Modeling of Radiative Properties of Polystyrene Foams Containing IR- Opacifiers

Transcription

1 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers Modeling of Radiative Properties of Polystyrene Foams Containing IR- Opacifiers M. Arduini*,1, J. Manara 1, and C. Vo 2 1 Bavarian Center for Applied Energy Research (ZAE Bayern), Am Galgenberg 87, Wuerzburg, Germany 2 Dow Europe GmbH, Bachtobelstrasse 3, CH-8810 Horgen, Switzerland Received: 15 June 2015, Accepted: 17 August 2015 SUMMARY The addition of opacifiers in foams considerably reduces the radiative thermal conductivity and consequently enhances the insulation performance of the foams. In this work two different methods were developed to calculate the spectral specific extinction coefficient of opacified extruded polystyrene (XPS) foam material. Cell morphology and thermal conductivity of two identical XPSfoams, one opacified with 3 wt% carbon black as opacifier and one without opacifier, were thoroughly characterized. The experimental results were in a good agreement with the theoretical results obtained from two different computing techniques. These methods allow a good prediction of the specific extinction coefficient of arbitrary opacified foam material. Keywords: Opacifiers, Porous materials, Complex refractive index, Radiative transfer, Radiative thermal conductivity, Extinction coefficient INTRODUCTION The consumption of energy in residential and tertiary buildings accounts for more than 40% of the European energy consumption on a monetary basis [1], generates about 33% of Green House Gases or more than 20% of total European CO 2 emissions. The most effective solution is the improvement of the building insulation: a better insulation of roofs and walls can cut the energy use in half and save the EU the equivalent of 3.3 million barrels of oil per day. * Corresponding author: Tel.: , fax: mariacarla.arduini@zae-bayern.de Smithers Information Ltd Cellular Polymers, Vol. 35, No. 2,

2 M. Arduini, J. Manara, and C. Vo These initiatives have been incorporated in the European Energy Performance of Buildings Directive (EPBD). The initial measures were initiated in 2002 and recently amended in May 2010 [2]. The recast of EPBD required a more demanding building energy performance and a more aggressive time line. The estimated impact of the recast is energy savings of Mtoe (Million Tonnes of Oil Equivalent) in 2020 or the total reduction of EU energy consumption by 5-6%. The cellular polymeric foams have been used since decades to insulate the buildings. A wide overview of modeling and properties of disperse systems can be found for instance in [3]. An improvement of the insulation performance of such foams would be an important step to efficiently reduce costs and energy consumption and will help to reduce the CO 2 emission in the atmosphere. Several routes have been investigated and one way to improve the thermal insulation performance of cellular polymeric foams is the addition of opacifiers [4]. Opacifiers are materials which strongly absorb or scatter the infrared radiation [5]. Their fine and well dispersion should reduce considerably the radiative thermal conductivity [6], [7]. However, there was no methodology that enables a prediction of impacts of these opacifiers in foams. Understanding their impact will help saving costs and resources, hence an efficient development and manufacturing of these opacified foams can be performed. The goal of this work is to find a method that allows an estimation of how the radiative thermal conductivity is affected by the opacifiers. Two different methods are developed and validated to calculate the spectral specific extinction coefficient of cellular polymeric foams to which different amounts of opacifiers are added. The first one is based on the model which has been developed and validated by Placido et al. [8] for non-opacified cellular plastic material, such as Polyurethane (PUR), extruded polystyrene (XPS) or expanded polystyrene (EPS) foams. This model was developed to predict the insulation performance of foams based on their density and geometrical structure. It was validated with experimental data for standard PS, XPS and PUR foams, but it was not tested for opacified foams. In this work the mentioned model of Placido et al. [8] was applied also to predict the extinction coefficient of opacified foams, as it will be described below. The second method focuses on the calculation of the specific extinction coefficient of opacified foams from the individual spectral extinction coefficient of each component constituting the cellular structure, i.e. polymers and opacifiers. MATERIAL DESCRIPTION Two extruded polystyrene (XPS) foams are obtained from the extrusion foaming process using carbon dioxide as the main blowing agent. The first foam, SFB-V, 50 Cellular Polymers, Vol. 35, No. 2, 2016

