Experimental and theoretical studies on high-temperature thermal properties of fibrous insulation

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Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 1309–1324 www.elsevier.com/locate/jqsrt

Experimental and theoretical studies on high-temperature thermal properties of fibrous insulation Bo-Ming Zhang, Shu-Yuan Zhao, Xiao-Dong He Center for Composite Materials and Structure, Harbin Institute of Technology, Harbin 150080, China Received 30 August 2006; received in revised form 16 September 2007; accepted 6 October 2007

Abstract In the present paper, an experimental apparatus has been developed to measure heat transfer through high-alumina fibrous insulation for thermal protection system. Effective thermal conductivities of the fibrous insulation were measured over a wide range of temperature (300–973 K) and pressure (102–105 Pa) using the developed apparatus. The specific heat and the transmittance spectra in the wavelength range of 2.5–25 mm were also measured. The spectral extinction coefficients and Rosseland mean extinction coefficients were obtained from transmittance data at various temperatures to investigate the radiative heat transfer in fibrous insulation. A one-dimensional finite volume numerical model combined radiation and conduction heat transfer was developed to predict the behavior of the effective thermal conductivity of the fibrous insulation at various temperatures and pressures. The two-flux approximation was used to model the radiation heat transfer through the insulation. The experimentally measured specific heat and Rosseland mean extinction coefficients were used in the numerical heat transfer model to calculate the effective thermal conductivity. The average deviation between the numerical results for different values of albedo of scattering and the experimental results was investigated. The numerical results for o ¼ 1 and experimental data were compared. It was found that the calculated values corresponded with the experimental values within an average of 13.5 percent. Numerical results were consistent with experimental results through the environmental conditions under examination. r 2007 Elsevier Ltd. All rights reserved. Keywords: Effective thermal conductivity; Transmission; Extinction coefficient; Albedo of scattering; Specific heat; Thermal physical properties

1. Introduction Metallic thermal protection systems (TPS) are an attractive technology to help meet the ambitious goals of future space transportation systems. Its thermal performance is of critical concern. There are many design options that have the potential to improve the thermal performance of metallic TPS. An obvious way to improve the thermal performance of metallic TPS is to develop more efficient non-load-bearing insulation. High-temperature fibrous insulation becomes the insulation candidate being considered for use in the metallic TPS on reusable launch vehicles, because it offers a low density, excellent heat-insulated property and high service temperature. Corresponding author. Tel.: +86 451 86418172; fax: +86 451 86402345.

E-mail address: [email protected] (S.-Y. Zhao). 0022-4073/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2007.10.008

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However, limited information is available on the thermal properties of insulation under the environmental conditions to which re-entry-type vehicles are exposed. Earth re-entry typically produces aerodynamic heating to a surface to the temperature as high as 1273 K in a pressure range from 1.333 to 1.013  105 Pa [1]. Heat transfer through a fibrous insulation involves combined modes of heat transfer: solid conduction through fibers, gas conduction and natural convection in spacers between fibers, and radiation interchange through participating media in the fibrous insulation. The relative contributions of the different heat transfer modes vary during reentry. The transport process of radiation in the fibrous insulation is complicated with increasing temperature because of the complex morphology and the inherent complexities associated with the transport mechanism itself. The complexity of the heat transfer makes the analysis and the design of insulation quite difficult. There has been extensive research on the heat transfer in porous media, experimentally and analytically. Lee and Cunnington [2] have provided a comprehensive review of heat transfer in general porous material. The researchers [3,4] have developed an effective thermal conductivity model based on superposition of gas, solid and apparent radiation thermal conductivities, based on optically thick assumption, and compared the results with measured effective thermal conductivities of samples subjected to certain temperature difference across the insulation thickness over a wide range of environmental pressures. Larkin and Churchill [5] modeled radiation heat transfer through fibrous insulations based on a two-flux approximation assuming a purely scattering medium. Their results were compared with the measured optical transmission through fibrous insulation samples. Tong et al. [6–8] used the two-flux model assuming a linearized anisotropic scattering to model radiation heat transfer through fibrous insulation, and the predicted fluxes that combined radiant and conduction were compared with measured data up to 450 K and at one atm. Ballis et al. [9] modeled heat transfer in open-cell carbon foams and stated that radiation was the primary source of heat transfer for temperature above 1000 K. A weighted spectral extinction coefficient was used to account for anisotropic scattering. C.Y. Zhao et al. [10] experimentally measured the radiative transfer in ultralight metal foams with high porosity and different cell sizes. The spectral transmittance and reflectance were measured at different infrared wavelengths ranging from 2.5 to 50 mm, which were subsequently used to determine the extinction coefficient and foam emissivity. An analytical model based on geometric optics laws, diffraction theory and metal foam morphology was developed to predict the radiative transfer. Close agreement between the predicted effective foam conductivity due to radiation alone and that measured was observed. The objective of this investigation was to measure the effective thermal conductivity of the fibrous insulation over a wide range of temperature (300–973 K) and pressure (102–105 Pa) using a developed apparatus. To investigate the radiative heat transfer through fibrous insulation, the spectral transmittance of fibrous insulation in the wavelength range of 2.5–25 mm was measured over the temperature range of 291–973 K, and subsequently used to determine the spectral extinction coefficient as well as Rosseland mean extinction coefficient. The experimentally measured specific heat and Rosseland mean extinction coefficient were used to calculate the effective thermal conductivity of high-alumina fibrous insulation for TPS from a numerical heat transfer model. The numerical model was validated by comparing the numerical results with the steady-state experimental results. 2. Sample description The insulation sample studied in this investigation is high-alumina fibrous insulation. Its primary components are alumina and silica. The nominal chemical composition of the insulation is shown in Table 1. The melting point of the fiber is around 2030 K. Continuous service temperature is 1473 K. The nominal density is 128 kg/m3. The SEM micrograph of fibrous insulation sample is shown in Fig. 1. From Fig. 1, the fibers are stratified randomly, in more or less parallel planes between the medium boundaries. The fiber diameter varies from 1 to 9 mm, approximately. Table 1 Nominal chemical composition of high-alumina fibrous insulation Chemical composition

