气凝胶-遮光剂复合材料辐射特性Radiative characteristics of opacifier-loaded silica aerogel composites

Radiative characteristics of opaci fier-loaded silica aerogel compositesXiao-Dong Wang a ,b ,⁎,Duo Sun a ,b ,Yuan-Yuan Duan c ,⁎⁎,Zi-Jun Hu daState Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources,North China Electric Power University,Beijing 102206,China bBeijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy,North China Electric Power University,Beijing 102206,China cKey Laboratory for Thermal Science and Power Engineering of MOE,Department of Thermal Engineering,Tsinghua University,Beijing 100084,China dNational Key Laboratory of Advanced Functional Composite Materials,Aerospace Research Institute of Materials and Processing Technology,Beijing 100076,Chinaa b s t r a c ta r t i c l e i n f o Article history:Received 28September 2012Received in revised form 27April 2013Available online xxxxKeywords:Opaci fier;Silica aerogel;Complex refractive index;Extinction coef ficient;Radiative thermal conductivityRadiative characteristics of opaci fier-loaded silica aerogel composites such as speci fic spectral extinction coef ficient and Rossland mean extinction coef ficient were usually calculated by the Fourier infrared spectral experiment and the Beer law.For the composites,it needs lots of experiments to find the proper opaci fier categories,contents,and sizes,hence,the optimal design becomes dif ficult.Based on this reason,this work pro-poses a theoretical method with four sub-models to evaluate the radiative characteristics of opaci fier-loaded sil-ica aerogel composites.First,the Fourier infrared spectral experiment and the modi fied Kramers –Krönig (K –K)relation are used to calculate the basic optical constants of the opaci fier (complex refractive index).Second,the extinction ef ficiency of a single opaci fier particle is calculated based on its complex refractive index.Third,the spectral and Rossland extinction coef ficients of opaci fier particle assemble are calculated by using extinction ef ficiency and mass fraction of opaci fier.Finally,the spectral and Rossland extinction coef ficients and radiative heat conductivity of the composite are obtained.The radiative characteristics of six kinds of opaci fiers with various particle diameters are investigated by using the present models.The results show that optimal opaci fier and its diameter are strongly temperature-dependent.The optimal diameter of opaci fier reduces with increased temperature,and SiC is the best choice due to its high-temperature stability.A gradient design of composite is proposed based on the temperature-dependent optimal opaci fier and its diameter,which signi ficantly reduces radiative heat transfer compared to the traditional design.©2013Elsevier B.V.All rights reserved.1.IntroductionSilica aerogels are open-cell,transparent,and nanoscale porous thermal insulation materials,which have high porosity (85%–99%),small average pore diameters (2–50nm),large speci fic surface area (500–1300m 2g −1),low density (30–150kg m −3)and lower thermal conductivity (0.01–0.02W m −1K −1)than air in room temperature,so they are also called super thermal insulation materials [1–7].With these advantages,silica aerogels are regarded as promising thermal insulation materials for aerospace,chemical engineering,metallurgy,energy saving and other applications [8–15].Pure silica aerogels are almost transparent for radiation wavelengths of 3–8μm,when they are used at high temperature conditions,the radiative heat transfer is signi fi-cantly enhanced,which limits their utilization as promising insulation materials.In order to improve the performance of silica aerogels at