Measurement model for near-infrared radiative properties of open-cell metallic foams based on transmittance spectra

International Journal of Heat and Mass Transfer 136 (2019) 365–372

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Measurement model for near-infrared radiative properties of open-cell metallic foams based on transmittance spectra Meng Liu, Qing Ai ⇑, Hai-Chao Yin, Xin-Lin Xia, He-Ping Tan School of Energy Science and Engineering, Harbin Institute of Technology, No. 92 West Dazhi Street, Harbin 150001, PR China

a r t i c l e

i n f o

Article history: Received 18 November 2018 Received in revised form 28 February 2019 Accepted 4 March 2019

Keywords: Metallic foams Three-layer structure Transmittance spectra Spectral radiative properties

a b s t r a c t Considering the existence of edge effect caused by the cut surface of struts or sintering points will affect the actual radiative transfer process in metallic foams, a three-layer structure model is constructed to obtain their spectral radiative properties. Using the double-thickness model for reference, the real spectral radiative properties of metallic foams are calculated based on transmittance spectra. To verify the feasibility of this method, transmittance spectra of nickel foams with different cell sizes and thicknesses are measured by FTIR spectrometer system in the near infrared band. Through comparing extinction coefficients obtained by traditional single-layer structure model and the three-layer structure model proposed in this paper, it is found that extinction coefficient will be underestimated by the latter method, while the deviation can reach 13% for 20ppi nickel foam and 8% for 40ppi nickel foam. Then absorption coefficient and scattering coefficient are calculated by a prediction model with parameters corrected. Based on this method, temperature influence on extinction coefficient of nickel foam is studied from 293 K to 773 K and it is found that extinction coefficient decreases with temperature in the researched band while the maximum changing rate is 2.1% in the researched band. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Metallic foams have well mechanical properties and thermal performance, such as low density, large specific surface area, high solid conductivity and flow mixing enhancement [1]. These outstanding properties make them have broad application prospects in many fields, such as solar collectors [2], heat exchangers [3] and electrodes for electric battery [4]. Most of these fields concern relatively high temperature, therefore thermal radiation propagation is a non-ignorable heat transfer mode [5]. As metallic foam is a semi-transparent medium for thermal radiation transfer and has the property of high-porosity, radiative heat transfer occupies a large share, therefore, it is vital to research on its thermal radiative properties. At present, there are mainly two ways to obtain thermal radiative properties of metallic foams, namely theoretical prediction [5– 9] and experimental identification [10–13]. Theoretical prediction means predicting thermal radiative properties from geometric parameters of metallic foams and corresponding radiative properties of base material. It has developed from macroscopic scale to microscopic scale. In earlier researches, the most common assump⇑ Corresponding author. E-mail address: [email protected] (Q. Ai). https://doi.org/10.1016/j.ijheatmasstransfer.2019.03.010 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

tion was regarding high porosity foams as continuous and homogeneous media, and modelling them as a random dispersion of particles whose contributions are summed up to obtain ‘‘effective radiative properties” [14,15]. Later, researchers recognized that the complex structure of foams would have effects on the practical radiation transfer process, then ordered polyhedral models were established to describe the cells of metallic foams. Glicksman et al. [16] adopted pentagon dodecahedron model which considered strut cross section was constant and occupied two thirds of the area of an equilateral triangle formed at the vertices, while Kuhn et al. [17] used infinitely long cylinders to describe the struts with the same geometric mean cross-section [18]. Then Doermann and Sacadura [19] considered two types of particles including struts with thickness varying along them and strut junctures formed from four struts intersecting. After that, the tetrakaidecahedric model generated by uniformly truncating the six corners of an octahedron is adopted, which contained eight regular hexagonal faces and six square faces [20]. This structure model was proposed by Lord Kelvin, and it could be considered a good compromise between the accuracy of the predictions and the simplicity of the structure of the foam [5]. As the development of science and technology, the real structure of foams can be obtained by scanning electron micrograph (SEM) or l-CT image, which can

