HIGHLY POROUS ISOTROPIC AND ANISOTROPIC FIBROUS MATERIALS
Following from: Spectral radiative properties of some important materials: Experimental data and theoretical models
Highly porous fibrous materials present thermal insulating performance for use in aerospace, automotive, marine, and building applications. Building thermal insulation is a main scope of applications (see the article Spectral radiative properties of some important materials: Experimental data and theoretical models). It can be noted that glass wools are the most widely sold materials for the building thermal insulation. The radiative properties of fibrous media containing glass–wool or carbon or aramid fibers can be modeled using the absorption and scattering characteristics of infinite circular cylinders, which can be randomly oriented or have particular orientations inducing an isotropic or anisotropic structure, respectively. The radiative properties of fibrous materials used for thermal insulation (mostly low-density materials) have long been investigated through theoretical modeling (Tong and Tien, 1980; Houston and Korpela, 1982; Lee, 1986, 1989, 1990, 1993; Boulet et al., 1993, 1994; Jeandel et al., 1993; Cunnington and Lee, 1996; Marschall and Milos, 1997; Lee and Cunnington, 1998, 2000; Heino et al., 2003; Kamdem Tagne and Baillis, 2005a). Experimental identification of porous fibrous materials has also been studied (Mathews et al., 1984; Nicolau et al., 1994; Milandri et al., 2002; Wan et al., 2009). Many fibrous materials, such as highly porous glass–wool insulation, have an anisotropic structure with fibers randomly oriented in a plane parallel to the boundaries. This can lead to anisotropic radiative properties of the material (Fig. 1), which is a particular source of complexity for radiative properties characterization. Therefore, recently particular attention has been paid to some simplified radiation models (Kamdem Tagne and Baillis, 2005b).
Figure 1. Scanning electron micrograph photograph of a glass–wool insulating media.
Radiative Properties Modeling
The fibers of ordinary thermal insulation, such as glass or carbon or aramid fibers, can be considered as being infinitely long circular cylinders because their length (a few millimeters) is much greater than the radiation wavelength and their diameter varies from 1 to 15 μm. Analysis of the radiative properties of highly porous fibrous materials based on the Mie theory for single cylinders was reported by Dombrovsky and Baillis (2010) [see also the article Radiative properties of particles and fibers (theoretical analysis)]. The dependence of the radiative properties on the incident radiation direction increases the complexity of the radiative transfer problem. For this reason, it is convenient to introduce the efficiency factors averaged over orientations and to model the radiative heat transfer in an equivalent isotropic medium. In addition, the scattering phase function obtained using the Mie theory may undergo strong angular oscillations and is characterized by a strong forward scattering peak. Therefore, it is also convenient to simplify the scattering function by the use of isotropic or Henyey–Greenstein approximations. Some reduced theoretical models have been proposed in the literature (Kamdem Tagne and Baillis, 2005b, 2010). These models can be classified into two groups in which the scattering phase function is considered to be either (1) isotropic or (2) anisotropic. Kamdem Tagne and Baillis (2010) showed that radiative heat transfer through anisotropic medium can be analyzed using simplified models of an anisotropic medium or an equivalent isotropic medium with average radiative properties and an isotropic or Henyey–Greenstein scattering phase function.
Identification of Radiative Properties
The radiative properties identification method requires experimental and theoretical data for both transmittance and reflectance obtained for the collimated incident radiation on a sample. For simplicity, the identification of fibrous media radiative properties is generally based on the assumption of the isotropic properties of the material (Yeh, 1986). Note that this assumption may result in some inaccuracies when the properties determined from normal incidence experiments are then used in practical calculations involving collimated irradiation of the material (Dombrovsky, 1998). The isotropic or Henyey–Greenstein scattering phase functions can be used in the inverse problem solution using the directional–hemispherical transmittance and reflectance only when the medium is optically thick and purely scattering (Kamdem Tagne, 2008). For absorbing anisotropic medium or optically thin medium, the use of these approximate scattering phase functions is not suitable due to the backscattering effect. The identification of radiative properties of a pure scattering optically thick fiber glass medium indicated that the estimated radiative properties assuming an equivalent isotropic medium and isotropic scattering correspond to the weighted mean radiative properties (Kamdem Tagne, 2008).
The spectral bi-directional transmittance and reflectance measurements can be used as information about the phase function. Nicolau et al. (1994) used the measurements of the spectral bi-directional transmittance and reflectance to determine the radiative properties of a fibrous material with silica fibers randomly oriented in plane. Figure 2 shows an apparatus enabling spectral bi-directional transmittance and reflectance measurements,developed by Nicolau et al. (1994). It consists basically of a Fourier transform infrared (FTIR) spectrometer coupled to a bi-directional attachment on which are the sample, with its front surface located on the rotation axis of the device, and the detector standing on the rotating arm. The sample was irradiated by a normally incident collimated radiation. The radiative transfer equation was solved using the discrete ordinates method with a fine angular quadrature (see the article Discrete ordinates and finite volume methods) to take into account the highly forward and backward peaked scattering observed for fibrous materials. The scattering phase function has been approximated by a combination of two Henyey–Greenstein (HG) functions coupled with an isotropic component. Such a phase function makes possible taking into account the highly phase forward peaked scattering combined with back scattering, which is observed for fibrous materials:
(1) |
where P_{HG,μ 1} is the forward HG function characterized by the positive asymmetry factor of scattering, μ_{1}; and P_{HG,μ 2} is the backward HG function characterized by the negative asymmetry factor of scattering, μ_{2}.
