Infrared Physics and Technology 93 (2018) 25–33 Contents lists available at ScienceDirect Infrared Physics & Technology journal homepage: www.elsevier.com/locate/infrared Regular article Scattering properties of solid rough surface of nickel skeleton ⁎ T Bo Liu, Xinlin Xia , Chuang Sun School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China A R T I C LE I N FO A B S T R A C T Keywords: Metal foam Micromorphology of solid skeleton Scattering properties FDTD The micromorphology of solid skeleton makes a signiﬁcant inﬂuence on the radiative transfer in metal foams. The modeling of complex morphology of rough surface and the prediction of bidirectional reﬂectance distribution function (BRDF) are the key issues for the radiative calculation of metal foam. In this work, a modeling method of analyzing and simplifying the complex rough surface using the SEM ﬁgures of metal foam’s skeleton is introduced. This method is also used to predict the scattering properties of nickel foam’s skeleton. Meanwhile, the hemispherical structures are used to replace the real convex structures on skeleton’s surface. These hemispherical convex are located on the surface randomly, and the radius of adapted hemispherical convex respect the distribution function extracted from SEM statistic data. Furthermore, the FDTD method is adapted to predict the reﬂectivity and BRDF of simpliﬁed rough surfaces models. The results show that the reﬂectivity is aﬀected by the size parameter obviously, while it’s changed slightly by diﬀerent incident angles. As for the BRDF of skeleton’s rough surfaces, the value of main peaks increase obviously and the scopes of main peaks decrease with the size parameter increasing. The sub-peaks locate diﬀerently surrounding the main peaks, and then fuse gradually with the increasing size parameter. The sub-peaks vanish completely at the size parameter up to 6. When it comes to the hemispherical space, the reﬂection vary from anisotropic reﬂection to isotropic reﬂection except the mirror direction. At a larger size parameter, the reﬂection of rough surfaces can be simpliﬁed as the combination of mirror reﬂection and diﬀuse reﬂection. While at a small size parameter, the BRDF of rough surface need to be calculated accurately to reduce the error in the prediction of radiative transfer. 1. Introduction Metal foam is an attracting wide focus on thermal applications, such as high-temperature exchangers [1,2], solar receivers [3,4] and so on. Radiative transfer is vital for the heat and mass transfer of metal foam. The structure inside metal foam has great inﬂuence on the radiative transfer. There are lots of excellent works for the research of the radiative transfer of metal foam. Rousseau [5] made the numerical prediction of the radiative behavior of metallic foams with the Monte Carlo Ray Tracing (MCRT) program. Li made the prediction of the spectral reﬂection behaviors of high-porosity metal foam sheets, bidirectional reﬂectance distribution function and directional-hemispherical reﬂectivity with the Monte Carlo ray tracing method [6], and a predictive relationship was established that links the homogenized apparent emissivity to the open porosity and the intrinsic emissivity of solid struts [7]. The modeling of the infrared surface temperature of opencell metallic foam with the discrete-scale approach (DSA) and continuous-scale approach based on Backward Monte Carlo simulation was investigated by Li and Xia [8]. However, in most researches about the radiative transfer in metal foam, the radiative properties of skeleton’s ⁎ Corresponding author. E-mail address: xiaxl@hit.edu.cn (X. Xia). https://doi.org/10.1016/j.infrared.2018.07.018 Received 17 April 2018; Accepted 11 July 2018 1350-4495/ © 2018 Elsevier B.V. All rights reserved. surfaces are usually from assumptions and empirical formulas. Actually, due to the complex morphology on the rough surface of skeleton, the radiative properties of metal foam’s skeleton are usually complex and cannot be described with uniform functions. For getting a better prediction of radiative transfer in metal foam, the radiative properties of skeleton’s rough surface need to be calculated accurately. The extreme complex morphology of skeleton’s surface produces large error between the actual anisotropic reﬂection and the regular expression of functions, and the complex morphology of