prediction of thermal conductivity of fiber/aerogel ...4)-13550.pdf · were added to the silica sol...

Journal of Porous Media, 18 (10): 971–984 (2015)

PREDICTION OF THERMAL CONDUCTIVITY OFFIBER/AEROGEL COMPOSITES FOR OPTIMALTHERMAL INSULATION

Jianming Yang,1 Huijun Wu,1,∗ Shiquan He,1 & Moran Wang2

1College of Civil Engineering, Guangzhou University, Guangzhou 510006, China2Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

∗Address all correspondence to Huijun Wu E-mail: [email protected]

Original Manuscript Submitted: 3/4/2015; Final Draft Received: 5/27/2015

A numerical model for predicting the effective thermal conductivity of fused silica fiber/aerogel composites by simul-taneously considering the effects of the fiber volume fraction and fiber diameter is presented. The predicted effectivethermal conductivity of the fiber/aerogel composites agreed well with the existing measured and predicted results. Theeffects of the volume fraction (0–25%) and diameter (0.3–10 µm) of fibers on the effective thermal conductivity of aerogelcomposites were investigated under a large range of temperatures (300–1300 K). The results indicated that the mini-mum effective thermal conductivity of the fiber/aerogel composites by simultaneously considering the optimized fibervolume fraction and diameter was significantly lower than when individually considering the optimized fiber volumefraction and diameter values. For instance, the minimum effective thermal conductivity by simultaneous optimizationwas 0.0262 W/m−1 K−1 at 1000 K, which was much lower than 0.0327 W/m−1 K−1 by individually optimizing thefiber volume fraction at a diameter of 8 µm and 0.0532 W/m−1 K−1 by individually optimizing the fiber diameter at avolume fraction of 3%. Moreover, the quantitative relations between the minimum effective thermal conductivity of thefiber/aerogel composites and the temperatures are presented, with the aim of identifying the optimal thermal insulationfor applications in aeronautics and astronautics, construction, and other industrial fields.

KEY WORDS: aerogels, composites, thermal conductivity, extinction coefficient, optimization

1. INTRODUCTION

Silica aerogels with a three-dimensional (3D) nanoporousstructure have well been acknowledged as one of the mostinteresting super-thermal insulating materials (Aegerteret al., 2011; Koebel et al., 2012) owing to their ex-tremely low thermal conductivity, which can be as lowas 0.01 W·m−1·K−1 (Kistler, 1932), and they have beenused in aeronautics and astronautics, construction, andother industrial fields (Papadopoulos, 2005; Schmidt andSchwertfeger, 1998). The super low thermal conductiv-ity of an aerogel is attributed to its nanopores (2–50 nm),high porosity (approximately 85–99%), and humongousshields to heat radiation (Han, 2013; Martinez-Gomezand Soria, 2000; Shan and Wang, 2013; Wang et al.,

2013; Zeng et al., 1995a). However, the weak strength andhigh transparency of aerogels at a wavelength of 3–8µmlimit their application in thermal insulation (Baetens etal., 2011; Zeng et al., 1995b). Reinforced fibers have re-cently been added to the aerogel matrix, both to improvethe strength and to reduce the radiative transfer, becausethe fibers have tremendous scattering or absorption, espe-cially at a wavelength of 3–8µm (Liao et al., 2012).

Very recently, there have been some studies report-ing calculations of the effective thermal conductivity offiber/aerogel composites when the fibers were distributedorderly, randomly in plane, or randomly in space withinthe aerogel composites. Wu et al. (2014) proposed aheat transfer model of multilayer-aligned fiber-reinforcedaerogel composites based on the unit cell of the sur-

1091–028X/15/$35.00 c⃝ 2015 by Begell House, Inc. 971

972 Yang et al.

NOMENCLATURE

a constant(= 8.07× 10−6 W·m2·kg−1·K−4/3)

an,I, an,Π,bn,Π

Mie coefficients

b constant(= 4.80× 10−14 W·m2·kg−1·K−4)

co light speed (= 2.998× 108 m·s−1)D diameter (µm)Dn recurrence relationd2F orientation distributionEb blackbody emissive power (W·m−2)Ebλ spectral blackbody emissive power

(W·m−2·µm−1)fi volume fraction of the components

H(1)n Ricatti–Hankel function of the first kind

h Planck’s constant(= 6.626× 10−34 m2·kg·s−1)

Im imaginary partJn Ricatti–Bessel function of the first kindk Boltzmann’s constant

(= 1.38× 10−23 J·K−1)kc conductive thermal conductivity of the

composite (W·m−1·K−1)kc,a conductive thermal conductivity of the

aerogel (W·m−1·K−1)kc,f conductive thermal conductivity of the

fiber (W·m−1·K−1)keff effective thermal conductivity of the

composite (W·m−1·K−1)kg gas phase thermal conductivity of the

aerogel (W·m−1·K−1)kr radiative thermal conductivity of the

composite (W·m−1·K−1)ks conductive thermal conductivity of the

nanoparticle (W·m−1·K−1)kunit conductive thermal conductivity of the

basic unit cell (W·m−1·K−1)m complex refraction indexN [r(Rf )] number size distributionn refractive indexnc,λ spectral refractive index of the compositeni,λ spectral refractive index of the

components

nT effective refractive index of the compositeenvironment

P Van de Huslst parameterQ extinction factorQabs,R absorption factor of Rayleigh scatteringQeλ,M,I extinction factor of Mie scattering (CaseI)Qeλ,M,Π extinction factor of Mie scattering (CaseΠ)Qsca,R scattering factor of Rayleigh scatteringRe real parts stage of the Sierpinsky spongeT temperature (K)x size factorYn Ricatti–Bessel function of the second kind

