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RADIATIVE PROPERTIES OF SEMITRANSPARENT FIBERS AT ARBITRARY ILLUMINATION

Following from: The scattering problem for cylindrical particles

Leading to: Near-infrared properties of quartz fibers

Disperse systems in nature and technology are very diverse. In many cases, when a form of particles is near to spherical, the model of spherical particles may be used in analysis of their radiative properties. But it is obvious that this model is inapplicable for elongated particles. In this case, a theoretical description of the radiative properties of a disperse system may be developed on the basis of the exact solution for infinite cylinders.

Filamentary particles are found in nature comparatively seldom; some aerosols are similar to such particles, bacteria and viruses, and elongated soot particles are formed sometimes in flames. At the same time, filamentary particles are well known in engineering as a component of numerous fibrous materials. The radiation heat transfer in highly porous fibrous materials is one of the important processes that determine their thermophysical properties. For this reason, a computational analysis of the radiative properties of single fibers and the corresponding disperse systems is of great interest in connection with heat transfer problems.

It is not realistic to present a complete list of references concerning the calculations of absorption and scattering characteristics of single fibers and fibrous materials here. Nevertheless, we have tried to give a representative set of references, which is expected to be useful for studying the specific features of the problem. One should first refer to the early papers by Tong and Tien, (1980, 1983), Tong et al. (1983, 1989), Wang et al. (1987), Swathi et al. (1991), and numerous papers by Lee with co-authors (Lee, 1986, 1988, 1989, 1990, 1992a,b,c, 1993, 1994, 1996; Lee et al., 1994; Lee and Grzesik, 1995; Cunnington and Lee, 1996, 1998; Lee and Cunnington, 1998, 2000), as well as the papers by Arduini-Schuster et al. (1991), Jeandel et al. (1993), Kumar and White (1995), Marschall and Milos (1997), and Dombrovsky (1994a,b, 1996a,b, 1998a,b). The properties of hollow and coated fibers have been studied both theoretically and experimentally in papers (Wang and Tien, 1983; McKay et al., 1984; Wang et al., 1987; Reiss et al., 1987; Tong et al., 1989; Büttner et al., 1989; Reiss, 1990; Swathi et al., 1991; Ebert et al., 1991; Caps et al., 1993; Lee, 1993; Dombrovsky, 1997, 1998a,b; Kleiman et al., 2007; Tian et al., 2007). A comprehensive review of theoretical models developed for radiative transfer in fibrous media was provided by Lee and Cunnington (1998). Note that specific cases of fibers with rough surface or noncircular cross-sectional shape were also considered in recent papers by Makino and Horiba (1999), Yamada and Kurosaki (2000), and Yamada (2002).

The hypothesis of independent scattering used in the majority of solutions presented is not obvious for disperse systems containing fibers. Particularly, one can expect considerable dependent scattering effects in the case of parallel fibers, especially at oblique illumination of the medium layer. Therefore, the dependent scattering effects have been analyzed in detail in some of the above-referenced papers. It was shown by Dombrovsky (1994a, 1996b, 1997, 1998a,b) that the independent scattering approximation allows obtaining the radiative characteristics of highly porous fibrous materials containing randomly oriented fibers. This conclusion was made on the basis of comparison of the theoretical predictions with the experimental data. Some of these results are also reported below.

As it was done for spherical particles, in the article Radiative properties of semi-transparent particles, we begin our analysis from the case of fibers of a weakly absorbing substance. This general analysis is applicable to various practical problems because fibrous thermal insulations are often composed of fibers that are semitransparent in the visible and near-infrared spectral ranges.

Homogeneous Fibers

Consider first the main characteristics of radiation absorption and extinction by a single homogeneous fiber of a weakly absorbing substance (κ << 1) at a normal incidence of radiation. The typical dependencies of efficiency factors of scattering Qs and extinction Qtr and asymmetry factor of scattering μ on diffraction parameter x are shown in Fig. 1. A comparison of Fig. 1 with Figs. 1 and 3 from the article Radiative properties of semitransparent spherical particles shows that the main oscillation of curves Qs(x), Qtr(x), and μ(x) for cylinders have greater amplitude than that for spheres. At the same period of oscillations, determined by the phase shift of the transmitted radiation, the extremums of the curves for cylinders are slightly displaced toward the low diffraction parameter. An average level of radiation scattering by cylinders is less than that by spheres. The effect of the incident wave polarization is considerable only at a small diffraction parameter. This result agrees with the Rayleigh scattering analysis (see article Rayleigh scattering). The value of Qs is greater for “E” polarization. The secondary oscillations (“ripple”) for orthogonal polarization are opposite in phase. The ripple on the scattering curve near the first main maximum takes place only at E polarization of the incident radiation.

