Solution Of Radiative Transfer Equation Research Articles

Multilayer modeling of radiative transfer by SLW and CW methods in non-isothermal gaseous medium

Non-isothermal gaseous medium is modeled using the multilayer approach, breaking the one-dimensional system into a series of isothermal layers. Spectral integration of the radiative transfer equation (RTE) is performed for the spectral line weighted-sum-of-gray-gases and cumulative wavenumber approaches to modeling the spectral nature of the gas radiation. An exact analytical solution of RTE is obtained for the layers with both black and gray walls. Predictions show high accuracy, even with surprisingly few layers.

Benchmark solutions of radiative transfer equation for three-dimensional rectangular homogeneous media

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.

Altitude and latitude distribution of atmospheric aerosol and water vapor from the narrow-band lunar eclipse photometry

This work contains the description of two narrow IR-bands observational data of the total lunar eclipse of March 3, 2007, one- and two-dimensioned procedures of radiative transfer equation solution. The results of the procedure are the extinction values for atmospheric aerosol and water vapor at different altitudes in the troposphere along the Earth's terminator crossing North America, Arctic, Siberia and South-Eastern Asia. The altitude range and possible latitude and altitude resolution of atmospheric remote sensing by the lunar eclipse observation are described. The results of water vapor retrieval are compared with data of space experiments, and the scale of vertical water vapor distribution is found.

Radiation-Transfer Calculations for the Diffuse Reflectance from Pigmented Coatings

We use a numerical solution of radiation transfer equations to calculate the diffuse reflectance of a pigmented film composed by a transparent pigment embedded in a latex resin. We introduce in the calculation experimentally determined values for the optical absorption of the resin as well as for the particle size distribution of the pigment, and analyze the sensitivity of the diffuse reflectance to the different parameters of the model.

Modeling the radiation of anisotropically scattering media by coupling Mie theory with finite volume method

A new mathematical model and code for radiative heat transfer of particulate media with anisotropic scattering for 2-D rectangular enclosure is developed. The model is based on the coupling of (i) finite volume method for the solution of radiative transfer equation with (ii) Mie equations for the evaluation of scattering phase function. It has not been done before to the authors’ best knowledge. The predictions were compared against the only found results, published 15 years ago. For those results the S-N discrete ordinates method for the solution of radiative transfer equation and the Legendre polynomials expansions for the evaluation of scattering phase function were used. The agreement between the results is very good. The advantages of new model and code are in their straight forward application to any given particles parameters without the need for previously designed analytical expression for scattering phase function. In addition, that analytical expression, with generated expansion coefficients, is restricted and can be used only for that particular case of particle parameters. The new model was applied to the solid particles of several various coals and of an ash and the series of 2-D predictions are performed. The effects of particle size parameter and of scattering albedo on radiative heat flux and on incident radiation were analyzed. It was found that the model developed is reliable and very accurate and thus suitable for extension towards: (i) 3-D geometries, (ii) mixtures of non-gray gases with particles as well as for (iii) incorporation in computational fluid dynamics codes.

Levantamento dos recursos de energia solar no Brasil com o emprego de satélite geoestacionário: o Projeto Swera

O conhecimento da radiação solar incidente na Terra desempenha papel fundamental em muitas atividades humanas como, por exemplo, a agricultura, a arquitetura e o planejamento energético. A radiação solar constitui uma opção limpa e renovável de produção de energia. Para melhor conhecimento da disponibilidade desta fonte de energia e seu aproveitamento, pode-se utilizar modelos computacionais que utilizam técnicas aproximadas na resolução numérica da equação de transferência radiativa para estimar o fluxo de energia solar na superfície. Este trabalho apresenta uma revisão sobre os princípios que estão por trás desses modelos tendo como exemplo aplicado o método do modelo BRASIL-SR. Esse modelo está sendo empregado no mapeamento do potencial energético solar da América Latina através do projeto SWERA (Solar and Wind Energy Resource Assessment), financiado pela Divisão de Ambiente Global (GEF) através do Programa das Nações Unidas para o Ambiente (UNEP).

Radiative transfer in moving media

We describe the solution of the one-dimensional radiative transfer equation using the discontinuous finite element method. Tests of this method prove its reliability for static media. The static solution is then applied to the solution of the transfer equation in a moving medium using a method which we call here the Local Lorentz Transformation (LLT) method. This new method is applicable also to non-monotonic velocity fields. Direct solution of the radiative transfer equation in the moving medium using the discontinuous finite element method is also presented for cases both with and without inclusion of the effect of aberration. All three techniques are compared and they give comparable results.

