# Experimental and numerical analysis of heat transfer and thermal deformation in small-dimension liquid mechanical seals

Amine Ben Abdelwahed, Shaolin Liu, Alain Sommier, David Reungoat

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## To cite this version:

Amine Ben Abdelwahed, Shaolin Liu, Alain Sommier, David Reungoat. Experimental and numerical analysis of heat transfer and thermal deformation in small-dimension liquid mechanical seals. International Journal of Heat and Mass Transfer, 2025, 241, pp.126754. 10.1016/j.ijheatmasstransfer.2025.126754. hal-04998078

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Submitted on 20 Mar 2025

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# Experimental and numerical analysis of heat transfer and thermal deformation in small-dimension liquid mechanical seals

Amine Ben-Abdelwahed, Shaolin Liu *, Alain Sommier, David Reungoat

Univ. Bordeaux, CNRS, INRAE, Bordeaux INP, I2M, UMR 5295, USC 1368, F-33400, Talence, France

Arts et Métiers Institute of Technology, CNRS, INRAE, Bordeaux INP, I2M, UMR 5295, F-33400, Talence, France

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## ARTICLE INFO

### Keywords:

Thermodynamic modeling

Mechanical seal

Numerical simulations

Experimental study

Lubrication

## ABSTRACT

This paper presents an experimental and numerical analysis of heat transfer and thermal deformation in small-dimension (1.4 mm) liquid mechanical seals operating in an unstable dynamic tracking mode. The studied non-contacting mechanical seal is used in a liquid pump for turbojets. The study aims to estimate the values of pressure, temperature, and thermal deformations that can prevent excessive wear of the sealing rings and control the increase in leakage rate or power loss during operation. Experimental investigations were conducted under a nominal inner pressure of 0.7 MPa, across a wide range of rotational speeds (from 1000 to 6000 rpm), and at low Reynolds numbers (Re < 70). Two high-viscosity fluids, glycerol and engine oil, were used as sealing fluids. Rotational speeds and inner pressure were set as boundary conditions in the simulations. Temperatures measured by thermocouples during the experiments were used to compare with the simulation results. Simulations were performed using the computational fluid dynamics (CFD) software COMSOL. The two-dimensional numerical models accounted for thermal transfers and face seal deformations, coupled with the pressure field in the lubrication fluid. The effects of various sealing fluids and rotational speeds on the time-dependent behavior of temperature, displacement, and pressure within the thin liquid lubricant film were investigated. Subsequent comparisons between experimental and numerical results, particularly for temperature data measured by thermocouples under various operating conditions, demonstrated strong consistency. The greatest discrepancy observed was less than 1.2 °C.

## 1. Introduction

Mechanical seals are used to contain pressurized fluids in rotating machinery such as pumps and compressors, where the operating conditions (pressure, temperature, and velocity) prevent the use of elastomeric seals [1]. A mechanical seal consists of a rotating ring (rotor) mounted on the shaft and a stationary ring (stator) fixed to the housing [2]. The contact between the two rings is maintained by elastic components (such as springs) and the pressure of the sealed liquid, while the secondary seals ensure proper sealing (Fig. 1). Optimal operating conditions are obtained when the sealing rings are slightly separated by a thin liquid film, only a few micrometers thick, which provides high wear resistance and a low leakage rate.

During the operation of the rotor, an important conversion of mechanical energy into heat occurs. As a result, heat flux is generated in the thin liquid film [3]. The heat flux is transferred to the sealing rings and then to the surrounding liquid. The amplitude of thermal phenomena depends on numerous factors, such as the correct alignment of the sealing rings [4], pressure of the sealed liquid [5], and heat

transfer conditions [6]. Furthermore, a high-temperature gradient is responsible for a change in the physical properties of the sealed liquid, which may lead to its vaporization in the thin liquid film and thermal deformations of the sealing rings. This will negatively affect the operating conditions for which the seal was designed. Therefore, the study of the temperature distribution in the sealing rings is of great importance in determining the performance of the mechanical seal [7,8].

Numerous studies on the thermal effects of non-contacting mechanical face seals have been conducted over the past few decades [9–12]. The main objective of these studies was to measure or theoretically determine the temperature distribution in the lubricant liquid, within the sealing rings, and the heat transfer in the surrounding fluid. A brief review is presented below. For instance, Lebeck [12] described the complexity of heat transfer mechanisms in mechanical seals. Because of the complex environment surrounding the seal, the heat transfer paths are multiple, leading to heat flow computation complications. In their studies, Buck [13] and Brunetiere et al. [14] confirmed the assumptions made by Lebeck [12], including the premise that most of

<footnote>* Correspondence to: Arts et Métiers Institute of Technology, 33400, Talence, France.
E-mail address: shaolin.liu@u-bordeaux.fr (S. Liu).</footnote>
Nomenclature

Latin symbols

<table><tr><td>c<sub>p</sub></td><td>Specific heat capacity, J kg<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>E</td><td>Young's modules, GPa</td></tr><tr><td>F<sub>V</sub></td><td>Contribution of the volume forces, N m<sup>-3</sup></td></tr><tr><td>h<sub>0</sub></td><td>Mean value of initial film thickness, m</td></tr><tr><td>I</td><td>Unit tensor</td></tr><tr><td>k</td><td>Thermal conductivity, W m<sup>-1</sup> K<sup>-1</sup></td></tr><tr><td>k<sub>s</sub></td><td>Spring stiffness, N m<sup>-1</sup></td></tr><tr><td>P<sub>o</sub></td><td>Outer pressure, MPa</td></tr><tr><td>P<sub>i</sub></td><td>Nominal inner pressure, MPa</td></tr><tr><td>Q</td><td>Additional heat sources, J</td></tr><tr><td>Q<sub>TE</sub></td><td>Thermoelastic damping, J</td></tr><tr><td>Q<sub>VD</sub></td><td>Viscous dissipation contribution, J</td></tr><tr><td>q</td><td>Heat flux by conduction, W m<sup>-2</sup></td></tr><tr><td>q<sub>v</sub></td><td>Leakage rates, mL h<sup>-1</sup></td></tr><tr><td>R</td><td>Radius, m</td></tr><tr><td>Re</td><td>Reynolds number,</td></tr><tr><td>S</td><td>Solid stress tensor</td></tr><tr><td>S<sub>EXT</sub></td><td>External stress field</td></tr><tr><td>S<sub>PK</sub></td><td>Second Piola-Kirchhoff tensor</td></tr><tr><td>S<sub>TH</sub></td><td>Stress field due to thermal effects</td></tr><tr><td>T</td><td>Temperature, K(°C)</td></tr><tr><td>u</td><td>Displacement fields</td></tr><tr><td>v</td><td>Velocity field of the fluid, m s<sup>-1</sup></td></tr></table>

