## 4 Model for Calculating Water Jet Slotting Depth  
By comprehensively considering the influence of the jet pressure, translation speed, and cutting times on the total cutting depth, the cutting depth can be expressed as follows:  
\[h=kv_{T}{}^{\alpha_{1}}J^{\alpha_{2}}(P-P_{c}), \tag{13}\]  
where \(h\) is the total cutting depth and \(k\) is the proportionality coefficient. By combining Equations (9) and (10), the model for calculating the slotting depth caused by a high-pressure water jet in a coal seam is expressed as follows:  
\[h=k(\frac{n\pi r}{30})^{\alpha_{1}}(nt)^{\alpha_{2}}(p-p_{c}). \tag{14}\]  
As expressed in Equation (14), the rotation speed has a significant effect on the slotting depth under fixed slotting time. As the rotation speed increases, the slotting depth becomes smaller each time but increases with the slotting repetition. Therefore, there exists an optimal rotation speed for the water jet slotting. This study conducted rotary slotting experiments to determine the relevant parameters in Equation (14) and obtain the optimal rotation speed. As shown in Figure 9, the experimental system mainly was comprised of a high-pressure pump, hydraulic drilling rig, and the sample. The drilling rig, with a rated torque of 1250 N\(\cdot\)m, was used to control the rotation speed in the water jet slotting. The slotting sample was shaped coal with a ratio of coal particle:cement:water = 1:0.5:1, size of \(1\times 1\times 1\) m, \(f\) coefficient of 0.21, and tensile strength of approximately 0.08 MPa. The experimental procedures are as follows: first, the nozzle diameter and initial standoff distance were set to 3 and 12.5 mm, and the jet pressure was controlled at 10 MPa. Then, the rotation speeds were set to 30, 40, 50, and 60 r/min, respectively. Additionally, the shaped coal sample rotary slotted by a high-pressure water jet and the slotting time was 0.5 min every time. Subsequently, the slotting depth was measured and a slot sample was taken again under the same slotting conditions until the slotting depth did not change. Finally, the rotation speed was changed and the abovementioned process was repeated. The results are presented in Figure 10. At the same rotation speed, the slotting depth increased with the slotting time, but the growth rate gradually decreased, then tended to be stable, and finally reached a certain limit value. As the rotation speed increased, the slotting depth became greater at the initial period and the limit depth was reached faster. Hence, the slotting depth was the combined result of rotation speed and slotting frequency. Although the relative moving speed was higher at  
Figure 9: System used in experiment of rotary slotting by water jet. a higher rotation speed, the slotting frequency per time increased, which resulted in deeper slotting depth at the initial period. Moreover, in the process of rotary slotting by the high-pressure water jet, the action mode of the water jet on the coal body changed. Similar to the oscillation effect of the pulsed water jet, the coal body is influenced by the water hammer pressure with periodic frequency, and thus the rock-breaking efficiency improves. In contrast, at lower rotation speed, the impact time of the single slotting coal becomes longer, and the hindrance of the water cushion becomes more severe. Therefore, in the early stages of slotting, the influence of slotting repetitions on the slotting depth is greater than that of a single impact. Increasing the rotation speed contributes to the improvement of the slotting efficiency and slotting depth. However, as the rotation speed increases, the limit slotting depth first increases and then decreases. This indicates that the influence of a single impact on the slotting depth is greater than that of numerous slotting repetitions at the later stages of jet slotting. This is attributed to the fact that the coal body is mainly destroyed by the quasi-static pressure of the water jet. To verify the proposed slotting depth model, the experimental data were fitted based on Equation (14).