# Influence of Particle Shape on Tortuosity of Non-Spherical Particle Packed Beds

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Tortuosity Models

## 3. Methods

#### 3.1. Particle Shape and Packing Generation

^{3}(Figure 1). Sphericity is calculated from the ratio of the equal-volume sphere surface area to the particle surface area,

#### 3.2. Domain and Thermal Simulations

^{3}from the inner region of the bed that was used for the thermal simulations. This domain was selected based on our previous work [49], where cubical particle-packed beds were simulated, and a domain study was carried out using additional micro-computed X-ray tomography (µ-CT) measurement results. The effect of the wall diminishes for $(R-r)/{d}_{p}$ > 3.22 at $D/{d}_{p}$ = 10.48 (${d}_{p}$ is the diameter of a sphere of equal volume as the particle with ${V}_{p}$ = 27 mm

^{3}), which is considered in this work. Regarding domain independence, 15 × 15 × 15 mm

^{3}provided the same results as a larger domain [49].

^{−4}W/(mK) for the particles and of ${\lambda}_{g}$ = 1 W/(mK) for the gas phase were used in the simulations; thus, the ratio ${k}_{p}={\lambda}_{p}/{\lambda}_{g}$ could be considered so small, as to completely neglect the conduction in the solid phase. The ICEM CFD toolbox from ANSYS was utilized for mesh generation. The mesh was generated based on the cell size of the interface between the phases. Finally, a maximum cell size of 0.1 mm (2.6% of ${d}_{p}$) was taken after grid independence studies for all shapes. These are discussed in Section 4.2.1. Temperature boundary values of 50 °C and 20 °C ($\Delta T$ = 30 K) were specified at two opposite faces of the computational domain, with all other faces at adiabatic conditions. Simulations were carried out for different shapes with three different realizations and in the two lateral directions, x and y (refer to Figure 2a). Lateral values were chosen because of the isotropic nature of the beds in respective directions, whereas minor differences may occur in the z-direction (axial direction), as observed in the previous work [49].

## 4. Results and Discussion

#### 4.1. Packing Results

^{3}was investigated. Figure 3b provides the average bed and domain porosities obtained from three different realizations with respect to different particle shapes, denoted by particle IDs. The domain porosities without the wall, top, and bottom effects were smaller than the overall bed porosities. For further study, these average domain values were considered.

#### 4.2. Thermal Results

#### 4.2.1. Mesh Independence Study for All Shapes

^{3}, as illustrated in Figure 2. The mesh sizing was based on the interface between the solid and gas phase, on which an unstructured octree mesh with a tetrahedral shape was placed. This interface mesh was then expanded to a volume mesh. Consequently, the maximum interface cell size is of importance for the ability of the simulation to resolve gaps between particles. This cell size varied from 0.09 to 0.15 mm for the study. This study was performed in the x-direction for all shapes, refer to Figure 2. The effective thermal conductivity with non-conducting particles, ${k}_{bed}({k}_{p}\to 0)$, which is equivalent to $\epsilon /\tau $ (Equation (1)), is calculated using the heat flux obtained from simulations. The results of the study are depicted in Figure 5, where it is seen that the results are independent from the mesh for a maximal surface cell size of 0.1 mm, which is 2.6% of ${d}_{p}$. The change in ${k}_{bed}({k}_{p}\to 0)$ was negligible from either cell size of 0.12 mm or at 0.1 mm. This was observed for all of the particle shapes; hence, 0.1 mm surface cell size was used for the simulations for every realization and also in the y-direction (the other lateral direction perpendicular to x). As mentioned, an average of three realizations in two directions for each shape is considered for further evaluation.

#### 4.2.2. Comparison of Results with Old Models

^{2}value and systematic deviations of the data points along the ZBS curve, which is an indication of particles with different shape parameters affecting the values. These discrepancies or dependencies could be avoided by introducing further physical parameters into the ZBS model (Equation (4)). Respective modification of the ZBS model equation is discussed in Section 4.4. Prior to that, the dependence of the tortuosity on the particle shape needs to be studied in Section 4.3.

#### 4.3. Effect of Particle Shape on Tortuosity

^{2}value of 0.9672, which shows good agreement for the sphericity range used in this work, i.e., $\varphi $ = 0.65–1. However, it will be further refined, as be seen in Section 4.4.

