## 2 Numerical Calculation Model of Four-Tooth Three-Stage Brush Seal
### Calculation Conclusions and Parameter Improvements
#### 3.3.2 Parameter Improvements
shown in Figure 35. Calculations show that the pressure drops in the first, second and third brush bundles of the prototype combined seal are 18.3%, 30.0% and 43.3%, respectively, while the pressure drops in the brush bundle of the improved combined seal are 31.8%, 33.3% and 26.0%, respectively. The highest pressure drop percentage of the brush bundle for the improved combination seal is reduced by 23%, and the interstage pressure drop distribution is significantly more balanced. For the improved combined seal, the risk of brush bundle premature failure caused by the unevenness characteristic of the pressure drop between stages is significantly reduced, and the pressure tolerance of the multi-stage brush bundle is significantly improved. Meanwhile, the leakage rate of the improved combined seal increased by 5.8% compared with the prototype due to the setting of radial clearance of the brush bundle in the rear stage, and the increase in leakage rate was small. The improved combined seal achieves the goal of sacrificing a smaller sealing performance for the balanced characteristics between the multi-stage brush bundle stages. Mechanism Analysis of Unevenness Characteristic of Interstage Pressure Drop of Multistage Brush Seals  
In the previous section, we analyzed the non-uniform pressure drop characteristics and leakage behavior of the four-tooth three-stage brush seal assembly. The influence of structural parameters on pressure drop distribution and leakage rate was revealed, with key parameters identified for effectively adjusting pressure drop non-uniformity. Structural parameter modifications were implemented to balance sealing performance and pressure drop uniformity. In this section, a mathematical relationship between pressure drops and entropy generation is derived from thermodynamic principles, aiming to introduce how entropy generation affects pressure drop behavior. As shown in Figure 36, assuming that an ideal gas flows steadily and adiabatically through a small segment of the brush bundle (brush wire clearance), according to the law of conservation of mass, the first law of thermodynamics, and the second law of thermodynamics, we obtain:  
\[\left\{ \begin{aligned} \dot{m}_{in}=\dot{m}_{out}\\ h_{in}=h_{out}\\ \dot{S}_{gen}=\dot{m}(s_{out}-s_{in})\end{aligned} \right. \tag{11}\]  
By the definitions of enthalpy and entropy:  
\[ \begin{array}{l}dh=Tds+vdP\\ ds=\frac{1}{2}dh-\frac{v}{2}dP\end{array}  \tag{12}\]  
Figure 35: A comparison of the calculation results of the improved combination seal and the prototype combination seal. (**a**) The pressure drop distribution of the interstage for improved and prototype combination seals, and (**b**) the leakage rates of the improved and prototype combination seals. For the brush bundle of a brush seal, **A** and **B** are constants at steady state. From (17) and (18) the unit flow fluid is given:  
\[S_{gen}=Rln\left(\frac{Av_{s}+Bv_{s}^{2}}{P_{out}}+1\right) \tag{19}\]  
Equations (18) and (19) indicate that the entropy increasing \(S_{gen}\) and pressure drop \(\Delta P\) increase with the increase in fluid velocity in the brush wire clearance. In a multistage brush seal, from the high-pressure upstream to the low-pressure downstream, high-energy fluid potential energy is gradually converted to kinetic energy; kinetic energy is dissipated into heat by the brush wire surface layer of the damping and the back of the brush wire is separated from the turbulence of the annihilation of inertial damping, and the entropy value is gradually increased. The differential pressure increases stage-by-stage, the pressure drop increases quickly with the increase in entropy, and then the multistage brush bundle emerges with an unevenness in the pressure drop stage-by-stage. As mentioned before, increasing the brush bundle thickness, decreasing the backplane height, and decreasing the brush bundle radial clearance can enhance the viscous and inertial damping dissipation of the gas flow within the brush bundle.