# Study on the Influence of Dust Removal Fan Structure Parameters on Dust Removal Performance
## 3 Numerical Simulation Scheme of Dust Removal Fan
### Governing Equation  
CFD software requires the solution of the mass conservation equation and the momentum conservation equation for all flow problems. The mass conservation equation, also known as the continuity equation, is expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\text{div}(\rho v)=0 \tag{1}\]  
where \(\rho\) represents density, \(t\) represents time, and \(v\) represents the velocity vector. The momentum conservation equation in the inertial reference frame, also known as the equation of motion, is expressed as  
\[ \begin{cases}\frac{\partial(\rho u)}{\partial t}+\text{div}\Big{(}\rho u \overset{\rightarrow}{u}\Big{)}=-\frac{\partial p}{\partial x}+\frac{ \partial r_{\text{rx}}}{\partial x}+\frac{\partial r_{\text{rx}}}{\partial y }+\frac{\partial r_{\text{rx}}}{\partial x}+F_{x}\\ \frac{\partial(\rho v)}{\partial t}+\text{div}\Big{(}\rho v\overset{ \rightarrow}{u}\Big{)}=-\frac{\partial p}{\partial y}+\frac{\partial r_{ \text{ry}}}{\partial x}+\frac{\partial r_{\text{ry}}}{\partial y}+\frac{ \partial r_{\text{ry}}}{\partial x}+F_{y}\\ \frac{\partial(\rho w)}{\partial t}+\text{div}\Big{(}\rho w\overset{ \rightarrow}{u}\Big{)}=-\frac{\partial p}{\partial z}+\frac{\partial r_{ \text{rx}}}{\partial x}+\frac{\partial r_{\text{yz}}}{\partial y}+\frac{ \partial r_{\text{zz}}}{\partial z}+F_{z}\end{cases}  \tag{2}\]  
where \(p\) represents the pressure on the microfluid; \(\tau_{ij}\) represents the component of viscous stress; and \(F_{x}\), \(F_{y}\), and \(F_{z}\) represent the volume force component per unit of fluid. The turbulence model in this paper adopts the Realizable K-\(\varepsilon\) model, and the model transport equations of turbulent kinetic energy k and dissipation rate \(\varepsilon\) are as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_ {j}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\left(u+\frac{u_{\text{t}}}{ \sigma_{k}}\right)\frac{\partial k}{\partial x_{j}}\Bigg{]}+G_{\text{k}}+G_{ \text{b}}-\rho\varepsilon-Y_{M} \tag{3}\]and  
\[\frac{\partial(\rho\epsilon)}{\partial t}+\frac{\partial(\rho\epsilon u_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}u+\frac{u_{t}}{ \sigma_{\epsilon}}\bigg{)}\frac{\partial\epsilon}{\partial x_{j}}\Bigg{]}+\rho C _{1}E\epsilon-\rho C_{2}\frac{\epsilon^{2}}{\dot{\kappa}+\sqrt{v\epsilon}}+C_ {1\epsilon}\frac{\epsilon}{\dot{\kappa}}C_{3\epsilon}C_{\rm{b}} \tag{4}\]  
where \(G_{k}\) represents the turbulent kinetic energy generated by the average velocity gradient. \(G_{b}\) represents the turbulent kinetic energy caused by buoyancy. \(Y_{M}\) represents the effect of pulsating expansion on the total dissipation rate in compressible turbulence. \(C_{1\epsilon}\), \(C_{3\epsilon}\), and \(C_{2}\) are empirical constants. The default values are \(C_{1\epsilon}=1.44\), \(C_{2\epsilon}=1.92\), \(C_{3\epsilon}=0.09\), and \(C_{2}=1.9\). \(\sigma_{k}\) and \(\sigma_{\epsilon}\) represent the Prandtl number corresponding to the turbulent kinetic energy and the turbulent dissipation rate respectively, and the default values are \(\sigma_{k}=1.0\) and \(\sigma_{\epsilon}=1.2\). The discrete phase model predicts the motion trajectory of discrete phase particles by integrating the force balance on the particles in the Lagrange reference frame. Its particle motion equation is expressed as follows:  
\[m_{p}\frac{\mathrm{d}U_{p}}{\mathrm{d}t}=m_{p}\frac{U-U_{p}}{\tau_{r}}+m_{p} \frac{g\big{(}\rho_{p}-\rho\big{)}}{\rho_{p}}+F \tag{5}\]  
where \(m_{p}\) represents the mass of the particle, \(U\) represents the velocity of the fluid phase, \(U_{p}\) represents the velocity of the particle, \(\rho\) represents fluid density; \(\rho_{p}\) represents particle density, \(F\) stands for additional force, and \(\tau_{r}\) represents the relaxation time of the droplet or particle.