Time-dependent Neutron Transport Equation Research Articles

High-Fidelity Transient Analysis with STREAM Based on the Predictor-Corrector Quasi-Static Method

The predictor-corrector quasi-static method (PCQM) is used to solve the transient problem in the STREAM code, a steady-state and transient reactor analysis code with the method of characteristics. In PCQM, the angular neutron flux undergoes a factorized split to form the product of shape and amplitude functions. The time-dependent neutron transport equation is solved to obtain the shape function whereas the amplitude function is obtained by resolving the exact point kinetics equations (EPKEs). A two-level coarse mesh finite difference technique is implemented to reduce the transient running time of the transport solution. Moreover, high-order polynomial interpolation is applied to the kinetics parameters utilized in EPKEs to reduce the error when the reactivity insertion is nonlinear. Several numerical benchmarks are solved to justify the application of the procedure, proving that the method maintains solution accuracy.

Application of stiffness confinement method within variational nodal method for solving time-dependent neutron transport equation

The stiffness confinement method (SCM) is implemented in the variational nodal method (VNM) framework to solve time-dependent multi-group neutron transport equations. With frequency-transform, the time-dependent neutron equation is decoupled from the precursor equation and transformed into a dynamic eigenvalue problem which can be solved with the in-house VNM code VITAS. An improved time-source iteration method within the VNM-SCM framework is proposed to accelerate constructing response matrices and reduces memory usage. Based on the VNM-SCM scheme, the kinetics module of VITAS is developed for the Cartesian and hexagonal-z geometry and verified by several benchmark problems.Numerical results demonstrate that VNM-SCM can perform accurate and efficient transient calculations with large time-step sizes. For the two-dimensional TWIGL benchmark, the VNM-SCM produces a maximum core power error of -0.57% with the time-step size of 20 ms. For the three-dimensional LMW benchmark, the VNM-SCM results in a maximum core power error of 0.95% with the time-step size of 2 s. The VNM-SCM also yields an accurate neutron flux prediction for the hexagonal-z geometry problem. In addition, the effect of the angular expansion is investigated systematically. Examinations with the three-dimensional LMW benchmark display an error reduction in the core power error from 0.131% to 0.011% from P1 to P3 interfacial angular expansion, with a substantial computing time increase of 6.14 hours. The study indicates that the high-order angular expansion significantly increases the computing time but contributes little to the accuracy of the core power. Besides, it was found that the initial steady-state calculation dominates the shape error of the flux in the transient calculation.

A Stable Method for Solving the Simplified Time-Dependent Neutron Transport Equation with Background Motion

The neutron transport equation is usually solved against a stationary background medium. When the background material is moving, the transport equation will need to be modified. Solving the transport equation with moving material is quite complicated, especially for the curved coordinate system because of the double angular redistributions. In this study, the discretization method of the simplified transport equation considering the moving-material effect is implemented in three-dimensional cylindrical geometry. Directly solving this modified transport equation with the standard solution technique is problematic since the advection term introduced by moving material may render the transport solver numerically unstable. The speed ratio λ is defined for stability analysis. A forced-stable method is proposed in this study to achieve good numerical stability for any material speeds and time-step sizes. The accuracy of this new method is verified using manufactured solutions. Steady numerical results demonstrate that the effects introduced by background motion cannot be neglected as the material speed starts to approach one-tenth of the neutron speed. Moreover, transient analysis indicates that the moving background has a considerable impact on the criticality of a system.

A new Monte Carlo approach for solution of the time dependent neutron transport equation based on nodal discretization to simulate the xenon oscillation with feedback

In this paper a probabilistic methodology based on core nodalization is proposed to estimate the core power in the presence of xenon oscillation. A time-dependent Monte Carlo neutron transport code named MCSP-NOD is developed for dynamic analysis in arbitrary 3D geometries to simulate xenon oscillations as well as sub-critical condition with feedbacks. The new code is based on the approach adopted in MCNP-NOD which was previously introduced as a tool for core transient analysis using the MCNPX platform. As before, the core is divided into nodes of arbitrary dimensions, and all terms of the transport equation e.g. interaction rates, leakage ratio are estimated using the MC techniques. However, as a new option the concentration of iodine and xenon are also estimated which enables us to predict the oscillatory behavior of reactor power following poison oscillation. These quantities are then employed within the time-dependent neutron transport equation for each node independently to compute the neutron population. Simulations prove the robustness of the method.

