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Effects of Contact Resistance on Heat Transfer Behaviors of Fibrous Insulation
Chinese Journal of Aeronautics 22(2009) 569-574
Chinese Journal of Aeronautics www.elsevier.com/locate/cja
Effects of Contact Resistance on Heat Transfer Behaviors of Fibrous Insulation Zhao Shuyuan*, Zhang Boming, Du Shanyi Center for Composite Materials and Structure, Harbin Institute of Technology, Harbin 150080, China Received 16 October 2008; accepted 3 March 2009
Abstract In this article, a numerical model combining conduction and radiation is developed based on two flux approximation to predict the heat transfer behavior of fibrous insulation used in thermal protection systems. Monte Carlo method is utilized to determine the modified radiative properties with experimentally measured transient external temperature as high as 1 000 K. It is found that the estimated radiative properties become time-independent after about t = 3 000 s. By comparing the predicted to the measured results in transient state, the contact resistance exerts significant influences upon the temperature distribution in the specimen. Results show that the averaged absolute deviation is 3.25% when contact resistance is neglected in heat transfer model, while 1.82% with no contact resistance. Keywords: heat transfer; Monte Carlo methods; radiative properties; effective thermal conductivity; contact resistance
1. Introduction1 Light-weight fibrous insulation has become an ideal material for use in the metallic thermal protection system on Reusable Launch Vehicles. It sustains severe aerodynamic heating during reentry into the atmosphere. Heat transfer inside fibrous insulation is composed of conduction, natural convection and radiation. Previous studies on heat transfer through fibrous media have displayed that radiation accounts for 40%-50% of the total heat transfer inside light-weight fibrous insulation at moderate temperatures[1]. The radiation process in fibrous insulation under aerodynamic heating becomes more complicated with temperature increasing, which makes the analysis and design of insulation quite difficult. Up to now there are some methods to measure effective thermal conductivity of fibrous insulation. However, the results of temperature distribution inside insulation during reentry into atmosphere calculated with effective thermal conductivity measured by the guarded hot plate method are found to be different from the experimentally measured temperature distribution—the predicted in-depth temperatures exceed the *Corresponding author. Tel: +86-13136640700. E-mail address: [email protected] Foundation item: National High-tech Research and Development Program (2006AA705317) 1000-9361/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(08)60143-0
measured data by 100 K and even more[2]. In this connection, limited accuracy could be resulted by means of the steady-state methods in transient environment, where surface temperature changes rapidly. The discrepancy might mostly be blamed for radiation at high temperatures[3]. Therefore, closer attention should be paid to the radiation in transient conditions to avoid significant errors. It has been accepted that numerical and analytical techniques can be used to calculate the combined radiation and conduction heat transfer in rarefied dispersed media on the basis of the radiation transfer equation in common with the energy conservation equation. But results are still uncertain because of the lack of wellknown spectral radiative properties. In view of this, extensive researches on the heat transfer in porous media have been performed experimentally and analytically[4-5]. T. W. Tong, et al.[6-7] used a two flux model that assumed linearized anisotropic scattering to model radiation heat transfer inside fibrous insulation and compared the predicted fluxes which associated radiation with conduction to the measured data up to 450 K and at one atmosphere pressure. K. Daryabeigi[8] modeled heat transfer in alumina fibrous insulation. A genetic algorithm-based parameter estimation technique was utilized to determine the relevant radiative properties of the fibrous insulation. W. Y. Walter, et al.[9] used a first-principle approach to analyze combined conduction/radiation heat transfer in a highly porous silica insulation material (LI900) used by the Space Shuttle.
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Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
Macroscopic average radiative properties, such as extinction coefficient, scattering albedo and phase function, were acquired by solving the radiation transfer equation for fibrous insulation using the measured value of the refractive index (n, k). The calculated transient temperatures were compared with experimental data. This article establishes a model, which combines radiation and conduction heat transfer inside a fibrous insulation based on the two flux approximation. The modified radiative properties are used to calculate the transient temperature responses. In order to assess contact resistance effect, two cases are investigated. Then the calculated results are compared with the measured data under transient condition. 2. Experimental Methodology By using graphite heating apparatus, the transient thermal test[10] could provide a time-dependent surface temperature during measuring the temperature distribution in fibrous insulation under transient heating condition. Fig.1 shows the schematic of the heat transfer model of the test setup. The slab-shaped setup, 300 mm
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square by 82.5 mm thick, was located between a septum plate and a water-cooled plate. The septum plate served as the hot side solid boundary. Mimicking the structure of launch vehicles, a 2.5 mm thick aluminum plate was installed in the middle of the test setup. A 60 mm thick aluminosilicate fibrous insulation was placed above the aluminum plate, and a 20 mm thick insulation below it. The nominal density of the fibrous insulation was 128 kg/m3. Guarded insulation boards were placed around the test setup to minimize heat loss. The setup was provided with thermocouples disposed at five sites with different distances of x (x is the spatial coordinate through the insulation thickness) from heating surface. The front (x = 0 mm) and back (x = 82.5 mm) surface temperatures and internal temperatures inside fibrous insulation at depths of 20.0, 40.0, 60.0 mm were to be measured. The hot side temperature rose from 288 K to 1 000 K in 600 s and was held at that temperature for 650 s. The insulation sample was gradually cooled to 340 K in another period. The maximum uncertainty of the temperature measurement was estimated about 23 K. Temperature distribution on the surface of the sample was uniform with an error of ±5%.