high temperatures,some mineral powders,such as SiC,TiO 2,ZrO 2,coal ash,Al 2O 3,carbon black,K 2Ti 6O 13,BN,Fe 3O 4,B 4C and ZrSiO 4(referredas to opaci fiers)are loaded into the aerogels to form opaci fier-loaded sil-ica aerogel composites [16–25].Generally,the extinction coef ficient is used to represent the radiative performance of materials,a high extinction coef ficient can signi ficantly reduce radiative heat transfer through materials.The extinction coef fi-cient of opaci fier is closely related to its usage temperature,as well as its category,shape,and size,hence,selecting a proper opaci fier is very important to reduce the radiative heat transfer of opaci fier-loaded aerogel composites.The extinction coef ficient of composites can be calculated by Beer law based on the infrared spectral experiment data [20–30].Kuhn et al.[20]tested the speci fic spectral extinction coef fi-cient of opaci fier-loaded (carbon black,SiC,TiO 2,Fe 3O 4and B 4C)silica aerogel composites with different particle sizes within wavelength range of 2.5–8.0μm.Wang et al.[21]tested the Rossland mean extinc-tion coef ficient of TiO 2-loaded silica aerogel composites with different TiO 2mass fractions.Feng et al.[25]tested the speci fic spectral extinc-tion coef ficient of opaci fier-loaded silica aerogel composites with differ-ent particle sizes and opaci fier mass fractions,where opaci fiers were selected as BN,SiC,K 2Ti 6O 13and ZrSiO 4and the wavelength was varied from 2.5μm to 7.0μm.These experiments measured the extinction coef ficient of the opaci fier-loaded aerogel composite only with the speci fic opaci fier category,size,and mass (or volume)fraction,so that once the component or structure of the composite changes,new testJournal of Non-Crystalline Solids 375(2013)31–39⁎Corresponding author.Tel./fax:+861062321277.⁎⁎Corresponding author.Tel./fax:+861062796318.E-mail addresses:wangxd99@ (X.-D.Wang),yyduan@ (Y.-Y.Duan).0022-3093/$–see front matter ©2013Elsevier B.V.All rights reserved./10.1016/j.jnoncrysol.2013.04.058Contents lists available at SciVerse ScienceDirectJournal of Non-Crystalline Solidsj o u r n a l h o m e p a ge :w w w.e l se v i e r.c o m/l o c a t e /j n o n c r y so lis needed.Thus,in order to predict the extinction coefficient of opacifier-loaded aerogel composites with various opacifier catego-ries,sizes,and mass fractions more effectively,a theoretical method is necessary indeed[25,30].However,due to the lack of complex re-fractive index data of opacifiers,such theoretical model has not been established yet.The complex refractive index m=n−iκis also referred to as optical constant.The real part,n,accounts for the radiation refraction and determines the phase velocity in the medium.The imaginary part,κ,accounts for the absorption and determines the radiation attenuation through the medium[31].For transparent materials,the absorption effect can be ignored withκ=0[32–36],however,for opacifiers and opacifier-loaded aerogel composites,both n andκare larger than zero.Several methods to calculate the complex refractive index have been proposed by previous researches[4,29,32,34].Lu et al.[4]calcu-lated the imaginary part of complex refractive index of pure aerogel directly through its effective extinction coefficient.Zeng et al.[32] calculated the complex refractive index by measuring the reflectance and transmittance of silica aerogel,due to low measurement accuracy of reflectance caused by the surface roughness of aerogel sample,the complex refractive index cannot be calculated accurately.Zeng et al.[32]proposed that complex refractive index can also be solved through the dielectric function,but the dielectric function is also not easy to get for opacifiers.Ruan et al.