Figure 2. Bi-directional transmittance and reflectance setup.
Nicolau et al. (1994) assumed an equivalent isotropic medium. The phase function proposed by these authors is more complete than the HG function but it requires four parameters (f_{1}, f_{2}, μ_{1}, μ_{2}) to be determined. As an example Fig. 3 represents forward and backward phase functions obtained for f_{1} = 0.9, g_{1} = 0.84, f_{2} = 0.95, and g_{2} = -0.6. As a result, to determine the six radiative characteristics (albedo, extinction coefficient, and the four parameters of phase function) a high condition number is induced (the condition number was larger than 10^{5}). Thus, the numerical simulation showed that it is impossible to determine the six parameters simultaneously. Since μ_{2} has the weakest sensitivity coefficient (which means that a variation of μ_{2} induces a smaller variation on the transmittances and reflectances than the variations of the other parameters), this parameter cannot be identified. Nicolau et al. (1994) decided to choose μ_{2} = μ_{1}. In our mind, it would be interesting to perform both hemispherical and bi-directional measurements and use them in the identification procedure for a more representative analysis of the material radiative properties.
Figure 3. Phase functions: f_{1} = 0.9, g_{1} = 0.84, f_{2} = 0.95, and g_{2} = -0.6.
REFERENCES
Boulet, P., Jeandel, G., and Morlot, G., Model of radiative transfer in fibrous media-Matrix method, Int. J. Heat Mass Transfer, vol. 36, no. 2, pp. 4287-4297. 1993.
Boulet, P., Jeandel, G., Morlot, G., Silberstein, A., and Dedianous, P., Study of the radiative behaviour of several fiberglass materials, in Thermal Conductivity, ed. Tong, T. W., Lancaster, PA: Technomic, pp. 749-759, 1994.
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., Infrared and microwave radiative properties of metal coated microfibres, Rev. Gen. Therm., vol. 37, no. 11, pp. 925-933, 1998.
Dombrovsky, L., and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Redding, CT: Begell House, 2010.
Heino, J , Arridge, S., Sikora, J., and Somersalo, E., Anisotropic effects in highly scattering media, Phys. Rev. E, vol. 68, pp. 031908.1-031908.8, 2003.
Houston, R. L. and Korpela, S. A., Heat transfer through fiberglass insulation, Proc. of 7th International Heat Transfer Conference, vol. 2, pp. 499-504, 1982.
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.
Kamdem Tagne, H. T., Etude du transfert thermique dans les mileux poreux anisotropies. Application aux isolants thermiques en fibres de silice, Ph.D. thesis, INSA, Lyon, France, 2008.
Kamdem Tagne, H. T. and D. Baillis, D., Radiative heat transfer using isotropic scaling approximation: Application to fibrous medium, ASME J. Heat Transfer, vol. 127, no. 10, pp. 1115-1123, 2005a.
Kamdem Tagne, H. T. and Baillis, D., Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence, J. Quant. Spectrosc. Radiat. Transf., vol. 93, no. 1-3, pp. 103-113, 2005b.
Kamdem Tagne, H. T. and Baillis, D., Reduced models for radiative heat transfer analysis through anisotropic fibrous medium, ASME J. Heat Transfer, vol. 132, no. 7, pp. 072403.1-072403.8, 2010.
Lee, S.-C., Radiative transfer through a fibrous medium: allowance for fiber orientation, J. Quant. Spectrosc. Radiat. Transf., vol. 36, no. 3, pp. 253-263, 1986.
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., Scattering phase function for fibrous media, Int. J. Heat Mass Transfer, vol. 33, no. 10, pp. 2183-2190, 1990.
Lee, S.-C., Enhanced thermal performance of fibrous insulation containing nonhomogeneous fibers, J. Quant. Spectrosc. Radiat. Transf., vol. 50, no. 2, pp. 199-209, 1993.
Lee S.-C. and Cunnington, G. R., Theoretical models for radiation heat transfer in fibrous media, in Annual Review of Heat Transfer, ed. Tien, C. L., vol. 9, 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.
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.
Mathews, L. K., Viskanta, R., and Incropera, F. P., Development of inverse methods for determining thermophysical and radiative properties of high temperature fibrous materials, Int. J. Heat Mass Transfer, vol. 27, no. 4, pp. 487-495, 1984.
