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An analysis of absorption and scattering of infrared radiation by quartz fibers is important for the understanding of heat-shielding properties of highly porous thermal insulations containing quartz or glass fibers. Similar materials are widely used in practice, and their spectral radiative properties in the visible and near-infrared spectral ranges are of great interest.

The index of refraction of fused quartz in the spectral range 0.21 ≤ λ ≤ 3.71 μm at room temperature was measured with high accuracy by Malitson (1965) and Brixner (1967). It is well described by using the three-term dispersion relation suggested by Malitson (1965),

(1)

where λ is expressed in microns. In Eq. (1), approximate values of coefficients are given. The index of refraction calculated by use of Eq. (1) varies from 1.53 to 1.40 with the wavelength in the spectral range cited above. Note that applicability of dispersion relation (1) has been confirmed in more recent papers by Tan (1998) and Tan and Arndt (2001). Data reviewed by Petrov (1979) also contain a temperature dependence, which can be approximated as follows:

(2)

where the temperature T is expressed in Kelvin. The refractive index of fused quartz is a weak function of temperature. Therefore, to simplify calculation in what follows, it is taken that n = n0. In our calculations, we will also ignore a temperature dependence of the index of absorption. In the short-wave range, we use the fused quartz index of absorption calculated from the transmittance data in a recent paper by Dombrovsky et al. (2005). One can see in Fig. 1 that spectral dependences κ(λ) determined in Dombrovsky et al. (2005) are in good agreement with published data by Beder et al. (1971), Touloukian and DeWitt (1972), and Khashan and Nassif (2001). The optical constants of quartz in the opacity region at temperatures up to 823 K are presented in graphs by Banner et al. (1989).

Figure 1. Index of absorption of fused quartz in the near infrared: 1, Beder et al. (1971); 2, Touloukian and DeWitt (1972); 3, Khashan and Nassif (2001); 4, Dombrovsky et al. (2005).

Ignoring the temperature dependence, one can use the following approximation in the wavelength range from 7.6 to 12μm (Dombrovsky, 1994, 1996a):

(3a)
(3b)

The wavelength in Eq. (3) is expressed in microns. At λ < 7.6 μm, we use Eq. (1) for the index of refraction. In the range 4 < λ < 7.6 μm, we calculate both n and κ by Eq. (3). The resulting approximations of the optical constants of fused quartz in the infrared are given in Fig. 2, where the main spectral features are evident: maximum of the index of refraction at λ = 9 μm and maximum of the index of absorption at λ = 9.5 μm, as well as the points of n = 1 at λ = 7.6 and 9 μm.

Figure 2. Infrared optical constants of fused quartz.

The calculated spectral properties of single fibers of fused quartz at normal incidence are presented in Fig. 3. Both absorption and scattering of infrared radiation by quartz fibers are quite different in various spectral regions. One can see very small absorption and relatively strong scattering in the range of semitransparency at λ < 5 μm. A sharp rise of absorption with the wavelength in the range from 5 to 7.6 μm corresponds to the strong increase of the index of absorption (see Fig. 2). The further increase of κ does not lead to an increase in the absorption efficiency factor. The scattering decreases dramatically in the wavelength range from 5 to 7.5μm due to transfer to the limiting case of optically soft fibers when the values of both n - 1 and κ are very small. The strong maximum of scattering is then reached at λ = 9 μm, exactly at another point of n = 1. It is a very interesting effect of scattering by absorption observed in the case of purely absorbing particles (Dombrovsky, 1996b). The second strong maximum of absorption takes place at λ ≈ 9.3 μm when κ ≈ 1. One can also see an additional absolute maximum of scattering for fibers of radius a = 2 μm at wavelength λ ≈ 10 μm. It is not difficult to show that the latter effect is caused by the coincidence of the Qstr main maximum position with the region of high refractive index and small index of absorption.

Figure 3. Absorption and scattering of randomly polarized radiation by quartz fibers at normal incidence.

