*3.2. Application of the Variant B of the Design Method*

For the geometry as per the variant B (described in Section 2.2.), numerical simulations of air flow for various air pressure ratios *pin/pout* from the range 2–2.8 were performed.

In the geometry with chambers of variable height the gas velocity field is more complex than in the geometry from the variant A. During the analysis of the distribution of the gas velocity, a better adaptation of chambers' size to the gas vortex can be observed, when compared with the variant A. The higher is the gas velocity in the clearance, the greater are the chambers' volume and vortex.

The gas velocity before clearances is significantly lower than in the variant A (Figure 14). Furthermore, a significantly weaker gas stream of a high velocity is observed in chambers behind clearances. When compared with the initial geometry and data from the variant A, in chambers no. 2, 4, 5, and 6 additional gas vortices were obtained in the area of the outlet from the clearance behind back walls of shorter teeth, which made the gas flow be more dissipative.

**Figure 13.** Leakage reduction for geometries 8t of the clearance height RC = 0.315, 0.542, and 0.77 mm depending on the pressure ratio for *Tin* = 300 K, *pout* = 10<sup>5</sup> Pa.

**Figure 14.** Velocity fields in the designed geometry as per variant B for *pin/pout* = 2.4, *Tin* = 300 K, *pout* = 10<sup>5</sup> Pa.

To sum up, more complicated flow results in a significant reduction of the leakage rate. Obtained results concerning the gas mass flow for the geometry in the variant B are summarized in Table 7.

**Table 7.** Variant B—change of the leakage rate by Equation (18) depending on the pressure ratio *pin/pout* for the initial geometry 8t and the improved one of the staggered seal RC = 0.315 mm, *Tin* = 300 K, *pout* = 105 Pa.


Application of the method for the improved spacing of teeth and changing the chambers depth enabled a significant reduction of the leakage rate by approx. 15.4%. The relative reduction of the leakage rate reveals a slightly decreasing trend for growing pressure ratio *pin/pout*. It is a good result for the leakage rate reduction when a slight change of the seal height with keeping the length unchanged is taken into consideration. It should be noted that in the designed geometry the gas flow field in clearances has not changed.

Pressure drop has no significant impact on the relative leakage decrease, therefore representative results for Variant A and B are summarized in Table 8.


**Table 8.** Summary of change of the leakage rate by Equations (17) and (18) for application of the design method for Variant A and B, for the initial geometry (8t) and the improved one 8t and 9t, *pin/pout* = 2 *Tin* = 300 K, *pout* = 10<sup>5</sup> Pa.

Variant A of the method for the geometry without change of teeth number (*t* = 8) brings improvement of leak-tightness *<sup>δ</sup>* . *m*(*t*) in the range from 3.3% to 3.5%. For geometry with the increased number of teeth (variant A), a significantly reduced leakage (*<sup>δ</sup>* . *m*(*t*, 8) was achieved in the range from 9.65% to 10.95%. Variant B enables a significant reduction of the leakage (even by 15.7%) without the increased number of teeth.