[29]and Liu et al.[34]calculated the complex refractive index based on Mie theory and K–K relation,in which only the transmittance of materials needs to be measured,but the theoretical calculation is too complicated and hard to obtain the exact solution of the complex refractive index.The above investigations mainly focused on the aerogels,however,to our best knowledge,theoretical predictions for the complex refractive index of opacifiers have not been reported in the open literatures.The purpose of this paper is to develop a theoretical model to predict radiative characteristics of opacifier-loaded aerogel composites. Firstly,a new and simple approach is developed to predict complex refractive index of opacifiers.Then the complex refractive index is used to calculate the extinction efficiency of a single opacifier particle based on Mie theory.Finally,a predictive model is proposed to calculate the specific extinction coefficient,and the radiative thermal conductiv-ity of opacifier-loaded silica aerogel composite based on the extinction efficiency and mass fraction of the opacifier.Based on the developed model,radiative characteristics of the six kinds of opacifiers are investi-gated systematically,and the optimal opacifier size and category in the composite are obtained under different temperatures.Furthermore,to reduce the radiative heat transfer further,a novel material design of opacifier-loaded aerogel composite is proposed,in which the optimal opacifier and its size are varied according to the internal temperature gradient of the composite.2.Modelplex refractive index of opacifiersOpacifier particles have both scattering effect and absorption effect to light,therefore,they are regarded as absorbing medium,hence, both the real part and the imaginary part of the complex refractive index should be calculated for opacifier particles.The transmittance of opacifier particles,τ,can be measured by Fourier infrared spectral experiment[29,32,34].In the experiment, opacifier particles with the specific size and mass fraction are uni-formly mixed with KBr particles.The mixture is dried in the oven and then is pressed about10s under pressure of50–100MPa by the pressure machine to form a thin circle disk(sample A).Another disk (sample B)with the same radius and thickness is also pressed using pure KBr particles.Two disks are put in the Fourier infrared spectrometer, respectively,to measure their transmittances.Since the volume fraction of opacifier particles in the sample A is very low,the transmittance of opacifier particles can be obtained by subtracting the transmittance of sample B from the transmittance of sample A.Based on the Beer law,the transmittance and specific extinction coefficient of opacifier meet the following relation[3,20,22,25,30]:τλðÞ¼e−eÃλðÞρhð1Þwhereλis the wavelength,e⁎is the specific extinction coefficient of opacifier,ρis the density of the sample A,and h is the sample thick-ness.It is noted thatτand e⁎are the transmittance and specific extinction coefficient,respectively of opacifier particles with specific volume fraction and size,so they are not the universal parameters. However,Zeng et al.proposed that for an absorbing homogeneous medium,such as opacifier in the opacifier-loaded aerogel composite, the imaginary part of complex refractive index can be derived from e⁎as follows[32]:eÃλðÞ¼4πκλðÞλρopað2Þwhereρopa is the density of opacifibining Eqs.(1)and(2),we have,κλðÞ¼λρopa ln1τλðÞ4πρh:ð3ÞThe real part and the imaginary part of complex refractive index meet the classic K–K relation[25,29,32,34]:nλðÞ¼1þ2λ2πP∫∞κλ0ðÞλ0λ−λ0ÀÁdλ0:ð4ÞRuan et al.