Study on Interstage Pressure Equalization of Differential Multi-Stage Finger Seal with Structural Design, Flow and Heat Transfer Characteristics

Abstract

To effectively address the issue of premature failure caused by the unbalanced distribution of pressure drops between the stages of a traditional two-stage finger seal, this study proposes a method to improve the pressure drop balance by increasing the protection height of the second stage back plate. We established a new numerical calculation model for a two-stage finger seal, based on the porous media model. After verifying the precision of the model, we conducted a numerical analysis to examine the impact of the protection height of the second stage back plate on the flow and heat transfer characteristics of the two-stage finger seal. We then conducted a differentiated structural design for each stage of the two-stage finger seal. The research results are as follows: the pressure drop at the second stage of the traditional two-stage finger seal exceeds that of the first stage; when the protection height of the second stage back plate of the traditional two-stage finger seal is increased from 1.5 mm to 1.57 mm, forming a two-stage pressure equalizing finger seal structure, the pressure drop between the two stages is balanced, but the leakage is greater than that of the traditional two-stage finger seal; a grate seal structure was arranged between the first and second stages of the two-stage pressure equalizing finger seal to form a two-stage pressure equalizing finger seal with grate teeth, which exhibits significantly lower leakage compared to the two-stage pressure equalizing finger seal. However, the proportion of pressure drop at the first and second stages of the two-stage pressure equalizing finger seal is 36.8% and 42.1%, respectively, while the grate tooth stage accounts for 21.1%, resulting in an imbalanced pressure drop once again. Increasing the protection height of the second stage back plate in the two-stage pressure equalizing finger seal with grate teeth to 1.6 mm results in a 37.5% pressure drop at the first and second stages, and a 25% pressure drop at the grate tooth stage. The two-stage finger seal balances the pressure drop and matches the leakage of the traditional two-stage finger seal. The maximum temperatures of the first and second stages of the finger seal are 0.7% lower and 2.6% higher compared to the traditional two-stage finger seal. This suggests that a differential multi-stage finger seal is the optimal structure.

1. Introduction

The sealing device in aeroengines plays a crucial role. Its sealing performance directly affects the stability, safety, reliability, service life, and economy of the engine [1]. With the continuous advancements in aeroengine technology, more stringent requirements are being placed on sealing devices, making the pursuit of more advanced sealing technologies an urgent issue that needs to be addressed [2]. The finger seal is a novel flexible sealing developed following the grate tooth seal and brush seal. It offers excellent sealing performance, a long service life cycle, low manufacturing costs, and a simple structure, thereby compensating for the shortcomings of the grate tooth seal and brush seal. This innovation is regarded as a revolutionary sealing technology [3]. Experimental studies have shown that compared with the grate tooth seal, the finger seal can reduce the airflow loss by 1% to 2%, thus reducing the fuel loss by 0.7% to 1.4% and operating costs by 0.35% to 0.70%, respectively. Compared with the brush seal, the manufacturing cost of the finger seal is less than 50% of that of the brush seal [4], and there is no excessive wear and tear due to the breakage of the brush filament in the finger seal [5].

Although a finger seal offers many advantages compared with a grate seal and a brush seal, in practical application, the traditional single-stage finger seal tends to experience increased leakage and serious wear under high axial pressure differences [6]. In order to effectively address the limited pressure-bearing capacity of the single-stage finger seal, a method using multi-stage finger seal instead of a single-stage finger seal has been proposed in recent years [7]. Research has shown that in traditional multi-stage finger seals, there is a phenomenon of imbalanced pressure drop across different stages, leading to premature failure of the last-stage finger seal due to excessive pressure [8]. This results in the overall failure of the finger seal and shortens its lifespan, which is detrimental to the performance and economic viability of the multi-stage finger seal. Therefore, studying multi-stage pressure equalizing finger seals is of utmost importance and value [9].

For the study of multi-stage finger seals, Zhao [6] employed a fluid–solid coupling iterative algorithm to investigate the leakage flow and interstage pressure drop characteristics of the two-stage finger seal, proposing improvements for mitigating interstage pressure drop imbalance by reducing the number of second stage finger sealing elements. Zhang [10] used the control variable method to systematically investigate the influence law of multiple structural and operating parameters on the pressure equalization characteristics and sealing performance of the two-stage finger seal by altering parameters individually. To summarize, the existing literature on multi-stage finger seals is primarily aimed at addressing the factors affecting interstage pressure drop distribution, with relatively limited theoretical analysis and research dedicated to resolving the pressure drop imbalance. Additionally, structural modifications in existing studies often overlook the impact of excessive temperatures at the end-stage finger foot and rotor contact surface on sealing performance. Moreover, no studies have been found that focus on the design of multi-stage finger seals aimed at achieving two-stage pressure drop equalization while simultaneously enhancing sealing performance and reducing friction-induced thermal effects.

To address this challenge, this paper presents a numerical investigation and theoretical analysis of the interstage pressure drop imbalance in traditional two-stage finger seals. We propose an approach to equalize the two-stage pressure drop and reduce leakage by increasing the protection height of the back plate in the second stage and incorporating a grate tooth sealing structure between the first and second stages of the finger seal. This design, aimed at achieving balanced pressure distribution across the two stages, enhances sealing performance and minimizes the impact of frictional heat generation. The findings of this study offer a valuable reference for the design and optimization of differential multi-stage finger seal structures.

2. Physical Model of the Finger Seal

The configuration of the finger seal is illustrated in Figure 1a. It primarily consists of a front plate (1), finger element (2), spacer (3), rivet (4), back plate (5), and rotor (6) [11]. The structure of the finger element is shown in Figure 1b. Numerous finger beams with identical shapes are uniformly arranged along the inner circumference of the finger seal to form a circular symmetrical structure [12]. A gap exists between the adjacent finger beam, and the circumferential width of the gap is narrower than that of a single finger beam [13]. This design ensures that upon assembly, the gaps between adjacent finger beams are fully covered by the finger beam of a neighboring finger element, thereby achieving an effective sealing performance [14].

The aforementioned structure represents a single-stage finger seal. Under conditions of high pressure difference, the single-stage finger seal is gradually being replaced by the two-stage finger seal because of its limited pressure-bearing capacity [15]. The traditional two-stage finger seal consists of two single-stage finger seals with identical structural parameters arranged in series, separated by a partition ring, as illustrated in Figure 2.

The axial structure of the traditional two-stage finger seal is depicted in Figure 3. The airflow inflow end represents the upstream high-pressure side, while the first stage, located near the inflow end, constitutes the first stage of the traditional two-stage finger seal. The corresponding area of the separation ring between the two stages forms the middle cavity, with the thickness of the separation ring denoted by S_t, the medium outflow end corresponding to the downstream low-pressure side, where the second stage situated near the outflow end represents the final stage of the traditional two-stage finger seal [16].