Figure 1. Scattering of randomly polarized (a) or linear polarized (b) radiation by a cylinder with m = 1.5 - 0.01i at normal incidence: 1, polarization plane is parallel to the cylinder axis (E polarization); 2, polarization plane is perpendicular to the cylinder axis (H polarization).

A graphic comparison of absorption efficiency factors for spherical and cylindrical particles is given in Fig. 2. At a not too small of a diffraction parameter, the radiation absorption by spherical particles is greater than that by cylinders, but this effect is small over the whole range of diffraction parameter. A detailed analysis of the scattering functions of spherical and cylindrical particles has been presented in a monograph by Dombrovsky (1996c), and it is not reproduced here. We give only a comparison for the transport efficiency factor of scattering. One can see in Fig. 3 that the curves Qstr(x) for two types of particles are similar, but the maximum values of Qstr for cylinders are significantly less than those for spherical particles.

Figure 2. Comparison of the absorption efficiency factors for cylinders at normal incidence of randomly polarized radiation (a) and for spheres of the same diffraction parameter (b) at n = 1.5: 1, κ = 0.002; 2, κ = 0.005; 3, κ = 0.01; 4, κ = 0.02; 5, κ = 0.05.

Figure 3. Transport efficiency factor of scattering for cylinders at normal incidence of randomly polarized radiation (a) and for spheres (b) at n = 1.5: 1, κ = 0.01; 2, κ = 0.02.

Consider now some special features of the radiation absorption and scattering by the same cylindrical particles at oblique incidence. A variation of the main characteristics Qa and Qstr with the angle α between the incident radiation direction and the normal to the cylinder axis is shown in Fig. 4. At oblique incidence and x > 2, the value of Qstr appears to be considerably less than that at normal incidence, whereas Qa decreases significantly with α only at a large diffraction parameter. Separately, in Fig. 5, an effect of the incident radiation polarization at α = π/3 is illustrated. At the same general character of dependences Qa(x) and Qstr(x), the effect of polarization on secondary oscillations is considerable. An evolution of the curves Qa(x), Qstr(x) with transfer to the extremely large angles of incidence may be dramatic because of high asymmetry of scattering. As a result, the scattering becomes negligible as compared with the absorption even at κ = 0.01 (see Fig. 6). It is interesting that resonance absorption with an almost periodic spectrum takes place at a large angle of incidence. This effect increases with decreasing of the index of absorption (see Fig. 7). It is not only an interesting physical effect; the absorption resonances should be taken into account in practical calculations by numerical integration of absorption characteristics of weakly absorbing fibers over the fiber sizes and orientations.

Figure 4. Absorption and scattering of randomly polarized radiation by a cylinder with m = 1.5 - 0.01i at normal and oblique incidence: 1, α = 0 (normal incidence); 2, α = π/4; 3, α = π/3.

Figure 5. Absorption and scattering of linear polarized radiation by a cylinder with m = 1.5 - 0.01i at oblique incidence (α = π/3): 1, E polarization; 2, H polarization.

Figure 6. Absorption and scattering of randomly polarized radiation by a cylinder with m = 1.5 - 0.01i at angle of incidence α = 85 deg.

Figure 7. Resonance absorption of randomly polarized radiation by a cylinder at angle of incidence α = 85 deg.

Hollow Fibers

Some characteristics typical of semitransparent hollow fibers at normal illumination are given in Fig. 8. The variation of the absorption efficiency factor with the relative internal radius δ = a'/a'' corresponds approximately to the variation of the absorbing material volume, i.e.,Qa ~ 1 -δ2. The deformation of curves Qa(x'') with δ increasing can be interpreted as an extension along the abscissa. A change of the value Qstr is more complex, and it is similar to that for spherical particles. One should note the appearance of new main oscillations of the curve Qstr(x'') at large values of δ.

Figure 8. Efficiency factor of absorption and transport efficiency factor of scattering for hollow cylinders with m = 1.5 - 0.01i at normal illumination by randomly polarized radiation: 1, δ = 0.5; 2, δ = 0.75; 3, δ = 0.9; 4, δ = 0.95.