Fig.1 Schematic of transient heat transfer model of test setup.
3. Radiative Properties In order to investigate the radiation heat transfer inside fibrous insulation, it is necessary to elucidate the spectral and temperature dependences of the extinction coefficient and the scattering albedo. A Fourier transform infrared spectrometer was used to measure the spectral transmittance of the fibrous insulation in the wavelength range of 2.5-25.0 m over the temperature range of 291-973 K[10]. The spectral extinction coefficients were calculated by using the measured transmittance data abiding by Beer’s law. The Rosseland mean extinction coefficient KR were obtained according to the following Equation[11] by assuming a = 2.5 m, b = 25.0 m. f 1 we b 1 we 1 bO bO dO | ³ dO (1) ³ 0 a KR E O web E O web where O is the wavelength, eb the blackbody emissive power, ebO the spectral blackbody emissive power and EO the spectral extinction coefficient. Fig.2 shows the
variation of Rosseland mean extinction coefficient KR with temperature T [10]. Taking into account anisotropic scattering by insulation, a weighted spectral extinction coefficient E O is adopted to replace the real spectral extinction coefficient. E O is defined as[12]
E O V aO V sO (1 g O ) E O V sO g O
(2)
where VaO is the spectral absorption coefficient, VsO the 1 1 spectral scattering coefficient, g O IO ( P ) P dP the 2 ³ 1 anisotropy factor, IO the phase function and P the cosine of the angle between the x-axis and the direction of radiative propagation. The weighted spectral albedo of scattering, which is defined as the ratio of the weighted scattering coefficient to the weighted extinction coefficient, is given by
ZO
V sO (1 g O ) E O V sO g O
(3)
Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
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From Eq.(2) and Eq.(3), the following can be obtained
E O
1 gO EO 1 g O ZO g O
(4)
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cific heat and kc the thermal conductivity of the fibrous insulation. Similarly, Ua, ca and ka are those of the aluminum plate. qrcc is the radiant heat flux, t the time. Eqs.(7)-(8) are subjected to the following initial and upper boundary conditions: T ( x,0) T0
(9) (10)
T (0, t ) T1 (t )
In order to assess the contact resistance effect, the lower boundary conditions are considered to be the following two cases: T ( L1 L2 L3 , t ) T5 (t )
Case 1 Case 2
Fig.2
Variation of Rosseland mean extinction coefficient vs temperature[10].
By substituting Eq.(4) into Eq.(1) and according to the first mean value theorem for integral[13], the equivalent Rosseland mean extinction coefficient K R and equivalent albedo of scattering Z * are K R
1 g[ 1 g[ Z[ g[
f[ K R , [ [ a, b ]
KR
Z ZK , K [a, b]
are the weighted spectral albedo of scattering at O =[ and O =K, respectively; f[ is defined as a modified factor of extinction coefficient, which represents the ratio of extinction coefficient with and without taking into account anisotropic scattering. For more details about the modification of radiative properties, refer to Ref.[14]. 4. Analytical Model
w § wT ¨ kc wx © wx
· wqrcc ¸ ¹ wx
(7)
in the fibrous insulation and wT wt
w wT ( ka ) wx wx
(8)
in the aluminum plate. Here, U is the density, c the spe-
1 wG 3K R wx
(13)
The incident radiation in the fibrous insulation is obtained by solving the following second order differential equation:
1 3( K R ) 2 (1
w 2G G Z ) wx 2
4V T 4
(14)
where V is the Stefan-Boltzmann constant. The boundary conditions for the second order differential equation governing the incident radiation are given by:
Generally, heat transfer inside fibrous insulation is composed of conduction, natural convection and radiation. However, C. Stark, et al.[15] pointed out that there is no natural convection in fibrous insulations having a density of larger than 20 kg/m3 because the fibers entrap the gas into sufficiently small pores. This allows us to neglect the effects of natural convective heat transfer in the on-going analysis. Then the energy conservation law for 1D heat transfer inside the insulation through conduction and radiation can be written into[16]
Ua ca
qrcc
(6)
where g[ is the anisotropy factor at O =[; Z[ and ZK
wT wt
