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# Narrow-Band Based Multiscale Full-Spectrum $k$-Distribution Method for Radiative Transfer in Inhomogeneous Gas Mixtures

[+] Author and Article Information
Liangyu Wang

The Pennsylvania State University, Department of Mechanical and Nuclear Engineering, University Park, PA 16802

Michael F. Modest1

The Pennsylvania State University, Department of Mechanical and Nuclear Engineering, University Park, PA 16802

1

Author to whom all correspondence should be addressed. Fax: (814) 863-4848; e-mail: MFModest@psu.edu

J. Heat Transfer 127(7), 740-748 (Dec 07, 2004) (9 pages) doi:10.1115/1.1925281 History: Received August 27, 2004; Revised December 07, 2004

## Abstract

The full-spectrum $k$-distribution (FSK) method has become the most promising model for radiative transfer in participating media since its introduction a few years ago. It achieves line-by-line (LBL) accuracy for homogeneous media with only a tiny fraction of LBL’s computational cost. Among the variants of the FSK method for dealing with inhomogeneous media, the multiscale FSK (MSFSK) method not only provides a strategy to treat the inhomogeneity problem by introducing an overlap coefficient, it also accommodates a solution to the so-called mixing problem (mixing of $k$-distributions for different gas species). The evaluation of MSFSK parameters, however, is tedious and excludes the MSFSK method from practical applications. In this paper a new scheme of evaluating $k$-distributions and overlap coefficients from a database of narrow-band $k$-distributions is formulated, treating each gas specie as a single scale. The new scheme makes the MSFSK method efficient and convenient for practical applications, and ready to accommodate nongray absorbing particles (such as soot) in the medium. The method virtually eliminates errors caused by uncorrelatedness due to independently varying species concentrations. It was also found that, in addition, breaking up a gas mixture into gas scales reduces the error caused by temperature inhomogeneities. The mathematical development of the new scheme is described and validated; the concept and the implication of the overlap coefficient are discussed. Sample calculations for inhomogeneous media with step changes in species mole fraction and temperature are performed to demonstrate the accuracy of the new scheme by comparison with LBL calculations.

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## Figures

Figure 1

Error introduced by Eq. 33 for a mixture of 10%CO2–10%H2O–80%N2 at 1 bar

Figure 2

Error introduced by Eq 34 for a mixture of 10%CO2–10%H2O–80%N2 at 1 bar

Figure 3

Validation of the new MSFSK scheme

Figure 4

λm−km relation for CO2

Figure 5

λm−km relation for H2O

Figure 6

Importance of neglect of overlap for one or all scales

Figure 7

k‐g and λ‐g distributions for water and carbon dioxide

Figure 8

Relative errors of the FSCK, FSSK, MSFSKdir, and MSFSKnb calculations for step changes in mole fraction, left layer: 20% CO2 and 2% H2O, right layer: 2% CO2 and 20% H2O

Figure 9

Relative errors of the FSCK, FSSK, MSFSKdir, and MSFSKnb calculations for step changes in mole fraction, left layer: 2% CO2 and 20% H2O, right layer: 20% CO2 and 2% H2O

Figure 10

Nondimensional heat flux leaving an inhomogeneous medium with step changes in species mole fraction and temperature

Figure 11

Nondimensional heat flux leaving an inhomogeneous medium with step changes in temperature only

## Errata

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