Al2O3

Al2O3+SiO2

Fe2O3

R2O

Content (wt.%)

52–55

98

0.25

0.25

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Fig. 1. SEM micrograph of fibrous insulation sample.

1 8

7 2 3 6

4

5

1. Carbon fabric insulation barrel; 2.Vacuum chamber; 3.Septum plate; 4. Test specimen; 5.Water-cooled plate; 6.Guarded insulation; 7. Graphite radiant heater; 8.Insulating screen Fig. 2. Schematic of the thermal conductivity apparatus.

3. Experimental methodology 3.1. Effective thermal conductivity Effective thermal conductivity, which combines the effects of heat transfer by conduction, convection and radiation, is one of the most important parameters to characterize the thermal performance of insulation. Effective thermal conductivity varies with temperature and pressure under the aerodynamic heating conditions. In order to investigate heat transfer through the fibrous insulation and obtain the effective thermal conductivity of insulation sample exactly and reliably, an apparatus was developed to measure the steady-state effective thermal conductivity of the insulation. A schematic of the apparatus is shown in Fig. 2. The radiant heat source is a graphite radiant heater that can reach a temperature as high as 1873 K. The sample is placed between a septum plate and a water-cooled plate maintained at room temperature. The water-cooled plate is placed at the bottom to reduce the effect of natural convection. Thermocouples are installed at strategic locations to monitor the change of temperature distribution throughout the specimen during heating. Heat flux gauges located on the water-cooled plate are used to measure the flux of heat energy flowing through the sample. The apparatus is located inside a vacuum chamber and the environmental pressure varies from 102 to 105 Pa. A pressure sensor is used to measure pressure in the vacuum chamber. Once the insulation sample is inserted into the test apparatus, the gas inside the vacuum chamber is removed. The primary reason as to why

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nitrogen is selected for use in this experiment is to eliminate water vapor in air from the chamber and reduce the oxidation of the graphite at higher temperature. Once all the thermocouples on the hot side and cold side and heat flux gages on the water-cooled plate are stabilized, data storage is initiated. Using the measured heat flux, the hot side temperature, the cold side temperature, and the sample thickness, the effective thermal conductivity of the insulation sample can be calculated from Fourier’s law of heat conduction: ke ¼

qL , T1  T2

(1)