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be used to analyze the radiative properties of media by MonteCarlo (M-C) method [8,15]. Considering the complexity and heavy calculation burden of theoretical method, experimental method is more rapid, more convenient and maybe matching more with the real radiation transfer process. Experimental method contains two parts, namely measurement and identification. Measurement aims at obtaining apparent radiation characteristics, such as directionalhemispherical transmittance/reflectance and bidirectional transmittance/reflectance, while identification aims at obtaining essential radiative properties, including extinction coefficient, absorption coefficient, scattering coefficient and scattering phase function. When adopting experimental method, the expression of scattering phase function is known in advance with several unknown parameters needed to be determined. Zhao et al. [11] obtained the spectral transmittance and reflectance of FeCrAlY foam and then the radiative properties are identified. Zhang et al. [10] measured the bidirectional reflectance distribution functions (BRDF) of a copper foam sheet for 6 different incident angles and the asymmetry parameter was calculated. In above-mentioned theoretical researches, some assumptions were adopted in the analysis of reflection behavior at the interface of metallic foams, which considered the reflection percentage at the interface was related to the porosity [8]. But as we can see from practical samples, by incision, surfaces of metallic foams exist not only cracked cells, but also sintering points, which are not considered in previous studies. Besides, for experimental researches, the effect of edge was not considered during the identification of radiative properties of metallic foams. The existence of edge layer makes radiation transfer in this structure is different from that of integrated cells, which makes the metallic foam sample as a three-layer structure with two edge layers and one complete layer. This structure is just like that of liquid samples packaged in a liquid cell. Radiative property researches for liquids have been operated by many researchers and the three-layer structure [21] considering two packaged windows are developed in recent years. Therefore, the concept of three-layer structure is introduced from the radiative property researches of liquid samples to our research on metallic foams for the first time. As it is difficult to describe this complex process of radiation transfer by simulation, we will carry out this research by experimental measurements. In this article, the three-layer structure model is first introduced from the radiative property determination of liquids to describe the radiation transfer in metallic foams, and the double-thickness method is introduced to determine thermal radiative properties based on transmission method. Then the solving model of determining the real spectral radiative properties of metallic foams are built based on the measured transmittance spectra of samples with different thicknesses and cell sizes. Based on this model, radiative properties of nickel foams with different cell sizes are calculated and the influence of edge effect and temperature on the calculation results is analyzed.

sk ¼

Researches on radiative properties of metallic foams usually regarded samples as homogeneous and continuous media, and the radiative transfer is described with RTE [22–24] as shown in Eq. (1).

Z 4p

Ik ðr; h0 ÞUðh0 ! hÞdh0 ð1Þ

where I is the radiation intensity, and Ib is the blackbody radiation, while a,rs and U stand for the absorption coefficient, scattering

Ik ðLÞ ¼ ebk L Ik ð0Þ

ð2Þ

where Ik ð0Þ stands for the incident radiation intensity and Ik ðLÞ stands for the remaining radiation intensity after attenuated by the foam whose thickness is L. In fact, there will exist edge layers with cracked cells at two sides of metallic foams after incision, whose radiative transfer is different from that in complete cells. As shown in Fig. 1, a part of lights incident on the edge layer are reflected by the fracture surface of struts or sintering points, which cannot enter into the sample inside. To determine radiative properties of metallic foams with actual structure, the three-layer structure model in researches of radiative properties of liquid samples is introduced, and the metallic foam sample is regarded as constituted by edge layer, complete layer and edge layer. As the radiative transfer mechanism of edge layer is different from that of complete layer, their radiative properties are different, and this research aims at establishing the theoretical model to determine actual radiative properties of metallic foams. Considering metallic foams with different thicknesses are the same materials with the same density and the same porosity, the thickness and properties of edge layer are the same and can be described uniformly. According to the three-layer structure established, the total transmittance can be expressed as Eq. (3).

