# Numerical and Experimental Analysis of Labyrinth Seals with Rhomboidal Cells
## 2 Numerical Analysis
### Thermalase Models Predictions
As can be seen, in Table 7, the sum of the two percentage fractions of the total mass flow rate, which flow respectively inside the restrictor and in the clearance, slightly exceed 100%. This is because the mass flow \(\dot{m}_{task}\) that bypasses the fin of the tip (Figure 12) includes a small amount of Coo air coming from the upper cavities. Changes were also made to the thermal model before performing the coupled fluid thermal analysis. These changes were necessary to take into account the effect of the new cell geometry on thermal conductivity and thus on heat transfer in the labyrinth seal region. The change of thermal conductivity does not represent a real variation of the thermo-physical properties since the material and the operating temperature range are the same used to test the honeycomb cell, but it is necessary to take into account the different solid/void ratio. Since only the solid parts of the cells contribute to conduction heat transfer, the scale factor reported in Equation (8) was applied to the thermal conductivity of the rhomboidal cell. \[f_{scale}=\frac{A_{s}}{A_{s}+\ A_{v}}\, \tag{8}\]  
However, due to the chosen cell thickness and solid/void ratio, the conductivity scale factor is quite similar to the honeycomb one, producing only a negligible effect on the heat transfer by conduction. For the convective heat transfer, the exchange coefficients were modified since, at the same height, the area of the rhomboidal cell is greater than the honeycomb one, and therefore, the portion of the shroud back-plate to be considered for the convective heat transfer is greater. After these modifications, two numerical simulations were carried out to examine the thermal behavior of the new seal. The settings used (Table 8) refer to two experimental tests (Test 3 and Test 4 [14]) previously used for tuning the numerical model with the honeycomb seal. Figure 14: Thermalcase model: Mass flow split in the sealing region with a Flow Path temperature of 873 K and a Cooling temperature in Case A (**left**) of 473 K and in Case B (**right**) of 573 K. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Test**} & \(\mathbf{T_{FP}}\) & \(\mathbf{p_{FF}}\) & \(\dot{m_{FP}}\) & \(\mathbf{T_{Coo}}\) & \(\mathbf{p_{Coo}}\) & \(\dot{m_{Coo}}\) & \(\mathbf{T_{ACC}}\) & \(\mathbf{p_{FF}}\) & \(\dot{m_{ACC}}\) \\  & **(K)** & **(bar)** & **(kg/S)** & **(K)** & **(bar)** & **(kg/s)** & **(K)** & **(bar)** & **(kg/s)** \\ \hline
3 & 796 & 1.8 & 0.438 & 461 & 2 & 0.043 & 329 & 1.04 & 0.013 \\
4 & 808 & 1.8 & 0.442 & 569 & 2 & 0.044 & 336 & 1.05 & 0.013 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Settings for the numerical simulations: temperatures, pressure and mass flow rates for the Flow Path, Cooling and Active Clearance Control. In analyzing the results, particular attention was paid to three metal (M#) and one air (T#) thermocouples (Figure 15) placed in the area of interest. In Tables 9 and 10, the temperatures resulting from the numerical simulations with the rhomboidal cells are compared with the numerical data previously obtained with the honeycomb cells [14]. It is possible to observe that the numerical temperatures obtained with the rhombi are only slightly lower than those recorded for the honeycomb and that this temperature difference can be considered negligible when compared with the high operating temperatures.