Study on the Thermal Deformation of Finger Seals Based on Local Thermal Non-Equilibrium in Porous Media

Abstract

Finger seals operate over extended periods under complex conditions involving high-pressure differentials, elevated rotational speeds, and rotor radial runout. Intense convective heat transfer arises within the seal, significantly impacting its structural deformation. To elucidate the influence of temperature on finger-seal deformation during convective heat transfer, the present study derives heat transfer energy equations for finger seals based on the Local Thermal Non-Equilibrium (LTNE) model. A three-dimensional porous-media flow-field model incorporating the LTNE framework, along with a solid thermal-deformation model, is developed. The effects of pressure differential and interference-fit magnitude on the structural deformation and average contact pressure of finger seals are analyzed under both the Local Thermal Equilibrium (LTE) and LTNE models. The results indicate that the LTNE model predicts a higher maximum seal temperature and a lower leakage rate compared to the LTE model. In both models, the deformation of individual seal-blade layers increases with rising pressure differentials and interference-fit magnitudes. Furthermore, the overall blade deformation is more pronounced under the LTNE model, suggesting a substantial thermal influence on sealing performance. The effects of pressure difference and interference fit on the thermal deformation of the seal plate are similar: both have the greatest impact on radial deformation, followed by circumferential deformation and axial deformation. Within the pressure difference range, the radial deformation of the third-layer seal plate in the LTNE model increases by 14.55%. When the interference fit increases from 0.05 mm to 0.2 mm, the radial deformation of each layer of the seal plate in the LTNE model increases by 0.18 mm. The average contact pressure increases with both pressure differential and interference-fit magnitude across both models. At a given pressure differential, the LTNE model yields a higher average contact pressure than the LTE model, with a maximum observed difference of 0.01 MPa. When the interference-fit magnitude is small, the pressure difference between the models remains minimal; however, at the maximum interference-fit, the difference reaches 0.08 MPa.

1. Introduction

With the continuous evolution of aeroengines toward higher thrust-to-weight ratios and improved efficiency, the thermodynamic parameters of core components—such as temperature and pressure—have progressively increased, resulting in increasingly severe internal operating environments. This escalation places more stringent demands on the performance stability and reliability of critical internal engine components [1]. As an emerging flexible contacting rotary seal, the finger seal has been extensively adopted in interstage sealing applications of compressors and turbines due to its favorable cost-effectiveness and superior sealing capability [2,3,4,5]. However, during actual engine operation, finger seals are subjected to prolonged exposure to extreme conditions, including high pressure differentials, elevated rotational speeds, and rotor radial runout [6]. The superposition of these complex factors significantly intensifies frictional interactions within the seal system, particularly between adjacent seal blades and between the blades and the backing plate [7]. This elevated friction not only induces considerable alterations in local flow and heat transfer characteristics within the sealing region—such as increased local heat flux density and non-uniform temperature distribution—but also causes non-uniform thermal deformation of the seal blades [8]. Such deformation directly undermines sealing effectiveness, accelerates component wear, and significantly reduces the service life of the seal, ultimately posing critical challenges to the long-term stable deployment of finger seals in high-performance aeroengines.

Extensive research on finger-seal technology has been carried out by academic and research institutions both domestically and internationally. Chen et al. [9,10,11] were the first to propose the concept of hysteresis ratio for finger seals and analyzed hysteresis behavior using the finite-element method. Bai [12] and Wang et al. [13] developed porous-media flow analysis models based on the structural characteristics of finger seals and investigated the effects of structural parameters, operating conditions, and frictional wear on leakage performance. Hu et al. [14] established a CFD-based porous-media numerical model to evaluate leakage characteristics under various operating conditions, and further validated the model through experiments involving pressure differentials and rotational speeds. Zhao et al. [15], utilizing the pressure-flow factor and the Hagen–Poiseuille law, proposed a permeability calculation method for the side-clearance porous-media structure of finger seals and subsequently developed a porous-media-based numerical model to analyze the side-clearance leakage performance. Their findings revealed that when the root-mean-square surface roughness of the finger blades is high, side-clearance leakage constitutes the primary component of total leakage; in contrast, under large rotor radial displacement excitations, the main flow passage dominates the overall leakage. Wang et al. [16] established a thermal analysis model for finger seals considering contact thermal resistance and studied the influence of structural parameters on heat transfer characteristics. Zhang et al. [17] constructed a coupled thermal-structural analysis model of the finger-seal system to investigate the influence of thermal effects on contact friction behavior.

In summary, existing research on finger seals predominantly focuses on flow and heat transfer characteristics, leakage behavior, and frictional wear, whereas investigations specifically addressing thermal deformation and its underlying mechanisms remain relatively limited. For instance, Chang et al. [18] conducted bidirectional fluid–structure interaction simulations on the clearance flow field of non-contact finger seals, incorporating thermal effects to obtain the deformation of lift-off pads under coupled thermal stresses. Zhu et al. [19,20] employed one-way coupled thermal analysis to determine the temperature distribution and deformation of the pneumatic shoe, demonstrating that thermal deformation increases with temperature and that gas viscosity further intensifies this deformation within finger seals. However, in practical applications, contact-type finger seals experience significantly more severe frictional heating, making thermal deformation a more critical concern. Based on thermal analysis, Su et al. [21] performed coupled thermal–structural simulations and found, through comparison with conditions neglecting thermal effects, that elevated system temperatures induced by higher rotational speeds result in increased thermal deformation. Wang et al. [22] developed a porous-media-based flow and heat transfer model for contact-type finger seals and utilized the thermal-stress module in ANSYS Workbench to evaluate thermal deformation under gas pressure and temperature fields. Their findings indicated that deformation primarily occurs in the toe region of the finger shoe and that structural optimization—by increasing the inner radius of the toe and reducing that of the heel—can effectively mitigate wear and leakage. Hu et al. [23] established a three-dimensional anisotropic flow and heat transfer model based on porous media, along with a solid model for thermal deformation analysis, revealing that maximum radial deformation increases with both rotational speed and interference fit. Du et al. [24] proposed a mathematical model for predicting the radial deformation limit of finger seals and investigated the influence of structural parameters on the radial limit deformation of arc-shaped finger seals.