The typical dependences of the efficiency factors Qa and Qstr on the angle of incidence are shown in Fig. 4. The ratios A = Qa(α)/Qa(0) and Str = Qstr(α)/ Qstr(0) for a quartz fiber at various wavelengths are plotted in Fig. 4.

Figure 4. Absorption and scattering characteristics of randomly polarized radiation by quartz fibers of radius a = 5 μm as functions of the incidence angle.

One can see that curves A(α) and Str(α) have numerous oscillations in the range of quartz semitransparency. The absorption resonances are especially strong. Contrarily, both A(α) and Str(α) are smooth and monotonic in the long-wave opacity range. The most simple and natural approximation of the angular dependences in the opacity range is

(4)

In this case, one can use Eq. (17) from the article The scattering problem for cylindrical particles to obtain the following relation between efficiency factors at normal incidence and the corresponding average values for fibers randomly oriented in space:

(5)

Going on to the properties of elementary volume of a fibrous material, it is appropriate to consider a dependence of the absorption coefficient and transport scattering coefficient on the orientation of fibers in more detail than in the article Radiative properties of semi-transparent fibers at arbitrary illumination. This question was discussed in some detail by Lee (1986, 1988), where the following three variants have been considered: random orientation of fibers in space, random orientation of fibers in parallel planes, and fibers arranged in specific directions (as in some spacecraft materials). We consider only the two first variants. The corresponding equations from the article The scattering problem for cylindrical particles are reproduced below.

(6)
(7)

Equation (6) is referred to as an isotropic disperse system, and Eq. (7) to as a transversally isotropic disperse system. At random orientation of fibers in space, the optical properties of a fibrous material are isotropic. An angular dependence of the values Qa, Qstr and the corresponding coefficients α, σtr in the radiative transfer equation (RTE) appears at layer-by-layer random orientation of fibers.

It was shown by Dombrovsky (1997) for the case of an anisotropic layered medium that one should use the following average values of absorption coefficient αλ and transport extinction coefficient βλtr = αλ + σλtr in the radiative transfer model based on P1 approximation:

(8)

The corresponding transport coefficient of scattering can be determined as

(9)

Thus, the calculation of the radiative properties of a transversely isotropic material can be conducted in the following succession: determination of single particle characteristics Qa, Qtr at various angles of incidence, calculation of average factors Qa(θ), Qtr(θ) by use of Eq. (7) and the corresponding spectral coefficients αλ(θ), βλtr(θ) and, finally, determination of average values αλ, βλtr by Eq. (8). To minimize computational difficulties, the succession of integration in Eqs. (7) and (8) can be changed. In so doing, it is convenient to introduce the following parameters:

(10)

At the same time, determination of Qstr as a difference between Qtr(θ) and Qa(θ) would be incorrect; the value σλtr can be obtained only after integration of Qa and Qtr over directions μ = cosθ. The radiative properties of an isotropic fibrous material are calculated in a considerably simpler manner, without integration over μ, since values Qa, Qtr according to (6) do not depend on the direction of illumination.

The effect of quartz fiber orientation on the average values of the absorption efficiency factor and transport efficiency factor of scattering is illustrated in Fig. 5. The coincidence of Qa for isotropic material and a for transversely isotropic material is not occasional. These values are exactly the same. But it is important that they are well approximated by the simple formula (5) over the infrared spectrum (i.e., the contribution of absorption resonances is small). The difference between Qstr for isotropic material and str for transversely isotropic material is considerable, but approximation (5) gives an intermediate result. It enables us to recommend the simple expression (5) for engineering estimates of average characteristics of quarts fibrous materials.

Figure 5. Efficiency factor of absorption and transport efficiency factor of scattering for quartz fibers of radius a = 5 μm illuminated by randomly polarized radiation: 1, average values Qa, Qstr for fibers randomly oriented in space; 2, average values a, str for fibers randomly oriented in parallel planes; 3, approximation (5) based on the data for normal incidence.

