# On the Effective Thermophysical Properties of Phase Change Materials Embedded in Metallic Lattice Structures with Generic Topological Parameters

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## Abstract

**:**

## 1. Introduction

## 2. Derivation of the Lattice Cell Geometric Properties

#### 2.1. Derivation of the Porosity

## 3. Effective Thermal Conductivity Model

- no convection or radiation is present;
- the thermophysical properties remain constant over the temperature interval across a unit cell;
- thermal equilibrium is assumed.

- All unit cells are symmetric in all three spatial directions lying on planes that pass through the center point. Thus, such symmetry is exploited to simplify the calculations.
- The thermal resistance of the strut intersections is modeled as a cuboid node (Figure 8). Since the cuboid is obtained as an equivalent volume of Steinmetz’s solid, the number and type of struts (i.e., face-centered or body-centered) that cross at a specific node are important in defining its resistance.
- Only the lattice structure is taken into account. The filler, i.e., PCM, is included as thermal resistance placed in parallel to one of the lattice unit cells.

#### 3.1. Three-Direction

#### 3.2. One- and Two-Direction

#### 3.3. Cell Topology

#### 3.3.1. Cuboid Prisms

#### 3.3.2. Hexagonal Prismatic Cells

#### 3.4. Simplified Models

## 4. Validation

#### 4.1. Porosity

#### 4.2. Effective Thermal Conductivity

#### 4.2.1. Finite Element Model

^{®}. The assumption is made that the only relevant heat transfer mode is heat conduction. A stationary and a transient model are developed. The stationary model is used to determine the ETC. The transient model is used to investigate the time-dependent behavior of the composite PCM and the effect of the cell size on it.

**Figure 12.**Simulation model for the calculation of the ETC and the applied boundary conditions, with a constant value of $\dot{q}$.

#### 4.2.2. Three-Direction

#### 4.2.3. One- and Two-Direction

^{®}Multiphysics. Figure 17 shows two plots where the ${f}_{2}ccz$ and the $bccz$ unit cells are validated against experimental results at a heating orientation opposite to gravity, so that the influence of natural convection can be neglected. While the maximum deviation amounts to 8.58%, this can be attributed to the approximation introduced by the apparent heat capacity method, rather than an error in the calculation of the thermophysical properties.

## 5. Results and Discussion

#### 5.1. Strut Radius

#### 5.2. Effect of Aspect Ratio (Angle $\phi $)

#### 5.3. Scale Variance: Effect of Cell Size and Number of Cells

## 6. Conclusions

- The cells exhibiting the highest effective thermal conductivity in the 3-direction, i.e., the direction of the z struts, at a given porosity, are the face-centered ones, i.e., ${f}_{2}cc$,${f}_{2}ccz$, $hpfcz$, $tpfcz$.
- In the 1,2-direction, the effective thermal conductivity is obviously reduced for all cells presenting a z strut, and the trends are partially inverted. Body-centered lattices exhibit higher effective thermal conductivity than face-centered ones. Hexagonal cells exhibit the peculiarity of showing the highest and the lowest effective thermal conductivities. The $hpfcz$ consistently shows the highest value in the 3-direction, while the $hpbcz$ exhibits the lowest value. In the 1,2-direction, the trend is reversed and the $hpbcz$ shows the highest effective thermal conductivity.
- In general, an increase in the aspect ratio, i.e., increasing the strut angle, leads to higher effective thermal conductivities for unit cells with porosity within ca. 75%.
- The effect of the cell size, also called scale variance, indirectly affects the accuracy of the proposed analytical equations. Indeed, if the domain volume is fixed, the variation in the cell size directly influences the number of cells that can fit within such a volume. The number of cells directly affects the transient thermal behavior. A minimum number of cells of 10 is identified as appropriate to describe the transient thermal behavior of the analyzed composites via homogenized models based on effective thermophysical properties. If this threshold is maintained, the cell size has no effect on the transient thermal behavior of the composite. Otherwise, a coupled effect of the reduction in the number of cells and of increasing the cell size can be evidenced. Additionally, the porosity has a direct influence on the scale variance. Indeed, at high porosities, the scale variance, related to the variable $\mathsf{\Gamma}$ of Equation (37), is reduced, while it increases with decreasing porosity.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PCM | Phase Change Material |