1.1. Research Gap and Objectives

An analysis of existing research reveals that while notable progress has been made in the study of flow heat transfer, leakage, and frictional wear in finger seals, a critical research gap concerning their thermal deformation under practical operating conditions remains. Specifically, existing research on flow and heat transfer in finger seals predominantly relies on porous media models that assume Local Thermal Equilibrium (LTE) between the solid components (including finger beams and shoes) and the fluid. This assumption implies that the temperatures of the finger beam, the finger shoe, and the gas within the porous-media region are identical. This assumption simplifies a complex physical process, which can result in inaccurate predictions of the finger seal’s temperature field. In practical operating conditions, however, significant temperature differences exist between the finger beams, shoes, and the airflow, necessitating the inclusion of convective heat transfer between these solids and the gas in the calculation. Consequently, to enable a more comprehensive investigation of the coupled heat-transfer phenomena within finger seals, a porous-media model based on local thermal non-equilibrium—namely, a dual-energy-equation model—is required.

The primary objective of this study is to address the research gap by developing a finger seal porous media model based on LTNE, thereby enabling a more reliable prediction of the flow heat transfer and thermal deformation.

1.2. Contributions to This Study

• A porous-media model for finger seals based on the Local Thermal Non-Equilibrium (LTNE) model is developed, and its accuracy in predicting the temperature and flow fields is validated through a comparative analysis with experimental results, thus establishing a reliable foundation for subsequent work.
• Based on the numerical results obtained from the LTNE model, pressure and temperature loads within the finger-seal region are extracted and incorporated as boundary conditions into the thermal deformation analysis model. This Multiphysics coupling approach enables a more realistic representation of the finger seal’s actual operating conditions.
• Considering the practical operating condition of rotor eccentric whirling in aeroengines, a comparative study of finger-seal thermal deformation under both the Local Thermal Equilibrium (LTE) model and the LTNE model is conducted. The analysis focuses on the effects of pressure differential and interference-fit magnitude on the radial, circumferential, and axial deformation characteristics of the finger seal, as well as the distribution patterns of contact pressure. The goal of this study is to provide a reliable theoretical basis and feasible technical approach for the structural optimization and performance enhancement of finger seals.

2. Physical Model and Numerical Method

2.1. Physical Model of the Finger Seal

The structural configuration of the finger seal is illustrated in Figure 1. It primarily consists of a front plate (1), spacer (2), finger element (3), rivet (4), back plate (5), and rotor (6). The lower section of the finger beam (7) forms the finger foot (8), which remains in contact with the rotor (6) to establish the sealing interface [25]. The structural parameters of the finger seal are listed in

Table 1. Structural parameters of finger seal.

Main Structural Parameters of Finger Seal	Value
The outer diameter of the finger seal $D_o$ (mm)	207
The diameter of the finger base $D_b$ (mm)	187
The diameter of the finger foot upper $D_f$ (mm)	163
The inner diameter of the finger seal $D_i$ (mm)	160
The diameter of the finger beam arcs’ centers $D_{cc}$ (mm)	43
The arc radius of finger beam $R_s$ (mm)	85
The finger foot repeat angle $a\prime$ (°)	4.7
The finger repeat angle $a$ (°)	5
The width of the gap between fingers $I_s$ (mm)	0.4
The thickness of the laminate $b$ (mm)	0.3

.

2.2. Governing Equations for the Porous-Media Model of the Finger Seal

2.2.1. Flow-Control Equations for the Porous-Media Region of the Finger Seal

The finger elements (3) (Figure 2) are composed of finger beams (7) that are periodically arranged in the circumferential direction, with identical gaps $I_s$ between adjacent beams. Adjacent finger elements (3) are staggered and clamped between the front plate (1) and the back plate (4) and are secured using rivets (5). Consequently, as illustrated in Figure 2, the regions corresponding to the finger beam (7) and finger foot (8) within the finger-seal structure are modeled as a porous medium [26].

A porous-media model is adopted to simulate the leakage-flow characteristics of the finger seal; some assumptions must be made.

• Neglecting the structural deformation of the finger beam and finger boot caused by aerodynamic forces, 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 in material physical property parameters with pressure and temperature, the physical property parameters under standard temperature and pressure conditions are used for the calculation.
• Given the significant pressure difference between the upstream and downstream sides of the sealing structure during operation, the density of the fluid flowing through the finger seal region will undergo significant changes. Therefore, the leaking fluid is assumed to be compressible ideal air, and the steady-state governing equations are adopted for the porous media region.

Continuity equation:

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

(1)

Momentum equation:

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

(2)

Energy equation:

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

(3)

Ideal-gas state equation:

$$p=\rho RT$$

(4)

In the above formula, $\rho$ denotes the fluid density in the porous-media region; $u_i$ represents the superficial velocity of the fluid in the $i$ ($i=1,2,3$) direction of the Cartesian coordinate system; $p$ is the fluid pressure; $\mu$ is the dynamic viscosity of the fluid; $c_f$ is the specific heat at constant pressure; $R$ is the ideal gas constant; $T$ denotes the temperature; and $k_{eff}$ is the effective thermal conductivity; In Equation (3), the left-hand side corresponds to the convective heat-transfer term induced by fluid motion, while the right-hand side accounts for the conductive heat-transfer contributions of the fluid and solid phases, respectively; $S_i$ denotes the additional momentum-loss source term in the $i$ direction, introduced by the flow-resistance effect of the finger beam or finger shoe within the porous-media domain [27].

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

(5)

In this formula, $1/\alpha$ and $C_2$ denote the viscous loss coefficient and inertial loss coefficient within the porous medium, respectively [28].

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{\alpha }}=5m{\displaystyle \frac{S^2}{{\epsilon }^3}}\\ C_2={\displaystyle \frac{n}{4}}{\displaystyle \frac{S}{{\epsilon }^3}}\end{array}\right.\left\{\begin{array}{l}m=0.1\\ n=0.03343+{\displaystyle \frac{0.0155}{3.0136\Delta P}}{e}^{{{\displaystyle \frac{-(\ln \frac{\Delta P}{0.1728})}{2.8908}}}^2}\end{array}\right.$$

(6)

In this formula, $m$ and $n$ are determined from experimental data [12]; $\epsilon$ and $S$ denote the porosity and the specific wetted surface area of the finger-seal porous medium, respectively; $\Delta P$ represents the pressure differential between the upstream and downstream sides of the finger-seal structure.

