Integro-differential Radiative Transfer Equation Research Articles

Unique solvability of a stationary radiative–conductive heat transfer problem in a semitransparent body with absolutely black inclusions

We consider a stationary boundary value problem describing a radiative–conductive heat transfer in a semitransparent body with absolutely black inclusions. To describe the radiative transfer, the integro-differential radiative transfer equation is used. We do not take into account the dependence of the radiation intensity and the properties of semitransparent materials on the radiation frequency. We proved at the first time the unique solvability of this problem. Besides, we proved the comparison theorems and established the results on improving the properties of solutions with increasing exponents of data summability.

Space-angle discontinuous Galerkin method for radiative transfer between concentric cylinders

The integro-differential radiative transfer equation (RTE) for concentric cylinders problem involving scattering, absorption and emission is solved using the discontinuous Galerkin (DG) finite element method (FEM). The space-angle DG method directly solves the cylindrically-symmetric RTE as a three-dimensional problem, where a 1D spatially domain in radial distance r is twice extruded in the cosine of polar angle (μ) and the difference in azimuthal angle (φ˜) directions. Thus, the method has a higher accuracy than hybrid FEM-Discrete Ordinate (SN) and FEM-Spherical Harmonic (PN) methods. This is reflected by numerically verified convergence rate of p+1 for smooth problems and space-angle polynomial interpolation order of p. The axisymmetric RTE formulation is more complicated than the plane-parallel formulation, for having two independent angle directions (μ and φ˜) and an extra derivative term with respect to φ˜ in the differential equation. This results in a complex characteristic structure in r−φ˜ plane with strong discontinuity lines in radiation intensity I. A method of characteristics is formulated and implemented to verify the DG formulation and demonstrate its accuracy when such strong discontinuities persist in the solution, specifically when there are no scattering and absorption terms. The relaxation of inter-element continuity constraint of continuous FEMs by this DG method implies its superiority in numerically capturing such discontinuities. Finally, a benchmark problem pertained to heat radiation in a gray gas and another one with nonzero phase function demonstrate the effectiveness of the method in modeling black-body and scattering angular integration terms.

Multi-hp adaptive discontinuous Galerkin methods for simplified PN approximations of 3D radiative transfer in non-gray media

In this paper we present a multi-hp adaptive discontinuous Galerkin method for 3D simplified PN approximations of radiative transfer in non-gray media capable of reaching accuracies superior to most of methods in the literature. The simplified PN models are a set of differential equations derived based on asymptotic expansions for the integro-differential radiative transfer equation. In a non-gray media the optical spectrum is divided into a finite set of bands with constant absorption coefficients and the simplified PN approximations are solved for each band in the spectrum. At high temperature, boundary layers with different magnitudes occur for each wavelength in the spectrum and developing a numerical solver to accurately capture them is challenging for the conventional finite element methods. Here we propose a class of high-order adaptive discontinuous Galerkin methods using space error estimators. The proposed method is able to solve problems where 3D meshes contain finite elements of different kind with the number of equations and polynomial orders of approximation varying locally on the finite element edges, faces, and interiors. The proposed method has also the potential to perform both isotropic and anisotropic adaptation for each band in the optical spectrum. Several numerical results are presented to illustrate the performance of the proposed method for 3D radiative simulations. The computed results confirm its capability to solve 3D simplified PN approximations of radiative transfer in non-gray media.

Numerical study of 3-D microscale heat transfer of a thin diamond slab under fix and moving laser heating

Laser heating is one of the most practical operations in the field of solid circuit production and thin condensed film treatment. The correct prediction of the heat propagation and flux into the micro/nanothin slab under laser heating has high practical importance. Many theoretical and numerical investigations have been performed for analysis of micro/nanoheat conduction based on one or 2-D approximations. For moving laser heating of thin films, with asymmetric paths, the one or 2-D analysis cannot be applied. The most appropriate equation for micro/ nanoheat transfer is the Boltzmann transport equation which predicts the phonon transport, precisely. In the present work, the 3-D microscale heat conduction of a diamond thin slab under fix or moving laser heating at very small time scales has been studied. Hence, the transient 3-D integro-differential equation of phonon radiative transfer has been derived from the Boltzmann equation transport and solved numerically to find the heat flux and temperature of thin slab. Regarding the boundary and interface scattering and the finite relaxation time in the equation of phonon radiative transfer, leads to more precise prediction than conventional Fourier law, especially for moving laser heating.

Unique Solvability of Stationary Radiative-Conductive Heat Transfer Problem in a System of Semitransparent Bodies

We consider the boundary value problem for a system of differential equations consisting of the stationary nonlinear heat equation and the integro-differential radiative transfer equation and describing stationary radiative-conductive heat transfer in a system of semitransparent bodies, taking into account the effects of reflection and refraction of radiation according to the Fresnel laws at the boundaries of bodies. We take into account the dependence of radiation intensity and optical properties of bodies on the radiation frequency. We establish the existence and uniqueness of a weak solution. We prove the comparison theorem and derive a priori estimates for weak solutions and obtain a regularity result.