s ¼ s0  s20 ¼ ebL  e2b0 L0 ¼ eðbL þ2b0 L0 Þ 0

0

ð3Þ

0

where b and L stand for the extinction coefficient and thickness of complete layer respectively, while b0 and L0 stand for those of edge layer. It can be seen from Eq. (3) that there are four unknown parameters, including b, b0 , L0 and L0 . As L0 and L0 meet L0 þ 2L0 ¼ L, where L is the sample thickness that can be obtained by measurement, unknown parameters decrease to three. According to transmittance expressions of metallic foams with different thicknesses, the influence of edge effect can be eliminated and the extinction coefficient of complete layer can be determined with Eq. (4). Then, the absorption coefficient and scattering coefficient of metallic foams can be calculated by the prediction model proposed by Zhao [11], as shown in Eqs. (5)–(7).

b¼

2. Radiative transfer mechanism

@Ik ðr; hÞ rsk ¼ ðak þ rsk ÞIk ðr;hÞ þ ak Ibk ½T ðr Þ þ 4p @r

coefficient and scattering phase function respectively. Note that the absorption coefficient a and scattering coefficient rs are related to extinction coefficient b ¼ a þ rs . Eq. (1) indicates that the incident intensity is attenuated by extinction (absorption and scattering) but enhanced by the emission of the medium and the scattering from all directions. In the research on radiative properties of metallic foams at room temperature, the self-emission is always ignored. For the determination of extinction coefficients [11,25], bi-directional transmittance measurements are enough and the exponential expression is adopted as shown in Eq. (2).

lnðs1 =s2 Þ L2  L1

ð4Þ

where L1 and L2 stand for thicknesses of different samples, and s1 and s2 stand for corresponding transmittances of these samples.

rsk ¼ C ðqk þ 1Þ

ð1  /Þn f ðkÞ 2ds

ð5Þ

ak ¼ C ð1  qk Þ 

ð1  /Þn f ðkÞ 2ds

ð6Þ

bk ¼ rsk þ ak ¼ C

ð1  /Þn f ðkÞ ds

ð7Þ

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Complete layer

Edge layer

Edge layer

Fig. 1. Schematic diagram of radiative transfer in metallic foams.

where qk is the spectral reflectivity of the solid material, / is the porosity of the metallic foam, f ðkÞ is the influence function about wavelength, while C and n are constants to be determined. As the research object in Ref. [11] is FeCrAlY foam, which is different from that in this research, the influence parameter f ðkÞ related to wavelength in above equation should be corrected according to the changing trend of experimental results. For the formula of f ðkÞ, we still adopt the form in Ref. [11] but with different constants, which is expressed in Eq. (8).

f ðkÞ ¼ 1 þ expða1  ðk þ a2 ÞÞ

ð8Þ

Considering the relationship between radiative properties and wavelength is presented by the function parameter f ðkÞ, and the metallic foam samples researched in this paper have the same porosity, the value of parameter n in the exponent position of (1-/) will not influence the trend of radiative properties. Therefore n is also set to be 1 in Eqs. (5)–(7) as the case of Ref. [11].

3. Experimental device Transmittance spectra of samples are measured with Bruker Vertex70 FTIR spectrometer, and the measurement schematic is shown in Fig. 2. Considering metallic foams are usually used in high-temperature fields, the interested band is near infrared band according to Wien displacement law. Therefore, this spectrometer is equipped with NIR light source and InGaAs detector with resolution ratio 4 cm1 and measuring band 0.83–2.2 lm. Besides, Michelson interferometer fixed in spectrometer adopts electromagnetic drive without bearing friction and optical compensation with permanent collimation. The sample cell manufactured by Specac Corporation is heatable from room temperature to 1000 K, which is controlled by a temperature controller with precision of