skeleton’s surface makes the radiative transfer in metal foam hard to predict. So the radiative properties of rough surface are the key issues for further study. For predicting the radiative behavior of rough surfaces, plenty of outstanding works have been fulﬁlled. Parviainen [9] studied the light scattering from self-aﬃne homogeneous isotropic random rough surfaces by using the ray-optics. Zhang [10] applied the updated conventional geometrical optics ray-tracing method by considering the eﬀect of the interference to study the radiative properties of a one-dimensional random rough surface and compared with the results by applied the FDTD method. Feng [11] measured the polarized BRDF of surfaces with an experiment and veriﬁed a hybrid model of polarized Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. BRDF for rough surfaces. Wang [12] established a ﬁve-parameter BRDF model of rough surface and made an experimental veriﬁcation within infrared band. Kang [13] modeled the micro-scaled rough surface respecting the Gaussian distribution and predicted the radiative properties with the FDTD method. Qi [14] analyzed the BRDF of fractal rough surface by applying the numerical calculation. The spectral reﬂectivity of a periodic surface in the thermal radiation wavelength range was predicted by Mendeleyev [15] and the inﬂuence of orientation of rough grooves were discussed. However, almost all the present works are focus on the rough surfaces of bulk material. What’s more, the surfaces are usually modeled by mathematical expression, such as the Gaussian surfaces and the fractal surfaces. However, there are obvious diﬀerences between the bulk material and the skeleton of metal foam due to the various fabrication process. Few studies consider the real morphology of skeleton’s surface, which can’t be described with simple mathematical formulas. The surface of skeletons are usually complex since the fabricating technique [16–18] and the real morphology need to be analyzed by applying advanced technique. Coquard [19] considered the inﬂuence of skeleton’s rough surface on the radiative properties of metal foam and predicted the radiative transfer with geometric optics since the characteristic sizes of the cells are almost always much greater than infrared radiation wavelengths, and the sizes of structures on the skeleton surface were still in the range of geometric optics. However, the application of geometric optics is limited by the size of structures. The Maxwell equations and other electromagnetic theories [20] are needed to study the radiative properties of skeleton rough surfaces when the characteristic sizes of structure are similar to the wavelength. The prediction of radiative properties of skeleton’s surfaces inside metal foam hasn’t attracted further attention. For getting the accurate radiative properties of micro-scaled structures, the numerical techniques should be adapted. The ﬁnite-diﬀerence time-domain (FDTD) method [21] is one of widely used methods for the radiative transfer of micro-scaled structures. Qiu [22] made an analysis of infrared radiative properties of one dimensional periodic aluminum surfaces with various micro-scale triangular gratings using the FDTD method. Chen [23] applied a computational model based on the ﬁnitediﬀerence time-domain method and the Wiener Chaos Expansion (WCE) method to calculate the near-ﬁeld radiative heat transfer between two plates with Gaussian type rough surfaces. Wang [24] studied the absorption characteristics of plasmonic metamaterials with an array of nanoshells with the FDTD method. The reliability of the FDTD method has been proved by plenty of works. In this work, by applying the SEM technique, a method is introduced for building the simpliﬁed model of rough surface of solid skeleton. And the skeleton’s surface of Ni foam is analyzed by applying the method. The FDTD method is adapted to get the reﬂectivity and BRDF of skeleton’s surface with various parameters. The results are compared and discussed. This work can be beneﬁt to the analysis of radiative properties for the radiative transfer in whole metal foam. The model adapted is introduced in Section 2. In Section 3, the numerical method and calculation condition are presented. The results and analyses are discussed in Section 4 and the conclusions are summarized in the following section. Fig. 1. The surface