Numerical design of dual-scale foams to enhance radiation absorption

Nomenclature: A_N: normal absorptance; d_c: diamter of cell, mm; d_h: diameter of concave hemisphere, mm; I: radiation intensity, W m⁻² sr⁻¹; I_b: blackbody radiation intensity, W m⁻² sr⁻¹; L_f: foam thickness, mm; l_r: reflection path, mm; l_e: extinction path, mm; n: refractive index; p_f: porosity of foam; p_s: porosity of strut; q: radiative heat flux, W m⁻²; R_NH: normal-hemispherical reflectance; $\mathop r\limits^{\rightharpoonup}$: position vector; $\mathop s\limits^{\rightharpoonup}$: propagation direction vector; T_NH: normal-hemispherical transmittance; ${V}_{\mathrm{s}}^{\mathrm{a}}$: apparent volume of strut, mm³; ${V}_{\mathrm{s}}^{\mathrm{r}}$: real volume of strut, mm³

Greek symbols: β: extinction coefficient of solid phase, mm⁻¹; γ: angle degree; κ: absorption coefficient of solid phase, mm⁻¹; λ: wavelength, μm; ξ: random number; ρ: reflectivity; σ_s: scattering coefficient of solid phase, mm⁻¹; Φ: scattering phase function of solid phase, sr⁻¹; Ω': solid angle, sr

Subscripts: a: absorbed; in: incident; r: reflected; s: strut; t: transmitted; λ: spectral

Acronyms: DSF: dual-scale foam; MCRT: Monte Carlo Ray-tracing; μ-CT: micro-computed tomography; GOA: geometric optics approximation

1.   Introduction

Thermal radiation plays a key role in energy applications at high temperature, since the radiative transfer becomes the considerable energy conversion mode due to temperature increasing ^[1,2,3]. To enhance the energy conversion, a kind of porous materials called open-cell foams are usually used in practice ^[4,5,6]. Open-cell foams are commonly composed of interconnected ligaments or called struts which are made of metals, ceramics, carbon and so on ^[7,8]. Thanks to the porous structure of foams, the incident radiation can be absorbed volumetrically throughout the whole material, which improves the efficiency of energy conversion ^[9,10,11].

The factors that affect the radiation absorption of foam materials can be attributed to porous structure and component material properties ^[12,13]. The porous structure is described by the structural parameters such as porosity, pore size (sometimes called pore density) and strut shape factor ^[14,15]. The way to obtain digital foam structure includes micro-computed tomography (μ-CT) technique and artificial reconstruction technique ^[16]. In recent years, more and more attention has been paid to artificial reconstruction technique which is more flexible and allows parameterized study ^[17]. With the development of advanced manufacturing technique such as 3D printing, foam materials with special designed structure (means having specific physical/chemical functions) are coming out ^[18]. Artificial reconstruction is based on different cell models ^[19], such as cube model, Kelvin model, Weaire-Phelan model, and Voronoi model. Among them, the Voronoi model is closer to the real foam structure, because its cell model is random and statistically consistent with the real foam ^[20]. In order to maximize the radiative heat transfer of foam materials in the tube, Pelanconi et al. ^[21] designed and 3D printed a set of alumina ceramic foams with radial gradient cells. Numerical and experimental results showed that the ceramic foams with radial shrinkage cells has better radiation absorption and higher temperature inside the materials. Furler et al. ^[22] designed and manufactured a set of CeO₂ foams with dual-scale pores, as shown in Figure 1, to accelerate the thermal cracking of CO₂. It was found that the reaction speed is increased by at least one order of magnitude due to the porous struts. In short, foam structure design has become a very valuable and promising research field.

In practice, the foams made of strongly absorbing materials such as silicon carbide ceramic are widely popular for radiation absorption ^[23,24]. However, different from silicon carbide ceramic ^[25], the alumina ceramic and zirconia ceramic are by its very nature weakly absorbing within visible and near infrared wavebands ^[26]. Therefore, such ceramic foams are rarely used in applications requiring strong absorption of radiation energy, such as solar absorbers ^[27,28]. In order to expand the potential of such foams, this work aims to digitally design dual-scale ceramic foams in the limit of geometric optics. The paper was organized as follows. In section 2.1, dual-scale foams were designed based on the Voronoi cell model, and then the porous struts were defined for parameterized study. In section 2.2, a Monte Carlo Ray-tracing (MCRT) method accelerated by octree algorithm was developed for radiative transfer calculation. In sections 3.1 and 3.2, the radiation absorption of alumina foam sheets and zirconia foam sheets was analyzed, respectively.