The resonance behavior of the curves Qa(x'') and Qstr(x'') at a large angle of incidence discussed above for homogeneous cylinders is very sensitive to the value of δ (compare Fig. 9 with Figs. 6 and 7). The scattering by a hollow cylinder is much greater than that by a homogeneous cylinder, and the period of absorption and scattering oscillations strongly increases with δ.

Figure 9. Resonance absorption and scattering of randomly polarized radiation by a hollow cylinder at angle of incidence α = 85 deg.

In the case of two-layer fibers, the additional parameters of the problem lead to a great variety of the physical results. Therefore, we do not consider the optical properties of various hypothetical two-layer fibers in this article. It seems more reasonable to analyze radiative properties of the real fibers important for industrial applications.

REFERENCES

Arduini-Schuster, M., Ebert, H.-P., Fricke, J., and Caps, R., Infrared-Optical Properties of Feathers and Downs, High Temp.-High Press., vol. 23, no. 2, pp. 135-142, 1991.

Büttner, D., Kreh, A., Fricke, J., and Reiss, H., Recent Advances in Thermal Superinsulations, High Temp.-High Press., vol. 21, no. 1, pp. 39-50, 1989.

Caps, R., Arduini-Schuster, M. C., Ebert, H. P., and Fricke, J., Improved Thermal Radiation Extinction in Metal Coated Polypropylene Microfibers, Int. J. Heat Mass Transfer, vol. 36, no. 11, pp. 2789-2794, 1993.

Cunnington, G. R. and Lee, S. C., Radiative Properties of Fibrous Insulations: Theory Versus Experiment, J. Thermophys. Heat Transfer, vol. 10, no. 3, pp. 460-466, 1996.

Cunnington, G. R. and Lee, S. C., Radiative Properties of Fiber-Reinforced Aerogel: Theory Versus Experiment, J. Thermophys. Heat Transfer, vol. 12, no. 1, pp. 17-22, 1998.

Dombrovsky, L. A., Quartz-Fiber Thermal Insulation: Calculation of Spectral Radiation Characteristics in the Infrared Region, High Temp., vol. 32, no. 2, pp. 209-215, 1994a.

Dombrovsky, L. A., Calculation of Infrared Radiative Properties of Carbon Fibers and Fibrous Materials, High Temp., vol. 32, no. 6, pp. 895-898, 1994b.

Dombrovsky, L. A., Analysis of Infrared Radiation Characteristics of Isotropic Fiberglass Materials in the Semitransparency Region, High Temp., vol. 34, no. 1, pp. 156-158, 1996a.

Dombrovsky, L. A., Quartz-Fiber Thermal Insulation: Infrared Radiative Properties and Calculation of Radiative-Conductive Heat Transfer, ASME J. Heat Transfer, vol. 118, no. 2, pp. 408-414, 1996b.

Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York and Redding, CT, 1996c.

Dombrovsky, L. A., Radiative Properties of Metalized-Fiber Thermal Insulation, High Temp., vol. 35, no. 2, pp. 275-282, 1997.

Dombrovsky, L. A., Calculation of Radiative Properties of Highly Porous Fibrous Materials, In: Heat Transfer in Modern Engineering, Inst. High Temp., pp. 279-291, 1998a (in Russian).

Dombrovsky, L. A., Infrared and Microwave Radiative Properties of Metal Coated Microfibers, Rev. Gener. Therm., vol. 37, no. 11, pp. 925-933, 1998b.

Ebert, H. P., Arduini-Schuster, M. C., Fricke, J., Caps, R., and Reiss, H., Infrared-Radiation Screens using Very Thin Metallized Glass Fibers, High Temp.-High Press., vol. 23, no. 2, pp. 143-148, 1991.

Jeandel, G., Boulet, P., and Morlot, G., Radiative Transfer through a Medium of Silica Fibers Oriented in Parallel Planes, Int. J. Heat Mass Transfer, vol. 36, no. 3, pp. 531-536, 1993.

Kleiman, M., Gurwich, I., and Shiloah, N., Enhanced Extinction of Electromagnetic Radiation by Metal-Coated Fibers, J. Quant. Spectr. Radiat. Transfer, vol. 106, no. 1-3, pp. 184-191, 2007.

