Optical and radiative properties of infrared opacifier particles loaded in silica aerogels for high temperature thermal insulation

International Journal of Thermal Sciences 70 (2013) 54e64

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Optical and radiative properties of infrared opacifier particles loaded in silica aerogels for high temperature thermal insulation Jun-Jie Zhao a, b, Yuan-Yuan Duan a, *, Xiao-Dong Wang c, **, Xue-Ren Zhang b, Yun-He Han b, Ya-Bin Gao b, Zhen-Hua Lv b, Hai-Tong Yu a, Bu-Xuan Wang a a b c

Key Laboratory of Thermal Science and Power Engineering of MOE, Beijing Key Laboratory for CO2 Utilization and Reduction Technology, Tsinghua University, Beijing 100084, China Guodian Power Development Company Limited, China Guodian Corporation, Beijing 100101, China State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 July 2012 Received in revised form 17 March 2013 Accepted 18 March 2013 Available online 21 April 2013

An inverse model based on the shooting method, Mie theory and the improved KramerseKronig (KK) relation was combined with FTIR and Abbe refractometer measurements to calculate the complex refractive indices of various infrared opacifiers. The effects of opacifier sizes, types and shapes were then analyzed based on the Rosseland mean extinction coefficient using Mie theory and anomalous diffraction theory (ADT). This model provides theoretical guidelines for designing materials with optimized parameters, such as size, type and shape of opacifiers, to improve the aerogel thermal insulation at high temperatures. The results show that the optimum diameter of SiC particles to minimize the radiation is 4 mm for T < 400 K and 3 mm for T > 400 K. Carbon black is the optimum opacifier for T < 600 K while SiC is the optimum opacifier to minimize the radiative heat transfer for T > 600 K among the investigated opacifiers of SiC, TiO2, ZrO2, amorphous SiO2 and carbon black. The infrared extinction ability for various shapes is largest for oblate spheroids and decreases for spheres, cubes, cylinders with small length-todiameter ratios, and then long, thin cylinders. Ó 2013 Elsevier Masson SAS. All rights reserved.

Keywords: Silica aerogel Opacifier Complex refractive index Extinction coefficient Radiative thermal conductivity High temperature thermal insulation

1. Introduction Silica aerogels are low-density nanoporous amorphous materials prepared by the solegel technique and supercritical drying [1e8]. Fig. 1(a) shows that silica aerogels consist of interconnected secondary nanoparticles in an open-cell pearl-necklace network with a mean pore size of 10e50 nm [9e18]. The aerogels have great potential in thermal insulation as they have much lower thermal conductivities than all other thermal insulation materials at ambient conditions [19e26]. The heat transfer modeling in super insulating materials is important to estimate the contributions of various heat transfer modes at various temperatures as well as to provide design strategies to minimize the thermal conductivity [26e38]. Bouquerel et al. [38] presented a complete review on heat transfer mechanisms in aerogels which include radiative transfer, solid conduction, gaseous conduction, and solidegas thermal coupling.

* Corresponding author. Tel./fax: þ86 10 6279 6318. ** Corresponding author. Tel./fax: þ86 10 6232 1277. E-mail addresses: [email protected] (Y.-Y. Duan), [email protected] (X.-D. Wang). 1290-0729/$ e see front matter Ó 2013 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.ijthermalsci.2013.03.020

Since the extinction coefficient of pure silica aerogels is extremely small for infrared wavelengths of 3e8 mm, the total thermal conductivity of aerogels dramatically increases with the temperature at higher temperatures [22e27]. Mineral powders, such as carbon black, TiO2 and SiC, are usually added into the aerogels as infrared opacifiers to reduce the radiative heat transfer and the total thermal conductivity at high temperatures [1e5,16]. Thus, the high-temperature thermal insulating properties can be improved by maximizing the extinction coefficients of the opacifiers [4,11,16]. Since the size, type and shape of opacifiers may significantly affect the extinction coefficient of the opacifier-loaded silica aerogel composite, these parameters need to be optimized to minimize the radiative heat transport [7,23]. The spectral extinction coefficient of an opacified aerogel sample can be obtained using Beer’s law based on the transmittance spectrum measured by a Fourier transform infrared (FTIR) spectrometer [19,24e27]. However, the experimental determinations of the opacifier extinction coefficient can only provide data for specific opacifier-loaded samples with specific opacifier compositions, sizes and morphologies [17e19]. Mie scattering theory can be used to calculate the extinction coefficients of opacifiers based on the complex refractive index

J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64

Nomenclature A absorbance a polar radius, m Mie scattering coefficient aj, bj b equatorial radius, m C1, C2, c1, c2 parameters d particle diameter, m e mass specific extinction coefficient, m2 kg1 Eb integral of the spectral blackbody emissive power, W m2 Ebl spectral blackbody emissive power, W m2 mm1 fv opacifier volume fraction g structure factor I radiation intensity, W m2 j1, j2, j3 parameters in Eq. (11) J first kind of cylindrical Bessel function k thermal conductivity, W m1 K1 l cylinder length, m L insulation thickness, m m complex refractive index (m ¼ n  ik) n real part of complex refractive index N number density of particles OF objective function q distance of the incidence point on the spheroid surface from its center Q extinction efficiency S area, m2 T temperature, K w opacifier mass fraction

