The spherical harmonics (PN) method, especially its lowest order, i.e., the P1 or differential approximation, enjoys great popularity because of its relative simplicity and compatibility with standard models for the solution of the (overall) energy equation. Low-order PN approximations perform poorly in the presence of strongly nonisotropic intensity distributions, especially in optically thin situations within nonisothermal enclosures (due to variation in surface radiosities across the enclosure surface, causing rapid change of irradiation over incoming directions). A previous modification of the PN approximation, i.e., the modified differential approximation (MDA), separates wall emission from medium emission to reduce the nonisotropy of intensity. Although successful, the major drawback of this method is that the intensity at the walls is set to zero into outward directions, while incoming intensity is nonzero, resulting in a discontinuity at grazing angles. To alleviate this problem, a new approach, termed here the “advanced differential approximation (ADA),” is developed, in which the directional gradient of the intensity at the wall is minimized. This makes the intensity distribution continuous for the P1 method and mostly continuous for higher-order PN methods. The new method is tested for a 1-D slab and concentric spheres, and for a 2-D medium. Results are compared with the exact analytical solutions for the 1-D slab as well as the Monte Carlo-based simulations for 2-D media.
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ASME 2009 Heat Transfer Summer Conference collocated with the InterPACK09 and 3rd Energy Sustainability Conferences
July 19–23, 2009
San Francisco, California, USA
Conference Sponsors:
- Heat Transfer Division
ISBN:
978-0-7918-4356-7
PROCEEDINGS PAPER
Advanced Differential Approximation Formulation of the PN Method for Radiative Transfer Available to Purchase
Michael F. Modest,
Michael F. Modest
Pennsylvania State University, University Park, PA
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Gopalendu Pal
Gopalendu Pal
Pennsylvania State University, University Park, PA
Search for other works by this author on:
Michael F. Modest
Pennsylvania State University, University Park, PA
Gopalendu Pal
Pennsylvania State University, University Park, PA
Paper No:
HT2009-88242, pp. 233-240; 8 pages
Published Online:
March 12, 2010
Citation
Modest, MF, & Pal, G. "Advanced Differential Approximation Formulation of the PN Method for Radiative Transfer." Proceedings of the ASME 2009 Heat Transfer Summer Conference collocated with the InterPACK09 and 3rd Energy Sustainability Conferences. Volume 1: Heat Transfer in Energy Systems; Thermophysical Properties; Heat Transfer Equipment; Heat Transfer in Electronic Equipment. San Francisco, California, USA. July 19–23, 2009. pp. 233-240. ASME. https://doi.org/10.1115/HT2009-88242
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