Three-dimensional transport theory: An analytical solution of an internal beam searchlight problem, III

We describe a method for obtaining analytical solutions and numerical results for three-dimensional one-speed neutron transport problems in a half-space containing a variety of source shapes which emit neutrons mono-directionally in the direction away from the surface. Thus this paper is a supplement to Williams [Williams, M.M.R., 2009, Three-dimensional transport theory: an analytical solution for the internal beam searchlight problem I. Annals of Nuclear Energy 36, 767–783]. For example, we consider a point source, a ring source and a disk source, and calculate the surface scalar flux as a function of the radial co-ordinate when the source is at a fixed distance from the surface. The results are in full agreement with the work of Ganapol and Kornreich [Ganapol, B.D., Kornreich, D.E., this issue. Three-dimensional transport theory: an analytical solution for the internal beam searchlight problem II. Annals of Nuclear Energy]. Diffusion theory results are also included.     

Three-dimensional transport theory: An analytical solution of an internal beam searchlight problem-I

We describe a number of methods for obtaining analytical solutions and numerical results for three-dimensional one-speed neutron transport problems in a half-space containing a variety of source shapes which emit neutrons mono-directionally. For example, we consider an off-centre point source, a ring source and a disk source, or any combination of these, and calculate the surface scalar flux as a function of the radial and angular co-ordinates. Fourier transforms in the transverse directions are used and a Laplace transform in the axial direction. This enables the Wiener–Hopf method to be employed, followed by an inverse Fourier–Hankel transform. Some additional transformations are introduced which enable the inverse Hankel transforms involving Bessel functions to be evaluated numerically more efficiently. A hybrid diffusion theory method is also described which is shown to be a useful guide to the general behaviour of the solutions of the transport equation.     

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 exact solution for the moving boiling boundary problem

An exact transient solution of the fluid velocity and temperature fields in a one-dimensional incompressible flow within a non-uniformly heated channel is presented. The first order partial differential equations for mass and energy conservation governing this problem are solved using Laplace transform technique. An analytical expression for the boiling boundary (BB) location as a function of time is derived from the temperature field solution when the saturation temperature is inserted. Results obtained reveal the interesting behavior of the temperature field and BB in space and time due to a step change in the fluid inlet mass flow rate.     

Simulation of isothermal fission gas release: An analytical solution

An analytical version of a previous diffusion model representing fission gas release during isothermal irradiation of UO₂ nuclear fuel is presented. The previous numerical version was successfully applied to a variety of experiments. The present model, although based on more restrictive assumptions, gives a quick but sufficiently accurate estimation, useful to predict experiments or more detailed calculations. The main new hypotheses are: constant fuel grain radius, constant gas generation rate and constant grain boundary gas content. The latter is met at the final stage of fuel irradiation, after grain boundary saturation, and partially met at the beginning of irradiation, when the grain boundary is nearly empty. Two analytical solutions are obtained and conveniently matched, yielding a unique solution representing the whole process. The difference between the analytical and numerical results for the fractional release is appreciable only near the matching. It is lower than 1% over 93% of the process duration for all the temperatures tested, ranging from 1250 to 2000 K and has no significant effect on the results at the end of life.     

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.     

An analytical solution for the general perturbative diffusion equation by integral transform techniques

In this work, we present analytical solutions for the eigenvalue problem of a neutron flux in a rectangular two dimensional geometry by a two step integral transform procedure. For a given effective multiplication factor _Keff$K_{\text{eff}}$ we consider a homogeneous problem for two energy groups, i.e. fast and thermal neutrons, respectively, where the problem is setup by two coupled bi-dimensional diffusion equations in agreement with general perturbation theory (GPT). These are solved in a two-fold way by integral transforms, in the sequence Laplace transform followed by GITT and vice versa. Although, the functional base and the employed integral transforms are the same for both sequences, the procedures differ. We verify the efficiency of the sequence on the solutions of such problems, further the results are compared to the solution obtained by the finite difference method.     

An approach to the solution of criticality problems in heterogeneous media by coupling the transport and diffusion theory

This work describes a new technique for the solution of criticality problems. Transport theory, in its integral form, is applied in the multiplying region, whilst diffusion theory is employed in the moderating region.The equations valid for a general geometry are obtained in two-group approximation. A numerical iterative scheme is developed for a multiplying reflected slab. The results obtained for a sample case are also described.     

Benchmark results for the critical slab and sphere problem in one-speed neutron transport theory

In this paper benchmark numerical results for the one-speed criticality problem with isotropic scattering for the slab and sphere are reported. The Fredholm integral equations of the second kind based on the Case eigenfunction formalism are numerically solved by Neumann iterations with the Double Exponential quadrature.     

An analytical approach for a nodal scheme of two-dimensional neutron transport problems

In this work, a solution for a two-dimensional neutron transport problem, in cartesian geometry, is proposed, on the basis of nodal schemes. In this context, one-dimensional equations are generated by an integration process of the multidimensional problem. Here, the integration is performed for the whole domain such that no iterative procedure between nodes is needed. The ADO method is used to develop analytical discrete ordinates solution for the one-dimensional integrated equations, such that final solutions are analytical in terms of the spatial variables. The ADO approach along with a level symmetric quadrature scheme, lead to a significant order reduction of the associated eigenvalues problems. Relations between the averaged fluxes and the unknown fluxes at the boundary are introduced as the usually needed, in nodal schemes, auxiliary equations. Numerical results are presented and compared with test problems.     

