# Simple Approach for Evaluation of Abrasive Mixing Efficiency for Abrasive Waterjet Rock Cutting
## 4 Experimental Results and Analysis
### DBP and Cutting Characteristics  
One of the objectives of this study was to identify a comprehensive relationship among variables affecting the mixing and acceleration, such as the water flow rate, the AFR, and the focus diameter. The energy required to cut a unit depth according to the DBP is shown in Figure 11. These results indicate both mixing efficiency and cutting performance. The DBP and energy required to cut a unit depth are correlated with \(\alpha=11.595\pm 0.63\) and \(\beta=-0.738\pm 0.02\) in Equation (10). The coefficient of determination is 0.9606, which shows a high goodness of fit and can be expressed as follows:  
\[E_{a}/D=\varepsilon=11.595\text{-}\text{DBP}^{-0.738}. \tag{11}\]  
Figure 10: Abrasive waterjet kinetic energy per second as functions of the AFR and the WFR.  
Figure 9: DBP as functions of the (**a**) focus diameter and (**b**) WFR.  
The colored mark in Figure 11 indicates the optimum AFR (\(\dot{m}_{a,o}\)) based on the WFR. In general, the larger the WFR, the larger the amount of energy required to cut 1 mm, which means that the cutting efficiency is higher at a lower WFR. Based on this result, the DBP corresponding to the optimum AFR at a high WFR was estimated to be less than approximately 0.6 mm. Using the relationship between the energy required to cut one unit depth and the DBP, the optimum AFR can be determined according to the WFR and the waterjet system.  
By substituting the energy required to cut one unit depth from Equation (10) and the DBP from Equation (7) into Equation (11), the cut depth can be expressed as follows:  
\[D=\frac{\dot{m}_{a}v_{t}^{2}}{2\cdot 11.595}\Bigg{(}\sqrt[3]{\frac{3}{2}d_{f}^{2}l_ {f}m_{p}/\dot{m}_{a}t_{f}}-d_{p}\Bigg{)}^{0.738}. \tag{12}\]  
This approach represents the characteristics of mixing variables such as the focus geometry, the abrasive particle diameter, the mass of a single abrasive particle, the AFR, and the water flow rate based on the terminal velocity. The cutting depth can be estimated under a wide range of waterjet system conditions by measuring \(\alpha\) and \(\beta\) from three or more points representing the relationship between the energy required to cut one unit depth and the DBP.  
Figure 12 presents the results of a parametric study, depicting the cutting depth with a given water flow rate according to the focus diameter at an AFR of 10 g/s (according to Equation (12)). The mixing variables were evaluated over a wide range, and it was found that the larger the focus diameter, the greater the cut depth. The effect of the focus diameter is larger at higher water flow rates. It can also be demonstrated that a high WFR results in a large reduction in efficiency because of the focus diameter. Increasing the water flow rate from 50 mL/s to 90 mL/s (i.e., by 80%) increases the cutting depth by 20%. Although the maximum cutting depth occurs at 90 mL/s, 50 mL/s of flow rate is considered to be economical. Thus, it is possible to design an economical water flow rate using Equation (12).  
Figure 13 illustrates the results of a parametric study, showing the effect of the focus diameter and the AFR on the cutting depth at a WFR of 50 mL/s. For a focus diameter less than 0.9 mm, the optimum AFR is close to 5 g/s (dashed dotted line), but for a larger diameter focus, the optimum AFR is close to 10 g/s (dashed line). This approach demonstrates the change in the optimum AFR according to the focus diameter. An increase in AFR increases the impact frequency (up arrow) while reducing the terminal velocity and cutting energy (down arrow), as indicated by Equation (3). In addition, a high AFR yields numerous collisions between abrasive particles, reducing the cutting efficiency. This  
Figure 11: Energy required to cut a unit depth versus DBP.  
parametric study demonstrates that the DBP in the focus enables optimal design of the focus diameter, the AFR, and the WFR.