# Numerical Study of Internal Flow Field in a Disc Stack Centrifuge Based on Mixture-PBM Model
## 2 Physical Models and Numerical Simulation
### Numerical Methods and Boundary Conditions  
In this paper, numerical simulation bowl components of the disc stack centrifuge are carried out based on ANSYS Fluent 2021 R1 software. The Mixture-PBM coupled model was used, the turbulence model was used for the coalescence model [26], and the Luo model was used for the breakup model [27]. The population balance equation is solved by the discrete method. The coupled method is chosen as the solution method, and the first-order windward format is chosen as the difference format. In addition, the RNG k-epsilon model is chosen as the turbulence model. The energy equation is used to describe the transfer and conversion of energy in the fluid, the wall is set as an adiabatic wall, and viscous heating is considered. The velocity inlet was chosen as the boundary condition for the inlet of the computational domain, and the static temperature of the inlet was 288.15 K; the volume content of the aqueous phase of the inlet was 5%. The direction of velocity at the inlet in the rotating coordinate system is perpendicular to the inlet boundary, and the two phases of oil and water at the inlet have the same velocity. Turbulence intensity and hydraulic diameter were chosen to describe the inlet turbulence, with a turbulence intensity of 5% and a hydraulic diameter of 22 mm. The surface tension coefficient was 0.018 N/m. The rest were the default values provided by ANSYS Fluent 2021 R1. The particle diameter of the water droplets at the inlet of the disc stack centrifuges was measured by the ML-1 oil-liquid abrasive particle detector, and the results showed that the oil-water mixture contained eight different diameters of water droplets ranging from 5 \(\upmu\)m to 200 \(\upmu\)m. The experimental measurements were fitted to a normal distribution function, as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{2}{c}{**Number of Grids**} & \multicolumn{2}{c}{**Number of Grids in the**} & \multicolumn{2}{c}{**Number of Grids in the**} \\  & **in the Circumferential Direction** & **Direction of the Generatrix** & **Normal Direction** & **Total Number of Grids** \\ \hline
1 & 24 & 61 & 2 & 6088 \\
2 & 29 & 73 & 3 & 11,249 \\
3 & 37 & 88 & 4 & 20,130 \\
4 & 44 & 106 & 5 & 35,800 \\
5 & 53 & 127 & 6 & 64,652 \\
6 & 60 & 147 & 7 & 123,480 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Meshing scheme. Figure 4: Calculation domain grid and local magnification. When the boundary condition for the outlet of the computational domain is a free exit boundary or a flow outlet boundary, this leads to a negative pressure field in the final result that does not match the actual situation. In reality, the liquid at the outlet of the computational domain under the action of centrifugal force will form a free liquid surface with the air, and this free liquid surface is parallel to the axis of rotation [28]. Therefore, the boundary conditions at the exit of the computational domain should be a pressure distribution that increases with the radius, and this pressure distribution conforms to the laws of the centrifugal hydraulic pressure equation, as shown in Equation (4). \[P(r)=\rho\omega^{2}\int_{r_{f}}^{r}r\ dr \tag{4}\]  
In the equation, \(\rho\) is the density of the liquid, and the unit is m\({}^{3}\)/kg; \(\omega\) is the rotational angular velocity, and the unit is rad/s; \(r\) is the radius at any position, and the unit is m; and \(r_{f}\) is the radius of the free liquid surface at the outlet of the oil phase or water phase, and the unit is m.  
To simplify the calculations, in this paper, the fluid domain of the air portion at the outlet of the oil phases and water phases is deleted, and the free liquid surface is treated as a wall boundary. Due to the low water content of the fluid in the computational domain, the radius of the free liquid surface at the outlet of the aqueous phase is considered to be the same as the radius of the inlet of the aqueous phase centripetal pump, which is taken to be \(r_{fh}=0.398D\).