[23,27,38]. Thus, the effects of the opacifier type and size on the radiative properties can be theoretically investigated assuming that the opacifiers are spherical and uniformly dispersed. However, optimized results cannot be provided due to the lack of spectral data of the complex refractive indices in the infrared region for various opacifiers [23,27]. Besides, to the best of the authors’ knowledge, there is limited information available on the fundamental relationships between the radiative properties and the shapes of opacifiers. Therefore, this paper improves the inverse model to obtain the complex refractive indices of various opacifiers based on FTIR and Abbe refractometer measurements. The inverse model is solved using a fast shooting method based on Mie theory and improved KramerseKronig (KK) relation while other studies [28,29] use one dimensional search method based on parabola fitting. The type and size of spherical opacifiers are then theoretically optimized using Mie theory to maximize the extinction coefficient. The opacifier shapes, which include spheres, spheroids, cubes and cylinders, are investigated using the modified anomalous diffraction theory (ADT) to show the shape effects on the opacifier radiative properties. This model provides theoretical guidelines for material insulation designs with optimized parameters of the opacifier size, type and shape. 2. Preparation and experiment After the sol was prepared using a tetraethylorthosilane (TEOS)ewatereethanol system, the opacifiers were well dispersed in the sol with an aqueous ammonia solution to base-catalyze the opacifieresol into the opacifieregel composite. The opacifierloaded aerogel composites shown in Fig. 1(b) were then prepared based on this two-step solegel process and subsequent

Y

55

second kind of cylindrical Bessel function

Greek symbols size parameter (a ¼ pd/l) extinction coefficient, m1 parametric angle difference incident angle orientation angle imaginary part of complex refractive index wavelength, m density, kg m3 phase delay StefaneBoltzmann constant (5.67  108 W m2 K4) transmittance RicattieHankel function RicattieBessel function wavelength, m

a b c D q g k l r r* s s xj jj j

Subscripts a aerogel ad ADT edge edge ext extinction opac opacifier r radiation ref reference total total l spectral

supercritical drying. Three common types of infrared opacifiers, SiC, TiO2 and ZrO2, were investigated in this study with the properties listed in Table 1. A Nicolet 6700 Fourier transform infrared (FTIR) spectrometer was used to measure the spectral absorbance of the opacifierloaded potassium bromide (KBr) powder composites, Al. FTIR spectra were obtained at a resolution of 4 cm1 over a spectral range of 400e4000 cm1 or 2.5e25 mm. As the KBr powder is transparent to infrared radiation [6], a thin circular KBr tablet was used as a support layer with a thickness of 1 mm and a diameter of 13 mm. The tablets were prepared by compacting and pressing an intimate mixture obtained by grinding 1 mg of opacifiers in 100 mg KBr. The spectral transmittance, sl, for wavelengths of 2.5e25 mm can be obtained by:

sl ¼ I=I0 ¼ 10Al

(1)

where I is the transmitted intensity and I0 is the incident intensity. The definitions of the spectral absorbance, Al, and extinction coefficient, bl, include the effects of both absorption and scattering. bl or the mass specific spectral extinction coefficient, el (m2 kg1), can be expressed using Beer’s law as [6,23,27e30]:

bl ¼ el $r ¼ lnðsl Þ=L

(2)

where L is the sample thickness and r is the density. Beer’s law or Eq. (2) is valid when no multiple scattering effects occur and the scattering is isotropic [4,39]. As the volume fraction of the opacifiers is usually much less than 5%, the multiple scattering effects are negligible and the scattering of the opacifiers can be assumed to be independent. As the opacifiers are all spherical and well dispersed with a uniform diameter d in this study, the

56

J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64 Table 1 Material types and properties. Type

Density, r/kg m3

Average particle diameter, d/mm

Thermal conductivity at ambient conditions, k/W m1 K1

SiC TiO2 (rutile type) ZrO2 Carbon black Amorphous SiO2 Silica aerogel

3100 4260 5890 1450 2200 110

2.5 2 2 / / /

83.6 4.26 2.0 4.18 1.38 0.018

study, Qext of a scattering spherical particle was calculated using Mie theory [31]:

2

Q ext ¼

aj ¼

bj ¼

3 N  X   Re4 2j þ 1 aj þ bj 5 2

2

a

(4)

j¼1

mjj ðmaÞj0j ðaÞ  j0j ðmaÞjj ðaÞ

(5)

0

mjj ðmaÞxj ðaÞ  j0j ðmaÞxj ðaÞ

jj ðmaÞj0j ðaÞ  mj0j ðmaÞjj ðaÞ jj ðmaÞx0j ðaÞ  mj0j ðmaÞxj ðaÞ

(6)

where aj and bj are the Mie scattering coefficients, jj is the Ricattie Bessel function and xj is the RicattieHankel function [31]:

rffiffiffiffiffiffiffi

jj ðaÞ ¼

pa 2

J jþ

1 ðaÞ 2

rffiffiffiffiffiffiffi

xj ðaÞ ¼ Fig. 1. SEM photograph: (a) pure silica aerogel; (b) SiC-loaded silica aerogel.

