# Optimizing the Auxiliary Air Channels of a Vortex Atomizer by 3D Printing Using the Taguchi Method
## 4 Results and Discussion
### CFD Simulation Validation  
For CFD simulations of the designed atomizers, a commercial software Fluent was used. As the flow involves two phases of water and air, the Volume of Fluid (VoF) model was adopted [27] through the discrete phase model (DPM) [28] setup. For the VoF model, [PERSON] et al. (2023) [29] have demonstrated its accuracy in two-phase flow simulations, including three-dimensional turbulent liquid jets. The VoF model has been shown to to be suitable for two-phase. In this study, the two-phase VoF, air, and water were the primary and secondary phases, respectively. In the DFM setup, the discrete phase was initiated with a particle number of 50,000.  
For the simulation of the present turbulent jets, the k-\(\omega\) shear stress turbulent model was applied. The inlet and outlet pressures of the water jet were set to 3.236 \(\times\) 10\({}^{6}\) Pa gauge and 0 Pa gauge (i.e., 1.033 \(\times\) 10\({}^{5}\) Pa), respectively. In contrast, the inlet and outlet pressures of the air were set to 0 Pa gauge; i.e., the air was entrained by the water jets. The SIMPLEC algorithm with the second order upwind discretization for the momentum equation and the first order upwind discretization for the turbulent kinetic energy was used in the simulation. All simulations were run on a desk computer of 12 cores and a clock frequency of about 3 GHz by a tetrahedral element with 7,112,845 elements. Typically, it took about 60 h to run a case.  
For comparion, the simulated spray patterns of the atomizers of Case 1 and the optimum case are shown in Figure 12, together with that of the experiment of the optimum atomizer. Figure 12a,b depicts the overall side views of the spray patterns of Case 1 and Case optimum atomizers, respectively. Qualitatively, the simulated patterns resemble those of the corresponding experiments. The jet spread opening angles are further examined by zooming in the regions close the jet exits marked by dashed squares in Figure 12a,b. These two closeups are shown correspondingly in Figure 12c,d. Quantitatively, the associated opening angles are about 60\({}^{\circ}\) and 45\({}^{\circ}\) for the atomizers of Case1 and Case optimum, respectively. That is, the optimum atomizer results in a smaller jet opening angle, thus leading to a more uniform spraying pattern as described in the previous experimental results. Moreover, for the optimum atomizer deduced from the Taguchi method, the simulated jet opening angle is about 42\({}^{\circ}\), close to that of the experiment as shown in Figure 12d,e. Namely, the optimum atomizer indeed performs the best among all the fabricated atomizers as being validated by both simulation and experiments.  
Figure 11: Droplet weight distributions with the optimum condition.  
Figure 12: Comparison of spray patterns of CFD simulations and experiments of two atomizers. (**a**) Spray distribution of Case1 atomizer (simulation) (**b**) Spray distribution of optimum atomizer (simulation). (**c**) Spray opening angle 60 \(\pm\) 1\({}^{\circ}\) of Case1 atomizer (simulation). (**d**) Spray opening angle 45 \(\pm\) 1\({}^{\circ}\) of optimum atomizer (simulation). (**e**) Spray opening angle 42\({}^{\circ}\) of optimum atomizer (experiment).  
In comparison, the results of [PERSON] et al. (2019) [PERSON] et al. (2019) [PERSON] et al. (2019) give the opening angles of about 32\({}^{\circ}\) (experiment) and 26\({}^{\circ}\) (simulation by Euler-Lagrange method), respectively, and are smaller than the present corresponding findings of 45\({}^{\circ}\) (simulation) and 42\({}^{\circ}\) (experiment). This is expected because the gravity in their case is parallel to the jet axis; hence, the opening angle will be reduced due to down pulling of the gravity. In contrast, in this study, the gravity is perpendicular to the jet axis; namely, the gravity will assist jet spreading.