ETC | Effective Thermal Conductivity |

BC | Body-Centered |

BCC | Body-Centered Cubic |

BCCZ | Body-Centered Cubic with Z Strut |

FC | Face-Centered |

F2CC | Face-Centered Cubic |

F2CCZ | Face-Centered Cubic with Z Strut |

F2BCC | (Combined) Face-Centered and Body-Centered Cubic |

F2BCCZ | (Combined) Face-Centered and Body-Centered Cubic with Z Strut |

HPFCZ | Hexagonal Prism Face-Centered with Z Strut |

HPBCZ | Hexagonal Prism Body-Centered with Z Strut |

TPFCZ | Triangular Prism Face-Centered with Z Strut |

## Nomenclature

$\Omega $ | Strut angle with respect to the horizontal |

$\phi $ | Aspect ratio angle of the unit cell |

h | Unit cell height |

L | Unit cell width |

r | Strut radius |

$\alpha $ | Angle generated by the intersection of two crossing struts |

$\epsilon $ | Porosity of the unit cell |

$\chi $ | Volume fraction of the unit cell |

${\delta}_{i}$ | Number of struts of type i in a unit cell (i.e., face-centered, body-centered, or z struts) |

${F}_{i}$ | i-th empirical factor for the calculation of the volume of the i-th Steinmetz’s solid |

${t}_{n}$ | Edge length of the equivalent prismatic volume of the Steinmetz’s solid for ETC calculation |

${R}_{i}$ | Thermal resistance of the i-th element (node or strut) of the unit cell |

${\sigma}_{i}$ | Empirical parameter for the calculation of the ETC |

$\lambda $ | Thermal conductivity |

$\rho $ | Density |

${C}_{p}$ | Specific heat capacity |

${H}_{f}$ | Latent heat of fusion |

## Appendix A. Material Properties

$\mathit{\lambda}$ [W/mK] | $\mathit{\rho}$ [kg/m${}^{3}$] | $\mathbf{Cp}$ [J/kgK] | Melting Point [°C] | Latent Heat [kJ/kg] | |
---|---|---|---|---|---|

n-Octadecane (solid) | 0.358 | 814 | 2150 | 29 | 244 |

n-Octadecane (liquid) | 0.152 | 774 | 2180 | - | - |

Al-6061 | 170 | 2700 | 1100 | - | - |

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**Figure 1.**Cubic unit cells. (

**a**) Face-centered cubic, ${f}_{2}cc$; (

**b**) face-centered cubic with z strut, ${f}_{2}ccz$; (

**c**) body-centered cubic, $bcc$; (

**d**) body-centered cubic with z strut, $bccz$; (

**e**) combined face-centered and body-centered cubic, ${f}_{2}bcc$; (

**f**) combined face-centered and body-centered cubic with z strut, ${f}_{2}bccz$.

**Figure 2.**The investigated hexagonal prism unit cells: (

**a**) hexagonal prism with face-centered struts and vertical struts, $hpfcz$; (

**b**) triangular prism with face-centered and vertical struts (i.e., the hexagon is split into triangles, on which the face-centered struts lie), $tpfcz$; (

**c**) hexagonal prism with body-centered and vertical struts, $hpbcz$.

**Figure 3.**Schematic diagram of the existing effective thermal conductivity models for cellular solids, i.e., foams and lattices. The model presented in this work belongs to the orientation-based layerwise methods.

**Figure 4.**Flowcharts introducing the framework of this study and its fundamental steps. (

**a**) Framework of the present study. (

**b**) Detailed description of the framework for the presented model.

**Figure 5.**Illustration of the unit cell geometric parameters based on a cell with face-centered struts. The 3-direction corresponds to the z strut direction.