scattering can be assumed to be isotropic. The spectral absorbance, Al, can be expressed as a function of the optical thickness of the sample, blL, so Al ¼ 0.4343blL based on Eqs. (1) and (2). Assuming isotropic scattering, the spectral extinction efficiency, Qext, can be expressed as [4,28]:

Qext ¼ 2dbl =ð3fv Þ

(3)

where fv is the volume fraction of the opacifiers. The distributions of particle diameters were measured using the laser particle size analyzer (Marvin Mastersizer 2000). The mean particle diameter, d, is then obtained by weighing the volume fraction of the corresponding particle diameter. As the opacifier particles have relatively uniform sizes, d can be used in Mie scattering calculations here. 3. Inverse model As the effects of scattering and absorption are both included here, Mie theory is used to describe the radiative properties of opacifiers based on m and a. a ¼ pd/l is the size parameter at wavelength l and m ¼ n  ik is the complex refractive index where the real part, n, is the refractive index and the imaginary part, k, is the absorptive index. If the scattering was neglected for a pure absorbing medium, the measured Qext or bl can be directly related to k (bl ¼ 4pk/l) with a much simpler model to calculate m. In this

rffiffiffiffiffiffiffi

pa 2

(7)

J jþ

1 ðaÞ þ i 2

pa 2

Y jþ

1 ðaÞ 2

(8)

where J is the first kind of cylindrical Bessel function and Y is the second kind of cylindrical Bessel function. J and Y can be calculated using the Bessel functions in the software MATLAB. The complex refractive index, m, has two variables, the refractive index n and the absorptive index k. As Mie theory can provide only one equation for m [29], another relationship between n and k is needed and provided by the KK relation [28e 30,32]:

nðlÞ ¼ 1 þ

2

2l

p

ZN 0

kðjÞ  dj j l2  j2 

(9)

As the classic KK relation in Eq. (9) tends to introduce some errors caused by the finite measured spectrum, the subtractive KK relation is more accurate [29,32]:

   2 l2ref  l2 ZN   nðlÞ ¼ n lref þ

p

0

jkðjÞ   dj l2  j2 l2ref  j2 (10)

where the reference wavelength, lref, is at the triple blue mercury line, so lref ¼ 0.4358 mm. The monochromatic blue light at the wavelength of 0.4358 mm is one of the three primary colors of light and, thus, can be accurately generated and identified. The refractive

J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64

57

Fig. 2. Inverse method used to calculate the opacifier complex refractive index.

index n(lref) was measured using a Hilger Watts Abbe refractometer [32]. In practical applications, the experimental data are usually measured over a finite spectral range of [lmin, lmax] with lmin ¼ 2.5 mm and lmax ¼ 25 mm here which corresponds to the measuring range of the FTIR spectrometer. The subtractive KK relation in Eq. (10) using the full wavelength integral needs to be extrapolated in both the shorter (0, lmin) and longer (lmax, N) spectral ranges. Thus, Eq. (10) can be rewritten as:





nðlÞ ¼ n lref þ j1 þ j2 þ j3

j1 ¼

j2 ¼

  2 2 Zlmin 2 lref  l

p



0

lmin

j1 ðlÞ ¼ p 8   l þ l  < l3ref   min ref  ln  lmin þ   2 2 lref  lmin  : 2 l  lref 9   l þ lmin = l3   ln   2 2 l  lmin ; 2 l l

jkðjÞ  dj l j l2ref  j2 2

2

For the measured spectral range of [lmin, lmax], Eq. (13) can be rewritten as:

2

jkðjÞ    dj 2 2 l j l2ref  j2

j3 ¼

p

lmax

p

j ¼ lmin

2l (13)



jkðjÞ  Dj l j l2ref  j2 2

2

(17)

The point at j ¼ l is a singular point. The limit for the integral at the singular point can be obtained using the Hilbert transform as:

p ¼

  2 2 ZN 2 lref  l

  2 2 lmax 2 lref  l X jsl

(12)

(16)

ref

j2 ðlÞ ¼

  lmax 2 2 Z 2 lref  l

p

(11)

  2 2 2C1 lref  l

lZþDl lDl

2l

2

p

2 kðjÞdj 2l   ref p j l2  j2





lZþDl lDl

kðjÞdj  j l2ref  j2 

kðl þ DlÞ kðl  DlÞ  ðl þ DlÞð2l þ DlÞ ðl  DlÞð2l  DlÞ



2

jkðjÞ  dj  2 2 l j l2ref  j2

(14)

where j1 is the extrapolation for the shorter wavelength limit, j2 is for the measured spectral range and j3 is the extrapolation for the longer wavelength limit. Neglecting the shorter wavelength modification, j1 in Eq. (11), tends to underestimate the spectral refractive index n up to 20% at shorter wavelengths for carbon black used as the standard material. The extrapolation of the shorter wavelength limit (0 < j < lmin) can be introduced using the dispersion theory as [28,29]:

 

k j ¼ C1 $j3 and C1 ¼ kðlmin Þ=l3min Combining with Eq. (15), Eq. (12) can be rewritten as:

