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In one of the earlier publications on the computational analysis of the radiative properties of fibrous materials by Wang and Tien (1983), it was shown that material made of aluminum fibers of about 0.1 μm diameter may have a very high absorption coefficient. A considerable reduction of the radiative conductivity of a material made of thin fibers with high electrical conductivity was also discussed by McKay et al. (1984). Because it is very difficult to manufacture metallic fibers of such thinness, attempts were later made at applying a metal coating to glass or polymer fibers (Reiss et al., 1987). Information on experimental investigations in this field can be found in the review papers by Büttner et al. (1989) and Reiss (1990). Among more recent works, experiments by Ebert et al. (1991) for fibers of diameter d ≈ 0.1 μm at aluminum coating thickness δc ≈ 50 nm should be mentioned, as well as the paper by Caps et al. (1993), in which some data for materials of metalized polypropylene fibers of diameter d = 2-3 μm were reported.

Following papers by Dombrovsky (1997, 1998a,b), we present the analysis of the main radiative properties of metalized fibers based on the most general statement of the scattering problem, allowing for the small thickness of the coating being applied. Simple estimates show that the radiative properties of metalized fibers in the infrared region should not differ substantially from the respective quantities for homogeneous metal fibers of the same radius. Indeed, the thickness of the skin layer, in which the field strength of an incident electromagnetic wave decreases by a factor of e, is determined by the following formula (Born and Wolf, 1999):

(1)

Hence it follows that, for example, at the wavelength of λ = 10 μm for typical values of the absorption index for metals κ ~ 20-100 (Ordal et al., 1983), the value of δsk is only several hundredths of a micron. In the case of thin fibers coated by a metal layer of thickness δ ~ 10 nm, one should, generally speaking, apply a rigorous solution for radiation scattering by a two-layer cylinder.

The effect of a thin surface layer of aluminum on the infrared absorption and scattering by randomly oriented polymer fibers is illustrated in Fig. 1. The assumed value of m = 1.7-0.02i corresponds approximately to polyester properties. The following range of values is used for aluminum: m2 = 31.2 - 104i 20.18-57.93i, from the experimental data by Lenham and Treherne (1966) cited by Brennon and Goldstein (1970); and m''2 = 31.2-104i, from the review by Ordal et al. (1983). The results of the calculation indicate that the optical constants of a polymer do not affect the properties of metalized fibers.

Figure 1. Absorption and scattering of randomly polarized radiation by randomly oriented fibers at the wavelength λ = 10 μm: (a) polymer fibers; (b) the same fibers with aluminum coating of thickness δc = 10 nm (1, 2) or 50 nm (3, 4) at complex index of refraction m''1 (1, 3) or m''2 (2, 4).

As is seen in Fig. 1, the uncertainty of the optical constants of aluminum has almost no effect on the scattering, even for very thin fibers, but it should be taken into account in the absorption calculations at fiber radius a < 0.3 μm, when the radiation absorption by metalized fibers becomes considerable. The increase in the aluminum layer thickness from 10 to 50 nm causes a decrease in the absorption by a factor of 3 or 4, with a minor increase in scattering. A gradual increase in Qa with the decrease in the fiber radius implies an approach to the zone of Rayleigh scattering, whose boundary is shifted to the region of extremely thin fibers because of the metal coating.

Data presented in Fig. 1 allow evaluating the region of degeneration of the solution for two-layer cylinders. One can see that extinction of the infrared radiation by metalized fibers of radius a > 0.4 μm coincides with that of homogeneous metal fibers, even for very thin aluminum coatings. Moreover, such fibers do not practically absorb the radiation, and the scattering depends very little on the complex index of refraction and can be calculated using the approximation of a perfectly conducting cylinder. The derived results make it possible to represent the optical properties of a single metalized fiber in a much more compact form, because only two independent variables remain for fibers that are not too thin: the diffraction parameter x and the incidence angle α. It is also convenient to consider separate functions Qstr(x,0), str(x), Qstr(x) and the angular dependencies of Qstr at fixed diffraction parameters instead of the function Qstr(x,α). The appropriate numerical data are presented in Fig. 2.

