Critical Slab Thickness Research Articles

The Slab Critical Thickness Problem with Reflecting Boundary Condition for the Anlı-Güngör Scattering Function

The criticality equation, which defines the relation between the secondary neutron number and the thickness of the slab, and the numerical solutions of this equation are investigated with reflecting boundary condition for the recently studied the Anlı-Güngör (AG) scattering function. The analytical calculations are performed by HN method. The numerical results are calculated with Wolfram Mathematica software for the varying secondary neutron number, the varying scattering parameter, and the varying reflection coefficient. The critical slab thickness values decrease for increasing reflection coefficient as expected.

Ionic Graphitization of Ultrathin Films of Ionic Compounds.

On the basis of ab initio density functional calculations, we performed a comprehensive investigation of the general graphitization tendency in rocksalt-type structures. In this paper, we determine the critical slab thickness for a range of ionic cubic crystal systems, below which a spontaneous conversion from a cubic to a layered graphitic-like structure occurs. This conversion is driven by surface energy reduction. Using only fundamental parameters of the compounds such as the Allen electronegativity and ionic radius of the metal atom, we also develop an analytical relation to estimate the critical number of layers.

Solving the criticality problem with the reflected boundary condition for the triplet anisotropic scattering with the modified FN method

Abstract One speed, time independent neutron transport equation in slab geometry can be solved with the triplet anisotropic scattering for the criticality problem with the reflected boundary condition. The value of the critical slab thickness is investigated numerically by using the Modified FN (HN) method. Case's eigenfunctions, normalization relation and orthogonality relations must be derived for this scattering in order to use the Modified FN method. Some selected values, which can be calculated from the criticality equation, can be tabulated. Also the term which comes from the Poincaré-Bertrand formula may be added to the orthogonality formula of the continuum eigenfunction.

Spontaneous graphitization of ultrathin cubic structures: a computational study.

Results based on ab initio density functional calculations indicate that cubic diamond, boron nitride, and many other cubic structures including rocksalt share a general graphitization tendency in ultrathin films terminated by close-packed (111) surfaces. Whereas such compounds often show an energy preference for cubic rather than layered atomic arrangements in the bulk, the surface energy of layered systems is commonly lower than that of their cubic counterparts. We determine the critical slab thickness for a range of systems, below which a spontaneous conversion from a cubic to a layered graphitic structure occurs, driven by surface energy reduction in surface-dominated structures.

3D Numerical Simulation of Thawing Frozen Wood Using Microwave Energy: Frequency Effect on the Applicability of the Beer–Lambert Law

In this article, the frequency effect on the applicability of Beer–Lambert's law for thawing frozen wood using the microwave energy was analyzed. To this end, we use Maxwell's equations to determine the absorbed power and characterize the critical slab thickness L crit of three Canadian eastern wood species: trembling aspen (Populus tremuloides Michx), yellow birch (Betula alleghaniensis), and sugar maple (Acer saccharum). The critical thickness L crit above which the Beer–Lambert law is valid is estimated as a hyperbolic function in the frequency domain: L crit = m/f n (f is the frequency of microwave radiation; m and n are adjustment constants). The nonlinear heat conduction problem involving phase changes such as wood freezing is solved by a three-dimensional volumetric specific enthalpy-based finite element method. The dielectric and thermophysical properties are functions of temperature and moisture content. The specific volumetric enthalpy approach is validated by experimental testing. For instance, we studied the frequency effect on the thawing of frozen trembling aspen wood.

TE Volume Modes in Anisotropic Single-Negative Slab with Negative Component in Permeability Tensor

Existence of backward TE volume modes which are to be identified as Magnetostatic Wave (MSW) in anisotropic single-negative slab with partly negative permeability tensor component have already been revealed by present authors. In this paper, detailed modal analysis has been carried out for this kind of TE volume modes to find out their novel and peculiar properties. From these numerical results, it has been clarified that dispersion curve of the lowest order mode for thicker slab has a frequency of turning point below which both forward and backward waves can be simultaneously observed and also there is a critical slab thickness for each order of TE volume modes to exist.

The critical slab problem for linearly anisotropic scattering and reflecting boundary conditions with the HN method

Abstract The HN method is used to calculate the critical slab thickness for a parallel media surrounded bay a reflector for linearly anisotropic scattering. It is shown that the method leads to concise equations and accurate numerical results even in the lowest order of approximation.

Variation of the critical slab thickness with the degree of strongly anisotropic scattering in one-speed neutron transport theory

The critical slab problem is studied in one-speed neutron transport theory using a linearly anisotropic kernel which combines forward and backward scattering. It is shown that, the recently observed non-monotonic variation of the thickness also exists in this strongly anisotropic case. In addition, the influence of the linear anisotropy on the critical thickness is analysed in detail. Numerical analysis for the critical thickness are performed using the spherical harmonics method and results are tabulated for selected illustrative cases as a function of different degrees of anisotropic scattering. Finally, some results are discussed and compared with those already obtained by other methods, the agreement is satisfactory. The spherical harmonic method gives generally accurate results in one dimensional geometry, and it is very suitable for the numerical solution of the neutron transport equation with linearly anisotropic scattering.