To facilitate the calculation, the structural parameters of the first and second stages of the traditional two-stage finger seal are assumed to be identical, and the finger beam profile is arc-shaped, as depicted in Figure 4. The primary structural parameters of the finger seal are presented in

Table 1. Main structural parameters of finger seal.

Structural Parameters		Numerical Value
Base circle diameter	$D_{cc}/\mathrm{m}\mathrm{m}$	43
Inner diameter of seal	$D_i/\mathrm{m}\mathrm{m}$	160
Diameter of upper end of finger foot	$D_f/\mathrm{m}\mathrm{m}$	163
Diameter of root circle of finger beam	$D_b/\mathrm{m}\mathrm{m}$	187
Installation hole uniformly distributed circumferential diameter	$D_e/\mathrm{m}\mathrm{m}$	197
Outer diameter of finger seal	$D_o/\mathrm{m}\mathrm{m}$	207
Arc radius of finger beam	$R_s/\mathrm{m}\mathrm{m}$	85
The angle occupied by a single finger	$\alpha /°$	5
Circumferential angle of finger foot	${\alpha }^{\prime }/°$	4.7
Finger thickness	$b/\mathrm{m}\mathrm{m}$	0.3
Width of finger beam gap	$I_s/\mathrm{m}\mathrm{m}$	0.4
Diameter of mounting hole	${\varphi }_d/\mathrm{m}\mathrm{m}$	4
Thickness of two-stage spacer ring	$S_t/\mathrm{m}\mathrm{m}$	2
Front plate protection height	$H_f/\mathrm{m}\mathrm{m}$	1.5
Back plate protection height	$H_b/\mathrm{m}\mathrm{m}$	1.5
Number of finger beams	$N$	72

.

3. Numerical Model of Finger Seal

3.1. Porous Medium Model

The enclosed section of Figure 5 illustrates the calculation domain of a traditional two-stage finger seal, which includes a high-pressure cavity adjacent to the upstream inlet of the first stage finger seal, a low-pressure cavity close to the downstream outlet of the second stage finger seal, intermediate cavities corresponding to the spacer rings of the two stages, a cavity region corresponding to the protective heights of the first and second stage finger seals’ front and back plate, a cavity region corresponding to a spacer, as well as the finger beam and finger foot portion [17].

The gap between the finger beam of the finger seal functions similarly to the pores of the porous media, facilitating the leakage of gas through the narrow gap in the finger beam under axial pressure, which exhibits the characteristics of porous media permeation flow. In this paper, the corresponding area of the finger beam and the finger foot is treated as a porous media model [18]. The two-stage finger seal demonstrates the structural characteristics of circular symmetry in the circumferential direction, with negligible differences in thermal conductivity and friction factor between the materials of the finger seal and the rotating shaft in this direction. Accordingly, this study establishes the axisymmetric rotation model, shown in Figure 6, to analyze the leakage flow and heat transfer characteristics of the two-stage finger seal [19].

In this paper, the finger seal material is GH605, and the rotor material is K477, whose properties [20] are like those of AMS5537 and Mar-M-247, respectively. The property parameters of the materials can be found in Reference [21].

When the pressure difference between the upstream and downstream is 0.216 MPa, the maximum velocity $\nu$ in the finger seal area is approximately 45 m/s, the finger seal thickness b is 0.3 mm, and the finger beam gap is 0.4 mm. According to the principle of equal area, the square cross-sectional area of the fluid flow through the finger beam gap is converted into an equivalent circular cross-sectional area. Then, the diameter of the circular cross-sectional is as follows:

$$d=2\sqrt{{\displaystyle \frac{I_s\cdot b}{\pi }}}\approx 0.39\left(\mathrm{mm}\right)$$

(1)

According to the formula of Reynolds number:

$$Re={\displaystyle \frac{\nu \times d\times {10}^{-3}}{\gamma }}\approx 1186$$

(2)

In the formula, $\gamma$ represents the dynamic viscosity of the gas in the standard state; the value is 1.48 × 10⁻⁵ m/s². Given that Re < 2000, it can be concluded that flow within the porous media area of the finger seal is laminar [22].

When the porous media model is employed to simulate the flow and heat transfer characteristics within the finger beam and finger foot area, according to Equation (2), it can be known that the flow state in the porous medium area of the finger seal is laminar. Neglecting the structural deformation of the finger beam and finger boot caused by aerodynamic forces and thermal stresses, it is considered that the porosity in the porous medium regions of the finger beam and finger boot is a constant value. Ignoring the variation of material physical property parameters with pressure and temperature, the physical property parameters under standard temperature and pressure conditions are used for the calculation [17].

Due to the large pressure difference between the upstream and downstream of the sealed structure during operation, the density of the fluid passing through the finger seal area will undergo significant changes. Therefore, the leakage fluid is assumed to be compressible ideal air, and it is assumed that steady-state control equations are used in the porous medium region.

Ideal gas state equation:

$$p=\rho RT$$

(3)

In the above formula, $p$ denotes the gas pressure, $\rho$ represents the density of the ideal gas, $R$ is the gas constant, and $T$ is the temperature.

Mass conservation equation:

$${\displaystyle \frac{\partial \left(\rho u_i\right)}{\partial x_i}}=0$$

(4)

In this formula, $x_i$ represents the flow direction, and $u_i$ denotes the velocity vector of the fluid in the direction of $x_i$ (i = 1, 2).

Momentum equation:

$${\displaystyle \frac{\partial \left(\rho u_i{u}_j\right)}{\partial x_j}}=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)\right]+S_i$$

(5)

In this formula, $\mu$ is the molecular viscosity, and S_i represents the additional momentum source term caused by the hindrance of the finger beam or finger foot to the fluid.

For the energy equation, the local thermal equilibrium model is applied because of the large ratio of the wetting area to volume and heat exchange capability of the finger beam, while the work done by the pressure and viscosity is ignored.

$${\displaystyle \frac{\partial \left(\rho c_p{u}_{i}T\right)}{\partial x_i}}={\displaystyle \frac{\partial }{\partial x_i}}\left(k_{eff}{\displaystyle \frac{\partial T}{\partial x_j}}\right)$$

(6)

Where, $c_p$ denotes the specific heat capacity, T represents the gas temperature, and $k_{eff}$ is the effective thermal conductivity [23].