(15)

2kðlÞDl  $  l l2ref  l2 2 3 2 l 4 ref 5 kðlÞDl ¼ 43 þ  2 pl lref  l2 

2lref

p

(18)

where Dl is a micro-interval. Neglecting the longer wavelength modification, j3 in Eq. (11), tends to overestimate the spectral refractive index n up to 30% at higher wavelengths for coal ash used as the standard material. The extrapolation of the longer wavelength limit (lmax < j < N) can be also introduced using the dispersion theory as [28,29]:

kðjÞ ¼ C2 =j and C2 ¼ kðlmax Þ$lmax

(19)

The extrapolations of the shorter and longer wavelength limits in Eqs. (15) and (19) using the dispersion theory can only be applied to

58

J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64

dispersive materials which have wavelength-dependent refractive and absorptive indices. Combining with Eq. (19), Eq. (14) can be rewritten as:

j3 ðlÞ ¼

2C2

p

"

1 2lref

   # l þ l  1 l þ lmax   ref max  ln   ln lref  lmax  2l l  lmax 

(20)

The expressions of the subtractive KK relation in Eqs. (11)e(20) are different from those in Ref. [29] as lref ¼ 0.4358 mm < lmin ¼ 2.5 mm here and lmin < lref < lmax in Ref. [29]. Besides, two singular points at j ¼ l and lref are shown in Ref. [29] while only one singular point at j ¼ l is shown in this model. Thus, the new developed equations of the subtractive KK relation in this study are more explicit and easier to solve than those in Ref. [29]. Fig. 2 shows the methodology to calculate the spectral complex refractive index, ml ¼ nl  ikl, for the spectral range of [lmin, lmax]. This approach uses a shooting method [40] based on Mie theory and the improved subtractive KK relation. Two groups of values for the imaginary part of the complex refractive index, kl, are first specified as the initial condition. The real part of the complex refractive index, nl, and the spectral extinction efficiency, Qext(l), can be expressed as a function of kl using Eqs. (4) and (11). One group of values for kl is set to give larger Qext(l) than Qext,exp(l) while the other group of values for kl is set to give lower Qext(l) than Qext,exp(l). The objective function is OF(l) ¼ jQext(l)  Qext,exp(l)j/Qext,exp(l), where Qext,exp(l) is the measured spectral extinction efficiency. OF(l) as the acceptable calculation accuracy at each wavelength is set to be 103 in this study. The two groups of values for kl are iteratively revised at each wavelength using the bisection method to shoot the target OF(l) ¼ 103 at each wavelength [40]. The average uncertainty of the identified complex refractive indices is estimated to be about 0.4% for carbon black with OF(l) ¼ 103. As Qext,exp is the only input parameter and m is the only output parameter, the parameter sensibility analysis is based on Qext,exp and m. Assuming the oscillation of Qext,exp to be 1% tends to cause a change of 8.3% for m. Thus, Qext,exp is a sensitive parameter to inversely calculate m. As the stability and repeatability of Qext,exp by FTIR measurements are good, the parameter sensibility effects of Qext,exp on m can be well reduced with acceptable precisions. The analysis of the uniqueness of solution is similar to discussions given by Ruan et al. [29]. When 0  k  (2 þ 2.4a)0.5 and 102 < a < 2, the uniqueness range of the complex refractive indices can be ensured, so m is unique based on the present inverse model. For l ¼ 3.2e 25 mm and d ¼ 2 mm, 0  k  (2 þ 2.4a)0.5 and 102 < a < 2 can be strictly satisfied, so m can be ensured to be unique.

" Q ext;spheroid ¼ 4Re

   # exp ir* 1  exp ir* 1 þ i 2 r* r*2

where Re represents the real part of a complex quantity and r* is the phase delay: *

r ¼

  4paspheroid m  1

lg

The shape significantly affects the radiative properties and the extinction performance of opacifiers. Classic Mie scattering theory is only applicable to spherical particles. For nonspherical particles, the T-matrix technique, the discrete dipole approximation and the method of moments can be used to calculate the radiative properties [41e43]. However, these methods are all complex, timeconsuming and do not converge well [42]. Thus, anomalous diffraction theory (ADT), which provides a much simpler approximation, can be used to predict the radiative properties of arbitrarily shaped particles within some acceptable degree of accuracy [44]. Spherical, spheroidal, cubical and cylindrical particles were investigated using ADT to show the effects of shapes on the opacifier radiative properties in this study. The extinction efficiency for a spheroid without considering the edge effect, Qad,spheroid, can be expressed by ADT as [42,44]:

and g ¼

cos q þ 2

a2spheroid b2spheroid

!0:5 $sin q 2

(22) where aspheroid is the polar radius, bspheroid is the equatorial radius and q is the incident angle between the vertical symmetry axis and the direction of the incident radiation. The structure factor, g, is defined as g ¼ q/bspheroid where q is distance of the incidence point on the spheroid surface from its center. Prolate spheroids have aspheroid/bspheroid > 1 while oblate spheroids have aspheroid/ bspheroid < 1. When aspheroid/bspheroid ¼ 1, the spheroids become spheres. For spheroids, the edge effect on the extinction efficiency, Qedge,spheroid, is [42]: 2=3

