Solution Of Neutron Transport Equation Research Articles

Theoretical Approach to Synthetic Accelerations for the Solution of Neutron Transport Equation

A discrete ordinates neutron transport solution usually consists of solving alternately the transport and the diffusion equation. Here, the diffusion equation which is a P₁ approximation to the transport equation is solved only for an acceleration of transport solution to achieve the computational efficiency. The derivation of various low-order accelerating equations are generalized by using an operator form of acceleration in this paper. The source iteration, P₀, P₁, and P₂ acceleration schemes are derived and their spectral radii are calculated and compared by using Fourier convergent analysis. The result is that the P₁ acceleration outperforms the other schemes in accelerating transport solution. These results confirm that one cannot simply assume that the replacement of P₁ with a higher order equation would enhance the acceleration efficiency.

Coupling of first collision source method with Lattice Boltzmann Method for the solution of neutron transport equation

Despite significant progress in the solution of Neutron Transport Equation (NTE) using Lattice Boltzmann Method (LBM), it suffers from the ray effect a problem similar to SN methods. The ray effect is prominent in the problems where source is localized and medium has low scattering cross-section. Due to ray effect spatial distortion in the scalar flux and angular flux is observed. This effect arises because NTE is solved in discrete directions, allowing neutrons to travel only in a few directions. Therefore, first collision source (FCS) method is applied for SN method. Presently, FCS method is applied to LBM and result are obtained for two-dimensional source problems. It is observed that by applying FCS to LBM, ray effect is mitigated. Compared to SN method, LBM approach is relatively simple in implementation and can be effectively used. Therefore, it has potential to provide easy and convenient solution of neutron transport equation.

Coupling of First Collision Source Method with Lattice Boltzmann Method for Solution of Neutron Transport Equation

Coupling of First Collision Source Method with Lattice Boltzmann Method for Solution of Neutron Transport Equation

STUDY OF THE EIGENVALUE SPECTRA OF THE NEUTRON TRANSPORT PROBLEM IN PN APPROXIMATION

The study of the steady-state solutions of neutron transport equation requires the introduction of appropriate eigenvalues: this can be done in various different ways by changing each of the operators in the transport equation; such modifications can be physically viewed as a variation of the corresponding macroscopic cross sections only, so making the different (generalized) eigenvalue problems non-equivalent. In this paper the eigenvalue problem associated to the time-dependent problem (α eigenvalue), also in the presence of delayed emissions is evaluated. The properties of associated spectra can give different insight into the physics of the problem.

Multi-group lattice Boltzmann method for criticality problems

In this paper, Lattice Boltzmann methodology (LBM) has been developed for the criticality search for the nuclear reactor systems. It is noted that the LBM has not been used for criticality calculations in optically thin (mean free path of neutrons close to system dimensions) and highly anisotropic systems. In order to solve such problems the number of angular directions need to be increased. Various criticality benchmarks with one group and multi-group cross-section including both isotropic and anisotropic scattering have been solved. The solution of these 1D benchmark problems not only verifies the solution methodology, but also demonstrates that multiplication factor of the critical systems having dimension close to mean free path of neutron travel can also be accurately predicted using higher angular discretization. Since LBM approach is relatively simple in implementation it can be effectively used and has a potential to provide easy and convenient solution of neutron transport equation.

Mesh-free method for numerical solution of the multi-group discrete ordinate neutron transport equation

The finite element method (FEM) is a widely-used method to solve neutron transport equation in an arbitrary domain, but in order to ensure the accuracy of solution a re-meshing process is often required and some regions of the domain need to be meshed with finer meshes and consequently the whole domain should be re-meshed again. To overcome this problem of the FEM we have applied Mesh Free (MFree) method for the solution of the neutron transport equation. In mesh free technique, there is no need to regenerate finer meshes to ensure the accuracy of the solution and only the number of nodes in local domains is increased. MFree method is flexible for distributing arbitrary and irregular nodes, therefore this method can be easily applied for high absorbent regions in the nuclear reactor problems. We have developed a program, named SNMF, which solves discrete ordinate neutron transport equation, using mesh free techniques based on Radial Point Interpolation Method (RPIM). To validate SNMF program some test problems have been modeled. For the first time, the application of MFree method is examined to solve neutron transport equation. Our results show that mesh free discrete ordinate method can be used as an efficient and accurate method to solve neutron transport equation without hardness and limitations of finite element methods.

