Time-dependent Transport Equation Research Articles

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.

An analytical solution for the time-dependent SN transport equation in a slab

Abstract In this work an analytical solution for the time-dependent SN transport equation in a slab using the double Laplace transform technique is reported. This procedure leads to an analytical solution for the angular flux in integral form. Numerical simulations are presented and compared with results in the literature.

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.

Mass conservation of finite element methods for coupled flow-transport problems

This paper gives an overview of the mass conservation properties of finite element discretisations applied to coupled flow-transport problems. The system is described by the instationary, incompressible Navier?Stokes equations and the time-dependent transport equation. Due to the incompressibility constraint, the weak solution of the transport equation satisfies a global mass conservation. Since the discretised velocity fulfils only a discrete incompressibility constraint, the global mass conservation is, in general, satisfied only approximately. Several discretisations which ensure the global mass conservation, also on the discrete level, will be studied.

Exact Controllability for the Time Dependent Transport Equation

Exact Controllability for the Time Dependent Transport Equation

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.

Chapman–Enskog-maximum entropy method on time-dependent neutron transport equation

The time-dependent neutron transport equation in semi and infinite medium with linear anisotropic and Rayleigh scattering is proposed. The problem is solved by means of the flux-limited, Chapman–Enskog-maximum entropy for obtaining the solution of the time-dependent neutron transport. The solution gives the neutron distribution density function which is used to compute numerically the radiant energy density E ( x , t ) , net flux F ( x , t ) and reflectivity R f . The behaviour of the approximate flux-limited maximum entropy neutron density function are compared with those found by other theories. Numerical calculations for the radiant energy, net flux and reflectivity of the proposed medium are calculated at different time and space.

Two-dimensional time dependent Riemann solvers for neutron transport

A two-dimensional Riemann solver is developed for the spherical harmonics approximation to the time dependent neutron transport equation. The eigenstructure of the resulting equations is explored, giving insight into both the spherical harmonics approximation and the Riemann solver. The classic Roe-type Riemann solver used here was developed for one-dimensional problems, but can be used in multidimensional problems by treating each face of a two-dimensional computation cell in a locally one-dimensional way. Several test problems are used to explore the capabilities of both the Riemann solver and the spherical harmonics approximation. The numerical solution for a simple line source problem is compared to the analytic solution to both the P 1 equation and the full transport solution. A lattice problem is used to test the method on a more challenging problem.

The Asymptotic Behavior of the Solution to a Cauchy Problem Governed by a Transport Operator

The purpose of this paper is to investigate some spectral properties of a time-dependent linear transport equation with diffusive boundary conditions arising in growing cell populations. After deriving the explicit expression of the strongly continuous semigroup {U K (t ): t ≥ 0} generated by the streaming operator, we establish the strict singularity of the operator BU K (s)B, s> 0, for a wide class of collision operators B. Making use of the Weis theorem, Theorem 4.1, this enables us to estimate the essential type of the transport semigroup from which the asymptotic behavior of the solution is derived.

Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation

An analytical expression of the particle distribution based on an analytical cumulant solution of the time-dependent elastic Boltzmann transport equation (BTE) is presented. This expression improves upon the previous second order cumulant solution of the BTE described by a Gaussian distribution in two aspects: (1) separating the ballistic component from the scattered component to ensure that the summation in expressions is convergent; and (2) enforcing the causality condition to ensure that no particle travels faster than the free speed of the particles. Time-resolved profiles obtained using the analytical form are compared with those obtained by the Monte Carlo simulation, for both transmission and backscattering. The calculating time using our analytical form is much faster than that using the Monte Carlo approach.

Time-dependent neutron transport in finite media using Pomraning–Eddington approximation

The time-dependent monoenergetic neutron transport equation in a finite plane-parallel medium with linear anisotropic scattering and with specular-reflecting boundaries is considered. The problem is solved using Pomraning–Eddington approximation and by using weight functions to force the boundary conditions to be fulfilled. Numerical calculations for the reflectivity and transmissivity of the proposed medium are calculated at different times and different thicknesses.

On the solution of time-dependent transport equation with time-varying cross sections

The time-dependent neutron transport equation in an infinite medium with time-varying cross sections has been solved by means of two techniques namely, flux-limited approach and maximum entropy method. The behaviour of the distribution function are shown graphically. Knowing the distribution function allows us to calculate directly some physical parametres of special interest such as the reflection function. The results reported in this article provide further evidence of the usefulness of both maximum entropy and flux limited methods for obtaining time-dependent solution of neutron transport in compact form.

Parallel Flux Sweep Algorithm for Neutron Transport on Unstructured Grid

Discontinuous finite element discrete ordinates (DFE-Sn) method is widely used to solve the time-dependent neutron transport equation for nuclear science and engineering applications. Most efficiently, the kernel is to iteratively sweep the neutron flux across the computational grid. However, for unstructured grid this will bring forward several challenges while implemented on distributed memory parallel computers where the grid are decomposed across processors. This paper presents a parallel flux sweep algorithm to improve the parallel scalability of this basic sweep algorithm on unstructured grid under 2-D cylindrical Lagrange coordinates system from two sides. One is to prioritize the sweep order of elements within each subdomain, another is to better decompose the unstructured grid across processors. With optimal combination of domain decomposition method and priority queuing algorithm, this parallel algorithm has successfully been incorporated into DFE-Sn method and has been implemented with MPI to solve the neutron and photon coupled transport equation for complex physics. Performance results for two different applications on hundreds of processors of two parallel computers are given in this paper. In particular, the parallel solver has respectively achieved speedup larger than 72 using 92 processors and 78 using 256 processors on these two machines.