3 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers is a standard XPS foam without any opacifier, while the second one, SFX-V, is produced with about 3 wt% carbon black (Thermax N990, particle size 300 nm) as opacifier. Both foams are commercial foams produced by DOW. Basic foam properties, measured at ZAE Bayern and at DOW, can be seen in Table 1. The total thermal conductivity of 25 mm thick samples was measured according to DIN EN by using a guarded hot plate apparatus with an incertitude of ± 5% [9]. Table 1. XPS foams properties N Foam Description Density (kg/m³) Average cell size (µm) Total thermal conductivity (mw/m K) 1 SFB-V Standard XPS not opacified 36.6 ± ± ± SFX-V XPS SFB-V + 3 wt% CarbonBlack 39.8 ± ± ± 1.6 At first, the infrared-optical properties of the two XPS-foam specimens have been thoroughly measured. The spectral effective specific extinction coefficient e* Λ was derived in the wavelength region between 1.4 µm and 35 µm. The total effective specific extinction coefficient was then calculated and the radiative thermal conductivity has been obtained. In a second step, the effective specific extinction coefficient of opacified foams was theoretically calculated using the two different methods mentioned above. Finally experimental values and theoretical results have been compared. EXPERIMENTAL DETERMINATION OF THE INFRARED OPTICAL PROPERTIES The two foams SFB-V and SFX-V were measured using a Fourier Transform Infrared (FTIR) spectrometer Bruker Vertex 70v in the wavelength range from 1.4 µm to 35 µm which is decisive for the radiative thermal transport at ambient temperature. Several thin slices of 20 mm 20 mm with different thicknesses (from about 1 mm to about 5 mm) were cut. The cutting slice is perpendicular to the specimen thickness and the thermal radiation flow. The samples are then placed in the opening of an integrating sphere which is coupled to the FTIR-spectrometer. Figure 1 shows the configurations of the integrating sphere for measuring transmittance and reflectance. The sample is irradiated normal to the surface and the radiation reflected into the front side hemisphere or transmitted into the rear side hemisphere is measured for Cellular Polymers, Vol. 35, No. 2,

4 M. Arduini, J. Manara, and C. Vo the transmittance or reflectance spectra, respectively. Several samples with different thicknesses were measured in order to guarantee a sufficiently good average measurement value. For calculating the spectral effective specific extinction coefficient e* Λ, the mass per area m of each sample was also determined. From the measured spectral directional-hemispherical transmittance T dh and reflectance R dh, the spectral effective specific extinction coefficient e* Λ and the spectral effective albedo ω* Λ of each specimen were calculated, using the three-flux solution of the equation of radiative transfer [5]. This three-flux solution enables a quantification of the radiative transfer through scattering and absorbing media as well as a determination of the spectral scattering and absorption coefficients of the investigated specimens. Figure 1. Integrating sphere setup for determining the directional-hemispherical transmittance T dh (on the right side) and reflectance R dh (on the left side) at ambient temperature DEFINITION TERMS OF RADIATIVE HEAT TRANSFER Spectral Extinction Coefficient The spectral specific extinction coefficient e Λ is a measure for the attenuation of radiation inside the samples. It includes scattering and absorption processes within the material. The radiative transfer is further affected by the anisotropic scattering, hence the effective quantities, marked with a star (s* Λ, e* Λ und ω* 0,Λ ), which are defined in detail in [10] and [11], are to be considered. The 52 Cellular Polymers, Vol. 35, No. 2, 2016

