Closed Cell Foams
Following from: Highly porous cellular foams
Baillis and Coquard (2008) and Dombrovsky and Baillis (2010) give comprehensive general overviews of experimental approaches and theoretical models used to determine the radiative properties of highly porous cell foams in their books. A synthesis is presented herein concerning closed cell foams. More details and specific results can be found in these text books.
Closed cell foams find applications in a large number of technological fields. They are notably used for packaging or mechanical protection due to their excellent mechanical resistance. However, thermal insulation is their main scope of application. As a matter of fact, numerous materials used in frigorific or building insulation have a closed cell structure. Polystyrene or polyurethane (PUR) foams are the most widely sold materials for building thermal insulation after glass wools. They are very convenient to manipulate due to their mechanical properties (lightness and stiffness) and they are relatively cheap.
In most of these applications, the knowledge and modeling of the thermal properties are of primary importance to improve their thermal performance. Thermal properties are greatly dependent on the type of foam associated with different cellular structures. The polystyrene foam structure varies depending on whether it is expanded or extruded. Extruded polystyrene foam (XPS) and polyurethane foam have a similar structure (see the article Classification of foam structures).
Highly porous cellular foams generally present a low density and, thus, the radiative heat transfer is significant. The radiative conductivity, k_{r}, is usually calculated by using the Rosseland approximation (Kuhn et al., 1992; Placido et al., 2005; Kaemmerlen et al., 2010):
(1) |
where T_{m} is the mean temperature in the medium, σ_{B} is the Stefan-Boltzmann constant, and β_{R} is the Rosseland average extinction coefficient. As scattering in foams is anisotropic, a transport extinction coefficient, β_{λ}^{tr}, is usually introduced in the calculation of β_{R} (Baillis and Coquard, 2008; Dombrovsky and Baillis, 2010).
The Rosseland average extinction coefficient is defined as follows:
(2) |
where β_{λ}^{tr} = κ_{λ} + σ_{λ} · (1 - μ_{λ}), I^{0}(T_{m}) is the blackbody intensity at temperature T_{m}, and I_{λ}(T_{m}) is the spectral blackbody intensity.
Therefore, determination of the transport extinction coefficient, β_{λ}^{tr}, as a function of the type of foam, cellular microstructure, cell diameter, and porosity, represents a considerable importance to researchers.
Expanded Polystyrene Foam
Expanded polystyrene (EPS) foams have a very low density (in the range of 10-35 kg/m^{3}). The mean diameter of the cells contained in EPS beads is generally between 100 and 300 μm. Note that the important wavelength range for radiative heat transfer at room temperature is centered around 10 μm. As a result, the height of the walls of the material cells is much larger than the wavelength. Thus, geometrical optics and diffraction approximations are usually used. Furthermore, the influence of diffraction by the walls can be neglected. Microscopic analysis of the cellular materials shows that the polymer is mainly concentrated in the cellular walls, whereas the struts can be neglected in the case of low-density EPS foams.
Coquard et al. (2009) have shown that a detailed consideration of the macrostructure of EPS foams (see Fig. 3 in the article Classification of foam structures) is not of primary importance. This means that EPS foams can be treated as homogeneous cellular closed cell foams. As a result, the radiative properties can be obtained on the basis of the model of a homogeneous cellular structure made of randomly oriented cell walls, assumed as infinite thin slabs, which scatter radiation independently of each other. The walls are usually assumed to have the same thickness and to be specularly reflecting. Several authors (Placido et al., 2005; Coquard and Baillis, 2006; Coquard et al., 2009) treated the radiation-cell wall interaction using the Fresnel relations. The contribution of forward scattering due to transmission can be neglected (Q_{s} = R). In this case, the extinction efficiency of an infinite thin slab whose normal makes an angle α with the incident direction (Fig. 1) becomes
(3) |
where R and T are the reflectance and transmittance of a thin slab, respectively. The radiative properties (extinction coefficient, β, and scattering coefficient, σ) can be determined as follows:
(4) |
(5) |
Figure 1. Schematic of specular reflection by a thin slab at oblique incidence α.
In the case of specular reflection, the scattering phase function P is (Siegel and Howell, 2002):
(6) |
The normalization of this function results in the following equation:
(7) |
The asymmetry factor of scattering, μ, and the spectral transport extinction coefficient, β^{tr}, are:
(8) |
(9) |
If we assume that the foams consist of monodisperse walls of thickness d_{w}, surface G_{wall}, and V _{wall}, then the following relation is true:
(10) |
where ρ_{0} is the bulk density. In this case, we obtain
(11) |
For foam without struts, each wall is shared by two neighboring cells and the following geometrical relation takes place:
(12) |
Geometrical characteristics of various polyhedron cells such as dodecahedron cells, cubic cells, or tetrakaidecahedron cells can be used to derive the following relations:
(13) |
where D_{cell} is the cell diameter as defined in the article Classification of foam structures.
