Radiative transfer in porous carbon-fiber materials for thermal protection systems

International Journal of Heat and Mass Transfer 144 (2019) 118582

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Radiative transfer in porous carbon-fiber materials for thermal protection systems Andrey V. Gusarov a,⇑, Erik Poloni b, Valery Shklover a, Alla Sologubenko b,c, Juerg Leuthold a, Susan White d, John Lawson d a

Institute of Electromagnetic Fields, ETH Zurich, Gloriastrasse 35, 8092 Zurich, Switzerland Complex Materials, ETH Zurich, Vladimir-Prelog-Weg 5, 8093 Zurich, Switzerland Scientific Center for Optical and Electron Microscopy, ETH Zurich, Otto-Stern-Weg 3, 8093 Zurich, Switzerland d NASA Ames Research Center, Moffett Field, CA 94035, USA b c

a r t i c l e

i n f o

Article history: Received 12 July 2019 Accepted 14 August 2019 Available online 6 September 2019 Keywords: Anisotropy Crystalline structure Microstructure Modeling Multiphase approach Radiative thermal conductivity Reflectance

a b s t r a c t Highly porous carbon-fiber materials are used in ablative heat shields for atmospheric re-entry. These carbon fiber based materials perform well against heat convection and conduction, and have high radiation absorption. During re-entry, spacecrafts are exposed to shock layer radiation, which provides a considerable share of the heat load faced by their shields. In this work, models for reflectance and heat transfer through FiberForm, a porous carbon-fiber material, are described. The models are based on the multiphase approach and were validated by comparison with experimental data on reflectance and on effective high-temperature thermal conductivity. The morphology of FiberForm was reconstructed from X-ray tomography data and was used as an input for the models. Transmission electron microscopy and diffraction analyses of FiberForm revealed the presence of an amorphous phase and graphite nanocrystalline grains smaller than 10 nm, with no preferential crystallographic orientation. A strong influence of anisotropy in fiber orientation was observed in measured reflectance and has been included in modeling. The porosity and the extinction coefficient were obtained from X-ray tomography images. The radiative transfer modeling indicates that 90% of the incident radiation at wavelengths around 1 lm is absorbed in the 200–300 lm thick outermost surface layer. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Highly-porous carbon-fiber materials of the kind FiberFormÒ [1] are excellent insulators and are used in ablative heat shields for the atmospheric re-entry of a spacecraft [2,3]. FiberForm consists of a mesh of carbon fibers mostly in-plane oriented and its nominal morphology parameters are [4]: fibers diameter of
5–20 lm, fibers length 1600 lm and bulk density of
0.18 g/cm3. The in-plane fiber orientation promotes an anisotropic thermal conductivity, displaying lower values in the throughthe-thickness direction (cf. experimental section [5] or simulation studies [4]). During atmospheric re-entry, a shock layer is formed at the front of a spacecraft, exposing their thermal protection systems

⇑ Corresponding author. E-mail addresses: [email protected] (A.V. Gusarov), erik.poloni@mat. ethz.ch (E. Poloni), [email protected] (V. Shklover), alla.sologubenko@scopem. ethz.ch (A. Sologubenko), [email protected] (J. Leuthold), susan.white@nasa. gov (S. White), [email protected] (J. Lawson). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118582 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

to high radiation rates [6]. The total heat transfer through a FiberForm-based heat shield is thus influenced by the reflectance of its surface and the radiative heat transfer in its voids. The radiative heat transfer in carbon-fiber materials was numerically calculated using the technique of configuration factors [7,8] and Monte Carlo ray-tracing (MCRT) [9]. The calculations were based on the structural models of straight fibers [7,9] and on the experimentally imaged FiberForm morphology [8]. Moreover, the thermal properties of similar fibrous structures [10,11] and open-pore foams [12,13] were experimentally and theoretically investigated. The morphologies of fiber- and foam-based porous materials vary considerably. Refs. [7–13] focused on modeling them. Simpler but more universal models for the radiative properties are based on the so-called multiphase approach (MPA) [14– 17], where the intensity of scattering at the solid-void interfaces is described by the specific surface of the porous material. The approach also allows considering statistically anisotropic structures [18,19]. The MPA applied in this work is the natural extension of the model of equivalent absorbing scattering medium (EASM)

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commonly applied to low-density dispersed media. While the radiative transfer in a heterogeneous medium is characterized by the overall average of the radiation intensity in EASM, in MPA the radiation is separately averaged over each phase, providing a more detailed description of the whole. Therefore, in the framework of MPA the volume occupied by scattering elements can be taken into account, allowing, for instance, far-field dependent scattering in dense dispersed media to be described [20]. On the other hand, MPA takes no account of the detailed morphology of scattering elements within the structure. This is advantageous for multiphase media with irregular phase interfaces, but may result in a systematic error for a dispersive system of high-symmetry scatterers. Such an error is estimated, for example, for spherical transparent inclusions in a transparent matrix [14]. Once the equations of MPA are obtained for a given multiphase medium, they can be expanded in series to evaluate the radiative thermal conductivity [15,18] or approximately solved by moment methods to evaluate the reflectance [21] and the attenuation coefficient [19]. Recently, it was proposed to include photonic reflectors to FiberForm-based thermal protection systems [22] to optimize their radiation shielding. The present work is devoted to the study of the radiative properties of FiberForm itself based on its morphology to define the penetration depth of the shock layer radiation at an arbitrary stage of ablation during atmospheric re-entry. An influence zone can then be defined as the volume where photonic reflectors would play a role in reducing the heat shield absorbance. Moreover, the possibility to reduce the radiative thermal conductivity of FiberForm is evaluated. The optical properties of the carbon fibers were estimated having regard to the presence of a crystalline structure and were used as input data for the proposed radiativetransfer models. The reflectance of FiberForm was experimentally measured and compared with the modeling results.

