## 3 Computational Model  
The flow of gas follows the laws of mass conservation, momentum conservation, and energy conservation. The fundamental equations of fluid mechanics are the continuity equation, momentum conservation equation, and energy conservation equation. The mass conservation equation states that the increase in mass within a fluid element per unit of time is equal to the net mass flowing into the element during the same time interval. The continuity equation is as follows:  
For steady flow, \(\frac{\partial\rho}{\partial t}=0\), the continuity equation is  
\[\frac{\partial\rho}{\partial t}+div(\rho u)=0 \tag{1}\]  
\[div(\rho u)=0. \tag{2}\]  
For incompressible fluids, \(\frac{D\rho}{\partial t}=0\), the equation is  
\[divu=0. \tag{3}\]  
In the above equations, \(\rho\) is density, \(t\) is time, and \(u\) is the velocity vector. The momentum conservation law states that the rate of change of momentum of a fluid element with respect to time is equal to the sum of the external forces acting on the element. That is  
\[\rho\frac{Du}{Dt}=\rho F_{b}+divP \tag{4}\]  
where \(P\) is the pressure on the fluid element and \(F_{b}\) is the body force on the element. The basic control equations are established based on laminar flow, but the mixing process is turbulent. To ensure good simulation results, the control equations need to be modified. Generally, turbulent flow is extremely complex, and the N-S model is a relatively accurate method for numerical calculation of flow. However, due to its high computational accuracy, it requires high CPU speed and memory size, making it generally unsuitable for engineering calculations. Common turbulence simulation methods suitable for engineering calculations include Large Eddy Simulation (LES) and Reynolds-Averaged Navier-Stokes (RANS). In this study, the standard \(k-\varepsilon\)[21] model was used. Due to its computational accuracy and convergence, it meets the general requirements of engineering calculations  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test ID**} & **Air Vane** & \multirow{2}{*}{**Rotation Dir. **} & **Gas Vane** & \multirow{2}{*}{**Rotation Dir. **} & **Helical Vane** & \multirow{2}{*}{**Rotation Dir. **} \\  & **Helix** & & **Helix** & & **(Mixing Zone)** & \\ \hline a & Yes & Counterclockwise & No & No & No & No \\ a1 & Yes & Counterclockwise & No & No & Yes & Counterclockwise \\ a2 & Yes & Counterclockwise & No & No & Yes & Clockwise \\ b & No & No & Yes & Clockwise & No & No \\ b1 & No & No & Yes & Clockwise & Yes & Counterclockwise \\ b2 & No & No & Yes & Clockwise & Yes & Clockwise \\ c & Yes & Counterclockwise & Yes & Clockwise & No & No \\ c1 & Yes & Counterclockwise & Yes & Clockwise & Yes & Counterclockwise \\ c2 & Yes & Counterclockwise & Yes & Clock  
and is widely used in engineering. This model uses the turbulence kinetic energy equation and the turbulence dissipation rate equation to close the turbulence control equations:  
\[\frac{\partial(\rho k)}{\partial t}+\
abla\bullet(\rho\overset{\rightarrow}{V} k)=\
abla\bullet(\Gamma_{k}\
abla k)+G_{k}+G_{b}-\rho\varepsilon-Y_{k}+S_{k} \tag{5}\]  
\[\frac{\partial(\rho k)}{\partial t}+\
abla\bullet\left(\rho\overset{ \rightarrow}{V}\varepsilon\right)=\
abla\bullet(\Gamma_{k}\
abla\varepsilon) +G_{1\varepsilon}\frac{\varepsilon}{k}(G_{K}+G_{3\varepsilon}G_{b})-G_{2 \varepsilon}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{6}\]  
where the model constants are \(C_{\mu}=0.09,C_{1\varepsilon}=1.44,C_{2\varepsilon}=1.92,\sigma_{k}=1.0, \sigma_{\varepsilon}=1.3\), and the diffusion coefficients are \(\Gamma_{k}=\mu+\frac{\mu_{t}}{\sigma_{\varepsilon}}\Gamma_{\varepsilon}=\mu+ \frac{\mu_{t}}{\sigma_{\varepsilon}}\).