# A Study on Shape-Dependent Settling of Single Particles with Equal Volume Using Surface Resolved Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

#### 2.1. Drag Coefficient

#### 2.2. Shape Parameter

#### 2.3. Particle Representation

#### 2.4. Drag Correlations for Non-Spherical Particles

## 3. Numerical and Statistical Methods

#### 3.1. Particle Generation

#### 3.2. Statistical Tools

## 4. Numerical Experiments

#### 4.1. Preparation

#### 4.2. Simulation Setup

#### 4.3. Validation

## 5. Results and Discussion

#### 5.1. Examination of Simulation Data

#### 5.1.1. Data Processing

#### 5.1.2. Shape Classes and Influence of Shape Parameters

#### 5.1.3. Analysis of Exceptions

#### 5.2. Regression Analysis

#### 5.2.1. Polynomial Regression Regarding the Drag Coefficient

#### 5.2.2. Polynomial Regression Regarding Terminal Settling Velocity

## 6. Conclusions

## Author Contributions

## Funding

`CAPES`) [

`DAAD/CAPES PROBRAL`88881.198766/2018-01; CAPES-Finance Code 001]-Brazil.

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HLBM | homogenized lattice Boltzmann method |

LBM | lattice Boltzmann method |

Roman | |

${a}_{\mathrm{I}}$ | intermediate half-axis |

${a}_{\mathrm{L}}$ | longest half-axis |

${a}_{\mathrm{S}}$ | shortest half-axis |

A | projected particle surface in direction of motion |

${A}_{\mathrm{p}}$ | particle surface |

${C}_{\mathrm{D}}$ | drag coefficient |

${C}_{\mathrm{D},\mathrm{BB}}$ | drag coefficient according to Bagheri and Bonadonna [52] |

${C}_{\mathrm{D},\mathrm{CG}}$ | drag coefficient according to Clift and Gauvin [17] |

${C}_{\mathrm{D},\mathrm{DM}}$ | drag coefficient according to Dioguardi and Mele [26] |

${C}_{\mathrm{D},\mathrm{G}}$ | drag coefficient according to Ganser [21] |

${C}_{\mathrm{D},\mathrm{HL}}$ | drag coefficient according to Haider and Levenspiel [28] |

${C}_{\mathrm{D},\mathrm{HS}}$ | drag coefficient according to Hölzer and Sommerfeld [23] |

${C}_{\mathrm{D},\mathrm{S}}$ | drag coefficient according to Stokes [15] |

${C}_{\mathrm{D},\mathrm{SN}}$ | drag coefficient according to Schiller and Naumann [16] |

${d}_{\mathrm{Cook}}$ | Cook’s distance [71] |

${d}_{\mathrm{eq}}$ | diameter of a volume equivalent sphere |

E | elongation |

${\mathit{F}}_{\mathrm{f}}$ | force acting on the fluid |

${\mathit{F}}_{\mathrm{p}}$ | force acting on the particle |

${\mathit{F}}^{\mathrm{BG}}$ | combined buoyancy and gravitational force |

${\mathit{F}}^{\mathrm{D}}$ | drag force |

${\mathit{F}}^{\mathrm{H}}$ | hydrodynamic force |

F | flatness |

${F}_{\mathrm{sample}}$ | score of an F-test |

g | gravitational acceleration |

${\mathit{J}}_{\mathrm{p}}$ | moment of inertia |

${K}_{\mathrm{N}}$ | drag correction factor for the Newton regime |

${K}_{\mathrm{S}}$ | drag correction factor for the Stokes regime |

${m}_{\mathrm{p}}$ | particle mass |

$\mathit{MI}$ | mutual information |

N | resolution related parameter (number of cells per ${d}_{\mathrm{eq}}$) |

p | pressure |

$\mathit{PI}$ | permutation importance |

r | residual |

${R}^{2}$ | coefficient of determination |

${R}_{a}^{2}$ | adjusted coefficient of determination |

$\mathrm{Re}$ | Reynolds number |

t | time |

${\mathbf{T}}_{\mathrm{p}}$ | torque |

${T}_{\mathrm{sample}}$ | score of a t-test |

${\mathit{u}}_{\mathrm{f}}$ | fluid velocity |

${u}_{\mathrm{max}}^{\mathrm{L}}$ | maximum lattice velocity in a simulation |

${u}_{\mathrm{ts}}$ | terminal settling velocity |

${u}_{\mathrm{ts},\mathrm{D}}$ | terminal settling velocity according to Dellino [25] |

${u}_{\mathrm{ts},\mathrm{HL}}$ | terminal settling velocity according to Haider and Levenspiel [28] |

${u}_{\mathrm{ts},\mathrm{S}}$ | terminal settling velocity according to Stokes [15] |

${\mathit{u}}_{\mathrm{p}}$ | particle velocity |

${V}_{\mathrm{p}}$ | particle volume |

$\mathit{VIF}$ | variance inflation factor |

Greek | |

$\delta t$ | temporal discretization parameter |

$\delta x$ | spatial discretization parameter |

${\kappa}_{\mathrm{con}}$ | convexity |

${\kappa}_{\mathrm{rnd}}$ | roundness |

${\lambda}_{\mathrm{CSF}}$ | Corey shape factor [18] |

${\lambda}_{\mathrm{H}}$ | Hofmann shape entropy [44] |

${\lambda}_{\mathrm{LR}}$ | Le Roux shape factor [46] |

$\nu $ | kinematic viscosity |

${\xi}_{1},{\xi}_{2}$ | exponents determining the shape of a superellipsoid |

${\rho}^{\prime}$ | ratio of particle to fluid density |

${\rho}_{\mathrm{f}}$ | fluid density |

${\rho}_{\mathrm{p}}$ | particle density |

$\tau $ | lattice relaxation time |

$\varphi $ | circularity |

$\psi $ | sphericity |

${\psi}_{\perp}$ | crosswise sphericity |

${\psi}_{\Vert}$ | lengthwise sphericity |

${\omega}_{\mathrm{p}}$ | particle angular velocity |

## Appendix A

ID | ${\mathit{a}}_{\mathbf{L}}$ in m | ${\mathit{a}}_{\mathit{I}}$ in m | ${\mathit{a}}_{\mathbf{S}}$ in m | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | E | F | ${\mathit{\rho}}_{\mathbf{p}}$ | ${\mathit{\kappa}}_{\mathbf{con}}$ | $\mathit{\psi}$ | ${\mathit{\psi}}_{\perp}$ | ${\mathit{\kappa}}_{\mathbf{rnd}}$ | ${\mathit{u}}_{\mathbf{ts}}$ | Re | ${\mathit{C}}_{\mathbf{D}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | $1.47\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $1.03\times {10}^{-4}$ | $8.0$ | $8.0$ | $1.0$ | $0.7$ | 2360 | $0.99$ | $0.87$ | $0.96$ | $0.18$ | $3.52\times {10}^{-2}$ | $11.16$ | $4.57$ |

2 | $2.17\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $1.21\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.7$ | $0.8$ | 2360 | $0.95$ | $0.88$ | $0.77$ | $0.15$ | $3.54\times {10}^{-2}$ | $11.23$ | $4.5$ |

3 | $1.83\times {10}^{-4}$ | $1.65\times {10}^{-4}$ | $1.32\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.9$ | $0.8$ | 2560 | $0.96$ | $0.91$ | $0.83$ | $0.15$ | $4.10\times {10}^{-2}$ | $12.98$ | $3.87$ |

4 | $2.08\times {10}^{-4}$ | $1.46\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.7$ | $0.9$ | 2660 | $0.95$ | $0.9$ | $0.83$ | $0.15$ | $4.24\times {10}^{-2}$ | $13.43$ | $3.85$ |

5 | $2.08\times {10}^{-4}$ | $1.46\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.7$ | $0.9$ | 2760 | $0.95$ | $0.9$ | $0.83$ | $0.15$ | $4.43\times {10}^{-2}$ | $14.03$ | $3.74$ |

6 | $1.90\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $1.37\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.8$ | $0.9$ | 2460 | $0.95$ | $0.9$ | $0.86$ | $0.15$ | $3.94\times {10}^{-2}$ | $12.48$ | $3.92$ |

7 | $1.90\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $1.37\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.8$ | $0.9$ | 2560 | $0.95$ | $0.9$ | $0.86$ | $0.15$ | $4.14\times {10}^{-2}$ | $13.12$ | $3.79$ |

8 | $1.90\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $1.37\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.8$ | $0.9$ | 2660 | $0.95$ | $0.9$ | $0.86$ | $0.15$ | $4.33\times {10}^{-2}$ | $13.74$ | $3.68$ |

9 | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.7$ | $1.0$ | 2360 | $0.95$ | $0.9$ | $0.89$ | $0.15$ | $3.74\times {10}^{-2}$ | $11.86$ | $4.04$ |

10 | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.7$ | $1.0$ | 2460 | $0.95$ | $0.9$ | $0.89$ | $0.15$ | $3.95\times {10}^{-2}$ | $12.52$ | $3.9$ |

11 | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.7$ | $1.0$ | 2760 | $0.95$ | $0.9$ | $0.89$ | $0.15$ | $4.55\times {10}^{-2}$ | $14.41$ | $3.54$ |

12 | $1.84\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.8$ | $1.0$ | 2360 | $0.96$ | $0.91$ | $0.93$ | $0.15$ | $3.81\times {10}^{-2}$ | $12.08$ | $3.89$ |

13 | $1.84\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.8$ | $1.0$ | 2560 | $0.96$ | $0.91$ | $0.93$ | $0.15$ | $4.23\times {10}^{-2}$ | $13.41$ | $3.62$ |

14 | $1.84\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.8$ | $1.0$ | 2660 | $0.96$ | $0.91$ | $0.93$ | $0.15$ | $4.44\times {10}^{-2}$ | $14.06$ | $3.51$ |

15 | $1.84\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $1.47\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.8$ | $1.0$ | 2760 | $0.96$ | $0.91$ | $0.93$ | $0.15$ | $4.63\times {10}^{-2}$ | $14.69$ | $3.41$ |

16 | $1.70\times {10}^{-4}$ | $1.53\times {10}^{-4}$ | $1.53\times {10}^{-4}$ | $2.0$ | $2.0$ | $0.9$ | $1.0$ | 2360 | $0.95$ | $0.92$ | $0.96$ | $0.15$ | $3.87\times {10}^{-2}$ | $12.28$ | $3.77$ |

17 | $2.92\times {10}^{-4}$ | $2.34\times {10}^{-4}$ | $1.17\times {10}^{-4}$ | $2.0$ | $1.0$ | $0.8$ | $0.5$ | 2760 | $0.91$ | $0.65$ | $0.39$ | $0.09$ | $3.24\times {10}^{-2}$ | $10.26$ | $7.16$ |

18 | $2.18\times {10}^{-4}$ | $2.18\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.6$ | 2760 | $0.97$ | $0.87$ | $0.84$ | $0.13$ | $4.91\times {10}^{-2}$ | $15.56$ | $3.57$ |

19 | $2.23\times {10}^{-4}$ | $2.00\times {10}^{-4}$ | $1.40\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $0.7$ | 2360 | $0.94$ | $0.87$ | $0.88$ | $0.13$ | $4.13\times {10}^{-2}$ | $13.09$ | $3.91$ |

20 | $2.23\times {10}^{-4}$ | $2.00\times {10}^{-4}$ | $1.40\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $0.7$ | 2460 | $0.94$ | $0.87$ | $0.88$ | $0.13$ | $4.35\times {10}^{-2}$ | $13.79$ | $3.78$ |

21 | $2.23\times {10}^{-4}$ | $2.00\times {10}^{-4}$ | $1.40\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $0.7$ | 2560 | $0.94$ | $0.87$ | $0.88$ | $0.13$ | $4.57\times {10}^{-2}$ | $14.47$ | $3.67$ |