A new approach for solution of time dependent neutron transport equation based on nodal discretization using MCNPX code with feedback

This paper proposes a new method for solving the time-dependent neutron transport equation based on nodal discretization using the MCNPX code. Most valid nodal codes are based on the diffusion theory with differences in approximating the leakage term until now. However, the Monte Carlo (MC) method is able to estimate transport parameters without approximations usual in diffusion method. Therefore, improving the nodal approach via the MC techniques can substantially reduce the errors caused by diffusion approximations. In the proposed method, the reactor core is divided into nodes of arbitrary dimensions, and all terms of the transport equation e.g. interaction rates and leakage ratio are estimated using MCNPX. They are then employed within the time-dependent neutron transport equation for each node independently to compute the neutron population. Based on this approach, a time-dependent code namely MCNP-NOD (MCNPX code based on a NODal discretization) was developed for solving time-dependent transport equation in an arbitrary geometry considering feed backs. The MCNP-NOD is able to simulate multi-group processes using appropriate libraries. Several test problems are examined to evaluate the method.

Fourier Convergence Analysis of the Infinite Homogenous Multigroup Time-Dependent Boltzmann Transport Equation Using Discrete Ordinates Formulation

Fourier analysis of the continuous infinite homogenous multigroup (MG) formulation is investigated in this paper for the time-dependent Boltzmann transport equation using discrete ordinates formulation. In addition, a continuous two-group (2G) and one-group (1G) Fourier analysis is performed to generate an analytical spectral radius and provide the basis for a theoretical analysis of the convergence. The discrete 1G formulation is then presented, and the theoretical analysis is found to predict the same spectral radius as the continuous 1G formulation. A typical pressurized water reactor pin cell problem with 47-group library is then homogenized with reflective boundary conditions, and the numerical spectral radius is calculated using the MPACT code. The theoretical predictions and the numerical results from the pin cell case agree very well and are found to have the following behavior: (1) The spectral radius is usually very close to unity for standard parameters for realistic transient application, (2) the spectral radius generally decreases as a function of inners per outer M, (3) the spectral radius generally decreases as a function of time-step size and then increases beyond unity for extremely small time steps, and (4) the spectral radius is almost constant as a function of the inserted reactivity. Good agreement is observed with the MG Fourier analysis. Finally, it is shown that the group sweeping coarse mesh finite difference method is theoretically and numerically very slow to converge the time-dependent neutron transport equation and that it is necessary to move the right-hand-side fission and transient source to the left-hand side and to solve the entire matrix form of the system.

Dynamic Monte Carlo calculation method by solving frequency domain transport equation using the complex-valued weight Monte Carlo method

A Monte Carlo method that calculates the transient behavior of neutron flux in the frequency domain is proposed in this study. A time-dependent neutron transport equation, in which the neutron source term varies over time while other properties remain constant, is Fourier transformed to obtain the transport equation in the frequency domain. The complex-valued transport equation in the frequency domain is subsequently solved with the complex-valued weight Monte Carlo method for each frequency contained in the Fourier transformed source intensity. The effect of delayed neutrons can be easily included in this frequency domain transport equation. Using the inverse Fourier transformation of the neutron flux in the frequency domain, we can obtain the time variation of the flux. This method is applicable to transient analyses of a subcritical system with a time-varying neutron source intensity. Several numerical examples indicate that the newly developed frequency domain calculation method provides good results compared to the time-dependent calculation method in the time domain. The computation time in the frequency domain is significantly shorter than that in the time domain, particularly for a nearly critical system.

Two-dimensional equations of the surface harmonics method for solving problems of spatial neutron kinetics in square-lattice reactors

Two-dimensional time-dependent finite-difference equations of the surface harmonics method (SHM) for the description of the neutron transport are derived for square-lattice reactors. These equations are implemented in the SUHAM-TD code. Verification of the derived equations and the developed code are performed by the example of known test problems, and the potential and efficiency of the SHM as applied to the solution of the time-dependent neutron transport equation in the diffusion approximation in two-dimensional geometry are demonstrated. These results show the substantial advantage of SHM over direct finite-difference modeling in computational costs.