where L1, L2 and L3 are the thicknesses of the upper layer fibrous insulation, the aluminum plate and the lower layer insulation, respectively. Energy conservation is adopted at the interface between the insulation and the aluminum plate. It can be stated that for Case 1 the effect of contact resistance at the interface between the insulation and the aluminum plate is neglected in the calculation of temperature field and estimation of radiative properties. The radiant heat flux is related to incident radiation, G, according to[17]
(5)
Uc
T ( L1 , t ) T4 (t )
(11) (12)
wG
2
H wx 3K R ( 1 ) 2 H1 2
3K R (
H2
2 H2
)
G
wG G wx
½ 4V T14 ° ° ° ¾ 4V T24 ° ° ° ¿
(15)
where H1 and T1 separately denote the emittance and temperature of the upper bounding surface, while H2 and T2 denote those of the lower bounding surface. The emittance of the septum plate is 0.8, and that of both the aluminum plate and the water-cooled plate is 0.3 in the present study. Here, the local thermal conductivity due to gas conduction and solid conduction inside fibrous insulation is given by[18] kc
f ( f 2 ks ) (1 f )kg
(16)
where f is the solid fraction ratio, kg the gas thermal
Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
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conductivity and ks* the variation of the thermal conductivity of the fiber parent material with temperature. 5. Parameter Estimation
The unknown modified factor of extinction coefficient f[ and equivalent albedo of scattering Z * are estimated by using parameter estimation techniques. The estimation strategy is based on the least-squares minimization of the difference between measured and calculated temperatures inside fibrous insulation: 2
§ Tm,i Tp,i ( f[ , Z ) · (17) ¸ ¦ ¨¨ ¸ Tm,i i 1© ¹ where Tp,i is the ith calculated temperature data, Tm,i the ith measured temperature data. Eq.(17) is subjected to the following physical constraints: 0.001 d f[ d 10 °½ (18) ¾ 0 d Z d 1 °¿ It can be stated that the modified factor of extinction coefficient varies between 0 and 2 according to Eq.(5). Taking into account the uncertainty of temperature measurements and the contact resistance effects during the thermal tests, the upper searching bound of the modified factor of extinction coefficient is enlarged to 10. It is assumed that the modified factor of extinction coefficient and the equivalent albedo of scattering are independent of temperature. Monte Carlo method is used for finding the set of parameters. For Case 1, the internal temperature responses at depths of 20.0, 40.0, 60.0 mm are used to estimate the radiative properties, and the total number of samples used in the parameter estimation routine, n, is 17 757 for the duration of 10 000 s under investigation. For Case 2, the measured temperature responses at depths of 20.0 mm and 40.0 mm are used and n is 11 838 for the same duration. n
S
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can be larger than 3.5 during heating period. This might be ascribed to the existence of multidimensional heat transfer and thermal contact resistance that generally decreases the rate of internal temperature rise in measurements, which apparently would diminish the “dynamic” effective thermal conductivity of fibrous insulation in experiments designed to set up 1D transient temperature response. Based on Rosseland diffusion approximation, the radiative conductivity of fibrous insulation can be obtained from kr 16V T 3 (3K R ) . The radiative conductivity declines with the rise of Rosseland mean extinction coefficient, so the effective thermal conductivity combining radiation and conduction heat transfer will decrease on condition that the thermal conductivity caused only by conduction be constant. There is no enough information about the properties for short heating time and at faster heating rate. Therefore, the estimated values of the modified factor of extinction coefficient by the developed inverse method are so large at the time before stopping heating as to exceed the reasonable reference range with specific physical meaning in whatever case. As can be expected, this effect gradually reduces with the increase of test time, and the estimated thermal radiative properties become nearly independent of time after t = 3 000 s. The steady value stands at about 0.2 for Case 1 while 1.2 in Case 2. As for the difference of the steady values, when the contact resistance is neglected in the heat transfer model of the test assembly, the apparent effective thermal conductivity has to be increased to compensate the existence of contact resistance. As a result, the estimated value of the modified factor of extinction coefficient for Case 1 is smaller than the value for Case 2.