Greek symbols

<table><tr><td>β<sub>1</sub></td><td>Initial stator taper</td></tr><tr><td>β<sub>2</sub></td><td>Initial rotor taper</td></tr><tr><td>λ</td><td>Coefficient of thermal expansion, K<sup>-1</sup></td></tr><tr><td>μ</td><td>Dynamic viscosity, Pa s<sup>-1</sup></td></tr><tr><td>ν</td><td>Poisson's ratio</td></tr><tr><td>ρ</td><td>Density, kg m<sup>-3</sup></td></tr><tr><td>τ</td><td>Viscous tensor</td></tr><tr><td>ω</td><td>Rotation speed of the rotor</td></tr></table>

Subscripts and superscripts

<table><tr><td>ir</td><td>Inner radius of the rotor</td></tr><tr><td>is</td><td>Inner radius of the stator</td></tr><tr><td>or</td><td>Outer radius of the rotor</td></tr><tr><td>os</td><td>Outer radius of the stator</td></tr></table>

<figure><img src="image_5.png" /><figcaption>Fig. 1. Operating principle of a mechanical face seals.</figcaption></figure>

the heat is evacuated through convection between the ring surfaces and the surrounding fluids. Several experimental investigations [15–17] have been carried out to quantify the magnitude and distribution of thermal phenomena at the interface of mechanical face seals. Due to viscous friction and asperities contacts in the sealing interface, the heat flux generated leads to a significant increase in temperature in the fluid thin film and the solid sealing rings. Consequently, the lubrication conditions are altered due to changes in fluid viscosity, and undesirable thermo-elastic distortions may lead to face contact [18,19].

Several authors have proposed analytical models for heat dissipation in mechanical face seals [10,20]. They conducted thermo-hydrodynamic (THD) analyses by assuming that the stator is completely thermally insulated and that the heat generated in the thin film is transferred through the rotor to the surrounding fluid. Blasiak et al. [9] presented a two-dimensional THD model where heat transfer within the sealing rings was solved analytically using the method of separation of variables. Other researchers [21,22] developed two-dimensional models where the temperature distribution in the sealing rings was solved using both analytical and numerical methods. In these studies, the heat generated in the thin fluid film was calculated analytically (as a boundary condition at the fluid-solid interface), and then the heat transfer in the sealing rings was solved numerically using finite element codes. Moreover, the convective heat transfer between the surfaces of the sealing rings and the surrounding fluid was determined using an empirical Nusselt number, as proposed in previous studies [23]. Subsequently, some experimental studies [24,25] allowed us to obtain more accurate values of convection coefficients. Therefore, experimental evaluations of the heat convection coefficients have been considered in Computed Fluid Dynamics (CFD) codes [3,4]. Some theoretical works [15,26] presented three-dimensional THD models of face seals where heat transfer in both rings was numerically solved. However, these numerical studies did not account for the thermo-elastic distortions of the sealing rings due to the increased temperature in the thin fluid film. Tournerie et al. [27] and Brunetière et al. [14,28] presented two-dimensional and three-dimensional thermal-elasto-hydrodynamic (TEHD) models, using external finite element codes to estimate thermal distortions. Through this TEHD modeling, extensive parametric analyses were performed for different seal parameters and heat transfer conditions, and these were successfully compared with experimental results obtained using an infrared camera [3,29]. More recently, Blasiak and Pawinska [10,30] proposed a complex theoretical method to solve thermoelastic problems in non-contacting face seals, where the solution was derived using analytical methods for both direct and inverse heat transfer problems. However, most of these approaches are designed for steady-state operating conditions and involve larger dimensions, as shown in Table 1, where the contact radius is consistently defined as the difference between the outer radius and the inner radius of the stator.

The aim of this work is to present an experimental and numerical study of the behavior of a small-dimension mechanical seal operating in an unstable dynamic tracking mode. The investigation focuses on heat transfer and thermal deformation in liquid mechanical seals specifically designed for small aero-engine applications. A review of the existing literature indicates limited experimental work addressing this type of application, particularly for seals with small dimensions (1.4 mm), high rotational speeds (6000 rpm), low pressure (0.7 MPa), and atmospheric conditions. At the beginning of the mechanical face seal operation, unstable dynamic tracking occurs and could last a few minutes before reaching a steady state operating condition. During this time, thermal deformations of the rings lead to considerable changes in the clearance height and, accordingly, disturbances in the dynamic properties of the whole sealing assembly, this may result in unstable operation of the seal, and consequently, ring collision, which is an undesirable phenomenon responsible for excessive wear of the elements in contact. By estimating the values of temperature and thermal deformations at the design stage, we can prevent excessive wear of the sealing rings and an uncontrolled increase in the leakage rate or power loss during
<table><caption>**Table 1**
Examples of contact radius, fluid pressures, rotational speeds, and applications examined in the literature.</caption><thead><tr><th>Year</th><th>Reference</th><th>Applications</th><th>Contact radius (mm)</th><th>Fluid pressure (MPa)</th><th>Speed (rpm)</th></tr></thead><tbody><tr><td>2003</td><td>Brunetière et al. [14]</td><td>pump</td><td>5</td><td>2.00</td><td>2000</td></tr><tr><td>2003, 2006, 2007</td><td>Brunetière et al. [28,31,32]</td><td>pump</td><td>40</td><td>15.3</td><td>0–3000</td></tr><tr><td>2009</td><td>Brunetière et al. [29]</td><td>pump</td><td>4</td><td>0.05</td><td>300–1500</td></tr><tr><td>2021</td><td>Zhao et al. [33]</td><td>aero-engines</td><td>20</td><td>0.5</td><td>9750</td></tr><tr><td>2022</td><td>Bai et al. [34]</td><td>aero-engines</td><td>3.5</td><td>0.1</td><td>0–20000</td></tr><tr><td>2022</td><td>Li et al. [35]</td><td>aero-engines</td><td>20</td><td>1.2</td><td>0–18300</td></tr><tr><td>2024</td><td>this work</td><td>aero-engines</td><td>1.4</td><td>0.2–0.7</td><td>1000–6000</td></tr></tbody></table>

operation.

