The applicability of an approximate expression for radiative heating

An analysis considering small departures from radiative equilibrium within a gas, for which radiative heating is approximately formulated in terms of a radiative response time, is compared with an exact solution. It is shown that the approximate formulation does not properly describe local departures from radiative equilibrium, although it is useful in a spatially averaged context.     

Tau method approximation for radiative transfer problems in a slab medium

An approximate method for solving the radiative transfer equation in a slab medium with an isotropic scattering is proposed. The method is based upon constructing the double Legendre series to approximate the required solution using Legendre tau method. The differential and integral expressions which arise in the radiative transfer equation are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. Numerical examples are included to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Evaluation of solution methods for radiative heat transfer in gaseous oxy-fuel combustion environments

The exact solution to radiative heat transfer in combusting flows is not possible analytically due to the complex nature of the integro-differential radiative transfer equation (RTE). Many different approximate solution methods for the solution of the RTE in multi-dimensional problems are available. In this paper, two of the principal methods, the spherical harmonics (P₁) and the discrete ordinates method (DOM) are used to calculate radiation. The radiative properties of the gases are calculated using a non-gray gas full spectrum k-distribution method and a gray method. Analysis of the effects of numerical quadrature in the DOM and its effect on computation time is performed. Results of different radiative property methods are compared with benchmark statistical narrow band (SNB) data for both cases that simulate air combustion and oxy-fuel combustion. For both cases, results of the non-gray full spectrum k-distribution method are in good agreement with the SNB data. In the case of oxy-fuel simulations with high partial pressures of carbon dioxide, use of gray method for the radiative properties may cause errors and should be avoided.     

Least-squares collocation meshless approach for radiative heat transfer in absorbing and scattering media

A least-squares collocation meshless method is employed for solving the radiative heat transfer in absorbing, emitting and scattering media. The least-squares collocation meshless method for radiative transfer is based on the discrete ordinates equation. A moving least-squares approximation is applied to construct the trial functions. Except for the collocation points which are used to construct the trial functions, a number of auxiliary points are also adopted to form the total residuals of the problem. The least-squares technique is used to obtain the solution of the problem by minimizing the summation of residuals of all collocation and auxiliary points. Three numerical examples are studied to illustrate the performance of this new solution method. The numerical results are compared with the other benchmark approximate solutions. By comparison, the results show that the least-squares collocation meshless method is efficient, accurate and stable, and can be used for solving the radiative heat transfer in absorbing, emitting and scattering media.     

Radiative source function for a slab: Approximate solution

Integral equations that govern the radiative source function in a slab are investigated. Closed-form, approximate solutions, which were obtained by using an exponential kernel approximation, are presented. The resulting simple algebraic expressions predict, to within 10%, the exact solution over a wide range of optical parameters. They also provide the advantage of calculating with ease the radiative field within the medium.     

Boundary conditions in the radiative relaxation problem

The problem of the dissipation of temperature perturbations in a finite homogeneous atmosphere is solved for the situation in which the temperature at one boundary is maintained constant (that is, the temperature perturbation is zero for all times) while energy can be freely radiated to space through the other boundary. Exact solutions are shown for the exponential-sum fit to the kernel of the basic integral equation. These solutions constitute the set of radiative eigenfunctions. Also, approximate solutions in terms of the radiative eigenfunctions in the diffusion approximation (one exponential term in the expansion of the kernel) are obtained. These, in turn, are used in the solution of an initial value problem. The constant temperature boundary condition simulates the interface between two regions in one of which the relaxation processes are much more rapid than the purely radiative relaxation of the other.     

Meshless approach for coupled radiative and conductive heat transfer in one-dimensional graded index medium

A meshless local Petrov-Galerkin (MLPG) approach is employed for solving the coupled radiative and conductive heat transfer in a one-dimensional slab with graded index media. The angular distribution term in discrete ordinate equation of radiative transfer within a one-dimensional graded index slab is discretized by a step scheme, and the meshless approach for radiative transfer is based on the discrete ordinate equation. A moving least-squares approximation is used to construct the shape function. Two particular test cases for coupled radiative and conductive heat transfer within a one-dimensional graded index slab are examined to verify this new approximate method. The temperatures and the radiative heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the MLPG approach has a good accuracy in solving the coupled radiative and conductive heat transfer in one-dimensional graded index media.     

Influence of the refractive index on the radiative source function of an isotropically scattering medium

The influence of refractive index on the radiative source function is presented for the case of emission from an isothermal, isotropically scattering medium. A closed-form, approximate solution is obtained and results are presented for finite and semi-infinite nonconservative cases. An increase in refractive index causes the source function to increase and that effect is more pronounced at higher scattering albedo and smaller optical depths.     

Boundary conditions for differential approximations

Low order spherical harmonic (P-N) approximations are applied to a radiative transfer Marshak wave problem. A modified Milne boundary condition is developed for the P-2 approximation, similar to one suggested earlier for the P-1 approximation. Comparison with exact Monte Carlo results suggests that this modified P-2 method may be an accurate and generally applicable differential approximation to the equation of transfer. The Monte Carlo results presented should be useful for testing other approximate formulations of radiative transfer and validating time dependent numerical solution methods for the equation of transfer.     

Near-infrared radiative properties of porous zirconia ceramics

Infrared radiative properties of zirconia ceramics of porosity about 16% are studied by means of the measurements of directional–hemispherical reflectance and transmittance in the wavelength range from 2.5 to 9 μm. The recently suggested modified two-flux approximation is examined as a simplified basis of the identification procedure. A comparison with the exact numerical solution confirms a good accuracy of this approach for identification of the absorption coefficient of ceramics. An analysis of the results for transport scattering coefficient showed that scattering is determined by isotropic pores with characteristic average radius about 1 μm. The corresponding approximate theoretical model of radiative properties of ceramics is suggested. The absorption coefficient of bulk zirconia in the semi-transparency range is obtained from the data for porous zirconia ceramics.     

