# Aircraft Turbine Engine Labyrinth Seal CFD Sensitivity Analysis
## 4 Calculation Discussion  
The aim is to obtain deeper understanding of the thermodynamics in a turbine engine labyrinth seal at different angular velocities \(\omega_{C}\), different radial clearances \(RC_{CORR}\), and different locations of the teeth. The total temperature, static pressure, total enthalpy, flow coefficient, kinetic energy, and density were chosen as evaluation parameters. Based on the following Formulas (4)-(10), all average values of thermodynamic parameters were plotted. * Total temperature \[T_{TCORR}=\frac{T_{T}}{T_{TF}}\] (4) where \(T_{T}\) [K] is the calculated total temperature and \(T_{TF}\) [K] is the reference total temperature. * Static pressure \[p_{SCORR}=\frac{p_{S}}{p_{SF}}\] (5) where \(p_{S}\) [Pa] is the calculated static pressure and \(p_{SF}\) [Pa] is the reference static pressure. * Total enthalpy \[h_{TCORR}=\frac{h_{T}}{h_{TF}}\] (6) where \(h_{T}\) [kJ/kg] is the total enthalpy and \(h_{TF}\) [kJ/kg] is the reference total enthalpy
* Flow coefficient \[\mu=\frac{Q}{Q_{C}}\] (7) where \(Q\) [kg/s] is the calculated mass flow and \(Q_{C}\) [kg/s] is the critical mass flow, defined by (8) \[Q_{C}=\pi\cdot RC^{2}\cdot\sqrt{\frac{2\cdot k}{k+1}\cdot r\cdot T_{0}}\cdot \left[\rho_{0}\left(\frac{2}{k+1}\right)^{\frac{1}{k+1}}\right]\] (8)  
Figure 8: Mass flow convergence. where \(k\) is the Poisson constant [-], \(r\) is the Gas constant [J/kg/K], \(T_{0}\) [K] is the ambient total temperature, and \(\rho_{0}\) is the ambient density [kg/m\({}^{3}\)]. Ambient conditions are values near the teeth. * Kinetic energy \[Ek_{CORR}=\frac{Ek}{Ek_{F}}\] (9) where \(Ek_{F}\) is [kJ] is the reference kinetic energy. * Density \[\rho_{CORR}=\frac{\rho}{\rho_{F}}\] (10) where \(\rho_{F}\) [kg/m\({}^{3}\)] is the reference density. A list of reference conditions (subscripted with \(F\)) is in Table 3:  
The above stated thermodynamic parameters are plotted in the following charts. The angular velocity \(\omega_{C}\) is plotted in the x-axis in all charts. The calculated data contains both the rotor teeth and stator teeth variants. The RC\({}_{CORR}\) values for the various teeth locations are listed in Table 1. The total temperatures and the statics pressure fields for RC\({}_{CORR}\) = 0.15 are presented in Appendix A. The same characteristics for RC\({}_{CORR}\) = 0.04 are presented in Appendix B. The font colors listed in Table 1 correspond to the colors used in the charts. The total temperature (based on (4)) trendlines are given in \[p_{SCORR}=\frac{p_{S}}{p_{SF}}\]0 Here, the trendlines with the identical radial clearances have similar development. The differences (of approximately 0.03) can be seen at high angular velocities at minimal radial clearance. With a small radial clearance, the teeth positions (stator or rotor) do not show any influence; the trendlines are similar. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline T\({}_{\text{TF}}\) & 288 & K \\ PS\({}_{\text{SF}}\) & 101325 & Pa \\ h\({}_{\text{TF}}\) & 260 & kJ/kg \\ Ek\({}_{\text{F}}\) & 200 & kJ \\ \(\rho_{\text{F}}\) & 1.029 & kg/m\({}^{3}\) \\ \(\omega_{\text{F}}\) & 3665 & rad/s \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reference conditions. Figure 9: Total temperature trendlines. It is possible to say that the dominant factor of the total temperature increase is the radial clearance value. In Figure 10, the static pressure (based on (5)) trendlines are presented. It is visible that the static pressure is of nearly constant value. Similar results appear for different angular velocities, where only a small difference (approx. 0.05) at radial clearance 0.15 at maximal angular velocity is registered. With small radial clearance, the static pressure trendlines are moved to higher value (from 1.35 to 1.63). The main geometric parameter of the labyrinth seal to control the static pressure development is the radial clearance. The total enthalpy trendlines (based on (6)) are given in Figure 11. They correspond to the temperature trendline results.