Qedge;spheroid ¼

2=3

4c1 aspheroid =bspheroid  2=3 p 2pbspheroid =l g2 

Zp=2 

4=3

sin2 c þ g 2 cos2 c

(23) dc

0

where c1 ¼ 0.996193 and c is the parametric angle. Considering the edge effect, the extinction efficiency for a spheroidal particle can then be modified from Qad,spheroid into Qext,spheroid [35,42]:

Qext;spheroid ¼ Q ad;spheroid þ

Q ad;spheroid h  i 2=Q edge;spheroid þ 1= c2 Q ad;spheroid þ 1 (24)

where c2 ¼ jm  1j. For cubical particles, the edge effect, Qedge,cubic, and the extinction efficiency without the edge effect, Qad,cubic, can also be combined using Eq. (24) to give the extinction efficiency with the edge effect:

" 4. Anomalous diffraction theory (ADT)

(21)



*

Qad;cubic ¼ Re 2  exp ir



þi

 # 1  exp ir*

(25)

r*

Qedge;cubic ¼ 2c1 =a2=3

(26)

where r* ¼ 8a(m  1)/3 is the phase delay for a cubical particle. For cylindrical particles, the edge effect, Qedge,cylinder, given in Ref. [42] and Qad,cylinder [34,35] can also be combined using Eq. (24) as:

Q ad;cylinder ¼ 2Re

8 p=2 > :

0



1  exp ir* $cos g

i

$cos g dg

9 > = > ;

(27)

where r* ¼ 2(m  1)a/sin q is the phase delay and the orientation angle, g, is the angle that describes the difference of the angle of the incident radiation from the cylinder axis.

J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64 Table 2 Spectral extinction coefficient, bl, expressed as a function of the spectral extinction efficiency, Qext, for various particle shapes.

bl as a function of Qext

Sphere [28] Spheroid Cube Cylinder [34,35]

bl bl bl bl

¼ ¼ ¼ ¼

3.0 TiO2

3fv Q ext =ð2dÞ 3fv Q ext =ð4bspheroid Þ fv Q ext =d 4fv Q ext =ðpdÞ

2.8

For all kinds of particle shapes, the spectral extinction coefficient, bl, can be expressed based on the spectral extinction efficiency, Q ext, as:

bl ¼ Sproject $Q ext $N

Real Part n

Shape

a

59

SiC

2.6 2.4

ZrO2

2.2

(28)

where Sproject is the head-on projected area and N is the number density of particles. For spherical, spheroidal, cubical and cylindrical particles per unit volume of the composite, bl can be expressed as a function of Qext as listed in Table 2 based on Eq. (28).

2.0

b 5. Results and discussion

0

5

10 15 Wavelength / m

SiC

Specific Extinction Coefficient e

2

m kg

SiC, dopac = 2.5 m

60

40 ZrO2, dopac = 2 m

Imaginary Part

0.15

80

25

0.20

5.1. Complex refractive index The properties of the investigated opacifiers of SiC, TiO2 and ZrO2 are listed in Table 1. Assuming that the opacifiers are spherical and uniformly dispersed in KBr powders, the mass specific extinction coefficients, el, of SiC, TiO2 and ZrO2 with volume fractions of 5e6% can be obtained based on FTIR measurements and Eqs. (1) and (2) as shown in Fig. 3. Fig. 3 shows that the measured el of SiC, TiO2 and ZrO2 vary significantly with the wavelength, especially in the spectral range of 2.5e15 mm. Thus, the extinction coefficients of opacifiers are dependent on the wavelength and temperature and, thus, cannot be simplified as constants [7]. The complex refractive indices, m ¼ n  ik, of SiC, TiO2 and ZrO2 shown in Fig. 4(a) and (b) were calculated based on the inverse method (Fig. 2) as well as the FTIR and Abbe refractometer measurements. Fig. 4(a) shows that the refractive indices, n, of these three opacifiers are in the range of 2e3. n of TiO2 is larger than n of SiC which is larger than n of ZrO2. Fig. 4(b) shows that the absorptive indices, k, of these three opacifiers are in the range of 0e 0.2. Thus, SiC, TiO2 and ZrO2 are strongly refractive opacifiers with small absorptive capabilities. The average uncertainty of the

20

ZrO2

0.10 TiO2

0.05

0.00 0

5

10

15

20

25

Wavelength / m Fig. 4. Complex refractive indices of opacifiers calculated based on the inverse model and experimental data: (a) real part, n; (b) imaginary part, k.

identified complex refractive indices is estimated to be about 4.2% in this study when the uncertainty of Qext,exp is estimated to be about 5  103 with OF(l) ¼ 103. Fig. 5 shows that carbon black is an ideal opacifier as carbon black has both strong refractive and absorptive capabilities due to the very large refractive index, n, and absorptive index, k. However, carbon black tends to oxidize in air atmosphere at temperatures larger than 600 K. Thus, SiC, TiO2 and ZrO2 are more suitable opacifiers loaded in aerogels to reduce the radiative heat transfer at higher temperatures. 5.2. Diameter effect

20

TiO2, dopac = 2 m

0 0

5

10 15 Wavelength / m

20

25

Fig. 3. Measured mass specific spectral extinction coefficient of opacifiers.

