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As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solutions, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solutions, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively. [ABSTRACT FROM AUTHOR]

In the present work, a compact, fast, and yet accurate algorithm is developed to calculate the numerical values of the Ambartsumian-Chandrasekhar's H -function for isotropic scattering and its moments on the basis of the double exponential (DE) formula of Takahashi and Mori (RIMS, Kyoto Univ., 9:721, 1974). The main improvement made in the new method is an elimination of the iterative procedure for automatic adjustment of the step-size of integrations carried out with the DE-formula. Instead, a set of optimal values for the upper limit of integration T max and the number of division points N T to specify the step-size of the quadrature is predetermined for calculations of the H -function with a 15-digit accuracy, and also another for the evaluations of the moments with an accuracy of 14-digits or better. FORTRAN90 subroutines HFISCA for the H -function and HFMOMENT for the moments of arbitrary degrees are subsequently constructed (their source codes and a driver together with a sample set of output are shown in Appendix of this paper). Tables of sample calculations of the H -function and its moments of degree −1 through 6 carried out by these programs are also presented. The routines HFISCA and HFMOMENT should prove useful not only in astrophysical applications but also in other disciplines of science such as the electron transports in condensed matter and remote-sensing data analyses. A request for a copy of the Fortran 90 source code of the program can be made by writing to kawabata@rs.kagu.tus.ac.jp. [ABSTRACT FROM AUTHOR]

The slab albedo problem is solved by using HN method. The albedo, transmission coefficient and angular flux are calculated for the isotropic scattering. The behavior of these parameters according to the changing of slab thickness is obtained. The angular flux distributions are good agreement with literature. It was shown that HN method leads to concise equations and to fast converging numerical results. [ABSTRACT FROM AUTHOR]

The nodal methods, as deterministic models, form a class of numerical methods developed to generate accurate numerical solutions of the Boltzmann equation for neutron transport. These methods are algebraically and computationally more laborious than the traditional deterministic fine-mesh methods like the Diamond Difference method ( D D ). However, their numerical solution for traditional coarse-mesh problems is more accurate. For this reason, the nodal methods and their algorithms for direct and iterative solution schemes have been the subject of extensive research. In this paper we propose a simpler methodology for the development of a method of spectral nodal class which is tested as an initial study of the solutions (spectral analysis) of the neutron transport equation in the formulation of discrete ordinates ( S N ), in one-dimensional geometry, multigroup energy approximation, isotropic scattering and considering homogeneous and heterogeneous domains. These results are compared with the traditional fine-mesh D D method and the spectral nodal methods, spectral Green's function ( S G F ) and Response Matrix ( R M ) to test their numerical accuracy, stability and consistency. [ABSTRACT FROM AUTHOR]

A 3-D geometry-based stochastic model (GBSM) based on two cylinders is proposed for high altitude platform (HAP) communication systems. The proposed 3-D GBSM uses the volume of inner cylinder to model isotropic scattering environment, and that between the inner cylinder and the outer cylinder to model non-isotropic scattering environment. In this letter, the time-variant parameters, including distances between the scatterers and receiver, azimuth angle of departure, and elevation angle of departure (EAOD), are investigated for HAP multiple-input multiple-output (MIMO) channel. The proposed model is validated by the spatial correlation function of measurement results, and the numerical results also demonstrate that the proposed 3-D GBSM with time-varying parameters is applicable to describe the 3-D HAP-MIMO channel. [ABSTRACT FROM PUBLISHER]

Applying the mean value theorem for definite integrals in the non-linear integral equation for Chandrasekhar’s H function describing conservative isotropic scattering, we have derived a new, simple analytic approximation for it, with a maximal relative error below 2.5%. With this new function as a starting-point, after a single iteration in the corresponding integral equation, we have obtained a new, highly accurate analytic approximation for the H function. As its maximal relative error is below 0.07%, it significantly surpasses the accuracy of other analytic approximations.