Figure 2. Transport efficiency factor of scattering for homogeneous perfectly conducting metal fibers illuminated by randomly polarized radiation: (a) dependences on diffraction parameter [1, at normal incidence; 2, 3, average values for fibers randomly oriented in parallel planes (2) or in space (3)]; (b) angular dependences Str(α) for single fibers.

Figure 2b indicates that the above-introduced angular function Str(α) is practically universal, i.e., independent of the diffraction parameter. As a result, the curves in Fig. 2a are almost similar to each other, and the radiation scattering by fibers randomly oriented in space can be evaluated as follows:

(2)

Here, k ≈ 0.6, while the value k ≈ π/4, derived above for Str(α) = cosα, is usually acceptable for dielectric fibers. The difference between the scattering efficiency factors for randomly oriented fibers and for fibers illuminated in the normal direction should be taken into account in radiation transfer calculations. This problem has been discussed in some detail by Lee (1989). Note that a smaller value of k for metalized fibers implies a more substantial effect of the orientation of fibers in the material.

It should be noted that a sharp decrease in Qstr when x is in the x < 0.3 range is replaced by the region of its approximate constancy, and even a local maximum is available at x ≈ 0.82 for Qstr(x,0), and at x ≈ 0.86 for str(x) and Qstr(x). High values of Qstr at small values of x represent the main advantage of metalized fibers, because the specific spectral extinction coefficient for a monodisperse material,

(3)

where ρf is the density of the material of fibers, increases considerably at small x. Naturally, a high density of fibrous material may cause a violation of the initial assumption about independent scattering of radiation by single fibers. This problem is a subject for special studies (Lee and Cunnington, 1998).

For wider practical implementation of metalized (especially aluminized) fibers, it is of interest to take into account an effect of metal oxidation. One can try to perform calculations for such three-layer fibers by use of the general solution for multilayer cylinders derived by Barabas (1987) and by Swathi et al. (1991) in the form of coupled algebraic equations for the Mie coefficients. But direct calculation with reasonable accuracy appears to be impossible due to very high values of indices n,κ in a thin metal layer. Fortunately, in the case of a sufficiently thick remaining layer of metal (for aluminum, δc > 10 nm), one can employ the solution for two-layer fiber: metal with oxide shell. Calculations at fixed external radius of the fiber showed that the maximum effect of the oxide layer takes place in the spectral range near the absorption band of oxide where n << 1, and κ is not too small. The corresponding results for an aluminized fiber with an oxide shell are given in Fig. 3. The following approximation of aluminum oxide optical properties from the review by Lingart et al. (1982) in the range 10.5 < λ < 11.5μm is used:

(4)

where λ is expressed in microns. Figure 3 indicates that an aluminum oxide layer of thickness δox > 10 nm results in a measurable increase in the extinction because of the radiation absorption in the oxide shell. It would be interesting to verify this theoretical result by comparison with a special experiment.

Figure 3. Effect of oxide layer on absorption and extinction of randomly polarized radiation by polymer fibers of radius a = 1 μm with opaque aluminum coating at normal incidence.

In Fig. 4, a comparison of the specific spectral extinction coefficient calculations for metalized polypropylene fibers (ρf = 900 kg/m3) with the experimental data by Caps et al. (1993) is presented. The thickness of the aluminum coating δc > 100 nm is sufficiently large to employ the homogeneous fiber approximation. The calculations have been performed for the monodisperse system with the fiber radius of a = 1.25 μm, which is the arithmetic mean for the 1 < a < 1.5 μm range, and for the polydisperse system with equal numbers of fibers having radii of a1 = 1 μm, a2 = 1.25 μm, and a3 = 1.5μm,

Figure 4. Specific transport coefficient of extinction for a material of aluminum-coated polypropylene fibers: 1, experimental data by Caps et al. (1993); 2, calculations for monodisperse system with the fiber radius a = 1.25 μm; 3, calculation for polydisperse system with the fiber size distribution (5); (a) EtrEtr (normal incidence);(b) Etr (fibers randomly oriented in parallel planes); (c) equivalent value Etr*.

(5)

This triple δ distribution function corresponds to information by Caps et al. (1993) on fiber radius distribution in the experimental sample of polypropylene-microfiber fleece.

