## 2 Numerical Calculation Model of Four-Tooth Three-Stage Brush Seal
### Numerical Modeling
#### 2.2.1 Fractal Geometry Porous Media Model  
(1) Theory of fractal geometry porous media model  
The interstitial spaces within brush seal bristle packs are intricate and microscale. The structural similarity between porous media and bristle packs has led to the widespread use of porous media models for simulating leakage flow in brush seals. However, the capillary diameter, tortuosity, and surface morphology of porous media pores exhibit inherent randomness and irregularity, making traditional theoretical methods inadequate for analyzing \begin{table}
\begin{tabular}{c c c} \hline
**Structural Parameters** & **Values/mm** & **Structural Parameters** & **Values/mm** \\ \hline Outside diameter & 158 & Thickness of third-stage brush bundle & 1.8/2.0/2.2/2.4 \\ Inside diameter & 138 & Height of backplane for first-stage brush bundle & 1.0/1.2 \\ Axial distance of first-stage brush bundle & 1.5/1.8/2.0 & Height of backplane for second-stage brush bundle & 1.0/1.2/1.4 \\ Axial distance of second-stage brush bundle & 1.5/1.8/2.0 & Height of backplane for third-stage brush bundle & 1.0/1.2/1.4 \\ Axial distance of third-stage brush bundle & 1.5/1.8/2.0 & number of grate teeth & 3/4 \\ Thickness of first-stage brush bundle & 2.0/2.2/2.4/2.6 & The inclination of grate teeth/\({}^{\circ}\) & 50/55/60 \\ Thickness of second-stage brush bundle & 2.0/2.2/2.4 & Radial clearance of grate teeth & 0.3/0.4/0.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of four teeth-three stage brush combination seals. Figure 2: Schematic diagram of the calculated fluid domain of a four-tooth three-stage brush seal combination. their seepage characteristics. Given the fractal nature of porous microstructures and pore size distributions [22; 23; 24], advancements in fractal geometry offer novel approaches to refine porous media models for brush seals (see Appendix A for details of the derivation). In the fractal geometry porous media equations, the effect of complex pores on the fluid is divided into viscous damped dissipation and inertial damped dissipation (see the Appendix A for details of the derivation). Viscous Damping Dissipation control equation is:  
\[\frac{\Delta p_{1}}{l}=\frac{32\mu}{L^{1-D_{T}}}\frac{3+D_{T}-D_{f}}{2-D_{f}} \frac{1-\mathcal{O}}{\mathcal{O}}\frac{v_{s}}{\lambda_{max}^{\gamma(1+D_{T})}} \tag{1}\]  
where \(v_{s}\) is the seepage velocity, m/s; \(\mu\) is the viscous coefficient of the fluid, Pa-s. Equation (1) (Equation (A25) in Appendix A) describes the pressure drop caused by viscous energy loss in porous media flow, i.e., viscous damping dissipation. Compared with the first term of the Ergun equation for porous media [4; 5], this equation eliminates the empirical constant \(C\) and reveals the factors influencing viscous damping-induced losses, including fluid viscosity, pore size, the fractal dimension of capillary tortuosity, and the fractal dimension of the pores. Equation (1) also indicates that the pressure drop becomes zero when porosity \(\mathcal{O}=1\), while approaching infinity as porosity tends to zero, which aligns with the physical reality of brush bundles. The inertial damping dissipation control equation is:  
\[\frac{\Delta p_{2}}{l}=\rho gh_{f}=\left(\frac{3}{2}+\frac{1}{\beta^{4}}-\frac {5}{2\beta^{2}}\right)\frac{\rho v_{s}^{2}}{2\mathcal{O}^{2}}\frac{1}{l}\Big{[} D_{T}L_{0}^{(D_{T}-1)}\overline{\lambda}^{(1-D_{T})}\Big{]}^{2} \tag{2}\]  
where \(l\) is the length of the pore throat, in mm, and \(v_{s}\) is the apparent velocity of the pore throat, in m/s. Equation (2) (Equation (A35) in Appendix A) represents the pressure loss caused by inertial damping dissipation. Compared with the second term of the Ergun equation [4; 5], this equation eliminates empirical constants, assigns explicit physical meanings to each term, and demonstrates that the pressure drop is proportional to the square of velocity.