Kumar, S. and White, S. M., Dependent Scattering Properties of Woven Fibrous Insulations for Normal Incidence, ASME J. Heat Transfer, vol. 117, no. 1, pp. 160-166, 1995.

Lee, S. C., Radiative Transfer through a Fibrous Medium: Allowance for Fiber Orientation, J. Quant. Spectr. Radiat. Transfer, vol. 36, no. 3, pp. 253-263, 1986.

Lee, S. C., Radiation Heat-Transfer Model for Fibers Oriented Parallel to Diffuse Boundaries, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 303-308, 1988.

Lee, S.-C., Effect of Fiber Orientation on Thermal Radiation in Fibrous Media, Int. J. Heat Mass Transfer, vol. 32, no. 2, pp. 311-319, 1989.

Lee, S.-C., Dependent Scattering of an Obliquely Incident Plane Wave by a Collection of Parallel Cylinders, J. Appl. Phys., vol. 68, no. 10, pp. 4952-4957, 1990.

Lee, S.-C., Effective Propagation Constant of Fibrous Media Containing Parallel Fibers in the Dependent Scattering Regime, ASME J. Heat Transfer, vol. 114, no. 2, pp. 473-478, 1992a.

Lee, S.-C., Scattering by Closely-Spaced Radially-Stratified Parallel Cylinders, J. Quant. Spectr. Radiat. Transfer, vol. 48, no. 2, pp. 119-130, 1992b.

Lee, S.-C., Dependent Scattering by Parallel Fibers: Effect of Multiple Scattering and Wave Interference, J. Thermophys. Heat Transfer, vol. 6, no. 4, pp. 589-595, 1992c.

Lee, S.-C., Enhanced Thermal Performance of Fibrous Insulation Containing Nonhomogeneous Fibers, J. Quant. Spectr. Radiat. Transfer, vol. 50, no. 2, pp. 199-209, 1993.

Lee, S.-C., Dependent vs Independent Scattering in Fibrous Composites Containing Parallel Fibers, J. Thermophys. Heat Transfer, vol. 8, no. 4, pp. 641-646, 1994.

Lee, S.-C., Angle of Incidence and Size Effects on Dependent Scattering in Fibrous Media, ASME J. Heat Transfer, vol. 118, no. 4, pp. 931-936, 1996.

Lee, S.-C., White, S., and Grzesik, J. A., Effective Radiative Properties of Fibrous Composites Containing Spherical Particles, J. Thermophys. Heat Transfer, vol. 8, no. 3, pp. 400-405, 1994.

Lee, S.-C. and Grzesik, J. A., Scattering Characteristics of Fibrous Media Containing Closely Spaced Parallel Fibers, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 403-409, 1995.

Lee, S.-C. and Cunnington, G. R., Theoretical Models for Radiative Transfer in Fibrous Media, Annual Review in Heat Transfer, vol. 9, C. L. Tien, (ed.), New York: Begell House, pp. 159-218, 1998.

Lee, S.-C. and Cunnington, G. R., Conduction and Radiation Heat Transfer in High-Porosity Fiber Thermal Insulation, J. Thermophys. Heat Transfer, vol. 14, no. 2, pp. 121-136, 2000.

Makino, T. and Horiba, J., Scattering of Radiation by a Fiber with a Rough Surface, Heat Transfer-Asian Res., vol. 28, no. 4, pp. 322-335, 1999.

Marschall, J. and Milos, F. S., The Calculation of Anisotropic Extinction Coefficients for Radiation Diffusion in Rigid Fibrous Ceramic Insulations, Int. J. Heat Mass Transfer, vol. 40, no. 3, pp. 627-634, 1997.

McKay, N. L., Timusk, T., and Farnworth, B., Determination of Optical Properties of Fibrous Thermal Insulation, J. Appl. Phys., vol. 55, no. 11, pp. 4064-4071, 1984.

Reiss, H., Radiative Transfer in Nontransparent Dispersed Media, High Temp.-High Press., vol. 22, no. 5, pp. 481-522, 1990.

Reiss, H., Schmaderer, F., Wahl, G., Ziegenbein, B., and Caps, R., Experimental Investigation of Extinction Properties and Thermal Conductivity of Metal-Coated Dielectric Fibers in Vacuum, Int. J. Thermophys., vol. 8, no. 2, pp. 263-280, 1987.