Fig. 6(a) shows that the spectral extinction coefficients of SiC with fv ¼ 3%, bl, vary significantly with the opacifier diameter, d. The maximum bl for 2.5e25 mm decreases with increasing the opacifier diameter while the wavelength corresponding to the maximum bl increases with the opacifier diameter. The optimum diameter for maximizing the opacifier extinction coefficient at various temperatures cannot be identified from Fig. 6(a). Thus, the Rosseland mean extinction coefficient, b, is defined to describe the

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J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64

Spectral Extinction Coefficient

/m

-1

a

/m Rosseland Mean Extinction Coefficient

Carbon black

Amorphous SiO2

4

10

SiC fv = 3% 3

d=3 m

10

Silica aerogel [2] -3 = 110 kg m a

2

10

0

b -1

Fig. 5. Complex refractive indices of carbon black and amorphous SiO2 [11].

ZrO2

5

10

5

TiO2

10 15 Wavelength / m

20

25

4

5x10

fv = 3%

4

4x10

d=3 m Carbon black

4

3x10

SiC 4

2x10

TiO2 Amorphous SiO2

ZrO2

4

1x10

Silica aerogel [33]

-3

a

= 110 kg m

0 300

400

500 600 Temperature

700

800

900

/

Fig. 7. Effects of the opacifier type on the radiative properties: (a) spectral extinction coefficient; (b) Rosseland mean extinction coefficient.

temperature-dependent average effect of the opacifier extinction coefficients [6,11,33e35]:

2

b¼ 4

ZN 0

Fig. 6. Diameter effect on the radiative properties of SiC: (a) spectral extinction coefficient; (b) Rosseland mean extinction coefficient.

31

1 vEbl 5 $ dl

bl vEb

2 6 z4

lmax Z lmin

31 1 vEbl 7 $ dl5

bl vEb

(29)

where Ebl is the spectral blackbody emissive power calculated using Planck’s law and Eb is the integral of the spectral blackbody emissive power. As the uncertainty of Qext is 0.1% in this study, the uncertainty of the derived extinction coefficient, b, is then 0.1% based on Eqs. (28) and (29) using the error transfer mechanism for independent variables. The studied spectral range is 2.5e25 mm while the errors from the lack of spectral data in the range of 1e2.4 mm are very small at low temperatures, but increase with the temperature. To reduce the errors within 10% due to the lack of data in the spectral range of 1e2.4 mm, the maximum temperature in this study should be no more than 900 K. Fig. 6(b) shows that the Rosseland mean extinction coefficient, b, of SiC with fv ¼ 3% varies significantly with the temperature, T, and the opacifier diameter, d. b for d ¼ 1 and 2 mm continuously increase with the temperature while b for

J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64

d ¼ 4e6 mm continuously decrease with increasing temperature. b for d ¼ 3 mm reaches a maximum at T ¼ 500 K and then decreases with increasing temperature for T > 500 K. Fig. 6(b) shows that the maximum b for SiC is at a diameter of 4 mm for T < 400 K and at a diameter of 3 mm for T > 400 K.

5.3. Effects of opacifier types Figs. 4 and 5 show that the complex refractive index, m, varies significantly with the material type. The Mie theory in Eq. (4) shows that the radiative properties depend on the complex refractive index. Thus, the radiative properties will vary significantly with the material type as shown in Fig. 7. Fig. 7(a) shows that the spectral extinction coefficients, bl, of SiC, TiO2, ZrO2 and carbon black calculated by the present model are 2e3 orders of magnitude larger than the measured bl of pure silica aerogels [2] for wavelengths of 3e8 mm. bl of SiC, TiO2 and ZrO2 are larger than bl of carbon black for wavelengths of 2e8 mm. bl of SiC, TiO2 and ZrO2 are much smaller than bl of carbon black for l > 8 mm. Fig. 7(b) shows that the Rosseland mean extinction coefficients, b, of SiC, TiO2, ZrO2, carbon black, amorphous SiO2 and silica aerogels all significantly vary with the temperature and cannot be

assumed to be constants. b of SiC, TiO2, ZrO2, carbon black and amorphous SiO2 increase with the temperature while b of silica aerogels decreases with increasing temperature. b of silica aerogels is much smaller than b of SiC, TiO2, ZrO2, carbon black and amorphous SiO2. For fv ¼ 3% and d ¼ 3 mm, b of SiC, TiO2 and ZrO2 are much smaller than b of carbon black, but are much larger than b of amorphous SiO2. Carbon black can be used as an ideal opacifier loaded in aerogels to minimize the radiative heat transfer for T < 600 K but not for T > 600 K due to the oxidization [3,7]. The radiative thermal conductivity, kr, through an optically thick sample can be expressed as [3,19,23,34]:

kr ¼ 16n2total sT 3 =ð3btotal Þ

(30)