The reflectance and transmittance of a thin dielectric slab are (Brewster, 1992):
(14) |
where r and t are the amplitude of the wave reflected and transmitted from the thin wall, respectively. They can be written as:
(15) |
where
(16) |
(17) |
The Snell law induces the refraction angle θ_{ref} from the incident angle α as follows:
(18) |
To calculate the values of R and T, wall thickness d_{w} is required, as well as the complex index of refraction, m, of the polystyrene used for the manufacturing of the foams. The wall thickness can be obtained from cell diameter D_{cell} and from the porosity using Eq. (12). Coquard et al. (2009) have determined the complex index of refraction in the infrared range using an identification method based on transmittance and reflectance measurements for thin (50 μm) and thick (600 μm) polystyrene films of melted EPS foams (Fig. 2). In this study, it appeared that the radiative properties (albedo, ω, extinction coefficient β, and asymmetry factor of scattering μ) are practically identical for the three polyhedral cells considered (Coquard et al., 2009). It was proven that the foam density and cell diameter are the most important morphological characteristics of these materials. The influences of these parameters on the global radiative properties (i.e., integrated over the whole wavelength spectrum) are illustrated in Figs. 3 and 4. As expected, when the foam is denser, a more important quantity of matter interacts with thermal radiation, leading notably to an increase of the extinction coefficient. The evolution of the radiative properties with the cell diameter is more complex than with the foam density since it actually presents an optimal cell diameter for which the radiation-matter interaction is maximum. This optimal cell diameter varies with the density of the foam and slightly more with the shape of the cells. The denser the foam is, the lower the optimal diameter. Finally, good agreement was found by the authors when comparing the experimental and theoretical transmittances and reflectances (Coquard et al., 2009).
Figure 2. Spectral optical constants of polystyrene: 1. Kaemmerlen et al. (2010); 2, Coquard et al. (2009).
Figure 3. Variation of the global radiative properties with the density for EPS foams with D_{cell} = 200 μm (Coquard et al., 2009).
Figure 4. Variation of the global radiative properties with the cell diameter for EPS foams with ρ = 8.95 kg/m^{3} (Coquard et al. 2009).
Extruded Polystyrene and Polyurethane Foams
Microscopic analyses of the XPS and PUR cellular foams show the presence of walls and struts. Glicksman and Torpey (1988) and Glicksman et al. (1990) first modeled the PUR foam structure as a set of randomly oriented opaque struts, assuming that the efficiency factor of absorption is equal to unity and neglecting the scattering by struts. The strut cross section was assumed to be triangular in shape with constant area along the strut. This area was assumed equal to two-thirds of the area of an equilateral triangle formed at the vertices. The cell membranes within the foam were so thin (approximately 1 μm) that they were assumed to be totally transparent to thermal radiation. According to Glicksman and Torpey (1988), the struts (constituting approximately 85% of the solid polymer contained in the foam) are much thicker than the cell membranes and are responsible for the attenuation of thermal radiation. The foam cells were assumed to be pentagon dodecahedron.
Therefore, the resulting extinction coefficient follows a simple relation:
(19) |
The total length of struts by volume unit, L_{v}, and the strut width, a (defined in Fig. 5), can be expressed as a function of the cell diameter and foam porosity from the geometrical characteristics of the dodecahedron cell (see the article Classification of foam structures) leading to the following relation (Glicksman and Torpey, 1988):
(20) |
A similar formalism can be found in the more recent work by Campo-Arnáiz et al. (2005) on polyolefin foams. These authors take into account the volume fraction of polymer contained in the struts of foams, f_{s}. The extinction coefficient based on the Glicksman approach, β_{G}, obeys the following equation:
(21) |
where K_{W} represents the mean absorption coefficient of the solid polymer.