of electrolytic Ni foam’s skeleton from SEM scanning. method is introduced for modeling the simpliﬁed surfaces, and then the method is adapted to model the skeleton’s surfaces of electrolytic Ni foam. The operations of the introduced method are shown as following. (a) Firstly, the ﬁgures of skeleton’s surface such as Fig. 1 are got by applying the SEM technique. The characteristics of the structures on the skeleton surfaces are observed and summarized. The regular structures with common shapes are adapted to simplify the microscale complex structures on the rough surfaces. The adapted simpliﬁed structures are approximate to the micro-scaled structures on the surfaces. (b) Secondly, the technique for analyzing the scanning pictures is used for getting the statistic data of surface’s micro-scaled structures. In this process, the sizes, amounts and other characteristics are counted and analyzed. The statistic data is used to model the microscaled structures, and the modeling of simpliﬁed rough surfaces is carried out later. (c) Furthermore, the simpliﬁed surfaces are modeled with the chosen regular structures. The sizes and locations of adapted simpliﬁed structures are from the statistic data above. The modeled rough surfaces are approximate to the scanning pictures of skeleton and own the statistical characteristics. (d) At last, the radiative properties of modeled rough surfaces can be predicted after calculating. The predicted radiative parameters are compared and analyzed. The predicted properties are the basis of further research about the radiative transfer of metal foam. For better understanding of the introduced method and further researches, the rough surfaces of electrolytic metal Ni foam’s skeleton are modeled by applying the method above. The details and results are shown in following sections in this work. After observing and analyzing Fig. 1, the skeleton’s surface is rough and made up of abundant random convex structures. The random micro-scaled structures are similar to part of sphere with diﬀerent radius. For simulating the real surfaces, the incomplete spheres are adapted to instead the convex structures on the surface. The adapted approximate hemispherical structure is shown in Fig. 2. The hemispherical structure is characterized by two features, including the bottom radius r and the height h as shown in Fig. 3. For better characterizing the surface’s morphology, the technique for analysis of scanning ﬁgure is used to analyze the scanning picture shown in Fig. 1, the bottom radius of diﬀerent convex structures and the distances between adjacent convex structures are measured and counted. For 500 convex structures are measured in this work, the statistic data of radius are shown as Fig. 4 followed. In Fig. 4, r is the bottom radius of convex structure, and N is the 2. Model For the radiative transfer in metal foam, the prediction for radiative properties of skeleton’s surface is a signiﬁcant issue. The surfaces of skeletons are usually complex in morphology, which aﬀect the radiative parameters of the whole foam. The structures on the skeleton’s surface are usually wavelength-scaled as shown in Fig. 1. Fig. 1 is the scanning picture of skeleton of electrolytic Ni foam with the SEM technique. The morphology of metal foam’s skeleton is complex and diﬃcult in constructing the real three-dimensional structure. Therefore, simpliﬁed rough surfaces are needed and the methods of simplifying the surfaces deserve further researches. In this work, a 26 Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. Fig. 2. The incomplete sphere structure adapted to model the convex. number of convex that has the corresponding radius r. From Fig. 4, the distribution of radius satisﬁes the Gaussian distribution. The least square method is used to ﬁt the statistic data and the ﬁtting curve is shown in Fig. 4. The occupying proportion P of convex structures with corresponding radius r can be expressed by the equation as followed P = 0.1804e (−((r − 0.975)/0.2954) 2) (1) From Eq. (1), the distribution of convex’ radius can be described by the Gaussian distribution function with mean value μ = 0.975 μm and standard deviation σ = 0.209. The function of occupying proportion shown in Eq. (1) is used to characterize the distribution of convex’ bottom radius on the rough surfaces discussed here. For each convex structure, the random