2.2.2. Porosity and Wetted Surface Area of the Finger-Seal Porous Medium

Porosity $\epsilon$ and wetted surface area $S$ are two fundamental characteristic parameters of porous media and are critical in determining fluid transport behavior within the porous structure [29]. The porosity ${\epsilon }_s$ of the finger seal is defined as the ratio of the gap volume $V_v$ of a single finger seal within the total volume $V_t$ of a single finger seal, i.e., ${\epsilon }_s=V_v/V_t$. The wetted surface area $S_s$ refers to the ratio of the surface area $S_v$ of a single finger seal within the total volume $V_t$ of a single finger seal, i.e., $S_s=S_v/V_t$.

Since the design methodologies for the finger beam and the finger foot differ during the development of the finger-seal blade, the porosity ${\epsilon }_s$ and wetted surface area $S_s$ of the finger beam and finger foot are calculated separately [30]. The corresponding calculation formulas are given as follows.

$${\epsilon }_s={\epsilon }_L+{\epsilon }_X={\displaystyle \frac{4L_{st}·I_s·N}{\pi \left(D_b^2-D_f^2\right)}}+{\displaystyle \frac{a-a^{\prime }}{a}}$$

(7)

$$S_s=S_L+S_X={\displaystyle \frac{8N·L_{st}\left(b-I_s\right)}{\pi \left(D_b^2-D_f^2\right)b}}+{\displaystyle \frac{2}{b}}+{\displaystyle \frac{\left(D_f-D_i\right)\times N}{\frac{1}{4}\pi \left(D_f^2-D_i^2\right)}}+{\displaystyle \frac{\pi D_i\times \frac{a^{\prime }}{a}}{\frac{1}{4}\pi \left(D_f^2-D_i^2\right)}}+{\displaystyle \frac{2a^{\prime }}{a\times b}}$$

(8)

$$L_{st}=R_s\left[\mathit{arccos}{\displaystyle \frac{D_{cc}^2+4R_s^2-D_b^2}{4D_{cc}{R}_s}}-\right.\left.\mathit{arccos}{\displaystyle \frac{D_{cc}^2+4R_s^2-D_f^2}{4D_{cc}{R}_s}}\right]$$

(9)

In these formula, ${\epsilon }_L$ is the porosity of the finger beam; ${\epsilon }_X$ is the porosity of the finger boot; $S_L$ is the wetted surface area of the finger beam; $S_X$ is the wetted surface area of the finger boot; and $L_{st}$ is the length of the finger beam.

2.3. Local Thermal Non-Equilibrium Model and Local Thermal-Equilibrium Model for Porous Media

Heat transfer within a porous medium can be categorized into two distinct regimes.

1.

When there is no flow or only slow flow within the porous medium, and when the temperature difference between the fluid and solid phases is negligible or small, it is assumed that the fluid and solid are at approximately the same temperature, i.e., $T_f\approx T_s$. Here, $T_f$ and $T_s$ denote the temperatures of the fluid and solid phases within the porous region, respectively [31]. Under this condition, the LTE model is adopted, and the energy equation of the porous medium can be expressed using an effective thermal conductivity as follows.

$$\left(\left(1-\epsilon \right){\rho }_s{c}_s+\epsilon {\rho }_f{c}_f\right){\displaystyle \frac{\partial T}{\partial t}}+{\rho }_f{c}_f{u}_f·\nabla T=\nabla ·\left(k_{eff}\nabla T\right)$$

(10)

In this formula, ${\rho }_s$ is the solid density; ${\rho }_f$ is the fluid density within the porous medium; $c_s$ and $c_f$ are the specific heats at constant pressure of the fluid and solid phases, respectively; $u_f$ is the fluid velocity vector, and $k_{eff}$ is the effective thermal conductivity of the porous medium. The first term on the left-hand side of the energy equation represents the energy accumulation, the second term accounts for energy transport due to fluid seepage through the porous matrix, and the right-hand side denotes internal heat exchange within the porous structure.

2.

When strong flow occurs within the porous medium and a significant temperature difference exists between the fluid and solid phases, i.e., $T_f\ne T_s$, forced convective heat transfer must be considered [32]. Under such conditions, the LTNE model is employed, and separate energy equations for the fluid and solid phase are formulated to characterize heat transfer within the porous medium.

Fluid-phase energy equation:

$$\left(\epsilon {\rho }_f{c}_f\right){\displaystyle \frac{\partial T_f}{\partial t}}+{\rho }_f{c}_f{u}_f·\nabla T_f=\nabla ·\left(k_{f,eff}\nabla T_f\right)+Q_{fs}$$

(11)

Solid-phase energy equation:

$$\left(1-\epsilon \right){\rho }_s{c}_s{\displaystyle \frac{\partial T_s}{\partial t}}=\nabla ·\left(k_{s,eff}\nabla T_s\right)+Q_{sf}$$

(12)

In these formula, $T_f$ is the fluid temperature within the porous medium; $T_s$ is the solid temperature; $k_{f,eff}$ and $k_{s,eff}$ are the effective thermal conductivities of the fluid and solid phases, respectively, and $Q_{fs}$ denotes the convective heat-transfer coefficient between the solid and fluid phases.

$$Q_{fs}=-Q_{sf}=h_{sf}{A}_{sf}\left(T_s-T_f\right)$$

(13)

In this formula, $h_{sf}$ is the convective heat-transfer coefficient between the solid and fluid phases within the porous medium, and $A_{sf}$ is the specific surface area of the porous medium, defined as the ratio of the interfacial contact area between the fluid and solid phases to the total volume of the porous region.

2.4. Frictional Heat-Generation Model for the Finger Seal

As a critical sealing component, the finger seal operates for extended durations under high-temperature, high-pressure, and high-rotational speed conditions. The interface between the bottom surface of the finger shoe and the rotor is configured with either a small clearance or an interference fit, resulting in intense friction during high-speed relative motion and generating substantial frictional heat [22]. This heat is conducted from the contact region toward the cooler finger beam and rotor, leading to the development of thermal stress and subsequent thermal deformation. Simultaneously, the working fluid undergoes convective heat transfer with the high-temperature finger blades, rotor, and the front and rear backing plates. In addition, conductive heat transfer occurs between adjacent finger blades and between the finger blades and the rear backing plate. A schematic illustration of frictional heat transfer within the finger seal is presented in Figure 3.