2. Experimental The morphology of FiberForm is imaged at a scanning electron microscope (Thermo Fischer Scientific, former FEI, Quanta200F) operated in low vacuum in combination with a large field detector as secondary electron detector. The microstructure of the carbon fibers is studied by HRTEM on a Cs-double-corrected JEOL GrandARM instrument operated with an acceleration voltage of 300 kV. The specimen for the HRTEM is produced by embedding a FiberForm sample in wax, slicing it with a microtome and placing the slices onto conventional TEM carbon grids. The samples were sliced in their in-plane directions, giving access to the long fiber axis. The crystalline structure is examined on a Scintag powder X-ray powder diffractometer using Cu radiation and a Ge detector. Two-dimensional virtual cross-sections are obtained by X-ray computer tomography (XCT) with the resolution of 2 lm using a Xradia MicroXCT-200 setup equipped with: a 150 Kv X-ray source (40–150 kV and 4–10 W range) or a 90 kV X-ray source (40–90 kV and 4–8 W range); 2X, 4X, 10X and 20X objective lenses (best spatial resolution ranges from
12 lm to
1.5 lm); a 2048  2048 CCD camera. Section 3.1 reports the obtained tomography results. For the reflectance measurements in the wavelength range from 0.25 to 2.5 mm, a Perkin Elmer Lambda 1050 UV/Vis spectrophotometer was used with a 15 cm diameter integrating sphere, internally coated with the diffuse broadband reflective material Spectralon. The integrating sphere collect the light scattered or reflected into all angles of the forward or backward hemisphere in order to get accurate total reflectance values needed for thermal calculations [23–25]. A set of calibration standards was used to verify performance. The sphere is configured to capture the total diffuse reflection, including the specular component. During a measurement using an integrating sphere, the repeated internal

reflection from the ideally diffuse surface allows integrating spheres to physically average the energy over all solid angles to directly measure hemispherical properties. Non-diffuse materials containing regularly structured components such as preferentially-oriented fibers, crystals and specular materials can exhibit directional features strong enough to overwhelm the averaging and giving misleading results [24,25]. For isotropic composite materials, the random orientation of fibers and/or particles can physically average out the orientation scattered components to measure the hemispherical values. 3. Results and discussion 3.1. Structure of FiberForm Fig. 1 shows the microstructure of FiberForm (Fiber Materials Inc. [1]) taken with scanning electron microscopy. The fibers display an in-plane preferred orientation and non-circular crosssections. Fig. 2 shows the aberration-corrected high-resolution TEM image of a carbon fiber and its Fast Fourier Transform (FFT). They both confirm that the carbon fibers are composed of randomly oriented nanocrystalline flakes. Therefore, there is no correlation between the orientation of the flakes and the fibers orientation. Two line profiles across two different flakes were plotted (Fig. 2, LP1 and LP2) and show their intensity modulations. The flakes consist of a few atomic layers with the interlayer separation d = 0.37 nm. This distance is close to the d002 of graphite. The flakes are 2–3 nm thick and about 10 nm long. Their spatial distribution is isotropic, which is also confirmed by FFTs (see representative FFT in the inset). The presence of amorphous carbon between the flakes is likely, though neither the HRTEM micrographs nor their corresponding FFTs evidence this clearly. Because of the in-plane fiber orientation in FiberForm (Fig. 1), the samples for XRD were cut out in three mutually perpendicular directions (S1, S2 and S3). The obtained diffractograms are shown in Fig. 3. There is no significant difference between the XRD patterns S1, S2 and S3. The patterns show the (0 0 2)g diffraction peak of graphite and reveal the presence of amorphous carbon. The sharp peaks at 2h around 80° and 97° probably belong to a nonidentified phase introduced during sample preparation. The graphite crystallite size was calculated using Scherrer’s equation [26] and the breadth of the (0 0 2)g diffraction peak. The result was found to be
7–10 nm, though many properties of the structure may influence the diffraction peak broadness and the applicability of Scherrer’s equation [27]. The diffraction peaks at 2h around 42.4°, 53.3° and 57.8° can be assigned to a graphite phase. 3.2. Porosity and penetration depth of incoming radiation into FiberForm Fig. 4a shows a virtual cross section of FiberForm obtained by XCT. The 300 lm leftmost band of this picture is thresholded via image analysis in Fig. 4b, where white pixels designate carbon and black pixels designate void. The estimated volume fraction of carbon f1 = 0.109 is obtained from Fig. 4b as the ratio of 8716 white pixels to the total number of pixels 166  480. The porosity is the complementary value f0 = 1  f1 = 0.891. Fig. 4c traces 240 parallel rays in the cross-section plane that represent the penetration of external radiation into FiberForm. The rays start on the left boundary and intercept the carbon phase. The black curve in Fig. 5 shows the cumulative penetration depth distribution of the rays. This distribution corresponds to the normalized count of rays versus their distance z from the input boundary. According to Bouguer’s law for homogeneous medium, this distribution can be approximated as

aebz

ð1Þ

A.V. Gusarov et al. / International Journal of Heat and Mass Transfer 144 (2019) 118582