**Figure 7.**Workflow to obtain the effective thermal conductivity. (

**a**) Determining the strut types and the quantity of each type, (

**b**) transforming the intersections into cubes and calculating the intersection volumes, (

**c**) modeling the struts in three layers and determining the resistance of each layer, (

**d**) calculating the overall strut resistance using thermal network calculations.

**Figure 9.**(

**a**) Two ${f}_{2}ccz$ cells stacked in 3-direction; (

**b**) two cells stacked in 1,2-direction.

**Figure 10.**Intersection of the struts at the cell corner of the hexagonal cells $hpfcz$ and $hpbcz$.

**Figure 11.**Comparison of the different models for calculation of the porosity of the cell type ${f}_{2}cc$. Either the radius r or the angle $\phi $ is varied. The height of all cells is ${h}_{cell}=10$ mm.

**Figure 13.**Convergence study of the stationary model for the ${f}_{2}bccz$ cell, which requires the most elements.

**Figure 14.**Comparison of the different models for calculation of the ETC in 3-direction of the cell type ${f}_{2}ccz$. The radius is varied between $r=0.1$ mm and $r=1.3$ mm. The height of all cells is $h=10$ mm, while the angle is specified in the caption under the plot. The model for aspect ratio 1 ($\phi ={45}^{\circ}$) is validated against the results of Hubert et al. [22]

**Figure 15.**Comparison of the different models for calculation of the ETC in 1,2-direction of the cell type $bccz$. The radius is varied between $r=0.1$ mm and $r=1.3$ mm. The height of all cells is $h=10$ mm, while the angle is specified below the plot. The model for aspect ratio 1 ($\phi ={45}^{\circ}$) is validated against the results of Hubert et al. [22].

**Figure 16.**Example of the overlap of the struts at high angles for the ${f}_{2}ccz$ cell. In (

**a**), the cell is shown from an isometric view; in (

**b**), a cross-section cut in the middle vertical plane is shown.

**Figure 17.**Comparison of the simulation results based on a homogenized composite and the experimental data from [7].

**Figure 18.**Effective thermal conductivity of all considered cells for varying porosity obtained by varying only the strut radius.

**Figure 19.**Effect of angle (i.e., aspect ratio) variation for the ${f}_{2}ccz$ unit cell at varying porosities, in the 3-direction.

**Figure 20.**Effect of varying the aspect ratio angle on the effective thermal conductivity (continuous lines) and on the porosity (dashed lines), respectively, in the 3-direction. Three different radii and thus different porosity ranges are considered. The cell size is fixed at 5 mm.

**Figure 21.**Effect of varying the aspect ratio angle on the effective thermal conductivity (continuous lines) and porosity (dashed lines) in the 1,2-direction. Three different radii and thus different porosity ranges are considered. The cell size is fixed at 5 mm.

**Figure 22.**Cell size effect on the effective thermal conductivity obtained by scaling the strut radius according to the cell size variation.

Intersection | $\mathit{\alpha}$ |
---|---|

2 fc struts | $\pi -2\phi $ |

2 bc struts | $\pi -2\Omega $ |

fc and z strut | $\pi /2-\phi $ |

bc and z strut | $\pi /2-\Omega $ |

**Table 2.**Coefficients for the empirical porosity calculation. The RMSE indicates the root mean squared error over the used parameter set.

Cell Type | ${\mathit{V}}_{\mathit{tot}}$ | ${\mathit{\delta}}_{\mathit{bc}}$ | ${\mathit{\delta}}_{\mathit{fc}}$ | ${\mathit{\delta}}_{\mathit{z}}$ | ${\mathit{F}}_{1}$ | ${\mathit{F}}_{2}$ | ${\mathit{F}}_{3}$ | ${\mathit{F}}_{4}$ | RMSE [%] |
---|---|---|---|---|---|---|---|---|---|

${f}_{2}cc$ | $\frac{{h}_{cell}^{3}}{{tan}^{2}\left(\phi \right)}$ | 0 | 4 | 0 | 3.061 | 1.954 | 0 | 0 | 0.38 |