Greek Symbolsβe,T Rosseland mean extinction

coefficient (m−1)βeλ,a spectral extinction coefficient of the

aerogel (m−1)βeλ,f spectral extinction coefficient of

the fiber (m−1)βeλ,t total spectral extinction coefficient (m−1)γ side ratioζ angular orientation of the fiberκ absorption indexλ wavelength (µm)ρa volume density of the aerogel (kg·m−3)ρf volume density of the fiber (kg·m−3)σ Stefan–Boltzmann constant

(= 5.67× 10−8 W·m−2·K−4)ω angular orientation of the fiber

Subscriptsa aerogelabs absorptionb blackbodyc conductiveeff effectiveext extinctionf fiberr radiativesca scatteringt total

Journal of Porous Media

Prediction for Optimal Thermal Insulation 973

face contact hollow cubic structure and the parallel lawof equivalent thermal resistance when the fibers wereorderly aligned. The effective thermal conductivities ofsilica fiber/aerogel composites were obtained at 0.0218,0.0373, and 0.0544 W·m−1·K−1 in three directions. Zhaoet al. (2012b) analyzed the effect of the fiber types onthe effective thermal conductivity of fiber/aerogel com-posites when the fiber was randomly distributed in planeby using the series/parallel hybrid model. They found thatthe thermal conductivity of fiber/aerogel composites canbe reduced by 90% compared to that of pure aerogel at1300 K when the volume fraction, diameter, and slant an-gle of the glass fiber are 3%, 4µm, and 5◦, respectively.Xie et al. (2013) investigated the effect of temperatureon the thermal conductivity of fiber/aerogel compositeswhen the fiber was randomly distributed in space by us-ing the Hamilton model. The effective thermal conductiv-ities of glass fiber/aerogel composites at 0.0313, 0.0488,and 0.0927 W m−1 K−1 were obtained at 300, 600, and900 K, respectively, when the fiber mass fraction was 30%and the fiber diameter was 8µm.

The fiber volume fraction and fiber diameter are twoimportant parameters that influence the thermal conduc-tivity of fibrous materials (Tong and Tien, 1983; Wanget al., 2007, 2009). The individual effects of the fibervolume/mass fraction and the fiber diameter on the ther-mal conductivity of fiber/aerogel composites have beeninvestigated with the aim of optimizing the thermal in-sulation of fiber/aerogel composites. Zeng et al. (1995b)investigated the effect of the mass content of a carbonopacifier on the thermal conductivity of aerogel compos-ites. The lowest thermal conductivity of carbon opaci-fied aerogel composites could be lowered by one-thirdthat of pure aerogels when the mass content of the car-bon opacifier is approximately 8% at ambient tempera-ture. Xie et al. (2013) investigated the effect of the contentof a SiC opacifier on the thermal conductivity of aerogels.They found that the preferable mass contents relevant tothe lowest thermal conductivities were 7 and 14% at 600and 900 K, respectively. Zhao et al. (2012c) explored theeffect of the fiber diameter (2–10µm) on the radiativeproperty and thermal conductivity of silicon fiber/aerogelcomposites. The results showed that the preferable diam-eters relevant to the lowest thermal conductivities were10 and 4–6µm at 300 and 1000 K, respectively, when thevolume fraction of the silicon fiber was 2%. Subsequently,in relation to amorphous SiO2 glass fiber/aerogel compos-ites, Zhao et al. (2012a) demonstrated that the preferablefiber diameter relevant to the lowest thermal conductivitywas of 4–8µm when the fiber volume fraction was 2%.

However, in the literature, there are very few studieson the thermal conductivity of fiber/aerogel compositesthat simultaneously consider the effect of the fiber volumefraction and the fiber diameter. In this paper, the thermalconductivities of fiber/aerogel composites are optimizedby simultaneously considering these two important pa-rameters, i.e., the fiber volume fraction and the fiber diam-eter, with the aim of obtaining the best thermal insulationfor fiber/aerogel composites.

2. THEORETICAL ANALYSIS

In this study, fused silica fibers with purity higher than99.5% SiO2 and a melting point higher than 1600 K(Bansal and Doremus, 1986; Lee and Cunnington, 2000)were used to reinforce the silica aerogel in order to formfiber/aerogel composites. The diameter of the fused silicafibers was 0.3–10µm and the ratio of the length to thediameter was greater than 1000. The complex refractionindex (m = n – iκ) of the fused silica fibers in the wave-length range of 0.5–25µm is shown in Fig. 1 (Kitamuraet al., 2007; Palik 1998), wheren andκ are the refractiveand absorption indices, respectively.

Fused silica fibers, approximately 10 mm in length,were added to the silica sol to obtain a fiber/silica gel.The as-obtained fiber/silica gels were surface modifiedand dried at ambient pressure in order to prepare thesilica fiber/aerogel composites. Detailed descriptions ofthe preparation procedures can be found in Liao et al.(2012) and Wu et al. (2014). Figure 2(a) shows an op-tical image of the fused silica fiber/aerogel compositesand a schematic drawing of the heat flow model, where

FIG. 1: Complex refractive index of the fused silica fiber

Volume 18, Number 10, 2015

974 Yang et al.

FIG. 2: Images of the fused silica fiber/aerogel composite:(a) optical image;(b) SEM image of the aerogel;(c) SEMimage of the fiber/aerogel composite;(d) scattering geometry for a single cylindrical fiber with oblique incidentradiation

the fibers are randomly placed in plane between the upperand lower surfaces. The upper and lower surfaces have aconstant temperature gradient and the other surfaces areassumed to be adiabatic. The cutting plane (A) is paral-lel to the upper and lower surfaces and perpendicular tothe heat flow. Figure 2(b) shows a scanning electron mi-croscope (SEM) image of the silica aerogel. It can be ob-served that the silica aerogel has a nanoporous structurewith particle sizes of 2–5 nm and pore sizes of 2–50 nm(Han, 2013), which are smaller than the mean-free pathof oxygen and nitrogen. Figure 2(c) shows a SEM im-age of the composite structure of the fused silica fiber andthe silica aerogel. It can be observed that the fibers arerandomly distributed in the silica aerogel matrix and areapproximately parallel to the cutting plane. Here, it canbe assumed that the fibers are perpendicular to the direc-tion of heat flow. It can also be observed from Fig. 2(c)that the fibers are wrapped by the aerogel without direct