3

Fig. 1. SEM images of FiberForm. The microstructure and an alignment along a preferred orientation of the fiber bundles can be seen. The fiber’s cross-section is non-round and has a diameter of about 15 lm. The porous morphology of carbon fibers and the presence of impurities due to FiberForm manufacturing can be seen.

where a is the absorbance and b is the extinction coefficient. The blue curve in Fig. 5 is a fit for the experimental distribution based on Eq. (1) with a = f0 (approximated value) and b = 10 mm1. The obtained parameters are summarized in Table 1. The extinction length 1/b (see Table 1) is defined as the distance at which the intensity of the external radiation decreases e times. Fig. 5 shows that the rays can penetrate to distances greater than the extinction length estimated as 100 lm, but 90% of the incident radiation is absorbed in the
200–300 lm outermost surface layer. The extinction coefficient b of a porous medium consisting of transparent void and opaque solid phases can be alternatively estimated from the two-point probability function S(r) as [28]

b¼


1 dS

; f 0 dr
r¼0

ð2Þ

where r is the distance between the two points. The two-point correlation function [4] 2

SðrÞ  f 1 WðrÞ ¼ ; f 0f 1

ð3Þ

was calculated in [4] from XCT images of FiberForm. The obtained values of -dW/dr were around 100 mm1. From Eq. (3) it is obtained that the corresponding value of dS/dr is around 10 mm1. Therefore, Eq. (2) results in b = 10 mm1. The same value is obtained in this work by ray tracing (see Table 1). 3.3. Radiative transfer in FiberForm The fiber material consists of two phases: the transparent matrix (void) and the fibers of an opaque material (carbon) with an approximated circular cross-section of diameter D. The volume

fractions of the matrix and fiber phases are f0 and f1, respectively, with f0 + f1 = 1. Let l be the total length of fibers in the unit volume of the material. Then, the volume of fibers in the unit volume of the fiber material is

f1 ¼

pD2 4

l:

ð4Þ

If the length of a fiber is much greater than its diameter, its total surface area is determined by the lateral surface area. In this case, the total surface of fibers in the unit volume of the composite is

A ¼ pDl

ð5Þ

The value of l can be excluded from Eqs. (4) and (5) to obtain the specific surface

A¼

4f 1 : D

ð6Þ

Two structure models are considered below. The first one is the statistically isotropic fiber medium where the fibers are randomly oriented. The second one is the medium where all the fibers are parallel, which is the extreme case of anisotropy. The direction of the fibers corresponds to a strong direction of FiberForm, while a perpendicular direction corresponds a weak one. This is considered for the sake of modeling, although FiberForm is composed of carbon fibers in-plane oriented, i.e., with no preferential orientation within their plane of alignment. In the considered case of the medium consisting of one transparent phase and one opaque phase, the MPA approach [14] is reduced to the conventional radiative transfer equation (RTE) for the angular radiation intensity I

X  rI ¼ aB  bI þ rUI;

ð7Þ

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Fig. 3. X-ray diffraction patterns of bulk FiberForm. The samples S1, S2 and S3 are cut out along three mutually perpendicular directions.

U : f ðXÞ !

1 4p

Z 4p

f ðX0 ÞUðX0 ; XÞdX0 ;

ð8Þ

where X0 is the incident direction, X the scattered direction and the kernel U(X0 , X) is the scattering phase function. For the model of statistically isotropic medium with randomly oriented fibers, the MPA evaluates the parameters of Eq. (8) as follows [14]:

A 4f 0

ð9Þ

r ¼ qb;

ð10Þ

a ¼ b  r;

ð11Þ

b¼

where q is the bi-hemispherical reflectance of the surface of the fiber. The scattering phase function is generally determined by the bi-directional reflectance distribution function. In the particular case of specular reflection independent of the angle of incidence,

U¼1

ð12Þ

At diffuse reflection independent of the angle of incidence

U¼

8 ðsinw  wcoswÞ; 3p

ð13Þ

where W is the scattering angle between vectors X0 and X. For the model of anisotropic medium with parallel fibers, the coefficients of extinction, scattering, and absorption depend on the direction X. The extinction coefficient is [18] Fig. 2. HRTEM micrograph of the cross-section of a carbon fiber in FiberForm and its FFT in the inset. The line profiles (LP1 and LP2) show intensity modulations across two nanocrystalline flakes. The modulations have the same periodicity of 0.37 nm. The radius of the inner FFT ring (bright diffuse spots disposed in a circle) corresponds to the same distance.

where X is the unit vector of direction, a is the effective absorption coefficient, b is the effective extinction coefficient, r is the effective scattering coefficient, B is the black body radiance and r is the nabla operator. Extinction coefficient b describes the attenuation of a collimated beam by absorption and scattering. Therefore, b = a + r [30]. The black body spectral radiance depending on temperature is given by Planck’s law [30]. The scattering operator U is defined as

b¼

A

pf 0

sinh;

ð14Þ

where h is the angle between X and the fiber axis, and the scattering and absorption coefficients are still given by Eqs. (10) and (11). At diffuse reflection from the surface of fibers, the scattering phase function is written in spherical coordinates with the axis parallel to the fiber axis [18]