6. Results and Discussion
The experimentally measured specific heat[10] and Rosseland mean extinction coefficient of fibrous insulation are substituted into numerical heat transfer model to calculate the transient thermal response in the test specimen. For the two cases under consideration, the same spatial grid and time steps for the estimation of thermal radiation parameters are used to avoid spurious discrepancy caused by the numerical solution machinery. Fig.3 lists the estimated parameters for different periods of time. This demonstrates that all the estimated values of the radiative property are independent of time after about t = 3 000 s. The equivalent albedo of scattering steadily stands at over 0.95, and the difference from the estimated results is very small for both cases. The modified factor of extinction coefficient is found to sustain relatively large variation before the heater is turned off. The estimated values in both cases
Fig.3
Estimated parameters during different periods of time.
The estimated results of the modified factor of extinction coefficient and equivalent albedo of scattering at t = 10 000 s, 0.204 2 and 0.999 9 for Case 1 versus 1.212 5 and 0.971 4 for Case 2, are used to calculate the transient temperature responses. Comparisons between calculated and measured transient thermal behavior of the test sample are shown in Fig.4. Fig.5 shows the temperature differences between the calculated temperature responses and the measured ones at different depths of the sample. Clearly, the variation of
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Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
temperature differences depends on both depth and period of time. As for in Case 1, the largest difference is found to be up to +40.59 K at t = 922 s and x = 40.0 mm. At t =1 247 s and x = 20.0 mm, the difference of 58.67 K reaches the minimum. However, the temperature difference in Case 2 varies between 21.99 K and +17.72 K with the minimum at t =1 309 s and x = 20.0 mm and the maximum at t =1 028 s and x = 40.0 mm. It is worth noting that the largest absolute difference appears near the heated surface where the temperature gradient is the largest. The contact resistance plays an important role in affecting the temperature distribution in the sample. Fig.6 shows the deviation of the calculated temperature from the measured
Fig.6
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Deviation of calculated temperatures from measured temperatures.
temperature against time as well as thermocouple location. As seen from it, the deviation is within ±12.30% for Case 1 and ±5.08% for Case 2. The averaged absolute deviations between the calculated and measured temperatures for the two cases are 3.25% and 1.82%, respectively. 7. Conclusions
Fig.4
Comparison between calculated and measured temperatures at different depths.
This article has investigated the thermal radiative properties and the heat transfer behavior of fibrous insulation widely used in modern metallic thermal protection systems. Based on two flux approximation, a numerical model combining radiation and conduction heat transfer has been developed to predict the transient thermal responses inside the fibrous insulation exposed to an external heating source. The modified radiative properties are determined by parameter estimation analysis using measured transient temperature data taken from the fibrous sample, of which one surface is heated by a graphite radiant heater. It is found that the estimated radiative properties become time-independent after about t = 3 000 s. Comparison has been carried out between calculated and measured temperatures at different depths and shows that the averaged absolute deviation with neglecting contact resistance in heat transfer model is 3.25% while 1.82% in the case with absence of contact resistance. Better agreement is achieved for the second case. References [1]
[2]
[3] Fig.5
Differences between calculated and measured temperatures. [4]
Stewart D A, Leiser D B, Kolodziej P, et al. Thermal response of integral, multicomponent composite thermal protection systems. Journal of Spacecraft & Rockets 1986; 23: 420-427. Lee S C, Cunnington G R. Conduction and radiation heat transfer in high porosity fiber thermal insulation. Journal of Thermophysics and Heat Transfer 2000; 14(2): 121-136. Petrov V A. Combined radiation and conduction heat transfer in high temperature fiber thermal insulation. International Journal of Heat and Mass Transfer 1997; 40(9): 2241-2247. Verschoor J D, Greebler P, Manville N J. Heat transfer by gas conduction and radiation in fibrous insulation.
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[5]
[6]
[7]
[8] [9]
[10]
[11]
[12] [13]
Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
Journal of Heat Transfer 1952; 74(6): 467-474. Ballis D, Raynaud M, Sacadura J F. Determination of spectral radiation properties of open cell foam: model validation. Journal of Thermophysics and Heat Transfer 2000; 14(2) :137-143. Tong T W, Tien C L. Radiative heat transfer in fibrous insulations—Part I: analytical study. Journal of Heat Transfer 1983; 105(1): 70-75. Tong T W, Yang S Q, Tien C L. Radiative heat transfer in fibrous insulations—Part II: experimental study. Journal of Heat Transfer 1983; 105(1): 76-81. Daryabeigi K. Heat transfer in high-temperature fibrous insulation. AIAA-2002-3332, 2002. Walter W Y, Ezra T, George C. Combined conductive/radiative heat transfer in high porosity fibrous insulation materials: theory and experiment. The 6th ASME-JSME Thermal Engineering Joint Conference, 2003. Zhang B M, Zhao S Y, He X D. Experimental and theoretical studies on high-temperature thermal properties of fibrous insulation. Journal of Quantitative Spectroscopy & Radiative Transfer 2008; 109(7): 1309-1324. Giaretto V, Miraldi E, Ruscica G. Simultaneous estimations of radiative and conductive properties in lightweight insulating materials. High Temperatures—High Pressure 1995-1996; 27-28(2): 191-204. Siegel R, Howell J R. Thermal radiation heat transfer. London: Taylor & Francis, 1992. Bernard J. On the mean value theorem for integrals.
