Simulation of reactor dynamics with temperature feedback

The finite element method, a well-known mathematical tool for the solution differential equations representing physical systems, is generally based on the Rayleigh-Ritz variational principle when the basic differential equation to be solved is self-adjoint. On the other hand, in the consideration of dynamical systems, it is more convenient to formulate the finite element method by using the weighted-residual method. The finite element method removes certain restrictions of classical discretization methods. Particularly, representation of complex geometries and an increase in the degree of the piecewise polynomial space make the finite element method preferable. Hence, in application, it makes modelling of systems possible with much less node numbers and reducing the load of computing by considerable amount as compared to, for example, the finite difference method. Although reactor dynamics, especially with temperature feedback, comprises many efforts in nuclear engineering, due to inefficiencies in the computing capabilities in the recent past, restrictions such as use of point-kinetics equations or reductions in space dimensions and/or energy groups were applied. Hence, many reactor dynamics studies with feedback in multi-dimensional geometry are limited to some extent as can followed from the OECD Data Bank. These studies differ in the mathematical tools and numerical methods used. Among these, SPARK [1] uses the quasistatic method in the solution of reactor dynamics equations, while DPOL3D [2], a two-group reactor dynamics program, uses finite difference method together with Alternating Direction Implicit method as time step algorithm. A similar application to that of DPOL3D is present in DYN3D/M2 [3], a three-dimensional two-group program using the nodal method, where the neutron flux is represented by expansion in Bessel functions and exponential transformation is used in the solution of time-dependent equations. Finally, very recent code QUARK [4] is a two-group three-dimensional reactor dynamics program that uses nodal method in the solution of reactor dynamics equations. Final aim of this work is to develop a general-purpose code with a controller and user interface for the analysis of a PWR core under normal or reactor trip conditions. Basically, the developed code simulates the two-group neutron dynamics equations coupled with six groups of delayed neutrons and temperature feedback in a three-dimensional Cartesian geometry by the use of the finite element method. Static eigenvalue computation part of the code is based on the work done by Oezgener's study [5]. In the approximation to the neutron flux bilinear polynomials are utilised and the space-dependent inner product operations are carried out by the Ritz method. In the space-time-dependent neutron dynamics formulation, however, discretization in space is based on the weighted-residual Galerkin method. In the discretization of the time domain the temporal sub domain method is utilised. The feedback effects are incorporated in such a way that the thermal-hydraulic channels and the thermal-hydraulic planes were defined with the use of finite elements where average fuel and average coolant temperatures will be calculated and included into the reactor dynamics equations. The developed program utilizes the finite element method fully in the space-time discretization of the reactor dynamics equations with temperature feedback. The symmetry property of half, quadrant or octant-core with any initial and/or boundary conditions can be used in the calculations. The thermal-hydraulic calculations are carried in separate axial hydraulic channels, each of which relates to one axial finite element. In the program, to represent a real reactor core by homogeneous regions with sufficient accuracy, fuel element number per each finite element, power generated in each finite element, number of thermal-hydraulic channels and mass flow rate per each thermal-hydraulic channel is calculated. In its present state, the code investigates systems response to step reactivity of control rod or any other changes, e.g., mass flow-rate or coolant inlet temperature. Both the moderator and fuel (Doppler) temperature feedback mechanisms were included into the system equations by a first-order perturbation approximation. The three-dimensional neutronics calculation is coupled with the one-dimensional thermal-hydraulic calculation where coolant flow is assumed upward and cross-flow between coolant channels is not considered. The code is implemented in the MATLAB language for its well-known capability of matrix and vector programming and especially due to its controller design toolbox for future development purposes. This feature especially allows running the code in Windows or Unix based operating systems without any need of modification