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# On Multilayer Modeling of Radiative Transfer for Use With the Multisource k-Distribution Method for Inhomogeneous Media

[+] Author and Article Information
John Tencer

Sandia National Laboratories,
1515 Eubank,
Albuquerque, NM 87123
e-mail: jtencer@sandia.gov

John R. Howell

University of Texas at Austin,
Austin, TX 78712
e-mail: jhowell@mail.utexas.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 5, 2012; final manuscript received January 21, 2014; published online March 10, 2014. Assoc. Editor: He-Ping Tan.

J. Heat Transfer 136(6), 062703 (Mar 10, 2014) (7 pages) Paper No: HT-12-1539; doi: 10.1115/1.4026554 History: Received October 05, 2012; Revised January 21, 2014

## Abstract

A nonisothermal medium is modeled using the multilayer approach in which the continuous temperature distribution in a one-dimensional system as modeled as being piecewise constant. This has been shown to provide accurate results for a surprisingly small number of layers. Analysis is performed on a nonisothermal gray medium to attempt to characterize the ways in which the errors introduced by the multilayer modeling change with various physical parameters namely, the optical thickness and the temperature or emissive power gradient. A demonstration is made of how the multisource k-distribution method is capable of evaluating the heat flux within a one-dimensional system with piecewise constant temperature distribution with line-by-line accuracy with a significant decrease in computational expense. The k-distribution method for treating the spectral properties of an absorbing–emitting medium represents a powerful alternative to line-by-line calculations by reducing the number of radiative transfer equation (RTE) evaluations from the order of a million to the order of 10 without any significant loss of accuracy. For problems where an appropriate reference temperature can be defined, the k-distribution method is formally exact. However, when no appropriate reference temperature can be defined, the method results in errors. The multisource k-distribution method extends the k-distribution method to problems with piecewise constant temperature and optical properties.

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## Figures

Fig. 1

Schematic illustration of multilayer system and geometric nomenclature [4]

Fig. 2

Local error in radiative flux normalized by the volume-averaged radiative flux as a function of optical depth (physical location) for a linear temperature distribution

Fig. 3

Local error in radiative flux normalized by the volume-averaged radiative flux as a function of optical depth (physical location) for a linear emissive power temperature profile

Fig. 4

Global error convergence for a variety of temperature profiles using layers distributed uniformly over the optical coordinate

Fig. 5

Global error convergence for a variety of temperature profiles using layers distributed uniformly throughout the range of emissive powers

Fig. 6

Effect of emissive power gradient on multilayer approximation error behavior

Fig. 7

k-Distribution demonstration [6]

Fig. 8

k-Distribution results for a homogeneous isothermal medium with cold black walls

Fig. 9

k-Distribution results for a purely absorbing medium subject to a known spectral boundary flux

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