# Simulation Analysis of Cyclone Separator for Separation of Cenospheres
## 2 Methods  
### Basic Equation  
In this study, the continuity and momentum N-S equations in the fluid domain are computed using the CFD model of Ansys Fluent. The two equations are used together to solve the dynamics of the fluid in the model.  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(V\rho)=0 \tag{1}\]  
\[\rho\left(\frac{\partial V}{\partial t}+V\cdot\
abla V\right)=-\
abla p+\mu \
abla^{2}+F \tag{2}\]  
\(\rho\): Density of the fluid.  
\(V\): The vector of velocity, usually expressed as \(V\) = (u,v,w), where u, v, and aw are the velocity components of the fluid in the x, y, and z directions, respectively.  
\(t\): Time.  
\(\
abla\cdot(V\rho)\): Scatter of the product of the velocity vector \(V\) and the density \(\rho\).  
\(p\): Fluid pressure.  
\(\mu\): Dynamic viscosity of a fluid, related to the viscosity of the fluid.  
\(F\): Vector of an external force acting on the fluid, e.g., gravity.  
Equation (1) is the continuity equation and Equation (2) is the N-S equation for the momentum equation in the three major coordinate systems. The RNG k-epsilon turbulence model is used for the flow state and viscosity settings. The RNG model has some advantages over the standard k-epsilon model: the RNG model has an additional term in its equations that improves the accuracy of the fast strain flow; the effect of eddies on the turbulence is included in the RNG model, which improves the accuracy of eddies; and the RNG theory provides an analytical formula for the turbulence Plummer number, whereas the standard model uses a user-specified constant value. It is an analytical formula, whereas the standard model uses user-specified constant values. Although the standard model is a high-Reynolds number model, the RNG theory provides an analytically derived equation for the effective viscosity differentiation that accounts for low-Reynolds number effects. However, the effective use of this feature depends on the appropriate treatment of the near-wall region. Equation (3) is the RNG turbulent kinetic energy transport equation and Equation (4) is the RNG model turbulent dissipation rate transport equation. The flow field and particle motion inside the cyclone separator were modeled using a combination of Lagrangian and Eulerian techniques. The solid particles were modeled as discrete phases, and the coupling between the gas and solid phases was observed.  
\[\frac{\partial}{\partial t}(\rho\kappa)+\
abla\cdot(\rho\kappa V)=P_{k}-\beta^ {*}\rho\epsilon+\
abla\cdot[(\mu+\sigma_{k}\mu_{t})\
abla k] \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\
abla\cdot(\rho V\epsilon)=\
abla \cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{\epsilon}}\right)\
abla\epsilon \right]+C_{1}\frac{\epsilon}{k}P_{k}-C_{2}\rho\frac{e^{2}}{k}+S_{\epsilon} \tag{4}\]  
\(\rho\): Density of the fluid.  
\(\epsilon\): Turbulent dissipation rate, which represents the rate of turbulent kinetic energy dissipation per unit mass of fluid.  
\(t\): Time.  
\(V\): The velocity vector of the fluid.  
\(P_{k}\): Turbulent kinetic energy generated by the mean velocity gradient.  
\(\beta^{*}\): Constants in the turbulent dissipation rate equation.  
\(\mu\): The molecular viscosity of a fluid.  
\(\mu_{t}\): Turbulent viscosity.  
\(\sigma_{k}\): Turbulent Prandtl number for turbulent kinetic energy.  
\(\sigma_{\epsilon}\): Turbulent Prandtl number for turbulent dissipation rate.  
\(C_{1}\) and \(C_{2}\): Model constant.  
\(S_{\epsilon}\): User-defined source item.