An improved mathematical model for a void fraction problem in nondestructive testing

Abstract

The aim of this article is to describe an improved mathematical model for the basic problem of evaluating the void fraction of a slab material by using the responses of a collimated detector from a typical neutron source-detector system. In contrast to previous work, the model reported here: (i) is able to treat explicitly and exactly a parallel neutron beam with normal incidence; (ii) allows for an arbitrary order of anisotropy of the neutron scattering phase function and (iii) provides us with a void fraction evaluation method that is free from computer overflow exceptions even for angular quadrature sets of high order and arbitrarily thick slabs. We perform a numerical experiment in order to illustrate the positive features of the mathematical model and the void fraction evaluation method reported here, and we conclude this article with closing remarks and directions for future work.

On the development of computational tools for the design of beam assemblies for boron neutron capture therapy

“…1-3, apart from computational finite arithmetic considerations and regardless of R, N and the dimensions of and materials for the layers. The ESGF method was developed by the present author in recent years, and has found application to more general radiation transport problems [21][22][23][24][25]. The basic equations of the ESGF method are the N boundary Eqs.…”

Material void fraction evaluation with mixed neutron beams: theory and numerical experiment

“…In addition, we have labelled the directions μ m so that μ m > 0 holds for m = 1 : N/2, μ m < 0 holds for m = N/2+1 : N, μ m-1 < μ m , m = 2 : N/2, and μ m+N/2 = -μ m , m = 1 : N/2. We remark that the nonnegative integer L in equations (11) means that the Legendre expansion of the scattering phase function has been truncated after (L+1) terms. Equations (11) are constrained by the discrete boundary conditions…”

“…However, the ZB model displays some theoretical/practical flaws due to the oversimplifying assumptions made in the design of its ingredients-the S N formulation and the SGF method. To do away with some of these flaws, we have recently developed and reported on an improved mathematical model [11], which is based on a more accurate S N formulation and on some analytic and numerical schemes adapted from a hybrid analytic/numerical method developed by the present author for more general radiation transport problems [12,13]. The testing equipment associated with this improved model differs from the one in BARROS et al [2] only in the beam shape.…”

“…The testing equipment associated with this improved model differs from the one in BARROS et al [2] only in the beam shape. In our earlier work [11], the neutron beam is treated more realistically than before: it is monodirectional and normally incident upon one side of the test slab. In contrast to the ZB model, the model described in DE ABREU [11]: i) is able to treat explicitly and exactly a monodirectional neutron beam with normal incidence; ii) allows for an arbitrary (Legendre) order of anisotropy of the neutron scattering phase function; and iii) provides us with a void fraction evaluation method that is free from computer overflow exceptions even for angular quadrature sets of high order and arbitrarily thick slabs.…”

“…In this article we take a step further, and while keeping the positive features (ii) and (iii) above, we incorporate into the model described in DE ABREU [11] the practical case of a mixed beam, i.e. a beam consisting of a monodirectional component with normal incidence upon the slab material (as we have thus far considered) and a component that is a smooth function of the angular variable (not necessarily isotropic).…”