Criticality eigenvalues of the one-speed transport equation with strong forward—backward scattering -relationship with rod model

The one—speed neutron transport equation with strong forward and backward scattering (characterized by α and β) has been solved analytically for an infinite slab with a finite order Sn method. It is shown that the solution changes smoothly as c(α + β) passes unity. For a fixed criticality factor, c, a finite thickness is obtained both for α + β = 1/c and = 1. When the number of directions goes to infinity, the critical slab thickness vanishes at c(α + β) = 1. The total flux is then asymptotically given by a cosine function going to zero at the boundary. This is in contrast to the integral equation, which gives a different flux distribution. The results are supported by numerical calculations using the S4, S16 and S96 approximations.

The effect of forward, backward and linear anisotropic scattering on the critical slab thickness

The critical slab problem is studied for, one-speed time independent, plane symmetric linear transport equation, using a scattering kernel which consists of linear combinations of backward, forward and linear anisotropic scattering. The moment equations of P N approximation written for anisotropic scattering kernel are transformed into moment equations for linear anisotropic scattering. This approach is based on the use of scale transformations. Moment solutions are connected with a subgroup of 2 × 2 reel symmetric matrices. Using the solution procedure given for linear anistropic scattering, the contribution of the combination of the linear anisotropy and δ-distributions to the critical slab thickness is obtained up to the P 15-approximation.

Microwave heating: an evaluation of power formulations

Microwave heating of food systems have conventionally been modeled based on a Lambert law formulation of the absorbed power. However this is strictly valid for semi-infinite samples only. The correct power dissipation must be computed from maxwell's equations. To determine the conditions of the approximate applicability of Lambert's law for finite slabs, we have compared it with the microwave heating predicted by Maxwell's equations. We have found that the critical slab thickness L crit (in cm) above which the Lambert law limit is valid can be estimated from L crit = 2.7 β −1 - 0.08, where the penetration depth, β −1, is the distance in cm from the sample surface where the field reduces to 1/ e of its incident intensity. Temperature profiles calculated with the Lambert law limit for slabs thicker than L crit are within 0.5% of those predicted with the power calculated from Maxwell's equations. Using Maxwell's equations we have developed a general formulation for power absorbed in a homogeneous, isotropic multilayered medium exposed to plane waves from both faces. We report temperature profiles obtained by solving the transient heat conduction equation with the microwave power as a source term. Thermal and dielectric properties are assumed to be temperature-independent.

TheF N method for a specified isotropic scattering model

The variation of the critical slab thickness in monoenergetic neutron transport theory is studied. TheFN method is used to solve, for the case of specified isotropic scattering, the critical problem. The results obtained for the critical thickness are tabulated for selected illustrative cases in comparison with results obtained using Case’s method. Numerical results given in the tables show that the critical thickness increases monotonically with increasing forward anisotropic scattering.

The critical slab with the backward-forward-isotropic scattering model

The variation of the critical slab thickness from backward to forward scattering is studied for specified isotropic scattering. Numerical results are obtained from the zeroth and first-order approximate analytical expressions, derived using the method of elementary solutions. It is shown that the critical slab thickness increases monotonically with increasing forward anisotropy.

A fictitious slab with vanishing critical thickness

Abstract In one-speed neutron transport theory, it is shown that a fictitious anisotropic fission and scattering kernel with an extreme forward bias would give a critical slab thickness varying non-monotonically with anisotropy. For this particular kernel, the critical thickness increases first with increasing forward bias but decreases later and vanishes at a pertain critical anisotropy. This is a geometrical effect which may be understood by considering the variation of the angular distribution with effectively increasing multiplicity.

Critical Experiments with Unmoderated Plutonium Oxide

The results and analyses presented are from the latest series of experiments in a continuing program for determining the critical parameters of plutonium mixtures having concentrations typical of wet powders, precipitates, slurries, and polymers. Previous measurements in this program were made on 15 H/Pu fuel having 240Pu isotopic concentrations of 2.2 and 8.08 wt% and on 5 H/Pu fuel having a 240Pu isotopic concentration of 11.46 wt%. This latest series of experiments was conducted with fuel in which the 240Pu isotopic content has been increased to 18.35 wt% and the H/Pu atomic ratio decreased to essentially zero. The minimum critical slab thickness for a water-reflected homogeneous PuO2-water system of this composition was determined to be 2.31 ± 0.06 cm as compared to only 1.15 ± 0.03 cm for Pu-water. Thus, having the plutonium in an oxide form at this degree of moderation results in an increase of about a factor of 2 in the critical thickness. For spherical geometry, the difference in critical mass bet...