## 2 Materials and Methods
### Numerical Models, Solution Methods and Boundary Conditions
#### 2.2.5 Validation of Numerical Model  
Firstly, a numerical simulation is conducted on the influence of treatment volume on the cavitation effect of the hydrodynamic cavitation generator. The impact pattern of handling capacity on the aeration effect of the aerator is studied using numerical simulation. By changing the handling capacity, the axial velocity of the fluid inside the aerator is essentially altered. With the rotational speed held constant, handling capacities of 0.5 m\({}^{3}\)/h, 1.0 m\({}^{3}\)/h, 1.5 m\({}^{3}\)/h, and 2.0 m\({}^{3}\)/h are selected in the simulation process. With the inlet cross-sectional diameter d = 30 mm, the inlet cross-sectional area is calculated to be 0.0007 m\({}^{2}\), resulting in inlet velocities of 0.2 m/s, 0.4 m/s, 0.6 m/s, and 0.8 m/s, respectively. As shown in Figure 4, at a speed of 1500 rpm, the volume fraction of cavitation bubbles decreases with the increase in treatment volume. It can be observed from Figure 5 that increasing the treatment volume leads to an increase in the inlet velocity, resulting in an increase in the axial velocity in the cavitation chamber and the radial velocity between teeth. This decrease in the time for the fluid to stay in each shear layer per revolution causes some fluid to reach the outlet without cavitation occurring in the cavitation chamber, thereby affecting the formation of cavitation bubbles. The rotational speed of the moving teeth varies while the initial parameters remain unchanged. During the numerical simulation, it was found that the local pressure in the hydrodynamic cavitation generator cannot be reduced below the saturated vapor pressure of the liquid at speeds below 1200 rpm. Therefore, the hydrodynamic cavitation generator hardly produces cavitation bubbles. When the rotational speed exceeds 2000 rpm, the increase in the volume fraction of cavitation bubbles is not obvious. Therefore, the rotational speeds of 1200 rpm, 1400 rpm, 1600 rpm, and 1800 rpm were chosen to compare the effect of different rotational speeds on the volume fraction of cavitation bubbles. The results are shown in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline
**Rotation** & **Hydrodynamic Cavitation** & **Single Row of** & **Pitch** & **Number of Teeth in a Row** & **Number of** \\
**Speed** & **Generator Diameter** & **Teeth Height** & **in a Row** & **Rows** \\ \hline
1500 rpm & 200 mm & 10 mm & 2 mm & 27 & 2 \\ \hline \end{tabular}
\end{table} 
Table 2: Initial parameter settings. Figure 3: Gas-phase cloud diagram. As shown in Figure 6, the volume fraction of cavitation bubbles increases with the increase in rotational speed at a treatment capacity of 1 m\({}^{3}\)/h. For Figure 7, the cavitation bubbles in the hydrodynamic cavitation generator occur with the fluid under high-speed rotation. Therefore, the higher the rotational speed, the higher the centrifugal force generated by the rotated teeth due to rotation. This results in a higher fluid flow rate inside the hydrodynamic cavitation generator, which results in a higher volume fraction of cavitation bubbles. Thus, the cavitation effect of the hydrodynamic cavitation generator increases with increasing speed within a convinced range. Figure 4: Volume fraction of cavitation bubbles at different treatment rates (\(n=1500\) rpm). Figure 5: Cloud diagram of cavitation bubble distribution at different treatment volume (\(n=1500\) rpm). Secondly, experiments are conducted to study the influence of treatment volume on the COD degradation rate to verify the accuracy and applicability of the numerical simulation results. A shear-type hydraulic cavitation performance test platform was constructed in the laboratory, using the printing and dyeing wastewater as the experimental subject. The cavitation performance of the cavitation device was reflected by measuring the difference in COD concentration in the printing and dyeing wastewater before and after cavitation, with the COD content of the samples determined using EPA-approved standard methods [32].