Prior studies have demonstrated that the heat generated between adjacent finger blades and between the finger blades and the rear backing plate is minimal and can generally be neglected [27]. Consequently, the primary source of frictional heat within the porous-media region of the finger seal is attributed to the friction at the interface between the bottom of the finger shoe and the rotor.

The frictional heat generated at the interface between the finger shoe and the rotor can be modeled as an annular heat source. The amount of heat $Q$ produced by friction is expressed as.

$$Q=\gamma f{k}_{i}sV$$

(14)

$$V={\displaystyle \frac{\pi n{D}_i}{60}}$$

(15)

$$q={\displaystyle \frac{Q}{A_i}}={\displaystyle \frac{\gamma f{k}_{i}sV}{A_i}}$$

(16)

In the above formula, $q$ is the frictional heat flux density; $\gamma$ is the friction-heat correction coefficient; $f$ is the coefficient of friction between the finger-shoe surface and the rotor surface, taken as 0.2; $k_i$ is the radial stiffness of a single finger beam, taken as 489.57 N/mm; $A_i$ is the contact area between a single finger shoe and the rotor surface, calculated from the structural parameters of the finger seal, equal to 1.965 $\mathrm{m}{\mathrm{m}}^2$; $s$ is the initial installation interference between the finger shoe and the rotor, taken as 0.1 mm; $V$ is the linear surface velocity of the rotor (m/s); and $n$ is the rotational speed of the rotor in rpm.

2.5. Local Thermal Non-Equilibrium Model for the Finger-Seal Porous Medium

From the above analysis, during operation, intense friction between the bottom of the finger shoe and the high-speed rotor generates substantial frictional heat, causing a rapid temperature increase in both the finger beam of the finger blade and the rotor surface. Furthermore, the high-pressure differential and high rotational speed conditions cause the internal fluid flow within the finger seal to be extremely rapid. A time-scale analysis reveals a mismatch between the time scale of fluid passage through the seal and the time scale of thermal diffusion in the finger beams and the rotor. This discrepancy leads to a significant temperature difference between the fluid and the finger beams, resulting in strong convective heat transfer between them. Therefore, the porous-media model of the finger seal in this study employs the LTNE approach, with the corresponding energy equations for the fluid phase and solid phase given by:

$$\begin{array}{l}\left({\epsilon }_s{\rho }_f{c}_f\right){\displaystyle \frac{\partial T_f}{\partial t}}+{\rho }_f{c}_f{u}_f·\nabla T_f=\left[{\displaystyle \frac{\partial }{\partial m}}\left(k_f{\displaystyle \frac{\partial T_f}{\partial m}}\right)\right.+\left.{\displaystyle \frac{\partial }{\partial n}}\left(k_{n,eff}{\displaystyle \frac{\partial T_f}{\partial n}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(k_{z,eff}{\displaystyle \frac{\partial T_f}{\partial z}}\right)\right]\\ -h_{sf}{A}_{sf}\left(T_s-T_f\right)\end{array}$$

(17)

$$\left(1-{\epsilon }_s\right){\rho }_s{c}_s{\displaystyle \frac{\partial T_s}{\partial t}}=\left[{\displaystyle \frac{\partial }{\partial m}}\left(k_s{\displaystyle \frac{\partial T_s}{\partial m}}\right)+{\displaystyle \frac{\partial }{\partial n}}\left(k_{n,eff}{\displaystyle \frac{\partial T_s}{\partial n}}\right)\right.\left.+{\displaystyle \frac{\partial }{\partial z}}\left(k_{z,eff}{\displaystyle \frac{\partial T_s}{\partial z}}\right)\right]+h_{sf}{A}_{sf}\left(T_s-T_f\right)$$

(18)

In the above formula, $k_f$ is the thermal conductivity of the fluid; $k_s$ is the thermal conductivity of the finger beam; $m$ denotes the direction along the finger beam; $n$ denotes the direction normal to the finger beam; $z$ represents the axial direction of the rotor; and $k_{z,eff}=k_{n,eff}$ is the effective thermal conductivity of the fluid in the direction normal to the finger beam.

Based on the actual operating conditions of the finger-seal system, when the fluid flows over the surface of the finger blades, the heat transfer between the fluid and the blade surface can be approximated as the convective heat-transfer process of external flow over a flat plate. Therefore, in this study, the surface convective heat-transfer coefficient $h_{sf}$ between the fluid and the finger blades within the porous-media region is determined using the heat-transfer correlation for external flow over a flat plate. According to the Schlichting criterion, it is given by the following expression [33]:

$$h_{sf}=\left\{\begin{array}{l}0.332{\displaystyle \frac{k_f}{l}}P{r}^{1/3}R{e}^{1/2 } (Re<5\times {10}^5)\hfill \\ {\displaystyle \frac{k_f}{l}}P{r}^{1/3}(0.037R{e}^{5/4}-850) (5\times {10}^5<Re<5\times {10}^7)\hfill \end{array}\right.$$

(19)

In this formula, $Pr$ denotes the Prandtl number; $l$ is the characteristic length of the finger seal; and $Re$ represents the Reynolds number of the fluid flow, calculated as:

$$Re={\displaystyle \frac{ul}{\eta }}$$

(20)

In this formula, $\eta$ is the kinematic viscosity of the fluid; and $u$ is the fluid-flow velocity.

The fluid-flow velocity can be calculated from:

$$u={\displaystyle \frac{\pi {\mu }_{\gamma }n{D}_i}{240}}$$

(21)

In this formula, $n$ is the rotor speed; and ${\mu }_{\gamma }$ denotes the fluid viscosity factor, which typically ranges 0.2~0.8; in this study, a value of 0.5 is adopted [33].

The Prandtl number can be calculated from:

$$Rr={\displaystyle \frac{c_s{\rho }_s\eta }{k_f}}$$

(22)

In this formula, $c_s$ is the specific heats at constant pressure of the solid phases; and ${\rho }_s$ is the solid density.