Fast multilevel radiative transfer

The vast majority of recent advances in the field of numerical radiative transfer relies on approximate operator methods better known in astrophysics as Accelerated Lambda-Iteration (ALI). A superior class of iterative schemes, in term of rates of convergence, such as Gauss-Seidel and successive overrelaxation methods were therefore quite naturally introduced in the field of radiative transfer by Trujillo Bueno and Fabiani Bendicho [A novel iterative scheme for the very fast and accurate solution of non-LTE radiative transfer problems. Astrophys J 1995;455:646]; it was thoroughly described for the non-LTE two-level atom case. We describe hereafter in details how such methods can be generalized when dealing with non-LTE unpolarised radiation transfer with multilevel atomic models, in monodimensional geometry.     

Microstructural characterization using diffuse backscatter: theory and experiment

A microstructure characterization technique is presented which utilizes the azimuthal moments of backscattered intensity to determine the Legendre moments of the microstructure's phase function. The technique is based on a late-time, diffuse approximate solution to the radiative transfer equation. Monte Carlo simulations are presented indicating that the technique is robust for the first azimuthal moment but less so for higher-order moments.     

Potential,problems, and prospects of experimental investigations of superfluidity of heated nuclei

The status of modern experiments aimed at determining level densities and radiative strength functions in complex nuclei from spectra of products of various reactions via recording a cascade or a single photon (nucleon) is analyzed. It is shown that the most precise data for these parameters and, accordingly, the best possibilities for studying the superfluidity of heated nuclei can be obtained only from spectra of two-step reactions, the detection of nucleons or light nuclei at the first step being mandatory.     

Radiation field in a semi-infinite homogeneous atmosphere with internal sources

The equation of radiative transfer in a semi-infinite homogeneous atmosphere with different internal sources is solved by the method of kernel approximation—the kernel in the equation for the Sobolev resolvent function is approximated by a Gauss-Legendre sum. Then the obtained approximate equation can be solved exactly and the solution is a weighted sum of exponentials. All the necessary coefficients of the solutions may be easily found. Since the resolvent function is closely connected with the Green function of the integral radiative transfer equation, the radiation field for different internal sources can be found by simple integration. For the considered cases the formulas for the radiation field are obtained and the respective accuracy estimated. The package of codes in Fortran-77 is given at http://www.aai.ee/∼viik/homogen.for.     

Size- and shape-dependent optical properties of assembled semiconductor spheres

The radiative corrections of nanostructures are investigated by applying a nonlocal theory to a model of semiconductor spheres which are assembled one-, two- and three-dimensionally. An approximate sum rule for radiative corrections and the relation between finite and infinite lattices are given. It is shown that radiative correction strongly depends on the shape as well as the size. For a linear chain, the radiative correction gets saturated when the system length becomes comparable to the wavelength of resonant light. For cubic systems, the radiative width deviates from the size linearity for a surprisingly small size. The two-dimensional case is marginal. This peculiar shape dependence originates from the dipole-dipole interaction.     

Application of the RBF meshless method to the solution of the radiative transport equation

In this paper, we introduce a radial basis function collocation method for computing solutions to the time-dependent radiative transfer equation. For these computations, we use finite differences to discretize the time coordinate, a discrete ordinate method to discretize the directional variable, and an expansion in radial basis functions to approximate the spatial dependence of the solution. The main advantages of the RBF method are that it does not require any mesh or grid, achieves spectral accuracy in multi-dimensions for arbitrary node layouts, and it is extremely simple to implement.     

Radiation field of an optically finite homogeneous atmosphere with internal sources

The equation of radiative transfer in an optically finite homogeneous atmosphere with different internal sources is solved using the method of kernel approximation the essence of which is to approximate the kernel in the equation for the Sobolev resolvent function by a Gauss-Legendre sum. This approximation allows to solve the equation exactly for the resolvent function while the solution is a weighted sum of exponents. Since the resolvent function is closely connected with the Green function of the integral radiative transfer equation, the radiation field for different internal sources can be found by simple integration. In order to simplify the obtained formulas we have defined the x and y functions as the generalization of the well-known Ambarzumian-Chandrasekhar X and Y functions.For some types of internal sources the package of codes in Fortran-77 can be found at http://www.aai.ee/∼viik/HOMOGEN.FOR.     

Transient heat transfer in a spherical protective material, submitted to flux and mixed boundary conditions: an investigation on zirconia

An analytic solution is presented for describing combined radiation and conduction heat transfer in a spherical fiber thermal protection exposed to combined radiative and convective heating. The solution includes the equation of radiative transfer within the material, coupled to a transient energy equation that contains both radiative and convective terms. At elevated temperatures radiative transfer becomes important, and if several hot surfaces view each other, the radiation exchange process must be considered carefully. Some thermal protections are partially transparent to thermal radiation. Hence, an exchange process is complicated by radiation penetrating into and coming out of material. The radiation leaving an area depends on the temperature distribution inside that area and that is unknown and is affected by the exchange process to other areas. The analysis has allowed for unlimited spectral detail but assumes that the various material properties do not vary significantly with temperature. Transient temperature distributions are obtained for the boundary conditions of external radiation and convection. The present analysis includes the influence of reflectivity, surface radiative properties and spectral properties on the temperature distributions.