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. Transf., vol. 74, no. 5, pp. 637-653, 2002.
Nicolau, V. P., Raynaud, M., and Sacadura, J.-F., Spectral radiative properties identification of fiber insulating materials, Int. J. Heat Mass Transfer, vol. 37, no. 1, pp. 311-324, 1994.
Tong, T. W. and Tien, C. L., Analytical models for thermal radiation in fibrous insulation, J. Therm. Insul., vol. 4, no. 7, pp. 27-44, 1980.
Wan, X., Fan, J., and Wu, H., Measurement of thermal radiative properties of penguin down and other fibrous materials using FTIR, Polym. Test., vol. 28, no. 7, pp. 673-679, 2009.
Yeh, H. Y., Radiative properties and heat transfer analysis of fibrous insulation, Ph.D. thesis, University of Mississippi, 1986.
Les références
- Boulet, P., Jeandel, G., and Morlot, G., Model of radiative transfer in fibrous media-Matrix method, Int. J. Heat Mass Transfer, vol. 36, no. 2, pp. 4287-4297. 1993.
- Boulet, P., Jeandel, G., Morlot, G., Silberstein, A., and Dedianous, P., Study of the radiative behaviour of several fiberglass materials, in Thermal Conductivity, ed. Tong, T. W., Lancaster, PA: Technomic, pp. 749-759, 1994.
- 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., Infrared and microwave radiative properties of metal coated microfibres, Rev. Gen. Therm., vol. 37, no. 11, pp. 925-933, 1998.
- Dombrovsky, L., and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Redding, CT: Begell House, 2010.
- Heino, J , Arridge, S., Sikora, J., and Somersalo, E., Anisotropic effects in highly scattering media, Phys. Rev. E, vol. 68, pp. 031908.1-031908.8, 2003.
- Houston, R. L. and Korpela, S. A., Heat transfer through fiberglass insulation, Proc. of 7th International Heat Transfer Conference, vol. 2, pp. 499-504, 1982.
- 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.
- Kamdem Tagne, H. T., Etude du transfert thermique dans les mileux poreux anisotropies. Application aux isolants thermiques en fibres de silice, Ph.D. thesis, INSA, Lyon, France, 2008.
- Kamdem Tagne, H. T. and D. Baillis, D., Radiative heat transfer using isotropic scaling approximation: Application to fibrous medium, ASME J. Heat Transfer, vol. 127, no. 10, pp. 1115-1123, 2005a.
- Kamdem Tagne, H. T. and Baillis, D., Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence, J. Quant. Spectrosc. Radiat. Transf., vol. 93, no. 1-3, pp. 103-113, 2005b.
- Kamdem Tagne, H. T. and Baillis, D., Reduced models for radiative heat transfer analysis through anisotropic fibrous medium, ASME J. Heat Transfer, vol. 132, no. 7, pp. 072403.1-072403.8, 2010.
- Lee, S.-C., Radiative transfer through a fibrous medium: allowance for fiber orientation, J. Quant. Spectrosc. Radiat. Transf., vol. 36, no. 3, pp. 253-263, 1986.
- 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., Scattering phase function for fibrous media, Int. J. Heat Mass Transfer, vol. 33, no. 10, pp. 2183-2190, 1990.
- Lee, S.-C., Enhanced thermal performance of fibrous insulation containing nonhomogeneous fibers, J. Quant. Spectrosc. Radiat. Transf., vol. 50, no. 2, pp. 199-209, 1993.
- Lee S.-C. and Cunnington, G. R., Theoretical models for radiation heat transfer in fibrous media, in Annual Review of Heat Transfer, ed. Tien, C. L., vol. 9, 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.
- 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.
- Mathews, L. K., Viskanta, R., and Incropera, F. P., Development of inverse methods for determining thermophysical and radiative properties of high temperature fibrous materials, Int. J. Heat Mass Transfer, vol. 27, no. 4, pp. 487-495, 1984.
- Milandri, A., Asllanaj, F., and Jeandel, G., Determination of radiative properties of fibrous media by an inverse methodâ€”Comparison with the Mie theory, J. Quant. Spectrosc. Radiat. Transf., vol. 74, no. 5, pp. 637-653, 2002.
- Nicolau, V. P., Raynaud, M., and Sacadura, J.-F., Spectral radiative properties identification of fiber insulating materials, Int. J. Heat Mass Transfer, vol. 37, no. 1, pp. 311-324, 1994.
- Tong, T. W. and Tien, C. L., Analytical models for thermal radiation in fibrous insulation, J. Therm. Insul., vol. 4, no. 7, pp. 27-44, 1980.
- Wan, X., Fan, J., and Wu, H., Measurement of thermal radiative properties of penguin down and other fibrous materials using FTIR, Polym. Test., vol. 28, no. 7, pp. 673-679, 2009.
- Yeh, H. Y., Radiative properties and heat transfer analysis of fibrous insulation, Ph.D. thesis, University of Mississippi, 1986.