INFRARED PROPERTIES OF CARBON FIBERS
Following from: The scattering problem for cylindrical particles
Spectral radiative properties of carbon fibers in the near-infrared spectral range are of interest for the calculation of heat transfer during thermal treatment of fibers in radiation-heating furnaces (Konkin, 1974; Morgan, 2005) and for the analysis of properties of some specific high-porosity fibrous materials (Alifanov et al., 1985).
The spectral dependences of both absorption and scattering characteristics of carbon fibers are much simpler than those for fibers of semitransparent substances. At the same time, there is a considerable uncertainty in optical constants of the material of fibers because of possible chemical impurities and complex microstructure of the fibers. Following the paper by Dombrovsky (1994), we use the spectral optical constants of soot obtained by Dalzell and Sarofim (1969) to calculate the radiative properties of carbon fibers. Some computational results for the above-introduced dimensionless characteristics of radiation absorption and scattering by fibers of radius a ≥ 2 μm are presented in Figs. 1a and 1b.
Figure 1. Absorption and scattering of infrared radiation by carbon fibers illuminated along the normal (a) and randomly oriented in space (b) in comparison to similar characteristics for spherical particles (c) (1, a = 2 μm; 2, a = 5 μm; 3, a = 20 μm).
At normal incidence, the spectral dependences Qa(λ) and Qstr(λ) for single fibers in the range of λ > 2 μm are monotonic: the absorption decreases and the scattering increases with the wavelength. Both the efficiency factor of absorption and the transport efficiency factor of scattering decrease with increasing fiber radius, but the dependences on the radius are comparatively weaker, especially for thick fibers (at a > 5 μm). The spectral characteristics Qa and Qstr of absorption and scattering of radiation by an elementary volume of the medium formed by randomly oriented fibers differ insignificantly from the values of Qa and Qstr calculated for single fibers illuminated along the normal (compare Figs. 1a and 1b). In the spectral region considered, we have
The change in the radiative characteristics of randomly oriented carbon fibers in the wavelength range 1 < λ < 10 μm for a ≥ 2 μm can be described (with an accuracy of 10%) by the following approximate relations (Dombrovsky, 1994):
where a and λ are expressed in microns. Equations (2) correspond to the adopted spectral dependence of optical constants. For a different spectral dependence of the complex index of refraction, the exact calculations should be used again. In the case of randomly oriented monodisperse fibers, the absorption coefficient and transport scattering coefficient of the fibrous material are expressed as
According to Eq. (3), the material having the same density (or volume fraction of fibers fv), but formed by thicker fibers, has smaller absorption and scattering coefficients (i.e., is more transparent for thermal radiation) because Qa and Qstr weakly depend on the fiber radius.
The calculations for randomly oriented fibers are rather complicated. Therefore, in would be interesting to analyze the possibility of using a simpler and more accessible theoretical model, i.e., a disperse system of spherical particles of the same radius. In this case, we have the following relations instead of (3):
A comparison of Figs. 1b and 1c shows that in spite of satisfactory agreement between spectral dependences, this model gives approximately doubled values, both for absorption and for scattering of radiation by the disperse system.
Alifanov, O. M., Gerasimov, B. P., Elizarova, T. G., Zantsev, V. K., Chetverushkin, B. N., and Shil'nikov, E. V., Mathematical Modeling of Combined Heat Transfer in Dispersed Materials, J. Eng. Phys. Thermophys., vol. 49, no. 5, pp. 781–791, 1985.
Dalzell, W. H. and Sarofim, A. P., Optical Constants of Soot and their Application to Heat-Flux Calculations, ASME J. Heat Transfer, vol. 91, no. 1, pp. 100–104, 1969.
Dombrovsky, L. A., Calculation of Infrared Radiative Properties of Carbon Fibers and Fibrous Materials, High Temp., vol. 32, no. 6, pp. 895–898, 1994.
Konkin, A. A., Carbonic and Some Others Heatproof Fibrous Materials, Khimiya, Moscow (in Russian), 1974.
Morgan, P., Carbon Fibers and Their Composites, CRC Press, New York, 2005.