Swathi, P. S., Tong, T. W., and Cunnington, Jr., G. R., Scattering of Electromagnetic Waves by Cylinders Coated with a Radially-Inhomogeneous Layers, J. Quant. Spectr. Radiat. Transfer, vol. 46, no. 4, pp. 281-292, 1991.

Tian, W., Huang, W., and Chiu, W. K. S., Thermal Radiative Properties of a Semitransparent Fiber Coated with a Thin Absorbing Film, ASME J. Heat Transfer, vol. 129, no. 6, pp. 763-767, 2007.

Tong, T. W. and Tien, C. L., Analytical Models for Thermal Radiation in Fibrous Insulations, J. Therm. Insul., vol. 4, no. 7, pp. 27-44, 1980.

Tong, T. W. and Tien, C. L., Radiative Heat Transfer in Fibrous Insulations - Part I: Analytical Study, ASME J. Heat Transfer, vol. 105, no. 1, pp. 70-75, 1983.

Tong, T. W., Yang, Q. S., and Tien, C. L., Radiative Heat Transfer in Fibrous Insulations - Part II: Experimental Study, ASME J. Heat Transfer, vol. 105, no. 1, pp. 76-81, 1983.

Tong, T. W., Swathi, P. S., and Cunnington, G. R., Reduction of Radiative Heat Transfer in Thermal Insulations by Use of Dielectric Coated Fibers, Int. Comm. Heat Mass Transfer, vol. 16, no. 6, pp. 851-856, 1989.

Wang, K. Y. and Tien, C. L., Radiative Heat Transfer through Opacified Fibers and Powders, J. Quant. Spectr. Radiat. Transfer, vol. 30, no. 3, pp. 213-223, 1983.

Wang, K. Y., Kumar, S., and Tien, C. L., Radiative Transfer in Thermal Insulations of Hollow and Coated Fibers, J. Thermophys. Heat Transfer, vol. 1, no. 4, pp. 289-295, 1987.

Yamada, J. and Kurosaki, Y., Radiative Characteristics of Fibers with a Large Size Parameter, Int. J. Heat Mass Transfer, vol. 43, no. 6, pp. 981-991, 2000.

Yamada, J., Radiative Properties of Fibers with Non-Circular Cross-Sectional Shape, J. Quant. Spectr. Radiat. Transfer, vol. 73, no. 2-5, pp. 261-272, 2002.

