RADIATIVE TRANSFER IN VACUUM THERMAL INSULATION OF SPACE VEHICLES
Following from: Radiative transfer problems in thermal protection of space vehicles
External heating of space vehicles during flight takes place due to direct solar irradiation, solar radiation reflected by planets, and infrared radiation of closely spaced planets. Various multilayer insulations (MLIs) are usually used to protect a vehicle from external heating. Vacuum insulations have obvious advantages such as relatively high thermal resistance and convenience use for surfaces with complex shapes. It may be interesting to compare MLIs with vacuum insulation panels (VIPs) widely used in building applications (Fricke et al., 2008; Baetens et al., 2010; Bouquerel et al., 2012; Alotaibi and Riffet, 2013). There are some general features of both MLIs and VIPs; however, the conditions encountered in space lead to additional strong restrictions.
A typical MLI resembles a set of thin metal screens having a thickness of ∼0.5–0.9 μm with a spacer between the screens. The spacer prevents contact between the screens. Simple estimates show that the use of several layers (screens with spacers) may lead to a significant reduction in the heat flux to the protected surface. MLIs with 10–30 shielding layers are usually used. The choice of material for the insulation depends on the expected temperature level. A polymer film coated with aluminum, silver, or gold can be used for screens in MLIs at working temperatures up to 423 K. Aluminum foil with spacers of fiberglass is used at higher temperatures up to 723 K. However, regardless of the material used for the layers and spacers, the principle behind MLIs is the same. In further calculations, aluminum foil screens and quartz fibrous spacers have been considered. Early thermal models, such as those studied by Alifanov et al. (2009), did not take into account the effect of spacers on heat transfer through the insulations. The present material investigated in this work is based on more recent studies (Gritsevich et al., 2014; Krainova et al., 2017).
Let us consider a single element of the MLI, which consists of two screens and a spacer made of semi-transparent highly porous fibrous material. The following assumptions are used in the heat transfer model: the radiative flux in the normal direction can be determined as a solution to a one-dimensional (1D) problem neglecting small two-dimensional effects; there is no thermal contact between the screens and the spacer, and thermal radiation is the only heat transfer mode to be considered; the normal emission and reflection are the most important properties of the surfaces, and the angular dependences of the radiative flux of these surfaces can be ignored; and the isothermal metal screens are totally opaque for thermal radiation. These assumptions enable us to use the following simple differential equations:
(1a) |
(1b) |
(1c) |
The heat capacity of each layer of the MLI is very small. This makes it possible to assume that a quasi-steady regime takes place in the case where the external heat transfer conditions vary slowly. This approach appears to be correct in practically all of the most important cases. The resulting simplifications are very important because it is sufficient to consider the balance equations for the spectral radiation fluxes:
(2a) |
(2b) |
where q_{i,λ}^{+} and q_{i,λ}^{–} are the spectral radiative fluxes in the gap with number i to the forward and backward hemispheres, respectively; ε_{1,λ} and ε_{2,λ} are the spectral hemispherical emittances of the screens at temperatures T_{1} and T_{2}, respectively; f_{λ} = πB_{λ}(T) is the blackbody spectral radiative flux; B_{λ}(T) is the Planck function; and R_{λ} and T_{λ} are the spectral hemispherical reflectance and transmittance of the spacer, respectively. According to Kirchhoff's law, the spectral hemispherical emittance of the spacer is expressed as ε_{sp,λ} = 1 – R_{λ} – T_{λ}.
In the case of a quasi-steady process, the resulting radiative fluxes in the two gaps are equal to each other:
(3) |
and Eqs. (2a) and (2b) are reduced to
(4) |
The corresponding integral radiative flux is determined as .
Both the transmittance and reflectance of radiation by highly porous fibrous spacers can be determined based on the independent scattering hypothesis and the Mie theory for infinite homogeneous cylinders (Bohren and Huffman, 1983; Dombrovsky and Baillis, 2010; Lee and Cunnington, 1998). It should be noted that typical fibrous spacers are made of fibers randomly oriented in the plane of the material layer. In the heat transfer problem, the transport efficiency factor of extinction (Q_{tr}), the efficiency factor of scattering (Q_{s}), and the efficiency factor of absorption (Q_{a}) at arbitrary illumination of the fibers can be obtained by neglecting the polarization effects:
(5) |
In the case of fibers randomly oriented in parallel planes, it is convenient to use the efficiency factors averaged over the orientations (Lee, 1986; Lee and Cunnington, 1998):
(6a) |
(6b) |
where
(6c) |
where α = |arcsin (sinθ sinψ)| is the incidence angle for a single fiber; θ is the incidence angle for the plane of fibers; and ψ is the angle between the incidence and normal planes [for more details, see chapter 2 in Dombrovsky and Baillis (2010)]. Additional information on the transport approximation and the use of this approach in solving radiative transfer problems can also be found in Dombrovsky (2012).