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American Mathematical Monthly 1982; 89(5): 300-301. [14] Zhao S Y, Zhang B M, Du S Y. Inverse identification of thermal properties of fibrous insulation from transient temperature measurements. Journal of Quantitative Spectroscopy and Radiative Transfer 2009; 110(13): 11111123. [15] Stark C, Fricke J. Improved heat-transfer models for fibrous insulation. International Journal of Heat and Mass Transfer 1993; 36(3): 617-625. [16] Sparrow E M, Cess R D. Radiation heat transfer, augmented edtion. New York: McGraw-Hill, 1978; 255271. [17] Ozisik M N. Radiative transfer and interactions with conduction and convection. New York: John Wiley & Sons, Inc., 1973. [18] Daryabeigi K. Thermal analysis and design of multilayer insulation for re-entry aerodynamic heating. AIAA-2001-2834, 2001.
Biography: Zhao Shuyuan Born in 1980, she is a Ph.D. candidate in Harbin Institute of Technology. Her main academic interests lie in thermal property determination of fibrous insulation with inverse method and probabilistic thermal analysis of thermal protection system. E-mail: [email protected]
Chinese Journal of Aeronautics www.elsevier.com/locate/cja
Effects of Contact Resistance on Heat Transfer Behaviors of Fibrous Insulation Zhao Shuyuan*, Zhang Boming, Du Shanyi Center for Composite Materials and Structure, Harbin Institute of Technology, Harbin 150080, China Received 16 October 2008; accepted 3 March 2009
Abstract In this article, a numerical model combining conduction and radiation is developed based on two flux approximation to predict the heat transfer behavior of fibrous insulation used in thermal protection systems. Monte Carlo method is utilized to determine the modified radiative properties with experimentally measured transient external temperature as high as 1 000 K. It is found that the estimated radiative properties become time-independent after about t = 3 000 s. By comparing the predicted to the measured results in transient state, the contact resistance exerts significant influences upon the temperature distribution in the specimen. Results show that the averaged absolute deviation is 3.25% when contact resistance is neglected in heat transfer model, while 1.82% with no contact resistance. Keywords: heat transfer; Monte Carlo methods; radiative properties; effective thermal conductivity; contact resistance
1. Introduction1 Light-weight fibrous insulation has become an ideal material for use in the metallic thermal protection system on Reusable Launch Vehicles. It sustains severe aerodynamic heating during reentry into the atmosphere. Heat transfer inside fibrous insulation is composed of conduction, natural convection and radiation. Previous studies on heat transfer through fibrous media have displayed that radiation accounts for 40%-50% of the total heat transfer inside light-weight fibrous insulation at moderate temperatures[1]. The radiation process in fibrous insulation under aerodynamic heating becomes more complicated with temperature increasing, which makes the analysis and design of insulation quite difficult. Up to now there are some methods to measure effective thermal conductivity of fibrous insulation. However, the results of temperature distribution inside insulation during reentry into atmosphere calculated with effective thermal conductivity measured by the guarded hot plate method are found to be different from the experimentally measured temperature distribution—the predicted in-depth temperatures exceed the *Corresponding author. Tel: +86-13136640700. E-mail address: [email protected] Foundation item: National High-tech Research and Development Program (2006AA705317) 1000-9361/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(08)60143-0
measured data by 100 K and even more[2]. In this connection, limited accuracy could be resulted by means of the steady-state methods in transient environment, where surface temperature changes rapidly. The discrepancy might mostly be blamed for radiation at high temperatures[3]. Therefore, closer attention should be paid to the radiation in transient conditions to avoid significant errors. It has been accepted that numerical and analytical techniques can be used to calculate the combined radiation and conduction heat transfer in rarefied dispersed media on the basis of the radiation transfer equation in common with the energy conservation equation. But results are still uncertain because of the lack of wellknown spectral radiative properties. In view of this, extensive researches on the heat transfer in porous media have been performed experimentally and analytically[4-5]. T. W. Tong, et al.[6-7] used a two flux model that assumed linearized anisotropic scattering to model radiation heat transfer inside fibrous insulation and compared the predicted fluxes which associated radiation with conduction to the measured data up to 450 K and at one atmosphere pressure. K. Daryabeigi[8] modeled heat transfer in alumina fibrous insulation. A genetic algorithm-based parameter estimation technique was utilized to determine the relevant radiative properties of the fibrous insulation. W. Y. Walter, et al.[9] used a first-principle approach to analyze combined conduction/radiation heat transfer in a highly porous silica insulation material (LI900) used by the Space Shuttle.