The above Equations (4)–(6) are collectively referred as fluid control equations. The momentum source term $S_i$ in Formula (5) comprises both a viscous loss term and an inertia loss term, which are expressed as:

$$S_i=-\left({\displaystyle \frac{\mu }{\alpha }}{\mu }_i+{\displaystyle \frac{1}{2}}{C}_2\rho \left|\mu \right|{\mu }_i\right)$$

(7)

In the formula, $1/\alpha$ represents the coefficient of viscous loss, and $C_2$ denotes the coefficient of inertia loss. As per Reference [23], according to the formula of brush seal momentum source term:

$$S_i=-\left(5m{\displaystyle \frac{S^2}{{\epsilon }^3}}\mu {\mu }_i+{\displaystyle \frac{n}{8}}{\displaystyle \frac{S}{{\epsilon }^3}}\rho \left|\mu \right|{\mu }_i\right)$$

(8)

Comparative expressions (7) and (8) are available:

$$\{\begin{array}{c}{\displaystyle \frac{1}{\alpha }}=5m{\displaystyle \frac{S^2}{{\epsilon }^3}}\hfill \\ C_2={\displaystyle \frac{n}{4}}{\displaystyle \frac{S}{{\epsilon }^3}}\hfill \end{array}$$

(9)

In Formula (9), m and n represent constants determined by comparison between experiments and numerical calculations [19], which can be expressed as follows:

$$\{\begin{array}{c}m=0.1\hfill \\ n=n_0+{\displaystyle \frac{A}{\sqrt{2\pi }\omega \cdot \mathsf{\Delta }p}}\cdot \exp \left\{\left[-\ln \left(\mathsf{\Delta }p/\mathsf{\Delta }{p}_c\right)/2{\omega }^2\right]\right\}\hfill \end{array}$$

(10)

In this formula, $n_0$ is 0.03343, $\mathsf{\Delta }{p}_c$ is 0.1728, $\omega$ is 1.220224, $A$ is 0.01549, and $\mathsf{\Delta }p$ represents the pressure difference between the upstream and downstream [19].

In Equation (9), $\epsilon$ and $S$ represent the porosity and wetting area per unit volume of porous media, respectively, which can be calculated by the following formula:

$$\{\begin{array}{c}\epsilon ={\displaystyle \frac{V_c}{V}}\hfill \\ S={\displaystyle \frac{S_f}{V}}\hfill \end{array}$$

(11)

In this formula, $V$ represents the volume of a single finger seal, $V_c$ denotes the gap volume between the finger beam of a single finger seal, and $S_f$ refers to the surface area of a single finger seal.

3.2. Heat Transfer Model

As illustrated in Figure 7, a significant amount of heat is generated by the friction between the finger foot and the rotor surface, which is mainly transmitted through the following seven mechanisms: ① Heat conduction between finger seal and rotor; ② Radial heat conduction within the finger seals; ③ Heat conduction between adjacent finger seals; ④ Heat conduction between the finger seal and the back plate; ⑤ Convective heat transfer between the rotor and the gas; ⑥ Convective heat transfer between finger seal and gas; ⑦ Convective heat transfer between front and back plates and gas. In addition, under the influence of axial pressure, secondary friction heat may also occur between the finger seals, the finger seals, and the back plates. However, this is not considered in the analysis due to the negligible amount of heat generated.

The finger foot and the rotor surface are set as the heat flow boundary, with the assumption that the frictional heat flux between them is evenly distributed, accounting for 50% each. The heat flux density $q$ of the friction pair between the finger foot and the rotor surface is expressed as follows [24]:

$$q={\displaystyle \frac{fV{k}_i\left(s+s_{\mathsf{\Delta }p}\right)}{A_i}}$$

(12)

According to the literature [25], in the formula, $f$ is the friction factor between the finger foot and the rotor surface, with a value of 0.2; $k_i$ is the radial stiffness of a single finger beam, with a value of 489.57 N/mm; $A_i$ is the contact area between a single finger foot and the surface of the rotor, with a value of 1.965 mm²; and $s$ is the interference of the fit between the finger foot and the rotor surface, with a value of 0.1 mm. $V$ represents the linear velocity on the surface of the rotor (m/s) and $s_{\mathsf{\Delta }p}$ is the radial displacement of a single finger foot caused by the pressure difference between the upstream and downstream. Due to the small protection height downstream of the finger seal, the radial displacement of the finger foot is of a negligible order of magnitude and can be ignored. Therefore, it can be considered that the friction heat between the finger foot and the rotor surface primarily arises from the interference between the finger foot and the rotor [25]. Consequently, the heat flux density $q$ of the friction pair between the finger foot and the rotor surface can be expressed as follows:

$$q={\displaystyle \frac{fV{k}_{i}s}{A_i}}$$

(13)

3.3. Gridding

Given the complexity of the flow conditions around the finger seals’ finger beam, finger foot, and the fluid domain near the rotor surface, the mesh in these regions is refined to improve the grid quality and calculation accuracy. In addition, in the axial direction, the flow situation in the downstream outlet fluid domain is more complex than that in the upstream inlet fluid domain. Therefore, the downstream outlet fluid domain is also finely meshed [26], as illustrated in Figure 8.

3.4. Boundary Conditions

In this paper, the finger seal region is calculated using Equation (3). It is known that the Reynolds number is small, and the laminar flow model is adopted. According to the actual working conditions of the finger seal, the pressure boundary condition is adopted at the inlet and outlet of the convective domain, with the outlet pressure of 0.1 MPa, and the inlet pressure is determined by the outlet pressure and the axial pressure difference, with the axial pressure difference of 0.216 MPa. The temperature of the upstream inlet and downstream outlet is set to 300 K [27]. Because the two-stage finger seal structure is a cyclic symmetrical structure, the front and rear surfaces are set as periodic boundaries, the contact surface between fingers and rotor is set as heat flow boundary, and the heat flux density is q. A velocity boundary condition is applied to the contact surface with the rotor, the other surfaces are modeled as non-slip wall, and the speed ω is specified. Moreover, k-ε RNG turbulence model and SIMPLE algorithm, based on velocity-pressure coupling, are selected for the numerical solution [28].

3.5. Verification of the Accuracy of Numerical Models

In this paper, the leakage rate obtained from the experiment is compared with the numerical results to validate the accuracy of the numerical model. The structural parameters of the finger seal are consistent with those provided in reference [10]. Figure 9 illustrates a comparison between the calculated results from the numerical model in this paper and the experimental results in reference [10].