#### 4.4. Modification of Tortuosity Equation—ZBS Model

- -
- Porosity for our range of simulations was found to have a weak correlation to tortuosity, which is also observed in Figure 6b. Hence, the effect of other parameters is essential.
- -
- When considering $F$, $E$, and $V$, these variables are strongly interrelated to each other but weakly related to tortuosity. With porosity, they are moderately inversely related. This effect is detected in Figure 4, where higher porosities can be seen for lower values of $F$, $E$, and $V$.
- -
- As already indicated, sphericity dominates the correlation coefficient analysis for tortuosity, with those variables being proportional to each other.

^{2}= 0.9562, similar to Equation (10). The proposed model is compared with the old model from Lanfrey et al. [46] in Figure 11a. This model had a sphericity relation with tortuosity, but it is completely off the present simulations. Figure 11b demonstrates the accuracy of the proposed model compared with simulation results and to the ZBS model without sphericity term. However, it should be noted that Equation (11) has been derived considering limited ranges of porosity and sphericity, $\epsilon $ = 0.3–0.4 and $\varphi $ = 0.65–1, respectively. In these ranges, sphericity has an at least equally strong influence as porosity. Large changes in porosity, however, may increase its influence compared with the influence of sphericity. Thus, this work should be regarded as a first step in deriving a universal model to predict tortuosity based on particle parameters. Adding more parameters, like aspect ratios and pore morphology, could provide a model usable for a wide range of porosity.

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$a,\text{}b,\text{}c$ | fitting coefficients, Equation (10) | - |

${A}_{p}$ | particle surface area | m^{2} |

${A}_{ss}$ | specific surface area | m^{−1} |

$C$ | adjusting parameter [39] | - |

$D$ | cylinder/bed diameter | m |

${D}_{0}$ | coefficient of unconfined diffusion | m^{2}/s |

${D}_{eff}$ | effective diffusion coefficient | m^{2}/s |

${d}_{p}$ | diameter of equivalent volume sphere | m |

$E$ | number of particle edges | - |

$F$ | number of particle faces | - |

${k}_{p}$ | ratio of particle-to-gas conductivity | - |

${k}_{bed}$ | ratio of bed-to-gas conductivity | - |

$m$ | empirical constant, Equation (11) | - |

$n$ | empirical constant, Equation (11) | - |

$N$ | number of mesh elements | - |

${q}_{x}$ | total directional heat flux | W/m^{2} |

${q}_{x,i}$ | directional heat flux of element | W/m^{2} |

$R$ | bed radius | m |

${v}_{i}$ | volume fraction of mesh element | - |

$V$ | number of particle vertices | - |

${V}_{p}$ | particle volume | m^{3} |

$\epsilon $ | porosity | - |

${\lambda}_{p}$ | particle conductivity | W/(mK) |

${\lambda}_{g}$ | gas conductivity | W/(mK) |

${\lambda}_{bed}$ | effective bed conductivity | W/(mK) |

$\tau $ | tortuosity | - |

${\tau}_{ZBS}$ | tortuosity from the ZBS model, Equation (4) [47] | - |

$\varphi $ | sphericity | - |

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**Figure 2.**(

**a**) Blender simulation of bed filling with particle array of 250 particles and (

**b**) completely packed bed. (

**c**) The smaller cubical domain highlighted in red was selected for thermal simulations, it is illustrated here with hexagonal prisms.

**Figure 3.**Porosities for different particle shapes: (

**a**) radial porosity profiles, red-dotted lines represent the domain considered for thermal simulations and (

**b**) comparison of bed and simulation domain porosities.

**Figure 4.**Variation of porosity with particle shape parameters for respective particle IDs: (

**a**) sphericity; (

**b**) number of faces; (

**c**) number of edges; and (

**d**) number of vertices.

**Figure 5.**Mesh independence study; dotted red represents the region where results are independent of the mesh size for all particle shapes.

**Figure 6.**Comparison of simulated results with prediction models: Equation (2) [34]; Equation (3) [36]; Equation (4) [47]; Equation (5) [39] with $C$ = 0.63 for cubes; Equation (7) [46] with $\varphi $ = 0.806 for cubes; and Equation (6) [41]. (

**a**) Effective residual thermal conductivity or effective diffusivity and (

**b**) tortuosity.

**Figure 7.**(

**a**) Comparison of residual conductivity results with the ZBS model, Equation (4). (

**b**) Expanded region of interest shown in green lines in (

**a**).

**Figure 8.**Comparison of bed tortuosity as a sphericity function; dotted line represents the correlation of simulation data according to Equation (10).