UðX0 ; XÞ ¼ Uh ðh0 ÞUu ðu0  uÞ;

ð15Þ

with

Uh ¼ sinh0 ;

ð16Þ

Uu ¼ jsinx  xcosxj;

ð17Þ

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Fig. 5. Normalized cumulative distributions of the radiation penetration depth. The experimental curve (black) was extracted from the ray tracing of Fig. 4c, while the fitted one (blue) was calculated with Bouguer’s law. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 1 FiberForm parameters estimated from XCT. Void fraction, f0

Solid fraction, f1

Extinction coefficient, b

Extinction length, 1/b

0.891

0.109

10 mm1

100 lm

Fig. 4. Penetration depth into FiberForm: (a) Cross section obtained by X-ray computer tomography (XCT); (b) Phase quantification by thresholding the 300 lm leftmost band of (a); (c) Ray tracing of (b).

where h0 is the polar angle of the incident ray, u0 and u are the azimuth angles of the incident and the scattered rays, respectively, and

8 0 0 > < u  u þ 2p; 2p 6 u  u < p 0 0 x¼ uu; p 6 u  u < p ; > : u  u0  2p; p 6 u  u0 6 2p

ð18Þ

is the normalized difference between the incident and scattered azimuth angles, which is always in the interval from –p to p. With such a definition of x, the azimuth part of the scattering phase function Uu is the 2 p -periodic function of u – u0 shown in Fig. 6. Scattering in the azimuth angle is highly backward directed because Uu attains maxima at u – u0 = p and zeros at u – u0 = 0. On the contrary, scattering in polar angle is isotropic because the polar part of the scattering phase function Uh is independent of polar scattering angle h (see Eq. (16)). The radiative thermal conductivity kr can be theoretically calculated by expanding the RTE, Eq. (7), at low thermal gradient. In the assumption of grey isotropic medium [19],

Fig. 6. Azimuth part of the scattering phase function Uu in the model of anisotropic fiber medium.

kr ¼ K;

ð19Þ

for the model of specular reflection from the surface of the fiber and

kr ¼

K ; 1 þ 49 q

ð20Þ

for the model of diffuse reflection, where 2

K¼

64f 0 rT 3 ; 3A

ð21Þ

and r is the Stefan-Boltzmann constant. In the model of anisotropic grey fiber medium with diffuse reflection from the fiber surface, the radiative thermal conductivity in the direction perpendicular to fibers is [19]

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kr ¼

3p2 K ; 32 1 þ p162 q

ð22Þ

Fig. 7 compares the radiative thermal conductivity calculated by Eqs. (19)–(22) with experimental data from [2]. Table 2 shows the accepted material parameters. The reflectance of fiber surface q is estimated at the wavelength of 1 mm, corresponding to the maximum of thermal emission at the temperature of about 3000 K, when the contribution of thermal radiation to heat transfer becomes significant. Section 3.1 revealed nanocrystalline graphite structure of fibers with no preferential orientation of crystals. Therefore, the value of q for crystalline calculations can be taken from the experiments with polycrystalline graphite [29] (see Table 2). On the other hand, the extremely small size of the crystallites may influence q and the presence of amorphous carbon has to be taken into account. The reflectance of amorphous glassy carbon [29] is thus used as the lower bound of q (see Table 2). The model of statistically isotropic fibrous medium with specular reflection at the fiber surface (Eq. (19)), gives the estimate for the superior value of the radiative thermal conductivity kr (see Fig. 7), which is independent of fiber reflectance q. The radiative thermal conductivity kr of the isotropic medium with diffuse reflection at the fiber surface is lower (Eq. (20)). The dependence on q is not significant (see curves for amorphous and crystalline carbon in Fig. 7). The inferior value of kr is given by the model of anisotropic fibrous medium for the weak direction according to Eq. (22). The dependence on q is not significant either (see Fig. 7). Total thermal conductivity is supposed to be the superposi-

tion of the radiative conductivity and the conductivity through solid phase. The calculated curves underestimate the experimental data [2] in Fig. 7. This indicates a considerable contribution of the thermal conduction through solid phase. However, the tendency of kr
T3 is clear at high temperatures. In view of Eqs. (19), (20) and (22), decreasing the radiative thermal conductivity by modifying the reflectance q does not seem to be promising. For re-entry applications, the surface of a heat shield can be exposed to a considerable radiative energy flux. Consider the problem of radiative transfer described by Eq. (7) in the fiber material occupying half-space z > 0 as shown in Fig. 8. Suppose that the incident radiation on the surface, z = 0, is diffuse. This means that the angular intensity of the incident radiation is uniform in the forward hemisphere of directions, X  ez > 0, where X is the unit vector of radiation propagation and ez the unit vector in the direction of the OZ axis. The boundary condition for Eq. (7) at z = 0 is specified in the forward hemisphere of directions as

IðXÞ ¼ I0 at X  ez > 0;