As depicted Figure 9, the leakage rates derived from the two numerical models and the experiment exhibit an increasing trend with the increase in the axial pressure difference across the two-stage finger seal. When the axial pressure difference is below 0.06 MPa, the experimentally observed leakage rate exceeds the rate predicted by the numerical model, and the variation in leakage rate shows a tendency to stabilize. This discrepancy is primarily attributed to the initial installation gap of 0.011 and 0.032 mm between the first stage and the second stage of the two-stage finger seal and the rotor, respectively [10], which leads to a persistent leakage gap between the finger seal and the rotor.

At the pressure difference of 0.02 MPa and 0.18 MPa, the deviation between the calculated experimental values reaches its maximum, with a relative error of 6.931% and 5.604%, respectively. The reason for this deviation lies in the fact that when the axial pressure difference is below 0.06 MPa, the drag coefficient in the numerical calculation is overestimated, resulting in a predicted leakage rate that is lower than the experimental value. Conversely, when the pressure difference exceeds 0.1 MPa, the experimental values become lower than the predicted ones as the axial pressure increases. This reversal is primarily due to the pressure course effect observed during the experiment, wherein the radial clearance between the finger seal and the rotor diminishes, reducing leakage as a result of rotor deformation induced by thermal effects. Consequently, the experimental leakage rate is less than the calculated leakage rate. However, across the entire range of pressure differences, the relative error in the leakage rate remains within 10%, thereby affirming the accuracy of the numerical method and the calculation model within an acceptable range.

Leakage and maximum temperature are important indicators for evaluating the sealing performance of finger seals. Figure 10 shows the variation of leakage and the maximum temperature of the two-stage finger seals with respect to the number of grids.

When the number of grids exceeds 80,000, the calculated results of leakage and maximum temperature tend to stabilize. Considering the quality of meshing, calculation accuracy, and efficiency, the number of grids is set to 80,000 for numerical calculation.

4. Design of Pressure Drop Equalization Structure Between Two-Stage Finger Seals

4.1. Effect of Front Plate Protection Height on Pressure Drop Equalization Between Two-Stage Finger Seals

Figure 11 shows the proportion of pressure drop at all levels of the two-stage finger seal when the rotor speed is 9000 r/min and the axial pressure difference is 0.216 MPa, the front plate protection height is 1.5 mm, 3 mm, 4.5 mm, 6 mm, 7.5 mm, respectively, while the back plate protection height is maintained at 1.5 mm. As illustrated in Figure 11, the axial pressure drop in the second stage of all five two-stage finger seal structures is greater than that in the first stage; the proportion of the first stage pressure drop ranges from 36.45~36.73%, while that of the second stage pressure drop is 63.26~63.54%. This indicates that the protection height of the front plate has no significant effect on the pressure drop equalization between the two-stage finger seals.

4.2. Effect of Front Plate Protection Height on Leakage of Two-Stage Finger Seal

Figure 12 illustrates the influence of the protection height of the front plate on the leakage of the two-stage finger seal when the rotor speed is 9000 r/min, and the axial pressure difference is 0.216 MPa. The curves a, b, c, d, and e represent the leakage rates of the two-stage finger seal when the protection height of the front plate is 1.5 mm, 3 mm, 4.5 mm, 6 mm, and 7.5 mm. As shown in the figure, the front plate protection height has no significant effect on the leakage of the two-stage finger seal. In conjunction with Section 3.1, in order to reduce the overall seal weight of the two-stage finger seal while ensuring its structural integrity, the front plate protection height is increased to 7.5 mm.

4.3. Theoretical Analysis of the Influence of Back Plate Protection Height on the Pressure Drop Equalization of Two-Stage Finger Seal

As shown in Figure 13, the fluid parameters at the inlet and outlet of the two-stage finger seal are depicted. Assuming that the ideal gas under study is not affected by gravity and flows steadily, it can be derived from the Bernoulli equation that the pressure drop of the first finger seal $\mathsf{\Delta }{p}_1$ is [9]:

$$\mathsf{\Delta }{p}_1={\displaystyle \frac{1}{2}}{\rho }_2{\nu }_2-{\displaystyle \frac{1}{2}}{\rho }_1{\nu }_1$$

(14)

The relationship between the mass flow rate and the volume flow rate of the gas in the finger seal structure is expressed as:

$$M=\rho Q=\rho \nu A$$

(15)

In the formula, $M$ is the mass flow rate of gas, $Q$ is the volume flow of the gas, and $A$ is the effective flow area.

By simultaneously solving Equations (14) and (15), the following can be obtained:

$$\{\begin{array}{c}\mathsf{\Delta }{p}_1={\displaystyle \frac{M}{2}}\left({\displaystyle \frac{Q_2}{A_2^2}}-{\displaystyle \frac{Q_1}{A_1^2}}\right)\hfill \\ \mathsf{\Delta }{p}_2={\displaystyle \frac{M}{2}}\left({\displaystyle \frac{Q_4}{A_4^2}}-{\displaystyle \frac{Q_3}{A_3^2}}\right)\hfill \end{array}$$

(16)

Among them, $\mathsf{\Delta }{p}_1$ represents the pressure drop across the first finger seal and $\mathsf{\Delta }{p}_2$ represents the pressure drop across the second finger seal, where A₁ = A₃ and A₂ = A₄.

The pressure drop equalization coefficient $C$ between the second stage and the first stage of the two-stage finger seal is defined as:

$$C={\displaystyle \frac{\mathsf{\Delta }{p}_2}{\mathsf{\Delta }{p}_1}}={\displaystyle \frac{{\left(\frac{A_1}{A_2}\right)}^2{Q}_4-Q_3}{{\left(\frac{A_1}{A_2}\right)}^2{Q}_2-Q_1}}$$

(17)

When the pressure drop distribution between the two stages is more balanced, the $C$ value is closer to 1.

The ratio of the entrance to outlet cross-sectional area of the two-stage finger seal is defined as proportional coefficient $Y$, which is expressed as follows:

$$Y={\displaystyle \frac{A_1}{A_2}}$$

(18)

We know that $Y$ is a number greater than 1. Therefore, Formula (17) can be expressed as:

$$C={\displaystyle \frac{\mathsf{\Delta }{p}_2}{\mathsf{\Delta }{p}_1}}={\displaystyle \frac{Y^2{Q}_4-Q_3}{Y^2{Q}_2-Q_1}}$$

(19)

The first stage gas volume flow increment $\mathsf{\Delta }{Q}_1$ in Figure 13 is expressed as:

$$\mathsf{\Delta }{Q}_1=Q_2-Q_1$$

(20)

Similarly, the second stage gas volume flow increment $\mathsf{\Delta }{Q}_2$ is expressed as:

$$\mathsf{\Delta }{Q}_2=Q_4-Q_3$$

(21)

Figure 14 illustrates the volume flow curve of each section of the two-stage finger seal when the axial pressure difference is 0.216 MPa and the rotor speed is 9000 r/min.

