# Numerical Study of Internal Flow Field in a Disc Stack Centrifuge Based on Mixture-PBM Model
## 3 Results and Discussion
### Three-Dimensional Flow Characteristics of the Internal Flow Field  
Figure 13 shows the surface streamline diagram at 50% height inside the separation channel. Combined with the velocity triangle in Figure 8, the relative velocity of the liquid can be obtained, with the colors of the streamlines representing the values of the relative velocity of the liquid. It can be seen that after the oil-water mixture enters the separation channel from the neutral pore, there is an inertial effect in the rotating coordinate system with a rotational speed of \(N_{d}\), and it flows first to the ribs at the position of the pressure surface and then towards the small end of the disc. During flow to the small end of the disc, there is a tendency to flow towards the ribs at the suction surface due to the effect of the Coriolis force, and the liquid flowing to the large end of the disc tends to deflect towards the ribs at the pressure surface. Figure 11: Water droplet content at the inlet of the rising channel and the inlet of the bowl. The vortex phenomenon occurs near the neutral pore due to the interaction of the water and oil phases; it gradually increases with the number of layers in the separation channel, and the vortex structure becomes simpler. However, at the large end of the disc, near the ribs of the pressure surface, a relative stagnation zone with low flow velocities occurs. The stagnation zone is larger in the separation channel of the middle layer. The viscous fluid motion in the separation channel has two velocity components: the radial velocity along the direction of the generatrix and the relative circumferential velocity due to the Coriolis force. Based on these two features, [PERSON] [28] obtained a dimensionless  
Figure 12: Particle size distribution of water droplets at different cross-sectional positions: (**a**) 1-1 cross-sectional position; (**b**) 2-2 cross-sectional position; (**c**) 3-3 cross-sectional position. Figure 13: Surface streamline at 50% height in different separation channels. (**a**) Separation channel on the lower layer. (**b**) Separation channel on the middle layer. (**c**) Separation channel on the top layer. number \(\lambda\) by solving the Navier-Stokes equations, which is used to describe the flow law of the liquid in the separation channel. \[\lambda=h\sqrt{\frac{\omega\cdot sin\alpha}{\
u}} \tag{8}\]  
In the equation, \(h\) is the height of the separation channel, and the unit is m; \(\omega\) is the disc stack centrifuge rotation speed, and the unit is rad/s; \(\alpha\) is the angle between the generatrix of the disc and the rotation axis, and the unit is \({}^{\circ}\); and \(\
u\) is the kinematic viscosity of the liquid, and the unit is m\({}^{2}\)/s. Under the influence of various velocity components, the trajectory of the liquid flowing from the neutral pore into the separation channel is not parallel to the generatrix of the disc. Figure 14 shows the streamline diagram near the neutral pore at different \(\lambda\) values. It can be seen that as the \(\lambda\) value increases, the deviation between the liquid motion trajectory and the direction of the busbar increases. Figure 15 shows the circumferential velocity distribution in the separation channels of layers 1-6 within the range of \(-1^{\circ}\) to \(1^{\circ}\) in the circumferential direction. It can be seen that the values of the circumferential velocity are different in the circumferential direction, which is due to the presence of vortex flow in the separation channel, and the circumferential velocity at the middle of the separation channel is higher than the circumferential velocity at the near-wall surface. Figure 14: Trajectory diagram of the liquid at the neutral pore: (**a**) \(\lambda=2.9\); (**b**) \(\lambda=4\); (**c**) \(\lambda=5\). Figure 15: Circumferential velocity distribution in the separation channel. Figure 16 shows the radial velocity distribution in the separation channels of layers 1-6 within the range of \(-1^{\circ}\) to \(1^{\circ}\) in the circumferential direction.