bottom radius satisﬁes the function of occupying proportion, and the ratio of height to radius r/h respects the uniform distribution from 0.1 to 0.9. According to the statistic results of the SEM ﬁgure 4, 100 convex structures are located on the surface with 25 × 25 μm. The modeled surface with the method above is shown in Fig. 5. The size parameter χ shown in Eq. (2) is used to characterize the surfaces roughness. The λ is the wavelength of incident light, and r is the mean value of bottom radius. χ= 2πr λ Fig. 4. The statistic data of convex structures’ radius on the skeleton’s surface. (2) In this work, the size parameters χ of discussed rough surfaces are 1, 3 and 6. When the sizes of convex structures are similar to the incident wavelength, the computational electromagnetic is needed for the prediction of radiative properties due to the fact that the geometrical optics is limited by the characteristic size. 3. Method The recent numerical calculating methods adapted widely usually need abundant computation resource and these method like the FDTD method, the method of moments and others are time consuming. Thus Fig. 5. The modeled surface with the method introduced above. Fig. 3. The parameters of modeled convex by using the incomplete sphere. 27 Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. N (ω) = ∫ Js (ω) exp(jkr ′·r ) ds′ the domain of computation is always limited in small area by applying numerical technique to solve the Maxwell equations. The work here for the prediction of rough surfaces’ radiative parameters is based on the FDTD method. For the non-magnetic material, the Maxwell equations and constitute equation are as followed: ∂D ∇×H= ∂t ∇ × E = −μ0 S L (ω) = ∫ Ms (ω) exp(jkr ′·r ) ds′ S where j = −1 , k is the wave number, r is the unit vector from the origin point to the far ﬁeld and r′ is the vector from the origin point to the source point. The electric ﬁeld in the far ﬁeld can be described as the following Eq. (11): (3) ∂H ∂t (4) Eθ = j exp(−jKR)(−ηNθ + Lϕ )/(2λR) Eϕ = j exp(−jKR)(−ηNϕ + Lθ )/(2λR) (5) D = ε0 εr E ∫ Sds = ∫ 12 Re[E × H ∗] ds a a (6) where H* is the complex conjugate of the magnetic ﬁeld vector, s is the area of the surface. In this work, the reﬂectivity and BRDF are discussed. The reﬂectivity R is deﬁned by R= Pr Pi (7) where Pr is the power of reﬂected light and Pi is the power of incident light. The BRDF in this work are deﬁned by equations as following BRDF (θr , φr ) = dPr (θr , φr ) Pi (θi , φi ) cos(θr ) dΩr (θr , φr ) 4. Results (8) where θi, φi are the zenith angle and azimuth angle of incident light, and the θr, φr denote the zenith angle and azimuth angle of reﬂected light. dΩi and dΩr are the detector solid angles of incident light and reﬂected light. The dΩ is shown in Fig. 6. For getting the BRDF of modeled rough surface of metal foam, the transformation from radiation in near-ﬁeld to far-ﬁeld is accomplished with the equivalence principle. For getting the radiative energy in far ﬁeld, a closed imaginary surface is located in the computation domain, and has the normal vector n. The equivalent electric current Js and magnetic current Ms can be get from the following equations. Js (ω) = n × H (ω) Ms (ω) = −n × E (ω) (11) where R is the distance from the source point, η is the wave impedance in free space. After getting the electric ﬁeld in far ﬁeld, the radiative power and BRDF can be calculated. The radiative parameters can be predicted after the calculation above. In this work, the incident light is non-polarization and the wavelengths are in the near infrared band. The boundary of calculating domain is period for the computation with the FDTD method. For verifying the accuracy and validity of the FDTD method adapted in this work, the absorptivity of tungsten grating calculated by the RCWA in reference [25] are calculated again by applying the adapted FDTD method in this work. The grating has periodic length of 1.6 μm, height of 0.2 μm and width of 1.28 μm. The results predicted by the FDTD method here are compared with the calculated results in reference [25] and depicted in Fig. 7. From the comparison in Fig. 7, the FDTD method used in this work is reliable for the prediction of radiative properties of discussed structures. where H, E and D represent the magnetic ﬁeld, the electric ﬁeld and the electric displacement vector. μ0 is the permeability of