It is illustrated in Figure 14 that the volume flow in the second stage of the two-stage finger seal is larger than that in the first stage, which can be expressed as:

$$\mathsf{\Delta }{Q}_2>\mathsf{\Delta }{Q}_1$$

(22)

By simultaneously solving Equations (20)–(22), we have:

$$Q_4-Q_2>Q_3-Q_1$$

(23)

Since Y is a number greater than 1, then:

$$Y^2\left(Q_4-Q_2\right)-\left(Q_3-Q_1\right)>0$$

(24)

That is:

$$Y^2{Q}_4-Q_3>Y^2{Q}_2-Q_1$$

(25)

By simultaneously solving Equations (19) and (24), we obtain the following:

$$\{\begin{array}{c}\mathsf{\Delta }{p}_2>\mathsf{\Delta }{p}_1\hfill \\ C>1\hfill \end{array}$$

(26)

That is, the pressure drop of the two-stage finger seal is unbalanced, with the second stage experiencing a larger pressure drop than the first stage. This imbalance leads to the premature failure of the second stage finger seal due to excessive pressure, thereby affecting the service life of the entire two-stage finger seal [29].

4.4. Pressure Equalizing Design of Two-Stage Finger Seal

As seen from Section 3.3 and Equation (16), with the increase in the cross-sectional area of the second stage outlet channel A₄, the second stage pressure drop Δp₂ decreases. According to the pressure drop equilibrium coefficient C = Δp₂/Δp₁ from Formula (17), the pressure drop equilibrium coefficient C approaches 1 from a value greater than 1, indicating that the pressure drop of the two stages tends to balance. Figure 15 illustrates the design scheme for the pressure equalizing structure of the two-stage finger seal. If the protection height of the first and second stage front plates are H_ff and H_sf respectively, and the back plate protection heights are H_fb and H_sb, then H_ff = H_sf, H_fb < H_sb. If the cross-sectional areas of the first and second stage finger seals are A₂ and A₄’, respectively, then A₂ = A₄ for the first stage, and for the second stage, A₄ seal A₄’ = A₄ + ΔA₄. Thus, A₄’ > A₂. It can be seen that, in order to balance the pressure drop across the two stages, the improved structure of the two-stage finger seal increases the protection height of the second stage back plate.

5. Analysis of Interstage Pressure Drop and Leakage Flow Characteristics of Two-Stage Pressure Equalizing Finger Seal

According to the theoretical analysis of Section 3.3, increasing the protection height of the second stage back plate in the two-stage finger seal can effectively mitigate the imbalance in the pressure drop between the two stages. Therefore, this paper constructs five kinds of two-stage finger seal structures with a gradual increase in the protection height of the second stage back plate, as shown in

Table 2. Two-stage finger seals with five different back plate protection heights.

Sealing Structure	First Stage Finger Seal Back Plate Protection Height/mm	Second Stage Finger Seal Back Plate Protection Height/mm
A	1.5	1.50
B	1.5	1.53
C	1.5	1.55
D	1.5	1.57
E	1.5	1.60

. The A structure represents a traditional two-stage finger seal structure.

Figure 16 presents the pressure drop equalization coefficient C for five two-stage finger seals listed in Table 2.

As observed in Figure 16, the pressure drop equalization coefficient C of the two-stage finger seal decreases gradually with the increase in the protection height of the back plate. The analysis indicates that increasing the protection height of the second stage back plate in the two-stage finger seal enhances the cross-sectional area of the outlet channel, reduces the increase in the volume flow in the second stage, and consequently reduces the difference in volume flow between the inlet and outlet, as well as the pressure drop, in the second stage. This potentially makes it even smaller than that in the first stage. Therefore, increasing the protection height of the second stage back plate can effectively balance the pressure drop between the two stages of the two-stage finger seal.

5.1. Two-Stage Pressure Drop Analysis of Two-Stage Finger Seal Structure

The proportion of interstage pressure drop is defined as the ratio of the first stage upstream and downstream pressure drop to the total pressure drop across the two stages. Figure 17 illustrates the interstage pressure drop ratio of the five two-stage finger seals, shown in Table 2, when the axial pressure difference is 0.216 MPa. As observed in Figure 17, with the increase in the protection height of the second stage back plate, the proportion of pressure drop in the first stage increases gradually, while that in the second stage decreases gradually. When the protection height of the second stage back plate is less than 1.57 mm, the proportion of pressure drop in the first stage is less than that of the second stage. However, when the protection height of the back plate increases to 1.6 mm, the proportion of the pressure drop of the first stage becomes larger than that in the second stage. When the protection height of the back plate is 1.57 mm, the pressure drop in the first and the second stages of the two-stage finger seal is exactly balanced, forming a two-stage pressure equalizing finger seal, which is designated as the D structure.

5.2. Analysis of Leakage Characteristics of Two-Stage Finger Seal Structure

Figure 18 illustrates the variation in leakage of the five two-stage finger seals, shown in Table 2, with respect to the axial pressure difference, when the rotor speed is 9000 r/min and the axial pressure difference is 0.216 MPa.

As observed in Figure 18, the leakage of A, B, C, D, and E structures increases approximately linearly with the increasing of the axial pressure difference across the entire pressure difference range. Under the same axial pressure difference, the leakage increases noticeably with the increase in the protection height of the second stage back plate. This trend occurs because, in a traditional two-stage finger seal, the height of the finger foot is equal to the protection height of the back plate, and the larger circumferential angle of the finger foot corresponds to a greater axial compressive stiffness compared to the finger beam part. However, when the protection height of the back plate increases slightly, the axial compressive stiffness of the finger foot in the corresponding area decreases significantly, resulting in a significant increase in leakage compared with that before the improvement of the back plate. Nevertheless, when the axial pressure difference is small, the axial compressive stiffness of the finger seal is insufficient to cause a difference in the leakage among the B, C, D, and E structures in the low-pressure stage.

Figure 19 shows a comparison of leakage between single-stage finger seals and two-stage finger seals with the same finger seal structural parameters. As seen in Figure 19, under the same axial pressure difference, the leakage of A and D structures is less than that of single-stage finger seal, and with the increases in axial pressure difference, the rate of leakage increase in these two structures is slower than that of the single-stage finger seal. Therefore, a two-stage finger seal can effectively reduce the leakage that occurs in a single-stage finger seal due to excessive axial pressure difference.

5.3. Velocity Field and Temperature Field Analysis of Two-Stage Finger Seal Structure

As a critical component of an aeroengine, the finger seal should not only provide an effective sealing performance, but also ensure a long service life. During operation, the friction between the finger seal and the high-speed rotor surface generates substantial heat, which can significantly impact the performance and longevity of the finger seal. Therefore, it is necessary to analyze the velocity field and temperature field of the two-stage finger seal.