Because the porous medium in the finger seal comprises multiple physical phases, its thermal conductivity is represented by an apparent thermal conductivity. The adjacent finger beams within a single finger blade are arranged nearly in parallel. When the direction of heat transfer aligns with the orientation of the finger beams, thermal conduction occurs simultaneously through both the solid beams and the fluid [34]. Accordingly, the effective thermal conductivity $k_{m,eff}$ is calculated by the following equation.

$$k_{m,eff}=\epsilon k_f+\left(1-\epsilon \right)k_s$$

(23)

If the direction of heat transfer is perpendicular to the finger beams, the beams and the fluid can be considered as two thermal resistances connected in series. The effective thermal conductivity $k_{n,eff}$ is then calculated by the following equation.

$$k_{n,eff}={\displaystyle \frac{k_f{k}_s}{\epsilon k_s+\left(1-\epsilon \right)k_f}}$$

(24)

In this study, the working fluid is treated as an ideal gas, and the finger-seal blades are constructed from GH605 alloy [35]. Therefore, $k_f$ and $k_s$ is calculated by the following equation, respectively.

$$\left\{\begin{array}{l}{k}_f=6.013·{10}^{-5}(T+273.15)+9.67·{10}^{-3}\\ k_s=0.0203(T+273.15)+3.46\end{array}\right.$$

(25)

3. Thermal-Deformation Coupling Method and Data Transfer

During the numerical computation of the finger seal’s porous-media model, the resistance coefficients within the porous medium are influenced by the structural deformation of the seal, making their variation difficult to predict. To address this, a one-way weak-coupling approach between the flow field and the solid field is employed to analyze the thermo-structural deformation of the finger seal [36,37,38]. Specifically, Fluent is used to compute the flow and heat transfer, from which pressure and temperature data are extracted and applied as boundary conditions to the thermal deformation analysis model of the finger seal. The structural deformation is then evaluated based on these applied loads.

The computation method and detailed procedure for thermal deformation are outlined as follows.

• The radial stiffness $k_i$ of a single finger beam is obtained through finite element analysis and substituted into Equation (16) to compute the frictional heat flux density $q$ between the seal and the rotor.
• The flow and heat transfer computational model of the finger seal, namely the porous-media model, is established based on the seal’s original geometric parameters. Parameters required for the LTNE porous-media model are computed according to the operating conditions and the physical properties of the finger-blade material. The frictional heat flux density $q$ is applied as a boundary condition, and Fluent is used to solve the governing equations for flow and heat transfer (Equations (1), (2), (4), (17), and (18)), yielding the flow field, temperature field, and other relevant results for the finger seal.
• The deformation analysis model for the finger seal is developed based on its structural characteristics. Using the Static Structure module in the Workbench platform, the pressure and temperature distributions of the seal-blade assembly obtained in step (2) are applied as boundary conditions to compute the contact pressure and thermal deformation of the seal blades under thermal stresses.
• Steps 2–3 are repeated iteratively until the results meet the convergence criteria.

A flowchart of the analysis procedure for studying the effects of thermal deformation is presented in Figure 4.

4. Computational Model and Boundary Conditions

4.1. LTNE Porous-Media Computational Model and Boundary Conditions for the Finger Seal

After approximating the finger seal as a porous medium and considering the circumferential cyclic symmetry of the seal’s structure, the computational model in this study is generated by rotating a two-dimensional axisymmetric model by 1° to reduce computational cost. The computational model is shown in Figure 5.

In this study, the commercial software ANSYS-Fluent 2022 is employed to simulate the LTNE porous-media model of the finger seal. The inlet boundary of the computational domain is defined as a pressure inlet with a specified temperature, while the outlet is set as a pressure outlet. The contact interface between the finger shoe and the rotor is subjected to a thermal boundary condition represented by a frictional heat flux. For the fluid region outside the porous-media zone, the RNG k-ε turbulence model is applied. Owing to the relatively low flow velocity and the corresponding low Reynolds number within the porous medium, a laminar flow model is adopted. The numerical solution is obtained using the SIMPLE algorithm. Table 1 presents the main structural parameters of the finger seal, and

Table 2. Operating parameters of the finger seal.

Boundary Name	Values	Work Unit
Pressure differential	0.077~0.216	MPa
Rotational speed	9000~21,000	r/min
Radial interference value	0.05~0.2	mm
Outlet pressure	0.1	MPa
Total inlet/outlet temperature	300	K

lists the main operating parameters of the finger seal.

4.2. Deformation Calculation Model and Boundary Conditions for the Finger Seal

Due to the circumferential cyclic symmetry and axial periodic stacking of the finger seal, its structural model is simplified—without compromising numerical accuracy—to reduce computational cost and enhance efficiency. During the modeling process, four axially staggered layers of seal blades are selected. Among these, the first and third layers each contain one complete finger beam and two half finger beams, while the second and fourth layers each consist of two complete finger beams.

Figure 6 presents the deformation calculation model of the finger seal along with its boundary conditions. The outer annular surfaces of the finger blades and the rear backing plate are subjected to fixed constraints, while the circumferential surfaces of the blade assembly and rear backing plate are assigned periodic cyclic boundaries. The rotor’s inner surface, perpendicular to the circumferential direction, is set as an axisymmetric boundary.

The material used for the finger seal in this study is GH605, and the rotor material is K477, whose performance characteristics are like AMS5537 and MAR-M247 [39], respectively. The material property parameters are shown in

Table 3. The physical properties of materials.

Physical Property	Fluid	Finger Seal	Rotor/Plate
Density $\rho$ (kg/m3)	Air ideal gas	9130	7950
$\mathrm{Specific} \mathrm{heat} \mathrm{capacity} c_p$ (J/kg·K)	0.132 T + 973	377	0.262 T + 350.1
Thermal conductivity $k$ (W/m·K)	6.03 × 10−5 T + 9.67 × 10−3	0.0203 T + 3.46	0.0215 T + 4.49
Viscosity $\mu$ (Pa·s)	3.42 × 10−8 T + 9.14 × 10−6	—	—
Thermal expansion coefficient $\beta$ (K−1)	—	1.38 × 10−5	1.33 × 10−5
Elastic modulus $E$ (GPa)	—	−0.0848 T + 258.4	−0.0841 T + 220.6
Poisson’s ratio $\upsilon$	—	4.52 × 10−5 T + 0.27	0.28

Note: The unit of T is K.

. Given the significant temperature variations, the specific heat capacity, thermal conductivity, and viscosity of the leakage fluid, as well as the thermal conductivity, Young’s modulus, and Poisson’s ratio of the solid, are all set as linear functions of temperature within the range of 293.15 K to 1073.15 K [22].

Although a temperature-linear elastic modulus is adopted in this study, future work should incorporate a more complex nonlinear material model to improve the prediction accuracy of deformation values.