The following assumptions are used to estimate both the reflectance and transmittance of a fibrous spacer: multiple scatterings by fibers in a thin semi-transparent spacer can be neglected and there are no dependent scattering effects for randomly oriented fibers in a spacer. Obviously, these assumptions are correct in the case of a thin, highly porous spacer. In this case, the following relations for the absorbance and reflectance of a monodisperse layer of fibers can be used:
(7) |
where p is the surface porosity of the spacer, and Q̃_{s}^{b} = Q̃_{s}^{tr}/2 is the efficiency factor of the backscattering.
Equation (4) shows that the mathematical model includes only the following combination of the reflectance and transmittance: U_{λ} = T_{λ} – R_{λ}. Therefore, it is convenient to use U_{λ} in further analysis. Taking into account that A_{λ} = 1 – R_{λ} – T_{λ}, one can obtain the following equation:
(8) |
It is known that a thin oxide film is formed on the surface of an aluminum foil under normal atmospheric conditions. It usually takes ∼2 hours to form an oxide film of 10 nm thickness (Bartl and Baranek, 2004). This oxide film protects the bulk of the aluminum from further oxidation. The normal emittance of oxidized aluminum foil can be calculated as suggested by Brannon and Goldstein (1970). According to Kirchhoff's law, the spectral normal emittance is equal to ε_{n,λ} = 1 – R_{n,λ}. The integral emittance is calculated as follows:
(9) |
It should be noted that for polished metals, when ε_{n} < 0.5, the hemispherical emittance is slightly greater than the normal emittance (Howell et al., 2010). At the same time, the oxidation of the metal foil leads to less pronounced angular dependence of the surface emittance. To simplify the theoretical model, taking into account the oxide film effect, one can use the following approximate relation:
(10) |
where ε_{n,λ}(0) and ε_{λ}(0) are the normal and hemispherical emittances of the non-oxidized film, respectively. The ratios of these values were taken from Howell et al. (2010), depending on the value of ε_{n,λ}(0). It should be noted that the thickness of the oxide films on both surfaces of the aluminum foil is originally the same due to the manufacturing conditions, and no subsequent variation of this thickness in space is expected. Details on the optical properties of the fused silica, aluminum, and aluminum oxide used in the calculations can be found in Gritsevich et al. (2014).
Some of radiative transfer calculations for a single layer of MLI at various surface porosities of the spacer and radii of the quartz fiber at realistic temperatures T_{1} = 500 K, T_{2} = 300 K, and T_{sp} = 400 K are presented in Fig. 1. The effect of the semi-transparent quartz-fiber spacer appears to be insignificant, even in the case of a relatively dense spacer. The minimum in the curve, q(a), at a = 1.2 μm and p = 0.2 is explained by the Mie scattering effect (Dombrovsky and Baillis, 2010).
Figure 1. Integral radiative flux as a function of the surface porosity of a fibrous spacer (a) and the average radius of the fibers (b)
At the same time, Fig. 2 indicates a considerable effect of the alumina film for one of the variants. The oxide film on aluminum foils at a film thickness greater than ∼1 μm should be taken into account. It should be noted that the upper estimates showed that a realistic spacer made of metalized fibers with aluminum coating of thickness ∼50 nm may lead to an almost two-fold decrease in the integral radiative flux (Gritsevich et al., 2014).
Figure 2. Effect of the alumina film thickness on the radiative flux at p = 0.2 and a = 1.2 μm
The generalized theoretical model suggested in Krainova et al. (2017) is free from the main assumption of the previously presented model on the absence of thermal contact between the spacer and the neighboring aluminum foil layers. At the same time, the assumptions for the small optical thickness of a fibrous spacer and the random orientation of fibers in planes remain unchanged. The latter is sufficient to neglect the temperature difference across the spacer. Of course, the maximum estimate of the effect of thermal contact between the foil and spacer can be obtained by ignoring possible local gaps between them. This means that the temperature of the fibrous spacer is assumed to be equal to the temperature of the adjacent aluminum foil. Strictly speaking, one can take into account the temperature difference across the spacer. This can be done by using the approaches developed in Dombrovsky (1996), Lee and Cunnington (2000), Zhao et al. (2009), Shuyuan et al. (2009), and Grinchuk (2014). At the same time, the maximum estimate is considered when this temperature difference is negligible because of the very small geometrical thickness of ordinary spacers. This is the main simplification, which was confirmed in Krainova et al. (2017) by comparison with the experimental data. It should be noted that the effect of an isothermal spacer is the increase in the spectral emissivity of the foil–spacer system, and the solution to the problem under consideration does not depend on which foil (relatively cold or hot) is in thermal contact with the spacer. It should also be noted that this is not the so-called thermal bridge effect studied in relation to VIPs (Mao et al., 2020).