where s ¼ 5.67  108 W m2 K4 is the StefaneBoltzmann constant, ntotal is the effective refractive index of the opacifier-loaded aerogel composite and btotal is the Rosseland mean extinction coefficient of the composite. As the uncertainty of b is 0.1% and the uncertainty of ntotal is 4.2%, the uncertainty of kr is then about 8.7% based on Eq. (30) using the error transfer mechanism for independent variables. ntotal can be expressed based on a volume averaging of the temperature-dependent Rosseland mean refractive indices of the aerogel, na, and the opacifier, nopac [34]:

fv = 3%

a

Pure silica aerogel

-1

d=3 m -3

= 110 kg m a

0.1

Amorphous SiO2-loaded aerogel

0.01 SiC-loaded aerogel TiO2-loaded aerogel

1E-3 Carbon black-loaded aerogel

1E-4

300

400

ZrO2-loaded aerogel

500 600 700 Temperature /

800

900

-1

/m

d=3 m

Pure silica aerogel -3

a

= 110 kg m

TiO2-loaded aerogel

0.1

SiC-loaded aerogel

0.01 ZrO2-loaded aerogel Amorphous SiO2-loaded aerogel

1E-3

1E-4

Carbon black-loaded aerogel

300

400

500 600 Temperature

700

800

900

/

Fig. 8. Effects of the opacifier type on the radiative thermal conductivity: (a) same opacifier volume fraction; (b) same opacifier mass fraction.

Rosseland Mean Extinction Coefficient

-1

Radiative Thermal Conductivity kr / W m K

-1

b w = 30%

Mie theory, d = 4 m

SiC fv = 3%

4

ADT, d = 4 m

6.0x10

ADT, d = 6 m

4

4.0x10

4

2.0x10

Mie theory, d = 6 m

0.0 0

b 1

4

8.0x10

/m

1

Spectral Extinction Coefficient

-1

Radiative Thermal Conductivity kr / W m K

-1

a

61

5

10 15 Wavelength / m

20

25

4

6x10

4

SiC fv = 3%

ADT, d = 4 m

5x10

Mie theory, d = 4 m 4

4x10

4

3x10

ADT, d = 6 m

4

2x10

Mie theory, d = 6 m 4

1x10

300

400

500 600 700 Temperature /

800

900

Fig. 9. Comparisons between Mie scattering theory and ADT for spherical particles: (a) spectral extinction coefficient; (b) Rosseland mean extinction coefficient.

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J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64

5

1.0x10

SiC

/m

-1

a

fv = 3%

Spectral Extinction Coefficient

4

8.0x10

d=4 m Sphere bspheroid= 2 m

4

4

4.0x10

Spheroid, aspheroid/bspheroid= 2

4

2.0x10

Cylinder, l/d = 50

0.0

-1

b /m

Cube

= 90°

6.0x10

0

Rosseland Mean Extinction Coefficient

Spheroid, aspheroid/bspheroid= 0.5

5

10 15 Wavelength / m

20

25

4

6x10

4

5x10

4

4x10

SiC fv = 3% bspheroid= 2 m

d=4 m Sphere

Spheroid, = 90° aspheroid/bspheroid= 0.5

Cube Spheroid, = 90° aspheroid/bspheroid= 2

4

3x10

4

2x10

Cylinder, l/d = 5, = 50° 4

1x10

300

400

Cylinder, l/d = 50, = 90° Cylinder, l/d = 5, = 90° Cylinder, l/d = 5, = 70°

500 600 700 Temperature /

800

900

Fig. 10. Shape effect on the radiative properties of SiC: (a) spectral extinction coefficient; (b) Rosseland mean extinction coefficient.

ntotal ¼ fv nopac þ ð1  fv Þna

(31)

where na and nopac are defined as:

2 na ¼ 4

ZN 0

2 nopac

¼ 4

31 1 vEbl 5 $ dl na;l vEb ZN 0

31 vEbl 5 $ dl nopac;l vEb 1

(32)

(33)

where the spectral refractive indices of silica aerogels, na,l, are given in Ref. [25]. The spectral refractive indices of opacifiers, nopac,l, are shown in Fig. 4(a).

As the opacifier volume fraction in the composite is usually less than 5% and the scattering by aerogels is negligible in the infrared region [11,14,25], the scattering can be assumed to be independent in the aerogel-based composite with well dispersed opacifiers. Thus, the total extinction coefficient of the opacifier-loaded aerogel composite, btotal, can be calculated from the components’ individual contributions [34]:

btotal ¼ bopac þ ba ð1  fv Þ

(34)

where bopac is the opacifier extinction coefficient with an opacifier volume fraction of fv and ba is the aerogel extinction coefficient. Fig. 8 shows that the radiative thermal conductivity of the pure silica aerogel, kr, increases dramatically with the temperature, from 0.002 W m1 K1 at 300 K to about 0.68 W m1 K1 at 900 K.