A Second-Order Finite Volume Discretization of the Time-Dependent Transport Equation on Arbitrary Quadrilaterals in R-Z Geometry

A second order, semi-implicit numerical method for solving the multigroup nonstationary transport equation and corresponding code is developed in two-dimensional R-Z geometry. Finite difference meshes are formed by arbitrary convex quadrangles. The conservative finite difference scheme is derived by the integro-interpolation method. The balance equation is augmented by linear approximations. The proposed additional relationships provide the second order of approximation at any side-visible cases using a corresponding choice of the weights of scheme. The number of additional relationships in spatial variables, as well as their form, depends on how many visible sides are under consideration. The additional relationships in time and angle variables are diamond-difference-like approximations relating the edge values to the cell-centered values.An analytical test problem is used to demonstrate the second order of spatial approximation of the proposed method. To test the algorithm for solving the stationary transport equation, we compare the numerical results, obtained by the developed technique, with the results produced by one-dimensional (1-D) codes such as KIN1D (The Keldysh Institute of Applied Mathematics, Russia) and ANISN (U.S.) by using spherical symmetrical 1-D problems. Special analytical benchmarks are developed to test the nonstationary technique. The tests have shown good agreement of the results.

Electron heat transport and ECRH modulation experiments in Tore Supra tokamak

An analytical solution has been found for the simplified time-dependent transport equation in a finite cylinder. This solution represents a decomposition of the heat source in eigenmodes. Their eigenvalues represent the effectiveness of the response to temperature for each eigenmode, which are different in the stationary regime: the higher the order, the lower the effectiveness. In this case, the response to temperature appears as non-local. On the other hand, the profile resilience mainly results from two effects: the first one is that the lower order eigenmodes are more favoured than the higher order; the second one (volume effect) is that the central source (Ohmic heating) is favoured with respect to the off-axis source (electron cyclotron resonance heating, ECRH) in the contribution to the temperature profile shape. This resilience effect on the temperature profile is a basic and natural property of the diffusion equation in cylindrical geometry. The above analytical solution has been used for the determination of the heat diffusivity χ and the damping time τd for the ECRH modulation experiments in the Tore Supra tokamak. The results obtained by this analytical method have been compared to those obtained by two other methods: the FFT and the power balance methods. The FFT is the easiest and the fastest method compared to the others, but it is too sensitive to perturbations such as the sawteeth, which may cause inaccurate results. The analytical solution is the most robust method, because it simulates the whole temperature modulation; hence, the results are less sensitive to other perturbations. Furthermore it contains more information than the FFT method. For 0.2 ⩽ r/a ⩽ 0.7, the values of χANA are very close to those of χPB during the ECRH phase. Generally, the values of χHP, which is more ‘transient’, are larger than those of χANA and χPB (ECRH).

Enclosure effects on flame spread over solid fuels in microgravity

Enclosure effects on the transition from a localized ignition to subsequent flame growth over a thermally thin solid fuel in microgravity are numerically investigated by solving the low Mach number time-dependent Navier-Stokes equations. The numerical model solves the two and three dimensional, time-dependent, convective/diffusive mass, and heat transport equations with a one-step global oxidation reaction in the gas phase coupled to a three-step global pyrolysis/oxidative reaction system in the solid phase. Cellulosic paper is used as the solid fuel and is placed in a slow imposed flow parallel to the surface. Ignition is initiated across the width of the sample or at a small circular area by an external thermal radiation source. Two cases are examined; an open configuration (i.e., without any enclosure) and the case with the test chamber used in our previous microgravity experiments. Numerical results show that the upstream centerline flame spread rate for the case with the enclosure is faster than that for the case without any enclosure. This is due to the confinement of the flow field and also thermal expansion initiated by heat and mass addition in the chamber. The confinement accelerates the flow in the chamber, which enhances oxygen transport into the flame. In the three-dimensional configuration with the spot ignition, the flame growth in the direction perpendicular to the flow is significantly enhanced by the confinement effects. The effect of the enclosure is most significant at the slowest flow condition investigated and the effect becomes less important with an increase in imposed flow velocity. The total heat release rate from the flame during a flame growth period increases significantly with the confinement and the enclosure effects should be accounted to avoid underestimating fire hazard in a spacecraft.

DIFFUSION OF PARTICLES IN THE PRESENCE OF STRONG FORWARD–BACKWARD ANISOTROPIC SCATTERING

ABSTRACT We consider one-speed transport problems with forward–backward-isotropic scattering in an infinite medium and study its relationship with the diffusion equation. Both time dependent and time independent transport equations are considered. We also obtain the Greens function for this scattering model.

THE LINEARIZED STABILITY FOR NONLINEAR TRANSPORT EQUATION

ABSTRACT The purpose of this paper is to investigate a class of time-dependent nonlinear neutron transport equation. The linearized stability of stationary solutions are given using the principle in (Nobuyuki Kato. A Principle of Linearized Stability for Nonlinear Evolution Equation. Trans. Amer. Math. Soci. 1995, 347(8), 2851–2868.) Especially, the case of a slab with various boundary and initial conditions are discussed.