# Modeling of Solid Particle Erosion for a Water–Sand Impingement System Using OpenFOAM

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## Abstract

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## 1. Introduction

_{2}O

_{3}, Ni–Al

_{2}O

_{3}–TiO

_{2}CoNiCrAlY [4] and WC–Co–Cr [5], deposited using different techniques such as High Velocity Air-Fuel (HVAF), Atmospheric plasma spray (APS), and High Velocity Oxygen Fuel (HVOF) [6]. In addition, the necessity for erosion testing and modeling extends beyond the realm of hydropower and has significant implications in various industries. For instance, anti-reflective coatings, such as those used in photovoltaic glass covers, are vulnerable to degradation by erosion [7,8]. Erosion-induced deterioration of these coatings can result in reduced optical and mechanical properties, impacting their overall performance. Therefore, understanding erosion phenomena and developing accurate predictive models are crucial not only for hydropower applications but also for enhancing the durability and performance of materials in diverse industries.

## 2. Materials and Methods

#### 2.1. Liquid Phase Modeling

- $u$= Fluid phase velocity
- $\rho $ = Fluid phase density
- $p$= Pressure
- $\mu $= Dynamic viscosity.
- ${f}_{i}$= External forces.

#### 2.2. Solid Phase Modeling

- ${F}_{p}$= Sum of forces acting over the particle,
- ${F}_{D}$= Drag Force for non-spherical particles,
- ${F}_{P}$= Pressure Gradient,
- ${F}_{g}$= Gravity and buoyancy force,
- ${F}_{A}$= Added mass or Virtual mass force.

#### 2.3. Erosion Modeling

#### Erosion Parameters

- $\alpha $= Impact angle,
- $E\left(\alpha \right)$= Erosion damage in $\left[\frac{{\mathrm{mm}}^{3}}{\mathrm{kg}}\right]$,
- $g\left(\alpha \right)$= Impact angle dependence of the normalized erosion,
- ${E}_{90}$= Erosion damage at a normal angle.

_{2-1}[7]. To align our erosion model with the characteristics of these sediment types, the suggested values of $s$ and $q$ from Table 1 have been adopted. These values have been derived from extensive research and are tailored to suit the distinctive properties of Ecuadorian river sediments. By incorporating these coefficients into the modeling framework, the aim is to enhance the precision and applicability of erosion predictions for this specific sediment class.

_{90}. This value corresponds to the erosion rate for 326 $\left[\mathsf{\mu}\mathrm{m}\right]$ particles impacting at a normal angle with a velocity of 104 $\left[\frac{\mathrm{m}}{\mathrm{s}}\right]$. Therefore, E

_{90}is determined as $82\left[\frac{{\mathrm{mm}}^{3}}{\mathrm{kg}}\right]$ for the material of interest.

_{90}from Equation (22), the values of a and b were taken from Figure 1a,b and plotted from the data in Oka and Yoshida. The value of these parameters was obtained by linear regression and extrapolation. Figure 1b shows the values of the b parameter for different metallic materials and the adjusted trend line. Figure 1c shows the values of ${\left(a\times Hv\right)}^{-b}$ plotted against the normal erosion damage for these materials. Equation (19) can be used to calculate the erosion damage at any given velocity or particle size. However, under the controlled conditions of the experimental setup, Equation (25) is proposed to determine the erosion damage at a normal impingement angle.

_{90}, a, and b for SS304, the reference erosion rate was calculated from Equation (22). The calculated values and the approximate values for the remaining parameters are shown in Table 2.

#### 2.4. Sediment Analysis

#### 2.5. Erosion Implementation in OpenFOAM

#### 2.5.1. Two-Way Simulation Approach

#### 2.5.2. The Significance of the Stokes Number

#### 2.6. Computational Domain

#### 2.6.1. Geometry

#### 2.6.2. Computational Domain Discretization

#### 2.7. Erosion Pattern Analysis

## 3. Results

#### 3.1. Liquid–Solid Phase Results

#### 3.2. Erosion Results

^{−4}mm

^{3}.

^{3}]), was divided by the area of the corresponding surface ($\left[{\mathrm{mm}}^{2}\right]$), which includes the region where the erosion value was obtained, as shown in Figure 9a.

## 4. Discussion

## 5. Conclusions

^{−4}mm

^{3}. This achievement allowed for a quantitative assessment of erosion depth and pattern, serving as a basis for comparative analysis with experimental data. This pseudo-stabilization provides a critical reference point for understanding the temporal dynamics of erosion in hydro-power applications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Results of sieve analysis on sediment samples from Ecuadorian rivers. This figure, obtained from Cruzatti’s investigation [30], illustrates the size distribution of sediments found in the rivers of Ecuador, providing valuable data for our study.

**Figure 3.**Comparison of sediment mineralogy across different regions. This figure offers a comparative view of sediment composition in various regions, providing insights into the differences in mineral content among these areas.

**Figure 4.**Computational domain and patch description. $\u201cD\u201d$ parameter represents the diameter of the nozzle, providing a key reference for understanding the dimensions within the computational domain.