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Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
Macroscopic average radiative properties, such as extinction coefficient, scattering albedo and phase function, were acquired by solving the radiation transfer equation for fibrous insulation using the measured value of the refractive index (n, k). The calculated transient temperatures were compared with experimental data. This article establishes a model, which combines radiation and conduction heat transfer inside a fibrous insulation based on the two flux approximation. The modified radiative properties are used to calculate the transient temperature responses. In order to assess contact resistance effect, two cases are investigated. Then the calculated results are compared with the measured data under transient condition. 2. Experimental Methodology By using graphite heating apparatus, the transient thermal test[10] could provide a time-dependent surface temperature during measuring the temperature distribution in fibrous insulation under transient heating condition. Fig.1 shows the schematic of the heat transfer model of the test setup. The slab-shaped setup, 300 mm
No.5
square by 82.5 mm thick, was located between a septum plate and a water-cooled plate. The septum plate served as the hot side solid boundary. Mimicking the structure of launch vehicles, a 2.5 mm thick aluminum plate was installed in the middle of the test setup. A 60 mm thick aluminosilicate fibrous insulation was placed above the aluminum plate, and a 20 mm thick insulation below it. The nominal density of the fibrous insulation was 128 kg/m3. Guarded insulation boards were placed around the test setup to minimize heat loss. The setup was provided with thermocouples disposed at five sites with different distances of x (x is the spatial coordinate through the insulation thickness) from heating surface. The front (x = 0 mm) and back (x = 82.5 mm) surface temperatures and internal temperatures inside fibrous insulation at depths of 20.0, 40.0, 60.0 mm were to be measured. The hot side temperature rose from 288 K to 1 000 K in 600 s and was held at that temperature for 650 s. The insulation sample was gradually cooled to 340 K in another period. The maximum uncertainty of the temperature measurement was estimated about 23 K. Temperature distribution on the surface of the sample was uniform with an error of ±5%.
Fig.1 Schematic of transient heat transfer model of test setup.
3. Radiative Properties In order to investigate the radiation heat transfer inside fibrous insulation, it is necessary to elucidate the spectral and temperature dependences of the extinction coefficient and the scattering albedo. A Fourier transform infrared spectrometer was used to measure the spectral transmittance of the fibrous insulation in the wavelength range of 2.5-25.0 m over the temperature range of 291-973 K[10]. The spectral extinction coefficients were calculated by using the measured transmittance data abiding by Beer’s law. The Rosseland mean extinction coefficient KR were obtained according to the following Equation[11] by assuming a = 2.5 m, b = 25.0 m. f 1 we b 1 we 1 bO bO dO | ³ dO (1) ³ 0 a KR E O web E O web where O is the wavelength, eb the blackbody emissive power, ebO the spectral blackbody emissive power and EO the spectral extinction coefficient. Fig.2 shows the
variation of Rosseland mean extinction coefficient KR with temperature T [10]. Taking into account anisotropic scattering by insulation, a weighted spectral extinction coefficient E O is adopted to replace the real spectral extinction coefficient. E O is defined as[12]
E O V aO V sO (1 g O ) E O V sO g O
(2)
where VaO is the spectral absorption coefficient, VsO the 1 1 spectral scattering coefficient, g O IO ( P ) P dP the 2 ³ 1 anisotropy factor, IO the phase function and P the cosine of the angle between the x-axis and the direction of radiative propagation. The weighted spectral albedo of scattering, which is defined as the ratio of the weighted scattering coefficient to the weighted extinction coefficient, is given by
ZO
V sO (1 g O ) E O V sO g O
(3)
Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
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From Eq.(2) and Eq.(3), the following can be obtained
E O
1 gO EO 1 g O ZO g O
(4)
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cific heat and kc the thermal conductivity of the fibrous insulation. Similarly, Ua, ca and ka are those of the aluminum plate. qrcc is the radiant heat flux, t the time. Eqs.(7)-(8) are subjected to the following initial and upper boundary conditions: T ( x,0) T0
(9) (10)
T (0, t ) T1 (t )
In order to assess the contact resistance effect, the lower boundary conditions are considered to be the following two cases: T ( L1 L2 L3 , t ) T5 (t )
Case 1 Case 2
Fig.2
Variation of Rosseland mean extinction coefficient vs temperature[10].