The structure of the article is organized as follows: Section 2 details the experimental setup, including the measurements taken and the physical properties of the materials used. It also details the principal dimensions and operating conditions related to temperature and pressure measurements within the experimental cell. Section 3 is dedicated to the governing equations and boundary conditions that are employed to simulate the behavior of a small-dimension (1.4 mm) liquid mechanical seal. This includes a discussion on the 2D axisymmetric structure of the experimental cell. In Section 4, the numerical results concerning the evolution of temperature, displacement, and pressure in the thin liquid lubricant film across various rotational speeds are discussed. A comparative analysis of these results with experimental results is also presented. Section 5 presents conclusions and offers insights into potential future work.

## 2. Experimental apparatus

In this section, the structure of the experimental cell of the mechanical face seal is described, along with the physical properties of the materials used. Subsequently, the industrial seal is placed in an experimental cell and integrated into the overall experimental system, which is presented by both a schematic diagram and a photograph of the complete experimental setup utilized in the study. Finally, a simple example of the nominal inner fluid pressure and nominal rotational speed cycle over time is presented, setting the stage for the detailed discussions that follow.

### 2.1. Experimental device

The experimental cell of the mechanical face seal is presented in Fig. 2. Specifically, Fig. 2(b) provides an enlarged view of the stator and rotor parts within the red box of Fig. 2(a). The AISI 4340 steel rotor Ⓐ is fixed on the shaft Ⓐ using a fixing key and O-ring. The stator Ⓐ, made of carbon, is mounted on an annular support Ⓑ which is maintained to the stator frame Ⓒ via a spring-washer. This design allows for dynamic tracking of any rotor misalignment by providing three degrees of freedom relative to the stator frame Ⓒ. The stator frame is attached to the bearing cage Ⓒ using fixing washers Ⓒ and O-rings Ⓓ. The lines in the bearing cage represent channels, which will be explained in more detail in Fig. 7. Combined needle roller bearings Ⓐ and ball bearings Ⓑ enable the motor shaft Ⓐ to rotate within the pump housing Ⓓ. The stator is pressed against the rotor by the pressurized liquid Ⓒ. The red point indicates the position of the thermocouples. Due to the small size of the seals (1.4 mm), thermocouples were placed at specific locations where space allowed. Air temperature measurements during both the experiment and simulation (see Section 4) were taken at these exact positions, ensuring consistency between the experimental data and numerical predictions.

The physical properties of the materials used in the seal components are provided in Table 2. The primary difference is that the rotor is made of Steel AISI 4340 and the stator is composed of carbon. Steel AISI 4340 exhibits higher thermal conductivity, density, coefficient of thermal expansion, and Young's modulus compared to carbon. Hydraulic equipment provides liquids (engine oil and glycerol) at controlled pressure and temperature. The characteristics of these liquids are presented in Table 3. The typical operating conditions and the main dimensions of the mechanical face seal are detailed in Table 4.

<table><caption>**Table 2**
Materials physical properties and assignment at 25 °C.</caption><thead><tr><th></th><th>Rotor</th><th>Stator</th></tr></thead><tbody><tr><td>Material</td><td>Steel AISI 4340</td><td>Carbon</td></tr><tr><td>Thermal conductivity k (W m<sup>-1</sup> K<sup>-1</sup>)</td><td>44.5</td><td>12</td></tr><tr><td>Density ρ (kg m<sup>-3</sup>)</td><td>7850</td><td>2250</td></tr><tr><td>Specific heat capacity c<sub>p</sub> (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>475</td><td>720</td></tr><tr><td>Coefficient of thermal expansion λ (K<sup>-1</sup>)</td><td>12 × 10<sup>-6</sup></td><td>6 × 10<sup>-6</sup></td></tr><tr><td>Young's modulus E (GPa)</td><td>205</td><td>16</td></tr><tr><td>Poisson's ratio ν</td><td>0.28</td><td>0.20</td></tr></tbody></table>

<table><caption>**Table 3**
Fluids (engine oil and glycerol) properties at 25 °C.</caption><thead><tr><th></th><th>Engine Oil</th><th>Glycerol</th></tr></thead><tbody><tr><td>Density ρ (kg m<sup>-3</sup>)</td><td>885</td><td>1260</td></tr><tr><td>Dynamic viscosity μ (Pas<sup>-1</sup>)</td><td>0.5</td><td>1.0</td></tr><tr><td>Specific heat capacity c<sub>p</sub> (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>1900</td><td>2420</td></tr><tr><td>Thermal conductivity k<sub>f</sub> (W m<sup>-1</sup> K<sup>-1</sup>)</td><td>0.14</td><td>0.286</td></tr></tbody></table>

<table><caption>**Table 4**
Principal dimensions and operating conditions at 25 °C.</caption><tbody><tr><td>Inner radius of the stator R<sub>ix</sub> (m)</td><td>6.25 × 10<sup>-3</sup></td></tr><tr><td>Outer radius of the stator R<sub>os</sub> (m)</td><td>7.65 × 10<sup>-3</sup></td></tr><tr><td>Inner radius of the rotor R<sub>ir</sub> (m)</td><td>4.80 × 10<sup>-3</sup></td></tr><tr><td>Outer radius of the rotor R<sub>or</sub> (m)</td><td>7.65 × 10<sup>-3</sup></td></tr><tr><td>Initial stator taper β<sub>1</sub> (rad)</td><td>1.0 × 10<sup>-4</sup></td></tr><tr><td>Initial rotor taper β<sub>2</sub> (rad)</td><td>0.0</td></tr><tr><td>Spring stiffness k<sub>s</sub> (N m<sup>-1</sup> K<sup>-1</sup>)</td><td>16 076</td></tr><tr><td>Mean value of Initial film thickness h<sub>0</sub> (m)</td><td>2.0 × 10<sup>-6</sup></td></tr><tr><td>Range of angular velocity (rpm)</td><td>1000 ≤ ω ≤ 6000</td></tr><tr><td>Nominal angular velocity ω (rpm)</td><td>4200</td></tr><tr><td>Outer pressure P<sub>o</sub> (MPa)</td><td>0.1</td></tr><tr><td>Nominal inner pressure P<sub>i</sub> (MPa)</td><td>0.7</td></tr></tbody></table>