As to verification of the general procedure, one can remember a comparison with experimental results by Kondratenko et al. (1991) and Moiseyev et al. (1992) for highly porous isotropic quartz insulation with known fiber size distribution and porosity 93.5% (the density is 144 kg/m3), and with measurements by Nicolau et al. (1994) for a fiberglass insulation of density 68 kg/m3. The computational results for the spectral radiation extinction and radiation diffusion coefficient in the wavelength range for 0.6 to 5 μm appeared to be in good agreement with these experiments. The details are not reproduced here but can be found in archive papers by Dombrovsky (1994, 1996a). A detailed work on verification of the computational model based on independent scattering hypothesis and the scattering problem solution for infinite cylinders has been conducted by Cunnington and Lee (1996). They considered practically the same fibrous material as that studied previously in Kondratenko et al. (1991), Moiseyev et al. (1992), and Dombrovsky (1994, 1996a). It was a bonded silica fiber rigid insulation that were used for the thermal protection tiles of the space shuttle Orbiter. This material has a typical bulk density of 145 kg/m3, but the samples were made of three different types of fibers. The hemispherical reflectance measurements were made over the wavelength interval from 1.5 to 10 μm, and the normal transmittance measurements in the wavelength interval from 1.5 to 5.5 μm. Both hemispherical reflectance and normal transmittance were compared with the theoretical predictions. A good agreement between theory and experimental data obtained by Cunnington and Lee (1996) demonstrated the validity of the theoretical model for the prediction of the radiative properties of high-porosity fibrous media having nearly random fiber orientation. One should also remember the qualitative agreement of computational results of Dombrovsky (1996b) with experimental data by Yeh and Roux (1988) for fiberglass insulation of density 10.9 kg/m3 in the opacity region.

After preparing this article, the author has read a recent experimental paper by Kitamura et al. (2007) concerning wide-range optical constants of silica glass. One can repeat the above calculations for more accurate spectral data for fused quartz optical constants. But there is no doubt that the qualitative results will be the same.

An additional confirmation of the applicability of the theoretical model based on the independent scattering approximation was obtained by Milandri et al. (2002). An inverse method based on experimental measurements of bidirectional reflection and transmission was used to determine the radiative properties of silica wool samples with an average fiber diameter of 8 μm and mass per unit area in the range from 50 to 80 g/m2. The results at wavelength λ = 4 μm were proven to be in good agreement with Mie theory calculations.

Another interesting estimate of the dependent scattering effects in a medium containing randomly oriented fibers in the geometrical optics limit have been recently reported by Coquard and Baillis (2006). The results of a Monte Carlo simulation were approximated by the following correlation for the extinction coefficient:

(11)

where βind is the extinction coefficient determined by use of an independent scattering approximation. It should be noted that the “geometrical” estimate indicates relatively small deviation from the independent scattering as compared with the case of opaque spherical particles (Coquard and Baillis, 2004).

REFERENCES

Banner, D., Klarsfeld, S., and Langlais, C., Temperature Dependence of the Optical Characteristics of Semitransparent Porous Media, High Temp.-High Press., vol. 21, no. 3, pp. 347-354, 1989.

Beder, E. C., Bass, C. D., and Shackleford, W. L., Transmissivity and Absorption of Fused Quartz Between 0.2μm and 3.5μm from Room Temperature to 1500 Degree C, Appl. Opt., vol. 10, no. 10, pp. 2263-2268, 1971.

Brixner, B., Refractive-Index Interpolation for Fused Silica, J. Opt. Soc. Am., vol. 57, no. 5, pp. 674-676, 1967.

Coquard, R. and Baillis, D., Radiative Properties of Dense Fibrous Medium Containing Fibers in the Geometrical Limit, ASME J. Heat Transfer, vol. 128, no. 10, pp. 1022-1030, 2006.

Coquard, R. and Baillis, D., Radiative Characteristics of Opaque Spherical Particles Beds: A New Method of Prediction, J. Thermophys. Heat Transfer, vol. 18, no. 2, pp. 178-186, 2004.