[29]and Dombrovsky et al.[37]used a more accurate modified K–K relation originally proposed by Ahrenkiel[38]:nλðÞ¼nλ1ðÞþ2λ21−λ2πP∫∞λ0kλ0ðÞÀÁ10ÀÁdλ0ð5Þwhereλ1=0.4358μm,is the wavelength at the triple blue mercury line,the refractive coefficient n(λ1)was measured using a Hilger Watts Abbe refractometer[37],and P is the Cauchy principal value of the integral.There is a Cauchy integral over all the wavelengths in Eq.(5),how-ever,we can only get afinite wavelength range[λmin,λmax]through the experiment,so that thefinite experimental data must be extrapolated into both long wavelength[λmax,+∞]and short wavelength[0,λmin] regions.Ruan et al.[29]used the following extrapolation method:λ≤λmin;κλðÞ¼C Lλ3λ≤λmax;κλðÞ¼C H=λð6Þwhere C L and C H are expressed by:C L¼κλminðÞλ3minC H¼κλmaxðÞ=λmax:ð7ÞAfter the extrapolation,the K–K relation can be expressed by[29]: nλðÞ¼nλ1ðÞþ2λ21−λ2π∫λminλmaxλ0kλ0ðÞλ−λ0ÀÁλ1−λ0ÀÁdλ0þN HþN Lð8Þ32X.-D.Wang et al./Journal of Non-Crystalline Solids375(2013)31–39where N H and N L are the Cauchy integrals in the longer and shorter ex-trapolated wavelength ranges,finally,they can be expressed by [29]:N L ¼C L λL þC L λ21þλ2 ln λ−λLλþλL 1þC L λ41⋅1⋅lnλ−λL λþλL −C L λ41λ2−λ21⋅12λ1⋅lnλ1−λL λ1þλL ð9ÞN H ¼C H λ2−λ21ÀÁ12λln λH þλλH −λ−12λ1ln λH þλ1λH −λ1 &':ð10ÞUsing the iterative method,the real part can be calculated by theimaginary part of complex refractive index.2.2.Extinction ef ficiency of a single opaci fier particleThe radiative characteristic of opaci fier-loaded aerogel composite is determined by the extinction coef ficient of aerogel,the extinction coef ficient of opaci fier,the volume fraction of opaci fier,and the tem-perature.This subsection will determine the extinction ef ficiency of a single opaci fier particle,which is a basis to determine the extinction coef ficient of opaci fier.The opaci fier particle is assumed to be a sphere with a diameter of D .Since opaci fier is uniformly distributed in the composite with a very small volume fraction,and its diameter has the same magnitude with the incident light wavelength,Mie theory can be used to describe the radiative behavior of a single opaci fier particle [34,35,39,40].The ex-tinction ef ficiency,Q ext ,scattering ef ficiency,Q sca ,and absorption ef fi-ciency,Q abs ,of a single opaci fier particle can be expressed by:Q ext¼2x Re X ∞j ¼12j þ1ðÞa j þb j 2435ð11ÞQ sca ¼2x 2X ∞j ¼12j þ1ðÞa j 2þb j 2 ð12ÞQ abs ¼Q ext −Q scað13Þwhere Re denotes the real part of a complex quantity,x =πD /λis the size parameter at the wavelength λ,and a j and b j are the Mie scattering coef ficients expressed by:a j ¼ψ′j mx ðÞψj x ðÞ−m ψj mx ðÞψ′j x ðÞψ′j mx ðÞξj x ðÞ−m ψj mx ðÞξ′j x ðÞð14Þb j ¼ψ′j mx ðÞψj x ðÞ−m ψj mx ðÞψ′j x ðÞψ′j mx ðÞξj x ðÞ−m ψj mx ðÞξ′j x ðÞð15Þwhere ξj =ψj −i χj ,ψj (x )and χj (x )are the Ricatti –Bessel functions,which meet the following recurrence relations:ψj þ1x ðÞ¼2j þ1x ψj x ðÞ−ψj −1x ðÞð16Þχj þ1x ðÞ¼2j þ1xχj x ðÞ−χj −1x ðÞð17Þandψ−1x ðÞ¼cos x ð18Þψ0x ðÞ¼sin x ð19Þχ−1x ðÞ¼−sin x ð20Þχ0x ðÞ¼cos x :ð21Þ2.3.Extinction coef ficients of opaci fier particles in compositeIt is noted that opaci fier particles are uniformly distributed in mono-lithic aerogel with a very small volume fraction,independent scatteringcondition is met,according to Mie theory and Beer law,the transmit-tance of opaci fier particles in the composite can be related to the extinc-tion ef ficiency of single opaci fier particle,or [34]:τ¼e−1πD 2Q ext NL ð22Þwhere N is the number density of opaci fier particles in the composite,and L is the composite thickness.The mass fraction of opaci fier in the composite,ω,can be expressed as:ω¼12πD 3N ρopaa V a 16N ρopað23Þwhere ρa is the aerogel density,and V a is the aerogel volume.From Eq.