4.3. Data Transfer in the Weakly Coupled Calculation

As previously described, the finger seal exhibits circumferential cyclic symmetry, with variations in internal parameters such as pressure and temperature being minimal and, thus, can be neglected. Consequently, the temperature distribution data of (1–6) (Figure 7) obtained from the LTNE model for the porous media of finger seals are loaded into (I–VI) (Figure 7) of the deformation calculation model, while the pressure distribution data at locations 2 and 6 are loaded into regions II and VI. Figure 7 illustrates the transfer of temperature and pressure data computed from the LTNE porous-media model, which are then imposed as boundary conditions in the thermal-deformation calculation model.

The application of boundary conditions in the finger-seal deformation calculation model is carried out in three steps: First, the pressure and temperature data obtained from Fluent simulations are loaded into the finger element. Second, a radial interference $s$ is imposed on the bottom surface of the finger foot to represent the actual rotor radial runout. Finally, after the finger beam returns to its original position, the thermal deformation of the finger blade is extracted.

4.4. Mesh Generation and Model Validation

4.4.1. Mesh Generation

Figure 8 presents the two-dimensional cross-section mesh of the three-dimensional porous-media model of the finger seal. Mesh refinement is applied near the rotor wall and in the vicinity of the porous-media region. The first-layer mesh height at the wall is set to 0.05 mm, with a growth factor of 1.2 mm.

In the deformation model of the finger seal, the interaction between the rotor’s eccentric whirling and the finger shoe must be considered, with deformation occurring primarily in the finger-shoe region. Therefore, mesh refinement is applied to the finger-shoe area, which is the main focus of the analysis. Since the rear backing plate and rotor do not directly contribute to the deformation behavior under evaluation, their mesh density is reduced accordingly. Figure 9 illustrates the mesh distribution of the deformation calculation model of the finger seal.

4.4.2. Mesh Independence Verification

During numerical simulation, both the mesh size and density directly influence the reliability of the results and the computational cost.

Eight mesh schemes with total cell counts of 30,000; 60,000; 110,000; 280,000; 480,000; 650,000; 830,000; and 1,080,000 were applied to the LTNE model for the porous media of finger seals. The leakage rate $m$ was monitored for each case. As shown in Figure 10, when the mesh count exceeds 480,000, the leakage rate $m$ variation stabilizes. Therefore, a mesh with approximately 480,000 cells is adopted for the LTNE model for the porous media of the finger seal.

For the thermal-deformation calculation model of the finger seal, the radial deformation of the first-layer seal element is used as the criterion for mesh-independence verification. As shown in Figure 11, when the mesh count exceeds 130,000, the radial deformation of the first-layer seal element reaches a stable value, and the mesh quality remains satisfactory. Consequently, the mesh count for the thermal-deformation computation model of the finger seal is set at approximately 130,000.

4.4.3. Validation of Computational Accuracy

To validate the accuracy of the LTNE porous-media finger seals computational model, the structural parameters and boundary conditions in this study are chosen to align with those reported in the literature, which are accompanied by available experimental data [28].

As shown in Figure 12, when the upstream–downstream pressure differential increases from 0.05 MPa to 0.25 MPa, the leakage rate increases approximately linearly. For pressure differentials below 0.133 MPa, the model-calculated values are slightly higher than the experimental values. At a pressure differential of 0.156 MPa, the model-calculated values and experimental values align closely, and for all subsequent pressure differentials, the deviation remains below 5%. Therefore, the LTNE porous-media computational model developed in this study demonstrates high predictive accuracy.

5. Results and Discussion

Based on the LTNE porous-media numerical model developed in this study, the maximum-temperature distribution and leakage characteristics of the finger seal are comparatively analyzed under different operating conditions, including pressure differential, rotational speed, and radial interference. The comparison between the LTE model and the LTNE model seeks to highlight the differences in sealing-performance evaluation results produced by the two approaches.

5.1. Analysis of Maximum Temperature and Leakage Characteristics of the Finger Seal Under the LTNE Model

5.1.1. Comparative Analysis of the Influence of Operating Parameters on Maximum Temperature Characteristics of Finger Seals

Figure 13 illustrates the variations in maximum temperature obtained from the porous-media LTE and LTNE models of the finger seal under an upstream–downstream pressure differential of 0.216 MPa, a radial interference of 0.1 mm, and a rotor speed of 21,000 r/min.

As illustrated in Figure 13, the maximum temperature predicted by both models increases with rising pressure differential, rotational speed, and radial interference. However, the LTNE model consistently yields higher maximum temperature values compared to the LTE model. This discrepancy arises primarily from the fundamental assumption of the LTE model, which considers the fluid and finger-seal blades to be at the same temperature. Under this assumption, the heat generated in the blades is instantaneously and entirely transferred to the fluid without any thermal resistance, effectively leading to an overestimation of the fluid’s thermal absorption and a corresponding underestimation of the blade temperature. In contrast, the LTNE model incorporates convective heat transfer between the fluid and the finger-seal blades, where thermal resistance at the blade-fluid interface introduces both transfer delay and energy dissipation. As a result, the rate of heat accumulation in the blades exceeds the rate of heat removal, leading to a higher predicted maximum temperature in the LTNE model relative to the thermal-equilibrium case.

Among the operating condition parameters of the finger seal system, rotational speed is regarded as the dominant factor affecting the maximum temperature of the finger seal. Specifically, in comparison with other operating condition parameters, the maximum temperature derived from the LNTE model increases significantly by 84.8% with the variation in rotational speed. The primary reason for this phenomenon is that as the rotational speed increases, frictional heat generation at the interface between the bottom of the finger boot and the rotor surface intensifies, which exceeds the heat dissipation capacity of the finger element structure and thus leads to significant temperature accumulation. The above research confirms that the LNTE model can capture the nonlinear thermal response of finger seals under high-speed operating conditions, providing a reliable theoretical basis for the performance optimization of finger seals.

5.1.2. Comparative Analysis of the Influence of Operating Parameters on Leakage Performance of Finger Seals

Figure 14 shows the leakage-rate variations in the finger seal under the operating parameters described in Section 5.1.1, comparing the LTE and LTNE porous-media models.

