# Simple Approach for Evaluation of Abrasive Mixing Efficiency for Abrasive Waterjet Rock Cutting
## 1 Introduction
Table 1 summarizes the literature on the effects of the focus geometry, the AFR, and the WFR on an abrasive waterjet. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Findings** & **Reference** \\ \hline Focus geometry & The focus geometry affects the momentum transfer efficiency. & [16] \\ Focus geometry–abrasive flow rate (AFR) & The optimal mixing efficiency is determined in the space in which mixing occurs. & [7] \\ Focus geometry–AFR & The optimum AFR is a function of pressure, and the focus diameter affects the power density. & [22] \\ Flow rate–focus geometry & The economically optimum focus diameter is three times the orifice diameter. & [28] \\ Focus–orifice geometry & The focus diameter affects the surface roughness and determines 98\% of the cutting width. & [25] \\ Focus geometry & The surface profile amplitude and surface roughness mainly depend on the focus diameter. & [11] \\ Focus geometry & The focus diameter is the dominant parameter affecting the cutting width and the surface roughness. & [18] \\ Focus geometry–AFR & A larger ratio between the diameters of the focus and abrasive particles reduces the acceleration energy. & [27] \\ Focus and orifice geometry–AFR & The AFR affects the mixing efficiency more strongly than the focus and the water flow rate (WFR). & [5] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Previous studies of the effects of the mixing variables on an abrasive waterjet. The mixing efficiency of an abrasive is determined by the abrasive-abrasive and abrasive-focus inner surface collisions. However, studies that comprehensively demonstrate the characteristics of the mixing efficiency considering the WFR, the AFR, and the focus diameter simultaneously are scarce. Under ideal mixing and acceleration, the abrasive is completely surrounded by water and accelerated without interference. Figure 1 provides a conceptual illustration of abrasive acceleration, showing the characteristics of the mixing variables. The AFR and the focus diameter affect the chance of collision as a function of the abrasive density in the focus. The WFR and the AFR determine the mixing time for which the abrasive remains in the focus because it determines the abrasive velocity. The longer the abrasive remains in the focus, the higher the collision frequency. Thus, the effects of the mixing variables on the geometric behavior of the abrasive were expressed. The distance between particles (DBP) in the focus can represent the mixing efficiency as a function of the collision chance. Figure 2 shows the characteristics of the DBP and mixing variables. As the AFR increases, the DBP decreases (e.g., DBP1-DBP2). As the focus diameter or WFR increases, the DBP increases (e.g., DBP2-DBP3 and DBP3-DBP4). A high DBP means that the abrasive particles are far from each other, so the chance of collision is low and the mixing efficiency is high. As such, the intuitive characteristics of the abrasive mixing efficiency can be expressed in terms of the DBP. In addition, because the concept of space occupied by a single particle was introduced to define the space in which the abrasives independently remain, not only abrasive-abrasive collisions but also abrasive-focus collisions can be considered. In this study, the mixing variables, developed using a theoretical model of the DBP, were experimentally evaluated, and the mixing efficiency was estimated based on the relationship between the rock cutting rate and the DBP. Through a parametric study of the developed model, the effects of the mixing variables were reviewed across wide ranges of the variables. The developed model using the DBP can intuitively represent the effects of the mixing variables. Using this model, the cutting performance can be predicted through  
Figure 1: Conceptual illustration of the distance between particles (DBP) in the focus. Figure 2: DBP according to AFR, WFR, and focus diameter. the relationship between the input energy and the mixing efficiency, and the optimum AFR can be estimated.