where q is the measured heat flux, L is the insulation thickness, and T1 and T2 are the measured temperatures on the hot side and the cold side, respectively. The uncertainty of the measurement is estimated from the error propagation equation:  2  2   !0:5 dk dq dL dDT 2 ¼ þ þ . (2) ke q L DT The uncertainty of the measured effective thermal conductivity is 8 percent approximately. In the present study, the sample has a square cross-sectional area with the sides being 450 mm. The nominal thickness of the sample is 40 mm. The effective thermal conductivity of the sample is measured with the nominal hot side temperatures set at 373, 473, 573, 673, 773, 873 and 973 K, and the nominal experimental pressures controlled at 1  102, 10, 1  102, 5  103, 5  104 and 1.013  105 Pa. Each measurement consists of setting the temperature of the septum plate on the top of the sample to the desired temperature, varying the nitrogen gas to the desired pressure, and allowing both temperature and pressure to reach a steady-state condition. 3.2. Specific heat Specific heat is one of the most important thermal physical properties of insulation which are used in heat energy calculations and designs. In the present study, specific heat is measured by the method of differential scanning calorimetry using a simultaneous thermal analyzer (NETZSCH STA 449C). The measuring principle is to compare the rate of heat flow with the sample and with a reference at the programmed temperature. Whether more or less heat must flow to the sample depends on whether the process is exothermic or endothermic. If the difference of heat flow between the sample and the reference is detected, from the mass of the sapphire standard, the known specific heat of sapphire, and the data signals collected from the DSC, the specific heat of sample can be computed: Cpsam ¼ Cpstd

ðDSC sam  DSC bsl Þ=M sam , ðDSC std  DSC bsl Þ=M std

(3)

where DSCbsl is the measured DSC signal of baseline, DSCstd is the measured DSC signal of standard (sapphire), DSCsam is the measured DSC signal of sample, M is mass, Cp is the specific heat. The sample was ground into powder before being measured. The specific heat of the sample was measured for the temperature range of 323–973 K at a heating rate of 3 K/min under argon atmosphere. The mass of the sample used for the experiment was 17.69 mg. 3.3. Rosseland mean extinction coefficient To study the thermal radiation within insulation and its local radiative property, the extinction coefficient is needed. Physically, the extinction coefficient represents the decay rate of the radiation intensity passing through the material. For engineering applications, the Rosseland mean extinction coefficient is a more commonly used material parameter than the spectral extinction coefficient, as the former represents the overall effect of energy decay in the material. The Rosseland mean extinction coefficient is defined as [11] R1 Z 1 ð1=bl Þðqebl =qTÞ dl 1 1 qebl ¼ ¼ 0 R1 dl, (4) KR bl qeb 0 0 ðqebl =qTÞ dl

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where l is the wavelength, T is the medium temperature, eb is the blackbody emissive power, ebl is the spectral blackbody emissive power and bl is the spectral extinction coefficient. The spectral extinction coefficients for thin sample can be obtained, using Beer’s law as [12]  Z L  T nl ¼ exp  bl dx , (5) 0

where Tnl is the spectral transmittance and L is the thickness of the sample. For a homogeneous sample, bl is independent of the sample thickness, then Eq. (5) can be reduced to lnðT nl Þ . (6) L For black bodies, the monochromatic emissive power was derived by Planck by introducing the quantum concept for electromagnetic energy as bl ¼ 

eb;l ¼

2phc2 , l5 ðexpðhc=KT lÞ  1Þ

(7)

where c is the speed of light, h is the Plank’s constant and K is the Boltzmann constant. Then Eq. (4) can be transformed to: Z 1 1 L 2ph2 c3 expðhc=KTlÞ ¼  dl. (8) KR lnðT nl Þ 4Ksl6 T 5 ½expðhc=KTlÞ  12 0 By integrating Planck’s law (Eq. (7)) over the wavelength, it has been established that over 50 percent of the thermal radiation lies in the wavelength range between 2.5 and 25 mm below 973 K. In this study, the spectral transmittance is measured for the wavelength range of 2.5–25 mm using a Fourier transform infrared spectrometer (Bruker IFS 66V/S, Germany). 4. Analytical model Existing techniques were utilized and combined to model the effective thermal conductivity of the insulation in the temperature and pressure range covered by the experiments. In the absence of natural convective heat transfer, the conservation of energy for one-dimensional heat transfer through the insulation by conduction and radiation yields the partial differential equation [13]   qT q qT qq00 rcP ¼ kc (9)  r qt qx qx qx subjected to the following initial and boundary conditions: Tðx; 0Þ ¼ T 0 ,

(10)

Tð0; tÞ ¼ T 1 ,

(11)