### 2.2. Measurements

Figs. 3 and 4 show the schematic diagram and photograph of the experimental setup used in our study, respectively. The pressure applied to the sealed liquid inside the experimental cell Ⓐ is generated by pressurizing nitrogen in the interface bottle Ⓑ. Air from the compressor, at a maximum pressure of 7 bars (0.7 MPa), is transmitted through the intensity/pressure converter Ⓒ, which controls the opening and closing of the gas regulator membrane Ⓒ. The outlet nitrogen pressure was monitored using a Keller PA-21S/100bar/80400 pressure transducer (operating range 0–100 bars). A thermocouple Alm 46 T-K-1 (operating range 0 °C–200 °C) was positioned within the mechanical seal, as indicated by the red points in Figs. 2 and 4. As shown in Fig. 7(a), it was placed in the outlet air flow of the aluminum bearing cage, thereby measuring the air temperature at that point. Pressure and temperature data were acquired through an indicator/sensor conditioner Ⓓ and a USB 6009 National Instruments data acquisition card Ⓒ. Rotational speeds of up to 6000 rpm are provided by a fully digital brushless motor Ⓒ integrated with a smart motion controller. Power is regulated by two 48 Vdc modules, supplying up to 500 W, and is managed through a power supply terminal. Transmission between the motor shaft and the mechanical seal shaft is achieved through mechanical trans-flex couplings and speed-up planetary gearing. Leakage during
<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Sectional view of the experimental cell and enlarged view of the stator and rotor parts: Ⓐ rotor (steel AISI 4340), Ⓑ stator (carbon), Ⓒ annular support and spring washer (steel), Ⓓ stator frame (steel), Ⓒ fixing washers (steel), Ⓓ O-ring (polytetrafluoroethylene), Ⓒ bearing cage (aluminum), Ⓓ pressurized liquid, Ⓓ rotating shaft (steel), Ⓓ① combined needle roller bearings (steel), Ⓒ ball bearings (steel), Ⓓ① pump housing (aluminum), red point: thermocouple position. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Outline of the experimental setup: (1) experimental cell of the mechanical seal, (2) interface bottle, (3) converter, (4) control membrane, (5) sensor conditioner, (6) data acquisition card, (7) brushless motor, (8) weighing balance, (9) LabVIEW interface.</figcaption></figure>

seal operation is measured using a METTLER TOLEDO balance Ⓒ with an accuracy of $10^{-3}$g and collected in a beaker. All measurements, including leakage $q_v$, temperature $T$, inner pressure $P$, and rotational speed $\omega$, are recorded and monitored over time via a LabVIEW interface Ⓓ.

## 2.3. Experimental conditions and validation for mechanical face seal

The experimental operating conditions of the mechanical face seal are defined to replicate the realistic operation of the system over time, focusing on fluid pressure and rotational speed. These conditions, applied to the rotor's moving walls, have been validated for use in the physical model (refer to Section 3). Fig. 5 represents an example of nominal inner fluid pressure (0.7 MPa) and nominal rotational speed (4200 rpm) cycle versus time that is applied to the mechanical face seal. The red dots represent experimentally measured inner fluid pressure values, while the blue dots denote measured rotational speeds. The solid red and blue lines accurately reflect the experimental measurements and represent the pressure and speed values used in the

simulations. The sealing fluid inside the experimental cell is pressurized at 0.7 MPa (inner pressure) at a temperature of 25 °C for 1400 s and two cycles of rotational speed with an angular velocity of about 4200 rpm applied to the rotating shaft.

Fig. 6(a) shows the experimental pressure and rotational speed cycle applied to the mechanical face seal, along with the corresponding air temperature differences (measured temperature minus room temperature) displayed in Fig. 6(b). The aim is to verify the accuracy and repeatability of the thermocouple temperature measurements in the experiment. The experimental tests maintained consistent operational conditions, with the fluid inner pressure $P_i$ set at 0.2 MPa for a preliminary test to validate the accuracy of the thermocouple temperature measurements, and the rotational speed $\omega$ at 4200 rpm. Fig. 6(b) shows the temperature evolution of the air $\Delta T$ across seven different tests. The temperature profiles from these tests show a remarkable consistency, underscoring the reproducibility of the measurements.

Throughout the experiments, the air temperature rapidly increased concomitant with the rotor's rotational speed. A peak temperature difference of around 8 °C was observed during engine operation, specifically at 460 s, marking the phase of highest thermal stress. This peak
<figure><img src="image_8.png" /><figcaption>Fig. 4. Photograph of the experimental setup.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Experimental and numerical liquid pressure/rotational speed versus time applied to the mechanical face seal.</figcaption></figure>

is indicative of the heat generated by friction at the interface between the rotating components and the seal, exacerbated by the elevated rotational speeds. Further analysis reveals that the experimental setup, characterized by its combination of film thickness, high-pressure gradient, smaller ring dimensions, and high rotational speeds, contributes to significant inertia effects. These effects are well-documented in the literature and are crucial for understanding the dynamic behaviors exhibited under such high-performance conditions. During the experiments, fluid leakage was monitored continuously during seal operation. Since new mechanical seals were used throughout the testing, no leakage was observed. A more detailed description of the surface condition analysis and measurement procedure can be found in Appendix.

### 3. Governing equations and boundary conditions

The schematic of the system being modeled (the 2D axisymmetric structure of the experimental cell shown in Fig. 2)) and the boundary conditions used are presented in Fig. 7. In Fig. 7(a), the boundary conditions across the entire 2D axisymmetric model are shown, with no-slip conditions and moving walls added at specific locations. The heat transfer within the 2D axisymmetric model is depicted in Fig. 7(b), which includes natural convection, forced convection, heat conduction, and thermal insulation. Fig. 7(c) shows the mechanical boundary conditions within the 2D axisymmetric geometry, focusing primarily on the stator, rotor, and their associated components. The diagram

<figure><img src="image_10.png" /><figcaption>Fig. 6. (a) Pressure and rotation speed cycle vs. time (b) Temperature evolution of the air measured by the thermocouple.</figcaption></figure>

includes representations of atmospheric pressure, fluid film pressure, fluid pressure surface loading, and fixed support, etc, highlighting the regions of interest within the mechanical seal area.