Kuhn et al. (1992), Placido et al. (2005), and Kaemmerlen et al. (2010) suggested a more complex model for expanded polystyrene and polyurethane foams. They investigated the dependence of the radiative conductivity on geometrical parameters characterizing the internal foam structure, such as the mean wall thickness, the mean strut, and the cell diameters. It was also assumed that infinitely long randomly oriented cylinders and infinitely large platelets can be considered for the modeling of the struts and the walls, respectively. The triangular cross section was converted into a circular one with the same geometrical mean area (Fig. 5). The geometrical optics approximation was employed for the computation of the radiative characteristics of the cell walls and the Mie theory was employed for the struts. The assumption of long struts was justified by joining the struts of contiguous cells. Independent scattering was assumed, and the foam spectral radiative properties were calculated by summing the contributions of two kinds of particles (struts and walls):
(22) |
(23) |
Figure 5. Illustration of the dodecahedral cell used in the models: (a) perspective view; (b) cross section of struts and walls (Placido et al., 2005).
Note that the volume fraction of polymer contained in the struts of XPS foams, f_{s}, was estimated with scanning electron microscopy (SEM) pictures to be between 15% and 40%. For polyurethane foams, it was about 60% (Kaemmerlen et al., 2010).
In the case of foam containing only randomly oriented walls, the radiative properties have already been calculated in the case of EPS foams [Eqs. (4)-(6) and (9)]. In the case of foam containing only randomly oriented struts, one can write
(24) |
where Q_{t}, Q_{s} are calculated from the Mie theory for infinite cylinders. These values depend on the complex index of refraction of polystyrene, strut diameter Φ_{s}, and angle of incidence ϕ (Fig. 6).
(25) |
Figure 6. Geometry of scattering by a fiber at oblique incidence.
In a recent study done by Kaemmerlen et al. (2010), the asymmetry factor of scattering, μ_{struts}, of a medium constituted only of struts was defined as follows (for more details, see Tagne and Baillis, 2005):
(26) |
where the function i(ϕ, η) is defined by the Mie theory and verifies:
(27) |
The transport extinction coefficient of the struts, β_{struts}^{tr} = κ_{struts}+σ_{struts}(1 - μ_{struts}), is thus deduced from the precedent equations:
(28) |
where
(29) |
Some geometrical relations can be used to correlate directly wall thickness d_{w} and strut volumetric fraction f_{s} to strut diameter Φ_{s}, bulk density ρ_{0}, foam density ρ_{f}, and the cell diameter (Dombrovsky and Baillis, 2010; Kaemmerlen et al., 2010).
Kuhn et al. (1992) and Placido et al. (2005), based on the Mie theory calculations, treated the struts as circular cylinders. However, the real cross section of the struts is a concave triangule and the cross section area is two-thirds of the area of the equilateral triangle formed at the strut vertices (Fig. 5), (see also Figs. 4 and 5 in the article Classification of foam structures). Therefore, Kaemmerlen et al. (2010) recently used the discrete dipole approximation (DDA) to estimate the radiative properties of the struts. The DDA is an accurate numerical method of computation of the radiative properties of absorbing particles with arbitrary shape, which consists of dividing the particle volume into a large number of electric dipoles that interact with the incident wave. Kaemmerlen et al. (2010) used the code DDSCAT (version 6.1) developed by Draine and Flatau (1994). Complete documentation of this code is freely available online (Draine and Flatau, 2004). The strut lengths were chosen to be long enough to satisfy the assumption of infinitely long struts. This was more convenient for comparison of the numerical results with the Mie theory. It was shown that the extinction coefficient of struts is overestimated of about 25% in the model of infinite circular cylinders of equivalent diameter when compared with the calculations for struts with a concave triangular cross section. The following correlation was suggested to correct the radiative properties calculated using the Mie theory for cylinders when the ratio λ/Φ_{s} ranges from 0.7 to 10:
(30) |
This correlation can be recommended as an alternative to DDA calculations, which is admittedly more accurate but very time consuming.
As an example, the results obtained by Kaemmerlen et al. (2010) for an XPS foam sample are presented. The complex index of refraction of polystyrene, required in the model, was determined by Kaemmerlen et al. (2010) from infrared transmittance and reflectance measurements performed on thin films of bulk polystyrene using the same method as that suggested by Coquard and Baillis (2006) for polystyrene films (Fig. 2). The small difference between the results obtained by Coquard and Baillis (2006) and Kaemmerlen et al. (2010) is explained by the slightly different composition of polystyrene used in these experiments.