±1 K and cooled by a water cooled circulator. The diagram of experimental device is shown in Fig. 3. During transmission measurements, the beam emitted by light source is modulated by Michelson interferometer, and then transmitted lights from the sample are collected by the detector, which can convert electrical signal to spectrograms displayed on the computer. Considering the complexity of geometric structures in metallic foams, there exists randomness in the transmission measurements, therefore this research operates twenty measurements for each side of samples, and then calculating the average of these results as the transmittance spectra of the sample with corresponding thickness. 4. Radiative properties analysis on nickel foam 4.1. Foam sample studied Two kinds of Nickel foams, having the same porosity (90%) but different cell sizes are studied, and their pictures are shown in Fig. 4. These samples are manufactured by Shanghai Zhongwei Novel Material Company with sintering method. Table 1 shows geometric parameters of researched nickel foams [26] measured by an image analysis software named Image-Pro Plus 6.0, where PPI means the number of pores per inch, qb means the density of metallic foams, / stands for porosity, while dn (=25.4/PPI), dp, ds and dc respectively stand for nominal diameter, pore diameter, strut diameter and cell diameter. 4.2. Transmittance spectra By experimental device in Section 3, the transmittance spectra of 20ppi and 40ppi nickel foams are measured and averaged

FTIR spectrometer

Beam splitter

Detector Sample

Light source

PC High-temperature sample cell

Water cooler circulator Temperature controller Fig. 2. Measurement schematic of transmittance spectra.

Fig. 3. Diagram of experimental device.

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40PPI

20PPI

Fig. 4. Picture of nickel foam.

Table 1 Parameters of Nickel foams [26]. PPI

qb (g/cm3)

/

dn (mm)

dp (mm)

ds (mm)

dc (mm)

20 40

0.9 0.9

0.90 0.90

1.27 0.635

1.168 0.872

0.461 0.366

2.433 1.720

respectively for different sample thicknesses, including 4 mm, 6 mm and 8 mm. The averaged transmittance spectra are drawn in Fig. 5. It can be seen from Fig. 5 that transmittance spectra of nickel foams decrease with the sample thickness increasing or the cell size decreasing. But the variable relationship between transmittance spectra and wavelength is not obvious. 4.3. Extinction coefficient In most researches by experimental method, extinction coefficient of foams is obtained directly by Beer law based on the measured transmittances, which is called the traditional single-layer structure model in this paper. If this method is adopted for nickel foam studied in this article, the calculation results are shown in Fig. 6. In theory, the extinction coefficient only depends on the structural characteristics of the foam [8], while it is independent of the sample thickness. But it can be found from Fig. 6 that extinction coefficients corresponding to different sample thicknesses

have different values, which indicates that extinction coefficients calculated by the single-layer structure model exist errors. Therefore, our study proposes the three-layer structure model to describe metallic foams, namely the edge layer-complete layeredge layer structure. Based on transmittance spectra of nickel foams, extinction coefficients of complete structure are calculated by Eq. (4). And the extinction coefficients are calculated with the transmittance spectra of 4 mm and 8 mm samples, which are plotted in Fig. 7(a). Comparing the calculated results of extinction coefficient with the single-layer structure and three-layer structure as shown in Fig. 7(a), we can find that it will underestimate the real extinction coefficient of metallic foams with the traditional method. Furthermore, the deviation between these two cases are calculated for 20ppi and 40ppi nickel foams respectively and drawn in Fig. 7(b). It can be seen that the deviation percentage for 20ppi is nearly 13%, while that for 40ppi is about 8%, which means the deviation of extinction coefficient calculated by these two different methods decreases with the increasing of cell size. At the same time, we can

20

15

L=4mm L=6mm L=8mm

L=4mm L=6mm L=8mm

10

10

τ (%)

τ (%)

15

5

5

0

0

0.9

1.2

1.5 λ (μm)

(a) 20ppi

1.8

2.1

0.9

1.2

1.5

1.8

λ (μm)

(b) 40ppi

Fig. 5. Transmittance spectra of nickel foams with different thicknesses in near infrared band.

2.1

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480

L=4mm L=6mm L=8mm average

650

β (m-1)

510

β (m-1)

700

L=4mm L=6mm L=8mm average

600

450

550 420

0.9

1.2

1.5

1.8

2.1

0.9

λ (μm)

1.2

(a) 20ppi

1.5

λ (μm)

1.8

2.1

(b) 40ppi

Fig. 6. Extinction coefficients of nickel foams obtained directly by Beer law.