**Figure 5.**Computational mesh and structural detail. (

**a**) Slice view of the computational mesh, showcasing its intricacies. (

**b**) Top view highlighting block-structured meshing details.

**Table 1.**Constant exponents ‘s’ and ‘q’ extracted from the study of Oka and Yoshida in 2005 [14] used as key parameters in our erosion modeling methodology.

Parameter | $\mathit{s}$ | $\mathit{q}$ |
---|---|---|

n_{1} | 0.71 | 0.14 |

n_{2} | 2.4 | −0.94 |

Parameter | Value | Unit |
---|---|---|

K | 65 | - |

k_{1} | −0.12 | - |

k_{2} | 2.36 | - |

k_{3} | 0.19 | - |

n_{1} | 0.78 | - |

n_{2} | 1.27 | - |

a | 0.0221 | - |

b | 0.45 | - |

104 | $\left[\frac{\mathrm{m}}{\mathrm{s}}\right]$ | |

326 | $\left[\mu \mathrm{m}\right]$ | |

Hv | 1.96 | $\left[\mathrm{GPa}\right]$ |

**Table 3.**Details of experimental setup. Sourced from Nguyen’s study [18], provides valuable information for the conducted experiments.

Parameter | Value | Units |
---|---|---|

Particle velocity | 30 | $\left[\frac{\mathrm{m}}{\mathrm{s}}\right]$ |

Particle diameter | 150 | $\left[\mathsf{\mu}\mathrm{m}\right]$ |

Nozzle diameter (D) | 6.4 | $\left[\mathrm{mm}\right]$ |

Particle volume fraction | 0.5 | $\left[\%\right]$ |

Plate dimensions | 25 × 25 × 5 | $\left[{\mathrm{mm}}^{3}\right]$ |

Standoff distance | 12.7 | $\left[\mathrm{mm}\right]$ |

**Table 4.**Computational Fluid Dynamics (CFD) setup. Summary of key parameters and settings for executing the erosion simulation.

Zone | Condition | Value | Units |
---|---|---|---|

Inlet | Velocity Inlet | 30 | $\left[\frac{\mathrm{m}}{\mathrm{s}}\right]$ |

Nozzle | Wall | No Slip | - |

Plate | Wall | No Slip | - |

Outlet | Pressure Outlet | 0 | $\left[\mathrm{Pa}\right]$ |

Discrete Phase | |||

$\mathrm{Density}\left(\rho \right)$ | 2719 | $\left[\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}\right]$ | |

Young Modulus | 600 | [GPa] | |

Composition | SiO2-1 | Silicon dioxide |

**Table 5.**Mesh independence study results. This table presents the outcomes of the mesh independence study, demonstrating the convergence of key simulation parameters with varying mesh sizes.

Nodes | Pressure (kPa) | Error (%) |
---|---|---|

299.785 | 66.59 | 1.41 |

496.936 | 67.52 | 1.38 |

770.959 | 68.52 | 1.46 |

1.114.943 | 69.12 | 0.88 |

Time | Depth (μm) | ||
---|---|---|---|

Center | Peak | Border | |

5 min | $-24.3\pm 2.1$ | $-95.6\pm 5.3$ | $-5\pm 1.2$ |

15 min | $-50.6\pm 3.8$ | $-291.0\pm 7.2$ | $-21\pm 2.5$ |

30 min | $-91\pm 4.5$ | $-580.5\pm 8.7$ | $-37\pm 3.1$ |

**Table 7.**Depth of the interest points [18].

Time | Depth (μm) | ||
---|---|---|---|

Center | Peak | Border | |

5 min | $-22.5$ | $-116.2$ | $-14.3$ |

15 min | $-46.6$ | $-280.3$ | $-16.8$ |

30 min | $-101.3$ | $-541.5$ | $-27.1$ |

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**MDPI and ACS Style**

Narváez, M.; Cruzatty, C.; Valencia, E.; Hidalgo, V.; Luo, X.; Torres, A.; Erazo, J.; Altamirano, G.; Cando, E.
Modeling of Solid Particle Erosion for a Water–Sand Impingement System Using OpenFOAM. *Coatings* **2023**, *13*, 2080.
https://doi.org/10.3390/coatings13122080

**AMA Style**

Narváez M, Cruzatty C, Valencia E, Hidalgo V, Luo X, Torres A, Erazo J, Altamirano G, Cando E.
Modeling of Solid Particle Erosion for a Water–Sand Impingement System Using OpenFOAM. *Coatings*. 2023; 13(12):2080.
https://doi.org/10.3390/coatings13122080

**Chicago/Turabian Style**

Narváez, Mateo, Cristian Cruzatty, Esteban Valencia, Víctor Hidalgo, Xianwu Luo, Alejandra Torres, José Erazo, Gonzalo Altamirano, and Edgar Cando.
2023. "Modeling of Solid Particle Erosion for a Water–Sand Impingement System Using OpenFOAM" *Coatings* 13, no. 12: 2080.
https://doi.org/10.3390/coatings13122080