5 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers spectral effective specific extinction coefficient e* Λ is given by the sum of the spectral effective specific scattering coefficient s* Λ and spectral specific absorption coefficient a Λ [12]: The reciprocal of the product of the spectral effective specific extinction coefficient e* Λ and the density ρ of the foam is the mean free path L Λ of thermal radiation in the medium, i.e. the path before scattering or absorption takes place: (1) (2) Spectral Albedo The spectral effective albedo ω* 0,Λ is the quotient of the spectral effective specific scattering coefficient s* Λ and the spectral effective specific extinction coefficient e* Λ [13]: The values of the albedo ω* 0 are between 0 and 1 (0 in the case of only absorption and 1 in the case of only scattering). (3) Spectral Scattering and Absorption Coefficients A complete description of the infrared-optical properties is given by either the extinction coefficient and the albedo or the scattering and the absorption coefficient. These four values are connected through Eq. (1) and Eq. (3). Total Extinction Coefficient In scattering and absorbing media the total effective specific extinction coefficient as a function of temperature e*(t ) is obtained by integrating the spectral effective specific extinction coefficient e* Λ using the Rosseland weight function f R (Λ,T ) [14]: Cellular Polymers, Vol. 35, No. 2,

6 M. Arduini, J. Manara, and C. Vo where the Rosseland function is the partial derivative of the spectral intensity i B (Λ,T ) emitted by a black body at a given wavelength Λ and temperature T with respect to the total intensity i Bt (T ) emitted by a black body at the same temperature: (4) The total effective specific extinction coefficient gives an impression of how the extinction coefficient changes with temperature. Chemical or structural changes due to a variation of the temperature are not included. The accuracy of the total effective specific extinction coefficient e* is also determined by the wavelength range of the spectral measurement. In this work, the foams were measured between 1.4 μm and 35 μm. This wavelength range covers 97 % of the total blackbody energy emitted at ambient temperature [15]. Moreover the values of the spectral directional-hemispherical transmittance T dh and reflectance R dh are measured with a relative uncertainty of 3% [15]. When applying three-flux calculation an additional uncertainty has to be taken into account due to the simplification by using the three-flux approximation as a solution of the equation of radiative transfer which lies around 6%. Using the law of propagation of uncertainties and taking the uncertainties due to the integration (Eq. (4)) and due to inhomogeneities of the measured samples into account, one gets an uncertainty of the total effective specific extinction coefficient e* of about 12% [16]. (5) Radiative Thermal Conductivity The radiative thermal conductivity λ rad for grey and optically thick samples can be approximated by the following formula [10, 11]: (6) e*(t ): ρ : total effective specific extinction coefficient, density of the foam, 54 Cellular Polymers, Vol. 35, No. 2, 2016

7 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers n : effective refractive index of the foam, σ : Stefan-Boltzmann-constant ( W/(m 2 K 4 )), T rad : radiative temperature (approx. mean temperature). THEORETICAL CALCULATION OF THE INFRARED OPTICAL PROPERTIES In this paragraph the theoretical method is described which was used to compute the spectral effective specific extinction e* Λ of opacified foams, taking into account its geometrical structure and the refractive index of the opacified bulk material [11]. At first, the refractive index of the opacified bulk material has to be determined. Secondly, it was assumed that the shape and the dimensions of the opacifier particles in the opacified bulk resin remain identical like in the opacified foam. Their concentration per volume in the bulk resin and in the foam is closely correlated with the density. Finally, using the theoretical model developed at ZAE for standard foams [8], it is then possible to predict the effective specific extinction and the thermal conductivities of the opacified foams. The real and the imaginary part n and k of the refractive index of opacified PS is determined from the transmittance and reflectance of thin foils obtained by melting opacified PS. They are calculated using the Fresnel s formula and the equation of radiative transfer for a plane, parallel slab with equal boundaries [13]. Different amounts of opacifiers in PS will result in different n and k. The surface of the thin foils reflects radiation predominantly specular. This has been proved by performing measurements of the directional-hemispherical reflectance using an integrating sphere as well as of the directional-directional reflectance using a specular reflection extension. As the derived values are in a good agreement Fresnel s formula can be applied for determining the complex refractive index of the bulk material. The relative uncertainty of the determined complex refractive index lies around 5% [20]. Also the complex refractive index of not opacified PS was determined at the same way from the transmittance and reflectance of thin foils obtained by melting not opacified PS and the results are in a good agreement with literature within the uncertainty specified above [21], [22], [23]. In this geometrical model, it is assumed that the cells are of a pentagonal dodecahedron shape and are separated in simpler elements: cylinders and platelets. The cylinders represent the struts, whose triangular cross section has been converted into circular one with the same geometrical mean cross section and diameter Φ strut ; the platelets model the walls assumed to have a Cellular Polymers, Vol. 35, No. 2,

