Research Papers: Radiative Heat Transfer

Statistical Approach to Radiative Transfer in the Heterogeneous Media of Thin-Wall Morphology—I: Theory

[+] Author and Article Information
A. V. Gusarov

Moscow State University of Technology,
“STANKIN”, Vadkovsky per. 3a,
Moscow 127055, Russia
e-mail: av.goussarov@gmail.com

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received November 3, 2017; final manuscript received May 25, 2018; published online July 23, 2018. Assoc. Editor: Xiulin Ruan.

J. Heat Transfer 140(11), 112701 (Jul 23, 2018) (11 pages) Paper No: HT-17-1656; doi: 10.1115/1.4040482 History: Received November 03, 2017; Revised May 25, 2018

Foams, three-dimensional (3D)-printed cellular and honeycomb structures, and very oblate particles dispersed in a matrix are the examples of heterogeneous media with thin-wall morphology. Phase boundaries can also be considered by this approach. Statistical description is proposed to estimate the effective radiative properties of such media. Three orientation models are studied: (i) isotropic, (ii) surface elements parallel to a plane, and (iii) surface elements parallel to an axis. Radiative transfer equations (RTEs) are obtained and analyzed in the framework of the homogeneous phase approach (HPA) and the multiphase approach (MPA). Analytical expressions are obtained for the absorption, extinction, and scattering coefficients, the scattering phase function, and the radiative thermal conductivity for very oblate particles dispersed in an absorbing scattering matrix. The reflective properties of the platelets and their preferential orientation can be used to optimize the radiative thermal conductivity.