${f}_{2}ccz$ | $\frac{{h}_{cell}^{3}}{{tan}^{2}\left(\phi \right)}$ | 0 | 4 | 1 | 2.935 | 3.667 | 0 | 0 | 0.18 |

$bcc$ | $\frac{{h}_{cell}^{3}}{{tan}^{2}\left(\phi \right)}$ | 4 | 0 | 0 | 0 | 0 | 2.993 | 3.340 | 0.13 |

$bccz$ | $\frac{{h}_{cell}^{3}}{{tan}^{2}\left(\phi \right)}$ | 4 | 0 | 1 | 0 | 0 | 3.137 | 4.923 | 0.33 |

${f}_{2}bcc$ | $\frac{{h}_{cell}^{3}}{{tan}^{2}\left(\phi \right)}$ | 4 | 4 | 1 | 3.940 | 4.380 | 3.706 | 4.190 | 0.07 |

${f}_{2}bccz$ | $\frac{{h}_{cell}^{3}}{{tan}^{2}\left(\phi \right)}$ | 4 | 4 | 1 | 3.741 | 5.874 | 3.340 | 4.779 | 0.61 |

$hpfcz$ | $\frac{3\sqrt{3}{h}_{cell}^{3}}{2{tan}^{2}\left(\phi \right)}$ | 0 | 6 | 2 | 5.133 | 4.756 | 0 | 0 | 0.98 |

$hpbcz$ | $\frac{3\sqrt{3}{h}_{cell}^{3}}{2{tan}^{2}\left(\phi \right)}$ | 6 | 0 | 2 | 0 | 0 | 5.093 | 8.334 | 0.18 |

$tpfcz$ | $\frac{3\sqrt{3}{h}_{cell}^{3}}{2{tan}^{2}\left(\phi \right)}$ | 0 | 18 | 3 | 12.907 | 20.254 | 0 | 0 | 0.28 |

**Table 3.**Coefficients to calculate the ETC in 3-direction for cubic cell types. The RMSE is calculated varying the topological cell setups.

Cell Type | ${\mathit{\delta}}_{\mathit{fc}}$ | ${\mathit{\delta}}_{\mathit{bc}}$ | ${\mathit{\delta}}_{\mathit{z}}$ | ${\mathit{n}}_{1,\mathit{fc}}$ | ${\mathit{n}}_{3,\mathit{fc}}$ | ${\mathit{n}}_{1,\mathit{bc}}$ | ${\mathit{n}}_{3,\mathit{bc}}$ | ${\mathit{\sigma}}_{1,\mathit{fc}}$ | ${\mathit{\sigma}}_{3,\mathit{fc}}$ | ${\mathit{\sigma}}_{1,\mathit{bc}}$ | ${\mathit{\sigma}}_{3,\mathit{bc}}$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

${f}_{2}cc$ | 4 | 0 | 0 | 4 | 2 | - | - | 2.024 | 1.793 | - | - | 2.09 |

${f}_{2}ccz$ | 4 | 0 | 1 | 4 | 2 | - | - | 1.448 | 1.168 | - | - | 1.65 |

$bcc$ | 0 | 4 | 0 | - | - | 4 | 4 | - | - | 1.903 | 1.903 | 1.59 |

$bccz$ | 0 | 4 | 1 | - | - | 4 | 4 | - | - | 1.537 | 1.537 | 1.85 |

${f}_{2}bcc$ | 4 | 4 | 0 | 8 | 2 | 8 | 4 | 2.145 | 1.311 | 2.145 | 1.311 | 1.01 |

${f}_{2}bccz$ | 4 | 4 | 1 | 8 | 2 | 8 | 4 | 1.654 | 1.185 | 1.654 | 1.133 | 1.00 |

**Table 4.**Coefficients to calculate the ETC in 1- and 2-direction for cuboid cell types. The RMSE is calculated with different topological cell setups. For this direction, high angles $\phi $ cannot be respected as it leads to large material accumulation at the intersections. For the cells marked with a *, the factor is multiplied by $tan\vartheta $ as in Equation (20).

