Novel Mixing Relations for Determining the Effective Thermal Conductivity of Open-Cell Foams

This paper proposes a new approach to relate the effective thermal conductivity of open-cell solid foams to their porosity. It is based on a recently published approach for estimating the dielectric permittivity of isotropic porous media. A comprehensive assessment was performed comparing the proposed mixing relation with published experimental data for thermal conductivity and with numerical data from state-of-the-art relations. The mixing relation for the estimation of thermal conductivities based on dodecahedrons as building blocks shows good agreement with experimental data over a wide range of porosity.


Introduction
Effective permittivity, ε eff , and effective thermal conductivity, k eff , of open-cell foams are important properties for the design and optimization of microwave-heated elements [1]. Determining both effective properties for foams with a reliable mixing relation based on foam porosity (ratio of the void volume to the total foam volume), bulk properties and skeletal morphology would greatly facilitate the design of systems using microwave-heated elements. Such "cross-property" relations are commonly used to interrelate changes in the effective values of various physical properties (e.g., thermal conductivity, elastic moduli, electrical conductivity and fluid permeability) [2][3][4][5] caused by microstructural features (e.g., pores and inclusions) of heterogeneous materials. Consequently, estimating effective properties turns out to be more complex than calculating a weighted volumetric mean. Predictive relations for ε eff or k eff (or any cross-property) are based on models describing the microstructure of heterogeneous materials. Reliable predictions can only be obtained if the considered structural model resembles features of the foam microstructure. The geometrical representation of the foam morphology in the literature varies from the less accurate idealized assumptions (e.g., ordered, random and symmetrical distributions of solid and void phases) to more complex 2D structures (e.g., hexagonal honeycombs) and 3D unit cell morphology approaches (e.g., simple cubes, Weaire-Phelan unit cell and truncated tetrakaidecahedrons) [5][6][7][8][9][10][11][12].
Historically, the first relations formulated from idealized distributions of solid and void phases are based on the effective medium approximation (EMA) and belongs to the class of mean-field theories [5]. More recent relations are based on geometrical models considering details of the material structure, as well as structures modeled by using the finite element method (FEM).
Recently, we proposed a numerical approach using FEM to derive predictive relations for the effective permittivity of open-cell foams based on two types of morphologies [16]. The first morphology corresponds to foams whose skeletons are based on Platonic solids (such as hexahedron, octahedron, icosahedron and dodecahedron) as building blocks, while the second morphology was reconstructed from micro-CT data of real open-cell ceramic foams. These relations based on Platonic solids and real open-cell foams are referred to as Platonic relation and OCF (open-cell foam) relation, respectively. Both relations agree well with the numerical data obtained from electromagnetic wave propagation calculations, as long as the foam behaves as an effective medium (effective medium approximation). In other words, on the macroscopic scale, foams behave similar to a homogenous medium. However, due to the lack of data, the relations are not yet validated at higher permittivity contrasts, where the EMA approach loses its applicability. Alternatively, the analogy of thermal and electrical networks can equally be used for the skeleton network of open-cell foams [9,17]. Here, extensive experimental and numerical data on the effective thermal conductivity of open-cell foams are available for validation. According to this analogy, it is reasonable to assume that any relation describing ε eff can be used to estimate k eff , or vice versa as a cross-property relation [5,12,15,18]. Please note that only thermal conductivities are considered in this study and not convective heat-transfer coefficients or overall heattransfer coefficients.
Following the thermal-electrical analogy, both Platonic and OCF relations can be used to estimate k eff . The Platonic relation for estimating k eff corresponds to the following: where g is a correlation parameter that represents topological details of the skeleton morphology; P is the porosity; and k s and k f are the thermal conductivities of the bulk materials, i.e., the solid skeleton and the medium that fills the voids of the skeleton, respectively. For complex-valued quantities, such as ε eff (ε eff = ε eff − jε eff ), the calculation of g also gives a complex-valued quantity (g = f g m , −jg m , g 0 − jg 0 ) [16]. In contrast to ε eff , k eff and, thus, also g are real-valued quantities (g = f (g m , g 0 )). As a result, g is calculated as follows: where g m and g 0 are expressed as follows: with Y = {g m ,g 0 }. The respective coefficients, a k , are summarized in Table A1 of the Appendix A. Note that the Platonic geometry is no longer preserved if the size of the struts exceeds a certain limit and struts overlap each other, causing the closure of the open-cell faces. The porosity range in which foams exhibit the Platonic geometry is listed in Table A1. This corresponds to ideal Platonic skeletons and no to real foam skeletons where cell faces close at higher porosities (P~0.5). The corresponding relation for estimating k eff using the OCF relation is as follows: The novelty of this work is the use mixing relations for k eff predictions developed by using FEM electromagnetic wave propagation calculations to estimate the effective permittivity ε eff of foams. Therefore, Equations (1) and (4) represent novel mixing relations for estimating k eff .
In the following assessment, predictions for the k eff of open-cell foams using Platonic and OCF relations, as well as predictions from selected relations from the literature relations are analyzed and compared with experimental and numerical data.

Relation
Expression Remarks This relation ignores k f because it was derived from a correlation for the tortuosity in periodic ordered structures, which is defined by the skeleton structure only.
Weighted arithmetic mean (WAM) [12] Weighted arithmetic mean of the Wiener bounds using the following: Calmidi [8] k Known also as the Bruggeman relation, non-symmetric.