TðL; tÞ ¼ T 2 ,

(12) q00r

where T is temperature, r is density, cp is specific heat, kc is thermal conductivity, is the radiant heat flux, t is time, x is the spatial coordinate through the insulation thickness and L is the insulation thickness. The gradient of radiant heat flux is given by qq00r ¼ bð1  oÞðG  4sT 4 Þ, (13) qx where G is the incident radiation, b is the extinction coefficient, the albedo of scattering, o, is defined as the ratio of the scattering coefficient to the extinction coefficient, and s is the Stefan–Boltzmann constant. The two-flux approximation, which is also known as the Milne–Eddington approximation, is used to model the radiation heat transfer through fibrous insulation. The radiant heat flux is related to incident radiation, G,

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according to [14]: q00r ¼ 

1 qG . 3b qx

(14)

The incident radiation in the fibrous insulation is obtained by solving the following second-order differential equation: 

1 q2 G þ G ¼ 4sT 4 . 3b2 ð1  oÞ qx2

(15)

The boundary conditions for the second-order differential equation governing the incident radiation are given by 

2 qG þ G ¼ 4sT 41 , 3bð1 =ð2  1 ÞÞ qx

2 qG þ G ¼ 4sT 42 , 3bð2 =ð2  2 ÞÞ qx

(16a)

(16b)

where e1 and T1 are the emittance and temperature of the bounding surface at x ¼ 0, while e2 and T2 are the emittance and temperature of the bounding surface at x ¼ L. e1 and e2 are, respectively, 0.8 and 0.3 in the present study. Several theories have been developed to describe the combined conduction thermal conductivity due to solid conduction and gas conduction. In the present study, two thermal resistances in a parallel arrangement are used for modeling heat transfer in fibrous insulation [15] kc ¼ fks þ ð1  f Þkg .

(17)

The gas thermal conductivity kg ¼

kg 1 þ 2ðð2  aÞ=aÞð2¯g=ð¯g þ 1ÞÞð1=PrÞðlm =Lc Þ

(18)

was based on the gas conduction model used by Daryabeigi, where a is the thermal accommodation coefficient, g¯ is the gas specific heat ratio, Pr is the Prandtl number, kg is the temperature-dependent gas thermal conductivity at an atmospheric pressure, Lc is the characteristic length for gas conduction in fibers and lm is the molecular mean free path which is given by KT lm ¼ pffiffiffi 2 , 2pd g P

(19)

where K is the Boltzmann constant, dg is the gas collision diameter, T and P are the gas temperature and pressure, and Lc is given by Lc ¼

p Df , 4 f

(20)

where f is the solid fraction ratio, which is defined as the ratio of density of fibrous insulation to the density of fiber parent material, and Df is the diameter of fiber. The solid thermal conductivity is defined as [16] ks ¼ f 2 ks ,

(21)

where ks is the variation of the thermal conductivity of the fiber parent material with temperature. The solution to the governing conservation of energy equation, Eq. (9), can be obtained using an iterative procedure. The distribution of incident radiation is obtained in the region by solving Eq. (15) subject to the boundary conditions in Eqs. (16a) and (16b), and the radiant flux is obtained according to Eq. (14). The gradient of radiant heat flux is obtained using Eq. (13), and substituted in the energy equation, Eq. (9), to solve for the temperature distribution in the domain. Eq. (9) was solved numerically using a finite difference

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technique and the discretized form yielded ðrcpj1 þ rcpj Þ þ

nþ1 n kj1 nþ1 kj Dx T j  T j 1 nþ1 ¼ ðT j1  T nþ1 ðT jþ1 ðG n  G nj Þ Þþ  T nþ1 Þþ j j 2 3bj1 Dx j1 Dt Dx Dx

1 ðG n  Gnj Þ, 3bj Dx jþ1

ð22Þ

where the superscript denotes the time step and the subscript denotes the spatial step. Boundary conditions applied were constant temperatures corresponding to the measured steady-state temperatures at the top and bottom of the sample. A linearly varying temperature distribution through the thickness of the sample was selected as the initial condition. The Rosseland mean extinction coefficient KR obtained in the above experiment was used to substitute for the extinction coefficient b in the calculation of radiant heat flux. The measured specific heat was also used in the calculation. Eq. (22) was solved numerically until the temperature at each volume element did not change with successive iterations and the steady-state conditions were achieved. The convergence criterion for the solution of the coupled system of the equation is met when the computed temperature behavior satisfies the following expression: jT nþ1  T ni j i o, nþ1 2pipm Ti max