Nonstationary Radiative Transfer: Evolution of a Spectrum by Multiple Compton Scattering

A new method is used for numerical solution of the nonlinear integro-differential radiative transfer equation for the evolution of a homogeneous emission spectrum owing to Compton scattering on equilibrium free electrons in an infinite uniform space. The temperature of the electron gas is assumed constant with no limits placed on it: the electrons can be both nonrelativistic and relativistic. The evolution of the spectrum is found to depend substantially on the initial dimensionless photon density. There is a bounday value for this density such that at lower values, there is a limiting equilibrium Bose-Einstein photon distribution, but not at higher values. In the latter case a quasi-line develops which shifts to shorter wavelengths with time while its width decreases and its maximum intensity increases. Calculations are carried out using two frequency redistribution functions of the photons, exact and simplified (assuming an isotropic distribution in the laboratory coordinate system). The results are compared with solutions of the Kompaneets equation.

Non-local radiative transfer in strongly inverted masers

Maser transitions are commonly observed in media exhibiting a large range of densities and temperatures. They can be used to obtain information on the dynamics and physical conditions of the observed regions. In order to obtain reliable constraints on the physical conditions prevailing in the masing regions, it is necessary to model the excitation mechanisms of the energy levels of the observed molecules. We present a numerical method that enables us to obtain self-consistent solutions for both the statistical equilibrium and radiative transfer equations. Using the standard maser theory, the method of Short Characteristics is extended to obtain the solution of the integro-differential radiative transfer equation, appropriate to the case of intense masing lines. We have applied our method to the maser lines of the H2O molecule and we compare with the results obtained with a less accurate approach. In the regime of large maser opacities we find large differences in the intensity of the maser lines that could be as high as several orders of magnitude. The comparison between the two methods shows, however, that the effect on the thermal lines is modest. Finally, the effect introduced by rate coefficients on the prediction of H2O masing lines and opacities is discussed, making use of various sets of rate coefficients involving He, o-H2 and p-H2. We find that the masing nature of a line is not affected by the selected collisional rates. However, from one set to the other the modelled line opacities and intensities can vary by up to a factor ~2 and ~10 respectively.

Comparison of discrete ordinate and Monte Carlo simulations of polarized radiative transfer in two coupled slabs with different refractive indices

A comparison is presented of two different methods for polarized radiative transfer in coupled media consisting of two adjacent slabs with different refractive indices, each slab being a stratified medium with no change in optical properties except in the direction of stratification. One of the methods is based on solving the integro-differential radiative transfer equation for the two coupled slabs using the discrete ordinate approximation. The other method is based on probabilistic and statistical concepts and simulates the propagation of polarized light using the Monte Carlo approach. The emphasis is on non-Rayleigh scattering for particles in the Mie regime. Comparisons with benchmark results available for a slab with constant refractive index show that both methods reproduce these benchmark results when the refractive index is set to be the same in the two slabs. Computed results for test cases with coupling (different refractive indices in the two slabs) show that the two methods produce essentially identical results for identical input in terms of absorption and scattering coefficients and scattering phase matrices.

EFFECT OF DUST ON Lyα PHOTON TRANSFER IN AN OPTICALLY THICK HALO

We investigate the effects of dust on Lyα photons emergent from an optically thick medium by solving the integro-differential equation of radiative transfer of resonant photons. To solve the differential equations numerically, we use the weighted essentially non-oscillatory method. Although the effects of dust on radiative transfer are well known, the resonant scattering of Lyα photons makes the problem non-trivial. For instance, if the medium has an optical depth of dust absorption and scattering of τa ≫ 1, τ ≫ 1, and τ ≫ τa, the effective absorption optical depth in a random walk scenario would be equal to . We show, however, that for a resonant scattering at frequency ν0, the effective absorption optical depth would be even larger than τ(ν0). If the cross section of dust scattering and absorption is frequency-independent, the double-peaked structure of the frequency profile given by the resonant scattering is basically dust-independent. That is, dust causes neither narrowing nor widening of the width of the double-peaked profile. One more result is that the timescales of the Lyα photon transfer in an optically thick halo are also basically independent of the dust scattering, even when the scattering is anisotropic. This is because those timescales are mainly determined by the transfer in the frequency space, while dust scattering, either isotropic or anisotropic, does not affect the behavior of the transfer in the frequency space when the cross section of scattering is wavelength-independent. This result does not support the speculation that dust will lead to the smoothing of the brightness distribution of a Lyα photon source with an optically thick halo.