Exact solution of the neutron transport equation in spherical geometry

Abstract Solution of the neutron transport equation in one dimensional slab geometry construct a basis for the solution of neutron transport equation in a curvilinear geometry. Therefore, in this work, we attempt to derive an exact analytical benchmark solution for both neutron transport equations in slab and spherical medium by using PN approximation which is widely used in neutron transport theory.

Solution of neutron transport equation by Method of Characteristics

A computer code based on Method of Characteristics (MOC) is developed to solve neutron transport equation for mainly assembly level lattice calculation with reflective and periodic boundary conditions and to some extent core level calculation with vacuum boundary condition. The code is able to simulate square, circular and hexagonal geometries and their combinations. Delaunay triangulation together with the Bower–Watson algorithm is used to divide the problem geometry into triangular meshes. Ray tracing technique is developed to draw characteristics lines along different directions over the geometry and the transport equation is solved over these lines to obtain neutron flux distribution and multiplication factor for the geometry. A number of benchmark problems available in literature are analyzed to demonstrate the capability and validity of the code.

Tracking algorithms for multi-hexagonal assemblies (2D and 3D)

BackgroundThere has been a continuous effort to design new reactors and study these reactors under different conditions. Some of these reactors have fuel pins arranged in hexagonal pitch. To study these reactors, development of computational methods and computer codes is required. For this purpose, we have developed algorithms to track two dimensional and three dimensional cluster geometries. These algorithms have been implemented in a subprogram HX7, that is implemented in the code DRAGON (Version 3.06F) to compute neutron flux distributions in these systems. MethodsComputation of the neutron flux distribution requires solution of neutron transport equation. While solving this equation, by using Carlvik’s method of collision probabilities, computation of tracks in the hexagonal geometries is required. In this paper we present equations that we have developed for the computation of tracks in two dimensional (2D) and three dimensional (3D) multi-hexagonal assemblies (with two rotational orientations). These equations have been implemented in a subprogram HX7, to compute tracks in seven hexagonal assemblies. The subprogram HX7 has been implemented in the NXT module of the DRAGON code, where tracks in the pins are computed. ResultsThe results of our algorithms NXT(+HX7) have been compared with the results obtained by the EXCELT module of DRAGON (Version 3.06F). ConclusionsWe find that all the fluxes in 2D and fluxes in the outer pin (3D) are converging to their 3rd decimal places, in both the modules EXCELT and NXT(+HX7). For other regions 3D fluxes, in general, are converging to their first decimal places. To improve convergence of fluxes, we plan to analyze more lattices in future and study some of the reasons behind convergence.

Nuclear Fuel Cell Calculation Using Collision Probability Method with Linear Non Flat Flux Approach

Nuclear fuel cell calculation is one of the most complicated steps of neutron transport problems in the reactor core. A few numerical methods use neutron flat flux (FF) approximation to solve this problem. In this approach, neutron flux spectrum is assumed constant in each region. The solution of neutron transport equation using collision probability (CP) method based on non flat flux (NFF) approximation by introducing linear spatial distribution function implemented to a simple cylindrical annular cell has been carried out. In this concept, neutron flux spectrum in each region is different each other because of an existing of the spatial function. Numerical calculation of the neutron flux in each region of the cell using NFF approach shows a fairly good agreement compared to those calculated using existing SRAC code and FF approach. Moreover, calculation of the neutron flux in each region of the nuclear fuel cell using NFF approach needs only 6 meshes which give equivalent result when it is calculated using 24 meshes in FF approach. This result indicates that NFF approach is more efficient to be used to calculate the neutron flux in the regions of the cell than FF approach.

Solution of neutron transport equation using Daubechies’ wavelet expansion in the angular discretization

Daubechies’ wavelet expansion is introduced to discretize the angular variables of the neutron transport equation when the neutron angular flux varies very acutely with the angular directions. An improvement is made by coupling one-dimensional wavelet expansion and discrete ordinate method to make two-dimensional angular discretization efficient and stable. The angular domain is divided into several subdomains for treating the vacuum boundary condition exactly in the unstructured geometry. A set of wavelet equations coupled with each other is obtained in each subdomain. An iterative method is utilized to decouple the wavelet moments. The numerical results of several benchmark problems demonstrate that the wavelet expansion method can provide more accurate results by lower-order expansion than other angular discretization methods.