vacuum, the ε0 is the permeability of vacuum. εr is the relative permittivity. The timeaveraged power ﬂow across a surface is deﬁned by p= (10) In this work, a method is introduced for analyzing the rough surfaces of skeleton and modeling the simpliﬁed surface. The method is carried out according to the statistic characteristics of micro-scaled structures on the metal foams’ skeleton. Then, the rough surfaces of Ni foam’s skeleton is modeled by the approach. The modeled rough surfaces are made up of various random hemispherical convex structures shown in Section 2. The radiative properties including the reﬂectivity and BRDF are predicted with the FDTD method. The complex refraction n + ik of metal Ni are shown in Fig. 8 [26]. In the following prediction for the radiative properties of modeled skeleton’s surfaces, ﬁve diﬀerent surfaces with same parameters are produced by the modeling method introduced above. The ﬁve surfaces are adapted to model the skeleton surface. The radiative parameters of the surfaces are calculated. The (9) where ω is the frequency of electromagnetic wave. The potential functions can be deﬁned as Eq. (10). Fig. 7. The comparison between the results from FDTD method and reference [25]. Fig. 6. The hemisphere space above the calculated surface. 28 Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. whole hemispherical space above the rough surfaces. The numbers of calculated positions for the prediction of BRDF are about 160,000 for diﬀerent cases, and these positions distribute over the discrete zenith angle and azimuth angle. The ﬁgures in following sections are depicted with points dropped since the calculated points are too dense. 4.1. The reﬂectivity The reﬂectivity of rough surfaces of Ni foam’s skeleton discussed above are shown in Fig. 9. The size parameters of modeled surfaces are 1, 3 and 6 with the incident angles are 0°, 30° and 60°. The reﬂectivity of diﬀerent cases with diﬀerent incident angles and size parameters are shown in Fig. 9. From Fig. 9, the reﬂectivity increases with decreasing size parameter. The reﬂectivity of diﬀerent incident angles are similar for a certain size parameter. The variety is the same as the various complex refraction shown in Fig. 8. For larger incident angle of 60°, the reﬂectivity decrease slightly for all the size parameters. The reﬂectivity is aﬀected by the size parameter obviously and change slightly with diﬀerent incident angles. Fig. 8. The complex refraction n + ik of metal Ni. 4.2. The BRDF The BRDF of rough surface is scattered over the whole hemisphere space. The BRDF in hemisphere space shows the distribution of reﬂected energy by the rough surface. In this work, the top views are shown in following discussions which are more convenient to be observed and analyzed in case that the distribution in whole space is hard to observe completely. The whole hemisphere space is shown in Fig. 10(a). What’s more, the BRDF in the incident plane shown in Fig. 10(b) is extracted for further analysis. The incident light and reﬂected light are shown in Fig. 10(b). In this plane, the +y direction correspond to the incident angle θi range from 0° to 90°, while the reﬂected angle θr is from −90° to 0°. 4.2.1. The rough surface of χ = 1 The rough surface’s BRDF in hemispherical space with size parameter χ = 1 are shown in Fig. 11. From Fig. 11, there is one main peak of BRDF in the mirror direction of incident light. And several sub-peaks exit surrounding the main peak. In other directions, the values of BRDF are very small compared with the main peaks and sub-peaks, and the BRDF of other directions are anisotropic obviously. The BRDF in the incident plane are shown in Fig. 11(d), the values of main peaks are similar under diﬀerent incident angles. Furthermore, the mirror reﬂections take the domination, and own most of reﬂected power. From Fig. 11(d), for the bigger incident angle, the location of main peak deviates slightly from the direction of mirror direction. Fig. 9. The reﬂectivity of metal Ni foam’s skeleton rough surface with various parameters. mean results of the ﬁve diﬀerent surfaces are used to characterize the properties of surfaces for removing the error produced by randomicity from the process of modeling surfaces. In the following sections, the predicted BRDF are scattered over the Fig. 10. The extracted incident surface for further analysis. 