(23),the number density of opaci fier particles is:N ¼6ρa ω1−ωðÞπD opa:ð24ÞThe spectral extinction coef ficient of opaci fier particles in the com-posite,βopa (λ),is connected to their transmittance,τ,according to Beer law [29,39]:τ¼I I 0¼e −βopa λðÞL ð25Þwhere I and I 0are the transmitting and incident intensity of light.By combining Eqs.(22),(24),and (25),the spectral extinction co-ef ficient of opaci fier particles in the composite can be expressed as:βopa λðÞ¼3ρa ωQ ext21−ωðÞD ρopa:ð26ÞThe speci fic spectral extinction coef ficient of opaci fier particles in the composite is de fined as a ratio of extinction coef ficient of opaci fier particles to the composite density.The composite density ρa −opa can be calculated by:ρa −opa ¼ρa1−ω:ð27ÞThus,the speci fic spectral extinction coef ficient of opaci fier particlesin the composite,e opc⁎(λ),can be expressed by:e ÃλðÞ¼βopa λðÞρa −opa ¼3ωQ ext2D ρopa:ð28ÞThe temperature-dependent Rossland extinction coef ficient ofopaci fier particles in the composite,βopa (T ),is integrated over the full wavelength range based on the spectral extinction coef ficient,βopa (λ),[26,42]:βopa T ðÞ¼∫∞1βopa λðÞ⋅∂Eb λbd λ"#−1ð29Þwhere E b λis the spectral blackbody emissive power calculated by Planck's law,E b is the integration of E b λover the wavelengths,or E b λT ðÞ¼C 1λ−5exp C 2=λT ðÞ−1ð30Þwhere C 1=3.743×108W μm 4m −2,C 2=1.439×104μm K.33X.-D.Wang et al./Journal of Non-Crystalline Solids 375(2013)31–392.4.Radiative characteristics of compositeIt is especially noted thatβopa(λ)andβopa(T)are the radiative characteristics of the ensemble of opacifier particles.Thus,with an independent scattering assumption due to a very small volume frac-tion of opacifier in the composite,the spectral and Rossland extinc-tion coefficient of the composite,βa−opa(λ)andβa−opa(T),can be calculated as follows,respectively:βa−opaλðÞ¼βopaλðÞþβaλðÞ1−f vðÞð31Þβa−opa TðÞ¼βopa TðÞþβa TðÞ1−f vðÞð32Þwhere,βa(T)is the Rossland extinction coefficient of aerogel,and f v is the volume fraction of opacifier particles in the composite,which is related to the mass fraction of opacifier as follows,f v¼ρa wρo:ð33ÞThus,the specific spectral extinction coefficient,eÃa−opa λðÞ,and the spe-cific temperature-dependent Rossland extinction coefficient,eÃa−opaTðÞare:eÃa−opa λðÞ¼βa−opaλðÞρa−opað34ÞeÃa−opa TðÞ¼βa−opa TðÞρa−opa:ð35ÞSince the opacifier-loaded aerogel composite slab is far thicker (several centimeters)than light wavelengths(2–100μm),the Rossland diffusion approximation can be used for the radiative heatflux [1–3,25,29,41,42],hence,the radiative thermal conductivity of the composite,k r,a−opa,can be calculated as:k r;a−opa¼16n2a−opaσ03βa−opa TðÞT3ð36Þwhere n a−opa is the effective refractive index of the composite,and σ0=5.67×10−8W m−2K−4is the Stefan–Boltzmann constant.The effective refractive index of the composite is calculated based on the volume-averaged refractive indexes of opacifier and aerogel: n a−opa¼f v n opaþ1−f vðÞn að37Þwhere n opa and n a are the refractive indexes of the opacifier and aerogel,respectively,n a is taken as1.04here[32].3.Transmittance of opacifiersSix kinds of opacifiers(SiC,TiO2,ZrO2,coal ash,carbon black,and Al2O3)are loaded within the pure silica aerogel to prepare the opacifier-loaded aerogel composites.The density,mass fraction and volume fraction of the opacifiers are listed in Table1.The transmittance of SiC,TiO2,and ZrO2opacifiers is measured by the Fourier infrared spectrometer(Nicolet6700,Thermo Fisher Company,The United States).The original experimental data are wave-number-dependent absorbance curves as shown in Fig.1. The corresponding transmittance can be calculated by the following relation,τλðÞ¼10−A1λðÞ:ð38ÞThe transmittances of coal ash and Al2O3come from Liu and Dai's experiments[34]as shown in Fig.2.However,the transmittance of carbon black is not found in the open literature,hence,its complex refractive index reported by Zeng et al.