22 | $2.23\times {10}^{-4}$ | $2.00\times {10}^{-4}$ | $1.40\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $0.7$ | 2660 | $0.94$ | $0.87$ | $0.89$ | $0.13$ | $4.78\times {10}^{-2}$ | $15.15$ | $3.56$ |

23 | $2.23\times {10}^{-4}$ | $2.00\times {10}^{-4}$ | $1.40\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $0.7$ | 2760 | $0.94$ | $0.87$ | $0.89$ | $0.13$ | $4.99\times {10}^{-2}$ | $15.83$ | $3.46$ |

24 | $2.07\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.45\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.7$ | 2460 | $0.96$ | $0.88$ | $0.9$ | $0.13$ | $4.40\times {10}^{-2}$ | $13.95$ | $3.7$ |

25 | $2.07\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.45\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.7$ | 2560 | $0.96$ | $0.88$ | $0.9$ | $0.13$ | $4.62\times {10}^{-2}$ | $14.65$ | $3.58$ |

26 | $2.07\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.45\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.7$ | 2660 | $0.96$ | $0.88$ | $0.9$ | $0.13$ | $4.84\times {10}^{-2}$ | $15.35$ | $3.47$ |

27 | $2.07\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.45\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.7$ | 2760 | $0.96$ | $0.88$ | $0.9$ | $0.13$ | $5.06\times {10}^{-2}$ | $16.04$ | $3.37$ |

28 | $2.13\times {10}^{-4}$ | $1.92\times {10}^{-4}$ | $1.53\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $0.8$ | 2460 | $0.94$ | $0.87$ | $0.85$ | $0.13$ | $4.44\times {10}^{-2}$ | $14.07$ | $3.63$ |

29 | $1.98\times {10}^{-4}$ | $1.98\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.8$ | 2360 | $0.97$ | $0.88$ | $1.03$ | $0.13$ | $4.31\times {10}^{-2}$ | $13.66$ | $3.59$ |

30 | $1.98\times {10}^{-4}$ | $1.98\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.8$ | 2460 | $0.97$ | $0.88$ | $1.03$ | $0.13$ | $4.54\times {10}^{-2}$ | $14.4$ | $3.46$ |

31 | $1.98\times {10}^{-4}$ | $1.98\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.8$ | 2560 | $0.97$ | $0.88$ | $1.03$ | $0.13$ | $4.78\times {10}^{-2}$ | $15.14$ | $3.35$ |

32 | $1.98\times {10}^{-4}$ | $1.98\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.8$ | 2760 | $0.97$ | $0.88$ | $1.03$ | $0.13$ | $5.23\times {10}^{-2}$ | $16.59$ | $3.15$ |

33 | $2.05\times {10}^{-4}$ | $1.84\times {10}^{-4}$ | $1.66\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $0.9$ | 2360 | $0.94$ | $0.87$ | $0.74$ | $0.12$ | $4.13\times {10}^{-2}$ | $13.1$ | $3.91$ |

34 | $1.91\times {10}^{-4}$ | $1.91\times {10}^{-4}$ | $1.72\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.9$ | 2360 | $0.97$ | $0.89$ | $1.06$ | $0.13$ | $4.27\times {10}^{-2}$ | $13.54$ | $3.65$ |

35 | $1.91\times {10}^{-4}$ | $1.91\times {10}^{-4}$ | $1.72\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.9$ | 2560 | $0.97$ | $0.89$ | $1.06$ | $0.13$ | $4.73\times {10}^{-2}$ | $15.0$ | $3.41$ |

36 | $1.91\times {10}^{-4}$ | $1.91\times {10}^{-4}$ | $1.72\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.9$ | 2660 | $0.97$ | $0.89$ | $1.06$ | $0.13$ | $4.96\times {10}^{-2}$ | $15.73$ | $3.3$ |

37 | $1.91\times {10}^{-4}$ | $1.91\times {10}^{-4}$ | $1.72\times {10}^{-4}$ | $1.0$ | $2.0$ | $1.0$ | $0.9$ | 2760 | $0.97$ | $0.89$ | $1.06$ | $0.13$ | $5.19\times {10}^{-2}$ | $16.44$ | $3.2$ |

38 | $1.98\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.0$ | $2.0$ | $0.9$ | $1.0$ | 2360 | $0.95$ | $0.87$ | $0.73$ | $0.12$ | $4.08\times {10}^{-2}$ | $12.95$ | $4.0$ |

39 | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ | 2360 | $1.0$ | $0.89$ | $0.87$ | $0.1$ | $3.66\times {10}^{-2}$ | $11.6$ | $4.22$ |

40 | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ | 2460 | $1.0$ | $0.89$ | $0.87$ | $0.1$ | $3.86\times {10}^{-2}$ | $12.24$ | $4.07$ |

41 | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ | 2560 | $1.0$ | $0.89$ | $0.87$ | $0.1$ | $4.06\times {10}^{-2}$ | $12.87$ | $3.93$ |

42 | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $1.0$ | $1.0$ | $1.0$ | $1.0$ | 2660 | $1.0$ | $0.89$ | $0.87$ | $0.1$ | $4.26\times {10}^{-2}$ | $13.49$ | $3.81$ |

43 | $2.31\times {10}^{-4}$ | $2.08\times {10}^{-4}$ | $1.46\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.9$ | $0.7$ | 2760 | $0.9$ | $0.85$ | $0.92$ | $0.12$ | $5.23\times {10}^{-2}$ | $16.57$ | $3.35$ |

44 | $2.61\times {10}^{-4}$ | $1.83\times {10}^{-4}$ | $1.46\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.7$ | $0.8$ | 2360 | $0.89$ | $0.82$ | $0.68$ | $0.11$ | $4.03\times {10}^{-2}$ | $12.78$ | $4.35$ |

45 | $2.61\times {10}^{-4}$ | $1.83\times {10}^{-4}$ | $1.46\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.7$ | $0.8$ | 2560 | $0.89$ | $0.82$ | $0.68$ | $0.11$ | $4.46\times {10}^{-2}$ | $14.14$ | $4.07$ |

46 | $2.61\times {10}^{-4}$ | $1.83\times {10}^{-4}$ | $1.46\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.7$ | $0.8$ | 2660 | $0.89$ | $0.82$ | $0.68$ | $0.11$ | $4.67\times {10}^{-2}$ | $14.8$ | $3.95$ |

47 | $2.61\times {10}^{-4}$ | $1.83\times {10}^{-4}$ | $1.46\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.7$ | $0.8$ | 2760 | $0.89$ | $0.82$ | $0.67$ | $0.11$ | $4.87\times {10}^{-2}$ | $15.44$ | $4.05$ |

48 | $2.21\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.9$ | $0.8$ | 2360 | $0.89$ | $0.85$ | $1.0$ | $0.12$ | $4.26\times {10}^{-2}$ | $13.5$ | $3.9$ |

49 | $2.21\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.9$ | $0.8$ | 2460 | $0.89$ | $0.85$ | $1.0$ | $0.12$ | $4.49\times {10}^{-2}$ | $14.22$ | $3.77$ |

50 | $2.21\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.9$ | $0.8$ | 2560 | $0.89$ | $0.85$ | $1.0$ | $0.12$ | $4.71\times {10}^{-2}$ | $14.93$ | $3.65$ |

51 | $2.21\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.9$ | $0.8$ | 2660 | $0.89$ | $0.85$ | $0.73$ | $0.12$ | $4.93\times {10}^{-2}$ | $15.63$ | $3.56$ |

52 | $2.21\times {10}^{-4}$ | $1.99\times {10}^{-4}$ | $1.59\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.9$ | $0.8$ | 2760 | $0.89$ | $0.85$ | $0.73$ | $0.12$ | $5.15\times {10}^{-2}$ | $16.31$ | $3.45$ |

53 | $2.06\times {10}^{-4}$ | $2.06\times {10}^{-4}$ | $1.65\times {10}^{-4}$ | $0.9$ | $2.0$ | $1.0$ | $0.8$ | 2560 | $0.89$ | $0.84$ | $1.03$ | $0.12$ | $4.90\times {10}^{-2}$ | $15.54$ | $3.4$ |

54 | $2.06\times {10}^{-4}$ | $2.06\times {10}^{-4}$ | $1.65\times {10}^{-4}$ | $0.9$ | $2.0$ | $1.0$ | $0.8$ | 2760 | $0.89$ | $0.84$ | $1.03$ | $0.12$ | $5.36\times {10}^{-2}$ | $17.0$ | $3.18$ |

55 | $2.30\times {10}^{-4}$ | $1.84\times {10}^{-4}$ | $1.66\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.8$ | $0.9$ | 2360 | $0.9$ | $0.83$ | $0.67$ | $0.12$ | $4.12\times {10}^{-2}$ | $13.05$ | $4.17$ |

56 | $2.30\times {10}^{-4}$ | $1.84\times {10}^{-4}$ | $1.66\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.8$ | $0.9$ | 2460 | $0.9$ | $0.83$ | $0.67$ | $0.12$ | $4.33\times {10}^{-2}$ | $13.74$ | $4.04$ |

57 | $2.30\times {10}^{-4}$ | $1.84\times {10}^{-4}$ | $1.66\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.8$ | $0.9$ | 2560 | $0.9$ | $0.83$ | $0.67$ | $0.12$ | $4.55\times {10}^{-2}$ | $14.42$ | $3.92$ |

58 | $2.30\times {10}^{-4}$ | $1.84\times {10}^{-4}$ | $1.66\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.8$ | $0.9$ | 2760 | $0.9$ | $0.83$ | $0.68$ | $0.12$ | $4.96\times {10}^{-2}$ | $15.73$ | $3.72$ |

59 | $2.13\times {10}^{-4}$ | $1.91\times {10}^{-4}$ | $1.72\times {10}^{-4}$ | $0.9$ | $2.0$ | $0.9$ | $0.9$ | 2460 | $0.9$ | $0.84$ | $0.7$ | $0.12$ | $4.43\times {10}^{-2}$ | $14.03$ | $3.87$ |

60 | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $1.22\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.6$ | 2360 | $0.91$ | $0.81$ | $1.11$ | $0.12$ | - | - | - |

ID | ${\mathit{a}}_{\mathbf{L}}$ in m | ${\mathit{a}}_{\mathit{I}}$ in m | ${\mathit{a}}_{\mathbf{S}}$ in m | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | E | F | ${\mathit{\rho}}_{\mathbf{p}}$ | ${\mathit{\kappa}}_{\mathbf{con}}$ | $\mathit{\psi}$ | ${\mathit{\psi}}_{\perp}$ | ${\mathit{\kappa}}_{\mathbf{rnd}}$ | ${\mathit{u}}_{\mathbf{ts}}$ | Re | ${\mathit{C}}_{\mathbf{D}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

61 | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $1.22\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.6$ | 2460 | $0.91$ | $0.81$ | $1.11$ | $0.12$ | - | - | - |

62 | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $1.22\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.6$ | 2560 | $0.91$ | $0.81$ | $1.11$ | $0.12$ | - | - | - |

63 | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $1.22\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.6$ | 2660 | $0.91$ | $0.81$ | $1.11$ | $0.12$ | - | - | - |

64 | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $1.22\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.6$ | 2760 | $0.91$ | $0.81$ | $1.11$ | $0.12$ | - | - | - |

65 | $1.78\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.60\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.9$ | 2360 | $0.93$ | $0.82$ | $0.97$ | $0.13$ | $5.62\times {10}^{-2}$ | $17.83$ | $3.34$ |