700

20ppi 40ppi

15

δ (%)

β (m-1)

20

20ppi, this model 20ppi, traditional model 40ppi, this model 40ppi, traditional model

600

10

500 0.9

1.2

1.5

λ (μm)

1.8

2.1

5

0.9

(a) extinction coefficient

1.2

1.5

λ (μm)

1.8

2.1

( b ) d e viat i on

Fig. 7. Comparison of extinction coefficients with the traditional single-layer structure model or the three-layer structure model for nickel foams in different porosity.

see that extinction coefficient changes little with wavelength in the researched band, and the value for 20ppi nickel foam is about 530 m1, while that for 40ppi nickel foam is about 650 m1. 4.4. Absorption and scattering coefficients

in good accordance with the calculated results as presented in Fig. 8(a) and (b). Table 3 gives the values of parameters to be determined.

f ð k Þ ¼ k b1

With the calculation model for radiative properties of metallic foams proposed by Zhao [11] in 2004, absorption and scattering coefficients can be calculated from extinction coefficient. Considering the researched metallic foam is different from that in the reference, parameters in Eqs. (5)–(7) need to be corrected. According to extinction coefficient spectra shown with triangles in Fig. 7(a), the extinction coefficient changes with the wavelength, which is not exact the case with Zhao’s model in reference [11] that f ðkÞ is a constant when the wavelength is shorter than 2.5 lm. Therefore the relationship between f ðkÞ and k [m] is fitted anew in this study. Adopting Eq. (8), the parameters to be determined are listed in Table 2 for 20ppi and 40ppi nickel foams by fitting the extinction coefficient. And the corresponding extinction coefficient spectra are drawn in Fig. 8(a) and (b). During the fitting process, we find that the form of f ðkÞ expressed as Eq. (9) can also make the fitted extinction coefficient

Table 2 Parameters in the radiative model with Eq. (7) (n = 1). PPI

C

a1

a2

20 40

0.3392 0.1786

7.605  103 2.569  104

2.386  104 9.638  105

ð9Þ

From Fig. 8(a) and (b), we can see that both of these two fitted equations are in good accordance with the calculated results from Eq. (4), but it seems that the fitted results from Eq. (9) (plotted as solid circle) behaves better. Then we calculated the deviations of extinction coefficients as shown in Fig. 8(c) and (d), which indicate that Eq. (9) gives a less deviation from the calculated results in most researched band. Therefore, we adopted Eq. (9) as the fitted formula of f ðkÞ in the following study. Absorption coefficient and scattering coefficient of metallic foams are related to the reflectivity of base metal material as expressed in Eqs. (5) and (6). The complex refractive index curves of metallic nickel [27] are plotted in Fig. 9, and the corresponding reflectivity curve is calculated by Fresnel equation and drawn in Fig. 10. By interpolation of reflectivity, scattering coefficient and absorption coefficient of metallic foams are calculated from Eqs. (5) and (6) and plotted in Fig. 11. As seen from Fig. 11, scattering effect plays the leading role in the researched wavelength band, and scattering coefficient increases slowly with the increasing of wavelength while absorption coefficient presents the opposite changing trend. For 20ppi nickel foam, scattering coefficient is located in the range of 445–492 m1 and absorption coefficient is located in the range of 40–82 m1, while for 40ppi nickel foam, these two ranges are 540–611 m1 and 50–100 m1.

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534

fitted with f (λ)=λb

fitted with f (λ)=λb

1

656

β (m-1)

532

β (m-1)

calculated fitted with f (λ)=1+exp(a1*(λ+a2))

664

calculated fitted with f (λ)=1+exp(a1*(λ+a2))

530

1

648

640 528

632 0.9

1.2

1.5 λ (μm)

1.8

2.1

(a) extinction coefficient, 20ppi

1.2

1.5

1.8

λ (μm)

2.1

(b) extinction coefficient, 40ppi

0.5

1.2 fitted with f (λ)=1+exp(a1*(λ+a2))

0.4

fitted with f (λ)=λb

fitted with f (λ)=1+exp(a1*(λ+a2))

0.9

fitted with f (λ)=λb

1

1

0.3

δ (%)

δ (%)

0.9

0.6

0.2 0.3

0.1 0.0

0.9

1.2

1.5

λ (μm)

1.8

2.1

0.0

0.9

(c) deviation, 20ppi

1.2

1.5

λ (μm)

1.8

2.1

(d) deviation, 40ppi

Fig. 8. Comparisons between fitted extinction coefficient and calculated results.