8 M. Arduini, J. Manara, and C. Vo constant thickness d wall. Such a transformation of a complex dodecahedron structure into simpler elements allows an accurate calculation of the interaction between the radiation and the single elements. One can then apply the Mietheory to long cylinders to calculate the effective specific extinction coefficient of the struts e* strut [24], [25] and the geometrical optics to infinite planes to calculate the effective specific extinction coefficient of the walls e* wall [26]. The absorption and the scattering of every single strut and wall will then be extended to an infinitesimal foam volume, composed of struts and walls randomly oriented [27]. For randomly oriented struts the effective specific extinction coefficient is: (7) where Q * strut is the spectral relative cross section calculated by Mie theory [24], [25], ρ 0 is the bulk density and Φ strut the strut diameter. For randomly oriented walls the effective specific extinction coefficient is: where α is the angle between the incident radiation and the plane direction, n is the real part of the refractive index, and k is the imaginary part; R(α, n, k) e T(α, n, k) are respectively the reflectance and transmittance of a plane, calculated by Fresnel equations [26]. The resulting spectral effective specific extinction coefficient e* for a given strut volumetric fraction f strut is computed from the one of the struts e* strut and the one of the walls e* wall [8]: (8) Assuming the uniform cell geometry, the cell volume V cell, the strut volume V strut and the wall volume V wall can be calculated for each cell: (9) (10) (11) (12) 56 Cellular Polymers, Vol. 35, No. 2, 2016

9 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers where Φ cell is the cell diameter and Φ strut the strut diameter. The strut volumetric fraction f strut, the ratio between the strut volume and the solid volume (strut volume + wall volume), is calculated as below: For a given cell diameter Φ cell, strut diameter Φ strut, bulk density ρ bulk and foam density ρ, the wall thickness d wall is derived by the assumed geometrical model: (13) The Eq. (14) is only valid for wall thicknesses which are below or equal to the strut diameter (d wall Φ strut ). The spectral effective specific extinction coefficient of the struts e* strut and the one of the walls e* wall depend on the geometry (diameter of the cells and of the struts, thickness of the walls) and on the density of the foam, bulk density, bulk thermal conductivity and on the complex refractive index (real part n and imaginary part k of the complex refractive index) of the polymer constituting the foam. The geometrical parameters of the XPS foams (mean cell diameter and strut diameter) have been determined using a Scanning Electron Microscope (SEM) (Figure 2 and Figure 3) and building an average value over about 30 cells. (14) Figure 2. SEM pictures of the foam SFB V Cellular Polymers, Vol. 35, No. 2,