As shown in Figure 14, the leakage rate predicted by both models increases with increasing pressure differential and decreases with increasing rotational speed and radial interference. With rising pressure differential, the maximum discrepancy in leakage rate between the LTE and LTNE models reaches 0.63 g/s. As the rotational speed increases, the difference gradually expands from 0.06 g/s to 0.23 g/s, while increasing radial interference leads to a slight rise in the difference from 0.17 g/s to 0.20 g/s. Under all operating conditions, the LTNE model consistently predicts a lower leakage rate than the LTE model.

These results demonstrate that the LTNE model takes into full account the effect of the temperature difference between fluid and solid on leakage rate, yielding predictions in better agreement with the actual behavior of the sealing system.

5.2. Thermal-Deformation Analysis of the Finger Seal

5.2.1. Comparative Analysis of Thermal Deformation of Each Seal-Element Layer Under the LTE and LTNE Models

Figure 15 presents the radial deformation contour plots of the finger-seal structure under an upstream–downstream pressure differential of 0.216 MPa, a radial interference of 0.05 mm, and a rotor speed of 21,000 r/min. All values are in SI units (mm).

As shown in Figure 15, radial deformation in both models is primarily concentrated at the front end of the finger foot. This is attributed to the combined effects of radial friction and the constraint provided by the rear backing plate, which limit the deformation of the outer circular section of the seal blade and the finger beam relative to the finger foot. Furthermore, the LTNE model predicts a larger radial deformation displacement for each element layer compared to the LTE model. This is mainly due to the thermal resistance associated with convective heat transfer between the fluid and the elements in the LTNE model, which results in the element temperature exceeding that of the fluid. On one hand, the elevated element temperature induces greater thermal expansion, leading to significantly increased thermal deformation. Simultaneously, the temperature rise reduces the elastic modulus of the element material, making it more susceptible to elastic deformation under identical mechanical loading and thereby further amplifying the radial deformation range. On the other hand, the lower fluid temperature increases its viscosity, which enhances the radial hydrodynamic pressure exerted on the element surface. This elevated pressure acts directly on the finger foot, further contributing to radial deformation.

5.2.2. Influence of Pressure Differential on the Thermal Deformation of the Finger Seal

Figure 16, Figure 17 and Figure 18 depict the thermal deformation characteristics of each seal-element layer under a rotor speed of 21,000 r/min, a radial interference of 0.05 mm, and upstream–downstream pressure differentials ranging from 0.077 MPa to 0.216 MPa.

Figure 16 shows the radial thermal deformation curves of each seal-element layer under both the LTE and LTNE models as a function of pressure differential. It is evident that the radial deformation in all element layers is greater under the LTNE model than under the LTE model. For both models, the deformation increases with rising pressure differential; however, the rate of increase is more pronounced in the LTNE model.

Under the LTNE model, the maximum radial deformation occurs in the third-layer seal element, while the minimum radial deformation occurs in the fourth-layer seal element. Under the LTE model, the radial deformation shows a different pattern: the maximum radial deformation occurs in the fourth-layer seal element, and the minimum radial deformation occurs in the fourth-layer seal element.

Figure 17 illustrates the circumferential thermal deformation curves of each seal-element layer under the LTE and LTNE models as the pressure differential varies. It is observed that the circumferential deformation in all element layers is consistently greater under the LTNE model than under the LTE model. In both models, deformation increases with rising pressure differential, but the variation is more pronounced under the LTNE model. Under identical operating conditions, both models predict the largest circumferential deformation in the fourth layer and the smallest in the first layer. This can be attributed to the mechanical constraints imposed on the elements: the first-layer elements are subjected to high-pressure airflow and relatively large interlayer preload, resulting in enhanced rigid constraint and restricted deformation; in contrast, the fourth-layer elements, positioned at the outermost layer, bear only residual aerodynamic loads transmitted from upstream layers and experience the weakest rigid constraint, allowing for the greatest deformation.

Under the LTNE model, the circumferential deformation of the first, second, and third layers exhibits significant variation with increasing pressure differential, while the fourth layer shows relatively moderate changes. In the LTE model, deformation across all layers remains comparatively stable. This difference arises primarily from the influence of the porous medium. In the LTNE model, the flow resistance characteristics promote load concentration in the first three layers, amplifying deformation differences among them. Meanwhile, in the LTE model, the efficient thermal conduction through the porous medium enhances the compensatory effect of thermal stress on the blades, leading to a more uniform deformation distribution across layers.

Within the specified pressure differential range, the circumferential deformation of the fourth-layer seal element exhibits a 12.18% increment in the LTNE model, as opposed to a 10.93% rise observed in the LTE counterpart.

Figure 18 displays the axial thermal deformation profiles of each seal-element layer as a function of pressure differential under both LTE and LTNE models. It is evident that the axial deformation predicted by both models increases with rising pressure differential.

Under identical pressure conditions, the LTNE model exhibits a non-monotonic “decrease-then-increase’’ trend in axial deformation from the first to the fourth layer, with the second layer showing the least deformation and the fourth layer the greatest. This is primarily attributed to the fact that the LTNE model incorporates the effect of temperature. Under the combined action of pressure differential and rotor radial runout, the temperature field distribution of the finger seal plates is non-uniform, which consequently results in disparate thermal deformation behaviors across individual seal layers, albeit with consistent overall variation trends.

In contrast, within the LTE model, the axial deformation of the seal elements decreases progressively—specifically, the axial deformation of the first-layer seal element is the largest, while that of the fourth-layer seal element is the smallest. This is because temperature effects are neglected in the LTE model; under pressure differential loading, the first-layer finger element bears the maximum axial pressure, thus undergoing greater axial deformation. Meanwhile, the back plate and rivet restrict the axial displacement of the last-layer finger seal element, which accounts for the minimal axial deformation observed in this layer.

Based on the analysis of Figure 16, Figure 17 and Figure 18, it can be concluded that the pressure differential exerts the most significant influence on the radial deformation of the finger-seal elements, followed by circumferential deformation, while its impact on axial deformation is relatively minor. Furthermore, the overall deformation of the finger-seal elements predicted by the LTNE model is consistently greater than that obtained from the LTE model.

5.2.3. Influence of Interference on the Thermal Deformation of the Finger Seal

Figure 19, Figure 20 and Figure 21 depict the thermal deformation characteristics of each seal-element layer under radial interference values ranging from 0.05 mm to 0.20 mm, with a pressure differential of 0.216 MPa and a rotor speed of 21,000 r/min.

