# Flow Field Analysis Inside and at the Outlet of the Abrasive Head

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background of AWJ Calculations

#### 1.2. Background of Abrasive Particle Velocity Measurement

#### 1.3. Background Summary

## 2. Methods

#### 2.1. Water Nozzle Geometrical Reconstruction Based on X-ray CT

#### 2.2. CFD Geometry of Computational Mesh, Model Settings, and Calculations

_{m}is the mixture density, x

_{i}is a position vector component in the ith direction, x

_{j}is a position vector component in the jth direction, u

_{mi}is a mixture velocity component in the ith direction, u

_{mj}is a mixture velocity component in the jth direction, p is the mixture pressure, g

_{i}is the gravity acceleration component in the ith direction, F

_{i}is the volume force component in the ith direction, μ

_{m}is the molecular viscosity of the mixture, μ

_{mt}is the turbulent viscosity of the mixture, k

_{m}is the turbulent kinetic energy of the mixture, and δ

_{ij}is the Kronecker delta. The mass equilibrium equation in continuum can be applied. The general form of the equation can be written as:

_{k}of the secondary component of the mixture is calculated from the following equation:

_{k}is the volume fraction of the k-component of the mixture, ρ

_{k}is the density of the k-component of the mixture, μ

_{k}is the viscosity of the k-component of the mixture, m

_{pk}is the mass transfer from the phase p to k, and m

_{kp}is the mass transfer from the phase k to p. The two-equation RANS shear stress transport (SST) k-ω model [31] was used for the turbulent flow of water and air description. The general form of these equations can be written as:

_{m}is a specific ratio of the dissipation and turbulent energy, Γ

_{k}represents the effective diffusivity of k

_{m}, Γ

_{ω}represents the effective diffusivity of ω

_{m}, G

_{k}represents production of k

_{m}, G

_{ω}represents the generation of ω

_{m}, Y

_{k}represents the dissipation of k

_{m}due to the fact of turbulence, Y

_{ω}represents the dissipation of ω

_{m}due to the fact of turbulence, D

_{ω}is the cross-diffusion term, S

_{k}is the user-defined source term of k

_{m}, and S

_{ω}is the user-defined source term of ω

_{m}. The above-mentioned equations solved the fluid flow of water and air as continuum using the Euler approach. The flow of abrasive particles in a continuous environment of water and air was solved based on the Lagrange approach, using the DPM. The abrasive particle trajectories were computed individually at specified intervals during water and air calculations. This approach is suitable when particle-particle interactions are neglected. This requires that the solid particle phase occupies a low volume fraction, even though high mass loading is acceptable. At the same time, the particles should be sufficiently small with respect to the solved domain size [31]. The abrasive particle movement was solved by the following equation:

_{pi}is the component of particle velocity in the ith direction, u

_{pj}is the component of particle velocity in the jth direction, F

_{D}is the drag force, ρ

_{p}is the particle density, F

_{pi}is the component of the additional force to the particle in the ith direction, C

_{D}is the drag coefficient, Re is the Reynolds number, and d

_{p}is the particle diameter. A two-way coupling method was used in the calculation of the particle interaction with the continuous phase [30]. The movement of abrasive particles affected the flow field shape of the continuous mixture given by water and air. Together with initial and boundary conditions (Figure 3), the above-mentioned equations formed a closed system that allowed for the calculation of an unambiguous solution of a turbulent flow field for the steady-state multiphase mixture in the solved domain. In particular, the effect of water compressibility was included in the calculation [32]. It was important to calculate the right shape of the velocity profiles at the outlet of the water nozzle, as the velocity profile results can differ significantly in incompressible and compressible fluid cases at pressures over 100 MPa. The water density was calculated using Equation (9):

_{ref}is the reference density, and p

_{ref}is the reference pressure. DPM was selected for the description of solid particle movement in air and water. The mutual interaction between abrasive particles and continuous phases was set in the model in order to obtain a better accuracy of the calculated velocity distribution. It has been shown that the particle movement influences the velocity distribution, pressure, and density of the continuous phases. In the computational model, various diameters of abrasive particles were described using a special function called the Rosin-Rammler distribution function (10) [33]:

_{d}is the volume fraction of particles with a diameter greater than d, d is the particle diameter, $\overline{d}$ is the average diameter of particles, and n is the spreading exponent. Constant values for the average particle diameter, $\overline{d}$ = 281 μm, and the spreading exponent, n = 5.9, were calculated for the 150–400 μm range of abrasive particle diameters. Figure 4 shows a comparison of the calculated and measured grain size curves. Apparently, the curves are almost similar. According to the measured grain size curves, the distribution of abrasives used in the calculation simulates the near-real situation. Measurement of the abrasive particle distribution was performed using a Fritsch Nanosizer Analysette NanoTec 22 (FRITSCH GmbH, Oberstein, Germany).