Equations of the surface harmonics method for solving time-dependent neutron transport problems and their verification

Time-dependent equations of the surface harmonics method (SHM) are obtained for planar one-dimensional geometry. The equations are verified by calculations of test problems from Benchmark Problem Book ANL-7416, and the capabilities and efficiency of applying the SHM for solving the time-dependent neutron transport equation in the diffusion approximation are demonstrated. The results of the work show that the implementation of the SHG for full-scale computations will make possible substantial progress in the efficient solution of time-dependent problems of neutron transport in nuclear reactors.

Numerical solution of the time dependent neutron transport equation by the method of the characteristics

This study presents three numerical algorithms to solve the time dependent neutron transport equation by the method of the characteristics. The algorithms have been developed taking into account delayed neutrons and they have been implemented into the novel MCART code, which solves the neutron transport equation for two-dimensional geometry and an arbitrary number of energy groups. The MCART code uses regular mesh for the representation of the spatial domain, it models up-scattering, and takes advantage of OPENMP and OPENGL algorithms for parallel computing and plotting, respectively. The code has been benchmarked with the multiplication factor results of a Boiling Water Reactor, with the analytical results for a prompt jump transient in an infinite medium, and with PARTISN and TDTORT results for cross section and source transients. The numerical simulations have shown that only two numerical algorithms are stable for small time steps.

Cellular neural network to the spherical harmonics approximation of neutron transport equation in x–y geometry: Part II: Transient simulation

In an earlier paper we utilized a novel method using cellular neural networks (CNNs) coupled with spherical harmonics method to solve the steady state neutron transport equation in x–y geometry. Here, the previous work is extended to the study of time-dependent neutron transport equation. To achieve this goal, an equivalent electrical circuit based on a second-order form of time-dependent neutron transport equation and one equivalent group of neutron precursor density is obtained by the CNN method. The CNN model is used to simulate step and ramp perturbation transients in a typical 2D core.

A neutron point kinetic model for fusion relevant calculations

In the framework of research activities on fusion reactors a great effort is dedicated by the scientific community to the development of tritium breeding blankets. One of the main goals is to assess the neutronic behaviour of such devices to analyse their tritium breeding performance and to evaluate the required data for their thermal–mechanic and thermal–hydraulic design. Many papers have been published on this topic considering some stationary condition to calculate such important quantities as heating power, gas production and dpa rates, tritium breeding ratio, etc., but not much attention has been focussed to neutronic transport analyses in transient conditions. The present paper proposes a simple model based on the point kinetics approximation, which has been set up deriving an alternative formulation of the time-dependent neutron transport equation. This approach allows to define some physical characteristics that can be interpreted in a statistical way, making possible to calculate these quantities numerically by the Monte Carlo method. The adoption of the aforementioned numerical method has the great advantage that complex geometries (as the fusion reactor's blankets are) can be analysed with acceptable computational times. Some simple cases have been investigated to implement the theoretical model proposed with MCNP5 code and to show its potentiality. Then, applications to fusion reactor ITER blanket module and to the Helium Cooled Test Blanket Module, to be tested in ITER, have been taken into account in order to assess their neutronic time-dependent behaviour and the results obtained have been critically discussed.

A Hybrid Transport Point-Kinetic method for simulating source transients in subcritical systems

Abstract This paper describes a novel hybrid technique for the solution of the time-dependent neutron transport equation in source switch-on transients. The Hybrid Transport Point Kinetics (HTPK) method is derived from a multi-collision description of the transport equation and is based on the observation that the external source mainly influences the first few neutron generations and, through collisions and multiplications, an ensemble behavior statistically emerges at higher order. The HTPK is designed to separately describe the propagation phenomenon from the ensemble response of the system, using both the rigorous transport equation and simplified kinetic models. Numerical simulations performed on a typical acceleration driven system configuration confirm the basic physical intuition of HTPK and show a significant gain in accuracy when compared to classical point kinetics. The proposed HTPK technique is also shown to be attractive because of its flexibility and competitive computational effort.

A response-based time-dependent neutron transport method

An efficient response-based solution to the time-dependent neutron transport equation in a semi-infinite slab is derived. The solution is based on polynomial expansions of the source terms and neutron flux in the time domain. The expansion coefficients of the flux solution are computed in terms of response functions, which are special cases of Green’s functions for arbitrary in-volume and surface sources. The resulting response equation, which is a convolution integral equation in time, is reduced to a linear algebraic system of equations in the expansion coefficients. Two example problems are solved using the response-based method, and the extension of the method to general (finite, heterogeneous) problems is discussed.