66 | $1.78\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.60\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.9$ | 2460 | $0.93$ | $0.82$ | $0.98$ | $0.13$ | $5.93\times {10}^{-2}$ | $18.78$ | $3.23$ |

67 | $1.78\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.60\times {10}^{-4}$ | $0.87$ | $8.0$ | $1.0$ | $0.9$ | 2660 | $0.93$ | $0.82$ | $0.99$ | $0.13$ | $6.51\times {10}^{-2}$ | $20.63$ | $3.04$ |

68 | $2.51\times {10}^{-4}$ | $2.26\times {10}^{-4}$ | $1.36\times {10}^{-4}$ | $0.83$ | $2.0$ | $0.9$ | $0.6$ | 2660 | $0.84$ | $0.82$ | $0.85$ | $0.11$ | $4.99\times {10}^{-2}$ | $15.81$ | $3.65$ |

69 | $2.51\times {10}^{-4}$ | $2.26\times {10}^{-4}$ | $1.36\times {10}^{-4}$ | $0.83$ | $2.0$ | $0.9$ | $0.6$ | 2760 | $0.84$ | $0.82$ | $0.85$ | $0.11$ | $5.21\times {10}^{-2}$ | $16.5$ | $3.55$ |

70 | $2.38\times {10}^{-4}$ | $2.15\times {10}^{-4}$ | $1.50\times {10}^{-4}$ | $0.83$ | $2.0$ | $0.9$ | $0.7$ | 2760 | $0.84$ | $0.82$ | $0.95$ | $0.11$ | $5.37\times {10}^{-2}$ | $17.02$ | $3.33$ |

71 | $2.28\times {10}^{-4}$ | $2.05\times {10}^{-4}$ | $1.64\times {10}^{-4}$ | $0.83$ | $2.0$ | $0.9$ | $0.8$ | 2360 | $0.84$ | $0.82$ | $0.68$ | $0.11$ | $4.32\times {10}^{-2}$ | $13.7$ | $3.99$ |

72 | $2.28\times {10}^{-4}$ | $2.05\times {10}^{-4}$ | $1.64\times {10}^{-4}$ | $0.83$ | $2.0$ | $0.9$ | $0.8$ | 2560 | $0.84$ | $0.82$ | $0.69$ | $0.11$ | $4.78\times {10}^{-2}$ | $15.15$ | $3.72$ |

73 | $3.94\times {10}^{-4}$ | $2.36\times {10}^{-4}$ | $1.65\times {10}^{-4}$ | $0.83$ | $1.0$ | $0.6$ | $0.7$ | 2360 | $0.84$ | $0.77$ | $0.57$ | $0.08$ | $3.04\times {10}^{-2}$ | $9.64$ | $6.4$ |

74 | $3.94\times {10}^{-4}$ | $2.36\times {10}^{-4}$ | $1.65\times {10}^{-4}$ | $0.83$ | $1.0$ | $0.6$ | $0.7$ | 2560 | $0.84$ | $0.77$ | $0.57$ | $0.08$ | $3.37\times {10}^{-2}$ | $10.69$ | $5.97$ |

75 | $3.94\times {10}^{-4}$ | $2.36\times {10}^{-4}$ | $1.65\times {10}^{-4}$ | $0.83$ | $1.0$ | $0.6$ | $0.7$ | 2660 | $0.84$ | $0.77$ | $0.57$ | $0.08$ | $3.53\times {10}^{-2}$ | $11.18$ | $5.81$ |

76 | $3.94\times {10}^{-4}$ | $2.36\times {10}^{-4}$ | $1.65\times {10}^{-4}$ | $0.83$ | $1.0$ | $0.6$ | $0.7$ | 2760 | $0.84$ | $0.77$ | $0.57$ | $0.08$ | $3.68\times {10}^{-2}$ | $11.67$ | $5.66$ |

77 | $3.40\times {10}^{-4}$ | $2.38\times {10}^{-4}$ | $1.90\times {10}^{-4}$ | $0.83$ | $1.0$ | $0.7$ | $0.8$ | 2460 | $0.84$ | $0.78$ | $0.63$ | $0.08$ | $3.47\times {10}^{-2}$ | $10.99$ | $5.28$ |

78 | $3.40\times {10}^{-4}$ | $2.38\times {10}^{-4}$ | $1.90\times {10}^{-4}$ | $0.83$ | $1.0$ | $0.7$ | $0.8$ | 2660 | $0.84$ | $0.78$ | $0.63$ | $0.08$ | $3.80\times {10}^{-2}$ | $12.06$ | $4.99$ |

79 | $3.62\times {10}^{-4}$ | $2.17\times {10}^{-4}$ | $1.95\times {10}^{-4}$ | $0.83$ | $1.0$ | $0.6$ | $0.9$ | 2460 | $0.85$ | $0.79$ | $0.67$ | $0.08$ | $3.47\times {10}^{-2}$ | $10.99$ | $5.29$ |

80 | $3.15\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $9.45\times {10}^{-5}$ | $0.8$ | $8.0$ | $0.6$ | $0.5$ | 2360 | $0.84$ | $0.78$ | $0.85$ | $0.12$ | - | - | - |

81 | $3.15\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $9.45\times {10}^{-5}$ | $0.8$ | $8.0$ | $0.6$ | $0.5$ | 2660 | $0.84$ | $0.78$ | $0.87$ | $0.12$ | $6.24\times {10}^{-2}$ | $19.77$ | $3.61$ |

82 | $3.15\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $9.45\times {10}^{-5}$ | $0.8$ | $8.0$ | $0.6$ | $0.5$ | 2760 | $0.84$ | $0.78$ | $0.86$ | $0.12$ | $6.52\times {10}^{-2}$ | $20.66$ | $3.51$ |

83 | $3.35\times {10}^{-4}$ | $1.67\times {10}^{-4}$ | $1.00\times {10}^{-4}$ | $0.8$ | $8.0$ | $0.5$ | $0.6$ | 2760 | $0.84$ | $0.78$ | $0.73$ | $0.12$ | $6.00\times {10}^{-2}$ | $19.02$ | $4.14$ |

84 | $2.67\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.12\times {10}^{-4}$ | $0.8$ | $8.0$ | $0.7$ | $0.6$ | 2360 | $0.84$ | $0.77$ | $1.02$ | $0.11$ | $5.48\times {10}^{-2}$ | $17.38$ | $3.83$ |

85 | $2.67\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.12\times {10}^{-4}$ | $0.8$ | $8.0$ | $0.7$ | $0.6$ | 2460 | $0.84$ | $0.77$ | $0.69$ | $0.11$ | $5.77\times {10}^{-2}$ | $18.28$ | $3.72$ |

86 | $2.67\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.12\times {10}^{-4}$ | $0.8$ | $8.0$ | $0.7$ | $0.6$ | 2560 | $0.84$ | $0.77$ | $0.68$ | $0.11$ | $6.05\times {10}^{-2}$ | $19.16$ | $3.61$ |

87 | $2.67\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.12\times {10}^{-4}$ | $0.8$ | $8.0$ | $0.7$ | $0.6$ | 2660 | $0.84$ | $0.77$ | $0.68$ | $0.11$ | $6.31\times {10}^{-2}$ | $20.01$ | $3.53$ |

88 | $2.67\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.12\times {10}^{-4}$ | $0.8$ | $8.0$ | $0.7$ | $0.6$ | 2760 | $0.84$ | $0.77$ | $0.69$ | $0.11$ | $6.57\times {10}^{-2}$ | $20.84$ | $3.45$ |

89 | $1.98\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.60\times {10}^{-4}$ | $0.8$ | $8.0$ | $0.9$ | $0.9$ | 2560 | $0.85$ | $0.77$ | $0.65$ | $0.11$ | $6.18\times {10}^{-2}$ | $19.59$ | $3.46$ |

90 | $4.02\times {10}^{-4}$ | $2.01\times {10}^{-4}$ | $1.00\times {10}^{-4}$ | $0.8$ | $2.0$ | $0.5$ | $0.5$ | 2460 | $0.85$ | $0.83$ | $0.68$ | $0.11$ | $3.92\times {10}^{-2}$ | $12.42$ | $5.36$ |

91 | $3.78\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $1.13\times {10}^{-4}$ | $0.8$ | $2.0$ | $0.5$ | $0.6$ | 2360 | $0.84$ | $0.83$ | $0.76$ | $0.11$ | $3.89\times {10}^{-2}$ | $12.33$ | $5.07$ |

92 | $3.78\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $1.13\times {10}^{-4}$ | $0.8$ | $2.0$ | $0.5$ | $0.6$ | 2460 | $0.84$ | $0.83$ | $0.76$ | $0.11$ | $4.08\times {10}^{-2}$ | $12.94$ | $4.94$ |

93 | $3.78\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $1.13\times {10}^{-4}$ | $0.8$ | $2.0$ | $0.5$ | $0.6$ | 2560 | $0.84$ | $0.83$ | $0.76$ | $0.11$ | $4.29\times {10}^{-2}$ | $13.6$ | $4.78$ |

94 | $3.78\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $1.13\times {10}^{-4}$ | $0.8$ | $2.0$ | $0.5$ | $0.6$ | 2660 | $0.84$ | $0.83$ | $0.76$ | $0.11$ | $4.49\times {10}^{-2}$ | $14.23$ | $4.65$ |

95 | $3.78\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $1.13\times {10}^{-4}$ | $0.8$ | $2.0$ | $0.5$ | $0.6$ | 2760 | $0.84$ | $0.83$ | $0.75$ | $0.11$ | $4.67\times {10}^{-2}$ | $14.81$ | $4.54$ |

96 | $3.63\times {10}^{-4}$ | $1.81\times {10}^{-4}$ | $9.07\times {10}^{-5}$ | $0.77$ | $8.0$ | $0.5$ | $0.5$ | 2360 | $0.85$ | $0.8$ | $0.85$ | $0.12$ | - | - | - |

97 | $4.08\times {10}^{-4}$ | $2.45\times {10}^{-4}$ | $1.72\times {10}^{-4}$ | $0.77$ | $1.0$ | $0.6$ | $0.7$ | 2760 | $0.78$ | $0.77$ | $0.59$ | $0.08$ | $3.78\times {10}^{-2}$ | $11.97$ | $5.55$ |

98 | $3.39\times {10}^{-4}$ | $2.37\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.77$ | $1.0$ | $0.7$ | $0.9$ | 2360 | $0.78$ | $0.76$ | $0.59$ | $0.08$ | $3.37\times {10}^{-2}$ | $10.67$ | $5.4$ |

99 | $3.39\times {10}^{-4}$ | $2.37\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.77$ | $1.0$ | $0.7$ | $0.9$ | 2660 | $0.78$ | $0.76$ | $0.58$ | $0.08$ | $3.88\times {10}^{-2}$ | $12.31$ | $4.96$ |

100 | $3.70\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $9.26\times {10}^{-5}$ | $0.73$ | $8.0$ | $0.5$ | $0.5$ | 2360 | $0.84$ | $0.8$ | $0.88$ | $0.11$ | - | - | - |

101 | $3.70\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $9.26\times {10}^{-5}$ | $0.73$ | $8.0$ | $0.5$ | $0.5$ | 2460 | $0.84$ | $0.8$ | $0.88$ | $0.11$ | - | - | - |

102 | $3.70\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $9.26\times {10}^{-5}$ | $0.73$ | $8.0$ | $0.5$ | $0.5$ | 2560 | $0.84$ | $0.8$ | $0.88$ | $0.11$ | - | - | - |

103 | $3.70\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $9.26\times {10}^{-5}$ | $0.73$ | $8.0$ | $0.5$ | $0.5$ | 2660 | $0.84$ | $0.8$ | $0.88$ | $0.11$ | - | - | - |