#### 2.3. High-Pressure Water Flow Monitoring in the Abrasive Head

#### 2.4. Measurement of Abrasive Particle Velocities at the Outlet of the Abrasive Head

## 3. Results and Discussions

#### 3.1. Water Nozzle Space

#### 3.2. Mixing Chamber and Abrasive Nozzle Space

#### 3.3. Outlet Space of Abrasive Head

## 4. Conclusions

- In the water nozzle, the calculated values of the volume flow rates were in very good conformity with the measured values under the determined feeding water pressures due to the exact geometry of the abrasive head that was included in the numerical model without simplifications. The exact geometry of the studied domain is thus a crucial part for creating and developing a numerical model which generates reasonable data;
- In the abrasive nozzle, a very good conformity of the pressure distribution in measurements and in calculations was observed. The shapes of dependences were similar with those published in other studies [38]. In the cylindrical part of the abrasive nozzle, the pressure decreased at first, then hit the minimum and, finally, grew up to the value of the atmospheric pressure. For both measurements and calculations, the increase in the high-speed water jet velocity caused a decrease in the minimum pressure;
- At the outlet of the abrasive head, a very good conformity of the abrasive particle velocity in measurements and calculations was reached due to the detailed description of abrasive particle diameters (from 150 to 400 μm) based on the function of the Rosin-Rammler particle size distribution used in calculations. For both measurements and calculations, the increase in the abrasive mass flow rate caused a decrease in the velocity of abrasive particles at the outlet of the abrasive head;
- A numerical model enabling a 3D multiphase steady-state turbulent visualisation of the flow in the space around the abrasive cutting head was developed. The numerical model provided stable and sufficiently accurate simulations in a short time without requiring extreme computing power.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

- All authors have participated in the conception and design or analysis and interpretation of the data;
- This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue;
- The authors have no affiliation with any organisation with a direct financial interest in the subject matter discussed in the manuscript.

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**Figure 2.**X-ray CT scheme: (

**a**) projection of the water nozzle using the CT method; (

**b**) 2D image of the water nozzle.

**Figure 4.**Comparison of the abrasive grain size curves obtained by measurement and the grain size curve used in the numerical model.

**Figure 7.**Image processing procedure: (

**a**) experimental image; (

**b**) background noise reduction; (

**c**) averaging of pixel intensities; (

**d**) tracking algorithm for abrasive particle velocity detection.

**Figure 9.**A graphical example of the calculated flow field shape around the water nozzle under the feeding pressure of 406 MPa: (

**a**) distribution of the absolute pressure (MPa); (

**b**) distribution of the density (kg/m

^{3}); (

**c**) distribution of the velocity vector (m/s); (

**d**) abrasive particle traces coloured by particle velocity magnitude (m/s)—abrasive particle mass flow rate of 100 g/min.

**Figure 10.**Calculated flow field in the mixing chamber and in the conical inlet part of the abrasive nozzle under the feeding pressure of 406 MPa: (

**a**) distribution of the absolute pressure (kPa); (

**b**) distribution of the density (kg/m

^{3}); (

**c**) distribution of the velocity vector (m/s); (

**d**) abrasive particle traces coloured by particle velocity magnitude (m/s)—abrasive particle mass flow rate of 100 g/min.

**Figure 11.**Calculated flow field in the abrasive nozzle under the feeding pressure of 406 MPa: (

**a**) distribution of the pressure (kPa (abs)); (

**b**) distribution of the density (kg/m

^{3}); (

**c**) distribution of the velocity vector (m/s); (

**d**) abrasive particle traces coloured by particle velocity magnitude (m/s)—abrasive particle mass flow rate of 100 g/min.