(23)

where e is a given strictly positive real. It is a relative termination tolerance. 5. Results and discussion 5.1. Experimental results 5.1.1. Effective thermal conductivity The heat transfer mechanisms in fibrous insulation include solid conduction, gas conduction and radiation, and convection heat transfer is negligible. The effective thermal conductivity as a function of the average sample temperature is shown in Fig. 3. The effective thermal conductivity increases non-linearly with

Effective thermal conductivity/W/ (m.K)

0.10

1×10-2 Pa

0.08

1×102 Pa 1.013×105 Pa 0.06

0.04

0.02

0.00 300

350

400

450

500 550 Temperature/K

600

650

Fig. 3. Variation of effective thermal conductivity with temperature.

700

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Effective thermal conductivity/W/ (m.K)

0.10

373K 673K 973K

0.08

0.06

0.04

0.02

0.00 10-3

10-2

10-1

100

101 102 Pressure/Pa

103

104

105

106

Fig. 4. Variation of effective thermal conductivity with pressure.

increasing sample average temperature, due to the fact that radiation heat transfer is related to the fourth power of temperature, and it is more dominant with the increase of temperature. To gain further insight into the influence of environmental pressure on the overall heat transfer, the variation of effective thermal conductivity with environmental pressure for the sample is shown in Fig. 4. Data are plotted for three different nominal hot side temperatures. The figure shows that the measured effective thermal conductivity for each sample average temperature increases with increasing pressure levels. As can be seen, the effective thermal conductivity increases somewhat below 10 Pa, rapidly between 1  102 and 5  103 Pa, and then stays relatively constant above 5  104 Pa. This trend is in accordance with the relevant experimental results in the paper [17]. As solid and radiative contributions are independent of gas pressure, changing pressure only affects the contribution of gas conduction to the effective thermal conductivity. It can be stated that gas conduction increases with increasing pressure. The same trends are observed for all other insulation sample. 5.1.2. Specific heat Fig. 5 presents the specific heat of fibrous insulation sample versus temperature. No sharp peak appears in the curve, which indicates that no phase transformation occurs for the investigated temperature range. The specific heat of the sample increases with increasing temperature. The rate increases rapidly below 800 K, and then increases slightly. The ability of heat absorption for insulation sample enhances with the rise of temperature. 5.1.3. Rosseland mean extinction coefficient The measured spectral transmittance is plotted as a function of wavelength in Fig. 6 for high-alumina fibrous insulation with different thicknesses. The results of Fig. 6 demonstrate that the spectral transmittance decreases as the media thickness increases. The insulation transmittance property is strongly dependent upon the spectral wavelength. The transmittance initially increases with increasing wavelength and decreases sharply to reach the first absorption peak at around 9 mm, and then increases somewhat before decreasing again. The second absorption peak is reached at around 12 mm. Afterward the transmittance gradually increases and reaches a plateau. It is noticed that all the three testing samples with different thicknesses exhibit similar dependence upon the wavelength.

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1.3

1.2

Specific heat/J/ (g.K)

1.1

1.0

0.9

0.8

0.7

0.6 200

300

400

500

600 700 Temperature/K

800

900

1000

Fig. 5. Variation of specific heat with temperature.

60

Spectral transmittance/ %

50

40

640μm

30

780μm 980μm

20

10

0 5

10

15

20

25

Wavelength/μm Fig. 6. Variation of spectral transmission of samples with different thickness with wavelength at room temperature.

Fig. 7 shows the variation of spectral transmittance of samples with wavelength at various temperatures. Given the fact that there is no obvious change in the physical and chemical properties of high-alumina fibrous insulation up to 973 K, the trends are similar for the sample at various temperatures. The maximum values of transmittance decrease and show a shift towards the direction of longer wavelength as temperature increases. It can be explained by the fact that spectral transmittance is the ratio of spectral intensity after and before the insulation sample of thickness. According to Planck’s law, emissive power increases and the maximum of

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50 291 K

Spectral transmittance/%

40

373 K 573 K

30 773 K 973 K

20

10

0 5

10

15

20

25

Wavelength/μm Fig. 7. Variation of spectral transmittance of samples with wavelength at various temperatures.