10 M. Arduini, J. Manara, and C. Vo Figure 3. SEM pictures of the foam SFX V For the theoretical calculations of a XPS foam opacified with 3 wt% carbon black powder, following densities and geometrical parameters are used: Table 2. Densities, bulk thermal conductivity and geometrical parameters used for the theoretical calculation of the effective specific extinction coefficient of the opacified XPS foam Foam XPS + carbon black Bulk thermal conductivity λ bulk mw/(m K) Bulk density ρ bulk kg/m 3 Foam density ρ kg/m 3 Mean cell diameter Φ cell µm Mean strut diameter Φ strut µm The wall thickness in the model strictly depends on the other parameters and it is automatically determined from Eq.(14). The density of the foam was measured at ZAE and at DOW with a relative uncertainty lower than 1%. The bulk density was determined at ZAE from slices of PS bulk material with a relative uncertainty of ± 1.5%. The value of the bulk thermal conductivity for PS is taken from literature [28] and it is assumed that the addition of 3 wt% opacifier does not influence significantly the bulk thermal conductivity of the polymer as no percolation path exists. For the cell diameter a logarithmic normal distribution with standard deviation σ equal to 0.8 is used in the model. 58 Cellular Polymers, Vol. 35, No. 2, 2016

11 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers RESULTS AND DISCUSSION Several foils of PS with different amounts of carbon black were melted and measured with a Fourier Transform Infrared (FTIR) spectrometer. The thickness of the foils is about 50 µm. The measured spectral reflectance and transmittance as well as the derived refractive indexes of PS opacified with different amounts of carbon black are presented in Figure 4, Figure 5 and Figure 6. The imaginary part of the complex index of refraction increases with increasing amount of added opacifier. The comparison of the measured spectral specific extinction of the opacified foam SFX-V, cut perpendicular to the direction of heat transfer and the modelled spectral specific extinction calculated from n and k of the PS foil opacified with 2.5 wt% carbon black is presented in Figure 7. There is a good agreement between measured and calculated spectral specific extinction, although the calculated one has a slightly lower e* due to a lower amount of opacifier in the bulk material (2.5 wt%) than the measured opacified foam SFX-V produced with 3 wt% carbon black. Attempt to produce the refractive index of PS opacified with more than 2.5 wt% carbon black in the whole spectral range between 1.4 µm and 35 µm was not successful Figure 4. Spectral transmittance of PS foils opacified with different percentages of carbon black powder Cellular Polymers, Vol. 35, No. 2,

12 M. Arduini, J. Manara, and C. Vo Figure 5. Spectral reflectance of PS foils opacified with different percentages of carbon black powder Figure 6. Complex refractive indexes of PS opacified with different percentages of carbon black powder in comparison with the refractive index of not opacified PS 60 Cellular Polymers, Vol. 35, No. 2, 2016

13 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers Figure 7. Measured and calculated effective specific extinction coefficient of foam SFX because the spectral transmittance of the opacified foils strongly decreases at short wavelengths down to zero as can be seen in Figure 4. Nevertheless it is possible to calculate the spectral specific extinction of opacified foams from its geometric structure and the refractive index of the opacified resins. The total extinction coefficient determined from the measured and calculated spectral extinction coefficient is reported in Table 3. Taking a relative uncertainty due to the determination of cells size, struts diameter and densities of about 3% into account, one gets an uncertainty of the modeled total effective specific extinction coefficient e* of about 10%. Table 3. Measured and modelled results of XPS foams at ambient temperature T = 293 K Foam Density kg/m³ Total spec. extinction coefficient m²/kg Radiative thermal conductivity mw/(m K) SFB V measured 36.6 ± ± 3 not opacified 7.7 ± 1.0 SFX V measured 39.8 ± ± 5 opacified with 4.3 ± 0.5 3% CB opacified XPS predicted method ± ± 3 opacified with 2.5% CB 5.8 ± 0.6 opacified XPS predicted method ± ± 4 opacified with 3% CB 4.3 ± 0.4 Cellular Polymers, Vol. 35, No. 2,