5.3.1. Velocity Field Analysis of Two-Stage Finger Seal

Figure 20 shows the velocity cloud diagram of the A and D structures when the axial pressure difference is 0.216 MPa and the rotor speed is 9000 r/min. As observed in Figure 20a, when the protection height of the second stage back plate is 1.5 mm, the maximum gas velocity driven by the rotor appears near the rotor surface and at the lower left corner of the second stage back plate. The two stages flow out in a jet shape at the lower left corner of the back plate. The gas flows out in a jet shape at the lower left corner of both the first and second stage back plates, with the jet velocity in the second stage being higher than that in the first stage. An obvious vortex forms between the intermediate cavity and the second stage finger seal, causing gas reflux at the outlet of the second stage. As can be seen in Figure 20b, when the protection height of the second stage back plate is increased to 1.57 mm, the gas still flows out in a jet shape at the lower left corner of the first and second stage back plate. However, unlike the results in Figure 20a, the maximum gas velocity appears at the lower left corner of the second stage back plate, and the gas reflux phenomenon increases at the second stage exit.

Table 3. Maximum velocity leakage flow of finger seals before and after the improvement of the second stage back plate.

Sealing Structure	The Maximum Velocity of Leakage Flow of the First Stage/(m/s)	The Maximum Velocity of Leakage Flow of the Second Stage/(m/s)
Traditional two-stage finger seal (A structure)	29.41	75.40
Two-stage pressure equalizing finger seal (D structure)	34.20	142.48

presents the maximum velocity leakage flow at all levels of the A and D structures. It is known that when the protection height of the second stage back plate increases from 1.5 mm to 1.57 mm, the maximum velocity leakage flow in the first stage finger seal increases from 29.41 m/s to 34.20 m/s, representing an increase of 16.29%. The maximum velocity leakage flow in the second stage finger seal increases from 75.40 m/s to 142.48 m/s, marking an 88.97% increase. As a result, the gas reflux phenomenon at the second stage exit is intensified.

5.3.2. Temperature Field Analysis of Two-Stage Finger Seal

Figure 21 shows the temperature distribution of the A and D structures of the two-stage finger seal when the axial pressure difference is 0.216 MPa, the inlet and outlet temperature is 300 K, the rotor speed is 9000 r/min, and the installation interference is 0.1 mm. As observed in Figure 21, the highest temperature occurs on the contact surface between the second stage finger foot and the rotor. This is primarily because the high-temperature gas, produced by the friction between the first stage finger foot and the rotor surface, flows to the second stage under the influence of axial pressure. After the calorimetric heat is produced by the friction between the second stage finger foot and the rotor surface superimposed on the heat from the first stage, the gas temperature on the downstream surface of the contact area between the second stage finger foot and the rotor becomes obviously higher than that of the first stage. As Figure 21a shows, when the protection height of the second stage back plate is 1.5 mm, the maximum gas temperature of the A structure shows a more regular ladder distribution in the radial direction. As shown in Figure 21b, when the back plate protection height increases to 1.57 mm, the maximum gas temperature distribution in the D structure gradually converges toward the rotor surface, forming a closed shape.

Table 4. Maximum gas temperature at all levels before and after the improvement of the second stage back plate.

Sealing Structure	The Maximum Temperature of the First Stage Finger Seal (K)	Maximum Temperature of Second Stage finger Seal (K)
Traditional two-stage finger seal (A structure)	560	636
Two-stage pressure equalizing finger seal (D structure)	541	658

shows the maximum gas temperature at the contact surface between the first stage and the second stage finger foot and rotor in the A and D structures. As observed in Table 4, when the protection height of the second stage back plate increases from 1.5 mm to 1.57 mm, the maximum temperature at the first stage finger seal contact surface decreases from 560 K to 541 K, a decrease of 3%, while the maximum contact surface temperature of the second stage finger seal increases from 636 K to 658 K, an increase of 3%. The analysis indicates that the increase in the protection height of the second stage back plate leads to an increase in the leakage at both the first stage and the second stage of the two-stage finger seal. The increased leakage in the first stage carries away more friction heat generated between the finger foot and the surface of the rotor, resulting in a decrease in the maximum temperature of the first stage finger seal. Conversely, the increased gas in the second stage flows toward the rotor surface, leading to an "encircling" effect of the downstream airflow in the contact area between the finger foot and the rotor. This effect hinders the discharge of high-temperature gas downstream of the second stage, thereby increasing the maximum temperature of the second stage finger seal.

6. Analysis of Leakage Flow Characteristics of Two-Stage Pressure Equalizing Finger Seal with Grate Teeth

6.1. Structural Design of Two-Stage Pressure Equalizing Finger Seal with Grate Teeth

From the above analysis, it can be observed that when the second stage back plate protection height H_sb of the two-stage finger seal is 1.57 mm, the pressure drop between the two stages is balanced. However, compared with a back plate protection height H_sb of 1.5 mm, the leakage increases significantly. This indicates that while increasing the second stage back plate protection height can balance the pressure drop between the two stages, it comes at the expense of sealing performance.

In order to realize a balanced distribution of pressure drop between stages, thereby improving the service life and the sealing performance, this paper draws inspiration from the design scheme of a multi-stage brush seal [30]. We replaced the spacer ring between the two-stage pressure equalizing finger seals (D structure) with a grate teeth structure to form the F structure, as shown in Figure 22. The design requirements of the grate teeth structure adopted in this paper are as follows: ① the oblique grate seal groove on the front-end face of the grate sealing ring is inclined toward the inner diameter of 45°, ② the axial spacing between the grate teeth and the first finger seal’s back plate and the second finger seal’s front bezel is 0.5 mm [30].

6.2. Analysis of Leakage Characteristics of Two-Stage Pressure Equalizing Finger Seal with Grate Teeth

Figure 23 shows the effect of axial pressure difference on the leakage characteristics of the F structure in a two-stage pressure equalizing finger seal with grate teeth. As observed from the figure, incorporating the grate teeth structure between the two stages of the two-stage pressure equalizing finger seal can effectively reduce the leakage and improve its sealing performance.

6.3. Pressure Drop Analysis of Two-Stage Pressure Equalizing Finger Seal with Grate Teeth

When the axial pressure difference is 0.216 MPa, Figure 24a,b show the axial pressure distribution of the two-stage pressure equalizing finger seal and the two-stage pressure equalizing finger seal with grate teeth, respectively.