contact between the fibers. Based on the heterogeneousanalysis of short fiber composites by Pan (1994), it canbe assumed that the fibers and aerogel are uniformly mix-ing and that the contact resistant between the fibers andaerogel is negligible. Figure 2(d) shows the scattering ina single cylindrical fiber with obliquely incident radiation.The inclination angle,ϕ, describes the extent of the fiberdirection deviating from the cutting plane (A); if the fiberis in the cutting plane (A) or the fiber axis is perpendicularto the direction of heat flow,ϕ equals zero.

2.1 Extinction Factor of Fused Silica Fiber

In order to calculate the extinction factor of fused silicafibers, some simplified approximations need to be clar-ified according to the Van de Huslst parameter,P =2πD |m− 1|/λ, whereD andλ are the diameter of thefused silica fiber and the wavelength, respectively. In this



paper, thermal radiation mainly occurs in the wavelengthrange of 0.5–25µm for the calculated temperature rangeof 300–1300 K, following Wien’s energy distribution law(Planck, 1901). Accordingly, parameterP can be calcu-lated in the range of [0.0009, 350]. Therefore, the sim-plified Mie and Rayleigh scattering theories were usedto calculate the extinction factor of the fused silica fibers(Barth, 1984).

The extinction factor of the fused silica fibers was cal-culated according to the Mie theory as follows by usingthe forward recursive calculation method in the MAT-LAB (The Mathworks, Massachusetts) program (Xie etal., 2013):

Qeλ,M,I =2

xRe

(bf0I + 2

∞∑n=1

bn1

)(1)

Qeλ,M,Π =2

xRe

(a0Π + 2

∞∑n=1

anΠ

)(2)

Qeλ,M =1

2(Qeλ,M,I +Qeλ,M,Π) (3)

where Re is the real part;x is the size factor de-fined asπD/λ; and an and bn are Mie coefficients,which are functions of the fiber complex refractive indexfor infinitely long cylindrial fibers expressed as follows(Bohren and Huffman, 1983):

anΠ (ξ = 90◦)

=mJ ′

n (x) Jn (mx)− Jn (x)J′n (mx)

mH(1)′n (x)Jn (mx)−H

(1)n (x)ϕ′

n (mx)(4)

bnI(ξ = 900

)=

J ′n (x) Jn (mx)−mJn (x)J

′n (mx)

H(1)′n (x) Jn (mx)−mH

(1)n (x) J ′

n (mx)(5)

whereJn is the Ricatti–Bessel function of the first kind;andH(1)

n = Jn + i · Yn is the Ricatti–Hankel function ofthe first kind, whereYn is the Ricatti–Bessel function ofthe second kind andJ ′

n andH(1)′n are the derivatives. The

recurrence relations can be expressed as

J ′n (α) = Jn−1 (α)−

n

αJn (α) (6)

Dn (α) = J ′n (α)/Jn (α) (7)

Dn−1 (α) =n− 1

α− 1

(n− z) +Dn (α)(8)

Then, the scattering coefficients can be transferred as

anΠ (ξ = 90◦)

=[Dn (mx) /m+ n/x] Jn (x)− Jn−1 (x)

[Dn (mx) /m+ n/x]H(1)n (x)−H

(1)n−1 (x)

(9)

bnΠ (ξ = 90◦)

=[mDn (mx) + n/x] Jn (x)− Jn−1 (x)

[mDn (mx) + n/x]H(1)n (x)−H

(1)n−1 (x)

(10)

Rayleigh scattering is more obvious in small fiber diam-eters, especially at short wavelengths, which is known asRayleigh’s inverse fourth-power law (Modest, 2013). Theextinction factor of Rayleigh scattering is calculated asfollows (Zhao and Hu, 2003):

Qsca,R =π2

8x3

(∣∣m2 − 1∣∣2 + 4

∣∣∣∣m2 − 1

m2 + 1

∣∣∣∣2)

(11)

Qabs,R = −πx

2Im

(m2 − 1

2+

m2 − 1

m2 + 1

)(12)

Qext,R = Qsca,R +Qabs,R (13)

whereQsca,R andQabs,R are the scattering and absorp-tion factors of Rayleigh scattering, respectively; and Imdenotes the imaginary part.

The total extinction factor is the integrative effect ofthe simplified Mie extinction and Rayleigh extinction the-ories of bridge technology, given as (Zhao et al., 2012c;Zhao and Hu, 2003):

Qeλ =

(Qext,R + |m− 1|Qc2

sca,R

)Qext,M(

Qsca,R + |m− 1|Qc2sca,R

)+Qext,M

(14)

wherec2 = 2|m−1|.