Kaemmerlen et al. (2010) and Dombrovsky and Baillis (2010) reported the experimental and theoretical results of transmittance and reflectance for an XPS foam sample with a mean cell size of Φ = 108 μm, foam density of ρ_{f} = 35 kg/m^{-3}, and with strut volume fractions f_{s} between 0 and 40%. The sample thickness was 1 mm. It was shown that the effect of the strut fraction on the hemispherical transmittance and reflectance spectra was very weak. In addition, it was shown by Kaemmerlen et al. (2010) that uncertainties in the input data, such as the mean cell diameter and foam density, cannot explain the significant discrepancy between the theoretical and experimental values of the hemispherical transmittance. The deviations could be explained by the uncertainty in the refractive index or by the limits of some assumptions of the models, such as the homogeneity and isotropy of the foam, but also the decomposition of the foam in two types of particles (infinitely large platelets of constant thickness and long struts).
Figure 7 shows the radiative conductivity versus the cell size for several XPS samples with the same density (35 kg/m^{-3}). The radiative conductivity is calculated with the Rosseland equations [Eqs. (1) and (2)], taking in account the radiative properties of struts and walls computed from the Kaemmerlen et al. (2010) model, and with the Glicksman formula [Eq. (21)]. For each cell diameter, different strut volume fractions f_{s} between 0 and 40% are considered. As for EPS foams, the evolution of the radiative properties with the cell diameter presents an optimal cell diameter for which the radiation-matter interaction is maximum, and thus the radiative conductivity is minimum. The lowest radiative conductivity for that range of porosity appears to be for a cell size around 140 μm. This optimum cell size is due to the spectral variations of the scattering coefficient depending on the wall thickness. Deviations from the Glicksman formula are observed, especially at small cell sizes.
Figure 7. Theoretical results for radiative conductivity.
In conclusion, the studies tend to show that, in the future, research efforts should focus on morphology modeling. Indeed, the theoretical results are very sensitive to the mean cell size determined from SEM pictures and are affected significantly by the dispersion of these measures. For that reason, considering a mean cell size appears inappropriate if better accuracy is expected in radiative conductivity modeling, especially if foams are anisotropic (cells elongated; wall thickness or strut diameter dispersions in the medium). In addition, all of the previously described models were based on some assumptions of idealized homogeneous cellular structures. The shapes of the model particles were generally simplified and it has been always assumed that independent scattering takes place. Today, the development of X-ray tomography allows envisaging a noticeable improvement in the morphological characterization of these materials and this can be used to achieve better numerical estimation of the radiative properties. To fulfill this objective, the recent work of Coquard et al. (2010), concerning the modeling of the radiative properties of closed cell foams, opens new perspectives. In this study, the spectral extinction coefficients, scattering albedo, and scattering phase function are computed from ray-tracing procedures inside three-dimensional meshes representing the real cellular structures of polymeric foams.
REFERENCES
Baillis, D. and Coquard, R., Radiative and conductive thermal properties of foams, in Cellular and Porous Materials: Thermal Properties Simulation and Prediction, eds. Öchsner, A., Murch, G. E., and de Lemos, M. J. S, Weinheim: Wiley-VCH, pp. 343–384, 2008.
Brewster, M. Q., Thermal Radiative Transfer and Properties, New York: Wiley, 1992.
Campo-Arnáiz, R. A., Rodriguez-Pérez, M. A., Calvo, B., and de Saja, J. A., Extinction coefficient of polyolefin foams, J. Polym. Sci., Part B: Polym. Phys., vol. 43, no. 13, pp. 1608-1617, 2005.
Coquard, R. and Baillis, D., Modeling of heat transfer in low-density EPS foams, ASME J. Heat Transfer, vol. 128, no. 6, pp. 538-549, 2006.
Coquard, R., Baillis, D., and Quenard, D., Radiative properties of expanded polystyrene foams, ASME J. Heat Transfer, vol. 131, no. 1, pp. 012702.1-012702.10, 2009.
Coquard, R., Baillis, D., and Maire E., Numerical investigation of the radiative properties of polymeric foams from tomographic images, AIAA J. Thermophys. Heat Transfer, vol. 24, no. 3, pp. 647-658, 2010.
Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Redding, CT: Begell House, 2010.
Draine, B. T. and Flatau, P. J., Discrete dipole approximation for scattering calculations, J. Opt. Soc. Am. A, vol. 11, no. 4, pp. 1491-1499, 1994.
Draine, B.T. and Flatau, P. J., User Guide to the Discrete Dipole Approximation Code DDSCAT 6.1., Available athttp://arxiv.org/abs/astro-ph/0409262v2, 2004.
Glicksman L. R. and Torpey, M. R., A study of radiative heat transfer through foam insulation, Subcontract Report No. 19X-09099C, Massachusetts Institute of Technology, 1988.