References

  1. Arduini-Schuster, M., Ebert, H.-P., Fricke, J., and Caps, R., Infrared-Optical Properties of Feathers and Downs, High Temp.-High Press., vol. 23, no. 2, pp. 135-142, 1991.
  2. Büttner, D., Kreh, A., Fricke, J., and Reiss, H., Recent Advances in Thermal Superinsulations, High Temp.-High Press., vol. 21, no. 1, pp. 39-50, 1989.
  3. Caps, R., Arduini-Schuster, M. C., Ebert, H. P., and Fricke, J., Improved Thermal Radiation Extinction in Metal Coated Polypropylene Microfibers, Int. J. Heat Mass Transfer, vol. 36, no. 11, pp. 2789-2794, 1993.
  4. Cunnington, G. R. and Lee, S. C., Radiative Properties of Fibrous Insulations: Theory Versus Experiment, J. Thermophys. Heat Transfer, vol. 10, no. 3, pp. 460-466, 1996.
  5. Cunnington, G. R. and Lee, S. C., Radiative Properties of Fiber-Reinforced Aerogel: Theory Versus Experiment, J. Thermophys. Heat Transfer, vol. 12, no. 1, pp. 17-22, 1998.
  6. Dombrovsky, L. A., Quartz-Fiber Thermal Insulation: Calculation of Spectral Radiation Characteristics in the Infrared Region, High Temp., vol. 32, no. 2, pp. 209-215, 1994a.
  7. Dombrovsky, L. A., Calculation of Infrared Radiative Properties of Carbon Fibers and Fibrous Materials, High Temp., vol. 32, no. 6, pp. 895-898, 1994b.
  8. Dombrovsky, L. A., Analysis of Infrared Radiation Characteristics of Isotropic Fiberglass Materials in the Semitransparency Region, High Temp., vol. 34, no. 1, pp. 156-158, 1996a.
  9. Dombrovsky, L. A., Quartz-Fiber Thermal Insulation: Infrared Radiative Properties and Calculation of Radiative-Conductive Heat Transfer, ASME J. Heat Transfer, vol. 118, no. 2, pp. 408-414, 1996b.
  10. Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York and Redding, CT, 1996c.
  11. Dombrovsky, L. A., Radiative Properties of Metalized-Fiber Thermal Insulation, High Temp., vol. 35, no. 2, pp. 275-282, 1997.
  12. Dombrovsky, L. A., Calculation of Radiative Properties of Highly Porous Fibrous Materials, In: Heat Transfer in Modern Engineering, Inst. High Temp., pp. 279-291, 1998a (in Russian).
  13. Dombrovsky, L. A., Infrared and Microwave Radiative Properties of Metal Coated Microfibers, Rev. Gener. Therm., vol. 37, no. 11, pp. 925-933, 1998b.
  14. Ebert, H. P., Arduini-Schuster, M. C., Fricke, J., Caps, R., and Reiss, H., Infrared-Radiation Screens using Very Thin Metallized Glass Fibers, High Temp.-High Press., vol. 23, no. 2, pp. 143-148, 1991.
  15. Jeandel, G., Boulet, P., and Morlot, G., Radiative Transfer through a Medium of Silica Fibers Oriented in Parallel Planes, Int. J. Heat Mass Transfer, vol. 36, no. 3, pp. 531-536, 1993.
  16. Kleiman, M., Gurwich, I., and Shiloah, N., Enhanced Extinction of Electromagnetic Radiation by Metal-Coated Fibers, J. Quant. Spectr. Radiat. Transfer, vol. 106, no. 1-3, pp. 184-191, 2007.
  17. Kumar, S. and White, S. M., Dependent Scattering Properties of Woven Fibrous Insulations for Normal Incidence, ASME J. Heat Transfer, vol. 117, no. 1, pp. 160-166, 1995.
  18. Lee, S. C., Radiative Transfer through a Fibrous Medium: Allowance for Fiber Orientation, J. Quant. Spectr. Radiat. Transfer, vol. 36, no. 3, pp. 253-263, 1986.
  19. Lee, S. C., Radiation Heat-Transfer Model for Fibers Oriented Parallel to Diffuse Boundaries, J. Thermophys. Heat Transfer, vol. 2, no. 4, pp. 303-308, 1988.
  20. Lee, S.-C., Effect of Fiber Orientation on Thermal Radiation in Fibrous Media, Int. J. Heat Mass Transfer, vol. 32, no. 2, pp. 311-319, 1989.
  21. Lee, S.-C., Dependent Scattering of an Obliquely Incident Plane Wave by a Collection of Parallel Cylinders, J. Appl. Phys., vol. 68, no. 10, pp. 4952-4957, 1990.
  22. Lee, S.-C., Effective Propagation Constant of Fibrous Media Containing Parallel Fibers in the Dependent Scattering Regime, ASME J. Heat Transfer, vol. 114, no. 2, pp. 473-478, 1992a.
  23. Lee, S.-C., Scattering by Closely-Spaced Radially-Stratified Parallel Cylinders, J. Quant. Spectr. Radiat. Transfer, vol. 48, no. 2, pp. 119-130, 1992b.
  24. Lee, S.-C., Dependent Scattering by Parallel Fibers: Effect of Multiple Scattering and Wave Interference, J. Thermophys. Heat Transfer, vol. 6, no. 4, pp. 589-595, 1992c.