104 | $2.24\times {10}^{-4}$ | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $0.73$ | $8.0$ | $0.9$ | $0.7$ | 2360 | $0.78$ | $0.75$ | $0.66$ | $0.11$ | $6.02\times {10}^{-2}$ | $19.09$ | $3.53$ |

105 | $2.24\times {10}^{-4}$ | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $0.73$ | $8.0$ | $0.9$ | $0.7$ | 2460 | $0.78$ | $0.75$ | $0.67$ | $0.11$ | $6.33\times {10}^{-2}$ | $20.06$ | $3.44$ |

106 | $2.24\times {10}^{-4}$ | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $0.73$ | $8.0$ | $0.9$ | $0.7$ | 2560 | $0.78$ | $0.75$ | $0.68$ | $0.11$ | $6.63\times {10}^{-2}$ | $21.02$ | $3.35$ |

107 | $2.24\times {10}^{-4}$ | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $0.73$ | $8.0$ | $0.9$ | $0.7$ | 2660 | $0.78$ | $0.75$ | $0.66$ | $0.11$ | $6.92\times {10}^{-2}$ | $21.95$ | $3.27$ |

108 | $2.24\times {10}^{-4}$ | $2.01\times {10}^{-4}$ | $1.41\times {10}^{-4}$ | $0.73$ | $8.0$ | $0.9$ | $0.7$ | 2760 | $0.78$ | $0.75$ | $0.66$ | $0.11$ | $7.21\times {10}^{-2}$ | $22.86$ | $3.19$ |

109 | $3.70\times {10}^{-4}$ | $2.22\times {10}^{-4}$ | $1.11\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.6$ | $0.5$ | 2360 | $0.78$ | $0.8$ | $0.73$ | $0.1$ | $4.09\times {10}^{-2}$ | $12.96$ | $4.94$ |

110 | $3.70\times {10}^{-4}$ | $2.22\times {10}^{-4}$ | $1.11\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.6$ | $0.5$ | 2460 | $0.78$ | $0.8$ | $0.73$ | $0.1$ | $4.30\times {10}^{-2}$ | $13.63$ | $4.79$ |

111 | $3.70\times {10}^{-4}$ | $2.22\times {10}^{-4}$ | $1.11\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.6$ | $0.5$ | 2560 | $0.78$ | $0.8$ | $0.73$ | $0.1$ | $4.53\times {10}^{-2}$ | $14.37$ | $4.6$ |

112 | $3.70\times {10}^{-4}$ | $2.22\times {10}^{-4}$ | $1.11\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.6$ | $0.5$ | 2660 | $0.78$ | $0.8$ | $0.73$ | $0.1$ | $4.74\times {10}^{-2}$ | $15.03$ | $4.47$ |

113 | $3.70\times {10}^{-4}$ | $2.22\times {10}^{-4}$ | $1.11\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.6$ | $0.5$ | 2760 | $0.78$ | $0.8$ | $0.73$ | $0.1$ | $4.91\times {10}^{-2}$ | $15.56$ | $4.43$ |

114 | $3.74\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.5$ | $0.7$ | 2360 | $0.79$ | $0.8$ | $0.63$ | $0.1$ | $3.89\times {10}^{-2}$ | $12.33$ | $5.45$ |

115 | $3.74\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.5$ | $0.7$ | 2460 | $0.79$ | $0.8$ | $0.63$ | $0.1$ | $4.09\times {10}^{-2}$ | $12.97$ | $5.29$ |

116 | $3.74\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.5$ | $0.7$ | 2560 | $0.79$ | $0.8$ | $0.63$ | $0.1$ | $4.29\times {10}^{-2}$ | $13.6$ | $5.14$ |

117 | $3.74\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.5$ | $0.7$ | 2660 | $0.79$ | $0.8$ | $0.63$ | $0.1$ | $4.48\times {10}^{-2}$ | $14.21$ | $5.0$ |

118 | $3.74\times {10}^{-4}$ | $1.87\times {10}^{-4}$ | $1.31\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.5$ | $0.7$ | 2760 | $0.79$ | $0.8$ | $0.63$ | $0.1$ | $4.67\times {10}^{-2}$ | $14.82$ | $4.89$ |

119 | $3.58\times {10}^{-4}$ | $1.79\times {10}^{-4}$ | $1.43\times {10}^{-4}$ | $0.73$ | $2.0$ | $0.5$ | $0.8$ | 2560 | $0.79$ | $0.78$ | $0.58$ | $0.1$ | $4.27\times {10}^{-2}$ | $13.54$ | $5.18$ |

120 | $3.57\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.6$ | 2460 | $0.78$ | $0.76$ | $0.68$ | $0.11$ | $5.63\times {10}^{-2}$ | $17.86$ | $4.61$ |

ID | ${\mathit{a}}_{\mathbf{L}}$ in m | ${\mathit{a}}_{\mathit{I}}$ in m | ${\mathit{a}}_{\mathbf{S}}$ in m | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | E | F | ${\mathit{\rho}}_{\mathbf{p}}$ | ${\mathit{\kappa}}_{\mathbf{con}}$ | $\mathit{\psi}$ | ${\mathit{\psi}}_{\perp}$ | ${\mathit{\kappa}}_{\mathbf{rnd}}$ | ${\mathit{u}}_{\mathbf{ts}}$ | Re | ${\mathit{C}}_{\mathbf{D}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

121 | $3.57\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.6$ | 2560 | $0.78$ | $0.76$ | $0.68$ | $0.11$ | $5.90\times {10}^{-2}$ | $18.71$ | $4.48$ |

122 | $3.57\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.6$ | 2660 | $0.78$ | $0.76$ | $0.68$ | $0.11$ | $6.16\times {10}^{-2}$ | $19.53$ | $4.39$ |

123 | $3.57\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.6$ | 2760 | $0.78$ | $0.76$ | $0.68$ | $0.11$ | $6.41\times {10}^{-2}$ | $20.33$ | $4.3$ |

124 | $3.39\times {10}^{-4}$ | $1.69\times {10}^{-4}$ | $1.19\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.7$ | 2360 | $0.79$ | $0.74$ | $0.61$ | $0.1$ | $5.34\times {10}^{-2}$ | $16.93$ | $4.79$ |

125 | $3.39\times {10}^{-4}$ | $1.69\times {10}^{-4}$ | $1.19\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.7$ | 2460 | $0.79$ | $0.74$ | $0.61$ | $0.1$ | $5.61\times {10}^{-2}$ | $17.79$ | $4.65$ |

126 | $3.39\times {10}^{-4}$ | $1.69\times {10}^{-4}$ | $1.19\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.7$ | 2560 | $0.79$ | $0.74$ | $0.61$ | $0.1$ | $5.87\times {10}^{-2}$ | $18.62$ | $4.54$ |

127 | $3.39\times {10}^{-4}$ | $1.69\times {10}^{-4}$ | $1.19\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.7$ | 2660 | $0.79$ | $0.74$ | $0.61$ | $0.1$ | $6.13\times {10}^{-2}$ | $19.42$ | $4.44$ |

128 | $3.39\times {10}^{-4}$ | $1.69\times {10}^{-4}$ | $1.19\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.5$ | $0.7$ | 2760 | $0.79$ | $0.74$ | $0.61$ | $0.1$ | $6.38\times {10}^{-2}$ | $20.22$ | $4.33$ |

129 | $2.37\times {10}^{-4}$ | $1.90\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.8$ | $0.8$ | 2360 | $0.73$ | $0.7$ | $0.61$ | $0.1$ | $5.95\times {10}^{-2}$ | $18.86$ | $3.86$ |

130 | $2.37\times {10}^{-4}$ | $1.90\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.8$ | $0.8$ | 2560 | $0.73$ | $0.7$ | $0.59$ | $0.1$ | $6.55\times {10}^{-2}$ | $20.75$ | $3.67$ |

131 | $2.37\times {10}^{-4}$ | $1.90\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.8$ | $0.8$ | 2660 | $0.73$ | $0.7$ | $0.59$ | $0.1$ | $6.82\times {10}^{-2}$ | $21.63$ | $3.58$ |

132 | $2.37\times {10}^{-4}$ | $1.90\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $0.7$ | $8.0$ | $0.8$ | $0.8$ | 2760 | $0.73$ | $0.7$ | $0.59$ | $0.1$ | $7.11\times {10}^{-2}$ | $22.54$ | $3.49$ |

133 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $2.0$ | $0.5$ | $0.5$ | 2360 | $0.78$ | $0.82$ | $0.73$ | $0.1$ | $4.05\times {10}^{-2}$ | $12.83$ | $5.27$ |

134 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $2.0$ | $0.5$ | $0.5$ | 2460 | $0.78$ | $0.82$ | $0.73$ | $0.1$ | $4.24\times {10}^{-2}$ | $13.44$ | $5.15$ |

135 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $2.0$ | $0.5$ | $0.5$ | 2560 | $0.78$ | $0.82$ | $0.73$ | $0.1$ | $4.46\times {10}^{-2}$ | $14.14$ | $4.97$ |

136 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $1.07\times {10}^{-4}$ | $0.7$ | $2.0$ | $0.5$ | $0.5$ | 2660 | $0.78$ | $0.82$ | $0.73$ | $0.1$ | $4.64\times {10}^{-2}$ | $14.71$ | $4.88$ |

137 | $4.78\times {10}^{-4}$ | $2.87\times {10}^{-4}$ | $1.43\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.5$ | 2660 | $0.74$ | $0.73$ | $0.52$ | $0.07$ | $3.34\times {10}^{-2}$ | $10.58$ | $6.99$ |

138 | $4.78\times {10}^{-4}$ | $2.87\times {10}^{-4}$ | $1.43\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.5$ | 2760 | $0.74$ | $0.73$ | $0.52$ | $0.07$ | $3.49\times {10}^{-2}$ | $11.05$ | $6.81$ |

139 | $4.31\times {10}^{-4}$ | $3.02\times {10}^{-4}$ | $1.51\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.5$ | 2460 | $0.74$ | $0.73$ | $0.52$ | $0.07$ | $3.17\times {10}^{-2}$ | $10.04$ | $6.83$ |

140 | $4.31\times {10}^{-4}$ | $3.02\times {10}^{-4}$ | $1.51\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.5$ | 2660 | $0.74$ | $0.73$ | $0.52$ | $0.07$ | $3.48\times {10}^{-2}$ | $11.04$ | $6.43$ |

141 | $4.31\times {10}^{-4}$ | $3.02\times {10}^{-4}$ | $1.51\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.5$ | 2760 | $0.74$ | $0.73$ | $0.53$ | $0.07$ | $3.61\times {10}^{-2}$ | $11.44$ | $6.36$ |

142 | $3.94\times {10}^{-4}$ | $3.15\times {10}^{-4}$ | $1.58\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $0.5$ | 2360 | $0.73$ | $0.73$ | $0.54$ | $0.07$ | $3.09\times {10}^{-2}$ | $9.81$ | $6.67$ |

143 | $3.94\times {10}^{-4}$ | $3.15\times {10}^{-4}$ | $1.58\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $0.5$ | 2460 | $0.73$ | $0.73$ | $0.54$ | $0.07$ | $3.25\times {10}^{-2}$ | $10.29$ | $6.5$ |

144 | $3.94\times {10}^{-4}$ | $3.15\times {10}^{-4}$ | $1.58\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $0.5$ | 2560 | $0.73$ | $0.73$ | $0.54$ | $0.07$ | $3.42\times {10}^{-2}$ | $10.86$ | $6.26$ |

145 | $3.94\times {10}^{-4}$ | $3.15\times {10}^{-4}$ | $1.58\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $0.5$ | 2660 | $0.73$ | $0.73$ | $0.54$ | $0.07$ | $3.58\times {10}^{-2}$ | $11.35$ | $6.09$ |