Radiative heat transfer is investigated for each of isotropic and backward scattering patterns to evaluate the radiative thermal conduction of vacuum insulation panels (VIPs). As a means of analysis, a new statistical formulation is proposed and the results are compared with those of diffusion approximation, Monte Carlo method and discrete ordinate interpolation method (DOIM). The results of the statistical formulation agree well with those of Monte Carlo method and DOIM within 1% error, so that it may be regarded as an exact solution method. The results show that the insulation performance of the VIP can be enhanced by increasing the optical thickness or decreasing the wall emissivity. Also, the backward scattering is about twice effective than the isotropic scattering. [ABSTRACT FROM AUTHOR]

In this paper, we implement and verify an algorithm based on the eigendecomposition method to solve the one-dimensional mono-energetic transport equation in a semi-infinite medium with isotropic scattering. Using our algorithm, we produce approximate closed-form expressions for the solution. Moreover, the algorithm does not utilize a Laplace transform, offers reasonable accuracy, and displays rapid convergence using a relatively small number of quadrature points. [ABSTRACT FROM AUTHOR]

Abstract: The exact 3-D radiative integral transfer equations (RITEs) for a rectangular absorbing, emitting and isotropically scattering, vertically segregated two-layered media are solved. The subtraction of singularity technique is used to handle singularities and to obtain the numerical solution of the RITEs. The surface and the volume integrals that arise in this method are evaluated analytically to handle the singular integrals and to reduce computational time. The enclosure is composed of two vertically-stacked layers in which the medium properties are discontinuous at the interface. The numerical integrations are carried out using sixth order Newton–Cotes quadratures along with the composite rule which assures highly accurate integration. In the numerical simulations, the optical thickness of layer-1 (τ zp =0.25, 0.5, 0.75mfp) and the medium properties of both layers are varied over a wide range, and the net partial radiative heat fluxes and the incident energy distributions are obtained along the vertical centerline and specified boundary lines of the enclosures for the extinction ratios of β r =0.25 and 4. For selected enclosures and medium, five-significant-digit accurate exact solutions for the incident energy and the partial heat fluxes are also provided in tabular forms for benchmarking purposes. [Copyright &y& Elsevier]

Highlights: [•] Radiative transfer in a pure scattering medium is analyzed with a Monte Carlo method. [•] The results for isotropic scattering case agree well with the diffusion solution. [•] The effects of radiation shields and opacifiers in VIPs have been verified. [•] The radiative thermal conductivity is reduced for the backward scattering medium. [ABSTRACT FROM AUTHOR]

A realistic fading channel simulator (FCS) is a key component for the development and faithful performance evaluation of wireless communication systems. This brief presents an efficient implementation of an FCS on a single field-programmable gate array. The FCS utilizes a configurable filter processor along with a multistage interpolator for a compact implementation. The FCS can be parameterized to accurately reproduce the statistical properties of both isotropic and nonisotropic scattering scenarios. [ABSTRACT FROM PUBLISHER]

New developments in the program package ATSAS (version 2.4) for the processing and analysis of isotropic small-angle X-ray and neutron scattering data are described. They include (i) multiplatform data manipulation and display tools, (ii) programs for automated data processing and calculation of overall parameters, (iii) improved usage of high- and low-resolution models from other structural methods, (iv) new algorithms to build three-dimensional models from weakly interacting oligomeric systems and complexes, and (v) enhanced tools to analyse data from mixtures and flexible systems. The new ATSAS release includes installers for current major platforms (Windows, Linux and Mac OSX) and provides improved indexed user documentation. The web-related developments, including a user discussion forum and a widened online access to run ATSAS programs, are also presented. [ABSTRACT FROM AUTHOR]

In this paper uniform convergences of the two-dimensional exponential integral functions are investigated. We also study the continuity, integrability and asymptotic behaviour of these functions. [ABSTRACT FROM AUTHOR]