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 ( P 1) 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.

Transient nonlinear optically-thick radiative–convective double-diffusive boundary layers in a Darcian porous medium adjacent to an impulsively started surface: Network simulation solutions

A boundary-layer model is described for the two-dimensional nonlinear transient thermal convection heat and mass transfer in an optically-thick fluid in a Darcian porous medium adjacent to an impulsively started vertical surface, in the presence of significant thermal radiation and buoyancy forces in an (X∗,Y∗,t∗) coordinate system. An algebraic approximation is employed to simplify the integro-differential equation of radiative transfer for unidirectional flux normal to the plate into the boundary-layer regime, by incorporating this flux term in the energy conservation equation. The conservation equations are non-dimensionalized into an (X,Y,T) coordinate system and solved using the Network Simulation Method (NSM), a robust numerical technique which demonstrates high efficiency and accuracy. The transient variation of non-dimensional streamwise velocity component (u) and temperature (T) and concentration (C) functions is computed for various selected values of Stark number (radiation–conduction interaction parameter) and Darcy number. Transient velocity (u) and steady-state local skin friction (τX) are also studied for various thermal Grashof number (Gr), species Grashof number (Gm), Schmidt number (Sc) and Stark number (N) values. These computations for the infinite permeability case (Da→∞) are compared with previous finite difference solutions [Prasad et al. Int J Therm Sci 2007;46(12):1251–8] and shown to be in excellent agreement. An increase in Darcy number is seen to accelerate the flow and boost velocity. A decrease in Stark number (corresponding to an increase in thermal radiation heat transfer contribution) is shown to increase the velocity values. Temperature function is observed to fall in value with a rise in Da and increase with decrease in N (corresponding to an increase in thermal radiation heat transfer contribution). Applications of the study include rocket combustion chambers, astrophysical flows, spacecraft thermal fluid dynamics in debris-laden environments (cosmic dust), heat transfer in forest fire spread, geochemical contamination and ceramic materials processing.

The Effect of Radiation on Flow in a Conical Diffuser

The effect of thermal radiation on the flow in a conical diffuser of half-cone angle 2.5° is studied numerically for constant thermophysical properties. The finite-volume method (FVM) is employed to solve both the Navier-Stokes equation (NSE) and the integro-differential radiative transfer equation (RTE). The effect of radiative parameters, e.g., optical diameter, scattering albedo and wall emissivity at two Reynolds numbers (Re = 500 and 1000) and Pr = 1 has been investigated for constant wall temperature of 1000 K and inlet temperature of 2000 K. The results show that radiation in a transparent or in a purely scattering medium does not affect the flow temperature profile and bulk mean temperature variation, whereas it increases the total Nusselt number; however, radiation in a absorbing-emitting participating medium significantly affects the temperature profile as well as Nusselt number, when compared with the pure convection case.

Coupled phenomenon of opposing mixed convection and radiation in presence of participating medium inside eccentric horizontal cylindrical annulus

In the present investigation, the coupled phenomenon of opposing mixed convection and radiation within differentially-heated eccentric horizontal cylindrical annulus has been numerically simulated. The radiation transfer contributed from the participating medium is obtained by solving the non-linear integro-differential radiative transfer equation using the discrete ordinate method. The participating grey medium is considered to be emitting, absorbing, and isotropically scattering. The walls of the annulus are considered to be opaque, diffuse, and grey. From the present investigation, it is found that substantial changes occur in the isotherms as well as the flow patterns, when the Richardson number is allowed to vary in the range of 0.01–1. The eccentricity of the inner cylinder has been varied adequately to illustrate the effect of same in a focused manner. The influence of radiative parameters on the interaction phenomenon has been delineated through the isotherm and streamline patterns.

Modélisation par éléments finis du chauffage infrarouge des membranes thermoplastiques semi-transparentes

Resume Dans cette etude, nous avons etabli par la methode des elements finis en trois dimensions une approche simplifiee pour analyser l'evolution de la temperature au sein d'une membrane semi-transparente de type polyethylene terephtalate (PET) amorphe, exposees a une source radiative. En se basant sur le comportement optique du PET, un transfert radiatif monodirectionnel, suivant l'epaisseur de la membrane est considere. Des simulations numeriques ont permis de confronter et de valider les resultats obtenus vis-a-vis des resultats analytiques et experimentaux. Comme application, nous avons etudie le chauffage d'une feuille thermoplastique en PET exposee a une source radiative infrarouge.

The contrast and brightness of halos in crystalline clouds

Abstract Numerical calculations of the halo brightness and contrast using the discrete ordinate method of the integro-differential radiative transfer equation solution have been carried out for a typical phase function of crystalline clouds exhibiting halo at 22 and 46°. The dependence of halo brightness and contrast on the cloud optical thickness has been investigated. A simple technique to determine the Ci cloud optical thickness from the halo contrast measurements is proposed.