By substituting Eq.(4) into Eq.(1) and according to the first mean value theorem for integral[13], the equivalent Rosseland mean extinction coefficient K R and equivalent albedo of scattering Z * are K R
1 g[ 1 g[ Z[ g[
f[ K R , [ [ a, b ]
KR
Z ZK , K [a, b]
are the weighted spectral albedo of scattering at O =[ and O =K, respectively; f[ is defined as a modified factor of extinction coefficient, which represents the ratio of extinction coefficient with and without taking into account anisotropic scattering. For more details about the modification of radiative properties, refer to Ref.[14]. 4. Analytical Model
w § wT ¨ kc wx © wx
· wqrcc ¸ ¹ wx
(7)
in the fibrous insulation and wT wt
w wT ( ka ) wx wx
(8)
in the aluminum plate. Here, U is the density, c the spe-
1 wG 3K R wx
(13)
The incident radiation in the fibrous insulation is obtained by solving the following second order differential equation:
1 3( K R ) 2 (1
w 2G G Z ) wx 2
4V T 4
(14)
where V is the Stefan-Boltzmann constant. The boundary conditions for the second order differential equation governing the incident radiation are given by:
Generally, heat transfer inside fibrous insulation is composed of conduction, natural convection and radiation. However, C. Stark, et al.[15] pointed out that there is no natural convection in fibrous insulations having a density of larger than 20 kg/m3 because the fibers entrap the gas into sufficiently small pores. This allows us to neglect the effects of natural convective heat transfer in the on-going analysis. Then the energy conservation law for 1D heat transfer inside the insulation through conduction and radiation can be written into[16]
Ua ca
qrcc
(6)
where g[ is the anisotropy factor at O =[; Z[ and ZK
wT wt
where L1, L2 and L3 are the thicknesses of the upper layer fibrous insulation, the aluminum plate and the lower layer insulation, respectively. Energy conservation is adopted at the interface between the insulation and the aluminum plate. It can be stated that for Case 1 the effect of contact resistance at the interface between the insulation and the aluminum plate is neglected in the calculation of temperature field and estimation of radiative properties. The radiant heat flux is related to incident radiation, G, according to[17]
(5)
Uc
T ( L1 , t ) T4 (t )
(11) (12)
wG
2
H wx 3K R ( 1 ) 2 H1 2
3K R (
H2
2 H2
)
G
wG G wx
½ 4V T14 ° ° ° ¾ 4V T24 ° ° ° ¿
(15)
where H1 and T1 separately denote the emittance and temperature of the upper bounding surface, while H2 and T2 denote those of the lower bounding surface. The emittance of the septum plate is 0.8, and that of both the aluminum plate and the water-cooled plate is 0.3 in the present study. Here, the local thermal conductivity due to gas conduction and solid conduction inside fibrous insulation is given by[18] kc
f ( f 2 ks ) (1 f )kg
(16)
where f is the solid fraction ratio, kg the gas thermal
Zhao Shuyuan et al. / Chinese Journal of Aeronautics 22(2009) 569-574
· 572 ·
conductivity and ks* the variation of the thermal conductivity of the fiber parent material with temperature. 5. Parameter Estimation
The unknown modified factor of extinction coefficient f[ and equivalent albedo of scattering Z * are estimated by using parameter estimation techniques. The estimation strategy is based on the least-squares minimization of the difference between measured and calculated temperatures inside fibrous insulation: 2
§ Tm,i Tp,i ( f[ , Z ) · (17) ¸ ¦ ¨¨ ¸ Tm,i i 1© ¹ where Tp,i is the ith calculated temperature data, Tm,i the ith measured temperature data. Eq.(17) is subjected to the following physical constraints: 0.001 d f[ d 10 °½ (18) ¾ 0 d Z d 1 °¿ It can be stated that the modified factor of extinction coefficient varies between 0 and 2 according to Eq.(5). Taking into account the uncertainty of temperature measurements and the contact resistance effects during the thermal tests, the upper searching bound of the modified factor of extinction coefficient is enlarged to 10. It is assumed that the modified factor of extinction coefficient and the equivalent albedo of scattering are independent of temperature. Monte Carlo method is used for finding the set of parameters. For Case 1, the internal temperature responses at depths of 20.0, 40.0, 60.0 mm are used to estimate the radiative properties, and the total number of samples used in the parameter estimation routine, n, is 17 757 for the duration of 10 000 s under investigation. For Case 2, the measured temperature responses at depths of 20.0 mm and 40.0 mm are used and n is 11 838 for the same duration. n
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can be larger than 3.5 during heating period. This might be ascribed to the existence of multidimensional heat transfer and thermal contact resistance that generally decreases the rate of internal temperature rise in measurements, which apparently would diminish the “dynamic” effective thermal conductivity of fibrous insulation in experiments designed to set up 1D transient temperature response. Based on Rosseland diffusion approximation, the radiative conductivity of fibrous insulation can be obtained from kr 16V T 3 (3K R ) . The radiative conductivity declines with the rise of Rosseland mean extinction coefficient, so the effective thermal conductivity combining radiation and conduction heat transfer will decrease on condition that the thermal conductivity caused only by conduction be constant. There is no enough information about the properties for short heating time and at faster heating rate. Therefore, the estimated values of the modified factor of extinction coefficient by the developed inverse method are so large at the time before stopping heating as to exceed the reasonable reference range with specific physical meaning in whatever case. As can be expected, this effect gradually reduces with the increase of test time, and the estimated thermal radiative properties become nearly independent of time after t = 3 000 s. The steady value stands at about 0.2 for Case 1 while 1.2 in Case 2. As for the difference of the steady values, when the contact resistance is neglected in the heat transfer model of the test assembly, the apparent effective thermal conductivity has to be increased to compensate the existence of contact resistance. As a result, the estimated value of the modified factor of extinction coefficient for Case 1 is smaller than the value for Case 2.