Results and Discussion
The relations inspired by the Platonic solids (bottom row of Figure 1) provide excellent predictions with only minor deviations from the experimental data (see Table 4). The results suggest that the dodecahedral structure (followed by the icosahedral) best mimics the skeleton of real foams. It is important to highlight that other researchers [10,22] have proposed the Kelvin tetrakaidecahedron (known to reproduce packings with low surface area) as the best geometrical element to mimic foams. However, the modeling of this polyhedron has been performed only in two dimensions, using ligaments corresponding to struts and thus lacks important features, such as the effect of the geometrical shape of joints. Accordingly, a better description of tetrakaidecahedron-inspired foams would be required for a fair comparison with the dodecahedral and the icosahedral relations from this study. The predictions from the OCF relation (see lower right graph of Figure 1) are similar to those of the DEM relation, which agrees with our previous study on the estimation of ε eff [16] but deviates-to a certain extent-from the experimental data of k eff . Figure 2 shows the k eff estimated from the relation of Bracconi [9] and from dodecahedron and OCF relations in comparison with those reported from numerical simulations [19]. Figure 1 (upper row) illustrates that the literature relations estimate better for → 1, and this is expected, because they were preferably developed for foams with higher porosity. The best predictions are obtained by the relation from Bracconi (see Table 4). However, it should be mentioned that the Bracconi relation accuracy decreases as increases, since the relation does not include a term. The relations inspired by the Platonic solids (bottom row of Figure 1) provide excellent predictions with only minor deviations from the experimental data (see Table 4). The results suggest that the dodecahedral structure (followed by the icosahedral) best mimics the skeleton of real foams. It is important to highlight that other researchers [10,22] have proposed the Kelvin tetrakaidecahedron (known to reproduce packings with low surface area) as the best geometrical element to mimic foams. However, the modeling of this polyhedron has been performed only in two dimensions, using ligaments corresponding to struts and thus lacks important features, such as the effect of the geometrical shape of joints. Accordingly, a better description of tetrakaidecahedron-inspired foams would be required for a fair comparison with the dodecahedral and the icosahedral relations from this study.
The predictions from the OCF relation (see lower right graph of Figure 1) are similar to those of the DEM relation, which agrees with our previous study on the estimation of [16] but deviates-to a certain extent-from the experimental data of . Figure 2 shows the estimated from the relation of Bracconi [9] and from dodecahedron and OCF relations in comparison with those reported from numerical simulations [19]. estimated from Bracconi, dodecahedron and OCF relations compared with those from numerical simulations of Al, Cu and Ni foams (left) and SS and PU foams (right). The errors bars correspond to the standard deviation as reported by August et al. [19]. Lines from Bracconi's relation are overlapping, as it does not consider the filling medium.  [19]. Lines from Bracconi's relation are overlapping, as it does not consider the filling medium. Figure 2 reveals that, in contrast to the experimental values, the best predictions for the simulated data are obtained from the OCF relation (RMSE OCF = 0.41 < RMSE Bracconi = 1.27 < RMSE Dodecahedron = 1.77). This is consistent with previous electromagnetic wave propagation calculations used for computing ε eff [16] and with calculations via the diffuse interface representation of the phase-field model used for computing k eff [19]. The structural models for computing ε eff were reconstructed from tomographic scans of open-cell foams, while the structural models for computing k eff correspond to synthetic foam structures Materials 2022, 15, 2168 7 of 10 using the algorithm proposed by August et al. [29]. Although the open-cell structural models used for the simulations are different, both are well represented by the OCF relation. Thus, it can be concluded that the simulation models are significantly different from real foams for the following reasons: 1.
For the simulations performed to calculate ε eff , the skeleton morphology was reconstructed from µCT-scans of samples with porosities of 0.90 ± 0.01. At this porosity, both experiments and simulations are well estimated by the OCF relation (see Figures 1 and 2). However, 3D erosion and dilation filters [16] were applied for generating models of different porosity, which-depending on the mesh resolution-may produce significant differences compared with the microstructure of real foams. 2.
The synthetic foam structures generated by August et al. [19] are not morphologically identical despite having the same porosity. Numerically computed and measured experimental values of k eff (as reported by August et al. [19]) only agree if the standard deviation is considered. This indicates that only a few of the synthetic structures are morphologically consistent with real foams.

Conclusions
Recently, relations have been derived to estimate the effective permittivity of opencell foams based on two approaches: (1) using Platonic solids as building blocks of foam skeletons and (2) using the morphology of foam samples extracted from tomographic scans. Based on the thermal-electrical analogy, these relations can be used to estimate cross-properties, such as the effective thermal conductivity. In this work, an assessment of the predictions of the effective thermal conductivity of open-cell foams from these new relations has been presented. The relations have been compared with experimental and numerical data from the literature as well as with predictions from available mixing relations. It has been shown that foam's thermal conductivity can be well estimated from the Platonic relation by using dodecahedrons (which describes foams based on dodecahedral building blocks).
The following recommendations can be derived from this work to properly select the most suitable relation:

1.
For foams with porosities ranging from 0.5 to 1.0 and low bulk thermal conductivities of the filling medium, the Bracconi relation [9] is recommended.

2.
Novel mixing relations are recommended for porosities ranging from 0.5 to 0.9, except for the Platonic relation based on dodecahedrons, which can be applied for porosities from 0.5 to 1.0. In addition, the Platonic relation based on dodecahedrons is recommended over the Bracconi relation for thermal conductivities of the filling medium higher than k f > 1 Wm −1 K −1 .

3.
The relations of Yao [11] and the weighted arithmetic mean approach with arithmetic coefficient proposed by Bhattacharya [14] are recommended if the porosity is higher than 0.85.
Finally, the simulated numerical data were well met by using the OCF relation. However, a significant difference was identified between the numerically predicted values and those from experiments.

Conflicts of Interest:
The authors declare that they are subject to no conflict of interest and that the funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.