# Study the Effect of Operating Temperature on Performance in Dry Scroll Vacuum Pump
## 2 Numerical Model
### Governing Equations  
As the shape and volume of the working chamber are constantly changing, the gas compression process is an unsteady compressible flow. In the Cartesian coordinate system, the governing equation with an unsteady compressible flow [30; 31; 32] can be expressed as Equation (1):  
\[ \begin{array}{c}\frac{d}{dt}\int\limits_{V(t)}\rho dV+\int\limits_{S}\rho \big{(}U_{j}-W_{j}\big{)}dn_{j}=0\\ \\ \frac{d}{dt}\int\limits_{V(t)}\rho U_{i}dV+\int\limits_{S}\rho\big{(}U_{j}-W_{ j}\big{)}U_{i}dn_{j}=-\int\limits_{S}pdn_{j}+\\ \\ \int\limits_{S}u_{eff}\Big{(}\frac{\partial U_{i}}{\partial x_{j}}+\frac{ \partial U_{j}}{\partial x_{i}}\Big{)}dn_{j}+\int\limits_{V}Su_{i}dV\end{array}  \tag{1}\]  
\[ \begin{array}{c}\frac{d}{dt}\int\limits_{V(t)}\rho\phi dV+\int\limits_{S}\rho \phi\big{(}U_{j}-W_{j}\big{)}dn_{j}=\int\limits_{S}\Gamma_{eff}\Big{(}\frac{ \partial\phi}{\partial x_{j}}\Big{)}dn_{j}+\int\limits_{V}S_{\phi}dV\end{array} \]  
where \(\rho\) is the fluid density, \(U\) is the velocity of the fluid, \(W\) is the velocity of the boundary of the control volume, \(\mu_{eff}\) is the viscosity coefficient containing turbulent viscosity, \(\Gamma_{eff}\) is diffusion coefficient containing turbulent diffusion, \(\varphi\) is a scalar, such as medium per unit mass enthalpy or internal energy, \(S_{U}\), \(S_{\varphi}\) are source terms, \(dn\) is the outer normal vector of the surface, \(i,j\) represents the axis in the Cartesian coordinate system. Dynamic mesh techniques can be applied to the simulation of computational domain changes caused by boundary deformation or motion [33]. When the dry scroll vacuum pump works, the shape and volume of the working chamber constantly change. Therefore, the actual working condition is simulated by the dynamic mesh method. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Items** & **First Stage** & **Second Stage** \\ \hline Radius of basic circle & 7.65 mm & 2.55 mm \\ Pitch of involute & 48 mm & 16 mm \\ Initial angle of involute & \(1/12\pi\) & \(1/4\pi\) \\ Thickness of scroll wrap & 4 mm & 4 mm \\ Height of scroll wrap & 20 mm & 16 mm \\ Loops of scroll & 2.20 & 5.25 \\ Radial clearance & 0.05 mm & 0.05 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structure parameters of the dry scroll vacuum pump. Figure 2: Geometric model of the flow area in dry scroll vacuum pump. As the theorem of geometric conservation law must be satisfied in the dynamic mesh method, the time derivative of the control volume [33; 34; 35] can be calculated by:  
\[\frac{d}{dt}\int\limits_{V(t)}dV=\int\limits_{S}W_{j}dn_{j} \tag{2}\]  
The renormalization group (RNG) \(k\)-\(\varepsilon\) model is based on the renormalization group analysis of the Navier-Stokes equations. This model represents the effects of small-scale turbulence by means of a random forcing function in the Navier-Stokes equation [30]. The transport equations for turbulence generation and dissipation are the same as those for the standard \(k\)-\(\varepsilon\) model, but the model constants are different. The transport equations for turbulence kinetic energy \(k\) and turbulence dissipation rate \(\varepsilon\) become [36]:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial}{\partial x_{i}}(pU_{i}k) =\frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+\frac{\mu_{t}}{\varepsilon_{k}} )\frac{\partial_{k}}{\partial x_{j}}\Bigg{]}+P_{k}-\rho\varepsilon \tag{3}\]  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial}{\partial x_{i}} (\rho U_{i}\varepsilon)=\frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+\frac{ \mu_{t}}{\varepsilon_{k}})\frac{\partial_{\varepsilon}}{\partial x_{j}} \Bigg{]}+\frac{\varepsilon}{k}(C_{\varepsilon 1}P_{k}-C_{\varepsilon 2}\rho\varepsilon) \tag{4}\]  
where \(C_{\varepsilon 2}\), \(\sigma_{k}\), and \(\sigma_{\varepsilon}\) are constants, \(C_{\varepsilon 2}=1.68\), \(\sigma_{k}=0.7179\) and \(\sigma_{\varepsilon}=0.7179\).