10000

Spectral extinction coefficient/1/m

9000 8000 7000 6000 5000 4000

640μm 780μm

3000

980μm

2000 1000 0 0

5

10

15

20

25

Wavelength/μm Fig. 8. Variation of spectral extinction coefficient of samples with different thickness with wavelength at room temperature.

emissive power shows a shift towards the direction of shorter wavelength with increasing temperature; therefore, spectral intensity after the insulation sample of thickness decreases and shows a shift towards the direction of longer wavelength with the rise of temperature. Spectral extinction coefficient can be obtained from Eq. (5) using the spectral transmittance. The spectral extinction coefficient thus determined is plotted as a function of wavelength in Figs. 8 and 9. Fig. 8 shows that the spectral extinction coefficients of the samples with different thicknesses show very large differences in the

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10000

Spectral extinction coefficient/1/m

9000 8000 7000 6000 5000 291K 4000

373K

3000

573K 773K

2000

973K

1000 0 0

5

10 15 Wavelength/μm

20

25

Fig. 9. Variation of spectral extinction coefficient of samples with wavelength at various temperatures.

wavelength range of 2.5–6 mm, partly because of the difference of scattering effects in the sample with different thickness. The spectral extinction coefficients coincide very well in the range of 6–8 mm. Little differences are shown in the spectral extinction coefficients of the three samples in the range of 8–25 mm. The fluctuations in the spectral extinction coefficients of samples with different thickness are also because of the error of measured thickness. On the whole, the assumption of spectral extinction coefficient independent of sample thickness is applicable for the samples in the present investigated range. Fig. 9 demonstrates that the spectral extinction coefficients of samples at various temperatures exhibit similar trend. Spectral extinction coefficients increase slightly as temperature increases. Rosseland mean extinction coefficient is an average extinction coefficient over the spectrum weighted by the emissive power. It represents the ability to eliminate thermal radiation at a certain temperature. The variation of Rosseland mean extinction coefficient with temperature obtained from Eq. (8) is shown in Fig. 10. The results demonstrate that the Rosseland mean extinction coefficient initially decreases with increasing temperature, reaching the minimum value of 4570/m at around 373 K, and then increases with the rise of temperature. 5.2. Numerical results In solving the incident radiation, the albedo of scattering for the fibrous insulation sample is unknown. Therefore, the effective thermal conductivities for different values of albedo of scattering are calculated in the present study. Variation of the average deviation between experimental results and numerical results with albedo of scattering is shown in Fig. 11. It demonstrates that in the investigated range, the average deviation between experimental results and numerical results is almost invariable for the albedo of scattering ranging from 0 to 0.9, and it stays around 14.8 percent. The average deviation increases slightly when the albedo of scattering increases from 0.9 to 0.99, and it reaches 15.1 percent at o ¼ 0.99. Then the average deviation decreases sharply to 13.5 percent at o ¼ 1. It can be stated that the variations of albedo of scattering from 0 to1 cause little effect on the average deviation between the calculated effective thermal conductivity and the experimental results. The variations of calculated effective thermal conductivity of fibrous insulation for different values of albedo of scattering with temperature and pressure are, respectively, shown in Figs. 12 and 13. Fig. 12 shows

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Rosseland mean extinction coefficient/ 1/m

9000

8000

7000

6000

5000

4000 200

300

400

500

600 700 Temperature/K

800

900

1000

1100

Fig. 10. Variation of Rosseland mean extinction coefficient with temperature.

16.0

15.5

Average deviation /%

15.0

14.5

14.0

13.5

13.0 0.0

0.1

0.2

0.3

0.4 0.5 0.6 Albedo of scattering

0.7

0.8

0.9

1.0

Fig. 11. Variation of average deviation between the experimental results and numerical results with albedo of scattering.

that the effective thermal conductivities are almost invariable for o ¼ 0, 0.5, 0.99 at low pressure, and the effective thermal conductivity for o ¼ 1 is obviously larger than that for the other three values of albedo of scattering. At high pressure, effective thermal conductivities for o ¼ 0, 0.5, 0.99 and 1 vary little. This phenomenon can also be seen in Fig. 13a. From Fig. 13b, we can see that effective thermal conductivity for o ¼ 1 is notably larger than that for o ¼ 0, 0.5, 0.99 at the nominal hot side temperature of 973 K. The difference between effective thermal conductivity for o ¼ 1 and that for any other albedo of scattering is about 4.7  103 W/(m K) at low pressure, and 1.5  103 W/(m K) at high pressure.