As observed in Figure 24a, the axial pressure drop of the two-stage pressure equalizing finger seal primarily occurs in the first stage area, with a larger pressure gradient near the back plate. In contrast, as shown in Figure 24b, the axial pressure drop of the two-stage pressure equalizing finger seal with grate teeth (F structure) occurs not only in the first stage and second stage area, but also in the grate teeth area.

When the axial pressure difference is 0.216 MPa, Figure 25 shows the proportion of the pressure drop of the two-stage pressure equalizing finger seal (D structure) and the two-stage pressure equalizing finger seal with grate teeth (F structure). As observed in Figure 25, when there is no grate seal structure between the two stages, the pressure drop of the first stage and the second stage each account for 50%, achieving a balanced pressure drop. However, when the grate teeth structure is added between the two stages, the pressure drop at the grate teeth accounts for 21.1% of the total pressure drop of the two-stage pressure equalizing finger seal, with the first stage finger seal pressure drop accounting for 36.8%, and the second stage finger seal pressure drop accounting for 42.1%. This indicates that while the addition of the grate seal structure can enhance the sealing performance of the two-stage pressure equalizing finger seal, it also disrupts the original balance of the two-stage pressure drop, causing the second stage pressure drop to become greater than that of the first stage.

6.4. Analysis of Velocity Field and Temperature Field of Two-Stage Pressure Equalizing Finger Seal with Grate Teeth

6.4.1. Velocity Field Analysis of Two-Stage Pressure Equalizing Finger Seal with Grate Teeth

When the axial pressure difference is 0.216 MPa and the rotor speed is 9000 r/min, Figure 26 shows the velocity cloud map of the two-stage pressure equalizing finger seal with grate teeth. As observed in the figure, the maximum velocity of leakage flow in the two-stage pressure equalizing finger seal with grate teeth appears at the gap between the grate tooth and the rotor, flowing out in a jet form between the top of the grate tooth and the surface of the rotor. Compared with the two-stage pressure equalizing finger seal (D structure), the two-stage pressure equalizing finger seal with grate teeth creates a more pronounced vortex between the grate tooth seal and the second stage finger seal, while the gas reflux phenomenon is weakened in the outlet area of the second stage finger seal.

Table 5. Maximum Velocity of Leakage Flow at Different Levels of D Structure and F Structure.

Sealing Structure	The Maximum Velocity of Leakage Flow of the First Stage/(m/s)	The Maximum Velocity of Leakage Flow of the Second Stage/(m/s)	The Maximum Velocity of Leakage Flow of the Grate Class/(m/s)
Two-stage pressure equalizing finger seal (D structure)	34.20	142.48	/
Two-stage pressure equalizing finger seal with grate teeth (F structure)	21.68	118.26	197.11

presents the maximum velocity of leakage flow at all levels of the two-stage pressure equalizing finger seal and two-stage pressure equalizing finger seal with grate teeth. It can be observed that with the introduction of the grate teeth structure, the actual cross-sectional area of the gas flow in the middle cavity is sharply reduced. The grate teeth structure exerts a certain "blocking" effect on the axial airflow between the first stage finger seal and the second stage finger seal, which hinders gas leakage in the two-stage finger seal. Consequently, the maximum velocity of leakage flow in the first stage finger seal decreases from 34.20 m/s to 21.68 m/s, a reduction of 36.61%. Similarly, the maximum velocity of leakage flow of the second stage finger seal decreases from 142.48 m/s to 118.26 m/s, indicating a reduction of 17%. The gas jet velocity at the surface of the grate tooth and the rotor reaches 197.11 m/s, and a jet gas forms an obvious vortex under the hindrance of the second stage finger seal, leading to the weakening of the gas reflux phenomenon in the outlet area due to the decrease in maximum airflow velocity of the second stage finger seal.

6.4.2. Temperature Field Analysis of Two-Stage Pressure Equalizing Finger Seal with Grate Teeth

Figure 27 shows the temperature distribution cloud diagram of the two-stage pressure equalizing finger seal with grate teeth, when the axial pressure difference is 0.216 MPa and the rotor speed is 9000 r/min. As observed in Figure 27, the maximum temperature of the two-stage pressure equalizing finger seal with grate teeth appears on the contact surface between the second stage finger foot and the rotor, and its maximum gas temperature distribution is similar to that of the two-stage pressure equalizing finger seal (D structure). However, the cavity temperature of the two-stage pressure equalizing finger seal with grate teeth near the front plate and rotor surface of the second stage finger seal is noticeably higher than that of the two-stage pressure equalizing finger seal. On the other hand, the highest gas temperature at the contact between the second stage finger foot and the rotor surface is lower than that of the two-stage pressure equalizing finger seal.

Table 6. Effect of grate structure on the maximum temperature of two-stage pressure equalizing finger seal.

Sealing Structure	The Maximum Temperature of the First Stage Finger Seal (K)	Maximum Temperature of Second Stage Finger Seal (K)
Two-stage pressure equalizing finger seal (D structure)	541	658
Two-stage pressure equalizing finger seal with grate teeth (F structure)	557	646

shows the maximum temperature at all levels of the two-stage pressure equalizing finger seal structure and the two-stage pressure equalizing finger seal with grate teeth structure. It is observed in the table that after adding the grate seal structure, the maximum temperature of the first stage increased from 541 K to 557 K, an increase of 3%, while the maximum temperature of the second stage decreased from 658 K to 646 K, a decrease of 2%. The analysis indicates that the grate seal structure can block the axial high-temperature airflow of the first stage, which hinders the discharge of the high-temperature gas generated by the friction between the first stage finger seal and the rotor surface, which increases the maximum temperature of the first stage. The high-temperature airflow produced by the first stage finger seal forms a stable and large vortex as it moves from the small cross-section jet between the grate seal and the rotor surface to the cavity area corresponding to the front plate and the rotor surface of the second stage finger seal. Under the influence of gas convection, the temperature in the cavity area near the front plate and the rotor surface of the second stage finger seal increases and the temperature of the airflow entering the second stage decreases, resulting in the lower maximum temperature of the second stage finger seal.

7. Analysis of Leakage and Flow Characteristics of Differential Multi-Stage Finger Seal

From the analysis in Section 5, it can be observed that incorporating a grate teeth structure into the two-stage pressure equalizing finger seal structure can improve the sealing performance. However, this also disrupts the balance of the pressure drop between the first and second stages. It has been found that when the protection height of the second stage back plate in a two-stage pressure equalizing finger seal H_sb is 1.6 mm, and a grate teeth structure is set up between the two stages, a differential multi-stage finger seal structure G is formed. This configuration can rebalance the pressure drop between the first stage and the second stage of the finger seal while also improving the sealing performance compared to the two-stage pressure equalizing finger seal.