146 | $4.50\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $1.62\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.6$ | 2360 | $0.74$ | $0.75$ | $0.57$ | $0.07$ | $3.08\times {10}^{-2}$ | $9.78$ | $6.72$ |

147 | $4.50\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $1.62\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.6$ | 2460 | $0.74$ | $0.75$ | $0.56$ | $0.07$ | $3.24\times {10}^{-2}$ | $10.27$ | $6.55$ |

148 | $4.50\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $1.62\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.6$ | 2560 | $0.74$ | $0.75$ | $0.56$ | $0.07$ | $3.40\times {10}^{-2}$ | $10.77$ | $6.36$ |

149 | $4.50\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $1.62\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.6$ | 2660 | $0.74$ | $0.75$ | $0.56$ | $0.07$ | $3.56\times {10}^{-2}$ | $11.3$ | $6.14$ |

150 | $4.50\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $1.62\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.6$ | 2760 | $0.74$ | $0.75$ | $0.57$ | $0.07$ | $3.70\times {10}^{-2}$ | $11.73$ | $6.05$ |

151 | $4.06\times {10}^{-4}$ | $2.84\times {10}^{-4}$ | $1.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.6$ | 2360 | $0.73$ | $0.74$ | $0.58$ | $0.07$ | $3.18\times {10}^{-2}$ | $10.09$ | $6.3$ |

152 | $4.06\times {10}^{-4}$ | $2.84\times {10}^{-4}$ | $1.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.6$ | 2460 | $0.73$ | $0.74$ | $0.58$ | $0.07$ | $3.36\times {10}^{-2}$ | $10.64$ | $6.09$ |

153 | $4.06\times {10}^{-4}$ | $2.84\times {10}^{-4}$ | $1.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.6$ | 2560 | $0.73$ | $0.74$ | $0.58$ | $0.07$ | $3.52\times {10}^{-2}$ | $11.17$ | $5.91$ |

154 | $4.06\times {10}^{-4}$ | $2.84\times {10}^{-4}$ | $1.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.6$ | 2660 | $0.73$ | $0.74$ | $0.58$ | $0.07$ | $3.69\times {10}^{-2}$ | $11.7$ | $5.74$ |

155 | $4.06\times {10}^{-4}$ | $2.84\times {10}^{-4}$ | $1.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.6$ | 2760 | $0.73$ | $0.74$ | $0.58$ | $0.07$ | $3.86\times {10}^{-2}$ | $12.22$ | $5.56$ |

156 | $4.08\times {10}^{-4}$ | $2.45\times {10}^{-4}$ | $1.96\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.8$ | 2360 | $0.74$ | $0.75$ | $0.6$ | $0.07$ | $3.29\times {10}^{-2}$ | $10.44$ | $5.9$ |

157 | $4.08\times {10}^{-4}$ | $2.45\times {10}^{-4}$ | $1.96\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.8$ | 2460 | $0.74$ | $0.75$ | $0.6$ | $0.07$ | $3.46\times {10}^{-2}$ | $10.97$ | $5.73$ |

158 | $4.08\times {10}^{-4}$ | $2.45\times {10}^{-4}$ | $1.96\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.8$ | 2560 | $0.74$ | $0.75$ | $0.59$ | $0.07$ | $3.63\times {10}^{-2}$ | $11.49$ | $5.57$ |

159 | $4.08\times {10}^{-4}$ | $2.45\times {10}^{-4}$ | $1.96\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.8$ | 2760 | $0.74$ | $0.75$ | $0.59$ | $0.07$ | $3.95\times {10}^{-2}$ | $12.51$ | $5.31$ |

160 | $3.45\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.03\times {10}^{-4}$ | $0.67$ | $8.0$ | $0.6$ | $0.5$ | 2360 | $0.73$ | $0.75$ | $0.7$ | $0.1$ | $5.85\times {10}^{-2}$ | $18.54$ | $4.3$ |

161 | $3.45\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.03\times {10}^{-4}$ | $0.67$ | $8.0$ | $0.6$ | $0.5$ | 2460 | $0.73$ | $0.75$ | $0.7$ | $0.1$ | $6.14\times {10}^{-2}$ | $19.46$ | $4.18$ |

162 | $3.45\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.03\times {10}^{-4}$ | $0.67$ | $8.0$ | $0.6$ | $0.5$ | 2560 | $0.73$ | $0.75$ | $0.7$ | $0.1$ | $6.43\times {10}^{-2}$ | $20.37$ | $4.08$ |

163 | $3.45\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.03\times {10}^{-4}$ | $0.67$ | $8.0$ | $0.6$ | $0.5$ | 2660 | $0.73$ | $0.75$ | $0.69$ | $0.1$ | $6.71\times {10}^{-2}$ | $21.26$ | $3.98$ |

164 | $3.45\times {10}^{-4}$ | $2.07\times {10}^{-4}$ | $1.03\times {10}^{-4}$ | $0.67$ | $8.0$ | $0.6$ | $0.5$ | 2760 | $0.73$ | $0.75$ | $0.7$ | $0.1$ | $6.99\times {10}^{-2}$ | $22.16$ | $3.89$ |

165 | $3.48\times {10}^{-4}$ | $1.74\times {10}^{-4}$ | $1.22\times {10}^{-4}$ | $0.67$ | $8.0$ | $0.5$ | $0.7$ | 2460 | $0.76$ | $0.73$ | $0.6$ | $0.1$ | $5.77\times {10}^{-2}$ | $18.29$ | $4.72$ |

166 | $3.09\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.6$ | $1.0$ | 2460 | $0.69$ | $0.69$ | $0.5$ | $0.08$ | $4.32\times {10}^{-2}$ | $13.69$ | $5.19$ |

167 | $3.09\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.6$ | $1.0$ | 2660 | $0.69$ | $0.69$ | $0.5$ | $0.08$ | $4.72\times {10}^{-2}$ | $14.96$ | $4.95$ |

168 | $2.79\times {10}^{-4}$ | $1.95\times {10}^{-4}$ | $1.95\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.7$ | $1.0$ | 2460 | $0.7$ | $0.69$ | $0.52$ | $0.08$ | $4.46\times {10}^{-2}$ | $14.15$ | $4.9$ |

169 | $2.79\times {10}^{-4}$ | $1.95\times {10}^{-4}$ | $1.95\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.7$ | $1.0$ | 2560 | $0.7$ | $0.69$ | $0.52$ | $0.08$ | $4.67\times {10}^{-2}$ | $14.82$ | $4.71$ |

170 | $2.55\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.8$ | $1.0$ | 2360 | $0.69$ | $0.7$ | $0.54$ | $0.08$ | $4.36\times {10}^{-2}$ | $13.81$ | $4.76$ |

171 | $2.55\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.8$ | $1.0$ | 2460 | $0.69$ | $0.7$ | $0.54$ | $0.08$ | $4.58\times {10}^{-2}$ | $14.53$ | $4.62$ |

172 | $2.55\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.8$ | $1.0$ | 2560 | $0.69$ | $0.7$ | $0.54$ | $0.08$ | $4.80\times {10}^{-2}$ | $15.23$ | $4.49$ |

173 | $2.55\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $0.67$ | $2.0$ | $0.8$ | $1.0$ | 2660 | $0.69$ | $0.7$ | $0.54$ | $0.08$ | $5.02\times {10}^{-2}$ | $15.91$ | $4.37$ |

174 | $3.28\times {10}^{-4}$ | $3.28\times {10}^{-4}$ | $1.97\times {10}^{-4}$ | $0.67$ | $1.0$ | $1.0$ | $0.6$ | 2760 | $0.67$ | $0.73$ | $0.65$ | $0.07$ | $4.24\times {10}^{-2}$ | $13.45$ | $4.74$ |

175 | $2.98\times {10}^{-4}$ | $2.98\times {10}^{-4}$ | $2.39\times {10}^{-4}$ | $0.67$ | $1.0$ | $1.0$ | $0.8$ | 2560 | $0.68$ | $0.69$ | $0.78$ | $0.07$ | $4.17\times {10}^{-2}$ | $13.21$ | $4.35$ |

176 | $3.33\times {10}^{-4}$ | $2.66\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $0.9$ | 2360 | $0.69$ | $0.7$ | $0.55$ | $0.07$ | $3.45\times {10}^{-2}$ | $10.95$ | $5.51$ |

177 | $3.33\times {10}^{-4}$ | $2.66\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $0.9$ | 2460 | $0.69$ | $0.7$ | $0.55$ | $0.07$ | $3.64\times {10}^{-2}$ | $11.53$ | $5.36$ |

178 | $3.33\times {10}^{-4}$ | $2.66\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $0.9$ | 2560 | $0.69$ | $0.7$ | $0.54$ | $0.07$ | $3.81\times {10}^{-2}$ | $12.09$ | $5.19$ |

179 | $3.33\times {10}^{-4}$ | $2.66\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $0.9$ | 2660 | $0.69$ | $0.7$ | $0.55$ | $0.07$ | $3.99\times {10}^{-2}$ | $12.64$ | $5.06$ |

180 | $3.08\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $2.49\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.9$ | $0.9$ | 2460 | $0.68$ | $0.69$ | $0.55$ | $0.07$ | $3.74\times {10}^{-2}$ | $11.85$ | $5.07$ |

ID | ${\mathit{a}}_{\mathbf{L}}$ in m | ${\mathit{a}}_{\mathit{I}}$ in m | ${\mathit{a}}_{\mathbf{S}}$ in m | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | E | F | ${\mathit{\rho}}_{\mathbf{p}}$ | ${\mathit{\kappa}}_{\mathbf{con}}$ | $\mathit{\psi}$ | ${\mathit{\psi}}_{\perp}$ | ${\mathit{\kappa}}_{\mathbf{rnd}}$ | ${\mathit{u}}_{\mathbf{ts}}$ | Re | ${\mathit{C}}_{\mathbf{D}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

181 | $3.08\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $2.49\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.9$ | $0.9$ | 2560 | $0.68$ | $0.69$ | $0.57$ | $0.07$ | $3.92\times {10}^{-2}$ | $12.42$ | $4.92$ |

182 | $3.08\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $2.49\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.9$ | $0.9$ | 2660 | $0.68$ | $0.69$ | $0.56$ | $0.07$ | $4.09\times {10}^{-2}$ | $12.98$ | $4.8$ |

183 | $3.08\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $2.49\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.9$ | $0.9$ | 2760 | $0.68$ | $0.69$ | $0.55$ | $0.07$ | $4.27\times {10}^{-2}$ | $13.52$ | $4.69$ |

184 | $2.87\times {10}^{-4}$ | $2.87\times {10}^{-4}$ | $2.58\times {10}^{-4}$ | $0.67$ | $1.0$ | $1.0$ | $0.9$ | 2360 | $0.68$ | $0.68$ | $0.72$ | $0.06$ | $3.69\times {10}^{-2}$ | $11.7$ | $4.82$ |

185 | $2.87\times {10}^{-4}$ | $2.87\times {10}^{-4}$ | $2.58\times {10}^{-4}$ | $0.67$ | $1.0$ | $1.0$ | $0.9$ | 2660 | $0.68$ | $0.68$ | $0.83$ | $0.06$ | $4.28\times {10}^{-2}$ | $13.58$ | $4.38$ |

186 | $2.87\times {10}^{-4}$ | $2.87\times {10}^{-4}$ | $2.58\times {10}^{-4}$ | $0.67$ | $1.0$ | $1.0$ | $0.9$ | 2760 | $0.68$ | $0.68$ | $0.83$ | $0.06$ | $4.47\times {10}^{-2}$ | $14.18$ | $4.25$ |

187 | $3.51\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.7$ | $1.0$ | 2360 | $0.69$ | $0.69$ | $0.53$ | $0.07$ | $3.32\times {10}^{-2}$ | $10.53$ | $5.97$ |

