Power series solution method for solving point kinetics equations with lumped model temperature and feedback

The numerical solution of the point kinetics equation using the matrix exponential method

Evaluating the variations of point kinetics parameters in pressurized heavy water reactor analyses

In this paper, the solution of the Neutron Point Kinetics model is presented, adding the effects of temperature and absorbers poisons within a historical and technical context to simulate the preliminary characteristics of the Chernobyl accident. The Point Kinetics model was able to extract physical information consistent with what was expected to predict the reactor situation until the accident. It was also possible to verify, given the results, that the Rosenbrock method was able to overcome the degree of stiffness of the ODE system, besides solving a non-linear problem. Thus, this study has contributed to highlighting the importance of temperature effects and especially absorbers poisons in the final power behavior, extremely relevant for decision making in the operation and safety of a nuclear power plant.

The numerical solution of the point kinetics equations in the presence of Newtonian temperature feedback has been a challenging issue for analyzing the reactor transients. Reactor point kinetics equations are a system of stiff ordinary differential equations which need special numerical treatments. Although a plethora of numerical intricacies have been introduced to solve the point kinetics equations over the years, some of the simple and straightforward methods still work very efficiently with extraordinary accuracy. As an example, it has been shown recently that the fundamental backward Euler finite difference algorithm with its simplicity has proven to be one of the most effective legacy methods. Complementing the back-ward Euler finite difference scheme, the present work demonstrates the application of ordinary differential equation suite available in the MATLAB software package to solve the stiff reactor point kinetics equations with Newtonian temperature feedback effects very effectively by analyzing various classic benchmark cases. Fair accuracy of the results implies the efficient application of MATLAB ordinary differential equation suite for solving the reactor point kinetics equations as an alternate method for future applications.

Fortran programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

Summary The exact solution to the simple pendulum problem has long been known in terms of Jacobi elliptic functions, of which an efficient numerical evaluation is standard in most scientific computing software packages. Alternatively, and as done in V. Fairén, López and L. Conde (Power series approximation to solutions of nonlinear systems of differential equation, Am. J. Phys. 56 (1988) 57–61], the pendulum equation can be analytically solved exactly by the power series solution method. Although recursive formulae for the series coefficients were provided in V. Fairén et al., the series itself—as well as the optimal location about which an expansion should be chosen to maximize the domain of convergence—has not yet been examined, and this is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the power series solution in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. We provide an exact resummation of the pendulum series that accelerates the series’ convergence uniformly from the top to the bottom of the trajectory. We also provide the formulae needed to extend the pendulum series for all time via symmetry. The pendulum problem, in its relative simplicity, provides an explicit demonstration of the effect of singularity structure and initial condition location on convergence properties of power series solutions.

In the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.

A new method for the solution of the point kinetics equations in the presence of Newtonian temperature feedback

Survey of recent applications of reactor noise techniques at Argonne National Laboratory

Fractional-space neutron point kinetics (F-SNPK) equations for nuclear reactor dynamics

Annular fuels are being studied to increase the power of advanced third-generation reactors by 50%. This paper aimed to analyze transient scenarios through a hybrid lumped parameter-finite difference model in a pressurized water reactor with annular fuel. The model used in this work is more detailed than the double lumped parameter one, but still simple enough to model some transients in PWR fuels, as rod ejection accident and cold water insertion accident. The heat transfer equations are solved by the numerical semi-implicit Crank-Nicolson method together with point kinetics equations with six groups of delayed neutrons and a lumped parameter model for the reactor coolant. The model takes into account in an approximate way the hot spot by using a composed peaking factor equal to 2.5. The reactivity feedback is taken into account by considering the Doppler effect of fuel temperature, and also moderator temperature variation. The results were compared with solid fuel performance and showed that the annular fuel reached considerable lower fuel temperature profiles even for 150% power, as compared to 100% power for solid fuel, thus showing that this kind of fuel has a better safety performance for the transients analyzed. The rod ejection accident showed that feedback effects can lead the reactor to a new safe steady state condition.

Numerical analysis of startup PWR with fractional neutron point kinetic equation

Simulation of over-power transients in tank-in-pool type research reactors

Physics-informed neural networks for the point kinetics equations for nuclear reactor dynamics

Fractional two energy groups matrix representation for nuclear reactor dynamics with an external source