Figure 19 illustrates the radial thermal deformation profiles of each seal-element layer under the LTE and LTNE models as the interference magnitude increases. As observed, the radial deformation in both models rises markedly with increasing interference. The primary reason is that larger radial interference leads to a greater radial displacement of the seal elements driven by the rotor, thereby significantly increasing the radial deformation of the seal elements.

When the radial interference increases from 0.05 mm to 0.20 mm, the absolute radial deformation increments under the LTNE model are 0.184 mm for the first layer, 0.186 mm for both the second and third layers, and 0.177 mm for the fourth layer. These results indicate that the first three layers exhibit nearly identical deformation increments, whereas the fourth layer shows a distinctly smaller increase. This reduction is mainly due to frictional constraints imposed by the rear back plate, which limit the deformation of the fourth-layer element. Under the LTE model, the radial deformation of each element layer increases smoothly and progressively with increasing interference, and the deformation magnitudes across layers remain relatively uniform, with no significant interlayer discrepancies observed.

Figure 20 displays the circumferential thermal deformation curves of each seal-element layer under varying interference magnitudes for both LTE and LTNE models. As shown, the circumferential deformation in all element layers increases with increasing radial interference in both models. Within the specified range of radial interference, although the rate of increase in circumferential deformation is lower under the LTNE model compared to the LTE model, the absolute deformation values are consistently greater.

Figure 21 illustrates the axial thermal deformation profiles of each seal-element layer under varying radial interference values for both LTE and LTNE models. As shown, the axial deformation of each element layer increases progressively with increasing interference in both models. Under the LTNE model, the axial deformation increases by 22.86%, 13.36%, 18,91%, and 13.7% for the first through fourth layers, respectively. In comparison, the corresponding increases under the LTE model are significantly lower, at 4.36%, 4.78%, 4.94%, and 4.83%. These results demonstrate that the LTNE model exhibits markedly higher sensitivity of axial deformation to radial interference, while the LTE model shows only marginal changes across all layers.

From the analysis of Figure 19, Figure 20 and Figure 21, it is evident that the influence of radial interference on finger element deformation exhibits a pattern similar to that of pressure differential: the most pronounced effect occurs in the radial direction, followed by circumferential deformation, with the axial direction being the least affected. Furthermore, the numerical results reveal substantial discrepancies between the LTE and LTNE models in predicting structural deformation, highlighting the critical role of thermal effects in determining the sealing performance of finger seals.

5.3. Analysis of Average Contact Pressure Under the LNTE Model for Finger Seals

5.3.1. Influence of Pressure Differential on the Average Contact Pressure of the Finger Seal

Figure 22 presents the variation curves of the average contact pressure at the finger foot under different pressure differentials for both LTE and LTNE models. As shown in Figure 22, the average contact pressure by both models increases with rising pressure differential. This trend is primarily attributed to the increase in fluid pressure, which elevates the downstream support height between the rear backing plate and the rotor. As a result, localized axial bending deformation of the finger beam occurs, leading to an increase in the contact pressure between the finger foot and the rotor surface. Furthermore, under the same pressure differential, the average contact pressure of the LTNE model is higher than that of the LTE model.

5.3.2. Influence of Interference on the Average Contact Pressure of the Finger Seal

As shown in Figure 23, the average contact pressure by both the LTE and LTNE models increases with rising interference. This behavior is primarily due to the greater axial deformation of the sealing blades induced by larger interference, which enhances the conformity between the finger shoe and the rotor surface, thereby increasing the contact pressure. Additionally, the results indicate that when the interference is small, the difference in average contact pressure between the two models is minimal. However, as the interference reaches its maximum value, the LTNE model predicts a higher average contact pressure, attributed to its more pronounced increase in axial deformation with increasing interference.

From the analysis of Figure 22 and Figure 23, it can be concluded that under varying pressure differentials, the difference in average contact pressure between the LTE and LTNE models is small, whereas under varying interference magnitudes, the difference increases to 0.08 MPa. This indicates that interference has a more pronounced influence on the average contact pressure of the finger seal compared to pressure differential. Accordingly, in the design of finger-seal structures for operating conditions characterized by significant rotor eccentricity, reducing the stiffness of the sealing material can effectively decrease the average contact pressure between the finger foot and the rotor surface. This reduction contributes to improved sealing performance and extended service life of the finger seal.

6. Conclusions

Based on the porous-media model under LTNE conditions, this study establishes heat transfer equations for the finger seal and constructs a three-dimensional flow and heat-transfer model, along with a solid model for thermal-deformation analysis, using computational fluid dynamics and the finite element method. The effects of pressure differential and radial interference on the structural deformation and average contact pressure of the finger seal are comparatively evaluated under both LTE and LTNE models, leading to the following conclusions.

• Due to the inclusion of convective heat transfer between the fluid and the finger-seal elements, the maximum temperature by the LTNE model is higher than that of the LTE model. Moreover, among the operating parameters, rotational speed has the most significant effect on the maximum temperature of the finger seal, with an increase of 84.8% observed in the LTNE model. Furthermore, convective heat exchange causes the blade temperature to exceed the fluid temperature, resulting in reduced fluid temperature, increased viscosity, and greater flow resistance. Consequently, the LTNE model yields a lower leakage rate than the LTE model.
• Under both models, the deformation of each blade layer increases with increasing pressure differential and radial interference. The LTNE model consistently calculated greater overall deformation than the LTE model, indicating that thermal effects have a significant impact on the sealing performance of finger seals.
• The influences of pressure differential and radial interference on thermal deformation exhibit similar trends: radial deformation is most strongly affected, followed by circumferential deformation, with axial deformation being least sensitive. Within the pressure difference range, the radial deformation of the third seal element in the LTNE model increases by 14.55%. As the radial interference of the rotor increases from 0.05 mm to 0.2 mm, the radial deformation of each seal element in the LTNE model increases by 0.18 mm.
• The average contact pressure increases with both pressure differential and interference in both models. At the same pressure differential, the LTNE model yields higher contact pressure than the LTE model. While the difference between models is minor at low radial interference levels, it reaches 0.08 MPa when the radial interference is at its highest value.

It is worth noting that the application of the LTNE model reveals that considering thermal effects leads to increased deformation, thereby reducing leakage, but higher operating temperatures may accelerate frictional wear and pose a risk of thermal instability. Future design optimizations must balance these conflicting factors to maximize the service life and reliability of the seal.