In the experiment by Caps et al. (1993), a thin sample of material was illuminated along a normal; therefore, the measured values of the specific extinction coefficient are expected to lie between the curve Etr(λ) for a normal incidence and the curve Etr(λ) for a transversally isotropic material. An evaluation of the relationship between specific extinction coefficient determined in the experiment and calculated parameters Etr, Etr can be derived from the approximate solution of the problem on reflection of the normally incident radiation by a layer of purely scattering transversally isotropic material with the angular-dependent scattering coefficient. In the P1 approximation, the expression for the directional-hemispherical reflectivity is (Dombrovsky, 1997)

(6)

where τλ = Eλρh is the optical thickness [ρ is the fibrous material density, h is the layer thickness), Dλ =Etr/(3Etr) is the dimensionless radiative diffusion coefficient, instead of 1/3 for isotropic material]. Note that Eq. (6) gives fairly accurate values of Rλ (with an error of <1%) in the case of isotropic material. It is easy to verify that the value Rλ at a small optical thickness of the specimen depends only on the normal incidence extinction coefficient. For an optically thick layer of material, we have

(7)

and the following value is actually determined in the experiment:

(8)

The substitution of the values of Eλ/Eλ from 1.34 to 1.37, derived for metalized fibers, yields

(9)

One can see in Fig. 4 that calculated spectral dependencies of the equivalent specific extinction coefficient Etr* are in good agreement with the experimental data by Caps et al. (1993). At the same time, the calculated local maximum of extinction is shifted relative to the experimental one to the long-wave region by Δλ = 1.5 μm. This effect can be explained by a fiber radius about 10% too high in the calculations. Comprehensive data for fiber size distribution and on their orientation in the material are necessary to make a more accurate analysis of the experimental data. Note that oxidation of the aluminum coating was not observed in the experiment, and the measured values of albedo over the spectrum were >0.9.

It is difficult to obtain reliable experimental data for the infrared extinction by extrafine metalized fibers because of the great optical thickness of the sample. For this reason, the value Etr = 4 m2/g, given by Ebert et al. (1991) for glass fibers of radius a = 50 nm with the aluminum coating of thickness δc = a, is treated by the authors as a minimum estimate for the spectral range of 2 < λ < 40 μm. The calculated values of Eλ for normal radiation incidence vary from 4.0 at λ = 2 μm to 11.2 m2/g at λ = 20 μm. This is in good agreement with the experimental estimate by Ebert et al. (1991). Note that the optical inhomogeneity of the metalized glass fiber may be not taken into account at a = δc = 50 nm, and the approximation of m = ∞ gives values of Etr only ~10% too large. Calculations by Dombrovsky (1997) for thinner fibers illustrate the possible effect of the optical inhomogeneity of fibers on the extinction coefficient.

The metal coating of fibers appears to be very effective for developing the materials with a high transport extinction coefficient. The same idea has been earlier discussed by Tien and Cunnington (1972) and Cunnington and Tien (1973). Their estimates showed that metal-coated hollow glass microspheres of sizes ranging from 15 to 150 μm could be a good solution for high-performance cryogenic thermal insulation. Compared to solid spheres, hollow spheres increase substantially the constriction resistance to conduction, and reduce heat capacity and weight. It is important that coated thin metal films provide effective radiation shielding with little increase in conduction.

It is interesting to treat the properties of metalized fibers in the microwave region, where the general solution does not degenerate into the limiting case of homogeneous fibers, or even to the asymptotic solution for m = ∞. The calculations of the radiative properties of these fibers in the millimeter and centimeter spectral ranges are of both theoretical and practical interest in view of the possibility of using light, highly porous materials with metalized fibers in microwave technology. As has been noted by Reiss (1990) and Ebert et al. (1991), such materials have advantages as radiation screens not only in the infrared. but also in the millimeter wavelength region and, for example, metal (or metalized) fibers woven into a glass fiber mat could make possible the production of fiber-reinforced materials for housing microwave components.

