An analytical solution to the hybrid diffusion-transport equation

A special integral equation was derived in previous work using a hybrid diffusion-transport theory method for calculating the flux distribution in slab lattices. In this paper an analytical solution of this equation has been carried out on a finite reactor lattice. The analytical results of disadvantage factors are shown to be accurate in comparison with the numerical results and accurate transport theory calculations.     

An alternative solution of the neutron diffusion equation in cylindrical symmetry

Alternative analytical solutions of the neutron diffusion equation for both infinite and finite cylinders of fissile material are formulated using the homotopy perturbation method. Zero flux boundary conditions are investigated on boundary as well as on extrapolated boundary. Numerical results are provided for one-speed fast neutrons in ²³⁵U. The results reveal that the homotopy perturbation method provides an accurate alternative to the Bessel function based solutions for these geometries.     

An analytical solution to time-dependent fission-product diffusion in an HTGR core

An analytical time-dependent fission-product diffusion model is solved for the fuel-moderator regions of a high temperature gas-cooled reactor (HTGR) during a hypothetical loss of forced circulation (LOFC) accident. A conservative approximate 1-D model is developed for the fuel and moderator regions, represented in cylindrical and slab geometries, from consideration of the hexagonal fuel-element symmetry. Transport is assumed along the shortest diffusion path and the concentration change across the fuel-moderator interface is approximated by a jump condition. The model is solved by construction of the Green's functions for the Laplace-transformed equations and identification of the pole structure. The concentration and current inverse Laplace transforms are obtained by the Cauchy residue theorem in each region for cubic piecewise polynomial initial conditions. A computer program was developed and validated to evaluate the solution, serve as a benchmark for more sophisticated numerical models and to investigate ⁹⁰Sr diffusion during a hypothetical LOFC.     

An exact general solution in spherical harmonics of the Boltzmann equation

A solution of the one-velocity kinetic Boltzmann equation is obtained as a series of spherical harmonics. General expressions are obtained for the terms of the series, derived without any approximately valid assumptions. As particular cases of this solution, we obtain formulas for the known P_N-approximations for the spherical-harmonic method.The exact general solution of the kinetic equation in the form of a series of spherical harmonics contains arbitrary functions which must depend on the formulation of boundary conditions. The general determination of the boundary conditions and the arbitrary functions is not considered. All the results of [4] remain valid for P_N-approximations.Translated from Atomnaya Énergiya, Vol. 18, No. 5, pp. 459–463, May, 1965     

Application of finite fourier transformation for the solution of the diffusion equation

Application of the finite Fourier transformation is discussed for the solution of the diffusion equation in one dimension, two dimensional x-y and triangular geometries. It can be shown that the equation by the Nodal Green's function method in Cartesian coordinate can be derived as a special case of the finite Fourier transformation method.     

A finite transform approach to the solution of radiation transport equation

Starting from the integral form of transport equation and defining a new transform G(x,μ,μ_m), the integration over space is carried out analytically resulting in an integral equation for F in μ. Use of Legendre polynomial for the solution of this equation is explored. The method is applicable to practical problems of radiation transport in multiregion, energy dependent systems with arbitrary degree of anisotropy. Its simplification for idealised systems and comparison with earlier work are indicated.     

Exact solution to the time-dependent one-speed multiregion neutron diffusion equation

Exact solutions are obtained to the time-dependent one-speed neutron diffusion equation in one-dimensional multiregion Cartesian and spherical geometries with multiplication and without delayed neutrons. These solutions enable the study of the one-speed space-time behavior of prompt neutrons in an arbitatry number of neutronically dissimilar material regions. Parametric benchmark calculations are presented.     

An integral equation arising in neutron transport theory

A classic problem in nuclear reactor physics is the calculation of the spatial distribution of fissile material to make the associated neutron flux distribution spatially constant. We examine a special case of that problem for an infinite slab of fissile material which is infinitely reflected on both sides by a non-multiplying material. The conditions for a constant flux are derived and lead to a singular integral equation. This equation is reduced analytically to a non-singular integral equation and the solution thereby obtained is compared with that from a direct numerical method. Some of the physical implications are examined. We also note that, contrary to a theorem for multi-group diffusion theory, the resulting total fissile loading of the system is not a minimum but rather a maximum. An important aspect of the present work is that transport theory is used and not diffusion theory. Indeed, we note that no solution exists for the corresponding diffusion theory model unless it is specially modified by the addition of generalised functions, and hence we note that the problem is intrinsically governed by transport effects.     

A solution of the neutron diffusion equation for a hemisphere containing a uniform source

An analytic solution of the diffusion equation for a hemisphere of fissile or non-fissile material is presented which contains a spatially uniform neutron source. Numerical results are given for the flux distribution for one-speed fast neutrons in ²³⁵U and also for a non-fissile element of similar scattering properties. We use these results to check the accuracy of the finite element code EVENT. The procedure is also developed for multigroup calculations. In an Appendix we outline the procedure required when the hemisphere contains a source and is also irradiated by an external current of neutrons.     

An approximate analytical solution to one-dimensional hydrogen diffusion in a temperature gradient and a field of traps

Exact analytical solutions are obtained for an approximate partial differential equation which describes one dimensional transport of hydrogen through a material layer having a uniform distribution of traps and a thermal gradient. For application to a specific material and thermal gradient the validity of the approximations underlying the differential equation can be verified with a few simple calculations. By making appropriate changes of variables (x → z(x), t → τ(t), and C(x,t) → G(z,τ)) diffusion profiles, G, are found to be slowly varying functions of two parameters, ρ and z_o, which are themselves functions of the activation energies and temperatures of interest in a particular application. Characterization of the solutions through these parameters emphasizes the similarities in the diffusion profiles for widely disparate material properties and thermal gradients. The generalized solutions for a limited range of ρ- and z_o-values thus allow construction of diffusion profiles and permeating fluxes for many materials of interest in fusion applications and for a broad range of fusion reactor operating scenarios. Graphical representations of the generalized profiles are presented for a useful range of ρ- and z_o-values.     