An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in Cartesian geometry

Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P_N approximation, the S_N method (discrete ordinates method) and their variants. In the discrete ordinates (S_N) formulations of the transport equation, it is assumed that the linearised Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this paper, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we describe the recent advances in the class of deterministic methods applied to one and two dimensional transport problems for photons and electrons in Cartesian Geometry. First, we describe the Laplace transform technique applied to S_N two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a mono-energetic electron beam in a rectangular domain. The main idea relies on applying the P_N approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x variable. Numerical results are given to illustrate the accuracy of deterministic methods presented.     

An overview of the Boltzmann transport equation solution for neutrons,photons and electrons in Cartesian geometry

Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P_N approximation, the S_N method (discrete ordinates method) and their variants. In the discrete ordinates (S_N) formulations of the transport equation, it is assumed that the linearised Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this paper, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we describe the recent advances in the class of deterministic methods applied to one and two dimensional transport problems for photons and electrons in Cartesian Geometry. First, we describe the Laplace transform technique applied to S_N two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker–Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a mono-energetic electron beam in a rectangular domain. The main idea relies on applying the P_N approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker–Planck equation and then applying the Laplace Transform in the spatial x variable. Numerical results are given to illustrate the accuracy of deterministic methods presented.     

A multi group finite-element solution of the neutron transport equation—II: R-Z geometry

The extension of a variational finite-element method of solving the neutron transport equation, to include multigroup-energy dependence in R-Z geometry, is evaluated. The method is implemented in a computer program called felicit. The solutions obtained by means of felicit to a number of axisymmetric test problems are given in numerous tables and graphs. A comparison is made between felicit, exact solutions and solutions obtained by other transport techniques. The ability of the method to handle problems characterized by difficult geometries is demonstrated; in particular by considering a fixed-source problem in a model Tokamak configuration.     

A two group analytical approximation solution for an external source problem without separation of space and time

This work presents the development of analytical approximation solutions for a space–time dependent neutron transport problem in two energy groups for a one dimensional system consisting of a homogenized medium with a localized external source. The approximation solutions are developed using Green’s functions, the influence of the delayed neutrons is not considered. Qualitative results for a given system are analyzed. A detailed comparison of the developed analytical approximation solutions with solutions with one energy group diffusion and P₁ equation without separation of space and time is given.     

Temperature decay in a reactor vessel subjected to thermal shock: An analytical solution

An analytical solution is derived for the time-dependent temperature profile in a reactor vessel subjected to the thermal shock of emergency cooling injection. Numerical results are given for a typical case, and it is shown how a simple correction can be used to improve the results of conventional calculational methods.     

An analytical approach to the rate theory for pulsed irradiation analyzing effects of point-defect bulk recombination

The concentrations of interstitials and vacancies created by a single irradiation pulse are calculated as a function of time. The underlying set of non-linear coupled rate equations is solved analytically by an approach which takes into account simultaneously point-defect absorption at sinks and their mutual recombination. The method allows the influence of both parameters — the dislocation density and the irradiation temperature — to be studied. The time-dependent point-defect concentrations thus derived are evaluated using materials data for stainless steel. These results are used to analyze analytically the effect of a single pulsing cycle on climb-glide creep. The climb velocity of an edge dislocation is calculated during the pulse on- and off-time whereupon we find the climb distance by integration. Inspecting it as a function of temperature reveals a broad peak of the climb distance at about 300°C, which is sensitive to the dislocation density as well as the time constants of the pulse.     

An analytic benchmark solution to the linear,time-dependent isothermal laminar flow fission-product plateout problem

The time-dependent convective-diffusion equation with radioactive decay is solved analytically in axisymmetric cylindrical geometry for laminar and slug velocity profiles under isothermal conditions. Concentration-dependent diffusion is neglected. The laminar flow solution is derived using the method of separation of variables and Frobenius' technique for constructing a series expansion about a regular singular point. These solutions, which describe the transport of fission products in a flowing stream, are then used to determine the pointwise and integrated concentrations of radioactive material deposited on a conduit wall using a standard mass-transfer model.Extensive parametric investigations have been conducted by varying the wall mass-transfer coefficient, diffusion coefficient, flow velocity, pipe radius and species half-life in the deposition models. The computational results indicate that the plateout estimates for the slug flow model are typically 5–100% greater than for the laminar model. The effect of axial diffusion is necessarily negligible for Péclet numbers greater than 100. Little increased plateout is observed for Péclet numbers less than 100; an additional 8% is predicted for a Péclet number of 20 if axial diffusion is included. Characteristic stream, wall and integrated deposition profiles are shown.Representative results from the analysis of fission-product deposition measurements for diffusion tubes in the Fort St Vrain high-temperature gas-cooled reactor plateout probe are presented. Using single-region slug and laminar models, the wall mass-transfer coefficients, diffusion coefficients and inlet concentrations are determined using least-squares analysis. The diffusion coefficients and inlet concentrations are consistent between tubes. The computed diffusion coefficients and wall mass-transfer coefficients are in agreement with known literature values. The analysis indicates that little difference can be discerned between computed laminar and slub flow-model parameters unless accurate data is obtained under strictly regulated conditions.