2.   Numerical methods

2.1.   Design of dual-scale foams

As shown in Figure 1, dual-scale foams have a classical porous morphology such as pores at several millimeters, but their struts are still also porous with pores at dozens of microns. Therefore, the structure construction of digital dual-scale foams contains two levels. The first order is the classical foam morphology including open cells and interconnected struts. The second order is the porous struts with smaller holes. The design of dual-scale foams are conducted by Voro++ and SolidWorks software. First, we use open source Voro++ software to perform the Voronoi tessellation to generate a cellular structure composed of solid struts. Then, the SolidWorks software with Application Program Interface (API) is used to modify the solid struts to obtain second-order pores. In the generation of first-order structure, as shown in Figure 2, a random dense packing of spheres with a log-normal distribution of sphere volumes are first generated and distributed irregularly within a given box. Then the Voronoi tessellation is performed to transform this cube box into a cellular structure. Last, open-cell foams are obtained by generating solid struts along the cell edges i.e., between the neighbor cell vertices. For typical open-cell foams, porosity p_f and cell diameter d_c are the main structural parameters in the generation of Voronoi foam, as defined in Ref. ^[19].

Since the present work focuses on the influence of porous struts, the cylindrical struts rather than the ones with complex shape as defined in our recent work ^[15] are used. Inspired by the porous struts shown in Figure 1, a set of porous struts were designed as shown in Figure 3. The struts were dug regularly with concave hemisphere. In order to satisfy geometric optics as much as possible, the diameter of concave hemisphere is given as a much bigger value d_h = 200 μm compared to the investigated wavebands. Furthermore, a structural parameter called 'strut porosity' p_s was defined to characterize the porous struts,

where ${V}_{\mathrm{s}}^{\mathrm{a}}$ is the apparent volume of a strut (containing the volume of concave hemisphere), and ${V}_{\mathrm{s}}^{\mathrm{r}}$ is the real volume of a strut (not containing the volume of concave hemisphere). Obviously, the quantity of strut porosity p_s has a physical sense similar to that of foams and varies from zero to a value less than 1 (in fact far less than 1 due to structural strength requirements).

2.2.   MCRT method for radiative transfer calculation

The assumptions and principles of the in-house MCRT code have been detailed in our recent papers ^[29,30]. Here is a brief introduction. The assumptions include geometric optics approximation (GOA), isothermal foam sheet, transparent void phase, absorbing-scattering semitransparent solid phase (i.e., strut), isotropic scattering inside the solid phase, and smooth solid surface. Based on these assumptions, the MCRT method considers the radiative transfer equation (RTE) in the direction $\mathop s\limits^{\rightharpoonup}$ as a field equation,

where $\mathop r \limits^{\rightharpoonup}$ is the position vector, $\mathop s\limits^{\rightharpoonup}$ is the propagation direction vector, ${\mathop s\limits^{\rightharpoonup} }^{\text{'}}$ is the incoming direction vector, Φ is the scattering phase function, ${{\mathit{\Omega}} }^{\text{'}}$ is the solid angle, I_λ is the spectral radiation intensity, ${I}_{b\lambda }$ is the blackbody intensity given by the Planck function, and ${\kappa }_{\lambda }$, ${\sigma }_{s\lambda }$ and ${\beta }_{\lambda }$ are respectively the spectral absorption, scattering and extinction coefficients, with ${\beta }_{\lambda } = {\kappa }_{\lambda }+{\sigma }_{s\lambda }$.

The physical model was illustrated in Figure 4. Special attention should be paid on a single strut in which the radiative transfer is quite complicated. Reflection and refraction occur at the semitransparent strut surface, and meanwhile, extinction, scattering and absorption happen inside the tiny strut. The calculation of reflection and refraction relies on the Fresnel reflection equation ^[31]. The calculation of extinction, scattering and absorption relies on the media radiation transfer theory ^[32]. The incident radiative flux (q_in, red rays) is assumed to be uniform, collimated and normal to the inlet face of the computational domain. This two-phase domain is compose of transparent void phase (Phase 1) and semitransparent solid phase (Phase 2). Since the refractive index of solid phase n₂ differs from that of void phase n₁, specular reflection and Snell-type refraction take place at the solid-void interface. The local specular reflectivity can be calculated using the Fresnel reflection equation,

where ${\gamma }_{in}$ is the incident angle, and ${\gamma }_{t}$ is the refractive angle. The refractive direction of the transmitted ray is given by Snell's law,

In the Monte Carlo tests, a random number ${\xi }_{\rho }$∈(0, 1) is generated to determine whether the ray is reflected or refracted,

When the ray enters into the semitransparent solid phase, the transmission, absorption and scattering events (as shown in insert of Figure 4) are determined according to the media radiation transfer theory. A possible transfer distance l_e2 is first calculated by the following probability expression,

where β₂ is the extinction coefficient of solid phase (β₂ = κ₂ + σ_s2), and ${\xi }_{\beta }$ is a random number which ranges from 0 to 1 uniformly. When the travel distance l_β2 is reached, another random number ${\xi }_{\omega }$ is generated to decide whether the ray is scattered or absorbed, with ω₂ = σ_s2/β₂ being the scattering albedo of solid phase,

where if the ray is scattered, it will continue to travel on into a new direction determined by the isotropic scattering phase function Φ₂ = 1; if absorbed, it is recorded and a new ray is emitted. The incident rays experience multiple reflection-refraction-scattering-absorption process inside the foams until they exit from the foam sheet through the upper surface or through the lower surface, or until they are absorbed by the solid phase.

