Numerical modelling consists of the following steps: micro-computed tomography data acquisition, generation of a three-dimensional model, triangulated surface mesh generation, volume mesh generation and FE-analysis. A detailed description of this process is given in [23].

Figure 1 shows the boundary conditions of the finite element model for thermal conductivity calculations in y-direction. The thermal conductivity can be determined in three perpendicular directions (i.e., x, y and z) by changing the constant temperature planes. All calculations are performed under steady state conditions and the four lateral surfaces are adiabatic where no thermal energy exchange occurs with the surrounding. The nodal heat reaction flux perpendicular to the surfaces with the constant temperature is calculated by the finite element analysis. The total heat flux is the sum of the single nodal heat fluxes through one of the boundary surfaces. Within this study, the thermal radiation effect on heat transfer is neglected since its contribution is very low for the relevant temperature range which is below 700 K [24]. The influence of thermal convection is also disregarded because sintered HSS represents mainly a closed-cell structure with a low volume fraction of interconnected porosity [2]. In this study, pores are assumed to be filled with air. Due to the low thermal conductivity of the gas mixture, conductive heat transfer inside the pore space is negligible. These assumptions simplify the mathematical description of the heat transfer and the effective thermal conductivity can be calculated according to Equation (1) with Fourier’s law.

The parameters of Fourier’s law are given in this paragraph: = heat flux, Δy = spatial distance between two opposing boundary surfaces, A = projected area, T₁ = temperature on the top, T₂ = temperature on the bottom. The geometry shown in Figure 1 consists of approximately 5 cells per edge length and consequently possesses about 5³ = 125 cells in total. This number exceeds the approximately 100 cells required for a representative volume [15]. The model size is 18 × 18 × 18 mm and is represented by 400³ voxels resulting in a resolution of 0.045 mm/voxel. The FE-model consists of roughly five million elements (i.e., hexahedral-, tetrahedral- and pentahedral elements).

Every sinterable material can be used in order to manufacture HSS with the powder based processing technique. Some thermal conductivities are given in the following text where the thermal conductivity is shown in brackets: titanium (6.9 W/m K), stainless steel (12.8 W/m K), high carbon steel (50 W/m K), pure molybdenum (138 W/m K) and aluminium (180 W/m K) [12]. Micro pores appear in the sphere wall due to the powder based manufacturing method and decrease thermal conductivity. Optical microscopy images are analysed of sintered HSS made of high carbon steel and a micro-porosity of p = 3.8% was detected. Based on this value, a correction factor C₁ is calculated with the Maxwell-type expression [15,25] in order to account for the decrease in conductivity due to micro-porosity. This model assumes that all micro-pores are of similar size and homogeneously distributed in the sphere wall.

The second influence on the effective thermal conductivity is the meso-structure which is denoted as C₂. Finally, the effective thermal conductivity can be calculated with Equation (3), which incorporates the thermal conductivity λ_B of the base material, the micro-porosity C₁ and meso-structure C₂.

The convergence analysis for the sintered HSS is performed on a smaller fraction of the calculation model (i.e., side length 7.5 mm) in order to reduce calculation time. The convergence analysis can be distinguished between geometrical convergence and numerical convergence. The geometric convergence requires a sufficient number of elements in order to accurately capture the complex geometry by using primitive geometric element shapes (i.e., tetrahedrons, pentahedrons and hexahedrons). Numerical convergence is achieved by an adequate number of nodes for the mathematical description of the temperature field. The results of the convergence analysis are shown in Table 1 along with the mesh density, which is the number of elements divided by the solid volume fraction of the model. The absolute volume deviation |ΔV| is used to quantify geometric convergence shown in Equation (4).

Variable V_3D is the solid volume of the micro-computed tomography data set and V_FE is the solid volume of the discretised calculation model. For perfect geometry the accuracy of the absolute volume deviation |ΔV| becomes zero. Numerical convergence is ensured for models with mesh densities of 0.88 or higher, i.e., the conductivity does not significantly change. Accordingly, the mesh density of 0.88 is chosen for all following computational models.

Figure 2 shows the five investigated geometries. All models exhibit different relative densities or wall thickness to radius ratios t_w/r. The radius and wall thickness are averaged measurement values based on evaluated computed tomography cross sectional images. Figure 2a shows the original geometry obtained by µCT images of a real HSS sample and Figure 2e the modified geometry with completely filled spheres. The intermediate modifications to the sphere wall thickness are shown in Figure 2(b–d) where the sphere wall thickness is progressively increased towards the inside of the spheres. This task was carried out by altering the wall thickness on each recorded micro-computed tomography image. Note, that the volume deviation (i.e., Equation (4)) between the 3D-object and the discretised finite element model is less than 0.7% for all models. This means that, the geometry is captured very well with the discretised numerical models. The difference between the thermal conductivities in three perpendicular directions (i.e., x, y and z) and their arithmetical mean is less than 2% in all cases. Therefore, the sintered HSS exhibit quasi-isotropic material behaviour in terms of the thermal conductivity. In contrast, other cellular materials such as m-pore^® and Alporas^® reveal anisotropic material behaviour [18]. Figure 4a shows the arithmetical mean (i.e., x, y and z) of the normalised thermal conductivity plotted as a function of the relative density ρ_r. It can be seen that the numerical data points follow very well a linear regression line. This is also shown with the coefficient of determination R² = 0.99 [9]. In addition, the results are in good agreement with a linear function by Fiedler et al. [16] formulated to predict the normalised conductivity for thin-walled structures. Finally, good agreement is obtained with a previous thermal analysis of sintered HSS based on micro-computed tomography [15] (see triangular marker in Figure 4a). Figure 4b shows the arithmetical mean (i.e., x, y and z) of the normalised thermal conductivity as a function of the ratio of wall thickness t_w to radius r. Numerical results are shown by filled squares and can be described very well with a 2nd order polynomial regression. The good agreement between numerical data points and polynomial regression is shown with the R² value of 0.99.

It must be mentioned that the packing density ϕ (i.e., the fraction of the volume occupied by the filled spheres) is ϕ = 0.744. It is interesting to compare this value with the maximum possible packing density ϕ_max ≈ 0.74 of spherical particles and typical packing densities of spherical packed beds ϕ = 0.64 [26]. The high value ϕ = 0.744 of the investigated geometry is slightly more than the theoretical maximum which can be explained by plastic deformation during the manufacturing process. The spheres are flattened on the diffusion bonded contact area (see Figure 3) between the hollow spheres resulting in good thermal contact of neighbouring spheres (as opposed to point contact). The current study aims to model the change of conductivity with increasing radius ratio t_w/r. However, the manufacturing of sintered HSS with thicker sphere walls is likely to yield less plastic deformation due to the rapidly increasing strength of the spheres. This results in smaller contact areas between neighbouring spheres. As a consequence, the numerical results of the current study are strictly valid for the considered model structures but may slightly overestimate the conductivities of real structures.