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0.040 0.035 ω=0 ω=0.5 ω=0.99 ω=1

0.030 0.025 0.020 0.015 0.010 0.005 0.000 300

350

400

450

500

550

600

650

700

Effective thermal conductivity/W/ (m.K)

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0.10 0.09

ω=0 ω=0.5 ω=0.99 ω=1

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 300

350

400

450

500

550

600

650

700

Temperature/K

Temperature/K

0.05 0.04 0.03

ω=0 ω=0.5 ω=0.99 ω=1

0.02 0.01 0.00 10-3 10-2 10-1 100 101 102 103 104 105 106 Pressure/Pa

Effective thermal conductivity/W/ (m.K)

Effective thermal conductivity/W/ (m.K)

Fig. 12. Variation of calculated effective thermal conductivity for different albedo of scattering with temperature at nominal pressures (a) 10 Pa; (b) 5  104 Pa.

0.08 0.07 0.06 0.05

ω=0 ω=0.5 ω=0.99 ω=1

0.04 0.03 0.02 0.01 0.00 10-3 10-2 10-1 100 101 102 103 104 105 106 Pressure/Pa

Fig. 13. Variation of calculated effective thermal conductivity for different albedo of scattering with pressure at nominal hot side temperatures (a) 200 K; (b) 700 K.

The comparisons of analytical predictions using the model described above for o ¼ 1 and experimental results of effective thermal conductivity at pressures of 1  102, 1  102, 5  103 and 1.013  105 Pa are shown in Figs. 14a–d. The effective thermal conductivity increases non-linearly with increasing average temperature of sample due to the non-linear nature of radiation heat transfer. Most of the numerical results underpredicted the experimental results for the sample. An average 13.5 percent difference was obtained between the calculated and measured values. The numerical results agree well with the experimental results at the nominal pressure of 1  102 Pa. The difference at this pressure is less than 4 percent. The comparisons of calculated and measured effective thermal conductivity of fibrous insulation sample at nominal hot side temperatures of 373, 473, 773 and 973 K are shown in Figs. 15a–d. As can be seen from these figures, the effective thermal conductivity at each nominal hot side temperature increases with increasing pressure levels. The largest difference occurs at high temperature and high pressure, which is almost 23 percent. As for the reasons for the difference, it can be stated that the uncertainties in the process of measuring specific heat and Rosseland mean extinction coefficient contributed to the differences between calculated and measured effective thermal

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0.04

Experimental Numerical

0.03 0.02 0.01 0.00 300 350 400 450 500 550 600 650 700 Temperature/K

0.12 0.10 0.08

Experimental Numerical

0.06 0.04 0.02 0.00 300 350 400 450 500 550 600 650 700 Temperature/K

0.06 0.05 0.04

Experimental Numerical

0.03 0.02 0.01 0.00 300 350 400 450 500 550 600 650 700 Temperature/K

Effective thermal conductivity/W (m.K)

Effective thermal conductivity/W/ (m.K) Effective thermal conductivity/W/ (m.K)

0.05

Effective thermal conductivity/W/ (m.K)

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0.12 0.10 0.08

Experimental Numerical

0.06 0.04 0.02 0.00 300 350 400 450 500 550 600 650 700 Temperature/K

Fig. 14. Comparison of calculated results for o ¼ 1 and experimental results of effective thermal conductivity at nominal pressures (a) 1  102 Pa; (b) 1  102 Pa; (c) 5  103 Pa; and (d) 1.013  105 Pa.

conductivity to a certain extent, and convective heat transfer was inevitably introduced in the process of the experimental measurement and slightly contributed to the total heat transfer. 6. Conclusions This work reports on experimental and numerical studies of thermal physical properties of high-alumina fibrous insulation for thermal protection systems. The effective thermal conductivities of the fibrous insulation were measured at gas pressures between 102 and 105 Pa and temperatures up to 973 K. The specific heat and the transmittance spectra of the fibrous insulation were also measured at the corresponding temperatures. The spectral extinction coefficients and Rosseland mean extinction coefficients were obtained based on transmittance data at various temperatures. Analytical models combining radiation and conduction heat transfer were developed to predict the behavior of the effective thermal conductivity of the fibrous insulation at various temperatures and pressures. The two-flux approximation was used to simulate the radiation heat transfer through the insulation. The experimentally measured specific heat and Rosseland mean extinction coefficient were used in the numerical heat transfer model to calculate the effective thermal conductivity. The average deviations between the numerical results for different values of albedo of scattering and the experimental results were investigated. It was found that the variation of albedo of scattering from 0 to1 cause little effect on the average deviation between the calculated results and the experimental results in the present investigated range. The numerical results for o ¼ 1 and experimental data were compared. The calculated