The spectral radiative flux in a plane-parallel single layer of vacuum insulation is determined as follows (hereafter, subscript λ for spectral quantities is omitted for brevity):
(11) |
For definiteness, it is assumed that the spacer is in contact with the hotter foil (T_{s} = T_{1}, where subscript s refers to the spacer). In this case, ε_{2} = ε_{f} (where subscript f refers to the foil), whereas the value of ε_{1}, taking into account the spacer effect, should be obtained.
The principal assumption of the model is that a continuous approach is applicable to the radiative transfer in a thin semi-transparent fibrous spacer. This assumption may lead to quantitative errors in the limit of a spacer containing only a few flat layers of fibers; however, the obvious advantage of the continuous approach, which enables one to use a simple differential approximation for radiative transfer, is too attractive to be ignored. In addition, the limiting case of a very thin and almost transparent spacer is not as interesting because of the small overall effect. This effect increases and deserves special consideration in the case of multiple scatterings of infrared radiation in a spacer. The latter circumstance explains the methodological choice made in the present study.
A schematic illustration of the problem under consideration is shown in Fig. 3. In many cases, the 1D problem for radiative transfer in an isothermal spacer can be rather accurately solved using the transport approximation for the scattering phase function and the traditional two-flux method (Dombrovsky and Baillis, 2010; Dombrovsky, 2012; Modest, 2013).
Figure 3. Schematic illustration of the problem
Perhaps the most detailed study of the errors of the two-flux (or Schuster–Schwarzschild) approximation for radiative transfer problems of this type is presented in the first chapter of the book by Dombrovsky and Baillis (2010) [see, also, chapter 15 in Modest (2013)], in which it was shown that this approach enables one to determine the radiative flux very accurately because of the integral nature (over the angles) of this quantity. This statement appears to be true for an arbitrary thickness of the medium layer. The typical error in the radiative flux is less than ∼5%. At the same time, it will be shown subsequently that some corrections to the boundary condition should be made in the optically thin limit and low emissivity of the boundary surface. The mathematical problem statement for a uniform isothermal spacer is similar to the following boundary-value problem for spectral irradiance g:
(12a) |
(12b) |
(12c) |
where z is the current coordinate across the spacer; d is the spacer thickness; D is the spectral radiation diffusion coefficient; β_{tr} = α + σ_{tr} is the transport extinction coefficient; α is the absorption coefficient; and σ_{tr} is the transport scattering coefficient. The spectral hemispherical emissivity (ε_{1}) of the open surface of the spacer is determined as follows:
(13) |
Omitting some transformations reported by Krainova et al. (2017), we obtain the following expression for the emissivity value, which is correct for spacers that are not too optically thin:
(14a) |
(14b) |
The typical dependences of spectral emissivity ε_{1} on the fiber radius and spacer thickness at two wavelengths are shown Fig. 4.
Figure 4(a) shows that the effect of the fiber radius is not significant and the natural uncertainty in the size distribution of the glass fibers does not lead to a considerable error in the value of ε_{1}. As one would expect, the spacer thickness is the main parameter of the problem [see Fig. 4(b)]. In the absorption band at λ = 9 μm, the optically thick limit is reached at d ≈ 0.08 mm, whereas the relatively thick spacer at d = 0.2 mm is not optically thick in the range of glass semi-transparencies (at λ = 6 μm).
Figure 4. Spectral hemispherical emissivity of aluminum foil covered by a fibrous spacer with a fiber volume fraction of f_{v} = 0.06: dependences on the fiber radius (a) and spacer thickness (b)
The calculations showed that the local thermal contacts between the spacer and one of the adjacent foils may considerably decrease the thermal resistance of the vacuum insulation. Fortunately, this situation is expected to be temporary and may take place only in some specific parts of the space vehicle trajectory. In all cases, the effect under consideration should be taken into account in the calculations of vehicle thermal protection. The computational results appear to be in good agreement with the laboratory thermal measurements taken under vacuum conditions (Krainova et al., 2017). This enables the authors to recommend the suggested analytical model for engineering estimates of the thermal effects of possible temporary contacts between the spacer and adjacent aluminum foils in relation to the quality of multi-layer thermal insulations under space flight conditions.