The complex index of refraction of a dielectric fiber in a microwave field may be taken to be equal to unity. For the metal of the fiber coating, the Hagen-Rubens relation can be used,

(10)

where σe = 3.72 × 107 S/m is the electric conductivity, ε0 = 8.86 × 10-12 A·s/(V·m) is the dielectric constant, and c = 3 × 108 m/s is the velocity of light. For aluminum, the resulting value of >m> varies from 1500 to 15,000 in the range of 1 < λ < 10 mm. Therefore, the microwave radiative properties of metalized fibers (even of micron radius) cannot be calculated by the Rayleigh approximation. Nevertheless, at fiber radius a = 1 μm we have >m>x < 10. For this reason, the numerical calculations for thin metalized fibers in the microwave range are comparably simple. Calculations with a variation in the coating thickness show that the homogeneous metal fiber approximation is not applicable even for δc = 50 nm (see Fig. 5): a decrease in δc from 50 to 10íì nm leads to significant absorption, which increases in the millimeter range. In other words, the absorption of radiation by a single metalized fiber can be controlled by varying the coating thickness. The angular dependences of the absorption efficiency factor shown in Fig. 5b have a maximum at oblique illumination of the fiber. As a result, the relationship between the average values tr, Qtr and the efficiency factor at normal incidence is not simple. Note that unusually high values of the absorption and scattering efficiency factors are associated with the incident radiation component polarized in the plane parallel to the fiber axis.

Figure 5. Efficiency factor of absorption and transport efficiency factor of scattering for fibers of radius a = 1 μm illuminated by randomly polarized radiation: (a) spectral dependences at normal incidence; (b) angular dependences at λ = 5 mm [1, homogeneous aluminum fiber; 2, 3, polymer fiber with aluminum coating of thickness δc = 50 nm (2) or δc = 10 nm (3)].

The calculated microwave characteristics of submicron fibers coated with aluminum are given in Fig. 6, where a is the radius of the dielectric fiber substrate. One can see that absorption predominates in almost the entire spectral range. It is interesting that in the Rayleigh region, the value of Qa does not depend on a, but it is proportional to the fiber shell thickness δc, in contrast to relation Qa ~ a for homogeneous metal fiber. As is seen from comparison of Figs. 6a and 6b, the spectral boundary of the Rayleigh region is also proportional to δc. It should be noted that there is a sharp increase of absorption in the millimeter range with a decrease in the fiber radius: the transition from a = 0.5 μm to 0.2 μm produces an increase in Qa at λ = 2 mm of 3.2 times for δc =10 nm and 7 times for δc = 50 nm. At the same time, Qtr increases by a factor of 8 or 17.5, respectively. Extremely high values of the absorption and scattering efficiency factors for metalized fibers in the microwave range bring to the fore the above-mentioned problem of electromagnetic interaction of closely spaced fibers.

Figure 6. Efficiency factor of absorption and transport efficiency factor of scattering for metalized fibers at normal incidence of randomly polarized radiation: (a) δc = 10 nm; (b) δc = 50 nm (1, a = 0.1 μm; 2, a = 0.2 μm; 3, a = 0.5 μm).

Microwave radiation, unlike infrared radiation, is, as a rule, polarized. For this reason, the numerical simulation of the microwave radiation transfer in materials containing metalized fibers, generally speaking, should be based on the vector radiative transfer equation, taking into account both the material anisotropy and a change of the polarization state by scattering.

REFERENCES

Barabas, M., Scattering of a Plane Wave by a Radially Stratified Tilted Cylinder, J. Opt. Soc. Am., vol. 4, no. 12, pp. 2240-2248, 1987.

Born, M. and Wolf, E., Principles of Optics, Seventh (expanded) edition, Cambridge University Press, New York, 1999.

Brennon, R. R. and Goldstein, R. J., Emittance of Oxide Layers on a Metal Substrate, ASME J. Heat Transfer, vol. 92, no. 2, pp. 257-263, 1970.

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.

Cunnington, G. R. and Tien, C. L., Heat Transfer in Microsphere Cryogenic Insulation, Advances in Cryogenic Engineering, vol. 18, Timmerhaus, K. D. (ed.), Plenum Press, New York, 1973.

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.

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, 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.

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

Lenham, A. P. and Treherne, D. M., Optical Constants of Single Crystals of Mg, Zn, Cd, Al, Ga, In, and White Sn, J. Opt. Soc. Am., vol. 56, no. 6, pp. 752-756, 1966.