[43]is used directly.4.Resultsplex refractive indexIn order to validate the approach to calculate the complex refractive index,complex refractive indexes for three different materials obtained by previous researches are compared with those predicted by the present model.Fig.2shows the variation of transmittance with wave-length for silica aerogel[32],coal ash[34],and cloud of aerosol particles [29],which were tested by using the Fourier infrared spectrometer.The complex refractive indexes for the three materials calculated by our model described in Section2.1are shown in Figs.3and4.Zeng et al.Table1Properties of opacifiers.Opacifier category Densityρ/kg m−3Mass fraction w Volume fraction f vSiC310025% 1.28%ZrO2589025%0.68%TiO2426025%0.93%Carbon black145025% 2.69%Al2O3397025% 1.00%Coal ash224525% 1.75%0500100015002000250030003500400020406080100k / cm-1ASiC D=2.5 mmTiO2D=1-3 mmZrO2D=1-3 mmFig.1.Variation of absorbance with wave number of light for SiC,TiO2and ZrO2 opacifiers.024681012141618202224260.00.10.20.30.40.50.60.70.80.91.0silica aerogel [32]coal ash [34]cloud of aerosol particles [29]Al2O3[34]/ μmτλFig.2.Spectral transmittivity of different materials.34X.-D.Wang et al./Journal of Non-Crystalline Solids375(2013)31–39[32],Liu et al.[34],and Ruan et al.[29]also calculated the complex refractive indexes of these materials by using different approaches.In Zeng et al.'s work [32],the re flectance and transmittance of silica aerogel were used to calculate κ,with consideration of lower test accu-racy of aerogel re flectance,n was calculated by the classical K –K relation.In Ruan et al.'s [29]and Liu et al.'s [34]works the complex refractive index was calculated based on the Mie theory and K –K relation,but the theoretical calculation is too complicated.Figs.3and 4compare n and κof silica aerogel,coal ash,and cloud of aerosol particles predicted by the present model with those by the previous approaches.Fig.3shows that κof silica aerogel and cloud of aerosol particles predicted by the present model are almost the same with Zeng et al.'s and Ruan et al.'s results across the whole wavelength range,κof coal ash predicted by the present model agrees well with Liu et al.'s results within the wavelength range of 1–7μm.Though there exists difference between the two approaches when the wavelength is larger than 7μm,the two curves change in the same tendency.As shown in Fig.4,the curves of n predicted by the present and previous approaches agree well when the wavelength is shorter than 12μm.The curves predicted by our model deviate from those predicted by the previous approaches by a small degree for wavelength larger than 12μparisons indicate that our model can predict the complex refractive index of materials very well,and it is a more simple approach excluding complex calculations.24681012141618202224260.000.010.020.030.040.050.060.070.080.09present work Zeng [32]a)246810121416182022240.00.10.20.30.40.50.60.70.80.91.0 present work Liu [34]b)24681012141618202224260.000.040.080.120.160.200.240.28present work Ruan [29]c)κκκ/ μmλ/ μmλ/ μmλFig.3.The imaginary part of complex refractive index predicted by the present model and previous approaches:(a)silica aerogel;(b)coal ash;and (c)cloud of aerosol particles.03691215182124270.960.981.001.021.041.061.08npresent work Zeng [32]a)036912151821241.11.21.31.41.51.61.71.81.92.02.1nb)present work Liu [34]03691215182124271.4251.4501.4751.5001.5251.5501.5751.6001.6251.6501.675c)npresent work Ruan [29]/ μmλ/ μmλ/ μmλFig.4.The real part of complex refractive index predicted by the present model and previous approaches:(a)silica aerogel;(b)coal ash;and (c)cloud of aerosol particles.35X.