Radiative transfer in finite cylindrical media using transformed integral equations

A modified direct integration method is presented to solve three-dimensional radiative transfer in emitting, absorbing and linear-anisotropic scattering finite cylindrical media. This scheme effectively avoids an integral singularity in the coupled Fredholm type integral equations of radiative transfer. The scheme leads to faster and more accurate results, which are needed in combined mode and non-gray problems. The calculated incident radiation and heat fluxes agree well with published results by discrete ordinates method. Using the transformed integral equations, the effects of boundary emission and reflection can also be easily handled.     

Radiative transfer in stochastically inhomogeneous media

New radiative transfer theory is developed for stochastically inhomogeneous scattering media. The three-dimensional shapes and large scale (compared to the mean free path) structures of the media are modeled by stochastic interfaces separating regions of different scattering properties. The small scale fluctuations are characterized by a pair-correlation function. The radiative transfer equation is extended to include individual scattering and propagation probabilities of a ray for each subregion as well as the probability for a ray to cross the interface between two subregions. The propagation probability is found to depend on the entire preceding path of the ray; the present formulation accounts for the two previous scatterings. A new adding/doubling algorithm is developed to solve this problem numerically. Transmission through a cloud layer and backward scattering seem to be particularly sensitive to inhomogeneities.     

Radiative transfer for a cylindrical beam scattered isotropically

Radiation intensities in an isotropically scattering slab irradiated by a cylindrical beam have been obtained by numerical solution of the radiative transport equation in two dimensions. In contrast to the usual one-dimensional solutions, solutions to the two-dimensional case provide a basis for the use of a narrow diagnostic beam to examine optically-thick media. It is shown that measurements of the intensity of radiation scattered from a narrow beam in an optically-thick slab, as well as of the transmitted beam, can be used to deduce the albedo and the optical thickness.     

Radiative transfer in finite participating atmospheric aerosol media

The properties of radiation transfer through a plane-parallel atmospheric aerosol medium has been studied. It has been done by employing Mie theory to calculate the radiation transfer scattering parameters of the medium in the form of extinction, scattering, and absorption efficiencies. Then, the equation of radiative transfer through a plane-parallel atmosphere of aerosol has been solved for partial heat fluxes using two different analytical techniques, namely, the Variational Pomraning -Eddington approximation and Galerkin technique. Average efficiencies over log-normal and modified gamma size distributions are calculated. Therefore, the radiative properties of Carbon, Anthracite, Bituminous, Lignite, and Fly ash have been calculated. The obtained numerical results show very good agreement with each other in addition to the previous published work.     

Radiative transfer in plane media without azimuthal symmetry

A rigorous integral theory is presented in this paper for the solution of radiative heat transfer problems in stratified media, when dependence on the azimuth of the propagating radiation must be taken into account. Anisotropy of scattering and specular and diffuse reflection from the bounding walls are incorporated in the final system of linear integral equations of Fredholm's type. A simple case of physical interest is considered in more detail, and solved explicitly by a constructive technique. Numerical results are reported and briefly discussed.     

Radiative transfer in absorbing, emitting and linearly anisotropic-scattering inhomogeneous cylindrical medium

Radiative integral transfer equations for one-dimensional solid cylinder with absorbing, emitting and linearly anisotropic-scattering inhomogeneous medium were derived by Abulwafa et al. (JQSRT 62 (1999) 755). The anisotropic terms in the integral equations and their results for anisotropic benchmark problems (JQSRT 66 (2000) 487) are inaccurate. In this study, the integral equations for absorbing, emitting and linearly anisotropic-scattering medium are rederived, and the integral equations for one-dimensional solid cylindrical medium are solved. The results are compared with selected cases using the discrete ordinates S₁₆ and the exact solutions available in the literature.     

Radiative transfer in spatially heterogeneous,two-dimensional,anisotropically scattering media

A method is presented for solving the radiative transfer equation for a general anisotropically scattering and emitting medium exposed to arbitrary boundary radiation conditions. The method allows, in principle, for quite arbitrary spatial variability in the scattering and extinction properties of the medium. We formulate the method in the context of 2-dimensional radiative transfer and describe general solution procedures, based on the principles of invariant imbedding, which are applied in the form of doubling algorithms to obtain solutions for optically thick media. Some selected results are shown to demonstrate the versatility of the approach.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

Radiative heat transfer in a cylindrical mixture of non-gray particulates and molecular gases

The interaction between particulate clouds and molecular gases with respect to radiative heat transfer in a one-dimensional cylindrical medium with black walls is investigated. The particulates are assumed to have a non-gray absorption coefficient with a power-law dependence on wavenumber. The absorption coefficient of the molecular gases is described by the exponential wide-band model. Heat transfer rates are found through the evaluation of internal and external total hemispherical emissivities of the particulate cloud and through evaluation of total band absorptances for the molecular gases, modified to account for the interaction with the particles. Simple, closed-form expressions for the total hemispherical emissivities and total band absorptance are also presented, resulting in good accuracy.     

Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous,anisotropically scattering media

Radiative transfer in a three-dimensional rectangular enclosure containing radiatively participating gases and particles is studied using the first- and third-order spherical harmonics approximations. Inhomogeneities in the radiative properties of the medium, as well as in the radiation characteristics of the boundaries, are allowed for. The scattering phase function is represented by the delta-Eddington approximation, and it is assumed to be a function of the location in order to account for density variation of the particles in the medium. Numerical solutions of the model equations are obtained using a finite difference scheme. For the purpose of validating the P₃-approximation, the results are compared with those based on Hottel's zonal method.     

Nonaxisymmetric radiative transfer in inhomogeneous cylindrical media with anisotropic scattering

In this paper, the control volume finite element method (CVFEM) is applied for the first time to solve nonaxisymmetric radiative transfer in inhomogeneous, emitting, absorbing and anisotropic scattering cylindrical media. Mathematical formulations as well as numerical implementation are given and the final discretized equations are based on similar meshes used for convective and conductive heat transfer in computational fluid dynamic analysis. In order to test the efficiency of the developed method, four nonaxisymmetric problems have been examined. Also, the grid dependence and the false scattering of the CVFEM are investigated and compared with the finite volume method and the discrete ordinates interpolation method.     

Radiative transfer in plant leaves

A two-layer model of light scattering and absorption in plant phytoelements is considered, which takes into account absorption of light by pigments and water and light scattering by particles of two types: chloroplasts and air cavities. An elementary light scattering event is described using the Mie theory. Multiple light scattering in a leaf is described within the framework of the theory of radiative transfer. The equation of radiative transfer with a strongly anisotropic phase function is solved using the method of addition of layers and the method of reduction to a medium with effective parameters depending on the propagation direction of light. The spectral dependences of reflection and transmission coefficients are calculated in the visible range as functions of the leaf structure.     

Radiative transfer in statistically heterogeneous mixtures

We solve three radiative-transfer problems in a two-phase medium for a Markov process.     

Radiative transfer in a moving medium

We illustrate first, for a one-dimensional medium, how to generalize the concept of the probability of quantum exit to the case of a moving atmosphere. The transfer of radiation takes place in a two-level atom with total redistribution in frequency at each scattering. The values of the reflection coefficient and of the source function at the surface of a semi-infinite medium with constant properties (especially concerning the velocity gradient) are given. The results are then extended to the three-dimensional case in the Appendix.     

Numerical solution of radiative and conductive heat transfer in concentric spherical and cylindrical media

A new technique is presented to improve the performance of the discrete ordinates method when solving the coupled conduction-radiation problems in spherical and cylindrical media. In this approach the angular derivative term of the discretized one-dimensional radiative transfer equation is derived from an expansion of the radiative intensity on the basis of Chebyshev polynomials. The set of resulting differential equations, obtained by the application of the S_N method, is numerically solved using the boundary value problem with the finite difference algorithm. Results are presented for the different independent parameters. Numerical results obtained using the Chebyshev transform method compare well with the benchmark approximate solutions. Moreover, the new technique can easily be applied to higher-order S_N calculations.     

Radiative transfer calculations in spheres and cylinders

Integral transformation techniques and the F_N method are used to solve, for the case of isotropic scattering, radiative transfer problems in spherical and cylindrical geometries. Numerical results accurate to five or six significant figures are given for selected cases basic to problems with internal heat generation and emitting and diffusely reflecting surfaces.     

Radiative transfer in mercury-sensitized photochemical vapor deposition

We analyze the problem of radiative transfer from a low-pressure mercury lamp inside parallel-plate reactors, which has been recently used to synthesize thin films by 253.7 nm mercury photosensitization of reactive molecules. The model takes into account (i) the hyperfine structure of the 253.7 nm radiation, (ii) Doppler and collision broadening of the absorption-line profile inside the reactor, (iii) self-absorption of the lamp-emission line-profile, (iv) imprisonment of the resonance radiation, (v) reflection by the film deposited on the substrate plate. Realistic experimental conditions are considered, both for parallel-beam and diffuse irradiation from the lamp. The case of finite lateral extension of the slab geometry is also approximated. Experimental parameters to control the photoabsorption efficiency are outlined.     

Radiative transfer in atmospheres with spherical symmetry

A new approximate method to solve the time-independent transfer equation in a system with spherical geometry is presented. No restriction is imposed on the radial variation of the opacity so that the method is applicable even when the opacity is tabulated. The approximation consists in solving the transfer equation only for the outward and inward radial directions and in using the corresponding intensities for the numerical computation of the mean intensity and flux. To obtain the integration weights, we make several hypotheses on the directional dependence of the intensity, taking into account the principal properties of radiation transfer in spherical symmetric systems. We point out the parallelism that exists between this approximation for spherical geometry and a similar one for the plane-parallel case. Results are presented for the conservative and non-conservative cases, and they are compared to those of other authors.