# Numerical Study of Internal Flow Field in a Disc Stack Centrifuge Based on Mixture-PBM Model
## 1 Introduction  
The disc stack centrifuge is a type of sedimentation centrifuge [1; 2] that uses a high-speed rotating bowl component to complete the separation, and there are multiple layers of discs inside the bowl component. The space between adjacent discs is the separation channel. The liquid in the separation channel under the action of the high-speed rotation of the bowl generates a strong centrifugal force field, allowing two different densities and immiscible liquids to obtain different centrifugal forces, thus achieving liquid-liquid separation in the separation channel of the bowl. It has the advantages of small size and high separation efficiency; therefore, it is widely used in the environmental protection, biopharmaceutical, ship, and chemical industries [3; 4; 5; 6; 7; 8; 9]. At present, the main research directions of disc stack centrifuges are focused on the finite element analysis of bowl stress [10; 11], rotor dynamics and dynamic balance research [12; 13], control system research [14], etc. However, there is less research on the theoretical analysis of the separation flow of the disc stack centrifuge, visualization of the flow field, and performance prediction. [PERSON] et al. [15] developed a mathematical model to characterize the separation performance of the separator at a low-concentration feed based on the fact that the separation performance is positively correlated with the centrifugal force and its action time. [PERSON] [16] used Lie group analysis to find the invariance of the set of two-dimensional boundary layer equations for thin-layer flow in the separation channel and obtained the relationship between the thickness of the laminar boundary layer of the disc stack centrifuge and the structural and physical parameters of the separator. However, these mathematical models are carried out based on certain assumptions, so they often have certain limitations. [PERSON] et al. [17] performed emulsion separation experiments using a laboratory-grade disc stack centrifuge and investigated the effect of rotational speed and separation time on critical diameter and separation efficiency. [PERSON] et al. [18] experimentally determined the transition of flow from laminar to turbulent in a separation channel. However, these experiments are time- and labor-intensive, and too little information on the flow field is obtained due to the limitations of the measurement's means of observation. With the rapid development of Computational Fluid Dynamics (CFD) technology, numerical simulation has become an important means to study the flow field inside a disc stack centrifuge. [PERSON] et al. [19] used a Mixture model to numerically simulate the two-dimensional model of the flow field of a rotating bowl and quantitatively evaluated the changes in the pressure and velocity of the flow field at different rotational speeds. [PERSON] et al. [20] used the Mixture model to analyze the effect of processing capacity and particle diameter on the separation performance under one particle diameter condition. [PERSON] et al. [21] used the Mixture model to analyze the velocity pressure field within a three-dimensional bowl and the effects of rotational speed and particle size on the separation efficiency. [PERSON] et al. [22] investigated the oil-water separation flow in the flow region formed by five-layer discs and in the sedimentation chamber of a rotary bowl and analyzed the velocity field and the volume distribution of the heavy phase (water) in the separation channel. [PERSON] et al. [23] used the Volume-of-Fluid (VOF) multiphase flow model and Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm to simulate the oil-water separation flow process inside the separator and compared and analyzed the calculation results of different initialization methods. [PERSON] et al.