Abstract: The ray tracing-node analyzing method (RTNAM) has been successfully developed to solve 1-D coupled heat transfer in isotropic and anisotropic scattering media in the past, and in this paper it is further extended to solving the 2-D coupled heat transfer in a rectangular isotropic scattering medium. Using the control-volume method, the partial transient energy equation is discretized in implicit scheme. The effect of radiation on heat transfer is considered as a radiative source term (RST) in the discretized energy equation, and in combination with spectral band model, the RST is calculated using the radiative transfer coefficients (RTCs), which are deduced by the ray tracing method. The Patankar’s linearization method is used to linearize the RST and the opaque boundary condition, and the linearized equations are solved by the ADI method. Before solving the RTCs for isotropic scattering media, the RTCs without considering scattering must be solved at first. And then, the RTCs without considering scattering are normalized according to their integrality relationships. In addition, the correctness of the results obtained by the RTNAM is validated, and effects of scattering albedo and refractive index on transient temperature distribution are investigated. [Copyright &y& Elsevier]

The present study addresses the mathematical modelling aspects of transport phenomena in steady, two-dimensional, laminar flow accompanied by heat transfer in a differentially heated cavity with moving walls in the presence of radiatively absorbing, emitting, and scattering grey medium. The discrete-ordinate method has been used in modelling the radiative transport equation, and finite-difference method based on finite-volume approach is used as the discretization technique. Two cases of opposing mixed convection arising out of vertical isothermal wall movement and horizontal adiabatic wall movement and their interaction with radiation have been critically examined. Buoyancy-opposed mixed convection interacting with radiation has been given emphasis in the present study. The effect of Richardson's number (i.e. Ri from 0.1 to 10) on fluid flow and energy transfer has been studied. In the presence of radiation, flow reversal is found to be one of the revealing characteristic features. [ABSTRACT FROM AUTHOR]

Abstract: The present numerical and experimental analysis addresses coupled conduction and radiation heat transfer (CCR) in differentially heated vertical isothermal walls and horizontal insulated walls of a square enclosure with absorbing, emitting and isotropic scattering participating gray medium. The P1 approximation solution is utilized as the input signal to the neuron model. The computational domain is treated by the neural-finite difference method (NFDM) with ray tracing technique of ray emission model (REM) for the development of improved differential approximation (IDA). The output results are validated with the results of DOM. The practical implementation of IDA for wide range of radiative parameters are illustrated and examined. Experiments have been performed in a square enclosure with solid isothermal walls made of aluminum and insulated walls with bakelite, thus forming air filled cavity. Finally, the consistence of isotherm pattern of the numerical work with the interferogram captured by Mach–Zehnder interferometer corroborates the IDA theory and its realistic approach. [Copyright &y& Elsevier]

In this paper, we show that the fractional transport equation can be reduced to a fractional linear differential equation system by using Chebyshev polynomials. We give the analytical solution of this system followed by the spectral approximation. [ABSTRACT FROM AUTHOR]

This study numerically simulates the transport phenomena in steady, two-dimensional, and laminar flow accompanied by heat transfer in a partially heated and partially cooled cavity in the presence of radiatively absorbing, emitting, and scattering gray medium. A portion of the left and right vertical walls is considered as the heater and cooler with constant temperature, respectively, whereas the rest of respective vertical wall is considered to be adiabatic. The horizontal walls are considered adiabatic, too. The discrete ordinate method has been used in modelling the radiative transfer equation followed with finite-volume method as discretization technique. The walls are considered opaque, diffusive, and gray. The effect of radiation, locations of both heater and cooler on fluid flow, and heat transfer has been studied. [ABSTRACT FROM AUTHOR]

Abstract: Passive diagnostics offer new ways of obtaining real-time data for the control and modeling of industrial furnaces. It has been proposed elsewhere that from the intensity profile between 3.8 and 4.7μm one may derive the temperature of a gas–particle medium and the particle emissivity (ε p) at 3.95μm. This technique applies to large columns of combustion products with enough CO2. The temperature is retrieved by finding the best fit between Planck''s function and the intensity profile between 4.56 and 4.7μm, which is that of a blackbody due to CO2 saturation. Here we consider the effect of particles on the intensity profile and, therefore, on the retrieved temperature and particle emissivity. We derive an analytic approximation of the effective emissivity for an optically thick gas–particle mixture that includes emission and absorption due to particles and gases, along with isotropic particle scattering. The derivation follows the method of embedded invariance and has been used already for particle-only clouds. It yields a spectral solution that is applicable in other infrared regions where gas and particle optical thicknesses are large. A key parameter (χ) is the ratio of the gas absorption coefficient to the particle extinction coefficient. For χ=1 and ε p=0.5, particle effects decrease the gas band profile by 5% from that of a blackbody. For χ<1 and ε p<0.5, particle effects on the calculated temperature and particle emissivity are noticeable and particle effects should be considered. If χ is known, an iterative procedure may be used to calculate temperature and particle emissivity. We illustrate this procedure with data from a coal-fired boiler. Accounting for particle effects, temperatures were 4% higher (at about 1500K) and particle emissivities 28% lower (for ε p within 0.3–05) than without considering these effects. [Copyright &y& Elsevier]