Table 3 Parameters in the radiative model with Eq. (8) (n = 1). PPI

C

b1

20 40

2.4376 2.3564

0.0088098 0.032966

0.84

ρ

0.80 10

0.76

n n, κ (m-1)

0.72

κ

8

0.68 6

0.9

1.2

1.5

1.8

2.1

λ (μm) Fig. 10. Reflectivity of nickel.

4

2

0.9

1.2

1.5 λ (μm)

1.8

2.1

Fig. 9. Complex refractive index of nickel [27].

4.5. Temperature influence Using the experimental device with the high-temperature sample cell, transmittance spectra of 40ppi nickel foam are measured

from room temperature to 773 K in the spectral range of 1.3– 2.2 lm, as shown in Fig. 12. The thicknesses of samples are chosen as 5 mm and 10 mm. It can be seen from Fig. 12 that transmittance spectra increase with temperature in the researched band. Then, extinction coefficients under different temperatures are calculated with the double-thickness transmission method based on three-layer structure model and plotted in Fig. 13. According to calculated results, extinction coefficients decrease with temperature and have no significant change with wavelength in the researched band. Within the temperature range from 293 K to 773 K, the relative change rate can reach 2.1% in maximum.

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1000

800

400

200

0

σs

800

α

σs (m-1), α (m-1)

σs (m-1), α (m-1)

σs

600

α

600 400 200

0.9

1.2

1.5

λ (μm)

1.8

0

2.1

0.9

1.2

1.5

1.8

2.1

λ (μm)

(a) 20ppi

(b) 40ppi

Fig. 11. Absorption and extinction coefficient of nickel foam.

0.20 293K 473K 673K

3.0

τ (%)

τ (%)

0.15

373K 573K 773K

2.7 293K 473K 673K

2.4

1.4

1.6

1.8

373K 573K 773K

2.0

0.10

2.2

0.05

1.4

1.6

1.8

λ (μm)

λ (μm)

(a) 5mm

(b) 10mm

2.0

2.2

Fig. 12. Transmittance spectra of 40ppi nickel foams under different temperatures.

750

293K 473K 673K

373K 573K 773K

β (m-1)

700

650

600 1.4

1.6

1.8

2.0

2.2

λ (μm) Fig. 13. Extinction coefficient of 40ppi nickel foam under different temperatures.

5. Conclusion A double-thickness transmission model based on the threelayer structure is introduced to determine the real spectral radiative properties of metallic foams, which can remove the edge effect caused by the cut surface of struts or sintering points. Transmittance spectra are measured with FTIR spectrometer and averaged over multiple data in the near infrared band from 0.83 lm to 2.2 lm. Comparing extinction coefficient calculated by the

traditional single-layer structure model and that obtained by this three-layer structure model, the traditional one will underestimate the extinction coefficient of metallic foams, and the deviation percentage for 20ppi nickel foam is nearly 13%, while that for 40ppi nickel foam is about 8%. By correcting parameters in the prediction model for spectral radiative properties of metallic foams, absorption coefficient and scattering coefficient are calculated, which indicates that scattering effect plays a dominate role in the researched band. Based on this three-layer structure model, temperature influence on the extinction coefficient of nickel foam is studied. By measurement and calculation, transmittance spectra increase with temperature in the researched band, while extinction coefficients decrease with temperature and have no significant change with wavelength from 293 K to 773 K in the researched band. This research supplies a relatively accurate model for determining radiative properties of metallic foams and obtains some basic data of radiative properties of nickel foams, which are useful for corresponding applications.

Conflict of interest The authors declare that there are no conflicts of interest. Acknowledgements This work was supported by the National Natural Science Foundation of China [grant numbers 51676055, 51536001].

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