ð23Þ

where I0 is the incident intensity. The boundary problem in the halfspace defined by Eqs. (7) and (23) is solved by the method of moments described in Appendix A. The calculations of the reflectance of FiberForm are accomplished for the wavelength of 1 lm, which is the average wavelength value of the shock-layer emission spectrum at re-entry [6]. Table 3 shows the reflectance calculated by two- and fourmoment methods for different models. The eigenvectors of Matrix M (see Appendix A) necessary for the four-moment method are calculated numerically and listed in Appendix B. The calculated reflectance for 1 mm slightly depends on the reflection model by the fiber surface (diffuse or specular) and on the calculation method (2-moment or 4-moment). The relative uncertainty of the moment method of calculation is estimated by the comparison of the 2-moment and 4-moment results (see Table 3) as 10%. The main uncertainty of the reflectance theoretical estimation is due to the uncertainty in the reflectance of the fiber surface q, used as the input parameter for the calculations. This value strongly

0

Fig. 7. Theoretically estimated radiative thermal conductivity of FiberForm (curves) compared with the experimentally measured effective thermal conductivity (data points) [2]. The three upper curves relate to the isotropic model, while the two lower ones, to the anisotropic model. Models considerations: medium isotropy/ anisotropy, fibers specular/diffuse reflection, amorphous/crystalline fibers structure.

Fig. 8. Half-space problem for reflection by fiber material.

Table 2 Parameters of FiberForm accepted for radiative transfer modeling. Parameter

Solid fraction, f1

Fiber diameter, D

Fiber reflectance at 1 lm, q Polycrystalline graphite

Amorphous glassy carbon

Value Reference

0.109 Table 1

15 lm Fig. 1

0.35 [29]

0.19 [29]

7

A.V. Gusarov et al. / International Journal of Heat and Mass Transfer 144 (2019) 118582 Table 3 Reflectance of FiberForm estimated by 2-moment (2M) and 4-moment (4M) methods. Anisotropy

NO

NO

YES, surface cut perpendicular to weak direction

YES, surface cut parallel to weak direction

Reflection at fiber surface Fiber structure Amorphous (q = 0.19) Crystalline (q = 0.35)

Diffuse 2M: 0.0679

Specular 2M: 0.0526 4M: 0.0463 2M: 0.1073 4M: 0.0950

Diffuse 2M: 0.0752

Diffuse 2M: 0.0526

2M: 0.1469

2M: 0.1073

2M: 0.1345

depends on the structure of carbon in the fibers (see Table 2). For the superior estimate of the reflectance q, the one of polycrystalline graphite, the calculated reflectance of FiberForm (see Table 3) is greater by a factor of around 2 when compared to its calculated reflectance for the inferior estimate of q, the one of amorphous glassy carbon. The model of anisotropic fiber medium results in an anisotropy in the reflectance. The reflectance of the FiberForm surface cut perpendicular to the weak direction is greater than the reflectance of a surface cut parallel to the weak direction by a factor of around 1.5 (see Table 3). The penetration depth of external radiation into FiberForm can be estimated from the profile of radiative energy flow q versus distance z from the surface. The radiative energy flow is defined by Eq. (A22) in Appendix A. Fig. 9 shows the profiles calculated by the moment method (see Appendix A). Each plot of Fig. 9 contains two curves of one color and two curves of another color. The curves of the same color represent calculations assuming two different structures of the fibers: amorphous and polycrystalline. The curves of the same color are hardly distinguishable. Essentially, the crystal structure does not influence the profile of the external radiation penetration. The penetration depth of external radiation into FiberForm can be estimated from the profile of radiative energy flow q versus distance z from the surface. The radiative energy flow is defined by Eq. (A22) in Appendix A. Fig. 9 shows the profiles calculated by the moment method (see Appendix A). Each plot of Fig. 9 contains two curves of one color and two curves of another color. The curves of the same color represent calculations assuming two different structures of the fibers: amorphous and polycrystalline. The curves of the same color are hardly distinguishable. Essentially, the crystal structure does not influence the profile of the external radiation penetration. Fig. 9a indicates that, at the depth above 100 lm, the 2-moment method underestimates the radiative energy flow relative to the 4-moment method by about 0.1 of the incident flow p (I0 – B). This estimates the relative uncertainty of the moment method as 10%. The same value is obtained above for the reflectance (see Table 3).

(a)

Fig. 9b compares the penetration profiles in the weak and strong directions calculated for the model of anisotropic medium. One can conclude that there is no anisotropy in the penetration profile. The calculated profiles in Fig. 9 are very similar to the cumulative extinction distribution obtained from the experimentally imaged structure (see Fig. 5). Thus, the moment-method calculations confirm that the external radiation at wavelengths around 1 lm doesn’t penetrate into FiberForm deeper than 200–300 lm. This is the influence zone in which reflectance modifications would alter the overall reflectance. However, as a heat shield material is constantly ablated during re-entry, its 200–300 lm thick outermost layer gets deeper over time. This means that the influence zone becomes thicker when ablation is important. Fig. 10 shows the hemispherical spectral reflectance measured by the spectrophotometer equipped with the integrating sphere. The FiberForm reflectance was measured and calculated in three mutually perpendicular axes: X, Y, and Z. The experimental curves and the calculations agree between 0.5 and 2.5 mm. The offset of the Z-axis reflectance in both experimental data and computational model underscores the dramatic effect that the fiber orientation has on the reflectance. The calculations become less reliable with increasing the wavelength when the wavelength approaches the fiber diameter of
15 lm, and wave effects such as diffraction become more important. The proposed reflectance models do not take wave effects into account. However, the comparison of the calculations with the experiments in Fig. 10 indicates that the model is acceptable below 2.5 lm wavelength. The dashed lines in Fig. 10 show the spectral reflectance calculated by the 2-moment method (see Appendix A). The model of anisotropic medium is accepted with weak axis Z and strong axes X and Y. The spectral dependence of fiber surface reflectance q necessary for the calculation is taken from Ref. [29] for amorphous glassy carbon and polycrystalline graphite and approximated by linear equation

q ¼ ðq1  q0 Þ

1

w þq lm 0

ð24Þ

(b)