Introducing the migration mode method for the solution of the space and energy dependent diffusion equation

The most usual method to take into account energy dependence in whole core spatial neutronics calculation is the multigroup method. In thermal spectrum reactors as PWR, two-group theory is sufficient to describe accurately the neutron spectrum variation among spatial regions. When the spectrum hardens the precision of two-group theory decreases and more groups are necessary to keep a good accuracy. The aim of the computation method presented here is to represent with a good accuracy the spectral transitions which appear in these situations, without increasing the number of unknowns (i.e. the number of energy groups). The neutron spectrum is considered as a combination of base shapes corresponding to the different modes of migration of the neutrons in the energy dimension. The resulting energy flux distribution is a continuous function that fits the real one. The spatial discretization leads to matrices having the same structure of the ones obtained with multi-group theory. Then the method can be easily applied to existing codes solving the diffusion equation on the whole core in 3D. A methodological comparison between the migration mode method and the multigroup (few-group) method as well as a numerical comparison is presented.     

An analytical solution for the consideration of the effect of adjacent fuel elements

A new analytical method is described to deal with the Leakage Environmental Effect – the influence of the adjacent fuel element on the cross-section preparation. The method is discussed and classified in comparison with other methods given in the literature. The new method is based on the analytical solution of the two group diffusion equation for two adjacent fuel elements. The specifics needed to create a highly efficient analytical solution are discussed. The very promising quality of the results for this highly efficient method is demonstrated on a homogeneous test case and on several heterogeneous combinations of two fuel elements described in the PWR MOX/UO2 CORE TRANSIENT BENCHMARK. One important advantage is the unproblematic extension of the solution to two-dimensional problems, since the analytical solution for each fuel element will be of the identical structure. Only the filled in data for the four fuel element quarters will vary. The coupling of the fuel elements does not affect the exponential solutions, only the constants attached to the single exponentials. Thus, the coupling will be solved in a system of linear equations.     

Convergence of the iteration scheme in the nodal expansion method for the solution of the diffusion equation

Convergence of the iteration scheme in the nodal expansion method for the solution of the diffusion equation has been established. The proof is applicable to 1-D, 2-D and 3-D problems with commonly occurring boundary conditions. It is restricted to square and cubic nodes and parabolic expansion of the flux over a node.     

Milne’s half-space problem: A numerical solution of the related integral equation

The one speed, isotropic scattering, planar Milne’s problem in neutron transport has been a test bed for many methods. We further explore here a numerical solution of the related integral equation. We find that with the use of singularity subtraction, high order quadratures, some interval transformations, and very high precision in all computations, it is possible to obtain results of benchmark accuracy in a straightforward manner.     

An analytical solution of the point kinetics equations with time-variable reactivity by the decomposition method

In this work, we report an analytical solution for the point kinetics equations by the decomposition method, assuming that the reactivity is an arbitrary function of time. The main idea initially consists in the determination of the point kinetics equations solution with constant reactivity by just using the well-known solution results of the first-order system of linear differential equations in matrix form with constant matrix entries. Applying the decomposition method, we are able to transform the point kinetics equations with time-variable reactivity into a set of recursive problems similar to the point kinetics equations with constant reactivity, which can be straightly solved by the mentioned technique. For illustration, we also report simulations for constant, linear and sinusoidal reactivity time functions as well comparisons with results in literature.     

An analytical solution of the point kinetics equations with time-variable reactivity by the decomposition method

In this work, we report an analytical solution for the point kinetics equations by the decomposition method, assuming that the reactivity is an arbitrary function of time. The main idea initially consists in the determination of the point kinetics equations solution with constant reactivity by just using the well-known solution results of the first-order system of linear differential equations in matrix form with constant matrix entries. Applying the decomposition method, we are able to transform the point kinetics equations with time-variable reactivity into a set of recursive problems similar to the point kinetics equations with constant reactivity, which can be straightly solved by the mentioned technique. For illustration, we also report simulations for constant, linear and sinusoidal reactivity time functions as well comparisons with results in literature.     

Fender: A finite element code for the solution of the diffusion equation in shield design applications

Diffusion theory remains an important method of calculation for shield design. Using adjusted coefficients the method provides an inexpensive solution of adequate accuracy for survey and optimization studies in two- and three-dimensional geometries. Solved in the adjoint mode, the method provides an estimate of the importance function which may be used for the acceleration of generalized-geometry Monte Carlo calculations.A number of computer codes exist to solve the diffusion equation by a finite difference approximation in one-, two- and three-dimensions. The mesh systems used in such codes usually impose restrictions on the accuracy of representation of shields with complicated geometries.The computer code FENDER solves the diffusion equation for neutron or gamma transport using the finite element technique. At present the code is written for a two-dimensional problem in which the geometry is specified as an array of triangular or rectangular elements. This permits a good representation to be made of shields containing curved surfaces. The variation of the calculated particle fluxes within an element is assumed to be quadratic.FENDER may take details of the element structure from an external mesh generating package but also contains a semi-automatic mesh generating routine for use as a stand-alone code. Multigroup diffusion parameters may be either input directly or generated from material compositions. The code is capable of handling problems with at least 1000 elements which is roughly equivalent in size and attenuation to 10,000 finite difference meshes. A variety of boundary conditions may be specified.The paper includes an example of application to demonstrate the potential usefulness of the method and the code. The case chosen is the calculation of neutron fluxes in a stylized fast reactor.