In Fig. 8, a close-up view of the 2D model is provided, highlighting the boundary conditions of the modeled region. The bottom-left image in Fig. 8 shows the seal dimensions, moving walls, fluid pressure, and thermocouple location, while the bottom-right image presents the heat transfer near the mechanical seal, including conduction (yellow arrows), natural convection (green arrows), forced convection (red arrows), and natural air convection (white dashed lines).

The fluid flow is assumed to be incompressible, Newtonian, and laminar, while the face plate material is considered linear-elastic and isotropic. The problem is treated as axisymmetric. Based on these
<figure><img src="image_11.png" /><figcaption>Fig. 7. Schematic of the system and boundary conditions.</figcaption></figure>

assumptions, a comprehensive model is presented that accounts for the effects of the working fluid and the deformation of the face plate. The following sections describe the main components of the model. The physical model integrates three coupled parts: fluid flow, heat transfer, and solid mechanics. The fluid modeling and solution steps involved solving the Navier-Stokes equations for non-isothermal flow, including the mass (Eq. (2)) and momentum (Eq. (3)) conservation equations, to characterize the behavior of the fluid in the mechanical seal. The next step was calculating the pressure and viscous stresses, which describe the internal forces within the fluid. These fluid stresses were then applied to the solid boundaries, including the stator and rotor, ensuring proper interaction between the fluid and solid domains. Finally, the displacements of the solid components were calculated by solving the equations of motion (Eq. (8)), considering the applied fluid stresses and thermal effects, allowing us to accurately model the deformation and behavior of the mechanical seal.

## 3.1. Fluid modeling

The seal is assumed to operate under full-film lubrication. The surfaces of the face plates are finely ground to achieve very low roughness. To characterize the flow regime, Owen and Rogers [36] recommended

using the following expression for the Reynolds number:

$$Re = \frac{\rho\omega R_{or}^2}{\mu} \qquad (1)$$

where $\rho$ is the fluid density, $\mu$ is the dynamic viscosity of the fluid, $\omega$ is the rotation speed of the rotor and $R_{or}$ is the outer radius of the rotor. For the present study, $Re < 70$ (calculated using the data in Table 3, where $\omega$ is converted to rad/s). The Reynolds number is 65 for engine oil and 46 for glycerol, indicating that the surrounding fluid flow can be assumed to be laminar within the thin flow domain confined by the two face plates. For incompressible flow, the continuity equation is given by:

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \qquad (2)$$

where $\mathbf{v}$ is the velocity field of the fluid. The momentum equation of the fluid is written as:

$$\rho \frac{\partial \mathbf{v}}{\partial t} + \rho (\mathbf{v} \cdot \nabla)\mathbf{v} = \nabla \cdot [-p\mathbf{I} + \mu(\nabla\mathbf{v} + (\nabla\mathbf{v})^T) - \frac{2}{3}\mu(\nabla \cdot \mathbf{v})\mathbf{I}] + \mathbf{F}_V \qquad (3)$$

where $p$ is the fluid pressure, $\mathbf{I}$ is the unit tensor and $\mathbf{F}_V$ is the contribution of the volume forces. Fluid slip boundary condition is imposed near the stator-moving walls ($\mathbf{v} = \omega \times \mathbf{r}$; where $\omega$ is the rotational speed and $\mathbf{r}$ is the radius coordinate).
<figure><img src="image_12.png" /><figcaption>Fig. 8. (A) dimensions of the seal, moving walls, fluid pressure and thermocouple location, (B) heat transfers near the mechanical seal: conduction (yellow arrows), natural convection (green arrows), forced convection (red arrows) and natural air convection (dashed white lines). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

## 3.2. Heat transfer modeling

With the Couette-Poiseuille flow, the fluid reaches very high levels of shearing within the small seal gap. This high shearing causes large viscous heat generation. As a result, the face plates and the entire support assembly deform due to the heat transferred from the fluid to the solid. The temperature distribution within the face plates (solid domains) can be calculated using the energy equation:

$$ \rho c_p \frac{\partial T}{\partial t} + \rho c_p \mathbf{v} \cdot \nabla T + \nabla \cdot \mathbf{q} = \mathbf{Q} + Q_{TE} \quad (4) $$

where $\rho$ is the solid density, $c_p$ is the solid specific heat, $\mathbf{v}$ is the velocity field, $\mathbf{q} = -k\nabla T$ is the heat flux by conduction (here $k$ is the solid thermal conductivity), $\mathbf{Q}$ contains additional heat sources and $Q_{TE}$ is the thermoelastic damping and accounts for thermoelastic effects in solids:

$$ Q_{TE} = -\lambda T : \frac{\partial S_{PK}}{\partial t} \quad (5) $$

where $\lambda$ is the solid coefficient of thermal expansion and $S_{PK}$ is the second Piola-Kirchhoff tensor.

Due to high shear rates, a large amount of energy is lost in the liquid in the form of viscous heat generation. This generated heat, when transferred over to the surrounding solid causes thermal expansion and resulting thermo-elastic distortions in the seal rings. The temperature distribution in the fluid domain is solved through the energy equation:

$$ \rho c_p \frac{\partial T}{\partial t} + \rho c_p \mathbf{v} \cdot \nabla T + \nabla \cdot \mathbf{q} = \mathbf{Q} + Q_{VD} \quad (6) $$

where $\rho$ is the fluid density, $c_p$ is the fluid specific heat, $\mathbf{v}$ is the velocity field, $\mathbf{q}$ is the heat flux by conduction, $\mathbf{Q}$ contains additional heat sources and $Q_{VD}$ is the viscous dissipation contribution:

$$ Q_{VD} = \tau : \nabla \mathbf{v} \quad (7) $$

where $\tau$ is the viscous tensor. Natural air convection is imposed as heat boundary condition at the external walls of the pump housing.

## 3.3. Solid modeling

In our study, only the displacement fields in the solid sealing rings were estimated. The stator and the rotor are assumed homogeneous

and isotropic. The displacement fields $\mathbf{u}$ are obtained by solving the equation of dynamics:

$$ \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} = \nabla \cdot \mathbf{S} + \mathbf{F}_V \quad (8) $$

where $\mathbf{F}_V$ is the contribution of the volume forces and $\mathbf{S}$ is the solid stress tensor:

$$ S = S_0 + S_{EXT} + S_{TH} + \mathbf{C} : \varepsilon_{EL} \quad (9) $$

where $S_0$ is the initial stress field, $S_{EXT}$ is the external stress field, $S_{TH}$ is the stress field due to thermal effects, $\mathbf{C}$ is the elasticity matrix and $\varepsilon_{EL}$ is the elastic strain field. The strain fields $\varepsilon$ in the solid sealing rings are defined as:

$$ \varepsilon = \frac{1}{2} [ (\nabla \mathbf{u})^T + \nabla \mathbf{u} ] \quad (10) $$

# 4. Results and discussion

This section presents the numerical results, which explore the impact of various sealing fluids and rotational speeds on the time-dependent behavior of temperature, displacement, and pressure within the thin liquid lubricant film. It also includes comparing these results with experimental data to validate the model's accuracy.