Calculation of time-dependent neutronic parameters using Monte Carlo method

A new code, named time-dependent Monte Carlo (TDMC), is developed for the calculation of neutronic parameters of multiplying systems. The code simulates the time-dependent neutron transport equation with explicit representation of delayed neutrons by using Monte Carlo method. Calculations with time-varying boundaries and/or cross sections are possible by this code. The resulting code is tested using benchmark problems and other methods that are available in the literature.

T N approximation on the critical size of time-dependent, one-speed and one-dimensional neutron transport problem with anisotropic scattering

The criticality problem is studied based on one-speed time-dependent neutron transport theory, for a uniform and finite slab, using the Marshak boundary condition. The time-dependent neutron transport equation is reduced to a stationary equation. The variation of the critical thickness of the time-dependent system is investigated by using the linear anisotropic scattering kernel together with the combination of forward and backward scattering. Numerical calculations for various combinations of the scattering parameters and selected values of the time decay constant and the reflection coefficient are performed by using the Chebyshev polynomials approximation method. The results are compared with those previously obtained by other methods which are available in the literature.

Numerical solution of the neutron transport equation using cellular neural networks

Various methods have been used for solving the neutron transport equation in the past, and a number of computer codes have been developed based on these solution methods. This paper describes a novel method for the solution of the steady-state and time-dependent neutron transport equation using the duality between neutronic parameters in the method of characteristic (MOC) and the electrical parameters in the cellular neural networks (CNN). The relevant electrical circuit can be simulated by professional electrical circuit simulator software, HSPICE. This software is used for numerical solution of the transport equation only by preparation of appropriate inputs. This method does not need inner and outer iterations, which is a necessary step in the other deterministic methods. One of the main applications of the proposed method may be the development of a new hardware by VLSI technology for online spatio-temporal calculations of the transport equation for nuclear reactor core. The accuracy and capability of this method are examined in a 2D steady-state problem for a BWR fuel assembly, and a 2D time-dependent TWIGL seed/blanket problem.

Cellular neural networks (CNN) simulation for the T N approximation of the time dependent neutron transport equation in slab geometry

This paper describes the application of a multilayer cellular neural network (CNN) to model and solve the time dependent one-speed neutron transport equation in slab geometry. We use a neutron angular flux in terms of the Chebyshev polynomials (T N) of the first kind and then we attempt to implement the equations in an equivalent electrical circuit. We apply this equivalent circuit to analyze the T N moments equation in a uniform finite slab using Marshak type vacuum boundary condition. The validity of the CNN results is evaluated with numerical solution of the steady state T N moments equations by MATLAB. Steady state, as well as transient simulations, shows a very good comparison between the two methods. We used our CNN model to simulate space–time response of total flux and its moments for various c (where c is the mean number of secondary neutrons per collision). The complete algorithm could be implemented using very large-scale integrated circuit (VLSI) circuitry. The efficiency of the calculation method makes it useful for neutron transport calculations.

Alternative forms of the time-dependent neutron transport equation

This work is devoted to the derivation of alternative formulations of the one-velocity neutron transport equation. The Peierl's integral form of the transport equation is considered as the starting point. Using a generalization of the mathematical procedure applied to derive the A N approximation, two formulations are obtained, both second-order differential in time and space, which can be considered as alternative forms of the A ∞ model. Some features and physical interpretations are investigated for the special case of plane geometry.

Modelling of fast source transients in PDS-XADS with VARIANT-KIN3D

In a system with an external neutron source such as an Accelerator Driven System (ADS), transients can be induced by a fast variation of the source amplitude. In this paper we present new capabilities of the VARIANT-KIN3D code for analyses of these transients. The code solves the time dependent neutron transport equation by using the spherical harmonics (PN) and simplified spherical harmonics (SPN) methods. Recently a more accurate time discretization scheme and an automatic time step control option have been implemented in KIN3D. In the paper we consider a fast transient in a 3D HEX-Z model representing the PDS-XADS (an ADS designed by ANSALDO). We consider responses of several detectors, placed at different axial and radial positions. The same transient is simulated with different options for angular discretization of the neutron transport equation. As expected, we observed that angular discretization influences the transient results. The influence is observed with respect to both time and spatial distributions of the detector rates. The new code version is able to deal with fast source induced transients and therefore may serve as an interesting tool for investigating the ADS physics.