Analysis of NEA C5MOX benchmark with computer code “cohint” based on interface current approach

Interface current approach to solution of neutron transport equation has been earlier used for LWR lattice problems. The analysis of the C5MOX benchmark is a opportunity to test its applicability to heterogeneous reactor problem. Computer code COHINT, which incorporates a routine for solution of neutron transport equation in X-Y geometry by interface current approach was used for this analysis by representing the fuel rod as a square one. Region interface angular fluxes are represented by a four-term expansion. It is found that the average pin power error is about 2.28 % (peak pin error 4.1 %) relative to reference calculations. Further improvement is possible by introduction of the capability to represent circular rods with in a square cell in COHINT.

Application of preconditioned GMRES to the numerical solution of the neutron transport equation

The generalized minimal residual (GMRES) method with right preconditioning is examined as an alternative to both standard and accelerated transport sweeps for the iterative solution of the diamond differenced discrete ordinates neutron transport equation. Incomplete factorization (ILU) type preconditioners are used to determine their effectiveness in accelerating GMRES for this application. ILU( τ), which requires the specification of a dropping criteria τ, proves to be a good choice for the types of problems examined in this paper. The combination of ILU( τ) and GMRES is compared with both DSA and unaccelerated transport sweeps for several model problems. It is found that the computational workload of the ILU( τ)-GMRES combination scales nonlinearly with the number of energy groups and quadrature order, making this technique most effective for problems with a small number of groups and discrete ordinates. However, the cost of preconditioner construction can be amortized over several calculations with different source and/or boundary values. Preconditioners built upon standard transport sweep algorithms are also evaluated as to their effectiveness in accelerating the convergence of GMRES. These preconditioners show better scaling with such problem parameters as the scattering ratio, the number of discrete ordinates, and the number of spatial meshes. These sweeps based preconditioners can also be cast in a matrix free form that greatly reduces storage requirements.

Asymptotic solutions of neutron transport equation and the limits of correct use of diffusion approximation for rocks

The diffusion approximation solution for neutron transport has been used in well-logging geophysics for calculating tool responses in boreholes, sometimes with success. The problem of the dimension of different materials to which it can be applied with success is important for the borehole environment. The results obtained show that the diffusion approximation can be used for distances greater than a few millimetre in some rock types. For iron, barium, and other highly absorbing media the use of the diffusion approximation is inappropriate even for large distances.

Implementation of certain analytical information in transport theory computation

Abstract This paper presents a new method for the solution of neutron transport equation in complex geometry. Using the Rvachev method, a solution structure satisfying exactly the boundary condition is derived. In such a way, information about the geometrical shape of the considered spatial domain is a priori analytically incorporated into the approximate solution. In particular, the construction of domain functions for the most used reactor lattice models (rectangular, triangular and cluster configurations) is given.

Solution of multi-dimensional neutron transport equation of the spherical harmonics method using the finite Fourier transformation and quadrature formula

Abstract A solution of the neutron transport equation in multi-dimensions of the spherical harmonics method is given by applying the finite Fourier transformation method. A quadrature formula of the form of the Gauss-Legendre one is derived using the associated Legendre functions. Using this quadrature formula for the integration over the polar angle and the finite Fourier transformation, equations for the spherical harmonics components of the angular flux at the region boundary of the constant cross sections are derived which are simpler than the previous ones. It is found that the characteristic roots of the differential equations which are satisfied by the spherical harmonics components of higher order are simply given by the roots of the associated Legendre functions.

Solution of Neutron Transport Equation by Superposition of Beams

The analytical solution of the isotropic, one-dimensional and time-dependent transport equation is expressed by the superposition of a newly defined Green function of the virgin neutron flow. As a result, the solution of the neutron transport equation is given by the superposition of beams possessing wave fronts. The present study has introduced the possibility of clearly explaining neutron transport behavior by beam processes, thus providing a further insight into the interpretation of the transport equation. This method should prove useful when the continuous mode is superior to discrete modes. The solution for the stationary case is also derived.

Solution of Neutron Transport Equation by Superposition of Beams

The analytical solution of the isotropic, one-dimensional and time-dependent transport equation is expressed by the superposition of a newly defined Green function of the virgin neutron flow. As a result, the solution of the neutron transport equation is given by the superposition of beams possessing wave fronts. The present study has introduced the possibility of clearly explaining neutron transport behavior by beam processes, thus providing a further insight into the interpretation of the transport equation. This method should prove useful when the continuous mode is superior to discrete modes. The solution for the stationary case is also derived.