## 4.1. Numerical results

The simulations were carried out using the computational fluid dynamics (CFD) code COMSOL Multiphysics [37,38]. This simulation platform offers fully coupled multiphysics modeling as well as single-physics modeling capabilities [39].

Fig. 9 shows the temperature profiles of the mechanical seal for glycerol at 4200 rpm and $t = 240$ s, corresponding to the time when the rotational speed stabilizes, as shown in Fig. 5. The top image displays the simulated 2D axisymmetric structure (as shown in Fig. 8). The bottom left image is an enlarged view of the area highlighted in red, where a contour map represents the temperature distribution. The red dots indicate the positions of the thermocouples. Temperature data collected at these points will be compared with experimental
<figure><img src="image_13.png" /><figcaption>Fig. 9. Temperature profiles in °C of the mechanical seal for glycerol at 4200 rpm (t = 240 s).</figcaption></figure>

results, which will be discussed in Section 4.2. As the focus is on the interface between the stator and rotor, this region within the simulated image has been further magnified (as indicated by the green box). In subsequent analyses, attention will be concentrated on the conditions at this interface.

### 4.1.1. Temperature and displacement evolution at a constant rotational speed

Fig. 10 shows the temporal evolutions of temperature *T* (in °C) and displacement *u* (in µm) within a mechanical seal employing glycerol as the sealing fluid, subjected to a rotational speed of 4200 rpm at designated times of 240 s, 420 s, and 1000 s. These times respectively correspond to the initial stabilization of the rotational speed, during the first cycle, and within the second cycle, as detailed in Fig. 5. Initially, at 240 s, temperatures are moderately elevated, with the highest temperature reaching 50.7 °C. By 420 s, there are noticeable increases in temperature, particularly near the inner radius of the stator (as shown in Fig. 8), with the highest temperature reaching 58.6 °C. By 1000 s, the temperature peaks at 62 °C across inner broader areas of the seal, indicating sustained thermal stress from continuous operation.

Displacement fields in the rotor area (right side) show clear increases over time, starting with minimal changes at 240 s with a maximum displacement of 1.36 µm. By 420 s, these values increase to 1.99 µm, especially near the rotor's outer radius. By 1000 s, thermoelastic displacement due to the thermal expansion of the steel rotor (AISI 4340) peaks at 2.24 µm, occurring predominantly near the rotor's outer radius, $R_{or}$. In contrast, the stator area (left side), made of carbon, exhibits virtually no displacement. This distribution is due to the different material properties of the rotor and stator, which influence how they respond to thermal stresses. The correlation between elevated temperatures and increased displacements suggests that thermal expansion influences the physical deformation of the seal components. The highest temperatures coincide with the largest displacements, particularly where thermal gradients are most pronounced.

### 4.1.2. Effect of rotational speed on temperature and displacement fields

The evolution of the temperature field (°C) of the mechanical seal and the displacement field (µm) of both the rotor and stator for different rotational speeds at *t* = 1000 s with glycerol as the sealing fluid are respectively displayed in Figs. 11 and 12 for rotational speeds of 1000 rpm, 3000 rpm, 4200 rpm, and 6000 rpm.

As shown in Fig. 11, with an increase in rotational speed from 1000 rpm to 6000 rpm, there is a rise in the temperature within the seal. At

the lower speed of 1000 rpm, temperatures are relatively moderate, peaking at 32.8 °C, and are uniformly distributed across the seal. As the speed increases to 3000 rpm and 4200 rpm, the central areas of the seal begin to experience higher temperatures, reaching up to 49.1 °C and 62 °C respectively. This trend continues, with temperatures going up to 87.2 °C at the highest speed of 6000 rpm, indicating intense heat generation due to increased friction at higher speeds. The highest temperature is concentrated at the contact interface between the rotor and stator. It should be noted that even at its highest temperature of 87.2 °C, glycerol does not reach its boiling point and continues to exist in liquid form [40]. Therefore, it is not necessary to consider its behavior in a gaseous state.

Corresponding to the temperature data, the displacement fields in Fig. 12 show minimal displacement at 1000 rpm, predominantly under 0.81 µm. As the rotational speed increases to 3000 rpm and 4200 rpm, the displacement becomes more pronounced, particularly near the outer radius of the rotor, rising to 2.24 µm. At the maximum speed of 6000 rpm, the displacement peaks at 3.24 µm, signaling substantial deformation of the seal material. This deformation is likely due to the combined effects of thermal expansion and mechanical stress resulting from the elevated operational speeds.

### 4.1.3. Effect of sealing fluid on temperature and displacement fields

To examine the effects of different sealing fluids on temperature and displacement fields, glycerol and engine oil were employed, with their fluid properties detailed in Table 3. Glycerol exhibits a higher density, dynamic viscosity, specific heat, and thermal conductivity compared to engine oil, with its dynamic viscosity being twice that of engine oil. At the same rotational speed, glycerol exhibits a lower Reynolds number.

A comparison of the temperature field (°C) and displacement field (µm) for these two fluids at *t* = 1000 s and *ω* = 4200 rpm is shown in Fig. 13.

It can be seen that, under the same time and rotational speed, switching the fluid from glycerol to engine oil reduces the maximum temperature from 62 °C to 54.7 °C and decreases the displacement field from 2.24 µm to 2.01 µm. This suggests that engine oil provides better thermal management than glycerol, resulting in lower temperatures and reduced thermal deformation of the mechanical seal. The lower maximum temperature with engine oil indicates its superior heat capacity or dissipation properties, which help maintain lower temperatures within the seal. Consequently, the reduced thermal expansion leads to less deformation of the rotor and stator surfaces, which can minimize wear and extend the seal's performance and lifespan.
<figure><img src="image_14.png" /><figcaption>Fig. 10. Evolution of temperature and displacement fields versus time of the mechanical seal for glycerol at 4200 rpm.</figcaption></figure>

### 4.1.4. Effect of rotational speed and sealing fluid on pressure of the thin liquid lubricant film

Fig. 14 illustrates the pressure of the thin liquid lubricant film (separating rotor and stator rings) under different conditions.