2.2 Extinction Coefficient and ThermalConductivity of Fiber/Aerogel Composites

Based on the extinction factor of the fused silica fiber,the spectral extinction coefficient of the fiber can be cal-culated for various spectral lengths by (Cunnington andLee, 1996)

βeλ =

∫ ωf2

ωf1

∫ ξf2

ξf1

∫ ∞

0

2rQeλN [r (Rf )] drd2F (15)

whered2F andN [r(Rf )] are the orientation and num-ber size distributions; and the limits of integration (ω, ζ)


976 Yang et al.

denote the range of the angular orientations of the fibers.Because the fused fibers are randomly distributed in planeand perpendicular to the heat flow, the incident angleθ

equals 90◦ and Eq. (4) is simplified to

βeλ,f = 4Qeλ/πD (16)

Therefore, the spectral extinction coefficient of thefiber/aerogel composites can be calculated as follows(Zhao et al., 2012a):

βeλ,t = βeλ,ffv + βeλ,a (1− fv) (17)

where βeλ,a is the spectral extinction coefficient ofthe aerogel (see Zeng et al., 1996). For optically thickfiber/aerogel composites, the Rosseland mean extinctioncoefficient can be calculated as follows (Wei et al., 2013):

βe,T =

∞∫0

1

βeλ,t

∂Ebλ

∂Ebdλ

−1

(18)

whereEbλ is the spectral blackbody emissive power cal-culated by Planck’s law; andEb is the blackbody emissivepower, defined as (Modest, 2013)

Ebλ (T, λ) =2πhc20

n2Tλ

5[ehc0/nTλkT − 1

] (19)

Eb = n2TσT

4 (20)

where σ = 5.67 × 10−8 W·m−2·K−4 is the Stefan–Boltzmann constant;h = 6.626× 10−34 m2·kg·s−1 isPlanck’s constant;co = 2.998× 108 m·s−1 is the lightspeed in vacuum; andk = 1.38 × 10−23 J·K−1 is theBoltzmann constant, wherenT is the recombination ef-fective refractive index of the composite environmentchanging with the wavelength and surrounding tempera-ture. A transformation to eliminate the direct relation be-tweennT andλ is needed to satisfy the previous formula.Since the wavelength is related to temperature via energyfraction, we can make an approximate calculation fornT

as follows:

nT =

∞∫0

nc,λ∂Ebλ

∂Ebdλ (21)

wherenc,λ is the spectral refractive index of the com-posite, which can be approximately calculated bync,λ =∑m

i=1 ni,λfi, whereni,λ is the spectral refractive index ofthe components andfi is the volume fraction of the com-ponents. A discussion on the spectral refractive index ofaerogel can be found in Zeng et al. (1996).

According to previous research (Lee and Cunnington,2000; Wei et al., 2011; Zhao et al., 2012a), aerogels andfiber/aerogel composites can be considered optically thicksince their thickness is great (e.g., 1–5 cm) when usedin thermal insulation applications. Based on this assump-tion, the radiative thermal conductivity of fiber/aerogelcomposites can be calculated as follows by the diffusionapproximation (Modest, 2013):

kr =16n2

TσT3

3βe,T(22)

The conductive thermal conductivity of composites is acombination of the conductive thermal conductivities ofthe aerogel and fiber. Because fibers are assumed to bepredominantly oriented in planes normal to the heat flowdirection and fibers in an aerogel matrix are uniformlymixed without contact between the fibers, the conduc-tive thermal conductivity can be calculated by the seriesmodel as follows (Wang and Pan, 2008):

kc =

[1− fvkc,a

+fvkc,f

]−1

(23)

wherekc,a is the gas/solid-coupled thermal conductivityof the aerogel, taking into consideration the convectionheat transfer of the restricted gas and the contact resis-tance; andkc,f is the conductive thermal conductivity offibers. The calculation forkc,a is the fractal-intersectingsphere model (Xie et al., 2013), briefly expressed as

k1ae=4kunit

(1−γ

2

)2+

4kunitkgγ(1− γ)

2kg(1−γ)+2kunitγ+kgγ

2 (24)

ksae=4ks−1ae

(1−γ

2

)2+

4ks−1ae kgγ(1− γ)

2kg(1−γ)+2ks−1ae γ

+kgγ2 (25)

wheres represents the stage of the Sierpinsky sponge;γ

is the side ratio, defined as the ratio of the length of theinternal pore to the length of the cubic array; andkg is thegas phase thermal conductivity. Here,kunit is the effectivethermal conductivity of the basic unit cell, written as

kunit =

{−2k2skgkg−ks

In

[1+M

(kgks

−1

)]−M

2kgksks − kg

}+ a2ks (26)

whereM = cos [arcsin (a)]; a represents the ratio of thecontact length to the diameter of the silica particle; andks is the conductive thermal conductivity of the nanopar-ticles.



In Eq. (13), a polynomial fit for the conductive thermalconductivity of the fused silica under a temperature vari-ation of 300–1600 K was given as follows (Incropera andDeWitt, 1981; Lee and Cunnington, 2000):

kf (T ) = ρf

(aT 1/3 + bT 3

)(27)

where the volume density of the fused silica (ρf ) is2200 kg·m−3; constanta is about 8.07× 10−6 W·m2

·kg−1·K−4/3; and constantb is about 4.8× 10−14 W·m2

·kg−1·K−4.The effective thermal conductivities of the fiber/

aerogel composites are the sum of the radiative and con-ductive thermal conductivities

keff = kc + kr (28)

3. RESULTS AND DISCUSSION

3.1 Validation of the Calculations

In the present prediction, the effective thermal conduc-tivity was calculated by summing the conductive ther-mal conductivity based on the fractal-intersecting spheremodel [Eqs. (24) and (25)] and the radiative thermalconductivity based on the optically thick assumption[Eq. (22)]. The spectral extinction coefficients of silicaaerogel were obtained by Wei et al. (2011) and the com-plex refraction indices of the fused silica fibers were ob-tained by Kitamura et al. (2007) and Palik (1998). Thecalculated effective thermal conductivity of the pure sil-ica aerogel matched well with the experimental data (Weiet al., 2011) as shown in Fig. 3(a).