29 Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. Fig. 11. The BRDF in hemispherical space and incident surface of rough surface with χ = 1 (a) the incident angle θi is 0°; (b) the incident angle θi is 30°; (c) the incident angle θi is 60°; (d) the BRDF extracted from the incident plane. 4.2.2. The rough surface of χ = 3 The rough surface’s BRDF in hemispherical space with size parameter χ = 3 are shown in Fig. 12, the BRDF in incident plane are shown in Fig. 12(d). As same as the case of χ = 1, there are main peaks in the mirror direction for incident angle are 0° and 30°, but the main peak deviates from the mirror location slightly for the larger incident angle of 60°. The main peaks own most reﬂected power, too. However, there are some diﬀerences for bigger size parameter, ﬁrstly, sub-peaks in diﬀerent directions fuse gradually, which causes the sub-peaks become obscure. Furthermore, the diﬀerence between the values of main peaks 30 Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. Fig. 12. The BRDF in hemispherical space and incident surface of rough surface with χ = 3 (a) the incident angle θi is 0°; (b) the incident angle θi is 30°; (c) the incident angle θi is 60°; (d) the BRDF extracted from the incident plane. 31 Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. Fig. 13. The BRDF in hemispherical space and incident surface of rough surface with χ = 6 (a) the incident angle θi is 0°; (b) the incident angle θi is 30°; (c) the incident angle θi is 60°; (d) the BRDF extracted from the incident plane. 4.2.3. The rough surface of χ = 6 The rough surface’s BRDF in hemispherical space with size parameter χ = 6 are shown in Fig. 13. Compared with the cases discussed above, the varieties of BRDF are similar. First of all, for largest size parameter of χ = 6, the reﬂections in mirror directions become more ever-dominant and concentrated. The scopes of main peaks are narrowed further. The values of main peaks are increased and the diﬀerences between diﬀerent incident angles are enhanced further. Secondly, under diﬀerent incident angles are enlarged, and the values of main peaks increase substantially while the scopes of the main peaks decreases obviously. The reﬂected power are concentrated further near the mirror directions. What’s more, the isotropy in other directions are enhanced. The reﬂections in other directions except the mirror direction are turning to the diﬀuse gradually. 32 Infrared Physics and Technology 93 (2018) 25–33 B. Liu et al. reﬂection. However, when the size parameters are small, the BRDF of rough surfaces need to be calculated accurately to reduce the error in the prediction of radiative transfer. The method introduced here is a choice for modeling the skeletons’ rough surfaces for further research. The results of BRDF distributions of discussed surfaces that are adapted to model the rough surfaces of Ni metal foam’s skeletons can be beneﬁt to the radiative transfer in metal foam further and the applications of foam materials for later studies. the sub-peaks in diﬀerent directions fuse completely. The sub-peaks develop to the whole region surrounding the main peaks. Thirdly, the reﬂections in other directions are almost isotropy completely. This part of reﬂected power can be treated as the diﬀuse reﬂection for the prediction of radiative transfer. Take all the conditions considered, the BRDF of modeled rough surfaces vary with increasing size parameters and incident angles. (1) There are main peaks near the mirror direction of incident light that own most of reﬂected power for all the cases. For a bigger incident angle, slight deviation of the location appears. With the size parameter increasing, the values of main peaks increase obviously and the scopes of main peaks decrease. (2) For smaller size parameter, the sub-peaks exit in diﬀerent locations surrounding the main peaks. These sub-peaks fuse gradually with increasing size parameter. When the size parameter become 6, the sub-peaks vanish completely and become whole regions with isotropic reﬂection surrounding the main peaks. (3) For other directions expect the mirror directions in the hemispherical space, the reﬂections vary from anisotropic reﬂection to isotropic reﬂection. For the largest size parameter discussed here, the reﬂections in other directions can be treated as the diﬀuse reﬂection. For larger size parameters, the reﬂections of rough surfaces can be simpliﬁed into the combination of mirror reﬂection and diﬀuse reﬂection. However, when the size parameters are small, the BRDF of rough surfaces need to be calculated accurately to reduce the error in the prediction of radiative transfer. Acknowledgement The work is supported by the National Natural Science Foundation of China (No. 51536001). