## 4 Methodology  
### Mathematical Modeling  
In this paper, the velocity and pressure distribution of the flow field are analyzed by varying the air inlet velocity, drum velocity, and spindle velocity of the spindle. An excellent range of air separation parameters is found. Thus, the film impurities are separated effectively.  
The fluid flow inside the cavity exhibits turbulent flow [28; 29]. The flow field within the film stalk separation device follows the principles of mass conservation and [PERSON]'s second law. Considering the characteristics of the drum's interior, the flow field is defined as unpressurized, constant, and isothermal. The accuracy of the numerical simulation primarily depends on the realism of the turbulence model. Different turbulence models can be applied to simulate flow fields with diverse flow characteristics. The turbulence model chosen for the numerical simulation of the flow field inside the roller screen of the residual film recycler is the standard \(\mathrm{k}-\varepsilon\) model, and the individual control equations in right-angle coordinates are as follows:  
1. The mass conservation equation: \[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0,\] (1) for incompressible fluids, the density is set to be constant, then Equation (1) can be simplified as: \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0,\] (2) In the equation, \(\rho\) is the density of the fluid, \(t\) is time, and \(v\) is the velocity vector of the fluid.
2. The momentum conservation equation (i.e., the \(N-S\) equation): \[\frac{\partial(\rho u)}{\partial t}+\mathrm{div}(\rho uu)=-\frac{\partial P}{ \partial x}+\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{yy}}{ \partial y}+\frac{\partial\tau_{xx}}{\partial z}+F_{X}\] (3) \[\frac{\partial(\rho u)}{\partial t}+\mathrm{div}(\rho vu)=-\frac{\partial P}{ \partial y}+\frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{ \partial y}+\frac{\partial\tau_{xy}}{\partial z}+F_{Y},\] (4) \[\frac{\partial(\rho w)}{\partial t}+\mathrm{div}(\rho wu)=-\frac{\partial P}{ \partial z}+\frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{ \partial y}+\frac{\partial\tau_{zz}}{\partial z}+F_{z}\] (5) 3. The are \(\mathrm{k}-\varepsilon\) turbulence equations: The standard \(\mathrm{k}-\varepsilon\) model is the most widely used two-equation model in engineering practice. The so-called standard \(\mathrm{k}-\varepsilon\) model because it introduces two parameters, \(k\) and \(\varepsilon\), into the equation respectively. The transport equations corresponding to these two parameters are as follows: \[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_ {i}}=\frac{\partial}{\partial x_{i}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{i}}{\delta _{k}}\bigg{)}\frac{\partial k}{\partial x_{i}}\Bigg{]}G_{K}-\rho\varepsilon \frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho u_{i})}{ \partial x_{i}}\mathrm{x}=\frac{\partial}{\partial x_{i}}\bigg{[}\bigg{(}\mu+ \frac{\mu_{i}}{\delta_{k}}\bigg{)}\frac{\partial\varepsilon}{\partial x_{i}} \bigg{]}+\frac{C_{1\varepsilon}\varepsilon}{K}G_{K}x-C_{2\varepsilon\rho} \frac{\varepsilon^{2}}{K},\] (6) In this equation, \(K\) represents turbulent kinetic energy, \(\varepsilon\) represents turbulent dissipation rate, and \(\mu\) represents eddy viscosity coefficient, all of which are related to fluid properties. \(i\) denotes the index of the velocity vector, while \(G_{K}\) denotes the production of turbulent kinetic energy caused by the mean velocity gradient. \(\mu_{i}=\rho C_{i}\frac{K^{2}}{\varepsilon}\) represents the turbulent viscous coefficient.  
The constants in the model are chosen according to the literature [30]\(\mathrm{C_{1\epsilon}}=1.44\), \(\mathrm{C_{2\epsilon}}=1.92\), \(\mathrm{\delta_{k}}=1.0\), \(\partial\epsilon=1.0\), A complete mathematical model is developed for further mathematical simulation and analysis.