FIG. 3: Comparison of predicted and measured thermal conductivity:(a) effective thermal conductivity of the silicaaerogels;(b) thermal conductivity of the fused silica in vacuum;(c) and(d) effective thermal conductivity of the fusedsilica/aerogel composites


978 Yang et al.

Figure 3(b) shows a comparison of the predicted ef-fective thermal conductivity of the fused silica fiber un-der the vacuum condition in the present model and themeasured effective thermal conductivity with a maximumexperimental uncertainty of 10% according to Lee andCunnington (2000). The predicted results were calculatedusing Eqs. (22), (27), and (28) when the fiber diameterand volume fraction were preset as 1.7µm and 0.0692,respectively. Figures 3(c) and 3(d) show comparisons ofthe effective thermal conductivities of the fused silicafiber/aerogel composites between the predicted values inthe present model and the measured values in Lee andCunnington (1998), with maximum experimental uncer-tainty of 11–13% when the fiber diameter was 2.38µm.It can be observed that the predicted and measured ef-fective thermal conductivities agree well with each other,although slight deviation exists. This slight deviation waspossibly caused by the interlacing connections betweenfibers in the silica fibers or fiber/aerogel composites. Inthe calculation for the extinction coefficients of the fibers,

the fibers were assumed to be independent, while in factinterlacing fibers always exist in thermal insulation appli-cations.

3.2 Effects of the Fiber Volume Fraction andFiber Diameter on the Thermal Propertiesof Fiber/Aerogel Composites

The extinction coefficients and thermal conductivities offiber/aerogel composites are calculated as the fiber vol-ume fraction varied in the range of 0–25% and the fiberdiameter varied in the range of 0.3–10µm at various tem-peratures (300–1300 K). In the calculation, the volumedensity of pure aerogel was preset as 110 kg·m−3.

3.2.1 Effect of the Fiber Volume Fraction

Figure 4 shows the predicted thermal properties (i.e., ex-tinction coefficient, radiative thermal conductivity, andeffective thermal conductivity) of the fiber/aerogel com-

FIG. 4: Effect of the fiber volume fraction on the thermal properties of the fiber/aerogel composites:(a) and(b) meanextinction coefficient;(c) radiative thermal conductivity;(d) effective thermal conductivity



posites for various fiber volume fractions when the fiberdiameter was 3µm. Figure 4(a) shows that the meanextinction coefficient of the pure aerogel decreases asthe temperature is increased. As a comparison, for thefiber/aerogel composites, their mean extinction coeffi-cients first decreased below 500 K and then significantlyincreased as the temperature increased. Moreover, the re-inforced fiber significantly improved the mean extinc-tion coefficient of the aerogel composites, especially athigh temperature. For example, the mean extinction coef-ficients of the fiber/aerogel composites were 13,480 and22,130 m−1 for 3 and 8%, respectively, compared to4560 m−1 for the pure aerogel at 300 K. At 1000 K, themean extinction coefficients of the fiber/aerogel compos-ites were 16,310 and 36,620 m−1 for 3 and 8%, respec-tively, compared to 780 m−1 for the pure aerogel. There-fore, the mean extinction coefficient of the fiber/aerogelcomposites increases as the fiber volume fraction is in-creased, which can also be observed in Fig. 4(b).

Figure 4(c) shows the radiative thermal conductivityof the fiber/aerogel composites for various fiber volumefractions at various temperatures. The radiative thermalconductivity of fiber/aerogel composites significantly de-creases as the fiber volume fraction is increased. Fig-ure 4(d) shows the effective thermal conductivity of thefiber/aerogel composites for various fiber volume frac-tions at various temperatures. Compared to the pure aero-gel from Fig. 3(a), the effective thermal conductivity ofthe fiber/aerogel composites is significantly decreased, es-pecially at high temperatures, which is attributed to thescattering and absorption of the fibers. For example, theeffective thermal conductivities of the fiber/aerogel com-posites with fiber volume fractions of 1 and 8% are 0.0664and 0.0283 W·m−1·K−1, respectively, compared to thepure aerogel (0.4204 W·m−1·K−1 at 1000 K). The effec-tive thermal conductivities of the fiber/aerogel compositeswith fiber volume fractions of 1 and 8% were 0.0174 and0.0180 W·m−1·K−1, respectively, compared to the pureaerogel (0.0181 W·m−1·K−1 at 300 K). The minimumeffective thermal conductivity of the fiber/aerogel com-posites at 0.0174 W·m−1·K−1 occurred when the fibervolume fraction was 3% at 300 K. This is because theradiative thermal conductivity of the aerogel compositesdecreases while the conductive thermal conductivity ofthe aerogel composites increases when the fiber volumefraction is increased. Therefore, the effective thermal con-ductivity of the fiber/aerogel composites would be mini-mized by simultaneously evaluating the effect of the fibervolume fraction on the radiative and conductive thermalconductivities at constant fiber diameters.

3.2.2 Effect of the Fiber Diameter

Figure 5 shows the predicted thermal properties of thefiber/aerogel composites for various fiber diameters whenthe fiber volume fraction was 2%. It can be observedfrom Fig. 5(a) that the mean extinction coefficient of thefiber/aerogel composites is significantly greater than thatof the pure aerogel. When the fiber diameters are in-creased from 0.3 to 3µm, the mean extinction coeffi-cients of the fiber/aerogel composites increase at temper-atures of 300–1300 K. However, when the fiber diametersare increased from 3 to 10µm, the mean extinction co-efficients of the fiber/aerogel composites decrease at hightemperatures. It can be deduced that the maximum extinc-tion coefficients of the fiber/aerogel composites at hightemperatures would occur at approximately 3µm. Thisis consistent with the maximum extinction coefficients ofthe fibers under the vacuum condition (7010, 8000, 9090,and 10150 m−1) obtained for fiber diameters of 3, 2.7,2.45, and 2.3µm at 1000, 1100, 1200, and 1300 K, re-spectively, as shown in Fig. 5(b).