7.1. Analysis of Interstage Pressure Drop of Differential Multi-Stage Finger Seal

Figure 28 shows the axial pressure distribution curves for the second stage back plate protection height H_sb of 1.5 mm and 1.57 mm for the two-stage finger seal structures A and D, when the axial pressure difference is 0.216 MPa and rotor speed is 9000 r/min. Additionally, it shows the curves for the second stage back plate protection height H_sb of 1.57 mm and 1.6 mm.

As observed in Figure 28, the pressure distribution in the front plate of the second stage finger seal and the corresponding cavity area of the rotor surface in the two types of grate tooth combination two-stage finger seals shows a pressure increase in the axial direction. Combined with the analysis of Figure 26, it is evident that this phenomenon occurs because the grate teeth structure constricts the gas flow channel downstream of the first stage. The gas is then expelled from the grate seal and the rotor micro-section as a higher-speed jet, which is subsequently redirected by the second stage finger seal, leading to rapid backflow due to the obstruction.

Figure 29 illustrates the distribution of pressure drop across the two-stage finger seals with different structures mentioned earlier. In the F structure, the pressure drop of the first stage finger seal and second stage finger seal each accounts for 36.8% and 42.1%, respectively, with the grate teeth accounting for 21.1%. When analyzed in conjunction with Figure 28, it is apparent that the axial pressure in both the first stage and second stage finger seals of the F structure is lower than that in the A structure. In the G structure, the pressure drop of the first stage finger seal and second stage finger seal each accounts for 37.5%, with the grate teeth accounting for 25%. At this point, the pressure drop between the first stage and second stage finger seals is effectively balanced.

7.2. Analysis of Leakage Characteristics of Differential Multi-Stage Finger Seal

Figure 30 presents the leakage curve of A, D, F, and G structures as a function axial pressure difference when the rotor speed is 9000 r/min.

As observed in Figure 30, the leakage rates of A, F, and G structures are almost identical under the same axial pressure difference, all of which are lower than that of the D structure. This indicates that the differential multi-stage finger seal G structure not only achieves a balanced pressure drop distribution between the two stages, but also maintains a leakage rate comparable to that of the A structure.

7.3. Velocity Field and Temperature Field Analysis of Differential Multi-Stage Finger Seal

7.3.1. Velocity Field Analysis of Differential Multi-Stage Finger Seal

Figure 31 shows the velocity distribution of the differential multi-stage finger seal. As observed in the figure, the velocity distribution of the differential multi-stage finger seal is similar to that of the two-stage pressure equalizing finger seal with grate teeth. The maximum velocity of leakage flows out in a jet form from the grate teeth and the rotor surface, forming a distinct vortex in the cavity area between the front plate of the second stage finger seal and the rotor surface. The jet gas then flows toward the rotor surface at the lower left corner of the back plate of the second stage finger seal.

Table 7. Maximum velocity of leakage flow at all levels of A, D, F, and G structures.

Sealing Structure	The Maximum Velocity of Leakage Flow of the First Stage/(m/s)	The Maximum Velocity of Leakage Flow of the Second Stage/(m/s)	The Maximum Velocity of Leakage Flow of the Grate Class/(m/s)
Traditional two-stage finger seal ( A structure)	29.41	75.40	/
Two-stage pressure equalizing finger seal (D structure)	34.20	142.48	/
Two-stage pressure equalizing finger seal with grate teeth (F structure)	21.68	118.26	197.11
Differential multi-stage finger seal (G structure)	28.98	124.21	207.01

shows the maximum velocity of leakage flow at all levels of A, D, F, and G structures. Comparing the A structure with the D structure and adding the protection height of the second stage finger seal results in an increase in the maximum gas velocity of both the first stage and the second stage finger seal. Through the comparison of the D structure, the F structure and the G structure, it is evident that the maximum gas velocity appears between the grate seal and the rotor after the introduction of the grate seal structure, leading to a reduction in the maximum gas velocity for both the first stage and the second stage finger seals. Notably, the maximum gas velocity in the G structure is slightly higher than that of the F structure.

7.3.2. Temperature Field Analysis of Differential Multi-Stage Finger Seal

Figure 32 shows the temperature distribution cloud map of the gas from inlet to outlet in the differential multi-stage finger seal G structure. The maximum temperature appears on the contact surface between the second stage finger foot and the rotor. As shown in Figure 33, it is the maximum temperature of the first stage and the second stage of structures A, D, F, and G. From the graph, it can be observed that the maximum temperature at the first stage of the G structure is lower than that first stage of the F structure, while the maximum temperature at the second stage of the G structure is higher than that of the second stage of the F structure. The analysis of the reasons from Table 7 shows that the maximum flow velocity of each stage of structure G is greater than that of structure F. In structure G, the increased flow velocity at the first stage takes away part of the frictional heat between the finger foot and the rotor surface, thereby reducing the maximum gas temperature at the first stage. However, due to the increase in the protection height of the back plate at the second stage, the jet velocity of the second stage seal toward the rotor surface is elevated, and the downstream gas containment effect is stronger, resulting in a slight increase in the maximum temperature of the second stage finger seal in the G structure.

8. Conclusions

This paper conducts a numerical study on the flow and heat transfer characteristics of the two-stage finger seal, validating the accuracy of the numerical model by comparing the numerical leakage results of the two-stage finger seal with existing experimental data. We investigate the impact of the back plate structure on the pressure drop, flow, and heat transfer characteristics of the two-stage finger seal under various operating conditions. We propose measures to improve the imbalance in the pressure drop of the two-stage finger seal. The major results are as follows:

• The traditional two-stage finger seal structure exhibits a significant imbalance in interstage pressure drops; the pressure drop proportions of the two stages are 36.7% and 63.3%, respectively. Under the same operating conditions, increasing the second stage back plate protection height can effectively improve the imbalance of the interstage pressure drop, but at the same time, it will increase the leakage of the seal structure.
• The maximum temperature and velocity are higher at the second stage compared to the first in all two-stage seal structures. The leakage rate of the F structure is significantly lower than that of the D structure.
• The protection height of the second stage back plate of the two-stage pressure equalizing finger seal with grate teeth (F structure) should be increased to 1.6 mm. This will form a differential multi-stage finger seal (G structure), where the pressure drop proportions of the first stage and the second stage are both 37.5%, and the proportion of the grate-stage is 25%, resulting in improved pressure drop equalization. The G structure shows no significant difference in leakage rate and the maximum temperature when compared to the traditional two-stage seal. The above research demonstrates the structural superiority of the differential multi-stage finger seal.