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.

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, 1994.

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, 1996a.

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

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

Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L., Use of Mie Theory to Analyze Experimental Data to Identify Infrared Properties of Fused Quartz Containing Bubbles, Appl. Opt., vol. 44, no. 33, pp. 7021-7031, 2005.

Khashan, M. A. and Nassif, A. Y., Dispersion of the Optical Constants of Quartz and Polymethyl Methacrylate Glasses in a Wide Spectral Range: 0.2-3μm, Optics Commun., vol. 188, no. 1-4, pp. 129-139, 2001.

Kitamura, R., Pilon, L., and Jonasz, M., Optical Constants of Silica Glass from Extreme Ultraviolet to Far Infrared at Near Room Temperatures, Appl. Opt., vol. 46, no. 33, pp. 8118-8133, 2007.

Kondratenko, A. V., Moiseyev, S. S., Petrov, V. A., and Stepanov, S. V., Experimental Determination of Optical Properties of Fiber Quartz Heat-Shielding Material, High Temp., vol. 29, no. 1, pp. 126-130, 1991.

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.

Malitson, I. H., Interspecimen Comparison of the Refractive Index of Fused Silica, J. Opt. Soc. Am., vol. 55, no. 10, pp. 1205-1209, 1965.

Milandri, A., Asllanaj, F., and Jeandel, G., Determination of Radiative Properties of Fibrous Media by an Inverse Method--Comparison with the Mie Theory, J. Quant. Spectrosc. Radiat. Transfer, vol. 74, no. 5, pp. 637-653, 2002.

Moiseyev, S. S., Petrov, V. A., and Stepanov, S. V., Optical Properties of High-Temperature Fibrous Silica Thermal Insulation, High Temp.-High Press., vol. 24, no. 8, pp. 391-402, 1992.

Nicolau, V. P., Raynaud, M., and Sacadura, J.-F., Spectral Radiative Properties Identification of Fiber Insulating Materials, Int. J. Heat Mass Transfer, vol. 37, Suppl. 1, pp. 311-324, 1994.

Petrov, V. A., Optical Properties of Silica Glasses at High Temperatures in the Region of Semi-Transparency, Rev. Thermophys. Propert. Subst., vol. 3, no. 17, pp. 30-72, 1979 (in Russian).

Tan, C. Z., Determination of Refractive Index of Silica Glass for Infrared Wavelength by IR Spectroscopy, J. Non-Cryst. Solids, vol. 223, no. 1-2, pp. 158-163, 1998.

Tan, C. Z. and Arndt, J., Refractive Index, Optical Dispersion, and Group Velocity of Infrared Waves in Silica Glass, J. Phys. Chem. Solids, vol. 62, no. 6, pp. 1087-1092, 2001.

Touloukian, Y. S. and DeWitt, D. P. (eds.), Thermal Radiative Properties: Nonmetallic Solids, vol. 8 of Thermopysical Properties of Matter, Plenum Press, New York, 1972.

Yeh, H. Y. and Roux, J. A., Spectral Radiative Properties of Fibreglass Insulation, J. Thermophys. Heat Transfer, vol. 2, no. 1, pp. 75-81, 1988.