Figure 5(c) shows the effective thermal conductivity ofthe fiber/aerogel composites for various fiber diameters atvarious temperatures. It can be observed that the optimaldiameters of the fiber/aerogel composites, with the aim offinding the minimum effective thermal conductivity, areabout 8µm at 300–500 K, 5µm at 500–800 K, and 3µmat 800–1300 K. For instance, the effective thermal con-ductivity of the composites at 5µm is higher than that at8 µm for the temperature range of 300–500 K, and alsohigher than that at 1 and 3µm for the temperature rangeof 800–1300 K. This is because the minimum effectivethermal conductivity of the composites under the optimaldiameter is attributed to the maximum extinction coeffi-cient of the fibers because the fiber diameter only influ-ences the radiative thermal extinction of the fibers. Also,in our calculation for the extinction factor (correspondingto the extinction coefficient) of the fibers, the maximumextinction factor for the infinite cylindrical fiber happenswhen the wavelength of the incident light is equal to thefiber diameter (namely,λ = D or X = π) for a fibercomplex refraction index of 1 (namely,m = 1), and thesize factor for the maximum extinction factor is slightlyvaried with the variation of the complex refraction index.Based on Wien’s displacement law (Planck, 1901), themajority of the light (peak wavelengthλmax) occurs ata wavelength of about 5µm for the temperature range of500–800 K, which is mainly the reason why the optimaldiameter of the composites is about 5µm at 500–800 K.Also, the effective thermal conductivity of the composites


980 Yang et al.

FIG. 5: Effect of the fiber diameters on the thermal properties:(a) mean extinction coefficient of the fiber/aerogelcomposites;(b) mean extinction coefficient of the pure fibers;(c) effective thermal conductivity of the fiber/aerogelcomposites

increases when the fiber diameter is away from the peakwavelength. For instance, the distance from 0.3, 0.5, and10µm to 5µm (peak wavelength for 500–800 K) is largerthan the distance from 8 to 5µm. It can be observed fromFig. 5(c) that the effective thermal conductivities of thecomposites at 0.3, 0.5, and 10µm are larger than that at8 µm for 500–800 K.

3.3 Optimal Design for the Best ThermalInsulation

The joint effect of the fiber volume fraction and the fiberdiameter was investigated with the aim of minimizing theeffective thermal conductivity of fiber/aerogel compositesin order to find the best thermal insulation.

Figures 6(a) and 6(b) show two-dimensional (2D) and3D variations of the effective thermal conductivity of thefiber/aerogel composites at 300 K for various fiber vol-ume fractions and fiber diameters, respectively. It canbe observed that the minimum effective thermal con-ductivity (kmin) of the fiber/aerogel composites (0.0170W·m−1·K−1 at 300 K) can be obtained when the fibervolume fraction and the fiber diameter are 1.3% and8.6 µm, respectively. Optimization of the fiber volumefraction and the fiber diameter in the present study iswithin one decimal of precision owing to the limitationof the calculation load.

Figures 7(a) and 7(b) show 2D and 3D variationsof the effective thermal conductivity of the fiber/aerogelcomposites at 1000 K for various fiber volume frac-tions and fiber diameters, respectively. It can be observed



FIG. 6: Effective thermal conductivity of the fiber/aerogel composites at 300 K:(a) 2D variations;(b) 3D variations

FIG. 7: Effective thermal conductivity of the fiber/aerogel composites at 1000 K:(a) 2D variations;(b) 3D variations

that for the fiber/aerogel composites akmin value of0.0262 W·m−1·K−1 at 1000 K can be obtained when thefiber diameter and fiber volume fraction are 3.5µm and16.3%, respectively.

Table 1 lists thekmin values of the fiber/aerogel com-posites and the relevant optimal fiber volume fractionsand optimal fiber diameters at temperatures in the rangeof 300–1300 K. Figure 8 shows the relations ofkmin, opti-mal fiber volume fraction, and optimal diameter at varioustemperatures. Figure 8(a) shows that the minimum effec-tive thermal conductivity of the fiber/aerogel compositesincreases when the environment temperature is increased.The relation of the minimum effective thermal conduc-

tivity and the temperature can be fitted by using a cubicequation with a maximum deviation of 0.4% as follows:

kmin = 1.3757× 10−11T 3 − 1.8317× 10−8T 2

+ 1.7811× 10−5T + 0.01296 (29)

Figure 8(b) shows that the optimal fiber volume fractionincreases, whereas the optimal fiber diameter decreasesas the environment temperature is increased. The opti-mal fiber volume fraction increases because the mean ex-tinction coefficient of the fiber increases and the thermalconductivity of the fiber/aerogel composites decreasesas the environment temperature increases, as shown in


982 Yang et al.

TABLE 1: Temperature-dependent optimal fiber volumefraction, fiber diameter, and minimum effective conduc-tivity

Temperature vf D λmin

(K) (%) (µm) (W·m−1·K−1)300 1.3 8.6 0.0170400 4.1 7.9 0.0180500 6.9 7 0.0190600 9.4 6.2 0.0200700 11.7 5.5 0.0212800 13.6 4.8 0.0225900 15.1 4.1 0.02421000 16.3 3.5 0.02621100 17.2 3.1 0.02871200 17.8 2.8 0.03171300 18.2 2.5 0.0354

FIG. 8: Optimal fiber diameter, volume fraction, and min-imum effective thermal conductivity versus temperature:(a) minimum effective thermal conductivity versus tem-perature;(b) optimal fiber volume fraction and diameterversus temperature

Figs. 4(b)–4(d). The optimal fiber diameter decreases be-cause the maximum mean extinction coefficient of thefiber or the composites and the minimum effective ther-mal conductivity move to a smaller diameter as the tem-perature increases, as shown in Figs. 5(a)–5(c). The rela-tion of the optimal fiber volume fraction and the temper-ature can be fitted by using a quadratic equation with amaximum deviation of 7.9% as follows:

vf = −1.5338× 10−5T 2 + 0.01458T − 9.948 (30)

The relation of the optimal fiber diameter and the temper-ature can be fitted by using a quadratic equation with amaximum deviation of 7.0% as follows:

D = 3.2634× 10−6T 2 + 0.01153T + 11.9 (31)

