Modeling of the Effective Permittivity of Open-Cell Ceramic Foams Inspired by Platonic Solids

Abstract

Open-cell solid foams are rigid skeletons that are permeable to fluids, and they are used as direct heaters or thermal dissipaters in many industrial applications. Using susceptors, such as dielectric materials, for the skeleton and exposing them to microwaves is an efficient way of heating them. The heating performance depends on the permittivity of the skeleton. However, generating a rigorous description of the effective permittivity is challenging and requires an appropriate consideration of the complex skeletal foam morphology. In this study, we propose that Platonic solids act as building elements of the open-cell skeletal structures, which explains their effective permittivity. The new, simplistic geometrical relation thus derived is used along with electromagnetic wave propagation calculations of models that represent real foams to obtain a geometrical, parameter-free relation, which is based only on foam porosity and the material’s permittivity. The derived relation facilitates an efficient and reliable estimation of the effective permittivity of open-cell foams over a large range of porosity.

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

“…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 [ 16 ] and with calculations via the diffuse interface representation of the phase-field model used for computing [ 19 ]. The structural models for computing were reconstructed from tomographic scans of open-cell foams, while the structural models for computing correspond to synthetic foam structures using the algorithm proposed by August et al [ 29 ].…”

“…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 .…”

“…At this porosity, both experiments and simulations are well estimated by the OCF relation (see Figure 1 and Figure 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. The synthetic foam structures generated by August et al [ 19 ] are not morphologically identical despite having the same porosity.…”

“…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.…”

“…The Platonic relation for estimating corresponds to the following: where is a correlation parameter that represents topological details of the skeleton morphology; is the porosity; and and 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 ( ), the calculation of also gives a complex-valued quantity ( ) [ 16 ]. In contrast to , and, thus, also are real-valued quantities ( ).…”