188 | $3.51\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.7$ | $1.0$ | 2460 | $0.69$ | $0.69$ | $0.53$ | $0.07$ | $3.50\times {10}^{-2}$ | $11.09$ | $5.79$ |

189 | $3.51\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.7$ | $1.0$ | 2560 | $0.69$ | $0.69$ | $0.53$ | $0.07$ | $3.67\times {10}^{-2}$ | $11.62$ | $5.63$ |

190 | $3.51\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $2.46\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.7$ | $1.0$ | 2660 | $0.69$ | $0.69$ | $0.52$ | $0.07$ | $3.83\times {10}^{-2}$ | $12.15$ | $5.48$ |

191 | $3.21\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $1.0$ | 2360 | $0.68$ | $0.68$ | $0.53$ | $0.07$ | $3.41\times {10}^{-2}$ | $10.82$ | $5.67$ |

192 | $3.21\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $1.0$ | 2460 | $0.68$ | $0.68$ | $0.53$ | $0.07$ | $3.59\times {10}^{-2}$ | $11.39$ | $5.48$ |

193 | $3.21\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $1.0$ | 2560 | $0.68$ | $0.68$ | $0.54$ | $0.07$ | $3.77\times {10}^{-2}$ | $11.95$ | $5.31$ |

194 | $3.21\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $1.0$ | 2660 | $0.68$ | $0.68$ | $0.53$ | $0.07$ | $3.94\times {10}^{-2}$ | $12.49$ | $5.16$ |

195 | $3.21\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $2.57\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.8$ | $1.0$ | 2760 | $0.68$ | $0.68$ | $0.54$ | $0.07$ | $4.11\times {10}^{-2}$ | $13.02$ | $5.06$ |

196 | $2.97\times {10}^{-4}$ | $2.67\times {10}^{-4}$ | $2.67\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.9$ | $1.0$ | 2460 | $0.68$ | $0.68$ | $0.54$ | $0.06$ | $3.69\times {10}^{-2}$ | $11.69$ | $5.21$ |

197 | $2.97\times {10}^{-4}$ | $2.67\times {10}^{-4}$ | $2.67\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.9$ | $1.0$ | 2660 | $0.68$ | $0.68$ | $0.56$ | $0.06$ | $4.04\times {10}^{-2}$ | $12.79$ | $4.95$ |

198 | $2.97\times {10}^{-4}$ | $2.67\times {10}^{-4}$ | $2.67\times {10}^{-4}$ | $0.67$ | $1.0$ | $0.9$ | $1.0$ | 2760 | $0.68$ | $0.68$ | $0.56$ | $0.06$ | $4.20\times {10}^{-2}$ | $13.33$ | $4.82$ |

199 | $2.77\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $0.67$ | $1.0$ | $1.0$ | $1.0$ | 2460 | $0.69$ | $0.67$ | $0.89$ | $0.06$ | $3.83\times {10}^{-2}$ | $12.14$ | $4.81$ |

200 | $2.77\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $2.77\times {10}^{-4}$ | $0.67$ | $1.0$ | $1.0$ | $1.0$ | 2560 | $0.69$ | $0.67$ | $0.89$ | $0.06$ | $4.02\times {10}^{-2}$ | $12.75$ | $4.67$ |

E | ${\mathit{\rho}}_{\mathbf{p}}$ | $\mathit{\psi}$ | ${\mathit{\kappa}}_{\mathbf{rnd}}$ | ${\mathit{\lambda}}_{\mathbf{H}}$ | Re | ${\mathit{C}}_{\mathbf{D}}$ | ${\mathit{u}}_{\mathbf{ts}}$ | |
---|---|---|---|---|---|---|---|---|

mean | $0.757$ | $2563.17$ | $0.786$ | $0.101$ | $-0.962$ | $14.380$ | $4.544$ | $0.045$ |

standard deviation | $0.166$ | $141.01$ | $0.072$ | $0.026$ | $0.038$ | $3.103$ | $0.970$ | $0.010$ |

## Appendix B

ID | ${\mathit{a}}_{\mathbf{L}}$ in m | ${\mathit{a}}_{\mathit{I}}$ in m | ${\mathit{a}}_{\mathbf{S}}$ in m | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | E | F | ${\mathit{\rho}}_{\mathbf{p}}$ | ${\mathit{\kappa}}_{\mathbf{con}}$ | $\mathit{\psi}$ | ${\mathit{\psi}}_{\perp}$ | ${\mathit{\kappa}}_{\mathbf{rnd}}$ | ${\mathit{u}}_{\mathbf{ts}}$ | Re | ${\mathit{C}}_{\mathbf{D}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | $5.39\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $1.35\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $0.5$ | 2460 | $0.75$ | $0.75$ | $0.51$ | $0.07$ | $2.89\times {10}^{-2}$ | $9.17$ | $8.21$ |

2 | $4.78\times {10}^{-4}$ | $2.87\times {10}^{-4}$ | $1.43\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.5$ | 2460 | $0.74$ | $0.73$ | $0.52$ | $0.07$ | $3.06\times {10}^{-2}$ | $9.71$ | $7.33$ |

3 | $3.65\times {10}^{-4}$ | $3.28\times {10}^{-4}$ | $1.64\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.9$ | $0.5$ | 2460 | $0.72$ | $0.73$ | $0.55$ | $0.07$ | $3.32\times {10}^{-2}$ | $10.54$ | $6.21$ |

4 | $3.40\times {10}^{-4}$ | $3.40\times {10}^{-4}$ | $1.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $0.5$ | 2460 | $0.72$ | $0.73$ | $0.56$ | $0.07$ | $3.40\times {10}^{-2}$ | $10.76$ | $5.96$ |

5 | $5.08\times {10}^{-4}$ | $2.54\times {10}^{-4}$ | $1.52\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $0.6$ | 2460 | $0.76$ | $0.77$ | $0.57$ | $0.08$ | $3.08\times {10}^{-2}$ | $9.75$ | $7.28$ |

6 | $3.71\times {10}^{-4}$ | $2.97\times {10}^{-4}$ | $1.78\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $0.6$ | 2460 | $0.72$ | $0.74$ | $0.59$ | $0.07$ | $3.45\times {10}^{-2}$ | $10.95$ | $5.75$ |

7 | $3.43\times {10}^{-4}$ | $3.09\times {10}^{-4}$ | $1.85\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.9$ | $0.6$ | 2460 | $0.71$ | $0.74$ | $0.61$ | $0.07$ | $3.54\times {10}^{-2}$ | $11.22$ | $5.5$ |

8 | $3.20\times {10}^{-4}$ | $3.20\times {10}^{-4}$ | $1.92\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $0.6$ | 2460 | $0.71$ | $0.73$ | $0.64$ | $0.07$ | $3.60\times {10}^{-2}$ | $11.4$ | $5.31$ |

9 | $4.82\times {10}^{-4}$ | $2.41\times {10}^{-4}$ | $1.69\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $0.7$ | 2460 | $0.75$ | $0.77$ | $0.61$ | $0.08$ | $3.24\times {10}^{-2}$ | $10.28$ | $6.53$ |

10 | $4.27\times {10}^{-4}$ | $2.56\times {10}^{-4}$ | $1.79\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.7$ | 2460 | $0.73$ | $0.76$ | $0.62$ | $0.07$ | $3.40\times {10}^{-2}$ | $10.79$ | $5.92$ |

11 | $3.85\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $1.89\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.7$ | 2460 | $0.72$ | $0.75$ | $0.63$ | $0.07$ | $3.54\times {10}^{-2}$ | $11.22$ | $5.48$ |

12 | $3.53\times {10}^{-4}$ | $2.82\times {10}^{-4}$ | $1.97\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $0.7$ | 2460 | $0.71$ | $0.74$ | $0.65$ | $0.07$ | $3.64\times {10}^{-2}$ | $11.53$ | $5.19$ |

13 | $3.26\times {10}^{-4}$ | $2.93\times {10}^{-4}$ | $2.05\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.9$ | $0.7$ | 2460 | $0.7$ | $0.73$ | $0.67$ | $0.07$ | $3.71\times {10}^{-2}$ | $11.77$ | $4.98$ |

14 | $3.04\times {10}^{-4}$ | $3.04\times {10}^{-4}$ | $2.13\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $0.7$ | 2460 | $0.7$ | $0.73$ | $0.69$ | $0.07$ | $3.76\times {10}^{-2}$ | $11.93$ | $4.85$ |

15 | $4.61\times {10}^{-4}$ | $2.31\times {10}^{-4}$ | $1.84\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $0.8$ | 2460 | $0.75$ | $0.77$ | $0.59$ | $0.08$ | $3.28\times {10}^{-2}$ | $10.4$ | $6.37$ |

16 | $3.69\times {10}^{-4}$ | $2.58\times {10}^{-4}$ | $2.06\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.8$ | 2460 | $0.72$ | $0.74$ | $0.61$ | $0.07$ | $3.61\times {10}^{-2}$ | $11.45$ | $5.27$ |

17 | $3.37\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $2.16\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $0.8$ | 2460 | $0.71$ | $0.74$ | $0.66$ | $0.07$ | $3.73\times {10}^{-2}$ | $11.83$ | $4.94$ |

18 | $3.12\times {10}^{-4}$ | $2.81\times {10}^{-4}$ | $2.24\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.9$ | $0.8$ | 2460 | $0.7$ | $0.72$ | $0.69$ | $0.07$ | $3.83\times {10}^{-2}$ | $12.15$ | $4.67$ |

19 | $2.91\times {10}^{-4}$ | $2.91\times {10}^{-4}$ | $2.32\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $0.8$ | 2460 | $0.7$ | $0.72$ | $0.75$ | $0.07$ | $3.96\times {10}^{-2}$ | $12.56$ | $4.38$ |

20 | $4.43\times {10}^{-4}$ | $2.22\times {10}^{-4}$ | $2.00\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $0.9$ | 2460 | $0.75$ | $0.75$ | $0.55$ | $0.07$ | $3.23\times {10}^{-2}$ | $10.24$ | $6.59$ |

21 | $3.93\times {10}^{-4}$ | $2.36\times {10}^{-4}$ | $2.12\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $0.9$ | 2460 | $0.72$ | $0.74$ | $0.56$ | $0.07$ | $3.39\times {10}^{-2}$ | $10.76$ | $5.96$ |

22 | $3.54\times {10}^{-4}$ | $2.48\times {10}^{-4}$ | $2.23\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $0.9$ | 2460 | $0.72$ | $0.73$ | $0.57$ | $0.07$ | $3.53\times {10}^{-2}$ | $11.17$ | $5.53$ |

23 | $2.79\times {10}^{-4}$ | $2.79\times {10}^{-4}$ | $2.51\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $0.9$ | 2460 | $0.71$ | $0.71$ | $0.76$ | $0.07$ | $3.88\times {10}^{-2}$ | $12.31$ | $4.54$ |

24 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $1.0$ | 2460 | $0.75$ | $0.73$ | $0.53$ | $0.07$ | $3.22\times {10}^{-2}$ | $10.2$ | $6.62$ |

25 | $3.79\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $1.0$ | 2460 | $0.73$ | $0.72$ | $0.54$ | $0.07$ | $3.38\times {10}^{-2}$ | $10.7$ | $6.02$ |

26 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $1.0$ | 2360 | $0.75$ | $0.73$ | $0.53$ | $0.07$ | $3.06\times {10}^{-2}$ | $9.7$ | $6.83$ |

27 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $1.0$ | 2460 | $0.75$ | $0.73$ | $0.53$ | $0.07$ | $3.22\times {10}^{-2}$ | $10.2$ | $6.62$ |