Cell Type | ${\mathit{\delta}}_{\mathit{fc}}$ | ${\mathit{\delta}}_{\mathit{bc}}$ | ${\mathit{n}}_{1,\mathit{fc}}$ | ${\mathit{n}}_{3,\mathit{fc}}$ | ${\mathit{n}}_{1,\mathit{bc}}$ | ${\mathit{n}}_{3,\mathit{bc}}$ | ${\mathit{\sigma}}_{1,\mathit{fc}}$ | ${\mathit{\sigma}}_{3,\mathit{fc}}$ | ${\mathit{\sigma}}_{1,\mathit{bc}}$ | ${\mathit{\sigma}}_{3,\mathit{bc}}$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|---|

${f}_{2}cc$ | 2 | 0 | 4 | 2 | - | - | 2.618 | 3.316 | - | - | 1.14 |

${f}_{2}ccz$ | 2 | 0 | 4 | 2 | - | - | 1.732 | 2.836 | - | - | 1.54 |

$bcc$ | 0 | 4 | - | - | 4 | 4 | - | - | 4.277 | 4.277 | 1.52 |

$bccz$ | 0 | 4 | - | - | 4 | 4 | - | - | 10.95 * | 4.073 | 1.61 |

${f}_{2}bcc$ | 4 | 4 | 8 | 2 | 8 | 4 | 10.86 * | 2.420 | 10.86 * | 3.840 | 1.87 |

${f}_{2}bccz$ | 4 | 4 | 8 | 2 | 8 | 4 | 9.576 * | 3.468 | 9.576 * | 4.564 | 2.38 |

**Table 5.**Coefficients to calculate the ETC in 3-direction for hexagonal cell types. The RMSE is calculated with different topological cell setups.

Cell Type | ${\mathit{\delta}}_{\mathit{fc}}$ | ${\mathit{\delta}}_{\mathit{bc}}$ | ${\mathit{\delta}}_{\mathit{z}}$ | ${\mathit{n}}_{1,\mathit{fc}}$ | ${\mathit{n}}_{3,\mathit{fc}}$ | ${\mathit{n}}_{1,\mathit{bc}}$ | ${\mathit{n}}_{3,\mathit{bc}}$ | ${\mathit{\sigma}}_{1,\mathit{fc}}$ | ${\mathit{\sigma}}_{3,\mathit{fc}}$ | ${\mathit{\sigma}}_{1,\mathit{bc}}$ | ${\mathit{\sigma}}_{3,\mathit{bc}}$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

$hpfcz$ | 6 | 0 | 2 | 3 | 2 | - | - | 1.027 | 1.803 | - | - | 1.27 |

$tpfcz$ | 18 | 0 | 3 | 6 | 2 | - | - | 1.671 | 1.178 | - | - | 2.75 |

$hpbcz$ | 0 | 6 | 2 | - | - | 3 | 6 | - | - | 1.484 | 1.754 | 2.93 |

**Table 6.**Coefficients to calculate the ETC in 1- and 2-direction for hexagonal cell types. The RMSE is calculated with different topological cell setups. For this direction, high angles $\phi $ cannot be respected as it leads to large material accumulation at the intersections. For the cells marked with a *, the factor is multiplied by $tan\vartheta $ as in Equation (20).

Cell Type | ${\mathit{\delta}}_{\mathit{fc}}$ | ${\mathit{\delta}}_{\mathit{bc}}$ | ${\mathit{n}}_{1,\mathit{fc}}$ | ${\mathit{n}}_{3,\mathit{fc}}$ | ${\mathit{n}}_{1,\mathit{bc}}$ | ${\mathit{n}}_{3,\mathit{bc}}$ | ${\mathit{\sigma}}_{1,\mathit{fc}}$ | ${\mathit{\sigma}}_{3,\mathit{fc}}$ | ${\mathit{\sigma}}_{1,\mathit{bc}}$ | ${\mathit{\sigma}}_{3,\mathit{bc}}$ | RMSE |
---|---|---|---|---|---|---|---|---|---|---|---|