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Brendel, H. , Seifert, G. , and Raether, F. , 2017, “ Heat Transfer Properties of Hollow-Fiber Insulation Materials at High Temperatures,” J. Thermophys. Heat Transfer, 31(2), pp. 463–472. [CrossRef]
Wang, B. X. , and Zhao, C. Y. , 2017, “ Effect of Anisotropy on Thermal Radiation Transport in Porous Ceramics,” Int. J. Therm. Sci., 111, pp. 301–309. [CrossRef]
Alkandry, H. , Boyd, I. D. , and Martin, A. , 2014, “ Comparison of Transport Properties Models for Flowfield Simulations of Ablative Heat Shields,” J. Thermophys. Heat Transfer, 28(4), pp. 569–582. [CrossRef]
Marti, J. , Roesle, M. , and Steinfeld, A. , 2014, “ Combined Experimental-Numerical Approach to Determine Radiation Properties of Particle Suspensions,” ASME J. Heat Transfer, 136(9), p. 092701. [CrossRef]
Boley, C. D. , Khairallah, S. A. , and Rubenchik, A. M. , 2015, “ Calculation of Laser Absorption by Metal Powders in Additive Manufacturing,” Appl. Opt., 54(9), pp. 2477–2482. [CrossRef] [PubMed]
Malinka, A. V. , 2014, “ Light Scattering in Porous Materials: Geometrical Optics and Stereological Approach,” J. Quant. Spectrosc. Radiat. Transfer, 141, pp. 14–23. [CrossRef]
Jacques, S. L. , 2013, “ Optical Properties of Biological Tissues: A Review,” Phys. Med. Biol., 58(11), pp. R37–R61. [CrossRef] [PubMed]
Mishchenko, M. I. , Dlugach, J. M. , Yurkin, M. A. , Bi, L. , Cairns, B. , Liu, L. , Panetta, R. L. , Travis, L. D. , Yang, P. , and Zakharova, N. T. , 2016, “ First-Principles Modeling of Electromagnetic Scattering by Discrete and Discretely Heterogeneous Random Media,” Phys. Rep., 632, pp. 1–75. [CrossRef] [PubMed]
van de Hulst, H. C. , 1981, Light Scattering by Small Particles, Dover, Mineola, NY. [PubMed] [PubMed]
Modest, M. F. , 2013, Radiative Heat Transfer, Academic Press, San Diego, CA.
Lee, S. C. , and Cunnington, G. R. , 2000, “ Conduction and Radiation Heat Transfer in High-Porosity Fiber Thermal Insulation,” J. Thermophys. Heat Transfer, 14(2), pp. 121–136. [CrossRef]
Wait, J. R. , 1955, “ Scattering of a Plane Wave From a Circular Dielectric Cylinder at Oblique Incidence,” Can. J. Phys., 33(5), pp. 189–195. [CrossRef]
Lind, A. C. , and Greenberg, J. M. , 1966, “ Electromagnetic Scattering by Obliquely Oriented Cylinders,” J. Appl. Phys., 37(8), pp. 3195–3203. [CrossRef]
Howell, J. R. , Siegel, R. , and Menguc, M. P. , 2011, Thermal Radiation Heat Transfer, CRC Press, Boca Raton, FL.
Shukla, D. K. , Kasisomayajula, S. V. , and Parameswaran, V. , 2008, “ Epoxy Composites Using Functionalized Platelets as Reinforcements,” Compos. Sci. Tech., 68(14), pp. 3055–3063. [CrossRef]
Sauceau, M. , Fages, J. , Common, A. , Nikitine, C. , and Rodier, E. , 2011, “ New Challenges in Polymer Foaming: A Review of Extrusion Processes Assisted by Supercritical Carbon Dioxide,” Prog. Polym. Sci., 36(6), pp. 749–766. [CrossRef]
Huo, W.-L. , Zhang, X.-Y. , Chen, Y.-G. , Lu, Y.-J. , Liu, W.-T. , Xi, X.-Q. , Wang, Y.-L. , Xu, J. , and Yang, J.-L. , 2016, “ Highly Porous Zirconia Ceramic Foams With Low Thermal Conductivity From Particle-Stabilized Foams,” J. Am. Ceram. Soc., 99(11), pp. 3512–3515. [CrossRef]
Shiomi, M. , Imagama, S. , Osakada, K. , and Matsumoto, R. , 2010, “ Fabrication of Aluminum Foams From Powder by Hot Extrusion and Foaming,” J. Mater. Process. Technol., 210(9), pp. 1203–1208. [CrossRef]
Li, J. E. , and Wang, B. , 2014, “ Equivalent Thermal Conductivity of Open-Cell Ceramic Foams at High Temperatures,” Int. J. Thermophys., 35(1), pp. 105–122. [CrossRef]
Cunsolo, S. , Coquard, R. , Baillis, D. , and Bianco, N. , 2016, “ Radiative Properties Modeling of Open Cell Solid Foam: Review and New Analytical Law,” Int. J. Therm. Sci., 104, pp. 122–134. [CrossRef]
Cunsolo, S. , Coquard, R. , Baillis, D. , Chiu, W. K. S. , and Bianco, N. , 2017, “ Radiative Properties of Irregular Open Cell Solid Foams,” Int. J. Therm. Sci., 117, pp. 77–89. [CrossRef]
Chantarapanich, N. , Laohaprapanon, A. , Wisutmethangoon, S. , Jiamwatthanachai, P. , Chalermkarnnon, P. , Sucharitpwatskul, S. , Puttawibul, P. , and Sitthiseripratip, K. , 2014, “ Fabrication of Three-Dimensional Honeycomb Structure for Aeronautical Applications Using Selective Laser Melting: A Preliminary Investigation,” Rapid Prototyping J., 20(6), pp. 551–558. [CrossRef]
Studart, A. R. , 2016, “ Additive Manufacturing of Biologically-Inspired Materials,” Chem. Soc. Rev., 45(2), pp. 359–376. [CrossRef] [PubMed]
