*3.1. Application of the Variant A of the Design Method*

In this chapter, the impact of the variant A application on the leakage rate for various number of the seal teeth is analyzed. Figure 10 presents the gas velocity contours for the initial geometry and the improved one.

For the geometries under consideration (Figure 10) of uniformly spaced teeth, the change of gas velocity along the seal length is great. The gas velocity in the first clearance is approx. 70 m/s, and in the last one it exceeds 180 m/s. The effect of gas expansion in clearances impacts the intensity of gas vortices in subsequent chambers. The vortex in the last chambers is significantly greater than in the first few chambers. In the designed geometry, it was observed that chamber lengths were adapting to the increasing gas velocity in clearances. In result, the gas velocity close to the last tooth (upstream the clearance) is significantly smaller for the designed geometry, when compared with the initial geometry for cases 8t, 9t, and 10t, being considered. When the distribution of the gas velocity in the optimized geometries of 9 and 10 teeth is analyzed, it can be observed that gas vortices occur within the whole chambers' volume. Table 3 summarizes obtained values of mass flow for geometries shown in Figure 10.

**Figure 10.** Distribution of air velocity in the seal of the initial geometry and of the designed one for the radial height RC = 0.315 mm comprising of (**a**) eight, (**b**) nine, and (**c**) ten teeth for boundary conditions *pin/pout* = 2.4, *Tin* = 300 K, *pout* = 105 Pa.

**Table 3.** Leakage rate for the seal of the initial geometry and of the designed one and their relative changes according to Equations (17) and (18), RC = 0.315 mm and of different number of teeth for *pin*/*pout* = 2, *Tin* = 300 K, *pout* = 10<sup>5</sup> Pa.


The design method for geometries 8t, 9t, and 10t, when compared to the geometry 8t with evenly spaced teeth (constant pitch length) LPconst (Figure 10a), improves the leaktightness of the relative value *<sup>δ</sup>* . *m*(*t*, 8) by 3.4%, 9.7%, and 15.5%, respectively. Comparing the relative difference of the mass flow (Table 3, Figure 11) obtained for the designed geometry . *mLPdm*(*t*) and the geometry with evenly spaced teeth . *mLPconst*(*t*) according to

the Equation (12), the leak-tightness was improved by 3.4%, 2.8%, and 2.2%, respectively, for the geometry 8t, 9t, and 10t.

**Figure 11.** Mass flow obtained for the seal of equal pitch and for the designed one depending on the number of teeth t, for the clearance RC = 3.15 mm and boundary conditions *pin/pout* = 2, *Tin* = 300 K, *pout* = 105 Pa.

Application of the design method and changing the number of teeth from eight to ten enable significant reduction of the leakage. There is a linear relationship between the number of teeth in the range from 8 and 10 and the relative reduction of the leakage rate for *pin/pout* = 2 (Figure 12). The limitation of the applied design method for the given seal length LS and many teeth is obtaining too small spaces between them. Therefore, the analysis has not been continued for a greater number of teeth.

**Figure 12.** Relative reduction of the integrity of the designed geometry *<sup>δ</sup>* . *m*(*t*, 8) [%] depending on the number of teeth t, for the clearance RC = 0.315 mm and the pressure ratio *pin/pout*, *Tin* = 300 K, *pout* = 105 Pa.

Distribution of flow and thermodynamic parameters of the seal is affected by the ratio of pressure upstream and downstream the sealing. For the ratio *pin/pout* equal to 2.4 and 2.8, the increase of leakage rate for the designed geometry with 10 teeth was observed (Figure 12). Within the frames of research work a series of numerical calculations was performed for the geometry of the clearance height RC = 0.315; 0.542 and 0.77 mm for the pressure ratio *pin/pout* ranging from 2 to 2.8. Tables 4–6 summarize the obtained values of air mass flow for the initial geometry and the designed one for various heights of the radial clearance RC.

**Table 4.** Change of the leakage rate by Equation (17) depending on the pressure ratio pin/pout for the initial 8t and the designed geometry 9t of the staggered seal, RC = 0.315 mm, *Tin* = 300 K, *pout* = 105 Pa.


**Table 5.** Change of the leakage rate by Equation (17) depending on the pressure ratio *pin/pout* for the initial 8t and the designed geometry 9t of the staggered seal, RC = 0. 542 mm, *Tin* = 300 K, *pout* = 105 Pa.


**Table 6.** Change of the leakage rate by Equations (17) and (18) depending on the pressure ratio *pin/pout* for the initial 8t and the designed geometry 9t of the staggered seal, RC = 0. 77 mm, *Tin* = 300 K, *pout* = 105 Pa.


The relative reduction of the leakage *<sup>δ</sup>* . *m*(*t*) for geometries of the staggered seal under consideration is slightly affected by the pressure ratio *pin/pout*. For geometries RC = 0.315, 0.542, 0.77 mm, it is included in the range 9.57–9.65%, 10.78–10.90%, and 10.76–10.97%, respectively. The leakage rate reduction depends on the clearance height. The greatest relative reduction of the leakage was obtained for the geometry RC = 0.542 and 0.77 mm, *t* = 8 (Tables 5 and 6).

For the increasing pressure ratio, the leakage reduction <sup>Δ</sup> . *m* is linear for the analyzed RC, (Figure 13).