28 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $1.0$ | 2560 | $0.75$ | $0.73$ | $0.53$ | $0.07$ | $3.37\times {10}^{-2}$ | $10.69$ | $6.44$ |

29 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $1.0$ | 2660 | $0.75$ | $0.73$ | $0.53$ | $0.07$ | $3.53\times {10}^{-2}$ | $11.18$ | $6.27$ |

30 | $4.28\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $2.14\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.5$ | $1.0$ | 2760 | $0.75$ | $0.73$ | $0.53$ | $0.07$ | $3.68\times {10}^{-2}$ | $11.66$ | $6.11$ |

31 | $3.79\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $1.0$ | 2360 | $0.73$ | $0.72$ | $0.53$ | $0.07$ | $3.21\times {10}^{-2}$ | $10.16$ | $6.23$ |

32 | $3.79\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $1.0$ | 2460 | $0.73$ | $0.72$ | $0.54$ | $0.07$ | - | - | - |

33 | $3.79\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $1.0$ | 2560 | $0.73$ | $0.72$ | $0.53$ | $0.07$ | $3.54\times {10}^{-2}$ | $11.21$ | $5.87$ |

34 | $3.79\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $1.0$ | 2660 | $0.73$ | $0.72$ | $0.53$ | $0.07$ | - | - | - |

35 | $3.79\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $2.28\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.6$ | $1.0$ | 2760 | $0.73$ | $0.72$ | $0.53$ | $0.07$ | - | - | - |

36 | $3.42\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $1.0$ | 2660 | $0.72$ | $0.71$ | $0.54$ | $0.07$ | - | - | - |

37 | $3.42\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $2.40\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.7$ | $1.0$ | 2760 | $0.72$ | $0.71$ | $0.55$ | $0.07$ | - | - | - |

38 | $3.13\times {10}^{-4}$ | $2.50\times {10}^{-4}$ | $2.50\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.8$ | $1.0$ | 2560 | $0.72$ | $0.71$ | $0.56$ | $0.07$ | $3.77\times {10}^{-2}$ | $11.94$ | $5.16$ |

39 | $2.89\times {10}^{-4}$ | $2.60\times {10}^{-4}$ | $2.60\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.9$ | $1.0$ | 2360 | $0.72$ | $0.7$ | $0.56$ | $0.07$ | $3.50\times {10}^{-2}$ | $11.09$ | $5.23$ |

40 | $2.89\times {10}^{-4}$ | $2.60\times {10}^{-4}$ | $2.60\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.9$ | $1.0$ | 2560 | $0.72$ | $0.7$ | $0.56$ | $0.07$ | - | - | - |

41 | $2.89\times {10}^{-4}$ | $2.60\times {10}^{-4}$ | $2.60\times {10}^{-4}$ | $0.7$ | $1.0$ | $0.9$ | $1.0$ | 2660 | $0.72$ | $0.7$ | $0.57$ | $0.07$ | - | - | - |

42 | $2.70\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $1.0$ | 2360 | $0.71$ | $0.69$ | $0.71$ | $0.07$ | - | - | - |

43 | $2.70\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $1.0$ | 2660 | $0.71$ | $0.69$ | $0.75$ | $0.07$ | $4.20\times {10}^{-2}$ | $13.32$ | $4.42$ |

44 | $2.70\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $0.7$ | $1.0$ | $1.0$ | $1.0$ | 2760 | $0.71$ | $0.69$ | $0.76$ | $0.07$ | $4.39\times {10}^{-2}$ | $13.91$ | $4.29$ |

## References

- Fu, X.; Huck, D.; Makein, L.; Armstrong, B.; Willen, U.; Freeman, T. Effect of particle shape and size on flow properties of lactose powders. Particuology
**2012**, 10, 203–208. [Google Scholar] [CrossRef] - Kashiwaya, K.; Noumachi, T.; Hiroyoshi, N.; Ito, M.; Tsunekawa, M. Effect of particle shape on hydrocyclone classification. Powder Technol.
**2012**, 226, 147–156. [Google Scholar] [CrossRef] [Green Version] - Champion, J.A.; Katare, Y.K.; Mitragotri, S. Particle shape: A new design parameter for micro- and nanoscale drug delivery carriers. J. Control. Release
**2007**, 121, 3–9. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Waldschläger, K.; Schüttrumpf, H. Effects of Particle Properties on the Settling and Rise Velocities of Microplastics in Freshwater under Laboratory Conditions. Environ. Sci. Technol.
**2019**, 53, 1958–1966. [Google Scholar] [CrossRef] [PubMed] - Zingg, T. Beitrag zur Schotteranalyse. Ph.D. Thesis, ETH, Zürich, Switzerland, 1935. [Google Scholar]
- Sneed, E.D.; Folk, R.L. Pebbles in the Lower Colorado River, Texas a Study in Particle Morphogenesis. J. Geol.
**1958**, 66, 114–150. [Google Scholar] [CrossRef] - ISO 9276-6:2008(E). Representation of Results of Particle Size Analysis—Part 6: Descriptive and Quantitative Representation of Particle Shape and Morphology; Standard; International Organization for Standardization: Geneva, Switzerland, 2008. [Google Scholar]
- Hentschel, M.L.; Page, N.W. Selection of Descriptors for Particle Shape Characterization. Part. Part. Syst. Charact.
**2003**, 20, 25–38. [Google Scholar] [CrossRef] - Blott, S.J.; Pye, K. Particle shape: A review and new methods of characterization and classification. Sedimentology
**2008**, 55, 31–63. [Google Scholar] [CrossRef] - Allen, J.R.L. Chapter 5 Orientation of Particles During Sedimentation: Shape-Fabrics. In Sedimentary Structures Their Character and Physical Basis Volume I; Elsevier: Amsterdam, The Netherlands, 1982; Volume 30, pp. 179–235. [Google Scholar] [CrossRef]
- Sheikh, M.Z.; Gustavsson, K.; Lopez, D.; Lévêque, E.; Mehlig, B.; Pumir, A.; Naso, A. Importance of fluid inertia for the orientation of spheroids settling in turbulent flow. J. Fluid Mech.
**2020**, 886. [Google Scholar] [CrossRef] [Green Version] - Bagheri, G.; Bonadonna, C. On the drag of freely falling non-spherical particles. Powder Technol.
**2016**, 301, 526–544. [Google Scholar] [CrossRef] [Green Version] - Komar, P.D.; Reimers, C.E. Grain Shape Effects on Settling Rates. J. Geol.
**1978**, 86, 193–209. [Google Scholar] [CrossRef] - Shao, B.; Liu, G.R.; Lin, T.; Xu, G.X.; Yan, X. Rotation and orientation of irregular particles in viscous fluids using the gradient smoothed method (GSM). Eng. Appl. Comput. Fluid Mech.
**2017**, 11, 557–575. [Google Scholar] [CrossRef] - Stokes, G.G. On the Effect of Internal Friction of Fluids on the Motion of Pendulums. Trans. Camb. Philos. Soc.
**1851**, 9, 8–106. [Google Scholar] [CrossRef] [Green Version] - Schiller, L.; Naumann, A. Über die grundlegenden Berechnungen bei der Schwerkraftaufbereitung. Z. Des Vereines Dtsch. Ingenieure
**1933**, 77, 318–320. [Google Scholar] - Clift, R.; Gauvin, W.H. Motion of entrained particles in gas streams. Can. J. Chem. Eng.
**1971**, 49, 439–448. [Google Scholar] [CrossRef] - McNown, J.S.; Malaika, J. Effects of particle shape on settling velocity at low Reynolds numbers. Trans. Am. Geophys. Union
**1950**, 31, 74. [Google Scholar] [CrossRef] - Sommerfeld, M.; Qadir, Z. Fluid dynamic forces acting on irregular shaped particles: Simulations by the Lattice–Boltzmann method. Int. J. Multiph. Flow
**2018**, 101, 212–222. [Google Scholar] [CrossRef] - Leith, D. Drag on Nonspherical Objects. Aerosol Sci. Technol.
**1987**, 6, 153–161. [Google Scholar] [CrossRef] [Green Version] - Ganser, G.H. A rational approach to drag prediction of spherical and nonspherical particles. Powder Technol.
**1993**, 77, 143–152. [Google Scholar] [CrossRef] - Loth, E. Drag of non-spherical solid particles of regular and irregular shape. Powder Technol.
**2008**, 182, 342–353. [Google Scholar] [CrossRef] - Hölzer, A.; Sommerfeld, M. New simple correlation formula for the drag coefficient of non-spherical particles. Powder Technol.
**2008**, 184, 361–365. [Google Scholar] [CrossRef] - Tran-Cong, S.; Gay, M.; Michaelides, E.E. Drag coefficients of irregularly shaped particles. Powder Technol.
**2004**, 139, 21–32. [Google Scholar] [CrossRef] - Dellino, P.; Mele, D.; Bonasia, R.; Braia, G.; La Volpe, L.; Sulpizio, R. The analysis of the influence of pumice shape on its terminal velocity. Geophys. Res. Lett.
**2005**, 32. [Google Scholar] [CrossRef] - Dioguardi, F.; Mele, D. A new shape dependent drag correlation formula for non-spherical rough particles. Experiments and results. Powder Technol.
**2015**, 277, 222–230. [Google Scholar] [CrossRef] - Hölzer, A.; Sommerfeld, M. Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles. Comput. Fluids
**2009**, 38, 572–589. [Google Scholar] [CrossRef] - Haider, A.; Levenspiel, O. Drag coefficient and terminal velocity of spherical and nonspherical particles. Powder Technol.
**1989**, 58, 63–70. [Google Scholar] [CrossRef] - Henn, T.; Thäter, G.; Dörfler, W.; Nirschl, H.; Krause, M.J. Parallel dilute particulate flow simulations in the human nasal cavity. Comput. Fluids
**2016**, 124, 197–207. [Google Scholar] [CrossRef] - Trunk, R.; Henn, T.; Dörfler, W.; Nirschl, H.; Krause, M.J. Inertial dilute particulate fluid flow simulations with an Euler–Euler lattice Boltzmann method. J. Comput. Sci.
**2016**, 17, 438–445. [Google Scholar] [CrossRef] - Maier, M.L.; Milles, S.; Schuhmann, S.; Guthausen, G.; Nirschl, H.; Krause, M.J. Fluid flow simulations verified by measurements to investigate adsorption processes in a static mixer. Comput. Math. Appl.
**2018**, 76, 2744–2757. [Google Scholar] [CrossRef] - Trunk, R.; Marquardt, J.; Thäter, G.; Nirschl, H.; Krause, M.J. Towards the simulation of arbitrarily shaped 3D particles using a homogenised lattice Boltzmann method. Comput. Fluids
**2018**, 172, 621–631. [Google Scholar] [CrossRef] - Krause, M.J.; Klemens, F.; Henn, T.; Trunk, R.; Nirschl, H. Particle flow simulations with homogenised lattice Boltzmann methods. Particuology
**2017**, 34, 1–13. [Google Scholar] [CrossRef] - Trunk, R.; Weckerle, T.; Hafen, N.; Thäter, G.; Nirschl, H.; Krause, M.J. Revisiting the Homogenized Lattice Boltzmann Method with Applications on Particulate Flows. Computation
**2021**, 9, 11. [Google Scholar] [CrossRef] - Krause, M.J.; Kummerländer, A.; Avis, S.J.; Kusumaatmaja, H.; Dapelo, D.; Klemens, F.; Gaedtke, M.; Hafen, N.; Mink, A.; Trunk, R.; et al. OpenLB—Open source lattice Boltzmann code. Comput. Math. Appl.
**2020**. [Google Scholar] [CrossRef] - Krause, M.; Avis, S.; Kusumaatmaja, H.; Dapelo, D.; Gaedtke, M.; Hafen, N.; Haußmann, M.; Jeppener-Haltenhoff, J.; Kronberg, L.; Kummerländer, A.; et al. OpenLB Release 1.4: Open Source Lattice Boltzmann Code. Comput. Math. Appl.
**2020**. [Google Scholar] [CrossRef] - Yow, H.N.; Pitt, M.J.; Salman, A.D. Drag correlations for particles of regular shape. Adv. Powder Technol.
**2005**, 16, 363–372. [Google Scholar] [CrossRef] - Dey, S.; Ali, S.Z.; Padhi, E. Terminal fall velocity: The legacy of Stokes from the perspective of fluvial hydraulics. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2019**, 475, 20190277. [Google Scholar] [CrossRef] [Green Version] - Bretl, C.; Trunk, R.; Nirschl, H.; Thäter, G.; Dorn, M.; Krause, M.J. Preliminary study of particle settling behaviour by shape parameters via lattice Boltzmann simulations. In High Performance Computing in Science and Engineering ’20; Nagel, W., Kröner, D., Resch, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]
- Wadell, H. Volume, Shape, and Roundness of Rock Particles. J. Geol.
**1932**, 40, 443–451. [Google Scholar] [CrossRef] - Krumbein, W.C. Measurement and Geological Significance of Shape and Roundness of Sedimentary Particles. SEPM J. Sediment. Res.
**1941**, 11. [Google Scholar] [CrossRef] - Hayakawa, Y.; Oguchi, T. Evaluation of gravel sphericity and roundness based on surface-area measurement with a laser scanner. Comput. Geosci.
**2005**, 31, 735–741. [Google Scholar] [CrossRef] - Dietrich, W.E. Settling velocity of natural particles. Water Resour. Res.
**1982**, 18, 1615–1626. [Google Scholar] [CrossRef] - Hofmann, H.J. Grain-shaped indices and isometric graphs. J. Sediment. Res.
**1994**, 64, 916–920. [Google Scholar] [CrossRef] - Le Roux, J.P. Application of the Hofmann shape entropy to determine the settling velocity of irregular, semi-ellipsoidal grains. Sediment. Geol.
**2002**, 149, 237–243. [Google Scholar] [CrossRef] - Le Roux, J. A Hydrodynamic Classification of Grain Shapes. J. Sediment. Res.
**2004**, 74, 135–143. [Google Scholar] [CrossRef] - Williams, J.R.; Pentland, A.P. Superquadrics and modal dynamics for discrete elements in interactive design. Eng. Comput.
**1992**, 9, 115–127. [Google Scholar] [CrossRef] - Barr, A.H. Superquadrics and Angle-Preserving Transformations. IEEE Comput. Graph. Appl.
**1981**, 1, 11–23. [Google Scholar] [CrossRef] [Green Version] - Wellmann, C.; Lillie, C.; Wriggers, P. A contact detection algorithm for superellipsoids based on the common-normal concept. Eng. Comput.
**2008**, 25, 432–442. [Google Scholar] [CrossRef] [Green Version] - Jaklič, A.; Leonardis, A.; Solina, F. Superquadrics and Their Geometric Properties. In Segmentation and Recovery of Superquadrics; Springer: Dordrecht, The Netherlands, 2000; pp. 13–39. [Google Scholar] [CrossRef]
- Jaklič, A.; Solina, F. Moments of superellipsoids and their application to range image registration. IEEE Trans. Syst. Man Cybern. Part B (Cybern.)
**2003**, 33, 648–657. [Google Scholar] [CrossRef] [Green Version] - Bagheri, G.; Bonadonna, C. On the Drag of Freely Falling Non-Spherical Particles. Available online: https://arxiv.org/abs/1810.08787 (accessed on 25 March 2021).
- Chrust, M.; Bouchet, G.; Dus̆ek, J. Numerical simulation of the dynamics of freely falling discs. Phys. Fluids
**2013**, 25, 044102. [Google Scholar] [CrossRef] - Seyed-Ahmadi, A.; Wachs, A. Dynamics and wakes of freely settling and rising cubes. Phys. Rev. Fluids
**2018**, 4. [Google Scholar] [CrossRef] - Rahmani, M.; Wachs, A. Free falling and rising of spherical and angular particles. Phys. Fluids
**2014**, 26, 083301. [Google Scholar] [CrossRef] - Kupershtokh, A.; Medvedev, D.; Karpov, D. On equations of state in a lattice Boltzmann method. Comput. Math. Appl.
**2009**, 58, 965–974. [Google Scholar] [CrossRef] [Green Version] - Wen, B.; Zhang, C.; Tu, Y.; Wang, C.; Fang, H. Galilean invariant fluid–solid interfacial dynamics in lattice Boltzmann simulations. J. Comput. Phys.
**2014**, 266, 161–170. [Google Scholar] [CrossRef] [Green Version] - Abraham, F.F. Functional Dependence of Drag Coefficient of a Sphere on Reynolds Number. Phys. Fluids
**1970**, 13, 2194. [Google Scholar] [CrossRef] - Krüger, T.; Kusumaatmaja, H.; Kuzmin, A.; Shardt, O.; Silva, G.; Viggen, E.M. The Lattice Boltzmann Method; Graduate Texts in Physics; Springer: Cham, Switzerland, 2017. [Google Scholar] [CrossRef]
- Young, D.S. Handbook of Regression Methods; Chapman and Hall/CRC: Boca Raton, FL, USA, 2017. [Google Scholar] [CrossRef]
- Pearson, K. VII. Mathematical contributions to the theory of evolution.—III. Regression, heredity, and panmixia. Philos. Trans. R. Soc. Lond. Ser. A Contain. Pap. A Math. Phys. Character
**1896**, 187, 253–318. [Google Scholar] [CrossRef] [Green Version] - Wisniewski, M. Applied Regression Analysis: A Research Tool. J. Oper. Res. Soc.
**1990**. [Google Scholar] [CrossRef] - Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Kraskov, A.; Stögbauer, H.; Grassberger, P. Estimating mutual information. Phys. Rev. E
**2004**, 69. [Google Scholar] [CrossRef] [Green Version] - Breiman, L. Random Forests. Mach. Learn.
**2001**, 45, 5–32. [Google Scholar] [CrossRef] [Green Version] - Horowitz, M.; Williamson, C.H.K. The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres. J. Fluid Mech.
**2010**, 651, 251–294. [Google Scholar] [CrossRef] - Szabó, T.; Domokos, G. A new classification system for pebble and crystal shapes based on static equilibrium points. Cent. Eur. Geol.
**2010**, 53, 1–19. [Google Scholar] [CrossRef] [Green Version] - Wilk, M.B.; Gnanadesikan, R. Probability plotting methods for the analysis for the analysis of data. Biometrika
**1968**, 55, 1–17. [Google Scholar] [CrossRef] - Thode, H.C. Testing For Normality; Taylor & Francis: Abingdon, UK, 2002. [Google Scholar] [CrossRef]
- Anscombe, F.J.; Tukey, J.W. The Examination and Analysis of Residuals. Technometrics
**1963**, 5, 141–160. [Google Scholar] [CrossRef] - Cook, R.D. Detection of Influential Observation in Linear Regression. Technometrics
**1977**, 19, 15. [Google Scholar] [CrossRef]