Where Fig. 14(a) shows the evolution of lubricant film pressure over time for glycerol as the sealing fluid at rotational speeds of 1000 rpm, 2000 rpm, 3000 rpm, 4200 rpm, 5000 rpm, and 6000 rpm. It is evident that as the rotational speed increases, the lubricant film pressure rises, reaching a maximum of around 83 MPa at 6000 rpm. It should be noted that under such high pressures, glycerol remains in a liquid state [40]. Furthermore, in our study of small-dimension seals (1.4 mm), the film thickness varies between 0.81 µm (with glycerol as the sealing fluid at 1000 rpm, Fig. 12) and 3.24 µm (at 6000 rpm), depending on the rotational speed. As shown in Fig. 14(a), even at the minimum speed of 1000 rpm, the pressure of the thin liquid lubricant film still reaches around 40 MPa.

To analyze the impact of different fluids on the pressure of the thin liquid lubricant film, the sealing fluid was switched from glycerol to engine oil, and measurements were taken at the minimum rotational speed of 1000 rpm. Fig. 14(b) presents the pressure evolution for both engine oil and glycerol at this speed, with engine oil exhibiting lower lubricant film pressures, reaching a maximum of around 30 MPa. Differences in thermophysical properties such as density and dynamic viscosity between engine oil and glycerol, as listed in Table 3, influence their respective Reynolds numbers. At the same rotational speed, engine oil, which has a higher Reynolds number, results in lower

lubricant film pressure. Although the data from Fig. 13, which provides results at 4200 rpm, serves primarily as a reference, it is observed that engine oil at the same speed leads to lower temperatures and displacement. These factors in turn contribute to lower lubricant film pressures. The relationship between lubricant film pressures, temperature, and displacement is yet to be explored further in subsequent studies. Overall, the study provides insights into how different sealing fluids and rotational speeds influence lubricant film pressure, aiding in the selection of optimal speeds and fluids to further reduce film pressures.

To provide a more specific information of the lubricant film pressure values, an in-depth analysis was conducted based on the case presented in Fig. 14(a), where glycerol is used as the sealing fluid, and the rotational speed is set to 4200 rpm. Fig. 15 shows the pressure of the thin liquid lubricant film over time as a function of the radial coordinate (Glycerol, 4200 rpm).

Six specific time points were selected: 0, 200, 240, 400, 800, and 1000 s. To highlight the pressure values at these different moments, they are displayed across six subfigures, where the horizontal axis represents the distance from the inner radius of the stator (6.25 mm) to the outer radius of the stator (7.65 mm), and the vertical axis represents the lubricant film pressure. At 0 s, corresponding to when the experiment has not yet started, the maximum pressure is equal to the set boundary condition of 0.7 MPa. Between 0 and 180 s, no change in pressure is observed. At 200 s, corresponding to the start of the experiment, the lubricant film pressure reaches a peak of around 57 MPa at the
<figure><img src="image_15.png" /><figcaption>Fig. 11. Impact of the rotational speed on the temperature field (°C) of the mechanical seal for glycerol at t = 1000 s.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 12. Impact of the rotational speed on the evolution of displacement field (µm) of the sealing rings at t = 1000 s for glycerol.</figcaption></figure>

middle region. By 240 s, the pressure reaches its maximum value, around 70 MPa. As time progresses, the maximum value decreases, for instance, reaching 62 MPa at 400 s. At 800 s, which corresponds to the second cycle, the maximum value shown in the subfigure is 34 MPa, and it gradually increases, eventually reaching around 62 MPa at 1000 s. Based on the analysis of the displacement field in Fig. 13, it can be inferred that the relative displacement between the stator and rotor significantly affects the pressure distribution in the lubricant film.

During dynamic operation, thermal expansion and mechanical loading lead to changes in the clearance between the stator and rotor, causing compression of the lubricant film in certain areas and resulting in high pressure peaks. This phenomenon is reflected in the time evolution of the pressure distribution, particularly at high rotational speeds, where displacement changes cause periodic variations in the film thickness, leading to notable pressure fluctuations. These pressure peaks can thus be attributed to local hydrodynamic effects induced by displacement
<figure><img src="image_17.png" /><figcaption>Fig. 13. Comparison of the evolution of temperature field (°C) and displacement field (µm) for glycerol and engine oil at *t* = 1000 s and *ω* = 4200 rpm.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 14. Pressure of the thin liquid lubricant film (separating rotor and stator rings) at different situations.</figcaption></figure>

changes in the mechanical seal.

## 4.2. Comparison with experimental results

Experimental and numerical results have been obtained for two fluids, engine oil, and glycerol, at various rotational speeds. Specifically, the air temperature values, measured by thermocouples and simulated at the same locations, were compared under different operating conditions. For example, Fig. 16 presents the temperature distribution on the seal's surface at six different times (0 s, 240 s, 420 s, 600 s, 1000 s, 1200 s) during a rotational speed of 4200 rpm, with glycerol as the sealing fluid. These timestamps correspond to various phases in the operation cycle, as detailed in Fig. 5: 0 s represents the initial stage, 240 s and 420 s fall within the first stable operational period, 600 s marks the pause between two cycles, 1000 s is during the second operational period, 1200 s occurs after the second cycle. From Fig. 16(a) to (f), the

peak temperatures do not consistently rise but vary as follows: 25 °C, 50.7 °C, 58.6 °C, 37.2 °C, 62 °C, and 41 °C. This pattern indicates that temperatures gradually increase during active periods and decrease during pauses, with the highest peak at 1000s in the second cycle (shown in Fig. 16(e)).

The positions marked by red dots in Fig. 16(d) correspond to the locations where thermocouples were placed experimentally. Here, we aim to highlight the positions of the thermocouples and extract data from these temperature distribution maps. Correspondingly, Fig. 17 displays the temperature profile at the thermocouple locations, where experimental results are represented by dots, while numerical results are shown as solid lines.