REFERENCES
Alifanov, O.M., Nenarokomov, A.V., and Gonzalez, V.M. (2009) Study of Multilayer Thermal Insulation by Inverse Problems Method, Acta Astronaut., 65(9–10): 1284–1291.
Alotaibi, S.S. and Riffet, S. (2013) Vacuum Insulated Panels for Sustainable Buildings: A Review of Research and Applications, Int. J. Energy Res., 38(1): 1–19.
Baetens, R., Jelle, B.P., Thue, J.V., Tenpierik, M.J., Grynning, S., Uvsløkk, S., and Gustavsen, A. (2010) Vacuum Insulation Panels for Building Applications: A Review and Beyond, Energy Build., 42(2): 147–172.
Bartl, J. and Baranek, M. (2004) Emissivity of Aluminium and Its Importance for Radiometric Measurements, Meas. Phys. Quant., 31: 31–36.
Bohren, C.F. and Huffman, D.R. (1983) Absorption and Scattering of Light by Small Particles, New York: Wiley.
Bouquerel, M., Duforestel, T., Baillis, D., and Rusaouen, G. (2012) Heat Transfer Modeling in Vacuum Insulation Panels Containing Nanoporous Silicas—A Review, Energy Build., 54: 320–336.
Brannon, R.R. and Goldstein, R.J. (1970) Emittance of Oxide Layers on a Metal Substrate, ASME J. Heat Transf., 92(2): 257–263.
Dombrovsky, L.A. (1996) Quartz-Fiber Thermal Insulation: Infrared Radiative Properties and Calculation of Radiative-Conductive Heat Transfer, ASME J. Heat Transf., 118(2): 408–414.
Dombrovsky, L.A. (2012) The Use of Transport Approximation and Diffusion-Based Models in Radiative Transfer Calculations, Comput. Therm. Sci., 4(4): 297–315.
Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, New York: Begell House.
Fricke, J., Heinemann, U., and Ebert, H.P. (2008) Vacuum Insulation Panels—From Research to Market, Vacuum, 82(7): 680–690.
Grinchuk, P.S. (2014) Contact Heat Conductivity under Conditions of High-Temperature Heat Transfer in Fibrous Heat-Insulating Materials, J. Eng. Phys. Thermophys., 87(2): 481–488.
Gritsevich, I.V., Dombrovsky, L.A., and Nenarokomov, A.V. (2014) Radiative Transfer in Vacuum Thermal Insulation of Space Vehicles, Comput. Therm. Sci., 6(2): 103–111.
Howell, J.R., Siegel, R., and Mengüç, M.P. (2010) Thermal Radiation Heat Transfer, New York: CRC Press.
Krainova, I.V., Dombrovsky, L.A., Nenarokomov, A.V., Budnik, S.A., Titov, D.M., and Alifanov, O.M. (2017) A Generalized Analytical Model for Radiative Transfer in Vacuum Thermal Insulation of Space Vehicles, J. Quant. Spectrosc. Radiat. Transf., 197: 166–172.
Lee, S.-C. (1986), Radiative Transfer through a Fibrous Medium: Allowance for Fiber Orientation, J. Quant. Spectrosc. Radiat. Transf., 36(3): 253–263.
Lee, S.-C. and Cunnington, G.R. (1998) Theoretical Models for Radiative Transfer in Fibrous Media, In C.L. Tien (Ed.), Annual Review in Heat Transfer, vol. 9, New York: Begell House, pp. 159–218.
Lee, S.-C. and Cunnington, G.R. (2000) Conduction and Radiation Heat Transfer in High-Porosity Fiber Thermal Insulation, AIAA J. Thermophys. Heat Transf., 14(2): 121–136.
Mao, S., Kan, A., and Wang, N. (2020) Numerical Analysis and Experimental Investigation on Thermal Bridge Effect of Vacuum Insulation Panel, Appl. Therm. Eng., 169: 114980.
Modest, M.F. (2013) Radiative Heat Transfer, 3rd ed., New York: Academic Press.
Shuyuan, Z., Boming, Z., and Shanyi, D. (2009) Effects of Contact Resistance on Heat Transfer Behaviors of Fibrous Insulation, Chin. J. Aeronaut., 22(5): 569–574.
Zhao, S.Y., Zhang, B.-M., and He, X.-D. (2009) Temperature and Pressure Dependent Effective Thermal Conductivity of Fibrous Insulation, Int. J. Therm. Sci., 48(2): 440–448.