Fig. 9. Penetration of 1 lm wavelength external radiation into FiberForm: (a) comparison of the two- and four-moment methods for the model of isotropic medium with specular reflection at the fiber surface; (b) two-moment approach for the model of anisotropic medium with diffuse reflection at the fiber surface. Two curves are given for each model assuming an amorphous (q = 0.19) and a crystalline (q = 0.35) structure of carbon.

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Fig. 10. Normal hemispherical reflectance of FiberForm measured and calculated in three mutually perpendicular axes. Solid lines: experimental results acquired in the UV/Vis/NIR spectra; dashed lines: calculated values for both crystalline and amorphous carbon in the Vis/NIR spectra. The Z axis direction is perpendicular to the in-plane orientation of the carbon fibers, allowing for higher reflectance.

Table 4 Parameters for the calculation of the spectral reflectance of carbon. Parameter

q1

q0

Glassy carbon (amorphous) Polycrystalline graphite

0.19 0.35

0.13 0.23

for the wavelength w in the range from 0.5 to 2.5 lm. Table 4 lists the parameters for Eq. (24) derived from experimental data [29]. The calculations assuming the crystalline structure of carbon fibers considerably overestimate the experimental data, while the calculations for amorphous carbon seem to be more reliable (see Fig. 10). A good agreement is observed for samples cut perpendicular to strong directions X and Y. The X and Y calculations are given by a single curve because the strong directions are equivalent in the model. Generally, the calculations confirm the experimentally observed strong anisotropy in reflectance. The Monte Carlo ray tracing [9] resulted in the same conclusion. The comparison of the experimental data with the calculations indicates that the reflectance of the carbon fiber surface corresponds better to that of amorphous carbon. This is in agreement with the HRTEM and the XRD observations, which indicated the presence of both a nanocrystalline and an amorphous carbon phase in FiberForm. The extremely small size of the identified crystals can explain why the reflectance of the fiber surface strongly differs from that of conventional polycrystalline graphite. 4. Conclusions The morphology study of the highly-porous carbon-fiber material FiberForm performed by SEM, HRTEM, XRD and XCT provided input for radiative transfer modeling. The SEM images indicate that the fibers have non-circular cross-sections and an in-plane preferred orientation. HRTEM revealed a fine nanocrystalline structure in the fibers consisting of 2–3 nm thick and 10 nm long flakes with no preferential crystallographic orientation relative to the fibers axes. The XRD patterns show strong graphite peaks and the presence of amorphous carbon. The graphite crystallite size estimated by its (0 0 2)g peak width is approximately 7–10 nm. From the analysis of the XCT data, the FiberForm porosity f0 = 0.891 and light extinction coefficient b = 10 mm1 were calculated. The spectral reflectance of FiberForm was measured and compared with calculations. An anisotropic model of its structure was proposed to take into account the influence of the preferential

orientation of the carbon fibers on the reflectance. The presence of either a nanocrystalline or an amorphous carbon phase in FiberForm was also considered in the model. The reflectance was independently calculated for both phases and the results corresponding to the presence of only amorphous carbon agree better with the measured reflectance than the ones corresponding to the presence of only polycrystalline graphite. Even if FiberForm was composed of both phases, the extremely small crystallites present in its fibers can explain why their reflectance strongly differs from that of conventional polycrystalline graphite. In addition, a half-space problem for RTE was approximately solved by 2- and 4-moment methods to estimate the reflectance of FiberForm. The relative uncertainty of the calculation results was about 10%. The reflectance type (diffuse or specular) at the fibers surface has a small influence on the reflectance of FiberForm, while the value of the fibers reflectance q is important. The radiative thermal conductivity kr was theoretically estimated by expanding the radiative transfer equation. The obtained dependence of kr on the reflectance type and on the fiber reflectance q is not significant. The anisotropy of the radiative thermal conductivity is not significant for kr either. The calculated radiative thermal conductivity is one component of the total thermal conductivity and therefore gives a lower value than the experimental data, indicating a considerable contribution of conduction through solid phase. The trend of kr
T3 consistent with modeling the radiative component is clear at high temperatures. The penetration depth of the external radiation into FiberForm is estimated by ray tracing in images taken by XCT. It indicates that 90% of the incident radiation is absorbed in the
200–300 lm thick outermost surface layer. Similar penetration depth profiles for wavelengths around 1 lm were also obtained from theoretical calculations. The modeling shows that the penetration depth is essentially independent of the carbon fibers reflectance q. The overall material reflectance at 1 lm in an arbitrary instant of time is only influenced by modifications within the penetration depth of the external radiation. Therefore, the addition of reflective particles would have to be focused on this influence zone, which can greatly change under the most extreme re-entry conditions. Declaration of Competing Interest The authors declare that they have no conflict of interest. Acknowledgements The authors are indebted to Swiss National Science Foundation for support (project 200021_160184), to Karsten Kunze (ScopeM, ETH Zurich, http://www.scopem.ethz.ch), for SEM measurements and to Thomas Weber (Röntgen Platform of Department of Materials of ETH Zurich, http://www.xray.mat.ethz.ch) for valuable comments. Appendix A. Moment method Radiative transfer equation

@I Xz ¼ aB  bI þ rUI; @z

ðA1Þ

is solved for angular radiation intensity I(X,z) in domain z  0 with boundary condition

IðX; 0Þ ¼ I0

at Xz > 0;

ðA2Þ

where scattering operator U is defined as

U : f ðXÞ !