J.-J. Zhao et al. / International Journal of Thermal Sciences 70 (2013) 54e64

Fig. 8(a) shows that the opacifiers of SiC, TiO2, ZrO2 and carbon black loaded in silica aerogels with fv ¼ 3% and d ¼ 3 mm can reduce kr by more than 2 orders of magnitude at high temperatures relative to the pure silica aerogel. Carbon black is the best infrared opacifier with the lowest radiative thermal conductivity for T < 600 K as carbon black tends to be oxidized for T > 600 K. In practical applications, the opacifiers are usually loaded into the aerogels for a given mass fraction. However, as the extinction of radiations is a volume effect, the extinction performance of the composite is directly related to the opacifier volume fraction. Thus, the opacifier density, ropac, can affect the extinction performance for various opacifier types with the same opacifier mass fraction as ropac varies greatly with the opacifier type as shown in Table 1. The opacifier volume fraction in the composite, fv, can be expressed as a function of the opacifier mass fraction, w:

  h i fv ¼ ra w= ropac 1  w þ ra w

(35)

where ra is the aerogel density. Since the density of SiC is much than those of TiO2 and ZrO2 as shown in Table 1, the volume fraction of SiC (fv ¼ 1.5%) is larger than that of TiO2 (fv ¼ 1.1%) and ZrO2 (fv ¼ 0.8%) for w ¼ 30% and ra ¼ 110 kg m3. Fig. 8(b) shows that SiC is the best opacifier among the investigated types to minimize the radiative heat transfer for T > 600 K. 5.4. Shape effect Anomalous diffraction theory (ADT) is an approximation method to calculate the nonspherical scattering based on the assumption of optically large and soft particles, or pd/l > 1 and jm  1j < 1 [44]. ADT tends to cause some errors when the wavelength, l, or the modulus of the complex refractive index, jmj, is large. Thus, the calculation accuracy of ADT needs to be evaluated compared with Mie theory while Mie theory is an accurate solution of Maxwell’s equations for spheres. Fig. 9(a) shows that the spectral extinction coefficients, bl, of SiC with fv ¼ 3% and d ¼ 4 and 6 mm predicted by Mie theory compare well with bl predicted by ADT for l < 12 mm. Fig. 9(b) shows that the difference between the Rosseland mean extinction coefficients predicted using Mie theory and ADT is about 12.7% for d ¼ 4 mm and about 7.5% for d ¼ 6 mm. Thus, ADT is a suitable approximation of Mie theory with an acceptable accuracy. ADT includes the effects of absorptions and interferences with the neglect of reflections and refractions compared with Mie theory [44]. Thus, the deviation between extinction coefficients from Mie theory and ADT is due to the neglect of reflections and refractions in ADT. ADT tends to overestimate the Rosseland mean extinction coefficient for arbitrarily shaped particles within some acceptable degree in this study. The calculation accuracy of ADT is mainly dependent on the equivalent particle size, wavelength and complex refractive index. Thus, for the same equivalent particle sizes and complex refractive indices, the deviation between ADT and Mie theory would not overturn the conclusions of the particle shape effects. Fig. 10(a) shows that the spectral extinction coefficient of SiC, bl, with fv ¼ 3% and d ¼ 4 mm varies significantly with the particle shape. The optimum opacifier shape was investigated to maximize the Rosseland mean extinction coefficient, b. Fig. 10(b) shows that spheroids with aspheroid/bspheroid ¼ 0.5 are better than spheres, and much better than cubes. Cylinders have the lowest b while spheroids with aspheroid/bspheroid ¼ 0.5 are the best shape for SiC among the investigated shapes. Fig. 10(b) also shows that the structural parameters significantly affect b. b of spheroids with

63

aspheroid/bspheroid ¼ 2 is smaller than b of spheres and much smaller than b of spheroids with aspheroid/bspheroid ¼ 0.5. Cylinders with smaller length-to-diameter ratios, l/d, have larger b. The cylinder incident angle, q, is the angle between the cylinder symmetry axis and the incident radiation. The cylinders with smaller incident angles, q, have smaller b at lower temperatures and larger b at higher temperatures as shown in Fig. 10(b). The best shape for maximizing the opacifier extinction coefficient is the oblate spheroid.

6. Conclusions Opacifier radiative properties were analyzed as functions of the opacifier size, volume fraction, shape and material type using Mie theory, anomalous diffraction theory (ADT) and the improved KramerseKronig (KK) relation. The results show that: (1) The inverse model based on the shooting method can be combined with FTIR and Abbe refractometer measurements to calculate the complex refractive indices of various infrared opacifiers. The opacifier extinction coefficients are strongly dependent on the wavelength and temperature. (2) The size, type and shape all significantly affect the opacifier extinction coefficients and the radiative heat transfer through the opacifier-loaded aerogel composites. The optimum diameter of spherical SiC particles to maximize the Rosseland mean extinction coefficient is 4 mm for T < 400 K while the optimum diameter is 3 mm for T > 400 K. Carbon black is the best opacifier for T < 600 K. SiC is the best opacifier to minimize the radiative heat transfer for T > 600 K among the investigated opacifiers of SiC, TiO2, ZrO2, amorphous SiO2 and carbon black. (3) The optimum shape to maximize the opacifier extinction coefficient is the oblate spheroid. The infrared extinction ability decreases from oblate spheroids to spheres to cubes to cylinders with small length-to-diameter ratios and then to long, thin cylinders.

Acknowledgments The authors acknowledge financial support from NSFC (Nos. 51236004 and 51076074) and SRFDP (No. 20100002110045).

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