finite element method was replaced by the exact wave solution of the modified four equations. And then the dynamic stiffness was deduced for the free vibration of pipe conveying fluid. The proposed method was validated by comparing the results of critical velocity with analytical solution for a simply supported pipe at both ends. In the example, the proposed method was applied to calculate the first three natural frequencies of a three span pipe with twelve meters long in three different cases. The results of natural frequency for the pipe conveying stationary fluid fitted well with that calculated by finite element software Abaqus. It was shown that the dynamic stiffness method can still hold high precision even though the element's size was quite large. And this is the predominant advantage of the proposed method comparing with conventional finite element method.     

The hexagonal bundle heat transfer and fluid flow experiment agathe hex

Problems of heat transfer and fluid flow in gas-cooled reactor fuel elements have been studied at the Swiss Federal Institute for Reactor Research (EIR) for 14 years. Since 1967, the activities have been directed toward gas-cooled fast breeder reactors (GCFRs). The aim of analytical and experimental studies has been to develop analytical models and comprehensive computer codes for the prediction of temperature and pressure distributions in GCFR fuel element configurations. The models developed at EIR are based on the results of specific experiments. Full-scale experiments in actual geometry are being carried out to verify the computer codes for a wide range of parameters. This paper describes the heat transfer loop and the test sections designed to verify GCFR thermohydraulic design codes.     

Three-dimensional h-adaptivity for the multigroup neutron diffusion equations

Adaptive mesh refinement (AMR) has been shown to allow solving partial differential equations to significantly higher accuracy at reduced numerical cost. This paper presents a state-of-the-art AMR algorithm applied to the multigroup neutron diffusion equation for reactor applications. In order to follow the physics closely, energy group-dependent meshes are employed. We present a novel algorithm for assembling the terms coupling shape functions from different meshes and show how it can be made efficient by deriving all meshes from a common coarse mesh by hierarchic refinement. Our methods are formulated using conforming finite elements of any order, for any number of energy groups. The spatial error distribution is assessed with a generalization of an error estimator originally derived for the Poisson equation.Our implementation of this algorithm is based on the widely used Open Source adaptive finite element library deal.II and is made available as part of this library's extensively documented tutorial. We illustrate our methods with results for 2-D and 3-D reactor simulations using 2 and 7 energy groups, and using conforming finite elements of polynomial degree up to 6.     

Measurement of Average Cross Section for 238U(n, 2n)237U Reaction

Accurate solution of the group diffusion equations for PWR cores requires explicit treatment of the non-homogeneous macroscopic parameters within each fuel assembly. It is argued that the response matrix approach is a convenient method to handle this problem provided all matrix elements for the non-homogeneous assemblies can be computed. This so called local problem is solved in this paper by a perturbation algorithm which leads to sensitivity coefficients for the power series expansions of the response matrix elements. A numerical study for 2 representative assemblies of the Indian Point Unit No. 2 (IP2) reactor is carried out and response matrices obtained by the perturbative method are compared with values computed by a finite difference program.     

Lie group analysis for the effects of temperature-dependent fluid viscosity and chemical reaction on MHD free convective heat and mass transfer with variable stream conditions

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching surface. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding three absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and two second-order ordinary differential equation corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. It is found that with the increase of magnetic field intensity the fluid velocity decreases but the temperature increases at a particular point of the heated stretching surface. Impact of chemical reaction in the presence of thermal radiation plays an important role on the concentration boundary layer. The results thus obtained are presented graphically and discussed.     

离心机流场非稳态过程的初步数值模拟

摘要：从非稳态线性离心机流场运动方程出发，利用有限体积法，采用交错网格得到离散方程。通过数值求解得到了存在温度驱动情况下，离心机内形成小扰动流场的过程。从形成小扰动流场的非稳态过程可以看出扰动流场在开始和稳定阶段的影响因素不同，从温度引起的轴向压强的不平衡引起流动逐渐过渡到压强分布不同而引起的轴向环流。数值模拟为更好地研究离心机内流场的非稳态过程提供了依据。     