Glicksman, L. R., Mozgowiec, M., and Torpey, M., Radiation heat transfer in foam insulation, Proc. of 9th International Heat Transfer Conference, Jerusalem, pp. 379-384, 1990.
Kaemmerlen, A., Vo, C., Asllanaj, F., Jeandel, G., and Baillis, D., Radiative properties of extruded polystyrene foams: Predictive models and experimental results, J. Quant. Spectrosc. Radiat. Transf., vol. 111, no. 6, pp. 865-877, 2010.
Kuhn, J., Ebert, H. P., Arduini-Schuster, M. C., Büttner, D., and Fricke, J., Thermal transport in polystyrene and polyurethane foam insulations, Int. J. Heat Mass Transfer, vol. 35, no. 7, pp. 1795-1801, 1992.
Placido, E., Arduini-Schuster, M. C., and Kuhn, J., Thermal properties predictive model for insulating foams, Infrared Phys. Technol., vol. 46, no. 3, pp. 219-231, 2005.
Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 4th ed., New York: Taylor & Francis, 2002.
Tagne, K. H. T and Baillis, D., Radiative heat transfer using isotropic scaling approximation: Application to fibrous medium, ASME J. Heat Transfer, vol. 127, no. 10, pp. 1115-1123, 2005.
Les références
- Baillis, D. and Coquard, R., Radiative and conductive thermal properties of foams, in Cellular and Porous Materials: Thermal Properties Simulation and Prediction, eds. Öchsner, A., Murch, G. E., and de Lemos, M. J. S, Weinheim: Wiley-VCH, pp. 343â€“384, 2008.
- Brewster, M. Q., Thermal Radiative Transfer and Properties, New York: Wiley, 1992.
- Campo-Arnáiz, R. A., Rodriguez-Pérez, M. A., Calvo, B., and de Saja, J. A., Extinction coefficient of polyolefin foams, J. Polym. Sci., Part B: Polym. Phys., vol. 43, no. 13, pp. 1608-1617, 2005.
- Coquard, R. and Baillis, D., Modeling of heat transfer in low-density EPS foams, ASME J. Heat Transfer, vol. 128, no. 6, pp. 538-549, 2006.
- Coquard, R., Baillis, D., and Quenard, D., Radiative properties of expanded polystyrene foams, ASME J. Heat Transfer, vol. 131, no. 1, pp. 012702.1-012702.10, 2009.
- Coquard, R., Baillis, D., and Maire E., Numerical investigation of the radiative properties of polymeric foams from tomographic images, AIAA J. Thermophys. Heat Transfer, vol. 24, no. 3, pp. 647-658, 2010.
- Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Redding, CT: Begell House, 2010.
- Draine, B. T. and Flatau, P. J., Discrete dipole approximation for scattering calculations, J. Opt. Soc. Am. A, vol. 11, no. 4, pp. 1491-1499, 1994.
- Draine, B.T. and Flatau, P. J., User Guide to the Discrete Dipole Approximation Code DDSCAT 6.1., Available athttp://arxiv.org/abs/astro-ph/0409262v2, 2004.
- Glicksman L. R. and Torpey, M. R., A study of radiative heat transfer through foam insulation, Subcontract Report No. 19X-09099C, Massachusetts Institute of Technology, 1988.
- Glicksman, L. R., Mozgowiec, M., and Torpey, M., Radiation heat transfer in foam insulation, Proc. of 9th International Heat Transfer Conference, Jerusalem, pp. 379-384, 1990.
- Kaemmerlen, A., Vo, C., Asllanaj, F., Jeandel, G., and Baillis, D., Radiative properties of extruded polystyrene foams: Predictive models and experimental results, J. Quant. Spectrosc. Radiat. Transf., vol. 111, no. 6, pp. 865-877, 2010.
- Kuhn, J., Ebert, H. P., Arduini-Schuster, M. C., Büttner, D., and Fricke, J., Thermal transport in polystyrene and polyurethane foam insulations, Int. J. Heat Mass Transfer, vol. 35, no. 7, pp. 1795-1801, 1992.
- Placido, E., Arduini-Schuster, M. C., and Kuhn, J., Thermal properties predictive model for insulating foams, Infrared Phys. Technol., vol. 46, no. 3, pp. 219-231, 2005.
- Siegel, R. and Howell, J. R., Thermal Radiation Heat Transfer, 4th ed., New York: Taylor & Francis, 2002.
- Tagne, K. H. T and Baillis, D., Radiative heat transfer using isotropic scaling approximation: Application to fibrous medium, ASME J. Heat Transfer, vol. 127, no. 10, pp. 1115-1123, 2005.