  25. Lee, S.-C., Enhanced Thermal Performance of Fibrous Insulation Containing Nonhomogeneous Fibers, J. Quant. Spectr. Radiat. Transfer, vol. 50, no. 2, pp. 199-209, 1993.
  26. Lee, S.-C., Dependent vs Independent Scattering in Fibrous Composites Containing Parallel Fibers, J. Thermophys. Heat Transfer, vol. 8, no. 4, pp. 641-646, 1994.
  27. Lee, S.-C., Angle of Incidence and Size Effects on Dependent Scattering in Fibrous Media, ASME J. Heat Transfer, vol. 118, no. 4, pp. 931-936, 1996.
  28. Lee, S.-C., White, S., and Grzesik, J. A., Effective Radiative Properties of Fibrous Composites Containing Spherical Particles, J. Thermophys. Heat Transfer, vol. 8, no. 3, pp. 400-405, 1994.
  29. Lee, S.-C. and Grzesik, J. A., Scattering Characteristics of Fibrous Media Containing Closely Spaced Parallel Fibers, J. Thermophys. Heat Transfer, vol. 9, no. 3, pp. 403-409, 1995.
  30. Lee, S.-C. and Cunnington, G. R., Theoretical Models for Radiative Transfer in Fibrous Media, Annual Review in Heat Transfer, vol. 9, C. L. Tien, (ed.), New York: Begell House, pp. 159-218, 1998.
  31. Lee, S.-C. and Cunnington, G. R., Conduction and Radiation Heat Transfer in High-Porosity Fiber Thermal Insulation, J. Thermophys. Heat Transfer, vol. 14, no. 2, pp. 121-136, 2000.
  32. Makino, T. and Horiba, J., Scattering of Radiation by a Fiber with a Rough Surface, Heat Transfer-Asian Res., vol. 28, no. 4, pp. 322-335, 1999.
  33. Marschall, J. and Milos, F. S., The Calculation of Anisotropic Extinction Coefficients for Radiation Diffusion in Rigid Fibrous Ceramic Insulations, Int. J. Heat Mass Transfer, vol. 40, no. 3, pp. 627-634, 1997.
  34. McKay, N. L., Timusk, T., and Farnworth, B., Determination of Optical Properties of Fibrous Thermal Insulation, J. Appl. Phys., vol. 55, no. 11, pp. 4064-4071, 1984.
  35. Reiss, H., Radiative Transfer in Nontransparent Dispersed Media, High Temp.-High Press., vol. 22, no. 5, pp. 481-522, 1990.
  36. Reiss, H., Schmaderer, F., Wahl, G., Ziegenbein, B., and Caps, R., Experimental Investigation of Extinction Properties and Thermal Conductivity of Metal-Coated Dielectric Fibers in Vacuum, Int. J. Thermophys., vol. 8, no. 2, pp. 263-280, 1987.
  37. Swathi, P. S., Tong, T. W., and Cunnington, Jr., G. R., Scattering of Electromagnetic Waves by Cylinders Coated with a Radially-Inhomogeneous Layers, J. Quant. Spectr. Radiat. Transfer, vol. 46, no. 4, pp. 281-292, 1991.
  38. Tian, W., Huang, W., and Chiu, W. K. S., Thermal Radiative Properties of a Semitransparent Fiber Coated with a Thin Absorbing Film, ASME J. Heat Transfer, vol. 129, no. 6, pp. 763-767, 2007.
  39. Tong, T. W. and Tien, C. L., Analytical Models for Thermal Radiation in Fibrous Insulations, J. Therm. Insul., vol. 4, no. 7, pp. 27-44, 1980.
  40. Tong, T. W. and Tien, C. L., Radiative Heat Transfer in Fibrous Insulations - Part I: Analytical Study, ASME J. Heat Transfer, vol. 105, no. 1, pp. 70-75, 1983.
  41. Tong, T. W., Yang, Q. S., and Tien, C. L., Radiative Heat Transfer in Fibrous Insulations - Part II: Experimental Study, ASME J. Heat Transfer, vol. 105, no. 1, pp. 76-81, 1983.
  42. Tong, T. W., Swathi, P. S., and Cunnington, G. R., Reduction of Radiative Heat Transfer in Thermal Insulations by Use of Dielectric Coated Fibers, Int. Comm. Heat Mass Transfer, vol. 16, no. 6, pp. 851-856, 1989.
  43. Wang, K. Y. and Tien, C. L., Radiative Heat Transfer through Opacified Fibers and Powders, J. Quant. Spectr. Radiat. Transfer, vol. 30, no. 3, pp. 213-223, 1983.
  44. Wang, K. Y., Kumar, S., and Tien, C. L., Radiative Transfer in Thermal Insulations of Hollow and Coated Fibers, J. Thermophys. Heat Transfer, vol. 1, no. 4, pp. 289-295, 1987.
  45. Yamada, J. and Kurosaki, Y., Radiative Characteristics of Fibers with a Large Size Parameter, Int. J. Heat Mass Transfer, vol. 43, no. 6, pp. 981-991, 2000.
  46. Yamada, J., Radiative Properties of Fibers with Non-Circular Cross-Sectional Shape, J. Quant. Spectr. Radiat. Transfer, vol. 73, no. 2-5, pp. 261-272, 2002.
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