1 4p

Z 4p

f ðX0 ÞUðX0 ; XÞdX0 ;

ðA3Þ

9

A.V. Gusarov et al. / International Journal of Heat and Mass Transfer 144 (2019) 118582

where scattering phase function U (X0 , X) is normalized to satisfy equation

U1 ¼ 1:

ðA4Þ

Absorption a, extinction b, and scattering r coefficients satisfy equation

b ¼ a þ r:

ðA5Þ

The medium is supposed to be uniform, so that radiation transfer parameters a, b, r, B, and U do not depend on z. In anisotropic medium, a, b, and r depend on direction X. The following relation is necessary to satisfy optical reciprocity:

rðXÞUðX0 ; XÞ ¼ rðX0 ÞUðX; X0 Þ

ðA6Þ

In isotropic medium, Eq. (A6) reduces to the condition of symmetry for the scattering phase function,

UðX0 ; XÞ ¼ UðX; X0 Þ

ðA7Þ

In the moment method, the solution is approximated by expansion

Iðz; XÞ ¼

N X

ðA8Þ

in basis functions sm depending on direction X with coefficients Cm depending on z, where N is the order of the method. The following moment equations are obtained by interior multiplication of Eq. (A1) by testing functions tn(X), n = 1 .. N:

ht n ; Xz sm i

m¼1

hWþ ; rUWþ i ; h1; rUWþ i

N X dC m ¼ Bhtn ; ai  htn ; bsm iC m dz m¼1

þ

and the fraction of collimated radiation scattered into the forward hemisphere

c¼

hWþ ; rUDþ i : h1; rUDþ i

ðA14Þ

Note, that the complementary values are

1h¼

hW ; rUWþ i hW ; rUDþ i and 1  c ¼ h1; rUWþ i h1; rUDþ i

N X

and for any function f(X)

h1; rUf ðXÞi ¼ hf ðXÞ; ri:

ðA16Þ

Eq. (A16) is proved by changing the order of integration in its RHS taking into account reciprocity, Eq. (A6), and normalizing, Eq. (A4). From Eqs. (A13)–(A16) one can obtain that

hWþ ; rUWþ i ¼ hhW ; ri;

hW ; rUWþ i ¼ ð1  hÞhW ; ri hW ; rUDþ i ¼ ð1  cÞhD ; ri

ðA17Þ ðA18Þ

Other four matrix elements are obtained by interchanging indices + and – in Eqs. (A17) and (A18). These matrix elements are evaluated in Table A2 for isotropic media. Table A3 shows the values of h and c calculated for different models. The other matrix elements are calculated for isotropic medium with the specular reflection model as

¼ hXz Wþ ; UD i ¼

htn ; rUsm iC m ; n ¼ 1 ::N;

ðA9Þ

ðA10Þ

4p

Below, the following sets of basis and testing functions are used in four-moment method:

s1 ¼ wþ ¼ hðXz Þ; s2 ¼ w ¼ 1  hðXz Þ; s3 ¼ Dþ ¼ dðXz  1Þ; s4 ¼ D ¼ dðXz þ 1Þ;

ðA11Þ

t 1 ¼ wþ ; t 2 ¼ w ; t 3 ¼ Xz wþ ; t 4 ¼ Xz w ;

ðA19Þ

1 ¼ hXz W ; UD i ¼  : 8

Z

f ðXÞgðXÞdX:

1 ; 8

hXz W ; UWþ i ¼ hXz W ; UW i ¼ hXz W ; UDþ i

where the inner product of functions f and g is defined as

1 hf ðXÞ; gðXÞi ¼ 4p

ðA15Þ

hXz Wþ ; UWþ i ¼ hXz Wþ ; UW i ¼ hXz Wþ ; UDþ i

m¼1

ðA12Þ

where h() is the Heaviside step function and d() is the Dirac delta function. In two-moment method, only two first functions are used in every set. Table A1 shows the multiplication table between the basis and testing functions obtained by direct integration according to Eq. (A10). Matrix elements of the scattering operator with the first two testing functions can be expressed through the fraction of radiation uniformly distributed over the forward hemisphere of directions that remains in this hemisphere after scattering

1

W+

W

D+

D

1/2 1/2 1/4 1/4 1/6 1/6

1/2 0 1/4 0 1/6 0

0 1/2 0 1/4 0 1/6

1/2 0 1/2 0 1/2 0

0 1/2 0 1/2 0 1/2

ðA20Þ

With the chosen set of basis functions, Eq. (A11), boundary conditions, Eq. (A2), become

C 1 ¼ I0 and C 3 ¼ 0 at z ¼ 0:

ðA21Þ

To estimate the depth of penetration of an external radiation into the medium, the radiative energy flow q [30] is calculated from angular radiation intensity I. Applying the notation introduced by Eq. (A10),

q ¼ 4phI; Xi:

ðA22Þ

In the studied problem, vector q is collinear to z-axis. Integration of Eq. (A8) gives its absolute value

q ¼ pðC 1  C 2 Þ þ 2pðC 3  C 4 Þ:

ðA23Þ

In two-moment approximation, the set of moment equations, Eqs. (A9), is

1 dC 1 ¼ haiB  hbiC 1 þ hri½hC 1 þ ð1  hÞC 2 ; 2 dx 

Table A1 Multiplication table.