Domain isolation concept for solution of radiative transfer in large-scale semitransparent media

Due to the volume effect of radiative transfer, theoretically, radiative transfer equation needs to be solved in the entire close cavity containing semitransparent media. In many engineering applications, the solution of radiative transfer equation in the entire cavity for many large systems is complicated and time consuming, but sometimes the interesting zone is only some local small domain. In this paper, a domain isolation approximation of radiative heat transfer is present, in which the local domain is enlarged by an isolation optical thickness and isolated by blackbody boundaries with local temperature, and the radiative transfer equation only needs to be solved in the isolated domains. Two examples are taken to verify this new approximation. The results show that the accuracy of domain isolation approximation is very well when the isolation optical thickness is larger than 2.0.

Numerical solution of radiative transfer equation coupled with nonlinear heat conduction equation

A new way of solving the steady‐state coupled radiative‐conductive problem in semi‐transparent media is proposed. An angular discretization technique is applied in order to express the radiative transfer equation (RTE) in an inhomogeneous system of linear differential equations associated with Dirichlet boundary conditions. The system is solved by a direct method, after diagonalizing the characteristic matrix of the medium. The RTE is coupled with the nonlinear heat conduction equation. A simulation of a real semi‐transparent medium composed of silica fibers is illustrated. Comparison with results of other methods validates the new model. Moreover, the general scheme is easy to code and fast. The algorithm proved to be robust and stable.

Transfer of electromagnetic radiation at a finite angle to the external magnetic field in a collisional plasma with large-scale irregularities

Using Monte Carlo simulations or numerical solution of radiation transfer equation in a random collisional plasma, we analyze the angular spectrum of a normal electromagnetic wave propagating at a finite angle to the external magnetic field. It was found that the anisotropy of absorption in the case of oblique wave propagation with respect to the external magnetic field leads to some specific effects which we call the anomalous growth in fluctuations.

Theoretical study of the role of ultraviolet radiation of thenon-equilibrium plasma in the dynamics of the microwavedischarge in molecular nitrogen

A self-consistent plasmachemical model describing the dynamics of the non-equilibrium microwave discharge in molecular nitrogen with consideration for kinetic, photochemical and electrodynamic phenomena is proposed. The photochemical block of the model accounts for the processes of photoexcitation and photoionization of nitrogen molecules in the ground and excited electronic states. Radiative emittance of the discharge plasma is conditioned by the processes of photorecombination of electrons and positive ions as well as by the processes of spontaneous radiation of electronically excited molecules. Solution of radiation transfer equations and calculation of photochemical constants were made with allowance for the vibrational-rotational structure of the corresponding radiative transitions. The calculations performed have shown that the velocities of the ionization front propagation in the microwave discharge that were observed experimentally may be explained without invoking considerations concerning the existence of easily ionizable admixtures in nitrogen. The phenomenon of electron generation in the photohalo region of the discharge is mainly conditioned by the processes of stepwise photoionization of nitrogen molecules. The mechanisms leading to the development of kinetic instabilities in the discharge region are analysed, and a satisfactory agreement between calculation and experimental results is noted.

Simulated emission spectra of samples at non-uniform temperature based on numerical solutions of the general radiative transfer equation

Infrared spectroscopic methods are increasingly being used for analysis in the food industry. A potentially attractive approach is that of infrared emission spectroscopy: in many food processes, the raw materials are subjected to heating, and once hot, act as sources of infrared radiation. If the emitted radiation is steered directly into a spectrometer with the conventional source removed, an emission spectrum may be obtained of the sample without any further light guiding requirement. However, for bulk samples, the spectra obtained are not very useful, due to the mechanism of self-absorption, whereby chemically different samples give rise to the same, featureless spectral profile. Promising strategies for overcoming this problem are Transient Infrared Emission and Transient Infrared Transmission Spectroscopies (TIRES and TIRTS), in which thermal gradients are induced at the sample surface, enabling composition-specific features to be seen in the emission spectrum. In this paper mathematical models of the effect of thermal gradients on emission spectra are presented, based on numerical solutions of the Radiative Transfer Equation (RTE) for the non-uniform temperature case, using parameters determined experimentally. This provides a method of relating the nature of thermal gradient within a sample to the information content expected in its emission spectrum. Thus, the conditions required to obtain useful emission spectra from bulk samples can be determined, and the potential of emission spectroscopy as an analysis tool in food production processes assessed.

Analytical solution of the non-discretized radiative transfer equation for a slab of finite optical depth

An analytical solution of the radiative transfer equation for a plane parallel slab of finite depth is presented without requiring discretization in space or angles. The solution, expressed in terms of the matrix hyperbolic tangent function, avoids the problem of exponentially increasing terms so that it is also suitable for numerical calculations. By using the resolvent of the transfer equation we obtain simple series expansions for the matrix elements.

The study on evaluation of atmospheric effect based on VTIR and MESSR of artificial satellite MOS-1

Abstract An attempt is made to retrieve aerosol number density based on the thermal infrared data obtained with VTIR (Visible and Thermal Infrared Radiometer) onboard MOS-1 (Marine Observation Satellite 1). It is found that in low moisture condition, aerosol number density can be retrieved from the thermal infrared remote sensing data as one of the maximum likelihood solutions of radiative transfer equation. The application of the method to MESSR data indicates a satisfactory result.