Tancrez, M. , and Taine, J. , 2004, “ Direct Identification of Absorption and Scattering Coefficients and Phase Function of a Porous Medium by a Monte Carlo Technique,” Int. J. Heat Mass Transfer, 47(2), pp. 373–383. [CrossRef]
Coquard, R. , and Baillis, D. , 2005, “ Radiative Characteristics of Beds of Spheres Containing an Absorbing and Scattering Medium,” J. Thermophys. Heat Transfer, 19(2), pp. 226–234. [CrossRef]
Rombouts, M. , Froyen, L. , Gusarov, A. V. , Bentefour, E. H. , and Glorieux, C. , 2005, “ Light Extinction in Metallic Powder Beds: Correlation With Powder Structure,” J. Appl. Phys., 98(1), p. 013533. [CrossRef]
Dauvois, Y. , Rochais, D. , Enguehard, F. , and Taine, J. , 2017, “ Statistical Radiative Modeling of a Porous Medium With Semi Transparent and Transparent Phases: Application to a Felt of Overlapping Fibers,” Int. J. Heat Mass Transfer, 106, pp. 601–618. [CrossRef]
Frankel, A. , Iaccarino, G. , and Mani, A. , 2016, “ Convergence of the Bouguer-Beer Law for Radiation Extinction in Particulate Media,” J. Quant. Spectrosc. Radiat. Transfer, 182, pp. 45–54. [CrossRef]
Gusarov, A. V. , 2008, “ Homogenization of Radiation Transfer in Two-Phase Media With Irregular Phase Boundaries,” Phys. Rev. B, 77(14), p. 144201. [CrossRef]
Randrianalisoa, J. , and Baillis, D. , 2010, “ Radiative Transfer in Dispersed Media: Comparison Between Homogeneous Phase and Multiphase Approaches,” ASME J. Heat Transfer, 132(2), p. 023405. [CrossRef]
Consalvi, J. L. , Porterie, B. , and Loraud, J. C. , 2002, “ A Formal Averaging Procedure for Radiation Heat Transfer in Particulate Media,” Int. J. Heat Mass Transfer, 45(13), pp. 2755–2768. [CrossRef]
Lipinski, W. , Petrasch, J. , and Haussener, S. , 2010, “ Application of the Spatial Averaging Theorem to Radiative Heat Transfer in Two-Phase Media,” J. Quant. Spectrosc. Radiat. Transfer, 111(1), pp. 253–258. [CrossRef]
Lipinski, W. , Keene, D. , Haussener, S. , and Petrasch, J. , 2010, “ Continuum Radiative Transfer Modelling in Media Consisting of Optically Distinct Components in the Limit of Geometric Optics,” J. Quant. Spectrosc. Radiat. Transfer, 111, pp. 2474–2480. [CrossRef]
Gusarov, A. V. , and Kruth, J.-P. , 2005, “ Modelling of Radiation Transfer in Metallic Powders at Laser Treatment,” Int. J. Heat Mass Transfer, 48(16), p. 3423. [CrossRef]
Gusarov, A. V. , and Smurov, I. , 2010, “ Homogenized Models of Radiation Transfer in Multiphase Media,” Integral Methods in Science and Engineering, Vol. 2, C. Constanda and M. E. Perez , eds., Birkhauser, Boston, MA, pp. 183–192. [CrossRef]
Gusarov, A. V. , 2011, “ Differential Approximations to the Radiation Transfer Equation by Chapman-Enskog Expansion,” ASME J. Heat Transfer, 133(8), p. 082701. [CrossRef]
Gusarov, A. V. , 2010, “ Model of Radiative Heat Transfer in Heterogeneous Multiphase Media,” Phys. Rev. B, 81(6), p. 064202. [CrossRef]
Gusarov, A. V. , “ Statistical Approach to Radiative Transfer in the Heterogeneous Media of Thin-Wall Morphology—II: Applications,” ASME J. Heat Transfer (accepted).
Lai, H. M. , Wong, W. Y. , and Wong, W. H. , 2004, “ Extinction Paradox and Actual Power Scattering in Light Beam Scattering: A Two-Dimensional Study,” J. Opt. Soc. Am. A, 21(12), pp. 2324–2333. [CrossRef]
Rosseland, S. , 1936, Theoretical Astrophysics, Oxford University Press, London.


Grahic Jump Location
Fig. 1

Thin-wall structures: (a) dispersed platelets and (b) continuous wall (thick line) virtually divided into platelets by hatches. Arrows are the normals to the platelets. The dashed rectangles are control volumes.

Grahic Jump Location
Fig. 2

Scattering by a platelet

Grahic Jump Location
Fig. 3

Normalized radiative thermal conductivity λ/Λ of gray platelets dispersed in a transparent matrix or gray structures with transparent pores: Thermal conductivity versus average diffuse reflectivity λ(ρd) at isotropic orientation; transversal thermal conductivity versus the average reflectivity for specularly λ(ρs) and diffusely λ(ρd) reflecting platelets or fibers parallel to an axis; and axial thermal conductivity versus the reflectivity of one face of the platelet at the completely absorbing other face λ||(ρf) and versus the reflectivity at the equivalent faces λ||(ρ) for platelets or oblate particles perpendicular to an axis



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