W+ W Xz W+ Xz W (Xz)2 W+ (Xz)2 W

ðA13Þ

hWþ ; rUDþ i ¼ chD ; ri; C m ðzÞsm ðXÞ;

m¼1

N X

h¼

ðA24Þ

1 dC 2 ¼ haiB  hbiC 2 þ hri½hC 2 þ ð1  hÞC 1 ; 2 dx

ðA25Þ

Table A2 Matrix elements ha; Ubi in isotropic media. ab

W+

W

D+

D

W W

h 2 1h 2

1h 2 h 2

c 2 1c 2

1c 2 c 2

+ -

10

A.V. Gusarov et al. / International Journal of Heat and Mass Transfer 144 (2019) 118582

Table A3 Parameters h and c for calculation the scattering matrix elements. Anisotropy

NO

NO

YES, surface cut perpendicular to weak direction

YES, surface cut parallel to weak direction

Reflection at fiber surface h c

Diffuse 1/3 [21] 1/6 [21]

Specular 1/2 [21] 1/2 [21]

Diffuse 1/4 –

Diffuse 1/2 –

l1 ¼

with angular-averaged coefficients of absorption hai ¼ h1; ai, extinction hbi ¼ h1; bi, and scattering hri ¼ h1; ri. The unique solution of Eqs. (A21), (A24) and (A25) bounded at z > 0 is

C 1 ¼ ðI0  BÞe2lz þ B C 2 ¼ ðI0  BÞ

b0  l

r0

ðA28Þ

d ¼ 64b2  104br þ 49r2 :

ðA37Þ

In the four-moment approximation, radiative energy flow, Eq. (A23), becomes

ðA29Þ

At z = 0, Eq. (A27) gives the intensity of radiation emitted from the surface

C 2 ¼ RI0  EB

pðI0  BÞ fv 3 ½u1  u2 þ 2ðu3  u4 Þel1 z u1 v 3  v 1 u3 u3 ½v 1  v 2 þ 2ðv 3  v 4 Þel2 z g:

q¼

ðA30Þ

with the reflectance of composite surface

r0

ðA36Þ

ðA38Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l ¼ b02  r02

R¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi m þ 2b d

80 1 9 0 1 1 0 1 C1 u1 v1 1 > > > > > > < = B1C C C BC C B B u v I  B 2 2 2 B C B C B C 0 B C B C¼ B Cv 3 el1 z  B Cu3 el2 z þ B CB; > @ C 3 A u1 v 3  v 1 u3 > @ v3 A @0A >@ u3 A > > > : ; C4 u4 v4 0

and attenuation coefficient

b0  l

1 2

0

ðA27Þ

r0 ¼ ð1  hÞhri

l2 ¼

Let the eigenvector corresponding to –l1 be (u1, u2, u3, u4) and that corresponding to –l2 be (v1, v2, v3, v4). Then, the solution of Eqs. (A21) and (A34) is

where coefficients

b0 ¼ hbi  hhri;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi m  2b d;

m ¼ 20b2  14br;

ðA26Þ

e2lz þ B;

1 2

;

Absorptance of the composite surface 1 – R is defined as energy flow at z = 0 divided by the non-equilibrium component of the incident radiative flux,

ðA31Þ

and emittance

1R¼

E¼1R

ðA32Þ

qð0Þ

pðI0  BÞ

:

ðA40Þ

Reflectance R is obtained from Eqs. (A39) and (A40) as

satisfying Kirchhoff’s law [30]. In the two-moment approximation, radiative energy flow, Eq. (A23), becomes

q ¼ pðI0  BÞEe2lz :

ðA39Þ

R¼

ðA33Þ

v 3 ðu2 þ 2u4 Þ  u3 ðv 2 þ 2v 4 Þ : u1 v 3  v 1 u3

ðA41Þ

Appendix B. Eigenvectors for the 4-moment method Component index, i Amorphous(q = 0.19)

Fiber structure

Crystalline (q = 0.35)

ui

vi

ui

vi

In four-moment approximation for isotropic medium with specular reflection, the set of moment equations, Eq. (A9), is

0

C1

1

0

C1

1

0

1

3

C B C C B 3 C dB C B C2 C B 2C B CaB; B C ¼ MB C þ B @ C 3 A @ 1=2 A dz @ C 3 A C4

ðA34Þ

1=2

C4

with matrix

0 B B M¼B B @

3 3b þ 32 r r 2 3 2r 3b  32 r b 2

 r4 r 4

r 3 2r 3 2

 r4

b  r4

 2b þ r4

r 4

1

r C 3 2rC 3 2

C:  r4 C A b þ r4

Eigenvalues of M are l1 and l2 , where

ðA35Þ

1

2

3

4

0.17361 0.96708 0.37079 0.96243

0.08869 0.03687 0.19440 0.07183

0.98078 0.25175 0.90801 0.26172

0.00728 0.00452 0.01567 0.00863

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