The minimum effective thermal conductivity of fiber/aerogel composites can be derived by simultaneously op-timizing the fiber volume fraction and fiber diameter. Afiber diameter of 8µm or a fiber volume fraction of 3%is commonly used to predict the effective thermal con-ductivity of composites. As a comparison, the minimumeffective thermal conductivity is predicted by individuallyoptimizing the fiber diameter of 8µm or the fiber volumefraction of 3%, as listed in Tables 2 and 3. It can be ob-served that the minimum effective thermal conductivity ofthe fiber/aerogel composites decreases by simultaneouslyoptimizing the fiber diameter and volume fraction com-pared to individually optimizing the fiber volume frac-tion or fiber diameter, which is also shown in Fig. 9. Forinstance, the minimum effective thermal conductivity ofthe fiber/aerogel composites by simultaneously optimiz-ing the fiber volume fraction and fiber diameter was

TABLE 2: Minimum effective conductivity of thefiber/aerogel composites for a fiber diameter of 8µm

Temperature D vf λmin

(K) (µm) (%) (W·m−1·K−1)300 8 1.7 0.0172400 8 4.7 0.0184500 8 5.9 0.0202600 8 10.6 0.0211700 8 13.3 0.0305800 8 15.7 0.0255900 8 17.9 0.02871000 8 19.7 0.03271100 8 21.2 0.03761200 8 22.4 0.04371300 8 23.4 0.0510



TABLE 3: Minimum effective conductivity of thefiber/aerogel composites for a fiber volume fraction of 3%

Temperature vf D λmin

(K) (%) (µm) (W·m−1·K−1)300 3 8.3 0.0182400 3 8.3 0.0209500 3 7.4 0.0247600 3 6.4 0.0293700 3 5.4 0.0346800 3 4.4 0.0405900 3 3.6 0.04661000 3 3.1 0.05321100 3 2.7 0.06031200 3 2.4 0.06801300 3 2.2 0.0766

FIG. 9: Minimum effective thermal conductivity of thefiber/aerogel composites via simultaneous or individualoptimization

0.0170 W·m−1·K−1, while it was 0.0182 or0.0172 W·m−1·K−1 by individually optimizing thefiber diameter or fiber volume fraction, respectively,at 300 K. Similarly, the minimum effective thermalconductivity of the fiber/aerogel composites by simulta-neously optimizing the fiber volume fraction and fiberdiameter was 0.0354 W·m−1·K−1, while it was 0.0766or 0.0510 W·m−1·K−1 by individually optimizing thefiber diameter or fiber volume fraction, respectively, at1300 K. It can be deduced that the minimum effectivethermal conductivity of fiber/aerogel composites bysimultaneously optimizing the fiber volume fractionand fiber diameter has better thermal insulation than by

individually optimizing the fiber volume fraction or fiberdiameter.

4. CONCLUSIONS

In this study, the effect of the fiber volume fraction andfiber diameter on the minimum effective thermal conduc-tivity was studied with the aim of optimizing the thermalinsulation of fiber/aerogel composites. The predicted ef-fective thermal conductivity of the fiber/aerogel compos-ites agreed well with previously measured and predicteddata. The predicted minimum effective thermal conduc-tivity of the fiber/aerogel composites by simultaneouslyoptimizing the fiber volume fraction and the fiber diame-ter was significantly lower than that by individually opti-mizing the fiber volume fraction or the fiber diameter. Theoptimal volume fraction of fused silica increased from1.3% at 300 K to 18.2% at 1300 K, whereas the opti-mal diameter decreased from 8.6µm at 300 K to 2.5µmat 1300 K when the temperature was increased. By vali-dation and analysis, we have proposed some quantitativerelations of the minimum effective thermal conductivity,optimal fiber volume fraction, and fiber diameter at vari-ous temperatures.

ACKNOWLEDGMENTS

This research was supported by the Guangdong ProvinceNatural Science Foundation for Distinguished Young Sci-entists, China (Grant No. S2013050014139), and theGuangdong Province Department of Education, China(Research Grant No. 2013KJCX0141).

REFERENCES

Aegerter, M. A., Leventis, N., and Koebel, M. M.,AerogelsHandbook, New York: Springer, 2011.

Baetens, R., Jelle, B. P., and Gustavsen, A., Aerogel insulationfor building applications: A state-of-the-art review,EnergyBuild., vol. 43, no. 4, pp. 761–769, 2011.

Bansal, N. P. and Doremus, R. H.,Handbook of Glass Proper-ties, Orlando, FL: Academic Press, 1986.

Barth, H. G.,Modern Methods of Particle Size Analysis, NewYork: Wiley, 1984.

Bohren, C. F. and Huffman, D. R.,Absorption and Scattering ofLight by Small Particles, New York: Wiley, 1983.

Cunnington, G. R. and Lee, S. C., Radiative properties of fibrousinsulations—Theory versus experiment,J. Thermophys. HeatTransfer, vol. 10, no. 3, pp. 460–466, 1996.


984 Yang et al.

Han, Y. F., Mechanism and characteristics of heat conductionin nanocomposites, Harbin Institute of Technology, 2013 (inChinese).

Incropera, F. and DeWitt, D.,Fundamentals of Heat Transfer,New York: Wiley, 1981.

Kistler, S. S., Coherent expanded-aerogels,J. Phys. Chem., vol.36, no. 1, pp. 52–64, 1932.

Kitamura, R., Pilon, L., and Jonasz, M., Optical constants ofsilica glass from extreme ultraviolet to far infrared at nearroom temperature,Appl. Opt., vol.46, no. 33, pp. 8118–8133,2007.

Koebel, M., Rigacci, A., and Achard, P., Aerogel-based thermalsuperinsulation: An overview,J. Sol-Gel Sci. Technol., vol.63, no. 3, pp. 315–339, 2012.