参考文献列表
- Alifanov, O.M., Nenarokomov, A.V., and Gonzalez, V.M. (2009) Study of Multilayer Thermal Insulation by Inverse Problems Method, Acta Astronaut., 65(9–10): 1284–1291.
- Alotaibi, S.S. and Riffet, S. (2013) Vacuum Insulated Panels for Sustainable Buildings: A Review of Research and Applications, Int. J. Energy Res., 38(1): 1–19.
- Baetens, R., Jelle, B.P., Thue, J.V., Tenpierik, M.J., Grynning, S., Uvsløkk, S., and Gustavsen, A. (2010) Vacuum Insulation Panels for Building Applications: A Review and Beyond, Energy Build., 42(2): 147–172.
- Bartl, J. and Baranek, M. (2004) Emissivity of Aluminium and Its Importance for Radiometric Measurements, Meas. Phys. Quant., 31: 31–36.
- Bohren, C.F. and Huffman, D.R. (1983) Absorption and Scattering of Light by Small Particles, New York: Wiley.
- Bouquerel, M., Duforestel, T., Baillis, D., and Rusaouen, G. (2012) Heat Transfer Modeling in Vacuum Insulation Panels Containing Nanoporous Silicas—A Review, Energy Build., 54: 320–336.
- Brannon, R.R. and Goldstein, R.J. (1970) Emittance of Oxide Layers on a Metal Substrate, ASME J. Heat Transf., 92(2): 257–263.
- Dombrovsky, L.A. (1996) Quartz-Fiber Thermal Insulation: Infrared Radiative Properties and Calculation of Radiative-Conductive Heat Transfer, ASME J. Heat Transf., 118(2): 408–414.
- Dombrovsky, L.A. (2012) The Use of Transport Approximation and Diffusion-Based Models in Radiative Transfer Calculations, Comput. Therm. Sci., 4(4): 297–315.
- Dombrovsky, L.A. and Baillis, D. (2010) Thermal Radiation in Disperse Systems: An Engineering Approach, New York: Begell House.
- Fricke, J., Heinemann, U., and Ebert, H.P. (2008) Vacuum Insulation Panels—From Research to Market, Vacuum, 82(7): 680–690.
- Grinchuk, P.S. (2014) Contact Heat Conductivity under Conditions of High-Temperature Heat Transfer in Fibrous Heat-Insulating Materials, J. Eng. Phys. Thermophys., 87(2): 481–488.
- Gritsevich, I.V., Dombrovsky, L.A., and Nenarokomov, A.V. (2014) Radiative Transfer in Vacuum Thermal Insulation of Space Vehicles, Comput. Therm. Sci., 6(2): 103–111.
- Howell, J.R., Siegel, R., and Mengüç, M.P. (2010) Thermal Radiation Heat Transfer, New York: CRC Press.
- Krainova, I.V., Dombrovsky, L.A., Nenarokomov, A.V., Budnik, S.A., Titov, D.M., and Alifanov, O.M. (2017) A Generalized Analytical Model for Radiative Transfer in Vacuum Thermal Insulation of Space Vehicles, J. Quant. Spectrosc. Radiat. Transf., 197: 166–172.
- Lee, S.-C. (1986), Radiative Transfer through a Fibrous Medium: Allowance for Fiber Orientation, J. Quant. Spectrosc. Radiat. Transf., 36(3): 253–263.
- Lee, S.-C. and Cunnington, G.R. (1998) Theoretical Models for Radiative Transfer in Fibrous Media, In C.L. Tien (Ed.), Annual Review in Heat Transfer, vol. 9, New York: Begell House, pp. 159–218.
- Lee, S.-C. and Cunnington, G.R. (2000) Conduction and Radiation Heat Transfer in High-Porosity Fiber Thermal Insulation, AIAA J. Thermophys. Heat Transf., 14(2): 121–136.
- Mao, S., Kan, A., and Wang, N. (2020) Numerical Analysis and Experimental Investigation on Thermal Bridge Effect of Vacuum Insulation Panel, Appl. Therm. Eng., 169: 114980.
- Modest, M.F. (2013) Radiative Heat Transfer, 3rd ed., New York: Academic Press.
- Shuyuan, Z., Boming, Z., and Shanyi, D. (2009) Effects of Contact Resistance on Heat Transfer Behaviors of Fibrous Insulation, Chin. J. Aeronaut., 22(5): 569–574.
- Zhao, S.Y., Zhang, B.-M., and He, X.-D. (2009) Temperature and Pressure Dependent Effective Thermal Conductivity of Fibrous Insulation, Int. J. Therm. Sci., 48(2): 440–448.