## 2 Waterjet Erosion Model of an Abrasive Particle
### Acceleration of an Abrasive Particle
The drag force (\(F_{2}\)) is a function of the drag coefficient (\(C_{d}\)), as the streamlined motion of spheres depends on the Reynolds number and the projection area (\(A_{d}\)) on which the force accelerating the particle is exerted, which is expressed as follows:  
\[F_{2}=\frac{1}{2}C_{d}\cdot\rho_{f}\cdot A_{d}\cdot\left(v_{f,t}-v_{p,t}\right) ^{2}. \tag{8}\]  
Under a constant force, the sum of the forces acting upon a particle is zero (\(F_{1}+F_{2}=0\)), i.e.,  
\[\frac{\pi\cdot d_{p}^{3}}{6}\cdot\frac{dv_{p,t}}{dt}\cdot\left(\rho_{p}-\rho_ {f}\right)+C_{d}\rho_{f}\cdot\frac{\pi\cdot d_{p}^{2}}{8}\cdot\left(v_{f,t}- v_{p,t}\right)^{2}=0, \tag{9}\]  
where \(d_{p}\) is the diameter of the abrasive particle. Rearranging the equation yields  
\[-\frac{dv_{p,t}}{\left(v_{f,t}-v_{p,t}\right)^{2}}=\frac{3}{4}\frac{\rho_{f}} {d_{p}\cdot\left(\rho_{p}-\rho_{f}\right)}\cdot C_{d}\cdot dt=\frac{\alpha}{d _{p}}\cdot dt, \tag{10}\]  
where \(\alpha\) is defined by the particle, fluid density, and drag coefficient and can be expressed as  
\[\alpha=\frac{3}{4}\cdot\frac{\rho_{f}}{\rho_{p}-\rho_{f}}\cdot C_{d}. \tag{11}\]  
Integrating Equations (10) and (11) over velocity and time, respectively, we can write  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]0  
This integration yields  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]1  
where \(C\) is the integral constant. As the initial velocity of the particle is zero, the integral constant (\(C\)) is obtained as follows:  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]2  
Upon substituting \(C\) from Equation (14) into Equation (13), the particle velocity at the acceleration time (\(t\)) becomes  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]3  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]4 Conceptual diagram showing particle motion through a fluid in a focus and the exit velocity after the particle exits the focus. Equation (15) can be expressed using the function of the fluid (\(v_{f,t}\)) and the particle velocity (\(v_{p,t}\)) at the acceleration time (\(t\)) and with the initial fluid velocity (\(v_{f,0}\)), as follows:  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]5  
By substituting the fluid velocity at time (\(t\)) from Equation (16) into Equation (15), the particle velocity at one moment (\(v_{p,t}\)) is given by  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]6  
Upon substituting the initial fluid velocity (\(v_{f,0}\)) from Equation (4) into Equation (17), the particle velocity at acceleration time (\(t\)) finally becomes  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]7  
By integrating the particle velocity (\(v_{p,t}\)) over time (\(t\)), the distance traveled by particle (\(l_{p,t}\)) during acceleration is  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]8  
where C\({}^{\prime}\) is the integral constant. As the traveled distance at \(t=0\) is zero, the integral constant (\(C^{\prime}\)) is obtained as follows:  
\[\frac{p_{f,p}}{\rho_{f}}=\frac{v_{f,0}^{2}}{2}. \tag{3}\]9  
The distance traveled by the abrasive particle (\(l_{p,t}\)) is given by  
\[v_{f,0}=\sqrt{\left(1-K\right)\frac{2p_{f,p}}{\rho_{f}}}, \tag{4}\]0  
Figure 3 shows the velocity of the abrasive particle and fluid and the distance traveled by the particle under the following conditions: a pump pressure of 320 MPa, a water flow rate of 29.71 g/s, and an abrasive flow rate of 7.5 g/s, with a particle diameter of 0.18 mm and a density of 0.00379 g/mm\({}^{3}\). These parameters were obtained using the velocity of the fluid and the abrasive particle (Equations (16) and (18)) and the model describing the distance traveled by the abrasive (Equation (21)), which was developed in this study. As the momentum of the fluid is transferred to the abrasive, the fluid velocity decreases, the abrasive velocity increases, and the distance traveled by the particle increases.