Lingart, Yu. K., Petrov, V. A., and Tikhonova, N. A., Optical Properties of Synthetic Sapphire at High Temperatures. II. Properties of Monocrystal in Opacity Region and Melt Properties, High Temp., vol. 20, no. 6, pp. 1085-1092, 1982.

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.

Ordal, M. A., Long, L. L., Bell, R. J., Bell, S. E., Bell, R. R., Alexander, Jr., R. W., and Ward, C. A., Optical Properties of the Metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the Infrared and Far Infrared, Appl. Opt., vol. 22, no. 7, pp. 1099-1119, 1983.

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. Spectrosc. Radiat. Transfer, vol. 46, no. 4, pp. 281-292, 1991.

Tien, C. L. and Cunnington, G. R., Recent Advances in High-Performance Cryogenic Thermal Insulations, Cryogenics, vol. 12, no. 6, pp. 419-421, 1972.

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

References

  1. Barabas, M., Scattering of a Plane Wave by a Radially Stratified Tilted Cylinder, J. Opt. Soc. Am., vol. 4, no. 12, pp. 2240-2248, 1987.
  2. Born, M. and Wolf, E., Principles of Optics, Seventh (expanded) edition, Cambridge University Press, New York, 1999.
  3. Brennon, R. R. and Goldstein, R. J., Emittance of Oxide Layers on a Metal Substrate, ASME J. Heat Transfer, vol. 92, no. 2, pp. 257-263, 1970.
  4. 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.
  5. Cunnington, G. R. and Tien, C. L., Heat Transfer in Microsphere Cryogenic Insulation, Advances in Cryogenic Engineering, vol. 18, Timmerhaus, K. D. (ed.), Plenum Press, New York, 1973.
  6. 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.
  7. Dombrovsky, L. A., Radiative Properties of Metalized-Fiber Thermal Insulation, High Temp., vol. 35, no. 2, pp. 275-282, 1997.
  8. 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).
  9. Dombrovsky, L. A., Infrared and Microwave Radiative Properties of Metal Coated Microfibers, Rev. Gener. Therm., vol. 37, no. 11, 925-933, 1998b.
  10. 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.
  11. 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.
  12. Lee, S.-C. and Cunnington, G. R., Theoretical Models for Radiative Transfer in Fibrous Media, Annual Review in Heat Transfer, vol. 9, Tien, C. L. (ed.), Begell House, New York and Redding, CT, pp. 159-218, 1998.
  13. Lenham, A. P. and Treherne, D. M., Optical Constants of Single Crystals of Mg, Zn, Cd, Al, Ga, In, and White Sn, J. Opt. Soc. Am., vol. 56, no. 6, pp. 752-756, 1966.
  14. Lingart, Yu. K., Petrov, V. A., and Tikhonova, N. A., Optical Properties of Synthetic Sapphire at High Temperatures. II. Properties of Monocrystal in Opacity Region and Melt Properties, High Temp., vol. 20, no. 6, pp. 1085-1092, 1982.
  15. 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.
  16. Ordal, M. A., Long, L. L., Bell, R. J., Bell, S. E., Bell, R. R., Alexander, Jr., R. W., and Ward, C. A., Optical Properties of the Metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the Infrared and Far Infrared, Appl. Opt., vol. 22, no. 7, pp. 1099-1119, 1983.
  17. Reiss, H., Radiative Transfer in Nontransparent Dispersed Media, High Temp.-High Press., vol. 22, no. 5, pp. 481-522, 1990.
  18. 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.
  19. 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. Spectrosc. Radiat. Transfer, vol. 46, no. 4, pp. 281-292, 1991.
  20. Tien, C. L. and Cunnington, G. R., Recent Advances in High-Performance Cryogenic Thermal Insulations, Cryogenics, vol. 12, no. 6, pp. 419-421, 1972.
  21. Wang, K. Y. and Tien, C. L., Radiative Heat Transfer Through Opacified Fibers and Powders, J. Quant. Spectrosc. Radiat. Transfer, vol. 30, no. 3, pp. 213-223, 1983.
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