## 3 Results and Discussion
### Mechanism Model of Droplet  
As described above, the droplets in the branch outlets are dragged by the vapor. Meanwhile there exist vortexes in the header and the branch outlets, resulting in the mainstream area being smaller than the actual cross-section area, which substantially increases the local mass flux. According to the droplet model proposed by [PERSON] [21] that is listed in Appendix A, the droplet diameter (\(d\)) is mainly affected by the mass flux that is relevant to the vortexes. Hence, the vortex has a great influence on the droplet entrainments. However, the vortexes are sometimes neglected when analyzing droplet [13]. In order to reveal the effect of the vortex on droplet characteristics, the force on the droplets is analyzed. Moveover, a critical vapor velocity for \"droplet entrainment\" (\(U_{\text{V,C}}\)) is also proposed. The following assumptions are made for the droplets: (1) they are spherical; (2) they have the same diameter in the same branch outlet. Taking \(B_{1}\) as an example, Figure 9 demonstrates \(d\) in the header and the vapor velocity in \(B_{1}\) with and without vortex. It is observed that an increase in \(\dot{m}_{\text{in}}\) leads to an enlarged droplet diameter and decreased vapor velocity simultaneously. In addition, the existence of the vortex decreases the droplet diameter by up to 1.76 times and increases the vapor velocity in the branch outlet by 35.10%, which significantly affects the forces of the droplet. The force balance of the droplet is further established and analyzed. The droplet in the header is generally subjected to four forces (Figure 10), they are the gravitation force (\(F_{\text{G}}\)), the drag force (\(F_{\text{D2}}\)) and the buoyancy force (\(F_{\text{B}}\)) in the vertical direction as well as the drag force (\(F_{\text{D1}}\)) in the axial direction. The details of these four forces are described in Appendix B. In these four forces, \(F_{\text{G}}\) and \(F_{\text{B}}\) are only dominated by \(d\). Moreover \(d\), the  
Figure 8: (**a**) Instructions and the vapor velocity distribution in (**b**) header and (**c**) branch outlet. drag force (\(F_{\rm D1}\) and \(F_{\rm D2}\)) is also depended on the velocity. The higher mass flux represents the smaller droplet diameter. The drag force under different \(\dot{m}_{\rm in}\) is a result of the vapor velocity competing with \(d\). As shown in Figure 11, the sum of forces in the vertical direction shows fluctuation with the increasing of \(\dot{m}_{\rm in}\) and the minimum value is \(2.04\times 10^{-4}\) N at \(\dot{m}_{\rm in}=12\) g/s, while the force in the axial increases monotonously. \(F_{\rm D1}\) and \(F_{\rm D2}\) increases with the increasing of \(\dot{m}_{\rm in}\), which indicates the increasing rate of the velocity is larger than the decreasing rate of \(d\). Meanwhile, \(F_{\rm G}\) and \(F_{\rm B}\) both decrease due to the droplet being smaller when increasing of \(\dot{m}_{\rm in}\). \(F_{\rm G}\) at \(\dot{m}_{\rm in}=21\) g/s is only \(1.02\%\) to that at \(\dot{m}_{\rm in}=12\) g/s. Since \(F_{\rm G}\) and \(F_{\rm B}\) are only related to the density difference, \(F_{\rm G}\) is constantly \(20.14\) times higher than \(F_{\rm B}\), which represents that in general, the force in the vertical upward is relatively small compared to the downward. Interestingly, the dominant force in the vertical direction switches from \(F_{\rm G}\) to \(F_{\rm D2}\) when \(\dot{m}_{\rm in}\) is higher than \(12\) g/s. When the droplets flow into the intersection of the header and the branch outlet, the axial distance for droplets to be entrained to the branch outlet is \(S_{\rm axial}\) and the vertical distance is the diameter of the branch outlet (\(d_{\rm B}\)), as shown in Figure 10. If the flowing time of droplet in the vertical direction \(t_{\rm vertical}\) is longer than that in the axial direction \(t_{\rm axial}\), the droplet will be entrained to the branch outlet. Therefore, the critical point for the droplet entrainment is defined at \(t_{\rm vertical}=t_{\rm axial}\).