$hpfcz$ | 2/2 | 0 | 3 | 2 | - | - | 2.815 | 2.483 | - | - | 2.5 |

$tpfcz$ | 6/2 | 0 | 6 | 2 | - | - | 8.880 * | 2.584 | - | - | 1.81 |

$hpbcz$ | 0 | 4 | - | - | 3 | 6 | - | - | 8.080 * | 8.192 | 2.86 |

Cell Type | ${\mathit{\delta}}_{\mathit{fc}}$ | ${\mathit{\delta}}_{\mathit{bc}}$ | ${\mathit{\delta}}_{\mathit{z}}$ | ETC |
---|---|---|---|---|

${f}_{2}cc$ | 4 | 0 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}{tan}^{2}\left(\phi \right)\pi \frac{{r}^{2}}{{h}^{2}}4sin\left(\phi \right)$ |

${f}_{2}ccz$ | 4 | 0 | 1 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}{tan}^{2}\left(\phi \right)\pi \frac{{r}^{2}}{{h}^{2}}\left(4sin\left(\phi \right)+1\right)$ |

$bcc$ | 0 | 4 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}{tan}^{2}\left(\phi \right)\pi \frac{{r}^{2}}{{h}^{2}}4sin(\Omega )$ |

$bccz$ | 0 | 4 | 1 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}{tan}^{2}\left(\phi \right)\pi \frac{{r}^{2}}{{h}^{2}}\left(4sin(\Omega )+1\right)$ |

${f}_{2}bcc$ | 4 | 4 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}{tan}^{2}\left(\phi \right)\pi \frac{{r}^{2}}{{h}^{2}}\left(4sin\left(\phi \right)+4sin(\Omega )\right)$ |

${f}_{2}bccz$ | 4 | 4 | 1 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}{tan}^{2}\left(\phi \right)\pi \frac{{r}^{2}}{{h}^{2}}\left(4sin\left(\phi \right)+4sin(\Omega )+1\right)$ |

$hpfcz$ | 6 | 0 | 2 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+2\frac{2\pi {r}^{2}tan{\phi}^{2}{\lambda}_{s}}{3\sqrt{3}{h}^{2}}+6\frac{2\pi {r}^{2}tan{\phi}^{2}{\lambda}_{s}}{3\sqrt{3}{h}^{2}}sin\phi $ |

$tpfcz$ | 18 | 0 | 3 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+3\frac{2\pi {r}^{2}tan{\phi}^{2}{\lambda}_{s}}{3\sqrt{3}{h}^{2}}+18\frac{2\pi {r}^{2}tan{\phi}^{2}{\lambda}_{s}}{3\sqrt{3}{h}^{2}}sin\phi $ |

$hpbcz$ | 0 | 6 | 2 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+2\frac{2\pi {r}^{2}tan{\phi}^{2}{\lambda}_{s}}{3\sqrt{3}{h}^{2}}+6\frac{2\pi {r}^{2}tan{\phi}^{2}{\lambda}_{s}}{3\sqrt{3}{h}^{2}}sin{\Omega}_{hex}$ |

Cell Type | ${\mathit{\delta}}_{\mathit{fc}}$ | ${\mathit{\delta}}_{\mathit{bc}}$ | ${\mathit{\delta}}_{\mathit{z}}$ | ETC |
---|---|---|---|---|

${f}_{2}cc$ | 2 | 0 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}\pi \frac{{r}^{2}}{{h}^{2}}2sin\left(\phi \right)$ |

${f}_{2}ccz$ | 2 | 0 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}\pi \frac{{r}^{2}}{{h}^{2}}2sin\left(\phi \right)$ |

$bcc$ | 0 | 4 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}\pi \frac{{r}^{2}}{{h}^{2}}4sin(\Omega )$ |

$bccz$ | 0 | 4 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}\pi \frac{{r}^{2}}{{h}^{2}}4sin(\Omega )$ |