**Figure 12.**Calculated velocity and pressure distribution along the axis of the abrasive head, mixing chamber, focusing tube, and outlet: (

**a**) abrasive mass flow rate of 100 g/min; (

**b**) abrasive mass flow rate of 200 g/min; (

**c**) abrasive mass flow rate of 300 g/min; (

**d**) abrasive mass flow rate of 400 g/min.

**Figure 13.**Abrasive particle velocities at the abrasive nozzle, measurement vs. computational models for given water pressure levels, and an abrasive particle mass flow rate of 100 g/min: (

**a**) 105 MPa; (

**b**) 194 MPa; (

**c**) 302 MPa; (

**d**) 406 MPa.

**Figure 14.**Abrasive particle velocities at the abrasive nozzle outlet, measurement vs. computational models for determined water pressure levels, and abrasive particle mass flow rate of 200 g/min: (

**a**) 105 MPa; (

**b**) 194 MPa; (

**c**) 302 MPa; (

**d**) 406 MPa.

**Figure 15.**Abrasive particle velocities at the abrasive nozzle outlet, measurement vs. computational models for determined water pressure levels, and an abrasive particle mass flow rate of 300 g/min: (

**a**) 105 MPa; (

**b**) 194 MPa; (

**c**) 302 MPa; (

**d**) 406 MPa.

**Figure 16.**Abrasive particle velocities at the abrasive nozzle outlet, measurement vs. computational models for determined water pressure levels, and an abrasive particle mass flow rate of 400 g/min: (

**a**) 105 MPa; (

**b**) 194 MPa; (

**c**) 302 MPa; (

**d**) 406 MPa.

**Table 1.**Results of the measured feeding water pressures, volume flow rates, and power distribution in the water nozzle.

p_{avg} (MPa) | SD (MPa) | SEM (MPa) | 95% CI (MPa) | Q_{avg} (L/min) | SD (L/min) | SEM (L/min) | 95% CI (L/min) | P_{avg} (kW) | SD (kW) | SEM (kW) | 95% CI (kW) |
---|---|---|---|---|---|---|---|---|---|---|---|

105.56 | 1.68 | 0.22 | 105.98 105.14 | 1.58 | 0.01 | 0.001 | 1.56 1.57 | 2.77 | 0.05 | 0.006 | 2.79 2.76 |

194.46 | 15.27 | 1.95 | 198.29 190.64 | 2.08 | 0.08 | 0.010 | 2.10 2.06 | 6.75 | 0.75 | 0.095 | 6.94 6.56 |

302.72 | 2.13 | 0.27 | 303.26 302.19 | 2.53 | 0.02 | 0.001 | 2.53 2.52 | 12.76 | 0.16 | 0.019 | 12.80 12.72 |

406.17 | 3.22 | 0.41 | 406.98 405.36 | 2.86 | 0.02 | 0.002 | 2.75 2.74 | 19.36 | 0.17 | 0.002 | 19.41 19.32 |

_{avg}: average value of the water pressure; Q

_{avg}: average value of the volume flow rate; P

_{avg}: average value of the power; SD: standard deviation; SEM: standard error of the mean; 95% CI: 95% confidence interval of values: upper limit and lower limit.

**Table 2.**Comparison of the average values of abrasive particle velocities from CFD calculations and PTV measurement for an abrasive mass flow rate of 100 g/min.

CFD Calculations | PTV Measurement | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

p (MPa) | N | v_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95%CI (m/s) | N | v_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95%CI (m/s) |

105.56 | 3059 | 296.07 | 113.79 | 2.05 | 300.10 292.04 | 1565 | 329.34 | 59.45 | 1.50 | 332.28 326.39 |

194.46 | 3060 | 400.09 | 159.71 | 2.89 | 405.75 394.42 | 2650 | 455.36 | 69.62 | 1.35 | 458.01 452.70 |

302.72 | 3054 | 493.39 | 205.39 | 3.72 | 500.76 486.19 | 1489 | 579.71 | 75.85 | 1.97 | 583.57 575.86 |

406.17 | 3056 | 566.06 | 242.27 | 4.38 | 574.65 557.47 | 2421 | 690.38 | 70.55 | 1.43 | 693.20 687.57 |

_{avg}: average velocity of abrasive particles; SD: standard deviation; SEM: standard error of the mean; 95%CI: 95% confidence interval of values: upper limit and lower limit.

**Table 3.**Comparison of the average values of abrasive particle velocities from CFD calculations and PTV measurement for an abrasive mass flow rate of 200 g/min.

