# Coupled 3D Neutron Kinetics/Thermal-hydraulics Calculation of PWR Core Using Nodal Green's Function Method on Neumann Boundary Condition

2023.9.5.

Chairman Kim Jong Il said:

"Along with this any scientific and technological problems relating to the construction of hydroelectric-and thermal-power stations and atomic power plants that are suited to the conditions in our country should be solved."

An accurate prediction of reactor core behavior in transients is very important for the design and safety evaluation of nuclear power reactors. It is necessary to perform the coupled neutron kinetics/thermal-hydraulics calculations in order to evaluate a short time transient of core such as the withdrawal of control rods. The spatial neutron kinetics calculation is referred to the numerical solution of the transient multi-group neutron diffusion equation which includes delayed neutron precursors.

Similarly to steady-state problem, space discretization procedures such as the finite difference method, the finite element method and nodal methods are also used in the numerical solution of the multi-dimensional transient neutron diffusion equation. In particular, nodal methods which were developed in the latter half of 1970's are widely used in reactor physics calculation due to its high efficiency and accuracy. In nodal methods, the multi-dimensional partial differential equation is commonly converted into a coupled set of the one dimensional (1D) ordinary differential equations through the transverse integration procedure, and these ordinary differential equations are related with each other by transverse leakage term. According to the concrete solving procedure of the one dimensional transverse integrated differential equation, nodal methods are divided into the nodal expansion method (NEM), the nodal Green's function method (NGFM), the analytic nodal method (ANM), and so on. In the time discretization procedure, the backward Euler method, theta methods, the Runge–Kutta method, the exponential transformation, and so on are used.

The thermal-hydraulic models which are coupled with spatial neutron kinetics are generally based on the single channel model. In this model, single-or two-phase coolant flows are considered in the core channels, and cross flow of coolant between core channels is neglected. Some codes such as PARCS obtain the temperature and flow field information by coupling with the thermal-hydraulic part of TRACE or RELAP5 codes.

We proposed a detailed numerical model to solve 3D neutron kinetics equation by NGFM on Neumann boundary condition and combined it with thermal-hydraulic model based on the single channel model, and then developed the coupled 3D neutron kinetics/thermal-hydraulics calculation code of PWR core. The research solution is as follows.

First, a numerical model to solve 3D neutron kinetics equation by NGFM on Neumann boundary condition was proposed. The nodal Green's function method under Neumann boundary condition is applied for the space discretization procedure, and the implicit difference scheme is used for the time differential. It is assumed that the fission source term in the delayed neutron precursor concentration equation is linearly changed within each time step.

Second, 3D core thermal-hydraulic calculation model based on the single channel model was proposed. In this model, the coolant is considered as a single phase incompressible fluid.

We proposed numerical solving models on the transient heat conduction equation in fuel rod and on the energy conservation equation of coolant. To evaluate the radial temperature distribution in the fuel rod, we obtained equations for the numerical calculation by using the central difference approximation for the space differential and the implicit difference approximation for the time differential, respectively. For the axial temperature distribution calculation of coolant, we also obtained the numerical calculation formula by using the central difference for the space differential and the implicit difference for the time differential respectively in the energy conservation equation.

Third, We developed the coupled 3D neutronics/thermal-hydraulics code to simulate the transient of PWR core based on above numerical models and verified its accuracy.

The model and the code proposed in here was tested by comparison with the 2D and 3D transient benchmarks for the PWR core. The numerical results are in good agreement with reference solutions.

Our results were published in the SCI International Journal "Progress in Nuclear Energy"(123 (2020) 103316) under the title of the "Coupled 3D neutron kinetics/thermal-hydraulics calculation of PWR core using nodal Green's function method on Neumann boundary condition"(https://doi.org/10.1016/j.pnucene.2020.103316).