Discrete-ordinates solution of radiative transfer equation in nonaxisymmetric cylindrical enclosures

The S4 discrete-ordinates approximation is used to solve the radiative transfer equation in nonasymmetric (i.e., three-dimensional) cylindrical enclosures containing absorbing-emitting and scattering media, with and without the temperature profile known a priori. Because neither detailed experimental data nor predictions from a zone or Monte-Carlo model for three-dimensional cylindrical enclosures are available, cylindrical equivalents of three-dimensional rectangular enclosures, for which zone model predictions of radiative transfer are available, are used in model evaluation. Limited evaluation of the model shows that the discrete-ordinates method provides acceptable predictions of radiative transfer in nonaxisymmetric cylindrical enclosures. 12 refs.

The Fn method for the one-angle radiative transfer equation applied to plant canopies

Solutions of the radiative transfer equation describing photon interactions with vegetation canopies are important in remote sensing since they provide the canopy reflectance distribution required in the interpretation of satellite-acquired information. Although the most widely used transport models such as the moments formulations collapse the photon directional information into a limited number of directions (usually two or four), the more sophisticated methods such as discrete ordinates can, in principle, employ an unlimited number of discrete directions. These discrete methods, however, also contain spatial discretization error and lack testing against more accurate Numerical formulations for specific vegetation canopy scattering kernels. In this article, we consider a semianalytical approach to the solution of the one-angle radiative transfer equation in slab geometry called the F n method. This method has a its basis two integral equations specifying the intensities exiting the vegetation canopy boundaries. The solution is then obtained through an expansion in a set of basis functions with expansion coefficients to be determined. These coefficients are obtained from a collocation procedure resulting in a set of algebraic equations solved by matrix inversion. The advantage of this method is that only discretization in the angular variable is required, thus avoiding spatial truncation error entirely. The paper begins by considering a canopy where all the leaves are oriented at the same angle. Lambertian scattering will be assumed with unequal leaf reflectance and transmittance. This simple model contains all the difficulties of the more complex model incorporating a general leaf angle distribution which is to be considered in the latter part of the presentation. A sensitivity analysis is performed including variation of the numerical and model parameters. In addition, discrete ordinates calculations as well as field measurements are compared to the F n results.

Modelling of radiation transfer in Earth’s hazy atmosphere and calculation of path radiances in IRS LISS-I Bands

A model (named ATMRAD) of radiation transfer in the Earth’s hazy atmosphere based on simplistic solution of radiation transfer equation and tropical model atmosphere is developed to estimate path radiance for correction of remote sensing multispectral data. The atmosphere is assumed to be cloud free and plane-parallel stratified with various quatities specified for 46 layers in its 100 km assumed thickness and model altitude profiles of haze aerosols specified for 23 km and 5 km visibility ranges. Altitude profiles of haze for any intermediate visibility ranges are deduced from linear mixing of the two profiles for 23 and 5 km visibilities. Deirmendjian size distribution function is assumed for aerosols. The radiation transfer calculations have been performed for 126 wavelengths in 0.25 to 3.0 μm range by considering Rayleigh and Mie Scatterings by air molecules and aerosols; and absorption by atmospheric gases. The model calculations have been carried out for path radiances in bands of IRS LISS-I sensor for different visibility ranges and solar illumination angles.

A model for retrieval of surface spectral reflectance from satellite radiance measurements using realistic atmospheric aerosol profiles

Abstract An effort has been made to retrieve surface spectral reflectance from satellite radiance measurements. For this purpose, a simple cubic equation with surface reflectance as one of its roots has been derived. Besides satellite measured radiance, the other required quantities in the equation are calculated by using the actual solution of radiative transfer equation by a discrete ordinate method, taking into account the inhomogeneous aerosol distribution. The method has been tested for internal consistency by numerical simulations using realistic aerosol profiles. Finally, it has been tested with a Landsat image, giving good agreement with ground observations.

An approximate analytical solution of the radiative transfer equation in hydrometeors accounting for scattering effects

An iterative method for the radiative transfer equation solution is suggested for the scattering hydrometeors. The method allows to find the layer scattering indicatrix. The method is shown to be more general as compared with the four-flux theory or the perturbation method and differs strongly from these ones because even in the case of the first iteration it gives the results which are very close to ones obtained by numerical methods for all rain scattering angles.

Pad� approximant calculations for dispersive, isotropic scattering, homogeneous media

Exact relations for radiation heat flux at the boundaries of a slab with diffusely reflecting boundary conditions and internal source are obtained in terms of the reflection and transmission coefficients of a source free slab with isotropic boundary conditions. The integral equation defining the radiation heat flux contains explicitly the internal source. So, the particular solution for radiative transfer equation is not required. Available exact values for albedos give exact values of radiation heat flux. Pade approximant technique is used to obtain numerical values for homogenous media.