Generating Function Method for the solution of point reactor kinetic equations

In this paper, an efficient numerical procedure for calculating the solution of the non–linear neutron point kinetics equations in the field of nuclear reactor dynamics is presented. The explicit finite difference method is applied to solve the non–linear neutron point kinetics equations with temperature feedback reactivity. The numerical solution obtained by explicit finite difference scheme is an approximate solution based on neutron density, the precursor's concentration of multi–group delayed neutrons, and the reactivity function. The method is investigated over data of nuclear reactor–type compositions, with appropriate initial conditions applying in three types of reactivity: step, ramp and temperature feedback. The computational results show that this numerical approximation method is straightforward and effective for solving the non–linear point kinetics equations. Numerical results addressing the time behaviour of neutron density and results of this method with different types of reactivity are presented graphically. The comparisons of results with other referred methods are also discussed.

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.

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

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

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

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

The operation of the IBR-2 pulsed reactor (1987-2006) showed that a total transfer coefficient of the power decreases during the operation of the reactor, therefore each reactor cycle has its own transfer coefficients [1] and stability margins. Hence, the study of dynamics of the new reactor IBR-2M (modernized version IBR-2, since 2011) is extremely important. In this paper, the results of an experimental and a modelling study of dynamics of the IBR-2M are presented, for the reliable and safe operation of reactor. For this purpose, model dynamics of the IBR-2M has been developed, which includes experimental data on reactor feedback parameters in period 2015-2017 at a mean power of 0.5 MW. Using the model dynamics of the IBR-2M, the stability margins of the reactor in different regimes its operation (self-regulating and automatic regulating) were estimated. It shown that during the analyzed time of the operation reactor with energy production up to 1200 MW·day significant changes in dynamics of reactor has not occurred. The reactor has been working in the stability region.

Authors

Solution of the nonlinear point nuclear reactor kinetics equations

A digital twin (DT) for nuclear reactor monitoring can be implemented using either a differential equations-based physics model or a data-driven machine learning model. The challenge of a physics-model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, whereas the challenge of a data-driven DT consists of extensive training requirements and a potential lack of predictive ability. We investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. We develop a PINN model to solve the point kinetic equations (PKEs), which are time-dependent, stiff, nonlinear, ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring spatial dependence of the neutron flux. The PINN model solution of PKEs is developed to monitor the start-up transient of Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. For the test cases considered, the extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.

Nuclear reactor safety and efficiency can be enhanced through the development of accurate and fast methods for prediction of reactor transient (RT) states. Physics informed neural networks (PINNs) leverage deep learning methods to provide an alternative approach to RT modeling. Applications of PINNs in monitoring of RTs for operator support requires near real-time model performance. However, as with all machine learning models, development of a PINN involves time-consuming model training. Here, we show that a transfer learning (TL-PINN) approach achieves significant performance gain, as measured by reduction of the number of iterations for model training. Using point kinetic equations (PKEs) model with six neutron precursor groups, constructed with experimental parameters of the Purdue University Reactor One (PUR-1) research reactor, we generated different RTs with experimentally relevant range of variables. The RTs were characterized using Hausdorff and Fréchet distance. We have demonstrated that pre-training TL-PINN on one RT results in up to two orders of magnitude acceleration in prediction of a different RT. The mean error for conventional PINN and TL-PINN models prediction of neutron densities is smaller than 1%. We have developed a correlation between TL-PINN performance acceleration and similarity measure of RTs, which can be used as a guide for application of TL-PINNs.

The filtration problem is one of the most relevant in the design of retaining hydraulic structures, water supply channels, drainage systems, in the drainage of the soil foundation, etc. Construction of transport tunnels and underground structures requires careful study of the soil properties and special work to prevent dangerous geological processes. The model of particle transport in the porous rock, which is based on the mechanical-geometric interaction of particles with a porous medium, is considered in the paper. The suspension particles pass freely through large pores and get stuck in small pores. The deposit concentration increases, the porosity and the permissible flow of particles through large pores changes. The model of one-dimensional filtration of a monodisperse suspension in a porous medium with variable porosity and fractional flow through accessible pores is determined by the quasi-linear equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth. This complex system of differential equations has no explicit analytical solution. An equivalent differential equation is used in the paper. The solution of this equation by the characteristics method yields a system of integral equations. Integration of the resulting equations leads to a cumbersome system of transcendental equations, which has no explicit solution. The system is solved numerically at the nodes of a rectangular grid. All calculations are performed for non-linear filtration coefficients obtained experimentally. It is shown that the solution of the transcendental system of equations and the numerical solution of the original hyperbolic system of partial differential equations by the finite difference method are very close. The obtained solution can be used to analyze the results of laboratory research and to optimize the grout composition pumped into the porous soil.

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

Inhomogeneous point kinetics equations and the source contribution

The singularly perturbed method (SPM) is proposed to obtain the analytical solution for the delayed supercritical process of nuclear reactor with temperature feedback and small step reactivity inserted. The relation between the reactivity and time is derived. Also, the neutron density (or power) and the average density of delayed neutron precursors as the function of reactivity are presented. The variations of neutron density (or power) and temperature with time are calculated and plotted and compared with those by accurate solution and other analytical methods. It is shown that the results by the SPM are valid and accurate in the large range and the SPM is simpler than those in the previous literature.