0
Research Papers: Radiative Heat Transfer

Advanced Differential Approximation Formulation of the PN Method for Radiative Transfer

[+] Author and Article Information
Gopalendu Pal

CD-Adapco,
Lebanon, NH 03766
e-mail: gopalendu.pal@cd-adapco.com

Michael F. Modest

School of Engineering,
University of California,
Merced, CA 95343
e-mail: mmodest@eng.ucmerced.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received August 13, 2013; final manuscript received February 2, 2015; published online March 24, 2015. Assoc. Editor: Zhuomin Zhang.

J. Heat Transfer 137(7), 072701 (Jul 01, 2015) (7 pages) Paper No: HT-13-1414; doi: 10.1115/1.4029814 History: Received August 13, 2013; Revised February 02, 2015; Online March 24, 2015

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 1D slab and concentric spheres and for a 2D medium. Results are compared with the exact analytical solutions for the 1D slab as well as the Monte Carlo-based simulations for 2D media.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Modest, M. F., 2003, Radiative Heat Transfer, 2nd ed., Academic Press, New York.
Chai, J. C., Lee, H. S., and Patankar, S. V., 1993, “Ray Effect and False Scattering in the Discrete Ordinates Method,” Numer. Heat Transfer, Part B, 24(2), pp. 373–389. [CrossRef]
Kofink, W., 1959, “Complete Spherical Harmonics Solution of the Boltzmann Equation for Neutron Transport in Homogeneous Media With Cylindrical Geometry,” Nucl. Sci. Eng., 6, pp. 473–486.
Bayazitoğlu, Y., and Higenyi, J., 1979, “The Higher-Order Differential Equations of Radiative Transfer: P3 Approximation,” AIAA J., 17(4), pp. 424–431. [CrossRef]
Mengüç, M. P., and Viskanta, R., 1985, “Radiative Transfer in Three-Dimensional Rectangular Enclosures Containing Inhomogeneous, Anisotropically Scattering Media,” J. Quant. Spectrosc. Radiat. Transfer, 33(6), pp. 533–549. [CrossRef]
Mengüç, M. P., and Viskanta, R., 1986, “Radiative Transfer in Axisymmetric, Finite Cylindrical Enclosures,” ASME J. Heat Transfer, 108(2), pp. 271–276. [CrossRef]
Tong, T. W., and Swathi, P. S., 1987, “Radiative Heat Transfer in Emitting–Absorbing–Scattering Spherical Media,” J. Thermophys. Heat Transfer, 1(2), pp. 162–170. [CrossRef]
Ou, S. C. S., and Liou, K. N., 1982, “Generalization of the Spherical Harmonic Method to Radiative Transfer in Multi-Dimensional Space,” J. Quant. Spectrosc. Radiat. Transfer, 28(4), pp. 271–288. [CrossRef]
Yang, J., and Modest, M. F., 2007, “High-Order P-N Approximation for Radiative Transfer in Arbitrary Geometries,” J. Quant. Spectrosc. Radiat. Transfer, 104(2), pp. 217–227. [CrossRef]
Olfe, D. B., 1967, “A Modification of the Differential Approximation for Radiative Transfer,” AIAA J., 5(4), pp. 638–643. [CrossRef]
Olfe, D. B., 1968, “Application of a Modified Differential Approximation to Radiative Transfer in a Gray Medium Between Concentric Sphere and Cylinders,” J. Quant. Spectrosc. Radiat. Transfer, 8(3), pp. 899–907. [CrossRef]
Olfe, D. B., 1970, “Radiative Equilibrium of a Gray Medium Bounded by Nonisothermal Walls,” Prog. Astronaut. Aeronaut., 23, pp. 295–317.
Olfe, D. B., 1973, “Radiative Equilibrium of a Gray Medium in a Rectangular Enclosure,” J. Quant. Spectrosc. Radiat. Transfer, 13(9), pp. 881–895. [CrossRef]
Modest, M. F., 1974, “Two-Dimensional Radiative Equilibrium of a Gray Medium in a Plane Layer Bounded by Gray Non-Isothermal Walls,” ASME J. Heat Transfer, 96(4), pp. 483–488. [CrossRef]
Modest, M. F., 1975, “Radiative Equilibrium in a Rectangular Enclosure Bounded by Gray Non-Isothermal Walls,” J. Quant. Spectrosc. Radiat. Transfer, 15(6), pp. 445–461. [CrossRef]
Modest, M. F., and Stevens, D., 1978, “Two Dimensional Radiative Equilibrium of a Gray Medium Between Concentric Cylinders,” J. Quant. Spectrosc. Radiat. Transfer, 19(3), pp. 353–365. [CrossRef]
Modest, M. F., and Yang, J., 2008, “Elliptic PDE Formulation and Boundary Conditions of the Spherical Harmonics Method of Arbitrary Order for General Three-Dimensional Geometries,” J. Quant. Spectrosc. Radiat. Transfer, 109(9), pp. 1641–1666. [CrossRef]
FlexPDE Software, PDE Solutions, Inc., Antioch, CA.

Figures

Grahic Jump Location
Fig. 1

Schematic diagram of a rectangular enclosure containing scattering medium with hot strip at the bottom center

Grahic Jump Location
Fig. 2

Comparison of nondimensional irradiation on surfaces, full bottom surface heated, optically thin case, τL = 0.1

Grahic Jump Location
Fig. 3

Comparison of nondimensional irradiation on surfaces, full bottom surface heated, optically intermediate case, τL = 1.0

Grahic Jump Location
Fig. 4

Comparison of nondimensional irradiation on surfaces, full bottom surface heated, optically thick case, τL = 5.0

Grahic Jump Location
Fig. 5

Comparison of nondimensional irradiation on surfaces, strip of the bottom surface heated, optically thin case, τL = 0.1

Grahic Jump Location
Fig. 6

Comparison of nondimensional irradiation on surfaces, strip of the bottom surface heated, optically intermediate case, τL = 1.0

Grahic Jump Location
Fig. 7

Comparison of nondimensional irradiation on surfaces, strip of the bottom surface heated, optically thick case, τL = 5.0

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In