**Figure 1.**Some examples of particle shapes considered via superellipsoids. Depicted (from left to right and top to bottom) are the particles with ID 5, 17, 28, 143, 166 and 200, according to Appendix A.

**Figure 3.**Voxel representation of the particle with the shortest half-axis length of $9.1\times {10}^{-5}$ $\mathrm{m}$ for $N=20$.

**Figure 4.**Coordinates of the center of mass of the particle, projected on a plane normal to the settling direction. Results printed for different grid spacings, for the particle with the shortest half-axis.

**Figure 5.**Plot of the angle for rotation around the y-axis over time during the settling. Results printed for different grid spacings, for the particle with the shortest half-axis.

**Figure 6.**Plot of the angle for rotation around the y-axis over time during the settling. Results printed for different grid spacings, for the particle with the longest half-axis.

**Figure 7.**Coordinates of the center of mass of the particle projected, on a plane normal to the settling direction. Results printed for different grid spacings, for the particle with the longest half-axis.

**Figure 9.**The particles considered in this study are plotted for a shape classification according to Zingg [5]. The four shape classes are 1: discs, 2: spheres, 3: blades and 4: rods.

**Figure 10.**The particles considered in this study are plotted for a shape classification according to Sneed and Folk [6]. They are also classified regarding the compactness, leading to the ten shape classes 1: compact, 2: compactly platy, 3: compactly bladed, 4: compactly elongated, 5: platy, 6: bladed, 7: elongated, 8: very platy, 9: very bladed and 10: very elongated.

**Figure 11.**The drag coefficient plotted against the Reynolds number, color with shape classification according to Zingg [5]. A spread is revealed, describable via the sphericity (marker style).

**Table 1.**Results of a chi-squared test for equal distribution of frequencies of different shape parameters.

Elongation | Flatness | Convexity | Sphericity | Density | |
---|---|---|---|---|---|

numberof bins | 6 | 6 | 6 | 5 | 5 |

p-value | $0.998$ | $0.995$ | $0.999$ | $1.000$ | $1.000$ |

**Table 2.**Absolute Values of the correlation coefficients according to Pearson [61] for the considered shape parameters.

${\mathit{a}}_{\mathbf{L}}$ | ${\mathit{a}}_{\mathit{I}}$ | ${\mathit{a}}_{\mathbf{S}}$ | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ | ${\mathit{\rho}}_{\mathbf{p}}$ | E | F | ${\mathit{\kappa}}_{\mathbf{con}}$ | $\mathit{\psi}$ | ${\mathit{\psi}}_{\perp}$ | ${\mathit{\kappa}}_{\mathbf{rnd}}$ | ${\mathit{\lambda}}_{\mathbf{CSF}}$ | ${\mathit{\lambda}}_{\mathbf{H}}$ | ${\mathit{\lambda}}_{\mathbf{LR}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${a}_{\mathrm{L}}$ | $1.0$ | $0.55$ | $0.04$ | $0.41$ | $0.17$ | $0.03$ | $0.74$ | $0.51$ | $0.65$ | $0.52$ | $0.64$ | $0.68$ | $0.67$ | $0.73$ | $0.75$ |

${a}_{\mathrm{I}}$ | $0.55$ | $1.0$ | $0.57$ | $0.39$ | $0.47$ | $0.08$ | $0.13$ | $0.13$ | $0.63$ | $0.61$ | $0.47$ | $0.85$ | $0.05$ | $0.04$ | $0.02$ |

${a}_{\mathrm{S}}$ | $0.04$ | $0.57$ | $1.0$ | $0.18$ | $0.57$ | $0.03$ | $0.44$ | $0.72$ | $0.41$ | $0.45$ | $0.31$ | $0.61$ | $0.73$ | $0.64$ | $0.65$ |

${\xi}_{1}$ | $0.41$ | $0.39$ | $0.18$ | $1.0$ | $0.02$ | $0.08$ | $0.19$ | $0.15$ | $0.47$ | $0.44$ | $0.31$ | $0.57$ | $0.18$ | $0.21$ | $0.21$ |

${\xi}_{2}$ | $0.17$ | $0.47$ | $0.57$ | $0.02$ | $1.0$ | $0.03$ | $0.16$ | $0.36$ | $0.07$ | $0.05$ | $0.23$ | $0.34$ | $0.34$ | $0.32$ | $0.31$ |

${\rho}_{\mathrm{p}}$ | $0.03$ | $0.08$ | $0.03$ | $0.08$ | $0.03$ | $1.0$ | $0.02$ | $0.1$ | $0.03$ | $0.05$ | $0.04$ | < |