Abstract: Three-dimensional steady-state radiative integral transfer equations (RITEs) for a cubic absorbing and isotropically scattering homogeneous medium are solved using the method of “subtraction of singularity”. Surface integrals and volume integrals are carried out analytically to eliminate singularities, to assure highly accurate solutions, and to reduce the computational time. The resulting system of linear equations for the incident energy is solved iteratively. Six benchmark problems for cold participating media subjected to various combinations of externally uniform/non-uniform diffuse radiation loads are considered. The solutions for the incident energy and the net heat flux components are given in tabular form for scattering albedos of ω=0, 0.5 and 1. [Copyright &y& Elsevier]

This paper' deals with the modeling, analysis, and simulation of multiple-input multiple-output (MIMO) narrowband fading channels for mobile-to-mobile communications. A stochastic MIMO mobile-to-mobile reference channel model is derived from the geometrical two-ring scattering model under the assumption that both the transmitter and the receiver are surrounded by an infinite number of local scatterers. Using a wave propagation model, the complex channel gains are derived and their statistical properties are studied. General analytical solutions are provided for the three-dimensional (3-D) space-time cross-correlation function (CCF). We show that this function can be expressed as the product of two 2D space-time cross-correlation function (CFs), called the transmit and receive CF. From the non-realizable reference model, a stochastic and a deterministic simulation model are derived using a double sum of complex exponential functions. It is shown how the parameters of the simulation model can be determined for any given distribution of the angle of departure (AOD) and angle of arrival (AOA). In case of isotropic scattering, we present a closed-form solution for the parameter computation problem and illustrate some numerical results concerning the transmit CF. The proposed procedure provides an important framework for designers of future mobile-to-mobile communication systems to verify new transmission concepts employing MIMO techniques under realistic propagation conditions. [ABSTRACT FROM PUBLISHER]

Abstract: We apply the semi-gray model of our previous paper to the particular case of the Earth''s atmosphere, in order to illustrate quantitatively the inverse problem associated with the direct problem we dealt with before. From given climatological values of the atmosphere''s spherical albedo and transmittance for visible radiation, the single-scattering albedo and the optical thickness in the visible are inferred, while the infrared optical thickness is deduced for given global average surface temperature. Eventually, temperature distributions in terms of the infrared optical depth will be shown for a terrestrial atmosphere assumed to be semi-gray and, locally, in radiative and thermodynamic equilibrium. [Copyright &y& Elsevier]

Abstract: We solve the radiative transfer equation for a semi-gray planetary atmosphere in radiative equilibrium, in an attempt to define an entirely analytical non-gray model atmosphere of finite optical thickness. The salient feature of the model is that the incident solar radiation is partitioned between two adjacent spectral domains—the “visible” and the “infrared”—in each of which the atmosphere''s (effective) opacity is assumed to be independent of frequency (the semi-gray assumption). We envisage a plane-parallel atmosphere illuminated by a beam of parallel radiation and bounded below by a partially reflecting and emitting ground. The former emits infrared radiation, induced by the absorption of radiation both visible and infrared, deriving from the external irradiation as well as from the emission of the planet''s surface layer. For an atmosphere with given single-scattering albedos and optical thicknesses in both the visible and infrared domains, we compute the temperature at every depth of the atmosphere, as well as the ground''s temperature. [Copyright &y& Elsevier]