The Weighted-Sum-of-Gray-Gases Model for Arbitrary Solution Methods in Radiative Transfer

1991 ◽  
Vol 113 (3) ◽  
pp. 650-656 ◽  
Author(s):  
M. F. Modest
Keyword(s):  
Radiative Transfer ◽  
Integral Equations ◽  
Solution Method ◽  
Weighted Sum ◽  
Arbitrary Solution ◽  
Absorption Coefficients ◽  
Zonal Method ◽  
Radiation Problem ◽  
Participating Media ◽  
Solution Methods

The weighted-sum-of-gray-gases approach for radiative transfer in nongray participating media, first developed by Hottel in the context of the zonal method, has been shown to be applicable to the general radiative equation of transfer. Within the limits of the weighted-sum-of-gray-gases model (nonscattering media within a black-walled enclosure), any nongray radiation problem can be solved by any desired solution method after replacing the medium by an equivalent small number of gray media with constant absorption coefficients. Some examples are presented for isothermal media and media at radiative equilibrium, using the exact integral equations as well as the popular P-I approximation for the equivalent gray media solutions. The results demonstrate the equivalency of the method with the quadrature of spectral results, as well as the tremendous computer times savings (by a minimum of 95 percent) that are achieved.

A Spectral Line-Based Weighted-Sum-of-Gray-Gases Model for Arbitrary RTE Solvers

1993 ◽  
Vol 115 (4) ◽  
pp. 1004-1012 ◽  
Author(s):  
M. K. Denison ◽  
B. W. Webb
Keyword(s):  
Spectral Line ◽  
Absorption Coefficient ◽  
Cross Section ◽  
Solution Method ◽  
Radiative Transfer Equation ◽  
Weighted Sum ◽  
Absorption Cross ◽  
Arbitrary Solution ◽  
Nonhomogeneous Media ◽  
Cross Section Spectrum

This paper presents an approach for generating weighted-sum-of-gray gases (WSGG) models directly from the line-by-line spectra of H2O. Emphasis is placed on obtaining detailed spectral division among the gray gases. Thus, for a given model spectrum, the gray gas weights are determined as blackbody fractional functions for specific subline spectral regions at all temperatures. The model allows the absorption coefficient to be the basic radiative property rather than a transmissivity or band absorptance, etc., and can be used with any arbitrary solution method for the Radiative Transfer Equation (RTE). A single absorption cross section spectrum is assumed over the entire spatial domain in order to fix the subline spectral regions associated with a single spectral calculation. The error associated with this assumption is evaluated by comparison with line-by-line benchmarks for problems of nonisothermal and nonhomogeneous media.

The Spectral Line-Based Weighted-Sum-of-Gray-Gases Model in Nonisothermal Nonhomogeneous Media

1995 ◽  
Vol 117 (2) ◽  
pp. 359-365 ◽  
Author(s):  
M. K. Denison ◽  
B. W. Webb
Keyword(s):  
Spectral Line ◽  
Cross Sections ◽  
Spatial Dependence ◽  
Radiative Transfer Equation ◽  
Gas Absorption ◽  
Weighted Sum ◽  
Absorption Cross ◽  
Arbitrary Solution ◽  
Nonhomogeneous Media ◽  
Solution Methods

An approach is developed to extend the previously developed spectral-line weighted-sum-of-gray-gases (SLW) model to nonisothermal, nonhomogeneous media. The distinguishing feature of the SLW gas property model is that it has been developed for use in arbitrary solution methods of the radiative transfer equation (RTE). A spatial dependence of the gray gas absorption cross sections on local temperature, pressure, and mole fraction is introduced through the absorption-line blackbody distribution function. Incorporating this spatial dependence results in significant improvement over the use of spatially uniform gray gas absorption cross sections in comparisons with line-byline benchmarks.

scholarly journals The Full-Spectrum Correlated-K Distribution and Its Relationship to the Weighted-Sum-of-Gray-Gases Method

2000 ◽  
Author(s):  
Michael F. Modest ◽  
Hongmei Zhang
Keyword(s):  
Radiative Transfer ◽  
Solution Method ◽  
Distribution Model ◽  
Weighted Sum ◽  
Full Spectrum ◽  
One Dimensional ◽  
Molecular Gases ◽  
Transfer Equations ◽  
Gray Medium ◽  
Radiative Transfer Equations

Abstract A new Full-Spectrum correlated-k distribution has been developed, which provides an efficient means for accurate radiative transfer calculations in absorbing/emitting molecular gases. The Full-Spectrum correlated-k distribution can be used together with any desired solution method to solve gray-medium radiative transfer equations for a small number of gray absorption coefficients, followed by numerical quadrature. It is shown that the Weighted-Sum-of-Gray-Gases model is effectively only a crude implementation of the Full-Spectrum correlated-k distribution approach. Within the limits of the Full-Spectrum correlated-k distribution model (i.e., an absorption coefficient obeying the so-called “scaling approximation”), the method is exact. This is demonstrated by comparison with line-by-line calculations for a one-dimensional CO2-N2 gas mixture with varying temperature and concentration fields.

scholarly journals Solution of Contact Problems for Nonlinear Gao Beam and Obstacle

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
J. Machalová ◽  
H. Netuka
Keyword(s):  
Solution Method ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Contact Problems ◽  
Beam Model ◽  
Contact Conditions ◽  
Beam Elements ◽  
Solution Methods ◽  
Euler Bernoulli Beam ◽  
Bernoulli Beam Model

Contact problem for a large deformed beam with an elastic obstacle is formulated, analyzed, and numerically solved. The beam model is governed by a nonlinear fourth-order differential equation developed by Gao, while the obstacle is considered as the elastic foundation of Winkler’s type in some distance under the beam. The problem is static without a friction and modeled either using Signorini conditions or by means of normal compliance contact conditions. The problems are then reformulated as optimal control problems which is useful both for theoretical aspects and for solution methods. Discretization is based on using the mixed finite element method with independent discretization and interpolations for foundation and beam elements. Numerical examples demonstrate usefulness of the presented solution method. Results for the nonlinear Gao beam are compared with results for the classical Euler-Bernoulli beam model.

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