**Figure 9.**Effect of particle shape parameters on tortuosity with fitted curves: (

**a**) number of faces; (

**b**) number of edges; and (

**c**) number of vertices.

**Figure 10.**Correlation matrix between various parameters, where red represents a strong correlation.

**Figure 11.**(

**a**) Comparison of the change in tortuosity with sphericity according to the proposed model and the Lanfrey model and (

**b**) comparison between simulated tortuosity values with ZBS predictions and with values predicted by the here developed model, according to Equation (11).

**Table 1.**Overview of sphericity and surface area of all particles together with assigned particle ID (${A}_{p}$ in mm

^{2}).

Particle ID | Particle Type | $\mathbf{Surface}\text{}\mathbf{Area},\text{}{\mathit{A}}_{\mathit{p}}$ | $\mathbf{Sphericity},\text{}\mathit{\varphi}$ |
---|---|---|---|

1 | Sphere | 43.54 | 1.000 |

2 | Icosahedron | 46.34 | 0.939 |

3 | Dodecahedron | 47.81 | 0.911 |

4 | Cylinder | 49.83 | 0.874 |

5 | Octahedron | 51.47 | 0.846 |

6 | Hexagonal prism | 51.59 | 0.844 |

7 | Cube | 54.00 | 0.806 |

8 | Square pyramid | 60.58 | 0.719 |

9 | Triangular prism | 60.79 | 0.716 |

10 | Tetrahedron | 64.85 | 0.671 |

**Table 2.**Results of $\epsilon $, $\epsilon /\tau $, and $\tau $ for different particle shapes and their deviation among three realizations.

Particle ID | Particle Type | $\mathit{\epsilon}$ | $\mathit{\epsilon}/\mathit{\tau}$ | $\mathit{\tau}$ |
---|---|---|---|---|

1 | Sphere | 0.364 ± 0.0084 | 0.248 ± 0.0062 | 1.470 ± 0.0105 |

2 | Icosahedron | 0.333 ± 0.0016 | 0.197 ± 0.0123 | 1.689 ± 0.0115 |

3 | Dodecahedron | 0.315 ± 0.0117 | 0.182 ± 0.0067 | 1.731 ± 0.1417 |

4 | Cylinder | 0.319 ± 0.0011 | 0.179 ± 0.0074 | 1.777 ± 0.0597 |

5 | Octahedron | 0.342 ± 0.0044 | 0.191 ± 0.0064 | 1.791 ± 0.0292 |

6 | Hexagonal prism | 0.344 ± 0.0135 | 0.194 ± 0.0080 | 1.769 ± 0.0129 |

7 | Cube | 0.356 ± 0.0030 | 0.194 ± 0.0049 | 1.836 ± 0.1276 |

8 | Square pyramid | 0.370 ± 0.0046 | 0.196 ± 0.0055 | 1.882 ± 0.0359 |

9 | Triangular prism | 0.332 ± 0.0075 | 0.206 ± 0.0116 | 1.838 ± 0.0465 |

10 | Tetrahedron | 0.379 ± 0.0098 | 0.180 ± 0.0009 | 1.848 ± 0.0296 |

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**MDPI and ACS Style**

Rodrigues, S.J.; Vorhauer-Huget, N.; Richter, T.; Tsotsas, E.
Influence of Particle Shape on Tortuosity of Non-Spherical Particle Packed Beds. *Processes* **2023**, *11*, 3.
https://doi.org/10.3390/pr11010003

**AMA Style**

Rodrigues SJ, Vorhauer-Huget N, Richter T, Tsotsas E.
Influence of Particle Shape on Tortuosity of Non-Spherical Particle Packed Beds. *Processes*. 2023; 11(1):3.
https://doi.org/10.3390/pr11010003

**Chicago/Turabian Style**

Rodrigues, Simson Julian, Nicole Vorhauer-Huget, Thomas Richter, and Evangelos Tsotsas.
2023. "Influence of Particle Shape on Tortuosity of Non-Spherical Particle Packed Beds" *Processes* 11, no. 1: 3.
https://doi.org/10.3390/pr11010003