In particular, it is assumed that the computational domain is symmetrically continued in the lateral directions, thus simulating an infinite foam slab, as claimed in Ref. ^[33]. Thus, the lateral sides are assumed to be symmetrical boundaries for radiation, which means 'limited specular reflection'. If the rays hit the lateral sides, a specular reflection will happen with limited reflection path l_r = l_e – l_r^', where l_e is the total extinction path of the current ray and l_r^' is the path from the starting point to the hitting point.

By tracking the rays, the normal-hemispherical transmittance T_NH, normal-hemispherical reflectance R_NH, and normal absorptance A_N of the foam sheet can be defined as,

where q presents the radiation energy contained by corresponding rays.

Since the present digital foam has an extremely complex structure, the MCRT calculation may consume quite a lot time. To accelerate the computation, an octree algorithm is combined with the MCRT algorithm, as illustrated in Figure 5. The computational domain is divided into eight equal parts in each partition operation. Accordingly, the solid phase, which means the meshes, is divided into different sub-box. When a ray transmission starts, a simple pre-calculation is firstly conducted to determine the potential path. The sub-boxes that are not in the potential path will not be considered in the MCRT calculation. In this way, the number of meshes to be calculated in MCRT will be greatly reduced. Thus, the computation speed can be accelerated the by nearly two orders of magnitude according to our work ^[34].

3.   Results and discussion

In the following, sections 3.1 and 3.2 investigate the apparent radiative properties of alumina and zirconia foam sheets, respectively. The enhance of radiation absorption due to dual-scale pores is particularly discussed. The optical and radiative properties of component materials (refractive index, absorption coefficient, and scattering coefficient) are necessary for radiation transport calculation. One uses the material properties of alumina ceramics from Yang et al. ^[35] ranging from wavelength 0.4 μm to 7 μm. The material properties of zirconia ceramics are taken from Eldridge et al. ^[36] ranging from wavelength 1.2 μm to 7.6 μm. In order to focus the research on the porous struts, the foam porosity, cell diameter and foam sheet thickness are given as a fixed value, where p_f = 0.85, d_c = 3 mm and L_f = 10 mm.

3.1.   Alumina foam sheets

Figure 6 shows the apparent radiative properties of alumina foam sheets with different strut porosities p_s. Let's focus on the influence of the second-order pores i.e., the porous struts. It can be found that the spectral absorptance increases with the increase of strut porosity, especially in the wavebands from 0.4 μm to 5.5 μm. At the same time, the spectral transmittance and reflectance decrease with the increase of strut porosity. Within the investigated wavebands from 0.4 μm to 7.0 μm, the total absorptance of foams with strut porosities p_s = 0.3, 0.2 and 0.1 (dual-scale-pore foam) are 62.7%, 42.2%, and 20.1% higher than that of p_s = 0 (single-scale-pore foam). The data within corresponding wavebands are listed in Table 1. The dual-scale pores (actually the porous struts) cause the incident radiation to undergo more reflection-refraction-scattering-absorption events in the foam volume, thus increasing the possibility of the radiation energy being absorbed. As a result, the radiation absorption is enhanced in the investigated wavebands.

3.2.   Zirconia foam sheets

Figure 7 shows the apparent radiative properties of zirconia foam sheets with different strut porosities p_s. It can be found that the spectral absorptance increases with the increase of strut porosity, especially in the wavebands from 1.2 μm to 6.5 μm. At the same time, the spectral transmittance and reflectance decrease with the increase of strut porosity. Within the investigated wavebands from 1.2 μm to 7.6 μm, the total absorptance of foams with strut porosities p_s = 0.3, 0.2 and 0.1 (dual-scale-pore foam) are 115.9%, 76.8%, and 36.2% higher than that of p_s = 0 (single-scale-pore foam). The statistics are listed in Table 1. The findings indicate that the dual-scale pores can obviously improve the radiation absorption of zirconia ceramic foams.

4.   Conclusions

To improve the radiation absorption, open-cell foams with dual-scale pores were digitally designed. The first-order pores are obtained by Voronoi tessellation technique and the second-order pores are described by the 'porous strut'. An in-house Monte Carlo Ray-tracing method is applied in the dual-scale foam structure for radiative transfer calculation. The parameterized study is conducted on the radiation absorption of alumina foam sheets and zirconia foam sheets, respectively.

The results show that the designed dual-scale alumina foams with strut porosity p_s = 0.3 can increases the normal absorptance within the wavebands from 0.4 μm to 7.0 μm by 62.7% than the foams with single-scale pores. For the zirconia foams, this increase is 115.9% within the wavebands from 1.2 μm to 7.6 μm. The findings can provide strong confidence on dual-scale foams to enhance the absorption of radiation energy in potential applications. Future work will focus on the design of open-cell foams with complex dual-scale pores and on the 3D-printing-based experiments.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Nos. 52006094, 52076064). It was also supported by talent scientific research fund of LSHU (Nos. 2019XJJL-027, 2020XJJL-006).

Conflict of interest

The authors declare no conflict of interest.