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0.04

Experimental Numerical

0.03 0.02 0.01 0.00 10-3 10-2 10-1 100 101 102 103 104 105 106 Pressure/Pa

0.05 0.04

Experimental Numerical

0.03 0.02 0.01 0.00 10-3 10-2 10-1 100 101 102 103 104 105 106 Pressure/Pa

Effective thermal conductivity/W/ (m.K)

0.05

Effective thermal conductivity/W/ (m.K)

Effective thermal conductivity/W/ (m.K)

Effective thermal conductivity/W/ (m.K)

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0.08 0.07 0.06

Experimental Numerical

0.05 0.04 0.03 0.02 0.01 0.00 10-3 10-2 10-1 100 101 102 103 104 105 106 Pressure/Pa

0.12 0.10 0.08

Experimental Numerical

0.06 0.04 0.02 0.00 10-3 10-2 10-1 100 101 102 103 104 105 106 Pressure/Pa

Fig. 15. Comparison of calculated results for o ¼ 1 and experimental results of effective thermal conductivity at nominal hot side temperatures (a) 373 K; (b) 473 K; (c) 773 K; and (d) 973 K.

values corresponded to the experimental values within an average of 13.5 percent. The model was consistent with experimental results through the temperatures and pressures under the investigation. Acknowledgments This work was supported by the National High Technology 863 Projects Foundation of China (No. 2004AA724031). The authors also gratefully acknowledge the financial support of the Center for Composite Materials and Structures of HIT in China. References [1] Daryabeigi K. Design of high-temperature multi-layer insulation for reusable launch vehicles. PhD dissertation, University of Virginia, 2000. [2] Lee SC, Cunnington GR. Conduction and radiation heat transfer in high porosity fiber thermal insulation. J Thermophys Heat Transfer 2000;14(2):121–36. [3] Verschoor JD, Greebler P, Manville NJ. Heat transfer by gas conduction and radiation in fibrous insulation. J Heat Transfer 1952;74: 467–74. [4] Hager NE, Steere RC. Radiant heat transfer in fibrous thermal insulation. J Appl Phys 1967;38(12):4663–8. [5] Larkin BK, Churchill SW. Heat transfer by radiant through porous insulations. Am Inst Chem Eng J 1959;4(5):467–74. [6] Tong TW, Tien CL. Radiative heat transfer in fibrous insulations—Part I: analytical study. J Heat Transfer 1983;105(1):70–5.

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[7] Tong TW, Yang SQ, Tien CL. Radiative heat transfer in fibrous insulations—Part II: experimental study. J Heat Transfer 1983; 105(1):70–5. [8] Daryabeigi K. Analysis and testing of high-temperature fibrous insulation for reusable launch vehicles. AIAA Paper 99-1044, January 1999. [9] Ballis D, Raynaud M, Sacadura JF. Determination of spectral radiation properties of open cell foam: model validation. J Thermophys Heat Transfer 2000;14(2). [10] Zhao CY, Lu TJ, Hodson HP. Thermal radiation in ultralight metal foams with open cells. Int J Heat Mass Transfer 2004;47: 2927–39. [11] Giaretto V, Miraldi E, Ruscica G. Simultaneous estimations of radiative and conductive properties in lightweight insulating materials. High Temp-High Pressures 1995/1996;27/28:191–204. [12] Siegel R, Howell JR. Thermal radiation heat transfer. London: Taylor & Francis; 1992. [13] Sparrow EM, and Cess RD. Radiation heat transfer, augmented ed. New York: McGraw-Hill; 1978. p. 255–71. [14] Ozisik MN. Radiative transfer and interactions with conduction and convection. New York: Wiley; 1973. [15] Daryabeigi K. Thermal analysis and design of multi-layer insulation for re-entry aerodynamic heating. AIAA Paper 2001-2834, June 2001. [16] Hager NE, Steere RC. Radiation heat transfer in fibrous thermal insulation. J Appl Phys 1967;38(12):4663–8. [17] Daryabeigi K. Effective thermal conductivity of high temperature insulation for reusable launch vehicle. NASA/TM-1999-208972, February 1999.