${f}_{2}bcc$ | 2 | 4 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}\pi \frac{{r}^{2}}{{h}^{2}}\left(2sin\left(\phi \right)+4sin(\Omega )\right)$ |

${f}_{2}bccz$ | 2 | 4 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}\pi \frac{{r}^{2}}{{h}^{2}}\left(2sin\left(\phi \right)+4sin(\Omega \right)$ |

$hpfcz$ | 2/2 | 0 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+{\lambda}_{s}\frac{2\sqrt{3}\pi {r}^{2}sin\left(\phi \right)}{3{h}^{2}}$ |

$tpfcz$ | 6/2 | 0 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+3{\lambda}_{s}\frac{2\sqrt{3}\pi {r}^{2}sin\left(\phi \right)}{3{h}^{2}}$ |

$hpbcz$ | 0 | 6 | 0 | ${\lambda}_{eff}=\epsilon {\lambda}_{f}+4{\lambda}_{s}\frac{2\sqrt{3}\pi {r}^{2}sin(\Omega )}{3{h}^{2}}$ |

**Table 9.**Unit cell geometries considered for experimental validation based on [7].

${\mathit{f}}_{2}\mathit{ccz}$ | $\mathit{bccz}$ | |
---|---|---|

Cell size | 5 mm | 5 mm |

Radius | 1 mm | 1 mm |

$\phi $ | 45° | 45° |

**Table 10.**Porosity of the unit cells considered in the stationary case, with cell size ranging from 5 to 30 mm and radius from 0.5 to 3 mm.

${\mathit{f}}_{2}\mathit{cc}$ | ${\mathit{f}}_{2}\mathit{ccz}$ | $\mathit{bcc}$ | $\mathit{bccz}$ | ${\mathit{f}}_{2}\mathit{bcc}$ | ${\mathit{f}}_{2}\mathit{bccz}$ | $\mathit{hpfcz}$ | $\mathit{hpbcz}$ | $\mathit{tpfcz}$ | |
---|---|---|---|---|---|---|---|---|---|

$\epsilon $ | 0.85 | 0.83 | 0.82 | 0.80 | 0.71 | 0.68 | 0.90 | 0.84 | 0.74 |

**Table 11.**Example of the geometric variables for the study of the scale variance considered for the case of a ${f}_{2}ccz$ cell with porosity 0.95.

Number of Cells | Cell Size [mm] | Radius [mm] | Aspect Ratio Angle $\mathit{\phi}$ [°] | $\mathit{\u03f5}$ |
---|---|---|---|---|

20 | 2.5 | 0.129 | 45 | 0.95 |

10 | 5 | 0.259 | 45 | 0.95 |

5 | 10 | 0.517 | 45 | 0.95 |

4 | 12.5 | 0.647 | 45 | 0.95 |

2 | 25 | 1.294 | 45 | 0.95 |

1 | 50 | 2.587 | 45 | 0.95 |

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**MDPI and ACS Style**

Piacquadio, S.; Soika, J.; Schirp, M.; Schröder, K.-U.; Filippeschi, S.
On the Effective Thermophysical Properties of Phase Change Materials Embedded in Metallic Lattice Structures with Generic Topological Parameters. *Thermo* **2023**, *3*, 566-592.
https://doi.org/10.3390/thermo3040034

**AMA Style**

Piacquadio S, Soika J, Schirp M, Schröder K-U, Filippeschi S.
On the Effective Thermophysical Properties of Phase Change Materials Embedded in Metallic Lattice Structures with Generic Topological Parameters. *Thermo*. 2023; 3(4):566-592.
https://doi.org/10.3390/thermo3040034

**Chicago/Turabian Style**

Piacquadio, Stefano, Johannes Soika, Maximilian Schirp, Kai-Uwe Schröder, and Sauro Filippeschi.
2023. "On the Effective Thermophysical Properties of Phase Change Materials Embedded in Metallic Lattice Structures with Generic Topological Parameters" *Thermo* 3, no. 4: 566-592.
https://doi.org/10.3390/thermo3040034