-D.Wang et al./Journal of Non-Crystalline Solids 375(2013)31–394.2.Speci fic extinction coef ficients of compositesAccording to Eqs.(31)–(35),to evaluate the extinction coef ficient of the opaci fier-loaded aerogel composite,the extinction coef ficient of silica aerogel must be known.The spectral extinction coef ficient of silica aerogel has been extensively tested in the previous researches [3,16,43]as shown in Fig.5.The results show that the spectral extinc-tion coef ficient tested by Wei et al.[3],Lu et al.[16],and Zeng et al.[43]has a large difference in the wavelength range of 2–8μm.More-over,their results also indicate that the silica aerogel has a poor extinc-tion performance in this wave band,hence,opaci fiers should be added into the aerogel to improve its extinction performance.In our calcula-tions,the spectral extinction coef ficient of aerogel tested by Wei et al.[3]is selected.Fig.6(a)compares the speci fic spectral extinction coef ficient of TiO 2-loaded aerogel composites predicted by the present model with experimental data [20].In our calculation,the diameter of TiO 2in the composite is taken to be 3.5μm,and the mass fraction of TiO 2is 10%,20%and 30%,respectively.The present model shows that the speci fic spectral extinction coef ficient increases as the mass fraction of TiO 2increases,which agrees with Kunh et al.'s experiment [20].The speci fic spectral extinction coef ficient of TiO 2-loaded aerogel composite predicted by the present model is slightly higher than ex-perimental results,the reason may be attributed to that it is hard to guarantee the uniform diameter and distribution of TiO 2in the com-posite during the experiment.The speci fic Rossland extinction coef ficient of TiO 2-loaded aerogel composites,predicted by the present model,is shown in Fig.6(b).The diameter of TiO 2opaci fier is 3.5μm with mass fractions of 10%,20%,and 30%,respectively.Wang et al's experimental data [21]are used to validate the present model.Wang et al.had presented that their data had an uncertainty of 15%[21].With consideration of this uncer-tainty,it can be regarded that the present predictions agree with Wang et al.'s results well.3691215182124102103104105a -1Lu et al. [16] Zeng et al. [43] Wei et al. [3]/ μmλ(λ) / m βFig.5.The extinction coef ficient of pure silica aerogel [3,16,43].2.0 2.53.0 3.54.0 4.55.0 5.56.0 6.57.07.58.08.520406080100120140160180200e *a -o p a a)Kuhn et al [20]10% TiO 2 20% TiO 230% TiO 2Present model10% TiO 220% TiO 230% TiO 2200400600800100012001400102030405060708090100b)T / KWang et al. [21]10% TiO 220% TiO 2 30% TiO 2present model10% TiO 220% TiO 230% TiO 2/ μmλ(λ)/ m 2 k g -1e *a -o p a (T )/ m 2 k g -1Fig.6.Speci fic extinction coef ficient of TiO 2-loaded aerogel composites:(a)speci fic spectral extinction coef ficient;and (b)speci fic Rossland extinction coef ficient.2004006008001000120014003000600090001200015000180002100024000270003000033000 SiC ZrO 2 TiO 2coal ash carbon black Al 2O 3T / Ka -o p a (T ) / m -1=25% D =4 mmωβFig.7.The Rossland extinction coef ficients of composites with various opaci fiers.2004006008001000120014000.000.020.040.060.080.10SiC ZrO 2 TiO 2coal ash carbon black Al 2O 3=25% D =4 μmT / Kk r ,a -o p a / W m -1 K -1ωFig.8.The radiative thermal conductivities of composites with various opaci fiers.36X.-D.Wang et al./Journal of Non-Crystalline Solids 375(2013)31–39。