Correlation of Measured and Computed Radiation Intensity Exiting a Packed Bed

1996 ◽  
Vol 118 (1) ◽  
pp. 94-102 ◽  
Author(s):  
P. D. Jones ◽  
D. G. McLeod ◽  
D. E. Dorai-Raj
Keyword(s):  
Radiative Transfer ◽  
Radiation Intensity ◽  
Transfer Equation ◽  
Scattering Theory ◽  
Radiative Transfer Equation ◽  
Packed Bed ◽  
Radiative Properties ◽  
Discrete Ordinates ◽  
Measured Intensity ◽  
Participating Media

The spectral and directional distribution of radiation intensity is measured, using a direct radiometric technique, at the exposed boundary of a packed bed of stainless steel spheres. The purpose of these measurements is to provide an experimental data base of radiation intensity with which to correlate intensity field solutions of the radiative transfer equation in participating media. The bed is considered to be one-dimensional, is optically thick, and has measured constant-temperature boundary conditions. Intensity exiting the bed is numerically simulated using a discrete ordinates solution to the radiative transfer equation, with combined mode radiation-conduction solution of the coupled energy conservation equation. Radiative properties for the bed are computed using the large size parameter correlated scattering theory derived by Kamiuto from the general theory of dependent scattering by Tien and others. The measured intensity results show good agreement with computed results in near-normal directions, though agreement in near-grazing directions is poor. This suggests that either radiative transfer near the boundaries of this medium might not be adequately represented by a continuous form of the radiative transfer equation, or that the properties derived from correlated scattering theory are insufficient. In either case, development of a more detailed radiation model for spherical packed beds appears warranted.

A General Closure Model for the Time-Averaged Radiative Transfer Equation in Turbulent Flows

2010 ◽  
Author(s):  
Pedro J. Coelho
Keyword(s):  
Radiative Transfer ◽  
Turbulent Flows ◽  
Radiation Intensity ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Mixture Fraction ◽  
Joint Probability ◽  
Turbulent Fluctuations ◽  
Proposed Model ◽  
Local Properties

The time-averaged form of the radiative transfer equation (RTE) includes emission and absorption correlations that need to be modeled. There is no general formulation to estimate the absorption coefficient-radiation intensity correlation, which is generally neglected (optically thin fluctuation approximation–OTFA). Here, a model to compute this correlation, as well as the other correlations in the time-averaged form of the RTE, is described. The formulation is based on the solution of two additional differential equations. The unclosed correlations in these equations are estimated assuming that the joint probability density function (pdf) of the radiation intensity and mixture fraction is a two-dimensional clipped Gaussian distribution. The model is applied to a turbulent jet diffusion flame, and a preliminary assessment of the model is reported. It is shown that fluctuations of the radiation intensity, caused by turbulence, imply the existence of a correlation between the radiation intensity and local properties. The assumption of the shape of the joint pdf of mixture fraction and radiation intensity yields satisfactory predictions if the turbulent fluctuations are moderate, but becomes inaccurate near the flame edge where turbulent fluctuations are very large. Nevertheless, the present results suggest that the proposed model may yield better predictions than the OTFA.

An Efficient Sparse Finite Element Solver for the Radiative Transfer Equation

2009 ◽  
Author(s):  
Gisela Widmer
Keyword(s):  
Linear System ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Degrees Of Freedom ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Five Dimensions ◽  
Computational Costs ◽  
Dimensional Domain

The stationary monochromatic radiative transfer equation (RTE) is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For non-scattering radiative transfer, sparse finite elements [1, 2] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared to the discrete ordinates method, they make it possible to significantly reduce the number of degrees of freedom N in the discretization with almost no loss of accuracy. However, using a direct solver to solve the resulting linear system requires O(N3) operations. In this paper, an efficient solver based on the conjugate gradient method (CG) with a subspace correction preconditioner is presented. Numerical experiments show that the linear system can be solved at computational costs that are nearly proportional to the number of degrees of freedom N in the discretization.

Propagation of Ultra-Short-Pulse Radiation in Participating Media: A Laguerre-Galerkin Solution

2007 ◽  
Author(s):  
Tuba Okutucu ◽  
Yaman Yener
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Transfer Problem ◽  
Short Pulse ◽  
Optical Tomography ◽  
Time Derivative ◽  
Product Analysis ◽  
Participating Media ◽  
Radiative Transfer Problem

Transient analysis of the radiative transfer problem in participating media has become essential due to the recent applications involving extremely small time scales. In classical radiation problems, the time derivative term in the radiative transfer equation has a negligible order of magnitude compared to the others. Lasers of pico- to femtosecond pulse durations are now being used to investigate the properties of scattering and absorbing media in such applications as, optical tomography, combustion product analysis, and remote sensing. For such applications, the time derivative in the radiative transfer equation can no longer be neglected. Numerous approaches such as, integral formulation, direct numerical approach, discrete ordinates method, Monte Carlo simulations, and Galerkin technique have been introduced for the solution of transient radiative transfer problems in participating media. In the present work, Laguerre-Galerkin solutions for both rectangular and Gaussian incident pulse profiles are presented.

Solution of the Radiative Transfer Equation in Three-Dimensional Participating Media Using a Hybrid Discrete Ordinates: Spherical Harmonics Method

2012 ◽  
Vol 134 (11) ◽  
Author(s):  
Maathangi Sankar ◽  
Sandip Mazumder
Keyword(s):  
Monte Carlo ◽  
Radiative Transfer ◽  
Spherical Harmonics ◽  
Hybrid Method ◽  
Transfer Equation ◽  
Large Range ◽  
Radiative Transfer Equation ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Medium Component

In this article, a new hybrid solution to the radiative transfer equation (RTE) is proposed. Following the modified differential approximation (MDA), the radiation intensity is first split into two components: a “wall” component, and a “medium” component. Traditionally, the wall component is determined using a viewfactor-based surface-to-surface exchange formulation, while the medium component is determined by invoking the first-order spherical harmonics (P1) approximation. Recent studies have shown that although the MDA approach is accurate over a large range of optical thicknesses, it is prohibitive for complex three-dimensional geometry with obstructions, both from a computational efficiency as well as memory standpoint. The inefficiency stems from the use of the viewfactor-based approach for determination of the wall-emitted component. In this work, instead, the wall component is determined directly using the control angle discrete ordinates method (CADOM). The new hybrid method was validated for both two-dimensional (2D) and three-dimensional (3D) geometries against benchmark Monte Carlo results for gray media in which the optical thickness was varied over a large range. In all cases, the accuracy of the hybrid method was found to be within a few percent of Monte Carlo results, and comparable to the solutions of the RTE obtained directly using CADOM. Finally, the new hybrid method was explored for 3D nongray media in the presence of reflecting walls and various scattering albedos. As a noteworthy advantage, irrespective of the conditions used, it was always found to be computationally more efficient than standalone CADOM and up to 15 times more efficient than standalone CADOM for optically thick media with strong scattering.

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