Air temperature profiles at rotational speeds of 1000, 3000, and 5000 rpm, with engine oil as the working fluid, are displayed in Fig. 17(a). The cases at 1000 rpm, as represented by the red dots and curve in Fig. 17(a), show both experimental and simulation data display
<figure><img src="image_19.png" /><figcaption>Fig. 15. Pressure of the thin liquid lubricant film over time as a function of the radial coordinate (Glycerol, 4200 rpm).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Fig. 16. Temperature distribution on the seal's surface at different moments at a rotational speed of 4200 rpm (glycerol).</figcaption></figure>

consistency in temperature rise, with differences noted to be less than 0.2 °C. From the start up to 200 s, the temperature is observed to remain nearly constant. During two cycles of rotational speed — from 200 to 450 s and from 800 to 1050 seconds — the temperature is seen gradually increasing. Outside these running periods, specifically from 450 to 800 s, and after 1050 s, a decrease in temperature is recorded. Further analysis across the three rotational speeds (1000, 3000, 5000 rpm) reveals that temperature increases with rotational speed at any given time. The highest temperature, recorded at 5000 rpm and 1050 s, measured 43.4 °C experimentally and 42.2 °C in simulations, indicating a difference of 1.2 °C. This highlights how rotational speed influences the thermal response of engine oil, and the close agreement between experimental and simulated data demonstrates the accuracy and reliability of the simulations in reflecting actual measurements.

Fig. 17(b) presents a comparative analysis of the measured and simulated air temperature profiles when engine oil and glycerol are used as the working fluids at a rotational speed of 4200 rpm. It is observed that the air temperature, when glycerol is used as the

sealing fluid, consistently remains higher than when engine oil is used throughout the experiment. For instance, the peak temperature reached by glycerol as the sealing fluid is 42.7 °C, whereas engine oil reaches a peak temperature of 41.2 °C under the same conditions, resulting in an air temperature difference of 1.5 °C. The higher air temperature observed when glycerol is used as the sealing fluid can be attributed to its higher specific heat capacity and thermal conductivity, indicating that glycerol absorbs and retains more heat compared to engine oil.

## 5. Conclusions

This paper presents an experimental and numerical study of the behavior of mechanical seals with small dimensions (1.4 mm) in unstable dynamic tracking modes used in turbojet engine liquid pumps. The aim is to estimate values of pressure, temperature, and thermal deformation that prevent excessive wear of the sealing rings and uncontrolled increases in leakage rate or power loss during seal operation.
<figure><img src="image_21.png" /><figcaption>Fig. 17. Comparison of experimental and numerical temperature profiles at the same location.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>Fig. 18. 3D surface profiles of the new seal and the seal after long-term use.</figcaption></figure>

Experimental investigations focused on rotating rings with small dimensions were conducted under a broad range of rotational speeds (1000 to 6000 rpm) and low Reynolds number conditions (< 70). During the experiments, two sealing fluids, glycerol and engine oil, were used, and parameters such as fluid leakage rate, temperature, fluid inner pressure, and rotational speed were recorded and monitored over time via a LabVIEW interface. The results indicate that when engine oil is used as the sealing fluid, the peak temperature measured by thermocouples increases from 28.4 °C at 1000 rpm to 43.4 °C at 5000 rpm. Furthermore, at a constant speed of 4200 rpm, the temperature when using glycerol as the sealing fluid is consistently 1.5 °C higher than when using engine oil. This demonstrates that glycerol tends to maintain higher temperatures under the same operational conditions.

Simulations were performed using the computational fluid dynamics (CFD) code COMSOL Multiphysics. The two-dimensional numerical modeling considered heat transfer and face seal deformations coupled with the pressure field in the lubrication fluid. Specifically, the physical model integrates three coupled aspects: fluid flow, heat transfer, and solid mechanics. The results revealed that: (1) At a constant rotational speed of 4200 rpm, the maximum temperatures and displacements were observed at the inner radius of the stator and the outer radius of the rotor, respectively. Thermoelastic displacement, resulting from the thermal expansion of the steel rotor, reached a peak of 2.24 µm after 1000 s of operation. (2) Increasing the rotational speed from 1000 to 6000 rpm raised the rotor's peak temperature from 32.8 °C to 87.2 °C, and correspondingly, the displacement increased from 0.81 µm to 3.24 µm. (3) At 4200 rpm, switching the sealing fluid from glycerol to engine oil reduced the peak temperature from 62 °C to 54.7 °C and decreased the displacement field from 2.24 µm to 2.01 µm. (4) For glycerol, as the rotational speed increased from 1000 to 6000 rpm,

the maximum lubricant film pressure rose from 40 MPa to 83 MPa. When switching the sealing fluid from glycerol to engine oil at 1000 rpm, the maximum lubricant film pressure decreased to around 30 MPa. The relationship between lubricant film pressure, temperature, and displacement requires further exploration in future studies.

Subsequent comparisons between experimental and numerical results, especially in temperature data measured by thermocouples under various operating conditions, showed strong consistency, with the largest difference being less than 1.2 °C.

### CRediT authorship contribution statement

**Amine Ben-Abdelwahed:** Data curation, Formal analysis, Investigation, Resources, Writing – original draft. **Shaolin Liu:** Formal analysis, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. **Alain Sommier:** Investigation, Validation, Writing – review & editing. **David Reungoat:** Resources, Supervision, Writing – review & editing.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

The authors gratefully acknowledge the support provided by the Région Aquitaine for this work under the SYRENA project.
# Appendix. Evaluation of surface deformation in mechanical seals

The industrial challenge addressed in this study concerns fluid leakage occurring in mechanical seals after long-term use. To investigate the causes of this leakage, the original aged seals were replaced with new, original mechanical seals for detailed examination. Initially, the surface condition of the carbon seals (stator) was measured. The device used for measuring the surface condition of the carbon stator is an AltiSurf 500 AltiMet profilometer, a modular metrology platform. Measurements were performed using two integrated probes: chromatic confocal optics, laser, mechanical stylus, zoom camera, and microforce sensor. Both new and aged mechanical seals were evaluated.

Fig. 18 presents the 3D surface profiles of the new seal and the seal after long-term use. As shown in Fig. 18(a), corresponding to the new mechanical seal, the entire surface is smoother and more uniform. In contrast, Fig. 18(b) shows the profile of the seal after 2000 h of use, where a distinct depression is visible on the left side of the surface. This deformation is the primary cause of fluid leakage. In the present work, new mechanical seals were used, which eliminated any leakage. Thus, the measured fluid leakage rate was zero.

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