14 M. Arduini, J. Manara, and C. Vo The differences between calculated and measured spectral specific extinction in the first method can be also explained by a) different% of opacifiers, b) homogeneous and isotropic cell structure versus complex and inhomogeneous real cell morphology. The second method computes the spectral effective specific extinction coefficient of the opacified foam by adding the measured spectral extinction coefficient of the opacifier in powder form to the measured spectral extinction coefficient of the foam without opacifier. For instance, if PS foam contains 3 wt% carbon black, the expected specific extinction coefficient of this opacified foam is, in first approximation, equal to the sum of 97% of the specific extinction coefficient of the same PS foam without opacifier plus 3% of the specific extinction coefficient of the opacifier itself. It is assumed that the addition of opacifiers does not modify significantly the structure of the foam. At first, the spectral specific extinction coefficient of the opacifier carbon black in powder form was measured (see Figure 8) and the total specific extinction coefficient was calculated according to Eq. (4). The total specific extinction coefficient of carbon black at ambient temperature (T = 293 K) is equal to 581 m²/kg. In a second step the specific extinction coefficient of foam SFX-V (XPS opacified with 3 wt% carbon black) was calculated as the sum of 97% extinction Figure 8. Measured effective specific extinction coefficient of carbon black Thermax N990, particle size 300 nm in powder form 62 Cellular Polymers, Vol. 35, No. 2, 2016

15 Modeling of Radiative Properties of Polystyrene Foams Containing IR-Opacifiers coefficient of foam SFB-V (control foam without opacifier) and 3% extinction coefficient of carbon black powder. A comparison of this calculated total extinction coefficient at T = 293 K with the measured extinction coefficient of foam SFX-V (opacified with 3 wt% carbon black) cut perpendicular to the direction of heat transfer is shown in Figure 9. The measured spectral specific extinction coefficient of the not opacified foam SFB-V is also shown in Figure 9. Figure 9. Comparison of the measured effective specific extinction coefficient of foams SFX-V (opacified with 3 wt% carbon black) and SFB-V (not opacified) and the calculated effective specific extinction coefficient from 97% SFB-V and 3% carbon black There is quite a good agreement between measured and calculated spectral extinction coefficient, particularly between 5 µm and 20 µm wavelength. At lower wavelength, the calculated values are slightly lower, and at high wavelength, they are slightly higher than measured ones. This depends on the spectral course of the extinction coefficient of the opacifier and of the reference foam. In spite of small variations at low and high wavelength regions, the total extinction coefficient agrees very well between measured and calculated ones, 44 m²/kg ± 5. The total extinction coefficient determined from the measured and from the calculated spectral extinction coefficient is also reported in Table 3. Cellular Polymers, Vol. 35, No. 2,

16 M. Arduini, J. Manara, and C. Vo CONCLUSIONS The addition of opacifiers to polystyrene foam strongly influences its radiative thermal conductivity and only marginally the solid thermal conductivity while the gaseous thermal conductivity of the foam does not change. Consequently, knowing the specific extinction coefficient of the opacified foam as well as the solid and gaseous thermal conductivities of the not opacified foam, the complete thermal insulation performance of the opacified foam can be predicted. The two methods to model the spectral specific extinction coefficient of opacified foams have been assessed. The first method consists of integrating the refractive index of the opacified bulk material into geometric cell structure. There is just a need to obtain the thin foils containing the exact concentration of opacifiers for the measurement in the whole spectral region. The second method consists of measuring the opacifiers in powder form and of adding their extinction coefficient to the one of the standard foam without opacifier. The second method appeared to deliver similar total spectral specific extinction coefficient, although there is a small deviation in the very low and very high wavelength. Understanding the impact of opacifiers on radiative heat transfer in foam materials will enable a further optimization of the performance of the insulating material. The possibility to predict the impact can considerably help in the development of opacified foams. Besides carbon black other opacifiers such as boron carbide, silicon carbide, graphite, aluminium and many others can also be added to a foam. ACKNOWLEDGMENT This work was partially supported by the European Union (EU) under the grant number REFERENCES 1. nutshell_en.pdf 2. Directive 2010/31/EU, European Parliament of the Council, Energy performance of buildings, (2010). 3. Dombrovsky L., Baillis D., Thermal Radiation in Disperse System, Redding Connecticut (2010). 64 Cellular Polymers, Vol. 35, No. 2, 2016

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