Mixed convection heat transfer in inclined rectangular ducts with radiation effects

A numerical study was carried out to investigate the radiation effect on the characteristics of the mixed convection fluid flow and heat transfer in inclined ducts. The three-dimensional Navier–Stokes equations and energy equation are solved simultaneously with the vorticity–velocity method. The integro-differential radiative transfer equation was solved by the discrete ordinates method. The effects of the thermal buoyancy and the radiative transfer on the distributions of the bulk fluid temperature, the friction factor and the Nusselt number are emphasized in detail. Results indicate that radiation effects have a considerable impact on the heat transfer and tend to reduce the thermal buoyancy effects. In addition, the development of the bulk fluid temperature is enhanced by the radiation effects.

Combined mixed convection and radiation heat transfer in rectangular ducts rotating about a parallel axis

The study of combined heat transfer of convection and radiation in rectangular ducts rotating in a parallel mode was investigated numerically in detail. The coupled momentum and energy equations are solved by the DuFort–Frankel numerical scheme to examine the interactions of convection with radiation. The integro-differential radiative transfer equation is solved by the discrete ordinates method. Results are presented over a wide range of the governing parameters. The present results reveal that the rotational effect in a square duct is more significant than that in a rectangular one. The predictions also demonstrate that the radiation presents significant effects on the axial distributions of the total Nusselt number, Nu t, and tends to reduce the centrifugal-buoyancy effects. The effect of rotation on the Nu t is restricted in the entrance region, however, the radiation affects the heat transfer through out the channel. Additionally, the Nu t increases with the decrease in the conduction-to-radiation parameter N C.

Mixed convection heat transfer in horizontal rectangular ducts with radiation effects

The study of mixed convection heat transfer in horizontal ducts with radiation effects has been numerically examined in detail. This work is primarily focused on the interaction of the thermal radiation with mixed convection for a gray fluid in rectangular horizontal ducts. The vorticity–velocity method is employed to solve the three-dimensional Navier–Stokes equations and energy equation simultaneously. The integro-differential radiative transfer equation was solved by the discrete ordinates method. The attention of the results is focused on the effects of thermal buoyancy and radiative transfer on the development of temperature, the friction factor and the Nusselt number. Results reveal that radiation effects have a considerable impact on the heat transfer and would reduce the thermal buoyancy effects. Besides, the development of temperature is accelerated by the radiation effects.

Coupled radiation‐convection heat transfer of high‐temperature participating medium in heated/cooled tubes

AbstractThe coupled radiation‐convection heat transfer of high‐temperature participating medium in heated/cooled tubes is investigated numerically. The medium flows in a laminar and fully developed state with a Poiseuille velocity distribution, but the thermal status is developing. By the discrete ordinate method, the nonlinear integrodifferential radiative transfer equation in a cylindrical coordinate form is solved to give the radiative source term in the energy equation of coupled heat transfer. The energy equation is solved by the control volume method. The local Nusselt number and wall heat flux of convection as well as the total wall heat flux are employed to evaluate the influence of radiation heat transfer on convection. The analysis shows that the radiation heat transfer weakens the convection effect, promotes the temperature development, and significantly shortens the tube length with obvious heated/cooled effect. There is an obvious difference between the coupled heat transfer in a heated tube and that in a cooled tube, even though the medium properties are kept constant. The wall emissivity, the medium thermal conductivity and scattering albedo have significant influences on the coupled heat transfer, but the effect of medium scattering phase function is small. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(1): 64–72, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/htj.10137

COMPARISON OF MONTE CARLO TECHNIQUES TO PREDICT THE PROPAGATION OF A COLLIMATED BEAM IN PARTICIPATING MEDIA

Monte Carlo (MC) techniques are the most versatile approaches in solving the integrodifferential radiative transfer equation (RTE). They are based on numerical simulation of propagation of radiant energy within absorbing, emitting, and scattering media [1-4]. MC simulations can easily be applied to multidimensional, nonhomogenous, highly forward-scattering media with time-dependent boundary conditions, where other techniques are almost impossible to implement. On the down side, statistical errors associated with these techniques can be significant if the number of photons accounted for in the simulation is not sufficiently large, yet the computational penalty increases considerably with increasing number of photons. Here, we consider three different MC approaches in solving the RTE in a planeparallel, absorbing and isotropically scattering medium subjected to a collimated light source. The collimated light source is assumed to be an impulse function impinged instantaneously on the upper boundary of the medium. The strength of the light source is not important in the simulations, since all the computed quantities will be normalized accordingly. Time-dependent as well as steady-state cases are considered. Three FORTRAN codes were developed to predict radiative reflectance and transmittance. They are compared in terms of speed of convergence and statistical accuracy.