Energy-dependent Problem Research Articles

The effect of stochastic perturbation of fuel distribution on the criticality of a one speed reactor and the development of multi-material multinomial line statistics

The effect of random fuel redistribution on the eigenvalue of a one-speed reactor is investigated. An ensemble of such reactors that are identical to a homogeneous reference critical reactor except for the fissile isotope density distribution is constructed such that it meets a set of well-posed redistribution requirements. The average eigenvalue, , is evaluated when the total fissile loading per ensemble element, or realization, is conserved. The perturbation is proven to increase the reactor criticality on average when it is uniformly distributed. The various causes of the change in reactivity, and their relative effects are identified and ranked. From this, a path towards identifying the causes. and relative effects of reactivity fluctuations for the energy dependent problem is pointed to. The perturbation method of using multinomial distributions for representing the perturbed reactor is developed. This method has some advantages that can be of use in other stochastic problems. Finally, some of the features of this perturbation problem are related to other techniques that have been used for addressing similar problems.

The effect of stochastic perturbation of fuel distribution on the criticality of a one speed reactor and the development of multi-material multinomial line statistics

The effect of random fuel redistribution on the eigenvalue of a one-speed reactor is investigated. An ensemble of such reactors that are identical to a homogeneous reference critical reactor except for the fissile isotope density distribution is constructed such that it meets a set of well-posed redistribution requirements. The average eigenvalue, < k eff>, is evaluated when the total fissile loading per ensemble element, or realization, is conserved. The perturbation is proven to increase the reactor criticality on average when it is uniformly distributed. The various causes of the change in reactivity, and their relative effects are identified and ranked. From this, a path towards identifying the causes, and relative effects of reactivity fluctuations for the energy dependent problem is pointed to. The perturbation method of using multinomial distributions for representing the perturbed reactor is developed. This method has some advantages that can be of use in other stochastic problems. Finally, some of the features of this perturbation problem are related to other techniques that have been used for addressing similar problems.

On the Energy-Dependent Neutron Transport Problem with Arbitrary Geometry

This paper is concerned with the stationary neutron transport problem which describes the transport of energy-dependent anisotropic scattered neutrons in a one, two, or three-dimensional inhomogeneous medium of arbitrary convex shape containing distributed source and incident neutrons. The approach to the problem is by successive approximation which leads to the existence of a unique solution as well as a recursion formula for the determination of the solution and error estimates for the approximations. In addition to the energy-dependent problem, a discussion is given to the constant-energy neutron transport and radiative transfer problems, including a slab problem. Some recursion formulas for calculating the solutions are obtained. All the formulas obtained in this paper involve only straightforward integration which offers a promising possibility for the calculation of numerical results by using a computer. A brief discussion on the calculation of approximate solutions and their errors is included.

Numerical Solution of the Linear Integral Boltzmann Equation

An iterative method is proposed for solving the homogeneous (i.e., critical) or inhomogeneous (i.e., source) linear irtegral Boltzmann equation for general geometry. By using successive approximations, these two classes of problems are shown to be mathematically equivalent. For the homogeneous problem, constraints on the algorithm regarding the existence of eigenvalues and the initial approximation are investigated. The algorithm is as the elementary interaction causing the pinning). For higher defect densities, the agreement with the experiments on niobium is better than with the previous theory. This method of correlations seemed suitable for studying the effect of cutting-off'' the small elementary intenactions and for the replacement of the Gauss distribution function by the Poisson distribution function for the number of defects in the elementary volumes. Both these give negative results with respect to the experiments; so far wrte are therefore not able to explain quartitatively the large please delete the above 9 lines======= applied to isotropically scattering slabs and spheres, and is compared with previously published results as well as an independent extrapolation method. For the inhomogeneous problem, an improvement over the normal successive collision method via the use of a Neumann series expansion is used to allow economic parametric studies. Constraintsmore » on the algorithm and methods of efficiently terminating the infinite Neumann series are investigated. The solution via the proposed method as applied to isotropically scattering slabs and spheres is provided in a compact form for a range of multiplication factors and optical dimensions. The shape of the scalar flux distribution is expiained. Extensions of the method to more complex problems are outlined; in particular, the solution to an energy-dependent problem in general geometry is obtained, and the implications of the results are discussed. (7 figures, 2 tables) (auth)« less

The thermal-neutron Milne problem with capture

A variational principle devised for the Milne problem without capture has been extended to include capture. The theory is at present restricted to isotropic scattering. The flux in the energy-dependent problem is expressed in terms of the corresponding one-velocity solution having the same asymptotic spatial dependence and the one-velocity half-space Green function for an isotropic plane source. Then a variational principle for a quantity related to the extrapolation length may be written down and the emergent angular distribution determined by iterating the trial function. Results are presented for some simple scattering models.

The thermal-neutron Milne problem

A variational principle devised originally for the isotropic Milne problem without capture has been developed for anisotropic scattering. The theory is described for linear anisotropy but the extension to higher-order anisotropy is, in principle, straightforward. The flux in the energy-dependent problem is expressed in terms of the corresponding one-velocity solution which is already well known; as a result some new one-velocity half-space Green functions are introduced. Then a variational principle for the extrapolation length may be written down and the angular distribution of emergent neutrons determined by iterating the trial function. Results are presented for both isotropic and anisotropic scattering models. Using linearly anisotropic scattering, there is excellent agreement with experiment, except at grazing incidence, for the spectrum emerging from a water surface.