Lee, S.-C. and Cunnington, G. R., Conduction and radiation heattransfer in high-porosity fiber thermal insulation,J. Thermo-phys. Heat Transfer, vol. 14, no. 2, pp. 121–136, 2000.

Lee, S.-C. and Cunnington, G. R., Fiber orientation effect onradiative heat transfer through fiber composites,Paper No.AIAA-1998-2840, 1998.

Liao, Y., Wu, H., Ding, Y., Yin, S., Wang, M., and Cao, A., En-gineering thermal and mechanical properties of flexible fiber-reinforced aerogel composites,J. Sol-Gel Sci. Technol., vol.63, no. 3, pp. 445–456, 2012.

Martinez-Gomez, R. E. and Soria, A., Effective conductivity forporous media: A Maxwellian approach,J. Porous Media, vol.3, no. 3, pp. 237–243, 2000.

Modest, M. F.,Radiative Heat Transfer, New York: AcademicPress, 2013.

Palik, E. D., Handbook of Optical Constants of Solids, SanDiego, CA: Academic Press, 1998.

Pan, N., Analytical characterization of the anisotropy and localheterogeneity of short fiber composites: Fiber fraction as avariable,J. Compos. Mater., vol. 28, no. 16, pp. 1500–1531,1994.

Papadopoulos, A. M., State of the art in thermal insulation ma-terials and aims for future developments,Energy Build., vol.37, no. 1, pp. 77–86, 2005.

Planck, M., On the law of distribution of energy in the normalspectrum,Ann. Phys., vol. 4, no. pp. 553–558, 1901.

Schmidt, M. and Schwertfeger, F., Applications for silica aero-gel products,J. Non-Cryst. Solids, vol. 225, pp. 364–368,1998.

Shan, X. and Wang, M., Understanding of thermal conductanceof thin gas layers,Adv. Mech. Eng., vol. 2013, pp. 1–7, 2013.

Tong, T. and Tien, C., Radiative heat transfer in fibrousinsulations—Part I: Analytical study,J. Heat Transfer, vol.105, no. 1, pp. 70–75, 1983.

Wang, M., He, J., Yu, J., and Pan, N., Lattice Boltzmann model-ing of the effective thermal conductivity for fibrous materials,

Int. J. Therm. Sci., vol. 46, no. 9, pp. 848–855, 2007.

Wang, M., Kang, Q., and Pan, N., Thermal conductivity en-hancement of carbon fiber composites,Appl. Therm. Eng.,vol. 29, no. 2-3, pp. 418–421, 2009.

Wang, M. and Pan, N., Predictions of effective physical prop-erties of complex multiphase materials,Mater. Sci. Eng., R,vol. 63, no. 1, pp. 1–30, 2008.

Wang, M., Wang, X., Wang, J., and Pan, N., Grain size effectson effective thermal conductivity of porous materials with in-ternal thermal contact resistance,J. Porous Media, vol. 16,no. 11, pp. 1043–1048, 2013.

Wei, G., Liu, Y., Zhang, X., and Du, X., Radiative heat transferstudy on silica aerogel and its composite insulation materials,J. Non-Cryst. Solids, vol. 362, pp. 231–236, 2013.

Wei, G., Liu, Y., Zhang, X., Yu, F., and Du, X., Thermal con-ductivities study on silica aerogel and its composite insula-tion materials,Int. J. Heat Mass Transfer, vol. 54, no. 11-12,pp. 2355–2366, 2011.

Wu, H., Liao, Y., Ding, Y., Wang, H., Peng, C., and Yin, S.,Engineering thermal and mechanical properties of multilayeraligned fiber-reinforced aerogel composites,Heat TransferEng., vol. 35, no. 11-12, pp. 1061–1070, 2014.

Xie, T., He, Y.-L., and Hu, Z.-J., Theoretical study on thermalconductivities of silica aerogel composite insulating material,Int. J. Heat Mass Transfer, vol. 58, no. 1-2, pp. 540–552,2013.

Zeng, J., Greif, R., Stevens, P., Ayers, M., and Hunt, A., Ef-fective optical constantsn andκ and extinction coefficientof silica aerogel,J. Mater. Res., vol. 11, no. 3, pp. 687–693,1996.

Zeng, S., Hunt, A., and Greif, R., Mean free path and apparentthermal conductivity of a gas in a porous medium,J. HeatTransfer, vol. 117, no. 3, pp. 758–761, 1995a.

Zeng, S., Hunt, A., and Greif, R., Theoretical modeling of car-bon content to minimize heat transfer in silica aerogel,J.Non-Cryst. Solids, vol. 186, pp. 271–277, 1995b.

Zhao, J.-J., Duan, Y.-Y., Wang, X.-D., and Wang, B.-X., An an-alytical model for combined radiative and conductive heattransfer in fiber-loaded silica aerogels,J. Non-Cryst. Solids,vol. 358, no. 10, pp. 1303–1312, 2012a.

Zhao, J.-J., Duan, Y.-Y., Wang, X.-D., and Wang, B.-X., Com-bined radiation and conduction heat transfer in nanocompos-ite insulation materials,J. Eng. Thermophys., vol. 33, no. 12,pp. 2185–2189, 2012b.

Zhao, J.-J., Duan, Y.-Y., Wang, X.-D., and Wang, B.-X., Radia-tive properties and heat transfer characteristics of fiber-loadedsilica aerogel composites for thermal insulation,Int. J. HeatMass Transfer, vol. 55, no. 19-20, pp. 5196–5204, 2012c.

Zhao, J.-Q. and Hu, Y.-Q., Bridging technique for calculatingthe extinction efficiency of arbitrary shaped particles,Appl.Opt., vol. 42, no. 24, pp. 4937–4945, 2003.


prediction of thermal conductivity of fiber/aerogel ...4)-13550.pdf · were added to the silica sol...

Documents