6. Results and Discussion
The experimentally measured specific heat[10] and Rosseland mean extinction coefficient of fibrous insulation are substituted into numerical heat transfer model to calculate the transient thermal response in the test specimen. For the two cases under consideration, the same spatial grid and time steps for the estimation of thermal radiation parameters are used to avoid spurious discrepancy caused by the numerical solution machinery. Fig.3 lists the estimated parameters for different periods of time. This demonstrates that all the estimated values of the radiative property are independent of time after about t = 3 000 s. The equivalent albedo of scattering steadily stands at over 0.95, and the difference from the estimated results is very small for both cases. The modified factor of extinction coefficient is found to sustain relatively large variation before the heater is turned off. The estimated values in both cases
Fig.3
Estimated parameters during different periods of time.
The estimated results of the modified factor of extinction coefficient and equivalent albedo of scattering at t = 10 000 s, 0.204 2 and 0.999 9 for Case 1 versus 1.212 5 and 0.971 4 for Case 2, are used to calculate the transient temperature responses. Comparisons between calculated and measured transient thermal behavior of the test sample are shown in Fig.4. Fig.5 shows the temperature differences between the calculated temperature responses and the measured ones at different depths of the sample. Clearly, the variation of
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temperature differences depends on both depth and period of time. As for in Case 1, the largest difference is found to be up to +40.59 K at t = 922 s and x = 40.0 mm. At t =1 247 s and x = 20.0 mm, the difference of 58.67 K reaches the minimum. However, the temperature difference in Case 2 varies between 21.99 K and +17.72 K with the minimum at t =1 309 s and x = 20.0 mm and the maximum at t =1 028 s and x = 40.0 mm. It is worth noting that the largest absolute difference appears near the heated surface where the temperature gradient is the largest. The contact resistance plays an important role in affecting the temperature distribution in the sample. Fig.6 shows the deviation of the calculated temperature from the measured
Fig.6
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Deviation of calculated temperatures from measured temperatures.
temperature against time as well as thermocouple location. As seen from it, the deviation is within ±12.30% for Case 1 and ±5.08% for Case 2. The averaged absolute deviations between the calculated and measured temperatures for the two cases are 3.25% and 1.82%, respectively. 7. Conclusions
Fig.4
Comparison between calculated and measured temperatures at different depths.
This article has investigated the thermal radiative properties and the heat transfer behavior of fibrous insulation widely used in modern metallic thermal protection systems. Based on two flux approximation, a numerical model combining radiation and conduction heat transfer has been developed to predict the transient thermal responses inside the fibrous insulation exposed to an external heating source. The modified radiative properties are determined by parameter estimation analysis using measured transient temperature data taken from the fibrous sample, of which one surface is heated by a graphite radiant heater. It is found that the estimated radiative properties become time-independent after about t = 3 000 s. Comparison has been carried out between calculated and measured temperatures at different depths and shows that the averaged absolute deviation with neglecting contact resistance in heat transfer model is 3.25% while 1.82% in the case with absence of contact resistance. Better agreement is achieved for the second case. References [1]
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[3] Fig.5
Differences between calculated and measured temperatures. [4]
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Biography: Zhao Shuyuan Born in 1980, she is a Ph.D. candidate in Harbin Institute of Technology. Her main academic interests lie in thermal property determination of fibrous insulation with inverse method and probabilistic thermal analysis of thermal protection system. E-mail: [email protected]