Theory of pump-induced pulsating coolant pressure in pressurized water reactors

The theory of pump-induced pulsating pressure distributions in a PWR coolant annulus is developed. The calculated pressure distribution can then be applied to predict the dynamic responses of the reactor internals. The mathematical analysis is formulated in accordance with the linearized Navier-Stokes' equations by assuming a compressible, inviscid liquid. These equations are combined to form a single equation in terms of the unknown pressure distribution. The boundary conditions are two concentric rigid walls in the radial direction and any combination of closed, open, and piston-spring supported end conditions in the axial direction. The pulsating pump pressure which induces the pressure fluctuation in the annulus is prescribed at a small opening of the outer cylindrical wall (pump inlet of the reactor).An approximate solution is obtained by introducing the concept of time-dependent body force in the governing differential equations. With this conceptual substitution for the actual loading, the time-dependent, mixed boundary value problem can be represented as a forced vibration problem with homogeneous boundary conditions. This problem can then be solved by the method of normal modes. Numerical examples are provided which give the pressure distribution in the axial and circumferential directions of the annulus for various configurations of one and/or several pumps.     

Finite element analysis of fluid–structure interaction in pipeline systems

Pipes used for transporting high velocity pressurized fluids often operate under time-varying conditions due to pump and valve operations. This can cause vibration problems. In the present work, a finite element formulation for the fully coupled dynamic equations of motion to include the effect of fluid–structure interaction (FSI) is introduced and applied to a pipeline system used in nuclear reactors. The fluid finite element model is based on flow velocity as the variable. The response of fluid filled pipelines to valve closure excitation has been studied. The model is validated with an experimental pipeline system.     

The why and how of finite elements

The ‘Why’ of the title reflects the desire in safety assessments for independent means of making realistic calculations for systems of complex shape. In principle the geometrical flexibility attainable by the deterministic finite element method being the equal of the stochastic Monte Carlo method.The development of the finite element method is traced to show how ideas from structural and fluid mechanics, the calculus of variations, functional analysis and the calculus of finite differences have been forged to provide a tool which minimizes the mismatch between the behaviour of a continuous system and that of a discrete model of the system assembled from finite elements. Geometrical flexibility of the model is achieved by the use of polygonal and curved elements. The behaviour of any point of an element is described in terms of its behaviour at discrete points or nodes of the element. In treating neutron transport the finite element method can be applied to phase-space, or as in this paper the spatial dependence can be treated by the use of finite elements in conjunction with expansions in orthogonal functions for the directional dependence.The mathematical formulation is based on a mixed parity form of the Boltzmann equation for one-group transport. The minimization of the mismatch between the system and its finite element model leads to a completely boundary-free maximum principle. This variational principle is also recast into a generalized least-squares principle. When the essential boundary conditions of the classical calculus of variations are imposed the well-known minimum and maximum principles for the even- and odd-parity second-order Boltzmann equations are obtained as special cases. The maximum principle for the second-order even-parity equation is used to demonstrate the precision and flexibility of the finite element method by solving the problems of a dog-legged duct in a shield and a cylindrical fuel element in a square lattice cell.The geometrical interpretation of the boundary-free maximum principle with the aid of a suitable Hilbert space then leads to completely boundary-free weighted residual or Galerkin schemes for both the first- and second-order forms of the Boltzmann equation. Imposing essential boundary conditions leads to classical schemes.The paper concludes with a sketch of finite element treatments of the multigroup Boltzmann equation.     

Three-Dimensional Numerical Calculations on Liquid-Metal Magnetohydrodynamic Flow in Magnetic-Field Inlet-Region

Three-dimensional numerical calculations have been performed on liquid-metal magnetohydrodynamic (MHD) flow through a rectangular channel in the inlet region of the applied magnetic field, including a region upstream the magnetic field section. The continuity equation, the momentum equation including the Lorentz force term and the induction equation have been solved numerically. The induction equation is derived from Maxwell's equations and Ohm's law in electromagnetism. The discretization of the equations is carried out by the finite difference method, and the solution procedure follows the MAC method. Along the flow axis (i.e. the channel axis), the pressure decreases slightly as normal non-MHD flow, increases once, thereafter decreases sharply and finally decreases as fully-developed MHD flow. The sharp decrease in the pressure, resulting in a large pressure drop, in the inlet region is due to increase in the induced electric current in this region comparing with that in the fully-developed region. In the inlet region, the flow velocity distribution changes from a parabolic profile of a laminar non-MHD flow to a flat profile of a fully-developed MHD flow.