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 for the internal beam searchlight problem, II

Multidimensional semi-analytical particle transport benchmarks to provide highly accurate standards of assessment are few and far between. Because of a well-established 1D theory for the analytical solution of the transport equation, it is sometimes possible however, to “bootstrap” 1D solutions to give more comprehensive solution representations. Here, we propose the internal searchlight problem in a half space, designated ISLP/HS, as a multidimensional benchmark to be constructed from 1D solutions. This is a variation of the usual SLP/HS where a source emits within the half space rather than striking its surface. Our primary interest is in the exiting intensity at the free surface established through a new Fn formulation. The benchmark features true 2/3D particle transport through integration of a point kernel to simulate 2/3D source emission. In this way, we accommodate a solid or hollow cylindrical source and a general line source in addition to the standard point, ring and disk sources featured in previous investigations.     

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.     

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 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.     

The transport and diffusion theory of a line source in an infinite half-space with internal reflection

A closed form analytical solution is obtained for a three-dimensional transport theory problem, namely that of a line source in a half-space in one-speed transport theory with an internally reflecting surface. The theory is developed by using Fourier transforms in the transverse directions and a Laplace transform in the axial direction, together with the Wiener–Hopf technique. Both specular and diffuse internal reflection are considered and it is shown that a closed form solution is not available for the specular case but is available for the diffuse case. The surface particle scalar intensity and current are obtained and their sensitivity to absorption and the reflection coefficient are assessed. We also obtain a number of analytic asymptotic estimates for the intensity valid at large distances from the source. Diffusion theory is also employed and compared numerically with the transport solution. In general, diffusion theory gives very satisfactory results within its regions of limitation. We also offer this paper as an exercise in analytical transport theory; an aspect of reactor physics which is not often seen in recent times.     

Rigorous solution to multiregion problems of monoenergetic neutron transport theory

A rigorous semianalytical algorithm is used, in the frame of the integral form of the transport equation, for the solution of some basic multilayer problems of monoenergetic neutron transport theory. The critical problem for a three region reactor is explicitly worked out, and numerical results are presented, in comparison with F_N calculations.     

Normal and anomalous transport across an interface: Monte Carlo and analytical approach

We present a Monte Carlo scheme to simulate particles going across an interface separating two layers of a medium characterized by different physical properties, together with an analytical formulation of the same problem, for both normal diffusive and subdiffusive regimes. We relate the Monte Carlo simulation parameters to the coefficients and boundary conditions appearing in the companion analytical equations. Under suitable physical hypotheses on the constraints to be imposed on such parameters, we show that the Monte Carlo simulation results are in good agreement with the corresponding analytical solutions. In particular, we remark that – while in the normal diffusion case the conservation of particle local velocities across the interface leads to a smoothly varying concentration profile – in the subdiffusive case the same condition leads to a neat jump in resident concentration.     

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.     

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 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.     

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 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 to fuel-and-cladding model of the rewetting of a nuclear fuel rod

An exact solution of the quasi-steady two-dimensional conduction equation for the rewetting of a nuclear fuel rod in water reactor emergency core cooling is obtained for a fuel-and-cladding model. A method of solving non-separable differential equations is presented, which is used in the present analysis. The recently developed theorem of orthogonality of piecewise continuous eigenfunctions is also used to handle the composite rod in the present model. The present analysis reveals that the wet front velocity increases with the increase of the gap resistance between the fuel and the cladding, and approaches a limiting value, which is equal to the wet front velocity of the tube of cladding alone, as the gap resistance becomes infinite. For convenience in practical application, the results of the present analysis are correlated in simple expressions.     

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 exact analytical solution of pool swell dynamics during depressurization by the method of characteristics

An exact analytical solution, based on the method of characteristics, has been obtained for the spatial and temporal variation of vapor volumetric (void) fraction in a depressurizing pool. Numerical evaluations have shown that the axial void profile is strongly dependent on the drift velocity formulation, and that wall heat flux plays only a minor role in the pool swell transient.     

An analytical solution to a two-dimensional model of the rewetting of a hot dry rod

The model considers a hot dry rod of infinite length cooled by a film of liquid moving along its surface. The heat transfer coefficient is assumed to be constant on the wet side and zero on the dry side of the rewetting front, and the liquid film is assumed to move at constant speed. We derive an analytical formula relating the temperature difference in the rod, the temperature at the rewetting front, the wet side heat transfer coefficient, and the rewetting speed. The formula is thought to apply to the rewetting of a fuel rod during emergency cooling by flooding.     

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.