CFD Calculations | PTV Measurement | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

p (MPa) | N | v_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95% CI m/s] | N | v_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95% CI (m/s) |

105.56 | 3060 | 285.01 | 107.33 | 1.94 | 288.81 281.21 | 3329 | 281.19 | 66.51 | 1.15 | 283.45 278.94 |

194.46 | 3058 | 388.61 | 153.81 | 2.78 | 394.07 383.16 | 3742 | 395.23 | 85.72 | 1.40 | 397.48 392.48 |

302.72 | 3059 | 488.95 | 193.68 | 3.50 | 495.81 565.62 | 3318 | 528.87 | 76.43 | 1.33 | 531.48 526.28 |

406.17 | 3059 | 557.36 | 233.34 | 4.22 | 565.63 549.07 | 4830 | 641.75 | 76.65 | 1.10 | 643.91 639.58 |

_{avg}: average velocity of abrasive particles; SD: standard deviation; SEM: standard error of the mean; 95% CI: 95% confidence interval of values: upper limit and lower limit.

**Table 4.**Comparison of the average values of abrasive particle velocities from CFD calculations and PTV measurement for an abrasive mass flow rate of 300 g/min.

CFD Calculations | PTV Measurement | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

p (MPa) | N | v_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95% CI (m/s) | N | v_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95% CI (m/s) |

105.56 | 3055 | 276.86 | 99.28 | 1.80 | 280.38 273.34 | 6483 | 251.81 | 63.55 | 0.79 | 253.36 250.27 |

194.46 | 3057 | 379.18 | 146.07 | 2.64 | 384.36 374.00 | 3350 | 365.40 | 82.14 | 1.42 | 368.18 362.62 |

302.72 | 3059 | 475.18 | 188.74 | 3.41 | 482.27 468.89 | 2218 | 479.52 | 86.59 | 1.84 | 483.13 475.92 |

406.17 | 3055 | 553.22 | 200.29 | 3.99 | 561.03 545.41 | 6372 | 601.70 | 80.28 | 1.01 | 603.68 599.73 |

_{avg}: average velocity of abrasive particles; SD: standard deviation; SEM: standard error of the mean; 95% CI: 95% confidence interval of values: upper limit and lower limit.

**Table 5.**Comparison of the average values of abrasive particle velocities from CFD calculations and PTV measurement—abrasive mass flow rate of 400 g/min.

CFD Calculations | PTV Measurement | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

p (MPa) | N | V_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95% CI(m/s) | N | V_{avg} (m/s) | SD (m/s) | SEM (m/s) | 95% CI (m/s) |

105.56 | 3055 | 269.31 | 92.54 | 1.67 | 272.59 266.02 | 5205 | 237.42 | 57.46 | 0.80 | 238.98 235.86 |

194.46 | 3059 | 375.86 | 133.85 | 2.42 | 380.60 371.12 | 4655 | 333.54 | 83.26 | 1.22 | 335.94 331.15 |

302.72 | 3056 | 473.57 | 174.53 | 3.15 | 479.75 467.38 | 2777 | 433.15 | 103.73 | 1.97 | 437.01 429.29 |

406.17 | 3055 | 549.16 | 208.90 | 3.78 | 556.57 541.76 | 6920 | 562.12 | 82.40 | 0.99 | 564.06 560.18 |

_{avg}: average velocity of abrasive particles, SD: standard deviation; SEM: standard error of the mean; 95% CI: 95% confidence interval of values: upper limit and lower limit.

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**MDPI and ACS Style**

Riha, Z.; Zelenak, M.; Soucek, K.; Hlavacek, A.
Flow Field Analysis Inside and at the Outlet of the Abrasive Head. *Materials* **2021**, *14*, 3919.
https://doi.org/10.3390/ma14143919

**AMA Style**

Riha Z, Zelenak M, Soucek K, Hlavacek A.
Flow Field Analysis Inside and at the Outlet of the Abrasive Head. *Materials*. 2021; 14(14):3919.
https://doi.org/10.3390/ma14143919

**Chicago/Turabian Style**

Riha, Zdenek, Michal Zelenak, Kamil Soucek, and Antonin Hlavacek.
2021. "Flow Field Analysis Inside and at the Outlet of the Abrasive Head" *Materials* 14, no. 14: 3919.
https://doi.org/10.3390/ma14143919