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.infrared.2018.07.018. References [1] A. Banerjee, R.B. Chandran, J.H. Davidson, Experimental investigation of a reticulated porous alumina heat exchanger for high temperature gas heat recovery, Appl. Therm. Eng. 75 (2015) 889–895. [2] H. Shokouhmand, F. Jam, M.R. Salimpour, The eﬀect of porous insert position on the enhanced heat transfer in partially ﬁlled channels, Int. Commun. Heat Mass Transf. 38 (8) (2011) 1162–1167. [3] A.L. Ávila-Marín, Volumetric receivers in solar thermal power plants with central receiver system technology: a review, Solar Energy 85 (5) (2011) 891–910. [4] M.I. Roldan, E. Zarza, J.L. Casas, Modeling and testing of a solar-receiver system applied to high-temperature processes, Renew. Energy 76 (2015) 608–618. [5] B. Rousseau, J.Y. Rolland, P. 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Hsu, FDTD analysis of infrared radiative properties of microscale structure aluminum surfaces, J. Quant. Spectrosc. Radiat. Transf. 111 (12) (2010) 1912–1920. [23] Y. Chen, Y. Xuan, The inﬂuence of surface roughness on nanoscale radiative heat ﬂux between two objects, J. Quant. Spectrosc. Radiat. Transf. 158 (6) (2015) 52–60. [24] Z. Wang, X. Quan, Z. Zhang, et al., Numerical studies on absorption characteristics of plasmonic metamaterials with an array of nanoshells, Int. Commun. Heat Mass Transf. 68 (2015) 172–177. [25] Y.B. Chen, Z.M. Zhang, Design of tungsten complex gratings for thermophotovoltaic radiators, Opt. Commun. 269 (2) (2007) 411–417. [26] E.D. Palik, E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, 1985. 5. Conclusion The radiative transfer progress in metal foam is complex and depends on the BRDF of skeleton surface strongly. The prediction of reﬂectivity and BRDF is an attractive issue in the radiative transfer calculation. In order to obtain the reﬂectivity and BRDF of skeleton surface, the model used to simulate the real rough surface and suitable method are crucial questions for the prediction of metal foam’s radiative properties. In this work, a method of modeling the skeleton rough surfaces is introduced to further analysis of radiative transfer in metal foam. This method is adapted to model and analyze the skeleton’s surface of electrolytic Ni metal foam then. The FDTD method is adapted to calculate the reﬂectivity and BRDF in the near infrared band. For complete analysis of the radiative properties of skeleton’s rough surfaces, diﬀerent size parameters of 1, 3 and 6 are compared and the incident angles range from 0° to 60°. The conclusions can be summarized as followed. 1. The reﬂectivity of skeleton’s rough surfaces increases with decreasing size parameter and the reﬂectivity of diﬀerent incident angles are similar for a certain size parameter. For a larger incident angle, the reﬂectivity decreases for all the size parameters. The reﬂectivity is aﬀected by the size parameter obviously and is changed slightly by diﬀerent incident angles. 2. For the BRDF of skeleton’s surfaces, there are main peaks that own most of reﬂected power near the mirror directions of incident lights. Slight deviations of location appear when incident angles become larger. With the size parameter increasing, the values of main peaks increase obviously while the scopes of main peaks decrease. The sub-peaks exit in diﬀerent locations surrounding the main peaks fuse gradually with increasing size parameter. When the size parameter become 6, the sub-peaks vanish completely and become whole regions with isotropic reﬂection surrounding the main peaks. 3. For other directions in the hemispherical space, the reﬂections vary from anisotropic reﬂection to isotropic reﬂection. For the largest size parameter discussed here, the reﬂection in other directions can be treated as the diﬀuse reﬂection. For larger size parameters, the reﬂection of rough surface can be simpliﬁed into the combination of mirror reﬂection and diﬀuse 33

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