References

  1. Banner, D., Klarsfeld, S., and Langlais, C., Temperature Dependence of the Optical Characteristics of Semitransparent Porous Media, High Temp.-High Press., vol. 21, no. 3, pp. 347-354, 1989.
  2. Beder, E. C., Bass, C. D., and Shackleford, W. L., Transmissivity and Absorption of Fused Quartz Between 0.2μm and 3.5μm from Room Temperature to 1500 Degree C, Appl. Opt., vol. 10, no. 10, pp. 2263-2268, 1971.
  3. Brixner, B., Refractive-Index Interpolation for Fused Silica, J. Opt. Soc. Am., vol. 57, no. 5, pp. 674-676, 1967.
  4. Coquard, R. and Baillis, D., Radiative Properties of Dense Fibrous Medium Containing Fibers in the Geometrical Limit, ASME J. Heat Transfer, vol. 128, no. 10, pp. 1022-1030, 2006.
  5. Coquard, R. and Baillis, D., Radiative Characteristics of Opaque Spherical Particles Beds: A New Method of Prediction, J. Thermophys. Heat Transfer, vol. 18, no. 2, pp. 178-186, 2004.
  6. 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.
  7. 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, 1994.
  8. 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, 1996a.
  9. Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, New York and Redding, CT, 1996b.
  10. Dombrovsky, L. A., Radiative Properties of Metalized-Fiber Thermal Insulation, High Temp., vol. 35, no. 2, pp. 275-282, 1997.
  11. Dombrovsky, L., Randrianalisoa, J., Baillis, D., and Pilon, L., Use of Mie Theory to Analyze Experimental Data to Identify Infrared Properties of Fused Quartz Containing Bubbles, Appl. Opt., vol. 44, no. 33, pp. 7021-7031, 2005.
  12. Khashan, M. A. and Nassif, A. Y., Dispersion of the Optical Constants of Quartz and Polymethyl Methacrylate Glasses in a Wide Spectral Range: 0.2-3μm, Optics Commun., vol. 188, no. 1-4, pp. 129-139, 2001.
  13. Kitamura, R., Pilon, L., and Jonasz, M., Optical Constants of Silica Glass from Extreme Ultraviolet to Far Infrared at Near Room Temperatures, Appl. Opt., vol. 46, no. 33, pp. 8118-8133, 2007.
  14. Kondratenko, A. V., Moiseyev, S. S., Petrov, V. A., and Stepanov, S. V., Experimental Determination of Optical Properties of Fiber Quartz Heat-Shielding Material, High Temp., vol. 29, no. 1, pp. 126-130, 1991.
  15. 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.
  16. 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.
  17. Malitson, I. H., Interspecimen Comparison of the Refractive Index of Fused Silica, J. Opt. Soc. Am., vol. 55, no. 10, pp. 1205-1209, 1965.
  18. Milandri, A., Asllanaj, F., and Jeandel, G., Determination of Radiative Properties of Fibrous Media by an Inverse Method--Comparison with the Mie Theory, J. Quant. Spectrosc. Radiat. Transfer, vol. 74, no. 5, pp. 637-653, 2002.
  19. Moiseyev, S. S., Petrov, V. A., and Stepanov, S. V., Optical Properties of High-Temperature Fibrous Silica Thermal Insulation, High Temp.-High Press., vol. 24, no. 8, pp. 391-402, 1992.
  20. Nicolau, V. P., Raynaud, M., and Sacadura, J.-F., Spectral Radiative Properties Identification of Fiber Insulating Materials, Int. J. Heat Mass Transfer, vol. 37, Suppl. 1, pp. 311-324, 1994.
  21. Petrov, V. A., Optical Properties of Silica Glasses at High Temperatures in the Region of Semi-Transparency, Rev. Thermophys. Propert. Subst., vol. 3, no. 17, pp. 30-72, 1979 (in Russian).
  22. Tan, C. Z., Determination of Refractive Index of Silica Glass for Infrared Wavelength by IR Spectroscopy, J. Non-Cryst. Solids, vol. 223, no. 1-2, pp. 158-163, 1998.
  23. Tan, C. Z. and Arndt, J., Refractive Index, Optical Dispersion, and Group Velocity of Infrared Waves in Silica Glass, J. Phys. Chem. Solids, vol. 62, no. 6, pp. 1087-1092, 2001.
  24. Touloukian, Y. S. and DeWitt, D. P. (eds.), Thermal Radiative Properties: Nonmetallic Solids, vol. 8 of Thermopysical Properties of Matter, Plenum Press, New York, 1972.
  25. Yeh, H. Y. and Roux, J. A., Spectral Radiative Properties of Fibreglass Insulation, J. Thermophys. Heat Transfer, vol. 2, no. 1, pp. 75-81, 1988.
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