# Analysis of Cyclone Spinning Effect with Different Guide Vane Heights
## 3 Numerical Calculations and Validation of Results
### Governing Equations and Boundary Conditions
Continuity Equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{1}\]  
Momentum Equation:  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{j}}(\rho u _{i}u_{j})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}[( \mu+\mu_{t})(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}})] \tag{2}\]  
Turbulent kinetic energy \(k\) equation:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho u_{i} k)=\frac{\partial}{\partial x_{i}}[(a_{k}\mu_{eff})\frac{\partial k}{\partial x_{i}}]+G_{k}-\rho\varepsilon \tag{3}\]  
Dissipation Rate Equation:  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}( \rho u_{i}\varepsilon)=\frac{\partial}{\partial x_{i}}[(a_{\varepsilon}\mu_{ eff})\frac{\partial\varepsilon}{\partial x_{i}}]+C^{*}_{1\varepsilon}\frac{ \varepsilon}{k}G_{k}-C_{2\varepsilon}\rho\frac{\varepsilon^{2}}{k} \tag{4}\]  
where  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon}, C_{\mu}=0.0845,\ \mu_{eff}=\mu+\mu_{t},\ a_{k}=a_{\varepsilon}=1.39,\ G_{k}=\mu_{t} \Big{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{i}}{\partial x _{i}}\Big{)}\frac{\partial u_{i}}{\partial x_{j}},\] \[C^{*}_{1\varepsilon}=C_{1\varepsilon}-\frac{\eta(1-\eta/\eta_{0 })}{1+\beta\eta^{3}},\ \eta_{0}=4.377,\ \beta=0.012,\] \[\eta=(2E_{ij};E_{ij})^{\frac{1}{2}}\frac{k}{\varepsilon},\ E_{ij}= \frac{1}{2}(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}),\ C_{1\varepsilon}=1.68,\ C_{2\varepsilon}=1.42,\]  
where \(t\) is the time; \(i\) and \(j\) = 1, 2, 3; \(u_{i}\) is the velocity component in the \(x_{i}\) direction; \(\rho\) is the density of the weighted average of volume fraction; \(p\) is the correction pressure; \(\mu\) is the molecular viscosity coefficient of the weighted average of volume fraction; \(a_{k},a_{\varepsilon}\) is the Plante number at turbulent kinetic energy \(k\) and dissipation rate \(\varepsilon\), respectively; \(\mu_{t}\) is the turbulent viscosity coefficient; and \(G_{k}\) is the turbulent kinetic energy generation term caused by the average velocity gradient [28]. In the case studied in this paper, the minimum Reynolds number is 105,419, which satisfies the high Reynolds number flow, and the default value of the turbulence model can be used [26]. In the simulation calculation, the inlet velocity of the model is set according to the Reynolds number Re = 105,419, 140,559, 175,699, and 210,838. The pressure outlet boundary is set at a pressure of 101,000 pa. Edge walls, such as the pipe wall and guide vane faces, are set to non-slip walls. Since the SIMPLEC algorithm has better convergence than the SIMPLE algorithm, the SIMPLEC algorithm is selected here. In order to meet the accuracy of the simulation, the convergence accuracy of each item is set to 0.0001. The standard initialization method is selected, and the flow field in the inlet part is initialized. The number of iterations (i.e., the calculation step) is set, and 1000 iterations are selected to ensure the accuracy of the calculations.